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High Performance Digital Control
IEEE APEC 2016
Long Beach
March, 2016
Hamish Laird
ELMG Digital Power
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Outline for today
●Introduction ● Component Parts of Power Converter Control ● Power Converter Control Blocks
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Digital Control System ● Introduction
● Power Converter Control Blocks
● Modulators ● Digital VPO performance ● Precision in VPOs – Extending, improving and correcting resolution ● Digital PWM performance ● Correcting in PWM ● Extending PWM precision.
● Compensators ● Unit Circle and Pole Zero Plots ● Useful pole zero combinations ● Pole zero placement – where to put them. ● Implementing these pole zero combinations ● A good filter structure.
● More compensators ● IIR Biquads – number of bits ● Filter coefficients from pole zero placements ● Digital Integrators
● Do I have precision and resolution problems? ● How to determine this?
● How to fix this resolution problem?
● Analogue to digital converters ● Single sample noise
● Anti aliasing filters ● Cutoff frequencies and orders
● Summary
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About Hamish Laird
●CTO at ELMG Digital Power
●Power Electronics Career ● Induction Motor Drives ● Induction Motor Starters ● DC drives (analogue) ● HVDC and SVC ● Medical Devices – motor drives ● Household appliances ● Active Filters ● Medium voltage ● UPS ● Brushless DC drives
●All digital control ● Except analogue ● Microprocessors ● DSP ● FPGA
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Modulators
●Modulator
●PWM
●VPO
●Any modulator is a Harmonic Generator
●Both PWM and VPO are nonlinear
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Compensator
●Filter that ensures that connecting the system output to the system is useful and stable
●Usually low pass
●Useful means error is small
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Analogue to Digital Converter
●Samples analogue.
●Multiplies two signals ● Train of impulses ● Feedback
●A2D is a modulator
●Aliasing can happen – spectrum out is not the same as the spectrum in
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Anti Aliasing Filter
●Low pass filter
●Prevents Aliasing
●Makes A2D linear
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●Typically has a low pass filter characteristic
●Prevents aliasing
Power Converter
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Modulators
●Two kinds covered today
●Variable Period Oscillation ● For resonant power supplies
●Pulse Width Modulator ● Pulse modulator
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●Quantization in modulators
Modulators – Quantization
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VPO
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Oscillators (VPO)
●Example ● 10 MHz clock ● 250kHz out ● Count is 40
●For count of 41
●Out is 243,902 Hz
●For count of 39
●Out is 256,410 Hz
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VPO- characteristics
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VPO- characteristics
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VPO – characteristics
●For 100MHz clock
●715 counts is 9.48 bits
Count Frequency out Hz
285 350,877
1000 100,000
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VPO – Characteristics
●For 1000MHz clock
●7143 counts is 12.8 bits
Count Frequency out Hz
2857 350,017
10000 100,000
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VPO – Characteristics
●Effect of VPO error in feedback control
●Dead-band non linearity
●Slip strike behavior ● Typically requires more bandwidth in compensator
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PWM – Characteristics
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PWM – Characteristics
●Clock Frequency 10MHz
●PWM carrier frequency 25kHz
●Counter maximum = 10MHz/25kHz = 400
●Number of bits is 8.64
On Time Counts Duty On time
400 100% 40us
274 68.5% 27.4us
1 0.25% 0.1us
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PWM – Characteristics
●Clock Frequency 100MHz
●PWM carrier frequency 25kHz
●Counter maximum = 100MHz/25kHz = 4000
●Number of bits 11.96
On Time Counts Duty On time
4000 100% 40us
2740 68.5% 27.4us
1 0.25% 0.01us
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PWM – Characteristics
●Clock Frequency 1000MHz
●PWM carrier frequency 25kHz
●Counter maximum = 100MHz/25kHz = 40000
●Number of bits 15.28 bits
On Time Counts Duty On time
40000 100% 40us
27400 68.5% 27.4us
1 0.025% 0.001us
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PWM Characteristics
●Clock Frequency 1MHz
●PWM carrier frequency 25kHz
●Counter maximum = 1MHz/25kHz = 40
●Number of bits is 5.32
On Time Counts Duty On time
40 100% 40us
27 67.5% 27us
1 2.5% 1us
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Modulators Characteristics
●Spectra of PWM or VPO with quantization
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PWM – Characteristics
●Effect of quantization resolution error in feedback system
●Step non linearity
●Creates slip strike behavior
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Modulators error correction
●Resolution correction gives precision extension
●Resolution in VPO and PWM from timer precision.
