CN104485866B - Motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator - Google Patents
Motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator Download PDFInfo
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Abstract
The present invention provides a kind of motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator, and the realization of this method includes the following steps：Step 1: establishing motor position servo system model；Step 2: motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator；Step 3: by adjusting the parameter in the robust controller based on finite time Interference Estimation so that system meets Control performance standard.Motor indirect self-adaptive robust output feedback control method proposed by the present invention based on high-order synovial membrane differentiator the characteristics of for motor position servo system, establishes motor position servo system model；The motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator of design, service system state estimate and is used for controller design, it avoids influence of the measurement noise to controller simultaneously, can effectively solve the problems, such as parameter uncertainty present in motor servo system and Uncertain nonlinear.
Description
Technical field
The present invention relates to a kind of control methods, and in particular to a kind of indirect self-adaptive robust of motor position servo system is defeated
Go out feedback.
Background technology
Electric energy is directly changed into linear motion mechanical energy by linear motor, does not need the transmission dress of any intermediate conversion mechanism
It sets.Have many advantages, such as that starting thrust is big, transmission stiffness is high, dynamic response is fast, positioning accuracy is high, stroke length is unrestricted, because
And it is widely used in the industry.
However, it is very difficult to design high performance controller for motor servo system, because designer is likely to
Encounter many model uncertainties, including parameter uncertainty and Uncertain nonlinear etc. do not model it is non-linear.These are not
Certainty factor may the control performance that can obtain of severe exacerbation, so as to cause low control accuracy, limit cycle concussion, even
Unstability.For known non-linear, feedback linearization technical finesse can be passed through.But no matter kinematic nonlinearity and ginseng
Number identification how accurate mathematical model, be impossible to obtain the entire non-linear behavior of practical nonlinear system and exact
Parameter, and then perfectly compensated.Always there is cannot enough specific functions do not model the parameter error simulated and
It is non-linear.
In order to improve the tracking performance of electric system, many advanced gamma controllers are applied to electric system control
In, such as Robust Adaptive Control, adaptive robust control (ARC), sliding formwork control etc..
It is not only needed in motion control however, all above methods are based on overall-finished housing development controller design
Position signal, it is also necessary to speed and/or acceleration signal.But in many real systems, by mechanical structure, volume, weight and
Cost limits, often only known to location information.In addition, even if speed and acceleration signal can obtain, there is also serious surveys
Noise is measured, and then deteriorates the performance that full-state feedback device can obtain.These realities in the presence of nonlinear Control application
Border problem results in PID control so far in Motor Control Field still in leading position.But in the new of modern industrial age
Under demand, PID is increasingly difficult to meet the high performance control increasingly pursued.Therefore, there is an urgent need to design nonlinear object feedback
Control strategy.In linear system, this problem can utilize separate design principle to solve, i.e., to considerable controllable linear system
System, separately designs state feedback controller and state observer is obtained with the output feedback controller of system.But non-thread
Sexual system is exactly one using the Stabilization of output feedback realization system and extremely difficult asks since separation principle is no longer set up
Topic, in recent years, the output-feedback control problem of nonlinear system has obtained extensive concern.Only system output is scalable
Under the conditions of how to realize that the calm of control system is one important problem of control theory.
Invention content
The present invention is known to only location status, for simultaneous parameter in motor position servo system
Certainty and Uncertain nonlinear problem propose that a kind of motor indirect self-adaptive robust output based on High-Order Sliding Mode differentiator is anti-
Present control method.
The above-mentioned purpose of the present invention realizes that dependent claims are to select else or have by the technical characteristic of independent claims
The mode of profit develops the technical characteristic of independent claims.
To reach above-mentioned purpose, the present invention proposes that a kind of motor indirect self-adaptive robust based on High-Order Sliding Mode differentiator is defeated
Go out feedback, realization includes the following steps：
Step 1: establishing motor position servo system model
According to Newton's second law, the kinetic model equation of motor inertia load is：
Y indicates that angular displacement, m indicate inertia load, k in formulafIndicate that torque coefficient, u are system control inputs, b represents viscous
Frottage coefficient, f represent other and do not model interference, including non-linear friction, external disturbance and Unmarried pregnancy.
