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**Closed-Loop Transfer Functions**

Introduction Stirred tank heating system Closed-loop block diagrams Closed-loop transfer functions Simulink example

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**Introduction Block diagrams Closed-loop transfer functions**

Convenient tool to represent closed-loop systems Also used to represent control systems in Simulink 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**

Control objective Drive outlet composition (x) to setpoint (xsp) by manipulating pure stream flow rate (w2) despite disturbances in flow rate (w1) and composition (x1) of other feed stream 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**

Mass balances for constant volume Linearized model Transfer function model

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**Control System Components**

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

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**Control System Components cont.**

Control valve – assume first-order dynamics Entire blending system

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**Closed-Loop Block Diagrams**

Gp(s) – process transfer function Gd(s) – disturbance transfer function Gv(s) – valve transfer function Gc(s) – controller transfer function Gm(s) – measurement transfer function Km – measurement gain Y(s) – controlled output U(s) – manipulated input D(s) – disturbance input P(s) – controller output E(s) – error signal Ysp(s) – setpoint Ym(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 Principle of superposition**

Open-loop transfer function Obtained by multiplying all transfer functions in feedback loop

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**General Method Closed-loop transfer function Setpoint change**

Z = any variable in feedback system Zi = any input variable in feedback system Z and Zi Pf = product of all transfer functions between Z and Zi Pe = product of all transfer functions in feedback loop Setpoint change 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: 15.93 s^ s 12.5 s^ s^ s

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