A discrete filled function method for the optimal control of switched systems in discrete time

A computational method for a class of non-standard time optimal control problems involving multiple time horizons

Inverse identification of inputs of a random utility based model using optimal control

A new discrete filled function method for finding global minimizer of the integer programming

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

Three layer control structure is proposed for optimal control of continuous fermentation processes. The start-up optimization problems are solved as a first step for optimization layer building. A steady state optimization problem is solved by a decomposition method using prediction principle. A discrete minimum time optimal control problem with state delay is formulated and a decomposition method, based on an augmented Lagrange's function is proposed to solve it. The problem is decomposed in time domain by a new coordinating vector. The obtained algorithms are used for minimum time optimal control calculation of Baker's Yeast fermentation process.

The author summarizes the theoretical results in "A discrete optimal control problem for descriptor systems," [IEEE Trans. Automat. Control, vol.34, pp.177-181, 1989] and "Authors' reply to 'Comments on 'A discrete optimal control problem for descriptor systems"," [ibid., to appear], and uses them in the computer-aided optimization problem for descriptor systems. An efficient algorithm for computing the optimal solution sequence of the optimization problem is developed and an example of the macroeconomic systems is shown.

In this paper, an optimization problem is performed to obtain an approximate solution for a class of fractional optimal control problems (FOCPs) with the initial and final conditions. The main characteristic of our approximation is to reduce the FOCP into a system of Volterra integral equations. Then by solving this new problem, based on minimization and control the total error, we transform the original FOCP into a discrete optimization problem. By obtaining the optimal solutions of this problem, we obtain the numerical solution of the original problem. This procedure not only simplifies the problem but also speeds up the computations. The numerical solutions obtained from the proposed approximation indicate that this approach is easy to implement and accurate when applied to FOCPs.

This paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main focus is on the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the “lack of compactness” in infinite dimensions, which leads to imposing certain “normal compactness” properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions.

Nonlinear discrete time and continuous time optimal control problems with terminal constraints are solved using a new evolutionary approach which seeks the control history directly by evolutionary computation. Unlike methods that use the first order necessary conditions to determine the optimum, the main advantage of the present method is that it does not require the development of a Hamiltonian formulation and consequently, it eliminates the requirement to solve the adjoint problem which usually leads to a difficult two-point boundary value problem. The method is verified on two benchmark problems. The first problem is the discrete time velocity direction programming problem with the effects of gravity, thrust and drag and a terminal constraint on the final vertical position. The second problem is a continuous time optimal control problem in rocket dynamics, the Goddard’s problem. The solutions of both problems compared favorably with published results based on gradient methods.

The optimization problem consists in finding an element that minimizes or maximizes the goal (or cost) functional on the set of admissible controls. The paper is devoted to the optimization problem for a pseudoparabolic equation with control in coefficients of the variational equality as well as on the right-hand side. We consider optimal control problem (parabolic initial boundary value problem) of a viscoelastic plate. The bending of a plate is described by means of the Kirchhoff model. We assume that a nonhomogeneous and anisotropic viscoelastic plate occupying a domain Ω × (—H, H) of the space ℝ3 is loaded be a transversal distributed force perpendicular to the middle plane of the plate. The viscoelastic reinforced plate with welded bars rests on a unilateral elastic foundation. The role of the control variables is played by the thickness of the viscoelastic plate and the stiffness characteristics of the elastic Winkler medium, respectively. We consider the desired moments field (the bending moments and torque) in time T as the cost functional. The control variables have to belong to a set of Lipschitz-continuous functions and to a set of continuously differentiable functions, respectively. The state problem is modelled by a pseudoparabolic variational equality, where the control variables influence both the coefficients of the linear strictly monotone operator and the Winkler functional. We derive there some apriori estimates of the state function which are useful for the problem of the existence of the optimal thickness function and the Winkler modulus, respectively, and for unicity of the optimal controls. On the basis of a general existence theorem for a class of pseudoparabolic optimization problems with the state variational equation, we prove the existence of at least one solution of the optimal control problem.

Optimal control for multistage uncertain random dynamic systems with multiple time delays

Decomposition method for solving two optimal control problems and one optimization problem in batch fermentation is proposed. The problems are formulated based on a nonstructured mathematical model with slowly varying parameters and a finite cost criterion of maximum end production. Dependence of the model parameters on one physical or chemical parameter, which could easily be used as a control input is introduced analytically in the model equations and three model descriptions are obtained by nonlinear difference equations. Sensitivity functions of state trajectories towards slowly varying coefficients are introduced to account for model uncertainties. Based on them extended optimal control and optimization problems are formulated.

Hierarchical two-level methods for solving state space constrained optimization problems are studied. The approach is a generalization of the methods suggested by the author and L. Hakkala in earlier papers. In this paper the original state space constrained problem is modified so that three control constrained problems result. The relationship between the original problem and the modified problems is studied by a dual approach which also gives a collection of algorithms for solving the original problem. The proposed approach is applied to discrete time optimal control problems and two numerical examples are given

This article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy.

In this paper, we consider a class of optimal switching control problems with multiple time-delays and a cost on changing control and subject to terminal state constraints. A computational method involving three stages is developed to solve this class of optimal control problems. First, by parameterizing the control function with piecewise-constant functions, the optimal switching control problem is approximated by a sequence of finite-dimensional optimization problems, where the original switching times, the control heights and the control switching times are decision variables. Second, by introducing new variables, the total variation of the control variables is transformed into an equivalently smooth function. Third, we convert the constrained optimization problem into one only with box constraints by an exact penalty function method. The gradients of the cost functional are then derived, which can be combined with any gradient-based optimization method to determine the optimal solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.