ui_observer - unknown input observer
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).
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);