Input-to-State Stable Coupled Oscillator Networks for Closed-form Model-based Control in Latent Space

General response to Reviewer Ys99

> Finally, it's the same story for deriving stability from the single oscillators to the network. Of course you can make the stiffness matrix negative definite to achieve a non-stable network but again, if all parameters are positive (definite), meaning defined so that you would expect the system to be stable, I think that the system is stable. I'd change me opinion if you can provide a counter example where the parameters are positive (definite) but would lead to an unstable system / network. We appreciate the direct nature of the Reviewer's comment, which rises indeed a compelling point with which however we still disagree. In our next response, we will try again to convince the Reviewer of our argument, although proving that something is _not trivial_ is quite a challenging task, as triviality is very subjective. So, before delving into it, we would like to take the chance of zooming out. Indeed, at the moment, we see the risk that we are debating on a relatively narrow point while it may be that we could essentially agree on several key aspects. Or at least, we believe we can work to find a common ground on these while we continue in parallel our discussion. 1. We believe that Theorem 2 is the main _stability proof_ contribution of the paper. To the best of our knowledge, we are the first in this community/problem setting to provide explicit input-to-state stability guarantees, including convergence rates into the region of attraction (Theorem 2). We believe that the proof of Theorem 2 is far from trivial. Does the Reviewer disagree on this point? 2. In this sense, we agree with the Reviewer that the proof of Theorem 1 is comparably simpler (although not trivial!) and less impactful than the one of Theorem 2. Would renaming "_Theorem 1_" as a "_Lemma_" or even as a "_Proposition_" better reflect the opinion of the Reviewer on the importance of this contribution? 3. Finally, we realize now that we did not make a great job, with the manuscript and with our previous answers, in conveying the message that: the relatively simplicity of the proof of Theorem 1 is no accident. The proof of Theorem 1 could be derived via _standard_ arguments from the control of mechanical systems, _because_ we designed the CON network in a particular way. So the Theorem 1 statement with its relative simplicity implicitly stresses this point. If the Reviewer agrees on this point, we are happy to include in the revision any remarks or changes that they see fit to better bring this point home. Specifically, the proof in its current form is possible because (a) a coordinate transformation exists that allows us to identify a Lyapunov candidate, and (b) the system has a potential energy. We see these two design choices a contribution by themselves as they allow the proof to be simpler as it could have been otherwise. We want to stress that already small modifications to the network formulation and the underlying assumptions in Theorem 1 would have made the proof much more difficult (or even impossible).

Technical response to Reviewer Ys99: Example 1

> Finally, it's the same story for deriving stability from the single oscillators to the network. Of course you can make the stiffness matrix negative definite to achieve a non-stable network but again, if all parameters are positive (definite), meaning defined so that you would expect the system to be stable, I think that the system is stable. I'd change me opinion if you can provide a counter example where the parameters are positive (definite) but would lead to an unstable system / network. We are now ready to go back to our technical discussion, with the discussed counterexamples. These, hopefully, also better illustrate our point #3. Essentially, our argument here is that even positive-definite stiffness & damping matrices are not sufficient to claim global asymptotic stability. This should answer the Reviewer's direct technical question. In the following, we will give two examples of modifications to the CON network (in the original coordinates) that would have, according to the Reviewer's argument, appeared to be stable but for which global stability proof would be hard to prove. ## Example 1: A system with multiple equilibria and without a valid potential energy We consider the slightly modified network dynamics $\ddot{x} + K x + D \dot{x} + W^{-1} \tanh(Wx + b) = 0$, where $K, D, W \succ 0$ are positive definite matrices. The equilibria of this system are given by the characteristic equation $K \bar{x} + W^{-1} \tanh(W \bar{x} + b) = 0$. For the system to be globally asymptotically stable, it would need to have a single equilibrium point. However, if we simulate the system with $K = [[5.0, -2.2], [-2.2, 1.0] \succ 0$ (positive-definite as symmetric and positive eigenvalues $0.027$ and $5.973$), $D = \mathrm{diag}(0.2, 0.2) \succ 0$, $W = [[1.0, -2.2], [-2.2, 5.0]] \succ 0$ (positive-definite as symmetric and positive eigenvalues $0.027$ and $5.973$), we notice that the system is actually bistable with two attractors at $\bar{x}_1 = [-212.6, -475.2]$ and $\bar{x}_2 = [212.6, 475.2]$. This is illustrated in the time series plot and phase portrait attached at https://anonymous.4open.science/r/neurips24-20062-rebuttal-7770. Therefore, the system is **not** globally asymptotically stable. For $\tau_\mathrm{pot} = -K x -W^{-1} \tanh(Wx + b)$ to be a valid potential force, it would need to satisfy the property $\frac{\partial \tau_\mathrm{pot}}{\partial x} = \left ( \frac{\partial \tau_\mathrm{pot}}{\partial x} \right )^\mathrm{T}$. Therefore, we derive $\frac{\partial \tau_\mathrm{pot}}{\partial x} = -K - W^{-1} \mathrm{diag}(\mathrm{sech}^2(Wx + b)) W$. Its transpose is given by $\left ( \frac{\partial \tau_\mathrm{pot}}{\partial x} \right )^\mathrm{T} = -K - W^\mathrm{T} \mathrm{diag}(\mathrm{sech}^2(Wx + b)) W^{-\mathrm{T}}$. Therefore, $\tau_\mathrm{pot}$ only stems from a potential iff $W^{-\mathrm{T}} = W \rightarrow W^\mathrm{T} W = \mathbb{I}$ (i.e., $W$ is orthogonal). This is not the case for a general positive-definite $W$. This example motivates that even if the matrices are positive-definite, the system can still have multiple equilibria (i.e., lose global asymptotic stability) and not have a valid potential energy function.

