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Basic Matrix Theory

Basic Matrix Notation

Consider the complex matrix $A\in \mathbb {C} ^{n\times m}$.

$A={\begin{bmatrix}a_{11}&\dots &a_{1m}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{n\times m}$

Transpose of a Matrix

The transpose of $A$, denoted as $A^{T}$ or $A'$ is:

$A^{T}={\begin{bmatrix}a_{11}&\dots &a_{n1}\\\vdots &\ddots &\vdots \\a_{1m}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $A$, denoted as $A^{*}$ is:

$A^{*}={\begin{bmatrix}a_{11}^{*}&\dots &a_{n1}^{*}\\\vdots &\ddots &\vdots \\a_{1m}^{*}&\dots &a_{nm}^{*}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$

Where $a_{nm}^{*}$ is the complex conjugate of matrix element $a_{nm}$.

Notice that for a real matrix $A\in \mathbb {R} ^{n\times m}$, $A^{*}=A^{T}$.

Important Properties of Matricies

Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $A\in \mathbb {C} ^{n\times n}$ is called Hermitian or self-adjoint if $A=A^{*}$.

If $A\in \mathbb {R} ^{n\times n}$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $A\in \mathbb {C} ^{n\times n}$ is called unitary if $A^{*}=A^{-1}$ or $A^{*}A=I$.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Notion of Matrix Positivity

Notation of Positivity

A symmetric matrix $A\in \mathbb {R} ^{n\times n}$ is defined to be:

positive semidefinite, $(A\geq 0)$, if $x^{T}Ax\geq 0$ for all $x\in \mathbb {R} ^{n},x\neq \mathbf {0}$.

positive definite, $(A>0)$, if $x^{T}Ax>0$ for all $x\in \mathbb {R} ^{n},x\neq \mathbf {0}$.

negative semidefinite, $(-A\geq 0)$.

negative definite, $(-A>0)$.

indefinite if $A$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

• For any matrix $M$, $M^{T}M>0$.
• Positive definite matricies are invertible and the inverse is also positive definite.
• A positive definite matrix $A>0$ has a square root, $A^{1/2}>0$, such that $A^{1/2}A^{1/2}=A$.
• For a positive definite matrix $A>0$ and invertible $M$, $M^{T}AM>0$.
• If $A>0$ and $M>0$, then $A+M>0$.
• If $A>0$ then $\mu A>0$ for any scalar $\mu >0$.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

KYP Lemma (Bounded Real Lemma)

KYP Lemma (Bounded Real Lemma)

The Kalman–Popov–Yakubovich (KYP) Lemma is a widely used lemma in control theory. It is sometimes also referred to as the Bounded Real Lemma. The KYP lemma can be used to determine the $H_{\infty }$ norm of a system and is also useful for proving many LMI results.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$, $u(t)\in \mathbb {R} ^{q}$, at any $t\in \mathbb {R}$.

The Data

The matrices $A,B,C,D$ are known.

The Optimization Problem

The following optimization problem must be solved.

${\begin{aligned}&{\underset {\gamma ,\;X}{\operatorname {minimize} }}\quad \gamma \\&\operatorname {subject\;to} &X>0\\&&{\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\gamma I\end{bmatrix}}+{\frac {1}{\gamma }}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}{\begin{bmatrix}C&D\end{bmatrix}}<0\\\end{aligned}}$

The LMI: The KYP or Bounded Real Lemma

Suppose ${\hat {G}}(s)(A,B,C,D)$ is the system. Then the following are equivalent.

$1)\quad \left\|G\right\|_{H_{\infty }}\leq \gamma$

$2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}$

${\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\gamma I\end{bmatrix}}+{\frac {1}{\gamma }}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}{\begin{bmatrix}C&D\end{bmatrix}}<0$

Conclusion:

The KYP Lemma can be used to find the bound $\gamma$ on the $H_{\infty }$ norm of a system. Note from the (1,1) block of the LMI we know that $A$ is Hurwitz.

Implementation

Since the KYP lemma shown above is nonlinear in gamma, in order to implement it in MATLAB we must first linearize it by using the Schur Complement to arrive at the dual formulation below:

${\begin{bmatrix}A^{T}X+XA&XB&C^{T}\\B^{T}X&-\gamma I&D^{T}\\C&D&-\gamma I\end{bmatrix}}<0$.

This dual KYP LMI is now linear in both $X$ and $\gamma$.

This implementation requires the use of Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/KYP_Lemma_LMI.m

Related LMIs

Positive Real Lemma

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$, $u(t)\in \mathbb {R} ^{q}$, at any $t\in \mathbb {R}$.

The Data

The matrices $A,B,C,D$ are known.

The LMI: The Positive Real Lemma

Suppose ${\hat {G}}(s)(A,B,C,D)$ is the system. Then the following are equivalent.

$1)\quad G\;{\text{is passive, i.e.}}\;\left\langle u,Gu\right\rangle _{L_{2}}\geq 0\;({\hat {G}}(s)+{\hat {G}}(s)^{*}\geq 0)$

$2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}$

${\begin{bmatrix}A^{T}X+XA&XB-C^{T}\\B^{T}X-C&-D^{T}-D\end{bmatrix}}\leq 0$

Conclusion:

The Positive Real Lemma can be used to determine if the system $G$ is passive. Note from the (1,1) block of the LMI we know that $A$ is Hurwitz.

Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m

Related LMIs

KYP Lemma (Bounded Real Lemma)

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

KYP Lemma for QSR Dissipative Systems

The Concept

In systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state $x(t)$, its input $u(t)$ and its output $y(t)$, the input-output correlation is given a supply rate $w(u(t),y(t))$. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function $V(x(t))$ such that $V(0)=0$, $V(x(t))\geq 0$ and

${\dot {V}}(x(t))\leq w(u(t),y(t))$

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate $w(u(t),y(t))=u(t)^{T}y(t)$.

The physical interpretation is that $V(x)$ is the energy stored in the system, whereas $w(u(t),y(t))$ is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.

The System

Consider a contiuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space realization (A, B, C, D), where ${\mathcal {A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

The Data

The matrices $A,B,C$ and $D$ which defines the state space of the system

The Optimization Problem

The system ${\mathcal {G}}$ is QSR disipative if

$\int _{0}^{T}(y^{T}(t)Qy(t)+2y^{T}Su(t)+u^{T}(t)Ru(t))\,dt\geq 0,\forall u\in {\mathcal {L}}_{2e},\forall T\geq 0$

where $u(t)$ is the input to ${\mathcal {G}},y(t)$ is the output of ${\mathcal {G}},Q\in {\mathcal {S}}^{p},S\in {\mathcal {R}}^{p\times m},$ and ${\mathcal {R}}\in {\mathcal {S}}^{m}$.

LMI : KYP Lemma for QSR Dissipative Systems

The system ${\mathcal {G}}$ is also QSR dissipative if and only if there exists $P\in {\mathcal {S}}^{n},$ where $P>0,$ such that

${\begin{bmatrix}PA+A^{T}P-C^{T}QC&PB-C^{T}S-C^{T}QD\\(PB-C^{T}S-C^{T}QD)^{T}&-D^{T}QD-(D^{T}S+S^{T}D)-R\end{bmatrix}}\leq 0.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is QSR dissipative.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2

KYP Lemma witout Feedthrough

The Concept

It is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)^-1B + D is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The System

Consider a contiuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, 0), where ${\mathcal {A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},$ and ${\mathcal {C}}\in {\mathcal {R}}^{m\times n},$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)\\\end{aligned}}$

The Data

The matrices The matrices $A,B$ and $C$

LMI : KYP Lemma without Feedthrough

The system ${\mathcal {G}}$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $p>0$ such that

${\begin{aligned}PA+A^{T}P\geq 0\\PB=C^{T}\end{aligned}}$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{aligned}AQ+QA^{T}\geq 0\\B=QC^{T}\end{aligned}}$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system ${\mathcal {G}}$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $p>0$ such that

${\begin{aligned}PA+A^{T}P<0\\PB=C^{T}\end{aligned}}$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{aligned}AQ+QA^{T}<0\\B=QC^{T}\end{aligned}}$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $\in {\mathcal {R}}_{>0}.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is Positive Real.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London

KYP Lemma for Descriptor Systems

The Concept

Descriptor system descriptions frequently appear when solving computational problems in the analysis and design of standard linear systems. The numerically reliable solution of many standard control problems like the solu­tion of Riccati equations, computation of system zeros, design of fault detection and isolation filters (FDI), etc. relies on using descriptor system techniques.

Many algorithm for standard systems as for example stabilization techniques, factorization methods, minimal realization, model reduction, etc. have been extended to the more general descriptor system descriptions. An important application of these algorithms is the numeri­cally reliable computation with rational and polynomial matrices via equivalent descriptor representations. Recall that each rational matrix R(s) can be seen as the transfer-function matrix of a continuous- or discrete-time descriptor system. Thus, each R(s) can be equivalently realized by a descriptor system quadruple (A-sE, B, C, D) satisfying R(S)= C(SE-A)^-1B+D

Many operations on standard matrices (e.g., finding the rank, determinant, inverse or generalized inverses), or the solution of linear matrix equa­tions can be performed for rational matrices as well using descriptor system techniques. Other important applications of descriptor techniques are the computation of inner-outer and spectral factorisations, or minimum degree and normal­ized coprime factorisations of polynomial and rational matrices. More explanation can be found in the website of Institute of System Dynamics and control

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (E, A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}E{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

The Data

The matrices The matrices $E,A,B,C$ and $D$

LMI : KYP Lemma for Descriptor Systems

The system ${\mathcal {G}}$ is extended strictly positive real (ESPR) if and only if there exists $X\in {\mathcal {R}}^{n\times n}$ and $W\in {\mathcal {R}}^{n\times m}$ such that

$E^{T}X=X^{T}E\geq 0$

$E^{T}W=0$

${\begin{bmatrix}X^{T}A+A^{T}X&A^{T}W+X^{T}B-C^{T}\\(A^{T}W+X^{T}B-C^{T})^{T}&W^{T}B+B^{T}W-(D^{T}+D)\end{bmatrix}}<0.$

The system is also ESPR if there exists ${\mathcal {X}}\in {\mathcal {R}}^{n\times n}$ such that

$E^{T}X=X^{T}E\geq 0$

${\begin{bmatrix}X^{T}A+A^{T}X&X^{T}B-C^{T}\\(X^{T}B-C^{T})^{T}&-(D^{T}+D)\end{bmatrix}}<0.$

Conclusion:

If there exist a X and W matrix satisfying above LMIs then the system ${\mathcal {G}}$ is Extended Strictly Positive Real.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
5. Numerical algorithms and software tools for analysis and modelling of descriptor systems. Prepr. of 2nd IFAC Workshop on System Structure and Control, Prague, Czechoslovakia, pp. 392-395, 1992.

Generealized KYP (GKYP) Lemma for conic Sectors

The Concept

The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

Also consider $\pi _{c}(a,b)\in {\mathcal {S}}^{m}$, which is defined as

$\pi _{c}(a,b)={\begin{bmatrix}-{\tfrac {1}{b}}I&{\tfrac {1}{2}}(1+{\tfrac {a}{b}})I\\({\tfrac {1}{2}}(1+{\tfrac {a}{b}})I)^{T}&-aI\end{bmatrix}}$,

where $a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$.

The Data

The matrices The matrices $A,B,C$ and $D$. The values of a and b

LMI : Generalized KYP (GKYP) Lemma for Conic Sectors

The following generalized KYP Lemmas give conditions for ${\mathcal {G}}$ to be inside the cone $[a,b]$ within finite frequency bandwidths.

1. (Low Frequency Range) The system ${\mathcal {G}}$ is inside the cone $[a,b]$ for all $\omega \in {\omega \in {\mathcal {R}}||\omega |<\omega _{1},det(j\omega I-A)\neq 0}$, where $\omega _{1}\in {\mathcal {R}}_{>0},a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$, if there exist $P,Q\in {\mathcal {S}}^{n}$ and ${\overline {\omega }}_{1}\in {\mathcal {R}}_{>0}$, where $Q\geq 0$, such that

${\begin{bmatrix}A&B\\I&0\end{bmatrix}}^{T}{\begin{bmatrix}-Q&P\\P^{T}&(\omega _{1}-{\overline {\omega }}_{1})^{2}Q\end{bmatrix}}{\begin{bmatrix}A&B\\I&0\end{bmatrix}}-{\begin{bmatrix}C&D\\0&I\end{bmatrix}}^{T}\pi _{c}(a,b){\begin{bmatrix}C&D\\0&I\end{bmatrix}}<0$.

If $\omega _{1}\rightarrow \infty ,P>0.$ and Q = 0, then the traditional Conic Sector Lemma is recovered. The parameter ${\overline {\omega }}_{1}$ is incuded in the above LMI to effectively transform $|\omega |\leq (\omega _{1}-{\overline {\omega }}_{1})$ into the strict inequality $|\omega |<\omega _{1}$

2. (Intermediate Frequency Range) The system ${\mathcal {G}}$ is inside the cone $[a,b]$ for all $\omega \in {\omega \in {\mathcal {R}}|\omega _{1}\leq |\omega |<\omega _{2},det(j\omega I-A)\neq 0}$, where $\omega _{1},\omega _{2}\in {\mathcal {R}}_{>0},a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$, if there exist $P,Q\in {\mathcal {C}}^{n}$ and ${\overline {\omega }}_{2}\in {\mathcal {R}}_{>0}$ and ${\hat {\omega }}_{2}=(\omega _{1}+{\tfrac {(\omega _{2}-{\hat {\omega }}_{2})}{2}}),$ where $P^{H}=P,Q^{H}=Q$ and $Q\geq 0$, such that

${\begin{bmatrix}A&B\\I&0\end{bmatrix}}^{T}{\begin{bmatrix}-Q&P+{\mathcal {j{\hat {\omega }}_{2}}}Q\\P-{\mathcal {j{\hat {\omega }}_{2}}}Q&\omega _{1}(\omega _{2}-{\hat {\omega }}-2)Q\end{bmatrix}}{\begin{bmatrix}A&B\\I&0\end{bmatrix}}-{\begin{bmatrix}C&D\\0&I\end{bmatrix}}^{T}\pi _{c}(a,b){\begin{bmatrix}C&D\\0&I\end{bmatrix}}<0$.

The parameter ${\overline {\omega }}_{2}$ is incuded in the above LMI to effectively transform $\omega _{1}\leq |\omega |\leq (\omega _{2}-{\overline {\omega }}_{2})$ into the strict inequality $\omega _{1}\leq |\omega |<\omega _{2}$.

3. (High Frequency Range) The system ${\mathcal {G}}$ is inside the cone $[a,b]$ for all $\omega \in {\omega \in {\mathcal {R}}|\omega _{2}<|\omega |,det(j\omega I-A)\neq 0}$, where $\omega _{2}\in {\mathcal {R}}_{>0},a\in {\mathcal {R}},b\in {\mathcal {R}}_{>0}$ and $a<b$, if there exist $P,Q\in {\mathcal {S}}^{n}$, where $Q\geq 0$, such that

${\begin{bmatrix}A&B\\I&0\end{bmatrix}}^{T}{\begin{bmatrix}-Q&P\\P^{T}&\omega _{2}^{2}Q\end{bmatrix}}{\begin{bmatrix}A&B\\I&0\end{bmatrix}}-{\begin{bmatrix}C&D\\0&I\end{bmatrix}}^{T}\pi _{c}(a,b){\begin{bmatrix}C&D\\0&I\end{bmatrix}}<0$.

If (A, B, C, D) is a minimal realization, then the matrix inequalities in all of the above LMI, then it can be nostrict.

Conclusion:

If there exist a positive definite $q$ matrix satisfying above LMIs for the given frequency bandwidths then the system ${\mathcal {G}}$ is inside the cone [a,b].

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
Exterior Conic Sector Lemma
Modified Exterior Conic Sector Lemma

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.

Discrete time Bounded Real Lemma

Discrete-Time Bounded Real Lemma

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Discrete-Time Bounded Real Lemma or the H∞ norm can be found by solving a LMI.

The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$
${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

The Data

The matrices: System $(A_{d},B_{d},C_{d},D_{d}),P$.

