A result on exponential tracking error convergence and persistent excitation

This work proposes a distributed composite adaptive synchronization algorithm for multiple uncertain EulerLagrange (EL) systems, where parameter convergence is achieved under a relaxed mathematical condition as compared to the state-of-the-art. Classical adaptive controllers require an analytical condition, called persistence of excitation (PE), to ensure parameter convergence, which results in better transient performance and robustness to disturbance. The PE condition is extended to Cooperative-PE (C-PE) condition for distributed adaptive controllers with cooperative estimation strategies. The PE and C-PE conditions are restrictive in nature since these conditions are not satisfied in most practical applications. Recent literature in adaptive control has relaxed the PE condition to Initial Excitation (IE), which is shown to be sufficient for parameter convergence. The IE condition is argued to be significantly milder than PE and can be satisfied in many practical setting. The proposed result further extends the IE condition to Cooperative-IE (C-IE) condition in distributed adaptive control architecture in the context of synchronizing multiple EL systems. It is established that the C-IE condition is milder than PE, IE, and C-PE conditions. Two-tier filter based estimation algorithm with strategic switching ensures parameter convergence under the C-IE condition and thereby provides exponential convergence of tracking and parameter estimation error to zero. Simulation results validate the efficacy of the proposed algorithm as compared to conventional distributed adaptive controllers in terms of superior tracking and estimation performance.

Combined Cooperative Adaptive Cruise Control using Collective Initial Excitation based Distributed Parameter Estimator

Parameter convergence is desirable in adaptive control as it brings several attractive features, including accurate online modelling, exponential tracking, and robust adaptation without parameter drift. However, a strong persistent‐excitation (PE) condition must be satisfied to guarantee parameter convergence in the conventional adaptive control. This study proposes a model reference composite learning control strategy to guarantee parameter convergence without the PE condition. In the composite learning, an integral at a moving‐time window is applied to construct a prediction error, an integral transformation is derived for avoiding the time derivation of plant states in the calculation of the prediction error, and both the tracking error and the prediction error are applied to update parametric estimates. Global exponential stability of the closed‐loop system is established under an interval‐excitation condition which is much weaker than the PE condition. Compared with a concurrent learning technique that has the same aim as this study, the proposed composite learning technique avoids the usage of singular value maximisation and fixed‐point smoothing resulting in a considerable reduction of computational cost. Numerical results have verified effectiveness and superiority of the proposed control strategy.

This work generalizes the recently developed initial excitation (IE) based adaptive control by introducing the notion of semi-initial excitation (semi-IE), a still relaxed version of IE. Typically, classical adaptive controllers guarantee Lyapunov stability of the overall error dynamics (tracking error + parameter estimation error) and asymptotic convergence of tracking error to zero, while requiring a restrictive condition of persistence of excitation (PE) for parameter convergence. The authors' recent work on a new adaptive control architecture based on the milder IE condition guarantees parameter convergence while ensuring exponential stability of the overall error dynamics once the online verifiable IE condition is satisfied. The current work further relaxes the excitation condition by proposing the concept of semi-IE. The proposed adaptive controller is proved to guarantee convergence of the parameter estimation error to a lower dimensional manifold under the weaker semi-IE condition, while the stronger condition of IE guarantees convergence of the parameter estimation error to zero. The designed adaptive algorithm has been shown to significantly improve the transient response of tracking error as compared to the conventional approaches.

Designing the optimal control for the practical systems is challenging due to the unknown system dynamics and unavoidable external disturbances. In this article, the $H_{\infty } $ optimal control problem is investigated for continuous-time linear systems with unknown dynamics. The existing reinforcement learning-based $H_{\infty } $ optimal control methods require persistence of excitation (PE) condition or data storage mechanism to guarantee the convergence of the algorithms. However, PE condition is hard to be monitored online and data storage mechanism requires to store huge amounts of past system data. In order to solve these problems, the initial excitation-based reinforcement learning algorithms are presented to learn the optimal control policy under an online-verifiable initial excitation condition. The properties of the initial excitation-based reinforcement learning algorithms are analyzed, which show that the presented algorithms converge to the optimum under the initial excitation condition. Numerical analysis is provided which demonstrates the correctness of the presented algorithms.

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.

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 /spl 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.

The problem of parameter estimation for salient permanent magnet synchronous motors (PMSMs) is considered in this paper. It is assumed that stator currents and voltages and rotor position are measurable signals and the only known motor parameter is the number of pole pairs. The proposed estimation algorithm uses dynamic operators applied to the measurable signals to get a linear regression model with unknown motor parameters. As proven, scalar estimators that are designed for each unknown motor parameter ensure finite time convergence of estimation errors to zero. The main benefit of the proposed approach is to relax the stringent assumptions imposed on regression functions such as condition of persistent excitation (PE) and non-square integrability that are often difficult to verify in practice.

