498 research outputs found
Backward Reachability Analysis of Perturbed Continuous-Time Linear Systems Using Set Propagation
Backward reachability analysis computes the set of states that reach a target
set under the competing influence of control input and disturbances. Depending
on their interplay, the backward reachable set either represents all states
that can be steered into the target set or all states that cannot avoid
entering it -- the corresponding solutions can be used for controller synthesis
and safety verification, respectively. A popular technique for backward
reachable set computation solves Hamilton-Jacobi-Isaacs equations, which scales
exponentially with the state dimension due to gridding the state space. In this
work, we instead use set propagation techniques to design backward reachability
algorithms for linear time-invariant systems. Crucially, the proposed
algorithms scale only polynomially with the state dimension. Our numerical
examples demonstrate the tightness of the obtained backward reachable sets and
show an overwhelming improvement of our proposed algorithms over
state-of-the-art methods regarding scalability, as systems with well over a
hundred states can now be analyzed.Comment: 16 page
Algorithmic Verification of Continuous and Hybrid Systems
We provide a tutorial introduction to reachability computation, a class of
computational techniques that exports verification technology toward continuous
and hybrid systems. For open under-determined systems, this technique can
sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661
Sampling-Based Robust Control of Autonomous Systems with Non-Gaussian Noise
Controllers for autonomous systems that operate in safetycritical settings must account for stochastic disturbances. Such disturbances are often modeled as process noise, and common assumptions are that the underlying distributions are known and/or Gaussian. In practice, however, these assumptions may be unrealistic and can lead to poor approximations of the true noise distribution. We present a novel planning method that does not rely on any explicit representation of the noise distributions. In particular, we address the problem of computing a controller that provides probabilistic guarantees on safely reaching a target. First, we abstract the continuous system into a discrete-state model that captures noise by probabilistic transitions between states. As a key contribution, we adapt tools from the scenario approach to compute probably approximately correct (PAC) bounds on these transition probabilities, based on a finite number of samples of the noise. We capture these bounds in the transition probability intervals of a socalled interval Markov decision process (iMDP). This iMDP is robust against uncertainty in the transition probabilities, and the tightness of the probability intervals can be controlled through the number of samples. We use state-of-the-art verification techniques to provide guarantees on the iMDP, and compute a controller for which these guarantees carry over to the autonomous system. Realistic benchmarks show the practical applicability of our method, even when the iMDP has millions of states or transitions.</p
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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