Co-evolutionary control of a class of coupled mixed-feedback systems

TL;DR

The paper addresses stabilizing a desired oscillatory pattern in fixed-structure networks of mixed-feedback oscillators by introducing two distributed, co-evolutionary controllers: a robust, full-information controller that cancels nonlinearities and an neuromodulation-inspired controller relying only on local error. The controllers are implemented as adaptive edges from an added controller node to the plant, enabling synchronization or rhythmic tracking across arbitrary topologies and time-varying adjacency. Theoretical results show global stability for the ideal controller under large integral gain and an $O(1/k)$ error bound for the neuromodulation-inspired controller, complemented by extensive simulations demonstrating robust performance in synchronization tasks and under dynamic network conditions. The work provides practical avenues for neuromorphic and brain-inspired control in systems where internal dynamics and connections are fixed, with potential extensions to spiking neurons and mismatched reference-plant configurations.

Abstract

Oscillatory behavior is ubiquitous in many natural and engineered systems, often emerging through self-regulating mechanisms. In this paper, we address the challenge of stabilizing a desired oscillatory pattern in a networked system where neither the internal dynamics nor the interconnections can be changed. To achieve this, we propose two distinct control strategies. The first requires the full knowledge of the system generating the desired oscillatory pattern, while the second only needs local error information. In addition, the controllers are implemented as co-evolutionary, or adaptive, rules of some edges in an extended plant-controller network. We validate our approach in several insightful scenarios, including synchronization and systems with time-varying network structures.

Figures (9)

• Figure 1: Network (upper) and block diagram (lower) representations of our control problem \ref{['Eq:ClosedLoop_General']}, in which elements of the reference, plant, and control are presented in blue, green, and red, respectively. For the networks, the effect of the controller on the plant is displayed in red, while the rest of the connections are provided for illustrative purposes only, as any topology is allowed in our methodology. For the block diagram, uppercase $({X},{Y})$ and lowercase $({x},{y})$ represent the variables of the reference and of the plant, respectively, from which the error variables $\tilde{x}\coloneq{X}-{x}$, and $\tilde{y}\coloneq{Y}-{y}$ are defined for our analysis.
• Figure 2: The first and second rows and panel (a) correspond to a simulation of \ref{['Eq:ClosedLoop_Ideal']} with $u_i^*$ is given by \ref{['eq:controlfull']} and for $4$ nodes; $x_i$ (plant) is dashed, and $X_i$ (reference) is solid. Panels (a), (b), (c), and (d) show, in logarithmic scale, the mean norm of the error, $\frac{1}{n}|\tilde{x}|$: (a) for $4$ nodes, (b) for $100$ nodes, (c) for $100$ nodes with time-varying adjacency matrices, and (d) for $100$ nodes with a mismatch in the reference's parameters used by the controller. For $t\in(0,300)$, the controller is off, and hence we see the open-loop response. At $t=300$, the controller is turned on, and from thereon, we observe the closed-loop response. We notice, naturally, that for $t>300$, the plant closely follows the reference. See more details in the main text and compare it with Figure \ref{['fig:ideal-hebbian']}.
• Figure 3: Simulation of \ref{['Eq:ClosedLoop_General']} with the neuromorphic inspired controller given by \ref{['eq:u_hebbian']}. The first and second rows and panel (a) correspond to a simulation with $4$ nodes; $x_i$ (plant) is dashed, and $X_i$ (reference) is solid. Panels (a), (b), and (c) show, in logarithmic scale, the mean norm of the error $\frac{1}{n}|\tilde{x}|$: (a) for $4$ nodes, (b) for $100$ nodes, (c) for $100$ nodes with time-varying adjacency matrices. For $t\in(0,300)$, the controller is off, and hence we see the open-loop response. At $t=300$, the controller is turned on, and from thereon, we observe the closed-loop response. We notice, naturally, that for $t>300$, the plant closely follows the reference. See more details in the main text and compare it with Figure \ref{['fig:u_ideal']}.
• Figure 4: Synchronization of the plant with respect to a reference node using the adaptation rule \ref{['eq:ideal_implementation']}-\ref{['eq:ai']}. The first and second rows show the time series of $X_1$ (solid) and $x_i$ (dashed) and the corresponding adaptive weights $a_i$ for $n=4$ nodes while panel (a) shows the corresponding error in logarithmic scale. Panel (b) shows a similar simulation but for $n=100$ nodes. In all simulations, we first show the system in open-loop for $t\in[0,300)$. At $t=300$ the controller is turned on, and we see how the error decreases. For $t\in[500,550]$ a random constant disturbance is added to each node of the plant, hence the observed increase in the error. After $t=550$ we see how the error again decreases, showcasing the recovery from the added disturbance.
• Figure 5: Synchronization of the plant with respect to a reference node using the Hebbian-like adaptation rule \ref{['eq:hebbian_implementation']}. The first and second rows show the time series of $X_1$ (solid) and $x_i$ (dashed) and the adaptive weights $a_i$ for $n=4$ nodes while panel (a) shows the corresponding error in logarithmic scale. Panel (b) shows a similar simulation but for $n=100$ nodes. In all simulations, we first show the system in open-loop for $t\in[0,300)$. At $t=300$ the controller is turned on, and we see how the error decreases. For $t\in[500,550]$, a random constant disturbance is added to each node of the plant, hence the observed increase in the error. After $t=550$, we see how the error again decreases, showcasing the recovery from the added disturbance. The most important distinction with Figure \ref{['fig:ideal_synch']} is that we observe periodic increases of the error, which coincide with the crossings of ${x^{c}}$ at the origin and due to the multiplicative effect of ${x^{c}}$ in the interconnection, namely the term $a_i{x^{c}}$ in each equation of the nodes.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 5
• Remark 6