CN104614994B - Robust self-adaptation control method for nonlinear system with input dead zone - Google Patents
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Abstract
The invention provides a robust self-adaptation control method for a nonlinear system with an input dead zone. The method includes the following steps: step1 building a mathematical model of a class of nonlinear system with the input dead zone; step2 designing a robust self-adaptation controller; step3 acquiring the performance of the robust self-adaptation controller. By means of the control method, the true control input is designed by introducing a dead zone inverse function, and the problem of poor controller design during full consideration of the dead zone model is solved. The designed robust self-adaptation controller can process parameter uncertainty and uncertainty nonlinearity of the system simultaneously, namely unknown parameters and other unknown parameters of the system in the dead zone can be estimated in the self-adaptation method, and certain robustness on the uncertainty nonlinearity is achieved.
Description
Technical field
The present invention relates to Control of Nonlinear Systems field, relates generally to a kind of nonlinear system robust containing input dead band certainly
Adaptive control method.
Background technology
Input dead band phenomenon is widely present in the utilizations such as hydraulic efficiency servo-valve, DC servo motor, mechanical connection, is being set up
If not taking in the presence in dead band during actual physics system model, it will the performance of severe exacerbation controller.Due in dead band
Parameter be often difficult to know that therefore for the design of controller is not that simply dead-time voltage function is inverted, this is to control
The design of device processed brings very big difficulty.Additionally, the nonlinear system of reality also to there are other parameters uncertain and not true
Qualitative non-linear, for the control problem of this nonlinear systems has very big challenge.
At present for the control method of the nonlinear system containing input dead band can substantially be divided into two categories below：The first kind is
The accurate model for considering dead-time voltage, designs adaptive controller so that system obtains good tracking with dead band inverse function
Performance.Wherein dead band inverse function can be divided into continuous inverse function and discontinuous inverse function.It is continuously contrafunctional to be obtained by
Serialization is carried out to discontinuous dead-zone model approximately to invert again acquisition, but so certainly exists certain approximate error,
The approximate error and the non-linear adaptive controller for causing to design in the past of systematic uncertainty can only be obtained and unanimously finally had
The tracking error on boundary.It is discontinuous due to its and discontinuous inverse function is acquisition of directly inverting to former discontinuous dead-zone model
Property, indirect self-adaptive robust control method designs indirect self-adaptive rule to estimate dead band by two discontinuity intervals to dead band
In unknown parameter to realize the accurate compensation to dead band, but indirect self-adaptive is based on there is no the uncertainties such as outer interference
Nonlinear premise design, the steady-state behaviour of asymptotic tracking can be obtained under the premise of here, and the system of reality all has one
Fixed is uncertain non-linear, transient performance and uniform ultimate bounded that the controller then can only ensure to determine in such case
Steady-state behaviour.The method of Equations of The Second Kind process input dead-time voltage is by the way that dead-time voltage model is approximately and control input
It is linear, approximate error be grouped into system it is uncertain non-linear in, redesign robust adaptive etc. controller ensures
The tracking performance of system.But, because without the accurate nonlinear model for considering dead band, designed controller can not be accurate
The impact in compensation dead band, system cannot obtain best tracking performance.
The content of the invention
For the problem for overcoming prior art to exist, the present invention is used for nonlinear system of the class containing input dead band
Continuous dead-zone model approaches actual discontinuous dead band, recycles continuous dead band inverse function to carry out Robust Adaptive Control
Device is designed.Robust adaptive controller designed by the present invention is restrained and using its company by dexterously designing nonlinear robust control
Continuous property that can be micro- carries out Backstepping design so that system exists concurrently with parameter uncertainty and uncertain nonlinear situation
Under still can obtain the steady-state behaviour of asymptotic tracking, enhance system for uncertain nonlinear robustness.
The technical solution for realizing the object of the invention is：A kind of nonlinear system robust adaptive control containing input dead band
Method processed, comprises the following steps：
Step 1, sets up the Mathematical Modeling of the nonlinear system containing input dead band；
Step 2, designs Robust adaptive controller；
Step 3, Robust adaptive controller performance and analysis.
Compared with prior art, its remarkable advantage is the present invention：The nonlinear model in input dead band is accurately considered, and is transported
Compensated with dead band inverse function, overcome impact of the input dead band to system tracking performance, and designed robust adaptive
Controller enhances the uncertain nonlinear robustness such as the external load disturbance of traditional Self Adaptive Control so that system is at the same time
Exist parameter uncertainty and it is uncertain it is nonlinear in the case of can still obtain the performance of asymptotic tracking.Simulation results show
Its validity.
The present invention is described further with reference to Figure of description.
Description of the drawings
Fig. 1 is nonlinear system Robust Adaptive Control principle schematic of the present invention containing input dead band；
Fig. 2 is the position command time history plot for expecting tracking；
Fig. 3 is the contrast tracking error curve of four kinds of controllers；
Fig. 4 is the contrast tracking error of four kinds of controllers in the simulation time of last 10s；
Fig. 5 is the time dependent curve of control input under RABC controller actions；
Fig. 6 is that the systematic parameter under RABC controller actions estimates time dependent curve；
Specific embodiment
With reference to the Robust Adaptive Control method of nonlinear system of Fig. 1 present invention containing input dead band, comprise the following steps：
Step 1, sets up the Mathematical Modeling of the nonlinear system containing input dead band；
The class strict feedback nonlinear systems containing input dead band that the present invention is considered have following state space form：
In formula (1)θ=[θ1,...θn]TFor the uncertain parameter in nonlinear system； For known continuously differentiable function；The uncertainty non-thread such as outer interference for system
Property；U is practical function in the control input of control object；V be real control input, the also as input in dead band；Y is system
Output.
It is as follows for the modeling of input dead band (Dead-Zone)：
Really dead-zone model is
M in formula (2)r,mlFor the slope of dead band both sides, br,blFor the value of the discontinuous flex point in dead band both sides, and mr,ml,br,bl
It is unknown parameter.Because the design for being directly controlled device using the dead-zone model in formula (2) can cause the controller of design
The discontinuous buffeting so as to cause control input, this is not allowed in practice.For this reason, it is necessary to dead-zone model is connected
Continuousization is approximate, and by approximate error be grouped into it is uncertain it is non-linear in.Dead band mould after serialization of the present invention is approximate
Type is：
U (v)=mr(v-br)φr(v)+ml(v-bl)φl(v) (3)
In formula (3)ε is the gain that can be aligned.
For ease of the design of Robust adaptive controller, linear parameterization is carried out to dead band as follows
In formula (4)
θd=[mr,mrbr,ml,mlbl]T, ω=[- σr(t)v,σr(t),-σl(t)v,σl(t)]T
Due to θdIt is unknown parameter, and ω cannot be obtained, therefore practical function is in control input u of control objectdCan be by such as
Lower thinking is designed：
In formula (5)
U and udBetween deviation be
D can be obtainedωThe upper bound be
By dωIt is grouped into uncertain non-linearIn, i.e.,Then original system equation (1)
Can be write as
The design object of system controller is：Given system reference signal yd(t)=x1dT (), designs the control of a bounded
System input u makes system export y=x1The reference signal of tracking system as much as possible.
For ease of controller design, there are hypothesis below and definition：
Assume 1：System reference command signal x1dT () is that n ranks are continuously differentiable, and x1dT the all-order derivative of () is bounded
, i.e.,
Assume 2：The uncertain non-linear of all passages of system equation is all bounded, but the value in the upper bound need not, it is known that
I.e.And meet
α in formula (9)1,...,αn-1Virtual controlling for each passage is restrained, Θ=[Θ1,...,Θn]TFor unknown constant.
Assume 3：The magnitude range of parameter uncertainty θ and the fluctuation range of upper bound Θ are, it is known that i.e.
Wherein θmin=[θ1min,...,θpmin]T,θmax=[θ1max,...,θpmax]T,Θmin=[Θ1min,...,Θnmin
]T, Θmax=[Θ1max,...,Θnmax]TIt is known.
