CN104730920A

CN104730920A - A Neural Network Adaptive Dynamic Surface Controller Structure and Design Method

Info

Publication number: CN104730920A
Application number: CN201510182386.7A
Authority: CN
Inventors: 彭周华; 刘陆; 王丹
Original assignee: Dalian Maritime University
Current assignee: Dalian Maritime University
Priority date: 2015-04-16
Filing date: 2015-04-16
Publication date: 2015-06-24
Anticipated expiration: 2035-04-16
Also published as: CN104730920B

Abstract

The invention discloses a neural network self-adaptive dynamic surface controller structure and a design method thereof. The predictor is introduced into the design of the controller, the on-line learning of the uncertainty of the controlled system is not based on the tracking error any more but on the prediction error, the influence of the larger tracking error in the initial stage on the transient performance of the control system can be overcome, and the control signal of the system is not easy to be saturated. The low-pass filter is introduced into the design of the controller, so that the problem of high-frequency oscillation of the input control signal caused by large adaptive parameters can be solved, and the control signal is favorably ensured to be within the execution frequency band range of the actuator. The invention adopts the differential tracker to replace a first-order filter in the traditional dynamic surface control, can ensure the finite time estimation of the virtual control signal and improve the estimation accuracy. The invention adopts the estimated state for feedback, which is beneficial to eliminating the interference of the measurement noise to the control system.

Description

A Neural Network Adaptive Dynamic Surface Controller Structure and Design Method

技术领域technical field

本发明涉及非线性系统控制领域，尤其涉及一类不确定严反馈非线性系统的神经网络自适应动态面控制器结构及设计方法。The invention relates to the field of nonlinear system control, in particular to a neural network self-adaptive dynamic surface controller structure and a design method of a kind of uncertain strict feedback nonlinear system.

背景技术Background technique

不确定严反馈非线性系统是一类典型的非线性系统，许多实际系统都满足不确定严反馈非线性系统的形式，如船舶自动舵系统、汽车主动悬架系统、飞行器系统和机器人系统等。因此对不确定严反馈非线性系统的控制研究具有重要的理论价值与广泛的实际意义。Uncertain strict feedback nonlinear system is a typical nonlinear system. Many practical systems satisfy the form of uncertain strict feedback nonlinear system, such as ship autopilot system, automobile active suspension system, aircraft system and robot system, etc. Therefore, the research on the control of uncertain strict feedback nonlinear systems has important theoretical value and extensive practical significance.

反步法(Backstepping)在处理不确定严反馈非线性系统控制问题时具有设计过程清晰，控制律容易推导等特点，在工业控制领域得到了广泛的重视和发展。然而，反步法在每步递推的过程中，需要对虚拟控制律进行求导，导致计算复杂性会随着被控系统阶数的增长呈爆炸性增长，从而制约了反步法的实际应用。为克服反步法设计的缺陷，Swaroop等人提出了一种动态面控制(DynamicSurface Control，DSC)设计方法，通过引入若干一阶滤波器，将复杂的求导运算转变成简单的代数运算，从而显著地降低了控制器的复杂性。王丹与黄捷等将动态面与神经网络自适应控制技术相结合，显著放松了以往对被控系统不确定性线性参数化的约束条件，解决了具有高度不确定性非线性系统的自适应控制设计问题。随后，该方法被学者们推广到各种基于在线逼近的自适应动态面控制设计中。Backstepping has the characteristics of clear design process and easy derivation of control law when dealing with uncertain strict feedback nonlinear system control problems, and has been widely valued and developed in the field of industrial control. However, the backstepping method needs to derive the virtual control law in each recursive process, resulting in an explosive increase in computational complexity as the order of the controlled system increases, thus restricting the practical application of the backstepping method . In order to overcome the defects of backstepping design, Swaroop et al. proposed a dynamic surface control (Dynamic Surface Control, DSC) design method, by introducing several first-order filters, the complex derivation operation is transformed into a simple algebraic operation, thus Significantly reduces the complexity of the controller. Wang Dan and Huang Jie combined the dynamic surface and neural network adaptive control technology, which significantly relaxed the previous constraints on the uncertain linear parameterization of the controlled system, and solved the adaptive control of highly uncertain nonlinear systems. control design issues. Subsequently, the method was extended by scholars to various adaptive dynamic surface control designs based on online approximation.

然而，在控制器结构和控制器设计方面，现有自适应动态面控制器设计技术仍存在下列不足：However, in terms of controller structure and controller design, the existing adaptive dynamic surface controller design technology still has the following deficiencies:

第一，现有技术均采用跟踪误差进行在线学习，由于在初始阶段跟踪误差通常较大，从而严重影响神经网络的暂态学习性能，进而使得控制器陷入饱和，最终会导致控制系统整体控制性能降低。First, the existing technologies all use tracking error for online learning. Since the tracking error is usually large in the initial stage, it seriously affects the transient learning performance of the neural network, which in turn causes the controller to fall into saturation, which will eventually lead to the overall control performance of the control system. reduce.

第二，现有技术中为了对被控系统不确定性进行快速学习，往往需要选取较大的自适应控制参数，但大的自适应控制参数会导致控制信号的高频振荡，控制系统的鲁棒性与稳定性相互制约，难以达到理想的控制效果。Second, in the prior art, in order to quickly learn the uncertainty of the controlled system, it is often necessary to select a large adaptive control parameter, but a large adaptive control parameter will cause high-frequency oscillation of the control signal, and the robustness of the control system Rodability and stability restrict each other, making it difficult to achieve the ideal control effect.

第三，现有技术中通常采用一阶滤波器得到虚拟控制信号及其导数的估计，由于采用线性滤波器形式，很难实现对导数的精确估计，并且需要通过增加滤波器的带宽来提高控制系统的跟踪性能。Third, in the prior art, the first-order filter is usually used to estimate the virtual control signal and its derivative. Due to the use of a linear filter, it is difficult to achieve an accurate estimation of the derivative, and it is necessary to improve the control by increasing the bandwidth of the filter. The tracking performance of the system.

第四，现有技术中均假设被控系统状态是精确可测的，但在实际应用中反馈状态不可避免地存在测量噪声，由于现有控制器设计方法无法消除噪声对控制系统的影响，因而很难保证理想的控制性能。Fourth, in the prior art, it is assumed that the state of the controlled system is accurate and measurable, but in practical applications, there is inevitably measurement noise in the feedback state, because the existing controller design method cannot eliminate the influence of noise on the control system, so It is difficult to guarantee ideal control performance.

发明内容Contents of the invention

为解决现有技术存在的上述问题，本发明要提出一种神经网络自适应动态面控制器结构及设计方法，不仅可以显著提高控制系统的暂态性能，避免控制器输入信号的高频振荡，并且能够提高对虚拟控制信号估计的精确性，同时能够消除测量噪声带来的影响，使得该设计方法更有利于实际的工程应用。In order to solve the above problems in the prior art, the present invention proposes a neural network adaptive dynamic surface controller structure and design method, which can not only significantly improve the transient performance of the control system, but also avoid high-frequency oscillation of the controller input signal, And it can improve the accuracy of virtual control signal estimation, and at the same time, it can eliminate the influence of measurement noise, making this design method more beneficial to practical engineering applications.

