CN108227679B - State estimation method of gap sandwich system with fault - Google Patents

Abstract

The invention discloses a state estimation method of a gap sandwich system with faults, which comprises the steps of firstly, constructing a non-smooth state space equation capable of accurately describing the gap sandwich system with the faults by using a key separation principle and a switching function and by referring to a constructed sandwich state space equation with dead zones and gaps from simple to complex; secondly, according to the non-smooth state space equation, when the system meets the existing condition of the observer, a switching proportional-integral observer capable of switching along with the change of the working interval of the system is constructed. The method has the advantages that: the system is more accurately described by introducing a switching function; compared with the traditional proportional-integral observer, the observer adopting the method can estimate the state and the fault of the system more accurately.

Description

State estimation method of gap sandwich system with fault

Technical Field

The invention belongs to the field of state estimation of nonlinear systems, and particularly relates to a state estimation method of a gap sandwich system with a fault.

Background

Backlash is typically present in gear mechanical transmission systems, electrically operated valves, digital circuits, sensors, hydraulic systems. For example, in a gear system, backlash between gear teeth can produce backlash. In practice, the gap does not exist alone, but is sandwiched between other links, i.e. the gap non-linearity is sandwiched between two linear dynamic subsystems, which can be described as a gap sandwich system. Faults often occur in mechanical and electronic systems. The presence of a fault may cause the system to deteriorate, vibrate, oscillate, and even become unstable. Therefore, in the actual controller design and fault tolerant control, it is important to accurately estimate the state and fault of the system.

Constructing a corresponding observer for a particular system has always been a research hotspot in the field of control engineering. The design theory and methodology for linear stationary system observers matured since the well-known Luenberger observer was proposed in the 70 s of this century d.j. But is different for the nonlinear system, firstly, the visibility of the nonlinear system is a local characteristic; secondly, the visibility of the linear system is independent of the system input and only depends on the structure of the system, while the visibility of the nonlinear system is not only related to the system structure but also related to the system input. Due to the complexity of the nonlinear system, it is difficult to find a unified observer constructing method for the nonlinear system, and a specific observer is often constructed for a certain type of nonlinear system. For example, morning light in 2017 and the like propose an extended state observer based on state compensation for a discrete system, which realizes more accurate state estimation and interference suppression. Soken H E in 2014 proposes a Lubang Kalman filtering method, and under the condition that a measuring system has faults, the attitude of the spacecraft is estimated. In 2015, an internal model observer is designed for a linear system with time delay by Efimov D and the like, and when a transfer function from fault input to error meets a specified H-infinity norm, the system is stable for dynamic estimation error.

Chinese patent CN105204332B discloses a state estimation method for a composite sandwich system containing dead zone and hysteresis based on a non-smooth observer. The invention estimates the state of the system under the condition that the sandwich system does not contain faults. However, in an actual system, a fault is often inevitable, and the state estimation of the system generates certain disturbance due to the existence of the fault, and even the estimation error is diverged, that is, the state of the system cannot be estimated. The invention also does not estimate system faults as the fault effects are ignored.

So far, no patent and literature has been found for simultaneously evaluating gap sandwich systems containing faults. The present invention proposes a new switching observer to do this. And introducing a switching item which can be switched along with the working interval of the system into the switching proportional-integral observer, and analyzing the state and the fault estimation error. And finally, giving the condition that the estimation error of the system state and the estimation error of the fault are bounded. The estimation effects of the switching proportional-integral observer and the conventional proportional-integral observer were compared by the embodiment. The results show that the switching proportional-integral observer is superior to the conventional proportional-integral observer. The state and fault estimates of the system may be used for future system control and fault tolerant control.

Disclosure of Invention

Aiming at the technical defects, the invention discloses a state estimation method for a gap sandwich system with faults based on a switching proportional-integral observer, the switching proportional-integral observer provided by the method comprises a switching vector which can change along with a system working interval, and compared with a traditional observer, the observer adopting the method can more accurately estimate the state and the faults of the system.

The technical scheme for realizing the purpose of the invention is as follows:

a method of estimating a state of a gap sandwich system including a fault, comprising the steps of:

step 1: constructing a non-smooth state space equation capable of accurately describing the gap sandwich system with the fault by using a key item separation principle and a switching function;

step 2: and (2) constructing a non-smooth state space equation of the clearance sandwich system containing the fault according to the step 1, constructing a switching proportional-integral observer capable of automatically switching along with the change of the working interval of the clearance sandwich system containing the fault when the system meets the existence condition of the observer, and giving the existence condition and the boundedness theorem of the corresponding switching proportional-integral observer.

