Optimal control of two-time-scale systems with state-variable inequality constraints
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Calise, A. J.↗
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Pole placement and order reduction in two-time-scale control systems through Riccati iteration
A transformation of variables taken from singular perturbations may be applied to two-time-scale linear systems in state space form to reduce the system to block-diagonal form with slow and fast modes decoupled. The transformation is easily computed by applying the new Riccati iteration. The iteration yields a solution to the nonsymmetric algebraic Riccati equation obtained by partitioning the original system matrix A. The numerical procedure is initiated with the trivial iterate L(0) = 0, and is globally convergent to the desired unique time scale decoupling solution. After transformation, the decoupled system may be used in controller design to achieve exact closed loop pole placement in the slow subsystem without altering the poles of the fast subsystem. The decoupled form may also be used to reduce system order by wetting a small parameter to zero. Provided the fast subsystem is stable, the order reduction can be expected to yield a good approximation to the original system. These methods are demonstrated using the 16th order linear model of a turbofan engine.
Anderson, L. R.↗
Optimal control of singularly perturbed nonlinear systems with state-variable inequality constraints
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-scale 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 variable. 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.
Calise, A. J.↗
Analytical solutions to constrained hypersonic flight trajectories
The flight trajectory of aerospace vehicles subject to a class of path constraints is considered. The constrained dynamics is shown to be a natural two-time-scale system. Asymptotic analytical solutions are obtained. Problems of trajectory optimization and guidance can be dramatically simplified with these solutions. Applications in trajectory design for an aerospace plane strongly support the theoretical development.
Lu, Ping↗
Analytical solutions to constrained hypersonic flight trajectories
The flight trajectory of aerospace vehicles subject to a class of path constraints is considered. The constrained dynamics is shown to be a natural two-time-scale system. Asymptotic analytical solutions are obtained. Problems of trajectory optimization and guidance can be dramatically simplified with these solutions. Applications in trajectory design for an aerospace plane strongly support the theoretical development.
Lu, Ping↗
Two-time scale stabilization of a flexible arm with output feedback
An approach is presented to designing a control system with output feedback for a lightweight flexible arm. A two-time-scale dynamic model is obtained which allows the adoption of a composite control strategy. First a slow control can be designed for the slow (rigid) subsystem, and then a fast stabilizing control for the fast (flexible) subsystem. The problem of the lack of full state measurements connected with the fast control design is solved. An output-feedback, low-order dynamic compensator whose optimal gains are computed by a convergent numerical algorithm is designed. A more robust design can be achieved if a loop transfer recovery procedure is introduced. The design procedure is tested by means of nonlinear simulation, and the results are discussed.
Siciliano, Bruno↗
Near-optimal output feedback regulation of ill-conditioned linear systems
A two-time scale approximation for the linear quadratic optimal output feedback regulator program is examined. Necessary conditions for optimality, as well as an algorithm for computing locally near-optimal gains are derived. If it is assumed that the slow and fast subsystem initial conditions are uniformly distributed, optimal gains for the two-time-scale problem provide a second-order approximation to optimal closed-loop performance in the unperturbed system. This is verified with a numerical example.
Moerder, Daniel D.↗
Aerospace plane guidance using time-scale decomposition - A geometric approach
A method is proposed for developing the necessary guidance logic to steer single-stage vehicles into orbit. The minimum-fuel ascent problem is first considered to analyze the effects of dynamic pressure, acceleration, and heating constraints on guidance systems to thereby develop the guidance logic. The optimal solution consists of behavior with two time scales, and the control law is used to develop near-optimal guidance. The solution uses the slow manifold to delineate the control for minimum-fuel reduced-order trajectory and a separate control for tracking the optimal reduced-order trajectory. A family of fast manifolds is then employed to resolve the tracking problem via the feedback linearization methodology from nonlinear geometric control theory. The two-time-scale decomposition is found to produce a near-optimal ascent by tracking the applicable state-constraint boundary, as well as to simplify the control-design task.
Van Buren, Mark A.↗
A new learning strategy for the two-time-scale neural controller with its application to the tracking control of rigid arms
A novel fast learning rule with fast weight identification is proposed for the two-time-scale neural controller, and a two-stage learning strategy is developed for the proposed neural controller. The results of the stability analysis show that both the tracking error and the fast weight error will be uniformly bounded and converge to a bounded region which depends only on the accuracy of the slow learning if the system is sufficiently excited. The efficiency of the two-stage learning is also demonstrated by a simulation of a two-link arm.
Cheng, W.↗