# Rhythmic neuromorphic control of a pendulum: A hybrid systems analysis

Elena Petri, Romain Postoyan, Maurice Heemels

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Elena Petri, Romain Postoyan, Maurice Heemels. Rhythmic neuromorphic control of a pendulum: A hybrid systems analysis. 64th IEEE Conference on Decision and Control, Dec 2025, Rio de Janeiro, Brazil. hal-05328841

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# Rhythmic neuromorphic control of a pendulum:
A hybrid systems analysis

E. Petri, R. Postoyan, W.P.M.H. Heemels

*Abstract*— Neuromorphic engineering is an emerging research domain that aims to realize important implementation advantages that brain-inspired technologies can offer over classical digital technologies, including energy efficiency, adaptability, and robustness. For the field of systems and control, neuromorphic controllers could potentially bring many benefits, but their advancement is hampered by lack of systematic analysis and design tools. In this paper, the objective is to show that hybrid systems methods can aid in filling this gap. We do this by formally analyzing rhythmic neuromorphic control of a pendulum system, which was recently proposed as a prototypical setup. The neuromorphic controller generates spikes, which we model as a Dirac delta pulse, whenever the pendulum angular position crosses its resting position, with the goal of inducing a stable limit cycle. This leads to modeling the closed-loop system as a hybrid dynamical system, which in between spikes evolves in open loop and where the jumps correspond to the spiking control actions. Exploiting the hybrid system model, we formally prove the existence, uniqueness, and a stability property of the hybrid limit cycle for the closed-loop system. We finally elaborate on a possible spiking adaptation mechanism on the pulse amplitude to generate a hybrid limit cycle of a desired maximal angular amplitude.

## I. INTRODUCTION

Inspired by brain computational principles, neuromorphic engineering consists of developing hardware and software that emulate biological neural systems [1]. Numerous research activities have been done in various engineering fields in the recent years, see, e.g., [2]–[6] and the references therein. The neuromorphic engineering field holds significant promise for control over classical digital technologies in terms of energy efficiency, real-time processing, adaptive learning, and robust integration of sensory actuators [1], [7], but until now very little has been done in the context of systems and control, although inspiring exceptions exist, see, e.g., [8]–[13]. It is fair to say that neuromorphic control is still in its early development, for which one of the main causes is the lack of methodological tools for its design and analysis.

Due to the event-based nature of biological spiking neurons, see, e.g., [10], [14], caused by the fact that information is communicated via the occurrence and the absence of spikes, we argue that theoretical hybrid systems tools as in, e.g., [15]–[17], are well suited to analyze and design

neuromorphic controllers. Indeed, they can be used model the spiking behavior with the discrete jumps and the continuous dynamics between spikes with the flows. To show this, following the preliminary path laid down in [18], where a stabilizing neuromorphic control is discussed for a simple scalar-state linear system, the aim of the present work is to show how the hybrid dynamical systems formalism in [15] can be exploited to study more complex dynamical scenarios in neuromorphic control. In particular, we formally investigate the scenario presented in [11] in which a mechanical pendulum system is sporadically controlled by bursting neurons to generate desirable oscillations. The idea in [11] is to activate a torque for a fixed amount of time using bursting neurons whenever the pendulum is at its resting position. In doing so, oscillations are generated [11], whose amplitude can be controlled by adapting the torque activation length through neuron parameters modifications, similar to neuromodulation in nature. This setting was deliberately chosen in [11] as it can be seen as a prototypical setup that demonstrates the idea of entraining a mechanical system with a neuro-inspired controller that connects the central pattern generators of neuroscience and the rhythmic controllers of legged robotics [19], [20]. The pendulum is therefore a simple yet insightful example to start studying rhythmic neuromorphic control systems. The objective of this work is to provide an analytical justification of the results presented in [11] using hybrid systems tools [15], [21], thereby underlining the potential of hybrid systems for providing formal analysis and development tools for neuromorphic controllers.

To study the pendulum setup in [11], we model the activation of bursting neurons as a Dirac delta pulse on the angular velocity (Section II). In this way, the duration of the activation is related to the amplitude of the pulse. After linearizing the pendulum dynamics, we present a hybrid model of the closed-loop system for pulses of a fixed amplitude (Section III), where a jump corresponds to the effect of a fix-amplitude pulse and the pendulum is in open loop during flows, i.e., between any two consecutive pulses. We then establish the existence of a unique hybrid limit cycle [21], and we prove that this hybrid limit cycle exhibits uniform exponential stability properties (Section IV). We exploit for this purpose the features of the problem, in particular expressions of the solutions to the hybrid system. Although our result relies on the explicit solution, we envision that the results can be extended to nonlinear pendulum models, still building on hybrid systems tools. The results are then numerically illustrated (Section V). We finally elaborate on

This work is partly funded by the ANR under grant OLYMPIA ANR-23-CE48-0006.

