Abatraet-Nonminimum-phase (NMP) zeros may cause severe transients, like undershoots and oscillations, in a step response. The discrete-time case is studied in which a design of a set-point feedforward by placement of additional zeros is proposed. Optimal placements of zeros are given that minimize the maximal undershoot when the system has one real NMP zero. A system having two complex NMP zeros can be compensated such that a monotonic response is guaranteed, even in presence of model uncertainty.
1. Introduction
In many standard textbooks in automatic control of discrete-time systems (e.g. Astrom and Wittenmark, 1984; Middleton and Goodwin, 1990, Landau, 1993), little or no attention is paid to compensation of the transient effects, like undershoot and oscillations, caused by nonminimum-phase zeros. Focus is mainly on the asymptotic properties and the feedback part of the design. The only commonly encountered design indication is that the nonminimum-phase zeros should not be canceled while closing the loop. They should be kept in the closed-loop transfer function to avoid unbounded control actions. In order to reduce the transients caused by NMP zeros, additional poles are chosen in the feedforward compensation by Middleton and Goodwin (1990) and Landau (1993). Landau (1993) chose the set-point feedforward as a low-pass filter, at the cost of a slow response. Middleton and Goodwin (1990) heuristically counterbalanced the NMP zeros by choosing feedforward poles corresponding to the NMP zeros reflected through the stability boundary. Application of the method does not make the response monotonic. The same idea was exploited by Haack and Tomizuka (1991) in the context of a tracking problem. Another approach (Jayasuriya and Tomizuka, 1993) in the same context relies on the principle of operator orthogonality. which is suitable in the case of a single real NMP zero. A different approach, not pursued here, is to perform the compensator design solely in the frequency domain (Ohashi et al., 1988).
The aim of this paper is to show that choosing additional zeros in the set-point feedforward path can compensate for the NMP zeros of the plant such that undershoots and oscillations are reduced and in some cases totally eliminated.
If the system has one positive real NMP zero, the step response will give an undershoot. This undershoot, however, can be reduced arbitrarily by adding zeros in the design. The more zeros are used to compensate for the undershoot zero *