Research on composite hierarchical anti-disturbance strategy for magnetic bearing rotor using H ∞ and observer

Abstract

The magnetic bearing rotor system is affected by internal uncertainties and various forms of external disturbances, which will affect the stability of the active magnetic bearing system. To address this issue, a composite hierarchical anti-disturbance strategy is used to improve the anti-disturbance ability of the magnetic bearing rotor system. The disturbance observer is used to compensate the system’s external disturbances, and the controller is used to suppress the system’s internal uncertainties. For this purpose, H_∞ controller, disturbance observer (DOB), and linear extended state observer (LESO) are designed, respectively. Then, H_∞ controller is combined with DOB and LESO to form H_∞-DOB and H_∞-LESO composite controllers and prove their stability. Finally, simulation and experiments show that H_∞-DOB and H_∞-LESO have better disturbance suppression effects than the H_∞ controller for external disturbances. And when the disturbance frequency or rotor rotation frequency of the system is high, H_∞-LESO controller has better disturbance suppression effect than H_∞-DOB.

Keywords

active magnetic bearing composite hierarchical anti-disturbance control anti-disturbance disturbance observer linear extended state observer

1. Introduction

Active magnetic bearing (AMB) is a new non-contact support structure that uses electromagnetic force to counteract the external force of rotor. Compared with traditional bearings, it has the advantages of high rotation speed, active control, etc. (Bleuler and Cole, 2009).

However, different disturbance sources under different application scenarios can affect the stability of AMB. When AMB is applied to vehicle flywheel (Hawkins et al., 2003) and spacecraft control moment gyroscope (Basaran et al., 2017), it will be affected by the base vibration. The main types of base vibration are harmonic excitation, impact excitation, and random excitation, with the low-frequency harmonic excitation of 20–60 Hz being the most common (Jugo et al., 2006, 2008; Kang et al., 2006). Therefore, for the magnetic bearing rotor system applied in different fields, the low-frequency sinusoidal signal is a common disturbance signal.

Because the internal uncertainties and external disturbances significantly affect the stability and robustness of the system, it is necessary to improve the anti-disturbance ability of the magnetic bearing rotor system. In addition to the traditional PID control, some advanced control schemes have been proposed, such as adaptive odd repetitive control (Zhang et al., 2023), adaptive backstepping control (Xu et al., 2021), robust control (Ran et al., 2018). Among these advanced control schemes, robust control methods, such as H_∞ control, have attracted increasing attention, becoming one of the research hotspots of magnetic bearing control. However, because determining the upper bound of disturbance is difficult, the robustness of the system is usually realized using a large gain and control voltage. Combining the controller with an observer is a more effective control method to overcome this shortcoming (Qu et al., 2020).

To improve the control accuracy, Guo et al. (2014) proposed a composite hierarchical anti-disturbance control strategy based on multisource disturbances classification modeling. The inner controller is the disturbance observer and the outer controller is the disturbance suppression controller. Disturbance observer is an effective method for estimating disturbances. It can effectively estimate and offset external disturbances, improving overall performance of the system in the presence of unknown but bounded disturbances. Existing research results include a combination of DOB and H_∞ control, a combination of DOB and sliding mode control, a combination of DOB and backstepping control (Wei et al., 2009, 2015, 2018). One of the advantages of using the disturbance observer is that the inner-loop controller is always designed to achieve a faster response speed than the outer-loop feedback controller, making it more effective than single-loop control schemes when dealing with disturbances. However, the observer cannot be used alone, an external loop feedback control is required to stabilize the system.

The traditional disturbance observer has high requirements for the accuracy of the system model, can only estimate the external disturbances to a limited extent, and poorly estimates the uncertainties inside the system (Noshadi et al., 2017). Therefore, Han proposed auto disturbance rejection controller (ADRC) based on modern control. However, ADRC has several adjustable parameters without any theoretical guidance, the application of ADRC is difficult. As a result, Gao (2003) proposed a linear auto disturbance rejection controller (LADRC) that converts the extended state observer (ESO) and state error feedback rate to linear mode, thereby greatly reducing the number of adjustable parameters of the LADRC. Jin et al. (2019) used LADRC to estimate various uncertain disturbances in the magnetic bearing system. By comparing simulation and experimental results, they found that LADRC has a better anti-disturbance effect than PID. Guan et al. (2020) used LESO in conjunction with all-coefficient adaptive control. Simulation and experiments showed that this method has better control effect than the original adaptive controller does. Li (2023) used the ADRC method to decouple the AMB system to solve the coupling of inertial and gyroscopic effects. And use LESO to suppress disturbances in the system, improving system stability. Because the LESO can expand the total disturbance into a new state variable of the system, all states including the original state variables and disturbances of the system can be reconstructed using the input and output of the system. Compared to the disturbance observer, LESO can estimate the total disturbances of the system, including all internal uncertainties and external disturbances, resulting in a better disturbance suppression effect.

