Showing papers by "Paulo Tabuada published in 2021"

Cloud-Based Quadratic Optimization With Partially Homomorphic Encryption

Abstract: This article develops a cloud-based protocol for a constrained quadratic optimization problem involving multiple parties, each holding private data. The protocol is based on the projected gradient ascent on the Lagrange dual problem and exploits partially homomorphic encryption and secure communication techniques. Using formal cryptographic definitions of indistinguishability, the protocol is shown to achieve computational privacy. We show the implementation results of the protocol and discuss its computational and communication complexity. We conclude this article with a discussion on privacy notions.

Control Barrier Function-Based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria

Abstract: Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have been used to develop provably safe controllers by means of quadratic programs (QPs), guaranteeing safety in the form of trajectory invariance with respect to a given set. In this letter, we show that this framework can introduce equilibrium points (particularly at the boundary of the safe set) other than the minimum of the Lyapunov function into the closed-loop system. We derive explicit conditions under which these undesired equilibria (which can even appear in the simple case of linear systems with just one convex unsafe set) are asymptotically stable. To address this issue, we propose an extension to the QP-based controller unifying CLFs and CBFs such that the resulting system trajectories avoid the undesirable equilibria problem on the boundary of the safe set. The solution is illustrated in the design of a collision-free controller.

An enhanced hierarchy for (robust) controlled invariance

Abstract: We revisit the problem of computing controlled invariant sets for controllable discrete-time linear systems. Inspired by previous work by the authors, our main idea works in two moves: the problem is lifted to a higher dimensional space, where we provide a closed-form expression for a set whose projection back onto the original space is proven to be controlled invariant. We propose two methods in which the key insight is computing controlled invariant sets by considering periodic control policies. The first method considers hyperboxes that are rendered recurrent and essentially improves computational performance of the authors' previous work, while computing the same sets. The second method relaxes the assumption of recurrent hyper-boxes and yields substantially larger controlled invariant sets as shown in case studies. These methods do not rely on iterative computations and their scalability is illustrated in several examples, which show that none of the methods is strictly better than the other.

Being correct is not enough: efficient verification using robust linear temporal logic.

Abstract: While most approaches in formal methods address system correctness, ensuring robustness has remained a challenge. In this paper we introduce the logic rLTL which provides a means to formally reason about both correctness and robustness in system design. Furthermore, we identify a large fragment of rLTL for which the verification problem can be efficiently solved, i.e., verification can be done by using an automaton, recognizing the behaviors described by the rLTL formula $\varphi$, of size at most $\mathcal{O} \left( 3^{ |\varphi|} \right)$, where $|\varphi|$ is the length of $\varphi$. This result improves upon the previously known bound of $\mathcal{O}\left(5^{|\varphi|} \right)$ for rLTL verification and is closer to the LTL bound of $\mathcal{O}\left( 2^{|\varphi|} \right)$. The usefulness of this fragment is demonstrated by a number of case studies showing its practical significance in terms of expressiveness, the ability to describe robustness, and the fine-grained information that rLTL brings to the process of system verification. Moreover, these advantages come at a low computational overhead with respect to LTL verification.

Symmetries and Isomorphisms for Privacy in Control Over the Cloud

Abstract: Cloud computing platforms are being increasingly used for closing feedback control loops, especially when computationally expensive algorithms, such as model-predictive control, are used to optimize performance. Outsourcing of control algorithms entails an exchange of data between the control system and the cloud and, naturally, raises concerns about the privacy of the control system's data (e.g., state trajectory and control objective). Moreover, any attempt at enforcing privacy needs to add minimal computational overhead to avoid degrading control performance. In this article, we propose several transformation-based methods for enforcing data privacy. We also quantify the amount of provided privacy and discuss how much privacy is lost when the adversary has access to side knowledge. We address three different scenarios: 1) the cloud has no knowledge about the system being controlled; 2) the cloud knows what sensors and actuators the system employs but not the system dynamics; and 3) the cloud knows the system dynamics, its sensors, and actuators. In all of these three scenarios, the proposed methods allow for the control over the cloud without compromising private information (which information is considered private depends on the considered scenario).

On the Computational Complexity of the Secure State-Reconstruction Problem.

