CN112769364B - Fast self-adaptive anti-interference control method of direct current motor servo system - Google Patents

Fast self-adaptive anti-interference control method of direct current motor servo system Download PDF

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CN112769364B
CN112769364B CN202011465917.0A CN202011465917A CN112769364B CN 112769364 B CN112769364 B CN 112769364B CN 202011465917 A CN202011465917 A CN 202011465917A CN 112769364 B CN112769364 B CN 112769364B
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CN112769364A (en
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姚建勇
方靖荃
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Nanjing University of Science and Technology
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/0003—Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control
    • H02P21/0017—Model reference adaptation, e.g. MRAS or MRAC, useful for control or parameter estimation
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/13—Observer control, e.g. using Luenberger observers or Kalman filters
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H02P21/18—Estimation of position or speed

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Abstract

The invention discloses a Fast Adaptive Disturbance Rejection (FADRC) control method of a direct current motor servo system, and provides a Fast Adaptive Disturbance Rejection (FADRC) control method based on finite time parameter estimation aiming at the problem of high-precision motion control of the direct current motor servo system under the combined action of parameter uncertainty and unmodeled disturbance. By introducing proper filtering operation, a novel parameter self-adaptive law which is compositely driven by a tracking error, a parameter estimation error and a system state estimation error is constructed, and the online quick update of unknown parameters is realized. Meanwhile, the extended state observer is designed to estimate and compensate unmodeled interference, and the robustness of the system is improved. The control method can ensure rapid parameter estimation performance and excellent tracking control effect.

Description

Fast self-adaptive anti-interference control method of direct current motor servo system
Technical Field
The invention relates to an electromechanical servo control technology, in particular to a Fast Adaptive Disturbance Rejection (FADRC) control method of a direct current motor servo system.
Background
The direct current motor servo system has wide application in the fields of industry and national defense, so the high-precision motion control problem is always a research hotspot in the related fields. The direct current servo motor can control the speed, has very accurate position precision, and can convert a voltage signal into torque and rotating speed to drive a control object. The rotation speed of the rotor of the servo motor is controlled by an input signal and can quickly respond, the servo motor is used as an actuating element in an automatic control system, has the characteristics of small electromechanical time constant, high linearity, starting voltage and the like, and can convert a received electric signal into angular displacement or angular speed on a motor shaft for output. However, because many model uncertainties exist in an actual system, including parameter uncertainties and unmodeled interferences (such as nonlinear friction and the like), the improvement of the control performance of the system is severely restricted, and a great challenge is brought to the high-precision motion control of a direct-current motor servo system.
To this end, researchers have proposed various control strategies. Specifically, the unknown parameters are updated on line through a parameter self-adaptive law, and the self-adaptive control can effectively process the uncertainty of the system parameters. For simultaneously processing system parameter uncertainty and unmodeled interference, Yao et al propose Adaptive Robust Control (ARC), theoretically ensuring given transient tracking performance and steady-state tracking accuracy, and when only parameter uncertainty exists, asymptotic tracking can be obtained. When the system is not modeled to have strong interference, the adaptive robust control can only ensure good control effect through high feedback gain, however, the high frequency dynamic of the system can be excited by the excessively high feedback gain, and the instability of the system is further caused. Korean Jingqing estimates generalized disturbance by using an Extended State Observer (ESO) in the book of active disturbance rejection control technology to realize disturbance compensation without too much system model information. However, the Active Disturbance Rejection Control (ADRC) does not accurately consider the parameter uncertainty of the system, and when the uncertainty of the system model mainly comes from uncertain parameters, the control performance of the ADRC is often weaker than that of the traditional adaptive control. Deng Wen Xiang combines the characteristics of adaptive control and compensation control based on interference observation in the text of electro-hydraulic position servo system tracking control based on interference observer, and provides an auto-disturbance-rejection adaptive control method (ADRAC) based on a model. The proposed controller overcomes their performance deficiencies while retaining the advantages of both control methods. However, since the parameter adaptive law in the control strategy is driven only by the tracking error and the system state estimation error, the convergence speed of the parameter estimation value is slow, and poor transient tracking performance of the system is caused.