To suppress the low-frequency sinusoidal disturbances and internal uncertainties in the magnetic bearing rotor system, this paper proposed a composite hierarchical anti-disturbance controller (H_∞-LESO) for the magnetic bearing rotor system, which reduces the workload of the controller while improving the anti-disturbance ability of AMB system. H_∞-LESO combines H_∞ controller and LESO. The latter is used to estimate and offset the internal uncertainties and external disturbances of the system to improve the robustness of the system.

The rest of the paper is arranged as follows: Section 2 describes the modeling of the magnetic bearing system. In Section 3, H_∞-DOB and H_∞-LESO composite controllers are designed and their stability is analyzed. The proposed control method is validated by simulation in Section 4. In Section 5, the simulation results are verified by experiments. Conclusions are drawn in Section 6.

2. Magnetic bearing rotor modeling

2.1. Electromagnet and rotor

The working speed of the magnetic bearing rotor system investigated in this paper is less than 0.7 times the first-order bending critical speed, indicating that it can be modeled as a rigid rotor. Without considering the coupling between different degree of freedom (DOF), the dynamic equation of 1-DOF magnetic levitation rotor can be written as follows using Newton’s second law

m \ddot{x} = k_{x} x + k_{i} i,

(1)

where

m

is rotor mass,

k_{x}

is displacement stiffness coefficient,

k_{i}

is current stiffness coefficient, i is control current, and x is displacement. Using Laplace transform, the transfer function of 1-DOF magnetic bearing rotor is written as

\frac{X (s)}{I (s)} = \frac{k_{i}}{m s^{2} - k_{x}} .

(2)

Equation (2) has two real poles, one on the positive real axis and the other on the negative real axis, making it an unstable second-order object that can only work stably with closed-loop control.

2.2. Displacement sensor

Because the magnetic bearing used in this paper has a radial protection clearance of 0.25 mm and the rotor displacement voltage signal output by the sensor ranges from 0 to 5 V, the proportional gain of the displacement sensor can be expressed as follows

k_{s} = \frac{U_{s_o u t}}{δ_{s_i n}} = 20000 V / m,

(3)

where

U_{s_o u t}

is the output displacement voltage of the sensor and

δ_{s_i n}

is the sensor input measurement gap.

2.3. Power amplifier

The power amplifier used in this paper can be regarded as a proportional link when it is working in the magnetic bearing system. It receives a voltage signal of −5–5 V, and the output current signal is 0–3.4 A. The transfer function of the power amplifier can be obtained as follows

k_{a} = 0.34 A / V .

(4)

Combining (2), (3), and (4), the transfer function of 1-DOF magnetic bearing rotor system can be obtained

\frac{X (s)}{I (s)} = \frac{k_{i} k_{a} k_{s}}{m s^{2} - k_{x}} .

(5)

The closed-loop control block diagram of the AMB system is shown in Figure 1. Assuming the rotor displacement is $x_{1}$ , its state-space equation is as follows

{\begin{cases} [\begin{array}{l} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{array}] = [\begin{array}{l} 0 & 1 \\ \frac{k_{x}}{m} & 0 \end{array}] [\begin{array}{l} x_{1} \\ x_{2} \end{array}] + [\begin{array}{l} 0 \\ \frac{k_{i} k_{a} k_{s}}{m} \end{array}] u \\ y = [\begin{array}{l} 1 & 0 \end{array}] [\begin{array}{l} x_{1} \\ x_{2} \end{array}] \end{cases} .

(6)

3. Controller design

3.1. H_∞ Controller design

In the magnetic bearing rotor system, H_∞ controller can improve the robustness of the system. The generalized controlled object model can be established and system uncertainties and external disturbances can be effectively suppressed by selecting appropriate weighting functions. Figure 2 depicts the structure diagram of H_∞ mixed sensitivity control, where u is the reference input, y is the system output, e is the tracking error, r is the control output of the controller, d is the external disturbance, $z_{1}$ , $z_{2}$ , and $z_{3}$ are the evaluation of output signal, $K (s)$ is the transfer function of controller, $G_{p} (s)$ is the transfer function of the controlled object, $W_{1} (s)$ is the sensitivity weighting function, $W_{2} (s)$ is the uncertainties weighting function, and $W_{3} (s)$ is the complementary sensitivity weighting function.

