arXiv subjects

Maryam Kamgarpour

Publications and source records attributed to Maryam Kamgarpour.

103 records · Page 6Linked to original sources

arXiv2018

From Uncertainty Data to Robust Policies for Temporal Logic Planning

We consider the problem of synthesizing robust disturbance feedback policies for systems performing complex tasks. We formulate the tasks as linear temporal logic specifications and encode them into an optimization framework via mixed-integer constraints. Both the system dynamics and the specifications are known but affected by uncertainty. The distribution of the uncertainty is unknown, however realizations can be obtained. We introduce a data-driven approach where the constraints are fulfilled for a set of realizations and provide probabilistic generalization guarantees as a function of the number of considered realizations. We use separate chance constraints for the satisfaction of the specification and operational constraints. This allows us to quantify their violation probabilities independently. We compute disturbance feedback policies as solutions of mixed-integer linear or quadratic optimization problems. By using feedback we can exploit information of past realizations and provide feasibility for a wider range of situations compared to static input sequences. We demonstrate the proposed method on two robust motion-planning case studies for autonomous driving.

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arXiv2018

Robust Control Policies given Formal Specifications in Uncertain Environments

We consider robust control synthesis for linear systems with complex specifications that are affected by uncertain disturbances. This work is motivated by autonomous systems interacting with partially known, time-varying environments. Given a specification in bounded linear temporal logic, we propose a method to synthesize control policies that fulfill the specification for all considered disturbances, where the disturbances affect the atomic propositions of the specification. Our approach relies on introducing affine disturbance feedback policies and casting the problem as a bilinear optimization problem. We introduce an inner approximation of the constraint set leading to a mixed-integer quadratic program that can be solved using general-purpose solvers. The framework is demonstrated on a numerical case study with applications to autonomous driving.

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arXiv2018

On Maximizing Safety in Stochastic Aircraft Trajectory Planning with Uncertain Thunderstorm Development

Dealing with meteorological uncertainty poses a major challenge in air traffic management (ATM). Convective weather (commonly referred to as storms or thunderstorms) in particular represents a significant safety hazard that is responsible for one quarter of weather-related ATM delays in the US. With commercial air traffic on the rise and the risk of potentially critical capacity bottlenecks looming, it is vital that future trajectory planning tools are able to account for meteorological uncertainty. We propose an approach to model the uncertainty inherent to forecasts of convective weather regions using statistical analysis of state-of-the-art forecast data. The developed stochastic storm model is tailored for use in an optimal control algorithm that maximizes the probability of reaching a waypoint while avoiding hazardous storm regions. Both the aircraft and the thunderstorms are modeled stochastically. The performance of the approach is illustrated and validated through simulated case studies based on recent nowcast data and storm observations.

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arXiv2018

Nash and Wardrop equilibria in aggregative games with coupling constraints

We consider the framework of aggregative games, in which the cost function of each agent depends on his own strategy and on the average population strategy. As first contribution, we investigate the relations between the concepts of Nash and Wardrop equilibrium. By exploiting a characterization of the two equilibria as solutions of variational inequalities, we bound their distance with a decreasing function of the population size. As second contribution, we propose two decentralized algorithms that converge to such equilibria and are capable of coping with constraints coupling the strategies of different agents. Finally, we study the applications of charging of electric vehicles and of route choice on a road network.

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arXiv2018

Scalable analysis of linear networked systems via chordal decomposition

This paper introduces a chordal decomposition approach for scalable analysis of linear networked systems, including stability, $\mathcal{H}_2$ and $\mathcal{H}_{\infty}$ performance. Our main strategy is to exploit any sparsity within these analysis problems and use chordal decomposition. We first show that Grone's and Agler's theorems can be generalized to block matrices with any partition. This facilitates networked systems analysis, allowing one to solely focus on the physical connections of networked systems to exploit scalability. Then, by choosing Lyapunov functions with appropriate sparsity patterns, we decompose large positive semidefinite constraints in all of the analysis problems into multiple smaller ones depending on the maximal cliques of the system graph. This makes the solutions more computationally efficient via a recent first-order algorithm. Numerical experiments demonstrate the efficiency and scalability of the proposed method.

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arXiv2017

The Linear Programming Approach to Reach-Avoid Problems for Markov Decision Processes

One of the most fundamental problems in Markov decision processes is analysis and control synthesis for safety and reachability specifications. We consider the stochastic reach-avoid problem, in which the objective is to synthesize a control policy to maximize the probability of reaching a target set at a given time, while staying in a safe set at all prior times. We characterize the solution to this problem through an infinite dimensional linear program. We then develop a tractable approximation to the infinite dimensional linear program through finite dimensional approximations of the decision space and constraints. For a large class of Markov decision processes modeled by Gaussian mixtures kernels we show that through a proper selection of the finite dimensional space, one can further reduce the computational complexity of the resulting linear program. We validate the proposed method and analyze its potential with a series of numerical case studies.

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arXiv2017

Information Structure Design in Team Decision Problems

We consider a problem of information structure design in team decision problems and team games. We propose simple, scalable greedy algorithms for adding a set of extra information links to optimize team performance and resilience to non-cooperative and adversarial agents. We show via a simple counterexample that the set function mapping additional information links to team performance is in general not supermodular. Although this implies that the greedy algorithm is not accompanied by worst-case performance guarantees, we illustrate through numerical experiments that it can produce effective and often optimal or near optimal information structure modifications.