●We know the quantization of the modulator
●So we know the error of each switching instant
●We can correct for the error on the next switching cycle
●In the long term we can apply the correct volt seconds.
●How to implement.
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How to correct errors
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Modulators error correction
●How it works
●1 sample at 14 and 1 sample at 15 for average of 14.5
Input Input + previous difference
Output Difference
14.5 14.5 14 0.5
14.5 15 15 -0.5
14.5 14.5 14 0.5
14.5 15 15 -0.5
14.5 14.5 14 0.5
14.5 15 15 -0.5
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Modulators error correction
●For a different value
●Three at 14 and one at 15 for an average of 14.25
Input Input + previous difference
Output Difference
14.25 14.25 14 0.25
14.25 14.5 14 0.25
14.25 14.75 14 0.25
14.25 15 15 -0.75
14.25 14.25 14 0.25
14.25 14.5 14 0.25
14.25 14.75 14 0.25
14.25 15 15 -0.75
14.25 14.25 15 0.25
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Modulators error correction
●Error correction or precision increase is spectral control
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Modulators error correction
●Error correction or precision increase is spectral control
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Modulators spectral effects
●Frequency components with error correction
●Without error correction
Input 0Hz Fs/2 Fs/4
14.5 14.5 0.636 0
14.25 14.25 0.318 0.448
Input 0Hz Fs/2 Fs/4
14.5 14 0 0
14.25 14 0 0
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Modulators –spectral control
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Modulators spectral control
●Simplest
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Modulators error correction
●How to code it
●difference[n]= input[n] – output[n]
●output[n] = input[n] + difference[n-1]
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●Increase filter order for less noise in the control band
Modulators - spectral control
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Modulators – spectral control
●H(z) = (1-z-1)N
●Often plotted on linear frequency scale
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Modulators – spectral control
●H(z) = (1-z-1)N
●Attenuates noise in the converter pass band
●Typically converter bandwidth target is 1/10th to 1/5th of the switching frequency
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Modulators – spectral control
●H(z) = (1-z-1)N
●Log Scale plot gives better insight
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Modulators error correction
●How many more bits?
●Frequency dependent ● At 0Hz get pre-quantization number of bits. ● At Fs/2 get no accuracy.
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Modulators – Summary
●Quantization in PWM and VPO modulators leads to white noise generation in control band.
●Fix this with a simple accumulate and correct.
●Use first order system with 1 delay. ● Typically need no more.
●Second or higher order if you really need to. ● Be careful – watch for its effect on the transient response.
●2nd order in our PWM IP Core block
●Because power converter, compensator and anti aliasing filter are all low pass ● High frequency noise is attenuated by the rest of the loop.
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Modulators error correction
●“This really works.” - Tomek Kotula
●Resolution corrected modulation: the practical realisation of ideal PWM waveforms. Boys and Handley - Electric Power Applications, IEE Proceedings B (Volume:139 , Issue: 4 ) 1992 (IET)
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Summary
●Modulators have low number of bits
●Increase the number of bits with precision extension
●Accumulate the error to correct the next switching
●Works with ● VPO for variable frequency converters ● PWM for pulse modulated converters
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Second Half Introduction
●Digital Control Design ● No analogue translation
●Unit circle
●Digital control blocks – a library
●Compensator form ● IIR Biquad ● Second order
●Design example ● Loop shaping ● Filter coefficients
●Things to watch out for
●Single sample noise
●Anti aliasing filters
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Digital controller design
●A way to design compensator frequency response directly in the digital domain.
●No analogue translation.
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Unit Circle – Poles and Zeros
●z Domain
●The unit circle
●Poles and Zeros in the digital domain
●Stability in z domain poles inside unit circle
●Zeros can be outside unit circle ● Avoid zeros outside unit circle.
