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

doi:10.1016/j.nahs.2021.101024

Nonlinear Analysis: Hybrid Systems

Volume 41, August 2021, 101024

https://doi.org/10.1016/j.nahs.2021.101024 Get rights and content

Abstract

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.

Introduction

Hybrid models, where continuous-time dynamics are admixed with discrete events, find a rapidly growing number of applications, especially in life science and medicine [1], [2]. Studying the complex nonlinear phenomena arising in such mathematical constructs in general terms is often too challenging and finding subclasses of hybrid systems that lend themselves to analytical treatment is an important task.

Periodicity (rhythmicity) is a prominent feature in living organisms and is captured mathematically in the concept of a dynamical oscillator. For instance, Kuramoto oscillator [3] and Goodwin’s oscillator [4], [5] are popular modeling paradigms for smooth periodic, quasi-periodic, and chaotic behaviors in neuroscience and biochemistry [6], [7], [8], [9].

In endocrine regulation, neural processes interact with the hormone kinetics thus giving rise to hybrid models with relatively slow continuous dynamics that are controlled through impulsive action of firing neurons [10], [11], [12]. A relatively simple example of such a system is testosterone regulation in the male that has been intensively studied by means of a specialized version of Goodwin’s oscillator called the Smith model [13], [14]. The mechanism of episodic secretion of the release hormones produced in the hypothalamus is typically described as pulse modulation, leading to hybrid model called the impulsive Goodwin’s oscillator [15].

The impulsive Goodwin’s oscillator is characterized by two main features: a cascade structure of the continuous part of the model and a frequency and amplitude pulse-modulated feedback. While most of the continuous nonlinear oscillators produce self-sustained periodical solutions through Andronov–Hopf equilibrium-destabilizing bifurcations, the impulsive Goodwin’s oscillator does not possess equilibria at all [15], [16]. Interestingly, the model has also been shown to describe actual endocrine data quite well, [17], [18]. The pulse-modulated feedback of the impulsive Goodwin’s oscillator is intrinsic, i.e. inaccessible for measurement, and poses a specific and seldom addressed in control theory problem of estimating discrete states of a hybrid system from only continuous measurements. This problem is more common in biomedical applications as measurements of discrete events are often not accessible in living organisms.

An observer making use of a single continuous output feedback for reconstructing the firing times and weights of the feedback pulses in the impulsive Goodwin’s oscillator is proposed in [15]. For observer design, the hybrid state estimation problem is recast as a synchronization problem for the impulsive sequence in the plant and that in the observer. While being completely functional and able to handle both periodic and chaotic solutions in the plant [19], this observer structure suffers from slow convergence, especially in the case of periodic solutions of low multiplicity. To enhance observer convergence, an additional feedback of the (continuous) output estimation error to the discrete state variable estimate is proposed in [20], [21]. Despite encouraging qualitative insights into the roles of the continuous and discrete feedback supported by simulation results, the observer in the latter publication still utilizes a momentaneous measurement of the plant output to calculate the next impulse firing. The introduction of a finite-memory convolution in the observer feedback enables taking into account the whole segment of the plant output in between the present firing and the previous one. This makes the correction term less sensitive to measurement disturbance and allows for faster convergence of the continuous state estimates. It is worth noticing that even the problem of state estimation in an impulsive system with known jump times is not trivial [22], whereas this paper considers estimating as well the latter.

The present paper further extends the synchronization approach to hybrid observation by examining in detail a general case of state estimation in a continuous linear chain under an intrinsic pulse-modulated feedback from continuous plant output measurements. This is in contrast with a classical observer design approach, where the observer structure is derived from an observability notion. Local stability of the synchronous mode is secured through assigning the spectral radius of the Jacobian of a pointwise mapping. The design approach is enabled by the obtained in this paper results on spectral decomposition of the Jacobian by means of solving non-symmetrical algebraic Riccati equations. A downside of this concept is that only local convergence results can be obtained. The numerical example is therefore focused on low-periodicity solutions that are known from [19] to be most challenging. Convergence to a synchronous mode is demonstrated by simulation for a wide range of initial conditions in the observer and at a rate that is clearly superior compared to what is obtained in [19].

