A Novel LQI-Based Speed Control of Switched Reluctance Motors for High Performance Applications

The increasing use of Switched Reluctance Machines (SRMs) across various applications has been driven by their notable advantages and advancements in power electronics. However, controlling SRMs remains inherently complex due to their nonlinear characteristics and parameter variations at different operating points. To address this challenge, this paper proposes the implementation of a Linear-Quadratic-Integral Regulator (LQI) for precise speed control of SRMs. The LQI controller, renowned for its optimal control capabilities, offers several benefits, such as favorable dynamic response, and robustness against disturbances and changing parameters. Unlike traditional controllers, the absence of an inner current control loop results in reduced torque ripple at the phase commutation frequency, simplifying design and implementation. By using PWM for voltage command, torque ripple at the switching frequency is reduced, and noise due to magnetostriction is minimized. A comprehensive design procedure for the proposed LQI speed control scheme is outlined, followed by a comparative analysis with established SRM controllers based on Proportional-Integral (PI) and Hysteresis (HC) controllers. Experimental validation of the proposed controller is provided, illustrating its superior performance across various operating points. The results demonstrate improved dynamic response, enhanced disturbance robustness, reduced torque ripple, a superior switching frequency spectrum, and minimal computational burden for embedded implementation.

SECTION I. Introduction

The Switched Reluctance Machine (SRM) has become a viable option compared to other electric machines due to its absence of windings or permanent magnets in the rotor. This characteristic allows for operation at high speeds and temperatures, resulting in simple construction, robustness, high efficiency, high torque density, and low maintenance requirements. Another advantage of the SRM is its ability to produce high torque at low speeds. The SRM has both stator and rotor salient poles, but only the stator poles have windings. The phase windings are electrically isolated from each other, which makes the SRM fault tolerant [1], [2]. However, the SRM has some drawbacks such as high torque ripple and audible noise, the need for a power converter, and the challenges in control and optimization due to its nonlinear characteristics. All these challenges can be mitigated by using adequate control strategies [1], [3]. All in all, the SRM is a good option for many applications, especially in hostile environments or variable speed systems [4]. The application of SRMs has been the focus of several studies in electric vehicles (EV), hybrid electric vehicles (HEV) [5], [6], [7], and wind generation [8], [9].

The SRM can be controlled in different ways. Commonly, a current controller is employed when the SRM operates below base speed to achieve higher efficiency and lower torque ripple [10]. The hysteresis current controller (HC) is widely employed [11], [12], [13] because it is model-independent and robust. However, this strategy presents variable switching frequency and may produce more torque ripple and acoustic noise due to its broad switching frequency band. The Proportional-Integral (PI) controller, which is the most used controller in the industry, was used by [14], [15]. Its fixed switching frequency has benefits [16], but achieving an excellent dynamic response at different operating points is challenging owing to the nonlinear SRM model, which varies according to the operating point. The work [17] proposed a Gain Scheduled PI (GSPI). The controller is tuned online with the stator current and rotor position to improve the current command following. Nonetheless, the design complexity and computational burden are increased. Alternatively, predictive, adaptive, or sliding mode current controllers have been applied to SRM by [18], [19], [20], and [21], respectively.

Usually, the current controller is internal to a speed or torque control loop. A PI controller is widely used in the speed loop [1], [22]. The authors of [12] used a proportional-integral-derivative controller (PID), which improves the phase margin of the system but only for one operating point since the SRM model parameters vary with speed and current. A speed-sensorless controller based on generalized proportional-integral (GPI) observers was proposed by [23]. It compensates for the unknown load torque while estimating the angular velocity. Nonlinear and robust speed controllers were also employed, such as fuzzy [15], Dahlin [5], and sliding mode controllers [6], [24].

The Linear-Quadratic-Integral Regulator (LQI) is an optimal controller that can be implemented through a servo system with integral action and provides a systematic way to compute the feedback control gain matrix [25]. The LQI can be designed to meet design requirements by tuning the weighting matrices in the cost function. By adjusting these weights, it becomes possible to prioritize certain aspects of system performance and achieve the desired response. It provides optimal control by minimizing a quadratic cost function that balances control effort and system performance. By incorporating state feedback, the LQI accounts for variations in system dynamics and ensures robust performance in the face of parameter changes or deviations. These features make the LQI suitable for applications that require optimal control and robustness.