The step 1 comprises the following steps:

(1) front-end linear subsystem L with faulty gap sandwich system₁The state space equation of (a) is:

(2) back end linear subsystem L with faulty gap sandwich system₂The state space equation of (a) is:

the above formula (1) and formula (2)

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，i＝1，2，x_1iAnd x_2iEach represents L₁And L₂In the (i) th state of (c),

in order to be a state transition matrix,

in order to input the matrix, the input matrix is,

to be the output matrix, the output matrix is,

for the failure matrix, u ∈ R^1×1As input, y ∈ R^1×1For output, f ∈ R^1×1Is a fault of the system, can be regarded as_fAs a factor of a failed link, b_fInput coefficient for faulty link, u_f∈R^1×1For a failed link that is the input to the failed link, assume u_fIs bounded, a_fIs less than 1, i.e. | a_f|<1, so the fault system is stable according to the linear system stability condition; n is_iDimension of the ith linear system; is provided with

And is

(3) The state space equation for the clearance subsystem:

in the gap with faultIn the therapeutic system, v₁(k) And v₂(k) Defining intermediate variables m (k) as the input and output of the interval respectively:

defining an intermediate variable w₁(k) Comprises the following steps:

w₁(k)＝m(k)(v₁(k)-D₁g₁(k)+D₂g₂(k))， (4)

wherein,

and is

The input-output relationship according to the gap can be:

v₂(k)＝w₁(k)+[v₂(k-1)-w₁(k)]g₃(k)＝(1-g₃(k))w₁(k)+g₃(k)v₂(k-1)， (5)

wherein

According to the formula (2), the formula (4) and the formula (5):

(4) the overall equation of state for the gap sandwich system containing the fault:

according to the formulae (1), (2), (6), and x_1n1(k)＝v₁(k) The state space equation of the system can be obtained as follows:

wherein

Wherein,

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

Wherein

0 is a zero matrix of the corresponding order

Then

Wherein h is_iIs the switching vector due to the presence of the gap.

The step 2 comprises the following steps:

(1) modeling of switching proportional-integral observer

According to the formula (8) in the step 1, a switching proportional integral observer shown as the following formula is established:

wherein

And

the ith working interval is proportional gain and integral gain respectively,

x (k), y (k), f (k), h_iWherein when i is j, j is 1,3,

when the j is 2, the sum of the j,

(2) estimation error analysis of proportional-integral observer

From the formula (9) and the formula (1), the following formula (10) and formula (11) can be obtained:

f(k+1)＝a_ff(k)+b_fu_f(k) (11)

equation (11) is subtracted from equation (10), and

and

then:

subtracting equation (8) from equation (9) and considering the interval estimation error is:

e(k+1)＝(A-K_pjC)e(k)+De_f(k)+Δη_os+ΔA_osx(k)。 (13)

definition of

ΔA_os＝A_j-A_i，

From equations (12) and (13), the state and fault estimation errors can be found as:

definition of

And is

Then it can be obtained:

e_t(k+1)＝A_eje_t(k)+Δ_t(k)。 (15)

let Δ_t(k) And initial estimation error e_t(1) The norm of (a) is bounded and all are less than phi_dI.e. phi_d(||Δ_t(k)||≤φ_dAnd | | | e (1) | | is less than or equal to phi |_d) When j is 1, 2, 3, an appropriate K is selected_pjAnd K_ijSo that A is_ejIs within the unit circle, the norms of the state and fault estimation errors are bounded and both are less than

Therefore, the condition that the switching proportional-integral observer has a bounded system state estimation error and fault estimation error is A_ejIs within the unit circle.

The invention has the beneficial effects that: according to the state estimation method of the gap sandwich system with the fault, which is provided by the invention, the gap sandwich system with the fault can be more accurately described by introducing the switching function, the model precision is higher, and compared with the traditional proportional-integral observer, the proportional-integral observer constructed by adopting the method can more accurately estimate the state and the fault of the system.

Drawings

FIG. 1 is a block diagram of a gap sandwich system containing a fault;

FIG. 2 is a schematic diagram of a servo hydraulic system;

FIG. 3 is a state result diagram of a switching proportional-integral observer under a step fault;

FIG. 4 is a state result diagram of a conventional proportional-integral observer under a step fault;

FIG. 5 is a comparison diagram of state errors of a switching proportional-integral observer and a conventional proportional-integral observer under a step fault;

FIG. 6 is a comparison graph of the fault estimation of the switching proportional-integral observer and the conventional proportional-integral observer under the condition of step fault;

FIG. 7 is a state result diagram of a switching proportional-integral observer under a sinusoidal fault with attenuated amplitude;

FIG. 8 is a state result diagram of a conventional proportional-integral observer under a sinusoidal fault with attenuated amplitude;

FIG. 9 is a comparison diagram of state errors of a switching proportional-integral observer and a conventional proportional-integral observer under a sinusoidal fault with attenuated amplitude;

fig. 10 is a comparison graph of the fault estimation of the switching proportional-integral observer and the conventional proportional-integral observer under the sinusoidal fault with the amplitude attenuation.

Detailed description of the preferred embodiments

The invention is further illustrated but not limited by the following figures and examples.

Example (b):

a method of estimating a state of a gap sandwich system including a fault, comprising the steps of:

step 1: constructing a non-smooth state space equation capable of accurately describing the gap sandwich system with the fault by using a key item separation principle and a switching function;

step 2: and (2) constructing a non-smooth state space equation of the clearance sandwich system containing the fault according to the step 1, constructing a switching proportional-integral observer capable of automatically switching along with the change of the working interval of the clearance sandwich system containing the fault when the system meets the existence condition of the observer, and giving the existence condition and the boundedness theorem of the corresponding switching proportional-integral observer.

The step 1 comprises the following steps:

(1) front-end linear subsystem L with faulty gap sandwich system₁The state space equation of (a) is:

(2) back end linear subsystem L with faulty gap sandwich system₂The state space equation of (a) is:

the above formula (1) and formula (2)

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,i＝1，2，x_1iAnd x_2iEach represents L₁And L₂In the (i) th state of (c),

in order to be a state transition matrix,

in order to input the matrix, the input matrix is,

to be the output matrix, the output matrix is,

for the failure matrix, u ∈ R^1×1Is input into，y∈R^1×1For output, f ∈ R^1×1Is a fault of the system, can be regarded as_fAs a factor of a failed link, b_fInput coefficient for faulty link, u_f∈R^1×1For a failed link that is the input to the failed link, assume u_fIs bounded, a_fIs less than 1, i.e. | a_f|<1, so the fault system is stable according to the linear system stability condition; n is_iDimension of the ith linear system; is provided with

And is

(3) The state space equation for the clearance subsystem:

in a gap-containing sandwich system with a fault, v is shown in FIG. 1₁(k) And v₂(k) Defining intermediate variables m (k) as the input and output of the interval respectively:

defining an intermediate variable w₁(k) Is composed of

w₁(k)＝m(k)(v₁(k)-D₁g₁(k)+D₂g₂(k))， (4)

Wherein,

and is

The input-output relationship according to the gap can be:

v₂(k)＝w₁(k)+[v₂(k-1)-w₁(k)]g₃(k)＝(1-g₃(k))w₁(k)+g₃(k)v₂(k-1)， (5)

wherein

The following formula (2), formula (4) and formula (5) can be obtained:

(4) the overall equation of state for the gap sandwich system containing the fault:

according to the formulae (1), (2), (6), and x_1n1(k)＝v₁(k) The state space equation of the system can be obtained as follows:

wherein

Wherein,

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

Wherein

0 is a zero matrix of the corresponding order

Then

η therein_iIs the switching vector due to the presence of the gap.

The step 2 comprises the following steps:

(1) modeling of switching proportional-integral observer

According to the formula (8) in the step 1, a switching proportional integral observer shown as the following formula is established:

wherein

And

the ith working interval is proportional gain and integral gain respectively,

x (k), y (k), f (k), η_iWherein when i is j, j is 1,3,

when the j is 2, the sum of the j,

(2) estimation error analysis of proportional-integral observer

From the formula (9) and the formula (1), the following formula (10) and formula (11) can be obtained:

f(k+1)＝a_ff(k)+b_fu_f(k) (11)

equation (11) is subtracted from equation (10), and

and

then

Subtracting equation (8) from equation (9) and considering the interval estimation error is:

e(k+1)＝(A-K_pjC)e(k)+De_f(k)+Δη_os+ΔA_osx(k) (13)

definition of

ΔA_os＝A_j-A_i，

From equations (12) and (13), the state and fault estimation errors can be found as:

definition of

And is

Then it can be obtained:

e_t(k+1)＝A_eje_t(k)+Δ_t(k) (13)

let Δ_t(k) And initial estimation error e_t(1) The norm of (a) is bounded and all are less than phi_dI.e. phi_d(||Δ_t(k)||≤φ_dAnd | | | e (1) | | is less than or equal to phi |_d) When j is 1, 2, 3, an appropriate K is selected_pjAnd K_ijSo that A is_ejIs within the unit circle, the norms of the state and fault estimation errors are bounded and both are less than

The condition existing in the proportional-integral observer, i.e. the condition that the system state estimation error and the fault estimation error are bounded, is a_ejIs within the unit circle.

