06_Feedback_Chap8S10_4

06_Feedback_Chap8S10_4 - P D I Chapter 8 Feedback...

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Unformatted text preview: P D I Chapter 8 Feedback Controllers: Week 1 Figure 8.1 Schematic diagram for a stirred-tank blending system. “Normal”, Kc < 0 “Reverse”, Kc > 0 p change wrt ym not wrt e = error 1 Although Eq. 8-1 indicates that the set point can be time-varying, in many process control problems it is kept constant for long periods of time. Basic Control Modes Next we consider the three basic control modes starting with the simplest mode, proportional control. For proportional control, the controller output is proportional to the error signal, Chapter 8 Chapter 8 Proportional Control Control In feedback control, the objective is to reduce the error signal to zero where e ( t ) = ysp ( t ) − ym ( t ) and e (t ) (8-1) = error signal ysp ( t ) = set point 2 (ysp and ym must have same units) p ( t ) = p + Kce ( t ) (8-2) where: p ( t ) = controller output p = bias (steady-state) value K c = controller gain (usually dimensionless) ym ( t ) = measured value of the controlled variable (or equivalent signal from the sensor/transmitter) 3 4 1 The key concepts behind proportional control are the following: Chapter 8 Chapter 8 1. The controller gain can be adjusted to make the controller output changes as sensitive as desired to deviations between set point and controlled variable; In order to derive the transfer function for an ideal proportional controller (without saturation limits), define a deviation variable p′ ( t ) as p′ ( t ) p ( t ) − p (8-4) Chapter 8 Chapter 8 (8-5) The transfer function for proportional-only control: = Kc 6 For integral control action, the controller output depends on the integral of the error signal over time, p (t ) = p + p′ ( t ) = K c e ( t ) E (s) Some controllers have a proportional band setting instead of a controller gain. The proportional band PB (in %) is defined as Th PB %) 100% 100 PB or K c = (8-3) Kc PB Integral Control Then Eq. 8-2 can be written as P′ ( s ) For proportional controllers, bias p can be adjusted, a procedure referred to as manual reset. For “PB”, Kc must be dimensionless, e.g. p, e in mA, each 4-20 mA range Example: if Kc = 2, a p change is twice the e change. Output p would change by 100% for an e change of 50% so PB = 50%. 5 Gc = 2. the sign of Kc can be chosed to make the controller output sign be chosed to make the controller output increase (or decrease) as the error signal increases. (8-6) An inherent disadvantage of proportional-only control is that a steady-state error usually occurs after a set-point change or a sustained disturbance. 7 1 τI ∫0 e ( t *)dt * t (8-7) where τ I , an adjustable parameter referred to as the integral time or reset time, has units of time. Integral control action is widely used because it provides an important practical advantage, the elimination of offset. Consequently, integral control action is normally used in conjunction with proportional control as the proportional-integral (PI) controller: ⎛ 1 p ( t ) = p + Kc ⎜ e ( t ) + τI ⎝ ⎞ ∫0 e ( t *) dt * ⎟ ⎠ t (8-8) 8 2 The corresponding transfer function for the PI controller in Eq. 8-8 is given by P′ ( s ) ⎛ ⎛ τI s +1 ⎞ 1⎞ = Kc ⎜1 + ⎟ = Kc ⎜ ⎟ E (s) ⎝ τI s ⎠ ⎝ τI s ⎠ • Further buildup of the integral term while the controller is saturated (“pegged at a limiting value by the controller”) is referred to as reset windup or integral windup. (8-9) Chapter 8 Chapter 8 Gc = • When a sustained error occurs, the integral term becomes quite large and the controller output eventually saturates. Some commercial controllers are calibrated in terms of 1/ τ I (repeats per minute) rather than τ I (minutes, or minutes per repeat). Reset Windup • An inherent disadvantage of integral control action is a phenomenon known as reset windup or integral windup. Derivative Control The function of derivative control action is to “anticipate” the future behavior of the error signal by considering its rate of change. An alternative description is that derivative action seeks to minimize the rate of change of the error de/dt. • The anticipatory strategy used by the experienced operator can be incorporated in automatic controllers by making the controller output proportional to the rate of change of the error signal or the controlled variable. • Recall that the integral mode causes the controller output to change as long as e(t*) ≠ 0 in Eq. 8-8. 9 10 • Thus, for ideal derivative action, p (t ) = p + τD de ( t ) dt • For analog controllers, the PD controller transfer function in (8-11) can be approximated by (8-10) Gc = For example, an ideal PD controller has the transfer function: Gc = P′ ( s ) E (s) = K c (1 + τ D s ) Chapter 8 Chapter 8 where τ D , the derivative time, has units of time. (8-11) • By providing anticipatory control action, the derivative mode providing anticipatory control action the derivative mode tends to stabilize the controlled process. P′ ( s ) E (s) ⎛ τDs ⎞ = K c ⎜1 + ⎟ ατ D s + 1 ⎠ ⎝ (8-12) where the constant α typically has a value between 0.05 and 0.2, with 0.1 being a common choice. • In Eq. 8-12 the derivative term includes a derivative mode filter (also called a derivative filter) that reduces the sensitivity of the control calculations to high-frequency noise in the measurement. • Unfortunately, the ideal proportional-derivative control algorithm in Eq. 8-10 is physically unrealizable because it cannot be implemented exactly. 