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

Output-Feedback Control of Nonlinear Systems using Control Contraction Metrics and Convex Optimization

Control contraction metrics (CCMs) are a new approach to nonlinear control design based on contraction theory. The resulting design problems are expressed as pointwise linear matrix inequalities and are and well-suited to solution via convex optimization. In this paper, we extend the theory on CCMs by showing that a pair of "dual" observer and controller problems can be solved using pointwise linear matrix inequalities, and that when a solution exists a separation principle holds. That is, a stabilizing output-feedback controller can be found. The procedure is demonstrated using a benchmark problem of nonlinear control: the Moore-Greitzer jet engine compressor model.Comment: Conference submissio

A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure

Fast, Optimal, and Safe Motion Planning for Bipedal Robots

Bipedal robots have the potential to traverse a wide range of unstructured environments, which are otherwise inaccessible to wheeled vehicles. Though roboticists have successfully constructed controllers for bipedal robots to walk over uneven terrain such as snow, sand, or even stairs, it has remained challenging to synthesize such controllers in an online fashion while guaranteeing their satisfactory performance. This is primarily due to the lack of numerical method that can accommodate the non-smooth dynamics, high degrees of freedom, and underactuation that characterize bipedal robots. This dissertation proposes and implements a family of numerical methods that begin to address these three challenges along three dimensions: optimality, safety, and computational speed. First, this dissertation develops a convex relaxation-based approach to solve optimal control for hybrid systems without a priori knowledge of the optimal sequence of transition. This is accomplished by formulating the problem in the space of relaxed controls, which gives rise to a linear program whose solution is proven to compute the globally optimal controller. This conceptual program is solved using a sequence of semidefinite programs whose solutions are proven to converge from below to the true optimal solution of the original optimal control problem. Moreover, a method to synthesize the optimal controller is developed. Using an array of examples, the performance of this method is validated on problems with known solutions and also compared to a commercial solver. Second, this dissertation constructs a method to generate safety-preserving controllers for a planar bipedal robot walking on flat ground by performing reachability analysis on simplified models under the assumption that the difference between the two models can be bounded. Subsequently, this dissertation describes how this reachable set can be incorporated into a Model Predictive Control framework to select controllers that result in safe walking on the biped in an online fashion. This method is validated on a 5-link planar model. Third, this dissertation proposes a novel parallel algorithm capable of finding guaranteed optimal solutions to polynomial optimization problems up to pre-specified tolerances. Formal proofs of bounds on the time and memory usage of such method are also given. Such algorithm is implemented in parallel on GPUs and compared against state-of-the-art solvers on a group of benchmark examples. An application of such method on a real-time trajectory-planning task of a mobile robot is also demonstrated. Fourth, this dissertation constructs an online Model Predictive Control framework that guarantees safety of a 3D bipedal robot walking in a forest of randomly-placed obstacles. Using numerical integration and interval arithmetic techniques, approximations to trajectories of the robot are constructed along with guaranteed bounds on the approximation error. Safety constraints are derived using these error bounds and incorporated in a Model Predictive Control framework whose feasible solutions keep the robot from falling over and from running into obstacles. To ensure that the bipedal robot is able to avoid falling for all time, a finite-time terminal constraint is added to the Model Predictive Control algorithm. The performance of this method is implemented and compared against a naive Model Predictive Control method on a biped model with 20 degrees of freedom. In summary, this dissertation presents four methods for control synthesis of bipedal robots with improvements in either optimality, safety guarantee, or computational speed. Furthermore, the performance of all proposed methods are compared with existing methods in the field.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162880/1/pczhao_1.pd