Time-optimal Control Systems Research Articles

Considerations on Two-Phase Averaging of Time-Optimal Control Systems

Averaging is a valuable technique to gain understanding of the long-term evolution of dynamical systems characterized by slow dynamics and fast periodic or quasi-periodic dynamics. Averaging the extremal flow of optimal control problems with one fast variable is possible if a special treatment of the adjoint to this fast variable is carried out. The present work extends these results by tackling averaging of time optimal systems with two fast variables, that is considerably more complex because of resonances. No general theory is presented, but rather a thorough treatement of an example, based on numerical experiments. After providing a justification of the possibility to use averaging techniques for this problem "away from resonances" and discussing compatibility conditions between adjoint variables of the original and averaged systems, we analyze numerically the impact of resonance crossings on the dynamics of adjoint variables. Resonant averaged forms are used to model the effect of resonances and cross them without loosing the accuracy of the averaging prediction.

Synthesis of Third-Order Time-Optimal Control System for Plants with Extremum Time Response

Synthesis of Third-Order Time-Optimal Control System for Plants with Extremum Time Response

Synthesis of a time-optimal control system for an extremal object

The paper presents the derivation of the synthesis method for the algorithm of the time-optimal controller for a third order dynamic system. A model with an extreme second-order transient response with delay was adopted as an object of research. The constant speed actuator is represented by an integrator. The synthesis is based on using the Pontryagin’s maximum principle and describing the dynamics of a system in the state space using canonical variables. The verification of the correctness of the result obtained by the theorem of Feldbaum A.A. on the number of switchings of the direction of movement of the regulating body during the control interval has been executed. To calculate the canonical state variables, it is proposed to use the position of the regulator, the controlled value and the derivative calculated from its values, measured on real objects.

Two-channel time-optimal control of nonstationary heat conductive process with account for response time of boundary control actions

The task of organization a closed time-optimal control system of linear object with distributed parameters of parabolic type is considered. The object has two lumped internal controls for the power of heat sources excited in the electromagnetic field of an inductor. The proposed method for the synthesis of optimal controllers uses an alternance method for calculating the optimal program controls for each of the control actions. An example of the construction of a quasi-optimal time control system for the process of periodic induction heating of a metal workpiece with constant values of the feedback coefficients calculated for the most characteristic initial spatial distribution is given.

Improved time optimal anti-swing control system based on low-pass filter for double pendulum crane system with distributed mass beam

Specific payload shape and hoisting mechanism will transform the crane to a complex double pendulum system with a distributed mass beams (DMB), which makes its control challenging. In this paper, the dynamic analysis of the double pendulum crane with DMB is conducted. Based on the low-pass filter, the system response law of the double pendulum system hoisting DMB is analyzed. On this basis, the previously proposed time optimal anti-swing control method is improved by utilizing low-pass filter to achieve the minimum maximum residual angle, as well as to smooth the velocity trajectory so that the jerk can be avoided. To evaluate the effectiveness of the proposed approach, simulations and experiments are performed. Results demonstrate that the proposed method can suppress and eliminate the DMB oscillation. The superiority of the proposed controller compared with the recently proposed controller is that it can achieve higher hook oscillation and DMB oscillation reduction. Besides, the advantage of the improved method is also illustrated in terms of low jerk and smoothness.

A Discrete Nonlinear Tracking-Differentiator and Its Application in Vibration Suppression of Maglev System

Vibration of the maglev train levitation system is harmful to riding comfort and safety. The signal processing method is effective in vibration control. In this paper, a novel kind of second-order nonlinear tracking differentiator is proposed and applied to suppress the vibration phenomenon. The switching curves of the second-order discrete time optimal control system are presented by the isochronous region method. A synthetic function is acquired depending on the cases whether a point in the phase plane can reach the switching curves within one sample step. The discrete form of tracking differentiator is constructed based on the position relationship between the state point and the characteristic curves. Numerical simulation shows that this discrete tracking differentiator can quickly track an input signal without overshoot and chattering and can produce a good differential signal. A test in a maglev test bench also demonstrates the effectiveness of the tracking differentiator in the suppression of the track-train vibration.

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.

Predictive time optimal algorithm for a third-order dynamical system with delay

AnnotationThere are given the results for the analytical synthesis of optimal control algorithm in a closed circuit for the third-order dynamical system, describing by two different groups of state variables. For improvement the quality of time optimal control system is offered to use the predictive output value, expected through the object’s dead-time. A comparative analysis of quality and sensitivity control systems with the time optimal and linear PID or PI controllers is done. Calculations were carried out on the example of the automatic control system of superheated steam temperature of strait-flow boiler TPP-210.

