# Modèle passif minimal d'instrument musical auto-oscillant à configuration variable en temps

Thomas Hélie, Maximino Linares, Guillaume Doras

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To cite this version:

Thomas Hélie, Maximino Linares, Guillaume Doras. Modèle passif minimal d'instrument musical auto-oscillant à configuration variable en temps. CFA 2025 - 17e Congrès Français d'Acoustique, Apr 2025, Paris, France. hal-05228704

HAL Id: hal-05228704

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Submitted on 28 Aug 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Modèle passif minimal d'instrument musical auto-oscillant à configuration variable en temps

Thomas Hélie, Maximino Linares, Guillaume Doras

Laboratoire STMS (IRCAM, CNRS, Sorbonne Université), UMR9912

Congrès Français d'Acoustique 2025

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Context & Motivation

1 PhD, Linares (2023-26)

[IRCAM, SU-SCAI]

Musical instrument synthesis using physics-consistent neural networks

2 ANR AVATARS (2022-27)

[IRCAM, LMA, GIPSA-lab, CHU-Liège]

Artificial Voice production:

Control of bio-inspired port-hAmilToniAn numeRical and mechatronic modelS

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1 PhD, Linares (2023-26)

[IRCAM, SU-SCAI]

Musical instrument synthesis using physics-consistent neural networks

CFA25

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database of mastered physical signals ?

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first tests on physics-constrained learning approaches ?

2 ANR AVATARS (2022-27)

[IRCAM, LMA, GIPSA-lab, CHU-Liège]

Artificial Voice production:
Control of bio-inspired port-hAmilToniAn numeRical and mechatronic modelS

CFA25

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energy-consistent keys in the regime control of the larynx ?

(with time-varying configurations of muscles)

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Toy model to study bifurcation & continuation method ?

(a small step towards LMA/LAUM [Silva, Terrien, Vergez et coll.])

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1 PhD, Linares (2023-26)

[IRCAM, SU-SCAI]

Musical instrument synthesis using physics-consistent neural networks

CFA25

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database of mastered physical signals ?

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first tests on physics-constrained learning approaches ?

2 ANR AVATARS (2022-27)

[IRCAM, LMA, GIPSA-lab, CHU-Liège]

Artificial Voice production:
Control of bio-inspired port-hAmilToniAn numeRical and mechatronic modelS

CFA25

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energy-consistent keys in the regime control of the larynx ?

(with time-varying configurations of muscles)

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Toy model to study bifurcation & continuation method ?

(a small step towards LMA/LAUM [Silva, Terrien, Vergez et coll.])

Physical grounds + Minimality = useful to understand, study and test

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[1] Benade (1976) Fundamentals of musical acoustics. Oxford Univ. Press.

[4] Kergomard (1995) Reed-instrument oscillations (...) *CISM*.

[2] Fletcher (1979) Air flow and sound generation (...). *Ann. Rev. Fluid Mech.*

[5] Debut & Kergomard (2004) Clarinet as Van der Pol oscillator (...) *JCA*.

[3] McIntyre et al. (1983) Oscillations (...) of musical instruments. *JASA*

[6] Chattopadhyay et al. (2018) Hannay angle (...) in dissipative systems. *Phys. Rev.*

Chaine et Kergomard (2008) Acoustique des instruments de musique. Belin.

p.400 **Généralités sur les auto-oscillations** (...) Les premières tentatives datent de (...) Helmholtz et Rayleigh (...)

p.401 En 1983, **Mc Intyre et coll.**
publièrent un article retentissant
tentant d'unifier les (...) types
d'instruments à l'aide de deux
éléments seulement

+see [1-5]

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[1] Benade (1976) Fundamentals of musical acoustics. Oxford Univ. Press.

[4] Kergomard (1995) Reed-instrument oscillations (...) *CISM*.

[2] Fletcher (1979) Air flow and sound generation (...). *Ann. Rev. Fluid Mech.*

[5] Debut & Kergomard (2004) Clarinet as Van der Pol oscillator (...) *JCA*.

[3] McIntyre et al. (1983) Oscillations (...) of musical instruments. *JASA*

[6] Chattopadhyay et al. (2018) Hannay angle (...) in dissipative systems. *Phys. Rev.*

Chaine et Kergomard (2008) Acoustique des instruments de musique. Belin.

p.400 Généralités sur les auto-oscillations (...) Les premières tentatives datent de (...) Helmholtz et Rayleigh (...)

p.401 En 1983, Mc Intyre et coll. publient un article retentissant tentant d'unifier les (...) types d'instruments à l'aide de deux éléments seulement

+see [1-5]

<figure><img src="image_57.png" /><figcaption>FIG. 1. Block diagram of a fairly general musical oscillator, idealized as an energetically active nonlinear element coupled to an energetically passive linear element.</figcaption></figure>

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[1] Benade (1976) Fundamentals of musical acoustics. Oxford Univ. Press.

[4] Kergomard (1995) Reed-instrument oscillations (...) *CISM*.

[2] Fletcher (1979) Air flow and sound generation (...). *Ann. Rev. Fluid Mech.*

[5] Debut & Kergomard (2004) Clarinet as Van der Pol oscillator (...) *JCA*.

[3] McIntyre et al. (1983) Oscillations (...) of musical instruments. *JASA*.

[6] Chattopadhyay et al. (2018) Hannay angle (...) in dissipative systems. *Phys. Rev.*

Chaigne et Kergomard (2008) Acoustique des instruments de musique. Belin.

p.400 Généralités sur les auto-oscillations (...) Les premières tentatives datent de (...) Helmholtz et Rayleigh (...)

p.401 En 1983, Mc Intyre et coll. publient un article retentissant tentant d'unifier les (...) types d'instruments à l'aide de deux éléments seulement

+see [1-5]

<figure><img src="image_71.png" /><figcaption>FIG. 1. Block diagram of a fairly general musical oscillator, idealized as an energetically active nonlinear element coupled to an energetically passive linear element.</figcaption></figure>

The simplest used oscillator = Van der Pol (VdP) [4-5 & many]

Passivity ? Hamiltonian modelling (wikipedia: Van der Pol & [6]) with 2+2 artificial states, $H \neq$ physical energy $\Rightarrow$ wrong way

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Issues & objectives

* Develop the **simplest**, energy-consistent (passive), **self-oscillating** system, as a **minimal**, yet physically meaningful, **toy model** of a **self-sustained** instrument

* Use the concept of a **toy model** to isolate **fundamental mechanisms**

* Investigate **time-varying configurations** within an **energy-consistent framework**, **to play notes** through a **physically meaningful interaction** with the musician

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## Issues & objectives

* Develop the **simplest, energy-consistent (passive), self-oscillating** system, as a **minimal, yet physically meaningful, toy model** of a **self-sustained** instrument

* Use the concept of a **toy model** to isolate **fundamental mechanisms**

* Investigate **time-varying configurations** within an **energy-consistent framework**, **to play notes** through a **physically meaningful interaction** with the musician

## Questions

Q1 What is the simplest passive **model** of a self-oscillating system?

Q2 Can this system be represented by a simple **electric circuit analogy**? (& interpretations in **aero-acoustics** or **vibro-acoustics**)

Q3 Can all **parameters** be varied over **time** while ensuring energy consistency (and how) ?

Q4 What are the **strategies for frequency variation**?

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1 Context and motivation

2 Energy-consistent tools: Port-Hamiltonian Systems (PHS) & Shifting

3 Simplest model and interpretations (Q1-Q2)

4 Time-varying parameters and strategies (Q3-Q4)

5 Conclusions and perspectives

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1. Context and motivation

2. Energy-consistent tools: Port-Hamiltonian Systems (PHS) & Shifting

3. Simplest model and interpretations (Q1-Q2)

4. Time-varying parameters and strategies (Q3-Q4)

5. Conclusions and perspectives

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A physical system is made of...

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A physical system is made of...

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(i) Energy-storing components

$$E = \sum_{n=1}^{N} e_n \geq 0$$

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A physical system is made of...

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(i) Energy-storing components

$$E = \sum_{n=1}^{N} e_n \geq 0$$

(ii) Memoryless passive components

$$P_{\text{diss}} = \sum_{m=1}^{M} d_m > 0 \text{ (dissipative) or } = 0 \text{ (conservative)}$$

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A physical system is made of...

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(i) Energy-storing components
$E = \sum_{n=1}^{N} e_n \ge 0$

(ii) Memoryless passive components
$P_{\text{diss}} = \sum_{m=1}^{M} d_m > 0$ (dissipative) or $= 0$ (conservative)

(iii) Environment (ports)
$P_{\text{env}} = \sum_{p=1}^{P} s_p$

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A physical system is made of...

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(i) Energy-storing components
$E = \sum_{n=1}^{N} e_n \ge 0$

(ii) Memoryless passive components
$P_{\text{diss}} = \sum_{m=1}^{M} d_m > 0$ (dissipative) or $= 0$ (conservative)

(iii) Environment (ports)
$P_{\text{env}} = \sum_{p=1}^{P} s_p$

+ Conservative connections

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A physical system is made of...

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(i) Energy-storing components

$$E = \sum_{n=1}^{N} e_n \geq 0$$

(ii) Memoryless passive components

$$P_{\text{diss}} = \sum_{m=1}^{M} d_m > 0 \text{ (dissipative) or } = 0 \text{ (conservative)}$$

(iii) Environment (ports)

$$P_{\text{env}} = \sum_{p=1}^{P} s_p$$

+ Conservative connections → sum of received powers is zero

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A physical system is made of...

→ receiver convention

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(i) Energy-storing components
$E = \sum_{n=1}^{N} e_n \ge 0$
→ store energy

(ii) Memoryless passive components
$P_{\text{diss}} = \sum_{m=1}^{M} d_m > 0$ (dissipative) or $= 0$ (conservative)
→ receive power

(iii) Environment (ports)
$P_{\text{env}} = \sum_{p=1}^{P} s_p$
+ Conservative connections → sum of received powers is zero
$P_{\text{stored}} + P_{\text{diss}} + P_{\text{env}} = 0$ with $P_{\text{stored}} = \dot{E}$ (power balance)
$\ge 0$

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A physical system is made of...

→ receiver convention

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(i) **Energy-storing components** → store energy
$E = \sum_{n=1}^{N} e_n \ge 0$

(ii) **Memoryless passive components** → receive power
$P_{\text{diss}} = \sum_{m=1}^{M} d_m > 0$ (dissipative) or $= 0$ (conservative)

(iii) **Environment (ports)** → receive power
$P_{\text{env}} = \sum_{p=1}^{P} s_p$
+ Conservative connections → sum of received powers is zero
$P_{\text{stored}} + P_{\text{diss}} + P_{\text{env}} = 0$ with $P_{\text{stored}} = \dot{E}$ (power balance)
$\ge 0$

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A physical system is made of...

→ receiver convention

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(i) Energy-storing components
$E = H(x) = \sum_{n=1}^{N} H_n(x_n) \ge 0$

→ store energy

(ii) Memoryless passive components
$P_{\text{dis}} = z(w)^T w = \sum_{m=1}^{M} z_m(w_m) w_m \ge 0$
(effort × flow : force × velocity, voltage × current, etc)

→ receive power

(iii) Environment (ports)
$P_{\text{env}} = u^T y = \sum_{p=1}^{P} u_p y_p$

→ receive power

+ Conservative connections → sum of received powers is zero
$$ \underbrace{\nabla H(x)^T \dot{x}}_{P_{\text{stored}}=dE/dt} + \underbrace{z(w)^T w + u^T y}_{\ge 0} = 0 \quad (\text{power balance}) $$

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A physical system is made of...

→ receiver convention

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(i) **Energy-storing components** → store energy

$$E = H(x) = \sum_{n=1}^{N} H_n(x_n) \ge 0$$

(ii) **Memoryless passive components** → receive power

$$P_{\text{pass}} = z(w)^T w = \sum_{m=1}^{M} z_m(w_m) w_m \ge 0$$

(effort × flow : force × velocity, voltage × current, etc)

(iii) **Environment (ports)** → receive power

$$P_{\text{env}} = u^T y = \sum_{p=1}^{P} u_p y_p$$

$$+ \text{ Conservative connections } \rightarrow \text{sum of received powers is zero} \\ \underbrace{\nabla H(x)^T \dot{x}}_{P_{\text{stored}}=dE/dt} + \underbrace{z(w)^T w + u^T y = 0}_{\text{(power balance)}} \ge 0$$

PHS: Input-State-Output representation

(S: interconnection matrix)

$$\underbrace{\begin{bmatrix} \dot{x} \\ \dot{w} \\ \dot{y} \end{bmatrix}}_{f} = \begin{bmatrix} S_{xx} & S_{xw} & S_{xu} \\ * & S_{ww} & S_{wu} \\ * & * & S_{yu} \end{bmatrix} \underbrace{\begin{bmatrix} \nabla H(x) \\ \mathbf{z}(w) \\ u \end{bmatrix}}_{e} \quad \begin{array}{ll} (i) & \text{storage} \rightarrow \text{differential eq.} \\ (ii) & \text{memoryless} \rightarrow \text{algebraic eq.} \\ (iii) & \text{ports} \rightarrow \text{physical signals} \end{array} \quad (1)$$

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PHS shifting

$$
\textcircled{1} \text{(PHS)} \quad \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}_{f(t)} = S \begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \end{bmatrix}_{e(t)}
$$

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PHS shifting

$$
\textcircled{1} \text{(PHS)} \quad \underbrace{\begin{bmatrix} \dot{\mathbf{x}} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}}_{f(t)} = \mathbf{S} \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \end{bmatrix}}_{e(t)}
$$

Effort

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Energy

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Effort

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Dissipated power

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PHS shifting

$$
\textcircled{1} \quad (\text{PHS}) \quad \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{y} \\ f(t) \end{bmatrix} = S \begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \\ \mathbf{e}(t) \end{bmatrix}
$$

$$
\textcircled{2} \quad \text{Equilibrium var}^* = \{\mathbf{u}^*, \mathbf{x}^*, \mathbf{w}^*, \mathbf{y}^*\}
$$

$$
(\text{PHS})^* \quad \underbrace{\begin{bmatrix} \dot{\mathbf{x}}^* = 0 \\ \mathbf{w}^* \\ \mathbf{y}^* \end{bmatrix}}_{f^*} = S \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}^*) \\ \mathbf{z}(\mathbf{w}^*) \\ \mathbf{u}^* \end{bmatrix}}_{\mathbf{e}^*}
$$

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PHS shifting

$$
\textcircled{1} \quad (\text{PHS}) \quad \begin{bmatrix} \dot{\mathbf{x}} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}_{f(t)} = \mathbf{S} \begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \end{bmatrix}_{e(t)}
$$

$$
\textcircled{2} \quad \text{Equilibrium } \mathrm{var}^* = \{\mathbf{u}^*, \mathbf{x}^*, \mathbf{w}^*, \mathbf{y}^*\}
$$

$$
(\text{PHS})^* \quad \underbrace{\begin{bmatrix} \dot{\mathbf{x}}^* = 0 \\ \mathbf{w}^* \\ \mathbf{y}^* \end{bmatrix}}_{f^*} = \mathbf{S} \begin{bmatrix} \nabla H(\mathbf{x}^*) \\ \mathbf{z}(\mathbf{w}^*) \\ \mathbf{u}^* \end{bmatrix}_{e^*}
$$

$$
\textcircled{3} \quad \text{Fluctuations } \tilde{\mathrm{var}}(t) = \mathrm{var}(t) - \mathrm{var}^*
$$

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PHS shifting

$$
\textcircled{1} \quad (\text{PHS}) \quad \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}_{\mathbf{f}(t)} = \mathbf{S} \begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \end{bmatrix}_{\mathbf{e}(t)}
$$

$$
\textcircled{2} \quad \text{Equilibrium } \mathrm{var}^* = \{\mathbf{u}^*, \mathbf{x}^*, \mathbf{w}^*, \mathbf{y}^*\}
$$

$$
(\text{PHS})^* \quad \underbrace{\begin{bmatrix} \dot{\mathbf{x}}^* = \mathbf{0} \\ \mathbf{w}^* \\ \mathbf{y}^* \end{bmatrix}}_{\mathbf{f}^*} = \mathbf{S} \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}^*) \\ \mathbf{z}(\mathbf{w}^*) \\ \mathbf{u}^* \end{bmatrix}}_{\mathbf{e}^*}
$$

$$
\textcircled{3} \quad \text{Fluctuations } \widetilde{\mathrm{var}}(t) = \mathrm{var}(t) - \mathrm{var}^*
$$
$$
(\mathrm{PHS}) \equiv (\mathrm{PHS}) - (\mathrm{PHS})^*
$$
$$
\mathbf{f}(t) - \mathbf{f}^* = \mathbf{S} (\mathbf{e}(t) - \mathbf{e}^*)
$$

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PHS shifting

$$
\textcircled{1} \text{(PHS)} \quad \underbrace{\begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}}_{f(t)} = S \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \end{bmatrix}}_{e(t)}
$$

$$
\textcircled{2} \text{ Equilibrium var}^* = \{\mathbf{u}^*, \mathbf{x}^*, \mathbf{w}^*, \mathbf{y}^*\}
$$

$$
(\text{PHS})^* \underbrace{\begin{bmatrix} \mathbf{x}^* = 0 \\ \mathbf{w}^* \\ \mathbf{y}^* \end{bmatrix}}_{f^*} = S \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}^*) \\ \mathbf{z}(\mathbf{w}^*) \\ \mathbf{u}^* \end{bmatrix}}_{e^*}
$$

$$
\textcircled{3} \text{ Fluctuations } \tilde{\operatorname{var}}(t) = \operatorname{var}(t) - \operatorname{var}^*
$$
$$
(\tilde{\operatorname{PHS}}) \equiv (\operatorname{PHS}) - (\operatorname{PHS})^*
$$
$$
f(t) - f^* = S (e(t) - e^*)
$$

$$
\underbrace{\begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}}_{\tilde{f}(t)} = S \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}^* + \tilde{\mathbf{x}}) - \nabla H(\mathbf{x}^*) \\ \mathbf{z}(\mathbf{w}^* + \tilde{\mathbf{w}}) - \mathbf{z}(\mathbf{w}^*) \\ \tilde{\mathbf{u}} \end{bmatrix}}_{\tilde{e}(t)}
$$

