Chapter 10 Stability Analysis and Controller Tuning

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Chapter 10 Stability Analysis and Controller
Tuning
? Bounded-input bounded-output (BIBO) stability
Ex. 10.1 A level process with P control
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(S1) Models (S2) Solution by Laplace transform
where
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• Note
• Stable if Kclt0
• Unstable Kcgt0
• Steady state performance by

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

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? Stability analysis
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Note Assume Gd(s) is stable.
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Stability of linearized closed-loop
systems Ex. 10.4 The series chemical reactors
with PI controller
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• _at_ Known values
• Process
• Controller

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_at_ Formulation stability
Stable
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? Criterion of stability ? Direct substitution
method
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The response of controlled output
P1. P2.
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P3.
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?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
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Example A.1 Known transfer functions
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Find (1) Ultimate gain (2) Ultimate
period S1. Characteristic eqn.
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S2. Let at KcKcu
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Example A.2
S1.
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S2.
S3.
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Example A.3 Find the following control loop (1)
Ultimate gain (2) Ultimate period
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S1. The characteristic eqn. for
H(s)KT/(?Ts1) S2. Gc-Kc to avoid the
negative gains in the characteristic eqn.
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S3. By direct substitution of at
KcKcu
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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
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Example A.4 Find the ultimate gain and frequency
of first-order plus dead-time process
S1. Closed-loop system with P control
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S2. Using Pade approximation
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S3. Using direct substitution method
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• Note
• The ultimate gain goes to infinite as the
  dead-time approach zero.
• The ultimate frequency increases as the dead time
  decreases.

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? Root locus
A graphical technique consists of roots of
characteristic equation and control loop
parameter changes.
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Definition Characteristic equation Open-loop
transfer function (OLTF) Generalized OLTF
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Example B.1 a characteristic equation is given
S1. Decide open-loop poles and zeros by OLTF
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S2. Depict by the polynomial (characteristic
equation) Kc1/3
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S3. Analysis
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Example B.2 a characteristic equation is given
S1. Decide poles and zeros
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S2. Depict by the polynomial (characteristic
equation)
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S3. Analysis
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Example B.3 a characteristic equation is given
S1. Decide poles and zeros
S2. Depict by the polynomial (characteristic
equation)
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S3. Analysis
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_at_ Review of complex number caib
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?Polar notations
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P1. Multiplication for two complex numbers (c,
p) P2. Division for two complex numbers (c, p)
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• _at_ Rules for root locus diagram
• Characteristic equation
• Magnitude and angle conditions

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Since
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• 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

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Example B.4 Depict the root locus of a
characteristic equation (heat exchanger control
loop with P control) S1. OLTF
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• 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.
• n3, three branches or loci are indicated.
• Because m0 (zeros), all loci approach infinity
  as Kc increases.
• Determine CG-0.155 and asymptotes with angles,
  ?60, 180 , 300 .
• Calculate the breakaway point by

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s 0.247 and 0.063
S3. Depict the possible root locus with ?u0.22
(direct substitution method) and Kcu24
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Example B.5 Depict the root locus of a
characteristic equation (heat exchanger control
loop with PI control) S1. OLTF
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S2. Following rules
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S3. Depict root locus
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Exercises
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Ans. 8.1
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Ans. 8.2
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Dynamic responses for various pole locations
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Which is good method for stability analysis
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• ? Bode method
• A brief review
• OLTF
• Frequency response

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? Stability criterion
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Frequency response stability criterion Determini
ng the frequency at which the phase angle of OLTF
is 180(p) and AR of OLTF at that frequency
Ex. C.1 Heat exchanger control system (Ex. A.1)
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S1. OLTF
S2. Find MR and ? S3. Bode plot in Fig.
9-2.3 to estimate ?0.219 by ? 180 and
decide MR0.0524
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S4. Decide Kc as AR1

Stability vs. controller gain In Bode plot,
as ? 180 both ? and MR are determined.
Moreover, ? ?u and Kcu can be obtained.
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Ex. C.2 Analysis of stability for a OTLF S1.
MR and ?
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S2. Show Bode plot (MR vs. ? ? vs. ? )
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S3. Find ?u and Kcu ?u0.16 by ? p
Kcu 12.8
Ex. C.3 The same process with PD
controller and ?0.1 (S1) OLTF
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(S2) By Fig. 9-2.5 ?u0.53 and
MR0.038 Kcu33 and ?u0.53
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Ex. 10.7
(S1) Bode plot (AR vs. ? ? vs. ? ) for Kc1
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(S2) Stability vs. controller gain Kc
Ex. 10.8 Determine whether this system is stable.
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(S1) Bode plot for Kc15 and TI1
(S2) Since the ARgt1 at , the
system is unstable.
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P1. Bode plot for the first-order system
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P2. Bode plot for the second-order system
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Ex. 10.9 Determine AR and ? of the following
transfer function at
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Controller tuning based on Z-N closed-loop
tuning method S1. Calculating ?c by setting
Kc1 and then determine Ku and Pu where
ARc
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S2. Controller tuning constants
Ex. 10.10 Calculate controller tuning constants
for a process, Gp(s)0039/(5s1)3, by uning the
Z-N method S1.
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S2. Bode plot
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S3. Tuning constants
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S4. Closed-loop test
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Ex. 10.14 Integral mode tend to destabilize the
control system
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_at_ 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 GM2 is the same as the Z-N
  tuning.

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(2) Phase margin (PM) Typical
specification PMgt45
Ex. D.1 Consider the same heat exchanger to tune
a P controller for specifications (Ex. C.2) (a)
While GM2
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(b) PM 45?? 135. By Fig. in Ex. C.2, we can
find and
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? 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
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Ex. E.2 Consider the amplitude ratio and the
phase angle angle of second-order lag are
given as
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Ex. E.3 Consider the second-order system with
tuning Kc
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Ex. E.4. Consider the amplitude ratio and the
phase angle angle of pure dead time system
are given as
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? Conformal mapping
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? Nyquist stability criterion (Nyquist plot)
Ex. E.5 Consider a closed-loop system, its OLTF
is given as
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Unstable stable Kcgt23.8
Marginal stable Kc23.8
stable Kclt23.8
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• Exercises
• Q.10.11
• Q.10.15