LMIs in Control/KYP Lemmas/KYP lemma for continous time QSR dissipative system

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 $D$

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

Kalman%E2%80%93Yakubovich%E2%80%93Popov_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