Reference for LPV controls

Question

I am looking for a good mathematical introduction to LPV (Linear Parameter Varying) methods in control theory. I would like it to be more on the mathematical side of things, instead of something aimed only at engineers, if possible. Does anybody have such a reference (better if available online)?

Answer 1

This may not be exactly what you're looking for, but this Ph.D. thesis explores the use of LPV methods in synthesizing gain scheduling controllers for nonlinear systems. Specifically, it uses LPV models as a means of describing non-linear systems and then designs controllers for the LPV models developed.

The thesis isn't too long and provides the research publications that it's based on.

Answer 2

For an LPV approach based on a set of linear models defined on a grid of parameter values you should look at:

1. F. Wu, Control of linear parameter varying systems. Ph.D. Dissertation, University of California at Berkeley, 1995. (http://www.mae.ncsu.edu/wu/paper/PhDthesis.ps)

For an LPV approach where the system can be modeled using a linear fractional representation you should look at the following papers:

1. A. Packard, "Gain scheduling via linear fractional transformations," Systems and Control Letters, vol. 22, no. 2, pp. 79-92, 1994.

2. P. Apkarian and P. Gahinet, "A convex characterization of gain-scheduled H∞ controllers," IEEE Transactions on Automatic Control, vol. 40, no. 5, 1995. (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.55.9110&rep=rep1&type=pdf)

3. P. Apkarian and P. Gahinet, "Erratum to "A convex characterization of gain-scheduled H∞ controllers"," IEEE Transactions on Automatic Control, vol. 40, no. 9, 1995.

The references above provide all the details you need to (1) understand how stability and performance are framed in these two approaches to LPV, and (2) implement algorithms to synthesize stabilizing LPV controllers.

An overview of techniques to derive LPV models from nonlinear systems is provided in this paper:

1. A. Marcos and G. Balas, "Development of Linear-Parameter-Varying Models for Aircraft," Journal of Guidance, Control, and Dynamics, Vol. 27, No. 2, March–April, 2004.

For details on model reduction for LPV systems, see:

1. G. Woods, "Control of parameter-dependent mechanical systems", Ph.D. Dissertation, University of Cambridge, 1995.

Finally, a recent survey of LPV modeling, identification and control can be found in the following book:

1. J. Mohammadpour and C. W. Scherer, eds., Control of Linear Parameter Varying Systems with Applications. Boston, MA: Springer US, 2012.

2. M. Lovera, M. Bergamasco, and F. Casella. "LPV modelling and identification: An overview." Robust Control and Linear Parameter Varying Approaches. Springer Berlin Heidelberg, 2013.

I hope this helps.