Observer-Based Remote Conductivity Variable-Parameter Sliding Mode Control for Water–Fertilizer Integration Machines Using Recursive Least Squares Adaptive Estimation

Abstract

In remote conductivity control for water–fertilizer integration systems, challenges such as long-distance nonlinearities and variable parameters can lead to fertilization inaccuracies, including over-irrigation and uneven distribution, affecting both productivity and environmental sustainability. To mitigate these issues, this study proposes a variable-parameter sliding mode control (VSMC) strategy, combined with an adaptive observer based on Recursive Least Squares (RLS) to estimate system inertia and load torque in real time. This allows for dynamic adjustment of the sliding surface parameters, ensuring robust control even under varying operating conditions. Two parameter derivation approaches—analytical modeling and data-driven fitting—are evaluated. Field tests demonstrate that VSMC outperforms the Proportional–Integral (PI) and conventional sliding mode control (SMC) methods in maintaining target electrical conductivity (EC) levels. Specifically, for a target EC of 1.4 mS/cm, VSMC stabilizes the system to within 1.18–1.60 mS/cm in 95 s, with a 14.3% overshoot, well within agronomic tolerance. In regional irrigation trials, VSMC significantly improves fertilizer uniformity, reducing the standard deviation of potassium nitrate distribution from 2.14 (PI) to 0.59. The simulation and experimental results validate the effectiveness and robustness of the proposed method, highlighting its potential to enhance agronomic efficiency and reduce environmental impact.

1. Introduction

1.1. Remote Conductivity Control in Water–Fertilizer Integration

The remote conductivity control system for water and fertilizer integration machines has garnered significant attention due to its application in various fields such as agriculture and horticulture [1,2]. This system plays a crucial role in optimizing water and fertilizer management, thereby enhancing crop yield and quality while conserving resources [3,4]. Comprising water supply, fertilization, drip irrigation, and monitoring components, this integrated system offers precise control over water and nutrient delivery, ensuring optimal plant growth [5,6].

1.2. Challenges in PMSM Servo Systems

The permanent magnet synchronous motor servo system (PMSM) has been widely applied in many fields such as control systems, agriculture, and engineering, and has attracted widespread attention from both industrial and academic communities [7,8]. The performance requirements for this system are becoming increasingly stringent [9,10]. Traditional PI control relies on precise mathematical models, and changes in operating conditions or motor parameters significantly affect its performance [11,12]. To enhance the robustness of servo systems, many modern control algorithms have been applied, such as adaptive control, fuzzy logic control (FLC), and sliding mode control (SMC) [13,14]. Recent studies comparing these techniques in PMSM applications show that while adaptive control improves parameter tuning under dynamic loads, FLC enhances nonlinear decision making, and SMC remains superior in terms of disturbance rejection and robustness under parameter uncertainties [15,16].

1.3. Advancements in Sliding Mode Control

In recent years, numerous scholars have combined observers with SMC to further enhance the performance of sliding mode controllers by observing disturbances [17,18]. For instance, some scholars have proposed methods to enhance the robustness of sliding mode controllers applied to permanent magnet linear synchronous motor servo systems [19]. Truong et al. designed an adaptive backstepping complementary sliding mode controller based on a disturbance observer to reduce the impact of parameter or load variations on system control performance [20]. Xu H. et al. used internal model control to achieve a fast and overshoot-free step velocity response for linear motors and combined a disturbance observer with a sliding mode controller to improve system robustness, though this method was structurally complex [21]. Xu B.L. et al. employed adaptive control to estimate the upper bound of uncertain system parameters, designing an adaptive non-singular fast terminal sliding mode to enhance system robustness, with a step position response able to quickly follow the set value in an overshoot-free manner [22].

In PMSM control systems, step-type or sinusoidal speed commands, or sinusoidal position commands, are typically used to verify control performance [23]. Numerous scholars have extensively studied enhancing the robustness of SMC under such conditions [24,25]. However, there is less research on step-type position response, and most studies have not verified the robustness of SMC against changes in rotational inertia [26]. For example, one study [27] used the disturbance estimates observed by a state observer as a feedforward compensation term to correct the output of SMC, improving the sliding mode control’s robustness against internal parameter perturbations. This enabled the PMSM servo system’s position response to better follow sinusoidal command signals even when rotational inertia varied, but there was often overshoot when following step command signals [28]. Another study employed the real-time adjustment of SMC’s output based on observed load torque values to enhance its robustness [29]. This method achieved good results in suppressing speed step response overshoot and speeding up response time, but it did not specify whether control performance would be affected by changes in rotational inertia [30,31].

1.4. Research Gap and Proposed Solution

Despite advancements in sliding mode control (SMC) for step-type position responses, a significant gap remains in addressing the robustness of SMC under varying rotational inertia conditions. Specifically, current research lacks comprehensive methodologies to ensure the consistent and reliable performance of SMC algorithms when applied to systems with dynamically changing rotational inertia. This paper aims to fill this gap by proposing a variable-parameter sliding mode control (VSMC) method that integrates inertia and load torque observations using a least squares observer. This approach allows for the adaptive adjustment of control parameters, enhancing the robustness of position control in systems where rotational inertia fluctuates.

