Showing papers on "Lyapunov equation published in 2011"

A Lyapunov Function for Economic Optimizing Model Predictive Control

Abstract: Standard model predictive control (MPC) yields an asymptotically stable steady-state solution using the following procedure. Given a dynamic model, a steady state of interest is selected, a stage cost is defined that measures deviation from this selected steady state, the controller cost function is a summation of this stage cost over a time horizon, and the optimal cost is shown to be a Lyapunov function for the closed-loop system. In this technical note, the stage cost is an arbitrary economic objective, which may not depend on a steady state, and the optimal cost is not a Lyapunov function for the closed-loop system. For a class of nonlinear systems and economic stage costs, this technical note constructs a suitable Lyapunov function, and the optimal steady-state solution of the economic stage cost is an asymptotically stable solution of the closed-loop system under economic MPC. Both finite and infinite horizons are treated. The class of nonlinear systems is defined by satisfaction of a strong duality property of the steady-state problem. This class includes linear systems with convex stage costs, generalizing previous stability results and providing a Lyapunov function for economic MPC or MPC with an unreachable setpoint and a linear model. A nonlinear chemical reactor example is provided illustrating these points.

Direct and Indirect Couplings in Coherent Feedback Control of Linear Quantum Systems

Abstract: The purpose of this paper is to study and design direct and indirect couplings for use in coherent feedback control of a class of linear quantum stochastic systems. A general physical model for a nominal linear quantum system coupled directly and indirectly to external systems is presented. Fundamental properties of stability, dissipation, passivity, and gain for this class of linear quantum models are presented and characterized using complex Lyapunov equations and linear matrix inequalities (LMIs). Coherent H∞ and LQG synthesis methods are extended to accommodate direct couplings using multistep optimization. Examples are given to illustrate the results.

Strict Lyapunov functions for semilinear parabolic partial differential equations

Abstract: For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

Lyapunov Differential Equation Approach to Elliptical Orbital Rendezvous with Constrained Controls

Abstract: and energy. A parametric Lyapunov differential equation approach is proposed in this paper to solve this constrained regulation problem. After establishing the fact that the Tschauner-Hempel equations are both null controllable with controls of bounded magnitude and energy, this paper proves that the proposed linear periodic controllersemigloballystabilizesthesystem.Equivalently,forany fixedinitialconditions,themagnitudeandenergy of the control can be made as small as desired by tuning some free parameters in the feedback laws. In comparison with the existing quadratic-regulation-based approach, which requires solutions to nonlinear Riccati differential equations,thenewapproachrequiresonlythesolutionoflinearperiodicLyapunovdifferentialequations,whichare investigated inthepaperbyusingtheperiodicgenerator approach.Numericalsimulationsofthe nonlinearmodelof the spacecraft rendezvous instead of a linearized one show that both the magnitude and energy of the control can be reduced to an arbitrarily small level by reducing the values of some parameters in the controller and that the rendezvous mission can be accomplished satisfactorily.

Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation

Abstract: For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for a long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit (ADI) recurrence. In this paper we develop a convergence analysis of the rational Krylov subspace method (RKSM) based on the Kronecker product formulation and on potential theory. Moreover, we propose new enlightening relations between this approach and the ADI method. Our results provide solid theoretical ground for recent numerical evidence of the superiority of RKSM over ADI when the involved parameters cannot be computed optimally, as is the case in many practical application problems.

Optimal Linear Estimators for Systems With Random Sensor Delays, Multiple Packet Dropouts and Uncertain Observations

Abstract: This paper is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with random sensor delays, packet dropouts and uncertain observations. We develop a unified model to describe the mixed uncertainties of random delays, packet dropouts and uncertain observations by three Bernoulli distributed random variables with known distributions. Based on the proposed model, the optimal linear estimators that only depend on probabilities are developed via an innovation analysis approach. Their solutions are given in terms of a Riccati equation and a Lyapunov equation. They can deal with the optimal linear filtering, prediction and smoothing for systems with random sensor delays, packet dropouts and uncertain observations in a unified framework. Simulation results show the effectiveness of the proposed optimal linear estimators.

