arXiv subjects

Claire J. Tomlin

Publications and source records attributed to Claire J. Tomlin.

At least 109 records · Page 6Linked to original sources

arXiv2016

Multiplayer Reach-Avoid Games via Pairwise Outcomes

A multiplayer reach-avoid game is a differential game between an attacking team with NA attackers and a defending team with ND defenders playing on a compact domain with obstacles. The attacking team aims to send M of the NA attackers to some target location, while the defending team aims to prevent that by capturing attackers or indefinitely delaying attackers from reaching the target. Although the analysis of this game plays an important role in many applications, the optimal solution to this game is computationally intractable when NA>1 or ND>1. In this paper, we present two approaches for the NA=ND=1 case to determine pairwise outcomes, and a graph theoretic maximum matching approach to merge these pairwise outcomes for an NA,ND>1 solution that provides guarantees on the performance of the defending team. We will show that the four-dimensional Hamilton-Jacobi-Isaacs approach allows for real-time updates to the maximum matching, and that the two-dimensional "path defense" approach is considerably more scalable with the number of players while maintaining defender performance guarantees.

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arXiv2016

Learning Quadrotor Dynamics Using Neural Network for Flight Control

Traditional learning approaches proposed for controlling quadrotors or helicopters have focused on improving performance for specific trajectories by iteratively improving upon a nominal controller, for example learning from demonstrations, iterative learning, and reinforcement learning. In these schemes, however, it is not clear how the information gathered from the training trajectories can be used to synthesize controllers for more general trajectories. Recently, the efficacy of deep learning in inferring helicopter dynamics has been shown. Motivated by the generalization capability of deep learning, this paper investigates whether a neural network based dynamics model can be employed to synthesize control for trajectories different than those used for training. To test this, we learn a quadrotor dynamics model using only translational and only rotational training trajectories, each of which can be controlled independently, and then use it to simultaneously control the yaw and position of a quadrotor, which is non-trivial because of nonlinear couplings between the two motions. We validate our approach in experiments on a quadrotor testbed.

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arXiv2016

Multi-Vehicle Collision Avoidance via Hamilton-Jacobi Reachability and Mixed Integer Programming

Multi-agent differential games are important and useful tools for analyzing many practical problems. With the recent surge of interest in using UAVs for civil purposes, the importance and urgency of developing tractable multi-agent analysis techniques that provide safety and performance guarantees is at an all-time high. Hamilton-Jacobi (HJ) reachability has successfully provided safety guarantees to small-scale systems and is flexible in terms of system dynamics. However, the exponential complexity scaling of HJ reachability prevents its direct application to large scale problems when the number of vehicles is greater than two. In this paper, we overcome the scalability limitations of HJ reachability by using a mixed integer program that exploits the properties of HJ solutions to provide higher-level control logic. Our proposed method provides safety guarantee for three-vehicle systems -- a previously intractable task for HJ reachability -- without incurring significant additional computation cost. Furthermore, our method is scalable beyond three vehicles and performs significantly better by several metrics than an extension of pairwise collision avoidance to multi-vehicle collision avoidance. We demonstrate our proposed method in simulations.

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arXiv2016

Exact and Efficient Hamilton-Jacobi-based Guaranteed Safety Analysis via System Decomposition

Hamilton-Jacobi (HJ) reachability is a method that provides rigorous analyses of the safety properties of dynamical systems. This method has been successfully applied to many low-dimensional dynamical system models such as coarse models of aircraft and quadrotors in order to provide safety guarantees in potentially dangerous scenarios. These guarantees can be provided by the computation of a backward reachable set (BRS), which represents the set of states from which the system may be driven into violating safety properties despite the system's best effort to remain safe. Unfortunately, HJ reachability is not practical for high-dimensional systems because the complexity of the BRS computation scales exponentially with the number of state dimensions. Although numerous approximation techniques are able to tractably provide conservative estimates of the BRS, they often require restrictive assumptions about system dynamics without providing an exact solution. In this paper we propose a general method for decomposing dynamical systems. Even when the resulting subsystems are coupled, relatively high-dimensional BRSs that were previously intractable or expensive to compute can now be quickly and exactly computed in lower-dimensional subspaces. As a result, the curse of dimensionality is alleviated to a large degree without sacrificing optimality. We demonstrate our theoretical results through two numerical examples: a 3D Dubins Car model and a 6D Acrobatic Quadrotor model.

