## 4 Benchmark Examples with Analytic Solution
### Example--Second-Order System  
Consider the second order nonlinear system [43]  
\[\dot{x}_{1} =-\frac{1}{8}\Big{(}29x_{1}+87x_{1}x_{2}^{2}\Big{)}-\frac{1}{4} \Big{(}2x_{2}+3x_{2}x_{1}^{2}\Big{)}+u_{1}+\frac{1}{2}d, \tag{45}\] \[\dot{x}_{2} =-\frac{1}{4}\Big{(}x_{1}+3x_{1}x_{2}^{2}\Big{)}+3u_{2}+d,\] \[\mathbf{z} = \begin{bmatrix}\sqrt{2}\big{(}2x_{1}+6x_{1}x_{2}^{2}\big{)}\\ \sqrt{2}\big{(}4x_{2}+6x_{1}^{2}x_{2}\big{)}\\ u_{1}\\ u_{2}\end{bmatrix} .\]  
If \(\gamma^{*}=1\) then by solving the corresponding HJI equation (for details see [43]), the analytical feedbacks are  
\[u_{1}^{*}(\mathbf{x}) =-x_{1}-3x_{1}x_{2}^{2}, \tag{46}\] \[u_{2}^{*}(\mathbf{x}) =-6x_{2}-9x_{1}^{2}x_{2},\] \[d^{*}(\mathbf{x}) =\frac{1}{2}x_{1}+2x_{2}+3x_{1}^{2}x_{2}+\frac{3}{2}x_{1}x_{2}^{2}.\]  
Approximation functions of control and uncertainty variables are chosen in the following form:  
\[\dot{u}_{1}(\mathbf{x}) =p_{1}^{1}x_{1}+p_{2}^{1}x_{2}+p_{3}^{1}x_{1}x_{2}+p_{4}^{1}x_{1}^{ 2}x_{2}+p_{5}^{1}x_{1}^{2}x_{2}^{2}+p_{6}^{1}x_{1}^{2}x_{2}^{2}+p_{7}^{1}x_{1}^ {2}+p_{8}^{1}x_{2}^{2}, \tag{47}\] \[\dot{u}_{2}(\mathbf{x}) =p_{1}^{2}x_{1}+p_{2}^{2}x_{2}+p_{3}^{2}x_{1}x_{2}+p_{4}^{2}x_{1}^ {2}x_{2}+p_{5}^{2}x_{1}x_{2}^{2}+p_{6}^{2}x_{1}^{2}x_{2}^{2}+p_{7}^{2}x_{1}^{2} +p_{8}^{2}x_{2}^{2},\] \[\dot{d}(\mathbf{x}) =r_{1}x_{1}+r_{2}x_{2}+r_{3}x_{1}x_{2}+r_{4}x_{1}^{2}x_{2}+r_{5}x _{1}x_{2}^{2}+r_{6}x_{1}^{2}x_{2}^{2}+r_{7}x_{1}^{2}+r_{8}x_{2}^{2},\]  
i.e., written in the form (12) and (13), we have  
\[\boldsymbol{\theta}^{1}(\mathbf{x}) =\boldsymbol{\theta}^{2}(\mathbf{x})=\boldsymbol{\psi}(\mathbf{ x})= \begin{bmatrix}x_{1}&x_{2}&x_{1}x_{2}&x_{1}^{2}x_{2}&x_{1}x_{2}^{2}&x_{1}^{2}x_{2} ^{2}&x_{1}^{2}&x_{2}^{2}\end{bmatrix} , \tag{48}\] \[\boldsymbol{\Theta}(\mathbf{x}) = \begin{bmatrix}\boldsymbol{\theta}^{1}(\mathbf{x})&\boldsymbol{0 }\\ \boldsymbol{0}&\boldsymbol{\theta}^{2}(\mathbf{x})\end{bmatrix} ,\ \ \mathbf{\Psi}(\mathbf{x})=\boldsymbol{\psi}(\mathbf{x}),\] \[\mathbf{p}^{1} = \begin{bmatrix}p_{1}^{1}\\ p_{2}^{1}\\ \vdots\\ p_{8}^{1}\end{bmatrix} ,\ \ \mathbf{p}^{2}= \begin{bmatrix}p_{2}^{2}\\ p_{2}^{2}\\ \vdots\\ p_{8}^{2}\end{bmatrix} ,\ \ \boldsymbol{\pi}= \begin{bmatrix}\mathbf{p}^{1}\\ \mathbf{p}^{2}\end{bmatrix} ,\ \ \boldsymbol{\rho}= \begin{bmatrix}r_{1}\\ r_{2}\\ \vdots\\ r_{8}\end{bmatrix} .\]  
In this example, we set the following values of the algorithm parameters: we divided the time interval from \(t_{0}=0\), to \(t_{f}=3\) [s] into \(N=5000\) equal subintervals, i.e., the discretization step is \(\tau=10^{-4}\) [s] and we applied the Adams method of the 4-th order by default; vector of initial conditions \(\mathbf{x}_{0}=[1\ \ 1]^{\text{T}}\); initial value of the parameter \(\mu_{0}=\gamma_{0}^{2}=5\); vectors of initial parameters of approximation functions \(\boldsymbol{\pi}_{0}=\boldsymbol{\rho}_{0}=\boldsymbol{1}\); stopping criterion of the quasi-Newton method \(\varepsilon=10^{-3}\); stopping criterion of conjugate-gradient method \(\epsilon=10^{-3}\), Dai-Yuan method is used by default, the initial numerical values of the conjugate gradient algorithm parameters are chosen as \(\eta_{0}=0.1\) and \(\beta_{0}=0.5\), coefficients \(c_{1}=10^{-3}\), \(c_{2}=0.9\) and \(\
u=0.8\). We obtain the following parameter \(\mu\) and parameters of approximation functions:  
\[\mu^{*}=1.0021,\quad\mathbf{p}^{1^{*}}= \begin{bmatrix}-0.9997\\ -0.0241\\ 0.0785\\ -0.0676\\ -3.0576\\ 0.0422\\ -0.0003\\ 0.0287\end{bmatrix} ,\quad\mathbf{p}^{2^{*}}= \begin{bmatrix}0.0004\\ -6.0319\\ 0.1224\\ -9.1208\\ -0.1533\\ 0.1016\\ -0.0009\\ 0.0825\end{bmatrix} ,\quad\mathbf{p}^{*}= \begin{bmatrix}0.4995\\ 2.0478\\ -0.1576\\ 3.1370\\ 1.6151\\ -0.0847\\ 0.0007\\ -0.0579\end{bmatrix} , \tag{49}\]  
i.e., the minimum \(\mathcal{L}_{2}\)-gain is \(\gamma^{*}=\sqrt{\mu^{*}}=1.0010\). Figure 4 shows the time dependence of the state variables, i.e., the response of the system (45) from the initial conditions \(\mathbf{x}_{0}=[1\;\;1]^{\mathrm{T}}\), where the control variables and uncertainty variable are of the form (47) with parameters (49).