Showing papers on "Lyapunov equation published in 1994"
TL;DR: This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle and Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived in analyzing the stability of equilibria of differential equations with discontinuous right-hand side.
Abstract: This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand side such as in nonsmooth dynamic systems or variable structure control. >
886 citations
14 Dec 1994
TL;DR: In this article, a Lyapunov function yielding a stable switching rule is shown to exist as long as there exists a stable convex combination of the system matrices, and the use of this stable combination for other control strategies is explored.
Abstract: This paper discusses the problem of stabilizing a pair of switched linear systems. A control law is developed using a Lyapunov function having a piecewise continuous derivative. A Lyapunov function yielding a stable switching rule is shown to exist as long as there exists a stable convex combination of the system matrices. The use of this stable combination for other control strategies is explored. >
344 citations
TL;DR: This paper considers several methods for calculating low-rank approximate solutions to large-scale Lyapunov equations of the form $AP + PA' + BB' = 0$.
Abstract: This paper considers several methods for calculating low-rank approximate solutions to large-scale Lyapunov equations of the form $AP + PA' + BB' = 0$. The interest in this problem stems from model reduction where the task is to approximate high-dimensional models by ones of lower order. The two recently developed Krylov subspace methods exploited in this paper are the Arnoldi method [Saad, Math. Comput., 37 (1981), pp. 105–126] and the Generalised Minimum Residual method (GMRES) [Saad and Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869]. Exact expressions for the approximation errors incurred are derived in both cases. The numerical solution of the low-dimensional linear matrix equation arising from the GMRES method is discussed and an algorithm for its solution is proposed. Low rank solutions of discrete time Lyapunov equations and continuous time algebraic Riccati equations are also considered. Throughout this paper, the authors tackle problems in which B has more than one column with the...
316 citations
TL;DR: Conditions of existence of a homogeneous polynomial Lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box-bounded uncertainty are derived.
Abstract: In this note, we derive conditions of existence of a homogeneous polynomial Lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box-bounded uncertainty. Verification of these conditions is reduced to solving a convex minimization problem. We produce numerical examples that demonstrate significant improvement in estimates of admissible uncertainty bounds compared with estimates obtained via the most commonly used quadratic Lyapunov functions. >
144 citations
TL;DR: The purpose of this note is to point out that Baksalary and Puntanen's main result is incorrect and a counterexample is presented.
Abstract: To obtain estimates of solutions of Lyapunov and Riccati equations which frequently occur in the stability analysis and optimal control design in linear control theory, many researchers have attempted to determine upper and lower bounds for the product of two matrices in terms of the trace of one matrix and the eigenvalues of the other. Baksalary and Puntanen claimed ("An inequality for the trace of matrix product", ibid., vol. 37, no. 2, p. 239-40, 1992) that they had obtained a better estimate for the trace of the product of two matrices. The purpose of this note is to point out that their main result is incorrect and a counterexample is presented. >
130 citations
TL;DR: In this paper, a generalized 2-D Lyapunov equation was proposed to find the gap between "sufficiency" and "necessity" for a state-space digital filter to be stable.
Abstract: This paper describes an approach to the stability analysis of two-dimensional (2-D) digital filters that are modeled in the Fornashni-Marchesini state space using a class of generalized 2-D Lyapunov equations, the generalization was made based on the constant 2-D Lyapunov equation proposed recently by Hinamoto (see ibid., vol. 40, no. 2, p. 102-10, 1993). It is shown that the use of the generalized Lyapunov equations narrows the gap between "sufficiency" and "necessity" for a state-space digital filter to be stable, which occurs in Hinamoto's Lyapunov theorem. Feasible methods for finding numerical solutions of the generalized 2-D Lyapunov equation are also proposed. An example is included to illustrate the main results of the paper. >
117 citations
TL;DR: In this paper, it was shown that the set of matrices whose characteristic polynomials also lie in a polytopic set is strictly positive real if the Lyapunov matrices solving the equations featuring in the Kalman-Yakubovic-Popov Lemma are multiaffinely parameterized.
Abstract: This paper has three contributions. The first involves polytopes of matrices whose characteristic polynomials also lie in a polytopic set (e.g. companion matrices). We show that this set is Hurwitz or Schur invariant if there exist multiaffinely parameterized positive definite, Lyapunov matrices that solve an augmented Lyapunov equation. The second result concerns uncertain transfer functions with denominator and numerator belonging to a polytopic set. We show all members of this set are strictly positive real if the Lyapunov matrices solving the equations featuring in the Kalman-Yakubovic-Popov Lemma are multiaffinely parameterized. Moreover, under an alternative characterization of the underlying polytopic sets, the Lyapunov matrices for both of these results admit affine parameterizations. Finally, we apply the Lyapunov equation results to derive stability conditions for a class of linear time varying systems. >
113 citations
14 Dec 1994
TL;DR: In this article, a robust stability/performance test for linear systems with uncertain real parameters is proposed, where the fixed quadratic Lyapunov function is replaced by a Lyapinov function with affine dependence on the uncertain parameters, which can be used for both fixed or time-varying uncertain parameters.
