\({\mathcal H}^\infty\) optimal terminal state control for linear systems with lumped and distributed time delays
The authors consider the linear time-varying delay system defined over a finite-time horizon of the form: \[ \dot x(t)= A(t)x(t)+\sum_{k=1}^{r}F_{k}(t)x(t-\tau_{k})+ \int_{-\tau}^{0}H(t,\sigma)x(t+\sigma) d\sigma+ B(t)u(t)+G(t)w(t), \] \[ x(s)=0, \quad s\in[-\tau,0], \quad t\in\Omega=[0,T], \] where \(x(t)\in\mathbb{R}^{n}\), \(u(t)\in\mathbb{R}^{m}\) are the state and control input, \(w(t)\in\mathbb{R}^{l}\) and \(w(\cdot)\) is unknown square integrable disturbance, i.e., \(w\in\mathcal L^{2}(\Omega)\). For any linear feedback control \(u=Kx\) the system is treated as a definition of the linear operator \(G:w(\cdot)\to(x(T),u(\cdot))=(x(T),Kx(\cdot))\in \mathbb R^{n}\times\mathcal L^{2}(\Omega)\). The \(\mathcal H^{\infty}\) norm of the operator \(G\) is defined as \(\|G\|=\sup\{\|(x(T),u)\|_{Q,R}/\|w\|_{2}, w\neq 0,w\in\mathcal L^{2}(\Omega)\}\), where \(\|w\|_{2}\) is a norm in \(\mathcal L^{2}(\Omega)\), \(\|(x(T),u)\|_{Q,R}\) is a weighted norm in \(\mathbb R^{n}\times\mathcal L^{2}(\Omega)\) with some positive definite matrices \(Q\) and \(R(t)\). The main result of the paper proclaims that the problem of finding a linear feedback control \(u=Kx\) which minimizes the \(\mathcal H^{\infty}\) norm of the operator \(G\) is solvable if and only if a certain finite-dimensional Riccati equation has a symmetric solution. A proof is reached by reducing of the problem to a differential game. An illustrative example is given.
scientific article; zbMATH DE number 1353733