Showing papers on "Lyapunov equation published in 2000"

Finite-Time Stability of Continuous Autonomous Systems

Abstract: Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

An integral inequality in the stability problem of time-delay systems

Abstract: An integral inequality is derived, and applied to the stability problem of time-delay systems using discretized Lyapunov functional formulation. As the result, a simpler stability criterion is derived.

A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions

Abstract: We consider dierential inclusions where a positive semidenite function of the solutions satises a class-KL estimate in terms of time and a second positive semidenite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-KL estimate, exists if and only if the class-KL estimate is robust, i.e., it holds for a larger, perturbed dierential inclusion. It remains an open question whether all class-KL estimates are robust. One sucient condition for robustness is that the original dierential inclusion is locally Lipschitz. Another sucient condition is that the two positive semidenite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for dierential equations and dierential inclusions that have appeared in the literature.

Lyapunov Characterizations of Input to Output Stability

Abstract: This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated.

Input-Output-to-State Stability

Abstract: This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case The main contribution of this work is to establish a complete equivalence between the IOSS property and the existence of a certain type of smooth Lyapunov function As corollaries, one shows the existence of "norm-estimators," and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates

Feedback Stabilization and Lyapunov Functions

Abstract: Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we employ it in order to construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A converse result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish that the feedback in question possesses a robustness property relative to measurement error, despite the fact that it may not be continuous.

Existence, uniqueness and boundedness results for impulsive delay differential equations

Abstract: This paper considers general impulsive delay differential equations. By utilizing a non-classical approach, the theory of existence and uniqueness of solutions are developed. Criteria on boundedness of solutions are also established through the use of Lyapunov functionals.

Stable adaptive control for a class of nonlinear systems using a modified Lyapunov function

Abstract: Investigates the adaptive control design for a class of nonlinear systems using Lyapunov's stability theory. The proposed method is developed based on a novel Lyapunov function, which removes the possible controller singularity problem in some of the existing adaptive control schemes using feedback linearization techniques. The resulting closed-loop system is proven to be globally stable, and the output tracking error converges to an adjustable neighborhood of zero.

Homogeneous State Feedback Stabilization of Homogenous Systems

Abstract: We show that for any asymptotically controllable homogeneous system in euclidean space (not necessarily Lipschitz at the origin) there exists a homogeneous control Lyapunov function and a homogeneous, possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system. If the system satisfies the usual local Lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate. If the system satisfies a global Lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate. The control Lyapunov function and the feedback are based on the Lyapunov exponents of a suitable auxiliary system and admit a numerical approximation.

A condition for the stability of switched nonlinear systems

Abstract: In this paper, we present a sufficient condition for the global asymptotic stability of a switched nonlinear system composed of a finite family of subsystems. We show that the global asymptotic stability of each subsystem and the pairwise commutation of the vector fields that define the subsystems (i.e., the Lie bracket of any pair of them is zero) are sufficient for the global asymptotic stability of the switched system. We also show that these conditions are sufficient for the existence of a common Lyapunov function.

On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities

Abstract: A sufficient condition for the stability of two-dimensional (2-D) systems described in the Fornasin-Marchesini model is presented in a succinct form. The condition is expressed as a Lyapunov-like matrix inequality involved with the parallel addition of two positive definite matrices.

Lyapunov exponents in random Boolean networks

Abstract: A new order parameter approximation to random boolean networks (RBN) is introduced, based on the concept of Boolean derivative. A statistical argument involving an annealed approximation is used, allowing to measure the order parameter in terms of the statistical properties of a random matrix. Using the same formalism, a Lyapunov exponent is calculated, allowing to provide the onset of damage spreading through the network and how sensitive it is to minimal perturbations. Finally, the Lyapunov exponents are obtained by means of dierent approximations: through distance method and a discrete variant of the Wolf’s method for continuous systems. c 2000 Elsevier Science B.V. All rights reserved.

