On stabilization of canonical nonlinear systems in a special case
Abstract
The paper addresses the stabilization problem for canonical nonlinear systems, which in the triangular case correspond to systems in p--normal form. These systems are inherently nonlinear because they are not feedback linearizable, and their linear approximation is degenerate and does not characterize the stability of the equilibrium at the origin. In the triangular case, backstepping is a standard feedback design tool due to its recursive nature. Backstepping provides a powerful framework for stabilizing a wide class of nonlinear systems in strict-feedback form. However, this recursive approach typically leads to feedback controllers of increasing complexity as the system dimension grows. This motivates the investigation of simpler control design methods. Several notable results on non-recursive stabilization have been obtained in recent years. The classes of polynomial control laws have been constructed for the case of strictly decreasing exponents of power terms in the right-hand side of the system. In particular, it has been shown that the system can be asymptotically stabilized by a linear feedback control law in the strictly decreasing case. Moreover, control coefficients can be chosen as arbitrary positive numbers. It is also established that it is possible to achieve stabilization in a certain special case of non-strictly decreasing exponents under additional conditions on the control coefficients. In this work, we solve the linear stabilization problem in a previously unexplored case of non-strictly decreasing exponents for a three-dimensional system. We propose constructive method of finding conditions for the control coefficients to guarantee asymptotic stabilization. Our approach is based on the Lyapunov function method. We also use stabilization through nonlinear approximation to generalize our result. The effectiveness of the proposed approach is demonstrated and confirmed by numerical examples.
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References
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