Technical communiqueEnhanced disturbance rejection for a predictor-based control of LTI systems with input delay☆
Introduction
The problem considered in this paper deals with possibly open-loop unstable disturbed LTI systems, defined by where is the state, is the control input, is an unknown input disturbance, is a known and constant input delay, and , 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 units of time ahead state predictor, (Artstein, 1982, Manitius and Olbrot, 1979): referred to as the conventional prediction in this paper. The variable is understood as the projection of the state starting at driven by the control history . In the absence of disturbances, the feedback law achieves asymptotic stabilization for any with a proper choice of .
However, in a disturbed system, an error is introduced in the prediction . 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, , and delayed predicted state, , is used to define a new prediction 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 . Such prediction is constructed from estimates of the disturbance and its derivatives up to the th order, which are obtained by means of a tracking differentiator. The predicted disturbance is used to define a new state prediction, denoted by , 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 is uniformly bounded by and it is -times continuously differentiable with .
Proposed predictor-based control
Let us assume that a future estimation of the disturbance , is available. Then, a new predicted state which considers the effect of the disturbance can be computed by The disturbance prediction error is defined as From (4), (6) and using the definition (7), the error of the new prediction is given by
Proposition 5 If , then the asymptotic convergence of the new prediction 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 leads to while the reduced system using the alternative prediction , proposed in Léchappé et al. (2015), is derived as 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), The simulation considers the same scenario that in Léchappé et al. (2015), with an input delay , the system starting from , a sinusoidal disturbance , and predictor-based control law given by with , . 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 , 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.