Convergence and frequency-domain analysis of a discrete first-order model reference adaptive controller

Abstract

We study the convergence properties of a direct model reference adaptive control system by applying techniques from numerical analysis. In particular, a first-order discrete system coupled to a minimal control synthesis algorithm discretized by the one-step one-stage zero-order-hold sampling is studied. This results in a strongly non-linear dynamic system owing to the adaptive mechanism where stability at steady state, i.e. at the operating point, equates to successful control. This paper focuses on the convergence analysis of the overall dynamical system for understanding accuracy, stability and performance at steadystate. The local stability of the steady state solution is considered by linearizing the system in the neighbourhood of an operating point when the input is a step function. This analysis allows us to specify two gain space domains which define the region of local stability. Moreover, both the accuracy and the frequency-domain analyses give insight into the range of adaptive control weightings that results in optimal performance of the minimal control synthesis algorithm and also highlights a possible approach to a priori selection of the time step and adaptive weighting values.The effectiveness of the proposed analysis is further demonstrated by simulations and experiments on a first-order plant.Adaptive control methods are generally divided into (i) direct and (ii) indirect methods. In the first case the adjustment rules provide directly how to update the controller parameters; in the indirect methods, the parameters of the unknown plant are estimated on-line, and the controller parameters are calculated on the base of these estimates. In the context of control theory some recent books and papers relevant to direct methods can be found [2,3]. In the context of structural control several applications have been made by several researches as well [4][5][6].When a direct reference model controller is used, it can be also applied to systems where the details of the plant cannot be fully known a priori or are varying with time. Using this type of algorithms without the knowledge of plant parameters, such that we assume zero initial conditions for the controller gains, has become known as the minimal control synthesis (MCS) approach [7]. Basing adaptive control schemes on a reference model enables the system to be controlled to behave like the model itself. This type of approach [8,9] has been applied to a wide range of systems including non-linear and chaotic systems [3,10]. As the approach is based primarily on linear control theory being the reference model usually a linear one, the effect of non-linearities and/or disturbances in non-linear systems is compensated for by the adaptive nature of the controller. Very recently in the field of dynamic and seismic testing [11], the MCS method was used in real-time substructuring, in order to minimize the error between displacements imposed by the actuators on the substructure and displacements at the interfaces of the numerical model [12,13]; and it was adop...

Stability and accuracy analysis of a discrete model reference adaptive controller without and with time delay

“…Integration formulas of different complexity can be found in the control literature, see for instance, the trapezoidal rule employed in optimal control algorithms [28]; and the collection of algorithms for nonstiff and stiff problems available in MATLAB [29]. Because ZOH sampling is not capable of filtering out the possible undesirable oscillations caused by bounded disturbances [17], we use an L-stable integrator originated by Rosenbrock [26]. Rosenbrock's methods belong to a large class of linearly implicit integrators that avoid the solution of non-linear systems by means of an updated Jacobian, J, which held constant within each t. Variants of these realtime integrators can be found in [30].…”

“…These are applied in the neighbourhood of the system operating point, while assuming an input excitation consisting of a unit step input. The first technique is based on physical insight and consists of substituting the values corresponding to the operating point into the amplification matrix [17]. The second technique is a more rigorous linearization, based upon a Taylor series expansion about the same operating point.…”

“…In what follows, we analyse the absolute stability and accuracy properties of control algorithms by means of techniques derived from the field of numerical analysis [17]. The MCS algorithm, when applied to the control of a first-order plant, is viewed as a non-linear dynamical continuous system, generated by control and state equations [18].…”

“…As a result, in order to show the reduction of both the absolute stability and the controller performance, ZOH sampling is employed for the integration of the discrete MCS controller when used with both single degree of freedom (SDoF) and two-degree of freedom (DoF) systems. Moreover, because ZOH sampling is not capable of filtering out undesirable oscillations caused by bounded disturbances [17], we have performed both stability analysis and simulations with a linearly implicit L-stable two-stage integration method (LSRT2) [25]. This approach is based on the Rosenbrock [26] schemes, with stability properties that allow the response of the stiff components, i.e.…”

Discrete-time integral MRAC with minimal controller synthesis and parameter projection

“…Moreover, the use of such a digital strategy prevents the onset of unwanted phenomena -instability and undesirable overshoots-that can arise when implementing continuous-time MCS algorithms via standard discretization methods [5].…”