Convergence in Networked Recursive Identification with Output Quantization

Proceedings,18th IFAC Symposium on System Identification Proceedings,18th IFAC Symposium System Identification July 9-11, 2018. Stockholm, Sweden on Proceedings,18th IFAC on System Proceedings,18th IFAC Symposium Symposium System Identification Identification July 9-11, 2018. Stockholm, Sweden on Available online at www.sciencedirect.com July 9-11, 9-11, 2018. 2018. Stockholm, Stockholm, Sweden on System Identification Proceedings,18th IFAC Symposium July Sweden July 9-11, 2018. Stockholm, Sweden

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IFAC PapersOnLine 51-15 (2018) 915–920

Convergence in Networked Recursive Convergence in Networked Recursive Convergence in Networked Recursive Convergence in Networked Recursive Identification with Output Quantization Convergence in Networked Recursive Identification with Output Quantization Identification Identification with with Output Output Quantization Quantization Identification with Sholeh Yasini andOutput Torbj¨ orn Quantization Wigren ∗∗∗ Sholeh Yasini and Torbj¨ orn Wigren

∗ Sholeh Yasini and Torbj¨ o rn Wigren Sholeh o Sholeh Yasini Yasini and and Torbj¨ Torbj¨ orn rn Wigren Wigren ∗∗ ∗ Yasini Torbj¨ orn Wigren of Systems and and Control, Department of Information ∗ Division Sholeh ∗ Division of Systems and Control, Department of Sweden Information ∗ Technology, Uppsala University, SE-751 05, Uppsala, (e-mails: ∗ Division of Systems and Control, Department of Information Division of Systems and Control, Department of Information Technology, Uppsala University, SE-751 05, Uppsala, Sweden (e-mails: ∗ sholeh.yasini, torbjorn.wigren@ it.uu.se). Technology, Uppsala University, SE-751 05, Uppsala, Sweden (e-mails:

Division of Systems and Control, of Sweden Information Technology, Uppsala University, SE-751Department 05, Uppsala, (e-mails: sholeh.yasini, torbjorn.wigren@ it.uu.se). sholeh.yasini, torbjorn.wigren@ it.uu.se). Technology, Uppsala University, SE-751 05, Uppsala, Sweden (e-mails: sholeh.yasini, torbjorn.wigren@ it.uu.se). sholeh.yasini, torbjorn.wigren@ it.uu.se). Abstract: This paper analyzes conditions for global parametric convergence of a networked Abstract: This paper analyzes conditions for global parametric convergence of a networked recursive identification algorithm. The FIR based algorithm accounts for networked delay and Abstract: This analyzes for parametric convergence of Abstract: This paper paper algorithm. analyzes conditions conditions for global global parametric convergence of aa networked networked recursive identification The FIR based algorithm accounts for networked delay signal quantization. The paper constructs counterexamples to parametric convergence using and low recursive identification algorithm. The FIR based algorithm accounts for networked delay and Abstract: This paper analyzes conditions for global parametric convergence of a networked recursive identification algorithm. The FIR based algorithm accounts forconvergence networked delay and signal quantization. The paper constructs counterexamples to parametric using low order dynamic models an constructs asymmetric binary quantizer. The associated ODEs ofdelay the first signal quantization. Theand paper constructs counterexamples to accounts parametric convergence using low recursive identification algorithm. The FIR based algorithm forconvergence networked and signal quantization. The paper counterexamples to parametric using low order model dynamic models and an asymmetric binary quantizer. The associated ODEs of the first aremodels analysed with analytical methods and by numerical simulation. In particular, the order dynamic an asymmetric binary quantizer. associated ODEs of the signal quantization. Theand paper constructs counterexamples to The parametric convergence low order dynamic models and ananalytical asymmetric binaryand quantizer. The associated ODEs ofusing the first first order model are analysed with methods by numerical simulation. In particular, the analysis proves that parametric convergence does not occur in case the input signal distribution model are analysed with analytical methods and by numerical simulation. In particular, the order dynamic models and an asymmetric binary quantizer. The associated ODEs of the first order model are that analysed with analytical methods and by numerical simulation. In particular, the analysis proves convergence does not occur in case the input distribution is discrete taking a parametric finitewith number of values, theand problem is or signal inIncase the input analysis proves parametric convergence does not occur in case the signal distribution order model are that analysed analytical methods by numerical particular, the analysis proves that parametric convergence does not occur in symmetric, case simulation. the input input signal distribution is discrete taking a finite number of values, the problem is symmetric, or in case the input signal distribution, gainconvergence and point areoccur such is that there no energy in an is discrete taking adynamic finite number number of switch values,does the not problem is symmetric, orsignal in case case the input input analysis proves thata parametric in case the is input signal distribution is discrete taking finite of values, the problem symmetric, or in the signal distribution, gain and switch point are such there is no in an arbitrary small neighborhood of any point. parametric convergence isenergy also signal distribution, dynamic gain and switch point are that there is signal energy in is discrete taking adynamic finite number ofswitch values, the Global problem isthat symmetric, orsignal in case theproved input signal distribution, dynamic gain and switch point are such such that thereconvergence is no no signal energy in an an arbitrary small neighborhood of any switch point. Global parametric is also proved for onedistribution, of small the low order cases. arbitrary neighborhood of any parametric also proved signal dynamic gain andswitch switchpoint. pointGlobal are such that thereconvergence is no signalis energy in an arbitrary small neighborhood of any switch point. Global parametric convergence is also proved for one of small the low order cases.of any switch point. Global parametric convergence is also proved for one the low order arbitrary neighborhood for one of of the(International low order cases. cases. © 2018, IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. for one1.ofINTRODUCTION the low order cases. accuracy at the expense of an increased computational 1. INTRODUCTION accuracy at the expense of an increased computational cost, as compared to the algorithm of (Wigren (1998)). In 1. INTRODUCTION INTRODUCTION accuracy at the the expense expense of an an increased increased computational 1. accuracy at of computational cost, as compared to the algorithm of (Wigren (1998)). In (Zhaoas al. FIR isof ascomputational well, followed cost, asetcompared compared toathe the algorithm ofused (Wigren (1998)). In 1. INTRODUCTION accuracy at (2007)) the expense ofmodel an increased cost, to algorithm (Wigren (1998)). In (Zhao et al. (2007)) a FIR model is used as well, followed Currently new fifth generation (5G) wireless systems are cost, by anas unknown nonlinearity. In that paper that (Zhao etcompared al. (2007)) (2007)) FIR model isofused used asconditions well, followed toaathe algorithm (Wigren (1998)). In et al. FIR model is well, followed Currently new fifth fifth The generation (5G) of wireless systems will are (Zhao by an identification unknown nonlinearity. In that paperasconditions that being standardized. low latency these systems Currently new generation (5G) wireless systems are allow using binary observations are studied by an unknown nonlinearity. In that paper conditions that (Zhao et al. (2007)) a FIR model is used as well, followed Currently new fifth generation (5G) wireless systems are by an identification unknown nonlinearity. In that paper conditions that being standardized. The lowoflatency latency of these systems will allow using binary observations are studied allow astandardized. massive deployment networked feedback control. being The low these systems will Currently new fifth generation (5G) of wireless systems are by and arebinary developed. Theconditions paper (Weyer allow identification using binary observations are studied studied anrecursive unknownmethods nonlinearity. In that paper that being standardized. The lowof latency of these systems will allow identification using observations are allow a massive deployment networked feedback control. and recursive methods are developed. The paper (Weyer New a relying networked force feedback allow massive of networked feedback control. being standardized. The lowon of these systems will allow et al.recursive (2009)) methods takesusing a are different approach and recursive methods are developed. The and paper (Weyer identification binary observations areperforms studied allow aapplications massive deployment deployment oflatency networked feedback control. and developed. The paper (Weyer New applications relying on networked force feedback et al. (2009))oftakes a different approach and quantizer performs control like the deployment tactile Internet and related virtual reality and New applications relying on networked force feedback allow aapplications massive of networked feedback control. identification confidence sets. Input signal et al.recursive (2009)) methods takes different approach and performs developed. The and paperperforms (Weyer New relying on networked force feedback et al. (2009)) takes aa are different approach control like the theare tactile Internet and related virtual reality identification of confidence sets. Input signal and quantizer functionality expected to emerge and enhance the control like tactile Internet and related virtual reality New applications relying on networked force feedback properties related to the present paper have also been identification of confidence sets. Input signal quantizer et al. (2009)) takes a different approach and performs control like the tactile Internet and related virtual reality of confidence sets. Input signal and also quantizer functionality are expected to emerge emerge and enhance enhance the identification properties related to the present paper have been mobile Internet experience (Rappaport (2012)). Therefore, functionality expected to and the control like theare tactile Internet related reality studied. The author’s (Bai andInput Reyland (2008), Baibeen and properties related to ofthe the present paper have also been of confidence sets. signal and also quantizer functionality are expected to and emerge andvirtual enhance the identification properties related to present paper have mobile Internet experience (Rappaport (2012)). Therefore, studied. The author’s of (Bai and Reyland (2008), Bai and as noted in (Samad (2016)), the introduction of 5G wireless mobile Internet experience (Rappaport (2012)). Therefore, functionality are expected to emerge and enhance the Reyland (2009)) focus on the minimal information that studied. The author’s of (Bai and Reyland (2008), Bai and properties related to the present paper have also been mobile experience (2012)). Therefore, studied. The author’s of on (Bai and Reylandinformation (2008), Bai that and as notedInternet in (Samad (2016)),(Rappaport the introduction of 5G wireless Reyland (2009)) focus the minimal requires that the automatic control community revisits the as noted in (Samad (2016)), the introduction of 5G wireless mobile experience (2012)). Therefore, needs toThe be known about the nonlinear output Reyland (2009)) focus on minimal information that author’s of on (Bai and Reylandinformation (2008), function Bai that and as notedInternet in (Samad (2016)),(Rappaport the introduction of 5G wireless Reyland focus the minimal requires that the automatic automatic control community revisits the studied. needs to (2009)) be known about the nonlinear output function handling of(Samad delay in(2016)), all aspects of networked requires the control community revisits the as noted that in the introduction ofcontrol. 5G wireless of a Wiener systems, to allow a successful identification, needs to be known about the nonlinear output function Reyland (2009)) focus on minimal information that requires that the automatic control community revisits the needs to be known about the anonlinear output function handlingthat of delay delay in all all aspects aspects of networked control. a Wiener systems, to allow successful identification, handling of in of requires the automatic control communitycontrol. revisits the needs aof topic is alsoabout touched upon in (Wigren (1995), of Wiener systems, to allow successful identification, to that be known the aanonlinear output function handling allsystem aspects of networked networked control. The sameofisdelay trueinfor identification, where algo- of aa Wiener systems, to allow successful identification, a topic that is also touched upon in (Wigren (1995), handling ofisdelay infor allsystem aspectsidentification, of networked control. Wigren (1998)). The reference (Casini et al. (2008)) is a topic that is also touched upon in (Wigren (1995), The same true where algoof a Wiener systems, to allow a successful identification, rithms ableis to support off-line and on-linewhere networked aWigren topic (1998)). that is also touched upon in et (Wigren (1995), The same true for system identification, algoThe reference (Casini al. (2008)) is The same is true for system identification, where algofocused on FIR models as well, and derives optimal input Wigren (1998)). The reference (Casini et al. (2008)) is rithms able to support off-line and on-line networked a topic that is also touched upon in (Wigren (1995), controller design, plant supervision, prediction need Wigren (1998)). The reference (Casini et optimal al. (2008)) is rithms able off-line and on-line networked The same is to truesupport for system identification, where algofocused on FIR models as well, and derives input rithms able to support off-line andand on-line networked controller design, plant supervision, and prediction need signals for the at In (Wang (2008)) focused on FIRproblem models ashand. well, and and derives optimal input Wigren (1998)). The reference (Casini etand al.Yin (2008)) is further able analysis. These system algorithms on FIR models as well, derives optimal input controller design, plant prediction need rithms to support off-lineidentification andand on-line networked signals for the problem at hand. In (Wang and Yin (2008)) controller design, plant supervision, supervision, and prediction need focused further analysis. These system identification algorithms the richness ofproblem the probing inputs investigated. signals for the problem at hand. In is (Wang and Yin Further (2008)) focused onthe FIR models ashand. well, and derives optimal input need to account for signal quantization and include delay signals for at In (Wang and Yin (2008)) further analysis. These system identification algorithms controller design, plant supervision, and prediction need the richness on of the probing further These system identification algorithms inputs is investigated. Further information problems the present paper can the richness ofproblem the probing inputs is investigated. Further need to analysis. account for signal quantization and include include delay for the at related hand. Intois (Wang and Yin (2008)) estimation. In addition, asystem massive deployment ofalgorithms on-line al- signals the richness of the probing inputs investigated. Further need to account for signal quantization and delay further analysis. These identification information on problems related to the present paper can need to account for signal quantization and include delay e.g. be found in (Wang et al. (2010)). information on problems related to the present paper can estimation. In addition, a massive deployment of on-line althe richness of the probing inputs is investigated. Further gorithms implies thatsignal the system identification to the present paper can estimation. In a deployment of on-line alneed to account for quantization and include delay e.g. be foundoninproblems (Wang etrelated al. (2010)). estimation. In addition, addition, a massive massive deployment ofalgorithms on-line al- information gorithms implies that the system identification algorithms e.g. be be found foundonin inproblems (Wang et etrelated al. (2010)). (2010)). to the present paper can need to be always perform as intended, e.g. (Wang gorithms implies that the the system identification estimation. Inguaranteed addition, a to massive deployment ofalgorithms on-line al- information As exemplified by (Wigrenal.(1998)) low complexity recurgorithms implies that identification need toneed be guaranteed tosystem always perform asalgorithms intended, e.g. be found inby (Wang et al.(1998)) (2010)). As exemplified (Wigren low complexity recuror the for manual support may become a concern. need to be guaranteed to always perform as intended, gorithms implies that the system identification algorithms sive stochastic gradient identification algorithms based on As exemplified by (Wigren (1998)) low complexity recurneed to be guaranteed to always perform as intended, exemplified by (Wigren (1998)) low complexity recuror therequirement need for manual manual support may become aintended, concern. As sive stochastic gradient identification algorithms based on This can be translated to global convergence or the need for support may become a concern. need to be guaranteed to always perform as FIR models can meet the above 5G deployment needs. sive stochastic gradient identification algorithms based on As exemplified by (Wigren (1998)) low complexity recuror the need for manual support may become a concern. sive stochastic gradient identification algorithms based on This requirement can be translated to global convergence FIR models can meet the above 5G deployment needs. in athesystem identification settingmay and to global stability sive This requirement can translated to global convergence or need for manual a concern. Suchstochastic methods allow output signal quantization, FIR models can meetgeneral the above 5Galgorithms deployment needs. gradient identification based on This requirement can be be support translated to become global convergence models can meet the above 5G deployment needs. in tracking a system identification setting and to global stability FIR SuchFIR methods