Abstract: In this paper, we discuss the computational complexity of reconstructing the state of a linear system from sensor measurements that have been corrupted by an adversary. The first result establishes that the problem is, in general, NP-hard. We then introduce the notion of eigenvalue observability and show that the state can be reconstructed in polynomial time when each eigenvalue is observable by at least $2s+1$ sensors and at most $s$ sensors are corrupted by an adversary. However, there is a gap between eigenvalue observability and the possibility of reconstructing the state despite attacks - this gap has been characterized in the literature by the notion of sparse observability. To better understand this, we show that when the $\mathbf{A}$ matrix of the linear system has unitary geometric multiplicity, the gap disappears, i.e., eigenvalue observability coincides with sparse observability, and there exists a polynomial time algorithm to reconstruct the state provided the state can be reconstructed.

Secure State-Reconstruction Over Networks Subject to Attacks

Abstract: Secure state-reconstruction is the problem of reconstructing the state of a linear time-invariant system from sensor measurements that have been corrupted by an adversary. Whereas most work focuses on attacks on sensors, we consider the more challenging case where attacks occur on sensors as well as on nodes and links of a network that transports sensor measurements to a receiver. In this letter we provide necessary and sufficient conditions for the secure state-reconstruction problem to be solvable in the presence of attacks on sensors and on the network.

Distortion-Based Lightweight Security for Cyber-Physical Systems

Abstract: In cyber-physical systems (CPS), inference based on communicated data is of critical significance as it can be used to manipulate or damage the control operations by adversaries. This calls for efficient mechanisms for secure transmission of data since control systems are becoming increasingly distributed over larger geographical areas. Distortion-based security, recently proposed as one candidate for secure transmissions in CPS, is not only more appropriate for these applications but also quite frugal in terms of prior requirements on shared keys. In this article, we propose distortion-based metrics to protect CPS communication and show that it is possible to confuse adversaries with just a few bits of preshared keys. In particular, we will show that a linear dynamical system can communicate its state in a manner that prevents an eavesdropper from accurately learning the state.

Learned Uncertainty Calibration for Visual Inertial Localization

Abstract: The widely-used Extended Kalman Filter (EKF) provides a straightforward recipe to estimate the mean and covariance of the state given all past measurements in a causal and recursive fashion. For a wide variety of applications, the EKF is known to produce accurate estimates of the mean and typically inaccurate estimates of the covariance. For applications in visual inertial localization, we show that inaccuracies in the covariance estimates are systematic, i.e. it is possible to learn a nonlinear map from the empirical ground truth to the estimated one. This is demonstrated on both a standard EKF in simulation and a Visual Inertial Odometry system on real-world data.

The Secure State Estimation Problem

Abstract: Sensors are the means by which cyber-physical systems perceive their own state as well as the state of their environment. Any attack on sensor measurements, or their transmission, has the potential to lead to catastrophic consequences since control actions would be based on an incorrect state estimate. In this chapter, we introduce the secure state estimation problem, discuss under which conditions it can be solved, and review existing algorithms.

Universal approximation power of deep residual neural networks via nonlinear control theory

Abstract: In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a general sufficient condition for a residual network to have the power of universal approximation by asking the activation function, or one of its derivatives, to satisfy a quadratic differential equation. Many activation functions used in practice satisfy this assumption, exactly or approximately, and we show this property to be sufficient for an adequately deep neural network with n+1 neurons per layer to approximate arbitrarily well, on a compact set and with respect to the supremum norm, any continuous function from Rn to Rn. We further show this result to hold for very simple architectures for which the weights only need to assume two values. The first key technical contribution consists of relating the universal approximation problem to controllability of an ensemble of control systems corresponding to a residual network and to leverage classical Lie algebraic techniques to characterize controllability. The second technical contribution is to identify monotonicity as the bridge between controllability of finite ensembles and uniform approximability on compact sets.