●Laplace transforms uniqueness does not apply to the z domain
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Pole Zero combinations
●In analogue control design we use known frequency responses to create compensator
●Integrators
●Lead Lags
●Low Pass filters
●High Pass filters
●Let’s just do the same in the z domain
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Digital control blocks
●Useful digital blocks ● Integrator ● Differentiator ● Low pass filter ● High pass filter ● Lead ● Lag
●Some Others perhaps not so useful ● Complex poles and zeros
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Integrator
●Fs =
●Filter response continues above Fs/2
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Integrator
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Differentiator
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Differentiator
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High Pass Filter
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High Pass Filter
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Low Pass Filter
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Low Pass Filter
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Low Pass Filter
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Low Pass Filter
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Low Pass Filter
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Low Pass Filter
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High Pass Filter
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High Pass Filter
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Lead Lag
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Lead Lag
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Lag
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Lag
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Complex Poles and Zeros
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Complex Poles and Zeros
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Complex Poles and Zeros
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Complex Poles and Zeros
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Complex
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Complex
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Compensator design
●Compensator design is now ● Loop frequency response shaping
●Use one, two or three of the cookbook blocks to implement ● Integrator ● Lead Lag ● Low pass
●Shape the loop
●Pole and zero positions give compensator coefficients
●Compensator form?
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Digital filter implementation
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Second order blocks
●Second order blocks ● Signal to noise is good for 16 bit mathematics. ● Easy to find precision, resolution or quantization
problems.
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Second order blocks
●Useful precision is retained
●Usually good enough for what power electronics control requires (with care)
●Set a0 to -1 normalizes the filter.
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No analogue design
●fs/2 = 10kHz
●Plant measurement
●Made using the digital controller
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Direct Digital Compensator
●Add the integrator
●Scale to close the loop with reasonable margin
●Need an integrator gain of 1/4
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Get the poles and zeros
●Total loop with the integrator added
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Get the poles and zeros
●Integrator form
●Scale the basic integrator form to move the crossover frequency
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Get the poles and zeros
●Lead Lag pole and zero
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Lead Lag
●Lead Lag also
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Complete response
●Frequency response of the complete system
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Coefficients
●Zeros at -1 and 0.975
●Poles at +1 and +0.97
●Multiply out z polynomial to second order
●Divide top and bottom through by z2 to get delayed operator z-1.
Coefficient Value
b0 0.25
b1 -0.725
b2 -0.24375
a1 -1.972
a2 0.97
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All done?
●Increase the precision of the modulator
●Measure the plant frequency response
●Use the block cookbook to design the loop compensator
●Integrator and Lead Lag
●Implement these in a second order IIR biquad.
• Once you have the coefficients your compensator filter design work is only about 1/8 done!
• Usually OK but there are some things to check.
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Things to watch
●There is no a0 choice.
●As many poles as zeros
●Or less zeros than poles
●Precision! Quantization!
●Make the code so that the coefficients can be changed. ● Software engineer creates code for IIR biquad! ● Control design engineer designs compensator coefficients! ● Put the coefficients into the biquad.
●Verify that the filter blocks do what you need. ● Use an automated test harness
●Never never never tweak coefficients. ● a and b do not match directly to pole or zero positions
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Numeric precision ●Timer precision
● Effect of low precision ● Extending precision ● Things to watch out for with precision extensions.
●Extending precision ● Sigma delta approach ● Bandwidth limits ● Intermodulation
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Precision in filters ●Narrow bandwidth filter
●Low pass filter example
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Low pass filter example
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Quantization
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Poles and Zeros
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Low pass filter example
●Consider the case where input range is 40.96 Amps full range with 12 bits. 10mA per bit.
●16 bit maths – round to 16 bits.
●Put the A2D 12 bits in the MSBs
●b0 and b1 =1247
●At what coefficient precision do you lose accuracy?
●What about loss of function?
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Low pass filter continued
●Your control design engineer now suggests these filters instead
●What do you say?
●b0 and b1 are 5
●b0 and b1 are 78
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Low pass filter example 2
Input Digital Equivalent
b0 product b1 product Product Output
10mA 8 40 40 80 0
20mA 16 80 80 160 0
60mA 48 240 240 480 0
100mA 80 400 400 800 0
200mA 160 800 800 1600 0
600mA 480 2400 2400 4800 0
1A 800 4000 4000 8000 0
2A 1600 8000 8000 16000 0
6A 4800 24000 24000 48000 1
10A 8000 40000 40000 80000 2
20A 16000 80000 80000 160000 4
40.96A 32768 163840 163840 327680 10
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Low pass filter example 2
Input Digital Equivalent
b0 product b1 product Sum of Products
Output
10mA 8 632 632 1264 0
20mA 16 1264 1264 2528 0
60mA 48 3792 3792 7584 0
100mA 80 6320 6320 12640 0
200mA 160 12640 12640 25280 0
600mA 480 37920 37920 75840 2
1A 800 63200 63200 126400 3
2A 1600 126400 126400 252800 7
6A 4800 379200 379200 758400 23
10A 8000 632000 632000 1264000 38
20A 16000 1264000 1264000 2528000 769
40.96A 32768 2588672 2588672 5177344 1580
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Poles and zeros
●Second order blocks can have any poles and zeros from transfer function
●Look out for poles and zeros very close together – could you remove them?
