Discrete Lyapunov Equation Research Articles

The Discrete Lyapunov Equation of The Orthogonal Matrix in Semiring

Semiring is an algebraic structure of (S, +, ×). Similar to a ring, but without the condition that each element must have an inverse to the adding operation. The forms (S, +) and (S, ×) are semigroups that satisfy the distributive law of multiplication and addition. In matrix theory, there is a term known as the Kronecker product. This operation transforms two matrices into a larger matrix containing all possible products of the entries in the two matrices. This Kronecker product has several properties often used to solve the complex problems of linear algebra and its applications. The Kronecker product is related to the Lyapunov equation of a linear system. Based on previous research in the Lyapunov equation in conventional linear algebra, this paper will describe the characteristics of the Lyapunov equation in a semiring linear system in terms of the Kronecker product.

Stability analysis of 2‐D discrete and continuous state‐space systems

This paper presents new stability conditions for two-dimensional (2-D) systems in state-space description. Both discrete and continuous systems are studied. These results are based on the criteria first presented by Huang, De Carlo, Strintzis, Murray, Delsarte, et al. and on the discrete Lyapunov equation with complex elements for 2-D systems. The stability properties of the Mansour matrix are also used for stability testing in state-space. Criteria for the VSHP property of 2-D polynomials are further presented using the continuous Lyapunov equation with complex elements and the stability properties of the Schwarz matrix form. The stability properties of the Schwarz matrix are also used for testing the VSHP property of 2-D polynomials in state-space. The proposed new criteria are non-conservative for the stability analysis of 2-D discrete and continuous systems and achieve the aim of reducing the original 2-D problem as much as possible to a set of 1-D stability tests. Numerical examples are given to illustrate the utility of the proposed conditions.

Closed Form Solutions of Lyapunov Equations Using the Vech and Veck Operators

New closed forms are presented of the solutions of the continuous and discrete Lyapunov equations using the vech and veck operators. The proposed solutions are faster than the classical solutions derived using the vec operator. The solutions via veck operator are faster than the solutions via vech operator

Inverse and Direct Solutions of the Discrete Time Lyapunov Equation With System Matrix in Companion Form

The discrete time Lyapunov equation is used in many applications and there is interest in its inverse and direct solutions. New methods are proposed to obtain solutions for cases where the system matrix is in controllable canonical form. The approach is based on the relationship between the discrete Lyapunov equation and the entries of one of the stability tables presented by Jury. It is shown that the inverse solution, which is based on this stability table, can be obtained using LDLt decomposition. Also the direct solution of the discrete Lyapunov equation can be obtained directly from the entries of this stability table. The proposed algorithms are illustrated by numerical examples.

Improved Bound Estimates for the Solution of the Discrete Lyapunov Equation

Improved Bound Estimates for the Solution of the Discrete Lyapunov Equation

Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

The paper is devoted to the discrete Lyapunov equation X - A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.

Frequency limited Gramians-based structure preserving model order reduction for discrete time second-order systems

ABSTRACTA new technique for frequency limited model order reduction of discrete time second-order systems is presented. Discrete time frequency limited Gramians (DFLGs) and corresponding discrete algebraic Lyapunov equations are developed. An efficient technique for the computation of DFLGs and their Cholesky factors is presented. Computed DFLGs are partitioned to obtain position and velocity Gramians. These Gramians are balanced with different combinations to obtain various balanced transformations that yield Hankel singular values (HSVs) for order reduction. Frequency limited discrete time balanced truncation framework is proposed and truncation based on magnitudes of HSVs is applied to obtain the reduced order model. Moreover, stability conditions for reduced order models are stated. Results of the proposed technique are compared with infinite Gramians balancing scheme in order to certify the usefulness of the presented technique for frequency limited applications.

Spectral decompositions for the solutions of Sylvester, Lyapunov, and Krein equations

Spectral decompositions for the solutions of Lyapunov equation obtained earlier are generalized to a more general class of solutions of Krein matrix equations including as a special case the standard Sylvester equation. Eigen parts of these decompositions are calculated using residues of matrix resolvents and their derivatives. In particular, spectral decompositions for the solutions of algebraic and discrete Lyapunov equations are obtained in a more general formulation. The practical significance of the obtained spectral expansions is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations into the asymptotic dynamics of the system perturbation energy.

Sufficient Conditions for Robust Stability of Discrete Large-Scale Interval Systems with Multiple Time Delays

The robust stability analysis for discrete large-scale uncertain systems with multiple time delays is addressed in this paper. We establish a method for selecting properly a positive definite matrix Q to derive a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach method with this upper bound, several sufficient conditions are presented to guarantee the robust stability of the overall systems. Comparisons between the proposed results with a previous one are also given.

Exponential dichotomy of systems of linear difference equations with periodic coefficients

We study the problem of exponential dichotomy for the systems of linear difference equations with periodic coefficients. Some criterion is established for exponential dichotomy in terms of solvability of a special boundary value problem for a system of discrete Lyapunov equations. We also give estimates for dichotomy parameters.

A unified approach of the measurement of solution bounds of the continuous and discrete algebraic Lyapunov equations

In this paper, an integrated study for measuring the solutions of the continuous and discrete Lyapunov equations is developed. A unified algebraic Lyapunov equation (UALE) will be given first. Then, by using a bilinear transformation, the mentioned equation can be transformed into the quasi-standard form of the discrete algebraic Lyapunov equation (DALE). Thus, those approaches appeared in the literature for solving the estimation problem of the solution of the Lyapunov equations may be used to solve the same problem for the UALE. By extending mainly the approach proposed in [20] for the continuous algebraic Lyapunov equation (CALE) associated with linear algebraic techniques, several upper and lower matrix bounds of the solution of the UALE are developed. Then, matrix bounds of the solutions of the CALE and DALE can be obtained directly by these obtained matrix bounds. Algorithms for getting better bounds of the solutions of the CALE and DALE are also given. By comparisons, the obtained solution bounds of the CALE and DALE are less restrictive or easier to be calculated than those parallel results appeared in the literature. Finally, numerical examples have given to demonstrate the merits of the proposed approach.

