Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 190463, 12 pages

http://dx.doi.org/10.1155/2015/190463

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

^{1}Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia^{2}Information Technology, Uppsala University, 75105 Uppsala, Sweden

Received 4 June 2015; Revised 19 October 2015; Accepted 20 October 2015

Academic Editor: Yan-Wu Wang

Copyright © 2015 Diana Yamalova et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Hormones mediate communication between organs and tissues through the bloodstream carrying chemical messages that regulate many aspects in the human body, that is, metabolism, growth as well as the sexual function and the reproductive processes. Hormones are secreted by endocrine glands directly into the bloodstream in continuous (basal) or pulsatile (nonbasal) manner. Endocrine glands, interacting via hormone concentrations in blood, build up dynamical control loops characterized by self-sustained oscillations of the involved physiological quantities [1].

The endocrine system of testosterone regulation in the male essentially consists of three hormones, namely, gonadotropin-releasing hormone (GnRH), luteinizing hormone (LH), and testosterone (Te). GnRH is produced in the hypothalamus of the brain and released in short pulses. Reaching the pituitary gland, GnRH stimulates production of LH, which in turn stimulates production of Te in the testes. Finally, both the GnRH outflow and the LH secretion are subject to feedback inhibition by Te [2]. However, the inhibition of LH has a relatively small effect on the dynamics of the closed-loop system and therefore not considered in this paper.

An impulsive mathematical model of testosterone regulation was proposed in [3] and is shown to comport with experimental data in [4]. It constitutes an impulsive version of Goodwin oscillator, a mathematical model that is well known in mathematical biology (see, e.g., [5–9]). The impulsive Goodwin oscillator consists of a continuous and an impulsive part [10], thus possessing hybrid dynamics and presenting a special version of an impulsive differential system [10–14]. It mathematically portrays the concept of pulsatile hormone regulation described in medical literature (see, e.g., [15]).

More recently, the impulsive Goodwin oscillator was augmented with a time delay in the continuous part of the system [16, 17], making it more aligned with the biological nature, as transport phenomena and biosynthesis are omnipresent in endocrine and metabolic systems [18–25]. With the time delay taken into account, the pulse-modulated model of endocrine regulation acquires an infinite-dimensional continuous part. The closed-loop dynamics become therefore both hybrid and infinite-dimensional, and this combination is mathematically challenging and so far rarely treated. However, the cascade structure of the continuous part, together with the impulsive feedback, allow application of the concept of finite-dimensional reducibility (FD-reducibility), [16, 17]. In particular, it was shown [17] that the dynamics of an impulsive time-delay system with an FD-reducible continuous part coincide on certain time intervals with the dynamics of a delay-free impulsive system. This idea plays a key role in the present study.

Concentrations of the hormones secreted in human hypothalamus that is located in the lower central part of the brain are not available for direct measurement due to ethical reasons and need to be estimated. It poses an unusual observation problem. A considerable number of papers is devoted to the observability of hybrid systems, for example, [26–28]. The discrete states of a system are usually assumed known, while observers for hybrid systems that are able to reconstruct discrete states from only continuous measurements are not so well covered in the literature.

In endocrine systems with pulsatile secretion, the highest degree of uncertainty is associated with the discrete (impulsive) part whose states have to be reconstructed from hormone concentration measurements. Two model-based estimation approaches are currently known. The first one is based on batch deconvolution techniques (blind system identification) [29, 30], while the relatively recent second one employs a state observer, whose estimates are corrected by output estimation error feedback [31, 32]. An extension of the observer scheme proposed in [31] to impulsive systems with time delay in continuous part was considered in [33]. Unlike the case treated in [33], the observer proposed here does not explicitly involve a delay but is rather based on a finite-dimensional plant model. Hence, the main contribution of the paper is in the novel structure and subsequent analysis of a hybrid observer exploiting a finite-dimensional model to reconstruct the states of the time-delay system.

Notice that impulsive feedback in the observer treated below is not contributed by design to achieve a performance objective but rather constitutes an integral and unmeasurable part of the plant model. On the contrary, in the impulsive observers for state estimation of linear and nonlinear continuous systems proposed in [34–37], the observer state is updated in an impulsive fashion in order to achieve, for example, faster convergence. This distinction results in a major complication in observer design for plants with intrinsic impulsive feedback as the timing and weights of the impulses are unknown and have to be estimated by the observer.

A preliminary version of the present material without proofs of the main statements was presented in [38].

The paper is organized as follows. First, an impulsive time-delay model is summarized and reduced to an equivalent delay-free one. Then, making use of the reduced model, a hybrid observer is proposed and a pointwise (impulse-to-impulse) mapping describing its dynamics is derived. Further, the properties of the mapping pertaining to the observer performance are investigated. Then the impulsive time-delay model of testosterone regulation is described. Numerical simulations and calculations illustrating the observer design performance are also provided.

