Hybrid Gauss Pseudospectral and Generalized Polynomial Chaos Algorithm to Solve Stochastic Trajectory Optimization Problems

Cottrill, Gerald C.; Harmon, Frederick G.

doi:10.2514/6.2011-6572

Cited by 15 publications

(12 citation statements)

References 18 publications

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“…(12)(13)(14)(15), which we accomplish using a conventional optimization technique such as the direct collocation method. We employ the pseudospectral method, a relatively new optimization technique that is categorized as a direct method [16][17][18].…”

Section: Solve Converted Problem By Direct Methodsmentioning

confidence: 99%

“…(12)(13)(14)(15) In the Legendre-Gauss pseudospectral method, the differential algebraic equations are collocated at N 0 Legendre-Gauss points τ 1 ; : : : ; τ N 0 that lie in the open interval (−1, 1). The initial time τ 0 −1 and terminal time τ N 0 1 1 are also added to the collocation points.…”

Section: Solve Converted Problem By Direct Methodsmentioning

confidence: 99%

“…Moreover, several researchers employed the gPC method for optimal control. Cottrill and Harmon [12] used both a pseudospectral method and the gPC technique to optimal control problems and examined the parameter sensitivity of the control output. Jia et al [13] used the gPC method to quantify the uncertainty propagation of systems, and Flanzer et al [14], who took the uncertainty of the five most sensitive parameters into account, employed it to obtain a robust trajectory for dynamic soaring.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Aircraft Control in Stochastic Severe Weather Conditions

Okamoto

Tsuchiya

2016

Journal of Guidance, Control, and Dynamics

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The purpose of this study is to apply a stochastic optimal control method to the guidance of an aircraft. The aircraft is given optimal control to make a final approach through a microburst, a hazardous weather phenomenon for lowflying aircraft that may cause accidents during landing. As the determination of the size and the strength of a microburst always involves uncertainty, its precise detection is difficult. To minimize the risk of accidents, we must account for system uncertainty, a difficult task using conventional optimization techniques, which cannot handle random variables. This paper aims to solve this problem by employing a stochastic optimization algorithm that incorporates the generalized polynomial chaos method into a conventional direct method. The generalized polynomial chaos method is a numerical technique for solving stochastic differential equations. To demonstrate the effectiveness of the proposed algorithm, we conduct a numerical simulation in which an aircraft flies through a microburst and attempts to land. The results of the simulation successfully verify its effectiveness. Nomenclature a = vector of static variables that are independent of timeacceleration, 9.81 m∕s i = index of random variables p j = index of set of collocation points and weight functions L = lift, N l = index of root of the Nth-degree Legendre orthogonal polynomial M = aircraft mass, kg m, n = indices indicating degree of orthogonal polynomials N = number of random variables N 0 = number of Legendre-Gauss collocation points P = maximum degree of N-variate orthogonal polynomial p = vector of random variables that are independent of time Q = total number of sets of collocation points and weight functions q i = number of sets of collocation points and weight functions of each random variable r p = microburst radius parameter, m S = wing surface, m 2 T = thrust, N t = time, s u = control variable u d = deterministic optimal control u m = microburst intensity parameter, m∕s u s = stochastic optimal control u = approximating function of control u = optimal control Var = variance v a = airspeed, m∕s w x = wind speed in x direction, m∕s w y = wind speed in y direction, m∕s w z = wind speed in z direction, m∕s x = horizontal position, m x c = microburst center parameter, m x = state variablẽ x = approximated state variable by generalized polynomial chaos methodx = approximatedx by Legendre-Gauss pseudospectral method z = altitude, m z m = microburst height parameter, where horizontal wind speed is strongest, m α = angle of attack, rad Γ = N-dimensional probability space γ = flight-path angle relative to air, rad γ g = flight-path angle relative to ground, rad ρ = probability density functions ρ air = density of air from standard atmosphere model, kg∕m 3 σ = standard deviation τ = normalized time τ T = time constant of thrust, s τ α = time constant of angle of attack, s Ψ = multidimensional orthogonal polynomials ψ = one-dimensional orthogonal polynomials Subscripts com = commanded value f = final nom = necessary value with no d...

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Section: Solve Converted Problem By Direct Methodsmentioning

confidence: 99%

Section: Solve Converted Problem By Direct Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal Aircraft Control in Stochastic Severe Weather Conditions

Okamoto

Tsuchiya

2016

Journal of Guidance, Control, and Dynamics

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“…In contrast, in studies of stochastic dynamic systems considering uncertainties, the primary focus has been on efficient algorithms for uncertainty propagation [14][15][16][17]21]. Only a few studies on dynamic system optimization design considering uncertainties can be found in the literature [22,23]. In [19,22], two different computational strategies are presented for trajectory optimization considering uncertainties.…”

Section: Introductionmentioning

confidence: 99%

“…Only a few studies on dynamic system optimization design considering uncertainties can be found in the literature [22,23]. In [19,22], two different computational strategies are presented for trajectory optimization considering uncertainties. A comparison between these two strategies is made in this study, which suggests that the method in [19] is very promising for practical applications.…”

Section: Introductionmentioning

confidence: 99%

Aircraft Robust Trajectory Optimization Using Nonintrusive Polynomial Chaos

Nair

Zhang

et al. 2014

Journal of Aircraft

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The development of algorithms for aircraft robust dynamic optimization considering uncertainties (for example, trajectory optimization) is relatively limited compared to aircraft robust static optimization (for example, configuration shape optimization). In this paper, an approach for dynamic optimization considering uncertainties is developed and applied to robust aircraft trajectory optimization. In the present approach, the nonintrusive polynomial chaos expansion scheme is employed to convert a robust trajectory optimization problem with stochastic ordinary differential equations into an equivalent deterministic trajectory optimization problem with deterministic ordinary differential equations. Two computational strategies for trajectory optimization considering uncertainties are compared. The performance of the developed method is studied by considering a classical deterministic trajectory optimization problem of supersonic aircraft short-time climb with uncertainties in the aerodynamic data. Detailed numerical studies are presented to illustrate the computational features of the proposed approach.

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