为了实现上述目的，本发明的技术方案如下：一种神经网络自适应动态面控制器结构，由n级子控制器组成；In order to achieve the above object, the technical scheme of the present invention is as follows: a neural network self-adaptive dynamic surface controller structure is composed of n-level sub-controllers;

第1级子控制器的输入端与外部参考信号y_r相连，还与测量机构的输出端x_1m,x_2m相连，第1级子控制器的输出端α₂与第2级子控制器的输入端相连，第1级子控制器的另一输出端分别与第1级到n级子控制器的输入端相连；The input terminal of the first-level sub-controller is connected with the external reference signal y _r , and also connected with the output terminals x _1m , x _2m of the measuring mechanism, the output terminal α ₂ of the first-level sub-controller is connected with the output terminal α2 of the second-level sub-controller The input terminal is connected to the other output terminal of the first stage sub-controller respectively connected to the input terminals of the first-level to n-level sub-controllers;

第i级子控制器的输入端与第i-1级子控制器的输入端α_i相连，并与前i级子控制器的输出端相连，还与测量机构的输出端x_im,x_(i+1)m相连，第i级子控制器的输出端α_i与第i+1级子控制器的输入端相连，第i级子控制器的另一输出端分别与第i级到n级子控制器的输入端相连；The input terminal of the i-th sub-controller is connected to the input terminal α _i of the i-1th sub-controller, and is connected to the output terminal of the previous i-level sub-controller is connected to the output terminals x _im , x _(i+1)m of the measurement mechanism, the output terminal α _i of the i-th sub-controller is connected to the input terminal of the i+1-th sub-controller, and the i-th sub-controller The other output of the controller respectively connected to the input terminals of the i-th to n-th level sub-controllers;

以此类推，第n级子控制器的输入端与第n-1级子控制器的输出端α_n相连，并分别与全部n级子控制器的输出端相连，还与测量机构的输出端x_nm相连，第n级子控制器的输出端u与被控系统的输入端相连；By analogy, the input terminal of the nth level sub-controller is connected to the output terminal α _n of the n-1th level sub-controller, and is respectively connected to the output terminals of all n-level sub-controllers connected, and also connected with the output terminal x _nm of the measuring mechanism, and the output terminal u of the nth sub-controller is connected with the input terminal of the controlled system;

所述的第1级子控制器由跟踪微分器、比较器1、比较器2、预估器、线性控制单元、逼近器、低通滤波器及求和器构成，跟踪微分器的输入端与外部参考信号y_r相连，跟踪微分器的输出端分别与比较器1和求和器的输入端相连；比较器1的另一输入端与预估器的输出端相连，比较器1的输出端与线性控制单元的输入端相连；比较器2的两个输入端分别与测量机构和预估器的输出端相连；预估器的输入端分别与测量机构和逼近器的输出端相连，预估器的输出端为第1级子控制器的一个输出端；逼近器的输入端与比较器2的输出端相连，另一输入端与预估器的输出端相连，逼近器的输出端还与低通滤波器的输入端相连；低通滤波器的输出端与求和器的输入端相连；求和器的另一输入端与线性控制单元的输出端相连，求和器的输出端为第1级子控制器的另一输出端；Described first stage sub-controller is made of tracking differentiator, comparator 1, comparator 2, predictor, linear control unit, approximator, low-pass filter and summator, and the input end of tracking differentiator and The external reference signal y _r is connected, and the output end of the tracking differentiator is respectively connected with the input end of the comparator 1 and the summer; the other input end of the comparator 1 is connected with the output end of the predictor, and the output end of the comparator 1 It is connected with the input end of the linear control unit; the two input ends of the comparator 2 are respectively connected with the output ends of the measuring mechanism and the predictor; the input ends of the predictor are respectively connected with the output ends of the measuring mechanism and the approximator, The output end of the device is an output end of the first stage sub-controller; the input end of the approximator is connected with the output end of comparator 2, and the other input end is connected with the output end of the predictor, and the output end of the approximator is also connected with The input end of the low-pass filter is connected; the output end of the low-pass filter is connected with the input end of the summer; the other input end of the summer is connected with the output end of the linear control unit, and the output end of the summer is the first The other output terminal of the level 1 sub-controller;

所述的第i级子控制器由跟踪微分器、比较器1、比较器2、预估器、线性控制单元、逼近器、低通滤波器及求和器构成，跟踪微分器的输入端与第i-1级子控制器的输出端相连，跟踪微分器的输出端分别与比较器1和求和器的输入端相连；比较器1的另一输入端与预估器的输出端相连，比较器1的输出端与线性控制单元的输入端相连；比较器2的两个输入端分别与测量机构和预估器的输出端相连；预估器的输入端分别与测量机构和逼近器的输出端相连，预估器的输出端为第i级子控制器的一个输出端；逼近器的输入端与比较器2的输出端相连，另一输入端与前i级子控制器的输出端相连，逼近器的输出端还与低通滤波器的输入端相连；低通滤波器的输出端与求和器的输入端相连；求和器的另一输入端与线性控制单元的输出端相连，求和器的输出端为第i级子控制器的另一输出端；The i-th level sub-controller is composed of tracking differentiator, comparator 1, comparator 2, predictor, linear control unit, approximator, low-pass filter and summer, and the input terminal of tracking differentiator is connected with The output terminals of the i-1th sub-controller are connected, and the output terminals of the tracking differentiator are respectively connected with the input terminals of the comparator 1 and the summer; the other input terminal of the comparator 1 is connected with the output terminal of the predictor, The output terminal of comparator 1 is connected with the input terminal of the linear control unit; the two input terminals of comparator 2 are respectively connected with the output terminals of the measuring mechanism and the predictor; the input terminals of the predictor are respectively connected with the measuring mechanism and the approximator The output terminal is connected, and the output terminal of the predictor is an output terminal of the i-th sub-controller; the input terminal of the approximator is connected with the output terminal of the comparator 2, and the other input terminal is connected with the output terminal of the previous i-level sub-controller The output of the approximator is also connected to the input of the low-pass filter; the output of the low-pass filter is connected to the input of the summer; the other input of the summer is connected to the output of the linear control unit , the output terminal of the summer is another output terminal of the i-th sub-controller;

所述的第n级子控制器由跟踪微分器、比较器1、比较器2、预估器、线性控制单元、逼近器、低通滤波器及求和器构成，跟踪微分器的输入端与第n-1级子控制器的输出端相连，跟踪微分器的输出端分别与比较器1和求和器的输入端相连；比较器1的另一输入端与预估器的输出端相连，比较器1的输出端与线性控制单元的输入端相连；比较器2的两个输入端分别与测量机构和预估器的输出端相连；预估器的输入端分别与测量机构，逼近器及求和单元的输出端相连；逼近器的输入端与比较器2的输出端相连，另一输入端与所有n级子控制器的输出端相连，逼近器的输出端还与低通滤波器的输入端相连；低通滤波器的输出端与求和器的输入端相连；求和器的另一输入端与线性控制单元的输出端相连，求和器的输出端为第n级子控制器的输出端，与被控系统的输入端相连；The nth stage sub-controller is composed of tracking differentiator, comparator 1, comparator 2, predictor, linear control unit, approximator, low-pass filter and summator, and the input terminal of tracking differentiator and The output terminals of the n-1th sub-controller are connected, and the output terminals of the tracking differentiator are respectively connected with the input terminals of the comparator 1 and the summer; the other input terminal of the comparator 1 is connected with the output terminal of the predictor, The output terminal of comparator 1 is connected with the input terminal of the linear control unit; the two input terminals of comparator 2 are respectively connected with the output terminals of the measuring mechanism and the estimator; the input terminals of the estimator are respectively connected with the measuring mechanism, the approximator and the The output end of the summation unit is connected; the input end of the approximator is connected with the output end of the comparator 2, and the other input end is connected with the output ends of all n-level sub-controllers, and the output end of the approximator is also connected with the output end of the low-pass filter The input terminal is connected; the output terminal of the low-pass filter is connected with the input terminal of the summer; the other input terminal of the summer is connected with the output terminal of the linear control unit, and the output terminal of the summer is the nth level sub-controller The output terminal is connected with the input terminal of the controlled system;

所述的被控系统由下列n阶不确定严反馈非线性系统形式来描述：The controlled system is described by the following n-order uncertain strict feedback nonlinear system form:

如果用R表示实数集合，那么式中x_i∈R表示被控系统第i级子系统的状态；u∈R表示被控系统的控制输入；为未知的非线性连续函数，y∈R为被控系统的输出。If R is used to represent the set of real numbers, then x _i ∈ R represents the state of the i-th subsystem of the controlled system; u ∈ R represents the control input of the controlled system; is an unknown nonlinear continuous function, y∈R is the output of the controlled system.

