The introduction of cardi nality constraint changes. Optimal controller design for 3d manipulation of buoyant. The optimal control is a nonlinear function of the current state and the initial state. The contribution of this paper is a iiovel approach to. Using the state separation theorem induced from its special structure, we derive the analytical solution for this class of problems. We derive two dual optimal penalties for the lq problem when the control space is unconstrained. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. A polynomial case of the cardinalityconstrained quadratic.
A linearquadratic optimal control problem of forward. Rawlings abstract this paper is a contribution to the theory of the in. In this paper we study the discretetime stochastic linearquadratic lq control optimal problem with conic control constraints and multiplicative. Cardinality constrained linearquadratic optimal control.
In this paper we study the solution to optimal control problems for constrained discretetime linear hybrid systems based on quadratic or linear performance criteria. This result can be used to solve the constrained dynamic meanvariance portfolio selection problem. Constrained linear quadratic optimal control where u 1 is the input used in the previous step of the receding horizon implementation, which has to be stored for use in the current xed horizon optimisation. Furthermore, the optimal control is easily calculated by solving an unconstrained lq control problem. Request pdf optimal control systems introduction classical and modern. This paper studies a class of continuoustime scalarstate stochastic linearquadratic lq optimal control problem with the linear control constraints. U, 2 where the cardinality constraint is removed and u is the set of indices of variables that have been branched up. For instance, in 31 the authors study the quadratic optimal control.
These two penalties, which are derived using value function and gradient methods, respectively, may be used to evaluate suboptimal. Control design objectives are formulated in terms of a cost criterion. Optimal control of linear systems with limited control actions. We propose in this paper a fixed parameter polynomial algorithm for the cardinalityconstrained quadratic optimization problem, which is nphard in general. Model predictive control of constrained piecewise affine.
The concepts are taken from the engineering theory of. We focus in this paper on exact solution methods for problem p. Furthermore, the optimal control is easily calculated by solving an unconstrained lq control problem together with an optimal parameter selection problem. The purpose of this paper is to design dynamic compensator 3 with proper dynamic order l for the system 1 such that the closedloop. For receding horizon control, an fhcoc problem is solved at each time step, aiid then only the initial value of the optimal input scquence is applied to the plant. Optimal control of constrained stochastic linearquadratic. For example, spectral method 10, timedomain decomposition iterative method 16, and etc. We consider the problem of stochastic finite and infinitehorizon linear quadratic control under power constraints. More specifically, we prove that, given a problem of size n the number of decision variables and s the cardinality, if the n. Optimal linearquadratic control is discussed in most graduate macroeconomics textbooks, e. The revealed optimal control policy is a piecewise affine function of system state. It presents the results in the context of finite and infinite horizon problems, and discusses a number of new and interesting issues. Linear quadratic optimal control 3745 where xct 2 rl l n is the state vector of the compensator, ac 2 rl l, bc 2 rl r, cc 2 rm l and d c 2 rm r are matrices of the dynamic compensator which are to be determined.
This paper considers a discretetime linear quadratic optimal control problem with a limited number of control implementation, which is termed as the cardinality constrained linearquadratic optimal control problem cclq. Under some mild conditions, the stationary optimal control policy can be also derived for this class of problem with infinite horizon. On continuoustime constrained stochastic linearquadratic. Lecture 4 continuous time linear quadratic regulator. Linear quadratic regulator lqr design is one of the most classical optimal control problems, whose. Linear quadratic methods dover books on engineering brian d. A explicit solution for constrained stochastic linear. Informationconstrained optimal control of distributed. Constrained optimal control of discretetime linear hybrid. This paper studies a class of continuoustime scalarstate stochastic linearquadratic lq optimal control problems with the linear control constraints. Optimal control theory georgia institute of technology. Derivation of a powerful solution method for cclq plays an essential role in developing an efficient solution scheme for linearquadratic optimal control.
Linear quadratic methods dover books on engineering kindle edition by anderson, brian d. Robust optimization has seen varied application, including antenna 11 and circuit 12 design, constrained stochastic linearquadratic control, and wireless channel power control 14. Gradient formulae for the cost functional of the optimal parameter selection problem is derived. As control implementation often incurs not only a variable cost associated with the magnitude or energy of the control, but also a setup cost, we consider a discretetime linearquadratic lq optimal control problem with a limited number of control implementations, termed in this technical note the cardinality constrained linearquadratic optimal control cclq. The rendezvous dynamics under linear quadratic optimal control. Linear quadratic methods dover books on engineering. Algorithm for cardinalityconstrained quadratic optimization. Model predictive control of constrained piecewise affine discrete. This paper considers a discretetime linear quadratic optimal control problem with a limited number of control implementation, which is termed as the cardinality constrained linearquadratic. In control theory, the linearquadratic gaussian lqg control problem is one of the most fundamental optimal control problems. For a linearquadratic problem, it can be shown that the optimal cost on the time interval. The optimal control law is the one which minimizes the cost criterion. More specifically, we develop solution algorithms for four specific cardinality constrained optimization problems, including i the cardinality constrained linearquadratic control problem, ii the optimal control problem of linear switched system with limited number of switching, iii the time cardinality constrained dynamic mean variance.
