arXiv subjects

Mohamadreza Ahmadi

Publications and source records attributed to Mohamadreza Ahmadi.

At least 37 records · Page 2Linked to original sources

arXiv2019

Optimal Safe Controller Synthesis: A Density Function Approach

This paper considers the synthesis of optimal safe controllers based on density functions. We present an algorithm for robust constrained optimal control synthesis using the duality relationship between the density function and the value function. The density function follows the Liouville equation and is the dual of the value function, which satisfies Bellman's optimality principle. Thanks to density functions, constraints over the distribution of states, such as safety constraints, can be posed straightforwardly in an optimal control problem. The constrained optimal control problem is then solved with a primal-dual algorithm. This formulation is extended to the case with external disturbances, and we show that the robust constrained optimal control can be solved with a modified primal-dual algorithm. We apply this formulation to the problem of finding the optimal safe controller that minimizes the cumulative intervention. An adaptive cruise control (ACC) example is used to demonstrate the efficacy of the proposed, wherein we compare the result of the density function approach with the conventional control barrier function (CBF) method.

math.OC↗

arXiv2019

Risk-Averse Planning Under Uncertainty

We consider the problem of designing policies for partially observable Markov decision processes (POMDPs) with dynamic coherent risk objectives. Synthesizing risk-averse optimal policies for POMDPs requires infinite memory and thus undecidable. To overcome this difficulty, we propose a method based on bounded policy iteration for designing stochastic but finite state (memory) controllers, which takes advantage of standard convex optimization methods. Given a memory budget and optimality criterion, the proposed method modifies the stochastic finite state controller leading to sub-optimal solutions with lower coherent risk.

cs.RO↗

arXiv2019

Safe Policy Synthesis in Multi-Agent POMDPs via Discrete-Time Barrier Functions

A multi-agent partially observable Markov decision process (MPOMDP) is a modeling paradigm used for high-level planning of heterogeneous autonomous agents subject to uncertainty and partial observation. Despite their modeling efficiency, MPOMDPs have not received significant attention in safety-critical settings. In this paper, we use barrier functions to design policies for MPOMDPs that ensure safety. Notably, our method does not rely on discretization of the belief space, or finite memory. To this end, we formulate sufficient and necessary conditions for the safety of a given set based on discrete-time barrier functions (DTBFs) and we demonstrate that our formulation also allows for Boolean compositions of DTBFs for representing more complicated safe sets. We show that the proposed method can be implemented online by a sequence of one-step greedy algorithms as a standalone safe controller or as a safety-filter given a nominal planning policy. We illustrate the efficiency of the proposed methodology based on DTBFs using a high-fidelity simulation of heterogeneous robots.

cs.RO↗

arXiv2019

Barrier Certificates for Assured Machine Teaching

Machine teaching can be viewed as optimal control for learning. Given a learner's model, machine teaching aims to determine the optimal training data to steer the learner towards a target hypothesis. In this paper, we are interested in providing assurances for machine teaching algorithms using control theory. In particular, we study a well-established learner's model in the machine teaching literature that is captured by the local preference over a version space. We interpret the problem of teaching a preference-based learner as solving a partially observable Markov decision process (POMDP). We then show that the POMDP formulation can be cast as a special hybrid system, i.e., a discrete-time switched system. Subsequently, we use barrier certificates to verify set-theoric properties of this special hybrid system. We show how the computation of the barrier certificate can be decomposed and numerically implemented as the solution to a sum-of-squares (SOS) program. For illustration, we show how the proposed framework based on control theory can be used to verify the teaching performance of two well-known machine teaching methods.

eess.SY↗

arXiv2019

Entropy-Regularized Stochastic Games

In two-player zero-sum stochastic games, where two competing players make decisions under uncertainty, a pair of optimal strategies is traditionally described by Nash equilibrium and computed under the assumption that the players have perfect information about the stochastic transition model of the environment. However, implementing such strategies may make the players vulnerable to unforeseen changes in the environment. In this paper, we introduce entropy-regularized stochastic games where each player aims to maximize the causal entropy of its strategy in addition to its expected payoff. The regularization term balances each player's rationality with its belief about the level of misinformation about the transition model. We consider both entropy-regularized $N$-stage and entropy-regularized discounted stochastic games, and establish the existence of a value in both games. Moreover, we prove the sufficiency of Markovian and stationary mixed strategies to attain the value, respectively, in $N$-stage and discounted games. Finally, we present algorithms, which are based on convex optimization problems, to compute the optimal strategies. In a numerical example, we demonstrate the proposed method on a motion planning scenario and illustrate the effect of the regularization term on the expected payoff.

math.OC↗

arXiv2019

Control Theory Meets POMDPs: A Hybrid Systems Approach

Partially observable Markov decision processes (POMDPs) provide a modeling framework for a variety of sequential decision making under uncertainty scenarios in artificial intelligence (AI). Since the states are not directly observable in a POMDP, decision making has to be performed based on the output of a Bayesian filter (continuous beliefs). Hence, POMDPs are often computationally intractable to solve exactly and researchers resort to approximate methods often using discretizations of the continuous belief space. These approximate solutions are, however, prone to discretization errors, which has made POMDPs ineffective in applications, wherein guarantees for safety, optimality, or performance are required. To overcome the complexity challenge of POMDPs, we apply notions from control theory. The goal is to determine the reachable belief space of a POMDP, that is, the set of all possible evolutions given an initial belief distribution over the states and a set of actions and observations. We begin by casting the problem of analyzing a POMDP into analyzing the behavior of a discrete-time switched system. For estimating the reachable belief space, we find over-approximations in terms of sub-level sets of Lyapunov functions. Furthermore, in order to verify safety and optimality requirements of a given POMDP, we formulate a barrier certificate theorem, wherein we show that if there exists a barrier certificate satisfying a set of inequalities along with the belief update equation of the POMDP, the safety and optimality properties are guaranteed to hold. In both cases, we show how the calculations can be decomposed into smaller problems that can be solved in parallel. The conditions we formulate can be computationally implemented as a set of sum-of-squares programs. We illustrate the applicability of our method by addressing two problems in active ad scheduling and machine teaching.

eess.SY↗

arXiv2018

The Partially Observable Games We Play for Cyber Deception

Progressively intricate cyber infiltration mechanisms have made conventional means of defense, such as firewalls and malware detectors, incompetent. These sophisticated infiltration mechanisms can study the defender's behavior, identify security caveats, and modify their actions adaptively. To tackle these security challenges, cyber-infrastructures require active defense techniques that incorporate cyber deception, in which the defender (deceiver) implements a strategy to mislead the infiltrator. To this end, we use a two-player partially observable stochastic game (POSG) framework, wherein the deceiver has full observability over the states of the POSG, and the infiltrator has partial observability. Then, the deception problem is to compute a strategy for the deceiver that minimizes the expected cost of deception against all strategies of the infiltrator. We first show that the underlying problem is a robust mixed-integer linear program, which is intractable to solve in general. Towards a scalable approach, we compute optimal finite-memory strategies for the infiltrator by a reduction to a series of synthesis problems for parametric Markov decision processes. We use these infiltration strategies to find robust strategies for the deceiver using mixed-integer linear programming. We illustrate the performance of our technique on a POSG model for network security. Our experiments demonstrate that the proposed approach handles scenarios considerably larger than those of the state-of-the-art methods.

cs.AI↗

arXiv2018

Cost-Bounded Active Classification Using Partially Observable Markov Decision Processes

Active classification, i.e., the sequential decision-making process aimed at data acquisition for classification purposes, arises naturally in many applications, including medical diagnosis, intrusion detection, and object tracking. In this work, we study the problem of actively classifying dynamical systems with a finite set of Markov decision process (MDP) models. We are interested in finding strategies that actively interact with the dynamical system, and observe its reactions so that the true model is determined efficiently with high confidence. To this end, we present a decision-theoretic framework based on partially observable Markov decision processes (POMDPs). The proposed framework relies on assigning a classification belief (a probability distribution) to each candidate MDP model. Given an initial belief, some misclassification probabilities, a cost bound, and a finite time horizon, we design POMDP strategies leading to classification decisions. We present two different approaches to find such strategies. The first approach computes the optimal strategy "exactly" using value iteration. To overcome the computational complexity of finding exact solutions, the second approach is based on adaptive sampling to approximate the optimal probability of reaching a classification decision. We illustrate the proposed methodology using two examples from medical diagnosis and intruder detection.