一种神经网络自适应动态面控制器设计方法，包括以下步骤：A neural network adaptive dynamic surface controller design method, comprising the following steps:

A、第1级子控制器设计：A. Level 1 sub-controller design:

A1、第1级跟踪微分器：第1级跟踪微分器接收外部参考信号y_r，经过下列变换：A1. The first-stage tracking differentiator: The first-stage tracking differentiator receives the external reference signal y _r and undergoes the following transformation:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11 r r} = = {x x}_{11 r r}^{d d} \\ {\overset{\cdot &Center Dot;}{x x}}_{11 r r}^{d d} = = - - {γ γ}_{11} sign sign (({x x}_{11 r r} - - {y the y}_{r r} + + \frac{{x x}_{11 r r}^{d d} | | {x x}_{11 r r}^{d d} | |}{{22 γ γ}_{11}})) \end{matrix} - - - - - - ((22))$

得到第1级跟踪微分器的输出信号为估计变量x_1r及其导数其中γ₁＞0为常值，是中间变量，sign(·)表示符号函数；The output signal of the first-level tracking differentiator is obtained as the estimated variable x _1r and its derivative Where γ ₁ >0 is a constant value, is an intermediate variable, and sign(·) represents a sign function;

A2、第1级比较器1：第1级比较器1分别接收跟踪微分器的输出信号估计变量x_1r和预估器的输出信号接收到的信号经过下列变换：A2. The first stage comparator 1: The first stage comparator 1 respectively receives the output signal of the tracking differentiator, the estimated variable x _1r and the output signal of the predictor The received signal undergoes the following transformations:

${\overset{^^}{S S}}_{11} = = {\overset{^^}{x x}}_{11} - - {x x}_{11 r r} - - - - - - ((33))$

得到第1级比较器1的输出信号 Get the output signal of the 1st stage comparator 1

A3、第1级比较器2：第1级比较器2分别接收由测量机构输出的被控系统第1级子系统的状态信号x_1m和预估器的输出信号接收到的信号经过下列变换得到比较器2的输出端信号 A3. First-level comparator 2: The first-level comparator 2 respectively receives the status signal x _1m of the first-level subsystem of the controlled system and the output signal of the predictor output by the measurement institution The received signal undergoes the following transformation to obtain the output signal of comparator 2

${\overset{~ ~}{x x}}_{11} = = {\overset{^^}{x x}}_{11} - - {x x}_{11 m m} - - - - - - ((44))$

其中x_1m＝x₁+w₁为含有测量噪声的第1级子系统的状态信号，w₁为测量噪声；Where x _1m =x ₁ +w ₁ is the status signal of the first-level subsystem containing measurement noise, and w ₁ is the measurement noise;

A4、第1级预估器：第1级预估器分别接收由测量机构输出的被控系统第1级子系统的状态信号x_1m和第2级子系统的状态信号x_2m，以及逼近器的输出信号α_w1，接收到的信号经过下列变换得到预估器的输出信号 A4. Level 1 predictor: The level 1 predictor respectively receives the state signal x _1m of the first level subsystem of the controlled system and the state signal x _2m of the second level subsystem output by the measurement institution, and the approximator The output signal α _w1 of the received signal undergoes the following transformation to obtain the output signal of the predictor

${\overset{\cdot &Center Dot;}{\overset{^^}{x x}}}_{11} = = {α α}_{w w 11} + + {x x}_{22 m m} - - (({k k}_{11} + + {κ κ}_{11})) (({\overset{^^}{x x}}_{11} - - {x x}_{11 m m})) - - - - - - ((55))$

其中k₁＞0，κ₁＞0为常数；预估器的输出信号同时也是第1级子控制器的一个输出信号；Where k ₁ >0, κ ₁ >0 are constants; the output signal of the predictor It is also an output signal of the first-level sub-controller;

A5、第1级线性控制单元：第1级线性控制单元接收比较器1的输出信号经过下列比例控制：A5. The first-level linear control unit: the first-level linear control unit receives the output signal of comparator 1 Controlled by the following proportions:

${α α}_{k k 11} = = {- - k k}_{11} {\overset{^^}{S S}}_{11} - - - - - - ((66))$

得到第1级线性控制单元的输出信号α_k1；Obtain the output signal α _k1 of the first-level linear control unit;

A6、第1级逼近器：第1级逼近器分别接收预估器的输出信号和比较器2的输出信号 A6. The first-stage approximator: the first-stage approximator receives the output signal of the predictor respectively and the output signal of Comparator 2

逼近器的作用是对被控系统中的未知非线性进行在线估计，所述的逼近器采用神经网络逼近方法，将表示为的形式，其中是逼近误差，满足是正常数；W₁∈R^s为未知的权值矩阵，并满足||W₁||_F≤W₁ ^*，W₁ ^*是正常数；为已知的激励函数矩阵，其形式为定义是W₁的估计，设计的更新率为：The role of the approximator is to estimate the unknown nonlinearity in the controlled system Carry out online estimation, described approximator adopts neural network approximation method, will Expressed as in the form of is the approximation error, satisfying is a normal number; W ₁ ∈ R ^s is an unknown weight matrix, and satisfies ||W ₁ || _F ≤ _{W 1} ^* , W ₁ ^* is a normal number; is a known activation function matrix, whose form is definition is an estimate of _W1 , the design The update rate is:

其中Γ_W1∈R，k_W1∈R是正常数；Where Γ _W1 ∈ R, k _W1 ∈ R is a positive constant;

最后得到第1级逼近器的输出信号α_w1：Finally, the output signal α _w1 of the first-stage approximator is obtained:

A7、第1级低通滤波器：第1级低通滤波器接收逼近器的输出信号α_w1，经过以下变换：A7. The first-stage low-pass filter: the first-stage low-pass filter receives the output signal α _w1 of the approximator, and undergoes the following transformation:

α_cw1＝C(s)α_w1 (9)得到第1级低通滤波器的输出信号α_cw1，其中C(s)表示滤波器；α _cw1 =C(s)α _w1 (9) to obtain the output signal α _cw1 of the first-stage low-pass filter, where C(s) represents the filter;

A8、第1级求和器：第1级求和器分别接收线性控制单元的输出信号α_k1、低通滤波器的输出信号α_cw1和跟踪微分器的输出信号估计变量的导数接收到的信号经过以下求和计算：A8. The first-stage summer: The first-stage summer respectively receives the output signal α _k1 of the linear control unit, the output signal α _cw1 of the low-pass filter, and the derivative of the estimated variable of the output signal of the tracking differentiator The received signals are summed as follows:

${α α}_{22} = = {α α}_{k k 11} + + {α α}_{cw cw 11} + + {\overset{\cdot &Center Dot;}{x x}}_{11 r r} - - - - - - ((1010))$

