Spectral Observer with Reduced Information DemandGyörgy Orosz, László Sujbert, Gábor Péceli

Department of Measurement and Information SystemsBudapest University of Technology and Economics, Budapest, Hungary

IEEE International Instrumentation and Measurement Technology Conference – I2MTC 2008Victoria, Vancouver Island, British Columbia, Canada, May 12-15, 2008

Motivations, background Wireless adaptive signal processing and wireless control systems

plant

sensor1

sensor2

sensorN

WirelessNetwork

Signalprocessing

Control signalsfeedback signals

Advantages of wireless communications: Flexible arrangement Easy installation

Typical problems of wireless systems: Data loss, unreliable data transfer Nodes with limited resources Bandwidth limit (approximately from ten to some hundred kilobit per second)

Lots of sensors Fast data acquisition and transmission

Goal: decreasing the amount of data in the feedback path Reduced bandwidth demand Increased number of nodes on the same channel is allowed Energy saving

Basic idea: adaptive and control algorithms are generally based on an error term en: Wn+1 = Wn + Φn∙en Hence the sign-error principle can be deployed:

Wn+1 = Wn + Φn∙sgn(en )

New application field: adaptive Fourier-analysis and control with resonator based observer and controller algorithms

Improvement: scalable structure Practical application: wireless active noise control (ANC) system

Goals and solutions

0if10if00if1

)sgn(xxx

x

The resonator based spectral observer (basic structure) The observer is based on the following signal model:

functionsbasis:];......[ ,,,,1nij

ninNninn ecccc c

signalperiodic:,, nn

L

Lininin xcy xc

Periodic signal: output of a linear system State variables: Fourier-coefficients The state variables are estimated by an observer

1z −1

1z −1

c1,n

c2,n

yncN,n1

z −1

scoeficientFourier:];......[ ,,,1 nnNninn xxx xxnx ,1

nx ,2

nNx ,

Possible design methods: Luenberger observer Kalman filter Minimization of squared error

The resonator based spectral observer (basic structure)

–yn

+en

1z −1

1z −1

g1,n

g2,n

c1,n

c2,n

y’ncN,n1

z −1

gN,n

nx ,1ˆ

nx ,2ˆ

nNx ,ˆ

22 '21

21)ˆ( nnnn yyeC x

2ˆ21)ˆ( nnnn yC xcx

nnnnnnn

n eyC HH )ˆ(ˆ

)ˆ( cxccxx

nnnnnnn

nnn eeC gxcx

xxxx

ˆˆˆ

)ˆ(ˆˆ H1

1z −1

gi,n ci,n

Qi(z)

i-th resonator channel

nix ,ˆ

Modification of the state variables with gradient method:

The resonator based spectral observer (basic structure)

1z −1

gi,n ci,n

Qi(z)

i-th resonator channel

nix ,ˆ

–yn

+en

1z −1

1z −1

g1,n

g2,n

c1,n

c2,n

y’ncN,n1

z −1

gN,n

nx ,1ˆ

nx ,2ˆ

nNx ,ˆ

Advantages: Recursive Fourier-decomposition Variable fundamental harmonic

(adaptive Fourier-analysis) Dead beat settling can be ensured Averaging for improving the noise

sensitivity Due to the internal loop, it

decreases the effect of quantization noise

Resonators: Infinite gain on the resonator

frequencies zero steady state error of the observation

The resonator based controller The plant to be controlled is in

the feedback loop Control algorithm:

–yn

+en

1z −1

1z −1

g1,n

g2,n

c1,n

c2,n

y”ncN,n1

z −1

gN,n

A(z)

y’n

nx ,1ˆ

nx ,2ˆ

nNx ,ˆ

)'(;ˆˆ 1 nnnnnnn yyee gxx

T*,

1 )(ˆniin czAg

Â−1(zi): inverse model of the plant α: convergence parameter

Settling time Disturbance rejection

Excellent properties for periodic signals

yn : reference signal

y’n: plant’s output

en = (yn –y’n ): error signal

If en→0: y’n →yn

nnny xc ˆ

The sign-error algorithm

The updating algorithm in the sign-error structure:

Reduces the computational complexity If the error is known on the sensor sgn(en) can be calculated The signum can be represented by few bits reduced amount of data

