Elsevier

Automatica

Volume 39, Issue 1, January 2003, Pages 75-80
Automatica

Brief Paper
Damping of harmonic disturbances in sampled-data systems—parameterization of all optimal controllers

https://doi.org/10.1016/S0005-1098(02)00172-3Get rights and content

Abstract

Optimal damping of harmonic disturbances of known frequencies is studied for sampled-data systems. A sampled-data output feedback controller is designed to minimize the intersample variations of the controlled variable. The set of all stabilizing optimal controllers is obtained in terms of the Youla parameterization and a set of interpolation conditions at the disturbance frequencies, which ensure that the stationary cost is minimized.

Introduction

There are many control problems where the objective is to reject harmonic disturbances of known frequencies. A typical example consists of vibrations in rotating machinery with known angular frequency. By the internal model principle (Francis & Wonham, 1976), the controller should include a model of the disturbance in order to achieve disturbance rejection. In the widely applied filtered-X LMS algorithm, which has been studied extensively in the signal processing literature, this is achieved by estimating the magnitudes and phases of a harmonically varying compensator by the least-squares method (Kuo & Morgan, 1996). The filtered-X LMS algorithm is equivalent to a controller with poles corresponding to the frequencies of the harmonic disturbance (Bodson, Sacks, & Khosla, 1994).

The filtered-X LMS algorithm is not applicable to more general cases, where the dimension of the controlled output exceeds the dimension of the control signal, or when the variations of the control signal should also be suppressed. One approach which can be applied to these cases, is to augment the plant model with a dynamic model of the disturbances, and to design an optimal controller for the augmented plant. This approach has been applied by Savkin and Petersen (1995) to a continuous-time, finite-horizon robust control problem with rejection of harmonic disturbances. An alternative approach is to exploit the fact that the stationary quadratic cost depends on the gain and phase of the controller at the disturbance frequencies only. This has been used by Lindquist and Yakubovich 1997a, Lindquist and Yakubovich 1997b to derive a parameterization of the set of all optimal regulators for the damping of harmonic disturbances for discrete systems.

In practice, it is common that continuous-time systems are controlled using sampled-data control. The design methods developed for the continuous-time and discrete-time cases do not address the hybrid continuous/discrete-time nature of sampled-data systems, as they do not describe the intersample behaviour correctly.

The wide-spread use of digital controllers to control continuous-time systems has motivated the study of sampled-data control systems (Chen & Francis, 1995), including frequency-domain techniques (Araki, Ito, & Hagiwara, 1996; Freudenberg, Middleton, & Braslavsky, 1995; Goodwin & Salgado, 1994; Yamamoto & Khargonekar, 1996; Lindgärde & Lennartson, 1998). A particular feature of the frequency response of sampled-data systems is the fact that the discrete-time components generate an infinite number of aliasing frequencies in the continuous-time signals.

In this paper the optimal damping of harmonic disturbances is studied for sampled-data systems. The objective is to design a discrete-time controller to control a continuous-time system, which is subject to a harmonic disturbance with known frequencies but unknown magnitudes and phases. In order to take the intersample behaviour into account, the control performance is defined in terms of an infinite-horizon continuous-time quadratic cost. The solution of the control problem is based on a frequency domain expression for the stationary cost. The cost, which consists of contributions from an infinite number of aliasing frequencies, is expressed compactly in terms of a discrete-time transfer function. As the cost depends on the frequency response at the disturbance frequencies only, the set of all stabilizing optimal controllers can be parameterized in terms of the Youla parameterization and a set of interpolation constraints.

The proposed procedure bears similarities to the optimal controller parameterization of Lindquist and Yakubovich 1997a, Lindquist and Yakubovich 1997b for the discrete-time case, although the controller parameterization is expressed in a different form. The suppression of periodic disturbances by sampled-data control has also been studied by Langari and Francis (1994), who consider a repetitive control problem in which the worst-case performance with respect to periodic disturbances with known period is minimized.

