disturbance decoupling
[Closed,F,G]=ddp(Sys,zeroed,B1,D1)
[Closed,F,G]=ddp(Sys,zeroed,B1,D1,flag,alfa,beta)
: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) :
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).
`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)