Time-optimal System Research Articles

A Time-optimal Problem of Stopping a Two-link Pendulum on a Movable Mount

We consider a time-optimal problem of stopping a two-link pendulum on a movable mount. Given nonlinear system with the lack of and limitation on the control resource is linearized in the neighborhood of a stable equilibrium position. For a linearized system the Pontryagin maximum principle is used and the optimal control law obtained. To estimate parameters of the law basing on finding out an orthogonal subspace was an algorithm developed for constructing a time-optimal system control. A software module has been developed in the Matlab environment with the help of which a comparison of the dynamics of linear and nonlinear systems with the obtained control law was made.

Estimation of proximity of controls synthesized on basis of maximum principle and ADAR method

Introduction. A special case of synthesizing the same electromechanical control system by the Pontryagin maximum principle and by the synergetic synthesis method is considered. The task was to solve the synthesis problem of the time optimal electromechanical position control system; herewith the travel resistance modulus linearly depended on the output coordinate of the system. This approach to the selection of the synthesis problem was because the synthesis of time optimal systems is one of the most widespread problems, and it is solved by increasing the efficiency of the existing control systems.Materials and Methods. Synthesis of the time optimal linear control system based on the maximum principle is a widely accepted problem in the modern control theory. However, the procedure of synergistic synthesis does not have such formalization. This being the case, the paper suggests an approach that brings together these two methods, which, in our opinion, will increase the efficiency of the synergistic synthesis method through adding some features of the synthesis methodology for optimal systems.Research Results. The paper formulates two key concepts. The first one is as follows: the application of the maximum principle for an object of the DC motor class when synthesizing the positioning algorithm under the conditions of linear loading functionally dependent on the engine rotation angle allows the time optimal system to be optimized. The second concept states that synthesis of a control system based on the synergistic approach enables to obtain a system close to optimal (quasioptimal), but after modifying the synergetic synthesis method itself. A hypothesis is formulated on the possible connection between the introduced (when implementing the procedure of state space extension in the synergetic synthesis method) time constants with the optimal switching time of control defined in the maximum method.Discussion and Conclusions. The synthesis through the maximum control technique and the ADAR method is performed. In virtue of the comparison of efficiency of these methods, a hypothesis is put forward on the possible compatibility of the studied methods.

Sufficient optimality conditions for controlling switched systems

The optimal control problem for a switched system whose state vector includes both continuous and discrete components is considered. The continuous part of the system is governed by differential equations, while its discrete part, which models the operation of a switching device with memory, is governed by recurrence inclusions. The discrete part switches the operation modes of the continuous part of the system, and is itself affected by the continuous part. The switching times and their number are not specified in advance. They are found as a result of optimizing a performance index. This problem is a generalization of a classical optimal control problem for continuous–discrete systems. It is shown that the cost function is constructed from certain auxiliary functions—the so-called conditional cost functions. Equations for the conditional cost functions are derived and sufficient optimality conditions are proved. The application of these conditions is demonstrated by the example of designing a time optimal switched system that is similar to A.A. Fel’dbaum’s classical example of the time optimal continuous system.

Structural-parametric synthesis of time-optimal distributed parameter control systems with interval uncertainty of the plant characteristics

This work is devoted to a possible way to construct time-optimal closed systems for incompletely determined linear models of distributed-parameter control plants of parabolic type, which provides real-time identification of their parametric characteristics according to the results of monitoring the state of the plants. The structural-parametric synthesis of the proposed controllers is based on the alternance method of designing optimal software controls. The paper gives an example (of independent interest) of constructing a time-optimal system for the control of the process of induction heating under typical conditions of interval uncertainty of the initial temperature and heat losses.

Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions

Abstract In this article, the authors study analytic and numerical solutions of nonlinear diffusion equations of Fisher’s type with the help of classical Lie symmetry method. Lie symmetries are used to reduce the equations into ordinary differential equations (ODEs). Lie group classification with respect to time dependent coefficient and optimal system of one-dimensional sub-algebras is obtained. Then sub-algebras are used to construct symmetry reduction and analytic solutions. Finally, numerical solutions of nonlinear diffusion equations are obtained by using one of the differential quadrature methods.

Applying bilinear time‐optimal control system in boost converters

A new development of time-optimal control in bilinear systems is introduced here and applied to a boost converter. By an analytic approach, time-optimal control system for output regulation is employed in bilinear systems and an optimal switching surface is obtained. Transient response of the time-optimal system is described to eliminate any output disturbances in minimum time. Some experimental and numerical results are illustrated that the time interval for output regulation and transient performances of a boost converter using the proposed technique in comparison with other classical method such as sliding mode control are effectively improved.

Time-optimal guidance control for an agricultural robot with orientation constraints

Time-optimal guidance system is suggested for an field mobile robot with orientation constraint.Time-optimal row guidance problem is formulated based on sonar sensors.Cascade substitute controller ...

