digital control 1

Digital Control SystemsCCS-532

Lecture - 10

Ref: Chapter 13: Nise, N. S. Control System Engineering

Chapter 13: Dorf, R. C. & Bishop, R. H. , Modern Control Systems

Chapter 2, 11: Benjamin C. Kuo. , Automatic Control Systems

Dr Pavan ChakrabortyIIIT-Allahabad

Indian Institute of Information Technology - Allahabad

Digital Control Systems

Digital/Discrete Control

• More useful for computer systems• Time is discrete

– denoted k instead of t

• Main tool is z-transform

– f(k) ® F(z) , where z is complex– Analogous to Laplace transform for s-domain

• Root-locus analysis has similar flavour– Insights are slightly different

0

)()()]([k

kzkfzFkfZ

Figure 13.1Conversion of antenna azimuth position control system from:a. analog control tob. digital control


1. Reduced Cost,

2. Flexibility in response to design changes

3. Noise immunity

Figure 13.2a. Placement of the digital computer within the loop;b. Detailed block diagram showing placement of A/D and D/A converters


Figure 13.3 Digital-to-analog conver ter (DAC)


ADC – 2 step process and is not instantaneous.1. Converted to a sampled signal.2. Converted to a sequence of binary numbers

DAC & ADC

c. conversion ofsamples to digitalnumbers

Figure 13.4Steps in ADC

a. analog signal;

Sampling Rate must be at least twice the bandwidth of the signal, or else there will be distortion.

This minimum sampling frequency is called the “Nyquist”sampling rate.


b. analog signal after sample-and-hold;


Figure 13.5 Two views of uniform-rate sampling:

a. Switch opening and closing;

k

WT TkTtukTtutftstftfW

)()()(*

b. Product of time waveform and sampling waveform


k


)()()(*

interval. sampling during constant )(tfTTW

])()([)(*

k


Since the Eq. (above) is a product of 2 time fn., taking the LAPLACE TRANSFORM in order to find a TRANSFER FUNCTION is not simple.

Simplification )()( kTftf

Small TW

kTs

k

sT

k

sTkTskTs

T es

ekTf

s

e

s

ekTfsF

ww

W

1)(*


kTs

k

sT

k

sTkTskTs

T es

ekTf

s

e

s

ekTfsF

ww

W

1)(*

kTs

k

ww

T es

sTsT

kTfsFW

......!2

11

)(

2

*

expantion, series with Replacing sTwe

Small TW

kTsw

k

kTs

k

wT eTkTfe

s

sTkTfsF

W

)(*

Inverse Laplace


Fig13.6 b. Product of

time waveform and sampling waveform

Figure 13.6Model of sampling with a uniform rectangular pulse train


Figure 13.7 Ideal sampling and the zero-order hold


Basic Concepts• Consider a sequence of values: {xk : k = 0,1,2,... }• These may be samples of a function x(t), sampled at

instants t = kT; thus xk = x(kT).

• The Z transform is simply a polynomial in z having the xk as coefficients:

0

)(k

kzkfkfZzF


Z Transform

Uses of Z-transform in this course

• Analysis of Discrete & Sampled-data Systems

• Transfer Function & Block Diagram Representations

• Exploit the z-plane– Dynamic Analysis– Root Locus Design


Fundamental Functions

• Define the impulse function: {k} = {1, 0, 0, 0,....}

1)( kZz

• Define the unit step function: {uk} = {1, 1, 1, 1,....}

11

11

0

z

z

zzuZzU

k

kk

(Convergent for |z| < 1)Indian Institute of Information Technology - Allahabad

Z-transform of

Consider: constant = and , akau kk ,2,1,0

az

z

azazaz

azzauZk

k

k

kkk

for 11

1 11

0

1

0

Recall:Geometric SeriesRecall:Geometric Series

11

1

0

xx

xk

k for

ka


Z-transform of

Consider: constant = and , akkau kk ,2,1,0

221

1

121

11

1

1

11

1}{

}{}{}{

az

az

az

az

zaza

azda

daaZ

da

da

ada

daZkaaZkaZuZ

k

kkkk

MATLAB: See ztrans and iztrans commands.

