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ddp Scilab Group Scilab Function ddp
NAME
ui_observer - unknown input observer
CALLING SEQUENCE
[UIobs,J,N]=ui_observer(Sys,reject,C1,D1)
[UIobs,J,N]=ui_observer(Sys,reject,C1,D1,flag,alfa,beta)
PARAMETERS
Sys : syslin list containing the matrices (A,B,C2,D2).
reject
: integer vector, indices of inputs of Sys which are unknown.
C1 : real matrix
D1 : real matrix. C1 and D1 have the same number of rows.
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
Unknown input observer. Sys: (w,u) --> y is a (A,B,C2,D2) syslin linear
system with two inputs w and u, w being the unknown input. The matrices
B and D2 of Sys are (implicitly) partitioned as: B=[B1,B2] and
D2=[D21,D22] with B1=B(:,reject) and D21=D2(:,reject) where reject =
indices of unknown inputs. The matrices C1 and D1 define z = C1 x + D1
(w,u), the to-be-estimated output.
The matrix D1 is (implicitly) partitioned as D1=[D11,D12] with
D11=D(:,reject)
The data (Sys, reject,C1, D1) define a 2-input 2-output system:
xdot = A x + B1 w + B2 u
z = C1 x + D11 w + D12 u
y = C2 x + D21 w + D22 u
An observer (u,y) --> zhat is looked for the output z.
flag='ge' no stability constraints flag='st' stable observer (default)
flag='pp' observer with pole placement alfa,beta = desired location of
closed loop poles (default -1, -2) J=y-output to x-state injection.
N=y-output to z-estimated output injection.
UIobs = linear system (u,y) --> zhat such that: The transfer function:
(w,u) --> z equals the composed transfer function: [0,I;
UIobs Sys] (w,u) -----> (u,y) -----> zhat i.e.
transfer function of system {A,B,C1,D1} equals transfer function
UIobs*[0,I; Sys]
Stability (resp. pole placement) requires detectability (resp.
observability) of (A,C2).
EXAMPLE
A=diag([3,-3,7,4,-4,8]);
B=[eye(3,3);zeros(3,3)];
C=[0,0,1,2,3,4;0,0,0,0,0,1];
D=[1,2,3;0,0,0];
rand('seed',0);w=ss2ss(syslin('c',A,B,C,D),rand(6,6));
[A,B,C,D]=abcd(w);
B=[B,matrix(1:18,6,3)];D=[D,matrix(-(1:6),2,3)];
reject=1:3;
Sys=syslin('c',A,B,C,D);
N1=[-2,-3];C1=-N1*C;D1=-N1*D;
nw=length(reject);nu=size(Sys('B'),2)-nw;
ny=size(Sys('C'),1);nz=size(C1,1);
[UIobs,J,N]=ui_observer(Sys,reject,C1,D1);
W=[zeros(nu,nw),eye(nu,nu);Sys];UIobsW=UIobs*W;
//(w,u) --> z=UIobs*[0,I;Sys](w,u)
clean(ss2tf(UIobsW));
wu_to_z=syslin('c',A,B,C1,D1);clean(ss2tf(wu_to_z));
clean(ss2tf(wu_to_z)-ss2tf(UIobsW),1.d-7)
/////2nd example//////
nx=2;ny=3;nwu=2;Sys=ssrand(ny,nwu,nx);
C1=rand(1,nx);D1=[0,1];
UIobs=ui_observer(Sys,1,C1,D1);
AUTHOR
F.D.
SEE ALSO
cainv, ddp, abinv
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