●Poles and zeros definitely on top of each other should be cancelled.
●If you have two biquads spread the poles and zeros to minimize the quantization and precision loss.
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Precision vs dynamic range
●Direct trade off between quantization and saturation
●How much precision is enough?
● Second order block precision rules of thumbs – it will mostly be OK but check.
●Noise effect of saturation is the same as precision loss ● Consider letting things saturate but be careful
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Numeric Precision summary
• Once you have the coefficients for your compensator your design is only about 1/8 done!
• Usually OK.
• But • Check narrow band filters • Poles close to unit circle • Frequency response • Use a ramp to find places where LPF have problems
• Always • Work with biggest signals and biggest coefficients if possible • Check filter’s performance • Always CHECK it.
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Digital Integrators
●Classic digital integrator from process control is an accumulator
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Possible Digital Integrators
●Frequency response
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Digital integrator choices
●Sum only integrator has pole zero map
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Digital Integrator
●Better integrator has pole zero map
●Integrator has one pole and one zero
●Implement in biquad.
●Have spare pole and zero for something else.
●Most compensators need only one biquad!!!
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Choosing the Best Integrator
●Better integrator
●Attenuation at Fs/2 – this is good
PAGE 104
Implementing the Digital Integrator
●Use the biquad
●Put the pole and zeros into the biquad.
PAGE 105
Resolution integrators
●Lower crossover frequency
●b0=0.001 b1 = 0.001, a0 = -1
PAGE 106
Integrator Example
Input Digital In b0 product b1 product Sum of Products
Output
10mA 8 256 256 512 0
20mA 16 512 512 1024 0
60mA 48 1536 1536 3072 0
100mA 80 2560 2560 5120 0
200mA 160 5120 5120 10240 0
600mA 480 15360 15360 30720 0
1A 800 25600 25600 51200 0
2A 1600 51200 51200 102400 0
6A 4800 153600 153600 307200 9
10A 8000 256000 256000 512000 15
20A 16000 512000 512000 1024000 31
40.96A 32768 1048576 1048576 2097152 64
PAGE 107
Fixing integrators
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Analogue to digital converters
●A2Ds are clocked systems
●Power electronic switching is noisy
●In a microprocessor system have multiple clocks
●Some are PLL’d to each other ●All clocks jitter all the time.
●Can get single sample noise from the A2Ds when all the clock edges line up.
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Single Sample Noise
●What does it look like?
●Impulse response of closed loop – part digital part analogue
●This response is almost completely unrecognizable behavior in the output
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Single Sample Noise
●A true digital impulse
●System has all available energy of the power converter to put into the output ● Single sample noise issues often cause converter destruction. ● It might be the cause of your blow up.
●Input to modulator must be low pass filtered!!!!
●If you have single sample noise ● hopefully get reasonable occurrence rate – not too high not too low
●Find problem by looking at the digital stream. ● Build this feature into your converter controller.
●Solve problem by ● Dealing with the correlation ● Electrostatic shielding ● Change the A2D – qualify the A2D before you begin.
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Anti Aliasing Filters
●Always use anti aliasing filters ● Always
•Always
●Always design the anti aliasing filter
●Some of the filtering from the power converter
●Get the rest by adding lowest order filter possible
PAGE 112
Anti aliasing cutoff frequency
●Best case design rule ● Make the sampled ripple at the A2D input the same size as 1 LSB
change.
●Realistic Design Rule ● Choose the phase shift of the anti aliasing filter for stability ● Power converter phase shift ● Anti aliasing filter phase shift
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Anti aliasing filter tips
●Transient response shorter than the sampling time if possible
●First order RC normally enough
●Even if you are oversampling and have digital anti-aliasing filters use an analogue anti-aliasing filter
●For higher orders - well damped is best.
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Summary
●System
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Summary
●Modulators ● Effective numeric precision ● Quantization ● Correcting the precision ● Frequency weighting
●Compensators ● Block set of useful pole zeros combinations ● Direct digital design ● No analogue translation ● Second order blocks IIR biquads ● Coefficients of IIR biquad filters from poles and zeros
●Anti Aliasing Filter ● How to Design
●Single Sample Noise ● How to find ● Effects ● How to diagnose
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