146 research outputs found
A Sum-of-Squares Approach to the Analysis of Zeno Stability in Polynomial Hybrid Systems
Hybrid dynamical systems can exhibit many unique phenomena, such as Zeno
behavior. Zeno behavior is the occurrence of infinite discrete transitions in
finite time. Zeno behavior has been likened to a form of finite-time asymptotic
stability, and corresponding Lyapunov theorems have been developed. In this
paper, we propose a method to construct Lyapunov functions to prove Zeno
stability of compact sets in cyclic hybrid systems with parametric
uncertainties in the vector fields, domains and guard sets, and reset maps
utilizing sum-of-squares programming. This technique can easily be applied to
cyclic hybrid systems without parametric uncertainties as well. Examples
illustrating the use of the proposed technique are also provided
Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares
In this paper, we explore the merits of various algorithms for polynomial
optimization problems, focusing on alternatives to sum of squares programming.
While we refer to advantages and disadvantages of Quantifier Elimination,
Reformulation Linear Techniques, Blossoming and Groebner basis methods, our
main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and
Handelman's theorem. We first formulate polynomial optimization problems as
verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's
algorithm, Bernstein's algorithm and Handelman's algorithm reduce the
intractable problem of feasibility of semi-algebraic sets to linear and/or
semi-definite programming. We apply these algorithms to different problems in
robust stability analysis and stability of nonlinear dynamical systems. As one
contribution of this paper, we apply Polya's algorithm to the problem of
H_infinity control of systems with parametric uncertainty. Numerical examples
are provided to compare the accuracy of these algorithms with other polynomial
optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series
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
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