Adaptive output regulation for linear systems via discrete-time identifiers

Abstract

The problem of output regulation for general multivariable linear systems has been solved in the 70s, in the seminal works of Francis, Wonham and Davison, under the assumption that the reference signals and the disturbances acting on the system are generated by a known exogenous linear system (the exosystem). The regulator is designed to embed an internal model of the exosystem, which ensures that asymptotic regulation is maintained under arbitrary plant perturbations that do not destroy linearity and closed-loop stability. This robustness property, however, is inexorably lost whenever the internal model does not match exactly the exosystem. In this paper we endow the linear regulator with a discrete-time adaptive unit that adapts the regulator's internal model on the basis of the closed-loop evolution. Compared to existing approaches, adaptation here is cast as an identification problem, and the corresponding optimal predictor is designed independently from the underlying control system. This permits to separate stabilization and adaptation, thus naturally handling general non-square multivariable non minimum-phase plants. Closed-loop stability is guaranteed and, if the dimension of the internal model is large enough and a persistency of excitation condition is fulfilled, asymptotic regulation is achieved for references and disturbances generated by an unknown exosystem. Robustness to parametric uncertainties is inherited by the linear regulator and robustness to additional unmodeled disturbances is proved to hold.

Adaptive Output Regulation for Multivariable Linear Systems via Slow Identifiers

“…Proposition 1 states that, in order to obtain asymptotic regulation along the persistently exciting solutions, the dynamics of the identifier have to be slow enough compared to the rest of the control system. In this respect it is worth comparing this result with the approach of [16], where the time-separation of the adaptation dynamics is obtained by means of a discrete-time identifier working on time instants that must be separated, on average, by a sufficiently large amoung of time.…”

“…The existence of such a set, when the transfer function of the plant has a finite number of zeros, has been proved in [16].…”

“…Instead of adaptation, immersion arguments have been used in [9], [10], [11], [12] and [13] for linear and some classes of nonlinear exosystems. More recently, a different approach based on identification techniques has been proposed in [14] for SISO normal forms, while in [15] a hybrid adaptive observer is designed for SISO stable plants ,and an adaptive design for multivariable linear systems, based on discretetime identification schemes, has been proposed in [16]. In this paper we consider the output regulation problem for general multivariable linear systems, with the reference signals and the disturbances that are generated by an unknown exosystem.…”

“…In this paper we consider the output regulation problem for general multivariable linear systems, with the reference signals and the disturbances that are generated by an unknown exosystem. On the heels of [16], we augment a canonical linear regulator with an identification unit (referred to as the "identifier") that adapts the internal model on the basis of the measurable data. Differently from [16], the identifier is continuous-time, and the asymptotic properties of the regulator are obtained thanks to a time-scale separation of the two units.…”

“…On the heels of [16], we augment a canonical linear regulator with an identification unit (referred to as the "identifier") that adapts the internal model on the basis of the measurable data. Differently from [16], the identifier is continuous-time, and the asymptotic properties of the regulator are obtained thanks to a time-scale separation of the two units. The identifier is designed to solve a least-squares optimization problem defined by the available measurements.…”