Introduction:

State$-$space models are models that use state variables to describe a system by means of a set of first$-$order differential equations. The model is conveniently expressed in a matrix format of $2$ equations, state equation and output equation.

In this exercise you will be using the active window in MATLAB to derive the unit step response for an electrical system and a mechanical system model in state space. You will be using the data obtained from MATLAB to ascertain the effect a system parameter has on the dynamic response of the systems. models that use state variables to describe a system by means of a set of firstorder differential equations. The model is conveniently expressed in a matrix format of $2$ equations, state equation and output equation.

Apparatus:

Computer with $M$

$\left[\begin{array}{c}{x}_1^{}\end{array}\right]$

$x_2=\left[\begin{array}{ccc}-\frac{1}{RC}& \frac{1}{C}& 0\end{array}\right]$

$-\frac{1}{L}\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2+\left[\begin{array}{c}0\end{array}\right]$

$\frac{1}{L}u$

$y=\left[\begin{array}{cc}\frac{1}{R}& 0\end{array}\right]\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2$;dots..$D=0$

Figure $2$

Procedure: Part A

For the electrical system shown in Figure $1,$ the state$-$space model relating input voltage and resistor current is shown in Figure $2.$

$\backslash$table$[[$Capacitor$,$ C $(F),1,2,10,20]]$

Table $1$

MATLAB Script format. You will need to enter the matrix values for A$,$ B$,$ C and D$.$

$A=$ place the derived matrix for A here

$B=$ place the derived matrix for $B$ here

$C=$ place the derived matrix for $C$ here

$D=$ place the derived matrix for $D$ here

sys $=ss(A,B,C,D)$

step $($sys$)$

$2.$ Write a MATLAB script to derive the unit step response for the system for each of the capacitor values indicated in Table $1$ given that $L=10mH$ and $R=100.$ Derive all the plots on the same graph and save it for placement it in your report. Discuss the effect of the capacitor value on the system response in terms of settling time, peak time, percentage overshoot.

Procedure: Part B

For the mechanical system shown in Figure $3$ the state$-$space model relating input force, $F(t)$ and mass displacement, $x(t)$ is shown in Figure $4.$

$I{n}_{m}atri{x}_f$orm:

$\left[\begin{array}{c}{x}_1^{}\end{array}\right]$

$x_2=\left[\begin{array}{ccc}0& 1& -\frac{c}{m}\end{array}\right]$

$-\frac{k}{m}\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2+\left[\begin{array}{c}0\end{array}\right]$

$\frac{1}{m}u.......stat{e}_e$quation

Since $y=x(t)=x_1$ :

$y=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$;$D=0$dots............. $Outpu{t}_e$quation

Figure $4$

$2.$ Write a MATLAB script to derive the unit step response for the system for the system parameters given. Capture this plot for placement in your report.

$3.$ By trial and error determine a value for the damper constant that will give the fastest settling time without an overshoot for a unit step input. Capture and save at least $4$ plots from your attempts including the best fit and place them in your report. Outline you approach and discuss your result.

$4.$ Outside of the lab session, use theory to derive the solution for the damper value that will produce the shortest settling time. Compare the results obtained from trail and error and that obtain from first principle.