## 3 Results
### Adaption Using Model-Following and Various Parameter Estimation (Benchmarks)  
Model-following involves the design of a control topology that effectively converts an unstable system response to a stable one, which prevents divergence of the output and allows for inputs to be tracked asymptotically. In addition, model-following can incorporate parameter estimation techniques like RLS, ELS, and Batch LS to iteratively adapt the designed control topology to the estimated process characteristics, which is known as a self-tuning regulator. Deterministic artificial intelligence (deterministic artificial intelligence) takes a different approach to input tracking, where an autonomous trajectory for output is defined for changes in input using a half-wave of a sinusoid. With the deterministic artificial intelligence modeling approach, natural system dynamics are asserted instead of modifying the existing process through control design as is done in model-following. In Figure 6a, a contrast between deterministic artificial intelligence and model-following performances exists in the transient responses with either approach. The transient with deterministic artificial intelligence has a comparably smaller peak, with a peak magnitude lower than that of RLS by 2.66%, ELS by 43.24%, and Batch LS by 67.18%. The discrete-time square wave depicted in Figure 6 was created with a sampling frequency of 1, displaying a slight slant visible in the plot at times of state change due to the short time axis length and nature of discrete (integer) sampling. The reference signal functionally acts as a square wave for discrete-time systems. As shown in Table 2, output with the deterministic artificial intelligence modeling approach results in a significant average input tracking error at least 40.88% higher than \begin{table}
\begin{tabular}{c c c c c} \hline
**Estimated Parameter** & **True Value** & **RLS** & **ELS** & **Batch LS** \\ \hline \(a_{1}\) & 1.9000 & 0.011/0.007 & 0.012/0.005 & 0.025/0.004 \\ \(a_{2}\) & \(-0.8800\) & 0.020/0.009 & 0.020/0.002 & 0.021/0.003 \\ \(b_{0}\) & 0.1065 & 0.002/0.001 & 0.0005/0.0002 & 0.001/0.001 \\ \(b_{1}\) & 0.0902 & 0.003/0.002 & 0.0004/0.0002 & 0.001/0.0007 \\ \hline \end{tabular}
\end{table} 
Table 1: Mean/Standard Deviation of Estimation Technique Errors. Figure 6: Results based on discretization of plant using difference equations in Equation (2) (**a**) Input tracking with model-following and parameter estimation on ordinate with time in seconds on the abscissa. (**b**) Input tracking with deterministic artificial intelligence and proportional plus derivative adaption of self-awareness statements on ordinate with time in seconds on the abscissa. error resulting from implementations of model-following with different parameter estimation techniques. The high error is also reflected in Figure 6a, in which the output with deterministic artificial intelligence visibly fluctuates heavily after each change in input. Error distributions between each implementation of model-following are similar, with each average error being within a 9% margin of each other. It is evident, however, that there is a sharp difference in tracking accuracy between model-following and deterministic artificial intelligence.