Uniform Global Asymptotic Stability Research Articles

A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems

A new infinitesimal sufficient condition is given for uniform global asymptotic stability (UGAS) for time-varying nonlinear systems. It is used to show that a certain relaxed persistency of excitation condition, called uniform /spl delta/-persistency of excitation (U/spl delta/-PE), is sufficient for uniform global asymptotic stability in certain situations. U/spl delta/-PE of the right-hand side of a time-varying differential equation is also shown to be necessary under a uniform Lipschitz condition. The infinitesimal sufficient condition for UGAS involves the inner products of the flow field with the gradients of a finite number of possibly sign-indefinite, locally Lipschitz Lyapunov-like functions. These inner products are supposed to be bounded by functions that have a certain nested, or triangular, negative semidefinite structure. This idea is reminiscent of a previous idea of Matrosov who supplemented a Lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the Lyapunov function is zero. For this reason, we call the main result a nested Matrosov theorem. The utility of our results on stability analysis is illustrated through the well-known case-study of the nonholonomic integrator.

Discussion on: “Uniform Parametric Convergence in the Adaptive Control of Mechanical Systems”

The paper by Loria et al. is an expanded and refined version of Ref. (9). The central result is a proof of uniform global asymptotic stability (UGAS) for the feedback loop consisting of a dynamical plant mod- eled using Euler-Lagrange equations of motion (5,11) and of the adaptive controller of Slotine and Li pro- posed in Ref. (13). It is stated that the property of UGAS also holds when the same plant is controlled with several others passivity-based adaptive con- trollers. The UGAS of the considered closed-loop systems is established based on the property of uniform � -persistency of excitation (8), which is a relaxation of the persistency of excitation condition as formulated in Ref. (12). The relaxed property is also used to derive sufficient conditions for uniform local exponential stability of the plant controlled by a general adaptive control algorithm. As appealing result, the authors point out that the paper establishes a mathematical basis for analytical evaluation of robustness of the considered adaptive control algo- rithms against exogenous disturbances (e.g., mea- surement noise). By presenting experimental results of a robot motion control using the adaptive control algorithm of Slotine and Li, the authors want to empirically illustrate the robustness of this control algorithm. The robustness, as they claim, follows from the UGAS property proven by theoretical analysis. The theoretical results presented in the paper deserve credit, since they extend the mathematical

OUTPUT FEEDBACK CONTROL VIA ADAPTIVE OBSERVERS WITH PERSISTENCY OF EXCITATION

We address the problem of adaptive observer design for nonlinear time-varying systems which can be transformed in the so-called output feedback form (linear in the unmeasured variables). The observer design follows up previous work on adaptive observers for linear systems and has the form of the classical Luenberger observers for linear systems except that the observer gain is time-varying. A specific form of persistency of excitation is imposed to guarantee the convergence of the (state and parameter) estimation errors. As for the output feedback loop, we proceed using a cascade approach, i.e., we impose the appropriate conditions so that the closed loop system has a cascaded structure. Uniform global asymptotic stability may then be concluded based on cascaded systems theory. Key Words—persistency of excitation, observers, time-varying systems, output feedback control.

OPTIMIZING NONLINEAR ADAPTIVE CONTROL ALLOCATION

A control-Lyapunov approach is used to develop an adaptive optimizing control allocation algorithm for over-actuated mechanical systems where the actuator model is affine in the uncertain parameters. Uniform global (asymptotic) stability is guaranteed by the control allocation defined by the dynamic update laws in combination with an exponentially stable controller.

UNIFORM SEMIGLOBAL ASYMPTOTIC STABILITY FOR TIME-VARYING NONLINEAR CASCADED SYSTEMS

In this paper we address the stability analysis problem for cascades of systems that are semiglobally uniformly asymptotically stable (USGPAS). That is, we establish that, under a uniform boundedness condition, the cascade of two USGPAS systems remains USGPAS. Our results generalize fundamental well known results for uniform global asymptotic stability for time-varying, and hence for autonomous, cascades.

Uniform Parametric Convergence in the Adaptive Control of Mechanical Systems

It is known in the practice of closed-loop identification of mechanical systems that, since the Lagrangian dynamic model is linear in the constant physical parameters, a sufficient condition for parameter convergence is that the system follows reference trajectories that render the regressor matrix persistently exciting. This was established many years ago for a particular robot adaptive controller by Slotine and Li. This result is based on adaptive linear systems theory and relies on the condition that the applied control law makes the robot asymptotically track the imposed rich reference trajectories. This paper revisits this problem and establishes a proof that this condition is not only sufficient but also necessary to establish the stronger property of uniform global asymptotic stability of the overall closed-loop system. Our proof is based on modern tools and a relaxed formulation of the condition of persistency of excitation, tailored for nonlinear systems. Hence, it is not limited to the controller of Slotine and Li but applies to many other reported controllers. Furthermore, we give sufficient conditions for exponential stability (hence, in particular for uniform exponential parameter convergence). Finally, we provide some experimental results, illustrative of our theoretical findings.

Input-to-state stability of networked control systems

A new class of Lyapunov uniformly globally asymptotically stable (UGAS) protocols in networked control systems (NCS) is considered. It is shown that if the controller is designed without taking into account the network so that it yields input-to-state stability (ISS) with respect to external disturbances (not necessarily with respect to the error that will come from the network implementation), then the same controller will achieve semi-global practical ISS for the NCS when implemented via the network with a Lyapunov UGAS protocol. Moreover, the ISS gain is preserved. The adjustable parameter with respect to which semi-global practical ISS is achieved is the maximal allowable transfer interval (MATI) between transmission times.

