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Robust dissipative control for linear systems with dissipative uncertainty. (English) Zbl 0930.93068

The authors consider a linear time-invariant system of the form

$\begin{array}{cc}\hfill \left({\Sigma }\right):\phantom{\rule{1.em}{0ex}}\stackrel{˙}{x}\left(t\right)& =Ax\left(t\right)+{B}_{1}\omega \left(t\right)+{B}_{2}u\left(t\right),\phantom{\rule{1.em}{0ex}}x\left(0\right)=0,\hfill \\ \hfill z\left(t\right)& ={C}_{1}x\left(t\right)+{D}_{11}\omega \left(t\right)+{D}_{12}u\left(t\right),\hfill \\ \hfill y\left(t\right)& ={C}_{2}x\left(t\right)+{D}_{21}\omega \left(t\right)+{B}_{2}u\left(t\right),\hfill \end{array}$

where $x\left(t\right)\in {ℝ}^{n}$ is the state, $\omega \left(t\right)\in {ℝ}^{q}$ is the exogenous input, $u\left(t\right)\in {ℝ}^{m}$ is the control input, $z\left(t\right)\in {ℝ}^{r}$ is the controlled output, $y\left(t\right)\in {ℝ}^{q}$ is the measured output, and $A,{B}_{1},{B}_{2},{C}_{1},{C}_{2},{D}_{11}$, ${D}_{12}$ and ${D}_{21}$ are known real constant matrices of appropriate dimensions. One associates with this system the quadratic energy supply function $E\left(\omega ,z,T\right)={〈z,Qz〉}_{T}+2{〈z,S\omega 〉}_{T}+{〈\omega ,R\omega 〉}_{T},$ defined by Hill and Moylan, where $Q,S$ and $R$ are real matrices of appropriate dimensions with $Q$ and $R$ symmetric. The quadratic dissipative control problem the authors address is as follows. Given matrices $Q,S$ and $R,$ design a linear feedback control law for the system $\left({\Sigma }\right)$ such that the resulting closed-loop system is asymptotically stable and strictly $\left(Q,S,R\right)$-dissipative. Systems without or with uncertainty are treated and both linear static state feedback and linear dynamic output feedback controllers are considered. For systems without uncertainty, the authors establish the equivalence between strict quadratic dissipativeness and a small-gain condition, and derive LMI-based necessary and sufficient conditions for the solution of the quadratic dissipative control problem. The robust dissipative control techniques for linear systems subject to quadratic dissipative uncertainty are also developed, where asymptotic stability and strict quadratic dissipativeness must be achieved irrespective of the uncertainty. It is shown that the robust dissipative control problem can be solved in terms of a quadratic dissipative control problem for a scaled linear system without uncertainty. The design of robust dissipative controllers using an LMI approach is also proposed. The obtained results unify existing results on (robust) ${ℋ}_{\infty }$ and positive real control and provide a more flexible and less conservative control design method, especially in applications where both phase and gain performances are considered.

MSC:
 93D21 Adaptive or robust stabilization 93B36 ${H}^{\infty }$-control 15A39 Linear inequalities of matrices 93D10 Popov-type stability of feedback systems