CN106980262B - Adaptive aircraft robust control method based on kernel recursive least square algorithm - Google Patents

Adaptive aircraft robust control method based on kernel recursive least square algorithm Download PDF

Info

Publication number
CN106980262B
CN106980262B CN201710171108.0A CN201710171108A CN106980262B CN 106980262 B CN106980262 B CN 106980262B CN 201710171108 A CN201710171108 A CN 201710171108A CN 106980262 B CN106980262 B CN 106980262B
Authority
CN
China
Prior art keywords
kernel
controller
matrix
output
recursive
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Expired - Fee Related
Application number
CN201710171108.0A
Other languages
Chinese (zh)
Other versions
CN106980262A (en
Inventor
赵广社
杨朝旭
荣海军
鲍容憬
高雷涛
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Xian Jiaotong University
Original Assignee
Xian Jiaotong University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Xian Jiaotong University filed Critical Xian Jiaotong University
Priority to CN201710171108.0A priority Critical patent/CN106980262B/en
Publication of CN106980262A publication Critical patent/CN106980262A/en
Application granted granted Critical
Publication of CN106980262B publication Critical patent/CN106980262B/en
Expired - Fee Related legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • G05B13/045—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance using a perturbation signal
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Software Systems (AREA)
  • Artificial Intelligence (AREA)
  • Mathematical Physics (AREA)
  • Computational Mathematics (AREA)
  • Mathematical Optimization (AREA)
  • Evolutionary Computation (AREA)
  • Automation & Control Theory (AREA)
  • Health & Medical Sciences (AREA)
  • Theoretical Computer Science (AREA)
  • Mathematical Analysis (AREA)
  • Medical Informatics (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Pure & Applied Mathematics (AREA)
  • Data Mining & Analysis (AREA)
  • Algebra (AREA)
  • Databases & Information Systems (AREA)
  • General Engineering & Computer Science (AREA)
  • Computing Systems (AREA)
  • Feedback Control In General (AREA)

Abstract

The adaptive aircraft robust control method based on the kernel recursive least square algorithm comprises the steps that a feedback regulation system is formed by an off-line identifier and an on-line controller to carry out adaptive regulation; an off-line identifier: on the basis of the input and output data of the actual model of the aircraft, a model equivalent to the measured system is identified off line by utilizing a kernel recursive least square algorithm; an online controller: on the basis of accurately identifying an actual model of the aircraft, the output quantity of the controller is adjusted on line by utilizing an improved kernel recursive least square algorithm according to system input and output data, and adaptive robust control is realized; the improved kernel recursive least square algorithm is based on characteristic value extraction, orthogonal decomposition of a kernel matrix is carried out according to principal component analysis and singular value decomposition respectively, compression and dimension reduction of a sample kernel function are achieved, some irrelevant redundant data information in the kernel function is removed, and the improved kernel recursive least square algorithm has good control precision and low operation burden.

