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Please note that the recommended version of Scilab is 2025.0.0. This page might be outdated.
See the recommended documentation of this function
obscont
observer based controller
Calling Sequence
[K]=obscont(P,Kc,Kf) [J,r]=obscont(P,Kc,Kf)
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 — linear system definition
- feedback — feedback operation
- observer — observer design
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