Jorge Cortés

Yingbo Zhao, Jorge Cortés

This paper studies Gramian-based reachability metrics for bilinear control systems. In the context of complex networks, bilinear systems capture scenarios where an actuator not only can affect the state of a node but also interconnections among nodes. Under the assumption that the input's infinity norm is bounded by some function of the network dynamic matrices, we derive a Gramian-based lower bound on the minimum input energy required to steer the state from the origin to any reachable target state. This result motivates our study of various objects associated to the reachability Gramian to quantify the ease of controllability of the bilinear network: the minimum eigenvalue (worst-case minimum input energy to reach a state), the trace (average minimum input energy to reach a state), and its determinant (volume of the ellipsoid containing the reachable states using control inputs with no more than unit energy). We establish an increasing returns property of the reachability Gramian as a function of the actuators, which in turn allows us to derive a general lower bound on the reachability metrics in terms of the aggregate contribution of the individual actuators. We conclude by examining the effect on the worst-case minimum input energy of the addition of bilinear inputs to difficult-to-control linear symmetric networks. We show that the bilinear networks resulting from the addition of either inputs at a finite number of interconnections or at all self loops with weight vanishing with the network scale remain difficult-to-control. Various examples illustrate our results.

Mohammad Javad Khojasteh, Pavankumar Tallapragada, Jorge Cortés et al.

We study event-triggered control for stabilization of unstable linear plants over rate-limited communication channels subject to unknown, bounded delay. On one hand, the timing of event triggering carries implicit information about the state of the plant. On the other hand, the delay in the communication channel causes information loss, as it makes the state information available at the controller out of date. Combining these two effects, we show a phase transition behavior in the transmission rate required for stabilization using a given event-triggering strategy. For small values of the delay, the timing information carried by the triggering events is substantial, and the system can be stabilized with any positive rate. When the delay exceeds a critical threshold, the timing information alone is not enough to achieve stabilization and the required rate grows. When the loss of information due to the communication delay perfectly compensates the implicit information carried by the triggering events, the delay equals the inverse of the entropy rate of the plant, and we obtain the same rate requirement prescribed by the data-rate theorem. When the delay is larger than this threshold, the required rate becomes larger than that required by the data-rate theorem. We also provide an explicit construction yielding a sufficient rate for stabilization, and generalize our results to vector systems. The results do not rely on any a priori probabilistic model of the delay or the initial conditions.

Erfan Nozari, Pavankumar Tallapragada, Jorge Cortés

This paper studies the problem of stabilization of a nonlinear system with time-varying delays in both sensing and actuation using event-triggered control. Our proposed strategy seeks to opportunistically minimize the number of control updates while guaranteeing stabilization and builds on predictor feedback to compensate for arbitrarily large known time-varying delays. We establish, using a Lyapunov approach, the global asymptotic stability of the closed-loop system as long as the open-loop system is globally input-to-state stabilizable in the absence of time delays and sampling. We further prove that the proposed event-triggered law has inter-event times that are uniformly lower bounded and hence does not exhibit Zeno behavior. For the particular case of a stabilizable linear system, we show global exponential stability of the closed-loop system and analyze the trade-off between the rate of exponential convergence and a bound on the sampling frequency. We illustrate these results in simulation and also examine the properties of the proposed event-triggered strategy beyond the class of systems for which stabilization can be guaranteed.

Pavankumar Tallapragada, Jorge Cortés

This paper considers the problem of coordinating the vehicular traffic at an intersection and on the branches leading to it for minimizing a combination of total travel time and energy consumption. We propose a provably safe hierarchical-distributed solution to balance computational complexity and optimality of the system operation. In our design, a central intersection manager communicates with vehicles heading towards the intersection, groups them into clusters (termed bubbles) as they appear, and determines an optimal schedule of passage through the intersection for each bubble. The vehicles in each bubble receive their schedule and implement local distributed control to ensure system-wide inter-vehicular safety while respecting speed and acceleration limits, conforming to the assigned schedule, and seeking to optimize their individual trajectories. Our analysis rigorously establishes that the different aspects of the hierarchical design operate in concert and that the safety guarantees provided by the proposed design are satisfied. We illustrate its execution in a suite of simulations and compare its performance to traditional signal-based coordination over a wide range of system parameters.

Yinzhuang Yi, Jorge Cortés, Nikolay Atanasov

We propose a control barrier function (CBF) formulation for enforcing equality and inequality constraints in variational inference. The key idea is to define a barrier functional on the space of probability density functions that encode the desired constraints imposed on the variational density. By leveraging the Liouville equation, we establish a connection between the time derivative of the variational density and the particle drift, which enables the systematic construction of corresponding CBFs associated to the particle drift. Enforcing these CBFs gives rise to the safe particle flow and ensures that the variational density satisfies the original constraints imposed by the barrier functional. This formulation provides a principled and computationally tractable solution to constrained variational inference, with theoretical guarantees of constraint satisfaction. The effectiveness of the method is demonstrated through numerical simulations.

