Generalized Lyapunov Equations Research Articles

Pareto Optimal Strategy Under H∞ Constraint for the Mean-Field Stochastic Systems in Infinite Horizon.

This article focuses on the mean-field linear-quadratic Pareto (MF-LQP) optimal strategy design for stochastic systems in infinite horizon, which is with the H∞ constraint when the system is disturbed by external interferences. The stochastic bounded real lemma (SBRL) with any initial state in infinite horizon is first investigated based on the stabilizing solution of the generalized algebraic Riccati equation (GARE). Then, by discussing the convexity of the cost functional, the stochastic indefinite MF-LQP control problem is defined and solved based on the MF-LQ theory and Pareto theory. When the worst case disturbance is considered in the collaborative multiplayer system, we show that the Pareto optimal strategy design with H∞ constraint [or robust Pareto optimal strategy, (RPOS)] can be given via solving two coupled GAREs. When the worst case disturbance and the Pareto efficient strategy work, all Pareto solutions are obtained by a generalized Lyapunov equation. Finally, a practical example shows that the obtained results are effective.

Balanced truncation reduced models of quadratic‐bilinear systems in time interval

AbstractAccumulative error along with time impedes the application of many computational methods on high‐precision simulation; therefore, time‐interval methods emerge. In this paper, a new model reduction method of quadratic‐bilinear (QB) systems based on time‐interval Gramians (TIGs) is presented. The conditions for generalized Lyapunov equations whose solutions are exactly TIGs to be solvable are derived. Lyapunov stability and error bound are discussed to demonstrate the succession and advance of time‐interval balanced truncation (TIBT). The accuracy and robustness are improved, as is illustrated in numerical results.

Robust weighted fusion Kalman estimators for systems with uncertain noise variances, multiplicative noises, missing measurements, packets dropouts and two‐step random measurement delays

AbstractFor multisensor networked systems with uncertain noise variances, multiplicative noises and multiple networked‐induced uncertainties including missing measurements, packets dropouts and two‐step random measurement delays, the robust weighted fusion estimation problem is addressed in this article. More precisely, the system noise variances are assumed to be uncertain but bounded, the other four uncertainties are compensated into fictitious white noise by the proposed model transformation method, which includes the augmented method and extended fictitious noise technique. Then local multi‐model system is obtained, for which robust local Kalman estimator is obtained based on the minimax robust estimation principle and unified estimation method. Based on this, the six robust weighted fusion time‐varying Kalman estimators are presented in a unified form, which include robust weighted fusers weighted by matrices, diagonal, scalars, and a robust covariance intersection (CI) fuser and two fast CI (FCI) fusers. The robustness proving method, including the extended Lyapunov equation approach with two kinds of generalized Lyapunov equations, non‐negative matrix factorization and elementary transformation of matrix, is presented to prove that the actual estimation error variances are guaranteed to have minimal upper bounds for all admissible uncertainties. The accuracy relations are proved. Further, the robust local and fused steady‐state Kalman estimators are presented. Finally, a simulation example applied to Internet‐based three tank water system is given to demonstrate effectiveness of the proposed results.

H2model order reduction of bilinear systems via linear matrix inequality approach

AbstractThis 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 H2norm 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.

Robust Finite-Time Zeroing Neural Networks With Fixed and Varying Parameters for Solving Dynamic Generalized Lyapunov Equation

For solving dynamic generalized Lyapunov equation, two robust finite-time zeroing neural network (RFTZNN) models with stationary and nonstationary parameters are generated through the usage of an improved sign-bi-power (SBP) activation function (AF). Taking differential errors and model implementation errors into account, two corresponding perturbed RFTZNN models are derived to facilitate the analyses of robustness on the two RFTZNN models. Theoretical analysis gives the quantitatively estimated upper bounds for the convergence time (UBs-CT) of the two derived models, implying a superiority of the convergence that varying parameter RFTZNN (VP-RFTZNN) possesses over the fixed parameter RFTZNN (FP-RFTZNN). When the coefficient matrices and perturbation matrices are uniformly bounded, residual error of FP-RFTZNN is bounded, whereas that of VP-RFTZNN monotonically decreases at a super-exponential rate after a finite time, and eventually converges to 0. When these matrices are bounded but not uniform, residual error of FP-RFTZNN is no longer bounded, but that of VP-RFTZNN still converges. These superiorities of VP-RFTZNN are illustrated by abundant comparative experiments, and its application value is further proved by an application to robot.