Based on the consideration, a Fast Adaptive Disturbance Rejection Control (FADRC) method is provided for a direct current motor servo system. By introducing proper filtering operation and auxiliary variables, a novel parameter self-adaptation law which is compositely driven by a tracking error, a parameter estimation error and a state estimation error is constructed, and is effectively fused with an extended state observer, so that the unknown parameter estimation index of the system can be converged, and meanwhile, the system can be actively compensated for the unmodeled interference. When the system has time-varying interference, the proposed control method can ensure consistent and finally bounded tracking performance, and when the system only has constant interference, excellent asymptotic tracking performance can be obtained. In addition, the transient performance of the system can be effectively improved thanks to a novel fast parameter self-adaptive law. The simulation results for the dc motor servo system verify the effectiveness of the proposed control method and the superiority over the existing methods.
Disclosure of Invention
The invention aims to provide a rapid self-adaptive anti-interference control method for a direct current motor servo system, which can actively compensate the non-modeling interference of the system while realizing the estimation index convergence of unknown parameters of the system and obtain excellent control performance;
the technical solution for realizing the purpose of the invention is as follows: a fast self-adaptive anti-interference control method of a direct current motor servo system comprises the following steps:
step 1, establishing a mathematical model of a direct current motor servo system, and turning to step 2.
And 2, designing a fast self-adaptive anti-interference controller based on a mathematical model of a direct current motor servo system, and turning to step 3.
And 3, performing rapid self-adaptive disturbance rejection controller stability verification by using the Lyapunov stability theory to obtain a system stability result.
Compared with the prior art, the invention has the remarkable advantages that: (1) the method has the advantages that symbol function gain self-adjustment is realized, uncertain parameter self-adaptation of a direct current servo motor system is realized by comprehensively considering tracking errors, parameter estimation errors and state estimation errors, and the parameter self-adaptation speed is high; (2) the method has the advantages that strong interference is resisted, when time-varying interference exists in the system, the provided control method can ensure consistent and finally bounded tracking performance, when the system only has constant interference, excellent asymptotic tracking performance can be obtained, and the effectiveness of the system is verified by a simulation result;
drawings
Fig. 1 is a schematic diagram of a Fast Adaptive Disturbance Rejection (FADRC) control method of a dc motor servo system according to the present invention.
Fig. 2 is a schematic architecture diagram of a dc motor servo system of the present invention.
FIG. 3 is a system reference position signal graph.
FIG. 4 is a comparison graph of control input curves of Fast Adaptive Disturbance Rejection (FADRC) control, Active Disturbance Rejection Adaptive (ADRAC) control and PID control under a non-interference working condition.
FIG. 5 is a comparison graph of Fast Adaptive Disturbance Rejection (FADRC) control and Adaptive Disturbance Rejection (ADRAC) control, PID control on-line parameter estimation curves under non-interference condition, where (a) in FIG. 5 is θ 1 A map of the parameter adaptation is generated,in FIG. 5, (b) is θ 2 A parametric adaptation map.
FIG. 6 is a comparison graph of tracking error curves of Fast Adaptive Disturbance Rejection (FADRC) control, Adaptive Disturbance Rejection and Adaptation (ADRAC) control and PID control under a non-interference working condition.
Fig. 7 is a graph comparing on-line parameter estimation curves of Fast Adaptive Disturbance Rejection (FADRC) control, Adaptive Disturbance Rejection Adaptive (ADRAC) control, and PID control when the system disturbance is d (x, t) ═ 0.3 × sin (0.5 pi t), where θ in fig. 7 is θ 1 A parameter adaptive graph, in which (b) in FIG. 7 is θ 2 A parametric adaptation map.
Fig. 8 is a graph comparing tracking error curves of the Fast Adaptive Disturbance Rejection (FADRC) control, the Auto Disturbance Rejection Adaptive (ADRAC) control, and the PID control when the system disturbance is d (x, t) ═ 0.3 × sin (0.5 pi t).
Fig. 9 is a graph comparing the on-line parameter estimation curves of the Fast Adaptive Disturbance Rejection (FADRC) control, the Adaptive Disturbance Rejection Adaptive (ADRAC) control, and the PID control when the system disturbance is d (x, t) ═ 0.2 × sin (1.5 pi t), where θ is θ in fig. 9 (a) 1 A parameter adaptive map, in FIG. 9, (b) is θ 2 A parametric adaptation map.
