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Lesson_28

Lesson_28 - Lesson28 Challenge27 Challenge28 Lesson 28...

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Lesson 28 Lesson 28 Challenge 27 Lesson 28  -  State variable models Challenge 28

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Lesson 28 Challenge 27 134kHz  If H(s)=(s-a)/((s-a) 2 ϖ 0 2 ), what is the ODE?
Lesson 28 Challenge 27 From           (s 2  -2as +(a 2 + ϖ o 2 ))V 0 (s) = (s-a)V(s) it must follow that: d 2 v 0 (t)/dt 2  -2a dv 0 (t)/dt + (a 2 + ϖ o 2 )v 0 (t) = dv(t)/dt – av(t) Suppose, for illustrative purposes, a=0.02,  ϖ 0 =10, then   H(s) = (s-.01)/(s 2  – 0.02s + (.01 2  +10 2 = (s-.01)/(s 2  – 0.02s + (100.0001)) » [H,w] = freqs([1,-.01],[1, 0.02, 100.0001]); » plot(w,abs(H)) » plot(w,360*angle(H)/(2*pi))

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Lesson 28 Challenge 27 90 ° -90 °
Lesson 28 Lesson 28 Have previously introduced transfer functions. H(s) and H(z) What are their limitations? State variables are information variables.   They reside at locations where the  system stores or saves information. Analog locations? Discrete-time locations?

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Lesson 28 Lesson 28 Core idea: The future states of a state-determined system can be computed if, all past state values are known, and the mathematical relationship (sometimes called the bonds on  interaction) between the state variables are known.  Current States Past States State Model System Inputs Future States Memory
Lesson 28 Lesson 28 The general form of a single-input single-output continuous-time  n -state state  model ( n  information storage locations) is: ( 29 Equation Output t dv t c t y Equation tate S t v t dt t d T ) ( ) ( ; ) 0 ( ); ( ) ( ) ( 0 + = = + = x x x b Ax x x 0 A b x (t) c T d y(t) v(t) x(t) an  n -vector of state variables

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Lesson 28 Lesson 28 Example:  RLC circuit The system states are the inductor current and capacitor voltage.   R L C v(t) i L v C ) ( ) ( 1 ) ( ) ( ) ( ) ( 0 t v dt t i C t v t v t Ri dt t di L C t L C L L = = + +
Lesson 28 Lesson 28 Define x 1 (t)=i L (t) and x 2 (t)=v C (t).

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Lesson_28 - Lesson28 Challenge27 Challenge28 Lesson 28...

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