High-energy orbit sliding mode control for nonlinear energy harvesting

Vibration energy harvesting has extensive application prospects in many significant occasions, such as mechanical structure health monitoring, vehicle tire pressure monitoring, IoT devices and human health monitoring. The nonlinearity is an effective method to improve the energy harvesting efficiency where there are low- and high-energy orbits in the multi-solution region of the system. The harvested power will be increased significantly when the system is guided from the low-energy orbit to the high-energy orbit. The sliding mode control is regarded as an easy, robust and adaptive method for orbit jump, but the implementation of this nonlinear control method has not been discussed. This paper proposes a high-energy sliding mode control method through rotatable magnets actuated by micro-motor. The electromechanical model of mono-stable and bi-stable systems with the identified nonlinear restoring force is established to design a sliding mode control algorithm for enhancing the energy harvesting performance. Simulation and experiment results demonstrate that the rotatable magnets with sliding mode control have a positive influence on reaching the high-energy orbit for both mono-stable and bi-stable systems within the multi-solution region. Moreover, the rotatable magnets method with a sliding mode control actuates the small magnets in the system for a short time with little theoretical consumption of energy. This research has provided a potential practical application of sliding mode control for high-energy orbit jump of the nonlinear energy harvesting.


Introduction
In recent years, with the rapid development of lowpower communication and microelectronics technologies, more and more low-power consumption sensors are applied widely in a variety of industrial occasions [1][2][3]. As a sustainable energy supply method, the vibration energy harvesting technology has attracted plenty of scholars to investigate the huge potential in many significant occasions, such as mechanical structure health monitoring, vehicle tire pressure monitoring, IoT and human health monitoring [4][5][6][7][8][9][10]. However, the traditional linear structures have a narrow resonance bandwidth under the ambient excitation, which poses a constraint to providing sufficient power for electronics.
To broaden the resonant bandwidth and improve the power density of the linear energy harvesters, nonlinearity is always introduced to the system with the use of various configured magnets [11][12][13][14] or mechanical structures [15]. Due to the compactness and stability, magnet configurations have attracted much attention. In this case, the magnetic force produced by external magnets can tune the stiffness of the energy harvesters, so that the multi-stable motion state can widen the resonant band of the piezoelectric cantilever beam. Many configurations have been applied to achieve the nonlinear multistable energy harvesters, normally including monostable system [16,17], bi-stable system [18,19] and tri-stable system [20]. Erturk et al. [21] designed a bistable piezomagnetoelastic energy capturing device by using two external bottom magnets and ferromagnetic beam, indicating that the open-circuit voltage amplitude has significantly increased than linear items. Stanton et al. [22] placed a magnet embedded cantilever beam between two external magnets and the influence of the position of two external magnets on the nonlinear behaviors was analyzed. Tang et al. [23] applied external oscillating magnets to improve the energy harvesting efficiency with the effective band broadened nearly 100%. Lan et al. [24] designed an enhanced bi-stable energy harvester by using a small magnet at the center of two fixed magnets to obtain a high output voltage at weak excitation. Zou et al. [25] enhanced the energy harvesting performance under low-frequency weak excitation conditions by means of magnetic stoppers, which shows that the compressivemode bi-stable vibration energy harvester is preferable under low frequency weak excitation. Yang et al. [26] investigated the bi-stable nonlinearity of designed double-beam piezo-magneto-elastic wind energy harvester for performance enhancement. In addition to bistable energy harvesters, tri-stable energy harvesters have also increasingly become another research focus. Zhou et al. [20,27] designed a piezoelectric vibration energy harvester with rotatable magnets to enhance the performance and different stable states can be performed by different angles of the external magnets. Cao et al. [28] studied the effect of the depth of multistable potential wells on its energy harvesting performance, and it shows that the restoring force of a multistable system is variable in stiffness due to the external magnets. Its movement between potential wells under certain initial conditions will cause a greater displacement of the cantilever beam, resulting in higher output voltage and power.
