Discussion on Laplace transform. We want to be able to switch from \(t\) to \(s\) domain and back. For tough ones, use tables. In Matlab use syms. Examples shown how to use syms in Matlab and obtain the Laplace and inverse Laplace transforms. Use of expand and simplify commands.
Closed loop can be used to improve performance. An example given of a mass spring damper. Steps of solution
Control engineers need a model to analyze. Example is \(mx^{\prime \prime }+cx^{\prime }+kx=u\left ( t\right ) \). We can always find transfer function TF assuming system is at rest initially. Need to know basic relation \(\mathcal{L}\)\(\frac{d^{k}y}{dt^{k}}=s^{k}Y\left ( s\right ) \) assuming all initial conditions are zero. Back to the above equation. Take Laplace transform we obtain\begin{equation} G\left ( s\right ) =\frac{Y\left ( s\right ) }{U\left ( s\right ) }=\frac{1}{ms^{2}+cs+k} \tag{1} \end{equation} We see if the open loop does what we want. If it does, no need for feedback. If open loop response is not good, then we use feedback to improve the response.
Behavior of open loop: Let \(u\left ( t\right ) =1\) (unit step). What is \(y\left ( t\right ) \)? From (1)\[ Y\left ( s\right ) =G\left ( s\right ) U\left ( s\right ) \] But \(\mathcal{L}\)\(\left \{ 1\right \} =\frac{1}{s}\) then the above becomes\[ Y\left ( s\right ) =\frac{1}{ms^{2}+cs+k}\frac{1}{s}\] To get \(y\left ( t\right ) \) we need to inverse Laplace the above. Let \(m=1,c=10,k=1\) then\[ y\left ( t\right ) =\mathcal{L}^{-1}\left \{ \frac{1}{s^{2}+10s+1}\frac{1}{s}\right \} \] There are 3 possibility of behavior of this system depending on roots of the denominator. Real roots implies \(y\left ( t\right ) \) involves only exponential. If the roots are complex, the result contain harmonics. The over solution will look like \(e^{-2t}\cos \left ( \cdots \right ) \).
Reader: For the above example, find \(y\left ( t\right ) \) for \(c=1,c=10.\) Use partial fractions to find the inverse Laplace transform, then sketch \(y\left ( t\right ) \) for each.
Now overview was given on using Simulink. Examples of using basic blocks explained.
Classical responses of system shown.