Sliding model control of active magnetic bearing rotor system based on state observer

Abstract

This paper addresses a novel sliding mode control based on state observer for active magnetic bearing rotor system. Firstly, the state-space model of a radial AMB rotor system is established with considering unbalance disturbance and gyro effect for a vertical flywheel energy storage system. Then a sliding mode function and switching surface are constructed based on an observer. Meanwhile, a separation and decoupling strategy based on Finsler’s lemma is proposed. Through this method, the constraint relationship between the controller gain, active magnetic bearing matrices and the Lyapunov variables is eliminated. After that a method for chattering reduction in the sliding-mode controller is raised. Relied on these techniques, new sufficient conditions for the stability of AMB rotor system are given in the framework of linear matrix inequalities. Finally, the effectiveness of the proposed sliding mode controller is validated on the experimental platform of the flywheel energy storage system.

Introduction

Active magnetic bearing (AMB) has been receiving increased attention in industry since the merits of no friction, no need of lubrication, maintenance reduction, long life span and high-velocity rotation.^1–3 In recent years, AMB has been widely used in diverse high-performance applications such as flywheel energy storage systems, bearing less motors, artificial heart pumps, vacuum pumps, and numerical control machines.^4–6

During the widely application of AMB system, rotor position control is the key and basic issue. AMB is an unstable system because of the relationship between the electromagnetic force and the length of the air gap. Moreover, external interference can also cause the rotor to deviate from its equilibrium point. Therefore, in order to stabilize and adjust the rotor position, diverse control methods have emerged. Proportional integral differential (PID) control, as a traditional control scheme, has been used in AMB systems due to its simple implementation long before.^7,8 However, due to the lack of robustness to external disturbances, various control methods are proposed, such as fuzzy control, model predictive control, adaptive control, sliding mode (SM), and so on.^9–13 Fuzzy control, which inevitably leads to poor control precision and dynamic quality. ⁹ Model predictive control requires high accuracy of system model. ¹⁰ Adaptive control may cause the stability of the system to become worse or even unstable in some cases.^11,12 Compared with the above methods, SM control is very appropriate for the control of AMB system with the advantages of strong robustness and no need for an accurate control object mode.^14–20 To the best of the authors’ knowledge, Rundell et al. ²¹ was one of the first attempts to use SM control in AMB system. Afterward, the authors designed an integral SM controller to solve the problem of parameter change and external disturbance of the magnetic suspension balance beam system. ²² A second-order integral SM control was addressed for the voltage-controlled three-pole AMB. ²³ A second-order SM control method was proposed to deal with the time-variation of rotor unbalance under a wide range of speed changes. ²⁴ However, because of high-frequency chattering, the traditional SM controller would weaken its ability to resist external disturbance or reduce the stability of the system.^25,26 In order to reduce the chattering phenomenon, a PID-SM controller was proposed for AMB system. ²⁷ The parameters of SM controller were optimized online by using the RBF network and adaptive law. ²⁸ Combined the benefits of SM control and neural networks, a new controller was advanced for a five-DOF AMB system. In this way, the model of the AMB system was adjusted online by the neural network. ²⁹ Although the above-mentioned methods can reduce chattering, there is a lack of method that can completely eliminate system chattering.

In summary, robust controller can be used to improve the anti-disturbance performance of the system both inside and outside the AMB system. ³⁰ However, it may lead to strong conservatism if the model is not accurate. To further improve the disturbance suppression performance of the robust controller, the extended state observer which proposed by Han et al.^31–33 is introduced to estimate and suppress disturbance. Combining the advantages of the state observer and SM controller, the optimal parameters of the observer and SM controller were obtained through linear quadratic method. ³⁴ Li et al. ³⁵ used the state observer to estimate the total disturbance including all the uncertainties and the external disturbance to stabilize the vehicle attitude to provide a good ride quality. The sliding mode observer was used to estimate the lumped disturbance caused by the system parameter perturbation, which effectively reduces the parameter sensitivity. ³⁶ Observer-based disturbance compensation schemes, parameter online identification, and model-free schemes for improving system robustness are reviewed.^37,38 The performance of the controller with and without state observer in magnetic bearings was compared by simulation and experiment. ³⁹ This paper will reveal a novel SM controller to deal with chattering based on state observer, which is very easy to understand and apply in practice. Furthermore, the outside disturbance, system uncertainty, and chattering are mostly rejected.

Motivated by the above analysis and previous studies, an observer-based SM control design method was given for the AMB rotor system. A novel type of sliding mode function and switching surface were constructed based on an observer. With the help of Lyapunov theory, sufficient conditions for the stable operation of the AMB rotor system were given. In order to obtain the conditions under the framework of linear matrix inequalities, a separation and decoupling strategy based on Finsler’s lemma was proposed. Through this method, the constraint relationship between the controller gain, AMB system matrices and the Lyapunov variables was eliminated. Then a method for chattering reduction in the sliding-mode control was raised, and the effectiveness of the method was verified by simulation. Finally, the effectiveness of the proposed SM controller was validated on the experimental platform of the flywheel energy storage system. The main contributions of this paper are as follows.

