Closed-loop nonlinear system identification via the vector optimal parameter search algorithm: Application to heart rate baroreflex control

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Abstract

The vector optimal parameter search (VOPS) and the constrained optimal parameter search (COPS) are recently developed algorithms for closed-loop linear system identification. We extend both algorithms to be applicable to a closed-loop nonlinear system, which is characterized by a vector nonlinear autoregressive model. Monte Carlo simulations of nonlinear closed-loop systems were performed to compare the performance of the VOPS to the widely utilized vector least squares (VLS), the COPS and the total least squares (TLS) approaches. The relative error and linear transfer functions are computed to determine the accuracy of each method. The comparative results show that both the VOPS and COPS algorithms provide far superior parameter estimates than does the VLS for all simulation examples considered. The TLS provides better estimates than the VOPS, COPS and VLS when there is only observation noise present in the data. However, the performance of the TLS degrades considerably when the data are corrupted by dynamic noise. The clinical applicability of the two extended methods is examined by applying them to a classical physiological closed-loop system, the heart rate baroreflex. It was found that while both control and blockade of parasympathetic system conditions are dominated by linear dynamics, more nonlinearity was observed in the latter. This observation is statistically supported by the calculation of the mutual information of the data and their surrogates.

Introduction

The prevailing evidence suggests that most, if not all, dynamics of physiological systems involve nonlinear control [1], [2]. For example, nonlinear feedback control mechanisms are important for maintaining homeostasis in both cardiovascular and renal regulatory systems. Investigation of physiological systems with feedback processes requires closed-loop system identification analyses where both feedforward and feedback coupling between various physiologic variables can be identified. Closed-loop system identification analyses have mostly involved linear techniques, on the assumption that the contributions of the nonlinear components are small when considering small fluctuations in the variables about their means. Small perturbations of the variable may justify the use of linear methods provided that the system is neither highly nonlinear nor chaotic. There has been a plethora of evidence pointing to possible deterministic chaotic behavior of heart rate dynamics and renal blood flow in hypertensive rats [2]. Consequently, controversies abound as to whether or not heart rate dynamics involve deterministic chaos [3]. If a system is chaotic, then even small perturbations will result in significant nonlinear dynamic changes. Therefore, if there is some probability that a system is chaotic, it is important to perform systematic nonlinear closed-loop identification to extract information that may not have been revealed using linear analyses. Another reason for the lack of application of nonlinear closed-loop identification methods to physiological systems modeling is the dearth of accurate nonlinear closed-loop parametric identification methods that are also computationally manageable and reasonably accurate.

In recognition of the lack of nonlinear closed-loop parametric identification methods, the major goal of this paper is to introduce a nonlinear vector optimal parameter search (VOPS) algorithm. Recently we have developed linear versions of closed-loop methods, the constrained optimal parameter search (COPS) and VOPS, which are more accurate than the most widely utilized method, the vector least squares (VLS), and in certain cases have been shown to be more accurate than the fast orthogonal search (FOS) algorithm [4], [5], [6]. The COPS and VOPS algorithms have been shown to be very robust in extracting only the significant parameters despite severe noise corruption and incorrect model selection [6], [7]. The extension of the FOS to linear closed-loop multivariate processes has been developed by Bagarinao and Sato [8]. These aforementioned algorithms are especially useful because they were specifically designed to select only the significant model terms from the initial pool of large model terms that are composed of both linear and nonlinear terms. The present work is to extend the linear versions of the COPS and VOPS to be appropriate for nonlinear modeling of closed-loop systems so that more accurate nonlinear model structure is obtained. Simulation examples are provided to demonstrate the feasibility of the nonlinear version of the VOPS and COPS algorithms for modeling nonlinear closed-loop systems. Furthermore, simulation examples are performed to compare the methods proposed to the vector versions of the least squares and the total least squares (TLS). The benefit of the TLS is more evident when the system is corrupted with observation noise, in which case the TLS provides more accurate results than does the least squares. Finally, the developed method is applied to heart rate and blood pressure data to test the feasibility of the method as well as to demonstrate the significance of nonlinearity in the closed-loop analysis of heart rate and blood pressure control.

Section snippets

Vector least squares applied to closed-loop nonlinear system identification

A two-channel closed-loop time-invariant nonlinear system with observation (ηx and ηy) and dynamic (ɛx and ɛy) noise can be illustrated as in Fig. 1.

The two channels in Fig. 1 can be described by the following model:

  • Channel 1:x(n)=f1(y,x,εx)=constx+i=1Maix(ni)+j=1Mbjy(nj)+i=1Mj=iMeijx(ni)x(nj)+i=1Mj=iMfijy(ni)y(nj)+i=1Mj=1Mgijx(ni)y(nj)++εx(n)x(n)=x(n)+ηx(n)

  • Channel 2:y(n)=f2(x,y,εy)=consty+i=1Mcix(ni)+j=1Mdjy(nj)+i=1Mj=iMhijx(ni)x(nj)+i=1Mj=iMlijy(ni)y(nj)+i=1Mj=1

Simulation examples

In this section, two VNAR models are simulated to assess the performance of the VOPS and COPS. The performance of these two methods is compared to the LS and TLS. In all simulation examples, the data length was set to 1024 data points. Monte Carlo simulations (100 realizations) were performed to investigate the differences in the performance of the various methods. For the VOPS, the multivariate AIC criterion is used first to determine the initial model order of the closed-loop nonlinear system

Application of the nonlinear VOPS to estimate heart rate baroreflex

In this section, we demonstrate the feasibility of the nonlinear VOPS method for estimating heart rate (HR) baroreflex. Many studies have been performed for estimating HR baroreflex including vector autoregressive modeling. However, all of these approaches were based on linear techniques [16]. Experimental evidence indicates that heart rate control exhibits nonlinear interactions between the sympathetic and parasympathetic nervous systems [17], which implies that HR baroreflex may involve

Discussion and conclusions

We demonstrated the feasibility of using the VOPS and COPS algorithms for closed-loop nonlinear system identification. Computer simulations have been performed to compare the proposed algorithms against some of the existing methods for closed-loop nonlinear identification. The comparative results show that both the VOPS or COPS algorithms are, in general, far superior to the VLS for all types of noise and they are superior to the TLS for dynamic noise. The TLS provides better results than

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