Optimal control of two-time-scale systems with state-variable inequality constraints

Abstract

The established necessary conditions for optimality in nonlinear control problems that involve state-variable inequality constraints are applied to a class of singularly perturbed systems. The distinguishing feature of this class of two-time-scal e systems is a transformation of the state-variable inequality constraint, present in the full-order problem, to a constraint involving states and controls in the reduced problem. It is shown that, when a state constraint is active in the reduced problem, the boundary-layer problem can be of finite time in the stretched time scale. Thus, the usual requirement for asymptotic stability of the boundary-layer system is not applicable and cannot be used to construct approximate boundary-layer solutions. Several alternative solution methods are explored and illustrated with simple examples. I. Introduction S TATE-VARIABLE inequality constraints are commonly encountered in the study of dynamic systems. The study of rigid body aircraft dynamics and control is certainly no exception. For instance, a maximum allowable value of dynamic pressure is usually prescribed for aircraft with supersonic capability. This limit is required to insure that the vehicle's structural integrity is maintained. Given a typical statespace description of the vehicle dynamics, this limit constitutes an inequality constraint on vehicle state. State inequality constraints have been studied extensively by researchers in the field of optimal control. First-order necessary conditions for optimality when general functions of state are constrained have been obtained.13 However, the construction of solutions via this set of conditions proves difficult. Most practitioners seeking an open-loop control solution rely on direct approaches to optimization that employ penalty functions for satisfaction of state inequality constraints.4 As a rule, algorithms employing such methods are computationally intense and slow to converge. Consequently, they are not well suited for real-time implementation. As discussed in the literature, the use of singular perturbation techniques in the study of aircraft trajectory optimization can, through order reduction, lead to both open- and closedloop solutions that are computationally efficient. These methods can sometimes be used to circumvent difficulties associated with enforcing a state inequality constraint as well.5 As an example, consider the minimum time intercept problem of Ref. 6 that employs a model of the F-8 aircraft. A nearoptimal feedback solution is obtained via singular perturbation theory that includes consideration of an inequality constraint on dynamic pressure. In the zero-order reduced solution, algebraic constraints are obtained when the perturbation parameter e, which premultiplies the so-called fast dynamic equations, is set to zero. These constraints can be used to eliminate the fast states (in this case altitude and flight-path angle) from the reduced problem. One can choose, however, to retain one or more of the fast states and to eliminate instead