CN112731809B - A State and Fault Estimation Method for Dead Zone Sandwich Systems - Google Patents

Abstract

The invention discloses a state and fault estimation method of a dead zone sandwich system. The method aims at the problem of unmeasurable intermediate variables of the dead zone nonlinear and non-smooth sandwich system. The invention discloses a switching proportional integral observer to accurately and quickly estimate system status and faults. First, using the key separation principle and switching function, a non-smooth state-space equation that can accurately describe the dead zone sandwich system is constructed. Secondly, according to the construction of the non-smooth state space equation, the SPIO that can switch with the change of the system working range is constructed. And by analyzing the estimation error of the observer, the condition of observer convergence is given, and the mathematical proof of the observer convergence theorem is given. The advantage of this method is that the switching function is introduced into the observer. Compared with the traditional proportional-integral observer, the observer using this method can more accurately estimate the state and fault of the system at the same time.

Description

A State and Fault Estimation Method for Dead Zone Sandwich Systems

technical field

The invention relates to the technical field of nonlinear system state estimation, in particular to a state and fault estimation method of a dead zone sandwich system.

Background technique

A class of dead zone sandwich systems exists in industry. The intermediate variables of this type of system cannot be measured or the measurement cost is very high, but if the intermediate state cannot be obtained, most state feedback control methods cannot be performed, because the state value of the system must be accurately known in state feedback control. At present, the most effective way to obtain the unknown state value is to design an observer, which uses the input and output variables of the system to estimate the unknown state of the system.

Chinese patent CN105204332A discloses a non-smooth observer-based method for estimating the state of a composite sandwich system with dead zones and hysteresis. The invention estimates the state of the system when the sandwich system does not contain faults. However, faults are often unavoidable in practical systems. Due to the existence of faults, the state estimation of the system will produce certain disturbances, and even cause the estimation error to diverge, that is, the state of the system cannot be estimated. The invention also does not estimate system failures since failure effects are ignored. So far, no patents and literatures have been found on a dead-zone sandwich system that simultaneously estimates state and faults.

The present invention proposes a new switched proportional-integral observer to accomplish this task. A switching item that can switch with the system working range is introduced in the switching proportional integral observer, and the state and fault estimation errors are analyzed. Finally, the condition that the system state estimation error and fault estimation error are bounded is given. The estimation effects of the switching proportional-integral observer and the traditional proportional-integral observer are compared through the embodiment. The results show that the switched proportional-integral observer is superior to the traditional proportional-integral observer, and this method can be used in future system control and fault-tolerant control.

Contents of the invention

The purpose of the present invention is to overcome the deficiencies of the prior art, and to provide a state and fault estimation method of a dead zone sandwich system, the switching proportional integral observer proposed by the method includes a switching vector that can vary with the system working range, which is different from the traditional Compared with the proportional-integral observer, this method can estimate the state and fault of the system at the same time, and the estimation error is smaller and the estimation speed is faster.

The technical scheme that realizes the object of the present invention is:

A state and fault estimation method for a dead zone sandwich system, comprising the following steps:

1) Using the principle of separation of key terms and switching functions, construct a non-smooth state-space equation that can accurately describe the gap sandwich system with faults. Due to the nonlinear characteristics of the dead zone, the size of the output is only related to the size of the input at the current moment. Regardless of the input and output at the previous moment, the state space equation of the system is established for the dead zone sandwich system with actuator faults, as follows:

1-1) Establish the state-space equation of the linear subsystem: According to the linear system theory, the state-space equation of the linear subsystem _L1 is as follows:

According to the linear system theory, the state space equation of the linear subsystem _L2 is as follows:

u∈R^{1 ×1}，y∈R^1×1，f∈R^1×1，u_f∈R^1×1，a_f∈R^1×1，b_f∈R^1×1，x₁₁和x₁₂分别代表L₁的第一个和第二个状态变量，x₂₁和x₂₂分别代表L₂的第一个和第二个状态变量，

分别为L₁、L₂的状态转移矩阵，

分别为L₁、L₂的输入矩阵，

分别为L₁、L₂的输出矩阵，

为故障矩阵，u∈R^1×1为输入，y∈R^1×1为输出，f∈R^1×1为系统的故障，a_f为故障环节的系数，b_f为未知的故障环节的输入系数；u_f∈R^1×1为故障环节的输入，是未知的；a_f和b_f由系统故障的先验知识获得，是已知的；假设u_f是有界的，a_f的范数小于1，即|a_f|＜1，因此根据线性系统稳定性条件，故障系统是稳定的，n_i为第i个线性系统的维数，设

