Chapter 9.6 - Extended Examples

9.6   EXTENDED EXAMPLES

9.6.1   PI Control of the Apache HTTP Server Using
Empirical Methods

This example extends Section 8.7.2 in which proportional control is used to manage
the Apache HTTP Server. As before, KeepAlive (the time that idle HTTP
connections are held) is manipulated so as to regulate CPU (CPU utilizations).
This example describes the use of a PI controller instead of the proportional
controller of Example 8.7.2.

Figure 9.26 displays a block diagram of the Apache HTTP Server with
noise that is modeled as an additive effect on CPU. The operating point is
= (0.58, 11). The input and output offsets are u(k) = KA(k) −
and y(k) = CPU(k) − , respectively. The controller has the transfer function
K_P + zK_I /(z − 1), and the transfer function of the target system is G(z) =
−0.014/(z − 0.59). Our focus is on noise rejection. The transfer function from
the noise to the output is

Note that F_N(1) = 0, so this system should have no steady-state error in response
to a step noise as long as K_P and K_I are chosen so that the system is stable in
closed loop.

Figure 9.27 explores how the gains K_P and K_I affect steady-state error, settling
time, and maximum overshoot. Since the noise directly affects the output, as
shown in Figure 9.26, the maximum change in the output value is always 100%
of the noise value. Also, the final value of the measured output is zero. That
is, y_ss = 0; or in the original system, CPU_ss = = 0.58. We also plot
the maximum overshoot to a reference input, as computed by simulation and as
estimated by the closed-loop poles. As already noted, e_ss = 0 as long as K_I ≠ 0
and the closed-loop system is stable. For settling time, this analysis predicts
that k_s is smallest when K_P = −43, although K_I influences when this minimum
occurs. We note that these estimates of settling time are approximate since F_N(z)
has two finite zeros, a situation not considered in Equation (8.7). The maximum
overshoot is smaller if K_I has a smaller magnitude.

Figure 9.28 displays the transient response of F_N(z) to a noise input of
magnitude 0.2. When the noise occurs, the output immediately increases by the

magnitude of the noise. The PI controller then adjusts the control input (here, KA)
based on the difference between the output and the reference. The dashed line is
the reference value. As predicted by analysis, there is no steady-state error.

9.6   EXTENDED EXAMPLES

9.6.1   PI Control of the Apache HTTP Server Using
Empirical Methods

This example extends Section 8.7.2 in which proportional control is used to manage
the Apache HTTP Server. As before, KeepAlive (the time that idle HTTP
connections are held) is manipulated so as to regulate CPU (CPU utilizations).
This example describes the use of a PI controller instead of the proportional
controller of Example 8.7.2.

Figure 9.26 displays a block diagram of the Apache HTTP Server with
noise that is modeled as an additive effect on CPU. The operating point is
= (0.58, 11). The input and output offsets are u(k) = KA(k) −
and y(k) = CPU(k) − , respectively. The controller has the transfer function
K_P + zK_I /(z − 1), and the transfer function of the target system is G(z) =
−0.014/(z − 0.59). Our focus is on noise rejection. The transfer function from
the noise to the output is

Note that F_N(1) = 0, so this system should have no steady-state error in response
to a step noise as...