Model Reference Adaptive Control

Abstract

Adaptive control methods present effective system‐theoretical tools in order to achieve closed‐loop system stability and performance in the presence of exogenous disturbances and system uncertainties, where they are generally classified as either direct or indirect. A well‐known class of direct adaptive control methods is model reference adaptive control architectures. In particular, these architectures employ two major components – a reference model and a parameter adjustment mechanism. A desired closed‐loop dynamical system response is captured by the reference model for which its response is compared with the response of the uncertain dynamical system. The system error signal resulting from this comparison drives the parameter adjustment mechanism. This mechanism then adjusts the controller parameters in a real‐time (i.e., online) manner for driving the trajectories of the uncertain dynamical system to the trajectories of the reference model. This article discusses these components in a basic state feedback setting in order to provide an introduction to the model reference adaptive control design procedure. The article also makes connections to several other, relatively advanced model reference adaptive control methods for interested readers.

Tracking and parameter identification for model reference adaptive control

“…which agrees with the integrator state dynamics from the work of Yucelen et al 23 when E p (t) is a nonzero constant matrix and E r = 0 n c ×n c ; see the work of Yucelen 29 and Remark 2 below for motivation for the integrator state. Then, (1) and…”

Experimental Results of a Model Reference Adaptive Control Approach on an Interconnected Uncertain Dynamical System

“…The parameter adjustment mechanism is then driven by the system error signal resulting from the comparison between the model reference system and the uncertain dynamical system. Through adjusting the controller parameters, the parameter adjustment mechanism is designed (asymptotically or approximately) to drive the trajectories of the uncertain dynamical system to the trajectories of the reference model [2].…”

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

“…In a standard MRAC architecture, the objective is to (approximately or asymptotically) drive the state vector x ( t ) of the uncertain dynamical system given by to the state vector of a reference model capturing an ideal desired closed‐loop dynamical system performance given by In , is a given uniformly continuous bounded command, is the Hurwitz reference system matrix, and is the command input matrix with and are respectively the nominal feedback and feedforward gain matrices. To this end, consider the feedback control algorithm given by In , is an estimate of W satisfying a projection operator–based weight update law …”

“…In a standard MRAC architecture, 21,24 the objective is to (approximately or asymptotically) drive the state vector x(t) of the uncertain dynamical system given by (1) to the state vector x ri (t) ∈ ℝ n of a reference model capturing an ideal desired closed-loop dynamical system performance given by .…”

“…with ∈ ℝ + being the learning rate, P ∈ ℝ n×n + is the unique solution to the Lyapunov equation 0 = A T r P + PA r + I n owing to A r being Hurwitz, and e s (t) ≜ x(t) − x ri (t) being the system error state vector for standard MRAC architecture. In addition,Ŵ s, max,i+( −1)n ∈ ℝ + denote symmetric elementwise projection bound defined such that |[Ŵ s (t)] i | ≤Ŵ s, max ,i+(j−1)n , *As it is known, 11,21,24 one needs to choose the nominal gain matrices K 1 and K 2 such that A r is Hurwitz and −EA −1 r B r = I m , where E ∈ ℝ m×n is a matrix that allows a user to select a subset of x(t) to be ideally followed by c(t). i = 1, … , n and j = 1, … , m. The following sufficient condition is needed for the stability of the standard MRAC architecture subject to actuator and unmodeled dynamics (see Section 3.1).…”