Uniform Global Asymptotic Stability of Differential Inclusions

Stability of differential inclusions defined by locally Lipschitz compact valued mappings is considered. It is shown that if such a differential inclusion is globally asymptotically stable, then, in fact, it is uniformly globally asymptotically stable (with respect to initial states in compacts). This statement is trivial for differential equations, but here we provide the extension to compact- (not necessarily convex-) valued differential inclusions. The main result is presented in a context which is useful for control-theoretic applications: a differential inclusion with two outputs is considered, and the result applies to the property of global error detectability.

Global asymptotic stabilization for chains of integrators with a delay in the input

The problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved. No limitation on the size of the delay is imposed. To validate the approach, a third-order example is presented.

Global uniform asymptotic stabilization of an underactuated surface vessel

Explicit formulas of smooth time-varying state feedbacks, which make the origin of an underactuated surface vessel globally uniformly asymptotically stable, are proposed. The construction of the feedback extensively relies on the backstepping approach. The feedbacks constructed are time-periodic functions.

Integral Characterizations of Uniform Asymptotic and Exponential Stability with Applications

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 Matrosov's 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.

INERTIA SHAPING TECHNIQUES FOR MARINE VESSELS USING ACCELERATION FEEDBACK

The concept of energy based Lyapunov control is extended to marine vessels with a nonsymmetric system inertia matrix. Acceleration feedback is used as the main tool to symmetrize the nonsymmetric part of the system inertia matrix. The main reason for a nonsymmetric mass distribution is hydrodynamic added mass which depend on the forward speed of the vessel and the frequency of the incoming waves. This is a well known phenomenon for marine vessels moving at positive speed in waves while low-speed applications like dynamic positioning systems are fairly well described with a symmetric system inertia matrix. The main contribution of the paper is a new Lyapunov-based design technique incorporating acceleration feedback to shape the kinetic energy of the system. Acceleration feedback is implemented in conjuncture with a nonlinear PID-controller derived from vectorial backstepping. The result is a uniformly globally asymptotically stable (UGAS) closed-loop control system applicable to marine vessels with nonsymmetric system inertia matrices. Typical applications are ships in maneuvering situations, vessels in transit and high speed craft where nonsymmetric added mass effects must be compensated for.

UGAS of Skew-symmetric Time-varying Systems: Application to Stabilization of Chained Form Systems

The contribution of this communication is twofold: firstly, we will present a theorem guaranteeing uniform global asymptotic stability of a class of stable nonlinear systems, known in the literature of nonholonomic systems as skew-symmetric. These are systems of the form x ˙ = − A ( t , x ) x where A ( . , . ) e R n × n is skew-symmetric except for the (1,1) element which is non-zero. Secondly, we extend previous results for the nonholonomic integrators to the case of systems with n>3 states. Our smooth time-varying controllers are called δ-persistently exciting since they are based on a property which generalizes for nonlinear functions of the state, the persistency of excitation concept originally defined for time functions, in the context of systems identification.

Integral Characterizations of Set UGAS of Differential Inclusions and Applications

We present an integral characterization of uniform asymptotic stability for differential equations and inclusions. This characterization is 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. We show the utility of our result by generalizing Matrosov's Theorem on the use of a differentiable auxiliary function. Also, we present an application to the stability analysis of time-varying systems.

UGAS of Skew-Symmetric Systems: Application to the Nonholonomic Integrator

The contribution of this communication is twofold: firstly, we will present a theorem guaranteeing uniform global asymptotic stability of a class of stable nonlinear systems, known in the literature of nonholonomic systems, as "skew-symmetric"(the use of quotes"" is motivated by having one non-zero element in the main diagonal). Secondly, we extend previous results for the nonholonomic integrators to the case of systems with n > 3 states. Our smooth time-varying controllers are called δ-persistently exciting since they are based on a property which generalizes, for nonlinear functions of the state, the persistency of excitation concept originally defined in the context of systems identification for time functions.

A theorem for UGAS and ULES of (passive) nonautonomous systems: robust control of mechanical systems and ships

AbstractThe main contribution of this paper is a theorem to guarantee uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES) for a class of nonlinear non‐autonomous systems which includes passive systems. These properties (and a uniform local Lipschitz condition) guarantee robustness of stability while weaker properties, like uniform global stability plus global convergence, do not. Our main result is then used in the tracking control problem of mechanical systems and ships. We use an adaptive backstepping design and prove UGAS of the closed‐loop tracking error system, in particular, we obtain that both the tracking and parameter estimation errors converge uniformly globally to zero. Copyright © 2001 John Wiley & Sons, Ltd.

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.

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.

Robust stability of multiple-time-delay uncertain systems with series nonlinearities

This paper considers the problem of output feedback stabilization of multiple time delay uncertain systems with series nonlinearities. We develop a delay-dependent method for designing linear dynamic output feedback controllers which ensures global uniform asymptotic stability for the range of allowable delay times. The proposed stabilization method, which is based on the comparison theorem and some analytical methods, is employed to treat this problem. The most striking feature of the results presented is their simplicity in testing the stabilization of multiple-time-delay systems with series nonlinearities and this can be used to estimate the size of the delay time. Examples are used to illustrate the less conservative results of the proposed methods compared with existing methods in the literature.

Stability Conditions for a Class of Neutral Systems with Multiple Time Delays

In this paper, some delay-dependent and delay-independent stability criteria are proposed to guarantee global uniform asymptotic stability for a class of neutral systems with multiple time delays. Two numerical examples are given to illustrate our main results.