Robust control of linear systems with generalized positive real uncertainty (Q1360491)

Considered is the uncertain system \[ \begin{aligned} \dot x(t) & = [A+\Delta A(t)]x(t)+ [B_1+\Delta B_1(t)]w(t)+ [B_2+\Delta B_2(t)]u(t),\\ z(t) & = [C_1+\Delta C_1(t)]x(t)+ [D_{11}+\Delta D_{11}(t)]w(t)+ [D_{12}+\Delta D_{12}(t)]u(t),\\ y(t) & = [C_2+\Delta C_2(t)]x(t)+ [D_{21}+\Delta D_{21}(t)]w(t)+ [D_{22}+\Delta D_{22}(t)]u(t),\end{aligned} \] where \(x(t)\in\mathbb{R}^n\) is the state, \(u(t)\in\mathbb{R}^m\) is the control input, \(w(t)\in\mathbb{R}^q\) is the exogeneous input (which belongs to \(L_2[0,\infty)\)), \(y(t)\in \mathbb{R}^r\) is the measured output, \(z(t)\in\mathbb{R}^p\) is the controlled output. In the system above \(A\), \(B_1,B_2,\dots\) are constant matrices and \(\Delta A,\dots\) represent time varying uncertainties. The problem of robust stability analysis, robust stabilization, and robust \(H_\infty\) control for this class of uncertain systems are studied. It is shown that the above robust control problems can be converted into associated scaled \(H_\infty\) control ones.