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In this paper, we consider finding a low-rank approximation to the solution of a large-scale generalized Lyapunov matrix equation in the form of A X M + M X A = C , where A and M are symmetric positive definite matrices. An algorithm called an Increasing Rank Riemannian Method for Generalized Lyapunov Equation (IRRLyap) is proposed by merging the increasing rank technique and Riemannian optimization techniques on the quotient manifold R ∗ n × p / O p . To efficiently solve the optimization problem on R ∗ n × p / O p , a line-search-based Riemannian inexact Newton’s method is developed with its global convergence and local superlinear convergence rate guaranteed. Moreover, we derive a preconditioner which takes M ≠ I into consideration. Numerical experiments show that the proposed Riemannian inexact Newton’s method and the preconditioner have superior performance, and that IRRLyap is preferable compared to the tested state-of-the-art methods when the lowest rank solution is desired. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes an H2‐optimal model order reduction (MOR) method for bilinear systems based on the linear matrix inequality (LMI) approach. In this method, to reduce the computational complexity, at first, a reduced middle‐order approximation of the system is derived based on common bilinear MOR methods. Next, the H2 norm of the error system is minimized to obtain the reduced‐order bilinear model. Generalized Lyapunov equations are added to the optimization problem as LMI constraints to guarantee the specification of type II Gramians of the bilinear system to improve accuracy. Besides, two stability conditions are included to the optimization problem as its constraints to preserve stability of reduced‐order bilinear model. One of advantages of the proposed method is the need for only one of the Gramians of controllability or observability. Since the proposed H2‐optimal MOR problem is a polynomial matrix inequality (PMI) problem, an iterative method is used to convert the PMI to the LMI problems and solve the optimization problem. Three bilinear test systems are considered to show the proposed method's efficiency, while its performance is compared with some classical methods. Results show that the proposed methods lead to a more accurate reduced‐order model than other MOR methods.

In the present article, we consider the problem on location of the matrix spectrum with respect to a parabola. In terms of solvability of a matrix Lyapunov type equation, we prove theorems on location of the matrix spectrum in certain domains (bounded by a parabola) and (lying outside the closure of ). A solution to the matrix equation is constructed. We use this equation and prove an analog of the Lyapunov–Krein theorem on dichotomy of the matrix spectrum with respect to a parabola. [ABSTRACT FROM AUTHOR]

This paper proposes an H2‐optimal model order reduction (MOR) method for bilinear systems based on the linear matrix inequality (LMI) approach. In this method, to reduce the computational complexity, at first, a reduced middle‐order approximation of the system is derived based on common bilinear MOR methods. Next, the H2 norm of the error system is minimized to obtain the reduced‐order bilinear model. Generalized Lyapunov equations are added to the optimization problem as LMI constraints to guarantee the specification of type II Gramians of the bilinear system to improve accuracy. Besides, two stability conditions are included to the optimization problem as its constraints to preserve stability of reduced‐order bilinear model. One of advantages of the proposed method is the need for only one of the Gramians of controllability or observability. Since the proposed H2‐optimal MOR problem is a polynomial matrix inequality (PMI) problem, an iterative method is used to convert the PMI to the LMI problems and solve the optimization problem. Three bilinear test systems are considered to show the proposed method's efficiency, while its performance is compared with some classical methods. Results show that the proposed methods lead to a more accurate reduced‐order model than other MOR methods. An H2‐optimal model‐order‐reduction (MOR) method proposed for bilinear systems. The H2 norm of the error system minimized for this purpose. Some linear matrix inequality (LMI) constraints added to guarantee the specification of the bilinear system Gramians. Since, proposed H2‐optimal MOR‐problem is a polynomial matrix inequality (PMI) problem, an iterative method proposed to convert the PMI to LMI‐problem and solve it. [ABSTRACT FROM AUTHOR]