allow general output signal quantization, in applications. low computational complexity a system system identification setting and to global stability This requirement can be A translated to global convergence the structure itself handles integer delays, while fracSuch methods allow general output signal quantization, FIR models can meet the above 5G deployment needs. in a identification setting and to global stability methods allow general output signal quantization, in tracking applications. A low low computational complexity the FIR structure itself handles integer delays, while fracalsotracking aidentification relevant requirement in any deploy- Such A complexity in a remains systemapplications. and to massive global stability tional delays can begeneral included e.g. by employing the FIR structure itself handles integer delays, while zerofracSuch methods allow output signal quantization, in tracking applications. A setting low computational computational complexity the FIR structure itself handles integer delays, while fracalso remains a relevant requirement in any massive deploytional delays can be included e.g. by employing zeroment. also remains a relevant requirement in any massive deployin tracking applications. A low computational complexity order hold sampling (Glad and Ljung (2000)). In addition, tional delays can itself be included included e.g. by bydelays, employing zerothe FIR structure handles integer while fracalso remains a relevant requirement in any massive deploytional delays can be e.g. employing zeroment. order hold sampling (Glad and Ljung (2000)). In addition, ment. also remains a relevant requirement in any massive deploy- order the averaging analysis of (Wigren (1998)) gives sufficient hold sampling (Glad and Ljung Ljung (2000)). In addition, addition, delays can be included e.g. by employing zeroment. Early publications on the subject of the present paper tional order hold sampling (Glad and (2000)). In the averaging analysis of (Wigren (1998)) gives sufficient ment. conditions for global convergence to a linear FIR model the averaging analysis of (Wigren (1998)) gives sufficient Early publications on the subject of the present paper order hold sampling (Glad and Ljung (2000)). In addition, includepublications e.g. (Wigrenon (1998)) that provided an analysis of the averaging of (Wigren to (1998)) gives sufficient Early the of paper conditions for analysis global convergence a linear FIR model Early publications on(1998)) the subject subject of the the present present paper thataveraging provides exact dynamic description conditions for an global convergence to a linear linear FIR model include e.g. (Wigren that provided an analysis of the analysis of (Wigreninput-output (1998)) gives sufficient the convergence properties of a recursive algorithm based conditions for global convergence to a FIR model include e.g. (Wigren (1998)) that provided an analysis of Early publications on the subject of the present paper that provides an exact dynamic input-output description include e.g. (Wigren (1998))of that provided an analysis of conditions the convergence properties a recursive algorithm based of the data. However, parametric convergence has not been that provides provides an exact dynamic input-output input-output description for an global convergence to a linear FIR model on Wiener with binary quantization. Addiexact dynamic description the convergence properties of that aoutput recursive algorithm based include e.g.models (Wigren (1998)) provided an analysis of that of the data. However, parametric convergence has not That been the convergence properties of a recursive algorithm based investigated, except in (Yasini and Wigren (2017a)). of the data. However, parametric convergence has not been on Wiener models with binary output quantization. Addithat provides an exact dynamic input-output description tional earlymodels workproperties include (Krishnamurthy (1995), Colinet the data. However, parametric convergence has not That been on Wiener with binary quantization. Addithe convergence of aoutput recursive algorithm based of investigated, except in (Yasini and Wigren (2017a)). on Wiener models with binary output quantization. Addipaper provesHowever, that a in monotone quantizer not investigated, except in (Yasini and and Wigrenis(2017a)). (2017a)). That tional early work work include (Krishnamurthy (1995), Colinet the data. parametric convergence hasnecessary not That been andWiener Juillard (2010), Jafari et al.output (2012)), that are also ap- of investigated, (Yasini Wigren tional early include (Krishnamurthy (1995), Colinet on with binary quantization. Addipaper proves except that a monotone quantizer ispaper not necessary tional earlymodels work include (Krishnamurthy (1995), Colinet and Juillard (2010), Jafari et al. (2012)), that are also apfor parametric convergence. The present therefore paper proves that a monotone quantizer is not necessary investigated, except in (Yasini and Wigren (2017a)). That plicable to recursive identification. The latter papers build proves that a monotone quantizer ispaper not necessary and Juillard (2010), Jafari et that are aptional early work include (1995), Colinet for parametric convergence. The present therefore and Juillard (2010), Jafari(Krishnamurthy et al. al. (2012)), (2012)), that are also also ap- paper plicable to recursive identification. The latter papers build contributes a further analysis of conditions for global for parametric convergence. The present paper therefore paper proveswith that a monotone quantizer ispaper not necessary on aJuillard refined definition of the least squares criterion. The parametric convergence. The present therefore plicable to recursive identification. The latter papers build and (2010), Jafari et al. (2012)), that are also ap- for contributes with a further analysis of conditions for global plicable to recursive identification. The latter papers build parametric convergence of (Wigren (1998)). Counterexamcontributes with a further analysis of conditions for global on a refined definition of the least squares criterion. The for parametric convergence. The present paper therefore paper (Godoy et al. (2011)) applied expectation maximizacontributes with a further analysis of conditions for global on a refined definition of the least squares The plicable to recursive identification. The lattercriterion. papers build parametric convergence of (Wigren (1998)). Counterexamon a refined definition of the least squares criterion. The ples are firstconvergence constructed low order FIR Counterexamsystems and a parametric convergence offor (Wigren (1998)). Counterexampaper (Godoy et al. (2011)) (2011)) applied expectation maximizawith a further analysis of conditions for global tiona (EM) techniques toofsolve aleast maximum (ML) parametric of (Wigren (1998)). paper (Godoy et applied expectation maximizaon refined definition the squareslikelihood criterion. The contributes ples arequantizer. first constructed for low analysis order FIR systems(1977), and a paper (Godoy et al. al. (2011)) applied expectation maximization (EM) techniques to solve a maximum likelihood (ML) binary The averaging of (Ljung ples are first constructed for low order FIR systems and parametric convergence of (Wigren (1998)). Counterexamproblem for the linear FIR model, obtaining enhanced ples are first constructed for low order FIR systems and aa tion (EM) techniques to solve a maximum likelihood (ML) paper (Godoy et al. (2011)) applied expectation maximizabinary quantizer. The averaging analysis of (Ljung (1977), tion (EM)for techniques to solve maximum likelihood (ML) ples problem the linear FIR amodel, obtaining enhanced binary quantizer. The averaging of (Ljung arequantizer. first constructed for low analysis order FIR systems(1977), and a binary The averaging analysis of (Ljung (1977), problem for the linear FIR model, obtaining enhanced tion (EM) techniques to solve a maximum likelihood (ML) problem for the linear FIR model, obtaining enhanced binary quantizer. The averaging analysis of (Ljung (1977), problem for the linear FIR model, obtaining enhanced Copyright 915 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © under 2018 IFAC 915 Control. Peer review responsibility of International Federation of Automatic Copyright © 915 Copyright © 2018 2018 IFAC IFAC 915 10.1016/j.ifacol.2018.09.077 Copyright © 2018 IFAC 915

