Elena Panteley

On global uniform asymptotic stability of nonlinear time-varying systems in cascade

Abstract In this short paper we deal with the stability analysis problem of nonautonomous nonlinear systems in cascade. In particular, we give sufficient conditions to guarantee that (i) a globally uniformly stable (GUS) nonlinear time-varying (NLTV) system remains GUS when it is perturbed by the output of a globally uniformly asymptotically stable (GUAS) NLTV system, under the assumption that the perturbing signal is absolutely integrable; (ii) if in addition the perturbed system is GUAS, it remains GUAS under the cascaded interconnection; (iii) two GUAS systems yield a GUAS cascaded system, under some growth restrictions over the Lyapunov function. Our proofs rely on the second method of Lyapunov, roughly speaking on a “δ-e stability analysis”.

Relaxed persistency of excitation for uniform asymptotic stability

The persistency of excitation property is crucial for the stability analysis of parameter identification algorithms and adaptive control loops. We propose a relaxed definition of persistency of excitation and we use it to establish uniform global asymptotic stability (UGAS) for a large class of nonlinear, time-varying systems. Our relaxed definition is conceptually equivalent to the definition introduced in Loria et al. (1999), but is easier to verify. It is useful for stability analysis of nonlinear systems that arise, for example, in nonlinear adaptive control, stabilization of nonholonomic systems, and tracking control. Our proof of UGAS relies on some integral characterizations of UGAS established in Teel et al. (2000). These characterizations also streamline the proof of UGAS in the presence of (uniform) classical persistency of excitation.

Uniform exponential stability of linear time-varying systems: revisited

Abstract Some classical results known in the adaptive control literature are often used as analysis tools for nonlinear systems by evaluating the nonlinear differential equations along trajectories. While this technique is widely used, as we remark through examples, one must take special care in the consideration of the initial conditions in order to conclude uniform convergence. One way of taking care explicitly of the initial conditions is to study parameterized linear time-varying systems. This paper re-establishes known results for linear time-varying systems via new techniques while stressing the importance of imposing that the formulated sufficient and necessary conditions must hold uniformly in the parameter. Our proofs are based on modern tools which can be interpreted as an “integral” version of Lyapunov theorems; rather than on the concept of uniform complete observability which is most common in the literature.

A separation principle for dynamic positioning of ships: theoretical and experimental results

Presents a globally asymptotically stabilizing (GAS) controller for regulation and dynamic positioning of ships, using only position measurements. It is assumed that these are corrupted with white noise hence a passive observer which reconstructs the rest of the states is applied. The observer produces noise-free estimates of the position, the slowly varying environmental disturbances and the velocity which are used in a proportional-derivative (PD)-type control law. The stability proof is based on a separation principle which holds for the nonlinear ship model. This separation principle is theoretically supported by results on cascaded nonlinear systems and standard Lyapunov theory, and it is validated in practice by experimentation with a model ship scale 1:70.

Brief Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems

In the last decade several results on cascaded autonomous systems have appeared in the literature. The sufficient conditions for the stability and stabilizability of these systems are often related to certain growth rate conditions on the functions which define the dynamics of the system. In this paper we analyze three complementary classes of nonlinear time-varying systems according to these growth rates. For each case, we give further results to guarantee uniform global asymptotic stability of the cascade and relate our contributions to input-to-state stability and other growth rate conditions previously reported. The advantage of this type of analysis is that it is often easier to verify such conditions than to find a Lyapunov function with a negative-definite derivative.

Straight Line Path Following for Formations of Underactuated Marine Surface Vessels

The problem of formation control and path following for underactuated 3-degrees-of-freedom (3-DOF) marine surface vessels is considered. The proposed decentralized controller makes the vessels asymptotically constitute a desired formation that follows a given straight line path with a given forward speed profile. The controller consists of a cross-track control law, based on line-of-sight guidance, and a nonlinear synchronization control law. The closed-loop dynamics of both the cross-track errors and the synchronization errors are analyzed in detail using nonlinear cascaded systems theory and the overall cascaded system is shown to be both uniformly globally asymptotically stable and uniformly globally exponentially stable under mild assumptions. The proposed control strategy is validated in experiments in both calm water and in waves using a scale model vessel.

Exponential tracking control of a mobile car using a cascaded approach

Abstract In this paper we address the problem of designing simple global tracking controllers for a kinematic model of a mobile robot and a simple dynamic model of a mobile robot. For this we use a cascaded systems approach, resulting into linear controllers that yield global K-exponential stability of the closed loop system.

Integral Characterizations of Uniform Asymptotic and Exponential Stability with Applications

Abstract. We present integral characterizations of uniform asymptotic stability and uniform exponential stability for differential equations and inclusions. These characterizations are used to establish new results on concluding uniform global asymptotic stability when uniform global stability is already known and uniform convergence must be established by additional arguments. In one case we generalize Matrosovs theorem on the use of a differentiable auxiliary function. In another case we draw conclusions from a system related to the original by suitable output injection.

An adaptive friction compensator for global tracking in robot manipulators

A novel adaptive friction compensator based on a dynamic model recently proposed in the literature is presented in this paper. The compensator ensures global position tracking when applied to an n degree of freedom robot manipulator perturbed by friction forces with only measurements of position and velocity, and all the system parameters (robot and friction model) unknown. Instrumental for the solution of the problem is the observation that friction compensation can be recasted as a disturbance rejection problem. The control signal is then designed in two steps, first a classical adaptive robot controller that (strictly) passifies the system, and then a relay-based outer-loop that rejects the disturbance.

/spl delta/-persistency of excitation: a necessary and sufficient condition for uniform attractivity

In previous papers we have introduced a sufficient condition for uniform attractivity of the origin of a class of nonlinear time-varying systems which is stated in terms of persistency of excitation (PE), a concept well known in the adaptive control and systems identification literature. The novelty of our condition, called uniform /spl delta/-PE, is that it is tailored for nonlinear functions of time and state and it allows us to prove uniform asymptotic stability. In this paper we present a new definition of u/spl delta/-PE which is conceptually equivalent to but technically different from its predecessors. We make connections between this property and similar properties previously used in the literature. We also show when this condition is necessary and sufficient for uniform (global) asymptotic stability for a large class of nonlinear time-varying systems.