Robust \(H_{\infty}\) control for linear Markovian jump systems with unknown nonlinearities (Q1399389)

The authors consider the problem of a class of nonlinear stochastic uncertain systems with Markovian jump parameters described by \[ {dx(t)\over dt} =A \bigl(r(t) \bigr)x(t)+ B\bigl(r(t) \bigr)u(t)+ F\bigl(x(t), u(t),r(t), w(t),t \bigr), x(0)=x_0,\;r(0)=r_0 \tag{1} \] \[ z(t)=C\bigl(r(t) \bigr)x(t) \tag{2} \] where \(\{r(t),t\in [0,T]\}\) is a homogeneous finite-state Markovian process with right continuous trajectories and taking values in a finite set \({\mathfrak I}= \{1, 2, \dots,s\}\), \(x\in\mathbb{R}^n\), and \(u\in\mathbb{R}^m\) are state and control vectors, respectively; \(w\in\mathbb{R}^q\) is the disturbance input which belongs to \({\mathcal L}_2 [0,T]\), \(z\in\mathbb{R}^l\) is the penalty variable related to some performance cost. \(A(r(t))\), \(B(r(t))\) and \(C(r(t))\) are known real constant matrices of appropriate dimensions, \(F(x(t),u(t),r(t), w(t),t)\in \mathbb{R}^n\) is the system uncertainty. The authors investigate the problems of stochastic stability and disturbance attenuation for the system (1), (2). They obtain a sufficient condition under which the closed-loop uncertain system is robustly stable with \(H_\infty\) norm bound \(\gamma\).