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## Background

The homework of Linear System Theory is challenging. Here is Problem 1:

Consider a linear system with a state transition matrix $\phi(t,t_0)$

$$\phi(t,t_0)=\displaystyle \left[\begin{matrix} e^{t} \cos{\left(2 t \right)} & e^{- 2 t} \sin{\left(2 t \right)}\\ -e^{t} \sin{\left(2 t \right)} & e^{- 2 t} \cos{\left(2 t \right)} \end{matrix}\right]$$

Compute A(t).

Since the given system is linear time variant system, by using some properties of $\phi$, we can easily compute $A(t)$. However, the expression is so complicate that i could not simplify it by hand. I call for some tools for help. The tools are Python and Matlab.

This post compare the difference of simplify function in Python and Matlab.

## Comparision

### Python

sympy.simplify.simplify.simplify(expr, ratio=1.7, measure=, rational=False, inverse=False). Link https://docs.sympy.org/latest/modules/simplify/simplify.html

This function does not work well, it did not simplify the expression, giving th following outcome.

$$\displaystyle \left[\begin{matrix}\frac{\left(- \left(\left(2 \sin{\left(2 t \right)} - \cos{\left(2 t \right)}\right) \left(\sin^{2}{\left(2 t_{0} \right)} - 1\right) e^{3 t} + \frac{\sqrt{2} \left(- \sin{\left(2 t - 4 t_{0} + \frac{\pi}{4} \right)} + \sin{\left(2 t + 4 t_{0} + \frac{\pi}{4} \right)}\right) e^{3 t_{0}}}{2}\right) \left(\left(\sin^{2}{\left(2 t \right)} - 1\right) e^{3 t_{0}} \cos{\left(2 t_{0} \right)} - \frac{\left(\cos{\left(4 t - 2 t_{0} \right)} - \cos{\left(4 t + 2 t_{0} \right)}\right) e^{3 t}}{4}\right) + \left(\left(2 \sin{\left(2 t \right)} - \cos{\left(2 t \right)}\right) e^{3 t} \sin{\left(2 t_{0} \right)} + 2 \sqrt{2} e^{3 t_{0}} \cos{\left(2 t_{0} \right)} \cos{\left(2 t + \frac{\pi}{4} \right)}\right) \left(\left(\sin^{2}{\left(2 t \right)} - 1\right) e^{3 t_{0}} \sin{\left(2 t_{0} \right)} + \frac{\left(\sin{\left(4 t - 2 t_{0} \right)} + \sin{\left(4 t + 2 t_{0} \right)}\right) e^{3 t}}{4}\right) \cos{\left(2 t_{0} \right)}\right) e^{- 3 t - 3 t_{0}}}{\cos{\left(2 t \right)} \cos{\left(2 t_{0} \right)}} & \frac{\left(\left(\left(2 \sin{\left(2 t \right)} - \cos{\left(2 t \right)}\right) \left(\sin^{2}{\left(2 t_{0} \right)} - 1\right) e^{3 t} + \frac{\sqrt{2} \left(- \sin{\left(2 t - 4 t_{0} + \frac{\pi}{4} \right)} + \sin{\left(2 t + 4 t_{0} + \frac{\pi}{4} \right)}\right) e^{3 t_{0}}}{2}\right) \left(e^{3 t} \sin{\left(2 t_{0} \right)} \cos{\left(2 t \right)} - e^{3 t_{0}} \sin{\left(2 t \right)} \cos{\left(2 t_{0} \right)}\right) + \left(\left(2 \sin{\left(2 t \right)} - \cos{\left(2 t \right)}\right) e^{3 t} \sin{\left(2 t_{0} \right)} + 2 \sqrt{2} e^{3 t_{0}} \cos{\left(2 t_{0} \right)} \cos{\left(2 t + \frac{\pi}{4} \right)}\right) \left(e^{3 t} \cos{\left(2 t \right)} \cos{\left(2 t_{0} \right)} + e^{3 t_{0}} \sin{\left(2 t \right)} \sin{\left(2 t_{0} \right)}\right) \cos{\left(2 t_{0} \right)}\right) e^{- 3 t - 3 t_{0}}}{\cos{\left(2 t_{0} \right)}}\\frac{\left(- \left(\left(\sin{\left(2 t \right)} + 2 \cos{\left(2 t \right)}\right) \left(\sin^{2}{\left(2 t_{0} \right)} - 1\right) e^{3 t} - \frac{\sqrt{2} \left(\cos{\left(2 t - 4 t_{0} + \frac{\pi}{4} \right)} - \cos{\left(2 t + 4 t_{0} + \frac{\pi}{4} \right)}\right) e^{3 t_{0}}}{2}\right) \left(\left(\sin^{2}{\left(2 t \right)} - 1\right) e^{3 t_{0}} \cos{\left(2 t_{0} \right)} - \frac{\left(\cos{\left(4 t - 2 t_{0} \right)} - \cos{\left(4 t + 2 t_{0} \right)}\right) e^{3 t}}{4}\right) + \left(\left(\sin{\left(2 t \right)} + 2 \cos{\left(2 t \right)}\right) e^{3 t} \sin{\left(2 t_{0} \right)} - 2 \sqrt{2} e^{3 t_{0}} \sin{\left(2 t + \frac{\pi}{4} \right)} \cos{\left(2 t_{0} \right)}\right) \left(\left(\sin^{2}{\left(2 t \right)} - 1\right) e^{3 t_{0}} \sin{\left(2 t_{0} \right)} + \frac{\left(\sin{\left(4 t - 2 t_{0} \right)} + \sin{\left(4 t + 2 t_{0} \right)}\right) e^{3 t}}{4}\right) \cos{\left(2 t_{0} \right)}\right) e^{- 3 t - 3 t_{0}}}{\cos{\left(2 t \right)} \cos{\left(2 t_{0} \right)}} & \frac{\left(\left(\left(\sin{\left(2 t \right)} + 2 \cos{\left(2 t \right)}\right) \left(\sin^{2}{\left(2 t_{0} \right)} - 1\right) e^{3 t} - \frac{\sqrt{2} \left(\cos{\left(2 t - 4 t_{0} + \frac{\pi}{4} \right)} - \cos{\left(2 t + 4 t_{0} + \frac{\pi}{4} \right)}\right) e^{3 t_{0}}}{2}\right) \left(e^{3 t} \sin{\left(2 t_{0} \right)} \cos{\left(2 t \right)} - e^{3 t_{0}} \sin{\left(2 t \right)} \cos{\left(2 t_{0} \right)}\right) + \left(\left(\sin{\left(2 t \right)} + 2 \cos{\left(2 t \right)}\right) e^{3 t} \sin{\left(2 t_{0} \right)} - 2 \sqrt{2} e^{3 t_{0}} \sin{\left(2 t + \frac{\pi}{4} \right)} \cos{\left(2 t_{0} \right)}\right) \left(e^{3 t} \cos{\left(2 t \right)} \cos{\left(2 t_{0} \right)} + e^{3 t_{0}} \sin{\left(2 t \right)} \sin{\left(2 t_{0} \right)}\right) \cos{\left(2 t_{0} \right)}\right) e^{- 3 t - 3 t_{0}}}{\cos{\left(2 t_{0} \right)}}\end{matrix}\right]$$

It’s rediculous!

### Matlab

S = simplify(expr) performs algebraic simplification of expr. If expr is a symbolic vector or matrix, this function simplifies each element of expr. Link https://www.mathworks.com/help/symbolic/simplify.html

This function works very well and much faster than Sympy.

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