得到第1级子控制器的另一输出端信号α₂；Obtain another output terminal signal α ₂ of the first-level sub-controller;

B、第i级子控制器设计：B. Design of i-level sub-controller:

B1、第i级跟踪微分器：第i级跟踪微分器接收第i-1级子控制器的输出信号α_i，经过下列变换：B1. The i-level tracking differentiator: the i-level tracking differentiator receives the output signal α _i of the i-1-th sub-controller, and undergoes the following transformation:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{ir ir} = = {x x}_{ir ir}^{d d} \\ {\overset{\cdot &Center Dot;}{x x}}_{ir ir}^{d d} = = - - {γ γ}_{i i} sign sign (({x x}_{ir ir} - - {α α}_{i i} + + \frac{{x x}_{ir ir}^{d d} | | {x x}_{ir ir}^{d d} | |}{{22 γ γ}_{i i}})) \end{matrix} - - - - - - ((1111))$

得到第i级滤波器单元的输出信号为估计变量x_ir及其导数其中γ_i＞0为常值，是中间变量；The output signal of the i-th filter unit is obtained as the estimated variable x _ir and its derivative Where γ _i >0 is a constant value, is an intermediate variable;

B2、第i级比较器1：第i级比较器1分别接收跟踪微分器的输出信号估计变量x_ir和预估器的输出信号接收到的信号经过下列变换B2, i-th level comparator 1: i-th level comparator 1 respectively receives the output signal of the tracking differentiator, the estimated variable x _ir and the output signal of the predictor The received signal undergoes the following transformation

${\overset{^^}{S S}}_{i i} = = {\overset{^^}{x x}}_{i i} - - {x x}_{ir ir} - - - - - - ((1212))$

得到第i级比较器1的输出信号 Get the output signal of comparator 1 in the i-th stage

B3、第i级比较器2：第i级比较器2分别接收由测量机构输出的被控系统第i级子系统的状态信号x_im和预估器的输出信号接收到的信号经过下列变换得到比较器2的输出端信号 B3. Comparator 2 of the i-th level: the comparator 2 of the i-th level respectively receives the state signal x _im of the i-th level subsystem of the controlled system output by the measurement institution and the output signal of the predictor The received signal undergoes the following transformation to obtain the output signal of comparator 2

${\overset{~ ~}{x x}}_{i i} = = {\overset{^^}{x x}}_{i i} - - {x x}_{im im} - - - - - - ((1313))$

其中x_im＝x_i+w_i为含有测量噪声的第i级子系统的状态信号，w_i为测量噪声；Where x _im = x _i + w _i is the status signal of the i-th subsystem containing measurement noise, and w _i is the measurement noise;

B4、第i级预估器：第i级预估器分别接收由测量机构输出的被控系统第i级子系统的状态信号x_im和第i+1级子系统的状态信号x_(i+1)m，以及逼近器的输出信号α_wi，接收到的信号经过下列变换得到预估器的输出信号 B4. The i-th level predictor: the i-th level predictor respectively receives the state signal x _im of the i-th level subsystem of the controlled system and the state signal x _{(i+ 1) m} , and the output signal α _wi of the approximator, the received signal undergoes the following transformation to obtain the output signal of the predictor

${\overset{\cdot &Center Dot;}{\overset{^^}{x x}}}_{11} = = {α α}_{wi wi} + + {x x}_{((i i + + 11)) m m} - - (({k k}_{i i} + + {κ κ}_{i i})) (({\overset{^^}{x x}}_{i i} - - {x x}_{im im})) - - - - - - ((1414))$

其中k_i＞0，κ_i＞0为常数；预估器的输出信号同时也是第i级子控制器的一个输出信号；Where k _i >0, κ _i >0 are constants; the output signal of the predictor It is also an output signal of the i-th sub-controller;

B5、第i级线性控制单元：第i级线性控制单元接收比较器1的输出信号经过下列比例控制：B5. The i-th level linear control unit: the i-th level linear control unit receives the output signal of the comparator 1 Controlled by the following proportions:

${α α}_{ki the ki} = = {- - k k}_{i i} {\overset{^^}{S S}}_{i i} - - - - - - ((1515))$

得到第i级线性控制单元的输出端信号α_ki；Obtain the output signal α _ki of the i-th linear control unit;

B6、第i级逼近器：第i级逼近器分别接收前i级子控制器的输出信号以及第i级比较器2的输出信号 B6. The i-th stage approximation device: the i-th stage approximation device respectively receives the output signal of the previous i-stage sub-controller and the output signal of comparator 2 of stage i

逼近器作用是对被控系统中的未知非线性进行在线估计，所述的逼近器采用神经网络逼近方法，将表示为的形式，其中是逼近误差，满足是正常数；W_i∈R^s为未知权值矩阵，并满足||W_i||_F≤W_i ^*，W_i ^*是正常数；为已知的激励函数矩阵，其形式为定义是W_i的估计，设计的更新率为The function of the approximator is to estimate the unknown nonlinearity in the controlled system Carry out online estimation, described approximator adopts neural network approximation method, will Expressed as in the form of is the approximation error, satisfying is a normal number; W _i ∈ R ^s is an unknown weight matrix, and satisfies ||W _i || _F ≤ _{W i} ^* , W _i ^* is a normal number; is a known activation function matrix, whose form is definition is the estimate of W _i , design The update rate of

其中Γ_Wi∈R，k_Wi∈R是正常数。where Γ _Wi ∈ R, k _Wi ∈ R are positive constants.

最后得到第i级逼近器的输出信号α_wi：Finally, the output signal α _wi of the i-th stage approximator is obtained:

B7、第i级低通滤波器：第i级低通滤波器接收逼近器的输出信号α_wi，经过以下变换：B7, the i-th stage low-pass filter: the i-th stage low-pass filter receives the output signal α _wi of the approximator, and undergoes the following transformation:

α_cwi＝C(s)α_wi (18)得到第i级低通滤波器的输出信号α_cwi；α _cwi =C(s)α _wi (18) obtains the output signal α _cwi of the i-level low-pass filter;

B8、第i级求和器：第i级求和器分别接收线性控制单元的输出信号α_ki、低通滤波器的输出信号α_cwi和跟踪微分器的输出信号估计变量的导数接收到的信号经过以下求和计算：B8. The i-th stage summer: the i-th stage summer respectively receives the output signal α _ki of the linear control unit, the output signal α _cwi of the low-pass filter and the derivative of the estimated variable of the output signal of the tracking differentiator The received signals are summed as follows:

${α α}_{((i i + + 11))} = = {α α}_{ki the ki} + + {α α}_{cwi cwi} + + {\overset{\cdot &Center Dot;}{x x}}_{ir ir} - - - - - - ((1919))$

得到第i级子控制器的另一输出端信号α_(i+1)；Obtain another output signal α _(i+1) of the i-th sub-controller;

重复步骤B1-B8递推设计得到第2级到第n-1级子控制器结构；Repeat steps B1-B8 for recursive design to obtain the sub-controller structure from level 2 to level n-1;

C、第n级子控制器设计：C. Design of the nth level sub-controller:

C1、第n级跟踪微分器：第n级跟踪微分器接收第n-1级子控制器的输出信号α_n，经过下列变换：C1. The nth level tracking differentiator: the nth level tracking differentiator receives the output signal α _n of the n-1th level sub-controller, and undergoes the following transformation:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{nr nr} = = {x x}_{nr nr}^{d d} \\ {\overset{\cdot &Center Dot;}{x x}}_{nr nr}^{d d} = = - - {γ γ}_{n no} sign sign (({x x}_{nr nr} - - {α α}_{n no} + + \frac{{x x}_{nr nr}^{d d} | | {x x}_{nr nr}^{d d} | |}{{22 γ γ}_{n no}})) \end{matrix} - - - - - - ((2020))$

