Optimal control of bilinear systems in a complex space setting. To the best of our knowledge, the study on optimal control concerned with coupled nonlinear schrodinger system is still lacking in mathematics literatures. Pdf optimal control of bilinear systems in a complex. In section vi we describe a class of bilinear systems for which complete analysis is possible, and we display the optimal, nonlinear, finitedimensional estimation equations for an example. In this chapter we consider optimal control problems for nonlinear. These burgeoning problems arise in diverse applications from quantum control and molecular imaging. Optimal bounded controls problem for bilinear systems. An optimal control for bilinear systems is considered here to describe the. For this class of system, we formulate an innite horizon optimal control problem and show that. Bilinear optimal control of the fokkerplanck equation. Based on the concept of relative order of the output with respect to the input, first we change a bilinear system to a pseudo linear system model through the coordinate transformation. We show that, in the same way in which the underlying dynamics can be well approximated by a reducedorder dynamics in the scale separation limit using classical homogenization results, the associated optimal.
A single shooting method is used to solve the boundary value problem in order to determine the control signal. The aim of this paper is to determine the feedforward and state feedback suboptimal time control for a subset of bilinear systems, namely, the control sequence and reaching time. Nearoptimal control of a bilinear, solarassisted heat. Optimal control problem for a class of bilinear systems via.
Bilinear optimal control for a wave equation mathematical. At last based on the theory of linear quadratic optimal control, an optimal control law which is used to eliminate the influence of the disturbances is derived from a riccati equation and matrix equations. Bilinear model predictive control of a hvac system using. The system is governed by a fourthorder parabolic operator. Fixedendpoint optimal control of bilinear ensemble systems. Part of the lecture notes in control and information sciences book series lncis, volume.
A system consisting of a flatplate solar collector, two heat storage tanks, a heat pump, and all necessary minor components is considered. Then based on the theory of linear quadratic optimal control, the optimal controller is designed by solving the. Bilinear system, feedforwardpd control, disturbance rejection, dynamic compensation 1. In particular, we find a necessary and sufficient condition for feedback stabilization in terms of the lyapunov spectrum. Optimal control for timedelay bilinear systems with. Suboptimal control for bilinear systems, optimal control. In this paper, we study an optimal control problem of bilinear type. This paper deals with an optimal control problem with quadratic cost for a class of bilinear systems using the orthogonal functions technique. Pdf stabilizing optimal control of bilinear system with a. Existence and uniqueness of optimal control for a distributed.
Paley, \global bilinearization and controllability of control a ne nonlinear systems. Optimal control theory is a branch of applied mathematics that deals with finding a control law for a dynamical system over a period of time such that an objective function is optimized. Time optimal control laws for bilinear systems hindawi. The main objective of this paper is to develop an optimal control design algorithm for a class of bilinear systems with a separated linear part. Optimal control of a bilinear system with a quadratic cost. Besides, optimal control is one of the most active subjects in the control theory. Linear optimal control of bilinear systems with applications to singular. The study of bilinear control systems began in the 1960s and has since developed into a fascinating field, vital for the solution of many challenging practical control problems. The optimal control of bilinear systems is considered and related to the lie algebra generated by the system matrices. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the.
Using adomians decomposition method, we shall first derive a functional expansion for the inputoutput map of the system, then transform the cost functional so that it yields the optimal control in a recursive manner. Feedback stabilization of bilinear control systems siam. Optimal control of bilinear systems in a complex space. We study linearquadratic stochastic optimal control problems with bilinear state dependence where the underlying stochastic differential equation sde has multiscale features. The main idea of this technique is that it reduces the problem to solving a system of algebraic equations, thus simplifying the problem. An iterative procedure for optimal control of bilinear systems.
Pdf stabilizing and optimizing feedback control policies are derived for the important class of bilinear systems with generalized quadratric. Timeoptimal control for bilinear nonnegativeincontrol. Linear optimal control of bilinear systems with applications to. For each major component a bilinear mathematical model is developed. By learning the theories and algorithms as well as exploring the examples in linear systems.
