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CHAPTER 8 Feedback Controllers. ERT 210 Process Control & dynamics. Anis Atikah binti Ahmad. Control objective: to keep the tank exit composition, x, at the desired value (set point) by adjusting the flow rate, w 2, via the control valve. Chapter 8. Chapter 8.
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CHAPTER 8 Feedback Controllers ERT 210Process Control & dynamics Anis Atikahbinti Ahmad
Control objective: to keep the tank exit composition, x, at the desired value (set point) by adjusting the flow rate, w2, via the control valve Chapter 8 Chapter 8 Figure 8.1: Schematic diagram for a stirred-tank blending system
Chapter 8 Figure 8.2: Flow control system
In feedback control, the objective is to reduce the error signal to zero where and Chapter 8 Chapter 8
Basic Control Modes highly endothermic reaction Three basic control modes: Proportional control costly requires very intensive energy Chapter 8 Chapter 8 Integral control Derivative control
Proportional Control For proportional control, the controller output is proportional to the error signal, Chapter 8 Chapter 8 where: • At time t = 25 min, e(25) = 60–56 = 4 • At time t = 40 min, e(40) = 60–62 = –2
Chapter 8 Chapter 8
The key concepts behind proportional control are the following: • the controller gain (Kc) can be adjusted to make the controller output changes as sensitive as desired to deviations between set point and controlled variable; • the sign of Kc can be chosen to make the controller output increase (or decrease) as the error signal increases. Chapter 8 Some controllers have a proportional band setting instead of a controller gain. The proportional band PB (in %) is defined as ≜ Large PB correspond to a smallvalue of Kcand vice versa
In order to derive the transfer function for an ideal proportional controller (without saturation limits), define a deviation variable as ≜ Then Eq. 8-2 can be written as Chapter 8 Chapter 8 The transfer function for proportional-only control: An inherent disadvantage of proportional-only control is that a steady-state error (or offset) occurs after a set-point change or a sustained disturbance.
Integral Control For integral control action, the controller output depends on the integral of the error signal over time, where ,= integral time/reset time 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.
Proportional-Integral (PI) Control Integral control action is normally used in conjunction with proportional control as the proportional-integral (PI) controller: The corresponding transfer function for the PI controller in Eq. 8-8 is given by
The integral mode causes the controller output to change as long as e(t*) ≠ 0 in Eq. 8-8 Disadvantage of Integral Action: Reset Windup Can grow very large If an error is large enough and/or persists long enough • Can produce p(t) that causes the final control element (FCE) to saturate. • That is, the controller drives the FCE (e.g. valve, pump, compressor) to its physical limit of fully open/on/maximum or fully closed/off/minimum. Chapter 8 If this extreme value is still not sufficient to eliminate the error • the integral term continue growing, • the controller command the FCE to move to 110%, then 120% and more • this command has no physical meaning (no impact on the process) *Antireset windup reduce the windup by temporarily halting the integral control action whenever the controller output saturates and resumes when the output is no longer saturates.
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 Chapter 8
Derivative Control • The function of derivative control action is to anticipate the future behavior of the error signal by considering its rate of change. • Controller output is proportional to the rate of change of the error signal or the controlled variable. • Thus, for ideal derivative action, Chapter 8 Chapter 8 where , the derivative time, has units of time.
Advantages of derivative action Chapter 8 Chapter 8
Derivative action always used in conjunction with proportional or proportional-integral control. • Unfortunately, the ideal proportional-derivative control algorithm in Eq. 8-11 is physically unrealizable because it cannot be implemented exactly. For example, an “ideal” PD controller has the transfer function: Physically unrealizable Chapter 8 Chapter 8
For “real” PD controller, the transfer function in (8-11) can be approximated by Chapter 8 Chapter 8 • 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.
Proportional-Integral-Derivative (PID) Control highly endothermic reaction 3 common PID control forms are: Parallel form costly requires very intensive energy Chapter 8 Chapter 8 Series form Expanded form
Parallel Form of PID Control The parallelform of the PID control algorithm (without a derivative filter) @ “Ideal” PID control is given by Chapter 8 The corresponding transfer function for “Ideal” PID control is:
The corresponding transfer function for “Real” PID control (parallel), with derivative filter is: Chapter 8 Chapter 8 Figure 8.8 Block diagram of the parallel form of PID control (without a derivative filter)
Series Form of PID Control • Constructed by having PI element and a PD element operated in series. • Commercial versions of the series-form controller have a derivative filter as indicated in Eq 8-15. PI PD Chapter 8 Chapter 8 Figure 8.9 Block diagram of the series form of PID control (without a derivative filter)
Used in MATLAB 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: Chapter 8 Chapter 8
Controller Comparison P - Simplest controller to tune (Kc). - Offset with sustained disturbance or setpoint change. PI - More complicated to tune (Kc, I) . - Better performance than P - No offset - Most popular FB controller Chapter 8 Chapter 8 PID - Most complicated to tune (Kc, I, D) . - Better performance than PI - No offset - Derivative action may be affected by noise
Features of PID Controllers Derivative and Proportional Kick • One disadvantage of the previous PID controllers : derivative kick : when there is a sudden change in set point (and hence the error, e) that will cause the derivative term to become very large. Chapter 8 • 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. • Replacing de/dt by –dym/dt gives Chapter 8 From a parallel form of PID control in Eq. 8-13
Reverse or Direct Action • The controller gain can be made either negative or positive. Chapter 8 Chapter 8
Reverse acting (Kc > 0) e(t)↑, p(t) ↑ ym(t)↓, p(t) ↑ Chapter 8 Chapter 8 Direct acting (Kc < 0) e(t) ↓, p(t) ↑ ym(t)↑, p(t) ↑
On-Off Controllers Synonyms: “two-position” or “bang-bang” controllers. ym(t)↓ ym(t)↑ Chapter 8 Chapter 8 eg: thermostat in home heating system. -if the temperature is too high, the thermostat turns the heater OFF. -If the temperature is too low, the thermostat turns the heater ON. Controller output has two possible values.
On-Off Controllers (continued) 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 • the disturbance variable) No control: the process slowly reaches a new steady state P – speed up the process response & reduces the offset PI – eliminate offset & the response more oscillatory PID – reduces degree of oscillation and the response time Chapter 8 Chapter 8 Figure 8.12. Typical process responses with feedback control.
Chapter 8 Chapter 8 Figure 8.13. Proportional control: effect of controller gain. • Increasing Kc tends to make the process response less sluggish (faster) • Too large of Kc, results in undesirable degree of oscillation or even become unstable • Intermediate value of Kc usually results in the best control.
Chapter 8 Chapter 8 Figure 8.14. PI control: (a) effect of reset time (b) effect of controller gain. • Increasing τI tends to make the process response more sluggish (slower) • Too large of τI, the controlled variable will return to the set point very slowly after a disturbance change @ set-point change occurs.
Chapter 8 Chapter 8 Figure 8.15. PID control: effect of derivative time. • Increasing τDtends to improve the process response by reducing the maximum deviation, response time and degree of oscillation. • Too large of τD: measurement noise is amplified and process response more oscillatory. • The intermediate value of τDis desirable.