1. A sliding mode surface based on a novel observer structure was designed. By extending the state observe and combining the SM control, the steady-state performance and anti-disturbance ability of the AMB rotor system are improved.

2. The SM controller gain and observer gain are calculated by the designed formulas, instead of being repeatedly adjusted in the simulation process. A new method for suppressing high-frequency chattering in SM control is introduced in the simulation process.

3. In order to ensure the sufficient conditions of the AMB rotor system in the frame work of linear matrix inequality, a lossless decoupling technique was proposed to reduce the conservatism.

AMB rotor system model

In this research, the radial four-DOF AMB system was explored for the vertical flywheel energy storage system. The rotor coordinate system c-xyz was established as shown in Figure 1.

Figure 1.

Schematic diagram of radial MB rotor.

In Figure 1, m is the rotor mass; c is the geometrical center of the rotor; L is the total length of the rotor; A and B denote the two centers of the two-DOF radial AMB respectively; a and b are the distances from upper radial AMB and lower radial AMB to the rotor center point c; J_x, J_y, J_z are the moments of inertia of the rotor in the X–, Y–, and Z–axes; x_a, x_b,y_a, y_b are the radial displacements; θ _xx and θ _yy are the rotor angular. The relation between rotor centroid displacement x_c, y_c, θ _xx , θ _yy and other parameters is as follows:

$$\{\begin{array}{c}{x}_c=\frac{a{x}_b+b{x}_a}{L}\hfill \\ y_c=\frac{a{y}_b+b{y}_a}{L}\hfill \\ {\theta }_{\mathrm{xx}}=\frac{y_a-y_b}{L}\hfill \\ {\theta }_{\mathrm{yy}}=\frac{x_b-x_a}{L}\hfill \end{array}$$

(1)

The AMB rotor system is affected by gravity, electromagnetic force and unbalanced disturbing force. According to the electromagnetic force linearization equation, the relationship between the electromagnetic force, displacement and current can be expressed as

$$F_{\mathrm{xaa}}=k_{\mathrm{ia}}{i}_{\mathrm{xa}}+k_{\mathrm{xa}}{x}_a$$

(2)

$$F_{\mathrm{xbb}}=k_{\mathrm{ib}}{i}_{\mathrm{xb}}+k_{\mathrm{xb}}{x}_b$$

(3)

$$F_{\mathrm{yaa}}=k_{\mathrm{ia}}{i}_{\mathrm{ya}}+k_{\mathrm{ya}}{y}_a$$

(4)

$$F_{\mathrm{ybb}}=k_{\mathrm{ib}}{i}_{\mathrm{yb}}+k_{\mathrm{yb}}{y}_b$$

(5)

where, F_xaa, F_xbb, F_yaa, F_ybb are the electromagnetic force; k_ia, k_ib and k_xa, k_xb, k_ya, k_yb are current stiffness coefficient and displacement stiffness coefficient. According to (2), (3), (4), (5), one can obtain:

$$F=K_{I}I+K_s{q}_c$$

(6)

where I=[i_xa i_xb i_ya i_yb]^T,

$$K_s=\mathrm{diag}(k_{\mathrm{xa}},k_{\mathrm{xb}},k_{\mathrm{ya}},k_{\mathrm{yb}}),$$

$$K_I=\mathrm{diag}(k_{\mathrm{ia}},k_{\mathrm{ib}},k_{\mathrm{ia}},k_{\mathrm{ib}}).$$

On the other hand, due to impurities in material or processing technology, the mass is unevenly distributed. Then, mass imbalance is caused by the uneven mass distribution. Therefore, when the AMB rotor system worked at high speed, the geometric center line of the rotor will not coincide with the inertia axis, which will cause unbalanced disturbance. According to the rotor mechanics principle of AMB system, the disturbance can be described as:

$${\overline{F}}_{\mathrm{ex}}=m\overline{e}{\omega }^2\cos (\omega t+{\theta }_1)$$

(7)

$${\overline{F}}_{\mathrm{ey}}=m\overline{e}{\omega }^2\sin (\omega t+{\theta }_1)$$

(8)

$${\overline{M}}_{\epsilon x}=(J_x-J_z)\overline{\epsilon }{\omega }^2\cos (\omega t+{\theta }_2)$$

(9)

$${\overline{M}}_{\epsilon y}=(J_y-J_z)\overline{\epsilon }{\omega }^2\sin (\omega t+{\theta }_2)$$

(10)

where, ${\overline{F}}_{\mathrm{ex}}$ and ${\overline{F}}_{\mathrm{ey}}$ are the unbalanced force, ${\overline{M}}_{\epsilon x}$ and ${\overline{M}}_{\epsilon y}$ are the unbalance torque, θ₁ and θ₂ are the phase angles of static unbalance force and dynamic unbalance force, $\omega$ is the rotational mechanical speed of the rotor, $\overline{e}$ is the eccentricity, $\overline{\epsilon }$ is the eccentric inclination angle.

