# LMIs in Control/pages/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, ${\displaystyle {\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}}$, with minimal state-space relization (E, A, B, C, D), where ${\displaystyle {\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},}$ and ${\displaystyle {\mathcal {D}}\in {\mathcal {R}}^{p\times m}}$.

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

## The Data

The matrices The matrices ${\displaystyle E,A,B,C}$ and ${\displaystyle D}$

## LMI : KYP Lemma for Descriptor Systems

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

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

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

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

## Conclusion:

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

## Implementation

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

## 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.