The sensitivity function (7), which is the closed-loop transfer function from d to y, represents the ability of the system to suppress external disturbances

S (s) = \frac{1}{1 + G_{p} (s) K (s)} .

(7)

By lowering the gain of the sensitivity function, the system’s ability to suppress disturbance can be improved.

The complementary sensitivity function (8), which is the transfer function from r to y, represents the ability of the system to suppress the uncertainties of the model

T (s) = \frac{G_{p} (s) K (s)}{1 + G_{p} (s) K (s)} .

(8)

Equation (8) represents the size of the perturbation in the multiplicative perturbation Δ of the system (I+Δ)G. The introduction of the complementary sensitivity function can effectively ensure the robustness of the system. It can be seen from (7) and (8)

S (s) + T (s) = I .

(9)

Let

R (s) = \frac{K (s)}{1 + G_{p} (s) K (s)} .

(10)

Equation (10) represents the size of the perturbation $Δ G$ in the additive perturbation $G + Δ G$ in the system and describes the low frequency parameter perturbation in the system.

$W_{1}$ , $W_{2}$ , and $W_{3}$ are introduced weighting functions. $W_{1}$ corresponds to the performance of the closed-loop system, that is, the weighting function for the sensitivity function S(s). $W_{2}$ is the linear weighting function for additive uncertainties, that is, the weighting function for R(s). $W_{3}$ is the weighting function for multiplicative uncertainties, that is, the weighting function of complementary sensitivity function T(s).

According to Zhang’s study (2010), the closed-loop transfer function from u to z is

T_{u z} (s) = G_{11} + G_{12} K {(I - G_{22} K)}^{- 1} G_{21} = [\begin{array}{l} W_{1} S \\ W_{2} K S \\ W_{3} T \end{array}],

(11)

G_{11} = {[\begin{array}{l} W_{1} & 0 & 0 \end{array}]}^{T}

G_{12} = {[\begin{array}{l} {- W}_{1} G_{p} & W_{2} & W_{3} G_{p} \end{array}]}^{T}

G_{21} = I

, and

G_{22} = - G_{p}

. In equation (12), the H_∞ mixed sensitivity controller is generally designed to select the appropriate weighting functions

W_{1}

W_{2}

, and

W_{3}

for finding the controller K that stabilizes the system and minimizes the H_∞ norm of the closed-loop transfer function

T_{u z} (s)

{‖ T_{u z} ‖ = ‖ \begin{array}{l} W_{1} S \\ W_{2} K S \\ W_{3} T \end{array} ‖}_{\infty} < Υ = 1 .

(12)

Because there is no clear theoretical guidance on how to select the weighting functions, it can only be debugged based on engineering experience, resulting in too much debugging time to achieve the desired control effect. In addition, if it is not selected properly, it affects the stability and performance of the controller. Therefore, in this paper, the H_∞ controller is used as an external loop feedback controller to stabilize the system and the observer is introduced as an internal structure to observe and offset the disturbances suffered by the system. By reducing the effect of the disturbances on the system output, the burden of the controller is reduced and the robustness of the system is improved.

3.2. Design of disturbance observer

The basic principle of the design of disturbance observer is that the external disturbances and model errors are equivalent to the control input and the observed equivalent disturbances are compensated to the system through the feed forward link to realize the complete suppression of the disturbance. The block diagram of disturbance observer is shown in Figure 3, where $\hat{d}$ is the estimated disturbance value, $u$ is control voltage value, $ε$ is the voltage value after removing the observed disturbance, $G_{p} (s)$ is the mathematical model of the system, $G_{n} (s)$ is the nominal model of the system, and $Q (s)$ is a low-pass filter.

According to Mason’s gain formula, it can be seen from Figure 3 that the transfer function from system control input u to system output y can be expressed as follows

G_{u y} = \frac{G_{p} (s) G_{n} (s)}{G_{n} (s) + [G_{p} (s) - G_{n} (s)] Q (s)} .

(13)

The transfer function from external disturbance d to system output y can be expressed as follows

G_{d y} = \frac{G_{p} (s) G_{n} (s) [1 - Q (s)]}{G_{n} (s) + [G_{p} (s) - G_{n} (s)] Q (s)} .

(14)

The transfer function from disturbance noise $ξ$ to system output y can be expressed as follows

G_{ξ y} = \frac{G_{p} (s) Q (s)}{G_{n} (s) + [G_{p} (s) - G_{n} (s)] Q (s)} .

(15)

The core of the design of disturbance observer is the design of low-pass filter Q(s). According to Figure 3, the design of Q(s) requires Q(s)G_n⁻¹(s) to be regular, that is, the order of denominator is greater than that of the numerator. As a feed forward link, the disturbance observer must be robust and have good anti-disturbance performance, which dictates the requirements for the design of Q(s) bandwidth of the low-pass filter. At present, the design principle of Q(s) is usually as follows: in the low frequency band, Q(s) = 1. In the high frequency band, Q(s) = 0.

