Towards Scalable Synthesis of Stochastic Control Systems

Formal control synthesis approaches over stochastic systems have received significant attention in the past few years, in view of their ability to provide provably correct controllers for complex logical specifications in an automated fashion. Examples of complex specifications of interest include properties expressed as formulae in linear temporal logic (LTL) or as automata on infinite strings. A general methodology to synthesize controllers for such properties resorts to symbolic abstractions of the given stochastic systems. Symbolic models are discrete abstractions of the given concrete systems with the property that a controller designed on the abstraction can be refined (or implemented) into a controller on the original system. Although the recent development of techniques for the construction of symbolic models has been quite encouraging, the general goal of formal synthesis over stochastic control systems is by no means solved. A fundamental issue with the existing techniques is the known "curse of dimensionality," which is due to the need to discretize state and input sets and that results in an exponential complexity over the number of state and input variables in the concrete system. In this work we propose a novel abstraction technique for incrementally stable stochastic control systems, which does not require state-space discretization but only input set discretization, and that can be potentially more efficient (and thus scalable) than existing approaches. We elucidate the effectiveness of the proposed approach by synthesizing a schedule for the coordination of two traffic lights under some safety and fairness requirements for a road traffic model. Further we argue that this 5-dimensional linear stochastic control system cannot be studied with existing approaches based on state-space discretization due to the very large number of generated discrete states.Comment: 22 pages, 3 figures. arXiv admin note: text overlap with arXiv:1407.273

Symbolic Models for Stochastic Switched Systems: A Discretization and a Discretization-Free Approach

Stochastic switched systems are a relevant class of stochastic hybrid systems with probabilistic evolution over a continuous domain and control-dependent discrete dynamics over a finite set of modes. In the past few years several different techniques have been developed to assist in the stability analysis of stochastic switched systems. However, more complex and challenging objectives related to the verification of and the controller synthesis for logic specifications have not been formally investigated for this class of systems as of yet. With logic specifications we mean properties expressed as formulae in linear temporal logic or as automata on infinite strings. This paper addresses these complex objectives by constructively deriving approximately equivalent (bisimilar) symbolic models of stochastic switched systems. More precisely, this paper provides two different symbolic abstraction techniques: one requires state space discretization, but the other one does not require any space discretization which can be potentially more efficient than the first one when dealing with higher dimensional stochastic switched systems. Both techniques provide finite symbolic models that are approximately bisimilar to stochastic switched systems under some stability assumptions on the concrete model. This allows formally synthesizing controllers (switching signals) that are valid for the concrete system over the finite symbolic model, by means of mature automata-theoretic techniques in the literature. The effectiveness of the results are illustrated by synthesizing switching signals enforcing logic specifications for two case studies including temperature control of a six-room building.Comment: 25 pages, 4 figures. arXiv admin note: text overlap with arXiv:1302.386

Large-Scale Invariant Sets for Safe Coordination of Thermostatic Loads

Extended version of ACC 2021 paper.Systems often face constraints at multiple levels. For example, in coordinating a collection of thermostatically controlled loads to provide grid services, the controller must ensure temperature constraints for each load (local constraints) and distribution network constraints (global constraints) are satisfied. In this paper, we leverage invariant sets to ensure safe coordination of systems with both local and global constraints. Specifically, we develop a method for constructing a controlled invariant set for a collection of subsystems, modeled as transition systems, to ensure they indefinitely satisfy the constraints, based on cycles in individual transition systems. Then, we develop a control algorithm that keeps the state inside the maximal controlled invariant set. We apply these algorithms to a demand response problem, specifically, the tracking of a power trajectory (e.g., a frequency regulation signal) by a population of homogeneous air conditioners. The algorithm simultaneously maintains local temperature requirements and aggregate power consumption limits, ensuring the control is nondisruptive to consumers and benign to the distribution network.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/166595/1/ACC2021_FInal_LongerVer.pdfSEL

Constructive Formal Control Synthesis through Abstraction and Decomposition

Control synthesis is the problem of automatically constructing a control strategy that induces a system to exhibit a declared behavior. Synthesis algorithms vary widely across different classes of system dynamics and specifications.While continuous optimization has traditionally been used to construct stabilizing controllers for physical systems modeled with differential equations, temporal logic synthesis for finite state machines heavily leverages discrete algorithms and data structures.Hybrid systems are a class of systems that exhibit both continuous and discrete behaviors, which are necessary to capture phenomena such as impacts for legged robots and congestion shockwaves in freeways. Tractable control synthesis remains elusive because hybrid systems violate many of the fundamental topological assumptions made by prior algorithms for purely continuous or discrete systems.This thesis exploits compositionality and system structure to provide a suite of algorithmic and theoretical techniques to tackle acute computational bottlenecks in hybrid control synthesis.The first half of this thesis provides a framework for engineers to model control systems and construct algorithmic pipelines for control synthesis.By explicitly capturing system structure, this framework gives users the flexibility to rapidly iterate over and leverage a library of optimizations for control synthesis.We demonstrate this framework in the context of abstraction-based control, a synthesis workflow that translates continuous systems into finite state machines by throwing away high precision information. Different optimization techniques such as multi-scale grids, lazy abstraction, and decomposed synthesis, can all be expressed as modifications to a computational pipeline. We demonstrate computational gains while synthesizing safe motion primitives for numerous robotic examples.The second half addresses distributed control synthesis where multiple controllers act as agents that seek to jointly satisfy a specification and are restricted by some communication topology. We introduce parametric assume-guarantee contracts as a formalism to derive guarantees about the closed loop behavior of a collection of interacting components. Dynamic contracts allow contract parameters to change at runtime and enable coordination of multiple interacting sub-systems.These results are demonstrated in the context of a freeway ramp meter and an adjacent arterial network