Infinite Terminal Time Research Articles

Pontryagin’s maximum principle of optimal control problems with time-delay

In this paper, we consider an optimal control problem with time-delay. The state and the control variables contain various constant time-delays. This allows us to represent the necessary conditions in an explicit form. Solution of this problem with infinite terminal time is also given.

Complete decomposition of sub-optimal regulators for singularly perturbed systems

The linear regulator problem is studied for systems with slow and fast modes. Using the descriptor variable method, the design of sub-optimal regulators for such systems is completely decomposed. It is shown that the sub-optimal regulator is composed of fast and slow regulators, and that both of these, mutually decoupled, can be designed independently. A relevant computational procedure is also presented. The procedure is applicable to designs of sub-optimal regulators with either finite or infinite terminal time, and for both singular and normal singularly perturbed systems.

Existence and transversality conditions for a general ‘unbounded-horizon’ model of the mining firm

This paper derives conditions under which optimal programs exist for a general model of the competitive or monopolistic mining firm. The model significantly extends and synthesizes previous formulations by including both resource extraction and discoveries of new reserves, with stock-dependent resource exploitation costs and incomplete resource exhaustion, endogenous technical innovation, and state-dependent resource demand. In addition, the analysis addresses complications arising from having optimal programs defined over an ‘unbounded horizon’, with the possibility of either a finite or infinite terminal time and a non-zero terminal (‘salvage’) valuation if the terminal time is finite. The necessity of the infinite-horizon transversality conditions also is established. The paper illustrates a strongly intuitive approach to existence questions for infinite-horizon or unbounded-horizon control problems, with weaker concavity and interiority assumptions than are often encountered in the literature.

Sub-Optimal Load Frequency Control for Interconnected Power System

This paper describes the application of infinite terminal time case to the load frequency control of a multiple-area interconnected power system. From the viewpoint of practical application, the linear feedback controller using the instantaneous values of output variables is proved to be effective.A number of different optimal or sub-optimal output feedback controllers can be designed by selecting different estimation functions of weighting factors, Q and R.The proposal sub-optimal output feedback controller is not significantly degraded by using the abbreviation of state covariance equation. And, it is deduced that this controller is promising as a candidate for on-line computer controls, because of the simplicity of the gain solution.

Low-sensitivity dynamic compensators for linear stochastic systems with random parameters

This paper presents procedures for the low-trajectory sensitivity design of reduced-order output feedback dynamic controllers for linear stochastic systems with uncertain parameters having zero-mean random variations around their expected values. Both the finite and infinite terminal time designs are discussed. Matrices are conveniently partitioned to reduce computational effort. Several special cases are considered, leading to simpler problem designs. Two numerical examples are included.

Numerical solution of the algebraic Riccati equation for single-input systems

A numerical method particularly suited to lightly damped systems is presented for solution of the linear regulator problem with infinite terminal time for single-input systems.

Linear Optimal Control of a Linear System With State and Control Dependent Noise

A quadratic minimization problem for a linear stochastic system is solved in this paper. Both the finite and infinite terminal time cases are considered. Also two precise representations of the controlled stochastic process are considered. In one representation we include the correction term [6] for the state and control-dependent noise and in the other we do not. For the case with no correction terms, the optimal control is shown to be a linear feedback of the system state variables. Uniqueness and stability conditions are presented for this problem. The case with correction term is much harder to solve and we only determine the linear optimal control. An example is included which illustrates many results of the paper.

Optimal stochastic control of linear systems with state- and control-dependent disturbances

A review of the solution of the linear regulator problem for linear systems with state- and control-dependent disturbances is presented. Both the finite and infinite terminal time cases are treated. The solution to the complete state feedback case is given in detail and that for the output feedback case is noted. The general conclusion is that control-dependent noise calls for conservative control (small gains) while state-dependent noise calls for vigorous control (large gains). Of course it is the degree of this behavior that is important and this is given explicitly by the algorithms in this paper.

On the characteristics of linear optimal systems

The characteristics of linear systems with constant inputs which are optimal with respect to an infinite terminal time, quadratic performance index are studied. An especially simple standard form of the algebraic Riccati equation is used to determine relationships between the performance index parameters and the normalized characteristic polynomial of the optimal system. The parameters corresponding to several standard forms of this polynomial are ascertained. Design guidelines are then given which allow the correlation of performance index parameters with the shape of the step function response.

Linear optimal stochastic control using instantaneous output feedback‡

Abstract The problem of determining the linear feedback control of the instantaneous system output which minimizes a quadratic performance measure for a linear system with state and control-dependent noise is solved in this paper. Both the finite and infinite terminal time versions of this problem are treated. For the latter case, a sufficient condition for the existence of an optimal control is obtained. For the finite terminal time problem, it is shown that a two-point boundary value problem must be solved to realize the optimal control. For the infinite terminal time case, two non-linear matrix equations must be solved to realize the optimal control. Some discussion on the numerical methods used by the author to solve these equations is included in the paper.

A New Approach to the Solution of Linear Optimal Control Problems

A simplified approach to the solution of linear optimal control problems with quadratic performance indexes is described in this paper. The phase-variable canonical form is used to develop a new type of optimal system equivalence. This concept leads to a substantial simplification of the matrix Riccati equation. The simplified matrix Riccati equation is of the same form for any problem of a given order, say, n, and contains only n nonzero forcing functions. That is, it always corresponds to a set of constant-coefficient scalar differential equations; in various nth-order problems the n nonzero forcing functions and the terminal conditions simply assume different forms. In a very strong sense, this simplified matrix Riccati equation is the simplest possible Riccati equation arising from optimization problems. The method is developed for general time-varying systems with finite terminal time. It is developed also for the important special case of time-invariant systems with infinite terminal time.