Jie Feng, Wenqi Cui, Jorge Cortés et al.

The increasing integration of renewable energy resources into power grids has led to time-varying system inertia and consequent degradation in frequency dynamics. A promising solution to alleviate performance degradation is using power electronics interfaced energy resources, such as renewable generators and battery energy storage for primary frequency control, by adjusting their power output set-points in response to frequency deviations. However, designing a frequency controller under time-varying inertia is challenging. Specifically, the stability or optimality of controllers designed for time-invariant systems can be compromised once applied to a time-varying system. We model the frequency dynamics under time-varying inertia as a nonlinear switching system, where the frequency dynamics under each mode are described by the nonlinear swing equations and different modes represent different inertia levels. We identify a key controller structure, named Neural Proportional-Integral (Neural-PI) controller, that guarantees exponential input-to-state stability for each mode. To further improve performance, we present an online event-triggered switching algorithm to select the most suitable controller from a set of Neural-PI controllers, each optimized for specific inertia levels. Simulations on the IEEE 39-bus system validate the effectiveness of the proposed online switching control method with stability guarantees and optimized performance for frequency control under time-varying inertia.

Masih Haseli, Jorge Cortés

Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.

Miguel Vaquero, Jorge Cortés, David Martín de Diego

This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether's Theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of $G$-invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether's Theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry towards simulating and learning problems.

Dhruv Shah, Jorge Cortés

The accuracy of Koopman operator approximations over finite-dimensional spaces relies critically on their invariance properties. These can be rigorously quantified via the principal angles between a candidate subspace and its image under the Koopman operator. This paper proposes a unified algebraic framework for subspace pruning designed to systematically refine the invariance error. We establish the geometric equivalence between consistency-based methods and principal-vector pruning, and build on this insight to introduce a hybrid strategy that balances between multiple and single principal vector pruning for improved numerical stability and scalability. We derive error bounds for the retention of approximate and external eigenfunctions, demonstrating that the multi-vector approach mitigates the numerical drift inherent to sequential pruning. To ensure scalability, we develop an efficient numerical update scheme based on rank-one modifications that reduces the computational complexity of tracking principal angles by an order of magnitude. Finally, we exploit the subspace obtained from the pruning algorithms to build a lifted linear model for state prediction that accounts for the trade-offs between improving invariance and minimizing state reconstruction error. Simulations demonstrate the effectiveness of our approach.

Miguel Vaquero, David Martín de Diego, Jorge Cortés

This paper presents a general method to construct Poisson integrators, i.e., integrators that preserve the underlying Poisson geometry. We assume the Poisson manifold is integrable, meaning there is a known local symplectic groupoid for which the Poisson manifold serves as the set of units. Our constructions build upon the correspondence between Poisson diffeomorphisms and Lagrangian bisections, which allows us to reformulate the design of Poisson integrators as solutions to a certain PDE (Hamilton-Jacobi). The main novelty of this work is to understand the Hamilton-Jacobi PDE as an optimization problem, whose solution can be easily approximated using machine learning related techniques. This research direction aligns with the current trend in the PDE and machine learning communities, as initiated by Physics- Informed Neural Networks, advocating for designs that combine both physical modeling (the Hamilton-Jacobi PDE) and data.

Yinzhuang Yi, Jorge Cortés, Nikolay Atanasov

This paper provides a formulation of the particle flow particle filter from the perspective of variational inference. We show that the transient density used to derive the particle flow particle filter follows a time-scaled trajectory of the Fisher-Rao gradient flow in the space of probability densities. The Fisher-Rao gradient flow is obtained as a continuous-time algorithm for variational inference, minimizing the Kullback-Leibler divergence between a variational density and the true posterior density.

Kehan Long, Vikas Dhiman, Melvin Leok et al.

This paper considers safe control synthesis for dynamical systems with either probabilistic or worst-case uncertainty in both the dynamics model and the safety constraints. We formulate novel probabilistic and robust (worst-case) control Lyapunov function (CLF) and control barrier function (CBF) constraints that take into account the effect of uncertainty in either case. We show that either the probabilistic or the robust (worst-case) formulation leads to a second-order cone program (SOCP), which enables efficient safe and stable control synthesis. We evaluate our approach in PyBullet simulations of an autonomous robot navigating in unknown environments and compare the performance with a baseline CLF-CBF quadratic programming approach.

