## 3 Robust Control Design of the FLHBR System
: _A nonlinear control system with the external disturbance input \(\overrightarrow{u}_{ip}\) is called to have the almost disturbance decoupling property if_  
_(a)_ _The nonlinear control system possesses the input-to-state stable performance._  
_(b)_ _The following two inequalities hold:_  
\[\Big{|}u_{op\_i}(t)-u_{op\_track}^{i}(t)\Big{|}\leq\gamma_{11}\Big{(}\| \dot{\overrightarrow{x}}_{se}(t_{0})\|,t-t_{0}\Big{)}+\frac{1}{\sqrt{\gamma_{ 22}}}\gamma_{33}\Bigg{(}\sup_{t_{0}\leq\tau\leq t}\|\dot{\overrightarrow{u}}_ {ip}(\tau)\|\Bigg{)} \tag{120}\]  
_and_  
\[\int\limits_{t_{0}}^{t}\Big{[}u_{op\_i}(\tau)-u_{op\_track}^{i}(\tau)\Big{]}^ {2}d\tau\leq\frac{1}{\gamma_{44}}\Bigg{[}\gamma_{55}\Big{(}\|\dot{ \overrightarrow{x}}_{se}(t_{0})\|\Big{)}+\int\limits_{t_{0}}^{t}\gamma_{33} \bigg{(}\|\dot{\overrightarrow{u}}_{ip}(\tau)\|^{2}\bigg{)}d\tau\Bigg{]} \tag{121}\]  
_where \(\dot{\overrightarrow{x}}_{se}(t_{0})\) denotes the initial state of the control system, \(\gamma_{11}\) is KL-class function, \(\gamma_{33}\), \(\gamma_{55}\) are K-class functions and \(\gamma_{22}>0,\gamma_{44}>0\)[25; 33]._  
It is worth mentioning that the aforementioned definition (hypothesis) of the almost disturbance decoupling property is more stringent in many ways when compared with the earlier definitions shown first for linear control engineering systems and then inherited for nonlinear control systems which are needed for closed-loop feedback systems:  
(Case 1) input-to-state stable performance when the initial state of control system is zero;  
(Case 2) globally asymptotical stability of the equilibrium point when the external disturbance input is set to be zero;  
Moreover, the above definition of the almost disturbance decoupling property possesses three features as follows [24]:  
The first feature of the above definition is the demand of input-to-state stable performance. In fact, while for linear control systems the input-to-state stable performance is implied by (120), this is not met for nonlinear control systems. The second feature is the appearance of function r33 for (120). While for earlier definitions the function is set to be r33 (x) = x, in fact, this flexibility is required only for special cases including linear cases. The third feature lies in the input-to-state stable performance that needs the asymptotical stability for the equilibrium point corresponding to the tracking signal and the origin point. As we shall see, for linear control systems, once the stabilization problem is addressed, the tracking problem is solved, this is not so for nonlinear ones. Based on the more stringent definition of the almost disturbance decoupling property, the robust ness of the proposed feedback linearization approach is stronger. Moreover, according to \(\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{m}^{2}}\leq|a_{1}|+|a_{2}|+\cdots+|a_{m}|\), it is easy to obtain  
\[\int\limits_{t_{0}}^{t}\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{m}^{2}}d\tau\leq \int\limits_{t_{0}}^{t}|a_{1}|+|a_{2}|+\cdots+|a_{m}|d\tau\]  
and  
\[\int\limits_{t_{0}}^{t}\sqrt{\left[u_{op\_1}(\tau)-u_{op\_track}^{1}(\tau) \right]^{2}+\cdots+\left[u_{op\_m}(\tau)-u_{op\_track}^{m}(\tau)\right]^{2}}d\tau\]  
Therefore, we can conclude the fact that the root mean square error can be implied by the almost disturbance decoupling condition (121). From (86), we obtain  
\[\left[\begin{array}{c}\dot{T}_{l\_1}^{i}-\dot{u}_{op\_track}^{i}\\ \dot{T}_{l\_d_{\ell},i}-\dot{u}_{op\_track}^{i(d_{\ell},i-1)}\\ +\left[ \begin{array}{c}\sum\limits_{j=1}^{p}\left(\frac{\partial}{\partial \dot{x}_{x}}L_{f}^{1-1}u_{op\_i}\right)\stackrel{{\rightarrow}}{{q} }_{noise\_j}^{*}\left(\Omega_{un\_j}+\Omega_{n\_j}\right)\\ \sum\limits_{j=1}^{p}\left(\frac{\partial}{\partial\dot{x}_{x}}L_{f}^{d_{ \ell},i-1}u_{op\_i}\right)\stackrel{{\rightarrow}}{{q}}_{noise\_ j}^{*}\left(\Omega_{un\_j}+\Omega_{n\_j}\right)\end{array} \right],d_{\ell,j}=2,\,i=1,\cdots,5 \tag{122}\]  
From (106), (108) and (122), we obtain  
\[\left[