Generalized Algebraic Riccati Equation Research Articles

Infinite horizon linear quadratic optimal control for discrete‐time stochastic systems

AbstractThis paper is concerned with the infinite horizon linear quadratic optimal control for discrete‐time stochastic systems with both state and control‐dependent noise. Under assumptions of stabilization and exact observability, it is shown that the optimal control law and optimal value exist, and the properties of the associated discrete generalized algebraic Riccati equation (GARE) are also discussed. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

A numerical algorithm for finding solution of sign-indefinite algebraic Riccati equations for general multiparameter singularly perturbed systems

In this paper, a computational algorithm for solving sign-indefinite general multiparameter algebraic Riccati equation (SIGMARE) that arises in the H ∞ filtering problem is investigated. After establishing the asymptotic structure of the solution of the SIGMARE, in order to solve the SIGMARE, Newton’s method and two fixed point algorithms are combined. As a result, the new iterative algorithm achieves the quadratic convergence property and succeeds in reducing the computing workspace dramatically. As another important feature, the convergence criteria for small parameters ε i is derived for the first time. Moreover, it is shown that the uniqueness and positive semidefiniteness of the convergence solutions are guaranteed in the neighborhood of the initial conditions.

Order reduction of discrete-time algebraic Riccati equations with singular closed loop matrix

We study the general discrete-time algebraic Riccati equation and deal with the case where the closed loop matrix corresponding to an arbitrary solution is singular. In this case the extended symplectic pencil associated with the DARE has 0 as a characteristic root and the corresponding spectral deflating subspace gives rise to a subspace where all solutions of the DARE coincide. This allows for a reduction of the original DARE to an equation of smaller size.

H ∞ control for descriptor systems: Controller reduction

State‐space formulas for reduced‐order H ∞ controllers for descriptor systems are given. The approach taken is mainly based on the solutions of two generalized algebraic Riccati equations (GARE), while exploiting the structure of the deflating subspace of the pencil {E, W ∞}, where W ∞ is an admissible solution to a GARE that is the descriptor systems counterpart of a certain ARE (algebraic Riccati equation) first developed by Petersen et al. (1991). This approach has the advantage that, by proper selection of the bases of the deflating space and suitable assumptions, the reduced‐order controller may be in a normal form, namely the E‐matrix of the controller is nonsingular.

A Numerical Method for a Generalized Algebraic Riccati Equation

In this paper we develop a numerical method for computing the semistabilizing solution of a generalized algebraic Riccati equation (GARE). The semistabilizing solution of such a GARE has been used to characterize the solvability of the $(J, J^\prime)$‐spectral factorization problem for general rational matrices which have poles and zeros on the extended imaginary axis. The main difficulty for solving such a GARE is that its associated skew‐Hamiltonian/Hamiltonian pencil has eigenvalues on the extended imaginary axis; consequently, it is not clear which eigenspace of the associated skew‐Hamiltonian/Hamiltonian pencil can characterize the desired semistabilizing solution; i.e., it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute, and hence the well‐known generalized eigenvalue approach for the classical algebraic Riccati equations cannot be directly employed for it. Our proposed method consists of computations of the eigendecomposition of the system pencil corresponding to the eigenvalues on the extended imaginary axis and the stable eigenspace of an augmented matrix pencil; hence, it is a generalization of the generalized eigenvalue approach for the classical algebraic Riccati equations.

The semi-stabilizing solution of generalized algebraic Riccati equation for descriptor systems

This paper considers the generalized algebraic Riccati equation (GARE) associated with a non-standard J-spectral factorization problem (NS-J-SFP) for descriptor systems, of which system matrices have full column rank. We derive necessary and sufficient conditions for the existence of a semi-stabilizing solution of the GARE based on the Hamiltonian matrix pencil approach and gives a parametrization of all the semi-stabilizing solutions. Then we provide a nonstandard J-spectral factor (NS-J-SF) in terms of a particular semi-stabilizing solution of the GARE. A simple numerical result is also included.

Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation – called a generalized algebraic Riccati equation (GARE) – is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a “stabilizing solution” to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.

Homogeneous Projective Transformation and Scaling of a General Quadratic Algebraic Matrix Riccati Equation

The matrix formulation and solution to the problems of homogeneous projective transformation and scaling of a generalized quadratic algebraic matrix Riccati equation is presented. The method presented is independent from the numerical algorithms used for the numerical solution of the equation under study. In the present paper, the proposed method is applied to probability-l homotopy algorithms for the numerical solution of the equation under study. The Advantages and disadvantages of the proposed method are discussed through a numerical example.

Numerical Solution of a General Quadratic Algebraic Matrix Riccati Equation via Probability-1 Homotopy Algorithms

The design of an algorithm for the numerical solution of a generalized quadratic algebraic matrix Riccati equation is presented. The approach is based on probability-1 homotopy methods. The algorithm is illustrated with numerical examples.

