Convergence analysis of a continuous-time nonlinear LMS-type algorithm using stochastic calculus

In several applications least mean square (LMS) has served as a good tool for estimating the parameters of linear models but the success of continuous-time in nonlinear models has not reached its height. In this paper, we have developed a nonlinear continuous-time LMS type algorithm that estimates parameters of nonlinear systems considering the noisy input–output relationship. The nonlinear system has been assumed to be memoryless and an additive Gaussian noise component to the system has been assumed. The mean squared error between the true system output and the estimated output, when the estimated output is modeled using the same form of the nonlinear function as the original system but with the parameters unknown, is minimized using the gradient scheme with the expectation removed. The result is a least mean square algorithm for nonlinear systems. In particular, we have performed a convergence analysis of the continuous-time nonlinear LMS algorithm applied to nonlinear systems when the time step goes to zero. The resulting algorithm then behaves as a stochastic differential equation, and the standard methods of Itô calculus and Fokker–Planck theory are applied to obtain statistical properties of the mean and covariance evolution of the parameter estimates. Computer simulations corroborate the theoretical results.     

Simultaneous and nonlinear identification of mechanical and reflexproperties of human elbow joint muscles

The naturally coexisting intrinsic mechanical and reflex properties of the human elbow joint were identified simultaneously using nonlinear, time-delay, continuous-time, and dynamic models. Angular random perturbations of small amplitude and low bandwidth were applied to the joint using a computer-controlled servomotor, while the subject maintained various levels of mean background muscle torque. Joint neuromuscular dynamics were identified from the measured elbow angle and torque. Stretch reflexes were modeled nonlinearly with both dynamic and static reflex gains. A continuous-time system identification method was developed to estimate parameters of the nonlinear models directly from sampled data while retaining realistic physical or physiological interpretations. Results from six subjects showed that dynamic stretch reflex gains, joint stiffness, and viscosity generally increased with mean background muscle torque; and that dynamic stretch reflex gain was higher during muscle stretch than that during muscle shortening. More importantly, the study provided realistic simultaneous estimates of the relative contributions of intrinsic mechanical and reflex actions to net joint torque. In particular, reflexively-mediated stiffness generated a significant portion of the total joint stiffness and the percentage varied systematically with background muscle torque     

Identification and control of continuous-time nonlinear systems via dynamic neural networks

In this paper, we present an algorithm for the online identification and adaptive control of a class of continuous-time nonlinear systems via dynamic neural networks. The plant considered is an unknown multi-input/multi-output continuous-time higher order nonlinear system. The control scheme includes two parts: a dynamic neural network is employed to perform system identification and a controller based on the proposed dynamic neural network is developed to track a reference trajectory. Stability analysis for the identification and the tracking errors is performed by means of Lyapunov stability criterion. Finally, we illustrate the effectiveness of these methods by computer simulations of the Duffing chaotic system and one-link rigid robot manipulator. The simulation results demonstrate that the model-based dynamic neural network control scheme is appropriate for control of unknown continuous-time nonlinear systems with output disturbance noise.     

Stochastic theory of continuous-time state-space identification

This paper presents theory, algorithms, and validation results for system identification of continuous-time state-space models from finite input-output sequences. The algorithms developed are methods of subspace model identification and stochastic realization adapted to the continuous-time context. The resulting model can be decomposed into an input-output model and a stochastic innovations model. Using the Riccati equation, we have designed a procedure to provide a reduced-order stochastic model that is minimal with respect to system order as well as the number of stochastic inputs, thereby avoiding several problems appearing in standard application of stochastic realization to the model validation problem     

Piecewise linear implementation of nonlinear dynamical systems: from theory to practice

It is shown how to implement by a mixed-signal circuit a continuous-time dynamical system. The chosen case study is the Hindmarsh-Rose model of a biological neuron, but the design strategy can be applied to a large class of continuous-time nonlinear dynamical systems. The system nonlinearities are first approximated by using piecewise-linear functions and then digitally implemented on a field programmable gate array. The linear part of the system is completely analogue and is implemented by using operational amplifiers. Measurement results show that the circuit can reproduce the main dynamics of a biologically plausible neuron.     

A least-squares parameter estimation algorithm for switched hammerstein systems with applications to the VOR

A "Multimode" or "switched" system is one that switches between various modes of operation. When a switch occurs from one mode to another, a discontinuity may result followed by a smooth evolution under the new regime. Characterizing the switching behavior of these systems is not well understood and, therefore, identification of multimode systems typically requires a preprocessing step to classify the observed data according to a mode of operation. A further consequence of the switched nature of these systems is that data available for parameter estimation of any subsystem may be inadequate. As such, identification and parameter estimation of multimode systems remains an unresolved problem. In this paper, we 1) show that the NARMAX model structure can be used to describe the impulsive-smooth behavior of switched systems, 2) propose a modified extended least squares (MELS) algorithm to estimate the coefficients of such models, and 3) demonstrate its applicability to simulated and real data from the Vestibulo-Ocular Reflex (VOR). The approach will also allow the identification of other nonlinear bio-systems, suspected of containing "hard" nonlinearities.     

A method for dynamics identification for haptic display of the operating feel in virtual environments

Realistic dynamics models are important for haptic display for virtual reality systems. Such dynamic models are desirably obtained via experimental identification. However, traditional dynamics identification methods normally require large sized training data sets, which maybe difficult to meet in many practical applications. To obtain the dynamics models, we present, in this paper, an identification method using support vector machines regression algorithm which is more effective than traditional methods for sparse training data. This method can be used as a generic learning machine or as a special learning technique that can make full use of the available knowledge about the dynamics structure. The experimental results show the application of our method for identifying friction models for haptic display.     

Nonlinear Modeling of DC/DC Converters Using the Hammerstein's Approach

This paper deals with the modelling of highly nonlinear switching power-electronics converters using black-box identification methods. The duty cycle and the output voltage are chosen, respectively, as the input and the output of the model. A nonlinear Hammerstein-type mathematical model, consisting of a static nonlinearity and a linear time-invariant model, is considered in order to cope with the well-known limitations of the more common small-signal models, i.e. the entity of the variations of the variables around a well-defined steady-state operating point and the incorrect reproduction of the steady-state behavior corresponding to input step variations from the above steady-state operating point. The static nonlinearity of the Hammerstein model is identified from input-output couples measured at steady state for constant inputs. The linear model is identified from input-output data relative to a transient generated by a suitable pseudorandom binary sequence constructed with two input values used to identify the nonlinearity. The identification procedure is, first, illustrated with reference to a boost DC/DC converter using results of simulations carried out in the PSpice environment as true experimental results. Then, the procedure is experimentally applied on a prototype of the above converter. In order to show the utility of the Hammerstein models, a PI controller is tuned for a nominal model. Simulation and experimental results are displayed with the aim of showing the peculiarities of the approach that is followed.     

Identification of electrically stimulated quadriceps muscles inparaplegic subjects

This work establishes a method for the noninvasive in vivo identification of parametric models of electrically stimulated muscle in paralyzed individuals, when significant inertial loads and/or load transitions are present. The method used differs from earlier work, in that both the pulse width and stimulus period (interpulse interval) modulation are considered. A Hill-type time series model, in which the output is the product of two factors (activation and torque-angle) is used. In this coupled model, the activation dynamics depend upon velocity. Sequential nonlinear least squares methods are used in the parameter identification. The ability of the model, using identified time-varying parameters, to accurately predict muscle torque outputs is evaluated, along with the variability of the identified parameters. This technique can be used to determine muscle parameter models for biomechanical computer simulations, and for real-time adaptive control and monitoring of muscle response variations such as fatigue     

Model conversions of uncertain linear systems using a scaling and squaring geometric series method

This paper proposes a scaling and squaring geometric series method along with the inverse-geometric series method for finding discrete-time (continuous-time) structured uncertain linear models from continuous-time (discrete-time) structured uncertain linear systems. The above methods allow the use of well-developed theorems and algorithms in the discrete-time (continuous-time) domain to indirectly solve the continuous-time (discretetime) domain problems. Moreover, these methods enhance the flexibility in modeling and control of a hybrid composite system. It has been shown that the commonly used bilinear approximation model is a specific class of the proposed geometric series model.     

Nonparametric identification of nonlinearities in block-oriented systems by orthogonal wavelets with compact support

The paper addresses the problem of identification of nonlinear characteristics in a certain class of discrete-time block-oriented systems. The systems are driven by random stationary white processes (independent and identically distributed input sequences) and disturbed by stationary, white, or colored random noise. The prior information about nonlinear characteristics is nonparametric. In order to construct identification algorithms, the orthogonal wavelets of compact support are applied, and a class of wavelet-based models is introduced and examined. It is shown that under moderate assumptions, the proposed models converge almost everywhere (in probability) to the identified nonlinear characteristics, irrespective of the noise model. The rule for optimum model-size selection is given and the asymptotic rate of convergence of the model error is established. It is demonstrated that, in some circumstances, the wavelet models are, in particular, superior to classical trigonometric and Hermite orthogonal series models worked out earlier.     

Stable identification of nonlinear systems using neural networks: theory and experiments

This paper presents an approach for stable identification of multivariable nonlinear system dynamics using a multilayer feedforward neural network. Unlike most of the previous neural network identifiers, the proposed identifier is based on a nonlinear-in-parameters neural network (NLPNN). Therefore, it is applicable to systems with higher degrees of nonlinearities. Both parallel and series-parallel models are used with no a priori knowledge about the system dynamics. The method can be considered both as an online identifier that can be used as a basis for designing a neural network controller as well as an offline learning scheme for monitoring the system states. A novel approach is proposed for the weight updating mechanism based on the modification of the backpropagation (BP) algorithm. The stability of the overall system is shown using Lyapunov's direct method. To demonstrate the performance of the proposed algorithm, an experimental setup consisting of a three-link macro-micro manipulator (M/sup 3/) is considered. The proposed approach is applied to identify the dynamics of the experimental robot. Experimental and simulation results are given to show the effectiveness of the proposed learning scheme.     

Modeling and Identification of Elastic Robot Joints With Hysteresis and Backlash

This paper presents a novel approach to the modeling and identification of elastic robot joints with hysteresis and backlash. The model captures the dynamic behavior of a rigid robotic manipulator with elastic joints. The model includes electromechanical submodels of the motor and gear from which the relationship between the applied torque and the joint torsion is identified. The friction behavior in both presliding and sliding regimes is captured by generalized Maxwell-slip model. The hysteresis is described by a Preisach operator. The distributed model parameters are identified from experimental data obtained from internal system signals and external angular encoder mounted to the second joint of a 6-DOF industrial robot. The validity of the identified model is confirmed by the agreement of its prediction with independent experimental data not previously used for model identification. The obtained models open an avenue for future advanced high-precision control of robotic manipulator dynamics.     

Nonparametric identification of population models: an MCMC approach

The paper deals with the nonparametric identification of population models, that is models that explain jointly the behavior of different subjects drawn from a population, e.g., responses of different patients to a drug. The average response of the population and the individual responses are modeled as continuous-time Gaussian processes with unknown hyperparameters. Within a Bayesian paradigm, the posterior expectation and variance of both the average and individual curves are computed by means of a Markov Chain Monte Carlo scheme. The model and the estimation procedure are tested on both simulated and experimental pharmacokinetic data.     

