Process Control & Instrumentation (CH 3040) -...
Transcript of Process Control & Instrumentation (CH 3040) -...
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Process Control & Instrumentation(CH 3040)
Arun K. Tangirala
Department of Chemical Engineering, IIT MadrasJanuary - April 2010
Lectures: Mon,Tue, Wed, Fri Extra class: Thu
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Closed-loop & Open-loop system
Open-loop system is obtained by cutting off the feedback to
the set-point comparator
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Gc GpR(s) Y(s)
D(s)
Ga
Sensor
Gsensor
+ ++-
Ym(s)
Controller Actuator Process
E(s) U(s) U*(s) Y*(s)
Closed-loop System
Gc GpR(s) Y(s)
D(s)
Ga
Gsensor
++
Ym(s)
Controller Actuator Process
E(s) U(s) U*(s) Y*(s)
Open-loop System
GCL(s) =Ym(s)
R(s)=
GcGaGpGsens
1 +GcGaGpGsens
GOL(s) =Ym(s)
R(s)= GcGaGpGsens
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Open-loop system for SISO controller design
The open-loop system consists of all the elements from the set-point to
the sensed signal (the true output Y*(s) is never (rarely) known!)
Assume controller has the form Gc(s) = KcGc0(s) where Kc is a control loop
parameter whose value is to be determined (tuned) and Gc0(s) is known
Then, the poles and zeros of the open-loop system are the same regardless
of the value of Kc (except zero). For design, we separate the part that does
not contain Kc
Thus,
The C.E. shows the effects of the parameter Kc on closed-loop stability
• What if Gc(s) is not in a factored form as above? (or Gc0(s) contains a tuning parameter?)
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G̃OL(s) = Gc0GaGpGsens
=! GCL(s) =Ym(s)
R(s)
!!!!c.l.
=KcG̃OL(s)
1 +KcG̃OL(s)
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Effective open-loop system
Suppose the controller has the form
Then, neglecting actuator and sensor dynamics, the closed-loop C.E. is:
If !I is known and Kc has to be tuned, then the o.l. system for design is:
If Kc is known and !I has to be tuned, the C.E. is re-written as
by comparison with the standard design equation.
Thus, w.r.t. the parameter !I, the open-loop t.f. is different (only for design!)
• In practice, therefore, GOL(s) is the effective open-loop transfer function with respect to the
tuning parameter
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Gc(s) = Kc
!1 +
1
!Is
"PI controller
1 +Kc
!1 +
1
!Is
"Gp(s) = 0
KcGp(s) + !I(1 +KcGp(s))s = 0
=! 1 + !I(1 +KcGp(s))s
KcGp(s)= 0
so that
G̃OL(s) =
!1 +
1
!Is
"Gp(s)
G̃OL(s) =s(1 +KcGp(s))
KcGp(s)
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Root Locus Plot
The root locus plot consists of the
locus of the poles of closed-loop
system when a control loop tuning
parameter is varied.
The basic equation that determines
the locus is the c.l. C.E.:
• Kc is a tuning parameter whose effect on
the closed-loop poles we wish to analyze
• is the equivalent open-loop
transfer function ( w.r.t. the
parameter Kc)
• The locus is the path of the roots of the
C.E. as Kc is varied.
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Root Locus plot
Gc(s) = Kc; Gp(s) =2
10s+ 1; Gsensor(s) = 1
1 +KcG̃OL(s) = 0
G̃OL(s)
The ‘x’ mark denotes the open-loop pole while the pink square denotes the closed-loop pole (here for Kc = 1)
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Compensated and Uncompensated system
Control strategies can be viewed as a concept of “compensation” for what
the process does not possess
• That is, the characteristics of process usually lack something that we desire it to have
• The controller in essence “compensates” for what the process does not have
Since the process characteristics are influenced by zeros and poles, the
compensation is in terms of adding zeros and poles to the open-loop
system
• The controller carries these additional zeros and poles.
In this sense, a process with a controller of the form Gc(s) = Kc is said to be
uncompensated
The controller design problem is that of choosing the gain, zeros and poles
of the compensator so that the stability and performance requirements of
the c.l. system are met!