Fig. 10 is a graph comparing tracking error curves of Fast Adaptive Disturbance Rejection (FADRC) control, Adaptive Disturbance Rejection (ADRAC) control, and PID control when the system disturbance is d (x, t) ═ 0.2 × sin (1.5 pi t).
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings and specific embodiments;
with reference to fig. 1 to 2, the fast adaptive disturbance rejection control method of the dc motor servo system of the present invention includes the following steps:
step 1, establishing a mathematical model of a direct drive motor system;
step 1-1, the direct current servo motor system is shown in figure 1, wherein a motor is controlled by a driver and directly drives an inertial load; considering that the electromagnetic time constant is much smaller than the mechanical time constant, and the current loop response speed is much larger than the speed and position loop response speeds, the current loop dynamics can be neglected, and the dynamics of the inertial load are listed by the following formula:
Figure GDA0003695757960000031
where y and m represent angular displacement and load inertia, respectively, K i Representing a torque constant, u representing a control input, B representing a viscous friction coefficient, f representing unmodeled disturbances, such as non-linear friction, external disturbances, etc., t representing a dc motor servo system time,
Figure GDA0003695757960000041
represents the derivative of y;
step 1-2, for any variable A, define
Figure GDA0003695757960000042
Which represents an estimate of the variable a, is,
Figure GDA0003695757960000043
which represents the error of the estimation of the variable a,
Figure GDA0003695757960000044
denotes the derivative of the variable a, for which all the same references apply hereinafter;
definable state variables of a DC motor servo system
Figure GDA00036957579600000411
Unknown parameter vector theta of direct current motor servo system is [ theta ═ theta [ [ theta ] 1 ,θ 2 ] T ＝[K i /m,B/m] T If the unmodeled disturbance d (x, t) of the dc motor servo system is f/m, the state equation is written as equation (1):
Figure GDA0003695757960000045
in the above formula
Figure GDA0003695757960000046
Respectively represent x 1 ,x 2 A derivative of (a);
for ease of controller design, assume the following:
assume that 1: system reference position signal x 1d The second order is continuous and differentiable, and each derivative of the second order is bounded;
assume 2: the range of the unknown parameter theta is known, i.e.
θ min ≤θ≤θ max (3)
In the formula [ theta ] min ＝[θ 1min ,θ 2min ] T ，θ max ＝[θ 1max ,θ 2max ] T Known lower and upper bounds, respectively, for unknown parameters;
turning to the step 2;
in step 2, designing a fast self-adaptive anti-interference controller based on a mathematical model of a direct current motor servo system;
step 2-1, before the controller is designed, discontinuous parameter mapping adopted by parameter self-adaptation is given:
order to
Figure GDA0003695757960000047
Figure GDA0003695757960000048
Representing an estimation of the unknown parameter theta of the system,
Figure GDA0003695757960000049
to ensure the stability of the adaptive control law for parameter estimation errors, the system-based parameter uncertainty is bounded, i.e. assume 2, defining the following parameter-adaptive discontinuous mapping
Figure GDA00036957579600000410
Figure GDA0003695757960000051
Wherein i is 1, 2; i represents the ith element of the vector.
Given the following parametersRate of adaptation
Figure GDA0003695757960000052
Figure GDA0003695757960000053
In the formula>0 is positive definite diagonal matrix, tau is self-adaptive law function, and for arbitrary self-adaptive law function tau the discontinuous projection mapping is used
Figure GDA0003695757960000054
Can ensure that:
Figure GDA0003695757960000055
the system Extended State Observer (ESO) can be constructed as follows:
Figure GDA0003695757960000056
in the formula
Figure GDA0003695757960000057
Represents a state x j Is estimated by the estimation of (a) a,
Figure GDA0003695757960000058
represents a state x j Is estimated error of, i.e.
