Advanced Adaptive Nonlinear Control with Deadbeat Observer for Permanent Magnet Synchronous Motor Drives

In the dynamic field of modern electrical engineering, it is vital to search for advanced control techniques for electrical machines constantly. Among these machines, the permanent magnet synchronous machine (PMSM) stands out for its high performance, compactness, and exceptional energy efficiency [1]. Its use extends from industrial applications to electric vehicles and renewable energies [2]. Besides, with the development of power electronics and the advance of highly integrated digital control units, several control strategies have been developed [3, 4]. Proportional-integral (PI) loop-based vector control is commonly the industrial standard for PMSM control, owing to its decoupling characteristics and simple structure, which stands out as a powerful and efficient tool that endows the PMSM with dynamic performance as satisfactory as DC machines. However, the PMSM control has a major problem related to the variation of parameters during operation and unknown disturbances, which may threaten the stability of the PMSM drive. Therefore, the control scheme design that can guarantee a safe and stable operation under the mentioned issues is challenging.

Numerous control strategies have been adopted for PMSM control, considering its parameters variation and various disturbances. For instance, an intelligent control strategy employing a recurrent Elman neural network based on the wavelet method (RWENN) is designed for PMSM’s position control, aiming to attain high tracking performance and manage the uncertainties' existence [4]. Indeed, ANNs are a simplified mathematical formulation of biological neurons, but the difficulty of interpreting the behavior of a neural network is a drawback when developing an application. It is also risky to generalize from previous experience and conclude or create rules about how neural networks work and behave. A fuzzy inferior is introduced to adjust the gains of a feed-forward PI controller (Pseudo-derivative feedback with feed-forward gain) [5]. The result of this approach is an appropriate placement of the transfer function closed-loop poles, which improves the ability to reject load torque disturbances. In the 1980s, the first direct torque control (DTC) was introduced [6]. The DTC is a technique applied to PMSMs, capable of providing good electromagnetic torque dynamics. This allows the possibility to achieve these objectives by selecting inverter output vectors from a switching table; stator flux and electromagnetic torque are controlled directly and independently. However, this strategy has significant drawbacks; the switching frequency is not controlled, the presence of large ripples in stator flux and torque, and variations in stator resistance due to temperature change or operation at low rotational speeds may degrade control performance [7]. Therefore, several advanced control techniques have been implemented to enhance the performance of PMSM drive, including the predictive control model (MPC) [8-10], robust control [11-13], sliding mode control (SMC) [14-16], backstepping technique [17-19], adaptive control [20-24], and deep learning algorithms-based control [25-27]. These non-linear control techniques have contributed to improving the PMSM system performance in some ways. On the other hand, recently, observer-based control techniques have gained significant attention for their ability to enhance robustness against load disturbances and parameter uncertainties in PMSMs, emerging as highly popular approaches adopted in the field of AC motor drives [28, 29]. Various observer-based control approaches have been proposed in the literature [30-36]. Although these methods can deliver acceptable performance under perturbation in load torque, the observers’ effectiveness, which depends on precise system models or parameters, tends to diminish when faced with parameter uncertainties.

In the study [30-32, 37], model-based reduced-order nonlinear observers-based controllers, including linear extended observer, extended-order nonlinear observer (ENO), and state-observer, have been developed to enhance robustness versus load torque perturbation with slow variation. Nevertheless, these observers-based control techniques exhibit sensitivity to parameter changes. In [33-35], reduced-order, linear extended state, and linear extended high-gain observers have been employed to accomplish satisfactory performance under load torque disturbances and provide strong robustness versus several parameter uncertainties. Nevertheless, some parameters still need to be precisely known.

