Lecture 5 –Sampled Time Control - web.stanford.eduweb.stanford.edu/.../ee/ee392m/ee392m.1056/Lecture5_Digital.pdf · – digital (sampled time) control system • Numerical integration

EE392m - Spring 2005Gorinevsky

Control Engineering 5-1

Lecture 5 –Sampled Time Control

• Sampled time vs. continuous time: implementation, aliasing

• Linear sampled systems modeling refresher, DSP• Sampled-time frequency analysis • Sampled-time implementation of the basic controllers

– I, PI, PD, PID – 80% (or more) of control loops in industry are digital PID



),,(),,()(

tuxgytuxfdtx

==+

Sampled Time Models • Time is often sampled because of the digital computer use

– digital (sampled time) control system

• Numerical integration of continuous-time ODE

• Time can be sampled because this is how a system works• Example: bank account balance

– x(t) - balance in the end of day t– u(t) - total of deposits and withdrawals that day– y(t) - displayed in a daily statement

( ) kdttuxfdtxdtx =⋅+≈+ ),,,()(

xytutxtx

=+=+ )()()1(



Sampling: continuous time view• Sampled and continuous time together• Continuous time physical system + digital controller

– ZOH = Zero Order Hold

Sensors

Controlcomputing

ActuatorsPhysicalsystem

D/A, ZOHA/D, Sample



Signal sampling, aliasing

• Nyquist frequency: ωN= ½ωS; ωS= 2π/T

• Frequency folding: kωS±ω map to the same frequency ω• Sampling Theorem: sampling is Ok if there are no frequency

components above ωN

• Practical approach to anti-aliasing: low pass filter (LPF) • Sampled→continuous: impostoring

Digitalcomputing

D/A, ZOHA/D, SampleLowPassFilter

LowPassFilter



Linear state space model • Generic state space model:

• LTI state space model– Physics-based linear system model– Obtained by sampling a continuous time model– Zero-order hold (ZOH)

• Matlab commands for model conversion: help ltimodels

)()()()()1(

tCxtytButAxtx

=+=+),,(

),,()1(tuxgy

tuxftx=

=+

)()()()()(

txCtytuBtxAtx

c

cc

=+=&

)()(

)()()(0

txCty

tudtBetxeTtx

c

B

c

T tA

A

TA cc

=

⎟⎠⎞⎜

⎝⎛+=+ ∫

443442143421



Read-write head control

Disk read-write head example• Sampled time model for T=0.0001

(use Matlab ss and c2d)

guyby =+ &&&

uss

y)20(

10+

=

{

[ ] xy

uxdtdx

C

BA

⋅=

⋅⎥⎦

⎤⎢⎣

⎡+⋅⎥⎦

⎤⎢⎣

⎡−=

321

43421

10

010

01020

10,20 == gb[ ] )(10)(

)(105

10)(

0100998.0

)1( 8

3

4

kxky

kukxkx

C

BA

⋅=

⋅⎥⎦

⎤⎢⎣

⎡

⋅+⋅⎥

⎦

⎤⎢⎣

⎡=+−

−

−

321

4342143421

)()()()()1(

kCxkytBukAxkx

=+=+



Sampled-time system stability

• Initial condition response0)0(),()1( xxkAxkx ==+

0)( xAkx k=

Example: % Take A, B, C % from the R-W head example>> eig(A)>> ans =

1.0000 0.9980

>>>> K = 10>> Ak = A - K*B*C;>> eig(Ak)>> ans =

0.9990 0.9990

>>>> N = 2000; >> x(:,1) = [1; 0]; >> for t=2:N, >> x(:,t) = Ak*x(:,t-1);>> end>>>> plot(1:N,x(2,:))

• Exponential stability condition)eig(}{ Aj =λ

1<jλ

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.01

0.02

0.03

0.04



Impulse response • Response to an input impulse

• Sampled time: t = 1, 2, ...• Control history = linear combination of the impulses ⇒

system response = linear combination of the impulse responses

( ) )(*)()()(

)()()(

0

0

tuhkukthty

kukttu

k

k

=−=

−=

∑

∑∞

=

∞

=

δ

)()( ⋅⎯→⎯⋅ hPδu

t

y

t



Convolution representation of a sampled-time system

• Convolution

• Impulse response

• Step response: u = 1 for t > 0

∑−∞=

−=t

k

kukthty )()()(

)()()()( thtyttu =⇒= δ

)1()()( −−= tgtgth

uhy *=signal processing notation

∑∑==

=−=t

j

t

k

jhkthtg00

)()()(

Matlab commands: g = cumsum(h); h = diff(g);



z-transform vs. Laplace transform• Discrete (z-transform) transfer function:

