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The motion of two interacting bodies along a straight line in a medium with quadratic resistance is considered. The force of interaction between the bodies is a control action, on which no restrictions are imposed. The problem of moving each of the bodies of the system over the given distance is solved, provided that the bodies are at rest at the beginning and at the end of the movement. In the constructed motion, moments of time at which the interaction force instantly changes the velocities of the bodies alternate with time intervals in which the interaction force is zero or provides the equality of the velocities of the bodies.

The problem of modal control by output for a fourth-order dynamic system with two control inputs and two measured outputs is considered, provided that the controllability index is not equal to the observability index. It is shown that this problem is solvable, despite the fact that the total dimension of the input and output vectors does not exceed the dimension of the state vector. Compact analytic solutions to this problem using matrix zero-divisors and solvability conditions of one-sided linear matrix equations are proposed. Theorems are proved that implement direct (control) and dual (observation) approaches to solve the problem for cases when the controllability index is correspondingly greater and less than the observability index. Examples are given confirming the efficiency of each approach.

This article considers a nonclassical optimal control problem, in which the dynamics of an object is described by a system of differential operator equations with a hysteresis converter on the right side. The hysteresis dependence is formalized using an analog of the Preisach converter with inverted threshold numbers, which reflects the nonlinear and multivalued dependence of consumer demand on the price of goods. This allows us to take into account the history of consumer relations over a finite time interval. The problem of optimal production, storage, and sales of products on a mono-commodity market under conditions of a hysteresis demand function is set and solved. The conditions of solvability of the problem under the conditions of applicability of the maximum principle of L.S. Pontryagin are given. The conditions under which the solution is unique are given. The article also presents the results of computational experiments in which the optimal control actions for the model case are identified.

We consider the set of multidimensional conservative systems that admits a family of single-frequency oscillations when taken as a unified system. The problem of aggregation of a set of systems into a coupled system with an attractive cycle close to the oscillation of uncoupled systems is solved. Weak universal coupling controls are applied. Previously, the problem was solved for identical reversible systems with one degree of freedom.

A locomotion system is considered in the form of a chain of a finite number of bodies (materi along points) moving in a straight line on a horizontal rough plane due to the forces of interaction between the bodies. These forces serve as the control variables. Dry Coulomb friction acts between the bodies and the plane. The necessary and sufficient conditions are obtained under which the nonreversible motion of all bodies of the system for the same distance is possible under the assumption that in the initial and final positions the velocities of all bodies are equal to zero. Nonreversible motion is understood as a motion in which none of the bodies changes the direction of their velocity in the process of moving.

An effective analytical method is proposed to solve the problem of modal control by output for a wide class of linear time-invariant systems in which the sum of inputs and outputs can be not only greater than or equal to but also less than the dimension of a state vector. The method is based on reducing the modal control by output to modal observation with fewer inputs. At the same time, it is not necessary to additionally ensure the solvability of the equation connecting the matrix of observer matrix and the desired matrix of controller by output. The reduction is performed by constructing a generalized dual canonical form of control using the operations of the block transpose and the rank decomposition of matrices. The method significantly expands the class of systems for which an analytical solution exists compared to the previously proposed approaches, since it is not strictly tied to the control system’s dimension and also does not require mandatory zeroing of the column and obtaining a system with a scalar input. Based on the proposed method, a strict algorithm for the analytical solution of problems from the considered class is formed. A simple and convenient necessary condition of reducibility of modal control by output to modal observation with fewer inputs is also obtained, which allows evaluating the possibility of analytical solution of the original problem basing only on its formulation. Examples of various problems of modal control by output in which the sum of inputs and outputs is less than or equal to the dimension of a state vector are considered in symbolic form. A detailed analytical solution of the considered examples demonstrates the efficiency of the proposed approach practical application.

We consider a linear controllable third-order mechanical system whose matrix has one positive eigenvalue. The system can serve as a mathematical model of a linearized inverse pendulum controlled by an active dynamic damper. A modulus constraint is imposed on the value of the control variable. Using Pontryagin’s maximum principle, the problem of synthesizing the optimal control that brings the system to a state of rest in the minimum time is solved.

In mathematical control theory, a Dubins car is a nonlinear motion model described by differential relations, in which the scalar control determines the instantaneous angular rate of rotation. The value of the linear velocity is assumed to be constant. The phase vector of the system is three-dimensional. It includes two coordinates of the geometric position and one coordinate having the meaning of the angle of inclination of the velocity vector. This model is popular and is used in various control tasks related to the motion of an aircraft in a horizontal plane, with a simplified description of the motion of a car, small surface and underwater vehicles, etc. Scalar control can be constrained either by a symmetric constraint (when the minimum rotation radii to the left and right are the same) or asymmetric constraint (when rotation is possible in both directions, but the minimum rotation radii are not the same). Usually, problems with symmetric and asymmetric constraints are considered separately. It is shown that when constructing the reachability set at the moment, the case of an asymmetric constraint can be reduced to a symmetric case.

The problem of stabilizing a chain of two integrators by a feedback in the form of nested sigmoids is considered. Such a feedback allows one to easily take into account boundedness of the control resource and ensure the fulfillment of desired characteristics of the transient process, such as a given exponential rate of the deviation decrease near the equilibrium state and the constraint on the maximum velocity. Global stability of the closed-loop system is proved by constructing its Lyapunov function.

A problem of modal control by output for fourth-order dynamical systems with two inputs and two outputs is presented. For a certain class of such systems, an approach is proposed for reducing the problem under consideration to a control (direct version) or observation (dual version) problem for a system with a single input. The approach is based on two successive similarity transformations of the closed-loop system with a controller by output, which make it possible to reset one of the rows of the controller matrix by state or one of the columns of the observer matrix. The class of systems for which the condition of such zeroing is simultaneously a condition for the existence of an output controller is studied. Theorems on the inequality of the indices of controllability and observability in the system under the conditions presented are proved. A variant of using the well-known Bass–Gura and Ackermann formulas is proposed, which significantly simplifies the symbolic expressions for the controller (observer) in the transformed system. Examples of the application of the proposed approach, both in the direct and in the dual version, are considered. Symbolic calculations in MATLAB validate the results.