A new method of identifying a systems based on the minimal quadratic discrepancy criterion for biophysics problems

A wide range of biophysical systems are described by nonlinear dynamic models mathematically presented as a set of ordinary differential equations in the Cauchy explicit form: [formula: see text] Fij(X1(t),..,XN(t),t), (i = 1,...,N, j = 1,...,M), where Fij (X1(t), ..., XN(t), t) is a set of basis functions satisfying the Lipschitz condition. We investigate the problem of evaluation of model constants aij (the system identification) using experimental data about the time dependence of the dynamic parameters of the system Xi(t). A new method of system identification for the class of similar nonlinear dynamic models is proposed. It is shown that the problem of identifying an initial nonlinear model can be reduced to the solution of a system of linear equations for the matrix of the dynamic model constants [aj]i. It is proposed to determine the set of dynamic model constants aij using the criterion of minimal quadratic discrepancy for the time dependence of the set of dynamic parameters Xi(t). An important special case of the nonlinear model, the quadratic model, is considered. Test problems of identification using this method are presented for two nonlinear systems: the Van der Pol type multiparametric nonlinear oscillator and the strange attractor of Ressler, a widely known example of dynamic systems showing the stochastic behavior.     

Application of a new method of nonlinear dynamical system identification to biochemical problems

The system identification method for a variety of nonlinear dynamic models is elaborated. The problem of identification of an original nonlinear model presented as a system of ordinary differential equations in the Cauchy explicit form with a polynomial right part reduces to the solution of the system of linear equations for the constants of the dynamical model. In other words, to construct an integral model of the complex system (phenomenon), it is enough to collect some data array characterizing the time-course of dynamical parameters of the system. Collection of such a data array has always been a problem. However difficulties emerging are, as a rule, not principal and may be overcome almost without exception. The potentialities of the method under discussion are demonstrated by the example of the test problem of multiparametric nonlinear oscillator identification. The identification method proposed may be applied to the study of different biological systems and in particular the enzyme kinetics of complex biochemical reactions.     

The Model Identification for Small Unmanned Aerial Rotorcraft Based on Adaptive Ant Colony Algorithm

This paper proposes a model identification method to get high performance dynamic model of a small unmanned aerial rotorcraft.With the analysis of flight characteristics,a linear dynamic model is constructed by the small perturbation theory.Using the micro guidance navigation and control module,the system can record the control signals of servos,the state information of attitude and velocity information in sequence.After the data preprocessing,an adaptive ant colony algorithm is proposed to get optimal parameters of the dynamic model.With the adaptive adjustment of the pheromone in the selection process,the proposed model identification method can escape from local minima traps and get the optimal solution quickly.Performance analysis and experiments are conducted to validate the effectiveness of the identified dynamic model.Compared with real flight data,the identified model generated by the proposed method has a better performance than the model generated by the adaptive genetic algorithm.Based on the identified dynamic model,the small unmanned aerial rotorcraft can generate suitable control parameters to realize stable hovering,turning,and straight flight.     

Estimation of the mechanical parameters of the human respiratory system

A numerical algorithm has been developed for the estimation of the mechanical parameters of the human respiratory system. In order to estimate the pulmonary resistance and the dynamic pulmonary elastance, the transpulmonary pressure and the airflow at the mouth or nose are expanded in Chebyshev series. The nonlinear mathematical lung model and a set of measurements for airflow and pressure are then handled by the numerical technique. The lung model includes a component to account for turbulent flow in the larynx and trachea. This contribution presents an alternative method for lung parameter estimation and differs from most existing methods in that it does not need measurements for the tidal volume. It therefore eliminates the use of a body plethysmograph. The method may also find potential application to various other parameter identification problems.     

Robust dynamic experiments for the precise estimation of respiration and fermentation parameters of fruit and vegetables

Dynamic models based on non-linear differential equations are increasingly being used in many biological applications. Highly informative dynamic experiments are valuable for the identification of these dynamic models. The storage of fresh fruit and vegetables is one such application where dynamic experimentation is gaining momentum. In this paper, we construct optimal O₂ and CO₂ gas input profiles to estimate the respiration and fermentation kinetics of pear fruit. The optimal input profiles, however, depend on the true values of the respiration and fermentation parameters. Locally optimal design of input profiles, which uses a single initial guess for the parameters, is the traditional method to deal with this issue. This method, however, is very sensitive to the initial values selected for the model parameters. Therefore, we present a robust experimental design approach that can handle uncertainty on the model parameters.     

Evolutionary optimization with data collocation for reverse engineering of biological networks

MOTIVATION: Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. RESULTS: We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.     

Dynamics of the head-neck complex in response to the trunk horizontal vibration: modeling and identification

Although many studies exist concerning the influence of seat vibration on the head in the seated human body, the dynamic response of the head-neck complex (HNC) to the trunk vibration has not been well investigated. Little quantitative knowledge exists about viscoelastic parameters of the neck. In this study, the dynamics of the HNC is identified when it is exposed to the trunk horizontal (fore-and-aft) vibration. The frequency response functions between the HNC angular velocity and the trunk horizontal acceleration, corresponding to four volunteers, are obtained in the frequency range of 0.5 Hz to 10 Hz. A fourth-order mathematical model, derived by considering a double-inverted-pendulum model for the HNC, is designed to simulate the dynamic response of the HNC to the trunk horizontal vibration. The frequency domain identification method is used to determine the coefficients of the mathematical model of the HNC. Good agreement has been obtained between experimental and simulation results. This indicates that the system, similar to the designed fourth-order model, has mainly two resonance frequencies. The viscoelastic parameters of the neck, including the spring and damping coefficients, are then obtained by use of the optimization method.     

Hybrid adaptive optimal control of anaerobic fluidized bed bioreactor for the de-icing waste treatment

Hybrid adaptive control strategy was developed and tested for the degradation of propylene glycol, a major component in de-icing waste, in an anaerobic fluidized bed bioreactor (AFBR). A linearized model with time-varying parameters was first employed to describe the dynamic behavior of the AFBR using a recursive off-line system identification method. A hybrid adaptive control strategy was then tested using a recursive off-line system identification routine followed by an on-line adaptive optimal control algorithm. The objective of the controller was to achieve the desired set point value of the propionate concentration (stand-alone control output variable) by manipulating the dilution rate (control input variable). To do so, the optimal control law was developed by minimizing a cost function with constraint equations. This novel idea was successfully applied to the underlying system for 200 h. The set point (700 mg HPrl(-1)) was achieved even in the case where the feed concentration suddenly increased by 50% (9000 mg HPrl(-1) to 13500 mg HPrl(-1)).     

Chaotic dynamics of the surface potential of human skeletal muscles determined in electromyography

A new method for differential evaluation of electromyographic data on straited muscles of human lower extremities was developed. This method is based on nonlinear dynamics and thermodynamics and can be used for identification of pathologies. The distance between two trajectories of the potential of two symmetric muscles was the main measured characteristic of coordinated muscle work. These data were used to determine the Lyapunov exponent and the time of forgetting initial conditions, which reflect the generally chaotic dynamics of muscle activity. Application of the theory of deterministic chaos to analysis of electromyographic patterns can improve the diagnosis of peripheral nervous system diseases and the efficacy of treatment control. Quantitation of nonlinear dynamic parameters of muscle activity, clear data representation, high prognostic information content of the Lyapunov exponent and Kolmogorov entropy are among the advantages of the new method.     

Adaptive optimization of a continuous culture with the use of a bilevel forgetting factor

An adaptive optimization algorithm using a dynamic identification scheme with a bilevel forgetting factor (BFF) has been developed. The simulation results show superiority of this method to other methods when applied to maximize the cellular productivity of a continuous culture of baker's yeast, Saccharomyces cerievisiae. Within the limited ranges of tuning parameters tested the BFF algorithm is found to be superior in terms of initial optimization speed and accuracy and reoptimization speed and accuracy when there is an external change and long term stability (removal of "blowing up" phenomena). Algorithms tested include those based on a constant forgetting factor, an adaptive variable forgetting factor (VFF) and moving window (MW) identification.     

Regularized EM algorithm for sparse parameter estimation in nonlinear dynamic systems with application to gene regulatory network inference

Parameter estimation in dynamic systems finds applications in various disciplines, including system biology. The well-known expectation-maximization (EM) algorithm is a popular method and has been widely used to solve system identification and parameter estimation problems. However, the conventional EM algorithm cannot exploit the sparsity. On the other hand, in gene regulatory network inference problems, the parameters to be estimated often exhibit sparse structure. In this paper, a regularized expectation-maximization (rEM) algorithm for sparse parameter estimation in nonlinear dynamic systems is proposed that is based on the maximum a posteriori (MAP) estimation and can incorporate the sparse prior. The expectation step involves the forward Gaussian approximation filtering and the backward Gaussian approximation smoothing. The maximization step employs a re-weighted iterative thresholding method. The proposed algorithm is then applied to gene regulatory network inference. Results based on both synthetic and real data show the effectiveness of the proposed algorithm.     

Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models

We use a technique from engineering (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005) to investigate the algebraic identifiability of a popular three-dimensional HIV/AIDS dynamic model containing six unknown parameters. We find that not all six parameters in the model can be identified if only the viral load is measured, instead only four parameters and the product of two parameters (N and λ) are identifiable. We introduce the concepts of an identification function and an identification equation and propose the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005). We show that the newly proposed MTP method has advantages over the HOD method in the practical implementation. We also discuss the effect of the initial values of state variables on the identifiability of unknown parameters. We conclude that the initial values of output (observable) variables are part of the data that can be used to estimate the unknown parameters, but the identifiability of unknown parameters is not affected by these initial values if the exact initial values are measured with error. These noisy initial values only increase the estimation error of the unknown parameters. However, having the initial values of the latent (unobservable) state variables exactly known may help to identify more parameters. In order to validate the identifiability results, simulation studies are performed to estimate the unknown parameters and initial values from simulated noisy data. We also apply the proposed methods to a clinical data set to estimate HIV dynamic parameters. Although we have developed the identifiability methods based on an HIV dynamic model, the proposed methodologies are generally applicable to any ordinary differential equation systems.     

