The model formats described above are usually directly obtainable for single block models.
In practical situations, a system model may be given by interconnected blocks, and the overall system model for the interconnected system structures can be obtained using the methods given in this section.
2.5.1 Series Connection
Consider the series connection of the two blocks shown in Figure 2.2(a). It can be seen that the input signal u(t ) travels through the first block G1(s), and the output of G1(s)is the input to the second block G2(s), which generates the output y(t ) of the overall system.
This kind of connection is referred to as a series, or cascade, connection and it is assumed that, in connecting a block, it does not “load” the previous block.
In the series connection, the overall transfer function of the whole system is given by G(s) = G2(s)G1(s). For SISO systems, the two blocks G1(s)and G2(s)are inter-changeable, i.e., G1G2= G2G1. For MIMO systems, however, the two blocks are generally not interchangeable.
Assume that the MATLAB description of the model G1(s)is represented in an LTI object G1, which is either tf, ss, or zpk, and that G2(s)is represented by G2. The overall system in a series connection can be simply obtained using the MATLAB statement
G=G2*G1 .
If the models G1and G2are given by symbolic variables, the above operation is also valid.
2.5.2 Parallel Connection
A typical parallel connection of two blocks G1(s)and G2(s)is shown in Figure 2.2(b), where the two blocks are subjected to the same input signal u(t). The outputs of the two blocks are summed up to form the output y(t) of the overall system. The overall transfer function of the parallel connection is then G(s)= G1(s)+ G2(s).
The LTI representation of the parallel connection can be obtained using the MATLAB statement G=G1+G2 , where G1and G2are LTI objects (tf,ss, orzpk) of G1(s)and G2(s), respectively. They can also be symbolic variables.
Example 2.8. It should be noted that, if G1(s)and G2(s)contain the same pole, then the result of the parallel manipulation may be simplified further in this case. Consider the two
G1(s) G2(s)
G1(s)
-(a) series connection
G2(s)
? 6
-(b) parallel connection
u(t) y(t) u(t) y(t)
Figure 2.2. Interconnections of blocks.
G1(s)
G2(s)
-(b) negative feedback
-
−6
G1(s)
G2(s)
-(a) positive feedback
- 6
u(t) y(t) u(t) y(t)
Figure 2.3. Feedback connections.
blocks G1(s)= 1/(s + 1)2and G2(s)= 1/(s + 1). The result of calling the appropriate MATLAB functions is given by the following MATLAB statements:
>> G1=zpk([],[-1,-1],1); G2=zpk([],[-1],1); G=G1+G2
As a result, the overall system obtained is G= (s + 2)(s + 1)/(s + 1)3. In fact, the overall transfer function can be simplified to G(s)= (s + 2)/(s + 1)2, since there is a common factor (s+ 1) in both the denominators of G1(s)and G2(s).
The minimum realization technique can be used to obtain the simplified model, and details of the technique will be given later in this chapter.
2.5.3 Feedback Connection
The simple feedback connection of two blocks G1(s)and G2(s)is shown in Figures 2.3(a) and (b), respectively. The two feedback connections in Figure 2.3 are different; the left one is called a system with positive feedback and the right a system with negative feedback. The overall transfer function of the positive feedback is G(s)= G1(s)[I −G2(s)G1(s)]−1, and for negative feedback it is G(s)= G1(s)[I + G2(s)G1(s)]−1.
A MATLAB functionfeedback()is provided in the Control Systems Toolbox to get the overall system model from the feedback connection with the syntax
G=feedback(G1,G2,Sign),
whereSignis used to identify the positive or negative feedback connection. IfSign=-1, the negative feedback structure is indicated. The Signvariable can be omitted in the function call in a negative feedback connection. The LTI objects in the forward path and feedback path are given by G1and G2, respectively.
A MATLAB function,feedback(), is also written for the models represented by symbolic variables:
function H=feedback(G1,G2,key) if nargin==2; key=-1; end,
H=G1/(sym(1)-key*G1*G2); H=simple(H);
This function should be placed under the@symdirectory under MATLAB path. This function is useful in theoretically deriving the overall model from more complicated sub-system configuration.
Example 2.9. Consider again the models in Example 2.8. If a negative feedback connection is assumed, one can find the overall transfer function by using the following MATLAB statements:
>> G1=tf(1,[1 2 1]); G2=tf(1,[1 1]); G=feedback(G1,G2)
and it can be found that
G(s)= s+ 1
s3+ 3s2+ 3s + 2.
For a positive feedback connection, the overall system model can be obtained from
>> G=feedback(G1,G2,+1)
where
G(s)= s+ 1 s3+ 3s2+ 3s.
2.5.4 More Complicated Connections
In the real world, a system structure can be very complex. In this section, we illustrate how to handle more complicated interconnections.
Consider the typical feedback control system structure shown in Figure 1.2. The overall system can be evaluated by assuming first that Gc(s)and G(s) are in series and then that they are connected to a negative feedback block H (s). The overall closed-loop transfer function of the typical feedback control system is
Gcl(s)= G(s)Gc(s)
1+ H (s)G(s)Gc(s). (2.22) The overall system object can be obtained using the MATLAB statement
Gcl=feedback(G*Gc,H).
