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Closed-Loop Transfer Functions

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Presentation on theme: "Closed-Loop Transfer Functions"— Presentation transcript:

1 Closed-Loop Transfer Functions
Introduction Stirred tank heating system Closed-loop block diagrams Closed-loop transfer functions Simulink example

2 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

3 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

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

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

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

7 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

8 Transfer Function for Setpoint Changes

9 Transfer Function for Disturbance Changes

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

11 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

12 Closed-Loop Transfer Function Example

13 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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