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

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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Parrilo, P., 2000, “Structured Semidefinite Programs and Semi-Algebraic Geometry Methods in Robustness and Optimization,” Ph.D. dissertation, California Institute of Technology, Pasadena, CA.
Jarvis-Wloszek, Z., and Packard, A., 2002, “An LMI Method to Demonstrate Simultaneous Stability Using Non-Quadratic Polynomial Lyapunov Functions,” Proc. IEEE Conf. Dec. Contr., pp. 287–292.
Tan, W., and Packard, A., 2008, “Stability Region Analysis Using Polynomial and Composite Polynomial Lyapunov Functions and Sum-of-Squares Programming,” IEEE Trans. Autom. Control, 53(2), pp. 565–571. [CrossRef]
Papachristodoulou, A., Peet, M. M., and Lall, S. K., 2009, “Analysis of Polynomial Systems With Time Delays via the Sum of Squares Decomposition,” IEEE Trans. Autom. Control, 54(5), pp. 1058–1064. [CrossRef]
Zheng, Q., and Wu, F., 2009, “Stabilization of Polynomial Nonlinear Systems Using Rational Lyapunov Functions,” Int. J. Control, 82(9), pp. 1605–1615. [CrossRef]
Narimani, M., and Lam, H., 2010, “SOS-Based Stability Analysis of Polynomial Fuzzy-Model-Based Control Systems via Polynomial Membership Functions,” IEEE Trans. Fuzzy Systems, 18(5), pp. 862–871. [CrossRef]
Franze, G., Famularo, D., and Casavola, A., 2012, “Constrained Nonlinear Polynomial Time-Delay Systems: A Sum-of-Squares Approach to Estimate the Domain of Attraction,” IEEE Trans. Autom. Control, 57(10), pp. 2673–2679. [CrossRef]
Prajna, S., 2005, “Optimization-Based Methods for Nonlinear and Hybrid Systems Verification,” Ph.D. Dissertation, California Institute of Technology, Pasadena, CA.
Topcu, U., and Packard, A., 2009, “Linearized Analysis Versus Optimization-Based Nonlinear Analysis for Nonlinear Systems,” Proceedings of the American Control Conference, pp. 790–795.
Summers, E., Chakraborty, A., Tan, W., Topcu, U., Seiler, P., Balas, G., and Packard, A., 2013, “Quantitative Local L2-Gain and Reachability Analysis for Nonlinear Systems”. Int. J. Robust Nonlinear Control, 23(10), pp. 1115–1135. [CrossRef]
Wu, F., and Prajna, S., 2005, “SOS-based Solution Approach For Polynomial LPV System Analysis and Synthesis Problems”. Int. J. Control, 78(8), pp. 600–611. [CrossRef]
Prajna, S., Papachristodoulou, A., and Wu, F., 2004, “Nonlinear Control Synthesis by Sum of Squares Optimization: A Lyapunov-Based Approach,” Proceedings of the Asian Control Conference, pp. 157–165.
Ebenbauer, C., Renz, J., and Allgower, F., 2005, “Polynomial Feedback and Observer Design Using Nonquadratic Lyapunov Functions,” Proc. Joint IEEE Conf. Dec. Contr. & Eur. Contr. Conf., pp. 7587–7592.
Ebenbauer, C., and Allgower, F., 2006, “Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares,” Comput. Chem. Eng., 30, pp. 1590–1602. [CrossRef]
Ichihara, H., 2008, “State Feedback Synthesis for Polynomial Systems With Bounded Disturbances,” Proc. IEEE Conf. Dec. Contr., pp. 2520–2525.
Ichihara, H., 2009, “Optimal Control for Polynomial Systems Using Matrix Sum of Squares Relaxations,” IEEE Trans. Autom. Control, 54(5), pp. 1048–1053. [CrossRef]
Lu, W.-M., and Doyle, J. C., 1995, “H∞ Control of Nonlinear Systems: A Convex Characterization,” IEEE Trans. Autom. Control, 40(9), pp. 1668–1675. [CrossRef]
Zheng, Q., and Wu, F., 2009, “Nonlinear H∞ Control Designs With Axi-Symmetric Rigid Body Spacecraft,” AIAA J. Guidance Nav. Control, 32(3), pp. 850–859. [CrossRef]
Prajna, S., Parrilo, P. A., and Rantzer, A., 2004, “Nonlinear Control Synthesis by Convex Optimization,” IEEE Trans. Autom. Control, 49(2), pp. 310–314. [CrossRef]
Bernstein, D. S., and Michel, A. N., 1995, “A Chronological Bibliography on Saturating Actuators,” Int. J. Robust Nonlinear Control, 5, pp. 375–380. [CrossRef]
