Sliding modes in systems with asymptotic state observers (Q1084054)

For a control system (1) \(\dot x=Ax+Bu\), \(y=Fx\), with constant matrices A,B,F of ranks \(n\times n\), \(n\times m\), \(r\times n\), resp. and (A,F) observable, the observed state x satisfies (2) \(\bar x\dot{\;}=A\bar x+Bu+L(F\bar x-y).\) The matrix L can be chosen so that \(A+LF\) is negatively definite and the system \({\dot \epsilon}=(A+LF)\epsilon\), where \(\epsilon =\bar x-x\), is asymptotically stable. Let (3) \(s_ 0=C\bar x=0\) be a linear manifold on which the control u switches its values. Then (4) \(\dot s_ 0=CA\bar x+CBu+CLF\epsilon =0.\) If the matrix CB has maximal rank m then (4) can be solved for u which rends (1) in a completely observable system. Moreover, n-m coordinates in (3) can be expressed in terms of the remaining ones and the dimension of (1) can be reduced by n-m. The same approach is used for a system with perturbations in the measuring device \(y=Fx\).