Description

Adaptive aircraft robust control method based on kernel recursive least square algorithm
Technical Field
The invention relates to the field of aircraft control, in particular to a self-adaptive aircraft robust control method based on a kernel recursive least square algorithm.
Background
The aerodynamic layout of a modern aircraft often cannot meet the requirement of stability performance, and along with complex air condition interference and system parameter change, an aircraft self-adaptive robust control system needs to be designed in order to obtain ideal flight quality. To address this problem, Model Reference Adaptive Control (MRAC) schemes design aircraft Adaptive robust Control systems to characterize the desired flight characteristics (implying good stability and maneuverability) with Reference models and to make the actual output asymptotically track the Reference output. In recent years, autopilots using this theoretical design have been successfully used on aircraft.
In MRAC control system design, high quality model identification and accurate online control are key to achieving good control performance. The Recursive Least Squares (RLS) algorithm is a linear algorithm that recursively finds tap weight coefficients by minimizing the squared error cost function. The recursive least squares algorithm uses a known time tap weight coefficient vector to find the time tap weight coefficient vector through simple iteration update.
At time i-1, a training sample set is known
Figure GDA0002262446830000011
Wherein xj∈RL×1Representing the input vector, yj∈R1The output quantity is represented. Recursive least square algorithm for finding optimal tap weight coefficient vector wi-1Minimizing the cost function value of equation (1):
Figure GDA0002262446830000012
as can be seen, when a new sample pair { x }i,yiWhen appearing, need to calculate new tap weight coefficient vector wiThe cost function is:
Figure GDA0002262446830000013
however, recalculation of w is not required in the RLS algorithmiBut can be represented by wi-1And the recursive calculation is obtained, so that the calculation amount is reduced, and the calculation efficiency is improved. Fig. 1 shows a structural block diagram of the RLS algorithm.
TABLE 1 RLS Algorithm update procedure
Figure GDA0002262446830000021
Due to PiThe dimension of the matrix is equal to the dimension L of the output vector and is kept unchanged, and the time and space complexity of each update of the recursive least square algorithm is equal to that of the output vector
Figure GDA0002262446830000022
The recursive least squares algorithm evenly divides the computational loadDistributed into each cycle, thereby having significant advantages in online time series prediction applications where sample sequences are obtained continuously.
The Kernel Recursive Least Square (KRLS) algorithm expands the processing capacity of the RLS algorithm on nonlinear data, maps the nonlinear data to a high-dimensional feature space through a Mercer Kernel function and converts the nonlinear data into a linear problem, and then linearly fits in the feature space by using the RLS algorithm. The mapping process of the Mercer kernel function is often called 'kernel technique', and the mapping process does not need to explicitly know the mapping form of the input sample in the feature space, and only needs to calculate the inner product of the mapping through the Mercer kernel function. The method replaces the complex calculation of mapping in a high-dimensional feature space by using simpler kernel function calculation.
Assume an input sample of xpInput samples are mapped in a special space as
Figure GDA0002262446830000023
And reduce it to
Figure GDA0002262446830000024
By "kernel trick" is meant that the computation of the inner product of the feature vectors in an arbitrary feature space can be replaced by a kernel function computation, i.e.
Figure GDA0002262446830000025
Wherein x isp，xqIn order to input the samples, the method,
Figure GDA0002262446830000026
is its corresponding feature vector in the feature space.
Will input samples
Figure GDA0002262446830000027
Mapping to a feature space results
Figure GDA0002262446830000028
The RLS cost function in the feature space:
Figure GDA0002262446830000029
given a feature matrix
Figure GDA00022624468300000210
Then the coefficient vector omegaiCan be linearly represented by the mapping of the input samples in the feature space as ωi＝ΦiαiWherein αi∈RiIs a coefficient vector in the feature space. Defining a kernel matrix
Figure GDA0002262446830000031
Wherein [ K ]i]p,q＝k(xp,xq) The advantage of this matrix is that it can be computed from the input samples using "kernel techniques".
Obtaining a new cost function of the KRLS algorithm according to the formula (4):
Figure GDA0002262446830000032
for off-line identification, an iteration process can be cancelled by an Extreme Learning Machine (ELM) training method, and the cost function (5) can be rewritten as:
Figure GDA0002262446830000033
Subject to:hjαi＝yj-ξj
in the formula, hjThe inner product of the j group of samples and the i group of total samples in the feature space mapping, hj＝ [k(xj,x1),…,k(xj,xi)]，ξjIs corresponding to sample xjTraining error of yjIs the target output value, and C is a constant.
Based on the KKT (Karush-Kuhn-Tucker) theory, the training problem is equivalent to the following double optimization problem:
Figure GDA0002262446830000034
in the formula, λjIs corresponding to sample xjLagrange operator of (2). This results in the KKT optimization condition of formula (6):
Figure GDA0002262446830000035
Figure GDA0002262446830000036
Figure GDA0002262446830000037
wherein λ ═ λ1,…,λi]T. By substituting formulae (7) and (8) for formula (9):
Figure GDA0002262446830000038
wherein y is [ y ═ y1,…,yi]T。
Considering an online control system, corresponding to the RLS algorithm, the KRLS algorithm aims to find the coefficient vector α that satisfies the minimum value of the cost function (5)iAnd need not be recalculated with each new sample, but instead is represented by αi-1α are calculated recursivelyiThe value of (c). To avoid matrix inversion operations at each update, a kernel inverse matrix is defined
Figure GDA0002262446830000041