The Optimization Problem

The following feasibility problem should be optimized:

$\gamma$ is minimized while obeying the LMI constraints.

The LMI:

Discrete-Time Bounded Real Lemma

The LMI formulation

H∞ norm < $\gamma$

${\begin{aligned}P\in {S^{n}};\gamma \in {R_{>0}}\;\\&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}&C_{d}^{T}\\*&B_{d}^{T}PB_{d}-\gamma I&D_{d}^{T}\\*&*&-\gamma I\end{bmatrix}}&<0,\end{aligned}}$

Conclusion:

The H∞ norm is the minimum value of $\gamma \in {R_{>0}}$that satisfies the LMI condition. If $(A_{d},B_{d},C_{d},D_{d})$ is the minimal realization then the inequalities can be non-strict.

Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIs

[1] - Continuous time KYP_Lemma_(Bounded_Real_Lemma)

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Discrete Time KYP Lemma for QSR Dissipative System

The Concept

In systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state $x(t)$, its input $u(t)$ and its output $y(t)$, the input-output correlation is given a supply rate $w(u(t),y(t))$. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function $V(x(t))$ such that $V(0)=0$, $V(x(t))\geq 0$ and

${\dot {V}}(x(t))\leq w(u(t),y(t))$

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate $w(u(t),y(t))=u(t)^{T}y(t)$.

The physical interpretation is that $V(x)$ is the energy stored in the system, whereas $w(u(t),y(t))$ is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.

The System

Consider a discrete-time LTI system, ${\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}$, with minimal state-space relization $({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})$, where ${\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}$.

$x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)$

$y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...$

The Data

The matrices ${\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}$ and ${\mathcal {D}}_{d}$

The Optimization Problem

The system ${\mathcal {G}}$ is QSR disipative if

$\sum _{i\mathop {=} 0}^{K}({\mathcal {y}}_{i}^{T}Q{\mathcal {y}}_{i}+2{\mathcal {y}}_{i}^{T}S{\mathcal {u}}_{i}+{\mathcal {u}}_{i}^{T}R{\mathcal {u}}_{i})\,dt\geq 0,\forall u\in {\mathcal {l}}_{2e},\forall k\in {\mathcal {Z}}\geq 0$

where ${\mathcal {u}}_{k}$ is the input to ${\mathcal {G}},{\mathcal {y}}_{k}$ is the output of ${\mathcal {G}},Q\in {\mathcal {S}}^{p},S\in {\mathcal {R}}^{p\times m},$ and ${\mathcal {R}}\in {\mathcal {S}}^{m}$.

LMI : Discrete-Time KYP Lemma for QSR Dissipative Systems

The system ${\mathcal {G}}$ is also QSR dissipative if and only if there exists $P\in {\mathcal {S}}^{n},$ where $P>0,$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P-C_{d}^{T}QC_{d}&A_{d}^{T}PB_{d}-C_{d}^{T}S-C_{d}^{T}QD_{d}\\(A_{d}^{T}PB_{d}-C_{d}^{T}S-C_{d}^{T}QD_{d})^{T}&B_{d}^{T}PB_{d}-D_{d}^{T}QD_{d}-(D_{d}^{T}S+S^{T}D_{d})-R\end{bmatrix}}\leq 0.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is QSR dissipative.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
KYP Lemma for continous Time QSR Dissipative system

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2

Discrete Time KYP Lemma with Feedthrough

The Concept

It is assumed in the Lemma that the state-space representation (A_d, B_d, C_d, D_d) is minimal. Then Positive Realness (PR) of the transfer function C_d(SI − A_d)^-1B_d + D_d is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A_d, B_d, C_d, D_d)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The System

Consider a discrete-time LTI system, ${\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}$, with minimal state-space relization $({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})$, where ${\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}$.

$x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)$

$y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...$

The Data

The matrices ${\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}$ and ${\mathcal {D}}_{d}$

LMI : Discrete-Time KYP Lemma with Feedthrough

The system ${\mathcal {G}}$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $P>0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.$

3. There exists $P\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.$

4. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system ${\mathcal {G}}$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ where $P>0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.$

2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.$

3. There exists $P\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.$

4. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that

${\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.$

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $\in {\mathcal {R}}_{>0}.$

Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is Positive Real.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
KYP Lemma without Feedthrough

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London

Schur Complement

An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.

The Schur Compliment

Consider the matricies $Q$, $M$, and $R$ where $Q$ and $M$ are self-adjoint. Then the following statements are equivalent:

1. $Q>0$ and $M-RQ^{-1}R^{*}>0$ both hold.
2. $M>0$ and $Q-R^{*}M^{-1}R>0$ both hold.
3. ${\begin{bmatrix}M&R\\R^{*}&Q\end{bmatrix}}>0$ is satisfied.

More concisely:

${\begin{bmatrix}M&R\\R^{*}&Q\end{bmatrix}}>0\iff {\begin{bmatrix}M&0\\0&Q-R^{*}M^{-1}R\end{bmatrix}}>0\iff {\begin{bmatrix}M-RQ^{-1}R^{*}&0\\0&Q\end{bmatrix}}>0$

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

LMI for Eigenvalue Minimization

LMI for Minimizing Eigenvalue of a Matrix

Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.

The System

Assume that we have a matrix function of variables $x$:

${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$

where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ to minimize the following cost function:

${\begin{aligned}J(x)=\lambda _{\text{max}}(A(x))\end{aligned}}$

where $J(x)$ is the cost function and $\lambda _{\text{max}}(.)$ indicates the maximim eigenvalue of a matrix.

According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent

${\begin{aligned}\lambda _{max}(A(x))\leq t\iff A(x)-tI\leq 0\end{aligned}}$

where $t$ is defined as the maximim eigenvalue of the matrix $A$.

The LMI: LMI for eigenvalue minimization

This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

${\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)-tI\leq 0\end{aligned}}$

Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem.

Moreover, we obtain the maximum eigenvalue, $t$, of the matrix $A(x)$.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix

Related LMIs

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External Links

• [2] - LMI in Control Systems Analysis, Design and Applications

• State-space Representation of a System

• Eigenvalues and Eigenvectors of a Matrix

LMI for Matrix Norm Minimization

LMI for Matrix Norm Minimization

This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an $H_{2}$ or an $H_{\infty }$ optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm.

The System

Assume that we have a matrix function of variables $x$:

${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$

where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ in order to minimize the following cost function:

${\begin{aligned}J(x)=||A(x)||_{2}\end{aligned}}$

where $J(x)$ is the cost function and $||.||_{2}$ indicates the norm of the matrix function $A$.

According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent:

${\begin{aligned}A^{T}A-t^{2}I\leq 0\iff {\begin{bmatrix}-tI&A\\A^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

The LMI: LMI for matrix norm minimization

This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

${\begin{aligned}&{\text{min}}\quad t&\\&{\text{s.t.}}\quad {\begin{bmatrix}-tI&A(x)\\A(x)^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem and we obtain $t$ that is the norm of matrix function $A(x)$.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

Related LMIs

LMI for Matrix Norm Minimization

LMI for Generalized Eigenvalue Problem

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External Links

A list of references documenting and validating the LMI.

• [3] - LMI in Control Systems Analysis, Design and Applications

LMI for Generalized Eigenvalue Problem

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like $A$ and $B$, to find the generalized eigenvector, $x$, and eigenvalues, $\lambda$, that satisfies $Ax=\lambda Bx$. If the matrix $B$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

The System

Assume that we have three matrice functions which are functions of variables $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}$ as follows:

$A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}$

$B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}$

$C(x)=C_{0}+C_{1}x_{1}+...+C_{n}x_{n}$

where are $A_{i}$, $B_{i}$, and $C_{i}$ ($i=1,2,...,n$) are the coefficient matrices.

The Data

The $A(x)$, $B(x)$, and $C(x)$ are matrix functions of appropriate dimensions which are all linear in the variable $x$ and $A_{i}$, $B_{i}$, $C_{i}$ are given matrix coefficients.

The Optimization Problem

The problem is to find ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ such that:

$A(x)<\lambda B(x)$, $B(x)>0$, and $C(x)<0$ are satisfied and $\lambda$ is a scalar variable.

The LMI: LMI for Schur stabilization

A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:

${\begin{aligned}&{\text{min}}\quad \lambda \\&{\text{s.t.}}\quad A(x)<\lambda B(x)\\&\quad \quad B(x)>0\\&\quad \quad C(x)<0\end{aligned}}$

Conclusion:

The solution for this LMI problem is the values of variables $x$ such that the scalar parameter, $\lambda$, is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIs

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

External Links

• [4] - LMI in Control Systems Analysis, Design and Applications

LMI for Linear Programming

LMI for Linear Programming

Linear programming has been known as a technique for the optimization of a linear objective function subject to linear equality or inequality constraints. The feasible region for this problem is a convex polytope. This region is defined as a set of the intersection of many finite half-spaces which are created by the inequality constraints. The solution for this problem is to find a point in the polytope of existing solutions where the objective function has its extremum (minimum or maximum) value.

The System

We define the objective function as:

$c_{1}x_{1}+c_{2}x_{2}+...c_{n}x_{n}$

and constraints of the problem as:

$a_{11}x_{1}+a_{12}x_{2}+...+a_{1n}x_{n}<b_{1}$

$a_{21}x_{1}+a_{22}x_{2}+...+a_{2n}x_{n}<b_{2}$

.

.

.

$a_{m1}x_{1}+a_{m2}x_{2}+...+a_{mn}x_{n}<b_{m}$

The Data

Suppose that $c_{j}\in \mathbb {R} ^{n}$, $a_{ij}\in \mathbb {R} ^{n}$, and $b_{i}\in \mathbb {R} ^{m}$ are given parameters where $i=1,2,...,m$ and $j=1,2,...,n$. Moreover, $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}$ is an $n\times 1$ vector of positive variables.

The Optimization Problem

The optimization problem is to minimize the objective function, $c^{\text{T}}x$ when the aforementioned linear constraints are satisfied.

The LMI: LMI for linear programming

The mathematical description of the optimization problem can be readily written in the following LMI formulation:

${\begin{aligned}&{\text{min}}\quad c^{\text{T}}x&\\&{\text{s.t.}}\quad a_{i}^{\text{T}}x\leq b_{i}x\\\end{aligned}}$

Conclusion:

Solving this problem results in the values of variables $x$ which minimize the objective function. It is also worthwhile to note that if $m\geq n$, the computational cost for solving this problem would be $mn^{2}$.

There does not exist an analytical formulation to solve a general linear programming problem. Nonetheless, there are some efficient algorithms, like the Simplex algorithm, for solving a linear programming problem.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Linear-Programming

Related LMIs

LMI for Feasibility Problem

External Links

• [5] - LMI in Control Systems Analysis, Design and Applications

LMI for Feasibility Problem

LMI for Feasibility Problem in Optimization

The feasibility problem is to find any feasible solutions for an optimization problem without regard to the objective value. This problem can be considered as a special case of an optimization problem where the objective value is the same for all the feasible solutions. Many optimization problems have to start from a feasible point in the range of all possible solutions. One way is to add a slack variable to the problem in order to relax the feasibility condition. By adding the slack variable the problem any start point would be a feasible solution. Then, the optimization problem is converted to find the minimum value for the slack variable until the feasibility is satisfied.

The System

Assume that we have two matrices as follows:

${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\quad i=1,2,...,n\end{aligned}}$

${\begin{aligned}B(x)=B_{0}+B_{1}x_{1}+...+B_{n}x_{n}\quad i=1,2,...,n\end{aligned}}$

which are matrix functions of variables $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}\in \mathbb {R} ^{n}$.

The Data

Suppose that the matrices ${\begin{aligned}A_{0},A_{1},...,A_{n}\quad \end{aligned}}$ and ${\begin{aligned}B_{0},B_{1},...,B_{n}\quad \end{aligned}}$ are given.

The Optimization Problem

The optimization problem is to find variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ such that the following constraint is satisfied:

${\begin{aligned}A(x)<B(x)\end{aligned}}$

The LMI: LMI for Feasibility Problem

This optimization problem can be converted to a standard LMI problem by adding a slack variable, $t$.

The mathematical description for this problem is to minimize $t$ in the following form of the LMI formulation:

${\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)<B(x)+tI\\\end{aligned}}$

Conclusion:

In this problem, $x$ and $t$ are decision variables of the LMI problem.

As a result, these variables are determined in the optimization problem such that the minimum value of $t$ is found while the inequality constraint is satisfied.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization

Related LMIs

LMI for Linear Programming

External Links

• [6] - LMI in Control Systems Analysis, Design and Applications

Structured Singular Value

User:ShakespeareFan00/Sandbox

The LMI can be used to find a $\Theta$ that belongs to the set of scalings $P\Theta$. Minimizing $\gamma$ allows to minimize the squared norm of $\Theta M\Theta ^{-}1$.

The System

${\begin{aligned}M{\text{ with transfer function }}{\hat {M}}(s)=C(sI-A)^{-1}B+D,&&{\hat {M}}\in H_{\infty }\end{aligned}}$

The Data

The matrices $A\in R^{n\times n},B\in R^{n\times m},C\in R^{o\times n},D\in R^{o\times m}$.

The Optimization Problem

${\begin{aligned}{\text{There exists }}\Theta \in \Theta {\text{ such that }}||\Theta M\Theta ^{-1}||^{2}<\gamma .\end{aligned}}$

The LMI:

${\begin{aligned}{\text{Find}}\;&X>0:\\{\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\Theta \end{bmatrix}}+\gamma ^{-2}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}\Theta {\begin{bmatrix}C&D\end{bmatrix}}<0\\\end{aligned}}$

Conclusion:

${\text{The optimization problem and the LMI are equivalent. }}\gamma {\text{ must be optimized using bisection.}}$

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Eigenvalue Problem

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

Eigenvalue Problem

User:ShakespeareFan00/Sandbox

The maximum eigenvalue of a matrix is going to have the most impact on system performance. This LMI allows for minimization of the maximum eigenvalue by minimizing $\gamma$.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)\end{aligned}}$

The Data

The matrices $A\in R^{n\times n},B\in R^{n\times m},C\in R^{o\times n}$.

The Optimization Problem

${\text{ Minimize }}\gamma {\text{ subject to the LMI below.}}$

The LMI:

${\begin{aligned}{\text{Find}}\;&P>0:\\{\begin{bmatrix}-A^{T}P-PA-C^{T}C&PB\\B^{T}P&\gamma I\end{bmatrix}}>0\\\end{aligned}}$

Conclusion:

The eigenvalue problem can be utilized to minimize the maximum eigenvalue of a matrix that depends affinely on a variable.

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Structured Singular Value

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

LMI for Minimizing Condition Number of Positive Definite Matrix

User:ShakespeareFan00/Sandbox

The System:

A related problem is minimizing the condition number of a positive-defnite matrix $M$ that depends affinely on the variable $x$, subject to the LMI constraint $F(x)$ > 0. This problem can be reformulated as the GEVP.

The Optimization Problem:

The GEVP can be formulated as follows:

minimize $\gamma$

subject to $F(x)$ > 0;

$\mu$>0;

$\mu I$< $M(x)$ < $\gamma \mu I$.

We can reformulate this GEVP as an EVP as follows. Suppose,

$M(x)$= $M_{0}$ +$\sum _{n=1}^{m}$$x_{i}M_{i}$ , $F(x)$= $F_{0}$+ $\sum _{n=1}^{m}$$x_{i}F_{i}$

The LMI:

Defining the new variables $\nu$=$1/\mu$ , ${\tilde {x}}$=$x/\mu$ we can express the previous formulation as the EVP with variables ${\tilde {x}},\nu$ and $\gamma$:

miminize$\gamma$

subject to $\nu F_{0}$+ $\sum _{n=1}^{m}$$x_{i}F_{i}$ >0; $I$ < $\nu M_{0}$+ $\sum _{n=1}^{m}$$x_{i}M_{i}$ < $\gamma I$

Conclusion:

The LMI is feasible.

References

• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Continuous Quadratic Stability

User:ShakespeareFan00/Sandbox

To study stability of a LTI system, we first ask whether all trajectories of system converge to zero as $t\rightarrow \infty$. A sufficient condition for this is the existence of a quadratic function $V(\xi )=\xi ^{T}P\xi$, $P>0$ that decreases along every nonzero trajectory of system . If there exists such a P, we say the system is quadratically stable and we call $V$ a quadratic Lyapunov function.

