Active Disturbance Rejection Control Method for Marine Permanent-Magnet Propulsion Motor Based on Improved ESO and Nonlinear Switching Function

Abstract

In the control of marine permanent-magnet propulsion motors, active disturbance rejection control has attracted much attention because it can deal with external load disturbances and uncertainties of motor parameters at the same time. However, the conventional second-order ADRC has the problem of slow disturbance observation speed. To this end, this paper proposes an improved third-order extended state observer using the proportional–integral disturbance update law to improve the tracking performance and anti-external disturbance ability of the motor control system. Then, aiming at the problem that the structure does not effectively use the current information, resulting in large speed fluctuations when the load changes, the measured value of the q-axis current is used as the disturbance feedforward compensation item to further improve the load disturbance suppression ability of the motor. Finally, in order to suppress the influence of the current periodic disturbance caused by unmodeled dynamics on the steady-state accuracy of the motor, a nonlinear switching function with bounded gain and an IIR low-pass filter are designed to suppress the periodic disturbance without affecting the dynamic performance of the system. Combined with the established ship propeller load model, the effectiveness of the method is verified on the motor experimental platform: When suddenly changing 75% of the propeller load, the motor speed decreases by about 20%, and the adjustment time is 0.1 s, which improves the performance by more than 70% compared to PI control and conventional ADRC methods.

1. Introduction

Under the background of carbon peaking and carbon neutrality goals, the issue of ship emission reduction and energy efficiency optimization is becoming more and more important. Among the different means of transportation, global maritime transport is responsible for 2–3% of global greenhouse gas (GHG) emissions, and it is predicted to increase to 17% by 2050 if no changes are adapted. Hence, the international maritime organization (IMO) has aimed to reach a 50% reduction in GHG emissions by 2050 compared to 2008. Hence, to meet these strict GHG emissions reduction targets, ships must use alternative power sources [1,2]. These strict regulations have brought new challenges to ship designers and ship owners, and have had a significant impact on the energy efficiency design indicators and energy efficiency operation indicators of ships. It is worth noting that by the end of the 20th century, with the rapid development of fast-switching semiconductor devices and the higher utilization of power converters, ship propulsion systems began to explore electrification [2,3]. In this stage, generators are used together with internal combustion engines, and the electrical energy obtained from the generator is used to run the propulsion motors. Therefore, regardless of the speed of internal combustion engine, the speed of propeller can be freely adjusted, thereby improving the overall energy efficiency [3]. Now, under the strict policy background, to further reduce ship emissions and achieve a more flexible energy distribution, full-electric propulsion systems using lithium batteries and fuel cells have begun to emerge; so, electric propulsion technology will be more advantageous. Therefore, electric propulsion ships are becoming the trend of future ships with their advantages of low emission, high efficiency and low noise. As the power source of an electric propulsion ship, the control performance and efficiency of the motor and its variable frequency drive system determine the maneuverability and energy efficiency of the ship; so, it is necessary to study its control strategy in depth.

Permanent-magnet synchronous motors (PMSMs) are widely used in propulsion devices for large ships and marine equipment due to their advantages of high efficiency and high power density. However, the permanent-magnet synchronous motor is a typical nonlinear and strongly coupled system [4,5], which is difficult to described with an accurate mathematical model. The permanent-magnet synchronous motor is susceptible to various load changes, external sea state changes and parameter changes, and the long-term operation of the motor system in a high-salt, high-temperature, and high-humidity marine environment will lead to problems such as the reduction of sensor accuracy [6,7]. The traditional PI control method cannot meet the requirements of high-performance control of marine propulsion motors; so, it is necessary to use nonlinear controllers to improve their tracking and anti-disturbance capabilities.

In recent years, relevant researchers have proposed a variety of control methods and applied them to the motor control, such as Active Disturbance Rejection Control (ADRC) [8,9,10] Model Predictive Control (MPC) [11,12,13], sliding mode control (SMC) [14], etc. Ref. [8] proposed the nonlinear ADRC controller, but due to the complexity of parameter setting and stability analysis of the nonlinear ADRC controller, this method was not widely used until [9] proposed a linear ADRC parameter-tuning method based on observer bandwidth. Ref. [10] used ADRC in permanent-magnet synchronous motor control and showed that the method had better anti-disturbance capabilities and parameter robustness. Extended State Observer (ESO) is the core link of ADRC, which can observe and estimate the system state and lumped disturbances. Ref. [13] adopted the extended state observer to solve the problem of steady-state error and poor robustness of parameters in the motor model predictive control. Ref. [14] combined the sliding mode control with the extended state observer and updated the permanent-magnet synchronous motor speed sliding mode control law in real time by using the lumped disturbances estimated by the observer to improve the speed tracking performance and reduce the steady-state speed tracking error. These control methods start from different practical problems, and use ESO’s effective state observation methods that do not depend on accurate models to improve the dynamic response performance, harmonic suppression ability, and stability and robustness of the system. These studies also show that ADRC has good disturbance suppression performance and parameter robustness, and is suitable for ship electric propulsion working conditions. For electric propulsion ships, permanent-magnet synchronous motors are usually used as propulsion motors to drive the propellers. During navigation, they often face complex ocean environments and disturbances such as wind, waves, and currents. At this time, the torque load of permanent-magnet synchronous motors is constantly changing, and a set of PI parameters is very difficult to adapt well to all working conditions. Hence, it is necessary to consider a more complete control strategy. If the changing load and friction are regarded as disturbances, the control problem of time-varying systems is equivalent to the suppression of disturbances. The problem to be solved by ADRC is the estimation and compensation of disturbances. In addition, there are unmodeled dynamics (such as Flux Harmonics, Dead-Time Effects, and Measurement Error Effects) in the motor itself [4], which cause torque and speed to fluctuate. Therefore, the control performance of disturbances and unmodeled dynamics determines the propulsion efficiency of the ship, and it is necessary to study the disturbance suppression strategy of the ship propulsion motor in depth.

Although ADRC has good control performance, it can still be improved for different control problems. To further improve the anti-disturbance capabilities of ADRC, researchers have proposed many improvement methods. First of all, although the linear ADRC parameter setting is simple and easy to use, and the suppression effect is better when there is a large disturbance, the nonlinear active disturbance rejection controller can achieve more accurate state estimation, and has a better suppression effect on small disturbances. To solve this problem, relevant scholars have proposed a series of linear/nonlinear active disturbance rejection switching control method to improve the system’s anti-disturbance capabilities. Ref. [15] analyzed the effective operating range and steady-state observation error of the linear/nonlinear ADRC controller, proposed a linear/nonlinear switching control method, and explained its necessity, but the direct switching method may cause current oscillation in the motor control. Therefore, Ref. [16] proposed a linear/nonlinear switching active disturbance rejection control method applied to permanent-magnet synchronous motor speed regulation and designed a hysteretic switching strategy, which achieved better anti-disturbance capabilities. Ref. [17] improved the nonlinear function and proposed a class of linear/nonlinear switching active disturbance rejection controllers. By designing the switching transition strategy, the smooth switching of the control variable under different disturbance amplitudes and the improvement of the anti-disturbance performance were achieved. However, the parameter design and stability analysis of the nonlinear method are complicated. Ref. [18] designed a new nonlinear function to combine the advantages of linear and nonlinear ADRC and apply it to rotor position estimation, which improved the system’s anti-disturbance capabilities and robustness. Ref. [19] adopted the observation disturbance optimization controller switching strategy and used it in the propeller propulsion motor control of the aircraft to enhances its speed response and anti-disturbance performances. Refs. [16,17,18,19] suggestions are all in the motor system, by switching the linear/nonlinear ADRC controller to improve the anti-disturbance capabilities, but this kind of method mainly combines the advantages of the linear/nonlinear ADRC controller; the parameter configuration and the stability analysis are still relatively complicated, and the control performance of the linear/nonlinear active disturbance rejection controller itself has not been improved in essence.

