Trigonometric approximation of optimal periodic control problems with variable delays

Trigonometric polynomials are used to approximate the state trajectory, the control and the functions in differential equations and in the criterion of the constrained optimal periodic control problem. Several discretized problems using such polynomials are proposed to approximate the problem considered. Sufficient conditions for the convergence of solutions of approximating problems to the optimal solution of the basic problem are given. The application to a class of optimal periodic control problems in chemical engineering is discussed.

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

Teleoperation system performance suffers from the delays in communication lines. These delays are usually variable time delays. The customary wave variable technique used for constant time delays cannot provide satisfactory results to stabilize the system under variable delays. Another problem is observed in the position tracking performance of the limited-workspace teleoperation systems. Although wave variable technique is proven to track position demands successfully under constant time delays given that there is no communication failure, position tracking performance also degrades under variable delays. In this paper, a new controller is proposed to stabilize the system and enhance position tracking performance under variable time delays. This proposed controller is then applied to an identical master-slave teleoperation configuration which qualifies as a limited-workspace teleoperation system.

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.

Mathematical formulation and numerical solutions of an optimal Dirichlet boundary control problem for the Boussinesq equations are considered. The solution of the optimal control problem is obtained by adjusting of the temperature on the boundary. We analyze finite element approximations. A gradient method for the solution of the discrete optimal control problem is presented and analyzed. Finally, the results of some computational experiments are presented.

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.

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schrodinger equations. The first one is an equivalence of the minimal norm and the minimal time control problems where the target sets are the origin of the state space. (The minimal time control problems are also called the first type of optimal time control problems.) The second one is an equivalence of three optimal control problems: optimal target control problems, optimal norm control problems, and the second type of optimal time control problems.

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.

An optimal control problem with an infinite horizon quadratic cost functional for a linear system with point-wise and distributed time delays in the state variables is considered. The case, where the cost functional does not contain a control cost, is analyzed. The latter means that the solution (an optimal control) of the considered problem is singular. This control problem is solved by a regularization method. Namely, it is associated with a new optimal control problem for the same equation of dynamics. The cost functional in this new problem is the sum of the original cost functional and an integral of the square of the control with a small positive weighting coefficient. Due to the smallness of this coefficient, the new problem is a cheap control problem. Using the singular perturbation technique, an asymptotic analysis of this cheap control problem is carried out. Based on this analysis, the infimum of the cost functional in the original problem is obtained, and a minimizing control sequence is constructed.

We present an optimal control approach to freeway traffic congestion control. A Link-Node Cell transmission model (LN-CTM) is used to represent the freeway traffic dynamics. The approach searches for solutions represented by a combination of ramp metering and variable speed control. The optimization problem corresponding to the optimal control problem based on the LN-CTM is non-convex and nonlinear. We relax the problem to a linear optimization problem, and propose an approach to map the solution of the linear optimization algorithm to the solution of the original optimal control problem. We prove that the solution derived from this approach is optimal for the original optimal control algorithm. Finally, we use a model predictive framework to demonstrate the optimal control formulation presented in this paper and discuss its potential use.

Distinctive features of the difference approximation for optimal periodic control (OPC) problems are considered. It is shown that the convergence conditions on approximate solutions imply consistency conditions on the time grid step and the accuracy of the periodicity constraint approximation such that the approximate discrete maximum principle holds for a sequence of discretized problems. A specific form of this principle when applied to the difference approximation method for OPC problems is analysed and computational methods are discussed. Application of this approach to a class of systems non-linearly dependent on control variables, and occurring in the chemical industry, is suggested.

Optimal Design and Control of Cyclically Operated Biological Wastewater Treatment Plants

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.

The problem of optimal control with a variable structure depending on the particular way on the parameter is described in the article. In the considered issue, the object’s motion equations are described by two differential equations depending on the small parameter and these equations are bound to each other. The study investigated the solution of the problem depending on the small parameter that gives a certain functional minimum in the range of solutions described by differential equations, with a small parameter and the necessary conditions for the optimization of the solution are given. The validity of the theorems obtained in this study was proved by applying the Lagrangian principle to the Lagrangian functions and the application of the Farm Theorem to the Lagrangian problem. At the same time, the problem of optimal control with periodic boundary conditions is also explored. It can also be used extensively in solving some practical issues, for example, model in oil extraction by gas—lift method for the case when the reciprocal value of well’s depth represents a small parameter is considered. Problem of optimal mode construction (i.e., construction of optimal program trajectories and controls) is reduced to the linear—quadratic optimal control problem with a small parameter. This issue is investigated using the Lagrange coefficient and the Lagrange principle for the Lagrange problem. Although this issue has been studied in numerous works, the specificity of the issue under consideration makes it possible to investigate it differently. The considered method can be applied to the solution of optimal control problems for the problems described by some nonlinear differential equations.