A specified time control method for a PMSM system with output constraint

Abstract

This invention discloses a specified-time control method for PMSM systems with output constraints, solving the specified-time control problem of permanent magnet synchronous motor systems with asymmetric output constraints. First, this invention proposes a novel specified-time Lyapunov stability theory for systems with asymmetric constraints and establishes a specified-time backstepping control design method based on a novel barrier Lyapunov function. This method, by introducing coordinate transformation and a time-varying gain function, and combining specified-time stability theory with backstepping control, designs a specified-time state feedback controller for the permanent magnet synchronous motor system. This controller can strictly limit the output of the permanent magnet synchronous motor system within the constraint boundaries and achieve convergence of the system to an arbitrarily small neighborhood near zero within any specified time. Finally, simulation examples verify the effectiveness of the controller.

Description

Technical Field

[0001] This invention pertains to backstepping control technology, specifically a time-defined control method for a PMSM (Permanent Magnet Synchronous Motors) system with output constraints. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs) are nonlinear systems with strong coupling, high system order, and numerous parameters. In recent years, due to the rapid development of permanent magnet material technology, their comprehensive performance has been greatly improved, leading to their widespread application in robot motion control and aerospace. With the development of control theory, many control methods have been introduced into the research of position servo control for PMSMs. Shen JX et al. (Variable parameter sliding mode control of PMSM based on recursive least squares observer. Proceedings of the CSEE, 2022, 42(18):6835-6845) addressed the chattering problem in traditional sliding mode control by designing a variable parameter sliding mode controller for PMSM based on a least squares observer. Huang et al. proposed in Active disturbances rejection controller for position servo control of PMSM. 2019 22nd International Conference on Electrical Machines and Systems (ICEMS). IEEE, 2019: 1-4 that they studied an active disturbance rejection controller for PMSMs to improve system dynamic performance. In addition, Chen ZY et al. proposed a backstepping controller for permanent magnet synchronous motors based on command filtering technology in their paper "Servo Control of Three-Phase Permanent Magnet Synchronous Motors Based on Filtered Backstepping Method. Control Theory and Applications, 2017, 34(4): 515-524", which improved the convergence accuracy.

[0003] The above control methods have improved the steady-state performance of permanent magnet synchronous motor systems to a certain extent, but most of the results are asymptotically convergent and cannot achieve high accuracy. However, in many engineering problems, the accuracy requirements of the system are extremely high. In order to pursue higher convergence accuracy, researchers have proposed the concept of finite-time stability. Zhang H L et al. studied the finite-time stability problem of permanent magnet synchronous motors and designed its finite-time controller in "Anti-disturbance control of permanent magnet synchronous motors based on dual-mode finite-time sliding mode" [J]. Packaging Engineering, 2024, 45(05): 188-196. However, the convergence time of finite-time control depends on the initial conditions, which makes it difficult to meet the fast response requirements under complex working conditions. In order to overcome this defect, researchers have proposed the concept of fixed-time stability. Wang KJ et al. studied and designed a fixed-time controller for permanent magnet synchronous motors in "Fixed-time synchronization research of permanent magnet synchronous motors with uncertain parameters" [J]. Journal of Shanghai University of Engineering Science, 2023, 37(03): 228-232. However, the convergence time of fixed-time control depends on numerous design parameters, making the adjustment methods complex and often leading to overestimation. As a generalization of fixed-time control, specified-time control can bring the system to equilibrium within any predetermined time, while also possessing strong anti-interference capabilities and fast response characteristics. Notably, Song Y et al., in their paper "Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time. Automatica, 2017, 83:243–251," studied and designed a time-varying gain function that can scale the system state variables, thereby achieving specified-time control of nonlinear systems. This has become a general method for handling specified-time control. However, as time approaches the specified time, the controller gain escapes to infinity, resulting in the problem of infinite gain in the controller. Therefore, infinite gain is also a noteworthy issue in specified-time control.

[0004] Due to physical limitations, safety requirements, and performance indicators, most machines in industry operate under certain constraints. Violating these constraints can lead to unpredictable dangers. Therefore, the constraint control problem of systems has become an important research topic. Existing methods for handling output constraints include model predictive control, reference governors, and the use of set invariance. In addition, the obstacle Lyapunov function is also a common approach. By constructing a function form that tends to infinity as the system state approaches the constraint boundary, it forces the control law to amplify its regulatory effect as the state approaches the constraint, thereby maintaining system stability while strictly preventing the state from going out of bounds, achieving a dual guarantee of safety and performance. In recent years, there has been an increasing amount of research on output constraints. For example, Xiao KW studied the finite-time adaptive controller under full-state constraints of nonlinear systems in his research on adaptive inversion control method for high-precision permanent magnet synchronous motors [D]. Guizhou University, 2023. Liu Y et al. proposed a finite-time controller for a permanent magnet synchronous motor system with output constraints based on an interval type II fuzzy system in their research on finite-time adaptive interval type II fuzzy output feedback PMSM servo control with output constraints [J]. Control and Decision, 2024, 39(04):1212-1222. At present, in the research on permanent magnet synchronous motors, some studies have simply solved the problem of output constraints, and some studies have further solved the problem of output constraints under finite-time stability, but there is almost no in-depth research on output constraints under specified time control. Summary of the Invention

[0005] To address the aforementioned problems, the present invention aims to propose a specified-time control method for a PMSM system with output constraints, thereby solving the infinite gain problem in current specified-time control and the high-performance control problem under output constraints in existing permanent magnet synchronous motor control strategies, and improving the dynamic performance and robustness of the motor system.

[0006] The above objectives are achieved through the following technical solutions:

[0007] A method for time-specified control of a PMSM system with output constraints, the method comprising the following steps:

[0008] S1. Based on the permanent magnet synchronous motor system, the dynamic equation is determined as follows: in, It is the angle of the motor rotor; It is the mechanical angular velocity of the rotor; It is the moment of inertia of the motor; It is the motor load torque; It is the coefficient of viscous friction; It is electromagnetic torque;

[0009] Define state variables The dynamic equations are rewritten as follows: in, For state variables Differentiate; For state variables beg; It is the system controller input and ; ; It is the number of pole pairs of the motor; It is the magnetic flux linkage of the permanent magnet rotor; Is the stator current in The components of the axis;

[0010] S2. Introducing State Coordinate Transformation in, yes State variables after coordinate transformation; yes State variables after coordinate transformation; It is the time-varying gain function to be designed, which satisfies in, It is a specified time. It is a time switching and satisfies ; and It is an intermediate variable, and satisfies , ;exist Within the interval, the system will converge to zero; hour, It converges to a constant rather than tending to infinity, thus avoiding the problem of infinite gain;

[0011] The following dynamic equations are obtained based on the introduced coordinate transformation: in, State variables after coordinate transformation Differentiate; State variables after coordinate transformation Differentiate; It is the time-varying gain function designed by S2. Differentiate;

[0012] S3. Define the system's constraint boundaries and introduce a specified-time backstepping control design method based on the novel Barrier Lyapunov Function (BLF).

[0013] S4. Using the backstepping control method designed in step S3, design a time controller for the permanent magnet synchronous motor system.

[0014] Furthermore, the system constraints are set as follows: in, It is a set that satisfies the constraints; and It is a positive integer, representing the set output boundary and ; It is a real number vector.

[0015] Furthermore, the novel barrier Lyapunov function BLF is introduced as follows: in, It is about A novel barrier Lyapunov function;

[0016] In the backstep control design process, let the initial barrier Lyapunov function be... . exist The surface is continuous and its derivative is also continuous. right Taking the derivative, in, It is about The initial barrier Lyapunov function; State variables after coordinate transformation The set of constraints; It is an intermediate constant and satisfies ; It was designed for future convenience. The function that satisfies and .

[0017] Furthermore, the designated time controller is designed as follows: , in, and Yes State variables after coordinate transformation; satisfy ; It is a design constant and is an integer. It is a sequence The number of terms; It is during the design process of backstep control. Virtual control law and ; It is an intermediate variable in the backstep control design process and ; A positive design constant; It is a design constant, and ; It is an intermediate variable in the backstep control design process and ; A positive design constant; , , , , It is an intermediate variable in the backstep control design process and ; ; ; ; ; ; ; It is an intermediate constant and ; , , , It is an intermediate variable in the backstep control design process and ; ; , ; , , , , , It is a positive design constant.

[0018] Compared with the prior art, the present invention has the following advantages:

[0019] First, this invention proposes a new method that can realize output constraint control and convergence within a specified time for nonlinear systems;

[0020] Secondly, this invention establishes a specified-time backstepping control design method based on a novel BLF (Brain Function-Based Function). By introducing a time-varying gain function and coordinate transformation, the specified-time control theory is combined with the backstepping control method to design a specified-time backstepping controller with output constraints for a permanent magnet synchronous motor system, ensuring system convergence and output constraint effectiveness within a specified time.

