Trajectory-based temporal logic analysis : inference, robust testing and controller synthesis

Abstract

Besides classifying finitely many trajectories in different sets, we also classify infinitely many trajectories with initial state variations and disturbances, which accounts for the spatial uncertainties; we also consider the time variations when the switching events occur in a switched system, which accounts for the temporal uncertainties. We design observation maps in the form of temporal logic formulas for fault detection and privacy preservation of cyber-physical systems with temporal and spatial uncertainties in a provably correct fashion. We implement the method on the simulation model of a smart building testbed for detecting the open window fault while preserving multiple privacy conditions of the room occupancy.

To combine the temporal logic inference and controller synthesis processes, we propose a method to iteratively learn (infer) and refine a set of advices from the trajectories generated in the successful and failed attempts in a task, with each advice in the form of advisory signal temporal logic (STL) formulas. Each advice consists of an advisory motion STL formula that characterizes the spatial-temporal pattern of the motion as a feature of success and an advisory selection STL formula as a criterion for the environment to select the advice. The advisory controller can advise or guide the human operators or the robots for better performance with the shared autonomy between the human operator and the controller. We implement the approach in two case studies to test the effectiveness of the advisory controller, one with a Baxter-On-Wheels simulator using the keyboard control and the other with two quadrotors in an experimental testbed using the joystick control in iteratively improving the success rates of completing the tasks with the help of the designed advisory controller.

For stochastic control systems, we propose the stochastic control bisimulation function, which bounds the divergence of the trajectories of the stochastic control system and the diffusionless deterministic control system in a probabilistic fashion. We design a feedforward controller by solving an optimization problem for the nominal trajectory of the deterministic control system with robustness against initial state variations and stochastic uncertainties. Then we learn a feedback control law from the state and input data of the feedforward controller. As an implementation, we apply the proposed approach in controlling a wind farm and an energy storage system for frequency regulation with provable probabilistic safety guarantees in the stochastic environment of wind power generation.

Finally, we present the controller synthesis for deterministic and stochastic systems with respect to temporal logic specifications. For deterministic systems, we use the functional gradient descent method and the feedback control law generation method to design the feedforward and feedback controller respectively. We apply the controller synthesis algorithm on a three-machine power system model with energy storage systems and we use metric temporal logic formulas to specify the requirements for power system frequency regulations. Both the feedforward and feedback controllers of the energy storage systems can guarantee that all the post-fault trajectories with the given fault clearing time uncertainties satisfy the metric temporal logic specifications.

Next, we present the robust testing of temporal logic specifications in hybrid systems with nonlinear continuous dynamics. We first present the algorithm of finding the robust neighbourhood around the nominal (simulated) trajectory by computing the bounded disturbance local discrepancy function for a general nonlinear system and then extend the methods to hybrid systems. Based on that, we introduce our robust testing algorithms in power systems cascading failures mitigations as power systems can be modeled as a hybrid system with different network topologies and relay dynamics. To test the effectiveness of our robust testing algorithms, we apply the algorithms on a three-machine power system model modified from the 2003 Italian blackout and the IEEE 39-bus benchmark system in two different scenarios: robust testing of various generator mechanical power dispatch schedules and robust testing of post-fault remedial actions based on quick-start storages.

This thesis is about the trajectory-based temporal logic analysis. In particular, three main themes are discussed in this thesis: temporal logic inference from data, robust testing of temporal logic specifications and controller synthesis for temporal logic specifications. The temporal logic inference from data refers to finding temporal logic formulas that can classify two given sets of trajectories (classification) or identifying temporal logic formulas that best fit a given set of trajectories with respect to certain fitness measures (identification). These obtained (inferred) temporal logic formulas can be used for more purposes such as fault detection, model discrimination, etc. The robust testing of temporal logic specifications refers to analyzing whether all the trajectories that start from an initial set satisfy the given temporal logic specification within certain time horizon. Specifically, we focus on the robust testing for trajectories simulated by hybrid systems with nonlinear continuous dynamics. The controller synthesis for temporal logic specifications refers to designing a controller such that the trajectories of the controlled deterministic or stochastic system satisfy the given temporal logic specification.

We first introduce the signal temporal logic and its inference from data. We utilize some a priori information to make the inferred temporal logic formulas more specific and useful according to user preferences. We apply this algorithm in the classification and identification of robot arm movements of Phantom Omni haptic devices. We also use the inference algorithm in model discrimination and we apply the method in discriminating between two competing mathematical models of extracellular signal-regulated kinase (ERK) responses to epidermal growth factor stimulation.

To extend the temporal logic inference approach to multi-agent systems, we define a new type of signal temporal logic specifically for multi-agent systems: census signal temporal logic (CensusSTL). The CensusSTL consists of an inner logic formula that characterizes a consistent, frequent and specific task and an outer logic formula that characterizes the pattern of the number of agents in certain subgroups whose behaviors satisfy the inner logic formula. We present a new inference algorithm that can infer both the subgroups and the CensusSTL formulas directly from individual agent trajectories and apply the inference algorithm to analyzing the strategies of a soccer game with the data of body sensors equipped on each player.