Linear Transfer Function Model Research Articles

Nonlinear modelling and control of electrically stimulated muscle: A local model network approach

Human muscle can be made to contract by electrical stimulation. There are many well established uses for this technique and new applications are being investigated. We are interested in two uses: restoration of function to paralysed limbs and reconfiguration of one of the back muscles to assist a failing heart. Muscle modelling has been widely studied in the past. Model types range from biophysical models, which are based on the structure and processes of actual muscles, through analogue models, to purely mathematical descriptions. The latter are derived only from the input (neural activation) and the output (mechanical response) signals. We are searching for mathematical models, which firstly can represent the complex nonlinear behaviour of muscle and secondly are controller oriented, that is that controller design based on such models is possible. We have developed a number of dynamic muscle models based on force measurements taken from isometric contracting muscles stimulated by an irregular pulse train. The results of experimental investigations have shown that a simple linear transfer-function model of muscle response can be rather inaccurate over the full operational range. The muscle response is significantly nonlinear with respect to the level of stimulation. However, the experiments with linear system identification provide results which can be used to choose structure parameters of a dynamic model. Nonlinear models based on local model networks (LMNs) are able to capture the nonlinear effects and to provide accuracy over a wide operational range. We investigate the use of an LMN with local linear models, describe the type of network model used, the training algorithm, and show some typical experimental results based on real measured data. We address the synthesis of feedback controllers for physical muscle. The simple linear transfer function model forms the basis for the design of a robust linear controller. The structure of the LMN derived previously is exploited as the basis of a non-linear control design: the local controller network. We compare the performance of both controller designs in simulation experiments.

Transfer Function Identification of an Electro-Rheological Actuator

A fluid clutch utilising an Electro-Rheological (ER) suspension provides a controlled torque coupling between input and output through the control of the applied electric field. If the input is driven at constant speed the device can be considered as an ER torque actuator and thus be used to drive robot links or other mechanisms requiring precise positioning. Such an ER torque actuator can replace a DC servo-motor in robotic applications with the benefits of low time constant and smooth output torque unaffected by cogging (i.e. variation in torque of a DC motor as the magnetic reluctance of the armature-stator path changes with rotation). Although the ER actuator has many benefits, it suffers from a non-linear and time varying relationship between input voltage and output torque. These undesirable characteristics can be mitigated by providing a local closed loop controller around the system. The design of such a controller requires a knowledge of the relationship between the applied voltage and output torque; i.e. the transfer function of the actuator. This transfer function has been determined by observing the response of an ER torque actuator in the frequency domain. It is shown that a linear transfer function model reasonable represents the actuator behaviour, that the actuator is a stable second order system and that the time constant of the clutch studied is sufficiently short to hold considerable promise for robotic applications. Furthermore, the maximum torque capability is shown to be sufficient for many medium scale industrial robots.

Statistical estimation of runoff characteristics of watersheds in central Chile

Abstract Estimation of monthly runoff statistical properties, such as monthly means and variances, is usually needed to design and evaluate water resource systems. If no local recorded data are available, a transfer of information through different alternative procedures can be used. In this paper, the use of linear Transfer Function (TF) models with precipitation series as inputs is proposed to estimate statistical properties of the resulting runoff series. Empirical relationships based on data from watersheds in the mountainous zone of central Chile are suggested to estimate parameters of low-order TF models and some of their properties.

Modelling the effects of spatial variability in rainfall on catchment response. 2. Experiments with distributed and lumped models

Computational experiments are carried out with the rainfall field model described in Part 1 of this paper and a physically based distributed modelling system, the Système Hydrologique Européen (SHE), to explore the interaction between spatial variability in rainfall and other factors controlling catchment response; both models have been calibrated for the small upland Wye catchment (area 10.55 km 2). Simulated rainfall fields are used to provide fully distributed rainfall inputs to the SHE; the corresponding ‘true’ catchment responses are then compared with those derived from incomplete sampling of the rainfall fields. The differences in simulated peak discharges and runoff volumes are assessed as a function of antecedent catchment conditions, network density and the level of spatial correlation in the rainfall input. A piecewise linear transfer function model with an averaged rainfall input is used to approximate SHE responses to fully distributed rainfall inputs, to provide insight into simple lumped model performance.

Water supply in monsoonal Asia: modelling and predicting small tank storage

Traditionally, small dams (tanks) have been both centres of village settlements and the source of irrigation in Asia. Being solely dependent on local rainfall characteristics and other climatic factors, the water storage behaviour closely follows the rainfall pattern. Carry-over of storage from year to year is rare because of natural evaporation and withdrawals for irrigation. The efficient allocation of storage, within any year, is analogous to the optimization of a time-varying exhaustible resource stock. This study attempts to model water storage of dams using weekly data with a view to intrayear optimization of storage. The behaviour of storage is approximated by a dynamic system driven by rainfall distribution and temperature, and characterized by a linear transfer function model following modification of rainfall to allow for losses. The order of the linear part of the model is identified and parameters estimated for three tanks in Sri Lanka using a recursive estimation procedure. This model can be used for the prediction of storage for a given rainfall scenario. This is demonstrated for past realistic rainfall experiences using stochastic simulation of storage.

A New Approach to Estimation of Signal Components for Fault Detection

A new algorithm for estimating the fundamental and harmonic components during normal and high impedance fault contents in a power system is proposed in this paper. The approach involves the use of a linear transfer function model for the fault induced signals embedded in noise, excited by a unit impulse input. A set of linear prediction equations is then formed from this model and is solved using partial total least squares approach, to extract the time varying fundamental and harmonic components during a high impedance fault on a power system. The proposed scheme is suitable for fault induced time varying signals with large noise input such as high impedance faults. Simulation results for the field recorded time varying waveforms are presented to illustrate this approach.

Decomposition of groundwater level fluctuations using transfer modelling in an area with shallow to deep unsaturated zones

Time series analysis of the fluctuations in shallow groundwater levels in the Netherlands lowlands have revealed a large-scale decline in head during recent decades as a result of an increase in land drainage and groundwater withdrawal. The situation is more ambiguous in large groundwater bodies located in the eastern part of the country, where the unsaturated zone increases from near zero along the edges to about 40 m in the centre of the area. As depth of the unsaturated zone increases, groundwater level reacts with an increasing delay to fluctuations in climate and influences of human activities. The aim of the present paper is to model groundwater level fluctuations in these areas using a linear stochastic transfer function model, relating groundwater levels to estimated precipitation excess, and to separate artificial components from the natural groundwater regime. In this way, the impact of groundwater withdrawal and the reclamation of a 1000 km 2 polder area on the groundwater levels in the adjoining higher ground could be assessed. It became evident that the linearity assumption of the transfer functions becomes a serious drawback in areas with the deepest groundwater levels, because of non-linear processes in the deep unsaturated zone and the non-synchronous arrival of recharge in the saturated zone. Comparison of the results from modelling the influence of reclamation with an analytical solution showed that the lowering of groundwater level is partly compensated by reduced discharge and therefore is less than expected.

Modelling, simulation and control of a crude oil preheating furnace

The energy necessary for the first fractionation of crude oil is supplied by a furnace which preheats and partially vaporizes the atmospheric column feed. This paper describes a dynamic nonlinear mathematical model for a typical preheating furnace. As a first approximation, all the crude volatile fractions are lumped into one phase, represented by n-decane. Analysis of the time scales associated with the furnace components leds to the conclusion that the long-term system dynamics are dominated by the thermal accumulation in the furnace walls, while the significant faster modes are contributed by thermal, mass and momentum accumulation in the flowing crude oil. The model is used principally to account for the effect of variations in the furnace fuel flow rate and the influence of changes in the feed temperature and feed flow rate on the furnace effluent temperature. Using the model, dynamic open-loop simulations are performed, and approximations based on parallel linear transfer function models are fitted to the responses. A robust feedback controller is designed to ensure that the exit temperature tracks its set point, on the basis of the variations in the parameters of the linear model relating fuel flow to the furnace exit temperature. Feedforward controllers are computed in order to gauge the level of disturbance rejection improvement possible. It is shown that both disturbances investigated can be eliminated within a reasonable time by manipulating the fuel fed to the burners. Comparing the combination feedforward-feedback control to feedback alone reveals that adding feedforward action is advantageous only in rejecting the temperature disturbance. This is because the significant dead time which precedes the effect of this disturbance on the effluent temperature gives the feedforward controller ample time to act.

Modeling and Control of Continuous Melt Polycondensation Reactors for the Synthesis of Thermoplastic Polymers

Abstract The dynamic models for continuous melt polycondensation of thermoplastic polyesters (poly(ethylene terephthalate)) have been developed. The prepolymerization reactor is a CSTR and the finishing polymerization reactor is a horizontal screw type reactor. A detailed kinetic model has been incorporated into these models for the prediction of polymer molecular weight, side product concentrations, and functional end group concentrations. A linear transfer function model has been developed for the prepolymerization reactor and steady state interactions between input and output variables are analyzed. The dynamic simulations of the finishing polymerization reactor indicate that variations in the feed prepolymer properties have significant effect on the final polymer properties.

A critical study of multivariable self‐tuning algorithms for distillation control

AbstractA control algorithm which has been acclaimed as the best algorithm for a real system may not be the best algorithm for a different real system. Therefore, various self‐tuning algorithms for real distillation columns have been evaluated, in order to compare their performances. A variable forgetting factor algorithm is modified using a filter which permits the employment of one instead of two covariance matrices for distillation control. A cautious self‐tuning control of SISO system is extended to MIMO system of distillation control. Multivariable self‐tuning regulator, multivariable self‐tuning controller and multivariable cautious self‐tuning controller are implemented with modified variable forgetting factor for linear transfer function model, Waller et al. column, and rigorous non‐linear model, Wood and Berry column. For distillation control, a multivariable cautious self‐tuning algorithm with modified variable forgetting factor is much simpler than earlier reported algorithms. This has produced better results and demonstrated its effectiveness, even in the presence of noise when other adaptive controllers give unsatisfactory performance.

Identification of a Frequency Response Model of Joint Rotation

A second order, linear oscillator transfer function model is fit to the measured transfer function relating the abduction-adduction rotation of the first finger to the applied moment. Nearly constant isometric contractions of the first palmar and dorsal interossei are maintained by the subjects during the measurements. The stiffness and damping components of the identified models increase significantly with increasing isometric contraction when compared to those recorded under relaxed contraction. Muscle fatigue causes the natural frequency, damping ratio and stiffness of the joint rotation to decrease under full isometric contraction, and it causes the natural frequency and stiffness to increase when the muscles are relaxed.

A simple method for the estimation of rational distributed lag models

In an earlier paper we suggested a method for the identification and estimation of linear transfer function models. The method was claimed to be especially suitable for polynomial transfer function models. In this paper we shall consider the case of rational transfer function models (distributed lag models) in more detail. A simple method for the estimation of the parameters of multiple input rational distributed lag models is suggested. The method is based on simple linear identities that the parameters always fulfill. The asymptotic distribution of the proposed estimator is derived. Two illustrative examples of the use of the new method are given.

Multiple-model adaptive control of blood pressure using sodium nitroprusside.

This paper presents a multiple-model adaptive control procedure for sodium nitroprusside regulation of arterial pressure. Pole-placement, via state-variable feedback, is included in the controller to achieve desired performance characteristics. A Smith predictor in the controller effectively removes the infusion delay time, thus simplifying the control analysis and design. Proportional-plus-integral control is used to achieve zero steady-state error. Computer simulations on linear transfer function models with varying gains, time constants, and infusion delays demonstrate the robustness of this multiple-model controller. Additional simulations on nonlinear, pulsatile-flow cardiovascular models lend further support to the ultimate use of this controller for blood-pressure regulation.

