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Chapter 10 Stability Analysis and Controller Tuning

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1 Chapter 10 Stability Analysis and Controller Tuning
※ Bounded-input bounded-output (BIBO) stability * Ex 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 The series chemical reactors with PI controller

8 @ Known values Process Controller

9 @ Formulation& stability

10 ◎ Criterion of stability
※ Direct substitution method

11 The response of controlled output:

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)

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

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= and asymptotes with angles, =60°, 180 °, 300 °. Calculate the breakaway point by

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

51 Example B.5 Depict the root locus of a characteristic equation (heat exchanger control loop with PI control) S1. OLTF

52 S2. Following rules

53 S3. Depict root locus

54 *Exercises

55 Ans. 8.1


57 Ans. 8.2




61 * Dynamic responses for various pole locations

62 * Which is good method for stability analysis

63 ※ Bode method A brief review: OLTF Frequency response

64 ◎ Stability criterion


66 * Frequency response stability criterion
Determining the frequency at which the phase angle of OLTF is –180°(–π) and AR of OLTF at that frequency Ex. C.1 Heat exchanger control system (Ex. A.1)

67 S1. OLTF S2. Find MR and θ S3. Bode plot in Fig to estimate ω=0.219 by θ= –180° and decide MR=0.0524


69 S4. Decide Kc as AR=1 * Stability vs. controller gain In Bode plot, as θ= –180° both ω and MR are determined. Moreover, ω = ωu and Kcu can be obtained.

70 Ex. C.2 Analysis of stability for a OTLF
S1. MR and θ

71 S2. Show Bode plot (MR vs.  &  vs.  )

72 S3. Find ωu and Kcu ωu=0.16 by = –π Kcu =12.8 Ex. C.3 The same process with PD controller and =0.1 (S1) OLTF

73 (S2) By Fig u=0.53 and MR=0.038 Kcu=33 and u=0.53


75 Ex. 10.7 (S1) Bode plot (AR vs.  &  vs.  ) for Kc=1

76 (S2) Stability vs. controller gain Kc
Ex Determine whether this system is stable.

77 (S1) Bode plot for Kc=15 and TI=1
(S2) Since the AR>1 at , the system is unstable.

78 P1. Bode plot for the first-order system

79 P2. Bode plot for the second-order system

80 Ex. 10.9 Determine AR and  of the following transfer function at


82 * Controller tuning based on Z-N closed-loop tuning method
S1. Calculating c by setting Kc=1 and then determine Ku and Pu where ARc=

83 S2. Controller tuning constants
Ex Calculate controller tuning constants for a process, Gp(s)=0039/(5s+1)3, by uning the Z-N method S1.

84 S2. Bode plot

85 S3. Tuning constants

86 S4. Closed-loop test

87 Ex. 10.14 Integral mode tend to destabilize the control system

88 @ Effect of modeling errors on stability
Gain margin (GM): Total loop gain increase to make the system just unstable. The controller gain that yields a gain margin * Typical specification: GM2 If P controller with GM=2 is the same as the Z-N tuning.

89 (2) Phase margin (PM): * Typical specification: PM>45° Ex. D.1 Consider the same heat exchanger to tune a P controller for specifications (Ex. C.2) (a) While GM=2

90 (b) PM= 45°θ= –135°. By Fig. in Ex. C.2, we can find

91 ※ Polar plot The polar plot is a graph of the complex-valued function G(i) as  goes from 0 to . Ex. E.1 Consider the amplitude ratio and the phase angle angle of first-order lag are given as


93 Ex. E.2 Consider the amplitude ratio and the phase angle angle of
second-order lag are given as


95 Ex. E.3 Consider the second-order system with tuning Kc

96 Ex. E.4. Consider the amplitude ratio and the phase angle angle of
pure dead time system are given as

97 ※ Conformal mapping

98 ※ Nyquist stability criterion (Nyquist plot)
Ex. E.5 Consider a closed-loop system, its OLTF is given as


100 Unstable stable Kc>23.8 Marginal stable Kc=23.8 stable Kc<23.8

101 Exercises: Q.10.11 Q.10.15

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