Process Control J.P. CORRIOU. Reaction and Process Engineering Laboratory University of Lorraine-CNRS, Nancy (France) Zhejiang University 2016

Size: px

Start display at page:

Download "Process Control J.P. CORRIOU. Reaction and Process Engineering Laboratory University of Lorraine-CNRS, Nancy (France) Zhejiang University 2016"

Abigayle Horn
5 years ago
Views:

1 Process Control J.P. CORRIOU Reaction and Process Engineering Laboratory University of Lorraine-CNRS, Nancy (France) Zhejiang University 206 J.P. Corriou (LRGP) Process Control Zhejiang University 206 / 37

2 Linear Feedback Control Most popular control method in industry is proportional-integral-derivative or PID controller, around 95% of industrial problems. Apparent simplicity of understanding and generally satisfactory performance. Based on action on the process according to the error between the set point and the measured output. Numerous variants of PID exist. Improvements possible by better tuning. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

3 Design of a Feedback Loop Process representation in open loop d(t) u(t) Process y(t) Open loop can work only if the process model is perfect and in the absence of disturbances, noise. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

4 Design of a Feedback Loop Process representation in closed loop Disturbance d(t) Controller Set point e(t) u a (t) u(t) Output Corrector Actuator Process y r (t) + y(t) y m (t) Measurement Objective : maintain the output y as close as possible to the desired set point y r for any disturbance. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

5 Design of a Feedback Loop The output is measured using a given measurement device ; the value indicated by the sensor is y m. This value is compared to the set point y r, giving the difference [set point measurement] to produce the error e = y r y m. The value of this error e is provided to the main corrector to modify the control variable u. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

6 General Types of Controllers A controller represents a control strategy, i.e. a set of rules providing a value of the control action when the output deviates from the set point. A controller can thus be constituted by an equation or an algorithm. In this first stage, only simple conventional controllers are considered. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

7 Proportional (P) Controller Operating output of the proportional controller is proportional to the error u a(t) = K c e(t)+u ab () where K c is the proportional gain of the controller. u ab is the bias signal of the actuator (u ab = operating signal when e(t) = 0). The gain K c can be positive or negative. The gain K c has dimensions. The higher the gain, the more sensitive the controller. A controller can saturate when its output u a(t) reaches a maximum u a,max or minimum u a,min value. Controller transfer function G c(s) = K c (2) simply equal to the controller gain. Philosophy : the integral action takes into account the present. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

8 Proportional-Integral (PI) Controller The operating output of the PI controller is proportional to the weighted sum of the magnitude and of the integral of the error u a (t) = K c ( e(t)+ τ I t τ I in the integral time constant. 0 ) e(x) dx + u ab (3) The integral action tends to modify the controller output u a (t) as long as an error exists in the process output. Transfer function of the PI controller is equal to ( G c (s) = K c + ) τ I s Philosophy : the integral action takes into account (integrates) the past. J.P. Corriou (LRGP) Process Control Zhejiang University / 37 (4)

9 Ideal Proportional-Derivative (PD) Controller The operating output of the ideal PD controller is proportional to the weighted sum of the magnitude and the time rate of change of the error ( ) de(t) u a (t) = K c e(t)+τ D + u ab (5) dt The derivative action is intended to anticipate future errors. τ D is the derivative time constant. Transfer function of the ideal PD controller G c (s) = K c (+τ D s) (6) Theoretical controller because the numerator degree of the controller transfer function G c (s) is larger than the denominator degree ; consequently, it is physically unrealizable. Philosophy : the derivative action takes into account (anticipates) the future. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

10 Ideal Proportional-Integral-Derivative (PID) Controller Combination of P, I and D actions ( de(t) u a (t) = K c e(t)+τ D + t dt τ I 0 ) e(x) dx + u ab (7) Transfer function of the ideal PID controller ( G c (s) = K c +τ D s + ) τ I s (8) Still derivative action is physical unrealizable. Philosophy : owing to the derivative action, the PID controller takes into account (anticipates) the future, and owing to the integral action, the PID takes into account (integrates) the past. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

