LMIs in Control/pages/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, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, 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 ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.

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

The Data

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

LMI : Exterior Conic Sector Lemma

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

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

${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq 0.$

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

${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)&0\\(PB-C^{T}(D-CI))^{T}&-(D-cI)^{T}(D-cI)&rI\\0&(rI)^{T}&-I\end{bmatrix}}\leq 0.$

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 ${\mathcal {G}}$ 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

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 Transactions 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.