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

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

The Data

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

LMI : Modified Exterior Conic Sector Lemma

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

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

$\left[\begin{array}{cc}PA+A^{T}P& PB-C^T(D-CI)\\ (PB-C^T(D-CI){)}^T& r^{2}I-(D-cI{)}^T(D-cI)\end{array}\right]\le 0.$

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

Therefore,

$\left[\begin{array}{cc}PA+A^{T}P-C^{T}C& PB-C^T(D-CI)\\ (PB-C^T(D-CI){)}^T& r^{2}I-(D-cI{)}^T(D-cI)\end{array}\right]\le \left[\begin{array}{cc}PA+A^{T}P& PB-C^T(D-CI)\\ (PB-C^T(D-CI){)}^T& r^{2}I-(D-cI{)}^T(D-cI)\end{array}\right]$

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

$\left[\begin{array}{ccc}PA+A^{T}P& PB-C^T(D-CI)& 0\\ (PB-C^T(D-CI){)}^T& -(D-cI{)}^T(D-cI)& rI\\ 0& (rI{)}^T& -I\end{array}\right]\le 0.$

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 $𝒢$ 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.