1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
|
ss2ss(1) Scilab Function ss2ss(1)
NAME
ss2ss - state-space to state-space conversion, feedback, injection
CALLING SEQUENCE
[Sl1,right,left]=ss2ss(Sl,T, [F, [G , [flag]]])
PARAMETERS
Sl : linear system (syslin list) in state-space form
T : square (non-singular) matrix
Sl1, right, left
: linear systems (syslin lists) in state-space form
F : real matrix (state feedback gain)
G : real matrix (output injection gain)
DESCRIPTION
Returns the linear system Sl1=[A1,B1,C1,D1] where A1=inv(T)*A*T,
B1=inv(T)*B, C1=C*T, D1=D.
Optional parameters F and G are state feedback and output injection respec-
tively.
For example, Sl1=ss2ss(Sl,T,F) returns Sl1 with:
| inv(T)*(A+B*F)*T , inv(T)*B |
Sl1 = | |
| (C+D*F)*T , D |
and right is a non singular linear system such that Sl1=Sl*right.
Sl1*inv(right) is a factorization of Sl.
Sl1=ss2ss(Sl,T,0*F,G) returns Sl1 with:
| inv(T)*(A+G*C)*T , inv(T)*(B+G*D) |
Sl1 = | |
| C*T , D |
and left is a non singular linear system such that Sl1=left*Sl (right=Id if
F=0).
When both F and G are given, Sl1=left*Sl*right.
- When flag is used and flag=1 an output injection as follows is used
| inv(T)*(A + GC)*T , inv(T)*(B+GD,-G) |
| C*T , (D , 0) |
and then a feedback is performed, F must be of size (m+p,n)
( x is in R^n , y in R^p, u in R^m ).
right and left have the following property:
Sl1 = left*sysdiag(sys,eye(p,p))*right
- When flag is used and flag=2 a feedback (F must be of size (m,n)) is
performed and then the above output injection is applied. right and
left have the following property:
Sl1 = left*sysdiag(sys*right,eye(p,p)))
EXAMPLE
Sl=ssrand(2,2,5); trzeros(Sl) // zeros are invariant:
Sl1=ss2ss(Sl,rand(5,5),rand(2,5),rand(5,2));
trzeros(Sl1), trzeros(rand(2,2)*Sl1*rand(2,2))
// output injection [ A + GC, (B+GD,-G)]
// [ C , (D , 0)]
p=1,m=2,n=2; sys=ssrand(p,m,n);
// feedback (m,n) first and then output injection.
F1=rand(m,n);
G=rand(n,p);
[sys1,right,left]=ss2ss(sys,rand(n,n),F1,G,2);
// Sl1 equiv left*sysdiag(sys*right,eye(p,p)))
res=clean(ss2tf(sys1) - ss2tf(left*sysdiag(sys*right,eye(p,p))))
// output injection then feedback (m+p,n)
F2=rand(p,n); F=[F1;F2];
[sys2,right,left]=ss2ss(sys,rand(n,n),F,G,1);
// Sl1 equiv left*sysdiag(sys,eye(p,p))*right
res=clean(ss2tf(sys2)-ss2tf(left*sysdiag(sys,eye(p,p))*right))
// when F2= 0; sys1 and sys2 are the same
F2=0*rand(p,n);F=[F1;F2];
[sys2,right,left]=ss2ss(sys,rand(n,n),F,G,1);
res=clean(ss2tf(sys2)-ss2tf(sys1))
SEE ALSO
projsl, feedback
|
