Chapter 9.2.2 - PI Control Design by Pole Placement

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

Fig. 9.9 Block diagram of a computing system G(z) with proportional–integral control.

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 .

Table 9.2

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 z² − 2r cos θz + r² = z²− 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 z² + [0.47(K_P + K_I) − 1.43]z + 0.43 − 0.47K_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

Fig. 9.10 PI control of the IBM Lotus Domino Server with KP = 0.15,KI = 0.76. The reference

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.

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