Tracking control of second‐order chained form systems by cascaded backstepping

Abstract

SUMMARYA design methodology is presented for tracking control of second-order chained form systems. The methodology separates the tracking-error dynamics, which are in cascade form, into two parts: a linear subsystem and a linear time-varying subsystem. The linear time-varying subsystem, after the first subsystem has converged, can be treated as a chain of integrators for the purposes of a backstepping controller. The two controllers are designed separately and the proof of stability is given by using a result for cascade systems. The method consists of three steps. In the first step we apply a stabilizing linear state feedback to the linear subsystem. In the second step the second subsystem is exponentially stabilized by applying a backstepping procedure. In the final step it is shown that the closed-loop tracking dynamics of the second-order chained form system are globally exponentially stable under a persistence of excitation condition on the reference trajectory. The control design methodology is illustrated by application to a second-order non-holonomic system. This planar manipulator with two translational and one rotational joint (PPR) is a special case of a second-order non-holonomic system. The simulation results show the effectiveness of our approach.

Output Based Control for an Underactuated System: Experimental Results

“…In (Aneke et al, 2003b) a cascaded backstepping approach has been used to stabilize the origin of the error dynamics. In this approach, the stabilization problem for (2) is decoupled into two separate stabilization designs for the subsystems ∆ 3 and (∆ 1 , ∆ 2 ), respectively.…”

“…Following the lines of the cascaded backstepping approach from (Aneke et al, 2003b), the controller/observer design is also decoupled into a fourth-order and a second-order design for the respective subsystems:…”

“…Hence the dual control system (12) can be stabilized by the linear state feedback u 2 (t) =K 1 (t)z(t) proposed in (Aneke et al, 2003b) where nowK 1 (t) is K 1 (t) computed along α(t) = u 1d (−t). Basically this amounts to using (3) in the dual setting.…”

“…The control of nonholonomic systems has received a lot of attention in the last decades (Kolmanovsky and McClamroch, 1995;Lefeber et al, 2000;Aneke et al, 2003b;Behal et al, 2002;Ma et al, 2002). Here attention will be drawn to underactuated mechanical systems with acceleration constraints (Reyhanoglu et al, 1999).…”

Robust control for uncertain nonlinear systems with state‐dependent control direction

“…Section 3 proposes the robust controller integrating with a bounding function and a Nussbaum-type gain, followed by the convergence analysis in Section 4. Then, the proposed controller is integrated with the backstepping design procedure [1,18,19] to deal with the cascade nonlinear systems in Section 5. The illustrative examples are given in Section 6.…”