Modeling and closed-loop displacement control method of shape memory alloy actuator

Abstract

The application discloses a kind of shape memory alloy driver's modeling and closed loop displacement control method, this method with inverse hysteresis feedforward compensation and SMC-PID feedback displacement control method as core, the positive model of shape memory alloy driver is established, and inverse model is obtained by inversion, adaptive SMC-PID feedback controller is designed, the positive model, inverse model and SMC-PID feedback controller are combined, and the control system for the closed loop displacement control of shape memory alloy driver is constructed;The control system designed is modeled and simulated test using Matlab / Simulink software, by selecting different tracking signals, observe the closed loop displacement control effect of shape memory alloy driver.Shape memory alloy driver's modeling and closed loop displacement control method of the application has excellent control effect under shape memory alloy driver back and forth, continuous heating deformation, continuous cooling deformation and other various motion conditions, with high tracking accuracy, small delay, realizes the strong robustness of system hysteresis and parameter uncertainty.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of intelligent materials and its mechanism modeling and control, and particularly relates to a modeling and closed-loop displacement control method of a shape memory alloy actuator. BACKGROUND

[0002] Shape memory alloy (SMA) has unique shape memory effect, super-elasticity, high damping and self-sensing characteristics. The shape memory alloy actuator designed based on these characteristics has the characteristics of high power-to-weight ratio, high strain stress and high driving frequency, and has been widely researched and applied in the fields of aerospace, automation and robots. For example, in the field of aerospace, shape memory alloy actuators are widely used in various driving mechanisms, such as satellite sail deployment mechanisms, variable wing surface control devices, fastening and unlocking mechanisms, engine intake and exhaust adjustment mechanisms, etc. In recent years, with the increase of application scenarios, the precise position control of shape memory alloy actuators has become increasingly prominent, and the precise position control of shape memory alloy actuators has become a research hotspot at home and abroad.

[0003] The action of the shape memory alloy actuator is derived from the phase change process of SMA. This process is reversible, highly nonlinear, has large hysteresis and slow cooling characteristics. The traditional "open-loop operation" control method has limited the further development of shape memory alloy actuators. Closed-loop control is the main control form of shape memory alloy actuator displacement control at present. Therefore, in order to solve the problem of complex hysteresis nonlinearity of shape memory alloy actuators and achieve higher precision displacement control of shape memory alloy actuators, more effective control strategies need to be proposed and controllers with better performance need to be designed. SUMMARY

[0004] In view of the characteristics of large hysteresis nonlinearity of shape memory alloy, in order to improve the displacement control precision and robustness of shape memory alloy actuators, the present application provides a modeling and closed-loop displacement control method of shape memory alloy actuator.

[0005] In order to achieve the above purpose, the present application adopts the following technical scheme:

[0006] A modeling and closed-loop displacement control method of shape memory alloy actuator, comprising the following steps:

[0007] S1: establishing a mechanism model of the shape memory alloy actuator based on the working principle of the shape memory alloy actuator, and establishing a mathematical model of the shape memory alloy actuator in a modular manner according to the mechanism model, i.e. a positive model, the positive model taking current as input and shape memory alloy wire strain as output, including heat transfer and temperature module, phase change hysteresis module, strain module and resistance module;

[0008] S2: an inverse model of the shape memory alloy actuator is established by inverting the forward model, the inverse model is derived in a modular manner, and the expected strain is input, and the required current calculated by the inverse model is output, through the inverse model of the shape memory alloy actuator, the high nonlinearity, large hysteresis and slow cooling hysteresis derived from the phase change process of the shape memory alloy are compensated before the shape memory alloy actuator is encountered, and the inverse model is thus used for inverse hysteresis feedforward compensation control;

[0009] S3: an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed as a feedback control part for realizing precise displacement control of the shape memory alloy actuator, wherein the control signal is composed of a sliding control signal and a sliding equivalent control signal, a proportional-integral-derivative tuning controller is used to approximate the sliding equivalent control signal, and the stability of the closed-loop feedback control system is proved by using Lyapunov's law;

[0010] S4: the forward model in step S1, the inverse model in step S2 and the adaptive sliding mode controller based on proportional-integral-derivative tuning in step S3 are combined to construct a control system for closed-loop displacement control of the shape memory alloy actuator;

[0011] S5: the control system in step S4 is simulated and tested by using Matlab / Simulink software, and the closed-loop displacement control effect of the shape memory alloy actuator is observed by selecting different tracking signals.

