Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finite-dimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems

Velocity control of a unicycle type of mobile robot using optimal pid controller

A unicycle model of control a mobile robot is a simplified modeling approach modified from the differential drive mobile robots. Instead of controlling the right speed, VR and the left speed, VL of the drive systems, the unicycle model is using u and ω as the controller parameters. Tracking is much easier in this model. In this paper, the dynamic of the robot parameter is controlled using two blocks of Proportional-Integral-Derivative(PID) controllers. The gains of the PID are firstly determined using particle swarm optimization(PSO) in offline mode. After the optimal gain is determined, the tracking of the robot’s trajectory is performed online with optimal PID controller. The achieved results of the proposed scheme are compared with those of dynamic model optimized with genetic algorithm(GA) and manually tuned PID controller gains. In the algorithm, the control parameters are computed by minimizing the fitness function defined by using the integral absolute error(IAE) performance index. The simulation results obtained reveal advantages of the proposed PSO-PID dynamic controller for trajectory tracking of a unicycle type of mobile robot. A MATLAB-Simulink program is used to simulate the designed system and the results are graphically plotted. In addition, numerical simulations using 8-shape as a reference trajectory with several numbers of iterations are reported to show the validity of the proposed scheme

Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures

In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the time-limited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) and provide finite dimensional approximations of this LP in terms of semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS. In contrast to traditional Lyapunov based approaches, which are non-convex and require feasible initialization, our approach is convex and does not require any form of initialization. The resulting time-varying controllers and approximated backwards reachable sets are well-suited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on four nonlinear systems.United States. Office of Naval Research. Multidisciplinary University Research Initiative (Grant N00014-09-1-1051)National Science Foundation (U.S.) (Contract IIS-1161679)Thomas and Stacey Siebel Foundatio

Formation control of mobile robots and unmanned aerial vehicles

In this dissertation, the nonlinear control of nonholonomic mobile robot formations and unmanned aerial vehicle (UAV) formations is undertaken and presented in six papers. In the first paper, an asymptotically stable combined kinematic/torque control law is developed for leader-follower based formation control of mobile robots using backstepping. A neural network (NN) is introduced along with robust integral of the sign of the error (RISE) feedback to approximate the dynamics of the follower as well as its leader using online weight tuning. Subsequently, in the second paper, a novel NN observer is designed to estimate the linear and angular velocities of both the follower and its leader robot and a NN output feedback control law is developed. On the other hand, in the third paper, a NN-based output feedback control law is presented for the control of an underactuated quad rotor UAV, and a NN virtual control input scheme is proposed which allows all six degrees of freedom to be controlled using only four control inputs. The results of this paper are extended to include the control of quadrotor UAV formations, and a novel three-dimensional leader-follower framework is proposed in the fourth paper. Next, in the fifth paper, the discrete-time nonlinear optimal control is undertaken using two online approximators (OLA\u27s) to solve the infinite horizon Hamilton-Jacobi-Bellman (HJB) equation forward-in-time to achieve nearly optimal regulation and tracking control. In contrast, paper six utilizes a single OLA to solve the infinite horizon HJB and Hamilton-Jacobi-Isaacs (HJI) equations forward-intime for the near optimal regulation and tracking control of continuous affine nonlinear systems. The effectiveness of the optimal tracking controllers proposed in the fifth and sixth papers are then demonstrated using nonholonomic mobile robot formation control --Abstract, page iv

Analysis of multi-agent systems under varying degrees of trust, cooperation, and competition

Multi-agent systems rely heavily on coordination and cooperation to achieve a variety of tasks. It is often assumed that these agents will be fully cooperative, or have reliable and equal performance among group members. Instead, we consider cooperation as a spectrum of possible interactions, ranging from performance variations within the group to adversarial agents. This thesis examines several scenarios where cooperation and performance are not guaranteed. Potential applications include sensor coverage, emergency response, wildlife management, tracking, and surveillance. We use geometric methods, such as Voronoi tessellations, for design insight and Lyapunov-based stability theory to analyze our proposed controllers. Performance is verified through simulations and experiments on a variety of ground and aerial robotic platforms. First, we consider the problem of Voronoi-based coverage control, where a group of robots must spread out over an environment to provide coverage. Our approach adapts online to sensing and actuation performance variations with the group. The robots have no prior knowledge of their relative performance, and in a distributed fashion, compensate by assigning weaker robots a smaller portion of the environment. Next, we consider the problem of multi-agent herding, akin to shepherding. Here, a group of dog-like robots must drive a herd of non-cooperative sheep-like agents around the environment. Our key insight in designing the control laws for the herders is to enforce geometrical relationships that allow for the combined system dynamics to reduce to a single nonholonomic vehicle. We also investigate the cooperative pursuit of an evader by a group of quadrotors in an environment with no-fly zones. While the pursuers cannot enter the no-fly zones, the evader moves freely through the zones to avoid capture. Using tools for Voronoi-based coverage control, we provide an algorithm to distribute the pursuers around the zone's boundary and minimize capture time once the evader emerges. Finally, we present an algorithm for the guaranteed capture of multiple evaders by one or more pursuers in a bounded, convex environment. The pursuers utilize properties of the evader's Voronoi cell to choose a control strategy that minimizes the safe-reachable area of the evader, which in turn leads to the evader's capture