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
|
// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) INRIA - F. Delebecque
//
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
function [Closed,F,G]=ddp(Sys,zeroed,B1,D1,flag,Alfa,Beta)
//--------------Exact disturbance decoupling----------
// 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.
//
// Sys = linear system {A,B2,C,D2} with one input and two outputs
// i.e Sys: u-->(z,y) in the following
//
// xdot = A x + B1 w + B2 u
// z = C1 x + D11 w + D12 u
// y = C2 x + D21 w + D22 u
//
// outputs of Sys are partitioned into (z,y) where z is to be zeroed,
// i.e. the matrices C and D2 are:
//
// C=[C1;C2] D2=[D12;D22]
// C1=C(zeroed,:) D12=D1(zeroed,:)
//
// The control is u=Fx+Gw and one
// looks for F,G such that the closed loop system: w-->z given by
//
// xdot= (A+B2*F) x + (B1 + B2*G) w
// z = (C1+D12F) x + (D11+D12*G) w
//
// has zero transfer transfer function.
//
// flag='ge' : no stability constraints
// ='st' : look for stable closed loop system (A+B2*F stable)
// ='pp' : eigenvalues of A+B2*F are assigned to Alfa and Beta
//
// Closed = w-->y closed loop system
//
// xdot= (A+B2*F) x + (B1 + B2*G) w
// y = (C2+D22*F) x + (D21+D22*G) w
//
// Stability (resp. pole placement) requires stabilizability
// (resp. controllability) of (A,B2).
//
[LHS,RHS]=argn(0);
if RHS==5 then Beta=-1;end
if RHS==4 then Beta=-1;Alfa=-1;end
if RHS==3 then Beta=-1;Alfa=-1;flag='st';end
if RHS==2 then Beta=-1;Alfa=-1;flag='st';D1=zeros(size(Sys('C'),1),size(B1,2));
end
if size(B1,1) ~= size(Sys('A'),1) then
error(msprintf(gettext("%s: Incompatible input arguments #%d and #%d: Same row dimensions expected.\n"),"ddp",1,3))
end
if size(D1,2) ~= size(B1,2) then
error(msprintf(gettext("%s: Incompatible input arguments #%d and #%d: Same row dimensions expected.\n"),"ddp",3,4))
end
Sys1=Sys(zeroed,:);
not_zeroed=1:size(Sys,1);not_zeroed(zeroed)=[];
[X,dims,F,U,k,Z]=abinv(Sys1,Alfa,Beta,flag);nv=dims(3);
Sys_new=ss2ss(Sys,X);Fnew=F*X;
B1new=X'*B1;B2new=Sys_new('B');
D11=D1(zeroed,:);D12=Sys1('D');
B21=B1new(nv+1:$,:);B22=B2new(nv+1:$,:);
// G s.t. B21+B22*G=0 D11+D12*G=0
G=lowlev();
[Anew,Bnew,Cnew,Dnew]=abcd(Sys_new);
Anew=Anew+B2new*Fnew;Cnew=Cnew+Dnew*Fnew;
B1new=B1new+B2new*G;
A11=Anew(1:nv,1:nv);C21=Cnew(not_zeroed,1:nv);
B11=B1new(1:nv,:);D21=D1(not_zeroed,:);
D22=Sys('D');D22=D22(not_zeroed,:);D21=D21+D22*G;
Closed=syslin(Sys('dt'),A11,B11,C21,D21);
endfunction
function G=lowlev()
ww=[B21 B22;D11 D12];
[xx,dd]=colcomp(ww);
K=kernel(ww);
rowG=size(B22,2);colG=size(B1,2);
if size(K,2) > colG then K=K(:,1:colG);end
Kup=K(1:size(K,2),:);
if rcond(Kup) <= 1.d-10 then
warning(msprintf(gettext("%s: Bad conditioning.\n"),"ddp"));
K1=K*pinv(Kup);G=K1(size(K,2)+1:$,:);return
end
K1=K*inv(Kup); //test conditioning here!
G=K1(size(K,2)+1:$,:);
endfunction
|