In terms of the structure of ADRC itself, some scholars have proposed some improvement methods for the problem of poor disturbance suppression performance in the middle frequency band of the controller, which suppresses the problems of speed, torque ripples, and motor noise caused by periodic disturbances in the motor. Ref. [20] proposed a control method that combines an improved vector resonant controller and an active disturbance rejection control controller for suppressing the current harmonics of permanent-magnet synchronous linear motors. Ref. [21] combined the quasi-resonant controller with the active disturbance rejection control law to suppress the first and second harmonics of the motor speed caused by current measurement errors, but this method is prone to stability problems under high-speed conditions. Ref. [22] designed an extended harmonic state observer (EHSO) for the estimation and attenuation of the selective periodic disturbance, which addresses the torque-ripple reduction in PMSM drives for smooth speed control. Moreover, a pole-placement strategy is proposed and analyzed through sensitivity function to improve the disturbance-rejection capability and the relative stability. Ref. [23] proposed a two-degree-of-freedom ADRC current-control method based on the improved extended state observer to simultaneously improve the current control’s dynamic response speed and steady-state control accuracy, eliminate the influence of ADRC tracking performance and disturbance suppression coupling on harmonic suppression. And, it is worth noting that the observed disturbance update law of the improved ESO is designed as a proportional–integral–repetitive control structure. Ref. [24] proposed an adaptive ADRC for the current loop of PMSM to suppress the uncertain periodic and aperiodic disturbances simultaneously, and it does not require the disturbance frequency information.

In summary, ADRC is a control method with strong anti-disturbance ability and good parameter robustness, which is suitable for the working conditions of ship electric propulsion [25]. However, for time-varying disturbances, the ADRC controller also has the problem of poor anti-disturbance capabilities, which needs to be further solved.

In the ship propulsion motor speed control system, the ADRC can be used to achieve direct speed control [16], and accurately estimate and compensate the system state variables in the presence of internal and external disturbances. It requires the observer to quickly and accurately deal with the lumped disturbances that deviate from the established standard integral-series controlled object [9], but in the actual system, the bandwidth of the controller is limited by the switching frequency of the inverter. The state tracking and speed estimation of ADRC based on integral observation structure are limited, and it is difficult to take the advantages of active disturbance rejection controllers. As a result, when the ADRC is used to control the speed in the permanent-magnet synchronous motor of the ship, it has a poor suppression effect on the speed fluctuation caused by the sudden change in the load torque, the first and second harmonics caused by the current measurement error and sixth harmonic caused by inverter dead-time effects and non-sinusoidal flux linkage.

In order to solve the above problems, this paper proposes a PMSM active disturbance rejection control method based on improved observer and current disturbance feedforward compensation. First, in order to improve the tracking performance and the disturbance suppression performance of ADRC, the state and disturbance observation laws of the extended state observer are improved to a proportional–integral control structure, which improves the dynamic response speed of the motor control system. Second, in order to further improve the system’s ability to suppress sudden changes in load torque and reduce speed fluctuations in the dynamic process, the measured current information is used to construct a disturbance feedforward compensation item to make the adjustment process faster and more stable. Third, in order to suppress the first, second, and sixth harmonics of the speed caused by the current measurement error and inverter nonlinearity, the continuous nonlinear function with bounded gain and the IIR low-pass filter are used to design dynamic filtering method, which can adjust the filtering weight according to the observation error, quickly track the current in the dynamic process and filter the current harmonics in the steady state, taking into account the dynamic response speed and steady-state control accuracy. Finally, this paper uses MATLAB/Simulink 2021a software to model the propeller load model of the ship’s electric propulsion system. On the permanent-magnet synchronous motor experimental platform, experiments such as motor load mutation, sudden current measurement error, ship staged start, and propeller torque mutation are carried out. Experimental results on motor speed and current verify the effectiveness and superiority of the proposed method. There are two main highlights in this paper:

(1)

In this study, the tracking performance and anti-disturbance performance of the controller are improved by reconstructing the structure of the extended state observer, which significantly enhances the dynamic response speed of the system.

(2)

On the basis of the original controller, a current disturbance compensation item with dynamic filtering is designed to reduce the speed fluctuation when the motor torque changes suddenly. The dynamic filtering link effectively filters the current harmonics, suppresses the influence of periodic disturbances such as current measurement errors of the motor system, and improves the current-control accuracy.

The remainder of the paper is organized as follows: Section 2 presents the mathematical model of the electrical propulsion system. Section 3 presents the Improved Active Disturbance Rejection Controller (IADRC) and the smooth switching strategy. Section 4 presents the stability and the performance analysis of the improved structure. Section 5 discusses the overall experimental results of the system. Finally, the conclusions are given in Section 6.

2. Ship Propulsion Motor and Its Propeller Load Model

2.1. Motor State Equation under the Framework of ADRC

Assuming that in the core saturation, eddy current and hysteresis losses are ignored, using the vector control strategy with i_d = 0, the surface-mounted permanent-magnet synchronous motor (SPMSM) motion equation, and voltage equation in the synchronous rotating coordinate system are expressed as follows:

$$\left\{\begin{array}{l}{T}_{\mathrm{e}}-T_{\mathrm{L}}=J{\dot{\omega }}_{\mathrm{m}}+B{\omega }_{\mathrm{m}}\\ T_{\mathrm{e}}=K_{\mathrm{t}}{i}_q=1.5n_{\mathrm{p}}{\lambda }_{\mathrm{r}}{i}_q\\ u_d=L_{\mathrm{s}}{\dot{i}}_d+R_{\mathrm{s}}{i}_d-L_{\mathrm{s}}{n}_{\mathrm{p}}{\omega }_{\mathrm{m}}{i}_q\\ u_q=L_{\mathrm{s}}{\dot{i}}_q+R_{\mathrm{s}}{i}_q+n_{\mathrm{p}}{\omega }_{\mathrm{m}}(L_{\mathrm{s}}{i}_d+{\lambda }_{\mathrm{r}})\end{array}\right.$$

(1)

where T_e and T_L are electromagnetic torque and load torque, respectively; J is the moment of inertia; ω_m is mechanical angular speed; B is friction coefficient; K_t is torque coefficient; i_d and i_q are d-axis and q-axis current, respectively; u_d and u_q are the d-axis and q-axis voltages; L_s and R_s are the stator inductance and resistance, respectively; n_p is the number of pole pairs of the PMSM; and λ_r is the magnetic flux of the permanent magnet.

