Unique Positive Definite Solution Research Articles

Modified Riccati iterative algorithms for stochastic coupled algebraic Riccati equations of linear stochastic Markovian jump systems

AbstractThis paper investigates the stochastic coupled algebraic Riccati equations (SCAREs) associated with linear stochastic Markovian jump systems. First, two existing algorithms utilized to solve the SCAREs are reviewed and discussed. Second, based on the analysis of these two algorithms, a modified Riccati iterative algorithm is proposed with a tuning parameter added. It is verified that the proposed algorithm with zero initial values converges to the unique positive definite solution of the SCAREs with fast speed. Furthermore, two numerical examples are given to verify the efficiency and superiority of the proposed algorithm.

Robust gradient‐based neural networks for solving online the discrete periodic Lyapunov matrix equations

AbstractHere, a gradient‐based neural network (GNN) model is constructed for solving the discrete periodic Lyapunov matrix equation (DPLME) associated with discrete‐time linear periodic systems. In practical applications, the recurrent neural network model should not only converge rapidly, but also be able to tolerate noise. However, the influence of noise on GNN models was seldom considered in the past. In order to improve the convergence and robustness of the GNN model, a novel type of non‐linear activation function is applied to the GNN model. Compared with the traditional activation functions, the activation function used here makes the GNN model to achieve fixed‐time convergence. Besides, when disturbed by bounded noise, the unique positive definite solution of the DPLME can still be obtained by using the GNN model. Finally, simulation experiment is performed to verify the effectiveness and superiority of the proposed GNN model.

Unique positive definite solution of non-linear matrix equation on relational metric spaces

Unique positive definite solution of non-linear matrix equation on relational metric spaces

Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products

<abstract><p>We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ \ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.</p></abstract>

Distributed Consensus Control of Vehicular Platooning Under Delay, Packet Dropout and Noise: Relative State and Relative Input-Output Control Strategies

In this article, we study the distributed control problem of vehicular platooning by taking into account network imperfections, namely the input delay and the packet dropout. The delay occurs within the inter network of each vehicle from its controller to its actuators and the packet dropout occurs within the intra network of vehicles when information is exchanged between vehicles. Relative state-feedback and relative input-output feedback control strategies are adopted for the vehicular platooning under network imperfection. To this end, and to avoid the requirement of relative state information, the relative state measurements are constructed for platoon systems based on a distributed observer via only relative input-output measurements and in the presence of delay, packet dropout and noise. Conditions for synchronization are provided in terms of the unique positive-definite solutions to some algebraic Riccati equations. Simulations validate the theoretical results.

H∞-Based Minimal Energy Adaptive Control With Preset Convergence Rate.

This work studies the H∞ -based minimal energy control with a preset convergence rate (PCR) problem for a class of disturbed linear time-invariant continuous-time systems with matched external disturbance. This problem aims to design an optimal controller so that the energy of the control input satisfies a predetermined requirement. Moreover, the closed-loop system asymptotic stability with PCR is ensured simultaneously. To deal with this problem, a modified game algebraic Riccati equation (MGARE) is proposed, which is different from the game algebraic Riccati equation in the traditional H∞ control problem due to the state cost being lost. Therefore, a unique positive-definite solution of the MGARE is theoretically analyzed with its existing conditions. In addition, based on this formulation, a novel approach is proposed to solve the actuator magnitude saturation problem with the system dynamics being exactly known. To relax the requirement of the knowledge of system dynamics, a model-free policy iteration approach is proposed to compute the solution of this problem. Finally, the effectiveness of the proposed approaches is verified through two simulation examples.

Linear–Quadratic Optimal Control for Discrete-Time Mean-Field Systems With Input Delay

The linear–quadratic (LQ) optimal control and stabilization problems for mean-field systems with input delay (MFSID) are investigated in this article. The necessary and sufficient solvability conditions for LQ control of MFSID are first given in terms of a convexity condition and the solvability of equilibrium conditions. Consequently, by solving the associated mean-field forward and backward stochastic difference equations, the optimal control is derived in terms of the solution of a modified Riccati equation. Furthermore, for the infinite-horizon case, the stabilization problem for MFSID is studied, and the necessary and sufficient stabilizability conditions are obtained. We show that MFSID can be mean square stabilizable if and only if a modified algebraic Riccati equation admits a unique positive-definite solution.

