On the relationship between the problem of anisotropy-based controller synthesis and classical optimal control problems

Consideration is given to the interrelation between different problems of \(\mathcal{H}_2 \)-and \(\mathcal{H}_\infty \)-optimization and their relationship with the theory of anisotropy-based control. It was shown that the anisotropy-based controller of the full order for the completely defined linear system minimizes the mean amount of the transmitted information between the input and output of the closed system.     

Method of state space expansion in the design of autonomous control

The results of constructive analysis and block design of the problems of autonomous control in the nonlinear dynamic systems were presented. As compared with the well-known results of the control theory, the method of state space expansion enabled augmentation of the class of systems that are controlled autonomously in the output variables. The state observers on sliding modes were used for the dataware of the basic algorithms. The procedures developed feature multilevel decomposition of the problem of high-dimensionality design into smaller independent subproblems.     

On approximation of the subdifferential of the nonsmooth penalty functional in the problems of optimal control

For the problem of optimal program control with terminal equality constraints which is considered as an “elementary operation” within the framework of the algorithm to calculate the optimal positional control, a smooth approximation the Frechét-nondifferentiable penalty functional was proposed. Examples of numerical experiments were presented.     

Adaptive suboptimal robust control of the first-order plant

For a linear first-order plant, consideration was given to minimization of the deviation of its output from the specified value in an uncertain environment. Both the parameters of the plant itself and the upper bounds of the external disturbances and operator perturbations in output and control were assumed to be unknown. The suboptimal adaptive control was constructed on the basis of an approximate solution of the problem of optimal identification from the measurement data where the performance of the problem of control was used as an identification criterion.     

Transformations of control systems for the investigation of pulse modes

It is considered a scheme of transformation of control system with unbounded velocity hodograph to a system of reduced order (called the derived system) via integral of the limit system which describes behavior of the original one under pulse (practically sufficiently large) control inputs. There are formulated the conditions (connected with controllability of the limit system on its integral manifold) when some solution of the derived system will be a generalized solution of the original one that is it can be approximated by a sequence of regular solutions. Methodical and applied examples are given.     

On polynomial solutions of the linear stationary control system

It is known that the controllable system x′ = Bx + Du, where the x is the n-dimensional vector, can be transferred from an arbitrary initial state x(0) = x ⁰ to an arbitrary finite state x(T) = x ^T by the control function u(t) in the form of the polynomial in degrees t. In this work, the minimum degree of the polynomial is revised: it is equal to 2p + 1, where the number (p ? 1) is a minimum number of matrices in the controllability matrix (Kalman criterion), whose rank is equal to n. A simpler and a more natural algorithm is obtained, which first brings to the discovery of coefficients of a certain polynomial from the system of algebraic equations with the Wronskian and then, with the aid of differentiation, to the construction of functions of state and control.     

Method to determine an optimal control in the minimum of resource consumption for the nonlinear stationary systems

An iterative method to solve the nonlinear problems of minimization of resource consumption was proposed. It generalizes to a more general class of nonlinear systems the method of resource minimization for the nonlinear systems with the control-linear right-hand side separated in state and control.     

Optimal digital filtering of continuous signals in time lag systems

The problem of the optimal estimation of continuous processes by discrete measurements in the presence of time lag (delay) is considered. On the basis of the theory of parametric transfer functions, an optimal, periodically nonstationary filter is developed, which affords a minimum of the estimation error variance at any instant of time. The comparison is performed of the obtained solution with the optimal stationary filter, which ensures a minimum of the mean (by continuous time) error variance. It is shown that in the problem of estimation of the Markov process of the first order, a simpler stationary filter with the fixer of order zero is insignificantly inferior to the optimal filter.     

Output control algorithm with the compensation of biased harmonic disturbances

The paper develops the methods of harmonic disturbance compensation using the measurements of plant output variable. An approach for biased harmonic disturbance compensation is proposed. This approach is superior to known ones in some characteristics.     

Attracting ellipsoids and synthesis of oscillatory regimes

An approach to stabilization of nonlinear oscillations in multidimensional spaces is proposed on the basis of the V.I. Zubov’s stability theory for invariant sets. As a special case, the derived controls make it possible to excite self-oscillating regimes in specified state subspaces R ^2k ? R ²ⁿ with simultaneous oscillation damping on Cartesian products R ^2n?2k .     

