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It was of some surprise, after listening to all the contributions, to conclude that nothing had been said about the theory and application of extremum-seeking devices based on Evolutionary Operation EVOP and its associated modifications such as SIMPLEX and COMPLEX. In many real control systems the values of the measured variables are obtained at discrete intervals of time, which may be of the order of minutes, and consequently there may be considerable delay in the evaluation of the Performance Index. Many of the continuous-perturbation methods proposed at the Symposium do not lend themselves to systems of the former type, and it is thus desirable to be aware of alternative methods of control.

For a rotary lime kiln used in the recausticizing process of a (kraft) paper mill, R mathematical model is presented. The form of the mathematical description is obtained on the basis of the mass and energy balances. The unknown coefficients in the mathematical model are determined by applying the stochastic approximation method. The identification scheme yields the optimal values of the parameters such that the best fit to the available discrete measurements is attained.

This paper deals with the computational methods for the state estimation problem of a large-Scale electric Power system. This state estimation problem can be defined as the power flow problem which has overdetermined boundary conditions, and non-linear function minimization techniques can be used to solve the problem. The mathematical foundation of proposed methods is based upon the perturbed Newton method which includes as special cases standerd function minimization techniques such as Newton-Gauss method, Fretcher-Powel1 method and steerest descent method. The proposed methods in this paper are considered as decomposition methods for the reason that the coefficient matrices of iteration schemes are decomposed into block diagonal forms from the view joints of a power system configuration or a sersitivity between variab1es. These methods are suitable for the state-estimation problem which has many state variables for the simplicity of the coefficient moatrix in non-linear function minimization techniques.

Summary This paper discusses the applicability of digital techniques to various optimization problems and not only demonstrates the utility of these techniques, but confirms that digital computers may be used for the study of plant behaviour. Results discussed in the early part of the paper show that analytical methods for the classes of problems dealt with involve mathematically intractable expressions; the analogue computational procedures also do not yield satisfactory results. The explicit digital relationships and the corresponding systems and structures developed in the paper overcome these difficulties. The digital methods are expected to prove most useful because of the availability of high-speed digital computers capable of handling increasingly large amounts of data. Ideas for further investigations are included.

This paper is concerned with optimal parameter selection in differential games. Necessary conditions and sufficient conditions are derived for the existence of a saddle point for a general two-person zero-sum form. The specified forms for the controls consist of weighted sums of the state variables the weighting factors being products of known timevarying functions and of constants to be determined in an optimal manner. The general results are applied to linear-quadratic games (games with linear dynamics and quadratic performance indices). For this class of differential games, an additional development is given to obtain control law parameters that are independent of initial conditions so that a saddle point with respect to the expected value of the performance index is obtained. The results are applied to specific scalar and vector dynamic systems, and numerical solutions are presented. The approach and the results of this paper are especially useful when only partial information about possible initial states is available to the players and when on-line computational facilities are limited. The results should broaden the class of problems to which differential game theory can be realistically applied. By restricting the form of feedback gains, it is possible in some cases to obtain control laws that are only nominally suboptimal, but are both computationally feasible and implementable in practice.

A differential game is a mathematical model of a competitive situation which evolves over time. Each player controls a different set of inputs to a single dynamic system and each player has an associated pay-off or cost functional. In the differential game considered in this paper, players participate for different intervals of time. The number of players may change during the course of the game. In particular, the situation where the number of players changes between one and two is considered. An example of a differential game with different intervals of play is a pursuitevasion game where the evader starts to evade only after the pursuer penetrates a certain region in state space. It is well known that nonzero-sum differential games may have a variety of acceptable solutions depending on the assumed rationality of the players. In this paper a Nash solution is considered. It is verified that a generalized principle of optimality holds for games with different intervals of play. This leads to a backward translation method for obtaining the strategies. Linear games with quadratic cost functionals over different intervals of play for two players are investigated in detail. The problem leads to a multiple point boundary value problem.

