Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system

Abstract

This paper studies a specific mechanical example that is representative of a class of underactuated, weakly coupled, unstable mechanical systems that are exceptionally difficult to stabilize. In particular, systems in this class are not stabilizable using static smooth feedback but are stabilizable using nonsmooth feedback. Although similar purely theoretical developments have been previously presented, we emphasize the physical basis and physical implications of the theoretical conclusions in the context of a specific example. The development in this paper is limited to a specific physical example, but the approach is applicable to a wide class of underactuated, weakly coupled, unstable mechanical systems.

Global uniform input-to-state stabilization of large-scale interconnections of MIMO generalized triangular form switched systems

“…The following three reasons justify these efforts: first, there are physical examples of the triangular systems in the singular case [17,29], second the backstepping approach appears to be applicable to the triangular form systems with singular inputoutput links [17,21,27,34,35], and third some coordinate-free criteria which extend the Respondek-Jakubczyk result were obtained in some important cases of triangular systems with uncontrollable linearization [5,28]. This promises a way to extend most results of the above-mentioned classical theory but such a theory has not been completed yet.…”

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

“…In the case when p i ¼ 1; i ¼ 1; …; n, system (1.1) reduces to a class of well-known feedforward systems including some practical systems such as the cart-pendulum system [6], the ball and beam system [1], the TORA system [14], etc. In the higher-order case (i.e., p i ≥1; i ¼ 1; …; n), system (1.1) can be regarded as the upper-triangular counter-part of a class of higher-order lower-triangular systems including the mechanical system with a nonlinear spring studied in [13]. 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.…”

Output feedback exponential stabilization of uncertain chained systems

“…In the past decade, many researchers have been attracted to the search for new control strategies. So far, several interesting solutions, such as open-loop periodic steering control, smooth or continuous time-varying feedback control, and discontinuous feedback control, have been found to overcome the abovedescribed obstruction in stabilizing a nonholonomic system, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

“…However, it has been known that for nonholonomic system a smooth time-varying state feedback law can be applied to achieve asymptotic stabilization but fails to meet the requirement of exponential convergence, while a continuous time-varying and/or discontinuous feedback law guarantees the exponential regulation of nonholonomic systems in chained form but fails to achieve asymptotic stabilization [3,4,[6][7][8]11,12,15,17,18,22]. More recently, Marchand and Alamir [19] obtained Lyapunov stability and exponential rate of convergence in the absence of disturbances.…”