With recursive updating of QiSimplifying the first recursion to update KiA complex process of matrix inversion. The KRLS algorithm iterative updating expression can be deduced by the cost function of the formula (5), and the basic updating process of the KRLS algorithm is as follows: 1) mapping the new input sample to the feature space, and calculating the inner product h of the ith group of new samples and the i-1 group of original samples in the feature spacei(ii) a 2) Calculating from the prior estimated value and the true valueTo derive a priori estimation error ei(ii) a 3) And 4) calculating adaptive control quantity z according to the kernel inverse matrix and the new input sampleiAnd ri5) adjusting α kernel adaptive filter coefficient vector according to adaptive control quantity and prior errori(ii) a 6) And iteratively updating the kernel inverse matrix according to the self-adaptive control quantity.
TABLE 2 KRLS Algorithm update flow
Figure GDA0002262446830000042
As can be seen from Table 2, the kernel inverse matrix QiIs equal to the input sample number i. Therefore, the KRLS algorithm has one-time update time and space complexity
Figure GDA0002262446830000043
I.e. its complexity increases with the amount of training samples.
Because the calculation complexity of the KRLS algorithm is continuously increased along with the iteration times of online operation, and the physical processing capacity of a computer is limited, the increase of the number of dimensions of a sample kernel function needs to be limited, and the calculation complexity is reduced. The method for reducing the kernel function dimension by using the characteristic value extraction can remove some irrelevant redundant data information of the kernel function in the iteration process, compress the kernel function dimension of the sample, and is more favorable for realizing the effects of improving the control precision and reducing the operation burden in a real-time control system.
Disclosure of Invention
The invention aims to provide a robust control method of an adaptive aircraft based on a kernel recursive least square algorithm aiming at the problems in the prior art, so that a control system has better control precision and lower operation burden.
In order to achieve the purpose, the adaptive aircraft robust control method based on the kernel recursive least square algorithm comprises the following steps: a feedback regulation system is formed by the offline identifier and the online controller to carry out self-adaptive regulation; an off-line identifier: on the basis of the input and output data of the actual model of the aircraft, a model equivalent to the measured system is identified off line by utilizing a kernel recursive least square algorithm; an online controller: on the basis of accurately identifying an actual model of the aircraft, the output quantity of the controller is adjusted on line by utilizing an improved kernel recursive least square algorithm according to system input and output data, and adaptive robust control is realized; the improved kernel recursive least square algorithm is based on characteristic value extraction, and orthogonal decomposition of a kernel matrix is carried out according to principal component analysis and singular value decomposition respectively, so that compression and dimension reduction of a sample kernel function are realized, and some irrelevant redundant data information in the kernel function is removed.
The calculation model of the off-line recognizer is built according to the following steps:
1.1) identifying model definition;
an aircraft system based on a parallel identification model is described as:
y(t)＝f(u(t),u(t-T),…,u(t-pT),y(t-T),…,y(t-nT)) (11)
wherein u (T) represents a system input signal, y (T) represents a system output signal, f (-) represents an unknown nonlinear function, and is composed of an output signal at the past n time and all input signals at the p time, and T is a sampling time;
select { u (T), u (T-T), …, u (T-pT), y (T-T), …, y (T-nT) } and y (T) as input x (T) to the identifiertAnd output ytThe formula (11) is represented as:
yt＝f(xt) (12)
1.2) designing an identifier;
based on a kernel recursive least squares algorithm, according to:
Figure GDA0002262446830000051
the output of the identifier is:
Figure GDA0002262446830000061
in the formula, ht＝[k(xt,x1),…,k(xt,xN)]，{x1,…,xN,y1,…,yNDenotes training samples, N denotes the number of samples, and the identification target is to make the output of the identification system and the output error of the actual system
Figure GDA0002262446830000062
Minimum;
according to the formula
Figure GDA0002262446830000063
Weight αtTaking:
Figure GDA0002262446830000064
where K is the Mercer kernel matrix, [ K ]]i,j＝k(xi,xj)，i＝1,…,N，j＝1,…,N，y＝[y1,…,yN]T。
The online controller is designed to obtain control system inputs u such that the aircraft system outputs y (t) are accurately tracked by the reference model RmAny desired output trajectory y produced*(t)。
The calculation model of the online controller is built according to the following steps:
2.1) control model definition;
the reference model is generated from flight quality criteria of the aircraft, and the reference model outputs y*(t) is expressed as a function of the input signal at time l in the past and the output signal at time m in the past:
y*(t)＝p[y*(t-T),…,y*(t-mT),r(t),r(t-T),…,r(t-lT)](15)
where p [. cndot. ] is a smooth continuous function, the controller system is described as:
u(t)＝g[y*(t),y(t-T),…,y(t-nT),u(t-T),…,u(t-pT)](16)
wherein, the mapping function g exists and is unique, and the target of the controller is to approach the unknown mapping function g;
substituting formula (15) for formula (16) to obtain:
u(t)＝g[p[y*(t-T),…,y*(t-mT),r(t),r(t-T),…,r(t-lT)],y(t-T),…,y(t -nT),u(t-T),…,u(t-pT)](17)
in the above formula, y is substituted for y*：
u(t)＝R[y(t-T),…,y(t-mT),r(t),r(t-T),…,r(t-lT)](18)
In the formula, R [. cndot.)]Is a smooth continuous function; selecting [ y (T-T), …, y (T-mT), r (T-T), …, r (T-lT)]And u (t) as input x to the controller at time ttAnd output ut：
ut＝R(xt) (19)；
2.2) designing a controller;
the mapping function R [. cndot. ] is unknown, the unknown R [. cndot. ] is approximated by adopting an improved kernel recursive least square algorithm, and the controller is constructed to calculate the controlled quantity u (t); the inputs to the controller are current and past reference signals and past aircraft system outputs;
