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 continuousmodels 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.


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][6][7][8][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][11][12][13][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][19][20][21][22][23][24][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.

Mathematical Problems in Engineering
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][27][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 finitedimensional 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][35][36][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-toimpulse) 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.

System Equations
Consider an impulsive time-delay model [16] given by the equations where In (1), z is the scalar controlled output, ỹ is the measurable output vector, x is the state vector, and  is a constant time delay.The amplitude modulation function (⋅) and frequency modulation function Φ(⋅) are continuous and bounded: (⋅) is nonincreasing and Φ(⋅) is nondecreasing.
Only the time-delay values that are less than the minimal distance between two consecutive impulses are considered: so that T >  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 that The notion of FD-reducibility is a formalization of the socalled "linear chain trick" originating from [39,40] for the system in question.

Reduction to a Delay-Free Impulsive System
Define the matrices  =  0 +  1 e − 0  ,  = e − e  0  B. 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 x(), () of systems ( 1), ( 4), respectively.Assume that  1 = t1 and ( 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 infinitedimensional 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).

A Hybrid Observer
The purpose of state observation in hybrid closed-loop system (1) is to produce estimates ( t , λ ) of the impulse parameters ( t , λ ).Notice that, unlike in the conventionally treated hybrid state estimation problem formulations, the jump times t 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 ( t , λ ) of the impulse parameters ( t , λ ) of ( 1) by exploiting the delay-free model in (4).To evaluate ( t , λ ), an estimate of the continuous state vector of (4), that is, () is produced by the hybrid observer: Notice that ẑ(), ŷ() are generally discontinuous in time.
The switched feedback gain K 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  ∈ R   ×  is chosen to satisfy the stability conditions derived in Section 8.

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 plantobserver 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,   ⩽ t0 <  +1 , for some integer  ⩾ 1.
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.

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 , 0 ⩽  ⩽ , define the sets Hence, each point ( x , t ) of the observer hybrid state belongs to one of the sets  , , that is, to each ( x , t ) one can uniquely match two points (  ,   ) and (  ,   ) of the observed system (if  = , these points coincide) such that Introduce For brevity sake, denote x = x( t−  ).
Theorem 2. Pointwise mapping ( 8) is given by the equations Proof.See Appendix A.
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.

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  left , and  Propagation of the observer hybrid dynamics through the firing times is described by iterations of the operator .The th iteration is defined as Theorem 4 implies that the operator  can be linearized in a vicinity of (  ,   ).
From Section 5, it follows that the synchronous mode with respect to () is completely characterized by the vector sequence where  is a number from the definition of a synchronous mode.
Then the Jacobian of  at the point q0  is calculated as where (  ) 12 =  +1 − e (  −) e   (  +   ) , By the chain rule, it follows that, for any  ⩾ 1, the Jacobian of the th iteration of the mapping is given by the expression

Local Stability of a Synchronous Mode with Respect to an 𝑚-Cycle
A solution of ( 4) is called -cycle if it is periodic with exactly  pulse modulation instants in the least period.The existence conditions of an -cycle in pulse-modulated timedelay system (4) were studied in [16,17].
Let ((),   ) be an -cycle of plant ( 4), where  is some integer,  ⩾ 1.The existence conditions of an -cycle of pulse-modulated system with time delay are readily derived in [17].Then  + ≡   ,  + ≡   ,  + ≡   .Consider a synchronous mode of observer (7) with respect to ((),   ) and let q0  be the corresponding vector sequence as in (17), such that q0 +1 = (q 0  ) is satisfied.Consider previously defined matrices   .Since  + ≡   , the sequence {  } ∞ =0 contains at most  distinct matrices, namely,  0 , . . .,  −1 .The theorem below provides a simple tool for checking local stability of observer (7).Theorem 5. Let the matrix product  0 ⋅ ⋅ ⋅  −1 be Schur stable; that is, all the eigenvalues of this matrix lie strictly inside the unit circle.Then the synchronous mode with respect to ((),   ) is locally asymptotically stable.
Proof.The result can be proved along the lines of Theorem 3 in [31].
Theorem 5 formulates a stability condition guiding the choice of the observer gain  that appears in the matrix  =  − .As pointed out above, the condition is local and depends not only on the coefficients of the system, but also on the parameters of the observed periodic mode.In particular, the multiplicity of the periodical solution in the plant has to be known.The spectral radius of Jacobian ( 18) reflects the local convergence rate of the linearized observer dynamics.To optimize the observer performance, the static gain  can be chosen numerically to fulfill the conditions of Theorem 5 while minimizing the spectral radius of the Jacobian.

