Types Of Optimal Control Problems Research Articles

Convergence of the forward-backward sweep method in optimal control

The Forward-Backward Sweep Method is a numerical technique for solving optimal control problems. The technique is one of the indirect methods in which the differential equations from the Maximum Principle are numerically solved. After the method is briefly reviewed, two convergence theorems are proved for a basic type of optimal control problem. The first shows that recursively solving the system of differential equations will produce a sequence of iterates converging to the solution of the system. The second theorem shows that a discretized implementation of the continuous system also converges as the iteration and number of subintervals increases. The hypotheses of the theorem are a combination of basic Lipschitz conditions and the length of the interval of integration. An example illustrates the performance of the method.

Modified Fletcher-Reeves and Dai-Yuan Conjugate Gradient Methods for Solving Optimal Control Problem of Monodomain Model

In this paper, we present the numerical solution for the optimal control problem of monodomain modelwith Rogers-modified FitzHugh-Nagumo ion kinetic. The monodomain model, which is a well-known mathematical model for simulation of cardiac electrical activity, appears as the constraint in our problem. Our control objective is to dampen the excitation wavefront of the transmembrane potential in the observation domain using optimal applied current. Various conjugate gradient methods have been applied by researchers for solving this type of optimal control problem. For the present paper, we adopt the modified Fletcher-Reeves method and modified Dai-Yuan methodfor computing the optimal applied current. Numerical results show that the excitation wavefront is successfully dampened out by the optimal applied current when the modified Fletcher-Reeves method is used. However, this is not the case when the modified Dai-Yuan method is employed. Numerical results indicate that the modified Dai-Yuan method failed to converge to the optimal solution when the Armijo line search is used.

Optimal performance analysis of a sodium cold trap

Cold traps are normally used to purify sodium, and their principle of operation is to cool sodium to temperature near its melting point, where soluble impurities contained in it are easy to precipitate. Two mathematical models are developed to analyze the processes that occur in sodium cold trap. One of them is used to analyze optimal passway shape of air cooler and the other is used to analyze optimal management of Stainless steel mesh. The models for analyzing optimal performance of sodium cold trap are typical optimal control problems. The problem are solved by means of Generalized Pontryagin's Minimum Principle. The numerical results show that air cooler of a cold trap should be two zones, that is, one is cooled by air and the other one is held in isothermal situation. The passway shape of air cooler should be a variable across section when performance of cold trap is optimal. The mesh management of a cold trap should be small surface in top and large surface in bottom when its precipitating performance is optimal and its working period is longer. The obtained conclusions are useful for practical engineering design and operation of a cold trap.

Optimal control problems with multiple characteristic time points in the objective and constraints

In this paper, we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points. These time points can be either fixed or variable. Using the control parametrization technique and a time scaling transformation, this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems. Each of these approximate problems can be viewed as a finite dimensional optimization problem. New gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be applied to solve each approximate optimal parameter selection problem. For illustration, two numerical examples are solved.

観測情報量制約下での障害物回避における準最適制御

This paper addresses a new type of optimal control problem for obstacle avoidance, where the observation of the allowable state region, the selection of the state waypoints of the controlled plant, and its motion between the waypoints are simultaneously optimized. In particular, the problem on the observation, that is, the problem of optimizing the available information on the allowable state region subject to information amount constraints, is formulated, and then an approximate solution method of the whole problem is presented. It is shown by simple simulations that the proposed method is useful.

Expressing optimal control problems as differential algebraic equations

The purpose of this paper is to present an approach to express certain types of optimal control problems in terms of a system of differential algebraic equations (DAEs). This system is obtained using calculus of variations to get the Kuhn–Tucker conditions. The inequalities associated with the complementarity conditions are converted to equalities by the addition of a new variable. Such systems of DAEs are well known in the Chemical Engineering literature, and there are a number of established numerical methods for their solution. Also, we introduce here the concept of the tractability index as a general purpose way of determining the index, by establishing which part of the system of DAEs must be differentiated and how many times. This provides a systematic way of determining the index, without needing to differentiate the whole system. Numerical examples from Chemical Engineering are used to illustrate the methodology.

Computing Switching Times in Bang-Bang Control

During the past half-century, many optimization problems have arisen in fields such as management, engineering, computer science, production, industry, and economics. Often one needs to optimize (minimize or maximize) certain objective subject to some constraints. Some optimization problems involve optimal control, which are more complex and involving a dynamic system. This paper introduces a special transformation of time scale for an oscillator problem (A typical optimal control problem) whose dynamic system could be described as a second order differential equation. The computational algorithms for these kinds of optimal control problems are described as well as the analysis of the result of the test example. The traditional problem solving in searching optimal control policy is replaced by computing the optimal switching times of the control in the algorithms. While the control switches in the optimal time intervals, the system reaches the optimum.