Tarbouriech, S., and Garcia, G., eds., 1997, Control of Uncertain Systems With Bounded Inputs, Springer, London.
Kapila, V., and Grigoriadis, K. M., eds., 2002, Actuator Saturation Control, Marcel Dekkar, New York.
Lin, Y., and Sontag, E. D., 1991, “A Universal Formula for Stabilization With Bounded Controls”. Syst. Control Lett., 16(6), pp. 393–397. [CrossRef]
Haddad, W. M., Fausz, J. L., and Chellaboina, V.-S., 1999, “Nonlinear Controllers for Nonlinear Systems With Input Nonlinearities,” J. Frank. Inst., 336, pp. 649–664. [CrossRef]
Castelan, E. B., Tarbouriech, S., and Queinnec, I., 2008, “Control Design for A Class of Nonlinear Continuous-Time Systems,” Automatica, 44(8), pp. 2034–2039. [CrossRef]
Valmorbida, G., Tarbouriech, S., and Garcia, G., 2013, “Design of Polynomial Control Laws for Polynomial Systems Subject to Actuator Saturation,” IEEE Trans. Autom. Control, 58(7), pp. 1758–1770. [CrossRef]
Brockett, R. W., 1973, “Lie algebras and Lie Groups in Control Theory,” Geometric Methods in System Theory, D. Q.Mayne and R. W.Brockett, eds., Dordrecht, Reidel, pp. 213–225.
Barkin, A., and Zelentsovsky, A., 1983, “Method of Power Transformations for Analysis and Stability of Nonlinear Control Systems,” Syst. Control Lett., 3, pp. 303–310. [CrossRef]
Zelentsovsky, A. L., 1994, “Nonquadratic Lyapunov Functions for Robust Stability Analysis of Linear Uncertain Systems,” IEEE Trans. Autom. Control, 39(1), pp. 135–138. [CrossRef]
Chesi, G., Garulli, A., Tesi, A., and Vicino, A., 2003, “Homogeneous Lyapunov Functions for Systems With Structured Uncertainties,” Automatica, 39(6), pp. 1027–1035. [CrossRef]
Choi, M. D., Lam, T. Y., and Reznick, B., 1995, “Sums of Squares of Real Polynomials,” Symp. Pure Math., 58, pp. 103–126.
Peet, M. M., and Papachristodoulou, A., 2012, “A Converse Sum of Squares Lyapunov Result With A Degree Bound,” IEEE Trans. Autom. Control, 57(9), pp. 2281–2293. [CrossRef]
Ahmadi, A. A., and Parrilo, P. A., 2001, “Converse Results on Existence of Sum of Squares Lyapunov Functions,” 50th IEEE CDC Conf. Dec. Control, pp. 6516–6521.
Prajna, S., Papachristodoulou, A., Seiler, P., and Parrilo, P. A., 2004, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, ver. 2, http://www.cds.caltech.edu/sostools.
Hu, T., Teel, A., and Zaccarian, L., 2006, “Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions,” IEEE Trans. Autom. Control, 51(11), pp. 1133–1168. [CrossRef]
Hu, T., Lin, Z., and Chen, B., 2002, “An Analysis and Design Method for Linear Systems Subject to Actuator Saturation and Disturbance,” Automatica, 38(2), pp. 351–359. [CrossRef]
Tsiotras, P., and Longuski, J. M., 1994, “Spin-Axis Stabilization of Symmetric Spacecraft With Two Control Torques,” Syst. Control Lett., 23, pp. 395–402. [CrossRef]
Conway, J. B., 1978, Functions of One Complex Variable, Springer, New York.


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Fig. 1

Simulation and comparison results

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Fig. 2

L2 gain comparison under different regional constraint sizes

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Fig. 3

Axi-symmetric rigid body with two controls

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Fig. 4

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

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Fig. 5

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

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Fig. 6

L2 gain comparison under different disturbance energy levels

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Fig. 7

Comparison of output and control input trajectories




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