A Finite-Time Sliding-Mode Controller Based on the Disturbance Observer and Neural Network for Hysteretic Systems with Application in Piezoelectric Actuators

Piezoelectric actuators (PEAs) have the benefits of a high-resolution and high-frequency response and are widely applied in the field of micro-/nano-high-precision positioning. However, PEAs undergo nonlinear hysteresis between input voltage and output displacement, owing to the properties of materials. In addition, the input frequency can also influence the hysteresis response of PEAs. Research on tracking the control of PEAs by using various adaptive controllers has been a hot topic. This paper presents a finite-time sliding-mode controller (SMC) based on the disturbance observer (DOB) and a radial basis function (RBF) neural network (NN) (RBF-NN). RBF-NN is used to replace the hysteresis model of the dynamic system, and a novel finite-time adaptive DOB is proposed to estimate the disturbances of the system. By using RBF-NN, it is no longer necessary to establish the hysteresis model. The proposed DOB does not rely on any priori knowledge of disturbances and has a simple structure. All the solutions of closed-loop systems are practical finite-time-stable, and tracking errors can converge to a small neighborhood of zero in a finite time. The proposed control method was compiled in C language in the VC++ environment. A series of comparative experiments were conducted on a platform of a commercial PEA to validate the method. According to the experimental results of the sinusoidal and triangular trajectories under the frequencies of 1, 50, 100, and 200 Hz, the proposed control method is feasible and effective in improving the tracking control accuracy of the PEA platform.

The RBF-NN is formulated as where W = [W 1 W 2 . . . W m ] T ∈ R m is the ideal weight, m is the number of the hidden layer of NN, h(X * ) = h 1 (X * ) h 2 (X * ) . . . h m (X * ) T ∈ R m , and X * is the input vector. h i (X * ) can be chosen as a Gaussian function, which is in the following , i = 1, 2, . . . , n, where µ = µ 1 µ 2 . . . µ n T is the central vector, ε is the width of the Gaussian function, and the range of The dynamic model of the system is defined as follows [24]: where m is the mass of PEA, and b is the damping coefficient. The term k is the electromechanical ratio, and u is the control input of the system. .
x and x are the velocity and position, respectively. The function f represents the hysteresis model, and d is the disturbance of the system.

Adaptive Finite-Time DOB and SMC Design
The dynamic model (4) is rewritten using Equation (5): where The tracking error is defined as follows: where x d is the desired position. Then, the derivative of error is defined as follows: To design adaptive finite-time DOB, the variable σ is introduced first as follows: where ω is designed as The estimated hysteresis modelf m is replaced by RBF-NN, which is given bŷ where h is the hidden layer. The hysteresis model f m in Equation (7) is replaced by RBF-NN, and defined as f m = W T h.Ŵ is the estimation of weight W, and the update ofŴ is defined as where δ 2 > 0.

Remark 1.
All the terms in Equation (5) are bounded. Then, the d m is bounded and denoted as |d m | ≤ τ.
The estimation d m is given bŷ where k 1 and k 2 are positive gains, andτ is the estimation of τ. The updateτ is designed as where δ 1 > 0. The estimated errors of the weights of RBF, disturbance, and τ are defined as follows: Lemma 3. Considering the nonlinear system (7), the adaptive finite-time DOB is designed as Equations (8)- (13). Then,d m will converge to the neighborhood of d m in a finite time.
Proof. Consider the Lyapunov function candidate for the system as follows: The time derivative V 1 is given by Note that Substituting (19) and (20) into (18) yields The terms − δ 1 (21) are quadratic functions and have the upper bounds λ 1 and λ 2 , respectively [32]. The upper bounds are defined as Using Lemma 1 and upper bounds, Equation (21) can be rewritten as follows: According to Lemma 2, Equation (24) is finite-time-stable within , 0 < θ 1 < 1, all terms in (17) are bounded. Therefore, d m can converge in a bounded set within a finite time; then, the proof is completed.
In the rest of this section, the finite-time SMC will be introduced. The sliding surface is defined as follows: β represents the integral gain. According to Lemma 3,d m will converge to the neighborhood of d m in a finite time, and W and d m are bounded. The observed valued m is then incorporated into the design of the control law to improve the disturbance rejection. The control input is designed as follows: where a 1 and a 2 are positive terms.