得到第n级滤波器单元的输出信号为估计变量x_nr及其导数其中γ_n＞0为常值，是中间变量；The output signal of the nth stage filter unit is obtained as the estimated variable x _nr and its derivative Where γ _n > 0 is a constant value, is an intermediate variable;

C2、第n级比较器1：第n级比较器1分别接收跟踪微分器的输出信号估计变量x_nr和预估器的输出信号接收到的信号经过下列变换：C2, nth level comparator 1: the nth level comparator 1 respectively receives the output signal of the tracking differentiator, the estimated variable x _nr and the output signal of the predictor The received signal undergoes the following transformations:

${\overset{^^}{S S}}_{n no} = = {\overset{^^}{x x}}_{n no} - - {x x}_{nr nr} - - - - - - ((21 twenty one))$

得到第n级比较器1的输出信号 Get the output signal of the nth stage comparator 1

C3、第n级比较器2：第n级比较器2分别接收由测量机构输出的被控系统第n级子系统的状态信号x_nm和预估器的输出信号接收到的信号经过下列变换得到比较器2的输出端信号 C3, nth level comparator 2: nth level comparator 2 respectively receives the state signal x _nm of the nth level subsystem of the controlled system and the output signal of the predictor output by the measurement institution The received signal undergoes the following transformation to obtain the output signal of comparator 2

${\overset{~ ~}{x x}}_{n no} = = {\overset{^^}{x x}}_{n no} - - {x x}_{nm nm} - - - - - - ((22 twenty two))$

其中x_nm＝x_n+w_n为含有测量噪声的第n级子系统的状态信号，w_n为测量噪声；Where x _nm =x _n +w _n is the state signal of the nth subsystem containing measurement noise, and w _n is the measurement noise;

C4、第n级预估器：第n级预估器分别接收由测量机构输出的被控系统第n级子系统的状态信号x_nm、逼近器的输出信号α_wn和求和器的输出信号u，接收到的信号经过下列变换得到预估器的输出信号 C4. The nth level predictor: the nth level predictor respectively receives the state signal x _nm of the nth level subsystem of the controlled system output by the measurement institution, the output signal α _wn of the approximator and the output signal of the summer u, the received signal undergoes the following transformation to obtain the output signal of the predictor

${\overset{\cdot &Center Dot;}{\overset{^^}{x x}}}_{n no} = = {α α}_{wn wn} + + u u - - (({k k}_{n no} + + {κ κ}_{n no})) (({\overset{^^}{x x}}_{n no} - - {x x}_{nm nm})) - - - - - - ((23 twenty three))$

其中k_n＞0，κ_n＞0为常数；Where k _n >0, κ _n >0 are constants;

C5、第n级线性控制单元：第n级线性控制单元接收比较器1的输出信号经过下列比例控制：C5. The nth level linear control unit: the nth level linear control unit receives the output signal of the comparator 1 Controlled by the following proportions:

${α α}_{kn k n} = = {- - k k}_{n no} {\overset{^^}{S S}}_{n no} - - - - - - ((24 twenty four))$

得到第n级线性控制单元的输出信号α_kn；Obtain the output signal α _kn of the nth level linear control unit;

C6、第n级逼近器：第n级逼近器分别接收所有n级子控制器的输出信号相连和第n级比较器2的输出信号 C6. The nth level approximator: the nth level approximator receives the output signals of all n level sub-controllers respectively connected to the output signal of the nth stage comparator 2

与步骤B6类似，接收到的信号经过下列逼近器：Similar to step B6, the received signal goes through the following approximators:

得到第n级逼近器的输出信号α_wn；Obtain the output signal α _wn of the nth stage approximator;

C7、第n级低通滤波器：第n级低通滤波器接收逼近器的输出信号α_wn，经过以下变换：C7, the n-th stage low-pass filter: the n-th stage low-pass filter receives the output signal α _wn of the approximator, and undergoes the following transformation:

α_cwn＝C(s)α_wn (27)得到第n级低通滤波器的输出信号α_cwn；α _cwn =C(s) α _wn (27) obtains the output signal α _cwn of the nth stage low-pass filter;

C8、第n级求和器：第n级求和器单元分别接收线性控制单元的输出信号α_kn、低通滤波器的输出信号α_cwn和跟踪微分器单元的输出信号估计变量的导数接收到的信号经过以下求和计算：C8, nth stage summer: the nth stage summer unit respectively receives the output signal α _kn of the linear control unit, the output signal α _cwn of the low-pass filter and the derivative of the estimated variable of the output signal of the tracking differentiator unit The received signals are summed as follows:

$u u = = {α α}_{kn k n} + + {α α}_{cwn cwn} + + {\overset{\cdot &Center Dot;}{x x}}_{nr nr} - - - - - - ((2828))$

得到被控系统总的控制输入u；选择合适参数，控制输入u能够使得被控系统的输出y跟踪给定的外部参考信号y_r。Get the total control input u of the controlled system; choose the appropriate parameters, the control input u can make the output y of the controlled system track the given external reference signal y _r .

与现有技术相比，本发明的有益效果是：Compared with prior art, the beneficial effect of the present invention is:

第一，本发明在控制器设计中引入预估器，对被控系统不确定性的在线学习不再基于跟踪误差，而是基于预估误差，通过选取κ_i控制预估误差的收敛速度，能够克服初始阶段跟踪误差较大对控制系统暂态性能的影响，从而使得被控系统具有较好的暂态输入，控制信号不容易陷入饱和。First, the present invention introduces an estimator in the design of the controller, and the online learning of the uncertainty of the controlled system is no longer based on the tracking error, but based on the estimation error. By selecting κ _i to control the convergence speed of the estimation error, It can overcome the influence of large tracking error on the transient performance of the control system in the initial stage, so that the controlled system has a better transient input, and the control signal is not easy to fall into saturation.

第二，本发明在控制器设计中引入低通滤波器，能够消除由于自适应参数较大引起的输入控制信号的高频振荡问题，因此可以有效地对控制信号的高频分量进行滤波，有利于保证控制信号在执行器的执行频带范围之内。Second, the present invention introduces a low-pass filter in the design of the controller, which can eliminate the high-frequency oscillation problem of the input control signal caused by the large adaptive parameter, so the high-frequency component of the control signal can be effectively filtered, and there is It is beneficial to ensure that the control signal is within the execution frequency band of the actuator.

第三，本发明采用微分跟踪器代替传统动态面控制中的一阶滤波器，在能够克服反步法中的“计算复杂性”问题的同时，还能够保证对虚拟控制信号的有限时间估计，提高估计的精确性，并且能够克服一阶滤波器需要通过增加带宽来提高跟踪性能的缺点。Third, the present invention uses a differential tracker to replace the first-order filter in the traditional dynamic surface control, while being able to overcome the "computational complexity" problem in the backstepping method, it can also ensure the finite time estimation of the virtual control signal, Improve the accuracy of estimation, and can overcome the disadvantage that the first-order filter needs to increase the bandwidth to improve the tracking performance.

第四，本发明采用预估的状态进行反馈，消除了测量噪声对控制系统的干扰，使得该控制器设计方法有利于实际的工程应用。Fourth, the present invention uses the estimated state for feedback, which eliminates the interference of measurement noise to the control system, making the controller design method beneficial to practical engineering applications.