To sum up, according to the rotor dynamics theory, the equations of motion to radial four-degree-of-freedom rotor were established as follows:

$$m{\stackrel{··}{x}}_c=F_{\mathrm{xaa}}+F_{\mathrm{xbb}}+{\overline{F}}_{\mathrm{ex}}$$

(11)

$$m{\stackrel{··}{y}}_c=F_{\mathrm{yaa}}+F_{\mathrm{ybb}}+{\overline{F}}_{\mathrm{ey}}$$

(12)

$$J_x{\stackrel{··}{\theta }}_{\mathrm{xx}}+J_z\omega {\stackrel{·}{\theta }}_{\mathrm{yy}}=a{F}_{\mathrm{yaa}}-b{F}_{\mathrm{ybb}}+{\overline{M}}_{\epsilon x}$$

(13)

$$J_y{\stackrel{··}{\theta }}_{\mathrm{yy}}-J_z\omega {\stackrel{·}{\theta }}_{\mathrm{xx}}=a{F}_{\mathrm{xbb}}-b{F}_{\mathrm{xaa}}+{\overline{M}}_{\epsilon y}$$

(14)

Because the shaft core of the magnetic bearing is parallel to the maglev line in the vertical flywheel energy storage system, the gravity of the rotor was not taken into account. Combining equations (11) to (14), we have

$$\overline{\mathrm{M}}{\stackrel{··}{\mathrm{q}}}_{\mathrm{c}}+\mathrm{G}{\stackrel{.}{\mathrm{q}}}_{\mathrm{c}}{+\mathrm{T}}_{\mathrm{L}}{\mathrm{F}}_{\mathrm{c}}{=\mathrm{F}}_{\mathrm{unb}}$$

(15)

where

$$\overline{M}=\left[\begin{array}{cccc}m\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& m\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & J_x\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & J_y\hfill \end{array}\right],$$

$$G=\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -J_z\omega \hfill \\ \multicolumn{1}{c}{0}& 0\hfill & J_z\omega \hfill & 0\hfill \end{array}\right],$$

$$T_L=\left[\begin{array}{cccc}-1\hfill & -1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -1\hfill & -1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -a\hfill & b\hfill \\ \multicolumn{1}{c}{b}& -a\hfill & 0\hfill & 0\hfill \end{array}\right],$$

$$q_c=\left[\begin{array}{c}{x}_c\hfill \\ \multicolumn{1}{c}{y_c}\\ \multicolumn{1}{c}{{\theta }_{\mathrm{xx}}}\\ \multicolumn{1}{c}{{\theta }_{\mathrm{yy}}}\end{array}\right],F_c=\left[\begin{array}{c}{F}_{\mathrm{xaa}}\hfill \\ \multicolumn{1}{c}{F_{\mathrm{xbb}}}\\ \multicolumn{1}{c}{F_{\mathrm{yaa}}}\\ \multicolumn{1}{c}{F_{\mathrm{ybb}}}\end{array}\right],F_{\mathrm{unb}}=\left[\begin{array}{c}{\overline{F}}_{\mathrm{ex}}\hfill \\ \multicolumn{1}{c}{{\overline{F}}_{\mathrm{ey}}}\\ \multicolumn{1}{c}{{\overline{M}}_{\mathrm{ex}}}\\ \multicolumn{1}{c}{{\overline{M}}_{e\mathrm{y}}}\end{array}\right].$$

Multiplying (15) with T_L⁻¹, we get

$$T_L^{-1}\overline{M}{\stackrel{··}{q}}_c+T_L^{-1}G{\stackrel{·}{q}}_c+F_c=T_L^{-1}{F}_{\mathrm{unb}}$$

(16)

Let $H_1=T_L^{-1}\overline{M}$ , $H_2=T_L^{-1}G$ , and bring in (6), (16) is rewritten as:

$${\stackrel{··}{q}}_c=-H_1^{-1}{H}_2{\stackrel{·}{q}}_c-H_1^{-1}{K}_s{q}_c-H_1^{-1}{K}_{I}I+H_1^{-1}{T}_L^{-1}{F}_{\mathrm{unb}}$$

(17)

Thus, the state-space model of the AMB rotor system with unbalance disturbance and gyro effect can be obtained

$$\begin{array}{c}\stackrel{.}{x}(t)=\overline{A}x(t)+\overline{B}u(t)+\overline{E}f(t)\hfill \\ y(t)=\overline{C}x(t)\hfill \end{array}$$