Based on the above analysis, a variable-parameter sliding mode controller (VSMC) based on a recursive least squares (RLS) observer is designed to enhance the robustness of PMSM servo systems, ensuring that these systems maintain an optimal step position response performance even when parameters such as rotational inertia change. VSMC uses the rotational inertia and load torque values observed by the RLS observer to adjust its sliding mode surface parameters, achieving variable parameter control. Simulations and experiments indicate that this method effectively improves the step position response performance of PMSM servo systems and is insensitive to changes in rotational inertia, demonstrating a strong robustness.

1.5. Objectives

To address the limitations of conventional PI and fixed-parameter sliding mode controllers under dynamic system conditions, this study aims to achieve the following:

• Develop a VSMC strategy to enhance the robustness of PMSM servo systems used in water–fertilizer integration equipment.
• Design a recursive least squares (RLS)-based observer to estimate real-time system parameters, including rotational inertia and load torque.
• Adaptively adjust the sliding surface parameters in the VSMC using the observer’s output to maintain control accuracy under varying operating conditions.
• Compare two parameter derivation strategies—analytical modeling and data-driven fitting—for determining sliding mode parameters.
• Validate the proposed method through simulations and field experiments by assessing its performance in maintaining target conductivity and improving fertilizer distribution uniformity.

2. Materials and Methods

2.1. Remote Conductivity Control System for Water and Fertilizer Integration Machine

The remote conductivity control system for a water and fertilizer integration machine is shown in Figure 1. The system consists of a control section and a monitoring section. The control section includes water supply components and fertilization components, equipped with response facilities such as filters, fertilizer storage tanks, pumps, pipelines, drip irrigation belts, and wireless electric control valves. The monitoring section includes remote conductivity sensors, pressure sensors, flow sensors, and others. Data transmission between the control and monitoring sections is realized through the LoRa wireless communication protocol, which provides long-distance, low-power, and stable data exchange capabilities, suitable for agricultural field environments.

The water and fertilizer integration system consists of multiple coordinated subsystems. The pumping unit uses a hose pump with a starting torque of 230 N·m, a maximum flow rate of 5310 L/h, and a maximum operating pressure of 0.8 MPa. The pump is driven by a frequency converter, enabling variable flow control according to real-time demands.

The fertilizer supply unit is designed for the proportional mixing and delivery of fertilizers such as urea, monoammonium phosphate, and potassium phosphate. It features a paddle stirring mechanism, driven by a three-phase asynchronous motor with a rated power of 0.75 kW. The stirring rod has a shaft diameter of 1 m and is equipped with two straight blades, ensuring an adequate mixing efficiency. Fertilizer is transported using a single-phase screw self-priming pump rated at 1.8 kW, with a suction range of 10–25 m and a flow rate of 9 m³/h. The unit supports a maximum stirring volume of 1.5 m³.

For flow measurement, an electromagnetic flow sensor with a measurement accuracy of ±0.5% is integrated into the system, ensuring precise fertilizer dosing.

The power supply unit converts on-site three-phase power into 220 V, 24 V, and 12 V outputs, providing stable power to the control, monitoring, and actuation components throughout the system.

An implementation diagram of the remote conductivity control system for the water and fertilizer integration machine is shown in Figure 2. The lower computer, based on a programmable logic controller (PLC), is responsible for monitoring sensor signals such as flow rate, pressure, and pipeline EC values, as well as controlling the water pump, fertilizer pump, and solenoid valves. The upper computer, developed using a combination of Python 3.8 and LabVIEW 2020 64-bit, communicates wirelessly with the cloud platform via the LoRa protocol, sends control signals to the lower computer, and uploads sensor data and component status in real time. The communication between the upper and lower computers adopts the Modbus RTU protocol, ensuring reliable and standardized data exchange.

2.2. Design of the Variable-Parameter Sliding Mode Position Controller

A control block diagram of the position, speed, and current three-loop control system is shown in Figure 3. Taking the PMSM as an example, the i_d = 0 control strategy is adopted. The main purpose of this paper is to investigate the robustness of the sliding mode position controller (outer loop) to parameter variations. To achieve this, it is necessary to ensure that the inner loops are minimally affected by parameter changes, thereby eliminating their influence on the robustness of the system’s position control. Therefore, a fuzzy logic controller, known for its strong robustness and fast dynamic response, is employed in the speed loop.

In the dq coordinate system, the mathematical model of the PMSM is shown in Equation (1), which describes the dynamic behavior of the motor in terms of stator voltages, currents, and rotor position. This model is based on the commonly adopted formulation presented in [32], which provides a detailed derivation of PMSM dynamics under field-oriented control assumptions.

$$\begin{array}{l}{u}_d=R_s(i_d^{\ast }-i_d)+L_d{\displaystyle \frac{\mathrm{d}(i_d^{\ast }-i_d)}{\mathrm{d}t}}-{\omega }_r{L}_q(i_q^{\ast }-i_q)\\ u_q=R_s(i_q^{\ast }-i_q)+L_q{\displaystyle \frac{\mathrm{d}(i_q^{\ast }-i_q)}{\mathrm{d}t}}+{\omega }_r{L}_d(i_d^{\ast }-i_d)+{\omega }_r{\psi }_f\\ T_e={\displaystyle \frac{3}{2}}p[{\psi }_f{i}_q+(L_d-L_q)i_d{i}_q]\\ J{\displaystyle \frac{\mathrm{d}{\omega }_m}{\mathrm{d}t}}=T_e-T_L-B_m{\omega }_m-T_f\mathrm{sgn}({\omega }_m)\\ {\omega }_e=\mathrm{d}\theta /\mathrm{d}t\end{array}$$

