# Model-Free Adaptive Control Based on Fractional Input-Output Data Model
## 1 Introduction  
Robust tracking and regulation in highly uncertain discrete-time nonlinear systems is a non-trivial and interesting problem. This paper proposes a fractional input-output data model that relies on a dynamic linearization between the fractional-order variation of the input function and the fractional-order variation of the output signal, in contrast to classical approaches that consider just local information related to first-order variations [1]. An intrinsic advantage of the proposed method is that the model is still valid when there are instantaneous changes in the output signal, while the variation in the controller is null. Another advantage is less sensitivity to measurement and input noise because more data are considered to synthesize the equivalent model. These features can be provided by the inherent memory properties of fractional-order operators [2; 3; 4; 5]. Modern engineering systems are composed of different elements with uncertain models. This motivates the study of model-free approaches that rely on less restrictive assumptions on the system response [6; 7; 8]. In addition, computing the controller during the implementation time is of preponderant importance to assure a good degree of robustness and performance [9]. In recent years, the paradigm of data-driven control has been considered for the case of complex and uncertain dynamic systems [10; 11; 12; 13; 14]. In particular, the concept of compact form dynamic linearization was investigated in [9], which proposed a data-driven model that relates the first-order variations of input and output signals by means of an instantaneous gain, also called the pseudo-partial derivative (PPD). The validity of the proposed model relies on assuring that the PPD is finite for any time instant, thus, the equivalent model is undefined if there is any variation of the output but the input remains at a constant value. In order to estimate the PPD, a fuzzy logic formulation was proposed in [15], where the universal approximation properties of fuzzy systems are considered to compute the equivalent model. Full and partial forms of dynamic linearizations were investigated in [16; 17], accounting for more data to produce a more accurate model representation. Nevertheless, this was done at the price of adjusting more parameters. A popular scheme for model-free implementation is, for instance, the proportional-integral-derivative (PID) control, which is very appealing in industrial applications due to its simplicity and reliability [18; 19]. In addition, the increasing in complexity of modern automation applications requires more robust and more flexible techniques to operate in uncertain conditions, thus, some advanced versions of the PID control consider sliding mode [20; 21], fractional-order [22], fuzzy logic [23], and adaptive schemes [24]. However, a structure for the dynamical model is still necessary to assure, at least analytically, an acceptable closed-loop response. In this paper, the inherent memory properties of fractional-order operators [25; 26] are considered to produce a discrete-time equivalent input-output data model, where the fractional PPD (FPPD) is computed by means of fuzzy rules emulating networks. The advantages of the proposed scheme consist of accounting for the history of the process, while searching for a single parameter through a fuzzy aggregation scheme. An interesting property of the proposed algorithm is that the model is still valid when the output changes in the presence of a locally constant output signal. It is important to note that some neural network structures can be considered [27; 28; 29], as an alternative to the fuzzy logic approach, to actively estimate the FPPD during implementation time. Nonetheless, the user experience on the system behaviors can be of great value when designing the fuzzy inference system [30; 31]. The remaining of this paper is organized as follows. The next section presents the problem formulation. The fractional input-output equivalent model is presented in Section 3.