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observer based controller

Calling Sequence




syslin list (nominal plant) in state-space form, continuous or discrete time


real matrix, (full state) controller gain


real matrix, filter gain


syslin list (controller)


syslin list (extended controller)


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)xr(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];


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
//Q is a stable parameter
spec(h_cl(P,K))  // closed-loop A matrix (should be stable);

See Also

  • ppol — pole placement
  • lqg — LQG compensator
  • lqr — LQ compensator (full state)
  • lqe — linear quadratic estimator (Kalman Filter)
  • h_inf — H-infinity (central) controller
  • lft — linear fractional transformation
  • syslin — definição de sistemas lineares
  • feedback — feedback operation
  • observer — observer design


F.D. ; ;

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Last updated:
Thu Mar 03 11:00:32 CET 2011