Lagrange Stability Research Articles

H∞ controller synthesis for pendulum-like systems

This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H ∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satisfied, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs).

Decentralized disturbance attenuating output-feedback trackers for large-scale nonlinear systems

The problem of decentralized output-feedback tracking with disturbance attenuation is addressed for a new class of large-scale and minimum-phase nonlinear systems. Common assumptions like matching and growth conditions are not required for the underlying decentralized system with a diagonal structure. An observer-based decentralized controller design is presented. The proposed decentralized output-feedback laws achieve asymptotic tracking and internal Lagrange stability when the disturbance inputs disappear, and, guarantee external stability in the presence of disturbance inputs. These external stability properties include Sontag's ISS and iISS conditions and standard L 2 -gain property.

Stability analysis of pulse-width-modulated feedback systems

In this paper, we present new Lyapunov and Lagrange stability results for pulse-width-modulated (PWM) feedback systems with linear and nonlinear plants. For systems with linear plants, we consider the noncritical case, where the poles of the transfer function of the plant are all in the left-half of the complex plane and the critical case, where one pole is at the origin while the remaining poles are all in the left-half of the complex plane. For these systems we apply the Direct Method of Lyapunov to establish new and improved results for both Lyapunov and Lagrange stability. As in most existing results for PWM feedback systems obtained by the Lyapunov method, we employ quadratic Lyapunov functions in our analysis. However, in the proofs we make use of different majorizations, requiring hypotheses that differ significantly from those used in the existing results. Additionally, and perhaps more importantly, we incorporate into our results optimization procedures that improve our results significantly. We demonstrate the applicability and quality of our results by means of five specific examples that are identical to examples presented in the literature. For PWM feedback systems with nonlinear plants we show that under reasonable conditions, the stability properties of the trivial solution of such systems can be deduced from the stability properties of the trivial solution of PWM feedback systems with corresponding linearized plant, for both noncritical and critical cases.

Moment stability of discontinuous stochastic dynamical systems

In this note, we establish new Lyapunov and Lagrange stability results in the pth mean for a class of discontinuous stochastic dynamical systems. We apply these results in the qualitative analysis of a class of digital feedback control systems that are subjected to multiplicative and additive disturbances in the plants. The present results constitute natural extensions of our earlier results for discontinuous deterministic dynamical systems.

Quasiperiodic Motions for Asymmetric Oscillators

We prove the existence of quasiperiodic solutions and Lagrange stability for a class of differential equations with jumping nonlinearity \({\ddot x}+ax^+-bx^-+\phi(x)=p(t)\), where a,b > 0, p(t) ∈C(ℝ/2πℤ) and φ : ℝ→ℝ is an unbounded function.

Lagrange stability for asymmetric Duffing equations

Lagrange stability for asymmetric Duffing equations

Robustness analysis of digital feedback control systems with time-varying sampling periods

In the present paper, we study robustness properties of a class of digital feedback control systems with time-varying sampling periods consisting of an interconnection of a continuous-time nonlinear plant (described by systems of first-order ordinary differential equations), a nonlinear digital controller (described by systems of first-order ordinary difference equations), and appropriate interface elements between the plant and controller (A/D and D/A converters). For such systems, we establish results for exponential stability of an equilibrium (in the Lyapunov sense) in the presence of vanishing perturbations and for the boundedness of solutions (i.e., Lagrange stability) under the influence of nonvanishing perturbations. We apply these results in the study of quantization effects.

Robust stabilizing control laws for a class of second-order switched systems

For a class of second-order switched systems consisting of two linear time-invariant (LTI) subsystems, we show that the so-called conic switching law proposed previously by the present authors is robust, not only in the sense that the control law is flexible (to be explained further), but also in the sense that the Lyapunov stability (resp., Lagrange stability) properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations (resp., nonvanishing perturbations). The analysis is possible since the conic switching laws always possess certain kinds of “quasi-periodic switching operations”. We also propose for a class of nonlinear second-order switched systems with time-invariant subsystems a switching control law which locally exponentially stabilizes the entire nonlinear switched system, provided that the conic switching law exponentially stabilizes the linearized switched systems (consisting of the linearization of each nonlinear subsystem). This switched control law is robust in the sense mentioned above.

