Discrete Algebraic Lyapunov Equation Research Articles

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.

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.

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 the Measurement of Lower Solution Bounds of the Discrete Algebraic Lyapunov Equation

In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE are presented. We also propose numerical algorithms to develop sharper solution bounds. The obtained bounds can give a supplement to those appeared in the literature.

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.

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.

Upper bounds for the solution of the discrete algebraic Lyapunov equation

New upper bounds for the solution of the discrete algebraic Lyapunov equation (DALE) P= APA T+ Q are presented. The only restriction on their applicability is that A be stable; there are no restrictions on the singular values of A nor on the diagonalizability of A. The new bounds relate the size of P to the radius of stability of A. The upper bounds are computable when the large dimension of A make direct solution of the DALE impossible. The new bounds are shown to reflect the dependence of P on A better than previously known upper bounds.

Bounds for solutions of the discrete algebraic Lyapunov equation

A family of sharp, arbitrarily tight upper and lower matrix bounds for solutions of the discrete algebraic Lyapunov equation are presented. The lower bounds are tighter than previously known ones. Unlike the majority of previously known upper hounds, those derived here have no restrictions on their applicability. Upper and Lower bounds for individual eigenvalues and for the trace of the solution are found using the new matrix bounds. Sharp trace bounds not derivable from the matrix bounds are also presented.

Bounds for the solution of the discrete algebraic Lyapunov equation

New bounds for solutions of the discrete algebraic Lyapunov equation P = APA T + Q are derived. The new bounds are compared to existing ones and found to be of particular interest when A is non-normal.

Upper and lower bounds of the solutions of the discrete algebraic Riccati and Lyapunov matrix equations

Matrix bounds, upper and lower, for the solution of the discrete algebraic Riccati and Lyapunov equations respectively, are proposed in this paper. They are new or sharper than the majority of existing results. By making use of these new matrix bounds, the corresponding bounds for each eigenvalue, the trace and the determinant of these solution matrices are also presented. Comparisons are made between these bounds and the majority of those appearing in the literature. It is shown that the obtained results are tighter. Finally, we give illustrative examples to show that these estimates appear to be considerably stronger than previously available results in many cases.

Lower bounds for the solution of the discrete algebraic Lyapunov equation

Lower bounds for the determinant and for the trace of the matrix solution of the discrete algebraic Lyapunov equation are presented. The determinant is tighter than bounds in the literature, and the trace supplements them.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Upper bounds for the eigenvalues of the solution of the discrete algebraic Lyapunov equation

Upper bounds for summations including the trace, and for products including the determinant, of the eigenvalues of the solution of the discrete algebraic Lyapunov equation (DALE) are presented. All bounds are derived from the matrix series solution of the DALE. The majority of the bounds are tighter than those in the literature.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>