0
MODELING FOR CONTROL

On Controlling an Uncertain System With Polynomial Chaos and H2 Control Design

[+] Author and Article Information
Brian A. Templeton1

Center for Vehicle Systems and Safety, Virginia Tech, Blacksburg, VA 24061batemple@vt.edu

David E. Cox

Dynamic Systems and Control Branch, NASA Langley Research Center, Hampton, VA 23681david.e.cox@nasa.gov

Sean P. Kenny

Dynamic Systems and Control Branch, NASA Langley Research Center, Hampton, VA 23681sean.p.kenny@nasa.gov

Mehdi Ahmadian

Center for Vehicle Systems and Safety, Virginia Tech, Blacksburg, VA 24061ahmadian@vt.edu

Steve C. Southward

Center for Vehicle Systems and Safety, Virginia Tech, Danville, VA 24540scsouth@vt.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 132(6), 061304 (Oct 29, 2010) (9 pages) doi:10.1115/1.4002474 History: Received September 18, 2008; Revised August 29, 2010; Published October 29, 2010; Online October 29, 2010

This paper applies the H2 norm along time and parameter domains. The norm is related to the probabilistic H2 problem. It is calculated using polynomial chaos to handle uncertainty in the plant model. The structure of expanded states resulting from Galerkin projections of a state space model with uncertain parameters is used to formulate cost functions in terms of mean performances of the states, as well as covariances. Also, bounds on the norm are described in terms of linear matrix inequalitys. The form of the gradient of the norm, which can be used in optimization, is given as a Lyapunov equation. Additionally, this approach can be used to solve the related probabilistic LQR problem. The legitimacy of the concept is demonstrated through two mechanical oscillator examples. These controllers could be easily implemented on physical systems without observing uncertain parameters.

FIGURES IN THIS ARTICLE
<>
Copyright © 2010 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Uncertain 1DOF and 2DOF system schematics

Grahic Jump Location
Figure 2

Plotted is log10(‖H‖22) in the stable region of the system. The weighting matrices are Q=diag[1 1] and R=diag[1 0.1]. Also, the line on the surface shows a gradient descent starting from K=[k1 k2]=[0.3 −2.5]. The gradient descent converges to K=[−0.3649 −1.4757]. Fourth order polynomials are used, thereby creating fifteen 2D basis functions.

Grahic Jump Location
Figure 3

Means and standard deviations of state responses as functions of time for the unit impulse into the fourth state δ4 are presented. The weighting matrices M̱Q=diag[1 1 1 1], M̱R=[1], C̱Q=diag[0.01 0.01 0.01 0.01], and C̱R=[1] are used. The determined gains are K=[−0.570 −0.285 −1.060 −0.550].

Grahic Jump Location
Figure 4

Means and standard deviations of state responses as functions of time for the unit impulse into the fourth state δ4 are presented. The weighting matrices M̱Q=diag[1 1 1 1], M̱R=[1], C̱Q=diag[0.01 0.01 100 0.01], and C̱R=[1] are used. The determined gains are K=[−1.225 0.149 −2.391 0.363]. Also, max std(F)=0.1187.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In