– function of complex variable z– z forward shift operator– analytical outside the circle |z|≥r– all poles are inside the circle– for a stable system r ≤ 1

k

k

zkhzH −∞

=∑=

0

)()(

• Laplace transform transfer function: – function of complex variable s– s differentiation operator

– analytical in a half plane Re s ≤ a– for a stable system a ≤ 1

∫∞

∞−

= dtethsH st)()(

Re s

Imag s

Re z

Imag z

1

i



Sampled time vs. continuous time

• Continuous time analysis (digital implementation of a continuous time controller)

– Tustin’s method = trapezoidal rule of integration for

– Matched Zero Pole: map each zero and a pole in accordance with

• Sampled time analysis (Sampling of continuous signals and system)

• Systems analysis is often performed continuous time -this is where the controlled plant is.

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⋅==→ −

−

1

1

112)()(

zz

TsHzHsH s

ssH 1)( =

zTes =



FIR model

• FIR = Finite Impulse Response • Cut off the trailing part of the impulse response to obtain FIR• FIR filter state x is the shift register content

h0

h1

h2

h3

u(t)

x1=u(t-1)

y(t)

x2=u(t-2)

x3=u(t-3)

z-1

z-1

z-1

( ) )(*)()()(0

tuhkukthty FIR

N

kFIR =−=∑

=

)()()()()1(

tCxtytButAxtx

=+=+

h

t



IIR model • IIR model:

• Filter states: y(t-1), …, y(t-na ), u(t-1), …, u(t-nb )

∑∑==

−+−−=ba n

kk

n

kk ktubktyaty

01

)()()(

u(t) b0

b1

b2

u(t-1)

u(t-2)

u(t-3)

z-1

z-1

z-1

-a1

-a2

y(t-1)

y(t-2)

y(t-3)

z-1

z-1

z-1

y(t)

b3-a3



IIR model • Matlab implementation of an IIR model: filter• Transfer function realization: unit delay operator z-1

)()()( tuzHty =

• FIR model is a special case of an IIR with A(z) =1 (or zN )

( ) ( ) )(...)(...1)(

110

)(

11 tuzbzbbtyzaza

zB

NN

zA

NN 4444 34444 214444 34444 21

−−−− +++=+++

NN

NN

NNN

NNN

zazazbzbb

azazbzbzb

zAzBzH

−−

−−

−

−

++++++=

++++++==

...1...

......

)()()(

11

110

11

110

Signal processing conventionMatlab filter etc

Control conventionMatlab tf, step etc



IIR approximation example• Low order IIR approximation of impulse response:

(prony in Matlab Signal Processing Toolbox) • Fewer parameters than a FIR model • Example: sideways heat transfer

– impulse response h(t)– approximation with IIR filter a = [a1 a2 ], b=[b0 b1 b2 b3 b4 ]

0 20 40 60 80 1000

0.02

0.04

0.06

TIME

IMPULSE RESPONSE

22

11

44

33

22

110

1)( −−

−−−−

++++++=

zazazbzbzbzbbzH



Linear state space model • LTI state space model

• Unit delay operator z-1: z-1 x(t) = x(t-1)

• Transfer function of an LTI model– defines an IIR representation

• Stability: the poles are inside the unit circle

( )[ ]( ) DBAIzCzH

uDBAIzy

+−=

⋅+−=−

−

1

1

)(

)()()()()()1(

tDutCxtytButAxtx

+=+=+



Stability analysis

• Transfer function poles tell you everything about stability• Model-based analysis for a simple feedback example:

)()(

dyyKuuzHy−−=

=dd yzLy

KzHKzHy )()(1

)( =+

=

• If H(z) is a rational transfer function describing an IIR model

• Then L(z) also is a rational transfer function describing an IIR model



[ ] )(

)()()()(

)(

)(

0001

)()()()(

010000100001

)1()1()1()1(

0

4

3

2

1

044033022011

4

3

2

13321

4

3

2

1

tub

txtxtxtx

babbabbabbabty

tu

txtxtxtxaaaa

txtxtxtx

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅−−−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −−−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++++

State space realization of IIR filter

y(t)

u(t)

a2a3a4 a1

b2b3b4 b1 b0

z-1 z-1 z-1 z-1

NN

NN

zazazbzbb

zAzBzH

tuzHty

−−

−−

++++++==

=

...1...