eess.SY↗

arXiv2018

Transfer Entropy in MDPs with Temporal Logic Specifications

Emerging applications in autonomy require control techniques that take into account uncertain environments, communication and sensing constraints, while satisfying highlevel mission specifications. Motivated by this need, we consider a class of Markov decision processes (MDPs), along with a transfer entropy cost function. In this context, we study highlevel mission specifications as co-safe linear temporal logic (LTL) formulae. We provide a method to synthesize a policy that minimizes the weighted sum of the transfer entropy and the probability of failure to satisfy the specification. We derive a set of coupled non-linear equations that an optimal policy must satisfy. We then use a modified Arimoto-Blahut algorithm to solve the non-linear equations. Finally, we demonstrated the proposed method on a navigation and path planning scenario of a Mars rover.

eess.SY↗

arXiv2018

Verification of Uncertain POMDPs Using Barrier Certificates

We consider a class of partially observable Markov decision processes (POMDPs) with uncertain transition and/or observation probabilities. The uncertainty takes the form of probability intervals. Such uncertain POMDPs can be used, for example, to model autonomous agents with sensors with limited accuracy, or agents undergoing a sudden component failure, or structural damage [1]. Given an uncertain POMDP representation of the autonomous agent, our goal is to propose a method for checking whether the system will satisfy an optimal performance, while not violating a safety requirement (e.g. fuel level, velocity, and etc.). To this end, we cast the POMDP problem into a switched system scenario. We then take advantage of this switched system characterization and propose a method based on barrier certificates for optimality and/or safety verification. We then show that the verification task can be carried out computationally by sum-of-squares programming. We illustrate the efficacy of our method by applying it to a Mars rover exploration example.

eess.SY↗

arXiv2018

Privacy Verification in POMDPs via Barrier Certificates

Privacy is an increasing concern in cyber-physical systems that operates over a shared network. In this paper, we propose a method for privacy verification of cyber- physical systems modeled by Markov decision processes (MDPs) and partially-observable Markov decision processes (POMDPs) based on barrier certificates. To this end, we consider an opacity-based notion of privacy, which is characterized by the beliefs in system states. We show that the belief update equations can be represented as discrete-time switched systems, for which we propose a set of conditions for privacy verification in terms of barrier certificates. We further demonstrate that, for MDPs and for POMDPs, privacy verification can be computationally implemented by solving a set of semi-definite programs and sum-of-squares programs, respectively. The method is illustrated by an application to privacy verification of an inventory management system.

eess.SY↗

arXiv2018

Compositional Analysis of Hybrid Systems Defined Over Finite Alphabets

We consider the stability and the input-output analysis problems of a class of large-scale hybrid systems composed of continuous dynamics coupled with discrete dynamics defined over finite alphabets, e.g., deterministic finite state machines (DFSMs). This class of hybrid systems can be used to model physical systems controlled by software. For such classes of systems, we use a method based on dissipativity theory for compositional analysis that allows us to study stability, passivity and input-output norms. We show that the certificates of the method based on dissipativity theory can be computed by solving a set of semi-definite programs. Nonetheless, the formulation based on semi-definite programs become computationally intractable for relatively large number of discrete and continuous states. We demonstrate that, for systems with large number of states consisting of an interconnection of smaller hybrid systems, accelerated alternating method of multipliers can be used to carry out the computations in a scalable and distributed manner. The proposed methodology is illustrated by an example of a system with 60 continuous states and 18 discrete states.

math.OC↗

arXiv2018

Distributed Synthesis Using Accelerated ADMM

We propose a convex distributed optimization algorithm for synthesizing robust controllers for large-scale continuous time systems subject to exogenous disturbances. Given a large scale system, instead of solving the larger centralized synthesis task, we decompose the problem into a set of smaller synthesis problems for the local subsystems with a given interconnection topology. Hence, the synthesis problem is constrained to the sparsity pattern dictated by the interconnection topology. To this end, for each subsystem, we solve a local dissipation inequality and then check a small-gain like condition for the overall system. To minimize the effect of disturbances, we consider the $\mathrm{H}_\infty$ synthesis problems. We instantiate the distributed synthesis method using accelerated alternating direction method of multipliers (ADMM) with convergence rate $O(\frac{1}{k^2})$ with $k$ being the number of iterations.