Figure GDA0003695757960000059
j＝1,2,3，ω o ＞0，ω o For the bandwidth of the ESO to be,
Figure GDA00036957579600000510
Figure GDA00036957579600000511
a process function vector for the construct; for a DC motor servo system, the model included in the systemUncertainty and parameter uncertainty
Figure GDA00036957579600000512
Related to unmodeled interference d (x, t); thus, there are two cases for the definition of the expanded state:
case 1: let the system expand state x 3 D (x, t), then constructing a system expansion state equation based on the system model as follows:
Figure GDA00036957579600000513
wherein h (t) represents x 3 With respect to the derivative of time,
Figure GDA00036957579600000514
denotes x 3 A derivative of (a);
the system state estimation error dynamics can be expressed as:
Figure GDA0003695757960000061
in the formula:
Figure GDA0003695757960000062
and the coefficient matrix is:
Figure GDA0003695757960000063
case 2: order to
Figure GDA0003695757960000064
The constructed system expansion state equation becomes:
Figure GDA0003695757960000065
the system state estimation error dynamics becomes:
Figure GDA0003695757960000066
since the matrix a is a Hurwitz matrix, there must be a positive definite matrix J that satisfies the Lyapunov equation:
A T J+JA＝-E (13)
wherein E represents an identity matrix. The matrix J is easily found:
Figure GDA0003695757960000067
by using the thought of a back stepping method, the fast self-adaptive disturbance rejection controller is designed, and the method specifically comprises the following steps:
error variables are defined as follows
Figure GDA0003695757960000068
In the formula z 1 For systematic tracking error, α 1 Is a state x 2 Virtual control law of z 2 Is the deviation between the two, x 1d Is a system position reference signal.
For error z 1 Derivation, we can obtain:
Figure GDA0003695757960000071
the virtual control law is designed as follows:
Figure GDA0003695757960000072
where the gain factor k is 1 > 0, from (15) to (17):
Figure GDA0003695757960000073
laplace transform is performed on the left and right sides of the above formula, and the following results are obtained:
Figure GDA0003695757960000074
wherein s represents a complex parameter variable;
the transfer function G(s) represented in the above formula is stable when z is 2 When approaching 0, z 1 Inevitably approaching 0, so the primary design objective is shifted to z 2 Approaching 0.
The adaptation law for the second channel is designed for the sake of theoretical derivation under the condition of case 1, i.e. x 3 D (x, t), assuming that the unmodeled disturbance of the system is constant, i.e. h (t) is 0, the second formula in equation (2) is modified as:
Figure GDA0003695757960000075
further, the above formula can be modified as follows:
Figure GDA0003695757960000076
the filtering operation is performed on each term at two sides of the above equation, and the following results are obtained:
Figure GDA0003695757960000077
wherein the filter operation coefficient k is more than 0,
Figure GDA0003695757960000078
representing state estimates
Figure GDA0003695757960000079
The filter variables obtained after the filtering are obtained,
Figure GDA00036957579600000710
representing errors in state estimation
Figure GDA00036957579600000711
Filtering variables obtained after filtering; in addition, note that
Figure GDA0003695757960000081
Therefore, when h (t) is 0,
Figure GDA0003695757960000082
can be derived from integration.
The following variables are defined:
Figure GDA0003695757960000083
in the above formula, P and Q are structural process parameters, the constant l is more than 0, and the above formula is integrated to obtain:
Figure GDA0003695757960000084
in the above formula, β represents an integral variable.
The following can be derived from the above formula:
Q＝Pθ (21)
defining a process matrix H such that it satisfies:
Figure GDA0003695757960000085
definition of lambda max () And λ min () The maximum and minimum eigenvalues of the corresponding matrix are respectively represented, and the following reasoning is provided.
Introduction 1: if matrix
Figure GDA0003695757960000086
If the continuous excitation condition is satisfied, the matrix P is positive, so that its minimum eigenvalue λ is min (P (t)) satisfies:
when T > T > 0 and the constant sigma > 0, there are
λ min (P(t))＞σ (23)
T represents a time constant;
further, a parameter adaptation function τ may be defined as:
Figure GDA0003695757960000087
in the formula, the gain coefficient C 1 ＞0,C 2 ＞0,C 3 And if the system state estimation error is larger than 0, epsilon and J are certain positive definite matrixes.
