# Hamiltonian-Based Hybrid Control of Permanent Magnet Synchronous Motor Drives for DC Microgrid Applications

Thong-In Suyata, Matheepot Phattanasak, Pongsiri Mungporn, Burin Yodwong, Ehsan Jamshidpour, Serge Pierfederici, Babak Nahid-Mobarakeh, Nicu Bizon, Phatiphat Thounthong

► To cite this version:

Thong-In Suyata, Matheepot Phattanasak, Pongsiri Mungporn, Burin Yodwong, Ehsan Jamshidpour, et al.. Hamiltonian-Based Hybrid Control of Permanent Magnet Synchronous Motor Drives for DC Microgrid Applications. IEEE Access, 2025, 13, pp.209117-209129. 10.1109/ACCESS.2025.3641642. hal-05418277

HAL Id: hal-05418277

https://hal.univ-lorraine.fr/hal-05418277v1

Submitted on 6 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Received 25 November 2025, accepted 2 December 2025, date of publication 8 December 2025, date of current version 15 December 2025.

Digital Object Identifier 10.1109/ACCESS.2025.3641642

RESEARCH ARTICLE

# Hamiltonian-Based Hybrid Control of Permanent Magnet Synchronous Motor Drives for DC Microgrid Applications

THONG-IN SUYATA<sup>1,3</sup>, (Member, IEEE), MATHEEPOT PHATTANASAK<sup>1</sup>, (Senior Member, IEEE), PONGSIRI MUNGPORN<sup>1,2</sup>, (Member, IEEE), BURIN YODWONG<sup>1,2</sup>, (Senior Member, IEEE), EHSAN JAMSHIDPOUR<sup>1,3</sup>, (Senior Member, IEEE), NOUREDDINE TAKORABET<sup>1,3</sup>, (Member, IEEE), SERGE PIERFEDERICI<sup>1,4</sup>, BABAK NAHID-MOBARAKEH<sup>1,5</sup>, (Fellow, IEEE), NICU BIZON<sup>1,6</sup>, (Senior Member, IEEE), AND PHATIPHAT THOUNTHONG<sup>1,2</sup>, (Senior Member, IEEE)

Corresponding author: Phatiphat Thounthong (phatiphat.t@fte.kmutnb.ac.th)

This work was supported in part by the PubArt Program of the National University of Science and Technology POLITEHNICA Bucharest, Piteşti University Centre, Piteşti, Romania; in part by the Framework Agreement between the National University of Science and Technology POLITEHNICA Bucharest, Piteşti University Centre, and the King Mongkut's University of Technology North Bangkok, Thailand; in part by the International Research Partnership "Electrical Engineering-Thai French Research Center (EE-TFRC)" through the French Programme d'Investissements d'Avenir (PIA) Project of Lorraine Université d'Excellence (LUE), in cooperation between Université de Lorraine, France, and the King Mongkut's University of Technology North Bangkok, under Grant ANR-15-IDEX-04-LUE; in part by the National Research Council of Thailand (NRCT) through the Research Team Promotion Grant (Senior Mathivichai) under Grant N42A680561; in part by the National Science and Research Fund (NSRF), Thailand, via the Program Management Unit for Human Resources and Institutional Development, Research and Innovation under Grant B41G680025; and in part by the Scholarship from the National Science and Technology Development Agency (NSTDA) through the Ministry of Higher Education, Science, Research, and Innovation (MHESI), Thailand.

**ABSTRACT** This paper presents a Hamiltonian-based hybrid control strategy of permanent magnet synchronous motor (PMSM) drives for dc microgrid applications. Conventional field-oriented and direct torque control schemes often struggle with constant power load (CPL) conditions, which introduce negative impedance and destabilize the dc bus. To overcome this, the proposed method integrates Hamiltonian energy control with second-order homogeneous tracking control. The Hamiltonian framework provides energy shaping and damping injections to ensure global stability, while the tracking controller eliminates steady-state speed errors and improves dynamic response. An extended Hamiltonian model with integral action is also introduced to guarantee accurate q-axis current regulation without relying on complex interconnection gains. Experimental tests on a real PMSM using a dSPACE MicroLabBox confirm the effectiveness of the approach, demonstrating smooth acceleration, robust speed regulation under load disturbances, and stable performance during speed reversal. These results highlight the practicality of Hamiltonian-based hybrid control for advanced EV and microgrid applications.

**INDEX TERMS** Constant power load (CPL), Hamiltonian energy control, Lyapunov stability, permanent-magnet synchronous motor (PMSM), second-order homogeneous tracking control.

The associate editor coordinating the review of this manuscript and approving it for publication was Zhengmao Li.

## I. INTRODUCTION

Electric vehicles (EVs) are a key pathway to reducing greenhouse gas emissions in road transport [1]. Their powertrains

<footnote n="1">¹Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand</footnote>

<footnote n="2">²Renewable Energy Research Centre, Thai-French Innovation Institute, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand</footnote>

<footnote n="3">³GREEN, Université de Lorraine, 54000 Nancy, France</footnote>

<footnote n="4">⁴Laboratoire Énergies and Mécanique Théorique et Appliquée, ENSEM, Université de Lorraine, 54054 Nancy, France</footnote>

<footnote n="5">⁵Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4L8, Canada</footnote>

<footnote n="6">⁶National University of Science and Technology POLITEHNICA Bucharest, Piteşti University Centre, 110040 Piteşti, Romania</footnote>
predominantly use permanent magnet synchronous motors (PMSMs) for traction due to high power density, efficiency, and compact size [2]. PMSM drives achieve 92–97% efficiency, far higher than induction or other motors, and maintain performance across wide speed ranges while producing less vibration and noise [3], [4]. These advantages have led to widespread adoption in commercial EVs [3]. Compared to induction motors (IMs) or switched reluctance motors (SRMs), PMSMs deliver greater torque density and drive-cycle efficiency [2], [4], while field-weakening control enables operation from standstill to highway speeds [5], [6]. Despite their benefits, PMSMs still face challenges such as torque ripple [7]. Thus, advanced control strategies are essential to ensure rapid dynamic response, stability, and efficiency under diverse driving conditions.

The EV's electrical system can be modeled as a DC microgrid where multiple sources and loads share a common DC bus, as illustrated in Fig.1 [8]. The main source is the rechargeable battery, often supported by auxiliary units such as ultracapacitors or fuel cells, which supply inverter-driven PMSM traction motors and auxiliary loads [9], [10], [11]. Power electronic converters (DC/DC and DC/AC) link these components, forming a network of bidirectional energy exchange. For instance, the battery and regenerative braking act as storage, while the propulsion inverter represents a controlled power load, with high voltage alternative current (HVAC) and lighting as additional loads. A typical configuration includes a battery, a DC/DC boost converter for bus voltage regulation, and an inverter for the PMSM motor [12]. This structure improves efficiency by reducing AC/DC stages and mirrors trends in DC microgrids for integrating batteries and renewable sources [9], [10].

A key stability challenge in EVs is that the traction inverter-motor system often behaves as a constant power load (CPL) from the DC bus perspective [13], [14]. When tightly regulated power is applied, the drive exhibits negative incremental impedance: a voltage drop prompts higher current draw, further reducing bus voltage. This constant power behavior can destabilize the system and require additional control and damping beyond what a simple source-load setup provides.

To control PMSM traction motors, the EV industry has long relied on field-oriented control (FOC) and direct torque control (DTC) [15]. FOC, or vector control, regulates the motor's *d-q* axis currents through coordinate transformation, enabling decoupled flux and torque control similar to a DC motor. It is typically implemented with cascaded PI loops: inner current controllers track the *dq* axis currents, while an outer PI speed loop regulates torque and speed. This structure provides fast, decoupled control and remains the standard in commercial EV drive [16], [17]. DTC, in contrast, directly controls torque and flux without fixed switching frequency, yielding rapid response but higher ripple. Both schemes dominate PMSM applications in EVs due to their proven reliability [18], [19].

Subsequent research has extended beyond conventional PI-based FOC to improve performance and robustness. Approaches such as proportional-resonant (PR) controllers, sliding-mode, model-predictive control (MPC), and iterative/ repetitive methods aim to reduce torque ripple, enhance dynamics, and improve disturbance rejection [20], [21], [22]. For example, Sangeetha et al. integrated a PI speed regulator with an adaptive proportional integral resonant (PIR) current regulator, achieving significant reductions in speed ripple compared to standard PI control [7].

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Typical DC microgrid configuration used in EV.</figcaption></figure>

While conventional controllers (e.g., PI-based cascades) effectively regulate PMSM drives under nominal conditions, they do not inherently address instability caused by constant power loads (CPLs) on the DC bus. When the inverter-motor subsystem behaves like a CPL, it introduces a negative incremental impedance that can trigger destabilizing feedback loops beyond the capability of linear controllers to counteract [23]. These controllers are typically designed assuming a stiff DC supply and are tuned for resistive or moderately dynamic loads, not for the nonlinear effects of regulated power draw [24]. Consequently, in weak DC buses with limited capacitance or source impedance, even well-tuned FOC systems may suffer from oscillations and poor damping. Prior studies confirm that traditional controllers may only “imperceptibly guarantee” stability under CPL conditions [25], and DC microgrid research shows CPLs often degrade bus voltage regulation and cause oscillatory transients unless special stabilization measures are implemented [26]. This limitation has motivated exploration of advanced control strategies capable of explicitly countering CPL-induced instabilities.

A promising solution is Hamiltonian-based control (HCL), also known as passivity-based control, which models the motor converter system as a port-Hamiltonian system and stabilizes it via energy shaping and damping injection. By leveraging natural passive dynamics, adding virtual
resistance or modifying interconnections, the closed loop can be made energy-dissipated, even with CPLs. Ortega et al. pioneered such controllers, demonstrating that regulating the system's total energy rather than only voltage or current ensure stability by dissipating excess energy from disturbances or negative impedance interaction [27].

Recent work increasingly applies Hamiltonian/passivity-based control in power electronics and drive systems. In DC microgrids with CPLs, these methods effectively mitigate voltage instability and improve dynamic response [28]. Virtual impedance injection has been shown to stabilize boost and interleaved converters under CPLs, where PI regulators fail [29]. EV applications also benefit, with passivity-based control improving onboard charger stability under grid disturbances [28]. For motor drives, the port-controlled Hamiltonian method enhances PMSM speed regulation, providing robustness against uncertainties [10]. Qiao et al. further extended this approach to include iron loss and saturation effects, achieving stable speed control within voltage limits [30]. Collectively, these studies highlight a clear trend: Hamiltonian-based control offers a robust solution for nonlinear effects and CPL challenges, motivating its use in advanced PMSM drive systems.

To demonstrate the advantages of the HCL, a performance comparison with the cascade PI controllers is illustrated in Fig. 2. This analysis considers a four-phase interleaved FC boost converter by Thounthong et al. [8]. In the constant resistive load (CRL) scenario shown in Fig. 2(a), both control strategies deliver comparable transient profiles due to finely tuned PI gains. Nonetheless, HCL exhibits better damping and smoother voltage transitions, contributing to enhanced stability margins. Under the CPL condition in Fig. 2(b), the PI controller suffers from pronounced oscillations caused by the CPL's negative incremental impedance, which can destabilize the system. In contrast, the HCL maintains a well-controlled and stable response, clearly demonstrating its robustness against nonlinear and destabilizing load conditions.

Unlike conventional PI control, the HCL design is derived from a Lyapunov stability framework that explicitly guarantees passivity and global asymptotic convergence. This Lyapunov-based formulation ensures that any deviation in the system's energy is dissipated through the defined interconnection and dissipation matrices, providing a mathematically verifiable stability margin that linear controllers cannot ensure.

Furthermore, when the same Hamiltonian principle is applied to all converters and drives in the DC microgrid configuration shown in Fig. 1, the entire networked system becomes inherently stable and energy balanced. This system-level stability property is a key advantage of the Hamiltonian approach over traditional decentralized PI regulation, where interactions among CPLs often lead to global instability.

This article presents an HCL for PMSM vector control systems integrated within a DC microgrid configuration.

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Performance comparison between a cascade PI control and a studied HCL by P. Thounthong [8] for DC bus voltage regulation at 110 V (FC input voltage: 50 V). (a) Load CRL: 12.1 Ω (1 kW) → 8.0667 Ω (1.5 kW), (b) Load CPL: 3.5 kW → 4.5 kW.</figcaption></figure>

The primary goal is to ensure stable and disturbance-free torque control despite fluctuations in the DC microgrid. To enhance torque and speed performance, the proposed HCL is hybridized with a second-order homogeneous tracking controller. This combination improves dynamic response and guarantees robust operation under dynamic conditions. The main contributions of this study are outlined as follows:

**C1:** An extended Hamiltonian model is proposed, incorporating an integral term that ensures the *q*-axis current precisely tracks its reference $i_{qREF}$, enabling steady-state accuracy.

**C2:** Zero steady-state speed errors in the PMSM drive system are achieved by generating the $i_{qREF}$ using a second-order homogeneous tracking control

**C3:** Development of an extended power plant model and a Hamiltonian control law that eliminates the need for the interconnection gain (denoted as $K^*$ [11], [31] or $K_I$ [9], [32] in prior works), whose computation is complex and may result in singularities.

The remainder of this paper is organized as follows. Section II introduces the fundamentals of Port-Hamiltonian modeling and the formulation of the Hamiltonian control law. Section III describes the PMSM drive system model and details the improved Hamiltonian control laws, the generation of the desired reference, the stability proof, and the overall control conclusion. Section IV presents simulation and experimental validation, including the test platforms and experimental results that demonstrate the effectiveness of the proposed strategy. Finally, Section V provides concluding remarks and highlights possible directions for future research.
## II. PORT-HAMILTONIAN MODEL AND CONTROL

### A. PORT-HAMILTONIAN MODELING

Port-Hamiltonian modeling provides a comprehensive framework for representing nonlinear systems. The general model is expressed as follows [27]:

$$ \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + \mathbf{g}(\mathbf{x}) \cdot \mathbf{u} + \xi \quad (1) $$

where $\mathbf{f}(\mathbf{x})$ denotes the nonlinear dynamics of the system, $\mathbf{g}(\mathbf{x})$ denotes input matrix, $\mathbf{u}$ denote control input, and $\xi$ represents external disturbances.

Following this, equation (1) is transformed into the standard Hamiltonian energy model, defining a system as a dissipative port-controlled Hamiltonian system, formulated as [11] and [27]:

$$ \dot{\mathbf{x}} = [\mathbf{J}(\mathbf{x}) - \mathbf{R}(\mathbf{x})] \cdot \frac{\partial \mathbf{H}(\mathbf{x})}{\partial x} + \mathbf{g}(\mathbf{x}) \cdot \mathbf{u} + \xi \quad (2) $$

In this representation, the Hamiltonian function $\mathbf{H}(\mathbf{x})$ quantifies the energy stored within the system, where $\mathbf{x}$ denotes the vector of state variables and $\mathbf{u}$ represents the vector of control variables. Additionally, $\xi$ signifies the vector of external disturbances, and $\mathbf{g}(\mathbf{x})$ is the input matrix. The matrix $\mathbf{J}(\mathbf{x})$ serves as the connection matrix, while $\mathbf{R}(\mathbf{x})$ represents the dissipation matrix subject to the condition $\mathbf{R}(\mathbf{x}) = \mathbf{R}^T(\mathbf{x}) \ge 0$.

### B. HAMILTONIAN CONTROL LAW

The main objective of the Hamiltonian control law is to establish an energy storage function linked to a specified equilibrium point $\mathbf{x}_d$ by adjusting the desired connection matrix $\mathbf{J}_d(\mathbf{e})$, the desired dissipation matrix $\mathbf{R}_d(\mathbf{e})$, and the reformulated control law. Ensuring system stability hinges on selecting a closed-loop Hamiltonian function $H_d(\mathbf{e})$, thereby configuring the closed-loop system as follows:

$$ \dot{\mathbf{x}} = [\mathbf{J}_d(\mathbf{e}) - \mathbf{R}_d(\mathbf{e})] \cdot \frac{\partial \mathbf{H}_d(\mathbf{e})}{\partial e} + \frac{d\mathbf{x}_d(t)}{dt} \quad (3) $$

with; $\mathbf{J}_d(\mathbf{x}) = -\mathbf{J}_d^T(\mathbf{e})$ and $(\mathbf{R}_d(\mathbf{e}) = \mathbf{R}_d^T(\mathbf{e}) \ge 0)$, therefore, integrating (2) and (3):

$$ \begin{aligned} & [\mathbf{J}_d(\mathbf{e}) - \mathbf{R}_d(\mathbf{e})] \cdot \frac{\partial \mathbf{H}_d(\mathbf{e})}{\partial e} + \frac{d\mathbf{x}_d(t)}{dt} \\ & = [\mathbf{J}(\mathbf{x}) - \mathbf{R}(\mathbf{x})] \cdot \frac{\partial \mathbf{H}(\mathbf{x})}{\partial x} + \mathbf{g}(\mathbf{x}) \cdot \mathbf{u} + \xi \end{aligned} \quad (4) $$

Finally, equation (4) allows the calculation of the state feedback control input $\mathbf{u}$:

$$ \mathbf{u} = \gamma(x) \quad (5) $$

where $\gamma(x)$ represents the state-feedback control law derived from (4), defining the mapping between the system state and the control input.