math.OC↗

arXiv2018

A Framework for Input-Output Analysis of Wall-Bounded Shear Flows

We propose a framework to understand input-output amplification properties of non- linear partial differential equation (PDE) models of wall-bounded shear flows, which are spatially invariant in one coordinate (e.g., streamwise-constant plane Couette flow). Our methodology is based on the notion of dissipation inequalities in control theory. In particular, we consider flows with body and other forcings, for which we study the input- to-output properties, including energy growth, worst-case disturbance amplification, and stability to persistent disturbances. The proposed method can be applied to a large class of flow configurations as long as the base flow is described by a polynomial. This includes many examples in both channel flows and pipe flows, e.g., plane Couette flow, and Hagen-Poiseuille flow. The methodology we use is numerically implemented as the solution of a (convex) optimization problem. We use the framework to study input-output amplification mechanisms in rotating Couette flow, plane Couette flow, plane Poiseuille flow, and Hagen-Poiseuille flow. In addition to showing that the application of the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature through linearization around the base flow, we demonstrate that the stability bounds to persistent forcings can be used as a means to predict transition to turbulence in wall-bounded shear flows.

physics.flu-dyn↗

arXiv2018

Controller Synthesis for Safety of Physically-Viable Data-Driven Models

We consider the problem of designing finite-horizon safe controllers for a dynamical system for which no explicit analytical model exists and limited data only along a single trajectory of the system are available. Given samples of the states and inputs of the system, and additional side information in terms of regularity of the evolution of the states and conservation laws, we synthesize a controller such that the evolution of the states avoid some pre-specified unsafe set over a given finite horizon. Motivated by recent results on Whitney's extension theorem, we use piecewise-polynomial approximations of the trajectories based on the data along with the regularity side information to formulate a data-driven differential inclusion model that can predict the evolution of the trajectories. For these classes of data- driven differential inclusions, we propose a safety analysis theorem based on barrier certificates. As a corollary of this theorem, we demonstrate that we can design controllers ensuring safety of the solutions to the data-driven differential inclusion over a finite horizon. From a computational standpoint, our results are cast into a set of sum-of-squares programs whenever the certificates are parametrized by polynomials of fixed degree and the sets are semi-algebraic. This computational method allows incorporating side information in terms of conservation laws and integral constraints using Positivstellensatz-type arguments.

math.OC↗

arXiv2017

Safety Verification for Distributed Parameter Systems Using Barrier Functionals

We study the safety verification problem for a class of distributed parameter systems described by partial differential equations (PDEs), i.e., the problem of checking whether the solutions of the PDE satisfy a set of constraints at a particular point in time. The proposed method is based on an extension of barrier certificates to infinite-dimensional systems. In this respect, we introduce barrier functionals, which are functionals of the dependent and independent variables. Given a set of initial conditions and an unsafe set, we demonstrate that if such a functional exists satisfying two (integral) inequalities, then the solutions of the system do not enter the unsafe set. Therefore, the proposed method does not require finite-dimensional approximations of the distributed parameter system. Furthermore, for PDEs with polynomial data, we solve the associated integral inequalities using semi-definite programming (SDP) based on a method that relies on a quadratic representation of the integrands of integral inequalities. The proposed method is illustrated through examples.

math.OC↗

arXiv2017

On Robust Stability of Switched Systems in the Context of Filippov Solutions

The stability problem of a class of nonlinear switched systems defined on compact sets with state-dependent switching is considered. Instead of the Caratheodory solutions, the general Filippov solutions are studied. This encapsulates solutions with infinite switching in finite time and sliding modes in the neighborhood of the switching surfaces. In this regard, a Lyapunov-like stability theorem, based on the theory of differential inclusions, is formulated. Additionally, the results are extended to switched systems with simplical uncertainty. It is also demonstrated that, for the special case of polynomial switched systems defined on semi-algebraic sets, stability analysis can be checked based on sum of squares programming techniques.

math.OC↗

arXiv2016

Barrier Functionals for the Analysis of Complex Systems: An Optimization-Based Approach

We propose a methodology to address two analysis problems concerning complex systems, namely bounding state functionals of stochastic differential equations (SDEs) and verifying set avoidance of systems described by partial differential equations (PDEs). The proposed method is based on barrier functionals, which are functionals of the states of the studied systems. The proposed method does not require the approximation of solutions nor the stability of trajectories. In the case of SDEs, the formulation relies on a generalized version of the Feynman-Kac formula and results in moments bounds for nonlinear SDEs. Furthermore, we show that the analysis problems can be cast as optimization problems and can be solved by semi-definite programming.

math.OC↗