Computer Control Systems

8/2/2019 Computer Control Systems

1/30

A very important category of real time systems are automatic control

systems. In fact all control systems are real-time systems because

they must react to external events within a specified amount of time.

The operation of computer control systems is usually synchronized by

a clock signal that determines the sampling period. This sampling

period specifies the maximum total amount of time that is available for

A/D and D/A conversions and control computations.

Computer control systemsComputer control systems


2/30

Control loop variables

y(t) ory(k) - controlled variable (temperature, pressure, water

level, flow, speed etc.)r(k) - reference or setpoint i.e. the desired value of the controlled

variable

e(k) - control error the difference between the desired value of the

controlled variable and its actual value e(k)=r(k) - y(k)u(t) oru(k) manipulated variable represents the action that is

used by the controller to change the controlled variable (control

valve position, power input of a heating element, speed of a

cooling fan, fuel flow to an engine or to a boiler)

Besides, there is usually also one or more disturbance variable(s)d(t) that are external influences affecting the controlled variable

(changing temperature of the environment, changing load of a

electric drive or of an engine etc.)


3/30

Examples:

Digital control

of an Air Heater:

Digital speed control

of a DC motor:


4/30

Two types of real-time control

systems:1. Embedded Systems

dedicated control systems the computer is an embedded part of

some piece equipment

microprocessors, real-time kernels, RTOS

aerospace, industrial robots, vehicular

systems

2. Industrial Control Systems

distributed control systems (DCS),

programmable controllers (PLC),Soft-PLCs

hierarchically organized, distributed

control systems

process industry, manufacturing industry,


5/30

Universal process controllerModular control system:


6/30

Actuators: Control valves


7/30


8/30

Control valves with electrical drive typically two phase

induction motor


9/30

Operation of the analog-to-digital converter (ADC), the digital-

to-analog converter (DAC) and the zero order hold (ZOH)

ADC performs two functions:

1. Analog signal sampling

Continuous signal is replacedwith a sequence of values

equally spaced in the time

domain

2. Quantization amplitude of

the signal is represented with a

discrete set of different values


10/30

Zero order hold is described by the formula

vvvTktkTkTyty )1()()( e!

The behavior of the output of standard DAC is that of zero order hold.

Discontinuous step changes at ZOH output can excite poorly damped

mechanical modes of the physical process and also cause wear in the

actuators of the system. A theoretically possible solution is higher order

hold circuits.

First order hold:

vvvv

v

v

vTktkTkTyTky

T

kTtkTyty )1()())1(()()( e

!

This form of FOH is not causal. Causal FOH can be obtained by

introducing a delay of one sampling period or using output prediction

based on extrapolation from previous sampling period.

vvvv

v

v

vTktkTTkykTy

T

kTtkTyty )1())1(()()()( e

!

First and higher order hold circuits are normally not used in control

systems because of the high phase shift they introduce.


11/30


12/30

The simplest approach to control: on-off and three

position control

Static characteristics of on-off and three position controllers


13/30

1. The higher is the control error the higher the control action

(manipulated variable) must be

)()( tertu o!Proportional orP controller r0proportional gain

Digital P/PI/PID controller

)s(EsTsT

r)s(Udt

)t(deTd)(e

T)t(er)t(u

d

i

od

t

i

o]

11[]

1[

0

!! XX

If the control error equals zero, the manipulated variable is also zero.

As a result of it, if the controlled plant is non-integrating (i.e. non-

zero plant input is necessary to have non-zero output), there will

always be some non-zero steady state error. The value of error will

be the smaller the higher the proportional gain will be.This is a significant drawback as the majority of the controlled plants

are non-integrating (some non-zero value of the plant input is

necessary to compensate for thermal losses, mechanical friction, load

torque etc. depending on the particular plant being controlled).


14/30

2. Proportional Integral or PI controller

)(1

)(]1

1[)(

])(1

)([)(0

sEsT

sTrsE

sTrsU

deT

tertu

i

io

i

o

t

i

o

!!

! XX Tiintegral time constant

Due to the presence of the integral term, the manipulated variable(=plant input) can be non-zero even if control error is zero.