(1) formula is write as state space form, it is as follows：
WhereinX=[x1,x2]TIndicate the state vector of position and speed.Parameter θ=[θ1,θ2]T,
Wherein θ1=kf/ m, θ2=b/m, d (x, t)=f/m tables indicate to concentrate interference；
It is assumed hereinafter that always set up：
Assuming that 1：Structural uncertainty θ meets：
Wherein θmin=[θ1min,θ2min]T,θmax=[θ1max,θ2max]T, they are all known, θ in addition1min＞ 0, θ2min
＞ 0.
Assuming that 2：D (x, t) is bounded, i.e.,
|d(x,t)|≤δd (4)
Wherein δdKnown quantity；
Step 2: motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator, specifically
Steps are as follows：
Step 2 (one), the projection adaptive law structure of tape speed limitation
It enablesIndicate the estimation of θ,Indicate the evaluated error of θ, i.e.,It is as follows to define a projection function：
Wherein ζ ∈ Rp,Γ(t)∈Rp×pIt is a positive definite symmetric matrices changed over time,WithIt indicates respectively
ΩθInside and boundary,It indicatesWhen other unit's normal vector；
For projection function (5) formula, in control parameter estimation procedure, preset adaptive limitation speed is used, because
And it is as follows to define a saturation function：
WhereinIt is a pre-set limiting speed, the structure that following lemma summarizes parameter estimation algorithm is special
Property：
Lemma 1：Assuming that using following projection type adaptive law and preset adaptive limiting speedMore new estimation is joined
Number
Wherein τ is auto-adaptive function, Γ (t) ＞ 0 be continuous can micro- just symmetrical adaptive rate matrix, thus adaptively
Rule, can obtain following ideal characterisitics：
P1) estimates of parameters is always in the Ω of known boundedθIn collection, i.e., for arbitrary t, always have thus by vacation
If 1 can obtain,
P2)
P3) Parameters variation rule is uniformly bounded：
Step 2 (two), the High-Order Sliding Mode differentiator for building motor, estimate system unknown state
First, system model (2) is re-converted into following form：
WhereinIt is a broad sense interference；
ByUnderstand that D (x, t) is also bounded with hypothesis 2, i.e.,：
Wherein θm=θmax-θmin,
One High-Order Sliding Mode differentiator is designed by formula (8), it is as follows：
Wherein x1, x2, x3Output position, speed and acceleration are indicated respectively,Respectively it is x1、x2、x3's
Estimated value, λ0, λ1, λ2For design parameter；
Lemma 1：There are a time T1, when time t is more than time constant T1When, wherein
Step 2 (three), motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator,
Include the following steps：
It is as follows to define one group of function：
Wherein z1=x1-x1d(t) it is output tracking error, k1>0 is a feedback oscillator；
Due to G (s)=z1(s)/z2(s)=1/ (s+k1) it is a stable transmission function, allow z1Very little levels off to zero
Exactly allow z2Very little levels off to zero；Therefore, controller design, which is transformed into, allows z2It is as small as possible or level off to zero；
To formula (11) differential, simultaneously wushu (8) is brought into, can be obtained：
Controller based on state estimation is as follows：
Wherein k2＞ 0 is a feedback oscillator；
Wushu (13) brings formula (12) into, can obtain z2Dynamical equation：
us2Meet following condition：
z2{us2-D}≤σ1 (15a)
z2us2≤0 (15b)
Wherein σ1＞ 0 is design parameter, the u of formula (15a) and formula (15b)s2Selection it is as follows：It is such as minor function to enable g
WhereinIt is the upper bound of D (x, t), thus designs following us2：
Wherein ks1For a non-linear gain；
Step 2 (four), design indirect parameter method of estimation
Assuming that 3：There is no interference, i.e. D=0 for system；
Rewriting system dynamic model can obtain following formula for parameter Estimation：
Introduce a low-pass filter Hf, and filter is applied in formula (18), it can obtain：
Wherein uf,x2fIndicate that input is u, x respectively2Filter HfOutput；
To carry out parameter Estimation, following defined parameters vector sum regression function：
Linear regression model (LRM) can be obtained by (20)：
