Papers by Alexander Churilov
Frequency Stability Criterion for Nonlinear Pulse Systems with Even Pulsation Law
Complex Dynamics and Hidden Attractors in Delayed Impulsive Systems
Emergence, Complexity and Computation, 2021
IFAC-PapersOnLine, 2018
An impulsive counterpart of the Goodwin biological oscillator with three discrete delays is consi... more An impulsive counterpart of the Goodwin biological oscillator with three discrete delays is considered. An impulse-to-impulse discrete map that captures dynamics of the impulsive delayed model is constructed.
Discrete-Time Mapping for an Impulsive Goodwin Oscillator with Three Delays
International Journal of Bifurcation and Chaos, 2017
A popular biomathematics model of the Goodwin oscillator has been previously generalized to a mor... more A popular biomathematics model of the Goodwin oscillator has been previously generalized to a more biologically plausible construct by introducing three time delays to portray the transport phenomena arising due to the spatial distribution of the model states. The present paper addresses a similar conversion of an impulsive version of the Goodwin oscillator that has found application in mathematical modeling, e.g. in endocrine systems with pulsatile hormone secretion. While the cascade structure of the linear continuous part pertinent to the Goodwin oscillator is preserved in the impulsive Goodwin oscillator, the static nonlinear feedback of the former is substituted with a pulse modulation mechanism thus resulting in hybrid dynamics of the closed-loop system. To facilitate the analysis of the mathematical model under investigation, a discrete mapping propagating the continuous state variables through the firing times of the impulsive feedback is derived. Due to the presence of mult...
A Review of Research on Dynamics of Impulsive Control Systems: the Impact of V. A. Yakubovich∗∗The authors were partly financed by by Saint Petersburg State University, Grant 6.38.230.2015. AG and AC were partly supported by the Russian Foundation for Basic Research, Grant 14-01-00107-a
IFAC-PapersOnLine, 2015
Abstract The paper gives a brief retrospective review of some research on impulsive control syste... more Abstract The paper gives a brief retrospective review of some research on impulsive control systems that was conducted at the Department of Theoretical Cybernetics of Saint Petersburg State University in 1960s—2010s. Until 2012 the department was headed and guided by V.A. Yakubovich.
Discrete-time map for an impulsive Goodwin oscillator with a distributed delay
Mathematics of Control, Signals, and Systems, 2016
A system of impulsive integro-differential equations representing a hybrid counterpart of the ren... more A system of impulsive integro-differential equations representing a hybrid counterpart of the renown Goodwin oscillator is considered. The continuous part of the system possesses a cascade structure and contains a distributed bounded delay. The impulses impacting the continuous part are modulated in amplitude and frequency by the continuous output thus implementing an impulsive feedback. This kind of mathematical models appears in mathematical biology and computational medicine. Applying a version of the linear chain trick, it is demonstrated that a discrete-time (Poincaré) map can be constructed to capture the main dynamical properties of the system in hand.
Delay-induced dynamical phenomena in impulsive Goodwin's oscillator: What we know so far
2015 54th IEEE Conference on Decision and Control (CDC), 2015
Impulsive Goodwin's oscillator model is introduced to capture the dynamics of sustained perio... more Impulsive Goodwin's oscillator model is introduced to capture the dynamics of sustained periodic processes in endocrine systems controlled by episodic pulses of hormones. The model is hybrid and comprises a continuous subsystem describing the hormone concentrations operating under a discrete pulse-modulated feedback implemented by firing neurons. Time delays appear in mathematical models of endocrine systems due to the significant transport phenomena but also because of the time necessary to produce releasable hormone quantities. From a biological point of view, the neural control should be robust against the time delay to ensure the loop functionality over a wide range of inter-individual variability. The paper provides an overview of the currently available results and contributes a generalization of a Poincaré mapping approach to study complex dynamics of impulsive Goodwin oscillator. Both pointwise and distributed time delays are considered in a general framework based on the Poincaré mapping. Bifurcation analysis is utilized to illustrate the analytical results.
Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay
The European Physical Journal Special Topics, 2015
ABSTRACT
Design Degrees of Freedom in a Hybrid Observer for a Continuous Plant under an Intrinsic Pulse-modulated Feedback∗∗A. Medvedev and D. Yamalova are in part financed by the European Research Council, Advanced Grant 247035 (SysTEAM) and Grant 2012-3153 from the Swedish Research Council. D. Yamalova ...