The System

${\begin{aligned}{\dot {x}}(t)&=A(\delta (t))x(t)\\\end{aligned}}$

The Data

The system coefficient matrix takes the form of

${\begin{aligned}{\dot {x}}(t)&=A_{0}+\Delta A(\delta (t))x(t)\\\end{aligned}}$

where $A_{0}\in \mathbb {R}$is a known matrix, which represents the nominal system matrix, while ${\begin{aligned}\Delta A(\delta (t))x(t)=\delta _{1}(t))A_{1}+\delta _{2}(t))A_{2}+...+\delta _{k}(t))A_{k}\end{aligned}}$ is the system matrix perturbation, where

$A_{i}\in \mathbb {R} ^{n\times n},i=1,2,..,k,$ are known matrices, which represent the perturbation matrices.

$\delta _{i}(t),i=1,2,...,k,$ which represent the uncertain parameters in the system.

$\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]^{T}$ is the uncertain parameter vector, which is often assumed to be within a certain compact and convex set : : $\Delta$ that is

$\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta ]^{T}\in \Delta$

The LMI: Continuous-Time Quadratic Stability

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+\Delta A(\delta (t))x(t))^{T}+P(A_{0}+\Delta A(\delta (t))x(t))<0\delta (t)\in \Delta \end{aligned}}$

The following statements can be made for particular sets of perturbations.

Case 1: Regular Polyhedron

Consider the case where the set of perturbation parameters is defined by a regular polyhedron as

${\begin{aligned}\Delta ={\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]\in \mathbb {R} ^{k}\mid \delta _{i}(t),{\underline {\delta _{i}}}(t),{\overline {\delta _{i}}}(t),{\underline {\delta _{i}}}\leq \delta _{i}(t)\leq {\overline {\delta _{i})}}}\end{aligned}}$

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+\Delta A(\delta (t))x(t))^{T}+P(A_{0}+\Delta A(\delta (t))x(t))<0\delta _{i}(t)\in {{\underline {\delta _{i}}},{\overline {\delta _{i}}}},i=1,2,...k.\end{aligned}}$

Case 2: Polytope

Consider the case where the set of perturbation parameters is defined by a polytope as

${\begin{aligned}\Delta ={\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]\in \mathbb {R} ^{k}\mid \delta _{i}(t)\in \mathbb {R} _{\geq 0}},\sum _{i=1}^{k}\delta _{i}(t)=1\end{aligned}}$

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+A_{i})^{T}P+P(A_{0}+A_{i})<0,i=1,2...,k.\end{aligned}}$

Conclusion:

If feasible, System is Quadratically stable for any $x\in \mathbb {R} ^{n}$

Implementation

https://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.

Exterior Conic Sector Lemma

The Concept

The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

The Data

The matrices The matrices $A,B,C$ and $D$

LMI : Exterior Conic Sector Lemma

The system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c (i.e. ${\mathcal {G}}\in$excone_r(c)), where $r\in {\mathcal {R}}_{>0}$ and $\in {\mathcal {R}}$, under either of the following equivalent necessary and sufficient conditions.

1. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that

${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq 0.$

2. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that

${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)&0\\(PB-C^{T}(D-CI))^{T}&-(D-cI)^{T}(D-cI)&rI\\0&(rI)^{T}&-I\end{bmatrix}}\leq 0.$

Proof, Applying the Schur complement lemma to the $r^{2}I$ terms in (1) gives (2).

Conclusion:

If there exist a positive definite $P$ matrix satisfying above LMIs then the system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transactions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.

Modified Exterior Conic Sector Lemma

The Concept

The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

The Data

The matrices The matrices $A,B,C$ and $D$

LMI : Modified Exterior Conic Sector Lemma

The system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c (i.e. ${\mathcal {G}}\in$excone_r(c)), where $r\in {\mathcal {R}}_{>0}$ and $\in {\mathcal {R}}$, under either of the following sufficient conditions.

1. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that

${\begin{bmatrix}PA+A^{T}P&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq 0.$

Proof. The term $-C^{T}C$ in the Actual Exterior Conic Sector Lemma makes the matrix inequality more neagtive definite.

Therefore,

${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq {\begin{bmatrix}PA+A^{T}P&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}$

2. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that

${\begin{bmatrix}PA+A^{T}P&PB-C^{T}(D-CI)&0\\(PB-C^{T}(D-CI))^{T}&-(D-cI)^{T}(D-cI)&rI\\0&(rI)^{T}&-I\end{bmatrix}}\leq 0.$

Proof. Applying the Schur complement lemma to the $r^{2}I$ terms in (1) gives (2).

Conclusion:

If there exist a positive definite $P$ matrix satisfying above LMIs then the system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
Exterior Conic Sector Lemma

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.

DC Gain of a Transfer Matrix

The continuous-time DC gain is the transfer function value at the frequency s = 0.

The System

Consider a square continuous time Linear Time invariant system, with the state space realization $(A,B,C,D)$ and $\gamma \in \mathbb {R} _{>0}$

${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}$

The Data

$A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},D\in \mathbb {R} ^{p\times m}$

The LMI: LMI for DC Gain of a Transfer Matrix

The transfer matrix is given by$G(s)=C(sI-A)^{-1}B+D$
The DC Gain of the system is strictly less than $\gamma$ if the following LMIs are satisfied.

${\begin{bmatrix}\gamma I&-CA^{-1}B+D\\(-CA^{-1}B+D)'&\gamma I\end{bmatrix}}$${\begin{aligned}>0\end{aligned}}$

OR

${\begin{bmatrix}\gamma I&-B^{T}A^{-T}C^{T}+DT\\(-B^{T}A^{-T}C^{T}+DT)'&\gamma I\end{bmatrix}}$${\begin{aligned}>0\end{aligned}}$

Conclusion

The DC Gain of the continuous-time LTI system, whose state space realization is give by ($A,B,C,D$), is
$K=D-CA^{-1}B$

• Upon implementation we can see that the value of $\gamma$ obtained from the LMI approach and the value of $K$ obtained from the above formula are the same

Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

External Links

• [7] - LMI in Control Systems Analysis, Design and Applications
• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• [8] -Mathworks reference to DC Gain

Discrete Time H2 Norm

Discrete-Time H2 Norm

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Discrete-Time LTI systems' H2 norm can be found by solving a LMI.

The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$
${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

The Data

The matrices: System $(A_{d},B_{d},C_{d},D_{d}),P,Z$.

The Optimization Problem

The following feasibility problem should be optimized:

$\mu$ is minimized while obeying the LMI constraints.

The LMI:

Discrete-Time Bounded Real Lemma

The LMI formulation

H2 norm < $\mu$

${\begin{aligned}P\in {S^{n}};Z\in {S^{p}};\mu \in {R_{>0}}\;\\&P>0,&Z>0\\{\begin{bmatrix}P&A_{d}P&B_{d}\\*&P&0\\*&*&I\end{bmatrix}}&>0,\\{\begin{bmatrix}Z&C_{d}P\\*&P\end{bmatrix}}&>0,\\trZ<\mu ^{2}\end{aligned}}$

Conclusion:

The H2 norm is the minimum value of $\mu \in {R_{>0}}$that satisfies the LMI condition.

Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIs

[9] - Continuous time H2 norm.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• [10] - MATLAB function for the H2 norm.

Discrete Time Minimum Gain Lemma

The Concept

The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:

$Y(s)=P(s)U(s)$

$U(s)=C(s)E(s)$

$E(s)=R(s)-F(s)Y(s).$

Solving for Y(s) in terms of R(s) gives

$Y(s)=\left({\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\right)R(s)=H(s)R(s).$

The expression $H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $|P(s)C(s)|\gg 1$, i.e., it has a large norm with each value of s, and if $|F(s)|\approx 1$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the ouput closely tracks the reference input.

The System

Consider a discrete-time LTI system, ${\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}$, with minimal state-space relization $({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})$, where ${\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}$.

$x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)$

$y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...$

The Data

The matrices ${\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}$ and ${\mathcal {D}}_{d}$

LMI : Discrete-Time Minimum Gain Lemma

The system ${\mathcal {G}}$ has minimium gain γ under any of the following equivalent sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ and γ $\in {\mathcal {R}}_{\geq 0}$ where $P\geq 0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P-C_{D}^{T}C_{D}&A_{d}^{T}PB_{d}-C_{d}^{T}D_{d}\\(A_{d}^{T}PB_{d}-C_{d}^{T}D_{d})^{T}&B_{d}^{T}PB_{d}+\gamma ^{2}I-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.$

2. There exists $P\in {\mathcal {S}}^{n},$ and $\gamma \in {\mathcal {R}}_{\geq 0}$ where $P\geq 0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P-C_{D}^{T}C_{D}&A_{d}^{T}PB_{d}-C_{d}^{T}D_{d}&0\\(A_{d}^{T}PB_{d}-C_{d}^{T}D_{d})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})&\gamma I\\0&\gamma I&I\end{bmatrix}}\leq 0.$

$proof$ : Applying the Schur complement lemma to the γ² term in equation 1 gives equation 2.

Conclusion:

If there exist a positive definite $P$ for the system ${\mathcal {G}}$, then the minimum gain of the system is γ can be obtaied from above defined LMIs.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
KYP Lemma without Feedthrough

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London

Modified Discrete Time Minimum Gain Lemma

The Concept

The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:

$Y(s)=P(s)U(s)$

$U(s)=C(s)E(s)$

$E(s)=R(s)-F(s)Y(s).$

Solving for Y(s) in terms of R(s) gives

$Y(s)=\left({\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\right)R(s)=H(s)R(s).$

The expression $H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $|P(s)C(s)|\gg 1$, i.e., it has a large norm with each value of s, and if $|F(s)|\approx 1$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the ouput closely tracks the reference input.

The System

Consider a discrete-time LTI system, ${\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}$, with minimal state-space relization $({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})$, where ${\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}$.

$x(k+1)={\mathcal {A}}_{d}x(k)+{\mathcal {B}}_{d}u(k)$

$y(k)={\mathcal {C}}_{d}x(k)+{\mathcal {D}}_{d}u(k),k=0,1...$

The Data

The matrices ${\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d}$ and ${\mathcal {D}}_{d}$

LMI : Discrete-Time Modified Minimum Gain Lemma

The system ${\mathcal {G}}$ has minimium gain γ under any of the following equivalent sufficient conditions.

1. There exists $P\in {\mathcal {S}}^{n},$ and γ $\in {\mathcal {R}}_{\geq 0}$ where $P\geq 0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}D_{d}\\(A_{d}^{T}PB_{d}-C_{d}^{T}D_{d})^{T}&B_{d}^{T}PB_{d}+\gamma ^{2}I-(D_{d}^{T}+D_{d})\end{bmatrix}}$

$Proof$. The term $-C_{d}^{T}C_{d}$ in Discrete Time Minimum Gain Lemma makes the matrix inequality more negative definite. Therefore,

${\begin{bmatrix}A_{d}^{T}PA_{d}-P-C_{D}^{T}C_{D}&A_{d}^{T}PB_{d}-C_{d}^{T}D_{d}\\(A_{d}^{T}PB_{d}-C_{d}^{T}D_{d})^{T}&B_{d}^{T}PB_{d}+\gamma ^{2}I-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq {\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}D_{d}\\(A_{d}^{T}PB_{d}-C_{d}^{T}D_{d})^{T}&B_{d}^{T}PB_{d}+\gamma ^{2}I-(D_{d}^{T}+D_{d})\end{bmatrix}}$

2. There exists $P\in {\mathcal {S}}^{n},$ and $\gamma \in {\mathcal {R}}_{\geq 0}$ where $P\geq 0$ such that

${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}D_{d}&0\\(A_{d}^{T}PB_{d}-C_{d}^{T}D_{d})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})&\gamma I\\0&\gamma I&I\end{bmatrix}}\leq 0.$

$proof$ : Applying the Schur complement lemma to the γ² term in equation 1 gives equation 2.

Conclusion:

If there exist a positive definite $P$ for the system ${\mathcal {G}}$, then the minimum gain of the system γ can be obtaied from above defined LMIs.

Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs

KYP Lemma
State Space Stability
KYP Lemma without Feedthrough
Discrete Time Minimum Gain Lemma

References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London

Discrete-Time Algebraic Riccati Equation

User:ShakespeareFan00/Sandbox

The System

Consider a Discrete-Time LTI system

${\begin{aligned}x_{k+1}=A_{d}x_{k}+B_{d}u_{k}\end{aligned}}$

${\begin{aligned}y_{k}=C_{d}x_{k}\end{aligned}}$

Consider A_d ∈ ${\begin{aligned}\mathbb {R} \end{aligned}}$^nxn ; B_d ∈ ${\begin{aligned}\mathbb {R} \end{aligned}}$^nxm

The Data

The Matrices A_d , B_d , C_d , Q, R are given

Q and R should necessarily be Hermitian Matrices.

A square matrix is Hermitian if it is equal to its complex conjugate transpose.

The Optimization Problem

Our aim is to find

P - Unique solution to the discrete-time algebraic Riccati equation, returned as a matrix.

K - State-feedback gain, returned as a matrix.

The algorithm used to evaluate the State-feedback gain is given by

${\begin{aligned}K=(R+B_{d}^{T}PB_{d})^{-1}B_{d}^{T}PA_{d}\end{aligned}}$

L - Closed-loop eigenvalues, returned as a matrix.

The LMI: Discrete-Time Algebraic Riccati Inequality (DARE)

An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time

The Discrete-Time Algebraic Riccati Inequality is given by

${\begin{aligned}A_{d}^{T}PA_{d}-A_{d}^{T}PB_{d}(R+B_{d}^{T}PB_{d})^{-1}B_{d}^{T}PA_{d}+Q-P\geq 0\end{aligned}}$

P , Q ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$ⁿ and R ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$^m where P > 0, Q ≥ 0, R > 0

P is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices.

The above equation can be rewritten using the Schur Complement Lemma as :

${\begin{bmatrix}A_{d}^{T}PA_{d}-P+Q&A_{d}^{T}PB_{d}\\B_{d}^{T}PA_{d}&R+B_{d}^{T}PB_{d}\end{bmatrix}}$${\begin{aligned}\geq 0\end{aligned}}$

Conclusion:

Algebraic Riccati Inequalities play a key role in LQR/LQG control, H2- and H∞ control and Kalman filtering. We try to find the unique stabilizing solution, if it exists. A solution is stabilizing, if controller of the system makes the closed loop system stable.

Equivalently, this Discrete-Time algebraic Riccati Inequality is satisfied under the following necessary and sufficient condition:

${\begin{bmatrix}Q&0&A_{d}^{T}P&P\\0&R&B_{d}^{T}P&0\\PA_{d}&PB_{d}&P&0\\P&0&0&P\end{bmatrix}}$${\begin{aligned}\geq 0\end{aligned}}$

Implementation

( X in the output corresponds to P in the LMI )

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

Related LMIs

LMI for Continuous-Time Algebraic Riccati Inequality

LMI for Schur Stabilization

External Links

A list of references documenting and validating the LMI.

• [11] - LMI in Control Systems Analysis, Design and Applications

Deduced LMI Conditions for H-infinity Index

H-infinity Index Deduced LMI

Although the KYP Lemma, also known as the Bounded Real Lemma, is a basic condition to evaluate an upper bound on the H_∞, the verification of the bound on the H_∞-gain of the system can be done via the deduced condition.

The System

A state-space representation of a linear system as given below:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bw(t)\\y(t)&=Cx(t)+Dw(t)\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $w(t)\in \mathbb {R} ^{r}$ are the system state, output, and the disturbance vector respectively. The transfer function of such a system can be evaluated as:

${\begin{aligned}G(s)=C(sI-A)^{-1}B+D\end{aligned}}$

The Data

Number of states n, number of outputs m and number of external noise channels r need to be known. Moreover, the system matrices A,B,C,D are also required to be known.