附图说明Description of drawings

本发明共有附图4张，其中：The present invention has 4 accompanying drawings, wherein:

图1是本发明的神经网络自适应动态面控制器结构示意图。Fig. 1 is a structural schematic diagram of the neural network adaptive dynamic surface controller of the present invention.

图2是本发明的神经网络自适应动态面控制器(PNDSC)与传统神经网络自适应动态面控制器(NDSC)输出响应的比较。Fig. 2 is a comparison of the output response of the neural network adaptive dynamic surface controller (PNDSC) of the present invention and the traditional neural network adaptive dynamic surface controller (NDSC).

图3是本发明的神经网络自适应动态面控制器与传统神经网络自适应动态面控制器的控制输入信号的比较。Fig. 3 is a comparison of control input signals between the neural network adaptive dynamic surface controller of the present invention and the traditional neural network adaptive dynamic surface controller.

图4是本发明的神经网络自适应动态面控制器与传统神经网络自适应动态面控制器的控制输入信号的幅度谱的比较。Fig. 4 is a comparison of the magnitude spectrum of the control input signal of the neural network adaptive dynamic surface controller of the present invention and the traditional neural network adaptive dynamic surface controller.

具体实施方式Detailed ways

下面结合附图以二阶不确定严反馈非线性系统为例对本发明进行进一步说明。The present invention will be further described below by taking the second-order uncertain strict feedback nonlinear system as an example in conjunction with the accompanying drawings.

考虑下列二阶不确定严反馈非线性系统Consider the following second-order uncertain strict-feedback nonlinear system

$\{\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = {f f}_{11} (({\overset{&OverBar; &OverBar;}{x x}}_{11})) + + {x x}_{22} \\ {\overset{\cdot \cdot}{x x}}_{22} = = {f f}_{22} (({\overset{&OverBar; &OverBar;}{x x}}_{22})) + + u u \\ y the y = = {x x}_{11} \end{matrix} - - - - - - ((2929))$

为仿真需要，设置与为For simulation needs, set and for

$\{\begin{matrix} {f f}_{11} (({\overset{&OverBar; &OverBar;}{x x}}_{11})) = = {22 x x}_{11}^{22} sin sin (({x x}_{11})) \\ {f f}_{22} (({\overset{&OverBar; &OverBar;}{x x}}_{22})) = = {x x}_{11}^{22} + + {x x}_{11} {x x}_{22} + + {x x}_{22} cos cos {x x}_{11} \end{matrix} - - - - - - ((3030))$

针对该系统，根据图1可以设计神经网络自适应动态面控制器如下：For this system, according to Figure 1, the neural network adaptive dynamic surface controller can be designed as follows:

第1级子控制器：Level 1 sub-controller:

第2级子控制器：Level 2 sub-controller:

选择如下控制器的参数：Select the parameters of the controller as follows:

k₁＝k₂＝5,γ₁＝γ₂＝0.005,κ₁＝κ₂＝100,Γ_W1＝Γ_W2＝10000,k_W1＝k_W2＝0.001；k ₁ =k ₂ =5, γ ₁ =γ ₂ =0.005, κ ₁ =κ ₂ =100, Γ _W1 =Γ _W2 =10000, k _W1 =k _W2 =0.001;

如采用传统神经网络自适应动态面控制设计方法，控制器结构如下：If the traditional neural network adaptive dynamic surface control design method is adopted, the controller structure is as follows:

第1级子控制器：Level 1 sub-controller:

第2级子控制器：Level 2 sub-controller:

控制器参数的选择与本发明的神经网络自适应动态面控制器中一致。The selection of controller parameters is consistent with the neural network adaptive dynamic surface controller of the present invention.

仿真结果如图2-4所示。图2所示的是控制系统的输出响应，从图中看出，本发明的神经网络自适应动态面控制器(PNDSC)与传统神经网络自适应动态面控制器(NDSC)均能使被控系统的输出有效跟踪参考信号y_r。图3所示的是本发明的神经网络自适应动态面控制器与传统神经网络自适应动态面控制器控制输入信号的比较示意图，图中能够看出，基于本发明的神经网络自适应动态面控制器的控制输入信号更加平滑，高频振荡明显减少。图4所示的是本发明的神经网络自适应动态面控制器与传统神经网络自适应动态面控制器控制输入信号在频域下的幅度谱的比较，能够明显看出，本发明的神经网络自适应动态面控制器显著改善了控制器的控制输入信号。综上所述，本发明提出的神经网络自适应动态面控制算法的控制性能明显优于现有的传统神经网络自适应动态面控制算法。The simulation results are shown in Figure 2-4. What Fig. 2 shows is the output response of control system, finds out from the figure, neural network adaptive dynamic surface controller (PNDSC) of the present invention and traditional neural network adaptive dynamic surface controller (NDSC) all can make controlled The output of the system effectively tracks the reference signal y _r . What Fig. 3 shows is the comparison schematic diagram of neural network adaptive dynamic surface controller of the present invention and traditional neural network adaptive dynamic surface controller control input signal, can find out in the figure, based on neural network adaptive dynamic surface of the present invention The control input signal of the controller is smoother and the high frequency oscillation is significantly reduced. Shown in Fig. 4 is the comparison of the amplitude spectrum of the control input signal of the neural network adaptive dynamic surface controller of the present invention and the traditional neural network adaptive dynamic surface controller under the frequency domain, it can be clearly seen that the neural network of the present invention The adaptive dynamic surface controller significantly improves the control input signal of the controller. In summary, the control performance of the neural network adaptive dynamic surface control algorithm proposed by the present invention is obviously better than the existing traditional neural network adaptive dynamic surface control algorithm.

Claims

1. A neural network adaptive dynamic surface controller structure is made up of n-level sub-controllers; it is characterized in that:

The input terminal of the first-level sub-controller is connected with the external reference signal y _r , and also connected with the output terminals x _1m , x _2m of the measuring mechanism, the output terminal α ₂ of the first-level sub-controller is connected with the output terminal α2 of the second-level sub-controller The input terminal is connected to the other output terminal of the first stage sub-controller respectively connected to the input terminals of the first-level to n-level sub-controllers;

The input terminal of the i-th sub-controller is connected to the input terminal α _i of the i-1th sub-controller, and is connected to the output terminal of the previous i-level sub-controller is connected to the output terminals x _im , x _(i+1)m of the measurement mechanism, the output terminal α _i of the i-th sub-controller is connected to the input terminal of the i+1-th sub-controller, and the i-th sub-controller The other output of the controller respectively connected to the input terminals of the i-th to n-th level sub-controllers;

By analogy, the input terminal of the nth level sub-controller is connected to the output terminal α _n of the n-1th level sub-controller, and is respectively connected to the output terminals of all n-level sub-controllers connected, and also connected with the output terminal x _nm of the measuring mechanism, and the output terminal u of the nth sub-controller is connected with the input terminal of the controlled system;

Described first stage sub-controller is made of tracking differentiator, comparator 1, comparator 2, predictor, linear control unit, approximator, low-pass filter and summator, and the input end of tracking differentiator and The external reference signal y _r is connected, and the output end of the tracking differentiator is respectively connected with the input end of the comparator 1 and the summer; the other input end of the comparator 1 is connected with the output end of the predictor, and the output end of the comparator 1 It is connected with the input end of the linear control unit; the two input ends of the comparator 2 are respectively connected with the output ends of the measuring mechanism and the predictor; the input ends of the predictor are respectively connected with the output ends of the measuring mechanism and the approximator, The output end of the device is an output end of the first stage sub-controller; the input end of the approximator is connected with the output end of comparator 2, and the other input end is connected with the output end of the predictor, and the output end of the approximator is also connected with The input end of the low-pass filter is connected; the output end of the low-pass filter is connected with the input end of the summer; the other input end of the summer is connected with the output end of the linear control unit, and the output end of the summer is the first The other output terminal of the level 1 sub-controller;