The second-order equation of the mechanical angular speed in PMSM can be obtained from Equation (1):

$${\ddot{\omega }}_{\mathrm{m}}=\frac{K_{\mathrm{t}}}{J{L}_{\mathrm{s}}}{u}_q-\frac{{\dot{T}}_{\mathrm{L}}}{J}-\frac{B{\dot{\omega }}_{\mathrm{m}}}{J}-\frac{n_{\mathrm{p}}{K}_{\mathrm{t}}{\lambda }_{\mathrm{r}}}{J{L}_{\mathrm{s}}}{\omega }_{\mathrm{m}}-\frac{K_{\mathrm{t}}{R}_{\mathrm{s}}}{J{L}_{\mathrm{s}}}{i}_q$$

(2)

Let

$$b=\frac{K_{\mathrm{t}}}{J{L}_{\mathrm{s}}}$$

(3)

$$f\left({\omega }_{\mathrm{m}},i_q,T_{\mathrm{L}}\right)=\left(b-b_0\right)u_q-\frac{{\dot{T}}_{\mathrm{L}}}{J}-\frac{B{\dot{\omega }}_{\mathrm{m}}}{J}-\frac{n_{\mathrm{p}}{K}_{\mathrm{t}}{\lambda }_{\mathrm{r}}}{J{L}_{\mathrm{s}}}{\omega }_{\mathrm{m}}-\frac{K_{\mathrm{t}}{R}_{\mathrm{s}}}{J{L}_{\mathrm{s}}}{i}_q$$

(4)

where b₀ is the estimated value of b; then, Equation (2) can be rewritten as follows:

$$\ddot{y}=b_{0}u+f\left({\omega }_{\mathrm{m}},i_q,T_{\mathrm{L}}\right)$$

(5)

where y and u represent ω_m and u_q, respectively; and f(ω_m, i_q, T_L) represents the lumped disturbances of PMSM. By estimating and compensating f(ω_m, i_q, T_L) in the control system, the influence of various disturbances and unmodeled dynamics on the system can be effectively suppressed, and the original complex permanent-magnet synchronous motor system control problem is transformed into the control of a unit gain double integrator. For this problem, the PD control method can achieve a good control performance and has strong robustness [9].

2.2. Mathematical Model of the Propeller Load

In the working condition of ship electric propulsion, the propulsion motor drives the propeller to overcome the seawater resistance moment to rotate, and at the same time, the seawater generates thrust on the propeller and transmits it to the hull to push the ship to sail [25,26,27]. This process requires the motor to provide electromagnetic torque for the propeller, and the motor works under complex conditions where the propeller interacts with the hull and seawater. The thrust P (unit is N) and torque T_p (unit is N·m) generated when the propeller rotates are

$$P=K_{\mathrm{p}}\rho D^2\left(v_{\mathrm{p}}^2+n^2{D}^2\right)$$

(6)

$$T_{\mathrm{p}}=K_{\mathrm{T}}\rho D^3\left(v_{\mathrm{p}}^2+n^2{D}^2\right)$$

(7)

where ρ is the density of seawater, generally 1025 kg/m³; D is the diameter of the propeller and the unit is m; and v_p is propeller velocity relative to water and the unit is m/s. n is the rotation speed of the propeller and the unit is r/s; K_p is the dimensionless coefficient of propeller thrust; and K_T is the dimensionless coefficient of drag torque, which are the functions of the advance ratio L′. According to the derivation process in Appendix A, when n > 0, K_p and K_T can be expressed approximately as an 8-order Chebyshev polynomial [28]:

$$K_{\mathrm{p}}=0.348-0.051L^{\prime }-1.224L^{\prime 2}+0.007L^{\prime 3}+3.618L^{\prime 4}-1.273L^{\prime 5}-4.670L^{\prime 6}+1.155L^{\prime 7}+1.944L^{\prime 8}$$

(8)

$$K_{\mathrm{T}}=0.047-0.012L^{\prime }-0.139L^{\prime 2}+0.068L^{\prime 3}+0.407L^{\prime 4}-0.306L^{\prime 5}-0.571L^{\prime 6}+0.228L^{\prime 7}+0.261L^{\prime 8}$$

(9)

When the propeller rotates at speed n to propel the ship forward, and the advance speed of the propeller is v_p, then the distance traveled by the propeller per revolution is

$$h_{\mathrm{p}}=\frac{v_{\mathrm{p}}}{n}$$

(10)

The ratio of the h_p to the propeller diameter D is called the advance ratio L of the propeller, also known as the advance speed coefficient. To avoid the situation that the values of K_p, K_T, and L are infinite when the propeller speed is close to 0 [27], a bounded form of advance speed ratio is adopted in this paper. The expression of the advance ratio L′ in bounded form is [28]

$$L^{\prime }=\frac{v_{\mathrm{p}}}{\sqrt{v_{\mathrm{p}}^2+n^2{D}^2}}$$

(11)

When the propeller rotates, it can cause additional resistance to the ship, and the impact of this resistance increase can be expressed by the thrust deduction coefficient t; so, the effective thrust P_e (unit is N) generated by the propeller is

$$P_{\mathrm{e}}=\left(1-t\right)P=\left(1-t\right)K_{\mathrm{p}}\rho D^2\left(v_p^2+n^2{D}^2\right)$$

(12)

As the ship sails through the water, the water around the ship is affected by the ship, which creates currents that flow with the ship. This phenomenon is called a wake. The wake will cause a difference between propeller forward speed v_p and ship speed v_s. The relationship between the two is

$$v_{\mathrm{p}}=\left(1-w\right)v_{\mathrm{s}}$$

(13)

where w is the wake coefficient; and the total resistance suffered by the ship is

$$R_{\mathrm{s}}=c{v}_{\mathrm{s}}^2$$

(14)

where c is the resistance coefficient; and the motion equation of the ship–machine–propeller system can be expressed as

$$\left(m+\Delta m\right)\frac{d{v}_{\mathrm{s}}}{dt}=P_{\mathrm{e}}-P_{\mathrm{s}}$$

(15)

where m is the mass of the hull and Δm is the mass of the attached water, which is generally taken as 5–15% of the mass of the hull. The experimental part of this paper adopts the ship parameters in

Table 1. Parameters of a ship.

Parameters	Value
Ship tonnage m	92 t
Propeller diameter D	0.9 m
Number of blades	4
Propeller screw pitch ratio H/D	0.965
Propeller disc square ratio A/Ad	0.45
Thrust deduction coefficient t	0.145
Wake coefficient w	0.157
Resistance coefficient c	694.2

.

According to the data in Table 1 and Equations (6)–(15), the propeller load model is built as shown in Figure 1:

Therefore, according to the motor speed and the propeller load model, the load torque generated by the propeller to the propulsion motor can be obtained.

3. Improved Active Disturbance Rejection Controller (IADRC)

In the ADRC method, the tracking differentiator (TD) generates tracking signals and differential tracking signals based on the input signals. The extended state observer (ESO) is used to observe the system state and compensate for lumped disturbances. The control law combines the deviation between the observation value and the given value and the disturbance compensation value, and the three parts work together to control the plant.

