Elsevier

Applied Soft Computing

Volume 85, December 2019, 105745
Applied Soft Computing

Efficient Volterra systems identification using hierarchical genetic algorithms

https://doi.org/10.1016/j.asoc.2019.105745Get rights and content

Highlights

  • Robust evolutionary identification of the active basis sets of Volterra series.

  • Heuristics for generating feasible initial solutions for system identification.

  • Parameters calibration evaluation

Abstract

The Volterra series consists of a powerful method for the identification of non-linear relationships. However, the identification of the series active basis sets requires intense research in order to reduce the computational burden of such a procedure. This is a result of a large number of measurements being required in order to produce an adequate estimate, due to overparameterization issues. In this work, we present a robust hierarchical evolutionary technique which employs a heuristic initialization and provides robustness against noise. The advanced solution is based on a genetic algorithm which improves on the computational complexity of existing methods without harming the identification accuracy. The impact of the parameters calibration is evaluated for different signal-to-noise levels and several nonlinear systems considered in the literature.

Introduction

The identification task can be interpreted as recovering the system hidden states by exploiting its input–output relationship data in the digital domain. Important applications of such a data-driven representation task are: (i) acoustic echo cancellation [1]; (ii) mitigation of intersymbolic interference by pre-distortion techniques [2]; (iii) spectral regrowth analysis [3]; and (iv) active noise control [4]. Note that some of these examples require the identification of nonlinearities, which is especially important when their impact on the single-valued output is not negligible or the input signal range is large [5]. Nonlinearities effects can be caused by (sometimes complex) physical nonlinear phenomena, such as operation near the saturation region [6], intermodulation distortion [7], and diffusion capacitance [8]. In practice, nonlinear systems can be accurately modeled with limited prior knowledge by Volterra1 series [9], [10], which are essentially a flexible (and always stable [11]) functional series expansion of a nonlinear time-invariant system. These series have the advantage of taking into account memory effects, in contrast to static nonlinear models [12], [13]. The generality of Volterra models can be shown either by interpreting them as discrete-time systems with fading memory or by the application of the Stone–Weierstrass theorem to the approximation of input/output finite-memory mappings [11], [14], [15]. The result published in [16] deserves mention, since it states the existence of a locally convergent power-series-like expansion of a large class of systems that contain an arbitrary (although finite) number of nonlinear elements.

It is noteworthy that Volterra modeling is an enduring research problem due to its wide range of applications [17]. Such models describe with conceptual simplicity the system output as a sum of a first-order operator, a second-order operator and higher-order operators, generalizing the convolution concept from linear time-invariant systems [5], [18], [19]. As the memory (or delays) and orders become larger, the number of the Volterra coefficients (each of them unequivocally associated to a kernel or basis waveform) increases geometrically. This makes the identification task a very challenging one, especially when there is a lack of knowledge about the operating principle and/or the structure of the device to be identified [5], [20]. In optical transmissions systems, for example, as the transmission capacity increases, the computational burden required for standard techniques to model the communication system may be unacceptable [21]. Due to these facts, it is important to identify such systems with a low computational burden. Furthermore, robustness against the ubiquitous noise is crucial.

The global search feature of evolutionary algorithms avoids the local minima trapping phenomenon in non-convex or NP-hard optimization problems, providing an effective search procedure for different research areas [22]. Such properties enable them to become a natural choice for the selection of proper basis Volterra sets. In general terms, an evolutionary algorithm deals with individuals, aiming to encounter a proper point that conveniently addresses the inherent trade-off between the exploration and exploitation abilities of the stochastic search [22]. Each individual is unequivocally mapped into a candidate solution of an objective function defined for optimization purposes. The value of such a function, evaluated using a properly mapped individual as an argument, is employed as a fitness evaluation of the candidate solution.

Several families of evolutionary schemes have been advanced, such as particle swarm optimization [23], multiobjective decomposition-based algorithm [24], genetic programming [25], reaction optimization [26], indicator-based algorithms [27], firefly algorithms [28], artificial bee or ant colony algorithms [29], differential evolution [30], learning automata-based selection [31]. To our knowledge, none of these evolutionary techniques was ever employed in order to address the identification of Volterra systems.

