We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index. In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case. We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls. Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example.

Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index

Hamilton-Jacobi equation for descriptor systems

Optimal control of causal differential–algebraic systems

This paper presents a new design approach to achieve decentralized optimal control of high-dimension complex singular systems with dynamic uncertainties. Based on robust adaptive dynamic programming &#x0028 robust ADP &#x0029 method, controllers for solving the singular systems optimal control problem are designed. The proposed algorithm can work well when the system model is not exactly known but the input and output data can be measured. The policy iteration of each controller only uses their own states and input information for learning, and do not need to know the whole system dynamics. Simulation results on the New England 10-machine 39-bus test system show the effectiveness of the designed controller.

Optimal control for a class of complex singular system based on adaptive dynamic programming

The work of Grantham and Vincent on the controllability problems of conventional systems is extended to the implicit singular system f(x, E[xdot], u) = 0, where E is a singular square constant matrix. The necessary conditions are given in the form of a controllability maximum principle, and the sufficient conditions are given with convexity and concavity. We also discuss the controllability problem for the nonlinear singular system E[xdot] = f(x, u) for which the hamiltonian function has the same form as that for conventional systems. Then we show that our principal results are equivalent to the results obtained previously by other authors for the linear time-invariant singular system E[xdot] = Ax + Bu, and we also present two numerical examples.

Approach to controllability problems for singular systems

Singular systems of the form E[xdot] = ƒ(x,u,t) are considered, where E is a square matrix and may be singular. It is assumed that for any ‘admissible’ initial state x(t 0), any control u(t) ∊ U yields one and only one continuous state x(t), and there is one and only one continuous adjoint state λ(t). The formulae for functional variation are derived; the necessary condition for optimality—the maximum principle—is obtained; the boundary conditions for the adjoint equations of the singular systems are given; and the necessary and sufficient condition for optimality of linear singular systems is derived.

Zi-Hou Yang

Papers

Optimal control problems for singular systems