Speed Control of Induction Motor Drives Based on Combining Slime Mold Optimization Algorithm and Sliding Mode Theory

Abstract

A robust speed controller integrating the slime mold algorithm (SMA) with sliding mode theory (SMT) is proposed for induction motor (IM) drives operating under field-oriented control (FOC). Unlike conventional controllers with fixed gain parameters, the proposed exponential reaching law sliding mode controller (ERLSMC) defines the sliding mode dynamic trajectory control gain, exponential reaching gain, and constant-speed reaching gain as the search space for the SMA. An adaptive fitness function based on the speed error and its rate of change is constructed to continuously evaluate and update these gain parameters, thereby determining the optimal controller gains according to the current operating state. Consequently, larger gain values are assigned when the system state is far from the sliding mode dynamic trajectory to accelerate the reaching process, whereas smaller gain values are adopted near the sliding mode dynamic trajectory to suppress chattering and reduce overshoot. Matlab/Simulink (2024b version) simulations are conducted to evaluate the proposed controller in an IM drive system and compare its performance with constant-speed reaching law sliding mode control, exponential reaching law sliding mode control, and zebra optimization algorithm (ZOA)-based ERLSMC methods. The simulation results demonstrate that the proposed controller achieves superior performance in both speed command tracking and load regulation response.

1. Introduction

Induction motors (IMs) have become indispensable in modern industrial automation owing to their robust mechanical construction, low maintenance overhead, and dependable operational characteristics [1]. However, conventional scalar control schemes are inadequate for the increasingly stringent speed regulation and dynamic precision demands of contemporary manufacturing, which necessitates more sophisticated control approaches. Consequently, the industry has widely adopted field-oriented control (FOC) [2] technology to achieve decoupled control of torque and flux. Traditional FOC typically uses proportional–integral (P-I) controllers. While P-I controllers are relatively easy to design, they struggle to balance speed command tracking and load regulation response simultaneously. Thus, when the system faces parameter inaccuracies or external disturbances, control performance is often insufficient.

Researchers have responded to these shortcomings with a variety of advanced strategies, each with its own trade-offs. Sliding mode control (SMC) [3], for example, delivers strong disturbance rejection, yet the high-frequency switching action inherent to the method introduces torque ripple through chattering near the sliding surface. Furthermore, the commonly used constant-speed reaching law lacks integral action, leading to steady-state errors, while the exponential reaching law can accelerate response but often causes significant overshoot. To enhance performance, research has attempted to combine different algorithms. For example, extension theory combined with the sliding mode controller (ETERLSMC) [3] can effectively suppress overshoot and improve tracking stability, but its performance relies heavily on the completeness of characteristic data; a lack of sufficient data samples leads to imprecise control. The neural network sliding mode controller (NNSMC) [4] can approximate nonlinear dynamics to reduce chattering, but its complex structure creates a massive computational burden, and performance depends on training convergence, making hardware implementation challenging. Extendable fuzzy theory combined with a two-degrees-of-freedom controller (2DOFC) [5] improves system flexibility, but rule switching during load recovery often causes transient oscillations, resulting in insufficient convergence smoothness.

To solve the optimization problem of sliding mode parameters, swarm intelligence algorithms have been introduced. However, particle swarm optimization (PSO) [6], while fast to converge, lacks a mutation mechanism and easily falls into local optima. Although the zebra optimization algorithm (ZOA) [7] employs a group defense mechanism to avoid local optima, it primarily relies on fixed-weight or linearly decreasing search strategies during the optimization process. Consequently, maintaining a proper balance between global exploration and local exploitation becomes difficult when dealing with high-dimensional coupled control parameters.

To calculate the current fitness, the proposed control strategy utilizes the speed difference between the induction motor’s speed command and actual speed, along with its rate of change, as features. Furthermore, the slime mold algorithm (SMA) [8] is employed to construct a three-dimensional search space, which defines the parameter ranges for the constant-speed reaching gain, exponential reaching gain, and sliding mode dynamic trajectory control gain in the ERLMC. Unlike traditional optimization algorithms that rely on fixed or linear weights, this study simulates the adaptive weight adjustment of slime mold under different food concentrations and its unique oscillation mechanism. This allows the algorithm to dynamically change step size based on environmental feedback, precisely finding the optimal position coordinates to assign to the three gain parameters of the exponential reaching law sliding mode velocity controller. Consequently, during IM operation, the most suitable gain combination is output in real time by comparing fitness values, improving speed command tracking response and overshoot caused by the exponential reaching law under different working conditions, and significantly enhancing system robustness under load variations.

2. Field-Oriented Control Architecture of IM System

FOC [9] achieves independent regulation of torque and flux by mapping the three-phase stator quantities onto a $\left(d,q\right)$ rotating reference frame that is synchronously aligned with the rotor flux, substantially reducing the complexity associated with IM controller design [10,11]. In this architecture, the speed controller compensates based on the error between the speed command and the feedback speed, while current controllers regulate the d- and q-axis currents respectively, achieving independent control of flux and torque. Finally, the voltage commands generated by the controllers control the duty cycle of the power semiconductor switches in the inverter [12] via space vector pulse width modulation (SVPWM) [13] to realize speed control of the IM drive.

2.1. Dynamic Equations of FOC

The fundamental objective of FOC is to provide the IM with control characteristics similar to those of a separately excited DC motor [14], thereby enabling independent regulation of flux and torque. Based on this principle, the dynamic equations of the squirrel-cage IM are formulated in the d-q rotating reference frame through the voltage and flux linkage relationships of the stator and rotor circuits. The stator and rotor voltage equations can be expressed as Equations (1)–(4) [10,11].

$$v_{ds}=R_s{i}_{ds}+L_{\sigma }{\displaystyle \frac{d{i}_{ds}}{dt}}-{\omega }_e{L}_{\sigma }{i}_{qs}+{\displaystyle \frac{L_m}{L_r}}{\displaystyle \frac{d{\varphi }_{dr}}{dt}}-{\displaystyle \frac{{\omega }_e{L}_m}{L_r}}{\varphi }_{qr}$$

(1)

$$v_{qs}={\omega }_e{L}_{\sigma }{i}_{ds}+R_s{i}_{qs}+L_{\sigma }{\displaystyle \frac{d{i}_{qs}}{dt}}+{\displaystyle \frac{{\omega }_e{L}_m}{L_r}}{\varphi }_{dr}+{\displaystyle \frac{L_m}{L_r}}{\displaystyle \frac{d{\varphi }_{qr}}{dt}}$$

(2)

$$0=-R_r{L}_m{i}_{ds}+R_r{\varphi }_{dr}+L_r{\displaystyle \frac{d{\varphi }_{dr}}{dt}}-({\omega }_e-{\omega }_r)L_r{\varphi }_{qr}$$

(3)

$$0=-R_r{L}_m{i}_{qs}+({\omega }_e-{\omega }_r)L_r{\varphi }_{dr}+R_r{\varphi }_{qr}+L_r{\displaystyle \frac{d{\varphi }_{qr}}{dt}}$$

(4)

where $R_s$ is the stator resistance, $R_r$ is the rotor resistance, $L_s$ is the stator inductance, $L_r$ is the rotor inductance, $L_m$ is the mutual inductance, ${\omega }_e$ is the synchronous angular velocity, ${\omega }_r$ is the rotor electrical angular velocity, $i_{ds}$ and $i_{qs}$ are the $d$- and $q$-axis stator currents, ${\varphi }_{dr}$ and ${\varphi }_{qr}$ are the rotor d- and q-axis fluxes, and $L_{\sigma }\triangleq L_s-L_m^2/L_r$ is the leakage inductance coefficient.

The electromagnetic torque and mechanical equations of the motor are expressed as in Equation (5).

$$T_e={\displaystyle \frac{3P}{4}}{\displaystyle \frac{L_m}{L_r}}(i_{qs}{\varphi }_{dr}-i_{ds}{\varphi }_{qr})=T_L+J{\displaystyle \frac{d{\omega }_{rm}}{dt}}+B{\omega }_{rm}$$

(5)

where $T_L$ is the load torque; J is the moment of inertia; B is the viscous friction coefficient; and the relationship between rotor electrical angular velocity, ${\omega }_r$, and mechanical angular velocity, ${\omega }_{rm}$, is ${\omega }_{rm}=(2/P){\omega }_r$, where P is the number of poles.

When the d-axis of the synchronous reference frame is continuously aligned with the rotor flux vector, the flux component along the q-axis vanishes identically such that ${\varphi }_{qr}=0$, concentrating the entire rotor flux magnitude on the d-axis alone as ${\varphi }_{dr}={\varphi }_r$. Consequently, Equations (1)–(4) can be rewritten as Equations (6)–(9) [11].

$${\displaystyle \frac{d{i}_{ds}}{dt}}=(-{\displaystyle \frac{R_s}{L_{\sigma }}}-{\displaystyle \frac{1-\sigma }{\sigma {\tau }_r}})i_{ds}+{\omega }_e{i}_{qs}+{\displaystyle \frac{1-\sigma }{\sigma {\tau }_r{L}_m}}{\varphi }_r+{\displaystyle \frac{v_{ds}}{L_{\sigma }}}$$

(6)

$${\displaystyle \frac{d{i}_{qs}}{dt}}=-{\displaystyle \frac{R_s}{L_{\sigma }}}{i}_{qs}-{\omega }_e{i}_{ds}-{\displaystyle \frac{1-\sigma }{\sigma L_m}}{\omega }_e{\varphi }_r+{\displaystyle \frac{v_{qs}}{L_{\sigma }}}$$

(7)

$${\displaystyle \frac{d{\varphi }_r}{dt}}=-{\displaystyle \frac{R_r}{L_r}}{\varphi }_r+R_r{\displaystyle \frac{L_m}{L_r}}{i}_{ds}$$

(8)

$$-R_r{\displaystyle \frac{L_m}{L_r}}{i}_{qs}+{\omega }_{sl}{\varphi }_r=0$$

(9)

where ${\tau }_r=L_r/L_m$ and $L_{\sigma }=\sigma L_s$ is the stator leakage inductance, $\sigma =1-L_m^2/L_s{L}_r$ is the leakage coefficient, and ${\omega }_{sl}={\omega }_e-{\omega }_r$ is the slip speed. Since ${\varphi }_{dr}={\varphi }_r$ and ${\varphi }_{qr}=0$, by substituting this condition into the electromagnetic torque Equation of (5), it can be rewritten as Equation (10). Furthermore, to describe the dynamic relationship between torque and speed, the mechanical equation of the motor can be further expressed as Equation (11).

