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ddp Scilab Group Scilab Function ddp
NAME
ddp - disturbance decoupling
CALLING SEQUENCE
[Closed,F,G]=ddp(Sys,zeroed,B1,D1)
[Closed,F,G]=ddp(Sys,zeroed,B1,D1,flag,alfa,beta)
PARAMETERS
Sys : syslin list containing the matrices (A,B2,C,D2).
zeroed
: integer vector, indices of outputs of Sys which are zeroed.
B1 : real matrix
D1 : real matrix. B1 and D1 have the same number of columns.
flag : string 'ge' or 'st' (default) or 'pp'.
alfa : real or complex vector (loc. of closed loop poles)
beta : real or complex vector (loc. of closed loop poles)
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.
Sys 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 Sys and B1,D1:
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=D2(zeroed,:)
The matrix D1 is partitioned similarly as D1=[D11;D21] with
D11=D1(zeroed,:). The control is u=Fx+Gw and one looks for matriced 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. flag='st' : look for stable closed
loop system (A+B2*F stable). flag='pp' : eigenvalues of A+B2*F are
assigned to alfa and beta.
Closed is a realization of the 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).
EXAMPLE
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)
AUTHOR
F.D.
SEE ALSO
abinv, ui_observer
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