## III. PMSM MODELLING AND CONTROL LAWS

### A. PMSM DRIVE SYSTEM MODEL

The PMSM drive system under study is illustrated in Fig. 3, comprising three-phase inverters interfaced with the

<figure><img src="image_4.png" /><figcaption>FIGURE 3. The proposed system for driving the PMSM system.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIGURE 4. Equivalent circuit of the driving PMSM system.</figcaption></figure>

PMSM and the load. The sinusoidal pulse-width modulation SPWM technique is employed in the inverter to generate an output voltage with minimal harmonics. Neglecting the effects of magnetic saturation, the equivalent circuit of the PMSM/inverter drive in the dq-synchronous rotating frames is depicted in Fig. 4. Furthermore, an extended model is applied to the system by incorporating $v_{td}$ and $v_{tq}$, which represent voltage drops associated with losses in the motor and inverter. The extended model is employed because a state observer, as presented in [33], is used to estimate the load torque. Based on this representation, the differential equations governing the dynamics of the PMSM/inverter system can be expressed as follows:

$$ \frac{di_d}{dt} = \frac{1}{L_d} (v_d - Ri_d + \omega_e L_q i_q - v_{td}) \quad (6) $$

$$ \frac{di_q}{dt} = \frac{1}{L_q} (v_q - Ri_q - \omega_e L_d i_d - \omega_e \omega_m - v_{tq}) \quad (7) $$

$$ \frac{d\omega_m}{dt} = \frac{1}{J} (T_e - B\omega_m - T_L) \quad (8) $$

with

$$ \begin{aligned} \omega_e &= nP\omega_m \\ T_e &= nPi_q (\varphi_m - (L_q - L_d) i_d) \end{aligned} \quad (9) $$

where

$v_d$ and $v_q$: the direct and quadrature motor voltages (V), respectively.
$i_d$ and $i_q$: the direct and quadrature motor currents (A), respectively.

$R$ is the static resistance of motor (Ω),

$L_d$ and $L_q$: d and q-axis inductance (H), respectively.

$nP$: number of pole pairs.

$\omega_e$ and $\omega_m$: electrical and angular frequency (rad/s) respectively.

$\varphi_m$: permanent magnet flux linkage (Wb).

$T_e$ and $T_L$: electromagnetic and load torque (Nm).

$B$: viscous friction coefficient (N·m·s/rad).

$J$ is the moment of inertia of the lumped rotor (kg·m²).

## B. IMPROVED HAMILTONIAN CONTROL LAWS

To achieve the 1st contribution (C1) of this work, which aims to ensure that the q-axis current ($i_q$) of non-salient pole synchronous motor drives equals to the q-axis current reference ($i_{qREF}$), we introduce a new variable, $\lambda_I$, representing the integral control action. This can be expressed as follows:

$$ \lambda_I = K_I \int (i_q - i_{qREF}) dt \quad (10) $$

where $K_I$ denotes the gain of the controller, while $i_{qREF}$ indicates the reference for the q-axis current.

Considering the simplification that $L$ equals both $L_d$ and $L_q$ for non-salient pole motors and in accordance with (6), (7), and (10), the extended PMSM drive system model can be reformulated in the Hamiltonian energy structure using (2), as follows:

$$ \begin{aligned} \mathbf{x} &= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} i_d \\ i_q \\ \lambda_I \end{bmatrix}, \quad \mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} = \begin{bmatrix} v_d \\ v_d \end{bmatrix}, \\ \xi &= \begin{bmatrix} -v_{td}/L \\ -(v_{tq} + nP\varphi_m\omega_m)/L \\ -K_I i_{qREF} \end{bmatrix} \end{aligned} \quad (11) $$

The mathematical expression for the Hamiltonian energy function is a quadratic form and can be derived as follows:

$$ \mathbf{H}(\mathbf{x}) = \frac{1}{2} \mathbf{x}^\mathrm{T} \mathbf{Q} \mathbf{x} \quad (12) $$

with

$$ \mathbf{Q} = \mathrm{diag}\{L, L, 1/K_I\} \quad (13) $$

Therefore, the energy function is expressed in the quadratic form as follows:

$$ H(x) = \frac{1}{2} [Lx_1^2 + Lx_2^2 + K_I^{-1}x_3^2] \quad (14) $$

and

$$ \frac{\partial H(x)}{\partial x} = \left[ Lx_1 Lx_2 K_I^{-1} x_3 \right]^T \quad (15) $$

The matrix $\mathbf{J}$, $\mathbf{R}$ and $\mathbf{g}$ are provided below:

$$ \mathbf{J}(\mathbf{x}) = \begin{bmatrix} 0 & \frac{nP\omega_m}{L} & 0 \\ -\frac{nP\omega_m}{L} & 0 & 0 \\ 0 & \frac{K_I}{L} & 0 \end{bmatrix} \quad (16) $$

$$ \mathbf{R}(\mathbf{x}) = \begin{bmatrix} \frac{R}{L^2} & 0 & 0 \\ 0 & \frac{R}{L^2} & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \mathbf{g}(\mathbf{x}) = \begin{bmatrix} \frac{1}{L} & 0 \\ 0 & \frac{1}{L} \\ 0 & 0 \end{bmatrix} \quad (17) $$

Subsequent, to implement the Hamiltonian control law as described in (3), the required reference variable ($x_d$) and the error variable ($e$) are obtained from:

$$ \begin{aligned} \mathbf{e} &= \begin{bmatrix} e_1 & e_2 & e_3 \end{bmatrix}^\mathrm{T} \\ &= \begin{bmatrix} x_1 - x_{1d} & x_2 - x_{2d} & x_3 \end{bmatrix}^\mathrm{T}; (x_{3d} = 0) \end{aligned} \quad (18) $$

Refer to (12), the desired energy of the Hamiltonian is as given:

$$ \mathbf{H}_d(\mathbf{e}) = \frac{1}{2} \mathbf{e}^\mathrm{T} \mathbf{Q} \mathbf{e} \quad (19) $$

Based on (13), (18), and (19), the $H_d(\mathbf{e})$, is determined in the following manner:

$$ H_d(\mathbf{e}) = \frac{1}{2} \begin{bmatrix} Le_1^2 + Le_2^2 + K_I^{-1}e_3^2 \end{bmatrix} \quad (20) $$

The gradient of $H_d(\mathbf{e})$ with respect to error ($e$) can be expressed as:

$$ \frac{\partial H_d(\mathbf{e})}{\partial e} = \begin{bmatrix} Le_1 Le_2 K_I^{-1} e_3 \end{bmatrix}^\mathrm{T} \quad (21) $$

Next, in conventional Port-Hamiltonian formulations [9], [32], an interconnection gain $K_J$ is often introduced to modify the system interconnection matrix $\mathbf{J}_C$ in order to ensure closed-loop passivity. However, $K_J$ is typically defined as a function of the tracking error and may become singular as the error approaches zero, resulting in discontinuities and instability in the controlled system.

To achieve the 3rd contribution (C3), the proposed approach systematically designs the controller connection matrix $\mathbf{J}_C$ to embed the interconnection effect directly into the closed-loop Hamiltonian structure, thereby eliminating the need for $K_J$. The controller connection matrix $\mathbf{J}_C$ and controller dissipation matrix $\mathbf{R}_C$ are represented as:

$$ \mathbf{J}_C = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -\frac{K_I}{L} \\ 0 & 0 & 0 \end{bmatrix} \quad (22) $$

$$ \mathbf{R}_C = \begin{bmatrix} \frac{K_{R1}}{L^2} & 0 & 0 \\ 0 & \frac{K_{R1}}{L^2} & 0 \\ 0 & 0 & 0 \end{bmatrix} \quad (23) $$

where $K_{R1}$ is the tuning controller parameters. Next, the desired controller connection matrix and controller dissipation matrix are defined as follows.

$$ \mathbf{J}_d = \mathbf{J} + \mathbf{J}_C \quad \text{and} \quad \mathbf{R}_d = \mathbf{R} + \mathbf{R}_C \quad (24) $$

The desired controller connection matrix $\mathbf{J}_d = -\mathbf{J}_d^T$ and controller dissipation matrix $\mathbf{R}_d = \mathbf{R}_d^T > 0$ are expressed as:

$$ \mathbf{J}_d = \begin{bmatrix} 0 & \frac{nP\omega_m}{L} & 0 \\ -\frac{nP\omega_m}{L} & 0 & -\frac{K_I}{L} \\ 0 & \frac{K_I}{L} & 0 \end{bmatrix} \quad (25) $$
$$ \mathbf{R}_d = \begin{bmatrix} \frac{(R+K_{R1})}{L^2} & 0 & 0 \\ 0 & \frac{(R+K_{R1})}{L^2} & 0 \\ 0 & 0 & 0 \end{bmatrix} \quad (26) $$