Assume 4：There is fully smooth positive integrable function δiT () meets following property：
Step 2, designs Robust adaptive controller, and step is as follows：
(2.1) the continuous projection mapping function that parameter adaptive is adopted first was provided before device design is controlled：
DefinitionFor the estimation of unknown parameter θ,For the error of parameter Estimation, i.e.,OrderWherein π
It is the successive projection mapping function of bounded.DefinitionFor the parameter estimating error after projection.Based on the assumption that 3, definition is such as
Under projection mapping function：
π (v)=[π1(v1),...,πp(vp)]T (12)
V=[v in formula1,...,vp]T.Take arbitrarily small arithmetic number vector εθ=[εθ1,...,εθp]T, then there is abundant light
Sliding nondecreasing function πiMeet following property：
And with n-1 order derivatives, therefore
In formula (14)WithAll it is compacting for bounded, μ=[μ1,...,μp]TFor a vector,For arithmetic number.DefinitionThe function that definition is known as below by formula (10) and (14) is positive definite：
Γ=diag { Γ in formula (15)1,...,ΓpIt is the diagonal adaptive gain matrix of positive definite, and defined in formula (14)
Function there is following property：
Therefore, for parameter estimation vectorThere is similar projection function to define and property, i.e.,
Successive projection function：
π (κ)=[π1(κ1),...,πn(κn)]T (17)
κ=[κ in formula (17)1,...,κn]T, take arbitrarily small arithmetic number vector εΘ=[εΘ1,...,εΘn]T, then have following
Property
In formula (18)And ΩπjAll it is compacting for bounded, λ=[λ1,...,λn]TFor a vector,For arithmetic number.It is fixed
JusticeAndThe matrix being defined as below：
With following property：
γ=diag { γ in formula (20)1,...,γnIt is the diagonal adaptive gain matrix of positive definite.
(2.2) based on contragradience (Backstepping) method for designing, z is defined1=x1-x1dFor the tracking error of system, z2=
x2-α1For x2With virtual controlling α1Between deviation, then can be obtained by first equation in formula (8)
Design virtual controlling rule α1For
K in formula (22)1,ks1For positive feedback oscillator, α1aIt is the feedforward control based on model for improving model compensation
Rule, α1sFor Robust Control Law and wherein α1s1For linear robust feedback term, α1s2It is used to overcome uncertainty for non linear robust item
The non-linear impact to systematic function.
Formula (22) is substituted into into formula (21) can obtain：
Choose liapunov functionAnd its derivation is obtained：
With formula (9) and inequality property 0≤η tanh (η/a)≤| η |,Can obtain：
(2.3) consider second equation of formula (8), define z3=x3-α2For x3With virtual controlling α2Between deviation
Due to restraining α to virtual controlling1Derivation, there is following expansion
In formula (27)ForIn computable part,ForIn incalculable part, therefore virtual controlling is restrained into
Can only use during row designModel compensation is carried out, andThen carry out robust process as indeterminate.
Design virtual controlling rule α2It is as follows：
K in formula (28)2,ks2For positive feedback oscillator.
Formula (28) is substituted into can be obtained in formula (26)：
Have chosen liapunov functionAnd its derivation is obtained：
Can obtain with formula (9)：
(2.4) for i-th equation of formula (8), 1≤i≤n-1.Define error variance zi+1=xi+1-αiFor xi+1With void
Intend control law αiBetween deviation, then
Due to restraining α to virtual controllingi-1Derivation, there is following expansion
In formula (33)ForIn computable part,ForIn incalculable part
Design virtual controlling rule αiIt is as follows
K in formulai,ksiFor positive feedback oscillator.
Formula (34) is substituted into can be obtained in formula (32)：
Choose liapunov functionAnd its derivation can be obtained：
(2.5) for n-th equation of formula (8), have
Design practical function is in control input u of control objectdFor
Formula (38) is substituted into into formula (37) can obtain：
With dead band inverse function, real control input v may be designed as
Step 3, the performance of Robust adaptive controller is specific as follows：
Controller performance：
For system unknown parameter includes θ and θd, and each unknown upper bound Θ in formula (9), using following self adaptation
Rule
Γ in formula (41), Υ, Μ are adaptive gain, and are all positive definite diagonal matrix.c1,...cnFor positive adjustable increasing
Benefit.
