Positive Semidefinite Solution Research Articles

Continuous Time Algebraic Riccati Equation Solution for Second Order Systems

Abstract The continuous time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second order structure, the ARE required to be solved to get the control values in standard control problems results in complex sub-equations in terms of the second order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second order system, due to the eigendecomposition of the 2n x 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n x 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, positive semi-definite solution matrix).

On discrete-time dissipative port-Hamiltonian (descriptor) systems

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman–Yakubovich–Popov inequalities is also studied.

Distributional Robustness in Minimax Linear Quadratic Control with Wasserstein Distance

To address the issue of inaccurate distributions in discrete-time stochastic systems, a minimax linear quadratic control method using the Wasserstein metric is proposed. Our method aims to construct a control policy that is robust against errors in an empirical distribution of underlying uncertainty by adopting an adversary that selects the worst-case distribution at each time. The opponent receives a Wasserstein penalty proportional to the amount of deviation from the empirical distribution. As a tractable solution, a closed-form expression of the optimal policy pair is derived using a Riccati equation. We identify nontrivial stabilizability and observability conditions under which the Riccati recursion converges to the unique positive semidefinite solution of an algebraic Riccati equation. Our method is shown to possess several salient features, including closed-loop stability, a guaranteed-cost property, and a probabilistic out-of-sample performance guarantee.

On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $

<abstract><p>Nonlinear matrix equation often arises in control theory, statistics, dynamic programming, ladder networks, and so on, so it has widely applied background. In this paper, the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ are discussed, where operator $ F $ are defined in the set of all $ n\times n $ positive semi-definite matrices, and $ Q $ is a positive definite matrix. Sufficient conditions for the existence and uniqueness of a positive semi-definite solution are derived based on some fixed point theorems. It is shown that under suitable conditions an iteration method converges to a positive semi-definite solution. Moreover, we consider the perturbation analysis for the solution of this class of nonlinear matrix equations, and obtain a perturbation bound of the solution. Finally, we give several examples to show how this works in particular cases, and some numerical results to specify the rationality of the results we have obtain.</p></abstract>

Positive semidefinite solution to matrix completion problem and matrix approximation problem

In this paper, firstly, we discuss the following matrix completion problem in the spectral norm: ?(A B B* X)?2 < 1 subject to (A B B* X) ? 0. The feasible condition for the above problem is established, in this case, the general positive semidefinite solution and its minimum rank are presented. Secondly, applying the result of the above problem, we also study the matrix approximation problem: ?A ? BXB*?2 < 1 subject to A ? BXB* ? 0, where A ? Cm?m?, B ? Cm?n, and X ? Cn?n?.

Optimal Preview Controller for Linear Discrete-Time Systems: A Virtual System Method

In this paper, a new method for the design of the preview controller for a class of discrete-time systems is proposed based on the virtual system. Firstly, by taking the known future reference signal as the output, the virtual system with similar structures to the controlled system is constructed. Then, the augmented error system is received by translating the controlled system to it and by integrating the error equation. Thus, the tracking problem of the controlled system is transformed into the regulation problem of the augmented error system. Secondly, in view of the minimum principle, the optimal controller of the augmented error system is acquired, and the preview controller of the controlled system is also gained. Further, by discussing the stabilizability and detectability of the augmented error system, the conditions for the existence of the unique positive semidefinite solution to an algebraic Riccati equation are obtained. By using the method in this paper, making difference and dimension expansion for the state equation in designing the augmented error system is avoided and the output can track the reference signals better. Finally, the numerical simulation shows the effectiveness of the proposed controller.

Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties

This note is devoted to a robust stability analysis, as well as to the problem of the robust stabilization of a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise type with unknown intensity. In order to effectively address the multi-perturbations case, we use scaling techniques. These techniques allow us to obtain an estimation of the lower bound of the stability radius. A first characterization of a lower bound of the stability radius is obtained in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations. A second characterization is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities. This second result is then exploited in order to solve a robust control synthesis problem.

An iterative algorithm for coupled Riccati equations in continuous-time Markovian jump linear systems

In this paper, a novel implicit iterative algorithm with some tuning parameters is developed to solve the coupled algebraic Riccati matrix equation arising in the continuous-time Markovian jump linear systems. By introducing some tuning parameters in the proposed iterative algorithm, the current estimation for unknown variables is updated by using the information not only in the last step but also in the current iterative step and previous iterative steps. These tuning parameters can be appropriately chosen such that the proposed algorithm has faster convergence performance than some previous algorithms. It is shown that the proposed algorithm with zero initial conditions can monotonically converge to the unique positive semidefinite solution of the coupled Riccati matrix equation if the corresponding Markovian jump system is stabilisable. Finally, an example is provided to show the effectiveness of the developed algorithm.

