A fast identification algorithm for Wiener-Hammerstein systems is proposed. The computational cost of separating the front and the back linear time-invariant (LTI) block dynamics is significantly improved by using discrete optimisation. The discrete optimisation is implemented as a genetic algorithm. Numerical results confirm the efficiency and accuracy of the proposed approach

Ferranti, Francesco

Rolain, Y

Schoukens, M

Vandersteen, G

English

Ghent University Academic Bibliography

Fast Identification of Wiener-HammersteinSystems using Discrete OptimizationM. Schoukens, G. Vandersteen, Y. Rolain, and F. FerrantiThis letter proposes a fast identification algorithm for Wiener-Hammerstein systems. The computational cost of separating the front andthe back linear time invariant block dynamics is significantly improved byusing discrete optimization. The discrete optimization is implemented as agenetic algorithm. Numerical results confirm the efficiency and accuracyof the proposed approach.Introduction: Block-oriented structures are useful to model a nonlinearsystem. Applications range from RF amplifiers over chemical processesto physiological systems [1]. A block-oriented model consists of twotypes of blocks: Linear Time Invariant (LTI) and static nonlinear blocks.The most simple block-oriented model structures are the Wiener (a LTIblock followed by a static nonlinear block) [1],[2] and the Hammerstein(static nonlinear block followed by a LTI block) [1],[3] model structure. Astraightforward extension of the Wiener and the Hammerstein structureis the Wiener-Hammerstein structure. A Wiener-Hammerstein modelstructure (see Figure 1) is given by a static nonlinearity that is sandwichedin between two LTI blocks [1],[4],[5]. In this letter we assume that theoutput y(t) of a Wiener-Hammerstein system is given by:y(t) = S(q) [f (H(q) [u(t)])] , (1)where q−1 denotes the backwards shift operator and u(t) is the noisefree input signal. H(q) and S(q) are rational forms in the backward shiftoperator q−1, and f(x) is assumed here to consist of a linear combinationof basis functions for the nonlinearity (e.g. polynomials):H(q) =D(q)C(q)=d0 + d1q−1 + . . .+ dndq−ndc0 + c1q−1 + . . .+ cncq−nc, (2)S(q) =B(q)A(q)=b0 + b1q−1 + . . .+ bnbq−nba0 + a1q−1 + . . .+ anaq−na, (3)r(t) = f(x(t)) =nw∑j=1wjfj(x(t)). (4)Obtaining an estimate for the two LTI-blocks is the main difficultyduring the identification of a Wiener-Hammerstein system. A brute-forceapproach is proposed in [4] to split the dynamics that are present at thefront and at the back of the Wiener-Hammerstein structure. This methodis quite straightforward and obtains good results. Its main drawback is thehigh computational time that is needed for the brute-force scan.A fractional power approach is proposed in [5] to avoid the time-consuming scanning procedure. All the poles and zeros are present in bothsubsystems H(q) and S(q) with an exponent of respectively α and 1 − α(0 ≤ α≤ 1):H(q) =D(q)C(q)=∏nzi=1(zi − q−1)αz,i∏npi=1(pi − q−1)αp,i, (5)S(q) =B(q)A(q)=∏nzi=1(zi − q−1)(1−αz,i)∏npi=1(pi − q−1)(1−αp,i), (6)where np and nz denote the number of poles and the number ofzeros respectively. This replaces the brute-force scan by a continuous(constrained) optimization over the fractional powers αp,i and αz,i of thepoles and zeros. This allows the poles and zeros to shift smoothly fromH(q) to S(q). The main disadvantage of this optimization scheme is thatit results in a fractional power representation of H(q) and S(q), which canbe a local optimum. This fractional power representation can be recast intoa power representation as in (2) and (3) in a second optimization step, butit makes the method more computationally involved.This letter proposes to use a discrete optimization approach to speedup the identification process, while keeping a high accuracy level andthe simplicity of the brute-force approach. The outline of the letter is asfollows. The first section summarizes the method that is presented in [4].Next, the discrete optimization approach is explained. Finally, a simulationexample shows the improvement in accuracy and efficiency that is obtainedusing the discrete optimization approach.Fig. 1 A Wiener-Hammerstein system: a static nonlinear block f(x)sandwiched in between two LTI blocks H(q) and S(q).Brute-force Identification by Pole-Zero Allocation Scan: The algorithmproposed in [4] starts with the estimation of the poles and zeros of theoverall dynamics that are present in the nonlinear system. This is doneusing the Best Linear Approximation (BLA) [6]. When input signalsbelonging to the Riemann equivalence class of asymptotically normallydistributed excitation signals are used, the BLA of a Wiener-Hammersteinsystem is given by [6]:GBLA = αH(q)S(q), (7)where α is a gain factor that depends on the power spectrum of the inputsignal and the Wiener-Hammerstein system under study. The poles andthe zeros of the parametrized BLA are also the poles and zeros of the LTIblocks H(q) and S(q) of the Wiener-Hammerstein system.These poles and zeros of the