Robust integrated sequential covariance intersection fusion Kalman filters and their convergence and stability for networked sensor systems with five uncertainties

AbstractFor networked sensor systems (NSSs) with hard and soft sensors including five uncertainties, two universal approaches of solving the robust fusion estimation problems are presented. It includes an integrated sequential covariance intersection (SCI) fusion minimax robust Kalman filtering approach with cross‐covariance information and a generalized Lyapunov equation approach with four pairs of Lyapunov equations. Applying them, the robust local and SCI fused time‐varying and steady‐state Kalman filters are presented in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds. The equivalent batch SCI fusers are also presented. Their robustness and accuracy relations are proved, and the sensitivity of the SCI fuser with respect to the fused orders of sensors is analyzed. Applying the dynamic error system analysis method and the dynamic variance error system analysis method, a new convergence and absolute asymptotic stability theory of robust fusion Kalman filtering is presented. The classical Kalman filtering convergence and stability theory is developed. Compared with the original covariance intersection fuser, they significantly reduced the computational complexity and burden. Compared with the optimal and conservative SCI fusers, they significantly improved the robust accuracies. They are suitable to deal with asynchronous or random delayed data and are suitable for real‐time applications. A simulation applied to the two‐mass spring damper mechanical system shows their effectiveness.

Indefinite Mean-Field Stochastic Cooperative Linear-Quadratic Dynamic Difference Game With Its Application to the Network Security Model.

In this article, we show how to obtain all of the Pareto optimal decision vectors and solutions for the finite horizon indefinite mean-field stochastic cooperative linear-quadratic (LQ) difference game. First, the equivalence between the solvability of the introduced N coupled generalized difference Riccati equations (GDREs) and the solvability of the multiobjective optimization problem is established. However, it is difficult to obtain Pareto optimal decision vectors based on the N coupled GDREs because the optimal joint strategy adopted by all players to optimize the performance criterion of some players in the game is different from the strategies of other players, which rely on the weighted matrices of cost functionals that may be different among players. Second, a necessary and sufficient condition is developed to guarantee the convexity of the costs, which makes the weighting technique not only sufficient but also necessary for searching Pareto optimal decision vectors. It is then shown that the mean-field Pareto optimality algorithm (MF-POA) is presented to identify, in principle, all of the Pareto optimal decision vectors and solutions via the solutions to the weighted coupled GDREs and the weighted coupled generalized difference Lyapunov equations (GDLEs), respectively. Finally, a cooperative network security game is reported to illustrate the results presented. Simulation results validate the solvability, correctness, and efficiency of the proposed algorithm.

Stochastic factors and string stability of traffic flow: Analytical investigation and numerical study based on car-following models

The emergence dynamics of traffic instability has always attracted particular attention. For several decades, researchers have studied the stability of traffic flow using deterministic traffic models, with less emphasis on the presence of stochastic factors. However, recent empirical and theoretical findings have demonstrated that the stochastic factors tend to destabilize traffic flow and stimulate the concave growth pattern of traffic oscillations. In this paper, we derive a string stability condition of a general stochastic continuous car-following model by the mean of the generalized Lyapunov equation. We have found, indeed, that the presence of stochasticity destabilizes the traffic flow. The impact of stochasticity depends on both the sensitivity to the gap and the sensitivity to the velocity difference. Numerical simulations of three typical car-following models have been carried out to validate our theoretical analysis. Finally, we have calibrated and validated the stochastic car-following models against empirical data. It is found that the stochastic car-following models reproduce the observed traffic instability and capture the concave growth pattern of traffic oscillations. Our results further highlight theoretically and numerically that the stochastic factors have a significant impact on traffic dynamics.