Abstract: A new test of robust stability/performance is proposed for linear systems with uncertain real parameters. This test is an extension of the notion of quadratic stability where the fixed quadratic Lyapunov function is replaced by a Lyapunov function with affine dependence on the uncertain parameters. Admittedly with some conservatism, the construction of such parameter-dependent Lyapunov functions can be reduced to an linear matrix inequality (LMI) problem, hence is numerically tractable. This LMI-based test can be used for both fixed or time-varying uncertain parameters and is always less conservative than the quadratic stability test whenever the parameters cannot vary arbitrarily fast. Its also completely bypasses the frequency sweep required in real /spl mu/-analysis. >
109 citations
TL;DR: In this paper, a new class of globally asymptotically stabilizing feedback control laws for the complete (i.e., dynamics and kinematics) attitude motion of a rotating rigid body is given in terms of two new parameterizations of the rotation group derived using stereographic projection.
103 citations
TL;DR: In this article, the authors established general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds, and proved that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli.
Abstract: We establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C∞ Riemannian metric whose geodesic flow is Bernoulli.
94 citations
TL;DR: This work addresses the issue of integrating symmetric Riccati and Lyapunov matrix differential equations by showing first that using a direct algorithm limits the order of the numerical method to one if the authors want to guarantee that the computed solution stays positive definite.
Abstract: In this work we address the issue of integrating symmetric Riccati and Lyapunov matrix differential equations. In many cases -- typical in applications -- the solutions are positive definite matrices. Our goal is to study when and how this property is maintained for a numerically computed solution. There are two classes of solution methods: direct and indirect algorithms. The first class consists of the schemes resulting from direct discretization of the equations. The second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy. We show first that using a direct algorithm -- a one-step scheme or a strictly stable multistep scheme (explicit or implicit) -- limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite. Then we show two ways to obtain positive definite higher order approximations by using indirect algorithms. The first is to apply a symplectic integrator to an associated Hamiltonian system. The other uses stepwise linearization.
01 Jan 1994
TL;DR: In this paper, a subjective account of the main results in this area is presented, along with a discussion of the use of quadratic Lyapunov functions for the robust analysis and stabilization of uncertain systems.
Abstract: In recent years, considerable progress has been achieved in the use of quadratic Lyapunov functions for the robust analysis and stabilization of uncertain systems. This paper presents a subjective account of some of the main results in this area.
TL;DR: This construction of a family of Lyapunov functions is used to analyze robust stability with H 2 performance bounds for state space systems and generalizes the classical discrete-time Popov criterion.
TL;DR: In this article, sufficient conditions for the stability of the trivial solution of a nonlinear Hill's equation were derived based on the twist theorem and the Birkhoff normal form, respectively.
Abstract: Sufficient conditions for the stability of the trivial solution of a nonlinear Hill’s equation are obtained. As a consequence, the classical Lyapunov’s criterion for stability is extended to certain nonlinear differential equations.The proofs are based on the computation of the corresponding Birkhoff normal forms together with an application of the twist theorem.
TL;DR: In this paper, a hierarchy of stability tests, including absolute stability, difference inclusions, and interval matrices, is presented, and it is shown that these tests are not, in general, polynomial-time tests.
Abstract: Stability concepts arising in the literature on absolute stability, difference inclusions and interval matrices are all shown to be equivalent to simultaneous asymptotic stability of a class of linear time-varying discrete systems and, in turn, to exponential stability. This enables the classification of an interval matrix stability test due to Bauer et al., a Lyapunov indicator test due to Barabanov and a constructive Lyapunov function test due to Brayton and Tong into a hierarchy of stability tests. Some applications of these tests are given and it is observed that they are not, in general, polynomial-time tests.
TL;DR: A method of selecting the weighting and covariance matrices such that the optimal LQG controller is internally asymptotically stable is presented.
Abstract: The optimal LQG controller is known to stabilize the closed loop if certain mild conditions are satisfied. The controller itself, however, may be unstable. The paper presents a method of selecting the weighting and covariance matrices such that the optimal controller is internally asymptotically stable. The method is very easy to apply and for stable open-loop system involves the solution of a single Lyapunov equation. >
TL;DR: This work gives a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2*2 real matrices using only two types of matrices, A and B, which are constructed according to a stochastic process.
Abstract: Despite significant work since the original paper by H Furstenberg(1963), explicit formulae for Lyapunov exponents of infinite products of random matrices are available only in a very few cases. In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2*2 real matrices. All these products are constructed using only two types of matrices, A and B, which are chosen according to a stochastic process. The matrix A is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the matrix B, which allows us to write the Lyapunov exponent as a sum of convergent series. The key point is the computation of all the integer powers of B, which is achieved by a suitable change of frame. The computation then follows by looking at each of the special types of B (hyperbolic, parabolic and elliptic). Finally, we show, with an example, that the Lyapunov exponent is a discontinuous function of the given parameter.
29 Jun 1994
TL;DR: In this article, a converse Lyapunov function theorem is established for stability of systems with disturbances. But the result applies only to closed, not necessarily compact, invariant sets.
