CN103116357B

CN103116357B - A kind of sliding-mode control with anti-interference fault freedom

Info

Publication number: CN103116357B
Application number: CN201310081166.6A
Authority: CN
Inventors: 郭雷; 雷燕婕; 乔建忠; 张培喜; 陈阳
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2013-03-14
Filing date: 2013-03-14
Publication date: 2016-05-11
Anticipated expiration: 2033-03-14
Also published as: CN103116357A

Abstract

A sliding mode control method with anti-disturbance and fault-tolerant performance. A sliding-mode controller with anti-disturbance and fault-tolerant performance is designed for systems with faults and disturbances. First, consider the faults and multi-source disturbances in the system to establish the system dynamic model; secondly, design fault diagnosis observer and disturbance observer to estimate the fault and modelable disturbance in the system; thirdly, solve the gain matrix of disturbance observer and fault diagnosis observer; then, design sliding mode controller, use The estimated value of the fault and disturbance compensates for the fault and disturbance respectively; finally, the stability of the controller is analyzed, and the sliding mode gain value is determined under the premise of system input saturation; this method ensures the anti-disturbance and fault-tolerant performance of the system, and is The non-modeling interference is robust, suitable for multi-source interference systems with limited input, improves the chattering phenomenon of sliding mode control, and can be used in attitude control systems in the fields of aviation, aerospace and deep space exploration.

Description

A sliding mode control method with anti-disturbance and fault-tolerant performance

技术领域technical field

本发明涉及一种具有抗干扰容错性能的滑模控制方法，该方法可用于输入受限系统的抗干扰容错控制，如受飞轮最大转速限制输入力矩的卫星、飞机等航空航天及深空探测领域的姿态控制子系统。The invention relates to a sliding mode control method with anti-interference and fault-tolerant performance, which can be used for anti-interference and fault-tolerant control of input-limited systems, such as satellites, aircrafts, and other aerospace and deep space exploration fields whose input torque is limited by the maximum rotational speed of a flywheel attitude control subsystem.

背景技术Background technique

随着航天器任务的复杂化，对姿态控制精度的要求也越来越高，航天器的高精度姿态控制成为国内外研究热点。航天器运行的空间环境复杂，受到外界及内部的多源干扰及未建模动态，且姿态控制系统故障发生概率较高。敏感器、飞轮等的故障会导致任务中断甚至失效，为提高系统的可靠性，必须采用精确地故障诊断与容错控制方法。同时，未建模动态、未知参数、随机干扰与其他等价干扰变量等多种因素导致飞行器姿态控制系统建模不精确，造成控制精度下降甚至失稳，航天器抗干扰姿态控制方法非常重要。另外，飞轮受到最大转速的限制，产生的输入力矩受限，系统存在输入饱和的问题。输入饱和会影响系统的控制性能，容易造成系统失稳，必须在控制器的设计过程中加以考虑。With the complexity of spacecraft missions, the requirements for attitude control accuracy are getting higher and higher, and the high-precision attitude control of spacecraft has become a research hotspot at home and abroad. The space environment in which the spacecraft operates is complex, subject to external and internal multi-source interference and unmodeled dynamics, and the attitude control system has a high probability of failure. Faults of sensors, flywheels, etc. will lead to task interruption or even failure. In order to improve the reliability of the system, accurate fault diagnosis and fault-tolerant control methods must be adopted. At the same time, various factors such as unmodeled dynamics, unknown parameters, random disturbances and other equivalent disturbance variables lead to inaccurate modeling of the aircraft attitude control system, resulting in a decrease in control accuracy or even instability. The spacecraft anti-interference attitude control method is very important. In addition, the flywheel is limited by the maximum rotational speed, the input torque generated is limited, and the system has the problem of input saturation. Input saturation will affect the control performance of the system and easily cause system instability, so it must be considered in the design process of the controller.

针对上述问题，国内外学者提出了很多有效的方法。当系统中存在只存在导数有界时变故障的情况下，故障诊断方法分为基于动态数学模型的方法、基于信号处理的方法和基于知识的方法，其中基于观测器的方法、基于神经网络的方法以及小波变换的研究非常广泛。系统中只存在可建模干扰的情况下，基于干扰观测器的控制可以对干扰估计并抵消，优点为结构简单，针对系统的不同性能要求可以结合不同的控制方法。以上方法都不能应用于存在故障、干扰和输入饱和限制的系统控制中。In response to the above problems, domestic and foreign scholars have proposed many effective methods. When there are only time-varying faults with bounded derivatives in the system, fault diagnosis methods are divided into methods based on dynamic mathematical models, methods based on signal processing and methods based on knowledge, among which methods based on observers, methods based on neural networks The research of method and wavelet transform is very extensive. When there are only modelable disturbances in the system, the control based on the disturbance observer can estimate and cancel the disturbance. The advantage is that the structure is simple, and different control methods can be combined for different performance requirements of the system. None of the above methods can be applied to the control of systems with faults, disturbances and input saturation limitations.

近年来，滑模控制因其所具有的优良特性而受到越来越高的重视，该方法对参数变化和扰动不敏感，结构简单，适用于卫星姿态控制系统的控制。由于姿态控制系统中的故障和干扰影响，滑模控制方法极易发生抖振现象，此前有学者设计干扰观测器抵消干扰的影响，改善滑模控制的抖振问题，实现高可靠性和高精度姿态控制，但没有考虑到故障。In recent years, sliding mode control has received more and more attention because of its excellent characteristics. This method is insensitive to parameter changes and disturbances, and has a simple structure, which is suitable for the control of satellite attitude control systems. Due to the influence of faults and disturbances in the attitude control system, the sliding mode control method is prone to chattering. Previously, some scholars designed a disturbance observer to offset the influence of disturbances, improve the chattering problem of sliding mode control, and achieve high reliability and precision. Attitude control, but failures are not taken into account.

发明内容Contents of the invention

本发明的技术解决问题是：针对输入饱和的多源干扰系统，提出一种具有故障补偿与干扰抵消和抑制性能的滑模控制方法。通过设计故障诊断观测器与干扰观测器，对系统中的故障与干扰进行估计与抵消，设计滑模控制器对参数不确定和扰动具有鲁棒性。The technical solution of the invention is to propose a sliding mode control method with fault compensation, interference cancellation and suppression performance for the input saturated multi-source interference system. By designing a fault diagnosis observer and a disturbance observer, the faults and disturbances in the system are estimated and offset, and the sliding mode controller is designed to be robust to parameter uncertainties and disturbances.

