Robust Sampled-Data Control: An Input Delay Approach

Modelling of continuous-time systems with digital control in the form of continuous-time systems with delayed control input was introduced by Mikheev, Sobolev & Fridman [19], Astrom & Wittenmark [1] and further developed by Fridman (1992). The digital control law may be represented as delayed control as follows: $$ u(t) = u_d (t_k ) = u_d (t - (t - t_k )) = u_d (t - \tau (t)),{\mathbf{ }}t_k < t < t_{k + 1} ,{\mathbf{ }}\tau (t) = t - t_k , $$ (1) where u d is a discrete-time control signal and the time-varying delay τ(t) = t−t k is piecewise-linear with derivative \( \dot \tau (t) = 1 \) for t ≠ t k. Moreover, τ ≤ t k+1 − t k. Based on such a model, for small enough sampling intervals t k+1−t k asymptotic approximations of the trajectory [19] and of the optimal solution to the sampled-data LQ finite horizon problem [7] were constructed.