Elena Petri and Maurice Heemels are with the Control Systems Technology (CST) Section of the Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands ({e.petri,m.heemels}@tue.nl).

Romain Postoyan is with the Université de Lorraine, CNRS, CRAN, 54000 Nancy, France (romain.postoyan@univ-lorraine.fr).
the possibility of adapting the pulse amplitude to generate a hybrid limit cycle of a desired maximum angular amplitude (Section VI). This reveals a simple but efficient learning mechanism illustrating some of the beneficial features of neuromorphic control. Finally, conclusions and directions for future work are discussed (Section VII). Proofs are omitted for space reasons and can be found in [22], together with the used notation and preliminaries. Interestingly, our work is developed in parallel with the complementary paper [23] also studying the same pendulum problem but using alternative methodological tools; see also the acknowledgements section at the end of the paper.

## II. PROBLEM STATEMENT

We consider the dimensionless pendulum model

$$ \ddot{q} + \alpha \dot{q} + \sin q = u, \quad (1) $$

where $q \in [-\pi, \pi]$ is the angular position of the pendulum, with $q = 0$ the resting position, $\alpha \in (0, 2)$ is the dimensionless damping and $u \in \mathbb{R}$ is the dimensionless torque used to control system (1).

Inspired by [11], a spiking torque, generated by a neuromorphic device, is applied whenever $q$ is at its resting position ($q = 0$) to make the pendulum swing. We denote by $\{t_i\}_{i \in \mathbb{Z}}$ with $\mathbb{Z} = \{1, \dots, N\} \subset \mathbb{Z}_{>0}$, where $N \in \mathbb{Z}_{>0}$ or $N = \infty$, the sequence of spiking instants, defined as, for any $i \in \mathbb{Z}$,

$$ t_{i+1} := \inf\{t > t_i : q(t_i) = 0\} \quad \text{with} \quad t_0 = 0, \quad (2) $$

with $t_{i+1} := \infty$ if $i+1 \notin \mathbb{Z}$. The spiking torque can be written as, for any $t \in [t_i, t_{i+1})$,

$$ u(t) = \sum_{k \in \mathbb{Z}} I \delta(t - t_k) w(t) \quad \text{with} \quad w(t) \in \text{SGN}(\dot{q}(t)), \quad (3) $$

where $I \in \mathbb{R}_{>0}$ is a design parameter. Note that the spiking torque $u$ in (3) is the input of system (1) and, at each spiking instant $t_i$, $i \in \mathbb{Z}_{>0}$, the effect of the spiking input can be modeled as a jump of fixed amplitude $I$ on the angular velocity of the pendulum, see Section III. Moreover, the direction of the spike depends on the sign of the angular velocity, where $\text{SGN}: \mathbb{R} \to \{-1, 1\}$ is defined as, for any $z \in \mathbb{R}$, $\text{SGN}(z) = \{1\}$ if $z > 0$, $\text{SGN}(z) = \{-1\}$ if $z < 0$, and $\text{SGN}(0) = \{-1, 1\}$. We will elaborate on the scenario where the value of $I$ is adapted to generate oscillations of desired maximal angular amplitude in Section VI, see also [23] for a more in-depth treatment. This torque can be approximately generated using bursting neurons, producing $I$ spikes of unitary amplitude [11]. With this spiking torque, we aim to generate a desirable limit cycle as experimentally done in [11].

The goal of this work is to analyse the existence and the stability properties of limit cycles for the closed-loop system (1), (3). We propose for this purpose to model the corresponding closed-loop system as a hybrid dynamical system, for which jumps correspond to the spiking control actions in $u$. We then establish the existence of so-called hybrid limit cycles as formalized later, which we prove to exhibit stability properties.

## III. HYBRID MODEL OF CLOSED-LOOP SYSTEM

In this section, we first introduce the variables needed to derive the desired hybrid model. We also linearize the dynamics (1) around the origin to facilitate computations. We then formalize the objective using hybrid systems terminology.

### A. Variables

We denote by $q_1 \in [-\pi, \pi]$ the pendulum angle and $q_2 \in \mathbb{R}$ the angular velocity for the sake of convenience. Note that from (3) it follows that in between spiking actions, we have $u = 0$ in (1). We thus derive from (1) that between two consecutive jumps

$$ \begin{aligned} \dot{q}_1 &= q_2 \\ \dot{q}_2 &= -\alpha q_2 - \sin(q_1). \end{aligned} \quad (4) $$

For ease of computations, we consider a linearized version of (4) given by

$$ \begin{aligned} \dot{q}_1 &= q_2 \\ \dot{q}_2 &= -\alpha q_2 - q_1, \end{aligned} \quad (5) $$

where we recall that $\alpha \in (0, 2)$. The linearization is performed for $q_1 = 0$, we will thus be concentrating on the behavior of the pendulum for “moderate” angular values. Hence, for the sake of convenience we will be considering $q_1 \in \mathbb{R}$ in the sequel thereby ignoring possible virtual “angular flips” when $q_1 \in \{-\pi, \pi\}$; otherwise a hybrid mechanism can very well be added to the hybrid model presented below.

When $q_1$ crosses the resting position, a spiking torque (Dirac pulse) is applied to the pendulum, which can be modeled in the hybrid systems framework of [15] as a jump. Thus, from (3), we have when $q_1 = 0$

$$ \begin{aligned} q_1^+ &= 0 \\ q_2^+ &= \{q_2 + Iz : z \in \text{SGN}(q_2)\}. \end{aligned} \quad (6) $$

Finally, we introduce an auxiliary logic variable $\sigma \in \{-1, 1\}$ to robustify the detection of the condition $q_1 = 0$ needed to activate the torque. On flows,

$$ \dot{\sigma} = 0, \quad (7) $$

and at jumps

$$ \sigma^+ \in \text{SGN}(q_2). \quad (8) $$

Let us give some insights regarding the choice of $\sigma$. We have seen that the spiking torque must occur whenever $q_1 = 0$ on the one hand, and flow should be allowed for any $q_1 \neq 0$. We may be tempted to define the flow set as $\{(q_1, q_2) : q_1 \neq 0\}$ and the jump set as $\{(q_1, q_2) : q_1 = 0\}$. However, such a reasoning leads to a flow set, which is not closed thereby violating one of the hybrid basic conditions, namely (iii) in [22, Section II]. Variable $\sigma$ allows overcoming this shortcoming. The idea is to enforce $\sigma$ to have the same sign as $q_1$, as depicted in Fig. 1. In this way, flow is only allowed when $\sigma q_1 \ge 0$ and the spike occurs only when $\sigma q_1 = 0$ and $\sigma q_2 \le 0$. As a result, we will see next that the obtained flow and jump sets enforce the spiking torque activation as desired when $q_1 = 0$, while working with closed flow and jump sets thereby satisfying one of the hybrid basic conditions.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Illustration of the pendulum.</figcaption></figure>

## B. Hybrid model

Given $I \in \mathbb{R}_{>0}$, we denote the concatenated state variable

$$x := (q_1, q_2, \sigma) \in \mathcal{X} := \mathbb{R} \times \mathbb{R} \times \{-1, 1\} \subset \mathbb{R}^{n_x} \quad (9)$$

with $n_x := 3$. We propose the hybrid model

$$\mathcal{H}_I : \left\{ \begin{array}{ll} \dot{x} = f(x) & x \in \mathcal{C} \\ x^+ \in G(x) & x \in \mathcal{D}, \end{array} \right. \quad (10)$$

with

$$\begin{aligned} \mathcal{C} &:= \{x \in \mathcal{X} : \sigma q_1 \ge 0\} \\ \mathcal{D} &:= \{x \in \mathcal{X} : \sigma q_1 = 0, \sigma q_2 \le 0\} \end{aligned} \quad (11)$$

and

$$\begin{aligned} f(x) &:= (q_2, -q_1 - \alpha q_2, 0) & x \in \mathcal{C} \\ G(x) &:= \{(0, q_2 + Iz, z) : z \in \text{SGN}(q_2)\} & x \in \mathcal{D}. \end{aligned} \quad (12)$$

As explained at the end of Section III-A, flow is only allowed when $\sigma q_1 \ge 0$. When $\sigma q_1 = 0$, a spiking torque occurs if in addition $\sigma q_2 \le 0$, see the definition of set $\mathcal{D}$. In this case, a jump of amplitude $I$ occurs on the angular velocity, which is positive or negative depending on the sign of $q_2$.

