Chapter 9.4 - PID Control

9.4 PID CONTROL

Proportional–integral–differential control (PID control) combines the three control
actions that we have studied thus far. Figure 9.24 contains a block diagram
of the PID controller. There is one parameter for each control action: K_P, K_I,
and K_D. Since PID controllers have more parameters, there is more flexibility in
design. However, there is more complexity as well.

Before continuing, we want to underscore the generality provided by the PID
controller. In Figure 9.24 the three control actions correspond to the three rows
of boxes in the PID controller. Observe that a proportional controller is a special
case of a PID controller in which K_I = K_D = 0. This is equivalent to deleting the
first and third rows of boxes inside the PID controller. Similarly, the PI controller
is constructed by having K_D = 0, which corresponds to deleting the third row in
the PID controller, and the PD controller is constructed by having K_I = 0, which
is obtained by deleting the first row of boxes in the PID controller.

The difference equation for a PID controller is

Figure 9.23

Figure 9.24

To find the transfer function of the PID controller, we first compute the difference
u(k) − u(k − 1), then take the Z-transform with zero initial conditions, to get

Similar to the PI and PD controllers, the PID controller can be written either in
a single transfer function form, highlighting the two zeros and two poles, or as
the sum of the three transfer functions for the P, I, and D controllers.

The poles added by the PID controller are at 0 and 1, as expected by the
integral and derivative terms. The two zeros are at the roots of the numerator
polynomial,

Depending on the relative magnitudes of K_p, K_I, and K_D, the zeros could be
either real or complex.

As expected by the presence of the integral term, a PID controller results in
zero steady-state error to both a constant reference and a constant disturbance
input, as long as the system is stable in closed loop. The calculations of these
errors are left as exercises for the reader.

Because there are three parameters in the PID controller, controller design
is more complicated. For a first-order system with PID control, there are three
closed-loop poles: one from the system and two added by the PID controller.
These three poles can be placed using the method of Section 9.2.2, abbreviated
here:

Compute the dominant poles based on the design goals.
Compute and expand the desired characteristic polynomial of the closedloop
system based on using the dominant poles.
Compute and expand the modeled characteristic polynomial of the closedloop
system, which will be a function of K_P, K_I, and K_D.
Solve for K_p, K_I, and K_D by matching coefficients between the desired
and modeled characteristic polynomials.
Verify the result (e.g., by simulation).

Typically, the two dominant poles can be chosen based on the design goals;
the third pole must be chosen smaller than the dominant one(s). For a secondorder
system with PID control, there are four closed-loop poles. Only three of
these can be arbitrarily placed using the three parameters in a PID controller.
Similarly, for higher-order systems, the method does not always yield a feasible
solution. There are also empirical design methods for PID controllers similar to
those discussed in Section 9.2.4; a good reference for these methods is [8].

It is worth mentioning that even if the derivative control law can help to add
certain predictability to the controller, it may also be sensitive to the stochastic
variations in the system output. This may become a serious problem for computing
systems because they typically have a significant stochastic component.
One way to solve this problem is to apply a low-pass filter to smooth the system
output so that the derivative control term will respond only to large system
changes, not to small stochastic variations. However, this additional filter may
slow down the system response, which is contrary to the purpose of introducing
the derivative control term. Hence, in practice, PI controllers are preferred over
PID controllers.

Example 9.8: PID control design by pole placement Consider the IBM Lotus
Domino Server, as in Example 9.5, with the same design goals. For a PID control
design, we must choose three closed-loop poles. We choose the dominant poles
p₁ and p₂ = 0.6e^±j0.6 = 0.5±0.34j. The third pole is chosen to have a smaller
magnitude than the dominant ones, p₃ = −0.3. As shown by the algebra, this
last pole must be chosen to be negative if all of the control gains are to be
positive. The desired characteristic polynomial is (z − p₁)(z − p₂)(z − p₃) =
(z² − z + 0.36)(z + 0.3) = z³ − 0.70z² + 0.063z + 0.11.

With PID control (as in Figure 9.24), the closed-loop transfer function from
the reference to the output is

Note that this closed-loop transfer function depends on K_P, K_I, and K_D.

To find the values of K_P, K_I, and K_D that result in this characteristic polynomial,
we match terms with the closed-loop transfer function of Equation (9.19),
to get three equations in three unknowns:

If the third pole p₃ had been chosen to be positive and real, the derivative gain
K_D would necessarily be negative. Negative gains are undesirable (except in the
case when the system transfer function has a negative gain—and then all control
gains should be negative).

Since the system model is only first order, there are no extra pole locations
to solve for. Simulation results in Figure 9.25 for a step reference of 10 show
that with this choice of gains, the design criteria are satisfied. The response to
a disturbance of magnitude 20 is also shown. The proportional, integral, and
derivative components of the control signal u(k) are shown individually, along
with their sum u(k) = u_P(k) + u_I(k) + u_D(k). Since the proportional gain is
small, the proportional control does not contribute very much to the response.
Also note that the derivative term is active only when the error changes abruptly,
but it does serve to speed up the response (comparing to Figure 9.10).

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