CN107894714A - The adaptive sliding-mode observer method of nonlinear system - Google Patents
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Abstract
The adaptive sliding-mode observer method of nonlinear system, belongs to artificial intelligence and control field, and for solving the problems, such as closed-loop control system Asymptotic Stability, technical essential is：Perfect condition feedback controller is approached to construct new feedback controller using LS SVM structures to the nonlinear system, it is compensated by the approximate error for imposing sliding formwork control to be returned to LS SVM and/or uncertain external disturbance, and weighting parameter vector is determined with adaptive rate, effect is：The nonlinear function approximation capability design of feedback Linearizing controller for making full use of LS SVM to return, the influence that the approximate error of sliding formwork control compensation LS SVM recurrence and uncertain external disturbance export to system is introduced, carries out the adjustment of LS SVM weighting parameters.
Description
Technical field
The invention belongs to artificial intelligence and control field, it is related to a kind of adaptive sliding-mode observer method of nonlinear system.
Background technology
The Sliding mode variable structure control of Nonlinear Uncertain Systems is always to control the focus of boundary's concern, and many scholars lead herein
Domain achieves achievement in research.The rough mathematical modeling of system known to the sliding formwork control of nonlinear system needs, therefore increase
Dependence of the sliding formwork control to system model.With the development of artificial intelligence theory, fuzzy logic and neutral net are introduced into cunning
Mould control design case, efficiently reduce dependence of the sliding formwork control to system model.Document [3] have studied to be observed based on high-gain
The nonlinear system Adaptive Fuzzy Sliding Mode Control of device, it is adaptive that document [4] have studied the nonlinear system based on neutral net
Sliding formwork control, they mainly utilize the ability of fuzzy logic or neutral net to any None-linear approximation.But fuzzy logic
With the problems such as algorithm is complicated, pace of learning is slow in Application of Neural Network be present, and least square method supporting vector machine (LS-SVM) solves
Determine above mentioned problem.LS-SVM maintains standard SVM powerful extensive and global optimum's ability, drastically increases training effect
Rate, while the Control of Nonlinear Systems research based on LS-SVM achieves abundant achievement].But LS-SVM and sliding formwork are become into knot
The Nonlinear Uncertain Systems analysis and the method for design that structure control is combined are then relatively fewer.
The content of the invention
In order to solve the problems, such as closed-loop control system Asymptotic Stability, the present invention proposes following scheme：
The adaptive sliding-mode observer method of nonlinear system, ideal is approached using LS-SVM structures to the nonlinear system
State feedback controller approaches mistake by imposing sliding formwork control to construct new feedback controller, to it with what is returned to LS-SVM
Poor and/or uncertain external disturbance compensation, and weighting parameter vector is determined with adaptive rate.
Beneficial effect：The present invention includes the nonlinear system of uncertain and unknown bounded external disturbance for one kind, carries
A kind of self-adaptive controlled sliding method of moulding is gone out.The nonlinear function approximation capability design that this method makes full use of LS-SVM to return is anti-
Linearization controller, the approximate error and uncertain external disturbance for introducing sliding formwork control compensation LS-SVM recurrence export to system
Influence, carry out LS-SVM weighting parameters adjustment, design is verified finally by a simulation example, explanation
The present invention can solve the problems, such as closed-loop control system Asymptotic Stability.
Brief description of the drawings
Fig. 1 is state and desired output schematic diagram；
Fig. 2 is state x2And desired output schematic diagram；
Fig. 3 is control input schematic diagram；
Fig. 4 is state x1And desired output schematic diagram；
Fig. 5 is state x2And desired output schematic diagram；
Fig. 6 is control input schematic diagram；
Fig. 7 is tracking error schematic diagram；
Fig. 8 is LS-SVM structural formulas.
Embodiment
Embodiment 1：The present embodiment includes the nonlinear system of uncertain and unknown bounded external disturbance for one kind, carries
A kind of self-adaptive controlled sliding method of moulding or system based on liapunov function are gone out, this method performs and makes full use of LS-SVM
The nonlinear function approximation capability design of feedback Linearizing controller of recurrence, introduce approaching for sliding formwork control compensation LS-SVM recurrence
The influence that error and uncertain external disturbance export to system, the tune of LS-SVM weighting parameters is carried out using Lyapunov functions
It is whole, design is verified finally by a simulation example.
1 problem describes
Consider Nonlinear Uncertain Systems
WhereinIt is unknown nonlinear function, b is unknown control gain, and d is that bounded is done
Disturb, u ∈ R and y ∈ R are the input and output of system respectively, and n is the exponent number of system mode.If
It is the state vector of system, acquisition can be measured.
Control targe be namely based on LS-SVM return realize STATE FEEDBACK CONTROL, so as to ensure closed-loop system uniform bound,
Tracking error is small.In order to realize target, hypothesis below is provided：
Assuming that 1.1 reference signal ymAndContinuous bounded, subscript m represent reference signal.DefinitionYm∈Ωm∈Rn(ΩmCompacted to be known), then output error is
That is e=ym- x,AndDefine K=(k1,k2,…,kn)TFor
Hurwitz vectors.
Assuming that 1.2 control gain b meet b >=bL＞ 0, bLFor b lower bound.Disturb d boundeds, it is assumed that its upper bound is D, i.e., | d |
≤ D, give D ＞ 0.
If function f (x) is known and interference d=0, state feedback controller are
It is calculated by formula (2) and formula (1)
e(n)+kne(n-1)+…+k1E=0 (3)
Formula (3) shows, by suitably selecting ki(i=1,2 ..., n), it is ensured that sn+knsn-1+…+k1=0 it is all
Root is all in complex plane Left half-plane, i.e. limt→∞e1(t)=0.
The 2 adaptive law designs returned based on LS-SVM
Least square line sexual system is introduced SVM by LS-SVM, is asked instead of traditional supporting vector using QUADRATIC PROGRAMMING METHOD FOR
Solution classification and Function Estimation problem, the derivation of algorithm is referring to document [5].
For u in approximant (2)*LS-SVM structures it is as shown in Figure 8.
Wherein：X=[x1 x2 … xn-1 xn]TFor input vector, the nodes of hidden layer are N+1, and N is input vector
Sample number.Wherein the 1st node definition is the deviation of hidden layer, wj(1 ..., N, N+1) it is power of the hidden layer to output layer
Value, Xj(j=1 ..., N, N+1) is supporting vector, K (Xj, x) (j=1 ..., N, N+1) it is kernel function.
The input/output relation that LS-SVM is returned is u (x, θ)=θTβ (4)
In formula：θ=[w1 w2 … wN+1]T, β=[1, K (X1,x),…,K(XN,x)]T。
Return to obtain u using LS-SVM*Be approximately For weighting parameter estimate vector.
If preferable weighting parameter vector is
In formulaAnd Ωx=x | | | x | |≤D2Be respectively weighting parameter and state vector bounded aggregate
Close, D1And D2It is the parameter designed by user.Then have
Wherein ε (x) is LS-SVM approximate error, to arbitrary constant Δ ε ＞ 0, is met | ε (x) |≤Δ ε.
OrderIt can obtain
Defining sliding-mode surface is
S=KTe (7)
Wherein kn=1, then
In formula,WithExpression is differentiated to variable s and vectorial e, e(i)(i=1 ..., n) represents e the i-th order derivative, and u is
Control input in system (1).
According to (6), based on sliding formwork control technology, the control input u of design system is
Wherein
Take
D is the d upper bound (see hypothesis 1.2) in formula, and η ＞ 0 are design parameter.
Weighting value parameter vector adaptive law be
In formula, Γθ＞ 0 is design parameter.
The Nonlinear Uncertain Systems that theorem describes for formula (1), it is approximant (2) using Fig. 1 LS-SVM regressive structures
In u*, control input is taken as formula (9), and weighting parameter vector adaptive law is (11), then all signal boundeds in closed-loop system.
