LMIs in Control/pages/Discrete Time QSR

The ConceptEdit

In systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state  , its input   and its output  , the input-output correlation is given a supply rate  . A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function   such that  ,   and

 

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate  .

The physical interpretation is that   is the energy stored in the system, whereas   is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.

The SystemEdit

Consider a discrete-time LTI system,  , with minimal state-space relization  , where   and  .

 
 

The DataEdit

The matrices   and  

The Optimization ProblemEdit

The system   is QSR disipative if

 

where   is the input to   is the output of   and  .


LMI : Discrete-Time KYP Lemma for QSR Dissipative SystemsEdit

The system   is also QSR dissipative if and only if there exists   where   such that

 

Conclusion:Edit

If there exist a positive definite   for the the selected Q,S and R matrices then the system   is QSR dissipative.

ImplementationEdit

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

Related LMIsEdit

KYP Lemma
KYP Lemma for continous Time QSR Dissipative system

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