Chapter 8 Feedback Controllers Chapter 8 Onoff Controllers

Chapter 8 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?

On-Off Controllers Chapter 8 Synonyms: “two-position” or “bang-bang” controllers. e = error = set point – measured variable Controller output has two possible values.

Chapter 8 Practical case (dead band) δ = tolerance system never reaches steady-state

Chapter 8

Chapter 8 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)

Since signals are time varying, Chapter 8 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 Standards (ISO/ISA) 3 – 15 psi 4 - 20 ma 0 – 10 VDC

Chapter 8 · Proportional Band, PB · Reverse or Direct Acting Controller · Kc can 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 Assume FT is direct-acting. Select sign of Kc so that Kc. Kv > 0 1. ) Air-to-open (fail close) valve ==> ? 2. ) Air-to-close (fail open) valve ==> ? • Consequences of wrong controller action? ?

Chapter 8 · 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

Proportional-Integral (PI) Control Chapter 8 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

· Some controllers are calibrated in 1/ I ("repeats per minute") instead of I. Chapter 8 · 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. Chapter 8 • Many variations of PID control are used in practice. • Next, we consider the three most common forms. Parallel Form of PID Control The parallel form of the PID control algorithm (without a derivative filter) is given by

The corresponding transfer function is: Chapter 8

Expanded Form of PID Control Chapter 8 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 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.

Chapter 8

Automatic and Manual Control Modes Chapter 8 • 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

Chapter 8

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 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 ~

Chapter 8 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.