本发明的技术解决方案为：一种具有抗干扰容错性能的滑模控制方法，其特征在于包括以下步骤：The technical solution of the present invention is: a sliding mode control method with anti-interference and fault-tolerant performance, which is characterized in that it comprises the following steps:

首先，考虑系统中的故障和多源干扰，建立系统的动力学模型；其次，设计故障诊断观测器和干扰观测器估计系统中的故障和可建模干扰；再次，求解干扰观测器与故障诊断观测器的增益矩阵；然后，设计滑模控制器，将干扰和故障估计值代入控制器中补偿等价干扰和故障；最后，分析控制器稳定性，在系统输入饱和的前提下求解滑模增益；具体步骤如下：First, consider the faults and multi-source disturbances in the system, and establish a dynamic model of the system; second, design a fault diagnosis observer and a disturbance observer to estimate the faults and modelable disturbances in the system; third, solve the disturbance observer and fault diagnosis The gain matrix of the observer; then, design the sliding mode controller, and substitute the disturbance and fault estimation values into the controller to compensate for the equivalent disturbance and fault; finally, analyze the stability of the controller, and solve the sliding mode gain under the premise of system input saturation ;Specific steps are as follows:

第一步，考虑系统中的故障和多源干扰，建立系统的动力学模型The first step is to establish a dynamic model of the system considering faults and multi-source disturbances in the system

搭建包含故障与干扰的系统动力学模型，如下所示：Build a system dynamics model including faults and disturbances, as follows:

$\{\begin{matrix} \overset{. .}{{x x}_{11}} ((t t)) = = {x x}_{22} ((t t)) \\ \overset{. .}{{x x}_{22}} ((t t)) = = {x x}_{33} ((t t)) \\ . . \\ . . \\ . . \\ \overset{. .}{{x x}_{n no}} ((t t)) = = - - {a a}_{00} {x x}_{n no} ((t t)) - - \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; - - {a a}_{n no - - 11} {x x}_{11} ((t t)) + + {b b}_{11} ((u u ((t t)) + + {d d}_{11} ((t t)) + + F f ((t t)))) + + {b b}_{22} {d d}_{22} ((t t)) \end{matrix}$

其中，x₁(t)，x₂(t)，…，x_n(t)为系统状态，n≥2为正整数，u(t)为控制输入，F(t)为变化率有界的时变故障，d₁(t)为可建模干扰，d₂(t)为不可建模随机干扰。a₀、a₁、…a_n-1与b₁、b₂均为系统内部参数。d₁(t)可由如下干扰模型∑₁表示：Among them, x ₁ (t), x ₂ (t), ..., x _n (t) is the system state, n≥2 is a positive integer, u(t) is the control input, F(t) is the bounded rate of change For time-varying faults, d ₁ (t) is a modelable disturbance, and d ₂ (t) is an unmodelable random disturbance. a ₀ , a ₁ , ... a _n-1 and b ₁ , b ₂ are all system internal parameters. d ₁ (t) can be represented by the following interference model Σ ₁ :

${Σ Σ}_{11} : : \{\begin{matrix} {d d}_{11} ((t t)) = = Vw Vw ((t t)) \\ \overset{\cdot &Center Dot;}{w w} ((t t)) = = Ww w ((t t)) + + {B B}_{33} δ δ ((t t)) \end{matrix}$

其中，w(t)为可建模干扰模型的状态变量，V为可建模干扰模型的输出矩阵，W表示可建模干扰模型的系统阵，B₃为不可建模随机干扰的增益阵，δ(t)为能量有界的不可建模随机干扰。Among them, w(t) is the state variable of the modelable interference model, V is the output matrix of the modelable interference model, W represents the system matrix of the modelable interference model, _B3 is the gain matrix of the non-modelable random interference, δ(t) is an unmodelable random disturbance with bounded energy.

选取状态变量X(t)＝[x₁(t)x₂(t)......x_n(t)]^T，写成状态空间表达式如下：Select the state variable X(t)＝[x ₁ (t)x ₂ (t)......x _n (t)] ^T , and write the state space expression as follows:

$\overset{\cdot \cdot}{X x} ((t t)) = = AX AX ((t t)) + + {B B}_{11} ((u u ((t t)) + + {d d}_{11} ((t t)) + + F f ((t t)))) + + {B B}_{22} {d d}_{22} ((t t))$

其中，X(t)为系统状态变量，A为系统阵，B₁为输入矩阵，B₂为不可建模随机干扰的增益阵。Among them, X(t) is the system state variable, A is the system matrix, B ₁ is the input matrix, and B ₂ is the gain matrix that cannot model random interference.

$B_{1} = {[\begin{matrix} 0 \\ 0 \\ 0 \\ \cdot \\ \cdot \\ \cdot \\ b_{1} \end{matrix}]}_{n \times 1},$ $B_{2} = {[\begin{matrix} 0 \\ 0 \\ 0 \\ . \\ . \\ . \\ b_{2} \end{matrix}]}_{n \times 1}$ $B_{1} = {[\begin{matrix} 0 \\ 0 \\ 0 \\ \cdot \\ \cdot \\ \cdot \\ b_{1} \end{matrix}]}_{no \times 1},$ $B_{2} = {[\begin{matrix} 0 \\ 0 \\ 0 \\ . \\ . \\ . \\ b_{2} \end{matrix}]}_{no \times 1}$

第二步，设计故障诊断观测器与干扰观测器分别估计故障与可建模干扰针对系统中的时变故障F(t)设计故障诊断观测器为：The second step is to design a fault diagnosis observer and a disturbance observer to estimate the fault and modelable disturbance respectively. For the time-varying fault F(t) in the system, the fault diagnosis observer is designed as:

$\{\begin{matrix} \overset{^^}{F f} ((t t)) = = ξ ξ ((t t)) - - KX KX ((t t)))) \\ \overset{. .}{ξ ξ} ((t t)) = = K K {B B}_{11} ((ξ ξ ((t t)) - - KX KX ((t t)))) + + K K [[AX AX ((t t)) + + {B B}_{11} u u ((t t)) + + {B B}_{11} \overset{^^}{{d d}_{11}} ((t t))))]] \end{matrix}$

针对系统中的可建模干扰d₁(t)设计干扰观测器为：The disturbance observer is designed for the modelable disturbance d ₁ (t) in the system as:

$\{\begin{matrix} \overset{^^}{{d d}_{11}} ((t t)) = = V V \overset{^^}{w w} ((t t)) \\ \overset{^^}{w w} ((t t)) = = r r ((t t)) - - LX LX ((t t)) \\ \overset{. .}{r r} ((t t)) = = ((W W + + L L {B B}_{11} V V)) ((r r ((t t)) - - LX LX ((t t)))) + + L L [[AX AX ((t t)) + + {B B}_{11} u u ((t t)) + + {B B}_{11} \overset{^^}{F f} ((t t))]] \end{matrix}$

其中，为对故障的估计值，为干扰估计值，为w(t)的估计值，ξ(t)和r(t)分别为故障诊断观测器与干扰观测器中的辅助变量，K和L分别为待定的故障诊断观测器增益矩阵和干扰观测器增益矩阵，由后续步骤3求得。in, is the estimated value of the failure, is the interference estimate, is the estimated value of w(t), ξ(t) and r(t) are auxiliary variables in the fault diagnosis observer and disturbance observer respectively, K and L are the undetermined fault diagnosis observer gain matrix and disturbance observer respectively The gain matrix is obtained by the subsequent step 3.