Definition prediction output errorWhereinIt can be obtained by (21) and predict error model as follows：
For static linear regression model (LRM), using with exponential forgetting factor and the preset least-squares estimation of covariance
Algorithm, the regressor for each group and corresponding unknown parameter vector, it is as follows to define adaptation rate matrix：
Wherein Γ (0)=ΓT(0) 0 ＞, α >=0 are forgetting factors,It is the preset time of covariance, that is, works as λmin(Γ
(t))=ρl, ρlTo preset the minimum limit value of Γ (t) and meeting 0 ＜ ρl＜ ρ0, λmin(Γ (t)) is the minimal eigenvalue of Γ (t),
I is unit matrix, and auto-adaptive function is as follows at this time：
υ >=0 is normalization factor in formula, and as υ=0, then auto-adaptive function at this time is non-standard adaptive letter
Number.It is not Persistent Excitation when returning device, Γ (t) may be unbounded, i.e. λ in the parameter Estimation containing forgetting factormax(Γ
(t)) → ∞ and the saturation for leading to estimator, such case generation, (23) formula is revised as in order to prevent：
ρ in formulaMIt is the upper bound of the norm of preset Γ (t), and λmax(Γ(t))≤ρM, and 0 ＜ ρl＜ ρ ＜ ρM；
Based on this improvement, it is ensured that ρlI ＜ Γ (t) ＜ ρMI。
As t >=T1When, d=0, if Persistent Excitation (PE) condition meets：
Then parameter converges to true value, i.e., as t → ∞,
Based on parameter update law defined above, following lemma can be obtained：
Lemma 2：By parameter update law (7), least square auto-adaptive function (23), formula (24) and prediction error model
(22) it can obtain：ε∈L2(0,∞)∩L∞(0,∞),
Step 3: adjusting the parameter k of u in the robust controller based on finite time Interference Estimation1,k2, a, v, λ0, λ1, λ2,
σ1So that system meets Control performance standard.
From the above technical solution of the present invention shows that, it is connected between the motor proposed by the present invention based on high-order synovial membrane differentiator
Robust output feedback control method is adapted to, the characteristics of for motor position servo system, establishes motor position servo system mould
Type；The motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator that the present invention designs, to system
System state carries out estimating and being used for controller design, avoids influence of the measurement noise to controller simultaneously, can effectively solve electricity
The parameter uncertainty and Uncertain nonlinear problem of machine servo system, the system control precision satisfaction property under above-mentioned disturbed condition
It can index；This invention simplifies controller design, simulation result shows its validity.
It should be appreciated that as long as aforementioned concepts and all combinations additionally conceived that describe in greater detail below are at this
Sample design it is not conflicting in the case of can be viewed as the disclosure subject matter a part.In addition, required guarantor
All combinations of the theme of shield are considered as a part for the subject matter of the disclosure.
Can be more fully appreciated from the following description in conjunction with attached drawing present invention teach that foregoing and other aspect, reality
Apply example and feature.The feature and/or advantageous effect of other additional aspects such as illustrative embodiments of the present invention will be below
Description in it is obvious, or by according to present invention teach that specific implementation mode practice in learn.
Description of the drawings
Attached drawing is not intended to drawn to scale.In the accompanying drawings, identical or approximately uniform group each of is shown in each figure
It can be indicated by the same numeral at part.For clarity, in each figure, not each component part is labeled.
Now, by example and the embodiments of various aspects of the invention will be described in reference to the drawings, wherein：
Fig. 1 is the typical structure schematic diagram of motor executive device.