11 Comparison of P, PI and PID Controllers P controller : It presents the drawback to create a deviation of the output with respect to the set point. PI controller : It presents the advantage of eliminating the deviation between the output and the set point owing to the integral action. However, this controller can produce oscillatory responses and diminishes the closed-loop system stability. Furthermore, the integral action can become undesirable when there is saturation. PID controller : It can stabilize the output with respect to the PI controller. It is sensitive to the noise. J.P. Corriou (LRGP) Process Control Zhejiang University 206 / 37

12 Sensors Without good measurement, it is hopeless to control the process well. Common sensors on chemical processes : Temperature sensors : thermocouples, platinum resistance probes, pyrometers. Modelled as first- or second-order model, sometimes with a time delay. Pressure sensors : classical manometers using bellows, Bourdon tube, membrane or electronic ones using strain gauges. Often represented by a second-order model. Flow rate sensors : for gases, thermal mass flow meters, variable area flow meters. For liquids, turbine flow meters, depression flow meters as venturi-type flow meters, vortex flow meters, electromagnetic flow meters, sonic flow meters, Coriolis effect flow meters. Fast dynamics. Often modelled by an equation of the form flow rate = a P J.P. Corriou (LRGP) Process Control Zhejiang University / 37

13 Sensors Level sensors : floats, displacement, conductivity, capacitance. Composition sensors : potentiometers, conductimeters, chromatographs, spectrometers. Sometimes, very long delay (chromatographs). In the absence of a measurement concerning a given variable, if a model of the process is available, it is possible to realize a state observer called a software sensor. Used in particular for composition estimations. In addition of the sensor itself, there is a transmitter, simple converter and simple gain (often transmitted in the range 4-20 ma). J.P. Corriou (LRGP) Process Control Zhejiang University / 37

14 Actuators Actuators constitute material elements allowing action by means of the control loop on the process Pneumatic valves. 00 Sliding stem Design position of a valve : either completely open (fail open, or air-to-open) or completely closed (fail closed, or air-to-close) when the air pressure is not ensured Many types of valves exist. 0 Valve dynamics is fast. A valve introduces a pressure drop in the pipe. Valves are freequently nonlinear and can saturate. For liquids, flow rate Q depends on the square root of the pressure drop P v of the valve Q = C v P d (9) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

15 Valves FIGURE : Influence of the aperture degree of a valve on its flow rate q = Q Q m x Linear (case ) : Butterfly (case 4) : Equal percentage (cases 2 and 3) : C v = C vs x C v = C vs ( cos( π 2 x)) C v = C vs R x v (0) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

16 Dynamics of Feedback-Controlled Processes FIGURE : Block scheme of the closed-loop process Process Disturbance D(s) Y r (s) K m Set point Controller Y rm (s) E(s) Corrector Actuator U a (s) U(s) G + c (s) Ga (s) Gp (s) G d (s) + + Y(s) Output Y G m (s) m (s) Measurement Process closed-loop response Y(s) Y(s) = G p(s) G a(s) G c(s) K m +G p(s) G a(s) G c(s) G Yr(s)+ G d (s) D(s) () m(s) +G p(s) G a(s) G c(s) G m(s) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

17 Dynamics of Feedback-Controlled Processes The closed-loop transfer function is equal to product of the transfer functions met on the path between an input and an output +the product of all transfer functions met in the loop sign + is indeed the opposite of the sign of comparison between measured output and set point. Two interesting closed loop transfer functions : between set point and output, influence of set point. between disturbance and output, influence of disturbance. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

18 Dynamics of Feedback-Controlled Processes FIGURE : How the "open loop" must be understood by opposition to the closed loop + G r G a G p G m "Open loop" and "closed loop" must be understood as electrician language. The closed-loop transfer function is equal to product of the transfer functions met on the path between an input and an output + transfer function of the open loop J.P. Corriou (LRGP) Process Control Zhejiang University / 37

19 Dynamics of Feedback-Controlled Processes Closed-loop transfer function for a set point variation G set point = G p (s) G a (s) G c (s) K m +G p (s) G a (s) G c (s) G m (s) (2) Closed-loop transfer function for a disturbance variation G disturbance = G d (s) +G p (s) G a (s) G c (s) G m (s) (3) Same denominators. Closed-loop transfer functions depend not only on the process dynamics, but also on the actuator, measurement device and the controller s own dynamics. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

