# Neural Robust Control for a Mobile Agent Leader-Follower System
## 2 Materials and Methods
### Recurrent High-Order Neural Networks Estimation  
Let us consider a neural network with \(n\) neurons and \(m\) inputs for our estimation purposes. The state of each neuron is governed by a differential equation of the following form:  
\[\dot{x}_{i}=-\lambda_{i}x_{i}+\mu_{i}\left[\sum_{k=1}^{L}w_{ik}\prod_{j\in I_{k} }y_{j}^{d_{j}(k)}\right], \tag{7}\]  
where \(x_{i}\) is the state of the \(i\)th neuron; \(L\) is the number of high-order product terms; \(\{I_{1},I_{2},\ldots,I_{L}\}\) is a collection of \(L\) unordered subsets of \(\{1,2,\ldots,m+n\}\); \(\lambda_{i},\mu_{i}>0\), \(\forall i=1,2,\ldots,n\) are real coefficients; \(w_{ik}\) are the adjustable real coefficients of the neural network; \(d_{j}(k)\) are non-negative integers; and \(y\) is a vector constructed from the inputs to each neuron, defined as  
\[y=\begin{bmatrix}y_{1}\\ \vdots\\ y_{n}\\ y_{n+1}\\ \vdots\\ y_{n+m}\end{bmatrix}=\begin{bmatrix}S(x_{1})\\ \vdots\\ S(x_{n})\\ \vdots\\ S(u_{m})\end{bmatrix}, \tag{8}\]  
where \(u=[u_{1}u_{2}\ldots u_{m}]^{T}\) is the vector of external inputs to the neural network. The function \(S(\bullet)\) is a smooth, monotonically increasing sigmoid [14].  
We define a vector of dimension \(L\) as follows:  
\[\zeta=\begin{bmatrix}\zeta_{1}\\ \xi_{2}\\ \vdots\\ \xi_{L}\end{bmatrix}=\begin{bmatrix}\prod_{j\in I_{1}}y_{j}^{d_{j}(1)}\\ \prod_{j\in I_{2}}y_{j}^{d_{j}(2)}\\ \vdots\\ \prod_{j\in I_{1}}y_{j}^{d_{j}(L)}\end{bmatrix} \tag{9}\]  
and thus, (7) can be expressed as follows:  
\[\dot{x}_{i}=-\lambda_{i}x_{i}+\mu_{i}\left[\sum_{k=1}^{L}w_{ik}\xi_{k}\right]. \tag{10}\]  
If the weight vector is defined as \(w_{i}=\mu_{i}[w_{i1}w_{i2}\ldots w_{iL}]^{T}\), then (10) can be written as  
\[\dot{x}_{i}=-\lambda_{i}x_{i}+w_{i}^{T}\zeta. \tag{11}\]  
Therefore, the RHONN model takes the following form:  
\[\dot{x}=\Lambda x+W^{T}\zeta, \tag{12}\]  
where \(x=[x_{1}\ldots x_{n}]^{T}\in\mathbb{R}^{n}\) ; \(W=[w_{1},w_{2},\ldots,w_{n}]^{T}\in\mathbb{R}^{n\times L}\) and \(\Lambda=diag(-\lambda_{1},-\lambda_{2},\ldots,\\ -\lambda_{n})\in\mathbb{R}^{n\times n}\), and it is assumed that \(\lambda_{i}>0\).  
Consider the identification problem of a nonlinear dynamic system of the form  
\[\dot{\chi}=F(\chi,u), \tag{13}\]  
where \(\chi\in\mathbb{R}^{n}\) is the vector state of the system, \(u\in\mathbb{R}^{m}\) is the system input, and \(F:\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n}\) is a smooth vector field defined on a compact set \(\gamma\in\mathbb{R}^{n+m}\). It was shown in [25] that the model (12) can approximate suitably any system of the form (13) as long as the number of connections \(L\) is selected adequately [26].  
Assuming that the system (13) and the model (12) are initialized at the same state \(x(0)=\chi(0)\), then for any \(\varepsilon>0\) and any finite \(T>0\), there exists an integer \(L\) and a matrix \(W^{*}\in\mathbb{R}^{L\times n}\) such that the state \(x(t)\) of the modeled RHONN in (12) with \(L\) high-order connections and weights \(W=W^{*}\) satisfies  
\[\sup_{0\leq t\leq T}|x(t)-\chi(t)|\leq\varepsilon. \tag{14}\]  
The weights \(w_{i}\), for \(i=1,2,\ldots,n\), are adjusted according to the learning law:  
\[\dot{w}_{i}=-\Gamma_{i}\zeta\varepsilon_{i}, \tag{15}\]  
where \(e_{i}:=x_{i}-\chi_{i}\) is the error between the state of the network and the state to be identified, \(\zeta\) is defined as in (9); and \(\Gamma_{i}\) is a positive definite matrix. In the special case where \(\Gamma_{i}=\gamma_{i}I\), where \(\gamma_{i}>0\) is a scalar, then \(\Gamma_{i}\) in (15) might be replaced by \(\gamma_{i}\)[14].