Singular Linear-quadratic Optimal Control Research Articles

Deterministic Optimality of the Steady-State Behavior of the Kalman–Bucy Filter

In this letter, we provide a deterministic characterization of optimality of the steady-state behavior of the Kalman–Bucy filter, via an inverse optimal control argument. The result is achieved in two steps, both interesting per se . First, a singular linear-quadratic (LQ) optimal control problem is formulated and solved with respect to the innovation term of a classic Luenberger observer, hence yielding a LQ optimal observer. Then, such a construction is employed to interpret the optimality of the steady-state behavior of the celebrated Kalman–Bucy filter in a purely deterministic sense.

On the reduction of the continuous-time generalized algebraic Riccati equation: An effective procedure for solving the singular LQ problem with smooth solutions

This paper presents a reduction technique for the continuous-time constrained generalized Riccati equation arising in the context of the singular Linear Quadratic (LQ) optimal control problem. This technique allows to express the solutions of the constrained generalized Riccati equation in terms of the solutions of a reduced-order standard Riccati equation. This result is used to provide a solution to the singular LQ problem with closed-loop stability in the case when the allowed controls are restricted to be regular for any initial condition.

Singular Linear Quadratic Optimal Control Problem for Stochastic Nonregular Descriptor Systems

AbstractThis paper is concerned with the singular linear quadratic (SLQ) optimal control problem for stochastic nonregular descriptor systems with time‐delay. By means of some reasonable assumptions and a series of equivalent transformations, the problem is finally transformed into a positive linear quadratic (LQ) problem for standard stochastic systems. Then dynamic programming principle is used to establish the solvability of the original problem, and the desired explicit presentation of the optimal controller is given in terms of matrix iterative form. The results due to Feng et al. are generalized and improved. As an application, a numerical example is presented to demonstrate the efficiency of the proposed approach.

On the generalized algebraic Riccati equations

Three hundred years have passed since Jacopo Francesco Riccati analyzed a quadratic differential equation that would have been of crucial importance in many fields of engineering and applied mathematics. Indeed, countless variations and generalizations of this equation have been considered as they proved to be the right mathematical tool to address important problems. This paper is focused on a generalized version of the matrix Riccati equation where the matrix that in the classical Riccati equation is inverted can be singular: we analyze the equation obtained by substituting the inverse operator with the Moore-Penrose pseudo-inverse. The equations obtained by this substitution are known as generalized Riccati equations. The relation between these equations — both in continuos-time and in discrete-time — and singular Linear Quadratic (LQ) optimal control problem are examined. A geometric characterization of the set of solutions of the generalized Riccati equation is illustrated. It is shown that in this general setting there are LQ optimal control problems for which the optimal closed-loop system is stable also in cases where the Riccati equation does not possess a stabilizing solution.

A Linear Quadratic Controller Design Incorporating a Parametric Sensitivity Constraint

The purpose of this paper is to propose a synthesis method of parametric sensitivity constrained linear quadratic (SCLQ) controller for an uncertain linear time invariant (LTI) system. System sensitivity to parameter variation is handled through an additional quadratic trajectory parametric sensitivity term in the standard LQ criterion to be minimized. The main purpose here is to find a suboptimal linear quadratic control taking explicitly into account the parametric uncertainties. The paper main contribution is threefold: 1) A descriptor system approach is used to show that the underlying singular linear-quadratic optimal control problem leads to a non-standard Riccati equation. 2) A solution to the proposed control problem is then given based on a connection to the so-called Lur’e matrix equations. 3) A synthesis method of multiple parametric SCLQ controllers is proposed to cover the whole parametric uncertainty while degrading as less as possible the intrinsic robustness properties of each local linear quadratic controller. Some examples are presented in order to illustrate the effectiveness of the approach.

Continuous-time singular linear–quadratic control: Necessary and sufficient conditions for the existence of regular solutions

The purpose of this paper is to provide a full understanding of the role that the constrained generalized continuous algebraic Riccati equation plays in singular linear–quadratic (LQ) optimal control. Indeed, in spite of the vast literature on LQ problems, only recently a sufficient condition for the existence of a non-impulsive optimal control has for the first time connected this equation with the singular LQ optimal control problem. In this paper, we establish four equivalent conditions providing a complete picture that connects the singular LQ problem with the constrained generalized continuous algebraic Riccati equation and with the geometric properties of the underlying system.

Zero-order Approximation of Three-time Scale Singular Linear-quadratic Optimal Control Problem

This paper is devoted to the construction of a zero-order approximation of the solution of a three-time scale singular perturbed linear-quadratic optimal control problem with the help of the direct scheme method. The algorithm of the method consists in immediate substituting a postulated asymptotic expansion of solution into the problem condition and constructing a family of control problems to define the terms of the asymptotic expansion. Asymptotic approximation of the solution contains regular functions and four boundary ones of exponential type which are determined from the five linearquadratic optimal control problems. It is shown, that the system of equations for a zero-order approximation appeared from control optimality conditions of the initial perturbed problem corresponds to control optimality conditions appeared in respective five optimal control problems constructed for finding zero-order asymptotic approximation with the help of the direct scheme method. An illustrative example is given.

