**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

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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,

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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

*p*_{1} and *p*_{2} = 0.6e^{±j0.6} = 0.5±0.34*j*. The third pole is chosen to have a smaller

magnitude than the dominant ones, *p*_{3} = −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* − *p*_{1})(*z* − *p*_{2})(*z* − *p*_{3}) =

(*z*^{2} − *z* + 0.36)(*z* + 0.3) = *z*^{3} − 0.70*z*^{2} + 0.063*z* + 0.11.

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

the reference to the output is

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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:

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If the third pole *p*_{3} 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).

**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

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To find the transfer function...

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