18 research outputs found
Convex Programs for Temporal Verification of Nonlinear Dynamical Systems
A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. The dual of this problem, i.e., the reachability problem, concerns proving the existence of a trajectory starting from the initial set that reaches another given set. Using insights from the linear programming duality appearing in the discrete shortest path problem, we show in this paper that reachability of continuous systems can also be verified through convex programming. Several convex programs for verifying safety and reachability, as well as other temporal properties such as eventuality, avoidance, and their combinations, are formulated. Some examples are provided to illustrate the application of the proposed methods. Finally, we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates
Input-to-State Safety With Control Barrier Functions
This letter presents a new notion of input-to-state safe control barrier
functions (ISSf-CBFs), which ensure safety of nonlinear dynamical systems under
input disturbances. Similar to how safety conditions are specified in terms of
forward invariance of a set, input-to-state safety (ISSf) conditions are
specified in terms of forward invariance of a slightly larger set. In this
context, invariance of the larger set implies that the states stay either
inside or very close to the smaller safe set; and this closeness is bounded by
the magnitude of the disturbances. The main contribution of the letter is the
methodology used for obtaining a valid ISSf-CBF, given a control barrier
function (CBF). The associated universal control law will also be provided.
Towards the end, we will study unified quadratic programs (QPs) that combine
control Lyapunov functions (CLFs) and ISSf-CBFs in order to obtain a single
control law that ensures both safety and stability in systems with input
disturbances.Comment: 7 pages, 7 figures; Final submitted versio
Optimal Stabilization using Lyapunov Measures
Numerical solutions for the optimal feedback stabilization of discrete time
dynamical systems is the focus of this paper. Set-theoretic notion of almost
everywhere stability introduced by the Lyapunov measure, weaker than
conventional Lyapunov function-based stabilization methods, is used for optimal
stabilization. The linear Perron-Frobenius transfer operator is used to pose
the optimal stabilization problem as an infinite dimensional linear program.
Set-oriented numerical methods are used to obtain the finite dimensional
approximation of the linear program. We provide conditions for the existence of
stabilizing feedback controls and show the optimal stabilizing feedback control
can be obtained as a solution of a finite dimensional linear program. The
approach is demonstrated on stabilization of period two orbit in a controlled
standard map
Converse Barrier Certificates for Finite-time Safety Verification of Continuous-time Perturbed Deterministic Systems
In this paper, we investigate the problem of verifying the finite-time safety
of continuous-time perturbed deterministic systems represented by ordinary
differential equations in the presence of measurable disturbances. Given a
finite time horizon, if the system is safe, it, starting from a compact initial
set, will remain within an open and bounded safe region throughout the
specified time horizon, regardless of the disturbances. The main contribution
of this work is to uncover that there exists a time-dependent barrier
certificate if and only if the system is safe. This barrier certificate
satisfies the following conditions: negativity over the initial set at the
initial time instant, non-negativity over the boundary of the safe set, and
non-increasing behavior along the system dynamics over the specified finite
time horizon. The existence problem is explored using a Hamilton-Jacobi
differential equation, which has a unique Lipschitz viscosity solution
Optimal Safe Controller Synthesis: A Density Function Approach
This paper considers the synthesis of optimal safe controllers based on density functions. We present an algorithm for robust constrained optimal control synthesis using the duality relationship between the density function and the value function. The density function follows the Liouville equation and is the dual of the value function, which satisfies Bellman’s optimality principle. Thanks to density functions, constraints over the distribution of states, such as safety constraints, can be posed straightforwardly in an optimal control problem. The constrained optimal control problem is then solved with a primal-dual algorithm. This formulation is extended to the case with external disturbances, and we show that the robust constrained optimal control can be solved with a modified primal-dual algorithm. We apply this formulation to the problem of finding the optimal safe controller that minimizes the cumulative intervention. An adaptive cruise control (ACC) example is used to demonstrate the efficacy of the proposed, wherein we compare the result of the density function approach with the conventional control barrier function (CBF) method
Temporal viability regulation for control affine systems with applications to mobile vehicle coordination under time-varying motion constraints
Controlled invariant set and viability regulation of dynamical control
systems have played important roles in many control and coordination
applications. In this paper we develop a temporal viability regulation theory
for general dynamical control systems, and in particular for control affine
systems. The time-varying viable set is parameterized by time-varying
constraint functions, with the aim to regulate a dynamical control system to be
invariant in the time-varying viable set so that temporal state-dependent
constraints are enforced. We consider both time-varying equality and inequality
constraints in defining a temporal viable set. We also present sufficient
conditions for the existence of feasible control input for the control affine
systems. The developed temporal viability regulation theory is applied to
mobile vehicle coordination.Comment: 7 pages, 3 figures. Submitted to a conference for publicatio
Input-to-State Safety with Control Barrier Functions
This letter presents a new notion of input-to-state safe control barrier functions (ISSf-CBFs), which ensure safety of nonlinear dynamical systems under input disturbances. Similar to how safety conditions are specified in terms of forward invariance of a set, input-to-state safety (ISSf) conditions are specified in terms of forward invariance of a slightly larger set. In this context, invariance of the larger set implies that the states stay either inside or very close to the smaller safe set; and this closeness is bounded by the magnitude of the disturbances. The main contribution of the letter is the methodology used for obtaining a valid ISSf-CBF, given a control barrier function (CBF). The associated universal control law will also be provided. Towards the end, we will study unified quadratic programs (QPs) that combine control Lyapunov functions (CLFs) and ISSf-CBFs in order to obtain a single control law that ensures both safety and stability in systems with input disturbances
A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates
This paper presents a methodology for safety verification of continuous and hybrid systems in the worst-case and stochastic settings. In the worst-case setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method