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A unifying Lyapunovbased framework for the eventtriggered control of nonlinear systems.
 In IEEE Conference on Decision and Control and European Control Conference,
, 2011
"... AbstractWe present a prescriptive framework for the eventtriggered control of nonlinear systems. Rather than closing the loop periodically, as traditionally done in digital control, in eventtriggered implementations the loop is closed according to a statedependent criterion. Eventtriggered con ..."
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Cited by 13 (7 self)
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AbstractWe present a prescriptive framework for the eventtriggered control of nonlinear systems. Rather than closing the loop periodically, as traditionally done in digital control, in eventtriggered implementations the loop is closed according to a statedependent criterion. Eventtriggered control is especially well suited for embedded systems and networked control systems since it reduces the amount of resources needed for control such as communication bandwidth. By modeling the eventtriggered implementations as hybrid systems, we provide Lyapunovbased conditions to guarantee the stability of the resulting closedloop system and explain how they can be utilized to synthesize eventtriggering rules. We illustrate the generality of the approach by showing how it encompasses several existing eventtriggering policies and by developing new strategies which further reduce the resources needed for control.
Control design along trajectories with sums of squares programming
, 2013
"... Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant “funnel” that leads to a predefined goal set. Our certificates of invariance ..."
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Cited by 13 (9 self)
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Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant “funnel” that leads to a predefined goal set. Our certificates of invariance are given in terms of sums of squares proofs of a set of appropriately defined Lyapunov inequalities. These certificates, together with our proposed polynomial controllers, can be efficiently obtained via semidefinite optimization. Our approach can handle timevarying dynamics resulting from tracking a given trajectory, input saturations (e.g. torque limits), and can be extended to deal with uncertainty in the dynamics and state. The resulting controllers can be used by spacefilling feedback motion planning algorithms to fill up the space with significantly fewer trajectories. We demonstrate our approach on a severely torque limited underactuated double pendulum (Acrobot) and provide extensive simulation and hardware validation.
Projection methods for conic feasibility problems, applications to polynomial sumofsquares decompositions
, 2009
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Eventtriggered and selftriggered stabilization of distributed networked control systems
, 2011
"... Eventtriggered and selftriggered control have recently been proposed as implementation strategies that considerably reduce the resources required for control. Although most of the work so far has focused on closing a single control loop, some researchers have started to investigate how these new ..."
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Cited by 10 (4 self)
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Eventtriggered and selftriggered control have recently been proposed as implementation strategies that considerably reduce the resources required for control. Although most of the work so far has focused on closing a single control loop, some researchers have started to investigate how these new implementation strategies can be applied when closing multiplefeedback loops in the presence of physically distributed sensors and actuators. In this paper, we consider a scenario where the distributed sensors, actuators, and controllers communicate via a shared wired channel. We use our recent prescriptive framework for the eventtriggered control of nonlinear systems to develop novel policies suitable for the considered distributed scenario. Afterwards, we explain how selftriggering rules can be deduced from the developed eventtriggered strategies.
Modeling and solving uncertain optimization problems in YALMIP
, 2008
"... A considerable amount of optimization problems arising in the control and systems theory field can be seen as special instances of robust optimization. Much of the modeling effort in these cases is spent on converting an uncertain problem to a robust counterpart without uncertainty. Since many of t ..."
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Cited by 9 (4 self)
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A considerable amount of optimization problems arising in the control and systems theory field can be seen as special instances of robust optimization. Much of the modeling effort in these cases is spent on converting an uncertain problem to a robust counterpart without uncertainty. Since many of these conversions follow standard procedures, it is amenable to software support. This paper presents the robust optimization framework in the modeling language YALMIP, which carries out the uncertainty elimination automatically, and allows the user to concentrate on the highlevel model instead.
Finding the maximum eigenvalue of essentially nonnegative symmetric tensors via sum of squares programming
, 2012
"... Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is nonnegative in the sense that all of its entries are nonnegative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on ..."
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Cited by 8 (8 self)
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Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is nonnegative in the sense that all of its entries are nonnegative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on a PerronFrobenius type theorem for nonnegative tensors. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in turn, can be equivalently rewritten as a semidefinite programming problem. Then, using this sum of squares programming problem, we also provide upper as well as lower estimate for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor.
Finitetime Regional Verification of Stochastic Nonlinear Systems
"... Abstract—Recent trends pushing robots into unstructured environments with limited sensors have motivated considerable work on planning under uncertainty and stochastic optimal control, but these methods typically do not provide guaranteed performance. Here we consider the problem of bounding the pro ..."
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Cited by 7 (4 self)
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Abstract—Recent trends pushing robots into unstructured environments with limited sensors have motivated considerable work on planning under uncertainty and stochastic optimal control, but these methods typically do not provide guaranteed performance. Here we consider the problem of bounding the probability of failure (defined as leaving a finite region of state space) over a finite time for stochastic nonlinear systems with continuous state. Our approach searches for exponential barrier functions that provide bounds using a variant of the classical supermartingale result. We provide a relaxation of this search to a semidefinite program, yielding an efficient algorithm that provides rigorous upper bounds on the probability of failure for the original nonlinear system. We give a number of numerical examples in both discrete and continuous time that demonstrate the effectiveness of the approach. I.
A facial reduction algorithm for finding sparse sos representations
 Operations Research Letters
"... Facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial ([3]) removes unnecessary monomials for an SOS representation. In this paper, we establish a relationship between a facial reduction algorithm and the eliminatio ..."
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Cited by 6 (3 self)
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Facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial ([3]) removes unnecessary monomials for an SOS representation. In this paper, we establish a relationship between a facial reduction algorithm and the elimination method for a sparse SOS polynomial.
Safety verification of reactive controllers for UAV flight in cluttered environments using barrier certificates
 In Proceedings of the 2012 IEEE International Conference on Robotics and Automation (ICRA
, 2012
"... Abstract — Unmanned aerial vehicles (UAVs) have a sofar untapped potential to operate at high speeds through cluttered environments. Many of these systems are limited by their adhoc reactive controllers using simple visual cues like optical flow. Here we consider the problem of formally verifying ..."
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Cited by 4 (3 self)
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Abstract — Unmanned aerial vehicles (UAVs) have a sofar untapped potential to operate at high speeds through cluttered environments. Many of these systems are limited by their adhoc reactive controllers using simple visual cues like optical flow. Here we consider the problem of formally verifying an outputfeedback controller for an aircraft operating in an unknown environment. Using recent advances in sumsofsquares programming that allow for efficient computation of barrier functions, we search for global certificates of safety for the closedloop system in a given environment. In contrast to previous work, we use rational functions to globally approximate nonsmooth dynamics and use multiple barrier functions to guard against more than one obstacle. We expect that these formal verification techniques will allow for the comparison, and ultimately optimization, of reactive controllers for robustness to varying initial conditions and environments. I.
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 ..."
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Cited by 3 (2 self)
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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.