13.3 The General Pairing Problem

Given a general multivariable process diagram, how do we select the pairings for a control structure that consists of multiple SISO loops? In the following sections, we use matrix vector notation, where y(s) is a vector of n outputs and u(s) is a vector of m inputs,

Equation 13.1

Equation 13.2

where each output can be represented by

Equation 13.3

To form a MVSISO system, we must ask the following:

Which output y_i should be paired with which input u_j to form an SISO control loop?

Two Input–Two Output Processes

Let us consider the relationship between y_i and u_j under a number of conditions. For simplicity, we will deal primarily with 2 x 2 (two input–two output) systems. The open-loop input-output relationships are

Equation 13.4

which is written in matrix form as

Equation 13.5

and we can think of the input output relationships as shown in Figure 13-7.

The corresponding feedback control system, if the pairings are u₁-y₁ and u₂-y₂, is shown in Figure 13-8.

Input 1–Output 1 Dynamic Behavior

For consistency in notation, we refer to loop 1 as the y₁-u₁ pairing and loop 2 as the y₂-u₂ pairing. We wish to consider how control loop 1 should be designed. We have focused on the use of input-output models for control system design. In the case of loop 1, we need to know how input 1 affects output 1.

Now, consider how u₁ affects y₁ if (a) loop 2 is open and if (b) loop 2 is closed.

(a) Loop 2 is open (loop 1 is also open). If the second input (u₂) is constant, then the outputs depend only on the first input (u₁), as shown in Figure 13-9.

The input-output relationship is

If we designed a SISO feedback control system for loop 1, we would use g₁₁(s) as the process transfer function for control system design.

(b) Loop 2 is closed. If the second loop (y₂-u₂) is closed, then the outputs depend on input 1 (u₁), as shown in Figure 13-10. Notice that there is an additional effect of input 1 on input 2 owing to the action of controller 2. Input 1 affects output 2, which then changes input 2 through the second controller. This change in input 2 then has an additional effect on output 1; this effect can be positive or negative.

Thus, the relationship between u₁(s) and y₁(s) is no longer just g₁₁(s). Indeed, we can derive the relationship in the fashion:

Equation 13.6

but (assuming no setpoint change in loop 2, r₂ = 0),

Equation 13.7

also

Equation 13.8

Combining Equations (13.7) and (13.8),

Equation 13.9

and solving for y₂(s),

Equation 13.10

Substituting this into Equation (13.7),

Equation 13.11

and substituting this result into Equation (13.6), we find

Equation 13.12

That is, y₁(s) = g_11,eff(s) u₁(s), where

Equation 13.13

g_11,eff(s) is the effective input-output relationship between u₁ and y₁, with loop 2 closed. This means that g_11,eff(s) should be used as the process transfer function to design controller 1 if loop 2 is closed. Unless g₁₂(s) or g₂₁(s) = 0, if the second control loop is closed, then the first control loop must be designed differently than if the second control loop were open. Naturally, the relationship between u₁ and y₁ is a function of the controller g_c2 when the second loop is closed.

Example 13.1, continued

Consider the expected effect of the second loop in the blending example posed earlier. In Figure 13-11 we compare the response of ethanol concentration due to a change in whiskey flow rate, under two conditions: with the flow controller (loop 2) open (solid), and with the flow controller closed (dashed). Notice the tremendous amplification of the effect of the whiskey flow rate. Here, we rationalize the effect of the flow loop being closed. When the whiskey flow rate is increased, it causes the total blend flow rate to increase. The total blend flow-rate controller then cuts back on the dilution water flow rate, which causes the ethanol concentration to increase even more. The flow rate controller has caused the whiskey stream to have roughly four times the effect on the ethanol concentration, compared to when the flow control loop is open.

Steady-State Effective Gain

Comment:

What we would really like is a method to determine the relationship between u₁ and y₁ with loop 2 closed, without knowing g_c2(s).

We can easily determine this relationship for a limiting case, the steady state. For the steady state, we simply let s 0. Assuming integral action is used in the controller, we find that as s 0, g_c2(s) . Also, from Equation (13.13) and g_c2(0) as s 0, we find the following relationship (where 0 represents s = 0, the steady state; do not confuse this with t = 0, the initial condition):

Equation 13.14

This can also be determined from the steady-state relationship

Assuming perfect control of y₂, that is, y₂ = 0,

Equation 13.15

Equation 13.16

and we can then find the following relationships:

Equation 13.17

We now have a steady-state effective gain relationship between u₁ and y₁ with the loop between u₂ and y₂ closed.