Input-output analysis of stochastic base flow uncertainty

We adopt an input-output approach to analyze the effect of persistent white-in-time structured stochastic base flow perturbations on the mean-square properties of the linearized Navier-Stokes equations. Such base flow variations enter the linearized dynamics as multiplicative sources of uncertainty that can alter the stability of the linearized dynamics and their receptivity to exogenous excitations. Our approach does not rely on costly stochastic simulations or adjoint-based sensitivity analysis. We provide verifiable conditions for mean-square stability and study the frequency response of the flow subject to additive and multiplicative sources of uncertainty using the solution to the generalized Lyapunov equation. For small-amplitude base flow perturbations, we bypass the need to solve large generalized Lyapunov equations by adopting a perturbation analysis. We use our framework to study the destabilizing effects of stochastic base flow variations in transitional parallel flows, and the reliability of numerically estimated mean velocity profiles in turbulent channel flows. We uncover the Reynolds number scaling of critically destabilizing perturbation variances and demonstrate how the wall-normal shape of base flow modulations can influence the amplification of various length scales. Furthermore, we explain the robust amplification of streamwise streaks in the presence of streamwise base flow variations by analyzing the dynamical structure of the governing equations as well as the Reynolds number dependence of the energy spectrum.

A New Method of Solving Special Solutions of Quaternion Generalized Lyapunov Matrix Equation

In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed and applied to the calculation of quaternion matrix equation. Using the H-representation method, the independent elements are extracted according to the structural characteristics of bisymmetric matrix and skew bisymmetric matrix, so as to simplify the operation process. Finally, it is compared with the real vector representation method of quaternion matrix equation to illustrate the effectiveness and superiority of the proposed method.

On the Solution of Generalized Lyapunov Equations for a Class of Continuous Bilinear Time-Varying Systems

On the Solution of Generalized Lyapunov Equations for a Class of Continuous Bilinear Time-Varying Systems

Numerical solution of stable generalized complex Lyapunov equations

Generalized Lyapunov equations are often encountered in systems theory, analysis and design of control systems, and in many applications, including balanced realization algorithms, procedures for reduced order models, or Newton methods for generalized algebraic Riccati equations. An important application is the computation of the Hankel singular values of a generalized dynamical system, whose behavior is defined by a regular matrix pencil (E, A), with E nonsingular. This application uses the controllability and observability Gramians of the system, given as the solutions of a pair of related generalized Lyapunov equations. For a stable system, the solutions of both equations are non-negative definite. The paper summarizes the numerical algorithms for complex continuous- and discrete-time generalized systems. Such solvers are not yet available in the SLICOT Library or MATLAB toolboxes, but could be an important addition. The developed solvers address the essential practical issues of reliability, accuracy, and efficiency.

Gramians, Energy Functionals, and Balanced Truncation for Linear Dynamical Systems With Quadratic Outputs

In this article, we investigate a balancing-based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. With this aim, we propose a new algebraic observability Gramian for the system based on the Hilbert space adjoint theory. We then show that the proposed Gramian satisfies a particular type of generalized Lyapunov equation and we investigate its connection to energy functionals. It allows one to find states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order models. Finally, based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_2$</tex-math></inline-formula> energy considerations, we derive error bounds, depending on the neglected singular values. We demonstrate the efficiency of the proposed methodology using a numerical example.

