Closed-loop Optimal Control Research Articles

Distributed Wiener-Poisson control †

A one-dimensional Wiener plus independent Poisson control problem, with the state governed by a partial differential equation, has an integrated discounted quadratic cost function and asymmetric bounds on the control, which is a function of the current state. A Bellman equation and the maximum principle for partial differential equations are used to obtain the optimal closed-loop control in bang-bang form. The finite and infinite integral quadratic cost functions are treated separately.

Optimal Control of Linear Stochastic Systems With Unknown Parameters by Multiple Hypothesis Testing

The optimal closed-loop control is estimated for linear stochastic systems with unknown parameters. The parameters are assumed to take values over a finite set (finite number of hypotheses) and are estimated by an on-line maximum likelihood algorithm. It is shown that the present optimal control is a function of the expected future improvement in the parameter estimates, even if there is no dual effect. (The control has a dual effect if it affects the identification algorithm's ability to estimate the parameters.)

Singular-perturbation method for discrete models of continuous systems in optimal control

The closed-loop and open-loop optimal controls of a singularly perturbed continuous system are considered by means of their discrete models. A singula-perturbation method is developed to obtain series solutions in terms of the outer, inner and intermediate series analogous to that in a continuous system. It is shown that the resulting matrix Riccati difference equation for closed-loop optimal control is not amenable to singular-perturbation analysis in its original form and has to be properly recast to fit into the framework of singular-perturbation theory. The discrete-model representation has the twin advantages of the reduction in order associated with singular perturbation and the reduction in the computation due to the recursive nature of the solutions in discrete systems. Series solutions are then possible for higher-order approximations with considerable reduction in computation. The method is illustrated by an example.

Singular perturbation method for a closed-loop optimal control problem

The closed-loop optimal control of a singularly perturbed linear system, with free-end-point conditions, gives rise to a matrix Riccati equation. A singular perturbation method is developed to analyse the Riccati equation. An algorithm is presented to indicate the sequence of steps involved in the actual application of the method for obtaining the zeroth-first- and second-order approximations. An example5,6,9 is provided to illustrate the method.

Singular-perturbation analysis of a closed-loop fixed-end-point optimal-control problem

The closed-loop optimal control of a singularly perturbed system with fixed-end-point conditions results in a special matrix Riccati equation. A singular-perturbation method is presented to analyse the Riccati equation. An algorithm is developed for obtaining the zeroth-, first- and second-order approximate solutions. An illustrative example is given. An integral-squared-error criterion is used to examine the closeness of the approximate solutions.

On the sensitivity of optimal control solutions

In solving the closed-loop optimal control problem, little knowledge exists regarding that solution's sensitivity to (a) the target values chosen, and (b) the forecasted values of exogenous variables. This note discusses a method by which those sensitivities may be analytically expressed and readily calculated.

Solution of the discrete-time stochastic optimal control problem in the 2-domain

The design of discrete-time optimal multivariate systems is considered in the z-domain. The constant plant can be non-square, unstable and/or non-minimum phase and feedback system dynamics can be modelled. The stationary coloured noise processes are assumed to be represented by discrete rational spectral densities. The system can contain transport delay elements and the effects of plant saturation can be limited by the choice of performance criterion. The system inputs are assumed to contain both stochastic and deterministic components. The two-stage design procedure is original and it enables the stochastic and deterministic control functions to be separated, A performance criterion is first defined which is insensitive to the deterministic signals and this defines the closed-loop optimal controller. The resulting closed-loop system acts as an optimum regulator to minimize the effects of stochastic disturbances. A second tracking error performance criterion is then specified which determines the optimal ...

Some properties of solutions to linear-exponential-Gaussian problems

In this correspondence, it is shown that the optimal control gains for the linear-exponential-Gaussian (LEG) problem can be computed without the "future observation program" at each time step. Furthermore, the connection between the closed-loop Optimal (CLO) control and the open-loop feedback optimal (OLFO) control is discussed.

Optimal closed-loop control stability and transient behaviour of a class of linear dynamic economic models

In this paper an attempt is made to apply results from linear systems theory and Liapunov stability theory to analyze a class of linear economic models. For a quadratic performance index, the Riccati equation can be used to derive the optimal control law in closed-loop form. Once the system is synthesized, the second method of Liapunov can be used to study the transient behaviour of the system. The approach is illustrated through two simple economic models.

Solution of the stochastic optimal control problem in the s-domain for systems with time delay

The design of linear stochastic optimal tracking and regulating systems is considered for systems with time delay. A transfer-function solution for the optimal closed-loop controller is obtained in the s-domain. This solution can be realised physically, but involves the complex-domain delay operator. The state-space realisation of the controller is also obtained. The optimal controllers for both tracking and regulating systems include a Kalman predictor and state estimate feedback. This shows that a form of the separation principle holds for this class of problem.The design technique applies to multivariable systems which may be unstable, nonminimum phase and non-square. The process and measuring system noise terms may be correlated and be coloured or white. The linear quadratic performance criterion to be minimised includes a crossproduct weighting term. The solution to the usual stochastic regulator problem is obtained first and this result is then used to obtain the solution to the optimal tracking problem. Time delays in the control and in the measurement system are shown to have the same effect on the controller design, provided that the performance criterion is chosen appropriately. An expression for the minimum value of the performance criterion, that shows how the minimum cost is increased by the presence of the time delays, is obtained.

