A compact version of the Variation Evolving Method (VEM) is developed for the optimal control computation. It follows the idea that originates from the continuous-time dynamics stability theory in the control field. The optimal solution is analogized to the equilibrium point of a dynamic system and is anticipated to be obtained in an asymptotically evolving way. With the introduction of a virtual dimension--the variation time, the Evolution Partial Differential Equation (EPDE), which describes the variation motion towards the optimal solution, is deduced from the Optimal Control Problem (OCP), and the equivalent optimality conditions with no employment of costates are established. In particular, it is found that theoretically the analytic feedback optimal control law does not exist for general OCPs because the optimal control is related to the future state. Since the derived EPDE is suitable to be solved with the semi-discrete method in the field of PDE numerical calculation, the resulting Initial-value Problems (IVPs) may be solved with mature Ordinary Differential Equation (ODE) numerical integration methods.
A dynamic backstepping control method is proposed for non-linear systems in the pure-feedback form, for which the traditional backstepping method suffers from solving the implicit non-linear algebraic equation. This method treats the implicit algebraic equation directly via a dynamic way, by augmenting the (virtual) controls as states during each recursive step. Compared with the traditional backstepping method, one more Lyapunov design is executed in each step. As new dynamics are included in the design, the resulting control law is in the dynamic feedback form. Under appropriate assumptions, the proposed control scheme achieves the uniformly asymptotic stability and the closed-loop system is local input-to-state stable for various disturbance. Moreover, the control law may be simplified to the inverse-free form by setting large gains, which will alleviate the problem of `explosion of terms’. The effectiveness of this method is illustrated by the stabilization and tracking numerical examples.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.