lqr

LQ compensator (full state)

Calling Sequence

[K,X]=lqr(P12)

Arguments

:P12 syslin list (state-space linear system) : :K,X two real matrices :

Description

lqr computes the linear optimal 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)).

The cost function is l2-norm of z’*z with z=C1 x + D12 u i.e. [x,u]’ * BigQ * [x;u] where

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

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'*X+Q-S*`inv`_(R)*S'=0
K=-`inv`_(R)*(B2'*X+S)

For a discrete plant:

X=A'*X*A-(A'*X*B2+C1'*D12)*`pinv`_(B2'*X*B2+D12'*D12)*(B2'*X*A+D12'*C1)+C1'*C1;
K=-`pinv`_(B2'*X*B2+D12'*D12)*(B2'*X*A+D12'*C1)

An equivalent form for X is

X=Abar'*`inv`_(`inv`_(X)+B2*`inv`_(r)*B2')*Abar+Qbar

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   0    0|   | A    0    B2|         |I   0   0|   | A    0    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|

Caution: It is assumed that matrix R is non singular. In particular, the plant must be tall (number of outputs >= number of inputs).

Examples

A=`rand`_(2,2);B=`rand`_(2,1);   //two states, one input
Q=`diag`_([2,5]);R=2;     //Usual notations x'Qx + u'Ru
Big=`sysdiag`_(Q,R);    //Now we calculate C1 and D12
[w,wp]=`fullrf`_(Big);C1=wp(:,1:2);D12=wp(:,3:$);   //[C1,D12]'*[C1,D12]=Big
P=`syslin`_('c',A,B,C1,D12);    //The plant (continuous-time)
[K,X]=lqr(P)
`spec`_(A+B*K)    //check stability
`norm`_(A'*X+X*A-X*B*`inv`_(R)*B'*X+Q,1)  //Riccati check
P=`syslin`_('d',A,B,C1,D12);    // Discrete time plant
[K,X]=lqr(P)
`spec`_(A+B*K)   //check stability
`norm`_(A'*X*A-(A'*X*B)*`pinv`_(B'*X*B+R)*(B'*X*A)+Q-X,1) //Riccati check

See Also

  • lqe linear quadratic estimator (Kalman Filter)
  • gcare Continuous time control Riccati equation
  • leqr H-infinity LQ gain (full state)

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