Driving mechanism simulator control method based on inertia compensation strategy

Abstract

The invention provides a driving mechanism simulator control method based on an inertia compensation strategy. The method comprises the following steps: S1, constructing an actual mechanical dynamic mathematical model of a driving mechanism; s2, constructing a driving mechanism simulator mathematical model applying a traditional inertia compensation strategy; s3, calculating a driving mechanism simulation system stability condition; s4, solving a torque deviation component generated by system time delay due to instability of the simulation system of the driving mechanism; and S5, designing a high-order filter to realize that the driving mechanism simulator accurately and stably simulates the mechanical dynamics of the large-rotational-inertia actual driving mechanism. According to the invention, on the basis of an existing driving mechanism simulator, an inertia compensation strategy is applied to realize variable inertia simulation of mechanical dynamic states of different types of driving mechanisms, and the defect of system instability of an original method is overcome by proposing an improved inertia compensation strategy applying a high-order filter; and the variable inertia simulation of the mechanical dynamics of different driving mechanisms by the driving mechanism simulator can be more stably realized.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of spacecraft sail drive mechanism simulation experiment, and particularly relates to a drive mechanism simulator control method based on an inertia compensation strategy. BACKGROUND

[0002] With the development of large-scale and diversified functions of spacecraft, the application of aircraft drive mechanisms is also more and more extensive. The drive mechanisms have characteristics such as large flexibility and large inertia, which causes large torque load of the drive mechanism in the movement process, and high change frequency, complex working conditions and the like. Due to the influence of the ground gravity and the volume of the drive mechanism and the solar sail, it is difficult to accurately simulate the on-orbit state of the drive mechanism on the ground. Meanwhile, different sails have different inertias due to different functions, and it is difficult to simulate the movement characteristics of multiple types of drive mechanisms by using one set of simulator.

[0003] Although the conventional inertia compensation strategy applied to the existing drive mechanism simulator can realize the simulation of the mechanical dynamics of the actual drive mechanism with variable inertia to a certain extent, there is a problem of system stability. Therefore, it is urgent to propose a control method capable of stably simulating the large rotational inertia drive mechanism by using the existing drive mechanism simulator. SUMMARY

[0004] In order to solve the problem that the existing drive mechanism simulator based on the conventional inertia compensation strategy cannot stably realize the simulation of the real mechanical dynamics of the large rotational inertia drive mechanism, the application applies the inertia compensation strategy based on the existing drive mechanism simulator to realize the simulation of the mechanical dynamics of different types of drive mechanisms with variable inertia. By proposing an improved inertia compensation strategy using a high-order filter, the system instability of the original method is overcome, and the mechanical dynamics of different drive mechanisms can be more stably simulated by the drive mechanism simulator with variable inertia. The specific technical solutions are as follows:

[0005] A drive mechanism simulator control method based on an inertia compensation strategy, comprising the following steps:

[0006] S1, constructing a mathematical model of the mechanical dynamics of the actual drive mechanism;

[0007] S2, constructing a mathematical model of the drive mechanism simulator applying the traditional inertia compensation strategy;

[0008] S3, calculating the stability condition of the drive mechanism simulation system;

[0009] S4, obtaining the torque deviation component generated by the system time delay as the reason for the instability of the drive mechanism simulation system;

[0010] S5, designing a high-order filter to accurately and stably simulate the mechanical dynamics of the actual drive mechanism with large rotational inertia by the drive mechanism simulator.

[0011] Further, the specific method of constructing the actual driving mechanism mechanical dynamic mathematical model in step S1 is as follows:

[0012] Suppose the actual driving mechanism system is connected through rigidity, the dynamic equation of the driving mechanism is obtained as follows:

[0013]

[0014] In the formula, T M is the motor output torque; i is the reduction ratio; J M , θ M are the rotation inertia and rotation angle of the driven side composed of the driving gear driven by the motor; J L , θ L are the rotation inertia and rotation angle of the driven side system composed of the driven gear and the sailboard and truss driven by the driven gear; T L is the torque acting on the driven side system; τ is the torque acting on the driving gear of the driven side, and γ is the torque acting between the driven gears, so it can be obtained that:

[0015]

[0016] In the formula, J t = J M / i+J L is the total rotation inertia converted to the driven side.

[0017] Further, the specific method of constructing the driving mechanism simulator mathematical model applying the traditional inertia compensation strategy in step S2 is as follows:

[0018] Based on the actual driving mechanism mathematical model of formula (1), the motion equation of the driving mechanism simulation system can be obtained as follows:

[0019]

[0020] In the formula, T s is the output torque of the driving mechanism simulator motor, J s is the total rotation inertia of the driving mechanism simulator, T N is the driven side torque of the driving mechanism simulator, and T L is the driven side torque of the actual driving mechanism. In order to make the driving mechanism simulator obtain the same mechanical dynamics as the actual driving mechanism, the driven side torques of the two are equal, i.e., T N

[0021] ​It can be seen that the driving simulation system is different from the actual driving mechanism in the moment of inertia of the sailboard and the truss system. The moment of inertia of the driving simulation system is much smaller than that of the actual sailboard and truss. Therefore, the moment of inertia compensation strategy is adopted to compensate the moment of inertia of the driving simulation system, so that the driving simulation system can accurately simulate the mechanical motion state of the actual sailboard driving mechanism. That is, by subtracting the above formula (2) from formula (3), the following formula (4) can be obtained

[0022]

[0023] In the formula, T c is the compensation torque.

[0024] Further, the specific method for calculating the stable condition of the driving mechanism simulation system in step S3 is as follows:

[0025] Since the torque control of the driving mechanism simulator is periodic, the rotational acceleration needs to be sampled and calculated with respect to the rotational speed, which will cause a one-step time delay in the acceleration compared to the actual driving mechanism. This delay is objectively unavoidable and occurs in the torque compensation loop to calculate the mechanism motion acceleration.

[0026] The driving mechanism acts on the servo side torque T L is affected by the elastic force and the damping force of the flexible link, which is simplified as a linear relationship, and a linear coefficient K L is introduced to obtain:

[0027]

[0028] Therefore, the transfer function of the driving mechanism simulator can be obtained as:

[0029]

[0030] According to the Routh criterion, the stable condition of the driving mechanism simulation system can be obtained as

[0031]

[0032] Since the value of T·k L is much smaller than the moment of inertia J t of the driving mechanism, the following formula (5) can be obtained: T / J s > 2, the system will be unstable.

[0033] Further, the specific method for obtaining the reason for instability of the driving mechanism simulation system in step S4 is as follows:

[0034] Because of the time delay of the drive mechanism simulator, a filter is designed to suppress the system instability caused by the delay, and the relationship of the filter is:

[0035]

[0036] The frequency domain expression of the filter is obtained as:

[0037]

[0038] The transmission model under the frequency domain response of the improved drive mechanism simulator is obtained, and the frequency domain response transfer function of the drive simulation system is:

[0039]

[0040] Therefore, according to the Routh criterion, the stability condition of the system is:

[0041]

[0042] Similarly, ignoring the influence of k L T / 2, we get:

[0043]

[0044] But the drive mechanism simulator is driven and controlled by bus instructions, and there is also a time delay in the acceleration observation calculation, so its actual time delay will be longer;

[0045] Assuming that the time delay of the drive mechanism simulator is τ, and the discrete step length of the drive system is v, we get

[0046] τ=k0·v..............(14)

[0047] Where k0 is the order of communication delay, k0=1, 2, 3…

[0048] In summary, the discrete mathematical model of the drive mechanism simulator is obtained, which takes torque △T as input, and the state equation of the drive mechanism simulator is:

[0049]

[0050] Let △T be a step input, and the time domain response expression of the simulator is calculated according to equation (15) as:

[0051]

[0052] It can be found that the time domain model of the drive mechanism simulator contains two components, where ​The second component is consistent with the response component of the actual driving mechanism is the deviation component affected by time delay, which leads to deviation from the actual driving mechanism model and even instability of the system.

[0053] Further, the step S5 designs a high-order filter to realize the specific method of the driving mechanism simulator to accurately and stably simulate the mechanical dynamics of the actual driving mechanism with large moment of inertia, and the specific method is as follows:

[0054] The torque deviation component obtained in the step S4 has the related property of k0+1 order, so an improved filter control strategy with k0+1 order is designed, and the frequency domain expression of the filter is

[0055]

[0056] Further, the model of the driving mechanism simulator is obtained, and the transfer function of the simulator is

[0057]

[0058] The poles of the transfer function are obtained as

[0059] When all the poles are in the unit circle, the simulator system is stable;

[0060] When the pole is at zero, i.e. The optimal solution of the system is obtained, and the calculation result is

[0061] At this time, the response speed of the simulation system is the fastest in the stable state, and the transfer function of the system is obtained as

[0062]

[0063] The z inverse transform of the expression is performed, and the time domain response expression of the system applying the designed k0+1 order filter is obtained as

[0064]

[0065] It is found that the obtained model still has the component affected by time delay This component, but the component u(k)-u(k-k0-1) always tends to zero when the time sequence k>k0+1, so the existence of the component does not affect the stability of the system.

[0066] Compared with the prior art, the present application has the following beneficial effects:

[0067] The present invention is based on the application of inertia compensation strategy in existing drive mechanism simulators to achieve variable inertia simulation of mechanical dynamics of different types of drive mechanisms. By proposing an improved inertia compensation strategy using high-order filters, the disadvantage of system instability in the original method is overcome, and the variable inertia simulation of mechanical dynamics of different drive mechanisms by the drive mechanism simulator can be achieved more stably. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0069] Figure 1 A schematic flow chart of a driving mechanism simulator control method based on an inertia compensation strategy provided by the present invention;

[0070] Figure 2 It is a schematic diagram of the mechanical model structure of the actual driving mechanism;

[0071] Figure 3 Schematic diagram of the mathematical model structure of the drive mechanism using the traditional inertia compensation strategy;

[0072] Figure 4 This is a schematic diagram of the discrete mathematical model structure of the driving mechanism simulator system;

[0073] Figure 5 Schematic diagram of the mathematical model structure of the drive mechanism simulator system using high-order filters;

[0074] Figure 6 This is a comparison chart of the speed curve when the simulated moment of inertia is 5 times;

[0075] Figure 7 This is a comparison chart of the speed curve when the simulated moment of inertia is 18 times.

[0076] Specific implementation methods

[0077] The present invention will be further described below with reference to the accompanying drawings.

[0078] Figure 1 A schematic flow chart of a drive mechanism simulator control method based on an inertia compensation strategy provided by the present invention.

[0079] A driving mechanism simulator control method based on an inertia compensation strategy, the method comprising:

[0080] S1. Construct a practical mechanical dynamic mathematical model of the driving mechanism;

[0081] The actual driving mechanism mechanical model is as follows Figure 2 As shown in Figure 2, assuming that the actual drive mechanism system is connected by rigidity, the dynamic equation of the drive mechanism is:

[0082]

[0083] Where T M is the motor output torque; i is the reduction ratio; J M ,θ M are the moment of inertia and angle of rotation of the driving gear driven by the motor; J L ,θ L are the moment of inertia and angle of rotation of the follower side system composed of the follower side gear and the sailboard and truss it drives; T L is the torque acting on the follower side system; τ is the torque acting on the driving gear on the drag side, and γ is the torque acting between the follower side gears, so we can get:

[0084]

[0085] Where J t =J M / i+J L , is the total moment of inertia converted to the follower side.

[0086] S2. Construct a mathematical model of a drive mechanism simulator that applies a traditional inertia compensation strategy;

[0087] Based on the mathematical model of the actual driving mechanism of formula (1), the motion equation of the driving mechanism simulation system can be obtained:

[0088]

[0089] Where T s is the output torque of the driving mechanism simulator motor, J s is the total moment of inertia of the driving mechanism simulator, T N is the follower side torque of the driving mechanism simulator. In order to make the driving mechanism simulator obtain the same mechanical dynamics as the actual driving mechanism, the follower side torques of the two are equal, that is, T L =T N ;

[0090] It can be seen from this that the difference between the driving simulation system and the actual driving mechanism lies in the different moments of inertia of the sailboard and truss system. The moment of inertia of the driving mechanism simulation system is much smaller than that of the actual sailboard and truss. Therefore, a moment of inertia compensation strategy is adopted to compensate the moment of inertia of the driving mechanism simulation system so that the driving mechanism simulation system can accurately simulate the mechanical motion state of the actual sailboard driving mechanism; that is, by subtracting the above formulas (2) and (3), we can get:

[0091]

[0092] T = T + T c is the compensation torque, i.e. the driving mechanism simulator can get the driving mechanism model as shown in Figure 3 The mathematical model of the driving mechanism simulator is shown in the following.

[0093] S3, calculating the stability condition of the driving mechanism simulation system;

[0094] The mathematical model in step S2 is discretized and the characteristic equation of the driving mechanism simulator is calculated, and the stability condition of the driving mechanism simulation system is calculated according to the Routh criterion.

[0095] The torque compensation feedback loop model is discretized, wherein

[0096]

[0097] α c (k) is the rotational acceleration of the driving mechanism simulator, and its expression is:

[0098]

[0099] Where T is the control period of the driving mechanism simulator, but since the torque control of the simulator is periodic and has discrete characteristics, it is discretized to obtain the rotational speed integral formula:

[0100]

[0101] Where α(k) is the average acceleration of the simulator between sampling points k and k+1, and formula (7) is substituted into formula (6) to obtain the rotational acceleration of the driving mechanism simulator:

[0102] α c (k) = α(k-1) (8)

[0103] It can be seen that the acceleration applied to the simulator has a one-step delay compared to the actual acceleration of the driving mechanism, which is generated in the torque compensation loop and objectively exists and cannot be avoided.

[0104] The driving mechanism acts on the servo side torque T L Affected by flexible elements such as elastic force and damping force, it is simplified as a linear relationship, and a linear coefficient K L is introduced to obtain:

[0105]

[0106] Therefore, the transfer function of the driving mechanism simulator can be obtained as:

[0107]

[0108] According to the Routh criterion, the stability condition of the drive mechanism simulation system is obtained as follows:

[0109]

[0110] Since the value of T·k L is far less than the rotational inertia J t of the drive mechanism, it is obtained that J T / J s >2, and the system is unstable.

[0111] S4, the reason for instability of the drive mechanism simulation system is that the torque deviation component caused by the system time delay is obtained;

[0112] Since the drive mechanism simulator exists the time delay of calculation, a filter is designed to suppress the system instability caused by the delay, and the relationship of the filter is

[0113]

[0114] The frequency domain expression of the filter is obtained as follows:

[0115]

[0116] The transmission model under the frequency domain response of the improved drive mechanism simulator is obtained, and the frequency domain response transfer function of the drive simulation system is:

[0117]

[0118] Therefore, according to the Routh criterion, the stability condition of the system is obtained as follows:

[0119]

[0120] Similarly, the influence of k L T / 2 is ignored, and it is obtained that:

[0121]

[0122] But the drive mechanism simulator is driven and controlled by bus instructions, and there is also time delay in the acceleration observation calculation, so its actual time delay will be longer.

[0123] Suppose the time delay of the drive mechanism simulator is τ, and the discretization step of the drive system is v, it is obtained that:

[0124] τ=k0·v.....................(18)

[0125] where k0 is the order of the communication delay, k0 = 1, 2, 3,...

[0126] The discrete mathematical model of the driving mechanism simulator is obtained as shown in Fig. 2, which takes torque △T as input, and the state equation of the driving mechanism simulator is obtained as: Figure 4

[0127] Let △T be a step input, and the time-domain response expression of the simulator is calculated according to formula (19) as:

[0128]

[0129] It can be found that the time-domain model of the driving mechanism simulator contains two components, wherein

[0130] the first component is consistent with the response component of the actual driving mechanism, and the second component is a deviation component affected by time delay, which causes deviation from the actual driving mechanism model and even instability of the system. S5, designing a high-order filter to realize accurate and stable simulation of the mechanical dynamics of the actual driving mechanism with large moment of inertia by the driving mechanism simulator;

[0131] The torque deviation component obtained in step S4 has a k0+1 order correlation, and therefore a k0+1 order improved filter control strategy is designed, and the frequency-domain expression of the filter is obtained as:

[0132]

[0133] The model of the driving mechanism simulator is further obtained as shown in Fig. 3, and the transfer function of the simulator is:

[0134] Figure 5

[0135] The poles of the transfer function can be obtained as

[0136] When all the poles fall within the unit circle, the simulator system is stable.

[0137] When the pole is at zero, i.e. the optimal solution of the system is obtained, and the calculation result is

[0138] At this time, the response speed of the simulation system is the fastest under stable state, and the transfer function of the system is:

[0139]

[0140] ​​​

[0141] The time-domain response expression of the system of the k0+1 order filter of the application design is obtained by performing the z inverse transform on the expression:

[0142]

[0143] It can be found that the obtained model still has the component affected by the time delay This component, however, is always tending to zero when k>k0+1, so the existence of the component does not affect the stability of the system.

[0144] In this embodiment, a direct current stepper motor is selected as the mechanism driving motor to construct the mathematical model of the driving mechanism and the model of the driving mechanism simulator of the control method proposed in the application, and simulation test research is performed on the model under different simulation multiples of the rotational inertia.

[0145] The key parameters and analysis data of the simulation test data of the application of the traditional inertia compensation strategy and the improved inertia compensation strategy are compared in Table 1, and it can be found that the driving mechanism simulator control method based on the inertia compensation strategy of the application can effectively overcome the problem of large mechanical dynamic error of the simulation of the actual driving mechanism by the traditional inertia compensation strategy.

[0146] Table 1 Simulation test data table of the driving mechanism simulator

[0147]

[0148] The simulation test results are shown in Figure 6 , 7 The curve of the rotational speed output of the driving mechanism simulator with the rotational inertia of 5 times compared with the actual driving mechanism output rotational speed is shown in Figure 6 , and the curve of the rotational speed output of the driving mechanism simulator with the rotational inertia of 18 times compared with the actual driving mechanism output rotational speed is shown in Figure 7 It can be seen that the driving mechanism simulator applying the control method proposed in the application can keep consistent with the mechanical dynamic of the actual driving mechanism in the steady state. The driving mechanism simulator control method based on the inertia compensation strategy of the application can stably realize the simulation of the mechanical dynamic of the driving mechanism with large rotational inertia by the driving motor with small inertia.

[0149] Although the present invention has been disclosed above with reference to preferred embodiments, this is not intended to limit the present invention. Any person skilled in the art may utilize the methods and techniques disclosed above to make possible changes and modifications to the technical solutions of the present invention without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent variations, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the scope of protection of the technical solutions of the present invention. Matters not described in detail in the present specification are common knowledge to those skilled in the art.

Claims

1. A driving mechanism simulator control method based on inertia compensation strategy, characterized in that: The method comprises the following steps: S1. Construct a practical mechanical dynamic mathematical model of the driving mechanism; S2. Construct a mathematical model of a drive mechanism simulator that applies a traditional inertia compensation strategy; S3. Calculate the stability conditions of the driving mechanism simulation system; S4. The reason for the instability of the driving mechanism simulation system is the torque deviation component caused by the system delay; S5. Design high-order filters to enable the drive mechanism simulator to accurately and stably simulate the mechanical dynamics of the actual drive mechanism with large rotational inertia.

2. The driving mechanism simulator control method based on inertia compensation strategy according to claim 1 is characterized in that: The specific method of constructing the actual mechanical dynamic mathematical model of the driving mechanism in step S1 is as follows: Assuming that the actual drive mechanism system is connected by rigidity, the dynamic equation of the drive mechanism is: Where T M is the motor output torque; i is the reduction ratio; J M ,θ M are the moment of inertia and angle of rotation of the driving gear driven by the motor; J L ,θ L are the moment of inertia and angle of rotation of the follower side system composed of the follower side gear and the sailboard and truss it drives; T L is the torque acting on the follower side system; τ is the torque acting on the driving gear on the drag side, and γ is the torque acting between the follower side gears, so we can get: Where J t =J M / i+J L , is the total moment of inertia converted to the follower side.

3. The driving mechanism simulator control method based on inertia compensation strategy according to claim 1 is characterized in that: The specific method of constructing the mathematical model of the driving mechanism simulator using the traditional inertia compensation strategy described in step S2 is as follows: Based on the mathematical model of the actual driving mechanism, the motion equation of the driving mechanism simulation system can be obtained: Where T s is the output torque of the driving mechanism simulator motor, J s is the total moment of inertia of the driving mechanism simulator, T N is the follower side torque of the driving mechanism simulator. In order to make the driving mechanism simulator obtain the same mechanical dynamics as the actual driving mechanism, the follower side torques of the two are equal, that is, T L =T N ; It can be seen from this that the difference between the driving simulation system and the actual driving mechanism lies in the different moments of inertia of the sailboard and truss system. The moment of inertia of the driving mechanism simulation system is much smaller than that of the actual sailboard and truss. Therefore, a moment of inertia compensation strategy is adopted to compensate the moment of inertia of the driving mechanism simulation system so that the driving mechanism simulation system can accurately simulate the mechanical motion state of the actual sailboard driving mechanism; that is, by subtracting the above formulas (2) and (3), we can get: Where: T c Expressed as compensation torque.

4. The driving mechanism simulator control method based on inertia compensation strategy according to claim 1 is characterized in that: The specific method for calculating the stability condition of the driving mechanism simulation system in step S3 is as follows: Because the torque control of the drive mechanism simulator is periodic, its rotational acceleration requires sampling and calculating the rotational speed, which results in a one-step time delay between the calculated acceleration and the actual drive mechanism acceleration. This delay is generated in the torque compensation loop when calculating the mechanism's motion acceleration and is objectively unavoidable. The driving mechanism acts on the follower side torque T L Affected by the flexible links such as elastic force and damping force, it is simplified into a linear relationship and the linear coefficient K is introduced L ,get: Therefore, the transfer function of the driving mechanism simulator can be obtained as: According to Routh's criterion, the stability condition of the driving mechanism simulation system can be obtained as follows: Because T.k L The value is much smaller than the moment of inertia J of the driving mechanism t , so we get J T / J s >2, the system will be unstable.

5. The driving mechanism simulator control method based on inertia compensation strategy according to claim 1 is characterized in that: The specific method for determining the torque deviation component caused by the system delay as the cause of the instability of the driving mechanism simulation system in step S4 is as follows: Since there is a driving mechanism simulator with calculation time delay, the design uses a filter to suppress the system instability caused by the delay. The relationship of the filter is: The frequency domain expression of the filter is obtained as: The transmission model of the improved drive mechanism simulator under the frequency domain response is obtained. The frequency domain response transfer function of the drive simulation system is: Therefore, according to the Routh criterion, the stability condition of the system is: Also ignore k L The effect of T / 2 gives: However, the driving mechanism simulator is driven and controlled through bus instructions, and there is a time delay in the acceleration observation calculation, so its actual time delay will be longer; Assuming that the time delay of the driving mechanism simulator is τ and the discretization step size of the driving system is v, we can get τ=k0·v............(14) Where k0 is the order of communication delay, k0 = 1, 2, 3...; In summary, the discrete mathematical model of the driving mechanism simulator is obtained. The simulator takes torque △T as input, and the state equation of the driving mechanism simulator is obtained as follows: Let △T be the step input, and the time domain response expression of the simulator is calculated according to formula (15): The time domain model of the drive mechanism simulator can be found There are two components in The second component is consistent with the actual drive mechanism response component. It is the deviation component affected by time delay, which causes it to deviate from the actual driving mechanism model and even leads to system instability.

6. The driving mechanism simulator control method based on inertia compensation strategy according to claim 1, characterized in that: Step S5: The specific method for designing a high-order filter to enable the drive mechanism simulator to accurately and stably simulate the mechanical dynamics of an actual drive mechanism with a large moment of inertia is as follows: For the torque deviation component obtained in step S4, since this component has a k0+1 order correlation property, an improved filter control strategy with k0+1 order is designed, and the frequency domain expression of the filter is obtained as follows: Then the model of the driving mechanism simulator is obtained, and the transfer function of the driving mechanism simulator is obtained as follows: The poles of the transfer function are found to be When these poles all fall within the unit circle, the simulator system is stable; When the pole is at zero, that is, Get the optimal solution of the system and calculate At this time, the simulation system responds fastest in a stable state, and the transfer function of the system is: Performing the inverse z transform on the expression, the time domain response expression of the system using the designed k0+1 order filter is obtained as follows: It was found that the obtained model was still affected by time delay. This component, but when the time series k>k0+1, u(k)-u(k-k0-1) always tends to zero, so the existence of this component will not affect the stability of the system.

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