20 Control problems Separation between two control problems : Regulation (or disturbance rejection) : Fixed set point (Y r (s) = 0 or δy r (t) = 0). Process subjected to disturbances. Tracking : Assumption that disturbance is constant (D(s) = 0 or δd(t) = 0). Set point is variable. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

21 Study of different actions Simplifications : G a =, G m =, K m = (4) Also, assumption that the orders of process transfer function G p and of the disturbance transfer function G d are equal. FIGURE : Block diagram for the study of the action of the different controllers d(t) G d y r (t) + G c u(t) G p y(t) Y(s) = Gp(s) Gc(s) +G p(s) G Yr(s)+ G d (s) D(s) (5) c(s) +G p(s) G c(s) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

22 Proportional Action Case of first-order systems G p(s) = Kp τ p s + ; G d (s) = K d τ d s + (6) Output Y(s) = K p K c τ p s + +K p K c Y r(s)+ K d τ d s + τ p s + τ p s + +K p K c D(s) (7) Final-value theorem : t s 0 gives closed-loop steady-state gain for the set point closed-loop steady-state gain for the disturbance K p and K d 0 if Kc increases. K p K d = Response faster in closed loop than in open loop τ p = = Kp Kc +K p K c (8) K d +K p K c (9) τ p +K p K c (20) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

23 Proportional Action y Example of studies / Tracking : Assume step set point step of amplitude A 0.9 Kc=5 Kc=2 Y r(s) = A s ; D(s) = 0 (2) Kc= Closed-loop response to a step set point is then equal to Y(s) = K p K c A τ p s + +K p K c s = Final value theorem A s K p τ p s + (22) Time lim t y(t) = lim s 0 sy(s) Kp Kc = A +K p K c (23) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

24 Proportional Action y 2/ Regulation : Assume disturbance point step of amplitude A 0.5 D(s) = A s ; Y r(s) = 0 (24) Closed-loop response to a step disturbance is then equal to Y(s) = K d τ d s + Final value theorem τ ps + τ ps + +K pk c A s (25) Time lim y(t) t = lim s 0 sy(s) K = d A +K p K c (26) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

25 Proportional Action Case of second-order systems Output Y(s) G p(s) = K p τ 2 p s 2 + 2ζ p τ p s + ; G d (s) = K d τ 2 d s2 + 2ζ d τ d s + (27) Y(s) = K p K c τp 2 s 2 Y r(s) + 2ζ p τ p s + +K p K c K + d τp 2 s2 + 2ζ p τ p s + τd 2 s2 + 2ζ d τ d s + τp 2 D(s) s 2 + 2ζ p τ p s + +K p K c (28) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

26 Proportional Action / Tracking Modification of period and damping factor τ p = τ p +Kp K c ζ p = ζ p +Kp K c (29) Steady-state gain K p = K p K c +K p K c (30) Same remarks as for first-order. a deviation between the set point and the asymptotic response exists which is all the more important as the gain is low. 2/ Regulation Same remarks as for first-order. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

27 Integral Action Even if integral action is never used alone, in order to characterize its influence, we first assume that the controller is pure integral G c(s) = Kc (3) τ I s Study of first-order systems. or K p K c +τ p s τ Y(s) = I s Y K p K r(s)+ c + + +τ p s τ I s Y(s) = τ p τ I s 2 + τ Y I r(s)+ s + K p K c K p K c Modification of the order in closed loop. K d τ d s + K d +τ d s K p +τ p s K c D(s) (32) τ I s (τ p s + )τ I s τ I τ p s 2 +τ I s + K p K c D(s) (33) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

28 Integral Action Y(s) = τ p τ I s 2 + τ Y I r(s)+ s + K p K c K p K c / Tracking, step set point of amplitude A K d τ d s + (τ p s + )τ I s τ I τ p s 2 +τ I s + K p K c D(s) (34) Y r(s) = A s (35) hence Final value theorem Y(s) = A τ p τ I s 2 + τ I s + K p K c K p K c s The integral action eliminates the asymptotic deviation. Response faster when gain is high, but oscillatory responses. (36) lim y(t) = lim[s Y(s)] = A (37) t s 0 J.P. Corriou (LRGP) Process Control Zhejiang University / 37

