1Assistant Professor, Department of Electrical Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran
2Professor, Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
In this paper, a new analytical method to find a near-optimal high gain controller for the non-minimum phase affine nonlinear systems is introduced. This controller is derived based on the closed form solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the cheap control problem. This methodology employs an algebraic equation with parametric coefficients for the systems with scalar internal dynamics and a differential equation for those systems with the internal dynamics of order higher than one. It is shown that 1) if the system starts from different initial conditions located in the close proximity of the origin the regulation error of the closed-loop system with the proposed controller is less than that of the closed-loop system with the high gain LQR, which is surely designed for the linearized system around the origin, 2). for the initial conditions located in a region far from the origin, the proposed controller significantly outperforms the LQR controller.
 J. B. Lasserre, C. Prieur, and D. Henrion.“Nonlinear optimal control: numerical approximation via moments and LMI relaxations.in joint,” IEEE Conference on Decision and Control and European Control Conference, 2005.  J. Arthur, E. Bryson, and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control: Hemisphere Publ. Corp., Washington D.C, 1975.  W. Grimm and A. Markl, “Adjoint estimation from a direct multiple shooting method,” Journal of Optimization Theory and Applications, vol. 92,No. 2, pp- 263–283, 1997.  J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, ed. T. Edition, New York: Springer- Verlag, 2002.  O. v. Stryk and R. Bulirsch, “Direct and indirect methods for trajectory optimization,” annals of operations research, vol. 37, No. 1, pp-357–373,1992.  R. Fletcher, Practical methods of optimization:Unconstrained optimization, John Wiley & Sons, Ltd., Chichester, 1980.  P. E. Gill, W. Murray, and M. H. Wright, Practical optimization. : Academic Press, Inc., London-New York, 1981.  D. E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall, Englewood Cliffs, NJ: Dover Publications, USA, 1998.  D. S. Naidu, Optimal Control Systems: CRC Press, Idaho State University, USA,, 2003.  S. Subchan and R. Zbikowski, computational Optimal Control: Tools and Practice, United Kingdom: John Wiley & Sons, 2009.  K. Zhou, J. C. Doyle, and K. Glover, Robust and optimal control: Prentice Hall, Englewood Cliffs, NJ, 1996.  A. Ben-Tal and A. Nemirovski, “Robust Convex Optimization,” Mathematics of operations research, vol. 23, No. 4, pp-769-805, 1998.  A. Ben-Tal and A. Nemirovski, “Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications,” MPSSIAM Series on Optimization. MPS-SIAM, Philadelphia, 2001.  Z. K. Nagy and R. Braatz, “Open-loop and closedloop robust optimal control of batch processes using distributional and worst-case analysis,”Journal of Process Control, vol. 14, No. 4, pp-411–422, 2004.  M. Diehl, H. G. Bock, and E. Kostina, “An approximation technique for robust nonlinear optimization,” Mathematical Programming, vol. 107, No. 1-2, pp-213–230, 2006.  S. Seshagiria and H. K. Khalilb, “Robust output feedback regulation of minimum-phase nonlinear systems usingconditional integrators,” Automatica,vol. 41, No. 1, pp-43 – 54, 2005.  H. Nogami and H. Maeda, “Robust Stabilization of Multivariable High Gain Feedback Systems,” Transactions of the Society of Instrument and Control Engineers, vol. E, No. 1, pp- 83-91, 2001.  W. Maas and A. v. d. Schaft. “Singular nonlinear Hinf optimal control by state feedback,” in 33th IEEE conference on decision & control. Lake Buena Vista, USA, 1994.  A. Astolfi. “singular Hinf control,” in 33th IEEE conference on decision and control. Lake Buena Vista, USA, 1994.  R. Marino, et al., “Nonlinear Hinf almost disturbance decoupling,” systems & control letters,vol. 23, No. 3, pp- 159-168, 1994.
 A. Isidori, nonlinear control systems. third ed, Berlin: Springer Verlag, 1995.  A. Isidori, “Global almost disturbance decoupling with stability for non minimumphase single-input single-output nonlinear systems,” systems & control letters, vol. 28, No. 2, pp-115-122, 1996.  B. Achwartz, A. Isidori, and T. Tarn. “Performance bounds for disturbance attenuation in nonlinear nonminimum-phase systems,” in European Control Conference Brussels, Belgium, 1997.  M. Seron, et al., Feedback limitations in nonlinear systems: From Bode integrals to cheap control, University of California: Santa Barbara, 1997.  M. M. Seron, et al., “Feedback limitations in nonlinear systems: From bode integrals to cheap control,” IEEE Transactions on Automatic Control, vol. 44, No. 4, pp-829–833, 1999.  M. Krstic', I. Kanellakopoulos, and P. Kokotovic', Nonlinear and adaptive control design: John Wiley & Sons, 1995.  J. W. Helton, et al., “Singularly perturbed control systems using non-commutative computer algebra,” International Journal of Robust and Nonlinear Control, Special Issue: GEORGE ZAMES COMMEMORATIVE ISSUE, vol. 10, No. 11-12, pp- 983–1003, 2000.  H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, New York, Chichester, Brisbane, Toronto: Wiley-Interscience, a division of John Wiley & Sons, Inc, 1972a.  H. Kwakernaak and R. Sivan, “The maximally achievable accuracy of linear optimal regulators and linear optimal filters,” IEEE Transactions on Automatic Control, vol. 17, No. 1, pp-79-86, 1972b.  U. Shaked, Singular and cheap optimal control: the minimum and nonminimum phase cases, National Research Institute for Mathematical Sciences: Pretoria, Republic of South Africa, 1980.  A. Jameson and R. E. O'Malley, “Cheap control of the time-invariant regulator,” applied mathematics & optimization, vol. 1, No. 4, pp-337-354, 1975.  R. Sepulchre, et al., Constructive nonlinear control, Springer, Editor: London, 1997.  P. V. Kokotovic, H. K. Khalil, and J. O’Reilly, Singular Perturbations Methods in Control: Analysis and Design, New York: Academic Press, 1986.