A relational-theoretic approach to get solution of nonlinear matrix equations

In this study, we consider a nonlinear matrix equation of the form mathcal{X}= mathcal{Q} + sum_{i=1}^{m} mathcal{A}_{i}^{*} mathcal{G} (mathcal{X})mathcal{A}_{i}, where mathcal{Q} is a Hermitian positive definite matrix, mathcal{A}_{i}^{*} stands for the conjugate transpose of an ntimes n matrix mathcal{A}_{i}, and mathcal{G} is an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that mathcal{G}(O)=O. We discuss sufficient conditions that ensure the existence of a unique positive definite solution of the given matrix equation. For this, we derive some fixed point results for Suzuki-FG contractive mappings on metric spaces (not necessarily complete) endowed with arbitrary binary relation (not necessarily a partial order). We provide adequate examples to validate the fixed-point results and the importance of related work, and the convergence analysis of nonlinear matrix equations through an illustration with graphical representations.

Pareto Optimality in Infinite Horizon Mean-Field Stochastic Cooperative Linear-Quadratic Difference Games

This paper is concerned with the mean-field stochastic cooperative linear-quadratic (LQ) dynamic difference game in an infinite time horizon. First, the necessary and sufficient conditions for the stability in the mean-square sense, and the stochastic Popov-Belevith-Hautus (PBH) eigenvector tests for exact observability and exact detectability of mean-field stochastic linear difference systems are derived by the <inline-formula><tex-math notation="LaTeX">$\mathscr {H}$</tex-math></inline-formula>-representation technique. Second, the relation between the solvability of the cross-coupled generalized Lyapunov equations&#x00A0;(CC-GLEs) and exact observability, exact detectability, and stability of the mean-field dynamic system is well characterized. It is then shown that the cross-coupled algebraic Riccati equations&#x00A0;(CC-AREs) admit a unique positive definite (positive semi-definite, respectively) solution under exact observability (exact detectability, respectively), which is also a feedback stabilizing solution. Furthermore, all Pareto optimal strategies and solutions can be respectively derived via the solutions to the weighted CC-AREs (WCC-AREs) and the weighted cross-coupled algebraic Lyapunov equations&#x00A0;(WCC-ALEs). Finally, a practical application on the computation offloading in the multi-access edge computing network (MECN) is presented to illustrate the proposed theoretical results.

Positive definite solution of non-linear matrix equations through fixed point technique

&lt;abstract&gt;&lt;p&gt;The goal of this study is to solve a non-linear matrix equation of the form $ \mathcal{X} = \mathcal{Q} + \sum\limits_{i = 1}^{m} \mathcal{B}_{i}^{*}\mathcal{G} (\mathcal{X})\mathcal{B}_{i} $, where $ \mathcal{Q} $ is a Hermitian positive definite matrix, $ \mathcal{B}_{i}^{*} $ stands for the conjugate transpose of an $ n\times n $ matrix $ \mathcal{B}_{i} $ and $ \mathcal{G} $ an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that $ \mathcal{G}(O) = O $. We explore the necessary and sufficient criteria for the existence of a unique positive definite solution to a particular matrix problem. For the said reason, we develop some fixed point results for $ \mathcal{FG} $-contractive mappings on complete metric spaces equipped with any binary relation (not necessarily a partial order). We give adequate examples to confirm the fixed-point results and compare them to early studies, as well as four instances that show the convergence analysis of non-linear matrix equations using graphical representations.&lt;/p&gt;&lt;/abstract&gt;

Optimal control and stabilization for linear continuous‐time mean‐field systems with delay

This paper studies optimal control and stabilization problems for continuous-time mean-field systems with input delay, which are the fundamental development of control and stabilization problems for mean-field systems. There are two main contributions: (1) To the best of the authors' knowledge, the present paper is the first to establish the necessary and sufficient solvability condition for this kind of optimal control problem with input delay, and to derive the analytical form of an optimal controller through overcoming the obstacle that separation principle no longer holds for multiplicative-noise systems. (2) For the stabilization problem, under the assumption of exact observability, it is strictly proven that the system is stabilizable if and only if the algebraic Riccati equation has a unique positive definite solution.

