LMIs in Control/pages/Exterior Conic Sector Lemma

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The Concept[edit]

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[edit]

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

The Data[edit]

The matrices The matrices and

LMI : Exterior Conic Sector Lemma[edit]

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

1. There exists P , where P , such that
2. There exists P , where P , such that

Proof, Applying the Schur complement lemma to the terms in (1) gives (2).


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


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

Related LMIs[edit]

KYP Lemma
State Space Stability


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.