The focus of this paper is on genetic algorithms [32], which can be regarded as a nature inspired meta heuristics that also enforces an evolutionary strategy. Accordingly, we propose a genetic algorithm that efficiently takes into account the idiosyncrasies of Volterra-based identification tasks. The first attempt to use genetic algorithms (GAs) for the identification of Volterra systems was devised in [33], which encoded the active kernels by binary chromosome representation. This paper (as well as the very similar approach of [34]) assumed a multi-objective performance criterion, combining both mean squared error (i.e., the 2-norm) and the maximum error (i.e., the -norm of the error). Work [35] proposed the usage of the least squares procedure for the estimation of the coefficients of the supposed-to-be active kernels. Reference [36] encoded the location of active kernels using B bits, which requires precautions against non-factible locations. The employment of genetic algorithms was also proposed by [37], which aims to capture the nonlinear relationships of functional link networks, consisting of high-order perceptrons that may be equivalently rewritten as Volterra models. The floating-point genetic algorithm presented in [38] combines the kernels selection and coefficients identification steps in one single evolutionary step. In [39] an adaptive variable-length GA was proposed whose chromosomes encode the selected candidate’s coefficients. The initialization procedure of this solution assumed to be active the basis functions where the correlation magnitude with the output was large.

This paper proposes an efficient genetic algorithm-based solution for the identification of time-domain Volterra series, suited to get a representation of complex nonlinear systems when a physically-based model is not available [5]. The memory length is assumed to be finite and upper bounded by a known value. The proposed algorithm takes into account the sparsity property that Volterra systems often present in practice [40], [41]. This avoids the need to estimate all kernel coefficients in each step, since often only a small number of them may contribute significantly to the output signal [35], [42]. Furthermore, an initialization procedure that chooses the most promising kernels ith higher probabilities is adopted. Sparsity-aware Volterra identification methods typically require a judicious pruning in order to reduce the basis set size [43], and the proposed method is not an exception.

The data structures of the proposed GA-based methodology were suitably selected to allow a customized hierarchical search of proper solutions in practical systems, spending little computational time for such an identification task. This hierarchical feature presents the potential to address large problems in an efficient way [44].

This paper is structured as follows. Section 2 presents the theoretical modeling regarding the Volterra series. Section 3 presents our advanced GA approach towards identifying the basis sets of time-domain Volterra series. Section 4 discusses the experimental setup and respective results obtained. Section 5 presents the main conclusions of this work.

Section snippets

Volterra series identification model

This paper focuses on the identification of single input single output nonlinear systems. In the case of continuous-time systems, one may write the output y(t) as a sum of response components xn(t) [45]: y(t)=n=1xn(t),where the nth component is described by: xn(t)n×h¯n(τ1,,τn)i=1nu(tτi)dτ1dτn,where h¯n(τ1,,τn) is the nth order Volterra kernel (or the n-th-order impulse response of the non-linear system [46]) and u(t) is the system input. Since the continuous-time modeling of 

Proposed approach

In this section we propose a method for the identification of time-domain Volterra series using genetic algorithms (GA), in order to conveniently address the existence of multiple local optima solutions. In brief, genetic algorithms operate on a population of individuals, where each individual is a potential solution to a problem. Frequently, GA-based approaches ensure higher convergence rates of the search procedure, when compared against conventional gradient-based techniques [54]. In this

Experimental results

In the forthcoming simulations, the measurement noise signal ν[k] is assumed to be a white Gaussian signal. Its variance is chosen accordingly to the considered signal-to-noise ratio (SNR), in dB, defined as: SNR (dB)10.log10E[y2[k]]E[ν2[k]].

The identification of the following three distinct nonlinear systems will be analyzed:

System I (considered in [64]): d[k]=0.6x[k]+1.2x2[k1]+0.8x[k1]x[k2]+ν[k]

System II (considered in [65], [66]): d[k]=x[k2]+0.08x2[k2]0.04x3[k1]+ν[k]

System III

Final remarks

This work has focused on developing a solution approach to identify in an efficient manner Volterra Systems. The proposed solution framework is based on genetic algorithms (GA) concepts, enhanced by two distinct heuristics and a hierarchical structured population.

Some features presented in this work are: (i) an efficient methodology for identification of Volterra series based on genetic algorithms; (ii) the introduction of two constructive heuristics; (iii) the reduction of the overall

Declaration of Competing Interest

No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105745.

Acknowledgments

This work has been supported by CNPq, Brazil, FAPERJ, Brazil and CAPES .

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