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ui_observer
unknown input observer
Syntax
[UIobs, J, N] = ui_observer(Sys, reject, C1, D1) [UIobs, J, N] = ui_observer(Sys, reject, C1, D1, flag, alfa, beta)
Arguments
- 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
andD1
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).
Examples
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);
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