LMIs in Control/pages/Modified Exterior Conic Sector Lemma

The ConceptEdit

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 SystemEdit

Consider a square, contiuous-time linear time-invariant (LTI) system,  , with minimal state-space relization (A, B, C, D), where   and  .

 

The DataEdit

The matrices The matrices   and  

LMI : Modified Exterior Conic Sector LemmaEdit

The system   is in the exterior cone of radius r centered at c (i.e.  exconer(c)), where   and  , under either of the following sufficient conditions.

1. There exists P  , where P  , such that
 
Proof. The term   in the Actual Exterior Conic Sector Lemma makes the matrix inequality more neagtive definite.

Therefore,

 
2. There exists P  , where P  , such that
 
Proof. Applying the Schur complement lemma to the   terms in (1) gives (2).

Conclusion:Edit

If there exist a positive definite   matrix satisfying above LMIs then the system   is in the exterior cone of radius r centered at c.

ImplementationEdit

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

Related LMIsEdit

KYP Lemma
State Space Stability
Exterior Conic Sector Lemma

ReferencesEdit

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.