We will use these two functions as generating functions.

MATLAB: See ztrans and iztrans commands.

We will use these two functions as generating functions.

kk kau


Generating common functions using

• a=1 yields the unit step function

• a=exp(+/-bT) yields the exponential function

ka

1z

z

bTez

z


Generating the Z-transform of common functions

• yields sinusoids

)}sin({)}cos({

cos2

sincos22

22

*

*

*

*

kTrjZkTrZ

rzTrz

TjrTrzz

rrerez

rezz

rez

rez

rez

zTjTj

Tj

Tj

Tj

Tj

Note: To ensure unambiguous poles on z-plane

wT<pi or

2pi/T=ws > 2w

the “sampling theorem”

Note: To ensure unambiguous poles on z-plane

wT<pi or

2pi/T=ws > 2w

the “sampling theorem”

Tjera


Generating common functions

using• a=1 yields the unit ramp function

•

21

z

TzkTZkTZ

kk kau

See (Appendix G of Dorf and Bishop) for the Properties of z-Transforms See (Appendix G of Dorf and Bishop) for the Properties of z-Transforms

2}{bT

bTbkT

ez

TzekTeZ

bTea


Z-Transform: Forward Shift or1-stage advance

0

001

10

)1(1

0

1)1(1

011

)(

}{

zuzzU

zuzuzuz

zuzzuz

zzuzuuZ

m

mm

m

mm

k

kk

k

kk

k

kkk


Z-Transform: Forward Shift or2-stage advance

02

12

10

112

)(

)(

}{}{

uzzuzUz

zuzuzzUz

zuuzZuZ kk

In general:In general:

1

0

)(}{n

m

mm

nnnk zuzzUzuZ


Delay/Shift Property• Let y(t) = x(t-T) (delayed by T and truncated at t = T)

yk = y(kT) = x(kT-T) = x((k-1)T) = xk-1 ; y0 = 0

1

11

)(k

kk

k

kkk zxzyyZzY

• Let j = k-1 ; k = j + 1

)()( 1

0

1

0

1 zXzzxzzxzYj

jj

j

jj

• The values in the sequence, the coefficients of the polynomial, slide one position to the right, shifting in a zero.

Indian Institute of Information Technology - AllahabadIndian Institute of Information Technology - Allahabad


Some Important Teorems of the z-Transform


No PolesNo Poles

Table 13.2 z-transform theorems


The Laplace Connection

• Consider the Laplace Transforms of x(t) and y(t): sXeTtxLtyLsY Ts

• Equate the transform domain delay operators:Tsez 1 Tsez

• Examine s-plane to z-plane mapping . . .


zeTs


S-Plane to Z-Plane Mapping

Anything in the Alias/Overlay region in the S-Plane will be overlaid on the Z-

Plane along with the contents of the strip between s=± j/T. In order to avoid aliasing, there must be nothing in this region, i.e. there must be no signals present with radian frequencies higher than f /T, or cyclic frequencies higher than f = 1/2T. Stated another way, the sampling frequency must be at least twice the highest frequency present (Nyquist rate).

Figure 13.13Mapping regions of the s-plane onto the z-plane


Mapping Poles and ZerosA point in the Z-plane r e jwill map to a point in the S-plane according to:

T

rs

lnRe

Ts

Im

Conjugate roots will generate a real valued polynomial in s of the form:

22 2 nnss

2ln

1

r

Tn

T

r

n ln


Example 1: Running Average Algorithm

Transfer Function

4

23321

4

1

4

1

z

zzzzX

zzzzXzY

4

23

4

1

z

zzzz

X

Y

4321

kkkkk

xxxxy

Note: Each [Z-1] block can be thought of as a memory cell, storing the previously applied value.