且

in

^{u∈R1 ×1} , y∈R1 ^×1 , f∈R1 ^{×1, uf∈R1×} ₁ ^, _af∈R1 ^×1 , _bf∈R1 ^×1 , _x11 and _x12 represent the first and second state variables of L ₁ , respectively, x ₂₁ and x ₂₂ represent the first and second state variables of L ₂ , respectively,

are the state transition matrices of L ₁ and L ₂ respectively,

are the input matrices of L ₁ and L ₂ respectively,

are the output matrices of L ₁ and L ₂ respectively,

is the fault matrix, u∈R ^1×1 is the input, y∈R ^1×1 is the output, f∈R ^1×1 is the fault of the system, a _f is the coefficient of the fault link, b _f is the input of the unknown fault link coefficient; u _f ∈ R ^1×1 is the input of the fault link, which is unknown; a _f and b _f are obtained from the prior knowledge of system faults, and are known; assuming that u _f is bounded, the range of a _f The number is less than 1, that is, |a _f |<1, so according to the linear system stability condition, the fault system is stable, and n _i is the dimension of the i-th linear system, let

and

1-2) Establish the state space equation of the dead zone subsystem: define the intermediate variables m(k), w ₁ (k) as:

m(k)=m ₁ +(m ₂ -m ₁ )h(k)

w ₁ (k)=m(k)(x(k)-D ₁ h ₁ (k)+D ₂ h ₂ (k))

in

为切换函数，

is the switching function,

According to the input-output relationship of the dead zone:

v ₂ (k)=w ₁ (k)−h ₃ (k)w ₁ (k)=(1−h ₃ (k))w ₁ (k),

in

It is also a switching function, when h ₃ (k)=0, the system works in the linear zone, and v ₂ (k)=w ₁ (k); when h ₃ (k)=1, the system works in the dead zone, And v ₂ (k)=w ₁ (k)-w ₁ (k)=0, according to the input and output characteristics of the dead zone:

将公式(2)代入公式(3)中，得：because

Substituting formula (2) into formula (3), we get:

则直线电机驱动系统的状态空间方程如下所示：1-3) Establish the overall state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and

Then the state space equation of the linear motor drive system is as follows:

in

其中

0是相应阶数的零矩阵，设According to the characteristics of the sandwich system, only the output y(k) of the system can be directly measured, then let

in

0 is the zero matrix of the corresponding order, let

则公式(5)写成如下形式：

Then formula (5) can be written as follows:

Among them, _ηi is the switching vector introduced by considering the nonlinear characteristics of the dead zone in the system;

2) According to the non-smooth state space equation of the fault-containing gap sandwich system constructed in step 1), when the system satisfies the existence condition of the observer, construct the switching ratio that can automatically switch with the change of the fault-containing gap sandwich system working interval Integral observer, and give the existence condition and boundedness theorem of corresponding switching proportional integral observer, including the following steps:

2-1) For the dead-zone sandwich system, construct a switching proportional-integral observer that can estimate the system state and fault at the same time. The details of constructing the switching proportional-integral observer for the dead-zone sandwich system are as follows:

According to the state-space equation (6) of the dead zone sandwich system, the following switched proportional-integral observer is constructed:

为第i个工作区间的比例增益，

为第i个工作区间的积分增益；

分别为x(k)、y(k)、f(k)、η_i的估计值，若j＝1、2、3时，i＝j，则

in

is the proportional gain of the i-th working interval,

is the integral gain of the i-th working interval;

are the estimated values of x(k), y(k), f(k), and η _i respectively, if j=1, 2, 3, i=j, then

f(k)的表达式，则得到下述公式(8)、公式(9)：According to formula (7) and formula (1) and

f(k), the following formula (8) and formula (9) are obtained:

f(k+1)＝a _f f(k)+b _f u _f (k) (9)

则得到下述公式(10)：Subtract formula (9) from formula (8), define

Then the following formula (10) is obtained:

Subtract formula (7) from formula (6) to obtain the expression of the estimated error as follows:

e(k+1)=(AK _pj C)e(k)+ _Def (k)+Δη _os +ΔA _os x(k) (11)

ΔA_os＝A_j-A_i；in

ΔA _os =A _j -A _i ;

Write formula (10) and formula (11) together in matrix form, that is, the matrix form of state and fault estimation error is as follows formula (12):

Simplify formula (12), let

Then formula (12) is written in the form of multiplication of rectangles, as shown in the following formula:

e _t (k+1)=A _ej e _t (k)+ _Δt (k) (13)

即切换比例积分观测器的收敛条件为A_ej的特征值在单位圆内；2-2) If the norm of Δ _t (k) and initial estimation error e _t (1) is bounded, they are both smaller than φ _d , that is, φ _d (||Δ _t (k)||≤φ _d and| |e(1)||≤φ _d ), when j=1, 2, 3, if appropriate K _pj and K _ij are selected so that the eigenvalues of A _ej are within the unit circle, the state and fault estimation errors The norms of are all bounded, less than

