On the exponential exponents of discrete linear systems

In this paper we introduce the concepts of exponential exponents of discrete linear time varying systems. It is shown that these exponents describe the possible changes in the Lyapunov exponents under perturbation decreasing at infinity at exponential rate. Finally we present formulas for the exponential exponents in terms of the transition matrix of the system.     

Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.      

Lyapunov stability of discontinuous dynamic systems

This paper derives necessary and sufficient conditions for thestability of dynamic systems in the sense of Lyapunov. Theseconditions are used to study the stability of discontinuousdynamic systems, which include fuzzy systems, hybrid systems,and impulsive differentia^! systems.     

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise affine function can be parameterized by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterized function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.     

Relations between Bohl exponents and general exponent of discrete linear time-varying systems

In the paper, properties of the upper Bohl exponents and senior upper general exponent of discrete linear time-varying systems are investigated. The relation of these exponents to uniform exponential stability is discussed. Moreover, an example of system, which is not uniformly exponentially stable but each trajectory tends uniformly and exponentially to zero is provided.     

Stabilization of nonlinear switched systems using control Lyapunov functions

In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution provides a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function; this is achieved with the converse Lyapunov theorem dedicated to switched systems. In the second part, an explicit construction of a common control Lyapunov function is addressed with respect to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.     

Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources

This paper deals with two classes of parabolic systems with localized nonlinear sources. The critical exponents as well as the estimates for blow-up rates and boundary layer profiles are determined.     

Exponential stability of simultaneously triangularizable switched systems with explicit calculation of a common Lyapunov function

In this note, a common quadratic Lyapunov function is explicitly calculated for a linear hybrid system described by a family of simultaneously triangularizable matrices. The explicit construction of such a function allows not only obtaining an estimate of the convergence rate of the exponential stability of the switched system under arbitrary switching but also calculating an upper bound for the output during its transient response. Furthermore, the presented result is then extended to the case where the system is affected by parametric uncertainty, providing the corresponding results in terms of the nominal matrices and uncertainty bounds.     

Adaptive control of uncertain nonlinear systems using mixed backstepping and Lyapunov redesign techniques

In this paper, a robust adaptive control law for a class of uncertain nonlinear systems is proposed. The proposed controller guarantees asymptotic output tracking of systems in the strict-feedback form with unknown static parameters, and matched and unmatched dynamic uncertainties. This controller takes advantages of a robust stability property of the Lyapunov redesign method and a systematic design procedure of the backstepping technique. In fact, the backstepping technique is employed to enrich the Lyapunov redesign method to compensate for not only matched - but also unmatched-uncertainties. On the other hand, using the Lyapunov redesign method in each step of the conventional backstepping technique makes backstepping robust. The suggested controller is designed through repeatedly utilizing the Lyapunov redesign method in each step of the backstepping technique. Simulation results reveal the efficiency of the Lyapunov redesign-based backstepping controller.     

A dynamic systems model of dyadic interaction

We develop a differential equation model of dyadic interaction that embodies the basic assumption that members of intimate couples form an interactive system in which the behavior of each member of a couple is influenced by the other's behavior and by goals that each person has for herself or himself. The dynamic solutions of this system suggest that when each person in the dyad is “cooperative”, then an equilibrium can be approached. The equilibrium represents a compromise position between the individuals’ own ideals and those of the partner. On the other hand, if one individual, or both, is uncooperative, then this system often, but not always, becomes unstable. One paradoxical deduction from the model is that, through mutual cooperation, couples can experience periods of stability, but such stable situations are not necessarily satisfying.     

Order reduction and determination of dominant state variables of nonlinear systems

The method presented can simplify nonlinear system models by reducing the number of state equations. Starting from a special state space representation, the main idea is to take over all nonlinear terms into the reduced system and to renew all couplings of state variables, input variables and nonlinear functions. The steady state performance can be influenced by additional measures which are discussed in detail and which are illustrated by a technical example. A dominance analysis is introduced which helps choosing the system order and the dominant state variables. All computations are based on proven algorithms and most of them are free of iterations.     

Stability and boundedness of nonlinear impulsive systems in terms of two measures via perturbing Lyapunov functions

This paper develops the concepts of stability, practical stability and boundedness in terms of two measures for nonlinear impulsive differential systems using the method of perturbing Lyapunov functions. The notion of perturbing Lyapunov functions enables us to discuss stability properties of solutions of nonlinear impulsive differential systems in terms of two measures under much weaker assumptions. The novel results offer a way to unify a variety of stability results found in the relative literature.     

Controllability of nonlinear systems

Some sufficient conditions are presented for the controllability of general nonlinear systems. First, the controllability problem is transformed into the problem of existence of fixed points for some operator; using Schauder's theorem, it is derived that a sufficient condition for controllability is the existence of a subsetS inC _n+m (T) which is invariant for a derived operator. Secondly, with the aid of the notion of comparison principle, the existence of the subsetS is guaranteed by the existence of solutions for some nonlinear integral inequality or equality equations. For example, one solution for such nonlinear integral equations is obtained under the assumption of the uniform boundedness for a nonlinear term of the differential equation.     

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger's control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.     

A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence

In this paper, by investigating an SIR epidemic model with nonlinear incidence, we present a new technique for proving the global stability of the endemic equilibrium, which consists of introducing a variable transformation and constructing a more general Lyapunov function. For the model we obtain the following results. The disease-free equilibrium is globally stable in the feasible region as the basic reproduction number is less than or equal to unity, and the endemic equilibrium is globally stable in the feasible region as the basic reproduction number is greater than unity.The generality of the technique is illustrated by considering certain nonlinear incidences and SIS and SIRS epidemic models.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

Singularity conditions for the non-existence of a common quadratic Lyapunov function for pairs of third order linear time invariant dynamic systems

The condition that a finite collection of stable matrices {A₁, … , A_M} has no common quadratic Lyapunov function (CQLF) is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters. These conditions are applied to the case of two stable 3 × 3 matrices, where they are used to derive necessary and sufficient conditions for the non-existence of a CQLF.     

Aggregation and stability analysis of nonlinear complex systems

The problem of stability of large-scale systems in critical cases is investigated. New form of aggregation for essentially nonlinear complex systems is suggested. With the help of this form the sufficient conditions of asymptotic stability are determined. The results obtained are used for the stability analysis of complex systems by the nonlinear approximation and for the investigation of absolute stability conditions for a certain class of nonlinear systems.     

Constructions of strict Lyapunov functions for discrete time and hybrid time-varying systems

We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov’s Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions.