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Technical Briefs

# Design of Disturbance Observers for Multi-Input Multi-Output Systems

[+] Author and Article Information
S. M. Shahruz

Berkeley Engineering Research Institute, P.O. Box 9984, Berkeley, CA 94709shahruz@cal.berkeley.edu

J. Vib. Acoust 130(3), 034502 (Apr 21, 2008) (7 pages) doi:10.1115/1.2890395 History: Received February 21, 2007; Revised December 17, 2007; Published April 21, 2008

## Abstract

In this paper, the long-standing problem of designing disturbance observers for multi-input multi-output (MIMO) systems is solved. The disturbance observer presented here has a simple structure equivalent to that of the internal model control (IMC), thereby there is no need for the system inversion. Techniques to design the proposed disturbance observer are given. Furthermore, the design procedure is illustrated via examples for different MIMO systems.

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## Figures

Figure 1

(a) The standard disturbance observer: P(s) is the transfer function of the system, Pn(s) is the transfer function of a mathematical model of the system, and d is a bounded disturbance. (b) A realization of the standard disturbance observer. The filter Q(s) is a low-pass filter of unity dc gain.

Figure 2

(a) A disturbance observer equivalent to that in Fig. 1; (b) a realization of the equivalent disturbance observer

Figure 3

A disturbance observer for MIMO systems

Figure 4

A MIMO feedback system with a disturbance observer and a controller G(s). The disturbance observer suppresses the effect of the disturbance d, whereas G(s) achieves desired control goals.

Figure 5

A disturbance observer for a MIMO system when the disturbance is applied at the output of the system

Figure 6

A two-degree-of-freedom system. There are possibly two bounded disturbances d1 and d2 and two control inputs u1 and u2 applied to the masses.

Figure 7

(a) The system outputs x1 and x2. It is clear that the outputs are close to zero. That is, the disturbance observer has suppressed the effect of disturbance. (b) The input u1 generated by the disturbance observer.

Figure 8

(a) The system outputs x1 and x2; (b) the inputs u1 and u2 generated by the disturbance observer

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