lqg2stan

lqg2stan — LQG to standard problem

Calling sequence

[P,r]=lqg2stan(P22,bigQ,bigR)

Parameters

`P22`	: `syslin` list (nominal plant) in state-space form
`bigQ`	: `[Q,S;S',N]` (symmetric) weighting matrix
`bigR`	: `[R,T;T',V]` (symmetric) covariance matrix
`r`	: `1`x`2` row vector = (number of measurements, number of inputs) (dimension of the 2,2 part of `P`)
`P`	: `syslin` list (augmented plant)

Description

lqg2stan returns the augmented plant for linear LQG (H2) controller design.

P22=syslin(dom,A,B2,C2) is the nominal plant; it can be in continuous time (dom='c') or discrete time (dom='d').



  . 
  x = Ax + w1 + B2u
  y = C2x + w2

for continuous time plant.



  x[n+1]= Ax[n] + w1 + B2u
      y = C2x + w2

for discrete time plant.

The (instantaneous) cost function is [x' u'] bigQ [x;u].

The covariance of [w1;w2] is E[w1;w2] [w1',w2'] = bigR

If [B1;D21] is a factor of bigQ, [C1,D12] is a factor of bigR and [A,B2,C2,D22] is a realization of P22, then P is a realization of [A,[B1,B2],[C1,-C2],[0,D12;D21,D22]. The (negative) feedback computed by lqg stabilizes P22, i.e. the poles of cl=P22/.K are stable.

Examples



ny=2;nu=3;nx=4;
P22=ssrand(ny,nu,nx);
bigQ=rand(nx+nu,nx+nu);bigQ=bigQ*bigQ';
bigR=rand(nx+ny,nx+ny);bigR=bigR*bigR';
[P,r]=lqg2stan(P22,bigQ,bigR);K=lqg(P,r);  //K=LQG-controller
spec(h_cl(P,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))

Author

F.D.