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arXiv2017

Sequential Linear Quadratic Optimal Control for Nonlinear Switched Systems

In this contribution, we introduce an efficient method for solving the optimal control problem for an unconstrained nonlinear switched system with an arbitrary cost function. We assume that the sequence of the switching modes are given but the switching time in between consecutive modes remains to be optimized. The proposed method uses a two-stage approach as introduced by Xu and Antsaklis (2004) where the original optimal control problem is transcribed into an equivalent problem parametrized by the switching times and the optimal control policy is obtained based on the solution of a two-point boundary value differential equation. The main contribution of this paper is to use a Sequential Linear Quadratic approach to synthesize the optimal controller instead of solving a boundary value problem. The proposed method is numerically more efficient and scales very well to the high dimensional problems. In order to evaluate its performance, we use two numerical examples as benchmarks to compare against the baseline algorithm. In the third numerical example, we apply the proposed algorithm to the Center of Mass control problem in a quadruped robot locomotion task.

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arXiv2016

Payoff-Based Approach to Learning Nash Equilibria in Convex Games

We consider multi-agent decision making, where each agent optimizes its cost function subject to constraints. Agents' actions belong to a compact convex Euclidean space and the agents' cost functions are coupled. We propose a distributed payoff-based algorithm to learn Nash equilibria in the game between agents. Each agent uses only information about its current cost value to compute its next action. We prove convergence of the proposed algorithm to a Nash equilibrium in the game leveraging established results on stochastic processes. The performance of the algorithm is analyzed with a numerical case study.

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arXiv2016

Exploring Vickrey-Clarke-Groves Mechanism for Electricity Markets

Control reserves are power generation or consumption entities that ensure balance of supply and demand of electricity in real-time. In many countries, they are operated through a market mechanism in which entities provide bids. The system operator determines the accepted bids based on an optimization algorithm. We develop the Vickrey-Clarke-Groves (VCG) mechanism for these electricity markets. We show that all advantages of the VCG mechanism including incentive compatibility of the equilibria and efficiency of the outcome can be guaranteed in these markets. Furthermore, we derive conditions to ensure collusion and shill bidding are not profitable. Our results are verified with numerical examples.

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arXiv2016

Robust Optimal Control with Adjustable Uncertainty Sets

In this paper, we develop a unified framework for studying constrained robust optimal control problems with adjustable uncertainty sets. In contrast to standard constrained robust optimal control problems with known uncertainty sets, we treat the uncertainty sets in our problems as additional decision variables. In particular, given a finite prediction horizon and a metric for adjusting the uncertainty sets, we address the question of determining the optimal size and shape of the uncertainty sets, while simultaneously ensuring the existence of a control policy that will keep the system within its constraints for all possible disturbance realizations inside the adjusted uncertainty set. Since our problem subsumes the classical constrained robust optimal control design problem, it is computationally intractable in general. We demonstrate that by restricting the families of admissible uncertainty sets and control policies, the problem can be formulated as a tractable convex optimization problem. We show that our framework captures several families of (convex) uncertainty sets of practical interest, and illustrate our approach on a demand response problem of providing control reserves for a power system.

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arXiv2015

Upper bounds for the reach-avoid probability via robust optimization

We consider finite horizon reach-avoid problems for discrete time stochastic systems. Our goal is to construct upper bound functions for the reach-avoid probability by means of tractable convex optimization problems. We achieve this by restricting attention to the span of Gaussian radial basis functions and imposing structural assumptions on the transition kernel of the stochastic processes as well as the target and safe sets of the reach-avoid problem. In particular, we require the kernel to be written as a Gaussian mixture density with each mean of the distribution being affine in the current state and input and the target and safe sets to be written as intersections of quadratic inequalities. Taking advantage of these structural assumptions, we formulate a recursion of semidefinite programs where each step provides an upper bound to the value function of the reach- avoid problem. The upper bounds provide a performance metric to which any suboptimal control policy can be compared, and can themselves be used to construct suboptimal control policies. We illustrate via numerical examples that even if the resulting bounds are conservative, the associated control policies achieve higher reach-avoid probabilities than heuristic controllers for problems of large state-input space dimensions (more than 20). The results presented in this paper, far exceed the limits of current approximation methods for reach-avoid problems in the specific class of stochastic systems considered.

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arXiv2012

Approximate Dynamic Programming via Sum of Squares Programming

We describe an approximate dynamic programming method for stochastic control problems on infinite state and input spaces. The optimal value function is approximated by a linear combination of basis functions with coefficients as decision variables. By relaxing the Bellman equation to an inequality, one obtains a linear program in the basis coefficients with an infinite set of constraints. We show that a recently introduced method, which obtains convex quadratic value function approximations, can be extended to higher order polynomial approximations via sum of squares programming techniques. An approximate value function can then be computed offline by solving a semidefinite program, without having to sample the infinite constraint. The policy is evaluated online by solving a polynomial optimization problem, which also turns out to be convex in some cases. We experimentally validate the method on an autonomous helicopter testbed using a 10-dimensional helicopter model.

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