$$T_e={\displaystyle \frac{3P}{4}}{\displaystyle \frac{L_m}{L_r}}{i}_{qs}{\varphi }_r$$

(10)

$${\displaystyle \frac{d{\omega }_{rm}}{dt}}={\displaystyle \frac{1}{J}}(T_e-T_L-B{\omega }_{rm})$$

(11)

Since the rotor flux cannot be measured directly, its value must be estimated through an appropriate flux estimation model. Therefore, by applying the Laplace transform to the rotor flux dynamic equation in Equation (8), the estimated rotor flux, ${\widehat{\varphi }}_r$, can be derived as expressed in Equation (12).

$${\widehat{\varphi }}_r={\displaystyle \frac{L_m{i}_{ds}}{\frac{L_r}{R_r}s+1}}$$

(12)

where ${\widehat{\varphi }}_r$ is the estimated flux and s is the Laplace operator.

From Equations (6) and (7), it can be seen that both the d- and q-axis current equations contain ordinary differential equation terms and nonlinear coupling terms. To eliminate the coupling effect and achieve system linearization, a control technique using feedforward compensation [15] is adopted. Using the feedback principle, the d-axis current error is passed through a P-I controller, and the resulting signal is Equation (13).

$${v_{ds}}^{\prime }=\left(K_{Pd}+{\displaystyle \frac{K_{Id}}{s}}\right)\left({i_{ds}}^{*}-i_{ds}\right)$$

(13)

This signal is then combined with the output signal of the d-axis current. After coordinate transformation from two axes to three phases, it forms the three-phase voltage command input to the inverter. To ensure that the current loop possesses linear control characteristics, appropriate compensation must be added to ${v_{ds}}^{\prime }$ before coordinate transformation. Therefore, the d-axis voltage command is expressed as Equation (14).

$${v_{ds}}^{*}=L_{\sigma }\left({v_{ds}}^{\prime }-{\omega }_e{i}_{qs}-{\displaystyle \frac{1-\sigma }{\sigma {\tau }_r{L}_m}}{\widehat{\varphi }}_r\right)$$

(14)

By substituting Equation (14) into $v_{ds}$ in Equation (6), it can be expressed as Equation (15).

$${\displaystyle \frac{d{i}_{ds}}{dt}}=(-{\displaystyle \frac{R_s}{L_{\sigma }}}+{\displaystyle \frac{1-\sigma }{\sigma {\tau }_r}})i_{ds}+{v_{ds}}^{\prime }$$

(15)

Similarly, the q-axis current signal and voltage are given by Equation (16) and Equation (17), respectively.

$${v_{qs}}^{\prime }=\left(K_{Pq}+{\displaystyle \frac{K_{Iq}}{s}}\right)\left({i_{qs}}^{*}-i_{qs}\right)$$

(16)

$${v_{qs}}^{*}=L_{\sigma }\left({v_{qs}}^{\prime }+{\omega }_e{i}_{ds}+{\displaystyle \frac{1-\sigma }{\sigma L_m}}{\omega }_e{\widehat{\varphi }}_r\right)$$

(17)

By substituting Equation (17) into $v_{qs}$ in Equation (7), it can be expressed as Equation (18).

$${\displaystyle \frac{d{i}_{qs}}{dt}}=-{\displaystyle \frac{R_s}{L_{\sigma }}}{i}_{qs}+{v_{qs}}^{\prime }$$

(18)

Within the speed control loop, the error between the reference speed and the actual motor speed obtained from the feedback signal is fed into a P-I controller to generate a torque command. This torque command is subsequently converted into a q-axis current command through the current control loop, thereby driving the motor to achieve speed regulation. The mechanical dynamics of the motor are determined by the moment of inertia and the viscous friction coefficient. Therefore, the Laplace transform [16] is applied to Equation (5) to establish the mathematical model of the speed loop. Based on this model, a controller is designed to provide good dynamic response and eliminate steady-state errors.

2.2. Sensorless FOC System Architecture of IM

The actual system still requires coordinate transformation and pulse width modulation (PWM) of the inverter to generate the corresponding three-phase voltage signals. In addition, this paper adopts a sensorless direct FOC strategy, so it is necessary to obtain the current flux and speed via a flux estimator [17] and a speed estimator [18]. In view of this, the complete architecture of the traditional sensorless FOC IM drive system is shown in Figure 1 [10].

3. The Proposed Novel Controller Algorithm

In traditional FOC of induction motors, the speed loop uses only a P-I controller, which often struggles to simultaneously balance the fast tracking of speed commands and regulation capability under load disturbances. Therefore, this paper proposes an exponential reaching law sliding mode controller combined with the SMA for dynamic gain parameter adjustment to replace the P-I controller in the speed loop. Specifically, taking the real-time speed error and the rate of change of the speed error as inputs, the SMA performs an optimization search to automatically correct the gains of the SMC. This equips the controller with self-adjustment capabilities and robustness, thereby effectively enhancing the overall performance of the IM speed control.

