On the design of optimal and sub-optimal control systems†

Though dynamic programming is an excellent and versatile algorithm, there are many cases in which we cannot apply this algorithm favourably to control system design because of the limited capacity of a digital computer and other difficulties.

Considering these facts, the successively optimizing method and the combined algorithm of dynamic programming and successively optimizing method are presented in this paper.

One of the objects of this paper is to present a practical method for obtaining the optimal solution of control systems with an integral form performance index by the use of dynamic programming.

The other is to present sub-optiinal solutions by the successively optimizing method and the combined algorithm.

These methods are very easy when compared with dynamic programming. Furthermore, these sub-optimal solutions can be made to converge to the optimum by using the recurrent equations repeatedly.     

On the design of optimal output feedback control systems†

For a linear control system with quadratic performance index the optimal control takes a feedback form of all state variables. However, if there are some states which are not fed in the control system, it is impossible to obtain the optimal feedback control by using the usual mathematical optimization technique such as dynamic programming or the maximum principle.

This paper presents the optimal control of output feedback systems for a quadratic performance index by using a new parameter optimization technique.

Since the optimal feedback gains depend on the initial states in the output feedback control system, two cases where (1) the initial states are known, and (2) the statistical properties of initial states such as mean and covariance matrices are known, are considered here. Furthermore, the proposed method for optimal output feedback control is also applied to sampled-data systems.     

Optimal design of sampled-data control systems by linear programming†

This paper is devoted to the design of digital compensators for a special class of fnite-settling-time, linear, sampled-data control systems which are required to track both step and ramp inputs with no steady-state error. A general procedure is given to design the digital compensator so that the system is optimum on the basis of the ITAE (integral of time-weighted absolute value of error) criterion, subject to the constraint that the peak overshoot of the system for a step input is less than a certain percentage. Linear programming is used to find the optimum solution. Suitable upper bounds are imposed on the variables involved so that the peak overshoot of the system for a unit step input may not exceed a given value. A method of incorporating a specification on the acceleration constant (K _a ) in the design is given, as also is a design example. For the case where finite settling time to a ramp input is not required, the design procedure is modified: a velocity constant (K _a specification is incorporated in the design in place of the K _a specification.     

Sensitivity design of optimal linear systems†

A measure of sensitivity is introduced into the performance index of the optimal linear system. The sensitivity vector is adjoined to the system state vector and the overall system is optimized. A procedure for selecting the weighting matrix elements is given. Finally a design procedure for obtaining approximately optimal feedback controllers with bounds on sensitivity is given. The methods are illustrated with a first-order example.     

Sensitivity of optimal control systems with bang-bang control†

The effects of small parameter variations on the performance index of optimal control systems with initial and final target manifolds, free end time, and bang-bang control are analysed in this paper. A new approach to the sensitivity equation is presented. This approach takes into account the pulse-shaped variation produced by the parameter change on the bang-bang control. An expression, that relates the variations of the performance index, the trajoctory, the final time, and the parameter, is derived. This expression extends to the class of optimal systems with bang-bang control, a result previously obtained by Courtin and Rootenberg (1971).     

On performance sensitivity of optimal control systems†

Formulae for the differential, with respect to plant parameters of an optimized performance index (performance sensitivity), are extended to include hard (instantaneous) as well as soft (isoperimetric) constraints on the control. In particular, this extends to discontinuous and bang-bang controls the Pagurek-Witscnhausen result on the equality of such differentials for open and closed-loop optimal systems     

Application of state space statistical linearization to optimal stochastic control of non-linear systems†

The formulas for different methods of state space version of statistical linearization are given and applied to sub-optimal stochastic control of non-linear systems, as recently suggested by Wonham and Cashman (1968). Comparison with their optimal results shows that there is no immediate reason to prefer one linearization method to another. Especially the conjecture of Kazakov does not hold in general, i.e. the mean coefficients of two different linearization methods must provide best results.     

Lyapunov design of time-shared sampled-data control systems†

This paper is concerned with the problem of sampled-data control of a multi-variable system when only one control variable may be used at a time. A quasi-optimal system for time-sharing control is designed via the direct method of Lyapunov. The design specifies control lows and the decision criterion for utilization of the controller. It is shown that even with simplified decision criteria, stability properties are preserved. The method is developed for continuous plants containing at most a single integration. The results also apply to the classical adaptive-sampling problem where the objective is simply to conserve the number of controlling sampling intervals for a single process. A numerical example is presented of the time-shared control of four second-order systems, each containing an integration.     