[0012] Further, step S1 comprises:

[0013] S1.1: the shape memory alloy actuator adopts spring bias driving, the shape memory alloy actuator is composed of a bias spring and a shape memory alloy wire, one end of the spring and the shape memory alloy wire is fixed at a fixed end, when the shape memory alloy wire is powered and heated, the material thereof shrinks due to the shape memory effect to generate a pulling force, when the pulling force exceeds the elastic force of the spring, the displacement of the end of the shape memory alloy wire changes accordingly, and when cooled, the shape memory alloy wire expands with the aid of the bias spring, and the purpose of precisely controlling the displacement of the shape memory alloy actuator is achieved by precisely controlling the input current of the shape memory alloy wire;

[0014] S1.2: the heat transfer and temperature module is used to describe the heat transfer problem of the shape memory alloy wire in the shape memory alloy actuator, and the heat transfer and temperature module model is given by the following expression (1) with current as input and temperature of the shape memory alloy wire as output:

[0015]

[0016] where R is the internal resistance of the alloy wire, d0, L0 are the cross-sectional diameter and length of the alloy wire in the undeformed state, T is the temperature of the alloy wire, and amb is the ambient temperature, p is the mass density of the alloy wire, c is the specific heat of the alloy wire, h is the convective heat transfer coefficient of the alloy wire, and c and h are functions of the temperature T;

[0017] S1.3: the phase transformation hysteresis module is configured to describe the relationship between the martensite volume fraction R of the shape memory alloy wire in the shape memory alloy actuator and the temperature T, and the phase transformation hysteresis module is described by the following expression (2) using the Duhem differential hysteresis model: m

[0018]

[0019] where g + / - is the Gaussian probability density function, h + / - is the integral of g + / - , and the subscripts + and - represent the increasing and decreasing curves, respectively, μ is the mean, and σ 2 is the variance used to control the shape of the hysteresis loop, is the error function when integrating the normal distribution;

[0020] S1.4: the strain module is configured to describe the relationship between the strain ε of the shape memory alloy wire in the shape memory alloy actuator and the martensite volume fraction R m , and the strain module is described by the following expression (3): m

[0021]

[0022] where ε0 is the strain of the alloy wire in the fully austenitic state caused by the pre-tightening load, and the values of k1, k2, and k3 are related to the specific shape memory alloy wire;

[0023] S1.5: the resistance module is configured to describe the resistance of the shape memory alloy wire in the shape memory alloy actuator, and is used to provide the temperature T, the martensite volume fraction R m , and the strain ε calculated by steps S1.2, S1.3, and S1.4 for calculation, and the resistance of the shape memory alloy wire is calculated by the following expression (4):

[0024]

[0025] where p a (T) is the resistivity of the austenite, p m (T) is the resistivity of the martensite, p1, p2, p3, q1, q2, and a i ​​(i = 1-9) are relevant constants.

[0026] Further, step S2 comprises:

[0027] Let the desired strain trajectory be ε rd (t), the inverse models of strain module, phase change hysteresis module, heat transfer and temperature module and resistance module are obtained in sequence according to the positive model of the shape memory alloy driver, as shown in the following formula (5), so as to obtain the inverse model of the shape memory alloy driver with the desired strain as the input and the required current calculated by the inverse model as the output. The inverse model is derived from the high nonlinearity, large hysteresis and slow cooling hysteresis of the phase change process inside the shape memory alloy, and the inverse model is thus used for inverse hysteresis feedforward compensation control.

[0028]

[0029] Wherein, δ>0 is an arbitrarily small constant, to avoid calculation overflow caused by zero denominator.

[0030] Further, step S3 comprises:

[0031] An adaptive sliding mode controller based on proportional-integral-derivative tuning is designed, and the control quantity u is composed of a sliding control signal and a sliding equivalent control signal: u = u s + u eq , wherein u s is the sliding control signal, which switches the system to the sliding surface; u eq is the sliding equivalent control; a proportional-integral-derivative tuning controller is used to approximate the sliding equivalent control signal, and the stability of the closed-loop feedback control system is proved by using Lyapunov law, and the specific design process is as follows:

[0032] S3.1: Mathematical description of shape memory alloy driver:

[0033]

[0034] Wherein, x = [x1(t), x2(t)] T represents the displacement and velocity of the end of the shape memory alloy wire in the shape memory alloy driver; b(x1, x2) is the input gain, which represents the internal resistance of the shape memory alloy; f(·) is a nominal parameter of the model, and Δf(·) represents the phase change of the shape memory alloy, which describes the hysteresis phenomenon of the shape memory alloy; d(t) represents external disturbance; u(t) = I is the input. Δf(·), d(t) and b(·) are all bounded variables, |Δf(·)|≤g, d(t)|≤α, b min ≤b(·)≤b max ; g is the maximum change of the hysteresis curve; α is the maximum value of the disturbance;

[0035] S3.2: Design of integral sliding surface σ

[0036] To improve the convergence performance of sliding surface, define the reference signal x r such that:

[0037]

[0038] where e represents the error between the desired displacement y d and the output y d ; K1 and K0 make the roots of s 2 + K1s + K0 = 0 in the left half of s-plane;

