Results 1  10
of
22
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 ..."
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Cited by 68 (21 self)
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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
LMI techniques for optimization over polynomials in control: a survey
 IEEE Transactions on Automatic Control
"... Abstract—Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the ..."
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Cited by 34 (17 self)
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Abstract—Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Pólya’s theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, timedelay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.
Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics
"... Abstract—This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models— systems with impulse effects—through control Lyapunov functions. The periodic orbit is assumed to lie in a C 1 submanifold Z that is contained in the zero set of an output func ..."
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Cited by 16 (12 self)
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Abstract—This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models— systems with impulse effects—through control Lyapunov functions. The periodic orbit is assumed to lie in a C 1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the fullorder dynamics of the system with impulse effects have relied on inputoutput linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the fullorder dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions. I.
Robust regionofattraction estimation
 IEEE Transactions on Automatic Control
, 2010
"... Abstract—We propose a method to compute invariant subsets of the regionofattraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty. Parameterindependent Lyapunov functions are used to characterize invariant subsets of the robust ..."
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Cited by 11 (1 self)
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Abstract—We propose a method to compute invariant subsets of the regionofattraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty. Parameterindependent Lyapunov functions are used to characterize invariant subsets of the robust regionofattraction. A branchandbound type refinement procedure reduces the conservatism. We demonstrate the method on an example from the literature and uncertain controlled shortperiod aircraft dynamics. Index Terms—Branchandbound, parameter uncertainty, regionofattraction (ROA).
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.
Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle
"... Abstract — This paper derives a differential contraction condition for the existence of an orbitallystable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimization tools such as s ..."
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Cited by 3 (0 self)
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Abstract — This paper derives a differential contraction condition for the existence of an orbitallystable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimization tools such as sumofsquares programming to be used to search for certificates of the existence of a stable limit cycle. Many desirable properties of contracting dynamics are extended to this context, including preservation of contraction under a broad class of interconnections. In addition, by introducing the concepts of differential dissipativity and transverse differential dissipativity, contraction and transverse contraction can be established for large scale systems via LMI conditions on component subsystems. I.
Gradient projection antiwindup scheme
, 2011
"... AbstractThe gradient projection antiwindup (GPAW) scheme was recently proposed as an antiwindup method for nonlinear multiinputmultioutput systems/controllers, the solution of which was recognized as a largely open problem in a recent survey paper. This paper analyzes the properties of the GP ..."
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Cited by 2 (1 self)
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AbstractThe gradient projection antiwindup (GPAW) scheme was recently proposed as an antiwindup method for nonlinear multiinputmultioutput systems/controllers, the solution of which was recognized as a largely open problem in a recent survey paper. This paper analyzes the properties of the GPAW scheme applied to an input constrained first order linear time invariant (LTI) system driven by a first order LTI controller, where the objective is to regulate the system state about the origin. We show that the GPAW compensated system is in fact a projected dynamical system (PDS), and use results in the PDS literature to assert existence and uniqueness of its solutions. The main result is that the GPAW scheme can only maintain/enlarge the exact region of attraction of the uncompensated system.
International Journal of Robust Nonlinear
"... Synthesis of
nonlinear
controller
for wind turbines
stability
when
providing grid support ..."
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Synthesis of
nonlinear
controller
for wind turbines
stability
when
providing grid support
ESTIMATING THE REGION OF ATTRACTION FOR UNCERTAIN POLYNOMIAL SYSTEMS USING POLYNOMIAL CHAOS FUNCTIONS AND SUM OF SQUARES METHOD
, 2016
"... We present a general formulation for estimation of the region of attraction (ROA) for nonlinear systems with parametric uncertainties using a combination of the polynomial chaos expansion (PCE) theorem and the sum of squares (SOS) method. The uncertain parameters in the nonlinear system are treate ..."
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We present a general formulation for estimation of the region of attraction (ROA) for nonlinear systems with parametric uncertainties using a combination of the polynomial chaos expansion (PCE) theorem and the sum of squares (SOS) method. The uncertain parameters in the nonlinear system are treated as random variables with a probability distribution. First, the decomposition of the uncertain nonlinear system under consideration is performed using polynomial chaos functions. This yields to a deterministic subsystem whose state variables correspond to the deterministic coefficient components of the random basis polynomials in PCE. This decomposed deterministic subsystem contains no uncertainty. Then, the ROA of the deterministic subsystem is derived using sum of squares method. Finally, the ROA of the original uncertain nonlinear system is derived by transforming the ROA spanned in the decomposed deterministic subsystem back to the original spatialtemporal space using PCE. This proposed framework on estimation of the robust ROA (RROA) is based on a combination of PCE and SOS and is specially useful, with appealing computation efficiency, for uncertain nonlinear systems when the uncertainties are nonaffine or when they are associated with a specific probability distribution. 1