(1)

In the equation, u_q and u_d represent the quadrature and direct-axis components of the stator voltage, respectively; i_q and i_d represent the quadrature and direct-axis components of the stator current, respectively; i_q^* and i_d^* represent the reference values of the d-axis and q-axis currents, respectively; L_d and L_q represent the direct and quadrature synchronous inductances, respectively; Rs is the stator winding resistance; ω_r is the electrical angular velocity; ω_m is the mechanical angular velocity, where ω_r = p*ω_m and p is the pole pair number; ψ_f is the excitation flux of the permanent magnet; B_m is the viscous friction coefficient; θ is the mechanical position angle of the motor rotor; J is the system’s moment of inertia; T_e is the electromagnetic torque; T_L is the load torque; and T_f is the friction torque.

The state variables of the PMSM system are selected as follows:

$$\left\{\begin{array}{l}{x}_1={\theta }_{\mathrm{ref}}-\theta \\ x_2={\dot{x}}_1={\dot{\theta }}_{\mathrm{ini}}-{\omega }_{\mathrm{m}}\end{array}\right.$$

(2)

In the equation, θ_ref is the position reference value and θ_ini is the instantaneous position error, which is the difference between θ_ref and θ at the instant when the reference undergoes a step change.

The sliding mode surface of the system is designed as follows:

$$s=c{x}_1+x_2=0$$

(3)

In the equation, c is the sliding mode surface parameter, which is a positive constant.

To enhance the quality of the reaching phase in sliding mode control, this paper adopts exponential reaching law, as follows:

$$\dot{s}=-\epsilon \mathrm{sgn}(s)-qs$$

(4)

In the equation, ε and q are positive constants. Sliding mode control may exhibit a chattering phenomenon, and in general, reducing the value of ε appropriately can mitigate chattering.

According to Equations (2) and (3), the expansion of $\dot{s}$ is as follows:

$$\dot{s}=c{\dot{x}}_1+{\dot{x}}_2=c{x}_2+{\dot{\theta }}_{\mathrm{ini}}-{\dot{\omega }}_{\mathrm{m}}$$

(5)

Substituting Equation (5) into Equation (4) yields the output of the fixed-parameter sliding mode position controller, i.e., the speed reference value is as follows:

$${\omega }_{\mathrm{mref}}={\dot{\theta }}_{\mathrm{ini}}+\int (c{x}_2+\epsilon \mathrm{sgn}(s)+qs)\mathrm{d}t$$

(6)

Because ${\dot{\theta }}_{\mathrm{ini}}$ is an impulse function, it becomes zero after the initial moment, thus, Equation (6) can be simplified.

$${\omega }_{\mathrm{mref}}=\int (c{x}_2+\epsilon \mathrm{sgn}(s)+qs)\mathrm{d}t$$

(7)

Clearly, ${\omega }_{\mathrm{mref}}$ is generated by the output of the integrator. The integrator acts as a filter to smooth the alternating signal, thus suppressing the chattering of SMC.

The above is the basic principle of fixed-parameter sliding mode position control. Next, the design idea of variable-parameter sliding mode control (VSMC) will be introduced.

Following this, the design concept of the system phase point (x₁, x₂) lies on the positive half-axis of x₁. Under sliding mode control, its trajectory is shown as the red line in Figure 4.

Let x₁(t₀) denote the value of the state variable x₁ when the phase point first reaches the sliding mode surface (corresponding to point A in Figure 2). From Equations (2) and (3), we can infer the trend of the changes in x₁ and x₂ during the sliding mode motion process as follows:

$$\left\{\begin{array}{l}{x}_1(t)=x_1(t_0){\mathrm{e}}^{-ct}\\ x_2(t)=-c{x}_1(t_0){\mathrm{e}}^{-ct}\end{array}\right.$$

(8)

From Equation (8), it can be observed that, by adjusting the parameter c, the speed at which x₁ approaches zero can be controlled. The larger the c, the faster the approach speed, meaning that the speed of the position response increases with the increase in c.

During the position step response process, the approximate variation trend of the given speed ω_mref and the actual speed ω_m is shown in Figure 5. Segment OB represents the approaching movement, and segment BC represents the sliding mode motion. Point B corresponds to point A in Figure 4. Usually, the stage where the actual speed decreases at the end (segment BC) is referred to as the positioning stage. The motion equation during the positioning stage is given by Equation (9), where the acceleration dω_m/dt is negative.

$${\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}={\displaystyle \frac{T_e-T_{\mathrm{L}}-B_{\mathrm{m}}{\omega }_{\mathrm{m}}-T_f\mathrm{sgn}({\omega }_{\mathrm{m}})}{J}}$$

(9)

The maximum rate of change in speed during this stage is given by Equation (10).

$${\left|{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}\right|}_{\max }={\displaystyle \frac{T_{e\max }+T_{\mathrm{L}\max }+B_{\mathrm{m}}{\omega }_{\mathrm{m}}+T_f}{J_{\min }}}$$

(10)

For typical motors, the comprehensive friction torque is much smaller compared to the maximum value of electromagnetic torque, so Equation (10) can be further simplified to the following:

$$|{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}}{|}_{\max }={\displaystyle \frac{T_{\mathrm{emax}}+T_{\mathrm{L}}}{J}}$$

(11)

As observed, it is limited by the maximum value of electromagnetic torque T_emax, the load torque T_L, and the system inertia J, and cannot increase indefinitely.

