lqg2stan

Arguments

P22

State space representation of the nominal plant (nu inputs, ny outputs, nx states).

Qxu

[Q,S;S',N] symmetric nx+nu by nx+nu weighting matrix.

Qwv

[R,T;T',V] symmetric nx+ny by nx+ny covariance matrix.

r

Row vector [ny nu].

P_aug

Augmented plant state space representation (see: syslin)

Description

lqg2stan returns the augmented plant for linear LQG (H2) controller design problem defined by:

• The nominal plant P22:described by

  

• The (instantaneous) cost function .

• The noises covariance matrix

Up to Scilab-5.5.2 lqg2stan returns wrong inverted values (see bug 13751) to obtain the good result one had to use [P,r]=lqg2stan(-P,Qxu,Qwv). This bug is fixed since Scilab-6.0.0, old codes must be modified accordingly.

Algorithm

If [B1;D21] is a factor of Qxu, [C1,D12] is a factor of Qwv (see: fullrf) then P_aug=syslin(P.dt,P.A,[B1,P.B],[C1;-P.C],[0,D12;D21,P.D])

Examples

ny=2;nu=3;nx=4;
P22=ssrand(ny,nu,nx);
Qxu=rand(nx+nu,nx+nu);Qxu=Qxu*Qxu';
Qwv=rand(nx+ny,nx+ny);Qwv=Qwv*Qwv';
[P_aug,r]=lqg2stan(P,Qxu,Qwv);
K=lqg(P_aug,r);  //K=LQG-controller
spec(h_cl(P_aug,r,K))      //Closed loop should be stable
//Same as Cl=P22/.K; spec(Cl('A'))

s=poly(0,'s')
lqg2stan(1/(s+2),eye(2,2),eye(2,2))

See also

• lqg — LQG compensator
• lqr — LQ compensator (full state)
• lqe — linear quadratic estimator (Kalman Filter)
• obscont — observer based controller
• h_inf — Continuous time H-infinity (central) controller
• augment — augmented plant
• fstabst — Youla's parametrization of continuous time linear dynamical systems
• feedback — feedback operation