Fig. 2 is the control of the motor indirect self-adaptive robust output feedback control method the present invention is based on high-order synovial membrane differentiator
Policy map processed.
Fig. 3 is the schematic diagram of the additional interference curve of system.
Fig. 4 is the schematic diagram of controller input voltage u-curve.
Fig. 5 is the schematic diagram of parameter Estimation curve.
Fig. 6 is the schematic diagram of location estimation and evaluated error curve.
Fig. 7 is the schematic diagram of velocity estimation and evaluated error curve.
Fig. 8 is the schematic diagram of acceleration estimation and evaluated error curve.
Fig. 9 is the schematic diagram for designing controller and PID controller tracking error curve.
Specific implementation mode
In order to know more about the technology contents of the present invention, spy lifts specific embodiment and institute's accompanying drawings is coordinated to be described as follows.
Various aspects with reference to the accompanying drawings to describe the present invention in the disclosure, shown in the drawings of the embodiment of many explanations.
It is not intended to cover all aspects of the invention for embodiment of the disclosure.It should be appreciated that a variety of designs and reality presented hereinbefore
Those of apply example, and describe in more detail below design and embodiment can in many ways in any one come it is real
It applies, this is to should be design disclosed in this invention to be not limited to any embodiment with embodiment.In addition, disclosed by the invention one
A little aspects can be used alone, or otherwise any appropriately combined be used with disclosed by the invention.
Illustrate present embodiment below in conjunction with the accompanying drawings, a kind of motor output based on state observation described in present embodiment is anti-
Feedback control method is as follows：
Step 1: establishing motor position servo system model；
Step 2: motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator；And
Step 3: by adjusting the parameter in the robust controller based on finite time Interference Estimation so that system meets
Control performance standard.
Below in conjunction with the accompanying drawings shown in 1-9, the exemplary realization process of aforementioned embodiments is described in detail.
Step 1: motor position servo system model is established, according to Newton's second law, the dynamics of motor inertia load
Model equation is：
Y indicates that angular displacement, m indicate inertia load, k in formulafIndicate that torque coefficient, u are system control inputs, b represents viscous
Frottage coefficient, f represent other and do not model interference, such as non-linear friction, external disturbance and Unmarried pregnancy.
(1) formula is write as state space form, it is as follows：
WhereinX=[x1,x2]TIndicate the state vector of position and speed.Parameter θ=[θ1,θ2]T,
Wherein θ1=kf/ m, θ2=b/m, d (x, t)=f/m indicate to concentrate interference.Under normal circumstances, due to systematic parameter m, kf, b is to become
Change, system is structural uncertainty, although we do not know the specifying information of system, the general information of system is can be with
Know.In addition, system also has unstructured uncertainty d (x, t), it is clear that it cannot clearly be modeled, but system does not model
Dynamic and interference always bounded.Thus, it is assumed hereinafter that always set up：
Assuming that 1：Structural uncertainty θ meets：
Wherein θmin=[θ1min,θ2min]T,θmax=[θ1max,θ2max]T, they are all known, θ in addition1min＞ 0, θ2min
＞ 0.
Assuming that 2：D (x, t) is bounded, i.e.,
|d(x,t)|≤δd (4)
Wherein δdIt is known.
Step 2: motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator is specific
Steps are as follows：
Step 2 (one), the projection adaptive law structure of tape speed limitation
It enablesIndicate the estimation of θ,Indicate the evaluated error of θ, i.e.,It is as follows to define a projection function
Wherein ζ ∈ Rp,Γ(t)∈Rp×pIt is a positive definite symmetric matrices changed over time,WithΩ is indicated respectivelyθ
Inside and boundary,It indicatesWhen other unit's normal vector.
For projection function (5) formula, in control parameter estimation procedure, preset adaptive limitation speed is used.Cause
And it is as follows to define a saturation function：
WhereinIt is a pre-set limiting speed.Following lemma summarizes the parameter that will be used herein and estimates
The architectural characteristic of calculating method.