Background and Objectives: This paper proposes a new Model Order Reduction (MOR) method based on the Bilinear Balanced Truncation (BBT) approach. In the BBT method, solving the generalized Lyapunov equations is necessary to determine the bilinear system's controllability and observability Gramians. Since the bilinear systems are generally of high order, the computation of the Gramians of controllability and observability have huge computational volumes. In addition, the accuracy of reduced-order model obtained by BT is relatively low. In fact, the balanced truncation method is only available for local energy bands due to the use of type I Gramians. In this paper, BBT based on type II controllability and observability Gramians would be considered to fix these drawbacks. Methods: At first, a new iterative method is proposed for determining the proper order for the reduced-order bilinear model, which is related to the number of Hankel singular values of the bilinear system whose real parts are closest to origin and have the most significant amount of energy. Then, the problem of determining of type II controllability and observability Gramians of the high-order bilinear system have been formulated as a constrained optimization problem with some Linear Matrix Inequality (LMI) constraints for an intermediate middle-order system. Then, the achieved Gramians are applied to the BBT method to determine the reduced-order model of the bilinear system. Next, the steady state accuracy of the reduced model would be improved via employing a tuning factor. Results: Using the concept of type II Gramians and via the proposed method, the accuracy of the proposed bilinear BT method is increased. For validation of the proposed method, three high-order bilinear models are approximated. The achieved results are compared with some well-known MOR approaches such as bilinear BT, bilinear Proper Orthogonal Decomposition (POD) and Bilinear Iterative Rational Krylov subspace Algorithm (BIRKA) methods. Conclusion: According to the obtained results, the proposed MOR method is superior to classical bilinear MOR methods, but is almost equivalent to BIRKA. It is out-performance respecting to BIRKA is its guaranteed stability and convergence. [ABSTRACT FROM AUTHOR]

The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation of the controllability and observability Gramians of continuous linear and bilinear stationary systems with many inputs and one output, based on the method of spectral expansion of the Gramians and the iterative method for computing the bilinear systems Gramians. An important feature of the concept is the idea of separability of the Gramians expansion: separate computation of the scalar and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation to calculation of numerical sequences of their elements. For the continuous linear systems with one output the method and the algorithm of the spectral decomposition of the controllability Gramian are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the Gramian matrices are obtained, and by making use of which the rest of the elements can be calculated. For continuous linear systems with many outputs the spectral decompositions of the Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly reduce the number of calculations. The obtained results are generalized for continuous bilinear systems with one output. Iterative spectral algorithms for computation of elements of Gramians of these systems are proposed. Examples are given that illustrate theoretical results. [ABSTRACT FROM AUTHOR]

The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order. [ABSTRACT FROM AUTHOR]

We study and compare two different model reduction techniques for bilinear systems, specifically generalized balancing and H2-based model reduction, and apply it to semi-discretized controlled Fokker-Planck and Liouville–von Neumann equations. For this class of transport equations, the control enters the dynamics as an advection term that leads to the bilinear form. A specific feature of the systems is that they are stable, but not asymptotically stable, and we discuss aspects regarding structure and stability preservation in some depth as these aspects are particularly relevant for the equations of interest. Another focus of this article is on the numerical implementation and a thorough comparison of the aforementioned model reduction methods. [ABSTRACT FROM AUTHOR]

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton–Jacobi–Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker–Planck equation is also provided. [ABSTRACT FROM AUTHOR]

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium. [ABSTRACT FROM AUTHOR]

We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics can be split into a fast and a slow subspace and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity. [ABSTRACT FROM AUTHOR]

We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In this paper, we describe a direct and an iterative method based on this idea. The direct method is applicable if the generalized Lyapunov operator is a low-rank perturbation of a standard Lyapunov operator; it is related to the Sherman–Morrison–Woodbury formula. The iterative method requires a stability assumption; it uses convergent regular splittings, an alternate direction implicit iteration as preconditioner, and Krylov subspace methods. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study generalized Lyapunov equations and present generalizations of Lyapunov stability theorems and some matrix inertia theorems for matrix pencils. We discuss applications of generalized Lyapunov equations with special right-hand sides in stability theory and control problems for descriptor systems. [Copyright &y& Elsevier]

We discuss the numerical solution and perturbation theory for the generalized continuous-time Lyapunov equation E*XA+A*XE=−G with a singular matrix E. If this equation has a solution, it is not unique. We generalize a Bartels–Stewart method and a Hammarling method to compute a partial solution of the generalized Lyapunov equation with a special right-hand side. A spectral condition number is introduced and perturbation bounds for such an equation are presented. Numerical examples are given. [Copyright &y& Elsevier]

We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method. This approach has already been proposed to solve standard Lyapunov equations in several publications. The extension to the generalized case is straightforward. We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor. The basic computational tools of the method are basic linear algebra operations that can be implemented efficiently on modern computer architectures and in particular on parallel computers. Hence, a considerable speed-up as compared to the Bartels–Stewart and Hammarling methods is to be expected. We compare the algorithms by performing a variety of numerical tests. [ABSTRACT FROM AUTHOR]