CN102570956B - Direct-current motor control method based on resonance suppression - Google Patents

Abstract

The invention discloses a direct-current motor control method based on resonance suppression. The method comprises five steps of: 1, drawing a baud diagram according to a transmission function of a dynamics simplified model of a direct-current motor with resonance; 2, calculating a resonance point on the baud diagram, which corresponds to the frequency of a resonance point according to a formula; 3, respectively designing a trap filter and a peak filter according to a design method for a resonance suppression filter; 4, connecting the trap filter as well as the peak filter with the simplified model of the direct-current motor to correct the simplified model of the direct-current motor, obtaining a nominal model corrected by the filters by fitting the baud diagram, and obtaining a dynamics equation of an actual object by considering a modeling uncertainty; and 5, designing a proportional-integral-derivative (PID) controller according to the model of the direct-current motor, which is corrected by the filters, so that high-precision control over the direct-current motor is realized. According to the direct-current motor control method, the problem about resonance produced by a loaded direct-current motor during running is solved; and on the premise of guaranteeing the dynamic performance of the system, high-precision control over the direct-current motor is realized.

Description

A kind of DC motor control method suppressing based on resonance

(1) technical field

The present invention relates to a kind of DC motor control method suppressing based on resonance, it is for the direct current machine dynamics simplified model with resonance, and a kind of proportional-integral-differential (Proportional-Integral-Derivative that suppresses resonance by filter of design, PID) control method, belongs to electric machines control technology field.

(2) background technology

At present, in DC MOTOR CONTROL field, resonance problems more and more comes into one's own.How suppressing resonance realization is the focus of DC MOTOR CONTROL area research to the high accuracy control of direct current machine.

In servo system, the output shaft of direct current machine is directly connected with bearing axle, and rotatable parts are fixed on bearing axle, is common single axle rotation.If motor and load are considered as a rigid body, be called single quality servo system, this system and actual characteristic have very big difference.For real system, although motor and load are direct-coupled, power transmission shaft, in transmission process, has certain elastic deformation, and transmission is flexible in essence, and bearing and framework are not exclusively all also rigidity.Under the effect of motor driving moment, mechanical axis can be subject to bending and distortion to a certain degree.Require large, rapidity and the high system of required precision or the system that moment of inertia is large, performance requirement is high for acceleration, elastic deformation can not be ignored the impact of systematic function.Due to the bending and distortion of power transmission shaft, in the time transmitting motion, contain energy-storage travelling wave tube.In the situation that speed damping is little, in its transmission characteristic, there will be higher mechanical resonant, this resonance may cause the deterioration of dynamic performance in some cases, even makes system unstable.This resonance is larger on the dynamic property impact of system.

(3) summary of the invention

1, object: in view of this, the object of this invention is to provide a kind of DC motor control method suppressing based on resonance, the problem of the resonance producing in loaded running with solution direct current machine, thereby ensureing, under the prerequisite of dynamic performance, to realize the high accuracy control to direct current machine.

2, technical scheme: for achieving the above object:

A kind of DC motor control method suppressing based on resonance of the present invention, the method comprises the following steps:

Step 1: according to the transfer function of the direct current machine dynamics simplified model with resonance, draw its baud (Bode) figure;

Step 2: calculate the frequency of resonance point according to formula, the resonance point on corresponding Bode figure;

Step 3: according to the method for designing of resonance rejects trap, design respectively notch filter and peak filter;

Step 4: notch filter and peak filter are connected with direct current machine simplified model, realize the correction to direct current machine simplified model, through Bode figure matching, obtain the nominal model after device correction after filtering, consider modeling uncertainty, obtain the kinetics equation of practical object;

Step 5: the model of the direct current machine after proofreading and correct according to device after filtering, design PID controller, realizes the high accuracy control to direct current machine.

