For a class of system with nonlinear hysteresis, this paper presents an adaptive hybrid controller based on the hybrid backstepping-sliding mode, and describes the controller analytically by the LuGre model. Both backstepping and the sliding mode techniques are based on the Lyapunov theory. Drawing on this common point, the authors developed a new controller combining the two control techniques with a recursive design. The design aims to achieve two effects: assuring the stability of the closed loop system, and improving the continuous performance of the tracking position trajectory. The performance of the proposed hybrid controller was verified by implementing the identified Piezo model. The results show that our controller can track the system output desirably with the reference trajectory.
Precision positioning applications increasingly rely on actuation technologies built on intelligent piezoelectric materials. They need compact devices or a small footprint for particular operations [1], [2], [3], [4]. Their high integrative power [5], [6], low heat [7], [8], and low noise levels [9], [10] can provide relatively large efforts, high reliability, and biocompatibility [11], [12], [13], [14]. Because of these benefits, the piezoelectric actuator has been widely used in a variety of industries, including space exploration [15], [16], active shutters, pulsed jets [17], [18], [19], vibration control [20], [21], [22], [23], [24], optical path control [25], [26], [27], [28], micro-motorization of instruments [29], [30], valves and pumps for implants [31], [32], [33], [34], [35], [36], magnetic resonance imaging (MRI) [37], [38], microsurgery [39], [40], [41], [42], and other micro-displacement techniques [43], [44], [45], [46], [47], [48]. Position control is severely hindered by the unique piezoelectric actuator structure, nonlinear hysteresis behaviors, and additional sources of positioning precision loss, such as creep drift and temperature effects [49], [50], [51], [52], [53]. Extensive study has been done for the modeling and control of the nonlinearity of hysteresis. Insofar as it enables the acquisition of a system representation from the input/output data, experimental modeling leading to a model of representation is particularly intriguing [54], [55], [56], [57], [58].
This work proposes the LuGre model, which represents the effects of the hysteresis of the piezoelectric actuator with precision and efficiency, and then identifies the control parameters experimentally by the evolutionary algorithm called particle swarm optimization (PSO) [59], [60], [61], [62], [63], [64], [65]. The control of piezoelectric actuators has received a lot of attention in recent years. A powerful method for stabilizing nonlinear systems and monitoring trajectory is called backstepping control. The basic goal of backstepping is to achieve Lyapunov cascade stability in equivalent loops in a subsystem of order one, which endows them with robustness and asymptotic overall stability [66], [67], [68], [69], [70], [71]. Another very well-known control method is sliding mode control, which is noted for its stability, short response times, and sensitivity to parameter fluctuations [72], [73], [74], [75], [76].
A current focus of research is the integration of sliding mode and backstepping control methods within the piezoelectric actuator. In this work, adaptive back stepping-sliding mode, one of the hybrid controls, is employed to enhance the piezoelectric actuator's trajectory tracking capability. The hybrid control technique is a controller design approach that ensures the stability of the control system. The combination between the two control methods is anticipated to result in a novel control algorithm that takes advantage of the strengths of the two methods. Simulation results demonstrate the strategy’s advantage in terms of lowering tracking error and improving tracking stability. The hybrid control exhibits great performance, compared to sliding mode, and adaptive backstepping.
The dynamic model of the piezoelectric actuator can be expressed as [77]:
where, M is the equivalent mass of the piezoelectric actuator; x is the displacement of the mechanism; $\dot{x}$ is the relative velocity; $\ddot{x}$ is the acceleration; D is the linear friction coefficient of the piezoelectric actuator;_{ FL }is the external load;_{ FH }is the function of the hysteresis friction force; u is the voltage applied to the piezoelectric actuator.
The hysteresis friction force _{FH} can be described by LuGre model [78]:
where,_{ σ0, σ1,} and_{ σ2 }are positive constants, which can be equivalently interpreted as the bristle stiffness, bristle damping and viscous coefficient, respectively. The Stribeck effect curve can be described by the function_{ }g(x_{2}):
where, _{fc} is the Coulomb friction; fs is the striction force; _{xs} is the Stribeck velocity. The complete electromechanical equations for the model can be expressed as [79]:
Eq. (4) shows that the piezoelectric actuator's dynamics is nonlinear as a function of the state variables_{ }x_{1 }and_{ }x_{2,} and that the relationship between position and control voltage must be monitored in order to obtain the parameters of the LuGre model. This paper utilizes the PSO-based identification approach, and runs tests to optimize the parameters of the algorithm and the model. Table 1 displays the values for the nine parameters.
Parameter | Value | Unit |
M | 4.119 | g |
k_{e} | 93.6 | N/v |
σ_{0} | 2.176e^{+06} | N/m |
σ_{1} | 4.257e^{+06} | N_{s}/m |
σ_{2} | -3.826e^{+06} | N_{s}/m |
F_{c} | 6224 | N |
F_{s} | -7.181e^{+04} | N_{s}/m |
x_{s} | 0.7621 | m/s |
D | 10.76 | N_{s}/m |
Figure 1 compares the tracking performance of LuGre model with that of experiments, when the piezoelectric positioning stage is driven by a sinusoidal input or a linear input. The comparison shows that the minimum and maximum errors are 4.492 nm and 144.65 nm, respectively. The experimental results agree well with the LuGre modeling results.
The control law can be generated according to two sequences. The first sequence adopts the backstepping technique to compute the virtual controls and the corresponding stabilization functions.
The nonlinear system can be described by:
where, φ_{i} R^{i }_{→}_{ }R^{p} _{is }a continuously differentiable vector of known nonlinear functions; θ $\in$ R^{P} _{-} a vector of constant coefficients (known or unknown); ψ_{n-}_{1} is a function_{ }_{ }$\neq 0 \forall x \in R^n$ the control. To track the desired trajectory x_{d} using the state x_{i,} then the backstepping algorithm can be used for the overall asymptotic stabilization of the system error e ϵ R^{n}. To better illustrate this technique, the second-order nonlinear system can be considered in the following form:
With x=[x_{1}, x_{2}], the state vector and system control input u(t), the problem is to determine the control u(t) that stabilizes the system at point (x_{1}, x_{2}) = (0,0) [80].
To achieve the desired trajectory and stabilize the entire system, it is preferable to use an adaptive law to estimate the system parameters by the backstepping technique. Suppose the output variable is denoted by x_{1} and the certain desired trajectory is denoted by x_{1d}, the quantity of the control can be selected in two steps:
A.1 First step
This first step consists in identifying the error and it is dynamic
$e_1=x_1-x_d$
The derivative of the error can be expressed as:
Considering the Lyapunov function:
The derivative can be obtained as:
From Eq. (9), it is known that if e_{2}(t) tends to zero, the derivative of V_{1}(t) will be less than or equal to zero. If V_{1}≤0, e_{1}(t) will converge to zero, and x_{1}(t) will converge to the reference point x_{d}. Consequently, the next step will design a controller u to make e_{2}(t) converge towards zero.
A.2 Second step
Considering the second system error: $e_2=x_2-\alpha_1-\dot{x_d}$
Alternatively, $\hat{k}_0, \hat{k}_1, \hat{k}_2$, and $\hat{k}_3$_{ }are the estimated values of k_{0}, k_{1}, k_{2}, and k_{3}, $\tilde{k}_0=\hat{k}_0-k_0, \tilde{k}_1=\hat{k}_1-k_1, \tilde{k}_2=\hat{k}_2-k_2$, and $\tilde{k}_3=\hat{k}_3-k_3$, respectively; sign (.) denotes the sign function:
The Lyapunov function V_{2} can be defined as:
The derivative can be obtained by:
With
c_{1} and c_{2} are positive defined constants. Then, the control law can be selected as:
With the following adaptation laws:
Substituting the adaptation laws (17), (18), (19), and (20) into Eq. (16), we have:
This means the equilibrium of the closed-loop system is globally asymptotically stable, and the error variables e_{1} and e_{2} converge towards zero.
B. Second sequence
The second sequence highlights the sliding mode technique to calculate the real controls in the final backstepping step. The goal is to make the errors between the virtual controls and their desired values converge to zero. The introduction of the sliding control reduces the effects of disturbances. The hybridization between the backstepping control and the sliding mode control is realized by changing a variable in the last step [81].
To being with, the linear sliding surface can be considered as:
With λ_{1} > 0, the derivative of Eq. (23) can be obtained as:
Therefore, the modified Lyapunov candidate can be expressed as:
Then, the derivative of V_{2} can be given by:
Substituting Eq. (10) into Eq. (25), we have:
From Eq. (16), the hybrid control law of the piezoelectric actuator can be expressed as:
Thus,
Thus,
The control law (29), and the adaptation laws in Eqns. (30-33) can be replaced to obtain:
The relationship (34) shows that, with the law of hybridization control (28) and the adaptation parameters (30-33), the variables e_{1}(t) and e_{2}(t) converge towards zero, which allows the exit in pursuit of system (1) following asymptotically the reference.
Figures 2, Figure 3, Figure 4 respectively display the tracking displacement, tracking voltage control error, and phase diagram of the simulations on the proposed hysteresis model of piezoelectric positioning mechanism, with an amplitude of the sinusoidal reference of 1 µm and a frequency of 0.5 Hz. It can be seen that the hybrid control managed to stabilize the closed-loop system, and the trajectory tracking effect.
Figure 5 illustrates the convergence of the control parameters (k_{0}, k_{1}, and k_{2}). It is clear that the hybrid controller can converge very quickly.
To verify the performance of different control laws for the piezoelectric actuator, a comparative analysis was carried out between the hybrid control, the adaptive backstepping control and the sliding mode control under the same conditions (e.g., the simulation time, the frequency and the input signal). The comparison criterion was defined as a function of the simulation error.
Figure 6 shows how the tracking error of the three techniques evolves. It can be seen that these techniques are valid for the piezoelectric actuator control. The errors suggest that the hybrid controller achieves better results than the adaptive backstepping controller and the sliding mode controller.
Table 2 compares the convergence time for the different control techniques.
Control technique | Convergence time (s) |
Hybrid | 0.35 |
Adaptive backstepping | 0.6 |
Sliding mode | 0.7 |
This paper proposes the LuGre model and the associated identification procedure, aiming to accurately depict the hysteresis behavior of the piezoelectric actuator. Besides, a hybrid control was implemented to validate the accuracy of the model. The results show that the proposed method can track the reference trajectory very precisely. The proposed control technique was found to improve the control performance, thanks to its merits like flexible selection of control gains, and the simplicity of forming the control law. In addition, three techniques were compared, including our technique, suggesting that our technique achieves better performance than the contrastive methods.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.