3.3. Design of linear extended state observer

The difference of LESO from the disturbance observer in part B is that the LESO can not only estimate the external disturbances of the system but can also estimates the uncertainties and modeling errors of the system. These advantages are relatively independent of the mathematical model of the controlled object and can provide better performance. Expand external disturbance, modeling error, and parameter uncertainty in the magnetic bearing rotor system to a new state variable $x_{3} (t)$ . The state-space equation of the magnetic bearing rotor system after expanding the new state variable can be expressed as follows

{\begin{cases} [\begin{array}{l} {\dot{x}}_{1} \\ {\dot{x}}_{2} \\ {\dot{x}}_{3} \end{array}] = [\begin{array}{l} 0 & 1 & 0 \\ \frac{k_{x}}{m} & 0 & 1 \\ 0 & 0 & 0 \end{array}] [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] + [\begin{array}{l} 0 \\ \frac{k_{i} k_{a} k_{s}}{m} \\ 0 \end{array}] u + [\begin{array}{l} 0 \\ 0 \\ 1 \end{array}] h (t) \\ y = [\begin{array}{l} 1 & 0 & 0 \end{array}] [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}] \end{cases} .

(16)

where

h (t)

is the first derivative of the generalized perturbation

x_{3} (t)

. According to the principle of LESO, the linear extended state-space equation of 1-DOF magnetic bearing rotor system can be written as follows

[\begin{array}{l} {\dot{\hat{x}}}_{1} \\ {\dot{\hat{x}}}_{2} \\ {\hat{x}}_{3} \end{array}] = [\begin{array}{l} 0 & 1 & 0 \\ \frac{k_{x}}{m} & 0 & 1 \\ 0 & 0 & 0 \end{array}] [\begin{array}{l} {\hat{x}}_{1} \\ {\hat{x}}_{2} \\ {\hat{x}}_{3} \end{array}] + [\begin{array}{l} β_{1} \\ β_{2} \\ β_{3} \end{array}] e + [\begin{array}{l} 0 \\ \frac{k_{i} k_{a} k_{s}}{m} \\ 0 \end{array}] u,

(17)

where

e = x_{i} - {\hat{x}}_{i}

represents the observation errors,

β_{1}, β_{2},

and

β_{3}

represent the respective feedback gain parameters estimated by LESO for rotor displacement, speed, and disturbances, y is the output displacement signal of the magnetic bearing rotor system, and u is the voltage signal output by the controller.

k_{x} {\hat{x}}_{1} / m

is the estimated value of displacement acceleration inherent in the magnetic bearing rotor system. Addition of this value into the LESO enables LESO to estimate the total disturbances caused by external disturbances, modeling errors, as well as parameter and structural uncertainties in the magnetic bearing rotor system. Moreover, the LESO’s estimation efficiency is improved. The error equation (18) is equal to the magnetic bearing rotor system’s state space equation (16) minus the state quantity equation (17) estimated by LESO

[\begin{array}{l} {\dot{e}}_{1} \\ {\dot{e}}_{2} \\ {\dot{e}}_{3} \end{array}] = [\begin{array}{l} {- β}_{1} & 1 & 0 \\ \frac{k_{x}}{m} - β_{2} & 0 & 1 \\ {- β}_{3} & 0 & 0 \end{array}] [\begin{array}{l} e_{1} \\ e_{2} \\ e_{3} \end{array}] + [\begin{array}{l} 0 \\ 0 \\ 1 \end{array}] h (t),

(18)

where

e_{1} = x_{1} - {\hat{x}}_{1}, e_{2} = x_{2} - {\hat{x}}_{2},

and

e_{3} = x_{3} - {\hat{x}}_{3}

. When the error observation reaches the steady state, that is, when the change rate of the three observation errors is 0, the LESO error can be calculated

{\begin{cases} e_{1} = \frac{h (t)}{β_{3}} \\ e_{2} = \frac{β_{1} h (t)}{β_{3}} \\ e_{3} = \frac{β_{2} h (t)}{β_{3}} - \frac{k_{x} h (t)}{{m β}_{3}} \end{cases} .

(19)

As shown in (19), if $β_{3} ≫ h (t)$ , the LESO error approaches 0, indicating that the LESO can accurately estimate the state quantity and total disturbances of the magnetic bearing rotor system.

