Optimal Speed Problem Research Articles

Controllability and optimal speed-in-action of linear systems with boundary conditions

The paper proposes a method for solving the problem of optimal performance for linear systems of ordinary differential equations in the presence of phase and integral restrictions, when the initial and final states of the system are elements of given convex closed sets, taking into account the control value restriction. The presented work refers to the mathematical theory of optimal processes from L.S. Pontryagin and his students and the theory of controllability of dynamic systems from R.E. Kalman. We study the problem of optimal speed for linear systems with boundary conditions from given sets close to the presence of phase and integral constraints, as well as constraints on the control value. A theory of the boundary value problem has been created and a method for solving it based on the study of solvability and the construction of a general solution to the Fredholm integral equation of the first kind has been developed. The main results are the distribution of all controls’ sets, each subject of which transfers the trajectory of the system from any initial state to any final state; reducing the initial boundary point to a special initial optimal control problem; constructing a system of algorithms for the gamma-algorithm study on the derivation of problems and rational execution with restrictions on the solution of the optimal speed’ problem with restrictions.

THE OPTIMAL CONTROL PROBLEM FOR SYSTEMS OF INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF-AXIS

This article is devoted to exploring the optimal control problem for a system of integro-differential equations on the infinite interval. Sufficient conditions for the existence of optimal controls and trajectories have been obtained in terms of right-hand sides and the quality criterion function. Integro-differential equation systems are the mathematical models for many natural science processes, such as those in fluid dynamics and kinetic chemistry, among others. Many of these equations have the control that minimizing specific functionals related to the dynamics of these processes. This work specifically focuses on deriving sufficient optimality conditions for integro-differential systems on the half-axis. The complexity of the research is in the following aspects: Firstly, the problem at hand involves optimal control with an infinite horizon, which makes the direct application of compactness criteria like the Arzela-Ascoli theorem impossible. Secondly, the problem is considered up to the moment $\tau$ when the solution reaches the boundary of the domain. This reach moment depends on the control $\tau = \tau(u)$. Hence, the solution to the problem is essentially represented by the triplet $(u^*, x^*, \tau^*)$ — the optimal control, the optimal trajectory, and the optimal exit time. We note that a particular case of this problem is the problem of optimal speed. The main idea of proving the existence of an optimal solution relies on a compactness approach and involves the following steps: identifying a weakly convergent minimizing sequence of admissible controls, extracting a strongly convergent subsequence of corresponding trajectories, and justifying boundary transitions in equations and the quality criterion. The work provides a problem statement, formulates, and proves the main result.

Construction of movement trajectories for objects based on the Dubins car problem, taking into account constant external influences

The paper considers an approach in terms of optimal speed problem for Dubins cars to the formation of control trajectories of moving objects (airplanes, ships), that have control restrictions, under external influences that are constant in magnitude and direction and constant control values at each part of the trajectory. Instead of solving the Pontryagin maximum principle, it is proposed to use a simple comparison of possible control strategies in order to determine the best among them in terms of speed. For each strategy, the calculation of control switching points on the trajectory is based on minimizing the difference between the specified coordinates of the endpoint and the coordinates of the point at which the trajectory comes, depending on the choice of the parameters of two intermediate control switching points. The problem of finding the best speed trajectory for an object from one point to another is solved using the Dubins approach, and their coordinates and heading angles are given for both points. All calculations were carried out taking into account wind and water disturbances, which are constant in magnitude and direction and distort the trajectory. The problem of finding the Dubins paths is reduced to finding the parameters of two intermediate points at which the control changes. Different possibilities for changing controls are considered, taking into account the existing restrictions. The lengths of the trajectories are calculated and the best travel time is selected. The proposed method considers several trajectories acceptable in terms of constraints, taking into account the external influences, from which the optimal path is selected by comparison. Having multiple feasible strategies is beneficial when choosing a trajectory depending on the environment. Instead of solving the problem of nonlinear optimization of the Pontryagin maximum principle, a simple comparison of possible control strategies is used in order to determine the best among them in terms of speed, each of the possible strategies is sought from the condition of minimizing the residual of the analytical solution and the boundary condition at the end of the trajectory. When searching for possible trajectories, control constraints, the influence of external influences, that are constant in magnitude and direction, and the constancy of the control value at each part of the trajectory are taken into account. And all these factors make it possible to sufficiently and adequately simulate the movement of the ship. Physically, restrictions on control (turning radius) are associated with a limited steering angle. Restrictions can be associated not only with restrictions on the turning radius, but also with geographical features of a specific area: for unmanned aerial vehicles this may be due to the buildings and terrain, and for ships this may be due to the coastline, shoals, islands, etc. In this regard, it may turn out that the solution found optimal in terms of speed cannot be realizable in practice. Then the method proposed in the work has the ability to choose another trajectory among the less optimal in terms of speed.