3.1. Slime Mold Algorithm

Introduced by Li et al. in 2020 [8], the slime mold algorithm (SMA) is a bio-inspired swarm intelligence optimization method inspired by the foraging behavior of Physarum polycephalum. Its search mechanism emulates several characteristic behaviors observed during the food acquisition process, including movement toward food sources guided by odor concentration gradients, adaptive vein contraction and expansion for resource allocation, and oscillatory behaviors generated by periodic vein contraction. Through these mechanisms, the SMA effectively balances exploration and exploitation during the optimization process, thereby enabling the algorithm to gradually converge toward the global optimum.

3.1.1. Approach Behavior

During the initial foraging phase, the slime mold extends an exploratory network radially in multiple directions to detect and approach potential food sources. If a higher concentration of a chemical odor is detected in a particular direction, the slime mold gradually concentrates energy to extend toward that direction, forming a temporary pathway. If the odor is insufficient or the direction is unclear, the slime mold changes direction and performs random drifting to avoid missing potential resources. This process, from rapid perception to approaching and then correcting direction, allows the slime mold to continuously update its movement path. Subsequently, in the iteration process, the movement of each slime mold depends on the fitness value corresponding to the food odor intensity and the best slime mold in the population. The mathematical expression for this approach behavior is represented by Equation (19).

$$\overrightarrow{X(t+1)}=\left\{\begin{array}{ll}\overrightarrow{X_b(t)}+\overrightarrow{vb}\cdot (\overrightarrow{W}\cdot \overrightarrow{X_A(t)}-\overrightarrow{X_B(t)})& , r<p\\ \overrightarrow{vc}+\overrightarrow{X(t)}& , r\ge p\end{array}\right.$$

(19)

where $\overrightarrow{X(t+1)}$ represents the new position for the approach food behavior, $\overrightarrow{X_b(t)}$ represents the optimal position of the slime mold individual at the t-th iteration, $\overrightarrow{vc}$ is a control vector that linearly decreases from one to zero, $\overrightarrow{X_A(t)}$ and $\overrightarrow{X_B(t)}$ represent the positions of two individuals randomly selected from the slime mold, $\overrightarrow{X(t)}$ is the current position, t is the current iteration count, and r is a random number in the range $\left[0,1\right]$.

In this process, in order to simulate the reaction of the slime mold to the strength of the odor, the SMA introduces a switching probability, p. When the fitness value of a certain individual is better than the average value, p will tend toward the positive direction, causing it to move and approach the best individual; conversely, it will bias toward the negative direction, causing it to stay or move away, thereby maintaining population diversity. The mathematical expression for p is shown in Equation (20).

$$p=\tanh \left|S(i)-DF\right|$$

(20)

where p is the switching probability, $S(i)$ is the fitness value of the i-th slime mold, and DF is the best fitness value found in all iterations.

Furthermore, to reflect the periodic contraction to expansion of the slime mold veins, the SMA designs an oscillation amplitude parameter, $\overrightarrow{vb}$. This vector has a larger range of values ($\left[-a,a\right]$) in the initial stage, enabling the slime mold to explore with large steps; as the iteration progresses, $a$ gradually approaches zero and the amplitude of $\overrightarrow{vb}$ converges synchronously, making the movement in the later stages more fine-grained. The mathematical expression for $\overrightarrow{vb}$ is represented by Equations (21) and (22).

$$\overrightarrow{vb}=[-a,a]$$

(21)

$$a=\arctan h\left(-\left({\displaystyle \frac{t}{T}}\right)+1\right)$$

(22)

where $\overrightarrow{vb}$ is the oscillation amplitude parameter and T is the maximum number of iterations.

In the mathematical modeling of the SMA, whether an individual is closer to the optimal solution depends not only on the position of the current best individual but is also influenced by its “relative fitness”. To reflect this difference, the algorithm introduces a weight, $\overrightarrow{W}$, the value of which is dynamically adjusted based on the individual’s ranking in the population (SmellIndex) and the difference between the best and worst fitness. When an individual’s performance is close to the optimal solution, $\overrightarrow{W}$ takes a larger positive value, making it converge more actively toward the optimal solution; conversely, individuals with poorer performance receive smaller or even lower weights, corresponding to a “weakening of the approach behavior”, to avoid the entire population concentrating too quickly. This design compresses the value range through a log function to ensure that $\overrightarrow{W}$ varies within a reasonable interval. The mathematical expression for $\overrightarrow{W}$ is represented by Equations (23) and (24).

$$\overrightarrow{W(SmellIndex(i))}=\left\{\begin{array}{l}1+r\cdot \log \left({\displaystyle \frac{bF-S(i)}{bF-wF}}+1\right),\hspace{1em}condition\\ 1-r\cdot \log \left({\displaystyle \frac{bF-S(i)}{bF-wF}}+1\right),\hspace{1em}other\end{array}\right.$$

(23)

$$SmellIndex=sort(S)$$

(24)

where bF is the best fitness value in the current iteration, wF is the worst fitness value in the current iteration, SmellIndex is the sequence of fitness values of the slime molds, and condition refers to the slime mold individuals ranked in the first half of fitness.

3.1.2. Wrap Behavior

When the slime mold finds food, it does not approach from a single direction alone; instead, it gradually contracts and strengthens the surrounding veins, ultimately wrapping around the food to become a center for cytoplasm energy transport. During this process, high-nutrient paths leading to the food gradually thicken, allowing a greater influx of material, whereas ineffective or nutrient-poor paths gradually contract until they vanish to conserve energy. Furthermore, a portion of the slime mold continues to diffuse outward, maintaining population diversity to ensure that other potential resources can be located should the current food source become insufficient. This mechanism of “concurrent concentration and contraction” enables the slime mold to pinpoint the food location while avoiding excessive convergence. The mathematical expression for this wrap food behavior is represented by Equation (25).

$$\overrightarrow{X^{*}}=\left\{\begin{array}{ll}rand\cdot \left(UB-LB\right)+LB& , rand < z\\ \overrightarrow{X_b(t)}+\overrightarrow{vb}\cdot (\overrightarrow{W}\cdot \overrightarrow{X_A(t)}-\overrightarrow{X_B(t)})& , r < p\\ \overrightarrow{vc}+\overrightarrow{X(t)}& , r\ge p\end{array}\right.$$

(25)

where UB is the upper bound of the search space, LB is the lower bound of the search space, rand is a random number in $\left[0,1\right]$, and z is the probability threshold.

3.1.3. Oscillation Behavior

In nature, even after the slime mold has found food, its veins still maintain periodic contraction and dilation. This “oscillation” behavior avoids local stagnation, ensures the continuous flow of cytoplasm, and allows for direction adjustment at any time. The random changes brought about by oscillation enable partial pathways to temporarily reverse or weaken, maintaining the population’s exploration capability. On the other hand, as time progresses, the oscillation amplitude gradually diminishes, allowing energy to ultimately concentrate on the best food source, completing the transition from “global” to “local”. In the mathematical modeling of the SMA, this oscillation behavior is not determined by a single factor but is realized through the combined action of three elements: the weight, $\overrightarrow{W}$, mentioned earlier that strengthens or weakens the contribution of individuals with high or low fitness; the stochastic vector, $\overrightarrow{vb}$, which simulates the randomness of vein flow direction switching; and the random control vector, $\overrightarrow{vc}$, whose range linearly decreases to zero with the total number of iterations. In other words, oscillation is the core mechanism for maintaining the balance between exploration and convergence during the evolutionary process of the SMA, enabling the algorithm to both escape local optima and gradually converge to the global optimum.

3.2. Sliding Mode Controller

Although P-I controllers deliver adequate performance under well-defined operating conditions, the strong nonlinearity and multivariable coupling inherent to IM dynamics fundamentally limit their effectiveness. Therefore, sliding mode control (SMC), as a form of variable structure control (VSC) [19], is adopted because it exhibits reduced dependence on accurate system models, providing strong anti-interference capability and robustness.

In speed tracking control, the objective is to make the error value between the speed command, ${\omega }_{rm}^{*}$, and the actual feedback speed, ${\omega }_{rm}$, zero. Therefore, the state variables of the IM system can be expressed as in Equation (26).

$$\left\{\begin{array}{l}{x}_1={\omega }_{rm}^{*}-{\omega }_{rm}\\ x_2={\dot{x}}_1=-{\dot{\omega }}_{rm}\end{array}\right.$$

(26)

where $x_1$ and $x_2$ are the state variable functions, representing the speed tracking error and the variation rate of the speed error, respectively.

The underlying principle of SMC is to steer the system state to move along a predefined sliding mode dynamic trajectory. This trajectory is formulated as a function composed of selected state variables and their corresponding control gains, as shown in Equation (27).

$$s_1=c{x}_1+x_2$$

(27)

where $s_1$ is the sliding mode dynamic trajectory function and $c$ is the sliding mode dynamic trajectory control gain.

To guarantee that the sliding mode dynamic trajectory function, $s_1$, converges to 0 at a certain time and to satisfy the stability requirements based on the Lyapunov second stability method [20], a suitable reaching law function [21] must be formulated. To adaptively adjust the convergence rate and suppress chattering, the exponential reaching law is adopted in this study, which is presented in Equation (28).

$${\dot{s}}_1=-\epsilon \mathrm{sgn}(s_1)-q{s}_1, \epsilon >0, q>0$$

(28)

where $\epsilon$ is the constant-speed reaching gain, $q$ is the exponential reaching gain, and $\mathrm{sgn}(s_1)$ is a function symbol.

By differentiating the sliding mode dynamic trajectory function, substituting the IM mechanical dynamics, and applying the exponential reaching law, the final controller output can be derived. Based on the fact that the controller output is the q-axis reference current, Equation (29) represents the q-axis reference current, $i_{qs}^{\ast }$, derived from the aforementioned control action.

$$i_{qs}^{\ast }={\displaystyle \frac{1}{D}}{\displaystyle \int \left[\left(c-\frac{B}{J}\right)x_2+\epsilon \mathrm{sgn}(s_1)+q{s}_1\right]}dt$$

(29)

where $D={\displaystyle \frac{1}{J}}\left({\displaystyle \frac{3P}{4}}{\displaystyle \frac{L_m}{L_r}}{\varphi }_r\right)$. Through the aforementioned design process, the speed-loop controller is completely formulated. This generated q-axis current command, $i_{qs}^{\ast }$, is subsequently fed into the inner current loop to achieve speed regulation of the IM system.

