Elsevier

Automatica

Volume 72, October 2016, Pages 205-208
Automatica

Technical communique
Enhanced disturbance rejection for a predictor-based control of LTI systems with input delay

https://doi.org/10.1016/j.automatica.2016.05.019Get rights and content

Abstract

In this paper, a new predictor-based control strategy for LTI systems with input delay and unknown disturbances is proposed. The disturbing signal and its derivatives up to the r-th order are estimated by means of an observer, and then used to construct a prediction of the disturbance. Such prediction allows defining a new predictive scheme that takes into account its effect. Also, a suitable transformation of the control input is presented and a performance analysis is carried out to show that, for a given controller, the proposed solution leads to better disturbance attenuation than previous approaches in the literature for smooth enough perturbations.

Introduction

The problem considered in this paper deals with possibly open-loop unstable disturbed LTI systems, defined by ẋ(t)=Ax(t)+B[u(th)+d(t)]u(t)=u0(t)t[h,0)x(0)=x0 where xRn is the state, uR is the control input, dR is an unknown input disturbance, h>0 is a known and constant input delay, and ARn×n, BR are known matrices.

The Smith Predictor (Smith, 1957), can be considered as the first predictor-based control for open-loop stable linear systems. Later, the same concept was extended for open-loop unstable systems by introducing an h units of time ahead state predictor, (Artstein, 1982, Manitius and Olbrot, 1979): xˆ1(t+h)eAhx(t)+thteA(ts)Bu(s)ds, referred to as the conventional prediction in this paper. The variable xˆ1(t+h) is understood as the projection of the state starting at x(t) driven by the control history u(t+s),s[h,0]. In the absence of disturbances, the feedback law u(t)=Kxˆ1(t+h) achieves asymptotic stabilization for any h>0 with a proper choice of K.

However, in a disturbed system, an error is introduced in the prediction xˆ1. Since there is always an error between the exact and the approximated predictions, it is not possible to remove constant disturbances even using integral action. Although this is an interesting topic from a practical point of view (Krstic, 2010), only few articles have addressed this problem. In an effort to predict the evolution of the disturbances, adaptive algorithms have drawn the attention of some researchers. For example, sinusoidal disturbances of unknown frequency are identified and rejected in Pyrkin, Smyshlyaev, Bekiaris-Liberis, and Krstic (2010) for LTI systems with known delay, and more recently in Basturk and Krstic (2015) for systems with matched uncertainties (see also the references therein). Also, adaptive schemes are used to estimate and reject constant disturbances for unknown input delay in Bresch-Pietri, Chauvin, and Petit (2012), and for known distributed delays in Bekiaris-Liberis, Jankovic, and Krstic (2013). Other works avoid any a priori knowledge of the disturbance structure. For example, a filtered version of the predicted state (2) is proved to minimize a cost functional involving the disturbance in Krstic, 2008, Krstic, 2010.

Recently in Léchappé, Moulay, Plestan, Glumineau, and Chriette (2015), a simple solution is considered, where additional feedback from the difference between the measured, x(t), and delayed predicted state, xˆ1(t), is used to define a new prediction xˆ2(t+h)=xˆ1(t+h)+[x(t)xˆ1(t)]. With this simple modification, it is proved that for a certain class of disturbing signals, the new prediction leads to better attenuation than the conventional one. However, perfect cancellation is only possible for constant disturbances, and the attenuation depends entirely on the characteristics of the disturbance.

The main contribution of this paper is the introduction of a new predictive scheme that takes into account a prediction of the disturbing signal, denoted by dˆ(t+h). Such prediction is constructed from estimates of the disturbance and its derivatives up to the rth order, which are obtained by means of a tracking differentiator. The predicted disturbance is used to define a new state prediction, denoted by xˆ3(t+h), allowing to compensate the effect of the disturbance in the overall system. A performance analysis based on Lyapunov’s theory is carried out to prove that the proposed scheme performs better than previous proposals in the literature, in the presence of smooth enough time-varying disturbances, achieving perfect cancellation in some particular cases.

Section snippets

Problem statement

Let us consider the system (1). Other than the accessibility to the full state, the following assumptions are taken:

Assumption 1

The pair (A,B) is controllable

Assumption 2

The unknown disturbance d(t) is uniformly bounded by |d(t)|D0 and it is (r+1)-times continuously differentiable with |d(r+1)(t)|Dr+1,t0.

From (1), it can be seen that the actual projection of the state at time t+h is given by x(t+h)=eAhx(t)+thteA(ts)B[u(s)+d(s+h)]ds. Although (4) cannot be used in practice because the disturbance is unknown,

Proposed predictor-based control

Let us assume that a future estimation of the disturbance dˆ(t+h), is available. Then, a new predicted state which considers the effect of the disturbance can be computed by xˆ3(t+h)eAhx(t)+thteA(ts)B[u(s)+dˆ(s+h)]ds. The disturbance prediction error is defined as σ(t)d(t)dˆ(t). From (4), (6) and using the definition (7), the error of the new prediction is given by x(t+h)xˆ3(t+h)=thteA(ts)Bσ(s+h)ds.

Proposition 5

If σ0, then the asymptotic convergence of the new prediction xˆ3 to zero implies the

Performance analysis

The Artstein’s reduction (Artstein, 1982), is a useful tool to analyze time delay systems as it transforms the original system into a delay-free one. It is easy to show that the reduction of system (1) with the conventional predicted variable z1(t)xˆ1(t+h) leads to ż1(t)=Az1(t)+Bu(t)+eAhBd(t) while the reduced system using the alternative prediction z2(t)xˆ2(t+h), proposed in Léchappé et al. (2015), is derived as ż2(t)=Az2(t)+Bu(t)+Bd(t)+eAhB[d(t)d(th)]. Similarly, considering the

Numerical validation

In order to validate the bounds derived in the previous section, let us consider the system (Léchappé et al., 2015), ẋ(t)=[0193]x(t)+[01]u(th)+[0d(t)]. The simulation considers the same scenario that in Léchappé et al. (2015), with an input delay h=0.5s, the system starting from x(0)=[1.5,1]T, a sinusoidal disturbance d(t)=3sin(0.5t), and predictor-based control law given by u(t)=[kp,kd]xˆ2(t+h) with kp=45, kd=18. The same control law is selected for the proposed scheme by computing (20)

Conclusion

A new predictive scheme to control time delay LTI systems with unknown input disturbances has been proposed. The new predicted variable xˆ3, for a given controller, leads to better disturbance attenuation than previous proposals, under some constraints in the disturbance signal. The new prediction also achieves perfect cancellation of a certain class of time-varying disturbances without having any knowledge of their structure.

Acknowledgments

The authors would like to thank the Associate Editor and anonymous reviewers for their constructive comments.

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This work was partially supported by projects TIN2014-56158-C4-4-P and PROMETEOII/2013/004. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Fouad Giri under the direction of Editor André L. Tits.

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