LMIs in Control/pages/Discrete Time KYP Lemma with Feedthrough

The Concept

It is assumed in the Lemma that the state-space representation (Ad, Bd, Cd, Dd) is minimal. Then Positive Realness (PR) of the transfer function Cd(SI − Ad)-1Bd + Dd is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (Ad, Bd, Cd, Dd)=(−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, ${\displaystyle {\mathcal {G}}:{\mathcal {l}}_{2e}\rightarrow {\mathcal {l}}_{2e}}$ , with minimal state-space relization ${\displaystyle ({\mathcal {A}}_{d},{\mathcal {B}}_{d},{\mathcal {C}}_{d},{\mathcal {D}}_{d})}$ , where ${\displaystyle {\mathcal {A}}_{d}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}_{d}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}_{d}\in {\mathcal {R}}^{p\times n},}$  and ${\displaystyle {\mathcal {D}}_{d}\in {\mathcal {R}}^{p\times m}}$ .

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

The Data

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

LMI : Discrete-Time KYP Lemma with Feedthrough

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

1. There exists ${\displaystyle P\in {\mathcal {S}}^{n},}$  where ${\displaystyle P>0}$  such that
${\displaystyle {\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.}$
2. There exists ${\displaystyle Q\in {\mathcal {S}}^{n},}$  where ${\displaystyle Q>0}$  such that
${\displaystyle {\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}\leq 0.}$
3. There exists ${\displaystyle P\in {\mathcal {S}}^{n},}$  where ${\displaystyle Q>0}$  such that
${\displaystyle {\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.}$
4. There exists ${\displaystyle Q\in {\mathcal {S}}^{n},}$  where ${\displaystyle Q>0}$  such that
${\displaystyle {\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}\geq 0.}$

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

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

1. There exists ${\displaystyle P\in {\mathcal {S}}^{n},}$  where ${\displaystyle P>0}$  such that
${\displaystyle {\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}-C_{d}^{T}\\(A_{d}^{T}PB_{d}-C_{d}^{T})^{T}&B_{d}^{T}PB_{d}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.}$
2. There exists ${\displaystyle Q\in {\mathcal {S}}^{n},}$  where ${\displaystyle Q>0}$  such that
${\displaystyle {\begin{bmatrix}A_{d}QA_{d}^{T}-Q&A_{d}QC_{d}^{T}-B_{d}\\(A_{d}QC_{d}^{T}-B_{d})^{T}&C_{d}PC_{d}^{T}-(D_{d}^{T}+D_{d})\end{bmatrix}}<0.}$
3. There exists ${\displaystyle P\in {\mathcal {S}}^{n},}$  where ${\displaystyle Q>0}$  such that
${\displaystyle {\begin{bmatrix}P&PA_{d}&PB_{d}\\(PA_{d})^{T}&P&C_{d}^{T}\\(PB_{d})^{T}&C_{d}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.}$
4. There exists ${\displaystyle Q\in {\mathcal {S}}^{n},}$  where ${\displaystyle Q>0}$  such that
${\displaystyle {\begin{bmatrix}Q&A_{d}Q&B_{d}\\(A_{d}Q)^{T}&Q&QC_{d}^{T}\\(B_{d})^{T}&(QC_{d}^{T})^{T}&D_{d}^{T}+D_{d}\end{bmatrix}}>0.}$

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

Conclusion:

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