Controller (38), the feedback oscillator k of (40)1,k2,...,knObtain the sufficiently large and suitable c of selection1,...cnWith
ks1,...ksnSo that the matrix Λ being defined as below is positive definite matrix：
The Robust adaptive controller for then designing can make the equal bounded of all signals in closed-loop system, and system obtains asymptotic defeated
Go out tracking performance, i.e., as t → ∞, z1→0。
Analysis to above-mentioned performance：
Choose liapunov function
Can obtain to above formula derivation and with formula (39), (9), (36) and (41)：
Z=[z in formula (44)1,...,zn]T。
(44) both sides integration can be obtained
Because the matrix of formula (42) definition is positive definite matrix, then W ＞ 0, by (45) V (t) ∈ L are understood∞, therefore By assuming that 1 understands that all states of system are all bounded, further according to assuming that 4 understand that v (t) is bounded, therefore closed loop
The all signals of system are all bounded.Can obtain further according to formula (23), (29), (35) and (39)Additionally, by (45)
Can obtain
Therefore W ∈ L2.Understood as t → ∞ by Barbalat ' s lemma, W → 0, you can push away to obtain conclusion：As t → ∞, z1→
0.Nonlinear system Robust Adaptive Control principle schematic containing input dead band is as shown in Figure 1.
Embodiment
For the designed controller performance of examination, following simulation example is given.
Consider the following first order nonlinear system containing input dead band
U is the output in dead band in formula (47), and d (t) is uncertain non-linear for outer interference etc..To simplify simulation process, examine
Consider the known i.e. m of the Slope Parameters in symmetrical dead band and dead bandr=ml=1, b=1 it is also known that.System other uncertain parameters it is true
Value is respectively br=-bl=1, a=1 and interference d (t)=0.6sint.Define θd=br, θ=a and Θ are the interference upper bound.
Expect that the instruction for tracking is：yd(t)=arctan (sint) [1-exp (- 0.01t3)], its schematic diagram such as Fig. 2 institutes
Show.To prove the validity of the controller designed by the present invention, following four controller is now contrasted：
1) Robust adaptive controller (RABC) designed by the present invention：Choose controller parameter：k1=100, c1=1,
ks1=1, ε=0.01, δ (t)=5000/ (t2+ 1), Parameter uncertainties scope is taken as：θmax=5, Θmax=30, θmin=0,
Θmin=0.The initial value of parameter Estimation is taken as：Parameter adaptive gain：Γ=
500, γ=500, Μ=100.
2) Robust adaptive controller (RAC) with following form：
Parameter update law：
Wherein
Choose controller parameter：h1=2, h=2, σ (t)=5000/ (t2+ 1), α=6, the initial value that parameter beta is estimated
3) adaptive controller (ABC)：The Self Adaptive Control is the part that RABC controllers remove non linear robust item, is
Ensure the fairness of contrast, its controller parameter is identical with the relevant parameter in RABC.
4) Adaptive variable control device (AVSC)：Smooth dead band inversion model pair is also used in the design of the controller
Dead band compensates, and practical function may be designed as in the control input of control object：
Parameter update law：
Wherein D is the upper bound of interference,For its estimate.And
Choose controller parameter：l1=100, Γa=200, Γb=40, Γd=30, ρ1=5 × 10-4。
Tracking error curve under four kinds of controller actions is as shown in Figure 3 and Figure 4.It can be seen that RABC controls
Device all shows optimal tracking performance in transient state or in stable state.Due to without model compensation, RAC controllers it is initial
Tracking error is quite big, and is being designed without considering that the impact in dead band causes its steady-state error also very big.The transient tracking of ABC
Error and RABC almost do not have difference, but due to lacking non linear robust item to weaken the impact of interference, its steady-state error ratio
RABC is big.The steady track error of Adaptive variable control device can be by ρ1Selection be adjusted, but ρ1Obtain too small meeting
Tracking error is caused to shake, the existing fine jitter of its tracking error that can read a book in figure.The control input of RABC controllers is at any time
Between change curve it is as shown in Figure 5.Systematic parameter estimation procedure is as shown in Figure 6.
