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
98
99
100
101
102
103
104
105
106
107
108
|
.TH ddp 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
ddp - disturbance decoupling
.SH CALLING SEQUENCE
.nf
[Closed,F,G]=ddp(Sys,zeroed,B1,D1)
[Closed,F,G]=ddp(Sys,zeroed,B1,D1,flag,alfa,beta)
.fi
.SH PARAMETERS
.TP
Sys
: \fVsyslin\fR list containing the matrices \fV(A,B2,C,D2)\fR.
.TP
zeroed
: integer vector, indices of outputs of \fVSys\fR which are zeroed.
.TP
B1
: real matrix
.TP
D1
: real matrix. \fVB1\fR and \fVD1\fR have the same number of columns.
.TP
flag
: string \fV'ge'\fR or \fV'st'\fR (default) or \fV'pp'\fR.
.TP
alfa
: real or complex vector (loc. of closed loop poles)
.TP
beta
: real or complex vector (loc. of closed loop poles)
.SH DESCRIPTION
Exact disturbance decoupling (output nulling algorithm).
Given a linear system, and a subset of outputs, z, which are to
be zeroed, characterize the inputs w of Sys such that the
transfer function from w to z is zero.
\fVSys\fR is a linear system {A,B2,C,D2} with one input and two outputs
( i.e. Sys: u-->(z,y) ), part the following system defined from \fVSys\fR
and \fVB1,D1\fR:
.nf
xdot = A x + B1 w + B2 u
z = C1 x + D11 w + D12 u
y = C2 x + D21 w + D22 u
.fi
outputs of Sys are partitioned into (z,y) where z is to be zeroed,
i.e. the matrices C and D2 are:
.nf
C=[C1;C2] D2=[D12;D22]
C1=C(zeroed,:) D12=D2(zeroed,:)
.fi
The matrix \fVD1\fR is partitioned similarly as \fVD1=[D11;D21]\fR
with \fVD11=D1(zeroed,:)\fR.
The control is u=Fx+Gw and one looks for matriced \fVF,G\fR such that the
closed loop system: w-->z given by
.nf
xdot= (A+B2*F) x + (B1 + B2*G) w
z = (C1+D12F) x + (D11+D12*G) w
.fi
has zero transfer transfer function.
.LP
\fVflag='ge'\fR : no stability constraints.
\fVflag='st'\fR : look for stable closed loop system (A+B2*F stable).
\fVflag='pp'\fR : eigenvalues of A+B2*F are assigned to \fValfa\fR and
\fVbeta\fR.
.LP
Closed is a realization of the \fVw-->y\fR closed loop system
.nf
xdot= (A+B2*F) x + (B1 + B2*G) w
y = (C2+D22*F) x + (D21+D22*G) w
.fi
.LP
Stability (resp. pole placement) requires stabilizability
(resp. controllability) of (A,B2).
.SH EXAMPLE
.nf
rand('seed',0);nx=6;nz=3;nu=2;ny=1;
A=diag(1:6);A(2,2)=-7;A(5,5)=-9;B2=[1,2;0,3;0,4;0,5;0,0;0,0];
C1=[zeros(nz,nz),eye(nz,nz)];D12=[0,1;0,2;0,3];
Sys12=syslin('c',A,B2,C1,D12);
C=[C1;rand(ny,nx)];D2=[D12;rand(ny,size(D12,2))];
Sys=syslin('c',A,B2,C,D2);
[A,B2,C1,D12]=abcd(Sys12); //The matrices of Sys12.
alfa=-1;beta=-2;flag='ge';
[X,dims,F,U,k,Z]=abinv(Sys12,alfa,beta,flag);
clean(X'*(A+B2*F)*X)
clean(X'*B2*U)
clean((C1+D12*F)*X)
clean(D12*U);
//Calculating an ad-hoc B1,D1
G1=rand(size(B2,2),3);
B1=-B2*G1;
D11=-D12*G1;
D1=[D11;rand(ny,size(B1,2))];
[Closed,F,G]=ddp(Sys,1:nz,B1,D1,'st',alfa,beta);
closed=syslin('c',A+B2*F,B1+B2*G,C1+D12*F,D11+D12*G);
ss2tf(closed)
.fi
.SH AUTHOR
F.D.
.SH SEE ALSO
abinv, ui_observer
|
