5.6 Effect of Tuning Parameters

We have shown how to use the closed-loop characteristic equation to determine the closed-loop stability of processes. Except for P-only control, we have not discussed the effect of controller tuning parameters on the feedback system performance (closed-loop response).

Our discussion here is based on open-loop stable processes. To better understand the effect of controller tuning parameters, you should experiment with a number of transfer functions. You can also determine the closed-loop poles as a function of the tuning parameters, for a particular process.

Table 5-1. Closed-Loop Poles as a Function of Proportional Gain for Example 5.3^[a]

kc	p1	p2	p3
0	–1.0000	–0.5000	–0.3333
0.1000	–1.0432	–0.3951–0.1402j	–0.3951+0.1402j
0.2500	–1.0926	–0.3704–0.2313j	–0.3704+0.2313j
0.5000	–1.1549	–0.3392–0.3184j	–0.3392+0.3184j
1.0000	–1.2452	–0.2940–0.4257j	–0.2940+0.4257j
3.0000	–1.4612	–0.1861–0.6493j	–0.1861+0.6493j
5.0000	–1.5991	–0.1171–0.7821j	–0.1171+0.7821j
7.0000	–1.7054	–0.0640–0.8819j	–0.0640+0.8819j
9.0000	–1.7938	–0.0198–0.9637j	–0.0198+0.9637j
10.0000	–1.8333	0.0000–1.0000j	0.0000+1.0000j
11.0000	–1.8704	0.0185–1.0339j	0.0185+1.0339j
13.0000	–1.9384	0.0525–1.0959j	0.0525+1.0959j
15.0000	–2.0000	0.0833–1.1517j	0.0833+1.1517j

^[a] The poles (p₁,p₂,p₃) are the roots of the characteristic equation, 6s³ + 11s² + 6s + 1 + k_c.

Effect of Controller Gain

An increase in controller gain (k_c) will speed up the closed-loop response. Except for first- and second-order processes with P-only control, too large a value of controller gain will destabilize the system.

Integral Time

An increase in integral time (t_I) tends to "slow down" the closed-loop response, while decreasing the integral time speeds up the response. Too small an integral time can cause the closed-loop system to be unstable.

Derivative Time

In this chapter we have not explicitly shown the effect of derivative action—examples are given in the next chapter. An increase in derivative time (t_D) tends to help stabilize the closed-loop response. A disadvantage of derivative action is that it causes the control system to be sensitive to measurement noise.

The next chapter presents some standard procedures for controller tuning.