)()()(

)()()(

11

110



Poles and Zeros System • …not quite so!• Example:

7.0)(

−==

zzuzHy

19

171819 0011400016280...4907.0)(z

. z.z.zzuzHy FIR+++++==

Impulse Response

Time (sec)

Ampl

itude

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Impulse Response

Time (sec)

Ampl

itude

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

• FIR model - truncated IIR



IIR/FIR example - cont’d • Feedback control:

• Closed loop:

)()(7.0

)(

dd yyyyKuz

zuzHy

−−=−−=−

==

Impulse Response

Time (sec)

Ampl

itude

0 5 10 15 20 250

0.2

0.4

0.6

0.8

uzLuzH

zHy )()(1

)( =+

=

uzLuzH

zHy FIRFIR

FIR )()(1

)( =+

=



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

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

IIR/FIR example - cont’dPoles and zeros

• Blue: Loop with IIR model poles xand zeros o

• Red: Loop with FIR model poles xand zeros o



Frequency domain description• Sinusoids are eigenfunctions of an LTI system:

LTIPlant

tiititi eeeze ωωωω == + )1(

• Frequency domain analysis system diagonalization

uzHy )(=

∑∑ =⇒= ti

y

kiti

kkkk eueHyeuu ω

ω

ωω

43421)(~

~)(~

k

ti

ue k

~ω

uPacket

of

sinusoids

Packet

of

sinusoids

)( ωieH y

ωiez →

k

ti

ye k

~ω



Frequency domain description• Bode plots:

tii

ti

eeHyeu

ωω

ω

)(==

• Example:

)(arg)(

)()(ω

ω

ωϕ

ωi

i

eH

eHM

=

=

7.01)(

−=

zzH

Bode Diagram

Frequency (rad/sec)

Phas

e (d

eg)

Mag

nitu

de (d

B)

-5

0

5

10

15

10-2

10-1

100

-180

-135

-90

-45

0

• |H| is often measured in dB



Fourier transform for sampled signals

• Fourier transform of a series

• Inverse Fourier transform∫

∑

−

∞

−∞=

=

=

πω

ω

ωωπ

ω

2

0

)(~21)(

)()(~

dextx

etxx

ti

t

ti

]2,0[]:1:[ π→∞−∞

]:1:[]2,0[ ∞−∞→π

• If x(t) is a smoothly varying signal, the sum approximates the integral for continuous FT

• approximates the continuos FT up to the Nyquist frequency of

• If sampling frequency is other than unity, need to scale

)(~ ωxπω =



Discrete Fourier Transform = DFT

• DFT

• Inverse DFTtiN

kk

k

N

t

tik

k

k

exN

tx

kN

etxx

ω

ω

ω

πωω

−−

=

=

∑

∑

=

==

1

0

1

)(~1)(

2,)()(~]2:)/2(:0[]:1:0[ ππ NN →

]:1:0[]2:)/2(:0[ NN →ππ

• DFT = FT of a sequence: …, 0, x(1), x(2), …, x(N), 0, 0, ...

• The sum in Inverse DFT approximates the infinite sequence inverse FT up to the ‘Nyquist time’ of N/2

DFT is computed by Matlab FFT

x=iFFT(f)

f=FFT(x)



Sampled-time LTI models - recap

• Linear system can be described by impulse response• Linear system can be described by frequency response =

Fourier transform of the impulse response • FIR, IIR, State-space models can be used to obtain close

approximations of sampled-time linear system • A pattern of poles and zeros can be very different for a

small change in an input-output model. • Impulse response, step response, and frequency response

do not change much for small model changes



Sampled time I control

• Step to step update:

• Closed-loop dynamics

• Deadbeat control:

[ ]dI ytyktvtvtututdtugty

−=−+−=

+⋅=

)()()1()1()(

)()()(

[ ]dI yy

zku

dugy

−−

=

+⋅=

1

dgkz

zygkz

gkyI

dI

I

+−−+

+−=

11

1

sampled timeintegrator

1=Igk ( )dzyzy d11 1 −− −+=



Run-to-run (R2R) control• Main APC (Advanced

Process Control) approach in semiconductor processes

• Modification of a product recipe between tool "runs"

Tool

Run-to-run control

Cell controller

Process

Runtime controller Metrology

system

Tunable

recipe

parameters

• Processes:– vapor phase epitaxy– lithography – chemical mechanical

planarization (CMP)– plasma etch

[ ]dI ytyktututdtugty

−−−=+⋅=

)()1()()()()(



Cascade loop interpretation:

PI control• P Control + Integrator for

cancelling steady state error

• Velocity form of the controller

)()()()()()1(

tektvktutetvtv

yye

PI

d

−−=+=+

−= Inner loop

Plant

kP

e(t)u(t)

ki∑

y-yd

PiI kkk ⋅=[ ])1()()()1()( −−−−−= tetektektutu PI



QoS Control Example• QoS control for video server• QoS = Quality of Service • Need to maintain constant delay

– for audio synchronization etc

• Control codec quality (processing rate)• Regulate buffer queue• Model: )()()()1(

)()1(twtTrtbtb

tutr+−=+

=+RateBuffer

AssumeT = 1

{ {

[ ] )(10)(

)(10

)(01

)(1100

)1(

kxky

kwkukxkx

C

FBA

⋅=

⋅⎥⎦

⎤⎢⎣

⎡+⋅⎥⎦

⎤⎢⎣

⎡−+⋅⎥⎦

⎤⎢⎣

⎡−

=+

321

43421



QoS Control Example, cont’d• Dominant integrator dynamics, use P control

)()()()()()1(

kCxkykFwkBukAxkx

=++=+ ( )dP ykykku −−= )()(

output yPlant

Control

disturbance w

-

auxillary output z

Matlab command: S=feedback(P,C,iI,iO);

Use tf to convert model25.0=Pk

0 5 10 15 20 250

1

2

3

4

Step Response to Disturbance

Time (sec)

Am

plitu

de



QoS Control Example, cont’d• P control provides poor disturbance rejection• Use PI control

)]1()([)()1()()()(

−−−−−=−=

kekekkekkukuykyke

PI

d

ez

zkku IP ⋅⎥⎦

⎤⎢⎣⎡

−+−=

1

• PI has better disturbance rejection

1.025.0

==

I

P

kk

0 5 10 15 20 25 30 35 40-1

0

1

2

3Step Response to Disturbance

Time (sec)

Am

plitu

de



PID Control

• PID: three-term control

• Sampled-time PID

• PID controller in velocity form– bumpless transfer between

manual and automatic

ekdtekeku

yye

DIP

d

&−⋅−−=

−=

∫( ) ezke

zkeku DIP ⋅−−

−−−= −

−1

1 11

1

futurepresent past

[ ] [ ])2()1(2)()1()()()1()( −+−−−−−−−−= tetetektetektektutu DPI

1

2

1 −−=∆

−∆−∆−=∆

zekekeku IPD



PID Controller Tuning

• Tune continuous-time PID controller, e.g. by Ziegler-Nichols rule, and set up the sampled time PID to approximate the continuous-time PID

• Cascaded loop design (continuous time structure) – Design P – Cascade I, retune P– Add D, retune PI

• Optimize the performance parameters by repeated simulation runs – search through the {P, I, D} space

• Loopshaping – Lectures 7-8 • Advanced control design – formal specs, other courses



Industrial PID Controller• A box, not an algorithm• Auto-tuning functionality:

– pre-tune– self-tune

• Manual/cascade mode switch• Bumpless transfer between

different modes, setpoint ramp• Loop alarms • Networked or serial port

connection



Plant Type

• Constant gain - I control• Integrator - P control • First order system - PI control • Double integrator or second order system - PD control• Generic response with delay - PID control

Documents

Lecture 5 –Sampled Time Control - web.stanford.eduweb.stanford.edu/.../ee/ee392m/ee392m.1056/Lecture5_Digital.pdf · – digital (sampled time) control system • Numerical integration