Achieving zero steady state error is therefore possible)


15/30

3. Proportional Integral Derivate PID controller

Td derivative time constant)(]1

1[)(

])(

)(1

)([)(0

sEsTsT

rsU

dt

tdeTde

T

tertu

d

i

o

d

t

io

!

!

XX

The derivative term does not affect the steady state behavior of the

control loop (derivative of constant is zero), however it can be used

to speed up and improve the dynamics response of the control loop


16/30

Digital PID controller

)s(EsT

sT

r)s(U

dt

)t(deTd)(e

T)t(er)t(u

d

i

od

t

i

o]

11[]

1[

0

!! XX

Individual terms are often discretized using different methods. The

proportional term requires no approximation because it is a purely

static part.

Integral term: rectangular rule, trapezoid rule

)k(I)i(eT

Tr)(e

T

rO

k

ii

vo

t

i

o !} !10

X

)k(I))i(e)i(e(T

Tr)(e

T

rL

k

ii

vot

i

o !} !10

12

X

Backward rectangular rule:

(it is better than forward rule as it

immediately reacts to setpoint changes)

Trapezoid rule:


17/30

))1()((2

)1()(finallyand

))1()((

2

)1()(

))1()((2

)1(

))1()((

2

)(

1

1

1

!

!

!

!

!

!

kekeT

TrkIkI

keke

T

TrkIkI

ieieT

TrkI

ieieT

TrkI

i

voLL

i

voLL

k

ii

voL

k

ii

voL

The formula for calculating the integral term includes summation

from the beginning (1 or0 depending on the particular rule that is

used). For implementation it is converted to the form of differenceequation that is updated recursively at each sampling instant. For

the trapezoid method this can be written in the following way


18/30

This simple approximation is very sensitive to noise and combined

effects of quantization and sampling may result in erratic behavior, in

particular if sampling time is small (the response to a slowly and

linearly growing signal e(t) is then not a small constant but a

sequence of short peaks with high magnitude)

Derivative term:

The simplest approach: backward difference

)())1()(( kDTkekeTrdt

deTr vdodo !}


19/30

Better Alternative:

Multipoint difference:

Derivative at time kTv is approximated with an average speed of controlerror change at several sampling intervals

Average error4

321 ! kkkkkeeee

e

v

kkkk

v

kk

v

kk

v

kk

v

kk

v

k

T

eeee

T

ee

T

ee

T

ee

T

ee

T

ekD

6

33

5,15,05,05,14

1)(

321

321

!

!

!(!


20/30

IntegralIntegral WindWind--upup

Internal computation of the integral term

is practically unlimited, while the physicalmanipulated variable is always limited

and the limits are hard. As a result of it

the value of the integral term can

significantly exceed the value of the

physically realizable manipulatedvariable. In such a situation, if the setpoint

is reached and the control error changes

sign, it takes the a long time before this

change can also be observed at

manipulated variable output. Long lasting

overshoots as can be seen from the figure

and other problems then result.


21/30

Wind-up can be prevented by using the dynamic limitation of the

integral term.

The procedure is as follows:

1. At each time step k compute the individual terms P(k),I(k) andD(k)

2. Sum these terms up to calculate the manipulated variable

u(k)=P(k)+I(k)+D(k)

3. If the value of the manipulated variable is within limits, it is

sent to the D/A converter. If not, manipulated variable remains at

one of its limits, the current value of the integral termI(k) is

discarded and replaced withI(k-1)

In the next sampling instant this procedure is repeated. Thus the

integral term is frozen for the whole time during which themanipulated variable is at its limits and if its value becomes

smaller after the change of control error sign the effect on

manipulate variable is immediate.

As the integral term is frozen at some static value that is not

specified a priori, this limitation is called dynamic.


22/30

AliasingAliasing)ftsin(A)t(u T2!Harmonic signal with frequency f

Sampling with period Tv

)fkTsin(A)k(uv

T2!

By sampling harmonic signal with frequencyv

nffs

)fkTsin(A)nsin()fkTcos(A)ncos()fkTsin(A

kT)nff(sinA)k(u

vv

vv

TTTTT

T

22222

2

!s!