(Non-Recursive)

Z Transform


Block Diagram

Example 2: Trapezoidal Integrator

211

Txxyy kkkk

2

11 TzXzzXzYzzY

21

1

21

11

1 T

z

zzX

T

z

zzXzY

(Recursive)

Z Transform

1

1

2 z

zT

zX

zY


Block Diagram Transfer Function

Ex. 2 (cont) Block Diagram Manipulation

Intuitive Structure

Equivalent Structure

Explicit representation of xk-1 and yk-1 has been lost, but memory element usage has been reduced from two to one.


Ex. 2 (cont) More Block Diagram Manipulation

1

1

2 z

zT

zX

zY


Note that the final form is equivalent to a rectangular integrator with an additive forward path. In a PI compensator, this path can be absorbed by the proportional term, so there is no advantage to be gained by implementing a trapezoidal integrator.

Table 13.1 Partial table of z- and s-transforms



Inverse z-Transform

Y(z) is y(kT); not y(t)

1. Partial-fraction expansion.

2. Power-series / Synthetic Division method.

3. The inverse formula

Common approaches for taking the inverse z-transform

Partial Fraction Expansion: decompose U(z) into a linear combination of easily inverted z-transforms.

• Power-series method / Synthetic Division: divide polynomials to obtain a power series in z-1.

MATLAB’s “residue” command computes the partial fraction expansion.

MATLAB’s “residue” command computes the partial fraction expansion.



1.Partial-fraction expansion.

If Y(z) has at least one zero at z=0, the paritial-fraction expansion of Y(z)/z should be performed.

Partial fraction expansion method

.1

)(

)()(

1

rr

r

m

jj

r

rj

i i

i

mz

a(z)r

a(z)i

Czz

zB

zz

zA

za

zbzU

r

ty multiplici has root

of roots repeated all includes

of roots distinct all includes

where

The coefficients (residues) {A}, {B} and C can be calculated by hand for low-order systems or by expanding U(z)/z in a partial fraction expansion using “residue” and multiplying by z in high-order systems.

The coefficients (residues) {A}, {B} and C can be calculated by hand for low-order systems or by expanding U(z)/z in a partial fraction expansion using “residue” and multiplying by z in high-order systems.


Power-series method / Synthetic division method

Represent U(z) as a power series in z-1:

Since,

22

110)( zuzuuzU

otherwise

for

0

1][1 mk

mkzZ m

]2[]1[][ 210 kukukuuk

N.B.: Synthetic division method is useful for a limited number of values in a sequence or as a quick check on other methods

N.B.: Synthetic division method is useful for a limited number of values in a sequence or as a quick check on other methods


2. Power-series method.

2-171



3. The inverse formula


Ap

plicati

on

of

the z

-Tra

nsfo

rm t

o t

he S

olu

tion

of

Lin

ear

Diff

ere

nce

Eq

uati

on

The Transfer FunctionDefinition: The transfer function of a system is the ratio of the transform of the output of the system to to that of its input with zero initial conditions.

nnn

nn

nn

nn

nn

nn

azaz

bzbzb

zaza

zbzbb

zE

zU

zEzbzEzbzEb

zUzazUzazU

11

210

11

110

110

11

1)(

)(

)()()(

)()()(

Using z-1 as the delay operator,

nknkknknkk ebebebuauau 11011


Relation between transfer function and pulse response

Consider H(z)=U(z)/E(z) with E(z) =1that is e[k]=1 for k=0 and zero elsewhere.Then, U(z) = H(z)

The z-transform of the unit-pulse response of a discrete system is its transfer function.


General input-output relation for linear, stationary discrete systems

U(z) = H(z) E(z)

H(z) is a rational function (ratio of two polynomials) of

the complex variable z.

For H(z) = a(z)/b(z):• Values of z for which a(z) = 0 are called zeros of H(z).

• Values of z for which b(z) = 0 are called poles of H(z).

