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A MacLaurin's series expansion is used to approximate the individual sheets of the characteristic gain function of a linear multivariable system in terms of its state space matrices, at the point at infinity on the frequfency plane for smlall values of characteristic gain. This yields the asymptotic root locus directions for systems with one or more sets of closed loop poles going to infinihy for large feedback gains. A bilinear transformation on the frequency variable is used to allow the approximation at any point on the frequency plane. This gives the angles of approach to finite zeros in terms of the state space matrices, and gives a simple way of calculating zeros for square systems. A bilinear transformation on the gain variable is then introduced allowing the characteristic gain function to be approximated at any values of frequency and gain. This enables the directions of departure from the open-loop poles to be determined in terms of the state space matrices of the system.

In this paper a class of self-tuning regulators is considered which combines a simple recursive least squares estimator with a pole/zero assignment design rule. Two such algorithms aTC shown to have the self-tuning property {i) detuned minimum variance regulators; (ii) pole-assignment regulators.The usc of these regulators in self-tuning can be of considerable benefit. In particular. case (i) is useful when minimum variance strategies need unrealistically high· loop gains. since they permit an engineering trade-off between optimality and practicality.Case {ii} is of use in regulating non-minimum phase systems or systems involving unknown time delays. Such systems are frequently encountered in discrete time control and cannot be handled by direct minimum variance methods.