The i-th level sub-controller is composed of tracking differentiator, comparator 1, comparator 2, predictor, linear control unit, approximator, low-pass filter and summer, and the input terminal of tracking differentiator is connected with The output terminals of the i-1th sub-controller are connected, and the output terminals of the tracking differentiator are respectively connected with the input terminals of the comparator 1 and the summer; the other input terminal of the comparator 1 is connected with the output terminal of the predictor, The output terminal of comparator 1 is connected with the input terminal of the linear control unit; the two input terminals of comparator 2 are respectively connected with the output terminals of the measuring mechanism and the predictor; the input terminals of the predictor are respectively connected with the measuring mechanism and the approximator The output terminal is connected, and the output terminal of the predictor is an output terminal of the i-th sub-controller; the input terminal of the approximator is connected with the output terminal of the comparator 2, and the other input terminal is connected with the output terminal of the previous i-level sub-controller The output of the approximator is also connected to the input of the low-pass filter; the output of the low-pass filter is connected to the input of the summer; the other input of the summer is connected to the output of the linear control unit , the output terminal of the summer is another output terminal of the i-th sub-controller;

The nth stage sub-controller is composed of tracking differentiator, comparator 1, comparator 2, predictor, linear control unit, approximator, low-pass filter and summator, and the input terminal of tracking differentiator and The output terminals of the n-1th sub-controller are connected, and the output terminals of the tracking differentiator are respectively connected with the input terminals of the comparator 1 and the summer; the other input terminal of the comparator 1 is connected with the output terminal of the predictor, The output terminal of comparator 1 is connected with the input terminal of the linear control unit; the two input terminals of comparator 2 are respectively connected with the output terminals of the measuring mechanism and the estimator; the input terminals of the estimator are respectively connected with the measuring mechanism, the approximator and the The output end of the summation unit is connected; the input end of the approximator is connected with the output end of the comparator 2, and the other input end is connected with the output ends of all n-level sub-controllers, and the output end of the approximator is also connected with the output end of the low-pass filter The input terminal is connected; the output terminal of the low-pass filter is connected with the input terminal of the summer; the other input terminal of the summer is connected with the output terminal of the linear control unit, and the output terminal of the summer is the nth level sub-controller The output terminal is connected with the input terminal of the controlled system;

The controlled system is described by the following n-order uncertain strict feedback nonlinear system form:

If R is used to represent the set of real numbers, then x _i ∈ R represents the state of the i-th subsystem of the controlled system; u ∈ R represents the control input of the controlled system; is an unknown nonlinear continuous function, y∈R is the output of the controlled system.

2. A neural network adaptive dynamic surface controller design method is characterized in that: comprise the following steps:

A. Level 1 sub-controller design:

A1. The first-stage tracking differentiator: The first-stage tracking differentiator receives the external reference signal y _r and undergoes the following transformation:

\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11 r r} = = {x x}_{11 r r}^{d d} \\ {\overset{\cdot \cdot}{x x}}_{11 r r}^{d d} = = - - {γ γ}_{11} sign sign (({x x}_{11 r r} - - {y the y}_{r r} + + \frac{{x x}_{11 r r}^{d d} | | {x x}_{11 r r}^{d d} | |}{22 {γ γ}_{11}})) \end{matrix} - - - - - - ((22))

The output signal of the first-level tracking differentiator is obtained as the estimated variable x _1r and its derivative Where γ ₁ >0 is a constant value, is an intermediate variable, and sign(·) represents a sign function;

A2. The first stage comparator 1: The first stage comparator 1 respectively receives the output signal of the tracking differentiator, the estimated variable x _1r and the output signal of the predictor The received signal undergoes the following transformations:

{\overset{^^}{S S}}_{11} = = {\overset{^^}{x x}}_{11} - - {x x}_{11 r r} - - - - - - ((33))

Get the output signal of the 1st stage comparator 1

A3. First-level comparator 2: The first-level comparator 2 respectively receives the status signal x _1m of the first-level subsystem of the controlled system and the output signal of the predictor output by the measurement institution The received signal undergoes the following transformation to obtain the output signal of comparator 2

{\overset{~ ~}{x x}}_{11} = = {\overset{^^}{x x}}_{11} - - {x x}_{11 m m} - - - - - - ((44))

Where x _1m =x ₁ +w ₁ is the status signal of the first-level subsystem containing measurement noise, and w ₁ is the measurement noise;

A4. Level 1 predictor: The level 1 predictor respectively receives the state signal x _1m of the first level subsystem of the controlled system and the state signal x _2m of the second level subsystem output by the measurement institution, and the approximator The output signal α _w1 of the received signal undergoes the following transformation to obtain the output signal of the predictor

{\overset{\overset{\cdot &Center Dot;}{^^}}{x x}}_{11} = = {α α}_{w w 11} + + {x x}_{22 m m} - - (({k k}_{11} + + {κ κ}_{11})) (({\overset{^^}{x x}}_{11} - - {x x}_{11 m m})) - - - - - - ((55))

Where k ₁ >0, κ ₁ >0 are constants; the output signal of the predictor It is also an output signal of the first-level sub-controller;

A5. The first-level linear control unit: the first-level linear control unit receives the output signal of comparator 1 Controlled by the following proportions:

{α α}_{k k 11} = = - - {k k}_{11} {\overset{^^}{S S}}_{11} - - - - - - ((66))

Obtain the output signal α _k1 of the first-level linear control unit;

A6. The first-stage approximator: the first-stage approximator receives the output signal of the predictor respectively and the output signal of Comparator 2

The role of the approximator is to estimate the unknown nonlinearity in the controlled system Carry out online estimation, described approximator adopts neural network approximation method, will Expressed as in the form of is the approximation error, satisfying is a normal number; W ₁ ∈ R ^s is an unknown weight matrix, and satisfies is a normal number; is a known activation function matrix, whose form is definition is an estimate of _W1 , the design The update rate is:

Where Γ _W1 ∈ R, k _W1 ∈ R is a positive constant;

Finally, the output signal α _w1 of the first-stage approximator is obtained:

A7. The first-stage low-pass filter: the first-stage low-pass filter receives the output signal α _w1 of the approximator, and undergoes the following transformation:

α _cw1 =C(s)α _w1 (9)

Obtain the output signal α _cw1 of the first-stage low-pass filter, where C(s) represents the filter;

A8. The first-stage summer: The first-stage summer respectively receives the output signal α _k1 of the linear control unit, the output signal α _cw1 of the low-pass filter, and the derivative of the estimated variable of the output signal of the tracking differentiator The received signals are summed as follows:

{α α}_{22} = = {α α}_{k k 11} + + {α α}_{cw cw 11} + + {\overset{\cdot &Center Dot;}{x x}}_{11 r r} - - - - - - ((1010))

Obtain another output terminal signal α ₂ of the first-level sub-controller;

B. The i-level sub-controller design:

B1. The i-level tracking differentiator: the i-level tracking differentiator receives the output signal α _i of the i-1-th sub-controller, and undergoes the following transformation:

\{\begin{matrix} {\overset{\cdot \cdot}{x x}}_{ir ir} = = {x x}_{ir ir}^{d d} \\ {\overset{\cdot &Center Dot;}{x x}}_{ir ir}^{d d} = = - - {γ γ}_{i i} sign sign (({x x}_{ir ir} - - {α α}_{i i} + + \frac{{x x}_{ir ir}^{d d} | | {x x}_{ir ir}^{d d} | |}{22 {γ γ}_{i i}})) \end{matrix} - - - - - - ((1111))

