Reconstruction of support of a measure from its moments

Jasour, Ashkan; Lagoa, Constantino

doi:10.1109/cdc.2014.7039677

Cited by 3 publications

(15 citation statements)

References 19 publications

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“…Now, consider the following theorem. Theorem 3: The sequence of optimal solutions to the finite SDP in (19) converges to the moment sequence of measures that are optimal to the infinite LP in (18). Hence, lim r→∞ P * r = P * measures .…”

Section: Semidefinite Programming Relaxations On Momentsmentioning

confidence: 96%

“…As in Theorem 2 and Theorem 3, if equivalent problem on measures has delta distribution solution µ * u , then problems on measures and moments in (18) and (19) are equivalent to the chance constraint problem (10) and the optimal distribution µ * u is a delta distribution whose mass is concentrated on the single point u * , i.e., its support is the singleton {u * }. Such distributions, have moment matrices with rank one.…”

Section: Semidefinite Programming Relaxations On Momentsmentioning

confidence: 97%

“…Proof: See Appendix II. In the next section, we provide the tractable finite relaxations to the problem (18).…”

Section: Equivalent Convex Problem On Measuresmentioning

confidence: 99%

See 2 more Smart Citations

Convex Chance Constrained Model Predictive Control

Jasour¹,

Lagoa²

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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We consider the Chance Constrained Model Predictive Control problem for polynomial systems subject to disturbances. In this problem, we aim at finding optimal control input for given disturbed dynamical system to minimize a given cost function subject to probabilistic constraints, over a finite horizon. The control laws provided have a predefined (low) risk of not reaching the desired target set. Building on the theory of measures and moments, a sequence of finite semidefinite programmings are provided, whose solution is shown to converge to the optimal solution of the original problem. Numerical examples are presented to illustrate the computational performance of the proposed approach.

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Section: Semidefinite Programming Relaxations On Momentsmentioning

confidence: 96%

Section: Semidefinite Programming Relaxations On Momentsmentioning

confidence: 97%

See 1 more Smart Citation

Convex Chance Constrained Model Predictive Control

Jasour¹,

Lagoa²

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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“…is the trigonometric moment of ω θ defined in (9) and can be computed in terms of its characteristic function as in (12). Similarly, E ω k r is the polynomial moment of ω r and can be computed in terms of its characteristic function as in (2).…”

Section: Nonlinear Uncertainty Transformationmentioning

confidence: 99%

“…Moments of uncertainties can be used in estimation, planning, control, and safety analysis of uncertain dynamical systems. For example, finite sequence of the moments are used in [1]- [4] for risk bounded control of probabilistic nonlinear and linear systems, in [5]- [7] to solve chance-constrained optimization problems, in [8], [9] for nonlinear risk estimation, in [10] to compute the probability density function of uncertainties, in [11], [12] to construct reachable sets, and in [13]- [15] for risk bounded motion planning problems (For more examples see [16]).…”

Section: Introductionmentioning

confidence: 99%

Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems

Jasour¹,

Wang²,

Williams³

2021

Preprint

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In this paper, we address the problem of uncertainty propagation through nonlinear stochastic dynamical systems. More precisely, given a discrete-time continuous-state probabilistic nonlinear dynamical system, we aim at finding the sequence of the moments of the probability distributions of the system states up to any desired order over the given planning horizon. Moments of uncertain states can be used in estimation, planning, control, and safety analysis of stochastic dynamical systems. Existing approaches to address moment propagation problems provide approximate descriptions of the moments and are mainly limited to particular set of uncertainties, e.g., Gaussian disturbances. In this paper, to describe the moments of uncertain states, we introduce trigonometric and also mixed-trigonometric-polynomial moments. Such moments allow us to obtain closed deterministic dynamical systems that describe the exact time evolution of the moments of uncertain states of an important class of autonomous and robotic systems including underwater, ground, and aerial vehicles, robotic arms and walking robots. Such obtained deterministic dynamical systems can be used, in a receding horizon fashion, to propagate the uncertainties over the planning horizon in real-time.To illustrate the performance of the proposed method, we benchmark our method against existing approaches including linear, unscented transformation, and sampling based uncertainty propagation methods that are widely used in estimation, prediction, planning, and control problems.

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Convex Constrained Semialgebraic Volume Optimization: Application in Systems and Control

Jasour,

Lagoa

2017

Preprint

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In this paper, we generalize the chance optimization problems and introduce constrained volume optimization where enables us to obtain convex formulation for challenging problems in systems and control. We show that many different problems can be cast as a particular cases of this framework. In constrained volume optimization, we aim at maximizing the volume of a semialgebraic set under some semialgebraic constraints. Building on the theory of measures and moments, a sequence of semidefinite programs are provided, whose sequence of optimal values is shown to converge to the optimal value of the original problem. We show that different problems in the area of systems and control that are known to be nonconvex can be reformulated as special cases of this framework. Particularly, in this work, we address the problems of probabilistic control of uncertain systems as well as inner approximation of region of attraction and invariant sets of polynomial systems. Numerical examples are presented to illustrate the computational performance of the proposed approach.

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Reconstruction of support of a measure from its moments

Cited by 3 publications

References 19 publications

Convex Chance Constrained Model Predictive Control

Convex Chance Constrained Model Predictive Control

Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems

Convex Constrained Semialgebraic Volume Optimization: Application in Systems and Control

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