Drawing 2025-10-24 11.54.34.excalidraw

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Excalidraw Data

Text Elements

m

Example.

Where here, we can sub in our chosen states…

Note that in general this is not LTI!

Using N2L

If we fix

We can find all equilibrium points which solve system for

The system that we want to solve is

Now solve this matrix for

which will yield a vector for

Physically these equilibrium points are when the mass is at the bottom And at the top

Embedded Files

ca93b89adcf5ce6aed11b1ad8088c0f1b28ea825: $ml{\theta }^{\prime \prime }=-mgsin(\theta )-D{\theta }^{\prime }+u$

60786c67b3ded4f241942e5705ecb5995108ae03: $\text{Choosing states}:x_1=\theta \text{, }{x}_2={\theta }^{\prime }$

ac6206a4a0836fb78b9d1fe10d6d27e90c8c9deb: $x^{\prime }=\left[\begin{array}{c}{\theta }^{\prime }\\ -\frac{g}{l}sin\theta -\frac{D}{ml}{\theta }^{\prime }+\frac{1}{ml}u\end{array}\right]$

275c938fc4bdf8debd0a8daf290fdde5aa795ec7: $\text{If we call our matrix}f(x,u)\text{ and}y=\theta =x_1$

e9d166415d5ca77eb32ef238357fe411e8eeabfd: $x^{\prime }=f(x,u)$

e4505b21c9cac68321a9a8f9005fcc93877c6e18: $y=g(x,u)$

4ae2471e0eb307de4aa9d7f4a88c7bc4c95e1876: $u(t)=\bar{u}\forall t>=0$

95696dea9e8f18de75677b1ee3a2ed1c858b0e58: $\bar{u}\in \mathbb{R}$

1e7c4ee7d951943969593a3a261e35c876c6e055: $u(t)=\bar{u}\forall t\ge 0$

6b732d7c498f5d13ad94178ef9e9627b52191e95: $f(x_e,\bar{u})=0\text{ for }{x}_e$

bbfda556420b7ca109eadff5730fc6479cc81c83: $f(x_e,\bar{u})=\left[\begin{array}{c}{x}_2^e\\ -\frac{g}{l}sin{x}_1^e-\frac{D}{ml}{x}_2^e+\frac{1}{ml}\bar{u}\end{array}\right]=0$

f8215b91f457e88f821f145f93e41655b015cec0: $x_1^e\text{ and }{x}_2^e$

61b54a4e1cdf50b236fde63aff8cb4fb037daed5: $x_e$