Sampled-Data Stabilization with Control Lyapunov Functions via Quadratically Constrained Quadratic Programs

Abstract: Controller design for nonlinear systems with Control Lyapunov Function (CLF) based quadratic programs has recently been successfully applied to a diverse set of difficult control tasks. These existing formulations do not address the gap between design with continuous time models and the discrete time sampled implementation of the resulting controllers, often leading to poor performance on hardware platforms. We propose an approach to close this gap by synthesizing sampled-data counterparts to these CLF-based controllers, specified as quadratically constrained quadratic programs (QCQPs). Assuming feedback linearizability and stable zero-dynamics of a system's continuous time model, we derive practical stability guarantees for the resulting sampled-data system. We demonstrate improved performance of the proposed approach over continuous time counterparts in simulation.

Decentralized Secure State-Tracking in Multi-Agent Systems

Abstract: This paper addresses the problem of decentralized state-tracking in the presence of sensor attacks. We consider a network of nodes where each node has the objective of tracking the state of a linear dynamical system based on its measurements and messages exchanged with neighboring nodes notwithstanding some measurements being spoofed by an adversary. We propose a novel decentralized attack-resilient state-tracking algorithm based on the simple observation that a compressed version of all the network measurements suffices to reconstruct the state. This motivates a 2-step solution to the decentralized secure state-tracking problem: (1) each node tracks the compressed version of all the network measurements, and (2) each node asymptotically reconstructs the state from the output of step (1). We prove that, under mild technical assumptions, our algorithm enables each node to track the state of the linear system and thus solves the decentralized secure state-tracking problem.

Automaton-based Implicit Controlled Invariant Set Computation for Discrete-Time Linear Systems.

Abstract: In this paper, we derive closed-form expressions for implicit controlled invariant sets for discrete-time controllable linear systems with measurable disturbances. In particular, a disturbance-reactive (or disturbance feedback) controller in the form of a parameterized finite automaton is considered. We show that, for a class of automata, the robust positively invariant sets of the corresponding closed-loop systems can be expressed by a set of linear inequality constraints in the joint space of system states and controller parameters. This leads to an implicit representation of the invariant set in a lifted space. We further show how the same parameterization can be used to compute invariant sets when the disturbance is not available for measurement.

Learned Uncertainty Calibration for Visual Inertial Localization.

Abstract: The widely-used Extended Kalman Filter (EKF) provides a straightforward recipe to estimate the mean and covariance of the state given all past measurements in a causal and recursive fashion. For a wide variety of applications, the EKF is known to produce accurate estimates of the mean and typically inaccurate estimates of the covariance. For applications in visual inertial localization, we show that inaccuracies in the covariance estimates are \emph{systematic}, i.e. it is possible to learn a nonlinear map from the empirical ground truth to the estimated one. This is demonstrated on both a standard EKF in simulation and a Visual Inertial Odometry system on real-world data.

Controlled invariant sets: implicit closed-form representations and applications.

Abstract: In this paper we revisit the problem of computing robust controlled invariant sets for discrete-time linear systems. The key idea is that by considering controllers that exhibit eventually periodic behavior, we obtain a closed-form expression for an implicit representation of a robust controlled invariant set in the space of states and finite input sequences. Due to the derived closed-form expression, our method is suitable for high dimensional systems. Optionally, one obtains an explicit robust controlled invariant set by projecting the implicit representation to the original state space. The proposed method is complete in the absence of disturbances, with a weak completeness result established when disturbances are present. Moreover, we show that a specific controller choice yields a hierarchy of robust controlled invariant sets. To validate the proposed method, we present thorough case studies illustrating that in safety-critical scenarios the implicit representation suffices in place of the explicit invariant set.

Joint Continuous and Discrete Model Selection via Submodularity.

Abstract: In model selection problems for machine learning, the desire for a well-performing model with meaningful structure is typically expressed through a regularized optimization problem. In many scenarios, however, the meaningful structure is specified in some discrete space, leading to difficult nonconvex optimization problems. In this paper, we relate the model selection problem with structure-promoting regularizers to submodular function minimization defined with continuous and discrete arguments. In particular, we leverage submodularity theory to identify a class of these problems that can be solved exactly and efficiently with an agnostic combination of discrete and continuous optimization routines. We show how simple continuous or discrete constraints can also be handled for certain problem classes, motivated by robust optimization. Finally, we numerically validate our theoretical results with several proof-of-concept examples, comparing against state-of-the-art algorithms.