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PHS shifting

$$
\textcircled{1} \text{(PHS)} \quad \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}_{\mathbf{f}(t)} = \mathbf{S} \begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \end{bmatrix}_{\mathbf{e}(t)}
$$

$$
\textcircled{2} \text{ Equilibrium } \mathrm{var}^* = \{\mathbf{u}^*, \mathbf{x}^*, \mathbf{w}^*, \mathbf{y}^*\}
$$

$$
(\text{PHS})^* = \underbrace{\begin{bmatrix} \mathbf{x}^* = 0 \\ \mathbf{w}^* \\ \mathbf{y}^* \end{bmatrix}}_{\mathbf{f}^*} = \mathbf{S} \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}^*) \\ \mathbf{z}(\mathbf{w}^*) \\ \mathbf{u}^* \end{bmatrix}}_{\mathbf{e}^*}
$$

$$
\textcircled{3} \text{ Fluctuations } \tilde{\mathrm{var}}(t) = \mathrm{var}(t) - \mathrm{var}^*
$$

$$
(\text{PHS}) \equiv (\text{PHS}) - (\text{PHS})^*
$$
$$
\mathbf{f}(t) - \mathbf{f}^* = \mathbf{S}(\mathbf{e}(t) - \mathbf{e}^*)
$$

$$
\underbrace{\begin{bmatrix} \tilde{\mathbf{x}} \\ \tilde{\mathbf{w}} \\ \tilde{\mathbf{y}} \end{bmatrix}}_{\tilde{\mathbf{f}}(t)} = \mathbf{S} \underbrace{\begin{bmatrix} \nabla \tilde{H}_{\mathbf{x}^*}(\tilde{\mathbf{x}}) \\ \tilde{z}_{\mathbf{w}^*}(\tilde{\mathbf{w}}) \\ \tilde{\mathbf{u}} \end{bmatrix}}_{\tilde{\mathbf{e}}(t)}
$$

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Shifted pHs with

$$
\tilde{H}_{x^*}(\tilde{x}) := H(\tilde{x} + \mathbf{x}^*) - \nabla H(\mathbf{x}^*)^\mathrm{T} \tilde{x} - H(\mathbf{x}^*)
$$

$$
\tilde{z}_{w^*}(\tilde{w}) := z(\tilde{w} + w^*) - z(w^*)
$$

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PHS shifting

$$
\textcircled{1} \quad (\text{PHS}) \quad \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{y} \end{bmatrix}_{\mathbf{f}(t)} = \mathbf{S} \begin{bmatrix} \nabla H(\mathbf{x}) \\ \mathbf{z}(\mathbf{w}) \\ \mathbf{u} \end{bmatrix}_{\mathbf{e}(t)}
$$

$$
\textcircled{2} \quad \text{Equilibrium } \mathrm{var}^* = \{\mathbf{u}^*, \mathbf{x}^*, \mathbf{w}^*, \mathbf{y}^*\}
$$

$$
(\text{PHS})^* \underbrace{\begin{bmatrix} \mathbf{x}^* = 0 \\ \mathbf{w}^* \\ \mathbf{y}^* \end{bmatrix}}_{\mathbf{f}^*} = \mathbf{S} \underbrace{\begin{bmatrix} \nabla H(\mathbf{x}^*) \\ \mathbf{z}(\mathbf{w}^*) \\ \mathbf{u}^* \end{bmatrix}}_{\mathbf{e}^*}
$$

$$
\textcircled{3} \quad \text{Fluctuations } \tilde{\mathrm{var}}(t) = \mathrm{var}(t) - \mathrm{var}^*
$$

$$
(\text{PHS}) \equiv (\text{PHS}) - (\text{PHS})^*
$$
$$
\mathbf{f}(t) - \mathbf{f}^* = \mathbf{S}(\mathbf{e}(t) - \mathbf{e}^*)
$$

$$
\begin{bmatrix} \tilde{\mathbf{x}} \\ \tilde{\mathbf{w}} \\ \tilde{\mathbf{y}} \end{bmatrix}_{\tilde{\mathbf{f}}(t)} = \mathbf{S} \begin{bmatrix} \nabla \tilde{H}_{\mathbf{x}^*}(\tilde{\mathbf{x}}) \\ \tilde{z}_{\mathbf{w}^*}(\tilde{\mathbf{w}}) \\ \tilde{\mathbf{u}} \end{bmatrix}_{\tilde{\mathbf{e}}(t)}
$$

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Shifted pHs with

$$
\tilde{H}_{x^*}(\tilde{x}) := H(\tilde{x} + x^*) - \nabla H(x^*)^T \tilde{x} - H(x^*)
$$

$$
\tilde{z}_{w^*}(\tilde{w}) := z(\tilde{w} + w^*) - z(w^*)
$$

Examples: gravity, battery, etc.

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1. Context and motivation

2. Energy-consistent tools: Port-Hamiltonian Systems (PHS) & Shifting

3. Simplest model and interpretations (Q1-Q2)

4. Time-varying parameters and strategies (Q3-Q4)

5. Conclusions and perspectives

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Simplest passive model of a self-oscillating system

(Q1)

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Requirements & Defining "the simplest"

H1a Linear resonator → linear energy-storing components

H1b Minimal state dimension → $dim X = 2$ (one resonance)

H2 NL simplest passive law → scalar, polynomial, minimal degree & $z(w)w \ge 0$

H3a Musician's gesture → physical input signal $u(t)$

H3b $\exists u^*$ s.t. $u(t) = u^*$ → sustained oscillation;

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Requirements & Defining "the simplest"

H1a Linear resonator → linear energy-storing components

H1b Minimal state dimension → $dim X = 2$ (one resonance)

H2 NL simplest passive law → scalar, polynomial, minimal degree & $z(w)w \ge 0$

H3a Musician's gesture → physical input signal $u(t)$

H3b $\exists u^*$ s.t. $u(t) = u^*$ → sustained oscillation;

Electronic circuit representations

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Simplest passive model of a self-oscillating system

(Q1)

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1: Capacitor of charge $q$

2: Coil of magnetic flux $p$

Total energy:

$$H(q, p) = \frac{q^2}{2C_1} + \frac{p^2}{2L_2}$$

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Simplest passive model of a self-oscillating system

(Q1)

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1: Capacitor of charge $q$

2: Coil of magnetic flux $p$

Total energy:

$$H(q, p) = \frac{q^2}{2C_1} + \frac{p^2}{2L_2}$$

PHS equation

$$ \begin{array}{l} \dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} \\ \dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} \\ w = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} \\ y = \begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix} \end{array} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ i_2 & -1 & -1 & 0 \\ i_3 & 0 & -1 & -1 \\ i_4 & 1 & 1 & u \end{pmatrix} \begin{array}{l} \dot{q}H = q/C_1 \\ \dot{p}H = p/L_2 \\ z(w): NL \text{ conductance} \\ u \end{array} $$

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Simplest passive model of a self-oscillating system

(Q1)

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1: Capacitor of charge $q$

2: Coil of magnetic flux $p$

Total energy:

$$H(q, p) = \frac{q^2}{2C_1} + \frac{p^2}{2L_2}$$

PHS equation

$$ \begin{array}{l} \dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} \\ \dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} \\ w = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} \\ y = \begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix} \end{array} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ 1 & -1 & -1 & 0 \\ 1 & 1 & -1 & -1 \\ 0 & 1 & 0 & 1 \end{pmatrix} \begin{array}{l} \dot{q}H = q/C_1 \\ \dot{p}H = p/L_2 \\ z(w): \text{NL conductance} \\ u \end{array} $$

Equilibrium : solve PHS for $\dot{x} = 0$ and $u(t) = u^*$

<table><tr><td>$q^* = 0$</td><td>$p^* = L_2 z(w^*)$</td><td>$w^* = u^*$</td><td>$y^* = z(w^*)$</td></tr></table>

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What nonlinear passive law $w \mapsto z(w)$?

$$z(w)w = vi = P_{\text{diss}} \ge 0$$

Conductance
linear

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Diode
exp-type

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Self-oscillation ?

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What nonlinear passive law $w \mapsto z(w)$ ?

$$z(w)w = vi = P_{\text{diss}} \ge 0$$

Conductance
linear

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Diode
exp-type

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Self-oscillation ?

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What nonlinear **passive** law $w \mapsto z(w)$ ?

$$z(w)w = vi = P_{\text{diss}} \ge 0$$

Conductance
linear

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Diode
exp-type

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Result

* at inflection point: Van der Pol Oscillator

* at any point: passive interpretation

Power-balanced simulation → videos

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<table><thead><tr><th>Domains</th><th colspan="2">flows and efforts</th><th>state q</th><th>parameter C</th><th>state p</th><th>parameter L</th></tr></thead><tbody><tr><td>Electric</td><td>i [A]<br>current</td><td>v [V]<br>voltage</td><td>q<sub>e</sub> [C]<br>charge</td><td>C<sub>e</sub> [F]<br>capacitance</td><td>p<sub>e</sub> [Wb]<br>magnetic flux</td><td>L<sub>e</sub> [H]<br>inductance</td></tr><tr><td>Acoustics</td><td>Q [m<sup>3</sup>.s<sup>-1</sup>]<br>airflow rate</td><td>P [Pa]<br>pressure</td><td>q<sub>a</sub> [m<sup>3</sup>]<br>volume load</td><td>C<sub>a</sub> = <sup>V<sub>0</sub></sup>&frasl;<sub>7P<sub>0</sub></sub> [m<sup>3</sup>.Pa<sup>-1</sup>]<br>acoustic compliance</td><td>p<sub>a</sub> [Pa]<br>acoustic momentum</td><td>L<sub>a</sub> = <sup>ρ<sub>0</sub>l<sub>0</sub></sup>&frasl;<sub>A<sub>0</sub></sub> [kg.m<sup>-4</sup>]<br>acoustic inertance</td></tr><tr><td>Mechanics</td><td>V [m.s<sup>-1</sup>]<br>velocity</td><td>F [N]<br>force</td><td>q<sub>e</sub> [m]<br>elongation</td><td>C<sub>m</sub> = <sup>1</sup>&frasl;<sub>K<sub>m</sub></sub> [m.N<sup>-1</sup>]<br>mechanical compliance</td><td>p = m<sub>0</sub>V [kg.m.s<sup>-1</sup>]<br>momentum</td><td>L<sub>m</sub> = m<sub>0</sub> [kg]<br>inertial mass</td></tr></tbody></table>

<figure><img src="image_474.png" /></figure>
<figure><img src="image_475.png" /></figure>

<figure><img src="image_476.png" /></figure>

<figure><img src="image_477.png" /></figure>

acoustics

<figure><img src="image_478.png" /></figure>

<table><thead><tr><th>Domains</th><th>flows and efforts</th><th>state q</th><th>parameter C</th><th>state p</th><th>parameter L</th></tr></thead><tbody><tr><td>Electric</td><td>i [A]<br>current</td><td>v [V]<br>voltage</td><td>q<sub>e</sub> [C]<br>charge</td><td>C<sub>e</sub> [F]<br>capacitance</td><td>p<sub>e</sub> [Wb]<br>magnetic flux</td><td>L<sub>e</sub> [H]<br>inductance</td></tr><tr><td>Acoustics</td><td>Q [m<sup>3</sup>.s<sup>-1</sup>]<br>airflow rate</td><td>P [Pa]<br>pressure</td><td>q<sub>a</sub> [m<sup>3</sup>]<br>volume load</td><td>C<sub>a</sub> = <sup>V<sub>0</sub></sup>&frasl;<sub>7P<sub>0</sub></sub> [m<sup>3</sup>.Pa<sup>-1</sup>]<br>acoustic compliance</td><td>p<sub>a</sub> [Pa]<br>acoustic momentum</td><td>L<sub>a</sub> = <sup>ρ<sub>0</sub>l<sub>0</sub></sup>&frasl;<sub>A<sub>0</sub></sub> [kg.m<sup>-4</sup>]<br>acoustic inertance</td></tr><tr><td>Mechanics</td><td>V [m.s<sup>-1</sup>]<br>velocity</td><td>F [N]<br>force</td><td>q<sub>e</sub> [m]<br>elongation</td><td>C<sub>m</sub> = <sup>1</sup>&frasl;<sub>K<sub>m</sub></sub> [m.N<sup>-1</sup>]<br>mechanical compliance</td><td>p = m<sub>0</sub>V [kg.m.s<sup>-1</sup>]<br>momentum</td><td>L<sub>m</sub> = m<sub>0</sub> [kg]<br>inertial mass</td></tr></tbody></table>