In terms of above nonlinear multi-stable energy harvesters, the chaos can bring the low-energy orbit and the high-energy orbit in the multiple-solutions regions. Under different initial conditions, these nonlinear energy harvesters exhibit different motion states that the high-energy orbit motion has the higher output voltage than low-energy orbit motion. Unfortunately, the ambient excitation is usually weak, thus making it difficult for the harvester to work in highenergy orbit. In order to reach high-energy orbit, many strategies have been developed for orbit jump, mainly including the electrical method and the mechanical method. For nonlinear piezoelectric energy harvesters, the piezoelectric transducer is excited by the voltage in a short time to achieve high-energy orbit. Sebald et al. [29,30] demonstrated that the energy harvester can jump from the lowest solution to the highest solution by applying a fast burst perturbation on the piezoelectric voltage. Lan et al. [31] put a voltage impulse perturbation to trigger high-energy orbit responses of piezoelectric energy harvesters based on negative resistance. Yan et al. [32] adopted the synchronized switch stiffness control technique to reach the lowcost orbit jump. Wang et al. [33] employed a bidirectional energy conversion circuit to get highenergy orbit by changing the direction of this circuit. As for nonlinear electromagnetic energy harvester, the electric load can be used as a perturbation to reach high-energy orbit. Mallick et al. [34] introduced a voltage perturbation to electromagnetic harvesting system obtaining high-energy orbit. Wang et al. [35] presented a load perturbation by disconnecting the electrical load to reach high-energy orbit.
Although above electrical methods are effective in strongly electromechanical coupled energy harvesters, for the weakly electromechanical coupled harvesters, the mechanical methods are more efficient. Erturk et al. [36] discovered the low-energy and high-energy orbit states of bi-stable energy harvesters and explored the conditions for achieving high-energy orbit by hand disturbance. Zhou et al. [37] proposed an impactinduced multi-stable nonlinear energy harvester and the method was verified experimentally. Huguet et al. [38] experimentally investigated the influence of the buckling level on the orbit jump by changing the length between the frame and the center of mass. Huang et al. [39] adjusted the buckling level to obtain the desired high-energy orbit by the two piezoelectric actuators. In addition, the Ott-Grebogi-Yorke (OGY) method shown by Ott et al. [40] is always used to achieve high-energy orbit by small time-dependent perturbations. Geiyer et al. [41,42] used the chaotic nonlinear phenomenon based on OGY and the intermittent control law to stabilize the system in the large output displacement orbit. Kumar et al. [43] designed a low-power controller of OGY to stabilize the unstable high-energy periodic orbits. However, this method may be not suitable for the occasion where the parameters of the system are not available. Therefore, the robust and adaptive sliding mode control has attracted much attention. Hosseinloo et al. [44] performed sliding mode control on the bi-stable energy harvester expressed by the Duffing equation and the simulation results proved the feasibility of the method to reach a high solution. Yousefpour et al. [45] presented a disturbance-observer-based terminal sliding mode control for the coexisting attractors for a bistable energy harvester, and the numerical results indicated that this method can successfully control and shift the different attractors with minimal control energy consumption. However, the researches on the sliding mode control are limited to the numerical simulation. During the simulation process, the assumed force on the system was selected as the control input to investigate the controlling performance of the sliding mode control for orbit jump, but the implementation method of applying the controlling force to the system has not been discussed. Therefore, this paper presents an implement method of sliding mode control by adjusting the rotational angle of external magnets. The variation of rotational angle for external magnets is selected to design the sliding mode controlling method. Numerical and experimental investigations into the high-energy orbit control method of nonlinear energy harvester based on sliding mode variable structure control algorithm are carried out. Results indicate that the rotatable magnets control method has a significant influence on reaching the high-energy orbit.
This paper is organized as follows: The model of energy harvester with the identified nonlinear restoring force is established in Sect. 2. The design of the sliding mode control and the numerical simulation are carried out in Sect. 3. Then, the experimental validation is implemented with the experimental platform in Sect. 4 and the conclusion is drawn in Sect. 5.