Pseudo-convexity and quasi-concavity in optimal discrete-time control systems†

Necessary and sufficient conditions in the form of a minimum principle are obtained for discrete-time control systems by non-linear programming methods. These results generalize previous ones in that they involve pseudo-convexity and quasi-convexity rather than convexity.     

Time optimal control of linear systems with delay†

Necessary conditions, analogous to those of Pontryagin's maximum principle are developed for the problem of reducing the state, of linear systems with delay, to zero in minimum time. An example is presented that demonstrates the technique of finding the time optimal control.     

On the optimal control for a class of distributed parameter systems†

An optimal control problem for a class of distributed parameter systems is discussed. A non-linear integral equation of a signum type is derived as a necessary and sufficient condition for the optimal control. Under very weak conditions, existence and unique ness of the solution to this non-linear integral equation are proved.     

On the optimal control of two classes of non-linear distributed parameter systems†

An optimal control problem, for two classes of non-linear distributed parameter systems, is formulated. Sufficient conditions on the optimal control are derived. These conditions form a two-point boundary value problem of partial differential equations.     

Sensitivity in the decoupling of non-linear control systems†

The problem of decoupling the inputs and outputs of non-linear control systems is considered for the situation where the system parameters are subject to variations or disturbances. The exact decoupling is obtained by an adaptive state feedback controller which requires full knowledge of the instantaneous values of the system parameters. By utilizing the parameter sensitivity concept an approximate adaptive decoupling controller is synthesized in the present paper which is shown to decouple the system to first (or higher) order in the variations of the system parameters from their known nominal values. The sensitivity equations which generate the state sensitivity function required for the synthesis are derived. The results of the paper actually provide the tools for reducing the system sensitivity to parameter variations when the final goal is the improvement of decoupling.     

Optimum design of linear multivariate sampled-data control systems†

This paper deals with the optimum design of linear multivariate sampled-data control systems,

Section 2 is concerned with the design of such control systems which are optimum on the basis of deadbeat performance. This design procedure is general and can be applied similarly to any deterministic inputs such as step, ramp, etc.

Section 3 is concerned with such control systems with respect to an integral form performance criterion. It is desired that the performance index takes the form of an integral, especially when the state variables of a control system are continuous with respect to time. For this reason, the minimum integral control is considered here.

The performance limits for such systems optimized for a deterministic input are considered in §4, and some interesting theorems are proved by using the w-transform.     

Integral action in the optimal control of linear systems with some inaccessible state variables†

Optimal feedback control for linear systems with the usual quadratic performance index which penalizes the state and the magnitude of the control is often difficult to apply because of the requirement of complete state measurement and lack of integral mode. A technique is presented, which introduces integral mode into the optimal control law and does not require the measurement of all of the state variables during the dynamic period. The control produced is equivalent to the original optimal Kalman feedback when external disturbance is absent, and during the period of constant disturbance the method yields a trajectory which can be made arbitrarily close to the optimal trajectory without disturbance.     

On finding switching times in time optimal control systems†

A system of non-linear algebraic equations is found which yields the switching times for the time optimal bang-bang control to the linear system [xdot] = Ax + bu. These equations can also yield switching curves in some instances.     

Time optimal control of second-order systems with transport lag†

The time optimal control of particular second–order systems with constant transport lags is determined by application of Pontryagin's Maximum Principle. A system with a delayed control and a system with a delayed state are considered. The optimal control for these systems is compared with the optimal control of a similar system which does not contain transport lag. It is found that the optimal control is essentially bang–bang but that it is non–unique under certain conditions.     

A discretization of the cost functional in linear-quadratic optimal control†

A discretized cost functional in linear-quadratic control is presented. Specifically, a hybrid linear-quadratic optimal control problem is analysed from the feedback, open-loop and existence-of-solutions points of view, methods being employed from the well established theory for the solution of linear tridiagonal systems of equations.     

Effects of controller dynamics on the stability of a class of optimal control systems†

In this paper, the effects of controller dynamics on the stability characteristics of a class of multivariate optimal control systems are investigated. The results of this investigation indicate that the asymptotic stability of systems of this class may well be lost because of the practical impossibility of implementing control laws derived on the basis of the classical optimal control theory of Kalman (1960),     

Observer theory in optimal control for systems with time delay†

An observer theory for estimating the state variables of deterministic linear time-invariant dynamic forced system is presented. The variation in the cost performance when some of the state variables are estimated by using a linear observer system is evaluated.