Positive Definite Solution Research Articles

ON INEXACT NEWTON METHODS FOR SOLVING TWO NONLINEAR MATRIX EQUATIONS

In this paper we consider inexact Newton methods for finding the largest positive definite solutions of two nonlinear matrix equations X+ A∗X−1A = Q and X − A∗X−1A = Q, respectively. Using Newton’s method for considered equations requires solving a Stein’s equation at each iteration. For solving the Stein’s equation, we use Smith type iterations instead of exact methods. Nonlocal convergence of the process is shown. Numerical experiments are included to illustrate the theory.

Data-driven adaptive optimal control for discrete-time linear time-invariant systems

In this paper, the problem of data-driven quadratic optimal control is investigated for discrete-time linear systems without the exact knowledge of system dynamics. A value iteration-based algorithm is proposed to obtain the optimal feedback gain. The main idea of the proposed approach is to obtain two sequences for the approximation of the unique positive definite solution of the corresponding algebraic Riccati matrix equation and the optimal feedback gain by using the collected data of the system states and control inputs. The proposed algorithm could be activated by an arbitrary bounded control input and a simple initial value of the Lyapunov matrix. An important feature of this algorithm is that an original stabilising feedback gain is not required. Finally, the effectiveness of the proposed approach is verified by two examples.

An accelerated zeroing neural network for solving continuous coupled Lyapunov matrix equations

AbstractIn this paper, an improved zeroing neural network (ZNN) model is proposed to obtain the positive definite solutions of the continuous coupled Lyapunov matrix equations (CLMEs) associated with continuous‐time Markovian jump (CMJ) systems. To achieve this, a general ZNN model is established by constructing a matrix‐valued error function. Then, to accelerate the convergence rate of the proposed ZNN model, the latest estimation is introduced to obtain an improved ZNN model. Some convergence conditions have been derived for the presented improved ZNN model through Lyapunov theory. Comparisons among the improved ZNN model and the existing results are conducted to illustrate the advantages of the proposed improved ZNN model in numerical examples.

Data‐driven adaptive optimal control for discrete‐time periodic systems

AbstractIn this paper, a problem of data‐driven optimal control is studied for discrete‐time periodic systems with unknown system matrices and input matrices. For this problem, a value iteration‐based adaptive dynamic programming algorithm is proposed to obtain the suboptimal controller. The core of the algorithm proposed in this paper is to obtain an approximation of the unique positive definite solution of the algebraic Riccati equation and the optimal feedback gain matrix by using the collected real‐time data of the system states and control inputs. Without an initial stabilizing feedback gain, the proposed algorithm could be activated by an arbitrary bounded control input. Finally, the effectiveness of the proposed approach is demonstrated by two examples.

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.

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.

On positive definite solutions of the matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $

<p>In this paper, we used the outstanding properties of the Thompson metric to conclusively demonstrate the existence of a unique positive definite solution for the nonlinear matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $ without any additional assumptions. Furthermore, we designed an iterative algorithm to compute this unique positive definite solution, and derive its corresponding error estimate formula. Additionally, we presented three refined existence intervals for positive definite solutions of this equation. Finally, numerical examples were employed to validate the practicability of our iterative algorithm.</p>

Suzuki-type fixed point theorems in relational metric spaces with applications

In this paper, we establish a relation-theoretic version of the results presented by Kim et al. (Journal of Nonlinear and Convex Analysis, 16 (9) (2015), 1779-1786). To showcase the versatility of our results, we furnish some illustrative examples. Furthermore, we exhibit an application of our results to establish sufficient conditions for the existence of a positive definite common solution to a pair of nonlinear matrix equations.

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

A new inversion-free iterative algorithm for the discrete algebraic Riccati equation

Abstract In this paper, by the transformation form of the discrete algebraic Riccati equation (DARE), we propose a new inverse-free iterative algorithm to obtain the positive definite solution of the DARE. Furthermore, the monotone convergence is proved and convergence rate analysis is presented for the derived algorithm. Compared with some existing algorithms, numerical examples demonstrate the feasibility and effectiveness of our algorithm.

Positive Definite Solution of a Class of Matrix Equations

Positive Definite Solution of a Class of Matrix Equations

On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations

In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation X+A∗X−1A=Q. We study the recent iterative methods for computing the solution to the above specific type of equations and propose more effective modifications of these iterative methods. In addition, we make comments and comparisons of the existing methods and show the effectiveness of our methods by illustration examples.

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.

Best Proximity Point and Existence of the Positive Definite Solution for Matrix Equations

In this research, α − ψ − θ contraction has been defined to find the best proximity point in partially ordered metric spaces. Proper support for the result has been given in the form of a suitable example. The third part is fully devoted to the positive definite solution of matrix equations.

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

A relation theoretic <i>m</i>-metric fixed point algorithm and related applications

<abstract><p>In this article, we introduce the concept of generalized rational type $ F $ -contractions on relation theoretic <italic>m</italic>-metric spaces (denoted as $ F_{R}^{m} $-contractions, where $ R $ is a binary relation) and some related fixed point theorems are provided. Then, we achieve some fixed point results for cyclic rational type $ F_{R}^{m} $- generalized contraction mappings. Moreover, we state some illustrative numerically examples to show our results are true and meaningful. As an application, we discuss a positive definite solution of a nonlinear matrix equation of the form $ \Lambda = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i} $.</p></abstract>

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>

Applying fixed point techniques for obtaining a positive definite solution to nonlinear matrix equations

<abstract><p>In this manuscript, the concept of rational-type multivalued $ F- $contraction mappings is investigated. In addition, some nice fixed point results are obtained using this concept in the setting of $ MM- $spaces and ordered $ MM- $spaces. Our findings extend, unify, and generalize a large body of work along the same lines. Moreover, to support and strengthen our results, non-trivial and extensive examples are presented. Ultimately, the theoretical results are involved in obtaining a positive, definite solution to nonlinear matrix equations as an application.</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.