Results 1  10
of
12
LQRTrees: Feedback motion planning via sums of squares verification
 International Journal of Robotics Research
, 2010
"... Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree ..."
Abstract

Cited by 68 (21 self)
 Add to MetaCart
(Show Context)
Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQRstabilized trajectories. The region of attraction of this nonlinear feedback policy “probabilistically covers ” the entire controllable subset of the state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic nonlinear feedback design algorithm on simple nonlinear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm. 1
A framework for worstcase and stochastic safety verification using barrier certificates
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2007
"... This paper presents a methodology for safety verification of continuous and hybrid systems in the worstcase and stochastic settings. In the worstcase 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 ..."
Abstract

Cited by 50 (1 self)
 Add to MetaCart
This paper presents a methodology for safety verification of continuous and hybrid systems in the worstcase and stochastic settings. In the worstcase 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.
Optimization of lyapunov invariants in verification of software systems
 MATTHIAS RUNGGER AND PAULO TABUADA 25
, 2013
"... The paper proposes a controltheoretic framework for verification of numerical software systems, and puts forward software verification as an important application of control and systems theory. The idea is to transfer Lyapunov functions and the associated computational techniques from control syste ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
The paper proposes a controltheoretic framework for verification of numerical software systems, and puts forward software verification as an important application of control and systems theory. The idea is to transfer Lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications. These include but are not limited to absence of overflow, absence of divisionbyzero, termination in finite time, presence of deadcode, and certain userspecified assertions. Central to this framework are Lyapunov invariants. These are properly constructed functions of the program variables, and satisfy certain properties—resembling those of Lyapunov functions—along the execution trace. The search for the invariants can be formulated as a convex optimization problem. If the associated optimization problem is feasible, the result is a certificate for the specification.
Lyapunov Analysis of Rigid Body Systems with Impacts and Friction via SumsofSquares
"... Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of conti ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, which can play a powerful role in motion planning and control design. Here, we present a method for applying sumsofsquares verification to rigid bodies with Coulomb friction undergoing discontinuous, inelastic impact events. The proposed algorithm explicitly generates Lyapunov certificates for stability, positive invariance, and reachability over admissible (nonpenetrating) states and contact forces. We leverage the complementarity formulation of contact, which naturally generates the semialgebraic constraints that define this admissible region. The approach is demonstrated on multiple robotics examples, including simple models of a walking robot and a perching aircraft.
Optimization on linear matrix inequalities for polynomial systems control
, 2013
"... ..."
(Show Context)
Measures and LMI for space launcher robust control validation
 Proceedings of the IFAC Symposium on Robust Control Design
, 2012
"... We describe a new temporal verification framework for safety and robustness analysis of nonlinear control laws, our target application being a space launcher vehicle. Robustness analysis, formulated as a nonconvex nonlinear optimization problem on admissible trajectories corresponding to piecewise p ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We describe a new temporal verification framework for safety and robustness analysis of nonlinear control laws, our target application being a space launcher vehicle. Robustness analysis, formulated as a nonconvex nonlinear optimization problem on admissible trajectories corresponding to piecewise polynomial dynamics, is relaxed into a convex linear programming problem on measures. This infinitedimensional problem is then formulated as a generalized moment problem, which allows for a numerical solution via a hierarchy of linear matrix inequality relaxations solved by semidefinite programming. The approach is illustrated on space launcher vehicle benchmark problems, in the presence of closedloop nonlinearities (saturations and deadzones) and axis coupling.
Optimal stabilization using Lyapunov measure,”
 in Proceedings of American Control Conference,
, 2008
"... Abstract Numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this paper. Settheoretic notion of almost everywhere stability introduced by the Lyapunov measure, which is weaker than conventional Lyapunov functionbased stabilization methods, ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract Numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this paper. Settheoretic notion of almost everywhere stability introduced by the Lyapunov measure, which is weaker than conventional Lyapunov functionbased stabilization methods, is used for optimal stabilization. The linear PerronFrobenius transfer operator is used to pose the optimal stabilization problem as an infinite dimensional linear program. Setoriented 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 controlled standard map.
Stability Analysis and Control of Rigid Body Systems with Impacts and Friction
"... Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of cont ..."
Abstract
 Add to MetaCart
Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sumsofsquares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, and can additionally be used to automatically synthesize stabilizing feedback controllers. Here, we present a method for applying sumsofsquares verification to rigid bodies with Coulomb friction undergoing discontinuous, inelastic impact events. The proposed algorithm explicitly generates Lyapunov certificates for stability, positive invariance, and reachability over admissible (nonpenetrating) states and contact forces. We leverage the complementarity formulation of contact, which naturally generates the semialgebraic constraints that define this admissible region. The approach is demonstrated on multiple robotics examples, including simple models of a walking robot, a perching aircraft, and control design of a balancing robot.
Operator theoretical methods for dynamical systems control and optimization
"... Nonconvex control and optimization problems for nonlinear dynamical systems can be approached with numerical methods inspired by operator theory. The workshop is an opportunity to present for the first time in a unified way two major operator theoretical approaches to nonlinear dynamical systems: • ..."
Abstract
 Add to MetaCart
(Show Context)
Nonconvex control and optimization problems for nonlinear dynamical systems can be approached with numerical methods inspired by operator theory. The workshop is an opportunity to present for the first time in a unified way two major operator theoretical approaches to nonlinear dynamical systems: • Koopman operator methods for dynamical systems, relying on Galerkin numerical discretization techniques; • polynomial optimization and optimal control formulated as generalized problems of moments, discretized by hierarchies of convex linear matrix inequalities, and solved numerically with semidefinite programming. These techniques are applied to compute regions of attraction and invariant sets, and to solve dynamical systems problems arising in neuroscience.