ddp

Arguments

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'.

alpha

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).

Examples

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.
my_alpha=-1;my_beta=-2;flag='ge';
[X,dims,F,U,k,Z]=abinv(Sys12,my_alpha,my_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',my_alpha,my_beta);
closed=syslin('c',A+B2*F,B1+B2*G,C1+D12*F,D11+D12*G);
ss2tf(closed)

See Also

• abinv — AB invariant subspace
• ui_observer — unknown input observer

Authors

F.D.