That is, the convergence condition of switching the proportional integral observer is that the eigenvalues of A _ej are within the unit circle;

3) To prove the convergence of switching proportional integral observer: for any k, ||Δ _t (k)||≤φ _d and φ _d is the upper bound constant, the norm operation is performed on both sides of the formula (13), and the matrix Norm triangle inequality theorem, get the following formula (14):

Among them, ||A _ej || is any norm of A _ej , where j=1, 2, 3,

总存在一个范数||·||，使得下述公式(15)成立：According to the spectral radius theorem of matrices, for A _ej ∈ ^{R n×n} ,

There is always a norm ||·||, so that the following formula (15) holds:

||A _ej ||≤ρ(A _ej )+ε (15)

According to the above theorem, due to the selection of appropriate K _pj and K _ij the eigenvalues are within the unit circle, that is, the spectral norm of the matrix ρ(A _ej )<1, if the selection of appropriate ε satisfies 0<ε<1-ρ( A _ej ), according to formula (15):

||A _ej ||＜1 (16)

Therefore, from formula (14) and formula (16), when k→∞, the norm of state and fault estimation errors ||e _t (k)||) is less than

4) Estimate the state and fault of the non-smooth sandwich system in the dead zone: the following formula (17) and formula (18) are obtained from formula (7):

From formula (17), formula (18) get

According to formula (7) and formula (19), the state and fault of the non-smooth sandwich system in dead zone are estimated according to the following process:

并计算

令

4-1) Initialize the estimated value of the state variable and the estimated value of the fault: make

and calculate

make

4-2) Let k=3;

4-3) Determine whether k is less than or equal to N, if k≤N, then perform step 4-4); otherwise end the operation;

则进行如下操作：4-4) If

Then proceed as follows:

则进行如下操作：like

Then proceed as follows:

则进行如下操作：like

Then proceed as follows:

4-5) Add 1 to the value of k, and repeat step 4-3).

In step 4), most of the actual systems are not first-order systems, but second-order systems or can be simplified to second-order systems. If the two subsystems L ₁ and L ₂ of the non-smooth sandwich system are second-order systems, then the entire The system is fourth-order, with 4 state variables, and z means that each calculation step of the system has 4 states, that is, the first state of the k-th step, the second state of the k-th step, and the third state of the k-th step and the 4th state of the kth step, therefore, a matrix x with N rows and 4 columns is used to save all states, and the 4 states of the kth step are stored in columns 1 to 4 of the kth row of the matrix x, written as x(k ,[1:4]), abbreviated as x(k,:).

三种情况下所进行的操作是A、η、K_p、K_i4个不同的矩阵。In step 4-4),

The operations performed in the three cases are 4 different matrices A, η, K _p , and K _i .

The state and fault estimation method of a dead zone sandwich system provided by the present invention, compared with the traditional proportional integral observer estimation method, this method can estimate the state and fault of the dead zone sandwich system at the same time, and the estimation error convergence speed is fast and the precision is high .

Description of drawings

Figure 1 is a structural block diagram of a dead zone sandwich system with faults;

Figure 2 is a Simulink model diagram of a non-smooth sandwich system;

Fig. 3 is the schematic diagram of linear motor drive system;

Figure 4 is a diagram showing the estimation effect of the system state by switching the proportional integral observer when the system contains a step fault;

Fig. 5 is the estimation effect diagram of the system state by the traditional proportional-integral observer when the system contains a step fault;

Figure 6 is a comparison diagram of the state estimation error between the switching proportional-integral observer and the traditional proportional-integral observer when the system contains a step fault;

Figure 7 is a comparison diagram of the fault estimation by switching the proportional integral observer and the traditional proportional integral observer when the system contains a step fault;

Fig. 8 is an estimation effect diagram of the system state by switching the proportional-integral observer when the system contains a sinusoidal signal with amplitude attenuation;

Fig. 9 is an estimation effect diagram of the system state by the traditional proportional-integral observer when the system contains a sinusoidal signal with amplitude attenuation;

Figure 10 is a comparison diagram of the system state estimation error between the two methods of switching the proportional integral observer and the traditional proportional integral observer when the system contains a sinusoidal signal with amplitude attenuation;

Fig. 11 is a comparison diagram of system fault estimation by switching proportional integral observer and traditional proportional integral observer when the system contains a sinusoidal signal with amplitude attenuation.

Detailed ways

The content of the present invention will be further elaborated below in conjunction with the accompanying drawings and embodiments, but the present invention is not limited thereto.

Example:

A state and fault estimation method for a dead zone sandwich system, comprising the following steps:

1) Using the principle of separation of key terms and switching functions, construct a non-smooth state-space equation that can accurately describe the gap sandwich system with faults. Due to the nonlinear characteristics of the dead zone, the size of the output is only related to the size of the input at the current moment. Regardless of the input and output at the previous moment, the state space equation of the system is established for the dead zone sandwich system with actuator faults, as follows:

As shown in Figure 1, the figure shows a dead-zone sandwich system with actuator faults, where u(k) and y(k) are measurable input and output variables, respectively, and v ₁ (k) and v ₂ ( k) is an unmeasurable intermediate variable, L ₁ represents the pre-Ray linear subsystem, L ₂ represents the post-Ray linear subsystem, D ₁ and D ₂ are the width of the dead zone, m ₁ and m ₂ are the slope of the linear region, f (k) represents the actuator failure of the system.

1-1) Establish the state-space equation of the linear subsystem: According to the linear system theory, the state-space equation of the linear subsystem _L1 is as follows:

According to the linear system theory, the state space equation of the linear subsystem _L2 is as follows:

u∈R^{1 ×1}，y∈R^1×1，f∈R^1×1，u_f∈R^1×1，a_f∈R^1×1，b_f∈R^1×1，x₁₁和x₁₂分别代表L₁的第一个和第二个状态变量，x₂₁和x₂₂分别代表L₂的第一个和第二个状态变量，

分别为L₁、L₂的状态转移矩阵，

分别为L₁、L₂的输入矩阵，

分别为L₁、L₂的输出矩阵，

为故障矩阵，u∈R^1×1为输入，y∈R^1×1为输出，f∈R^1×1为系统的故障，a_f为故障环节的系数，b_f为未知的故障环节的输入系数；u_f∈R^1×1为故障环节的输入，是未知的；a_f和b_f由系统故障的先验知识获得，是已知的；假设u_f是有界的，a_f的范数小于1，即|a_f|＜1，因此根据线性系统稳定性条件，故障系统是稳定的，n_i为第i个线性系统的维数，设

且

in

^{u∈R1 ×1} , y∈R1 ^×1 , f∈R1 ^{×1, uf∈R1×} ₁ ^, _af∈R1 ^×1 , _bf∈R1 ^×1 , _x11 and _x12 represent the first and second state variables of L ₁ , respectively, and x ₂₁ and x ₂₂ represent the first and second state variables of L ₂ , respectively,

are the state transition matrices of L ₁ and L ₂ respectively,

are the input matrices of L ₁ and L ₂ respectively,

are the output matrices of L ₁ and L ₂ respectively,

is the fault matrix, u∈R ^1×1 is the input, y∈R ^1×1 is the output, f∈R ^1×1 is the fault of the system, a _f is the coefficient of the fault link, b _f is the input of the unknown fault link coefficient; u _f ∈ R ^1×1 is the input of the fault link, which is unknown; a _f and b _f are obtained from the prior knowledge of system faults, and are known; assuming that u _f is bounded, the range of a _f The number is less than 1, that is, |a _f |<1, so according to the linear system stability condition, the fault system is stable, and n _i is the dimension of the i-th linear system, let

and

1-2) Establish the state space equation of the dead zone subsystem: define the intermediate variables m(k), w ₁ (k) as:

m(k)=m ₁ +(m ₂ -m ₁ )h(k)

w ₁ (k)=m(k)(x(k)-D ₁ h ₁ (k)+D ₂ h ₂ (k))

in

为切换函数，

is the switching function,

According to the input-output relationship of the dead zone:

v ₂ (k)=w ₁ (k)−h ₃ (k)w ₁ (k)=(1−h ₃ (k))w ₁ (k),

in

It is also a switching function, when h ₃ (k)=0, the system works in the linear zone, and v ₂ (k)=w ₁ (k); when h ₃ (k)=1, the system works in the dead zone, And v ₂ (k)=w ₁ (k)-w ₁ (k)=0, according to the input and output characteristics of the dead zone:

将公式(2)代入公式(3)中，得：because

Substituting formula (2) into formula (3), we get:

则直线电机驱动系统的状态空间方程如下所示：1-3) Establish the overall state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and

Then the state space equation of the linear motor drive system is as follows:

in

其中

0是相应阶数的零矩阵，设According to the characteristics of the sandwich system, only the output y(k) of the system can be directly measured, then let

in

0 is the zero matrix of the corresponding order, let

则公式(5)写成如下形式：

Then formula (5) can be written as follows:

Among them, _ηi is the switching vector introduced by considering the nonlinear characteristics of the dead zone in the system;

2) According to the non-smooth state space equation of the fault-containing gap sandwich system constructed in step 1), when the system satisfies the existence condition of the observer, construct the switching ratio that can automatically switch with the change of the fault-containing gap sandwich system working interval Integral observer, and give the existence condition and boundedness theorem of corresponding switching proportional integral observer, including the following steps:

2-1) For the dead-zone sandwich system, construct a switching proportional-integral observer that can estimate the system state and fault at the same time. The details of constructing the switching proportional-integral observer for the dead-zone sandwich system are as follows:

According to the state-space equation (6) of the dead zone sandwich system, the following switched proportional-integral observer is constructed:

为第i个工作区间的比例增益，

为第i个工作区间的积分增益；

分别为x(k)、y(k)、f(k)、η_i的估计值，若j＝1、2、3时，i＝j，则

in

is the proportional gain of the i-th working interval,

is the integral gain of the i-th working interval;

are the estimated values of x(k), y(k), f(k), and η _i respectively, if j=1, 2, 3, i=j, then

f(k)的表达式，则得到下述公式(8)、公式(9)：According to formula (7) and formula (1) and

f(k), the following formula (8) and formula (9) are obtained:

f(k+1)＝a _f f(k)+b _f u _f (k) (9)

则得到下述公式(10)：Subtract formula (9) from formula (8), define

Then the following formula (10) is obtained:

Subtract formula (7) from formula (6) to obtain the expression of the estimated error as follows:

e(k+1)=(AK _pj C)e(k)+ _Def (k)+Δη _os +ΔA _os x(k) (11)

ΔA_os＝A_j-A_i；in

ΔA _os =A _j -A _i ;

Write formula (10) and formula (11) together in matrix form, that is, the matrix form of state and fault estimation error is as follows formula (12):

Simplify formula (12), let

Then formula (12) is written in the form of multiplication of rectangles, as shown in the following formula:

e _t (k+1)=A _ej e _t (k)+ _Δt (k) (13)

即切换比例积分观测器的收敛条件为A_ej的特征值在单位圆内；2-2) If the norm of Δ _t (k) and initial estimation error e _t (1) is bounded, they are both smaller than φ _d , that is, φ _d (||Δ _t (k)||≤φ _d and| |e(1)||≤φ _d ), when j=1, 2, 3, if appropriate K _pj and K _ij are selected so that the eigenvalues of A _ej are within the unit circle, the state and fault estimation errors The norms of are all bounded, less than

That is, the convergence condition of switching the proportional-integral observer is that the eigenvalues of A _ej are within the unit circle;

3) To prove the convergence of switching proportional integral observers: for any k, || _Δt (k)|| _≤φd and _φd is the upper bound constant, then the norm operation is performed on both sides of formula (13), and the matrix Norm triangle inequality theorem, get the following formula (14):

Among them, ||A _e j|| is any norm of A _ej , where j=1, 2, 3,

总存在一个范数||·||，使得下述公式(15)成立：According to the spectral radius theorem of matrices, for A _ej ∈ ^{R n×n} ,

There is always a norm ||·||, so that the following formula (15) holds:

||A _ej ||≤ρ(A _ej )+ε (15)

According to the above theorem, due to the selection of appropriate K _pj and K _ij the eigenvalues are within the unit circle, that is, the spectral norm of the matrix ρ(A _ej )<1, if the selection of appropriate ε satisfies 0<ε<1-ρ( A _ej ), according to formula (15):

||A _ej ||＜1 (16)

Therefore, from formula (14) and formula (16), when k→∞, the norm of state and fault estimation errors ||e _t (k)||) is less than

As shown in Figure 2, the Simulink model of a non-smooth sandwich system is mainly composed of ideal input, summation module, controller, L1 subsystem, dead zone link, L2 subsystem, feedback gain and connecting lines, and may also include fault signals , such as L1 fault signal, L2 fault signal and dead zone increase module.

4) Estimate the state and fault of the non-smooth sandwich system in the dead zone: the following formula (17) and formula (18) are obtained from formula (7):

From formula (17), formula (18) get

According to formula (7) and formula (19), the state and fault of the non-smooth sandwich system in dead zone are estimated according to the following process:

并计算

令

4-1) Initialize the estimated value of the state variable and the estimated value of the fault: make

and calculate

make

4-2) Let k=3;

4-3) Determine whether k is less than or equal to N, if k≤N, then perform step 4-4); otherwise end the operation;

则进行如下操作：4-4) If

Then proceed as follows:

则进行如下操作：like

Then proceed as follows:

则进行如下操作：like

Then proceed as follows:

4-5) Add 1 to the value of k, and repeat step 4-3).

In step 4), most of the actual systems are not first-order systems, but second-order systems or can be simplified to second-order systems. If the two subsystems L ₁ and L ₂ of the non-smooth sandwich system are second-order systems, then the entire The system is fourth-order, with 4 state variables, and z means that each calculation step of the system has 4 states, that is, the first state of the k-th step, the second state of the k-th step, and the third state of the k-th step and the 4th state of the kth step, therefore, a matrix x with N rows and 4 columns is used to save all states, and the 4 states of the kth step are stored in columns 1 to 4 of the kth row of the matrix x, written as x(k ,[1:4]), abbreviated as x(k,:).

三种情况下所进行的操作是A、η、K_p、K_i4个不同的矩阵。In step 4-4),

The operations performed in the three cases are 4 different matrices A, η, K _p , and K _i .

The above method is used to estimate the state and fault of the DC motor drive simulation system, and compared with the results of state and fault estimation using the traditional proportional-integral observer, the details are as follows:

As shown in Figure 3, the schematic diagram of the linear motor drive system, in which the control circuit includes circuit amplifiers and filters, etc., can be regarded as a linear subsystem L ₁ ; the load can be regarded as a linear subsystem L ₂ ; the linear motor is composed of a magnetic field winding and an electrical Composed of pivot windings, brackets, etc., there is Coulomb friction in it; the linear motor is equivalent to a dead zone link, which is represented by DZ; therefore, the entire linear motor drive system can be regarded as a dead zone sandwich system; this system is mainly used for requiring output linear Displacement and automatic control system.

The simulation model of the linear motor drive system is as follows:

The input signal is u=20sin(0.8t), the sampling time is 80s, the sampling period is 0.01s, and the initial values of all states and faults are set to 0; then:

Linear subsystem L ₁ :

Linear subsystem L ₂ :

dead zone:

Table 1 shows the four state variables x ₁₁ , x ₁₂ , x ₂₁ , x ₂₂ of the linear motor drive system and their respective physical meanings.

Table 1 State variables of the linear motor drive system

Therefore, with the state space equation of the system and the system parameters, the corresponding matrix can be obtained as follows.

C=[0 0 0 00], and D=[0.04107 0 0 0] ^T .

The traditional observer regards the dead zone link as a linear link and replaces it with a proportional link, ignoring the switching between the linear zone and the dead zone. Therefore, the switching item is not included in the traditional proportional-integral observer, and the specific expression is as follows:

in

Now simulate two types of faults:

The first type of fault, assuming there is a step fault at 10, represents a sudden fault in practical applications, a _f =0.85, b _f =1, when 0≤t≤10, u _f (t)=0, that is, the former There is no fault in 10 seconds; when 10<t≤80, u _f (t) is 0.2, the sampling frequency is 100Hz, according to the convergence condition of switching proportional integral observer, select K _pj =[0.1, 0.1, 0.1, 0.1] ^T and K _ij =1, when j=1, 3, the characteristic value of Ae is [0.4480, 0.8075, 0.7975+0.0406i, 0.7975-0.0406i, 0.9494] ^T ; when j=2, the characteristic value of Ae is [0.4500, 0.8000, 0.9500, 0.8000+0.0316i, 0.8000-0.0316i]] ^T are all within the unit circle, that is, whether the system works in the linear zone or the dead zone, the eigenvalues of Ae are all within the unit circle.

By comparing Figure 4 and Figure 5, it can be clearly seen that the switched proportional-integral observer can estimate the state of the system more accurately than the traditional proportional-integral observer; the state estimation error between the switched proportional-integral observer and the traditional observer is shown in Figure 6 , from Fig. 6, it can be seen that the estimation error of the switched proportional-integral observer is smaller than that of the traditional proportional-integral observer. When there is a step fault at 10s, the state estimation effects of the switched proportional-integral observer and the traditional observer are shown in Fig. 4 and As shown in Figure 5, the fault estimation effect of the switched proportional-integral observer and the traditional observer is shown in Figure 7. From Figure 7, it can be seen that the switched proportional-integral observer can track the fault signal in time and accurately, but the traditional proportional-integral observer has a little The fault signal cannot be tracked either; in conclusion, the switched proportional-integral observer has better performance than the conventional proportional-integral observer in terms of state estimation and fault estimation.

