Norms of solutions to the Lur'e-Riccati matrix equations as criteria of the quality of stabilizability and detectability
The optimal control system (1) \(x'= Ax+ Bu\), minimizing \[ J= 2|A|\int^\infty_0 (|x(t)|^2+ |Bu(t)|^2) dt, \] leads to the feedback control \(u(t)= - (B* B)^{- 1} B* \Lambda x(t)\), where \(\Lambda\) is constant, symmetric and positive definite solution of the Lur'e-Riccati matrix equation. System (1) (or the pair \(A\), \(B\)) is stabilizable, if for any \(x(0)\) there exists a control \(u(t)\) such that \(u(t)\to 0\) and \(x(t)\to 0\), as \(t\to \infty\). In this paper, certain estimates of the growth of the solutions \(|x(t)|\) of equation (1) are established. These depend on the matrix \(\Lambda\). Therefore, its norm \(|\Lambda|\) is proposed as a measure to characterize the ``stabilizability'' of system (1), in the sense that the larger is \(|\Lambda|\), the larger is the control necessary to stabilize the system. Of course, \(\Lambda\) can be calculated as solution of the Lur'e-Riccati equation. Similar estimates are also obtained for related Hamiltonian systems. These results also are extended to detectability, controllability and observability. Finally, an algorithm is given for solving algebraic quadratic matrix equations of the Lur'e-Riccati type.
scientific article; zbMATH DE number 917966