Research Papers

Polynomial Chaos-Based Controller Design for Uncertain Linear Systems With State and Control Constraints

[+] Author and Article Information
Souransu Nandi

Department of Mechanical Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: souransu@buffalo.edu

Victor Migeon

Department of Mechanical Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: victor.migeon@orange.fr

Tarunraj Singh

Department of Mechanical Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: tsingh@buffalo.edu

Puneet Singla

Department of Mechanical Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: psingla@buffalo.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 3, 2017; final manuscript received December 13, 2017; published online January 24, 2018. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 140(7), 071009 (Jan 24, 2018) (12 pages) Paper No: DS-17-1237; doi: 10.1115/1.4038800 History: Received May 03, 2017; Revised December 13, 2017

For linear dynamic systems with uncertain parameters, design of controllers which drive a system from an initial condition to a desired final state, limited by state constraints during the transition is a nontrivial problem. This paper presents a methodology to design a state constrained controller, which is robust to time invariant uncertain variables. Polynomial chaos (PC) expansion, a spectral expansion, is used to parameterize the uncertain variables permitting the evolution of the uncertain states to be written as a polynomial function of the uncertain variables. The coefficients of the truncated PC expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A transformation of PC polynomial space to the Bernstein polynomial space permits determination of bounds on the evolving states of interest. Linear programming (LP) is then used on the deterministic set of equations with constraints on the bounds of the states to determine the controller. Numerical examples are used to illustrate the benefit of the proposed technique for the design of a rest-to-rest controller subject to deformation constraints and which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass-damper system.

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Fig. 1

Two mass spring damper system

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Fig. 2

MC simulations (10,000), MC mean and PC mean for mass 1 position—one uncertain variable

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Fig. 3

Bernstein bounds: (a) definite bounds from Bernstein coefficients (1D) and (b) definite bounds from Bernstein coefficients two-dimensional (2D)

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Fig. 4

Convex hulls: (a) time = 11, (b) time = 17, and (c) time = 29

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Fig. 5

Three-dimensional convex hulls: (a) time = 17, (b) time = 19, and (c) time = 27

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Fig. 6

Comparison of bounds

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Fig. 7

Split Bernstein coefficients and split convex hulls: (a) split coefficients at time = 29 and hull (1D) and (b) split coefficients at time = 27 and hull (2D)

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Fig. 8

Residual sensitivities and state distributions for unsplit bounds: (a) residual energy sensitivity, (b) residual state distribution: l form, and (c) residual state distribution: l1 form

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Fig. 9

(a) MC simulations showing all trajectories meet the state constraints, (b) and (c) comparison of residual energy sensitivities: (a) relative displacement with Bernstein envelope, (b) l formulation, and (c) l1 formulation




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