Distributed non‐linear robust consensus‐based sensor calibration for networked control systems

IET Control Theory & ApplicationsVolume 14, Issue 9 p. 1200-1208 Research ArticleFree Access Distributed non-linear robust consensus-based sensor calibration for networked control systems Maja S. Stanković, Corresponding Author Maja S. Stanković majasta@telekom.rs orcid.org/0000-0001-6114-6028 Faculty of Electronics, University of Niš, Niš, Serbia Telekom Srbija, Belgrade, SerbiaSearch for more papers by this authorDragan S. Antić, Dragan S. Antić Faculty of Electronics, University of Niš, Niš, SerbiaSearch for more papers by this author Maja S. Stanković, Corresponding Author Maja S. Stanković majasta@telekom.rs orcid.org/0000-0001-6114-6028 Faculty of Electronics, University of Niš, Niš, Serbia Telekom Srbija, Belgrade, SerbiaSearch for more papers by this authorDragan S. Antić, Dragan S. Antić Faculty of Electronics, University of Niš, Niš, SerbiaSearch for more papers by this author First published: 28 April 2020 https://doi.org/10.1049/iet-cta.2019.0672AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study a new algorithm is proposed for distributed blind sensor macro-calibration in networked control systems robust to noise. The proposed distributed algorithm for estimation of gain and offset correction parameters is of stochastic approximation type, with local non-linear transformations of residuals. Convergence of the algorithm in mean-square and with probability one to consensus is proved for a large class of non-linear transformations, network properties and communication and measurement noise characteristics. The choice of the introduced non-linear transformations in accordance with the theory of robust statistics leads to the proposal of new calibration algorithms robustified w.r.t. noise. It is demonstrated by Monte Carlo simulation that the proposed algorithms are very efficient in the presence of large outliers from the point of view of both achievement of high convergence rate and adequate values of convergence points, outperforming the existing linear algorithms. 1 Introduction Sensor calibration represents one of very important problems for wide deployment of large networked control systems containing a large number of sensors and actuators in industry, robotics, agriculture, monitoring and surveillance of wide areas, etc. (e.g. [1–3]). In particular, one can place the calibration problem in the context of distributed fault detection [4] and state estimation [5] in large-scale systems, using wireless sensor networks (WSN's) [6]. The available literature covers the main problems related to calibration of networked control systems and sensor networks and contains different practical solutions starting from different a priori assumptions and using different methodologies (parameter estimation, optimisation, etc.) e.g. [7–11]. However, all of these approaches require some forms of centralised actions at the network level. It is very important, both conceptually and practically, to develop efficient decentralised algorithms for blind calibration, without requiring information about ground-truth (stimuli). A methodologically consistent approach to distributed blind calibration has been proposed in [12, 13] and further developed in [14, 15]. The calibration algorithm is of distributed gradient-type, providing equal corrected sensor outputs by ensuring convergence to consensus of all the corrected sensor gains and offsets, without requiring any type of centralisation. Assuming asynchronous communications, a specific linear gradient-type stochastic approximation algorithm has been proposed recently in [16]. It has been proved that the algorithm provides convergence to consensus, in the sense that all the corrected gains and offsets converge to equal values. This algorithm, together with a similar consensus-based blind calibration algorithms from [12–15] have been presented recently in [15]. In general, it has been found that one of possible problems for practical applications of the existing linear calibration algorithms may lie in the influence of communication and measurement noise, which affects not only the practical convergence rate to consensus, but also the asymptotic values of the corrected gains and offsets, which depend on entire noise realisations [16]. These effects are characteristic for large sensor networks deployed in wide and sometimes hostile environments (agriculture, monitoring, fault detection), when measurement noise may unexpectedly become temporarily large for some sensors, and where communication noise may contain impulsive behaviour, e.g. [17–20]. As a consequence, in spite of the theoretical convergence to consensus, the whole setting may become somewhat artificial and far from practical usefulness. In an attempt to provide a solution to the identified problem, in this paper a new algorithm is proposed for distributed blind calibration of sensor networks under the presence of measurement and communication noise. As in the previous work [16, 21], the algorithm is composed of two distributed recursions for separate correction of sensor gains and offsets, first of which is completely autonomous, while the second is derived from the corrected sensor outputs and utilises the current gain correction estimates. The main new quality of the proposed algorithm consists of the introduction of nonlinear transformations of residuals, following the classical papers devoted to the stochastic approximation algorithms [22–25], on one hand, and the recent papers related to stochastic