Quadratic Control Problem Research Articles

LMI based output-feedback controllers: γ-optimal versus linear quadratic

Solution to the linear quadratic control problem is given in the class of linear dynamic output-feedback full order controllers. Necessary and sufficient conditions for existence of such an optimal controller are stated in terms of linear matrix inequalities provided that initial conditions for controller states to be zero. It is shown that parameters of the optimal controller depend on an initial plant state. As an alternative we introduce γ-optimal controller which minimizes the maximal ratio of the performance index and square of the norm of the initial plant state. Numerical comparison for two kinds of these controllers is presented for inverted and double inverted pendulums.

Why “state” feedback?

We study the linear quadratic control problem from a representation-free point of view, and we show that this formulation brings forth two self-contained and original proofs of the optimality of state feedback control laws; these proofs which do not depend on an a priori state-space representation. Moreover, we show an orthogonality property characterizing the set of optimal trajectory of a LQ-control problem.

A Quadratic Numerical Scheme for Fractional Optimal Control Problems

This paper presents a quadratic numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. For a linear system, this method results into a set of linear simultaneous equations, which can be solved using a direct or an iterative scheme. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

A discretized algorithm for the solution of a constrained, continuous quadratic control problem

Numerical solution techniques such as Function space algorithm (FSA), Extended conjugate gradient method (ECGM) and Imbedding extended conjugate gradient method (MECGM) are common techniques for solving optimal control problems. However, these techniques are computationally expensive and iteratively time consuming. In this paper, a Discretized constrained algorithm (DCA) with an associated operator which replaces the integral features of these techniques by a series of summation is developed. Illustrative examples are presented. The results obtained show that the Discretized constrained algorithm (DCA) is much more accurate and more efficient than some of these techniques, particularly the FSA. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 295-300

Finite-difference discretizations of quadratic control problems governed by ordinary elliptic differential equations

In this paper, we analyze finite difference discretizations for a class of control constrained elliptic optimal control problems. If the optimal control has a derivative of bounded variation, we show discrete quadratic convergence in terms of the mesh size h of the discrete optimal controls. Furthermore, based on the optimality conditions, we construct a new discrete control for which we derive continuous error estimates of order h 2.

A result on output feedback linear quadratic control

In this note we consider the static output feedback linear quadratic control problem. We present both necessary and sufficient conditions under which this problem has a solution in case the involved cost depends only on the output and control variables. This result is used to present both necessary and sufficient conditions under which the corresponding linear quadratic differential game has a Nash equilibrium in case the players use static output feedback control. Another consequence of this result is that the conditions also provide sufficient conditions for the static output stabilizability problem. Of course, in case these conditions are not met this does not mean that the system is not stabilizable via static output feedback.

A Note on Cooperative Linear Quadratic Control

In this note we consider the cooperative linear quadratic control problem. That is, the problem where a number of players, all facing a (different) linear quadratic control problem, decide to cooperate in order to optimize their performance.It is well-known, in case the performance criteria are positive definite, how one can determine the set of Pareto efficient equilibria for these games.In this note we generalize this result for indefinite criteria.

Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented. It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation. The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.

Mean–variance optimization problems for an accumulation phase in a defined benefit plan

In this paper we deal with contribution rate and asset allocation strategies in a pre-retirement accumulation phase. We consider a single cohort of workers and investigate a retirement plan of a defined benefit type in which an accumulated fund is converted into a life annuity. Due to the random evolution of a mortality intensity, the future price of an annuity, and as a result, the liability of the fund, is uncertain. A manager has control over a contribution rate and an investment strategy and is concerned with covering the random claim. We consider two mean–variance optimization problems, which are quadratic control problems with an additional constraint on the expected value of the terminal surplus of the fund. This functional objectives can be related to the well-established financial theory of claim hedging. The financial market consists of a risk-free asset with a constant force of interest and a risky asset whose price is driven by a Lévy noise, whereas the evolution of a mortality intensity is described by a stochastic differential equation driven by a Brownian motion. Techniques from the stochastic control theory are applied in order to find optimal strategies.

Optimal control of entanglement via quantum feedback

It has recently been shown that finding the optimal measurement on the environment for stationary Linear Quadratic Gaussian control problems is a semi-definite program. We apply this technique to the control of the EPR-correlations between two bosonic modes interacting via a parametric Hamiltonian at steady state. The optimal measurement turns out to be nonlocal homodyne measurement -- the outputs of the two modes must be combined before measurement. We also find the optimal local measurement and control technique. This gives the same degree of entanglement but a higher degree of purity than the local technique previously considered [S. Mancini, Phys. Rev. A {\bf 73}, 010304(R) (2006)].

Linear quadratic control problem with a terminal convex constraint for discrete-time distributed systems

The present work deals with the linear quadratic control problem for a discrete distributed system with terminal convex constraint. Using techniques of perturbation by feedback, it is shown that the resolution of the considered problem is equivalent to that of a controllability, one so-called Extended Exact Controllability with time-varying operators. The Hilbert uniqueness method approach is then extended to this case to provide an explicit form for the optimal control. In the same framework, the inequality constraint case is examined for which a practical numerical resolution is given. Finally, the results obtained are used to treat a minimum-time reachability problem.

