Parametric Identification of Linear Systems Followed by Non-Invertible Piecewise Nonlinearities

The aim of the given paper is the development of an approach for parametric identification of Wiener systems with static non-invertible function, i.e., when the linear part with unknown parameters is followed by piecewise linear nonlinearity with negative slopes. It is shown here that the problem of identification of a nonlinear Wiener system could be reduced to a linear parametric estimation problem by a simple input-output data reordering and by a following data partition into three data sets. A technique based on ordinary least squares (LS) is proposed here for the separate estimation of parameters of linear and nonlinear parts of the Wiener system, including the unknown threshold of piecewise nonlinearity, by processing respective particles of input-output observations. The simulation results are given. Introduction A lot of physical systems are naturally described as Wiener systems, i.e., when the linear system is followed by a static nonlinearity [1 – 3]. Frequently, nonlinearities of actuator devices occur on the output of the system to be controlled that limit the system performance considerably [2, 3]. Therefore, Wiener models, consisting of a linear dynamic block followed by a static nonlinear one, are considered to be suitable for a broad spectrum of nonlinearities [4]. A special class of such systems is piecewise affine Wiener systems, consisting of some subsystems, between which occasional switchings happen at different time moments [3]. Assuming the nonlinearity to be piecewise linear, one could let the linear part of the Wiener system be represented by different regression functions with some parameters that are unknown. In such a case, observations of an output of the Wiener system could be partitioned into distinct data sets according to different descriptions. However, the boundaries the set of observations depend on the value of the unknown threshold d – observations are divided into regimes subject to whether the some observed threshold variable is smaller or larger than d. Thus, there arises a problem, first, to find a way to partition available data, second, to calculate the estimates of parameters of regression functions by processing particles of observations to be determined, and, third, to get the unknown threshold d. It is known, that various compensators have been tried to adjust the performance of control systems by reducing parasitic effects of a nonlinearity. On the other hand, we describe here the approach based on reconstruction of the unmeasurable internal intermediate signal, acting between both blocks of the Wiener system, without designing special and complex enough compensators [5]. Afterwards, instead of measurable output of the Wiener system, affected by the piecewise nonlinearity, the reconstructed signal, free of the parasitic effects, could be used for parametric identification of the linear time invariant (LTI) system. Here the same problem of the Wiener system is considered as in the article [6], except for recursive identification. The identification method in [6] is a direct application of the well-known recursive LS (RLS) algorithm [4], extended with the estimation of internal variables, some of which appear both linearly and nonlinearly. International Frontier Science Letters Online: 2015-12-29 ISSN: 2349-4484, Vol. 6, pp 16-27 doi:10.18052/www.scipress.com/IFSL.6.16 © 2015 SciPress Ltd., Switzerland SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ In this paper, the initial data partition allows us to separate the parametric identification problem into two parts that are related by the internal signal to be restored. Thus, the nonlinearity of variables and some known problems related with it are avoided. In the first part of the article the parameters of the FIR model are estimated. Then, the unknown intermediate signal between two blocks is reconstructed. Finally, the parameters of the linear block based on the samples of the reconstructed signal are evaluated. In the second part the values of negative linear segment slopes are calculated by processing respective segments of partitioned input-output data with missing observations. At last, it is shown here how to improve the initial estimate of the threshold. The recursive method proposed in [6] enables the on-line estimation of the parameters of linear block transfer function and the parameters characterizing the non-invertible piecewise-linear nonlinearity and their changes during the process. In our paper we do not use the recursive expressions. However, forgetting factors applied to reduce an old information in [6] can be used here, too. Moreover, it is clearly shown that it is possible to identify the Wiener system parametrically, even if the static piecewise nonlinearity is non-invertible. It is obvious, that such a method based on the data partition can be applied to find initial parameter estimates based on short-length input-output measurements. They can be used by the recursive parametric identification of Wiener systems. In Section 2, a statement of the problem is presented. In Section 3, the general method is given for determining an auxiliary signal that corresponds to the extracted version of the internal one. Section 4 presents the simulation results of the Wiener system to be identified parametrically. Section 5 contains conclusions. Statement of the problem. The Wiener system consists of a linear part followed by a static non-invertible nonlinearity ) , ( η  f with the vector of parameters . η The linear part of Wiener system is dynamic, time invariant, causal, and stable. It can be represented by LTI (linear, time-invariant) dynamic system with the transfer function ) , ( 1 Θ  q G as a rational function of the form , 1 ) , ) , ( 1 ( 1 ) , ( 1 1 1 1 1 1 a b Θ                q A q B q a q a q b b q G m m m m q   (1) with a finite number of parameters ), ( , , ( ), , , ( ), , , , , , ( ) , 1 1 1 1 m T m T m m T T T a a b b a a b b         a b a b Θ (2) that are determined from the set  of permissible parameter values . Θ Here 1  q is a time-shift operator, the set  is restricted by conditions on the stability of the respective difference equation. The unmeasurable intermediate signal ), ( ) , ( 1 ) , ( ) ( 1 1 k u q A q B k x a b     (3) generated by the linear part of the Wiener system N k , 1   as a response to the input N k k u , 1 ) (   is acting on the static non-invertible nonlinear part ) , ( η  f as follows ). ( ) ), ( ( ) ( k e k x f k y   η (4) Here e(k) is a measurement noise. e(k) u(k) x(k)  y(k) Fig. 1. A Wiener system, consisting of LTI system with ) , ( 1 Θ  q G (1) with parameters (2) and a nonlinearity ) , ( η  f (5), (Fig. 2) [6, 7]. The signal {x(k)} is acting between the LTI system and ) , ( η  f . Only samples of signals {u(k)} and noisy {y(k)} are available. ) , ( 1 Θ  q G ) , ( η  f International Frontier Science Letters Vol. 6 17


Introduction
A lot of physical systems are naturally described as Wiener systems, i.e., when the linear system is followed by a static nonlinearity [1 -3].Frequently, nonlinearities of actuator devices occur on the output of the system to be controlled that limit the system performance considerably [2,3].Therefore, Wiener models, consisting of a linear dynamic block followed by a static nonlinear one, are considered to be suitable for a broad spectrum of nonlinearities [4].A special class of such systems is piecewise affine Wiener systems, consisting of some subsystems, between which occasional switchings happen at different time moments [3].Assuming the nonlinearity to be piecewise linear, one could let the linear part of the Wiener system be represented by different regression functions with some parameters that are unknown.In such a case, observations of an output of the Wiener system could be partitioned into distinct data sets according to different descriptions.However, the boundaries the set of observations depend on the value of the unknown threshold d -observations are divided into regimes subject to whether the some observed threshold variable is smaller or larger than d.Thus, there arises a problem, first, to find a way to partition available data, second, to calculate the estimates of parameters of regression functions by processing particles of observations to be determined, and, third, to get the unknown threshold d.It is known, that various compensators have been tried to adjust the performance of control systems by reducing parasitic effects of a nonlinearity.On the other hand, we describe here the approach based on reconstruction of the unmeasurable internal intermediate signal, acting between both blocks of the Wiener system, without designing special and complex enough compensators [5].Afterwards, instead of measurable output of the Wiener system, affected by the piecewise nonlinearity, the reconstructed signal, free of the parasitic effects, could be used for parametric identification of the linear time invariant (LTI) system.
Here the same problem of the Wiener system is considered as in the article [6], except for recursive identification.The identification method in [6] is a direct application of the well-known recursive LS (RLS) algorithm [4], extended with the estimation of internal variables, some of which appear both linearly and nonlinearly.
In this paper, the initial data partition allows us to separate the parametric identification problem into two parts that are related by the internal signal to be restored.Thus, the nonlinearity of variables and some known problems related with it are avoided.In the first part of the article the parameters of the FIR model are estimated.Then, the unknown intermediate signal between two blocks is reconstructed.Finally, the parameters of the linear block based on the samples of the reconstructed signal are evaluated.In the second part the values of negative linear segment slopes are calculated by processing respective segments of partitioned input-output data with missing observations.At last, it is shown here how to improve the initial estimate of the threshold.The recursive method proposed in [6] enables the on-line estimation of the parameters of linear block transfer function and the parameters characterizing the non-invertible piecewise-linear nonlinearity and their changes during the process.In our paper we do not use the recursive expressions.However, forgetting factors applied to reduce an old information in [6] can be used here, too.Moreover, it is clearly shown that it is possible to identify the Wiener system parametrically, even if the static piecewise nonlinearity is non-invertible.It is obvious, that such a method based on the data partition can be applied to find initial parameter estimates based on short-length input-output measurements.They can be used by the recursive parametric identification of Wiener systems.
In Section 2, a statement of the problem is presented.In Section 3, the general method is given for determining an auxiliary signal that corresponds to the extracted version of the internal one.Section 4 presents the simulation results of the Wiener system to be identified parametrically.Section 5 contains conclusions.
Statement of the problem.The Wiener system consists of a linear part followed by a static non-invertible nonlinearity ) , with the vector of parameters .η The linear part of Wiener system is dynamic, time invariant, causal, and stable.It can be represented by LTI (linear, time-invariant) dynamic system with the transfer function ) , ( as a rational function of the form , , ( with a finite number of parameters ), ( that are determined from the set  of permissible parameter values .Θ Here 1  q is a time-shift operator, the set  is restricted by conditions on the stability of the respective difference equation.
The unmeasurable intermediate signal The nonlinear part of the Wiener system is a non-invertible piecewise linear nonlinearity with negative slopes as follows [7,8] that could be partitioned into three functions: ), ( ) ), ( ( ). ( ) ), ( ( Note that the function The measurement noise The aim of the given paper is to estimate parameters of the linear and nonlinear parts (1), (5), respectively, avoiding the parasitic effects of nonlinear distortions, induced by the non-invertible nonlinearity (5) (Fig. 2), that appear in the noisy output )} ( { k y of the Wiener system (see Fig. 1).
The input-output data reordering.To calculate estimates it is needed to determine an auxiliary signal ) having no parasitic distortions.To solve such a problem one could approximate the linear part (1) of the Wiener system by the finite impulse response (FIR) model of the form [8,9] ) is the full rank regression matrix is the vector of unknown parameters of FIR model (7);  is the order of FIR filter (7) that can be arbitrarily large, but fixed (  can be detected experimentally by simulation).The reason for the use of the FIR model is as follows: the dependence of some regressors on the process output will be facilitated, and the assumption of ordinary least squares (LS) that the regressors depend only on the non-noisy input signal, will be satisfied.
It could be emphasized that from the engineering point of view it is assumed here that no less than 50 persentage observations {y(k)} are concentrated on the middle-set that corresponds to the condition and approximately by 25 percentage or less on any side set with conditions respectively.Let us rearrange now the true output data an ascending order of their values, assuming that measurement noise is absent, parameters, and the threshold d of non-invertible nonlinearity (5) are known.We could do that by interchanging equations in the initial system (8).Note that the interchange of equations does not influence the accuracy of LS parameter estimates ) ( ˆ, , Then, assuming that a static nonlinearity is present in the given Wiener system (Fig. 1), the vector V and the matrix Π should be partitioned into three data sets: the left-hand data set , and the right-hand data set , d Thus, initial system ( 7) is reordered into a system by simply interchanging equations in the initial system of linear equations (8).Here vectors, respectively, and matrices, correspondingly.Hence, the observations with the highest and positive values will be concentrated on the righthand side set, while the observations with the lowest and negative values on the left-hand side one.It could be noted that on boundaries the small portions of observations of the middle data set of are mixed together with some portions of observations of the left-hand side and right-hand side data sets, respectively, due to negative slopes of the nonlinearity (5).In general case (a noisy environment, unknown parameters Θ and η and the threshold d) it is imperative for the efficient parametric identification of the Wiener system that such ambiguities are resolved.with small portions of missing observations within it that belong to the left-hand and right-hand side sets of the reordered data.To estimate the parameters ) , , , ( of FIR model ( 7), we can use the expression of the form respectively.Then, the estimate avoiding parasitic influence of the nonlinearity (5).Here the true values vectors of the estimates of parameters (2), respectively; of the internal signal is calculated by that can be compared with its previous version (14).Estimates of the parameters , , β α and η like vector of estimates Θ ˆare calculated by the ordinary LS, too.In such a case, we can use two systems of linear equations as follows: first system The estimates  19) and (21) as follows International Frontier Science Letters Vol. 6  18), will be applied for the parametric identification of the basic LTI system.It is easy to understand that equations ( 13), (15) as well as ( 23) -( 26) can be transformed into recursive form.In such a case, the on-line solution is possible, too [ 9,10].

Simulation results. The sum of sinusoids
and white Gaussian noise with variance 1 were generated as inputs to the linear block of the Wiener system (see Fig. 1) respectively [7].Here the true values of parameters (1) are: .7 .0 , 1 passes the noninvertible nonlinearity of the form [6,7] that produces the output (4).First of all, N=100 data samples have been generated without additive measurement noise.The initial estimate of the static nonlinearity, calculated by processing is reconstructed according to Eq. ( 14).In such a case, the initial estimate of the threshold d can be determined, too.It follows from Fig. 3, where the output The abovementioned signals to be used are shown in Fig. 4. In Table 1 estimates are shown for different inputs.In the first experiment (second and third columns in Table 1) they were calculated by means of ordinary LS by processing 100 samples of (see Fig. 1).
In the second experiment (fourth and fifth columns in Table 1) observations of  It is obvious (see Table 1), that the nonlinearity was influenced on the accuracy of estimates, significantly, for example, 1 â changed even sign.Afterwards, the LS problem (13) was solved using 42 and 55 rearranged samples of the output of the Wiener system for both inputs (see Figures 5  8), excluding zeros.In such a case, the whole number of FIR filter (7) parameters   14 has been chosen experimentally by multifold simulation.Values of their estimates are given in Table 2 in the absence of additive noise.respectively (see Table 3).

International Frontier Science Letters Vol. 6
In order to determine how realizations of different process-and measurement noises affect the accuracy of estimation of unknown parameters, we have used the Monte Carlo simulation with 10 data samples, each containing 100 pairs of input-noisy-output observations.10 experiments with the different realizations of the measurement noise 100 , 1 ) (  k k e have been carried out.The intensity of noise was assured by choosing respective signal-to-noise ratios (SNR) (the square root of the ratio of signal and noise variances).For the noise SNR is defined as , log 10 The Monte Carlo simulation implies that the accuracy of parametric identification of the Wiener system with static non-invertible nonlinearity depends on the intensity of measurement noise.  . 0.02 -0.70  0.01 -1.08  0.02 -0.09  0.01 1.10  0.03 -0.10  0.01

IFSL Volume 6
Conclusions.The approach is presented here, based on the extraction of an unknown internal intermediate signal, acting between linear and nonlinear blocks of the Wiener system with a static non-invertible piecewise nonlinearity, avoiding complex enough compensators [9 -10].It is shown here that a problem of identification of Wiener systems (Fig. 1) could be essentially reduced by using FIR model, and a simple data rearrangement in an ascending order according to their values.Thus, the available data are partitioned into three data sets that correspond to distinct threshold regression models.Afterwards, the estimates of unknown parameters of linear regression models can be calculated by processing respective sets of the rearranged output and associated input observations.A technique, based on ordinary LS, is proposed here for estimating the parameters of linear and nonlinear parts of the Wiener system, including the unknown threshold of the piecewise nonlinearity, too.It is shown here (see Fig. 3) how at the beginning the initial estimate of the unknown threshold can be determined.During successive steps the unknown intermediate signal is reconstructed and the missing values of observations of output data particles are replaced by their estimates.Note that missing values can be retrieved by the approach given in [12].Various results of numerical simulation (Fig. 2  8), including that of Monte Carlo (Table 4) prove the efficiency of the proposed approach for the parametric identification of LTI systems followed by static noninvertible piecewise nonlinearity in a noisy frame.
Finally, we can state that the successful parametric identification of the non-invertible piecewiselinear nonlinearity is a new element in the known approach (see [8][9][10]).It is obvious, that the proposed here data partition method can be used for different types of complex nonlinearities with piecewise-linear segments.

1 N 2 N 3 N
Wiener system in an unnoisy or slightly noisy frame should be partitioned into three data sets: left-hand side data set ( samples) with values less than or equal to negative d, middle data set ( samples) with values higher than negative d but lower or equal to d, and right-hand side data set ( samples) with values higher than d.Here , any integer k rearranged in an ascending order, dependent on the reordered values of observations ) calculated by Eq. (13).Note that the result of this step is the auxiliary signal (14) that is a reconstructed version of the intermediate unmeasurable signal )}, ( { k x that acts between linear and nonlinear parts of the Wiener system.Now, let us calculate the estimates of the parameters of the transfer function ) , ( 1 solving eq. (
column of Table2).Then, the estimates of the parameters of the transfer function )The accuracy of the estimates of the intermediate signal, calculated by formulas (14) and (18), more or less similar except for the first 15 samples, when the FIR model(7) was used.been obtained, then it is simple to separate different particles of samples that belong to distinct side-sets.Then, the estimates


is a variance of the additive Gaussian noise (see Fig.1).As inputs for given nonlinearity white Gaussian noise with variance 1 was chosen.Then, in each experiment 100 Ndata samples have been generated with additive measurement noise according to , values of  was chosen so that SNR for the measurement noise was equal to 50.In each experiment the estimates of parameters were calculated.During the Monte Carlo simulation averaged values of estimates of the parameters and of the threshold and their confidence

Table 2 .
Estimates of 10-th parameters of FIR system for distinct input signals.Estimates Input is of the form (28) Input is white Gaussian noise

Table 4 .
Estimates of parameters