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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

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

## Abstract

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.

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## Figures

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.

Figure 1

Uncertain 1DOF and 2DOF system schematics

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.

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].

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