# LMIs in Control/pages/KYP Lemma QSR

**The Concept**Edit

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 System**Edit

Consider a contiuous-time LTI system, , with minimal state-space realization **(A, B, C, D)**, where and .

**The Data**Edit

The matrices and which defines the state space of the system

**The Optimization Problem**Edit

The system is **QSR** disipative if

where is the input to is the output of and .

**LMI : KYP Lemma for QSR Dissipative Systems**Edit

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.

**Implementation**Edit

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

**Related LMIs**Edit

**References**Edit

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