Controllable Neural Architectures for Multi-Task Control

TL;DR

This work proposes a brain-inspired multi-task control framework where a single nonlinear neural controller with fixed connectivity $W$ and task-dependent bias vectors $d$ regulates multiple linear plants. By exploiting the diagonal per-task linearization $D_i$ realized via bias modulation, the approach aims to approximate a family of linear controllers around equilibria using a compact, gradient-trained model; the optimization minimizes the $\mathcal{H}_2$-norm differences between target and realized linearized dynamics. The paper derives explicit Gramian-based gradients for training, and establishes upper and lower bounds that connect approximation quality to controller size, mode diversity, and system similarity, complemented by numerical demonstrations. The results highlight a scalable, context-adaptive control architecture with theoretical guarantees that inform design trade-offs between controller order and task complexity.

Abstract

This paper studies a multi-task control problem where multiple linear systems are to be regulated by a single non-linear controller. In particular, motivated by recent advances in multi-task learning and the design of brain-inspired architectures, we consider a neural controller with (smooth) ReLU activation function. The parameters of the controller are a connectivity matrix and a bias vector: although both parameters can be designed, the connectivity matrix is constant while the bias vector can be varied and is used to adapt the controller across different control tasks. The bias vector determines the equilibrium of the neural controller and, consequently, of its linearized dynamics. Our multi-task control strategy consists of designing the connectivity matrix and a set of bias vectors in a way that the linearized dynamics of the neural controller for the different bias vectors provide a good approximation of a set of desired controllers. We show that, by properly choosing the bias vector, the linearized dynamics of the neural controller can replicate the dynamics of any single, linear controller. Further, we design gradient-based algorithms to train the parameters of the neural controller, and we provide upper and lower bounds for the performance of our neural controller. Finally, we validate our results using different numerical examples.

Figures (5)

• Figure 1: An illustration of the multi-task control problem considered in this paper, where a set of linear systems is to be regulated by a single non-linear controller. This paper proposes a neural controller with (smooth) ReLU activation function and two parameters. The connectivity matrix $W$ is typically large and remains constant across the different tasks. The bias vector $d$, instead, is low-dimensional and is used to adapt the performance of the neural controller to different control tasks. See Section \ref{['sec:prelim']} for a detailed explanation of the neural controller and our multi-task control problem, and Section \ref{['sec:grad']} for a numerical study of this multi-task control example.
• Figure 2: This figure displays the impulse responses of the feedback interconnection between the desired controllers $\Sigma_i^{\rm D}$ (or their implementations via the neural controller \ref{['eq:appSys']}) and the plants described in Fig. \ref{['fig:figExample']}. The closed-loop impulse responses associated with the approximating controllers, which are the result of linearizing our neural controller, are highlighted in red. In contrast, the impulse responses of the closed-loop systems governed by the desired controllers are illustrated in blue.
• Figure 3: This figure illustrates a box plot summarizing the final cost \ref{['eq:problem2']} across $17$ simulations with distinct sets $(M = 5)$ of randomly generated systems $\Sigma_i^{\rm D}$, as the dimension $N$ of the neural controller increases from $1$ to $8$. The median of the simulations is marked by the central red line in each box, while the bottom and top edges of the box delineate the 25th and 75th percentiles, respectively. The whiskers extend to cover approximately $99.3\%$ of the data, indicating the range within which most observations lie.
• Figure 4: This figure shows a box plot (with the format introduced in Fig. \ref{['fig:figSim5']}) aggregating the final cost \ref{['eq:problem2']} across $100$ simulations for $100$ randomly selected sets of $\Sigma_i^{\rm D}$. For each set, we calculate the cost \ref{['eq:problem2']}, considering increasing subsets of systems to be approximated, with the number of systems $M$ increasing from $3$ to $10$.
• Figure 5: The figure illustrates two distinct curves: the red one represents the evolution of the minimum of the cost function \ref{['eq:problem2']} derived via a gradient descent algorithm that utilizes the gradient discussed in Section \ref{['sec:grad']}; the blue curve shows the upper bound \ref{['eq:upperBound']}. These outcomes are averaged across the same $100$ simulations of Fig. \ref{['fig:figSim4']}.

Theorems & Definitions (12)

• Theorem II.1
• proof
• Remark 1
• Remark 2
• Theorem III.1
• proof
• Theorem IV.1
• proof
• Theorem IV.2
• proof