Standard J-spectral factorization of rational matrices via generalized algebraic riccati equation

This paper considers a J-spectral factorization problem for a rational spectral matrix. The problem is called the standard J-spectral factorization problem (S-J-SEP) since it is applicable to the standard H∞ control for a descriptor system (DS). A solution to the S-J-SFP is provided in terms of a stabilizing solution of an indefinite generalized algebraic Riccati equation (GARE). The main result of this paper includes the well-known result on the canonical J-spectral factorization problem (C-J-SFP).

On the generalized algebraic Riccati equation for continuous-time descriptor systems

In this paper we consider the generalized algebraic Riccati equation (GARE) for a continuous-time descriptor system. Necessary and sufficient conditions for the existence of stabilizing solutions of the GARE are derived based on the Hamiltonian matrix pencil approach. A parametrization of all stabilizing solutions is also provided. The main result has a potential applicability to a wide class of control problems for a descriptor system, including H 2/ H ∞ controls.

On the Strong Solutions and Maximal Solutions of Generalized Algebraic Riccati Equations

A comparison theorem for the solutions of two generalized algebraic Riccati equations (GAREs) coming from two different systems is presented in this paper. It is then shown that the so-called strong solutions, whose related pencils have all their finite eigenvalues in the closed left half plane, are maximal. The results obtained in this paper generalize the existing monotonicity results of algebraic Riccati equations. An application of the results is the derivation of the parameterization of all strong solutions of the GARE related to the singular spectral factorization of a proper transfer function with finite and infinite imaginary axis zeros.

Standard J-Spectral Factorization of Rational Matrices via Generalized Algebraic Riccati Equation

In this paper we consider a J-spectral factorization problem for a rational spectral matrix. The problem considered here is called the standard J-spectral factorization problem (S-J-SFP) since it is applicable to the standard H∞ control problem for a descriptor system (DS). Based on DS representation of the spectral matrix, we derive a solution to the S-J-SFP using a stabilizing solution of an indefinite generalized algebraic Riccati equation (GARE). The main result of this paper includes the well-known solution to the canonical J-spectral factorization problem (C-J-SFP).

The Dissipation Inequality and Generalized Algebraic Riccati Equation for Linear Quadratic Control Problem of Descriptor System

We consider the infinite-horizon linear quadratic control problem for a descriptor system (DLQCP) based on the theory of dissipative system. First, we derive a dissipative inequality for the descriptor system satisfied by the optimal cost of DLQCP. This implies that the optimal cost is characterized by the solution of a linear matrix iniquality (LMI). Secondly, we show that the solution of LMI characterizing the optimal cost also satisfies a related generalized algebraic Riccati equation (GARE) if DLQCP is regular. Finally, we derive the optimal solution to DLQCP with fixed terminal condition.

(J, J?)-Spectral factorization and conjugation for discrete-time descriptor system

This paper considers a (J, J?)-spectral factorization of a signed spectral matrix associated with a discrete-time improper transfer matrix by using the descriptor system representation and the generalized algebraic Riccati equation (GARE). Under the assumption that the improper transfer matrix is stable, a numerical algorithm for (J, J?)-spectral factorization is developed based on solutions of the generalized eigenvalue problem (GEP) and a related matrix equation. For an unstable transfer matrix, aJ-unitary conjugation method is applied to obtain a stable transfer matrix generating the same signed spectral matrix. Then a Hankel norm model reduction problem is briefly discussed. A simple example is also included to illustrate the numerical procedure.

The Dissipation Inequality and Generalized Algebraic Riccati Equation for Linear Quadratic Control Problem of Descriptor System

We consider the infinite-horizon linear quadratic control problems (LQCPs) for a descriptor system based on the theory of dissipative system. We derive a dissipation inequality (DI) for the descriptor system satisfied by the optimal costs. It is shown that the optimal costs can be characterized by the solution of the LMI and that a related generalized algebraic Riccati equation (GARE) has solutions under the regularity condition. Finally, the optimal solution to the LQCP with zero terminal condition is derived.

On the existence of solutions of the general algebraic Riccati equation

This paper addresses the problem of real symmetric solutions of the general algebraic Riccati equation (GARE) with an indefinite quadratic term. The GARE arises in linear quadratic differential games, in the stabilization of uncertain systems, robust optimal control and disturbance attenuation problems. Using the properties of the solutions of the differential equation corresponding to the GARE and the related conclusions of differential games, we have established the main results of this paper. Theorems 1 and 2

Design of coupled decentralized output regulators

Decentralized static output regulators are designed via projective controls in cases where the controllers are directly coupled, i.e. when the controls of other decision-makers appear in the local outputs. For two decision-makers, the solution is reduced to the problem of solving a general (non-square) algebraic Riccati equation. The existence and uniqueness of the solution is proved and then determined analytically. These results are then generalized to an arbitrary number of decision-makers. Finally, the possibilities of the sequential design of static and dynamic regulators for this type of decentralized information structure are investigated. A numerical example illustrates the design procedure.