Bounded backstepping control and robustness analysis for time-varying systems under converging-input-converging-state conditions

Abstract

We provide new bounded backstepping results that ensure global asymptotic stability for a large class of partially linear systems with an arbitrarily large number of integrators. We use a dynamic extension that contains one artificial delay, and a converging-input-converging-state assumption. When the nonlinear subsystem is control affine, we provide sufficient conditions for our converging-input-converging-state assumption to hold. We also show input-to-state stability with respect to a large class of model uncertainties, and robustness to delays in the measurements of the state of the nonlinear subsystem. We illustrate our result using an example that is beyond the scope of classical backstepping.

Stabilization and Robustness Analysis for a Chain of Saturating Integrators With Imprecise Measurements

“…A significantly different backstepping design was proposed in [13] and [14]. It used artificial pointwise delays in the control, which circumvent the problem of determining Lie derivatives of the fictitious controls.…”

“…The advantages of [13] motivate the present work, which adapts the approach from [13] to a control problem for a chain of saturating integrators for an important dynamics with outputs that arises in the vision based [6] landing of aircraft; see (1)-(2) below. A key difficulty is that only imprecise measurements of the two first states are available, so we cannot apply the semi-global or regional stability results mentioned above, nor [13] or its extension in [14]. Thus, we propose a new control, which extends [13] and [14].…”

“…A key difficulty is that only imprecise measurements of the two first states are available, so we cannot apply the semi-global or regional stability results mentioned above, nor [13] or its extension in [14]. Thus, we propose a new control, which extends [13] and [14]. Our theory is inspired by forwarding theory in [15].…”

Stabilization and Robustness Analysis for a Chain of Saturating Integrators Arising in the Visual Landing of Aircraft

“…This paper continues our search for more effective feedback stabilization methods for cases where only imprecise output measurements are available for use in the control. This led to our novel backstepping approach in [14] and [15] where pointwise delays are present in the feedback even if current output values are available, and then our work [11], [12] that uses the preceding backstepping approach to solve a feedback control problem for a chain of saturated integrators with imprecise output measurements using an unbounded control. In the present work, we use our backstepping approach to solve a stabilization problem for a chain of saturating integrators with imprecise measurements using dynamic output feedback controls of arbitrarily small amplitude; see Section II for more on the potential advantages of this work as compared with the method in [11], [12].…”

“…The backstepping designs proposed in [14] and [15] are significantly different from prior backstepping methods, because they circumvent the problem of determining Lie derivatives of the fictitious controls by introducing artificial delays in the control (which are called artificial because they are present even if current state values are available for measurement). The artificial delays approach relaxes the smoothness requirement on the fictitious control that was present in previous backstepping approaches.…”