obscont

Arguments

P

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

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

Description

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];

Examples

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);

See also

• ppol — pole placement
• lqg — LQG compensator
• lqr — LQ compensator (full state)
• lqe — linear quadratic estimator (Kalman Filter)
• h_inf — Continuous time H-infinity (central) controller
• lft — linear fractional transformation
• syslin — définition d'un système dynamique linéaire
• feedback — feedback operation
• observer — observer design