Class 7: Block Diagrams
Dynamic Behavior and Stability of Closed-Loop Control Systems We no ant to consider the dynamic behavior of processes that are operated using feedback control. The combination of the process, the feedback controller, and the instrumentation is referred to as a feedback control loop or a closed-loop system. Block Diagram Representation To illustrate the development of a block diagram, e return to a previous eample, the stirred-tank blending process considered in earlier chapters.
Composition control system for a stirred-tank blending process (Fig..) (V,, and assumed to be constant) CT CC Controlled variable: Measured variable: Manipulated variable: Disturbance variable: Outlet concentration () Outlet concentration () Flo rate ( ) Inlet concentration ( ) 3
4 Process In section 4.3 the approimate dynamic model of a stirred-tank blending system as developed: dt d V ) ( ρ ) ( ) ( f f f f ss ss ss variables
5 here s W s X s X s V dt d V ) ( ρ ρ Combining partial fractions and deviation variables, s W s K s X s K s X τ τ K and K V,, ρ τ
Disturbance Controlled Variable Manipulated Variable Process Figure. Block diagram of the process. 6
Wanted: Transfer function for each piece of equipment - Please try to label variables, then transfer functions 7
Standard Labels D Y sp Y ~ sp - E P U Y u Y d Y Y m 8
Definitions Y controlled variable U manipulated variable D P disturbance variable (also referred to as load variable) controller output E error signal Y m Y sp measured value of Y set point Y u Y d G c G v G p G d G m K m change in Y due to U change in Y due to D controller transfer function transfer function for final control element (including K IP, if required) process transfer function disturbance transfer function transfer function for measuring element and transmitter steady-state gain for G m Y % sp internal set point (used by the controller) 9
Transfer Functions D G d Y sp Y ~ sp E P U Y u K m G c G v G P - Y d Y Y m G m 0
No back to our problem (Blending Tank) Need transfer functions for: CT CC
Modified Block Diagram D G d Y sp Y ~ sp E P U Y u K m G c G ip G v G P - P t Y d Y Y m G m Notes:. All variables are in deviation variables ecept E. All variables are in Laplace coordinates (i.e., Y (s)) 3. Pink boes need transfer functions
Composition Sensor-Transmitter (Analyzer) We assume that the dynamic behavior of the composition sensortransmitter can be approimated by a first-order transfer function: X m ( s) Km G m (-3) X s τ s Controller Suppose that an electronic proportional plus integral controller is used. From Chapter 8, the controller transfer function is G c m P s Kc (-4) E s τi s here P s and E(s) are the Laplace transforms of the controller output p ( t) and the error signal e(t). Note that p and e are electrical signals that have units of ma, hile K c is dimensionless. The error signal is epressed as 3
% (-5) e t t t or after taking Laplace transforms, sp m % (-6) E s X s X s sp The symbol % sp t denotes the internal set-point composition epressed as an equivalent electrical current signal. This signal is used internally by the controller. % sp ( t) is related to the actual composition set point sp ( t) by the composition sensortransmitter gain K m : m Thus K m sp (-7) % t K t X% sp X sp ( s) ( s) m sp K m (-8) 4
Current-to-Pressure (I/P) Transducer Because transducers are usually designed to have linear characteristics and negligible (fast) dynamics, e assume that the transducer transfer function merely consists of a steady-state gain K IP : Pt ( s) G ip KIP (-9) P s Control Valve As discussed in Section 9., control valves are usually designed so that the flo rate through the valve is a nearly linear function of the signal to the valve actuator. Therefore, a first-order transfer function usually provides an adequate model for operation of an installed valve in the vicinity of a nominal steady state. Thus, e assume that the control valve can be modeled as G v W s Kv P s τ s t v (-0) 5
Block diagram for the entire blending process composition control system (Fig.7) Done! 6
What about PID ith derivative on measurement? P P t c τ I 0 ( t) K e( t) e( t) dy dt ( s) K E( s) K τ sy ( s) c τ Is K c C τ D G (s) G (s) D m m ( s) Y ~ sp Ym E Y sp Y ~ sp E K m G - P G Y m 7
PID /Derivative on Measurement D G d Y sp Y ~ sp E P U Y u K m G PI G ip G v G P - P t Y d Y G deriv Y m G m 8
Your Homeork Problem. 9
Manipulated variable Problem. Disturbance Want to control c, not c TL c CT CC. First identify the controlled variable, manipulated variable, and disturbance variable. 0
Prob..
Problem. Want to control c, not c TL c CT Set point CC Derivative on measurement. Dra a block diagram similar to the one e did in class. My diagram has 9 boes for transfer functions, including the unit conversion on the set point (K m ). Also, you ill have a transfer function G TL for the transport delay in the transfer line. The time delay bo should be in the feedback loop, since it only represents the delay in measurement.