3.1. Tracking Differentiator (TD)

The mathematical model of the tracking differentiator is expressed as

$$\left\{\begin{array}{l}{\dot{\omega }}_1={\omega }_2\\ {\dot{\omega }}_2=fhan({\omega }_1-{\omega }_{\mathrm{ref}},{\omega }_2,r,h)\end{array}\right.$$

(16)

where ω_ref is the reference input speed; ω₁ is the tracking signal of the reference speed; ω₂ is the differential of the tracking signal; r is the speed factor; h is the control step; and fhan(f₁,f₂,r,h) is the optimal tracking function. They are given by

$$\left\{\begin{array}{l}d=r{h}^2\\ y=f_1+h{f}_2\\ a_1=\sqrt{d(d+8|y|)}\\ a_2=h{f}_2+\mathrm{sign}(y)(a_1-d)/2\\ a_3=(\mathrm{sign}(y+d)-\mathrm{sign}(y-d))/2\\ a_4=(h{f}_2+y-a_2)a_3+a_2\\ a_5=(\mathrm{sign}(a_4+d)-\mathrm{sign}(a_4-d))/2\\ fhan(f_1,f_2,r,h)=-r(\frac{a_4}{d}-\mathrm{sign}(a_4))a_5-r\mathrm{sign}(a_4)\end{array}\right.$$

(17)

For the fhan function, the selection of r and h affects the performance of the tracking differentiator, and the differential signal helps improve the speed of motor speed regulation. Taking r = 50,000 and h = 0.0001, the TD tracking characteristics are shown in Figure 2:

Figure 2 shows that the traditional fhan function has different tracking performance for different rotational speed tracking values, but after a simple transformation,

$$\left\{\begin{array}{l}{\dot{\omega }}_1={\omega }_2,{\omega }_{r1}={\omega }_1/\lambda ,{\omega }_{r2}={\omega }_2/\lambda \\ {\dot{\omega }}_2=fhan({\omega }_1-\lambda {\omega }_{\mathrm{ref}},{\omega }_2,r,h)\end{array}\right.$$

(18)

For different given signals, the same tracking performance can be obtained, and the transformed function tracking characteristics are shown in Figure 3:

The tracking speed can be adjusted by adjusting λ to obtain the desired tracking effect.

3.2. Extended State Observer (ESO)

According to the motor motion equation, a third-order extended state observer (as shown in Figure 4) is designed, and then an ADRC controller is designed. According to the controller design method [16], the Linear ADRC controller directly outputs the reference voltage to adjust the motor speed, and the q-axis current feedback link is canceled, reducing the dependence of the motor control system on the current sensor.

According to Equation (5), the state equation of PMSM can be expressed as follows:

$$\left\{\begin{array}{l}y=x_1={\omega }_{\mathrm{m}}\\ {\dot{x}}_{{}_1}=x_2={\dot{\omega }}_{\mathrm{m}}\\ {\dot{x}}_2=x_3+b_{0}u\\ {\dot{x}}_3=\dot{f}\left({\omega }_{\mathrm{m}},i_q,T_{\mathrm{L}}\right)\end{array}\right.$$

(19)

According to the state Equation (19) of the motor, the third-order extended state observer is designed as follows:

$$\left\{\begin{array}{l}e={\omega }_{\mathrm{m}}-z_1=y-\widehat{y}\\ {\dot{z}}_1=z_2+{\beta }_{1}e\\ {\dot{z}}_2=z_3+{\beta }_{2}e+b_{0}u\\ {\dot{z}}_3={\beta }_{3}e\end{array}\right.$$

(20)

where z₁, z₂ and z₃ are the observed values of x₁, x₂ and x₃, respectively. And the control law u can be designed as

$$u_0=k_1\left({\omega }_r{}_1-z_1\right)+k_2\left({\omega }_{r2}-z_2\right)$$

(21)

$$u=\frac{u_0-z_3}{b_0}$$

(22)

When the tracking transition process ends, ω_r1 becomes a given signal value ω_ref, ω_r2 is 0, and Equation (22) becomes:

$$u=\frac{k_1\left({\omega }_r{}_1-z_1\right)-k_2{z}_2-z_3}{b_0}$$

(23)

Extended State Observer (ESO) is the core of the Active Disturbance Rejection Controller (ADRC), which is used to estimate the state variables and lumped disturbance f(ω_m, i_q, T_L) of the system. Therefore, the operating characteristics of the ESO play a key role in the control performance of the ADRC.

3.3. Improved Extended State Observer (IESO)

Refs. [23,24,25,26,27,28,29] show that the use of the proportional–integral disturbance update law can improve the anti-disturbance performance of the second-order extended state observer, which is a part of the first-order ADRC controller. In this paper, to improve the state tracking and estimation capabilities of the third-order observer, this section reconstructs the structure of the third-order extended state observer. The state equation of the improved extended state observer is as follows:

$$\left\{\begin{array}{l}e={\omega }_{\mathrm{m}}-z_1=y-\widehat{y}\\ {\dot{z}}_1=z_2\\ z_2=\frac{\left(z_3+b_{0}u\right)}{s}+{\beta }_{1}e\\ z_3={\beta }_{2}e+\frac{{\beta }_{3}e}{s}\end{array}\right.$$

(24)

where s is the Laplace operator. Compared with the structure in Figure 4, the z₂ and z₃ observe update laws of the IESO are designed as a proportional–integral control structure. The structure of the improved ADRC controller is shown in Figure 5:

The control law u can be designed as

$$u_0=k_1^{\prime }\left({\omega }_r{}_1-z_1\right)+k_2^{\prime }\left({\omega }_{r2}-z_2\right)$$

(25)

$$u=\frac{u_0-z_3}{b_0}$$

(26)

When the tracking transition process ends, Equations (25) and (26) become

$$\begin{array}{l}{u}_0=k_1^{\prime }\left({\omega }_r{}_1-z_1\right)-k_2^{\prime }{z}_2\\ u=\frac{k_1^{\prime }\left({\omega }_r{}_1-z_1\right)-k_2^{\prime }{z}_2-z_3}{b_0}\end{array}$$

(27)

3.4. q-Axis Current Disturbance Compensation Item

It can be seen from Figure 4 and Figure 5 that the original active disturbance rejection controller regards the motor current disturbance item as a part of the lumped disturbance, only estimates compensation through the disturbance observation link, and cancels the q-axis current feedback, which can avoid current measurement errors for the impact. However, when the load torque changes greatly, the proportion of this item in the aggregate disturbance is larger. At this time, it is difficult to obtain excellent anti-disturbance performance by only relying on the observation link to deal with these disturbances. In addition, according to Equations (3)–(5), the coefficient of the current disturbance term and the ADRC gain coefficient b only differ by the resistance value R_S. This is a parameter value that is convenient to obtain; so, it is necessary to introduce a current disturbance compensation term in the system.

Rewrite Equation (5) as

$${\ddot{\omega }}_{\mathrm{m}}=b_0{u}_q+f\left({\omega }_{\mathrm{m}},i_q,T_{\mathrm{L}}\right)-b_0{R}_{\mathrm{s}}{i}_q$$

(28)

Then, the lumped disturbances become

$$f^{\prime }\left({\omega }_{\mathrm{m}},i_q,T_{\mathrm{L}}\right)=\left(b-b_0\right)u_q-\frac{{\dot{T}}_{\mathrm{L}}}{J}-\frac{B{\dot{\omega }}_{\mathrm{m}}}{J}-\frac{n_{\mathrm{p}}{K}_{\mathrm{t}}{\lambda }_{\mathrm{r}}}{J{L}_{\mathrm{s}}}{\omega }_{\mathrm{m}}-\left(b-b_0\right)R_{\mathrm{s}}{i}_q$$

(29)

In Equations (28) and (29), the q-axis current disturbance item in the total disturbance is separated and compensated into the disturbance observation item z₃. Through feedforward compensation, this item can quickly compensate the disturbance caused by the change in the motor load torque, and further improve the system tracking and anti-disturbance performance. The designed compensation structure (IADRC + iq, “iq” represents the q-axis current disturbance compensation term, IADRC with the q-axis current disturbance compensation item) is shown in Section 3.5 (except for the smooth switching strategy).

