inverted%20pendulum

# inverted%20pendulum - Example 1 Consider the circuit shown...

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Example 1 Consider the circuit shown in Figure 1 R 1 = 1 W L = 2H C 1 = 1F C 2 = 1F R 1 = 3 W + _ + _ Figure 1 An electric circuit with three states The three states of the circuit are defined as: ) ( ) ( ) ( ) ( ) ( ) ( 3 2 1 1 2 t i t x t v t x t v t x L c c (1) The Kirchhoff’s voltage law (KVL) around the second loop is ) ( 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 1 2 2 1 2 2 1 t x t x t x C R t x t v t v t v t x R c c (2) Therefore we have: ) ( 3 1 ) ( 3 1 ) ( 2 1 2 t x t x t x (3) The Kirchhoff’s current law (KCL) at node (1) yields ) ( ) ( ) ( ) ( ) ( 1 2 1 2 2 1 3 t x t x t x C t x C t x (4) Therefore: ) ( ) ( 3 1 ) ( 3 1 ) ( ) ( ) ( 3 2 1 2 3 1 t x t x t x t x t x t x (5) Finally writing the KVL around the first loop results in ) ( ) ( ) ( ) ( ) ( 2 2 2 3 3 1 t u t x R t x t x L t x R (6) ) ( t u ) ( t i L ) ( 1 t v c + ) ( ) ( 2 t y t v c + _ _

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2 Hence we have: ) ( 2 1 ) ( 2 1 ) ( 2 1 ) ( 2 1 ) ( 2 3 ) ( 2 1 ) ( 2 1 ) ( 3 1 2 2 3 3 t u t x t x t u t x t x t x t x (7) Equations (3), (5), and (7) can be written in matrix form as: ) ( 2 1 0 0 ) ( ) ( ) ( 2 1 0 2 1 0 3 1 3 1 1 3 1 3 1 ) ( ) ( ) ( 3 2 1 3 2 1 t u t x t x t x t x t x t x (8)
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## This note was uploaded on 11/10/2008 for the course ME 0304.870.3 taught by Professor Sohaildainat during the Fall '08 term at RIT.

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inverted%20pendulum - Example 1 Consider the circuit shown...

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