Feedback identification of continuous microbial growth systems

The fundamental problem of dynamic modeling of continuous culture systems for process control and optimization is addressed. Forcing a system to bifurcation via feedback control is a very promising method for model discrimination and identification. Dynamic information is obtained by using this technique, the dynamic behavior of the chemostat as predicted by unstructured models, the model with delay, and a structured model has been analyzed. The method exposes significant differences in the nonlinear dynamic structure of the various models and can be implemented to discriminate between various possible models for a continuous culture system.     

A Linear Domain System Identification for Small Unmanned Aerial Rotorcraft Based on Adaptive Genetic Algorithm

<正> This paper proposes a new adaptive linear domain system identification method for small unmanned aerial rotorcraft.Byusing the flash memory integrated into the micro guide navigation control module, system records the data sequences of flighttests as inputs (control signals for servos) and outputs (aircraft's attitude and velocity information).After data preprocessing, thesystem constructs the horizontal and vertical dynamic model for the small unmanned aerial rotorcraft using adaptive geneticalgorithm.The identified model is verified by a series of simulations and tests.Comparison between flight data and the one-stepprediction data obtained from the identification model shows that the dynamic model has a good estimation for real unmannedaerial rotorcraft system.Based on the proposed dynamic model, the small unmanned aerial rotorcraft can perform hovering,turning, and straight flight tasks in real flight tests.     

Computation of a stabilizing set of feedback matrices of a large-scale nonlinear musculoskeletal dynamic model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.     

Computation of a Stabilizing Set of Feedback Matrices bf a Large-scale Nonlinear Musculoskeletal Dynamic Model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/ output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.     

Parameter identification of the human lower limb under dynamic, transient torsional loading

The response of the lower limb to dynamic, transient torsional loading applied at the foot has been measured for a male test subject. The dynamic loading was provided by a computer controlled pneumatic system which applied single haversine (i.e. half cycle of a sine wave) axial moment pulses of variable amplitude (0-100 Nm) and duration (50-600 ms). Potentiometers measured the absolute rotations of the three leg segments. Test variables included rotation direction, weight bearing and joint flexion. Two approaches were explored for specifying parameters (i.e. inertia, damping, stiffness) of a three degree-of-freedom dynamic system model which best duplicated the measured response. One approach involved identification of linear parameters by means of optimization while the other approach entailed estimation. Parameter estimates, which included non-linear, asymmetric stiffness functions, were derived from the literature. The optimization was undertaken so as to identify parameter dependence on test variables. Results indicate that parameter values are influenced by test variables. Results also indicate that the non-linear, estimated model better approximates the experimental data than the linear, identified model. In addition to identifying parameters of a three degree-of-freedom model, parameters were also identified for a single degree-of-freedom model where the motion variable was intended to indicate the rotation of the in vivo knee. It is concluded that the simpler model offers good accuracy in predicting both magnitude and time of occurrence of peak knee axial rotations. Model motion fails to track the measured knee rotation subsequent to the peak, however.     

A computationally efficient optimisation-based method for parameter identification of kinematically determinate and over-determinate biomechanical systems

This paper introduces a general optimisation-based method for identification of biomechanically relevant parameters in kinematically determinate and over-determinate systems from a given motion. The method is designed to find a set of parameters that is constant over the time frame of interest as well as the time-varying system coordinates, and it is particularly relevant for biomechanical motion analysis where the system parameters can be difficult to accurately determine by direct measurements. Although the parameter identification problem results in a large-scale optimisation problem, we show that, due to a special structure in the linearised Karush–Kuhn–Tucker optimality conditions, the solution can be found very efficiently. The method is applied to a set of test problems relevant for gait analysis. These involve determining the local coordinates of markers placed on the model, segment lengths and joint axes of rotation from both gait and range of motion experiments.     

A discrete method for the identification of parameters of a deterministic epidemic model

We study a problem of identification of the parameters for a deterministic epidemic model of the Kermack-McKendrick type. Particular emphasis is put on the analysis of the conditions of numerical stability of the method of integration used to calculate the solutions of the system of differential equations which describe the model. The numerical method can be regarded as a discrete model which reproduces the basic qualitative properties of the continuous model, which are positivity of the solutions, points of equilibrium, and the “threshold theorem.” This allows us to identify the parameters with good reliability, by means of an iterative procedure to minimize the functional which is the measure of discrepancy between the data observed and the data obtained from the discrete model. The initial estimate of the parameters is obtained by a direct method applied to the discretized system of equations.