Pulse-modulated Feedback Research Articles

Cycles in Impulsive Goodwin’s Oscillators of arbitrary order

Existence of periodical solutions, i.e. cycles, in the Impulsive Goodwin’s Oscillator (IGO) with the continuous part of an arbitrary order m is considered. The original IGO with a third-order continuous part is a hybrid model that portrays a chemical or biochemical system composed of three substances represented by their concentrations and arranged in a cascade. The first substance in the chain is introduced via an impulsive feedback where both the impulse frequency and weights are modulated by the measured output of the continuous part. It is shown that, under the standard assumptions on the IGO, a positive periodic solution with one firing of the pulse-modulated feedback in the least period also exists in models with any m≥1. Furthermore, the uniqueness of this 1-cycle is proved for the IGO with m≤10 whereas, for m>10, the uniqueness can still be guaranteed under mild assumptions on the frequency modulation function.

Design of the Impulsive Goodwin's Oscillator in 1-cycle

This paper presents a systematic approach to design a hybrid oscillator that admits an orbitally stable periodic solution of a certain type with pre-defined parameters. The parsimonious structure of the Impulsive Goodwin's oscillator (IGO) is selected for the implementation due to its well-researched rich nonlinear dynamics. The IGO is a feedback interconnection of a positive third-order continuous-time LTI system and a nonlinear frequency and amplitude impulsive modulator. A design algorithm based on solving a bilinear matrix inequality is proposed yielding the slope values of the modulation functions that guarantee stability of the fixed point defining the designed periodic solution. Further, assuming Hill function parameterizaton of the pulse-modulated feedback, the parameters of those rendering the desired stationary properties are calculated. The character of perturbed solutions in vicinity of the fixed point is controlled through localization of the multipliers. The proposed design approach is illustrated by a numerical example. Bifurcation analysis of the resulting oscillator is performed to explore the nonlinear phenomena in vicinity of the designed dynamics.

Doubling of a closed invariant curve in an impulsive Goodwin’s oscillator with delay

In the present paper, we focus on the doubling of closed invariant curves associated with quasiperiodic dynamics. We consider a 5D map derived from a hybrid model originating from systems biology and containing a continuous part with time delay and pulse-modulated feedback. Using numerical bifurcation analysis, we show that doubling bifurcation takes place on a closed 2D invariant manifold. We explain how such a configuration of the phase space can be created and highlight the role of delay.

Hybrid observer with finite-memory output error correction for linear systems under intrinsic impulsive feedback

A novel hybrid observer that estimates the states of an oscillating system composed of a linear chain structure and an intrinsic pulse-modulated feedback is considered. This particular type of plant model appears in e.g. endocrine systems with pulsatile hormone secretion. The observer reconstructs the continuous states of the model as well as the firing times and weights of the feedback impulses. Since the pulse-modulated feedback is intrinsic, no measurements of the discrete part of the plant are available to the observer. For a periodical plant solution, to reconstruct the hybrid state, the impulses in the observer have to be synchronized with those in the plant. The observer is equipped with two feedback loops driven by the output estimation error. One of these is utilized to correct the estimates of the continuous states. In contrast with previous observer designs, the estimate of the next impulse firing time is implemented by means of a finite-memory convolution operator. A pointwise mapping capturing the propagation of the continuous plant and observer states through the discrete cumulative sequence of the feedback firing instants is derived. Local stability properties of the synchronous mode are related to the spectral radius of the Jacobian of the pointwise mapping. The observer design is based on assigning a guaranteed convergence rate to the local dynamics of a synchronous mode through the output error feedback gains to the continuous and discrete part of the observer. The observation of a stable m-cycle in the plant is treated to establish a general scenario, whereas the special case of 1-cycle is worked out in detail as the most common one. A numerical example illustrates the observer performance in the case of periodic modes of low multiplicity in an impulsive model of testosterone regulation in the male. Despite the local nature of the design approach, convergence to a synchronous mode is observed for a wide range of initial conditions for the discrete state estimate in the observer.