To enhance the performance of controllers based on observer that relies on accurately known parameters, techniques insensitive to system parameters have been adopted [38-49]. Such methods not only offer strong robustness under load torque disturbance but also enhance performance with system parameter uncertainties. In [38, 39], extended sliding-mode and terminal sliding-mode observers have been introduced to estimate load torque and mechanical PMSM parameters. As electrical parameters can be measured more easily by employing sensors or instruments compared to mechanical parameters, the focus is primarily on estimating the mechanical parameters. However, electrical parameters often change with time due to factors such as temperature fluctuations, cross-saturation effects, and magnetic saturation, making them difficult to accurately determine during operation [40-42]. A combination of SMC and a disturbance observer has been adopted to tackle time-varying disturbances and parameters [43]. However, the controllers’ and observers’ parameters tuning is complicated and time-demanding, and it also lacks validation under parameter uncertainties. In [44, 45], disturbance observer-based techniques have been designed to improve the PMSM control performance under disturbances and uncertainties, but test results for parameter uncertainties have not been provided. In [46, 47], nonlinear disturbance and extended nonlinear observers have been introduced for estimating specific parameters and load torque, but practical results were missing. A design of an automatic disturbances’ rejection controller based on an extended state observer has been explored for high-performance PMSM robust motion control, focusing on the control of position [48]. An innovative control structure utilizing dual proportional-integral (PI) observers has been developed to maintain operation with high performance in the presence of parameter uncertainties and slowly varying disturbances [49]. In [34, 50, 51], a high-order sliding mode observer, a comprehensive disturbance observer, and an adaptive observer of stator current and disturbance have been proposed to improve the PMSM control performance, ensuring high robustness under disturbances and parameter uncertainties. Beyond the aforementioned techniques, a nonlinear adaptive control approach incorporating high-gain states and an observer of perturbation has been successfully implemented in various applications [52, 53]. This method ensures acceptable performance even versus model or parameter uncertainties, as well as load disturbances.

Furthermore, to improve the robustness of the SMC, a design of a sliding mode speed controller considering a novel variable rate exponential reaching law is suggested [54]. Subsequently, an extended state observer is employed to extract and compensate for parameter mismatches and external disturbances. This approach results in a novel sliding mode predictive current control strategy, which not only maintains the fast dynamic response of the current control but also significantly improves the system's robustness.

Considering this literature review, the primary objective of our research is to develop a control law for the PMSM that enhances performance in trajectory tracking, disturbance rejection, stability, robustness against parameter uncertainties, adherence to physical, and computational efficiency while preserving the inherent non-linear characteristics of the system. In addition, it is known that the control by linearization between outputs strategy suffers from a lack of robustness to external disturbances and model imperfections. Therefore, a novel hybrid adaptive nonlinear control with a specifically tailored DO technique is proposed in this paper, aiming to enhance the control performance of PMSMs. The key innovation lies in the integration of linear and nonlinear control methodologies, facilitated by the utilization of a DO for real-time state estimation. Unlike traditional control approaches, which often struggle to adapt to the dynamic and nonlinear characteristics of PMSMs, the proposed method aims to provide a more effective and robust solution. The novelty of our approach is represented by its ability to dynamically adapt to system variations and disturbances while maintaining precise control over PMSM. By leveraging the real-time state estimation capabilities of the DO, the proposed strategy can generate control signals that accurately track desired references, even in the presence of changing load conditions and speed fluctuations. This adaptive nature of the proposed strategy makes it particularly well-suited for applications where precise control of PMSMs is essential, such as electric vehicles, industrial automation, and RESs.

In the context of the control of complex dynamic systems, such as the system of a PMSM, the application of the proposed non-linear adaptive control with a DO offers a robust and efficient solution. This approach, specifically adapted to the PMSM, enables the control parameters to be dynamically adjusted to suit variations in the system while guaranteeing fast and accurate estimation of the internal states, thanks to the DO. By combining these advanced techniques, it is possible to achieve optimum performance even in dynamic and non-linear environments in automotive electric propulsion. In the proposed scheme, the adaptive nonlinear controller combines the nonlinear technique of input-output looping linearization and the adaptive linear control method to ensure the tracking of system reference trajectories with parametric uncertainties.