Abstract: Establishes a converse Lyapunov function theorem useful for studying stability of systems with disturbances. The result applies to global stability with respect to closed, not necessarily compact, invariant sets, and the resulting Lyapunov functions are smooth.
TL;DR: The method is based on the power method and matrix-vector multiplications and is particularly suitable for problems where those multiplications can be done efficiently, such as where the coefficient matrices are large and sparse or low-rank.
Abstract: Describes a simple method for efficiently estimating the dominant eigenvalues and eigenvectors of the solution to a Lyapunov equation, without first solving the equation explicitly. The method is based on the power method and matrix-vector multiplications and is particularly suitable for problems where those multiplications can be done efficiently, such as where the coefficient matrices are large and sparse or low-rank. The same idea is directly applicable to balanced-truncation order reduction of linear systems. >
29 Jun 1994
TL;DR: A new least-squares estimator with nonlinear data weighting is developed and used to adaptively control a simple discrete-time nonlinear system without assuming any growth conditions on the nonlinearities.
Abstract: A new least-squares estimator with nonlinear data weighting is developed and used to adaptively control a simple discrete-time nonlinear system. A global stability result is obtained through Lyapunov analysis without assuming any growth conditions on the nonlinearities.
TL;DR: In this article, a new control design method for the control of flexible systems that not only guarantees closed-loop asymptotic stability but also effectively suppresses vibration is presented.
Abstract: This paper presents a new control design method for the control of flexible systems that not only guarantees closed-loop asymptotic stability but also effectively suppresses vibration. This method allows integrated determination of actuator/sensor locations and feedback gain via minimization of an energy criterion, which is chosen as the integrated total energy stored in the system. The energy criterion is determined via an efficient solution of the Lyapunov equation and minimized with a quasi-Newton or recursive quadratic programming algorithm. The prerequisite for this optimal design method is that the controlled system be asymptotically stable. This study shows that when the controller structure is a collocated direct velocity feedback design with positive definite feedback gain, the number and placement of actuators/sensors are the only factors needed to determine necessary and sufficient conditions for ensuring closed-loop asymptotic stability. The application of this method to a simple flexible structure confirms the direct relationship between our optimization criterion and effectiveness in vibration suppression.
08 May 1994
TL;DR: Algorithms for continuous-time quadratic optimization of impedance control are presented and system stability is investigated according to Lyapunov function theory, and it is shown that global asymptotic stability holds.
Abstract: This paper presents algorithms for continuous-time quadratic optimization of impedance control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are found by solving an algebraic matrix equation. System stability is investigated according to Lyapunov function theory, and it is shown that global asymptotic stability holds. The solution results in design parameters in the form of square weighting matrices or impedance matrices as known from linear quadratic optimal control. The proposed optimal control is useful both for motion control and force control. >
TL;DR: Asymptotically stable linear systems subject to unstructured time varying perturbations are considered andable perturbation bounds are obtained such that the perturbed systems remain stable.
Abstract: Asymptotically stable linear systems subject to unstructured time varying perturbations are considered. Allowable perturbation bounds are obtained such that the perturbed systems remain stable. These bounds are derived iteratively by means of adjusting a sequence of Lyapunov matrices. In comparison with existing methods, less conservative quantitative measures of robustness are obtained. >
TL;DR: In this article, a simple method for designing stabilizing controllers for linear delay equations containing unmatched uncertainty is presented, which is based on differential inequalities rather than on Lyapunov stability theory.
Abstract: We present a simple method for designing stabilizing controllers for linear delay equations containing unmatched uncertainty. The methodology employed is based on differential inequalities rather than on Lyapunov stability theory.
TL;DR: Two methods are proposed for efficient numerical evaluation of the exact complex perturbation bound ν, which combines Byers' bisection method with a three-point-pattern optimization technique to compute ν.
Abstract: In this paper, we study the problem of stability robustness of two-dimensional discrete systems in a local state-space setting. Two methods are proposed for efficient numerical evaluation of the exact complex perturbation bound ν. The first method combines Byers' bisection method with a three-point-pattern optimization technique to compute ν. The second method utilizes a direct optimization technique to find the bound. In addition, a 2-D Lyapunov approach is proposed to obtain two lower bounds of ν, and numerical techniques for solving the constant 2-D Lyapunov equation involved are presented. The paper is concluded with an example illustrating the main results obtained.
29 Jun 1994
TL;DR: In this paper, the authors provide a self-contained, unified and extended treatment of the stability of matrix second-order systems, in addition to obtaining necessary and sufficient conditions for Lyapunov and asymptotic stability.
Abstract: This paper provides a self-contained, unified and extended treatment of the stability of matrix second-order systems. The results we obtain encompass numerous results from prior literature in addition'to several new results. Specifically, in addition to obtaining necessary and sufficient conditions for Lyapunov and asymptotic stability, we consider the case of semistability. Semistability is of particular interest in the analysis of vibrating systems in that it represents the case of "damped rigid body modes", that is, systems that eventually come to rest, although not necessarily at a specified equilibrium point. This paper presents the first treatment of semistability for matrix second-order systems.