定义故障估计误差 $e_{F} (t) = F (t) - \hat{F} (t),$ 干扰观测误差 $e_{w} (t) = w (t) - \hat{w} (t);$ Define Fault Estimation Error $e_{f} (t) = f (t) - \hat{f} (t),$ interference observation error $e_{w} (t) = w (t) - \hat{w} (t);$

根据故障诊断观测器的表达式可得故障估计误差方程为：According to the expression of the fault diagnosis observer, the fault estimation error equation can be obtained as:

${\overset{. .}{e e}}_{F f} ((t t)) = = K K {B B}_{11} {e e}_{F f} ((t t)) + + K K {B B}_{11} V V {e e}_{w w} ((t t)) + + K K {B B}_{22} {d d}_{22} ((t t)) + + \overset{. .}{F f} ((t t))$

根据干扰观测器的表达式可得干扰估计误差方程为：According to the expression of the interference observer, the interference estimation error equation can be obtained as:

${\overset{. .}{e e}}_{w w} ((t t)) = = L L {B B}_{11} V V {e e}_{w w} ((t t)) + + L L {B B}_{11} {e e}_{F f} ((t t)) + + L L {B B}_{22} {d d}_{22} ((t t)) + + {B B}_{33} δ δ ((t t))$

第三步，故障诊断观测器增益矩阵与干扰观测器增益矩阵求解The third step is to solve the gain matrix of the fault diagnosis observer and the gain matrix of the disturbance observer

联列第二步中的可建模干扰的估计误差方程和故障的估计误差方程如下：The estimated error equations of the modelable disturbances and faults in the second step of the concatenation are as follows:

$\{\begin{matrix} \overset{. .}{e e} ((t t)) = = (({W W}_{11} + + N N {B B}_{11} E E.)) e e ((t t)) + + N N {B B}_{22} {d d}_{22} ((t t)) + + {H h}_{11} \overset{. .}{F f} ((t t)) + + {H h}_{33} δ δ ((t t)) \\ {z z}_{\infty \infty} ((t t)) = = Ce Ce ((t t)) \end{matrix}$

其中 $e (t) = [\begin{matrix} e_{w} (t) \\ e_{F} (t) \end{matrix}],$ $W_{1} = [\begin{matrix} W & 0 \\ 0 & 0 \end{matrix}],$ $N = [\begin{matrix} L \\ K \end{matrix}],$ E＝[VI]， $H_{1} = [\begin{matrix} 0 \\ I \end{matrix}],$ $H_{3} = [\begin{matrix} B_{3} \\ 0 \end{matrix}] .$ in $e (t) = [\begin{matrix} e_{w} (t) \\ e_{f} (t) \end{matrix}],$ $W_{1} = [\begin{matrix} W & 0 \\ 0 & 0 \end{matrix}],$ $N = [\begin{matrix} L \\ K \end{matrix}],$ E=[VI], $h_{1} = [\begin{matrix} 0 \\ I \end{matrix}],$ $h_{3} = [\begin{matrix} B_{3} \\ 0 \end{matrix}] .$

z_∞(t)为H_∞性能参考输出，C为H_∞性能可调输出矩阵。z _∞ (t) is the H _∞ performance reference output, and C is the H _∞ performance adjustable output matrix.

利用凸优化算法求解多源干扰系统的可建模干扰观测器增益矩阵和故障诊断观测器增益矩阵；给定初始值e_w(0)和e_F(0)，可调输出矩阵C，干扰抑制度γ₁、γ_２和γ_３，求解以下凸优化问题：Use convex optimization algorithm to solve the modelable disturbance observer gain matrix and fault diagnosis observer gain matrix of multi-source interference system; given initial values e _w (0) and e _F (0), adjustable output matrix C, interference suppression degrees γ ₁ , γ ₂ and γ ₃ , solve the following convex optimization problem:

min(e^T(0)Pe(0))min(e ^T (0)Pe(0))

$Φ Φ = = [\begin{matrix} sym sym ((P P {W W}_{11} + + R R {B B}_{11} E E.)) & P P {H h}_{33} & P P {H h}_{11} & R R {B B}_{22} & {C C}^{T T} \\ * * & {- - γ γ}_{11}^{22} I I & 00 & 00 & 00 \\ * * & * * & - - {γ γ}_{22}^{22} I I & 00 & 00 \\ * * & * * & * * & - - {γ γ}_{33}^{22} I I & 00 \\ * * & * * & * * & * * & - - I I \end{matrix}] < < 00$

其中，符号*表示对称矩阵中相应部分的对称块，sym(PW₁+RB₁E)表达式如下：sym(PW₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T。Among them, the symbol * represents the symmetrical block of the corresponding part in the symmetrical matrix, and the expression of sym(PW ₁ +RB ₁ E) is as follows: sym(PW ₁ +RB ₁ E)=(PW ₁ +RB ₁ E)+(PW ₁ + RB ₁ E) ^T .

求解上式得P、R，观测器增益矩阵 $[\begin{matrix} L \\ K \end{matrix}] = P^{- 1} R .$ Solve the above formula to get P, R, observer gain matrix $[\begin{matrix} L \\ K \end{matrix}] = P^{- 1} R .$

第四步，设计滑模控制器，使用故障与干扰的估计值分别补偿故障和干扰滑模控制器的设计步骤如下：The fourth step is to design the sliding mode controller, and use the estimated values of the fault and disturbance to compensate the fault and disturbance respectively. The design steps of the sliding mode controller are as follows:

1)设计滑模面s(t)1) Design the sliding surface s(t)

滑模面通常的设计方法如下：The usual design method of the sliding surface is as follows:

$s (t) = Σ_{i = 1}^{n - 1} k_{i} x_{i} (t) + x_{n} (t),$ 其中k_i＞0，i＝1，2，…，n-1。 $the s (t) = Σ_{i = 1}^{no - 1} k_{i} x_{i} (t) + x_{no} (t),$ Where k _i> 0, i=1, 2, . . . , n-1.

2)设计滑模控制律2) Design sliding mode control law

采用函数切换控制律，包括等价输入与切换输入两部分，等价输入由求得。控制律设计如下：The function switching control law is adopted, including two parts of equivalent input and switching input, and the equivalent input is composed of Get it. The control law is designed as follows:

u(t)＝u_eq(t)+u_vs(t)u(t)＝u _eq (t)+u _vs (t)

其中u_eq(t)为系统的等价控制量，u_vs(t)为开关控制量。Among them, u _eq (t) is the equivalent control quantity of the system, and u _vs (t) is the switch control quantity.