Analyzing Figure 4 and Figure 5, point B in Figure 5 corresponds to point A in Figure 4, i.e., during the positioning stage, the phase point undergoes sliding mode motion. Combining Equations (2) and (8), let the average rate of decrease in the given speed during the positioning stage be as follows:

$$|{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{mref}}}{\mathrm{d}t}}{|}_{\mathrm{avg}}={\displaystyle \frac{c^2{x}_1(t_0)}{2}}$$

(12)

From Equation (12), it is evident that, the larger the sliding mode surface parameter c, the greater the average rate of decrease in the given speed. When this rate exceeds the maximum rate of change in speed, the actual speed cannot follow the given speed, resulting in overshoot in the position response when the given speed is equal to zero (${\omega }_{\mathrm{mref}}$ = 0).

The objective of servo system control is to achieve a fast and overshoot-free position response to the given value. From the previous analysis, it is clear that a larger c leads to a faster response, but excessively large c values can cause overshoot. The optimal value of c is the maximum value that ensures no overshoot in the position response, and this optimal value is constrained by the maximum rate of change in speed, which is positively correlated with it. Additionally, Equation (11) indicates that, under constant conditions, the maximum rate of change in speed during the positioning stage is related to T_L and J, so the optimal value of parameter c is also related to them.

In a system using fixed-parameter SMC, the value of c remains constant (i.e., the average rate of decrease in the given speed remains constant). When variations in T_L and J lead to a maximum rate of change in speed smaller than this average rate, overshoot occurs in the position response. Conversely, the position response cannot follow the given value at the fastest rate. Therefore, fixed-parameter SMC cannot achieve optimal control under any operating conditions; its performance is significantly affected by T_L and J, and its robustness is poor. Therefore, this paper proposes a variable-parameter sliding mode position controller, in which the parameter c is dynamically adjusted according to T_L, J, and θ_ini. Additionally, when the position reference undergoes a step change, the smaller the θ_ini, the smaller the value of x₁(t₀). From Equation (12), it is apparent that under constant average rate conditions, reducing x₁(t₀) allows the value of c to be appropriately increased. Thus, the parameter c should also vary with changes in θ_ini.

In summary, the VSMC proposed in this paper dynamically adjusts its sliding mode surface parameter c according to the values of T_L, J, and θ_ini to achieve optimal control under any operating conditions. Clearly, θ_ini can be directly calculated, while T_L and J are unknown quantities. Therefore, this paper designs a recursive least squares observer to observe them. The following section will briefly introduce how to utilize this method to obtain their observed values.

2.3. Design of the Recursive Least Squares Observer

Incorporating the mathematical model of the motor, let us briefly introduce the principle of the recursive least squares (RLS) observer. By performing the Laplace transform on the motion equation of the motor, the motion equation of the motor in the frequency domain can be obtained as follows:

$${\omega }_{\mathrm{m}}(s)={\displaystyle \frac{1}{Js+B_{\mathrm{m}}}}(T_e(s)-T_{\mathrm{L}}(s)-h\cdot T_f(s))$$

(13)

In the equation, $h=sgn({\omega }_m)$.

After introducing the zero-order hold element, performing the Z-transform on the motion equation of the motor yields its expression, as shown in Equation (14) as follows:

$$\left\{\begin{array}{l}A={\displaystyle \frac{1-{\mathrm{e}}^{-B_{\mathrm{m}}{T}_s/J}}{B_{\mathrm{m}}}}\\ B=-{\mathrm{e}}^{-B_{\mathrm{m}}{T}_s/J}\\ C={\displaystyle \frac{1-{\mathrm{e}}^{-B_{\mathrm{m}}{T}_s/J}}{B_{\mathrm{m}}}}{T}_{\mathrm{L}}\\ D={\displaystyle \frac{1-{\mathrm{e}}^{-B_{\mathrm{m}}{T}_s/J}}{B_{\mathrm{m}}}}{T}_f\end{array}\right.$$

(14)

The above equation can be solved to obtain the following:

$$\begin{array}{cc}\hfill {\omega }_{\mathrm{m}}(z)& ={\displaystyle \frac{z^{-1}(1-{\mathrm{e}}^{-B_{\mathrm{m}}{T}_s/J})}{B_{\mathrm{m}}(1-z^{-1}{\mathrm{e}}^{-B_{\mathrm{m}}{T}_s/J})}}\hfill \\ & \times (T_e(z)-T_{\mathrm{L}}(z)-h\cdot T_f(z))\hfill \end{array}$$

(15)

where T_s represents the sampling period of the identification algorithm, assuming that within one sampling period, the values of ω_m and T_e remain constant.

$${\omega }_{\mathrm{m}(k)}=A{T}_{e(k-1)}-B{\omega }_{\mathrm{m}(k-1)}-C-h\cdot D$$

