The resulting bilinear system is then utilized for optimal control with the help of pontryagins principle and the corresponding twopoint boundaryvalue problem. An overview of the available control strategies for bilinear systems can be found in 15. This book is designed to be a comprehensive treatment of linear methods to optimal control of bilinear systems. The system control problem is divided into steadystate and dynamic cases, and the optimal steadystate solution is found. Optimal control of a class of positive markovian bilinear. This is a rather weak action of the controller on the system, usually called of bilinear type, since the control takes action as a coef. The given simulations validate the effectiveness of the approach. In this paper, we prove several new results that give new insights into bilinear systems. An iterative procedure for optimal control of bilinear systems arxiv.
Linear quadratic stochastic control, bilinear systems, slowfast dynamics, model reduction, forwardbackward stochastic di erential equations, least squares monte carlo. Optimal control of discretetime bilinear systems with applications to. On the optimal control problem for a class of monotone. Optimal control of a constrained bilinear dynamic system. Pdf regional optimal control of a class of bilinear systems. As we know optimal control problem for the bilinear. 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. Given a control law, find all performance indices for which this control law is optimal. While we focus on quantum optimal control problems we argue that many of the results of this paper can be extended to general timedependent bilinear control problems. This thesis concerns modeling and control of bilinear systems bls.
We consider the bilinear optimal control of an advectionreactiondiffusion system, where the control arises as the velocity field in the advection term. Section 4 relates the stability of a positive bilinear control system to the generalized spectral radius of its transition matrix. Controllability for distributed bilinear systems siam journal on. This paper proposes a practical algorithm that provably obtains the globally optimal solution to a class of bilinear. Stabilizing optimal control of bilinear systems with a generalized cost. For many semiactive devices, physical considerations constrain the actual damping force such that it can only resist the structural motion in the damper anchors. Bilinear optimal control of the velocity term in a kirckkoff. We consider a bilinear optimal control problem where the state equation is a. 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. In 14, the expandability of the optimal control for nonlinear analytic and di erentiable systems is analyzed in detail. Then the optimal control law of the bilinear system 1 is.
This book represents a comprehensive overview of the current state of knowledge of both the recursive approach and the hamiltonian approach to weakly coupled linear and bilinear optimal control systems. 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. The obtained results allow to steer a bilinear system by an optimal control to a desired state on a subregion of the system domain. Optimal control for timedelay bilinear systems with. Time optimal control laws for bilinear systems hindawi. An overview of the available control strategies for bilinear systems can be found in 619. The parallel algorithms presented in this book are applicable to a wider class of practical systems than those served by traditional methods for large scale singularly perturbed and weakly coupled systems. M2pa laboratory, university of sidi mohamed ben abdellah, fez, morocco. This motivates the optimal control problem of maximizing the spectral radius of the transition matrix. Optimizing the performance of the feedback controller for. Regional optimal control of a class of bilinear systems. Aug 01, 2016 the switched system is embedded in a special class of discretetime bilinear control systems. Questions are still open, this is the case for bilinear systems with unbounded control operator.
Bangbang control, bilinear systems, mechatronic systems, minimumtime control, optimal control. In this motivation, bradley and lenhart 1 studied the bilinear optimal control problem for a kirchho. The book devises unique powerful methods whose core results are re. Stochastic optimal control is one of the important elds in mathematics which has attracted the attention of both pure and applied mathematicians 57,27. Optimal control problem for a class of bilinear systems via. Optimal control of distributed bilinear systems sciencedirect. Optimal control of infinite dimensional bilinear systems.
Feedback linearization optimal control approach for bilinear. Pdf numerical optimal control for bilinear hyperbolic. Optimal control problem for a class of bilinear systems. The purpose of this paper is to formulate, study, and in certain cases resolve the inverse problem of optimal control theory, which is the following. 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. Nice reachability for planar bilinear control systems with. The optimal cost function is shown to be a lyapunov function for the bilinear. Quadratic cost functionals with and without quadratic control penalization are modified by the inclusion of additional nonnegative state penalizing functions in the respective cost integrands.