Claims (3)
1. it is a kind of containing the nonlinear system Robust Adaptive Control method for being input into dead band, it is characterised in that to comprise the following steps：
Step 1, sets up the Mathematical Modeling of the nonlinear systems containing input dead band；
Step 2, designs Robust adaptive controller；
Step 3, the performance of Robust adaptive controller；
The Mathematical Modeling of the nonlinear systems containing input dead band is set up described in step 1, it is specific as follows：
Step 1.1, a class strict feedback nonlinear systems of the design containing input dead band have following state space form：
In formula (1)θ=[θ1,...θp]TFor the uncertain parameter in nonlinear system, p is the dimension of θ；For known continuously differentiable function；For the outer interference uncertainty nonlinear function of system；U is actual work
For the control input of control object；V is real control input；Y is system output；
Step 1.2, sets up dead-zone model
M in formula (2)r,mlFor the slope of dead band both sides, br,blFor the value of the discontinuous flex point in dead band both sides, DZ () is dead band letter
Number；
Step 1.3, it is approximate to carry out serialization to dead-zone model, and by approximate error be grouped into it is uncertain it is non-linear in, connected
Dead-zone model after continuousization is approximate：
U (v)=mr(v-br)φr(v)+ml(v-bl)φl(v) (3)
In formula (3)ε is the gain that can be aligned；
Step 1.4, to dead band linear parameterization is carried out
In formula (4)
Step 1.5, calculates practical function in design load u of control input u of control objectd
In formula (5)
Respectively mr,mrbr,ml,mlblEstimate；
Step 1.6, u and udBetween deviation be
In formula (6)For θdEvaluated error
D can be obtainedωThe upper bound be
Step 1.7, by dωIt is grouped into uncertain non-linearIn, i.e.,Then original system equation
(1) can be write as
Step 1.8, for controller design, it is assumed that as follows：
Assume 1, system reference command signal x1dT () is that n ranks are continuously differentiable, and x1dT the all-order derivative of () is bounded, i.e.,
Assume 2, the uncertain non-linear of all passages of system equation is all bounded, but the value in the upper bound need not be, it is known that i.e.And meet
α in formula (9)1,...,αn-1Virtual controlling for each passage is restrained, Θ=[Θ1,...,Θn]TFor unknown constant；
Hypothesis 3, the magnitude range of uncertain parameter θ and the fluctuation range of upper bound Θ are, it is known that i.e.
Wherein θmin=[θ1min,...,θpmin]T,θmax=[θ1max,...,θpmax]T,Θmin=[Θ1min,...,Θnmin]T,
Θmax=[Θ1max,...,Θn max]TIt is known；
Assume 4, there is fully smooth positive integrable function δiT () meets following property：
For δiThe jth order derivative of (t)；For positive constant, δ is representediThe upper bound of the jth order derivative of (t)；Also for just
Constant, represent δiThe upper bound of the time integral of (t).
2. according to claim 1 containing the nonlinear system Robust Adaptive Control method for being input into dead band, it is characterised in that
Robust adaptive controller is designed described in step 2, step is as follows：
Step 2.1, before device design is controlled the continuous projection mapping function that parameter adaptive is adopted first was given：
DefinitionFor the estimation of unknown parameter θ,For the error of parameter Estimation, i.e.,OrderWherein π is that have
The successive projection mapping function on boundary, definitionFor the parameter estimating error after projection, based on condition (3), it is defined as follows
Projection mapping function：
π (v)=[π1(v1),...,πp(vp)]T (12)
V=[v in formula1,...,vp]TFor the independent variable of mapping function π；
Step 2.2, takes arbitrarily small arithmetic number vector εθ=[εθ1,...,εθp]T, then there is fully smooth nondecreasing function πiIt is full
Be enough to lower property：
And with n-1 order derivatives, therefore
In formula (14)For functionJth order derivative,WithAll it is compacting for bounded, μ=[μ1,...,μp]T
For a vector, cπijFor arithmetic number；
Step 2.3, definitionThe function that definition is known as below by formula (10) and (14) is positive definite：
Γ=diag { Γ in formula (15)1,...,ΓpIt is the diagonal adaptive gain matrix of positive definite, and the letter defined in formula (14)
Number has following property：
Therefore, for parameter estimation vectorThere is similar projection function to define and property, i.e.,
Successive projection function：
π (κ)=[π1(κ1),...,πn(κn)]T (17)
κ=[κ in formula (17)1,...,κn]TThe independent variable of the mapping function π to redefine, takes arbitrarily small arithmetic number vector
εΘ=[εΘ1,...,εΘn]T, then have following property