Finally, by substituting (11), (15)-(17), (21), and (25)-(26) into (4) and conducting algebraic manipulations, we derive a set of four equations. Solving (4) enables us to obtain estimates for $u_1$ and $u_2$ as follows:

$$ u_1 = v_{td} + L\dot{x}_{1d} - K_{R1}e_1 + Rx_{1d} - nPL\omega_m x_{2d} \quad (27) $$

$$ \begin{aligned} u_2 = & vtq + L\dot{x}_{2d} - x_3 - K_{R1}e_2 + Rx_{2d} \\ & + nP\varphi_m\omega_m + nPL\omega_m x_{1d} \end{aligned} \quad (28) $$

### C. GENERATION OF DESIRED REFERENCE

To achieve the 2nd contribution (C2) related to speed regulation by generating the q-axis current reference ($x_{2d} = i_{qREF}$) using a second-order homogeneous tracking control to eliminate speed errors. In this approach, the speed command ($\omega_{COM}$) is used to generate the reference speed profile ($\omega_{REF}$) for motion trajectory planning, as expressed in the following form:

$$ \frac{\omega_{\text{REF}}(s)}{\omega_{\text{COM}}(s)} = \frac{\omega_{n1}^2}{s^2 + 2\xi_1\omega_{n1}s + \omega_{n1}^2} \quad (29) $$

where $\xi_1$ and $\omega_{n1}$ is the desired dominant damping ratio and natural frequency, respectively, of the motion trajectory planning.

Then the new variable of the tracking control can be set as follows:

$$ \mu = \dot{\omega}_{\text{REF}} + K_1 (\omega_{\text{REF}} - \omega_{\text{m}}) + K_2 \int (\omega_{\text{REF}} - \omega_{\text{m}}) dt \quad (30) $$

where

$$ K_1 = 2\xi_2\omega_{n2}; \quad K_2 = \omega_{n2}^2 $$

where $\xi_2$ represents the desired dominant damping, while $\omega_{n2}$ indicates the desired natural frequency of the second-order homogeneous tracking control.

Finally, by combining (8), (9), and (30), the q-axis current reference ($x_{2d}$) can be written as follows:

$$ x_{2d} = i_{qREF} = (J\mu + B\omega_{m} + T_{L})/nP\varphi_{m} \quad (31) $$

### D. STABILITY PROOF

To establish the stability of the proposed dynamical system, define the Lyapunov function as:

$$ V = \sum_{i=1}^{n} V_i \quad (32) $$

Next, let $V_i$ be the Lyapunov candidate function for each controller in the proposed system. The Lyapunov candidate of the closed-loop Hamiltonian system is given by (20) and can therefore be expressed as:

$$ V_1 = H_d(\mathbf{e}) > 0 \quad (33) $$

The time derivative of $V_1$ can be obtained as:

$$ \begin{aligned} \dot{V}_1 &= \frac{dH_d(\mathbf{e})}{de} = \left(\frac{\partial H_d(\mathbf{e})}{\partial e}\right)^T \cdot \dot{\mathbf{x}} \\ &= -\left(\frac{\partial H_d(\mathbf{e})}{\partial e}\right)^T \mathbf{R}_d \frac{\partial H_d(\mathbf{e})}{\partial e} < 0 \end{aligned} \quad (34) $$

Similarly, following the second-order homogeneous tracking control in (30), we define $e_y = (\omega_m - \omega_{REF})$ and $X_y = \int (\omega_m - \omega_{REF})dt$, so that the Lyapunov function is given as:

$$ V_2 = \frac{1}{2} J e_y^2 + \frac{1}{2} J K_2 X_y^2 \geq 0 \quad (35) $$

The derivative of the function $V_2$ is given by:

$$ \dot{V}_2 = J e_y \dot{e}_y + J K_2 X_y \dot{X}_y \quad (36) $$

The time derivatives of $e_y$ and $X_y$ are obtained (30), which capture the error dynamics of the speed control loop, as summarized in (37).

$$ \left\{ \begin{aligned} \dot{e}_y &= -K_1 e_y - K_2 X_y \\ \dot{X}_y &= e_y \end{aligned} \right. \quad (37) $$

Substituting (37) in (36) yields (38):

$$ \dot{V}_2 = -Je_yK_1e_y - Je_yK_2X_y + J K_2X_ye_y < 0 \quad (38) $$

Finally, substituting (34) and (38) into (32) establishes the stability of the entire system via a Lyapunov argument, thereby ensuring global convergence to the desired equilibrium point.

$$ \dot{V} = - \left( \frac{\partial H_d(\mathbf{e})}{\partial e} \right)^T \mathbf{R}_d \frac{\partial H_d(\mathbf{e})}{\partial e} - J K_1 e_y^2 \quad (39) $$

<figure><img src="image_6.png" /><figcaption>FIGURE 5. Proposed Hamiltonian energy function based on speed/torque control with state observer of a PMSM drive.</figcaption></figure>

Since $V > 0$ & $\dot{V} < 0$, the Lyapunov function $V$ satisfies the conditions for large-signal stability. Consequently, the system is guaranteed to converge globally and asymptotically to the equilibrium point, thereby confirming the robustness of the proposed control law.

The Lyapunov-based formulation used in this proof forms the foundation of the proposed Hamiltonian control law. It ensures that the system's energy remains bound and monotonically decreasing, a property later verified experimentally
<figure><img src="image_7.png" /><figcaption>FIGURE 6. Experimental test-bench for testing the drive system.</figcaption></figure>

in Section IV, where the drive maintains smooth, oscillation-free performance under CPL disturbances.

This theoretical experimental consistency highlights the key advantage of Hamiltonian control over conventional PI regulation, which cannot guarantee global asymptotic stability [8].

### E. CONTROL CONCLUSION

The control diagram of the proposed vector control based on the Hamiltonian control law is shown in Fig. 5. In this scheme, the *q*-axis current reference ($x_{2d}$) is determined using the second-order homogeneous tracking control described in Subsection C. In parallel, the voltage variations on the *dq*-axis, denoted as $\hat{v}_{td}$ and $\hat{v}_{tq}$, together with the load torque ($\hat{T}_L$) of the PMSM drive system, are estimated using Lyapunov-based state observers as presented in [33]. For non-salient pole machines with equal inductances ($L_d = L_q$), the *d*-axis current reference ($i_{dREF} = x_{1d}$) is set to zero. Moreover, the *q*-axis current reference ($x_{2d}$) is integrated according to (10) to achieve the first contribution (C1). The integrator term plays a crucial role in ensuring that the measured *q*-axis current ($i_q$) accurately tracks its reference ($x_{2d}$). Finally, the proposed Hamiltonian control algorithm processes all of this information to compute the control voltages $v_d$ and $v_q$ for the PMSM drive system.

Although this study focuses on a PMSM drive, the proposed Hamiltonian-based hybrid control law can be readily applied to other port-Hamiltonian systems such as DC/DC converters, fuel-cell sources, and hybrid energy storage units, since its formulation depends on the system's energy and interconnection structure rather than on a specific model.

## IV. SIMULATION AND EXPERIMENTAL VALIDATION

### A. EXPERIMENTAL TEST PLATFORMS

To evaluate the effectiveness of the proposed improved vector control strategy, which is based on the Hamiltonian control law and employs a Lyapunov observer [33] for the PMSM drive system, an experimental platform was developed.