29 Integral Action Y(s) = τ p τ I s 2 + τ Y I r(s)+ s + K p K c K p K c 2/ Regulation, step disturbance of amplitude A K d τ d s + (τ p s + )τ I s τ I τ p s 2 +τ I s + K p K c D(s) (38) D(s) = A s (39) hence Final value theorem Y(s) = K d τ d s + The integral action rejects step-like disturbances. (τ p s + )τ I s τ I τ p s 2 +τ I s + K p K c A s (40) lim y(t) = lim[s Y(s)] = 0 (4) t s 0 J.P. Corriou (LRGP) Process Control Zhejiang University / 37

30 Derivative Action Transfer function of a pure ideal derivative controller is equal to G c(s) = K c τ D s (42) Improper transfer function (numerator degree larger than denominator degree). Block diagram of the ideal PID controller Block diagram of a real PID controller E(s) Kc τis τds + Ua(s) + + E(s) + Kc τis + τds τd N s+ Ua(s) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

31 Ideal Derivative Action Response Y(s) to a set point or disturbance variation Y(s) = = K p τ p s + Kc τ D s Y K r(s)+ p + τ p s + Kc τ D s + K p K c τ D s (τ p + K p K c τ D )s + Yr(s)+ K d τ d s + K d τ d s + K p τ p s + Kc τ D s D(s) τ p s + (τ p + K p K c τ D )s + D(s) (43) No influence on the system order. Closed-loop time constant τ p = τp + Kp Kc τ D (44) Closed-loop response will be slower than the open-loop one. Stabilization of the process. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

32 PI Controller Transfer function of PI Controller ( G c(s) = K c + ) τ I s (45) Output Y(s) = + ( K p +τ p s Kc + τ I s K p +τ p s Kc ) ( + τ I s ) Y r(s)+ + K d +τ d s K p +τ p s Kc ( + τ I s ) D(s) (46) or Y(s) = τ I s + Y τ p τ I s 2 K p K c + r(s)+ +τ I s + K p K c K p K c τ I s(+τ p s) K p K c K d τ d s + τ p τ I s 2 K p K c + +τ I s + K p K c K p K c Conclusions for the Integral action are still valid. D(s) (47) J.P. Corriou (LRGP) Process Control Zhejiang University / 37

33 Summary of Controller Characteristics P controller : one tuning parameter K r, deviation from the set point, decreased by increasing gain. PI controller : elimination of the deviation between the asymptotic state and the set point, faster response, possible oscillations, possible saturation of control variable u. PID controller : same interest as PI, stabilizing effect, sensitivity to measurement noise. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

34 Influence of controller on first-order system y y y Response of a first-order system (K p = 5,τ p = 0) to a set point unit step. P controller PI controller Real PID controller Time Time Time K c = 2 K c = 2 τ I = 20 K c = 2 τ I = 20 τ D = β = 0. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

35 Influence of controller on second-order system y y y response of a second-order system (K p = 5,τ p = 0,ζ p = 0.5) to a set point unit step. P controller PI controller Real PID controller Time Time Time K c = 2 K c = 2 τ I = 20 K c = 2 τ I = 20 τ D = β = 0. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

36 Influence of controller on first-order system y y y Response of a first-order system (K p = 5,τ p = 0 : K d = 2,τ d = 2) to a disturbance unit step. P controller PI controller Real PID controller Time Time Time K c = 2 K c = 2 τ I = 20 K c = 2 τ I = 20 τ D = β = 0. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

37 Influence of controller on second-order system y y y Response of a second-order system (K p = 5,τ p = 0,ζ p = 0.5 : K d = 2,τ d = 2,ζ d = 0.25) to a disturbance unit step. P controller PI controller Real PID controller Time Time Time K c = 2 K c = 2 τ I = 20 K c = 2 τ I = 20 τ D = β = 0. J.P. Corriou (LRGP) Process Control Zhejiang University / 37

Chapter 2 Linear Feedback Control

Chapter 2 Linear Feedback Control The by far most used control method in industry is the proportional-integralderivative or PID controller. It is currently claimed that 90 to 95% of industrial problems