2 Electromechanical model of piezoelectric energy harvester Figure 1 shows the schematic diagram of the nonlinear piezoelectric energy harvester considered in this paper. The system mainly includes an elastic cantilever beam, tip magnet, external rotatable magnets and two piezoelectric ceramics attached to the root of the cantilever beam. When the external excitation x a (t) along x direction is given to the system, the cantilever beam will oscillate based on the resonance theory. Due to the existence of magnetic force between the external magnets and the tip magnet of the cantilever beam, the interactive magnetic force will make this system show the nonlinearity. The magnetic interaction can be changed by adjusting the system parameters h, d, h as shown in Fig. 1, so that the different types of configurations including monostable and bi-stable can be obtained. In this design, in order to achieve high-energy orbit from low-energy orbit, the servos are applied to adjust the rotational angle h of external magnets to change the nonlinear  restoring force. As a result, this controlling method can guide the system from the low-energy orbit to the high-energy orbit.
In this piezoelectric energy harvester, the angle of the external magnets connected to the servos is changeable while other parameters are always constant. The center distance d of the selected external magnets is 0.035 m; the distance h of the tip magnet and the external magnets is 0.02 m. The polarization direction of these magnets is shown in Fig. 1. The sizes of tip magnet and external magnets are 4 mm 9 10 mm 9 10 mm and 10 mm 9 10 mm 9 10 mm, respectively, with the material of N35.
According to the research of previous works based on Hamilton principle and Lagrange equation [46,47], the electromechanical dynamic equations considering the first mode can be obtained as where m, c, k and f are the equivalent mass, damping, stiffness and electromechanical coupling coefficient, respectively; C p is the equivalent capacitance of the PZT; x(t) is the displacement of the tip magnet along the x-direction; R is the load resistance; V(t) is the voltage across the R; F(t) is the external excitation; F m (x, h) is the magnetic force exerted on the tip magnet.
The combination of linear elastic force kx(t) and magnetic force F m (x, h) is normally defined as nonlinear restoring force Due to the small displacement of cantilever beam, the linear elastic force can be calculated based on the Mechanics of Materials where E is Young's modulus of elasticity; I is the inertia moment of beam; and l is the length of cantilever beam. It should be noted that the servos are used to change the rotational angle h of external magnets. Hence, the function of nonlinear restoring force for different h should be obtained before controlling method design. The theoretical expression of nonlinear restoring force based on rotational magnetic charge method [48] is applied in this research to obtain the influence of rotational angle h on the restoring force and potential energy as shown in Figs. 2 and 3, and the detailed expression of nonlinear restoring force can be seen in Appendix 1. When the rotational angle h of the magnets is 0 to 0.2p, the system is the bi-stable configuration. When the rotational angle h is 0.3p to 0.5p, the system is the mono-stable configuration. As shown in Fig. 3, bi-stable systems have two potential wells, while mono-stable systems have one potential well. For bi-stable systems, the potential well is becoming deeper and wider with the decrease of the rotational angle h.
In order to obtain an accurate dynamic model of the rotatable magnets energy harvester, the magnetic charge method is used to calculate the data points of the nonlinear restoring force with displacement when the rotational angle of the external magnet is 0, p/6, p/ 3, p/2. However, since the original expression of the magnetic restoring force is complex, it may be difficult to be applied in the state equation and controlling design. Therefore, the simple polynomial is used to fit the magnetic restoring force calculated by the original expression for easily designing the sliding mode control. Therefore, Matlab's Cftool toolbox is applied to simulate nonlinear restoring force polynomial fitting surface as shown in Fig. 4.
According the results of nonlinear restoring force by using magnetic charge method, the polynomial expression of nonlinear restoring force F r (x, h) can be   (1) can be rewritten as the following polynomial form: where p 00 = 6.19 9 10 -9 , p 10 = -5.609, p 01 = -1. The RMS error of the fitting is 0.72%. The order of the polynomial is selected by considering the accuracy and the complexity. When the order is higher than the present order, the accuracy cannot be significantly improved.

Controller design and simulation
In this specific problem, the sliding mode control is selected as the controlling method for its desirable advantages compared with other control methods, including easy implementation, good robustness, fast dynamic response, and insensitivity to variations of plant parameters and external disturbance. It is suitable for the motion control, in particular the control system with accurate mathematical models.
In this nonlinear energy harvesters, in order to achieve orbit jump from the low-energy orbit to the high-energy orbit, the trajectory of high-energy orbit in certain excitation is regarded as the desired value. Therefore, the sliding surface is defined by the tracking error and derivative of tracking error between the trajectory of high-energy orbit and the state position. It should be noted that the trajectory data of the high-energy orbit for certain frequency should be obtained in advance and input into the controller. Then, in order to reduce the chattering, the reaching law is designed to make the state point reach the sliding surface steadily, thus entering into the highenergy orbit. More importantly, the rotational angle of external magnets is selected as the controlling law. During the process of sliding mode controlling, the rotational angle can be adjusted adaptively to change the nonlinear restoring force of energy harvesters and meet the controlling law.   In terms of the process of pushing the low-energy orbit to the high-energy orbit, the desired trajectory of high-energy orbit in certain frequency is the controlling goal. When collecting the real-time trajectory of the tip magnet, it needs to reduce the tracking error between the real-time trajectory and the desired trajectory. Therefore, the controller will drive the servo actuator to change the rotational angle of external magnet adaptively. If the rotational angle of external magnet is changed, the stiffness of the system will change accordingly, thus affecting the dynamic response of the controlled system.

Slide mode controller
From Eq. (4), assuming h = h i ? u, where h i is the initial angle of the external magnets, u is the angle when the external magnets rotate (control law), this equation can be then transformed into the following form: The equation above can be further simplified as follows by introducing c m = c/m, r m = r/m, f m = f/m, The state equation is required to design the sliding mode controller for position tracking based on the reaching law, therefore the first-order dynamics from Eq. (6) can be transformed as The first step of sliding mode controller is to design the sliding surface based on the trajectory tracking between the desired trajectory and the state trajectory. The sliding surface is defined as where c e is the positive constant; e is the tracking error and _ e is the derivative of tracking error. Provided that r is the desired trajectory of highenergy orbit in the certain excitation, the tracking error is The derivative of tracking error is Therefore, the sliding surface is where slaw is the reaching law that determines the velocity from the state position to the sliding surface.
Here, the exponential reaching law is adopted because its varying reaching velocity can accelerate the approaching time and weaken the chattering. Hence, the reaching law is where k and e are the parameters to define the reaching velocity from the state position to the sliding surface. When the state point is far from the switching surface, the approaching velocity mainly depends on the -ks, otherwise approaching velocity relies on the esgn s ð Þ. The increase of k can make the state point approach the area of switching surface rapidly. The decrease of e can reduce the approaching velocity near the switching surface, thus weakening the chattering.
As previous defined, u is the variation of rotational angle for external magnet, so that the variation u of rotational angle is selected as the controlled object. Therefore, control law can be expressed as Since the external magnets rotate within a limited range of angles, and variation u of rotational angle has the limited value. In the simulation and experiment, the saturation module is used to limit the control input. Therefore, the control law should meet where k 1 and k 2 are constants, determined by the initial condition of mono-stable or bi-stable. Hence, the control law is assumed as where k 0 = 1 2 ðk 1 þ k 2 Þ, k 00 ¼ 1 2 ðk 1 À k 2 Þ. From the sliding surface Eq. (11), it has Considering the s [ 0 region of phase plane, then it has Therefore, the upper boundary of sliding mode region is The lower boundary is

Numerical simulation
The feasibility of guiding the trajectory to the highenergy orbit based on the sliding mode control is investigated by numerical simulation.

Mono-stable energy harvester
When the initial angle of the external magnets is set to p/3, the system is a mono-stable system. Under the excitation of up-sweep and down-sweep of 0.1 g (g is the gravity acceleration), the output displacement curve of the tip magnet is shown in Fig. 5. It can be seen that its multi-solution region range is 6.14-8.18 Hz. Here, an arbitrary excitation frequency of 7 Hz is selected from the multi-solution region to simulate the characteristics from the low-energy orbit to the high-energy orbit. The simulation process is as follows: Firstly, the desired trajectory of high-energy orbit under the 0.1 g and 7 Hz is obtained when the initial speed is 0.2 m/s. The amplitude of the high-energy orbit trajectory of the tip magnet is 0.008801 m.
Secondly, the desired trajectory of high-energy orbit is put into the sliding mode controlling, setting the simulation time of 30 s with the excitation condition of 0.1 g, 7 Hz. The initial conditions of the system are [0, 0, 0], which means that the initial displacement, velocity, and voltage are all zero, respectively. The initial values of reaching law to determine the motion velocity are k = 0.01 and e = 1000. In the initial 10 s, the system oscillates in a low-energy orbit due to the weak excitation condition. Then, after the sliding mode controller performs between 10 s and 10.1 s, the system is verified to be stable on a high-energy orbit without control as shown in Fig. 6a.
In detail, as can be seen from the time history of the displacement curve and control angle shown in Fig. 6b, c, when the sliding mode controlling is applied to the system, the system changes its original  Figure 7 gives the velocitydisplacement phase diagram of the mono-stable system, the trajectory can be guided from the low-energy orbit to the high-energy orbit within 0.1 s. In addition, as shown in Fig. 8, the amplitude of system output voltage is 1.77 V in the low-energy orbit state, while the output voltage amplitude is 8.98 V after entering the high-energy orbit by sliding mode control, increasing by 407%.   In order to investigate the controlling process of sliding mode control for mono-stable energy harvester, the different reaching velocities are considered. Figure 9 shows the trajectory and the control angle of the system under different reaching speeds e = 500, 200, 100 and 10. It can be seen that when e = 500 and 200, the controlling processes are much similar. They both can be guided from the low-energy orbit to the high-energy orbit, but e = 500 shows the more chattering. Besides, the control angle of external magnet is different between 10.07 s and 10.1 s. As for e = 100 and 10, they both cannot be guided to the high-energy orbit. The controlling process can be regarded as an excitation, and the more reaching speed can bring the more excitation. Therefore, e = 100 can see the higher displacement than e = 10 after controlling process. Afterward, they both return back to the low-energy orbit.
In addition, the influence of different controlling time of 0.05 s and 1 s on the controlling process is investigated as shown in Fig. 10. It indicates that when the controlling time is 0.05 s, this system cannot be guided to the high-energy orbit as shown in Fig. 10a. This is because the trajectory cannot enter the highenergy orbit when finishing the controlling process. In contrast, when the controlling time is 1 s shown in Fig. 10b, the system can not only complete the jump process from the low-energy orbit to the high-energy orbit, but also maintain the high-energy orbit until the controlling process is finished. The trajectory of 1 s Fig. 9 The influence of reaching speed on controlling mono-stable system controlling time in Fig. 10b can cover the trajectory of 0.05 s controlling time in Fig. 10a. In the beginning of entering the high-energy orbit, there exists the significant chattering, after which the trajectory can be controlled in the high-energy orbit steadily.
The beginning of controlling time in above analysis is always set to 10 s, which is in the maximal amplitude of displacement. Figure 11 depicts the trajectory and the control angle beginning in 10.0357 s in the maximal velocity. In this circumstance, it needs 1.75 cycles to enter the high-energy orbit steadily, while Fig. 10b requires 2 cycles to enter the high-energy orbit steadily. This may mean that it needs more time to drive the system from the lowenergy orbit to the high-energy orbit when the beginning of controlling process is in the maximal velocity.

Bi-stable energy harvester
When the initial angle of the external magnets for the system is set to p/6, the bi-stable system can be obtained. Under the excitation of up-sweep and downsweep of 0.2 g, the displacement curve of the tip magnet is shown in Fig. 12, and its multi-solution region range is 4.14-8.06 Hz. Hence, the excitation  First, the trajectory of high-energy orbit under the excitation of 0.2 g and 5 Hz should be obtained. Here, the initial condition of the high-energy orbit is [0.0106, 0, 0] which means that the initial displacement is 0.0106 m. Then, the initial controlling time of sliding mode controller is between 10 s and 10.02 s, and the initial values of reaching law to determine the motion velocity are k = 0.01 and e = 1000. After controlling, the system is verified to be stable on a high-energy orbit without control as shown in Fig. 13a, and the trajectory of tip magnet is guided from the local intra-well oscillation to the inter-well oscillation by 0.02 s sliding mode control. The detailed change of the displacement can be seen in Fig. 13b that indicates the displacement of the tip magnet can have a sharp change in the 10 s when the sliding mode control is applied immediately. During the controlling process, the motion of the tip magnet is approaching the high-energy orbit. In addition, Fig. 13c depicts the change of control angle. It is clear that the rotational angle decreases by 0.523 rapidly to the lower boundary at 10 s. After 0.012 s, the angle change increases to -0.284 and then rises significantly to the upper boundary of 1.047 in 10.02 s until the controlling is finished. The velocity-displacement phase diagram of the bi-stable system is illustrated in Fig. 14, it can be seen that when the control is given, the trajectory of tip magnet can be guided to the high-energy orbit. Figure 15 shows the system output voltage with the amplitude of 2.41 V in the low-energy orbit state, while the output voltage amplitude is 15.14 V after entering the high-energy orbit, increasing by 528%. In order to investigate the influence of reaching speed on the sliding mode controlling process, the different values of e with 500, 50, 5, 0.001, respectively, are selected shown in Fig. 16. It can be seen that when e is 500, the system can be directly guided to the high-energy orbit during the controlling period. In contrast, for other three occasions, the system cannot be guided to the high-energy orbit until the controlling is finished. This means that when e is 50, 5 and 0.001, the system just receives the perturbation from the controlling process. In detail, the system can experience the chaos before reaching the high-energy orbit with e of 5 and 0.001 after the controlling process. However, when e is 0.001, despite the slight chaos, the system cannot reach the high-energy orbit.
In addition, the influence of controlling time on the controlling process for bi-stable energy harvester is investigated as shown in Fig. 17. It can be seen that when the controlling time is 0.002 s, the system cannot break the barrier of its own potential well, even there is no chaotic motion. With the controlling time increasing to 0.005 s, there is a 2.2-s chaos after the controlling process, after which this system reaches to the high-energy orbit. In terms of 0.03 s controlling time in Fig. 17c, when the controlling process is finished, the system is in the high-energy orbit. If the controlling time is continuously extended to 1 s as shown in Fig. 17d, the trajectory can be controlled in the high-energy orbit steadily. It is of interest that when the system just enters the high-energy orbit, there is a significant chattering. In addition, the trajectory of the stable high-energy orbit driven by the controller contains some burrs, which is not as smooth as the high-energy orbit of free motion.
Because the constant frequency excitation in the environment is usually mixed with noise, this paper takes the system with the initial state as a bistable state (h = p/6) as an example to explore the effect of noise interference on the sliding mode control algorithm of the rotating magnets active control energy harvester. The simulation conditions of the system are the same as the above, the excitation condition is 0.2 g, 5 Hz, the initial condition is [0.0106, 0, 0], the difference is that Gaussian white noise is added to the excitation signal and the signalto-noise ratio is 1 dB. It can be seen from velocitydisplacement phase diagram under the white noise interference excitation shown in Fig. 18 that the system has good stability under the fixed frequency excitation doped with white noise interference. The sliding mode controller can still guide the tip magnet from local intra-well oscillations to global inter-well oscillations and the amplitude of the system output voltage increases by 5 times as shown in Fig. 19.
Since the rotatable magnets active control energy harvester consumes a little energy theoretically during the control process, the energy capturing efficiency of the system is greatly improved after the system is guided from the low-energy orbit to the high-energy orbit in a short time. Taking the system with the initial state as a bi-stable state and 0.02 s controlling time as an example, the energy consumption ratio is analyzed, as shown in Fig. 20. At time t 1 , the sliding mode control is applied to the system, and the energy consumption quickly increases. With the removal of sliding mode control, the energy consumption remains unchanged as the red curve shows. After the sliding mode control, the power of the energy capture system is greatly increased, resulting in a rapid increase in the harvested energy as the blue curve shown. Eventually, at time t 2 , the control energy consumption is equal to the energy harvested by the system. After that, the system will be in a state of positive energy gain for a long time stably. Considering the maximum force of the external magnets and the magnet of the cantilever beam is 0.5 N, then the recovery time of the energy The output voltage with control of the bi-stable system consumed by the controller can be estimated to be 800 s, which is similar to the result presented by Hosseinloo et al. [44] and Yousefpour et al. [45]. This means that the energy harvested by the system can compensate for the energy consumed by the control process after 800 s.

Experimental setup
The experimental system is set up to verify the effectiveness of the sliding mode control algorithm shown in Fig. 21. A laser displacement sensor (model HL-G1 from Panasonic) is used to measure the tip displacement of the cantilever, as a feedback signal of the control system. The servo (model MG90S from HS) attached to the external magnets is controlled by the controller (model RTU-BOX from Rtunit). The external excitation force is applied by the shaker (model E-JZK-50 from ECON). All the experimental data are acquired by an oscilloscope (model MSOX3052A from Agilent). The metal substrate is made of stainless steel. Two PZT-51 piezo-ceramics connected to metal substrate in parallel have the dimension of 12 9 10 9 0.8 mm 3 . NdFeB magnets are N35 with B 1.2 T. The tip magnet has a dimension Fig. 16 The influence of reaching speed on controlling bi-stable system of 10 9 10 9 4 mm 3 , and the external magnets have a dimension of 10 9 10 9 10 mm 3 . During the experiment, as the angle of the external magnets rotates, the nonlinear restoring force will change. The nonlinear restoring force at different angles is measured as the magnets angle is adjusted. And then, the measured restoring force is used to identify the nonlinear force function for control systems. The corresponding discrete sliding mode control algorithm is applied for experiments. The control program is written by Simulink software and downloaded to the RTU_BOX controller. The displacement signal collected by the laser displacement sensor is as the feedback on the controller through the AD module, and the controller outputs the control signal to the servos. The external excitation force is applied by the shaker constantly.
During the experiment, with the rotation of the angle of the external magnet, the nonlinear restoring force will change with the change of its angle. Adjusting the servo angle, the nonlinear restoring force at different angles is measured and the nonlinear force polynomial is identified, which is applied to the control system. The center distance of the selected external magnets d is 0.03 m; the center distance of the tip magnet and the external magnets h is 0.015 m; the polarization direction of the magnets as shown in Fig. 1. The measured value and fitting curve of Fig. 17 The influence of controlling time on controlling bi-stable system nonlinear restoring force are shown in Fig. 22, and the potential energy curve calculated from the integral of nonlinear restoring force is shown in Fig. 23. It indicates that the energy harvester is the bi-stable system with the angle of p/2 and p/3, while the energy harvester is the mono-stable system with the angle of p/6 and 0.

Experimental validation of monostable harvester
Firstly, experimental verification of sliding mode control of a mono-stable system is performed. The simulation results described in the above section show that sliding mode control can significantly improve the efficiency of the energy harvester system in the multisolution region. To verify the effectiveness of control method, a sliding mode control experiment of the nonlinear energy harvester is designed in this paper. The excitation level is 0.3 g at the frequency 10 Hz, and the initial state of the system is mono-stable, with initial angle p/3. At a certain time, the servos rotate a certain angle with the sliding mode controller working. Then, the system's motion trajectory moves from low-energy orbit to high-energy orbit and sliding mode control is stopped after entering the high-energy orbit. The system still runs in high-energy orbit stably after the sliding mode controller off as shown in Fig. 24. In high-energy orbits, the peak output displacement increased from 2.77 mm to 6.04 mm, and the output voltage of the system increased significantly from 0.44 V to 0.96 V in the low-energy orbit. Figure 25 illustrates the experimental velocitydisplacement phase diagram of the mono-stable system, and it shows the process of entering high-energy orbit after control. The power of the system has increased by approximate 4.54 times, significantly increasing the output power of the energy harvester.

