Nonlinear feedback control with global stabilization
Results from the second author, Int. J. Robust Nonlinear Control, No. 3, 211-229 (1993; Zbl 0795.93041) are specialized to affine control systems \(\dot x= f(x)+ g(x) u\). A cost of the form \[ J(x_0, u)= \int_0^{+\infty} (L_1 (x)+ L_2 (x) u+ u^t Ru)dt \] with \(L_2 (0)=0\), is minimized by a stabilizing feedback of the form \[ U(x)=- (1/2) R^{-1} (L^t_2 (x)+ g^t (x) \text{grad} (V) (x)) \] provided there exists a Lyapunov function \(V\) satisfying suitable conditions. This feedback is derived by finding an extremum of the Hamiltonian \(H_u (x,p)\) associated to the optimal control problem; the moment \(p\) is taken as \(p= \text{grad} (V)\). This extremum is a minimum provided \[ L_1 (x)= U^t (x) RU(x)- \text{grad} (V)^t (x) f(x) \] in which case \[ H_u (x,\text{grad} (V))= (u- U(x))^t R(u- U(x)). \] The stabilization can be made global if \(V(x)\) tends to \(+\infty\) when \(|x|\) tends to \(+\infty\). This approach allows to replace the condition \(L_f (V)\leq 0\) of Jurdjevic-Quinn type systems by the cost dependent condition \[ L_f (V)\leq (1/2) L_{g(x) R^{-1} L^t_2 (x)} (V). \] For a class of minimum phase systems with minimal relative degree, this condition is satisfied by choosing an \(L_2\) depending on a Lyapunov function \(V_0\) for the zero dynamics; \(V= V_0+ h^t Ph\) (\(h\) is the output map and \(P>0\)), will be a Lyapunov function for the closed-loop system. Such systems are shown to be feedback equivalent to Jurdjevic-Quinn type systems. The design is used for the global stabilization of the Lorenz equation and of the angular velocity equations of a rigid spacecraft with less than three actuators. It generalizes previous results.
scientific article; zbMATH DE number 836190
scientific article