3.4. Design of composite hierarchical anti-disturbance controller

3.4.1. Composite hierarchical anti-disturbance controller based on disturbance observer (H_∞-DOB)

In the H_∞-DOB, the disturbance observer is a part of the overall closed-loop feedback system. Therefore, it is necessary to analyze the system’s stability. Figure 4 shows the basic structure of a composite hierarchical anti-disturbance controller based on disturbance observer. Because the disturbance observer can achieve its robustness by selecting an appropriate low-pass filter Q(s), it can be seen from (13) and (14) that when $G_{p} (s) \approx G_{n} (s)$ , the disturbance observer does not affect the entire closed-loop system. Therefore, as long as the nominal model is close to the actual model of the system, the outer controller (H_∞ controller) and inner controller (disturbance observer) can be designed independently.

Assuming uncertainty is modeled by a multiplicative perturbation model

G_{P} (s) = G_{n} (s) (1 + Δ (s)),

(20)

where

Δ (s)

represents the multiplicative uncertainties in the system. Considering the uncertainties of the controlled system and in order to analyze the stability of the whole closed-loop feedback system, Figure 4 can be simplified to Figure 5. K(s) is outer feedback controller, Q(s) is low-pass filter, and

G_{n} (s)

is the system’s nominal model.

As can be seen from Figure 5, the transfer function from uncertain input to output can be expressed as

T_{u n} (s) = \frac{K (s) G_{n} (s) + Q (s)}{1 + K (s) G_{n} (s)} .

(21)

According to the small gain theorem, the overall closed-loop system in (21) is robustly stable against uncertainties if (22) is satisfied

| W_{T} (s) T_{u n} (s) | < 1, | Δ (s) | \leq | W_{T} (s) |, \forall s = j ω .

(22)

where

W_{T} (s)

is the upper bound of the uncertainties of the controlled system. Equations (21) and (22) show that the stability of the whole closed-loop system depends not only on the design of low-pass filter Q(s) but also on the outer feedback controller K(s). However, Q(s) must meet the robustly stable of the whole closed-loop system to parameter changes. In order to test the robustly stable of the whole closed-loop system, (22) can be expressed as

| W_{T} (s) \frac{K (s) G_{n} (s) + Q (s)}{1 + K (s) G_{n} (s)} | < 1, \forall s = j ω .

(23)

| Q (s) | < | \frac{1 + K (s) G_{n} (s)}{W_{T} (s)} |, \forall s = j ω .

(24)

The design of low-pass filter Q(s) should make the disturbance observer have a good cancellation effect on external disturbances. Equation (24) can be used as a sufficient condition to verify whether the designed low-pass filter Q(s) makes the whole closed-loop system have robustly stable.

3.4.2. Composite hierarchical anti-disturbance controller based on LESO (H_∞-LESO)

The basic structure of H_∞-LESO for magnetic bearing rotor system is shown in Figure 6.

The H_∞ controller of the system is designed as u = Ky, and its state space can be expressed as

{\begin{cases} \dot{\hat{x}} = A_{k} \hat{x} + B_{k} y \\ u = C_{k} \hat{x} + D_{k} y \end{cases} .

(25)

\dot{\hat{x}}

is the controller state quantity, and

A_{k}, B_{k}, C_{k}, D_{k}

is the parameter matrix. According to Zhang’s study (2010), when the H_∞ controller is applied to the system (equation (16)), the state space of the closed-loop system can be expressed as

{\begin{cases} \dot{ξ} = A_{w} ξ + B_{w} h \\ y = C_{w} ξ \end{cases},

(26)

where

A_{w} = [\begin{array}{l} A + B_{1} D_{k} C & B_{1} C_{k} \\ B_{k} C & A_{k} \end{array}], B_{w} = [\begin{array}{l} B_{2} \\ 0 \end{array}], C_{w} = [\begin{array}{l} C & 0 \end{array}]

A = [\begin{array}{l} 0 & 1 & 0 \\ \frac{k_{x}}{m} & 0 & 1 \\ 0 & 0 & 0 \end{array}], B_{1} = [\begin{array}{l} 0 \\ \frac{k_{i} k_{a} k_{s}}{m} \\ 0 \end{array}], B_{2} = [\begin{array}{l} 0 \\ 0 \\ 1 \end{array}], C = [\begin{array}{l} 1 & 0 & 0 \end{array}]

(27)

The state variable $ξ$ can be expressed as

ξ = [\begin{array}{l} x \\ \hat{x} \end{array}], x = [\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}], \hat{x} = [\begin{array}{l} {\hat{x}}_{1} \\ {\hat{x}}_{2} \\ {\hat{x}}_{3} \end{array}]

(28)

When the LESO is applied to the controlled system, the state space equation of the closed-loop feedback system can be expressed as

