Coupled Lyapunov Matrix Equations Research Articles

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.

A Newton iterative method for coupled Lyapunov matrix equations

A Newton iterative method for coupled Lyapunov matrix equations

Improved SOR iterative method for coupled Lyapunov matrix equations

Improved SOR iterative method for coupled Lyapunov matrix equations

A relaxed MSIO iteration algorithm for solving coupled discrete Markovian jump Lyapunov equations

In this paper, combining the multi-step Smith-inner-outer (MSIO) iteration framework with some tunable parameters, a relaxed MSIO iteration method is proposed for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) in the discrete-time jump linear systems with Markovian transitions. The convergence properties of the relaxed MSIO iteration method are investigated, and the choices of the parameters are also discussed. In order to accelerate the convergence rate of the relaxed MSIO iteration method for solving the CLMEs, a current-estimation-based and a weighted relaxed MSIO iteration algorithms are presented, respectively. Finally, several numerical examples are given to verify the superiorities of the proposed relaxed algorithms.

An SOR-type algorithm based on IO iteration for solving coupled discrete Markovian jump Lyapunov equations

In this paper, based on the inner-outer (IO) iteration framework [17], by introducing some tunable parameters, an SOR-type IO (SIO) iteration method is proposed for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) in the discrete-time jump linear systems with Markovian transitions. Fisrtly, the SIO iteration algorithm for solving the discrete Sylvester matrix equation is developed, its convergence property is analyzed and the choices of the parameters are also discussed. Next, the SIO iteration algorithm is used to solve the CLMEs. Moreover, by using the latest estimations, a current-estimation-based SIO (CSIO) iteration algorithms are also constructed for solving the CLMEs, respectively. The boundedness and monotonicity of the iteration sequence derived from the proposed algorithm with zero initial conditions are established. Finally, several numerical examples are implemented to illustrate the superiorities of the proposed iteration algorithms.

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.

A multi-step Smith-inner-outer iteration algorithm for solving coupled continuous Markovian jump Lyapunov matrix equations

In this paper, combining the Smith method [30] with inner-outer (IO) iteration [4], a multi-step Smith-IO (MSIO) iteration algorithm for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) is proposed. Moreover, some convergence conditions and comparison results are also given for the MSIO iteration algorithm. Finally, several numerical examples are used to verify the effectiveness of the proposed numerical algorithms.

Explicit Iterative Algorithms for Continuous Coupled Lyapunov Matrix Equations

In this article, two iterative algorithms are proposed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. First, an explicit iterative algorithm is presented to solve the considered matrix equations via introducing a tuning parameter. A necessary and sufficient condition is established for the proposed algorithm to be convergent. Furthermore, the optimal value of the tuning parameter achieving the fastest convergence rate is given explicitly. Second, a modified version of the previous explicit algorithm is established by using a new information update scheme. Similarly, a necessary and sufficient condition is provided for the modified explicit iterative algorithm. Based on this convergence condition, some easily verifiable convergence results are further derived, and the optimal tuning parameter is given explicitly. Finally, two examples are presented to validate the obtained theoretical results.

A gradient-based iterative algorithm for solving coupled Lyapunov equations of continuous-time Markovian jump systems

ABSTRACTIn this paper, a new gradient-based iterative algorithm is proposed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. A necessary and sufficient condition is established for the proposed gradient-based iterative algorithm to be convergent. In addition, the optimal value of the tunable parameter achieving the fastest convergence rate of the proposed algorithm is given explicitly. Finally, some numerical simulations are given to validate the obtained theoretical results.

Stability and stabilisation of Itô stochastic systems with piecewise homogeneous Markov jumps

ABSTRACTThis paper focuses on the problem of stability and stabilisation for continuous-time Itô stochastic Markovian jump linear systems with time-varying transition rates. The time-varying property of the transition rates is considered to be finite piecewise homogeneous. Firstly, the stability conditions of the piecewise homogeneous Itô stochastic Markovian jump linear systems are given in terms of linear matrix inequalities (LMIs). Especially, a novel stability criterion is developed for the considered systems by the existence of the unique positive-definite solution of the corresponding coupled Lyapunov matrix equations. Secondly, two state-feedback controllers are designed via LMIs to stabilise the systems. Finally, a practical example is provided to illustrate the effectiveness of the presented theoretical results.

New explicit iteration algorithms for solving coupled continuous Markovian jump Lyapunov matrix equations

In this paper, based on the Smith iteration (Smith, 1968), an inner-outer (IO) iteration algorithm for solving the coupled Lyapunov matrix equations (CLMEs) is presented. First, the IO iteration algorithm for solving the Sylvester matrix equation is proposed, and its convergence is analyzed in detail. Second, the IO iteration algorithm for solving the CLMEs is constructed. By utilizing the latest estimation, a current-estimation-based and two weighted IO iteration algorithms are also given for solving the CLMEs, respectively. Convergence analyses indicate that the iteration solutions generated by these algorithms always converge to the unique solutions to the CLMEs for any initial conditions. Finally, Several numerical examples are provided to show the superiority of the proposed numerical algorithms.

Stochastic stability analysis of Markovian jump linear systems with incomplete transition descriptions

In this study, the authors focus on the stability analysis for Markovian jump linear systems with partly known transition rates in the continuous-time domain and partly known transition probabilities in the discrete-time domain. By using the properties of the transition rates and transition probabilities, two new sufficient conditions are derived for the stochastic stability of the continuous-time and discrete-time Markovian jump linear systems, respectively. The main advantage of the proposed stability conditions is that the total number of linear matrix inequalities (LMIs) in the proposed stability conditions is much less than that in some existing results. Based on the presented stability conditions, two state feedback controllers are designed for the considered systems in terms of LMIs. In addition, two kinds of stability criteria are developed for the stochastic stability of the considered systems with incomplete transition descriptions by the existence of the unique positive definite solution of the coupled Lyapunov matrix equations. Finally, two numerical examples and one practical example are provided to verify the correctness of the theoretical results.

