CN109507870B - Structure-adaptive fractional order proportional integral or proportional differential controller design method - Google Patents
Structure-adaptive fractional order proportional integral or proportional differential controller design method Download PDFInfo
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Abstract
The invention discloses a design method of a structure self-adaptive fractional order proportional integral or proportional derivative controller; firstly, inputting a sinusoidal signal and a capture output signal to a controlled object to obtain the amplitude and the phase of an open loop system of the controlled object at a specified gain crossover frequency; acquiring the slope of a phase-frequency curve of the controlled object by inputting another sinusoidal signal and capturing an output signal to the controlled object; and substituting the obtained three experimental data into the relationship among the amplitude, the phase and the phase slope of the constructed controlled object transfer function and three parameters of the fractional order controller, and obtaining the three parameters of the fractional order controller through calculation, wherein the three parameters comprise a proportionality coefficient, a time constant and a micro-integral order, and the micro-integral order determines whether the fractional order controller is a proportional integral or proportional differential structure. The method does not depend on a mathematical model of the controlled object, can automatically calculate the controller parameters through the self-tuning process of the controller parameters, and determines the controller structure.
Description
Technical Field
The invention relates to the field of automation, in particular to a design method of a structure-adaptive fractional order proportional integral or proportional derivative controller.
Background
In a closed-loop controller in an industrial process, a proportional-integral (PI) control or proportional-derivative (PD) control method having a simple structure is widely used. With the increasing control performance requirements of industrial processes, the requirements of traditional integer-order PI or PD control are difficult to meet. The engineering technicians are forced to search for the control method with more excellent performance, and the control method is simple and easy to use and convenient for engineering realization.
Fractional calculus theory has been in history for over 300 years. Through continuous improvement of the theory and great progress of scientific technology of people in recent years, the fractional calculus theory is gradually applied to the control field. And a fractional order calculus link is used for replacing an integer order calculus link in PI control and PD control, so that the fractional order PI control and the fractional order PD control are derived. Many scholars verify from theoretical analysis and practical application that the fractional order PI or PD control method can determine better control effect.
In the conventional controller structure design, regardless of an integer-order or fractional-order controller, a certain knowledge of the physical characteristics of a controlled object is required, and then a decision is made as to whether PI control or PD control is adopted. In practical application, especially in some large-scale industrial processes with subsystem interaction, the structure and the operation mechanism are complex, it is usually difficult to obtain accurate dynamic characteristics and priori knowledge, it cannot be determined whether PI control or PD control is adopted, and parameters of the controller are also difficult to determine, and a large number of experiments are required for trial and error, which brings great difficulty to the structural design of the controller and increases the design complexity of the controller.
Disclosure of Invention
In view of this, the present invention provides a method for designing a structure-adaptive fractional order PI or PD controller, which does not depend on a mathematical model of a controlled object, and can automatically calculate parameters of the controller through a self-tuning process of the parameters of the controller, and determine the structure of the controller.
In order to solve the technical problem, the specific design process of the controller of the invention is as follows:
the transfer function expression of the fractional order PI or PD controller is expressed in the form of equation (1).
Hc(s)＝Kp(1+Tis-λ) (1)
In the formula (1), Hc(s) is a transfer function of the controller, and comprises three parameters, wherein T is a time constant, KpIs a proportionality coefficient, and λ is a real number, representing the order of calculus. When lambda is less than 0, the controller described by the formula (1) is a fractional order PD controller; when lambda is>At 0, the controller described by equation (1) is a fractional order PI controller, and it can be seen that the sign of the parameter λ determines the specific structure of the controller. The structure of the controller and the parameters of the controller can be determined by automatically setting the sizes of the three parameters.
In the invention, three equations are obtained by satisfying the following three conditions, and then three parameters T in the controller can be solved according to the obtained three equationsi，Kp，λ。
Condition 1: the controlled object open loop system has a specified gain crossover frequency omegagc
The gain crossover frequency is the angular frequency corresponding to the amplitude crossing 0 dB line on the Bode diagram of the transfer function of the controlled system. Designing a larger gain crossover frequency effectively shortens the settling time of the closed-loop control system. In order to obtain a specified gain crossover frequency for the controlled closed loop system, the condition described by equation (2) must be satisfied.
|H(jωgc)|＝|Hc(jωgc)|·|P(jωgc)|＝1 (2)
The transfer functions in equation (2) are all described in the frequency domain, where H (j ω) isgc) Is the open-loop transfer function of the system to be controlled, Hc(jωgc) Is the transfer function of the controller, P (j ω)gc) It is the transfer function of the controlled object.
Condition 2: open loop systems having a specified phase margin
The phase margin is the difference between the value of a phase frequency curve corresponding to the gain crossover frequency of the open-loop system and-180 degrees, and the magnitude of the phase margin is related to the overshoot of the step response of the closed-loop system. In general, a reasonable phase margin is selected within a range of about 45-65 °. In order for the closed loop system to achieve a specified phase margin, the condition described by equation (3) must be satisfied.
Symbol ≦ in equation (3) represents a phase angle.
Condition 3: equal damping characteristic at gain crossover frequency
The equal damping characteristic means that on an open-loop bode diagram of a controlled system, a phase-frequency curve is flat, namely, the corresponding phase slope is 0 at a specified frequency. This means that the controller is robust to changes in the open loop gain of the controlled object. In order for the system to obtain equal damping characteristics at the gain crossover frequency, the condition described by equation (4) must be satisfied.
In the formula (4), ω represents an angular frequency, and the three equations of the formulas (2), (3) and (4) can be obtained according to the three conditions.
Substituting s-j ω for equation (1) gives a complex description of the transfer function of the fractional order controller in the frequency domain:
then, the amplitude | H of the fractional order controller at the frequency ω can be calculatedC(j ω) | and phase ∠ HC(j ω) as shown in formula (6) and formula (7), respectively:
when formula (6) is substituted for formula (2), it is possible to obtain:
by substituting formula (7) for formula (3), it is possible to obtain:
by substituting formula (7) for formula (4), it is possible to obtain:
the above equations (8), (9) and (10) are the controlled object transfer function P (j ω)gc) Amplitude | P, (jωgc) I, phase ∠ P (j ωgc) And phase slope
And the relation between three parameters of the fractional order controller. As long as the controlled object can be calculated to be located at the gain crossover frequency omegagcAmplitude | P (j ω) ofgc) I, phase ∠ P (j ωgc) And phase slope
The parameters of the controller of equation (1) can be calculated by the following specific steps:
step 1: calculating the crossover frequency omega of the controlled object open loop system at the appointed gaingcAmplitude M and phase phi of
Inputting a sinusoidal signal u (t) to the controlled object:
u(t)＝Aisin(ωgct) (11)
wherein the amplitude of the input signal u (t) is AiAngular frequency is the gain crossover frequency omegagcAnd t is time.
Under the excitation of the sinusoidal signals, the controlled object outputs sinusoidal signals with the same period and different transition time and amplitude. The output signal of the controlled object is defined as y (t), and the amplitude of the output signal is AoThe transition time between the input and output signals is tau, tau being ti-toWherein, tiIs the peak time, t, of a peak x in the input signal u (t)oThe peak time of the same peak x in the output signal y (t). Then the gain crossover frequency omega of the open loop system can be obtainedgcAmplitude M and phase phi:
φ＝∠P(jωgc)＝ωgcτ＝ωgc(ti-to) (13)
step 2: calculating the slope of the Bode graph phase-frequency curve of the controlled object, and expressing the slope as
Inputting a sine signal u to the controlled object again1(t)：
u1(t)＝Ai1sin(ω1t) (14)
Wherein the input signal u1(t) has an amplitude of Ai1Preferably A is preferredi1＝AiAngular frequency of ω1＝ωgc(1+ α), α is a given positive real number less than 0.1, preferably α is 0.01. the purpose of setting a difference α is to approximate the derivative of equation (16) so that there is a difference in the angular frequency of the two signals and this difference approaches 0, so a small value α is set to meet this requirement.
The controlled object is located at the angular frequency ω1Phase phi of1Comprises the following steps:
φ1＝∠P(jω1)＝ω1τ1＝ω1(ti1-to1) (15)
in the above formula (15), ti1For an input signal u1(t) time of peak x1, to1Is composed of u1(t) the excited output signal y1(t) peak time of the same peak x 1.
According to the basic definition of the derivative, when the constant α takes a smaller value, the controlled object can be approximately calculated to be located at the gain crossover frequency ωgcSlope of the phase frequency curve at:
and step 3: calculating controller parameters:
experimental data calculated according to the equations (12), (13) and (16) include a controlled object transfer function P (j ω)gc) Amplitude of | P (j ω)gc) I, phase ∠ P (j ωgc) And phase slope
The three parameters T of the controller can be solved by being taken into the formulas (8), (9) and (10)i、Kpλ, λ. Wherein the parameter TiAnd KpAll real numbers are more than zero, and lambda is any real number, if lambda is less than 0, the controller is essentially a fractional order PD controller; if λ>0, the controller is essentially a fractional order PI controller.
Has the advantages that:
(1) the design method of the structure self-adaptive fractional order proportional integral or proportional derivative controller provided by the invention does not need to determine the specific structure of the controller in advance, and can determine whether PI control or PD control is adopted according to the result of automatic parameter setting.
(2) The controller design method provided by the invention does not need to acquire a mathematical model and related parameters of the controller in advance, and can calculate the parameters of the controller only through an experimental method.
Drawings
FIG. 1 is a closed loop system architecture for a fractional order controller.
Fig. 2 shows the input and output signals of an open-loop sinusoidal experiment.
Fig. 3 is a first open loop experiment at a given gain crossover frequency.
Fig. 4 is a second open loop experiment at a given gain crossover frequency.
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
The controlled object is a first-order time-lag process, namely
In the formula, K ═ 1 is an open loop gain, T ═ 3 is a time constant, and L ═ 0.1 is a dead time.
Designing the above controlled object with a fractional order controller as shown in FIG. 1, wherein R(s) is the transfer function of the input signal, E(s) is the transfer function of the error signalP(s) is the transfer function of the controlled object, U(s) is the transfer function of the control signal, Y(s) is the transfer function of the output signal, KPIs the proportional part, T, of the controlleris-λIs a differential or integral part of the transfer function of the controller.
In the present embodiment, it is assumed that the structural parameters K, T, L of the controlled object are unknown. Since the controlled object is a linear system with hysteresis, a sinusoidal excitation signal is input to the controlled object, and a sinusoidal signal with the same period and different transition time and amplitude is obtained, as shown in fig. 2.
In the controller design, the following control performance indexes are designed, namely: gain crossover frequency omegagc1rad/s, phase margin
The specific controller design process is as follows:
step 1: calculating amplitude M and phase phi at gain crossover frequency
Inputting a sinusoidal signal u (t) sin (t) with an amplitude A to the controlled objectiAngular frequency is the gain crossover frequency omega 1gc1 rad/s. Under the excitation of the sinusoidal signal, a sinusoidal signal with the same period and different transition time and amplitude as shown in fig. 3 can be obtained. Input signal u (t) having a peak time ti32.987s, the corresponding output signal y (t) has a peak time to34.336s, amplitude Ao0.3163. The amplitude of the controlled object at the gain crossover frequency can be obtained as
Phase phi ═ omegagc·(ti-to)＝-77.29°。
Step 2: calculating the slope of the phase frequency curve:
let α be 0.01, and input sine signal u (t) sin (1.01 × t) to the controlled object, the input and output signals shown in fig. 4 are obtained, where the input signal u is1(t) a peak time of ti1＝32.660sCorresponding to this is the output signal y1(t) peak time to134.000s, the phase of the controlled object at the frequency can be obtained as phi1＝ω1(ti1-to1) -77.54 o. Phase slope of controlled object at gain crossover frequency
And step 3: calculating controller parameters:
the amplitude | P (j ω) obtained by the experimentgc) M, phase ∠ P (j ω |)gc) Phi and phase slope
Substituted into the following three formulas
Wherein the content of the first and second substances,
is the phase margin of the given open loop system.
Three nonlinear equations to be solved can be obtained after substitution, and three parameters of the controller can be solved by utilizing calculation tools such as Matlab and the like as follows:
Kp＝1.9667,Ti1.1921, λ 0.9563. Thus, a set fractional order controller is obtained:
it can be seen that the controller is in the form of a fractional order PI structure.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (1)
1. A design method of a structure-adaptive fractional order proportional integral or proportional derivative controller is characterized by comprising the following steps:
step 1: obtaining the open-loop system cross-over frequency omega of the controlled object at the specified gaingcThe amplitude M and the phase phi of (d);
inputting a sinusoidal signal u (t) to the controlled object, wherein the amplitude of u (t) is AiAngular frequency being said gain crossover frequency omegagc(ii) a At this time, the controlled object outputs a sinusoidal signal y (t) with the same period and with a transition time tau and an amplitude difference, and the amplitude of y (t) is Ao(ii) a Then the process of the first step is carried out,
open loop system at gain crossover frequency omegagcAmplitude of (d)
Open loop system at gain crossover frequency omegagcPhase phi of (j ω) ∠ P (j ω)gc)＝ωgcτ (2)
The transition time τ ═ ti-toWherein, tiIs the peak time, t, of a peak x in the input sinusoidal signal u (t)oThe peak time of the same peak x in the output signal y (t);
step 2: acquiring Bode graph phase frequency curve slope of controlled object
Inputting an angular frequency omega to a controlled object1＝ωgcSinusoidal signal u of (1+ α)1(t), α ═ 0.01, yield u1(t) corresponding output signal y1(t) a transition time of τ1(ii) a Then the gain crossover frequency omega of the controlled object can be calculatedgcThe slope of the phase-frequency curve at (1) is:
wherein phi is1To output a signal y1(t) in a sinusoidal input signal u1(t) angular frequency ω1Phase of (phi)1＝ω1τ1；
And step 3: calculating fractional order controller Hc(s)＝Kp(1+Tis-λ) Parameters in (1), including the proportionality coefficient KpTime constant TiAnd a calculus order λ;
in this step, a controlled object transfer function P (j ω) is constructedgc) Amplitude of | P (j ω)gc) I, phase ∠ P (j ωgc) And phase slope
Relation (4), (5) and (6) between three parameters of the fractional order controller:
wherein, in formula (5)
Phase margin for a given open loop system;
then, ∠ P (j ω) obtained by the formulas (1), (2) and (3) is addedgc)、|P(jωgc) I and
substituting the relation expressions (4), (5) and (6) to calculate three parameters of the fractional controller.
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