Lemma 1：Assuming that using following projection type adaptive law and preset adaptive limiting speedMore new estimation is joined
Number
Wherein τ is auto-adaptive function, and Γ (t) ＞ 0 are continuous can micro- just symmetrical adaptive rate matrix.Thus adaptive
Rule, can obtain following ideal characterisitics：
P1) estimates of parameters is always in the Ω of known boundedθIn collection, i.e., for arbitrary t, always have thus by vacation
If 1 can obtain,
P2)
P3) Parameters variation rule is uniformly bounded.
Step 2 (two), the High-Order Sliding Mode differentiator for building motor, estimate system unknown state.
First, system model (2) can be written as form again：
WhereinIt is a broad sense interference.
ByUnderstand that D (x, t) is also bounded with hypothesis 2.I.e.
Wherein θm=θmax-θmin,
A High-Order Sliding Mode differentiator can be designed by (8), it is as follows：
Wherein x1, x2, x3Output position, speed and acceleration are indicated respectively,Respectively it is x1, x2, x3's
Estimated value, λ0, λ1, λ2For design parameter.
Lemma 1：There are a time T1, when time t is more than time constant T1When, wherein
Step 2 (three), motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator are such as
Under：
It is as follows to define one group of function
Wherein z1=x1-x1d(t) it is output tracking error, k1＞ 0 is a feedback oscillator.Due to G (s)=z1(s)/z2
(s)=1/ (s+k1) it is a stable transmission function, allow z1Very little or to level off to zero be exactly to allow z2Very little levels off to zero.Cause
This, controller design, which is transformed into, allows z2It is as small as possible or level off to zero.To formula (11) differential, simultaneously wushu (8) is brought into, can be obtained：
Controller based on state estimation is as follows：
Wherein k2＞ 0 is a feedback oscillator.
Wushu (13) brings formula (12) into, can obtain z2Dynamical equation：
us2Meet following condition：
z2{us2-D}≤σ1 (15a)
z2us2≤0 (15b)
Wherein σ1＞ 0 is a design parameter, how to choose the u for meeting formula (15a) and formula (15b)s2, one is provided here
Example, it is as follows：It is such as minor function to enable g
WhereinIt is the upper bound of D (x, t).Thus following u is designeds2
Wherein ks1For a non-linear gain.
Step 2 (four), design indirect parameter algorithm for estimating
The main task of this part is to establish a suitable parameter estimation algorithm, improves can get for parameter Estimation
Property.
Assuming that 3：There is no interference, i.e. D=0 for system.
Following formula can be obtained for parameter Estimation in rewriting system dynamic model：
Introduce a low-pass filter Hf, and filter is applied in formula (18), it can obtain
Wherein uf,x2fIndicate that input is u, x respectively2Filter HfOutput.To carry out parameter Estimation, ginseng of such as giving a definition
Number vector and regression function：
Linear regression model (LRM) can be obtained by (20)
Definition prediction output errorWhereinIt can be obtained by (21) and predict error model as follows：
For the linear regression model (LRM) of this static state, unknown parameter is may be used to determine there are many different algorithm for estimating,
Our uses are a kind of with exponential forgetting factor and the preset least-squares estimation algorithm of covariance.For each group of recurrence
Amount and corresponding unknown parameter vector, the adaptation rate matrix that we can define are as follows：
Wherein Γ (0)=ΓT(0) 0 ＞, α >=0 are forgetting factorsIt is the preset time of covariance, that is, works as λmin(Γ
(t))=ρl, ρlTo preset the minimum limit value of Γ (t) and meeting 0 ＜ ρl＜ ρ0, λmin(Γ (t)) is the minimal eigenvalue of Γ (t),
I is unit matrix.Auto-adaptive function is as follows at this time：
υ >=0 is normalization factor in formula, and as υ=0, then auto-adaptive function at this time is non-standard adaptive letter
Number.It is not Persistent Excitation when returning device, Γ (t) may but in the implementation of the parameter Estimation containing forgetting factor realization
It can be unbounded.That is λmax(Γ (t)) → ∞ and the saturation for leading to estimator.Such case occurs in order to prevent, and (23) formula is changed
For：
ρ in formulaMIt is the upper bound of the norm of preset Γ (t), and λmax(Γ(t))≤ρM, and 0 ＜ ρl＜ ρ ＜ ρM.Based on this
Kind is improved, we can ensure ρlI ＜ Γ (t) ＜ ρMI。
As t >=T1When, d=0.If Persistent Excitation (PE) condition meets：
Then parameter converges to true value, i.e., as t → ∞,
Based on parameter update law defined above, following lemma can be obtained：
Lemma 2：By parameter update law (7), least square auto-adaptive function (23), formula (24) and prediction error model
(22) it can obtain：ε∈L2(0,∞)∩L∞(0,∞),
Step 3: the parameter k by adjusting u in the robust controller based on finite time Interference Estimation1,k2, α, v, λ0,
λ1, λ2, σ1So that system meets Control performance standard.