(16)

We define the output variable y_(k), input column vector x_(k), and parameter column vector λ as expressed in Equation (16).

$$\left\{\begin{array}{l}{y}_{(k)}={\omega }_{m(k)}\\ x_{(k)}={\left[T_{e(k-1)},-{\omega }_{m(k-1)},-1,-h\right]}^T\\ \lambda ={[A,B,C,D]}^T\end{array}\right.$$

(17)

The relationship between them can be derived from Equation (15), as follows:

$$y_{(k)}=x_{(k)}^{\mathrm{T}}\lambda$$

(18)

According to the recursive least squares principle, the identification formula for the observer is given by the following:

$$\left\{\begin{array}{l}{K}_{(N)}={\displaystyle \frac{{\mathit{P}}_{(N-1)}{x}_{(N)}}{1+x_{(N)}^{\mathrm{T}}{\mathit{P}}_{(N-1)}{x}_{(N)}}}\\ {\widehat{y}}_{(N)}=x_{(N)}^{\mathrm{T}}{\widehat{\lambda }}_{(N-1)}\\ e_{(N)}=y_{(N)}-{\widehat{y}}_{(N)}\\ {\widehat{\lambda }}_{(N)}={\widehat{\lambda }}_{(N-1)}+K_{(N)}{e}_{(N)}\\ {\mathit{P}}_{(N)}={\mathit{P}}_{(N-1)}-{\mathit{K}}_{(N)}{\mathit{x}}_{(N)}^{\mathrm{T}}{\mathit{P}}_{(N-1)}\end{array}\right.$$

(19)

In the equation, K_(N) is a 4-dimensional column vector; P_(N) is a 4 × 4 covariance matrix; ${\widehat{y}}_{\left(N\right)}$ is the N-th estimate of the output variable; ${\widehat{\lambda }}_{\left(N\right)}$ is the N-th estimate of the parameter column vector; and e_(N) represents the deviation caused by using the parameter vector from the previous time step for prediction at the current time step.

During the identification process, the correction value K_(N)e_(N) is continuously used to adjust the parameter vector estimate. The error between the estimated value and the actual value of parameter vector ${\widehat{\lambda }}_{\left(N\right)}$ gradually converges to zero. Eventually, using Equation (19), parameters ${\widehat{T}}_{\mathrm{L}}$ and $\widehat{J}$ can be inversely solved from $\widehat{A}$, $\widehat{B}$, and $\widehat{C}$.

$$\left\{\begin{array}{l}\widehat{J}={\displaystyle \frac{-(1+\widehat{B})T_{\mathrm{s}}}{\widehat{A}\ln (-\widehat{B})}}\\ {\widehat{T}}_{\mathrm{L}}=\widehat{C}/\widehat{A}\end{array}\right.$$

(20)

Using the observed values ${\widehat{T}}_{\mathrm{L}}$ and $\widehat{J}$, combined with θ_ini, the sliding mode control (VSMC) parameter c can be dynamically adjusted. A fitting method is employed to establish the relationship between the three values (${\widehat{T}}_{\mathrm{L}}$, $\widehat{J}$, and θ_ini), and the parameter c. The fitting process involves conducting a series of simulations under different conditions of T_L, J, and θ_ini to obtain the optimal value of c, where the position response does not exhibit overshoot (i.e., the system’s position response can quickly follow the reference value without overshooting).

Once the optimal value is determined, curve fitting is performed using existing data to derive the relationship between T_L, J, θ_ini, and c. To simplify the fitting process, θ_ini is initially kept constant to obtain the relationship between c and J under different inertia conditions. Then, keeping J constant, the relationship between c and θ_ini under different inertia conditions is determined. Finally, the two relationships are combined to derive the overall fitting equation.

When θ_ini rad and J are 1.75 × 10⁻⁵ kg·m² and 2.85 × 10⁻⁵ kg·m², respectively, the relationship between c and θ_ini determined by the fitting method is shown in

Table 1. Relationship between parameter c and load T.

TL/Nm	c
	J = 1.75 × 10−5 kg·m2	J = 2.85 × 10−5 kg·m2
0	242	198
0.01	255	206
0.02	268	212
0.03	282	222
0.04	286	232
0.05	305	240
0.06	318	249

. The relationship between c and T_L, J can be expressed as Equation (21).

$$\left\{\begin{array}{c}c=100000{(T_{\mathrm{L}})}^3-9400{(T_{\mathrm{L}})}^2+1240T_{\mathrm{L}}+242\\ (J=1.75\times {10}^{-5}\mathrm{kg}\cdot {\mathrm{m}}^2)\\ c=-100000{(T_{\mathrm{L}})}^3+{(100T_{\mathrm{L}})}^2+550(T_{\mathrm{L}})+200\\ (J=2.85\times {10}^{-5}\mathrm{kg}\cdot {\mathrm{m}}^2)\end{array}\right.$$

(21)

At different moments of inertia, when T_L = 0 N·m and J is constant, the relationship between c and θ_ini is expressed as Equation (22).

$$c=\sqrt{{\left[c_0\right]}^2/{\theta }_{\mathrm{ini}}}$$

(22)

