Global stabilization of a class of upper‐triangular systems with unbounded or uncontrollable linearizations

Abstract

SUMMARYIn this paper, we consider the problem of global stabilization for a class of upper-triangular systems which have unbounded or uncontrollable linearizations around the origin. The explicit formula of the control law is designed in two steps: First, we use the generalized adding a power integrator technique to design a homogeneous controller which locally stabilizes the upper-triangular systems. Then, we integrate a series of nested saturation functions with the homogeneous controller and adjust the saturation level to ensure global asymptotic stability of the closed-loop systems. Owing to the versatility of the generalized adding a power integrator technique, our controller not only can be used to stabilize more general uppertriangular systems by relaxing the current conditions used in existing results, but also is able to lead to a stronger result of finite-time stabilization under appropriate conditions.

Global asymptotic stabilization of stochastic feedforward nonlinear systems with input time-delay

“…Step 1: Choosing ξ 1 = η 1 and V 1 (η 1 ) = 1 4 ξ 4 1 , by (3) and (6), we have LV 1 = κξ 3 1 η 2 . The first virtual controller η * 2 = −c 11 ξ 1 =: −α 1 ξ 1 , c 11 > 0,…”

“…As discussed in stochastic feedforward references [12,14,15] and the deterministic feedforward references [3,18,25,[27][28][29], Assumption 1 is a general and frequently used condition.…”

“…On the other hand, many practical physical devices can be modeled by equations with the feedforward form, such as the cart-pendulum system in [4], the ball-beam with a friction term in [19] and the PVTOL aircraft in [21]. Therefore, the stabilization problem of feedforward systems rapidly became an interesting research field and received many important results, see, e.g., [1,[3][4][5][6][7]9,18,24,25] and the references therein.…”

Global smooth stabilization of a class of feedforward systems under the framework of generalized homogeneity with monotone degrees

“…However, compared with the global stabilization results for lower-triangular nonlinear systems, fewer results exist in the literature for solving the global stabilization problem of upper-triangular systems (1.1), especially in the higher-order case. As a matter of fact, the global stabilization problem of higher-order upper-triangular systems is very challenging and has only been considered in very few works, e.g., [5,16,2,3].…”

“…In the existing literature, the nested saturation control method [1,[15][16][17]2,3], the forwarding method based on Lyapunov functions [7,8] and the homogeneous domination approach [9,11] are the main strategies to solve the control problems for these upper-triangular systems. In a recent work [3], the authors proposed a new design method to solve the global stabilization problem of Eq.…”