LMIs in Control/pages/Generalized KYP Lemma Conic Sector

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  .

 

Also consider  , which is defined as

 ,

where   and  .

The DataEdit

The matrices The matrices   and  . The values of a and b

LMI : Generalized KYP (GKYP) Lemma for Conic SectorsEdit

The following generalized KYP Lemmas give conditions for   to be inside the cone   within finite frequency bandwidths.

1. (Low Frequency Range) The system   is inside the cone   for all  , where   and  , if there exist   and  , where  , such that
 .
If   and Q = 0, then the traditional Conic Sector Lemma is recovered. The parameter   is incuded in the above LMI to effectively transform   into the strict inequality  
2. (Intermediate Frequency Range) The system   is inside the cone   for all  , where   and  , if there exist   and   and   where   and  , such that
 .
The parameter   is incuded in the above LMI to effectively transform   into the strict inequality  .
3. (High Frequency Range) The system   is inside the cone   for all  , where   and  , if there exist  , where  , such that
 .

If (A, B, C, D) is a minimal realization, then the matrix inequalities in all of the above LMI, then it can be nostrict.

Conclusion:Edit

If there exist a positive definite   matrix satisfying above LMIs for the given frequency bandwidths then the system   is inside the cone [a,b].

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
Modified 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.