Response to Reviewer W9L3 Part 1

> _RW9L3:_ I agree with the authors' view that their method might underperform for systems with highly discontinuous dynamics. Since this paper is framed as a method for prediction and control of systems, with soft robots being only one application area (rather than a method for the control of soft robots), I would appreciate a lucid analysis of classes of systems/application areas in which the method might perform strongly/weakly (including a clear reasoning why; and potentially a remedy left to future work). Please consider including this, both from a systems theory and an application standpoint into the paper. We appreciate the Reviewer's feedback and suggestion. We fully agree with the Reviewer that specific categories or examples of systems for which the proposed method might be suitable or unsuitable should be discussed in the paper. In the following, we will detail the planned changes. ## Planned changes to the Introduction First, we will add the following sentence to the end of the introduction (i.e., Section 1): > The proposed methodology is particularly well-suited for learning the latent dynamics of mechanical systems with continuous dynamics, dissipation, and a single, attractive equilibrium point. Examples of such systems include many soft robots, deformable objects with dominant elastic behavior, other mechanical structures with elasticity, or locally for other mechanical systems such as robotic manipulators, legged robots etc. For these systems, we can fully leverage the structural prior of the proposed latent dynamics including the integrated stability guarantees. If the system is actuated, the learned dynamics can be subsequently exploited for model-based control, as demonstrated in Section 5. ## Planned changes to the _Limitations_ section We will also extend the _Limitations_ section (i.e., Section 6): > **Limitations.** While we think our proposed method shows great potential and opens interesting avenues for future research, there are currently certain limitations. For example, the proposed method of learning (latent) dynamics implicitly assumes that the underlying system adheres to the Markov property (e.g., the full state of the system is observable), that it can be approximated by a system with mechanical structure, and that it has an isolated, globally asymptotically stable equilibrium. This is, for example, the case for many mechanical systems (e.g., some continuum soft robots, deformable objects, and elastic structures) with continuous dynamics, convex elastic behaviour, dissipation and whose time-dependent effects (e.g., viscoelastcity, hysteresis) are negligible. Even if these conditions are not met globally, the method can be applied to model the local behavior around an asymptotic equilibrium point of the system (e.g., robotic manipulators, legged robots) with added stability benefits for out-of-distribution samples. Alternatively, the method could be extended to relax some of these assumptions, e.g., by allowing for multiple equilibria, zero damping, or by incorporating additional terms to capture discontinuous dynamics (e.g., stick-slip models) or period motions (e.g., limit cycles such as the Van der Pol oscillator). For some physical systems, such as nonholonomic systems, partially observable systems, or systems with non-Markovian properties, the proposed method might not be suitable. Examples of such systems include mobile robots and systems with hidden states or delayed observations. Finally, the application of this method to non-physical systems, such as financial systems, social networks, or other complex systems, is out of the scope of this work. _THE DISCUSSION OF OTHER LIMITATIONS OF THIS PAPER WHICH ARE ALREADY MENTIONED IN THE INITIAL PAPER SUBMISSION CONTINUES HERE..._

Technical response to Reviewer Ys99: Example 2

## Example 2: A system where the W-coordinate transformation does not help The proof underlying Theorem 1 is only (relatively) simple as the coordinate transformation $x_\mathrm{w} = W x$ helps us to identify the potential energy function in the $\mathcal{W}$ coordinates, where the hyperbolic nonlinearity operates elementwise. However, this is not always possible. For example, consider the system $\ddot{x} + K x + D \dot{x} + W^{\mathrm{T}} \tanh(Wx + b) = 0$, where $K$, $D$, $W \succ 0$ are positive definite matrices. Unfortunately, the unactuated dynamics in the \mathcal{W} coordinates are now given by $M_w \ddot{x}_w + K_w x_w + D_w \dot{x}_w+ W^T \tanh(x_w + b) = 0$, for which we cannot easily derive a potential energy function (i.e., integrate), as the hyperbolic term $W^\mathrm{T} \tanh(x_\mathrm{w} + b)$ is not (easily) separable. Remark: In contrast to Example 1, this system actually has a single equilibrium point. This example motivates why we chose the proposed CON network architecture such that we can (easily) derive the kinetic and potential energy terms, subsequently prove GAS & ISS, and perform model-based control.