• If z0 is a pole of H(z) and (z-z0)p.H(z) has neither a

pole or zero at z0, H(z) is said to have a pole of order p

at z0.


Figure 13.8Sampled-datasystems:a. Continuous;b. Sampled input;c. sampled inputand output


Pulse Transfer Function

nTtnTrtrn

0

* nTtgnTrtcn

0

k

k

zkTczC

0


nTtnTrtrn

0

*

nTtgnTrtcn

0

k

k

zkTczC

0

nTkTgnTrkTcn

0

k

k n

zTnkgnTrzC

0 0

)(

)()(00

0 0

zRzGznTrzmTg

zmTgnTrzC

n

mm

m

nm

nm n

Letting m=k-n we find

Figure 13.9 Sampled-data systems and their z-transforms


Block Diagram Reduction

Figure 13.10Steps in blockdiagram reductionof a sampled-data system


Figure 13.11 Digital system for Skill-Assessment Exercise 13.4


Problem: Find T(z)=C(z)/R(z) Digital system for Skill-Assessment Exercise 13.4

Block diagram descriptions of discrete systemsTransfer function representations of discrete systems permit the use of block diagrams to describe discrete systems in a manner analogous to continuous system representation.

• Add transfer functions in parallel.

• Multiply transfer functions in series (cascade).

• A simple feedback loop reduces to the forward path transfer function divided by one minus the open-loop transfer function.

• Block diagram manipulation and Mason’s rule apply without change.


The discrete convolution sumConsider a linear, stationary discrete system.

• If its response to a unit pulse is h[k], then its response to a pulse of amplitude e0 is e0*h[k] since the system is linear.

• Since the system is stationary a delay of the input will cause an equal delay in the response.

• The effect of a sequence of pulses is the sum of their individual effect. For an infinite sequence:

k. sample at output the on i sample at

pulse input an of effect the is where ik

iikik

h

heu

N.B. k >i for a

causal system

N.B. k >i for a causal system


State space descriptions of discrete systems

Any linear, stationary discrete system can be represented in the following format:

dimension. eappropriat of matrices are J and H,,

and vector, state the is x

vectors output and input the are and where

yu

kuJkxHky

kukxkx

][][][

][][]1[

N.B. The matrices A,B,C, and D are often used to describe the state space representation of a system.

N.B. The matrices A,B,C, and D are often used to describe the state space representation of a system.


Example of a state space representation

e[k][k] x] [kx

[k] x] [kx

][k xe[k] [k] x] [kx

e[k][k]]-x[k x )-(z

(z)/E(z) X

12

21

211

1121

1

1

12

21

1 or

Consider G(z)=U(z)/E(z)=K(z+1)/(z2-1)Let:

Then,


[k]xK[k]x K

[k]xK][kxKu[k]

)K(z(z)U(z)/X

**

**

21

11

1

1

1

or

Example of a state space representation (cont.)

][0][][

][1

0][

01

10]1[

2

1

2

1

2

1

kekx

xKKku

kekx

xk

x

x


Figure 13.12Computer-controlled torches cut thicksheets of metal used in construction

© BlairSteitz/Photo Researchers.


Correspondence with Continuous Signals

Consider the discrete signal to be generated by sampling a continuous signal, We can then exploit our knowledge of the s-plane

features by transferring them to the z-plane.


Mapping from the s-plane to the z-plane

Consider an arbitrary point on the s-plane;

js Which maps onto the z-plane as:

TjTTj eeez

We will consider several lines on the s-plane and their

map onto the z-plane. Recall ωT < prevents ambiguity, I.e., permits a one-to-one mapping between the two planes.


Figure 13.13Mapping regions of the s-plane onto the z-plane


S-plane real axis s z e T

• For Re{s}>0 : Re{z}>1

• For Re{s}<0 : 0<Re{z}<1

• For s=0: z=1

Real axis of s-plane maps to positive-real axis of z-planeReal axis of s-plane maps to positive-real axis of z-plane


S-plane imaginary axis &arbitrary point in primary strip

• For s=j ω : z=exp{j ω T}A line of unit length making an angle of wT radians withthe real axis.