This paper studies neural learning control with predefined tracking error bound for a marine surface vessel whose accurate dynamics could not be obtained a priori . With the introduction of an error transformation function, the constrained tracking control of the original vessel is transformed into the stabilization of an unconstrained system. A filtered tracking error is introduced based on the error transformation, and radial basis function (RBF) neural networks (NNs) are employed to approximate unknown vessel dynamics. Subsequently, stable adaptive NN control is proposed to ensure ultimate boundedness of all the signals in the closed-loop system and to guarantee prescribed tracking performances. Under persistent excitation (PE) condition, the proposed adaptive NN control is shown to be capable of acquiring knowledge on the vessel dynamics and of storing the learned knowledge in memory. The stored knowledge is reused to develop neural learning control such that the improved control performance with faster tracking convergence rate and less computational burden could be achieved, while prescribed transient and steady-state tracking control performances are guaranteed. Simulation studies are performed to demonstrate the effectiveness of the proposed design techniques.

This paper presents a composite fuzzy learning finite-time prescribed performance control (PPC) approach for uncertain nonlinear systems with dead-zone inputs. First, a finite-time performance function is constructed by a quartic polynomial. Subsequently, with the help of an error transformation function, the restriction problem of the tracking error performance is transformed into a stability problem of an equivalent transformation system. In order to ensure that all signals of the closed-loop system are bounded, a finite-time PPC method combined with fuzzy logic systems (FLSs) is proposed. Although the tracking error can be guaranteed to be limited within a predefined range, the proposed finite-time PPC method only uses instantaneous data, which cannot guarantee the accurate estimation of unknown functions under the influence of dead-zone inputs. Therefore, based on the persistent excitation (PE) condition, a predictive error is defined by using online recorded data and instantaneous data, and a corresponding composite learning finite-time PPC method with parameter updating the law, which can not only achieve the control aim of the former method but also improve the control effect, is designed. The simulation results verified that the composite learning finite-time PPC method is more effective than the finite-time PPC method without learning.

This paper concerns adaptive estimation of dynamic systems which are nonlinearly parameterized. A majority of adaptive algorithms employ a gradient approach to determine the direction of adjustment, which ensures stable estimation when parameters occur linearly. These algorithms, however, do not suffice for estimation in systems with nonlinear parameterization. We introduce in this paper a new algorithm for such systems and show that it leads to globally stable estimation by employing a different regression vector and selecting a suitable step size. Both concave/convex parameterizations as well as general nonlinear parameterizations are considered. Stable estimation in the presence of both nonlinear parameters and linear parameters which may appear multiplicatively is established. For the case of concave/convex parameterizations, parameter convergence is shown to result under certain conditions of persistent excitation.

Many practical adaptive feedforward systems are overparametrized and, for this reason, will not satisfy persistent excitation (PE) conditions. For these systems, a weaker PE condition is proposed under which it is shown that the while parameters may not converge, the cancellation error (the error between the desired and estimated outputs) still converges exponentially. Bounds are given on the exponential rate of convergence useful for understanding the various tradeoffs and for systematic optimization and design purposes. The convergence rate is determined by properties of the confluence matrix (defined herein) that plays a role similar to that played by the autocorrelation matrix for fully PE systems. As a case study, the structure of the confluence matrix is examined in detail for adaptive systems with a tap delay line (TDL) regressor and sinusoid excitation.

This paper presents an adaptive observer design for nonlinear systems that have parametric uncertainties in the unmeasured state dynamics. Without persistency of excitation (PE) the convergence of the state estimation error to zero is proved. Under PE conditions, uniform global asymptotic stability of the origin of the state and parameter error is shown. Furthermore, if the regressor is sufficiently smooth, uniform local asymptotic stability of the origin is proved. Two simulation examples are shown to illustrate the results.

An important property of adaptive laws without robustness modifications and in the absence of plant uncertainties is the exponential convergence of the estimated parameters to their true ones under some persistent excitation (PE) condition on the information signal vector. When the adaptive laws are modified for robustness this ideal property of parameter convergence may no longer be maintained due to the effect of the modifications on the convergence properties of the adaptive law. In this paper we examine the effect of the switching-σ modification on the exponential convergence of parameter estimates for two different adaptive laws. We show that the switching-σ modification does not affect the convergence properties of the estimated parameters but instead improves the speed of convergence.

This paper addresses the problem of distributed cooperative adaptive identification for time-varying static parametric model systems with a new distributed event-triggered schedule. In the present framework, the cooperative persistent exciting (PE) condition is introduced to remove the existing assumption that the feasible solution of the controllers depended on the solving of line matrix inequality. By considering the communication resources constraints, a new event-triggered strategy is proposed to relax the requirement of continuous monitoring for each system. It is shown that, if the proposed assumpations are statisfied, the closed-loop adaptive systems are uniformly globally exponentially stable (UGES), the parameter estimation error converge to zero uniformly exponentially under a cooperative persistent excitation (PE) condition, and the Zeno behaviour is avoided by intoruding appropriate assumption that removes the state dependence requirement for each system.