Response to Reviewer W9L3 Part 2

## Planned additions to the Appendix Furthermore, we could envision adding a more detailed discussion (similar to the one below), or alternatively a table, to the Appendix: ### Systems for which we would expect the proposed method to work - **Mechanical systems with continuous dynamics, dissipation, and a single, attractive equilibrium point:** The proposed method is a very good fit for mechanical systems with continuous dynamics, dissipation, and a single, attractive equilibrium point. In this case, the real system and the latent dynamics share both the energetic structure and stability guarantees. Examples of such systems include many soft robots, deformable objects with dominant elastic behavior, other mechanical structures with elasticity, etc. - **Local modeling of (mechanical) systems that do not meet the global assumptions:** Even if the global assumptions of the proposed method are not met, the method can still be applied to model the local behavior around a local asymptotic equilibrium point of the system (i.e., in the case of multi-stability). For example, the method could be used to model the behavior of a robotic leg locally in contact with the ground, a cobot's interaction with its environment, etc. ### Systems for which we could envision the proposed method to work under (minor) modifications - **Mechanical systems without dissipation:** The proposed method would currently not work well for mechanical systems without any dissipation, as (a) the original system will likely not have a globally asymptotically stable equilibrium point, and more importantly, (b) we currently force the damping learned in latent space to be positive definite. However, these systems are not common in practice as friction and other dissipation mechanisms are omnipresent, and the proposed method can learn very small damping values (e.g., the mass-spring+friction system). A possible remedy could be to relax the positive definiteness of the damping matrix in the latent space, allowing for zero damping. This would allow the method to work for systems without dissipation, such as conservative systems. Examples of such systems include a mass-spring system without damping, the n-body problem, etc. - **(Mechanical) systems with discontinuous dynamics:** The proposed method might underperform for systems with highly discontinuous dynamics, such as systems with impacts, friction, or other discontinuities. In these cases, the latent dynamics might not capture the real system's behavior accurately, and the control performance of feedforward + feedback will very likely be worse than pure feedback. Again, the method should be able to capture local behavior well. A possible remedy for learning global dynamics could be to augment the latent dynamics with additional terms that capture the discontinuities, such as contact and friction models (e.g., stick-slip friction). - **(Mechanical) systems with multiple equilibrium points:** The original system having multiple equilibria conflicts with the stability assumptions underlying the proposed CON latent dynamics. In this case, as, for example, seen on the pendulum+friction and double pendulum + friction results, the method might work locally but will not be able to capture the global behavior of the system. A possible remedy could be to relax the global stability assumptions of the CON network. For example, the latent dynamics could be learned in the original coordinates of CON while allowing $W$ also to be negative definite. This would allow the system to have multiple equilibria & attractors. Examples of such systems include a robotic arm under gravity, pendula under gravity, etc. - **(Mechanical) systems with periodic behavior:** The proposed method will likely not work well for systems with periodic behavior, as they do not have a single, attractive equilibrium point. Examples of such systems include a mass-spring system with a periodic external force, a pendulum with a periodic external force, some chemical reactions, etc. Again, it is likely possible to apply the presented method to learning a local behavior (i.e., not completing the full orbit). A possible remedy could be to augment the latent dynamics with additional terms that capture the periodic behavior, such as substituting the harmonic oscillators with Van der Pol oscillators to establish a limit cycle or a supercritical Hopf bifurcation.

Response to Reviewer W9L3 Part 3

### Systems for which we would not expect the proposed method to work - **Nonholonomic systems:** The proposed method likely would not work well for nonholonomic systems, as both structure (e.g., physical constraints) and stability characteristics would not be shared between the real system and the latent dynamics. Examples of such systems include vehicles, a ball rolling on a surface, and many mobile robots. - **Partially observable and non-markovian systems:** As the CON dynamics are evaluated based on the latent position and velocity encoded by the observation of the current time step and the observation-space velocity, we implicitly assume that the system is (a) fully observable, and (b) fulfills the Markov property. This assumption might not hold for partially observable systems, such as systems with hidden states or systems with delayed observations. Examples of such cases include settings where the system is partially occluded or in situations without sufficient (camera) perspectives covering the system. Furthermore, time-dependent material properties, such as viscoelasticity or hysteresis, that are present and significant in some soft robots and deformable objects are not captured by the method in its current formulation.