BLA need to be assigned to either H(q) orS(q). All possible partitions of poles/zeros Ĥk(q) and Ŝk(q) are createdsuch that:GBLA = αkĤk(q)Ŝk(q), (8)where k is the combination index and αk is a gain factor, which is includedin the estimation of the static nonlinearity.Next, a static nonlinearity is estimated for every combination k. Thisproblem is linear in the parameters if the static nonlinearity f̂k(x(t)) isdescribed by a linear combination of basis functions. It is solved usinglinear least squares regression, resulting in the estimate f̂k.Finally, all the estimated models are ranked based on their mean squareserror ek:ek =1NN∑t=1(y(t) − ŷk(t))2 , (9)ŷk(t) = Ŝk(q)[f̂k(Ĥk(q) [u(t)])], (10)where N is the number of data points in the measured input-output record.The model with the lowest error is selected.The total number of combinations k depends on the number of polesand the number of zeros, np and nz respectively. It equals the totalnumber of least squares regressions that needs to be performed. The totalnumber of combinations is minimum 2np+nz2 and maximum 2np+nz .The minimum number of combinations is obtained when all poles andzeros are part of a complex conjugate pair, the maximum number ofcombinations is obtained when all poles and zeros are real. This numberincreases very rapidly with the model order, which makes the brute-forcescan approach in [4] computationally expensive. It is important to notethat each complex conjugate pole or zero pair is assigned either to S(q)or H(q) in the combinations. The single elements of the pair are neverassigned separately. This allows the LTI blocks to be transfer functionswith real coefficients.A similar approach is also used for the identification of parallel Wiener-Hammerstein systems [7]. A parallel Wiener-Hammerstein system consistsof several Wiener-Hammerstein subsystems that share the same input. Theoutput of a parallel Wiener-Hammerstein system is given by the sum of theoutputs of the Wiener-Hammerstein subsystems. The computational costof the pole-zero partitioning step is even higher in this case since the brute-force scan is performed for all the parallel branches simultaneously.Pole-Zero Allocation using Discrete Optimization: The discreteoptimization-based approach proposed in this letter speeds up the processof finding the best pole and zero allocation in a Wiener-Hammersteinmodel. The position of each pole and zero is represented by a binary value.The value 1 indicates that the pole or zero belongs to the front LTI blockH(q), the value 0 that it belongs to the back LTI block S(q). This resultsin a binary parameter vector θ. The optimization of the binary parametervector is performed using a genetic algorithm. Genetic algorithms belongto the larger class of evolutionary algorithms, which generate solutionsto optimization problems using techniques inspired by natural evolution,such as mutation, selection and crossover [8],[9]. Genetic algorithmsELECTRONICS LETTERS 14th August 2014 Vol. 00 No. 00Table 1: Settings of the Matlab Optimization toolbox function ’ga’.Setting Name settingPopulationType bitstringCreationFcn gacreationuniformCrossoverFcn crossoverscatteredMutationFcn mutationuniformGenerations 50PopulationSize see Table 2StallGenLimit 5TolFun 1e-20Table 2: Monte Carlo simulation results: average time and success rates.na = nb = nc = nd 5 6 7 8Population Size 200 400 600 800brute-force Scan 8.73s 12.9s 149s 247sDiscrete optimization (GA) 4.12s 7.30s 14.7s 18.9sSuccess Rate 98% 97% 95% 95%are used in several fields such as bioinformatics, computational science,engineering, economics, chemistry, mathematics and physics. The staticnonlinearity is estimated each time the cost function is evaluated for acertain value of the parameter vector Θ, as explained in the previoussection.Instead of evaluating all possible pole/zero combinations, only a subsetwill be evaluated using the genetic algorithm. This results in a significantspeedup of the identification algorithm.Simulation Example: A Monte Carlo simulation is performed to show thegood results in accuracy and efficiency that are obtained using the discreteoptimization approach. In each simulation a Wiener-Hammerstein systemis created with a polynomial static nonlinearity of degree 3:r= 3x+ w2x2 + w3x3. (11)The coefficient of the linear term is equal to 3, the 2nd and 3rd degreecoefficientsw2 andw3 are uniformly distributed in the range [−0.25, 0.25].The simulation is performed for different LTI orders. The front LTI blockand the back LTI block are Chebychev filters of the same order (na =nb = nc = nd = [5, 6, 7, 8]). The front LTI block is a Chebychev type 1filter with an in band ripple of 3 dB and a cut-off frequency that isuniformly distributed in the range [0.025fs, 0.125fs]. The back LTI blockis a Chebychev type 2 filter with a stop band ripple at 50 dB and a cut-off frequency that is uniformly distributed in the range [0.025fs, 0.125fs].The cut-off frequencies of the LTI blocks are independent of each other.The brute-force algorithm and the discrete optimization approach