3.3. Speed Control of SMA Combined with SMC

In light of the preceding analysis, the exponential reaching law sliding mode controller (ERLSMC) is employed as the primary speed controller for the IM drive system. To further suppress speed overshoot and improve the tracking speed and stability, the SMA is incorporated to dynamically adjust the three gain parameters of the controller. Specifically, the parameter search domain is determined by the predefined upper and lower limits of the three controller gains. The initial fitness evaluation is established according to the speed tracking error between the reference command and the actual motor speed together with its rate of change. Thereafter, the population fitness condition is assessed to obtain the current optimal fitness and solution position. By means of the second term in Equation (25), the wrap mechanism is performed to direct the population toward the identified best solution while enhancing local exploitation capability. In addition, the third term of Equation (25) introduces an oscillatory search behavior that prevents the optimization process from converging prematurely to local optima. As a result, the algorithm progressively approaches the global optimum and determines the appropriate gain parameters for the exponential reaching law sliding mode controller. The adaptive adjustment procedure of the SMA is described in the following section, while the corresponding control flowchart is presented in Figure 2.

Step 1:

Set relevant parameters, such as slime mold population size, N; dimension, dim; maximum number of iterations, T; the search space range [$l{b}_j,u{b}_j$]; random numbers r and rand; the probability threshold, z; and weights $\alpha (W_1)$ and $\beta (W_2)$.

Step 2:

Initialize the position, X, and fitness value, F, of each slime mold.

Step 3:

First, a slime mold individual is randomly selected as the initial search slime mold, $S{M}_j$, and its position coordinate values are directly input into the exponential reaching law sliding mode controller. Subsequently, the integral of absolute error (IAE) is adopted to calculate the system feedback speed error, e, and the rate of change of the speed error, $\dot{e}$, and the initial fitness value is calculated based on the set weight values, $\alpha (W_1)$ and $\beta (W_2)$. The weight values represent the relative importance of each feature quantity. According to the analysis of the sliding mode dynamic trajectory function in Equation (23), it is known that this controller possesses compensation and correction effects regarding the rate of change of the speed error; therefore, in the initial stage, $\alpha (W_1)=0.7$ and $\beta (W_2)=0.3$ are set, with $\alpha \left(W_1\right)+\beta \left(W_2\right)=1$. In subsequent iterations, the weight update mechanism of the SMA in Equation (23) of Step 5 is used to dynamically adjust the aforementioned weights and, based on this, the IAE is utilized to calculate the fitness value, $F_i$, of the search slime mold, as shown in Equation (30).

$$F_i=\alpha (W_1)\cdot {\displaystyle {\displaystyle \sum _{t=1}^T}\left|e_{i,t}\right|}+\beta (W_2)\cdot {\displaystyle {\displaystyle \sum _{t=1}^T}\left|{\dot{e}}_{i,t}\right|}$$

(30)

where $e_{i,t}$ and ${\dot{e}}_{i,t}$ are the error and the rate of change of the error of the i-th slime mold in the t-th iteration, respectively.

Step 4:

Based on the fitness values calculated in Step 3, all slime molds are sorted. The best fitness, bF; the worst fitness, wF; and the global best position, $X_{best}$, in the current population are recorded. This sorting result will serve as the basis for the subsequent calculation of weights.

Figure 2. Control flowchart of the proposed novel SMA.

Figure 2. Control flowchart of the proposed novel SMA.

Step 5:

At the current iteration number, t; for each slime mold individual, $X_i$, first calculate the approach food probability, p, using Equation (20) based on its current fitness value, $S(i)$, and the best fitness, bF, in the population so far. Then, update the range, a, of the oscillation control parameter according to Equation (22) to determine the oscillation vector, $\overrightarrow{vb}$, and simultaneously update the linearly decreasing control vector, $\overrightarrow{vc}$. Finally, calculate the corresponding adaptive weight, $\overrightarrow{W}$, via Equation (23) to provide the proportion for regulating exploration and exploitation behaviors during the subsequent position update. Furthermore, feed this weight back into the fitness function to serve as the basis for dynamically adjusting the weight coefficients, $\alpha (W_1)$ and $\beta (W_2)$, within the integral of absolute error (IAE) formula.

Step 6:

For each dimension, j, of each slime mold individual, $X_i$, calculate its candidate position, $X_i^{new}$, according to the single integrated position update Equation (25) of the SMA. This mechanism first determines whether to execute the first term of Equation (25) by judging if rand > z; if the random distribution is not triggered, it decides whether to execute the second term of Equation (25) based on r < p; otherwise, it executes the third term of Equation (25). After completing the position update, use it as the basis for the subsequent fitness evaluation and population sorting.

Step 7:

After all slime mold individuals have completed their position updates, sort the population fitness, and update the worst fitness, wF, and the best fitness, bF, to the global best record values achieved so far.

Step 8:

Update and record the best fitness value and best position obtained so far. If the iteration count has reached the preset maximum number of iterations, stop the iteration and output the coordinate values of the best position to adjust the three gain parameters of the exponential reaching law sliding mode controller and determine the final output, $i_{qs}^{*}$, of the controller via Equation (29).

