Constrained optimal control of stochastic switched affine systems using randomization

We consider a finite-horizon optimal control problem for a switched affine system with controlled switches, affected by uncertainty and subject to input and/or state constraints. We show how the logical statements that govern the underlying switching mechanism can be transformed into robust mixed-integer inequalities, leading to an infinite dimensional linear program with robust constraints. Following a randomized methodology, based on enforcing the constraints only on a finite number of uncertainty instances/scenarios, we relax the infinite dimensional program to a mixed-integer linear program, which is amenable to existing numerical tools. We establish a probabilistic link between the infinite dimensional robust program and its scenario-based relaxation, showing that the optimal solution of the latter is feasible, in a probabilistic sense, for the former

A scenario approach for non-convex control design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in terms of performance and require high sample complexity to achieve the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control- design problems that can be addressed via randomization, we apply our scenario approach to randomized Model Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro

A New Simulation Metric to Determine Safe Environments and Controllers for Systems with Unknown Dynamics

We consider the problem of extracting safe environments and controllers for reach-avoid objectives for systems with known state and control spaces, but unknown dynamics. In a given environment, a common approach is to synthesize a controller from an abstraction or a model of the system (potentially learned from data). However, in many situations, the relationship between the dynamics of the model and the \textit{actual system} is not known; and hence it is difficult to provide safety guarantees for the system. In such cases, the Standard Simulation Metric (SSM), defined as the worst-case norm distance between the model and the system output trajectories, can be used to modify a reach-avoid specification for the system into a more stringent specification for the abstraction. Nevertheless, the obtained distance, and hence the modified specification, can be quite conservative. This limits the set of environments for which a safe controller can be obtained. We propose SPEC, a specification-centric simulation metric, which overcomes these limitations by computing the distance using only the trajectories that violate the specification for the system. We show that modifying a reach-avoid specification with SPEC allows us to synthesize a safe controller for a larger set of environments compared to SSM. We also propose a probabilistic method to compute SPEC for a general class of systems. Case studies using simulators for quadrotors and autonomous cars illustrate the advantages of the proposed metric for determining safe environment sets and controllers.Comment: 22nd ACM International Conference on Hybrid Systems: Computation and Control (2019

Sequential Randomized Algorithms for Convex Optimization in the Presence of Uncertainty

In this paper, we propose new sequential randomized algorithms for convex optimization problems in the presence of uncertainty. A rigorous analysis of the theoretical properties of the solutions obtained by these algorithms, for full constraint satisfaction and partial constraint satisfaction, respectively, is given. The proposed methods allow to enlarge the applicability of the existing randomized methods to real-world applications involving a large number of design variables. Since the proposed approach does not provide a priori bounds on the sample complexity, extensive numerical simulations, dealing with an application to hard-disk drive servo design, are provided. These simulations testify the goodness of the proposed solution.Comment: 18 pages, Submitted for publication to IEEE Transactions on Automatic Contro

Decentralized Resource Allocation through Constrained Centroidal Voronoi Tessellations

The advancements in the fields of microelectronics facilitate incorporating team elements like coordination into engineering systems through advanced computing power. Such incorporation is useful since many engineering systems can be characterized as a collection of interacting subsystems each having access to local information, making local decisions, interacting with neighbors, and seeking to optimize local objectives that may well conflict with other subsystems, while also trying to optimize certain global objective. In this dissertation, we take advantage of such technological advancements to explore the problem of resource allocation through different aspects of the decentralized architecture like information structure in a team. Introduced in 1968 as a toy example in the field of team decision theory to demonstrate the significance of information structure within a team, the Witsenhausen counterexample remained unsolved until the analytical person-by-person optimal solution was developed within the past decade. We develop a numerical method to implement the optimal laws and show that our laws coincide with the optimal affine laws. For the region where the optimal laws are non-linear, we show that our laws result in the lowest costs when compared with previously reported costs. Recognizing that, in the framework of team decision theory, the difficulties arising from the non-classical information structure within a team currently limit its applicability in real-world applications, we move on to investigating Centroidal Voronoi Tessellations (CVTs) to solve the resource allocation problem. In one-dimensional spaces, a line communication network is sufficient to obtain CVTs in a decentralized manner, while being scalable to any number of agents in the team. We first solve the static resource allocation problem where the amount of resource is fixed. Using such static allocation solution as an initialization step, we solve the dynamic resource allocation problem in a truly decentralized manner. Furthermore, we allow for flexibility in agents\u27 embedding their local preferences through what we call a civility model. We end the dissertation by revisiting the application of Demand-response in smart grids and demonstrate the developed decentralized dynamic resource allocation method to solve the problem of power allocation in a group of building loads