Semidefinite Relaxations for Stochastic Optimal Control Policies

Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation have led to the discovery of a formulation of the value function as a linear Partial Differential Equation (PDE) for stochastic nonlinear systems with a mild constraint on their disturbances. This has yielded promising directions for research in the planning and control of nonlinear systems. This work proposes a new method obtaining approximate solutions to these linear stochastic optimal control (SOC) problems. A candidate polynomial with variable coefficients is proposed as the solution to the SOC problem. A Sum of Squares (SOS) relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in Portland, Oregon. 7 pages, colo

Complexity Reduction for Parameter-Dependent Linear Systems

We present a complexity reduction algorithm for a family of parameter-dependent linear systems when the system parameters belong to a compact semi-algebraic set. This algorithm potentially describes the underlying dynamical system with fewer parameters or state variables. To do so, it minimizes the distance (i.e., H-infinity-norm of the difference) between the original system and its reduced version. We present a sub-optimal solution to this problem using sum-of-squares optimization methods. We present the results for both continuous-time and discrete-time systems. Lastly, we illustrate the applicability of our proposed algorithm on numerical examples

An Extended Kalman Filter with a Computed Mean Square Error Bound

The paper proposes a new recursive filter for non-linear systems that inherently computes a valid bound on the mean square estimation error. The proposed filter, bound based extended Kalman, (BEKF) is in the form of an extended Kalman filter. The main difference of the proposed filter from the conventional extended Kalman filter is in the use of a computed mean square error bound matrix, to calculate the filter gain, and to serve as bound on the actual mean square error. The paper shows that when the system is linear the proposed filtering algorithm reduces to the conventional Kalman filter. The theory presented in the paper is applicable to a wide class of systems, but if the system is polynomial, then the recently developed theory of positive polynomials considerably simplifies the filter's implementation.Comment: 7 pages, 1 figur

Global Singularity-Free Aerodynamic Model for Algorithmic Flight Control of Tail Sitters

This paper addresses fundamental issues in tail-sitting and transition flight aerodynamics modeling in view of sumof- squares (SOS) algorithmic guidance and control design. A novel approach, called ϕ theory, for modeling aerodynamic forces and moments is introduced herein. It yields polynomial-like differential equations of motion that are well suited to SOS solvers for real-time algorithmic guidance and control law synthesis. The proposed ϕ theory allows for first principles model parameter identification and captures dominant dynamical features over the entire flight envelope. Furthermore, ϕ theory yields numerically stable and consistent models for 360 deg angles of attack and sideslip. Additionally, an algorithm is provided for analytically computing all feasible longitudinal flight operating points. Finally, to establish ϕ-theory validity, predicted trim points and wind-tunnel experiments are compared

Optimization-based Framework for Stability and Robustness of Bipedal Walking Robots

As robots become more sophisticated and move out of the laboratory, they need to be able to reliably traverse difficult and rugged environments. Legged robots -- as inspired by nature -- are most suitable for navigating through terrain too rough or irregular for wheels. However, control design and stability analysis is inherently difficult since their dynamics are highly nonlinear, hybrid (mixing continuous dynamics with discrete impact events), and the target motion is a limit cycle (or more complex trajectory), rather than an equilibrium. For such walkers, stability and robustness analysis of even stable walking on flat ground is difficult. This thesis proposes new theoretical methods to analyse the stability and robustness of periodic walking motions. The methods are implemented as a series of pointwise linear matrix inequalities (LMI), enabling the use of convex optimization tools such as sum-of-squares programming in verifying the stability and robustness of the walker. To ensure computational tractability of the resulting optimization program, construction of a novel reduced coordinate system is proposed and implemented. To validate theoretic and algorithmic developments in this thesis, a custom-built “Compass gait” walking robot is used to demonstrate the efficacy of the proposed methods. The hardware setup, system identification and walking controller are discussed. Using the proposed analysis tools, the stability property of the hardware walker was successfully verified, which corroborated with the computational results