Linear quadratic (LQ) servo is command following regarding to the reference input. An nth-order system having r inputs and m outputs: x(t) = [xr (t); yp (t)J, where yp (t) e Rmxl is output and xr (t) e R(n~m)"1 is rest of system state, e(t) = r(t) - y(t) e Rm/l is the error vector with r(t) e Rmxl is plan output and y(t) e Rmxl is reference. Consider state space model:
with y(t) = Cpx(t) , xr (t) = Dpx(t) , and Cp is [°my(n-m) Jmym ] , Dp is \_I(n-m)y(n-m) 0(n-m)xm J . Consider control input u(t) = -Gx(t), and control gain G consists of G = _Gy Gr ] , where the gain vector is Gy e Rmxm and Gr e Rmx(n-m) . The control law than became:
substitute (8) into (7), we have differential equation of close-loop system as follows:
Solution of equation (9) could be solving by using Runga-Kutta 4th order approximation:
, „ _ (kl + 2k 2 + 2k 3 + k 4) , s x(t +1) = x(t ) +----(10)
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