In formula (18)For functionJth order derivative,And ΩπjAll it is compacting for bounded, λ=[λ1,...,λn
]TFor a vector, λiFor i-th element of λ, sπijFor arithmetic number, definition AndThe matrix being defined as follows：
With following property：
In formula (20)ForNth elements, γ=diag { γ1,...,γnIt is the diagonal adaptive gain of positive definite
Matrix；
Step 2.4, based on Backstepping design method, defines z1=x1-x1dFor the tracking error of system, z2=x2-α1For x2With it is virtual
Control law α1Between deviation, then can be obtained by first equation in formula (8)
Design virtual controlling rule α1For
K in formula (22)1,ks1For positive feedback oscillator, α1aFor α1In for improving the feedforward control based on model of model compensation
Rule, α1sFor Robust Control Law and wherein α1s1For linear robust feedback term, α1s2It is used to overcome uncertainty for non linear robust item
The non-linear impact to systematic function, δ1T () is to meet the function that 4 are assumed in claim 1, tanh (z1/δ1(t)) it is from change
Measure as z1/δ1The hyperbolic tangent function of (t)；
Step 2.5, it is considered to second equation of formula (8), defines z3=x3-α2For x3α is restrained with virtual controlling2Between deviation
In formulaDue to restraining α to virtual controlling1Derivation, there is following expansion
In formula (24)ForIn computable part,ForIn incalculable part, therefore to virtual controlling rule set
Timing can only be usedModel compensation is carried out, andThen robust process is carried out as indeterminate,
Design virtual controlling rule α2It is as follows：
K in formula (25)2,ks2For positive feedback oscillator, α2aFor α2In model compensation item, α2sFor Robust Control Law, α therein2s1
For linear robust Feedback Control Laws, α2s2For nonlinear robust control rule, δ2T () is to meet the function that 4 are assumed in claim 1,
tanh(z2/δ2(t)) it is that independent variable is z2/δ2The hyperbolic tangent function of (t)；
Step 2.6, for i-th equation of formula (8), 1≤i≤n-1；Define error variance zi+1=xi+1-αiFor xi+1With it is virtual
Control law αiBetween deviation, then
Due to restraining α to virtual controllingi-1Derivation, there is following expansion
In formula (27)ForIn computable part,ForIn incalculable part
Design virtual controlling rule αiIt is as follows
K in formulai,ksiFor positive feedback oscillator, αiaFor αiIn model compensation item, αisFor Robust Control Law, α thereinis1For line
Property robust feedback control rule, αis2For nonlinear robust control rule, δiT () is to meet the function that 4 are assumed in claim 1, tanh
(zi/δi(t)) it is that independent variable is zi/δiThe hyperbolic tangent function of (t)；
Step 2.7, for n-th equation of formula (8), has
Design practical function is in design load u of control input u of control objectdFor
K in formulan,ksnFor positive feedback oscillator, udaFor udIn model compensation item, udsFor Robust Control Law, u thereinds1For
Linear robust Feedback Control Laws, uds2For nonlinear robust control rule, δnT () is to meet the function that 4 are assumed in claim 1,
tanh(zn/δn(t)) it is that independent variable is zn/δnThe hyperbolic tangent function of (t)；
Adaptive law is designed as：
Γ in formula (31), Υ, Μ are adaptive gain, and are all positive definite diagonal matrix；c1,...cnFor positive adjustable gain；
With dead band inverse function, real control input v may be designed as
In formula (32)Respectively mr,mrbr,ml,mlblEstimate, and
3. according to claim 2 containing the nonlinear system Robust Adaptive Control method for being input into dead band, it is characterised in that
The performance of Robust adaptive controller described in step 3, it is specific as follows：
For system unknown parameter includes θ and θd, and each unknown upper bound Θ in formula (9), using adaptive law (31), control
Device (30) processed, the feedback oscillator k of (32)1,k2,...,knObtain the sufficiently large and suitable c of selection1,...cnAnd ks1,...ksn
So that the matrix Λ being defined as below is positive definite matrix：
The Robust adaptive controller for then designing can make the equal bounded of all signals in closed-loop system, and system obtain asymptotic output with
Track performance, i.e., as t → ∞, z1→0。
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| CN103197562B (en) * | 2013-04-11 | 2015-04-22 | 浙江工业大学 | Rotary-table servo system neural network control method |
| CN104216284A (en) * | 2014-08-14 | 2014-12-17 | 浙江工业大学 | Limit time cooperative control method of mechanical arm servo system |
| CN104333280B (en) * | 2014-09-17 | 2017-05-17 | 南京理工大学 | Robustness adaptive control (RAC) method of direct driving motor system |
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