[\begin{array}{l} \dot{ξ} \\ \dot{e} \end{array}] = [\begin{array}{l} A_{w} & B_{w} \\ 0 & A - L C \end{array}] [\begin{array}{l} ξ \\ e \end{array}] + [\begin{array}{l} B_{w} \\ B_{2} \end{array}] h

(29)

The system matrix shown in (29) is in the form of an upper triangle, and its eigenvalues are determined by the eigenvalues of matrices $A_{w}$ and (A−LC). Assuming that the matrices $A_{w}$ and $B_{w}$ are controllable and the matrices A and C are observable, the error stability of equation (29) can be ensured by the appropriate pole assignment of the controller and observer. Under the condition that h is bounded, the linear system described in equation (29) is stable and the system has bounded input and bounded output. When the change rate of internal disturbance d is 0, that is, h = 0, the composite system is asymptotically stable. If the rate of change of d fluctuates in a small range, the composite system satisfies asymptotic stability. Therefore, the composite system achieves asymptotic stability and working conditions studied in this paper.

4. Simulation analysis

In this section, the MATLAB/Simulink simulation platform is used to build closed-loop system models under H_∞, H_∞-DOB and H_∞-LESO controller, and different sine voltage signals with amplitude of 2.5 V are added as external disturbance to the system. Among them, the H_∞ controller is compared as a conventional control method to verify the anti-interference performance of H_∞-DOB and H_∞-LESO controller under different conditions. The parameters used in the simulation and experiment are listed in Table 1.

Table 1.

Some parameters about simulation and experiment.

Symbol	Quantity	Value
m	Rotor mass	2.067 kg
S _r	Magnetic pole area	2 × 10⁻⁴ m²
N ₀	Unipolar coil turns	150
δ ₀	Unilateral clearance	0.4 mm
δ _min	Protect the unilateral clearance	0.125 mm
I ₀	Bias current	1.7 A
$μ_{0}$	Vacuum permeability	$4 π \times 10^{- 7} Vs / Am$
$ω$	Bandwidth	5500

For the H_∞ controller used in this paper, (30) is the selected weight function, and the required H_∞ controller transfer function (31) is calculated using MATLAB. The transfer function of the low-pass filter $Q (s)$ in the disturbance observer is given by (32)

{\begin{cases} W_{1} (s) = \frac{4000}{s + 10} \\ W_{2} (s) = 2.5 \times 10^{- 6} \\ W_{3} (s) = \frac{0.5 s}{5 s + 5000} \end{cases} .

(30)

K (s) = \frac{3005.4 (s + 457.1) (s + 478.7) (s + 997.7)}{(s + 5.339 \times 10^{4}) (s + 1170) (s + 7.982)} .

(31)

Q (s) = \frac{3 \times 10^{- 4} s + 1}{1 \times 10^{- 12} s^{3} + 3 \times 10^{- 8} s^{2} + 3 \times 10^{- 4} s + 1} .

(32)

Figure 7 shows the vibration displacement spectrum curve of the magnetic bearing rotor system at different rotation frequencies under the control of H_∞ and H_∞-DOB controllers when sinusoidal disturbance of 30 Hz and 60 Hz is applied to the magnetic bearing rotor system, respectively. As shown in Figure 7, when the frequency and amplitude of the disturbances are constant, the higher the rotation frequency of the system, the vibration amplitude caused by the disturbances also increases slightly. However, the rotation frequency has a small influence on the control effect. When the rotation frequency is constant, the vibration amplitude of the system increases with the increase of the disturbance frequency. The vibration displacement of the system under H_∞-DOB control is reduced by 90% when compared to the H_∞ controller, whereas the control effect remains unchanged, which demonstrates that the disturbance observer as the inner controller can effectively offset the external disturbances.

Similarly, Figure 8 shows the vibration displacement spectrum curve of H_∞ and H_∞-LESO controllers. As shown in Figure 8, when frequency and amplitude of disturbances are constant, the rotation frequency has a small influence on the control effect. When the rotation frequency is constant, the vibration amplitude of the system increases with increasing disturbance frequency and the control effect of H_∞-LESO decreases gradually. When compared to the H_∞ controller, the reduction of the vibration amplitude of the system by the H_∞-LESO controller decreases from 90% to 80%, indicating that the LESO as an inner controller can effectively offset the external disturbances.