A comparison theory for the stability analysis of general hybrid systems

In some of our recent work, a general model for hybrid dynamical systems was proposed whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time . By transforming this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but defined on real time, we established the Principal Lyapunov Stability Results as well as Lagrange Stability Results. Making use of stability preserving mappings, results which comprise a general comparison theory for hybrid dynamical systems can be developed and these results can be specialized by employing vector Lyapunov functions. It then can be shown that the latter results can be used to yield the Principal Lyapunov Stability Results discussed above as special cases. Due to space limitations, only a summary of the paper is presented.

On the Lagrange and Hill Stability of the Motion of Certain Systems with Newtonian Potential

The Lagrange and Hill stability of the motion of point masses moving under the influence of their mutual gravitational attraction are studied. A modified model of the restricted three-body problem is proposed. In accordance with this model, the smallest of the masses is not infinitesimal. This simplifies the study of the Lagrange and Hill stability of the motion. In particular, it is shown that the correctness of the statement of the restricted three-body problem by itself provides the Hill stability of the small-mass point m3.

Towards a stability theory of general hybrid dynamical systems

In recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time. For such systems we introduced the usual concepts of Lyapunov and Lagrange stability. We showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but defined on real time R +=[0, ∞) . The motions of this class of systems are in general discontinuous. This class of systems may be finite or infinite dimensional. For the above discontinuous dynamical systems (and hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Stability Theorems as well as Lagrange Stability Theorems. For some of these, we also established converse theorems. We demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems. In the present paper we continue the work described above. In doing so, we first develop a general comparison theory for the class of hybrid dynamical systems (resp., discontinuous dynamical systems) considered herein, making use of stability preserving mappings. We then show how these results can be applied to establish some of the Principal Lyaponov Stability Theorems. For the latter, we also state and prove a converse theorem not considered previously. Finally, to demonstrate the applicability of our results, we consider specific examples throughout the paper.

Stability analysis of a class of nonlinear multirate digital control systems

We consider multirate digital control systems which consist of an interconnection of a continuous-time nonlinear plant (described by ordinary differential equations) and a digital lifted controller (described by ordinary difference equations). The input to the digital controller consists of the multirate sampled output of the plant and the input to the continuous-time plant consists of the multirate hold output of the digital controller. We show that when quantizer nonlinearities are neglected, then under reasonable conditions (which exclude the critical cases), the stability properties (in the Lyapunov sense) of the trivial solution of the nonlinear multirate digital control system can be deduced from the stability properties of the trivial solution of its linearization. When quantizer nonlinearities are not neglected, corresponding results can be established for Lagrange stability rather than Lyapunov stability of an equilibrium.

A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics

This paper presents a constructive robust adaptive nonlinear control scheme which can be regarded as a robustification of the now popular adaptive backstepping algorithm. The allowed class of uncertainties includes nonlinearly appearing parametric uncertainty, uncertain nonlinearities, and unmeasured input-to-state stable dynamics. The adaptive control laws proposed in this paper do not require any dynamic dominating signal to guarantee the robustness property of Lagrange stability. The numerical example of a simple pendulum with unknown parameters and without velocity measurement illustrates our theoretical results.

Control of nonlinear non-minimum phase systems using dynamic sliding mode

A newly developed dynamic sliding mode control technique for multiple input systems is shown to be useful in the control of nonlinear, non-minimum phase systems where the zero dynamics have no finite escape time. The system may not be dynamic feedback linearizable. To achieve asymptotic performance, unbounded control may be necessary as determined by the zero dynamics. As long as the growth rate of the zero dynamics is no more than exponential, ultimate bounded performance can be achieved with finite control effort. Lagrange stability analysis of the closed-loop system resulting from the proposed variable structure scheme is performed. Essentially a thin layer is introduced around the sliding surface. Outside the layer, the sliding mode controller is used; inside the layer, the controller is designed to asymptotically (exponentially) stabilize the dynamic compensator. It is shown that there is a trade-off between control performance and control effort. The method is illustrated by the control of the Inverted Double Pendulum which is not dynamic-feedback linearizable and is non-minimum phase and thus constitutes a testing example for the proposed scheme.