The main contribution of this work is in introducing a finite-memory convolution operator in the output error feedback of the hybrid observer and demonstrating the performance benefits of it. This observer structure allows for balancing the action of the continuous and discrete parts of the observer for achieving superior overall (hybrid) convergence. Previously, a high continuous feedback gain was considered as a viable design solution. Further, the studied observer structure is presently the only one that solves the hybrid state estimation problem with guaranteed performance from only continuous plant output measurements.

The rest of the paper is composed as follows. First, the hybrid plant model in hand is specified. Then, the observer structure under investigation is revisited and the observer design problem is reduced to synchronization of the impulsive feedback sequences in the plant and the observer by means of output error feedback. Further, 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 of the synchronous observer mode is analyzed and shown to be guaranteed by the Jacobian matrix of the mapping being Schur-stable. A method for achieving a certain pre-assigned spectral radius of the Jacobian by means of selecting the continuous and discrete gains of the observer is proposed. Finally, numerical simulations illustrating the theoretical contribution of the paper are provided using the mathematical model of testosterone regulation given by the impulsive Goodwin’s oscillator as a meaningful example.

Section snippets

Plant equations

Consider an impulsive feedback system given by $\dot{x} (t) = A x (t), z (t) = C x (t), y (t) = L x (t),$ $x (t_{n}^{+}) = x (t_{n}^{-}) + λ_{n} B, t_{n + 1} = t_{n} + T_{n},$ $T_{n} = Φ (z (t_{n})), λ_{n} = F (z (t_{n})),$ where $A \in R^{n_{x} \times n_{x}}$ , $B \in R^{n_{x}}$ , $C \in R^{1 \times n_{x}}$ , $L \in R^{n_{y} \times n_{x}}$ , $z$ is the scalar controlled output, $y$ is the vector measurable output, and $x$ is the state vector of (1). The minus and plus in a superscript denote the left-sided and a right-sided limit, respectively. The matrix $A$ is Metzler and Hurwitz stable, the matrix pair $(A, L)$ is observable, and the relationships $C B = 0, L B = 0$ apply.

Observer equations

The main characteristic of the state estimation problem in system (1) is that the firing times $t_{0}, t_{1}, t_{2}, \dots$ produced by intrinsic impulsive feedback (2) are not known to the observer. In particular, the initial firing time $t_{0}$ is unknown.

In [21], for plant (1), (2), the following hybrid observer with modulated correction is introduced $\dot{\hat{x}} (t) = A \hat{x} (t) + K (y (t) - \hat{y} (t)), \hat{y} (t) = L \hat{x} (t),$ $\hat{z} (t) = C \hat{x} (t), {\hat{t}}_{n} < t < {\hat{t}}_{n + 1},$ $\hat{x} (t_{n}^{+}) = \hat{x} (t_{n}^{-}) + {\hat{λ}}_{n} B, {\hat{t}}_{n + 1} = {\hat{t}}_{n} + {\hat{T}}_{n}, {\hat{λ}}_{n} = F (\hat{z} (t_{n})),$ where the matrix $K$ is a constant feedback gain

Synchronous mode and its stability

Let $(x (t), t_{n})$ be a solution of plant equations (1), (2) with the parameters $λ_{k}$ , $T_{k}$ , and $x_{k} ≜ x (t_{k}^{-})$ . Suppose that the hybrid state $(x (t), t_{n})$ is reconstructed by means of observer (7)–(9). Without loss of generality, assume that $t_{0} ⩽ {\hat{t}}_{0} < t_{1}$ .

It is easy to see that if ${\hat{t}}_{n} = t_{n}$ , $\hat{x} (t_{n}^{-}) = x_{n}$ , and $\hat{x} (θ) = x (θ)$ for $θ \in [{\hat{t}}_{n} - ν, {\hat{t}}_{n}]$ , then ${\hat{T}}_{n} = Φ (\hat{z} ({\hat{t}}_{n})) + Ψ (0) = T_{n}$ because $Ψ (0) = 0$ . Hence, the solution $(\hat{x} (t), {\hat{t}}_{n})$ of observer Eqs. (7)–(9) subject to the initial conditions ${\hat{t}}_{0} = t_{0}$ , $\hat{x} ({\hat{t}}_{0}^{-}) = x (t_{0}^{-})$ , yields $\hat{x} (t) = x (t)$ for

Properties of the Jacobian in the case of 1-cycle

Consider a solution $(x (t), t_{n})$ of system (1)–(2) satisfying $x_{n + 1} \equiv x_{n}$ , $λ_{n + 1} \equiv λ_{n}$ , $T_{n + 1} \equiv T_{n}$ (1-cycle) and a synchronous mode of observer (7)–(9) with respect to it. Since $J_{i + 1} \equiv J_{i}$ , all the matrices in the sequence ${J_{i}}_{i = 0}^{\infty}$ are equal, i.e. $J_{i} = J_{0}$ .

