Negative-Definite Domain Approach for Nonlinear Dynamic Output Feedback Stabilization With DOA Estimation

This technical note deals with asymptotic stabilization of general nonlinear discrete-time systems by dynamic output feedback control. First, a sufficient condition for the dynamic output feedback stabilization with the closed-loop domain of attraction (DOA) estimation is given. For a given Lyapunov function of the plant state and the controller state, the negative-definite domain (NDD) in the $(z_+$-$u$-$z$-$y)$-space is proposed, where $z_+,u,z$, and $y$ denote the next controller state, the current control input, the current controller state and the current plant output, respectively. All points $(z_+;u;z;y)$ in the NDD can make the time difference of the Lyapunov function negative-definite for the controller state $z$ and all plant states in a set of plant states, which is determined by the controller state $z$ and the plant output $y$. The NDD in the $(z_+$-$u$-$z$-$y)$-space could be viewed as an unstructured dynamic output feedback stabilization controller set. And any invariant subset of the projection of the NDD in the $(z_+$-$u$-$z$-$y)$-space onto the $(z$-$x)$-space is an estimate of the DOA for the closed-loop, where $x$ denotes the current plant state. Second, based on the NDD, the controller design problem is converted to a set approximation problem in Euclidean space. Using numerical set approximation methods, dealing with strong nonlinearities is no longer the bottleneck. An estimation method for the unstructured dynamic output feedback stabilization controller set is proposed based on interval analysis, which is a kind of guaranteed numerical method for approximating sets.