Elsevier

Automatica

Volume 120, October 2020, 109134
Automatica

Technical communique
Model matching of input/state asynchronous sequential machines with actuator saturation and bounded delays

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Abstract

This paper presents model matching of input/state asynchronous sequential machines (ASMs) in which the corrective controller has the restriction of actuator saturation. As this restriction gives rise to a decrease in the length of control input sequences, a tighter existence condition is needed to achieve the control goal. We also propose the delayed model matching problem in association with actuator saturation, where the stable-state behavior of the closed-loop system is made equivalent with that of a reference model within a finite delay. The delay-bound controllability condition is analyzed and the design procedure for an appropriate model matching controller is addressed in the framework of corrective control theory. An illustrative example is provided to demonstrate the proposed notions and controller synthesis.

Introduction

Corrective control for model matching aims at rendering the stable-state behavior of an asynchronous sequential machine (ASMs) equivalent with a model. Insofar as the machine has stable reachability toward desired states, a corrective controller provides proper control inputs so that the closed-loop system can have instantaneous transitions to the states without violating the nominal operation. This compensation is made possible since the interaction between the controller and machine is very fast under asynchrony.

While various promising results on corrective control have been reported (Murphy et al., 2003, Wang et al., 2018, Xu and Hong, 2013), two important problems still remain to be tackled. First, in all the prior work any chain of controlled transitions is assumed to end before further change of the external input. In reality, however, each step of the correction procedure takes non-negligible time (see Yang & Kwak, 2019 for experimental verification). Hence if too long a control input sequence is utilized, the external input may change before the transmission of the input sequence, leading to incorrect control outcomes. It would be more so if the external input is generated by independent entities with high transmission frequency. To accommodate fast changes of the external input, it is required to set the limit on the length of control input sequences, in other words, actuator saturation must be considered in the design of corrective controllers.

Next, no study has been done on an alternative to the failure of (immediate) model matching, which could be exacerbated by actuator saturation. We pay attention to a class of control problems for ASMs where temporary violation of model matching is allowed. In the case of error counters in satellite systems (Karp & Gilbert, 1993) that will be discussed in this paper, failure of counting some error signals is often tolerated as long as the counter catches up with desirable count values until receiving a finite number of error signals. To solve such a problem, we present delayed model matching for ASMs wherein the closed-loop system is made stably equivalent with the model within a finite delay as measured with changes of the external input.

We first address the existence condition for a corrective controller that achieves immediate model matching with the constraint that the length of input sequences is bounded by a prescribed limit. Since full reachability of the machine is not utilized, a tighter condition is needed to ensure the success of control. Under actuator saturation, we then investigate the delay-bound controllability condition and address a design algorithm for an appropriate controller in the framework of corrective control theory. A series of examples on a practical system runs through the paper.

Dealing with actuator saturation has been a topic of active research mostly in continuous-time nonlinear systems (Sakthivel et al., 2018, Selvaraj et al., 2017, Selvaraj et al., 2018). Compared with these studies, a major contribution of this work is that instead of being merely regarded as inherent disturbance, actuator saturation is actively exploited for quantifying realistic constraint on controller synthesis. On the other hand, corrective control with bounded-delay was first studied in Yang (2017). The present work is distinctive from Yang (2017) in that (i) the control objective is extended to model matching; and (ii) the controller synthesis tolerates the constraint of actuator saturation.

Section snippets

Main result

Example 1

Let us start with a practical example to illustrate the motivation for our study. Fig. 1 shows two input/state ASMs Σ and Σ that represent asynchronous error counters used for the error detection and correction (EDAC) module of satellites (Karp & Gilbert, 1993). Σ counts up to six levels of errors ai, i=1,,6, by transferring to the corresponding states xi%6+1, and Σ counts not only ai’s but also a different kind of errors with four levels b1,,b4. The goal is to apply corrective control to Σ

Example: controller synthesis

From Example 6, Example 7, three entries (1,4), (4,3), and (6,5) violate condition (2) for immediate model matching. Consider the case of (1,4) as an example. Referring to Fig. 1, (1,4) in Σ corresponds to s(x1,b1)=x4. Applying (6), we first compute Pj(x1) and Qj(x1) until an empty Pj(x1) is obtained. For instance, since R1(x1)={x1,x2,x4} but S1,1(2)(Σ)=S1,2(2)(Σ)=1 and S1,4(2)(Σ)=0, P1(x1)={x4} and Q1(x1)={x1,x2} and so on. It turns out that P2(x1) and P3(x1)=. Referring to

Conclusion

Model matching of input/state ASMs has been addressed under actuator saturation. We have quantified the design restraint of the corrective controller in terms of the length of the used input sequences. Under actuator saturation, immediate model matching needs a tighter existence condition for a controller. As an alternative control objective for accommodating actuator saturation, delayed model matching has been discussed. We have elucidated the condition under which the closed-loop system can

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This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2018R1D1A1A09082016), and in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1A5A1025137). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Christoforos Hadjicostis under the direction of Editor André L. Tits.

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