<table><caption>TABLE 1. Circuit parameters.</caption><thead><tr><th>Variable</th><th>Quantity</th><th>Value/Unit</th></tr></thead><tbody><tr><td>P<sub>rated</sub></td><td>Rated Power</td><td>4 kW</td></tr><tr><td>n<sub>rated</sub></td><td>Rated Speed</td><td>3000 r/min</td></tr><tr><td>T<sub>rated</sub></td><td>Rated Torque</td><td>28 N·m</td></tr><tr><td>n<sub>P</sub></td><td>Number of Pole Pair</td><td>8</td></tr><tr><td>R</td><td>Resistance (Motor+Inverter)</td><td>0.71 Ω</td></tr><tr><td>L</td><td>Stator inductance (L<sub>d</sub>=L<sub>q</sub>)</td><td>0.46 mH</td></tr><tr><td>φ<sub>m</sub></td><td>Magnetic flux</td><td>0.44 Wb</td></tr><tr><td>B</td><td>Fiction coefficient</td><td>0.013 N·m·s/rad</td></tr><tr><td>J</td><td>Equivalent inertia</td><td>0.130 kg·m<sup>2</sup></td></tr><tr><td>V<sub>BUS</sub></td><td>DC bus voltage</td><td>300 V</td></tr><tr><td>f<sub>s</sub></td><td>Switching frequency</td><td>16 kHz</td></tr></tbody></table>

<table><caption>TABLE 2. Control parameters.</caption><thead><tr><th>Variable</th><th>Quantity</th><th>Value/Unit</th></tr></thead><tbody><tr><td>K<sub>R1</sub></td><td>Gain tuning parameter</td><td>10 Ω</td></tr><tr><td>K<sub>I</sub></td><td>Gain controller</td><td>2.25</td></tr><tr><td>ξ<sub>1</sub></td><td>Damping ratio of trajectory profiles</td><td>1 pu</td></tr><tr><td>ω<sub>n1</sub></td><td>Natural frequency of trajectory profiles</td><td>4 rad·s<sup>-1</sup></td></tr><tr><td>ξ<sub>2</sub></td><td>Damping ratio of tracking control</td><td>1 pu</td></tr><tr><td>ω<sub>n2</sub></td><td>Natural frequency of tracking control</td><td>40 rad·s<sup>-1</sup></td></tr><tr><td>I<sub>qMin</sub></td><td>Minimum quadrature motor current</td><td>-40 A</td></tr><tr><td>I<sub>qMax</sub></td><td>Maximum quadrature motor current</td><td>+40 A</td></tr></tbody></table>

Fig. 6 shows the implemented test setup, which consists of the following equipment: 1) a real-time DS1202 dSPACE Micro-LabBox, operated via the dSPACE ControlDesk software; 2) a TSA600 DC programmable power supply from Magna-Power; 3) a custom-designed three-phase inverter; and 4) a PMSM coupled with a power crust and powder brake serving as the motor load. The specifications of the PMSM drive system and the controller settings are summarized in Tables 1 and 2, respectively.

Additionally, voltage and current were measured using LEM LV 25-P and LA 55-P closed-loop Hall-effect transducers, with total errors $\le \pm 0.9\%$ and $\pm 0.65\%$, respectively.

### B. EXPERIMENTAL RESULTS

To assess the control law's performance in dynamic scenarios, an experimental test was conducted in three different dynamic operational settings. These settings were intentionally designed to present significant differences in acceleration control, speed regulation control, and the reverse direction of the PMSM.

#### 1) PMSM DRIVE ACCELERATION TEST

Firstly, the acceleration test was conducted by applying a step speed command $n_{COM}$ from 0 to 600 rpm. The experimental results are shown in Fig. 7. At the initial condition, the PMSM drive system operated with an estimated external load torque $\hat{T}_L = 0$ N·m at $n_{COM} = 0$ rpm. At $t = 0.5$ s, the speed command $n_{COM}$ (Ch4) stepped from 0 to 600 rpm. The speed reference $n_{REF}$ (Ch1) exhibited a gradual increase due to the second-order filter implemented in the trajectory profile (29). Consequently, the measured speed $n_M$ (Ch2) closely tracked $n_{REF}$, showing a smooth transition with a settling time of
approximately 1.5 s. Simultaneously, the tracking control variable $\mu$ (Ch3) rapidly increased to $500 \text{ rad} \cdot \text{s}^{-1}$ to compensate for the speed error by increasing the q-axis current reference ($x_{2d}$) as given in (31), before gradually decreasing to zero once $n_M$ equaled $n_{REF}$. As a result, the q-axis current $i_q$ (Ch7) rose to 40 A during the transition and stabilized at 19 A in steady state. Concurrently, the estimated load torque $\hat{T}_L$ (Ch9), and q-axis voltage losses $\hat{v}_{tq}$ (Ch10) peaked at 46 N·m and -23 V, respectively, before settling to 32 N·m and -11 V at 600 rpm. Meanwhile, the d-axis voltage loss $\hat{v}_{td}$ (Ch11) was observed at 4 V, showing minor fluctuations during the speed transition. On the other side, the q-axis voltage $v_q$ increased from zero to 118 V, while the d-axis voltage $v_d$ decreased to -28 V, in order to control the inverter operating at a speed of 600 rpm. Finally, the d-axis current $i_d$ (Ch8) was regulated at 0 A owing to the non-salient pole control. The proposed controller achieved a settling time of approximately 1.5 s for the speed step from 0 to 600 rpm, with no observable overshoot (< 1%). The q-axis current peak of 40 A smoothly decayed to the steady-state value of 19 A, indicating fast torque buildup with minimal transient stress and confirming effective energy shaping during acceleration.

## 2) LOAD TORQUE DISTURBANCE TEST

Secondly, the speed command $n_{COM}$ and d-axis current reference $i_{dREF}$ were set to 600 rpm and 0 A, respectively, while the external load torque was varied. An external load torque steps change from 5 N·m to 50 N·m, as depicted in Fig. 8. Based on the obtained experimental results, when the external load torque changed at $t = 0.5$ s, the estimated load torque $\hat{T}_L$ (Ch 9) increased to 50 N·m. Meanwhile, the estimated q-axis voltage losses $\hat{v}_{tq}$ (Ch10) and estimated d-axis voltage losses $\hat{v}_{td}$ (Ch11) were estimated at -18 and 10 V, respectively. Consequently, the q-axis current $i_q$ (Ch7) rose and remained constant at 30 A, mirroring the behavior of the phase current $i_A$ (Ch4), while the d-axis current $i_d$ (Ch8) was regulated to 0 A. It's noteworthy that the speed measurement $n_M$ (Ch2) displayed minimal variations before stabilizing at $t = 1.5$ s. The tracking control variable $\mu$ (Ch3) actively compensated for errors in the PMSM drive system only during dynamic tests, being equal to zero during steady-state operation. Similarly, to the initial experimental tests, the q-axis voltage $v_q$ increased from 110 V to 120 V, while the d-axis voltage $v_d$ decreased to -38 V. This adjustment was necessary to control the specifically designed inverter, operating at a speed of 600 rpm, under an external load torque of 50 N·m. When the external torque increased from 5 N·m to 50 N·m, the speed deviation was less than $\pm 2\%$ and recovered within 1.2 s, while the q-axis current response remained well damped with oscillation.

## 3) REVERSING SPEED TEST

The final experimental test was performed to evaluate the performance of the proposed Hamiltonian-based hybrid control strategy under a severe operating condition involving speed reversal of the PMSM drive. As shown in Fig. 9, the

<figure><img src="image_8.png" /><figcaption>FIGURE 7. Experimental results: acceleration performance from 0 rpm to 600 rpm.</figcaption></figure>

speed command $n_{COM}$ (Ch4) was stepped from -600 rpm to +600 rpm at $t = 0.5$ s, while the d-axis current reference was maintained at 0 A. The speed reference $n_{REF}$ (Ch1) transitioned smoothly due to the second-order trajectory filter, and the measured speed $n_M$ (Ch2) closely tracked this profile, achieving steady-state convergence within approximately 1.5 s. During the transient, the tracking control
<figure><img src="image_9.png" /><figcaption>FIGURE 8. Experimental results: the speed response at $n_{COM} = 600$ rpm, and a stepped load torque from 5 N·m to 50 N·m.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIGURE 9. Experimental results: the performance of speed response during speed command stepped from –600 rpm to 600 rpm.</figcaption></figure>

variable $\mu$ (Ch3) rapidly increased to about $630 \text{ rad·s}^{-1}$ to compensate for the large speed error by raising the q-axis current reference, and then gradually returned to zero once $n_M$ equaled $n_{REF}$. Correspondingly, the q-axis current $i_q$ (Ch7) surged to nearly 40 A to provide the acceleration torque required for reversal, before stabilizing at around 1 A during steady-state operation, while the

d-axis current $i_d$ (Ch8) remained regulated at 0 A due to the non-salient pole machine design. The inverter voltages also reflected the reversal process, with the q-axis voltage $v_q$ (Ch5) switching polarity from approximately –106 V to +106 V and the d-axis voltage $v_d$ (Ch6) changing from +8 V to –8 V. Furthermore, the estimated load torque $\hat{T}_L$ (Ch9) varied from –1.5 N·m to 30 N·m during the transient
and settled at 1.5 N·m in steady-state operation, consistent with the reversal of rotational direction. Meanwhile, the estimated voltage losses $\hat{v}_{tq}$ (Ch10) and $\hat{v}_{td}$ (Ch11) remained relatively small, at approximately –4 V and –6 V, respectively, exhibiting only minor fluctuations throughout the transition. The system tracked the command step from –600 rpm to +600 rpm within 1.5 s, with smooth current and voltage transitions and no overshoot in speed or torque reversal. The measured q-axis current exhibited a controlled peak of 40 A during reversal and rapidly settled to approximately 1 A in steady-state, confirming effective damping injection and symmetric dynamic performance in both rotational directions.