The output signal of the i-th filter unit is obtained as the estimated variable x _ir and its derivative Where γ _i >0 is a constant value, is an intermediate variable;

B2, i-th level comparator 1: i-th level comparator 1 respectively receives the output signal of the tracking differentiator, the estimated variable x _ir and the output signal of the predictor The received signal undergoes the following transformation

{\overset{^^}{S S}}_{i i} = = {\overset{^^}{x x}}_{i i} - - {x x}_{ir ir} - - - - - - ((1212))

Get the output signal of comparator 1 in the i-th stage

B3. Comparator 2 of the i-th level: the comparator 2 of the i-th level respectively receives the state signal x _im of the i-th level subsystem of the controlled system output by the measurement institution and the output signal of the predictor The received signal undergoes the following transformation to obtain the output signal of comparator 2

{\overset{~ ~}{x x}}_{i i} = = {\overset{^^}{x x}}_{i i} - - {x x}_{im im} - - - - - - ((1313))

Where x _im = x _i + w _i is the status signal of the i-th subsystem containing measurement noise, and w _i is the measurement noise;

B4. The i-th level predictor: the i-th level predictor respectively receives the state signal x _im of the i-th level subsystem of the controlled system and the state signal x _{(i+ 1) m} , and the output signal α _wi of the approximator, the received signal undergoes the following transformation to obtain the output signal of the predictor

{\overset{\overset{\cdot &Center Dot;}{^^}}{x x}}_{i i} = = {α α}_{wi wi} + + {x x}_{((i i + + 11)) m m} - - (({k k}_{i i} + + {κ κ}_{i i})) (({\overset{^^}{x x}}_{i i} - - {x x}_{im im})) - - - - - - ((1414))

Where k _i >0, κ _i >0 are constants; the output signal of the predictor It is also an output signal of the i-th sub-controller;

B5. The i-th level linear control unit: the i-th level linear control unit receives the output signal of the comparator 1 Controlled by the following proportions:

{α α}_{ki the ki} = = - - {k k}_{i i} {\overset{^^}{S S}}_{i i} - - - - - - ((1515))

Obtain the output signal α _ki of the i-th linear control unit;

B6. The i-th stage approximation device: the i-th stage approximation device respectively receives the output signal of the previous i-stage sub-controller and the output signal of comparator 2 of stage i

The function of the approximator is to estimate the unknown nonlinearity in the controlled system Carry out online estimation, described approximator adopts neural network approximation method, will Expressed as in the form of is the approximation error, satisfying is a normal constant; W _i ∈ R ^s is an unknown weight matrix, and satisfies is a normal number; is a known activation function matrix, whose form is definition is the estimate of W _i , design The update rate of

where Γ _Wi ∈ R, k _Wi ∈ R are positive constants;

Finally, the output signal α _wi of the i-th stage approximator is obtained:

B7, the i-th stage low-pass filter: the i-th stage low-pass filter receives the output signal α _wi of the approximator, and undergoes the following transformation:

α _cwi =C(s)α _wi (18)

Obtain the output signal α _cwi of the i-th stage low-pass filter;

B8. The i-th stage summer: the i-th stage summer respectively receives the output signal α _ki of the linear control unit, the output signal α _cwi of the low-pass filter and the derivative of the estimated variable of the output signal of the tracking differentiator The received signals are summed as follows:

{α α}_{((i i + + 11))} = = {α α}_{ki the ki} + + {α α}_{cwi cwi} + + {\overset{\cdot &Center Dot;}{x x}}_{ir ir} - - - - - - ((1919))

Obtain another output signal α _(i+1) of the i-th sub-controller;

Repeat steps B1-B8 for recursive design to obtain the sub-controller structure from level 2 to level n-1;

C. Design of the nth level sub-controller:

C1. The nth level tracking differentiator: the nth level tracking differentiator receives the output signal α _n of the n-1th level sub-controller, and undergoes the following transformation:

\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{nr nr} = = {x x}_{nr nr}^{d d} \\ {\overset{\cdot &Center Dot;}{x x}}_{nr nr}^{d d} = = - - {γ γ}_{n no} sign sign (({x x}_{nr nr} - - {α α}_{n no} + + \frac{{x x}_{nr nr}^{d d} | | {x x}_{nr nr}^{d d} | |}{22 {γ γ}_{n no}})) \end{matrix} - - - - - - ((2020))

The output signal of the nth stage filter unit is obtained as the estimated variable x _nr and its derivative Where γ _n > 0 is a constant value, is an intermediate variable;

C2, nth level comparator 1: the nth level comparator 1 respectively receives the output signal of the tracking differentiator, the estimated variable x _nr and the output signal of the predictor The received signal undergoes the following transformations:

{\overset{^^}{S S}}_{n no} = = {\overset{^^}{x x}}_{n no} - - {x x}_{nr nr} - - - - - - ((21 twenty one))

Get the output signal of the nth stage comparator 1

C3, nth level comparator 2: nth level comparator 2 respectively receives the state signal x _nm of the nth level subsystem of the controlled system and the output signal of the predictor output by the measurement institution The received signal undergoes the following transformation to obtain the output signal of comparator 2

{\overset{~ ~}{x x}}_{n no} = = {\overset{^^}{x x}}_{n no} - - {x x}_{nm nm} - - - - - - ((22 twenty two))

Where x _nm =x _n +w _n is the state signal of the nth subsystem containing measurement noise, and w _n is the measurement noise;

C4. The nth level predictor: the nth level predictor respectively receives the state signal x _nm of the nth level subsystem of the controlled system output by the measurement institution, the output signal α _wn of the approximator and the output signal of the summer u, the received signal undergoes the following transformation to obtain the output signal of the predictor

{\overset{\overset{\cdot \cdot}{^^}}{x x}}_{n no} = = {α α}_{wn wn} + + u u - - (({k k}_{n no} + + {κ κ}_{n no})) (({\overset{^^}{x x}}_{n no} - - {x x}_{nm nm})) - - - - - - ((23 twenty three))

Where k _n >0, κ _n >0 are constants;

C5. The nth level linear control unit: the nth level linear control unit receives the output signal of the comparator 1 Controlled by the following proportions:

{α α}_{kn k n} = = - - {k k}_{n no} {\overset{^^}{S S}}_{n no} - - - - - - ((24 twenty four))

Obtain the output signal α _kn of the nth level linear control unit;

C6. The nth level approximator: the nth level approximator receives the output signals of all n level sub-controllers respectively connected to the output signal of the nth stage comparator 2

Similar to step B6, the received signal goes through the following approximators:

Obtain the output signal α _wn of the nth stage approximator;

C7, the n-th stage low-pass filter: the n-th stage low-pass filter receives the output signal α _wn of the approximator, and undergoes the following transformation:

α _cwn =C(s)α _wn (27)

Obtain the output signal α _cwn of the nth stage low-pass filter;

C8, nth stage summer: the nth stage summer unit respectively receives the output signal α _kn of the linear control unit, the output signal α _cwn of the low-pass filter and the derivative of the estimated variable of the output signal of the tracking differentiator unit The received signals are summed as follows:

u u = = {α α}_{kn k n} + + {α α}_{cwn cwn} + + {\overset{\cdot &Center Dot;}{x x}}_{nr nr} - - - - - - ((2828))

Get the total control input u of the controlled system; choose the appropriate parameters, the control input u can make the output y of the controlled system track the given external reference signal y _r .