LMIs in Control/pages/KYP Lemma for Descriptor Systems

The ConceptEdit

Descriptor system descriptions frequently appear when solving computational problems in the analysis and design of standard linear systems. The numerically reliable solution of many standard control problems like the solu­tion of Riccati equations, computation of system zeros, design of fault detection and isolation filters (FDI), etc. relies on using descriptor system techniques.

Many algorithm for standard systems as for example stabilization techniques, factorization methods, minimal realization, model reduction, etc. have been extended to the more general descriptor system descriptions. An important application of these algorithms is the numeri­cally reliable computation with rational and polynomial matrices via equivalent descriptor representations. Recall that each rational matrix R(s) can be seen as the transfer-function matrix of a continuous- or discrete-time descriptor system. Thus, each R(s) can be equivalently realized by a descriptor system quadruple (A-sE, B, C, D) satisfying R(S)= C(SE-A)-1B+D

Many operations on standard matrices (e.g., finding the rank, determinant, inverse or generalized inverses), or the solution of linear matrix equa­tions can be performed for rational matrices as well using descriptor system techniques. Other important applications of descriptor techniques are the computation of inner-outer and spectral factorisations, or minimum degree and normal­ized coprime factorisations of polynomial and rational matrices. More explanation can be found in the website of Institute of System Dynamics and control

The SystemEdit

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

 

The DataEdit

The matrices The matrices   and  

LMI : KYP Lemma for Descriptor SystemsEdit

The system   is extended strictly positive real (ESPR) if and only if there exists   and   such that

 
 
 

The system is also ESPR if there exists   such that

 
 

Conclusion:Edit

If there exist a X and W matrix satisfying above LMIs then the system   is Extended Strictly Positive Real.

ImplementationEdit

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

Related LMIsEdit

KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough

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. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
5. Numerical algorithms and software tools for analysis and modelling of descriptor systems. Prepr. of 2nd IFAC Workshop on System Structure and Control, Prague, Czechoslovakia, pp. 392-395, 1992.