Fall 2009/JonathanS - Revision history

Jonathan.schreven: /* Asymptotes & Break Points */

2009-12-10T02:17:14Z

Asymptotes & Break Points

← Older revision		Revision as of 19:17, 9 December 2009
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	From our original input we know that <math>\omega=3</math> so		From our original input we know that <math>\omega=3</math> so
	:<math>H(j\omega)={1 \over {\sqrt{3^2+1}\sqrt{3^2-1}}}</math>		:<math>H(j\omega)={1 \over {\sqrt{3^2+1}\sqrt{3^2-1}}}</math>
	:<math>H(j\omega)=0.1118</math>		:<math>H(j\omega)=0.1118\,</math>
	Now we can plug that into our log function		Now we can plug that into our log function
	:<math>\begin{alignat}{3}		:<math>\begin{alignat}{3}
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	We get asymptotes at s=3j and s=4.47j from the denominator. For our purposes we can just ignore the imaginary part. We should also have increases of 20db/decade at s=0 and s=3.01 which we find from the numerator.		We get asymptotes at s=3j and s=4.47j from the denominator. For our purposes we can just ignore the imaginary part. We should also have increases of 20db/decade at s=0 and s=3.01 which we find from the numerator.


	== Convolution ==		== Convolution ==

2009-12-10T01:12:22Z

2009-12-09T23:07:13Z

Solving with Matrix Exponentials

← Older revision		Revision as of 16:07, 9 December 2009
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	1		1
	\end{bmatrix}cos(3\theta)</math>		\end{bmatrix}cos(3\theta)</math>

	~~== Solving with Matrix Exponentials ==~~

2009-12-09T23:07:04Z

State Space Equations

← Older revision		Revision as of 16:07, 9 December 2009
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	== State Space Equations ==		== State Space Equations ==

	Now we can transfer the functions we already have into state space. This can make ~~them~~ easier to work with.		Now we can transfer the functions we already have into state space. This is simply taking our original differential equation and putting it in terms of vectors and matrices. This can make the equations easier to work with.

	<math>\begin{bmatrix}		<math>\begin{bmatrix}

2009-11-30T06:36:15Z

State Space Equations

2009-11-30T06:34:28Z

← Older revision		Revision as of 23:34, 29 November 2009
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	Now we can transfer the functions we already have into state space. This can make them easier to work with.		Now we can transfer the functions we already have into state space. This can make them easier to work with.

			<math>\begin{bmatrix}
			\dot{\theta} \\
			\ddot{\theta}
			\end{bmatrix}
			=
			\begin{bmatrix}
			1 & 0 \\
			{-g\over \ell} & 0
			\end{bmatrix}

			\begin{bmatrix}
			\theta \\
			\dot{\theta}
			\end{bmatrix}
			+
			\begin{bmatrix}
			0 \\
			1
			\end{bmatrix}cos(3\theta)</math>

	== Solving with Matrix Exponentials ==		== Solving with Matrix Exponentials ==

2009-11-30T04:44:49Z

← Older revision		Revision as of 21:44, 29 November 2009
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	Convolution, by definition, states that H(s)=Y(s)/X(s) where H(s) is the transfer function, Y(s) is the output function, and X(s) is the input function. We would like to find the output of the system, so for our purposes we will rearrange the above equation like this.		Convolution, by definition, states that H(s)=Y(s)/X(s) where H(s) is the transfer function, Y(s) is the output function, and X(s) is the input function. We would like to find the output of the system, so for our purposes we will rearrange the above equation like this.
	<math>Y(s)=H(s)X(s)</math>		:<math>\begin{alignat}{3}
			Y(s) & = H(s)X(s) \\
			& = \bigg({12s^3+109s\over {s^4+28.62s^2+176.58}}\bigg)\bigg({s\over {s^2+9}}\bigg) \\
			& = {2s^2(12s^2+109)\over {(s^2+9)^2(2s^2+39.24)}} \\
			\end{alignat}</math>

			<math></math>

	== State Space Equations ==		== State Space Equations ==

2009-11-30T04:24:05Z

Convolution

← Older revision		Revision as of 21:24, 29 November 2009
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	== Convolution ==		== Convolution ==

	~~Now we will use convolution~~ to find the output of our ~~system~~.		Convolution, by definition, states that H(s)=Y(s)/X(s) where H(s) is the transfer function, Y(s) is the output function, and X(s) is the input function. We would like to find the output of the system, so for our purposes we will rearrange the above equation like this.
			<math>Y(s)=H(s)X(s)</math>

	== State Space Equations ==		== State Space Equations ==

2009-11-30T04:13:46Z

State Space Equations

← Older revision		Revision as of 21:13, 29 November 2009
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	Now we can transfer the functions we already have into state space. This can make them easier to work with.		Now we can transfer the functions we already have into state space. This can make them easier to work with.

			== Solving with Matrix Exponentials ==

2009-11-30T03:57:44Z

Convolution

← Older revision		Revision as of 20:57, 29 November 2009
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	Now we will use convolution to find the output of our system.		Now we will use convolution to find the output of our system.

			== State Space Equations ==

			Now we can transfer the functions we already have into state space. This can make them easier to work with.