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(30.24)
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Inserting the appropriate elements of the G, matrix [I$. (30.21)] and the G, matrix in Eq. (30.24) gives after considerable simplification Kl G, = s + 3 2;2 0 s+5 I (decoupled system) (30.25)
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The block diagram for this decoupled MIMO system is shown in Fig. 30.11. Assuming that the measurement matrix G, is a unit diagonal matrix, the diagram
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FIGURJI 30.2.
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Block diagram for decoupled system in Example
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STATE-SPACE METHODS
FIGURE 30-12
Simplified block diagram for Example 30.2. in Fig. 30.11 can be simplified to the unity feedback diagram of Fig. 30.12. From Fig. 30.12, we may write directly
C = G,E E = R - C
therefore C = G,R - G,C or
Go22
I[ 1
From this expression, we may write
Cl = GollRl - GollCl C2 = Go292 - Go22C2
Solving for Cl(s) gives
Inserting Got1 from Eq. (30.25) gives
cl(s) = s + 3 Rl(s) K1
(30.26)
In a similar way, one can show that
2K2 C2b)
:;,
R2b)
,(30.27)
1+s+5 The result shows that the cross-controllers of Eqs. (30.22) and (30.23) give two separate noninteracting control loops as shown in Fig. 30.13. The response of the control system of Fig. 30.10 is shown in Fig. 30.14 for a unit-step change in RI. In Fig. 30.14u, no cross-controllers am present in the matrix G,. In Fig. 30.14b, cross-controllers having the transfer functions given by Eq. (30.22) and (30.23) am present. As expected, for the case of no crosscontrollers, one sees from Fig. 30.14~ that a request for a unit-step change in r 1
MUIXIVARIABLE
CONTROL
causes both cl and c2 to change. For the case where cross-controllers are present, one sees from Fig. 30.14b that a change in r i does not affect c2 as demanded by a decoupled system. To avoid the offset associated with proportional control, we can use PI controllers for the primary controllers for the decoupled system. To study the effect of PI controllers for the decoupled system, let and Gc22
For this case, the cross-controller transfer functions may be obtained from, Eqs. (30.14) and (30.15); the results are Gcl2
-4Ws
s(s +
-2K1(s + 1) s(s + 3)
A simulation using these four controller transfer functions with K1 = K2 = 4 is shown in Fig. 30.15. From the transient response, we see that ci moves toward the set point of 1.0 and that c2 does not change, as is expected for a decoupled system.
l.Or
;;I-, .,i0 FIGURE 30-14 1.5 3' '0 1.5 3'
Response for control system in Example 30.2 for R1 = l/s, Rz = 0, Ccl1 = K1 = 4, GC22 = KZ = 4. (a) no cross-controllers, (b) cross-controllers present.
1.0,
STATE-SPACE METHODS
Cougnanowr
RI = l/s, R2 = 0.
Response of decoupled control system in Example 30.2 for PI primary controllers: Gcll = Gc22 = 4(1 + l/s),
STABILITY OF MULTIVARIABLE SYSTEMS
Determining the stability for a multivariable control system, such as the one in Fig. 30.4 or Fig. 30.5, can be much more complicated than for an SISO system. The transfer function for the closed-loop response of an MIMO system is given by Eq. (30.11): C = [I + G,G,]- G,R To invert this expression, we write
c = a4 [I + GG,lG,R II+ G&z 1
(30.28)
The numerator of this expression is an n X n matrix; the denominator is a nth order polynomial. To simplify the following argument, let the matrix in Eq. (30.28) be 2 X 2. Let the elements of the numerator, after expansion, be written as follows:
adj[I + GGJGoR =
Let the elements of G,G, be written as follows:
(30.29)
(30.30) Expansion of the determinant in Eq. (30.28), using Eq. (30.30), is shown below II+ G,G,[ = + a11(s) 1 yl;z ,s)
cyZl(S)
11 + GGnI = [l + ~II(s)IU +
a22G)l - ~12(sb21(s)
(30.31)
Equation (30.31) is a polynomial expression, for which the order will depend on the order of the transfer functions in G, and G,. Equation (30.28) can now be written in terms of the expansions shown in Eqs. (30.29) and (30.31) as follows:
MUU NARIABLE
CONTROL 465
Since each term contains the polynomial II + G,G, I in the denominator, the stability of the multivariable system will depend on the roots of the polynomial equation II+G,G,) = o (characteristic equation) (30.32)
Equation (30.32) is the characteristic equation of the multivariable system. Although Eq. (30.32) has been derived here for the case where G,G, is a 2 X 2 matrix, one can show that Eq. (30.32) applies to the general MIMO system of Fig. 30.7 in which G,G, is a matrix of any size (n X n). If the roots of the characteristic equation are in the left half of the complex plane, we know that the system is stable. One method to be used for examining the stability of a multivariable system is to apply the Routh test to the characteristic equation of Eq. (30.32). In practice, the characteristic equation can be of high order for a simple 2 X 2 multivariable control system. Example 30.3 illustrates the determination of stability for a multivariable control system.
Example 30.3. For the control system of Example 30.2, which is shown in Fig. 30.10, determine stability for the case where G,.tl = K1, Gc22 = K2, and there are no cross-controllers present (i.e., Gct2 = Gc21 = 0) also let G, and G, be unit matrices. From Example 30.1, we have for the elements of G, S+5 G1l = (s + l)(s + 7) 4 G21 = (s + l)(s + 7) 4 G12 = (s + I)(s + 7) 2(s + 3) G22 = (s + l)(s + 7)
Since G, = I, G, = G,G,. Since G, = I, the characteristic equation of Eq. (30.32) can now be written as II+G,G,I = 0 (30.33)
Introducing the elements of the matrices G, and G, into Eq. (30.33) gives, after expansion of the determinant
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