This paper addresses the problem of organizing data and formulating questions to be answered for the purpose of making planning and development decisions. The problem is separated into five distinct parts. Each part is discussed in the context of the planning process and each leads logically to the optimization of development decisions. The five parts are goal definition, establishment of criteria, criteria weighting, alternative rating, and alternative ranking. The feature of making development decisions which distinguishes them from other optimization problems is what has been called “fuzziness”. In any realistic problem formulation, the criteria are not precisely defined; they are fuzzy. The relative importance of each criterion is also fuzzy. Indeed, even when one attempts to rate a particular possible solution, he must deal with fuzzy information. The technique proposed in the paper accounts for this uncertainty in all aspects of the problem and yields probabilistic answers. Thus, when various alternative solutions are proposed for a development problem, the technique yields a probabilistic ranking of the alternatives. Sharper results are obtained if less uncertainty is present in certain parts of the data. However, even in the presence of great uncertainty, realistic problem solutions are obtained. Alternative solutions are rated independent of all others and only after the (fuzzy) ratings are complete are comparative rankings accomplished. Throughout the procedure the realistic uncertainties remain a prominent feature of the procedure.

This paper is intended to solve a scheduling control problem of a manufacturing process which is interdisciplinary area of process control and business management. Scheduling is the activity that determines the overall production for the next several periods. The whole process is assumed to consist of several subprojects, and each subproject is serially connected in time. The objective function to be minimized is an aggregate quadratic loss which is derived from the resource-periods required and from the movement of resources. The mathematical model of the overall scheduling problem is obtained as a 2-level hierarchical quadratic programming problem. The dual problem is formulated applying Fenchel's duality principle to this scheduling problem. The iterative algorithm to solve the dual problem is easily obtained by taking into account the hierarchical structure of the primal problem. The algorithm obtained is called the dual algorithm in this paper. This results lead to realize the actual scheduling systems control problem with much less computational efforts and also with much less computer memory in comparison to solving the original primal problem numerically. Some numerical examples are included.

Summary Plants with distributed parameters are widespread in many branches of engineering. The problem of designing control systems which are the best in a given sense for such plants requires the development of new techniques of optimal control. Control systems with distributed parameters have a number of specific properties that make investigation into them very complex compared with lumped-constant systems. This paper gives a statement of optimal control problems on systems with distributed parameters for a fairly broad class of systems described by non-linear integral equations under arbitrary constraints. It formulates a maximum principle for this case, which gives the necessary conditions for determining optimal controls. The paper also considers approximate methods of solving optimal control problems on systems with distributed parameters, based on approximating the equations for the distribution functions by ordinary differential equations.

The development of automatic operation control of power systems makes it necessary to devote much attention to the algorithmical aspects of thie problem. The algorithms of control applied now have usually non-flexible structure. It results in their complexity and hinders control. When planning control system as a system where the human operator has a key position one is able to make the algorithms much simpler, to increase their reliability and to decrease the time of response. The prospects of such approach to the planning and operation control of the modern developing power systems and some special conditions for the algorithms used in this system are discussed in the paper. The efficiency of above mentioned principles in application to power systems is demonstrated on the problem of finding optimal combination of units to be loaded, when the startup costs should be taken into consideration, it is shown in the paper that optimal operation can be found by means of a simple algorithm, in which the startup costs are not taken into consideration and several logical checks performed by an operator. The characteristics of these checks depend on the time at his disposal, his experience and competence. The advantages of this kind of algorithms are especially obvious in their application to planning of medium voltage complex ramified transmission lines. It is shown in the paper that by means of superposition of the image of electrical network on the map and step by step correction of the original scheme of the network one is able to seek optimal route of transmission line. The algorithms for finding the optimal way of passage over the points and for finding optimal points of branching are applied during each step of correction. The functions of logical operators are performed by a man.

This paper deals with the linear-quadratic-Gaussian multilevel control problem with hierarchical information pattern, which is a typical information pattern to ease the information processing and the system operation of large-complex system. It is shown that the optimal control laws for this problem depend not only on the information but on the periods of control action. Under some constraints one of the optimal control laws is shown to satisfy the separation theorem. The design method of the hierarchical information pattern is also discussed.

