Computer Control: An Overview Wittenmark, strm, rzn

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Computer Control An Overview Wittenmark,
Åström, Årzén

• Computer controlled Systems
• The sampling process
• Approximation of continuous time controllers
• Aliasing and new Frequencies Anti-aliasing
  filters
• PID Control Discretization, integrator windup
• Real time implementation

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Computer controlled systems
Quantization in time and magnitude
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The sampling process

• Periodic sampling
• Sampling instants times when the measured
  physical variables are converted into digital
  form
• Sampling period time between two sampling
  instants (denoted by h or T)

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Sampling and Reconstruction

• Sampler A device that converts a continuous time
  signal into a sequence of numbers, e.g., A/D
  converter
• A/D 8-16 bits ? 28 to 216 levels of
  quantization, normally higher than the precision
  of physical sensor
• Reconstructor Converts numbers from the computer
  to analog signals
• D/A converter combined with a hold circuit (Zero
  Order Hold)

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Discretization of continuous controllers
k
J
Disk drive system controller design Analysis
done on board
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Sampling continuous signals
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Aliasing and new frequencies

• Can information be lost by sampling a continuous
  time signal?
• ? sin(2p 0.9t) sin(2p 0.1t)
• Are sampled values obtained from a low frequency
  signal or a high frequency signal? ? It may be
  possible that some continuous signal CANNOT be
  recovered from the sampled signal
• ? May have loss of information

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Shannons Sampling Theorem (1949)

• If a signal contains no frequencies above w0,
  then the continuous time signal can be uniquely
  reconstructed from a periodically sampled
  sequence provided the sampling frequency is
  higher than 2w0 (Nyquist rate)

Can recover 0.1 Hz but not 0.9 Hz
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Sampling Theory- Example

• Can recover 0.1 Hz but not 0.9 Hz
• What should be the sampling frequency for
  recovering the 0.9 Hz sinusoidal?

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Aliasing Anti-aliasing Filters

• Aliasing (or frequency folding) High frequency
  signal interpreted as a low frequency signal when
  sampled at lower than the Nyquist rate (the
  0.91-0.1 Hz signal)

w1
ws
-w1
ws-w1
wsw1
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anti-aliasing filters

• To prevent aliasing
• Remove all frequencies above the Nyquist
  frequency before sampling the signal.
• Antialiasing filter (low pass analog filter)
• High attenuation at and above Nyquist frequency
• Bessel/Butterworth filters (order 2-6)

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Example Pre-filtering

• Square wave 0.9 Hz sine disturbance

Filtered by a 0.25Hz LPF.
Sampled values
Sampled at 1 Hz (alias freq 0.1 Hz)
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Summary

• Information can be lost through sampling if the
  signal contains frequencies higher than the
  Nyquist frequency.
• Sampling creates new frequencies (aliases).

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summary

• Aliasing makes it necessary to filter the
• signals before they are sampled.
• Effect of the anti-aliasing filters must
• be considered when designing controllers.
• ? The dynamics can be neglected if the sampling
• period is sufficiently short.

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Computer control (Wittenmark, Astrom, Arzen)

• PID Control Discretization, saturation and
  windup
• Other practical issues

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PID Control

• Most common controller
• e error, K gain, Ti integration time, Td
  derivative time
• System error as K 1/Ti , stability improves
  as Td , increasing 1/Ti reduces stability
• Originally implemented using analog technology

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PID Control Discretization

• Modifications to PID
• Derivative term
• Derivative not acting on the command signal

N gain at high freq 3-20
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Discretization of PID Controller

• Proportional part needs no approximation
• Integral part
• ? can be approximated as

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.. Discretization of PID PID

• Derivative part D(t)
• ? use backward differences for approximation

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Integrator windup

• Actuators can become saturated due to large
  inputs
• Examples of saturated actuators
• - throttle position in an automobile
• - amplifier output voltage
• - aircraft control surfaces

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A taste of the practical world

• If error is large, Integral Control can saturate
  the actuator.

uc
reference
u
System
y
PID

error
-

• Integrator output can become large (windup) with
  no effect on the output (due to saturation).

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integrator windup

• Integrator output winds up until the sign of e(t)
  changes and the integration turns the other way
  around ?Solution Reset integrator when actuator
  is saturated

Using anti-windup
Desired output
Without anti windup
y
uc
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PID implementation
Signals
States
Parameters double Kp, Ki, Kd double Ts
//Sampling time double ul, uh //output limits

double uc // set point double y // measured
variable double v // control output double u
//limited control output
double I 0 //Integral part double D 0
//Derivative part double yold0 //delayed
measured variable
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PID class
PID Signals signals States states Parameters
par double calculateOutput(uc, y) void
updateState(double u)

oo
Calculate, Perform integrator anti-windup
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Main program

• main( )
• double uc, y, u
• PID pid new PID( ) // Can pass parameters
  here
• long time getCurrentTimeMillis( ) // Get
  current time
• while (true)
• y readY( ) uc readUc( )
• u pid.calculateOutput(uc,y)
• writeU(u)
• pid.updateState(u)
• time time pid.par.Ts1000 // Increment time
• waitUntil(time) // Wait until "time- RTOS call

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void updateState(double u)

• void updateState(double u)
• states.I states.I par.bi(signals.uc -
  signals.y) par.ar(u - signals.v)
• states.yold signals.y

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calculateOutput( )

• double calculateOutput(double uc, double y)
• .
• double P par.K(par.buc - y)
• states.D par.adstates.D - par.bd(y -
  states.yold)
• signals.v P states.I states.D
• if (signals.v lt par.ulow)
• signals.u par.ulow
• else
• if (signals.v gt par.uhigh)
• signals.u par.uhigh
• else
• signals.u signals.v
• return signals.u

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Summary of PID

• PID is a useful controller.
• Things to consider
• Filtering of the derivative term
• Updating the integrator
• Anti windup compensation

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Some practical issues

• Numerical accuracy is improved if more bits are
  used
• float coeff vs double coeff (64 bits)
• A/D and D/A ? Much less accuracy
• Typical resolutions 8, 12, 16 bits
• Better to have a good resolution at the A/D
  converter than D/A
• A control system is less crucial for a quantized
  signal applied to the input of the plant.

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practical issues

• Nonlinearities such as quantization by A/D can
  result in oscillations
• Do simulations to find out if A/D, D/A
  resolutions can improve performance
• Selection of the sampling period
• A common rule wT 0.1 to 0.6,
• w desired BW of the closed-loop system

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practical issues

• Anti-aliasing filters
• Must provide attenuation at Nyquist frequency
• Anti-reset windup is important to include in the
  controller.