Taking the servo hydraulic system shown in fig. 2 as an example, in the servo hydraulic system shown in fig. 2, the dc motor can be regarded as a front-end linear subsystem L₁The load can be regarded as a back-end linear subsystem L₂The gear transmission system composed of the components such as the gear, the screw rod and the nut has the clearance characteristic BL, so that the servo hydraulic system can be regarded as a clearance sandwich system. In practical applications, the effect of the servo hydraulic system is mainly a force amplification. The input signal u (t) of the system in the simulation is 2sin (0.4 pi t), the sampling time of the first fault and the second fault is 120s and 240s respectively, the sampling period is 0.01s, and all initial values of the state and the fault are set to be 0.

Linear subsystem L₁：

Linear subsystem L₂:

Gap BL:

x₁₁representing the angular velocity of rotation of the main valve, in rad/s,

x₁₂representing the main valve rotation angle in rad, corresponding to v in FIG. 2₁，

x₂₁Representing the moving speed of the piston, with the unit of m/s,

x₂₂and represents the piston movement displacement in m, corresponding to y in fig. 2.

Thus, from equations (6) and (14), the corresponding matrices can be derived as follows:

C＝[0 0 0 1]，D＝[0.004107 0 0 0]^T。

the traditional proportional-integral observer considers the gap as a proportional link and ignores the switching of the system, so that the traditional proportional-integral observer does not comprise a switching term, and a specific expression is as follows:

two types of faults are simulated respectively. The first type of fault, assuming that there is a step fault at 30s, represents a sudden fault, Af is 0.85, bf is 1, and when t is 0 ≦ t ≦ 30, u_f(t) 0, i.e. no fault for the first 30 seconds; when 30 is turned into<When t is less than or equal to 120, u_f(t) is 0.2, the sampling frequency is 100Hz, and Kpj is selected to be [1.5532, 0.7079, 1.9574, 0.5045 ] according to the observer existence theorem]^T，K_ijWhen j is 1,3, the characteristic value of Ae is [0.4520, 0.5358, 0.9061, 0.9417, 0.9899]^T(ii) a When j is 2, the characteristic value of Ae is [0.9900, 0.4500, 0.9900, 0.5517, 0.8438%]^TAll within the unit circle.

The second type of fault is a sinusoidal signal with attenuated amplitude, which in practical applications represents a slowly varying fault. af is 0.99 and bf is1. When t is more than or equal to 0 and less than or equal to 30, u_f(t) ═ 0, i.e., the first 30 seconds were without failure. When 30 is turned into<When t is less than or equal to 240, the fault is u_f(t)＝0.05e^{(-(t-30)/100)}sin (π (t-30)/50+ 1). The sampling frequency is also 100 Hz. Choose Kpj ═ 1.5532, 0.7079, 1.9574, 0.5045]^T，K _ij1. When j is 1,3, the characteristic value of Ae is [0.4520, 0.5358, 0.9061, 0.9417, 0.9899]^T(ii) a When j is 2, the characteristic value of Ae is [0.9900, 0.4500, 0.9900, 0.5517, 0.8438%]^TAll within the unit circle.

By comparing fig. 3 and 4, it can be clearly seen that the switching proportional-integral observer can estimate the state of the system more accurately than the conventional proportional-integral observer; the switching proportional-integral observer and conventional observer state estimation errors are shown in fig. 5, from which fig. 5 it can be derived that the estimation errors of the switching proportional-integral observer are smaller than those of the conventional proportional-integral observer. When there is a step fault at 30s, the state estimation effect of the switching proportional-integral observer and the conventional observer is shown in fig. 3 and 4, and the fault estimation effect of the switching proportional-integral observer and the conventional observer is shown in fig. 6, and it can be seen from fig. 6 that the switching proportional-integral observer can accurately track the fault signal in time. However, the conventional proportional-integral observer cannot track the fault signal at all, and in short, the switching proportional-integral observer has better performance than the conventional proportional-integral observer in the aspects of state estimation and fault estimation.