(18)

where, $x(t)$ is the state variable, $u(t)$ is the input control, $f(t)$ is the unbalanced disturbance, $y(t)$ is the output displacement, $x(t)={\left[\begin{array}{cc}{q}_c& {\stackrel{.}{q}}_c\end{array}\right]}^T,$

$$\begin{array}{l}f(t)=[\begin{array}{cccc}{\omega }^2\cos (\omega t+{\theta }_1)& {\omega }^2\sin (\omega t+{\theta }_1)& {\omega }^2\cos (\omega t+{\theta }_2)& {\omega }^2\sin (\omega t+{\theta }_2)\end{array}]\\ \overline{A}=\left[\begin{array}{cc}{0}_{4\times 4}& I_{4\times 4}\\ -H_1^{-1}{K}_s& -H_1^{-1}{H}_2\end{array}\right],\overline{B}=\left[\begin{array}{c}{0}_{4\times 4}\\ -H_1^{-1}{K}_I\end{array}\right],\end{array}$$

$$\overline{E}=\left[\begin{array}{c}{0}_{4\times 4}\hfill \\ \multicolumn{1}{c}{-H_1^{-1}{T}_L^{-1}}\end{array}\right]N,$$

$$N=\left[\begin{array}{cccc}m\overline{e}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& m\overline{e}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & (J-J_Z)\overline{\epsilon }\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & (J-J_Z)\overline{\epsilon }\hfill \end{array}\right],$$

$$\overline{C}=\left[\begin{array}{cc}{T}_L^T\hfill & 0_{4\times 4}\hfill \end{array}\right].$$

Before the specific controller design, one lemma was introduced firstly.

• Lemma 1. ⁴⁰ For given $\Im ={\Im }^T\in R^{n\times n}$ and any matrix $\aleph \in R^{m\times n}$ , the following inequalities are equal.

(Ai) If $\overline{V}\ne 0$ and $\aleph \overline{V}=0$ , ${\overline{V}}^T\Im \overline{V}<0$ ;

(Bi) ${{\aleph }^{\perp }}^T\Im {\aleph }^{\perp }<0$ ;

(Ci) ∃ $\overline{S}\in R^{n\times m}$ , $\Im +\mathrm{He}(\overline{S}\aleph )<0$ .

Controller development

In order to effectively suppress the disturbance caused by mass unbalance and gyro effect in the AMB rotor system, an observer-based integral SM controller is constructed to ensure the accessibility of the sliding mode surface and the sliding mode motion of the state trajectory on the basis of the state-space equation (18) in Figure 2.

Figure 2.

The block diagram of the AMB rotor system based on the proposed SM controller with state observer.

where $r(t)$ , $y(t)$ are the given displacement and the target displacement signals of the AMB rotor system, $u(t)$ is the input control, $\hat{x}(t)$ is observer state variable.

Sliding surface design

First of all, the integral sliding mode surface function is constructed as follows:

$$s(t)={\overline{B}}^{T}X\hat{x}(t)-{\int }_0^t{\overline{B}}^{T}X(\overline{A}+\overline{B}K)\hat{x}(\tau )d\tau$$

(19)

where $X\in R^{m\times n}$ and $K\in R^{m\times n}$ are matrix variables that will be calculated later, $\hat{x}(t)$ is observer state variable. The state observer structure is as follows:

$$\stackrel{.}{\hat{x}}(t)=\overline{A}\hat{x}(t)+\overline{B}u(t)+\overline{B}\overline{L}(y(t)-\overline{C}\hat{x}(t))$$

(20)

where $\overline{L}$ is the gain matrix of the state observer.

• Remark 1. In the existing literature, most observers used the classical Romberg observer, such as $\overline{L}(y(t)-\overline{C}\hat{x}(t))$ . However, in this paper the observer structure $\overline{B}\overline{L}(y(t)-\overline{C}\hat{x}(t))$ is adopted. The advantage of this structure is that the observer gain L can be adjusted by using the parameter B of the AMB rotor system.

Calculating the derivative of equation (19) gives

$$\stackrel{.}{s}(t)={\overline{B}}^{T}X\overline{B}[u(t)+\overline{L}(y(t)-\overline{C}\hat{x}(t))-K\hat{x}(t)]$$

(21)

In accordance with the equivalent control principle of SM control, when the controller is properly designed, the sliding mode function can reach onto the sliding surface $s(t)=0$ in a finite time.

SM controller law design scheme

In this part, on the basis of the existing sliding mode surface function (19), appropriate rules used in SM control will be selected to ensure the stability of the AMB rotor system (18) under the action of the SM controller and meet the H∞ performance index γ.

• Theorem 1. Considering the AMB rotor system (18) and the sliding mode function (19), the rules used in SM control are designed as follows:

$$\{\begin{array}{c}u(t)=u_1(t)+u_2(t)\hfill \\ \multicolumn{1}{c}{u_1(t)=-\theta s+K\hat{x}(t)}\\ \multicolumn{1}{c}{u_2(t)=-\rho \mathrm{sign}(s)}\end{array}$$

(22)

where θ is a smaller positive number, and $\rho \ge \Vert \overline{L}\Vert (\Vert y-C\hat{x}(t)\Vert )$ .Then state trajectories can reach onto the sliding surface in a finite time.