11 12 3 The corresponding transfer function is: Proportional-Integral-Derivative (PID) Control Now we consider the combination of the proportional, integral, and derivative control modes as a PID controller. Gc = • Next, we consider the three most common forms. Chapter 8 Chapter 8 • Many variations of PID control are used in practice. Parallel Form of PID Control The parallel form of the PID control algorithm (without a derivative filter) is given by ⎡ 1 p ( t ) = p + K c ⎢e ( t ) + τI ⎣ ∫0 e ( t *) dt * + τ D t de ( t ) ⎤ ⎥ dt ⎦ (8-13) P′ ( s ) ⎡ ⎤ 1 = K c ⎢1 + + τDs⎥ E (s) ⎣ τI s ⎦ (8-14) Series Form of PID Control Form of PID Control Historically, it was convenient to construct early analog controllers (both electronic and pneumatic) so that a PI element and a PD element operated in series. Commercial versions of the series-form controller have a derivative filter that is applied to either the derivative term, as in derivative filter that is applied to either the derivative term, as in Eq. 8-12, or to the PD term, as in Eq. 8-15: P′ ( s ) ⎛ τ s + 1 ⎞⎛ τ D s + 1 ⎞ = Kc ⎜ I ⎟⎜ ⎟ E (s) ⎝ τ I s ⎠⎝ ατ D s + 1 ⎠ (8-15) 13 14 • This sudden change is undesirable and can be avoided by basing the derivative action on the measurement, ym, rather than on the error signal, e, but then the controller has two inputs: e and ym. 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: de ( t ) 0 dt Chapter 8 • We illustrate the elimination of derivative kick by considering the parallel form of PID control in Eq. 8-13. (8-16) Chapter 8 t p ( t ) = p + K c e ( t ) + K I ∫ e ( t *) dt * + K D Features of PID Controllers Elimination of Derivative and Proportional Kick • One disadvantage of the previous PID controllers is that a disadvantage of the previous PID controllers is that 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. 15 • Replacing de/dt by –dym/dt gives ⎡ 1 p ( t ) = p + K c ⎢e ( t ) + τI ⎣ ∫0 e ( t *) dt * −τ D t dym ( t ) ⎤ ⎥ dt ⎦ (8-17) Reverse or Direct Action • The controller gain can be made either negative or positive. 16 4 • For proportional control, when Kc > 0, the controller output p(t) increases as its input signal ym(t) decreases, as can be seen by combining Eqs. 8-2 and 8-1: (8-22) • This controller is an example of a reverse-acting controller. Figure 8.11 Reverse and direct-acting proportional controllers. (a) reverse acting (Kc > 0. (b) direct acting (Kc < 0) Chapter 8 Chapter 8 p ( t ) − p = K c ⎡ ysp ( t ) − ym ( t ) ⎤ ⎣ ⎦ • When Kc < 0, the controller is said to be direct acting because the controller output p increases as the ym “input” increases. • Equations 8-2 through 8-16 describe how controllers perform during the automatic mode of operation. • However, in certain situations the plant operator may decide to override the automatic mode and adjust the controller output manually = controller is “on manual”. 17 18 • Example: Flow Control Loop Automatic and Manual Control Modes Chapter 8 Chapter 8 • Assume FT is direct-acting, what kind of P controller (direct or reverse acting) should be used for the two kinds of valves ? th ki 1. Air-to-open (fail close) valve ==> ? 2. Air-to-close (fail open) valve ==> ? 19 Automatic Mode Controller output, p(t), depends on e(t), controller yp 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) 20 5 Example: Liquid Level Control • Control valves are air-to-open • Level transmitters are direct acting Chapter 8 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. Question: Type of controller action? 21 22 On-Off Controllers (continued) Practical case (dead band) = no change unless “large” change in e Chapter 8 Chapter 8 Synonyms: “two-position” or “bang-bang” controllers. Controller output has two possible values. 23 24 6 P PID Controller D I • Ideal controller Chapter 8 • Transfer function (ideal) t ⎡ 1 de ⎤ p(t ) = p + K c ⎢e(t ) + ∫ e(t ′)dt ′ + τ D ⎥ τI 0 dt ⎦ ⎣ ⎛ ⎞ P′(s) 1 = K c ⎜1 + +τ Ds ⎟ E(s) τIs ⎝ ⎠ • Transfer function (actual) ⎛ τ s +1⎞ ⎛ τ Ds +1 ⎞ P′(s) = Kc ⎜ I ⎟⎜ ⎟ E(s) ⎝ τ I s ⎠ ⎝ ατ D s + 1 ⎠ “Normal”, Kc < 0 “Reverse”, Kc > 0 p change wrt ym not wrt e = error lead / lag units α = small number (0.05 to 0.20, 0.05 used in CHE461-OSU) 25 Typical Response of Feedback Control Systems Consider response of a controlled system after a sustained disturbance occurs (e.g., step change in the disturbance variable) Controller Comparison - Simplest controller to tune (Kc). with sustained disturbance or setpoint - Offset with sustained disturbance or setpoint change. PI - More complicated to tune (Kc, τI) . Better performance than P No offset Most popular FB controller PID - Most complicated to tune (Kc, τI, τD) . Better performance than PI No offset Derivative action may be affected by noise Chapter 8 Chapter 8 P 26 y Figure 8.12. Typical process responses with feedback control. 27 28 7 ...
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