7 research outputs found
Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets
We consider the problem of overbounding and underbounding both the backward
and forward reachable set for a given polynomial vector field, nonlinear in
both state and input, with a given semialgebriac set of initial conditions and
with inputs constrained pointwise to lie in a semialgebraic set. Specifically,
we represent the forward reachable set using the value function which gives the
optimal cost to go of an optimal control problems and if smooth satisfies the
Hamilton-Jacobi- Bellman PDE. We then show that there exist polynomial upper
and lower bounds to this value function and furthermore, these polynomial
sub-value and super-value functions provide provable upper and lower bounds to
the forward reachable set. Finally, by minimizing the distance between these
sub-value and super-value functions in the L1-norm, we are able to construct
inner and outer bounds for the reachable set and show numerically on several
examples that for relatively small degree, the Hausdorff distance between these
bounds is negligible
Learning to Satisfy Unknown Constraints in Iterative MPC
We propose a control design method for linear time-invariant systems that
iteratively learns to satisfy unknown polyhedral state constraints. At each
iteration of a repetitive task, the method constructs an estimate of the
unknown environment constraints using collected closed-loop trajectory data.
This estimated constraint set is improved iteratively upon collection of
additional data. An MPC controller is then designed to robustly satisfy the
estimated constraint set. This paper presents the details of the proposed
approach, and provides robust and probabilistic guarantees of constraint
satisfaction as a function of the number of executed task iterations. We
demonstrate the safety of the proposed framework and explore the safety vs.
performance trade-off in a detailed numerical example.Comment: Long version of the final paper for IEEE-CDC 2020. First two authors
contributed equall
Robust Control Synthesis and Verification for Wire-Borne Underactuated Brachiating Robots Using Sum-of-Squares Optimization
Control of wire-borne underactuated brachiating robots requires a robust
feedback control design that can deal with dynamic uncertainties, actuator
constraints and unmeasurable states. In this paper, we develop a robust
feedback control for brachiating on flexible cables, building on previous work
on optimal trajectory generation and time-varying LQR controller design. We
propose a novel simplified model for approximation of the flexible cable
dynamics, which enables inclusion of parametric model uncertainties in the
system. We then use semidefinite programming (SDP) and sum-of-squares (SOS)
optimization to synthesize a time-varying feedback control with formal
robustness guarantees to account for model uncertainties and unmeasurable
states in the system. Through simulation, hardware experiments and comparison
with a time-varying LQR controller, it is shown that the proposed robust
controller results in relatively large robust backward reachable sets and is
able to reliably track a pre-generated optimal trajectory and achieve the
desired brachiating motion in the presence of parametric model uncertainties,
actuator limits, and unobservable states.Comment: 8 pages, 12 figures, 2020 IEEE/RSJ International Conference on
Intelligent Robots and Systems (IROS
Sum-of-squares Flight Control Synthesis for Deep-stall Recovery
Under review for publication in the Journal of Guidance, Control, and Dynamics.In lieu of extensive Monte-Carlo simulations for flight control verification, sum-of-squares programming techniques provide an algebraic approach to the problem of nonlinear control synthesis and analysis. However, their reliance on polynomial models has hitherto limited the applicability to aeronautical control problems. Taking advantage of recently proposed piecewise polynomial models, this paper revisits sum-of-squares techniques for recovery of an aircraft from deep-stall conditions using a realistic yet tractable aerodynamic model. Local stability analysis of classical controllers is presented as well as synthesis of polynomial feedback laws with the objective of enlarging their nonlinear region of attraction. A newly developed synthesis algorithm for backwards-reachability facilitates the design of recovery control laws, ensuring stable recovery by design. The paper's results motivate future research in aeronautical sum-of-squares applications