Piezoelectric driving system constraint control method based on predetermined time observer

Abstract

The invention discloses a piezoelectric driving system constraint control method based on a predetermined time observer, and the method comprises the steps: designing a predetermined time interference observer in combination with a predetermined time performance function, and guaranteeing that an interference estimation error is converged to a bounded range within a predetermined time; a state conversion function is designed based on the obstacle Lyapunov function, so that the system output can also realize preset time output constraint under the condition that the initial state does not meet the constraint condition; a predetermined time adaptive controller is designed by using a neural network approximation technology, a dynamic surface control technology, a constraint control technology and a predetermined time interference observer technology, so that a closed-loop system realizes predetermined time stability. The problem that an existing disturbance observer cannot be used for preset time control is solved, and the assumption that initial conditions of constrained variables need to meet constraint conditions in constraint control is avoided.

Description

Technical Field

[0001] This invention relates to the field of anti-interference control technology, and in particular to a constraint control method for a piezoelectric drive system based on a predetermined time observer. Background Technology

[0002] In actual industrial manufacturing, drive systems composed of piezoelectric ceramics as the basic material are subject to a variety of factors that affect their accuracy. The inherent hysteresis nonlinearity of piezoelectric ceramics and external interference of the drive system are the most widespread influencing factors. The proper handling of these influencing factors has been a top priority in improving the accuracy of piezoelectric drive systems in recent years. On the other hand, system response time has always been an important indicator of controller performance, especially in piezoelectric drive systems. How to ensure that the drive system can respond quickly while also being able to converge quickly from any position to a suitable operating range is an urgent problem to be solved. Summary of the Invention

[0003] This invention aims to at least partially solve one of the technical problems in related technologies. To this end, the first objective of this invention is to propose a constraint control method for a piezoelectric drive system based on a predetermined time observer, addressing the problem that existing disturbance observers cannot be used for predetermined time control, and avoiding the assumption in constraint control that the initial conditions of the constrained variables must satisfy the constraint conditions.

[0004] To achieve the above objectives, a first aspect of the present invention proposes: 1. A constraint control method for a piezoelectric drive system based on a predetermined time observer, the method comprising: S1, transform the piezoelectric drive system model into a standard form of state equation, and define the control objectives of the system based on the state equation. The control objectives include the system output meeting the preset constraints within a predetermined time and the closed-loop system remaining stable within a predetermined time. S2, for the unknown composite disturbance in the state equation, combine neural network approximation technology with a predetermined time performance function to construct a predetermined time disturbance observer so that the estimation error of the unknown composite disturbance converges to a bounded region near zero within a predetermined time. S3. To address the system's output constraints, a state transition function is designed based on the obstacle Lyapunov function. This function performs a state transition on the first subsystem of the state equation, mapping the constrained original output tracking error variable to an unconstrained new state variable. This transforms the output constraint control problem into a stable control problem for the new state variable. The state transition function is configured to allow the initial state of the system output to exceed the preset constraint boundary and ensures that the transformed system satisfies the output constraints within a predetermined time. S4. Based on the disturbance estimate provided by the predetermined time disturbance observer in step S2 and the unconstrained new state variable obtained after transformation in step S3, and by integrating dynamic surface control technology to handle system nonlinearity, a predetermined time adaptive controller is designed. The predetermined time adaptive controller drives the piezoelectric drive system through the generated control signal, so that all signals of the entire closed-loop system are bounded within a predetermined time, and the system output stabilizes and satisfies preset constraints within the predetermined time set by the user.

[0005] In addition, the piezoelectric drive system constraint control method based on a predetermined time observer according to the above embodiments of the present invention may also have the following additional technical features: According to one embodiment of the present invention, the piezoelectric drive system model is as follows: (1) in, For the quality of the operating mechanism, For the stiffness of the running mechanism, To output displacement, For speed, For acceleration, The damping coefficient is... The transformation ratio of the electromechanical converter; , This represents the sum of all piezoelectric ceramic capacitors. For a voltage amplifier with a fixed gain, This represents the actuator output affected by hysteresis nonlinearity. To control the input signal, This is due to unknown external interference. make , Then formula (1) becomes: (2) in, , .

[0006] According to an embodiment of the present invention, in step S2, constructing a predetermined time interference observer includes the following steps: S21, Combining with a neural network, construct a composite interference, simplifying the second subsystem in formula (2) into the following form: (3) in, Composite interference , , This represents the optimal weight vector for the neural network. A vector of basis functions; The approximation error of the neural network is a bounded variable. The error between the hysteresis output and the control input, and the composite disturbance. Its first derivative is bounded, that is, it satisfies , It is an unknown positive number.