<figure><img src="image_479.png" /></figure>
<figure><img src="image_480.png" /><figcaption>acoustics</figcaption></figure>

<figure><img src="image_481.png" /><figcaption>mechanics</figcaption></figure>

<figure><img src="image_482.png" /></figure>

<figure><img src="image_483.png" /></figure>

<table><thead><tr><th>Domains</th><th>flows and efforts</th><th>state q</th><th>parameter C</th><th>state p</th><th>parameter L</th></tr></thead><tbody><tr><td>Electric</td><td>i [A]<br>current</td><td>v [V]<br>voltage</td><td>q<sub>e</sub> [C]<br>charge</td><td>C<sub>e</sub> [F]<br>capacitance</td><td>p<sub>e</sub> [Wb]<br>magnetic flux</td><td>L<sub>e</sub> [H]<br>inductance</td></tr><tr><td>Acoustics</td><td>Q [m<sup>3</sup>.s<sup>-1</sup>]<br>airflow rate</td><td>P [Pa]<br>pressure</td><td>q<sub>a</sub> [m<sup>3</sup>]<br>volume load</td><td>C<sub>a</sub> = <sup>V<sub>0</sub></sup>&frasl;<sub>7P<sub>0</sub></sub> [m<sup>3</sup>.Pa<sup>-1</sup>]<br>acoustic compliance</td><td>p<sub>a</sub> [Pa]<br>acoustic momentum</td><td>L<sub>a</sub> = <sup>ρ<sub>0</sub>l<sub>0</sub></sup>&frasl;<sub>A<sub>0</sub></sub> [kg.m<sup>-4</sup>]<br>acoustic inertance</td></tr><tr><td>Mechanics</td><td>V [m.s<sup>-1</sup>]<br>velocity</td><td>F [N]<br>force</td><td>q<sub>e</sub> [m]<br>elongation</td><td>C<sub>m</sub> = <sup>1</sup>&frasl;<sub>K<sub>m</sub></sub> [m.N<sup>-1</sup>]<br>mechanical compliance</td><td>p = m<sub>0</sub>V [kg.m.s<sup>-1</sup>]<br>momentum</td><td>L<sub>m</sub> = m<sub>0</sub> [kg]<br>inertial mass</td></tr></tbody></table>

<figure><img src="image_484.png" /></figure>
1. Context and motivation

2. Energy-consistent tools: Port-Hamiltonian Systems (PHS) & Shifting

3. Simplest model and interpretations (Q1-Q2)

4. Time-varying parameters and strategies (Q3-Q4)

5. Conclusions and perspectives

<figure><img src="image_485.png" /></figure>

<figure><img src="image_486.png" /></figure>

<figure><img src="image_487.png" /></figure>

<figure><img src="image_488.png" /></figure>

<figure><img src="image_489.png" /></figure>

<figure><img src="image_490.png" /></figure>

<figure><img src="image_491.png" /></figure>
Time-varying parameters

(Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} \dot{i}_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix} \quad \partial_q H = q/C_1 \\
\dot{x}_2 = \dot{p} \begin{pmatrix} \dot{i}_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \quad \partial_p H = p/L_2 \\
w = \dot{q} \begin{pmatrix} \dot{i}_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \quad z(w): \text{NL conductance} \\
y = \dot{p} \begin{pmatrix} \dot{i}_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \quad u
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L2}{=} \frac{1}{C_1} \dot{q} \stackrel{L1}{=} \frac{1}{C_1} \left( -\frac{p}{L_2} - z(w) \right) \stackrel{L3}{=} \frac{1}{C_1} \left( -\frac{p}{L_2} - z \underbrace{\left( \frac{q}{C_1} - u \right)}_{\stackrel{L2}{=} \dot{p}} \right)
$$

<figure><img src="image_492.png" /></figure>

<figure><img src="image_493.png" /></figure>

<figure><img src="image_494.png" /></figure>

<figure><img src="image_495.png" /></figure>

<figure><img src="image_496.png" /></figure>

<figure><img src="image_497.png" /></figure>

<figure><img src="image_498.png" /></figure>
Time-varying parameters (Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} \dot{i}_1 \\ v_1 \end{pmatrix} \\
\dot{x}_2 = \dot{p} \begin{pmatrix} v_2 \\ v_2 \end{pmatrix} \\
w = \begin{pmatrix} v_3 \\ v_3 \end{pmatrix} \\
y = \begin{pmatrix} i_4 \\ i_4 \end{pmatrix}
\end{array}
=
\begin{pmatrix}
v_1 & i_2 & i_3 & v_4 \\
-1 & -1 & 0 & 0 \\
1 & 0 & -1 & 0 \\
0 & 1 & 0 & 1
\end{pmatrix}
\begin{array}{l}
\left( \begin{array}{c} v_1 \\ i_2 \\ i_3 \\ v_4 \end{array} \right) \quad \partial_q H = q/C_1 \\
\partial_p H = p/L_2 \\
z(w): \text{NL conductance} \\
u
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L2}{=} \frac{1}{C_1} \dot{q} \stackrel{L1}{=} \frac{1}{C_1} \left( -p/L_2 - z(w) \right) \stackrel{L3}{=} \frac{1}{C_1} \left( -p/L_2 - z \underbrace{\left( q/C_1 - u \right)}_{L2=p} \right)
$$

$$
\bullet \quad \ddot{p} + \frac{1}{C_1} z(\dot{p} - u) + \frac{1}{C_1 L_2} p = 0
$$

<figure><img src="image_499.png" /></figure>
Time-varying parameters

(Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad \partial_q H = q/C_1 \\ \\
\dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad \partial_p H = p/L_2 \\ \\
w = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad z(w) \text{: NL conductance} \\ \\
y = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad u
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L2}{=} \frac{1}{C_1} \dot{q} \stackrel{L1}{=} \frac{1}{C_1} \left( -p/L_2 - z(w) \right) \stackrel{L3}{=} \frac{1}{C_1} \left( -p/L_2 - z\left( \underbrace{q/C_1}_{L2} - u \right) \right)
$$

$$
\bullet \quad \ddot{p} + \frac{1}{C_1} z(\dot{p} - u) + \frac{1}{C_1 L_2} p = 0
$$