• Proof: Consider the following Lyapunov-Krasovskii functional candidate:

$$V_1(t)=\frac{1}{2}s(t{)}^T\overline{X}s(t)$$

(23)

where $\overline{X}=({\overline{B}}^{T}X\overline{B}{)}^{-1}$

Considering the differential of (23) and $\stackrel{.}{s}(t)$ , we can get

$${\stackrel{·}{V}}_1(t)=s(t{)}^T(\overline{X}{)}^{-1}\overline{X}[u(t)+\overline{L}(y-\overline{C}\hat{x}(t))-K\hat{x}(t)]$$

(24)

By substituting (22) into (24), we can get

$${\stackrel{·}{V}}_1(t)=-\theta s(t{)}^{T}s(t)-\rho |s(t)|+s(t{)}^T\overline{L}(y-\overline{C}\hat{x}(t))$$

(25)

In order to ensure ${\stackrel{·}{V}}_1(t)<0$ , the following formula must be satisfied

$$-\rho |s(t)|+s(t{)}^T\overline{L}(y-\overline{C}\hat{x}(t))\le 0$$

(26)

Through the norm calculation, the left side of (26) is scaled as follows:

$$\begin{array}{c}\multicolumn{1}{c}{-\rho |s(t)|s(t{)}^T\overline{L}(y-\overline{C}\hat{x}(t))\le -\rho |s(t)|}\\ +\Vert s(t)\Vert \Vert \overline{L}\Vert \Vert y-\overline{C}\hat{x}(t)\Vert \hfill \end{array}$$

(27)

Considering $|s_i(t)|\ge \Vert s_i(t)\Vert$ , to make sure (27) holds, we can get $\rho \ge \Vert \overline{L}\Vert (\Vert y-\overline{C}\hat{x}(t)\Vert )$

This completes the proof.

• Remark 2. Chattering is an ordinary phenomenon in the use of SM control. In order to avoid this phenomenon, $-\rho \mathrm{sign}(s)$ is replaced by $-\rho \frac{s(t)}{\Vert s(t)\Vert +\iota }$ in the following simulation process to effectively overcome the chattering.

The above sliding mode surface function (19) has been proved to be reachable. Next, the main research is to stabilize the closed-loop AMB rotor system based on the controller (22).

In order to make the sliding mode function reach onto the sliding surface $s(t)=0$ and $\stackrel{.}{s}(t)=0$ in a finite time, the equivalent control law can be derived as follows:

$$u_{\mathrm{eq}}(t)=K\hat{x}(t)-\overline{L}(y(t)-\overline{C}\hat{x}(t))$$

(28)

Substituting (28) into (18), the state track can be obtained:

$$\stackrel{.}{\hat{x}}(t)=(\overline{A}+\overline{B}K)\hat{x}(t)$$

(29)

The observer error is defined as follows:

$$\overline{e}(t)=x(t)-\hat{x}(t)$$

(30)

and

$$\stackrel{.}{\overline{e}}(t)=(\overline{A}-\overline{B}\overline{L}\overline{C})e(t)+\overline{E}f(t)$$

(31)

In the following, the sufficient conditions for the stability of the AMB rotor system satisfying the H_∞ performance index γ are given. The controller gain K and observer gain L can be obtained by this condition.

Stability analysis

• Theorem 2. For given scalars τ₁, τ₂, and γ > 0, if there exist symmetric matrices X > 0 and matrices H, M, U, Y, V, R, W with appropriate dimension, satisfying the following LMIs:

$$\left[\begin{array}{ccccc}\nabla 11\hfill & 0\hfill & 0\hfill & X\overline{B}-{\tau }_1\overline{B}U\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& \nabla 22\hfill & X\overline{E}\hfill & 0\hfill & X\overline{B}-{\tau }_2\overline{B}V\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }^{2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \mathrm{He}({\tau }_{1}U)\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & \mathrm{He}({\tau }_{2}V)\hfill \end{array}\right]<0$$

(32)

$$\nabla 11=\mathrm{He}(X\overline{A}+{\tau }_1\overline{B}W),$$

$$\nabla 22=\mathrm{He}(X\overline{A}-{\tau }_2\overline{B}Y\overline{C}),$$

the AMB rotor system (18) will operate stably and meet the H_∞ performance index γ with the controller (22).

Moreover, controller gain matrices K and $\overline{L}$ can be calculated as K = U⁻¹W, $\overline{L}$  = −V⁻¹Y.