According to the comparison between Figures 7 and 8, both H_∞-DOB and H_∞-LESO controllers have better interference suppression effect than H_∞ controller. However, because the disturbance observer in this paper is based on the nominal model for disturbance offset, there will be no model errors in the simulation. Thus, the anti-disturbance effect of H_∞-DOB is better than H_∞-LESO at different rotation and disturbance frequencies. However, H_∞-DOB requires an accurate system model because there will be some errors between the theoretical and actual models, and the model will change constantly in the actual process, resulting in some model errors in the actual H_∞-DOB controller. The H_∞-LESO controller has no specific requirements for the system model and can effectively offset the disturbance of the system at various rotation frequencies and disturbance frequencies.

5. Experiments

5.1. Magnetic bearing rotor system experimental platform

The experimental bench used in this article, depicted in Figure 9, primarily consists of an AMB test rig, sensor circuits, a power amplifier, an oscilloscope, an adapter box, an inverter, a dSPACE1202, and a computer. It should be noted that the disturbances targeted in this paper result from different working conditions. Likewise, their manifestations can be equivalent to low-frequency sinusoidal disturbances.

5.2. Experiment of H_∞-DOB

When the magnetic bearing rotor system rotates at constant speeds of 200, 300, 400, and 500 Hz, sinusoidal disturbance signals with a frequency of 30, 40, 50, and 60 Hz and amplitudes of 0.1 V are applied to it, respectively, and its vibration displacement spectrum curve compared with H_∞ controller is shown in Figure 10. The specific experimental data are shown in Tables 2 –5, and the meanings of the parameters in the table are explained later in Table 5. Because of the requirements of journal format, only 30 Hz and 60 Hz pictures are shown here, the experimental data of 40 Hz and 50 Hz are shown in the following table.

Table 2.

Displacement of different control methods under 30 Hz disturbance.

Rotation frequency (Hz)	$x_{1}$ (μm)	$x_{2}$ (μm)	$x_{3}$ (μm)	$e_{1}$ (%)	$e_{2}$ (%)
200	4.21	2.32	2.53	44.9	39.9
300	4.76	2.86	2.88	39.9	39.5
400	5.08	3.37	3.09	33.7	39.2
500	5.43	3.89	3.35	28.4	38.3

Table 3.

Displacement of different control methods under 40 Hz disturbance.

Rotation frequency (Hz)	$x_{1}$ (μm)	$x_{2}$ (μm)	$x_{3}$ (μm)	$e_{1}$ (%)	$e_{2}$ (%)
200	6.15	3.41	3.71	44.6	39.7
300	6.53	4.05	3.96	38.0	39.4
400	6.78	4.69	4.12	30.8	39.2
500	6.95	5.08	4.33	26.9	37.7

Table 4.

Displacement of different control methods under 50 Hz disturbance.

Rotation frequency (Hz)	$x_{1}$ (μm)	$x_{2}$ (μm)	$x_{3}$ (μm)	$e_{1}$ (%)	$e_{2}$ (%)
200	8.19	4.66	5.03	43.1	38.6
300	8.71	5.41	5.36	37.9	38.5
400	9.21	6.19	5.69	32.8	38.2
500	9.49	7.06	5.97	25.6	37.1

Table 5.

Displacement of different control methods under 60 Hz disturbance.

Rotation frequency (Hz)	$x_{1}$ (μm)	$x_{2}$ (μm)	$x_{3}$ (μm)	$e_{1}$ (%)	$e_{2}$ (%)
200	6.49	5.12	4.56	21.1	29.7
300	6.92	5.74	4.88	17.1	29.4
400	7.47	6.49	5.31	13.1	28.9
500	7.73	7.01	5.61	9.3	27.4

$x_{1}$ : Maximum amplitude under H_∞ control.

$x_{2}$ : Maximum amplitude under H_∞-DOB control.

$x_{3}$ : Maximum amplitude under H_∞-LESO control.

$e_{1}$ : Percentage of maximum amplitude decrease of H_∞-DOB control over H_∞ control.

$e_{2}$ : Percentage of maximum amplitude decrease of H_∞-LESO control over H_∞ control.

According to Figure 10, when external sinusoidal disturbances of a certain frequency are applied to the magnetic bearing rotor system, the vibration amplitude generated by the system is increased as the rotation frequency increases. As the rotational speed increases, when the system applies interference frequencies of 30, 40, and 50 Hz, the amplitude reduction rate of H_∞-DOB compared to the H_∞ controller varies by 44.9% −28.4%, 44.6% −26.9%, and 43.1% −25.6%, respectively. When 60 Hz interference is applied, the amplitude decreases from 21.1% to 9.3%. This is because the observation accuracy of the disturbance observer is associated with the accuracy of the established nominal model. The higher the model’s accuracy, the more accurate is the disturbance estimation. The mathematical model of the system changes in real time as the system rotates. The greater the error between the actual and nominal models, the lower the estimation accuracy of the disturbance observer’s external disturbances. Therefore, with increasing rotating frequency, the suppression ability of the H_∞-DOB controller to the disturbance reduces. When the rotation frequency of the system is constant, with increasing disturbance frequency, the disturbance suppression ability of H_∞-DOB decreases slightly but decreases sharply at 60 Hz disturbance, with a decrease of approximately 20%. Through comparison with the H_∞ controller, it is found that the H_∞-DOB controller can improve the control performance by 9.3%–44.9%.