In this example, it is further advanced by and chooses Lyapunov Equation to the controller of previous designs and the stability of system
Carry out verification explanation.
Verify system stability：
Theorem 1：By High-Order Sliding Mode differentiator (10), the motor indirect self-adaptive Shandong based on High-Order Sliding Mode differentiator of design
Stick output feedback controller (13) has following property：
A. at a time T1Later, the state of High-Order Sliding Mode differentiator estimation is accurate, i.e.,Definition is such as
Under Lyapunov Equation
Meet following inequality
It proves：To formula (27) differential, and wushu (14) is brought into and can be obtained
Wushu (15a) is brought (29) into and can be obtained
Inequality (28) can be obtained to formula (30) both ends integral.Therefore controller is convergent, and system is stable.
It is tested under simulated environment with reference to some specific examples：
Following parameter is taken to model system in simulations：M=0.01kgm2,kf=5, b=1.25Ns/m.
Take controller parameter k1=300, k2=500, σ1=1 × 105, v=22, α=2, λ0=5, λ1=5, λ2=6；θmin
=[0,0]T,θmax=[200,1000]T,SelectedIt is away from the true value of parameter, certainly with examination
The effect of suitable solution rule.
Position angle input signal y=0.2sin (π t) [1-e-0.01t3], unit rad.
Outer interference added by system is as shown in Figure 3.
Control law function and effect：
Fig. 4 is controller input voltage u-curve under interference effect, the input of controller input voltage satisfaction -10V~+10V
Range meets practical application.
Fig. 5 is parameter Estimation curve.
Fig. 6 is location estimation and evaluated error curve.
Fig. 7 is velocity estimation and evaluated error curve.
Fig. 8 is acceleration estimation and evaluated error curve.
Fig. 9 is design controller and PID controller tracking error curve.
By comparison illustrated above it is found that method proposed by the present invention is under simulated environment can accurately estimate to do well
Value, more accurately estimates systematic parameter.Compared to more traditional PID controller, the controller that the present invention designs can obtain
Good control accuracy.Simulation result shows that under the influence of parameter uncertainty and Uncertain nonlinear, the present embodiment proposes
Method disclosure satisfy that performance indicator.
Although the present invention has been disclosed as a preferred embodiment, however, it is not to limit the invention.Skill belonging to the present invention
Has usually intellectual in art field, without departing from the spirit and scope of the present invention, when can be used for a variety of modifications and variations.Cause
This, the scope of protection of the present invention is defined by those of the claims.
