CN110879526A - Fractional order controller and parameter setting method thereof - Google Patents

Fractional order controller and parameter setting method thereof Download PDF

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CN110879526A
CN110879526A CN201911180530.8A CN201911180530A CN110879526A CN 110879526 A CN110879526 A CN 110879526A CN 201911180530 A CN201911180530 A CN 201911180530A CN 110879526 A CN110879526 A CN 110879526A
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fractional order
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郑伟佳
邓敏
李欣
罗映
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Foshan University
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Abstract

The invention discloses a fractional order controller and a parameter setting method of the fractional order controller, wherein the fractional order controller improves an open loop target transfer function of a control system applied by the fractional order controller, and expresses the transfer function C(s) of the fractional order controller as an expression shown in formula 9, so that an amplitude-frequency characteristic curve of the control system applied by the fractional order controller has higher slopes in a high frequency band and a low frequency band, and the control system applied by the fractional order controller has better steady-state tracking performance and load disturbance resistance performance.

Description

Fractional order controller and parameter setting method thereof
Technical Field
The invention relates to the technical field of controllers, in particular to a fractional order controller and a parameter setting method thereof.
Background
At present, a feedback control mode based on an output error is widely used in a servo system, and the used controller is mainly a traditional PID controller. The PID control has the advantages of simple structure, easy realization and the like, but the problems of overlarge overshoot, overlong adjusting time and the like easily occur to a system with model uncertainty, and the performance index requirements of a high-performance motion control system can not be met. Therefore, the scholars have proposed a fractional order PID controller and other controller forms based on the fractional order PID controller based on the traditional PID controller. Compared with the traditional integer order PID controller, the fractional order controller has the advantages of larger value range and wider performance regulation range, so that the fractional order controller adopted by an actual system can obtain better control performance than the integer order PID controller. But at the same time the design of the fractional order controller is also more complex than the design of the integer order PID controller.
A fractional order controller based on Bode Ideal Transfer Function (BITF) is a new controller form based on a PID controller, and the Bode Ideal Transfer Function is:
Figure BDA0002291127580000011
where s represents the Laplace operator, ωcRepresenting the open loop cutoff frequency, ξ representing the fractional order, ξ satisfies ξ e (1, 2).
The frequency characteristic of the Bode ideal transfer function has the following characteristics:
first, GB(j ω) the amplitude-frequency characteristic crosses the 0dB line with a slope of-20 ξ dB/dec;
second, GBThe phase margin of (j ω) is constant, and its specific value is
Figure BDA0002291127580000012
Setting a transfer function of a controlled object as follows:
Figure BDA0002291127580000021
where a, b, c, K, α, and β are all object model parameters.
The design method of the controller based on the Bode ideal transfer function in the prior art is as follows:
firstly, according to the design index of a control system, setting an open-loop cut-off frequency omegacAnd phase margin
Figure BDA0002291127580000022
The fractional order ξ is calculated according to the following equation:
Figure BDA0002291127580000023
assuming the controller transfer function as C(s), taking Bode ideal transfer function as the open-loop target transfer function of the control system, the following equations are listed:
Figure BDA0002291127580000024
the controller transfer function based on the Bode ideal transfer function is obtained as follows:
Figure BDA0002291127580000025
from the above expressions, the controller based on the Bode ideal transfer function includes a fractional PID controller and a fractional integrator.
When the existing controller based on the Bode ideal transfer function is applied to a motion control system, the load disturbance resistance of the system is poor, for example, the controller based on the Bode ideal transfer function is used as a feedback controller to control the rotating speed of a motor, and when the load of the system suddenly changes, the rotating speed of the motor cannot be recovered to a set value in a short time.
Disclosure of Invention
The present invention is directed to a fractional controller and a parameter tuning method for the fractional controller, so as to solve one or more technical problems in the prior art and provide at least one useful choice or creation condition.
The technical scheme adopted for solving the technical problems is as follows:
a fractional order controller is provided, wherein a transfer function of the fractional order controller is set as C(s), an open-loop target transfer function of a control system applied by the fractional order controller is set as L(s), and a transfer function of a controlled object of the control system is set as G(s), wherein s represents a Laplace operator;
let the open-loop target transfer function of the control system be expressed as:
Figure BDA0002291127580000031
wherein G isf(s) denotes a PI controller, ωcRepresenting the open-loop cut-off frequency, G, of the control systemBICO(s) represents Bode ideal cut filter, ξ represents first fractional order;
the PI controller Gf(s) is expressed as:
Figure BDA0002291127580000032
wherein T is1Represents an integration time constant;
the Bode ideal cut-off filter GBICO(s) is expressed as:
Figure BDA0002291127580000033
wherein ω iscutoffRepresents the truncation frequency, gamma represents the second fractional order, and meets gamma epsilon (0, 1);
let the transfer function of the controlled object be G(s) as:
Figure BDA0002291127580000034
wherein a, b, c, K, α, and β are object model parameters;
according to the relation l(s) ═ c(s) g(s), the transfer function c(s) of the fractional order controller is expressed as:
Figure BDA0002291127580000035
ξ denotes a first fractional order, the first fractional order ξ expression is as follows:
Figure BDA0002291127580000036
wherein Arg [ G ]f(jωc)]Represents the PI controller Gf(s) at open loop cut-off frequency ωcAt the phase Arg [ G ]BICO(jωc)]Representing the Bode ideal cut-off filter GBICO(s) at open loop cut-off frequency ωcThe phase of (a) is determined,
Figure BDA0002291127580000041
representing a phase margin of the control system.
The application also discloses a parameter setting method of the fractional order controller, which comprises the following steps:
step 100, setting each object model parameter of the transfer function G(s) of the controlled object;
step 200, selecting a PI controller G according to the performance requirement of the control system applied by the fractional order controllerfTime constant integral T of(s)1；
Step 300, selecting a Bode ideal cut-off filter G according to the performance requirement of the control system applied by the fractional order controllerBICO(s) cutoff frequency ωcutoffAnd a second fractional order γ;
step 400 of setting the open loop cut-off frequency ω of the control system to which said fractional order controller appliescAnd phase margin
Figure BDA0002291127580000042
Step 500, cutting off the open loop cut-off frequency omegacAnd the phase margin
Figure BDA0002291127580000043
Substituting the following equation to obtain a first fractional order ξ for the control system:
Figure BDA0002291127580000044
wherein Arg [ G ]f(jωc)]Represents the PI controller Gf(s) at open loop cut-off frequency ωcAt the phase Arg [ G ]BICO(jωc)]Representing the Bode ideal cut-off filter GBICO(s) at open loop cut-off frequency ωcThe phase of (d);
step 600, the PI controller G and each object model parameter of the transfer function of the controlled objectfTime constant integral T of(s)1Said Bode ideal cut-off filter GBICO(s) cutoff frequency ωcutoffAnd a second fractional order gamma, the open loop cut-off frequency omega of the control systemcAnd phase margin
Figure BDA0002291127580000045
And substituting the first fractional order ξ of the control system into equation 9 to obtain the transfer function c(s) of the fractional order controller.
The invention has the beneficial effects that: according to the technical scheme, the open-loop target transfer function of the control system applied to the fractional order controller is improved, and the transfer function C(s) of the fractional order controller is expressed as an expression shown in a formula 9, so that an amplitude-frequency characteristic curve of the control system applied to the fractional order controller has higher slope in a high-frequency band and a low-frequency band, and the control system applied to the fractional order controller has better steady-state tracking performance and load disturbance resistance performance.
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The invention is further described with reference to the accompanying drawings and examples;
FIG. 1 is a schematic diagram showing the comparison of the amplitude-frequency characteristics between the control system applied to the fractional order controller of the present invention and the control system applied to the existing fractional order controller;
FIG. 2 is a flow chart of a design method of the fractional order controller of the present invention.
Detailed Description
Reference will now be made in detail to the present preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to like elements throughout.
In the description of the present invention, it should be understood that the orientation or positional relationship referred to in the description of the orientation, such as the upper, lower, front, rear, left, right, etc., is based on the orientation or positional relationship shown in the drawings, and is only for convenience of description and simplification of description, and does not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus, should not be construed as limiting the present invention.
In the description of the present invention, if words such as "a plurality" are described, the meaning is one or more, the meaning of a plurality is two or more, more than, less than, more than, etc. are understood as excluding the present number, and more than, less than, etc. are understood as including the present number.
In the description of the present invention, unless otherwise explicitly limited, terms such as arrangement, installation, connection and the like should be understood in a broad sense, and those skilled in the art can reasonably determine the specific meanings of the above terms in the present invention in combination with the specific contents of the technical solutions.
The application discloses a fractional order controller, wherein in a first embodiment of the fractional order controller, a transfer function of the fractional order controller is set as C(s), an open-loop target transfer function of a control system applied by the fractional order controller is set as L(s), and a transfer function of a controlled object of the control system is set as G(s), wherein s represents a Laplace operator;
let the open-loop target transfer function of the control system be expressed as:
Figure BDA0002291127580000061
wherein G isf(s) denotes a PI controller, ωcRepresenting the open-loop cut-off frequency, G, of the control systemBICO(s) represents Bode ideal cut filter, ξ represents first fractional order;
the PI controller Gf(s) is represented by：
Figure BDA0002291127580000062
Wherein T is1Represents an integration time constant;
the Bode ideal cut-off filter GBICO(s) is expressed as:
Figure BDA0002291127580000063
wherein ω iscutoffRepresents the truncation frequency, gamma represents the second fractional order, and meets gamma epsilon (0, 1);
let the transfer function of the controlled object be G(s) as:
Figure BDA0002291127580000064
wherein a, b, c, K, α, and β are object model parameters;
according to the relation l(s) ═ c(s) g(s), the transfer function of the fractional order controller is calculated as c(s), and expressed as:
Figure BDA0002291127580000065
ξ denotes a first fractional order, the first fractional order ξ expression is as follows:
Figure BDA0002291127580000066
wherein Arg [ G ]f(jωc)]Represents the PI controller Gf(s) at open loop cut-off frequency ωcAt the phase Arg [ G ]BICO(jωc)]Representing the Bode ideal cut-off filter GBICO(s) at open loop cut-off frequency ωcThe phase of (a) is determined,
Figure BDA0002291127580000071
representing phases of said control systemA bit margin.
According to the technical scheme, the open-loop target transfer function of the control system applied to the fractional order controller is improved, and the transfer function C(s) of the fractional order controller is expressed as an expression shown in a formula 9, so that an amplitude-frequency characteristic curve of the control system applied to the fractional order controller has higher slope in a high-frequency band and a low-frequency band, and the control system applied to the fractional order controller has better steady-state tracking performance and load disturbance resistance performance.
The fractional order controller disclosed in the present application will be described below by referring to a specific control system, compared with the effect of the prior art.
Now, the index parameter of the control system to which the fractional order controller is applied is set, and the open loop cut-off frequency ω of the control system is setcIs set to omegac30rad/s, adjusting the phase margin of the control system
Figure BDA0002291127580000072
Is set as
Figure BDA0002291127580000073
The fractional order controller is set according to the prior art in the background art, the obtained amplitude-frequency characteristic curve of the applied control system is shown as a curve shown by a solid line in fig. 1, the fractional order controller is set according to the technical scheme of the embodiment, and the obtained amplitude-frequency characteristic curve of the applied control system is shown as a curve shown by a dotted line in fig. 1.
As can be seen from the two curves in fig. 1, in the vicinity of the intermediate frequency band, the amplitude-frequency characteristics of the control system applied to the fractional controller according to the prior art are very close to those of the control system applied to the fractional controller according to the present embodiment, and in the low frequency band and the high frequency band, the slope of the amplitude-frequency characteristics of the control system applied to the fractional controller according to the present embodiment is larger, in the low frequency band, the slope of the amplitude-frequency characteristics of the control system applied to the fractional controller according to the present embodiment is-20 (ξ +1) dB/dec, while in the low frequency band, the slope of the amplitude-frequency characteristics of the control system applied to the fractional controller according to the prior art is-20 (20 ξ dB/dec, increased by-20 dB/dec, in the high frequency band, and in the control system applied to the fractional controller according to the present embodiment, the slope of the amplitude-frequency characteristics of the control system is-20 (ξ + γ) dB/dec, whereas in the control system applied to the fractional controller according to the prior art, the slope of the amplitude-frequency characteristics of the fractional controller according to the present embodiment is increased by-20 (20 ξ dB/dec), and the control system has better performance against the steady-20/dec.
Referring to fig. 2, the present application also discloses a method for tuning a parameter of a fractional order controller, wherein a first embodiment of the method comprises the following steps:
step 100, setting each object model parameter of the transfer function G(s) of the controlled object;
step 200, selecting a PI controller G according to the performance requirement of the control system applied by the fractional order controllerfTime constant integral T of(s)1；
Step 300, selecting a Bode ideal cut-off filter G according to the performance requirement of the control system applied by the fractional order controllerBICO(s) cutoff frequency ωcutoffAnd a second fractional order γ;
step 400 of setting the open loop cut-off frequency ω of the control system to which said fractional order controller appliescAnd phase margin
Figure BDA0002291127580000081
Step 500, cutting off the open loop cut-off frequency omegacAnd the phase margin
Figure BDA0002291127580000082
Substituting the following equation to obtain a first fractional order ξ for the control system:
Figure BDA0002291127580000083
wherein Arg [ G ]f(jωc)]To representThe PI controller Gf(s) at open loop cut-off frequency ωcAt the phase Arg [ G ]BICO(jωc)]Representing the Bode ideal cut-off filter GBICO(s) at open loop cut-off frequency ωcThe phase of (d);
step 600, the PI controller G and each object model parameter of the transfer function of the controlled objectfTime constant integral T of(s)1Said Bode ideal cut-off filter GBICO(s) cutoff frequency ωcutoffAnd a second fractional order gamma, the open loop cut-off frequency omega of the control systemcAnd phase margin
Figure BDA0002291127580000084
And substituting the first fractional order ξ of the control system into equation 9 to obtain the transfer function c(s) of the fractional order controller.
The specific steps of the method for tuning the parameters of the fractional order controller according to this embodiment will be described below by taking a permanent magnet synchronous motor servo system for a machine tool as an example.
According to step 100 in the method for tuning the fractional order controller parameter, firstly, a transfer function G(s) of a controlled object is set according to the controlled object of a servo system as follows:
Figure BDA0002291127580000091
step 200, selecting a PI controller G according to the performance requirement of the control system applied by the fractional order controllerfTime constant integral T of(s)10.2s, PI controller Gf(s) is represented as follows:
Figure BDA0002291127580000092
step 300, selecting a Bode ideal cut-off filter G according to the performance requirement of the control system applied by the fractional order controllerBICO(s) cutoff frequency ωcutoff300rad/s and a second fractional order γ of 0.6, thus yielding Bode ideal cut-off filteringDevice GBICO(s) the following:
Figure BDA0002291127580000093
step 400 of setting the open loop cut-off frequency ω of the control system to which said fractional order controller appliesc35rad/s and phase margin
Figure BDA0002291127580000094
Step 500, cutting off the open loop cut-off frequency omegacAnd the phase margin
Figure BDA0002291127580000095
Substituting the following equation to obtain a first fractional order ξ for the control system:
Figure BDA0002291127580000096
step 600, the PI controller G and each object model parameter of the transfer function of the controlled objectfTime constant integral T of(s)1Said Bode ideal cut-off filter GBICO(s) cutoff frequency ωcutoffAnd a second fractional order gamma, the open loop cut-off frequency omega of the control systemcAnd phase margin
Figure BDA0002291127580000097
And substituting the first fractional order ξ of the control system into equation 9, to obtain the transfer function c(s) of the fractional order controller as follows:
Figure BDA0002291127580000098
while the preferred embodiments of the present invention have been illustrated and described, it will be understood by those skilled in the art that the present invention is not limited to the details of the embodiments shown and described, but is capable of numerous equivalents and substitutions without departing from the spirit of the invention as set forth in the claims appended hereto.

Claims (2)

1. A fractional order controller, characterized by:
setting a transfer function of the fractional order controller as C(s), setting an open-loop target transfer function of a control system applied by the fractional order controller as L(s), and setting a transfer function of a controlled object of the control system as G(s), wherein s represents a Laplace operator;
let the open-loop target transfer function of the control system be expressed as:
Figure FDA0002291127570000011
wherein G isf(s) denotes a PI controller, ωcRepresenting the open-loop cut-off frequency, G, of the control systemBICO(s) represents Bode ideal cut filter, ξ represents first fractional order;
the PI controller Gf(s) is expressed as:
Figure FDA0002291127570000012
wherein T is1Represents an integration time constant;
the Bode ideal cut-off filter GBICO(s) is expressed as:
Figure FDA0002291127570000013
wherein ω iscutoffRepresents the truncation frequency, gamma represents the second fractional order, and meets gamma epsilon (0, 1);
let the transfer function of the controlled object be G(s) as:
Figure FDA0002291127570000014
wherein a, b, c, K, α, and β are object model parameters;
according to the relation l(s) ═ c(s) g(s), the transfer function c(s) of the fractional order controller is expressed as:
Figure FDA0002291127570000015
ξ denotes a first fractional order, the first fractional order ξ expression is as follows:
Figure FDA0002291127570000021
wherein Arg [ G ]f(jωc)]Represents the PI controller Gf(s) at open loop cut-off frequency ωcAt the phase Arg [ G ]BICO(jωc)]Representing the Bode ideal cut-off filter GBICO(s) at open loop cut-off frequency ωcThe phase of (a) is determined,
Figure FDA0002291127570000022
representing a phase margin of the control system.
2. A method for tuning parameters of a fractional order controller is characterized by comprising the following steps:
step 100, setting each object model parameter of the transfer function G(s) of the controlled object;
step 200, selecting a PI controller G according to the performance requirement of the control system applied by the fractional order controllerfTime constant integral T of(s)1；
Step 300, selecting a Bode ideal cut-off filter G according to the performance requirement of the control system applied by the fractional order controllerBICO(s) cutoff frequency ωcutoffAnd a second fractional order γ;
step 400 of setting the open loop cut-off frequency ω of the control system to which said fractional order controller appliescAnd phase margin
Figure FDA0002291127570000023
Step 500, cutting off the open loop cut-off frequency omegacAnd the phase margin
Figure FDA0002291127570000024
Substituting the following equation to obtain a first fractional order ξ for the control system:
Figure FDA0002291127570000025
wherein Arg [ G ]f(jωc)]Represents the PI controller Gf(s) at open loop cut-off frequency ωcAt the phase Arg [ G ]BICO(jωc)]Representing the Bode ideal cut-off filter GBICO(s) at open loop cut-off frequency ωcThe phase of (d);
step 600, the PI controller G and each object model parameter of the transfer function of the controlled objectfTime constant integral T of(s)1Said Bode ideal cut-off filter GBICO(s) cutoff frequency ωcutoffAnd a second fractional order gamma, the open loop cut-off frequency omega of the control systemcAnd phase margin
Figure FDA0002291127570000026
And substituting the first fractional order ξ of the control system into equation 9 to obtain the transfer function c(s) of the fractional order controller.
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