Robust control of networks under discrete disturbances and controls.

International audienceWe consider dynamic networks where the disturbances and control actions take discrete values. We briefly survey some of our recent results establishing necessary and sufficient conditions for the existence of robustly globally invariant (hyper box) sets, as well as sufficient conditions for global attractivity of such sets.We then establish connections between these results and existing results in the literature for the setup where all the inputs are analog. Finally, we derive tight upper and lower bounds on the smallest such set in the special case of a degenerate network

Formal Synthesis of Control Strategies for Positive Monotone Systems

We design controllers from formal specifications for positive discrete-time monotone systems that are subject to bounded disturbances. Such systems are widely used to model the dynamics of transportation and biological networks. The specifications are described using signal temporal logic (STL), which can express a broad range of temporal properties. We formulate the problem as a mixed-integer linear program (MILP) and show that under the assumptions made in this paper, which are not restrictive for traffic applications, the existence of open-loop control policies is sufficient and almost necessary to ensure the satisfaction of STL formulas. We establish a relation between satisfaction of STL formulas in infinite time and set-invariance theories and provide an efficient method to compute robust control invariant sets in high dimensions. We also develop a robust model predictive framework to plan controls optimally while ensuring the satisfaction of the specification. Illustrative examples and a traffic management case study are included.Comment: To appear in IEEE Transactions on Automatic Control (TAC) (2018), 16 pages, double colum

Robust Temporal Logic Model Predictive Control

Control synthesis from temporal logic specifications has gained popularity in recent years. In this paper, we use a model predictive approach to control discrete time linear systems with additive bounded disturbances subject to constraints given as formulas of signal temporal logic (STL). We introduce a (conservative) computationally efficient framework to synthesize control strategies based on mixed integer programs. The designed controllers satisfy the temporal logic requirements, are robust to all possible realizations of the disturbances, and optimal with respect to a cost function. In case the temporal logic constraint is infeasible, the controller satisfies a relaxed, minimally violating constraint. An illustrative case study is included.Comment: This work has been accepted to appear in the proceedings of 53rd Annual Allerton Conference on Communication, Control and Computing, Urbana-Champaign, IL (2015

Finite Alphabet Control of Logistic Networks with Discrete Uncertainty

We consider logistic networks in which the control and disturbance inputs take values in finite sets. We derive a necessary and sufficient condition for the existence of robustly control invariant (hyperbox) sets. We show that a stronger version of this condition is sufficient to guarantee robust global attractivity, and we construct a counterexample demonstrating that it is not necessary. Being constructive, our proofs of sufficiency allow us to extract the corresponding robust control laws and to establish the invariance of certain sets. Finally, we highlight parallels between our results and existing results in the literature, and we conclude our study with two simple illustrative examples

Robust distributed linear programming

This paper presents a robust, distributed algorithm to solve general linear programs. The algorithm design builds on the characterization of the solutions of the linear program as saddle points of a modified Lagrangian function. We show that the resulting continuous-time saddle-point algorithm is provably correct but, in general, not distributed because of a global parameter associated with the nonsmooth exact penalty function employed to encode the inequality constraints of the linear program. This motivates the design of a discontinuous saddle-point dynamics that, while enjoying the same convergence guarantees, is fully distributed and scalable with the dimension of the solution vector. We also characterize the robustness against disturbances and link failures of the proposed dynamics. Specifically, we show that it is integral-input-to-state stable but not input-to-state stable. The latter fact is a consequence of a more general result, that we also establish, which states that no algorithmic solution for linear programming is input-to-state stable when uncertainty in the problem data affects the dynamics as a disturbance. Our results allow us to establish the resilience of the proposed distributed dynamics to disturbances of finite variation and recurrently disconnected communication among the agents. Simulations in an optimal control application illustrate the results

Sparse and Constrained Stochastic Predictive Control for Networked Systems

This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. The control channel is assumed to have i.i.d. Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. Our proposed class of policies is affine in the past dropouts and saturated values of the past disturbances. We further consider a regularization term in a quadratic performance index to promote sparsity in control. We demonstrate how to augment the underlying optimization problem with a constant negative drift constraint to ensure mean-square boundedness of the closed-loop states, yielding a convex quadratic program to be solved periodically online. The states of the closed-loop plant under the receding horizon implementation of the proposed class of policies are mean square bounded for any positive bound on the control and any non-zero probability of successful transmission

Safety control of monotone systems with bounded uncertainties

Monotone systems are prevalent in models of engineering applications such as transportation and biological networks. In this paper, we investigate the problem of finding a control strategy for a discrete time positive monotone system with bounded uncertainties such that the evolution of the system is guaranteed to be confined to a safe set in the state space for all times. By exploiting monotonicity, we propose an approach to this problem which is based on constraint programming. We find control strategies that are based on repetitions of finite sequences of control actions. We show that, under assumptions made in the paper, safety control of monotone systems does not require state measurement. We demonstrate the results on a signalized urban traffic network, where the safety objective is to keep the traffic flow free of congestion.This work was partially supported by the NSF under grants CPS-1446151 and CMMI-1400167. (CPS-1446151 - NSF; CMMI-1400167 - NSF

Distributed Robust Set-Invariance for Interconnected Linear Systems

We introduce a class of distributed control policies for networks of discrete-time linear systems with polytopic additive disturbances. The objective is to restrict the network-level state and controls to user-specified polyhedral sets for all times. This problem arises in many safety-critical applications. We consider two problems. First, given a communication graph characterizing the structure of the information flow in the network, we find the optimal distributed control policy by solving a single linear program. Second, we find the sparsest communication graph required for the existence of a distributed invariance-inducing control policy. Illustrative examples, including one on platooning, are presented.Comment: 8 Pages. Submitted to American Control Conference (ACC), 201