A Second Order Nonlinear Time-Varying Controller Which Provides Asymptotic Stabilization

In this paper we construct a two-dimensional nonlinear time-varying (NLTV) controller which asymptotically stabilizes every finite-dimensional, stabilizable and detectable, linear time-invariant (LTI) discrete-time plant (with a fixed number of inputs and outputs). We first show that for every stabilizable and detectable plant there exists a one-dimensional linear time-varying (LTV) compensator which provides exponential stabilization; we then construct a two-dimensional NLTV controller which asymptotically stabilizes every admissible plant by switching between a countable number of such LTV compensators. We also briefly discuss how this can be extended to the continuous-time case.