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Research Papers

Control of Polynomial Nonlinear Systems Using Higher Degree Lyapunov Functions

[+] Author and Article Information
Shuowei Yang

Department of Mechanical and
Aerospace Engineering,
North Carolina State University,
Raleigh, NC 27695
e-mail: syang9@ncsu.edu

Fen Wu

Department of Mechanical and
Aerospace Engineering,
North Carolina State University,
Raleigh, NC 27695
e-mail: fwu@eos.ncsu.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 8, 2013; final manuscript received November 18, 2013; published online February 24, 2014. Assoc. Editor: Sergey Nersesov.

J. Dyn. Sys., Meas., Control 136(3), 031018 (Feb 24, 2014) (13 pages) Paper No: DS-13-1109; doi: 10.1115/1.4026172 History: Received March 08, 2013; Revised November 18, 2013

Abstract

In this paper, we propose a new control design approach for polynomial nonlinear systems based on higher degree Lyapunov functions. To derive higher degree Lyapunov functions and polynomial nonlinear controllers effectively, the original nonlinear systems are augmented under the rule of power transformation. The augmented systems have more state variables and the additional variables represent higher order combinations of the original ones. As a result, the stabilization and $L2$ gain control problems with higher degree Lyapunov functions can be recast to the search of quadratic Lyapunov functions for augmented nonlinear systems. The sum-of-squares (SOS) programming is then used to solve the quadratic Lyapunov function of augmented state variables (higher degree in terms of original states) and its associated nonlinear controllers through convex optimization problems. The proposed control approach has also been extended to polynomial nonlinear systems subject to actuator saturations for better performance including domain of attraction (DOA) expansion and regional $L2$ gain minimization. Several examples are used to illustrate the advantages and benefits of the proposed approach for unsaturated and saturated polynomial nonlinear systems.

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Figures

Fig. 4

Comparison of trajectories using 2nd and 4th degree Lyapunov functions

Fig. 1

Simulation and comparison results

Fig. 2

L2 gain comparison under different regional constraint sizes

Fig. 3

Axi-symmetric rigid body with two controls

Fig. 7

Comparison of output and control input trajectories

Fig. 5

Estimated DOA as level sets of V ≤ 1: 2nd-degree Lyapunov function (dashed), 4th-degree Lyapunov function (solid)

Fig. 6

L2 gain comparison under different disturbance energy levels

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