[0039] Design the integral sliding surface σ = x r - x2, substitute into equation (7), σ can be rewritten as When the sliding mode occurs, σ tends to zero, and the introduction of integral element and high-order form in the sliding mode function can effectively improve the system control performance;

[0040] S3.3: Stability analysis of closed-loop feedback control system:

[0041] Define Lyapunov function as V = 1 / 2σ 2 , then To realize the stability of the system Design the control u to satisfy K > 0, that is; combined with equation (6), the derivative of sliding surface σ = x r - x2 can be expressed as:

[0042]

[0043] Therefore, the control quantity is:

[0044]

[0045] The control quantity u can be divided into two parts: u = u s + u eq . u s is the sliding control signal, which switches the system to the sliding surface, u eq is the sliding equivalent control,

[0046] S3.4: Design of sliding equivalent control signal u eq

[0047] Use PID tuning controller to approximate the equivalent control:

[0048]

[0049] ​For the tuning of PID parameters, the following objective function is selected And the gradient descent method is used to obtain the update rule of PID parameters:

[0050]

[0051] Since when ε u →0, there is σ→0, the above PID parameter update rule can be optimized as:

[0052]

[0053] S3.5: Design of sliding control signal u s

[0054] From equations (8), (9) and (10), we can deduce that:

[0055]

[0056] Where u PID is a positive term, so if K is selected to satisfy the following condition, the system is certainly asymptotically stable The equality holds if and only if σ=0:

[0057]

[0058] In order to weaken the inherent high-frequency chattering of the control input, the saturation function sat(σ / δ) is selected to replace sgn(σ), δ is the width of the boundary layer, and the sliding surface function σ with any initial value will reach and remain in the boundary layer σ|<δ, and the sliding control signal u s becomes:

[0059]

[0060] Further, step S4 comprises:

[0061] S4.1: The output of the inverse model of the shape memory alloy actuator established in step S2 is the current required by the shape memory alloy actuator to achieve the desired strain, which is input into the positive model in the control system as a feedforward signal, compensating for the influence of the highly nonlinear, large hysteresis and slow cooling hysteresis from the phase change process inside the shape memory alloy on the shape memory alloy actuator;

[0062] S4.2: In step S3, an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed, and the control quantity u is composed of a sliding control signal and a sliding mode equivalent control signal: u=u s +u eq ; this part is input into the mathematical model of the shape memory alloy actuator as a feedback control signal;

[0063] ​S4.3: combine the positive model described in step S1, the inverse model described in step S2 and the adaptive sliding mode controller based on proportional-integral-derivative tuning described in step S3 to build a control system for closed-loop displacement control of the shape memory alloy driver, for precise control of the displacement of the shape memory alloy driver.

[0064] Further, step S5 comprises:

[0065] S5.1: according to the derivation of steps S1, S2, S3 and S4, build a Simulink simulation platform;

[0066] S5.2: input a sine signal, simulate the sine tracking working condition of the shape memory alloy driver, and observe the tracking of the output displacement of the shape memory alloy driver;

[0067] S5.3: input a continuous descending step signal, simulate the tracking working condition of the shape memory alloy in the heating shrinkage process to the step command signal, and observe the tracking of the output displacement of the shape memory alloy driver;

[0068] S5.4: input a continuous ascending step signal, simulate the tracking working condition of the shape memory alloy in the recovery process to the step command signal, and observe the tracking of the output displacement of the shape memory alloy driver.

[0069] Compared with the prior art, the present application has the following beneficial effects:

[0070] The control scheme proposed by the present application has a simple structure and includes a feedforward part and a feedback part. The feedforward part is realized based on an inverse model, and the required inverse model is obtained by inverting each module of the mathematical model of the shape memory alloy driver, which is easy to obtain. The feedback part adopts a proportional-integral-derivative (PID) parameter adaptive sliding mode controller (SMC), which can update the gain parameters of the PID online through an adaptive law, and uses a tuned PID controller to approximate the sliding mode equivalent control signal, thereby weakening the high-frequency chattering of the sliding mode, and the feedback part has a simple control rate form.

[0071] The designed control method has better control effect than the traditional control method in various motion working conditions of the shape memory alloy driver, such as back-and-forth reciprocating, continuous heating deformation, continuous cooling deformation, etc., has high tracking accuracy and small delay, and realizes strong robustness to system hysteresis and parameter uncertainty. BRIEF DESCRIPTION OF DRAWINGS

[0072] The above and / or additional aspects and advantages of the present application will become apparent and more readily appreciated from the following description of the embodiments, taken in conjunction with the following drawings of which:

[0073] Figure 1A flowchart of displacement control of a shape memory alloy driver in a shape of a shape memory alloy wire-spring is realized by taking the displacement control method of inverse hysteresis feedforward compensation and SMC-PID feedback as the core in the embodiment of the present application.

