LMIs in Control/pages/Dissipativity of Systems

Dissipativity of Systems

The dissipativity of systems is associated with their supply function. In general, a linear system is dissipative if the accumulated sum (integration) of the supply function is non-negative over all the duration of $$ T \geq 0 $$.

The System
A state-space representation of a linear system as given below:



\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t) \end{align}$$

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$u(t)\in \R^r$$ are the system state, output, and the input vector respectively. A, B, C and D are system coefficient matrices of appropriate dimensions. The control input u is restricted to be a piece-wise continuous vector function defined of $$[0,\infty)$$.

The transfer function of such a system can be evaluated as:



\begin{align} G(s)=C(sI-A)^{-1}B+D \end{align}$$

For such a system, a general quadratic supply function is defined as:



\begin{align} s(u,y) &= \begin{bmatrix} y & u \end{bmatrix} Q \begin{bmatrix} y \\ u \end{bmatrix} \end{align}$$

where Q is a real symmetric matrix of (m+r) dimensions. Q need not be either symmetric positive or negative definite.

The Data
Number of states n, number of outputs m and number of control inputs r need to be known. Moreover, the system matrices A,B,C,D are also required to be known. The system should also be controllable.

The Feasibility LMI
The system $$ G(s) $$ defined can be evaluated to be dissipative with respect to a supply function $$ s(u,y) $$ iff there exist $$P \geq 0 $$ and a $$Q$$ (defining $$s(u,y)$$) such that the following is feasible:



\begin{align} \text{Find} \; & P, Q:\\ & P \geq 0 \\ \begin{bmatrix} A^\top P + PA & PB \\ B^\top P     & 0 \end{bmatrix} & - \begin{bmatrix} C & D \\ 0 & I \end{bmatrix}^\top Q \begin{bmatrix} C & D \\ 0 & I \end{bmatrix} \leq 0. \end{align}$$

Conclusion:
If there is a feasible solution to the aforementioned LMI, then there exists a supply function $$ s(u,y) $$ for which the system $$ G(s) $$ is dissipative. Since the assumption of the system being controllable is required for it to be dissipative, this check can be used of as a sufficient condition to check the controllability of the linear system, just like the feasibility for Lyapunov stability.

Implementation
To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Dissipativity_example.m

Related LMIs
Continuous_Time_Lyapunov_Inequality