令有代入系统的动力学模型可得make Have Substituting into the dynamic model of the system, we can get

$- - {a a}_{00} {x x}_{n no} ((t t)) - - \cdot \cdot \cdot \cdot \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; \cdot \cdot - - {a a}_{n no - - 11} {x x}_{11} ((t t)) + + {b b}_{11} ((u u ((t t)) + + {d d}_{11} ((t t)) + + F f ((t t)))) + + {b b}_{22} {d d}_{22} ((t t)) = = - - {Σ Σ}_{i i = = 11}^{n no - - 11} {k k}_{i i} {\overset{. .}{x x}}_{i i} ((t t))$

由上式求出的u(t)即为等价控制量，进而可得：The u(t) obtained from the above formula is the equivalent control quantity, and then it can be obtained:

${u u}_{eq eq} ((t t)) = = 11 / / {b b}_{11} (({Σ Σ}_{i i = = 11}^{n no} {a a}_{n no - - i i} {x x}_{i i} ((t t)) - - {Σ Σ}_{i i = = 22}^{n no} {k k}_{i i - - 11} {x x}_{i i} ((t t)))) - - {d d}_{11} ((t t)) - - F f ((t t));;$

使用可建模干扰与故障的估计值分别代替实际值d₁(t)、F(t)，求得 $u_{eq} (t) = 1 / b_{1} (Σ_{i = 1}^{n} a_{n - i} x_{i} (t) - Σ_{i = 2}^{n} k_{i - 1} x_{i} (t)) - {\hat{d}}_{1} (t) - \hat{F} (t);$ Use estimates for modelable disturbances and failures Substitute the actual values d ₁ (t) and F(t) respectively to obtain $u_{eq} (t) = 1 / b_{1} (Σ_{i = 1}^{no} a_{no - i} x_{i} (t) - Σ_{i = 2}^{no} k_{i - 1} x_{i} (t)) - {\hat{d}}_{1} (t) - \hat{f} (t);$

开关控制量设计为u_vs(t)＝-T_psgn(s(t))。其中，T_p为滑模增益，由第五步求得；sgn(s(t))为开关函数，用如下形式表示：The switch control quantity is designed as u _vs (t)=-T _p sgn(s(t)). Among them, T _p is the sliding mode gain, obtained from the fifth step; sgn(s(t)) is the switching function, expressed in the following form:

$sgn sgn ((s the s ((t t)))) = = \{\begin{matrix} 11 & s the s ((t t)) > > 00 \\ 00 & s the s ((t t)) = = 00 \\ - - 11 & s the s ((t t)) < < 00 \end{matrix}$

则控制输入表达式为：Then the control input expression is:

$u u ((t t)) = = {u u}_{eq eq} ((t t)) + + {u u}_{vs vs} ((t t)) = = 11 / / {b b}_{11} (({Σ Σ}_{i i = = 11}^{n no} {a a}_{n no - - i i} {x x}_{i i} ((t t)) - - {Σ Σ}_{i i = = 22}^{n no} {k k}_{i i - - 11} {x x}_{i i} ((t t)))) - - {\overset{^^}{d d}}_{11} ((t t)) - - \overset{^^}{F f} ((t t)) - - {T T}_{p p} sgn sgn ((s the s ((t t))))$

第五步，求解滑模增益，保证系统稳定The fifth step is to solve the sliding mode gain to ensure the stability of the system

李雅普诺夫函数设计为 The Lyapunov function is designed as

由s(t)的定义及系统的动力学模型，可得From the definition of s(t) and the dynamic model of the system, we can get

$\overset{. .}{s the s} ((t t)) = = {b b}_{11} ((u u ((t t)) + + {d d}_{11} ((t t)) + + F f ((t t)))) + + {b b}_{22} {d d}_{22} ((t t)) - - {Σ Σ}_{i i = = 11}^{n no} {a a}_{n no - - i i} {x x}_{i i} ((t t)) + + {Σ Σ}_{i i = = 22}^{n no} {k k}_{i i - - 11} {x x}_{i i} ((t t))$

将第四步求得的控制输入表达式代入上式，有Substituting the control input expression obtained in the fourth step into the above formula, we have

$\overset{. .}{G G} ((t t)) = = {s the s}^{T T} ((t t)) \overset{. .}{s the s} ((t t)) = = {s the s}^{T T} ((t t)) (({b b}_{11} {e e}_{F f} ((t t)) + + {b b}_{11} V V {e e}_{w w} ((t t)) + + {b b}_{22} {d d}_{22} ((t t)) - - {T T}_{p p} sgn sgn ((s the s ((t t))))))$

根据李雅普诺夫定理，当成立，证明系统能够达到滑模面，且滑动模态平面是渐近稳定的。记α＝||b₁e_F(t)+b₁Ve_w(t)+b₂d₂(t)||，显然，需要满足T_p≥α，则有系统能够达到滑模面，并达到渐近一致稳定状态。According to Lyapunov's theorem, when established, it proves that the system can reach the sliding mode surface, and the sliding mode plane is asymptotically stable. Denote α＝||b ₁ e _F (t)+b ₁ Ve _w (t)+b ₂ d ₂ (t)||, obviously, it is necessary to satisfy T _p ≥ α, then we have The system can reach the sliding mode surface and reach an asymptotically uniform stable state.

考虑到系统的饱和输入问题，T_p＝max(α，u_om)。其中u_om为系统的饱和输入值， $\max (α, u_{om}) = \{\begin{matrix} α & α > u_{om} \\ u_{om} & α \leq u_{om} \end{matrix} .$ Considering the saturated input problem of the system, T _p =max(α, u _om ). where u _om is the saturation input value of the system, $\max (α, u_{om}) = \{\begin{matrix} α & α > u_{om} \\ u_{om} & α \leq u_{om} \end{matrix} .$

本发明与现有技术相比的优点在于：The advantage of the present invention compared with prior art is:

本发明一种具有抗干扰容错性能的滑模控制方法采用滑模控制器与故障诊断观测器和干扰观测器结合，适用于输入饱和系统的抗干扰容错控制。故障诊断观测器和干扰观测器保证了系统的容错抗干扰性能，对故障和干扰进行估计和抵消，滑模控制器对于扰动具有鲁棒性，滑模增益的设计考虑到了系统的饱和输入值。控制方法解决了输入饱和系统的容错抗干扰问题。同时，设计两个观测器对系统中的故障和干扰进行抵消，减弱了滑模控制的抖振现象，提高姿态控制系统中滑模控制方法的精度和可靠性。A sliding mode control method with anti-jamming and fault-tolerant performance adopts a sliding-mode controller combined with a fault diagnosis observer and a disturbance observer, and is suitable for anti-jamming and fault-tolerant control of an input saturation system. The fault diagnosis observer and disturbance observer ensure the fault-tolerant and anti-disturbance performance of the system, and estimate and offset faults and disturbances. The sliding mode controller is robust to disturbances, and the design of the sliding mode gain takes into account the saturation input value of the system. The control method solves the fault-tolerant and anti-jamming problem of the input saturated system. At the same time, two observers are designed to offset the faults and disturbances in the system, weaken the chattering phenomenon of sliding mode control, and improve the accuracy and reliability of the sliding mode control method in the attitude control system.

附图说明Description of drawings

图1为本发明一种具有抗干扰容错性能的滑模控制方法的设计流程图。Fig. 1 is a design flow chart of a sliding mode control method with anti-interference and fault-tolerant performance in the present invention.

