Subspace identification of Hammerstein systems using B-splines.

This paper presents an algorithm for the identification of Hammerstein cascades with hard nonlinearities. The nonlinearity of the cascade is described using a B-spline basis with fixed knot locations; the linear dynamics are described using a state-space model. The algorithm automatically estimates both the order of the linear system and the number and locations of the knots used to characterize the nonlinearity. Therefore, it significantly reduces the a priori knowledge about the underlying system required for identification. A simulation study on a model of reflex stiffness shows that the new method estimates the nonlinearity accurately in the presence of output noise.

Subspace methods for identification of human ankle joint stiffness.

Joint stiffness, the dynamic relationship between the angular position of a joint and the torque acting about it, describes the dynamic, mechanical behavior of a joint during posture and movement. Joint stiffness arises from both intrinsic and reflex mechanisms, but the torques due to these mechanisms cannot be measured separately experimentally, since they appear and change together. Therefore, the direct estimation of the intrinsic and reflex stiffnesses is difficult. In this paper, we present a new, two-step procedure to estimate the intrinsic and reflex components of ankle stiffness. In the first step, a discrete-time, subspace-based method is used to estimate a state-space model for overall stiffness from the measured overall torque and then predict the intrinsic and reflex torques. In the second step, continuous-time models for the intrinsic and reflex stiffnesses are estimated from the predicted intrinsic and reflex torques. Simulations and experimental results demonstrate that the algorithm estimates the intrinsic and reflex stiffnesses accurately. The new subspace-based algorithm has three advantages over previous algorithms: 1) It does not require iteration, and therefore, will always converge to an optimal solution; 2) it provides better estimates for data with high noise or short sample lengths; and 3) it provides much more accurate results for data acquired under the closed-loop conditions, that prevail when subjects interact with compliant loads.

Analysis of canine model of gastric electrical uncoupling using recurrence quantification analysis.

This study reports the results of recurrence quantification analysis (RQA) applied to internal gastric electrical activity (GEA) and cutaneous electrogastrographic (EGG) digital recordings obtained from acute canine models. The purpose of this chaos analysis is to differentiate three states--normal, mild, and severe induced electrical uncoupling--utilizing five quantities associated with RQA: percent recurrence, percent determinism, maximum deterministic line, entropy, and trend. The results indicate that percent recurrence and trend are the only quantities that relate the three states of gastric electrical uncoupling in any meaningful way. The ability of EGG to detect mild electrical uncoupling in the stomach appears to be limited due to the impact of numerous external factors on the signals, even if multichannel recordings are utilized.

Separable least squares identification of nonlinear Hammerstein models: application to stretch reflex dynamics.

The Hammerstein cascade, consisting of a zero-memory nonlinearity followed by a linear filter, is often used to model nonlinear biological systems. This structure can represent some high-order nonlinear systems accurately with relatively few parameters. However, it is not possible, in general, to estimate the parameters of a Hammerstein cascade in closed form. The most effective method available to date uses an iterative approach, which alternates between estimating the linear element from a crosscorrelation, and then fitting a polynomial to the nonlinearity via linear regression. This paper proposes the use of separable least squares optimization methods to estimate the linear and nonlinear elements simultaneously in a least squares framework. A separable least squares algorithm for the identification of Hammerstein cascades is developed and used to analyze stretch reflex electromyogram data from two experimental subjects. The results show that in each case the proposed algorithm produced a better model, in that it predicted the system's response to novel inputs more accurately, than did models estimated using the traditional iterative algorithm. Monte-Carlo simulations demonstrated that when the input is a non-Gaussian, nonwhite signal, as is often the case experimentally, the traditional iterative identification approach produces biased models, whereas the separable least squares approach proposed in this paper does not.

Fast, robust identification of nonlinear physiological systems using an implicit basis expansion.

Because the number of parameters required by a Volterra series grows rapidly with both the length of its memory and the order of its nonlinearity, methods for identifying these models from measurements of input/output data are limited to low-order systems with relatively short memories. To deal with these computational and storage requirements one can either make extensive use of the structure of the Volterra series estimation problem to eliminate redundant storage and computations (e.g., the fast orthogonal algorithm), or apply a basis expansion, such as a Laguerre expansion, which seeks to reduce the number of model parameters, and hence, the size of the estimation problem. The use of an appropriate expansion basis can also decrease the noise sensitivity of the estimates. In this paper, we show how fast orthogonalization techniques can be combined with an expansion onto an arbitrary basis. We further demonstrate that the orthogonalization and expansion may be performed independently of each other. Thus, the results from a single application of the fast orthogonal algorithm can be used to generate multiple basis expansions. Simulations, using a simple nonlinear model of peripheral auditory processing, show the equivalence between the kernels estimated using a direct basis expansion, and those computed using the fast, implicit basis expansion technique which we propose. Running times for this new algorithm are compared to those for existing techniques.

Parametric and nonparametric nonlinear system identification of lung tissue strip mechanics.