Bifurcation analysis for non-local design of a hybrid observer for the impulsive Goodwin\u2019s oscillator

The impulsive Goodwin’s oscillator is a mathematical model capturing the dynamics arising in a closed-loop system, where a third-order linear time-invariant plant is subject to an intrinsic pulse-modulated feedback. Originally, the model was motivated by pulsatile regulation in endocrine systems but also has other potential applications. The asymptotic estimation of the hybrid state of the impulsive Goodwin’s oscillator is considered in this paper. A hybrid observer makes use of the continuous plant output to correct the estimates of the state vector through two output error feedbacks: a continuous and a discrete one. When the hybrid state estimation error is zero, the observer is in a synchronous mode characterized by the firings of the impulses in the observer feedback and those of the plant occurring simultaneously. The synchronous mode thus corresponds to an equilibrium point of the hybrid state error dynamics. To guarantee asymptotic convergence of the observer to the synchronous mode, the basin of attraction of the equilibrium has to include all feasible initial deviations of the state estimates. To guarantee the above properties, a numerical observer design approach based on bifurcation analysis of a discrete map capturing the observer state transitions from one impulse firing to another is proposed and its efficacy is demonstrated in simulation.

Structural Identifiability of a Third-order Continuous System under Impulsive Feedback

Structural identifiablity of a third-order continuous time-invariant linear plant under an intrinsic pulse-modulated feedback is analyzed. The model represents a biomedical system, where the input signal to the continuous plant is immeasurable and the feedback modulation functions have to be identified along with the continuous dynamics. It is shown that two eigenvalues of the continuous plant system matrix (i.e. time constants), along with the times and weights of impulses occurring during a finite time interval, are identifiable from the measurement of one continuous system state over the interval in question. When an infinite time horizon is considered, all parameters are identifiable, up to gain scaling and linear block permutations.

Stability analysis of Lur’e systems with a pulse-modulated feedback

A nonlinear system with a sector bound nonlinearity is considered. The system is subject to a stabilizing sampled feedback with finite width impulses. An impulsive counterpart of the circle criterion for absolute stability is obtained with the help of the Gelig’s averaging method.

Hybrid Observer for an Intrinsic Impulsive Feedback System

This paper deals with continuous plants subject to intrinsic pulse-modulated feedback, thus exhibiting hybrid closed-loop dynamics. The system structure implements a hybrid oscillator and arises in living organisms, e.g. when episodically firing neurons control the production of hormones in endocrine glands. Hybrid observers reconstructing both the continuous and discrete states of the hybrid plant from only continuous measured outputs are considered. They excel over the existing solutions through the introduction of two co-ordinated feedbacks of the output estimation error: one correcting the continuous state estimates and another adjusting the discrete ones. Different types of the feedback operator to the discrete estimates are analyzed. The observer design problem is reduced to synchronization of the observer solution with that of the plant. The synchronous mode of the observer is rendered locally stable by the selection of the feedback gains. Numerical illustration of the design procedure and observer performance with respect to a pulse-modulated model of testosterone regulation is provided.

Design Degrees of Freedom in a Hybrid Observer for a Continuous Plant under an Intrinsic Pulse-modulated Feedback

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.

Finite-Dimensional Hybrid Observer for Delayed Impulsive Model of Testosterone Regulation

The paper deals with the model-based estimation of hormone concentrations that are inaccessible for direct measurement in the blood stream. Previous research demonstrated that the dynamics of nonbasal endocrine regulation can be closely captured by linear continuous models with time delays under a pulse-modulated feedback. The presence of continuous time delays is inevitable in such a model due to transport phenomena and the time necessary for an endocrine gland to produce a certain hormone quantity. Yet, thanks to the finite-dimensional reducibility of the linear time-delay part of the system, a finite-dimensional model can be used to reconstruct both the continuous and discrete states of the hybrid time-delay plant. A hybrid observer exploiting this possibility is suggested and analyzed by means of a discrete impulse-to-impulse mapping.

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.

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.