Example 2.10. If the three blocks in the above typical feedback structure are given by
G(s)= s3+ 7s2+ 24s + 24
s4+ 10s3+ 35s2+ 50s + 24, Gc(s)= 10s+ 5
s , H (s)= 1
0.01s+ 1, the overall transfer function of the closed-loop system can be obtained by the following MATLAB statements:
>> G=tf([1 7 24 24],[1,10,35,50,24]); Gc=tf([10,5],[1,0]);
H=tf([1],[0.01,1]); G_cl=feedback(G*Gc,H)
and it can be found that
Gcl(s)= 0.1s5+ 10.75s4+ 77.75s3+ 278.6s2+ 361.2s + 120 0.01s6+ 1.1s5+ 20.35s4+ 110.5s3+ 325.2s2+ 384s + 120.
1
1+0.01s - - - 130
s 0.212
?
0.1 1+0.01s 0.0044
1+0.01s
6
6
-
-- 1+0.17s
0.085s 1 1+0.01s
- 1+0.15s
0.051s
70 1+0.0067s
- - - 0.21
1+0.15s
-g1(s) g2(s) g3(s) g4(s) g5(s) g6(s) g7(s)
g8(s)
g9(s) g10(s)
Figure 2.4. An example of DC electric drive system.
Unfortunately, the overall system evaluation is not always as simple as in this case, where one can perform the calculation by hand. For more complicated structures, evalua-tions by hand is laborious and computer aids are very useful.
Example 2.11. Consider the structure of the DC motor drive system, shown in Figure 2.4.
It can be seen that the overall system model is not easily evaluated. From the block diagram, it can be seen that the difficulty lies in the interconnections among paths 6, 7, 8, and 9. Rearranging path 9 so that it starts from the output signal, we see that the equiv-alent transfer function then becomes g91(s) = g9/g7. The overall system can then be constructed using the following MATLAB statements:
>> g1=tf(1,[0.01,1]); g2=tf([0.17,1],[0.085,0]); g3=g1;
g4=tf([0.15,1],[0.051,0]); g5=tf(70,[0.0067,1]); g7=tf(130,[1,0]);
g6=tf(0.21,[0.15,1]); g8=0.212; g9=tf(0.1,[0.01,1]); g91=g9/g7;
g10=0.0044*g1; gg1=feedback(g7*g6,g8); %paths 6-8 gg2=feedback(gg1*g5*g4,g91); %paths 4-9
G=feedback(gg2*g3*g2,g10)*g1; minreal(zpk(G)), %overall system
where the overall model can be simply obtained as
G(s)= 111852502194.908(s+ 6.667)(s + 5.882)
(s+ 180.9)(s + 84.1)(s + 48.2)(s2+ 15.2s + 74.3)(s2+ 27.57s + 354). One can also perform symbolic manipulations to the models. When the following are given, the results follow immediately.
>> syms g1 g2 g3 g4 g5 g6 g7 g8 g9 g10
g91=g9/g7; gg1=feedback(g7*g6,g8); gg2=feedback(gg1*g5*g4,g91);
G=feedback(gg2*g3*g2,g10)*g1
thus the overall model can then be obtained as
G(s)= g2g3g6g7g5g4g1
1+ g7g6g8+ g6g5g4g9+ g4g5g7g6g3g2g10.
Example 2.12. Consider the motor drive system shown in Figure 2.5, where there are two inputs r(t ), M(t), and one output n(t).
c1 ka kr
Figure 2.5. Block diagram of a motor control system.
ka
Figure 2.6. Equivalent block diagram when M(t) is applied alone.
Let us first consider the modeling problem with set-point input r(t) only. The overall system model can be obtained easily with the following statements:
>> syms Ka Kr c1 c2 c Ra T1 T2 Km Kb s % symbolic declaration Ga=feedback(1/Ra/(T1*s+1)*Km*1/c/(T2*s+1),Kb); % inner loop g1=c1*feedback(Ka*Kr*Ga/s,c2); g1=collect(g1,s)
The transfer function is then derived from
g1(s)= c1kmkakr
RacT1T2s3+ (RacT1+ RacT2)s2+ (kmkb+ Rac)s+ kakrkmc2
. If the load disturbance M(t) is used alone, the original structure of the system can be rearranged as shown in Figure 2.6, and the following statements can be used to find the overall model
>> g2=feedback(1/c/(T2*s+1)/s, Km/Ra/(T1*s+1)*(Kb*s+c2*Ka*Kr));
g2=collect(simplify(g2),s)
and it can be found that
g2(s)= (T1s+ 1)Ra
cRaT2T1s3+ (cRaT1+ cRaT2)s2+ (kbkm+ cRa)s+ kmc2kakr
. The transfer function matrix of the system is G(s)= [g1(s), g2(s)].