Computation of the constrained infinite time linear. This control policy can be computed efficiently by solving two riccati equations offline. Solving of time varying quadratic optimal control problems. A unified framework of regularization methods for degenerate non.
The convergence properties of a smoothly parametrized curve, known as the central trajectory, is studied. As for the deterministic lq control problem, gao et al. More generally, we may require to impose state constraints of the form. The linear quadratic optimal control problems are a class of optimal control and there is an extensive literature on them. A similar problem arises in optimal linearquadratic control with cardinalityconstrained input 14 see also 20 for optimal control with sparse statefeedback gains. Cardinality constrained robust optimization applied to a. The approach is also extended to the case of constraints on systems states,inputs, and. Optimization of constrained stochastic linearquadratic control on. Cardinality constrained linearquadratic optimal control, ieee transactions on. Furthermore, a polynomially solvable case of the cardinality constrained quadratic optimization problem was identi. Algorithm for cardinalityconstrained quadratic optimization the relaxation we solve at each node is. Finite horizon optimal control is shown to be a linear nonstationary feedback control with a gain matrix generated by a backward differential matrix riccati equation. The theory optimal control theory is a mature mathematical discipline which provides algorithms to solve various control problems the elaborate mathematical machinery behind optimal control models is rarely exposed to computer animation community most controllers designed in practice are theoretically suboptimal.
Unfortunately, when some control constraints are involved, except for a few special cases, there is hardly a closedform control policy for the constrained lq optimal control model. The calculations of the optimal control law can be done offline as in the classical linear quadratic gaussian control theory using dynamic programming, which turns out to be a special case of the new theory developed in this technical note. Chamon, alexandre amice, and alejandro ribeiro abstractthis work considers the problem of scheduling actuators to minimize the linear quadratic regulator lqr objective. One of the most remarkable results in linear control theory and design. Fullstate feedback 1 linear quadratic optimization is a basic method for designing controllers for linear and often nonlinear dynamical systems and is actually frequently used in practice, for example in aerospace applications. We consider magnetic actuation and control of a spherical neutrally buoyant magnetic microrobot via magnetic coil setups and seek to design an optimal controller to reduce the required energy and coils currents.
Rodriguez and athanasios sideris abstract a sequential quadratic pr ogramming method is pr oposed for solving nonlinear optimal contr ol pr oblems subject to general path constraints including mixed state contr ol and stateonly constraints. In this paper, we consider a path following algorithm for solving infinite quadratic programming problems. It concerns linear systems driven by additive white gaussian noise. We show that the points of this curve converge to the optimal solution of the problem, so by approximating this curve, solutions arbitrarily close to the optimal solution can. Matroidconstrained approximately supermodular optimization for nearoptimal actuator scheduling luiz f. Use features like bookmarks, note taking and highlighting while reading optimal control.
Cardinality constrained linearquadratic optimal control was investigated in 28. We derive closedform solutions for the linearquadratic lq optimal control problem subject to integral quadraticconstraints. Tight miqp reformulations for semicontinuous quadratic. The cost functional is considered both for finite and infinite horizons. Rakovic optimal control of constrained piecewise affine discrete time systems using reverse transformation 1546 1551. A sufficient solvability condition is given by means of an appropriately stated linear boundary value problem bvp and by discussing the special structure of the regular differential system inherent in this bvp. We showed that in currently employed setups, where the actuation frequency is few tens of hertz, the nonlinear dynamics of the system can be well approximated by a set of linear. Indeed, in 6 a linear quadratic gaussian team problem is constructed with a nonclassical. In general, this problem is nphard and its solution. We deal with linearquadratic optimal control problems for timevarying descriptor systems in a hilbert space setting. This book gathers the most essential results, including recent ones, on linearquadratic optimal control problems, which represent an important aspect of stochastic control. On linearquadratic optimal control problems for time.
It explores linear optimal control theory from an engineering. Applying the state separation theorem induced from its special structure, we develop the explicit solution for this class of problem. This augmented edition of a respected text teaches the reader how to use linear quadratic gaussian methods effectively for the design of control systems. Linearquadratic control and information relaxations. Linear quadratic regulator lqr is a fundamental one 1. Part of this work first appeared in the 1995 ucla school of engineering and applied science internal report no.
Linear quadratic optimal control in this chapter, we study a di. On the infinite horizon constrained switched lqr problem oxford. There are many papers which their authors have given methods for solving linear quadratic optimal control problems. On constrained infinitetime linear quadratic optimal control.
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