Repetitive control system: a new type servo system for periodic exogenous signals

TL;DR: In this article, a control scheme called repetitive control is proposed, in which the controlled variables follow periodic reference commands, and a high-accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period L into the closed-loop system.

Abstract: A control scheme called repetitive control is proposed, in which the controlled variables follow periodic reference commands. A high-accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period L into the closed-loop system. Sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small-gain theorem and the stability theorem for time-lag systems. Synthesis algorithms are presented by both the state-space approach and the factorization approach. In the former approach, the technique of the Kalman filter and perfect regulation is utilized, while coprime factorization over the matrix ring of proper stable rational functions and the solution of the Hankel norm approximation are used in the latter one. >

Summary

I. INTRODUCTION

• NE of the basic requirements in control systems is that they 0 have the ability to regulate the controlled variables to reference commands without a steady-state error against unknown and unmeasurable disturbance inputs.
• It is natural to expect, in view of the internal model principle, that the desired asymptotic tracking for all periodic exogenous signals of period L may be achieved by implementing a model which generates such periodic signals.
• On the other hand, the same initial condition is assumed in every trial in the betterment process, hence, the iterative action is discrete and it is enough to assure not only the stability but the convergence of the error.
• The authors introduce the repetitive control system and state the basic principle in Section 11.
• Synthesis algorithms are presented both by the state-space approach and the factorization approach in Section IV.

U. PNNCIPLE OF REPETITIVE CONTROL

• Any periodic signal with period L can be generated by the free time-delay system shown in (1 -exp ( -Ls)) into the closed-loop system.
• The authors now consider the repetitive control system with the model a(s) + exp (-Ls)/(lexp (-Ls)) depicted in Fig. 2 , where R(s), Y,(s), and E (s) are the Laplace transforms of the reference command r(t), controlled output y,(t), and error signal e(t), respectively.
• G(s) E R;"" denotes the transfer matrix for the compensated plant, and a(s) is an appropriate proper stable rational function.

Theorem I :

• Further, Theorem 1 assures asymptotic stability or uniform convergence (3.6) while Proposition 1 guarantees only the L2 input/output stability (2.10).
• Since lowfrequency band is dominant in any reference signal, this will virtually satisfy any practical demands.
• The authors now estimate the tracking error in such a case where the reference signals have the frequency band lower than w,.
• Suppose that the hypotheses of Theorem 1 are satisfied.
• The same result holds also for uniform convergence.

IV. SYNTHESIS OF REPETITIVE CONTROL SYSTEMS

• The authors consider the synthesis of modified repetitive control systems both by the state-space approach and by the factorization approach.
• For simplicity, the authors investigate only the case a(s) = 1. However, a similar discussion can be carried out for the general case [ 191.
• The compensated plant G(s) and the repetitive controller N(s) are expressed as.

=I-N X -NKl = -NK1

• (4.17) and then condition 2) in Corollary 1 is reduced to the existence of Kl(s) E RYXP such that Note that this condition does not depend on the free paramete: K2(s), but only on Kl(s).
• The authors see that the system can be stabilized by using a low-pass filter q(s) near 1.
• In other words, a modified repetitive control system with arbitrarily small steady-state error can be constructed.
• The authors now clarify the class of q(s) satisfying (4.4) or (4.18) under the following assumptions: A similar discussion can be carried out for the single freedom case.
• The parametrization can also be obtained via the Nevanlinna-Pick theory [ 191.

[A Synthesis Algorithm by the Factorization Approach]:

• Find the class of SI@) [respectively, Sz(s)] satisfying (4.20) [respectively, (4.23)], which is parametrized by a strictly bounded real matrix and choose an appropriate free parameter Z(s), also known as Step 2.
• This parametrizes the class of stabilizing controllers with one-degree-of-freedom.

V. CONCLUSION

• A new control scheme named repetitive control has been proposed.
• The authors have derived sufficient conditions for the stability of repetitive and modified repetitive control systems by applying the small gain theorem and the stability theorem for time-lag systems.
• Synthesis algorithms are presented both by the statespace approach and the factorization approach, and the class of stabilizing controllers and the low-pass filters has been characterized using the technique of the Hankel norm approximation.
• Furthermore, it is also useful for periodic disturbance inputs.
• The application to the attenuation of rotational fluctuations synthesized with the motor speed has been shown in [24] .