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## Chapter 8

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**Feedback Controllers**Chapter 8**On-off Controllers**• Simple • Cheap • Used In residential heating and domestic refrigerators • Limited use in process control due to continuous cycling of controlled variable excessive wear on control valve. • Examples • Batch process control (PLC = programmable logic controller) • Solenoid in home heating unit • Sprinkler systems • Cruise control? Chapter 8**On-Off Controllers**Synonyms: “two-position” or “bang-bang” controllers. e = error = set point – measured variable Chapter 8 Controller output has two possible values.**Practical case (dead band)**Chapter 8 δ = tolerance system never reaches steady-state**Three Mode (PID) Controller**• Proportional • Integral • Derivative • Proportional Control • Define an error signal, e, by e = Ysp – Ym • where • Ysp = set point • Ym = measured value of the controlled variable • (or equivalent signal from transmitter) Chapter 8**Since signals are time varying,**• e(t) = Ysp(t) - Ym(t) • n.b. Watch units!! • For proportional control: • where, • p(t) = controller output • = bias value (adjustable) • Kc = controller gain (dimensionless, adjustable) Chapter 8**Figures 8.4, 8.5**in Text Chapter 8 Standards (ISO/ISA) 3 – 15 psi 4 - 20 ma 0 – 10 VDC**Proportional Band, PB**• Reverse or Direct Acting Controller • Kccan be made positive or negative • Recall for proportional FB control: • or • Direct-Acting (Kc < 0) • “output increases as input increases" • p(t) Ym(t) • Reverse-Acting(Kc > 0) • “output increases as input decreases" Chapter 8**Example 2:Flow Control Loop**Chapter 8 • Assume FT is direct-acting. Select sign of Kc so that KcKv > 0 • 1.) Air-to-open (fail close) valve ==> ? • 2.) Air-to-close (fail open) valve ==> ? • Consequences of wrong controller action??**Transfer Function for Proportional Control:**• Let • Then controller input/output relation can written as • Take Laplace transform of each side, • or • INTEGRAL CONTROL ACTION • Synonyms: "reset", "floating control" • I reset time (or integral time) - adjustable Chapter 8**Proportional-Integral (PI) Control**integral provides memory of e most popular controller • Response to unit step change in e: Chapter 8**Integral action eliminates steady-state error**• (i.e., offset) Why??? e 0 p is changing with • time until e = 0, where p reaches steady state. • Transfer function for PI control Chapter 8**Some controllers are calibrated in 1/I**("repeats per minute") instead of I . • For PI controllers, is not adjustable. • Derivative Control Action • Ideal derivative action • Used to improve dynamic response of the controlled variable • Derivative kick (use -dym/dt ) • Use alone? Chapter 8**Proportional-Integral-Derivative (PID) Control**Now we consider the combination of the proportional, integral, and derivative control modes as a PID controller. • Many variations of PID control are used in practice. • Next, we consider the three most common forms. Parallel Form of PID Control The parallelform of the PID control algorithm (without a derivative filter) is given by Chapter 8**The corresponding transfer function is:**Chapter 8**Expanded Form of PID Control**In addition to the well-known series and parallel forms, the expanded form of PID control in Eq. 8-16 is sometimes used: Features of PID Controllers Chapter 8 Elimination of Derivative and Proportional Kick • One disadvantage of the previous PID controllers is that a sudden change in set point (and hence the error, e) will cause the derivative term momentarily to become very large and thus provide a derivative kick to the final control element.**Automatic and Manual Control Modes**• Automatic Mode • Controller output, p(t), depends on e(t), controller • constants, and type of controller used. • ( PI vs. PID etc.) • Manual Mode • Controller output, p(t), is adjusted manually. • Manual Mode is very useful when unusual conditions exist: • plant start-up • plant shut-down • emergencies • Percentage of controllers "on manual” ?? • (30% in 2001, Honeywell survey) Chapter 8**Digital PID Controller**• finite difference approximation • where, • = the sampling period (the time between • successive samples of the controlled variable) • = controller output at the nth sampling • instant, n=1,2,… • = error at the nth sampling unit • velocity form - see Equation (8-19) • (pn)- incremental change I D Chapter 8**Typical Response of Feedback Control Systems**• Consider response of a controlled system after a • sustained disturbance occurs (e.g., step change in • disturbance variable); y > 0 is off-spec. Chapter 8**Figure 8.13 Proportional control: effect of Controller gain**Chapter 8 Figure 8.15 PID control: effect of derivative time**Chapter 8**Figure 8.14 Proportional-integral control: (a) effect of integral time, (b) effect of controller gain integral action ~**Summary of the Characteristics of the Most Commonly Used**Controller Modes • 1. Two Position: • Inexpensive. • Extremely simple. • 2. Proportional: • Simple. • Inherently stable when properly tuned. • Easy to tune. • Experiences offset at steady state. (OK for level control) • 3. Proportional plus integral: • No offset. • Better dynamic response than reset alone. • Possibilities exist for instability due to lag • introduced. Chapter 8**4. Proportional plus derivative:**• Stable. • Less offset than proportional alone (use of • higher gain possible). • Reduces lags, i.e., more rapid response. • 5. Proportional plus integral plus derivative: • Most complex • Rapid response • No offset. • Best control if properly tuned. Chapter 8**Example 3:Liquid Level Control**• Control valves are air-to-open • Level transmitters are direct acting Chapter 8**Question:**• 1. Type of controller action? Select Kc so that • air-to-open valve: sign of Kv? • sign of process gain? Chapter 8**Chapter 8**Previous chapter Next chapter