Singular linear quadratic optimal control for singular stochastic discrete‐time systems

SUMMARYThe finite time horizon singular linear quadratic (LQ) optimal control problem is investigated for singular stochastic discrete‐time systems. The problem is transformed into positive LQ one for standard stochastic systems via two equivalent transformations. It is proved that the singular LQ optimal control problem is solvable under two reasonable rank conditions. Via dynamic programming principle, the desired optimal controller is presented in terms of matrix iterative form. One simulation is provided to show the effectiveness of the proposed approaches. Copyright © 2012 John Wiley & Sons, Ltd.

A numerical algorithm for singular optimal LQ control systems

A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear–quadratic optimal control problems is presented. The algorithm is based on the Presymplectic Constraint Algorithm (PCA) by Gotay-Nester (Gotay et al., J Math Phys 19:2388–2399, 1978; Volckaert and Aeyels 1999) that allows to solve presymplectic Hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems (Dirac, Can J Math 2:129–148, 1950). The numerical implementation of the algorithm is based on the singular value decomposition that, on each step, allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 2 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour.

Generalized solutions for singular optimal control problems

This text presents a complete theory of existence/uniqueness and the structure of generalized solutions for singular linear-quadratic optimal control problems. Generalized optimal controls are distributions of order r and the corresponding generalized trajectories are distributions of order (r − 1). r is the “order of singularity” of the problem, an integer no greater than the dimension of the state space. Its value is obtained through a certain reduction procedure. In the final section, some perspectives and partial results concerning the extension of these results to nonlinear problems are briefly discussed.

Singular linear-quadratic optimal control problem for a class of discrete singular systems with multiple time-delays

The singular linear-quadratic optimal control problem for a class of discrete-time linear singular systems with multiple time-delays is investigated. The equivalent relationship between this problem and the non-singular linear-quadratic problem for standard state-space discrete-time linear systems without time-delay is derived, and the necessary and sufficient condition guaranteeing this equivalent relationship is also given. The optimal control is synthesized as state feedback, and the closed-loop system is regular, impulse-free and without time-delay. An example is given to show the validity of the proposed method.

Singular optimal control problem of linear singular systems with linear-quadratic cost *

This paper investigates the singular linear-quadratic optimal control problem of a kind of linear time-in variant singular systems containing impulse mode. The relation between this problem and the linear-quadratic optimal control problem of standard state-space linear systems is established by using an equivalent transformation based on the Weierstrass canonical form of singular systems. The existence and uniqueness of the solution, the method to compute the optimal control and corresponding state trajectory, and the feedback synthesis of the optimal control of this problem is also given

Singular Linear-Quadratic Optimal Control for Single-Input Linear Digital Control Systems.

This paper discusses the singular linear-quadratic (LQ) optimal control for the single-input linear digital system. Since the performance index does not contain the input explicitly, the optimal control becomes singular. First, by using the property of the linear digital system, the singular performance index is transformed into a nonsingular (normal) one. Then, singular LQ optimal controls for the free-end-point problem, for the fixed-end-point problem, and for the infinite control time problem are obtained. They are expressed by the state feedback with the bounded gains. Furthermore, the sufficiency of the singular LQ optimal control is shown. Finally, the relationship between the singular LQ optimal control and the nonsingular one is described.

The Quadratic Matrix Inequality in Singular $H_\infty $ Control with State Feedback

In this paper the standard $H_\infty $ control problem using state feedback is considered. Given a linear, time-invariant, finite-dimensional system, this problem consists of finding a static state feedback such that the resulting closed-loop transfer matrix has $H_\infty $ norm smaller than some a priori given upper bound. In addition it is required that the closed-loop system is internally stable. Conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality, reminiscent of the dissipation inequality of singular linear quadratic optimal control. Where the direct feedthrough matrix of the control input is injective, the results presented here specialize to known results in terms of solvability of a certain indefinite algebraic Riccati equation.

Existence, uniqueness, and stability of solutions to singular linear quadratic optimal control problems

We consider general continuous and discrete time linear quadratic optimal control problems, where the dynamics of the system is a linear differential or difference algebraic equation with constant coefficient matrices and the cost functionals are defined by infinite integrals or sums. These problems occur in numerous applications in many different fields, such as economics, operations research, electrical engineering, model and system design, biology, and chemistry, since they form two of the simplest optimal control problems, and they are also used as linear approximations for many nonlinear problems, which are much harder to solve. We extend some of the known results to these general problems.

Riccati equation solution of the minimum time output control problem

By taking a geometrical approach, and by using a type of orthogonality in constructing bases for the k th controllable subspaces, a general solution is derived for the multivariable minimum time output control problem. The solution is easier to implement than solutions derived by traditional approaches to output control, and in fact it is shown that the optimal control sequence may be constructed from the solution to a specialized Riccati equation. This new approach thus relates geometrical approaches to the problem to the previously known result that output control is just a special case of the singular linear quadratic optimal control problem.