Event-triggered Control of Uncertain Nonlinear Discrete-time Systems with Extended State Observer

This paper is dedicated to extended state observer-based event-triggered model free iterative learning control (ET-MFILC) for nonlinear systems with disturbances. A modiﬁed MFILC scheme is proposed by using the estimated uncertainties for the disturbed system. With the help of the estimated errors, an iterative extended state observer (IESO) is constructed to estimate the unknown uncertainties. New triggering mechanism integrated true tracking errors, the estimated errors and the estimated uncertainties is designed for multiple inputs and multiple outputs (MIMO) systems. Suﬃcient conditions are proposed to make the resultant ET-MFILC tracking error systems be uniformly ultimately bounded. An illustrative example is presented to demonstrate the eﬀectiveness of the proposed scheme.


Introduction
Since network control systems are affected by the limited bandwidth, packet losses and time-delay occur frequently in practical applications [1][2][3][4]. Considering the negative influence of these phenomena to system performance, event-triggered control (ETC) has been adopted widely as an effective method to save the communication resources [5][6][7][8][9][10][11]. Specially, different from the event-triggered mechanism in [5] requiring special hardwares to measure the states of system continuously, [6,7] put forward a periodic ETC strategy by combining ETC and traditional periodic sampling data control. In [8], a self-triggered control (STC) scheme is presented instead of checking the triggering condition passively.
Employing the estimated errors to adjust the triggering condition, this STC is further extended to an adaptive paradigm by [9]. Contrasted to these static triggering rules, a novel dynamic triggering mechanism is supplied in [10] via introducing an internal dynamic variable.
Note that the above mentioned ETC methods require systems with accurate model. However, this requirement is unrealistic because of complex production process. Therefore, how to construct controllers independent of system models is a meaningful topic in control community. Data-driven control (DDC) refers that the design of controller employs system inputs and outputs merely. Based on this concept, there are some DDC methods called differently, such as modelfree adaptive control (MFAC) [11,12], unfalsified control (UC) [13,14], virtual reference feedback tuning (VRFT) [15,16]. Among them, MFAC is discussed in [17][18][19][20]. In particular, by introducing pseudo partial derivative (PPD) parameter and dynamic linearization technology, an equivalent data model is proposed in [17]. Motivated by this idea, [18] presents an ETC strategy based on MFAC by using the tracking error.

Problem Formulation
Consider the repetitive disturbed MIMO nonlinear system as below: where y(Ξ, δ) ∈ R m denotes the system output, u(Ξ, δ) ∈ R m denotes the system input and d(Ξ, δ) ∈ R m denotes the disturbance. The iteration direction is presented by Ξ ∈ {0, 1, 2...} and δ ∈ {0, 1, ...T } is the time domain index. c y and c u are the unknown order. Nonlinear function f (·) is continuously differentiable.
Some assumptions and lemmas are presented for the following derivation.

Remark 1 (2) bridges the relationship from inputs and
uncertainties to outputs without practical mechanism meaning. As [16], the subsequent derivation depends on ⊖(Ξ, δ) with slow change in the iterative direction. Refering to the MFILC method proposed in [17], a MFILC scheme for disturbed MIMO systems is modified as below.

Main Results
In this part, the ET-MFILC scheme based on IESO for MIMO systems is proposed firstly. Then the boundness of⊖(Ξ, δ) and tracking error e(Ξ, δ) under the proposed scheme is established by means of the Lyapunov method.
. Therefore, the event-triggered iterative learning parameter estimation law is presented as below: In the ET-MFILC scheme, iterative learning control law (4) is modified as: By applying Lyapunov theory to the tracking error on the iterative domain, the event-triggering rule is proposed for MIMO system as below: . D(·) is a conditional function, namely, if |e(Ξ, δ + 1)| > H e , then D(e 2 ET (Ξ, δ + 1)) = e 2 ET (Ξ, δ+1); Otherwise, D(e 2 ET (Ξ, δ+1)) = 0 in which H e is a predetermined positive scalar.
With the above preparations, the ET-MFILC scheme is summarized as below.

Remark 3 One can sees from
Next, the boundness of⊖(Ξ, δ) is established in Theorem 1.
Combining Case I and Case II, it means that the tracking error e(Ξ, δ) under (10)-(13) is ultimate bounded.

Numerical example
Consider the following MIMO nonlinear system in [14]: where It is seen from Fig. 1