Mathematical Model of Testosterone Regulation
To model testosterone regulation in the human male [17], the case of a third-order system (1) with the matrices is considered.Here  1 ,  2 ,  3 ,  1 ,  2 are given positive parameters reflecting the kinetics of the involved hormones.
From the biology of the system, one has   ̸ =   for  ̸ = .The elements of  correspond to the concentrations of GnRH ( 1 ), LH ( 2 ), and Te ( 3 ).
The presence of a constant time delay  in closed loop relates to a delay in the hormone transport in the blood stream and a delay occurring in hormone synthesis prior to secretion [41].The contribution of the transport delays is relatively smaller than that due to synthesis of testosterone.In the simulations, the delay value is selected so that the minimal distance between two consecutive impulses does not exceed the sum of the testosterone synthesis and hormone transport delays, that is,  < 40.This is in line with the data provided in [18,20].
The concentrations of Te and LH can be measured in the blood, while the concentration of GnRH is typically not available in humans and has to be estimated.Nonetheless, the level of testosterone is usually overly more noisy than the level of LH (see, e.g., [4]), and it is difficult to distinguish between the basal and pulsatile components.Thus the structure of the output row vector  is chosen so that only the measurement of LH concentration is taken into account.
Within a feedback construct, pulsatile secretion of a hormone gives rise to a dynamic system where amplitude and frequency modulation are employed to control the concentrations of other hormones, ostensibly in order to induce sustained oscillations in the closed-loop system.
As the amplitude modulation function (⋅) and frequency modulation function Φ(⋅) Hill functions with the following (continuous) parameterizations are chosen where  1 ,  2 ,  3 ,  4 , ℎ,  are positive parameters and  is integer.It is easy to check that the functions (⋅) and Φ(⋅) are smooth, strictly monotonic and bounded.
The matrix exponentials are given by ] , where Introduce the numbers Then it can be easily seen that The state vector x() of system (1) with matrix coeffecients (21) experiences jumps at the times  =   , portraying nonbasal (episodic) release of GnRH.However, because of the matrix relationship  = 0, the assumptions of Theorem 4 are valid and the impulse-to-impulse mapping is smooth.
Below the observer design and performance are exemplified by two cases of periodical solutions in the plant arising for different values of the time delay within the considered interval.Notice that, for the numerical values in question, the multiplicity of the periodical solutions in the plant decreases with increasing delay.
where ⋅ T denotes transpose.
Choose the observer feedback gain in the form with Hence, the characteristic polynomial of  is independent of  and equal to Since  1 ,  2 ,  3 ,  1 are positive and  1 ,  2 are nonnegative,  is Hurwitz stable.
To ensure the (locally) fastest convergence rate, find  1 ,  2 for which the synchronous mode is locally asymptotically stable and the spectral radius of  0  1  2  3 is minimal.By inspection of Figures 2, 3  This criterion somehow captures the most demanding state estimation error in the hybrid observer since all the information regarding the discrete state in (1) comes from the continuous measurements.The relationship between the value of the threshold in (33) and P(  ) is depicted in Figure 5.A search for  1 ,  2 that render a locally asymptotically stable synchronous mode and minimal spectral radius of  0  1 gives (see Figures 6, 7, and 8): The relationship between the value of the threshold in (33) and P(  ) for a certain stabilizing observer gain is depicted in Figure 9.

Conclusions
A state estimation problem motivated by unmeasured hormone concentrations in the system of nonbasal testosterone regulation in the human male is considered.The system dynamics are modeled by a linear continuous time-delay system under intrinsic pulse-modulated feedback.The continuous part of the model is known to possess the property of finite-dimensional reducibility that opens up for the use of a finite-dimensional (delay-free) model for the reconstruction of the discrete and continuous states of the process model.A hybrid observer exploiting this possibility is introduced and analyzed by means of a discrete impulse-to-impulse mapping.where at all the points , where () has no jumps.Derive explicit formulas for the map (8).Introduce a number  ⩾ 0 such that  =  + .

Figure 1 :
Figure 1: A neighborhood of the point (  ,   ) in the axes  and .

Figure 7 :Figure 8 :Figure 9 :
Figure 7: Two-cycle with  = 30.The dependence of the spectral radius of the product  0  1 on  1 ( 2 = 6.85).The values less than one correspond to stability.

Figure 10 :
Figure 10: The firing times of the plant and the observer.