To illustrate the configuration of the overall control architecture, the proposed slime mold algorithm combined with the exponential reaching law sliding mode controller (SMAERLSMC) is integrated into a traditional sensorless FOC IM drive system. The parameters adopted by the SMA within the proposed control scheme are listed in

Table 1. Parameter values adopted for the SMA of the proposed control scheme.

Parameters	Value
Number of slime molds (N)	10
Maximum number of iterations (T)	30
Search space dimension (dim)	3
Range of sliding mode dynamic trajectory control gain [$c_{min},c_{\max }$]	[350, 2000]
Range of exponential reaching gain [$q_{min},q_{max}$]	[0.1, 5]
Range of constant-speed reaching gain [${\epsilon }_{min},{\epsilon }_{max}$]	[0.1, 1]
Range of random number ($r,rand$)	[0, 1]
Random reset probability (z)	0.03
Initial weight coefficient of speed error ($\alpha (W_1)$)	0.7
Initial weight coefficient of rate of change in speed error ($\beta (W_2)$)	0.3

. The overall block diagram is shown in Figure 3.

3.4. Computational Complexity Analysis

To justify the feasibility of true online adaptation, the computational burden of the proposed SMAERLSMC is evaluated. Based on the configuration in Table 1, the SMA utilizes a small population size of N = 10 and a low iteration count of T = 30, resulting in a maximum of 300 fitness evaluations per control cycle. Because the algorithm relies exclusively on basic floating-point arithmetic rather than massive matrix computations, its overall computational complexity is strictly bounded by N*dim*T. For a standard industrial digital signal processor (e.g., the 150 MHz TMS320F28335 with a hardware floating-point unit), these basic operations consume a negligible fraction of the processing capacity. Consequently, the entire optimization process can be seamlessly completed within a single speed-loop interrupt cycle, guaranteeing immediate gain updates without causing any control delay.

4. Simulation Results

To evaluate the speed regulation capability of the proposed robust controller in the IM drive system, simulation analysis is carried out in the Matlab/Simulink environment. The corresponding motor specifications are provided in

Table 2. Specifications of the adopted IM.

Electrical Specifications	Values
Three-phase rated voltage	AC 180 V
Three-phase rated current	AC 7.65 A
Rated power	1.9 kW
Rated speed	4750 rpm
Operating frequency	80 Hz
Number of poles	2
Stator resistance	0.6797 Ω
Rotor resistance	0.6327 Ω
Armature inductance	53.423 mH
Mutual inductance	51.75 mH
Moment of inertia	0.003 $\mathrm{kg}\cdot {\mathrm{m}}^2$
Viscous friction coefficient	0.0001 $\mathrm{N}\cdot \mathrm{m}\cdot \mathrm{s}$

. Under the same testing conditions, the IM drive performance is examined using four different control strategies: the slime mold algorithm combined with the exponential reaching law sliding mode controller (SMAERLSMC), the constant-speed reaching law sliding mode controller (CSRLSMC), the exponential reaching law sliding mode controller (ERLSMC), and the zebra optimization algorithm combined with the exponential reaching law sliding mode controller (ZOAERLSMC). Figure 4a–d and Figure 5a–d present the simulation results of the speed command tracking response for the IM under loads of 1 N-m and 2 N-m, respectively, with the speed command rising from 0 to different speed commands (1000 rpm, 2000 rpm, 3000 rpm, and 4000 rpm). Figure 6a–d show the simulation results of the load regulation response for the IM under different speed commands (1100 rpm, 2100 rpm, 3100 rpm, and 4100 rpm), where the load torque variation is $T_L$: 0$\to$1$\to$0$\to$2$\to$0 N-m. The simulation results indicate that without an integral term in its design, the CSRLSMC struggles to maintain stable convergence toward the reference speed during operation, thereby prolonging the overall tracking duration; furthermore, in terms of load regulation response, its speed recovery time is longer, and it exhibits a larger speed drop. While the ERLSMC presents a faster response in speed command tracking, it easily produces overshoot during the dynamic tracking process, leading to concerns regarding system stability. From Figure 4 and Figure 5, it can be observed that although the ZOAERLSMC can effectively suppress overshoot during the dynamic response process compared to the pure ERLSMC, this controller can still be further optimized in the balance between speed tracking and load regulation. Specifically, under load variations, its transient response and speed recovery capability remain insufficient. When the SMA is employed to dynamically adjust the three gain parameters of the ERLSMC, the overshoot produced by the conventional ERLSMC can be effectively reduced. In addition, the proposed controller exhibits superior load regulation response compared with the other three control methods. Compared to other control strategies, the SMAERLSMC proposed in this paper can simultaneously balance the capabilities of command tracking speed and overshoot suppression, and regarding load regulation response, its overall control performance is superior to the other three types of controllers.

Table 3. Speed tracking comparison for load = 1 N-m.

