Continuous-time Lyapunov Equations Research Articles

Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

Distributed Optimization Approach for Solving Continuous-Time Lyapunov Equations With Exponential Rate of Convergence

This article establishes an approach, based on distributed optimization, for solving continuous-time Lyapunov equations (CTLE) over multiagent networks. Each agent in the network knows partial information of the CTLE and has a dynamical system to estimate exact or least-squares solutions. The aim of agents is to find a solution to CTLE by sharing information with connected agents over a network. This article develops distributed algorithms with an exponential rate of convergence for CTLE via the convex optimization design. Finally, this article presents numerical simulations to show the efficacy of the main results.

Positive solution to a generalized Lyapunov equation via a coupled fixed point theorem in a metric space endowedwith a partial order

We consider the generalized continuous-time Lyapunov equation: A*XB + B*XA = -Q, where Q is an N x N Hermitian positive definite matrix and A,B are arbitrary N x N matrices. Under certain conditions, using a coupled fixed point theorem du to Bhaskar and Lakshmikantham combined with the Schauder fixed point theorem, we establish an existence and uniqueness result of Hermitian positive definite solution to such equation. Moreover, we provide an iteration method to find convergent sequences which converge to the solution if one exists. Numerical experiments are presented to illustrate our theoretical results.

Noniterative Improvement of Trace Bounds for the Lyapunov Equation Solution

The improvement problem for available lower and upper trace bounds for the solution of the continuous-time Lyapunov equation (LE) is investigated. It is shown how an arbitrary positive (semi)-definite lower matrix solution bound can be used to get always tighter trace estimates.

A Parameter Free Iterative Method for Solving Projected Generalized Lyapunov Equations

This paper is devoted to the numerical solution of projected generalized continuous-time Lyapunov equations with low-rank right-hand sides. Such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. A parameter free iterative method is proposed. This method is based upon a combination of an approximate power method and a generalized ADI method. Numerical experiments presented in this paper show the effectiveness of the proposed method.

Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

This paper studies the iterative solutions of Lyapunov matrix equations associated with Itô stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms.

On the sensitivity of the coupled continuous-time Lyapunov equation

The sensitivity of coupled continuous-time Lyapunov equations is analyzed through three types of norms of the inverse Lyapunov-like operator. This analysis leads to bounds on the relative perturbation of the solution. The results extend the bounds obtained for the standard Lyapunov equation in Hewcer et al. (1988).

On the Positive Definite Solutions to the 2-D Continuous-timeLyapunov Equation

The very strict positive real lemma is further developed for nonminimal 1-D continuous-time systems and is used to study the 2-D continuous-time Lyapunov equation. Based on it, an extended condition for the bivariate characteristic polynomial of a matrix to be very strict Hurwitz is proposed for general 2-D analog systems with characteristic polynomials involving 1-D factor polynomials. It is also shown that in such a case the bivariate polynomial can be decomposed into a 2-D bivariate polynomial with the corresponding matrix satisfying certain controllability and observability conditions and into up to two 1-D polynomials. Further, two algorithms for computing the positive definite solutions to the 2-D Lyapunov equation are presented.

Sensitivity of the stable discrete-time Lyapunov equation

The sensitivity of the stable discrete-time Lyapunov equation is analyzed through the spectral norm of the inverse Lyapunov operator. This leads to a directly computable easy-to-interpret sensitivity measure, which involves only the open-loop state matrix and also provides insight into the connection between sensitivity, stability radius, and conditioning of the eigenproblem of the open-loop state matrix. These results are an extension, to the discrete-time case, of analogous results for the continuous-time Lyapunov equation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>