lqr - LQ compensator (full state)
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