In prior studies, a Model Predictive Controller (MPC), known as a finite horizon Linear-Quadratic Regulator (LQR), was employed for current regulation [26]. An adaptive Pseudo-Derivative Feedback controller (PDF), and an LQI were employed for speed control in [27] and [28], respectively. However, in the latter study, increased torque ripple was observed due to the constant current reference generated by the LQI in steady-state, and the HC introduces a wider switching frequency spectrum, potentially leading to increased current and torque ripple. Furthermore, the use of a search algorithm like the Genetic Algorithm (GA) in [28] increases the complexity of the LQI design procedure. In contrast, the proposed LQI speed control strategy directly outputs the voltage command, eliminating the need for an internal current control loop. This innovative approach achieves reduced torque ripple, fixed switching frequency, and improved overall control performance. The design procedure presented in this work offers a simpler approach to achieving desired performance metrics, including settling time, maximum overshoot, and control effort.

This work proposes a novel SRM speed control method based on LQI. A servo system with states feedback and integral action is employed. Since a voltage command is directly generated, no inner current control loop is needed. The main contributions are enumerated as follows:

1. The LQI controller is applied for direct speed control of the SRM, eliminating the need for an inner current control loop;

2. A design procedure for the LQI controller in SRM speed control is proposed, with a detailed explanation of the linearized state-space model used in the design process;

3. A comprehensive comparison with established controllers is presented. The proposed strategy demonstrated performance improvements in several areas, including consistent dynamic response across different operating conditions, enhanced disturbance robustness, reduced torque ripple, fixed switching frequency, and lower acoustic noise.

The proposed controller is evaluated experimentally under different operation conditions and disturbances. The proposed LQI is compared to two SRM speed control schemes. The first comprises a PI speed controller and an inner HC current controller, forming the PI-HC controller. The second control scheme used for comparison comprises PI controllers in the speed and current control loops, named PI-PI controller.

This work is organized as follows. Section II presents the working principles and the linearized state-space model of the SRM. Section III introduces the principles of LQI. The LQI design procedure is detailed in Section IV. Section V presents the experimental results and performed comparisons. Lastly, Section VI provides the conclusion of this paper.

SECTION II. Switched Reluctance Motor

In an SRM, the Magnetomotive Force (MMF) is generated by energizing the stator phase windings. The electromagnetic torque ($T_{e}$ ) is produced when the magnetic flux through the rotor pole attracts the rotor towards the stator pole.

Neglecting magnetic saturation, the electromagnetic torque can be obtained by [29]:\begin{equation*} T_{e} = \frac {1}{2} i^{2} \frac {\text {d}L(\theta)}{\text {d}\theta }, \tag {1}\end{equation*} where L is the phase winding inductance, $\theta $ is the rotor position, and i is the phase electric current.

Despise the phases coupling, the phase voltage (v) is given by:\begin{equation*} v = R i + L(\theta)\frac {\text {d}i}{\text {d}t} + i \omega \frac {\text {d}L(\theta)}{\text {d}\theta }, \tag {2}\end{equation*} where R is the electrical resistance of the stator phase winding, and $\omega $ is the rotor angular speed.

In (2), the three terms on the right represent the voltage drop across the resistor, the voltage drop across the inductance, and the back electromotive force (back-EMF or $e_{b}$ ) across the phase, respectively.

Applying the electromagnetic torque given by (1), the mechanical equation can be written as:\begin{equation*} \frac {1}{2} i^{2} \frac {\text {d}L(\theta)}{\text {d}\theta } - T_{l} = B\omega + J\frac {\text {d}\omega }{\text {d}t}, \tag {3}\end{equation*} where $T_{l}$ is the load torque, B is the viscosity friction coefficient, and J is the inertia moment.