**Remark 1.** It can be noticed that when $x$ is initialized in $\{0\} \times \{0\} \times \{-1, 1\}$, $\mathcal{H}_I$ admits multiple solutions, i.e., we have non-uniqueness of solutions from this initial condition. Moreover, some of these solutions may remain there for all future times without ever experiencing a jump and thus without converging to a limit cycle. This case will be ruled out in the sequel by focusing on solutions not initialized in $\{0\} \times \{0\} \times \{-1, 1\}$, see Section IV-A.

System $\mathcal{H}_I$ satisfies the next properties.

**Proposition 1.** Given any $I \in \mathbb{R}_{>0}$, the following holds for system $\mathcal{H}_I$ in (10).

(i) The hybrid basic conditions are satisfied.

(ii) Maximal solutions are complete.

## C. Hybrid limit cycles

Our objective is to establish the existence and the stability of hybrid limit cycles for system $\mathcal{H}_I$ for any given $I \in \mathbb{R}_{>0}$ as formalized next using the terminology introduced in [21], which we revisit and adapt to our purpose.

**Definition 1.** Given any $I \in \mathbb{R}_{>0}$, a complete solution $x^*$ to $\mathcal{H}_I$ is $(T^*, N^*)$-periodic if $(T^*, N^*) \in \mathbb{R}_{\ge 0} \times \mathbb{Z}_{\ge 0}$ is such that the following holds.

(i) $T^* + N^* > 0$.

(ii) For all $(t, j) \in \text{dom } x^*$, $(t + T^*, j + N^*) \in \text{dom } x^*$.

(iii) For all $(t, j) \in \text{dom } x^*$, $x^*(t + T^*, j + N^*) = x^*(t, j)$. We call $\mathcal{O}_I := \text{rge } x^* = \{x^*(t, j) : (t, j) \in \text{dom } x^*\}$ a hybrid limit cycle.

The next definition formalizes the stability property we are interested in for the hybrid limit cycle generated by $\mathcal{H}_I$.

**Definition 2.** Given any $I \in \mathbb{R}_{>0}$, consider system $\mathcal{H}_I$ and a set $\mathcal{X}_0 \subset \mathcal{C} \cup \mathcal{D}$. We say that the hybrid limit cycle $\mathcal{O}_I$ is uniformly exponentially stable on $\mathcal{X}_0$ when the following properties hold.

(i) $\mathcal{X}_0$ is forward invariant, i.e., $\text{rge}(x) \subset \mathcal{X}_0$ for all solutions $x$ initialized in $\mathcal{X}_0$.

(ii) There exist $\alpha \in K_\infty$ and $\gamma \in \mathbb{R}_{>0}$ such that all solutions $x$ initialized in $\mathcal{X}_0$ verify $|x(t, j)|_{\mathcal{O}_I} \le e^{-\gamma(t+j)}\alpha(|x(0, 0)|_{\mathcal{O}_I})$ for all $(t, j) \in \text{dom } x$, where $|x|_{\mathcal{O}_I} = \inf\{|x - z| : z \in \mathcal{O}_I\}$.

## IV. EXISTENCE AND STABILITY OF A HYBRID LIMIT CYCLE

Unfortunately, we cannot invoke the existing results of the literature on the existence and the stability of limit cycles for hybrid systems to prove the desired properties for system $\mathcal{H}_I$ given $I \in \mathbb{R}_{>0}$. Indeed, as we will see in this section, the periodic solutions do not exhibit the right number of jumps over one "hybrid period" required by<fnref n="2" /> [21], [24] for system $\mathcal{H}_I$. Moreover, we aim at concluding stronger properties, namely an exponential stability property as in Definition 2. We provide instead an analysis tailored to system $\mathcal{H}_I$.

### A. Forward invariant set $\mathcal{X}_0$

We define the set $\mathcal{X}_0$ as

$$\mathcal{X}_0 := (\mathcal{C} \cup \mathcal{D}) \setminus (\{0\} \times \{0\} \times \{-1, 1\}). \quad (13)$$

The set $\mathcal{X}_0 \subset \mathcal{X} \subset \mathbb{R}^{n_x}$ simply excludes $(q_1, q_2) = (0, 0)$, see Remark 1. The next proposition establishes that $\mathcal{X}_0$ is forward invariant for system $\mathcal{H}_I$ for any given $I \in \mathbb{R}_{>0}$, as required by Definition 2(i). Therefore, it guarantees that, for any given $I \in \mathbb{R}_{>0}$, all solutions initialized in $\mathcal{X}_0$ remain in $\mathcal{X}_0$ and thus will not reach $(q_1, q_2) = (0, 0)$. Excluding this point is essential in the sequel to prove the existence of a unique limit cycle for any given $I \in \mathbb{R}_{>0}$.

