The regulator problem with robust stability
The system considered is \(\dot x= Ax+ Bu\), \(y= Cx\) where the exosubsystem \(\dot x_2= A_{22} x_2\) has its eigenvalues on the stability boundary \(\text{Re}(s)= 0\). The compensator is \(\dot z= Fz+ Gy\), \(u= Hz+ Jy\). It is supposed that \((A_{11}, B)\) is stabilizable, \((C, A)\) is detectable and that the plant zeros do not coincide with the exosubsystem poles. With the controlled subplant \((A_{11}, B, C_1)\), the plant \((A, B, C)\) and with the controller \((F, G, H, J)\) are associated some subspace-valued complex-variable functions \(P(s)\), \(M(s)\), \(C(s)\), which are some generalizations of the Nyquist curve. The robust stability measure is introduced following Gohberg and Krein. Then, a necessary and sufficient condition about the stability of the closed-loop system is given. Then the regulator problem with a given robust stability margin is posed. Necessary and sufficient conditions for this problem are given. They are the main result of this paper. The method is illustrated for a scalar regulated plant under constant disturbance. The approach of Cevik and Schumacher combines finely Kučera's purely algebraic approach giving the general parametrization of the stabilizing controller with the complex variable approaches adopted from complex interpolation theory generalizing the classical, Nyquist curve based approaches. The article testifies that linear control theory reached its full maturity.
scientific article; zbMATH DE number 819357
scientific article