Wherein, the transfer function of the direct current machine dynamics simplified model with resonance described in step 1 is:

$G\left(s\right)=\frac{\frac{1}{{110}^2}{s}^2+\frac{0.02}{110}s+1}{(\frac{1}{{68}^2}{s}^2+\frac{0.07}{68}s+1)(\frac{1}{{130}^2}{s}^2+\frac{0.1}{130}s+1)}.$

In formula, s=σ+j ω is complex variable.

Wherein, the frequency computation part formula of the resonance point described in step 2 is the transfer function of oscillation element the damping ratio (0 < ζ < 1) that in formula, ζ is oscillation element, ω _nthe natural frequency of oscillation of oscillation element.Calculate successively G ₁, G ₂, G ₃three corresponding resonance point frequencies of link.

Wherein, described in step 3, the method for designing of resonance rejects trap is as follows:

For the model with resonance

$G_p\left(s\right)=\underset{m=1}{\overset{n}{\mathrm{\&Pi;}}}\frac{s^2+2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{1}s+{{\mathrm{\&omega;}}_1}^2}{s^2+2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{2}s+{{\mathrm{\&omega;}}_2}^2}$

In order to suppress resonance, notch filter is proposed and peak filter method for designing is as follows

$C_{\mathrm{notch}\_\mathrm{filters}}=\underset{m=1}{\overset{n_1}{\mathrm{\&Pi;}}}\frac{s^2+2{\mathrm{\&zeta;}}_{n_1}{\mathrm{\&omega;}}_{n_1}s+{{\mathrm{\&omega;}}_{n_1}}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}{\mathrm{\&omega;}}_{n_2}s+{{\mathrm{\&omega;}}_{n_2}}^2}$

$C_{peak\_\mathrm{filters}}=\underset{m=1}{\overset{n_2}{\mathrm{\&Pi;}}}\frac{s^2+2{\mathrm{\&zeta;}}_{p_1}{\mathrm{\&omega;}}_{p_1}s+{{\mathrm{\&omega;}}_{p_1}}^2}{s^2+2{\mathrm{\&zeta;}}_{p_2}{\mathrm{\&omega;}}_{p_2}s+{{\mathrm{\&omega;}}_{p_2}}^2}$

N in formula ₁, n ₂be respectively the number of trap and peak value resonance, with be respectively the damping ratio of notch filter and peak filter, with be respectively resonance point frequency.Conventionally, get

Wherein, the nominal model after the correction of the device after filtering described in step 4 is the kinetics equation of practical object is

Wherein, the design of the PID controller described in step 5 is as follows:

PID controller is a kind of linear controller, and it is according to set-point y _d(t) form control deviation with real output value y (t)

e(t)＝y _d(t)-y(t)

The control law of PID is

$u\left(t\right)=k_P[e\left(t\right)+\frac{1}{T_I}{\&Integral;}_0^{t}e\left(t\right)\mathrm{dt}+\frac{T_D\mathrm{de}\left(t\right)}{\mathrm{dt}}]=k_{P}e\left(t\right)+k_I{\&Integral;}_0^{t}e\left(t\right)\mathrm{dt}+\frac{k_D\mathrm{de}\left(t\right)}{\mathrm{dt}}$

In formula, k _pfor proportionality coefficient; T _ifor integration time constant; T _dfor derivative time constant.

The model of the direct current machine after proofreading and correct taking device after filtering, as controlled device, is got k _p, k _i, k _d,, input instruction y _d.Simulation time is set.Finally obtain simulation data and can be good at following the tracks of input signal, be i.e. set-point y _d(t) form control deviation e (t)=y with real output value y (t) _d(t)-y (t)=0, has realized the direct current machine high accuracy control with resonance.

3, advantage and effect: a kind of DC motor control method suppressing based on resonance of the present invention, its advantage is:

(1) combine the frequency domain analysis method that used, the method for designing of the method for designing of resonance rejects trap and PID control law, calculate the frequency of multiple resonance points, design process and the result of resonance filter are provided, use PID design of control law method, realized the direct current machine high accuracy control with resonance.

(2) can, for the situation of the one or more resonance that produce in motor operation course, effectively suppress by designing filter.

(3) solved the control problem with the direct current machine of resonance, had simple, to be easy to realization feature, universality is good, and can improve the dynamic property with the direct current machine of resonance.

(4) brief description of the drawings

Fig. 1: closed-loop control system structural representation of the present invention;

Fig. 2: the FB(flow block) of the DC motor control method suppressing based on resonance of the present invention;

Fig. 3: the present invention is with direct current machine-power transmission shaft-load module schematic diagram of resonance;

Fig. 4: the present invention is with the direct current machine dynamics simplified model Bode figure of resonance;

Fig. 5: the Bode figure of three resonance filter series connection of the present invention;

Fig. 6: the model of the present invention after resonance filter is proofreaied and correct and the Bode figure of matching thereof;

Fig. 7: the step response schematic diagram that the present invention proofreaies and correct without resonance filter;

Fig. 8: the step response schematic diagram of the present invention after resonance filter is proofreaied and correct.

Label, symbol and lines etc. in figure are described as follows:

In Fig. 3, u _afor motor input voltage, ω _mfor the rotating speed of motor, T _mfor the torque of motor, J _afor the moment of inertia of power transmission shaft, θ _mand θ _lbe respectively the corner of motor and load, J _lfor the moment of inertia of load, b _mand b _lbe respectively the viscous damping coefficient of motor and load, K _lfor the coupling stiffness coefficient between motor and framework, T _mLfor load end output torque square.

In Fig. 4, upper figure is log magnitude-frequency characteristics curve, and figure below is corresponding with it logarithmic phase frequency response curve.

In Fig. 5, upper figure is log magnitude-frequency characteristics curve, and figure below is corresponding with it logarithmic phase frequency response curve.

In Fig. 6, upper figure is log magnitude-frequency characteristics curve, and figure below is corresponding with it logarithmic phase frequency response curve.

(5) embodiment

For making the object, technical solutions and advantages of the present invention express clearlyer, below in conjunction with drawings and the specific embodiments, the present invention is further described in more detail.

Basic ideas of the present invention are according to the transfer function of the direct current machine dynamics simplified model with resonance, draw its Bode figure; Calculate the frequency of resonance point according to formula, the resonance point on corresponding Bode figure; According to the method for designing of resonance rejects trap, design respectively notch filter and peak filter; Notch filter and peak filter are connected with direct current machine simplified model, realize the correction to direct current machine simplified model, through Bode figure matching, obtain the nominal model after device correction after filtering, consider modeling uncertainty, obtain the kinetics equation of practical object; The model of the direct current machine after proofreading and correct according to device after filtering, design PID controller, realizes the high accuracy control to direct current machine.

Hardware system aspect, is with loaded general-purpose machine.Software systems aspect, system emulation is taking software MATLABR2008b as foundational development.Fig. 1 is closed-loop control system structural representation of the present invention.

Describe with an example below, the DC motor model with resonance is set as follows: its structure as shown in Figure 3.See Fig. 2, a kind of DC motor control method suppressing based on resonance of the present invention, it comprises the following steps:

Step 1: according to the transfer function of the direct current machine dynamics simplified model with resonance, draw its baud (Bode) figure;

Transfer function with the direct current machine dynamics simplified model of resonance is:

$G\left(s\right)=\frac{\frac{1}{{110}^2}{s}^2+\frac{0.02}{110}s+1}{(\frac{1}{{68}^2}{s}^2+\frac{0.07}{68}s+1)(\frac{1}{{130}^2}{s}^2+\frac{0.1}{130}s+1)}$

Controlled device G (s) can decompose as follows:

G＝G ₁G ₂G ₃

Wherein $G_1=\frac{{68}^2}{s^2+68\&times;0.07s+{68}^2},$ $G_2=\frac{1}{{110}^2}{s}^2+\frac{0.02}{110}s+1,$ $G_3=\frac{{130}^2}{s^2+130\&times;0.1s+{130}^2}.$

In formula, s=σ+j ω is complex variable.Bode figure is Fig. 4.