→ Oscillator with damping term $z/C_1$ and pulsation $\omega = 1/\sqrt{C_1 L_2}$

<figure><img src="image_500.png" /></figure>
Time-varying parameters

(Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \end{pmatrix} \quad \partial_q H = q/C_1 \\
\dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} -1 & -1 & 0 & 0 \end{pmatrix} \quad \partial_p H = p/L_2 \\
w = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} 1 & 1 & -1 & 0 \end{pmatrix} \quad z(w) = \text{NL conductance} \\
y = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 & 1 \end{pmatrix} \quad u
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L2}{=} \frac{1}{C_1} \dot{q} \stackrel{L1}{=} \frac{1}{C_1} \left( -p/L_2 - z(w) \right) \stackrel{L3}{=} \frac{1}{C_1} \left( -p/L_2 - z\left( \underbrace{q/C_1}_{L2} - u \right) \right)
$$

$$
\bullet \quad \underbrace{\ddot{p} + \frac{1}{C_1} z(\dot{p} - u) + \frac{1}{C_1 L_2} p}_{= 0}
$$

$$
\rightarrow \text{Oscillator with damping term } z/C_1 \text{ and pulsation } \omega = 1/\sqrt{C_1 L_2}
$$

Q3: Can all parameters be varied over time in this ODE ?

$C_1(t) \mid L_2(t) \mid$ parameters $\theta(t)$ in $z$

<figure><img src="image_501.png" /></figure>
Time-varying parameters

(Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \end{pmatrix} \quad \partial_q H = q/C_1 \\
\dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} -1 & -1 & 0 & 0 \end{pmatrix} \quad \partial_p H = p/L_2 \\
w = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} 1 & 1 & -1 & 0 \end{pmatrix} \quad z(w): \text{NL conductance} \\
y = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 & 1 \end{pmatrix} \quad u
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L2}{=} \frac{1}{C_1} \dot{q} \stackrel{L1}{=} \frac{1}{C_1} \left( -p/L_2 - z(w) \right) \stackrel{L3}{=} \frac{1}{C_1} \left( -p/L_2 - z\left( \underbrace{q/C_1}_{L2=\dot{p}} - u \right) \right)
$$

$$
\bullet \quad \underbrace{\ddot{p} + \frac{1}{C_1} z(\dot{p} - u) + \frac{1}{C_1 L_2} p}_{p=0} = 0
$$

$$
\rightarrow \text{Oscillator with damping term } z/C_1 \text{ and pulsation } \omega = 1/\sqrt{C_1 L_2}
$$

Q3: Can all parameters be varied over time in this ODE ?

$C_1(t) \mid L_2(t) \mid \text{parameters } \theta(t) \text{ in } z$
yes

<figure><img src="image_502.png" /></figure>
Time-varying parameters (Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \end{pmatrix} \quad \partial_q H = q/C_1 \\
\dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} -1 & -1 & 0 & 0 \end{pmatrix} \quad \partial_p H = p/L_2 \\
w = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} 1 & 1 & -1 & 0 \end{pmatrix} \quad z(w): \text{NL conductance} \\
y = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 & 1 \end{pmatrix} \quad u
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L_2}{=} \frac{1}{C_1} \dot{q} \stackrel{L_1}{=} \frac{1}{C_1} \left( -p/L_2 - z(w) \right) \stackrel{L_3}{=} \frac{1}{C_1} \left( -p/L_2 - z\left( \underbrace{q/C_1}_{L_2=p} - u \right) \right)
$$

$$
\bullet \quad \ddot{p} + \frac{1}{C_1} z(\dot{p} - u) + \frac{1}{C_1 L_2} p = 0
$$

$$
\rightarrow \text{Oscillator with damping term } z/C_1 \text{ and pulsation } \omega = 1/\sqrt{C_1 L_2}
$$

Q3: Can all parameters be varied over time in this ODE ?

$C_1(t) \quad | \quad L_2(t) \quad | \quad \text{parameters } \theta(t) \text{ in } z$

**no** \quad | \quad **no** \quad | \quad **yes**

<figure><img src="image_503.png" /></figure>
Time-varying parameters

(Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad \partial_q H = q/C_1 \\
\dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad \partial_p H = p/L_2 \\
w = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad z(w) = \text{NL conductance} \\
y = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad u = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix}
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L_2}{=} \frac{1}{C_1} \dot{q} \stackrel{L_1}{=} \frac{1}{C_1} \left( -p/L_2 - z(w) \right) \stackrel{L_3}{=} \frac{1}{C_1} \left( -p/L_2 - z \underbrace{\left( q/C_1 - u \right)}_{L_2 \stackrel{\cdot}{=} p} \right)
$$

$$
\bullet \quad \ddot{p} + \frac{1}{C_1} z(\dot{p} - u) + \frac{1}{C_1 L_2} p = 0
$$

$$
\rightarrow \text{Oscillator with damping term } z/C_1 \text{ and pulsation } \omega = 1/\sqrt{C_1 L_2}
$$

Q3: Can all parameters be varied over time in this ODE ?

$$
\begin{array}{l@{\hspace{1em}}c@{\hspace{1em}}c@{\hspace{1em}}c@{\hspace{1em}}c@{\hspace{1em}}c}
C_1(t) & L_2(t) & \text{parameters } \theta(t) \text{ in } z & \quad & \quad & \\
\text{no} & \text{no} & \text{yes} & \leftarrow & \text{Power balance compatibility} & \\
\text{energy-storing} & \text{power modulation} & & & & \\
\end{array}
$$

<figure><img src="image_504.png" /></figure>
Time-varying parameters

(Q3)

$$
\begin{array}{l}
\dot{x}_1 = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 1 & 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad \partial_q H = q/C_1 \\
\dot{x}_2 = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 1 & 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad \partial_p H = p/L_2 \\
w = \dot{q} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 1 & 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad z(w): \text{NL conductance} \\
y = \dot{p} \begin{pmatrix} i_1 \\ v_2 \\ v_3 \\ i_4 \end{pmatrix} = \begin{pmatrix} v_1 & i_2 & i_3 & v_4 \\ -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & 0 \\ 1 & 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} v_1 \\ i_2 \\ i_3 \\ v_4 \end{pmatrix}, \quad u
\end{array}
$$

ODE

$$
\bullet \quad \ddot{p} \stackrel{L_2}{=} \frac{1}{C_1} \dot{q} \stackrel{L_1}{=} \frac{1}{C_1} \left( -p/L_2 - z(w) \right) \stackrel{L_3}{=} \frac{1}{C_1} \left( -p/L_2 - z \underbrace{\left( q/C_1 - u \right)}_{L_2 \stackrel{\cdot}{=} p} \right)
$$

$$
\bullet \quad \ddot{p} + \frac{1}{C_1} z(\dot{p} - u) + \frac{1}{C_1 L_2} p = 0
$$

$$
\rightarrow \text{Oscillator with damping term } z/C_1 \text{ and pulsation } \omega = 1/\sqrt{C_1 L_2}
$$

Q3: Can all parameters be varied over time in this ODE ?