The Feasibility LMI

For an arbitrary $\gamma$, the transfer function G(s) satisfies

$\left\|G(s)\right\|_{\infty }<\gamma$

if and only if there exists a symmetric matrix X > 0 and a matrix $\Omega .$ such that:

${\begin{aligned}{\text{Find}}\;X,\Omega :&\\\Theta +\Phi ^{\top }\Omega \Psi +\Psi ^{\top }\Omega ^{\top }\Phi <0\end{aligned}}$

where:

${\begin{aligned}\Theta &={\begin{bmatrix}0&X&0&0&0\\X&-X&0&0&0\\0&0&-\gamma I_{m}&0&0\\0&0&0&-X&0\\0&0&0&0&\gamma I_{r}\end{bmatrix}}\\\Phi &={\begin{bmatrix}-I_{n}&A^{\top }&C^{\top }&I_{n}&0\\0&B^{\top }&D^{\top }&0&-\gamma I_{r}\end{bmatrix}}\\\Psi &={\begin{bmatrix}I_{n}&0&0&0&0\\0&0&0&0&I_{r}\end{bmatrix}}\end{aligned}}$

The above LMI can be combined with the bisection method to find minimum $\gamma$ to find the minimum upper bound on the H_∞ gain of $G(s)$.

Conclusion:

If there is a feasible solution to the aforementioned LMI, then the $\gamma$ upper bounds the infinity norm of the system G(s).

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Deduced_hinf_example.m

Related LMIs

Bounded Real Lemma

External Links

A list of references documenting and validating the LMI.

• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 5.2.2 pp. 153–156.

Deduced LMI Conditions for H₂ Index

H₂ Index Deduced LMI

Although there are ways to evaluate an upper bound on the H₂, the verification of the bound on the H₂-gain of the system can be done via the deduced condition.

The System

We consider the generalized Continuous-Time LTI system with the state space realization of $(A,B,C,D)$

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $u(t)\in \mathbb {R} ^{r}$ are the system state, output, and the input vectors respectively.
The transfer function of such a system can be evaluated as:

${\begin{aligned}G(s)=C(sI-A)^{-1}B+D\end{aligned}}$

The Data

The system matrices $A,B,C$ are known.

The Optimization Problem

For an arbitrary $\gamma >0$(a given scalar), the transfer function satisfies

$\left\|C(sI-A)^{-1}B+D\right\|_{2}<\gamma$

The H₂-norm condition on Transfer function holds only when the matrix A is stable. And this can be conveniently converted to an LMI problem

if and only if 1. There exists a symmetric matrix $X>0$ such that:
$AX+XA^{T}+BB^{T}<0$, $trace(CXC^{T})<\gamma ^{2}$

2. There exists a symmetric matrix $Y>0$ such that:
$AY+YA^{T}+C^{T}C<0$, $trace(B^{T}YB)<\gamma ^{2}$

The LMI - Deduced Conditions for H2-norm

These deduced condition can be derived from the above equations. According to this

For an arbitrary $\gamma >0$(a given scalar), the transfer function satisfies

$\left\|G(s)\right\|_{2}<\gamma$

if and only if there exists symmetric matrices $Z$ and $P$; and a matrix $V$ such that
$trace(Z)<\gamma ^{2}$
${\begin{bmatrix}-Z&B^{T}\\B&-P\end{bmatrix}}$${\begin{aligned}<0\end{aligned}}$
${\begin{bmatrix}-(V+V^{T})&V^{T}A^{T}+P&V^{T}C^{T}&V^{T}\\AV+P&-P&0&0\\CV&0&-I&0\\V&0&0&-P\end{bmatrix}}$${\begin{aligned}<0\end{aligned}}$

The above LMI can be combined with the bisection method to find minimum $\gamma$ to find the minimum upper bound on the H₂ gain of $G(s)$.

Conclusion:

If there is a feasible solution to the aforementioned LMI, then the $\gamma$ upper bounds the norm of the system G(s).

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the problem, and SeDuMi or MOSEK is required to solve the problem. The following link showcases an example of the problem:

https://github.com/yashgvd/ygovada

Related LMIs

Bounded Real Lemma
Deduced LMIs for H-infinity index

External Links

A list of references documenting and validating the LMI.

• [12] - LMI in Control Systems Analysis, Design and Applications
• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.

Dissipativity of Systems

Dissipativity of Systems

The dissipativity of systems is associated with their supply function. In general, a linear system is dissipative if the accumulated sum (integration) of the supply function is non-negative over all the duration of $T\geq 0$.

The System

A state-space representation of a linear system as given below:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $u(t)\in \mathbb {R} ^{r}$ are the system state, output, and the input vector respectively. A, B, C and D are system coefficient matrices of appropriate dimensions. The control input u is restricted to be a piece-wise continuous vector function defined of $[0,\infty )$.

The transfer function of such a system can be evaluated as:

${\begin{aligned}G(s)=C(sI-A)^{-1}B+D\end{aligned}}$

For such a system, a general quadratic supply function is defined as:

${\begin{aligned}s(u,y)&={\begin{bmatrix}y&u\end{bmatrix}}Q{\begin{bmatrix}y\\u\end{bmatrix}}\end{aligned}}$

where Q is a real symmetric matrix of (m+r) dimensions. Q need not be either symmetric positive or negative definite.

The Data

Number of states n, number of outputs m and number of control inputs r need to be known. Moreover, the system matrices A,B,C,D are also required to be known. The system should also be controllable.

The Feasibility LMI

The system $G(s)$ defined can be evaluated to be dissipative with respect to a supply function $s(u,y)$ iff there exist $P\geq 0$ and a $Q$ (defining $s(u,y)$) such that the following is feasible:

${\begin{aligned}{\text{Find}}\;&P,Q:\\&P\geq 0\\{\begin{bmatrix}A^{\top }P+PA&PB\\B^{\top }P&0\end{bmatrix}}&-{\begin{bmatrix}C&D\\0&I\end{bmatrix}}^{\top }Q{\begin{bmatrix}C&D\\0&I\end{bmatrix}}\leq 0.\end{aligned}}$

Conclusion:

If there is a feasible solution to the aforementioned LMI, then there exists a supply function $s(u,y)$ for which the system $G(s)$ is dissipative. Since the assumption of the system being controllable is required for it to be dissipative, this check can be used of as a sufficient condition to check the controllability of the linear system, just like the feasibility for Lyapunov stability.

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Dissipativity_example.m

Related LMIs

Continuous_Time_Lyapunov_Inequality

External Links

A list of references documenting and validating the LMI.

• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.3 pp. 178–184.

D-Stabilization

$\mathbb {D} _{(q,r)}$-Stabilization

There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of $\mathbb {D} _{(q,r)}$-Stabilization, a form of $\mathbb {D}$-Stabilization where the closed-loop poles are located on the left-half of the complex plane.

The System

For this particular problem, suppose that we were given a linear system in the form of:

${\begin{aligned}\rho x&=Ax+Bu,\\\end{aligned}}$

where $x\in \mathbb {R} ^{n}$, $u\in \mathbb {R} ^{r}$, and $\rho$ represents either the differential operator (in the continuous-time case) or the one-step forward operator (for the discrete-time system case). Then the LMI for determining the $\mathbb {D} _{(q,r)}$-stabilization case would be obtained as described below.

The Data

In order to obtain the LMI, we need the following 2 matrices: $A{\text{ and }}B$.

The Optimization Problem

Suppose - for the linear system given above - we were asked to design a state-feedback control law where $u=Kx$ such that the closed-loop system:

${\begin{aligned}\rho x&=(A+BK)x\\\end{aligned}}$

is $\mathbb {D} _{(q,r)}$-stable, then the system would be stabilized as follows.

The LMI: $\mathbb {D} _{(q,r)}$-Stabilization

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix $W$ and a symmetric matrix $P$ that satisfies the following:

${\begin{aligned}{\begin{bmatrix}-rP&&qP+AP+BW\\qP+PA^{T}+{W^{T}}{B^{T}}&&-rP\end{bmatrix}}<0\end{aligned}}$

Conclusion:

Given the resulting controller matrix $K=WP^{-1}$, it can be observed that the matrix is $\mathbb {D} _{(q,r)}$-stable.

Implementation

• Example Code - A GitHub link that contains code (titled "DStability.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

• H stabilization - Equivalent LMI for $\mathbb {H} _{(\alpha ,\beta )}$-stabilization.
• Continuous Time D-Stability Controller - LMI for deriving a Controller using D-Stability.
• Continuous Time D-Stability Observer - LMI for deriving an Observer using D-Stability.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.

H-Stabilization

$\mathbb {H} _{(\alpha ,\beta )}$-Stabilization

There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of $\mathbb {H} _{(\alpha ,\beta )}$-Stabilization, a form of $\mathbb {D}$-Stabilization where the real components are located on the left-half of the complex plane.

The System

For this particular problem, suppose that we were given a linear system in the form of:

${\begin{aligned}{\dot {x}}&=Ax+Bu,\\\end{aligned}}$

where $x\in \mathbb {R} ^{n}$ and $u\in \mathbb {R} ^{r}$. Then the LMI for determining the $\mathbb {H} _{(\alpha ,\beta )}$-stabilization case would be obtained as described below.

The Data

In order to obtain the LMI, we need the following 2 matrices: $A{\text{ and }}B$.

The Optimization Problem

Suppose - for the linear system given above - we were asked to design a state-feedback control law where $u=Kx$ such that the closed-loop system:

${\begin{aligned}{\dot {x}}&=(A+BK)x\\\end{aligned}}$

is $\mathbb {H} _{(\alpha ,\beta )}$ stable, then the system would be stabilized as follows.

The LMI: $\mathbb {H} _{(\alpha ,\beta )}$-Stabilization

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix $W$ and a symmetric matrix $P>0$ that satisfy the following:

${\begin{aligned}{\begin{cases}AP+P{A^{T}}+BW+{W^{T}}{B^{T}}+2{\alpha }P&<0\\-AP-P{A^{T}}-BW-{W^{T}}{B^{T}}-2{\beta }P&<0\end{cases}}\end{aligned}}$

Conclusion:

Given the resulting controller matrix $K=WX^{-1}$, it can be observed that the matrix is $\mathbb {H} _{(\alpha ,\beta )}$-stable.

Implementation

• Example Code - A GitHub link that contains code (titled "HStability.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

• D stabilization - Equivalent LMI for $\mathbb {D} _{(q,r)}$-stabilization.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.

H-2 Norm of the System

${H_{2}}$-norm of System

The ${H_{2}}$-norm is conceptually identical to the Frobenius (aka Euclidean) norm on a matrix. It can be used to determine whether the system representation can be reduced to its simplest form, thereby allowing its use in performing effective block-diagram algebra.

The System

Suppose we define the state-space system$G:{L_{2}}\rightarrow {L_{2}}{\text{ by }}y=Gu$ if:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

where $A\in \mathbb {R} ^{mxm}$, $B\in \mathbb {R} ^{mxn}$, $C\in \mathbb {R} ^{pxm}$, and $D\in \mathbb {R} ^{qxn}$ for any $t\in \mathbb {R}$. Then the ${H_{2}}$-norm of the system can be determined as described below.

The Data

In order to determine the ${H_{2}}$-norm of the system, we need the matrices ${A}$, ${B}$, and ${C}$.

The Optimization Problem

Suppose we wanted to to infer properties of the system behaviour (which is represented in the form (A,B,C,D)). Then it becomes necessary to ensure that the overall system forms an algebra, as the standard use of block-diagram algebra would otherwise be invalid. The only way this is possible is by calculating ${H_{2}}$ and/or ${H_{\infty }}$-norms - both of which are signal norms that (in a certain sense) measure the size of the transfer function.

The LMI: The $H_{2}$ Norm

Assuming that ${\hat {P}}(s)=C(sI-A)^{-1}B$, this means that the following are equivalent:

$1)\quad A{\text{ is Hurwitz and }}||{\hat {P}}||_{H_{2}}^{2}<\gamma$

$2){\begin{aligned}{\begin{cases}trace(CXC^{T})&<\gamma \\AX+XA^{T}+BB^{T}&<0\\X&>0\end{cases}}\end{aligned}}$

Conclusion:

The LMI can be used to minimize the ${H_{2}}$-norm of the system. It is worth noting that a finite ${H_{2}}$-norm does not guarantee finite ${H_{\infty }}$-norm, and that in order for the block diagram algebra to be valid, ${H_{\infty }}$-norm must be finite.

Implementation

• Example Code - A GitHub link that contains code (titled "H2Norm.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

• $H_{2}$-Filtering - LMI for $H_{2}$-Filtering
• Discrete-Time $H_{2}$ Norm - LMI for $H_{2}$-norm in the Discrete-Time case.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Algebraic Riccati Equation

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Riccti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\\end{aligned}}$

The Data

Following matrices are needed as Inputs:.

$A,B,N$.

The Optimization Problem

In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find

The LMI: Algebraic Riccati Inequality

Title and mathematical description of the LMI formulation.

${\textstyle {\begin{aligned}P>0,\\Q>0,\\R>0.\\{\text{ The Algebraic Riccati inequality is given by}}\;\\A^{T}P+PA-(PB+N^{T})R^{-}1(B^{T}P+N)+Q&\geq 0\\\\{\text{ can be written using the Schur complement lemma as}}\;\\{\begin{bmatrix}A^{T}P+PA+Q&&PB+N^{T}\\\star &&R\\\end{bmatrix}}&\geq 0\end{aligned}}}$

Conclusion:

If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

Implementation:

Matlab code for this LMI in the Github repository:

1. REDIRECT [[13]]- CODE

External links

• [14]-Optimal Solution to Matrix Riccati Equation
• https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory.- - A List of LMIs by Ryan Caverly and James Forbes

System Zeros without feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: ${\begin{aligned}H(s)={\frac {N(s)}{D(s)}}\\\end{aligned}}$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $N(s)=0$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros without feedthrough, we take the assumption that $D=0$.

The System

Consider a continuous-time LTI system, $G$ , with minimal statespace representation $(A,B,C,0)$

${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y(t)=Cx(t)\end{aligned}}$

The Data

The matrices:

${\begin{aligned}A\in \mathbb {R} ^{n\times n}\\M\in \mathbb {R} ^{n\times q}\\N\in \mathbb {R} ^{q\times n}\\\end{aligned}}$

The LMI: System Zeros without feedthrough

The transmission zeros of $G(s)=C(sI-A)^{-}1B$ are the eigenvalues of $NAM$, where $N\in \mathbb {R} ^{q\times n},M\in \mathbb {R} ^{n\times q},CM=0,NB=0,NM=1.$. Therefore , $G(s)$ is a minimum phase if and only if there exists $P\in \mathbb {S} ^{q}$, where $P>0$ such that

${\begin{aligned}PNAM+M^{T}A^{T}N^{T}P<0\end{aligned}}$

Conclusion:

If P exists, it ensures non-minimum phase. Eigenvalues of NAM then gives the zeros of the system.

Implementation

https://github.com/Ricky-10/coding107/blob/master/Systemzeroswithoutfeedthrough

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• Control_Systems/Poles_and_Zeros

System zeros with feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: ${\begin{aligned}H(s)={\frac {N(s)}{D(s)}}\\\end{aligned}}$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $N(s)=0$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros with feedthrough, we take $D$ as full rank.

The System

Consider a continuous-time LTI system, $G$ , with minimal statespace representation $(A,B,C,D)$

${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y(t)=Cx(t)+Du\end{aligned}}$

The Data

The matrices needed as inputs are:

${\begin{aligned}A\in \mathbb {R} ^{n\times n}\\B\in \mathbb {R} ^{n\times m}\\N\in \mathbb {R} ^{p\times n}\\\end{aligned}}$

In this case, $m\leq p$

The LMI: System Zeros with feedthrough

The transmission zeros of $G(s)=C(sI-A)^{-}1B+D$ are the eigenvalues of $A-B(D^{T}D)^{-1}D^{T}C$. Therefore , $G(s)$ is a minimum phase if and only if there exists $P\in \mathbb {S} ^{q}$, where $P>0$ such that

${\begin{aligned}P(A-B(D^{T}D)^{-1}D^{T}C)+(A-B(D^{T}D)^{-1}D^{T}C)^{T}P<0\\\end{aligned}}$

Conclusion:

If P exists, it ensures non-minimum phase. Eigenvalues of $A-B(D^{T}D)^{-1}D^{T}C$ then gives the zeros of the system.

