Zeros of the closed loop

In the method presented above, the zeros of the closed loop transfer function for command changes

are obtained automatically. In fact, the zeros of the plant, i.e. the roots of , can be considered during the choice of the pole distribution and may be compensated, but the polynomial arises not in the design and must possibly be compensated after this step. This can be done by introducing a pre-filter in the feed-forward path according to 10.6a with a transfer function

Figure 10.6: Compensation of the plant zeros (a) with a controller in the feed-forward path and (b) in the feedback path

The zeros of the controller and plant are compensated in this way. For stability reasons, this is only possible for left-half-plane zeros. If and are polynomials with only left-half-plane zeros and and the corresponding polynomials with only right-half-plane zeros including the imaginary axis, the polynomials of and can be factorised as

with

and

For the case that and , and and do not have common divisors, i.e.  the controller does not compensate plant poles and zeros, the denominator polynomial of the pre-filter can be determined as

The transfer function for a command input is then

If both, the controller and the plant, show minimum phase behaviour and their transfer functions do not have zeros on the imaginary axis, all zeros of the closed loop can be compensated, such that one obtains instead of Eq. (10.31)

If the closed-loop transfer function also contains given zeros, the transfer function should have a corresponding numerator polynomial. The coefficient in the numerator is used to make the gain of the closed-loop transfer function equal to 1. From Eq. (10.31) it therefore follows that

The expression in the denominator is the first coefficient of the characteristic polynomial , and therefore with Eq. (10.33)

For a controller with integral action the coefficient is zero and according to Eq. (10.22) . From Eqs. (10.33) and (10.24) to (10.28) and (10.29) it follows directly that

When the controller is inserted into the feedback path according to Figure 10.6b the inherent closed-loop dynamics will not be changed compared with the configuration according to Figure 10.6a, because the denominator polynomial of the transfer function, and therefore the characteristic equation of the closed loop, are preserved. Indeed, the zeros of the controller transfer function do no longer arise, but their poles as zeros in the closed-loop transfer function. Analogous considerations for lead to

whereby the polynomial contains the poles of the controller and the plant zeros in the left-half plane. The transfer function

is the same as for the case of a stable controller and a minimum-phase plant according to Eq. (10.32).

The constant for a proportional controller is

For an integral controller in the feedback loop a feed-forward path is not realisable.