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Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Cascade and Feedback compensations
Compensations can occur as (i) cascade (series) and / or (ii) feedback
Three types of compensators exist: lag, lead and lag-lead
Among the cascade compensators, the PI, PD and PID controllers are commonly used
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K GpR(s) Y(s)
Unity feedback uncompensated system
+
- Gc GpR(s) Y(s)+
-
Unity feedback compensated system
K GpR(s) Y(s)
Feedback compensated system
+
-
H
Gc GpR(s) Y(s)
Cascade and feedback compensated system
+
-
H
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Consider a simple first-order process with two different controllers
• The offset due to a P controller is true regardless of the value of Kc, however large it may be (but finite)
The unit step change in r(t) => R(s) = 1/s (a pole at the origin) - this should be
contained somewhere in the loop, either with the process or the controller
• Since the process does not contain this pole, it is compensated by the PI controller -
hence the name!
Notion of compensation
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Gp(s) =2
10s+ 1Gc(s) = Kc = 2
A unit step in the set-point is introduced
Offset = 0.2
The uncompensated system (with the P controller) produces an offset to step changes in s.p.
The PI controller produces no offset (so long as it produces a stable closed-loop system!)
Gc(s) = Kc
!1 +
1
!Is
"; Kc = 2; !I = 0.5
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Relative Stability
The relative stability of a system is the distance into the LHP from the
imaginary axis to the nearest characteristic root or roots
• Example: G(s) = 1/(s+2)(s+4) has a relative stability of 2 units
For a system’s natural response to decay as quickly as e-"t, a system must
have a relative stability of at least " units.
• The characteristic roots must be on or to the left of the line Re(s) = -"
The relative stability is a useful way of stating the stability and performance
requirements in a controller design problem
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-"
Regions of greater
relative stability
Im(s)
Re(s)
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Effects of adding zeros / poles on stability
The root locus plot can be used to study the effect of adding a zero and
pole, to the open-loop system - particularly with a PI controller
• We shall assume all feedback systems to be of unity feedback (unless otherwise stated)
Example:
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Gp(s) =2
(5s+ 1)(s+ 1)
Gc(s) = Kc
!1 +
1
!Is
"= Kc
!s+ a
s
"
a = 2
Gc(s) = Kc
The closed-loop poles shown correspond to Kc = 2
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Effects of adding zeros / poles … contd.
Clearly the addition of a pole (at the origin) has tremendous impact on the closed-loop
stability and reduces the admissible values of Kc
The uncompensated gain has to be reduced to give some stability margin
Changing the zero location to a position between the poles has removed stability
concerns, but certainly reduced the relative stability (w.r.t. the uncompensated system)
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Kc = 0.4, a = 2 Kc = 2, a = 0.5
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Tuning the zero location in a PI
The effect of zero-location (choice of ) for a fixed Kc can be studied by
constructing the root-locus for the effective open-loop transfer function
For the example, w.r.t the tuning parameter !I, the effective o.l. system is
where K*c is the controller gain for the uncompensated system
• Observe that the effective open-loop system is improper! - not useful for analysis
Instead, we can easily arrive at the effective o.l. system based on the zero
location s = -a,
• The effective o.l. system is
The effective open-loop system can then be analyzed for the effect of the
zero location.
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G̃OL(s) =s(5s2 + 6s+ 1 + 2K!
c )
2K!c
G̃OL(s) =2K!
c
s(5s2 + 6s+ 1 + 2K!c )
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Presence of delays
Root locus plots give very useful insights into the time-domain behaviour of
a control loop system as a single parameter is varied
• We can easily understand the limitations of a controller in terms of stability & performance
• It is easy to understand how the addition of a zero and/or pole through the compensator,
influences the allowable values of a tuning parameter w.r.t. stability
However, the presence of delays in systems pose tremendous challenges to
the construction of root locus plots
Pade’s (and other) approximations of delays could be used in
accommodating delays during the root locus construction
• However, such approximations serve in a very limited way.
• Any root locus or performance analysis may only be trusted for very small delays
Frequency response methods provide excellent support in handling delays!
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Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Frequency Response Analysis
The frequency response analysis approach to controller design for LTI
systems is the most powerful approach
• The approach can handle delays without any approximations and unstable systems.
• Provides measures for margins of stability (how “far” the system is from instability)
• Tells us over which ranges the controllers can effectively deliver good performance
• Provide the same information as the time-domain methods
There are mainly two approaches based on two different criteria
• Bode’s stability criterion - makes use of Bode plots of FRF
• Nyquist’s stability criterion - makes use of Nyquist plots of FRF
The Bode plots are easier to use but unstable systems cannot be handled
The Nyquist criterion is the most powerful stability criterion for LTI systems
and can handle unstable systems as well.