Related LMIs

LMIs_in_Controls/pages/systemzeroswithoutfeedthrough

Implementation

https://github.com/Ricky-10/coding107/blob/master/systemzeroswithfeedthrough

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• Control_Systems/Poles_and_Zeros

Negative Imaginary Lemma

User:ShakespeareFan00/Sandbox Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".

The System

Consider a square continuous time Linear Time invariant system, with the state space realization $(A,B,C,D)$

${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}$

The Data

$A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},D\in \mathbb {S} ^{m}$

The LMI: LMI for Negative Imaginary Lemma

According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions

• There exists a $P$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$ⁿ,where $P\geq 0$, such that,

${\begin{bmatrix}A^{T}P+PA&PB-A^{T}C^{T}\\B^{T}P-CA&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$

• There exists a $Q$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$ⁿ,where $Q\geq 0$, such that,

${\begin{bmatrix}QA^{T}+AQ&B-QA^{T}C^{T}\\B^{T}-CAQ&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$

Conclusion

The system is strictly negative if det($A$) $\neq$ 0 and either of the above LMIs are feasible with resulting $P>0$ or $Q>0$

Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

Related LMIs

Positive Real Lemma

External Links

• [15] - LMI in Control Systems Analysis, Design and Applications
• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• Xiong, Junlin & Petersen, Ian & Lanzon, Alexander. (2010). A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems.

Small Gain Theorem

LMIs in Control/Matrix and LMI Properties and Tools/Small Gain Theorem

The Small Gain Theorem provides a sufficient condition for the stability of a feedback connection.

Theorem

Suppose $B$ is a Banach Algebra and $Q\in B$. If $\|Q\|<1$, then $(I-Q)^{-1}$ exists, and furthermore,

                   $(I-Q)^{-1}=\sum _{k=0}^{\infty }Q^{k}$ 

Proof

Assuming we have an interconnected system $(G,K)$:

$y_{1}=G(u_{1}-y_{2})$ and, $y_{2}=K(u_{2}-y_{1})$

The above equations can be represented in matrix form as

${\begin{aligned}{\begin{bmatrix}I&0\\0&I\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}&={\begin{bmatrix}\ 0&-G\\\ -K&0\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}+{\begin{bmatrix}G&0\\0&K\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}\end{aligned}}$

Making ${\begin{bmatrix}y_{1}&y_{2}\end{bmatrix}}^{T}$ the subject, we then have:

${\begin{aligned}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}&={\begin{bmatrix}\ I&G\\\ K&I\\\end{bmatrix}}^{-1}{\begin{bmatrix}G&0\\0&K\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}&={\begin{bmatrix}\ (I-GK)^{-1}G&-G(I-KG)^{-1}K\\\ -K(I-GK)^{-1}G&(I-KG)^{-1}K\\\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}\end{aligned}}$

If $(I-GK)^{-1}$ is well-behaved, then the interconnection is stable. For $(I-GK)^{-1}$ to be well-behaved, $\|(I-GK)^{-1}\|$ must be finite.

Hence, we have $\|(I-GK)^{-1}\|<\infty$

$\|G\|\|K\|=\|Q\|$ and $\|Q\|<I$ for the higher exponents of $\|Q\|$ to converge to $0.$

Conclusion

If $\|Q\|<1$, then this implies stability, since the higher exponents of $Q$ in the summation of $\sum _{k=0}^{\infty }Q^{k}$ will converge to $0$, instead of blowing up to infinity.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Tangential Nevanlinna-Pick Interpolation

Tangential Nevanlinna-Pick

The Tangential Nevanlinna-Pick arises in multi-input, multi-output (MIMO) control theory, particularly $H_{\infty }$ robust and optimal control.

The problem is to try and find a function $H:C\to C^{pxq}$ which is analytic in $C_{+}$ and satisfies
‌ $H({\lambda }_{i})u_{i}=v_{i},$     $i=1,...,m$       with $||H||_{\infty }\leq 1$               $(1)$

The System

$N_{(ij)}$ is a set of $pxq$ matrix valued Nevanlinna functions. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if $Im(H({\lambda }))\geq 0$ $({\lambda }\in {\pi }^{+})$.

The Data

Given:
‌ Initial sequence of data points on real axis ${\lambda }_{1},...,{\lambda }_{m}$ with ${\lambda }_{i}\in C_{+}{\widehat {=}}(s|Re(s)>0)$,
‌ And two sequences of row vectors containing distinct target points $u_{1},....,u_{m}$ with $u_{i}\in C^{q}$, and $v_{1},...,v_{m}$ with $v_{i}\in C^{p},i=1,...,m$.

The LMI: Tangential Nevanlinna- Pick

Problem (1) has a solution if and only if the following is true:

Nevanlinna-Pick Approach

     $N_{ij}=$ ${\frac {u_{i}^{\ast }u_{j}-v_{i}^{\ast }v_{j}}{\ {\lambda }_{i}^{\ast }+{\lambda }_{j}}}$
‌

Lyapunov Approach

N can also be found using the Lyapunov equation:

     $A^{\ast }N+NA-(U^{\ast }U-V^{\ast }V)=0$

where $A=diag({\lambda }_{1},...,{\lambda }_{m}),U=[u_{1}...u_{m}],V=[v1...vm]$

The tangential Nevanlinna-Pick problem is then solved by confirming that $N\geq 0$.

Conclusion:

If $N_{(ij)}$ is positive (semi)-definite, then there exists a norm-bounded analytic function, $H$ which satisfies $H({\lambda }_{i})u_{i}=v_{i},$    $i=1,...,m$       with $||H||_{\infty }\leq 1$

Implementation

Implementation requires YALMIP and a linear solver such as sedumi. [16] - MATLAB code for Tangential Nevanlinna-Pick Problem.

Related LMIs

Nevalinna-Pick Interpolation with Scaling

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• Generalized Interpolation in $H_{\infty }$ by Donald Sarason.
• Tangential Nevanlinna-Pick Interpolation Problem With Boundary Nodes in the Nevanlinna Class And The Related Moment Problem by Yong Jian Hu and Xiu Ping Zhang.

Nevanlinna-Pick Interpolation with Scaling

Nevanlinna-Pick Interpolation with Scaling

The Nevanlinna-Pick problem arises in multi-input, multi-output (MIMO) control theory, particularly $H_{\infty }$ robust and optimal controller synthesis with structured perturbations.

The problem is to try and find ${\gamma }_{opt}=inf(||DHD^{-1}||_{\infty })$ such that $H$ is analytic in $C_{+},$   $D=D^{\ast }>0,$ and $D\in \mathbb {D}$ define the scaling, and finally,
‌ $H({\lambda }_{i})u_{i}=v_{i}$    $i=1,...,m$         $(1)$

The System

The scaling factor $\mathbb {D}$ is given as a set of $mxm$ block-diagonal matrices with specified block structure. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if $Im(H({\lambda }))\geq 0$ $({\lambda }\in {\pi }^{+})$. The Nevanlinna LMI matrix $N$ is defined as $N=G_{in}-G_{out}$. The matrix $A$ is a diagonal matrix of the given sequence of data points ${\lambda }_{i}\in \mathbb {C} (A=diag({\lambda }_{1},...,{\lambda }_{m})$

The Data

Given:
‌ Initial sequence of data points in the complex plane ${\lambda }_{1},...,{\lambda }_{m}$ with ${\lambda }_{i}\in C_{+}{\widehat {=}}(s|Re(s)>0)$.
‌ Two sequences of row vectors containing distinct target points $u_{1},....,u_{m}$ with $u_{i}\in C^{q}$, and $v_{1},...,v_{m}$ with $v_{i}\in C^{p},i=1,...,m$.

The LMI: Nevanlinna- Pick Interpolation with Scaling

First, implement a change of variables for $P=D^{\ast }D$ and $N=G_{in}-G_{out}$.

From this substitution it can be concluded that ${\gamma }_{opt}$ is the smallest positive ${\gamma }$ such that there exists a $P>0,P\in \mathbb {D}$ such that the following is true:

      $A^{\ast }G_{in}+G_{in}A-U^{\ast }PU=0$,

      $A^{\ast }G_{out}+G_{out}A-V^{\ast }PV=0$,

      ${\gamma }^{2}G_{in}-G_{out}\geq 0$

Conclusion:

If the LMI constraints are met, then there exists a $H_{\infty }$ norm-bounded optimal gain ${\gamma }$ which satisfies the scaled Nevanlinna-Pick interpolation objective defined above in Problem (1).

Implementation

Implementation requires YALMIP and Mosek. [17] - MATLAB code for Nevanlinna-Pick Interpolation.

Related LMIs

Nevalinna-Pick Interpolation

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• Generalized Interpolation in $H_{\infty }$.

Generalized $H_{2}$ Norm

Generalized $H_{2}$ Norm

The $H_{2}$ norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented $D$ matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, $G$ as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

The System

Consider a continuous-time, linear, time-invariant system $G:L_{2e}\to L_{2e}$ with state space realization $(A,B,C,0)$ where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, amd $A$ is Hurwitz. The generalized $H_{2}$ norm of $G$ is:

$\left\|G\right\|_{2,{\infty }}=\sup _{u\in L_{2},{u}\neq {\text{zero}}}$ ${\frac {\left\|Gu\right\|_{\infty }}{\ \left\|u\right\|_{2}}}$

The Data

The transfer function $G$, and system matrices $A$, $B$, $C$ are known and $A$ is Hurwitz.

The LMI: Generalized $H_{2}$ Norm LMIs

The inequality $\left\|G\right\|_{2,{\infty }}<\mu$ holds under the following conditions:

1. There exists $P\in \mathbb {S} ^{n}$ and $\mu \in \mathbb {R} _{>0}$ where $P>0$ such that:

${\begin{bmatrix}A^{T}P+PA&PB\\*&-\mu 1\end{bmatrix}}<0$.

${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$.

2. There exists $Q\in \mathbb {S} ^{n}$ and $\mu \in \mathbb {R} _{>0}$ where $Q>0$ such that:

${\begin{bmatrix}QA^{T}+AQ&B\\*&-\mu 1\end{bmatrix}}<0$.

${\begin{bmatrix}Q&QC^{T}\\*&\mu 1\end{bmatrix}}>0$.

3. There exists $P\in \mathbb {S} ^{n},V\in \mathbb {R} ^{n\times n}$ and $\mu \in \mathbb {R} _{>0}$ where $P>0$ such that:

${\begin{bmatrix}-(V+V^{T})&V^{T}A+P&V^{T}B&V^{T}\\*&-P&0&0\\*&**-\mu 1&0\\*&*&*&-P\end{bmatrix}}<0$.

${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$.

Conclusion:

The generalized $H_{2}$ norm of $G$ is the minimum value of $\mu \in \mathbb {R} _{>0}$ that satisfies the LMIs presented in this page.

Implementation

This implementation requires Yalmip and Sedumi.

Generalized $H_{2}$ Norm - MATLAB code for Generalized $H_{2}$ Norm.

Related LMIs

LMI for System H_{2} Norm

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

Passivity and Positive Realness

This section deals with passivity of a system.

The System

Given a state-space representation of a linear system

${\begin{aligned}\ {\dot {x}}=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}$

$x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}$ are the state, output and input vectors respectively.

The Data

$A,B,C,D$ are system matrices.

Definition

The linear system with the same number of input and output variables is called passive if

${\begin{aligned}\int \limits _{0}^{T}u^{T}y(t)dt\geq 0\\\end{aligned}}$

(1)

hold for any arbitrary $T\geq 0$, arbitrary input $u(t)$, and the corresponding solution $y(t)$ of the system with $x(0)=0$. In addition, the transfer function matrix

${\begin{aligned}G(s)&=C(sI-A)^{-1}B+D\\\end{aligned}}$

(2)

of system is called is positive real if it is square and satisfies

${\begin{aligned}\ G^{H}(s)+G(s)\geq 0\forall s\in \mathbb {C} ,Re(s)>0\\\end{aligned}}$

(3)

LMI Condition

Let the linear system be controllable. Then, the system is passive if an only if there exists $P>0$ such that

${\begin{aligned}\ {\begin{bmatrix}A^{T}P+PA&PB-C^{T}\\B^{T}P-C&-D^{T}-D\end{bmatrix}}\leq 0\\\end{aligned}}$

(4)

Implementation

This implementation requires Yalmip and Mosek.

• https://github.com/ShenoyVaradaraya/LMI--master/blob/main/Passivity.m

Conclusion

Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013

Non-expansivity and Bounded Realness

This section studies the non-expansivity and bounded-realness of a system.

The System

Given a state-space representation of a linear system

${\begin{aligned}\ {\dot {x}}=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}$

$x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}$ are the state, output and input vectors respectively.

The Data

$A,B,C,D$ are system matrices.

Definition

The linear system with the same number of input and output variables is called non-expansive if

${\begin{aligned}\int \limits _{0}^{T}y^{T}y(t)dt\geq \int \limits _{0}^{T}u^{T}u(t)dt\\\end{aligned}}$

(1)

hold for any arbitrary $T\geq 0$, arbitrary input $u(t)$, and the corresponding solution $y(t)$ of the system with $x(0)=0$. In addition, the transfer function matrix

${\begin{aligned}G(s)&=C(sI-A)^{-1}B+D\\\end{aligned}}$

(2)

of system is called is positive real if it is square and satisfies

${\begin{aligned}\ G^{H}(s)+G(s)\geq I\forall s\in \mathbb {C} ,Re(s)>0\\\end{aligned}}$

(3)

LMI Condition

Let the linear system be controllable. Then, the system is bounded-real if an only if there exists $P>0$ such that

${\begin{aligned}\ {\begin{bmatrix}A^{T}P+PA&PB&C^{T}\\B^{T}P&-I&D^{T}\\C&D&-I\end{bmatrix}}<0\\\end{aligned}}$

(4)

and

${\begin{aligned}\ {\begin{bmatrix}PA+A^{T}P+C^{T}C&PB+C^{T}D\\B^{T}P+D^{T}C&D^{T}D-I\end{bmatrix}}<0\\\end{aligned}}$

(5)

Implementation

This implementation requires Yalmip and Mosek.

• https://github.com/ShenoyVaradaraya/LMI--master/blob/main/bounded_realness.m

Conclusion:

Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013

Change of Subject

LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

A Bilinear Matrix Inequality (BMI) can sometimes be converted into a Linear Matrix Inequality (LMI) using a change of variables. This is a basic mathematical technique of changing the position of variables with respect to equal signs and the inequality operators. The change of subject will be demonstrated by the example below.

Example

Consider $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},K\in \mathbb {R} ^{m\times n}$, and $Q\in \mathbb {S} ^{n}$, where $Q>0$.

The matrix inequality given by:

$QA^{T}+AQ-QK^{T}B{T}-BKQ<0$ is bilinear in the variables $Q$ and $K$.

Defining a change of variable as $F=KQ$ to obtain

$QA^{T}+AQ+-F^{T}B^{T}-BF<0$,

which is an LMI in the variables $Q$ and $F$.

Once this LMI is solved, the original variable can be recovered by $K=FQ^{-1}$.

Conclusion

It is important that a change of variables is chosen to be a one-to-one mapping in order for the new matrix inequality to be equivalent to the original matrix inequality. The change of variable $F=KQ$ from the above example is a one-to-one mapping since $Q^{-1}$ is invertible, which gives a unique solution for the reverse change of variable $K=FQ^{-1}$.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.

Congruence Transformation

LMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation

This methods uses change of variable and some matrix properties to transform Bilinear Matrix Inequalities to Linear Matrix Inequalities. This method preserves the definiteness of the matrices that undergo the transformation.

Theorem

Consider $Q\in \mathbb {S} ^{n},W\in \mathbb {R} ^{n\times n}$, where $\operatorname {rank} (W)=n$. The matrix inequality $Q<0$ is satisfied if and only if $WQW^{T}<0$ or equivalently, $W^{T}QW<0$.