The turn-on angle ($\theta _{on}$ ) can be chosen so that the current (i) reaches its reference value ($i^{*}$ ) at the beginning of the positive derivative region of the inductance. Two types of torque ripple can be observed. The first and most significant occurs during phase commutation ($T_{rip,com}$ ), at a frequency $f_{Trip,com} = N_{\Phi } \omega _{e}$ , i.e., 24 times each rotation of the rotor of the 12/8 three-phase SRM. The second type, high-frequency torque ripple ($T_{rip,hf}$ ), occurs due to switching from current regulation, the frequency of this oscillation equals the switching frequency and depends on the current control method employed.

A static converter is required to drive the SRM. Since the current in the SRM is unidirectional, the Asymmetric Half-Bridge Converter (AHB) is commonly used [30].

In this work, the angles $\theta _{on}$ and $\theta _{off}$ are selected from a Look-up Table (LUT) based on the operation point (speed and load torque), aiming for a tradeoff between increased efficiency and minimized torque ripple. The proposed LQI speed controller delivers the phase voltage command, whereas, in the compared controllers, the PI for the speed loop gives the phase current reference. The SRM mathematical model is presented below.

A. SRM Model

The SRM plant state variables are rotor angular speed ($\omega $ ) and phase current (i), while the inputs are phase voltage (v) and load torque ($T_{l}$ ). In the voltage equation, (2), there is a product of states, resulting in a nonlinear system. Therefore, the goal is to obtain the linearized plant to use linear control techniques.

Due to pole salience and magnetic saturation, the inductance depends on both rotor position and phase current, denoted as $L(\theta,i)$ , along with its derivative $\text {d}L(\theta,i)/\text {d}\theta $ . For simplicity, the inductance and its derivative are assumed to be constant, referred to as L and $\text {d}L/\text {d}\theta $ , respectively. These values are computed as the average between the unaligned position ($\theta = 22.5^{\circ }$ ) and the aligned position ($\theta = 45^{\circ }$ ) at the design phase current [17]. Fig. 1 depicts the surfaces of $L(\theta,i)$ and $\frac {dL(\theta,i)}{d\theta }$ for the investigated 12/8 SRM. The average values at the design current are also presented.

FIGURE 1.

Inductance and inductance derivative profiles.

Perturbing the system around a steady-state operating point with a small signal, the system states and inputs become:\begin{equation*} i = \bar {i} + \tilde {i}\quad \omega = \bar {\omega } + \tilde {\omega }~v = \bar {v} + \tilde {v}~T_{l} = \bar {T_{l}} + \tilde {T_{l}} \tag {4}\end{equation*} where $\bar {}$ and $\tilde {}$ are used to represent the steady-state and the small-signal values, respectively. Substituting the perturbed variables into (2) and (3), the steady-state terms cancel, and the multiplication of small signal terms is neglected, resulting in:\begin{align*} \frac {\text {d}\tilde {i}}{\text {d}t} & = \left ({{ -\frac {R}{L} -\frac {1}{L}\frac {\text {d}L}{\text {d}\theta } \bar {\omega } }}\right) \tilde {i} - \frac {1}{L}\frac {\text {d}L}{\text {d}\theta } \bar {i} \tilde {\omega } + \frac {1}{L}\tilde {v} \tag {5}\\ \frac {\text {d}\tilde {\omega }}{\text {d}t}& = \left ({{ \frac {1}{J}\frac {\text {d}L}{\text {d}\theta }\bar {i} }}\right) \tilde {i} - \frac {B}{J} \tilde {\omega } -\frac {1}{J} \tilde {T_{l}} \tag {6}\end{align*}

The following variables are defined for later substitution:\begin{equation*} R_{eq} = R + \frac {\text {d}L}{\text {d}\theta } \bar {\omega } \quad K_{b} = \frac {\text {d}L}{\text {d}\theta } \bar {i}~\tilde {e_{b}} = K_{b} \tilde {\omega } \tag {7}\end{equation*} where $R_{eq}$ is the equivalent resistance, $K_{b}$ is the back-EMF constant and $\tilde {e_{b}}$ is the induced EMF. The load torque can be included in the model as incremental friction rather than as a perturbation. One can write $B_{t} = B + B_{l}$ , where $B_{t}$ , B, and $B_{l}$ are the total, machine, and load viscosity friction coefficients, respectively.