B 1 ＝[0,1,0] T (ii) a Based on the obtained parameter estimation values, the actual control input u is designed as:
Figure GDA0003695757960000091
in the formula k 2 > 0 is the gain factor, u a Is a model compensation term, u, adjusted based on-line parameter estimation and system state estimation s Is a linear robust feedback term.
Under the action of the controller (25) and the parameter adaptive law (5), the DC motor servo system meets the following theorem, and specific theoretical proof is given in step 3.
Theorem 1: when h (t) is 0, all signals of the closed-loop system are bounded, and the tracking error converges asymptotically to zero;
theorem 2: when h (t) ≠ 0, all signals of the closed-loop system are bounded;
theorem 3: system parameter estimation error
Figure GDA0003695757960000092
Converge to zero within a finite time;
and (5) turning to the step 3.
Step 3, a stability certification is carried out on the rapid self-adaptive anti-interference control method of the direct current motor servo system by applying the Lyapunov stability theory, and a system stability result is obtained;
theorem 1 proves that: under the condition of case 1, the Lyapunov function is defined as follows:
Figure GDA0003695757960000093
assuming that h (t) is 0, the derivation of the above equation is obtained:
Figure GDA0003695757960000094
defining:
Z＝[z 1 ,z 2 ,ε 1 ,ε 2 ,ε 3 ] T ，
Figure GDA0003695757960000101
by adjusting the parameter k 1 ,k 2 ,C 1 ,C 2 ,C 3 ,ω o Making the symmetric matrix Λ 1 Is positive, so:
Figure GDA0003695757960000102
in the formula of min (Λ 1 ) Representation matrix Λ 1 W is a positive function; thus obtaining V 1 ∈L ∞ ，W∈L 2 Error signal Z and
Figure GDA0003695757960000103
are bounded; from hypothesis 1, the system state x 1 ,x 2 And an expanded state x 3 Both by themselves and their estimates are bounded; additionally, considering hypothesis 2, the control input u of the derivable system is bounded; in summary, all signals are bounded for the closed loop system.
On the basis of the analysis of the error,
Figure GDA0003695757960000106
is bounded, so W has consistent continuity; when the time approaches infinity, the tracking error of the system approaches zero by applying the barbalt theorem, and theorem 1 proves.
Theorem 2 proves that: when the unmodeled interference term d (x, t) is a time-varying function, case 2 is applied for analysis; assuming h (t) is bounded (i.e., there is a constant ζ > 0) such that | h (t) | ≦ ζ), the Lyapunov function is defined as follows:
Figure GDA0003695757960000104
similar to the process of theorem 1, the derivation of the above equation can be obtained:
Figure GDA0003695757960000105
wherein η is defined as:
Figure GDA0003695757960000111
defining:
Figure GDA0003695757960000112
by adjusting the parameter k 1 ,k 2 ,C 1 ,C 2 ,C 3 ,ω o Make the symmetric matrix Λ 2 Positive, then:
Figure GDA0003695757960000113
integrating the above equation yields:
Figure GDA0003695757960000114
from the above derivation, all signals are bounded for the closed loop system, and the tracking error z of the system is 1 It is stable with the theorem 2.
Theorem 3 proves that: to prove parameter estimation error
Figure GDA0003695757960000115
Can converge to zero in a finite time, and defines the Lyapunov function as follows:
Figure GDA0003695757960000116
derivation of the above equation yields:
Figure GDA0003695757960000117
due to z 2 → 0 and
Figure GDA0003695757960000118
is bounded, so there is a certain time t f Is greater than 0, when t is greater than t f At a time there is
Figure GDA0003695757960000121
Thus, parameter estimation error
Figure GDA0003695757960000122
Can converge to zero within a limited time, theorem 3 proving.
Examples
In order to verify the excellent performance of the designed FADRC controller, simulation comparison under different working conditions is carried out based on Matlab/Simulink.
The parameters of the DC motor servo system are set as follows: load inertia m is 0.02kg m 2 (ii) a Force ofMoment constant K i ＝5N·m·V -1 (ii) a Viscous friction coefficient B10 N.m.rad -1 ·s -1 (ii) a The validity of the designed controller was verified by comparing the following three controllers.