A computationally efficient Lyapunov-based scheduling procedure for control of nonlinear systems with stability guarantees

Abstract: We propose an alternative to gain scheduling for stabilization of nonlinear systems. For a useful class of nonlinear systems, the characterization of a region of stability based on a control Lyapunov function is computationally tractable, in the sense that computation times vary polynomially with the state dimension for a fixed number of scheduling variables. Using this fact, we develop a procedure to expand the region of stability by constructing control Lyapunov functions to various trim points of the system. A Lyapunov-based control synthesis algorithm is used to construct a control law that guarantees closed-loop stability for initial conditions in the expanded region of state space. This control asymptotically recovers the optimal stability margin in the sense of a Lyapunov derivative, which in turn can be seen as a performance measure. Robustness to bounded disturbances and stabilization under bounded control are easily incorporated into this framework. In the worst case, the computational complexity of the analysis problem that develops in the new method is increased by an exponential in the disturbance dimension. Similarly, we can handle control constraints with an increase in computational complexity of no more than an exponential in the control dimension. We demonstrate the new control design procedure on an example.

On Assigning the Derivative of a Disturbance Attenuation Control Lyapunov Function

Abstract: We consider feedback design for nonlinear, multi-input affine control systems with disturbances and present results on assigning, by choice of feedback, a desirable upper bound to a given control Lyapunov function (clf) candidate's derivative along closed-loop trajectories. Specific choices for the upper bound are motivated by ℒ2 and ℒ∞ disturbance attenuation problems. The main result leads to corollaries on “backstepping” locally Lipschitz disturbance attenuation control laws that are perhaps implicitly defined through a locally Lipschitz equation. The results emphasize that only rough information about the clf is needed to synthesize a suitable controller. A dynamic control strategy for linear systems with bounded controls is discussed in detail.

Integral manifolds of singularly perturbed systems with application to rigid-link flexible-joint multibody systems

Abstract: In this paper, we first review results of integral manifolds of singularly perturbed non-linear differential equations. We then outline the basic elements of the integral manifold method in the context of control system design, namely, the existence of an integral manifold, its attractivity, and stability of the equilibrium while the dynamics are restricted to the manifold. Toward this end, we use the composite Lyapunov method and propose a new exponential stability result which gives, as a by-product, an explicit range of the small parameter for which exponential stability is guaranteed. The results are applied to the control problem of multibody systems with rigid links and flexible joints in which the inverse of joint stiffness plays the role of the small parameter. The proposed controller is a composite control law that consists of a fast component, as well as a slow component that was designed based on the integral manifold approach. We show that the proposed composite controller has the following properties: (i) it enables the exact characterization and computation of an integral manifold, (ii) it makes the manifold exponentially attractive, and (iii) it forces the dynamics of the reduced flexible system on the integral manifold to coincide with the dynamics of the corresponding rigid system (i.e. the one obtained by making stiffness very large) implying that any control law that stabilizes the rigid system would stabilize the dynamics of the flexible system on the manifold. We finally present a detailed stability analysis and give an explicit range of the joint stiffness, in terms of system parameters and controller gains, for which the established exponential stability is guaranteed.

On input-to-state stability for time varying nonlinear systems

Abstract: Input-to-state stability was introduced about 10 years ago. This notion is nowadays a central concept in the analysis of nonlinear systems. However, most theoretical developments dealt mainly with time invariant systems. In this work we study the Lyapunov characterizations of input-to-state stability for time varying nonlinear systems, and in particular, for periodic time varying systems. We also present a small gain theorem for time varying nonlinear systems.

Solving algebriac Riccati equations on parallel computers using Newton's method with exact line search

Abstract: We investigate the numerical solution of continuous-time algebraic Riccati equations via Newton's method on serial and parallel computers with distributed memory. We apply and extend the available theory for Newton's method endowed with exact line search to accelerate convergence. We also discuss a new stopping criterion based on recent observations regarding condition and error estimates. In each iteration step of Newton's method a stable Lyapunov equation has to be solved. We propose to solve these Lyapunov equations using iterative schemes for computing the matrix sign function. This approach can be efficiently implemented on parallel computers using ScaLAPACK. Numerical experiments on an ibm sp 2 multicomputer report on the accuracy, scalability, and speed-up of the implemented algorithms.

