Chapter 10 Stability Analysis and Controller Tuning

1. Chapter 10 Stability Analysis and Controller Tuning ※ Bounded-input bounded-output (BIBO) stability ＊ Ex. 10.1 A level process with P control

2. (S1) Models (S2) Solution by Laplace transform where

3. Note: • Stable if Kc<0 • Unstable Kc>0 • Steady state performance by

4. ＊ Ex. 10.3 A level process without control • Response to a sine flow disturbance • Response to a step flow disturbance

5. ※ Stability analysis

6. Note: Assume Gd(s) is stable.

7. ＊ Stability of linearized closed-loop systems Ex. 10.4 The series chemical reactors with PI controller

8. @ Known values • Process • Controller

9. @ Formulation& stability Stable

10. ◎ Criterion of stability ※ Direct substitution method

11. The response of controlled output: P1. P2.

12. P3.

13. ﹪Ultimate gain (Kcu): The controller gain at which this point of marginal instability is reached ﹪Ultimate period (Tu): It shows the period of the oscillation at the ultimate gain ＊ Using the direct substitution method by in the characteristic equation

15. Example A.1 Known transfer functions

16. Find: (1) Ultimate gain (2) Ultimate period S1. Characteristic eqn.

17. S2. Let at Kc=Kcu

19. Example A.2 S1.

20. S2. S3.

21. Example A.3 Find the following control loop: (1) Ultimate gain (2) Ultimate period

22. S1. The characteristic eqn. for H(s)=KT/(Ts+1) S2. Gc=-Kc to avoid the negative gains in the characteristic eqn.

23. S3. By direct substitution of at Kc=Kcu

24. ＊ Dead-time Since the direct substitution method fails when any of blocks on the loop contains deadt-ime term, an approximation to the dead-time transfer function is used. First-order Padé approximation:

25. Example A.4 Find the ultimate gain and frequency of first-order plus dead-time process S1. Closed-loop system with P control

26. S2. Using Pade approximation

27. S3. Using direct substitution method

28. Note: • The ultimate gain goes to infinite as the dead-time approach zero. • The ultimate frequency increases as the dead time decreases.

29. ※ Root locus A graphical technique consists of roots of characteristic equation and control loop parameter changes.

30. ＊Definition: Characteristic equation: Open-loop transfer function (OLTF): Generalized OLTF:

31. Example B.1: a characteristic equation is given S1. Decide open-loop poles and zeros by OLTF

32. S2. Depict by the polynomial (characteristic equation) Kc:1/3

33. S3. Analysis

34. Example B.2: a characteristic equation is given S1. Decide poles and zeros

35. S2. Depict by the polynomial (characteristic equation)

36. S3. Analysis

37. Example B.3: a characteristic equation is given S1. Decide poles and zeros S2. Depict by the polynomial (characteristic equation)

39. S3. Analysis

40. @ Review of complex number c=a+ib

41. Polar notations

42. P1. Multiplication for two complex numbers (c, p) P2. Division for two complex numbers (c, p)

43. @ Rules for root locus diagram • Characteristic equation • Magnitude and angle conditions

44. Since

45. Rule for searching roots of characteristic equation • Ex. A system have two OLTF poles (x) and one OLTF zero (o) • Note: If the angle condition is satisfied, then the point s1 is the part of the root locus

48. Example B.4 Depict the root locus of a characteristic equation (heat exchanger control loop with P control) S1. OLTF

49. S2. Rule for root locus • From rule 1 where the root locus exists are indicated. • From rule 2 indicate that the root locus is originated at the poles of OLTF. • n=3, three branches or loci are indicated. • Because m=0 (zeros), all loci approach infinity as Kc increases. • Determine CG=-0.155 and asymptotes with angles, =60°, 180 °, 300 °. • Calculate the breakaway point by

50. s= – 0.247 and –0.063 S3. Depict the possible root locus with ωu=0.22 (direct substitution method) and Kcu=24