Spectral Observer with Reduced Information Demand György Orosz, László Sujbert, Gábor Péceli Department of Measurement and Information Systems Budapest University of Technology and Economics, Budapest, Hungary IEEE International Instrumentation and Measurement Technology Conference – I 2 MTC 2008 Victoria, Vancouver Island, British Columbia, Canada, May 12-15, 2008

Motivations, background  Wireless adaptive signal processing and wireless control systems plant sensor 1 sensor 2 sensor N Wireless Network Wireless Network Signal processing Control signals feedback 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 e n : W n+1 = W n + Φ n ∙e n Hence the sign-error principle can be deployed: W n+1 = W n + Φ n ∙sgn(e n )  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

The resonator based spectral observer (basic structure)  The observer is based on the following signal model:  Periodic signal: output of a linear system  State variables: Fourier-coefficients  The state variables are estimated by an observer 1 z −1 1 c 1,n c 2,n ynyn cN,ncN,n 1 z −1

 Possible design methods: Luenberger observer Kalman filter Minimization of squared error The resonator based spectral observer (basic structure) – ynyn + enen 1 z −1 1 g 1,n g 2,n c 1,n c 2,n y’ n cN,ncN,n 1 z −1 gN,ngN,n 1 g i,n c i,n Qi(z)Qi(z) i-th resonator channel  Modification of the state variables with gradient method:

The resonator based spectral observer (basic structure) 1 z −1 g i,n c i,n Qi(z)Qi(z) i-th resonator channel – ynyn + enen 1 z −1 1 g 1,n g 2,n c 1,n c 2,n y’ n cN,ncN,n 1 z −1 gN,ngN,n  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: – ynyn + enen 1 z −1 1 g 1,n g 2,n c 1,n c 2,n y” n cN,ncN,n 1 z −1 gN,ngN,n A(z)A(z) y’ n  Â −1 (z i ): inverse model of the plant  α : convergence parameter Settling time Disturbance rejection  Excellent properties for periodic signals y n : reference signal y’ n : plant’s output e n = (y n –y’ n ) : error signal If e n →0: y’ n → y n

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(e n ) 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 ν −ν−ν enen – ynyn + enen y” n Q1(z)Q1(z) Q2(z)Q2(z) QN(z)QN(z) A(z)A(z) y’ n

The sign-error algorithm ν −ν−ν enen – ynyn + enen y” n Q1(z)Q1(z) Q2(z)Q2(z) QN(z)QN(z) A(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(e n ) 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 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 basic algorithm

sign-error algorithm  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 Properties of the sign-error algorithm

The normalized sign-error algorithm normalized sign-error algorithm V=3  Utilization of the norm of the error signal: e m = [e m e m −1 … e m −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

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: y n = noise: reference signal y’ n = anti-noise: output of the plant e n = y n − y’ n = remaining noise: superposition of the noise and anti-noise The error signal e n is directly measured by the sensor: signum can be calculated! sensor DSP gateway Noise source − y’n− y’n ynyn radio communication sgn(e n ); ||e m || 1 [+++−−…−−−+]

Practical application sensor DSP gateway Noise source − y’n− y’n ynyn radio communication sgn(e n ); ||e m || 1 t enen [+++−−…−−−+] − − … − − − + 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  Parameters were set to ensure similar steady state error  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

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

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