Lung parenchyma is a soft biological material composed of many interacting elements such as the interstitial cells, extracellular collagen-elastin fiber network, and proteoglycan ground substance. The mechanical behavior of this delicate structure is complex showing several mild but distinct types of nonlinearities and a fractal-like long memory stress relaxation characterized by a power-law function. To characterize tissue nonlinearity in the presence of such long memory, we investigated the robustness and predictive ability of several nonlinear system identification techniques on stress-strain data obtained from lung tissue strips with various input wave forms. We found that in general, for a mildly nonlinear system with long memory, a nonparametric nonlinear system identification in the frequency domain is preferred over time-domain techniques. More importantly, if a suitable parametric nonlinear model is available that captures the long memory of the system with only a few parameters, high predictive ability with substantially increased robustness can be achieved. The results provide evidence that the first-order kernel of the stress-strain relationship is consistent with a fractal-type long memory stress relaxation and the nonlinearity can be described as a Wiener-type nonlinear structure for displacements mimicking tidal breathing.

Sensitivity analysis of kernel estimates: implications in nonlinear physiological system identification.

Many techniques have been developed for the estimation of the Volterra/Wiener kernels of nonlinear systems, and have found extensive application in the study of various physiological systems. To date, however, we are not aware of methods for estimating the reliability of these kernels from single data records. In this study, we develop a formal analysis of variance for least-squares based nonlinear system identification algorithms. Expressions are developed for the variance of the estimated kernel coefficients and are used to place confidence bounds around both kernel estimates and output predictions. Specific bounds are developed for two such identification algorithms: Korenberg's fast orthogonal algorithm and the Laguerre expansion technique. Simulations, employing a model representative of the peripheral auditory system, are used to validate the theoretical derivations, and to explore their sensitivity to assumptions regarding the system and data. The simulations show excellent agreement between the variances of kernel coefficients and output predictions as estimated from the results of a single trial compared to the same quantities computed from an ensemble of 1000 Monte Carlo runs. These techniques were validated with white and nonwhite Gaussian inputs and with white Gaussian and nonwhite non-Gaussian measurement noise on the output, provided that the output noise source was independent of the test input.

Factors affecting Volterra kernel estimation: emphasis on lung tissue viscoelasticity.

The goal of this study is to quantitatively investigate how the memory length, order of nonlinearity, type of input, and measurement noise can affect the identification of the Volterra kernels of a nonlinear viscoelastic system, and hence the inference on system structure. We explored these aspects with emphasis on nonlinear lung tissue mechanics around breathing frequencies, where the memory length issue can be critical and a ventilatory input is clinically demanded. We adopted and examined Korenberg's fast orthogonal algorithm since it is a least-squares technique that does not demand white Gaussian noise input and makes no presumptions on the kernel shape and system structure. We then propose a memory autosearch method, which incorporates Akaike's final production error criterion into Korenberg's fast orthogonal algorithm to identify the memory length simultaneously with the kernels. Finally, we designed a special ventilatory flow input and evaluated its potential for the kernel identification of the nonlinear systems requiring oscillatory forcing. We found that the long memory associated with soft tissue viscoelasticity may prohibit correct identification of the higher-order kernels of the lung. However, the key characteristics of the first-order kernel may be revealed through averaging over multiple experiments and estimations.

Generalized eigenvector algorithm for nonlinear system identification with non-white inputs.

Traditional methods for nonlinear system identification require a white, Gaussian, test input, a restriction that has limited their usability in many fields. In this study, we address the problem of identifying the dynamics of a nonlinear system when the input is highly colored-a restriction commonly encountered in the study of physiological systems. An extension of the parallel cascade method is developed that is optimal in a constrained minimum mean squared error sense and exactly corrects for the distortion induced by the non-white input spectrum. However, this correction is a deconvolution, which may become extremely ill-conditioned if the input spectrum departs significantly from whiteness; to confront this, we develop a low-rank projection operation that stabilizes the deconvolution. The overall algorithm is robust and places few requirements on the nature of the test input. Practical application of this new method is demonstrated by using it to identify a known analog nonlinear system from experimental data.

Identification of physiological systems: a robust method for non-parametric impulse response estimation.

The identification of non-parametric impulse response functions (IRFs) from noisy finite-length data records is analysed using the techniques of matrix perturbation theory. Based on these findings, a method for IRF estimation is developed that is more robust than existing techniques, particularly when the input is non-white. Furthermore, methods are developed for computing confidence bounds on the resulting IRF estimates. Monte Carlo simulations are used to assess the capabilities of this new method and to demonstrate its superiority over classical techniques. An application to the identification of dynamic ankle stiffness in humans is presented.

A new algorithm for the identification of multiple input Wiener systems.

Multiple-input Wiener systems consist of two or more linear dynamic elements, whose outputs are transformed by a multiple-input static non-linearity. Korenberg (1985) demonstrated that the linear elements of these systems can be estimated using either a first order input-output cross-covariance or a slice of the second, or higher, order input-output cross-covariance function. Korenberg's work used a multiple input LNL structure, in which the output of the static nonlinearity was then filtered by a linear dynamic system. In this paper we show that by restricting our study to the slightly simpler Wiener structure, it is possible to improve the linear subsystem estimates obtained from the measured cross-covariance functions. Three algorithms, which taken together can identify any multiple-input Wiener system, have been developed. We present the theory underlying these algorithms and detail their implementation. Simulation results are then presented which demonstrate that the algorithms are robust in the presence of output noise, and provide good estimates of the system dynamics under a wide set of conditions.