Proposition 4

The following properties of the Jacobian $J_{0}$ hold:

1.
The matrix $J_{0}$ can be decomposed as $J_{0} = V_{0} (K, K_{d}) + W_{0} (K) .$
2.
There exists a nonsingular transformation matrix $T$ such that the matrices ${\tilde{V}}_{0} (K, K_{d}) = T^{- 1} V_{0} (K, K_{d}) T$ and ${\tilde{W}}_{0} (K) = T^{- 1} W_{0} (K) T$ have the following structure: ${\tilde{V}}_{0} (K, K_{d}) = [\begin{bmatrix} O_{n_{x}} & O_{n} \end{bmatrix}]$

Properties of the Jacobian in the case of $m$ -cycle

Consider a synchronous mode of observer (7)–(9) with respect to an $m$ -cycle $(x (t), t_{n})$ of system (1)–(2), i.e. $x_{n + m} \equiv x_{n}$ , $λ_{n + m} \equiv λ_{n}$ , $T_{n + m} \equiv T_{n}$ . Selecting a fixed point of the cycle $x_{0}$ , a closed orbit $x_{0}, x_{1}, \dots, x_{m - 1}$ , $x_{m} \equiv x_{0}$ is obtained. Each of the distinct vectors $x_{i}, i = 0, \dots, m - 1$ defines the corresponding Jacobian matrix $J_{i}$ . Proposition 3 provides a necessary and sufficient local asymptotical stability condition for a synchronous mode with respect to the $m$ -cycle $σ (\prod_{i = 0}^{m - 1} J_{i}) \subset D,$ where $D$ is an open unit disk.

State estimation in a mathematical model of testosterone regulation

Testosterone regulation in the human male is modeled in [15] by a third-order system in the form of (1)–(2) with the matrices $A = [\begin{bmatrix} - b_{1} & 0 & 0 \\ g_{1} & - b_{2} & 0 \\ 0 & g_{2} & - b_{3} \end{bmatrix}], B = [\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}], L = {[\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}]}^{T}, C = {[\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}]}^{T},$ where $b_{1}$ , $b_{2}$ , $b_{3}$ , $g_{1}$ , $g_{2}$ are positive constants, reflecting the kinetics of the involved hormones. The elements of $x$ correspond to the concentrations of gonadotropin-releasing hormone (GnRH) – $x_{1}$ , luteinizing hormone (LH) – $x_{2}$ , and testosterone (Te) – $x_{3}$ . From the biology of the system, one has $b_{i} \neq b_{j}$ for $i \neq j$ . The state vector $x (t$

Conclusions

The problem of reconstructing a periodic solution of a linear continuous system subject to an intrinsic pulse-modulated feedback from continuous measurements of the plant output is considered. A novel observer offering structural improvements over the existing ones through the introduction of a frequency modulation laws making use of a finite-memory convolution operator is proposed. Due to the impulsive nature of the plant, the observer dynamics are highly nonlinear. A pointwise mapping

CRediT authorship contribution statement

Diana Yamalova: Conceptualization, Methodology, Software, Formal analysis, Data curation, Writing - original draft, Visualization. Alexander Medvedev: Conceptualization, Methodology, Formal analysis, Investigation, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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^☆: A. Medvedev and D. Yamalova were in part financed under Grant 2015-05256 from the Swedish Research Council . A. Medvedev is in part supported under Grant 2019-04451 from the same funding organization.

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

Abstract

Introduction

Section snippets

Plant equations

Observer equations

Synchronous mode and its stability

Properties of the Jacobian in the case of 1-cycle

Properties of the Jacobian in the case of m-cycle

State estimation in a mathematical model of testosterone regulation

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

J. Math. Biol.

Methods Enzymol.

Nonlinear Anal. Hybrid Syst.

Bull. Math. Biol.

Automatica

IFAC-PapersOnLine

Automatica

A note on semi-discrete modeling in the life sciences

Phil. Trans. R. Soc. A

Theory of hybrid dynamical systems and its applications to biological and medical systems

Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.

Self-entrainment of a population of coupled non-linear oscillators