Summary Control in closed systems, with incomplete a priori information about the plant, should, in the general case, include two types of action on the plant: (a) tentative actions, ‘experiments’, with the controlled plant, whose problem consists in studying it; (b) directing actions, whose task consists in adjusting the plant to the required operating conditions. The controlling action in optimum control, should, generally speaking, realize both of the above-mentioned tasks simultaneously, and because of this, has a dual character. Dual control problems are considered in this paper. The problem is set up, and its general solution given, for discrete-time systems, when a change in the plant's characteristics represents a discrete Markov process. Various examples are considered.

A hierarchical system is usually a system of subordination levels structured as a directed graph without loops. The paper introduces the concept of simple generalized diagrams of resource allocation. Formalized descriptions of hierarchical resource allocation systems consisting of simple and/or generalized allocation diagrams are proposed. Specific examples include the problem of optimal long-range planning of oil or gas output in large fields; the problem of optimal gas flow control in a pipeline system; the problem of optimal raw material allocation in gas refineries.

This paper deals with an optimal control problem of a discrete-time linear stochastic system with two controllers, each of which can make certain noisy measurements of the state of the system. These two controllers cooperate in trying to minimize the same performance index. Certain constraints are imposed on control structure, and so our problem is to determine the optimal values of the structure parameters of the controllers. Both the cases with and without information exchange between the two controllers are treated. Weak separability is introduced as an extended concept of the separation theorem and sufficient conditions for the problem to be weakly separable are obtained. Usefulness of the concept is illustrated by several numerical examples.

The paper is concerned with construction of a control system of low sensitivity to changes in the plant parameters. The requirement of dynamic property stability in the control system is met by introducing the functions of the plant sensitivity relative a certain parameter vector into the quadratic functional of performance. The optimal synthesis problem is solved with due regard for the specifics of initial conditions for sensitivity functions. Allowance for zero initial conditions of the sensitivity function led to an optimal control law in a form that its implementation does not require the knowledge of current values of the plant sensitivity functions.

The goal of this paper is to present an approach to the problem of the approximation of a complex plant which must to be controlled. Starting from input-output data of the actual system, the proposed procedure leads to determine an optimal reduced model, on which it is very easy to design a min-max control policy for the plant which guarantees a bound of the performance degradation.

An attempt will be made to apply an optimal stochastic control policy to a realistically flavored dynamical system differing to some extent from the system assumed in the development of the policy. The justification for this will rest on the quality of performance of the resulting system. This paper is a preliminary report on a study of this kind. (Author)

It has been shown in several cases that linear stochastic control theory can be used successfully to design regulators for the steady state control of industrial processes. To use this theory it is necessary to have mathematical models of the system dynamics and the disturbances. In practice it is thus necessary to go through the steps of identification, computation of control strategies and implementation. It might also be necessary to repeat the identification if the system dynamics or the disturbances are changing. In this paper we discuss algorithms that combine the steps of identification and control. The special case of discrete time single-input single-output systems is considered. It is assumed that the disturbances can be characterized as filtered white noise. The main result is a theorem which states that if the algorithm converges it will in fact converge to a minimum variance regulator. The behaviour of the algorithm is illustrated with several examples which indicate that it has nice convergence properties. The result has several practical implications. It can be used to construct self-adjusting regulators for direct digital control. It can be interpreted as a real-time maximum likelihood identification scheme. If the control problem discussed is reformulated as a stochastic control problem the theorem implies that asymptotically there is a finite dimensional sufficient statistic. The implementation of the algorithm on process computers is also discussed. It is shown that it is feasible to implement it on snail computers.

Summary Non-linear programming problems may be met with in the selection of optimal automatic control systems. This paper presents a method of solving a problem in non-linear programming. The essence of the method consists in reducing the set problem to a repeated search for a solution of a linear programming problem and the choice of values for certain additional parameters that are introduced.

The paper treats a method for developing a control strategy for a continuously operating nonlinear process. The strategy is shown to be optimal in the steady state and for small disturbances. It is near optimal for larger disturbances.