• For s = j ω s/2=jπ/T : z =e xp{jπ} = -1

• For s=a+j ω : z = exp{aT+j ω T} = exp{aT}*exp{j ω T}

A line of length, exp{aT}, making an angle of ω T radianswith the real axis.

Note that - π < ω T<π for there to be a one-to-onemap between the s-plane and the z-plane, I.e., -π/T< ω <π/Tor - ω s/2< ω < ω s/2. (Sampling Theorem) The region on the s-plane for ω s/2< ω < ω s/2 is called the primary strip.

Note that - π < ω T<π for there to be a one-to-onemap between the s-plane and the z-plane, I.e., -π/T< ω <π/Tor - ω s/2< ω < ω s/2. (Sampling Theorem) The region on the s-plane for ω s/2< ω < ω s/2 is called the primary strip.


S-plane line of constant damping

s j

z e e e

n n

sT T j Tn n

1 2

1 2

11

2

2

1 2

n n

j

TT

z e e

Consider:

Let:

Then,Logarithmic spirals on the z-plane for constant damping ratio.

Logarithmic spirals on the z-plane for constant damping ratio.


S-plane line of constant natural frequency

s j

z e e e

n n

sT T j Tn n

1 2

1 2

z e e for

T

n nT j T

n

1 2

0 1 and

Again consider:

Contours of constant damping ratio and natural frequency on the z-plane are drawn by the MATLAB command ‘zgrid’.

Contours of constant damping ratio and natural frequency on the z-plane are drawn by the MATLAB command ‘zgrid’.


-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-20

-15

-10

-5

0

5

10

15

20S-PLANE USING "SGRID" - lines of constant damping ratio & natural frequency

LINES OF CONSTANT DAMPING RATIO

0.1

0.9

LINES OF CONSTANTNATURAL FREQUENCY

10

5


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Z-PLANE USING "ZGRID" - lines of constant damping ratio & natural frequency

LINES OFCONSTANTNATURAL FREQUENCY

LINES OF CONSTANTDAMPING RATIO

PI/10T 9PI/10T

0.1

0.3

Upper plane is divided into 10 segments so wn*T=pi/10 or

wn=pi/10T per segment.

Upper plane is divided into 10 segments so wn*T=pi/10 or

wn=pi/10T per segment.


Figure 13.14Finding stability ofa missile controlsystem:a. missile;b. conceptual blockdiagram;c. block diagram;d. block diagramwith equivalent singlesampler


Figure 13.15Digital system for

Example 13.7


Internal & external stability

• Internal stability is concerned with the responses of all the internal (state) variables of a system.

• External stability is concerned with the response of the output variables of a system such as described by the transfer function or impulse response model.

• They differ in that some of the internal modes of the system may not be connected to the input and output of a given system.


Bounded-Input Bounded-Output (BIBO) stability

For external stability a common definition of an appropriate response is that for every bounded input the output should also be bounded.

A necessary and sufficient condition for BIBO stability is

iikh

Note: A rational transfer function can be expanded in a partial fraction expansion so its pulse response will be the sum of its terms. Thus, if all poles are inside the unit circle, the system is stable. If at least one pole is on or outside the unit circle the system is not BIBO stable.

Note: A rational transfer function can be expanded in a partial fraction expansion so its pulse response will be the sum of its terms. Thus, if all poles are inside the unit circle, the system is stable. If at least one pole is on or outside the unit circle the system is not BIBO stable.


Proof:

.

.

i-k

k

i

h if bounded is output the Thus,

u Then,

all for e Let

ikikiiki hMhehe

iM1) Sufficiency

2) Necessity

BIBO not is system the true is condition the unless Thus,

u

:is at output The input. thisApply

h for

h for e :input bounded the Consider

0

i-

i-i

ii

i i

i

iii

i

i

hh

hhe

k

h

h

2

0

00

0


MATLAB commands defining discrete systems as objects

• tfSYS = tf(NUM,DEN,TS) creates a discrete-time transfer function with

sample time TS (set TS=-1 if the sample time is undetermined).

