On the equivalence relations of detectability and PE conditions with applications to stability analysis of time-varying systems

We study the problem of stability analysis for certain nonlinear systems. Our contributions are new tools to guarantee uniform global asymptotic stability (UGAS) of nonlinear time-varying (NLTV) systems. First, we provide new definitions of persistency of excitation (PE). In particular, we give a new definition of uniform /spl delta/-PE (uG-PE) which, though conceptually equivalent to the original one we introduced (1999), is mathematically less conservative. We also provide with some properties of /spl delta/-PE pairs and contribute with a result which establishes UGAS of NLTV systems under u/spl delta/-PE.

Persistent excitation (PE) conditions have been widely used to analyze stability properties of various parameter identification algorithms and to establish uniform global asymptotic stability (UGAS) for a large class of nonlinear time-varying systems. In order to generalize such conditions to a more general setting, a new PE condition is proposed with three basic ingredients: a signal set to represent a family of time functions (e.g., trajectories); a pseudo distance measure to describe the convergence; and some binary relations (e.g., state-to-output relations). Closely related to detectability, this PE condition is a necessary condition to guarantee UGAS. Under uniform global stability and an integral inequality, it becomes a sufficient condition of UGAS. A novel concept: M-pair, which aims at simplifying the checking of the PE condition, is introduced. By using M-pair, it is possible to simplify the structure of the referred signal set (in the spirit of the classic Krasovskii-LaSalle theorem) and to extend the dimension of the reference signal set (similar to the Matrosov theorem). Thus, the framework of M-pair not only unifies these well-known results, but also generates more flexibility in checking the PE conditions. When applied to nonlinear switched systems, three new tools to verify the PE condition are obtained. Finally, an example illustrates that a nonlinear time-varying switched system with arbitrary switching can be shown to be UGAS without using a common Lyapunov function.

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.

The convergence of parameters in model reference adaptive control (MRAC) requires that a restrictive persistence of excitation (PE) condition be satisfied. A recent data driven approach, concurrent learning, uses past input-output data in conjunction with standard adaptive laws to ensure parameter convergence without needing the PE condition. However, the concurrent learning method assumes the knowledge of the state derivative, which is a limitation. This paper combines a state derivative estimator with concurrent learning to guarantee parameter convergence, thus eliminating the need for both the PE condition and the knowledge of the state derivative. Simulation results are presented to demonstrate the effectiveness of the proposed control method.

Satisfying the persistence of excitation (PE) condition is an important, yet challenging problem in system identification and adaptive control. In this paper, it is shown that a regressor vector consisting of radial basis functions can satisfy the PE condition. Specifically, for radial basis function networks (RBFN) constructed on a regular lattice, any periodic orbit that stays within the regular lattice can lead to the satisfaction of a partial PE condition. The significance of this result is that, with the partial PE condition satisfied, accurate RBFN approximation of unknown system dynamics can be achieved in a local region along the periodic orbit. This result will be very useful in identification, control and recognition of nonlinear systems using RBFN.

This article addresses the adaptive control problem for strict-feedback nonlinear time-varying systems with unknown control coefficients and external disturbances. To solve the problem of multiple unknown control coefficients, a new lemma based on a general class of Nussbaum functions rather than a specific Nussbaum function is proposed. By lumping all the time-varying parameters and control coefficients into an augmented time-varying vector, and combining the congelation of variables approach, a robust adaptive controller, without the restriction of the persistent excitation (PE) conditions, is constructed for the systems with external disturbances to guarantee the boundedness of all closed-loop variables. Then, by proper transformation, a novel adaptive control scheme is developed to achieve asymptotic stability for strict-feedback nonlinear time-varying systems without external disturbances. In addition, a tracking control method is acquired from the proposed adaptive control method. Finally, two simulations and an actual experiment demonstrate the applicability of the proposed schemes.

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.

