observer based controller
[K]=obscont(P,Kc,Kf)
[J,r]=obscont(P,Kc,Kf)
: :Kc real matrix, (full state) controller gain : :Kf real matrix, filter gain : :K syslin list (controller) : :J syslin list (extended controller) : :r 1x2 row vector :
obscont returns the observer-based controller associated with a nominal plant P with matrices [A,B,C,D] ( syslin list).
The full-state control gain is Kc and the filter gain is Kf. These gains can be computed, for example, by pole placement.
A+B*Kc and A+Kf*C are (usually) assumed stable.
K is a state-space representation of the compensator K: y->u in:
xdot = A x + B u, y=C x + D u, zdot= (A + Kf C)z -Kf y +B u, u=Kc z
K is a linear system ( syslin list) with matrices given by: K=[A+B*Kc+Kf*C+Kf*D*Kc,Kf,-Kc].
The closed loop feedback system Cl: v ->y with (negative) feedback K (i.e. y = P u, u = v - K y, or
xdot = A x + B u,
y = C x + D u,
zdot = (A + Kf C) z - Kf y + B u,
u = v -F z
) is given by Cl = P/.(-K)
The poles of Cl ( spec(cl(‘A’))) are located at the eigenvalues of A+B*Kc and A+Kf*C.
Invoked with two output arguments obscont returns a (square) linear system K which parametrizes all the stabilizing feedbacks via a LFT.
Let Q an arbitrary stable linear system of dimension r(2)`x `r(1) i.e. number of inputs x number of outputs in P. Then any stabilizing controller K for P can be expressed as K=lft(J,r,Q). The controller which corresponds to Q=0 is K=J(1:nu,1:ny) (this K is returned by K=obscont(P,Kc,Kf)). r is size(P) i.e the vector [number of outputs, number of inputs];
ny=2;nu=3;nx=4;P=`ssrand`_(ny,nu,nx);[A,B,C,D]=`abcd`_(P);
Kc=-`ppol`_(A,B,[-1,-1,-1,-1]); //Controller gain
Kf=-`ppol`_(A',C',[-2,-2,-2,-2]);Kf=Kf'; //Observer gain
cl=P/.(-obscont(P,Kc,Kf));`spec`_(cl('A')) //closed loop system
[J,r]=obscont(P,Kc,Kf);
Q=`ssrand`_(nu,ny,3);Q('A')=Q('A')-(`max`_(`real`_(`spec`_(Q('A'))))+0.5)*`eye`_(Q('A'))
//Q is a stable parameter
K=`lft`_(J,r,Q);
`spec`_(`h_cl`_(P,K)) // closed-loop A matrix (should be stable);