Theorem 1.
For the PEA control system (5), Equations (11) and (12) are the adaptive law of RBF-NN weight and disturbance estimation, respectively. Equation (25) is chosen as the sliding surface, and Equation (26) is the designed control input. If l d > d m and a 1 > W T h , the control system is bounded.
Proof. Choose the following Lyapunov function: The time derivative of (27) is The convergence time of the PEA system-based SMC is finite, so the proof is completed.
The saturation function is adopted to replace the sign function (sgn) in (26) and is defined as where z is input, and φ denotes the boundary layer thickness. The saturation function ensures the sliding surface is always bounded by φ.

Experimental Setup
The diagram of the experimental platform is shown in Figure 1. The platform consisted of a PC with a PCI board, a piezoelectric controller with a voltage amplifier and strain gauge sensor, a PEA with strain gauge sensor, ADC, and DAC. The gain of the piezoelectric controller (PEC) was 15, PEC (E01.B1, Coremorrow, Inc., Harbin, China), which could amplify the input voltage of DAC (PCI-9302, OLP, Inc., Chengdu, China) from 0-10 V to 0-150 V, and the maximum displacement of PEA (PSt20VS12, Coremorrow, Inc.) was 20 µm at input voltage 150 V. The PEA displacement 0-20 µm was transferred by the PEC position sensor as voltage and sampled by ADC (PCI-9203, OLP, Inc.) from 0 to 10 V.
where z is input, and ϕ denotes the boundary layer thickness. The saturation function ensures the sliding surface is always bounded by ϕ .

Experimental Setup
The diagram of the experimental platform is shown in Figure 1. The platform consisted of a PC with a PCI board, a piezoelectric controller with a voltage amplifier and strain gauge sensor, a PEA with strain gauge sensor, ADC, and DAC. The gain of the piezoelectric controller (PEC) was 15, PEC (E01.B1, Coremorrow, Inc., Harbin, China), which could amplify the input voltage of DAC (PCI-9302, OLP, Inc., Chengdu, China) from 0-10 V to 0-150 V, and the maximum displacement of PEA (PSt20VS12, Coremorrow, Inc.) was 20 µm at input voltage 150 V. The PEA displacement 0-20 µm was transferred by the PEC position sensor as voltage and sampled by ADC (PCI-9203, OLP, Inc.) from 0 to 10 V. [ ] For the adaptive laws, . The parameters of the controller are designed as The scheme of the proposed method is shown in Figure 2. The parameters of RBF were selected as X * = e . e , µ = 0 0 T , ε = 2.0. For the adaptive laws, δ 1 = 0.2, δ 2 = 0.25, The parameters of the controller are designed as β = 0.01, a 2 = 0.1, (a) (b) . The parameters of the controller are designed as The maximum absolute error and average absolute error are defined (30) and (31), respectively, and abbreviated as MAE and AAE in this paper. The maximum absolute error and average absolute error are defined as Equations (30) and (31), respectively, and abbreviated as MAE and AAE in this paper.
where x d (t) is the desired displacement, x(t) is the actual displacement generated by PEA, and N is the number of data.

Experimental Verification
To verify the performance of the proposed method, we adopted different control methods, namely [24] (Con.1) and [38] (Con.2) for comparison. The proposed control method and the other methods were all compiled in C language in the VC++ environment. The stiffness and capacitance of the actuator were 60 N/µm and 1.8 µF, respectively. The sampling frequency of the ADC was 200 kHz, and the control voltages calculated using the control method were generated in VC++ and then transferred to the voltage amplifier of PEC through the PCI-9302 card. Experiments were implemented for the following two cases to evaluate the effectiveness of the proposed method: tracking sinusoidal and triangular trajectories. The frequencies of the trajectories were 1, 50, 100, and 200 Hz.
The tracking results of the sinusoidal and triangular trajectories with the 12 µm peakto-peak amplitude and 1 Hz frequency are shown in Figures 3 and 4. The AAE of the proposed method under sinusoidal and triangular trajectories were 0.0063 and 0.0132 µm, respectively. The MAEs of the proposed method were 0.0238 and 0.0478 µm, which was less than those of Con.1 and Con.2. The tracking results of the sinusoidal and triangular trajectories with the 12 µm peakto-peak amplitude and 1 Hz frequency are shown in Figures 3 and 4. The AAE of the proposed method under sinusoidal and triangular trajectories were 0.0063 and 0.0132 µm, respectively. The MAEs of the proposed method were 0.0238 and 0.0478 µm, which was less than those of Con.1 and Con.2.   The results of tracking the 50 Hz sinusoidal and triangular trajectories are shown in Figures 5 and 6. The AAE of the proposed method tracking in sinusoidal trajectory was 0.0067 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 74% and 73%, respectively. The AAE tracking control in triangular trajectory was reduced by 54% and 59%, respectively, compared with Con.1 and Con.2. The MAEs cases to evaluate the effectiveness of the proposed method: tracking sinusoidal and triangular trajectories. The frequencies of the trajectories were 1, 50, 100, and 200 Hz.
The tracking results of the sinusoidal and triangular trajectories with the 12 µm peakto-peak amplitude and 1 Hz frequency are shown in Figures 3 and 4. The AAE of the proposed method under sinusoidal and triangular trajectories were 0.0063 and 0.0132 µm, respectively. The MAEs of the proposed method were 0.0238 and 0.0478 µm, which was less than those of Con.1 and Con.2.
(a) (b)  The results of tracking the 50 Hz sinusoidal and triangular trajectories are shown in Figures 5 and 6. The AAE of the proposed method tracking in sinusoidal trajectory was 0.0067 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 74% and 73%, respectively. The AAE tracking control in triangular trajectory was reduced by 54% and 59%, respectively, compared with Con.1 and Con.2. The MAEs The results of tracking the 50 Hz sinusoidal and triangular trajectories are shown in Figures 5 and 6. The AAE of the proposed method tracking in sinusoidal trajectory was 0.0067 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 74% and 73%, respectively. The AAE tracking control in triangular trajectory was reduced by 54% and 59%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were 0.0421 and 0.0604 µm, which were less than the other control methods.
Sensors 2023, 23, x FOR PEER REVIEW 9 of 13 of the proposed method were 0.0421 and 0.0604 µm, which were less than the other control methods.
(a) (b)  The results of the tracking of 100 Hz sinusoidal and triangular trajectories are shown in Figures 7 and 8. The AAE of the proposed method tracking in sinusoidal trajectory is 0.0257 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 41% and 37%, respectively. The AAE tracking control in triangular trajectory was 0.0142 µm. The AAE was reduced by 73% and 74%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were 0.0504 and 0.0535 µm. We can observe that the tracking errors of the proposed method were less than those of Con.1 and Con.2.  The results of the tracking of 100 Hz sinusoidal and triangular trajectories are shown in Figures 7 and 8. The AAE of the proposed method tracking in sinusoidal trajectory is 0.0257 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 41% and 37%, respectively. The AAE tracking control in triangular trajectory was 0.0142 µm. The AAE was reduced by 73% and 74%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were 0.0504 and 0.0535 µm. We can observe that the tracking errors of the proposed method were less than those of Con.1 and Con.2. The results of the tracking of 100 Hz sinusoidal and triangular trajectories are shown in Figures 7 and 8. The AAE of the proposed method tracking in sinusoidal trajectory is 0.0257 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 41% and 37%, respectively. The AAE tracking control in triangular trajectory was 0.0142 µm. The AAE was reduced by 73% and 74%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were 0.0504 and 0.0535 µm. We can observe that the tracking errors of the proposed method were less than those of Con.1 and Con.2.  The frequency was further increased to 200 Hz, the sinusoidal and triangular trajectories tracking results are shown in Figures 9 and 10. The AAE of the proposed method tracking in sinusoidal trajectory was 0.0268 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 39% and 38%, respectively. The AAE tracking control in triangular trajectory was 0.0261 µm. The AAE was reduced by 50% and 51%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were The frequency was further increased to 200 Hz, the sinusoidal and triangular trajectories tracking results are shown in Figures 9 and 10. The AAE of the proposed method tracking in sinusoidal trajectory was 0.0268 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 39% and 38%, respectively. The AAE tracking control in triangular trajectory was 0.0261 µm. The AAE was reduced by 50% and 51%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were 0.1006 and 0.1025 µm, which were less than Con.1 and Con.2. The frequency was further increased to 200 Hz, the sinusoidal and triangular trajectories tracking results are shown in Figures 9 and 10. The AAE of the proposed method tracking in sinusoidal trajectory was 0.0268 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 39% and 38%, respectively. The AAE tracking control in triangular trajectory was 0.0261 µm. The AAE was reduced by 50% and 51%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were 0.1006 and 0.1025 µm, which were less than Con.1 and Con.2.   The frequency was further increased to 200 Hz, the sinusoidal and triangular trajectories tracking results are shown in Figures 9 and 10. The AAE of the proposed method tracking in sinusoidal trajectory was 0.0268 µm. Compared with Con.1 and Con.2, the AAE of the proposed method was reduced by 39% and 38%, respectively. The AAE tracking control in triangular trajectory was 0.0261 µm. The AAE was reduced by 50% and 51%, respectively, compared with Con.1 and Con.2. The MAEs of the proposed method were 0.1006 and 0.1025 µm, which were less than Con.1 and Con.2.    Tables 1 and 2, respectively. Tables 3 and 4 provide a list of the AAE and MAE values of the tracking triangular trajectories. The control law with finite-time stability in this study can enable the developed control method to achieve better performance than that of the ultimately bounded approach. DOB assists the control method in compensating nonlinear disturbances, especially at high frequencies. The results of the comparative experiments validate that the proposed control method has superior performance in improving the accuracy of tracking the sinusoidal and triangular trajectories under different frequencies. According to the experimental results of tracking control of sinusoidal and triangular trajectories with 200 Hz frequency, the proposed control method can feasibly and effectively realize the high-precision tracking control of the PEA platform under high-frequency conditions.

Conclusions
In this paper, a finite-time SMC based on the DOB and RBF-NN was proposed. The proposed DOB has the advantage of a simple structure and does not rely on any priori knowledge of disturbances. By using the RBF-NN to replace the hysteresis model of the PEA dynamic system, it is no longer necessary to establish the hysteresis model. The proposed finite-time SMC-based DOB and RBF-NN can ensure that the system errors converge to a small neighborhood of zero in a finite time. The control method with finitetime stability can achieve better performance than that of the ultimately bounded approach. DOB in the control method can compensate for nonlinear disturbances at high frequencies.
The experiments of tracking the sinusoidal and triangular trajectories are validated on a commercial PEA platform. The experimental tracking results of the sinusoidal trajectories under the frequencies of 1, 50, 100, and 200 Hz show that the AAE values of the proposed control method were 0.0063, 0.0067, 0.0257, and 0.0268 µm, respectively. In the comparative experiments of tracking control triangular trajectories under the frequencies of 1, 50, 100, and 200 Hz, the AAE values of the proposed hybrid controller were 0.0132, 0.0141, 0.0142, and 0.0261 µm, respectively. All experimental results show that the proposed control method can feasibly and effectively realize the high-precision tracking control of the PEA platform under high and low frequencies.