the output of the controller is equal to:
Figure GDA0002262446830000071
in the formula (I), the compound is shown in the specification,
Figure GDA0002262446830000072
the method is to improve the output of the core recursive least square algorithm at the t moment, and the design of the controller aims to ensure that the actual output of an aircraft system and the error of the output of the improved core recursive least square algorithm controller
Figure GDA0002262446830000073
And minimum.
In the improved kernel recursive least squares controller, assuming the identification error is negligible, the method comprises the following steps:
Figure GDA0002262446830000074
obtaining:
Figure GDA0002262446830000075
in the formula (I), the compound is shown in the specification,
Figure GDA0002262446830000076
the output of the aircraft control system is the sum of the output quantities of a PID-like controller and an improved kernel recursive least square algorithm controller:
Figure GDA0002262446830000077
and (3) improving the design of a kernel recursive least square algorithm:
2.3.1) for a given matrix A ∈ Rm×mCalculating the Karhunen-Loeve transformation matrix of the matrix A
Figure GDA0002262446830000078
a) Carrying out normalization processing on the matrix A to obtain a matrix B, so that:
Figure GDA0002262446830000079
b) calculating the covariance matrix of the matrix B:
C＝cov(B) (25)
c) calculating an eigenvector V and an eigenvalue M of the matrix C:
V-1CV＝M (26)
wherein M is a diagonal matrix of eigenvalues;
4) sorting the eigenvalues M in a descending order, and simultaneously sorting the eigenvectors correspondingly to obtain a sorted eigenvector matrix
Figure GDA0002262446830000081
5) According to the characteristic values, the first n 'main components are selected, and n' is less than or equal to n, so that:
Figure GDA0002262446830000082
2.3.2) for the matrix A ∈ Rm×nThere is an orthogonal matrix U ═ U1,…,um]∈Rm×mAnd an orthogonal matrix V ═ V1,…,vn]∈Rn×nMake UTAV＝diag(σ1,…,σp)∈Rm×n，p＝min{m,n}，σ1≥σ2≥…≥σp≥0，σiIs the eigenvalue of the matrix A, vector uiAnd viThe ith left singular vector and the ith right singular vector are respectively; according to the Golub-Reinsch algorithm, the operation complexity is
Figure GDA0002262446830000083
Suppose m>n, for U (: 1: n) ═ U1,…,un]The complexity of the operation becomes
Figure GDA0002262446830000084
If r < rank (A), let:
Figure GDA0002262446830000085
then:
Figure GDA0002262446830000086
A*＝U(:,1：r)SV(:,1：r)Twherein S ═ diag (σ)1,…,σr)，U(:,1：r)＝[u1,…,ur]， V(:,1：r)＝[v1,…,vr](ii) a Definition x ═ x1,…,xn]Is a vector where all elements are independent of each other, then Ax ≈ Bz, where B ═ U (: 1: r) S, z ═ V (: 1: r)Tx is an r x 1 vector, and the approximation of Ax is achieved using Bz.
Calculating a feature vector V and a feature value M, selecting the feature value exceeding 10% of the highest feature value as a reserved component, L being the number of the reserved components, and expressing the reserved feature vector as
Figure GDA0002262446830000087
The weight of the components after dimensionality reduction becomes:
Figure GDA0002262446830000088
the t kernel functions are replaced by new L kernel functions, and the kernel functions after dimensionality reduction become:
h’(xt)＝h(xt)V1(32)
selecting a linear kernel function:
Figure GDA0002262446830000091
h’(xt) Is represented by [ k (x) ]t,x1′),…,k(xt,xL′)]Wherein x isi′＝{[x1,…,xt]V1}(:,i)；
Figure GDA0002262446830000092
Decreases from t to L, and becomes, after the component weight dimensionality reduction:
Qt′＝US (34)
at the same time, the target output ytThe following steps are changed:
yt′＝V2(:,1：L)Tyt(35)
the final approximation result is:
f＝K’(xt)Qt′yt′ (36)。
when the online controller measures the kernel function similarity, two kernel functions are given: h isold＝[k(xt-1,x1),…,k(xt-1,xN)]Information representing the last time, hnew＝[k(xt,x1),…,k(xt,xN)]Representing the information of the current time, the comparison value is defined as:
Figure GDA0002262446830000093
in the formula (I), the compound is shown in the specification,<hold,hnew>Frepresents holdAnd hnewFrobenius norm of (a), KA (h)old,hnew) For measuring holdAnd hnewSimilarity of two kernel functions; KA (h)old,hnew) The larger the more similar, and vice versa.
Since in MRAC control system design, high quality model identification and accurate online control are key to achieving good control performance. Compared with the prior art, the identification model established by the kernel recursive least square algorithm adopted by the invention can be accurately matched with the aircraft nonlinear model, and the minimum structural difference between the identification model and the controlled object is realized. On the basis, the invention adopts the improved kernel recursive least square algorithm for reducing the kernel function dimension based on the characteristic extraction, solves the inherent defect that the sample kernel function dimension increases along with the iteration times, compresses the sample kernel function dimension and self-adaptively adjusts the output quantity of the controller on line, and realizes the effects of improving the control precision and reducing the operation burden in a real-time control system. When the model reference adaptive aircraft robust control system is designed, the defect of real-time increase of the algorithm operation complexity is considered, and the kernel function dimension reduction is carried out by utilizing the characteristic value extraction, so that the operation burden is reduced, and the control effect is improved. The invention constructs a model reference adaptive null value robust control system and carries out simulation verification on the model reference adaptive null value robust control system, and the result shows that the control system has better control effect and strong real-time property.
Drawings
FIG. 1 shows a block diagram of a recursive least squares RLS structure;
FIG. 2 is a block diagram of a model reference adaptive robust control system according to the present invention;
FIG. 3 is a block diagram of a kernel recursive least squares algorithm identifier;
FIG. 4 is a block diagram of an improved controller for a kernel recursive least squares algorithm of the present invention;