Pareto Optimality in Infinite Horizon Mean-Field Stochastic Cooperative Linear-Quadratic Difference Games

This paper is concerned with the mean-field stochastic cooperative linear-quadratic (LQ) dynamic difference game in an infinite time horizon. First, the necessary and sufficient conditions for the stability in the mean-square sense, and the stochastic Popov-Belevith-Hautus (PBH) eigenvector tests for exact observability and exact detectability of mean-field stochastic linear difference systems are derived by the <inline-formula><tex-math notation="LaTeX">$\mathscr {H}$</tex-math></inline-formula>-representation technique. Second, the relation between the solvability of the cross-coupled generalized Lyapunov equations&#x00A0;(CC-GLEs) and exact observability, exact detectability, and stability of the mean-field dynamic system is well characterized. It is then shown that the cross-coupled algebraic Riccati equations&#x00A0;(CC-AREs) admit a unique positive definite (positive semi-definite, respectively) solution under exact observability (exact detectability, respectively), which is also a feedback stabilizing solution. Furthermore, all Pareto optimal strategies and solutions can be respectively derived via the solutions to the weighted CC-AREs (WCC-AREs) and the weighted cross-coupled algebraic Lyapunov equations&#x00A0;(WCC-ALEs). Finally, a practical application on the computation offloading in the multi-access edge computing network (MECN) is presented to illustrate the proposed theoretical results.

On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be divided into parts (sub-Gramians) that characterize the measure of these generalized modes and their interactions. Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability and observability. We also propose an algorithm for computing the Gramians and sub-Gramians based on the element-wise computation of the solution matrix. Based on the proposed algorithm, a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed, which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear equations. Examples are provided that illustrate the application and practical use of the considered spectral decompositions.

Matrix iteration algorithms for solving the generalized Lyapunov matrix equation

In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms.

The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation

In this paper, we propose the generalized modified Hermitian and skew-Hermitian splitting (GMHSS) approach for computing the generalized Lyapunov equation. The GMHSS iteration is convergent to the unique solution of the generalized Lyapunov equation. Moreover, we discuss the convergence analysis of the GMHSS algorithm. Further, the inexact version of the GMHSS (IGMHSS) method is formulated to improve the GMHSS method. Finally, some numerical experiments are carried out to demonstrate the effectiveness and competitiveness of the derived methods

Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations

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.

H∞ Control of Discrete-Time Stochastic Systems With Borel-Measurable Markov Jumps

This paper is concerned with a kind of discrete-time stochastic systems with Markov jump parameters taking values in a Borel measurable set. First, both strong exponential stability and exponential stability in the mean square sense are introduced for the considered systems. Based on generalized Lyapunov equation and inequality, necessary and sufficient conditions are derived for the strong exponential stability. By use of the given stability criteria, it is shown that strong exponential stability can lead to exponential stability and further to stochastic stability. Moreover, strong exponential stability can guarantee the so-called $l_{2}$ input-state stability, which characterizes the asymptotic behavior of system state influenced by exogenous disturbance with finite energy. Second, $H_{\infty }$ performance is analyzed for the perturbed dynamic models over finite and infinite horizons, respectively. For a prescribed disturbance attenuation level, stochastic bound real lemmas are presented in terms of Riccati equations or linear matrix inequalities. As a direct application, the infinite-horizon $H_{\infty }$ control problem is settled and the state-feedback controller is constructed. Numerical simulations are conducted to illustrate the validity of the proposed results.

Analysis of a bilinear model of an electric power system using spectral decompositions of Lyapunov functions

In this paper, the spectral decompositions of Lyapunov functions are applied for the first time to the analysis of the behavior of a bilinear model of a two-area electric power system. In contrast to the technique of normal forms and modal series methods, we consider the spectral decomposition not for the dynamics of state variables, but for Lyapunov functions, which characterize the L2-norms of variables or signals in the time domain. The solution of the generalized Lyapunov equation for a bilinear system is represented as the sum of Hermitian matrices corresponding to individual eigenvalues of the system or their pairwise combinations. An iterative algorithm for calculating spectral terms is developed for stable bilinear systems. In a test experiment for the purpose of transient stability analysis, we evaluate the value of individual eigenmodes and their pairwise combinations depending on the magnitude of bilinear terms. The obtained results are consistent with an intuitive interpretation derived from the model equations and eigenvalue analysis. The spectral decompositions of Lyapunov functions allowed us to indicate the range of applicability of linear model and to reveal dominant eigenmodes in the analysis of transient stability of electric power system.