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Ch71-I044963.fm Page 349 Tuesday, August 1, 2006 4:45 PM
Page 349
Tuesday, August
4:45 PM
Ch71-I044963.fm
1, 2006
349
349
The transfer function from the controller force F to the load velocity x 2 is the following:
bs + k
(1)
F jS' +b(m l+m 2)s 2 +k(m l+m 2)s
where b is the damping constant, k is the spring constant, and m\ and ni2 are motor and load mass,
respectively. Tn theory, the conventional linear controller (Pl/PTD) can suppress the vibration of the
load in the linear system. There are small gain margins in the root locus where the system is stable.
However, when controlling the load by a simple PT controller, the velocity becomes unstable very
quickly when the gains are increased. The physical linear motor application is also highly non-linear,
and therefore conventional controllers fail in the suppression.
Due to the instability problems it is therefore necessary to have other control strategies than those
based on a PI corrector. In the proposed controller the load acceleration compensator is added to a
conventional velocity PI controller in order to reduce mechanical vibration, which can be assumed to
be a disturbance force added to a flexible load. The advantage of the proposed method is that it
suppresses vibrations without degrading the overall velocity control performance. In Figure 2, there is
the structure of the proposed controller. K m and K a in the figure are the motor constant and the
compensation gain, respectively.

Figure 2: Control system diagram.

The force reference of the controller is the following

(2)

where vi is the motor velocity, a, is the load acceleration estimation, K p and K{ are the proportional-
and integral gains of the velocity controller and K a is the compensation gain. The values are introduced
in Table 1 in the appendix.

The classical control system theory assumes that all state variables are available for feedback. In
practice, however, not all state variables are available for feedback. Therefore, we need to estimate the
unavailable state variables. There are several methods to estimate unmeasurable state variables without
a differentiation process. The acceleration of the load in the controller is estimated using the Kalman
filter (Kalman, 1960). The use of the estimated acceleration is based on the fact that the estimated
acceleration is preferable (delayless and noiseless) to the measured and filtered signal. The Kalman
filter is an optimum observer, meaning that the observer gain, here called the Kalman gain, is
optimally chosen, whereas with a linear observer the gains are positioned arbitrarily.