When J = 1.75 × 10⁻⁵ kg·m², c₀ = 245, and when J = 2.85 × 10⁻⁵ kg·m², c₀ = 200. Combining Equations (21) and (22), the overall relationship between parameter c and T_L, J, θ_ini can be obtained.

$$\left\{\begin{array}{l}J=1.75\times {10}^{-5}\mathrm{kg}\cdot {\mathrm{m}}^2\\ \sqrt{{\left[c_0\right]}^2/{\theta }_{\mathrm{ini}}}=100000{(T_{\mathrm{L}})}^3-9400{(T_{\mathrm{L}})}^2\\ \begin{array}{cccc}& & & \end{array}+1240T_{\mathrm{L}}+242\end{array}\right.$$

(23)

$$\left\{\begin{array}{l}J=2.85\times {10}^{-5}\mathrm{kg}\cdot {\mathrm{m}}^2\\ \sqrt{{\left[c_0\right]}^2/{\theta }_{\mathrm{ini}}}=-100000{(T_{\mathrm{L}})}^3\\ \begin{array}{cccc}& & & \end{array}+{(100T_{\mathrm{L}})}^2+550(T_{\mathrm{L}})+200\end{array}\right.$$

(24)

Under different moments of inertia conditions, the relationship surface between c, T_L, and θ_ini determined by the fitting method or analytical analysis is shown in Figure 6.

2.4. Software and Versions

All numerical simulations and control algorithm implementations were performed using MATLAB (version R2021a, MathWorks Inc., Natick, MA, USA). Figures were generated using MATLAB’s built-in plotting functions.

3. Results and Discussion

3.1. Simulation Analysis

3.1.1. Response Speed Simulation Analysis

In this study, Matlab was utilized to simulate the position servo system, aiming to validate the effectiveness of both the recursive least squares (RLS) observer and the variable-parameter sliding mode control (VSMC). To showcase the efficiency of the designed controllers, a step position of 15 radians was initiated, with the motor operating under no-load conditions. Figure 7 illustrates the position tracking curves for the three control methods, along with an enlarged view spanning from 0.05 to 0.10 s. Upon closer examination, it is evident that the time required for the Proportional–Integral (PI) control to achieve the desired position was approximately 0.14 s, while for SMC, it was 0.12 s, and for VSMC, it was 0.07 s.

While the VSMC strategy demonstrated a superior response speed and shorter settling time compared to the other control methods, it is important to note that the simulation was conducted under no-load conditions. In real applications, external disturbances and varying load conditions can affect the performance of the VSMC strategy. This limitation should be considered, as the control system might need further tuning or compensation mechanisms when applied in real-world scenarios. The impact of load variations on the response speed and accuracy of VSMC in practical applications is an area for future research.

Furthermore, the stability and robustness of the VSMC strategy were examined in this study. The VSMC controller, due to its sliding mode nature, offers robustness against parameter variations and external disturbances. This is because the sliding mode control inherently drives the system to the sliding surface, where the system’s behavior is less sensitive to changes in the system’s parameters. However, while the simulation results demonstrated a good robustness under ideal conditions, the real-world implementation of VSMC may require further validation, particularly under conditions with more significant parameter uncertainties and external disturbances.

3.1.2. Motor Rotation Response Simulation Analysis

Given the rotational angle as θr = 20 × sin(t) rad/s, the load torque changes from 0 N·m to 4 N·m at 5 s. Figure 8 shows the simulation curves of the rotational angle response for each control method, with an enlarged view demonstrating the motor’s rotational angle response around 5 s. It can be observed from Figure 8 that after adding the load disturbance, the VSMC method exhibits the response that is closest to the reference θr curve compared to the PI and SMC methods.

Figure 9 displays the simulation curves of position response errors for each control method. The initial response error of PI control is approximately 0.18 rad, with a tracking error range from approximately −0.09 to 0.09 rad; the initial response error of SMC is 0.15 rad, with a response error range from approximately −0.075 to 0.075 rad. The initial response error of VSMC is 0.1 rad, with a tracking error range from approximately −0.05 to 0.05 rad. These results show that VSMC reduces both the initial response error and the error range compared to the PI and SMC methods.

Figure 10 shows the disturbance simulation curve, where the observer can rapidly estimate the change in disturbance. From the enlarged view, it can be observed that the response time of the observer to the load change at 5 s is 0.0085 s.

Given the rotational angle as θr = 20×sint rad, as shown in Figure 11, the load torque changes from 0.5 N·m to 1 N·m at 10 s. The motor’s position response, shown in Figure 12, indicates that the initial tracking error of PI control is approximately 0.08 rad, with a response error range from approximately −0.08 to 0.08 rad. The initial response error of SMC is 0.075 rad, with a response error range from approximately −0.08 to 0.08 rad. The initial response error of VSMC is 0.07 rad, with a response error range from approximately −0.07 to 0.07 rad. Adopting the VSMC strategy yields a more accurate rotational response precision and stronger disturbance rejection capability. Figure 13 depicts the angle response curve, indicating that after the load increases, the speed curve of the control strategy proposed in this paper is smoother. Figure 14 illustrates the timely estimation of load changes using the variable-parameter sliding mode control strategy based on the recursive least squares observer when the load changes. Comparing Figure 7, Figure 8 and Figure 9, it can be seen that the experimental results are consistent with the simulation results, which proves the feasibility of the proposed MTPA-ESO control strategy.

