CN112731809B - State and fault estimation method for dead zone sandwich system - Google Patents
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Abstract
The invention discloses a state and fault estimation method of a dead zone sandwich system, which aims at the problem that intermediate variables of a dead zone nonlinear non-smooth sandwich system cannot be measured. Firstly, a key separation principle and a switching function are utilized to construct a non-smooth state space equation capable of accurately describing a dead zone sandwich system. And secondly, constructing the SPIO which can be switched along with the change of the system working interval according to the non-smooth state space equation. And by analyzing the estimation error of the observer, the condition of observer convergence is given, and a mathematical proof is given to the observer convergence theorem. The method has the advantage that the switching function is introduced into the observer, and compared with the traditional proportional-integral observer, the observer adopting the method can estimate the state and the fault of the system more accurately and simultaneously.
Description
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
There is a class of dead zone sandwich systems in the industry. The intermediate variables of the system cannot be measured or are very expensive to measure, but if the intermediate state cannot be obtained, most state feedback control methods cannot be performed, because the state values of the system must be accurately known in the state feedback control. The most effective method for obtaining the unknown state value is to design an observer and estimate the unknown state of the system by using input and output variables of the system.
Chinese patent CN105204332A discloses a state estimation method of a composite sandwich system with dead zones and hysteresis based on a non-smooth observer, and the method is used for estimating the state of the system under the condition that the sandwich system does not have faults. However, in an actual system, a fault is often unavoidable, and the state estimation of the system generates certain disturbance due to the existence of the fault, and even results in divergence of estimation errors, 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 patents and literature have been found for dead zone sandwich systems that estimate both state and fault.
The present invention proposes a new switching proportional-integral observer to accomplish this. Switching items which can be switched along with the working range of the system are introduced into the switching proportional-integral observer, and state and fault estimation errors are analyzed. 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 result shows that the switching proportional-integral observer is superior to the traditional proportional-integral observer, and the method can be used for future system control and fault tolerance control.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a state and fault estimation method of a dead zone sandwich system.
The technical scheme for realizing the purpose of the invention is as follows:
a state and fault estimation method of a dead zone sandwich system comprises 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
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 ，x 11 And x 12 Each represents L 1 First and second state variables, x 21 And x 22 Each represents L 2 The first and second state variables of (a),
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)(x(k)-D 1 h 1 (k)+D 2 h 2 (k))
wherein
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 (2) into the formula (3) to obtain:
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
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
For the proportional gain of the ith operating interval,
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
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)
is defined by subtracting equation (9) from equation (8)
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-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 equation (12):
the formula (12) is simplified 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) And initial estimation error e t (1) The norm of (a) is bounded and all are less than phi d I.e. phi d (||Δ t (k)||≤φ d And e (1) is less than or equal to phi d ) When j =1,2,3, 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 Is within the unit circle;
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 Wherein 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 the proper K is selected pj 、K ij So that the characteristic value 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 → ∞ is satisfied, norm | | | e of the state and fault estimation errors t (k) I) 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:
4-5) adding 1 to the value of k, repeating steps 4-3).
In step 4), most of the actual systems are not first-order systems, but second-order systems or two subsystems L which can be simplified into second-order systems, if not smooth sandwich systems 1 、L 2 The system is a second-order system, the whole system is a fourth-order system and has 4 state variables, and z is a system with 4 states in each step of calculation, namely the 1 st state and the k th step in the k stepThe 2 nd state, the 3 rd state at the k-th step and the 4 th state at the k-th step, therefore, all the states are stored by the matrix x of N rows and 4 columns, and the 4 states at the k-th step are stored in the 1 to 4 columns of the k-th row of the matrix x and written as x (k, [ 1:4)]) Abbreviated as x (k,: in short).
In the step 4-4), the step of,
the operation carried out in three cases is A, eta, K p 、K i 4 different matrices.
Compared with the traditional proportional-integral observer estimation method, the method for estimating the state and the fault of the dead zone sandwich system can estimate the state and the fault of the dead zone sandwich system at the same time, and has the advantages of high estimation error convergence speed and high precision.
