Pole-zero placement controllers and self-tuning regulators with better set-point tracking S.C. Puthenpura, BTech, PhD Prof. J.F. MacGregor, MSc, PhD Indexing terms: Digital control, Closed-loop systems, Self-tuning regulators, Poles and zeros, Simulation Abstract: A new procedure for designing control- lers and explicit self-tuning regulators is presented. This method allows the placement of closed-loop poles of a given system arbitrarily like other exist- ing methods. The additional attractive feature of the new procedure is the elimination of tracking error for a given set of input (reference) signals. This is done by placing the zeros of the 'error transfer function' at the modes of reference signals. This is very useful in some practical applications like radar tracking and robotics. Some results of simulations are included which illustrate these aspects of the proposed method. 1 Introduction During the past couple of decades, a lot of attention has been given to the problem of designing pole-placement controllers and self-tuning regulators. Comparatively, only very little importance is given to zeros because they are considered to be less crucial than poles. Most of the discussion on zeros are centred around the choice of the sampling interval so that the resulting system is invert- ible. However, it is important to note that zeros have a vital role in the 'tracking error' corresponding to a given set of input signals. Therefore, it is useful to modify the existing pole-placement design of controllers and self- tuning regulators to incorporate zero placement in the 'error transfer function'. It is to be noted that here we are not dealing with the zeros of the overall (closed-loop) transfer function. There- fore, this approach is quite different from the discussion by Astrom and Wittenmark [1], where they consider the retention or cancellation of the original (open-loop) system zeros in the closed loop. This paper is organised as follows. First we discuss how zero placement in the error transfer function can eliminate tracking error corresponding to a given set of reference signals. Next we look into the feasibility of incorporating this idea into a standard pole-placement algorithm, where the system model is known. In the final stage we examine how the proposed algorithm works in an explicit self-tuning mode. Paper 501 ID (C8), first received 30th May 1985 and in revised form 25th July 1986 S.C. Puthenpura was formerly with the Department of Electrical and Computer Engineering (currently with AT & T Bell Laboratories, Holmdel, New Jersey, USA) and J.F. MacGregor is with the Depart- ment of Chemical Engineering at McM aster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7 26 2 Pole-zero placement design 2.1 Notation For ease of argument, let us consider single-input/single- output (SISO) systems. A typical SISO system can be represented in the transfer-function form as B(z~ l )z where a 2 z 2 -2 1 +b,z~ 2 + a n z " • + b m z~ m (la) (lb) and b represents the process delay as an integral multiple of the sampling interval. The design of a pole-placement controller is to evaluate the controller transfer function (2a) where and vy? ) = go + g\z + g 2 z + •• • + g q z H (zb) such that the overall system (closed-loop) will have pre- specified poles. The configuration is shown in Fig. 1. 2.2 Offline design In this Section we assume that the system model is already identified, and we ignore the presence of noise in the design of the controller. However, the performance of this controller will be evaluated (with the help of an example) in a noisy environment. Consider the system described by Fig. 1. The closed- loop (overall) transfer function can be evaluated as B(z~')G(z-')z r\z) B(z- 1 )G(z~ 1 )z- (3) Now, the overall system characteristic equation (closed- loop poles) is to be prespecified. Therefore, we have to choose F(z -1 ) and G(z~ l ) so that the denominator of eqn. 3 has specified roots. This is essentially the pole- placement problem. A widely used solution can be found in Astrom and Wittenmark [2], where G(z~ l ) and F(z~ l ) noise n(t) y (t) output Fig. 1 System-controller configuration IEE PROCEEDINGS, Vol. 134, Pt. D, No. 1, JANUARY 1987