Prove：Select following Lyapunov functions
Making V differentiate the time has
It can be obtained by formula (11)
η ＞ Δ ε ＞ 0 are taken, can be obtained using formula (10)
Understand that closed-loop system is asymptotically stable.
3 simulation studies
Consider Nonlinear Uncertain Systems
In formula,B=1.5+0.5sin (5t), d=12cos (t)
The adaptive sliding-mode observer returned based on LS-SVM is realized first.The input for taking LS-SVM to return is x=[x1 x2
]T, export as u*.Choose KT=(k1,k2)=(2,1), controller parameter Γθ, η, D and bLRespectively 2,0.5,12 and 1.Control
Amount u takes white noise signal (average 0, variance 0.01), obtains state x=[x1 x2]TMeasurement data.Selected from u and x data
Select 100 pairs and be used as training sample, meanwhile, 40 pairs of data therein are taken as test sample.With the mean square error of system output errors
Difference is evaluation index, and the hyper parameter of LS-SVM recurrence is tried to achieve using cross validation optimization.The hyper parameter obtained using optimization, again
Learnt and trained, obtain the initial parameter values of the nonlinear feedback controller based on LS-SVM regression fits.Selection system is joined
It is y to examine signalm(t)=sin (t), original state x=[0 1]T, applying equation (9) is to system progress in-circuit emulation experiment.System
State x1(t)、x2(t) and controlled quentity controlled variable u simulation curve as shown in Figure 1, Figure 2 and Figure 3.From simulation result it can be seen that the design
Method achieves more satisfactory control effect.
Then the adaptive sliding-mode observer based on neutral net is realized.Nerve network controller structure and parameter chooses reference
Document [11].The simulation result of adaptive sliding-mode observer based on neutral net such as Fig. 4, Fig. 5 and Fig. 7.Wherein, Fig. 7 is two kinds
The tracking error curve of control method.Contrast tracking error curve understand, the AME based on LS-SVM methods for-
0.0093, the AME based on neural net method is -0.0207, shows the present embodiment control method control accuracy more
It is high.
4 conclusions
The present embodiment have studied based on the adaptive of the LS-SVM a kind of single-input single-output Nonlinear Uncertain Systems returned
Answer sliding formwork control problem.In the design of control system, returned using the feedback linearization technology and LS-SVM of nonlinear system
Any Nonlinear Function approximation capability construction feedback controller, the robust of control system is improved by sliding formwork control technology
Property, and demonstrate proposed control program and can ensure closed-loop control system Asymptotic Stability.Simulation results show this method
Validity.
Bibliography (References)
[1]Cong S,Liang Y Y.Adaptive Sliding Mode Tracking Control of
Nonlinear System with Time-varying Uncertainty[J].Control Engineering
ofChina,2009, 16(4):383-387.
[2]Koshkouei A J,Burnham K J.Adaptive Backstepping Sliding Mode
Control for Feedforward Uncertain Systems[J].International Journal of Systems
Sciece,2011,42(12):1935-1946.
[3] Liu Yunfeng, Peng Yunhui, Yang little Gang, nonlinear systems of the flat of Miao Dong, Yuan Run based on High-gain observer are adaptive
Fuzzy sliding mode tracking control [J] system engineerings and electronic technology, 2009,31 (7):1723-1727.
[4]Park B S,Yoo S J,Park J B,et al.Adaptive Neural Sliding Mode
Control of Nonholonomic Wheeled Mobile Robots with Model Uncertainty[J].IEEE
Transactions on Control Systems Technology,2009,17(1):207-214.
[5]Suykens J A K.Nonlinear Modeling and Support Vector Machines[A],
Proc of the 18th IEEE Conf on Instrumentation and Measurement Technolog[C].
Budapest,2001:287-294.
[6]Yuan X F,Wang Y N,Wu L H.Adaptive Inverse Control of Excitation
System with Actuator Uncertainty[J].WSEAS Transactions on Systems and
Control,2007,8(2):419-427.
[7] long range predictive identifications of Guo Zhenkai, Song Zhaoqing, the Mao Jianqin based on least square method supporting vector machine
[J] is controlled and decision-making, 2009,24 (4):520-525.
