Continuous Algebraic Riccati Equation Research Articles

A unified study of the redundant control input problem with different conditions in the stabilizable case

This paper investigates the redundant control input problem in the sense of linear quadratic regulator optimal control. Increasing redundant control inputs leads to a decrease in the quadratic performance index. However, in some cases, this also increases the norm of the feedback matrix, resulting in actuator saturation. To solve the problem, a comparative analysis of the latest research results on the redundant control input problem under different conditions is first presented. Secondly, new tighter upper bounds of the positive (semi-)definite solution of the continuous algebraic Riccati equation (CARE) corresponding to the redundant control input problem are studied uniformly in the stabilizable case. Thirdly, based on them, a class of sufficient conditions to guarantee that the norm of the feedback matrix decreases is presented. Finally, numerical experiments demonstrate the effectiveness and superiority of our conclusions. Meanwhile, a simulation experiment demonstrates that systems with redundant control inputs can achieve good control performance.

High-performance LQR control for continuous time systems with redundant control inputs

This paper investigates the linear quadratic regulator (LQR) problem for continuous-time linear systems with redundant control inputs. The LQR control problem has been widely studied, but due to technical difficulties, there is little advance in the study of adding input redundancies. To address this problem, a convergent matrix series, the limit of a monotonically decreasing sequence, is first presented as an upper bound of the symmetric positive definite solution of a class of continuous algebraic Riccati equations (CAREs). Compared with the existing studies on this topic, the obtained bounds are more precise. Moreover, by utilising these upper bounds in the LQR problem of adding input redundancies, a class of sufficient conditions is presented to decrease the quadratic performance index. Finally, the corresponding numerical examples are given to illustrate the effectiveness of our results and compare them with the existing results. Meanwhile, some simulation experiments show that systems with redundant control inputs have good control performance.

New Upper Solution Bounds of a Class of the Continuous Algebraic Riccati Equations and Their Application in Redundant Control Inputs Systems

New Upper Solution Bounds of a Class of the Continuous Algebraic Riccati Equations and Their Application in Redundant Control Inputs Systems

The Continuous-Time Singular LQR Problem and the Riddle of Nonautonomous Hamiltonian Systems: A Behavioral Solution

In this article, we deal with the continuous-time singular linear quadratic regulator (LQR) problems, which give rise to nonautonomous Hamiltonian systems. This case arises when the system’s transfer function matrix is not left-invertible. A special case of this problem can be solved using the constrained generalized continuous algebraic Riccati equation (CGCARE), when a certain condition on the input-cardinality of the Hamiltonian is satisfied. However, this condition is only a special case among many other possible cases. On the other hand, singular LQR problems with autonomous Hamiltonian systems have been well studied in the literature. In this article, we apply behavioral theoretic techniques to show that the general case of the singular LQR problem with nonautonomous Hamiltonian can be solved by a direct sum decomposition of the plant behavior, where one of the direct summands can be solved via CGCARE, while the other gives rise to an autonomous Hamiltonian system.

On an upper bound for the eigenvalues of the solution of the continuous algebraic Riccati equation

Under the assumption that A+AT is positive semi-definite, Liu et al. (2010) proposed an upper bound for the eigenvalues of the solution K of the continuous algebraic Riccati equation ATK+KA−KRK=−Q,where Q is positive semi-definite and R is positive definite. In this paper, we complement Liu et al.’s result without assuming the positivity of A+AT. A possible improvement is also proposed and a numerical example is presented to illustrate the new result.

Modeling and Estimation of Multivariate Discrete and Continuous Time Stationary Processes

In this paper, we give an autoregressive model of order 1 type of characterization covering all multivariate strictly stationary processes indexed by the set of integers. Consequently, under square integrability, we derive continuous time algebraic Riccati equations for the parameter matrix of the characterization. This provides us with a natural way to define the corresponding estimator. In addition, we show that the estimator inherits consistency from autocovariances of the stationary process. Furthermore, the limiting distribution is given by a linear function of the limiting distribution of the autocovariances. We also present the corresponding existing results of the continuous time setting paralleling them to the discrete case treated in this paper.

Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

New estimates of upper bounds for the solutions of the continuous algebraic Riccati equation and the redundant control inputs problems

In this paper, according to the properties of the given matrix of the continuous algebraic Riccati equation (CARE), a new positive semi-definite matrix is constructed. And then basing on some significant properties of the matrix eigenvalue and eigenvalue inequality, using inequality techniques and formula transformation and the new positive semi-definite matrix, a class of new upper bounds of the CARE is obtained. Consequently, in redundant control inputs system, by the derived bounds, inequality techniques, and the important property of the matrix eigenvalue and singular value, we present some new sufficient conditions to guarantee the decrease of controller gain when increasing the control input, which are more adaptable than previous results. Finally, we show the effectiveness of the results by numerical examples.

The Structure-preserving Doubling Numerical Algorithm of the Continuous Coupled Algebraic Riccati Equation

In the paper, we apply a structure-preserving doubling algorithm to solve the continuous coupled algebraic Riccati equation (CCARE). Using the existence and uniqueness of the CCARE, we show that the iteration solution of the CCARE are positive semi-definite, symmetric, and unique. Further, we discuss the convergent analysis of the structure-preserving doubling algorithm. Moreover, we present two modified structure-preserving doubling algorithms. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived numerical algorithms.

KRYLOV SUBSPACE METHODS OF HESSENBERG BASED FOR ALGEBRAIC RICCATI EQUATION

In this paper, we propose a class of special Krylov subspace methods to solve continuous algebraic Riccati equation (CARE), i.e., the Hessenberg-based methods. The presented approaches can obtain efficiently the solution of algebraic Riccati equation to some extent. The main idea is to apply Kleinman-Newton's method to transform the process of solving algebraic Riccati equation into Lyapunov equation at every inner iteration. Further, the Hessenberg process of pivoting strategy combined with Petrov-Galerkin condition and minimal norm condition is discussed for solving the Lyapunov equation in detail, then we get two methods, namely global generalized Hessenberg (GHESS) and changing minimal residual methods based on the Hessenberg process (CMRH) for solving CARE, respectively. Numerical experiments illustrate the efficiency of the provided methods.

New solution bounds of the continuous algebraic Riccati equation and their applications in redundant control input systems

Redundant control inputs have wide applications in engineering, and have significant effects in quadratic performance optimal control, H2 control, tracking control, and many uncertain systems. In this paper, using some properties related to the inequality of the special matrix and the transformation of the matrix equation, we present new bounds of the solution of the continuous algebraic Riccati equation. The new bounds are more precise than those proposed in some existing studies on this topic. Subsequently, two lower bounds are presented for the optimal controller gain of the system (1), and a new upper bound is obtained for the optimal controller gain of the extended control system (3). Several examples are provided to demonstrate that our bounds of the optimal controller gain are superior to some previous results. Subsequently, we provide the applications of the new bounds in relevant redundant control problems and obtain good results. Finally, the corresponding numerical examples are provided to illustrate the effectiveness of our results and to compare them with the existing results.

Constrained Generalized Continuous Algebraic Riccati Equations (CGCAREs) Are Generically Unsolvable

In this letter, we show that a constrained generalized continuous algebraic Riccati equation (CGCARE), arising in the context of a singular infinite-horizon linear quadratic regulator (LQR) problem, is generically 1 unsolvable. We achieve this result in three steps: first, we provide a set of necessary and sufficient conditions for the solvability of a CGCARE. Then, using these conditions, we show that the determinant of a certain matrix pencil being identically zero is necessary for a CGCARE to be solvable. Finally, we show that this necessary condition is generically false. It is well-known that CGCARE solvability is a necessary and sufficient condition for a singular infinite-horizon LQR problem to admit a solution that is implementable as a static state-feedback control law. Thus, our main result reveals that a singular infinite-horizon LQR problem generically disallows solution by a static state-feedback law.

The geometry of the generalized algebraic Riccati equation and of the singular Hamiltonian system

This paper analyses the properties of the solutions of the generalized continuous algebraic Riccati equation from a geometric perspective. This analysis reveals the presence of a subspace that may provide an appropriate degree of freedom to stabilize the system in the related optimal control problem even in cases where the Riccati equation does not admit a stabilizing solution.

