Peyman Mohajerin Esfahani

Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs—in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification.

Cyber attack in a two-area power system: Impact identification using reachability

This paper presents new results on the applications of reachability methods and computational tools to a two-area power system in the case of a cyber attack. In the VIKING research project a novel concept to assess the vulnerabilities introduced by the interaction between the IT infrastructure and power systems is proposed. Here we develop a new framework and define a systematic methodology, based on reachability, for identifying the impact that an intrusion might have in the Automatic Generation Control loop, which regulates the frequency and the power exchange between the controlled areas. The numerical results reveal the weaknesses of the system and indicate possible policies that an attacker could use to disturb it.

Symbolic Control of Stochastic Systems via Approximately Bisimilar Finite Abstractions

Symbolic approaches for control design construct finite-state abstract models that are related to the original systems, then use techniques from finite-state synthesis to compute controllers satisfying specifications given in a temporal logic, and finally translate the synthesized schemes back as controllers for the original systems. Such approaches have been successfully developed and implemented for the synthesis of controllers over non-probabilistic control systems. In this paper, we extend the technique to probabilistic control systems modelled by controlled stochastic differential equations. We show that for every stochastic control system satisfying a probabilistic variant of incremental input-to-state stability, and for every given precision ε > 0, a finite-state transition system can be constructed, which is ε-approximately bisimilar to the original stochastic control system. Moreover, we provide results relating stochastic control systems to their corresponding finite-state transition systems in terms of probabilistic bisimulation relations known in the literature. We demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic. Our technique enables automated, correct-by-construction, controller synthesis for stochastic control systems, which are common mathematical models employed in many safety critical systems subject to structured uncertainty.

A robust policy for Automatic Generation Control cyber attack in two area power network

This paper develops methodologies to robustly destabilize a two-area power system in the case of a cyber attack in the Automatic Generation Control (AGC). In earlier work reachability methods were used to establish conditions under which an attacker can cause undesirable behavior by interrupting the AGC signals and introducing an appropriate fake signal. In this paper we investigate how to robustify this approach to deal with practical situations where the attacker only has partial information about the parameters of the power system and the values of its states. We first propose an open loop procedure, based on Markov Chain Monte Carlo optimization, to identify an optimal attack signal. Motivated by the fact that the results are very sensitive to parameter uncertainty, we develop a systematic algorithm, based on feedback linearization, to construct a feedback policy that an intruder may use to disrupt the network. The numerical simulations demonstrate the effectiveness of the resulting policy, as well as its robustness with respect to modeling uncertainty and imperfect state information.

Performance Bounds for the Scenario Approach and an Extension to a Class of Non-Convex Programs

We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilistic bridge from the optimal value of SCP to the optimal values of RCP and CCP in which the uncertainty takes values in a general, possibly infinite dimensional, metric space. We then extend our results to a certain class of non-convex problems that includes, for example, binary decision variables. In the process, we also settle a measurability issue for a general class of scenario programs, which to date has been addressed by an assumption. Finally, we demonstrate the applicability of our results on a benchmark problem and a problem in fault detection and isolation.

On a problem of stochastic reach-avoid set characterization

We develop a novel framework for formulating a class of stochastic reachability problems with state constraints as a stochastic optimal control problem. Previous approaches to solving these problems are either confined to the deterministic setting or address almost-sure stochastic notions. In contrast, we propose a new methodology to tackle probabilistic specifications that are less specific than almost sure requirements. To this end, we first establish a connection between two stochastic reach-avoid problems and a class of stochastic optimal control problems for diffusions with discontinuous payoff functions. We then derive a weak version of dynamic programming principle (DPP) for the value function. Moreover, based on our DPP, we give an alternate characterization of the value function as the solution to a partial differential equation in the sense of discontinuous viscosity solutions. Finally we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.

Adaptively constrained Stochastic Model Predictive Control for closed-loop constraint satisfaction

Stochastic Model Predictive Control (SMPC) for discrete-time linear systems subject to additive disturbances with chance constraints on the states and hard constraints on the inputs is considered. Current chance constrained MPC methods-based on analytic reformulations or on sampling approaches-tend to be conservative partly because they fail to exploit the predefined violation level in closed-loop. For many practical applications, this conservatism can lead to a loss in performance. We propose an adaptive SMPC scheme that starts with a standard conservative chance constrained formulation and then on-line adapts the formulation of constraints based on the experienced violation frequency. Using martingale theory we establish guarantees of convergence to the desired level of constraint violation in closed-loop for a special class of linear systems. Comments are given on how to extend this to a broader class of (non-)linear systems. The developed methodology is demonstrated with an illustrative example.

Cyber-security of SCADA systems

After a general introduction of the VIKING EU FP7 project two specific cyber-attack mechanisms, which have been analyzed in the VIKING project, will be discussed in more detail. Firstly an attack and its consequences on the Automatic Generation Control (AGC) in a power system are investigated, and secondly the cyber security of State Estimators in SCADA systems is scrutinized.

LQG Control With Minimum Directed Information: Semidefinite Programming Approach

We consider a discrete-time linear–quadratic–Gaussian (LQG) control problem, in which Masseys directed information from the observed output of the plant to the control input is minimized, while required control performance is attainable. This problem arises in several different contexts, including joint encoder and controller design for data-rate minimization in networked control systems. We show that the optimal control law is a linear–Gaussian randomized policy. We also identify the state-space realization of the optimal policy, which can be synthesized by an efficient algorithm based on semidefinite programming. Our structural result indicates that the filter–controller separation principle from the LQG control theory and the sensor–filter separation principle from the zero-delay rate-distortion theory for Gauss–Markov sources hold simultaneously in the considered problem. A connection to the data-rate theorem for mean-square stability by Nair and Evans is also established.

The stochastic reach-avoid problem and set characterization for diffusions

In this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective. Preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost-sure stochastic requirements. In contrast, we propose a methodology to tackle problems with less stringent requirements than almost sure. To this end, we first establish a connection between two distinct stochastic reach-avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions. Subsequently, we focus on solutions of one of the classes of stochastic optimal control problems-the exit-time problem, which solves both the two reach-avoid problems mentioned above. We then derive a weak version of a dynamic programming principle (DPP) for the corresponding value function; in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions. Moreover, based on our DPP, we provide an alternative characterization of the value function as a solution of a partial differential equation (PDE) in the sense of discontinuous viscosity solutions, along with boundary conditions both in Dirichlet and viscosity senses. Theoretical justifications are also discussed to pave the way for deployment of off-the-shelf PDE solvers for numerical computations. Finally, we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.