User:Egm6321.f2010.team5.riveros/hw3p6

= Problem 6 - Obtaining SC-L1-ODE-CC with int. factor method = From Meeting 15, p. 15-1

Given
We are given a system of coupled linear first order ODEs with constant coefficients,
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A part of the solution to this system is the state transition matrix, $$\underline{\Phi}$$, which has the properties
 * style="width:95%" | $$\displaystyle \underline{\dot{x}}(t)=\underline{A}\underline{x}(t)+\underline{B}\underline{u}(t)$$
 * (6.1)
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and
 * style="width:95%" | $$\displaystyle \frac{d}{dt}\underline{\phi}(t,t_0)=\underline{A}\underline{\phi}(t,t_0)$$
 * (6.2)
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where $$I$$ is the identity matrix.
 * style="width:95%" | $$\displaystyle \underline{\phi}(t_0,t_0)=I,$$
 * (6.3)
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Find
Use equations (6.1), (6.2), and (6.3), to obtain the following equation:
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$$\displaystyle \underline{x}(t)=\underbrace{\left[{\rm exp}\underline{A}(t-t_0)\right]}_{\underline{\phi}(t,t_0)}\underline{x}(t_0)+\int\limits_{t_0}^{t}{\underbrace{\left[{\rm exp}\underline{A}(t-\tau)\right]}_{\underline{\phi}(t,\tau)}\underline{B}\underline{u}(\tau)\,d\tau} $$
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 * (6.4)
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Solution
We can rearrange equation (6.1),
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$$\displaystyle N\dot{x}(t)-M=\dot{x}(t)-Ax(t)=Bu(t)$$ (6.5) Now we substitute $$\displaystyle x(t)$$ for $$\underline{\phi}(t,t_0)$$,
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$$\displaystyle \dot{\underline{\phi}}(t,t_0)-A\underline{\phi}(t,t_0)=Bu(t)$$ (6.6) to show that it meets the first condition of exactness. The equation, however, does not satisfy the second condition of exactness. We must apply the Euler integrating factor method to make the equation exact by multiplying both sides of equation (6.5) by an integrating factor $$\displaystyle h(t)$$. The integrating factor $$\displaystyle h(t)$$ is found to be,
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$$\displaystyle h(t)=e^{-\underline{A}(t-t_0)}$$ (6.7) We multiply $$h(t)$$ to both sides of equation (6.6)
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$$\displaystyle e^{-\underline{A}(t-t_0)}\dot{\underline{\phi}}(t,t_0)-e^{-\underline{A}(t-t_0)}A\underline{\phi}(t,t_0)=e^{-\underline{A}(t-t_0)}Bu(t)$$ (6.8) Integrating both sides of equation (6.8) from $$\displaystyle t_0$$ to $$\displaystyle t$$ and solving for $$\underline{\phi}(t,t_0)$$ yeilds,
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$$\displaystyle \underline{\phi} (t,t_0)=e^{\underline{A}(t-t_0)}\underline{\phi}(t_0,t_0)+\int\limits_{t_0}^{t}e^{\underline{A}(t-\tau)}\underline{B}\underline{u}(t)d\tau $$ (6.9) Substituting $$\displaystyle x(t)$$ into $$\underline{\phi}(t,t_0)$$ yeilds equation (6.4).
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