Optimal and Robust Control

Assigned (compulsory) reading

Chapters 13 of the manuscript [1]:

[1] S. Gros, M. Diehl. Numerical Optimal Control. Draft manuscript. May 15, 2020. [ONLINE] downloadable at https://syscop.de/files/2020ss/NOC/book-NOCSE.pdf.

Note that as this is obviously a draft of a future book, some prelimiary version circulate on the internet (I may have linked one too in some previous lectures). If have already downloaded the text previously, just check that you really have the latest version.

There are also corresponding video lectures (#18 and perhaps #14 nd 15 too) in Diehl's course accessible at his (online) course webpage https://syscop.de/teaching/ss2020/numerical-optimal-control-online.

Recommended (not compulsory) further reading (and videos)

The indirect methods are well described in the classic and affordable [2]. The direct methods are described in a number of online resources. A description of the collocation method can be found in the original [3]. But some tutorial papers are also available, such as [4] and [5]. The author of the later has also published a full book [6] on the topic (the third edition has just been published, just in case you consider buying).

Brief intro into the direct techniques can also be found in the course by R. Tedrake on Underactuated robotics at MIT, namely Chapter 10 of his online lecture notes. The text is rather short but have a look at the

Another nice introductory material is by Matthew Kelly. First, there is a tutorial paper [7]. The author has also created a very illustrative visual explanation of the difference between several numerical concepts such as shooting, multiple shooting and collocation at http://www.matthewpeterkelly.com/tutorials/trajectoryOptimization/canon.html. He even keeps a dedicated tutorial webpage for the whole topic. There is also a video at

You may also want to have a look at the "motivating lecture" on trajectory optimization given by a research engineer from Boston Dynamics:

[2] D. E. Kirk. Optimal Control Theory - An Introduction. Dover, 2004.

[3] C. R. Hargraves, S. W. Paris. Direct Trajectory Optimization using Nonlinear Programming and Collocation, 1986. [download from oen of the author's ResearchGate]

[4] A. V. Rao. A Survey of Numerical Methods for Optimal Control. Advances in the Astronautical Sciences, 2009.

[5] John T. Betts.  Survey of Numerical Methods for Trajectory Optimization, Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2 (1998), pp. 193-207. http://dx.doi.org/10.2514/2.4231.

[6] John T. Betts. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed., SIAM, 2010.

[7] Matthew. Kelly, “An Introduction to Trajectory Optimization:  How to Do Your Own Direct Collocation,” SIAM Rev., vol. 59, no. 4, pp. 849–904, Jan. 2017, doi: 10.1137/16M1062569.