On the Theory of Trajectory Tubes — A Mathematical Formalism for Uncertain Dynamics, Viability and Control

Kurzhanski, Alexander B.; Filippova, Tatiana F.

doi:10.1007/978-1-4612-0349-0_4

Cited by 102 publications

(44 citation statements)

References 24 publications

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“…In subsequent sections, it will be useful to interpret the first argument of J as a state trajectory tube [7] ft ↦ xt; x 0 : x 0 ∈ suppx 0 g, whereas its second argument is a single control trajectory u·. Thus, we can now define a preliminary Riemann-Stieltjes optimal control problem as the problem to determine the conditionally deterministic decision pair x·; ·; u·, which minimizes the J functional subject to the dynamic constraint given by Eq.…”

Section: Development Of a Preliminary Mathematical Problem Formulmentioning

confidence: 99%

“…(61) and (62)] that was implemented at Honeywell was a form of tychastic control (or "robust" control). Standard minimax optimal control problems are connected to the computation of viability kernels and capture basins [7,47,48], which form the central concepts in viability theory.…”

Section: Further Applications and Open Problems Inmentioning

confidence: 99%

“…Viability theory was initiated in the late 1970s [7,49] to mathematically address the dynamics of socioeconomic, biological and other organizational systems that evolve in a Darwinian manner. The evolutionary engine is modeled as a differential inclusion _ x ∈ F x.…”

Section: Further Applications and Open Problems Inmentioning

confidence: 99%

See 2 more Smart Citations

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Ross

Proulx

Karpenko

et al. 2015

Journal of Guidance, Control, and Dynamics

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Motivated by uncertain parameters in nonlinear dynamic systems, we define a nonclassical optimal control problem where the cost functional is given by a Riemann-Stieltjes "functional of a functional." Using the properties of Riemann-Stieltjes sums, a minimum principle is generated from the limit of a semidiscretization. The optimal control minimizes a Riemann-Stieltjes integral of the Pontryagin Hamiltonian. The challenges associated with addressing the noncommutative operations of integration and minimization are addressed via cubature techniques leading to the concept of hyper-pseudospectral points. These ideas are then applied to address the practical uncertainties in control moment gyroscopes that drive an agile spacecraft. Ground test results conducted at Honeywell demonstrate the new principles. The Riemann-Stieltjes optimal control problem is a generalization of the unscented optimal control problem. It can be connected to many independently developed ideas across several disciplines: search theory, viability theory, quantum control and many other applications involving tychastic differential equations.

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Section: Development Of a Preliminary Mathematical Problem Formulmentioning

confidence: 99%

Section: Further Applications and Open Problems Inmentioning

confidence: 99%

See 1 more Smart Citation

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Ross

Proulx

Karpenko

et al. 2015

Journal of Guidance, Control, and Dynamics

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“…Assume that we find an ellipsoid E + such that X(T ;U, {0}) ⊆ E + , where X(T ;U, {0}) is defined in (9) for X 0 = {0}. Then applying well-known formulas for calculating the external ellipsoidal estimate for the sum of two ellipsoids E(r * , R * ) and E + [2,11] we can find the resulting external ellipsoid for X(T ;U, X 0 )) in (10). So the main difficulty now is in constructing the outer ellipsoidal estimate for X(T ;U{0}).…”

Section: Theoremmentioning

confidence: 99%

“…Now we consider the main scheme of construction the external ellipsoidal estimates of reachable sets X(t;Ũ, X 0 ) of the system (17)-(19). This scheme is based on the idea [10] of transformation of the problem with state constraints into the problem without coordinates restrictions. This transformation is made by including the special matrix of parameters in the dynamical system.…”

Section: Theorem 4 the Following Inclusion Holds X(t;u X 0 ) ⊆ X(t;mentioning

confidence: 99%

Estimation problem for impulsive control systems under ellipsoidal state bounds and with cone constraint on the control

Matviychuk¹

2012

AIP Conference Proceedings

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Articles you may be interested inAbstract. The paper deals with the state estimation problem for the linear control system containing impulsive control terms (or measures). The problem is studied here under uncertainty conditions when the initial system state is unknown but bounded, with given bound. It is assumed also that the system states should belong to the given ellipsoid in the state space. So the main problem of estimating the reachable set of the control system is studied here under more complicated assumption related to the case of state constraints. It is assumed additionally that impulsive controls in the dynamical system must belong to the intersection of a special cone with a generalized ellipsoid both taken in the space of functions of bounded variation. The last constraint is motivated by problems of impulsive control theory and by models from applied areas when not every direction of control impulses is acceptable in the system. We present here the state estimation algorithms that use the special structure of the control system and take into account additional restrictions on states and controls. The algorithms are based on ellipsoidal techniques for estimating the trajectory tubes of uncertain dynamical systems. Numerical simulation results related to proposed procedures are also given.

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Reliable Robot Localization

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On the Theory of Trajectory Tubes — A Mathematical Formalism for Uncertain Dynamics, Viability and Control

Cited by 102 publications

References 24 publications

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Estimation problem for impulsive control systems under ellipsoidal state bounds and with cone constraint on the control

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