Meanwhile, the DO is introduced to ensure an accurate estimation of the current dq components and the PMSM speed using the actual measured currents.

In the following subsections, the design of the nonlinear controller is described in detail, considering variations of the system's parameters and load disturbances for non-adaptive and adaptive controller cases. In addition, the operation principle of the adopted DO is discussed.

3.1 Design of the adaptive nonlinear controller

The design of the nonlinear controller is established by considering the uncertainty of the different parameters. The first case considers a linear uncertainty of stator resistance and load torque. The second one inspects the inductances and moment of inertia uncertainty, which are non-linear, as they are difficult to measure exactly. In these studied cases, both non-adaptive and adaptive controllers are investigated.

3.1.1 Case of variations on R and T_r

Here, the aim is to design an adaptive non-linear controller that guarantees speed regulation of the PMSM with constant but unknown stator resistance R_s and resistive torque T_r. To achieve this, we start by designing a controller using the linearization method applied to the input-output dynamics of the nominal PMSM model. Then, we derive the adaptation law to extract the vector of uncertain parameters. The vector of uncertain parameters, λ, is considered as follows:

Then, the system of (1) can be rewritten in the expression given in (3), which proposes an adaptive scheme for estimating R_s and T_r. By replacing (2) in (1), we get:

$[\lambda]=\left[\begin{array}{l}\lambda_1 \\ \lambda_2\end{array}\right]=\left[\begin{array}{l}R-R_n \\ T_r-T_{r n}\end{array}\right]$             (2)

$\left\{\begin{array}{c}\frac{d}{d t} I_d=-\frac{R_n}{L_d} I_d+\lambda_1 \frac{I_d}{L_d}+p \omega_r \frac{L_q}{L_d} I_q+\frac{u_d}{L_d} \\ \frac{d}{d t} I_q=-\frac{R_s}{L_q} I_q-\lambda_1 \frac{I_d}{L_d}-\boldsymbol{p} \omega_r \frac{L_d}{L_q} I_d-\frac{p \omega_r}{L_q} \varphi_f+\frac{u_q}{L_q} \\ \frac{d}{d t} \omega_r=\frac{K_T}{J} I_q+\frac{K_T\left(L_d-L_q\right)}{J \varphi_f} I_d I_q-\frac{f}{J} \omega_r-\frac{T_{r n}}{J}-\frac{\lambda_2}{J}\end{array}\right.$            (3)

where, the subscript n denotes nominal values.

In a compact form:

$\dot{X}=f(X, \lambda)+\sum_{i=1}^2 g_i(X, \lambda) u_i$           (4)

$\dot{X}=f_0(X)+\sum_{i=1}^2 g_i(X, \lambda) u_i+\sum_{i=1}^2 \lambda_j f_{\lambda_j}(x)$           (5)

Non-adaptive controller case

This subsection provides the design of the controller’s non-adaptive version, where the vector of uncertain parameters is known. The objective is to ensure PMSM speed control while maintaining operation at maximum torque (I_d= 0). To this end, a linearization law in the input-output directions is applied to its model, which guarantees total decoupling between inputs and outputs. The system outputs are the mechanical speed ω_r and the current I_d, which are defined as follows:

$\left\{\begin{array}{l}y_1=h_1=I_d \\ y_2=h_2=\omega_r\end{array}\right.$                  (6)

These two outputs should follow the reference trajectories. For the current (I_d), it is zero (I_d = 0), while the speed reference may be a step or any trajectory.

To establish the decoupling matrix D(x), we start by determining the relative degree of each output to be controlled.

1. Relative degree: For current $I_d$ with $y_1=h_1=I_d$, $\nabla h_1=\left[\begin{array}{lll}1 & 0 & 0\end{array}\right]$, using (4), we get:

$\dot{y}_1=L_f h_1(x)+\sum_{i=1}^2 L_{g_i} h_1(x) u_i$                 (7)

Note that the $u_d$ input appears in (7); stop here and take the relative degrees of this output $r_1=1$.