[0007] S22, Construct a predetermined time interference observer. According to the form of formula (3), construct a predetermined time interference observer in the following form: (4) in, To interfere with the observer gain at a predetermined time, , This is a vector composed of the optimal weight estimates of the neural network. This is the estimated value of the composite interference. As an auxiliary variable for the interference observer, A time-performance function is defined as follows: (5) in, ,therefore ,and It is monotonically decreasing and twice continuously differentiable. The expression for the first derivative is: (6) The expression for the second derivative is: (7) therefore, It is bounded. It is also bounded.

[0008] According to an embodiment of the present invention, in step S3, the state transition includes: For formula (2), for the state variables Perform the following transformations: definition: (8) in, This represents a second-order continuously differentiable reference signal, and Both its first and second derivatives are bounded; and Represents a constant constraint, and , ; The definition of is: therefore, It is a monotonically increasing function that is continuously differentiable of the second order.

[0009] According to an embodiment of the present invention, in step S4, designing a predetermined time adaptive controller includes the following steps: S41, define the following coordinate transformation: (9) in, For state error, It is the filter output signal. It is both a virtual controller and a filter input signal. It is filtering error. and It is the optimal weight vector of the neural network. and These are the estimated values ​​of the optimal weight vector of the neural network. and It is the estimation error of the optimal weight vector of the neural network. This is the estimation error of the observer that interferes at the predetermined time; For the first subsystem of formula (2), combined with the state transition in step S3, we get: (10) in, , ; Find the formula (10) The first-order time derivative is obtained as follows: (11) in, ; Combining neural networks, formula (11) simplifies to: (12) in, , For the basis function vectors of the neural network, This represents the approximation error of the neural network. Design a virtual controller in the following form. and adaptive update law : (13) in, and For design parameters, ; Due to virtual controller The derivative calculation is quite complex. To avoid this calculation, the following filter is introduced: (14) in, and For design parameters, ; The Lyapunov function is defined as follows: (15) Taking the first-order time derivative of formula (15) and substituting formulas (12)-(14) into it, we get: (16) According to Yang's inequality, we obtain the following inequality: (17) in, This is the upper bound of the neural network approximation error. for The upper bound; Substituting formula (17) into formula (16) yields: (18) in, ; S42, according to formula (3) and formula (9), we get: (19) Based on the predetermined time interference observer designed in step S22, we obtain: (20) therefore: (twenty one) According to Yang's inequality, we get: (twenty two) Therefore, formula (21) simplifies to: (twenty three) in, ; Design a controller and an adaptive update law of the following form: (twenty four) The Lyapunov function is defined as follows: (25) Differentiating formula (25) and combining it with formulas (19)-(24), we get: (26) in, and For the parameters to be designed, , ; Combining Young's inequality, we get: (27) Substituting formula (27) into formula (26) yields: (28) in, .

[0010] According to one embodiment of the present invention, proving the stability of a closed-loop system at a predetermined time includes the following steps: S43, Analysis and verification of time-stability, defined by the following Lyapunov function: (29) Differentiating equation (29) and substituting the virtual controller, controller, and adaptive update law, we obtain the following result: (30) in, ; definition: (31) Then formula (30) simplifies to the following form: (32) The selection of parameters satisfies ; Based on the final simplified result of formula (32), the following is performed: Taking its first-order time derivative, we get: (33) Integrating both sides simultaneously, we get: (34) Therefore, when hour, , Therefore, the closed-loop system will complete the task at the predetermined time. The variable converges inward to a bounded range. The scheduled time is stable; due to It is a constant vector. It is a bounded variable, so It also has a stable scheduled time; and virtual control. It is about The function, so The scheduled time is stable; according to ,and It is about The function, so the variable It also has a stable scheduled time; controller It is about The function, so the controller It is also time-stable; therefore, all variables within the closed-loop system are time-stable; according to The stability of the scheduled time, It will be within the scope of constraints, regardless of Does the initial condition satisfy the constraint condition? hour, It will always remain within the constraints; therefore, the control objective is achieved.