A description of the reachability set in space by a phase change is given. An extremal control is constructed that transfers the initial position to the boundary of the reachability set as a solution to the corresponding optimal speed problem. Numerical examples are given. When conducting

A description of the reachability set in space by a phase change is given. An extremal control is constructed that transfers the initial position to the boundary of the reachability set as a solution to the corresponding optimal speed problem. Numerical examples are given. When conducting

Synthesis of Linear Control Systems with Maximum Speed and Given Overshoot

W e st udy the solution of the so-called maximum speed problem of a linear control system, in which, unlike the classical optimal speed problem with a relay-type control, a linear control algorithm is determined for a linear object that provides the maximum speed of the system. It is of practical importance due to the widespread practical application of linear co ntrol laws. The problem is formulated in relation to continuous, one-dimensional high-order control objects described by the corresponding transfer function or an equivalent system of ordinary differential equations in a certain phase space. The time of the transition process t tp of the designed system is understood in the sense of the classical theory of automatic control and is determined using the zone D = s pr 4,321 %, equal to the given (desired) value of the overshoot of the synthe si zed s ystem. This overshoot corresponds to an oscillatory link with a damping coefficient √ 2/2 — the Butterworth filter of the second order. Accordingly, the maximum performance problem is put in the following formulation: it is required to fi nd a linear feedback algorithm that provides the closed control system with a given astatism order n a and transferring the control object from the initial state to the final one, determined by the constant signal of the regulator’s task, with the minimum value of transient time t tp and set value overshoot s = s pr when performing a constraint on the control signal |u(t)| m u max . At present, this problem has been solved by an approximately algebraic method of synthesizing linear control systems in determining the desired transfer function of a projected closed system based on typical (reference) normalized transfer functions (NTF). The approximate nature of the decision is determined by the fact that the NTF used in the synthesis of high-speed systems are established empirically. This paper proposes a mathematically sound solution to the problem of maximum speed using the theory of analytical design of optimal controllers (ADOC). The maximum speed and set limits on the overshoot and the value of the control signal in the synthesized system are provided by the proposed method for selecting the weights of the quadratic quality functional. We emphasize that the proposed method for the synthesis of high-speed systems, in contrast to the algebraic method, is applicable to a wider class of control objects: both minimal-phase and non-minimal-phase, as containing zeroes, and not. The method is illustrated by an example of the synthesis of a high-speed fourth-order control system containing the results of its simulation.

On the Optimal Speed Problem for the Class of Linear Autonomous Infinite-Dimensional Discrete-Time Systems with Bounded Control and Degenerate Operator

The optimal speed problem for the class of linear discrete-time systems with the infinite-dimensional state vector and degenerate operator is solved. Some properties of convex sets are formulated and proved. Necessary and sufficient conditions under which this problem has a solution in the case of the zero point located on the boundary of the reachability set are established. The optimality conditions are written as a discrete-time maximum principle. For the inner points, the degenerate character of the maximum principle is demonstrated. For an inner point, the optimal speed problem is solved by developing an algorithm with reduction to the admissible case of the boundary point. Some examples are given.

Numerical solution of the optimal speed problem with a phase constraint for one parabolic equation

An algorithm for the numerical solution of the optimal speed problem with phase constraint for a parabolic equation describing the heat conduction processes in inhomogeneous media is proposed. To solve the problems with the use of first-order optimization methods and finite differences on non-uniform grids, analytical formulas are obtained for the components of the gradient of the functional with respect to controllable functions. A method is proposed for selecting initial approximations for optimal controls and a step in time at each iteration, which makes it possible to accelerate the computation process. To achieve the specified accuracy, the speed problem requires 6 iterations and . Based on the analysis of the results of numerical experiments, the influence of various parameters on the iterative process is investigated and recommendations are developed on the use of the proposed algorithm. In optimal control problems, the total number of iterations in option a by the conditional gradient method is 110 and the gradient projection method is 108. In option b, the total number of iterations is CGM – 81, GPM – 64, i. e., the total number of iterations in the optimal control problem in method b the choice of the initial approximation is much less than in variant a. The optimal speed control, obtained by both methods, is close enough to test controls. Numerical experiments are also carried out in the case when the control-optimal controls have two switching points. However, the nature of the results obtained does not change. The proposed algorithm can be used to determine the optimal regime and time of thermal conductivity processes in inhomogeneous media.

The Problem of Optimal Operation Speed of Positional Hydromechanical Drive Systems

The paper solves the problem of optimal operation of the positional hydrodrive dynamic system with hydraulic communication lines. The developed generalized structure of the hydrodrive dynamic system makes it possible to introduce the methods to solve the problems of optimal speed of operation and evaluation of output factors (based on transfer time, supplied power, positional accuracy, etc.).The use of the consequent simplication method of the multimass mathematical model results in finding a rational solution of the multicriteria problem of structural and parametric optimization with minimal timetable and capital inputs. The paper reviews the results of the experimental investigation which prove the adequacy of the proposed hydrodrive model and suboptimal trajectories of its output member displacement when managed by hydromechanic control units.

Method of Quasi-Optimal Synthesis Using Invariants

This paper studies the problem of synthesis of terminal control of a dynamic system. It is shown that to solve the problem the convolution product of objective functional and Gaussian constraint can be used as expanded functional. The use of needle variation allows to get a required condition of the objective functional minimum and to reduce the optimization problem to the boundary problem, which can be solved in closed form. The quasi-optimal solution to the problem of optimal speed of operation is obtained. The modes of operation with choosing the control law parameters were studied.

On the problem of optimal speed for the discrete linear system with bounded scalar control on the basis of 0-controllability sets

For a system with discrete time and bounded scalar control, consideration was given to the problem of optimal control in a minimal number of steps. The control is based on the 0-controllability sets.

Asymptotic solution of the optimal speed problem for the quasilinear system under Euclidean constraint on control

For the system with a small parameter under nonlinearities, consideration was given to the problem of optimal speed. The values of the multivariable control actions are bounded in the Euclidean norm. The theorem of existence and asymptotic properties of the solution of this problem was proved. It underlies the algorithm to construct asymptotic arbitrary-order approximations to the optimal control.