For speed $\omega_r$ with $y_2=h_2=\omega_r$ and $\nabla h_2=\left[\begin{array}{lll}\mathbf{0} & \mathbf{0} & 1\end{array}\right] ;$

Note that if the problem involves finding a linear relationship between input and output, derive the output until at least one input appears by using the following expression:

$\begin{gathered}y_j^{\left(r_j\right)}=L_j^{\left(r_j\right)} h_j(x)+\sum_{i=1}^p L_{g_i}\left(L_j^{\left(r_j-1\right)} h_j(x)\right) u_i ; i=1,2, \ldots \ldots p\end{gathered}$       (8)

If $L_j^{\left(r_j-1\right)} h_j(x) \neq 0$, so $u_i$ does not appear in the $y_j, \dot{y}_j, \ldots . y_j^{\left(r_j-1\right)}$ and appears in the equation of $y_j^{\left(r_j\right)}$.

In this case, $\left(r_j\right)$ is called the system relative degree. According to the following definition, using (9), we get:

$\dot{y}_2=L_f h_2(x)+\sum_{i=1}^2 L_{g_i} h_2(x) u_i$      (9)

Note that no input appears in (10), so we are forced to derive another time:

From (10), $r_2=2$ for this output.

The total relative degree (global or vector) $r_1+r_2=r=3$, so the system is exactly linearizable for non-linear state feedback. There are no internal or unobservable zero dynamics to consider. By putting together (7) and (10), we obtain the following form

$\ddot{y}_2=L^2{ }_f h_2(x)+\sum_{i=1}^2 L_{g_i} h_2(x) u_i$       (10)

$\left[\begin{array}{ll}\dot{y}_1 & \ddot{y}_2\end{array}\right]^T=\zeta(x)+D(x) u$    (11)

where, the decoupling matrix is invertible if the determinant det (D(x) ≠ 0), which equals:

$|D(x)|=\frac{3 p}{2 J L_q L_d}\left(\varphi_f+I_d\left(L_d-L_q\right) \neq 0\right.$      (12)

The linearizing control law that guarantees decoupling between inputs and outputs (i.e., a linear relationship) is:

$\left[\begin{array}{l}u_d \\ u_q\end{array}\right]=D^{-1}(\mathrm{x})\left(\left[\begin{array}{l}v_1 \\ v_2\end{array}\right]-\boldsymbol{\zeta}(\boldsymbol{x})\right)$        (13)

Replacing the linearizing control law (13) in (10) gives:

$\left[\begin{array}{l}\dot{y}_1 \\ \ddot{y}_2\end{array}\right]=\left[\begin{array}{c}\dot{I}_d \\ \ddot{\omega}_{r_2}\end{array}\right]=\left[\begin{array}{l}v_1 \\ v_2\end{array}\right]$               (14)

So, in the case of a trajectory tracking problem, the new control vector is designed.

2. Conception of the new control vector: For the reference trajectory tracking problem, the vector v must satisfy:

$\begin{aligned} v_j=y_{d j}^{\left(r_j\right)}+ & K_{r j}\left(y_{d j}^{\left(r_j-1\right)}-y_j^{\left(r_j-1\right)}\right)+\cdots  +K_1\left(y_{d_j}-y_j\right) 1 \leq j \leq p\end{aligned}$          (15)

where, the vectors $\left(y_{d_j}, y_{d_j}^{\cdot}, \ldots, y_{d j}^{\left(r_j-1\right)}, y_{d j}^{\left(r_j\right)}\right)$ are the reference trajectories imposed for each output.

