ui_observer - Unknown input observer (original) (raw)
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See the recommended documentation of this function
Scilab help >> CACSD > ui_observer
Calling Sequence
[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 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 constraintsflag='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=-N1C;D1=-N1D; 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;
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)
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);