## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
\begin{array}{l}a_{0}=1>0,\\ P_{ce}(1)>0\Rightarrow K_{p}T_{s}+2K_{d}>0,\\ (-1)^{2}P_{ce}(-1)>0\Rightarrow 4+K_{p}T_{s}^{2}>0\Rightarrow K_{p}>-\frac{4}{T ^{2}},\\ |a_{2}|<1\Rightarrow\left|1+K_{p}T_{s}^{2}-K_{d}T_{s}\right|<1\Rightarrow\left\{ \begin{array}{l}K_{p}T_{s}^{2}-K_{d}T_{s}<0,\\ 2+K_{p}T_{s}^{2}-K_{d}T_{s}>0.\end{array}\right.\end{array} \right. \tag{31}\]

## 3 Nonlinear Systems

In order to present a solid comparison on the Model Free laws described in Section 2, three nonlinear systems with distinct dynamics were considered. These are represented by a BLDC motor [40], a nonminimum phase nonlinear system, used in [7] that has time-varying structure and time-varying order. The third system is a nonlinear system with a time-varying parameter, also used in [7]. ### BLDC Motor

The brushless direct current (BLDC) motor is an advanced electric machine where rotor rotation is achieved through electronic commutation. Unlike traditional brushed motors, the BLDC motor features coils located in the stator, and rotation is controlled by manipulating the magnetic field generated by these coils. This design eliminates the need for brushes, enhancing reliability and reducing maintenance requirements. Additionally, BLDC motors are capable of delivering high power output as they can operate continuously at maximum torque. Their precise control over torque and speed further minimizes energy losses, making them highly efficient machines. The dynamics of a BLDC motor can be described by the following equations for each armature winding:

\[\left\{ \begin{array}{l}V_{a}=I_{a}R+L\frac{dI_{a}}{dt}+E_{a},\\ V_{b}=I_{b}R+L\frac{dI_{b}}{dt}+E_{b},\\ V_{c}=I_{c}R+L\frac{dI_{c}}{dt}+E_{c},\end{array} \right. \tag{32}\]

where it is assumed that the mutual inductance is constant while the motor is revolving, \(L_{a}=L_{b}=L_{c}=L\), are the armature inductance, \(R_{a}=R_{b}=R_{c}=R\), represents the armature resistance for each phase, \(V\) is the terminal voltage, \(I\) is the phase current, and \(E\) is representing the back-emf of the active commutated phase. Due to the fact that a three-phase BLDC motor is used, the back-emf for each phase can be described as:

\[\left\{ \begin{array}{l}E_{a}(t)=K_{E}\theta\omega(t),\\ E_{b}(t)=K_{E}\big{(}\theta-\frac{2\pi}{3}\big{)}\omega(t),\\ E_{c}(t)=K_{E}\big{(}\theta+\frac{2\pi}{3}\big{)}\omega(t),\end{array} \right. \tag{33}\]

where \(K_{E}\) represents the back-emf constant of one phase, while each phase has \(120^{\circ}\) phase angle difference, while \(\theta\) is the rotor position. Considering constant torque \(K_{T}(\theta)\) satisfying \(K_{T}(\theta)\propto K_{E}(\theta)\), results in torque output of each phase as:\[\left\{ \begin{array}{l}T_{a}(t)=K_{T}(\theta)I_{a}(t),\\ T_{b}(t)=K_{T}\big{(}\theta-\frac{2\pi}{3}\big{)}I_{b}(t),\\ T_{c}(t)=K_{T}\big{(}\theta+\frac{2\pi}{3}\big{)}I_{c}(t),\end{array} \right. \tag{34}\]

where \(K_{T}\) is the torque constant of one phase. With a reduced torque reluctance, the total torque of the motor can be written as:

\[T_{E}=T_{a}+T_{b}+T_{c} \tag{35}\]

The equation of mechanical component may be represented as:

\[T_{E}-T_{L}=J\frac{d\omega}{dt}+B\omega \tag{36}\]

where \(T_{L}\) is the torque load, \(J\) represents the rotor inertia, while \(B\) is the damping constant. The conventional scheme for controlling a three-phase BLDC motor involves an inverter comprising a voltage source and a diode bridge, which exhibit nonlinear behavior. The control signal represented by the voltage value applied to the inverter is converted into pulse-width modulation (PWM) signals. These signals control the currents in the three phases, ultimately producing the desired torque and speed. The back-emf wave form derivated from the rotor speed, determining the switching of the inverter bridge gates. The parameters of the BLDC motor used in this study are consistent with those presented in [40]. ### Nonminimum Phase Nonlinear System

To demonstrate the advantages of MFAC-CFDL law, [7] uses a non-minimum phase nonlinear system as described by Equation (37).