Characterizing and Computing the ${\cal H}_{2}$ Norm of Time-Delay Systems by Solving the Delay Lyapunov Equation

Abstract: It is widely known that the solutions of Lyapunov equations can be used to compute the H2 norm of linear time-invariant (LTI) dynamical systems. In this paper, we show how this theory extends to dynamical systems with delays. The first result is that the H2 norm can be computed from the solution of a generalization of the Lyapunov equation, which is known as the delay Lyapunov equation. From the relation with the delay Lyapunov equation we can prove an explicit formula for the H2 norm if the system has commensurate delays, here meaning that the delays are all integer multiples of a basic delay. The formula is explicit and contains only elementary linear algebra operations applied to matrices of finite dimension. The delay Lyapunov equations are matrix boundary value problems. We show how to apply a spectral discretization scheme to these equations for the general, not necessarily commensurate, case. The convergence of spectral methods typically depends on the smoothness of the solution. To this end we describe the smoothness of the solution to the delay Lyapunov equations, for the commensurate as well as for the non-commensurate case. The smoothness properties allow us to completely predict the convergence order of the spectral method.

Analytical and numerical Lyapunov functions for SISO linear control systems with first‐order reset elements

Abstract: In this paper we provide analytical and numerical Lyapunov functions that prove stability and performance of a First-order Reset Element (FORE) in feedback interconnection with a SISO linear plant. The Lyapunov functions also allow to establish finite gain ℒ2 stability from a disturbance input acting at the input of the plant to the plant output. ℒ2 stability is established by giving a bound on the corresponding ℒ2 gains. The proof of stability and performance is carried out by showing that the Lyapunov functions constructed here satisfy the sufficient conditions in the main results of Nesic et al. (Automatica 2008; 44(8):2019–2026). In the paper we also point out and illustrate via a counterexample an analysis subtlety overlooked in the preliminary results of Zaccarian et al. (American Control Conference, Portland, OR, U.S.A., 2005). Copyright © 2010 John Wiley & Sons, Ltd.

A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function

Abstract: We give a simple, explicit example of a two-dimensional polynomial vector field that is globally asymptotically stable but does not admit a polynomial Lyapunov function.

Linear programming based Lyapunov function computation for differential inclusions

Abstract: We present a numerical algorithm for computing Lyapunov functions for a class of strongly asymptotically stable nonlinear differential inclusions which includes spatially switched systems and systems with uncertain parameters The method relies on techniques from nonsmooth analysis and linear programming and constructs a piecewise affine Lyapunov function We provide necessary background material from nonsmooth analysis and a thorough analysis of the method which in particular shows that whenever a Lyapunov function exists then the algorithm is in principle able to compute it Two numerical examples illustrate our method

Convergence analysis of the extended Krylov subspace method for the Lyapunov equation

Abstract: The extended Krylov subspace method has recently arisen as a competitive method for solving large-scale Lyapunov equations. Using the theoretical framework of orthogonal rational functions, in this paper we provide a general a priori error estimate when the known term has rank-one. Special cases, such as symmetric coefficient matrix, are also treated. Numerical experiments confirm the proved theoretical assertions.

Fractional differential equations and Lyapunov functionals

Abstract: We consider a scalar fractional differential equation, write it as an integral equation, and construct several Lyapunov functionals yielding qualitative results about the solution. It turns out that the kernel is convex with a singularity and it is also completely monotone, as is the resolvent kernel. While the kernel is not integrable, the resolvent kernel is positive and integrable with an integral value of one. These kernels give rise to essentially different types of Lyapunov functionals. It is to be stressed that the Lyapunov functionals are explicitly given in terms of known functions and they are differentiated using Leibniz’s rule. The results are readily accessible to anyone with a background of elementary calculus.

An Error Analysis for Rational Galerkin Projection Applied to the Sylvester Equation

Abstract: In this paper we suggest a new formula for the residual of Galerkin projection onto rational Krylov spaces applied to a Sylvester equation, and establish a relation to three different underlying extremal problems for rational functions. These extremal problems enable us to compare the size of the residual for the above method with that obtained by ADI. In addition, we deduce several new a priori error estimates for Galerkin projection onto rational Krylov spaces, both for the Sylvester and for the Lyapunov equation.