When the fault is a sinusoidal signal with attenuated amplitude, the state estimation effects of the switching proportional-integral observer and the conventional observer are as shown in fig. 7 and 8. By analogy with fig. 3-6, it can be seen from fig. 7-10 that the switching proportional-integral observer is more accurate than the conventional observer model, and therefore, the switching proportional-integral observer is more effective than the conventional observer state estimation.

Claims (2)

1. A method for estimating a state of a gap sandwich system including a fault, comprising the steps of:

step 1: constructing a non-smooth state space equation capable of accurately describing the gap sandwich system with the fault by using a key item separation principle and a switching function;

step 2: constructing a non-smooth state space equation of the clearance sandwich system with the fault according to the step 1, constructing a switching proportional-integral observer capable of automatically switching along with the change of the working interval of the clearance sandwich system with the fault when the system meets the existence condition of the observer, and giving the existence condition and the boundedness theorem of the corresponding switching proportional-integral observer;

the step 1 comprises the following steps:

(1) front-end linear subsystem L with faulty gap sandwich system₁The state space equation of (a) is:

(2) back end linear subsystem L with faulty gap sandwich system₂The state space equation of (a) is:

the above formula (1) and formula (2)

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，i＝1，2，x_1iAnd x_2iEach represents L₁And L₂In the (i) th state of (c),

in order to be a state transition matrix,

in order to input the matrix, the input matrix is,

to be the output matrix, the output matrix is,

for the failure matrix, u ∈ R^1×1As input, y ∈ R^1×1For output, f ∈ R^1×1Is a fault of the system, can be regarded as_fAs a factor of a failed link, b_fInput coefficient for faulty link, u_f∈R^1×1For a failed link that is the input to the failed link, assume u_fIs bounded, a_fIs less than 1, i.e. | a_fI < 1, so the fault system is stable according to the linear system stability condition; n is_iDimension of the ith linear system; is provided with

And is

(3) The state space equation for the clearance subsystem:

in gap sandwich systems with faults, v₁(k) And v₂(k) Defining intermediate variables m (k) as the input and output of the interval respectively:

defining an intermediate variable w₁(k) Comprises the following steps:

w₁(k)＝m(k)(v₁(k)-D₁g₁(k)+D₂g₂(k))， (4)

wherein,

and is

The input-output relationship according to the gap can be:

v₂(k)＝w₁(k)+[v₂(k-1)-w₁(k)]g₃(k)＝(1-g₃(k))w₁(k)+g₃(k)v₂(k-1)， (5)

wherein

According to the formula (2), the formula (4) and the formula (5):

(4) the overall equation of state for the gap sandwich system containing the fault:

according to the formulae (1), (2), (6), and

the state space equation of the system can be obtained as follows:

wherein

Wherein,

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

Wherein

0 is a zero matrix of the corresponding order

Then

η therein_iIs the switching vector due to the presence of the gap.

2. The method according to claim 1, wherein step 2 comprises the steps of:

(1) modeling of switching proportional-integral observer

According to the formula (8) in the step 1, a switching proportional integral observer shown as the following formula is established:

wherein

And

the ith working interval is proportional gain and integral gain respectively,

x (k), y (k), f (k), η_iWherein when i is j, j is 1,3,

when the j is 2, the sum of the j,

(2) estimation error analysis of proportional-integral observer

From the formula (9) and the formula (1), the following formula (10) and formula (11) can be obtained:

f(k+1)＝a_ff(k)+b_fu_f(k) (11)

equation (11) is subtracted from equation (10), and

and

then:

subtracting equation (8) from equation (9) and considering the interval estimation error is:

e(k+1)＝(A-K_pjC)e(k)+De_f(k)+Δη_os+ΔA_osx(k) (13)

definition of

ΔAo_s＝A_j-A_i，

From equations (12) and (13), the state and fault estimation errors can be found as:

definition of

And is

Then it can be obtained:

e_t(k+1)＝A_eje_t(k)+Δ_t(k) (15)

let Δ_t(k) And initial estimation error e_t(1) The norm of (a) is bounded and all are less than phi_dI.e. phi_d(||Δ_t(k)||≤φ_dAnd | | | e (1) | | is less than or equal to phi |_d) When j is 1, 2, 3, an appropriate K is selected_pjAnd K_ijSo that A is_ejIs within the unit circle, the norms of the state and fault estimation errors are bounded and both are less than

The condition for the switching proportional-integral observer to exist, i.e. the condition that the system state estimation error and the fault estimation error are bounded, is a_ejIs within the unit circle.

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