Stabilizing optimal control of bilinear systems with a. It should be noted that we have obtained an openloop control depending on the initial value of the statex 0. In 4, a bilinear quadratic optimal control problem was defined for a homogeneous bilinear system, unconstrained control forces and a quadratic. Pdf an iterative procedure for optimal control of bilinear. Feedback synthesis of optimal constrained controls for singleinput bilinear systems is considered. As in the other works on this topic, an important assumption is the local stabilizability of the underlying system. Thus, in its most general form, bilinear programming subsumes diverse sub.
Chapter 1 is concerned with the problems of systems analysis of bilinear processes. Section 2 states the optimal control problem for bilinear systems. Section 4 presents an example illustrating the efficiency of control provided by the obtained optimal regulator for bilinear systems. Optimal control of discretetime bilinear systems with.
An overview of bilinear system theory and applications ieee xplore. Koopman bilinearization and optimal control of a control. The control problem is turned into a modal control problem by way of. Reachability energy estimates using the same gramian have only been local so far. An iterative procedure for optimal control of bilinear systems. A single shooting method is used to solve the boundary value problem in order to determine the control signal. The optimal control problem for a bilinear distributed parameter system it is shown that the subject to a quadratic cost functional is solved. Nearoptimal control of a bilinear, solarassisted heat pump. Bilinearquadratic control problem, optimal control. Contributions are analyzed in optimal control, controllability, stnucture, and identification and realization.
Such a problem is generally challenging from both theoretical analysis and algorithmic design perspectives mainly because the state variable depends nonlinearly on the. Stabilization of bilinear systems via linear state. The controller design requires the solution of a convex optimization problem involving linear matrix. Optimal control of weakly coupled systems and applications.
A system consisting of a flatplate solar collector, two heat storage tanks, a heat pump, and all necessary minor components is considered. The aim of this paper is to study the optimal control problem for finite dimensional bilinear systems with bounded controls. Quadratic optimal control and feedback stabilization of. Bilinearization, controllability and optimal control of control a ne nonlinear systems debdipta goswami1, derek a. It has successful applications is many disciplines, economics, environment, management, engineering etc. Pdf bilinear optimal control of an advectionreaction. In this method, by introducing a recursive process, the optimal control law is determined in the form of infinite series with easycomputable terms. A control system is called bilinear if it is described by linear differential equations in which the control inputs appear as coefficients. We show under which conditions a bilinear system is asymptotically stable. Abstract we consider a class of positive systems in which the control signal enters bilinearly with the state. In this chapter we consider optimal control problems for nonlinear. 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. Introduction in this paper we derive no gap second order optimality conditions for optimal control problems governed by a bilinear system being anelinear in the control and with pointwise constraints on the control. Feedback linearization optimal control approach for.
Bilinear systems, optimal control, orthogonal functions, shifted legendre polynomials. For each major component a bilinear mathematical model is developed. We further show that under a controllability inequality, the obtained optimal control guarantees the strong stability of the resulting system. Jul 14, 2006 2017 regional optimal control problem of a class of infinitedimensional bi linear systems. The main difficulty in solving a quadratic optimal control for bilinear systems is the non. Finally, performance of the obtained optimal control for bilinear systems with a quadratic criterion is verified in the cstr chemical reactor example. Such dynamics arise naturally, for example, from modeling the.
The optimal control of bilinear systems is obtained by considering the lie algebra generated by the system matrices. The optimal control of bilinear systems is considered and related to the lie algebra generated by the system matrices. Besides, optimal control is one of the most active subjects in the control theory. Quadratic optimal control for discretetime infinite. Optimal bilinear control problem related to a chemo. Controllability for distributed bilinear systems siam. Concerning stabilizability, several studies have addressed. Problems, applications and computational methods, 156163.
In order to derive an aprropriate optimal control law, a dynamic optimization subjected to inequality constraints is required. This paper considers the optimal control problem for the bilinear system based on state feedback. Key words bilinear systems stabilizing optimal control policy linear state. Optimal control for a bilinear model with recruiting agent.
