The vast majority of today's critical infrastructure is supported by numerous feedback control loops and an attack on these control loops can have disastrous consequences. This is a major concern since modern control systems are becoming large and decentralized and thus more vulnerable to attacks. This paper is concerned with the estimation and control of linear systems when some of the sensors or actuators are corrupted by an attacker.In the first part we look at the estimation problem where we characterize the resilience of a system to attacks and study the possibility of increasing its resilience by a change of parameters. We then propose an efficient algorithm to estimate the state despite the attacks and we characterize its performance. Our approach is inspired from the areas of error-correction over the reals and compressed sensing.In the second part we consider the problem of designing output-feedback controllers that stabilize the system despite attacks. We show that a principle of separation between estimation and control holds and that the design of resilient output feedback controllers can be reduced to the design of resilient state estimators.for example the sensors communicate their measurements to the controllers, the controllers use 2 this information to compute the control input, and the control input is then sent to the actuators so that it can be physically implemented. In order for this communication to take place, a communication network is usually deployed across the plant to be controlled. Although wired networks have been traditionally used for this purpose, an increasing number of control systems now use wireless networks since they are easier to deploy and to maintain. In addition, these networks are sometimes connected to the corporate intranet, and in some cases even to the Internet. Consequently, modern control systems are becoming more open to the cyber-world, and as such, are more vulnerable to attacks that can cause faults and failures in the physical process even though launched in the cyber-domain. This realization led to the emergence of new security challenges that are distinct from traditional cyber security as highlighted in [1], [2]. Real-world attacks on control systems have in fact occurred in the past decade and have in some cases caused significant damage to the targeted physical processes. Perhaps one of the most popular examples is the attack on Maroochy Shire Council's sewage control system in Queensland, Australia that happened in January 2000 [3], [4]. In this incident, an attacker managed to hack into some controllers that activate and deactivate valves and, by doing so, caused flooding of the grounds of a hotel, a park, and a river with a million liters of sewage [3]. Another well publicized example of an attack launched on physical systems is the very recent StuxNet virus that targeted Siemens' supervisory control and data acquisition systems which are used in many industrial processes [5]. Other cases of attacks have been reported in the past years, and we refer ...

The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations.

Let M be a p-by-q matrix with nonnegative entries. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $A_i, B_j$ of size $k \times k$ such that $M_{ij} = \text{trace}(A_i B_j)$. The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.Comment: 35 page