Jing Shuang Li

Yaozhi Du, Jing Shuang Li

This paper provides a novel approach for finding sparse state-space realizations of linear systems (e.g., controllers). Sparse controllers are commonly used in distributed control, where a controller is synthesized with some sparsity penalty. Here, motivated by a modeling problem in sensorimotor neuroscience, we study a complementary question: given a linear time-invariant system (e.g., controller) in transfer function form and a desired sparsity pattern, can we find a suitably sparse state-space realization for the transfer function? This problem is highly nonconvex, but we propose an exact method to solve it. We show that the problem reduces to finding an appropriate similarity transform from the modal realization, which in turn reduces to solving a system of multivariate polynomial equations. Finally, we leverage tools from algebraic geometry (namely, the Gröbner basis) to solve this problem exactly. We provide algorithms to find real- and complex-valued sparse realizations and demonstrate their efficacy on several examples.

Yaozhi Du, Jing Shuang Li

This paper designs H2 and H-infinity distributed controllers with local communication and local disturbance rejection. We propose a two-step procedure: first, select closed-loop poles; then, optimize over parameterized controllers. We build on the system level synthesis (SLS) parameterization -- primarily used in the discrete-time setting -- and extend it to the general continuous-time setting. We verify our approach in simulation on a 9-node grid governed by linearized swing equations, where our distributed controllers achieve performance comparable to that of optimal centralized controllers while facilitating local disturbance rejection.

Qin He, Jing Shuang Li

Humans learn and form memories in stochastic environments. Auto-associative memory systems model these processes by storing patterns and later recovering them from corrupted versions. Here, memories are learned by associating each pattern with an attractor in a latent space. After learning, when (possibly corrupted) patterns are presented to the system, latent dynamics facilitate retrieval of the appropriate uncorrupted pattern. In this work, we propose a novel online auto-associative memory system. In contrast to existing works, our system supports sequential memory formation and provides formal guarantees of robust memory retrieval via region-of-attraction analysis. We use a threshold-linear network as latent space dynamics in combination with an encoder, decoder, and controller. We show in simulation that the memory system successfully reconstructs patterns from corrupted inputs.

Jaidev Gill, Jing Shuang Li

In this work, we study the identifiability of network structures (i.e., topologies) for networked nonlinear systems when partial measurements of the nodal dynamics are taken. We explore scenarios where different candidate structures can yield similar measurements, thus limiting identifiability. To do so, we apply the contraction theory framework to facilitate comparisons between different networks. We show that semicontraction in the observable space is a sufficient condition for two systems to become indistinguishable from one another based on partial measurements. We apply this framework to study networks of Kuramoto oscillators, and discuss scenarios in which different network structures (both connected and disconnected) become indistinguishable.

Jaidev Gill, Jing Shuang Li

This work studies the limitations of uniquely identifying the structure (i.e., topology) of a networked linear system from partial measurements of its nodal dynamics. In general, many networks can be consistent with these measurements; this is a consideration often neglected by standard network inference methods. We show that the space of these networks are related through the nullspace of the observability matrix for the true network. We establish relevant metrics to investigate this space, including an analytic characterization of the most structurally dissimilar network that can be inferred, as well as the possibility of mis-inferring presence or absence of edges. In simulations, we find that when observing over 6\% of nodes in random network models (e.g., Erd\H os-R\' enyi and Watts-Strogatz), approximately 99\% of edges are correctly classified. Extending this discussion, we construct a family of networks that keep measurements $ε$-close to each other, and connect the identifiability of these networks to the spectral properties of an augmented observability Gramian.

Pengyang Wu, Jing Shuang Li

This paper studies the co-design of actuators, sensors, and communication in the distributed setting, where a networked plant is partitioned into subsystems each equipped with a sub-controller interacting with other sub-controllers. The objective is to jointly minimize control cost (measured by LQ cost) and material cost (measured by the number of actuators, sensors, and communication links used). We approach this using an evolutionary algorithm to selectively prune a baseline dense LQR controller. We provide convergence and stability analyses for this algorithm. For unstable plants, controller pruning is more likely to induce instability; we provide an algorithm modification to address this. The proposed methods is validated in simulations. One key result is that co-design of a 98-state swing equation model can be done on a standard laptop in seconds; the co-design outperforms naive controller pruning by over 50%.

Jaidev Gill, Jing Shuang Li

Human and animal brains perform planning to enable complex movements and behaviors. This process can be effectively described using model predictive control (MPC); that is, brains can be thought of as implementing some version of MPC. How is this done? In this work, we translate model predictive controllers into firing rate neural networks, offering insights into the nonlinear neural dynamics that underpin planning. This is done by first applying the projected gradient method to the dual problem, then generating alternative networks through factorization and contraction analysis. This allows us to explore many biologically plausible implementations of MPC. We present a series of numerical simulations to study different neural networks performing MPC to balance an inverted pendulum on a cart (i.e., balancing a stick on a hand). We illustrate that sparse neural networks can effectively implement MPC; this observation aligns with the sparse nature of the brain.