Nonlinear parametric identification of magnetic bearings

This paper proposes a new electromagnetic force model and its parameter identification method. As a case study, the parameters of the proposed model for an experimental electromagnetic bearing system are obtained using extended Kalman filter (EKF). The experimental setup includes a symmetric rigid rotor which is disturbed by the electromagnet of a magnetic bearing. Experimental results show that the system response to harmonic excitation includes super-harmonic terms which are not shown by the well-known conventional electromagnetic force model. This shortcoming necessitates an investigation to propose a more realistic electromagnetic force model. Based on the observations of the system response, a novel parametric model is presented in the form of a nonlinear Mathieu–Duffing equation with unknown coefficients. Then in the operating frequency range, a random input is synthesized and applied to the experimental system as a persistent excitation and the response of the system is recorded. In order to estimate the states and parameters of the model, the EKF method has been applied to the recorded input–output data. To validate the identification results the outputs of estimated and experimental models are compared in time and frequency domains. The results show a notable improvement in modeling of magnetic force. The proposed model and the method for identifying its parameters are applicable for all magnetic fields.     

All-digital impulse radio with multiuser detection for wireless cellular systems

Impulse radio is an ultrawideband system with attractive features for baseband asynchronous multiple-access, multimedia services, and tactical wireless communications. Implemented with analog components, the continuous-time impulse radio multiple-access model utilizes pulse-position modulation and random time-hopping codes to alleviate multipath effects and suppress multiuser interference. We introduce a novel continuous-time impulse radio transmitter model and deduce from it an approximate one with lower complexity. We also develop a time-division duplex access protocol along with orthogonal user codes to enable impulse radio as a radio link for wireless cellular systems. Relying on this protocol, we then derive a multiple-input/multiple-output equivalent model for full continuous-time model and a single-input/single-output model, for the approximate one. Based on these models, we finally develop design composite linear/nonlinear receivers for the downlink. The linear step eliminates multiuser interference deterministically and accounts for frequency-selective multipath while a maximum-likelihood receiver performs symbol detection. Simulations are provided to compare performance of the different receivers.     

Identification of linear continuous-time multivariable systems using linear multistep methods

It is shown that linear multistep methods, commonly used to integrate ordinary differential equations numerically, can be used for the identification of linear continuous-time multivariable systems from samples of input/output data. An example problem is given to illustrate the use of multistep methods in system identification, and the results are compared with those obtained by the use of a discrete-time model.<>     

Identification of Continuous-Time Systems Using Instrumental Variables with Application to an Industrial Robot

A new method of constructing instrumental variables for identification is introduced. Its usefulness in the identification of continuous-time systems is investigated. The technique is then applied for modeling the arm of an industrial robot used for welding purposes. Results showed that the proposed method of using instrumental variables is computationally simple and at the same time gives better performance in the presence of measurement noise as compared to existing methods.     

Discrete-Time Sliding-Mode State and Unknown Input Estimations for Nonlinear Systems

This paper addresses the problems of discrete-time state and unknown input/fault estimation for continuous-time nonlinear systems with multiple unknown inputs. Taylor series expansion and a nonlinear transformation are used to convert the nonlinear continuous-time system into a discrete-time model. The conditions for the observability of unknown inputs w.r.t. outputs are discussed. The novelty lies in the formulation of multiple sliding-mode estimator for the states that are directly influenced by unknown inputs, which cannot be decoupled by nonlinear transformation. This framework allows for the estimation of unknown inputs from the multiple sliding modes. The existence of discrete-time sliding mode is guaranteed, and the relation between the boundary layer thickness and the sliding-mode gain design that will eliminate chattering and the boundedness conditions is obtained. The proposed technique can be applied for fault detection and isolation. Simulation results with application to three-phase motor are given to demonstrate the effectiveness of the proposed method.      

机敏约束层阻尼结构动态模型辨识

针对机敏约束层阻尼(SCLD)非线性系统的动力学建模问题,系统辨识是一种简便有效的方法.该文以NARX网络表征待辨识模型,并采用串并联与并联相结合的方法训练网络,根据实验数据辨识出非线性系统的动力学模型.通过对SCLD薄板结构外扰通道和控制通道的建模研究,证明了NARX网络良好的辨识性能及该文研究方法的正确性.为进一步验证该文建模方法的有效性和可行性,将NARX网络用于SCLD复杂车厢结构的动态模型辨识,并取得了较满意的效果.