• zpk SYS = zpk(Z,P,K,Ts) creates a discrete-time ZPK model with sample

time Ts (set Ts=-1 if the sample time is undetermined).

• ss SYS = ss(A,B,C,D,Ts) creates a discrete-time SS model with sample

time Ts (set Ts=-1 if the sample time is undetermined).


Table 13.3Routh table for Example 13.8


Figure 13.16Digital system forSkill-Assessment

Exercise 13.5


Figure 13.18Constant damping ratio, normalizedsettling time, and normalized peak time plots on the z-plane


Figure 13.19The s-plane

sketch of constant

percentovershoot line


Figure 13.20Generic digital

feedback controlsystem


Figure 13.21Digital feedback

control forExample 13.10


Root Locus analysis of Discrete Systems

• Stability boundary: |z|=1 (Unit circle)

• Settling time = distance from Origin

• Speed = location relative to Im axis– Right half = slower– Left half = faster

Effect of discrete poles

|z|=1

Longer settling time

Re(s)

Im(s)

Unstable

Stable

Higher-frequencyresponse

Tsez :Intuition

Figure 13.22Root locus

for the systemof Figure

13.21


Figure 13.23Root locus for the system of Figure

13.21 with constant

0.7 damping ratiocurve


Figure 13.24Sampled step

response of thesystem of

Figure 13.21 withK = 0.0627


Figure 13.25a. Digital controlsystem showing

the digital computerperforming

compensation;b. continuous

systemused for design;c. transformed

digitalsystem


Figure 13.26Closed-loop

responsefor the

compensatedsystem of Example

13.12 showing effect

of three differentsampling

frequencies


Figure 13.27Block diagram

showing computeremulation of a digital

compensator


Figure 13.28 Flowchart for a second-orderdigital compensator


Figure 13.29 Flowchart to implement


Figure 13.30 Antenna control system:a. analog implementation;b. digital implementation


Figure 13.31Analog antenna azimuth position

control system converted to a digital system


Figure 13.32Root locussuperimposed

over constant damping ratio curve


Figure 13.33Sampled step

response of theantenna azimuthposition control

system


Figure 13.34Simplified block

diagram of antennaazimuth control

system


Figure 13.35Closed-loop digitalstep response forantenna controlsystem with a leadcompensator.


Figure 13.36Flowchart for digital lead compensator


Figure P13-1 (p. 839)

Figure P13.2

Figure P13-3 (p. 840)

Figure P13-4 (p. 840)


Figure P13.5


Figure P13.6


Figure P13.7


Figure P13.8


Figure P13.9Simplified blockdiagram for robot

swing motion


Figure P13.10Simplified block

diagram of a floppydisk drive


z-Transforms of Common Functions

Name f(t) F(z)

Impulse

Step

Ramp

Exponential

Sine

1

1z

z

2)1( z

z

aez

z

1)(Cos2

Sin2 zaz

az

1)( tf

ttf )(

atetf )(

)sin()( ttf

F(s)

1

s

1

2

1

s

as 1

22

1

s

00

01)(

t

ttf

System ID for Admission Control

M(z)

G(z) N(z) S(z)

Controller Notes Server

Sensor

R(z)+

E(z)U(z)

-

Q(z)

ARMA Models

Control Law

Transfer Functions

zz

zK

cz

dzd

az

zbzGzSzN i 1

1)()()(

1

10

1

0

Open-Loop:

zz

zKzG

cz

dzdzS

az

zbzN

i 1

1)(

)(

)(

1

10

1

0

)()1()(

)1()()1()(

)()1()(

101

01

teKtutu

tqdtqdtmctm

tubtqatq

i

Root Locus Analysis of Admission Control

Predictions:•Ki small => No controller-induced oscillations•Ki large => Some oscillations•Ki v. large => unstable system (d=2)•Usable range of Ki for d=2 is small

Experimental Results

Control(MaxUsers)

Response(queue length)

Good

Slow

Bad

Useless

Advanced Control Topics

• Robust Control– Can the system tolerate noise?