3.5. Design of Smooth Switching Filtering Strategy

After introducing the q-axis current disturbance item, although the system anti-disturbance performance is further improved, the harmonics in the current also directly participate in the system control, resulting in the increase in the first, second, and sixth harmonic content of the motor current. Since the low-pass filtering effect of the system’s rotational inertia, the sixth harmonic has little effect on the motor speed.

At present, the notch filter is mainly used to suppress the current harmonics, but the increased resonance term makes it easy to reduce the system phase margin, resulting in system instability. Since these harmonics are introduced by compensation, the first, second, and sixth harmonics existing in the q-axis current can be filtered out by using an IIR low-pass filter, which fundamentally solves this problem. However, when the filter cutoff frequency is low, it will affect the effect of the current compensation item in the dynamic process (filter delay), and then affect the anti-disturbance performance of the motor. Therefore, it is necessary to adjust the filtering effect according to the working condition of the motor.

To solve the above problems, this paper obtains a nonlinear switching function that is convenient for parameter adjustment by modifying the fal function. Equation (30) of the fal function is as follows:

$$fal\left(e,\alpha ,\delta \right)=\left\{\begin{array}{l}\begin{array}{cc}\frac{e}{{\delta }^{1-\alpha }}& \left|e\right|\le \delta \end{array}\\ \begin{array}{cc}|e{|}^{\alpha }\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(e\right)& \left|e\right|>\delta \end{array}\end{array}\right.$$

(30)

Figure 6 is the fal function characteristic curve when α = 0.01; δ takes different values and δ = 0.01, and α takes different values.

Because the fal function gain item is a continuous nonlinear function with bounded gain, and by adjusting the parameter values α, δ can easily obtain different nonlinear characteristics; so, it is suitable to be transformed into a smooth transition function for the switching process. Equation (31) of the designed transition function f is as follows:

$$f\left(e,\alpha ,\delta \right)=\left\{\begin{array}{l}\begin{array}{cc}1& \left|e\right|\end{array}\le \delta \\ \begin{array}{cc}\frac{|e{|}^{\alpha }\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(e\right){\delta }^{1-\alpha }}{e}& \left|e\right|>\delta \end{array}\end{array}\right.$$

(31)

Figure 7 is the f function characteristic curve when α = 0.01, δ takes different values and δ = 0.2, while α takes different values.

The output of the f function is the weight of the motor current item after filtering, and 1 − f is the weight of the original current item. Scaling the observed deviation e in Equation (24) can match the gain range of the switching function. Through experiments, the fluctuation range of the error e during the steady-state operation of the motor can be easily obtained. At this time, the output of the f function should be 1. During the operation of the motor, the weight is dynamically adjusted according to the working conditions. Since the f function is continuous, there will be no sudden change in the current during the adjustment process, and the dynamic and steady-state performance of the system will be taken into account. The effectiveness of the method is verified by experiments.

The improved controller structure (IFADRC + iq, IADRC+iq controller with dynamic filtering method) is shown in Figure 8:

Figure 8 shows that as the observe error e changes, the current filter weights change accordingly. In the dynamic adjustment process, the current in the compensation item is closer to the real current to ensure the response speed; in the steady state process, all the compensation items are the filtered current, and the current harmonic content is very small when the cut-off frequency is very low, which avoids the current and speed ripples in the steady state.

4. IADRC Stability and Frequency Domain Performance Analysis

4.1. IADRC Stability Analysis

The state equation of the motor in Equation (19) can be written as

$$\begin{array}{l}\dot{x}=Ax+Bu+H\\ y=Cx\end{array}$$

(32)

where

$$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& -{\beta }_1& 1\\ 0& -{\beta }_2& 0\end{array}\right],B=\left[\begin{array}{l}0\\ b_0\\ 0\end{array}\right],C=[\begin{array}{ccc}1& 0& 0\end{array}],H=\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right]\dot{f}\left({\omega }_{\mathrm{m}},i_q,T_{\mathrm{L}}\right)+\left[\begin{array}{ccc}0& 0& 0\\ 0& {\beta }_1& 0\\ 0& {\beta }_2& 0\end{array}\right]x$$

The improved observer state equation in Equation (24) can be written as

$$\begin{array}{l}\dot{z}=Az+Bu+L\left(y-\widehat{y}\right)+W\\ \widehat{y}=Cz\end{array}$$

(33)

where

$$L=\left[\begin{array}{l}0\\ 0\\ {\beta }_3\end{array}\right],W=\left[\begin{array}{l}0\\ {\beta }_1\\ {\beta }_2\end{array}\right]{\dot{\omega }}_{\mathrm{m}}$$

Let e_i = x_i − z_i, I = 1, 2, 3. and subtract Equation (33) from Equation (32), the error equation can be written as

$$\dot{e}=A_{e}e+H-W$$

(34)

where

$$A_e=A-LC=\left[\begin{array}{ccc}0& 1& 0\\ 0& -{\beta }_1& 1\\ -{\beta }_3& -{\beta }_2& 0\end{array}\right]$$

The characteristic equation of the matrix A_e is

$$\left|sI-A_e\right|=\left|\begin{array}{ccc}s& -1& 0\\ 0& s+{\beta }_1& -1\\ {\beta }_3& {\beta }_2& s\end{array}\right|=s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3$$

(35)

when H-W is bounded and the root of the characteristic equation is on the left side of the complex plane, the IESO is bounded-input bounded-output (BIBO) stable [9]. Let s³ + β₁s² + β₂s + β₃ = (s + ω_o)³. When β₁ = 3ω_o, β₂ = 3ω_o², β₃ = ω_o³, and ω_o > 0, the observer is stable and has a faster convergence speed.

Theorem 1. 

The IADRC in Figure 5 yields a BIBO stable closed-loop system if the observer in Equation (33) and the control law Equation (27) for the plant are stable.

Proof of Theorem 1. 

Equation (27) can be written in a state feedback form of u = (1/b₀)[−$k_1^{\prime }$ − $k_2^{\prime }$ − 1]z = Fz, where F = (1/b₀)[−$k_1^{\prime }$ −$k_2^{\prime }$ − 1]. The closed-loop system is then represented by the state-space equation of

$$\left[\begin{array}{l}\dot{x}\\ \dot{z}\end{array}\right]=\left[\begin{array}{cc}A& BF\\ LC& A-LC+BF\end{array}\right]\left[\begin{array}{l}x\\ z\end{array}\right]+\left[\begin{array}{l}G\\ G\end{array}\right]{\omega }_{r1}+\left[\begin{array}{l}H\\ W\end{array}\right]$$

(36)

where G = $k_1^{\prime }$B/b₀. Equation (36) is BIBO stable if its eigenvalues are in the left half plane. By applying elementary row and column operations, it is obvious that the closed-loop eigenvalues satisfy

$$\begin{array}{l}eig\left(\left[\begin{array}{cc}A& BF\\ LC& A-LC+BF\end{array}\right]\right)=eig\left(\left[\begin{array}{cc}A+BF& BF\\ 0& A-LC\end{array}\right]\right)\\ =eig\left(A+BF\right)\cup eig\left(A-LC\right)\\ =\{\begin{array}{ccc}\mathrm{root}& \mathrm{of}& s^2+\left(k_2^{\prime }+{\beta }_1\right)\end{array}s+k_2^{\prime }\}\cup \left\{\begin{array}{ccc}\mathrm{root}& \mathrm{of}& s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3\end{array}\right\}\begin{array}{cc}& \mathrm{Q}.\mathrm{E}.\mathrm{D}.\end{array}\end{array}$$

Since ω_r1, as the reference signal, is always bounded, the only nontrivial condition on the plant is that H and W are bounded. In other words, the system state x₁, x₂, and disturbance f(ω_m,i_q,T_L) must be differentiable, which is a reasonable assumption. □

4.2. IADRC Frequency Domain Performance Analysis

Figure 9 is the frequency domain model of permanent-magnet synchronous motor using IADRC controller (the IADRC structure is presented in Figure 5 and Figure 8). From Figure 9, it can be seen that there are various disturbances and unmodeled dynamics in the motor-speed control system, these factors will affect the motor control effect. For the controller used in this paper, unmodeled dynamics such as these external loads and current measurement errors, flux harmonics, etc., can be regarded as disturbance items in the feedforward channel of the control loop; so, it is very important to improve the anti-disturbance capabilities of the controller.

