Sensitivity analysis for parametric control problems with control-state constraints

Malanowski, K.; Maurer, Helmut

doi:10.1007/bf00248267

Cited by 93 publications

(60 citation statements)

References 18 publications

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“…This idea of parametrizing the trajectories by their switching times also extends to the case where there are state constraints, and has been carried out in [58,73,74,75,76], and in these references the test of positivity of the resulting quadratic form permits to derive sufficient conditions for local optimality. The concept of conjugate point is however not so straightforward to define, since one has to investigate stability properties of the structure.…”

Section: Generalizations Open Problems and Challengesmentioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

186

179

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This article surveys the classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the classical tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 80's and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric re-entry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low cost interplanetary space missions. The article ends with open problems and perspectives.

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Section: Generalizations Open Problems and Challengesmentioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

186

179

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“…the optimal inputs are continuously differentiable with respect to the parametric perturbation (4.1) in a neighborhood of η = 0. Sufficient conditions under which these assumptions hold have been discussed extensively in the literature, e.g., [21,29,26,28,22]; see also [10] for a derivation, based on the theory of neighboring extremals [5,30], of the first-order variations of the optimal inputs with respect to parametric perturbations in optimal control problems with mixed control-state constraints.…”

Section: Implications Of Selective Input Adaptation On Optimality Lossmentioning

confidence: 99%

“…Here again, approximations of the perturbed optimal solutions can be obtained via sensitivity analysis. In particular, extensive work has been done regarding the sensitivity analysis of parametric optimal control problems over the last decades, including the differentiability of the optimal inputs with respect to parameter values [21,29,26,28,22]. Other ways of approximating the solution of parametric constrained optimization problems include spectral finite-element methods [1].…”

Section: Introduction a Large Class Of Industrial Processes Is Transmentioning

confidence: 99%

Directional Input Adaptation in Parametric Optimal Control Problems

Deshpande¹,

Bonvin²,

Chachuat³

2012

SIAM J. Control Optim.

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Abstract. This paper deals with input adaptation in dynamic processes in order to guarantee feasible and optimal operation despite the presence of uncertainty. For those optimal control problems having terminal and mixed control-state path constraints, two orthogonal sets of adaptation directions can be distinguished in the input space: the sensitivity-seeking directions, along which a small variation from an optimal nominal solution will not affect the respective active constraints, and the complementary constraint-seeking directions, along which a variation will affect the respective constraints. It follows that selective input adaptation strategies can be defined, namely, adaptation in the sensitivity-and constraint-seeking directions. This paper proves the important result that, for small parametric perturbations, the cost variation resulting from adaptation in the sensitivity-seeking directions (over no input adaptation) is typically smaller than the cost variation due to adaptation in the constraint-seeking directions. It is also established that no selective input adaptation along a sensitivity-seeking direction can reduce the dominant, first-order term in the optimality gap; adaptation along a constraint-seeking direction is necessary to cancel it out. These results are illustrated with two numerical case studies.Key words. parametric optimal control problem, terminal constraints, mixed control-state constraints, sensitivity-seeking directions, constraint-seeking directions, linear integral equations, selective input adaptation, cost variation, optimality loss AMS subject classifications. 49K15, 49K40, 49M05, 49M25DOI. 10.1137/110820646 Introduction. A large class of industrial processes is transient in nature.For instance, many processes in the resource industries are either discontinuous or operated in an unsteady-state manner. Batch and semibatch processes in the chemical process industry are other examples of processes that are characterized by the absence of a steady state. Many alternative technologies in the energy sector also exhibit discontinuous operation, and transient energy-generation systems are expected to play a key role in years to come.In engineering applications, the optimization of transient processes aims at computing time profiles of certain process variables to improve process performancemeasured in terms of profit or product quality-subject to tight operational constraints, including strict safety requirements and environmental regulations. Many numerical methods have been developed over the years to solve such optimal control problems on the basis of a process model; see, e.g., [3,4,16]. However, in applying the computed optimal profiles open-loop to the process, both plant-model mismatch and process disturbances typically result in suboptimal process operation or, which is worse, infeasible operation.A common practice of dealing with uncertainty is to represent it in the form of parametric perturbations. The optimal input profiles are computed off-line for the

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“…The key assumptions throughout the paper are that (i) the nominal optimal inputs u * consist of finitely many arcs, and (ii) the second-order sufficiency conditions (SOSC) for differentiability of the optimal solution, as described in [3] and [4], hold. In particular, the Hamiltonian function is assumed to be regular, which implies that the optimal inputs u * are continuous in [t 0 , t f ].…”

Section: Parametric Optimal Control Problemmentioning

confidence: 99%

Parametric sensitivity of path-constrained optimal control: Towards selective input adaptation

Deshpande

Chachuat

Bonvin

2009

2009 American Control Conference

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Abstract-In the context of dynamic optimization, plant variations necessitate adaptation of the input profiles in order to guarantee both feasible and optimal operation. For those problems having path constraints, two sets of directions can be distinguished in the input space at each time instant: the so-called sensitivity-seeking directions, along which a small input variation does not affect the active path constraints; the complementary constraint-seeking directions, along which a variation affects the path constraints. Hence, three selective input adaptations are possible, namely, adaptation along each set of input directions and adaptation of the switching times between arcs. This paper considers parametric variations around a nominal optimal solution and quantifies the influence of these variations on each type of input adaptation.Index Terms-Parametric optimal control, change in optimal inputs, second-order sufficient conditions, constraint-seeking and sensitivity-seeking directions, change in switching times.

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Sensitivity analysis for parametric control problems with control-state constraints

Cited by 93 publications

References 18 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

Directional Input Adaptation in Parametric Optimal Control Problems

Parametric sensitivity of path-constrained optimal control: Towards selective input adaptation

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