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

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

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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. Example 2 Figure 8.2: Flow control system

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Chapter 8 In feedback control, the objective is to reduce the error signal to zero where and Chapter 8

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Basic Control Modes Three basic control modes: Chapter 8 requires very intensive energy costly highly endothermic reaction Derivative control Proportional control Integral control

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Proportional Control Chapter 8 For proportional control, the controller output is proportional to the error signal, where: At time t = 25 min, e(25) = 60–56 = 4 At time t = 40 min, e(40) = 60–62 = –2

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Chapter 8

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The key concepts behind proportional control are the following: 1.the controller gain ( K c ) can be adjusted to make the controller output changes as sensitive as desired to deviations between set point and controlled variable; 2.the sign of K c can be chosen to make the controller output increase (or decrease) as the error signal increases. 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 K c and vice versa

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Chapter 8 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 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. Chapter 8

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Chapter 8 For integral control action, the controller output depends on the integral of the error signal over time, where,= integral time/reset time 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. Integral Control

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The corresponding transfer function for the PI controller in Eq. 8-8 is given by Integral control action is normally used in conjunction with proportional control as the proportional-integral (PI) controller: Proportional-Integral (PI) Control

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The integral mode causes the controller output to change as long as e(t*) 0 in Eq. 8-8 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. If an error is large enough and/or persists long enough 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) If this extreme value is still not sufficient to eliminate the error Chapter 8 *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. Disadvantage of Integral Action: Reset Windup Can grow very large

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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

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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, where, the derivative time, has units of time. Chapter 8 Derivative Control

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Chapter 8 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 Advantages of derivative action

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Chapter 8 For example, an ideal PD controller has the transfer function: Derivative action always used in conjunction with proportional or proportional-integral control. Unfortunately, the ideal proportional-derivative control algorithm in Eq is physically unrealizable because it cannot be implemented exactly. Chapter 8 Physically unrealizable

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Chapter 8 For real PD controller, the transfer function in (8-11) can be approximated by where the constant α typically has a value between 0.05 and 0.2, with 0.1 being a common choice. In Eq 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

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

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Parallel Form of PID Control The parallel form of the PID control algorithm (without a derivative Ideal PID control is given by The corresponding transfer function for Ideal PID control is: Chapter 8

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The corresponding transfer function for Real PID control (parallel), with derivative filter is: Figure 8.8Block diagram of the parallel form of PID control (without a derivative filter) Chapter 8

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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 Chapter 8 Figure 8.9Block diagram of the series form of PID control (without a derivative filter) Chapter 8 PI PD

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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 is sometimes used: Chapter 8 Used in MATLAB

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PID- Most complicated to tune (K c, I, D ). - Better performance than PI - No offset - Derivative action may be affected by noise PI- More complicated to tune (K c, I ). - Better performance than P - No offset - Most popular FB controller P- Simplest controller to tune (K c ). - Offset with sustained disturbance or setpoint change. Controller Comparison Chapter 8

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This sudden change is undesirable and can be avoided by basing the derivative action on the measurement, y m, rather than on the error signal, e. Replacing de/dt by –dy m /dt gives One disadvantage of the previous PID controllers : Derivative and Proportional Kick Features of PID Controllers Chapter 8 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. From a parallel form of PID control in Eq. 8-13

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

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Chapter 8 Reverse acting ( K c > 0) e(t), p(t) y m (t), p(t) Reverse acting ( K c > 0) e(t), p(t) y m (t), p(t) Direct acting ( K c < 0) e(t), p(t) y m (t), p(t) Direct acting ( K c < 0) e(t), p(t) y m (t), p(t)

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On-Off Controllers Synonyms: two-position or bang-bang controllers. Controller output has two possible values. Chapter 8 y m (t) 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. 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.

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

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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) Chapter 8 Figure Typical process responses with feedback control. 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

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Figure Proportional control: effect of controller gain. Chapter 8 -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

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-Increasing τ I tends to make the process response more sluggish (slower) -Too large of τ I, the controlled variable will return to the set point very slowly after a disturbance set-point change occurs. Figure PI control: (a) effect of reset time (b) effect of controller gain. Chapter 8

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Figure PID control: effect of derivative time. -Increasing τ D tends to improve the process response by reducing the maximum deviation, response time and degree of oscillation. -Too large of τ D : measurement noise is amplified and process response more oscillatory. -The intermediate value of τ D is desirable. Chapter 8

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