Figure 5:

Error in applying the optimal control policy occurs at the zero-crosses of the ipPRC. For a sampling time of 4 h, the zero-cross occurs in the second step t₁ < t⁰ < t₂, and so u₂ will necessarily violate the continuous-time optimal control law. For the cases shown here, Δϕ₂(u_max) = Δϕ₂(u_min) = 0, and the resulting phase at the end of step 2 is identical due to cancellation of the advance by the delay. Finally, it is noteworthy that this single pathological step consumes most of the positive region of the ipPRC, suggesting that τ = 4 h is a poor choice for sampling time if phase advances are desirable.