On linear copositive Lyapunov functions for switched positive systems

Abstract: This paper addresses the existence of a common linear copositive Lyapunov function for discrete-time switched positive systems. Our results reveal that the existence of such a function is equivalent to the Schur stability of a kind of special matrices, these matrices consist of column vector of system matrices in an appropriate manner. A simple example is provided to illustrate the implication of our results.

Lyapunov Balancing for Passivity-Preserving Model Reduction of RC Circuits ∗

Abstract: We apply a Lyapunov-based balanced truncation model reduction method to differential-algebraic equations arising in modeling of RC circuits. This method is based on diagonalizing the solution of one projected Lyapunov equation. It is shown that this method preserves passivity and delivers an error bound. By making use of the special structure of circuit equations, we can reduce the numerical effort for balanced truncation drastically.

Analysis of the joint spectral radius via lyapunov functions on path-complete graphs

Abstract: We study the problem of approximating the joint spectral radius (JSR) of a finite set of matrices. Our approach is based on the analysis of the underlying switched linear system via inequalities imposed between multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, maximum/minimum-of-quadratics Lyapunov functions. We characterize all path-complete graphs consisting of two nodes on an alphabet of two matrices and compare their performance. For the general case of any set of n x n matrices we propose semidefinite programs of modest size that approximate the JSR within a multiplicative factor of 1/4√n of the true value. We establish a notion of duality among path-complete graphs and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

On the Hermitian Positive Definite Solutions of Nonlinear Matrix Equation

Abstract: Nonlinear matrix equation 𝑋 𝑠 + 𝐴 ∗ 𝑋 − 𝑡 1 𝐴 + 𝐵 ∗ 𝑋 − 𝑡 2 𝐵 = 𝑄 has many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equation 𝑋 𝑠 + 𝐴 ∗ 𝑋 − 𝑡 1 𝐴 + 𝐵 ∗ 𝑋 − 𝑡 2 𝐵 = 𝑄 are considered, where 𝑄 is a Hermitian positive definite matrix, 𝐴 , 𝐵 are nonsingular complex matrices, 𝑠 is a positive number, and 0 𝑡 𝑖 ≤ 1 , 𝑖 = 1 , 2 . Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and numerical example is given to show the efficiency of the proposed iterative method.

Converse results on existence of sum of squares Lyapunov functions

Abstract: Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming can fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a homogeneous polynomial vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos. The latter result is extended to develop a converse sos Lyapunov theorem for robust stability of switched linear systems.

Examples of pipeline monitoring with nonlinear observers and real-data validation

Abstract: This article shows how nonlinear observers can be used as tools for the monitoring of pipelines. In particular two observer approaches for two different applications are presented: a one-leak detection and isolation problem on the one the hand, and the same problem with friction estimation in addition on the other hand. In the first case, the system which represents the pipeline with a leak satisfies some uniform observability condition allowing for the design of a classical high gain observer (with a static Lyapunov equation). In the second case, the system is no longer uniformly observable, but still satisfies the observability rank condition, and an Extended Kalman Filter is proposed, under the use of exciting inputs. In both cases, experimental results are provided.

Lyapunov stability analysis of higher-order 2-D systems

Abstract: In this paper we prove a necessary and sufficient condition for the asymptotic stability of a 2-D system described by a system of higher-order linear partial difference equations. We show that asymptotic stability is equivalent to the existence of a vector Lyapunov functional satisfying certain positivity conditions together with its divergence along the system trajectories. We use the behavioral framework and the calculus of quadratic difference forms based on four-variable polynomial algebra.

Discrete-time nonlinear systems inverse optimal control: A control Lyapunov function approach

Abstract: This paper presents an inverse optimal control approach for exponential stabilization of discrete-time nonlinear systems, avoiding to solve the associated Hamilton-Jacobi-Bellman (HJB) equation, and minimizing a meaningful cost function. This stabilizing optimal controller is based on a discrete-time control Lyapunov function. The applicability of the proposed approach is illustrated via simulations by stabilization of an example.