CN106980262B - Adaptive aircraft robust control method based on kernel recursive least square algorithm - Google Patents
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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
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 knownWherein 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):
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:
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
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 vectorThe 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 asAnd reduce it toBy "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.
Wherein x isp,xqIn order to input the samples, the method,is its corresponding feature vector in the feature space.
given a feature matrixThen 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 matrixWherein [ 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):
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:
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:
in the formula, λjIs corresponding to sample xjLagrange operator of (2). This results in the KKT optimization condition of formula (6):
wherein λ ═ λ1,…,λi]T. By substituting formulae (7) and (8) for formula (9):
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 definedWith 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
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 complexityI.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:
the output of the identifier is:
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 systemMinimum;
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:
in the formula (I), the compound is shown in the specification,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 controllerAnd minimum.
In the improved kernel recursive least squares controller, assuming the identification error is negligible, the method comprises the following steps:
obtaining:
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:
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
a) Carrying out normalization processing on the matrix A to obtain a matrix B, so that:
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
5) According to the characteristic values, the first n 'main components are selected, and n' is less than or equal to n, so that:
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
If r < rank (A), let:
then:
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 asThe weight of the components after dimensionality reduction becomes:
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:
h’(xt) Is represented by [ k (x) ]t,x1′),…,k(xt,xL′)]Wherein x isi′={[x1,…,xt]V1}(:,i);
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:
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:
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 systemAnd minimum.
According to equation (10), the weight αtTaking:
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:
in the formula (I), the compound is shown in the specification,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 controllerAnd 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,
can be easily obtained by the user, and can be easily obtained,
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:
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 AThe method mainly comprises the following steps:
1) carrying out normalization processing on the matrix A to obtain a matrix B, so that:
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
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:
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 isSuppose m>n, for U (: 1: n) ═ U1,…,un]The complexity of the operation becomes
If r < rank (A), let:
then it is determined that,
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:
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 asThe weight of the components after dimensionality reduction becomes:
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:
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 toRepresents 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,after SVD dimension reduction processing, the existence of an orthogonal array U epsilon RL×LAnd V2∈Rt×tAnd a diagonal matrix S ∈ RL×L。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
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:
in the formula (I), the compound is shown in the specification,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:
the simulation is carried out according to the first-level flight quality requirement, and the transfer function corresponding to the pitch angle speed instruction isThe 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:
the output of the identifier is:
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 systemMinimum;
according to the formula:
weight αtTaking:
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:
in the formula (I), the compound is shown in the specification,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 controllerMinimum;
in the improved kernel recursive least squares controller, assuming the identification error is negligible, the method comprises the following steps:
obtaining:
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:
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
a) Carrying out normalization processing on the matrix A to obtain a matrix B, so that:
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
5) According to the characteristic values, the first n 'main components are selected, and n' is less than or equal to n, so that:
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
If r < rank (A), let:
then:
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 asThe weight of the components after dimensionality reduction becomes:
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:
h’(xt) Is represented by [ k (x) ]t,x1′),…,k(xt,xL′)]Wherein x isi′={[x1,…,xt]V1}(:,i);
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:
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.
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