On the critical case for global stability of adaptive nonlinear stochastic control

Global stability and instability of a class of discrete-time adaptive nonlinear stochastic control systems are investigated. The systems to be controlled may exhibit chaotic behavior and are assumed to be linear in unknown parameters but nonlinear in output dynamics, which are characterized by a nonlinear function (say f(x)). It is found and proved that there is a critical case for global stability of least-squares (LS) based adaptive control systems. To be specific, let the growth rate of f(x) be f(x)=O(||x||^b) with b⩾O, then it is found that b=4 is a critical value for global stability, i.e. the closed-loop adaptive system is globally stable if b<4, and is unstable in general if b⩾4. As a consequence, we find an interesting phenomenon that the linear case does not have: For some LS-based certainty equivalence adaptive controls, even if the LS parameter estimates are strongly consistent, the closed-loop systems may still be unstable. This paper also indicates that adaptive nonlinear stochastic control that is designed based on for example, Taylor expansion (or Weierstrass approximation) for nonlinear models may not be feasible´ in general