On the convergence and stability of fractional singular Kalman filter and Riccati equation

In our recent paper [1], we have developed a simple algorithm for the state estimation of discrete-time linear stochastic fractional-order singular (FOS) systems based on the deterministic least squares method. In this paper, we shall consider the asymptotic properties of the obtained generalized Riccati equation and the associated state estimator. The approach is based on the analysis in detail of the derived filter and algebraic Riccati equation by generalizing significantly the results for standard causal systems, in which provide us conditions for the stability and convergence of the fractional singular Kalman filter (FSKF). Also, conditions under which the Riccati equation has a unique positive semi-definite (PSD) solution are given. Towards this, stability studies of discrete-time FOS state-space systems are investigated, and detectability and stabilizability criterions have been derived. Some examples are performed and verified using numerical simulations in order to illustrate the given analysis.

Positive Realness of Second-order and High-order Descriptor Systems

This paper is concerned with the positive realness problem for second-order and high-order descriptor systems. First, without any linearization, necessary and sufficient conditions are established under which the second-order descriptor systems are strictly positive real and extended strictly positive real, respectively. Applying the relations between the positive realness and the optimal control theory, the solutions of the proposed positive real lemma equations can be represented by the symmetric positive semi-definite solutions for the second-order generalized Riccati equations. Then, employing polynomial matrix decomposition techniques, the extended strictly positive real lemma of high-order descriptor systems is also presented based on the original coefficient matrices of the system. Furthermore, linear matrix inequality conditions are given that can effectively test the positive realness and the extended strictly positive realness of the system. Finally, three numerical examples are provided to verify the effectiveness of the developed theoretical results.

Minimum rank positive semidefinite solution to the matrix approximation problem in the spectral norm

In this paper, we discuss the following minimum rank matrix approximation problem in the spectral norm: minX⩾0r(X)subject to‖A−BXB∗‖2=min, where A∈ℂ⩾m×m and B∈ℂm×n. By using the positive-semidefinite-type generalized singular value decomposition, we derive the expressions of the minimum rank and the minimum rank positive semidefinite solution to the above matrix approximation problem.

Iterative Algorithms for Symmetric Positive Semidefinite Solutions of the Lyapunov Matrix Equations

It is well-known that the stability of a first-order autonomous system can be determined by testing the symmetric positive definite solutions of associated Lyapunov matrix equations. However, the research on the constrained solutions of the Lyapunov matrix equations is quite few. In this paper, we present three iterative algorithms for symmetric positive semidefinite solutions of the Lyapunov matrix equations. The first and second iterative algorithms are based on the relaxed proximal point algorithm (RPPA) and the Peaceman–Rachford splitting method (PRSM), respectively, and their global convergence can be ensured by corresponding results in the literature. The third iterative algorithm is based on the famous alternating direction method of multipliers (ADMM), and its convergence is subsequently discussed in detail. Finally, numerical simulation results illustrate the effectiveness of the proposed iterative algorithms.

Stabilization Control for Linear Continuous-Time Mean-Field Systems

This paper investigates the stabilization and control problems for linear continuous-time mean-field systems (MFS). Under standard assumptions, necessary and sufficient conditions to stabilize the mean-field systems in the mean square sense are explored for the first time. It is shown that, under the assumption of exact detectability (exact observability), the mean-field system is stabilizable if and only if a coupled algebraic Riccati equation (ARE) admits a unique positive semi-definite solution (positive definite solution), which coincides with the classical stabilization results for standard deterministic systems and stochastic systems. One of the key techniques in the paper is the obtained solution to the forward and backward stochastic differential equation (FBSDE) associated with the maximum principle for an optimal control problem. Actually, with the analytical FBSDE solution, a necessary and sufficient solvability condition of the optimal control, under mild conditions, is derived. Accordingly, the stabilization condition is presented by defining an Lyaponuv functional via the solution to the FBSDE and the optimal cost function. It is worth of pointing out that the presented results are different from the previous works for stabilization and also different from the works on optimal control.

