Closed-Loop Transfer Functions 1. Introduction 2. Stirred tank heating system 3. Closed-loop block diagrams 4. Closed-loop transfer functions 5. Simulink example Introduction  Block diagrams » 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 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 currentpressure converter (I/P) to drive valve Blending Process Model  Mass balances for constant volume 0  w1  w2  w  d ( Vx )  w1 x1  w2 x2  wx  dt  w  w1  w2 dx w1 ( x1  x)  w2 ( x2  x)   f ( x, x1 , w2 ) dt V Linearized model dx'  w x '  w1 x1'  (1  x ) w2'  dt V  Transfer function model w1 w (1  x ) w ' K1 K2 ' ' X ( s)  X 1 ( s)  W2 ( s)  X 1 ( s)  W2' ( s) V V s  1 s  1 s 1 s 1 w w ' Control System Components  Composition analyzer – assume first-order dynamics  Controller – assume PI controller  I/P converter – assume negligible dynamics Control System Components cont.   Control valve – assume first-order dynamics Entire blending system 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 Transfer Function for Setpoint Changes Y  Yu  Yd  Yu  G pU  G p Gv P  G p Gv Gc E ~ E  Ysp  Ym  K mYsp  GmY Y  G p Gv Gc E  G p Gv Gc K mYsp  GmY  K mGc Gv G p Y  Ysp 1  Gc Gv G p Gm Transfer Function for Disturbance Changes Y  Yu  Yd  G pU  Gd D  G p Gv P  Gd D  G p Gv Gc E  Gd D ~ E  Ysp  Ym  Ym  GmY Y  G p Gv Gc E  G p Gv Gc  GmY   Gd D Gd Y  D 1  Gc Gv G p Gm Simultaneous Changes  Principle of superposition K mGcGvG p Gd Y Ysp  D 1  GcGvG pGm 1  GcGvG pGm  Open-loop transfer function » Obtained by multiplying all transfer functions in feedback loop GOL  Gc Gv G p Gm Y K mGc Gv G p 1  GOL Gd Ysp  D 1  GOL General Method  Closed-loop transfer function Pf Z  Zi 1  P e » » » »  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 P f  K mGcGvG p  P e  GcGvG p Gm  GOL Disturbance change P f  Gd P e  GcGvG p Gm  GOL Closed-Loop Transfer Function Example Simulink Example 0.05 Inlet flow 0 6.37 50 PID Setpoint Km Add PID Controller 0.12 0.0103 Kip Kv y 5s+1 Add 1 Tank Level 50 Km1 >> gp=tf([6.37],[5 1]); >> gcl=gp/(1+gc*kv*gp*km) >> kv=0.0103; >> kip=0.12; >> km=50; >> gc=tf([2.5 5],[0.5 0]); Disturbance transfer function: 15.93 s^2 + 3.185 s ----------------------------------12.5 s^3 + 46.01 s^2 + 90.72 s + 16.4