Claims (2)
1. a kind of motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator, which is characterized in that
The realization of this method includes the following steps：
Step 1: establishing motor position servo system model, following steps are specifically included：
According to Newton's second law, the kinetic model equation of motor inertia load is：
Y indicates that angular displacement, m indicate inertia load, k in formulafIndicate that torque coefficient, u are system control inputs, b represents viscous friction
Coefficient, f represent other and do not model interference, including non-linear friction, external disturbance and Unmarried pregnancy；
(1) formula is write as state space form, it is as follows：
WhereinX=[x1,x2]TIndicate the state vector of position and speed, parameter θ=[θ1,θ2]T, wherein
θ1=kf/ m, θ2=b/m, d (x, t)=f/m indicate to concentrate interference；
It is assumed hereinafter that always set up：
Assuming that 1：Structural uncertainty θ meets：
Wherein θmin=[θ1min,θ2min]T,θmax=[θ1max,θ2max]T, they are all known, θ in addition1min＞ 0, θ2min＞ 0；
Assuming that 2：D (x, t) is bounded, i.e.,
|d(x,t)|≤δd (4)
Wherein δdKnown quantity；
Step 2: motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator, specific steps
It is as follows：
Step 2 (one), the projection adaptive law structure of tape speed limitation
It enablesIndicate the estimation of θ,Indicate the evaluated error of θ, i.e.,It is as follows to define a projection function：
Wherein ζ ∈ Rp,Γ(t)∈Rp×pIt is a positive definite symmetric matrices changed over time,WithΩ is indicated respectivelyθIt is interior
Portion and boundary,It indicatesWhen other unit's normal vector；
For projection function (5) formula, in control parameter estimation procedure, preset adaptive limitation speed is used, thus, it is fixed
An adopted saturation function is as follows：
WhereinIt is a pre-set adaptive limiting speed, the structure that following lemma summarizes parameter estimation algorithm is special
Property：
Lemma 1：Assuming that using following projection type adaptive law and preset adaptive limiting speedMore new estimation parameter
Wherein τ is auto-adaptive function, Γ (t) ＞ 0 be continuous can micro- just symmetrical adaptive rate matrix, thus adaptive law can
Obtain following ideal characterisitics：
P1) estimates of parameters is always in the Ω of known boundedθIn collection, i.e., for arbitrary t, always haveThus by assuming that 1 can
,
P3) Parameters variation rule is uniformly bounded：
Step 2 (two), the High-Order Sliding Mode differentiator for building motor, estimate system unknown state
First, formula (2) is re-converted into following form：
WhereinIt is a broad sense interference；
ByUnderstand that D (x, t) is also bounded with hypothesis 2, i.e.,：
Wherein θm=θmax-θmin,
One High-Order Sliding Mode differentiator is designed by formula (8), it is as follows：
Wherein x1, x2, x3Output position, speed and acceleration are indicated respectively,Respectively x1、x2、x3Estimated value,
λ0, λ1, λ2For design parameter；
Lemma 1：There are a time T1, when time t is more than time constant T1When,Wherein
Step 2 (three), motor indirect self-adaptive robust output feedback control device of the design based on High-Order Sliding Mode differentiator, including
Following steps：
It is as follows to define one group of function：
Wherein z1=x1-x1d(t) it is output tracking error, x1d(t) physical location, k are indicated1＞ 0 is a feedback oscillator；
Due to G (s)=z1(s)/z2(s)=1/ (s+k1) it is a stable transmission function, allow z1Very little levels off to and zero is exactly
Allow z2Very little levels off to zero；Therefore, controller design, which is transformed into, allows z2It is as small as possible or level off to zero；
To formula (11) differential, simultaneously wushu (8) is brought into, can be obtained：
Controller based on state estimation is as follows：
Wherein k2＞ 0 is a feedback oscillator；
Wushu (13) brings formula (12) into, can obtain z2Dynamical equation：
us2Meet following condition：
z2{us2-D}≤σ1 (15a)
z2us2≤0 (15b)
Wherein σ1＞ 0 is design parameter, the u of formula (15a) and formula (15b)s2Selection it is as follows：It is such as minor function to enable g
WhereinIt is the upper bound of D (x, t), thus designs following us2：
