- Ultimate solution is to find direct funcional dependancy between t_rise = f1(Q,R,N), t_settling = f2(Q,R,N), So far no direct no obvious natural math. standard expressible form found. There are exist know well expressed formulas for t_rise = f3(zetha, natural_frequency), f3 is known for 2n order also next formula found: zeta = cos(atan2(pi,-log(overshoot/100))). Then to find how zetha, natural_frequency, overshoot are afffected mathematically by Q,R,N values through state-space to s laplace or z-domain transition via K (gain) and ARE. Also what can be found in S and P from: [K,S,P] = lqr(sys,Q,R,N)
2.So far they used Bryson's Rule, GA, PSO optimization algorithms to find optimal the ARE and associated LQR controller K gain for most robust, optimal closed loop functioning (which means optimal t_rise, t_settling, overshoot, damping, phase and gain margins - basic closed loop agent+environment(plant+controller) performance characteristics).