Example

Consider $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},K\in \mathbb {R} ^{m\times p},C^{T}\in \mathbb {R} ^{n\times p},P\in \mathbb {S} ^{n}$ and $V\in \mathbb {S} ^{p}$, where $P>0$ and $V>0$. The matrix inequality given by

${\begin{aligned}\ Q&={\begin{bmatrix}\ A^{T}P+PA&-PBK+C^{T}V\\\ *&-2V\\\end{bmatrix}}\ <0,\end{aligned}}$

is linear in variable $V$ and bilinear in the variable pair $(P.K)$. Choose the matrix $\operatorname {diag} (P^{-1},V^{-1})$ to obtain the equivalent BMI given by

${\begin{aligned}\ WQW^{T}&={\begin{bmatrix}\ P^{-1}A^{T}+AP^{-1}&-BKV^{-1}+P^{-1}C^{T}\\\ *&-2V^{-1}\\\end{bmatrix}}\ <0,\end{aligned}}$

Using a change of variable $X=P^{-1},U=V^{-1}$ and $F=KV^{-1}$, the above equation becomes

${\begin{aligned}\ WQW^{T}&={\begin{bmatrix}\ XA^{T}+AX&-BF+XC^{T}\\\ *&-2U\\\end{bmatrix}}\ <0,\end{aligned}}$

which is an LMI of variables $X,U$ and $F$. The original variable $K$ is recovered by doing a reverse change of variable $K=FU^{-1}$.

Conclusion

A congruence transformation preserves the definiteness of a matrix by ensuring that $Q<0$ and $W^{T}QW<0$ are equivalent. A congruence transformation is related, but not equivalent to a similarity transformation $TQT^{-1}$, which preserves not only the definiteness, but also the eigenvalues of a matrix. A congruence transformation is equivalent to a similarity transformation in the special case when $W^{T}=W^{-1}$.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.

Finsler's Lemma

LMIs in Control/Matrix and LMI Properties and Tools/Finsler's Lemma

This method It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. It is equivalent to other lemmas used in optimization and control theory, such as Yakubovich's S-lemma, Finsler's lemma and it is wedely used in Linear Matrix Inequalities

Theorem

Consider $\Psi \in \mathbb {S} ^{n},G\in \mathbb {R} ^{n\times m},\mathrm {A} \in \mathbb {R} ^{m\times p},H\in \mathbb {R} ^{n\times p}$and $\sigma \in \mathbb {R}$. There exists $\mathrm {A}$ such that

$\Psi +G\mathrm {A} H^{T}+H\mathrm {A} ^{T}G^{T}<0,$

if and only if there exists $\sigma$ such that

$\Psi -\sigma GG^{T}<0$

$\Psi -\sigma HH^{T}<0$

Alternative Forms of Finsler's Lemma

Consider $\Psi \in \mathbb {S} ^{n},Z\in \mathbb {R} ^{p\times n},x\in \mathbb {R} ^{n}$and $\sigma \in \mathbb {R} _{>0}$. If there exists $Z$ such that

$x^{T}\Psi x,0$

holds for all $x$ ≠ $0$ satisfying $Zx=0$, then there exists $\sigma$ such that

$\Psi -\sigma Z^{T}Z<0$

Modified Finsler's Lemma

Consider $\Psi \in \mathbb {S} ^{n},G\in \mathbb {R} ^{n\times m},\mathrm {A} \in \mathbb {R} ^{m\times p},H\in \mathbb {R} ^{n\times p}$and $\epsilon \in \mathbb {R} _{>0}$, where $\mathrm {A} ^{T}\mathrm {A}$ is less that on equal to $\mathbb {R}$, and $R>0$. There exists $\mathrm {A}$ such that

$\Psi +G\mathrm {A} H^{T}+H\mathrm {A} ^{T}G^{T}{T}<0,$

there exists $\epsilon$ such that

$\Psi +\epsilon ^{-1}GG^{T}+\epsilon HRH^{T}<0.$

Conclusion

In summary, a number of identical methods have been stated above to determine the positive definiteness of LMIs.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.

D-Stability

1. Continuous Time D-Stability Observer

Time-Delay Systems

1. Delay Dependent Time-Delay Stabilization
2. Delay Independent Time-Delay Stabilization

Parametric, Norm-Bounded Uncertain System Quadratic Stability

User:ShakespeareFan00/Sandbox

Given a system with matrices A,M,N,Q the quadratic stability of the system with parametric, norm-bounded uncertainty can be determined by the following LMI. The feasibility of the LMI tells if the system is quadratically stable or not.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in \mathbf {\Delta } \;:=\{\Delta \in \mathbb {R} ^{n\times n}:\|\Delta \|\leq 1\}\\\end{aligned}}$

The Data

The matrices $A,M,N,Q$.

The LMI:

${\begin{aligned}{\text{Find}}\;&P>0,\mu \geq 0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+\mu {\begin{bmatrix}MM^{T}&MQ^{T}\\QM^{T}&QQ^{T}-I\end{bmatrix}}<0\\\end{aligned}}$

Conclusion:

The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching

Quadratic Stability Margins

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

Stability of Structured, Norm-Bounded Uncertainty

User:ShakespeareFan00/Sandbox

Given a system with matrices A,M,N,Q with structured, norm-bounded uncertainty, the stability of the system can be found using the following LMI. The LMI takes variables P and $\Theta$ and checks for a feasible solution.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in {\bf {{\Delta }\;,||\Delta ||\leq 1}}\\\end{aligned}}$

The Data

The matrices $A,M,N,Q$.

The LMI:

${\begin{aligned}{\text{Find}}\;&P>0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+{\begin{bmatrix}M\Theta M^{T}&M\Theta Q^{T}\\Q\Theta M^{T}&Q\Theta Q^{T}-\Theta \end{bmatrix}}<0\\\end{aligned}}$

Conclusion:

${\begin{aligned}{\text{The system above is quadratically stable if and only if there exists some }}\Theta \in P\Theta {\text{ and }}P>0{\text{ such that the LMI is feasible.}}\end{aligned}}$

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability under Arbitrary Switching

Quadratic Stability Margins

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

Stability under Arbitrary Switching

LMIs in Control/Stability Analysis/Continuous Time/Stability under Arbitrary Switching

Using the LMI below, find a P matrix that fits the constraints. If there exists one, then the system can switch between subsystems $A_{1}$ and $A_{2}$ arbitrarily and remain stable.

The System

${\begin{aligned}{\dot {x}}(t)\in \{A_{1}x(t),A_{2}x(t)\}\end{aligned}}$

The Data

The matrices $A_{1}\in R^{n\times n},A_{2}\in R^{n\times n}$.

The LMI

${\begin{aligned}{\text{Find}}\;&P>0:\\A_{1}^{T}P+PA_{1}<0{\text{ and }}\\A_{2}^{T}P+PA_{2}<0\end{aligned}}$

Conclusion

The switched system is stable under arbitrary switching if there exists some P > 0 such that the LMIs hold.

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability of Structured, Norm-Bounded Uncertainty

Quadratic Stability Margins

External links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

Quadratic Stability Margins

User:ShakespeareFan00/Sandbox

${\begin{aligned}{\text{The quadratic stability margin of the system is defined as the largest }}\alpha \geq 0{\text{ for which the system is quadratically stable.}}{\text{This LMI applies for systems with norm-bounded uncertainty.}}\end{aligned}}$

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t),&&p^{T}p\leq \alpha ^{2}x^{T}C_{q}^{T}C_{q}x\\\end{aligned}}$

The Data

The matrices $A,B_{p},C_{q}$.

The Optimization Problem

${\text{Maximize }}\beta =\alpha ^{2}{\text{ subject to the LMI constraint.}}$

The LMI:

${\begin{aligned}{\text{Find}}\;&P,\lambda ,{\text{ and }}\beta =\alpha ^{2}:\\{\begin{bmatrix}A^{T}P+PA+\beta \lambda C_{q}^{T}C_{q}&PB_{p}\\B_{p}^{T}P&-\lambda I\end{bmatrix}}<0\\\end{aligned}}$

Conclusion:

If there exists an $\alpha \geq 0$ then the system is quadratically stable, and the stability margin is the largest such $\alpha$.

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.

• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Stability of Linear Delayed Differential Equations

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+\sum _{i=1}^{L}A_{i}x(t-\tau _{i}),\\\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$ and $\tau _{i}>0$.

The Data

The matrices $A,\{A_{i}.\tau _{i}\}_{i=1}^{L}$.

The LMI:

Solve the following LMIP

${\begin{aligned}&{\text{Find}}\{P\succ 0,P_{1}\succ 0,\dots ,P_{L}\succ 0\}:\\&\quad s.t.{\begin{bmatrix}A^{\top }P+PA+\sum _{i=1}^{L}P_{i}&PA_{1}&\dots &PA_{L}\\A_{1}^{\top }P&-P_{1}&\dots &0\\\vdots &\vdots &\ddots &\vdots \\A_{L}^{\top }P&0&\dots &-P_{L}\end{bmatrix}}\prec 0,P_{1}\succ 0,\dots ,P_{L}\succ 0.\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/50fc71737b69f2cf57d15634f2f19d091bf37d02

Conclusion

The stability of the above linear delayed differential equation is proved, using Lyapunov functionals of the form $V(x,t)=x(t)^{\top }Px(t)+\sum _{i=1}^{L}\int _{0}^{L}x^{\top }(t-s)P_{i}x(t-s)\ ds$, if the provided LMIP is feasible.

Remark

The techniques for proving stability of norm-bound LDIs [Boyd, ch.5] can also be used.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

H infinity Norm for Affine Parametric Varying Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{w}w(t),\\z(t)&=C_{z}(\theta )x(t)+D_{zw}(\theta )w(t),\end{aligned}}$

where $C_{z}$ and $D_{zw}$ depend affinity on parameter $\theta \in \mathbb {R} ^{p}$.

The Data

The matrices $A,B_{w},C_{z}(.),D_{zw}(.)$.

The Optimization Problem:

Solve the following semi-definite program

${\begin{aligned}&\min _{\{P\succ 0,\gamma \geq 0\}}\gamma \\&\quad s.t.{\begin{bmatrix}A^{\top }P+PA&PB_{w}\\B_{w}^{\top }P&-\gamma ^{2}I\end{bmatrix}}+{\begin{bmatrix}C_{z}^{\top }(\theta )\\D_{zw}^{\top }(\theta )\end{bmatrix}}{\begin{bmatrix}C_{z}(\theta )&D_{zw}(\theta )\end{bmatrix}}\preceq 0.\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/5462bc1dc441bc298d50a2c35075e9466eba8355

Conclusion

The value function of the above semi-definite program returns the ${\mathcal {H}}_{\infty }$ norm of the system.

Remark

It is assumed that $A$ is stable and $(A,B_{w})$ is controllable and the semi-infinite convex constraint $\|H_{\theta }(j\omega )\|<\gamma$ for all $\omega \in \mathbb {R}$, is converted into a finite-dimensional convex LMI.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Entropy Bond for Affine Parametric Varying Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{w}w(t),\\z(t)&=C_{z}(\theta )x(t)+D_{zw}(\theta )w(t),\end{aligned}}$

where $C_{z}$ and $D_{zw}$ depend affinely on parameter $\theta \in \mathbb {R} ^{p}$.

The Data

The matrices $A,B_{w},C_{z}(.),D_{zw}(.)$.

The Optimization Problem:

Solve the following semi-definite program

${\begin{aligned}&\min _{\{P\succ 0,\gamma ^{2},\lambda ,\theta \}}\gamma ^{2}\\&\quad s.t.\quad D_{zw}(\theta )=0,{\begin{bmatrix}A^{\top }P+PA&PB_{w}&C_{z}(\theta )^{\top }\\B_{w}^{\top }P&-\gamma ^{2}I&0\\C_{z}(\theta )&0&-I\end{bmatrix}}\preceq 0,\quad {\rm {{Tr}(B_{w}^{\top }PB_{w})\leq \lambda .}}\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1

Conclusion

The value function of the above semi-definite program returns a bound for $\gamma$-entropy of the system, which is defined as

${\begin{aligned}I_{\gamma }(H_{\theta })\triangleq {\begin{cases}{\frac {-\gamma ^{2}}{2\pi }}\int _{-\infty }^{\infty }\log \det(I-\gamma ^{2}H_{\theta }(i\omega )H_{\theta }(i\omega )^{*})d\omega ,\quad {\text{if}}\ \|H_{\theta }\|_{\infty }<\gamma \\\infty ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{otherwise.}}\end{cases}}\end{aligned}}$

Remark

When it is finite, $I_{\gamma }(H_{\theta })$ is given by ${\rm {{Tr}(B_{w}^{\top }PB_{w})}}$ where $P$, is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Dissipativity of Affine Parametric Varying Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{w}w(t),\\z(t)&=C_{z}(\theta )x(t)+D_{zw}(\theta )w(t),\end{aligned}}$

where $C_{z}$ and $D_{zw}$ depend affinely on parameter $\theta \in \mathbb {R} ^{p}$.

The Data

The matrices $A,B_{w},C_{z}(.),D_{zw}(.)$.

The Optimization Problem:

Solve the following semi-definite program

${\begin{aligned}&\min _{\{P\succ 0,\gamma ,\theta \}}\gamma \\&\quad s.t.\quad {\begin{bmatrix}A^{\top }P+PA&PB_{w}-C_{z}(\theta )^{\top }\\B_{w}^{\top }P-C_{z}(\theta )&2\gamma I-D_{zw}(\theta )-D_{zw}(\theta )^{\top }\end{bmatrix}}\preceq 0.\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa

Conclusion

The dissipativity of $H_{\theta }$ (see [Boyd,eq:6.59]) exceeds ${\gamma }$ if and only if the above LMI holds and the value function returns the minimum provable dissipativity.

Remark

It is worth noticing that passivity corresponds to zero dissipativity.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Hankel Norm of Affine Parameter Varying Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{w}w(t),\\z(t)&=C_{z}(\theta )x(t)+D_{zw}(\theta )w(t),\end{aligned}}$

where $C_{z}$ and $D_{zw}$ depend affinely on parameter $\theta \in \mathbb {R} ^{p}$.

The Data

The matrices $A,B_{w},C_{z}(.),D_{zw}(.)$.

The Optimization Problem:

Solve the following semi-definite program

${\begin{aligned}&\min _{\{Q\succeq 0,\gamma ^{2},\theta \}}\gamma ^{2}\\&\quad s.t.\quad D_{zw}(\theta )=0,\quad A^{\top }Q+QA+C_{z}(\theta )C_{z}(\theta )\preceq 0,\quad \gamma ^{2}I-W_{c}^{1/2}QW_{c}^{1/2}\succeq 0,\end{aligned}}$

where $W_{c}$ is the controllability Gramian, i.e., $W_{c}\triangleq \int _{0}^{\infty }e^{At}B_{w}B_{w}^{\top }e^{A^{\top }t}dt$.

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35

Conclusion

The Hanakel norm (i.e., the square root of the maximum eigenvalue) of $H_{\theta }$ is less than ${\gamma }$ if and only if the above LMI holds and the value function returns the maximum provable Hankel norm.

Remark

$D_{zw}$ is assumed to be zero.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Positive Orthant Stabilizability

Positive Orthant Stabilizability

The positive orthant stability of a linear system refers to the property of the system states being real and positive for all $t\geq 0$ and decaying down to zero over time. In this section, the feasibility problem for systems to be positive orthant stable, and the stabilizability conditions to make the system positive orthant stable will be covered.

The System

Consider a linear state-space representation of a system as:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$ and $u(t)\in \mathbb {R} ^{r}$ are the system state and the input vector respectively. A and B are system coefficient matrices of appropriate dimensions.

The Data

Number of states n and number of control inputs r need to be known. Moreover, the system matrices A,B are also required to be known.

The Feasibility LMI

An LTI system is positive orthant stable if $x(0)\in \mathbb {R} _{+}^{n}$ implies that $\forall t\geq 0,x(t)\in \mathbb {R} _{+}^{n}$. Moreover, as $t\rightarrow \infty$, $x(t)\rightarrow 0$. This is possible if and only if the following conditions hold:

${\begin{aligned}A_{ij}&\geq 0,\forall i\neq j,\\\exists P&>0{\text{ s.t. }}PA^{\top }+AP<0,\end{aligned}}$

The above LMI feasibility is the positive orthant stability criteria. To convert it into a positive orthant stabilizability check, the problem can be modified so that we check if ${\dot {x}}=(A+BK)x$ is positive orthant stable. As $K$ is also a design variable here, the second inequality in the above LMI will result in bilinearity. A simple change of variables can overcome that to result in the following LMI feasibility problem for checking positive orthant stabilizability of the LTI system:

${\begin{aligned}{\text{Find }}Q,Y&{\text{ subj. to:}}\\&Q>0,(AQ+BY)_{ij}\geq 0,\forall i\neq j,\\&{\text{ s.t. }}QA^{\top }+AQ+BY+Y^{\top }B^{\top }<0,\end{aligned}}$

If the above LMI is feasible, the LTI system is stabilizable with controller $K=YQ^{-1}$.

