LQ compensator (full state)
[K,X]=lqr(P12)
:P12 syslin list (state-space linear system) : :K,X two real matrices :
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).
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