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Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Basic idea
Recall how the root locus plots showed the presence of a crossover point on
the IA whenever a locus entered the RHP (regions of instability)
• Thus, it is sufficient to determine the presence of a crossover frequency
The presence of a crossover frequency can be easily determined by checking
for a value of Kc (call it Kcu) such that a pair of roots are imaginary
• Assuming that the root locus is monotonic (i.e., values > Kcu will take the locus into RHP)
Thus, Kcu (ultimate gain, sometimes denoted by Ku) satisfies
• The frequency that satisfies the above equation is known as the crossover frequency #c
The equation above can be written in two parts (as in root locus analysis)
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1 +KcuG̃OL(j!) = 0
Kcu =1
|G̃OL(j!)|!G̃OL(j!) = !180!
Magnitude criterion (weaker condition)(gives gain crossover frequency)
Phase criterion (stronger condition)(gives phase crossover frequency)
AssumeKcu > 0
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Bode’s stability criterion
It provides a necessary and sufficient condition for closed-loop stability
under the conditions stated above
The criterion can easily handle the presence of delays
In case the open-loop system does not satisfy the above conditions, the
Nyquist’s stability criterion can be invoked
Systems with multiple (phase and gain) crossover frequencies, some
modifications are required (see Hahn et al’s work)
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Consider an open-loop transfer function GOL = GcGaGpGsens that is strictly proper(more poles than zeros) and has no poles located on or to the right of imaginaryaxis with the exception of a single pole at the origin. Assume that the open-loopfrequency response has only a single phase crossover frequency !c and single gaincrossover frequency !g. Then the closed-loop system is stable if AROL(!c) < 1.Otherwise, it is unstable.
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Examples
First-order open-loop systems are always stable under unity feedback as well as with PI compensation
• The phase never reaches -180º (phase crossover frequency) - however, additional elements (such as delays, lags) can cause instability
Gain crossover frequency exists, but that is not a sufficient condition for instability
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Gc(s) = Kc; Gp(s) =2
10s+ 1
Gain crossover frequency
Gc(s) = Kc
!1 +
1
0.5s
"; Gp(s) =
2
10s+ 1
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Second-order system
Consider the same second-order process that we analyzed earlier
• The closed-loop system on the left is stable and can accommodate an additional phase lag of
104º before becoming unstable
• The closed-loop system on the right is unstable since AR > 1 at critical frequency
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Gp(s) =2
(5s+ 1)(s+ 1)Gc(s) = Kc = 1 Gc(s) = Kc
!1 +
1
!Is
"; Kc = 1 !I = 0.5
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Gain Margin
The margin is calculated as
• GM can also be expressed in decibels (dB)
Clearly, if GM > 1 (or GMdB > 0), then
closed-loop system is stable
• Expressed in dB, GM should always be positive for
stable closed-loop systems
The GM is also a measure of all gain
uncertainties that can be tolerated and
additional gain elements that can be included
before the c.l. system turns unstable
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The margin left for the gain (Kc) to increase with respect to AR = 1 at the phase crossover frequency (where ! = -180º)
GM =1
|G̃OL(!c)|=
1
AR(!c)
Gp(s) =2
(10s+ 1)(4s+ 1)(2s+ 1)
Gc(s) = Kc ; GM = 16 dB stable closed-loop system
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Phase Margin
The margin is calculated as
Clearly, if PM > 0, then the closed-loop
system is stable
• PM = 0 => marginally unstable
The PM is a measure of all phase
uncertainties that can be tolerated and
the additional phase (lag/lead) elements
that can be included before the c.l. system
turns unstable
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Gp(s) =2
(10s+ 1)(4s+ 1)(2s+ 1)
Gc(s) = Kc ; PM = 82.3º stable closed-loop system
The margin left for the phase (!) to decrease with respect to ! = -180º at the gain crossover frequency (where AR = 1)
PM = !G̃OL(!g)! (!180!)
= 180! + "G̃OL(!g)
Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation January-April 2010
Stability of Control Systems Root Locus Analysis & Design, Frequency Response Analysis & Design
Effects of delays
To the same first-order system, consider the inclusion of a delay term
The plot on the left shows the minimum stability margins (gain and phase margins)
The plot on the right shows all stability margins (the phase crosses -180º multiple
times due to the presence of delay!)
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Gp(s) =2
10s+ 1e!2s Gc(s) = Kc