An Approach to Control Design for Nonlinear Systems via Two-stage Formal Linearization and Two-type LQ Controls

In this paper we consider a nonlinear control design for nonlinear systems by using two-stage formal linearization and twotype LQ controls. The ordinary LQ control is designed on almost linear region around the steady state point. On the other region, another control is derived as follows. This derivation is based on coordinate transformation twice with respect to linearization functions which are defined by polynomials. The linearized systems can be made up by using Taylor expansion considered up to the higher order. To the resulting formal linear system, the LQ control theory is applied to obtain another LQ control. Finally these two-type LQ controls are smoothly united to form a single nonlinear control. Numerical experiments indicate that this control show remarkable performances for a nonlinear system.

Authors:

• Kazuo Komatsu
• Hitoshi Takata

Keywords:

• Formal Linearization
• LQ Control
• Nonlinear Control
• Taylor Expansion
• Zero Function.

References:

[1] R. W. Brockett, "Feedback Invariants for Nonlinear Systems," in Proc. of
IFAC Congress, Helsinki, 1978, pp.1115-1120.
[2] B. Jakubczyk and W. Respondek, "On the Linearization of Control
Systems," Bull. Acad. Polon. Sci. Ser. Math., Vol.28, pp.517-522, 1980.
[3] A. J. Krener, "Approximate Linearization by State Feedback and Coordinate
Change," Systems and Control Letters, Vol.5, pp.181-185, 1984.
[4] R. Marino, "On the Largest Feedback Linearizable Subsystem," Systems
and Control Letters, Vol.6, pp.345-351, 1986.
[5] R. R. Kadiyala, "A Tool Box for Approximate Linearization on Nonlinear
Systems," IEEE Control Systems, Vol.13, No.2, pp.47-57, 1993.
[6] A. Ishidori, Nonlinear Control Systems, An Introduction, 3rd ed., Berlin:
Springer-Verlag, 1995.
[7] H. K. Khalil, Nonlinear Systems, 3rd ed., New Jersey: Prentice Hall, 2002.
[8] W. T. Baumann and W. J. Rugh, "Feedback Control of Nonlinear Systems
by Extended Linearization," IEEE Trans., AC-31, 1, pp.40-46, 1986.
[9] H. Takata," Transformation of a Nonlinear System into an Augmented
Linear System," IEEE Trans. on Automatic Control, Vol.AC-24, No.5,
pp.736-741, 1979.
[10] K. Komatsu and H. Takata, "A Computation Method of Formal Linearization
for Time-Variant Nonlinear Systems via Chebyshev Interpolation,"
in Proc. of the IEEE CDC, Las Vegas, 2002, pp.4173-4178.
[11] K. Komatsu and H. Takata, "Computer Algorithms of Formal Linearization
and Estimation for Time-Variant Nonlinear Systems via Chebyshev
Expansion," Journal of Signal Processing, Vol.7, No.1, pp.23-29, 2003.
[12] K. Komatsu and H. Takata, "A Formal Linearization for a General Class
of Time-Varying Nonlinear Systems and Its Applications," IEICE Trans.
, Vol.E87-A, No.9, pp.2203-2209, 2004.
[13] K. Komatsu and H. Takata, "Design of Formal Linearization and
Observer for Time-Delay Nonlinear Systems," in Proc. of NOLTA, Italy,
2006, pp.907-910.
[14] A. P. Sage and C. C. White III, Optimum Systems Control, 2nd ed.,
New Jersey:Prentice Hall, 1977.