Showing papers on "Lyapunov equation published in 2008"
01 Dec 2008
TL;DR: The introduction of a Lyapunov function allows not only to study more deeply the known properties of finite time convergence and robustness to strong perturbations, but also to improve the performance by adding linear correction terms to the algorithm.
Abstract: In this paper a strong Lyapunov function is obtained, for the first time, for the super twisting algorithm, an important class of second order sliding modes (SOSM). This algorithm is widely used in the sliding modes literature to design controllers, observers and exact differentiators. The introduction of a Lyapunov function allows not only to study more deeply the known properties of finite time convergence and robustness to strong perturbations, but also to improve the performance by adding linear correction terms to the algorithm. These modification allows the system to deal with linearly growing perturbations, that are not endured by the basic super twisting algorithm. Moreover, the introduction of Lyapunov functions opens many new analysis and design tools to the higher order sliding modes research area.
843 citations
TL;DR: A new unifying framework for the optimal $\mathcal{H}_2$ approximation problem is developed using best approximation properties in the underlying Hilbert space and leads to a new set of local optimality conditions taking the form of a structured orthogonality condition.
Abstract: The optimal $\mathcal{H}_2$ model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal $\mathcal{H}_2$ approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov- and interpolation-based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation-based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for $\mathcal{H}_2$ model reduction. The formulation is based on finding a reduced order model that satisfies interpolation-based first-order necessary conditions for $\cHtwo$ optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.
607 citations
Book•
19 May 2008
TL;DR: In this article, the stability of linear systems variance of linear stochastic systems quadratic performance measure book organization is discussed, as well as robustness and sensitivity of the Lyapunov equation.
Abstract: Part 1 Introduction: stability of linear systems variance of linear stochastic systems quadratic performance measure book organization. Part 2 Continuous algebraic Lyapunov equation: explicit solutions solution sounds numerical solutions. Part 3 Discrete algebraic Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions. Part 4 Differential and difference Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions singularly perturbed and weakly coupled systems coupled differential equations. Part 5 Algebraic Lyapunov equation with small parameters: singularly perturbed continuous Lyapunov equation weakly coupled continuous Lyapunov equation singularly perturbed discrete systems recursive methods for weakly coupled discrete systems. Part 6 Robustness and sensitivity of the Lyapunov equation: stability robustness sensitivity of algebraic Lyapunov equation. Part 7 Iterative methods and parallel algorithms: Smith's algorithm ADI iterative method SOR iterative method parallel algorithms parallel algorithms for coupled Lyapunov equations. Part 8 Lyapunov iterations: Kleinman algorithm for Riccati equation Lyapunov iterations for jump linear systems Lyapunov iterations for Nash differential games Lyapunov iterations for output feedback control. Part 9 Concluding remarks: Sylvester equations related topics applications. Appendix: matrix inequalities.
375 citations
TL;DR: N numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linear‐quadratic optimal control problems, which arise from such systems with a sparse or structured state matrix and a relatively small number of inputs and outputs are studied.
Abstract: We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments.
335 citations
27 Mar 2008
TL;DR: In this article, a class of global Lyapunov functions is revisited and used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of multi-group models in mathematical epidemiology.
Abstract: A class of global Lyapunov functions is revisited and used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multi-group models in mathematical epidemiology. We show how the group structure of the models, as manifested in the derivatives of the Lyapunov function, can be completely described using graph theory.
303 citations
TL;DR: Finite-time stability and stabilization of retarded-type functional differential equations are developed and an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input is given.
279 citations
TL;DR: In this paper, the relationship between two classical approaches for quantitative ergodic properties, Lyapunov type controls and functional inequalities (of Poincare type), is studied. And explicit examples for diffusion processes are studied.
253 citations
TL;DR: This note proposes an alternative approach to low gain feedback design based on the solution of a parametric Lyapunov equation, which possesses the advantages of both the eigenstructure assignment approach and the ARE-based approach.
Abstract: Low gain feedback has found several applications in constrained control systems, robust control and nonlinear control. Low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero. Such feedback gains can be constructed either by an eigenstructure assignment algorithm or through the solution of a parametric algebraic Riccati equation (ARE). The eigenstructure assignment approach leads to feedback gains in the form of a matrix polynomial in the parameter, while the ARE approach requires the solution of an ARE for each value of the parameter. This note proposes an alternative approach to low gain feedback design based on the solution of a parametric Lyapunov equation. Such an approach possesses the advantages of both the eigenstructure assignment approach and the ARE-based approach. It also avoids the possible numerical stiffness in solving a parametric ARE and the structural decomposition of the open loop system that is required by the eigenstructure assignment approach.
220 citations
TL;DR: The key element in this work is the employment of an infinite-dimensional "backstepping" transformation, and the resulting complete Lyapunov function, for the infinite dimensional systems consisting of the state of the ODE plant and the delay state, which allows to establish inverse optimality of the modified feedback and its disturbance attenuation properties.
214 citations
TL;DR: It is proved that the kernel of the optimal feedback of each subsystem decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the system.