3.2. Experimental Validation

To test the control performance of various methods for remote conductivity control in a water and fertilizer integration machine, a potassium nitrate fertilizer solution was used. The testing device was positioned approximately 110 m from the field sensor section, with sensor sampling every 5 s, resulting in 180 sampling points over a total of 15 min. The EC target values were set at 1.4 mS/cm, 1.8 mS/cm, and 2.2 mS/cm in three separate groups. The control algorithms tested were PI, SMC, and VSMC. The initial EC value was approximately 1.0 mS/cm. Statistical analysis was performed using MATLAB to calculate steady-state EC ranges, settling times, and overshoot.

The comprehensive experimental data (

Table 2. Performance metrics of control algorithms for EC regulation.

Control Algorithm	Set EC (mS/cm)	Steady-State EC (mS/cm)	Steady-State Time (s)	Overshoot (%)
PI	1.4	1.25~1.56	160	19.3
	1.8	1.65~1.93	175	22.2
	2.2	2.10~2.30	190	25.1
SMC	1.4	1.30~1.50	130	15.6
	1.8	1.67~1.90	120	16.7
	2.2	2.12~2.28	130	17.2
VSMC	1.4	1.18~1.60	95	14.3
	1.8	1.60~1.98	100	15.5
	2.2	2.06~2.35	120	16.1

) shows that in terms of remote conductivity control in water and fertilizer integration systems, the VSMC method performed the best. At a target electrical conductivity (EC) of 1.4 mS/cm, it achieved the smallest EC fluctuation range (0.204 mS/cm), shortest steady-state time (95 s), and lowest overshoot (14.3%). The SMC method exhibited an intermediate performance, with better results than PI but not as good as VSMC. For example, at a target EC of 1.4 mS/cm, the SMC method achieved an EC fluctuation range of 0.181 mS/cm, a steady-state time of 130 s, and an overshoot of 15.6%. These results suggest that SMC offers a balanced trade-off between the higher steady-state time and larger fluctuation observed with PI and the better performance seen with VSMC. In contrast, the PI control method performed the worst, exhibiting the largest EC fluctuation range (0.543 mS/cm), longest steady-state time (190 s), and highest overshoot (25.1%) at a target EC of 2.2 mS/cm.

The experimental results of the control performance for the PI, SMC, and VSMC methods are shown in Figure 15, Figure 16 and Figure 17, respectively.

The PI control method demonstrated a varied performance across different target EC values. For a target EC of 1.4 mS/cm, the system achieved a steady-state EC range from 1.25 to 1.56 mS/cm, with a steady-state time of 160 s and an overshoot of 19.3%. When the target EC was set to 1.8 mS/cm, the steady-state EC ranged from 1.65 to 1.93 mS/cm, the steady-state time increased to 175 s, and the overshoot was 22.2%. At the highest target EC of 2.2 mS/cm, the steady-state EC varied from 2.10 to 2.30 mS/cm, with a steady-state time of 190 s and an overshoot of 25.1%.

The SMC method showed an improved performance compared to the PI control method. For a target EC of 1.4 mS/cm, the steady-state EC ranged from 1.30 to 1.50 mS/cm, with a steady-state time of 130 s and an overshoot of 15.6%. When the target EC was 1.8 mS/cm, the steady-state EC ranged from 1.67 to 1.90 mS/cm, with a reduced steady-state time of 120 s and an overshoot of 16.7%. At the highest target EC of 2.2 mS/cm, the steady-state EC ranged from 2.12 to 2.28 mS/cm, the steady-state time was 130 s, and the overshoot was 17.2%.

The VSMC method demonstrated the best overall performance among the three control algorithms. At a target EC of 1.4 mS/cm, the system reached a steady-state EC range from 1.18 to 1.60 mS/cm, with a steady-state time of 95 s and an overshoot of 14.3%. For a target EC of 1.8 mS/cm, the steady-state EC varied from 1.60 to 1.98 mS/cm, with a steady-state time of 100 s and an overshoot of 15.5%. At the highest target EC of 2.2 mS/cm, the steady-state EC ranged from 2.06 to 2.35 mS/cm, with a steady-state time of 120 s and an overshoot of 16.1%.

Analyzing the results, it is evident that the VSMC method outperformed both the PI and SMC methods in terms of response speed, steady-state time, and overshoot across all tested target EC values. The improved performance of the VSMC method can be attributed to its ability to dynamically adjust the sliding mode surface parameters based on observed values of rotational inertia and load torque. This adaptive capability allows the VSMC method to handle disturbances and changes in system parameters more effectively, leading to more accurate and stable control.

The field experiments further validate these findings, demonstrating that the VSMC method consistently achieved faster steady-state times, lower overshoot percentages, and smaller EC fluctuation ranges. This enhanced performance is particularly significant for practical agricultural applications, where the precise and stable control of water and fertilizer integration is crucial for optimizing crop yields and conserving resources.

The superior performance of the VSMC method highlights its potential as a robust and efficient solution for managing the dynamic and nonlinear characteristics inherent in water and fertilizer integration systems. Its ability to achieve rapid convergence to set points, minimal overshoot, and effective disturbance rejection makes it an ideal choice for improving the efficiency and effectiveness of agricultural irrigation and fertilization processes.