Rewriting the model in state space, one gets:\begin{equation*} \dot {x} = \mathbb {A} x + \mathbb {B} u \quad y = \mathbb {C} x + \mathbb {D} u \tag {8}\end{equation*} where\begin{align*} \mathbb {A} & = \left [{{\begin{array}{cc} - R_{eq}/{L} & \quad -K_{b}/{L} \\ {K_{b}}/{J} & \quad -{B_{t}}/{J} \\ \end{array} }}\right ] \quad \mathbb {B} = \left [{{\begin{array}{c} {1}/{L} \\ 0 \\ \end{array} }}\right ] \tag {9}\\ \mathbb {C} & = \left [{{\begin{array}{cc} 0 & \quad 1 \\ \end{array} }}\right ] \quad \mathbb {D} = \left [{{\begin{array}{c} 0 \\ \end{array} }}\right ] \tag {10}\\ x & = \left [{{\begin{array}{c} \tilde {i} \\ \tilde {\omega } \\ \end{array} }}\right ] \quad u = \left [{{ \tilde {v} }}\right ]. \tag {11}\end{align*}

SECTION III. Linear-Quadratic-Integral Regulator

The LQR is a method that provides the optimal gain matrix to bring the state variables to zero with minimum control effort (u). It is implemented by a servo system where the states variables (x) are feedback through a gain K. Conversely, LQI is implemented by a servo system, in which the error ($\dot {\xi }$ ) of the controlled variable (y) is integrated and amplified by a gain $K_{i}$ , and the states (x) are fed back through a gain K. Fig. 2 shows the LQI block diagram. The system shown in the figure can be represented using augmented matrices, as described by [25], where the error integral ($\xi $ ) is included as a state.

FIGURE 2.

LQI block diagram based on servo system with state feedback and integral action.

The LQI is an alternative to the pole placement method for obtaining the gains $K_{i}$ and K. However, system poles are allocated indirectly through the minimization of the quadratic performance index $\mathbb {J}$ , given by (12), and the solution of the Riccati Equation [25].\begin{equation*} \mathbb {J} = \int ^{\infty } _{k=0} \left ({{ x^{T}_{k} \mathbb {Q} x_{k} + u^{T}_{k} \mathbb {R} u_{k} }}\right)dt, \tag {12}\end{equation*}

In (12), $\mathbb {Q}$ and $\mathbb {R}$ are the state and control penalty matrices. The selection of these matrices for the SRM speed control problem is detailed in Section IV-A.

SECTION IV. Controllers Design Procedure

The SRM control strategy with LQI and the design procedure is shown below. In sequence, the same is done for the strategies that employ Proportional-Integral (PI) and Hysteresis controllers (HC), chosen for comparison. In this work, the angles $\theta _{on}$ and $\theta _{off}$ were previously obtained for each operating point (torque and speed) to minimize torque ripple and input power. The optimization result is stored in a LUT, which provides the driving angles. Table 1 shows the parameters of the SRM used at the operation point chosen for controller design. The inductance and inductance derivative surfaces obtained through the magnetization curves, as per [31], for the studied 12/8 SRM are presented in Fig. 1. The design point is highlighted.

TABLE 1 Three-Phase 12/8 SRM Parameters at the Operation Point Chosen for Controller Design

A. LQI Controller

The SRM speed control strategy based on a servo system with integral action and state feedback is shown in Fig. 3a. The gains are calculated by LQR theory. The controller generates the voltage command for each phase ($v^{*}_{abc}$ ). After that, the AHB switching pulses are generated by the Pulse Width Modulator (PWM) according to the soft-chopping switching logic during the conduction period defined by $\theta _{on}$ and $\theta _{off}$ .

FIGURE 3.

SRM speed control strategies.

The gain matrices $K_{i}$ and K are calculated using the function lqr() in software MATLAB. The function arguments are the penalty matrices ($\mathbb {Q}$ and $\mathbb {R}$ ), and the augmented matrices ($\widehat {\mathbb {A}}$ and $\widehat {\mathbb {B}}$ ) that include the integral error ($\xi $ ) as a state variable, calculated according to (13) [25]. The result of the operation is the matrix $\begin{aligned} \widehat {K} = \left [{{\begin{array}{cc} K & -K_{i} \\ \end{array} }}\right] \end{aligned}$ .\begin{align*} \widehat {\mathbb {A}} = \left [{{\begin{array}{cc} \mathbb {A} & \quad 0 \\ \mathbb {-C} & \quad 0 \\ \end{array} }}\right ] \quad \widehat {\mathbb {B}} = \left [{{\begin{array}{c} \mathbb {B } \\ 0 \\ \end{array} }}\right ] \tag {13}\end{align*}

The state penalty matrix ($\mathbb {Q}$ ) is a diagonal matrix that contains weighting elements corresponding to each state variable ($Q_{\tilde {i}}$ , $Q_{\tilde {\omega }}$ , and $Q_{\xi }$ ), as shown in Table 2. These elements determine the relative importance of each state variable. On the other hand, the input penalty matrix ($\mathbb {R}$ ) is also a diagonal matrix with weights assigned to each input ($R_{\tilde {v}}$ ). By setting a high value for the weighting element corresponding to a particular input, a low actuation effort will be generated for that input, resulting in a longer settling time.

TABLE 2 LQI Controller Parameters

The coefficients adopted for the penalty matrices, the resulting gains, and the performance indices for the unit step response are summarized in Table 2.

In this paper, the following approach is proposed to determine the weighting elements for the LQI design in this application:

1. A total of four weighting elements must be determined. Since only the relative differences between the elements matter, the element $Q_{\tilde {\omega }}$ is set to 1.

2. The remaining elements ($Q_{\tilde {i}}$ , $Q_{\xi }$ , and $R_{\tilde {v}}$ ) are swept, and performance indices such as settling time, control effort, and overshoot for the unit step response are analyzed, resulting in three 4-D datasets, as shown in Fig. 4a, 4b, and 4c, respectively.

3. The desired settling time is 0.5 s. As shown in Fig. 4a, this characteristic is highly sensitive to the $R_{\tilde {v}}$ element. The combinations of coordinates ($Q_{\tilde {i}}$ , $Q_{\xi }$ , $R_{\tilde {v}}$ ) that achieve the desired settling time are marked with red dots in Fig. 4.

4. Among the red dots, i.e., the coordinates corresponding to the desired settling time, the resulting control effort is examined in Fig. 4b, and maximum overshoot in Fig. 4c.

5. To achieve low overshoot, minimize control effort, and avoid saturation events, the lowest values of $Q_{\tilde {i}}$ and $Q_{\xi }$ that meet the desired settling time are selected. This point, referred to as the Design Point, is represented by the black dot in Fig. 4.

FIGURE 4.

Study about the influence of the $\mathbb {Q}$ and $\mathbb {R}$ matrices over the LQI speed servo controller.

The closed-loop system matrix $\mathbb {A}_{cl}$ is calculated by (14). Its computed eigenvalues are −8983.2 and $-6.5\pm \text { i}4.4$ . Since all the eigenvalues have negative real parts, the system is strictly stable under the LQI controller [25].\begin{align*} \mathbb {A}_{cl} = \left [{{\begin{array}{cc} \mathbb {A}-\mathbb {B} K & \quad \mathbb {B} K_{i} \\ \mathbb {-C} & \quad 0 \\ \end{array} }}\right ] \tag {14}\end{align*}

Fig. 5a presents a comparison of the Bode diagrams of the Sensitivity and Complementary Sensitivity functions for the LQI and PI (designed hereafter) speed controllers. Both controllers exhibit the same bandwidth, indicating that their ability to track reference signals and reject disturbances over a specific frequency range is comparable. In the low-frequency range, the slightly lower sensitivity gain of the PI controller suggests it provides marginally better disturbance rejection than the LQI controller under constant plant conditions. In the high-frequency region, the LQI controller demonstrates a significantly lower complementary sensitivity gain compared to the PI controller. This highlights the LQI controller’s superior noise rejection capabilities, making it more robust against high-frequency disturbances. In the transition region between low and high frequencies, the smooth transition without peaks observed for both controllers is crucial to ensuring system stability and preventing overshoot or oscillations. The step response of the designed LQI controller is shown in Fig. 6a, where the desired settling time of 0.5 s is achieved.