!s!

In other words, it is not possible to distinguish betweensampled signal with frequency fand vs nfff O!

Signal 50 Hz sampled with sampling frequency a) 49 Hz b) 51 Hz


23/30

2v

ff

Shannon-Kotlnikov theorem

This theorem cannot be satisfied for noise signals anti-aliasingfilter is necessary

Usually analogue low-pass filter is combined with digital filter that

works with smaller sampling time that is the sampling time of thecontrol algorithm.

fnfffvs"!Then and aliasing does not occur


24/30

PID Controller Output with PWM modulation

Tc cycle time


25/30

Tuning ofPID Controllers

The behavior of the PID controllers depends on 3 parameters:

r0 - proportional gain

Ti -integral time constant or reset time

Td - derivative time constant or rate time

Their values must be chosen appropriately in order to make the

control loop:1) stable

2) well tuned with respect to dynamic responses to setpoint and/or

external disturbance changes. Typically, these responses should be

fast and without big over- or undershoots


26/30

If the dynamics ofw(t) and d(t) changes is significantly different (and

typically it will be different: setpoint is often changed in steps resulting in stepchanges of control error while the disturbances usually cause gradual changes

of the controlled variable and control error), the controller tuning must be

optimized either with respect to setpoint tracking or with respect to

disturbance rejection but it is not possible to achieve both objectives with the

same controller tuning.

)()()()()(1

1)()()(

)(

)()(1

)()(

)()(1

)()()(

sDsGsWsGsG

sYsWsE

sD

sGsG

sGsW

sGsG

sGsGsY

d

RS

RS

d

RS

RS

!!

!

Two different objectives ofPID controllers tuning:

a) Set-point tracking

b) Disturbance rejection


27/30

Ziegler Nicholsmethod. This method is suitable for

disturbance rejection. It exists in two variants.

1. Variant Ultimate gain method: only P controller is used, I and Dterms are switched off. Proportional gain is gradually increased until

the stability limit is reached (undamped oscillations with constant

magnitude). This value of the gain is called the ultimate gain rkand the

period of oscillations is denoted as the ultimate period Tk. Using these

two pieces of information the recommended controller tuning can becalculated using the table below.

Controller ro Ti TdP 0,5rk

PI 0,45rk 0,85Tk

PID 0,6rk 0,5Tk 0,125Tk

Ziegler-Nichols tuning rules ultimate gain method


28/30

2. Variant - Step response method Tangent in the inflection point of

the response (the point where the

slope of the response is

maximum)

Tu dead time

Tn rise time

K steady state gain

)(

)(

g(

g(

! u

y

K

n

u

T

T!5

Normalized dead

timeController ro Ti Td

P 1/(K5

PI 0,9/(K5 3Tu

PID 1,2/(K5 2 Tu 0,5 Tu

Ziegler Nichols tuning rules step response method


29/30

Chien Hrones Reswick method

Based on the first order plus time delay approximation of the controlled

plant dynamics. Two possible choices: aperiodic closed loop response and

closed loop response with ca 20% overshoot. This method gives tuningformulae both for the setpoint tracking and for the disturbance rejection.


30/30

)1()(

!

s

KesG

DsT

sX

Controller No overshoot Max 20% overshoot

Set-point tracking Disturbancerejection

Set-point tracking Disturbancerejection

P ro=0,3X/(KTD) ro=0,3X/(KTD) ro=0,7X/(KTD) ro=0,7X/(KTD)

PI ro=0,35X/(KTD)Ti=1,2X

ro=0,6X/(KTD)Ti=4TD

ro=0,6X/(KTD)Ti=X

ro=0,7X/(KTD)Ti=2,3TD

PID ro=0,6X/(KTD)Ti=X

Td=0,5TD

ro=0,95X/(KTD)Ti=2,4TDTd=0,42TD

ro=0,95X/(KTD)Ti=1,35XTd=0,47TD

ro=1,2X/(KTD)Ti=2TD

Td=0,42TD

Chien Hrones Reswick tuning rules

Documents

Computer Control Systems