Chapter 9.2.4 - PI Control Design Using Empirical Methods

9.2.4 PI Control Design Using Empirical Methods

In many cases, the system model G(z) is unknown. One approach is to use
the system identification techniques described in Chapter 2 and then apply pole
placement design or root locus analysis to the estimated transfer function. Here,

Fig. 9.12 Predicted settling times and maximum overshoot for the IBM Lotus Domino Server

we describe an alternative approach in which the controller gains are computed
in a more direct manner.

The starting point is to obtain the step response of the target system by applying
a bump test, a step change in the control input. For computing systems, multiple
replications of the bump test are needed because of stochastics. Also, the step
input should be scaled by a factor u_ss so that the output response covers the
desired operating region of the system.

Fig. 9.13 PI control of the IBM Lotus Domino Server with KP = 0.6, KI = 0.9. The reference is

Our focus is the CHR controller design method developed by Hrones and
others [14]. (A good description of this method can also be found in [8].) This
method assumes that system dynamics can be approximated by a first-order model
with a delay of n sample times. That is, y(k + n) = ay(k) + bu(k). These
parameters are determined from a plot of the results of the bump test. However,
instead of using a, b, and n, it is more common to use an alternative set of three
parameters: L, R, and T. The definitions of these three parameters are shown

Fig. 9.14 Step response curve for the CHR design method. The tangent line to the output

in Figure 9.14. A tangent line with maximum slope is drawn at the inflection
point of the system response curve. The slope of the tangent line is R, and
the intersection of the tangent line with the time axis is L, or the time lag
of the system. The time constant T is the time needed by the system to reach
1 − e⁻¹ ≈ 0.63 of its final value after it starts to react. The system reaches this
point at time T + L. Note that T and L are expressed in terms of samples and
that R has the same units as the output.

The parameters of the PI controller are computed directly from R, L, and T for
a specific control objective and design goal using the formulas in Table 9.3. The
approach taken is to minimize settling time subject to constraints on overshoot for

a change in either the reference or disturbance inputs.¹ The table considers two
control objectives: disturbance rejection and reference tracking. The design goal is
expressed in terms of maximum overshoot, which is either 0% or 20%. K_P and K_I
are chosen so that the closed-loop system has the shortest settling time within the
overshoot constraints. Note that the controller gains are larger if a 20% overshoot
is permitted, which will result shorter settling times. Since all of the integral
control gains are nonzero, there is a zero steady-state error in response to a step
change in either the disturbance or reference inputs. The closed-loop system
will be stable if the first-order system with delay is a reasonable approximation
to the actual system. If the open-loop response is qualitatively different than that
shown in Table 9.3, a stability analysis should be performed before an empirically
designed controller is implemented.

Example 9.7: Controller design using the CHR rules Figure 9.15 displays
the results of an experiment conducted on a fourth-order system in open loop by
applying a unit step to the control input u(k). From the data plotted, it appears that
the steady-state output is approximately 15, and the line of steepest slope (which
is drawn on the plot) is approximately R = 15/9 = 1.7. This line intersects the
time axis at L = 4. Since the final value is 15 and the initial value is 0, the
time constant is found at the time when the response reaches 0.63(15) = 10.

Fig. 9.15 Open-loop response of a fourth-order system to a unit step input u(k). The estimates

This happens at approximately k = 9. Thus, L + T = 9 and the time constant is
estimated to be T = 5.

Using these estimates of R, L, and T, we can compute the control gains for a
PI controller from Table 9.3. The results are shown in Table 9.4. Figure 9.16 displays
the closed-loop response to a five-unit-step change in the reference input.
Observe that the reference tracking controllers RT:00 and RT:20 have shorter

Table 9.4 & Figure 9.16

Fig. 9.17 Closed-loop response of PI controllers to a 50-unit-step increase in the disturbance

settling times and smaller overshoot than the disturbance rejection controllers
DR:00 and DR:20. Also observe that the effect of permitting a larger overshoot
is to reach the steady-state value faster. This is the case in part because the true
system is a fourth-order system as opposed to the first-order system that is
assumed by the CHR method. Figure 9.17 plots the closed-loop response to a
50-unit-step change in the disturbance input. Here, controllers DR:00 and DR:20,
which designed for disturbance rejection, have shorter settling times and smaller
overshoot than RT:00 and RT:20. As before, the effect of permitting a larger maximum
overshoot is that the system reaches its steady-state value faster, although
this does not necessarily mean that settling times are shorter.

___________________________________________
¹[14] extends the definition of overshoot to apply to changes in either the reference or the disturbance
inputs.

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