Optimal preposterior observation of dynamic systems

Consideration was given to the problem of optimal observation which accompanies the problem of optimal control under multiple uncertainties using feedforward, feedback and combined, closable and closed, loops. Sets of the initial and current preposterior distributions of the terminal state of the dynamic system were introduced. Problems of linear programming were formulated to evaluate these sets. Positional solution of the optimal observation problem which may be used for positional solution of the optimal control problem under multiple uncertainties was established relying on the current preposterior distribution. A method of positional solution of the optimal observation problem using the optimal estimator which in real time generates the current values of the positional solution was proposed. The results obtained were illustrated by an example.     

Stabilization of linear autonomous systems of differential equations with distributed delay

Consideration is given to the problem of optimal stabilization of differential equation systems with distributed delay. The optimal stabilizing control is formed according to the principle of feedback. The formulation of the problem in the functional space of states is used. It was shown that coefficients of the optimal stabilizing control are defined by algebraic and functional-differential Riccati equations. To find solutions to Riccati equations, the method of successive approximations is used. The problem for this control law and performance criterion is to find coefficients of a differential equation system with distributed delay, for which the chosen control is a control of optimal stabilization. A class of control laws for which the posed problem admits an analytic solution is described.     

On the difference approximation of the delay control system

Consideration was given to the difference approximation of the control system obeying a differential inclusion with retarded argument. Approximation of an ensemble of the trajectories of a differential inclusion by a sequence of ensembles of the trajectories of the corresponding difference (discrete) inclusions was investigated. Estimates enabling one to judge whether the approximating sequences converge were obtained. The possibility of applying the results obtained to one minimax problem of optimal control of the system under consideration was demonstrated.     

Optimal pulse observation of one type of systems with delay

Consideration is given to a linear problem of optimal pulse observation of a non-stationary dynamic system with delay in an equation of its mathematical model. To compute estimates of an unknown vector parameter of the initial state of the system, fast direct and dual methods are proposed. The main role belongs to quasi-reduction of the fundamental matrix of solutions to systems with delay. As is shown, to perform iterations of the method, integration of auxiliary systems of ordinary differential equations on small time intervals is sufficient. An algorithm of the operation of an optimal estimator—device for computing estimates of current states—is described. The results are illustrated by the problem of optimal observation of the fourth-order system with delay.     

Robust-stabilizing controller horizons on the basis of predictive model with truncated pulse transient function

Consideration was given to the problem of determining the robust-stabilizing values of the control (M) and prediction (P) horizons for the systems with predictive controllers. The specified z-image of the controller with predictive model allows one to examine the asymptotic characteristics of the closed loop and determine the condition for system robust stability in the case of astatic delay plants. The advantage of the proposed method of determination of M and P was demonstrated by way of example.     

Design of the optimal control of periodic processes in unlimited time

Consideration was given to the design of optimal control of periodic and oscillatory processes which are characteristics of the space applications where the time of control is not strictly limited and the limited resource is represented by the stock of the jet working fluid. The methods of the theory of singular relaxation extensions and the degenerate problems were used. In this case, solution has the form of the estimated minimizing sequence of positional controls which is constructed in the original class by approximating the permissible modes of the generalized solution, the cyclic sliding mode.     

Experience from design and application of the top-level system of the process control system of nuclear power-plant

Consideration was given to the design of the top-level system of the nuclear power-plant process control system with regard for the impact on its safety. This system is aimed at centralizing process instrumentation and control. The top-level system is a sophisticated hardware complex supported by the computer-aided design and adjustment system and intended for integrating all subsystems of the process control system. The top-level system executes the informational, control, service, and auxiliary functions of the nuclear power-plant process control system. Consideration was given to the structure of the top-level system, the structure of its software, and the principles of design providing high performance of the top-level system.     

On some sufficient conditions for optimality of the pursuit time in the differential game with multiple pursuers

Consideration was given to the problem of optimal pursuit of one object by multiple objects. The player’s moves obey the ordinary differential equations. Geometrical constraints are imposed on the player controls. Sufficient conditions were obtained for optimality of the pursuit time, and the optimal player strategies were constructed.     

Optimal controllability norm for the power cost function

Consideration was given to a model of the multilevel hierarchical structure controlling a “production line” (successive business process) where the manager’s costs are defined by a power function depending on the intensity of the subprocess controlled by this manager. For the model under study, the optimal hierarchy of control was determined, and the dependence of the optimal controllability norm (the number of “deputies” of one manager) on the model parameters (for example, the power which may be interpreted as the degree of environmental instability) was analyzed.     

Analysis of the problem of continuous-product inventory control under deterministic lead time

Consideration was given to the control of the continuous-product inventory in the storage system delivering to the end consumers. In the system, the product is replenished at a deterministic lead time of duration obeying the volume of order. The system behavior is described by a regenerative random or deterministic process. The general task of control lies in determining the optimal time to place the replenishment order. Functionals of the mean specific income and mean specific costs were constructed, and a theorem of solution existence and uniqueness was given. For several variants of the model, explicit solutions of the problem of control under different inter-parameters relations were obtained. A number of examples of particular economic systems obeying the model under study were discussed. A procedure for seeking numerical parameters of the optimal control was analyzed in detail.     

Two algorithms for approximate construction of the set of positional absorption in the game problem of pursuit

Consideration is given to the differential game on the finite period of time in which the first player has to guarantee that the phase point will entry the terminal set and the second one must ensure the evasion of the terminal set on this interval. Is proposed a method for approximate construction of the set of positional absorption, i.e, the set of all initial points for which the problem of pursuit posed for the first player is solvable. The method is based on unification constructions and structures close to them. Relations defining the system of sets that approximates the set of positional absorption are written out. The convergence of the approximating system to the set of positional absorption is proved.     

Variational optimality condition in the problem of control of initial boundary conditions for semilinear hyperbolic systems

Consideration is given to the problem of optimal control of a system of semilinear hyperbolic equations at finite-dimensional (pointwise) relations between initial boundary states of the system and control actions. In this case, the optimality condition in the form of a pointwise maximum principle is invalid. A nonclassical optimality condition of a variational type is proved. The sense of the obtained result is in that an optimal boundary or start control nearly at each point is a solution to a special problem of control of initial conditions for the system of ordinary differential equations constructed on the family of characteristics of the initial hyperbolic system. Constructions of the indicated optimality condition are illustrated by two examples. An iterative method based on the obtained result is stated. The method does not require additional assumptions about the differentiability of parameters of the control problem and convexity of the set of admissible controls necessary in using gradient methods.     

Principle of maximum in the problem of control under limited phase coordinates

Consideration was given to the autonomous problem of optimal control with phase constraints for which the necessary optimality conditions in the form of the Pontryagin principle of maximum were obtained. The result obtained features completeness of the conditions of the maximum principle and their nondegeneracy.     

Bilateral Estimation of the Bellman Function in the Problems of Optimal Stochastic Control of Discrete Systems by the Probabilistic Performance Criterion

Consideration was given to the optimal control of discrete stochastic systems by the probabilistic quality criterion. The new characteristics of the Bellman equation for this class of problems were examined, and the two-sided estimate of the Bellman function was determined. The problem of optimal control of the security portfolio with one riskless and a given number of risk assets was considered by way of example. The class of strategies featuring asymptotic optimality was established using the two-sided estimate of the Bellman function.     

Degenerate problem of stabilization of a specified trajectory

Consideration is given to the problem of stabilization of a specified (supporting) mode as a degenerate problem of minimization of the root-mean-square deviation from it where there are no deviations of control actions. This formulation makes it possible to completely use available limited control resources for stabilization. We propose a method of approximate optimal synthesis in the neighborhood of a turnpike manifold that is obtained as a solution in the form of optimal synthesis of linearly quadratic problem with unbounded linear control. The investigation is performed for a linearized discrete model of the controlled system; however, the obtained stabilizing control can be applied in the initial nonlinear system directly or after the correction by regular iterations.     

Generalized solutions in one problem of optimal control for the Tobin macroeconomic model

Consideration was given to the Tobin macromodel of economic growth with real and money markets. This model was used to study the optimization of the monetary policy which is degenerate from the mathematical standpoint because the linearly occurring control (rate of money offering) is unlimited. That is why in reality an extended problem of optimal control with generalized discontinuous trajectories was solved. Extension was made by a special transformation of the original problem. The solution was eventually obtained in the form of a quasioptimal pulse process because the optimal solution does not exist even in the class of pulse processes.     

Control of Systems on Self-Organizing Models

Consideration was given to the control of the nonlinear system in the absence of its mathematical model. Solution was based on the Krasovskii self-organizing optimal controller with extrapolation. The algorithm of control of a multivariable plant was used to solve the problems of control of the overhead crane and calculation of the cost of option for buy (sell) of shares. The self-organizing optimal controller with extrapolation enables one to adapt solutions of the mathematical models to physical plants.     

Design of speed-optimal continuous linear controllers

Consideration was given to the design of a continuous controller which is optimal in the time of control and meets additional requirements on the performance indices of the control system. The time of control was defined as in the theory of automatic control. Solution was based on the method of desirable transfer functions defined in terms of the standard transfer functions in the Vyshnegradskii form.     

Necessary conditions for optimality of stochastic systems with uncontrollable flow of failures and constraints of the equality and inequality type

Consideration was given to the necessary conditions for optimal control of the nonlinear random-structure stochastic systems conditioned by the uncontrollable flow of failures with constraints of the equality and inequality on the system parameters, control functions, and state vector describing various requirements on the system.     

On one problem of optimal control at the impact phase and unification of the interaction end instants

Consideration was given to the problem of optimal control of a mechanical system at the impact phase and the influence of different instants of phase end on the performance criterion.