Per Mattsson

Periodical Solutions in a Pulse-Modulated Model of Endocrine Regulation With Time-Delay

A hybrid mathematical model of endocrine regulation obtained by augmenting the classical continuous Smith model with a pulse-modulated feedback to describe episodic (pulsatile) secretion is considered. Conditions for existence and local orbital stability of periodical solutions with m impulses in the least period ( m-cycles) are derived. An important implication of the performed analysis is that the nonlinear dynamics of the pulse-modulated system and not the delay itself cause the sustained closed-loop oscillations. Furthermore, simulation and bifurcation analysis indicate that increasing the time delay in the system in hand typically, but not always, leads to less complex dynamic pattern in the closed-loop system by giving rise to stable cycles of lower periodicity.

Finite-dimensional reducibility of time-delay systems under pulse-modulated feedback

The paper deals with further development and refinement of recent results on hybrid systems with impulsive feedback and a time delay in the continuous part. Such a closed-loop system can be considered as an impulsive and time-delayed version of the Goodwin oscillator, which is well known in mathematical biology. It also arises in mathematical modeling of non-basal regulation of endocrine systems as the pulsatile Smith model. The key property in the analysis of the system dynamics is finite-dimensional (FD) reducibility of the linear infinite-dimensional part that is studied in depth in the present paper. Under FD-reducibility of the linear continuous part, reductions of the hybrid dynamics of the considered system to a discrete and a delay-free formulation are suggested. Conditions for the existence and local stability of periodic solutions with m impulses in the least period (m-cycles) are derived.

Modeling of testosterone regulation by pulse-modulated feedback: An experimental data study

The continuous part of a hybrid (pulse-modulated) model of testosterone feedback regulation is extended with infinite-dimensional and nonlinear dynamics, to better explain the testosterone concentration profiles observed in clinical data. A linear least-squares based optimization algorithm is developed for the purpose of detecting impulses of gonadotropin-realsing hormone from measured concentration of luteinizing hormone. The parameters in the model are estimated from hormone concentration measured in human males, and simulation results from the full closed-loop system are provided.

Modeling of testosterone regulation by pulse-modulated feedback

The continuous part of a hybrid (pulse-modulated) model of testosterone (Te) feedback regulation in the human male is extended with infinite-dimensional and nonlinear blocks, to obtain the dynamics that better agree with the hormone concentration profiles observed in clinical data. A linear least-squares based optimization algorithm is developed for the purpose of detecting impulses of gonadotropin-releasing hormone (GnRH) from measured concentration of luteinizing hormone (LH). The estimated impulse parameters are instrumental in evaluating the frequency and amplitude modulation functions parameterizing the pulse-modulated feedback. The proposed approach allows for the identification of all model parameters from the hormone concentrations of Te and LH. Simulation results of the complete estimated closed-loop system exhibiting similar to the clinical data behavior are provided.

Convergence analysis for recursive Hammerstein identification

This paper derives a recursive prediction error identification method based on the Hammerstein model structure. The convergence properties of the algorithm are analysed by application of Ljungs associated differential equation method. It is proved that the algorithm can only converge to stable stationary points of the associated ordinary differential equation. General conditions for local convergence to the true parameter vector are given, and the cases with piecewise linear and polynomial nonlinearities are treated in detail. The derived identification method is illustrated in a numerical study that treats identification of a subsystem of a cement mill.

Discrete-time modeling of a hereditary impulsive feedback system

The paper deals with a broad class of hybrid systems where a linear hereditary plant with a cascade structure operates under impulsive feedback. The plant incorporates a distinct infinite-dimensional block that might be a pointwise or distributed time delay. Mathematical models that belong to this class of systems appear in mathematical biology and computational medicine. A discrete time (Poincaré) map is constructed to capture the system dynamics and investigate its periodic solutions. Simulation results indicate that the effects on the system dynamics incurred by distributed delays are quite similar to those previously observed for pointwise delays. Generally, it appears that the complexity of the nonlinear dynamics does not increase with an increasing delay value.

Recursive Identification Method for Piecewise ARX Models: A Sparse Estimation Approach

This paper deals with the identification of nonlinear systems using piecewise linear models. By means of a sparse over-parameterization, this challenging problem is turned into a convex optimization problem. The proposed method uses a likelihood-based methodology which adaptively penalizes model complexity and directly leads to a recursive implementation. In this sparse estimation approach, the tuning of user parameters is avoided, and the computational complexity is kept linear in the number of data samples. Numerical examples with both simulated and experimental data are presented and the results are compared with previously published methods.

Analysis of a pulse-modulated model of endocrine regulation with time-delay

The effects of time delay in the continuous part of a pulse-modulated model of non-basal endocrine feedback regulation are investigated. Conditions for the existence and local orbital stability of periodical solutions with m impulses in the least period (m-cycles) are derived. Applied to the case of testosterone regulation in the human male, the model indicates that increasing time delays in the system in hand typically lead to less complex dynamic pattern in the closed-loop system by giving rise to stable cycles of lower periodicity.

Recursive nonlinear-system identification using latent variables

In this paper we develop a method for learning nonlinear systems with multiple outputs and inputs. We begin by modelling the errors of a nominal predictor of the system using a latent variable framework. Then using the maximum likelihood principle we derive a criterion for learning the model. The resulting optimization problem is tackled using a majorization-minimization approach. Finally, we develop a convex majorization technique and show that it enables a recursive identification method. The method learns parsimonious predictive models and is tested on both synthetic and real nonlinear systems.

Recursive identification of Hammerstein models

The nonlinear Hammerstein model, which consists of a static nonlinear block followed by a linear dynamic block, is considered. A recursive prediction error algorithm is derived. The linear block is modelled as a single-input single-output transfer function, and the nonlinearity as a linear combination of basis functions. The case when the nonlinear block is modelled as a piecewise linear function is studied in detail. A direct computation of the gradient allows the number of estimated parameters to be minimized, a fact that is crucial when small data sets are used. Numerical examples validate the algorithm, and shows that the scheme successfully identifies a biomedical model from 200 measurements.