## 4 Discussion  
Predictive controllers (both continuous and sampled-data) derived from the outset to be optimal by first solving an optimization problem with the governing dynamic equations of motion lead to several controllers (including a controller that twice invokes optimization to formulate robust, predictive control, which was referred to as a two degree-of-freedom controller). These controllers were compared to each other with noise and modeling errors in Table 3. Many figures of merit are used: tracking error and rate error deviations and means, in addition to total mean cost. Robustness was evaluated using Monte Carlo analysis where plant parameters were randomly assumed to be incorrectly modeled. Six instances of controllers were compared against these methods and interpretations allow engineers to select a tailored control for their given circumstances. Optimal control, predictive control, discrete (sampled-data) control, proportional-derivative control, feedforward + feedback control, and a proposed 2 DOF control were all derived with the articulation of an individual optimization problem. Control structures were then designed to enforce optimality, rather than the ubiquitous methodology of first asserting the control structure, then subsequently seeking to optimize the imposed structure. The proposed comparison paradigm utilized six disparate types of control that were compared against each other's ability to meet commanded end conditions at minimum cost in the face of both noise and modeling errors.  
The optimal open loop controller established the baseline cost (being the mathematical minimum possible), but was not realistically capable of implementation due to lacking feedback. Continuous and discrete (sampled-data) predictive controls proved very good at achieving the commanded final position with very low cost, but revealed a tolerable, but non-trivial, susceptibility to noise and mismodeling only with regard to difficulties achieving the final commanded velocity state. Very much akin the classical PD controller, the proportional-derivative controller developed under the new design paradigm was very good at state control (achieving the best rate control), but had the highest cost by a very wide margin compared to all of the other control options. Akin to classical feedforward + feedback controllers, the controller designed under the new paradigm achieved the very best position control at very good cost. However, it has some slight susceptibility to noise and  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Controller** & **Benefits** & **Weakness** \\ \hline Optimal open loop & Establishes optimal case & Not realistically implementable \\ Continuous predictive * & Good cost and position & High rate error and deviation \\ Sampled predictive * & Good cost and position & High rate error and deviation \\ Proportional-derivative (PD) * & Best rate control & Worst cost \\ Feedforward + feedback & Best position control & Slight rate error and deviation \\
2 DOF * & All-around good & Slightly higher cost than optimal \\ \hline \hline \end{tabular}  
* Large generalizations;
* _Not_ the ubiquitous PD topology.  
\end{table}
Table 3: Summary 1 of the deterministic plant.  
modeling errors with regard to the rate control. The proposed 2 DOF controller was the all-around best controller, achieving near-perfect state controls with very low (suboptimal) cost. The controller had no performance degradation due to noise or modeling errors.  
_Proposed 2 DOF control designed foremost as an optimization problem: The open-loop optimal control is used as a feedforward, while the optimal states derived from a time-parameterized optimal control are compared to the feedback signal to generate the error fed to the feedback controller whose gains are a reparameterization of the optimal solution._