Search

This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the K-exponential stability is achieved based on our approach. Index Terms--Detectability, nonholonomic systems, nonlinear time-varying systems, reduced limiting systems, uniform asymptotic stability.

This note investigates the fast tracking control problem of underactuated ships via persistent excitation (PE) conditions. By combining a novel transformation with the computed torque method, a decoupling controller related to the surge force is given first to decompose the error model into two cascade subsystems. Then, a stabilizing controller involving the yaw moment is designed. With the help of the proposed cascaded structure, a weaker PE condition than those given in past literature can be used to verify an integral detectability and guarantee global [kappa]-exponential convergence by employing several newly developed stability criteria. A new feature of the obtained results is that only one of these reference signals is needed to satisfy the usual PE condition. Simulation results are provided to validate the effectiveness of the proposed scheme. Index Terms--Cascaded systems, global [kappa]-exponential stability, persistent excitation (PE) condition, underactuated ships.

This note investigates equivalent relations of detectability and persistency excitation (PE) for general nonlinear time-varying systems. Three new definitions relating to the detectability are proposed and discussed. As a preliminary result, we show that these detectability conditions are all equivalent under a mild assumption. Furthermore, two PE conditions that are a nonlinear-version extension of the PE proposed in present literature are defined. Very interestingly, it can be shown that these nonlinear-version PE conditions are both equivalent to the proposed detectability conditions under the same assumption. To the best of our knowledge, this is a first result about the connection between detectability and PE in the area of nonlinear time-varying systems. Moreover, by combining a newly developed stability criterion with the proposed results, an interesting example is given to illustrate how the classic PE conditions for linear systems can be extended and used in nonlinear systems. From this example, it can also be seen that the uniformly asymptotic stability can be guaranteed either using the detectability conditions or by the PE conditions based on our approach. Index Terms--Adaptive systems, detectability, equivalent relations, persistency excitation (PE), uniformly asymptotic stability.

Uniform asymptotic stability for general nonlinear time-varying systems is investigated from the output-to-state viewpoint. A unified criterion is proposed using a modified detectability condition and certain integral inequalities. The criterion extends the Krasovskii-LaSalle theorem to nonperiodic time-varying systems. An example taken from the tracking control of mobile robots is given to illustrate that, much as the invariance principle of LaSalle is used in studying the stability of time-invariant systems, our results can be used to study the stability of nonperiodic time-varying systems by checking the modified detectability condition. Index Terms--Detectability, nonlinear time-varying systems, uniform asymptotic stability.

A general invariance principle, from the output-to-state point of view, is proposed for the dynamical analysis of nonlinear time-varying systems. This is achieved by the construction of a simple and intuitive criterion using integral inequality of the output function and modified detectability conditions. The proposed scheme can be viewed as an extension of the integral invariance principle (Byrnes and Martin, 1995) for time-invariant systems to time-varying systems. Such extension is nontrivial and can be used in various research areas such as adaptive control, tracking control and the control of driftless systems. An application to global tracking control of four-wheeled mobile robots is given to demonstrate the feasibility and validity of the proposed approach. Index Terms--Invariance principle, mobile robots, time-varying systems.

This paper investigates the regulation problem for a class of nonholonomic systems that includes power form systems and an approximated system of the rolling sphere as special cases. The basic idea is first to introduce a virtual periodic moving trajectory that satisfies certain persistent excitation condition (PE) and has the zero value at some time instants. Based on LaSalle invariance principle, the associated tracking problem is then solved under a necessary condition for stabilization and particularly true for the power form systems and the rolling sphere. With the help of virtual trajectory, the achieved tracking result is applied to the regulation problem and used to guarantee practical stability. The proposed controllers have a simple and explicit form, and hence are easily implemented. Simultaneously, fast convergence is guaranteed, thanks to the $\boldsymbol {\mathcal {K}}$ -exponential convergence. More interestingly, the used approach is adding sufficiently exciting signals to the systems by considering virtual tracking signals so that the attractivity of the origin can be guaranteed based on LaSalle invariance principle. Thus, it is possible to extend the proposed results to more general systems. To verify the effectiveness of the proposed scheme, interesting simulation results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we extend Veillette's results (1995) to the study of reliable linear-quadratic regulator problem for nonlinear systems. This is achieved by employing the Hamilton-Jacobi inequality in the nonlinear case instead of algebraic Riccati equation in the linear one. The proposed state-feedback controllers are shown to be able to tolerate the outage of actuators within a prespecified subset of actuators. Both the gain margins of guaranteeing system stability and retaining a performance bound are estimated. Index Terms--Algebraic Riccati equation, Hamilton-Jacobi inequality, linear-quadratic regulator problem.