## 3 Results
### Monte Carlo Analysis on a Mismodeled Plant with Noise  
Similar to Section 3.1, this section summarizes the simulation analysis of a mismodeled plant (the double integrator with an inertia-velocity control dependence). In this section, comments are restricted to changes in controller performance due to the mismodeled plant. Generally, the controllers maintained the characteristics demonstrated with the deterministic plant (e.g., the PD controller and the 2 DOF PD controller did not demonstrate susceptibility to \(\omega\) uncertainty like the other controllers). Applying these controllers to a mismodeled plant, notice in Table 2 that many of the controllers still perform very well (surprising, since the controls were based in optimization of analytic dynamics that were not correct). The controllers experienced slightly degraded performance overall, that, in most cases, was well within expected tolerances for most simple applications.  
If the situation demands precise attainment of the final conditions with mismodeled plants, we can deduce four tiers of controllers: (1) 2 DOF PD controller, (2) PD controller and Feedforward/Feedback PD controller, and (3) Continuous, and Sampled Predictive controllers, and (4) Optimal Open Loop Controller. The first tier (the 2 DOF PD controller) met the stated end conditions within a miniscule error and maintained a very low-cost number (while not optimum). The second tier (PD and Feedforward/Feedback PD) each did well at meeting one of the stated end conditions, but showed  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Controller** & **Deviation, \(\sigma_{\theta}\)** & **Mean Error, \(\mu_{\theta}\)** & **Deviation, \(\sigma_{\omega}\)** & **Mean Error, \(\mu_{\omega}\)** & **Mean Cost, \(J\)** \\ \hline Optimal open loop & **0.034** & **0.044** & **0.0283** & **0.0484** & **6.004** \\ Continuous predictive & 0.0015 & 0.0019 & 0.02 & 0.025 & 6.0137 \\ Sampled predictive & 0.0029 & 0.007 & 0.0222 & 0.0307 & 6.0183 \\ Proportional-derivative (PD) & \(9.89\times 10^{-4}\) & 0.014 & \(\textbf{3.12}\times\textbf{10}^{-16}\) & \(\textbf{1.02}\times\textbf{10}^{-15}\) & **59.3244** \\ Feedforward + feedback & \(7.36\times 10^{-16}\) & \(9.69\times 10^{-16}\) & 0.0058 & 0.0086 & 8.9637 \\
2 DOF & \(\textbf{6.42}\times\textbf{10}^{-16}\) & \(\textbf{9.02}\times\textbf{10}^{-16}\) & \(1.90\times 10^{-15}\) & \(2.56\times 10^{-15}\) & 7.8284 \\ \hline \hline \end{tabular}  
* note highest and lowest values in bold font.  
\end{table}
Table 2: Statistical Analysis or mis-modeled plant 1.  
Figure 10: Monte Carlo analysis of a deterministic plan with noise with \(\theta\) (radians) on the abscissa and \(\omega\) (rad/s) on the ordinate.  
increased error in the other end condition. The third tier (Continuous/Sampled Optimal Controllers) had increased error in both stated end conditions, while the final tier (the open loop controller) was inappropriate for application due to high errors and deviation in end conditions.  
If the requirements on stated end conditions can be relaxed sufficiently to emphasize minimum cost, the tiers of results would change. The optimum open loop controller would join the first tier due to its minimum cost, but such a re-ranking will not be done in this case due to the large error achieving end conditions. The first tier would include the continuous and the sample optimum controllers. They achieved minimum cost while also closely meeting the end conditions. Next, the second tier would include feedback/feedforward and 2 DOF PD controllers. They had slightly higher costs while very precisely meeting the stated end conditions. Third, the PD controller had very high cost, but also did a good job meeting the end conditions.