A Servo System Controller Design Method

Abstract

The present invention relates to control system design and specifically discloses a servo system controller design method, including: setting the error transfer function to 0 and listing the feedforward controller transfer function; designing a feedback controller closed-loop transfer function that does not include a feedforward link, the steps being as follows: designing five poles in the closed-loop transfer function of the position loop, with S1 and S2 being conjugate poles and S3, S4, and S5 being negative real poles; performing pole configuration on the denominator of the position loop transfer function based on the five poles, and listing the position loop transfer function with the pole configuration; solving for the fifth-order pole configuration of the feedback controller, solving for the gain coefficient solutions of the proportional, integral, and differential links of the PID controller, and determining the transfer function of the feedback controller; combining the feedforward controller with the feedback controller and applying them to the controlled object to improve the resonant mode. The two-degree-of-freedom closed-loop servo system designed in this paper has good dynamic performance and vibration suppression effect compared to a filter feedback control system.

Description

Technical Field

[0001] The present invention belongs to the technical field of control systems, and in particular relates to a servo system controller design method. Background Art

[0002] Due to mechanical design and economic cost issues, flexible links in servo systems are inevitable. Flexible links not only cause system vibration, but also reduce the system's resonant frequency, affect the system bandwidth, and make system control more difficult.

[0003] In order to reduce the impact of system vibration, a feedback link is usually added to the control system in the existing technology. However, feedback control can only solve the problems of system stability and interference suppression characteristics, and does not improve the trajectory tracking performance of the servo system. Poor trajectory tracking performance means poor system stability.

[0004] Therefore, current high-precision servo systems usually adopt two-degree-of-freedom control that combines feedforward and feedback. That is, two independent controllers are designed with two independent sets of parameters. The feedback controller is used to solve the interference suppression problem of the servo system, and the feedforward controller is used to optimize the following performance of the system.

[0005] However, when using two-degree-of-freedom control, the mechanical system will produce resonant modes, which will affect the dynamic performance of the system. In order to reduce the impact of the resonant modes, the existing technology generally adds a filter inside the system and outside the controller. However, using this method, the feedback filter controller is complex in design and has a slow dynamic response. The improvement effect of the resonant mode is limited, and the dynamic performance of the system and the vibration suppression effect are not good. Summary of the Invention

[0006] The present invention provides a servo system controller design method for solving the problem that the current resonance mode affects the dynamic performance and response speed of the system.

[0007] In order to solve the above technical problems, the technical solution of the present invention is: a servo system controller design method, which includes the following steps:

[0008] Step 1: Set the error transfer function to 0, design the feedforward controller, and list the feedforward controller transfer function;

[0009] Step 2: Design the closed-loop transfer function of the feedback controller without the feedforward link. The steps are as follows:

[0010] Design five poles S1, S2, S3, S4 and S5 in the closed-loop transfer function of the position loop, where S1 and S2 are conjugate poles and S3, S4 and S5 are negative real poles. Based on the five poles, perform pole configuration on the denominator in the transfer function of the position loop and list the position loop transfer function with the pole configuration.

[0011] Based on the closed-loop transfer function of the position loop and the position loop transfer function of the pole configuration, the fifth-order pole configuration of the feedback controller is solved, the gain coefficient solutions of the proportional, integral, and differential links of the PID controller are solved, and the transfer function of the feedback controller is determined;

[0012] Step 3: Combine the feedforward controller transfer function in Step 1 with the feedback controller transfer function in Step 2 and apply it to the controlled object to improve the resonant mode.

[0013] In a preferred embodiment of the present invention, in step 1, the error transfer function Formula (1):

[0014] (1);

[0015] in, is the position loop controlled object, is the resonance mode, is the system controller transfer function, is the feedforward controller transfer function;

[0016] The feedforward controller function is formula (2):

[0017] (2);

[0018] in, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, is the corresponding damping ratio.

[0019] In a preferred embodiment of the present invention, the position loop closed-loop transfer function excluding the feedforward link in Step 2 is formula (3):

[0020] (3);

[0021] in, is the position loop controlled object, is the resonance mode, is the system controller transfer function, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, is the corresponding damping ratio.