IFAC-PapersOnLine, 2015
Abstract A hybrid observer for a linear time-invariant continuous plant under an intrinsic pulse-... more Abstract A hybrid observer for a linear time-invariant continuous plant under an intrinsic pulse-modulated feedback is considered. The fring times of the feedback representing the discrete state of the hybrid system to be observed are thus unknown. The observer possesses two feedback gains making use of the continuous output estimation error in order to correct the estimates of the continuous and discrete states, respectively. By driving the hybrid state estimation error to zero, the observer solves a synchronization problem between the fring times of the pulse-modulated feedback of the plant and that of the observer. Equivalence between a synchronous mode in the observer and zero output estimation error is proved. The influence of the observer design degrees of freedom on the observer performance is investigated by extensive numerical experiments. The introduction of the discrete observer gain clearly improves the observer performance for low-multiplicity periodic solutions in the plant but not for those with high multiplicity. In the latter case, suficiently fast observer convergence is achieved even for a zero value of the discrete gain.
Periodic modes in a mathematical model of tesrosterone regulation
ABSTRACT
LMI approach to stabilization of a linear plant by a pulse modulated signal
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Time delay induced multistability and complex dynamics in an impulsive model of endocrine regulation
2014 European Control Conference (ECC), 2014
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Complex dynamics and chaos in a scalar linear continuous system with impulsive feedback
2012 American Control Conference (ACC), 2012
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An impulse-to-impulse discrete-time mapping for a time-delay impulsive system
Automatica, 2014
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2007 IEEE International Conference on Control Applications, 2007
A mathematical model of the pulse-modulated regulation of non-basal testosterone secretion in the... more A mathematical model of the pulse-modulated regulation of non-basal testosterone secretion in the male is introduced. The suggested model is of third order, reflecting the three most significant hormones in the regulation loop, but yet is 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. The periodic mode with two GnRH pulses in the least period has not yet been described in medical literature but is supported by experimental data.
Conditional Stability of a State Observer for a Low-order Hybrid Plant
IFAC Proceedings Volumes, 2013
ABSTRACT
Hybrid State Observer for Time-Delay Systems under Intrinsic Impulsive Feedback
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Forced Periodic Oscillations. Method of Equations of Periods
Stability and Oscillations of Nonlinear Pulse-Modulated Systems, 1998
Consider the control system shown in Fig. 1.5. Let us discuss how a notion of a periodic mode can... more Consider the control system shown in Fig. 1.5. Let us discuss how a notion of a periodic mode can be properly introduced. First assume that a state-space model is used to describe a continuous linear part (CLP). Then the system is given by the equations $$ f = M\sigma $$ (5.1) $$ \frac{{dx}}{{dt}} = Ax + bf, \sigma = {c^*} + \psi $$ (5.2) Here M is an operator that describes a pulse modulator (at first we do not specify its form), ψ(t) is a given function (an external action), and the other notationa are clarified in Section 1.4.
Mathematical Description of Pulse-Modulated Systems
Stability and Oscillations of Nonlinear Pulse-Modulated Systems, 1998
We begin with the mathematical description of pulse-modulated systems. Basically, we shall follow... more We begin with the mathematical description of pulse-modulated systems. Basically, we shall follow the classification given in [TP73]. The principal element of an impulsive system is a pulse modulator. In mathematical terms, it is described by a nonlinear operator that maps an input function σ(t) to an output function f (t) (both functions are defined for \(t \geqslant 0\) and have real scalar values). The specific form of this operator and the restrictions imposed the type of modulation and on the mathematical model accepted. The most general property of a pulse modulator is that it produces an increasing sequence of time moments t0 = 0 < t1 < t2 <… called sampling moments. The time interval [tn, tn+1) is called the nth sampling interval. For some types of modulation it is supposed that t n+1 -t n = T = const. (the value T is called a sampling period), whereas for the other types the value t n+1 -t n depends on an input function.
Stability of Processes. Averaging Method
Stability and Oscillations of Nonlinear Pulse-Modulated Systems, 1998
In the preceding sections we studied the stability of the zero equilibrium x = 0 of the system wh... more In the preceding sections we studied the stability of the zero equilibrium x = 0 of the system whose CLP is given by (3.1). However, the technique developed there enables us to investigate the stability of an arbitrary solution (a process) of the system $$ \frac{{dx}}{{dt}} = Ax + bf + q\left( t \right),\sigma = {c^*}x + \psi \left( t \right),f = M\sigma $$ (4.1) with a given function ψ(t) and a given vector function q(t) (these are external disturbances), and the other notation is the same as in (3.1) where operator M describes a modulation law. Indeed, suppose that xi (t) and x2 (t) are solutions of (4.1) with the initial values x1(0) and x2(0), respectively, \( {\sigma _i}\left( t \right) + \psi \left( t \right) \) and \( {f_i}\left( t \right) = M{\sigma _i}\left( t \right) \) ( x = 1,2)Then the deviations \( {x_d} = {x_1} - {x_2},{\sigma _d} = {\sigma _1} - {\sigma _2}, \) and \( {f_d} = {f_1} - {f_2} \) satisfy the equations $$ \frac{{d{x_d}}}{{dt}} = A{x_d} + b{f_d},\sigma d = {c^*}xd,{f_d} = M{\sigma _1} - M{\sigma _2}. $$ (4.2)
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Papers by Alexander Churilov