Chapter 10. Closed-Loop Control Systems

Transcription

1 hapter 0 loed-loop ontrol Sytem

2 ontrol Diagram of a Typical ontrol Loop Actuator Sytem F F 2 T T 2 ontroller T Senor Sytem T TT

3 omponent and Signal of a Typical ontrol Loop F F 2 T Air 3-5 pig 4-20 ma I/P D/A T 2 Thermowell T Thermocouple millivolt ignal Operator onole T p DS ontrol omputer A/D 4-20 ma Tranmitter

4 Example: Two ontrol Loop For Steam Heated Stirred Tank F in,t in TT T IP Temperature ontrol Level ontrol LT P Steam L IP F,T ondenate Feedback control ytem: Valve i manipulated to increae flow of team to control tank temperature loed-loop proce: ontroller and proce are interconnected

5 ontrol Objective: Feedback ontrol maintain a certain outlet temperature and tank level Feedback ontrol: temperature i meaured uing a thermocouple level i meaured uing differential preure probe undeirable temperature trigger a change in upply team preure fluctuation in level trigger a change in outlet flow Note: level and temperature information i meaured at outlet of proce/ change reult from inlet flow or temperature diturbance inlet flow change UST affect proce before an adjutment i made

6 Feedback ontrol: Baic component of control ytem : require enor, actuator and controller: e.g. Temperature ontrol Loop T R + - e Feedback ontrol ontroller Actuator (Valve) A eaurement (Thermocouple) T in, F T out ontroller: oftware component implement math hardware component provide calibrated ignal for actuator Actuator: phyical (with dynamic) proce triggered by controller directly affect proce Senor: monitor ome property of ytem and tranmit ignal back to controller P Proce

7 Study of proce dynamic focued on uncontrolled or Open-loop procee Oberve proce behavior a a reult of pecific input ignal U() p Y() In proce control, we are concerned with the dynamic behavior of a controlled or loed-loop proce D() d Final ontroller Element Proce Y SP () + V() Y() or c U() + v p + R() E() - Ym() m ontroller i dynamic ytem that interact with the proce and the proce hardware to yield a pecific behaviour

8 p () - Proce Tranfer Function c () - ontroller Tranfer Function m () - Senor Tranfer Function v () - Actuator Tranfer Function d() -Load Tranfer Function D Diturbance or load variable Y controlled variable Yp etpoint or deired or reference variable Ym meaured variable E error V controller output variable U manipulated variable

9 loed-loop Tranfer Function For control, we need to identify cloed-loop dynamic due to: - Setpoint change Servo - Diturbance Regulatory. loed-loop Servo Repone tranfer function relating Y() and R() when D()=0 Iolate Y() Y( ) Y( ) ( ) V ( ) p Y( ) ( ) ( ) U ( ) p Y( ) ( ) ( ) ( ) E( ) v p v c Y( ) ( ) ( ) R( ) Y ( ) p v c( ) m Y( S) ( ) ( ) ( ) R( ) ( ) Y( ) p v c m p( ) v ( ) c ( ) R ( ( ) ( ) ( ) ( ) ) p v c m

10 2. loed-loop Regulatory Repone with no diturbance dynamic (ie d =0) Tranfer Function relating D() to Y() at R()=0 Y( ) D( ) ( ) V ( ) Y( ) D( ) ( ) ( ) U ( ) Y( ) D( ) ( ) ( ) ( ) E( ) Y( ) D( ) ( ) ( ) ( ) 0 Y ( ) Y( ) D( ) ( ) ( ) ( ) 0 ( ) Y( ) Iolating Y() p p v p v c p v c m p v c m Y( ) ( ) ( ) ( ) ( ) D ( ) p v c m

11 loed-loop Tranfer Function 2. Regulatory Repone with Diturbance Dynamic Y( ) d ( ) D ( ( ) ( ) ( ) ( ) ) p v c m 3. Overall loed-loop Tranfer Function Servo Y( ) p( ) v ( ) c ( ) R ( ) p( ) v ( ) c( ) m( ) d ( ) D ( ) p( ) v ( ) c( ) m( ) Regulatory

12 Example: Blending Proce

13 Tranfer function for each of the four element in the feedback control loop. For the ake of implicity, flow rate w i aumed to be contant, and the ytem i initially operating at the nominal teady rate. Proce Dynamic model of a tirred-tank blending ytem 2 X X W2 (-) τ τ where Vρ w x,, and 2 (-2) w w w

14

15 (-5) e t x t x t or after taking Laplace tranform, p m (-6) E X X p The ymbol xp t denote the internal et-point compoition expreed a an equivalent electrical current ignal. Thi ignal i ued internally by the controller. xp t i related to the actual compoition et point xp t by the compoition enortranmitter gain m : m Thu p (-7) x t x t X p X p m p m (-8)

16 urrent-to-preure (I/P) Tranducer Becaue tranducer are uually deigned to have linear characteritic and negligible (fat) dynamic, we aume that the tranducer tranfer function merely conit of a teady-tate gain IP : Pt (-9) P IP ontrol Valve ontrol valve are uually deigned o that the flow rate through the valve i a nearly linear function of the ignal to the valve actuator. Therefore, a firt-order tranfer function uually provide an adequate model for operation of an intalled valve in the vicinity of a nominal teady tate. Thu, we aume that the control valve can be modeled a t W2 v P τ v (-0)

17 ompoition Senor-Tranmitter (Analyzer) We aume that the dynamic behavior of the compoition enortranmitter can be approximated by a firt-order tranfer function: Xm m (-3) X τ ontroller Suppoe that an electronic proportional plu integral controller i ued. A will be hown later, the controller tranfer function i where P output p t m P c (-4) E τi and E() are the Laplace tranform of the controller and the error ignal e(t).

18

19 . Summer 2. omparator 3. Block Y() ()X() Block in Serie are equivalent to... Y

20 In general, the tandard cloed loop block diagram i

21 loed-loop Tranfer Function Indicate dynamic behavior of the controlled proce (i.e., proce plu controller, tranmitter, valve etc.) Set-point hange ( Servo Problem ) Aume Yp 0 and D = 0 (et-point change while diturbance change i zero) Diturbance hange ( Regulator Problem ) Aume D 0 and Y p = 0 (contant et-point) *Note ame denominator for Y/D, Y/Y p. P V P V p Y Y ) ( ) ( P V L D Y ) ( ) (

22

23 Example: Draw block diagram for thi Proce (PI control of liquid level regulatory loop)

24 In general

25 Aumption. q, varie with time; q 2 i contant. 2. ontant denity and x-ectional area of tank, A. q 3 f (h) 3. (for uncontrolled proce) 4. The tranmitter and control valve have negligible dynamic (compared with dynamic of tank). 5. Ideal PI controller i ued (direct-acting). For thee aumption, the tranfer function are: V P L () () V () A () A I 0 ()

26 P V L Q H D Y V I A A D Y v I V I I A D Y v I V I A 2 () 2 2 The cloed-loop tranfer function i: Subtitute, Simplify, haracteritic Equation: Recall the tandard 2 nd Order Tranfer Function: (2) (3) (4) (5)

27 To place Eqn. (4) in the ame form a the denominator of the T.F. in Eqn. (5), divide by c, V, : AI 2 V 2 V A I 0 omparing coefficient (5) and (6) give: 2 2 Subtitute, I AI V I I 2 AI 0 For 0 < <, cloed-loop repone i ocillatory. Thu decreaed degree of ocillation by increaing c or I (for contant v,, and A). V