5.3. Experiment of H_∞-LESO

Like part B, sinusoidal disturbances of different frequencies are applied to the magnetic bearing rotor system at various rotation frequencies. The vibration displacement under H_∞-LESO control is compared with the H_∞ control in Figure 11. The specific experimental data are shown in Tables 2 –5.

Through comparison, it is found that compared with the H_∞ controller, the H_∞-LESO controller can improve the control performance by 27.4%–39.9%. When compared to H_∞ control at 30–50 Hz disturbance frequency, increasing the rotation frequency reduces the amplitude of H_∞-LESO controller decreases from 40% to 37%, and from 29.7% to 27.4% under 60 Hz disturbance. When the rotation frequency is the same, compared with the H_∞ controller, the disturbance suppression ability of the H_∞-LESO controller decreases by less than 2% at the disturbance frequency of 30-50 Hz. Under 60 Hz disturbance, the suppression ability of the H_∞-LESO controller for disturbance decreases approximately 10%.

It can be seen from the comparison between Figures 10 and 11 that H_∞-DOB and H_∞-LESO controllers have better disturbance suppression effect than H_∞ controller. At low rotation frequency, the H_∞-DOB controller has a better disturbance suppression effect. However, with increasing rotation frequency, the disturbance suppression effect of the H_∞-DOB controller decreases significantly, whereas the disturbance suppression effect of H_∞-LESO changes little. In particular, under 60 Hz disturbance, the disturbance suppression effect of the two controllers decreases significantly; however, the effect of H_∞-LESO is better than H_∞-DOB.

6. Conclusions

To achieve refined anti-disturbance in the magnetic bearing rotor system, this paper is based on the theory of composite hierarchical anti-disturbance. Combining the H_∞ controller and LESO, the H_∞-LESO composite controller is proposed, and then it is compared with the H_∞ and H_∞-DOB controllers through simulation and experiment. The conclusions are as follows:

1. When compared to the H_∞ controller, both H_∞-DOB and H_∞-LESO composite controllers can suppress the low-frequency sinusoidal disturbance.

2. When the rotation frequency of the system is low, the anti-disturbance performance of the H_∞-DOB controller is slightly better than that of H_∞-LESO. However, H_∞-DOB controller depends more on the mathematical model of the system than the H_∞-LESO controller does, when the disturbance frequency and system rotation frequency are high, H_∞-LESO controller has better suppression effect on disturbance.

3. In general, when compared to the H_∞ and H_∞-DOB controllers, the H_∞-LESO controller proposed in this paper has better disturbance suppression effect and can effectively suppress the low-frequency disturbance of the magnetic suspension rotor.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under project 51875275, Six Talent Peaks Project in Jiangsu Province under project JNHB-041 and Key R&D Program of Jiangsu Province under project BE2019122, National Natural Science Foundation of China (52275059) and Science Center for Gas Turbine under project P2022-B-Ⅲ-004-001.

ORCID iDs

Chaowu Jin

Jin Zhou

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Research on composite hierarchical anti-disturbance strategy for magnetic bearing rotor using H ∞ and observer

Abstract

Keywords

1. Introduction

2. Magnetic bearing rotor modeling

2.1. Electromagnet and rotor

2.2. Displacement sensor

2.3. Power amplifier

3. Controller design

3.1. H∞ Controller design

3.2. Design of disturbance observer

3.3. Design of linear extended state observer

3.4. Design of composite hierarchical anti-disturbance controller

3.4.1. Composite hierarchical anti-disturbance controller based on disturbance observer (H∞-DOB)

3.4.2. Composite hierarchical anti-disturbance controller based on LESO (H∞-LESO)

4. Simulation analysis

5. Experiments

5.1. Magnetic bearing rotor system experimental platform

5.2. Experiment of H∞-DOB

5.3. Experiment of H∞-LESO

6. Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

3.1. H_∞ Controller design

3.4.1. Composite hierarchical anti-disturbance controller based on disturbance observer (H_∞-DOB)

3.4.2. Composite hierarchical anti-disturbance controller based on LESO (H_∞-LESO)

5.2. Experiment of H_∞-DOB

5.3. Experiment of H_∞-LESO