Goddard problem in presence of a dynamic pressure limit

Seywald, Hans; Cliff, Eugene M.

doi:10.2514/3.21080

Cited by 44 publications

(31 citation statements)

References 6 publications

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“…The maximum altitude h(T ) and the end time T for ε = 10 Particularly interesting is the additional singular arc of the thrust u(t) appearing at t ≈ 0.075, which is due to the additional dynamic pressure constraint. This instance is explained in more details by Seywald and Cliff (1992). Although the singular arc at t ≈ 0.075 only appears for penalty parameters ε < 10 −10 and barely influences the final cost, it shows the applicability of the saturation function approach and the accuracy of the collocation method for solving the BVP (29)-(32).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…However, the consideration of the constraints (4), (5) and the singular arcs of the Goddard problem require to analytically investigate the switching structure of the optimal solution and lead to interior boundary conditions (Seywald and Cliff, 1992).…”

Section: Consideration Of Thrust Constraintmentioning

confidence: 99%

“…with the altitude h from the center of Earth, the velocity v, and the mass m as the mass of the rocket (Seywald and Cliff, 1992;Bryson, 1999). The states h, v, m, the thrust u as the input of the system, and the time t are commonly normalized and dimension-free.…”

Section: Model Equationsmentioning

confidence: 99%

“…A particularly interesting formulation of the Goddard problem is given by Seywald and Cliff (Seywald and Cliff, 1992), who considered both thrust and dynamic pressure constraints. This significantly complicates the Goddard problem, since the dynamic pressure constraint represents a first-order state constraint (in the sense of (Bryson and Ho, 1969, Ch.…”

Section: Introductionmentioning

confidence: 99%

“…In this contribution, the Goddard problem with thrust and dynamic pressure constraints (Seywald and Cliff, 1992) is used as an example to present a new method for handling constraints in optimal control by means of saturation functions. Following the ideas in (Graichen, 2006) originally developed in the context of feedforward control design, the saturation function approach takes advantage of the fact that the thrust and dynamic pressure constraints have different orders.…”

Section: Introductionmentioning

confidence: 99%

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Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions

Graichen

Petit

2008

IFAC Proceedings Volumes

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This paper addresses the well-known Goddard problem in the formulation of Seywald and Cliff with the objective to maximize the altitude of a vertically ascending rocket subject to dynamic pressure and thrust constraints. The Goddard problem is used to propose a new method to systematically incorporate the constraints into the system dynamics by means of saturation functions. This procedure results in an unconstrained and penalized optimal control problem which strictly satisfies the constraints. The approach requires no knowledge of the switching structure of the optimal solution and avoids the explicit consideration of singular arcs. A collocation method is used to solve the BVPs derived from the optimality conditions and demonstrates the applicability of the method to constrained optimal control problems.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Consideration Of Thrust Constraintmentioning

confidence: 99%

Section: Model Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions

Graichen

Petit

2008

IFAC Proceedings Volumes

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An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

Malisani

Chaplais

Petit

2014

Optim Control Appl Methods

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SUMMARYThis paper exposes a methodology to solve state and input constrained optimal control problems for nonlinear systems. In the presented 'interior penalty' approach, constraints are penalized in a way that guarantees the strict interiority of the approaching solutions. This property allows one to invoke simple (without constraints) stationarity conditions to characterize the unknowns. A constructive choice for the penalty functions is exhibited. The property of interiority is established, and practical guidelines for implementation are given. A numerical benchmark example is given for illustration.

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