FIG. 5 is a diagram of an aircraft expectation plot;
figure 6 is a graph of the effect of the control system tracking.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
The model reference adaptive aircraft robust control system based on the kernel recursive least square algorithm is built according to the following method:
KRLS off-line identifier;
1.1 identifying model definition;
the parallel identification model is a classic method for establishing a Nonlinear dynamic system model, namely an externally input Nonlinear autoregressive Moving Average with exogenous estimation models (NARMAX). The model is Bounded Input Bounded Output (BIBO) stable for all input sequences used in the recognition process. The aircraft system may be described as:
y(t)＝f(u(t),u(t-T),…,u(t-pT),y(t-T),…,y(t-nT)) (11)
where u (T) represents the system input signal, y (T) represents the system output signal, f (-) represents the unknown nonlinear function, which consists of the output signal at the past time n and all input signals at time p, and T is the sampling time. As shown in FIG. 4, { u (T), u (T-T), …, u (T-pT), y (T-T), …, y (T-nT) } and y (T) are selected as inputs x (T) to the identifiertAnd output yt，
Equation (11) can be expressed as:
yt＝f(xt) (12)
1.2 designing an identifier;
the purpose of designing the identifier is to approximate an unknown nonlinear function f (·).
Based on the KRLS algorithm, according to equation (5), the identifier output is:
Figure GDA0002262446830000111
in the formula, ht＝[k(xt,x1),…,k(xt,xN)]，{x1,…,xN,y1,…,yNDenotes training samples, and N denotes the number of samples. The identification target is to make the output of the identification system and the output error of the actual system
Figure GDA0002262446830000112
And minimum.
According to equation (10), the weight αtTaking:
Figure GDA0002262446830000113
where K is the Mercer kernel matrix, [ K ]]i,j＝k(xi,xj)，i＝1,…,N，j＝1,…,N，y＝[y1,…,yN]T。
An M-KRLS online controller;
2.1 control model definition;
the purpose of the controller design is to obtain control system inputs u so that the aircraft system outputs y (t) can be accurately tracked by the reference model RmAny desired output trajectory y produced*(t) of (d). The reference model is generated from aircraft flight quality standards.
Since the reference model itself is a dynamic system, the reference model outputs y*(t) may be expressed as a function of the input signal at time i in the past and the output signal at time m in the past:
y*(t)＝p[y*(t-T),…,y*(t-mT),r(t),r(t-T),…,r(t-lT)](15)
where p [. cndot. ] is a smooth continuous function. The controller system may be described as:
u(t)＝g[y*(t),y(t-T),…,y(t-nT),u(t-T),…,u(t-pT)](16)
in the formula, the mapping function g [. cndot. ] exists and is unique. The goal of the controller is therefore to approximate the unknown mapping function g.
Obtained by substituting formula (15) for formula (16),
u(t)＝g[p[y*(t-T),…,y*(t-mT),r(t),r(t-T),…,r(t-lT)],y(t-T),…,y(t-nT),u(t -T),…,u(t-pT)](17)
obviously, the control inputs u are with respect to the aircraft control response y and the reference signal y*Y. If the aircraft can accurately track any trajectory, then y*Y. Thus, we can replace y with y in the above formula*，
u(t)＝R[y(t-T),…,y(t-mT),r(t),r(t-T),…,r(t-lT)](18)
Wherein R is a smooth continuous function.
As shown in FIG. 5, [ y (T-T), …, y (T-mT), r (T), r (T-T), …, r (T-lT)]And u (t) as input x to the controller at time ttAnd output ut：
ut＝R(xt) (19)
2.2 controller design:
the mapping function R is unknown, M-KRLS is used to approximate the unknown R, and the controller is constructed to calculate the control quantity u (t). The inputs to the controller are the current and past reference signals and past aircraft system outputs. The output of the controller is equal to:
Figure GDA0002262446830000121
in the formula (I), the compound is shown in the specification,
Figure GDA0002262446830000122
is the output of the M-KRLS algorithm at time t. This means that the goal of the controller design is to make the error of the actual output of the aircraft system and the output of the M-KRLS controller
Figure GDA0002262446830000123
And minimum.
The difficulty in controller design is the desired control quantity utIs unknown and cannot directly obtain the approximation error DeltautCorresponding to e in Table 2t. In the M-KRLS controller, the tracking error Δ u is calculated by the identifier, assuming that the identification error is negligible.
In a clear view of the above, it is known that,
Figure GDA0002262446830000124
can be easily obtained by the user, and can be easily obtained,
Figure GDA0002262446830000125
in the formula (I), the compound is shown in the specification,
Figure GDA0002262446830000126
the linear PID-like controller is used for controlling the linear dynamic of the system within a certain determined working range. The aircraft control system output can then be viewed as the sum of the PID-like controller and M-KRLS controller outputs:
Figure GDA0002262446830000127
2.3M-KRLS algorithm;
2.3.1PCA method;
PCA is a nonlinear feature extraction method commonly used in unsupervised dimension reduction. PCA transforms raw data into a set of linearly independent representations of dimensions by linear transformation, which can be used to extract the principal feature components of data, which is often used for dimensionality reduction of high-dimensional data. For a given matrix A ∈ Rm×mThe goal of PCA is to compute the Karhunen-Loeve transformation matrix for matrix A
Figure GDA0002262446830000131
The method mainly comprises the following steps:
1) carrying out normalization processing on the matrix A to obtain a matrix B, so that:
Figure GDA0002262446830000132
2) calculating the covariance matrix of the matrix B:
C＝cov(B) (25)
3) calculating an eigenvector V and an eigenvalue M of the matrix C:
V-1CV＝M (26)
where M is a diagonal matrix of eigenvalues.
4) Sorting the eigenvalues M in a descending order, and simultaneously sorting the eigenvectors correspondingly to obtain a sorted eigenvector matrix
Figure GDA0002262446830000133
5) And selecting the first n 'main components according to the characteristic values, wherein n' is less than or equal to n. It is thus possible to obtain:
Figure GDA0002262446830000134
2.3.2SVD method;
the matrix A ∈ Rm×nThere is an orthogonal matrix U ═ U1,…,um]∈Rm×mAnd an orthogonal matrix V ═ V1,…,vn]∈Rn×nSo that U isTAV＝diag(σ1,…,σp)∈Rm×n，p＝min{m,n}，σ1≥σ2≥…≥σp≥0。σiIs the eigenvalue of the matrix A, vector uiAnd viThe ith vector is used as a left singular vector and a right singular vector. The process of finding the matrices U and V is known as Singular Value Decomposition (SVD).
According to the Golub-Reinsch algorithm, the operation complexity is
Figure GDA0002262446830000135
Suppose m>n, for U (: 1: n) ═ U1,…,un]The complexity of the operation becomes
Figure GDA0002262446830000136
If r < rank (A), let:
Figure GDA0002262446830000141
then it is determined that,
Figure GDA0002262446830000142
in other words, A*Is the best 2-norm approximation of the matrix a. A. the*＝U(:,1：r)SV(:,1：r)TWherein S ═ diag (σ)1,…,σr)，U(:,1：r)＝[u1,…,ur]，V(:,1：r)＝[v1,…,vr]. Definition x ═ x1,…,xn]Is a vector where all elements are independent of each other, then Ax ≈ Bz, where B ═ U (: 1: r) S, z ═ V (: 1: r)Tx is an r x 1 vector. It can be seen that the dimension of the matrix B is far lower than that of the original matrix A, and the approximation of Ax can be realized by utilizing Bz.
2.3.3 kernel function similarity criterion;
measuring kernel function similarity is a key technology for judging data redundancy. Commonly used information-based and distance-based methods tend to lose the location information of the kernel function. The invention provides a kernel alignment method (KA). Given two kernel functions, hold＝[k(xt-1,x1),…,k(xt-1,xN)]Information representing the last time, hnew＝ [k(xt,x1),…,k(xt,xN)]Representing the information of the current time, the comparison value is defined as:
Figure GDA0002262446830000143
in the formula (I), the compound is shown in the specification,<hold,hnew>Frepresents holdAnd hnewFrobenius norm of (1). KA (h)old,hnew) For measuring holdAnd hnewSimilarity of two kernel functions. KA (h)old,hnew) The larger the more similar, and vice versa. KA (h)old,hnew) Shows hnewThe discrimination capability of the kernel function is evaluated at the same time.
2.3.4M-KRLS algorithm flow; KRLS has a great computational burden with the increase of the iteration step size, and finally presents a 'fat' kernelA small two-times algorithm. At the same time, the kernel matrix KtAnd its modification QtThe generalization capability of the nuclear least squares algorithm is rather weakened by containing more and more irrelevant information. In the past algorithm, the higher the kernel function dimension is, the higher the approximation precision is, and obviously, the more kernels generated by the input data stream will bring better approximation effect. However, due to the physical storage limitations of computers, it is difficult to obtain optimal control performance with too many cores. In the invention, an unsupervised characteristic value extraction and dimension reduction method is introduced, data redundant information is deleted, and the operation complexity is reduced. When the 'obese' kernel least square algorithm is processed into the 'slim' kernel least square algorithm through PCA (principal component analysis), the reduced-dimension QtThe next iteration can be performed without losing the original kernel matrix information. Meanwhile, the KRLS method can autonomously delete and update the kernel function.
PCA processing for Qt∈Rt×tRepresents the kernel function htAnd the component weight of the link between the output f. We can use the linear dependence and the existing bias in these weights to reduce the number of kernel functions. Assigning a value L to QtDecreases from t to L. During the PCA dimension reduction processing, according to 2.3.1, the eigenvector V and the eigenvalue M can be calculated. And selecting the characteristic value which exceeds 10% of the highest characteristic value as the component to be reserved, wherein L is the quantity of the reserved components. The retained feature vectors are represented as
Figure GDA0002262446830000151
The weight of the components after dimensionality reduction becomes:
Figure GDA0002262446830000152
then, the original t kernel functions are replaced by new L kernel functions, and the kernel functions after dimensionality reduction become:
h’(xt)＝h(xt)V1(32)
in our work, a linear kernel function was chosen:
Figure GDA0002262446830000153
the 'kernel technique' only needs to directly calculate the inner product of different sample data under linear condition. Thus, h' (x)t) Can be expressed as [ k (x) ]t,x1′),…,k(xt,xL′)]Wherein x isi′＝{[x1,…,xt]V1}(:,i)。
SVD processing is directed to
Figure GDA0002262446830000154
Represents the vector y of target output valuestAnd the component weight of the link between the output f. We can use the linear dependence and the existing bias in these weights to reduce the number of kernel functions. With reference to figure 2.3.2,
Figure GDA0002262446830000155
after SVD dimension reduction processing, the existence of an orthogonal array U epsilon RL×LAnd V2∈Rt×tAnd a diagonal matrix S ∈ RL×L。
Figure GDA0002262446830000156
Decreases from t to L, and becomes, after the component weight dimensionality reduction:
Qt′＝US (34)
at the same time, the target output ytThe following steps are changed:
yt′＝V2(:,1：)L)Tyt(35)
the final approximation is:
f＝K’(xt)Qt′yt′ (36)；
TABLE 3M-KRLS Algorithm update procedure
Figure GDA0002262446830000161
Simulation example
Consider the vertical axis control problem of a high performance fighter. When a fighter is flying at 2000ft high altitude at 150ft/s and an angle of attack of 15deg, the local nonlinear perturbation equation for an aircraft can be described as:
Figure GDA0002262446830000162
in the formula (I), the compound is shown in the specification,
Figure GDA0002262446830000163
B＝[0 0 0 020]T， C＝[0 0 1 0 0]d ═ 0, where x ═ Vt,α,q,θ,δe]∈R5Representing speed, angle of attack, pitch rate, elevator yaw, respectively. g (x) is a smooth and bounded nonlinear function represented by:
Figure GDA0002262446830000164
the simulation is carried out according to the first-level flight quality requirement, and the transfer function corresponding to the pitch angle speed instruction is
Figure GDA0002262446830000165
The aircraft expects the corresponding as shown in figure 5. As can be seen from FIG. 6, the control system can meet the control requirements of the aircraft, achieves good control effect and has high real-time performance.

