# One-Degree of Freedom Internal Model Control

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## 3.2 Properties of IMC

#### 3.2.1 Transfer Functions

An easy way to develop the transfer functions between the inputs d and r and the process output y is to first redraw Figure 3.1 as a simple feedback system, as shown in Figure 3.2, and then apply the following rule: Figure 3.2. Alternate IMC Configuration.

The transfer function between any input and the output of a single-loop feedback system is the forward path transmission from the input to the output divided by one plus the loop transmission for negative feedback.

For the feedback controller c(s) of Figure 3.2, the rule gives

Equation 3.1

The negative term in the denominator of Eq. (3.1) arises from the positive feedback around q(s).

The input-output relationships for Figure 3.2 are given by

Equation 3.2

Equation 3.3

Equation 3.4

Equation 3.5

Substituting Eq. (3.1) into Equations (3.2) and (3.3) and clearing fractions gives

Equation 3.6a

Equation 3.6b

#### 3.2.2 No Offset Property of IMC

The steady-state gain of any stable transfer function can be obtained by replacing the Laplace variable s with zero (see Chapter 2). If Equations (3.6a) and (3.6b) are stable, and if we choose the steady-state gain of the controller q(0) to be the inverse of the model gain , then the gain of the denominator of Equations (3.6a) and (3.6b) is p(0)q(0). Thus the gain between the setpoint r(s) and y(s) is one; the gain between the disturbance d(s) and y(s) is zero, and there is no steady-state deviation of the process output from the setpoint.

An ideal control system would force the process output to track its setpoint instantaneously and perfectly suppress all disturbances so that they do not affect the output. That is, the ideal controller would accomplish

Equation 3.7a

and

Equation 3.7b

From Equations (3.6a) and (3.6b), the above requires that

Equation 3.8

Thus, for perfect control, we need a perfect model, and from Eq. (3.8), the controller must perfectly invert that perfect model. Unfortunately, one never has a perfect model, and if the model has any dynamics at all (i.e., it is not just a gain), no controller can perfectly invert the process model. A controller can, however, come very close to inverting the process model. Just how close it can come is the subject of our next section, where we discuss controller design assuming perfect models. The design methodology always incorporates a tuning parameter that can slow down the control system response sufficiently to accommodate most modeling errors for inherently stable processes. Calculation of the tuning parameter for various descriptions of anticipated modeling errors is treated in Chapter 7.

The next section is limited to controller design of stable processes. IMC controller design for unstable processes is discussed in Chapter 4.