**Proposition 2.** Given any $I \in \mathbb{R}_{>0}$, consider system $\mathcal{H}_I$ in (10). The set $\mathcal{X}_0$ in (13) is forward invariant, i.e., $\text{rge}(x) \subset \mathcal{X}_0$ for all solution $x$ to $\mathcal{H}_I$ initialized in $\mathcal{X}_0$.

In addition to Proposition 2, for any initial condition in $\mathcal{X}_0$, there exists a unique solution to $\mathcal{H}_I$, as formalized next.

**Lemma 1.** Given any $I \in \mathbb{R}_{>0}$, consider system $\mathcal{H}_I$ in (10). For any $x_0 \in \mathcal{X}_0$, there exists a unique maximal solution initialized at $x_0$.

<footnote n="1">¹Function $\alpha$ may not be linear, as customary when referring to exponential stability in the literature. As the decaying term is exponential, we argue that the stated stability property can still be qualified as being uniform and exponential.</footnote>

<footnote n="2">²By (11), taking $h(x) = \sigma q_1$, we have $\mathcal{C} = \{x \in \mathcal{X} : h(x) \ge 0\}$ and $\mathcal{D} = \{x \in \mathcal{X} : h(x) = 0, L_f h(x) \le 0\}$ with the notation of [21], [24]. In this case, these references require $N^* = 1$, while we will see in Theorem 1 that $N^* = 2$.</footnote>
Note that, if $x_0 \notin \mathbb{X}_0$, then solutions are not-unique, see
Remark 1.

## B. Solution expression

The flow map of system (10) is a linear and time-invariant
system, and the jump map generates increases or decreases
of $q_2$ depending on its sign of fixed amplitude. Given these
two properties, we can provide expressions of the solution
to system (10) for any given $I$, which will be useful for
determining the existence and stability of a hybrid limit cycle
in the sequel. Note that, the property guaranteed by Lemma 1
is important even if the results we prove in the sequel rely
on the explicit expression of the solution. Indeed, Lemma 1
ensures that the solution computed in the next proposition is
unique.

**Proposition 3.** Given any $I \in \mathbb{R}_{>0}$, consider system $\mathcal{H}_I$ in (10). Let $x = (q_1, q_2, \sigma)$ be a solution initialized in $\mathbb{X}_0$ as defined in (13). For any $(t, j) \in \text{dom } x$, for all $s \in [0, t_1]$,

$$
\begin{align}
q_1(s, 0) &= q_1(0, 0)e^{as} \cos(b s) + c_0(x(0, 0))e^{as} \sin(b s) \nonumber \\
q_2(s, 0) &= q_2(0, 0)e^{as} \cos(b s) \nonumber \\
&\quad + (c_0(x(0, 0))a - q_1(0, 0)b)e^{as} \sin(b s) \nonumber \\
\sigma(s, 0) &\in \text{SGN}(q_1(0, 0)), \tag{14}
\end{align}
$$

and, for each $i \in \{1, 2, \dots, j\}$, and all $s \in [t_i, t_{i+1}]$,

$$
\begin{align}
q_1(s, i) &= c(q_2(t_i, i-1))e^{a(s-t_i)} \sin(b(s-t_i)) \nonumber \\
q_2(s, i) &= c(q_2(t_i, i-1))e^{a(s-t_i)} [a \sin(b(s-t_i)) \nonumber \\
&\phantom{=} + b \cos(b(s-t_i))] \nonumber \\
\sigma(s, i) &= \begin{cases} (-1)^t \sigma(0, 0) & \text{if } (q_1(0, 0), q_2(0, 0)) \neq 0 \\ (-1)^{i-1} \text{sign}(q_2(0, 1)) & \text{otherwise} \end{cases} \tag{15}
\end{align}
$$

with $0 = t_0 \le t_1 \le \dots \le t_{j+1} = t$ satisfy dom $x \cap ([0, t] \times \{0, 1, \dots, j\}) = \bigcup_{i=0}^{j} [t_i, t_{i+1}] \times \{i\}$, $a := -\frac{\alpha}{2} \in (-1, 0)$, $b := \frac{\sqrt{4-\alpha^2}}{2} \in (0, 1)$, $c_0(x(0, 0)) = \frac{q_2(0, 0) - q_1(0, 0)}{b} a \in \mathbb{R}$ and $c(z) := \frac{z+\text{Isign}(z)}{b} \in \mathbb{R}\{0\}$ for any $z \in \mathbb{R}$.