Optimal Stabilization Control for Discrete-Time Mean-Field Stochastic Systems

This paper will investigate the stabilization and optimal linear quadratic (LQ) control problems for infinite horizon discrete-time mean-field systems. Unlike the previous works, for the first time, the necessary and sufficient stabilization conditions are explored under mild conditions, and the optimal LQ controller for infinite horizon is designed with a coupled algebraic Riccati equation (ARE). More specifically, we show that under the exact detectability (exact observability) assumption, the mean-field system is stabilizable in the mean square sense with the optimal controller if and only if a coupled ARE has a unique positive semidefinite (positive definite) solution. The presented results are parallel to the classical results for the standard LQ control.

A Positive Real Lemma Type-Result with Respect to the Anisotropic Norm for Stochastic Systems with Multiplicative Noise

Abstract The passivity of discrete–time linear stochastic systems with multiplicative noise is considered, where the systems inputs have bounded anisotropy. Using first principles analysis applying completing the square arguments, it is proved that such systems are passive if a specific Riccati equation has a stabilizing positive semidefinite solution satisfying two additional conditions. The theoretical results are illustrated by a simple numerical example.

A bounded real lemma type-result with respect to the anisotropic norm setup for stochastic systems with multiplicative noise

The anisotropic norm of discrete-time linear stochastic systems with state dependent noise is considered. Using first principles analysis applying completing the square arguments, it is proved that the anisotropic norm of such systems is upper bounded by a given positive real scalar, if a specific Riccati equation has a stabilizing positive semidefinite solution satisfying two additional conditions. It is shown that these conditions are sufficient and necessary for the boundedness of the anisotropic norm. Numerical algorithms to determine the stabilizing solution of this Riccati equation allowing thus to compute the anisotropic norm of stochastic systems with multiplicative noise are also presented. The theoretical results are illustrated by numerical examples.

Filtering [formula omitted]-coupled algebraic Riccati equations for discrete-time Markov jump systems

This paper deals with a set of S-coupled algebraic Riccati equations that arises in the study of filtering of discrete-time linear jump systems with the Markov chain in a general Borel space S. By S-coupled it is meant that the algebraic Riccati equations are coupled via an integral over S. Conditions for the existence and uniqueness of a positive semi-definite solution to the filtering S-coupled algebraic Riccati equations are obtained in terms of the concepts of stochastic detectability and stochastic stabilizability. This result is then applied to solve the infinite horizon minimum mean square linear Markov jump filtering problem. The obtained results generalize previous ones in the literature, which considered only the case of the Markov chain taking values in a finite state space.

Comments on “On the Necessity of Diffusive Couplings in Linear Synchronization Problems With Quadratic Cost”

In this note, we want to comment on the recent paper “On the necessity of diffusive couplings in linear synchronization problems with quadratic cost” ( IEEE Transactions on Automatic Control , vol. 60, pp. 3029–3034, 2015) by Montenbruck et al. on the necessity of diffusiveness of optimal control laws in linear quadratic control problems in the context of synchronization. In the above paper, the authors concentrate on the optimal feedback law associated with the maximal solution of the underlying algebraic Riccati equation (ARE). In this comment, we argue that it is more natural to use the smallest positive semidefinite solution of the ARE. Our approach generalizes the results in the above paper to the case in which the agent dynamics is allowed to have eigenvalues in the open right half plane. Moreover, our proof considerably simplifies the one in the above paper, as it avoids the analysis of the Riccati differential equation. In addition, we propose and solve the zero endpoint version of the linear quadratic problem studied in the above paper.

Stabilizing Solution and Parameter Dependence of Modified Algebraic Riccati Equation With Application to Discrete-Time Network Synchronization

This technical note deals with a modified algebraic Riccati equation (MARE) and its corresponding inequality and difference equation, which arise in modified optimal control and filtering problems and are introduced into the cooperative control problems recently. The stabilizing property of the solution to MARE is presented. Then, the uniqueness is proved for the almost stabilizing and positive semi-definite solution. Next, the parameter dependence of MARE is analyzed. An obtained parameter dependence result is finally applied to the study of semi-global synchronization of leader-following networks with discrete-time linear dynamics subject to actuator saturation.

Dykstra’s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations

Dykstra’s alternating projection algorithm was proposed to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. In this paper, we first apply Dykstra’s alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations AXB = E, CXD = F. If we choose the initial iterative matrix X0 = 0, the least Frobenius norm symmetric positive semidefinite solution of these matrix equations is obtained. A numerical example shows that the new algorithm is feasible and effective.