### C. SIMULATION ANALYSIS OF PMSM DRIVE WITH LC-FILTERED DC BUS UNDER CPL DISTURBANCES

Although the experimental results presented in Section IV-B confirm the practical effectiveness of the proposed HCL, they do not fully clarify the mechanism underlying instability when PMSM drives behave as CPLs. In many DC microgrid implementations, particularly those employing LC-filtered DC interfaces, the interaction between the filter dynamics and the CPL's negative incremental impedance during load transients can induce oscillatory behavior and compromise system stability. To replicate this phenomenon and allow controlled comparison, an LC-filtered DC supply was modeled in the simulation study.

Therefore, a dedicated simulation study was performed using an LC-filtered DC supply to gain deeper physical insight into this phenomenon and to complement the experimental observations. This setup enables controlled evaluation of CPL disturbances, allowing direct comparison between the conventional PI controller and the proposed HCL under both small (500 W to 1000 W) and large (1500 W to 3900 W) step load changes at a constant speed of 600 rpm.

For a fair comparison, the PI parameters were tuned according to widely adopted guidelines to achieve comparable closed-loop response characteristics under nominal operating conditions, thereby minimizing the influence of tuning. The resulting gains were set to $K_{pi} = 21.3$ and $K_{ii} = 571$ for the current loop, and $K_{p\omega} = 50.49$ and $K_{i\omega} = 12118$ for the speed loop.

Fig. 10 illustrates the system response under a small CPL disturbance. When using the classical PI controller, the DC-bus voltage exhibits oscillations of approximately 11 V peak-to-peak, indicating the presence of LC dynamics. In contrast, the proposed HCL reduces the oscillation amplitude to about 6 V, corresponding to an improvement of roughly 45% in voltage ripple suppression. It should be noted that, under this low-stress operating condition, both controllers maintain satisfactory performance in terms of motor speed regulation. The PI controller presents small deviations from reference speed, while the HCL achieves slightly tighter tracking with negligible disturbance-induced variation.

Under a severe CPL disturbance, the destabilizing effect becomes significantly more pronounced, as illustrated in

<figure><img src="image_11.png" /><figcaption>FIGURE 10. Dynamic response of the PMSM drive supplied through an LC filter during a step load change from 500 W to 1000 W at a constant speed of 600 rpm.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIGURE 11. Dynamic response of the PMSM drive supplied through an LC filter during a step load change from 1500 W to 3900 W at a constant speed of 600 rpm.</figcaption></figure>

Fig. 11. When regulated by the conventional PI controller, the DC bus voltage exhibits large and sustained oscillations of approximately 315 V peak to peak, with no clear damping trend. This persistence of high amplitude oscillations indicates that the system is driven toward marginal or even unstable behavior, as the PI control loop fails to counteract the negative incremental impedance characteristic of the CPL. In contrast, the proposed HCL achieves a substantial reduction in oscillation amplitude to around 110 V, corresponding to an approximate 65% improvement, while also providing a noticeably faster transient decay. Additionally, the motor speed under PI control shows observable fluctuation consistent with the voltage disturbance, whereas the HCL maintains nearly constant speed throughout the event. These results demonstrate that the PI-controlled system does not preserve stability under severe disturbance conditions, whereas the proposed HCL maintains a convergent and stable dynamic
response, effectively protecting the system's stability margin. This behavior is consistent with the passivity-based Hamiltonian design approach.

Overall, the simulation results confirm that the proposed HCL offers superior damping, improved voltage stability, and more reliable speed regulation than the conventional PI approach. These findings support the experimental observations in Section IV-B and provide a solid theoretical and practical foundation for the extended discussion in Section IV-D.

## D. DISCUSSION ON CONSTANT POWER LOAD EFFECT

The simulation results in Figs. 10 and 11 provide a deeper understanding of how CPL characteristics affect PMSM drive performance when interfaced through an LC-filtered DC bus. While both control strategies maintain stable operation under mild disturbances, only the HCL preserves stability under severe CPL-induced loading conditions. This outcome demonstrates the controller's inherent robustness in handling negative incremental impedance effects.

The experimental results are shown in Figs. 7 to 9 in Section IV-B, where the proposed HCL maintained smooth torque regulation and stable speed tracking under large-signal disturbances. From a system perspective, CPL-induced instability arises from rapid energy extraction during transients that excites the LC filter dynamics. The Hamiltonian-based control method actively mitigates this through energy shaping and passivity enforcement, whereas conventional PI control lacks the capacity to guarantee stability under such nonlinear interactions.

Furthermore, both simulation and experimental evidence confirm the advantage of the proposed Hamiltonian-based hybrid control from a Lyapunov stability perspective. Unlike the PI controller, which ensures only local stability, the Hamiltonian framework injects appropriate damping and shapes system energy, enabling global asymptotic stability even under severe CPL disturbances. Within the broader DC-microgrid structure shown in Fig. 1, adopting Hamiltonian-based control across multiple converters would promote energy balance and improve overall system stability.

## V. CONCLUSION

This paper has presented a Hamiltonian-based hybrid control strategy for permanent magnet synchronous motor (PMSM) drives operating in DC microgrids. By combining the Hamiltonian control law with a second-order homogeneous tracking controller, the proposed method achieves both global stability and precise speed regulation, even under the destabilizing effects of constant power loads (CPLs). An extended Hamiltonian model incorporating integral action further enhances current regulation without the need for complex interconnection gains.

Experimental validation was conducted through three representative tests: acceleration, load torque disturbance, and speed reversal. The results demonstrated smooth dynamic response, accurate speed tracking, and effective torque

regulation across all operating conditions. The controller successfully mitigated transient overshoot, ensured robustness against parameter variations, and maintained steady-state stability despite large-signal disturbances. In particular, the bounded behavior of estimated torque and voltage losses confirmed the controller's ability to suppress the negative impedance effects typically introduced by CPLs.

As a future direction, further refinement of the proposed control strategy will aim to reduce the remaining DC-bus voltage oscillations observed under certain operating conditions. Although the Hamiltonian-based hybrid controller significantly improves damping and mitigates CPL-induced instability, a small degree of DC-bus ripple is still present due to the interaction between the LC filter dynamics and the CPL characteristics. Addressing this limitation through enhanced energy-shaping mechanisms or coordinated multi-converter control will be an important step toward achieving fully ripple-free DC-bus performance. In addition, integrating the PMSM drive system, governed by the proposed algorithms, with a bi-directional battery converter in an electric vehicle (EV) powertrain represents a promising direction for improving overall power-flow management and ensuring stable operation in more complex DC microgrid environments.

## ACKNOWLEDGMENT

The authors would like to express their sincere appreciation to the Research, Development, and Technological Transfer Department of the Thai-French Innovation Institute, KMUTNB, for providing the research equipment used in this study.

## REFERENCES

[1] U. Fesli and M. B. Ozdemir, "Electric vehicles: A comprehensive review of technologies, integration, adoption, and optimization," *IEEE Access*, vol. 12, pp. 140908-140931, 2024, doi: 10.1109/ACCESS.2024.3469054.

[2] Z. Wang, T. W. Ching, S. Huang, H. Wang, and T. Xu, "Challenges faced by electric vehicle motors and their solutions," *IEEE Access*, vol. 9, pp. 5228-5249, 2021, doi: 10.1109/ACCESS.2020.3045716.

[3] M. Alzayed and H. Chaoui, "Direct voltage MTPA speed control of IPMSM-based electric vehicles," *IEEE Access*, vol. 11, pp. 33858-33871, 2023, doi: 10.1109/ACCESS.2023.3263815.

[4] M. S. Mubarok and T.-H. Liu, "An adjustable wide-range speed-control method for sensorless IPMSM drive systems," *IEEE Access*, vol. 10, pp. 42727-42738, 2022, doi: 10.1109/ACCESS.2022.3168390.

[5] X. Wan, S. Yang, Y. Li, Y. Shi, and J. Lou, "Minimization of cogging torque for V-type IPMSM by the asymmetric auxiliary slots on the rotor," *IEEE Access*, vol. 10, pp. 89428-89436, 2022, doi: 10.1109/ACCESS.2022.3201246.

[6] T. T. Nguyen, T. H. Nguyen, and J. W. Jeon, "Explicit model predictive speed control for permanent magnet synchronous motor with torque ripple minimization," *IEEE Access*, vol. 11, pp. 134199-134210, 2023, doi: 10.1109/ACCESS.2023.3335992.

[7] E. Sangeetha and V. P. Ramachandran, "Speed and current harmonics reduction using an adaptive proportional integral resonant controller for PMSM based electric vehicle drives," *Sci. Rep.*, vol. 15, no. 1, p. 26514, Jul. 2025, doi: 10.1038/s41598-025-09981-1.