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Fig. 1. The networked system identification architecture. u is the plant control signal, y is the output signal of the plant and yˆ is the model output. Ljung and S¨ oderstr¨om (1983), Wigren (1994)) is then applied and the right hand side of the associated differential equations (ODE) of the algorithm of each counterexample is computed analytically. The stationary points are analysed analytically and with numerical integration. Restrictions on the sufficient conditions of (Wigren (1998)), for proving global parametric convergence are then formulated. In addition, global parametric convergence is proved for one of the low order models.

sent to the controller node in the uplink direction. The identification algorithm processes quantized control sigu(t)) and quantized output feedback nals u(t) = fn,ε,u (¯ signals yn (t) = fn,ε,y (y(t − T )). The network interface, the wireless interface and the transmit buffers in the wireless transmission nodes contribute to the loop delay T .

It is stressed that the purpose of the paper is to study fundamental conditions for parametric convergence, to provide an enhanced understanding of the problem. Since that is done by counterexamples it is enough to find a single such example to formulate necessary restrictions. It is then not limiting to focus on a low order example. This in fact appears to be needed since even the first order example of the paper shows a very complicated asymptotic dynamics. The difficulties with finding Lyapunov functions for analytic convergence analysis are expected to increase in a general high dimensional case. The fact that the first order FIR model is used of course prevents handling of delay, however, that is not the primary scope of the paper.

In the convergence analysis the above quantities are also subscripted by ε , as in (Wigren (1998)). The subscript ε indicates a smoothening of quantizer corners that are introduced to meet certain regularity conditions.

The paper is organized as follows. The networked 5G identification problem, the algorithm, the method of analysis and the previous result on global convergence are reviewed in Section 2. The main contributions of the paper follow in Sections 3-5. The conclusions are given in Section 6. 2. REVIEW OF THE IDENTIFICATION METHOD AND ITS CONVERGENCE PROPERTIES 2.1 Networked Identification The 5G networked control and identification architecture appears in Fig. 1. The networked controller and identification algorithm are located in a controller node. Each controller is connected to multiple wireless transmission nodes over a network interface which can be a wired or wireless Internet interface. Each wireless transmission node connects to the controlled plant over a wireless 5G interface (Middleton et al. (2017)). The control signals are quantized in the controller node by the quantizer function fn,ε,u (¯ u(t)) where u ¯(t) denotes the control signal, and sent to the plant in the downlink direction. The feedback signal is quantized in the plant node by the quantizer fn,ε,y (y(t)), where y(t) denotes the output signal of the plant, and 916

2.2 Recursive Identification Method

To review the algorithm of (Wigren (1998)), note that the u(t)) in Fig. 1 is quantized control signal u(t) = fn,ε,u (¯ filtered by the FIR filter to produce the model output (1) yˆ(t, θ) = θ ϕ(t) ϕ(t) = (u(t − 1) . . . u(t − m))



(2)

θ = (b1 . . . bm ) , (3) where θ is the unknown parameter vector. After transformation by the quantizer, the output of the model becomes y (t, θ)) = fn,ε,ˆy (θ ϕ(t)). (4) yˆn,ε (t, θ) = fn,ε,ˆy (ˆ The quantized output feedback signal is yn,ε (t) = fn,ε,y (y(t − T )) + w(t)   (5) = fn,ε,y θ0 ϕ(t) + w(t), where y(t) denotes the output signal of the plant in Fig. 1 and w(t) is a zero mean correlated disturbance independent of u(t). The subscript n denotes nonlinear while the subscript y and yˆ refer to the independent variable, to keep notational consistency with (Wigren (1998)). The identification algorithm is then derived by minimization of the criterion 1 2 V (θ) = E [n,ε (t, θ)] , (6) 2 where n,ε (t, θ) = yn,ε (t)− yˆn,ε (t, θ) is the prediction error. The negative gradient ψn,ε (t, θ) of n,ε (t, θ) with respect to the parameter vector θ is required in the algorithm and it contains the gradient of the quantizer. However, the gradient of the quantized output would be identically zero except at the quantization steps where it would contain Dirac pulses, implying an algorithm based on the

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exact gradient would fail to work in practice. To sidestep this problem, a continuously differentiable approximate gradient kn,ˆy (ˆ y (t, θ)) ≈ dfn,ε,ˆy (ˆ y (t, θ))/dˆ y is used instead. Proceeding as in (Wigren (1995), Wigren (1998)), the normalized stochastic gradient algorithm is  ˆ = θ(t ˆ − 1) θ(t)

 µ(t)  1 kn,ˆy (ˆ y (t))ϕ(t) (yn,ε (t) − yˆn,ε (t)) t r(t − 1) DM  r(t) = r(t − 1)  µ(t)  2 + kn,ˆy (ˆ y (t))ϕ (t)ϕ(t) − r(t − 1) t DM  ϕ(t + 1) = (u(t) . . . u(t − m + 1)) yˆ(t + 1) = θˆ (t)ϕ(t + 1) +

y (t + 1)), yˆn,ε (t + 1) = fn,ε,ˆy (ˆ (7) ˆ where θ(t) is the parameter estimate, µ(t)/t is the gain sequence, and DM indicates a projection in order to ensure that the estimates remain in the compact model set DM ={(θ, r)||bi | ≤ C < ∞, i = 1, . . . , m, (8) 0 < δ1 ≤ r ≤ C < ∞}, irrespective of any instability mechanism that could affect the unprojected nonlinear algorithm (Ljung (1977)). 2.3 Associated ODE Analysis As shown in (Ljung (1977)) the convergence properties of an algorithm like (7) are tied to the stability of an associated deterministic ODE. It is e.g. proved that the asymptotic path of the recursive algorithm will follow the trajectories of the associated ODE. The right hand side of the associated ODE consists of the average updating direction of the algorithm that is computed for a fixed value of the parameter vector. It is proved in (Ljung (1977)) that the local (global) stability of the associated ODE implies local (global) convergence of the algorithm.