Step 2: calculate the frequency of resonance point according to formula, the resonance point on corresponding Bode figure;

The frequency computation part formula of resonance point is the transfer function of oscillation element the damping ratio (0 < ζ < 1) that in formula, ζ is oscillation element, ω _nthe natural frequency of oscillation of oscillation element.Calculate successively G ₁, G ₂, G ₃three corresponding resonance point frequencies of link are 68rad/s, 110rad/s and 130rad/s.

Step 3: according to the method for designing of resonance rejects trap, design respectively notch filter and peak filter;

The method for designing of resonance rejects trap is as follows:

For the model with resonance

$G_p\left(s\right)=\underset{m=1}{\overset{n}{\mathrm{\&Pi;}}}\frac{s^2+2{\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{1}s+{{\mathrm{\&omega;}}_1}^2}{s^2+2{\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{2}s+{{\mathrm{\&omega;}}_2}^2}$

In order to suppress resonance, notch filter is proposed and peak filter method for designing is as follows

$C_{\mathrm{notch}\_\mathrm{filters}}=\underset{m=1}{\overset{n_1}{\mathrm{\&Pi;}}}\frac{s^2+2{\mathrm{\&zeta;}}_{n_1}{\mathrm{\&omega;}}_{n_1}s+{{\mathrm{\&omega;}}_{n_1}}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}{\mathrm{\&omega;}}_{n_2}s+{{\mathrm{\&omega;}}_{n_2}}^2}$

$C_{\mathrm{peak}\_\mathrm{filters}}=\underset{m=1}{\overset{n_2}{\mathrm{\&Pi;}}}\frac{s^2+2{\mathrm{\&zeta;}}_{p_1}{\mathrm{\&omega;}}_{p_1}s+{{\mathrm{\&omega;}}_{p_1}}^2}{s^2+2{\mathrm{\&zeta;}}_{p_2}{\mathrm{\&omega;}}_{p_2}s+{{\mathrm{\&omega;}}_{p_2}}^2}$

N in formula ₁, n ₂be respectively the number of trap and peak value resonance, with be respectively the damping ratio of notch filter and peak filter, with be respectively resonance point frequency.Conventionally, get

Designing respectively notch filter and peak filter: Bode schemes as Fig. 5.

(1) for 68rad/s place by the peak value resonance producing, design notch filter is offset peak value resonance, should get notch filter is

$C_{\mathrm{notch}\_\mathrm{filter}1}=\frac{s^2+2{\mathrm{\&zeta;}}_{n_1}{\mathrm{\&omega;}}_{n_1}s+{{\mathrm{\&omega;}}_{n_1}}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}{\mathrm{\&omega;}}_{n_2}s+{{\mathrm{\&omega;}}_{n_2}}^2}=\frac{s^2+0.07\&times;68s+{68}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}\&times;68s+{68}^2}$

? $G_1\&times;C_{\mathrm{notch}\_\mathrm{filter}1}=\frac{{68}^2}{s^2+68\&times;0.07s+{68}^2}\&times;\frac{s^2+0.07\&times;68s+{68}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}\&times;68s+{68}^2}=\frac{{68}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}\&times;68s+{68}^2}$

Above formula amplitude-frequency is $L\left(\mathrm{\&omega;}\right)=201g\left|H\left(\mathrm{j\&omega;}\right)\right|=201g\frac{1}{2{\mathrm{\&zeta;}}_{n_2}\sqrt{1-{\mathrm{\&zeta;}}_{n_2}^2}}.$

Because 68rad/s is in low-frequency range, in order to make Bode scheme that amplitude-frequency is unattenuated herein, get can obtain

(2) for 130rad/s place by the peak value resonance producing, design notch filter is offset peak value resonance, should get notch filter is

$C_{\mathrm{notch}\_\mathrm{filter}2}=\frac{s^2+2{\mathrm{\&zeta;}}_{n_1}{\mathrm{\&omega;}}_{n_1}s+{{\mathrm{\&omega;}}_{n_1}}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}{\mathrm{\&omega;}}_{n_2}s+{{\mathrm{\&omega;}}_{n_2}}^2}=\frac{s^2+0.10\&times;130s+{130}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}\&times;130s+{130}^2}$

? $G_3\&times;C_{\mathrm{notch}\_\mathrm{filter}2}=\frac{{130}^2}{s^2+130\&times;0.10s+130^2}\&times;\frac{s^2+0.10\&times;130s+{130}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}\&times;130s+{130}^2}=\frac{{130}^2}{s^2+2{\mathrm{\&zeta;}}_{n_2}\&times;130s+{130}^2}$

Above formula amplitude is $L\left(\mathrm{\&omega;}\right)=201g\left|H\left(\mathrm{j\&omega;}\right)\right|=201g\frac{1}{2{\mathrm{\&zeta;}}_{n_2}\sqrt{1-{\mathrm{\&zeta;}}_{n_2}^2}}$

Because 130rad/s is in Mid Frequency, for make Bode scheme herein amplitude-frequency be monotonic decay without peak value, should get get can obtain $C_{\mathrm{notch}\_\mathrm{filter}2}=\frac{s^2+0.10\&times;130s+{130}^2}{s^2+1.5\&times;130s+{130}^2}.$

(3) for the trap value at 110rad/s place be by $G_2=\frac{1}{110^2}{s}^2+\frac{0.02}{110}s+1=\frac{s^2+0.02\&times;110s+{110}^2}{{110}^2}$ Produce, design peak filter, gets peak filter is

$C_{\mathrm{peak}\_\mathrm{filter}}=\frac{s^2+2{\mathrm{\&zeta;}}_{p_1}{\mathrm{\&omega;}}_{p_1}s+{{\mathrm{\&omega;}}_{p_1}}^2}{s^2+2{\mathrm{\&zeta;}}_{p_2}{\mathrm{\&omega;}}_{p_2}s+{{\mathrm{\&omega;}}_{p_2}}^2}=\frac{s^2+2{\mathrm{\&zeta;}}_{p_1}\&times;110s+{110}^2}{s^2+0.02\&times;110s+{110}^{\text{2}}}$

? $G_2\&times;C_{\mathrm{peak}\_\mathrm{filter}}=\frac{s^2+0.02\&times;110s+{110}^2}{{110}^2}\&times;\frac{s^2+2{\mathrm{\&zeta;}}_{p_1}\&times;110s+{110}^2}{s^2+0.02\&times;110s+{110}^2}=\frac{s^2+2{\mathrm{\&zeta;}}_{p_1}\&times;110s+{110}^2}{{110}^2}$

Above formula amplitude-frequency is $L\left(\mathrm{\&omega;}\right)=201g\left|H\left(\mathrm{j\&omega;}\right)\right|=201g\left(2{\mathrm{\&zeta;}}_{p_1}\sqrt{1-{\mathrm{\&zeta;}}_{p_1}^2}\right).$

Because 110rad/s is in Mid Frequency, for make Bode scheme herein amplitude-frequency be monotonic decay without peak value, should get get can obtain $C_{\mathrm{peak}\_\mathrm{filter}}=\frac{s^2+1.5\&times;110s+{110}^2}{s^2+0.02\&times;110s+{110}^2}.$

Step 4: notch filter and peak filter are connected with direct current machine simplified model, realize the correction to direct current machine simplified model, through Bode figure matching, obtain the nominal model after device correction after filtering, consider modeling uncertainty, obtain the kinetics equation of practical object;

Nominal model after device correction is after filtering the kinetics equation of practical object is wherein a ₁=90.26, a ₂=6346, b=6392.As shown in Figure 6.

Step 5: the model of the direct current machine after proofreading and correct according to device after filtering, design PID controller, realizes the high accuracy control to direct current machine.