So, according to (15), we have:

The Laplace transformation of (16) leads to:

$v_1=I_d^*+K_d\left(I_d^*-I_d\right)$        (16)

$v_2=\ddot{\omega}_r^*+K_2\left(\dot{\omega}_r^*-\dot{\omega}_r\right)+K_1\left(\omega_r^*-\omega_r\right)$        (17)

$s+K_d=0$        (18)

Also, applying the Laplace transformation to (17) gives:

$s^2+K_{2 s}+K_1=0$      (19)

To determine the coefficients $K_d, K_1$, and $K_{2 S}$ we use the pole placement method so that the system is stable and its output responds without any overshoot. If the reference trajectory (the imposed trajectory) is a step, we have $I_d^*=$ $\ddot{\omega}_r^*=\dot{\omega}_r^*=0$, and the control vector becomes:

$\begin{gathered}v_1=K_d\left(I_d^*-I_d\right) \\ v_2=-K_2 \dot{\omega}_r+K_1\left(\omega_r^*-\omega_r\right)\end{gathered}$           (20)

Accordingly, the block diagram of the linearized closed-loop model can be derived, as shown in Figure 2.

2.png

Figure 2. Block diagram of the linearized closed-loop model

Adaptive controller case

Since $\lambda$ is unknown, it is replaced by its estimate $\hat{\lambda}$, which is proposed for the design. Accordingly, the output vectors are:

$\left[\begin{array}{ll}y_1 & y_2\end{array}\right]^T=\left[\begin{array}{ll}I_d & \omega_r\end{array}\right]^T=\left[\begin{array}{ll}\varphi_1 & \varphi_2\end{array}\right]^T=\left[\begin{array}{ll}h_1 & h_2\end{array}\right]^T$          (21)

The change of coordinates according to $\hat{\lambda}$ is defined as follows:

$[\hat{z}]=\left[\begin{array}{c}L_{f_0} \varphi_1 \\ L_{f_0} \varphi_2 \\ L_{f_0}^2 \varphi_3\end{array}\right]+\left[\begin{array}{cc}L_{g_1} \varphi_1 & 0 \\ 0 & 0 \\ L_{g_1} L_{f_0} \varphi_2 & L_{g_2} L_{f_0} \varphi_2\end{array}\right]\left[\begin{array}{l}u_d \\ u_q\end{array}\right]$          (22)

According to expression (22), we have:

$\left\{\begin{array}{l}R=\lambda_1+R_n \\ T_r=\lambda_2+T_{r n}\end{array}\right.$       (23)

By replacing Eq. (22) in (23), we obtain the dynamics of Z as follows:

$\left[\begin{array}{c}\hat{\dot{Z}}_1 \\ \hat{\dot{Z}}_2 \\ \hat{\dot{Z}}_3\end{array}\right]=\left[\begin{array}{c}L_{f_0} \varphi_1+\lambda_1 L_{f_{\delta_1}} \varphi_1 \\ L_{f_0} \varphi_2+\lambda_2 L_{f_{\delta_2}} \varphi_2 \\ L_{f_0}^2 \varphi_3+\lambda_1 L_{f_{\lambda_1}} L_{f_0} \varphi_2+\lambda_2 L_{f_{\lambda_2}} L_{f_0} \varphi_2+\hat{\lambda}_2 L_{f_{\delta_2}} \varphi_2\end{array}\right]+\left[\begin{array}{cc}L_{g_1} \varphi_1 & 0 \\ 0 & 0 \\ L_{g_1} L_{f_0} \varphi_2 & L_{g_2} L_{f_0} \varphi_2\end{array}\right]\left[\begin{array}{l}u_d \\ u_q\end{array}\right]$                   (24)

where, the decoupling matrix is given by:

$L_{f_{\lambda_1}} \varphi_1=-\frac{1}{L_d} I_d$           (25)

Our controller depends on $\lambda$, which is not yet determined, so the linearizing law becomes, in this case:

$\left[\begin{array}{l}\hat{u}_d \\ \hat{u}_q\end{array}\right]=D^{-1}(x)\left(-\zeta_0(x)-\zeta_\delta(x, \hat{\lambda}, \hat{\lambda})+\hat{v}\right)$           (26)

To adjust our adaptive controller using the Lyapunov-based adjustment mechanism, we need to define the state error vector, which is given by:

That should satisfy the following condition:

$e=\left|\begin{array}{c}I_d-I_d^* \\ \omega_r-\omega_r^* \\ \omega_r-\dot{\omega}_r^*\end{array}\right|=\left[\begin{array}{l}e_1 \\ e_2 \\ e_3\end{array}\right]$      (27)

$e=K \dot{e}+W_1 \Delta \lambda$       (28)

where, K is a matrix that depends on the desired poles, and from (24), we define $W_1$ as follows:

$W_1=\left[\begin{array}{cc}-\frac{1}{L_d} I_d & 0 \\ 0 & -\frac{1}{\mathrm{~J}} \\ -\left(\frac{K_T}{J L_q} I_q+\frac{K_T\left(L_d-L_q\right)}{J \varphi_f L_q} I_q I_d\right)-\left(\frac{K_T\left(L_d-L_q\right)}{J \varphi_f L_d} I_q I_d\right) & \frac{f}{J^2}\end{array}\right]$       (29)

Thus, the adaptation law can be determined using the following Lyapunov function:

$V=e^T P e+\Delta \lambda^T \Gamma \Delta \lambda$      (30)

where, its derivative is given by:

$\frac{d V}{d t}=-e^T Q_e+2 \Delta \lambda^T\left(W_1^T P e+\Gamma \Delta \dot{\lambda}\right)$     (31)

and Γ and P are positive-definite matrices defining the adaptation gain and the solution to the Lyapunov equation, respectively.

So that $\frac{d V}{d t} \leq 0$ it is necessary:

$W_1 P e+\Gamma \Delta \dot{\lambda}=0 \rightarrow \Delta \dot{\lambda}=-\Gamma^{-1} W_1^T P e$      (32)

where, $\Delta \dot{\lambda}$ represents the adaptation act, which ensures:

$\frac{d V}{d t}=-e^T Q_e \leq 0$      (33)

In this case, according to the Barbalat lemma, we have:

$\lim _{t \rightarrow \infty}\|e(t)\|=0$      (34)

So, now it remains to determine the matrix P, which verifies the following equation:

$K^T P+P K=-Q$           (35)

where, Q is a positive-definite symmetric matrix that is generally an identity matrix.

3.1.2 Case of variations on (L_d = L_q) and J

In this case, the steps to design the controller are similar to those in the previous section, considering L and J variations, which are involved within the PMSM model in a non-linear way. Thus, the vector of uncertain parameters is considered as follows:

$\lambda=\left[\begin{array}{l}\lambda_1 \\ \lambda_2\end{array}\right]=\left[\begin{array}{l}\frac{1}{L}-\frac{1}{L_n} \\ \frac{1}{J}-\frac{1}{J_n}\end{array}\right]$         (36)

$\left\{\begin{array}{l}L=\frac{L_n}{L_n \lambda_1+1} \\ J=\frac{J_n}{J_n \lambda_2+1}\end{array}\right.$        (37)

Note that from (36), the vector $\lambda$ denotes the difference between the uncertain parameters and their nominal values, so to obtain the values of $L$ and $J$ from $\lambda_1$ and $\lambda_2$, the following expressions can be used:

3.2 Deadbeat state observer

The DO is designed to estimate the unmeasured states of the system at each instant. In the case of the PMSM, the states typically estimated are the stator currents, $\hat{I}_d$ and $\hat{I}_q$, and the rotational speed $\widehat{\omega}_r$. The state estimates equations of the DO can be formulated as follows:

$\left\{\begin{array}{c}\hat{x}(k+1)=A \hat{x}(k)+B u(k)+L(y(k)-C \hat{x}(k)) \\ \hat{x}(0)=x_0\end{array}\right.$          (38)

with: $\hat{x}(k)$: the vector of the estimated states at time $k ; A$: the system state matrix; $B$: the control matrix; $u(k)$: the control vector at time $k ; L$: the DO gain matrix; $y(k)$: the output vector measures at time $k$; and $C$: The output matrix.