Time-optimal feedback control for small disturbances

This paper introduces a new technique for the analysis of time-optimal systems having multidimensional inputs. The method provides a cellular decomposition of the controllable set for small response times for a new generic class of systems called "minimally controllable systems." The decomposition permits a complete study of the time-optimal flow near the origin and forms the basis for the synthesis of closed-loop optimal control. The method, while generally applicable, is restricted here to the simplest, nontrivial case of third-order systems with two-dimensional controls.

Optimal maintenance, repairmen, and control strategies for systems with breakdowns

Optimal preventive maintenance and repairmen policies for a system of machines subject to degradation with age and intermittent breakdowns and repairs are derived using optimal control theory. When this system of machines forms part of a dynamic process control system, a method of obtaining optimal closed-loop control law is indicated.

Synthesis of Digital Control Systems for Nuclear Reactors, (I)

The present work is an attempt to synthesize an optimal closed-loop control system for controlling the transition from one power level to another in nuclear reactors. This first report covers a study on optimal solutions, undertaken with a view to gaining a better understanding of the basic characteristics possessed by such solutions, and to providing useful guides to the synthesis of closed loop control systems. The reactor system model adopted here is a three-dimensional nonlinear discrete-time system of about twenty sampling intervals, which makes it impracticable to apply the conventional dynamic programming method, on account of the excessive demands on computer time and core memories. To overcome this difficulty, a method termed “regionwise dynamic programming” is adopted, to handle the nonlinear multi-stage discrete time system. Some discussions are presented concerning the accuracies obtained and the range of applicability of the present method. The numerical solutions solved by this proposed method can be considered almost exact and physically reasonable. These solutions shall be utilized in a forthcoming paper for synthesizing control systems.

Synthesis of Digital Control Systems for Nuclear Reactors, (I)

The present work is an attempt to synthesize an optimal closed-loop control system for controlling the transition from one power level to another in nuclear reactors. This first report covers a study on optimal solutions, undertaken with a view to gaining a better understanding of the basic characteristics possessed by such solutions, and to providing useful guides to the synthesis of closed loop control systems. The reactor system model adopted here is a three-dimensional nonlinear discrete-time system of about twenty sampling intervals, which makes it impracticable to apply the conventional dynamic programming method, on account of the excessive demands on computer time and core memories. To overcome this difficulty, a method termed “regionwise dynamic programming” is adopted, to handle the nonlinear multi-stage discrete time system. Some discussions are presented concerning the accuracies obtained and the range of applicability of the present method. The numerical solutions solved by this proposed method can be considered almost exact and physically reasonable. These solutions shall be utilized in a forthcoming paper for synthesizing control systems.

Eigenvalue spectra of optimal single-input closed-loop linear control systems

Abstract Simple optimality conditions are derived which must be satisfied by the eigenvalue spectra of single-input closed-loop linear control systems.

A Note on Optimal Open-Loop and Closed-Loop Control

Optimal open-loop and closed-loop controls are defined in a consistent fashion. It is shown that under certain conditions an optimal closed-loop control generates an optimal open-loop one and that optimal closed-loop control can be synthesized from optimal open-loop one.

A probabilistic likelihood approach to trajectory sensitivity

Comparative trajectory sensitivity is investigated with respect to small probabilistic perturbations in initial state and plant parameters when a neighboring feedback control rather than the nominally equivalent open-loop control is used. The uncertainty in initial state and plant parameters is modeled by jointly normally distributed random variables with known mean and known covariance matrix. The nominal control is assumed to yield a satisfactory nominal trajectory and it is desired to preserve this shape in non-nominal situations. The likelihood of the state provides a measure of insensitivity at time t. Using the system equations linearized around the nominal trajectory, the joint distributions of the incremental state are computed in both open-loop and closed-loop cases. It is established that the augmented nominal state when a neighboring optimal feedback control is used is at least as likely as that when the nominally equivalent open-loop control is used. The domain where the augmented incremental state for the optimal closed-loop control is more densely distributed than that corresponding to open-loop control is shown to be a hyper-hyperboloid. This domain is unbounded in certain directions, thus pointing out a potential disadvantage in using optimal feedback control.

Optimal SAM defense system: An application of optimal control concept to operations research

An operations research problem concerning the optimal surface-to-air-missile (SAM) firing pattern to defend a surface target is solved via applications of the concept of closed-loop (feedback) and open-loop optimal control. The SAM defense problem is formulated as a Markov decision process with the number of SAM's in each salvo as the decision variable. Interesting cases, including the presence of imperfect sensor observation and a bound on the number of SAM's available are considered. The principle of dynamic programming and the technique of nonlinear integer programming are applied to reach closed-loop and open-loop solutions. Numerical examples are given for illustration.

A dynamic observer for a synchronous machine†

The closed-loop optimal control law obtained for a linear time-invariant system always requires the entire state vector to be available for direct measurement. It is seldom that all the state variables of a physical system are available for measurement. A compatible dynamic observer of the Luenberger type is designed to reconstruct the entire state vector for a synchronous machine-infinite bus system. The performance of the system with the observer cascaded to the optimal controller is compared with the performance of the system when all the state variables are available for feedback. The transfer matrix relating the output and input is derived.

Closed-Loop Optimal Excitation Control for Power System Stability

An optimal excitation control for the stability of synchronous machines is found in a closed form using Pontryagin's minimum principle. While the optimal control is bang bang, in practice it is necessary to compromise with normal voltage regulator action to produce a sub-optimal control. The time required for stabilization is slightly increased by this. The results obtained for the single machine- infinite bus case indicate that such control provides major benefits in removing transients.