Process Control & Instrumentation (CH 3040) - Yolaarunkt.yolasite.com/resources/ch3040/lectures/controldesign1... · Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation

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



Closed-loop & Open-loop system

Open-loop system is obtained by cutting off the feedback to

the set-point comparator

2

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



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?)

3

G̃OL(s) = Gc0GaGpGsens

=! GCL(s) =Ym(s)

R(s)

!!!!c.l.

=KcG̃OL(s)

1 +KcG̃OL(s)



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

4

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)



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.

5

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)



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!

6



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

7

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



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

8

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



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

9

-"

Regions of greater

relative stability

Im(s)

Re(s)



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:

10

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



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)

11

Kc = 0.4, a = 2 Kc = 2, a = 0.5



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.

12

G̃OL(s) =s(5s2 + 6s+ 1 + 2K!

c )

2K!c

G̃OL(s) =2K!

c

s(5s2 + 6s+ 1 + 2K!c )



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!

13



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.

14



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)

15

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



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)

16

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.



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

17

Gc(s) = Kc; Gp(s) =2

10s+ 1

Gain crossover frequency

Gc(s) = Kc

!1 +

1

0.5s

"; Gp(s) =

2

10s+ 1



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

18

Gp(s) =2

(5s+ 1)(s+ 1)Gc(s) = Kc = 1 Gc(s) = Kc

!1 +

1

!Is

"; Kc = 1 !I = 0.5



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

19

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



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

20

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)



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!)

21

Gp(s) =2

10s+ 1e!2s Gc(s) = Kc

Documents

Process Control & Instrumentation (CH 3040) - Yolaarunkt.yolasite.com/resources/ch3040/lectures/controldesign1... · Arun K. Tangirala (IIT Madras) CH 3040: Process Control & Instrumentation