[0011] Beneficial effects: 1. Compared with the prior art, the significant advantage of this invention lies in the design of a predetermined time disturbance observer, which can be used in the design of asymptotic, finite time, fixed time and predetermined time controllers. Its disturbance estimation error will converge to a bounded range within a predetermined time. In addition, this invention integrates external disturbances, hysteresis nonlinearity and neural network approximation error of the piezoelectric drive system into a composite disturbance and uses the predetermined time disturbance observer designed in this invention to estimate it. The estimation result is embedded into the controller design, which can effectively reduce the adverse effects of composite disturbances on the piezoelectric drive system and obtain better control performance.

[0012] 2. By constructing new state transitions, the control method designed in this invention can achieve output constraints within a predetermined time even when the initial value of the output does not meet the constraint conditions. Therefore, compared with existing technologies, the control method of this invention has better applicability.

[0013] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0014] Figure 1 This is a flowchart of a constraint control method for a piezoelectric drive system based on a predetermined time observer according to an embodiment of the present invention; Figure 2 This is a flowchart illustrating the design of a constraint control method for a piezoelectric drive system based on a predetermined time observer, according to an embodiment of the present invention. Figure 3 The tracking curve (a) of a simulation system according to an embodiment of the present invention; Figure 4 The tracking error curve (a) of a simulation system according to an embodiment of the present invention; Figure 5 The state variables of a simulation system according to an embodiment of the present invention A curve graph; Figure 6This is a control input curve of a simulation system according to an embodiment of the present invention; Figure 7 These are the estimated values ​​of the composite interference and predetermined time interference observers in the simulation system of one embodiment of the present invention; Figure 8 This refers to the interference estimation error of a predetermined time interference observer in a simulation system according to an embodiment of the present invention. Figure 9 The tracking curve (b) of the simulation system according to an embodiment of the present invention; Figure 10 The tracking error curve (b) of a simulation system according to an embodiment of the present invention is shown. Detailed Implementation

[0015] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0016] The following description, with reference to the accompanying drawings, describes a constraint control method for a piezoelectric drive system based on a predetermined time observer, as proposed in an embodiment of the present invention.

[0017] like Figure 1 As shown, the piezoelectric drive system constraint control method based on a predetermined time observer according to an embodiment of the present invention may include the following steps: S1 transforms the piezoelectric drive system model into a standard form of state equations, and defines the control objectives of the system based on the state equations. The control objectives include the system output meeting preset constraints within a predetermined time and the closed-loop system remaining stable within a predetermined time.

[0018] According to one embodiment of the present invention, the piezoelectric drive system model is as follows: (1) in, For the quality of the operating mechanism, For the stiffness of the running mechanism, To output displacement, For speed, For acceleration, The damping coefficient is... The transformation ratio of the electromechanical converter; , This represents the sum of all piezoelectric ceramic capacitors. For a voltage amplifier with a fixed gain, This represents the actuator output affected by hysteresis nonlinearity. To control the input signal, This is due to unknown external interference. make , Then formula (1) becomes: (2) in, , .

[0019] Specifically, this invention proposes a constraint control method for a piezoelectric drive system based on a predetermined-time observer. The key challenge is designing a predetermined-time disturbance observer that satisfies stability at a predetermined time and implementing output constraints without initial condition limitations. The control method of this invention aims to achieve the following control objectives for the piezoelectric drive system under conditions of external disturbances and hysteresis nonlinearity: Objective 1: The closed-loop system achieves stability within a predetermined time. Objective 2: The unknown composite disturbance is estimated by the disturbance observer and compensated into the controller, and the disturbance estimation error converges to a bounded range within a predetermined time. Objective 3: The system output can achieve the predetermined time output constraint even if the initial state does not meet the constraint conditions.

[0020] S2, for the unknown composite disturbance in the state equation, combine neural network approximation technique with a predetermined time performance function to construct a predetermined time disturbance observer so that the estimation error of the unknown composite disturbance converges to a bounded region near zero within a predetermined time.

[0021] According to an embodiment of the present invention, in step S2, constructing a predetermined time interference observer includes the following steps: S21, Combining with a neural network, construct a composite interference, simplifying the second subsystem in formula (2) into the following form: (3) in, Composite interference , , This represents the optimal weight vector for the neural network. A vector of basis functions; The approximation error of the neural network is a bounded variable. The error between the hysteresis output and the control input, and the composite disturbance. Its first derivative is bounded, that is, it satisfies , It is an unknown positive number.

[0022] S22, Construct a predetermined time interference observer. According to the form of formula (3), construct a predetermined time interference observer in the following form: (4) in, To interfere with the observer gain at a predetermined time, , This is a vector composed of the optimal weight estimates of the neural network. This is the estimated value of the composite interference. As an auxiliary variable for the interference observer, A time-performance function is defined as follows: (5) in, ,therefore ,and It is monotonically decreasing and twice continuously differentiable. The expression for the first derivative is: (6) The expression for the second derivative is: (7) therefore, It is bounded. It is also bounded.

[0023] S3, for the output constraint requirements of the system, a state transition function is designed based on the obstacle Lyapunov function, and the state transition is performed on the first subsystem of the state equation, mapping the constrained original output tracking error variable to an unconstrained new state variable, thereby transforming the output constraint control problem into a stable control problem of the new state variable; the state transition function is configured to allow the initial state of the system output to exceed the preset constraint boundary, and to ensure that the transformed system satisfies the output constraint within a predetermined time.

[0024] According to an embodiment of the present invention, in step S3, the state transition includes: For formula (2), for the state variables Perform the following transformations: definition: (8) in, This represents a second-order continuously differentiable reference signal, and Both its first and second derivatives are bounded; and Represents a constant constraint, and , ; The definition of is: therefore, It is a monotonically increasing function that is continuously differentiable of the second order.

[0025] When variables When it is proven to be bounded, Will be in and Within the constrained range. In previous constraint controls, the constrained variables needed to be... The initial conditions are satisfied beforehand, while in this invention, when... When the initial value does not satisfy the constraint conditions, because Properties of functions It will satisfy the constraints, and because Therefore, in time hour, Therefore, the output quantity The constraints will be met, meaning that if the initial value of the constrained variable does not meet the constraints, the event will occur within a predetermined time. And for the time thereafter, the constrained variable will always satisfy the constraint conditions.

[0026] S4. Based on the disturbance estimate provided by the predetermined time disturbance observer in step S2 and the unconstrained new state variable obtained after transformation in step S3, and by integrating dynamic surface control technology to handle system nonlinearity, a predetermined time adaptive controller is designed. The predetermined time adaptive controller drives the piezoelectric drive system through the generated control signal, so that all signals of the entire closed-loop system are bounded within a predetermined time, and the system output stabilizes and satisfies preset constraints within the predetermined time set by the user.

[0027] According to an embodiment of the present invention, in step S4, designing a predetermined time adaptive controller includes the following steps: S41, define the following coordinate transformation: (9) in, For state error, It is the filter output signal. It is both a virtual controller and a filter input signal. It is filtering error. and It is the optimal weight vector of the neural network. and These are the estimated values ​​of the optimal weight vector of the neural network. and It is the estimation error of the optimal weight vector of the neural network. This is the estimation error of the observer that interferes at the predetermined time; For the first subsystem of formula (2), combined with the state transition in step S3, we get: (10) Find the formula (10) The first-order time derivative is obtained as follows: (11) in, ; Combining neural networks, formula (11) simplifies to: (12) in, , For the basis function vectors of the neural network, This represents the approximation error of the neural network. Design a virtual controller in the following form. and adaptive update law : (13) in, and For design parameters, ; Due to virtual controller The derivative calculation is quite complex. To avoid this calculation, the following filter is introduced: (14) in, and For design parameters, ; The Lyapunov function is defined as follows: (15) Taking the first-order time derivative of formula (15) and substituting formulas (12)-(14) into it, we get: (16) According to Yang's inequality, we obtain the following inequality: (17) in, This is the upper bound of the neural network approximation error. for The upper bound; Substituting formula (17) into formula (16) yields: (18) in, ; S42, according to formula (3) and formula (9), we get: (19) Based on the predetermined time interference observer designed in step S22, we obtain: (20) therefore: (twenty one) According to Yang's inequality, we get: (twenty two) Therefore, formula (21) simplifies to: (twenty three) in, ; Design a controller and an adaptive update law of the following form: (twenty four) The Lyapunov function is defined as follows: (25) Differentiating formula (25) and combining it with formulas (19)-(24), we get: (26) in, and For the parameters to be designed, , ; Combining Young's inequality, we get: (27) Substituting formula (27) into formula (26) yields: (28) in, .

