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FIGURE 30-1 Single-input single-output system (SISO): (a) two-tank interacting level system, (b) block diagram for SISO system.
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FIGURE 30-2 Multiple-input multiple-output system (M&IO): (a) level process, (b) block diagram.
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the result that a change in flow to tank 2 would not affect H 1.) If both H 1 and Hz are to be controlled, a single control loop will not be sufficient; in this case two control loops are needed. The addition of control loops to the interacting system will be considered in the next section.
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CONTROL OF INTERACTING SYSTEMS
The problem of controlling the outputs of an MIMO system will be discussed by means of a 2 X 2 system showri in Fig. 30.3. The problem can be extended to the case of more than two pairs of inputs and outputs by the same procedure described here. The control objective is to control C 1 and C2 independently, in spite of changes in Ml and A42 or other load variables not shown. Two control loops are added to the diagram of Fig. 30.3 as shown in Fig. 30.4. Each loop has a block for the controller, the valve, and the measuring element. In principle, the multiloop control system of Fig. 30.4 will maintain control of Cl and CT. However, because of the interaction present in the system, a change in RI will also cause C2 to vary because a disturbance enters the lower loop through the transfer function G21. Because-of interaction, both outputs (Cl and C2) will change if a change is made in either input alone. If G21 and G12 provide weak interaction, the two-controller scheme of Fig. 30.4 will give satisfactory control. In the extreme, if G12 = G21 = 0, we have no interaction and the two control loops are isolated from each other. To completely eliminate the interaction between outputs and set points, two more controllers (cross-controllers) are added to the diagram of Fig. 30.4 to give the diagram shown in Fig. 30.5. In principle, these cross-controllers can be designed to eliminate interaction. The following analysis, which is expressed in matrix form, will lead to the method of design for cross-controllers that will eliminate interaction.
j-z--F
O--l ,
322
FIGURE303
MIMO system for two pairs of inputs and outputs.
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I c2
FIGURE 30-4 Multiloop control system with two controllers.
Response of Multiloop Control System
From Fig. 30.5, we may write by direct observation the following relationships in the form of the matrix expression
C = G,M
We also may write from Fig. 30.5
MI = GvlGcllEl + GvlGcnE2 M2 = Gv2Gc21El + Gv~Gcm%
(30.2) (30.3)
FIGURE. 30-5 Multiloop control system with two primary controllers and two cross-controllers,
MULTIVARIABLE
CONTROL
where G,i and Gv2 are the transfer functions for the valves. Equations (30.2) and (30.3) may be written in matrix form as
where G, = G, =
0 G cl1 Gc21 El E2 I
M = G,G,E
(30.4)
(valve
matrix) matrix)
Cc12 Gc22
(controller
From Fig. 30.5, we write directly
El = RI -G,lCl E2 = R2 -Gm2C2
(30.5) (30.6)
where El and E2 are the error signals from the comparators. Equations (30.5) and (30.6) can be written in the matrix form
E = R-G&
(30.7)
matrix)
where G,
=[G;l
(measuring
element
From Eqs. (30.1) and (30.4), we obtain
C = G,G,G,E
(30.8)
If we let G, = G,G,G,, Eq. (30.8) becomes
C = G,E
(30.9)
Combining Eqs. (30.7) and (30.9) gives C = G,R - G,G,C We may now solve Eq. (30.10) for C to obtain
C = [I + G,G,]- G,R
(30.10)
(30.11)
Notice that the closed-loop behavior expressed by this matrix equation is analogous to the closed-loop response of a SISO system, which may be written
C(s) =
Go(s) R(s) 1 + G&G&)
(30.12)
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:C FIGURE 30-6 Block diagram for MIMO control system in terms of matrix l blocks.
The matrix term [I+G,G,]- is equivalent to the scalar term l/[l +G,(s)G,(s)]. A block diagram equivalent to the diagram for the MIMO control system in Fig. 30.5 is shown in Fig. 30.6. In this diagram, the blocks are filled with the matrices in Eqs. (30.1), (30.4), and (30.7). The double line indicates that more than one variable is being transmitted. Each block contains a matrix of transfer functions that relates an output vector to an input vector. The diagram can be simplified by multiplying the three matrices in the forward loop together and calling the result G,, as was done to obtain Eq. (30.9). The simplified diagram is shown in Fig. 30.7.
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