FIGURE 5.

Bode diagrams analysis.

FIGURE 6.

Step response.

B. PI Controllers

The SRM control strategy with PI controllers is shown in Fig. 3b. An outer speed control loop is composed of a $PI_{\omega }$ controller, which produces the phase current commands ($i^{*}_{abc}$ ). The inner controller ($PI_{i}$ ) regulates the current and delivers the voltage commands ($v^{*}_{abc}$ ) that are modulated via PWM. As the inner loop has a response time three times smaller than that of the outer loop, the response of one loop can be considered constant with respect to the other, and the controller design can be performed separately. In addition, the dynamics of PWM were neglected in this project.

From the SRM linearized model written in (8), the transfer functions of speed, $G_{\omega }(s)$ , and current plants, $G_{i}(s)$ , can be derived.\begin{align*} G_{\omega }(s) & = \frac {\tilde {\omega }(s)}{\tilde {i}(s)} = \frac {\frac {K_{b}}{B_{t}}}{1+ s \frac {J}{B_{t}}} \tag {15}\\ G_{i}(s) & = \frac {\tilde {i}(s)}{\tilde {v}(s)} = \frac {\frac {1}{L}\left ({{s + \frac {B_{t}}{J}}}\right)}{s^{2} + s\left ({{\frac {R_{eq}}{L} + \frac {B_{t}}{J} }}\right) + \frac {R_{eq} B_{t} + K_{b}^{2}}{L J} } \tag {16}\end{align*}

For the $PI_{\omega }$ tuning, a gain crossover frequency of 1 Hz was chosen to achieve a settling time similar to that of the LQI controller. The zero of the PI was set to half this frequency. The uncompensated and compensated open-loop bode diagrams are shown in Fig. 5b, where the uncompensated diagram represents the open-loop response of the plant without the PI controller, and the compensated diagram illustrates the frequency response after the PI tuning, achieving the desired bandwidth, low-frequency amplification, and high-frequency attenuation. Fig. 6a compares the step responses of the LQI and PI speed controllers, showing similar settling times with no overshoot observed.

In the current controller ($PI_{i}$ ), the gain crossover frequency was set to a decade below the switching frequency, resulting in 3 kHz. The zero of this PI was allocated by reducing another decade, that is, 300 Hz. Figs. 5c and 6b show the Bode diagram of the compensated and uncompensated plant and the unit step response, respectively.

Table 3 summarizes the design choices, the resulting transfer function, the stability margins, and the performance indices obtained for the PI controllers.

TABLE 3 PI Controllers Parameters

C. HC Controller

Fig. 3c shows the SRM control strategy using a Hysteresis current controller. The same PI speed controller mentioned above is employed in this application. The hysteresis bandwidth is set to 0.3 A, and the sampling frequency is $f_{s} = 30$ kHz.

SECTION V. Experimental Results and Discussion

All the experimental results were initially reproduced in a MATLAB/Simulink simulation environment. Since the results were consistent, we focused on presenting only the experimental findings. The suitability of the proposed LQI speed controller was verified through experiments that were conducted using the setup shown in Fig. 7. The setup comprises a three-phase 12/8 SRM and a Direct Current Machine (DCM) used to produce the load torque. The grid CA voltages are rectified and reduced by the Bidirectional DC-DC Converters (BDC) to provide the supply voltage for the AHB and H-bridge converters, which serve as the SRM and DCM drivers. The signal conditioning boards, equipped with the DSP TMS320F28335, receive the signals from the absolute encoder and sensors, implement the control, and deliver the switching signals. Moreover, human-machine interfaces (HMI) were developed in the LabVIEW environment for system handling. Note that the control and power circuits for the SRM and DCM are kept separate.

FIGURE 7.

Experimental setup.

