3,416 research outputs found
Scalable Safety-Preserving Robust Control Synthesis for Continuous-Time Linear Systems
We present a scalable set-valued safety-preserving controller for constrained
continuous-time linear time-invariant (LTI) systems subject to additive,
unknown but bounded disturbance or uncertainty. The approach relies upon a
conservative approximation of the discriminating kernel using robust maximal
reachable sets---an extension of our earlier work on computation of the
viability kernel for high-dimensional systems. Based on ellipsoidal techniques
for reachability, a piecewise ellipsoidal algorithm with polynomial complexity
is described that under-approximates the discriminating kernel under LTI
dynamics. This precomputed piecewise ellipsoidal set is then used online to
synthesize a permissive state-feedback safety-preserving controller. The
controller is modeled as a hybrid automaton and can be formulated such that
under certain conditions the resulting control signal is continuous across its
transitions. We show the performance of the controller on a twelve-dimensional
flight envelope protection problem for a quadrotor with actuation saturation
and unknown wind disturbances
Underapproximation of Reach-Avoid Sets for Discrete-Time Stochastic Systems via Lagrangian Methods
We examine Lagrangian techniques for computing underapproximations of
finite-time horizon, stochastic reach-avoid level-sets for discrete-time,
nonlinear systems. We use the concept of reachability of a target tube in the
control literature to define robust reach-avoid sets which are parameterized by
the target set, safe set, and the set in which the disturbance is drawn from.
We unify two existing Lagrangian approaches to compute these sets and establish
that there exists an optimal control policy of the robust reach-avoid sets
which is a Markov policy. Based on these results, we characterize the subset of
the disturbance space whose corresponding robust reach-avoid set for the given
target and safe set is a guaranteed underapproximation of the stochastic
reach-avoid level-set of interest. The proposed approach dramatically improves
the computational efficiency for obtaining an underapproximation of stochastic
reach-avoid level-sets when compared to the traditional approaches based on
gridding. Our method, while conservative, does not rely on a grid, implying
scalability as permitted by the known computational geometry constraints. We
demonstrate the method on two examples: a simple two-dimensional integrator,
and a space vehicle rendezvous-docking problem.Comment: Submitted to CDC 201
Inner-Approximating Reachable Sets for Polynomial Systems with Time-Varying Uncertainties
In this paper we propose a convex programming based method to address a
long-standing problem of inner-approximating backward reachable sets of
state-constrained polynomial systems subject to time-varying uncertainties. The
backward reachable set is a set of states, from which all trajectories starting
will surely enter a target region at the end of a given time horizon without
violating a set of state constraints in spite of the actions of uncertainties.
It is equal to the zero sub-level set of the unique Lipschitz viscosity
solution to a Hamilton-Jacobi partial differential equation (HJE). We show that
inner-approximations of the backward reachable set can be formed by zero
sub-level sets of its viscosity super-solutions. Consequently, we reduce the
inner-approximation problem to a problem of synthesizing polynomial viscosity
super-solutions to this HJE. Such a polynomial solution in our method is
synthesized by solving a single semi-definite program. We also prove that
polynomial solutions to the formulated semi-definite program exist and can
produce a convergent sequence of inner-approximations to the interior of the
backward reachable set in measure under appropriate assumptions. This is the
main contribution of this work. Several illustrative examples demonstrate the
merits of our approach.Comment: Accepted by IEEE TAC[Volume 65 (2020), Issue 4 (April)
Verification for Machine Learning, Autonomy, and Neural Networks Survey
This survey presents an overview of verification techniques for autonomous
systems, with a focus on safety-critical autonomous cyber-physical systems
(CPS) and subcomponents thereof. Autonomy in CPS is enabling by recent advances
in artificial intelligence (AI) and machine learning (ML) through approaches
such as deep neural networks (DNNs), embedded in so-called learning enabled
components (LECs) that accomplish tasks from classification to control.