#### 2. System Equations

Consider an impulsive time-delay model [16] given by the equationswhere , , , , are constant matrices and .

In (1), is the scalar controlled output, is the measurable output vector, is the state vector, and is a constant time delay. The amplitude modulation function and frequency modulation function are continuous and bounded: is nonincreasing and is nondecreasing.

System (1) is considered for subject to the initial condition , , where is a continuous initial vector function. The state vector of system (1) experiences jumps at the times , . The condition ensures that the modulating signal is continuous in time.

Only the time-delay values that are less than the minimal distance between two consecutive impulses are considered:so that for all . This condition implies that only one firing of the pulse-modulated feedback in (1) is possible within a time interval whose length is equal to the time-delay value.

Suppose that the linear part of the system possesses the property of finite dimensional (FD) reducibility [16, 17], implying thatThe notion of FD-reducibility is a formalization of the so-called “linear chain trick” originating from [39, 40] for the system in question.

#### 3. Reduction to a Delay-Free Impulsive System

Define the matrices , . Introduce a delay-free impulsive system:

The following lemma obtained in [17] reveals the relationship between the solutions of system (1) and those of system (4).

Lemma 1. *Consider solutions , of systems (1), (4), respectively. Assume that and . Then it holds that , and for all . Moreover,At the same time, generally, the solutions do not coincide entirely*

The result above will be exploited further in the paper to design a finite-dimensional observer for the infinite-dimensional hybrid system in (1). Note that the value of the time delay in the delay-free impulsive system still influences the system dynamics as affects the matrix coefficients , of (4).

#### 4. A Hybrid Observer

The purpose of state observation in hybrid closed-loop system (1) is to produce estimates of the impulse parameters . Notice that, unlike in the conventionally treated hybrid state estimation problem formulations, the jump times are considered to be unmeasurable in (1) and require estimation. In fact, the problem solved by the proposed observer is synchronization of the firings in the feedback of the plant representing its discrete state and those of the observer.

From Lemma 1, it follows that one can produce estimates of the impulse parameters of (1) by exploiting the delay-free model in (4). To evaluate , an estimate of the continuous state vector of (4), that is, is produced by the hybrid observer:

Notice that , are generally discontinuous in time.

The switched feedback gain is zero in the time intervals where the solutions of system (1) and those of system (2) do not coincide, while the static feedback gain is chosen to satisfy the stability conditions derived in Section 8.

#### 5. Synchronous Mode

Keeping in mind that the purpose of the hybrid estimation here is essentially synchronization, and following [31], introduce the notion of a synchronous mode for the plant-observer system (4), (7). Let be a solution of plant equations (4) with the parameters , , and . Suppose that the plant is already running at the moment when the observer is initiated, that is, , for some integer .

Consider the solution of observer equations (7) subject to the initial conditions , , that yields , , , , and for . Such a solution will be called* a synchronous mode* with respect to . Thus, a synchronous mode is a null solution of the state estimation error dynamics of hybrid observer (7) on .

Following [31], a synchronous mode will be called* locally asymptotically stable* if, for any solution of (7) such that the initial estimation errors and are sufficiently small, it follows that and as . The latter implies as . In the definition above, the operator stands for any vector norm.

To ensure practical usefulness of the observer, stability properties of the synchronous mode have to be investigated. By choice of , the synchronous mode has to be rendered asymptotically stable with a suitable convergence rate and domain of attraction.

#### 6. Pointwise Mapping and Its Properties

Consider the pointwise mapping describing the evolution of the observer hybrid state from one firing of the impulsive part in (7) to the next one:

For any integer numbers and , , define the sets

Hence, each point of the observer hybrid state belongs to one of the sets , that is, to each one can uniquely match two points and of the observed system (if , these points coincide) such that , .

Introducewhere . Note that the functions are piecewise continuous due to the definition of .

Define at withFor brevity sake, denote

Theorem 2. *Pointwise mapping (8) is given by the equations*

*Proof. *See Appendix A.

Theorem 3. *The mapping is continuous.*

*Proof. *See Appendix B.

It will be shown in the next section that the mapping is not continuously differentiable in the whole state space. However, due to its local differentiability, local stability properties of mapping (12) characterizing the dynamics of the observer state can be investigated via linearization.

#### 7. Linearization of the Discrete-Time Map

The behaviors of pointwise mapping (12) in vicinity of the points will be studied with respect to local stability of a synchronous mode.

To show the smoothness of the mapping introduced below at the points , divide each set for all into two subsets and by the hyperplanes , that is, , where

Note that the set () is not divided into subsets and due to the assumption on delay in (2) and thus implying and .

Consider a point for some . It can be seen that the closures of the four sets , , , intersect at the point . Moreover, . For a sufficiently small neighborhood of the point , the mapping can only take the values , where and is one of the four sets , , , ; see Figure 1.