The electromagnetic torque produced in phase k is estimated through a LUT $T_{k}(i_{k},\theta _{k})$ . Then, the total torque is the sum of the phases’ contributions. The torque LUT was experimentally obtained according to the procedure shown in [31]. Moreover, the phase magnetic flux is estimated online through $\lambda _{k} = \int (v_{k}-i_{k} R)dt $ . A drift correction was employed to avoid the cumulative error of the digital integrator.

The previously designed controllers chosen for comparison, PI-HC and PI-PI, are also implemented. The table 4 summarizes the comparison of processor requirements. The digitalized controllers are achieved using the Euler transformation associated with a saturation, resulting in a conditional integration anti-windup action. Evaluating the computational cost is important as these controllers are implemented in microcontrollers. In this regard, the LQI implementation requires an intermediate amount of operations (processing time), while the PI-HC is the technique that performs the fewest operations. Regarding the memory required, the LQI is the controller that uses the least number of stored variables.

TABLE 4 Processor Requirements Comparison

The following results are divided into: speed dynamics, current dynamics, efficiency and torque ripple analysis.

A. Speed Dynamic Response

SRM is a non-linear machine with variable model parameters according to the operating point, so the controller must be robust to maintain dynamic performance. Tests were performed with a torque load of 2 Nm, applying several speed command steps, as illustrated in Fig. 8a. Additional tests were conducted for a speed command step ranging from 20 rad/s to 60 rad/s under varying load torque conditions, with the results shown in Fig. 8b. The objective of this investigation is to validate the consistency of the speed controller across various conditions, including various speed reference steps and load conditions. The study also assesses the controller’s robustness to parameter variations, considering the inherent changes in the SRM plant with variations in speed and load levels. The results indicate that the LQI consistently delivers a reliable speed response across diverse conditions. Notably, the settling time and overshoot levels remain nearly unchanged, irrespective of the speed step or load. The settling times for the LQI closely match the design settling time of 0.5 s. In contrast, strategies employing PI speed controllers exhibit longer settling times under high-speed or heavy-load conditions (approaching 1 s). Conversely, for low-speed or light-load conditions, PI controllers demonstrate shorter settling times (less than 0.5 s) but with the occurrence of overshoot (up to 8.3%).

FIGURE 8.

Step response.

Fig. 9 shows the controllers’ response to a ramp speed command that increases by 20 rad/s per second. In this test, the load torque was set to 4 Nm. The strategies with PI speed controllers demonstrated slightly better ramp reference tracking capability. As shown in Table 5, the Root Mean Square Error (RMSE) is less than 2% of the design angular speed (100 rad/s) for all controllers. The electromagnetic torque produced can also be observed. It must be sufficient to overcome the load torque and ensure acceleration. Note that the torque wrap is smaller with LQI and larger with PI-HC, in accordance with the torque ripple map shown in Fig. 13b.

TABLE 5 Speed RMSE Comparison for Speed Ramp Command Test

FIGURE 9.

Comparison of controller response to a speed command ramp ($T_{l} = 4$ Nm).

FIGURE 10.

Perturbance robustness analysis.

FIGURE 11.

Comparison of phase current, phase electromagnetic torque and flux for sampling frequency of 30 kHz (black line) or 15 kHz (red line) at two operating points.

FIGURE 12.

Comparison of the switching pulse frequency spectrum ($\omega ^{*} = 40$ rad/s and $T_{l}=2$ Nm).

FIGURE 13.

Efficiency and torque ripple map comparison.

In Fig. 10a, the load torque disturbance rejection is compared. The speed command is set to 100 rad/s, and load torque steps are applied: a positive step (from 2 Nm to 4 Nm, represented by continuous lines) and a negative step (from 4 Nm to 2 Nm, represented by dashed lines). It can be observed that the LQI controller achieves a significantly faster response with minimal undershoot or overshoot. Likewise, when a disturbance is applied to the supply voltage (changed from 80 V to 40 V, as displayed in Fig. 10b) the LQI had a 64.5% faster response than the PI-HC controller. In that case, the PI-PI controller failed to track the reference. The performance indexes for these tests are summarized in Table 6.