1) FADRC: namely the fast adaptive immunity controller designed above; controlling input gain k 1 ＝250,k 2 200 parts of a total weight; bandwidth omega of system extended state observer o 10; the magnitude of the system parameter theta is set to theta min ＝[10,10] T ,θ max ＝[1000,1000] T (ii) a Initial estimated value of parameter is set as
Figure GDA0003695757960000123
The adaptive law matrix is set to Γ ═ diag {28,10.5 }; adaptive law gain coefficient set to C 1 ＝10,C 2 ＝10,C 3 10; the filter operation gain coefficient is set to 1 and k is set to 0.05.
2) ADRAC: namely a traditional active disturbance rejection adaptive controller; the parameter self-adaptive law only comprises a tracking error and a system state estimation error, so that the gain coefficient of the self-adaptive law is adjusted to be C 1 ＝10,C 2 ＝0，C 3 Other parameters are consistent with FADRC 10.
3) PID: widely used proportional integral derivative controllers; three gains are set to k p ＝100,k i ＝100,k d ＝0。
Reference position signal x 1d (t)＝1*sin(0.5πt)*(1-e -0.5t ) As shown in fig. 3; the simulation is divided into three working conditions of no interference, low-frequency interference and high-frequency interference.
Working condition 1: no interference (i.e., d (x, t) ═ 0); FIGS. 4, 5, and 6 show the control input, online parameter estimation, and tracking error for different control methods under this condition; compared with simulation results, analysis on tracking errors shows that the FADRC transient performance is superior to that of ADRAC and PID, the convergence rate is obviously higher than that of the ADRAC and PID, and the steady-state error is more ideal than that of the ADRAC and PID; analysis is carried out on the aspect of online parameter estimation, a novel parameter adaptive law which introduces a parameter estimation error term and is proposed in the FADRC enables a parameter estimation value to be rapidly converged to a true value within a limited time, and the parameter estimation in the traditional ADRAC is not only slow in convergence speed and obvious in oscillation in the estimation process, but also inaccurate in estimation value; the control inputs u under all three control methods are continuous and bounded.
Working condition 2: low-frequency interference (i.e., d (x, t) ═ 0.3 × sin (0.5 π t)); the online parameter estimation and tracking errors under different control methods are shown in fig. 7 and 8; under the working condition, compared with the ADRAC and the PID, the FADRC transient performance is still optimal, the convergence speed is still far higher than that of the ADRAC and the PID, the steady-state error is relatively free of interference, the working condition is reduced to some extent, the jitter is slightly increased, and the steady-state error is better than that of the ADRAC and the PID; in the presence of low-frequency interference, FADRC on-line parameter estimation can still ensure that the estimated value is rapidly converged to the true value within a limited time, and compared with the non-interference working condition, the parameter estimation under the traditional ADRAC not only increases the jitter further, but also further deviates the estimated value from the true value.
Working condition 3: high frequency interference (i.e., d (x, t) ═ 0.2 × sin (1.5 π t)); the online parameter estimation and tracking error under different control methods are shown in fig. 9 and 10; it can be easily found that the steady-state error and the parameter estimation value of the FADRC are further increased compared with the jitter of the low-frequency interference condition under the influence of high-frequency interference, but compared with the ADRAC and the PID, the FADRC still has the most ideal performance in terms of tracking error and online parameter estimation.
In conclusion, compared with the ADRAC controller and the PID controller, the designed FADRC controller can obviously improve the convergence speed of tracking errors and online parameter estimation under the working conditions of no interference, low-frequency interference or high-frequency interference, obtain more excellent control performance, and fully verify the effectiveness and superiority of the FADRC controller compared with the existing method.

