Output feedback based event-triggered sliding mode control for delta operator systems

TL;DR

This paper tackles the control of delta operator systems under high sampling rates where discrete-time representations become numerically ill-conditioned. It introduces a two-rate MRSE framework with a new observability matrix $C_o^\delta$ that converges to the continuous-time observability as $\Delta\to0$, enabling accurate output feedback. A sliding mode controller is designed using MRSE estimates, with a proven condition $\epsilon> d_m+f_m$ ensuring quasi-sliding motion, and the stability analysis yields bounded trajectories within a predicted region. To further reduce computational and communication load, the authors integrate MRSE with event-triggered control, deriving a triggering rule that avoids Zeno behavior and yields a practical quasi-sliding band; simulations on a ball-and-beam system and a numeric example demonstrate improved observability, reduced update frequency, and robust performance under matched disturbances.

Abstract

In this paper, we present an output feedback based design of event-triggered sliding mode control for delta operator systems. For discrete-time systems, multi-rate output sampling based state estimation technique is very useful if the output information is available. But at high sampling rates, the discrete-time representation of the system using shift operator becomes numerically ill-conditioned and as a result, the observability matrix becomes singular as the sampling period tends to zero. Here, a new formulation of multi-rate state estimation (MRSE) for a small sampling period is presented. We first propose a new observability matrix and then discuss its relationship with the observability matrix defined in the conventional sense. For the delta operator system with matched uncertainty, we have presented the design of MRSE based sliding mode control (SMC). Additionally, to make the control efficient in terms of resource utilization, MRSE based event-triggered SMC is proposed. The absence of Zeno phenomenon is guaranteed as the control input is inherently discrete in nature. Finally, the effectiveness of the proposed method is illustrated through numerical simulations, considering a ball and beam system and a general linear system as a numerical example.

Figures (11)

• Figure 1: System architecture
• Figure 2: State trajectories
• Figure 3: Sliding trajectory
• Figure 4: Control input
• Figure 5: Evolution of inter-event time

Theorems & Definitions (17)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• Definition 1
• Remark 4
• Theorem 1
• Remark 5
• Theorem 2
• Proposition 1