1. CHE517Advanced Process Control Prof. Shi-Shang Jang Chemical Engineering Department National Tsing-Hua University Hsin Chu, Taiwan

2. Course Description • Course: CHE517 Advanced Process Control • Instructor: Professor Shi-Shang Jang • Text: Seborg, D.E., Process Dynamics and Control, 2nd Ed., Wiley, USA, 2003. • Course Objective: To study the application of advanced control methods to chemical and electronic manufacturing processes • Course Policies: One Exam(40%), a final project (30%) and biweekly homework(30%)

3. Course Outline • Review of Feedback Control System • Dynamic Simulation Using MATLAB and Simu-link • Feedforward Control and Cascade Control • Selective Control System • Time Delay Compensation • Multivariable Control

4. Course Outline - Continued 7. Computer Process Control 8. Model Predictive Control 9. R2R Process Control

5. Chapter 1 Review of Feedback Control Systems • Feedback Control • Terminology • Modeling • Transfer Functions • P, PI, PID Controllers • Block Diagram Analysis • Stability • Frequency Response • Stability in Frequency Domain

6. Transmitter Controller Set point Temp sensor Heat loss stream condensate Feedback Control Examples: • Room temperature control • Automatic cruise control • Steering an automobile • Supply and demand of chemical engineers

7. Feedback Control-block diagram Manipulated variable Terminology: • Set point • Manipulated variable (MV) • Controlled variable (CV) • Disturbance or load (DV) • Process • controller disturbance error process Controlled variable Controller + Σ Set point - Sensor + transmitter Measured value

8. Transmitter Controller Set point Temp sensor Heat loss stream condensate Instrumentation • Signal Transmission: Pneumatic 3-15psig, safe longer time lags, reliable • Electronic 4-20mA, current, fast, easy to interface with computers, may be sensitive to magnetic and/or electric fields • Transducers: to transform the signals between two types of signals, I/P: current to pneumatic, P/I, pneumatic to current

9. Q=UA(T-T0) Mass M Cp T Q Modeling Rate of accumulation = Input – output + generation – consumption At steady state : let T = TS and Q = QS  0 = QS– UA(TS - T0S) Deviation variables : let T = TS+Td , Q = QS+Qd , T0 = T0s+T0d Then : If system is at steady state initially Td(0) = 0

10. + ∑ qd(S) Td(S) + Tod(S) Transfer Functions M Cp S Td(S) = qd(S) - U A (Td(S) – Tod(S)) Or Laplace Transforming:

11. F0 ρ0 CA0 T0 A  B rA = - KCA mol/ft3 K = αe-E/RT F ρ CA T T V ρ CA CB steam condensate Non-isothermal CSTR • Total mass balance: • Mass balance : • Energy balance : • Initial conditions : V(t=0) = Vi , T(t=0) = Ti , CA(t=0) = CAi • Input variables : F0 , CA0 , T0 ,F

12. F(X) aX+b X X0 -△ X0 X0+△ -△ 0 △ Linearization of a Function

13. Linearization

14. Linearization of Non-isothermal CSTR

15. Common Transfer FunctionsK=Gain; τ=time constant; ζ=damping factor; D=delay • First Order System • Second Order System • First Order Plus Time Delay • Second Order Plus Time Delay

16. e(s) m(s) Kc e(s) e(s) m(s) m(s) Transfer Functions of Controllers m(s) = Kc[ e(s) ] e = Tspt - T • Proportional Control (P) • Proportional Integral Control (PI) • Proportional-Integral-Derivative Control (PID)

17. The Stability of a Linear System • Given a linear system y(s)/u(s)= G(s)=N(s)/D(s) where N, D are polynomials • A linear system is stable if and only if all the roots of D(s) is at LHS, i.e., the real parts of the roots of D(s) are negative.

18. Stability in a Complex Plane Im Exponential Decay with oscillatory Purdy oscillatory Exponential growth with oscillatory Fast Exponential growth Exponential Decay Re Fast Decay Slow Decay Slow growth Purdy oscillatory Stable (LHP) Unstable (RHP)

19. Partial Proof of the Theory • For example: y(s)/u(s)=K/(τs+1) • The root of D(s)=-1/τ • In time domain: τy’+y=ku(t) • The solution of this ODE can be derived by y(t)=e-t/τ [∫e1/τku(t)dt+c] • It is clear that if τ<0, limt→∞y →∞.