So far the problem of optimization of large-scale processes is not solved on a general and effective basis, although such a basis exists in connection with the adaptivity of the living organisms. The subject of this paper is such a method of learning optimization which simulates the living adaptivity to a great extent.

The paper is concerned with the problem of adaptation in nonstationary conditions, in other words the problem of dynamic adaptation. A method to obtain discrete dynamic adaptation algorithms for the estimation of the trajectory of the movement of the minimum loss function point is given, which characterizes the quality of the adaptive system. The known learning algorithms in nonstationary conditions are obtained as a particular case from general algorithms of dynamic adaptation. The possibilities of obtaining the optimal algorithms of dynamic adaptation are discussed. The fields of application for dynamic adaptation and directions of their development are pointed up to.

A method of designing a feedback controller for a linear dynamic system disturbed by additive white noise treats the noise as a maximizing (vindictive) opponent and minimaximizes a quadratic performance criterion which is the difference between the original criterion, for the system performance, and a quadratic function of the maximizing player (noise). This procedure, though well known, is viewed with some suspicion as it appears to be a very pessimistic design method in that it assumes that the noise will “conspire” to oppose the controller in the most perverse way. We show, however, in this paper that the controller obtained in this manner is the same as that obtained by minimizing the expected value of an exponential function of the given quadratic performance criterion. When looked at from this viewpoint the “worst case design” does not appear to be too pessimistic since the exponential criterion is rather appealing. Of theoretical interest is the fact that existence of a solution to the zero sum differential game (which arises in the worst case formulation) implies and is implied by existence of the expected value of the exponential performance criterion.

This paper presents the formulation of a class of optimization problems dealing with selecting, at each instant of time, one measurement provided by one out of many sensors. Each measurement has an associated measurement cost. The basic problem is then to select an optimal measurement policy, during a specified observation time interval, so that a weighted combination of ''prediction accuracy'' and accumulated ''observation cost'' is optimized. The current analysis is limited to the class of linear stochastic dynamic systems and measurement subsystems. The problem of selecting the optimal measurement strategy can be transformed into a deterministic optimal control problem. An iterative digital computer algorithm is suggested for obtaining numerical results. It is shown that the optimal measurement policy and the associated ''matched'' Kalman-type filter can be precomputed, i.e. specified before the measurements actually occur. Numerical results for a third-order system with two possible measurements are presented.

In the present paper, the problem of controlling large urban area traffic system is formulated as an optimal control problem and a new urban-area-traffic-control-system is proposed. A dynamic model for area traffic system with a larger unit of time is constructed, a control performance criterion is defined mathematically, and a control policy is sought which minimizes this criterion. In constructing the traffic model, we assumed a new manipulating variable and a slightly more knowledge of the state of the traffic flow than is generally available at present.

This paper describes the work done in evaluating the application of linear programming techniques to on-line cement raw materials blending. The cement making process is briefly described and a generalized blending system is shown. The method of operation of the blending system is outlined with particular reference to the use of the linear program. The choice of linear programming algorithm is discussed and the blending problem is defined. Two different formulations of the problem are set down and an example is given using data taken from a cement plant. Several cases, using both formulations, are run on two different computers each using a different linear programming system. The results are discussed and lead to the selection of the MINMAX formulation for on-line use.

Three distinct types of hillclimbing methods are studied in this paper; variations on the conventional discrete search as proposed by Bocharov and Fel’dbaum(2), Kushner’s method(3), and the piecewise. cubic method(8), presented herein. The concept of testing various methods on a large number of randomly selected hills is proposed, and a class of such test hills is presented. This approach is then used to evaluate the performance of the three types of hillclimber with respect to five criteria.

This paper describes a feasible method for estimating the gradient of the cost function with respect to the parameters of the controller in order to improve or, possibly to optimise the control of non-linear discretetime stochastic systems. When the disturbances have few discrete values a gradient method for determining the optimal control sequences for each possible outcome of disturbances is presented. For more complex problems a Monte Carlo method is proposed.