A6) fn,ε,ˆy (ˆ y ) is increasing (not necessarily strictly increasing) and kn,ˆy (ˆ y ) ≥ δ2 > 0 or and fn,ε,ˆy (ˆ y ) is decreasing (not necessarily strictly decreasing), and kn,ˆy (ˆ y ) ≤ −δ2 < 0.

A1 is needed to ensure that the model is not underparametrized, i.e., the system is exactly described by the model. A2 means that the system is in the model set. A3 ensures that the regularity conditions of (Ljung (1977) and Wigren (1994)) hold while A4 is standard in convergence analysis. A5 summerizes technical conditions for the averaging analysis to be valid; cf., (Ljung (1977)). Then consider the following set derived in (Wigren (1998)) DCθ = {θ| lim E[|ˆ y (t, θ) − y(t)| t→∞

· |fn,ε,ˆy (ˆ y (t, θ)) − fn,ε,y y(t − T )|]θ=θD = 0}. (9) DCθ defines the possible convergence points of the algorithm, obtained from the choice of Lyapunov function V (θ, r) = (θ − θ0 ) (θ − θ0 ). (10) The set DCθ disregards the fact that the algorithm cannot converge to locally unstable stationary points of the ODE, as stated in (Ljung (1977)). This fact will become important in Section 3.3 where the local convergence theorem of (Wigren (1994)) that builds on (Ljung (1977)) is needed. Therefore the result of Lemma 1 below is enhanced to account also for local stability properties. Lemma 1. Assume that A1)-A6) hold for (7). Assume that there exists a twice differentiable, positive definite function V (θ, r) such that dV (θD (τ ), rD (τ ))/dτ ≤ 0 for (θD , rD ) ∈ DM \∂DM , when evaluated along the solutions to d µ θD (τ ) = f (θD (τ )) dτ rD (τ ) (11) d rD (τ ) = µ(g(θD (τ )) − rD (τ )), dτ where f (θD ) and g(θD ) are defined as   f (θD ) = lim E kn,ˆy (ˆ y (t, θ))(yn,ε (t) − yˆn,ε (t, θ))ϕ(t) t→∞

2 g(θD ) = lim E[kn,ˆ y (t, θ))ϕ (t)ϕ(t)]|θ=θD > 0. y (ˆ

2.4 Known Convergence Properties

917

|θ=θD

t→∞

Global output error convergence for the algorithm (7) is proved in (Wigren (1998)). That analysis requires the following assumptions A1) m ≥ m0 , where m0 = dim(θ0 ) and m = dim(θ) A2) θ0 = (b01 . . . b0m0 0 . . . 0) ∈ DM \∂DM where θ0 has been filled out with zeros to make m = m0 . A3) fn,ε,ˆy (ˆ y ) and kn,ˆy (ˆ y ) are a priori known, continuous, and continuously differentiable w.r.t yˆ such that for some C < ∞ and θ ∈ DM , |fn,ε,ˆy (ˆ y )| ≤ C(1 + |ˆ y |), |dfn,ε,ˆy (ˆ y )/dˆ y | + |kn,ˆy (ˆ y )| + |dkn,ˆy (ˆ y )/dˆ y | ≤ C. A4) limt→∞ µ(t) = µ > 0, and 0 ≤ µ(t) < C < ∞. A5) u(t) and w(t) are realizations of bounded stationary stochastic processes. u(t) is exponentially uncorrelated in the sense that for each t, s, t > s, there exists a random vector u0s (t) (u0s (s) = 0) that belongs to the σ−algebra generated by (u(0) . . . u(t))   but is independent of (u(0) . . . u(s)) such that E |u(t) − u0s (t)|4 ≤ Cλt−s , C < ∞, λ < 1. w(t) is exponentially uncorrelated in the sense above, has zero mean and is independent of u(t). 917

(12)

ˆ y (t, θ)) = Then, θ(t) → DCθ ⊂ {θ ∈ DM \∂DM |fn,ε,ˆy (ˆ fn,ε,y (y(t − T )), eigi (∂f (θ))/∂θ) ≤ 0, i = 1, ..., m, } w.p.1 as t → ∞, or there is a cluster point on ∂DM . Proof 1. The result follows from Corollary 1 of (Wigren (1998)), using the fact that all conditions of (Wigren (1994)) hold. The condition eigi (∂f (θ))/∂θ ≤ 0 follow since the algorithm renders a scalar rD (τ ) > 0. eigi (.) denotes the ith eigenvalue of a matrix. 3. GLOBAL PARAMETRIC CONVERGENCE To investigate the possibilities for proving parametric convergence of (7), the approach of the paper is now to study low order FIR systems in combination with a binary output quantizer. Below the subscript ε is dropped, and the delay T = 0. 3.1 Low Order Test Models The following FIR systems are treated:

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i) Single parameter FIR system with uniformly distributed independently identically distributed (i.i.d.) input u ∼ U (−u0 , u0 ), Gaussian i.i.d. disturbance w ∼ N (0, σ 2 ) with cut tails to meet assumption A5, binary quantizer with levels -1 and 1 and switch point f0 . The system and model are yn (t) = fn,y (b01 u(t − 1)) + w(t) (13) yˆn (t, θ) = fn,ˆy (b1 u(t − 1)) where b01 = 1.0 and kn,ˆy (ˆ y (t, θ)) = k0 = 1.0. ii) Single parameter FIR system with discretely distributed i.i.d. input  with probability p u0 (14) u(t) = −u0 with probability 1 − p, Gaussian i.i.d. disturbance w ∼ N (0, σ 2 ) with cut tails to meet Assumption A5, binary quantizer with levels -1 and 1 and switch point f0 . The system and model are yn (t) = fn,y (b01 u(t − 1)) + w(t) (15) yˆn (t, θ) = fn,ˆy (b1 u(t − 1)),

where b01 = 1.0 and kn,ˆy (ˆ y (t, θ)) = k0 = 1.0.

In the report (Yasini and Wigren (2017b)), a two parameter system is studied. 3.2 Associated ODEs The associated ODEs corresponding to the test models are now derived by evaluation of the average updating direction (12). i) The average updating direction for (13) becomes f1 (θ) =k0 lim Eu(t − 1)fn,y (u(t − 1)) t→∞

− k0 lim Eu(t − 1)fn,ˆy (b1 u(t − 1)) t→∞   u0  k 0  u0 = ufn,y (u)du − ufn,ˆy (b1 u)du , 2u0 −u0 −u     0   T1

T2

(16)

where

T1 =   0       0  

 0

|f0 | > u0  1  2 , 2  u − f0 |f0 | ≤ u0 2u0 0 b1 = 0

(17)

f0 | > u0 b1   f0 f0 1 T2 = , u20 − ( )2 b1 > 0, f0 = 0, | | ≤ u0   2u b b  0 1 1      1 f0 2 f0  2  u0 − ( ) b1 < 0, f0 = 0, | | ≤ u0 − 2u0 b1 b1 (18) 2 2 k u (19) g1 (θ) = k02 lim Eu2 (t) = 0 0 . t→∞ 3 b1 = 0, f0 = 0, |

ii) The average updating direction for the model (15) with discretely distributed input is determined as follows f2 (θ) =k0 lim Eu(t − 1)fn,y (u(t − 1)) t→∞    T¯1

− k0 lim Eu(t − 1)fn,ˆy (b1 u(t − 1)), t→∞   

(20)

T¯2

918

where

 u0 (2p − 1) T¯1 = u0  −u0 (2p − 1)  u0 (2p − 1) b1      −u0 (2p − 1) b1       b1 u0 (2p − 1)    T¯2 = u0 b1       −u0 b1        −u (2p − 1) b 0

f0 < −u0 |f0 | ≤ u0 , f0 > u0

= 0, f0 < 0 = 0, f0 > 0 f0 < −u0 = 0, b1 f0 > 0, | | ≤ u0 , b1 f0 < 0, | | ≤ u0 b1 f0 > u0 1 = 0, b1 2 2 g2 (θ) = k0 u0 .

(21)

(22)

(23)

3.3 Analytic Parametric Convergence Analysis The present subsection provides an analytical global parametric convergence result for the test model i). The proof is a sketch because of the page limitation. It follows from Lemma 1 that only stable stationary points of (11) are possible convergence points of (7). Therefore, consider the stationary points of (11) which are given by ∗ ∗ f (θD ) = 0, g(θD ) = rD . (24) For the model (13) it first follows from (19) that 1 k 2 u2 ∗ ∗ rD (θD (25) ) = 0 0 = > 0. ) = g1 (θD 3 3 ∗ The remaining equation f (θD ) = 0 is then considered. In case b1 = 0 and the switch point is outside the support of ∗ the input signal it follows that T1 = T2 = 0 ⇒ f (θD ) = 0. This means that this point is a stationary point. In case the switch point is inside the support of the input signal f1 (θ) = T1 > 0 and b1 will move away from 0, which in this case is an unstable stationary point. Then, in case b1 = 0, and the switch point |f0 /b1 | > u0 then T2 = 0 and f1 (θ) = T1 . Again, in case |f0 | > u0 it follows that f1 (θ) = 0 and this represents a set of stable stationary points. In case |f0 | ≤ u0 the average updating direction component becomes f1 (θ) = (u20 − f02 )/(2u0 ) > 0 so b1 starts to increase with constant rate. Eventually the trajectory will reach the point where |f0 /b1 | = u0 , entering the region of the last cases where |f0 /b1 | ≤ u0 . If |f0 | > u0 and b1 > 0 then f1 (θ) = −(u20 − (f0 /b1 )2 )/(2u0 ) < 0 and the parameter starts to decrease until |f0 /b1 | = u0 where it will be stuck. In case b1 < 0, the parameter starts to increase until |f0 /b1 | = u0 . In case |f0 | ≤ u0 and b1 > 0 follows that the stationary points are given by   2   1 1  2 f0 2 2 f1 (θ) = u0 − u − f0 − 2u0 0 2u0 b1   2 1 f − 1 = 0. (26) = 0 2u0 b21 Introducing the assumption that |f0 | > 0 implies (27) b1 = 1 or b1 = −1. Since (27) then directly leads to f2 1 df1 (θ) = − 0 3. (28) dθ u 0 b1

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It follows that b1 = 1 (29) is the only stable stationary point of (26). For the case b1 < 0, there is one positive and one negative solution to f1 (θ) = 0. However, the positive solution is not consistent with b1 < 0 and the negative solution is unstable. A check of the conditions A1-A6 reveal that A1, A2, A5 and A6 are immediately implied by the definition of the test model i). This leads to the following result Proposition 1. Assume that A3) and A4) hold for the algorithm (7). Then, if 0 < |f0 | ≤ u0 holds for i), ˆb1 (t) → b0 = 1.0 w.p.1 as t → ∞, or there is cluster 1 point on (8).