Rank-one LMIs and Lyapunov's inequality

Abstract: We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semidefinite programming. Links are established between the Lyapunov matrix, rank-one linear matrix inequalities (LMI), and the Lagrange multiplier arising in duality theory.

Novel energy-based Lyapunov function for controlled power systems

Abstract: This letter first derives the Hamiltonian equation for a power system with excitation control. A modified energy-based Lyapunov function is then presented for fast transient stability analysis of the controlled power systems.

Control Lyapunov functions for homogeneous “Jurdjevic-Quinn” systems

Abstract: This paper presents a method to design explicit control Lyapunov functions for affine and homogeneous systems that satisfy the so-called “Jurdjevic-Quinn conditions”. For these systems a positive definite function V 0 is known that can only be made non increasing by feedback. We describe how a control Lyapunov function can be obtained via a deformation of this “weak” Lyapunov function. Some examples are presented, and the linear quadratic situation is treated as an illustration.

Exponential stability and periodic solutions of delayed cellular neural networks

Abstract: A set of criteria are presented for the global exponential stability and the existence of periodic solutions of delayed cellular neural networks ( DCNNs) by constructing suitable Lyapunov functionals . introducing many parameters q ij * , r ij * , q ij , r ij ∈ R and w i > 0 ( i , j = 1 , 2 , ... , n ) and combining them with the elementary inequality 2 ab ≦ a 2 + b 2 technique. These criteria have important significance in the design and applications of globally stable DCNNs and periodic oscillatory DCNNs. In addition, the results in literature are extended and improved. Two examples are given to illustrate the theory.

Relaxed stability conditions for Takagi-Sugeno fuzzy systems

Abstract: This paper discusses conditions on stability and stabilization of continuous T-S fuzzy systems. Stability analysis is derived via a non-quadratic Lyapunov function technique and LMIs (linear matrix inequalities) formulation to obtain an efficient solution. The non-quadratic Lyapunov function is built by inference of quadratic Lyapunov function of each local model. We show that the stability condition of the open-loop T-S systems is assured under certain restrictions on the rate of change of state variables. Following a similar approach, the stabilization of closed-loop continuous T-S fuzzy systems using the well-known PDC (parallel distributed compensation) technique is investigated. The design methodology is illustrated by numerical examples.

Decentralized control of nonlinear large-scale systems using dynamic output feedback

Abstract: In this paper, a class of nonlinear large-scale systems with similar subsystems is studied. Both matched and unmatched uncertainties are considered by utilizing their bounding functions, and the interconnections take a more general form than considered previously. Based on a constrained Lyapunov equation, a nonlinear dynamic output feedback decentralized controller is presented. Unlike existing results, matched uncertainties are considered in the control design; by using a decomposition of the interconnections, the known and uncertain interconnections are treated separately; thus, the robustness is improved and conservativeness is reduced significantly. The computation effort for solving the Lyapunov equation is greatly reduced by taking into account the similar subsystem structure. Finally, simulation is used to illustrate the effectiveness of our results.

A parameter-dependent Lyapunov function for a polytope of matrices

Abstract: A new sufficient condition for a polytope of matrices to be Hurwitz-stable is presented. The stability is a consequence of the existence of a parameter-dependent quadratic Lyapunov function, which is assured by a certain linear constraint for generating extreme matrices of the polytope. The condition can be regarded as a duality of the known extreme point result on quadratic stability of matrix polytopes, where a fixed quadratic Lyapunov function plays the role. The obtained results are applied to a polytope of second-degree polynomials for illustration.

Stability of limit cycles in hybrid systems using discrete-time Lyapunov techniques

Abstract: This paper concerns stability analysis of limit cycles in hybrid systems. Continuous-time hybrid systems are modeled in a discrete-time affine framework. The discrete-time approach is shown to be appropriate in order to find a Lyapunov formulation for the stability of a hybrid limit cycle. Multiple Lyapunov functions are associated with the transitions in the hybrid system so that the trajectory is shown to converge to the switch points of the limit cycle. The results are formulated in linear matrix inequalities (LMI) which gives a constructive way to find the Lyapunov functions using efficient algorithms. The results are applied to a two-tank example with discrete valued actuators.