[0021] Third, the design method used in this invention is the first to achieve specified time control of a permanent magnet synchronous motor system with output constraints. Attached Figure Description

[0022] Figure 1 This is a flowchart illustrating the design of a time-defined state feedback control method for a permanent magnet synchronous motor system with output constraints, according to the present invention.

[0023] Figure 2 , Figure 3 , Figure 4 The simulation group (a) of this invention is respectively , , The convergence trajectory.

[0024] Figure 5 , Figure 6 , Figure 7 The simulation group (b) of this invention is respectively , , The convergence trajectory.

[0025] Figure 8 , Figure 9 , Figure 10 These are simulation groups (c) of the present invention. , , The convergence trajectory. Detailed Implementation

[0026] To make the design concept and theory of this invention clearer, the following will provide a detailed explanation of the establishment, design, principle and proof of the state feedback controller. The following will be explained in conjunction with the accompanying drawings.

[0027] like Figure 1 As shown, this invention provides a time-defined state feedback control method for a permanent magnet synchronous motor system with output constraints. According to the design of this invention, the state feedback controller designed for the permanent magnet synchronous motor system can achieve system convergence to zero within a specified time without violating the output constraints. The specific steps of the technical solution of this invention are as follows:

[0028] Step 1: Based on the permanent magnet synchronous motor system, determine the dynamic equation as follows: in, It is the angle of the motor rotor; It is the mechanical angular velocity of the rotor; It is the moment of inertia of the motor; It is the motor load torque; It is the coefficient of viscous friction; It is electromagnetic torque.

[0029] Define state variables The dynamic equations are rewritten as follows: in, For state variables Differentiate; For state variables beg; It is the system controller input and ; ; It is the number of pole pairs of the motor; It is the magnetic flux linkage of the permanent magnet rotor; Is the stator current in The components of the axis.

[0030] Step 2: Introduce state coordinate transformation: in, yes State variables after coordinate transformation; yes State variables after coordinate transformation; It is the time-varying gain function to be designed, which satisfies in, It is a specified time. It is a time switching and satisfies ; and It is an intermediate variable, and satisfies ; ;exist Within the interval, the system will converge to zero; hour, It converges to a constant rather than tending to infinity, thus avoiding the problem of infinite gain.

[0031] The following dynamic equations are obtained based on the introduced coordinate transformation: in, State variables after coordinate transformation Differentiate; State variables after coordinate transformation Differentiate; It is the designed time-varying gain function Find the derivative.

[0032] Step 3: Set system constraints:

[0033] in, It is a set that satisfies the constraints; and It is a positive integer, representing the set output boundary and ; It is a real number vector; a new type of barrier Lyapunov function is introduced as follows: in, It is about A novel barrier Lyapunov function. In the backstep control design process, the initial barrier Lyapunov function is set... . exist The surface is continuous and its derivative is also continuous. right Taking the derivative, in, It is about The initial barrier Lyapunov function; State variables after coordinate transformation The set of constraints; It is an intermediate constant and satisfies ; It was designed for future convenience. The function that satisfies and .

[0034] Step 4: For system (5), the initial value is set to...

[0035] in, It is the initial state vector of the system; and These are the values ​​of the system state variables at the initial moment; It is the set of constraints for the initial state; This is a positive integer. The specified time controller is designed as follows: , in, and Yes State variables after coordinate transformation; satisfy ; It is a design constant and is an integer. It is a sequence The number of terms; It is during the design process of backstep control. Virtual control law and ; It is an intermediate variable in the backstep control design process and ; A positive design constant; It is a design constant, and ; It is an intermediate variable in the backstep control design process and ; A positive design constant; , , , , It is an intermediate variable in the backstep control design process and ; ; ; ; ; ; ; It is an intermediate constant and ; , , , It is an intermediate variable in the backstep control design process and ; ; , ; , , , , , It is a positive design constant.

[0036] To better illustrate the technology of this invention, a proof will be provided for the design of a specified time controller. The following lemmas and theorems will be used in the proof.

[0037] Lemma 1, if , , For any real variable , Non-negative continuous functions ,exist (11)

[0038] Lemma 2, if For any real number , ,exist (12)

[0039] Lemma 3, if For any real number , ,have (13) Lemma 4. Consider a system with asymmetric output constraints. ,in It is the system's state variable. In the opening episode and and It is any positive number. for Other state variables, yes An open set in a 3D state space. System dynamic equations. about Piecewise continuity, for Satisfies local Lipschitz continuity, and regarding Consistent. Assume existence.

[0040] 1) It is a positive definite continuously differentiable function, when or hour, .

[0041] 2) satisfy ,in for Function-like.

[0042] 3) Composite Lyapunov function .

[0043] 4) In the opening episode Inside.

[0044] If Then for ,state Always satisfied .

[0045] Lemma 5. Consider the system with asymmetric output constraints described in Lemma 4. Assume that there exists

[0046] 1) It is a positive definite continuously differentiable function, when or hour, .

[0047] 2) satisfy ,in for Function-like.

[0048] 3) Composite Lyapunov function .

[0049] 4) In the opening episode Inside.

[0050] If so: (14) in , , ; It is a positive time-varying gain function and .

[0051] The following conclusions can be drawn.

[0052] (i) State Always satisfy constraints .

[0053] (ii) The system is stable at the specified time.

[0054] The proof is as follows:

[0055] Step 1: Definition , According to (8), The derivative along system (5) can be calculated as

[0056] Differentiating (4), we get

[0057] Based on (4) and (16), it is easy to obtain Substituting into equation (15) yields

[0058] In equation (17) Convert to ,

[0059] Introducing virtual controllers ,

[0060] In order to eliminate irrelevant terms, we define... , And choose virtual control law as (20) Where nonnegative function , , , This is a design constant. Substituting into equation (19) yields

[0061] Step Two: Consider and

[0062] in, It is the initial barrier Lyapunov function defined above. It is the integral increment term of the barrier Lyapunov function. Clearly, It is a valid barrier Lyapunov function. The derivative along system (5) is

[0063] First calculate , :

[0064] Substituting (5), (21), and (24) into (23), we get

[0065] Arrange (26) to get

[0066] Next, we focus on (27) and scale its terms. According to Lemma 2,

[0067] According to Lemma 1 and (28) in , .according to ,

[0068] Take the absolute value of both sides. in .right Scaling / Scaling in According to (3) and (31), in According to (25), (28) and (32), in According to Lemma 1 and , in , According to Lemmas 1, (3), (33) and , in , According to (34), And Lemma 1, in , According to (31), (34) and Lemma 1,

[0069] For (38) According to Lemma 1, in , For (35) According to Lemma 1, in , .

[0070] make , And substitute (39) and (40) into (38).

[0071] Substituting (29), (36), (37), (38), and (41) into (27), we get

[0072] At this point, select control input. in , , Substituting (43) into (42), we get in , .

[0073] Step 3: Stability analysis.

[0074] Theorem 1: For the dynamic system model (1) of the permanent magnet synchronous motor, after simplification (2), construction transformation (3), and design of virtual control law (20) and input controller (43), then...

[0075] (i) State It does not violate output constraints;

[0076] (ii) The closed-loop system is stable for a specified time.

[0077] prove:

[0078] (i) Initial state of system (1) Selection under constraints In the middle, according to (3), then Also constrained According to (44), and combined with Lemma 4, we can obtain... Therefore That is, it does not violate the output constraints, and thus the proof is complete.

[0079] (ii) According to (7) and (22), for ,

[0080] According to (28)

[0081] In equation (45), , Next, we will process (44). in , According to Lemma 3,

[0082] In equation (47), , Because (45), there are

[0083] Substitute (48) into (47).

[0084] in , Obviously , According to Lemma 5, the system is stable at a given time.

[0085] Example

[0086] To verify the effectiveness of the time controller proposed in this invention, a simulation verification of a permanent magnet synchronous motor system is performed. The state equation of the permanent magnet synchronous motor system is selected as (1), the controller is selected as (10), and the parameters are selected as follows: , V·s, 1.86e-5 kg·m 2 , 0.8e-3 N·s / m 0.05 N·m. The constraint conditions are set as follows: Set the controller parameters as follows: =1; =0.6; =0.2; =2.5; =3; =2; =5; ; =0.5; =0.3; =0.4; =0.25; =0.1; =0.2. Next, the initial state parameters are divided into three groups: (a), (b), and (c):

[0087] (a) The first group selects a specified time. ;Keep ,Change The values ​​are respectively , , .

[0088] (b) The second group still chooses a designated time. ;Keep ,Change The values ​​are respectively , , .

[0089] (c) The third group maintains ; Change the value of the specified time respectively , , .

[0090] The simulation results for (a), (b) and (c) are shown in Figure 2-10 . Figure 2-4 Group (a) represents the different initial positions of the rotors under the same conditions of specified time and initial velocity. , , The convergence trajectory; Figure 5-7 Group (b) represents the change. After setting the value, under the condition that the initial position of different rotors is the same at the specified time and initial velocity, the values ​​are... , , The convergence trajectory; Figure 8-10 Group (c) represents the different convergence times given all identical initial states. , , The convergence trajectory.

[0091] from Figures 2 to 4 It is clear that at the specified time At any time, regardless Regardless of the input value chosen, the system can converge to an arbitrarily small neighborhood near zero within two seconds by adjusting the input control via the controller. Furthermore, it is easy to see that the initial rotor position... Increasing the value of will not affect the convergence time of the system. Similarly, Figures 5 to 7 This indicates that the system can still converge within two seconds, and the convergence time remains essentially unchanged. Figures 8 to 10 This indicates that the system's convergence speed changes with the specified time. The shorter the specified time, the faster the system converges, representing the controller's adaptability to the specified time. In summary, the controller designed in this invention has better convergence performance than a finite-time controller.

[0092] This invention introduces a novel barrier Lyapunov function for permanent magnet synchronous motor (PMSM) systems. Based on this function, a specified-time backstepping control design method is established, and the design process of this specified-time controller is presented. Stability analysis shows that this invention's design method achieves, for the first time, specified-time control of a PMSM with asymmetric output constraints, and the convergence time is unaffected by the system's initial state. How to apply this result to systems with unknown motor parameters and high disturbance tolerance is a future research area.

[0093] It should be understood that the above description is only a general procedure of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention shall be protected within the scope of protection of the present invention.

Claims

1. A specified time control method of a PMSM system with output constraint, characterized by, The method includes the following steps: S1. Based on the permanent magnet synchronous motor system, the dynamic equation is determined as follows: in, It is the angle of the motor rotor; It is the mechanical angular velocity of the rotor; It is the moment of inertia of the motor; It is the motor load torque; It is the coefficient of viscous friction; It is electromagnetic torque; Define state variables The dynamic equations are rewritten as follows: in, For state variables Differentiate; For state variables Differentiate; It is the system controller input and ; ; It is the number of pole pairs of the motor; It is the magnetic flux linkage of the permanent magnet rotor; Is the stator current in The components of the axis; S2. Introducing state coordinate transformation: in, yes State variables after coordinate transformation; yes State variables after coordinate transformation; It is the time-varying gain function to be designed, which satisfies in, It is a specified time. It is a time switching and satisfies ; and It is an intermediate variable, and satisfies , ; exist Within the interval, the system will converge to zero; hour, It converges to a constant rather than tending to infinity, thus avoiding the problem of infinite gain; The following dynamic equations are obtained based on the introduced coordinate transformation: in, State variables after coordinate transformation Differentiate; State variables after coordinate transformation Differentiate; It is the time-varying gain function designed by S2. Differentiate; S3. Set the system constraints and introduce a specified-time backstepping control design method based on the obstacle Lyapunov function (BLF). S4. Using the specified time backstepping control design method designed in step S3, design the specified time controller of the PMSM system. The system constraints set in step S3 are as follows: in, It is a set that satisfies the constraints; and It is a positive integer, representing the set output boundary and ; It is a real number vector; The introduced barrier Lyapunov function BLF is: in, It is about The barrier Lyapunov function; In the backstep control design process, let the initial barrier Lyapunov function be... ; exist The surface is continuous and its derivative is also continuous. right Taking the derivative, in, It is about The initial barrier Lyapunov function; State variables after coordinate transformation The set of constraints; It is an intermediate constant and satisfies ; It was designed for future convenience. The function that satisfies and .

2. The specified time control method for a PMSM system with output constraints according to claim 1, characterized in that, The specified time controller is designed as follows: in, and Yes State variables after coordinate transformation; satisfy ; It is a design constant and is an integer. It is a sequence The number of terms; It is during the design process of backstep control. Virtual control law and ; It is an intermediate variable in the backstep control design process and ; A positive design constant; It is a design constant, and ; It is an intermediate variable in the backstep control design process and ; A positive design constant; , , , , It is an intermediate variable in the backstep control design process and ; ; ; ; ; ; ; It is an intermediate constant and ; , , , It is an intermediate variable in the backstep control design process and ; ; , ; , , , , , It is a positive design constant.

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