To further validate the effectiveness of the VSMC method in regional irrigation applications, a comparative experiment was conducted between the conventional PI control method and the proposed VSMC-based remote conductivity method. The experiments were performed on the same plot, with a one-month interval between two operations. To minimize external disturbances, artificial irrigation and natural rainfall were avoided for three days prior to each trial. The experimental field was divided into 45 subplots, labeled using a combination of letters (A–E) and numbers (1–9). The deviations between the target and actual applied potassium nitrate fertilizer content across different regions were analyzed (as shown in Figure 18).

The distribution of soil potassium nitrate fertilizer content before and after irrigation was compared to assess the performance of both methods in controlling irrigation uniformity. As illustrated in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18a, the conventional PI control method exhibited the largest over-application in subplot B8, where 7.48 kg/ha of excess potassium nitrate was applied. The most under-applied region was B2, with a deficit of 6.52 kg/ha. The overall standard deviation of the PI control method was 2.14. In contrast, as shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18b, the VSMC method exhibited a significantly improved uniformity. The largest over-application occurred in subplot C4, with an excess of 1.06 kg/ha, while the most under-applied region was A4, with a deficit of 2.01 kg/ha. The overall standard deviation of the VSMC method was reduced to 0.59, demonstrating its superior precision in achieving the desired fertilizer distribution.

3.3. Field Performance and Adaptive Control Challenges

The experimental and simulation results demonstrate the superiority of the proposed VSMC method with RLS-based adaptive estimation over PI and conventional SMC in terms of response speed, steady-state accuracy, and robustness to load disturbances. However, several practical limitations and uncertainties should be acknowledged.

The control performance relies on the accuracy of the system model, particularly the relationship between motor dynamics and fertilizer flow. While the RLS observer effectively estimates inertia and load torque, unmodeled nonlinearities (e.g., hydraulic resistance, pump wear, or sensor drift) may degrade performance over time. Future work could integrate hybrid modeling (e.g., physics-based and data-driven) to improve adaptability.

The RLS algorithm assumes quasi-static parameter variations, making it less effective under highly dynamic conditions (e.g., sudden pipe blockages or rapid EC fluctuations). In such cases, the observer’s convergence time (0.0085 s in simulations) may lag behind real-world disturbances. Combining RLS with robust filtering (e.g., Kalman or H∞) could mitigate this issue.

The experiments were conducted under controlled disturbances (step torque changes), but actual field conditions involve stochastic factors (e.g., soil heterogeneity and varying water pressure). While VSMC reduced potassium nitrate deviation (SD: 0.59 vs. PI’s 2.14), long-term validation across diverse terrains is needed.

VSMC’s adaptive gains improve responsiveness but may amplify noise in low-signal environments (e.g., noisy EC sensors). A sensitivity analysis of observer tuning parameters (forgetting factor and covariance matrix) would help to balance robustness and noise immunity.

Despite these limitations, the VSMC method significantly advances precision in water–fertilizer integration by dynamically adjusting to operational changes. Its real-time adaptability makes it a promising solution for sustainable agriculture, though further field trials are recommended to refine robustness under extreme conditions.

4. Conclusions

This study developed and validated a variable-structure model control (VSMC) method for water and fertilizer integration systems, effectively addressing nonlinear remote conductivity disturbances and system parameter variability. By incorporating a recursive least squares observer, the proposed system dynamically adjusts electrical conductivity (EC) control, significantly improving the precision of irrigation and fertilization processes.

Simulations revealed that VSMC outperformed the conventional PI and standard SMC methods in response speed, settling time, and disturbance rejection. In step response tests, VSMC achieved the target EC in 0.07 s, compared to 0.14 and 0.12 s for PI and SMC, respectively. VSMC also showed a higher tracking accuracy and better robustness under variable load conditions, markedly reducing steady-state errors.

Field experiments further confirmed these findings. When tested at target EC values of 1.4, 1.8, and 2.2 mS/cm, VSMC consistently achieved more stable and accurate EC regulation than PI and SMC. Regional irrigation trials demonstrated that VSMC significantly improved fertilizer distribution uniformity, lowering the standard deviation of EC from 2.14 (PI control) to 0.59. This mitigated fertilizer over- and under-application across subplots, supporting more sustainable crop management.

Regarding practical implementation, large-scale deployment of the VSMC system requires several prerequisites, as follows: (1) integration with high-precision EC sensors and stable actuator systems, (2) real-time communication infrastructure for feedback control, and (3) adaptive parameter tuning tools for varying soil and environmental conditions. Furthermore, the modularization of control units and compatibility with existing irrigation infrastructure will be key to scalability.

Beyond demonstrating a superior control performance, this research offers valuable insights for designing advanced adaptive control strategies in precision agriculture. Future studies will focus on multi-zone irrigation applications, real-time soil moisture integration, energy optimization, and the application of machine learning to enhance sliding surface parameter adaptation.

In conclusion, the VSMC approach will significantly enhance EC regulation in integrated water and fertilizer systems. It presents a scalable and adaptive solution with a high potential for real-world agricultural deployment, aiming to improve crop productivity, optimize input usage, and reduce environmental footprint.