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.

The H∞ Algebraic Riccati Equations Arising in Linear Systems with Markov Jump

We consider a set of continuous algebraic Riccati equations with indefinite quadratic parts that connected to the stochastic controlled system with the continuous Markov process. We consider iterative methods to compute the stabilizing solution of a set of Riccati equations in order to find the equilibrium point of the two person zero-sum stochastic differential games. Our consideration can be considered as an extension of the methods derived in [1, 2]. The convergence properties of introduced methods are investigated and they are numerically compared .... Keywords: H∞ Optimal Control Problem, Indefinite Sign, Stabilizing Solution.

Inexact Newton’s method with inner implicit preconditioning for algebraic Riccati equations

Continuous algebraic Riccati equations (CARE) appear in several important applications. A suitable approach for solving CARE, in the large-scale case, is to apply Kleinman–Newton’s method which involves the solution of a Lyapunov equation at every inner iteration. Lyapunov equations are linear, nevertheless, solving them requires specialized techniques. Different numerical methods have been designed to solve them, including ADI and Krylov-type iterative projection methods. For these iterative schemes, preconditioning is always a difficult task that can significantly accelerate the convergence. We present and analyze a strategy for solving CARE based on the use of inexact Kleinman–Newton iterations with an implicit preconditioning strategy for solving the Lyapunov equations at each inner step. One advantage is that the Newton direction is approximated implicitly, avoiding the explicit knowledge of the given matrices. Only the effect of the matrix–matrix products with the given matrices is required. We present illustrative numerical experiments on some test problems.

A preconditioned block Arnoldi method for large scale Lyapunov and algebraic Riccati equations

In the present paper, we propose a preconditioned Newton---Block Arnoldi method for solving large continuous time algebraic Riccati equations. Such equations appear in control theory, model reduction, circuit simulation amongst other problems. At each step of the Newton process, we solve a large Lyapunov matrix equation with a low rank right hand side. These equations are solved by using the block Arnoldi process associated with a preconditioner based on the alternating direction implicit iteration method. We give some theoretical results and report numerical tests to show the effectiveness of the proposed approach.

<i>H</i><sub>&#8734</sub> Optimal Control Problems for Jump

We consider a set of continuous algebraic Riccati equations with indefinite quadratic parts that arise in H￥ control problems. It is well known that the approach for solving such type of equations is proposed in the literature. Two matrix sequences are constructed. Three effective methods are described for computing the matrices of the second sequence, where each matrix is the stabilizing solution of the set of Riccati equations with definite quadratic parts. The acceleration modifications of the described methods are presented and applied. Computer realizations of the presented methods are numerically compared. In addition, a second iterative method is proposed. It constructs one matrix sequence which converges to the stabilizing solution to the given set of Riccati equations with indefinite quadratic parts. The convergence properties of the second method are commented. The iterative methods are numerically compared and investigated.

On the detectability and observability of continuous stochastic Markov jump linear systems

This paper is concerned with the detectability and observability of continuous linear stochastic systems subject to Markov jump. By introducing some operators, we investigate the relationship between some different concepts of detectability and observability, and obtain their sufficient and necessary conditions. Continuous-time stochastic Lyapunov equations and stochastic continuous algebraic Riccati equations are also discussed to serve as analysis tools.

Two new upper bounds of the solution for the continuous algebraic Riccati equation and their application

In this paper, for the solution of the continuous algebraic Riccati equation (CARE), we derived two new upper matrix bounds. Compared with the existing results, the newly obtained bounds are less conservative and more practical, which means that the condition for the existence of the upper bounds derived here is much weaker. The advantage of the results is shown by theoretical analysis and numerical examples. Moreover, in redundant optimal control, when we increase the columns of the input matrix, some sufficient conditions are presented to strictly decrease the largest singular value of the feedback matrix by utilizing these upper bounds. We also give some examples to illustrate the effectiveness of these sufficient conditions.