[0028] According to one embodiment of the present invention, proving the stability of a closed-loop system at a predetermined time includes the following steps: S43, Analysis and verification of time-stability, defined by the following Lyapunov function: (29) Differentiating equation (29) and substituting the virtual controller, controller, and adaptive update law, we obtain the following result: (30) in, ; definition: (31) Then formula (30) simplifies to the following form: (32) The selection of parameters satisfies ; Based on the final simplified result of formula (32), the following is performed: Taking its first-order time derivative, we get: (33) Integrating both sides simultaneously, we get: (34) Therefore, when hour, , Therefore, the closed-loop system will complete the task at the predetermined time. The variable converges inward to a bounded range. The scheduled time is stable; due to It is a constant vector. It is a bounded variable, so It also has a stable scheduled time; and virtual control. It is about The function, so The scheduled time is stable; according to ,and It is about The function, so the variable It also has a stable scheduled time; controller It is about The function, so the controller It is also time-stable; therefore, all variables within the closed-loop system are time-stable; according to The stability of the scheduled time, It will be within the scope of constraints, regardless of Does the initial condition satisfy the constraint condition? hour, It will always remain within the constraints; therefore, the control objective is achieved. Thus, all the control objectives mentioned in step S1 of this invention have been achieved.

[0029] The effectiveness of the proposed control method will now be verified using a piezoelectric drive system. The parameters selected are: control gain... Neural network weight update parameters Predetermined time interference observer gain Filter parameters The relevant parameters for stable system scheduled times are set as follows: The system output constraint's predetermined time is The output constraints are set to Reference signal ,interference In the simulation of this invention, the choice of hysteresis nonlinearity is as follows: The initial value of the variable is , .

[0030] The design block diagram of the control method of this invention is as follows: Figure 2 As shown, the simulation results are as follows: Figure 3-10 , Figure 3 and Figure 9 These are the initial output values. and The tracking curve Figure 4 and Figure 10 The tracking error is due to two different initial values. Simulation results show that even when the initial value of the system output does not meet the constraints, the system output can still achieve the expected result within the predetermined time. It enters the constraint range. Figure 5 It is a system state variable The curve, Figure 6 It is the curve of the system control input. Figure 7 This is the operating curve of the predetermined time interference observer designed in this invention. Figure 8 It is interference estimation error. Figure 7 and Figure 8 It can be seen that this predetermined time disturbance observer has the ability to estimate unknown composite disturbances and the disturbance estimation error is stable within the predetermined time. From all the simulation results, it can be seen that the closed-loop system is stable within the predetermined time, and the control objective of this invention can be achieved.

[0031] In the description of this specification, references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0032] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0033] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components, unless otherwise explicitly limited. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0034] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. A method for constrained control of a piezoelectric driving system based on a predetermined time observer, characterized by, The method comprises: S1, converting a piezoelectric driving system model into a state equation in a standard form, and determining a control target of the system based on the state equation, the control target comprising that a system output satisfies a preset constraint within a predetermined time and a closed-loop system is stable within the predetermined time; S2, constructing a predetermined time disturbance observer by combining a neural network approximation technique and a predetermined time performance function for unknown compound disturbances existing in the state equation, so that an estimation error of the unknown compound disturbances converges to a bounded region near zero within the predetermined time; S3, designing a state conversion function based on a barrier Lyapunov function for an output constraint requirement of the system, and performing state conversion on a first subsystem of the state equation, so as to map a constrained original output tracking error variable to an unconstrained new state variable, thereby converting an output constraint control problem into a stable control problem of the new state variable; the state conversion function is configured to allow an initial state of the system output to exceed the preset constraint boundary and ensure that the converted system satisfies the output constraint within the predetermined time; S4, designing a predetermined time adaptive controller based on disturbance estimation values provided by the predetermined time disturbance observer in step S2, the unconstrained new state variable obtained after conversion in step S3, and fusion of a dynamic surface control technique to process system nonlinearity; the predetermined time adaptive controller drives the piezoelectric driving system through a generated control signal, so that all signals of the entire closed-loop system are bounded within the predetermined time, and the system output realizes stability and satisfies the preset constraint within the predetermined time set by a user in advance.

2. The piezoelectric driving system constrained control method based on a predetermined time observer according to claim 1, characterized by, The piezoelectric driving system model is: （1） wherein, is the mass of the operating mechanism, is the stiffness of the operating mechanism, is the output displacement, is the velocity, is the acceleration, is the damping coefficient, is the voltage ratio possessed by the electromechanical converter; , represents the sum of all piezoceramic capacitances, is the fixed gain of the voltage amplifier, denotes the actuator output affected by the hysteresis nonlinearity, is the control input signal, is the unknown external disturbance; Let , , then equation (1) transforms into: （2） wherein , .