Optimal control for multiinput bilinear systems with an. Successive approximation procedure for steadystate. Interesting results obtain when this lie algebra is nilpotent. Globally optimal bilinear programming for computer vision. On the optimal control of bilinear systems and its relation. Moreover, we provide a global characterization of reachability in bilinear systems based on a certain gramian. 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. This paper proposes a method that uses block pulse functions as an orthogonal base. The unified theme of this book is the use of dynamic programming in order to simplify and decompose required computations for the optimal control of bilinearquadratic systems. Bilinear optimal control of the fokkerplanck equation. In addition, they derive an optimality system through a penalty. An overview of the available control strategies for bilinear systems can be found in 67.
Bilinear systems are a special class of nonlinear systems, in which nonlinear terms are constructed by multiplication of control vector and state vector. We have already proved one, in reference 2 also in 1, actually with. Optimal control of bilinear systems in a complex space setting. Bilinear systems, optimal control, homotopy perturbation method. In the present paper, we consider the quadratic optimal control problem not necessarily in a feedback form for an infinitedimensional discretetime stochastic bilinear system subject to an. Optimal control with bilinear inequality constraints. Consequently, and although more complicated structures cannot. Optimal control of a bilinear system with a quadratic cost. Chapter 2 solves the problem of optimal control of bilinear systems on the basis of di. Stabilization of bilinear systems via linear state feedback.
As a function of time and space, the control needs to belong to an appropriate banach space. It is shown that the optimal control can be characterized in terms of a convergent power series as well. Stabilization of bilinear systems via linear state feedback control. Regional optimal control of a class of bilinear systems ima. Besides, optimal control is one of the most active subjects in the control. Researcharticle time optimal control laws for bilinear systems. It is shown that the optimal control is given by a convergent power series in the state with tensor coefficients. A distributed parameter system is considered and a bilinear control is applied to the system. 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. Timeoptimal control for bilinear nonnegativeincontrol. Jan 18, 2021 then we provide a class of bilinear systems, including the commutative case, for which the optimal control can be expressed as a time. Also, for more information about modeling and control of bilinear systems, we can see the thesis 5 and the references therein. The solution to the optimal control problem and the proof of the obtained results, based on the maximum principle are given in section 3.
For solving nonlinear optimal control problems, indirect methods often lead to solving a. Optimal control problem for a class of bilinear systems via shifted. Bilinear systems in the frequency domain and an application. Rm are the state and control vectors respectively, a, b and n j are constant matrices of appropriate dimensions with n m n j r. The mathematical integration is transformed into a product of matrices. 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. Parallel algorithms for optimal control of large scale linear systems is a comprehensive presentation for both linear and bilinear systems. Taylor expansions of the value function associated with a. Paley2 1department of electrical engineering 2department of aerospace engineering institute for systems research university of maryland, college park icteam seminar, uc louvain, august 27, 2019 goswami, paley.
This paper deals with an optimal control problem with quadratic cost for a class of bilinear systems using the orthogonal functions technique. Optimal control of bilinear systems in a complex space. Aug 12, 2020 this article is concerned with the design and performance optimization of feedback controllers for state. The objective is to find the optimal control law u. Quadratic optimal control and feedback stabilization of bilinear systems. Conclusionin this paper, based on the hpm, an efficient iterative method has been introduced for optimization of bilinear control systems. Optimal bounded controls problem for bilinear systems. On the optimal control of bilinear systems and its relation to lie algebras. This paper, however, ignores the possibility of optimal controls that are not bb.
Optimal control for a bilinear model with recruiting agent in. Singularly perturbed forwardbackward stochastic di. On the optimal control problem for a class of monotone bilinear systems marcello colombino, neil k. On the optimal control of affine nonlinear systems emis. In many real life applications of optimal control problems with constraints system governed by a hyperbolic equation with three types of controls together viz. An optimal control for bilinear systems is considered here to describe the. Indeed, few controllability results are known for such a kind of control system e. Then based on the theory of linear quadratic optimal control, the optimal controller is designed by solving the. Optimal control of discretetime bilinear systems with applications to switched linear stochastic systems author links. This paper considers the optimal control problem for timedelay bilinear systems affected by sinusoidal disturbances with known frequency and measurable amplitude and phase. Pdf is controlled through the advection term of equation 2. The optimal control problem for a bilinear distributed parameter system subject to a quadratic cost functional is solved. Particular attention is paid to control of open physical processes functioning in a nonequilibrium mode.
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