Direct transcription solution of higher-index optimal control problems and the virtual index

Engelsone, A.; Campbell, Stephen L.; Betts, John T.

doi:10.1016/j.apnum.2006.03.012

Cited by 23 publications

(19 citation statements)

References 12 publications

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“…However, a different choice of algebraic state and control from the algebraic variables could produce a lower index DAE. The least such index is called the virtual index of the DAE [10,13,25]. A key earlier observation was that when working with DT the important thing is the virtual index and not what the user thinks the index is.…”

Section: Virtual Indexmentioning

confidence: 98%

“…The next problem from [13] is a more challenging problem where the index in terms of the control is 3 while the virtual index is one. While this problem is linear for a given control it is nonlinear due to the bL and bx 2 terms.…”

Section: Problem 1 Consider the Following Controlled Daementioning

confidence: 99%

“…This is consistent with Figs. 7 and 8 in [13] where this same case was solved with SOCX and there were oscillations in the angle of the blade of the hockey stick even though the center of the hockey stick seemed to move smoothly.…”

Section: Problem 1 Consider the Following Controlled Daementioning

confidence: 99%

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Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches

Campbell

Betts

2016

Numer Algor

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There have been a number of results in the literature showing that a direct transcription numerical approach to optimal control problems could have a number of surprising and desirable behaviors that were not always predicted by the existing theory especially when differential algebraic equations are involved. Most of these results were developed with an implicit Runge-Kutta (IRK) discretization being used. It is important to know which of these observations hold for direct transcription software using different types of discretizations and which are discretization specific. This paper reexamines several of these questions but using an hp-pseudospectral code. It is seen that while philosophically the results are often similar to the IRK results, there are some differences that should be understood by users solving constrained optimal control problems. This paper also discusses a type of regularization for higher index problems that does not reduce the index.

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Section: Virtual Indexmentioning

confidence: 98%

Section: Problem 1 Consider the Following Controlled Daementioning

confidence: 99%

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Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches

Campbell

Betts

2016

Numer Algor

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“…However, to SOS they are just algebraic variables. This enables SOS to solve some problems with DAEs that the user thinks are higher index [17]. SOS can solve a wide variety of delayed state optimal control problems [7,9,10].…”

Section: General Direct Transcription Algorithmmentioning

confidence: 99%

Solving optimal control problems with control delays using direct transcription

Betts

Campbell

Thompson

2016

Applied Numerical Mathematics

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The numerical treatment of optimal control problems with state and control delays is important in a wide variety of scientific and technical applications. Solutions to these types of problems are difficult to obtain via analytic techniques since the system may be nonlinear and subjected to complicated inputs and constraints. There are several numerical methods available to compute the solutions of optimal control problems without delays. One such popular method is direct transcription. Although the numerical solutions of optimal control delay problems are important, less literature and software exists in this area. A general purpose industrial grade direct transcription code that can handle optimal control problems with both state and control constraints and delays is under development. Control delays pose a special challenge. A new technique for treating control delays when using a direct transcription approach is investigated in this paper.

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“…Due to the many connections, some researchers in the numerical DAE community have begun to consider control/optimization issues while at the same time some from the control/optimization community have begun to examine DAE issues [7,8,10,13,14,15,22,23,34,36].…”

Section: Recent Developments and Open Problemsmentioning

confidence: 99%

Control and Optimization with Differential-Algebraic Constraints

Biegler¹,

Campbell²,

Mehrmann³

2012

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Differential Algebraic Equations (DAEs) are mixed systems of differential and algebraic equations. It has been recognized for some time now that they have great potential both theoretically and in applications. DAEs form one of the most elegant and simple ways to model a physical system because they allow for the creation of separate models for subcomponents that can then be pasted together via a network. As a consequence, this concept is used in many modern CAD/modeling systems like SIMULINK, Scicos and DYMOLA, although most software packages cannot fully exploit the full potential of DAE models.But this nice feature of DAEs for modeling has also a disadvantage, since it shifts all of the difficulties of a system onto the analysis and the numerical methods. For this reason in recent years much effort has been spent to analyze general DAEs and to derive suitable numerical methods either for general DAEs, see e.g. [9,19,18,20,29,38] or for special DAEs arising in applications, see e.g. [6,12,28,37].This analytical and numerical work so far has been primarily driven by the simulation community, where the desire was to simulate the behavior of a complex system which could be electrical, mechanical, chemical, or all three. Due to the ever growing complexity of models which pose new challenges, the field is developing rather rapidly including now also hybrid [2,3,4,11,21,31,39] and delay systems [16,17,25,26,27].Once a system can be modeled and simulated, there arises the need to control the process or optimize its performance. The control of physical processes is an important task in many applications and over the last two decades there have been tremendous advances in the theory and applications of control in almost all disciplines of science and technologies. Note that this includes not only the obvious applications such as designing a more efficient process, but also determining what are the control mechanisms inherent in complex biological systems and fitting models to data. Recent Developments and Open ProblemsAs the two topics, simulation of DAEs and control/optimization, have evolved in recent years, there has been a growing awareness of interconnections which arise in a number of ways. One obvious way is the control of DAE modeled systems. Optimality conditions and a maximum principle for general DAEs have only recently been obtained [30] and the results are far from complete. But there are other more subtle connections in that the necessary conditions for an optimal control problem typically form a DAE and the solution of some control problems in the presence of constraints or invariants also involve DAEs. An open problem is in 1

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Direct transcription solution of higher-index optimal control problems and the virtual index

Cited by 23 publications

References 12 publications

Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches

Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches

Solving optimal control problems with control delays using direct transcription

Control and Optimization with Differential-Algebraic Constraints

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