Time optimal control problem for second-order linear time-invariant systems

Previous article Next article Time-Optimal Control of Solutions of Operational Differenital EquationsH. O. FattoriniH. O. Fattorinihttps://doi.org/10.1137/0302005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Einar Hille and , Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957xii+808 MR0089373 0392.46001 Google Scholar[2] S. Bochner and , A. E. Taylor, Linear functionals on certain spaces of abstractly valued functions, Ann. of Math. (2), 39 (1938), 913–944 MR1503445 CrossrefGoogle Scholar[3] R. Bellman, , I. Glicksberg and , O. Gross, On the “bang-bang” control problem, Quart. Appl. Math., 14 (1956), 11–18 MR0078516 0073.11501 CrossrefGoogle Scholar[4] R. S. Phillips, Perturbation theory for semi-groups of linear operators, Trans. Amer. Math. Soc., 74 (1953), 199–221 MR0054167 CrossrefISIGoogle Scholar[5] Tosio Kato, On linear differential equations in Banach spaces, Comm. Pure Appl. Math., 9 (1956), 479–486 MR0086986 0070.34602 CrossrefISIGoogle Scholar[6] L. S. Pontryagin, , V. G. Boltyanskii, , R. V. Gamkrelidze and , E. F. Mishchenko, The mathematical theory of optimal processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962viii+360 MR0166037 0102.32001 Google Scholar[7] Ju. V. Egorov, Optimal control in a Banach space, Dokl. Akad. Nauk SSSR, 150 (1963), 241–244, translated in Soviet Mathematics, 4 (1963). 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We study the time-optimal control problem for an unmanned aerial vehicle (drone) moving in the plane of a constant altitude. A kinematic model is considered where the angular velocity is a control. Such a system is described by Markov-Dubins equations; a large number of works are devoted to solving different optimal and admissible control and stabilization problems for such models. In the papers [T. Maillot, U. Boscain, J.-P. Gauthier, U. Serres, Lyapunov and minimum-time path planning for drones, J. Dyn. Control Syst., V. 21 (2015)] and [M.A.~Lagache, U. Serres, V. Andrieu, Minimal time synthesis for a kinematic drone model, Mathematical Control and Related Fields, V. 7 (2017)] the time optimal control problem is solved where the drone must reach a given unit circle in the minimal possible time and stay on this circle rotating counterclockwise. In particular, in the mentioned works it is shown that is this case the problem is simplified; namely, the problem becomes two-dimensional. In the present paper we consider a natural generalization of the formulation mentioned above: in our problem, the drone must reach a given unit circle in the minimal possible time and stay on this circle, however, both rotating directions are admissible. That is, the drone can rotate clockwise or counterclockwise; the direction is chosen for reasons of minimizing the time of movement. Such a reformulation leads to the time-optimal control problem with two final points. In the paper, we obtain a complete solution of this time-optimal control problem. In particular, we show that the optimal control takes the values $\pm1$ or $0$ and has no more than two switchings. If the optimal control is singular, i.e., contains a piece $u=0$, then this piece is unique and the duration of the last piece equals $\pi/3$; moreover, in this case the optimal control ins non-unique and the final point can be $(0,1)$ as well as $(0,-1)$. If the optimal control is non-singular, i.e., takes the values $\pm1$, then it is unique (except the case when the duration of the last piece equals $\pi/3$) and the optimal trajectory entirely lies in the upper or lower semi-plane. Also, we give a solution of the optimal synthesis problem.

Time optimal sampled-data controls for the heat equation

Notice of Violation of IEEE Publication Principles<br><br>"Hybrid Time-Optimal Predictive Control for Mechanical Systems with Backlash Nonlinearity"<br>by Dong Lingxun, Dou Lihua, Feng Heping<br>in the Proceeding of the 2008 IEEE/ASME International Conference on Mechatronics and Embedded Systems and Applications, 12-15 October 2008<br><br>After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE's Publication Principles.<br><br>This paper contains significant portions of original text from the papers cited below. The original text was copied without attribution (including appropriate references to the original authors and/or paper titles) and without permission.<br><br>Due to the nature of this violation, reasonable effort should be made to remove all past references to this paper, and future references should be made to the following article:<br><br>"Hybrid Theory-Based Time-Optimal Control of an Electronic Throttle,"<br>by Mario Vasak, Mato Baotic, Ivan Petrovic, and Nedjeljko Peric <br>in IEEE Transactions on Electronics, Vol 54, No 3, June 2007, pp. 1483-1494<br><br>"Controlling Mechanical Systems with Backlash-A Survey"<br>by M. Nordin and P.-O. Gutman <br>in Automatica Volume 38, Issue 10, October 2002, pp. 1633-1649<br><br>"A Hybrid Approach to Modeling, Control and State Estimation of Mechanical Systems with Backlash"<br>by Ph. Rostalski, Th. Besselmann, M. Baric, Femke von Belzen, M. Morari<br> in International Journal of Control, vol. 80, no. 11, pp. 1729-1740<br><br> <br/> The mechanical system with backlash nonlinearity is distinguished between a "backlash mode" and a "contact mode". The inherent switch between the two operating modes makes the system a prime example of hybrid system. In this paper, a piecewise affine (PWA) model of the mechanical system with backlash nonlinearity is built. A constrained time-optimal predictive control problem based on the hybrid PWA model is formulated for solving the optimal control problem of mechanical system with backlash nonlinearity. The proposed constrained time-optimal predictive control strategy consists of two parts, that is, control invariant set computation and constrained time-optimal control law computation. For the purpose of reducing computation of online implementation, the control law is precomputed offline for the range of model states and references by combining dynamic programming strategy with the reachability analysis for the PWA model. In the simulation of tracking the reference speed, it is demonstrated that the constrained time-optimal predictive control approach has better tracking control effect and less computation time compared with the constrained finite time optimal control approach. The resulting control law achieves that for any reference the tracking error remains within a small bounded set, furthermore, the online implementation becomes simpler by the offline computation in the former step, thereby reducing computation burden and being implemented on low-cost hardware for systems with small sampling time.

