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Published byLexi Siford Modified over 2 years ago

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Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example

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Introduction l Block diagrams »Convenient tool to represent closed-loop systems »Also used to represent control systems in Simulink l Closed-loop transfer functions »Transfer function between any two signals in a closed-loop system »Usually involve setpoint or disturbance as the closed-loop input and the controlled output as the closed-loop output »Conveniently derived from block diagram »Can be derived automatically in Simulink »Used to analyze closed-loop stability and compute closed-loop responses

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Stirred Tank Blending System l Control objective »Drive outlet composition (x) to setpoint (x sp ) by manipulating pure stream flow rate (w 2 ) despite disturbances in flow rate (w 1 ) and composition (x 1 ) of other feed stream l Control system »Measure x with composition analyzer (AT) »Perform calculation with composition controller (AC) »Convert controller output to pneumatic signal with current- pressure converter (I/P) to drive valve

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Blending Process Model l Mass balances for constant volume l Linearized model l Transfer function model

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Control System Components l Composition analyzer – assume first-order dynamics l Controller – assume PI controller l I/P converter – assume negligible dynamics

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Control System Components cont. l Control valve – assume first-order dynamics l Entire blending system

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Closed-Loop Block Diagrams l G p (s) – process transfer function l G d (s) – disturbance transfer function l G v (s) – valve transfer function l G c (s) – controller transfer function l G m (s) – measurement transfer function l K m – measurement gain l Y(s) – controlled output l U(s) – manipulated input l D(s) – disturbance input l P(s) – controller output l E(s) – error signal l Y sp (s) – setpoint l Y m (s) – measurement

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Transfer Function for Setpoint Changes

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Transfer Function for Disturbance Changes

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Simultaneous Changes l Principle of superposition l Open-loop transfer function »Obtained by multiplying all transfer functions in feedback loop

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General Method l Closed-loop transfer function »Z = any variable in feedback system »Z i = any input variable in feedback system Z and Z i » f = product of all transfer functions between Z and Z i » e = product of all transfer functions in feedback loop l Setpoint change l Disturbance change

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Closed-Loop Transfer Function Example

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Simulink Example >> gp=tf([6.37],[5 1]); >> kv=0.0103; >> kip=0.12; >> km=50; >> gc=tf([2.5 5],[0.5 0]); >> gcl=gp/(1+gc*kv*gp*km) Disturbance transfer function: s^ s s^ s^ s

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