Claims (1)

1. A self-adaptive aircraft robust control method based on a kernel recursive least square algorithm is characterized in that: a feedback regulation system is formed by the offline identifier and the online controller to carry out self-adaptive regulation; an off-line identifier: on the basis of the input and output data of the actual model of the aircraft, a model equivalent to the measured system is identified off line by utilizing a kernel recursive least square algorithm; an online controller: on the basis of accurately identifying an actual model of the aircraft, the output quantity of the controller is adjusted on line by utilizing an improved kernel recursive least square algorithm according to system input and output data, and adaptive robust control is realized; the improved kernel recursive least square algorithm is based on characteristic value extraction, and orthogonal decomposition of a kernel matrix is carried out according to principal component analysis and singular value decomposition respectively, so that compression and dimension reduction of a sample kernel function are realized, and some irrelevant redundant data information in the kernel function is removed;
the calculation model of the off-line recognizer is built according to the following steps:
1.1) identifying model definition;
an aircraft system based on a parallel identification model is described as:
y(t)＝f(u(t),u(t-T),…,u(t-pT),y(t-T),…,y(t-nT)) (11)
wherein u (T) represents a system input signal, y (T) represents a system output signal, f (-) represents an unknown nonlinear function, and is composed of an output signal at the past n time and all input signals at the p time, and T is a sampling time;
select { u (T), u (T-T), …, u (T-pT), y (T-T), …, y (T-nT) } and y (T) as input x (T) to the identifiertAnd output ytThe formula (11) is represented as:
yt＝f(xt) (12)
1.2) designing an identifier;
based on a kernel recursive least squares algorithm, according to:
Figure FDA0002262446820000011
the output of the identifier is:
Figure FDA0002262446820000012
in the formula, ht＝[k(xt,x1),…,k(xt,xN)]，{x1,…,xN,y1,…,yNDenotes training samples, N denotes the number of samples, and the identification target is to make the output of the identification system and the output error of the actual system
Figure FDA0002262446820000013
Minimum;
according to the formula:
Figure FDA0002262446820000021
weight αtTaking:
Figure FDA0002262446820000022
where K is the Mercer kernel matrix, [ K ]]i,j＝k(xi,xj)，i＝1,…,N，j＝1,…,N，y＝[y1,…,yN]T；
αiIs a coefficient vector in the feature space, C is a constant, KiIs a kernel matrix;
the online controller is designed to obtain control system inputs u such that the aircraft system outputs y (t) are accurately tracked by the reference model RmAny desired output trajectory y produced*(t)；
The calculation model of the online controller is built according to the following steps:
2.1) control model definition;
the reference model is generated from flight quality criteria of the aircraft, and the reference model outputs y*(t) is expressed as a function of the input signal at time l in the past and the output signal at time m in the past:
y*(t)＝p[y*(t-T),…,y*(t-mT),r(t),r(t-T),…,r(t-lT)](15)
where p [. cndot. ] is a smooth continuous function, the controller system is described as:
u(t)＝g[y*(t),y(t-T),…,y(t-nT),u(t-T),…,u(t-pT)](16)
wherein, the mapping function g exists and is unique, and the target of the controller is to approach the unknown mapping function g;
substituting formula (15) for formula (16) to obtain:
u(t)＝g[p[y*(t-T),…,y*(t-mT),r(t),r(t-T),…,r(t-lT)],y(t-T),…,y(t-nT),u(t-T),…,u(t-pT)](17)
in the above formula, y is substituted for y*：
u(t)＝R[y(t-T),…,y(t-mT),r(t),r(t-T),…,r(t-lT)](18)
In the formula, R [. cndot.)]Is a smooth continuous function; selecting [ y (T-T), …, y (T-mT), r (T-T), …, r (T-lT)]And u (t) as input x to the controller at time ttAnd output ut：
ut＝R(xt) (19)；
2.2) designing a controller;
the mapping function R [. cndot. ] is unknown, the unknown R [. cndot. ] is approximated by adopting an improved kernel recursive least square algorithm, and the controller is constructed to calculate the controlled quantity u (t); the inputs to the controller are current and past reference signals and past aircraft system outputs;
the output of the controller is equal to:
Figure FDA0002262446820000031
in the formula (I), the compound is shown in the specification,
Figure FDA0002262446820000032
the method is to improve the output of the core recursive least square algorithm at the t moment, and the design of the controller aims to ensure that the actual output of an aircraft system and the error of the output of the improved core recursive least square algorithm controller
Figure FDA0002262446820000033
Minimum;
in the improved kernel recursive least squares controller, assuming the identification error is negligible, the method comprises the following steps:
Figure FDA0002262446820000034
obtaining:
Figure FDA0002262446820000035
in the formula (I), the compound is shown in the specification,
Figure FDA0002262446820000036
the output of the aircraft control system is the sum of the output quantities of a PID-like controller and an improved kernel recursive least square algorithm controller:
Figure FDA0002262446820000037
and (3) improving the design of a kernel recursive least square algorithm:
2.3.1) for a given matrix A ∈ Rm×mCalculating the Karhunen-Loeve transformation matrix of the matrix A
Figure FDA0002262446820000038
a) Carrying out normalization processing on the matrix A to obtain a matrix B, so that:
Figure FDA0002262446820000039
b) calculating the covariance matrix of the matrix B:
C＝cov(B) (25)
c) calculating an eigenvector V and an eigenvalue M of the matrix C:
V-1CV＝M (26)
wherein M is a diagonal matrix of eigenvalues;
4) sorting the eigenvalues M in a descending order, and simultaneously sorting the eigenvectors correspondingly to obtain a sorted eigenvector matrix
Figure FDA0002262446820000049
5) According to the characteristic values, the first n 'main components are selected, and n' is less than or equal to n, so that:
Figure FDA0002262446820000041
2.3.2) for the matrix A ∈ Rm×nThere is an orthogonal matrix U ═ U1,…,um]∈Rm×mAnd an orthogonal matrix V ═ V1,…,vn]∈Rn×nMake UTAV＝diag(σ1,…,σp)∈Rm×n，p＝min{m,n}，σ1≥σ2≥…≥σp≥0，σiIs the eigenvalue of the matrix A, vector uiAnd viThe ith left singular vector and the ith right singular vector are respectively; according to the Golub-Reinsch algorithm, the operation complexity is
Figure FDA0002262446820000042
Suppose m>n, for U (: 1: n) ═ U1,…,un]The complexity of the operation becomes
Figure FDA0002262446820000043
If r < rank (A), let:
Figure FDA0002262446820000044
then:
Figure FDA0002262446820000045
A*＝U(:,1：r)SV(:,1：r)Twherein S ═ diag (σ)1,…,σr)，U(:,1：r)＝[u1,…,ur]，V(:,1：r)＝[v1,…,vr](ii) a Definition x ═ x1,…,xn]Is a vector where all elements are independent of each other, then Ax ≈ Bz, where B ═ U (: 1: r) S, z ═ V (: 1: r)Tx is an r x 1 vector, and the approximation of Ax is realized by Bz;
calculating the characteristic vector V and the characteristic value M, and selecting the characteristic value which exceeds the highest characteristic value by 10 percent as a guaranteeThe remaining components, L the number of remaining components, the remaining feature vectors are represented as
Figure FDA0002262446820000046
The weight of the components after dimensionality reduction becomes:
Figure FDA0002262446820000047
the t kernel functions are replaced by new L kernel functions, and the kernel functions after dimensionality reduction become:
h’(xt)＝h(xt)V1(32)
selecting a linear kernel function:
Figure FDA0002262446820000048
h’(xt) Is represented by [ k (x) ]t,x1′),…,k(xt,xL′)]Wherein x isi′＝{[x1,…,xt]V1}(:,i)；
Figure FDA0002262446820000051
Decreases from t to L, and becomes, after the component weight dimensionality reduction:
Qt′＝US (34)
at the same time, the target output ytThe following steps are changed:
yt′＝V2(:,1：L)Tyt(35)
the final approximation result is:
f＝K’(xt)Qt′yt′ (36)
when the online controller measures the kernel function similarity, two kernel functions are given: h isold＝[k(xt-1,x1),…,k(xt-1,xN)]Information representing the last time, hnew＝[k(xt,x1),…,k(xt,xN)]Representing the information of the current time, the comparison value is defined as:
Figure FDA0002262446820000052
in the formula (I), the compound is shown in the specification,<hold,hnew>Frepresents holdAnd hnewFrobenius norm of (a), KA (h)old,hnew) For measuring holdAnd hnewSimilarity of two kernel functions; KA (h)old,hnew) The larger the more similar, and vice versa.
CN201710171108.0A 2017-03-21 2017-03-21 Adaptive aircraft robust control method based on kernel recursive least square algorithm Expired - Fee Related CN106980262B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201710171108.0A CN106980262B (en) 2017-03-21 2017-03-21 Adaptive aircraft robust control method based on kernel recursive least square algorithm