具体实施方式detailed description

如图1所示，本发明具体实现步骤如下(以三轴稳定卫星姿态控制系统为例来说明方法的具体实现)：As shown in Figure 1, the specific implementation steps of the present invention are as follows (taking the three-axis stable satellite attitude control system as an example to illustrate the specific implementation of the method):

1、考虑卫星姿态控制系统中的故障与多源干扰，搭建系统动力学模型1. Considering the faults and multi-source interference in the satellite attitude control system, build a system dynamics model

三轴稳定卫星本体坐标系和轨道坐标系之间的欧拉角很小，将卫星姿态动力学和运动学模型线性化可得：The Euler angle between the three-axis stable satellite body coordinate system and the orbital coordinate system is very small, and the linearization of the satellite attitude dynamics and kinematics model can be obtained as follows:

$\{\begin{matrix} {J J}_{11} \overset{. . . .}{φ φ} - - {ω ω}_{00} (({J J}_{11} - - {J J}_{22} + + {J J}_{33})) \overset{. .}{ψ ψ} + + 44 {ω ω}_{00}^{22} (({J J}_{22} - - {J J}_{33})) φ φ = = {u u}_{11} + + {T T}_{d d 11} \\ {J J}_{22} \overset{. . . .}{θ θ} + + 33 {ω ω}_{00}^{22} (({J J}_{11} - - {J J}_{33})) θ θ = = {u u}_{22} + + F f ((t t)) + + {T T}_{d d 22} \\ {J J}_{33} \overset{. . . .}{ψ ψ} + + {ω ω}_{00} (({J J}_{11} - - {J J}_{22} + + {J J}_{33})) \overset{. .}{φ φ} + + {ω ω}_{00}^{22} (({J J}_{22} - - {J J}_{11})) ψ ψ = = {u u}_{33} + + {T T}_{d d 33} \end{matrix}$

上式三个方程依次为卫星横滚轴、俯仰轴、偏航轴三个轴向的姿态动力学方程。J₁，J₂，J₃分别为三轴转动惯量，φ，θ，ψ分别为卫星本体坐标系和轨道坐标系之间的三轴欧拉角；分别为三轴欧拉角速率；分别为三轴欧拉角加速度；u₁，u₂，u₃分别为三轴控制力矩；ω₀为卫星轨道角速度；F(t)为时变故障，T_d1，T_d2，T_d3分别为三轴的干扰力矩(包括敏感器和执行机构带来的干扰力矩)；The three equations in the above formula are the attitude dynamic equations of the satellite roll axis, pitch axis, and yaw axis in turn. J ₁ , J ₂ , J ₃ are the three-axis moment of inertia respectively, φ, θ, ψ are the three-axis Euler angles between the satellite body coordinate system and the orbit coordinate system; are the three-axis Euler angle rates; are three-axis Euler angular acceleration; u ₁ , u ₂ , u ₃ are three-axis control torque; ω ₀ is satellite orbital angular velocity; F(t) is time-varying fault, T _d1 , T _d2 , T _d3 are Triaxial disturbance torque (including disturbance torque caused by sensors and actuators);

以下步骤以卫星俯仰通道动力学模型为例设计具有抗干扰容错性能的滑模控制器，横滚轴与偏航轴设计方法相同。The following steps take the dynamic model of the satellite pitch channel as an example to design a sliding mode controller with anti-interference and fault-tolerant performance. The design method of the roll axis and the yaw axis is the same.

三轴稳定卫星的期望姿态角、角速度和角加速度分别记为θ_c(t)、和且均为零。可得误差系统方程如下：The expected attitude angle, angular velocity and angular acceleration of the three-axis stabilized satellite are respectively denoted as θ _c (t), and And both are zero. The error system equation can be obtained as follows:

${J J}_{22} {\overset{. . . .}{e e}}_{θ θ} + + 33 {ω ω}_{00}^{22} (({J J}_{11} - - {J J}_{33})) {e e}_{θ θ} = = {u u}_{22} + + F f ((t t)) + + {T T}_{d d 22}$

其中e_θ(t)＝θ(t)-θ_c(t)＝θ(t)为误差角，为误差角加速度。Where e _θ (t)=θ(t)-θ _c (t)=θ(t) is the error angle, is the error angular acceleration.

将俯仰通道姿态误差模型写成状态空间形式如下：The pitch channel attitude error model is written in the form of state space as follows:

$\overset{. .}{X x} ((t t)) = = AX AX ((t t)) + + {B B}_{11} ((u u ((t t)) + + {d d}_{11} ((t t)) + + F f ((t t)))) + + {B B}_{22} {d d}_{22} ((t t))$

其中，多源干扰系统状态变量u(t)为控制输入，F(t)为时变故障，d₁(t)为可建模干扰，d₂(t)为不可建模干扰，d₁(t)与d₂(t)组成T_d2。A、B₁和B₂如下所示：Among them, the multi-source interference system state variable u(t) is the control input, F(t) is the time-varying fault, d ₁ (t) is the disturbance that can be modeled, d ₂ (t) is the disturbance that cannot be modeled, d ₁ (t) and d ₂ (t) Composition T _d2 . A, B ₁ and B ₂ look like this:

$A A = = [\begin{matrix} 00 & 11 \\ - - 33 {ω ω}_{00}^{22} (({J J}_{11} - - {J J}_{33})) / / {J J}_{22} & 00 \end{matrix}],,$ ${B B}_{11} = = [\begin{matrix} 00 \\ {J J}_{22}^{- - 11} \end{matrix}],,$ ${B B}_{22} = = [\begin{matrix} 00 \\ {J J}_{22}^{- - 11} \end{matrix}] . .$

外部模型描述干扰d₁(t)由下列外部干扰模型∑₁表示：External model description Disturbance d ₁ (t) is represented by the following external disturbance model Σ ₁ :

${Σ Σ}_{11} : : \{\begin{matrix} {d d}_{11} ((t t)) = = Vw Vw ((t t)) \\ \overset{. .}{w w} ((t t)) = = Ww w ((t t)) + + {B B}_{33} δ δ ((t t)) \end{matrix}$

其中，w(t)为可建模干扰模型的状态变量，V为可建模干扰模型的输出矩阵，W表示可建模干扰模型的系统阵，δ(t)为能量有界的不可建模随机(即L₂范数有界)干扰，B₃为不可建模干扰的增益阵。where w(t) is the state variable of the modelable disturbance model, V is the output matrix of the modelable disturbance model, W represents the system matrix of the modelable disturbance model, and δ(t) is the non-modelable random ₍ i.e. the L2 norm Bounded) interference, B ₃ is the gain array of interference that cannot be modeled.

2、设计故障诊断观测器与干扰观测器分别估计故障与可建模干扰2. Design a fault diagnosis observer and a disturbance observer to estimate faults and modelable disturbances respectively

针对系统中的时变故障F(t)设计故障诊断观测器为：The fault diagnosis observer designed for the time-varying fault F(t) in the system is:

其中，为故障的估计值，为干扰估计值，为w(t)的估计值，ξ(t)和r(t)分别为故障诊断观测器与干扰观测器中的辅助变量，K和L分别为待定的故障诊断观测器增益矩阵和干扰观测器增益矩阵，由后续步骤3求得。in, is the estimated value of the failure, is the interference estimate, is the estimated value of w(t), ξ(t) and r(t) are auxiliary variables in the fault diagnosis observer and disturbance observer respectively, K and L are the undetermined fault diagnosis observer gain matrix and disturbance observer respectively The gain matrix is obtained by the subsequent step 3.

定义故障估计误差 $e_{F} (t) = F (t) - \hat{F} (t),$ 干扰估计误差 $e_{w} (t) = w (t) - \hat{w} (t);$ Define Fault Estimation Error $e_{f} (t) = f (t) - \hat{f} (t),$ interference estimation error $e_{w} (t) = w (t) - \hat{w} (t);$

根据故障诊断观测器的表达式，可得故障估计误差方程为：According to the expression of the fault diagnosis observer, the fault estimation error equation can be obtained as:

根据干扰观测器的表达式，可得干扰估计误差方程为：According to the expression of the interference observer, the interference estimation error equation can be obtained as:

3、故障诊断观测器增益矩阵与干扰观测器增益矩阵求解3. Solving the gain matrix of the fault diagnosis observer and the gain matrix of the disturbance observer

联列第二步中的可建模干扰的估计误差方程和故障的估计误差方程如下：The estimation error equation of the modelable disturbance and the estimation error equation of the fault in the second step of the concatenation are as follows:

其中 $e (t) = [\begin{matrix} e_{w} (t) \\ e_{F} (t) \end{matrix}],$ $W_{1} = [\begin{matrix} W & 0 \\ 0 & 0 \end{matrix}],$ $N = [\begin{matrix} L \\ K \end{matrix}],$ E＝[VI]， $H_{1} = [\begin{matrix} 0 \\ I \end{matrix}],$ $H_{3} = [\begin{matrix} B_{3} \\ 0 \end{matrix}] .$ I为单位阵，z_∞(t)为H_∞性能参考输出，C为H_∞性能输出矩阵。in $e (t) = [\begin{matrix} e_{w} (t) \\ e_{f} (t) \end{matrix}],$ $W_{1} = [\begin{matrix} W & 0 \\ 0 & 0 \end{matrix}],$ $N = [\begin{matrix} L \\ K \end{matrix}],$ E=[VI], $h_{1} = [\begin{matrix} 0 \\ I \end{matrix}],$ $h_{3} = [\begin{matrix} B_{3} \\ 0 \end{matrix}] .$ I is the identity matrix, z _∞ (t) is the H _∞ performance reference output, and C is the H _∞ performance output matrix.

利用凸优化算法求解多源干扰系统的容错抗干扰控制器增益阵；给定初始值e_w(0)和e_F(0)，输出矩阵C，干扰抑制度γ₁、γ₂和γ₃，求解以下凸优化问题：Using the convex optimization algorithm to solve the gain matrix of the fault-tolerant and anti-interference controller of the multi-source interference system; given the initial values e _w (0) and e _F (0), the output matrix C, the interference suppression degrees γ ₁ , γ ₂ and γ ₃ , Solve the following convex optimization problem:

min(e^T(0)Pe(0))min(e ^T (0)Pe(0))

上式中，符号*表示对称矩阵中相应部分的对称块，sym(PW₁+RB₁E)表达式如下：sym(PW₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T。In the above formula, the symbol * represents the symmetrical block of the corresponding part in the symmetrical matrix, and the expression of sym(PW ₁ +RB ₁ E) is as follows: sym(PW ₁ +RB ₁ E)=(PW ₁ +RB ₁ E)+(PW ₁ +RB ₁ E) ^T .

4、设计滑模控制器，使用故障与干扰的估计值分别补偿故障和干扰4. Design a sliding mode controller to compensate for faults and disturbances, respectively, using estimated values of faults and disturbances

1)切换函数设计如下：1) The switching function is designed as follows:

$s the s ((t t)) = = {e e}_{θ θ} ((t t)) + + {\overset{. .}{e e}}_{θ θ} ((t t))$

其中，e_θ(t)＝θ(t)-θ_c(t)，三轴稳定卫星期望姿态角θ_c(t)＝0，故有Among them, e _θ (t)=θ(t)-θ _c (t), the desired attitude angle of the three-axis stabilized satellite θ _c (t)=0, so

$s the s ((t t)) = = θ θ ((t t)) + + \overset{. .}{θ θ} ((t t))$

2)滑动模态控制律设计如下：2) The sliding mode control law is designed as follows:

u(t)＝u_eq(t)+u_vs(t)u(t)＝u _eq (t)+u _vs (t)

其中，u_eq(t)为等价控制量，u_vs(t)为开关控制量。Among them, u _eq (t) is the equivalent control quantity, u _vs (t) is the switch control quantity.

令有代入俯仰轴的动力学方程可得make Have Substituting into the dynamic equation of the pitch axis, we can get

$- - {J J}_{22} {k k}_{11} \overset{. .}{θ θ} ((t t)) = = - - 33 {ω ω}_{00}^{22} (({J J}_{11} - - {J J}_{33})) θ θ ((t t)) + + {u u}_{eq eq} ((t t)) + + {d d}_{11} ((t t)) + + F f ((t t)) . .$

使用可建模干扰与故障的估计值分别代替实际值d₁(t)、F(t)，求得 $u_{eq} (t) = 3 {ω_{0}}^{2} (J_{1} - J_{3}) θ (t) - J_{2} k_{1} \dot{θ} (t) - {\hat{d}}_{1} (t) - \hat{F} (t) .$ Use estimates for modelable disturbances and failures Substitute the actual values d ₁ (t) and F(t) respectively to obtain $u_{eq} (t) = 3 {ω_{0}}^{2} (J_{1} - J_{3}) θ (t) - J_{2} k_{1} \dot{θ} (t) - {\hat{d}}_{1} (t) - \hat{f} (t) .$

开关控制量设计为u_vs(t)＝-T_psgn(s(t))。其中T_p为滑模增益，由后续步骤5求得，sgn(s(t))为开关函数，用如下形式表示：The switch control quantity is designed as u _vs (t)=-T _p sgn(s(t)). Among them, T _p is the sliding mode gain, which is obtained from the subsequent step 5, and sgn(s(t)) is the switching function, expressed in the following form:

系统控制输入如下：The system control inputs are as follows:

$u u ((t t)) = = {u u}_{eq eq} ((t t)) + + {u u}_{vs vs} ((t t)) = = 33 {ω ω}_{00}^{22} (({J J}_{11} - - {J J}_{33})) θ θ ((t t)) - - {J J}_{22} {k k}_{11} \overset{. .}{θ θ} ((t t)) - - {\overset{^^}{d d}}_{11} ((t t)) - - \overset{^^}{F f} ((t t)) - - {T T}_{p p} sgn sgn ((s the s ((t t))))$

5、求解滑模增益，保证系统稳定5. Solve the sliding mode gain to ensure the stability of the system

李雅普诺夫函数设计为 $G (t) = \frac{1}{2} s^{T} (t) J_{2} s (t) &GreaterEqual; 0 .$ The Lyapunov function is designed as $G (t) = \frac{1}{2} {the s}^{T} (t) J_{2} the s (t) &Greater Equal; 0 .$