consensus seeking [19, 20, 26–30], on the other. General properties of the new non-linear calibration algorithm are interesting, per se. However, the main motivation for the introduction of non-linearities is the idea to choose their form in such a way as to cope with the noise influence, following the principles of robust statistics, e.g. [18, 31–33]. In this sense, besides the proposal itself of a general non-linear distributed consensus-based calibration algorithm, the paper provides two major contributions: (i) the first is theoretical, giving an insight into the asymptotic behaviour of the algorithm for a large class of non-linearities, (ii) the second is more practical, showing that a specific choice of the introduced non-linearities in accordance with the principles of robust statistics provides obvious advantage over the existing linear methods. The theoretical part provides a rigorous proof of convergence of the corrected gains and offsets to consensus in mean square and with probability one. The proof is non-trivial and in many aspects methodologically original, but essentially based on the classical results from [23, 31]. In particular, generalising the results from [31] by using the methodology of [27, 30, 34–36] in the context of network characteristics expressed through weighted Laplacians of the underlying graphs, it is possible to demonstrate convergence to consensus using the general results from [34, 37]. The derivations are in line with the very recent approach to non-linear stochastic consensus [26]; however, one should bear in mind that the calibration algorithm is a specific and very complex consensus based scheme, containing two interrelated recursions and unknown correlated random processes representing the stimuli. It should be pointed out that the obtained results clarify intrinsic features of the limit values at consensus, sensitive to both communication and measurement noise realizations; the derived features are represented and tested in the experimental part of the work. Applying the methodology of robust statistics to the definition of the non-linear functions characterising the proposed calibration algorithms, two types of robustifying non-linearities aimed at the suppression of outliers in communication and measurement noise sequences are considered: (i) non-linearity of saturation type, and (ii) sign non-linearity. Simulation results show high efficiency of the proposed robustified scheme, drastically reducing the effect of large outliers w.r.t. all the existing linear algorithms. Formally, the paper is organised as follows. Section 2 deals with the problem formulation and presentation of the proposed algorithm. Section 3 contains the convergence analysis. Section 4 is devoted to the design of robust calibration algorithms, while Section 5 contains simulation results. 2 Algorithm 2.1 Problem formulation Following [12, 14], we assume a stochastic setting. We consider n distributed sensors measuring a signal at discrete time instants k, , where is a set of positive integers. Sensor outputs are defined, as usually, by a simple model (1)where the gain (drift) and the offset are unknown constants, and the additive measurement noise, . Local calibration of each sensor i is done by introducing the following affine calibration transformation (2)where is a corrected sensor output, and are the gain (drift) correction and offset correction parameters, respectively, (calibration parameters). It follows from (2) that represents the corrected gain (drift) and the corrected offset. Remark 1.The simple idea behind (2) is to achieve and by a proper choice of the calibration parameters and . However, the fact that the sensor parameters and are unknown, together with the lack of information about the measured signal , makes the blind calibration problem highly non-trivial. The sensors are assumed to be interconnected, forming a sensor network with the topology represented by a connected undirected graph , where is the set of nodes (sensors) and the set of links (i,j). Let be the set of neighbouring nodes of i th node. Each message between nodes j and i is corrupted by additive communication noise . As stated in Section 1, starting from the above setting, the papers [16, 21] have treated the problem of distributed real-time estimation of the correction parameters and by a specific gradient-type stochastic approximation algorithm, consisting of one autonomous recursion for real-time estimation of the gain correction parameters and a separate recursion for estimation of the offset correction parameters , assuming that the estimates of are given. Such an algorithm structure has its roots in analogous recursive schemes for distributed time synchronisation, e.g. [38–40]. The crucial general property of this algorithm is that all the corrected gains and offsets become asymptotically equal. In such a way, majority of sensors with good characteristics lead to better global performance; also, one locally calibrated reference sensor is sufficient for obtaining identical performance of all sensors (see [12, 14, 16, 21] for additional considerations). In this paper, we shall develop further the main idea about the algorithm structure from [16, 21] in the following directions: (i) proposal of a novel non-linear calibration algorithm, inspired formally by the recent results in the domain of non-linear consensus seeking [17, 26, 28], (ii) an original theoretical proof of convergence of the proposed algorithm to consensus, and (iii) application of the methodology of robust statistics to the definition of the introduced non-linear transformations which make the algorithm much more insensitive to additive communication and measurement noise than the existing linear algorithms. 