• Proof: Considering the following Lyapunov-Krasovskii functional candidate:

$$V_2(x(t))=\hat{x}(t{)}^{T}X\hat{x}(t)+e(t{)}^T\mathrm{Xe}(t)$$

(33)

where X > 0

Then, the derivative of $V_2(x(t))$ is obtained as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2(x(t))=2\hat{x}(t{)}^{T}X\stackrel{.}{\hat{x}}(t)+2e(t{)}^{T}X\stackrel{.}{e}(t)}\\ =2\hat{x}(t{)}^{T}X(\overline{A}+\overline{B}K)\hat{x}(t)\hfill \\ +2e(t{)}^{T}X(\overline{A}-\overline{B}L\overline{C})e(t)\hfill \\ +2e(t{)}^{T}X\overline{E}f(t)\hfill \\ =\hat{x}(t{)}^T\mathrm{He}(X(\overline{A}+\overline{B}K))\hat{x}(t)\hfill \\ +e(t{)}^T\mathrm{He}(X(\overline{A}-\overline{B}\overline{L}\overline{C}))e(t)\hfill \\ +2e(t{)}^{T}X\overline{E}f(t)\hfill \end{array}$$

(34)

To guarantee the asymptotic stability of the system (18) with H_∞ performance, one needs

$${\stackrel{·}{V}}_2(x(t))+z(t{)}^{T}z(t)-{\gamma }^{2}f(t{)}^{T}f(t)={{\varsigma }_t}^T\mathrm{\Xi }{\varsigma }_t<0$$

(35)

where

$${\varsigma }_t(t)=[\begin{array}{ccc}\hat{x}{(t)}^T\hfill & e{(t)}^T\hfill & f{(t)}^T\hfill \end{array}{]}^T,$$

$\mathrm{\Xi }=\left[\begin{array}{ccc}\mathrm{He}(X(\overline{A}+\overline{B}K))\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& \mathrm{He}(X(\overline{A}-\overline{B}\overline{L}\overline{C}))\hfill & X\overline{E}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }^{2}I\hfill \end{array}\right]$ Let $K=U^{-1}W$ and $L=-V^{-1}Y$ , $X\overline{B}K$ and $X\overline{B}\overline{L}\overline{C}$ in $\mathrm{\Xi }$ can be rewritten as,

$$\begin{array}{c}\multicolumn{1}{c}{X\overline{B}K=X\overline{B}{U}^{-1}W}\\ =(X\overline{B}-\overline{B}U)U^{-1}W+\overline{B}W\hfill \\ -X\overline{B}\overline{L}\overline{C}=X\overline{B}{V}^{-1}Y\overline{C}\hfill \end{array}$$

(36)

$$=(X\overline{B}-\overline{B}V)V^{-1}Y\overline{C}+\overline{B}Y\overline{C}$$

(37)

On the other hand, $\mathrm{\Xi }$ is equivalent to

$$\mathrm{\Xi }={H^{\perp }}^{T}P{H}^{\perp }<0$$

(38)

where

$$\begin{array}{c}P=\left[\begin{array}{ccccc}\mathrm{He}(X(\overline{A}+\overline{B}K))\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& \mathrm{He}(X(\overline{A}-\overline{B}\overline{L}\overline{C}))\hfill & X\overline{E}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }^{2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & 0\hfill \end{array}\right],\hfill \\ H^{\perp }=\left[\begin{array}{c}I\hfill \\ \multicolumn{1}{c}{H}\end{array}\right],\hfill \end{array}$$

$$H=\left[\begin{array}{ccccc}K\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\overline{L}\overline{C}\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]$$

Applying Lemma 1 to (38), we get

$$P+\mathrm{He}(\overline{S}\aleph )<0$$

(39)

where

$$\begin{array}{c}\overline{S}=\left[\begin{array}{ccccc}{\tau }_1\overline{B}U-X\overline{B}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\tau }_2\overline{B}V-X\overline{B}\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right],\hfill \\ \aleph =\left[\begin{array}{ccccc}K\hfill & 0\hfill & 0\hfill & -I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\overline{L}\overline{C}\hfill & 0\hfill & 0\hfill & -I\hfill \end{array}\right]\hfill \end{array}$$

which is equivalent to condition in (32). So if the equation (39) holds, it can be ensure that $\mathrm{\Xi }$  < 0. That is to say the system (18) would be stable with the proposed SM controller (28). This completes the proof.

• Remark 3: Compared with the SM control for unbalance disturbance suppression of AMB system in the existing literature, the merit is that the controller gain K and observer gain $\overline{L}$ could be directly calculated by $K=U^{-1}W$ and $\overline{L}=-V^{-1}Y$ , instead of being repeatedly adjusted in the simulation process.

• Remark 4: In this chapter, a lossless decoupling method is proposed to decouple the $X\overline{B}K$ term and the $X\overline{B}\overline{L}\overline{C}$ term, so as to ensure that the sufficient conditions in Theorem 2 are linear matrix inequalities. Through this decoupling technique and Finsler Lemma, the constraint relations between the controller gain K, observer gain $\overline{L}$ , Lyapunov matrix variable X and AMB system matrix $\overline{B}$ and $\overline{C}$ is eliminated. Therefore, the proposed Theorem 2 has potential to reduce the conservatism considerably.