3. The piezoelectric driving system constrained control method based on a predetermined time observer according to claim 2, characterized by, In step S2, the predetermined time disturbance observer is constructed, comprising the following steps: S21, a compound disturbance is constructed by combining a neural network, and the second subsystem in formula (2) is simplified into the following form: （3） where , composite disturbance , , is the optimal weight vector of the neural network, is the basis function vector; is the approximation error of the neural network, which is a bounded variable; is the error between the hysteresis output and the control input, the composite disturbance and its first derivative are bounded, that is, they satisfy , is an unknown positive constant. S22, a predetermined time disturbance observer is constructed, and according to the form of formula (3), a predetermined time disturbance observer in the following form is constructed: （4） where is the scheduled time disturbance observer gain, , is the vector of neural network optimal weight estimates, is the composite disturbance estimate, is the auxiliary variable of the disturbance observer, is a scheduled time performance function defined as follows: （5） wherein Thus And is monotonically decreasing and twice continuously differentiable, The expression of the first derivative of （6） The expression for the second derivative is: （7） Thus, is bounded, is also bounded.

4. The piezoelectric driving system constrained control method based on a predetermined time observer according to claim 3, characterized by, In step S3, the state conversion comprises: For equation (2), the following transformation is made to the state variable ​ Definition: （8） wherein, represents a second order continuously derivable reference signal, and and its first and second derivatives are bounded; and represents a constant constraint, and , ; is defined as: Thus, is a second order continuously differentiable monotonically increasing function.

5. The piezoelectric driving system constrained control method based on a predetermined time observer according to claim 4, characterized by, In step S4, the predetermined time adaptive controller is designed, comprising the following steps: S41, the following coordinate transformation is defined: （9） wherein, is a state error, is a filter output signal, is a virtual controller and also a filter input signal, is a filtering error, and is a neural network optimal weight vector, and are, respectively, an estimate of the neural network optimal weight vector, and is an estimation error of the neural network optimal weight vector, is an estimation error of the predetermined time disturbance observer; For the first subsystem of formula (2), combined with the state transformation in step S3, the following is obtained: （10） wherein , ; Taking the first time derivative of equation (10) gives: ​ （11） wherein ; Combined with a neural network, formula (11) is simplified as: （12） wherein, , is a neural network basis function vector, is a neural network approximation error; A virtual controller is designed in the form and an adaptive update law : （13） wherein and are design parameters, ; Due to the derivative calculation of the virtual controller is complex, in order to avoid this calculation, the following filter is introduced: （14） wherein and are design parameters, ; The following Lyapunov function is defined: （15） The first-order time derivative of formula (15) is obtained and formula (12)-(14) is substituted into it, to obtain: （16） According to Young's inequality, the following inequality is obtained: （17） wherein, is an upper bound on the neural network approximation error, is an upper bound on the neural network approximation error. Substitute formula (17) into formula (16) to obtain: （18） wherein ; According to formula (3) and formula (9), the following is obtained: （19） According to the predetermined time disturbance observer designed in step S22, the following is obtained: （20） Therefore: （21） According to Young's inequality, the following is obtained: （22） Therefore, formula (21) is simplified as: （23） wherein ; The following controller and adaptive update law are designed: （24） The following Lyapunov function is defined: （25） The derivative of formula (25) is obtained and combined with formula (19)-(24) to obtain: （26） wherein and are parameters to be designed, , ; Combined with Young's inequality, the following is obtained: （27） Substitute formula (27) into formula (26) to obtain: （28） wherein .

6. The piezoelectric driving system constrained control method based on a predetermined time observer according to claim 5, characterized by, The proof of the predetermined time stability of the closed loop system includes the following steps: S43, the analysis verification of the predetermined time stability, defines the Lyapunov function as follows: （29） Derivate formula (29) and substitute the virtual controller, the controller and the adaptive update law, and then arrange and calculate to obtain: （30） wherein ; Definition: （31） Then formula (30) is simplified as follows: （32） wherein the parameters are chosen to satisfy ; According to the final simplified result of formula (32), the first time derivative of is obtained as follows: （33） Integrate both sides to obtain: （34） Thus, when , , , the closed-loop system converges to a bounded range within a predetermined time , and the variable is predetermined time stable; since is a constant vector, is a bounded variable, so is also predetermined time stable; and the virtual control is a function of , so is predetermined time stable; according to , and is a function of , so the variable is also predetermined time stable; the controller is a function of , so the controller is also predetermined time stable; thus, all variables within the closed-loop system are predetermined time stable; according to the predetermined time stability of , will be within the constraint range, regardless of whether the initial condition of satisfies the constraint condition, when , will always be within the constraint range; thus, the control objective is achieved.

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