Free Terminal State Research Articles

A symplectic local pseudospectral method for solving nonlinear state‐delayed optimal control problems with inequality constraints

SummaryIn this paper, a symplectic local pseudospectral (PS) method for solving nonlinear state‐delayed optimal control problems with inequality constraints is proposed. We first convert the original nonlinear problem into a sequence of linear quadratic optimal control problems using quasi‐linearization techniques. Then, based on local Legendre‐Gauss‐Lobatto PS methods and the dual variational principle, a PS method to solve these converted linear quadratic constrained optimal control problems is developed. The developed method transforms the converted problems into a coupling of a system of linear algebraic equations and a linear complementarity problem. The coefficient matrix involved is sparse and symmetric due to the benefit of the dual variational principle. Converged solutions can be obtained with few iterations because of the local PS method and quasi‐linearization techniques are used. The proposed method can be applied to problems with fixed terminal states or free terminal states, and the boundary conditions and constraints are strictly satisfied. Numerical simulations show that the developed method is highly efficient and accurate.

Positional strengthenings of the maximum principle and sufficient optimality conditions

We derive nonlocal necessary optimality conditions, which efficiently strengthen the classical Pontryagin maximum principle and its modification obtained by B. Kaśkosz and S. Łojasiewicz as well as our previous result of a similar kind named the “feedback minimum principle.” The strengthening of the feedback minimum principle (and, hence, of the Pontryagin principle) is owing to the employment of two types of feedback controls “compatible” with a reference trajectory (i.e., producing this trajectory as a Carath´eodory solution). In each of the versions, the strengthened feedback minimum principle states that the optimality of a reference process implies the optimality of its trajectory in a certain family of variational problems generated by cotrajectories of the original and compatible controls. The basic construction of the feedback minimum principle—a perturbation of a solution to the adjoint system—is employed to prove an exact formula for the increment of the cost functional. We use this formula to obtain sufficient conditions for the strong and global minimum of Pontryagin’s extremals. These conditions are much milder than their known analogs, which require the convexity in the state variable of the functional and of the lower Hamiltonian. Our study is focused on a nonlinear smooth Mayer problem with free terminal states. All assertions are illustrated by examples.

A symplectic sequence iteration approach for nonlinear optimal control problems with state-control constraints

A sequence iteration numerical method with symplectic discretization form for solving nonlinear constrained optimal control problems with given terminal or free terminal states is presented. The nonlinear optimal problem with state-control constraints is firstly transcribed into a sequence of constraint linear quadratic (LQ) optimal control problems. Then, the symplectic discretization form is applied to the constrained LQ optimal control problem and then the explicit linear complementary problem can be derived. Finally, the solutions of nonlinear constrained optimal control problems can be obtained by employing a standard linear complementary solver. Because the proposed numerical method is symplectic preserved and the state-control constraints are imposed at the discrete points of time interval, the high accuracy and high efficiency numerical solutions can be obtained with large time interval and small iteration times. Meanwhile, the state-control constraints and the terminal states constraints can be satisfied exactly for given accuracy. Numerical simulations show than various types of optimal control problems are investigated to demonstrate the effectiveness of the proposed approximation scheme. The method is robust, easy to implement and provides very accurate results.

Optimal control of the temperature in a solar furnace

SummaryThis article describes the application of optimal control to a solar furnace that is used to perform temperature stress cycling tests in material samples. This process is characterized by having nonlinearities that depend on the sample properties and relate the temperature of the sample with the solar energy fluctuations and the position of the furnace shutter. An optimal control problem with fix terminal time and free terminal state and control constraints is addressed in continuous time domain. The solution is approximated using discretized state and costate equations and applied to the furnace according to a receding horizon strategy. The performance of the overall system is evaluated from computer simulations which show that the controller is able to tolerate, up to some degree, the presence of parameter uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.

Discrete-Time Linear Quadratic Optimal Control with Fixed and Free Terminal State via Double Generating Functions

A double generating functions based method for the discrete-time linear quadratic optimal control problem with fixed and free terminal state, i.e., hard constraint problem and soft constraint problem, is developed in this paper. By this method, the optimal state and input can be expressed by functions only in terms of the boundary conditions and pre-computed coefficients. Accordingly, the whole optimal computation can be divided into two parts in which the off-line part calculates the coefficients and the on-line part generates the optimal solutions by these proposed functions. In view of this, the developed method is useful in the on-line iterative computation for optimal control problem with a number of different boundary conditions. Examples demonstrate the effectiveness of the double generating functions method for the both hard and soft constraint problems with different boundary conditions.

Optimal Birth Control for an Age-Dependent Competition System of N Species

In this paper, we investigate optimal policies for an age-dependent n-dimensional competition system, which is controlled by fertility. By using Dubovitskii-Milyutin’s general theory, the maximum principles are obtained for the problems with free terminal states, infinite horizon, and target sets, respectively.

Optimal birth control for competition system of three species with age-structure

In this paper, we investigate optimal policies for three age-dependent populations in a competition system, which is controlled by fertility. By using Dubovitskii-Milyutin's general theory, the maximum principles are obtained for problems with free terminal states, infinite horizon and target sets, respectively.

