Week 8 Classroom
Eigenvalues of a state space matrix tells you about stability. The following is best done as a live script (start the editor and select 'new livescript')
>> syms omega zeta
Create a second order system in state space
>> A=[0 1; -omega^2 -2*zeta*omega]
Compute the Eigenvalues
>> ev=eig(A)
and simplify
>> simplify(ev)
If you are not using livescripts you can pretty print the result
>> pretty(simplify(ev))
Look at the equations. Is it possible for any of the real parts of the Eigenvalues to become postive? If so why
A pendulum linearised around the lowest point of the swing has the the following equation for $\omega$
\[ \omega=\sqrt{\frac{g}{l}} \]Since $\omega=2\pi f$ where $f$ is the frequency of the pendulum, calculate the length of the pendulum needed for a period of 1 second.
Set this up as a numerical simulation
>> g=9.8 % metres per second^2
>> omegaSim=sqrt(g/l)
>> zetaSim=0;
>> A=[0 1; -omegaSim^2 0]
Set up the function to integrate as a state space
>> fn=@(t,x)A*x
>> [t,y]=ode45(fn,[0 10],[.2;0]);
>> plot(t,y)
Is it a one second period? What are the Eigenvalues? Is it stable?
Now try a more practical pendulum where the amplitude will decay
>> zetaSim=somevalue
>> A=[0 1; -omegaSim^2 -2*zetaSim*omegaSim]
Recompute the function, integrate and plot. What happens if zetaSim is large? What are the Eigenvalues?
How about an impossible pendulum. Set zetaSim to a small or a big negative number. Recompute the function, integrate and plot. What are the Eigenvalues?
How do the Eigenvalues predict stability?
The non-linear equation for a pendulum with damping $B$ is given by
\begin{equation} \ddot\theta=-\frac{g}{l}\sin\theta -\frac{B}{m}\dot\theta \label{eq:pend1} \end{equation}Write the state space equation of the damped non-linear pendulum equation, in the form
\[ \begin{bmatrix}\dot\theta\\\dot\omega\end{bmatrix} = f(\begin{bmatrix}??\\\omega\end{bmatrix}) \]You will need to fill in the ?? and work out what the states are.