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139 lines
2.3 KiB
Markdown
139 lines
2.3 KiB
Markdown
# Excersices
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Find a numerical solution to the following differential equations with the associated initial conditions. Expand the requested time horizon until the solution reaches a steady state. Show a plot of the states ($x(t)$ and/or $y(t)$). Report the final value of each state as $t \rightarrow \infty$.
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## Problem 1:
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$$\frac{dy(t)}{dt} = -y(t)+1 $$
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with,
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$$ y(0)=0$$
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This equation can be solved using the separation of variables method:
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$$y(t) = 1-e^{-t}$$
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```python
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import numpy as np
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from scipy.integrate import odeint
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import matplotlib.pyplot as plt
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# function that returns dy/dt
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def model(y,t):
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dydt = -y + 1.0
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return dydt
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# initial condition
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y0 = 0
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# time points
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t = np.linspace(0,5)
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# solve ODE
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y = odeint(model,y0,t)
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ySol = 1-np.exp(-t)
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# plot results
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plt.plot(t,y,'ok', label='ODE')
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plt.plot(t,ySol, '.:r', label='ySol')
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plt.legend()
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plt.xlabel('time')
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plt.ylabel('y(t)')
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plt.show()
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```
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# Problem 2
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$$5\frac{dy}{dt}=-y(t)+u(t)$$
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with,
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$$y(0)=1$$
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and $u$ step at $t=2$
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```python
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# function that returns dy/dt
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def model(y,t):
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# u steps from 0 to 2 at t=10
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if t<10.0:
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u = 0
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else:
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u = 2
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dydt = (-y + u)/5.0
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return dydt
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# initial condition
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y0 = 1
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# time points
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n = 40
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t = np.linspace(0,n-1,n)
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# solve ODE
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y = odeint(model,y0,t)
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# plot results
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plt.plot(t,y,'r-',label='Output (y(t))')
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plt.plot([0,10,10,40],[0,0,2,2],'b:',label='Input (u(t))')
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plt.ylabel('values')
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plt.xlabel('time')
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plt.legend(loc='best')
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plt.show()
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```
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```python
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# Define the step function u(t)
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def u(t):
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return 0 if t < 10.0 else 2 # Step from 0 to 2 at t = 10
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# function that returns dy/dt
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def model(y,t):
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dydt = (-y + u(t))/5.0
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return dydt
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# initial condition
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y0 = 1
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# time points
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n = 20
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t = np.linspace(0,n-1,n)
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# solve ODE
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y = odeint(model,y0,t)
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# Compute u(t) for all values in t
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u_values = np.array([u(ti) for ti in t]) # Evaluate u at each time point
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# plot results
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plt.plot(t,y,'.:r',label='Output (y(t))')
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plt.plot(t, u_values, 'g-', linewidth=2, label="u(t)")
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plt.axvline(x=10, color='r', linestyle='--', label="Step at t=10")
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plt.ylabel('values')
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plt.xlabel('time')
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plt.legend(loc='best')
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plt.show()
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```
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```python
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```
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