Speed Command Change	SMAERLSMC		CSRLSMC		ERLSMC		ZOAERLSMC
	Overshoot	Settling Time	Overshoot	Settling TIME	Overshoot	Settling TIME	Overshoot	Settling TIME
0→1000 rpm	0 rpm	0.09 s	0 rpm	0.34 s	342 rpm	0.32 s	0 rpm	0.12 s
0→2000 rpm	0 rpm	0.10 s	0 rpm	0.35 s	1020 rpm	0.40 s	0 rpm	0.14 s
0→3000 rpm	0 rpm	0.12 s	0 rpm	0.37 s	1684 rpm	0.49 s	0 rpm	0.17 s
0→4000 rpm	0 rpm	0.14 s	0 rpm	0.39 s	2150 rpm	0.56 s	0 rpm	0.19 s

and

Table 4. Speed tracking comparison for load = 2 N-m.

Speed Command Change	SMAERLSMC		CSRLSMC		ERLSMC		ZOAERLSMC
	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time	Overshoot	Settling Time
0→1000 rpm	0 rpm	0.11 s	0 rpm	0.36 s	366 rpm	0.34 s	0 rpm	0.13 s
0→2000 rpm	0 rpm	0.13 s	0 rpm	0.38 s	1060 rpm	0.42 s	0 rpm	0.15 s
0→3000 rpm	0 rpm	0.15 s	0 rpm	0.42 s	1720 rpm	0.52 s	0 rpm	0.18 s
0→4000 rpm	0 rpm	0.18 s	0 rpm	0.44 s	2262 rpm	0.64 s	0 rpm	0.21 s

present the speed tracking performance of the four controllers under loads of 1 N-m and 2 N-m, respectively, where the speed command rises from 0 rpm to 1000, 2000, 3000, and 4000 rpm. The key metrics compared are overshoot and settling time.

Table 5. Load regulation comparison for load = 0→1→0 N-m.

Speed Command	SMAERLSMC		CSRLSMC		ERLSMC		ZOAERLSMC
	Speed Drop	Recovery Time	Speed Drop	Recovery Time	Speed Drop	Recovery Time	Speed Drop	Recovery Time
1100 rpm	2.64 rpm	0.053 s	25.86 rpm	0.28 s	15 rpm	0.23 s	4.1 rpm	0.07 s
2100 rpm	2.76 rpm	0.058 s	26.92 rpm	0.32 s	16.7 rpm	0.27s	5.2 rpm	0.11 s
3100 rpm	2.82 rpm	0.062 s	27.98 rpm	0.36 s	18.5 rpm	0.31 s	6.4 rpm	0.16 s
4100 rpm	2.94 rpm	0.068 s	29.04 rpm	0.41 s	20.1 rpm	0.36 s	7.5 rpm	0.19 s

and

Table 6. Load regulation comparison for load = 0→2→0 N-m.

Speed Command	SMAERLSMC		CSRLSMC		ERLSMC		ZOAERLSMC
	Speed Drop	Recovery Time	Speed Drop	Recovery Time	Speed Drop	Recovery Time	Speed Drop	Recovery Time
1100 rpm	2.86 rpm	0.063 s	28.02 rpm	0.38 s	19.08 rpm	0.33 s	6.5 rpm	0.17 s
2100 rpm	3.01 rpm	0.069 s	29.52 rpm	0.43 s	21.09rpm	0.38 s	6.9 rpm	0.21 s
3100 rpm	3.16 rpm	0.072 s	30.02 rpm	0.48 s	22.82 rpm	0.42 s	7.3 rpm	0.27 s
4100 rpm	3.28 rpm	0.081 s	31.52 rpm	0.52 s	23.94 rpm	0.48 s	7.9 rpm	0.31 s

demonstrate the load regulation performance under speed commands of 1100, 2100, 3100, and 4100 rpm for 1 N-m and 2 N-m load steps, respectively, where each load is applied at 2 s and 3 s and unloaded at 2.5 s and 3.5 s; the metrics compared are speed drop and recovery time. Simulation results confirm that the proposed SMAERLSMC outperforms the other three controllers in both speed tracking and load regulation, achieving zero overshoot and the fastest settling and recovery times across all test conditions.

5. Conclusions

This paper presents a SMAERLSMC to replace the conventional P-I controller used in traditional FOC for IM speed control. Different from conventional fixed-weight or linear decreasing strategies, the proposed method incorporates the dynamic weighting mechanism of the slime mold algorithm. This mechanism requires very low additional computational cost to utilize feedback on individual quality, realizing adaptive adjustment of the fitness function, thereby altering the search convergence path to precisely optimize the gains of the sliding mode controller. This control strategy not only effectively suppresses the speed overshoot caused by the ERLSMC but also overcomes the issue of poor load regulation response performance found in the CSRLSMC. Simultaneously, simulation results demonstrate that the robust controller proposed in this paper is superior to the ZOAERLSMC and two common reaching law sliding mode controllers in terms of speed tracking and load regulation performance. In particular, through adaptive weight calculation, it exhibits better robustness under load variations. Additionally, the proposed controller possesses a compact structure and low computational complexity and does not depend on extensive training data, thereby facilitating practical implementation. To broaden the application context beyond simulation studies, future research will prioritize the practical implementation and experimental validation of the proposed methodology. Specifically, the control strategy will be deployed on a real hardware system featuring a physical induction motor drive and a digital signal processor (DSP). This hardware-based investigation will be crucial to further assess the controller’s real-time execution performance and its overall robustness under actual industrial operating conditions.