Comparative analysis of the Lyapunov function for different strategies of analogue circuits design

Generalized methodology has been developed for analogue circuits design based on applying the optimal control theory. The problem of time optimal system design was defined as a classical functional optimization problem of the optimal design theory. In this case the process of analogue circuits design is mathematically defined as a dynamic control system. In this context the minimization problem of the processor time of designing can be formulated as a minimization problem of the response time of dynamic system. In order to analyze the properties of such a system, it is proposed to use the concept of the Lyapunov dynamic system function. Using this function and its time derivative a special function has been built that allows us to predict the total processor time of circuit design by using the characteristics of the initial period of designing. Numerical results indicate the possibility of predicting the processor time of different design strategies in terms of the special function behavior.

Switching curve deformation in a time-optimal closed-loop system

The dynamics of a closed-loop system resulting from the solution of a time-optimal problem of a two-dimensional differential inclusion are discussed. It is shown that internal uncertainties (such as variations in resistances or parameter changes) cause trajectories of the closed-loop motion to (1) spiral convergently around the target and reach it in a finite time; (2) spiral divergently around the target and reach a limit curve; and (3) reach the target in a finite time in a sliding mode. Some suggestion for practical applications are given. >

Automatic Near Time-Optimal Control of Plants with Two Controlling Elements or Parameters

An approach for synthesis of automatic near time-optimal system with two controlling elements or parameters is proposed in the present paper. The problem of synthesizing a time-optimal controller of plants with two controls is treated in a theoretical and supplementary aspect. In the paper is formulated the principle of optimality through the quickest alteration of the optimal Lyapunov's function. The application for certain completely controllable and observable third order plants is illustrated in examples, where the corresponding optimal control functions are synthesized. In the paper is shown that it is possible to synthesize a near time-optimal controller of a two control variables system through the method of equivalent signum-functions. The equivalent control structure of a controller with two control circuits is obtained, belonging to the most simple structures known, which is realized by minimum number of linear and relay elements. It is possible to realize an equivalent structure of near timeoptimal controller with two controls using only linear and relay elements.

Sensitivity to switching-function variations in a time-optimal positional system

The paper deals with the sensitivity to switching-function variations of a time–optimal system which is controlling a positional mechanism. The model chosen has a second-order differential equation with non-linear and discontinuous right–hand side. Such a model covers a large class of motion resistances, in particular all types of friction. Differential inequality theory and the theory of discontinuous differential equations, and in particular a method of generalized jT solutions, are used to show that time-optimal switching-function variations cause the origin to become an unstable equilibrium point of the state-plane and the control system to generate limit–cycles. Some suggestions for applications are given, as is a practical example.

Time-Optimal Control Of A Stepping Motor

An original time-optimal system of a stepping motor is discussed. The proper switching moment for each phase of the motor is dependent on the normalized mechanical velocity wM, the normalized electromagnetic time constant Te and the normalized internal damping ration.

Time optimal control and indifference curves of certain second-order systems

Abstract By using an explicit solution of the adjoint equations for a time optimal second-order system in terms of the state variables, it is shown that the first two lobes of the switching curve generated by backward time solution are optimal switching curves

Synthesis of a tim-optimal control system†

Abstract A second-order system has been taken for the synthesis of time-optimal control. Important elements in the closed-loop system are the relay element, the controlled element and the controlling device performing time-optimal control. An equation for the switching boundary has been obtained by the phase-space technique and verified by a state-variable approach. The realized time-optimal system is found to give better response than any other non-optimal system.

Discontinuous variational problem of the optimization of control processes: PMM vol. 36, n≗2, 1972, pp.357–360

On the basis of the results obtained in [1 – 3] we develop a generalized approach to the solution of discontinuous variational problems of the optimization of control processes. We have obtained necessary optimality conditions (the Weierstrass conditions). We have solved the problem of the synthesis of a second-order time-optimal system with constraints of the second type leading to discontinuities in the phase coordinates. We have shown how constraints of the second type, influencing the process to be synthesized, may be taken into account.

Dynamic optimisation of a boiler

Optimal and suboptimal digital controllers are developed for the control of a linear, oil-fired boiler model specified by discrete state variables and transition functions.The model is based on design and steady-state test data and the resulting equations are transformed into a normal co-ordinate reference frame for isolating the time-dependent modes of the system. Analytical techniques are developed for reducing the system order, and the open-loop behaviour of the reduced system is shown to be similar to that of the original system.Controller design is based upon measurement and subsequent minimisation of a quadratic performance index related to the displacement of the discrete state variables.For suboptimal control, minimisation is considered at each successive forward sampling point with input amplitude constraints, and this is shown to produce an approximately time-optimal system. The concepts of dynamic programming are used for investigation of the optimal controller. This requires a complex iteration procedure, and the response of the controlled system does not appear to be superior to that of the simplified suboptimal system.The use of Lyapunov functions in the performance index are also investigated, and these are shown to produce local stability of the controlled system.