10 Analysis of Some PID Tuning Techniques

Case 1: r > 1.2. In this case, we have 0 < fcp < ^ Then the stabilizing set is given either by Fig. 8.6(a) or by Fig. 8.6(b). Notice from (10.6) that the parameter kd is always less than j as illustrated in Fig. 10.2. The derivative gain value provided by the Ziegler-Nichols method is robust in the sense that it is not close to the stability boundary j . Following the same principle, we would like to guarantee that the integral gain value is also far away from the stability boundary. __ Let xi be the fc^-coordinate of the point where the line kd = kd intersects the line kd = miki-\-bi. From Fig. 10.2, we now find the conditions under
k
பைடு நூலகம்
-i
- a i sin(Q;i) — cos(ai)
T
(10.4)
where ai is now the solution of the equation tan(a) = ^ ^ a 1+ r
in the interval (0, TT). We now compare kp k and kupp k as functions of the parameter r. As can be seen from Fig. 10.1, the proportional gain value given by the Ziegler-Nichols step response method is always less than the upper bound kupp- Thus_this tuning technique always provides a feasible proportional gain value kp. We now set kp = kp and consider two cases, requiring different treatments according to the results of Chapter 8. Moreover, for clarity of presentation, let us rewrite the parameters ki and kd in (10.2) as h kd = = j -J^ . (10.5) (10-6)
10
Analysis of Some PID Tuning Techniques
In this chapter we present an analysis of some PID tuning techniques that are based on first-order models with time delays. Using the characterization of all stabilizing PID controllers derived in Chapter 8, each tuning rule is analyzed to first determine if the proportional gain value dictated by that rule lies inside the range of admissible proportional gains. Then the integral and derivative gain values are examined to determine conditions under which the tuning rule exhibits robustness with respect to controller parameter perturbations.
1 \ Kp \
_ l T
Kupp •— J
- T Q I s i n ( a i ) — COS(Q;I) JL
where ai is the solution of the equation
tan(a) =
'YTI^
T
in the interval (0,7r). Let us rewrite the upper bound on kp as a function of the parameter r:
5=i-
<'°-^'
10.2 The Ziegler-Nichols Step Response Method
225
Since A > 0 and r > 0 (the plant is open-loop stable), then kp > 0. From : Theorem 8.1, the range of kp values for which a given open-loop stable plant, with transfer function G(s) as in (10.1), can be stabilized using a PID controller is given by 1
where k is the steady-state gain, L is the apparent time delay, and T is the apparent time constant. A simple way to determine the parameters of a PID controller based on step response data was developed by Ziegler and Nichols in 1942. This method first characterizes the plant by the parameters L and a, where the parameter a is defined as a = k— . T Once these parameters are determined, the PID controller parameters are given in terms of L and a by the following formulas:
224
10. Analysis of Some PID Tuning Techniques
troller that is to be practically implemented must necessarily be nonfragile or the system must be controller-robust. Four tuning techniques will be discussed: (1) the classical Ziegler-Nichols step response method, (2) the Chien, Hrones, and Reswick (CHR) method, (3) the Cohen-Coon method, and (4) the internal model controller (IMC) design technique. In each case, we will first study if the proposed proportional gain value lies inside the allowable range determined in Chapter 8. We then analyze for this fixed proportional gain the possible location of the integral and derivative gains inside the stabilizing region. This procedure will allow us to determine conditions under which each tuning technique provides a good parametric stability margin in the space of the controller parameters. In this way, we will avoid undesirable scenarios such as PID controller parameters that are dangerously close to instability.
jr^Li
1.2
_ _ _ 0.6
t;=^ ki ~
—
E = —— . K>d
7^_
0 ^
(10.2)
a aL a This tuning rule was developed by empirical simulations of many different systems and is only applicable to open-loop stable plants. We now define the parameter r as the ratio of the apparent time delay to the apparent time constant of the plant, that is, L First we focus on the proportional gain expression given in (10.2) and rewrite it as a function of r: