Predefined time sliding mode control method based on improved cascade nonlinear extended state observer

Abstract

The invention discloses a predefined time sliding mode control method based on an improved cascade non-linear extended state observer, so as to realize higher-performance and more reliable control of an uncertain non-linear system, and the method comprises the following steps: 1, providing an inequality based on PTC based on PTC basic definition, and aiding the design of a PTC controller with an adjustable characteristic; 2, aiming at the inequality proposed in the step 1, designing an adjustable PTSMC in combination with a sliding mode control algorithm; and step 3, for non-matching disturbance widely existing in a non-linear system, providing an improved non-linear extended state observer, and compensating an adjustable PTSMC algorithm. The invention provides a novel PTC (Positive Temperature Coefficient)-based inequality and auxiliary controller design. And meanwhile, the designed controller has a better convergence speed. An adjustable predefined time sliding mode controller is designed based on an inequality, and the control performance of the system is effectively improved. The design of ICNLESO can observe and compensate system disturbance in real time, and the anti-interference performance of the system is enhanced.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of automatic control, in particular to a predefined time sliding mode control method for uncertain nonlinear systems, and especially to a design method of a predefined time sliding mode controller based on an improved cascade nonlinear extended state observer. BACKGROUND

[0002] With the increasing complexity of modern industrial systems, such as robot manipulators, aerospace vehicles, precision servo systems, etc., their dynamic models often exhibit high nonlinearity and uncertainty (including parameter perturbation, unmodeled dynamics and external disturbances). How to achieve high precision, high robustness and fast convergence of the system under these uncertainties has always been a difficult point in the research of control field.

[0003] To cope with the above challenges, many advanced control strategies have been proposed, among which the most representative strategies are two: sliding mode control (SMC) and extended state observer (ESO). SMC is known for its inherent strong robustness to matching uncertainties. The traditional SMC designs a sliding surface and uses a reaching law to drive the system state to converge to the sliding surface in finite time, and then slides along the sliding surface to the equilibrium point. This traditional SMC has two main defects: first, the convergence time depends on the initial state of the system, and it is impossible to achieve consistent and pre-set convergence performance; second, there is an inherent "chattering" problem, and high-frequency switching control will excite unmodeled dynamics, damage the actuator and affect the control accuracy. The second is the extended state observer (ESO). As the core of active disturbance rejection control (ADRC), ESO regards internal uncertainties and external disturbances as a "total disturbance" and estimates and compensates it in real time. This method effectively reduces the dependence on accurate mathematical models, but the estimation performance is limited, and it is difficult to balance the convergence accuracy and estimation accuracy, and the estimation error converges gradually rather than in finite time. Although there are subsequent finite time ESO (FTESO) and fixed ESO (FxTESO), there is still room for improvement in the convergence speed, accuracy of disturbance estimation and ability to cope with complex time-varying disturbances (especially high-order disturbances).

[0004] In recent years, researchers have tried to combine ESO and SMC, use ESO to observe and compensate the total disturbance, and reduce the switching gain of SMC to effectively suppress chattering. However, based on the defects of the ESO and SMC mentioned above, the final control performance of the system will be poor. Therefore, it is urgent to design a new type of control algorithm with better convergence speed and superior disturbance rejection performance. The present application aims to propose a predefined time sliding mode control method based on an improved cascade nonlinear extended state observer to achieve higher performance and more reliable control of uncertain nonlinear systems. SUMMARY

[0005] The present invention aims to propose a predefined time sliding mode control method based on an improved cascaded nonlinear extended state observer, so as to achieve higher performance and more reliable control of uncertain nonlinear systems.

[0006] Its technology includes the following steps:

[0007] Step 1: To address the discrepancy between theoretical and practical CT in existing PTC (Performance Computation) methods, a novel PTC-based inequality is proposed based on the fundamental definitions of PTC. This can assist in the design of a PTC controller with adjustable characteristics.

[0008] Consider the following nonlinear system:

[0009]

[0010] in For system status, It is a nonlinear continuous function, and the origin is assumed to be the system. The balance point.