Proposition 3 allows concluding that, after a first jump, the amount of continuous time between two consecutive jumps is fixed and the same for any solution to system (10) initialized in $\mathbb{X}_0$. Therefore, the pulses occurs periodically. However, the event-based nature of the neuromorphic controller is preserved because the events are generated whenever the pendulum crosses the resting position, which is therefore not a time-based controller.

**Lemma 2.** Given any $I \in \mathbb{R}_{>0}$, consider system $\mathcal{H}_I$ in (10). For any solution $x$ initialized in $\mathbb{X}_0$ and any $(t, j)$, $(t', j) \in \text{dom } x$ such that $j \ge 1$ and $(t, j-1)$, $(t', j+1) \in \text{dom } x$, it holds that $t' - t = \frac{\pi}{b}$.

## C. Existence and uniqueness of hybrid limit cycle

In the next theorem we establish the existence of a periodic
solution for system (10) according to Definition 1.

**Theorem 1.** Given any $I \in \mathbb{R}_{>0}$, system $\mathcal{H}_I$ in (10) generates a $(2\frac{\pi}{b}, 2)$-periodic solution, which defines a compact hybrid limit cycle denoted $\mathcal{O}_I$.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Angular position (red), velocity (blue) and $\sigma$ (yellow) with nonlinear dynamics and initial condition $(\frac{\pi}{3}, 2, -1)$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Hybrid limit cycle with nonlinear dynamics with different initial conditions.</figcaption></figure>

Not only system $\mathcal{H}_I$ generates $(2\frac{\pi}{b}, 2)$-periodic solutions for any given $I \in \mathbb{R}_{>0}$ as established in Theorem 1, but all such periodic solutions that are initialized in $\mathbb{X}_0$ have the same range and define the same, unique hybrid limit cycle $\mathcal{O}_I$, as shown in the next theorem.

**Theorem 2.** Given any $I \in \mathbb{R}_{>0}$, consider system $\mathcal{H}_I$. The set $\mathcal{O}_I$ in Theorem 1 is the unique hybrid limit cycle in $\mathbb{X}_0$.

## D. Stability properties of the hybrid limit cycle

In the next theorem we prove the existence and a stability
property for a hybrid limit cycle of system (10).

**Theorem 3.** Given any $I \in \mathbb{R}_{>0}$, consider system $\mathcal{H}_I$. The hybrid limit cycle $\mathcal{O}_I$ in Theorem 1 is uniformly exponentially stable on $\mathbb{X}_0$ in (13).

## V. NUMERICAL ILLUSTRATION

We consider the nonlinear dimensionless pendulum dynamics in (1), with $\alpha = 0.5$ and spiking input $u$ defined in (3) with $I = 0.1$. In particular, we consider system (10) where the nonlinear dynamics (4) is considered in the flow map (12), instead of (5). The simulation results obtained with initial conditions $q_1(0, 0) = \frac{\pi}{3}$, $q_2(0, 0) = 2$ and $\sigma(0, 0) = -1$ are depicted in Fig. 2. Fig. 2 shows that the considered solution to the closed-loop pendulum system converges to a hybrid limit cycle, as guaranteed by Theorem 3. Moreover, we have run three simulations considering three different initial conditions. The results are depicted in Fig. 3, where in the blue plot we used $(\frac{\pi}{3}, 2, -1)$, in the red one $(\frac{\pi}{4}, -2, 1)$ and in the yellow plot we considered $(-\frac{\pi}{6}, 1, -1)$. Fig. 3 shows that, for all the three considered initial conditions, the solutions converge to the same hybrid limit cycle consistently with Theorem 3.
## VI. DISCUSSION ON PULSE AMPLITUDE ADAPTATION

A next relevant challenge, mentioned in [11] as well as in the companion paper [23], is to control the maximal amplitude of the hybrid limit cycle by introducing an adaptation loop on the control parameter $I \in \mathbb{R}_{>0}$. In [11], [23] this is done by tuning the neurons activation and analytical guarantees are provided. In this section, we briefly elaborate on this research direction by adding an adaptation loop for $I$ to system $\mathcal{H}_I$ in (10), together with associated numerical simulations showing interesting and promising results. A formal analysis of this augmented hybrid system, investigating how the adaptation loop affects the properties ensured in the previous sections, is left for future work.

