This chapter focuses on the adaptive neural control of a class of uncertain multi-input multi-output (MIMO) nonlinear time-delay non-integer order systems with unmeasured states, unknown control direction, and unknown asymmetric saturation actuator. The design of the controller follows a number of steps. Firstly, based on the semi-group property of fractional order derivative, the system is transformed into a normalized fractional order system by means of a state transformation in order to facilitate the control design. Then, a simple linear state observer is constructed to estimate the unmeasured states of the transformed system. A neural network is incorporated to approximate the unknown nonlinear functions while a Nussbaum function is used to deal with the unknown control direction. In addition, the strictly positive real (SPR) condition, the Razumikhin lemma, the frequency distributed model, and the Lyapunov method are utilized to derive the parameter adaptive laws and to perform the stability proof.
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In the control literature, strict-feedback, pure-feedback and stochastic nonlinear systems are frequently encountered (Yu et al., 2016; Shi, 2015; Cui et al., 2015; Yu & Du, 2011; Chen et al., 2010). In addition, there exist several real systems described by non-integer order differential equations such as (Boulkroune et al., 2016a): regular variation in thermodynamics, viscoelastic systems, dielectric polarization, electrical circuits, biological and financial systems, electromagnetic waves, heat conduction in a semi-infinite slab, robotics, biophysics, and so on. It is worth noting that integer order systems are a special case of non-integer order systems (Boulkroune et al., 2016a; Miao & Li, 2015; Shi, 2015a; Bouzeriba et al.,2016a; Boulkroune et al., 2012a; Boulkroune et al., 2012b; Boulkroune et al., 2014a; Boulkroune & M'Saad, 2011; Boulkroune et al., 2014b; Sui et al., 2015; Iqbal et al., 2015; Wang et al., 2015a; Wang et al., 2015b; Liu et al., 2015a; Du & Chen, 2009; Mizumoto et al., 2015; Shahnazi, 2016; Wei et al., 2015; Lan & Zhou, 2013; N'Doye & Laleg-Kirati, 2015; Li & Wang, 2014; Vargas-De-León, 2015; Chen & Chen, 2015; Stamova & Stamov, 2014; Stamova & Stamov, 2013). Additionally, it has been proved that the fractional calculus is an excellent mathematical tool for accurate descriptions of memory and hereditary properties of several materials and processes (Boulkroune et al., 2016a). Compared with integer order systems, there is very little research dealing with multi-input multi-output (MIMO) fractional-order systems (Yu et al., 2016; Stamova & Stamov, 2013; Lan et al., 2012; Zhang et al., 2015; Ladaci et al., 2009; Farges et al., 2010; Lazarević & Spasić, 2009; Lim et al., 2013; Domek & Dworak, 2016; Luo & Liu, 2014; Liu & Jiang, 2013a; Li et al., 2015a; Li et al., 2015b; Li et al.,2015c; McGarry et al., 2007; Rădac et al., 2014; Yacoub et al., 2014; Yan et al., 2016; Liu et al., 2016a; Gao & Liu, 2016; Liu & Tong, 2013b; Liu & Tong, 2014; Liu & Tong, 2015b; Bouzeriba et al., 2016b; Chen et al., 2016; Wang et al., 2017; Ibeas, & de la Sen, 2007; Tabatabaei & Arefi, 2016; Arefi et al., 2014a; Arefi et al., 2014b; Petras, 2011; Boulkroune et al., 2010; Ioannou & Sun, 1996; Boulkroune et al., 2016b; Yue & Li, 2012; Liu et al., 2016b). This fact can be explained by the specificity of MIMO systems and the difficulties with the extension of the approaches employed for integer order systems to fractional ones (Boulkroune et al., 2012a; Shahnazi, 2016; Domek & Dworak, 2016; Li et al., 2015b; Yan et al., 2016; Liu & Tong, 2014; Tabatabaei & Arefi, 2016).