Researcharticle time optimal control laws for bilinear systems. An overview of the available control strategies for bilinear systems can be found in 15. The resulting bilinear system is then utilized for optimal control with the help of pontryagins principle and the corresponding twopoint boundaryvalue problem. Also, for more information about modeling and control of bilinear systems, we can see the thesis 5 and the references therein. We characterize an optimal control that minimizes a quadratic cost functional using pontryagins minimum principle, we derive sufficient conditions of uniqueness from the fixed point theorem, and we develop an algorithm that allows to compute the optimal control and the.
Optimal control problem for a class of bilinear systems via shifted legendre polynomials international journal of scientific and innovative mathematical research ijsimr page 3. A bilinear control is used to bring the state solutions close to a desired profile under a quadratic cost of control. Hvac system we focus on, then derives loworder models for the temperature dynamics and energy costs. Bilinear systems are a special class of nonlinear systems, in which nonlinear terms are constructed by multiplication of control vector and state vector.
Freeendpoint optimal control of inhomogeneous bilinear. Optimal control of discretetime bilinear systems with. Pdf stabilizing optimal control of bilinear system with. Request pdf optimal control of a constrained bilinear dynamic system in this paper, an optimal feedback, for a free vibrating semiactive controlled plant, is derived. Koopman bilinearization and optimal control of a control. Consequently, and although more complicated structures cannot.
Pdf continuoustime singularly pertrbed bilinear systems. The optimal switching strategy is proposed using geometric arguments and veri ed using numerical simulations and experiments with a laboratory platform for noncontact magnetic manipulation. On the optimal control of bilinear systems and its relation. However, effective methods for solving emerging optimal control problems involving an ensemble of deterministic or stochastic bilinear systems are underdeveloped. The switched system is embedded in a special class of discretetime bilinear control systems. Feedback linearization optimal control approach for bilinear. The lietheoretic significance of these results is also discussed.
An overview of bilinear system theory and applications ieee xplore. Section 3 outlines the mpc control algorithm and presents a tailored sequential quadratic programming sqp method which exploits the bilinear structure of the optimal control problem. In this method, first the original optimal control problem is transformed into a nonlinear twopoint boundary value problem tpbvp via the. Controlling nonlinear pdes using lowdimensional bilinear approximations obtained from data sebastian peitz1 1. An overview of the available control strategies for bilinear systems can be found in 619. The derivative of slps is 20 9 where if is odd and if is even. At last, the simulation results in chemi cal reactor show that the proposed approach is valid and easy to implement, the controller has a good convergence effect.
Inverse optimal control and construction of clf for. This paper presents a new and straightforward procedure for solving bilinear quadratic optimal control problem. As an application we detail the case when the state equation is the schrodinger one, with pointwise constraints on the bilinear control. A continuoustime bilinear control system with metzler matrices has been considered in margaliot and branicky, 2009. In mpc, an openloop optimal control is computed repeatedly on a nitetime horizon using a model of the system dynamics.
Feedback linearization optimal control approach for. Controlling nonlinear pdes using lowdimensional bilinear. In addition, it shows how a blend of sliding mode control and h. Pdf stabilizing and optimizing feedback control policies are derived for the important class of bilinear systems with generalized quadratric cost. On the optimal control of bilinear systems and its relation to lie algebras. An iterative procedure for optimal control of bilinear systems arxiv. We shall consider the minimisation of the cost functional. Optimal and robust control, students will be able to better understand and ultimately better manage engineering processes and systems. In section vii the use in estimation system design of harmonic. This paper proposes a method that uses block pulse functions as an orthogonal base. To make use of the orthogonal functions properties and mainly the operational matrix of. The design method of this bilinear suboptimal control system is presented. The algorithm is based on the hpm, where an iterative process is proposed to find a solution sequence of the nonlinear tpbvp.
We consider the problem of optimal control of a wave equation. Pdf regional optimal control of a class of bilinear systems. Optimal control problem for a class of bilinear systems. The optimal tracking problem of the probability density function of a stochastic process can be expressed in term of an optimal bilinear control problem for the fokkerplanck equation, with the control in the coef. Optimal control of infinite dimensional bilinear systems. We establish the existence of solutions of the underlying initial boundaryvalue problem and of an optimal control that minimizes the cost functional. In this paper, we study the region in which a bilinear control system is feedback stabilizable. These multiplicative controls yield bilinear systems bls. I convergence of koopman bilinear form i reachability in koopman bilinear form kbf. Then the system with timedelay in control variable is transformed to a linear controllable system without delay using model transformation. Under suitable hypotheses, it is shown that there exists an optimal control u and it satisfies an appropriate optimality system. Optimal disturbance rejection via feedforwardpd for bilinear.