[8] Mu Chaoxu, Zhang Ruimin, grandson grow silver-colored nonlinear system least square method supporting vector machines of the based on particle group optimizing
Forecast Control Algorithm [J] control theories and application, 2010,27 (2):164-168.
[9] research [D] of nonlinear system self-adaptation control methods of the Xie Chunli based on least square method supporting vector machine
Dalian：Dalian University of Technology, 2011.
[10] nonlinear systems of Xie Chunli, Shao Cheng, Zhao Dan pellet mono- are direct adaptive based on least square method supporting vector machine
[J] controls and decision-making, 2010,25 (8) should be controlled:1261-1264.
[11]Yang Y S,Wang X F.Adaptive HBB∞BBtracking control for a class
ofuncertain nonlinear systems using radial basis function neural networks[J].
Neurocomputing,2007,70(4-6):932-941.
Embodiment 2, the system performed as method in embodiment 1, the present embodiment include following scheme：
A kind of adaptive sliding-mode observer system of nonlinear system, is stored with a plurality of instruction, and the instruction is suitable to processor
Load and perform：
Perfect condition feedback controller is approached to construct new feedback control using LS-SVM structures to the nonlinear system
Device processed；
It is compensated by the approximate error for imposing sliding formwork control to be returned to LS-SVM and/or uncertain external disturbance；
Weighting parameter vector is determined with adaptive rate.
The nonlinear system approaches perfect condition feedback controller to construct based on following manner using LS-SVM structures
New feedback controller
The nonlinear system
Wherein：It is unknown nonlinear function, b is unknown control gain, and d is bounded
Interference, u ∈ R and y ∈ R are the input and output of system respectively, and n is the exponent number of system mode, if
It is the state vector of system；
Assuming that reference signal ymAndContinuous bounded, subscript m represent reference signal, definitionYm∈Ωm∈Rn, ΩmCompacted to be known, output error isAndDefine K=
(k1,k2,…,kn)TFor Hurwitz vectors；
Assuming that control gain b meets b >=bL＞ 0, bLFor b lower bound.Disturb d boundeds, it is assumed that its upper bound is D, i.e., | d |≤
D, give D ＞ 0；
If function f (x) is known and interference d=0, state feedback controller are
It is calculated by formula (2) and formula (1)
e(n)+kne(n-1)+…+k1E=0 (3)
Formula (3) shows, by suitably selecting ki(i=1,2 ..., n), can guarantee that sn+knsn-1+…+k1All of=0
All in complex plane Left half-plane, make limt→∞e1(t)=0；
The LS-SVM structures are as shown in Figure 8.
Wherein：X=[x1 x2 … xn-1 xn]TFor input vector, the nodes of hidden layer are N+1, and N is input vector
Sample number.Wherein the 1st node definition is the deviation of hidden layer, wj(1 ..., N, N+1) it is power of the hidden layer to output layer
Value, Xj(j=1 ..., N, N+1) is supporting vector, K (Xj, x) (j=1 ..., N, N+1) it is kernel function；
The input/output relation of LS-SVM structural regressions is u (x, θ)=θTβ (4)
In formula：θ=[w1 w2 … wN+1]T, β=[1, K (X1,x),…,K(XN,x)]T；
U is obtained using LS-SVM structural regressions*Be approximately For weighting parameter estimate vector.
If preferable weighting parameter vector is
In formulaAnd Ωx=x | | | x | |≤D2Be respectively weighting parameter and state vector bounded aggregate
Close, D1And D2It is the parameter designed by user, then has
Wherein ε (x) is the approximate error of LS-SVM structures, to arbitrary constant Δ ε ＞ 0, is met | ε (x) |≤Δ ε.
To nonlinear system by the approximate error for imposing sliding formwork control to be returned to LS-SVM and/or uncertain external disturbance
Compensation is realized by following manner：Define sliding-mode surface s
S=KTe (7)
The control input u of nonlinear system is
Wherein
Take
D is the d upper bound in formula, and η ＞ 0 are design parameter.
