spectral observer with reduced information demand györgy orosz, lászló sujbert, gábor péceli...
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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!
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