Next, the frequency domain characteristics of the control system are analyzed, and Equations (20) and (23) are regarded as a dual-input single-output system with two inputs y, ω_r1 and output u, and the transfer function from y and ω_r1 to u is derived as follows [30]:

$$\frac{u\left(s\right)}{{\omega }_{\mathrm{m}}\left(s\right)}=-\frac{\left({\beta }_1{k}_1+{\beta }_2{k}_2+{\beta }_3\right)s^2+\left({\beta }_2{k}_1+{\beta }_3{k}_2\right)s+{\beta }_3{k}_1}{s^3+({\beta }_1+k_2)s^2+\left({\beta }_1{k}_2+{\beta }_2+k_1\right)s}\frac{1}{b_0}$$

(37)

$$\frac{u\left(s\right)}{{\omega }_{r1}\left(s\right)}=\frac{k_1{s}^3+k_1{\beta }_1{s}^2+k_1{\beta }_{2}s+k_1{\beta }_3}{s^3+({\beta }_1+k_2)s^2+\left({\beta }_1{k}_2+{\beta }_2+k_1\right)s}\frac{1}{b_0}$$

(38)

Neglecting the sampling delay, the frequency domain model of the permanent-magnet synchronous motor speed control system under ADRC is shown in Figure 10:

where

$$\begin{array}{l} G_f\left(s\right)=-\frac{{\omega }_{\mathrm{m}}\left(s\right)}{u\left(s\right)}\frac{u\left(s\right)}{{\omega }_{r1}\left(s\right)}=\frac{k_1{s}^3+k_1{\beta }_1{s}^2+k_1{\beta }_{2}s+k_1{\beta }_3}{\left({\beta }_1{k}_1+{\beta }_2{k}_2+{\beta }_3\right)s^2+\left({\beta }_2{k}_1+{\beta }_3{k}_2\right)s+{\beta }_3{k}_1}\\ G_c\left(s\right)=-\frac{b_{0}u\left(s\right)}{{\omega }_{\mathrm{m}}\left(s\right)}=\frac{\left({\beta }_1{k}_1+{\beta }_2{k}_2+{\beta }_3\right)s^2+\left({\beta }_2{k}_1+{\beta }_3{k}_2\right)s+{\beta }_3{k}_1}{s^3+({\beta }_1+k_2)s^2+\left({\beta }_1{k}_2+{\beta }_2+k_1\right)s}\end{array}$$

(39)

when using IADRC,

$$\begin{array}{l}{G}_{f\mathrm{I}}\left(s\right)=\frac{k_1^{\prime }{s}^3+k_1^{\prime }{\beta }_1{s}^2+k_1^{\prime }{\beta }_{2}s+k_1^{\prime }{\beta }_3}{\left({\beta }_1{k}_2^{\prime }+{\beta }_2\right)s^3+\left({\beta }_1{k}_1^{\prime }+{\beta }_2{k}_2^{\prime }+{\beta }_3\right)s^2+\left({\beta }_2{k}_1^{\prime }+{\beta }_3{k}_2^{\prime }\right)s+{\beta }_3{k}_1^{\prime }}\\ G_{c\mathrm{I}}\left(s\right)=\frac{\left({\beta }_1{k}_2^{\prime }+{\beta }_2\right)s^3+\left({\beta }_1{k}_1^{\prime }+{\beta }_2{k}_2^{\prime }+{\beta }_3\right)s^2+\left({\beta }_2{k}_1^{\prime }+{\beta }_3{k}_2^{\prime }\right)s+{\beta }_3{k}_1^{\prime }}{s^3+({\beta }_1+k_2^{\prime })s^2+k_1^{\prime }s}\end{array}$$

(40)

Figure 10 shows that the disturbances in the system are mainly concentrated in two places: current loop feedforward channel disturbances (Dead-Time Effects, etc.), and speed loop feedforward channel disturbances (load torque, flux harmonics, etc.). These disturbances include both DC disturbances and periodic disturbances; so, it is necessary to analyze the frequency domain anti-disturbance capabilities of the control system for different disturbance locations.

When using ADRC and IADRC, respectively, the anti-disturbance transfer function of the current loop feedforward channel is:

$$G_{di}\left(s\right)=\frac{G_i\left(s\right)G_s\left(s\right)}{1+G_c\left(s\right)G_i\left(s\right)G_s\left(s\right)/b_0+G_i\left(s\right)G_s\left(s\right)n_{\mathrm{p}}{\lambda }_{\mathrm{r}}}$$

(41)

$$G_{di\mathrm{I}}\left(s\right)=\frac{G_i\left(s\right)G_s\left(s\right)}{1+G_{c\mathrm{I}}\left(s\right)G_i\left(s\right)G_s\left(s\right)/b_0+G_i\left(s\right)G_s\left(s\right)n_{\mathrm{p}}{\lambda }_{\mathrm{r}}}$$

(42)

Therefore, the amplitude–frequency curve can be obtained as in Figure 11.

When using ADRC and IADRC, respectively, the anti-disturbance transfer function of the speed loop feedforward channel is

$$G_{ds}\left(s\right)=\frac{G_s\left(s\right)}{1+G_c\left(s\right)G_i\left(s\right)G_s\left(s\right)/b_0+G_i\left(s\right)G_s\left(s\right)n_{\mathrm{p}}{\lambda }_{\mathrm{r}}}$$

(43)

$$G_{ds\mathrm{I}}\left(s\right)=\frac{G_s\left(s\right)}{1+G_{c\mathrm{I}}\left(s\right)G_i\left(s\right)G_s\left(s\right)/b_0+G_i\left(s\right)G_s\left(s\right)n_{\mathrm{p}}{\lambda }_{\mathrm{r}}}$$

(44)

The amplitude–frequency curve can be obtained as in Figure 12.

The amplitude–frequency characteristic curve in Figure 11 and Figure 12 shows that the improved control structure can effectively improve the anti-disturbance capabilities of the motor in the low-frequency range.

5. Experimental and Simulation Analysis

To verify the effectiveness of the proposed method, this paper conducts experiments on the Teknic-2310P motor platform produced by the TEXAS INSTRUMENTS company and uses a TMS320F28379D dual-core microprocessor to execute the researched control algorithm. Another PMSM is mechanically coupled with the tested motor as a load. The experimental platform is shown in Figure 13:

In the experiment, under the FOC framework, the improved active disturbance rejection direct speed control method proposed in this paper is used to drive the power motor, and the torque output of the load motor is controlled according to the load model of the ship electric propulsion system. The overall structure of the ship’s propulsion motor vector control model is shown in Figure 14:

The propeller load parameters, motor parameters, and controller parameters used in the experiment are shown in Table 1,

Table 2. Main parameters of PMSM.