2.2 Non-linear calibration recursions The gain correction algorithm is derived using the following model for increments of the measured signal (3)where and (see [16, 21]). The basic linear algorithm has been derived in [16] using the methodology of instrumental variables introduced for coping with the problem of measurement noise correlatedness in explicit gradient-based schemes [41]. We shall not repeat the whole reasoning here. Rather, we shall present only the final form of a new algorithm for estimation of the local gain correction parameters , which is substantially different from the one in [16] by the introduction of non-linear transformations of residuals : (4)where is an estimate of , is the step size, are a priori chosen constant weights specifying relative importance of the neighbouring nodes of node i, , (5) , represents the additive communication noise that occurs when is sent from node j to node i and represents the delay introduced by the instrumental variables (see [16] for more details); notice that for one obtains a gradient-type algorithm, adequate in the case of absence of measurement noise. Functions are non-linear real functions to be specified below. It is adopted that [16]. Procedurally, the algorithm is extremely simple: (i) measurement increments are obtained at node i and the nodes from its neighbourhood , (ii) increments from the neighbourhood are sent to node i and received with superposed communication noise, (ii) the terms are computed according to (5), (iv) the previous measured increment is taken from the local memory, (v) updating of the previous estimate is done according to (4). Remark 2.Introduction of the measurement noise into the basic model (1) has been done in [14, 16] in order to be as close as possible to reality, having in mind even possible sensor deployments in hostile and time varying environments (see the discussion below). Existence of measurement noise causes difficult problems in the algorithm (4), requiring the introduction of instrumental variables, which leads, itself, to a deterioration of the convergence rate w.r.t. the gradient scheme. It will be seen later that the proof of convergence of the algorithm in the presence of measurement noise is faced with additional technical difficulties. Assuming that is given by (4), the following distributed algorithm for generating the estimates of in (2) is proposed: (6)where (7) and represents the communication noise that occurs when is sent from node j to node i. Real functions are to be specified below. It will be adopted that . Procedurally, we have the following main steps: (1) measurements are obtained at node i and the neighbouring nodes, (2) measurements from the neighbouring nodes are sent to node i and received with superposed communication noise, (3) the terms are computed at node i, (4) updating of is done according to (6). Obviously, the recursions (4) and (6) reduce to the corresponding linear recursions from [16] by assuming that and . 2.3 Global calibration model Obviously, it is possible, to derive from (4) and (6) (for the sake of convergence analysis) the corresponding local recursions for the corrected gains and corrected offsets . Since our immediate aim is to analyse the behaviour of the algorithm at the network level, we shall formulate directly the global models for estimation of all corrected gains and offsets. All the recursions (4), are represented by the following global model: (8)where: ; ; ; ; is i th row vector of the matrix ; ; ; . Similarly, the global model derived from all the recursions defined by (6) is (9)where ; ; ; ; , ; . According to (4) and (6), and . 3 Convergence analysis Convergence analysis of the proposed algorithm will be done for non-linear transformations satisfying general assumptions, starting from the basic results in [26, 31, 34]. A detailed proof is given for the autonomous gain correction algorithm. Convergence proof for the offset algorithm is presented shortly, having in mind paper size constraints and the fact that it follows the same methodological lines of thought as in the case of the gain correction algorithm. Assume the following: (A1) Graph is connected. (A2) Random sequences and and are mutually independent i.i.d., with symmetric distribution functions, . (A3) Real functions and are (a) odd and continuous, excluding, possibly, a set of points of Lebesgue measure zero, (b) monotone non-decreasing, having a common growth point with the distribution functions of and , respectively, (see [31] for details). (A4) , , , . (A5) Functions and are differentiable at zero, where (10) and representing the distribution functions of and , respectively. (A6) . (A7) Sequence is stationary, bounded a.s. and independent of and ; , where is the minimal -algebra generated by the measurements up to the instant , . Remark 3.Assumptions (A1) and (A2) are standard. Assumptions (A3)–(A5) are related to the introduced non-linearities; more technical details can be found in [31] (Lemmas 1-6 from [31] are very important for all the derivations presented below). It should be noticed that, practically, the non-linearities have to be sign preserving, e.g. [29]. Notice that (A5) is somewhat milder than the analogous assumption in [26]; this fact has a purely theoretical meaning. Assumption (A6) is introduced only for the sake of a better transparency of the analysis, and a direct connection with the asymptotically optimal algorithms presented in [23, 32] (see Section 4). Assumptions in (A7) concerning the measured signal are very mild; essentially, insufficiently exciting signals are eliminated. As a consequence of (A1), matrix has one eigenvalue at the origin, and the remaining ones in the left half plane [12, 36, 42]. Consequently, we define a non-singular matrix T in the form , where is an orthogonal matrix satisfying and giving (11)where is a symmetric negative definite matrix [14]. 