Abstract: In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant-dependent coupling functions over arbitrary graphs. We study the structural properties of optimal control problems with infinite-horizon linear quadratic criteria, by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD). These operators are a generalization of translation invariant operators used in the study of spatially invariant systems. We prove that given a control system with a state-space representation consisting of SD operators, the solution of operator Lyapunov and Riccati equations are SD. Furthermore, we show that the kernel of the optimal state feedback for each subsystem decays in the spatial domain, with the type of decay (e.g., exponential, polynomial or logarithmic) depending on the type of coupling between subsystems.
212 citations
TL;DR: The converse Lyapunov theorems are used to establish semiglobal practical robustness to slowly varying, weakly jumping parameters, to temporal regularization, and to the insertion of jumps according to an ldquoaverage dwell-timerdquo rule.
Abstract: It is shown that (pre)asymptotic stability, which generalizes asymptotic stability, of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth Lyapunov function. This result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small, state-dependent perturbations. As a special case, we state a converse Lyapunov theorem for systems with logic variables and use this result to establish input-to-state stabilization using hybrid feedback control. The converse Lyapunov theorems are also used to establish semiglobal practical robustness to slowly varying, weakly jumping parameters, to temporal regularization, to the insertion of jumps according to an ldquoaverage dwell-timerdquo rule, and to the insertion of flow according to a ldquoreverse average dwell-timerdquo rule.
TL;DR: This paper summarizes three Lyapunov functions in controlling closed quantum systems and their controller designing processes and gives the generalized condition of the largest invariant set in the original reference and the construction method of the imaginary mechanical quantity.
TL;DR: In this paper, the generalized Lyapunov equation approach is used to study stochastic stabilization/detectability with state-multiplicative noise and some practical test criteria for stoChastic stabilization and detectability are obtained.
Abstract: In this paper, the generalized Lyapunov equation approach is used to study stochastic stabilization/detectability with state-multiplicative noise. Some practical test criteria for stochastic stabilization and detectability, such as stochastic Popov-Belevitch-Hautus criterion for exact detectability, are obtained. Moreover, useful properties of the generalized Lyapunov equation are derived based on critical stability and exact detectability introduced in this paper. As applications, first, the stochastic linear quadratic regulator as well as the related generalized algebraic Riccati equation are discussed extensively. Second, the infinite horizon stochastic H 2/H infin control with state- and control-dependent noise is also investigated, which extends and improves the recently published results.
TL;DR: An efficient linear system reduction method that computes approximations to the controllability and observability gramians of large sparse power system descriptor models by exploiting the fact that a Lyapunov equation solution can be decomposed into low-rank Choleski factors, which are determined by the alternating direction implicit (ADI) method.
Abstract: This paper presents an efficient linear system reduction method that computes approximations to the controllability and observability gramians of large sparse power system descriptor models. The method exploits the fact that a Lyapunov equation solution can be decomposed into low-rank Choleski factors, which are determined by the alternating direction implicit (ADI) method. Advantages of the method are that it operates on the sparse descriptor matrices and does not require the computation of spectral projections onto deflating subspaces of finite eigenvalues, which are needed by other techniques applied to descriptor models. The gramians, which are never explicitly formed, are used to compute reduced-order state-space models for large-scale systems. Numerical results for small-signal stability power system descriptor models show that the new method is more efficient than other existing approaches.
TL;DR: This article discusses the efficient implementation of Hammarling's method and proposes among other algorithmic improvements a block variant, which is demonstrated to perform significantly better than existing implementations.
Abstract: This article is concerned with the efficient numerical solution of the Lyapunov equation ATX + XA = -C with a stable matrix A and a symmetric positive semidefinite matrix C of possibly small rank. We discuss the efficient implementation of Hammarling's method and propose among other algorithmic improvements a block variant, which is demonstrated to perform significantly better than existing implementations. An extension to the discrete-time Lyapunov equation ATXA - X = -C is also described.
TL;DR: A Lyapunov measure is proposed, shown to be a stochastic counterpart of stability just as an invariant measure is a counterpart of the attractor (recurrence) and useful for the study of more general (weaker and set-wise) notions of stability.
Abstract: This paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system. Based upon a stochastic representation of deterministic dynamics, a Lyapunov measure is proposed for these purposes. This measure is shown to be a stochastic counterpart of stability (transience) just as an invariant measure is a counterpart of the attractor (recurrence). It is a dual of the Lyapunov function and is useful for the study of more general (weaker and set-wise) notions of stability. In addition to the theoretical framework, constructive methods for computing approximations to the Lyapunov measures are presented. These methods are based upon set-oriented numerical approaches. Several equivalent descriptions, including a series formula and a system of linear inequalities, are provided for computational purposes. These descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets. Finally, in certain cases, the exact relationship between Lyapunov functions and Lyapunov measures is also given.
01 Dec 2008
TL;DR: It is shown that when non-monotonic Lyapunov functions exist, one can construct a more complicated function that decreases monotonically and it will be shown that tighter bounds on the joint spectral radius can be obtained using the technique.