Optimal Control Systems

1R6. Optimal Control Systems. - DS Naidu (Idaho State Univ, Pocatello ID). CRC Press LLC, Boca Raton FL. 2003. 433 pp. ISBN 0-8493-0892-5. $99.95.Reviewed by I Kolmanovsky (Sci Res Lab, MD-2036, Ford Motor Co, 2101 Village Rd, Dearborn MI 48124).The book, Optimal Control Systems, by DS Naidu provides an introduction to key concepts and methods of the optimal control theory for deterministic continuous-time and discrete-time finite-dimensional dynamic systems. The material of the book is based on the author’s experience teaching graduate level optimal control courses and has grown out of the lecture notes used by the author in these courses. After an Introduction (the first chapter of the book), the second chapter describes in detail the main ideas in the calculus of variations (culminating in the study of Euler-Lagrange equations and second variation-based Legendre conditions) and seamlessly transitions to the treatment of the optimal control problems. The Hamiltonian function is introduced early on which subsequently features prominently in the treatment of the Pontryagin’s maximum principle. The end result of Chapter 2 is a solution to several types of optimal control problems (such as fixed end-time, with no control constraints) using the principles of the calculus of variations but stated in the formalism that will later be useful for more general optimal control problems. These developments are already sufficient for the in-depth treatment of Linear Quadratic Optimal Control problem for continuous-time systems, which is one of the central tools in the control theory. This treatment is a subject of Chapters 3 and 4 where the finite horizon and infinite horizon cases, set point and trajectory tracking, fixed-end point regulator system, Linear Quadratic Regulator with a specified degree of stability, and other topics are considered in detail. Frequency domain properties of the linear quadratic regulator are analyzed and the celebrated results on one half to infinity gain margin and 60 degrees phase margin are derived. Chapter 5 treats similar issues as in Chapters 2-4 but for discrete-time systems. In Chapter 6, the treatment of constrained control problems begins with the introduction of the Pontryagin maximum principle for these systems (using the notion of admissible variations) and dynamic programming techniques. The latter are also used to illustrate an alternative approach to deriving continuous-time and discrete-time linear quadratic regulator solutions. Time-optimal control problems are treated in Chapter 7 including the analysis of the number of switchings in the optimal “bang-bang” control laws for linear systems. Other special optimal control problems (energy and fuel-optimal control problems with integral costs) including the case of singular controls are touched upon as well. The developments are completed by considering the point wise-in-time state constraints and describing how they can be treated using either the penalty function method or the slack variable method. The appendices help make the book self-contained by reviewing some of the key results in linear algebra and state space analysis of linear systems, and by providing the listings of the relevant Matlab files (available electronically from the author). The book is very readable and careful attention has certainly been given to provide the reader with a comprehensive set of tools without overburdening with the complex mathematical details and notations (although the key mathematical ideas are, indeed, well exposed and the logic underlying the developments is transparent and thorough). It should be accessible to a wide audience including graduate students and practitioners from engineering and other fields. Optimal Control Systems can be utilized both as a textbook in the introductory courses in the optimal control theory, or as a quick reference by practicing engineers. The use of Matlab/Simulink to treat the examples (including the snippets of the actual code), historical remarks about the scientists behind the major discoveries in the optimal control theory, recipe-style summaries of the key methodologies, exercise problems at the end of each of the chapters are undoubtedly the strong points of this book which significantly contribute to its pedagogical value. On a somewhat more critical side, most of the examples in the book are, in fact, of relatively low order linear systems. This can actually be viewed as an advantage in that the in-depth treatment of these examples is possible so that the main ideas can be illustrated rather well. At the same time, the treatment of more of higher order nonlinear system examples in combination with a more extended introduction into the numerical methods of the optimal control could have been beneficial for the readers to gain an additional insight into how the underlying techniques can be applied.

A New Merging Method of Optimal Control Synthesis for Distributed Parameter Systems.

A new convenient method to synthesize optimal control for distributed parameter systems is presented. Merging the two conventional optimal control synthesis schemes based on partial differential and integral equations, the new method generates the optimal control when applied to an unsteady state tubular reactor model. For this type of optimal control problems, we found that the new method is more efficient than the conventional methods in computing procedure and in computing time.

A Bounded Derivative Model for Prior Ignorance about a Real‐valued Parameter

A new method is proposed for drawing coherent statistical inferences about a real‐valued parameter in problems where there is little or no prior information. Prior ignorance about the parameter is modelled by the set of all continuous probability density functions for which the derivative of the log‐density is bounded by a positive constant. This set is translation‐invariant, it contains density functions with a wide variety of shapes and tail behaviour, and it generates prior probabilities that are highly imprecise. Statistical inferences can be calculated by solving a simple type of optimal control problem whose general solution is characterized. Detailed results are given for the problems of calculating posterior upper and lower means, variances, distribution functions and probabilities of intervals. In general, posterior upper and lower expectations are achieved by prior density functions that are piecewise exponential. The results are illustrated by normal and binomial examples

Revised simplex algorithm for finite Markov decision processes

We introduce a revised simplex algorithm for solving a typical type of dynamic programming equation arising from a class of finite Markov decision processes. The algorithm also applies to several types of optimal control problems with diffusion models after discretization. It is based on the regular simplex algorithm, the duality concept in linear programming, and certain special features of the dynamic programming equation itself. Convergence is established for the new algorithm. The algorithm has favorable potential applicability when the number of actions is very large or even infinite.

