# Neural Robust Control for a Mobile Agent Leader-Follower System
## 3 Neuro-Robust Control
### Stability Analysis  
From (27), the RHONN identifier for the unknown parameters, taking into account input control and modeling error, can be rewritten as  
\[\dot{\underline{z}}=\Lambda\hat{\underline{z}}(t)+W\zeta(z(t))+\
u_{\sigma}(t), \tag{29}\]  
where \(\
u_{\sigma}(t)\in\mathbb{R}^{n}\) is the modeling error, and \(W\in\mathbb{R}^{n\times L}\) is a matrix with values of the online estimated neural network weights, which minimizes \(\
u_{\sigma}(t)\). We define trajectory tracking error and identification error, respectively, as  
\[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\] \[e(t) = z(t)-\hat{\underline{z}}(t). \tag{31}\]  
Assumption 1 [18]. For every \(w_{j}\in W\), a bounded set, the system (29) is bounded for every bounded state \(\hat{\underline{z}}\). Assumption 2 [18]. There exist unknown constant weights \(W^{*}\in L_{\infty}\) such that the plant (21) and (22) is described without modeling error, by the neural network of the form  
\[\hat{\underline{z}}(t)=\Lambda z(t)+W^{*}\zeta(z(t)). \tag{32}\]  
In (32), we consider \(\hat{\underline{z}}(t)\triangleq\dot{\underline{z}}(t)\). In order to analyze the error dynamics between the identified and the linearized system, we first obtain the identification error from (32) and (27) as in [14]  
\[\dot{e}(t) = \dot{\underline{z}}(t)-\dot{\underline{z}}(t) \tag{33}\] \[= \Lambda e(t)+\tilde{W}\zeta(z(t)),\]  
where \(\tilde{W}=W^{*}-W\). As in [14], the weight adaptation law is selected as  
\[tr\Big{\{}\dot{\tilde{W}}^{T}\tilde{W}\Big{\}}=-\gamma e(t)^{T}\tilde{W} \zeta(z(t)), \tag{34}\]  
which can be written element by element as  
\[\tilde{w}_{ij}=-\gamma e(t)^{T}\zeta_{j}(z(t)) \tag{35}\] \[i=1,2,\ldots,n\;\;j=1,2,\ldots,L.\]  
The adaptation law (35) guarantees that \(\dot{e}(t)\to 0\) as \(t\to\infty\), and also \(\tilde{W}(t)\to 0\) as \(t\to\infty\). On the other hand, we can write the expression for the derivative of the trajectory tracking error:  
\[\dot{z}_{e}(t) = \dot{\underline{z}}(t)-\dot{\underline{z}}_{r}(t) \tag{36}\] \[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]0where \(\tilde{W}_{r}=W^{*}-W_{r}\). Now, we are prepared to demonstrate asymptotic stability. **Theorem 1**. : _For the linearized parametric system (21) and (22) under Assumptions 1 and 2, using the online learning control laws (25) and (26) the tracking error (36) is asymptotically stable at \(z_{e}(t)=0\)._  
Proof. : The trajectory tracking problem for (23) reduces to a stabilization problem for (36), where the dynamics of identification error are included. Note that \([z_{e}(t),e(t),\tilde{W}]=[0,0,0]\) is an equilibrium point of the undisturbed autonomous system. For stability analysis, the candidate Lyapunov function is defined as  
\[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]1  
which fulfills the needed characteristics of radially unbounded and \(V(0)=0\) of Barbashin-Krasovskii theorem [28], so that we demonstrate regulation for each of the three error dynamics (trajectory tracking error, identification error, and weight matrix error). The time derivative of (37) along the trajectories of (36) is  
\[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]2 \[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]3 \[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]4  
Then, we substitute the learning law (34) with \(\gamma=2\) into (38) and make a change of variable from (31) to obtain  
\[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]5 \[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]6 \[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]7 \[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]8 \[z_{e}(t) = z(t)-z_{r}(t), \tag{30}\]9  
Now, if we factorize the elements on the right-hand side of (39),  
\[e(t) = z(t)-\hat{\underline{z}}(t). \tag{31}\]0 \[e(t) = z(t)-\hat{\underline{z}}(t). \tag{31}\]1  
If we prove that the right-hand side of (40) is negative, \(V_{T}\) is decreasing and bounded from below by \(V_{T}(0)\). Thus, directly from Assumption 1, (25) and (26), one can conclude that \(z_{e}(t),w_{rij}(t),u\in L_{1}\), where \(L_{1}\) is the space of Lebesgue integrable functions; this means that weights remain bounded.