Conclusion:

The feasibility of the above LMIs guarantees that the system is positive orthant stable if the first LMI is feasible or stabilizable with a controller if the second LMI holds.

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Positive_Orthant_LMI.m

External Links

A list of references documenting and validating the LMI.

• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control

LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control

The LMI in this page gives the feasibility conditions which, if satisfied, imply that the correstponding system can be stabilized.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+BSat(u(t)),\\x(0)&=x_{0},\\Sat(u)_{i}=,\end{aligned}}$

where $x\in \|R^{n}$ is the state, $u\in \|R^{m}$ is the control input.

For the system given as above, its symmetrical saturated control form can be derived by following the procedure in the original article. The new system will have the form:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+{\tilde {B}}sat(z(t))+Ew\end{aligned}}$

where $w_{i}=u_{i}-{\frac {\alpha _{i}-\beta _{i}}{2}},z_{i}=w_{i}{\frac {2}{\alpha _{i}+\beta _{i}}}$

The Data

The system matrices $(A,{\tilde {B}},E)$, the saturation bounds $(\alpha _{i},\beta _{i})$ of the control inputs. Positive scalars $\rho ,\eta$.

The LMI: The Stabilization Feasibility Condition

${\begin{aligned}&{\text{Find}}\;X,Y,Z:\\&{\text{subj. to: }}X>0,\\&\quad {\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}+{\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}^{\top }+{\frac {\eta }{\rho }}X+{\frac {1}{\eta }}EE^{\top }<0,\\{\begin{bmatrix}\mu &Z_{i}\\*X\end{bmatrix}}>0,i=1,...,{\bar {m}}\end{aligned}}$

Here $D_{s}$ is a diagonal matrix with a component either 0 or 1, and $D_{s}+D_{s}^{-}={\frac {\Lambda +\mathrm {T} }{2}}$ and ${\hat {D}}_{s}^{-}=e_{f_{m}}\times D_{s}^{-}$

Conclusion:

The feasibility of the given LMI implies that the system is stabilizable with control gains $K=YX^{-1},H=ZX^{-1}$.

Implementation

A link to CodeOcean or other online implementation of the LMI

External Links

• Stabilization of Systems with Unsymmetrical Saturated Control: An LMI Approach Link to the original article.

LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays

LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays

For systems experiencing time-varying delays where the delays are bounded, the feasibility LMI in this section can be used to determine if the system is $\alpha$-exponentially stable.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Dx(t-h(t)),&t\in \mathbb {R} ^{+},\\x(t)&=\phi (t),&t\in [-h_{2},0],\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$ is the state, $A,D\in \mathbb {R} ^{n\times n}$ are the matrices of delay dynamics, and $\phi (t)\in \mathbb {R} ^{n}$ is the initial function with norm $\|\phi \|=sup_{-{\bar {h}}\leq t\leq 0}\{\|\phi (t)\|,\|{\dot {\phi }}(t)\|\}$ and it is continuously differentiable function on $[-h_{2},0]$. The tyime-varying delay function $h(t)$ satisfies:

$0\leq h_{1}\leq h(t)\leq h_{2},t\in \mathbb {R} ^{+},$

The Data

The matrices $(A,D)$ are known, as well as the bounds $(h_{1},h_{2})$ of the time-varying delay.

The Optimization Problem

For a given $\alpha >0$, the zero solution of the system described above is $\alpha$-exponentially stable if there exists a positive number $N>0$ such that every solution $x(t,\phi )$ satisfies the following condition:

$\|x(t,\phi )\|\leq Ne^{-\alpha t}\|\phi \|,\forall t\in \mathbb {R} ^{+}$

The LMI: $\alpha$-Stability Condition

The following feasibility LMI can be used to check if the system is $\alpha$-exponentially stable or not for a given $\alpha >0$:

${\begin{aligned}&{\text{Find}}\;P,Q,R,U,S_{i},{\text{ where }}i=1,2,...,5:\\&\quad \quad {\begin{bmatrix}M_{11}&M_{12}&M_{13}&M_{14}&M_{15}\\*&M_{22}&0&M_{24}&S_{2}\\*&*&M_{33}&M_{34}&S_{3}\\*&*&*&M_{44}&S_{4}-S_{5}D\\*&*&*&*&M_{55}\end{bmatrix}}<0\\&{\text{where:}}\\&\quad \quad M_{11}=A^{\top }P+PA+2\alpha P-(e^{-2\alpha h_{1}}+e^{-2\alpha h_{2}})R+0.5S_{1}(I-A)+0.5(I-A^{\top })S_{1}^{\top }+2Q,\\&\quad \quad M_{12}=e^{-2\alpha h_{1}}R-S_{2}A,\quad M_{13}=e^{-2\alpha h_{2}}R-S_{3}A,\\&\quad \quad M_{14}=PD-S_{1}D-S_{4}A,\quad M_{15}=S_{1}-S_{5}A,\\&\quad \quad M_{22}=-e^{-2\alpha h_{1}}(Q+R),\quad M_{24}=S_{2}D+e^{-2\alpha h_{2}}U,\\&\quad \quad M_{33}=-e^{-2\alpha h_{1}}(Q+R+U),\quad M_{34}=-S_{3}D+e^{-2\alpha h_{2}}U,\\&\quad \quad M_{44}=0.5(S_{4}D+D^{\top }S_{4}^{\top })-e^{-2\alpha h_{2}}U,\\&\quad \quad M_{55}=S_{5}+S_{5}^{\top }+(h_{1}^{2}+h_{2}^{2})R+(h_{2}-h_{1})^{2}U,\end{aligned}}$

The above LMI can be combined with the bisection method to find $\alpha$.

Conclusion:

For systems with time-varying delays with intervals, the LMI in this section can be used to check if the system is exponentially stable with a certain $\alpha$. The bisection algorithm can be additionally used to compute $\alpha$.

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Intervaled_Delay_Sys_Stability_example.m

External Links

A list of references documenting and validating the LMI.

• LMI approach to exponential stability of linear systems with interval time-varying delays - Original Article by Phat et al.

Conic Sector Lemma

Conic Sector Lemma

For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization $(A,B,C,D)$, where $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},$ and $D\in {\mathcal {R}}^{m\times m}$. The state-space representation is:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $u(t)\in \mathbb {R} ^{m}$ are the system state, output, and the input vector respectively.

The Data

The system coefficient matrices $(A,B,C,D)$ are required. Optionally, the parameters to define a cone, either in the form of $[a,b]$ where $a,b\in \mathbb {R} ,a<b$ or a radius $r\in \mathbb {R} _{+}$ and ceter $c\in \mathbb {R}$.

The Feasibility LMI

The system ${\mathcal {G}}$ is inside the given cone $[a,b]$ if the following is feasible:

${\begin{aligned}{\text{Find: }}&P\\{\text{subj. to: }}&P>0\\&{\begin{bmatrix}PA+A^{\top }P+C^{\top }C&PB-{\frac {a+b}{2}}C^{\top }+C^{\top }D\\(PB-{\frac {a+b}{2}}C^{\top }+C^{\top }D)^{\top }&D^{\top }D-{\frac {a+b}{2}}(D+D^{\top })+abI\end{bmatrix}}\leq 0.\end{aligned}}$

The above LMI can be used to also determine the cone parameters by setting $a$ as a variable along with the condition $a<b$, and use the bisection method to find $b$.

If the given cone is represented by a center $c$ and radius $r$, then the following feasibility problem can be evaluated to check if ${\mathcal {G}}$ is inside the given cone:

${\begin{aligned}{\text{Find: }}&P\\{\text{subj. to: }}&P>0\\&{\begin{bmatrix}PA+A^{\top }P+C^{\top }C&PB-cC^{\top }+C^{\top }D\\(PB-cC^{\top }+C^{\top }D)^{\top }&D^{\top }D-c(D+D^{\top })+(c^{2}-r^{2})I\end{bmatrix}}\leq 0.\end{aligned}}$

In order to also find the cone parameters, substituting $\gamma =r^{2}$ as a decision variable with additional constraint $\gamma \geq 0$ and then solving for $c$ via the bisection method will give the cone in which the system ${\mathcal {G}}$ resides if the problem is feasible.

Conclusion:

The aforementioned LMIs can be utilized to either check if ${\mathcal {G}}$ is in the specified cone or not, or can be used to check the stability of ${\mathcal {G}}$ by finding if a feasible cone can be obtained that encloses ${\mathcal {G}}$. An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:

$Q=-1,S=frac{a+b}{2}I=cI,R=-abI=(r^{2}-c^{2})I$.

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Conic_sector_example.m

Related LMIs

Exterior Conic Sector Lemma.

KYP Lemma

External Links

A list of references documenting and validating the LMI.

• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.

Polytopic Quadratic Stability

An important result to determine the stability of the system with uncertainties

The System:

Consider the system with Affine Time-Varying uncertainty (No input)

${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}$

where

${\begin{aligned}\Delta A(t)=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\end{aligned}}$

where $\delta _{i}(t)$ lies in either the intervals

${\begin{aligned}\delta _{i}\in [\delta _{i}^{-},\delta _{i}^{+}]\\\end{aligned}}$

or the simplex

${\begin{aligned}\delta (t)\in {\delta :\Sigma \alpha _{i}=1,\alpha \geq 0}\end{aligned}}$

where $x\in \mathbb {R} ^{m}$ and $A\in \mathbb {R} ^{mxm}$

The Data

The matrix A and $\Delta _{A(t)}$ are known

The Optimization

The Definitions: Quadratic Stability for Dynamic Uncertainty

The system

${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}$

is Quadraticallly Stable over $\Delta$ if there exists a P > 0

Theorem
$(A+\Delta ,{\boldsymbol {\Delta }})$ is quadratically stable over ${\boldsymbol {\Delta }}:=Co(A_{1},...,A_{k})$ if and only if there exists a P > 0 such that

${\begin{aligned}(A+A_{i})^{T}P+P(A+A_{i})<0\quad for\quad all\quad i=1,....,k\end{aligned}}$

The theorem says the LMI only needs to hold at the EXTREMAL POINTS or VERTICES of the polytope.

• Quadratic Stability MUST be expressed as an LMI

The LMI

${\begin{aligned}(A+\Delta )^{T}P+P(A+\Delta )<0\quad for\quad all\quad \Delta \in {\boldsymbol {\Delta }}\end{aligned}}$

Conclusion:

Quadratic Stability Implies Stability of trajectories for any $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$ for all $t\geq 0$
Quadratic Stability is CONSERVATIVE.
There are Stable System which are not Quadratically stable.
Quadratic Stability is sometimes referred to as an "infinite-dimensional LMI"

• Meaning it represents an infinite number of LMI constraints.
• One for each possible value $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$
• Also called a parameterized LMI
• Such LMIs are not tractable.
• For polytopic sets, the LMI can be made finite.

Implementation

A link to implementation of the LMI
https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m

Related LMIs

• Parametric Norm Bounded Uncertain System Quadratic Stability

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Mu Analysis

Mu Synthesis. The technique of $\mu$ synthesis extends the methods of $H\infty$ synthesis to design a robust controller for an uncertain plant. You can perform $\mu$ synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the "musyn" command in MATLAB. $\mu$ analysis is an extremely powerful multivariable technique which has been applied to many problems in the almost every industry including Aerospace, process industry etc.

The System:

Consider the continuous-time generalized LTI plant with minimal states-space realization

${\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\\\end{aligned}}$

where it is assumed that $D$ is Invertible.

The Data

The matrices needed as inputs are only, $A$ and $D$.

The LMI: $\mu$- Analysis

The inequality ${\overline {\sigma }}(DAD^{-1})<\gamma$ holds if and only if there exist $X\in \mathbb {S} ^{n}$ and $\gamma \in \mathbb {R} _{>0}$, where $X>0$, satisfying:

${\begin{aligned}A^{T}XA-\gamma ^{2}X<0\end{aligned}}$

Conclusion:

The inequality ${\overline {\sigma }}(DAD^{-1})<\gamma$ holds for $D=X^{1/2},$ where X satisfies the above Inequality.

Implementation

• https://github.com/Ricky-10/coding107/blob/master/Mu%20Analysis - $\mu$-Analysis

External links

• [18]-MATLAB $\mu$-synthesis Implementation
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.

Optimization Over Affine Family of Linear Systems

Optimization over an Affine Family of Linear Systems

Presented in this page is a general framework for optimizing various convex functionals for a system which depends affinely, or linearly, on a parameter using linear matrix inequalities. The optimization problem presented on this page generalizes an LMI which can be applied to various problems within linear systems and control. Some examples of these applications are finding the $H_{2}$ and $H_{\infty }$ norms, entropy, dissipativity, and the Hankel norm of an affinely parametric system.

The System

Consider a family of linear systems
‌ ${\dot {x}}=Ax+B_{w}w,$
‌ $z=C_{z}({\theta })x+D_{zw}({\theta })w$
‌with state space realization $(A,B_{w},C_{z},D_{zw})$ where $C_{z}$ and $D_{zw}$ depend affinely on the parameter ${\theta }\in \mathbb {R} ^{p}$.

We assume $A$ is stable and $(A,B_{w})$ is controllable.

The transfer function, $H_{\theta }(s){\widehat {=}}C_{z}({\theta })(sI-A)^{-1}B_{w}+D_{zw}({\theta })$ depends affinely on ${\theta }$.

The Data

The transfer function $H$, and system matrices $A$, $B_{w}$, $C_{z}$, $D_{zw}$ are known. ${\varphi }$ represents the convex functionals, ${\alpha }$ and ${\psi }$ represent some auxiliary variables dependent on the problem being solved.

The LMI:Generalized Optimization for Affine Linear Systems

Several control theory problems, mentioned earlier, take the following form:
‌minimize ${\psi }_{0}(H_{\theta })$
‌subject to ${\psi }_{i}(H_{\theta })<{\alpha }_{i},i=1,...,p$

Problems of this nature can be formulated as an LMI by representing ${\psi }_{i}(H_{\theta })<{\alpha }_{i}$ as an LMI in ${\theta },{\alpha }_{i},$ and possibly ${\psi }_{i}$ such that $F_{i}({\theta },{\alpha }_{i},{\psi }_{i})>0$

Thus, the general optimization problem to be applied to an affine family of linear systems is as follows:
‌minimize ${\alpha }_{0}$
‌subject to $F_{i}({\theta },{\alpha }_{i},{\psi }_{i})>0,i=0,1,....,P$

Conclusion:

The LMI for this generalized optimization problem may be extended to various convex functionals for affine parametric systems. For extensions of this LMI, see the related LMIs section.

Implementation

Implementation of LMI's of this form require Yalmip and a linear solver such as Sedumi or SDPT3.

$H_{\infty }$ Norm for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

Entropy Bond for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.

LMI can be applied to other extensions in stability and controller analysis. Please see the related LMI pages in the section below.

Related LMIs

$H_{\infty }$ Norm for Affine Parametric Systems

Entropy Bond for Affine Parametric Systems

Dissipativity for Affine Parametric Systems

for Affine Parametric Systems

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

Return to Main Page:

LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control

Hurwitz Stabilizability

This section studies the stabilizability properties of the control systems.

The System

Given a state-space representation of a linear system

${\begin{aligned}\ \rho x=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}$

Where $\rho$ represents the differential operator ( when the system is continuous-time) or one-step forward shift operator ( Discrete-Time system). $x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}$ are the state, output and input vectors respectively.

The Data

$A,B,C,D$ are system matrices.

