**9.2.2 PI Control Design by Pole Placement**

Consider the closed-loop system with PI control in Figure 9.9. We have four

design goals for the PI controller: (1) the closed-loop system is stable; (2) steady-

state error is minimized; (3) settling time does not exceed; and (4) maximum

overshoot does not exceed . The first design goal is achieved by ensuring that

all poles lie within the unit circle. The second goal is achieved by using a PI

controller, at least for a step change in the reference and/or disturbance inputs.

Thus, the design problem is reduced to goals 3 and 4. These control goals can

be achieved by properly selecting the parameters *K*_{P} and *K*_{I} of the PI controller.

Our approach assumes that *G*(*z*) is a first-order system. If *G*(*z*) is a higher-order

system, we can use Equation (3.30) to construct a first-order approximation

of *G*(*z*). The case of *G*(*z*) having order 0 is considered in the problems at the end

of the chapter. Note that if *G*(*z*) is first order, the closed-loop system is second

order since the PI controller is a first-order system. Hence, the closed-loop system

has two poles.

Table 9.2 details the steps in our procedure for pole placement design. The

first step is to compute the desired poles of the closed-loop system based on

and . We assume that the poles are complex conjugates *re*^{±jθ}. From

Equation (8.7) we know that *k*_{s} < −4/ log *r*. Thus, an upper bound for *r* is

| |

Equation (8.8) relates *M*_{P} to *θ* as *M*_{P} ≈ r^{π/θ} (for *θ* ≥ 0) so

Note that both *r* and *θ* are constructed so that smaller (absolute) values will also

satisfy the design goals.

The next step is to construct the *desired characteristic polynomial*, which

is the characteristic polynomial that we want for the closed-loop system. The

desired characteristic polynomial is

The third step is to construct the *modeled characteristic polynomial*, which is

the denominator of the transfer function of the closed-loop system. In Figure 9.9

this is the denominator of

In the fourth step we solve for *K*_{P} and *K*_{I} so that the desired characteristic

polynomial is the same as the modeled characteristic polynomial. This is done

by equating the coefficient of each power of *z* in the desired characteristic polynomial

with the coefficient of the same power of *z* in the modeled characteristic

polynomial. The result is two linear equations in the two unknowns *K*_{P} and *K*_{I}.

Having assigned values to *K*_{P} and *K*_{I}, we now verify that the design goals are

achieved. First, we confirm that the poles of the closed-loop transfer function lie

within the unit circle. Next, we simulate the transient response to confirm that

settling times do not exceed and the maximum overshoot does not exceed .

Below we give an example of applying the procedure in Table 9.2.

**Example 9.5: PI control design by pole placement ** Consider the IBM Lotus

Domino Server, with transfer function

Recall that *y*(*k*) is the offset of RPC’s in the system (RIS) from the operating

point, and *u*(*k*) is the offset of MaxUsers from the operating point. We use the

procedure in Table 9.2 to design a PI controller so that = 10 and = 10%.

- Compute the dominant poles. Using Equation (9.9), we have
*r* = *e*^{−4/10} = 0.67.

Using Equation (9.10), we determine that *θ* = π(ln *r*/ ln 0.1) = 0.70.

To be conservative, we round this to *r* = 0.6 and *θ* = 0.6. - Construct and expand the desired characteristic polynomial. The desired

characteristic polynomial is *z*^{2} − 2*r *cos *θz* +* r*^{2} = *z*^{2}− *z* + 0.36. - Construct and expand the modeled characteristic polynomial. With PI control

(as in Figure 9.9), the closed-loop transfer function from the reference

input to the measured output is

The modeled characteristic polynomial is the denominator of Equation (9.13),

which is *z*^{2} + [0.47(*K*_{P} + *K*_{I}) − 1.43]*z* + 0.43 − 0.47*K*_{P}. - Solve for
*K*_{P} and *K*_{I}. We want the desired characteristic polynomial to equal

the modeled characteristic polynomial. That is,

This is true if

Solving this system of equations, we have

- Verify the result. Substituting into Equation (9.13), we have

As expected, the poles of *F*_{R}(*z*) are 0.5 ± 0.33, so the system is stable.

*F*_{R}(1) = 1 and hence there is no steady-state error to a step change in the

reference or disturbance inputs. Figure 9.10 displays simulation results to

a step increase of 10 in the reference input and an increase of 20 in the

disturbance input. We see that the design criteria are satisfied in that settling

times are well under the objective of 10, and the maximum overshoot is well

under 10%. Also shown in the figure are the magnitudes of the proportional

and integral components of the control signal *u*(*k*). Note that the integral

controller has the most effect on *u*(*k*).

Recall that the foregoing procedure handles higher-order *G*(*z*) by using a first-order

approximation. Another approach is to increase the number of controller

parameters so that they are equal to the order of the system. This approach is

used in Chapter 10 in the discussion of pole placement design for state-space

feedback control.

**9.2.2 PI Control Design by Pole Placement**

Consider the closed-loop system with PI control in Figure 9.9. We have four

design goals for the PI controller: (1) the closed-loop system is stable; (2) steady-

state error is minimized; (3) settling time does not exceed; and (4) maximum

overshoot does not exceed . The first design goal is achieved by ensuring that

all poles lie within the unit circle. The second goal is achieved by using a PI

controller, at least for a step change in the reference and/or disturbance inputs.

Thus, the design problem is reduced to goals 3 and 4. These control goals can

be achieved by properly selecting the parameters *K*_{P} and *K*_{I} of the PI controller.

Our approach assumes that *G*(*z*) is a first-order system. If *G*(*z*) is a higher-order

system, we can use Equation (3.30) to construct a first-order approximation

of *G*(*z*). The case of *G*(*z*) having order 0 is considered in the problems at the end

of the chapter. Note that if *G*(*z*) is first order, the closed-loop system is second

order since the PI controller...

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