Smooth Lyapunov Functions for Discontinuous Stable Systems

It has been proved that a differential system dx/dt = f(t,x)with a discontinuous right- hand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This paper focuses on the smoothness of such a Lyapunov function. An example of an (asymptotically) stable system for which there does not exist any (even weak) Lyapunov functions of class C 1 is given. In the more general context of differential inclusions, the existence of a weak Lyapunov function of class C 1 (or C ! ) is shown to be equivalent to the robust stability of some perturbed system obtained in introducing measurement error with respect to x and t .T his condition is proved to be satisf ied by most of the robustly stable systems encountered in the literature. Analogous results are given for the Lagrange stability. As an application to the study of the links between internal and external stability for control systems, an extension of a result by Bacciotti and Beccari is obtained by means of a smooth Lyapunov function associated with a robustly Lagrange stable system. Mathematics Subject Classifications (1991): 34A60, 34D20, 93D05, 93D25.

Recent trends in the stability analysis of hybrid dynamical systems

Dramatic progress in computing capabilities has resulted in the synthesis and implementation of increasingly complex dynamical systems. Since such systems frequently exhibit simultaneously several kinds of dynamic behavior in different parts of the system, they are referred to as hybrid dynamical systems. Such systems frequently defy traditional modeling and analysis techniques since the different system components may evolve along different notions of including real time, discrete time, and discrete events. Most investigations of such systems to date involve ad hoc models and tailor-made analysis results. In some recent work, however, a general model has been proposed which contains most of the different classes of hybrid dynamical systems considered in the literature as special cases. At the core of this general model of hybrid dynamical system, which is defined on an arbitrary metric space, is a notion of generalized time. For this class of general hybrid dynamical systems, a variety of Lyapunov and Lagrange stability results have been established and made public in a scattering of publications and workshop records. Our objective in this paper is to present a unified overview of the more important aspects of this work.

Liapunov and lagrange stability: Inverse theorems for discontinuous systems

The main result of this paper is a converse Liapunov theorem which applies to systems of ordinary differential equations with a discontinuous righthand side. We treat both the problem of local stability of an equilibrium position and the problem of boundedness of solutions. In particular, we show that in order to achieve a necessary and sufficient condition in terms of continuous Liapunov functions, the classical definitions need to be strengthened in a convenient way. This work was motivated by the recently renewed interest in stabilization by discontinuous feedback and analysis of the state evolution with respect to bounded inputs. To achieve a more general treatment, the exposition is developed in the framework of differential inclusions theory.

Stability theory for hybrid dynamical systems

We first formulate a model for hybrid dynamical systems which covers a very large class of systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of an invariant set for hybrid dynamical systems and we define several types of (Lyapunov-like) stability concepts for an invariant set. We then establish sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). In addition to the above, we also establish sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability). To demonstrate the applicability of the developed theory, we present specific examples of hybrid dynamical systems and we conduct a stability analysis of some of these examples.

Multiple Lyapunov functions and other analysis tools for switched and hybrid systems

We introduce some analysis tools for and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz vector fields, allowing limit cycle analysis of a class of continuous switched systems.

Modeling and Qualitative Theory for General Hybrid Dynamical and Control Systems

Hybrid dynamical systems which are capable of exhibiting simultaneously several kinds of dynamic behavior, such as continuous-time dynamics, discrete-time dynamics, jump phenomena, switching and logic commands, discrete events, and the like, are of great current interest. In the present paper we employ a general model of dynamical system suitable in the qualitative analysis of such systems. This model of dynamical system allows discontinuous motions, and convergence of motions is relative to generalized time. For the model of hybrid dynamical systems described above, results are established for the principal Lyapunov stability of invariant sets and the principal Lagrange stability of motions. Some of the results of the present paper are applied in the analysis of a specific class of systems.