leqr - H-infinity LQ gain (full state)

Calling Sequence

[K,X,err]=leqr(P12,Vx)

Parameters

• P12 : syslin list
• Vx : symmetric nonnegative matrix (should be small enough)
• K,X : two real matrices
• err : a real number (l1 norm of LHS of Riccati equation)

Description

leqr computes the linear suboptimal H-infinity LQ full-state gain for the plant P12=[A,B2,C1,D12] in continuous or discrete time.

P12 is a syslin list (e.g. P12=syslin('c',A,B2,C1,D12)).

      [C1' ]               [Q  S]
      [    ]  * [C1 D12] = [    ]
      [D12']               [S' R]
   

Vx is related to the variance matrix of the noise w perturbing x; (usually Vx=gama^-2*B1*B1').

The gain K is such that A + B2*K is stable.

X is the stabilizing solution of the Riccati equation.

For a continuous plant:

(A-B2*inv(R)*S')'*X+X*(A-B2*inv(R)*S')-X*(B2*inv(R)*B2'-Vx)*X+Q-S*inv(R)*S'=0
   

K=-inv(R)*(B2'*X+S)
   

For a discrete time plant:

X-(Abar'*inv((inv(X)+B2*inv(R)*B2'-Vx))*Abar+Qbar=0
   

K=-inv(R)*(B2'*inv(inv(X)+B2*inv(R)*B2'-Vx)*Abar+S')
   

with Abar=A-B2*inv(R)*S' and Qbar=Q-S*inv(R)*S'

The 3-blocks matrix pencils associated with these Riccati equations are:

               discrete                        continuous
   |I  -Vx  0|   | A    0    B2|       |I   0   0|   | A    Vx    B2|
  z|0   A'  0| - |-Q    I    -S|      s|0   I   0| - |-Q   -A'   -S |
   |0   B2' 0|   | S'   0     R|       |0   0   0|   | S'   -B2'   R|
   

See Also

lqr  

Author

F.D.;