TABLE 6 Comparison of Controllers in Perturbance Tests

B. Current Dynamic Response

As expected, the three compared controllers produce different shapes of phase current. Hence, the phase electromagnetic torque and flux have different characteristics. As shown in Figs. 11a and 11b, and Table 7, the current reference tracking is worse for higher load torque and speed. The PI-HC reaches better current RMSE but produces more current, torque, and flux ripple. Since the LQI does not have an inner current control loop, its current does not follow a constant reference. For protection, the state −1 or demagnetizing state (negative DC-link voltage being applied to the phase) is applied to the phase if the current exceeds the maximum allowed value. The RMS current and the average flux are almost the same regardless of the controller, these values are correlated with copper and iron losses, respectively.

TABLE 7 Comparison of Controllers in Current Cycle ( $f_{s}=30$ kHz)

The sampling frequency ($f_{s}$ ) was also the object of comparison. Reducing it from 30 kHz to 15 kHz, as shown in Fig. 11a and 11b, yields increased current ripple in both the PI-HC and PI-PI. Since the LQI does not have an inner current control loop at high frequency, it makes LQI suitable to be used with low sampling frequencies without prejudice to its performance. The ripple current for the PI-HC controller is highly affected by the reduction of $f_{s}$ .

The frequency spectrum of the triggering pulse of the AHB switches for each control strategy is shown in Fig. 12. The LQI and PI-PI strategies that use PWM have a fixed switching frequency of 30 kHz. Thus, the observed pulse frequencies are concentrated in the switching frequency ($f_{sw}$ ), in the electrical frequency ($\omega _{e}$ ), and in their multiples. For the hysteresis current controller, a higher number of switching events occurs within the audible frequency range (less than 20 kHz). Therefore, LQI and PI-PI controllers produce less acoustic noise through magnetostriction.

C. Efficiency and Torque Ripple Analysis

The efficiency and the torque ripple were investigated for various speeds and torque loads, which yielded enough data to constitute the maps from Fig. 13a and 13b. The efficiency was calculated by dividing the mechanical output power ($P_{mec}=T_{l} \omega $ ) by the DC supply power, so the efficiency of both AHB and SRM is regarded. The torque ripple was calculated by $T_{rip}=(T_{max}-T_{min})/T_{avg}$ , where $T_{max}$ , $T_{min}$ , and $T_{avg}$ are, respectively, the maximum, minimum, and average torque. Note that the efficiency is almost the same regardless of the controller. However, the LQI controller produced distinctly less torque ripple. This is because in the LQI strategy, the current is not forced to follow a constant reference during the conduction period, and the shape of the current produced contributes to delivering a more constant torque. In addition, the high-frequency torque ripple component is negligible for controllers with PWM, i.e., LQI and PI-PI. It was observed that the PI-PI controller was not able to reach the maximum power point of the map ($\omega ^{*} = 120$ rad/s and $T_{l}=6$ Nm).

Table 8 summarizes the comparison of the controllers, highlighting the best controllers for each aspect analyzed.

TABLE 8 Best Controller in Each Comparison Criteria

SECTION VI. Conclusion

This work proposes using LQI to speed control the SRM without an inner current control loop. The SRM modeling and the LQI design procedure were also presented. PI and Hysteresis controllers are widely used in SRM drivers and were chosen for comparison. Several tests were performed in the experimental setup to validate the control strategy and design developed. The LQI controller presented greater robustness to the model parameters variations (which occurs for different speeds and load torque), to the disturbance in the load and in the supply voltage, and to the reduction of the control sampling frequency. The SRM efficiency was almost equal for all controllers. However, the LQI produced less torque ripple at all operating points. The LQI also reduces the production of audible acoustic noise by concentrating the switching at a high frequency. Furthermore, in microcontroller implementation, the LQI controller requires less memory usage, and its processing time is intermediate between PI-HC and PI-PI. The results show that the LQI controller is suitable for the control of SRM and can be used in several applications, such as ventilation, traction of electric and hybrid vehicles, starter motors, automation, and pumping.

ACKNOWLEDGMENT

For open access purposes, the authors have assigned the Creative Commons CC BY license to any accepted version of the article.