As a function of time and space, the control needs to belong to an appropriate banach space. The aim of this paper is to propose a method for solving a regional control problem with minimum energy for a system governed by a wave equation via. The mathematical integration is transformed into a product of matrices. Suboptimal control for bilinear systems is discussed by use of an extension of the linear.
Using adomians decomposition method, we shall first derive a functional expansion for the inputoutput map of the system, then transform the cost functional so that it. The purpose of this paper is to study regional optimal control of a class of distributed bilinear systems in both cases of unbounded and bounded controls. Contributions are analyzed in optimal control, controllability, stnucture, and identification and realization. This system together with specified constraints is treated as an optimization problem. Paley, \global bilinearization and controllability of controla ne nonlinear systems. It has numerous applications in both science and engineering.
For the optimal bilinear control problem governed by following grosspitaevskii equation 1. Design of linear feedback for bilinear control system was cosidered in belozyorov,2002. The system control problem is divided into steadystate and dynamic cases, and the optimal steadystate solution is found. With applications to singular perturbations and weak coupling lecture notes in control and information sciences aganovic, zijad, gajic, zoran on. Pdf in this paper we discuss optimality conditions for optimal control problems with a semigroup structure. This paper considers the optimal control problem for the bilinear system based on state feedback. A knowledge of linear systems provides a firm foundation for the study of optimal control theory and many areas of system theory and signal processing.
International journal of control, automation, and systems vol. The aim of this paper is to study the optimal control problem for finite dimensional bilinear systems with bounded controls. Such dynamics arise naturally, for example, from modeling the evolutionary dynamics of hiv in the presence of drug therapy. On the optimal control of bilinear systems and its. Control inputs enter through coupling operators and results in a bilinear control system.
Inverse optimal control and construction of clf for bilinear. This allows us to apply the variational approach to the bilinear control system associated with a mayertype optimal control problem, and a secondorder necessary optimality condition is derived. On the optimal control problem for a class of monotone. Interesting results obtain when this lie algebra is nilpotent. A control system is called bilinear if it is described by linear differential equations in which the control inputs appear as coefficients. Optimal control of a constrained bilinear dynamic system article online only version available in journal of optimization theory and applications 1745 september 2017 with 57 reads. The simulation results show that this suboptimal control system functions very well.
Stabilization of bilinear control systems with applications. Pdf an iterative procedure for optimal control of bilinear. Summary in view of their ability to model many interesting and useful processes and the relative ease by which their behaviour may be analysed, bilinear control systems have provi. Firstly, using the differential homeomorphism, a timedelay bilinear system affected by sinusoidal disturbances is changed to a timedelay pseudo linear system through the coordinate transformation. Bangbang control, bilinear systems, mechatronic systems, minimumtime control, optimal control. Control method for bilinear systems beomsoo kim and myotaeg lim abstract.
Feedback control laws with switching term are developed for the orbit tracking and the performance of the feedback control laws is demonstrated by a stable and accurate numerical integration of the closedloop system. Optimal control of infinite dimensional bilinear systems 3 this. With applications to singular perturbations and weak coupling lecture notes in control and information. The model and analysis of doublelayer vibration suppression bilinear system is briefly described in section 2. These burgeoning problems arise in diverse applications from quantum control and molecular imaging to. In this paper we discuss optimality conditions for optimal control problems with a semigroup structure. Schattler, optimal bangbang controls for a twocompartment model in cancer chemotherapy, journal of optimization theory and applications. Introduction stability is an important performance for the control system, but there are various external disturbances affecting the performance of control system. Statespace techniques developed since the early sixties have been proved to be very effective. Products of state and control take part in bls, which means that state and control are linear separately. The precise optimal controller is designed by solving the riccati equation and introducing state feedback with state prediction.
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