Modeling and simulation of control system

The effectiveness of the proposed observer-based SM control for the AMB rotor system will be verified in the following sections. The main parameters of the AMB rotor system using in the simulation process are shown in Table 1.

Table 1.

Simulation parameters of AMB rotor system.

Parameter	Value	Parameter	Value
m	14.8 kg	$\overline{e}$	1 × 10−2 m m
$a$	0. 065 m	$\overline{\epsilon }$	5 × 10−5rad
$b$	0. 065 m	Jz	0.183 kg·m2
Jx	0.093 kg·m2	Ks	−2.08 × 106 N/m
Jy	0.093 kg·m2	Ki	1.69 × 103 N/m

During the simulation, it is assumed that x₀ = [0 −0.05 0.05 0.03 0 0 0 0 ] ^T , $\hat{x}$  = [0 0 0 0 0 0 0 0] ^T , and ω = 20,000 rpm under the initial state. Other parameters are selected as θ = 0.01, ι = 0.01, τ ₁  = τ ₂  = 1. Based on Theorem 2, the controller parameter K and observer parameter $\overline{L}$ are obtained as follows:

$$\begin{array}{c}K=[\begin{array}{cccc}-0.9334\hfill & -3.7834\hfill & -6.8611\hfill & -3.6552\hfill \\ \multicolumn{1}{c}{-0.9214}& -9.4531\hfill & -7.2314\hfill & -7.2314\hfill \\ \multicolumn{1}{c}{-3.7821}& -4.8106\hfill & -11.4550\hfill & -11.4550\hfill \\ \multicolumn{1}{c}{-3.8387}& -3.8371\hfill & -10.4099\hfill & -10.4099\hfill \end{array}\hfill \\ \begin{array}{c}\begin{array}{cccc}-5.7876\hfill & -0.9334\hfill & -8.3163\hfill & -12.5028\hfill \\ \multicolumn{1}{c}{-12.9334}& -4.5326\hfill & -6.1858\hfill & -11.9334\hfill \\ \multicolumn{1}{c}{-8.6520}& -5.9334\hfill & -5.776\hfill & -8.9530\hfill \\ \multicolumn{1}{c}{-3.1122}& -14.9334\hfill & -5.3301\hfill & -13.2604\hfill \end{array}]\hfill \end{array}\hfill \end{array}$$

$$\overline{L}=\left[\begin{array}{cc}0.9268\hfill & 0.8711\hfill \\ \multicolumn{1}{c}{0.8148}& 1.0156\hfill \end{array}\right],$$

Substituting the above parameter K and state observer gain $\overline{L}$ into the equivalent SM control law (28), the sliding mode function curve, system controller input curve and state response curve of the AMB rotor system are simulated, as presented in Figures 3 to 5.

Figure 3.

The curve of sliding mode function.

Figure 4.

The curve of system input control.

Figure 5.

The response curves of system states.

As can be seen from Figures 3 and 4, the chattering was effectively overcome by substituting $-\rho \frac{s(t)}{\Vert s(t)\Vert +\iota }$ for $-\rho \mathrm{sign}(s)$ in the simulation process. Figure 5 describes the response curves of the AMB rotor system under the designed observer-based SMC. And $x_1(t)$ , $x_2(t)$ , $x_3(t)$ , and $x_4(t)$ represent the displacement of the upper magnetic levitation bearing A and the lower magnetic levitation bearing B in the x-axis and y-axis directions respectively. As can be seen from Figure 5, the AMB rotor system (18) works steadily. This method can effectively suppress unbalance disturbance and gyro effect.

Under the same conditions, comparative simulation are also carried out for the AMB rotor system using the proposed sliding mode controller with state observer and the traditional sliding mode controller without state observer. The displacements of the upper magnetic levitation bearing A and the lower magnetic levitation bearing B in the x-axis and y-axis directions are presented in Figure 6.

Figure 6.

Rotor position under the proposed SM controller with state observer and the traditional SM controller without state observer: (a) the response curve of state x₁(t), (b) the response curve of state x₂(t), (c) the response curve of state x₃(t), and (d) the response curve of state x₄(t).

Both the proposed SM controller with state observer and the traditional SM controller without state observer can make the rotor of the bearing quickly adjusted from the initial to the fixed position and remained stable in Figure 6. The speed response curves based on the proposed SM controller with state observer are faster and better than the conventional existing SM controller without state observer. As a result, the proposed SM controller has better dynamic characteristics than the traditional control strategy. Meanwhile, the number of decision variables involved in the proposed SM controller with state observer is 18.5n² + (m + 5.5)n + 1, while the number of decision variables involved in the traditional SM controller without state observer is 11n² + 12n.

Experiment platform and results

Establishment of experimental platform

In order to validate the effectiveness of the proposed new SM controller, the experimental platform of the flywheel energy storage system was established. The components of the flywheel energy storage system and its controller are shown in Figure 7. It is mainly consisted of the flywheel battery, flywheel battery control system, uninterrupted power supply (UPS), dissipation resistance box, Labview monitoring system, and vacuum pump.