Analysis of community interactions using linear transfer function models

Interactions among populations simulated with several ecosystem models and the interactions among populations in a sealed aquatic microcosm were analyzed using linear transfer function models. A transfer function is composed of the sum of three components: a deterministic transfer function ( F), a stochastic input function ( G), and a white-noise error time series ( e). The model has the form: y(t) = F{x(t), y ̂ (t−1)} + G{n(t), r ̂ (t−1)} + e(t) , where y( t) is an output time series, x( t) is an input time series, ^ y( t−1) is the estimate of y( t−1) based on F alone, n( t) is an unobserved white-noise time series, and r ̂ (t−1) is the estimate of the residual series r(t−1) = y(t−1) − y ̂ (t−1) based on G alone. The deterministic transfer function accounts for the pattern in the output time series that is attributable to the interaction between the input and output variables. The stochastic input function accounts for the remaining pattern in the output time series. The deterministic portion of these simple linear models could recover most of the pattern in complex modeled systems ( r-square generally greater than 0.9). Deterministic interactions were not as readily detected in the microcosm although a significant portion of the pattern could often be recovered with the stochastic input function. The ability to detect relationships among variables in modeled systems but not in the living system suggests that may extant ecological models may be poorly posed. Temporal correlation between variables is a necessary (but not sufficient) condition for a mechanistic model based on the interaction between these variables to be successful. The transfer function estimation procedure, therefore, can be a useful tool for identifying potentially important interactions in poorly understood systems before more mechanistic models are developed.

Analysis of community interactions using linear transfer function models : Rastetter, E.B., 1987. Ecol. Model., 36(1–2):101–117

Analysis of community interactions using linear transfer function models : Rastetter, E.B., 1987. Ecol. Model., 36(1–2):101–117

Time series analysis of daily growth in Dicentrarchuslabrax L. otoliths

Variations in the thickness of daily growth increments in the otoliths ( n = 40) of sea bass ( Dicentrarchus labrax L.) hatched and reared in the laboratory and ranging in age from 2 to 116 days were studied. The complete increment series and three partial series corresponding to distinct stages of development were also analysed by means of time series. The relationships between otolith growth and water temperature fluctuations during the period studied were analysed using transfer functions. In the experimental conditions the major uncontrolled external variable affecting growth was water temperature, observed daily at noon. Time series analysis of the daily increments was carried out by estimated auto-correlation and partial autocorrelation functions. An IMA model was identified, estimated, and verified. The bivariate time series process — water temperature and otolith growth increments — was analyzed by means of a linear transfer function model. The examination of the cross-correlation function showed dependence between both series. The influence of water temperature on otolith growth was adequately described by an undelayed third-order transfer function model, with a steady state of −0.47μm·day −1·°C −1. An updating forecast of otolith growth was made using the transfer function model. As otolith size and body length were closely related, the predicted values of otolith growth could be used for body growth estimations. This paper presents evidence suggesting that otolith growth in sea bass is a conservative process, with growth on any given day depending upon growth and the disturbances occurring on the day of deposition, ( t, and the day before, t − 1. Water temperature exerted an effect on the growth the same day, and thereafter the effect decreased exponentially every 3 days. Multidaily growth patterns were not present in sea bass otoliths. This lack of periodicity is probably related to the lack of periodic cycles in water temperature.

Interpretation of the coherence function when using pseudorandom inputs to identify nonlinear systems.

Use of certain types of pseudorandom input stimuli to identify linear transfer function models of biological or physiological systems can lead to bias errors if the system is nonlinear. Moreover, use of the coherence function to assess the linearity of the system may yield misleading results. Results from a computer simulation are presented, along with example experimental data.

Selecting the best linear transfer function model

Statistics currently used to determine the best model order based on the covariance of the parameter estimates or the instrumental product-moment matrix are critically examined, particularly for their applicability to situations where input and output units differ or where there is a large or small steady-state gain. A normalization process is advocated which extends the usefulness and reliability of the statistics in these situations. A procedure for comparing determinant statistics for models of different orders is described, which simplifies current methods. The improved procedures and statistics are compared using simulated data and an instrumental variable estimation method, confirming their worth.

A Two-Step Least Squares Parameter Estimation Method: Test Case Examples

A method for the parameter estimation of linear discrete time transfer function models is described and examples of its performance demonstrated. The data used in the examples is test case data proposed for the IFAC Symposium, The Hague, 1973 and the IFAC Congress, Boston, 1975. The 2-step method is shown to give improved estimates over other known results for these test cases.

Compensatory tracking performance of children in terms of linear operator theory

Two groups of children of ages 8 and 12 respectively and a group of young adults were trained at a continuous visual-input manual-output tracking task. Performance was analysed in terms of conformity to a linear transfer function model and the McRuer crossover model. The operator linearity and operational characteristics of the linear model showed a considerable improvement with age. The pure time delay of the crossover model showed a monotonic decrease with increasing age, and an attempt was made to relate this to developmental changes in reaction time. The age dependence of control stability was also considered.