A $$black box$$ system is driven with a unit step input $($ref$.$ Figure $2\mathrm{.}1\mathrm{.}5,$ Palm$),$ u$($t$)=$ us$($t$).$ The output response x$($t$)$ is measured and compared with the response with the two candidate models given in Table $1($ref$.$ Table $2\mathrm{.}1\mathrm{.}1,$ Palm$).$ Note that the data that is supplied to you comes from a stable system. This means that the system output will go to a constant final value when driven by a unit step input.

Using the curve fitting tool, $$cftool$,$ in Matlab and the following models, fit the output response data $($x$\_$out$$ versus $$time$)$ supplied in BlackBoxStepResponse$\_$Lab$1.$mat. Adjust the Fit Options until a $$best fit$$ is obtained for each model. Additional tips for curve fitting are provided under $$Notes for cftool$$ below. The following models $($equations given in Table $1)$ and Fit Option constraints should be considered:

$5$th Order Polynomial $-$ no constraints

$2$nd Order Exponential with Sinusoid $-$b$$ and $$c$$ must be positive

Table $1.$ Reference Equations

y$($t$)$

$1$

y$($t$)=$p$1*$t$5+$p$2*$t$4+$p$3*$t$3+$p$4*$t$2+$p$5*$t$+$p$6$

$2$

y$($t$)=$a$*$e$(-$b$*$t$)*$sin$($c$*$t$+$d$)+$e

For each model, determine the constant coefficients that give the best fit and report the values, along with the corresponding R$-$square and RMSE performance metrics, in a table. Discuss how the R$-$square and RMSE metrics compare to the observed fits.

In MATLAB, create an output vector $($y$($t$))$ for each model using the results of the cftool curve fits where the time vector is extended to $0\mathrm{.}5$ s$,$ with a step size of $0\mathrm{.}01$ s$.$ Plot y$($t$)$ vs$.$ t for the models on a single plot. On the same plot, plot the black box response output supplied in BlackBoxStepResponseFinal$\_$Lab$1.$mat, which is the original data response extended to $0\mathrm{.}5$ seconds. Note: Use appropriate labels on the plot and dashed lines if signals completely overlap so that you can clearly see all signals.

How you report all the different graphs is up to your group. Remember, you only have one page so you need to be succinct, yet clear.

Discuss the accuracy of the different curve fits, particularly when the time vector was extended to a longer time interval. Also discuss which model you feel most accurately represents the data for the black box system and why you believe that is the best model. In general, are polynomials good models for dynamic systems? Explain your answer.