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lec02_print - MAE143b Linear Control - Theory and...

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MAE143b Linear Control - Theory and Applications Lecture 2, Thursday Aug. 6, 2:00-4:50pm Prof. R.A. de Callafon callafon@ucsd.edu CONTENTS OF THIS LECTURE Dynamic Models ( continue chap 2, textbook ) Notion of di±erential equation & transfer function models Properties of transfer function (proper, strictly proper, series, parallel and feedback connections) Transfer function of ideal electrical components Examples: RC-netwok and DC-motor with load State space models ( section 7.1-7.2, textbook ) Examples: mass/damper spring & DC motor Properties of state space models (non-uniqueness & transfer function) MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 2, Page 1 Dynamic models - di±erential equations RECAP: Relationship between input u and output y in y = Gu for a linear dynamic system is described by a linear ordinary di±erential equation (ODE): d n d t n y ( t )+ a n 1 d n 1 d t n 1 y ( t ··· + a 1 d d t y ( t a 0 y ( t )= = b n d n d t n u ( t b n 1 d n 1 d t n 1 u ( t + b 1 d d t u ( t b 0 u ( t ) Shorthand notation of ODE: replace d n dt n by s n . Following this notation, also replace y ( t )by y ( s ) and u ( t u ( s ) : y ( s b n s n + b n 1 s n 1 + + b 1 s + b 0 s n + a n 1 s n 1 + + a 1 s + a 0 u ( s ) , or y ( s G ( s ) u ( s ) and G ( s ) is the transfer function of the dynamic system G MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 2, Page 2
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Dynamic models - transfer function properties The transfer function notation of the ODE: y ( s )= b n s n + b n 1 s n 1 + ··· + b 1 s + b 0 s n + a n 1 s n 1 + + a 1 s + a 0 u ( s ) , or y ( s G ( s ) u ( s ) is an ‘algebraic’ relation between y ( s )and u ( s ) Mathematically, this expression is found by Laplace transform of the ODE (more about this later) Transfer function G ( s ) usually follows from a higher order ( n th order) ODE shorthand notation: G ( s num( s ) den( s ) Typically, degree num( s ) degree den( s ). If degree num( s ) den( s ), G ( s )i ssa idtobe proper . If degree num( s ) < degree den( s ), G ( s )issa strictly proper . MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 2, Page 3 Dynamic models - transfer function example Example : From transfer function to ODE Consider strictly proper G ( s )w ith G ( s 1 s + τ This corresponds to the ODE: y ( s G ( s ) u ( s ) sy ( s )+ τy ( s u ( s ) d d t y ( t ( t u ( t ) y ( T Z T t =0 { u ( t ) ( t ) } d t We have knowledge of u ( t ), y ( t )upto t<T and y ( t )at t = T is found by computing the ‘surface under’ the function u ( t ) ( t ) from t = 0 till t T MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 2, Page 4
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Dynamic models - transfer function example Consider non-proper transfer function G ( s )w ith G ( s )= s + τ 1 This corresponds to the ODE: y ( s G ( s ) u ( s ) y ( s su ( s )+ τu ( s ) y ( t d d t u ( t ( t ) y ( T d d t u ( t ) t = T + ( T ) Hence y ( t )a t t = T equals u ( t )p lu s τ times the derivative of u ( t t t = T . Unfortunately, we only have knowledge of u ( t ) and y ( t )upto t<T ,sode r ivateo f u ( t )at t = T not known.
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This note was uploaded on 05/27/2010 for the course MAE 143B taught by Professor Paoc.chau during the Spring '06 term at UCSD.

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lec02_print - MAE143b Linear Control - Theory and...

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