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arXiv2016

Plug-and-Play Model Predictive Control for Load Shaping and Voltage Control in Smart Grids

This paper presents a predictive controller for handling plug-and-play (P&P) charging requests of flexible loads in a distribution system. We define two types of flexible loads: (i) deferrable loads that have a fixed power profile but can be deferred in time and (ii) shapeable loads that have flexible power profiles but fixed energy requests, such as Plug-in Electric Vehicles (PEVs). The proposed method uses a hierarchical control scheme based on a model predictive control (MPC) formulation for minimizing the global system cost. The first stage computes a reachable reference that trades off deviation from the nominal voltage with the required generation control. The second stage uses a price-based objective to aggregate flexible loads and provide load shaping services, while satisfying system constraints and users' preferences at all times. It is shown that the proposed controller is recursively feasible under specific conditions, i.e. the flexible load demands are satisfied and bus voltages remain within the desired limits. Finally, the proposed scheme is illustrated on a 55 bus radial distribution network.

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arXiv2016

Secure Estimation for Unmanned Aerial Vehicles against Adversarial Cyber Attacks

In the coming years, usage of Unmanned Aerial Vehicles (UAVs) is expected to grow tremendously. Maintaining security of UAVs under cyber attacks is an important yet challenging task, as these attacks are often erratic and difficult to predict. Secure estimation problems study how to estimate the states of a dynamical system from a set of noisy and maliciously corrupted sensor measurements. The fewer assumptions that an estimator makes about the attacker, the larger the set of attacks it can protect the system against. In this paper, we focus on sensor attacks on UAVs and attempt to design a secure estimator for linear time-invariant systems based on as few assumptions about the attackers as possible. We propose a computationally efficient estimator that protects the system against arbitrary and unbounded attacks, where the set of attacked sensors can also change over time. In addition, we propose to combine our secure estimator with a Kalman Filter for improved practical performance and demonstrate its effectiveness through simulations of two scenarios where an UAV is under adversarial cyber attack.

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arXiv2016

Secure Estimation based Kalman Filter for Cyber-Physical Systems against Adversarial Attacks

Cyber-physical systems are found in many applications such as power networks, manufacturing processes, and air and ground transportation systems. Maintaining security of these systems under cyber attacks is an important and challenging task, since these attacks can be erratic and thus difficult to model. Secure estimation problems study how to estimate the true system states when measurements are corrupted and/or control inputs are compromised by attackers. The authors in [1] proposed a secure estimation method when the set of attacked nodes (sensors, controllers) is fixed. In this paper, we extend these results to scenarios in which the set of attacked nodes can change over time. We formulate this secure estimation problem into the classical error correction problem [2] and we show that accurate decoding can be guaranteed under a certain condition. Furthermore, we propose a combined secure estimation method with our proposed secure estimator and the Kalman Filter for improved practical performance. Finally, we demonstrate the performance of our method through simulations of two scenarios where an unmanned aerial vehicle is under adversarial attack.

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arXiv2016

Building Model Identification during Regular Operation - Empirical Results and Challenges

The inter-temporal consumption flexibility of commercial buildings can be harnessed to improve the energy efficiency of buildings, or to provide ancillary service to the power grid. To do so, a predictive model of the building's thermal dynamics is required. In this paper, we identify a physics-based model of a multi-purpose commercial building including its heating, ventilation and air conditioning system during regular operation. We present our empirical results and show that large uncertainties in internal heat gains, due to occupancy and equipment, present several challenges in utilizing the building model for long-term prediction. In addition, we show that by learning these uncertain loads online and dynamically updating the building model, prediction accuracy is improved significantly.

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arXiv2016

Secure State Estimation for Nonlinear Power Systems under Cyber Attacks

This paper focuses on securely estimating the state of a nonlinear dynamical system from a set of corrupted measurements. In particular, we consider two broad classes of nonlinear systems, and propose a technique which enables us to perform secure state estimation for such nonlinear systems. We then provide guarantees on the achievable state estimation error against arbitrary corruptions, and analytically characterize the number of errors that can be perfectly corrected by a decoder. To illustrate how the proposed nonlinear estimation approach can be applied to practical systems, we focus on secure estimation for the wide area control of an interconnected power system under cyber-physical attacks and communication failures, and propose a secure estimator for the power system. Finally, we numerically show that the proposed secure estimation algorithm enables us to reconstruct the attack signals accurately.