Wherein ks1For a non-linear gain；
Step 2 (four), design indirect parameter method of estimation
Assuming that 3：There is no interference, i.e. D=0 for system；
Rewriting system dynamic model can obtain following formula for parameter Estimation：
Introduce a low-pass filter Hf, and filter is applied in formula (18), it can obtain：
Wherein uf,x2fIndicate that input is u, x respectively2Filter HfOutput；
To carry out parameter Estimation, following defined parameters vector sum regression function：
Linear regression model (LRM) can be obtained by (20)：
Definition prediction output errorWhereinIt can be obtained by (21) and predict error model as follows：
For static linear regression model (LRM), calculated using with exponential forgetting factor and the preset least-squares estimation of covariance
Method, the regressor for each group and corresponding unknown parameter vector, it is as follows to define adaptation rate matrix：
Wherein Γ (0)=ΓT(0) 0 ＞, α >=0 are forgetting factors,It is the preset time of covariance, ρ ＞ 0 are constant value, Γ (t)
For adaptive gain matrix, that is, work as λmin(Γ (t))=ρl, ρlTo preset the minimum limit value of Γ (t) and meeting 0 ＜ ρl＜ ρ, λmin
(Γ (t)) is the minimal eigenvalue of Γ (t), and I is unit matrix, and auto-adaptive function is as follows at this time：
υ >=0 is normalization factor in formula, and as υ=0, then auto-adaptive function at this time is non-standard auto-adaptive function,
It is not Persistent Excitation when returning device, Γ (t) may be unbounded, i.e. λ in parameter Estimation containing forgetting factormax(Γ(t))
→ ∞ and the saturation for leading to estimator, such case generation, (23) formula is revised as in order to prevent：
ρ in formulaMIt is the upper bound of the norm of preset Γ (t), and λmax(Γ(t))≤ρM, and 0 ＜ ρl＜ ρ ＜ ρM；
Based on this improvement, it is ensured that ρlI ＜ Γ (t) ＜ ρMI；
As t >=T1When, d=0, if persistent excitation condition meets：
Then parameter converges to true value, i.e., as t → ∞,
IpUnit matrix is tieed up for p；
Based on parameter update law defined above, following lemma can be obtained：
Lemma 2：By parameter update law (7), least square auto-adaptive function (23), formula (24) and prediction error model (22)
It can obtain：Wherein, L2The model of representing matrix
Number, L∞The Infinite Norm of representing matrix；
Step 3: by adjusting the parameter in the robust controller based on finite time Interference Estimation so that system meets control
Performance indicator.
2. the motor indirect self-adaptive robust output feedback control side according to claim 1 based on High-Order Sliding Mode differentiator
Method, which is characterized in that the realization of abovementioned steps three includes：By adjusting in the robust controller based on finite time Interference Estimation
The parameter k of u1,k2, α, v, λ0, λ1, λ2, σ1So that system meets Control performance standard.
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| CN106066604B (en) * | 2015-04-21 | 2019-01-18 | 南京理工大学 | Implementation method based on adaptive and expansion error symbol integral robust motor servo system positioner |
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| CN109227528B (en) * | 2018-10-16 | 2020-06-30 | 工极智能科技（苏州）有限公司 | Self-adaptive sliding mode control method based on information concentration estimation |
| CN109739084B (en) * | 2018-12-11 | 2022-08-23 | 曲阜师范大学 | System sliding mode anti-interference model obtaining method, system, controller and control method |
| CN110262246B (en) * | 2019-07-04 | 2022-05-20 | 南京理工大学 | Control method of vertical material conveying device based on sliding mode adaptive robust control |
| CN110879589A (en) * | 2019-12-12 | 2020-03-13 | 山东大学 | Mechanical arm fault-tolerant control method and system based on backstepping strategy and sliding mode strategy |
| CN113791240B (en) * | 2021-08-30 | 2023-07-25 | 暨南大学 | Acceleration estimation method, system, equipment and medium based on high-order synovial membrane tracking differentiator |
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| CN114326432B (en) * | 2021-12-10 | 2023-09-05 | 南京理工大学 | Special vehicle rapid leveling method based on cross coupling control |
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