Abstract: We relax the monotonicity requirement of Lyapunov?s theorem to enlarge the class of functions that can provide certificates of stability. To this end, we propose two new sufficient conditions for global asymptotic stability that allow the Lyapunov functions to increase locally, but guarantee an average decrease every few steps. Our first condition is non-convex, but allows an intuitive interpretation. The second condition, which includes the first one as a special case, is convex and can be cast as a semidefinite program. We show that when non-monotonic Lyapunov functions exist, one can construct a more complicated function that decreases monotonically. We demonstrate the strength of our methodology over standard Lyapunov theory through examples from three different classes of dynamical systems. First, we consider polynomial dynamics where we utilize techniques from sum-of-squares programming. Second, analysis of piecewise affine systems is performed. Here, connections to the method of piecewise quadratic Lyapunov functions are made. Finally, we examine systems with arbitrary switching between a finite set of matrices. It will be shown that tighter bounds on the joint spectral radius can be obtained using our technique.
TL;DR: In this article, stability properties of positive descriptor systems in the continuous-time as well as in the discrete-time case were discussed and characterisations of positivity and established generalised stability criteria for the case of positive descriptors.
TL;DR: For any s > 0, the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent as mentioned in this paper.
Abstract: We prove that for any s > 0 the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense subset of cocycles and, actually, vanishing Lyapunov exponents correspond to codimension-1. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation.
TL;DR: An unbiased optimal filter is developed in the linear least-mean-square sense, whose solution depends on the recursion of a Riccati equation and a Lyapunov equation.
Abstract: This paper is concerned with the optimal filtering problem for discrete-time stochastic linear systems with multiple packet dropouts, where the number of consecutive packet dropouts is limited by a known upper bound. Without resorting to state augmentation, the system is converted to one with measurement delays and a moving average (MV) colored measurement noise. An unbiased optimal filter is developed in the linear least-mean-square sense. Its solution depends on the recursion of a Riccati equation and a Lyapunov equation. A numerical example shows the effectiveness of the proposed filter.
TL;DR: A direct and an iterative method based on the Sherman–Morrison–Woodbury formula for linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator are described.
Abstract: 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.
TL;DR: In this article, the stability conditions of linear uncertain systems with time-varying delay are derived based on a generalized discretized Lyapunov functional approach, and the kernel of the functional, which is a function of two variables, is chosen as piecewise linear.
Abstract: This paper investigates the stability of linear uncertain systems with time-varying delay. Stability criteria are derived based on a generalized discretized Lyapunov functional approach. The kernel of the functional, which is a function of two variables, is chosen as piecewise linear. The stability conditions are written in the form of linear matrix inequalities. Numerical examples indicate significant improvements over the existing results.
01 Oct 2008
TL;DR: It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed and obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible.
Abstract: This paper is concerned with stability analysis and H infin decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of J interconnected discrete-time Takagi-Sugeno (T-S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The H infin controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.
TL;DR: An explicit parametrization of a finite-dimensional subset of the cone of Lyapunov functions is given, enforced using sum-of-squares polynomial matrices, which allows the computation to be formulated as a semidefinite program.
Abstract: We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.
TL;DR: In this article, it was shown that there exists no quadratic Lyapunov function for a certain class of linear agreement/consensus algorithms, a fact that had been numerically verified in.
Abstract: We provide an example proving that there exists no quadratic Lyapunov function for a certain class of linear agreement/consensus algorithms, a fact that had been numerically verified in . We also briefly discuss sufficient conditions for the existence of such a Lyapunov function.
TL;DR: Characterizations of internal notions of output stability for uncertain time-varying systems described by retarded functional differential equations are provided and results have been developed for systems with outputs in abstract normed linear spaces in order to allow outputs with no delay.
TL;DR: An extension of the well-known Filippov's solution concept, that is appropriate for 'open' systems so as to allow interconnections of DDS, is proposed, proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS and a (small gain) ISS interconnection theorem is derived.
TL;DR: In this article, a Lyapunov-based approach for the trajectory generation of an $N$-dimensional Schr{\"o}dinger equation in whole $\RR^N$ is proposed.
Abstract: A Lyapunov-based approach for the trajectory generation of an $N$-dimensional Schr{\"o}dinger equation in whole $\RR^N$ is proposed. For the case of a quantum particle in an $N$-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.
TL;DR: It is shown that control Lyapunov functions (CLFs) may be constructed on the domain of asymptotic null-controllability as viscosity solutions of a first order PDE that generalizes Zubov's equation.
Abstract: For finite-dimensional nonlinear control systems we study the relation between asymptotic null-controllability and control Lyapunov functions It is shown that control Lyapunov functions (CLFs) may be constructed on the domain of asymptotic null-controllability as viscosity solutions of a first order PDE that generalizes Zubov's equation The solution is also given as the value function of an optimal control problem from which several regularity results may be obtained
TL;DR: The proposed output feedback control law asymptotically stabilizes the open loop system with a larger domain of attraction and achieves better disturbance attenuation under energy and magnitude bounded disturbances.