Experimental validation of bi-stable harvester
The above numerical simulation results show that the control method is more efficient for the system when the initial state is bi-stable, improving the output voltage several times. When the excitation level is 0.3 g at 5 Hz, the initial state of the system is bistable and the initial angle is set to p/6. At a certain  Fig. 20 The schematic diagram of the system energy point in time, the servos also rotate a certain angle with the sliding mode controller working. Then, the tip magnet's running trajectory is guided from local intrawell oscillations to global inter-well oscillations.
When the system enters into the high-energy orbit, sliding mode controller stops and the system still runs   in high-energy orbit stably shown in Fig. 26. In highenergy orbits, the peak output displacement increased from local intra-well oscillations to 9.78 mm, and the output voltage of the system increased significantly from 0.24 V to 1.22 V in the low-energy orbit. Figure 27 illustrates the experimental velocity-displacement phase diagram of the bi-stable system, and it shows the process of entering high-energy orbit after control. The power of the system has increased by approximate 26.14 times, significantly increasing the output power of the energy harvester.

Conclusion
In this paper, a high-energy control method achieved by adjusting rotatable magnets based on sliding mode control is proposed experimentally. The method is validated by numerical simulation and experimental application. The nonlinear restoring force function is identified, and the electromechanical model is established. The sliding mode control algorithm was designed based on the electromechanical model. Then, the designed sliding mode control system was used to lead the mono-stable and bi-stable systems to highenergy orbit within the multi-solution region. Both the simulation and experimental results indicate that the rotating magnets with sliding mode control have a significant influence on reaching the high-energy orbit, and the output power has been increased for 4.54 times and 26.14 times in the experimental monostable and bi-stable systems, respectively. The rotating magnets method with sliding mode control costs a little energy theoretically for it just actuates the small magnets in the system for a short time. This novel control method makes the system have tunable performance and widens the bandwidth of efficient harvesting, improving the energy harvesting efficiency of the system. However, the changing rotational angle of external magnet provides the potential method to achieve orbit jump, but in the present prototype, the experimental energy consumption may be very large, such as the laser displacement sensor, controller and computer with the power in the level of W. Therefore, the experiment configuration could be improved in the further consideration. For example, the voltage signal of piezoelectric transducer can be used as the desired trajectory, without the extra sensor to consume energy. Besides, the low-powered microcontroller can be applied to collect voltage signal and then drive the servo actuator. This may be the potential solution in the future investigation.
In addition, since this sliding mode control requires the trajectory of high-energy orbit to put into the controller, it may be only suitable for the application of multi-solution occasion. As for the stochastic excitation, the high-energy orbit may be not obvious. When giving the arbitrary desired trajectory much higher than original response, the system can be guided to the higher output but cannot reach the desired trajectory. Besides, when the control is  Fig. 27 Experimental velocity-displacement phase diagram of the bi-stable system removed, the trajectory can degenerate to the original response. Therefore, this method may be unable to control the system under the stochastic excitation to maintain the stable high-energy orbit.
According to the research [48] on the restoring force modeling, the force condition of tip magnet in given configuration is shown in Fig. 28. The nonlinear restoring force here is along y direction.
The governing equation of tip magnet along y and z directions can be obtained as: where F my, F mz are magnetic force along y direction and z direction, respectively; F e1 is elastic force along y direction; F e2 is the elastic force of cantilever beam along axial direction; F g is the gravity force of tip magnets; F r is the nonlinear restoring force. In order to solve Eq. (20) to obtain the nonlinear restoring force, it is necessary to calculate the magnetic force between the tip magnet and the external magnets. The detailed expression of magnetic force between two cubic magnets can be seen in the research [49]. In terms of two cubic magnets with the magnetization J, the size is a 9 b 9 c and a 0 9 b 0 9 c 0 along x, y and z directions, respectively. The relative coordinates between these two magnets are (x 01 , y 01 , z 01 ), and relative rotational angle along x direction is h.
The magnetic force along z direction is