W2 - 3 2 SYSTEMS AND SIGNALS JAN 6, 2010 SYSTEMS INPUT...

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3 2 SYSTEMS AND SIGNALS JAN 6, 2010 SYSTEMS INPUT OUTPUT DESCRIPTION CLASSIFICATION TIME INVARIANT SYSTEMS TIME VARYING SYSTEMS LINEAR SYSTEMS Printed by Mathematica for Students
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INPUT - OUTPUT RELATION STIMULUS - RESPONSE RELATION The Dynamic relation - ship of output and input. Examples First Order Linear TimeVarying System : a 1 H t L dv H t L dt + a 0 H t L v H t L = b 1 H t L u H t L , t > t 0 What do we want to know ? Can we determine the output v H t L at time t > t 0 , knowing our input u H s L , t 0 < s < t? For this we need to "solve" the First Order Time Varying Ordinary Differential Equation in one dimension as a function of t. To begin with we consider a simpler form : we assume the coefficients a 0 H t L , a 1 H t L do not depend on t : we specialise to the TimeInvariant case. Then we have a 1 dv H t L dt + a 0 v H t L = b 1 u H t L , t > t 0 t 0 being the ' start time ' for your input Assume a 1 NOT zero . If it is , the order of the equation is zero and the solution is immediate : v H t L = b 1 a 0 u H t L and of course if a 0 is also zero, the system is undefined ! Dividing by a 1 we can write it in the form : dv H t L dt + Α v H t L = u H t L , t > t 0 absorbing the constant b 1 a 0 into the input. ~~ Book p .3 For Electrical Example ~~ So we proceed to ' solve ' this equation : Multiply both sides by the non - zero ' Integrating Factor ' e Α t : e ± 0 t Α ± s and obtain : e Α t dv H t L dt + e Α t Α v H t L = e Α t u H t L t > t 0 2 W2.nb Printed by Mathematica for Students
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Observe that the leftside can be expressed d dt I e Α t v H t LM . Hence we have d dt I e Α t v H t LM = e Α t dv H t L dt + e Α t Α v H t L = e Α t u H t L , t > t 0 Note that if necessary we can rewrite this with another variable : d ds H e Α s v H s LL = e Α s u H s L , s > t 0 if it makes it any easier for you to : Integrate both sides from t 0 to t : ± t 0 t d ds H e Α s v H s LL ± s = ± t 0 t e Α s u H s L ± s The left side = e Α t v H t L - e Α t 0 v H t 0 L Hence e Α t v H t L = e Α t 0 v H t 0 L + ± t 0 t e Α s u H s L ± s Multiplying both sides by e t we get v H t L = e Α t 0 e t v H t 0 L + e t ± t 0 t e Α s u H s L ± s which we can rewrite as : v H t L = e H t - t 0 L v H t 0 L + ± t 0 t e H t - s L u H s L ± s, t > t 0 And equivalently as : v H Τ + t 0 L = e -ΑΤ v H t 0 L + ± t 0 Τ+ t 0 e H Τ+ t 0 - s L u H s L ± s, Τ > 0 H This can also be obtained by ' method of variation of parameters ' L Hence we can make the important conclusion : W2.nb 3 Printed by Mathematica for Students
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the response at time t 0 + Τ can be broken down as the response which depends on v H t 0 L Plus the response to the input in the time interval from t 0 to t 0 + Τ . We may call t 0 the initial time and v H t 0 L the ' initial condition ' - or initial STATE - More on the latter later - but here let us note first that the response is NOT determined unless you know the initial state. The output is not determined by the Input alone ! It depends on the state of the system when you started your input. The initial state represents the ' past ' of the system. We will revisit this important conceptual result for Linear Systems many times in the future
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This note was uploaded on 09/03/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.

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W2 - 3 2 SYSTEMS AND SIGNALS JAN 6, 2010 SYSTEMS INPUT...

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