Parameters	Value
motor pole pairs np	4
stator resistance Rs	0.36 Ω
stator inductor Ls	0.2 mH
permanent magnet flux λr	0.0064 Wb
inverter input voltage Vdc	30 V
motor rated current Ir	7 A

and

Table 3. Parameters of the controller.

Parameters	ADRC/IADRC
r	50,000
h	0.0001
λ	0.8
k1	400,000
k2	4000
$k_2^{\prime }$	20,000
$k_2^{\prime }$	1000
b0	1.36 × 10−7
ωo	4000 rad/s
β1	12,000 rad/s
β2	4.8 × 107 rad/s
β3	6.4 × 1010 rad/s

, respectively. Among them, the PI controller adopts the parameters set by Matlab example for the platform, and the parameters of the extended state observer adopt the bandwidth method (in Section 4) configuration.

5.1. IADRC Disturbance Suppression Experiment

To verify the superiority of the improved method of ADRC proposed, firstly, the three control methods of PI, ADRC, and IADRC are used to test the anti-disturbance capabilities of the speed control system when the external load is suddenly added. The experimental results are shown in Figure 15:

In Figure 15a, when PI is used, the speed drops by 21.2 rpm, and the adjustment time is 0.14 s; when ADRC is used, the speed drops by 20.2 rpm, and the adjustment time is 0.15 s; and when IADRC is used, the speed drops by 10.4 rpm, and the adjustment time is 0.04 s. In Figure 15b, when PI is used, the speed drops by 49.6 rpm, and the adjustment time is 0.2 s; when ADRC is used, the speed drops by 48 rpm, and the adjustment time is 0.22 s; and when IADRC is used, the speed drops by 24.2 rpm, and the adjustment time is 0.05 s. In Figure 15c, when PI is used, the speed drops by 93.5 rpm, and the adjustment time is 0.23 s; when ADRC is used, the speed drops by 92 rpm, and the adjustment time is 0.26 s; and when IADRC is used, the speed drops by 48 rpm, and the adjustment time is 0.07 s. In Figure 15d, under the same load of 1A, IADRC has a faster start-up speed than ADRC. To show the experimental results more clearly,

Table 4. Dynamic performance comparison of the PMSM (Figure 15).

Performance Metrics		PI	ADRC	IADRC
Δωm (rpm)	1A	21.2	20.2	10.4
	2A	49.6	48	24.2
	3A	93.5	92	48
Δt (s)	1A	0.14	0.15	0.04
	2A	0.2	0.22	0.05
	3A	0.23	0.26	0.07
Start-up time t(s)		-	0.25	0.09

summarizes these data.

The results in Table 4 indicate that when IADRC is used, compared with PI and ADRC, the speed fluctuation of the motor is reduced by about 50%, the adjustment time is shortened by more than 60%, and the start-up time is shortened by 64%. These results show that the tracking performance is significantly improved after the observer structure is improved, and the disturbance suppression ability is enhanced.

5.2. Experimental Analysis after Introducing q-Axis Current Disturbance Compensation Item

From the analysis in Section 3.4, it can be known that the current feedback value contains the information of the q-axis current disturbance item, which is missing in the original structure, which greatly limits the anti-disturbance capabilities of the system, and this problem is more serious when the load changes greatly. To solve this problem, this paper introduces the current disturbance term in the observation value z₃. To verify the effectiveness of this method, the anti-disturbance experiment of the traditional ADRC and IADRC before and after the introduction of the current disturbance item, when the system suddenly adds an external load; the dynamic response performance of the four methods is shown in Figure 16:

In Figure 16a, when using ADRC, the speed drops by 47 rpm, and the adjustment time is 0.21 s; and when using traditional ADRC + iq, the speed drops by 24 rpm, and the adjustment time is 0.09 s. When using IADRC, the speed drops by 20 rpm, and the adjustment time is 0.04 s; and when using IADRC + iq, the speed drops by 7 rpm, and the adjustment time is 0.02 s. In Figure 16b, the speed drops by 89 rpm when ADRC is used, and the adjustment time is 0.25 s; and when ADRC + iq is used, the speed drops by 43 rpm, and the adjustment time is 0.12 s. When using IADRC, the speed drops by 41 rpm, and the adjustment time is 0.06 s; and when using IADRC + iq, the speed drops by 15 rpm, and the adjustment time is 0.03 s. To show the experimental results more clearly,

Table 5. Dynamic performance comparison of the PMSM (Figure 16).

Performance Metrics		ADRC	ADRC + iq	IADRC	IADRC + iq
Δωm (rpm)	2A	47	24	20	7
	3A	89	43	41	15
Δt (s)	2A	0.21	0.09	0.04	0.02
	3A	0.25	0.12	0.06	0.03

summarizes these data.

The results in Table 5 show that introducing the q-axis current disturbance item in the ADRC and IADRC controller can reduce the speed fluctuation caused by the external load change, which can further improve the load-disturbance-suppression ability of the motor.

However, this is only an improvement on the disturbance suppression performance, and does not indicate the impact on the steady-state performance of the system. To further analyze the influence of the introduction of the q-axis current disturbance compensation item on the system, the following experiment is carried out. After the motor is started, the phase current offset error (i_{A_offset} = 0.1A, i_{B_offset} 0.04A) is added. The steady-state phase current waveform is shown in Figure 17a, and the response waveforms of IADRC and IADRC + iq (speed and phase current) are shown in Figure 17b.

In Figure 17a, when using PI, the phase current measurement errors will cause significant offset in motor phase current. However, phase current measurement error has little effect on ADRC and IADRC, because these two control structures do not introduce q-axis current, thus avoiding this problem.

Then, Figure 17b shows that when IADRC+iq is adopted, the phase currents are distorted and the motor speed ripples increase because the current harmonics will be introduced into the control system along with the current. Next, FFT analysis is carried out on the motor speed and phase current. The results show that the 1st, 2nd, 6th, and 12th harmonics in the speed and the 2nd, 3rd, 5th, and 7th harmonics in the phase current have content increase; the harmonic content values are shown in

Table 6. Harmonic analysis of Figure 17b.

Harmonics of Speed	1st	2nd	6th	12th	THD
IADRC	1.22%	0.37%	0.03%	0.02%	1.45%
IADRC + iq	2.35%	0.43%	0.28%	0.04%	2.48%
Harmonics of Current	2nd	3rd	5th	7th	THD
IADRC	2.68%	0.65%	1.44%	1.36%	3.59%
IADRC + iq	6.27%	0.95%	5.13%	4.92%	8.92%

.

The FFT (Fast Fourier Transform Algorithm) analysis results in Table 6 show that when IADRC + iq is used (compared with IADRC), the motor speed and current harmonic total distortion increase by 1.03% and 5.33%, respectively. Which indicate that the q-axis current disturbance compensation item will have an adverse effect on the steady-state accuracy of the system. To overcome this problem, this paper adopts the method of dynamic filtering to deal with the q-axis current harmonics, and it will be verified experimentally in the next section.