Determine that weighting parameter vector is realized by following manner with adaptive rate：Weighting value parameter vector adaptive law be
In formula, Γθ＞ 0 is design parameter.
It is described above, the only preferable embodiment of the invention, but the protection domain of the invention is not
This is confined to, any one skilled in the art is in the technical scope that the invention discloses, according to the present invention
The technical scheme of creation and its inventive concept are subject to equivalent substitution or change, should all cover the invention protection domain it
It is interior.
Claims (8)
1. a kind of adaptive sliding-mode observer method of nonlinear system, it is characterised in that LS- is used to the nonlinear system
SVM structures approach perfect condition feedback controller to construct new feedback controller, to it by imposing sliding formwork control with to LS-
The approximate error and/or the compensation of uncertain external disturbance that SVM is returned, and weighting parameter vector is determined with adaptive rate.
2. the adaptive sliding-mode observer method of nonlinear system as claimed in claim 1, it is characterised in that
The nonlinear system
Wherein：It is unknown nonlinear function, b is unknown control gain, and d is BOUNDED DISTURBANCES,
U ∈ R and y ∈ R are the input and output of system respectively, and n is the exponent number of system mode, ifIt is to be
The state vector of system；
Assuming that reference signal ymAndContinuous bounded, subscript m represent reference signal, definitionYm
∈Ωm∈Rn, ΩmCompacted to be known, output error is
AndDefine K=(k1,k2,…,kn)TFor Hurwitz vectors；
Assuming that control gain b meets b >=bL＞ 0, bLFor b lower bound.Disturb d boundeds, it is assumed that its upper bound is D, i.e., | d |≤D, give
Determine D ＞ 0；
If function f (x) is known and interference d=0, state feedback controller are
3. the adaptive sliding-mode observer method of nonlinear system as claimed in claim 2, it is characterised in that
It is calculated by formula (2) and formula (1)
e(n)+kne(n-1)+…+k1E=0 (3)
Formula (3) shows, by suitably selecting ki(i=1,2 ..., n), can guarantee that sn+knsn-1+…+k1All of=0 are multiple
Plane Left half-plane, makes limt→∞e1(t)=0.
4. the adaptive sliding-mode observer method of nonlinear system as claimed in claim 2, it is characterised in that
Wherein：X=[x1 x2 … xn-1 xn]TFor input vector, the nodes of hidden layer are N+1, and N is the sample of input vector
Number.Wherein the 1st node definition is the deviation of hidden layer, wj(1 ..., N, N+1) is hidden layer to the weights of output layer, Xj(j=
1 ..., N, N+1) it is supporting vector, K (Xj, x) (j=1 ..., N, N+1) it is kernel function.
5. the adaptive sliding-mode observer method of nonlinear system as claimed in claim 4, it is characterised in that
The input/output relation of LS-SVM structural regressions is u (x, θ)=θTβ (4)
In formula：θ=[w1 w2 … wN+1]T, β=[1, K (X1,x),…,K(XN,x)]T；
U is obtained using LS-SVM structural regressions*Be approximately For weighting parameter estimate vector.
6. the adaptive sliding-mode observer method of nonlinear system as claimed in claim 5, it is characterised in that
If preferable weighting parameter vector is
In formulaAnd Ωx=x | | | x | |≤D2It is respectively weighting parameter and the bounded set of state vector, D1
And D2It is the parameter designed by user, then has
Wherein ε (x) is the approximate error of LS-SVM structures, to arbitrary constant Δ ε ＞ 0, is met | ε (x) |≤Δ ε.
7. the adaptive sliding-mode observer method of nonlinear system as claimed in claim 5, it is characterised in that
Define sliding-mode surface s
S=KTe (7)
The control input u of nonlinear system is
Wherein
Take
D is the d upper bound in formula, and η ＞ 0 are design parameter.
8. the adaptive sliding-mode observer method of nonlinear system as claimed in claim 7, it is characterised in that weighting value parameter to
The adaptive law of amount is
In formula, Γθ＞ 0 is design parameter.
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