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lec03_print - MAE143b Linear Control Theory and...

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MAE143b Linear Control - Theory and Applications Lecture 3, Tuesday Aug. 11, 2:00-4:50pm Prof. R.A. de Callafon [email protected] CONTENTS OF THIS LECTURE More on state space models ( section 7.1-7.3, textbook ) – state space model transfer function model – block diagrams of transfer function and state space models – connection of state space models Examples Functions in matlab (with demo) Dynamic Response ( section 3.1-3.2 & 7.4, textbook ) – time response using State Space models – time response using Transfer Function models MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 3, Page 1 Dynamic models - state space transfer function Recall: state space model ˙ x ( t )= Fx ( t )+ Gu ( t ) y ( t Hx ( t Du ( t ) Conversion to transfer function G ( s ) via Laplace transform: sx ( s ( s Gu ( s ) y ( s ( s ( s ) So we can solve ±rst equation for x ( s ): ( sI F ) x ( s Gu ( s ) x ( s )=( sI F ) 1 Gu ( s ) where I = identity matrix. Now substitute solution into second equation: y ( s ( s ( s ) y ( s [ H ( sI F ) 1 G + D ] | {z } G ( s ) u ( s ) Important formulae: G ( s )=[ H ( sI F ) 1 G + D ] MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 3, Page 2
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Dynamic models - state space transfer function example Example with two-dimensional state vector: m F y d k State space model with x 1 ( t )= y ( t ) and x 2 ( t )=˙ y ( t ) : " ˙ x 1 ( t ) ˙ x 2 ( t ) # = " 01 k m d m # " x 1 ( t ) x 2 ( t ) # + " 0 1 m # F ( t ) y ( t h 10 i " x 1 ( t ) x 2 ( t ) # + h 0 i F ( t ) Computation of ( sI F ) 1 : ( sI F ) 1 = " s 0 0 s # " k m d m # 1 = " s 1 k m s + d m # 1 MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 3, Page 3 Dynamic models - state space transfer function example Using the simple inversion rule for 2 × 2 matrices: A = " a 1 a 2 a 3 a 4 # ,A 1 = 1 det ( A ) " a 4 a 2 a 3 a 1 # we can compute " s 1 k m s + d m # 1 = 1 s 2 + d m s + k m " s + d m 1 k m s # As a result G ( s )=[ H ( sI F ) 1 G + D ] now becomes h i | {z } H 1 s 2 + d m s + k m " s + d m 1 k m s # | {z } ( sI F ) 1 " 0 1 m # | {z } G +0 = 1 m s 2 + d m s + k m The transfer function G ( s 1 m s 2 + d m s + k m is what we found before for the mass/spring/damper system. MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 3, Page 4
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Dynamic models - transfer function state space From transfer function to state space is easy for systems with 1 input and 1 output. Outline of procedure: 1. Covert transfer function to corresponding n th order ODE 2. Choose state variables x ( t ) according to the n derivatives occuring in the ODE 3. Derive the state space model from the ODE MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 3, Page 5 Dynamic models - transfer function state space example Simple example - 1st order transfer function: G ( s )= g s f With y ( s G ( s ) u ( s )wehave sy ( s ) fy ( s gu ( s ) Now choose (only 1 state needed!) x ( s y ( s )then sx ( s fx ( s )+ ( s ) which completes the state space realization: ˙ x ( t ( t ( t ) y ( t hx ( t du ( t ) where h =1and d =0 MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 3, Page 6
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Dynamic models - transfer function state space example 2nd order transfer function: G ( s )= 1 m s 2 + d m s + k m .
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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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lec03_print - MAE143b Linear Control Theory and...

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