LMIs in Control/Click here to continue/LMIs in system and stability Theory/Generalized KYP 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.