Lyapunov differential equations and inequalities for stability and stabilization of linear time-varying systems

This paper studies exponential stability and stabilization of linear time-varying systems without assuming that the coefficient matrix is bounded, which is generally necessary in the existing literature. First, by introducing the notions of uniformly exponentially stable and uniformly exponentially expanding functions, some Lyapunov differential inequalities based characterizations for exponential stability of linear time-varying systems are established. Second, it is shown that the linear time-varying system is both uniformly exponentially stable and uniformly exponentially expanding if and only if the corresponding Lyapunov differential equation has a unique positive definite solution. Third, some degenerated Lyapunov differential equations, where the Q matrix is only positive semi-definite, are also re-examined. Finally, a weighted controllability Gramian, which contains a weighting time-varying function that can be properly designed to reveal a trade-off between the convergence rate/regulation time of the closed-loop system and the control effort, based stabilization approach is established to ensure that the closed-loop system is both uniformly exponentially stable and uniformly exponentially expanding. The effectiveness of the proposed approach is illustrated by the design of the spacecraft attitude control system with magnetic actuation.

Some Fixed Point Results on Relational Quasi Partial Metric Spaces and Application to Non-Linear Matrix Equations

We introduce a qϱ-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation U=B+∑i=1mAi*G(U)Ai, where B is an n×n Hermitian positive definite matrix, A1, A2, … Am are n×n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.

Gradient-based neural networks for online solutions of coupled Lyapunov matrix equations

This study investigates gradient-based neural networks (GNNs) for solving coupled Lyapunov matrix equations arising in the stability analysis of continuous-time Markovian jump linear systems. First, based on the gradient descent principle, a general framework of GNNs is presented to solve the considered equations by following the ideas in some existing results, which are used to solve the standard Lyapunov matrix equation. Subsequently, a new GNN solver is developed for finding the online solution of the coupled Lyapunov equations. Compared with the general GNN solver, a main advantage of the proposed GNN solver is that its convergence can be directly proven through theoretical analysis. To be specific, according to Lyapunov theory, it is shown that the state of the presented GNN solver with appropriately activation functions can globally converge to the unique positive definite solution of the coupled Lyapunov equations. In addition, to accelerate its convergence rate, an improved version of the proposed GNN is established. Simulation results are given to substantiate the effectiveness of the theoretical results and the superior of the developed GNN solvers compared with the originally obtained general GNN solvers.

Linear quadratic optimal control for a class of continuous-time nonhomogeneous Markovian jump linear systems in infinite time horizon

In this paper, the infinite time horizon linear quadratic optimal control problem is investigated for the continuous-time Itô stochastic Markovian jump linear systems (MJLSs) with time-varying transition rates. It is assumed that the time-varying transition rates of the MJLSs possess piecewise homogeneous time-varying property, which implies that the transition rates are time-varying in the whole time domain but they are time-invariant in some small time intervals. The variations of the transition rates in these small time intervals are considered to be in two cases: arbitrary variation and stochastic variation. With this, the considered system becomes a piecewise homogeneous Itô stochastic MJLS. The main contribution of this paper is that two linear quadratic optimal controllers in infinite time horizon are proposed for the above modeled continuous-time piecewise homogeneous Itô stochastic MJLS in the sense of arbitrary variation and stochastic variation, respectively. Moreover, the sufficient and necessary conditions for the existence of the designed controllers are established based on the existence of the unique positive definite solution of two coupled algebraic Riccati matrix equations. Finally, a simulation example is provided to illustrate the effectiveness of the proposed linear quadratic optimal controllers.

