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# ddp

disturbance decoupling

### Syntax

[Closed,F,G]=ddp(Sys,zeroed,B1,D1) [Closed,F,G]=ddp(Sys,zeroed,B1,D1,flag,alfa,beta)

### 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

## Comments

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