Figure 7.

Test platform for magnetic levitation flywheel energy storage system.

The flywheel battery system is the core component of the system. It is composed of flywheel, motor, magnetic suspension bearing and auxiliary bearing. The main function is to charge and discharge the flywheel battery.

The main task of flywheel battery control cabinet is to complete the start, stop and parameter setting of magnetic levitation bearing and motor.

The UPS can not only supply power to the flywheel energy storage system, but also ensure that the system will not be damaged due to external interference caused by sudden power failure.

The dissipative resistance box, taken as the load of the flywheel energy storage system, consumes the electric energy emitted by the flywheel battery. The Labview monitor computer is used to display the output displacement curves of the rotor. In order to ensure that the flywheel rotor is not subjected to air resistance during operation, vacuum pumps are used.

Experiment results

In this part, the proposed SM controller is utilized to guarantee the rotor displacement with the experimental platform in Figure 7. In this experiment platform, the mass of the flywheel AMB rotor systems is 58 kg, the external diameter and height are 318 and 815 mm. The stiffness of the AMB is 600 N/μm. Its charging and discharging times are 133.29 and 110.10 s, respectively. During the test, sampling frequency, nominal air gap, the barrier of the rotor displacement, and the rotor speed were set to 33.3 kHz, 0.5 mm, 0.25 mm, and 10,020 r/min.

The rotor trajectories used the proposed SM controller with state observer in this article and conventional SM controller without state observer are given Figures 8 and 9.

Figure 8.

Axis position of static suspension MB with the proposed controller (a) the trajectory of the upper axis and (b) the trajectory of the lower axis.

Figure 9.

Axis position of static suspension MB with the conventional SM controller (a) the trajectory of the upper axis and (b) the trajectory of the lower axis.

The voltage of the displacement sensor was transformed into the actual position of the rotor, and the experimental results of the two controllers were compared. As can be seen from Figures 8 and 9, all the axis positions changed within the range of 50 μm under the action of the SM controller. The designed SM controller with state observer could stabilize the AMB and was superior to the traditional sliding mode controller without state observer in terms of stability time and rotor position fluctuation, as shown in Table 2. In summary, it can be seen that the performances of rotor displacement orbits under the proposed controller was better than using conventional SM controller.

Table 2.

Comparison of experimental results.

Controller	Stabilization time	Trajectory range
Proposed SM controller with state observer	0.06	40  $\mu \mathrm{m}$
Conventional SM controller without state observer	0.1 s	50  $\mu \mathrm{m}$

In the meantime, the experimental results of the radial-axial output displacement of suspension AMB system applied the proposed controller are present in Figure 10 at 10,020 r/min. Figure 11 gives the corresponding control currents and other parameters.

Figure 10.

The radial-axial displacement of MB at 10,020 r/min.

Figure 11.

The parameters of flywheel battery control panel at 10,020 r/min.

From Figure 10, it can be observed that the peak-to-peak value in the rotor trajectory was controlled in the range of 120μm. It indicates that the performance with the proposed SM controller based on state observer is resultful. As can be seen from Figure 11, the energy stored by the flywheel energy storage system at this time is 27 W.H. The control currents 1, 2, 3, and 4 in x/y directions of the upper radial AMB are 2620, 2261, 2346, and 2529 mA. The control currents 5, 6, 7, and 8 in the x/y directions of the lower radial AMB are 2583, 2339, 2581, and 2309 mA. The control current 9 of the upper axial maglev bearing is 4378 mA, and the control current 10 of the lower axial maglev bearing is 3023 mA. Compared with static suspension, the radial control current of each degree of freedom in dynamic suspension increased, while the axial control current decreased. Due to the large mass of the flywheel rotor, the control currents of the axial AMB changes more than that of the radial AMB. In summary, the flywheel energy storage system realized the task of energy storage, and the static and dynamic suspension processes of the AMB are relatively stable. All of these demonstrate that the proposed method has significant effectiveness in the rotor imbalance vibration suppression.

Conclusion

In this article, an improved SM controller with the introduction of a state observer was presented for suppression of unbalanced disturbances in AMB rotor system. Firstly, the state-space model of a vertical radial AMB rotor system with unbalance disturbance and gyro effect is presented. Then, an observer based sliding mode function and switching surface were constructed. Unlike traditional observer structures, the gain L of the observer can be designed based on the parameters of the magnetic levitation bearing rotor system. Meanwhile, in order to provide sufficient conditions for the AMB rotor system in the form of linear matrix inequalities, a separation method based on Finsler lemma were proposed to eliminate the coupling among controller parameters and Lyapunov variables. Finally, simulation and experimental results are supplied to confirm the validity of the presented schemes. How to extend the proposed schemes for the AMB rotor systems with time-delay will be conducted in the future.