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arXiv2016

Safe Sequential Path Planning of Multi-Vehicle Systems via Double-Obstacle Hamilton-Jacobi-Isaacs Variational Inequality

We consider the problem of planning trajectories for a group of $N$ vehicles, each aiming to reach its own target set while avoiding danger zones of other vehicles. The analysis of problems like this is extremely important practically, especially given the growing interest in utilizing unmanned aircraft systems for civil purposes. The direct solution of this problem by solving a single-obstacle Hamilton-Jacobi-Isaacs (HJI) variational inequality (VI) is numerically intractable due to the exponential scaling of computation complexity with problem dimensionality. Furthermore, the single-obstacle HJI VI cannot directly handle situations in which vehicles do not have a common scheduled arrival time. Instead, we perform sequential path planning by considering vehicles in order of priority, modeling higher-priority vehicles as time-varying obstacles for lower-priority vehicles. To do this, we solve a double-obstacle HJI VI which allows us to obtain the reach-avoid set, defined as the set of states from which a vehicle can reach its target while staying within a time-varying state constraint set. From the solution of the double-obstacle HJI VI, we can also extract the latest start time and the optimal control for each vehicle. This is a first application of the double-obstacle HJI VI which can handle systems with time-varying dynamics, target sets, and state constraint sets, and results in computation complexity that scales linearly, as opposed to exponentially, with the number of vehicles in consideration.

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arXiv2016

Safe Platooning of Unmanned Aerial Vehicles via Reachability

Recently, there has been immense interest in using unmanned aerial vehicles (UAVs) for civilian operations such as package delivery, firefighting, and fast disaster response. As a result, UAV traffic management systems are needed to support potentially thousands of UAVs flying simultaneously in the airspace, in order to ensure their liveness and safety requirements are met. Hamilton-Jacobi (HJ) reachability is a powerful framework for providing conditions under which these requirements can be met, and for synthesizing the optimal controller for meeting them. However, due to the curse of dimensionality, HJ reachability is only tractable for a small number of vehicles if their set of maneuvers is unrestricted. In this paper, we define a platoon to be a group of UAVs in a single-file formation. We model each vehicle as a hybrid system with modes corresponding to its role in the platoon, and specify the set of allowed maneuvers in each mode to make the analysis tractable. We propose several liveness controllers based on HJ reachability, and wrap a safety controller, also based on HJ reachability, around the liveness controllers. For a single altitude range, our approach guarantees safety for one safety breach; in the unlikely event of multiple safety breaches, safety can be guaranteed over multiple altitude ranges. We demonstrate the satisfaction of liveness and safety requirements through simulations of three common scenarios.

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arXiv2016

Exact and Efficient Hamilton-Jacobi Reachability for Decoupled Systems

Reachability analysis is important for studying optimal control problems and differential games, which are powerful theoretical tools for analyzing and modeling many practical problems in robotics, aircraft control, among other application areas. In reachability analysis, one is interested in computing the reachable set, defined as the set of states from which there exists a control, despite the worst disturbance, that can drive the system into a set of target states. The target states can be used to model either unsafe or desirable configurations, depending on the application. Many Hamilton-Jacobi formulations allow the computation of reachable sets; however, due to the exponential complexity scaling in computation time and space, problems involving approximately 5 dimensions become intractable. A number of methods that compute an approximate solution exist in the literature, but these methods trade off complexity for optimality. In this paper, we eliminate complexity-optimality trade-offs for time-invariant decoupled systems using a decoupled Hamilton-Jacobi formulation that enables the exact reconstruction of high dimensional solutions via low dimensional solutions of the decoupled subsystems. Our formulation is compatible with existing numerical tools, and we show the accuracy, computation benefits, and an application of our novel approach using two numerical examples.

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arXiv2016

Model Comparison of a Data-Driven and a Physical Model for Simulating HVAC Systems

Commercial buildings are responsible for a large fraction of energy consumption in developed countries, and therefore are targets of energy efficiency programs. Motivated by the large inherent thermal inertia of buildings, the power consumption can be flexibly scheduled without compromising occupant comfort. This temporal flexibility offers opportunities for the provision of frequency regulation to support grid stability. To realize energy savings and frequency regulation, it is of prime importance to identify a realistic model for the temperature dynamics of a building. We identify a low- dimensional data-driven model and a high-dimensional physics- based model for different spatial granularities and temporal seasons based on a case study of an entire floor of Sutardja Dai Hall, an office building on the University of California, Berkeley campus. A comparison of these contrasting models shows that, despite the higher forecasting accuracy of the physics-based model, both models perform almost equally well for energy efficient control. We conclude that the data-driven model is more amenable to controller design due to its low complexity, and could serve as a substitution for highly complex physics- based models with an insignificant loss of prediction accuracy for many applications. On the other hand, our physics-based approach is more suitable for modeling buildings with finer spatial granularities.