Accelerated smith iterative algorithms for coupled Lyapunov matrix equations

In this paper, a kind of explicit iterative algorithm is studied by using Smith methods for solving coupled Lyapunov matrix equations arising in continuous-time Markovian jump linear systems. First, by introducing the latest estimation, a current-estimation-based Smith iterative algorithm is presented to solve the considered matrix equations. The monotonicity and boundedness of the sequence generated by this algorithm with zero initial conditions are shown by the mathematical induction, and a convergence result is accordingly obtained based on these two properties. To further improve the convergence performance, an accelerated Smith iterative algorithm is proposed. Convergence analysis indicates the iterative solutions generated by the developed algorithms always converge to the unique solutions of the coupled Lyapunov matrix equations for any initial conditions. A numerical example is provided to show the effectiveness of the presented iterative algorithms.

Implicit Iterative Algorithms for Continuous Markovian Jump Lyapunov Equations

In this technical note, implicit iterative algorithms with some tunable parameters are developed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. A significant feature of the proposed algorithms is that the iterative sequences are updated by using not only the information in the last step, but also the information in the current step and the previous steps. Also the convergence rate of the proposed algorithms can be significantly improved by choosing appropriate parameters in the algorithms.

Balanced truncation-rational Krylov methods for model reduction in large scale dynamical systems

In this paper, we consider the balanced truncation method for model reductions in large-scale linear and time-independent dynamical systems with multi-inputs and multi-outputs. The method is based on the solutions of two large coupled Lyapunov matrix equations when the system is stable or on the computation of stabilizing positive and semi-definite solutions of some continuous-time algebraic Riccati equations when the dynamical system is not stable. Using the rational block Arnoldi, we show how to compute approximate solutions to these large Lyapunov or algebraic Riccati equations. The obtained approximate solutions are given in a factored form and used to build the reduced order model. We give some theoretical results and present numerical examples with some benchmark models.

An implicit sequential algorithm for solving coupled Lyapunov equations of continuous-time Markovian jump systems

In this paper, an implicit sequential algorithm is presented for solving coupled Lyapunov matrix equations of continuous-time Markovian jump linear systems. First, some existing iterative algorithms which can be utilized to solve the coupled Lyapunov matrix equations are reviewed and discussed. Next, based on the existing parallel iterative algorithm, an implicit sequential algorithm is proposed by using the latest updated information. The proposed algorithm fills the current gap of implicit algorithms for solving continuous coupled Lyapunov matrix equations. It is shown that the proposed algorithm with zero initial conditions can monotonically converge to the unique positive definite solutions of the coupled Lyapunov matrix equations if the associated Markovian jump system is stochastically stable. Moreover, a necessary and sufficient condition is established for the proposed algorithm to be convergent. The algorithm presented in this paper has much better convergence performance than other existing iterative algorithms and requires less storage capacity. Finally, a numerical example is given to show the effectiveness of the proposed algorithm.

New Iterative Algorithms for Solving Coupled Markovian Jump Lyapunov Equations

In this technical note, two iterative algorithms are developed for solving the coupled Lyapunov matrix equations associated with discrete-time Markovian jump linear systems. Some convergence results have been obtained for these algorithms. The new algorithms are different from the previous results in nature and they can be regarded as sequential versions of the parallel algorithm reported in the work of Borno and Gajic. Numerical experiments show that the proposed algorithms are faster in terms of the number of iterations to achieve the desired accuracy.

Asymptotic stability condition for stochastic Markovian systems of differential equations

Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by ${\rm d} X(t) = A(\xi (t))X(t) {\rm d} t + H(\xi (t))X(t) {\rm d} w(t)$, where $\xi (t)$ is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.

Model reduction in large scale MIMO dynamical systems via the block Lanczos method

In the present paper, we propose a numerical method for solving the coupled Lyapunov matrix equations A P + P AT + B BT = 0 and AT Q + Q A + CT C = 0 where A is an n ×n real matrix and B, CT are n × s real matrices with rank(B) = rank(C) = s and s << n . Such equations appear in control problems. The proposed method is a Krylov subspace method based on the nonsymmetric block Lanczos process. We use this process to produce low rank approximate solutions to the coupled Lyapunov matrix equations. We give some theoretical results such as an upper bound for the residual norms and perturbation results. By approximating the matrix transfer function F(z) = C (z In - A)-1 B of a Linear Time Invariant (LTI) system of order n by another one Fm(z) = Cm (z Im - Am)-1 Bm of order m, where m is much smaller than n , we will construct a reduced order model of the original LTI system. We conclude this work by reporting some numerical experiments to show the numerical behavior of the proposed method.

Matrix Krylov subspace methods for large scale model reduction problems

The present paper considers matrix Krylov subspace methods for solving large coupled Lyapunov matrix equations of the form AP + PA T + BB T = 0 and A T Q + QA + C T C = 0 where A is a real n × n matrix, B and C T are real n × s and n × r matrices, respectively, with s ≪ n and r ≪ n. Such problems appear in many areas of control theory such as the computation of the controllability and observability Gramians of a stable Linear Time Invariant (LTI) system. The proposed methods are based on the global Arnoldi and Lanczos processes. In the second part, we show how to use matrix Krylov subspace techniques to obtain a reduced order model for LTI systems. This will be done by approximating the corresponding transfer functions. Finally, numerical experiments are reported to illustrate the behavior and the effectiveness of the global Arnoldi and Lanczos processes when applied to solve some large coupled Lyapunov equations and to approximate some transfer functions.