5.3. Experimental Analysis of Dynamic Filtering Method

From the comprehensive Figure 8, Figure 16 and Figure 17, it can be seen that after the introduction of the q-axis current disturbance term, although the system’s anti-disturbance capabilities are further improved, the harmonics in the current are also feedforward-compensated to the control law, directly participating in the calculation of the voltage vector, and will lead to an increase in the harmonic content in the current. According to the analysis in Section 3.5, using a low-pass filter and designing a smooth switching strategy can solve this problem better; the weights are dynamically adjusted according to the working conditions when the motor is running, taking into account both dynamic and steady-state performance. After the current measurement error is introduced, the disturbance-suppression experiment results and phase current waveforms under the IADRC and IFADRC + iq methods are shown in Figure 18:

Figure 18a shows that after adopting the smooth switching filter strategy, the anti-disturbance capabilities of the motor are still significantly improved compared with IADRC, and the regulation process is smoother. The speed drops of the IADRC and IFADRC (IFADRC + iq) are 41 rpm and 16 rpm, respectively. The FFT analysis results of the phase current (Figure 18b) is shown in

Table 7. Harmonic analysis of the phase current in Figure 18b.

FFT of Phase Current	3rd	5th	7th	11th
IADRC + iq	0.95%	5.13%	4.92%	0.18%
IFADRC + iq	0.79%	1.51%	1.44%	0.05%

.

The FFT results in Table 7 show that when IFADRC + iq is used, in other words, the dynamic filtering method, the third, fifth, seventh and eleventh harmonics are reduced by 0.16%, 3.62%, 3.48%, and 0.13%, respectively. In addition, the phase current waveforms (Figure 18b) show a significant improvement in current distortion.

This section completes the design of the controller proposed in this paper, which can be divided into three parts:

(1)

IADRC: Active Disturbance Rejection Controller with Improved Observation Structure;

(2)

The q-axis current disturbance compensation item;

(3)

Dynamic filtering strategy.

This method has the advantages of fast response, strong anti-disturbance capabilities, and low steady-state ripples.

5.4. Ship–Machine–Propeller Speed Control System Experiment

To further verify the starting performance of the proposed method under the condition of ship electric propulsion, the motor starting experiment under the propeller load was carried out (multiply the calculated propeller torque by 1/8000 to match the experimental motor load range); when the actual ship sailed, to avoid excessive torque, the method of staged starting is usually adopted. The experimental results of the staged starting of the propulsion motor under the three control methods are shown in Figure 19.

To make the comparison effect more obvious, this paper also conducted a set of direct starting experiments, and the results are shown in Figure 20:

The experimental results in Figure 20 show that the propulsion motor has a faster starting speed when using IFADRC (IFADRC + iq), which shows that the method has a stronger anti-disturbance ability when the load torque is constantly changing, and the motor starting performance and ship maneuverability are better. When PI, ADRC and IFADRC are used, the motor starting times are 0.19 s, 0.39 s and 0.39 s, respectively.

To further verify the control performance of the proposed method for the step load disturbance under the ship’s electric propulsion condition, the motor-speed control experiment under the propeller load was carried out. Ref. [31] indicated that the propeller may be exposed to the water surface in wind and waves, and the propeller torque of the ship will change significantly. Therefore, by changing 75% of the propeller torque to verify the anti-disturbance performance of the control system when the propeller is exposed to the sea due to severe sea conditions, the experimental results are shown in Figure 21:

The experimental results in

Table 8. Motor dynamic response performance in Figure 21.

Performance Metrics		PI	ADRC	IFADRC + iq
Δωm (rpm)	up	278	279	64
	drop	186	158	58
	total	464	437	122
Δt (s)	up	0.24	0.26	0.10
	drop	0.34	0.41	0.15

show that when PI control is used, the speed fluctuations are 278 rpm and −186 rpm, respectively; and when ADRC is used, the speed fluctuations are 279 rpm and −158 rpm, respectively. It is difficult to quickly suppress the disturbance effect when the propeller torque changes greatly, and the speed fluctuation is large. It is not conducive to ship maneuvering and stable navigation. When using IFADRC, the speed fluctuations are 64 rpm and −58 rpm, respectively, and the adjustment time is shorter, and the current response is fast without fluctuation. Compared with PI and ADRC, the total speed fluctuation of IFADRC is reduced by 73.7% and 72.1%, respectively. This method can still maintain good anti-disturbance capabilities under large disturbances.

To sum up, when the improved ADRC is used to propel the motor, the speed and torque can be adjusted quickly and smoothly. It can overcome various uncertainties and the influence of internal and external disturbances, and is suitable for ship electric propulsion conditions with high mobility requirements and time-varying disturbances.

5.5. Simulation Analysis of Another Ship Propulsion Motor

In order to better demonstrate the effectiveness of the method proposed in this paper, the motor parameters of another electric propulsion ship are used for simulation. The ship has a total length of 21.1 m, an empty ship weight of 29 t, and is equipped with an AC propulsion motor with a power level of 100 kw. The simulation was built in the MATLAB/simulink environment and carried out according to the parameter design method mentioned in the paper.

Table 9. Parameters of the PMSM in simulation.

Parameters	Value
motor pole pairs np	12
stator resistance Rs	0.36 Ω
stator inductor Ls	3.074 H
permanent magnet flux λr	0.305 Wb
Moment of inertia J	0.8051 kg·m2

shows the motor parameters used in the simulation.

To show the simulation results more clearly, the results of Figure 22 are shown in

Table 10. Simulation results.

Performance Metrics		ADRC	ADRC + iq	IADRC	IFADRC + iq
Δωm (rpm)	50%-rated load	175	136	71	42
	100%-rated load	336	243	150	96
Start-up time (50% load) t(s)		0.12	-	0.26	-

.

The simulation results in Table 10 show that the method proposed in this paper is also suitable for high-power motor control. The control method proposed in this paper can still improve the tracking speed and disturbance suppression ability of the high-power propulsion motor system.

6. Conclusions and Outlook

This article proposes an improved ADRC method for ship-propulsion motors, which effectively improves the tracking performance, disturbance rejection performance, and steady-state control accuracy of the propulsion motor system.

(1) In order to improve the disturbance observation speed of the third-order ESO, the observer structure is reconstructed, and then the IADRC method is proposed, which has the advantages of fast-tracking speed and strong disturbance-suppression ability. In the 200 rpm load start-up and 3A anti-load disturbance experiments, the index of IADRC is about 0.14 s and 44 rpm smaller than ADRC, respectively.

(2) To utilize the measured current information, a current disturbance compensation item is built in the controller, which further improves the anti-disturbance performance of the system. The results of anti-disturbance experiments show that adding current compensation items to ADRC can reduce about 50% of the speed drop and 53% of the adjustment time; adding current compensation items to IADRC can reduce about 64% of the speed drop and 50% of the adjustment time. However, current experiments show that the current compensation term introduces current harmonics, and in IADRC, the fifth and seventh harmonics cause significant current distortion and degrade the current quality.

(3) To solve the above problems, this paper designs a dynamic filtering method, including IIR low-pass filter and nonlinear smooth switching function. While ensuring the dynamic performance, the harmonic content of the steady-state current is reduced. Current FFT analysis shows that the fifth and seventh harmonic content are reduced by about 70% after using the filtering method. The phase current waveforms show that the current distortion is significantly improved.

(4) Combined with the built propeller load, the effectiveness of the proposed method is verified. The results of the start-up experiment with propellers show that the start-up time of IFADRC is 0.2 s shorter than that of ADRC. In the experiment where the propeller torque changes by 75%, the speed fluctuation of IFADRC is more than 70% smaller than that of PI and ADRC, and the adjustment time is shortened by more than 50%.

Our future work will verify the effectiveness of the proposed algorithm in actual ships. This will significantly improve the anti-disturbance performance of propulsion motors, and enhance ship maneuverability and stability.