3.1 Gain correction Theorem 1.Let Assumptions (A1)–(A7) hold. Then converges to consensus in the sense that in the mean-square sense and w.p.1, where is a random variable satisfying (a.s.) and , dim . Moreover, (12)for some , and (13)where . Proof.Part A: Communication noise. We shall first analyse the case when there is only communication noise and when, consequently, in (4) (the algorithm reduces to the gradient form), in order to indicate more clearly the main effects of the introduced non-linearities and themselves.Rewrite (8) in the following way (14)where , , ( denotes the conditional expectation given the previous gain estimate at the instant ); , with , , and .It is important to notice immediately that we have explicitly , making the analysis entirely compatible with the setting from [31]. I. Consider (15)where is the conditional distribution of given and . Using [31], Theorem 1, we infer that , since by (A7). Therefore, the equation , defining the set of possible convergence points of (8), holds iff . Let , where T is defined in (11) and . Then, we have (16)where and are and matrices, respectively. As , according to (11) (see also [34]), the set D is defined by , leaving arbitrary. II. Introduce the Lyapunov function (17)tending to when , having in mind (11). As , we conclude directly that (18)for , . III. Furthermore, we define Following methodologically [34], using (A2)–(A6) we obtain, after some technicalities, that (19)where is a constant. Therefore, according to [34], we infer that tends w.p.1 to a point belonging to the set D. IV. Define matrix , where (20)According to (A3)–(A6) and the convergence w.p.1 of to a point in D, together with the properties of the functions in (10) (see [31] for details), it follows that for there exists such that (21)for having in mind that , where is the i th row of the matrix (notice that , , having in mind that exists and is finite, owing to (A5) and Lemma 6 from [31]). According to (21), we have (22)where . V. Let . After introducing the expression for in (14) and performing a standard majorisation procedure according to [34], Lemma 2.1, Chapter 6, we obtain that, for k large enough, (23)where . After iterating back to the initial condition, we get (24)wherefrom we obtain that converges to zero in the mean square sense following (12), with (for more technical details see Lemma 2.1, Chapter 6, in [34]). VI. In order to prove (13), we introduce the following auxiliary function: (25)where and [34]. We directly obtain (26)For k large enough and , we obtain (27). It follows that is a supermartingale, and, consequently, that exists w.p.1 and is equal to zero. Thus, (13) holds. VII. Introducing in (14), we write , and obtain, using (20), the following recursion: (28)where ; ; is the first row of the matrix ; ; is the first component of . After iterating back to the initial condition, we obtain (29)where and Boundedness of by (21) implies that as a consequence of (13). On the other hand, (a.s.), having in mind that is a martingale difference sequence with bounded second moment. VIII. According to VII, we have , (a.s.). Then, we have (a.s.), where . Thus, the result. Part B: Measurement noise. In the case, when both the communication and measurement noise exist, the general model (8) holds with all its stochastic components, and the analysis becomes more complex. One can observe immediately that there are two terms depending on the measurement noise in (8): (1) additional component in the argument of the functions (the term in , and (2) 'exogenous' noise term in the form of an additional multiplier of (the term in (8)). As far the term (1) is concerned, it is very important to notice that in the case of the measurement noise existence, the instrumental variables should be introduced, implying in (8) (in order to decorrelate and - see [16] for more details). In order to be able to apply the methodology of Theorem 1, one should follow the general idea from [12, 16] and to introduce a convenient modification of (14) in the form of a time-shift of the term in (8), so that it becomes . This modification is aimed at achieving the desired noise decorrelation w.r.t. previous estimate , and defines (14) formally as a set of nested recursions of stochastic approximation type (see, e.g. [40]). Starting from this step, we can proceed following the same line of thought as in Part A and obtain analogous results, taking care of a number of non-trivial additional technical issues. For example, one should not forget that the covariance of the noise term becomes time varying, requiring additional assumptions concerning function .Term (2) can be handled using standard stochastic approximation techniques, e.g. [34, 37].We shall not be involved in this paper with the formal aspects of the corresponding derivations; instead, we shall show in the experimental part of the paper that the convergence to consensus is still successfully achieved even in the presence of high level outliers. □ 3.2 Offset correction Convergence analysis of the algorithm for offset correction (9) can be done following