Alexander N. Churilov

Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback

A parsimonious mathematical model of pulse modulated regulation of non-basal testosterone secretion in the male is developed. The model is of the third differential order, reflecting the three most significant hormones in the regulation loop, but is yet shown to be capable of sustaining periodic solutions with one or two pulses of gonadotropin-releasing hormone (GnRH) in each period. Lack of stable periodic solutions is otherwise a main shortcoming of existing low-order hormone regulation models. Existence and stability of periodic solutions are studied. The periodic mode with two GnRH pulses in the least period has not been described in medical literature, but is found to explain experimental data well.

Feedback Kalman-Yakubovich lemma and its applications to adaptive control

In this paper we give a survey of results related to the feedback Kalman-Yakubovich lemma (FKYL) giving necessary and sufficient solvability conditions for some class of bilinear matrix inequalities or conditions of feedback passivity of linear systems. Applications to adaptive and variable structure control systems are also discussed.

Bifurcation phenomena in an impulsive model of non-basal testosterone regulation

Complex nonlinear dynamics in a recent mathematical model of non-basal testosterone regulation are investigated. In agreement with biological evidence, the pulsatile (non-basal) secretion of testosterone is modeled by frequency and amplitude modulated feedback. It is shown that, in addition to already known periodic motions with one and two pulses in the least period of a closed-loop system solution, cycles of higher periodicity and chaos are present in the model in hand. The broad range of exhibited dynamic behaviors makes the model highly promising in model-based signal processing of hormone data.

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.

Brief paper: A state observer for continuous oscillating systems under intrinsic pulse-modulated feedback

A static gain observer for linear continuous plants with intrinsic pulse-modulated feedback is analyzed. The purpose of the observer is to asymptotically drive the state estimation error to zero and synchronize the sequence of pulse modulation instants estimated by the observer with that of the plant. Conditions on the observer gain matrix locally stabilizing the observer error along an arbitrary periodic plant solution are derived and illustrated by simulation for the case of pulsatile testosterone regulation.

An impulse-to-impulse discrete-time mapping for a time-delay impulsive system

It is shown that an impulsive system with a time-delay in the continuous part can be equivalently represented by discrete dynamics under less restrictive conditions on the time-delay value than considered previously.

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.

Frequency methods in the theory of pulse-modulated control systems

The frequency methods of studying stability of the nonlinear control systems with various kinds of pulse modulation were reviewed.

Hybrid State Observer with Modulated Correction for Periodic Systems under Intrinsic Impulsive Feedback

Abstract A new observer structure for hybrid systems comprised of a linear time-invariant subsystem and immeasurable (intrinsic) impulsive feedback is suggested. It brings about significant improvements in the state estimation error settling time in comparison with that of an earlier considered static feedback observer by making use of the weighted output estimation error in the calculation of the discrete part of the observer.

Periodic Modes and Bistability in an Impulsive Goodwin Oscillator with Large Delay

Abstract The impact of time delay on the dynamics of a hybrid model of pulsatile feedback endocrine regulation is investigated. The model in hand can be seen as an impulsive and delayed version of the popular in computational biology Goodwin oscillator, where the feedback is implemented by means of pulse modulation. The value of the time delay is related to the duration of the time interval between the firing times of the feedback impulses. Under the assumption of a cascade structure of the continuous part of the model, the hybrid dynamics of the closed-loop system are shown to be governed by a discrete mapping propagating through the firing times of the impulsive feedback. Conditions for existence and stability of periodic solutions of the model are obtained. Bifurcation analysis of the mapping reveals the phenomenon of bistability arising for larger time delay values but not observed for the smaller ones.