On spectral decompositions of solutions to discrete Lyapunov equations

A new approach to solving discrete Lyapunov matrix algebraic equations is based on methods for spectral decomposition of their solutions. Assuming that all eigenvalues of the matrices on the left-hand side of the equation lie inside the unit disk, it is shown that the matrix of the solution to the equation can be calculated as a finite sum of matrix bilinear quadratic forms made up by products of Faddeev matrices obtained by decomposing the resolvents of the matrices of the Lyapunov equation. For a linear autonomous stochastic discrete dynamic system, analytical expressions are obtained for the decomposition of the asymptotic variance matrix of system’s states.

Evaluation of a Power System Stability Based on Usage of Virtual Digital Analyzer

For early detection of the threat of a cascading failure we need to use the immune information system, the main subsystem of which is a virtual stability analyzer. We propose to implement the analyzer by usage of discrete linearized models of energy systems and apparatus of Lyapunov direct method, based on the study of discrete Lyapunov equation solutions, called Gramians. The functional of stability loss risk is equal to the square of H2-norm of the matrix transfer function, which is proportional to energy of the system. We propose to use a functional risk assessment through spectral decomposition of controllability and observability Gramians for evaluating of the energy accumulated in combinational dominant eigenvalues of the system. In comparison with the method of modal analysis the proposed approach allows to evaluate the impact of synergies between weak-stable eigenvalues and their combinations to the system stability loss risk and to detect the potential source of stability loss.

Robust Stability of Discrete Large-Scale Interval Time-Delay Systems

This paper addresses the robust stability analysis for discrete large-scale interval systems subjected to time delays. We establish a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach associated with this upper bound and some linear algebraic techniques, several simple criteria are presented to guarantee the robust stability of the overall systems. The featuresof these present results arethat they are independent of any Lyapunov equation although the Lyapunov equation approach is adopted.

Lowest-rank solutions of continuous and discrete Lyapunov equations over symmetric cone

The low-rank solutions of continuous and discrete Lyapunov equations are of great importance but generally difficult to compute in control system analysis and design. Fortunately, Mesbahi and Papavassilopoulos (1997) [30] showed that with the semidefinite cone constraint, the lowest-rank solutions of the discrete Lyapunov inequality can be efficiently solved by a linear semidefinite programming. In this paper, we further show that the lowest-rank solutions of both the continuous and discrete Lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations, the symmetric cone linear programming problems. Therefore, they are polynomial-time solvable. Since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case, our results also answer an open question proposed by Recht, Fazel and Parrilo (2010) [36].

Extended Hamiltonian algorithm for the solution of discrete algebraic Lyapunov equations

In this paper, we use a second-order learning algorithm for solving the numerical solution of the discrete algebraic Lyapunov equation. Specifically, Extended Hamiltonian algorithm based on the manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm and the two traditional iteration methods. Simulation examples show that the convergence speed of the Extended Hamiltonian algorithm is the fastest one among these algorithms.

A natural gradient descent algorithm for the solution of discrete algebraic Lyapunov equations based on the geodesic distance

A new framework to calculate the numerical solution of the discrete algebraic Lyapunov equation is proposed by using the geometric structures on the Riemannian manifold. Specifically, two algorithms based on the manifold of positive definite symmetric matrices are provided. One is a gradient descent algorithm with an objective function of the classical Euclidean distance. The other is a natural gradient descent algorithm with an objective function of the geodesic distance on the curved Riemannian manifold. Furthermore, these two algorithms are compared with a traditional iteration method. Simulation examples show that the convergence speed of the natural gradient descent algorithm is the fastest one among three algorithms.

The generalised discrete algebraic Riccati equation in linear-quadratic optimal control

This paper investigates the properties of the solutions of the generalised discrete algebraic Riccati equation arising from the classic infinite-horizon linear quadratic (LQ) control problem. In particular, a geometric analysis is used to study the relationship existing between the solutions of the generalised Riccati equation and the output-nulling subspaces of the underlying system and the corresponding reachability subspaces. This analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem, which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil. In establishing the main results of this paper, several ancillary problems on the discrete Lyapunov equation and spectral factorisation are also addressed and solved.

On the numerical solution of large-scale sparse discrete-time Riccati equations

We discuss the numerical solution of large-scale discrete-time algebraic Riccati equations (DAREs) as they arise, e.g., in fully discretized linear-quadratic optimal control problems for parabolic partial differential equations (PDEs). We employ variants of Newton’s method that allow to compute an approximate low-rank factor of the solution of the DARE. The principal computation in the Newton iteration is the numerical solution of a Stein (aka discrete Lyapunov) equation in each step. For this purpose, we present a low-rank Smith method as well as a low-rank alternating-direction-implicit (ADI) iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.

Nonequilibrium Phase Transition in a Periodically DrivenXYSpin Chain

We present a general formulation of Floquet states of periodically time-dependent open Markovian quasifree fermionic many-body systems in terms of a discrete Lyapunov equation. Illustrating the technique, we analyze periodically kicked XY spin-½ chain which is coupled to a pair of Lindblad reservoirs at its ends. A complex phase diagram is reported with reentrant phases of long range and exponentially decaying spin-spin correlations as some of the system's parameters are varied. The structure of phase diagram is reproduced in terms of counting nontrivial stationary points of Floquet quasiparticle dispersion relation.