Claims (2)

1. A fast self-adaptive disturbance rejection control method of a direct current motor servo system is characterized by comprising the following steps:
step 1, establishing a mathematical model of a direct current motor servo system, which comprises the following specific steps:
step 1-1, a motor in the direct current servo motor system is controlled by a driver and directly drives an inertial load; neglecting current loop dynamics, the dynamics of inertial loads are as follows:
Figure FDA0003695757950000011
where y and m represent angular displacement and load inertia, respectively, K i Representing the moment constant, u representing the control input, B representing the viscous friction coefficient, f representing the unmodeled disturbance, t representing the dc motor servo system time,
Figure FDA0003695757950000012
represents the derivative of y;
Figure FDA0003695757950000013
represents the second derivative of y;
step 1-2, for any variable A, define
Figure FDA0003695757950000014
Which represents an estimate of the variable a, is,
Figure FDA0003695757950000015
which represents the error of the estimation of the variable a,
Figure FDA0003695757950000016
represents the derivative of variable a, for which all subsequent identical designations apply;
defining state variables of a DC motor servo system
Figure FDA0003695757950000017
Unknown parameter vector theta of direct current motor servo system is [ theta ═ theta [ ] 1 ,θ 2 ] T ＝[K i /m,B/m] T If the unmodeled disturbance d (x, t) of the dc motor servo system is f/m, the state equation is written as equation (1):
Figure FDA0003695757950000018
in the above formula
Figure FDA0003695757950000019
Respectively represent x 1 ,x 2 A derivative of (a);
for ease of controller design, assume the following:
assume that 1: system reference position signal x 1d The second order is continuously differentiable, and each derivative is bounded;
assume 2: the range of the unknown parameter vector theta is known, i.e.
θ min ≤θ≤θ max (3)
In the formula [ theta ] min ＝[θ 1min ,θ 2min ] T Is a known lower bound, θ, of the unknown parameter vector max ＝[θ 1max ,θ 2max ] T A known upper bound for the unknown parameter vector;
turning to the step 2;
step 2, designing a fast self-adaptive disturbance rejection controller based on a mathematical model of a direct current motor servo system, and specifically comprising the following steps:
step 2-1, before the controller is designed, discontinuous parameter mapping adopted by parameter self-adaptation is given:
order to
Figure FDA0003695757950000021
Figure FDA0003695757950000022
Representing an estimate of the system unknown parameter vector theta,
Figure FDA0003695757950000023
to ensure the stability of the adaptive control law for parameter estimation errors, the system-based parameter uncertainty is bounded, i.e. assume 2, defining the following parameter-adaptive discontinuous mapping
Figure FDA0003695757950000024
Figure FDA0003695757950000025
Wherein i is 1, 2; i represents the ith element of the vector;
the adaptation rate is given by the following parameters
Figure FDA0003695757950000026
Figure FDA0003695757950000027
In the formula of>0 is positive definite diagonal matrix, tau is self-adaptive law function, and for arbitrary self-adaptive law function tau the discontinuous projection mapping is used
Figure FDA0003695757950000028
Can ensure that:
Figure FDA0003695757950000029
the system extended state observer is constructed as follows:
Figure FDA00036957579500000210
in the formula
Figure FDA00036957579500000211
Represents a state x j Is estimated by the estimation of (a) a,
Figure FDA00036957579500000212
represents a state x j The error of the estimation of (2) is,namely, it is
Figure FDA00036957579500000213
ω o ＞0，ω o For the bandwidth of the ESO to be,
Figure FDA00036957579500000214
Figure FDA00036957579500000215
a process function vector for the construct;
the following two scenarios are defined for the expanded state:
case 1: let the system expand state x 3 D (x, t), then constructing a system expansion state equation based on the system model as follows:
Figure FDA0003695757950000031
in the above formula
Figure FDA0003695757950000032
Denotes x 3 H (t) represents the derivative of the dilated state with respect to the system time t;
case 2: order to
Figure FDA0003695757950000033
The constructed system expansion equation of state becomes:
Figure FDA0003695757950000034
by using the idea of a back-stepping method, a fast self-adaptive disturbance rejection controller is designed, and the method specifically comprises the following steps:
error variables are defined as follows
Figure FDA0003695757950000035
In the formula z 1 For systematic tracking error, α 1 Is a state x 2 Virtual control law of z 2 Is the deviation between the two, x 1d Referencing a position signal for the system;
to z 1 And (5) obtaining a derivative:
Figure FDA0003695757950000036
the virtual control law is designed as follows:
Figure FDA0003695757950000037
where the gain factor k is 1 > 0, according to the formulae (10) to (12):
Figure FDA0003695757950000038
and performing Laplace transform on the left side and the right side of the formula to obtain:
Figure FDA0003695757950000039
wherein s represents a complex parameter variable;
the transfer function G(s) represented by the formula (14) is stable when z is 2 When approaching 0, z 1 Inevitably approaching 0, so the primary design objective is shifted to z 2 Approaching to 0;
the adaptation law for the second channel is designed for the sake of theoretical derivation under the condition of case 1, i.e. x 3 D (x, t), assuming that the unmodeled disturbance of the system is constant, i.e. h (t) is 0, the second formula in equation (2) is modified as:
Figure FDA0003695757950000041
further, the above formula can be modified as follows:
Figure FDA0003695757950000042
and carrying out filtering operation on each term on two sides of the above formula to obtain:
Figure FDA0003695757950000043
wherein the filter operation coefficient k is more than 0,
Figure FDA0003695757950000044
representing state estimates
Figure FDA0003695757950000045
The filter variables obtained after the filtering are obtained,
Figure FDA0003695757950000046
representing errors in state estimation
Figure FDA0003695757950000047
Filtering variables obtained after filtering; x is the number of 2f (0) When t is 0, x 2f Corresponding values, for which all subsequent identical identifiers apply;
in addition, note that
Figure FDA0003695757950000048
Therefore, when h (t) is 0,
Figure FDA0003695757950000049
obtained by integration;
the following variables are defined:
Figure FDA00036957579500000410
in the above formula, P and Q are structural process parameters, the constant l is more than 0, and the above formula is integrated to obtain:
Figure FDA00036957579500000411
in the above formula, β represents an integral variable;
derived from the above equation:
Q＝Pθ (21)
defining a process matrix H such that it satisfies:
Figure FDA0003695757950000051
definition of lambda max () And λ min () Respectively representing the maximum and minimum eigenvalues of the corresponding matrix, the following reasoning is provided;
introduction 1: if matrix
Figure FDA0003695757950000052
If the continuous excitation condition is satisfied, the matrix P is positive, so that its minimum eigenvalue λ is min (P (t)) satisfies:
when T > T > 0 and the constant sigma > 0, there are
λ min (P(t))＞σ (23)
T represents a time constant;
further, a parameter adaptation function τ is defined as:
Figure FDA0003695757950000053
in the formula, the gain coefficient C 1 ＞0,C 2 ＞0,C 3 If the system state estimation error is larger than 0, the epsilon system state estimation error is larger than 0, and J is a positive definite matrix; b is 1 ＝[0,1,0] T ；
Based on the obtained parameter estimation values, the actual control input u is designed as:
Figure FDA0003695757950000054
in the formula k 2 > 0 is the gain factor, u a Is a model compensation term, u, adjusted based on-line parameter estimation and system state estimation s Is a linear robust feedback term;
turning to the step 3;
and 3, performing rapid self-adaptive disturbance rejection controller stability verification by using the Lyapunov stability theory to obtain a system stability result.
2. The fast adaptive disturbance rejection control method of the direct current motor servo system according to claim 1, wherein a stability verification is performed on the fast adaptive disturbance rejection control method of the direct current motor servo system by using a lyapunov stability theory, and a system stability result is obtained, specifically as follows:
under the condition of case 1, the Lyapunov function is defined as follows:
Figure FDA0003695757950000055
Figure FDA0003695757950000061
for a closed loop system, all signals are bounded and, based on an analysis of the error,
Figure FDA0003695757950000066
is bounded, so W has consistent continuity; by applying the Barbalt's theorem, the time approaches infinityWhen the system tracking error approaches zero;
when the unmodeled interference term d (x, t) is a time-varying function, analysis is performed using case 2, assuming that h (t) is bounded, i.e., there is a constant ζ > 0 such that | h (t) | ≦ ζ, defining the Lyapunov function as follows:
Figure FDA0003695757950000062
for this closed loop system, all signals are bounded, and the tracking error z of the system is 1 The bounding is stable;
to prove parameter estimation error
Figure FDA0003695757950000063
Can converge to zero in a finite time, and defines the Lyapunov function as follows:
Figure FDA0003695757950000064
by derivation, parameter estimation error
Figure FDA0003695757950000065
Can converge to zero within a limited time.
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