## 2 Materials and Methods  
This section derives and describes the implementation of the controllers in this study including Optimum Open Loop, Continuous Predictive Optimum, Sampled Predictive Optimum,proportional-derivative (PD), Feedforward/Feedback PD, and 2 DOF control. The context of the already commonly defined term \"feedforward\" used here (differently and more restrictively) is the context of deterministic artificial intelligence, where analytical self-awareness statements are strictly determined by the governing physics (of mechanics in this case, e.g. [PERSON], [PERSON], and [PERSON]). When the feedforward is combined with proportional-derivative, \"PD\" feedback control (developed from the optimization statement at the outset using a novel re-parameterization of the time-optimal results from time-to-state parameterization), the resulting controller is referred to here as \"2 DOF\" or two degrees-of-freedom to indicate the twice invocation of optimization at the genesis of the feedforward and the feedback, respectively.  
First, a brief derivation of the optimum control for our double integrator assumes a quadratic cost function. Using a simple double-integrator plant (modeling translational motion in an inertial reference frame described by [PERSON] and [PERSON]), controllers were developed in accordance with the new design techniques to travel from the initial conditions \((\theta(0)=0,\omega(0)=0)\) to the final conditions \((\theta(1)=1,\omega(1)=0)\), where \(\theta\) is the translational motion state and \(\omega\) is the translational velocity state. The controllers were then applied to a mismodeled plant, which contained a \(0.005[I]\big{[}\omega^{2}\big{]}\) dependence (where \(I\) is the mass inertia and \(\omega\) is angular velocity). Additionally, the inertia of a body is difficult to know exactly what adds \"inertia noise,\" which is also referred to as modeling errors. After witnessing the deleterious impact of noise on an open loop controller's robustness, the remaining controllers studied in this research were closed loop. Feedback was instituted to deal with noise. Figures of merit address two key issues of control: achieving desired end conditions and minimizing the required control effort. Accordingly, controllers in this research report will be compared via nominal error norms (for \(\theta\), \(\omega\), and cost) and via statistical analysis of random variation (mean error, deviation, and mean cost).  
Quadratic cost was evaluated for each controller, where every controller's model coded in MATLAB/SIMULINK is depicted in the Appendix A to aide repeatability for the reader. End conditions were compared via error norms (specifically \(\ell^{2}\), also known as least squares). The controllers were designed to both meet the end conditions and minimize the cost function (often opposing goals) from the outset. Designs were iterated to get as close as possible to the end conditions while keeping cost low (a ubiquitous trade-off). In the main body of the manuscript, each controller is designed with descriptions to address the key issues of control in addition to how well they perform in a Monte Carlo analysis.  
It will be shown that the optimum control for our double integrator is \(u=at+b\), and this is simply a generalized form that can be narrowed to \(u=K_{p}(1-t)\theta+K_{v}(1-t)\dot{\theta}\) (looks somewhat like a classical PD controller). This research implements the optimum control in several manners: Open loop, Continuous-predictive, Sampled-Predictive, Feedforward/feedback, and 2 Degree-of-freedom (2 DOF). Thus, the predictive controllers are derived from optimality rather than the opposite.