function [P,A_u,A_w,k] = ppp_are (A,B,C,D,Q,R,A_type)
## usage: [P,A_u,A_w] = ppp_are (A,B,C,D,Q,R,A_type)
##
##
if nargin<1
disp("usage: [P,A_u,A_w] = ppp_are (A,B,C,D,Q,R,A_type)");
return
endif
if nargin<7
A_type = "feedback";
endif
[n_x,n_u,n_y] = abcddim(A,B,C,D); # Dimensions
n_q = is_square(Q); # Size of Q
if n_q==0
error("Q must be square");
endif
## Steady-state Linear Quadratic solution
## using Algebraic Riccati equation (ARE)
if n_q==n_y # Output weight
Q_x = C'*Q*C; # Weighting on x
elseif n_q==n_x # State weight
Q_x = Q;
else
error(sprintf("Q (%ix%i) must be %ix%i or %ix%i",n_q,n_q,n_y,n_y,n_x,n_x));
endif
[k, P, poles] = lqr (A, B, Q_x, R); # Algebraic Riccati solution
## Basis functions
if strcmp(A_type,"companion")
A_u = compan(poly(poles));
elseif strcmp(A_type,"feedback")
A_u = A-B*k;
else
error(sprintf("A_type must be %s, not %s", "companion or feedback", A_type));
endif
## Avoid spurious imag parts due to rounding
A_u = real(A_u);
## Setpoint basis functions
A_w = 0;
endfunction