bothuse the same cost function. The brute-force algorithm evaluates this costfunction for every possible parameter vector θ. The discrete optimizationapproach evaluates this cost function for a much smaller set of parametervectors. The genetic algorithm that is present in the Matlab Optimizationtoolbox (’ga’ function name) is used. The optimization settings used forthe simulations are shown in Table 1. An important setting of the geneticalgorithm is the population size. If the population size is too small, thealgorithm might get stuck in a local optimum, a too large populationsize slows down the algorithm since it needs to perform many functionevaluations.The true zeros and poles of the LTI blocks are used, instead of the zerosand poles of the estimated BLA, to simplify the Monte Carlo simulation.The cost function is evaluated with 1 period of a periodic Gaussian noisesignal with a standard deviation equal to one. One period contains 4096points. 100 simulations are performed for each model order. The lowdegree of the nonlinearity in the system, and the limited number of datapoints that is used during the cost function evaluation ensure that the MonteCarlo simulation can be performed in a reasonable amount of time.The discrete optimization approach clearly speeds up the identificationprocess, as is shown in Table 2. A speed-up factor equal to 10 is obtainedfor the higher model orders, while a speed-up factor equal to 2 (a bit lessthan 2 for the model order 12 case) is obtained for the lower model orders.The success rate shows the percentage of the cases where the discreteoptimization and the brute-force scan ended up in the same minimum valueof the cost function. A success rate of 95% to 100% is achieved in all cases.An even higher speed-up can be achieved, but this comes at the cost of alower succes rate.Conclusion: We have presented a fast identification algorithm for Wiener-Hammerstein systems. The problem is reformulated such that a binaryparameter vector can be used to represent the combinations of poles andzeros in the front and the back LTI blocks. A genetic algorithm is used toperform the discrete optimization. The simulation results show that theproposed method is able to identify a Wiener-Hammerstein system tentimes faster than the brute-force scanning approach for the high ordermodels. The proposed method is still almost twice as fast as the brute-force method for lower order models. The proposed method is moreefficient than the brute-force scan approach, while it achieves a very similaraccuracy level. The extension of the proposed algorithm towards parallelWiener-Hammerstein systems will be investigated as future work.Acknowledgment: This work was supported in part by the VUB(SRP-19), the Fund for Scientific Research (FWO-Vlaanderen), theMethusalem grant of the Flemish Government (METH-1), by theBelgian Government through the Inter university Poles of AttractionIAP VII/19 DYSCO program, and the ERC advanced grant SNLSID,under contract 320378. F. Ferranti is currently a Post-DoctoralResearch Fellow of FWO-Vlaanderen. M. Schoukens is currently anFWO Aspirant, supported by FWO-Vlaanderen.M. Schoukens, G. Vandersteen, and Y. Rolain (Department Elec, VrijeUniversiteit Brussel (VUB), Pleinlaan 2, B-1050 Brussel, Belgium)F. Ferranti (Department of Information Technology, Ghent University-iMinds, Gaston Crommenlaan 8/201, B-9050 Gent, Belgium)E-mail: maarten.schoukens@vub.ac.beReferences1 Giri, F. and Bai, E.W. (editors): ‘Block-oriented Nonlinear SystemIdentification’, Volume 404 of Lecture Notes in Control and InformationSciences, 2010, Berlin, Heidelberg.2 Billings, S.A. and Fakhouri, S.Y.: ‘Identification of nonlinear systems usingthe Wiener model’, Electronics Letters, 1977, 13(17), pp. 502-504.3 Crama, P., Schoukens, J. and Pintelon, R.: ’Generation of enhanced initialestimates for Hammerstein systems’, Automatica, 2004, 40(7), pp. 1269-1273.4 Sjöberg, J. and Schoukens, J.: ‘Initializing Wiener-Hammerstein modelsbased on partitioning of the best linear approximation’, Automatica, 2012,48(2), pp. 353-359.5 Vanbeylen, L.: ‘A fractional approach to identify Wiener-Hammersteinsystems’, Automatica, 2014, 50(3), pp. 903-909.6 Pintelon, R. and Schoukens, J.: ’System Identification: A FrequencyDomain Approach’, Wiley-IEEE Press, 2012, Hoboken, New Jersey.7 Schoukens, M., Vandersteen, G. and Rolain, Y.: ’An identification algorithmfor parallel Wiener-Hammerstein systems’, 52nd IEEE Conference onDecision and Control (CDC), 2013, 4907-4912.8 Bäck, T.: ’Evolutionary algorithms in theory and practice evolutionstrategies, evolutionary programming, genetic algorithms’, OxfordUniversity Press, 1996, New York.9 Ahn, C.W.: ’Advances in evolutionary algorithms theory, design andpractice’, Springer, 2006, Berlin, New York.2

Fast identification of Wiener-Hammerstein systems using discrete optimisation

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Fast identification of Wiener-Hammerstein systems using discrete optimisation

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