Optimal birth control for an age-dependent n-dimensional food chain model

In this paper, we investigate optimal policies for an age-dependent n-dimensional food chain model, which is controlled by fertility. By using Dubovitskii–Milyutin's general theory, the maximum principles are obtained for problems with free terminal states, infinite horizon and target sets, respectively.

Optimal birth control for predator–prey system of three species with age-structure

In this paper, we investigate optimal policies for three age-dependent populations in a predator–prey system, which is controlled by fertility. By using Dubovitskii–Milyutin's general theory, the maximum principles are obtained for problems with free terminal states, infinite horizon and target sets, respectively.

On the Selection of One Feedback Nash Equilibrium in Discounted Linear-Quadratic Games

We study a selection method for a Nash feedback equilibrium of a one-dimensional linear-quadratic nonzero sum game over an infinite horizon : by introducing a change in the time variable, one obtains an associated game over a finite horizon T > 0 and with free terminal state. This associated game admits a unique solution which converges to a particular Nash feedback equilibrium of the original problem as the horizon T goes to infinity. Key Words. Linear-quadratic games. Nonzero sum differential games. Nash equilibria. Infinite horizon.

New Condition of Optimality for Multi-Input Linear Digital Regulator with Fixed Terminal State.

This paper discusses the regulator problem with the fixed terminal state for multi-input linear digital systems. First, for the general additive performance index, the fundamental structure of the regulator which drives the state to zero within a finite time is obtained. This problem is divided into three phases: the new free terminal state problem in the first phase, the new free terminal state problem for the time-variant system with the time-variant structure of the new input in the second phase, and the constant gain special deadbeat control in the third phase. That is, the new condition of optimality of the regulator problem with the fixed terminal state is expressed by the special deadbeat control and the induced regulator problem with the free new terminal state. The new solution of the deadbeat linear-quadratic optimal control problem is obtained to show the effectiveness of the new condition of optimality

Fundamental Structure of Single-Input Accurate Digital Regulator and New Solution of Deadbeat Linear-Quadratic Optimal Control.

This paper discusses the regulator problem with the zero terminal state for a single-input linear time-invariant discrete-time system. First, for the general additive performance index, the fundamental structure of the accurate digital regulator which drives the state to zero within a finite time is obtained. This problem is divided into two sections; the new free terminal state problem in the former section, and the constant gain deadbeat control in the latter section. By using this fundamental structure, the new solution of the deadbeat LQ optimal control problem is obtained. Furthermore, the sufficiency of the optimal solution of the deadbeat LQ optimal control problem is shown by using the property of the linear-quadratic form.

Spectral theory of the linear-quadratic optimal control problem: Discrete-time single-input case

The single-input discrete-time linear-quadratic optimal control problem, together with the classical time-domain and frequency domain conditions, is reviewed and treated from a novel point of view. We show that when the quadratic cost is not positive semidefinite, the problem of the boundedness of the infimum in both the zero and the free terminal state cases is related to the positivity of a self-adjoint Hilbert space operator. The structure of the spectrum of the operator in question, in both cases, is investigated, leading to a clarification of the linkage between the time-domain and the frequency-domain conditions for boundedness. More over, it is shown that the spectra of the operators in question reflect the relevant properties of the control problem.

Necessary conditions for minimum in problems with nonstandard cost functionals

Usual formulation of optimal control problems involves the minimization of a cost functional which is of the form of a definite integral. In this chapter we develop necessary conditions for an optimal control in the case of problems in which the cost functional is either a quotient or a product of definite integrals. We call such functionals nonstandard, and these naturally arise in Chapters 2,4, and 5 during the computation of the performance of a suboptimal H∞ controller. In Chapters 2,4, and 5 a criterion for the evaluation of the cost functional will be presented in the specialized case of linear systems and quadratic integrands. Preliminary results for problems having a fixed final time and free terminal state are in [1]. Related results can also be found in [2,3]. In this monograph, we consider only fixed final time problems. Problems in which the final time is free are treated in [4]. In Section 5, we discuss the relation of our results to those in [2,3].

On the Optimal Control of Systems of Quadratic and Bilinear Differential Equations

Sufficient conditions are given for the optimal feedback controller to be linear, in the problem of minimizing a quadratic performance criterion subject to a dynamic system constraint of quadratic and bilinear terms. The cases of free terminal state infinite-time, and fixed terminal state specified terminal time, are considered. In addition, a two-dimensional example is presented.

On the imbedding solution of a class of optimal control problems

This paper is concerned with the application of the “imbedding” method to optimal control problems with free terminal state and fixed final time. The associated two-point boundary value problem is imbedded into a class of initial value problems by introducing a parameter κε[0, 1] in the final process time. Modified Newton's iterations are used to improve on the missing initial values of the costates. A continuous stirred-tank reactor and a strip winding process examples illustrate the method.

On line solution of optimal nonstationary linear systems

The gradient acceleration technique is described as a means for solving optimal nonstationary linear systems with unbounded control. A numerical example with a free terminal state is solved. Also, the usefulness of this method for nonlinear and bounded control linear systems is indicated.