Definition

The system , or the matrix pair $(A,B)$ is Hurwitz Stabilizable if there exists a real matrix $K$ such that $(A+BK)$ is Hurwitz Stable. The condition for Hurwitz Stabilizability of a given matrix pair (A,B) is given by the PBH criterion:

${\begin{aligned}\ rank{\begin{bmatrix}sI-A&B\end{bmatrix}}&=n,\forall s\in \mathbb {C} ,Re(s)\geq 0\\\end{aligned}}$

(1)

The PBH criterion shows that the system is Hurwitz stabilizable if all uncontrollable modes are Hurwitz stable.

LMI Condition

The system, or matrix pair $(A,B)$ is Hurwitz stabilizable if and only if there exists symmetric positive definite matrix $P$ and $W$ such that:

${\begin{aligned}AP+PA^{T}+BW+W^{T}B^{T}<0\\\end{aligned}}$

(2)

Following definition of Hurwitz Stabilizability and Lyapunov Stability theory, the PBH criterion is true if and only if , a matrix $K$ and a matrix $P>0$ satisfying:

${\begin{aligned}(A+BK)P+P(A+BK)^{T}<0\\\end{aligned}}$

(3)

Letting

${\begin{aligned}W=KP\\\end{aligned}}$

(4)

Putting (4) in (3) gives us (2).

Implementation

This implementation requires Yalmip and Mosek.

• https://github.com/ShenoyVaradaraya/LMI--master/blob/main/Hurwitz_Stabilizability.m

Conclusion

Compared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability.

References

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013

Quadratic Hurwitz Stabilization for Polytopic Systems

This section studies the Quadratic Hurwitz stabilization for polytopic systems.

The System

Given a state-space representation of a linear system

${\begin{aligned}\ {\dot {x}}(t)=A(\delta (t)x(t)+B\delta (t)u(t)\\A(\delta (t))=A_{0}+\delta _{1}(t)A_{1}+\delta _{2}(t)A_{2}\\B(\delta (t))=B_{0}+\delta _{1}(t)B_{1}+\delta _{2}(t)B_{2}\end{aligned}}$

(1)

LMI Condition

With $\Delta =\Delta _{p}$ , the quadratic Hurwitz Stabilization problem has a solution if and only if there exists a symmetric positive definite matrix $P$ and a matrix $W$ satisfying the below LMI :

${\begin{aligned}(A_{0}+A_{i})P+P(A_{0}+A_{i})^{T}+(B_{0}+B_{i})W+W(B_{0}+B_{i})^{T}<0,i&=1,2,....k\end{aligned}}$

(2)

In this case, a solution to the problem is given by

${\begin{aligned}K=WP^{-1}\\\end{aligned}}$

(3)

Conclusion

Stability of a system does not guarantee quadratic stability. Since quadratic stability can represent infinite LMI constraints, it is not tractable. Therefore, to make it feasible and tractable, polytopic sets are helpful.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013

Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Stability of DT LTI systems can be determined by solving Lyapunov Inequality.

The System

Discrete-Time System

${\begin{aligned}x(t)_{k+1}&=A_{d}x(t)_{k},&A_{d}\in {\bf {{R^{n*n}}\;}}\\\end{aligned}}$

The Data

The matrices: System $A_{d},P$.

The Optimization Problem

The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:

Discrete-Time Bounded Real Lemma

The LMI formulation

${\begin{aligned}P\in {\bf {{S^{n}}\;}}\\{\text{Find}}\;&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P\end{bmatrix}}&<0\end{aligned}}$

Conclusion:

If there exists a $P\in {\bf {S^{n}}}$ satisfying the LMI then, $|\lambda _{i}(A_{d})|\leq 1,\forall i=1,2,...,n;$ and the equilibrium point ${\bar {x}}=0$ of the system is Lyapunov stable.

Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIs

Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

LMI for Schur Stabilization

LMI for Schur Stabilization

Similar to the stability of continuous-time systems, one can analyze the stability of discrete-time systems. A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems and a linear time-invariant system with this property is called a Schur stable system.

The System

We consider the following system:

${\begin{aligned}x(k+1)=Ax(k)+Bu(k)\end{aligned}}$

where the matrices $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times r}$, $x\in \mathbb {R} ^{n}$, and $u\in \mathbb {R} ^{r}$ are the state matrix, input matrix, state vector, and the input vector, respectively.

Moreover, $k$ represents time in the discrete-time system and $k+1$ is the next time step.

The state feedback control law is defined as follows:

${\begin{aligned}u(k)=Kx(k)\end{aligned}}$

where $K\in \mathbb {R} ^{n\times r}$ is the controller gain. Thus, the closed-loop system is given by:

${\begin{aligned}x(k+1)=(A+BK)x(k)\end{aligned}}$

The Data

The matrices $A$ and $B$ are given.

We define the scalar as $\gamma$ with the range of $0<\gamma \leq 1$.

The Optimization Problem

The optimization problem is to find a matrix ${\begin{aligned}K\in \mathbb {R} ^{r\times n}\end{aligned}}$ such that:

${\begin{aligned}||A+BK||_{2}<\gamma \end{aligned}}$

According to the definition of the spectral norms of matrices, this condition becomes equivalent to:

${\begin{aligned}(A+BK)^{T}(A+BK)<\gamma ^{2}I\end{aligned}}$

Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:

${\begin{aligned}{\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

The LMI: LMI for Schur stabilization

The LMI for Schur stabilization can be written as minimization of the scalar, $\gamma$, in the following constraints:

${\begin{aligned}&{\text{min}}\quad \gamma \\&{\text{s.t.}}\quad {\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

Conclusion:

After solving the LMI problem, we obtain the controller gain $K$ and the minimized parameter $\gamma$. This problem is a special case of Intensive Disk Region Design (page 230 in [1]). This problem may not have a solution even when the system is stabilizable. In other words, once there exists a solution, the solution is robust in the sense that when there are parameter perturbations, the closed-loop system's eigenvalues are not easy to go outside of a circle region within the unit circle [1].

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIs

LMI for Hurwitz stability

External Links

• [19] - LMI in Control Systems Analysis, Design and Applications

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LMIs in Control/Tools

L2-Gain of Systems with Multiplicative noise

The System

${\begin{aligned}x(k+1)&=Ax(k)+B_{w}w(k)+\sum _{i=1}^{L}(A_{i}x(k)+B_{w,i}w(k))p_{i}(k),\quad x(0)=0,\\z(k)&=C_{z}x(k)+D_{zw}w(k)+\sum _{i=1}^{L}(C_{z,i}x(k)+D_{zw,i}w(k))p_{i}(k),\end{aligned}}$

where $p(0),p(1),\dots$, are independent, identically distributed random variables with $Ep(k)=0,Ep(k)p^{\top }(k)=\Sigma =diag(\sigma _{1},\dots ,\sigma _{L})$ and $x(0)$ is independent of the process $p$.

The Data

The matrices $A,B_{w},\{A_{i}.B_{w,i}\}_{i=1}^{L},C_{z},D_{zw},\{C_{z,i},D_{zw,i}\}_{i=1}^{L},\{\sigma _{i}\}_{i=1}^{L}$.

The LMI:

${\begin{aligned}&\min _{\{P\succ 0,\gamma ^{2}\}}\gamma ^{2}\\&\quad s.t.{\begin{bmatrix}A&B_{w}\\C_{z}&D_{zw}\end{bmatrix}}^{\top }{\begin{bmatrix}P&0\\0&I\end{bmatrix}}{\begin{bmatrix}A&B_{w}\\C_{z}&D_{zw}\end{bmatrix}}-{\begin{bmatrix}P&0\\0&\gamma ^{2}I\end{bmatrix}}+\sum _{i=1}^{L}\sigma _{i}^{2}{\begin{bmatrix}A_{i}&B_{w,i}\\C_{z,i}&D_{zw,i}\end{bmatrix}}^{\top }{\begin{bmatrix}P&0\\0&I\end{bmatrix}}{\begin{bmatrix}A_{i}&B_{w,i}\\C_{z,i}&D_{zw,i}\end{bmatrix}}^{\top }\preceq 0\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/a34713575cd8ae9831cb5b7eb759d0fd45a8c37f

Conclusion

The optimal $\gamma$ returns an upper bound on the ${\mathcal {L}}_{2}$ gain of the system. .

Remark

It is straightforward to apply scaling method [Boyd, sec.6.3.4] to obtain component-wise results.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Discrete-Time Quadratic Stability

Discrete-Time Quadratic Stability

Stability is an important property, stability analysis is necessary for control theory. For robust control, this criterion is applicable for the uncertain discrete-time linear system. It is based on the Discrete Time Lyapunov Stability.

The System

${\begin{aligned}x_{k+1}&=A_{d}(\alpha )x_{k}\\Where:\\&A_{d}(\alpha )=A_{d}+\Delta A_{d}(\delta (t))\\&\Delta A_{d}(\delta (t))=\sum _{k=1}^{n}\delta _{k}(t)A_{d;k}\in \mathrm {R} ^{n\times n}\\&\delta (t)=[\delta _{1}(t),...\delta _{n}(t)]-{\text{The set of perturbation parameters}}\\&\delta (t)\in \mathrm {R} \;\;\;A_{d;i}\in \mathrm {R} ^{n\times n}\end{aligned}}$

The Data

The matrices $A\in R^{n\times n}\;A_{d;i}\in R^{n\times n}$.

The Optimization Problem

The following feasibility problem should be solved:

${\begin{aligned}{\text{Find}}\;&P>0:\\&(A_{d;0}+\Delta A_{d}(\delta (t)))^{T}P(A_{d;0}+\Delta A_{d}(\delta (t)))-P<0{\text{ for all }}\delta \end{aligned}}$

Where $P\in R^{n,n}$.

In case of polytopic uncertainty:

${\begin{aligned}{\text{Find}}\;&P>0:\\&(A_{d;0}+A_{d;i})^{T}P(A_{d;0}+A_{d;i})-P<0{\text{ for all }}i=1,...n\end{aligned}}$

Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

Implementation:

• [20] - Matlab implementation using the YALMIP framework and Mosek solver

Related LMIs: =

• - Discrete Time Stabilizability
• Polytopic stability for continuous time case
• Quadratic polytopic stabilization
• Discrete Time Lyapunov Stability

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (3.20.2 page 64)
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Return to Main Page:

• Main Page.

Stability of Lure's Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}$

The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z}$.

The LMI: The Lure's System's Stability

The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.

${\begin{aligned}{\text{Find}}\;&P>0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0:\\&{\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\prec 0\\\end{aligned}}$

Implementation

https://codeocean.com/capsule/0232754/tree

Conclusion

If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.

Remark

The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system . It is also necessary when there is only one nonlinearity, i.e., when $n_{p}=1$.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

L2 Gain of Lure's Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}$

The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z}$.

The Optimization Problem:

The following semi-definite problem should be solved.

${\begin{aligned}&\min _{\{P\succ 0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0\}}\gamma ^{2}\;\\&\quad \quad \quad \quad \quad \quad \quad \quad s.t.\quad {\begin{bmatrix}A^{\top }P+PA+C_{z}^{\top }C_{z}&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T&PB_{w}\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T&\Lambda C_{q}B_{w}\\B_{w}^{\top }P&B_{w}^{\top }C_{q}^{\top }\Lambda &-\gamma ^{2}I\end{bmatrix}}\preceq 0\\\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/12a7039f9e3d966e24b43fd58a3cce3725282ed2

Conclusion

The value function returns the square of the smallest provable upper bound on the ${\mathcal {L}}_{2}$ gain of the Lure's system.

Remark

The Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Output Energy Bound for Lure's Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q(t)&=C_{q}x(t),\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}$

The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z},x(0)$.

The Optimization Problem:

The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.

${\begin{aligned}&\min _{P\succ 0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0}x^{\top }(0)(P+C_{q}^{\top }\Lambda C_{q})x(0)\\&\quad \quad \quad \quad \quad \quad \quad {\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\preceq 0\\\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/blob/master/LMIs%20for%20Output%20Energy%20Bounds%20of%20Lure's%20Systems

Conclusion

The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., $J=\int _{0}^{\infty }z^{\top }z\ dt$ with initial conditions $x(0)$.

Remark

The key step in the proof is to satisfy ${\frac {d}{dt}}V(x)+z^{\top }z\leq 0$, where $V(.)$ is Lyapunov function in a special form.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Stability of Quadratic Constrained Systems

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{u}u(t)+B_{w}w(t),\\q(t)&=C_{q}x(t)+D_{qp}p(t)+D_{qu}u(t)+D_{qw}w(t),\\z(t)&=C_{z}x(t)+D_{zp}p(t)+D_{zu}u(t)+D_{zw}w(t)\\\int _{0}^{t}&p^{\top }(\tau )p(\tau )\ d\tau \leq \int _{0}^{t}q^{\top }(\tau )q(\tau )\ d\tau .\end{aligned}}$

The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z},D_{qp},D_{zw}$.

The LMI:

The following feasibility problem should be solved.

${\begin{aligned}{\text{Find}}\;&\{P\succ 0,\lambda \geq 0\}:\\&s.t.\quad {\begin{bmatrix}A^{\top }P+PA+\lambda C_{q}^{\top }C_{q}&PB_{p}+\lambda C_{q}^{\top }D_{qp}\\(PB_{p}+\lambda C_{q}^{\top }D_{qp})^{\top }&\lambda (I-D_{qp}^{\top }D_{qp})\end{bmatrix}}\prec 0.\end{aligned}}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/38f3b55ca7060a1260384a96e9dc31142af07a9a

Conclusion

The integral quadratic constrained system is stable if the provided LMI is feasible

Remark

The key point of the proof is to satisfy ${\dot {V}}<0$ whenever $p^{\top }p\leq q^{\top }q$, using ${\mathcal {S}}$-procedure.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Conic Sector Lemma

Conic Sector Lemma

For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.

The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization $(A,B,C,D)$, where $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},$ and $D\in {\mathcal {R}}^{m\times m}$. The state-space representation is:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $u(t)\in \mathbb {R} ^{m}$ are the system state, output, and the input vector respectively.

The Data

The system coefficient matrices $(A,B,C,D)$ are required. Optionally, the parameters to define a cone, either in the form of $[a,b]$ where $a,b\in \mathbb {R} ,a<b$ or a radius $r\in \mathbb {R} _{+}$ and ceter $c\in \mathbb {R}$.

The Feasibility LMI

The system ${\mathcal {G}}$ is inside the given cone $[a,b]$ if the following is feasible:

${\begin{aligned}{\text{Find: }}&P\\{\text{subj. to: }}&P>0\\&{\begin{bmatrix}PA+A^{\top }P+C^{\top }C&PB-{\frac {a+b}{2}}C^{\top }+C^{\top }D\\(PB-{\frac {a+b}{2}}C^{\top }+C^{\top }D)^{\top }&D^{\top }D-{\frac {a+b}{2}}(D+D^{\top })+abI\end{bmatrix}}\leq 0.\end{aligned}}$

The above LMI can be used to also determine the cone parameters by setting $a$ as a variable along with the condition $a<b$, and use the bisection method to find $b$.

If the given cone is represented by a center $c$ and radius $r$, then the following feasibility problem can be evaluated to check if ${\mathcal {G}}$ is inside the given cone:

${\begin{aligned}{\text{Find: }}&P\\{\text{subj. to: }}&P>0\\&{\begin{bmatrix}PA+A^{\top }P+C^{\top }C&PB-cC^{\top }+C^{\top }D\\(PB-cC^{\top }+C^{\top }D)^{\top }&D^{\top }D-c(D+D^{\top })+(c^{2}-r^{2})I\end{bmatrix}}\leq 0.\end{aligned}}$

In order to also find the cone parameters, substituting $\gamma =r^{2}$ as a decision variable with additional constraint $\gamma \geq 0$ and then solving for $c$ via the bisection method will give the cone in which the system ${\mathcal {G}}$ resides if the problem is feasible.

Conclusion:

The aforementioned LMIs can be utilized to either check if ${\mathcal {G}}$ is in the specified cone or not, or can be used to check the stability of ${\mathcal {G}}$ by finding if a feasible cone can be obtained that encloses ${\mathcal {G}}$. An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:

$Q=-1,S=frac{a+b}{2}I=cI,R=-abI=(r^{2}-c^{2})I$.

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Conic_sector_example.m

Related LMIs

Exterior Conic Sector Lemma.

KYP Lemma

External Links

A list of references documenting and validating the LMI.

• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.