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201710171108.0A CN106980262B (en) 2017-03-21 2017-03-21 Adaptive aircraft robust control method based on kernel recursive least square algorithm

Publications (2)

Publication Number Publication Date
CN106980262A CN106980262A (en) 2017-07-25
CN106980262B true CN106980262B (en) 2020-03-17

Family

ID=59338284

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201710171108.0A Expired - Fee Related CN106980262B (en) 2017-03-21 2017-03-21 Adaptive aircraft robust control method based on kernel recursive least square algorithm

Country Status (1)

Country Link
CN (1) CN106980262B (en)

Families Citing this family (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108279567B (en) * 2017-12-29 2021-04-09 浙江中控软件技术有限公司 System identification method for robust control
CN108717505A (en) * 2018-05-29 2018-10-30 广东工业大学 A kind of solidification thermal process space-time modeling method based on K-RVFL
CN109510610B (en) * 2018-11-07 2020-08-25 电子科技大学 Nuclear self-adaptive filtering method based on soft projection weighted nuclear recursive least squares
DE102019201557A1 (en) * 2019-02-07 2020-08-13 Zf Friedrichshafen Ag Method and device for the automated identification of a product defect in a product and / or for the automated identification of a product defect cause of the product defect
CN111709297A (en) * 2020-05-18 2020-09-25 武汉工程大学 Intelligent diagnosis and fault prediction method and system based on NARMAX-FRF and PCA

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR101096113B1 (en) * 2009-05-26 2011-12-30 한국교통연구원 Navigation system having enhanced reliability and method thereof
CN104766313B (en) * 2015-03-27 2017-10-31 哈尔滨工程大学 One kind uses the recursive EO-1 hyperion rapid abnormal detection method of core

Also Published As

Publication number Publication date
CN106980262A (en) 2017-07-25

Similar Documents

Publication Publication Date Title
CN106980262B (en) Adaptive aircraft robust control method based on kernel recursive least square algorithm
CN106600059B (en) Intelligent power grid short-term load prediction method based on improved RBF neural network
Shi et al. Quantized learning control for flexible air-breathing hypersonic vehicle with limited actuator bandwidth and prescribed performance
KR101887267B1 (en) Neural Networks Training Method Using Subnetwork Training algorithms for multi-class prediction neural networks on high level image recognition
Grande et al. Nonparametric adaptive control using Gaussian processes with online hyperparameter estimation
CN112859898A (en) Aircraft trajectory prediction method based on two-channel bidirectional neural network
Erdogmus et al. Recursive principal components analysis using eigenvector matrix perturbation
CN116880538B (en) High subsonic unmanned plane large maneuvering flight control system and method thereof
CN108647726B (en) Image clustering method
CN110398942B (en) Parameter identification method for industrial production process control
CN110610188A (en) Markov distance-based shadow rough fuzzy clustering method
CN115455670B (en) Non-Gaussian noise model building method based on Gaussian mixture model
Hug et al. B\'ezier Curve Gaussian Processes
CN111160464B (en) Industrial high-order dynamic process soft measurement method based on multi-hidden-layer weighted dynamic model
Bartecki PCA-based approximation of a class of distributed parameter systems: classical vs. neural network approach
Saleem et al. Application and comparison of kernel functions for linear parameter varying model approximation of nonlinear systems
CN115102868A (en) Web service QoS prediction method based on SOM clustering and depth self-encoder
You et al. Iterative learning optimization for UAV path planning with avoidance zones
Zhang et al. Multi-dimensional local weighted regression ship motion identification modeling based on particle swarm optimization
CN114037931A (en) Multi-view discrimination method of self-adaptive weight
Bartecki Neural network-based PCA: An application to approximation of a distributed parameter system
CN113159233A (en) Spectral clustering acceleration method, system, computer equipment and storage medium
Sanei et al. Improving the precision of KNN classifier using nonlinear weighting method based on the spline interpolation
Uddin System Identification of Two-Wheeled Robot Dynamics Using Neural Networks
Ferdaus et al. Evolving fuzzy inference system based online identification and control of a quadcopter unmanned aerial vehicle

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant
CF01 Termination of patent right due to non-payment of annual fee

Granted publication date: 20200317

CF01 Termination of patent right due to non-payment of annual fee