• Adaptive Control– Controller changes over time (adapts)

• MIMO Control– Multiple inputs and/or outputs

• Stochastic Control– Controller minimizes variance

• Optimal Control– Controller minimizes a cost function of error and control energy

• Nonlinear systems– Neuro-fuzzy control– Challenging to derive analytic results

Issues for Computer Science

• Most systems are non-linear– But linear approximations may do

• eg, fluid approximations

• First-principles modeling is difficult– Use empirical techniques

• Control objectives are different– Optimization rather than regulation

• Multiple Controls– State-space techniques– Advanced non-linear techniques (eg, NNs)

Selected Bibliography

• Control Theory Basics– G. Franklin, J. Powell and A. Emami-Naeini. “Feedback Control of Dynamic Systems, 3rd ed”.

Addison-Wesley, 1994.– K. Ogata. “Modern Control Engineering, 3rd ed”. Prentice-Hall, 1997.– K. Ogata. “Discrete-Time Control Systems, 2nd ed”. Prentice-Hall, 1995.

• Applications in Computer Science– C. Hollot et al. “Control-Theoretic Analysis of RED”. IEEE Infocom 2001 (to appear).– C. Lu, et al. “A Feedback Control Approach for Guaranteeing Relative Delays in Web Servers”.

IEEE Real-Time Technology and Applications Symposium, June 2001.– S. Parekh et al. “Using Control Theory to Achieve Service-level Objectives in Performance

Management”. Int’l Symposium on Integrated Network Management, May 2001– Y. Lu et al. “Differentiated Caching Services: A Control-Theoretic Approach”. Int’l Conf on

Distributed Computing Systems, Apr 2001– S. Mascolo. “Classical Control Theory for Congestion Avoidance in High-speed Internet”.

Proc. 38th Conference on Decision & Control, Dec 1999– S. Keshav. “A Control-Theoretic Approach to Flow Control”. Proc. ACM SIGCOMM, Sep

1991– D. Chiu and R. Jain. “Analysis of the Increase and Decrease Algorithms for Congestion

Avoidance in Computer Networks”. Computer Networks and ISDN Systems, 17(1), Jun 1989

Effect of additional zero or pole on discrete step response

• The major effect of an additional zero to the step response of a discrete system is to substantially change the percent overshoot. See FP&W Figures 4.29 through 4.31.• The major effect of an additional pole is to increase the rise time. See Figure 4.32.• These effects are conveniently demonstrated using MATLAB’s ltiview GUI. Consider the system:theta=18*pi/180; zeta=0.5; %Figure 4.29 r=exp(-zeta*theta/sqrt(1-zeta^2))sys=tf(1,[1 -2*r*cos(theta) r^2],1);sys=sys/dcgain(sys),ltiview


Discrete frequency response•Let: G(z)=Y(z)/U(z) with U(z) a sinusoid.

•Then, U(z)=z/(z-exp{jwT)) and Y(z)=G(z)z/(z-exp(jwT))=N(z)z/D(z)(z-exp(jwT)) Y(z) = G(exp(jwT))*z/(z-exp(jwT)+Transient response

•For a stable system, the transient response will die out withincreasing time and the steady-state output will be:

y_steady-state(kT) = G(z=exp(jwT))*exp(jkwT)= |G(exp(jwT))|*exp(jwT+arg(G(exp(jwT)))

• G(exp(jwT)) with -pi<wT<=pi is the discrete system’s frequency response. G(exp(jwT)) for 0=<wT=<pi is computed or measured and G(exp(-jwT)) =conj(G(exp(jwT))).


digital control 1

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