It is assumed that the second type of fault is a sinusoidal signal with attenuated amplitude, which represents a slowly changing fault in practical applications. The sampling frequency is also 100Hz, af=0.8 and bf=1, when 0≤t≤10, u _f (t)=0, that is, there is no fault in the first 10 seconds; when 10<t≤80, the fault For u _f (t)=e ^(-(t-30)/10) sin(π(t-10)/10+1)+4, choose Kpj=[0.0575, -0.0998, -0.0960, 0.0201] ^T , K _ij =1; when j=1, 3, the characteristic value of Ae is [0.4517, 0.8930, 0.7776+0.0235i, 0.7776-0.0235i, 0.8301] ^T ; when j=2, the characteristic value of Ae is [0.4500 , 0.8000, 0.8000, 0.8905, 0.7894] ^T , all within the unit circle.

When the fault is a sinusoidal signal with amplitude attenuation, the state estimation effects of switching the proportional-integral observer and the traditional observer are shown in Fig. 8 and Fig. 9 . From Fig. 8-11, it can be seen that the switched proportional-integral observer has higher precision than the traditional observer model, therefore, the switched proportional-integral observer has a better state estimation effect than the traditional observer.

Claims (1)

1. A method for estimating the state and fault of a dead zone sandwich system is characterized by comprising the following steps:

1) The method comprises the following steps of constructing a non-smooth state space equation capable of accurately describing a gap sandwich system with faults by using a key item separation principle and a switching function, wherein the characteristic of the nonlinear characteristic of a dead zone is that the size of output is only related to the size of input at the current moment and is unrelated to the input and the output at the previous moment, and aiming at the dead zone sandwich system with actuator faults, the state space equation of the system is established, and the method specifically comprises the following steps:

1-1) establishing a state space equation of the linear subsystem: according to the linear system theory, the linear subsystem L ₁ The state space equation of (a) is as follows:

according to the linear system theory, the linear subsystem L ₂ The state space equation of (a) is as follows:

wherein

For the first linear subsystem L ₁ The state variable of (a) is changed,

for the second linear subsystem L ₂ The state variable of (a) is changed,

u∈R ^1×1 ，y∈R ^1×1 ，f∈R ^1×1 ，u _f ∈R ^1×1 ，a _f ∈R ^1×1 ，b _f ∈R ^1×1 ，

are respectively L ₁ 、L ₂ The state transition matrix of (a) is,

are respectively L ₁ 、L ₂ The input matrix of (a) is selected,

are respectively L ₁ 、L ₂ The output matrix of (a) is obtained,

for the failure matrix, u ∈ R ^1×1 As input, y ∈ R ^1×1 For output, f ∈ R ^1×1 Is a failure of the system, a _f As a factor of a failed link, b _f Input coefficients for unknown fault links; u. of _f ∈R ^1×1 The input for the failed link is unknown; a is _f And b _f Obtained from a priori knowledge of system faults, known; suppose u _f Is bounded, a _f Is less than 1, i.e. | a _f L < 1, so the fault system is stable according to the linear system stability condition, n _i For the dimension of the i-th linear system, let

And is

1-2) establishing a state space equation of the dead zone subsystem: defining intermediate variables m (k), w ₁ (k) Comprises the following steps:

m(k)＝m ₁ +(m ₂ -m ₁ )h(k)

w ₁ (k)＝m(k)(v ₁ (k)-D ₁ h ₁ (k)+D ₂ h ₂ (k))

wherein the parameter D ₁ To a positive dead zone width, D ₂ Is the negative dead zone width, m ₁ Is a positive linear region slope, m ₂ Is the slope of the negative linear region,

in order to be a function of the switching,

obtaining the following according to the input-output relation of the dead zone:

v ₂ (k)＝w ₁ (k)-h ₃ (k)w ₁ (k)＝(1-h ₃ (k))w ₁ (k)，

wherein

Also as a switching function, when h ₃ (k) When =0, the system operates in the linear region, and v ₂ (k)＝w ₁ (k) (ii) a When h is generated ₃ (k) When =1, the system is operating in a dead zone, and v ₂ (k)＝w ₁ (k)-w ₁ (k) =0, the input/output characteristic according to the dead zone is obtained:

due to the fact that

Substituting the formula (3) into the formula (2) to obtain:

wherein, the symbol "·" is a multiplication sign, and the function of the symbol "·" is multiplication operation;

1-3) establishing an integral state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and

the state space equation of the linear motor drive system is as follows:

wherein

According to the characteristics of the sandwich system, only the output y (k) of the system can be directly measured, so that

Wherein

0 is a zero matrix of the corresponding order

Equation (5) is written as follows:

wherein eta _i Switching vectors introduced for considering the existence of dead zone non-linear characteristics of the system;

2) According to the non-smooth state space equation of the gap sandwich system with the fault, which is constructed in the step 1), when the system meets the existence condition of the observer, a switching proportional-integral observer capable of automatically switching along with the change of the working interval of the gap sandwich system with the fault is constructed, and the existence condition and the bounded theorem of the corresponding switching proportional-integral observer are given, which comprises the following steps:

2-1) aiming at the dead zone sandwich system, a switching proportional-integral observer capable of estimating the state and the fault of the system simultaneously is constructed, and the switching proportional-integral observer for constructing the dead zone sandwich system is specifically as follows:

according to the state space equation (6) of the dead zone sandwich system, the following switching proportional-integral observer is constructed:

wherein

Proportional gain for the ith operating interval, K _ij ∈R ^1x1 Is the integral gain of the ith working interval;

x (k), y (k), f (k), eta, respectively _i If j =1,2,3, i = j, then

v ₁ (k) As input to the first linear subsystem, according to equation (7) and equation (1) and

f (k), the following formula (8) and formula (9) are obtained:

f(k+1)＝a _f f(k)+b _f u _f (k) (9)

wherein, K _i In order to integrate the gain of the gain,

subtracting equation (9) from equation (8) defines

The following formula (10) is obtained:

equation (7) is subtracted from equation (6) to obtain the following expression for the estimation error:

e(k+1)＝(A _j -K _pj C)e(k)+De _f (k)+Δη _os +ΔA _os x(k) (11)

wherein

ΔA _os ＝A _j -A _i ；

The equations (10) and (11) are written together in a matrix form, that is, the matrix form of the state and fault estimation errors is as follows in equation (12):

simplifying the formula (12) by

Equation (12) is written in the form of a rectangular multiplication, as shown in the following equation:

e _t (k+1)＝A _ej e _t (k)+Δ _t (k) (13)

2-2) setting Δ _t (k) Norm sum e of _t (1) The norm of (a) is bounded upward, and the bounded upward of the norms of both (b) is smaller than phi _d Expressed as | | Δ by mathematical expression _t (k)||≤φ _d ，||e _t (1)||≤φ _d Wherein e (1) is the initial estimation error, | | Δ _t (k) | | is Δ _t (k) Arbitrary norm, | | e _t (1) | is e _t (1) When j =1,2,3, if the appropriate K is selected _pj And K _ij So that A is _ej Is within the unit circle, the norms of the state and fault estimation errors are bounded and both are less than

I.e. switching the convergence condition of the proportional-integral observer to A _ej Characteristic value of (2) in unit circleInternal;

3) The convergence of the switching proportional-integral observer is proved: for any k, | | Δ _t (k)||≤φ _d And phi is _d If the matrix norm is an upper bound constant, then norm operations are performed on two sides of the formula (13), and the following formula (14) is obtained by the trigonometric inequality theorem of the matrix norm:

wherein, | | A _ej I is A _ej Where j =1,2,3,

according to the theorem of the spectral radius of the matrix, for A _ej ∈R ^n×n ，

There is always a norm | · | |, such that the following equation (15) holds:

||A _ej ||≤ρ(A _ej )+ε (15)

according to the above theorem, since a suitable K is selected _pj 、K _ij So that A is _ej Is within the unit circle, i.e. the spectral norm ρ (a) of the matrix _ej ) If the proper epsilon is selected, the epsilon is more than 0 and less than 1-rho (A) _ej ) According to equation (15), there are:

||A _ej ||＜1 (16)

therefore, from equation (14) and equation (16), when k → ∞ the norm of the state and fault estimation errors | | | e _t (k) Less than | |

4) Estimating the state and fault of the dead zone non-smooth sandwich system: the following formula (17) and formula (18) are obtained from formula (7):

is obtained by the formula (17) and the formula (18)

According to the formula (7) and the formula (19), the estimation of the state and the fault of the dead zone non-smooth sandwich system is carried out according to the following process:

4-1) initializing the estimated value of the state variable and the estimated value of the fault: order to

And calculate

Order to

4-2) let k =3;

4-3) judging whether k is less than or equal to N, and if k is less than or equal to N, executing the step 4-4); otherwise, ending the operation;

4-4) if

The following operations are performed:

if it is

The following operations are performed:

if it is

The following operations are performed:

wherein the parameters

Meaning the estimated values of all states in step k,

two subsystems L of a non-smooth sandwich system ₁ 、L ₂ The system is a second-order system, the whole system is a fourth-order system and has 4 state variables, and in the process of solving the system differential equation by adopting an iterative method, 4 state quantities of the system are calculated in each step, namely the 1 st state of the k step, the 2 nd state of the k step, the 3 rd state of the k step and the 4 th state of the k step, so that all the states are stored by using a matrix x with N rows and 4 columns, and the 4 states of the k step are stored in 1 to 4 columns of the kth row of the matrix x and written into x (k, [ 1:4)]) Abbreviated as x (k,: in this case),

4-5) adding 1 to the value of k, repeating steps 4-3).

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