Possibility for increasing the number of nodes or the sampling frequency

)sgn(ˆˆ 1 nnnn egxx T*,

1 )(ˆniin czAg

ν

−νen

–yn

+

en

y”n

Q1(z)

Q2(z)

QN(z)

A(z)

y’n

The sign-error algorithm

ν

−νen

–yn

+

en

y”n

Q1(z)

Q2(z)

QN(z)

A(z)

y’n

sensor

The updating algorithm in the sign-error structure:

Reduces the computational complexity If the error is known on the sensor sgn(en) can be calculated The signum can be represented by few bits reduced amount of data

Possibility for increasing the number of nodes or the sampling frequency

)sgn(ˆˆ 1 nnnn egxx T*,

1 )(ˆniin czAg

controller

Properties of the sign-error algorithm In the original algorithm

the step size depends on the amplitude of the error

Direction is basically influenced by the signum

2x̂

1x̂

basic algorithm2

21)ˆ( eC x

nn eHˆ cx

2x̂

1x̂

sign-error algorithm2

21)ˆ( eC x

)sgn(ˆ Hnn ecx

The updating depends only on the signum

α doesn’t influence the stability

Steady state error:

Settling time:

N: number of harmonics Contradictory conditions for

choosing α

22

1)(21

0

Nn

en

nEn

kka

x

NM

x

Properties of the sign-error algorithm

The normalized sign-error algorithm

1x̂

2x̂ normalized sign-error algorithm2

21)ˆ( eC x V=3

Utilization of the norm of the error signal:em = [em em−1 … em−V+1]T

V: the length of intervals the norms of which is calculated

m: time instant where the norm is calculated

Increased amount of data better properties

V allows the tuning of the properties: V=1 : original observer V∞: sign-error observer

1)sgn(ˆ H

mnn e ecx

11 )sgn(ˆˆ Hmnnnn e ecxx

Practical application Wireless active noise control system (ANC)

Goal: decreasing the power of acoustic noise Noise sensing by wireless sensor nodes

Why is critical? Relatively high sampling frequency (1-2 kHz) Lots of sensors Real time operation of the system

Notations: yn = noise: reference signal y’n = anti-noise: output of the plant en= yn− y’n = remaining noise: superposition of the noise and anti-noise The error signal en is directly measured by the sensor: signum can be calculated!

sensor

DSP gateway

Noise source

− y’n

ynradio communication

sgn(en); ||em||1[+++−−…−−−+]

Practical application

sensor

DSP gateway

Noise source

− y’n

ynradio communication

sgn(en); ||em||1

t

en

[+++−−…−−−+]+ + + − − … − − − +

V=32

Preprocessing of the signal on the sensor: Calculation of the signum Calculation of the norm of the error

Wireless sensors: Berkeley micaz motes 8 bit microcontroller 250kbps ZigBee radio

DSP: Analog Devices ADSP21364 (Sharc) 32 bit floating point 330 MHz CPU

In normalized sign error structure norm is calculated for 32 samples long periods Data reduction compared to the original algorithm:

Simple sign-error controller : 12.5% of the original amount of data Normalized sign-error controller: 16.6% of the original amount of data

Measurement results: steady state

0 200 400 600

-100

-80

-60

-40

-20

0

[Hz]

original

ampl

itude

[dB

]

0 200 400 600

-100

-80

-60

-40

-20

0

[Hz]

sign error

0 200 400 600

-100

-80

-60

-40

-20

0

[Hz]

normalized sign error

ANC offANC on

ANC offANC on

ANC offANC on

Parameters were set to ensure similar steady state error 20-30 dB noise suppression

Measurement results: transient

Settling times: Fastest transient: 0.3 sec (original) Slowest transient: 25 sec (simple sign-error) Normalized sign-error: 1 sec

0.5 1 1.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8original

[sec]

ampl

itude

0 10 20 30-0.5

0

0.5sign error

[sec]0 0.5 1 1.5 2 2.5

-0.5

0

0.5normalized sign error

[sec]

Results and future plans Development of the sign-error spectral observer Investigation of the properties of the algorithm Improvement of the sign-error algorithm Implementation of the sign-error and normalized

sign-error controller in a practical application: wireless ANC system

Plans: Further investigation of these structures Expansion of the theory to MIMO systems

Thank You for the attention!