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arXiv2015

Approximation Algorithms for Optimization of Combinatorial Dynamical Systems

This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function. The performance and utility of the proposed algorithms are demonstrated with the ON/OFF control problems of interdependent refrigeration systems.

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arXiv2014

Reach-Avoid Problems with Time-Varying Dynamics, Targets and Constraints

We consider a reach-avoid differential game, in which one of the players aims to steer the system into a target set without violating a set of state constraints, while the other player tries to prevent the first from succeeding; the system dynamics, target set, and state constraints may all be time-varying. The analysis of this problem plays an important role in collision avoidance, motion planning and aircraft control, among other applications. Previous methods for computing the guaranteed winning initial conditions and strategies for each player have either required augmenting the state vector to include time, or have been limited to problems with either no state constraints or entirely static targets, constraints and dynamics. To incorporate time-varying dynamics, targets and constraints without the need for state augmentation, we propose a modified Hamilton-Jacobi-Isaacs equation in the form of a double-obstacle variational inequality, and prove that the zero sublevel set of its viscosity solution characterizes the capture basin for the target under the state constraints. Through this formulation, our method can compute the capture basin and winning strategies for time-varying games at no additional computational cost with respect to the time-invariant case. We provide an implementation of this method based on well-known numerical schemes and show its convergence through a simple example; we include a second example in which our method substantially outperforms the state augmentation approach.

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arXiv2014

Risk-Limiting Dynamic Contracts for Direct Load Control

This paper proposes a novel continuous-time dynamic contract framework that has a risk-limiting capability. If a principal and an agent enter into such a contract, the principal can optimally manage its performance and risk with a guarantee that the agent's risk is less than or equal to a pre-specified level and that the agent's expected payoff is greater than or equal to another pre-specified threshold. We achieve such risk-management capabilities by formulating the contract design problem as mean-variance constrained risk-sensitive control. A dynamic programming-based method is developed to solve the problem. The key idea of our proposed solution method is to reformulate the inequality constraints on the mean and the variance of the agent's payoff as dynamical system constraints by introducing new state and control variables. The reformulations use the martingale representation theorem. The proposed contract method enables us to develop a new direct load control method that provides the load-serving entity with financial risk management solutions in real-time electricity markets. We also propose an approximate decomposition of the optimal contract design problem for multiple customers into multiple low-dimensional contract problems for one customer. This allows the direct load control program to work with a large number of customers without any scalability issues. Furthermore, the contract design procedure can be completely parallelized. The performance and usefulness of the proposed contract method and its application to direct load control are demonstrated using data on the electric energy consumption of customers in Austin, Texas as well as the Electricity Reliability Council of Texas' locational marginal price data.

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arXiv2014

Path integral formulation of stochastic optimal control with generalized costs

Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated Hamilton-Jacobi-Bellman (HJB) equation. The MC approach avoids the need for a global grid of the domain of the HJB equation and, therefore, path integral control is in principle applicable to control problems of moderate to large dimension. The class of problems path integral control can solve, however, is defined by requirements on the cost function, the noise covariance matrix and the control input matrix. We relax the requirements on the cost function by introducing a new state that represents an augmented running cost. In our new formulation the cost function can contain stochastic integral terms and linear control costs, which are important in applications in engineering, economics and finance. We find an efficient numerical implementation of our grid-free MC approach and demonstrate its performance and usefulness in examples from hierarchical electric load management. The dimension of one of our examples is large enough to make classical grid-based HJB solvers impractical.

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arXiv2014

A sampling-based approach to scalable constraint satisfaction in linear sampled-data systems---Part I: Computation

Sampled-data (SD) systems, which are composed of both discrete- and continuous-time components, are arguably one of the most common classes of cyberphysical systems in practice; most modern controllers are implemented on digital platforms while the plant dynamics that are being controlled evolve continuously in time. As with all cyberphysical systems, ensuring hard constraint satisfaction is key in the safe operation of SD systems. A powerful analytical tool for guaranteeing such constraint satisfaction is the viability kernel: the set of all initial conditions for which a safety-preserving control law (that is, a control law that satisfies all input and state constraints) exists. In this paper we present a novel sampling-based algorithm that tightly approximates the viability kernel for high-dimensional sampled-data linear time-invariant (LTI) systems. Unlike prior work in this area, our algorithm formally handles both the discrete and continuous characteristics of SD systems. We prove the correctness and convergence of our approximation technique, provide discussions on heuristic methods to optimally bias the sampling process, and demonstrate the results on a twelve-dimensional flight envelope protection problem.

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