Towards -stability of discrete-time reset control systems via dissipativity theory

Abstract

This paper proposes conditions on input-output stability of discrete-time reset systems by using some key dissipativity properties. In the continuoustime setting, dissipativity of the base linear system is preserved under reset actions if the storage function is decreasing at reset times. Indeed, when the reset system is full reset, the dissipativity of the base linear system ensures the dissipativity of the reset system. However, in the discrete-time setting, this condition on the storage function is not enough to ensure the dissipativity of the base linear system. We define some dissipativity properties of discretetime reset systems and give an appropriate definition of the reset system in order to preserve the (Q, S, R)−dissipativity of the base linear system under reset actions. As a result, ℓ 2 -stability of feedback interconnected dissipative control reset systems is obtained.

LMI‐based reset unknown input observer for state estimation of linear uncertain systems

“…To prove (30), two possibilities should be considered: 1. After a jump, the error trajectory is in the flow set.…”

“…The proof of the Theorem 2 is valid for both finite and infinite N . If N is finite, from (29) and (30) the system states remains bounded and after the last jump since the nominal error dynamics is asymptotically stable, the error will go to zero asymptotically. Similarly, if N is infinite, since (29) and (30) hold for every t k , k → ∞, the system (25) is asymptotically stable.…”

“…Since then on, this field has been an appealing research field and a number of international research groups have been working actively in this area and it turns to an attractive control design method with a significant potential for practical applications [26,27,28,29]. The stability problem and performance issue of the reset control systems is addressed in [30,31]. Based on the well-posedness of reset instants, the necessary and sufficient conditions for existence and uniqueness of solutions is developed in [32].…”

Robust control and data‐driven tuning of a hybrid integrator‐gain system with applications to wafer scanners

“…Notice that, for industrial motion systems nonparametric models, ie, frequency response functions (FRFs), are highly preferred over parametric models, because, in the wafer scanner industry, FRFs are quickly measured with sufficient accuracy. Alternatives in the given context are given by other works [20][21][22] using multiple Lyapunov functions, the works of Carrasco et al 23,24 using passivity and dissipativity concepts, and the works of Yu et al 25 and Briant 26 considering dwell-time related stability conditions. As a possible disadvantage, a circle-criterion-like approach may render a rather conservative estimate on closed-loop stability, providing only limited direction toward robust performance and parameter tuning.…”

Stability analysis of periodic triggering reset control systems

“…With zero-crossing resets, many results are obtained in the literature that including the H β -condition guaranteeing quadratic stability of closed loop system [5], delaydependent and delay-independent stability of reset systems [6,7], passivity and dissipativity properties of reset control systems [8,9], practical applications [10,11], more results can be found in [12]. A class of novel reset model under the hybrid systems framework of [13] was proposed in [14], which allows flows and jumps on more complicated closed sets.…”