Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part19

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part19

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 5 7 Copyright © Orchard Publications Solution of Single State Equations We will now prove that the solution of the first state equation in (5.22) is (5.24) Proof: First, we must show that (5.24) satisfies the initial condition of (5.23). This is done by substitu- tion of in (5.24). Then, (5.25) The first term in the right side of (5.25) reduces to since (5.26) The second term of (5.25) is zero since the upper and lower limits of integration are the same. Therefore, (5.25) reduces to and thus the initial condition is satisfied. Next, we must prove that (5.24) satisfies also the first equation in (5.22). To prove this, we dif- ferentiate (5.24) with respect to and we obtain or or (5.27) We observe that the bracketed terms of (5.27) are the same as the right side of the assumed solu- tion of (5.24). Therefore, and this is the same as the first equation of (5.22). In summary, if and are scalar constants, the solution of (5.28) xt () e α tt 0 x 0 e α t e ατ β u τ ()τ d t 0 t + = 0 = 0 e α t 0 t 0 x 0 e α t e α τ β u τ d t 0 t 0 + = x 0 e α t 0 t 0 x 0 e 0 x 0 x 0 == 0 x 0 = t x · t d dt ---- e α 0 x 0 d α t e β u τ d t 0 t    + = x · t α e α 0 x 0 α e α t e β u τ e α t e β u τ [] τ t = + d t 0 t + = α e α 0 x 0 e α t e β u τ d t 0 t + e α t e α t β ut + = x · t α e α 0 x 0 e α t τ β u τ d t 0 t + β + = x · α x β u + = αβ x · α x β u + =
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Chapter 5 State Variables and State Equations 5 8 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications with initial condition (5.29) is obtained from the relation (5.30) Example 5.5 Use (5.28) through (5.30) to find the capacitor voltage of the circuit of Figure 5.6 for , given that the initial condition is Figure 5.6. Circuit for Example 5.5 Solution: From (5.20) of Example 5.3, Page 5 5, and by comparison with (5.28), and Then, from (5.30), or (5.31) Assuming that the output is the capacitor voltage, the output state equation is x 0 xt 0 () = e α tt 0 x 0 e α t e α τ β u τ ()τ d t 0 t + = v C t t0 > v C 0 1 V = + + 0.5 F R 2u 0 t v C t 2 C x · 1 RC -------x v S u 0 t + = α 1 ------- 1 20 . 5 × ---------------- 1 == = β 2 = e α 0 x 0 e α t e α τ β u τ d t 0 t + e 1 1e t e τ τ d 0 t + e t 2e t e τ τ d 0 t + e t t e τ [] 0 t + e t t e t 1 + = v C t t u 0 t y
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 5 9 Copyright © Orchard Publications The State Transition Matrix (5.32) 5.3 The State Transition Matrix In Section 5.1, relation (5.14), we defined the state equations pair (5.33)
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This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part19

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