Section snippets

Problem statement

We consider the sampled-data control system depicted in Fig. 1. The plant Gc is a time-invariant finite-dimensional linear continuous-time system with the transfer function matrixGc(s)=Gzv(s)Gzu(s)Gyv(s)Gyu(s)=AB1B2C10D12C200.Here v(t)∈Rm1 is the disturbance, uc(t)∈Rm2 is the control signal, and z(t)∈Rp1 and yc(t)∈Rp2 are the controlled and measurement outputs, respectively. The output yc is sampled at the discrete sampling instants kh. It is assumed that the plant description (1) includes the

Closed-loop performance

In order to derive the set of optimal controllers, a closed-form expression for the cost (7) will be derived. The closed-loop response is harmonic, and it consists of the disturbance frequencies ωi as well as the associated alias frequencies ωi+2πl/h, l=±1,±2,…. The cost can therefore be expressed in terms of the frequency components of the output z(t).

Define the discrete closed-loop system transfer functionT(z)=K(z)(I−Gd(z)K(z))−1associated with (8), where Gd denotes the discrete transfer

Optimal sampled-data controller

By Theorem 3.2, the stationary cost depends on the values of the closed-loop transfer function T(z) at the disturbance frequencies ωi only. It is therefore possible to parameterize all optimal controllers in terms of interpolation constraints defined at these frequencies. For this purpose, introduce the doubly coprime factorization Gd=NM−1=M̃−1Ñ, where N,M,M̃,Ñ are stable discrete transfer function matrices in the set RH of complex-valued, real-rational bounded analytic functions in |z|⩾1,

A numerical example

To illustrate the procedure, we consider an experimental vibration process consisting of an excentric mass attached to a rotating motor shaft (Diaz, Fischer, & Medvedev, 1998), which can be described by (1) withA=−175.40411.15−298.20−37.46,B1=0200,B2=−8.57964.2265,C1=C2=−10.3774.8568,D12=0.The output is measured according to (2), and the control signal is generated by a zero-order hold, uc(t)=uk,t∈[kh,kh+h). The sampling interval is h=0.005s, corresponding to the sampling frequency ωs=1257rad/s

Conclusion

Optimal damping of harmonic disturbances in sampled-data systems has been studied. The set of all stabilizing optimal controllers has been expressed in terms of the Youla parameterization and a set of interpolation constraints at the disturbance frequencies. The parameterization can be regarded as a sampled-data counterpart of the approach of Lindquist and Yakubovich 1997a, Lindquist and Yakubovich 1997b.

Acknowledgements

This work was supported by the Volvo Research Foundation and the Swedish Research Council for Engineering Sciences.

Hannu T. Toivonen was born in Turku, Finland, in 1952. He received the Ph.D. degree in Chemical Engineering in 1981 from Abo Akademi University, Turku (Abo). He is currently a professor at the Department of Chemical Engineering at Abo Akademi University. His research interests include sampled-data systems, optimal and robust control, and control applications.

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Hannu T. Toivonen was born in Turku, Finland, in 1952. He received the Ph.D. degree in Chemical Engineering in 1981 from Abo Akademi University, Turku (Abo). He is currently a professor at the Department of Chemical Engineering at Abo Akademi University. His research interests include sampled-data systems, optimal and robust control, and control applications.

Alexander Medvedev was born in Leningrad (St. Petersburg), USSR (Russia), in 1958. He received his M.Sc. (Honors) and Ph. D. degrees in control engineering from Leningrad Electrical Engineering Institute (LEEI) in 1981 and 1987, respectively. From 1981 to 1991, he subsequently kept positions as a system programmer, Assistant Professor, and Associate Professor (docent) in the Department of Automation and Control Science at the LEEI. He was with the Process Control Laboratory, Abo Akademi, Finland during a long research visit in 1990-91. In 1991 he has received Docent degree from the LEEI. In 1991, he joined the Computer Science and System Engineering Department at Lulea University of Technology, Sweden as an Associate Professor in the Control Engineering Group (CEG). In February 1996 he has been promoted to Docent. From October 1996 to December 1997, he served as Acting Professor of Automatic Control in the CEG. Since January 1998, he has been Full Professor of Automatic Control at the same institution.

Starting October 2001, he is also Professor of Automatic Control at Uppsala University, Sweden. He is presently involved in research on fault detection, time-delay systems, time-varying and alternative parameterization methods in analysis and design of dynamic systems. He as well has established and supervises activities within the Center for Process and System Automation (ProSA) at LTU.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Per-Olof Gutman under the direction of Editor Tamer Baser.

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