Masih Haseli, Jorge Cortés

This paper tackles the data-driven approximation of unknown dynamical systems using Koopman-operator methods. Given a dictionary of functions, these methods approximate the projection of the action of the operator on the finite-dimensional subspace spanned by the dictionary. We propose the Tunable Symmetric Subspace Decomposition algorithm to refine the dictionary, balancing its expressiveness and accuracy. Expressiveness corresponds to the ability of the dictionary to describe the evolution of as many observables as possible and accuracy corresponds to the ability to correctly predict their evolution. Based on the observation that Koopman-invariant subspaces give rise to exact predictions, we reason that prediction accuracy is a function of the degree of invariance of the subspace generated by the dictionary and provide a data-driven measure to measure invariance proximity. The proposed algorithm iteratively prunes the initial functional space to identify a refined dictionary of functions that satisfies the desired level of accuracy while retaining as much of the original expressiveness as possible. We provide a full characterization of the algorithm properties and show that it generalizes both Extended Dynamic Mode Decomposition and Symmetric Subspace Decomposition. Simulations on planar systems show the effectiveness of the proposed methods in producing Koopman approximations of tunable accuracy that capture relevant information about the dynamical system.

Kehan Long, Cheng Qian, Jorge Cortés et al.

Control barrier functions are widely used to enforce safety properties in robot motion planning and control. However, the problem of constructing barrier functions online and synthesizing safe controllers that can deal with the associated uncertainty has received little attention. This paper investigates safe navigation in unknown environments, using onboard range sensing to construct control barrier functions online. To represent different objects in the environment, we use the distance measurements to train neural network approximations of the signed distance functions incrementally with replay memory. This allows us to formulate a novel robust control barrier safety constraint which takes into account the error in the estimated distance fields and its gradient. Our formulation leads to a second-order cone program, enabling safe and stable control synthesis in a priori unknown environments.

Erfan Nozari, Jorge Cortés

Goal-driven selective attention (GDSA) refers to the brain's function of prioritizing the activity of a task-relevant subset of its overall network to efficiently process relevant information while inhibiting the effects of distractions. Despite decades of research in neuroscience, a comprehensive understanding of GDSA is still lacking. We propose a novel framework using concepts and tools from control theory as well as insights and structures from neuroscience. Central to this framework is an information-processing hierarchy with two main components: selective inhibition of task-irrelevant activity and top-down recruitment of task-relevant activity. We analyze the internal dynamics of each layer of the hierarchy described as a network with linear-threshold dynamics and derive conditions on its structure to guarantee existence and uniqueness of equilibria, asymptotic stability, and boundedness of trajectories. We also provide mechanisms that enforce selective inhibition using the biologically-inspired schemes of feedforward and feedback inhibition. Despite their differences, both lead to the same conclusion: the intrinsic dynamical properties of the (not-inhibited) task-relevant subnetworks are the sole determiner of the dynamical properties that are achievable under selective inhibition.

Erfan Nozari, Jorge Cortés

Goal-driven selective attention (GDSA) is a remarkable function that allows the complex dynamical networks of the brain to support coherent perception and cognition. Part I of this two-part paper proposes a new control-theoretic framework, termed hierarchical selective recruitment (HSR), to rigorously explain the emergence of GDSA from the brain's network structure and dynamics. This part completes the development of HSR by deriving conditions on the joint structure of the hierarchical subnetworks that guarantee top-down recruitment of the task-relevant part of each subnetwork by the subnetwork at the layer immediately above, while inhibiting the activity of task-irrelevant subnetworks at all the hierarchical layers. To further verify the merit and applicability of this framework, we carry out a comprehensive case study of selective listening in rodents and show that a small network with HSR-based structure and minimal size can explain the data with remarkable accuracy while satisfying the theoretical requirements of HSR. Our technical approach relies on the theory of switched systems and provides a novel converse Lyapunov theorem for state-dependent switched affine systems that is of independent interest.

Erfan Nozari, Yingbo Zhao, Jorge Cortés

We consider linear time-invariant networks with unknown topology where only a manifest subset of the nodes can be directly actuated and measured while the state of the remaining latent nodes and their number are unknown. Our goal is to identify the transfer function of the manifest subnetwork and determine whether interactions between manifest nodes are direct or mediated by latent nodes. We show that, if there are no inputs to the latent nodes, the manifest transfer function can be approximated arbitrarily well in the H-infinity norm sense by the transfer function of an auto-regressive model and present a least-squares estimation method to construct the auto-regressive model from measured data. We show that the least-squares auto-regressive method guarantees an arbitrarily small H-infinity norm error in the approximation of the manifest transfer function, exponentially decaying once the model order exceeds a certain threshold. Finally, we show that when the latent subnetwork is acyclic, the proposed method achieves perfect identification of the manifest transfer function above a specific model order as the length of the data increases. Various examples illustrate our results.