# An Improved Model-Free Adaptive Nonlinear Control and Its Automatic Application
## 4 Simulation Experiment and Analysis
### Experiment 1: Numerical Simulation  
In order to verify the performance of the proposed algorithm in nonlinear tracking problems, the MIMO-FFDL-MFAC and MIMO-FFDL-MFAC-OER algorithms are comparedand simulated under time-varying signals by selecting typical nonlinear controlled objects. The first set of simulation object is the following MIMO system:  
\[\left\{\begin{array}{l}y_{1}(t+1)=1.5\times y_{1}(t)+0.12\times{y_{1}}^{2}(t-1 )+\\ 0.8\times u_{1}(t)+0.3\times u_{2}(t)+0.2\times u_{3}(t)+0.5\times d(t)\\ y_{2}(t+1)=1.6\times y_{2}(t)+0.14\times y_{2}^{2}(t-1)+\\ 0.3\times u_{1}(t)+0.8\times u_{2}(t)+0.2\times u_{3}(t)+0.5\times d(t)\\ y_{3}(t+1)=1.6\times y_{3}(t)+0.15\times y_{3}^{2}(t-1)+\\ 0.2\times u_{1}(t)+0.3\times u_{2}(t)+0.8\times u_{3}(t)+0.5\times d(t)\\ \end{array}\right. \tag{34}\]  
where \(d(t)\) is a white noise signal with mean 0 and variance 1. It can be seen from the specific structure of the model that the system is a coupling system with three inputs and three outputs. The desired trajectory is set as follows:  
\[y_{d}=\left\{\begin{array}{l}\text{linspace}(0,50,100)\times 5,0<t\leq 100\\ \text{linspace}(250,250,50),100<t\leq 150\\ \text{linspace}(250,290,50),150<t\leq 200\\ \text{linspace}(290,290,525),200<t\leq 725\\ \text{linspace}(290,0,175),725<t\leq 900\\ \end{array}\right. \tag{35}\]  
where the function of the \(\text{linspace}(x_{1},x_{2},N)\) is to generate \(N\) point row vectors between \(x_{1},x_{2}\), and the adjacent data span is the same; \(x_{1},x_{2}\) and \(N\) are the starting value, the ending value, and the number of elements, respectively.  
The initial condition of the system is set to \(y_{1}(1)=y_{2}(1)=y_{3}(1)=y_{d}\), \(\mathbf{\Phi}_{1}(1)=\mathbf{\Phi}_{2}(1)=diag\{0.5,0.5,0.5\}\). The MFAC and MFAC-OER controller parameters are set to \(\lambda=1\), \(\rho=0.85\), \(\mu=1\), \(\beta=0.2\), \(\theta=0.2\)  
Figure 2 is the system output curve of the MFAC and MFAC-OER methods. Figure 3 is the corresponding output error diagram. From the local zoom of the output curves, it can be seen that, when the expected output changes, the MFAC-OER method can reach the steady state within 10 s, while MFAC takes more than 20 s. For the rise time, the MFAC-OER method is almost half of the MFAC. In addition, from the error change curve, it can be seen that the MFAC-OER not only has the fast tracking characteristic, but can also effectively suppress the influence of external interference, and the overall tracking effect is better than the traditional MFAC.  
Figure 2: Two control methods’ system output curves: **(a)** MFAC output curve; and **(b)** MFAC-OER output curve.  
From the error-related indexes in Figure 3 and Table 1, it can be seen that the proposed MFAC-OER method has better tracking performance than the traditional MFAC method regardless of whether the given signal is constant or changing. The MAE and MSE values of MFAC-OER are half to one-third of those of traditional MFAC. In summary, MFAC-OER not only has fast tracking characteristics but has stronger robustness.  
(1) Mean absolute error performance (MAE)  
\[\text{MAE}=\frac{1}{mT}\sum_{i=1}^{m}\sum_{t=1}^{T}|y_{d}-y_{i}(t)|. \tag{36}\]  
(2) Mean square error performance (MSE)  
\[\text{MSE}=\frac{1}{mT}\sum_{i=1}^{m}\sum_{t=1}^{T}|y_{d}-y_{i}(t)|^{2}. \tag{37}\]