| Control System Toolbox™ | |
sysc = connect(sysa, sysb, ..., inputs, outputs) sysc
= connect(sys,Q,inputs,outputs)
sysc = connect(sysa, sysb, ..., inputs, outputs) sysc = connect(sys,Q,inputs,outputs)
connect constructs the aggregate model for a given block diagram interconnection of LTI models. You can specify the block diagram connectivity in two ways, name-based and index-based.
In this approach, you name the input and output signals of all LTI blocks sys1, sys2,... in the block diagram, including the summation blocks. The aggregate model sys is then built by
sys = connect (sys1,sys2,...,inputs,outputs)
where inputs and outputs are the names of the block diagram external I/Os, specified as strings or cell arrays of strings.
Given LTI models C and G, and referring to this block diagram,
you can construct the closed-loop transfer T from r to y as follows
C.InputName = e; C.OutputName = u;
G.InputName = u; G.OutputName = y;
Sum = ss([1,-1],'InputName',{'r','y'},'OutputName','e');
T = connect(G,G,Sum,'r','y')
In this approach, first combine all LTI blocks into an aggregate, unconnected model blksys using append. Then construct a matrix Q where each row specifies one of the connections or summing junctions in terms of the input vector u and output vector y of blksys. For example, the row
[3 2 0 0]
indicates that y(2) feeds into u(3), while the row
[7 2 -15 6]
indicates that y(2)-y(15)+y(6) feeds into u(7). The aggregate model sys is then obtained by
sys = connect(blksys,Q,inputs,outputs)
where inputs and outputs are index vectors into u and y selecting the block diagram external I/Os.
You can construct the closed-loop model T for the block diagram above as follows:
blksys = append(C,G); % u = inputs to C,G. y = outputs of C,G. % Here y(1) feeds into u(2) and -y(2) feeds into u(1) Q = [2 1; 1 -2]; % External I/Os: r drives u(1) and y is y(2) T = connect(blksys,Q,1,2)
Since it is easy to make a mistake entering all the data required for a large model, be sure to verify your model in as many ways as you can. Here are some suggestions:
Make sure the poles of the unconnected model sys match the poles of the various blocks in the diagram.
Check that the final poles and DC gains are reasonable.
Plot the step and bode responses of sysc and compare them with your expectations.
The connect function supports I/O and internal delays. See Time Delays for more information and examples.
Consider the following block diagram
.
Given the matrices of the state-space model sys2
A = [ -9.0201 17.7791
-1.6943 3.2138 ];
B = [ -.5112 .5362
-.002 -1.8470];
C = [ -3.2897 2.4544
-13.5009 18.0745];
D = [-.5476 -.1410
-.6459 .2958 ];
Define the three blocks as individual LTI models.
sys1 = tf(10,[1 5],'inputname','uc')
sys2 = ss(A,B,C,D,'inputname',{'u1' 'u2'},...
'outputname',{'y1' 'y2'})
sys3 = zpk(-1,-2,2)
Next append these blocks to form the unconnected model sys.
sys = append(sys1,sys2,sys3)
This produces the block-diagonal model
sys
a =
x1 x2 x3 x4
x1 -5 0 0 0
x2 0 -9.0201 17.779 0
x3 0 -1.6943 3.2138 0
x4 0 0 0 -2
b =
uc u1 u2 ?
x1 4 0 0 0
x2 0 -0.5112 0.5362 0
x3 0 -0.002 -1.847 0
x4 0 0 0 1.4142
c =
x1 x2 x3 x4
? 2.5 0 0 0
y1 0 -3.2897 2.4544 0
y2 0 -13.501 18.075 0
? 0 0 0 -1.4142
d =
uc u1 u2 ?
? 0 0 0 0
y1 0 -0.5476 -0.141 0
y2 0 -0.6459 0.2958 0
? 0 0 0 2
Continuous-time system.
Note that the ordering of the inputs and outputs is the same as the block ordering you chose. Unnamed inputs or outputs are denoted b.
To derive the overall block diagram model from sys, specify the interconnections and the external inputs and outputs. You need to connect outputs 1 and 4 into input 3 (u2), and output 3 (y2) into input 4. The interconnection matrix Q is therefore
Q = [3 1 -4
4 3 0];
Note that the second row of Q has been padded with a trailing zero. The block diagram has two external inputs uc and u1 (inputs 1 and 2 of sys), and two external outputs y1 and y2 (outputs 2 and 3 of sys). Accordingly, set inputs and outputs as follows.
inputs = [1 2]; outputs = [2 3];
You can obtain a state-space model for the overall interconnection by typing
sysc = connect(sys,Q,inputs,outputs)
a =
x1 x2 x3 x4
x1 -5 0 0 0
x2 0.84223 0.076636 5.6007 0.47644
x3 -2.9012 -33.029 45.164 -1.6411
x4 0.65708 -11.996 16.06 -1.6283
b =
uc u1
x1 4 0
x2 0 -0.076001
x3 0 -1.5011
x4 0 -0.57391
c =
x1 x2 x3 x4
y1 -0.22148 -5.6818 5.6568 -0.12529
y2 0.46463 -8.4826 11.356 0.26283
d =
uc u1
y1 0 -0.66204
y2 0 -0.40582
Continuous-time system.
Note that the inputs and outputs are as desired.
[1] Edwards, J.W., "A Fortran Program for the Analysis of Linear Continuous and Sampled-Data Systems," NASA Report TM X56038, Dryden Research Center, 1976.
append, feedback, minreal, parallel, series, lft
| conj | covar | |
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