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).

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.
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)
 

See Also

abinv  ui_observer  

Author

F.D.