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
|
cainv(1) Scilab Function cainv(1)
NAME
cainv - Dual of abinv
CALLING SEQUENCE
[X,dims,J,Y,k,Z]=cainv(Sl,alfa,beta)
PARAMETERS
sl : syslin list containing the matrices [A,B,C,D].
alfa : real number or vector (possibly complex, location of closed
loop poles)
alfa : real number or vector (possibly complex, location of closed
loop poles)
X : orthogonal matrix of size nx (dim of state space).
dims : integer row vector dims=[nd1,nu1,dimS,dimSg,dimN] (5 entries,
nondecreasing order)
J : real matrix (output injection)
Y : orthogonal matrix of size ny (dim of output space).
k : integer (normal rank of Sl)
Z : non-singular linear system (syslin list)
DESCRIPTION
cainv finds a bases (X,Y) (of state space and output space resp.) and out-
put injection matrix J such that the matrices of Sl in bases (X,Y) are
displayed as:
[A11,*,*,*,*,*] [*]
[0,A22,*,*,*,*] [*]
X'*(A+J*C)*X = [0,0,A33,*,*,*] X'*(B+J*D) = [*]
[0,0,0,A44,*,*] [0]
[0,0,0,0,A55,*] [0]
[0,0,0,0,0,A66] [0]
Y*C*X = [0,0,C13,*,*,*] Y*D = [*]
[0,0,0,0,0,C26] [0]
The partition of X is defined by the vector dims=[nd1,nu1,dimS,dimSg,dimN]
and the partition of Y is determined by k.
Eigenvalues of A11 (nd1 x nd1) are unstable. Eigenvalues of A22 (nu1-nd1 x
nu1-nd1) are stable.
The pair (A33, C13) (dimS-nu1 x dimS-nu1, k x dimS-nu1) is observable, and
eigenvalues of A33 are set to alfa.
Matrix A44 (dimSg-dimS x dimSg-dimS) is unstable. Matrix A55 (dimN-
dimSg,dimN-dimSg) is stable
The pair (A66,C26) (nx-dimN x nx-dimN) is observable, and eigenvalues of
A66 set to beta.
The dimS first columns of X span S= smallest (C,A) invariant subspace which
contains Im(B), dimSg first columns of X span Sg the maximal "complementary
detectability subspace" of Sl
The dimN first columns of X span the maximal "complementary observability
subspace" of Sl. (dimS=0 iff B(ker(D))=0).
This function can be used to calculate an unknown input observer:
// DDEP: dot(x)=A x + Bu + Gd
// y= Cx (observation)
// z= Hx (z=variable to be estimated, d=disturbance)
// Find: dot(w) = Fw + Ey + Ru such that
// zhat = Mw + Ny
// z-Hx goes to zero at infinity
// Solution exists iff Ker H contains Sg(A,C,G) inter KerC
//i.e. H is such that:
// For any W which makes a column compression of [Xp(1:dimSg,:);C]
// with Xp=X' and [X,dims,J,Y,k,Z]=cainv(syslin('c',A,G,C));
// [Xp(1:dimSg,:);C]*W = [0 | *] one has
// H*W = [0 | *] (with at least as many aero columns as above).
SEE ALSO
abinv, dt_ility
|