$C_1(t) \begin{array}{c|c} L_2(t) & \text{parameters } \theta(t) \text{ in } z \\ \text{no} & \text{no} \\ \text{energy-storing} & \text{power modulation} \end{array}$
$\text{yes}$
$\leftarrow$
Power balance compatibility

Physics needs power (through port variables) to modify C₁ or L₂! Why/How ?

<figure><img src="image_505.png" /></figure>
Strategy 1: example on $C_1$ (similar for $L_2$) (Q3-Q4)

**Why?** If $C_1 \neq$ physical constant, it is (or depends on) a configuration variable

<figure><img src="image_506.png" /></figure>

<figure><img src="image_507.png" /></figure>

<figure><img src="image_508.png" /></figure>

<figure><img src="image_509.png" /></figure>

<figure><img src="image_510.png" /></figure>

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**Why?** If $C_1 \neq$ physical constant, it is (or depends on) a configuration variable

Acoustics example: Sliding-piston Helmholtz resonator

* Compliance: $C_1 = \frac{V_0}{\gamma P_0} \rightarrow C_1(l_1) = \frac{A_0 l_1}{\gamma P_0}$

* Energy function: $H_1(q_1, l_1) = \frac{q_1^2}{2C_1(l_1)}$

* Flows: $\dot{q}_1$ (acoustic flow rate), $\dot{l}_1$ (piston velocity)

* Efforts: $\partial_q H_1$ (acoustic pressure), $\partial_l H_1$ (piston restoring force)

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**Why?** If $C_1 \neq$ physical constant, it is (or depends on) a configuration variable

**Acoustics example:** Sliding-piston Helmholtz resonator

* Compliance: $C_1 = \frac{V_0}{\gamma P_0} \rightarrow C_1(l_1) = \frac{A_0 l_1}{\gamma P_0}$

* Energy function: $H_1(q_1, l_1) = \frac{q_1^2}{2C_1(l_1)}$

* Flows: $\dot{q}_1$ (acoustic flow rate), $\dot{l}_1$ (piston velocity)

* Efforts: $\partial_q H_1$ (acoustic pressure), $\partial_l H_1$ (piston restoring force)

**How?** Complete PHS

$$ V_{\text{piston}} \begin{pmatrix} \dot{q} \\ \dot{l} \\ \dot{p} \\ w \\ y \\ y_{\ell} \end{pmatrix} = \begin{pmatrix} \partial_q H & \partial_l H & \partial_p H & z(w) & u & u_{\ell} \\ -1 & -1 & 1 & 1 & & \\ 1 & & & & & \\ 1 & & & & & \\ 0 & 1 & & & & \\ -1 & & & & & \end{pmatrix} \begin{pmatrix} \partial_q H \\ \partial_l H \\ \partial_p H \\ z(w) \\ u \\ u_{\ell} \end{pmatrix} F_{\text{piston} \rightarrow \text{Helmh.}} \\ F_{\text{piston} \rightarrow \text{ext}} \end{pmatrix} $$

The new input $u_{\ell}$ exchanges (mechanical) power to tune $C_1$

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## Strategy 2 to modify the frequency (Q4)

* Nonlinear energy-storing component

+ Shift the equilibrium to locally match the target parameter

### Spring stiffness

<table><thead><tr><th>System</th><th>Energy<br>ℓ ↦ H(ℓ) [m→J]</th><th>Force<br>H'(ℓ) [N=J/m]</th><th>Local stiffness<br>H'' [N/m]</th></tr></thead><tbody><tr><td>Linear</td><td>Kℓ²/2</td><td>Kℓ</td><td>K</td></tr><tr><td>Duffing</td><td>K(ℓ²/2 + ℓ⁴/4ℓ²₀)</td><td>Kℓ(1 + ℓ²/ℓ²₀)</td><td>K(1 + 3ℓ²/ℓ²₀)</td></tr><tr><td>Other laws</td><td>...</td><td></td><td></td></tr></tbody></table>

### Example

Many natural cases. An expected application:

* Larynx/vocal fold configuration

* Muscle modelling (e.g. Hill)

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1. Context and motivation

2. Energy-consistent tools: Port-Hamiltonian Systems (PHS) & Shifting

3. Simplest model and interpretations (Q1-Q2)

4. Time-varying parameters and strategies (Q3-Q4)

5. Conclusions and perspectives

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Answers to the Questions

Q1 **What is the simplest passive model of a self-oscillating system?**

→ A passive system that can be shifted to Van der Pol dynamics.

→ Other non-monotonic nonlinearities are also possible.

Q2 **Can this system be represented by a simple electric circuit analogy?**
(with interpretations in aeroacoustics or vibroacoustics)

→ A Helmholtz resonator (or mass-spring system) excited by a generator combined with a non-monotonic dissipative law.

Q3 **How can parameters be varied over time while ensuring energy consistency?**

→ Only the parameters of passive instantaneous laws can be varied.

→ Otherwise, consider them as configuration states.

Q4 **What are the strategies for frequency variation?**

→ Modify the configuration of energy-storing components via port variables.

→ Use nonlinear energy-storing components combined with equilibrium tuning.

Some Perspectives

● Extension to higher-dimensional systems.

● Control of laryngeal configuration: Hill-type muscle models, nonlinear elasticity of the vocal folds [L. Bailly].

● Integration of a power-balanced perspective into continuation methods.

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Merci !

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