由定义得 $\dot{s} (t) = k_{1} \dot{θ} (t) + \overset{. .}{θ} (t) = k_{1} \dot{θ} (t) + (- 3 {ω_{0}}^{2} (J_{1} - J_{3}) θ (t) + u (t) + d_{1} (t) + F (t)) / J_{2}$ by definition $\dot{the s} (t) = k_{1} \dot{θ} (t) + \overset{. .}{θ} (t) = k_{1} \dot{θ} (t) + (- 3 {ω_{0}}^{2} (J_{1} - J_{3}) θ (t) + u (t) + d_{1} (t) + f (t)) / J_{2}$

将控制输入表达式代入上式，可得Substituting the control input expression into the above formula, we can get

$\overset{. .}{G G} ((t t)) = = {s the s}^{T T} ((t t)) {J J}_{22} \overset{. .}{s the s} ((t t)) = = {s the s}^{T T} ((t t)) (({b b}_{11} {e e}_{F f} ((t t)) + + {b b}_{11} V V {e e}_{w w} ((t t)) + + {b b}_{22} {d d}_{22} ((t t)) - - {T T}_{p p} sgn sgn ((s the s ((t t))))))$

根据李雅普诺夫定理，当成立，证明系统能够达到滑模面，且滑动模态平面是渐近稳定的；According to Lyapunov's theorem, when Established, it proves that the system can reach the sliding mode surface, and the sliding mode plane is asymptotically stable;

记α＝||b₁e_F(t)+b₁Ve_w(t)+b₂d₂(t)||。显然，需要满足T_p≥α，则有系统能够达到滑模面，并达到渐近一致稳定状态。Write α＝||b ₁ e _F (t)+b ₁ Ve _w (t)+b ₂ d ₂ (t)||. Obviously, T _p ≥ α needs to be satisfied, then The system can reach the sliding mode surface and reach an asymptotically uniform stable state.

考虑到系统的饱和输入问题，T_p＝max(α，u_om)。其中u_om已知，为飞轮提供的最大输入力矩， $\max (α, u_{om}) = \{\begin{matrix} α & α > u_{om} \\ u_{om} & α \leq u_{om} \end{matrix} .$ Considering the saturated input problem of the system, T _p =max(α, u _om ). where u _om is known, is the maximum input torque provided by the flywheel, $\max (α, u_{om}) = \{\begin{matrix} α & α > u_{om} \\ u_{om} & α \leq u_{om} \end{matrix} .$

为避免滑模控制器输出的抖振现象，采用饱和函数sat(s(t))代替开关函数sgn(s(t))。sat(s(t))表达式如下：In order to avoid chattering in the output of the sliding mode controller, the saturation function sat(s(t)) is used instead of the switching function sgn(s(t)). The sat(s(t)) expression is as follows:

$sat sat ((s the s ((t t)))) = = \{\begin{matrix} sgn sgn ((s the s ((t t)))) & | | s the s ((t t)) | | > > σ σ \\ s the s ((t t)) / / | | σ σ | | & | | s the s ((t t)) | | \leq \leq σ σ \end{matrix}$

其中，σ为消颤因子，目的在于既有效消除抖振，又保证系统快速收敛，取值在(0.02，0.08)范围内。Among them, σ is the defibrillation factor, the purpose is not only to effectively eliminate chattering, but also to ensure the rapid convergence of the system, and the value is in the range of (0.02, 0.08).

本发明说明书中未作详细描述的内容属于本领域专业技术人员公知的现有技术。The contents not described in detail in the description of the present invention belong to the prior art known to those skilled in the art.

Claims

1. A sliding mode control method with anti-interference fault-tolerant performance is characterized by comprising the following steps: firstly, establishing a dynamic model of a system; secondly, designing a fault diagnosis observer and a disturbance observer to estimate faults and modelable disturbance in a system; thirdly, solving a gain matrix of the interference observer and the fault diagnosis observer; then designing a sliding mode controller, substituting the interference and fault estimation values into the sliding mode controller to compensate equivalent interference and faults; finally, solving the sliding mode gain on the premise of system input saturation to ensure the stability of the system; the method comprises the following specific steps:

firstly, establishing a system dynamics model

A system dynamics model containing interference and faults is built, and the following steps are included:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} (t) = x_{2} (t) \\ {\overset{\cdot}{x}}_{2} (t) = x_{3} (t) \\ \cdot \\ \cdot \\ \cdot \\ {\overset{\cdot}{x}}_{n} (t) = - a_{0} x_{n} (t) - ... ... - a_{n - 1} x_{1} (t) + b_{1} (u (t) + d_{1} (t) + F (t)) + b_{2} d_{2} (t) \end{matrix}

wherein x is₁(t),x₂(t),…,x_n(t) is the system state, n ≧ 2 is a positive integer, u (t) is the control input, F (t) is the rate-of-change bounded time-varying fault, d₁(t) modelable interference, d₂(t) unmoldable random interference, a₀、a₁、…a_n-1And b₁、b₂Are all system internal parameters; d₁(t) from the following external interference model ∑₁Represents:

Σ_{1} : \{\begin{matrix} d_{1} (t) = V w (t) \\ \overset{\cdot}{w} (t) = W w (t) + B_{3} δ (t) \end{matrix}

wherein W (t) is a state variable of the modelable interference model, V is an output matrix of the modelable interference model, W represents a system matrix of the modelable interference model, B₃A gain array that is unmoldable random interference, (t) unmoldable random interference that is energy bounded;

selecting a system state variable X (t) ═ x₁(t)x₂(t)……x_n(t)]^TWritten as a state space expression is as follows:

\overset{\cdot}{X} (t) = A X (t) + B_{1} (u (t) + d_{1} (t) + F (t)) + B_{2} d_{2} (t)

wherein X (t) is system state variable, A is system array, B₁As an input matrix, B₂A gain array for unmoldable random interference;

designing a fault diagnosis observer and a disturbance observer to respectively estimate faults and modelable disturbance

Designing a fault diagnosis observer aiming at the time-varying fault F (t) in the system as follows:

\{\begin{matrix} \hat{F} (t) = ξ (t) - K X (t) \\ \overset{\cdot}{ξ} (t) = {KB}_{1} (ξ (t) - K X (t)) + K [A X (t) + B_{1} u (t) + B_{1} {\hat{d}}_{1} (t)] \end{matrix}

for modelable interference d in a system₁(t) designing a disturbance observer as follows:

\{\begin{matrix} {\hat{d}}_{1} (t) = V \hat{w} (t) \\ \hat{w} (t) = r (t) - L X (t) \\ \overset{\cdot}{r} (t) = (W + L B_{1} V) (r (t) - L X (t)) + L [A X (t) + B_{1} u (t) + B_{1} \hat{F} (t)] \end{matrix}

wherein,is an estimate of the fault that is,in order to be an interference estimate,the estimated value of w (t), ξ (t) and r (t) are respectively auxiliary variables in the fault diagnosis observer and the disturbance observer, K and L are respectively a gain matrix of the fault diagnosis observer to be determined and a gain matrix of the disturbance observer, and the estimated value is obtained by the subsequent third step;

defining a fault estimation error asInterference observation error is

e_{w} (t) = w (t) - \hat{w} (t);