3.2.1 Design of the control parameters for the DO

The system state matrix can be derived from its dynamic model in the dq frame, which is defined as follows:

$\begin{aligned} \frac{d}{d t}\left[\begin{array}{c}I_d \\ I_q \\ \omega_r\end{array}\right]= & {\left[\begin{array}{ccc}-\frac{R}{L_d} & \omega_r & 0 \\ -\omega_r & -\frac{R}{L_q} & \frac{1}{L_q} \varphi_f \\ 0 & -\frac{1}{J} \frac{d L}{d \omega_r} & \frac{-f}{J}\end{array}\right]\left[\begin{array}{c}I_d \\ I_q \\ \omega_r\end{array}\right] } +\left[\begin{array}{cc}\frac{1}{L_d} & 0 \\ 0 & \frac{1}{L_q} \\ 0 & 0\end{array}\right]\left[\begin{array}{l}u_d \\ u_q\end{array}\right]\end{aligned}$          (39)

The state matrix A is, therefore, the part of the dynamic model that determines the evolution of system states over time. Using the equations above, we can extract the elements of the A matrix:

$A=\left[\begin{array}{ccc}-\frac{R}{L_d} & \omega_r & 0 \\ -\omega_r & -\frac{R}{L_q} & \frac{1}{L_q} \varphi_f \\ 0 & -\frac{1}{J} \frac{d L}{d \omega_r} & \frac{-f}{J}\end{array}\right]$                  (40)

For a PMSM, the state vector estimates $\hat{x}$ and the matrix B can be obtained as:

$\hat{x}=\left[\begin{array}{l}\hat{I}_d \\ \hat{I}_q \\ \widehat{\omega}_r\end{array}\right], B=\left[\begin{array}{cc}\frac{1}{L_d} & 0 \\ 0 & \frac{1}{L_q} \\ 0 & 0\end{array}\right]$            (41)

In the matrix B, each column represents the effect of the input voltage on the corresponding states of the system. Rows 1 and 2 correspond to the currents $L_d$ and $L_q$, respectively, while row 3 corresponds to the rotational speed $\omega_r$. The non-zero elements of this matrix B indicate how the input voltage components influence the different state variables of the system. For example, the input voltage $u_d$ affects only the current $(I)$, so only the first column of matrix $B$ has non-zero elements corresponding to $L_d$. Similarly, the input voltage $u_q$ affects only the current, so only the second column of matrix $B$ has non-zero elements corresponding to $L_q$. The rotational speed $\omega_r$ is not directly affected by the input voltage, so the third column of matrix $B$ is zero. The matrix $B$ is important in designing control laws for PMSMs because it indicates how the system inputs (the voltages $u_d$ and $q$ in this case) influence the system states, which is essential for designing an effective controller.

Observation matrix C

The matrix C is used to select the states that can be measured. In the case of the PMSM, let's assume we can directly measure the currents as well as the mechanical rotational speed $\omega_r$. Then, the matrix C  would be:

$C=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$       (42)

Design of the gain matrix L of the DO

To design the gain matrix L of the DO for a PMSM, we can use the classical state observer design method, such as the pole placement method. This method consists of placing the poles of the estimation error dynamics where we want them to obtain a fast and stable observer’s convergence. The estimation error dynamics is given by the equation:

$\tilde{\dot{x}}=(A-L C) \tilde{x}$         (43)

In which $\tilde{x}$ defines the estimation error vector:

$\tilde{x}=x-\hat{x}$        (44)

where, $\boldsymbol{x}$ and $\hat{x}$ are the actual state vector and the state estimate vector.

The matrix L can be calculated for an n-dimensional system using:

$L=(A-L C)^{-1} B$        (45)

Thus, to calculate the matrix L, we need to follow the following steps:

Choose the desired eigenvalues $\lambda_1, \lambda_2 \ldots \lambda_{\mathrm{n}}$:

• Calculate the matrix (A-LC) by substituting the values of A and assuming L as an unknown matrix;
• Calculate the inverse of (A-LC). Suppose we want to place the observer poles at specific positions. We choose the desired eigenvalues $\lambda_1, \lambda_2 \cdot \lambda_3$ and form the characteristic matrix (zI- (A-LC));
• Next, we equalize the determinant of the characteristic matrix to zero to obtain the characteristic equations. Solving these equations, we obtain the elements of the matrix L.