Conditional Stability of a State Observer for a Low-order Hybrid Plant

Dynamical behaviors in a recently proposed observer for linear time-invariant systems with intrinsic pulse-modulated feedback are studied. A special case of scalar linear dynamics is considered as it is not covered by the previously presented mathematical analysis. Notably, the lowest non-trivial dimension of the continuous part of the plant results in more complex dynamics of the observer. In fact, the convergence of the observer becomes dependent on the observer initialization, which phenomenon does not exist in the case of the second and higher dimension continuous dynamics.

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.

State Observer for Continuous Oscillating Systems with Pulsatile Feedback⋆

AbstractThe paper introduces a static gain state observer for linear continuous systems under sustained oscillations driven by intrinsic and inaccessible for measurement pulse-modulated feedback. This kind of hybrid dynamics arises in e.g. modeling of endocrine systems with pulsatile regulation. Conditions under which the observer state estimation error converges to zero and the sequence of pulse modulation instances estimated by the observer asymptotically synchronizes with that of the plant are presented.

Explicit renormalisation in piecewise linear bimodal maps

We discuss 2 two-parameter families of piecewise linear bimodal (two-hump) maps. The first example arises naturally in the study of bimodal maps: any bimodal map is semi-conjugate to a member of this family and topological entropy is preserved. The second example is a symmetric subcase of a map derived from a simple pulse-modulated feedback system. In each case we define an explicit map of parameter space which governs the effect of renormalisation, and use it to describe the non-wandering set.

Analysis and Design of Pulse Ratio Modulated Feedback Control Systems

An exact method of analysis is presented for a class of pulse-modulated feedback control systems which can be modeled by a multiple-loop nonlinear feedback system. The nonlinearity has a hysteresis characteristic. By means of the Tsypkin-Hamel criterion, suitably modified, a set of equations is obtained, determining the switching frequency, duty cycle, and steady-state dc error. Based on the graphical form of these equations, a design procedure is set forth. Specific cases have been examined and peculiar phenomena are evidenced; experimental data verify both the chosen model and the suggested procedures. A simplified pulse ratio modulator is also presented.

Frequency domain stability and boundedness criteria of pulse-modulated feedback systems

Abstract Stability and boundedness conditions of pulse modulated feedback systems are derived in this paper. These conditions, based on the frequency domain approach, extend the range of applicability of this approach to systems without the strict sector non-linearity. Geometric interpretations of the results in the Popov plane are also presented.

Generalized pulse-modulated feedback systems: Norms, gains, Lipschitz constants, and stability

Sufficient conditions for the asymptotic stability in the large of pulse-modulated feedback systems are developed from the operator theoretic viewpoint. Stability here requires that the pulse-modulated feedback system be a Lipschitz continuous operator on the extended space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{1e}</tex> . This strong definition of stability is motivated by an examination of a first-order pulsewidth-modulated system. To provide a unified format for the main development two distinct general classes of pulse modulators are defined. Type I includes the pulsewidth modulator and more general pulsewidth frequency modulators that contain a sampler. Type II includes, for example, the integral pulse frequency modulator and its generalizations. For elements of Type I conditions are derived to bound the incremental gain (on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{1e}</tex> ) of the modulator in cascade with a linear element; a standard transformation of the feedback loop similar to that used in the derivation of the Popov criterion yields sufficient conditions for stability of the feedback system in the above strong sense. Type II modulators are discontinuous on any normed linear space and thus, only conditions for boundedness of the closed-loop system as an operator on L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> are given for this case.

Generalized weighting function and restricted stability of a linear pulse-modulated error feedback system

A generalized weighting function is obtained for a linear feedback system with a pulse-modulated error signal. This is expressed in the form of a matrix operator acting on an input vector, the components of which are the first R − 1 derivatives of the input, where R is the order of the unmodulated closed-loop system. In addition, since the system operator takes the form of a finite dimensional matrix, it has been possible to make and realize more stringent conditions on the transient stability; namely, a preassigned bounded output after a preassigned time. Although the solution has been obtained generally for non-uniform pulsing, the latter stability investigation has been made only for the uniform case.