Persistent Excitation is Unnecessary for On-line Exponential Parameter Estimation: A New Algorithm that Overcomes this Obstacle

Abstract

In this paper we prove that it is possible to estimate on-line the parameters of a classical vector linear regression equation Y = Ωθ, where Y ∈ R n , Ω ∈ R n×q are bounded, measurable signals and θ ∈ R q is a constant vector of unknown parameters, even when the regressor Ω is not persistently exciting. Moreover, the convergence of the new parameter estimator is global and exponential and is given for both, continuous-time and discrete-time implementations. As an illustration example we consider the problem of parameter estimation of a linear timeinvariant system, when the input signal is not sufficiently exciting, which is known to be a necessary and sufficient condition for the solution of the problem with standard gradient or least-squares adaptation algorithms.

A novel concurrent learning‐based sliding‐mode observer for second‐order multivariable systems with a time‐varying coefficient: An application to machine vision

“…The initial observer estimate is chosen as η = [0.25, 0, 0, 0, 0, 0] ⊤ . In accordance with the sliding-mode differentiator F I G U R E 2 Comparison of estimation error obtained with proposed observer (black, solid), complete-order observer 9 (red, dashed), reduced-order observer 10 (yellow, dashed-dotted) and LRE 26 (blue, dashed) for example 1.…”

“…Comparison of RMS error values obtained with the proposed scheme, the complete-order observer 9 , the reduced-order observer 10 and the LRE 26 .…”

“…Comparison of estimation error obtained with proposed observer (black, solid), complete-order observer 9 (red, dashed), reduced-order observer 10 (yellow, dashed-dotted), and LRE 26 (blue, dashed) for example 2.…”

“…Thus, in recent years, linear regression estimator (LRE) techniques have been developed that rely on the less-restrictive interval excitation (IE) condition 24 to ensure global exponential convergence of constant (time-invariant) parameters to the ground truth. 25,26 A related study 27,28 considers the synthesis of an LRE that has the additional feature of finite-time convergence, however a potential drawback of this approach pertains to the difficulty of choice of initial conditions for the estimator that may adversely affect observer performance. Thus, the idea of leveraging the interval excitation condition, which is a finite-time observability condition for system (1), for ensuring global finite-time estimation of the time-varying signal z 2 (t) has not been considered in these studies.…”

“…It is shown that the inclusion of the CL term enables the sliding mode observer to generate accurate estimates of the unknown states for both these cases, including during the time interval over which the PE condition is violated. A detailed performance comparison study with the approaches considered in 9,10,26 is also undertaken to demonstrate superior performance of the proposed scheme and illustrate the benefits of leveraging concurrent learning within the proposed framework. Thus, the most important contribution of this study is the synthesis of a provably stable multivariable sliding-mode observer that relies on a relaxed observability condition to realize accurate and robust performance in the presence of system uncertainty in practical applications such as visual navigation and visual servo control.…”