Asymptotic Tracking for Systems with Structured and Unstructured Uncertainties

Abstract

Abstract-The control of systems with uncertain nonlinear dynamics has been a decades-long mainstream area of focus. The general trend for previous control strategies developed for uncertain nonlinear systems is that the more unstructured the system uncertainty, the more control effort (i.e., high gain or high-frequency feedback) is required to cope with the uncertainty, and the resulting stability and performance of the system is diminished (e.g., uniformly ultimately bounded stability). This brief illustrates how the amalgamation of an adaptive model-based feedforward term (for linearly parameterized uncertainty) with a robust integral of the sign of the error (RISE) feedback term (for additive bounded disturbances) can be used to yield an asymptotic tracking result for Euler-Lagrange systems that have mixed unstructured and structured uncertainty. Experimental results are provided that illustrate a reduced root-mean-squared tracking error with reduced control effort.

Asymptotic tracking by a reinforcement learning-based adaptive critic controller

“…The first controller (denoted by NN+RISE) employs a standard NN gradientbased weight update law that is affine in the tracking error, given asẆ a x a )). The proposed AC-based controller (denoted by AC+RISE) uses a composite weight update law, consisting of a gradient-based term and a reinforcement-based term, as in (20), where the reinforcement term is generated from the critic architecture in (34). For the NN+RISE controller, the initial weights of the NN,Ŵ a (0) is chosen to be zero, whereasV a (0) is randomly initialized in [−1, 1], such that it forms a basis [46].…”

“…6 Stability analysis Theorem 1 The RISE-based AC controller given in (15) and (34) along with the weight update laws for the action and critic NN given in (20) and (36), respectively, ensure that all system signals are bounded under closed-loop operation and that the tracking error is regulated in the sense that e 1 (t) → 0 as t → ∞ provided the control gains k a and k c are selected sufficiently large based on the initial conditions of the states, α n−1 , α n , β 2 , and k c , are chosen according to the following sufficient conditions…”

“…where μ a (t) ∈ R m denotes the RISE feedback term defined as [34,35] μ a (k a + 1)e n (t) − (k a + 1)e n (0) + v, (16) where v(t) ∈ R m is the generalized solution (in Filippov's sense [36]) tȯ v = (k a + 1)α n e n + β 1 sgn(e n ), v(0) = 0, (17) where k a , β 1 ∈ R are constant positive control gains, and sgn( · ) denotes a vector signum function.…”

“…Typically, the presence of bounded disturbances and NN approximation errors lead to a uniformly ultimately bounded (UUB) result. The main contribution of this paper is the use of a recently developed continuous feedback technique, robust integral of sign of the error (RISE) [34,35], in conjunction with the AC architecture to yield asymptotic tracking of an unknown nonlinear system subjected to bounded external disturbances. The use of RISE in conjunction with the action NN makes the design of the critic NN architecture challenging from a stability standpoint.…”

Nonlinear tracking control of a human limb via neuromuscular electrical stimulation

“…Recently, a new continuous feedback method (coined RISE for Robust Integral of the Sign of the Error in [19], [20]) has been developed that was proven to yield asymptotic tracking of nonlinear systems with unstructured uncertainty and bounded additive disturbances. The contribution of this paper is to illustrate how the RISE controller can be applied for NMES systems.…”