### A. Adaptation rule

To control the amplitude of the pendulum oscillations we introduce in this section a “spiky” adaptation loop on the parameter $I \in \mathbb{R}_{>0}$ in (3). Recall that, as discussed in Section II, the spiking torque can be approximately generated using bursting neurons, producing $I$ spikes of unitary amplitude [11]. Therefore, adapting the parameter $I$ corresponds to modifying the duration of the neuronal burst, while keeping the spikes’ amplitude constant. By the pendulum dynamics (4) and the explicit solution in Proposition 3, the maximal amplitude of the oscillations is obtained whenever the angular velocity $q_2$ changes sign, which, according to [11], [23], can be measured using neuromorphic devices. Based on the value of the actual maximal amplitude and the desired one, the parameter $I \in \mathbb{R}_{>0}$ is adapted. In particular, when $q_2$ is equal to 0, a spiking adaptive action with amplitude $\varepsilon \in \mathbb{R}_{>0}$ on the control parameter $I$ is generated, whose sign depends on the sign of the difference between the measured amplitude of the oscillations, given by $\sigma_1 q_1 \in (0, \pi]$, where $\sigma_1$ denotes variable $\sigma$ in the previous sections, and the desired one, denoted by $q_1^* \in (0, \pi)$, i.e., in terms of the hybrid systems’ notation in [22, Section II],

$$I^+ \in \{I - \varepsilon w : w \in \text{SGN}(\sigma q_1 - q_1^*)\} := G_I(q_1, \sigma, I), \quad (16)$$

and, on flows, $\dot{I} = 0$. This learning (adaptation) loop is “spiky” in nature and thus sparse (efficient communications). To add this adaptation to the hybrid system $\mathcal{H}_I$ in a robust way (especially the jump (16) related to $q_2 = 0$), similarly to (7) and (8) we introduce an auxiliary logic variable $\sigma_2 \in \{-1, 1\}$ to robustify the detection of the condition $q_2 = 0$. In particular, on flows, $\dot{\sigma}_2 = 0$, and at jumps, when $q_2 = 0$, $\sigma_2^+ = -\sigma_2$. When $q_2$ changes sign and thus the parameter $I \in \mathbb{R}_{>0}$ is updated according to (16), the pendulum angular position, velocity and the auxiliary signal $\sigma$, which in the augmented hybrid model we call $\sigma_1$, are not modified, i.e., $q_1^+ = q_1$, $q_2^+ = q_2$ and $\sigma_1^+ = \sigma_1$.

Using (4), (6), (7), (8) and the equations above in the next section we define an augmented hybrid model, where a jump corresponds to a spiking torque action, as in (10), or to an update of the parameter $I \in \mathbb{R}_{>0}$, according to (16).

### B. Augmented hybrid model

Contrary to $\mathcal{H}_I$, we consider the nonlinear pendulum dynamics during flow (4) and not the linearized one (5), although (5) can be used similarly.

Given a desired maximal angular amplitude $q_1^* \in (0, \pi)$, we denote the concatenated state variable

$$\begin{aligned} \bar{x} &:= (q_1, q_2, \sigma_1, \sigma_2, I) \in \bar{X} \subset \mathbb{R}^{n_{\bar{x}}} \\ \bar{X} &:= \mathbb{R}^2 \times \{-1, 1\}^2 \times \mathbb{R}_{\ge 0}, \end{aligned} \quad (17)$$

with $n_{\bar{x}} := 5$. We propose the hybrid model

$$\bar{H} : \left\{ \begin{array}{ll} \dot{\bar{x}} = \bar{f}(\bar{x}) & \bar{x} \in \bar{C} \\ \bar{x}^+ \in \bar{G}(\bar{x}) & \bar{x} \in \bar{D}, \end{array} \right. \quad (18)$$

with

$$\left\{ \begin{aligned} \bar{C} &:= \{\bar{x} \in \bar{X} : \sigma_1 q_1 \ge 0, \sigma_2 q_2 \ge 0\} \\ \bar{D} &:= \bar{D}_1 \cup \bar{D}_2 \\ \bar{D}_1 &:= \{\bar{x} \in \bar{X} : \sigma_1 q_1 = 0, \sigma_1 q_2 \le 0, \sigma_2 q_2 \ge 0\} \\ \bar{D}_2 &:= \{\bar{x} \in \bar{X} : \sigma_1 q_1 \ge 0, \sigma_2 q_2 = 0, \\ & \quad -\sigma_2 \sin(q_1) \le 0\} \end{aligned} \right.$$

