Direct methods with maximal lower bound for mixed-integer optimal control problems

Säger, Sebastian; Bock, Hans Georg; Reinelt, Gerhard

doi:10.1007/s10107-007-0185-6

Cited by 97 publications

(113 citation statements)

References 42 publications

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“…Even if global MINLP methods are applied, one does not know how good the solution really is, as the underlying control discretization grid might be too coarse in some regions or simply not hit the optimal switching points. Only recently the connection between rigorous bounds on the optimal integer solution value and results of relaxed, continuous control problems has been made, [32,36].…”

Section: Theory For Control-affine Systemsmentioning

confidence: 99%

“…We will see later how the special case (9) relates to the more general problem (1) that we are really interested in. One of the observations in [32,36] was that the optimal solution of the relaxation of control problem (9) yields the exact lower bound for (1), i.e., the value that can either be reached or be approximated arbitrarily close by an integer control. However, the proof used arguments from functional analysis and hence this result does not apply to a finite number of switches.…”

Section: Theory For Control-affine Systemsmentioning

confidence: 99%

“…The reformulation yields additional nonconvexities in the optimization space. Even if the optimization problem is convex in the optimization variables resulting from a constant discretization of the control function v(·), the reformulated problem may be nonconvex, compare [32].…”

Section: Switching Time Optimizationmentioning

confidence: 99%

“…The mentioned drawbacks of the switching time optimization approach can be overcome, though, if it is combined with a bunch of other concepts, compare [32,17]. This includes good initial values, a strategy to deal with diminishing stage lengths and a direct all-at-once approach like direct multiple shooting that helps when dealing with nonconvexities as discussed in [32].…”

Section: Switching Time Optimizationmentioning

confidence: 99%

“…We will use the term integer control for (1e), while binary control refers to the special case ω(t) ∈ {0, 1} nω . We use the expression relaxed, whenever a restriction v(·) ∈ Ω is relaxed to a convex control set with a recently proposed outer convex relaxation [32] that we define as follows. For every element v i of Ω a binary control function ω i (·) is introduced.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

Säger

2009

Journal of Process Control

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117

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Section: Theory For Control-affine Systemsmentioning

confidence: 99%

Section: Theory For Control-affine Systemsmentioning

confidence: 99%

Section: Switching Time Optimizationmentioning

confidence: 99%

Section: Switching Time Optimizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

Säger

2009

Journal of Process Control

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117

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State‐of‐the‐art and progress in the optimization‐based simultaneous design and control for chemical processes

et al. 2012

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Significant progress in the area of simultaneous design and control for chemical processes has been achieved and various methodologies have been put forward to address this issue over the last several decades. These methods can be classified in two categories (1) controllability indicator‐based frameworks that are capable of screening alternative designs, and (2) optimization‐based frameworks that integrate the process design and control system design. The major objective is to give an up‐to‐date review of the state‐of‐the‐art and progress in the challenging area of optimization‐based simultaneous design and control. First, motivations and significances of simultaneous design and control are illustrated. Second, a general classification of existing methodologies of optimization‐based simultaneous design and control is outlined. Subsequently, the mathematical formulations and relevant theoretical solution algorithms, their merits, strengths and shortcomings are highlighted. Last, based on the recent advances in this field, challenges and future research directions are discussed briefly. An attempt is made with the help of this review article to stimulate further research and disseminate the simultaneous design methods to challenging problem areas. In particular, the application of optimization‐based simultaneous design and control methods to large‐scale systems with highly inherent nonlinear dynamics often the case in industrial chemical processes remains a challenging task and yet to be solved. © 2012 American Institute of Chemical Engineers AIChE J, 58: 1640–1659, 2012

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Time‐optimal control of automobile test drives with gear shifts

Kirches

Säger

Bock

et al. 2009

Optim Control Appl Methods

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SUMMARYWe present a numerical method and results for a recently published benchmark problem (Optim. Contr. Appl. Met. 2005; 26:1-18; Optim. Contr. Appl. Met. 2006; 27(3):169-182) in mixed-integer optimal control. The problem has its origin in automobile test-driving and involves discrete controls for the choice of gears. Our approach is based on a convexification and relaxation of the integer controls constraint. Using the direct multiple shooting method we solve the reformulated benchmark problem for two cases: we were able to reduce the overall computing time considerably, applying our algorithm. We give theoretical evidence on why our convex reformulation is highly beneficial in the case of time-optimal mixed-integer control problems as the chosen benchmark problem basically is (neglecting a small regularization term).

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Direct methods with maximal lower bound for mixed-integer optimal control problems

Cited by 97 publications

References 42 publications

Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

State‐of‐the‐art and progress in the optimization‐based simultaneous design and control for chemical processes

Time‐optimal control of automobile test drives with gear shifts

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