Supervisory control of the structured dynamic discrete-event systems

A new model is developed and studied in this paper—structured dynamic discrete event systems as a theoretical foundation for designing the supervisory control over the set of autonomous components. The composition of the model is defined and the question of a supervisor’s existence (the controllability of the given specification)is studied. Besides, there were stated the basic stages of the design technology in the framework of the introduced controllability analysis model and the supervisors synthesis method.     

Optimization of boundary control at one end of a string in the presence of a model nonlocal condition

The paper is devoted to problems of optimal boundary control of string vibrations under nonlocal conditions that relate values of string displacement or its derivative at the boundary point with their values at the interior point. The solutions for all problems of boundary control are obtained in explicit analytic form.     

Parametric synthesis method of integral systems on the basis of root locus curves of Kharitonov polynomials

The method is developed for the synthesis of a stable set of characteristic polynomials of an interval dynamic system on the basis of the initial unstable set with the use of the system model in the form of a free root portrait. The synthesis is carried out by changing the boundaries (adjustment) of the interval of uncertainty of a free polynomial term, which enables affording the stability without a change of the configuration of the root portrait of the system. As a criterion of the adjustment, the distance is assigned that is measured along trajectories of the root portrait; in particular, a new polynomial can be defined by the nearest polynomial to the prescribed one with due regard for requirements of the quality of the system.     

Constructive methods of control optimization in nonlinear systems

The paper investigates methods of optimal program and point-to-point control in nonlinear differential systems. The proposed methods are based on the idea of discretizing the continuous-time problem. In case of program control, the method of projected Lagrangian is used, which involves solution of an auxiliary problem with linearized constraints by the reduced gradient method; in case of point-to-point control, Bellman’s optimality principle is employed for a “grid” over an approximating solvability tube of the system for a specified goal set with approximation of the point-to-point control by families of controlling function or parameters. An example is given, where the optimal point-to-point control is calculated for a model of maneuvering aircraft.     

The order of singularity of pulse optimal controls in degenerate linear-quadratic optimization problems with time delay

In the paper, we study a problem of minimization of degenerate integral quadratic functional of special delayed structure (a subintegral function contains items concentrated at the moment and items with delay in phase coordinates) on trajectories of a linear system of differential equations with time delay in phase coordinates considered on a finite period of time. Sufficient conditions for the existence of optimal program control in space of generalized functions of the lth order of singularity are established and an optimal control is obtained. A special feature of the problem is that the optimal program contains pulse components concentrated at the ends of the control gap.     

On the method of dynamic programming for linear-quadratic problems of optimal control in hybrid systems

For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid linear-quadratic optimal control based on the recently suggested principle of maximum. Interrelations between the hybrid principle of maximum and the method of dynamic programming for the systems of this class were discussed. The classical formalism was extended, the corresponding Riccati equations were obtained, and discontinuity of the “hybrid” Riccati matrix was proved. The computational aspects of the established theoretical results were considered.     

Optimality conditions of sliding modes and the maximum principle for control problems with the scalar argument

Various types of optimality criteria and conditions, which define the set of admissible solution, are brought to the canonical form. For the problem set in this form, the optimality conditions of sliding modes are stated. It is shown that the optimality conditions emerge from these conditions in the form of the maximum principle for problems with a scalar argument and an arbitrary combination of the optimality criterion and constraints.     

Approximate solution of two-point boundary value problems for linear in control systems

The two-point boundary value problem for nonlinear systems, linear in unbounded control, is considered. For three-dimensional systems we present a local algorithm of approximate solution to the control problem. On the basis of this algorithm were constructed algorithms of approximate solution to the control problem on the 3-dimensional integral manifold (orbit) of n-dimensional systems with 2-dimensional control. Demonstrating examples are presented.     

Output control of linear systems with unmodeled dynamics

Recently, an output control law for linear parametrically uncertain plants was proposed by one of the authors of this paper. Here, the feasibility of this approach is substantiated in the presence of unmodeled asymptotically stable dynamics. A theorem is proved which formulates the feasibility conditions for the control law for systems with unmodeled dynamics.