The fault estimation error equation can be obtained according to the expression of the fault diagnosis observer as follows:

{\overset{\cdot}{e}}_{F} (t) = {KB}_{1} e_{F} (t) + {KB}_{1} {Ve}_{w} (t) + {KB}_{2} d_{2} (t) + \overset{\cdot}{F} (t)

the disturbance estimation error equation can be derived from the expression of the disturbance observer as follows:

{\overset{\cdot}{e}}_{w} (t) = {LB}_{1} {Ve}_{w} (t) + {LB}_{1} e_{F} (t) + {LB}_{2} d_{2} (t) + B_{3} δ (t)

thirdly, solving the gain matrix of the fault diagnosis observer and the gain matrix of the interference observer

The interference estimation error equation and the fault estimation error equation in the second step of the cascade are as follows:

\{\begin{matrix} \overset{\cdot}{e} (t) = (W_{1} + {NB}_{1} E) e (t) + {NB}_{2} d_{2} (t) + H_{1} \overset{\cdot}{F} (t) + H_{3} δ (t) \\ z_{\infty} (t) = C e (t) \end{matrix}

wherein

e (t) = [\begin{matrix} e_{w} (t) \\ e_{F} (t) \end{matrix}], W_{1} = [\begin{matrix} W & 0 \\ 0 & 0 \end{matrix}], N = [\begin{matrix} L \\ K \end{matrix}], E = [\begin{matrix} V & I \end{matrix}], H_{1} = [\begin{matrix} 0 \\ I \end{matrix}], H_{3} = [\begin{matrix} B_{3} \\ 0 \end{matrix}];

I is a unit array, z_∞(t) is H_∞Performance reference output, C is H_∞A performance adjustable output matrix;

solving a modelable interference observer gain matrix and a fault diagnosis observer gain matrix of the multi-source interference system by using a convex optimization algorithm; given an initial value e_w(0) And e_F(0) Adjustable output matrix C, interference suppression degree gamma₁、γ₂And gamma₃Solving the following convex optimization problem:

min(e^T(0)Pe(0))

Φ = [\begin{matrix} s y m ({PW}_{1} + {RB}_{1} E) & {PH}_{3} & {PH}_{1} & {RB}_{2} & C^{T} \\ * & - {γ_{1}}^{2} & 0 & 0 & 0 \\ * & * & - γ_{2}^{2} & 0 & 0 \\ * & * & * & - γ_{3}^{2} & 0 \\ * & * & * & * & - I \end{matrix}] < 0

wherein symbols denote symmetric blocks, sym (PW) of the corresponding part of the symmetric matrix₁+RB₁E) The expression is as follows:

sym(PW₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T；

p, R, observer gain matrix, is solved for the above equation

[\begin{matrix} L \\ K \end{matrix}] = P^{- 1} R;

Fourthly, designing a sliding mode controller, and respectively compensating the fault and the disturbance by using the estimated values of the fault and the disturbance, wherein the design steps of the sliding mode controller are as follows:

1) design sliding mode surface s (t)

The design method of the sliding mode surface comprises the following steps:

s (t) = Σ_{i = 1}^{n - 1} k_{i} x_{i} (t) + x_{n} (t),

wherein k is_i＞0,i＝1,2,…,n-1；

2) Design sliding mode control law

Adopting function switching control law, including equivalent input and switching input, the equivalent input is formed fromObtaining; the control law is designed as follows:

u(t)＝u_eq(t)+u_vs(t)

wherein u is_eq(t) is the equivalent control quantity of the system, u_vs(t) is a switching control quantity;

order toIs provided withThe kinetic model of the substitution system can be obtained

- a_{0} x_{n} (t) - ... ... - a_{n - 1} x_{1} (t) + b_{1} (u (t) + d_{1} (t) + F (t)) + b_{2} d_{2} (t) = - Σ_{i = 1}^{n - 1} k_{i} {\overset{\cdot}{x}}_{i} (t)

U (t) obtained from the above equation is an equivalent control amount, and further:

u_{e q} (t) = 1 / b_{1} (Σ_{i = 1}^{n} a_{n - i} x_{i} (t) - Σ_{i = 2}^{n} k_{i - 1} x_{i} (t)) - d_{1} (t) - F (t);

using estimates of modelable disturbances and faultsRespectively replacing the actual value d₁(t), F (t), have

u_{e q} (t) = 1 / b_{1} (Σ_{i = 1}^{n} a_{n - i} x_{i} (t) - Σ_{i = 2}^{n} k_{i - 1} x_{i} (t)) - {\hat{d}}_{1} (t) - \hat{F} (t);

The switch control quantity is designed as u_vs(t)＝-T_psgn (s (t)); wherein, T_pObtaining the sliding mode gain through the fifth step; sgn (s (t)) is a switching function and is expressed in the following form:

s g n (s (t)) = \{\begin{matrix} 1 & s (t) > 0 \\ 0 & s (t) = 0 \\ - 1 & s (t) < 0 \end{matrix}

the control input expression is:

u (t) = u_{e q} (t) + u_{v s} (t) = 1 / b_{1} (Σ_{i = 1}^{n} a_{n - i} x_{i} (t) - Σ_{i = 2}^{n} k_{i - 1} x_{i} (t)) - {\hat{d}}_{1} (t) - \hat{F} (t) - T_{p} sgn (s (t))

fifthly, solving the sliding mode gain to ensure the system stability

The Lyapunov function is designed as

From the definition of s (t) and the kinetic model of the system, it can be obtained

\overset{\cdot}{s} (t) = b_{1} (u (t) + d_{1} (t) + F (t)) + b_{2} d_{2} (t) - Σ_{i = 1}^{n} a_{n - i} x_{i} (t) + Σ_{i = 2}^{n} k_{i - 1} x_{i} (t)

Substituting the control input expression obtained in the fourth step into the formula

\overset{\cdot}{G} (t) = s^{T} (t) \overset{\cdot}{s} (t) = s^{T} (t) (b_{1} e_{F} (t) + b_{1} {Ve}_{w} (t) + b_{2} d_{2} (t) - T_{p} sgn (s (t)))

According to Lyapunov's theorem, whenThe establishment proves that the system can reach the sliding mode surface, and the sliding mode plane is asymptotically stable;

remember α | | b₁e_F(t)+b₁Ve_w(t)+b₂d₂(T) |, it is obvious that only T needs to be satisfied_pNot less than α, there areThe system can reach the sliding mode surface and reach a stable state of asymptotic consistency;

considering the saturated input problem of the system, the switching value T_p＝max(α,u_om) (ii) a Wherein u is_omIs the value of the saturated input to the system,

m a x (α, u_{o m}) = \{\begin{matrix} α & α > u_{o m} \\ u_{o m} & α \leq u_{o m} \end{matrix} .