Robust adaptive output feedback control of nonlinear systems without persistence of excitation

Parameter convergence and minimal internal model with an adaptive output regulation problem

Composite adaptive control for bilateral teleoperation systems without persistency of excitation

In this paper, a novel synchronous Q-learning method is proposed for solving discrete-time linear quadratic regulator (LQR) problems. To begin with, the Bellman equation corresponding to the optimal Q-function is reformulated into a consistency equation on the parameters of the optimal Q-function and the optimal controller. Then an actor-critic structure is introduced to learn the optimal Q-function and the optimal controller online in real time by using the state samples generated by the behavior policy. Particularly, the proposed synchronous Q-learning scheme simultaneously updates the Q-function approximation and the optimal controller approximation, rather than iterating between policy evaluation and policy improvement. The proposed control scheme is proved to be uniformly ultimately bounded (UUB) under appropriate learning rates, provided that certain persistence of excitation (PE) conditions are satisfied. Besides, the PE conditions can be easily met by injecting appropriate exploration noise into the behavior policy without causing any excitation noise bias. Finally, one simulation example is provided to verify the effectiveness of the proposed synchronous Q-learning method.

Parameter convergence is of great importance as it enhances the overall stability and robustness properties of adaptive control systems. However, a stringent persistent-excitation (PE) condition usually has to be satisfied to achieve parameter convergence in adaptive control. In this paper, a least-squares learning control strategy without regressor filtering is presented to achieve parameter convergence at the absence of the PE condition. An additional modified modeling error that utilizes online recorded data is constructed to update parametric estimates, and an integral transformation is derived to avoid the time differentiation of plant states in the computation of the modified modeling error. An indirect adaptive control law equipped with a novel filtering-free least-squares estimation is proposed to guarantee exponential convergence of both tracking errors and parameter estimation errors by an interval-excitation (IE) condition which is much weaker than the PE condition. An illustrative example has verified effectiveness of the proposed approach.

SummaryThis paper studies adaptive parameter estimation and control for nonlinear robotic systems based on parameter estimation errors. A framework to obtain an expression of the parameter estimation error is proposed first by introducing a set of auxiliary filtered variables. Then three novel adaptive laws driven by the estimation error are presented, where exponential error convergence is proved under the conventional persistent excitation (PE) condition; the direct measurement of the time derivatives of the system states are avoided. The adaptive laws are modified via a sliding mode technique to achieve finite‐time convergence, and an online verification of the alternative PE condition is introduced. Leakage terms, functions of the estimation error, are incorporated into the adaptation laws to avoid windup of the adaptation algorithms. The adaptive algorithm applied to robotic systems permits that tracking control and exact parameter estimation are achieved simultaneously in finite time using a terminal sliding mode (TSM) control law. In this case, the PE condition can be replaced with a sufficient richness requirement of the command signals and thus is verifiable a priori. The potential singularity problem encountered in TSM controls is remedied by introducing a two‐phase control procedure. The robustness of the proposed methods against disturbances is investigated. Simulations based on the ‘Bristol‐Elumotion‐Robotic‐Torso II’ (BERT II) are provided to validate the efficacy of the introduced methods. Copyright © 2014 John Wiley & Sons, Ltd.

Strict Lyapunov functions for time-varying systems with persistency of excitation

Adaptive observer design deals with online estimation of states using input-output information of a dynamical system in the presence of parametric uncertainty in the dynamics. It works with the principle of simultaneous estimation of states and the uncertain parameters using suitable online update routines to ensure stability of the estimation error dynamics. Conventional adaptive observers rely on the richness of input-output signals to satisfy the persistence of excitation (PE) condition for parameter convergence. The PE condition is restrictive since it demands sufficient energy of the signal for the entire time span and the condition depends on the future behavior of the signal, which poses difficulty in online verification. In contrast to conventional designs, the proposed work develops a switched adaptive observer which ensures uniformly ultimately bounded (UUB) stability of the estimation error dynamics without requiring the stringent PE condition, while imposing an online-verifiable condition of initial excitation (IE) on the regressor signal. The IE condition is significantly milder than PE, since it demands sufficient energy/richness of the signal only in the initial time-window. Strategic introduction of multiple switching in the parameter estimator ensures the ultimate bound to be arbitrarily reducible by appropriate choice of the design parameters.