[0074] Figure 2 A mechanism model diagram of a shape memory alloy driver;

[0075] Figure 3 A mathematical model (forward model) block diagram of a shape memory alloy driver provided by the embodiment of the present application;

[0076] Figure 4 An inverse hysteresis model block diagram of a shape memory alloy driver provided by the embodiment of the present application;

[0077] Figure 5 A control block diagram of feedforward inverse hysteresis and SMC-PID feedback combination of a shape memory alloy driver provided by the embodiment of the present application;

[0078] Figure 6 A displacement tracking curve of a shape memory alloy driver to a continuous sinusoidal command signal provided by the embodiment of the present application;

[0079] Figure 7 A displacement tracking curve of a shape memory alloy driver to a step command signal in a heating shrinkage process provided by the embodiment of the present application;

[0080] Figure 8 A displacement tracking curve of a shape memory alloy driver to a step command signal in a recovery (elongation) process provided by the embodiment of the present application. DETAILED DESCRIPTION

[0081] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be described in more detail below by combining theoretical formula, drawings in the specification and specific embodiments. It should be noted that the described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the present application.

[0082] The present application provides a modeling and closed-loop displacement control method of a shape memory alloy driver. In order to meet the high-precision displacement control requirement of a shape memory alloy driver, such as Figure 1 As shown in the figure, the displacement control method of inverse hysteresis feedforward compensation and SMC-PID feedback based on proportional-integral-derivative tuning is taken as the core in the embodiment of the present application, a displacement control of a driver in a shape of a shape memory alloy wire-spring is realized, and the method comprises:

[0083] S1: a mechanism model of the shape memory alloy actuator is established based on the working principle of the shape memory alloy actuator, and a mathematical model, i.e. a positive model, of the shape memory alloy actuator is established in a modular manner according to the mechanism model, the positive model taking current as input and shape memory alloy wire strain as output, and including a heat transfer and temperature module, a phase change hysteresis module, a strain module and a resistance module;

[0084] S2: an inverse model of the shape memory alloy actuator is established by inverting the positive model, the inverse model being derived in a modular manner and taking desired strain as input and required current calculated by the inverse model as output, so that the highly nonlinear, large hysteresis and slow cooling hysteresis of the phase change process originating from the shape memory alloy are compensated before the shape memory alloy actuator encounters them, and the inverse model is thus used for inverse hysteresis feedforward compensation control;

[0085] S3: an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed as a feedback control part for realizing precise displacement control of the shape memory alloy actuator, wherein the control signal is composed of a sliding control signal and a sliding equivalent control signal, a proportional-integral-derivative tuning controller is used to approximate the sliding equivalent control signal, and the stability of the closed-loop feedback control system is proved by using Lyapunov's law;

[0086] S4: the positive model in step S1, the inverse model in step S2 and the adaptive sliding mode controller based on proportional-integral-derivative tuning in step S3 are combined to construct a control system for closed-loop displacement control of the shape memory alloy actuator;

[0087] S5: the control system in step S4 is simulated and built and tested using Matlab / Simulink software, and the closed-loop displacement control effect of the shape memory alloy actuator is observed by selecting different tracking signals.

[0088] In the step S1, the following steps are included:

[0089] S1.1: the shape memory alloy actuator adopts spring bias driving, and the working principle is simplified as shown in Figure 2 The shape memory alloy actuator is composed of a bias spring and a shape memory alloy wire, one end of the spring and the shape memory alloy wire is fixed at a fixed end, when the shape memory alloy wire is heated by power, the material thereof shrinks due to shape memory effect to generate tension. When the tension exceeds the spring force, the displacement of the end of the shape memory alloy wire changes accordingly. When cooled, the shape memory alloy wire expands with the aid of the bias spring, and the purpose of precisely controlling the displacement of the shape memory alloy actuator is achieved by precisely controlling the current input to the shape memory alloy wire. Figure 2L0 is the length of the alloy wire in the undeformed state, ε0 is the strain of the alloy wire caused by the pre-tightening load in the fully austenitic state, k is the elastic constant of the bias spring, Δ is the deformation of the spring in the fully austenitic state, ε r is the strain of the alloy wire caused by the transformation from austenite to martensite;

[0090] S1.2: The heat transfer and temperature module describes the heat transfer problem of the shape memory alloy wire in the shape memory alloy actuator, and the heat transfer and temperature module model is given by the following expression, taking the current as the input and the temperature of the shape memory alloy wire as the output:

[0091]

[0092] where R is the internal resistance of the alloy wire, which is provided by the resistance module calculated in step S1.5, d0 and L0 are the cross-sectional diameter and length of the undeformed alloy wire, respectively, T amb is the ambient temperature, p is the mass density of the alloy wire, c is the specific heat of the alloy wire, and h is the convective heat transfer coefficient of the alloy wire. c and h are functions of temperature T, and each parameter is a constant obtained by experimental calibration and fitting.

[0093] S1.3: The phase transformation hysteresis module describes the relationship between the martensite volume fraction R m of the shape memory alloy wire in the shape memory alloy actuator and the temperature T, and adopts the Duhem differential hysteresis model. The phase transformation hysteresis model is given by the following expression:

[0094]

[0095] where g + / - is the Gaussian probability density function, describing the slope of the hysteresis loop. h + / - is the integral of g + / - , describing the shape of the hysteresis loop. Subscripts + and - represent increasing and decreasing curves, respectively, μ is the mean, and σ 2 is the variance, used to control the shape of the hysteresis loop. is the error function when integrating the normal distribution.

[0096] S1.4: The strain module describes the relationship between the strain ε of the shape memory alloy wire in the shape memory alloy actuator and the martensite volume fraction R m , which is specifically given by the following formula. The total strain ε can be approximated as a polynomial with respect to R m .

[0097]

[0098] Where ε0 is the strain caused by preload on the alloy wire in the fully austenitic state, and the values ​​of k1, k2, and k3 are related to the specific shape memory alloy wire and can be identified through parameters.

[0099] S1.5: The resistance module describes the resistance of the shape memory alloy wire in the shape memory alloy actuator, which is provided for calculation in step S1.2. Specifically, it is given by the following formula, calculated from steps S1.2, S1.3, and S1.4: temperature T, martensite volume fraction R. m The resistance of the shape memory alloy wire can be calculated using strain ε:

[0100]

[0101] Where, ρ a (T) is the resistivity of austenite, ρ m (T) is the resistivity of martensite. p1, p2, p3, q1, q2, and α i (i = 1 - 9) are relevant constants.

[0102] In step S2, the inverse model of each module is obtained by inverting the forward model in S1, which serves as a feedforward compensation for the nonlinearity and hysteresis of the shape memory alloy actuator. The specific process is as follows:

[0103] Similar to the forward model structure in S1, the inverse model of the shape memory alloy actuator is derived in a modular manner, as shown in the block diagram below. Figure 4 As shown. Let the desired strain trajectory be ε. rd Based on the modular mathematical model (forward model) of the shape memory alloy actuator, the inverse models of each module of the shape memory alloy actuator can be obtained sequentially, as shown in the following equation, which describes the inverse models of the strain module, phase change hysteresis module, heat transfer and temperature module, and resistance module in sequence. Thus, the inverse model of the shape memory alloy actuator with the desired strain as input and the required current calculated by the inverse model as output is obtained, which is used as a feedforward compensation for the hysteresis nonlinearity of the SMA.

[0104]

[0105] In the second equation, the inverse model of the phase transition hysteresis module is the inverse of the Duhem differential model, where δ > 0 is an arbitrarily small constant to avoid computational overflow caused by a zero denominator. The meanings of the remaining symbols are consistent with those of the forward model in step S1.

[0106] In step S3, an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed. The control quantity u consists of the sliding control signal and the sliding mode equivalent control signal: u = u s +u eq u sis the sliding control signal, which switches the system to the sliding surface; u eq is the equivalent control of the sliding mode. A proportional-integral-derivative tuning controller is used to approximate the equivalent control signal of the sliding mode, and the stability of the closed-loop feedback control system is proved by using Lyapunov's law. The specific design process is as follows:

[0107] S3.1: The mathematical description of the shape memory alloy driver is shown in the following formula (6):

[0108]

[0109] where x = [x1(t), x2(t)] T represents the displacement and velocity of the end of the shape memory alloy wire in the shape memory alloy driver; b(x1, x2) is the input gain, which represents the internal resistance of the SMA; f(·) is a nominal parameter of the model, and Δf(·) represents the phase change of the SMA, which describes the hysteresis phenomenon of the SMA; d(t) represents external disturbance; u(t) = I is the input. Δf(·), d(t) and b(·) are all bounded variables, |Δf(·)|≤g, |d(t)|≤α, b min ≤b(·)≤b max ; g is the maximum change of the hysteresis curve; α is the maximum value of the disturbance.

[0110] S3.2: The integral sliding surface σ is designed as

[0111] In order to improve the convergence performance of the sliding surface, the reference signal x r is defined as:

[0112]

[0113] where e represents the error between the expected displacement y d and the output y, e = y d -y; K1 and K0 make the roots of s 2 +K1s+K0=0 in the left half of the s plane;

[0114] The integral sliding surface σ = x r -x2 is designed, which can be rewritten as When the sliding mode occurs, σ tends to zero, and the introduction of the integral element and the high-order form in the sliding mode function can effectively improve the control performance of the system;

[0115] S3.3: Stability analysis of the closed-loop feedback control system:

[0116] Define the Lyapunov function as V = 1 / 2σ 2 , then In order to realize the stability of the system , the control u is designed to satisfy K>0 is satisfied; in combination with equation (6), the sliding surface is σ = x r The derivative of -x2 is given by:

[0117]

[0118] Therefore, the control variable is given by:

[0119]

[0120] The control variable u can be divided into two parts: u = u s + u eq . u s is the sliding control signal that switches the system to the sliding surface, u eq is the sliding equivalent control,

[0121] S3.4: Design of the sliding equivalent control signal u eq

[0122] A PID tuning controller is used to approximate the equivalent control:

[0123]

[0124] For the tuning of the PID parameters, the following objective function is selected and the gradient descent method is used to obtain the update rule for the PID parameters:

[0125]

[0126] Since when ε u → 0, there is σ → 0, the above PID parameter update rule can be optimized as:

[0127]

[0128] S3.5: Design of the sliding control signal u s

[0129] From equations (8), (9), and (10), we can derive:

[0130]

[0131] where u PID is a positive term, so if K is selected to satisfy the following condition, the system is certainly asymptotically stable The equality holds if and only if σ = 0:

[0132]

[0133] ​​In order to weaken the high frequency chattering inherent in the control input, a saturation function sat(σ / δ) is selected to replace sgn(σ), δ is the width of the boundary layer, and the sliding surface function σ with an arbitrary initial value will reach and remain in the boundary layer |σ|<δ, and the sliding control signal u s becomes:

[0134]

[0135] In step S4, as Figure 5 shown, the control quantity designed in steps S2 and S3 is introduced into the closed-loop system, and a control mode combining feedforward and feedback is adopted to accurately control the displacement of the shape memory alloy driver. The specific steps are as follows:

[0136] S4.1: The output of the inverse model of the shape memory alloy driver established in step S2 is the current required by the shape memory alloy driver to achieve the expected strain, which is input into the positive model in the control system as a feedforward signal to compensate for the influence of the high nonlinearity, large hysteresis and slow cooling hysteresis from the phase change process in the shape memory alloy on the shape memory alloy driver.

[0137] S4.2: In step S3, an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed, and the control quantity u is composed of a sliding control signal and a sliding mode equivalent control signal: u=u s +u eq This part is input into the mathematical model of the shape memory alloy driver as a feedback control signal.

[0138] S4.3: The positive model in step S1, the inverse model in step S2 and the adaptive sliding mode controller based on proportional-integral-derivative tuning in step S3 are combined to build a control system for closed-loop displacement control of the shape memory alloy driver, which is used to accurately control the displacement of the shape memory alloy driver.

[0139] In step S5, according to steps S1, S2, S3 and S4, the designed control system is modeled and simulated using Matlab / Simulink software, the output displacement is tracked by selecting different tracking signals, and the control effect is observed by comparing different controllers. The specific steps are as follows:

[0140] S5.1: According to the derivation of steps S1, S2, S3 and S4, a Simulink simulation platform is built;

[0141] S5.2: The input is a sine signal to simulate the sine tracking working condition of the shape memory alloy driver, and the tracking of the output displacement of the shape memory alloy driver is observed.

[0142] S5.3: input is a continuous descending step signal, analog shape memory alloy driver in the heating shrinkage process to track the tracking conditions of the step command signal, observe the tracking of the output displacement of the shape memory alloy driver.

[0143] S5.4: input is a continuous ascending step signal, analog shape memory alloy driver in the recovery (elongation) process to track the tracking conditions of the step command signal, observe the tracking of the output displacement of the shape memory alloy driver.

[0144] Embodiment:

[0145] The parameters used in the simulation process are listed in Table 1.

[0146] Table 1 Parameters of the mathematical model of the shape memory alloy driver

[0147]

[0148]

[0149] The parameters of the SMC-PID controller designed in the application are as follows: the sliding surface parameters are K0=0.1s, K1=10s; the PID gain update rate is η1=1, η2=0.2, η3=0.1; the sliding mode control signal related parameters are K=0.84A, δ=0.3. -1 max For comparative analysis, the control effects of the classic PID and feedforward inverse compensation PID are adjusted to the best, and the parameters of the control group controller are: K=10, K=K=0.4. P I D

[0150] The displacement tracking of the shape memory alloy driver of the application under the conditions of the command signal being a sine signal, a continuous descending step signal and a continuous ascending step signal is shown in Figure 6 , Figure 7 and Figure 8 . For comparative analysis, the control effects of the classic PID and feedforward inverse compensation PID of the comparison group are adjusted to the best, and it can be seen from Figure 6 , Figure 7 and Figure 8 that the modeling and closed-loop displacement control method of the shape memory alloy driver described in the application has the fastest response speed and the smallest tracking error. Therefore, from the running results of the embodiment, it can be seen that the control method designed in the application has the optimal control effect compared with the classic PID and feedforward inverse compensation PID in various working conditions such as back and forth, continuous heating deformation, continuous cooling deformation, etc., and realizes strong robustness to system hysteresis and parameter uncertainty.

[0151] ​​​​It should be noted that the above-described embodiments are merely preferred embodiments of the present application. For those skilled in the art, some modifications, improvements and equivalent replacements can be made to the present application without departing from the principles of the present application, and these modifications, improvements and equivalent replacements are also considered to fall within the protection scope of the claims of the present application.

Claims

1. A modeling and closed-loop displacement control method of a shape memory alloy actuator, comprising the following steps: S1: establishing a mechanism model of the shape memory alloy actuator based on the working principle of the shape memory alloy actuator, and establishing a mathematical model, i.e., a forward model, of the shape memory alloy actuator in a modular manner according to the mechanism model, wherein the forward model takes current as input and shape memory alloy wire strain as output, and comprises a heat transfer and temperature module, a phase change hysteresis module, a strain module and a resistance module; S2: establishing an inverse model of the shape memory alloy actuator by inverting the forward model, wherein the inverse model is derived in a modular manner, takes expected strain as input and calculates the required current as output, and the high nonlinearity, large hysteresis and slow cooling hysteresis of the phase change process in the shape memory alloy are compensated before the shape memory alloy actuator meets them, so that the inverse model is used for inverse hysteresis feedforward compensation control; S3: design an adaptive sliding mode controller based on proportional-integral-derivative tuning as a feedback control part for realizing precise displacement control of the shape memory alloy actuator, wherein, The control signal is composed of a sliding control signal and a sliding equivalent control signal, a proportional-integral-derivative tuning controller is used to approximate the sliding equivalent control signal, and the stability of the closed-loop feedback control system is proved by using Lyapunov's law; S4: combining the forward model in step S1, the inverse model in step S2 and the adaptive sliding mode controller based on proportional-integral-derivative tuning in step S3 to build a control system for closed-loop displacement control of the shape memory alloy actuator; S5: using Matlab / Simulink software to simulate and build the control system in step S4 and test it, and observing the closed-loop displacement control effect of the shape memory alloy actuator by selecting different tracking signals.

2. The method of modeling and closed loop displacement control of a shape memory alloy actuator of claim 1, wherein, The step S1 comprises: S1.1: The shape memory alloy actuator adopts spring bias driving, and is composed of a bias spring and a shape memory alloy wire, one end of the spring and the shape memory alloy wire is fixed in a fixed end, when the shape memory alloy wire is powered and heated, the material thereof shrinks due to shape memory effect to generate tension, when the tension exceeds the spring force, the displacement of the end of the shape memory alloy wire changes accordingly, and when cooled, the shape memory alloy wire expands with the aid of the bias spring, and the purpose of accurately controlling the displacement of the shape memory alloy actuator is achieved by accurately controlling the current input to the shape memory alloy wire; S1.2: The heat transfer and temperature module is used to describe the heat transfer problem of the shape memory alloy wire in the shape memory alloy actuator, and the heat transfer and temperature module model is given by the following expression (1), wherein the current is input and the temperature of the shape memory alloy wire is output: （1） wherein is the internal resistance of the alloy wire, , are the cross-sectional diameter and the length of the alloy wire, respectively, before deformation, is the ambient temperature, is the mass density of the alloy wire, is the specific heat of the alloy wire, is the convective heat transfer coefficient of the alloy wire, and are functions of the temperature . S1.3: The phase transformation hysteresis module is used to describe the martensite volume fraction of the shape memory alloy wire in the shape memory alloy actuator The relationship between the temperature and the stress, using the Duhem differential hysteresis model, the phase transformation hysteresis model is given by the following expression (2): （2） wherein is a Gaussian probability density function describing the slope of the hysteresis loop, is the integral of describing the shape of the hysteresis loop, the indices + and - denote the increasing and decreasing curve, respectively, is the mean value, is the variance, which is used to control the shape of the hysteresis loop, is the error function when integrating the normal distribution. S1.4: the strain module is used to describe the strain of the shape memory alloy wire in the shape memory alloy actuator The relationship between the martensite volume fraction The total strain can be approximated as a polynomial in ​ （3） wherein the strain induced by the pre-tension load in the fully austenitic state of the alloy wire, the value of which is related to the specific shape memory alloy wire; S1.5: said electrical resistance module is used to describe the electrical resistance of the shape memory alloy wire in the shape memory alloy actuator, for providing to step S1.2 to perform the calculation, given by the following equation (4), the temperature calculated from steps S1.2, S1.3 and S1.4 , the martensite volume fraction and the strain the electrical resistance of the shape memory alloy wire is calculated: （4） in, It is the resistivity of austenite. It is the resistivity of martensite. , , , , and ( ) is a relevant constant.

3. The method of modeling and closed loop displacement control of a shape memory alloy actuator of claim 1, wherein, The step S2 comprises: Let the desired strain trajectory be , the inverse model of the shape memory alloy driver is obtained by sequentially inverting the positive model of the shape memory alloy driver to obtain the inverse models of the strain module, the phase change hysteresis module, the heat transfer and temperature module and the resistance module, as shown in the following formula (5), thereby obtaining the inverse model of the shape memory alloy driver with the desired strain as the input and the required current calculated by the inverse model as the output, through the inverse model of the shape memory alloy driver, the high nonlinearity, large hysteresis and slow cooling hysteresis from the phase change process inside the shape memory alloy will be compensated before the shape memory alloy driver encounters, and the inverse model is thus used for inverse hysteresis feedforward compensation control; （5） wherein is any small constant to avoid computational overflow caused by a zero denominator.

4. The method of modeling and closed loop displacement control of a shape memory alloy actuator of claim 1, wherein, The step S3 comprises: An adaptive sliding mode controller based on proportional-integral-derivative tuning is designed, and the control variable is composed of a sliding control signal and a sliding equivalent control signal: wherein, is a sliding control signal, which switches the system to a sliding surface; is a sliding equivalent control; a proportional-integral-derivative tuning controller is used to approximate the sliding equivalent control signal, and the stability of the closed-loop feedback control system is proved by using Lyapunov's law. The specific design process is: S3.1: Mathematical description of the shape memory alloy actuator: （6） wherein denotes the displacement and velocity of the end of the shape memory alloy wire in the shape memory alloy actuator; denotes the change of the internal resistance of the shape memory alloy and is the gain factor of the input ; is a nominal parameter of the model, denotes the phase transition of the shape memory alloy and describes the hysteresis of the shape memory alloy; denotes an external disturbance; is the input; , and are bounded variables, , , ; is the maximum change of the hysteresis curve; is the maximum value of the disturbance; S3.2: Designing the integral sliding surface To improve the convergence performance of the sliding mode surface, a reference signal is defined such that: （7） wherein, represents the desired displacement error, ; ; and the roots of are in the left half-plane of the s-plane; Design integral sliding mode surface , into equation (7), Can be rewritten as When the sliding mode occurs, Tends to zero, and the introduction of integral link and high order form in sliding mode function can effectively improve the system control performance; S3.3: Stability analysis of the closed-loop feedback control system: The Lyapunov function is defined as Then In order to realize the system stability , the control u is designed to satisfy That is, the derivative of the sliding surface can be expressed as: （8） Therefore, the control quantity is: （9） Control quantity u Can be divided into two parts: ; Is the sliding control signal, switching the system to the sliding surface, ; Is the sliding mode equivalent control, ; S3.4: Sliding mode equivalent control signal Design: The PID tuning controller is used to approximate the equivalent control: （10） For the tuning of PID parameters, the following objective function is selected And the update rule of PID parameters is obtained by using gradient descent method: (11) Since when there exists Therefore, the PID parameter updating rule can be optimized as follows: (12) S3.5: slide control signal Design: From equations (8), (9) and (10), it can be deduced that: （13） wherein, is positive, so if one chooses K to satisfy the following condition, then the system is certainly asymptotically stable iff the equality holds. （14） In order to weaken the high frequency chattering inherent in the control input, a saturation function is selected substitution , is the width of the boundary layer, the slip surface function with arbitrary initial value will reach and remain in the boundary layer , the slip control signal becomes: （15）。 5. The method of modeling and closed loop displacement control of a shape memory alloy actuator of claim 1, wherein, The step S4 comprises: S4.1: the output of the inverse model of the shape memory alloy actuator established in step S2 is the current required by the shape memory alloy actuator to achieve the desired strain, which is input into the positive model in the control system as a feedforward signal to compensate for the influence of the highly nonlinear, large hysteresis and slow cooling hysteresis originating from the phase change process of the shape memory alloy on the shape memory alloy actuator; S4.2: Step S3 is designed based on the proportion-integral-differential setting of adaptive sliding mode controller, and the control quantity u is composed of sliding control signal and sliding mode equivalent control signal: ; this part is input into the mathematical model of the shape memory alloy driver as a feedback control signal; S4.3: combine the positive model in step S1, the inverse model in step S2 and the adaptive sliding mode controller based on proportional-integral-derivative tuning in step S3 to build a control system for closed-loop displacement control of the shape memory alloy actuator for precise control of the displacement of the shape memory alloy actuator.

6. The method of modeling and closed loop displacement control of a shape memory alloy actuator of claim 1, wherein, The step S5 comprises: S5.1: according to the derivation of steps S1, S2, S3 and S4, build a Simulink simulation platform; S5.2: input a sinusoidal signal to simulate the sinusoidal tracking condition of the shape memory alloy actuator and observe the tracking of the output displacement of the shape memory alloy actuator; S5.3: input a continuous descending step signal to simulate the tracking condition of the shape memory alloy to the step command signal in the heating and shrinking process and observe the tracking of the output displacement of the shape memory alloy actuator; S5.4: input a continuous ascending step signal to simulate the tracking condition of the shape memory alloy to the step command signal in the recovery process and observe the tracking of the output displacement of the shape memory alloy actuator.

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