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Convex programs for temporal verification of nonlinear dynamical systems
 SIAM J. Control Optim
"... Abstract. 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 se ..."
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Abstract. 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.
Analysis of decentralized potential field based multiagent navigation via primaldual Lyapunov theory
 in Proc. IEEE Conf. Decision and Control
"... Abstract — We use a combination of primal and dual Lyapunov theory for almost global asymptotic stabilization and collision avoidance in multiagent systems. Previous work provided local analysis around the critical points with the use of the dual Lyapunov technique. This paper provides analysis o ..."
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Abstract — We use a combination of primal and dual Lyapunov theory for almost global asymptotic stabilization and collision avoidance in multiagent systems. Previous work provided local analysis around the critical points with the use of the dual Lyapunov technique. This paper provides analysis of the whole workspace with the use of the recently introduced combined use of primal and dual Lyapunov functions. I.
Analysis of Robot Navigation Schemes using Rantzer’s Dual Lyapunov Theorem
"... Abstract — When robots are driven by the negative gradient of a potential field that consists of the sum of an attractive and a repulsive term, convergence to the desired configuration cannot be guaranteed by traditional Lyapunov techniques. In this paper, sufficient conditions for convergence of su ..."
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Abstract — When robots are driven by the negative gradient of a potential field that consists of the sum of an attractive and a repulsive term, convergence to the desired configuration cannot be guaranteed by traditional Lyapunov techniques. In this paper, sufficient conditions for convergence of such systems are provided instead with the use of Rantzer’s Dual Lyapunov Theorem. In particular, a condition that involves the trace of the Hessian matrix of the potential function is derived and then applied to the cases of navigation of a single robot and of multirobot formation stabilization. The main result of the paper states that a sufficient condition for convergence to a desired configuration in both cases is that the attractive potential admits a sufficiently large gain. A lower bound on the attractive potential is computed. Computer simulations that support the new results are provided. I.
1Sufficient Conditions for Decentralized Potential Functions Based Controllers using Canonical Vector Fields
"... A combination of dual Lyapunov analysis and properties of decentralized navigation function based controllers is used to check the stability properties of a certain class of decentralized controllers for navigation and collision avoidance in multiagent systems. The derived results yield a less cons ..."
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A combination of dual Lyapunov analysis and properties of decentralized navigation function based controllers is used to check the stability properties of a certain class of decentralized controllers for navigation and collision avoidance in multiagent systems. The derived results yield a less conservative condition from previous approaches, which relates to the negativity of the sum of the minimum eigenvalues of the Hessian matrices at the critical points, instead of requiring each of the eigenvalues to be negative itself. This provides an improved characterization of the reachable set of this class of decentralized navigation function based controllers, which is less conservative than the previous results for the same class of controllers. I.
Sufficient Conditions for Decentralized Navigation Functions Based Controllers using Canonical Vector Fields
"... Abstract — A combination of dual Lyapunov analysis and properties of decentralized navigation function based controllers is used to check the stability properties of a certain class of decentralized controllers for navigation and collision avoidance in multiagent systems. The derived results yield ..."
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Abstract — A combination of dual Lyapunov analysis and properties of decentralized navigation function based controllers is used to check the stability properties of a certain class of decentralized controllers for navigation and collision avoidance in multiagent systems. The derived results yield a less conservative condition from previous approaches, which relates to the negativity of the sum of the minimum eigenvalues of the Hessian matrices at the critical points, instead of requiring each of the eigenvalues to be negative itself. This provides an improved characterization of the reachable set of this class of decentralized navigation function based controllers, which is less conservative than the previous results for the same class of controllers. I.
Scalable SafetyPreserving Robust Control Synthesis for ContinuousTime Linear Systems∗
"... We present a scalable setvalued safetypreserving controller for constrained continuoustime linear timeinvariant (LTI) systems subject to additive, unknown but bounded disturbance or uncertainty. The approach relies upon a conservative approximation of the discriminating kernel using robust maxi ..."
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We present a scalable setvalued safetypreserving controller for constrained continuoustime linear timeinvariant (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 highdimensional systems. Based on ellipsoidal techniques for reachability, a piecewise ellipsoidal algorithm with polynomial complexity is described that underapproximates the discriminating kernel under LTI dynamics. This precomputed piecewise ellipsoidal set is then used online to synthesize a permissive statefeedback safetypreserving 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 twelvedimensional flight envelope protection problem for a quadrotor with actuation saturation and unknown wind disturbances. 1
Analysis of Decentralized Potential Field Based MultiAgent Navigation via PrimalDual Lyapunov Theory
, 2015
"... Analysis of decentralized potential field based multiagent navigation via primaldual Lyapunov theory Citation Dimarogonas, Dimos V., and Emilio Frazzoli. “Analysis of decentralized potential field based multiagent navigation via ..."
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Analysis of decentralized potential field based multiagent navigation via primaldual Lyapunov theory Citation Dimarogonas, Dimos V., and Emilio Frazzoli. “Analysis of decentralized potential field based multiagent navigation via
PROBABILISTIC METHODS FOR MODEL VALIDATION
, 2014
"... This dissertation develops a probabilistic method for validation and verification (V&V) of uncertain nonlinear systems. Existing systemscontrol literature on model and controller V&V either deal with linear systems with normbounded uncertainties, or consider nonlinear systems in setbased ..."
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This dissertation develops a probabilistic method for validation and verification (V&V) of uncertain nonlinear systems. Existing systemscontrol literature on model and controller V&V either deal with linear systems with normbounded uncertainties, or consider nonlinear systems in setbased and moment based framework. These existing methods deal with model invalidation or falsification, rather than assessing the quality of a model with respect to measured data. In this dissertation, an axiomatic framework for model validation is proposed in probabilistically relaxed sense, that instead of simply invalidating a model, seeks to quantify the “degree of validation”. To develop this framework, novel algorithms for uncertainty propagation have been proposed for both deterministic and stochastic nonlinear systems in continuous time. For the deterministic flow, we compute the timevarying joint probability density functions over the state space, by solving the Liouville equation via methodofcharacteristics. For the stochastic flow, we propose an approximation algorithm that combines the methodofcharacteristics solution of Liouville equation with the