ERT 210 Process Control & dynamics CHAPTER 8 Feedback Controllers Anis Atikah binti Ahmad

Chapter 8 Feedback Controller Chapter 8 Example 1 Corrective action occurs when the controlled variable deviates from the set point Example 1 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 Variables Controlled variable: exit composition, x Manipulated variable: flow rate of A,w2 Figure 8.1: Schematic diagram for a stirred-tank blending system

Example 2 The flow rate of a process stream is measured and transmitted electronically to a flow controller (FC) The controller compares the measured value to the set point value and takes appropriate corrective action by sending an output signal to the current-to-pressure (I/P) transducer. I/P transducer sends a corresponding pneumatic signal to the control valve. 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 Chapter 8 Chapter 8 Three basic control modes: highly endothermic reaction Proportional control Chapter 8 Chapter 8 costly Integral control requires very intensive energy Derivative control

Proportional Control Chapter 8 Chapter 8 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 small value of Kc and 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 Chapter 8 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

Chapter 8 Disadvantage of Integral Action: Reset Windup The integral mode causes the controller output to change as long as e(t*) ≠ 0 in Eq. 8-8 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.

Chapter 8 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 Chapter 8

Derivative Control Chapter 8 Chapter 8 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 By providing anticipatory control action, the derivative mode tends to stabilize the controlled process. Derivative control action tends to improve dynamic response of the controlled variable Chapter 8 Chapter 8

Physically unrealizable 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.

Chapter 8 Proportional-Integral-Derivative (PID) Control Chapter 8 3 common PID control forms are: highly endothermic reaction Parallel form costly Chapter 8 Chapter 8 Series form requires very intensive energy Expanded form

Chapter 8 Parallel Form of PID Control The parallel form 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)

Chapter 8 Chapter 8 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)

Chapter 8 Chapter 8 Expanded Form of PID Control 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

Chapter 8 Controller Comparison Chapter 8 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

From a parallel form of PID control in Eq. 8-13 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 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 From a parallel form of PID control in Eq. 8-13

Chapter 8 Reverse or Direct Action Chapter 8 The controller gain can be made either negative or positive. Chapter 8 Chapter 8 Direct (Kc < 0) Reverse (Kc > 0) Controller output p(t) increases as the input signal ym(t) increases Controller output p(t) increases as its input signal ym(t) decreases

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 Chapter 8 Chapter 8 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) Common use: residential heating domestic refrigerators Advantages Simple Cheap Disadvantages continual cycling of controlled variable produce excessive wear on control valve. Chapter 8 Chapter 8

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 τD tends 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 τD is desirable.