and $\bar{f}(\bar{x}) := (q_2, -\sin(q_1) - \alpha q_2, 0, 0, 0)$ for any $\bar{x} \in \bar{C}$, $\bar{G}(\bar{x}) := \bar{G}_1(\bar{x}) \cup \bar{G}_2(\bar{x})$ for any $\bar{x} \in \bar{D}$,

$$\begin{aligned} \bar{G}_1(\bar{x}) := & \left\{ \begin{array}{l} \{(q_1, q_2 + Iz, z, \sigma_2, I) : z \in \text{SGN}(q_2)\}, \quad \bar{x} \in \bar{D}_1 \\ \emptyset, \quad \bar{x} \notin \bar{D}_1 \end{array} \right\} \\ \bar{G}_2(\bar{x}) := & \left\{ \begin{array}{l} \{(q_1, q_2, \sigma, -\sigma_2, I - \varepsilon w : w \in \text{SGN}(\sigma q_1 - q_1^*)\}, \\ \emptyset, \quad \bar{x} \in \bar{D}_2 \\ \emptyset, \quad \bar{x} \notin \bar{D}_2. \end{array} \right\} \end{aligned}$$

The next result follows similarly as Proposition 1.

**Proposition 4.** Given any $q_1^* \in (0, \pi)$, the following hold for system $\bar{H}$ in (18).

(i) The hybrid basic conditions are satisfied.

(ii) Maximal solutions are complete.

### C. Simulation results

We consider system (18) with $\alpha = 0.5$, $q_1^* = \frac{\pi}{6}$ and $\varepsilon = 0.02$. The simulation results obtained for the pendulum angular position and velocity with initial conditions $q_1(0, 0) = \frac{\pi}{3}$, $q_2(0, 0) = 2$, $\sigma_1(0, 0) = 1$, $\sigma_2(0, 0) = 1$ and $I(0, 0) = 0.1$ are shown in Fig. 4, together with the angular position obtained without adaptation mechanism, i.e., simulating system $\mathcal{H}_I$ with nonlinear flow dynamics from the same initial conditions, with $I \equiv 0.1$. Fig. 4 also depicts the adaptation of the parameter $I \in \mathbb{R}_{>0}$, which is linked to the amplitude of the spiking control actions inducing the limit cycle, and is thus used to control the maximal amplitude of the oscillations. Fig. 4 shows that including the spiking adaptation loop allows generating a hybrid limit cycle with a desired maximal angular amplitude, and thus illustrates the potential of hybrid system models and tools for this purpose. Note that, in view of the fixed amplitude $\varepsilon \in \mathbb{R}_{>0}$ of the spiking adaptation mechanism in (16), we expect a practical convergence property of the maximal amplitude of the oscillations $\sigma q_1$ to the desired one $q_1^*$, as illustrated in the zoom in Fig. 4. A formal analysis of system (18) is left for future work.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Angular position (blue) and velocity (red) with nonlinear dynamics and spiking adaptation loop to adapt the amplitude of the spikes *I*. Angular position (black) without adaptation (constant *I* = 0.1).</figcaption></figure>

## VII. CONCLUSION

We have illustrated how hybrid system theoretic tools can be exploited to show the existence of a limit cycle and analyze its stability property for a pendulum controlled by a neuromorphic device. We relied on two main assumptions for this purpose: we have linearized the pendulum dynamics and abstracted the effect of bursting neurons, see [11], as Dirac impulses. We plan to relax these assumptions in the future by notably exploiting monotonocity properties of the effect of the bursting neurons activation on the angular velocity. We also plan to formalize the results hinted in Section VI by exploiting hybrid singularly perturbed techniques (see, e.g., [25], [26]) thereby leading to a rigorous learning mechanism of the amplitude *I*, which is related to the length of activation of the bursting neurons in [11].

## ACKNOWLEDGEMENTS

We would like to thank Alessio Franci, Fernando Castaños, and Taisia Medvedeva for discussions on the complementary paper [23] and for providing their insights on the present study. We also thank Mattia Giaccagli and Pietro Lorenzetti for initial discussions, which have led to this paper.

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