[8] P. Thounthong, P. Mungporn, S. Pierfederici, D. Guilbert, N. Takorabet, B. Nahid-Mobarakeh, Y. Hu, N. Bizon, Y. Huangfu, P. Kumam, and P. Burikham, "Robust Hamiltonian energy control based on Lyapunov function for four-phase parallel fuel cell boost converter for DC microgrid applications," *IEEE Trans. Sustain. Energy*, vol. 12, no. 3, pp. 1500-1511, Jul. 2021, doi: 10.1109/TSTE.2021.3050783.
[9] P. Mungporn, U. Kamnarn, B. Yodwong, S. Khomfoi, S. Pierfederici, B. Nahid-Mobarakeh, G. Vitale, N. Bizon, and P. Thounthong, "Hamiltonian-based approach to enhance the stability of hybrid fuel cell and supercapacitor sources," *IEEE Open J. Ind. Appl.*, vol. 6, pp. 565-578, 2025, doi: 10.1109/OJIA.2025.3591144.

[10] Y. Zhao and H. Yu, "Cooperative control of deadbeat predictive and state error port-controlled Hamiltonian method for permanent magnet synchronous motor drives," *IET Electr. Power Appl.*, vol. 15, no. 10, pp. 1343-1357, Oct. 2021, doi: 10.1049/elp2.12104.

[11] S. Pang, S. A. Hashjin, B. Nahid-Mobarakeh, S. Pierfederici, Y. Huangfu, G. Luo, and F. Gao, "Large-signal stabilization of power converters cascaded input filter using adaptive energy shaping control," *IEEE Trans. Transport. Electrific.*, vol. 7, no. 2, pp. 838-853, Jun. 2021, doi: 10.1109/TTE.2020.3021954.

[12] T. S. Babu, K. R. Vasudevan, V. K. Ramachandaramurthy, S. B. Sani, S. Chemud, and R. M. Lajim, "A comprehensive review of hybrid energy storage systems: Converter topologies, control strategies and future prospects," *IEEE Access*, vol. 8, pp. 148702-148721, 2020, doi: 10.1109/ACCESS.2020.3015919.

[13] M. A. Hassan and Y. He, "Constant power load stabilization in DC microgrid systems using passivity-based control with nonlinear disturbance observer," *IEEE Access*, vol. 8, pp. 92393-92406, 2020, doi: 10.1109/ACCESS.2020.2992780.

[14] A. Emadi, A. Khaligh, C. H. Rivetta, and G. A. Williamson, "Constant power loads and negative impedance instability in automotive systems: Definition, modeling, stability, and control of power electronic converters and motor drives," *IEEE Trans. Veh. Technol.*, vol. 55, no. 4, pp. 1112-1125, Jul. 2006, doi: 10.1109/TVT.2006.877483.

[15] A. Kumar, A. Chandekar, P. W. Deshmukh, and R. T. Ugale, "Development of electric vehicle with permanent magnet synchronous motor and its analysis with drive cycles in MATLAB/Simulink," *Mater. Today, Proc.*, vol. 72, pp. 643-651, Jan. 2023, doi: 10.1016/j.matpr.2022.08.304.

[16] C. Attaiamese, M. D'Arpino, M. Di Monaco, and L. Pio Di Noia, "Experimental implementation of an algorithm for real time compensation of phase current sensors DC offset in field oriented controlled PMSM drives," *IEEE Access*, vol. 13, pp. 14392-14403, 2025, doi: 10.1109/ACCESS.2025.3532220.

[17] W. Song, J. Li, C. Ma, Y. Xia, and B. Yu, "A simple tuning method of PI regulators in FOC for PMSM drives based on deadbeat predictive conception," *IEEE Trans. Transport. Electrific.*, vol. 10, no. 4, pp. 9852-9863, Dec. 2024, doi: 10.1109/TTE.2024.3385114.

[18] P. K. Kesavan, U. Subramaniam, and D. J. Almakhles, "Laboratory implementation of direct torque controller based speed loop pseudo derivative feedforward controller for PMSM drive," *CES Trans. Electr. Mach. Syst.*, vol. 8, no. 1, pp. 12-21, Mar. 2024, doi: 10.30941/cestems.2024.00004.

[19] Y. Luo, K. Yang, and Y. Zheng, "Feedback linearization-based direct torque control for asymmetrical six-phase PMSM motor with back EMF harmonics compensation," *IEEE J. Emerg. Sel. Topics Power Electron.*, vol. 11, no. 5, pp. 5145-5155, Oct. 2023, doi: 10.1109/JESTPE.2023.3292922.

[20] J. Kim, S.-W. Ryu, M. S. Rafaq, H. H. Choi, and J.-W. Jung, "Improved torque ripple minimization technique with enhanced efficiency for surface-mounted PMSM drives," *IEEE Access*, vol. 8, pp. 115017-115027, 2020, doi: 10.1109/ACCESS.2020.3004042.

[21] Y. Cho, K.-B. Lee, J.-H. Song, and Y. I. Lee, "Torque-ripple minimization and fast dynamic scheme for torque predictive control of permanent-magnet synchronous motors," *IEEE Trans. Power Electron.*, vol. 30, no. 4, pp. 2182-2190, Apr. 2015, doi: 10.1109/TPEL.2014.2326192.

[22] Y.-S. Choi, H. H. Choi, and J.-W. Jung, "Feedback linearization direct torque control with reduced torque and flux ripples for IPMSM drives," *IEEE Trans. Power Electron.*, vol. 31, no. 5, pp. 3728-3737, May 2016, doi: 10.1109/TPEL.2015.2460249.

[23] M. Boukerdja, A. Chouder, L. Hassaine, B. O. Bouamama, W. Issa, and K. Louassaa, "$H_\infty$ based control of a DC/DC buck converter feeding a constant power load in uncertain DC microgrid system," *ISA Trans.*, vol. 105, pp. 278-295, Oct. 2020, doi: 10.1016/j.isatra.2020.05.031.

[24] S. Singh, D. Fulwani, and V. Kumar, "Robust sliding-mode control of DC/DC boost converter feeding a constant power load," *IET Power Electron.*, vol. 8, no. 7, pp. 1230-1237, Jul. 2015, doi: 10.1049/iet-pel.2014.0534.

[25] G. Sulligoi, D. Bosich, G. Giadrossi, L. Zhu, M. Cupelli, and A. Monti, "Multiconverter medium voltage DC power systems on ships: Constant-power loads instability solution using linearization via state feedback control," *IEEE Trans. Smart Grid*, vol. 5, no. 5, pp. 2543-2552, Sep. 2014, doi: 10.1109/TSG.2014.2305904.

[26] Y. Guan, Y. Xie, Y. Wang, Y. Liang, and X. Wang, "An active damping strategy for input impedance of bidirectional dual active bridge DC-DC converter: Modeling, shaping, design, and experiment," *IEEE Trans. Ind. Electron.*, vol. 68, no. 2, pp. 1263-1274, Feb. 2021, doi: 10.1109/TIE.2020.2969126.

[27] R. Ortega and M. W. Spong, "Adaptive motion control of rigid robots: A tutorial," *Automatica*, vol. 25, no. 6, pp. 877-888, Nov. 1989, doi: 10.1016/0005-1098(89)90054-x.

[28] M. A. Hassan, E.-p. Li, X. Li, T. Li, C. Duan, and S. Chi, "Adaptive passivity-based control of DC-DC buck power converter with constant power load in DC microgrid systems," *IEEE J. Emerg. Sel. Topics Power Electron.*, vol. 7, no. 3, pp. 2029-2040, Sep. 2019, doi: 10.1109/JESTPE.2018.2874449.

[29] D. Jeltsema and J. M. A. Scherpen, "Tuning of passivity-preserving controllers for switched-mode power converters," *IEEE Trans. Autom. Control*, vol. 49, no. 8, pp. 1333-1344, Aug. 2004, doi: 10.1109/TAC.2004.832236.

[30] C. Qiao, W. Sun, and Q. Zhang, "A Hamiltonian-based method for PMSM in electric vehicle considering iron loss and saturation," *Int. J. Control, Autom. Syst.*, vol. 22, no. 8, pp. 2385-2393, Aug. 2024, doi: 10.1007/s12555-022-0927-5.

[31] S. Pang, B. Nahid-Mobarakeh, S. A. Hashjin, S. Pierfederici, J.-P. Martin, Y. Liu, Y. Huangfu, G. Luo, and F. Gao, "Stability improvement of cascaded power conversion systems based on Hamiltonian energy control theory," *IEEE Trans. Ind. Appl.*, vol. 57, no. 1, pp. 1081-1093, Jan. 2021, doi: 10.1109/TIA.2020.3038355.

