Periodic disturbance cancellation with uncertain frequency

doi:10.1016/j.automatica.2003.10.024

Automatica

Volume 40, Issue 4, April 2004, Pages 631-637

https://doi.org/10.1016/j.automatica.2003.10.024 Get rights and content

Abstract

This paper presents a new algorithm to cancel periodic disturbances with uncertain frequency. The disturbances are cancelled using an internal model structure with adaptive frequency, in parallel with a stabilizing controller. The time-varying internal model controller's states, in steady state, can be mapped to two time-invariant variables: the magnitude or energy of the internal model and frequency of the disturbance. An additional integral controller then can be used to reduce the difference between the internal model controller (IMC) and disturbance frequency to zero. The stability of the feedback control system with this algorithm and convergence of the algorithm to the correct frequency with exact disturbance cancellation are justified by singular perturbation and averaging theories. The algorithm is locally exponentially stable, rather than asymptotically stable. Simulations demonstrate the performance of the algorithm, the ability of this algorithm to identify the frequency of periodic disturbances and to reject periodic disturbances with uncertain frequency.

Introduction

The problem of rejecting periodic disturbances appears in a variety of applications including optical and magnetic disk drives (Sacks, Bodson, & Khosla, 1996), rotating mechanical systems, and active noise control (Bodson, Jensen, & Douglas, 2001), etc. There are several methods to design control systems for eliminating periodic disturbances. Francis and Wonham (1976) proposed the Internal Model Principle (IMP) in 1976. Francis and Wonham stated that a suitably reduplicated model of the dynamic structure of the disturbance or reference signals should be incorporated in the feedback loop for perfect disturbance rejection. For sinusoidal disturbances, this means that the controller must have a pair of poles on the jω-axis at a location corresponding to the frequency of the disturbances. When exactly cancelling periodic disturbances, it is very important to identify the exact frequency, magnitude, and phase of the periodic disturbances. Therefore, there is a need for controllers that can identify and track this frequency.

There is a vast literature covering the topic of identifying and cancelling periodic disturbances. Some recent examples include Bodson et al. (2001); Tsao, Qian, and Nemani (2000); Arcara, Bittanti, and Lovera (2000) and Hsu, Ortega, and Damm (1999). A review of other approaches is given in Kwok and Jones (2000) and Sievers and Flotow (1992). Identification and cancellation of the class of disturbances considered in this paper with traditional adaptive control was considered by Tsypkin (1991).

In Section 2 of this paper, an internal model is constructed in a state-space form in parallel with a traditional controller in a feedback control system. We develop a function that maps the time-varying states of the internal model to the time-invariant frequency of periodic disturbances and introduce an integral controller to update the parameter of the internal model. In Section 3, we give the proof of convergence of our frequency estimates and stability of the feedback control system using singular perturbation and averaging theory. Section 4 shows simulations which demonstrate the behavior and performance of the algorithm. The performance is then compared to two recently developed algorithms, presented in Bodson et al. (2001) and Tsao et al. (2000), that achieve exact disturbance rejection even with uncertainty in the frequency. It is seen that the proposed algorithm has advantages in speed, complexity, and/or disturbance rejection capability.

Section snippets

Adaptive algorithm

The block diagram of an internal model controller is shown in Fig. 1, in which, C(s) can be chosen as a common controller such as PI or PID controller. G(s) is the plant which we want to control. If $v=a sin (ω_{c} t+ϕ)$ , then an estimate of the frequency ω_c is $ω ̂_{c} =ω− ωK_{f} ex_{1} (ωx_{1})^{2} +x^{2}_{2} .$ This estimate will be derived in the following:

Convergence and stability analysis

In this section, we apply singular perturbation theory to verify that the control system including our algorithm is locally exponentially stable and that ω converges to ω_c exponentially. Stability of the reduced or slow system will be shown by applying averaging theory.

Simulations

In this section, the performance of our adaptive algorithm is examined via simulation. We will compare our algorithm with other algorithms for sinusoidal and periodic inputs. All simulations have been performed in Mathworks Simulink with the integrator algorithm ODE45.

Conclusions

An adaptive algorithm based on the state-space form of the internal model is presented in this paper for identifying and tracking the uncertain frequency of a periodic disturbance and cancelling this disturbance. The internal model controller has two functions. One is to provide the information for frequency identification, and the other is to cancel the disturbance. If the initial value of ω is close enough to one of the frequencies of the periodic disturbance, our algorithm will converge to

Lyndon J. Brown (M’87) received the B.A.Sc. degree in electrical enginering from the University of Waterloo, Waterloo, ON, Canada, in 1988 and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign, in 1991 and 1996, respectively. He was with E. I. DuPont de Nemours, Newark, DE, from 1996 to 1999 and joined the Faculty of Engineering, University of Western Ontario, London, ON, in 1999. His research interests include adaptive control, periodic

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Qing Zhang received the B.A.Sc. degree in electronic engineering from Xi'an Institute of Technology, Xi'an, China, in 1991, the M.E.Sc. degree in electrical engineering from Xi'an Jiaotong University in 1994, and the M.E.Sc. degree in electrical and computer engineering from the University of Western Ontario, London, ON, Canada, in 2001. He is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering at the University of Western Ontario. His research interests include signal identification and processing, adaptive and nonlinear system control, and real-time and embedded control systems.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Thomas Parisini under the direction of Editor Robert R. Bitmead.

¹: Supported by Natural Science and Engineering Research Council and Auto 21 NCE.

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Brief paperPeriodic disturbance cancellation with uncertain frequency☆

Abstract

Introduction

Section snippets

Adaptive algorithm

Convergence and stability analysis

Simulations

Conclusions

Automatica

Automatica

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IEEE Transactions on Control Systems Technology

Active noise control for periodic disturbances

IEEE Transactions on Control Systems Technology

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Brief paper
Periodic disturbance cancellation with uncertain frequency☆