Stability and Optimization Analyses of the Generalized Brain-State-in-a-Box Neural Network Model

Anderson's brain-state-in-a-box (BSB) neural network model may be viewed as a version of Hopfield's two-state neural model with continuous rather than discrete states and synchronous updating. A generalization of the BSB model which allows each model neuron in the system to have a bias and its own maximum and minimum firing rate is considered. Moreover, the generalization does not require that the matrix of connection weights be symmetric. A method of choosing the step constant of the algorithm is then described which guarantees that the generalized discrete-time BSB model: (i) approaches the largest set of system equilibrium points, and (ii) minimizes the Boltzmann machine quadratic energy function. A simple test (essentially a vector by matrix multiplication) for checking for asymptotically stable equilibrium points is also provided. Finally, a special case of the generalized BSB model is shown to be applicable to a particular type of optimal control problem.

Extreme Points for Linear Optimal Control Problems with Diagonal Structure

This paper discusses the extreme points of the feasible set for a certain type of optimal control problem with constraints on both the state and control variables. The problem is posed as a continuous-time linear program in the space of bounded measurable functions. The extreme points can be characterized by a certain full rank condition.

A vector implementation of an ODE code for multi-point-boundary-value problems

When applying the multiple-shooting code BNDSCO [7]to boundary-value problems (BVPs) on a vector computer, a vectorized version of a numerical integration method for ordinary differential equations (ODEs) has to be developed. Since the dominating part of the computing time is spent in solving a series of initial value problems (IVPs), the computing time can be considerably reduced by the use of a vectorized Runge-Kutta method. The method presented is applicable for BVPs with nearly all kinds of discontinuities in the solution as well as in the right-hand side at a finite number of interior points and is therefore applicable for many types of optimal-control problems (OCPs) with constraints.

General applications of optimal control theory in cancer chemotherapy.

There are an increasing number of clinical papers dealing with the continuous delivery of an anticancer drug to patients. In this paper, two basic theoretical models are presented of cancer growth under the action of a continuously delivered anticancer drug. The therapeutic objective is to obtain the nature of the control agent that can drive the tumour population to a desired level so as to penalize excessive usage of the drug and to keep deviations of the tumour population from the desired level to a minimum. This gives rise to a certain type of optimal control problem. The nature of the optimal controller for each model is presented. The optimal control approach leads to a therapeutic strategy that may be of relevance to the clinical studies and this is investigated.

Blade Design of Axial-Flow Compressors by the Method of Optimal Control Theory—Application of Pontryagin’s Maximum Principles, a Sample Calculation and Its Results

Using the continual transformation technique [3] and the augmented penalty function method, the typical optimal control problem with various constraints proposed in the paper [2] has been converted to a new equivalent optimal control problem with no constraint. This enables the application of Pontryagin ’s maximum principle. Further, by means of the conjugate gradient method an example of the calculation is shown and the corresponding program is developed. A satisfactory optimal diffusion factor distribution has been obtained.

Some remarks on the numerical solution of bang-bang type optimal control problems

This paper is concerned with the numerical solution of optimal control problems for which each optimal control is bang-bang. Especially, the results apply to parabolic boundary control Problems. Starting from a sequence of feasible solutions converging to an optimal control u, a sequence of bang-bang controls converging to u is constructed. Bang-bang approximations of u are desirable for certain numerical reasons. Sequences of arbitrary feasible controls converging to u may be obtained by discretization or by a descent method. Numerical examples are also given.

Optimal control of a class of strongly nonlinear parabolic systems

This paper considers some typical optimal control problems for a class of strongly nonlinear parabolic systems. After some necessary preparation, it is shown that the family of admissible trajectories is a weakly closed and weakly sequentially compact subset of a reflexive Banach space and that the set of attainable states at any given time is a weakly compact subset of a Hilbert space. Using these basic results, proofs of existence of optimal controls are presented. A terminal control problem, a special Bolza problem, and a time optimal control problem are solved, and the necessary conditions of optimality for the corresponding control problems are given.

Analog computer solution of a third-order Pontryagin optimum control system

Pontryagin's maximum principle is used to develop a method for solving several types of optimal control problems on the analog computer. In particular, this is applied to a third-order control system, and the results shown. The restrictions on the problem and the analog computer are shown, and a procedure for setting up and using the analog mechanization is given.