[32] P. Thounthong, P. Mungporn, B. Nahid-Mobarakeh, N. Bizon, S. Pierfederici, and D. Guilbert, "Improved adaptive Hamiltonian control law for constant power load stability issue in DC microgrid: Case study for multiphase interleaved fuel cell boost converter," *Sustainability*, vol. 13, no. 14, p. 8093, Jul. 2021, doi: 10.3390/su13148093.

[33] P. Thounthong, S. Sikkabut, N. Poonnoy, P. Mungporn, B. Yodwong, P. Kumam, N. Bizon, B. Nahid-Mobarakeh, and S. Pierfederici, "Non-linear differential flatness-based speed/torque control with state-observers of permanent magnet synchronous motor drives," *IEEE Trans. Ind. Appl.*, vol. 54, no. 3, pp. 2874-2884, May 2018, doi: 10.1109/TIA.2018.2800678.

<figure><img src="image_13.png" /></figure>

**THONG-IN SUYATA** (Member, IEEE) received the M.E. degree in electrical engineering from the King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, in 2015. He is currently pursuing the joint Ph.D. degree in electrical engineering with the Research Center Group of Research in Electrical Engineering of Nancy (GREEN), Université de Lorraine, Nancy, France, and the King Mongkut's University of Technology North Bangkok (KMUTNB), Bangkok. His main research interests include power converters, electrical drives, electric vehicles, nonlinear control, and observers.

<figure><img src="image_14.png" /></figure>

**MATHEEPOT PHATTANASAK** (Senior Member, IEEE) received the B.Sc. and M.E. degrees in electrical engineering from the King Mongkut's Institute of Technology North Bangkok (KMITNB), Bangkok, Thailand, in 1996 and 2004, respectively, and the Ph.D. degree in electrical engineering from Université de Lorraine, Nancy, France, in 2012. Currently, he is a Full Professor with the Department of Teacher Training in Electrical Engineering (TE), King Mongkut's University of Technology North Bangkok (KMUTNB). His current research interests include power electronics and their controllers.
<figure><img src="image_15.png" /></figure>

**PONGSIRI MUNGPORN** (Member, IEEE) received the B.S. and M.E. degrees in electrical engineering from the King Mongkut's University of Technology North Bangkok (KMUTNB), Bangkok, Thailand. He is currently pursuing the Ph.D. degree in electrical engineering with the King Mongkut's Institute of Technology Ladkrabang, Bangkok. His current research interests include power electronics, encompassing areas such as fuel cells, solar cells, batteries, and supercapacitors.

<figure><img src="image_16.png" /></figure>

**BURIN YODWONG** (Senior Member, IEEE) was born in Chiang Rai, Thailand, in 1990. He received the B.S. and M.S. degrees in electrical engineering from the King Mongkut's University of Technology North Bangkok (KMUTNB), Bangkok, Thailand, in 2014 and 2016, respectively, and the Ph.D. degree in electrical engineering from Université de Lorraine (UL), Nancy, France, in 2022. Currently, he is a Researcher with the Thai-French Innovation Institute, KMUTNB.

His current research interests include power electronics and their control laws for electrolyzer applications, renewable energy source systems, electric vehicle applications, and the modeling of proton exchange membrane electrolyzers.

<figure><img src="image_17.png" /></figure>

**EHSAN JAMSHIDPOUR** (Senior Member, IEEE) received the M.S. degree in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 2001, and the Ph.D. degree in electrical engineering from Université de Lorraine, Nancy, France, in 2014. From 2003 to 2010, he was a Lecturer with the Institute for Energy and Hydro Technology, Kermanshah, Iran. From 2016 to 2020, he was a Senior Lecturer with Icam Strasbourg Europe and was affiliated with the iCube Laboratory, University of Strasbourg. Since September 2020, he has been an Associate Professor with École Nationale Supérieure d'Électricité et de Mécanique (ENSEM), Université de Lorraine, where he is also a member of the Group of Research in Electrical Engineering of Nancy (GREEN). His research interests include electric motor drives, fault detection and fault tolerance in power electronic converters, and renewable energy systems. He serves as an Associate Editor for *IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION*, since 2021, and *Power Electronics and Drives* (PEAD).

<figure><img src="image_18.png" /></figure>

**NOUREDDINE TAKORABET** (Member, IEEE) received the Dipl.Ing. degree from ENP, Algiers, in 1993, the M.Sc. degree in electrical engineering from University Nancy I, France, in 1994, and the Ph.D. degree from INPL, in 1996. He is currently a Professor with Université de Lorraine and a member of GREEN. His main research activities deal with electromagnetic devices modeling and optimization. His teaching activities concern electrical machines, numerical simulation, and design of electromagnetic devices.

<figure><img src="image_19.png" /></figure>

**SERGE PIERFEDERICI** received the Dipl.-Ing. degree in electrical engineering from École Nationale Supérieure d'Electricité et de Mécanique (ENSEM), Institut National Polytechnique de Lorraine (INPL), Nancy, France, in 1994, and the Ph.D. degree from INPL, in 1998. Since 1999, he has been with Université de Lorraine, Nancy, where he is currently a Full Professor. His research activities with LEMTA Laboratory, CNRS, Université de Lorraine, deal with the stability study of distributed power systems, the control of dc and ac microgrids, and the design of power electronic converters for specific applications, such as fuel cell systems and electrolyzers.

<figure><img src="image_20.png" /></figure>

**BABAK NAHID-MOBARAKEH** (Fellow, IEEE) received the Ph.D. degree in electrical engineering from Institut National Polytechnique de Lorraine, Nancy, France, in 2001. From 2001 to 2006, he was with the Centre de Robotique, Electrotechnique et Automatique, University of Picardie, Amiens, France. In September 2006, he joined École Nationale Supérieure d'Électricité et de Mécanique, Université de Lorraine, Nancy, where he was a Professor, till December 2019. Since January 2020, he has been a Professor with McMaster University, Hamilton, ON, Canada. He has authored or co-authored more than 250 international peer-reviewed journals and conference papers, and several book chapters and patents. His research interests include nonlinear and robust control design of power converters and drives, fault detection and fault-tolerant control of electric systems, and design, control, and stabilization of microgrids.

<figure><img src="image_21.png" /></figure>

**NICU BIZON** (Senior Member, IEEE) was born in Albeștii de Muscel, Argeș, Romania, in 1961. He received the B.S. degree in electronic engineering and the Ph.D. degree in automatic systems and control from the University Politehnica of Bucharest, Romania, in 1986 and 1996, respectively. From 1996 to 1989, he was in hardware design with Dacia Renault SA, Romania. Since 2000, he has been a Professor with the National University of Science and Technology POLITEHNICA Bucharest, Pitești University Centre, Romania. His current research interests include power electronic converters, fuel cell and electric vehicles, renewable energy, energy storage systems, microgrids, and control and optimization of these systems. He received two awards from Romanian Academy, in 2013 and 2016.

<figure><img src="image_22.png" /></figure>

**PHATIPHAT THOUNTHONG** (Senior Member, IEEE) was born in Phatthalung, Thailand, in December 1974. He received the B.S. and M.E. degrees in electrical engineering from the King Mongkut's Institute of Technology North Bangkok (KMITNB), Bangkok, Thailand, in 1996 and 2001, respectively, the Ph.D. degree in electrical engineering from Institut National Polytechnique de Lorraine, Nancy, France, in 2005, and the Habilitation à Diriger des Recherches (H.D.R.) degree in electrical engineering from Université de Lorraine, Nancy, France, in 2020.

Since 2012, he has been a Full Professor with the Department of Teacher Training in Electrical Engineering, King Mongkut's Institute of Technology North Bangkok. Since 2021, he has led the International Research Partnership "Electrical Engineering-Thai French Research Center (EE-TFRC)" under the French PIA Program of Lorraine Université d'Excellence (LUE). Since 2024, he has been a Visiting Professor with Université de Lorraine. He has authored more than 321 scientific publications, including 27 articles in IEEE Transactions and Magazines. According to the Scopus database, his work has received more than 9300 citations, with an H-index of 50. He is also active in nonlinear control design and observer theory. His research interests include power electronics, electric drives, electric vehicles, and renewable energy systems, with a focus on fuel cells, photovoltaic systems, wind turbines, batteries, and supercapacitors.

Dr. Thounthong was a two-time recipient of Thailand's prestigious Senior Research Fellow (Senior Mathivichai) National Research Grant, from 2021 to 2024 and from 2025 to 2028. In 2025, he received Thailand Best Researcher Award from the National Research Council of Thailand (NRCT).

<figure><img src="image_23.png" /></figure>