More precisely, for a nonlinear dynamic system such as that of a PMSM, the eigenvalues λ  of the matrix A-LC determine the dynamics of the state observer's estimation errors:

$\operatorname{det}(z I-(A-L C))=0$         (46)

$\tilde{\dot{x}}=(A-L C) \tilde{x}$       (47)

If the eigenvalues λ have a negative real part, this means that the estimation errors will decrease over time, indicating stable convergence of the estimation to the real values of the system states. In other words, the PMSM will tend to track the real values of its states quickly and accurately with a well-designed observer.

By judiciously choosing the eigenvalues λ, we can control the observer's performance, such as speed of convergence and robustness to disturbances. For example, if faster response times are required, we can choose eigenvalues with a more negative real part, which will speed up the observer's convergence. The approach proposed in this article enables the matrix L to be calculated as a function of the specific PMSM’s parameters and the observer’s specifications, providing an accurate estimate of the internal states of the PMSM for fast, stable convergence of the observer.

3.2.2 Application of the nonlinear adaptive deadbeat control

Before delving into the integration process, it's crucial to understand the context of designing and applying nonlinear adaptive input-output feedback control with DO for PMSMs. This control approach aims to regulate PMSM performance effectively using advanced control and state estimation methods. The system relies on a DO to estimate unmeasured states, enabling precise and rapid feedback. The integration of this control occurs in four distinct steps, each playing a crucial role in the overall dynamics of the system. The main steps of the DO are given as follows.

• Step 01: State Estimation: At each instant k, the DO is used to estimate the state of the system from the measurements of the output y(k) and the input u(k):

$\widehat{\boldsymbol{x}}(k+1)=A \widehat{\boldsymbol{x}}(k)+B u(k)+L(y(k)-C)$           (48)

• Step 02: Linearized control calculation: The linearized control signal v(k) is computed using the estimated state $\hat{x}(k)$ and the PI controller gains:

$v(k)=-\Gamma \hat{\boldsymbol{x}}(k)$          (49)

where, Γ is the adaptive gain matrix cited in the above part.

• Step 03: Control calculation: The control u(k) is then computed from the linearized control signal v(k), the position set point $y_{r e f}(k)$, and the integrals of past errors:

$\begin{aligned} u(k)= & -K_p\left(y(k)-y_{r e f}(k)\right) -K_i \int\left(y(k)-y_{r e f}(k)\right) d t  +v(k)\end{aligned}$            (50)

• Step 4: Updating adaptive parameters contribution: The adaptive parameters of the matrix Γ are updated using the Least Mean Squares (LMS) algorithm:

$\left\{\begin{aligned} \Gamma(k+1)= & \Gamma(k)+\mu\left(y(k)-y_{r e f}(k) \widehat{x}(k)^T\right. \\ & \varepsilon=y(k)-y_{\text {ref }}(k)\end{aligned}\right.$           (51)

where, $\mu$ is the learning factor.

The equation for the learning factor (μ) in the LMS with the stochastic gradient descent (SGD) algorithm for updating the adaptive parameters of the Γ(x) matrix is as follows:

$\mu=\delta /\|\hat{x}(k)\|^2$               (52)

with δ is a positive parameter that controls the learning speed and $\|\hat{x}(k)\|^2$ is the Euclidean norm of the state estimation vector $\hat{x}(k)$.

Note that δ controls the learning speed. A larger δ means a faster update of the adaptive parameters, while a smaller δ means a slower update. While $\|\hat{x}(k)\|^2$ is used to normalize the updating of adaptive parameters. This prevents the adaptive parameters from becoming too large or too small.