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 control of one and double one winding VCM motor

Abstract This article presents the time optimal control system adopted to control double winding VCM motor. This kind of control is widely used in hard disk drive servo for head positioning. Mathematical model of double winding VCM motor is presented, and its implementation in MATLAB/Simulink is shown. The extended time optimal control algorithm is implemented on dSpace DS1104 board. The results obtained from simulation and real measurements are compared and discussed

Minimum Time Regulation of DC-DC Converters in Damping Mode with an Optimal Adjusted Sliding Mode Controller

In this paper, a new development in the time optimal control theory in sliding mode control systems for multi-quadrant buck converters with a variable load is presented. In general, the closed-loop time optimal control system is applied to multi-quadrant buck converters for output regulation, so that an optimal switching surface is obtained. Moreover, an adjusted optimal sliding mode controller is suggested which adjusts the controller parameters to give an optimal switching surface. In addition, a description of the transient response of the closed-loop system is proposed and used to damp any output or input disturbances in minimum time. Numerical simulations and experimental results are employed to demonstrate that the output regulation time and transient performances of dc/dc converters using the proposed technique are improved effectively when compared to the classical sliding mode control method.

Time-optimal sliding-mode control for multi-quadrant buck converters

A new development of time-optimal control systems in multi-quadrant buck converters for output regulation in minimum time is presented. Pontryagen's minimum principle is employed to obtain time-optimal control law for linear constrained input control systems. Applying this method, transient performance of system variables is improved for output regulation of any disturbance. Some numerical simulations and experimental results are shown to prove the effectiveness of the proposed technique in comparison to the classical sliding-mode control method.

Двухпараметрические T-системы функций и их применение для исследования оптимальных по быстродействию линейных нестационарных управляемых систем

Двухпараметрические T-системы функций и их применение для исследования оптимальных по быстродействию линейных нестационарных управляемых систем

On the curvature of two-dimensional optimal control systems and Zermelo’s navigation problem

The goal of this paper is to extend the notion of Gaussian curvature of two-dimensional surfaces to nonlinear time-optimal control systems in the plane by applying the moving frame method. This notion of curvature was introduced earlier by A. A. Agrachev and R. V. Gamkrelidze by means of Jacobi curves. Here we give a self-contained presentation of its two-dimensional version and apply the results to the well-known Zermelo navigation problem.

A DESIGN OF QUASI TIME-OPTIMAL FUZZY CONTROL SYSTEMS

The problems of quasi time-optimal digital control are discussed. A new design methodology of quasi time-optimal fuzzy controllers based on approximation of prototype discrete controller is suggested. Four kinds of practicable structures for fuzzy controllers are considered. Examples of computer design of quasi time-optimal fuzzy control systems are given.

Non-Euclidean Dubins' Problem

The problem of Dubins on a Riemannian manifold consists in finding a curve of minimal length which has bounded first curvature and satisfies some arbitrarily given first order conditions on its initial and terminal points. We study Dubins' problem on two-dimensional homogeneous spaces of constant curvature. In the Euclidean case it is known that optimal curves are necessarily concatenations of arcs of circles C and line segments L. Such concatenations are of at most three pieces, and they follow what we have called the pattern of Dubins, that is, they appear as CCC or CLC. We prove that Dubins' pattern appears also in non-Euclidean cases, with Cdenoting a constant curvature arc and L a geodesic. In the Euclidean case we provide a new proof for the nonoptimality of concatenations of four arcs of circles. We derive our results by applying the maximum principle to a time optimal control system determined on the connected component of the identity of the isometry group of the base manifold.

Adaptive Time Optimal Model Reference and Sensitivity Based Control of Servosystems

A structure of a self-tuning time optimal control system which achieves a good and fast adaptation by using a reference model and a sensitivity model is described- A procedure of determination fhe time optimal control algorithm parameters by using the reference model and the sensitivity model is elaborated. A way of including an unmodelled dynamics and external disturbance is proposed. Simulation results indicate that the proposed structure provides very good adaptation to variations of system parameters within only two to three stepwise changes of the referent input.

An Application of Time-Optimal Control by Using Switching Surface

The authors have previously proposed a design method of time-optimal feedback control system by using switching hypersurface. In the present paper, the proposed method is applied to construct the control system of the training ship Ohshimamaru 330 GT, and then the practical applicability is shown in the sea trial.The dynamics of a ship is presented by a complicated mathematical model with some nonlinear elements and dead time elements. To make the design of the dual mode controller tractable, the model of ship dynamics is simplified from the practical point of view. And dead time problem is evaded by making use of predictive control. In the simulated course changing control using the nonlinear mathematical model as a controlled object, it is confirmed that the time-optimal control is approximately realized, It is demonstrated that the proposed method is efective showing the resemblance of the results in the sea trial, with those in the simulation.

Time optimal control for induction motor servo system

A time-optimal position control scheme based on Pontryagin's minimum principle is proposed for a current-source-inverter (CSI) fed induction motor system. The induction motor with field-oriented control is modeled with a fast time-optimal position control system of second order which has two modes of operation: the first bang-bang mode operates until the position is close to target, followed by second DC braking mode which gives asymptotic stability. In order to realize this aim, the time-optimal controller is added instead of the speed controller to the conventional field oriented control loop of the CSI-fed induction motor. The validity of the time-optimal control solution has been verified by an experimental test. Experimental results are in close agreement with a digital simulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Measurement stability of time-optimal feedback control of two-input strictly normal linear systems

The concept of “stability with respect to measurement” was introduced by H. Hermes to characterize those feedback control systems which are tolerant of errors in state variable measurement within the feedback loop. This paper considers autonomous linear strictly normal systems with two-dimensional controls and develops necessary and sufficient conditions for local measurement stability of the time-optimal feedback control system.