Iterative algorithms with the latest update for Riccati matrix equations in Itô Markov jump systems

This study is concerned with the problem to solve the continuous coupled Riccati matrix equations in Ito Markov jump systems. A new iterative algorithm is developed by using the latest estimation information and introducing a tuning parameter. The iterative solution obtained by the proposed algorithm with zero initial conditions converges to the unique positive definite solution of the considered equations. The convergence rate of the algorithm is dependent on the adjustable parameter. Furthermore, a numerical example is provided to show the effectiveness of the presented algorithms.

Solvability for Two Forms of Nonlinear Matrix Equations

In this paper, we study nonlinear matrix equations $$\begin{aligned} X^p=A+\sum \limits _{i=1}^m M_i^T(X\#B)M_i \end{aligned}$$ and $$\begin{aligned} X^p=A+\sum \limits _{i=1}^j M_i^T(X\#B)M_i+\sum \limits _{i=j+1}^m M_i^T(X^{-1}\#B)M_i, \end{aligned}$$ where p, m, j are positive integers, $$1\le j\le m$$ , A, B are $$n\times n$$ positive definite matrices and $$M_i(i=1,2,3,\ldots ,m)$$ are $$n\times n$$ nonsingular real matrices. Based on some fixed point theorems for monotone and mixed monotone operators in ordered Banach spaces and some properties of cone, we prove that these equations always have a unique positive definite solution. In addition, an iterative sequence can be given to approximate the unique positive definite solution by employing a multi-step stationary iterative method.

Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation $$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$

In this paper, we study the nonlinear matrix equation $$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$ , where $$A,Q \in {{\mathbb {C}}}^{n\times n}$$ , Q is a Hermitian positive definite matrix and $$X \in {{\mathbb {C}}}^{n\times n}$$ is an unknown matrix. We prove that the equation always has a unique Hermitian positive definite solution. We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation, and the convergence theories are established. Finally, we show the effectiveness of the algorithms by numerical experiments.

On the nonlinear matrix equation Xp = A#t (MT (X –1 + B)–1 M)

ABSTRACT In this paper, the nonlinear matrix equation , where is a positive integer, , M is an nonsingular matrix, A is a positive definite matrix and B is a positive semidefinite matrix, is considered. The notation is the t-weighted geometric mean of the positive definite matrices A and B. Based on the properties of the Thompson metric, we prove that the nonlinear matrix equation always has a unique positive definite solution and we compare it with the unique positive definite solution of the equation , which has been studied in Jung et al. [On the solution of the nonlinear matrix equation . Linear Algebra Appl. 2009;430:2042–2052]; Meng and Kim [The positive definite solution of the nonlinear matrix equation . J Comput Appl Math. 2017;322:139–147]. A fixed-point iteration method for obtaining the unique positive definite solution and an elegant estimate of the solution are given. Perturbation analysis of the unique positive definite solution is presented.

On linear quadratic optimal control of discrete‐time complex‐valued linear systems

SummaryWe study in this paper the linear quadratic optimal control (linear quadratic regulation, LQR for short) for discrete‐time complex‐valued linear systems, which have several potential applications in control theory. Firstly, an iterative algorithm was proposed to solve the discrete‐time bimatrix Riccati equation associated with the LQR problem. It is shown that the proposed algorithm converges to the unique positive definite solution (bimatrix) to the bimatrix Riccati equation with appropriate initial conditions. With the help of this iterative algorithm, the LQR problem for the antilinear system, which is a special case of complex‐valued linear system, was carefully examined and three different Riccati equations–based approaches were provided, namely, bimatrix Riccati equation, anti‐Riccati equation, and normal Riccati equation. The established approach is then used to solve the LQR problem for a discrete‐time time‐delay system with one‐step state delay, and a numerical example was used to illustrate the effectiveness of the proposed methods.