methodologically Theorem 1; due to space limitation it will not be given here in extenso. In the following, we shall point out only specific features of (9), connected to the external influence of the gain correction estimates generated by (8). Obviously, the structure of (9) is more complex than the one of (8), since it contains two non-linear terms and , in which the arguments depend on both and generated by (8). Notice, immediately, that in the case when , where is a random variable, i.e. when has already achieved consensus, we have and . The term depending on in (9) becomes a white noise term multiplied by , and the term depending on receives the same structure as the term depending on in (8). Consequently, the result of Theorem 1 can be directly applied, having in mind that both and satisfy (A3)–(A5), by assumption. In general, for some , the asymptotic relation (13) for the algorithm (8), implies Coming back to algorithm (9), we have as a result addition of to the expressions for the arguments in and . Such terms can be neglected for k large enough with respect to the other terms in relations (24) and (27), so that the results of Theorems 1 can be applied to (9) with minor changes. Remark 4.Convergence analysis of the proposed distributed calibration algorithm is far from being a trivial task and represents a challenge. The main ideas regarding the methodology can be found in the classical book on stochastic approximation by Nevel'son and Khasminsky [34] and in the papers by Polyak and Tsypkin [23–25, 31, 32, 43], on one hand, and in the papers related to stochastic consensus seeking [26, 27, 30], on the other. Also, the algorithms for distributed calibration proposed in [12, 14, 16] represent the specific linear schemes of consensus type adding new insights that can be at least indirectly utilised. Stankovíc et al. [26] devoted to the general non-linear stochastic consensus seeking, represents an immediate precursor of this paper in this sense. However, the above given proof is by far more intrinsic, having in mind that (i) the algorithm contains two interconnected recursions of consensus type, (ii) the unknown measured signal (which is correlated) is incorporated in both recursions, (iii) two types of noise sequences are incorporated, implying additional complexity and time variability of the overall noise terms. In this sense, the proof itself can be considered as an important theoretical contribution of this paper. 4 Robust calibration algorithms The main purpose of the introduction of non-linear transformations in the calibration algorithm (8) and (9) is, as stated above, robustification w.r.t. communication and measurement noise. This section provides a necessary link between the theory of robust estimation and a precise definition of robustified distributed blind calibration algorithms. 4.1 General aspects Robust behaviour of statistical estimation methods has been extensively studied on the basis of the Huber's methodology of robust statistics [31, 33, 44, 45]. Very shortly, starting from the assumption that the noise density p belongs to a given class , one introduces the following criterion function: (30)where is the so-called influence function, and define the optimal function from the saddle point conditions (31)It has been shown that the solution to this problem can be obtained by finding the least convenient density by minimising the Fisher information over , and taking (32) The following examples of classes and the solutions of the corresponding variational problems of minimising are important for practise [31]: continuous at – the largest class of arbitrary non-singular densities. Then, is the Laplace density, and , i.e. we obtain the sign algorithm. . Then, is normal, and . The linear algorithm is optimal at the class of noise with uniformly bounded variance. is arbitrary – the class of approximately normal densities. Then, for and for , (), i.e. we obtain the proportional algorithm with saturation. In their fundamental papers, Tsypkin and Polyak [31] connected robust statistics with asymptotic optimality of the recursive stochastic approximation schemes. They have demonstrated that the optimal choice in the sense of the minimal mean-square error of the non-linear function (or ) in non-linear gradient-based stochastic approximation algorithms follows from the noise density p according to (32). This indicates that once one finds the least convenient distribution on a given class, the corresponding non-linear transformation in the algorithm coincides with the influence function from the theory of robust estimation. It has been also shown that in the case of scalar step size and linear regression function , optimality in the sense of minimal asymptotic mean-square error is achieved when , where is Fisher information , is the additive noise density and . This fact has been one of the motivations to perform the general convergence analysis of the proposed algorithm under the assumption (A6) that the step size is in the form (see (4) and (6)). Not going into details exposed in the vast literature related to the subject, we shall concentrate in the sequel on the so-called contaminated distributions, where the nominal density (mostly normal) is contaminated by a density , typically characterised by a large variance, according to (33)where a small 'contamination' parameter. Such a model is adequate for numerous real situations in which the measurements can contain rare, but sometimes very large outliers. In this situation, algorithms optimised for the nominal normal distribution ca