The design of PID controller is as follows:

PID controller is a kind of linear controller, and it is according to set-point y _d(t) form control deviation with real output value y (t)

e(t)＝y _d(t)-y(t)

The control law of PID is

$u\left(t\right)=k_P[e\left(t\right)+\frac{1}{T_I}{\&Integral;}_0^{t}e\left(t\right)\mathrm{dt}+\frac{T_D\mathrm{de}\left(t\right)}{\mathrm{dt}}]=k_{P}e\left(t\right)+k_I{\&Integral;}_0^{t}e\left(t\right)\mathrm{dt}+\frac{k_D\mathrm{de}\left(t\right)}{\mathrm{dt}}$

In formula, k _pfor proportionality coefficient; T _ifor integration time constant; T _dfor derivative time constant.

The model of the direct current machine after proofreading and correct taking device after filtering, as controlled device, using unit step signal as input, is got k _p=5, k _i=5, k _d=0.1,, input instruction y _d=1 (t), wherein t>=0.Simulation time is 10s.Finally obtain simulation data and can be good at following the tracks of input signal, be i.e. set-point y _d(t) form control deviation e (t)=y with real output value y (t) _d(t)-y (t)=0, has realized the direct current machine high accuracy control with resonance.Unit step response before and after filtering as shown in Figure 7, Figure 8.

Claims (1)

1. the DC motor control method suppressing based on resonance, is characterized in that: the method comprises the following steps:

Step 1: according to the transfer function of the direct current machine dynamics simplified model with resonance, draw its baud (Bode) figure;

Transfer function with the direct current machine dynamics simplified model of resonance is:

$G\left(s\right)=\frac{\frac{1}{{110}^2}{s}^2+\frac{0.02}{110}s+1}{(\frac{1}{{68}^2}{s}^2+\frac{0.07}{68}s+1)(\frac{1}{{130}^2}{s}^2+\frac{0.1}{130}s+1)}$

Controlled device G (s) decomposes as follows:

G＝G ₁G ₂G ₃

Wherein $G_1=\frac{{68}^2}{s^2+68\&times;0.07s+{68}^2},G_2=\frac{1}{{110}^2}{s}^2+\frac{0.02}{110}s+1,G_3=\frac{{130}^2}{s^2+130\&times;0.1s+{130}^2};$

In formula, s=σ+j ω is complex variable;

Step 2: calculate the frequency of resonance point according to formula, the resonance point on corresponding Bode figure;

The frequency computation part formula of resonance point is the transfer function of oscillation element the damping ratio (0< ζ <1) that in formula, ζ is oscillation element, ω _nthe natural frequency of oscillation of oscillation element; Calculate successively G ₁, G ₂, G ₃three corresponding resonance point frequencies of link are 68rad/s, 110rad/s and 130rad/s;

Step 3: according to the method for designing of resonance rejects trap, design respectively notch filter and peak filter;

The method for designing of resonance rejects trap is as follows:

For the model with resonance

$G_p\left(s\right)=\underset{m=1}{\overset{n}{\mathrm{\&Pi;}}}\frac{s^2+{2\mathrm{\&zeta;}}_1{\mathrm{\&omega;}}_{1}s+{{\mathrm{\&omega;}}_1}^2}{s^2+{2\mathrm{\&zeta;}}_2{\mathrm{\&omega;}}_{2}s+{{\mathrm{\&omega;}}_2}^2}$

In order to suppress resonance, notch filter is proposed and peak filter method for designing is as follows

$C_{\mathrm{notch}\_\mathrm{filters}}=\underset{m=1}{\overset{n_1}{\mathrm{\&Pi;}}}\frac{s^2+{2\mathrm{\&zeta;}}_{n_1}{\mathrm{\&omega;}}_{n_1}s+{{\mathrm{\&omega;}}_{n_1}}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}{\mathrm{\&omega;}}_{n_2}s+{{\mathrm{\&omega;}}_{n_2}}^2}$