[0022] In a preferred embodiment of the present invention, in step 2, the expressions of the five extreme points S1, S2, S3, S4 and S5 are as follows:

[0023] (4)

[0024] (5)

[0025] (6)

[0026] (7);

[0027] in, is the damping coefficient, is the undamped natural oscillation frequency, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5.

[0028] In a preferred embodiment of the present invention, in step 2, the mathematical expression after the pole configuration of the denominator in the transfer function of the position loop is formula (8):

[0029] (8);

[0030] Where, is the damping coefficient, is the undamped natural oscillation frequency, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5.

[0031] In a preferred embodiment of the present invention, in step 3, the process of solving the gain coefficients of the proportional, integral, and differential links of the PID controller is as follows:

[0032] Combine equation (2) and equation (8) to form a system of equations;

[0033] Take the proportional coefficient of the pole S3 closest to the imaginary axis For a fixed value, list 、 about Expressions of

[0034] Substituting the above expressions into the derived system of equations, we obtain 、 、 Solve the three-variable linear equations and find the gain coefficient solutions of the proportional, integral and differential links of the PID controller.

[0035] In a preferred embodiment of the present invention, the equation group formed by equation (2) and equation (8) is equation (9):

[0036] (9);

[0037] in, is the damping coefficient, is the undamped natural oscillation frequency, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency.

[0038] In a preferred embodiment of the present invention, 、 about The expression of is formula (10):

[0039] (10);

[0040] in, is the corresponding damping ratio, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, M and N are intermediate quantities.

[0041] In a preferred embodiment of the present invention, ∈[0.6, 0.8], =1.

[0042] In a preferred embodiment of the present invention, after simplifying equation (9), equation group (11) is obtained:

[0043] (11);

[0044] The gain coefficients of the proportional, integral, and differential links are solved as Equations (12) to (14):

[0045] (12)

[0046] (13)

[0047] (14);

[0048] Where A~F are the intermediate quantities in equations (12) to (14), as follows:

[0049] (15)

[0050] (16)

[0051] (17)

[0052] (18)

[0053] (19)

[0054] (20);

[0055] in, is the damping coefficient, is the corresponding damping ratio, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency.

[0056] The technical solution provided by this invention offers the following advantages over existing technologies: It proposes a novel method for designing a two-degree-of-freedom controller that considers the resonant modes of a servo system. By combining a feedforward controller with a feedback controller, the resonant modes of the servo system are incorporated into the controlled object. The feedforward controller is designed based on the invariance principle, while a PID feedback controller is designed using a novel fifth-order pole configuration. Specific parameter design methods are also provided. Finally, simulation experiments demonstrate that the designed two-degree-of-freedom closed-loop servo system exhibits superior dynamic performance and vibration suppression compared to a filter feedback control system. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] 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 only some embodiments of the present invention. For ordinary technicians in this field, without paying any creative work, they can also obtain drawings of other embodiments based on these drawings.

[0058] Figure 1 is a Bode diagram of a first-order resonance mode of a servo system controller design method according to an embodiment of the present invention;

[0059] Figure 2 is a closed-loop pole position under the influence of a resonance mode of a servo system controller design method according to an embodiment of the present invention;

[0060] Figure 3 is a structural diagram of a two-degree-of-freedom controller of a servo system controller design method described in one embodiment of the present invention;

[0061] Figure 4 is a system expected pole position diagram of a servo system controller design method according to an embodiment of the present invention;

[0062] Figure 5 is the position error under the filter controller of a servo system controller design method described in one embodiment of the present invention;

[0063] Figure 6 It is the position error under a two-degree-of-freedom controller of a servo system controller design method described in one embodiment of the present invention. DETAILED DESCRIPTION

[0064] For ease of understanding, a servo system controller design method is described below in conjunction with embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention.

[0065] In the description of the present invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicating orientations or positional relationships, are based on the orientations and positional relationships shown in the accompanying drawings and are intended solely to facilitate and simplify the description of the present invention. They are not intended to indicate or imply that the devices or components referred to must have, be constructed, or operate in a specific orientation, and therefore should not be construed as limitations on the present invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0066] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood broadly. For example, they may refer to fixed, detachable, or integral connections; mechanical or electrical connections; direct or indirect connections through an intermediary; and internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention.

[0067] To facilitate understanding of the present invention, the present invention will be described more fully below with reference to the accompanying drawings. The accompanying drawings illustrate preferred embodiments of the present invention. However, the present invention may be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a more thorough and comprehensive understanding of the present disclosure.

[0068] 1. List the mathematical model of the position loop controlled object in the mechanical system

[0069] Since the servo system usually has a three-loop control structure, namely the current loop, the speed loop, and the position loop, the controller design of the present invention is mainly based on the regulator of the position loop, and the speed loop is replaced by its closed-loop transfer function. The speed loop is equivalent to a link that can reflect the main characteristics of the position loop. When the cutoff frequency of the position loop is much smaller than the cutoff frequency of the speed loop, the closed-loop transfer function of the speed loop can be equivalent to an inertia link. The position loop is the integral of the speed loop, so the mathematical model of the controlled object of the position loop can be expressed as formula (1):

[0070] (1);

[0071] in, is the position loop object gain, is the damping coefficient of the position loop object.

[0072] 2. List a mathematical model of the resonance mode

[0073] Resonance refers to the phenomenon that when the frequency of the excitation to a mechanical system is close to a certain order of the system's natural frequency, the amplitude of the system increases significantly. The order of a system's resonant mode is determined by the system's mechanical structure, and a system often has multiple resonant modes. In the controller design process, usually only one resonant mode with the lowest resonant frequency and the largest resonant amplitude needs to be considered. At this time, the system is most susceptible to resonance caused by external excitation, which has the greatest impact on the system's performance. The mathematical model of a resonant mode is the product of a second-order differential link and an oscillation link. Its mathematical model is formula (2):

[0074] (2);

[0075] in, is the anti-resonance frequency, is the resonant frequency, is the corresponding damping ratio. Damping ratio and resonance amplitude , anti-resonance amplitude The relationship is as follows:

[0076] (3)

[0077] (4)

[0078] At this time, the Bode diagram of the resonance mode expressed by equation (2) is as follows: Figure 1 shown.

[0079] like Figure 2 As shown in the figure, due to the simplified modeling of the current loop and speed loop, the object model of the position loop is only an approximate model. At this time, even if the feedforward link of the system is designed according to the invariance principle, the closed-loop system will not obtain good dynamic and static performance.

[0080] 3. List the closed-loop transfer function and error transfer function of the system

[0081] The two-degree-of-freedom controller of the servo system is designed based on the mathematical model of the controlled object that already includes the resonance mode. The feedforward controller is designed based on the invariance principle. The structure of the two-degree-of-freedom composite control is as follows: Figure 3 As shown, Given by the system, is the controlled quantity of the system, is the position ring object, is the resonance mode, is the system controller transfer function, is the feedforward controller transfer function, then the closed-loop transfer function of the system is:

[0082] (5)

[0083] in, Given by the system, is the controlled quantity of the system, is the position loop controlled object, is the resonance mode, is the system controller transfer function, is the feedforward controller transfer function.

[0084] The error transfer function at this time It can be expressed as:

[0085] (6);

[0086] in, is the position loop controlled object, is the resonance mode, is the system controller transfer function, is the feedforward controller transfer function.

[0087] 4. List the feedforward controller transfer function

[0088] The feedforward controller is designed according to the invariance principle, that is, the system can completely reproduce the input signal, and the transient error and steady-state error are both zero. From formula (6), we can see that when When , the error transfer function of the system is zero, and the feedforward controller transfer function is (7):

[0089] (7);

[0090] in, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, is the corresponding damping ratio.

[0091] exist and When the mathematical models of are known, the mathematical model of the feedforward controller can be obtained by formula (7). and The mathematical model of can be obtained by the method of system identification, so it can be considered that the parameters Known.

[0092] 5. List the transfer function of the PID controller

[0093] The transfer function of the PID controller can be expressed as:

[0094] (8)

[0095] in, 、 、 are the gain coefficients of the proportional, integral, and differential links respectively.

[0096] 6. List the position loop closed-loop transfer function excluding the feedforward link

[0097] Reference Figure 3 As shown in the two-degree-of-freedom controller structure diagram, in order to make the closed-loop system have ideal dynamic and static performance, the position loop transfer function in the feedback link does not include the position loop controlled object in the feedforward link.

[0098] The closed-loop transfer function of the position loop is:

[0099] (9);

[0100] in, is the position loop controlled object, is the resonance mode, is the system controller transfer function, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, is the corresponding damping ratio.

[0101] 7. Design five extreme points and list the mathematical expressions

[0102] Since the denominator of the closed-loop transfer function is fifth order, the corresponding closed-loop system also has five poles. Pole configuration is a PID controller design method that configures the closed-loop poles of the servo system to the desired pole positions to obtain good system performance indicators. According to the concept of dominant poles, that is, the farther the pole is from the imaginary axis, the less influence it has on the dynamic response of the system, when configuring multi-order poles, it is usually simplified to a pair of conjugate poles for analysis. Based on the traditional pole configuration method with three closed-loop poles, the present invention assigns these five pole positions according to Figure 4 Configuration shown.

[0103] exist Figure 4 In the equation (10), S1 and S2 are a pair of conjugate poles, and S3, S4, and S5 are negative real poles. The mathematical expressions are shown in equations (10) to (13):

[0104] (10)

[0105] (11)

[0106] (12)

[0107] (13);

[0108] in, is the damping coefficient, is the undamped natural oscillation frequency, 、 、 are the proportional coefficients of the three negative real poles.

[0109] 8. Perform pole placement on the denominator in the position loop transfer function

[0110] The expression after pole configuration of the denominator of the position loop closed-loop transfer function is formula (14):

[0111] (14);

[0112] in, is the damping coefficient, is the undamped natural oscillation frequency, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5.

[0113] 9. Determine the gain coefficients of the proportional, integral, and differential components of the PID controller

[0114] 9.1 Comparing Equations (9) and (14), we can obtain the equation system shown in Equation (15): (15).

[0115] 9.2 According to the servo system performance requirements, take the proportional coefficient of the pole S3 closest to the imaginary axis is a constant. At this time, the unknown quantity in equation (15) is 、 、 、 、 , we can simplify it to get:

[0116] (16)

[0117] (17)

[0118] Formula (16) is about 、 The nonlinear equations of Eq. (17) are about 、 、 A system of three linear equations.

[0119] 9.3 When the remaining parameters are known, solving Equations (16) and (17) can complete the fifth-order pole configuration of the PID feedback controller. The solution expressions are shown in Equations (18) to (20).

[0120] (18)

[0121] (19)

[0122] (20);

[0123] in:

[0124] (twenty one)

[0125] (twenty two)

[0126] (twenty three)

[0127] (twenty four)

[0128] (25)

[0129] (26).

[0130] 10. Combine the feedforward controller transfer function in S3 with the feedback controller transfer function in S6 and apply it to the controlled plant to improve the resonant modes.

[0131] In the actual control system design, the dynamic and static characteristics of the system and the characteristics of the transmission components should be given. 、 and ,in, The larger the value, the faster the system responds, but too large a value will cause system instability. In order to ensure that the system has sufficient phase margin and good response speed, The value is generally between 0.6 and 0.8. In this embodiment, The value is 0.707, The value is 1.

[0132] In the present invention, the effectiveness of the proposed two-degree-of-freedom controller design considering the resonance mode can be verified by means of SIMULINK simulation experiments. The simulation data are as follows: =3055, =20, =90π, =0.707, =1; the mechanical resonance mode of the system is an anti-resonance with an amplitude of -20dB at 250Hz and a resonance with an amplitude of 20dB at 380Hz.

[0133] In order to better verify the performance of the two-degree-of-freedom controller proposed in this paper under the influence of resonant modes, we use the same S-curve acceleration command to simulate the position error of the controller designed in this paper and compare it with the position error of the controller filtered by a band-stop filter. Figure 5 is the position error curve under the filter controller, Figure 6 This is the position error curve under the two-degree-of-freedom controller proposed in this paper.

[0134] Depend on Figure 5 and Figure 6 From the comparison, it can be seen that the two-degree-of-freedom controller proposed in this paper enables the servo system to have better vibration suppression ability and faster dynamic response.

[0135] This paper proposes a new approach to designing a two-degree-of-freedom controller that considers the resonant modes of a servo system. By incorporating the resonant modes of the servo system into the controlled object, a feedforward controller is designed based on the invariance principle, and a PID feedback controller is designed using a new fifth-order pole configuration. Specific parameter design methods are also provided. Finally, simulation experiments verify that the designed two-degree-of-freedom closed-loop servo system exhibits superior dynamic performance and vibration suppression compared to a filter feedback control system.

[0136] Finally, it should be noted that the above embodiments are intended only to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art will appreciate that the technical solutions described in the above embodiments may be modified or some or all of the technical features thereof may be replaced with equivalents, and that such modifications or replacements do not deviate from the essence of the corresponding technical solutions within the scope of the various embodiments of the present invention.

Claims

1. A servo system controller design method, characterized in that: The following steps are involved: Step 1: Set the error transfer function to 0, design a feedforward controller, and list the feedforward controller transfer function; Step 2: Design the closed-loop transfer function of the feedback controller without the feedforward link. The steps are as follows: Design five poles S1, S2, S3, S4 and S5 in the closed-loop transfer function of the position loop, where S1 and S2 are conjugate poles and S3, S4 and S5 are negative real poles. Based on the five poles, perform pole configuration on the denominator in the transfer function of the position loop and list the position loop transfer function with the pole configuration. Based on the closed-loop transfer function of the position loop and the position loop transfer function of the pole configuration, the fifth-order pole configuration of the feedback controller is solved, the gain coefficient solutions of the proportional, integral, and differential links of the PID controller are solved, and the transfer function of the feedback controller is determined; Step 3: Combine the feedforward controller transfer function in Step 1 with the feedback controller transfer function in Step 2 and apply them to the controlled object to improve the resonant mode. In step 1, the error transfer function Formula (1): （1） in, is the position loop controlled object, is the resonance mode, is the system controller transfer function, is the feedforward controller transfer function; The feedforward controller function is formula (2): (2)； in, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, is the corresponding damping ratio; The position loop closed-loop transfer function excluding the feedforward link in Step 2 is formula (3): (3)； in, is the position loop controlled object, is the resonance mode, is the system controller transfer function, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, is the corresponding damping ratio; In Step 2, the expressions of the five extreme points S1, S2, S3, S4 and S5 are as follows: (4) (5) (6) (7)； in, is the damping coefficient, is the undamped natural oscillation frequency, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5; In step 2, the mathematical expression after the pole configuration of the denominator in the transfer function of the position loop is formula (8): （8）； in, is the damping coefficient, is the undamped natural oscillation frequency, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5; In step 3, the process of solving the gain coefficients of the proportional, integral, and differential links of the PID controller is as follows: Combine equation (2) and equation (8) to form a system of equations; Take the proportional coefficient of the pole S3 closest to the imaginary axis For a fixed value, list 、 about Expressions of Substituting the expression into the resulting system of equations, we get 、 、 The three-variable linear equation system is used to find the gain coefficient solutions of the proportional, integral and differential links of the PID controller; The combined equations of equation (2) and equation (8) are equation (9): （9） in, is the damping coefficient, is the corresponding damping ratio, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency; 、 about The expression of is formula (10): (10)； in, is the corresponding damping ratio, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency, M and N are intermediate quantities.

2. A servo system controller design method according to claim 1, characterized in that: ∈[0.6，0.8]， =1。 3. A servo system controller design method according to claim 1, characterized in that: After simplifying formula (9), we get the equation group (11): （11）； The gain coefficients of the proportional, integral, and differential links are solved as Equations (12) to (14): （12） （13） （14）； Where A~F are the intermediate quantities in equations (12) to (14), as follows: （15） （16） （17） （18） （19） （20）； in, is the damping coefficient, is the corresponding damping ratio, is the proportional coefficient of the negative real point S3, is the proportional coefficient of the negative real point S4, is the proportional coefficient of the negative real point S5, is the gain coefficient of the proportional link, is the gain coefficient of the integral link, is the gain coefficient of the differential link, is the position loop object gain, is the damping coefficient of the position loop object, is the anti-resonance frequency, is the resonant frequency.

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