[0011] Definition 1: If the system The equilibrium point is stable over a fixed time, and the stable time function is... Conditions met:

[0012]

[0013] In the formula: And it has an explicit relationship with the system parameters. Then the system... The equilibrium point at this point is considered to be the adjustable PTS. For a predefined time (PT).

[0014] Theorem 1: If for the system There exist continuous positive definite radially unbounded functions that satisfy the following conditions. :

[0015]

[0016] In the formula: , So the system The equilibrium point is PTS, where PT is .

[0017] Step 2: Based on the inequality proposed in Step 2, an adjustable PTSMC was designed using the sliding mode control algorithm;

[0018] Based on the proposed inequality A novel integral sliding surface with adjustable PTS was constructed:

[0019] (4)

[0020] In the formula, , .

[0021] Based on the basic principles of SMC (i.e.) and Therefore, we differentiate (4):

[0022] (5)

[0023] The equivalent control component in SMC can be obtained as follows:

[0024] (6)

[0025] Then, a sliding mode convergence law with adjustable PTS was designed:

[0026] (7)

[0027] In the formula, , .

[0028] Therefore, the final control law can be obtained as follows:

[0029] (8)

[0030] Step 3: To address the widespread mismatched disturbances in nonlinear systems, an improved nonlinear extended state observer and a compensated adjustable PTSMC algorithm are proposed.

[0031] Traditional nonlinear ESO uses NLF as Its expression is as follows:

[0032] (9)

[0033] In the formula For smaller positive constants, for The positive constants between . Using the function in (9) not only ensures that the error converges in a finite time, but also reduces the steady-state error of the system. Choosing a smaller value can prevent the controller from reacting too aggressively, and the controller's sensitivity to errors is concentrated on small errors. This means it can quickly eliminate small errors and improve steady-state accuracy. However, the feedback effect weakens at large errors, leading to a slower convergence speed. Therefore, a nonlinear function is constructed. Its expression is as follows:

[0034] (10)

[0035] In the formula and It is a positive number.

[0036] To further reduce the impact of total disturbance on system performance, a cascaded ESO is introduced on the basis of the traditional nonlinear ESO. Simultaneously, an ICNLESO is proposed in conjunction with the constructed NLF. Its mathematical expression is shown in (11). Used to track initial disturbances (preliminary disturbance) Used to track remaining disturbances (remaining disturbance). and Used to observe rotational speed.

[0037] (11)

[0038] Therefore, the final observed disturbance is .

[0039] Substituting this into equation (8), we can obtain the final controller output as:

[0040] (12).

[0041] The technical effects and advantages of this invention are as follows:

[0042] (1) A novel PTC-based inequality is proposed to assist in controller design. The designed controller also has a superior convergence speed.

[0043] (2) An adjustable predefined time sliding mode controller was designed based on inequalities, which effectively improved the control performance of the system.

[0044] (3) ICNLESO’s design can observe and compensate for system disturbances in real time, enhancing the system’s anti-disturbance performance.

[0045] (4) The adjustable PTSMC based on ICNLESO is simple to implement and can be widely used in various control systems. Attached Figure Description

[0046] Figure 1 The structural framework of ICNLESO;

[0047] Figure 2The following is a schematic diagram of the experimental platform for the example, wherein: (a) PMSM experimental platform, (b) Host Computer (MATLAB / Simulink);

[0048] Figure 3 The experimental results under different controllers are compared, including: (a) velocity response curve, (b) q-axis current response curve, and (c) d-axis current response curve.

[0049] Figures 4-6 The experimental results for the four algorithms under different perturbations are as follows:

[0050] Figure 4 . Experimental results for the following four algorithms: (a) PI, (b) PTSMC+NDOB, (c) PTSMC+ESO, (d) PTSMC+ICNLESO;

[0051] Figure 5 . Experimental results for the following four algorithms: (a) PI, (b) PTSMC+NDOB, (c) PTSMC+ESO, (d) PTSMC+ICNLESO;

[0052] Figure 6 . Experimental results for the following four algorithms: (a) PI, (b) PTSMC+NDOB, (c) PTSMC+ESO, (d) PTSMC+ICNLESO;

[0053] Figure 7 For performance metrics comparison, including: (a) Response time, (b) Speed ​​drop, and (c) Speed ​​fluctuation. Detailed Implementation

[0054] Example (Experimental Study):

[0055] A PMSM-based towing platform was used as an example for algorithm verification. The dual PMSM towing platform used is shown in Figure 2. This experimental platform consists of four parts: motors, a host computer, an integrated control board, and a power supply. The motor towing system consists of two identical PMSMs (model: 42JSF630AS-1000), with a TMS320F28379D motion control board as the control core and two BOOSTXL-DRV8305 driver boards as the hardware drive circuit for the towing platform. One motor acts as the controlled object, and the other as the load torque generator. Furthermore, since the PMSMs used are surface-mount, a... The PMSM vector control method is described. The code is implemented by building a control system model using MATLAB / Simulink, then automatically converting it into C code and burning it into the main control chip for execution. The parameters of PMSM are shown in Table 1.

[0056]

[0057] Table 1. PMSM Parameters

[0058] (1) Step response performance

[0059] First, the effectiveness of the proposed PTSMC algorithm is verified. To highlight the superiority of the proposed algorithm, comparative experiments are conducted with the proposed algorithm, the classic PI algorithm, finite-time SMC (FTSMC), and fixed-time SMC (FxTSMC).

[0060] 1) FTSMC Design:

[0061] (13)

[0062] In the formula ,in To ensure the effectiveness of the algorithm comparison, the parameter selection and formula... Maintain consistency. That is, have , , , , , .

[0063] 2) FxTSMC Design:

[0064] (14)

[0065] In the formula Similarly, to ensure the validity of the comparative experiment, the parameters were kept consistent. , , , . , , , .

[0066] Figures 3(a)-(c) show the comparison results of the speed, q-axis current, and d-axis current response curves of the PMSM under four different algorithms. As shown in Figure 3(a), the convergence times of the motor under PI, FTSMC, FxTSMC, and PTSMC control are 0.952 s, 1.195 s, 1.64 s, and 0.444 s, respectively. The proposed algorithm and FxTSMC do not exhibit overshoot, but FxTSMC's convergence time is too long. Meanwhile, although the PI convergence time is second only to PTSMC, its overshoot is as high as 24%. Furthermore, the overshoot of FTSMC is 8.3%. Therefore, the proposed algorithm has the advantages of no overshoot and fast convergence speed. Observing Figures 3(b) and (c), the change in the speed loop controller also has a certain impact on the dq-axis current response. That is, compared with the PI controller, the adopted SMC control scheme can reduce the d-axis current fluctuation to a certain extent. In summary, the proposed PTSMC has excellent dynamic response performance.

[0067] (2) Immunity test

[0068] To further verify the effectiveness of the algorithm, this section combines PTSMC with ICNLESO. The disturbance rejection algorithms compared include traditional PI, PTSMC+NDOB, and PTSMC+ESO. The time-varying load torque function applied in the comparative experiment is as follows:

[0069]

[0070] Figures 4-6 show the disturbance rejection experimental results of PMSM under the four algorithms mentioned above. Several observations can be drawn from the figures: (1) The PMSM speed convergence time is the shortest under the proposed PTSMC+ICNLESO control, and there is no overshoot. (2) During the startup phase, the peak motor current under the proposed algorithm control is smaller than that of other schemes, and the dynamic performance is better. (3) Among the four algorithms, the PMSM speed overshoot is the largest and the convergence time is the longest under the PI controller. Moreover, its speed deviation is the largest for sudden disturbances. (4) Under sudden load and time-varying load, the PMSM speed deviation under the proposed algorithm control is the smallest, that is, the ability to suppress disturbances is the best. In summary, the proposed algorithm has the advantages of fast convergence speed and strong robustness.

[0071] Figure 7 presents a comparison of the performance indicators of four different controllers under varying load torque. First, observing Figure 7(a), we can see that the response times increase sequentially from PTSMC+ICNLESO, PTSMC+ESO, PTSMC+NDOB, to PI: 0.4 s, 0.437 s, 0.51 s, and 0.952 s, respectively. However, Figure 7(b) shows that with a sudden increase of 0.2... After loading, the speed drops for PI, PTSMC+NDOB, and PTSMC+ESO were 117.3 rpm, 109.6 rpm, and 100 rpm, respectively. The proposed algorithm, however, only reduced the speed by 55.2 rpm. This demonstrates that the proposed algorithm effectively improves the system's immunity to sudden fixed-value disturbances. Finally, observing Figure 7.(c), after adding... After time-varying disturbances, the rotational speed fluctuations of PI and PTSMC+NDOB were 64.7 rpm and 65.5 rpm, respectively, while that of PTSMC+ESO was 62.73 rpm. The proposed algorithm, however, exhibited a fluctuation of only 34.2 rpm. This demonstrates that the proposed algorithm also possesses superior resistance to time-varying disturbances. In summary, the proposed algorithm effectively accelerates the convergence speed of system state errors, while also improving the system's robustness.

[0072] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A pre-defined time sliding mode control method based on improved cascaded nonlinear extended state observer to achieve higher performance and more reliable control of uncertain nonlinear systems, characterized in that, Comprising the following steps: Step 1, a PTC-based inequality is proposed based on the PTC basic definition, which can assist in designing a PTC controller with adjustable characteristics; Step 2, for the inequality proposed in step 1, an adjustable PTSMC is designed combined with a sliding mode control algorithm; Step 3, for the non-matching disturbance widely existing in the nonlinear system, an improved nonlinear extended state observer is proposed to compensate the adjustable PTSMC algorithm.

2. The pre-defined time sliding mode control method based on the improved cascade nonlinear extended state observer according to claim 1, wherein: In step 1, a PTC-based inequality is proposed based on the PTC basic definition, which can assist in designing a PTC controller with adjustable characteristics; Consider the following nonlinear system: （1） wherein is the system state, is a nonlinear continuous function and the origin is assumed to be the equilibrium point of system (1); Definition 1: If the equilibrium point of system (1) is fixed-time stable, and the stable time function satisfies the condition: （2） In the formula: , and has an explicit relationship with the system parameters; the equilibrium point of the system (1) is then considered to be a tunable PTS, is a predefined time (PT); Theorem 1: If for system (1) there exists a continuous positive definite radial unbounded function V(x) satisfying the following conditions : （3） In the formula: , ; then the equilibrium point of the system (1) is PTS, PT is .

3. The pre-defined time sliding mode control method based on the improved cascade nonlinear extended state observer according to claim 2, wherein: In step 2, for the inequality proposed in step 1, an adjustable PTSMC is designed combined with a sliding mode control algorithm; Based on the proposed inequality (3), a new adjustable PTS integral sliding surface is constructed: （4） In the formulae, , ; According to the basic principle of SMC (i.e. and ), the derivative of (4) is taken: （5） The equivalent control part in SMC can be obtained as: （6） Then a adjustable PTS sliding mode reaching law is designed: （7） In the formulae, , ; Therefore, the final control law is: （8）。 4. The pre-defined time sliding mode control method based on the improved cascade nonlinear extended state observer according to claim 3, wherein: In step 3, for the non-matching disturbance widely existing in the nonlinear system, an improved nonlinear extended state observer is proposed to compensate the adjustable PTSMC algorithm; The NLF used by the conventional nonlinear ESO is The expression is as follows: （9） wherein is a small positive constant, is is a positive constant between 0 and 1; using the function in (9) not only guarantees the convergence of the error in a finite time, but also reduces the steady-state error of the system; wherein The smaller the selected, the more the controller can avoid overacting, and the sensitivity of the controller to the error is concentrated on small errors; that is, to quickly eliminate small errors and improve steady-state accuracy; but the feedback effect is weakened at large errors, which will lead to a decrease in convergence speed; To this end, a non-linear function is constructed, the expression of which is as follows: （4） In the formula with is positive; In order to further reduce the influence of total disturbance on system performance, a cascade ESO is introduced on the basis of the traditional nonlinear ESO; At the same time, combined with the constructed NLF, an ICNLESO is proposed; The mathematical expression is shown as (11); wherein for tracking the initial disturbance (preliminary disturbance), for tracking the remaining disturbance (remaining disturbance); and for observing the rotation speed; （11） Therefore, the observed perturbation is ; Then substitute it into equation (8), the final controller output is: （12）。

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