Fig. 2. a) Trajectories of the ODE (11) and b) asymptotic paths of the algorithm for the model (15).

The condition 0 < |f0 | ≤ u0 requires that the switch point of the quantizer is not zero. The condition can be intuitively motivated by consideration of another ramp input signal u(t) = αt, α > 0, in a noise free case. By observation of the input signal level us that causes a switch, it follows that b1 us = f0 . In case f0 = 0 the equation cannot be used to determine b1 , while |f0 | > 0 implies that b1 = f0 /us . The problem is likely to be caused by the symmetry between the quantizer and the input signal distribution. Note also that a quick check of the calculations above show that Proposition 1 holds for arbitrary system parameter values, provided that the system is such that there is signal energy in the switch point, c.f. Section 4.2. and (Wigren (1995)). 4. NUMERICAL SIMULATION OF THE ASSOCIATED ODES

Fig. 3. ODE Trajectories (11) for the model (13) when the switch approaches zero.

In this section, numerical counterexamples to parameter convergence are studied using the software of (Wigren (2007)). 4.1 Counterexample 1 - Discretely Distributed Input Signal In the first set of simulation, numerical solutions of the associated ODE (11) are presented for the model (15). The probability p = 0.5 and u0 = 1 were selected. The quantizer had a switch point f0 = 0.5. White Gaussian measurement noise with a standard deviation of 0.5 was added to the output signal when the algorithm was simulated. The trajectories of the associated ODE are depicted in Fig. 2 a). Correspondingly initialized simulations of the algorithm, with µ(t) = 2/t, running from t = 10000 to 110000 are depicted in Fig. 2 b). It can be seen that the paths of the algorithm stay close to the ODE trajectories. However, there is no single stationary point when the input is discretely distributed. Starting from different initial conditions, all the trajectories converge towards the line where rD = g3 (θD ) = k02 u20 = 1. Note that for the case where b1 > 0.5, it follows from (20) that f2 (θ) = 0 and thus the trajectories remain in the initial parameter vector. 4.2 Counterexample 2 - Quantizer Approaching Symmetry Next, solutions of the ODE are obtained numerically for switch point values that are selected increasingly close to zero. The same simulation assumptions as in Section 4.1 are applied. For the model (13) the switch points of the 919

output quantizer were selected as f0 = 0.7, 0.5, 0.3, 0.1. Fig. 3 and Fig. 4 show the trajectories of the associated ODEs and the asymptotic paths of the algorithm, respectively. The algorithm was run for 30000000 steps, with µ = 10/t. Since u(t) has zero mean, Proposition 1 states that it is only possible to identify the static gain by observation of the output signal when the switch point is not zero. As can be seen, the trajectories behave in a more singular manner when the switch point approaches zero, approaching a one dimensional asymptotic low gain convergence. This shows that the static gain cannot be estimated when f0 = 0. 5. NECESSARY RESTRICTIONS It is clear from Section 4.1 that parametric convergence is not generally possible in case the input signal distribution is discrete and takes a finite number of values, c.f. Fig. 2. At the same time Proposition 1 shows that for a continuous distribution with a probability density function of y(t) that is positive in a neighborhood around the switch point, parametric convergence can be proved. As illustrated by Proposition 1 and the counterexample of Section 4.3 a symmetric situation between the quantizer and the distributions of u(t) and y(t) also leads to a failure of parametric convergence. Since such symmetry can only exist with respect to one switch point, the binary quantizer requires an additional restriction. The findings are formulated as:

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Fig. 4. Asymptotic paths of the algorithm (7) for the model (13) when the switch point approaches zero. Proposition 2. Without the following restrictions, A1-A6 cannot be sufficient for the global parametric convergence: R1) The set Su of input signal distributions, selected without information of the system, does not contain discrete distributions with a finite number of values. R2) The probability distribution py (Y ) of y(t) fulfills py (Y ) > δ > 0 in a neighborhood of at least one switch point of the quantizer. R3) In the binary quantizer case, Ey(t) = f0 , for symmetric input distributions, where f0 is the switch point. 6. CONCLUSION The paper derived restrictions on previous sufficient conditions for input-output convergence, to enable global parametric convergence in identification of linear systems subject to output quantization. These restrictions are believed to be fundamental and valid for related identification problems. Global parametric convergence w.p.1 was established in a one dimensional case. REFERENCES Bai, E.W. and Reyland, J. (2008). Towards identification of Wiener systems with the least amount of a priori information on the nonlinearity. Automatica, 44, 910– 919. Bai, E.W. and Reyland, J. (2009). Towards identification of wiener systems with the least amount of a priori information on the nonlinearity: IIR cases. Automatica, 45, 956–964. Casini, M., Garulli, A., and Vicino, A. (2008). Optimal input design for identification of systems with quantized measurements. IEEE Conf. Decision and Control, Cancun, Mexico, 5506–5512. Colinet, E. and Juillard, J. (2010). A weighted leastsquares approach to parameter estimation problems based on binary measurements. IEEE Trans. Automatic Control, 2, 148–152. Glad, T. and Ljung, L. (2000). Control Theory. Taylor and Francis, Bodmin, UK. Godoy, B.I., Goodwin, G.C., Aguero, J.C., Marelli, D., and Wigren, T. (2011). On identification of FIR systems 920

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