Norm saturating property of time optimal controls for wave-type equations

Time optimal control function and optimal trajectories solved for second order linear system with constant time delay

There are several methods to solve an initial value problem of second-order homogenous linear systems of differential equations with constant coefficients. That are elimination method and matrix method. Whereas to solve nonhomogenous systems, used undetermined coefficient method and variation of parameter method, that through some difficulties and complex procceses. But then, there is an alternative method to solve it. It is Matrix Laplace Transform Method. The goals of the research are to explain Matrix Laplace Transform Method and use it to solve initial value problems of second-order homogenous linear systems of differential equations with constant coefficients. All of matrix entries are constant. The result of the research is be obtained solutions of second-order linear systems of differential equations with constant coefficients use Matrix Laplace Transform Method

Numerical methods are considered for solving optimal control problems in linear systems, namely, terminal control problems with control and phase constraints and time-optimal control problems. Several algorithms with various computer storage requirements are proposed for solving these problems. The algorithms are intended for finding an optimal control in linear systems having certain features, for example, when the reachable set of a system has flat faces.

Distributed fault detection for interconnected second-order systems

This paper studies the commutativity of second-order linear time-varying systems. We present new and simple explicit commutative theories and conditions for second-order linear time varying systems, the commutative theories and conditions are derived and solved to simplify the use of commutativity for practical and industrial uses. Some second-order systems are commutative, some are not commutative, while some are commutative under certain conditions. Moreover, some second-order systems are sensitive toward change in initial conditions or parameters while orders are not, the effect due to disturbance also varies within systems depending on the systems. Furthermore, the stability and robustness of second-order systems has so many issues. But the proposed method provides simple explicit commutative theories and conditions for second-order systems that tackle these issues, simple in computations and accuracy. This finding will help to simplify and investigate the commutativity, sensitivity, and effects due to disturbance on the second-order linear-time varying systems, which has great important in science and engineering. An example is given to support our results.

In this study, the problem of σ -stabilising a general class of second-order linear time invariant systems with dead-time using a proportional–integral–retarded (PIR) controller is considered. The σ -stability of a system determines the exponential decay in its response. Here the σ -stability of the closed-loop system is ensured by assigning up to four dominant real roots at − σ . For the tuning of the controller gains an extensive analysis of all the possible allocations of the gains according to the response of the closed-loop system is presented. The D-partition method is used to provide important insight into the problem. As a consequence of this analysis, to achieve the desired decay rate, exact analytic expressions for tuning the PIR controller parameters are given. To illustrate the theoretical results obtained, the under-actuated mechanical system called inverted pendulum is used.

This paper focuses on online time optimal control of nonlinear systems. This is achieved by approximating the results of time optimal control problems (TOCP) with deep neural networks (DNN) depending on the initial and terminal system state. In general, solving a TOCP for nonlinear systems is a computationally challenging task. Especially in the context of time optimal nonlinear model predictive control (TMPC) with hard real time constraints successful termination of a TOCP within sample times suitable for controlling mechanical systems cannot be guaranteed. Therefore, our approach is to train three DNNs with different aspects of numerical solutions of TOCPs with random initial and terminal state. These networks can then be used to approximate the TMPC by a one step model predictive control scheme with a significantly simpler structure and decreased calculation time. In order to verify this procedure by simulation, it is applied to a prove of concept example as well as the model of an industrial robot.

Stochastic Galerkin method and port-Hamiltonian form for linear dynamical systems of second order

Moving a dynamic system in minimum time from a given initial state to a desired final state on a prescribed path is one of the oldest and most enduring technological dreams of the scientific and industrial communities. In this research, the problem of bounded-input time optimal control for applied multi-body dynamic systems subject to a full nonlinear dynamical model is solved. To solve the problem, an innovative method, called the ‘floating-time’ method is introduced and utilized. Compared to traditional methods, the floating-time method is an applied method not based on variational calculus. It can be applied to the full nonlinear model of the dynamical system and can handle static and dynamic constraints defined by differential or algebraic equations. The problem of time optimal control is as follows. Find the control law of bounded inputs that drive a given multi-body dynamic system (such as the gripper of a manipulator) along a pre-specified trajectory (in either configuration space or generalized coordinate space) from a given initial position to a given final position, minimizing the time of the motion as a performance index. Using variable time increments, the equations of motion of the system will be reduced to a set of algebraic equations. Searching for a set of time increments (floating-times) that make the equations to exert the maximum available effort produces the minimum possible floating-times, and minimizes the total time of motion. The applicability of the method will be shown by using three examples: a point mass sliding on a rough surface, a 2R robotic manipulator, and the well-known Brachistochrone.

This chapter presents relationship between the cost parameters of the Linear Quadratic Regulator (LQR)-based control technique and the gains of Proportional-Integral-Derivative type control technique. The relationships have been derived mathematically by comparing the characteristic equations in the frequency domain with their counterparts in time domain. The scope of this chapter is first- and second-order linear time invariant systems in standard forms. Examples have been provided in the chapter to demonstrate how closed-loop dynamics such as natural frequency, damping ratio, and time constant can be related to the values of cost parameters in LQR. This chapter serves as a bridge between modern and classical control theory that enables the control designer to foresee the effects of the selection of cost parameters related to the LQR on the dynamics of the closed-loop system.