Recently, the formal methods and formal verification community has developed
methods to characterize behaviors in these LECs with eventual goals of formally
verifying specifications for LECs, and this article presents a survey of many
of these recent approaches
Computing Probabilistic Controlled Invariant Sets
This paper investigates stochastic invariance for control systems through
probabilistic controlled invariant sets (PCISs). As a natural complement to
robust controlled invariant sets~(RCISs), we propose finite- and
infinite-horizon PCISs, and explore their relation to RICSs. We design
iterative algorithms to compute the PCIS within a given set. For systems with
discrete spaces, the computations of the finite- and infinite-horizon PCISs at
each iteration are based on linear programming and mixed integer linear
programming, respectively. The algorithms are computationally tractable and
terminate in a finite number of steps. For systems with continuous spaces, we
show how to discretize the spaces and prove the convergence of the
approximation when computing the finite-horizon PCISs. In addition, it is shown
that an infinite-horizon PCIS can be computed by the stochastic backward
reachable set from the RCIS contained in it. These PCIS algorithms are
applicable to practical control systems. Simulations are given to illustrate
the effectiveness of the theoretical results for motion planning.Comment: Journal article in the IEEE Transactions on Automatic Control
(Volume: 66, Issue: 7, July 2021
Dynamic Polytopic Template Approach to Robust Transient Stability Assessment
Transient stability assessment of power systems needs to account for
increased risk from uncertainties due to the integration of renewables and
distributed generators. The uncertain operating condition of the power grid
hinders reliable assessment of transient stability. Conventional approaches
such as time-domain simulations and direct energy methods are computationally
expensive to take account of uncertainties. This paper proposes a reachability
analysis approach that computes bounds of the possible trajectories from
uncertain initial conditions. The eigenvalue decomposition is used to construct
a polytopic template with a scalable number of hyperplanes that is guaranteed
to converge near the equilibrium. The proposed algorithm bounds the possible
states at a given time with a polytopic template and solves the evolution of
the polytope over time. The problem is solved with linear programming
relaxation based on outer-approximations of nonlinear functions, which is
scalable for large scale systems. We demonstrate our method on IEEE test cases
to certify the stability and bound the state trajectories
Stochastic reachability of a target tube: Theory and computation
Probabilistic guarantees of safety and performance are important in
constrained dynamical systems with stochastic uncertainty. We consider the
stochastic reachability problem, which maximizes the probability that the state
remains within time-varying state constraints (i.e., a ``target tube''),
despite bounded control authority. This problem subsumes the stochastic
viability and terminal hitting-time stochastic reach-avoid problems. Of special
interest is the stochastic reach set, the set of all initial states from which
it is possible to stay in the target tube with a probability above a desired
threshold. We provide sufficient conditions under which the stochastic reach
set is closed, compact, and convex, and provide an underapproximative
interpolation technique for stochastic reach sets. Utilizing convex
optimization, we propose a scalable and grid-free algorithm that computes a
polytopic underapproximation of the stochastic reach set and synthesizes an
open-loop controller. This algorithm is anytime, i.e., it produces a valid
output even on early termination. We demonstrate the efficacy and scalability
of our approach on several numerical examples, and show that our algorithm
outperforms existing software tools for verification of linear systems
Lagrangian Approximations for Stochastic Reachability of a Target Tube
In this paper we examine how Lagrangian techniques can be used to compute
underapproximations and overapproximation of the finite-time horizon,
stochastic reach-avoid level sets for discrete-time, nonlinear systems. This
approach is applicable for a generic nonlinear system without any convexity
assumptions on the safe and target sets. We examine and apply our methods on
the reachability of a target tube problem, a more generalized version of the
finite-time horizon reach-avoid problem. Because these methods utilize a
Lagrangian (set theoretic) approach, we eliminate the necessity to grid the
state, input, and disturbance spaces allowing for increased scalability and
faster computation. The methods scalability are currently limited by the
computational requirements for performing the necessary set operations by
current computational geometry tools. The primary trade-off for this improved
extensibility is conservative approximations of actual stochastic reach set. We
demonstrate these methods on several examples including the standard
double-integrator, a chain of integrators, and a 4-dimensional space vehicle
rendezvous docking problem
Safe Control under Uncertainty
Controller synthesis for hybrid systems that satisfy temporal specifications
expressing various system properties is a challenging problem that has drawn
the attention of many researchers. However, making the assumption that such
temporal properties are deterministic is far from the reality. For example,
many of the properties the controller has to satisfy are learned through
machine learning techniques based on sensor input data. In this paper, we
propose a new logic, Probabilistic Signal Temporal Logic (PrSTL), as an
expressive language to define the stochastic properties, and enforce
probabilistic guarantees on them. We further show how to synthesize safe
controllers using this logic for cyber-physical systems under the assumption
that the stochastic properties are based on a set of Gaussian random variables.
One of the key distinguishing features of PrSTL is that the encoded logic is
adaptive and changes as the system encounters additional data and updates its
beliefs about the latent random variables that define the safety properties. We
demonstrate our approach by synthesizing safe controllers under the PrSTL
specifications for multiple case studies including control of quadrotors and
autonomous vehicles in dynamic environments.Comment: 10 pages, 6 figures, Submitted to HSCC 201
Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis
We introduce sparse polynomial zonotopes, a new set representation for formal
verification of hybrid systems. Sparse polynomial zonotopes can represent
non-convex sets and are generalizations of zonotopes and Taylor models.
Operations like Minkowski sum, quadratic mapping, and reduction of the
representation size can be computed with polynomial complexity w.r.t. the
dimension of the system. In particular, for the reachability analysis of
nonlinear systems, the wrapping effect is substantially reduced using sparse
polynomial zonotopes as demonstrated by numerical examples. In addition, we can
significantly reduce the computation time compared to zonotopes
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