## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
By adding the element dependent on the value of the previous control signal, \(u_{k-1}\), the nonlinear system also acquires a non-minimum phase characteristic that increases its complexity and makes it suitable to be controlled by DDC methods. Previously, neural networks for nonlinear self-tuning adaptive control [41] were used for such nonlinear systems. \[y_{k+1}=\frac{2.5y_{k}y_{k-1}}{1+y_{k}^{2}+y_{k-1}^{2}}+1.2u_{k}+1.4u_{k-1}+0.7 \sin(0.5(y_{k}+y_{k-1}))\cos(y_{k}+y_{k-1}). \tag{37}\]

### Nonlinear System with Time-Varying Parameter

Another nonlinear system presented in [7] is the one described by Equation (38). This is a system used as an example for the MFAC-PFDL law as it is a nonlinear time-varying system. Due to these characteristics, the system is suitable for DDC type methods, and it was previously controlled with recurrent neural networks in [42]. \[y_{k+1}=\frac{y_{k}y_{k-1}y_{k-2}u_{k-1}(y_{k}-1)+(1+a_{k})u_{k}}{1+y_{k}^{2}+ y_{k-1}^{2}+y_{k-2}^{2}}. \tag{38}\]

where the signal notations described in Section 3. 2 are preserved, while \(a_{k}\) represents the time-varying parameter. ## 4 Model Free Laws Applied to Nonlinear Systems

The Model Free laws described in Sections 2.1 and 2.2 are amenable to applications on nonlinear systems, as [1,8] extensively describes. Therefore, in the following, Algorithms 1 and 2 are analyzed to obtain the initial values of PPD and PG for the three nonlinear systems described in Section 3. After obtaining the values that provide the best transient time performance, the obtained parameter sets are used in a comparison with iP and iPID laws for the same nonlinear systems. ### Applications of Using Algorithm 1 for Nonlinear Systems

#### 4.1.1 Speed Control of BLDC Motor

BLDC motor speed control application uses the MFAC-CFDL law (8) with the starting parameters: \(\mu=0.1\), \(\eta=0.1\), \(\rho=0.1\), \(\lambda=50\), \(\varepsilon=0.00001\), \(\hat{\phi}_{1}^{0}=13\) and the sampling period \(T_{s}=0.0001\). With this set of parameters, optimization Algorithm 1 was applied. In Figure 1 are shown the variations of the average absolute error (\(aEs\)), as the number of iterations during the optimization process, as described in Equation (25), for the step-type references described in detail previously. The minimum value of \(aEs\) is obtained for \(\hat{\phi}_{1}^{*}=12.1607\). The comparative results between the starting version of CFDL (denoted CFDL\({}_{0}\)) and the optimized version (denoted CFDL\({}^{*}\)) during the simulation time of 1.5 s are shown in Figure 2.

\[|\overline{e}_{k}|=\frac{\sum_{i=1}^{N}|e_{k}|}{N}. \tag{39}\]

Figure 1: Average absolute error for steps variation during Algorithm 1 for BLDC motor. Figure 2: BLDC motor speed for CFDL optimization algorithm. Figure 2 shows how the output signal of CFDL* shows a better settling time the three step signals applied as system references. Also, the overshoot values are lower compared to the CFDL\({}_{0}\) variant, by 4.5%. Another remarkable characteristic is related to the oscillations around the constant value of the reference signal, which are lower than the CFDL\({}_{0}\) variant. The numeric analysis of performance is presented in Table 1.

#### 4.1.2 Control of Nonminimum Phase Nonlinear System

Algorithm 1 was applied for the non-minimum phase nonlinear system described by Equation (37). Figure 3 shows how the error varies for step-type references (_aEs_) during the 30 iterations of the Algorithm 1. The values of the starting parameters used in the case of this system were: \(\mu=1\), \(\eta=2\), \(\rho=0.2\), \(\lambda=10\), \(\epsilon=0.00001\), \(\hat{\Phi}_{1}^{0}=0.1\), and the sampling period \(T_{s}=0.01\) was chosen. Following the application of Algorithm 1, an optimized value for \(\hat{\Phi}_{1}^{*}=3.8056\) was obtained.