20. G1(S) U1(S) X1(S) + X(S) Σ + G2(S) U2(S) X2(S) Transfer functions in parallel X(S)= G1(S)*U1(S) + G2(S)*U2(S) X1(S) X2(S)

21. + Tset Kc Td Σ QS - control process Measuring device 1 Transfer function Block diagram Proportional control No measurement lags

22. Block Diagram Analysis L(S) GL(S) e + + Xs X(S) ∑ Gc(S) GP(S) ∑ m X1 + - Gm(S) Xm e = Xs – Xm m = Gc (S) e(s) = Gc e X1 = Gp m = Gp Gc e X = GL L + X1 = GL L + Gp Gc e Xm = Gm X = Gm GL L + Gp Gc e X = GL L + Gp Gc[Xs – Xm] = GL L + Gp Gc [Xs] – Gp Gc [Xm] =GL L + Gp Gc Xs – Gp Gc Gm X

23. Stability of a Closed Loop System • A closed loop system is stable if and only of the roots of its characteristic equation : 1+Gc(s)Gp(s)Gm(s)=0 are all in LHP

24. Level System

25. The jacketed CSTR W Set Point TRC FC 2A  B Tc Wc T, Ca

26. A Nonisothermal Jacketed CSTR • (i) Material balance of species A • (ii) Energy balance of the jacket • (iii) Energy balance for the reactor • (iv) Dependence of the rate constant on temperature

27. Linearization of Nonisothermal CSTR • CV=T(t) • MV=Wc(t) • It can be shown that

28. Process output Maximum slope △C τ D Dead time time Time constant A Practical Example –Temperature Control of a CSTRMethod of Reaction Curve

29. Ziegler-Nichols Reaction Curve Tuning Rule

30. △C D τ △m Kc= -579.2079 τi =3.33 D= 1 τ =13 k = -0.0202

32. setpoint

34. Ziegler-Nichols Ultimate Gain TuningFind the ultimate gain of the process Ku. The period of the oscillation is called ultimate period Pu

35. Measuring Controller Performance

36. Upper Limit of Designed Controller Parameters of PID Controllers • Q: Given a plant with a transfer function G(s), one implements a PID controller for closed loop control, what is the upper limit of its parameters? • A: The upper limit of a controller should be bounded at its closed loop stability.

37. Approaches • Direct Substitution for Kc • Root Locus method for Kc • Frequency Analysis for all parameters

38. ＋ ○ Kc － An Example

39. 1. Stability Limit by Direct Substitution • At the stability limit (maximum value of Kc permissible), roots cross over to the RHP. Hence when Kc=Ku, there are two roots on the imaginary axis s=±iω • (s+1)(s+2)(s+3)+Ku=0, and set s= ±iω, we have (iω+1)(iω+2)(iω+3)+Ku= 0, i.e. (6+Ku-6ω2)+i(11ω-ω3)=0. This can be true only if both real and imaginary parts vanishes: 11ω-ω3=0→ ω= ±√11 ; 6+Ku-6×11=0 →Ku=60

40. 2. Method of Root Locus Rlocus (sys,k) k(12) ans =69.6706

41. 3. Frequency Domain Analysis • Definitions: Given a transfer function G(s)=y(s)/x(s); Given x(t)=Asinωt; we have y(t) →Bsin(ωt+ψ) • We denote Amplitude Ratio=AR(ω) =B/A; Phase Angle=ψ(ω) • Both AR and ψ are function of frequency ω; we hence define AR and ψ is the frequency response of system G(s)

42. A sin(wt) B = sin(wt+f) An Example

43. Frequency Response of a first order system

44. Basic Theorem • Given a process with transfer function G(s); • AR(ω)=︳G(iω)︳ • φ(ω)=∠ G(iω) • Basically, G(iω)=a+ib

45. Example: First Order System

46. Corollary • If G(s)=G1(s)G2(s)G3(s) • Then AR(G)=AR(G1) AR(G2) AR(G3) • φ(G)=φ (G1) +φ (G2)+φ (G3) • Proof: Omitted

47. Example

48. Bode Plot: An exampleG(s)=1/(s+1)(s+2)(s+3)where db=20log10(AR)

49. Nyquist Plotsys=tf(num,den)NYQUIST(sys,{wmin,wmax}))

50. Nyquist Stability Criteria • Given G(iω), assume that at a frequency ωu, such that φ=-180° and one has AR(ωu), the sufficient and necessary condition of the stability of the closed loop of G(s) is such that: AR(ωu) ≦1