In this assignment you will determine parameters for a controller design based on linearized

EOM and determine the efficacy of the design by simulating the non$-$linear EOM with the

controller implemented. Specifically:

Create a python program that uses the parameters $($SI units$)M=100,m=10,L=2,$ and

$g=10\mathrm{.}0$ to calculate $K_P$ and $K_D$ given $=0\mathrm{.}7$ and ${}_n=0\mathrm{.}5.$

Within the program create a function that provides the state$-$space representation of the

nonlinear EOM. Use the state variables vec$(x)=\left[\begin{array}{cccc}& {}^{}& x& x^{}\end{array}\right]$ and implement the controller as

$F=u=-K_P-K_D{}^{}.$ Use the double pendulum example as a template for the state$-$

space representation.

Simulate the system given the initial condition vec$(x)(0)=\left[\begin{array}{cccc}5& 0& 0& 0\end{array}\right]($remember to convert

to radians$).$ Congratulate yourself on the response for $$ before noticing the unacceptable

response for $x.$ Simulate for $75$ seconds.

Simulate the system given the initial condition vec$(x)(0)=[450000]$ to see if the

controller will still bring $0$ despite the initial conditions being so far from the

reference value about which you linearized. Conclude that maybe the other state variable

needs to be feed back.

Try $F=u=-K_P-K_D{}^{}-K_{x}x-K_v{x}^{}$ using guesses for $K_x$ and $K_v$ until frustration

sets in then try $F=u=-3K_P-9K_D{}^{}+\mathrm{.}03K_{P}x+0\mathrm{.}5K_D{x}^{}$ and the initial

condition vec$(x)(0)=\left[\begin{array}{cccc}45& 0& 0& 0\end{array}\right]$ and then congratulate yourself on the overall response.

Observe that even though the initial condition is far from the state about which you

linearized, the controller still works. This course will be$,$ in large part, about techniques

for designing the controller without relying completely on trial and error.