Discrete time linear optimal multi-periodic repetitive control: a benchmark tracking solution

The use of optimal control tracking methodologies for the development of feedback control schemes for arbitrary discrete-time linear systems is considered for the special case when the demand signal is multi-periodic, i.e. can be expressed as a finite sum of periodic signals of different but known frequencies. By embedding the reference signal into plant dynamics, the problem is reduced to a classical linear quadratic control problem which yields an explicit formula for a globally stabilizing tracking controller. Asymptotic perfect tracking is guaranteed by the presence of the familiar internal model of the demand signal dynamics within the control structure. Transfer function analysis of the controller indicates that it consists of an inner state feedback loop plus an error actuated forward path control element containing the internal model.

A communication mix for an event planning: a linear quadratic approach

The communication mix is a relevant decision issue for an organization that plans the advertising campaign for a fixed future event. It is assumed that the objectives of the organization are to minimize the cost of the advertising campaign and to drive the final demand as close as possible to a target value. Two different advertising channels are available: the first affects deterministically the consumers’ demand, whereas the second presents some stochastic aspects which are out of decision-maker’s control. Some recent mathematical developments on the stochastic linear quadratic control problem allow to formulate and solve some interesting instances of the problem. A comparative analysis of the efficiency of deterministic and stochastic controls is done and the optimal feedback policies are discussed. The trade-off between efficiency and risk of an advertising channel is essential to understand the features of the optimal solutions.

Optimality Conditions for Infinite Order Hyperbolic Differential System

In this paper, first, we establish the solvability of (n × n)-systems of hyperbolic type with inhomogeneous mixed Neumann conditions involving operator of infinite order. Using theorems of J. L. Lions, quadratic boundary control problem for this system is considered. The necessary and sufficient conditions for the optimality of the control is obtained and the set of inequalities that characterize these conditions is formulated. Finally, by applying the Dubovitskii-Milyutin theorem of W. Kotarski, necessary and sufficient conditions of optimality are derived for the same problem, where the performance index is more general than the quadratic one and has an integral form.

Viscosity solution of linear regulator quadratic for degenerate diffusions

The paper studied a linear regulator quadratic control problem for degenerate Hamilton-Jacobi-Bellman (HJB) equation. We showed the existence of viscosity properties and established a unique viscosity solution of the degenerate HJB equation associated with this problem by the technique of viscosity solutions.

Reducing the dimensionality of linear quadratic control problems

In linear-quadratic control (LQC) problems with singularities in the control cost and/or the transition matrices, we derive a reduction of the dimension of the Riccati matrix, simplifying iteration and solution. Employing a novel transformation, we show that, under a certain rank condition, the matrix of optimal feedback coefficients is linear in the reduced Riccati matrix. For a substantive class of problems, our technique permits scalar iteration, leading to simple analytical solution.

Generalized Riccati equations arising in stochastic games

We study a class of rational matrix differential equations that generalize the Riccati differential equations. The generalization involves replacing positive definite “weighting” matrices in the usual Riccati equations with either semidefinite or indefinite matrices that arise in linear quadratic control problems and differential games-both stochastic and deterministic. The purpose of this paper is to prove some fundamental properties such as comparison, monotonicity and existence theorems. These properties are well known for classical Riccati differential equations when certain matrices are assumed definite. As applications, we obtain conditions for the existence of solutions to the algebraic Riccati equation and to equations with periodic coefficients.

A FORMULATION AND A NUMERICAL SCHEME FOR FRACTIONAL OPTIMAL CONTROL PROBLEMS

This paper presents a general formulation and a general numerical scheme for a class of Fractional Optimal Control Problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fractional Differential Equations (FDEs). The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An iterative numerical scheme is presented to find the approximate numerical solution of the resulting equations. For a linear system, this method results into a set of linear simultaneous equations, which can be solved directly. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control

Bellman equations of ergodic type related to risk-sensitive control are considered. We treat the case that the nonlinear term is positive quadratic form on first-order partial derivatives of solution, which includes linear exponential quadratic Gaussian control problem. In this paper we prove that the equation in general has multiple solutions. We shall specify the set of all the classical solutions and classify the solutions by a global behavior of the diffusion process associated with the given solution. The solution associated with ergodic diffusion process plays particular role. We shall also prove the uniqueness of such solution. Furthermore, the solution which gives us ergodicity is stable under perturbation of coefficients. Finally, we have a representation result for the solution corresponding to the ergodic diffusion.

A Result on Output Feedback Linear Quadratic Control

In this note we consider the static output feedback linear quadratic control problem. We present both necessary and sufficient conditions under which this problem has a solution in case the involved cost depend only on the output and control variables. This result is used to present both necessary and sufficient conditions under which the corresponding linear quadratic differential game has a Nash equilibrium in case the players use static output feedback control.