$C_{\mathrm{peak}\_\mathrm{filters}}=\underset{m=1}{\overset{n_2}{\mathrm{\&Pi;}}}\frac{s^2+{2\mathrm{\&zeta;}}_{p_1}{\mathrm{\&omega;}}_{p_1}s+{{\mathrm{\&omega;}}_{p_1}}^2}{s^2+{2\mathrm{\&zeta;}}_{p_2}{\mathrm{\&omega;}}_{p_2}s+{{\mathrm{\&omega;}}_{p_2}}^2}$

N in formula ₁, n ₂be respectively the number of trap and peak value resonance, with be respectively the damping ratio of notch filter and peak filter, with be respectively resonance point frequency; Get

Design respectively notch filter and peak filter:

(1) for 68rad/s place by the peak value resonance producing, design notch filter is offset peak value resonance, should get notch filter is

$C_{\mathrm{notch}\_\mathrm{rilter}1}=\frac{s^2+{2\mathrm{\&zeta;}}_{n_1}{\mathrm{\&omega;}}_{n_1}s+{{\mathrm{\&omega;}}_{n_1}}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}{\mathrm{\&omega;}}_{n_2}s+{{\mathrm{\&omega;}}_{n_2}}^2}=\frac{s^2+0.07\&times;68s+{68}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}\&times;68s+{68}^2}$

? $G_1\&times;C_{\mathrm{notch}\_\mathrm{filter}1}=\frac{{68}^2}{s^2+68\&times;0.07s+{68}^2}\&times;\frac{s^2+0.07\&times;68s+{68}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}\&times;68s+{68}^2}=\frac{{68}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}\&times;68s+{68}^2}$

Above formula amplitude-frequency is $L\left(\mathrm{\&omega;}\right)=20\lg \left|H\left(\mathrm{j\&omega;}\right)\right|=20\lg \frac{1}{{2\mathrm{\&zeta;}}_{n_2}\sqrt{1-{\mathrm{\&zeta;}}_{n_2}^2}};$

Because 68rad/s is in low-frequency range, in order to make Bode scheme that amplitude-frequency is unattenuated herein, get obtain $C_{\mathrm{notch}\_\mathrm{filter}1}=\frac{s^2+0.07\&times;68s+{68}^2}{s^2+1.1\&times;68s+{68}^2};$

(2) for 130rad/s place by the peak value resonance producing, design notch filter is offset peak value resonance, should get notch filter is

$C_{\mathrm{notch}\_\mathrm{filter}2}=\frac{s^2+{2\mathrm{\&zeta;}}_{n_1}{\mathrm{\&omega;}}_{n_1}s+{{\mathrm{\&omega;}}_{n_1}}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}{\mathrm{\&omega;}}_{n_2}s+{{\mathrm{\&omega;}}_{n_2}}^2}=\frac{s^2+0.10\&times;130s+{130}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}\&times;130s+{130}^2}$

? $G_3\&times;C_{\mathrm{notch}\_\mathrm{filter}2}=\frac{{130}^2}{s^2+130\&times;0.10s+{130}^2}\&times;\frac{s^2+0.10\&times;130s+{130}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}\&times;130s+{130}^2}=\frac{{130}^2}{s^2+{2\mathrm{\&zeta;}}_{n_2}\&times;130s+{130}^2}$

Above formula amplitude-frequency is $L\left(\mathrm{\&omega;}\right)=20\lg \left|H\left(\mathrm{j\&omega;}\right)\right|=20\lg \frac{1}{{2\mathrm{\&zeta;}}_{n_2}\sqrt{1-{\mathrm{\&zeta;}}_{n_2}^2}};$

Because 130rad/s is in Mid Frequency, for make Bode scheme herein amplitude-frequency be monotonic decay without peak value, get ${\mathrm{\&zeta;}}_{n_2}=0.75,$ Obtain $C_{\mathrm{notch}\_\mathrm{filter}2}=\frac{s^2+0.10\&times;130s+{130}^2}{s^2+1.5\&times;130s+{130}^2};$

(3) for the trap value at 110rad/s place be by $G_2=\frac{1}{{110}^2}{s}^2+\frac{0.02}{110}s+1=\frac{s^2+0.02\&times;110s+{110}^2}{{110}^2}$ Produce, design peak filter, gets peak filter is $C_{\mathrm{peak}\_\mathrm{filter}}=\frac{s^2+{2\mathrm{\&zeta;}}_{p_1}{\mathrm{\&omega;}}_{p_1}s+{{\mathrm{\&omega;}}_{p_1}}^2}{s^2+{2\mathrm{\&zeta;}}_{p_2}{\mathrm{\&omega;}}_{p_2}s+{{\mathrm{\&omega;}}_{p_2}}^2}=\frac{s^2+{2\mathrm{\&zeta;}}_{p_1}\&times;110s+{110}^2}{s^2+0.02\&times;110s+{110}^2}$

? $G_2\&times;C_{\mathrm{peak}\_\mathrm{filter}}=\frac{s^2+0.02\&times;110s+{110}^2}{{110}^2}\&times;\frac{s^2+{2\mathrm{\&zeta;}}_{p_1}\&times;110s+{110}^2}{s^2+0.02\&times;110s+{110}^2}=\frac{s^2+{2\mathrm{\&zeta;}}_{p_1}\&times;110s+{110}^2}{{110}^2}$

Above formula amplitude-frequency is $L\left(\mathrm{\&omega;}\right)=20\lg \left|H\left(\mathrm{j\&omega;}\right)\right|=20\lg \left({2\mathrm{\&zeta;}}_{p_1}\sqrt{1-{\mathrm{\&zeta;}}_{p_1}^2}\right);$

Because 110rad/s is in Mid Frequency, for make Bode scheme herein amplitude-frequency be monotonic decay without peak value, get obtain $C_{\mathrm{peak}\_\mathrm{filter}}=\frac{s^2+1.5\&times;110s+{110}^2}{s^2+0.02\&times;110s+{110}^2};$

Step 4: notch filter and peak filter are connected with direct current machine dynamics simplified model, realize the correction to direct current machine dynamics simplified model, through Bode figure matching, obtain the nominal model after device correction after filtering, consider modeling uncertainty, obtain the kinetics equation of practical object;

Nominal model after device correction is after filtering the kinetics equation of practical object is wherein a ₁=90.26, a ₂=6346, b=6392;

Step 5: the direct current machine dynamics simplified model after proofreading and correct according to device after filtering, design PID controller, realizes the high accuracy control to direct current machine;

The design of PID controller is as follows:

PID controller is a kind of linear controller, and it is according to set-point y _d(t) form control deviation with real output value y (t)

e(t)＝y _d(t)-y(t)

The control law of PID is

$u\left(t\right)=k_P[e\left(t\right)+\frac{1}{T_1}{\&Integral;}_0^{t}e\left(t\right)\mathrm{dt}+\frac{T_D\mathrm{de}\left(t\right)}{\mathrm{dt}}=k_{p}e\left(t\right)+k_I{\&Integral;}_0^{t}e\left(t\right)\mathrm{dt}+\frac{k_D\mathrm{de}\left(t\right)}{\mathrm{dt}}$

In formula, k _pfor proportionality coefficient; T _ifor integration time constant; T _dfor derivative time constant;

Direct current machine dynamics simplified model after proofreading and correct taking device after filtering, as controlled device, using unit step signal as input, is got k _p=5, k _i=5, k _d=0.1, input instruction y _d(t)=1, wherein t>=0; Simulation time is 10s; Finally obtain simulation data and can be good at following the tracks of input signal, be i.e. set-point y _d(t) form control deviation e (t)=y with real output value y (t) _d(t)-y (t)=0, has realized the direct current machine high accuracy control with resonance.