Summary Dynamic programming theory will solve, in principle, many of the problems connected with optimal systems synthesis, but the number of complex problems actually solved with this method is not large, because the calculations become more and more difficult the more complicated the problem becomes. Introduction of ‘sufficient coordinates’, on which the risk function depends, eases the situation, the ‘sufficient coordinates’ forming the space in which the Bellman equation is considered. In the paper a non-trivial example illustrates the effectiveness of introducing sufficient coordinates. In the example the sufficient coordinates are a combination of a posteriori probabilities and the dynamic variable of the controlled plant. A practical result of the introduction of sufficient coordinates is hat the optimal control breaks down in two units, one producing he sufficient coordinates, the other providing optimal control.

The paper is devoted to the development of an optimizing control system for a group of single-reactor continuous chemical proceses. A generalized model is developed based on the heat and mass balance equations. The least squares estimators of the process output variables are presented. The optimizing problem is formulated as a chance constrained programming problem under uncertainty and its confidence equivalent is proposed. The structure of the optimizing system algorithm for a group of formaldehyde production process is given.

This paper considers the implementation of optimal control using two degrees of freedom, i.e., a combination of the open- and closed-loop configurations. It has been shown that with this arrangement, one can minimize the integral state sensitivity, while at the same time, either the performance sensitivity, or the terminal-state sensitivity may be reduced to zero. In practice, some compromise must be accepted between the reductions in the various sensitivities. An example of a linear system with an integral quadratic cost function is used to illustrate the procedure.

Quality control is very important in the economy of developing nations. The problem can be stated in terms of system approach. The problem is solved by using a comrehensive model of a specific industry. Up-to-date identification methods lead to a modes of a complicated industrial process. In the paper the problem of quality control is solved on the knowledge of identification results. A solution is sought for a line consisting of a number of interrelated plants. The desired quality at the line output is ensured by measuring the quality of input variables. Equations to obtain input characteristics of each plant of the line are obtained. A special case is determining the optimal input characteristics by the techno-economic index. Two examples are given. The first illustrates the application to a roller bearing production line, the second, to an oil processing facility.

This paper proposes and explores a new approach to the design of state estimators for systems with large, but bounded uncertainties in plant and measurement noise covariances. A linear estimator with unspecified gain is chosen a priori. Useful sensitivity measures for this filter are its total mean square estimation error (S"1), and the deviation of this error from the optimum, minimum estimation error in either an absolute (S"2) or relative (S"3) sense. These sensitivity measures are a function of the uncertain statistics and the unspecified filter gain. If a particular measure is first maximized over the set of uncertain covariances and then minimized with respect to the adjustable gains, a filter is obtained which yields a least upper bound on the actual measure regardless of the exact value of the statistics. Minimax filter design for plants with constant, but uncertain plant and measurement noise covariances is fully explored. First, for the S"1 measure, it is shown that min-max equals max-min. Thus the minimax problem is replaced with a simple maximization of the optimal mean square error over the uncertain parameter set and the S"1 filter is simply the Kalman filter for the maximizing noise statistics. Several properties of the required maximization for the infinite time case are then developed. Next the S"2 and S"3 filters are shown to be unique and optimal for at least one point in the set of uncertain parameters. Min-max does not equal max-min for the S"2 and S"3 measures. However, the convexity of S"2 and S"3 in the uncertain statistics is used to show that the maximum of these sensitivity measures is attained over a finite set of points. Three short examples are presented to illustrate the properties of minimax filters and the utility of the minimax design approach.

A number of specific and complicated optimization problems arise in the designing of modern computer systems. These problems by their nature are the problems of automatic control. The function of the control unit in such systems is carried out by the program which may be as part of software or hardware. This paper deals with two optimal problems. The first one is the problem of optimal processor time allocation in time-sharing systems and the second is the optimal storage allocation problem. The main interest is concentrated on the latter problem.

The paper is concerned with optimal control problems for a complex dynamic plant consisting of a number of subsystems having the same final objective and the same constraint on control actions. Complex nonlinear relationships betweem plant coordinates and controls are assumed. A linear programming computing algorithm for approximate solution of such problems is proposed.