Drawings
FIG. 1 is a block diagram of a dead band sandwich system with a fault;
FIG. 2 is a Simulink model diagram of a non-smooth sandwich system;
FIG. 3 is a schematic view of a linear motor drive system;
FIG. 4 is a diagram showing the effect of switching the PID observer on the estimation of the system state when the system has a step fault;
FIG. 5 is a diagram of the effect of a conventional proportional-integral observer on the estimation of the state of a system when the system has a step fault;
FIG. 6 is a comparison graph of state estimation errors of the two methods of switching the proportional-integral observer and the conventional proportional-integral observer when the system has step faults;
FIG. 7 is a comparison graph of two methods of switching the proportional-integral observer and the conventional proportional-integral observer to the fault estimation when the system has step fault;
FIG. 8 is a diagram showing the effect of switching the proportional-integral observer on the estimation of the system state when the system contains a sinusoidal signal with attenuated amplitude;
FIG. 9 is a diagram showing the effect of a conventional proportional-integral observer on the estimation of the state of a system when the system contains a sinusoidal signal with an attenuated amplitude;
FIG. 10 is a diagram showing the comparison of the system state estimation error by the two methods of switching the proportional-integral observer and the conventional proportional-integral observer when the system contains a sinusoidal signal with attenuated amplitude;
fig. 11 is a comparison diagram of system fault estimation by two methods of switching the proportional-integral observer and the conventional proportional-integral observer when the system contains sinusoidal signals with attenuated amplitudes.
Detailed Description
The invention will be further elucidated with reference to the drawings and examples, without however being limited thereto.
Example (b):
a state and fault estimation method of a dead zone sandwich system comprises 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:
as shown in FIG. 1, a dead-zone sandwich system is shown with actuator faults, where u (k) and y (k) are measurable input and output variables, respectively, and v (k) is a measurable variable 1 (k) And v 2 (k) Is an unmeasurable intermediate variable, L 1 Representing a pre-cursor linear subsystem, L 2 Representing a post-Rake linear subsystem, D 1 And D 2 Is the width of the dead zone, m 1 And m 2 Is the slope of the linear region, and f (k) represents the actuator failure of the system.
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
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 ，x 11 And x 12 Each represents L 1 First and second state variables, x 21 And x 22 Each represents L 2 The first and second state variables of (a),
are respectively L 1 、L 2 The state transition matrix of (a) is,
are each 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 Is the ith linear systemDimension of system, set
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)(x(k)-D 1 h 1 (k)+D 2 h 2 (k))
wherein
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) =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 (2) into the formula (3) to obtain:
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
wherein eta i To account for the presence of dead band non-of the systemLinear characteristic induced switching vector;
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, a switching proportional-integral observer is constructed as follows:
wherein
For the proportional gain of the i-th operating interval,
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
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)
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-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 equation (12):
the formula (12) is simplified 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) And initial estimation errorsDifference e t (1) The norm of (a) is bounded and all are less than phi d I.e. phi d (||Δ t (k)||≤φ d And | | | e (1) | | is less than or equal to phi | d ) When j =1,2,3, 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 Is within the unit circle;
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 e j | | | is A ej Wherein 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 the proper K is selected pj 、K ij So that the characteristic value of is within the unit circle, i.e. the spectral norm ρ (A) of the matrix ej ) Less than 1, if an appropriate ε is selected to satisfy 0 < ε < 1- ρ (A) ej ) According to equation (15), there are:
||A ej ||＜1 (16)
therefore, from equation (14) and equation (16), when k → ∞ the state and fault estimation is mistakenPoor norm e t (k) I) less than
As shown in fig. 2, the Simulink model of the non-smooth sandwich system mainly comprises an ideal input, a summation module, a controller, an L1 subsystem, a dead zone link, an L2 subsystem, a feedback gain and a connection line, and may further include fault signals, such as an L1 fault signal, an L2 fault signal and a dead zone increasing module.
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:
4-5) adding 1 to the value of k, repeating steps 4-3).
In step 4), most of the actual systems are not first-order systems, but second-order systems or two subsystems L which can be simplified into second-order systems, if not smooth sandwich systems 1 、L 2 The system is a second-order system, the whole system is a fourth-order system, 4 state variables exist, z is the 4 states in each step of the system, namely the 1 st state in the k step, the 2 nd state in the k step, the 3 rd state in the k step and the 4 th state in the k step, therefore, a matrix x with N rows and 4 columns is used for storing all the states, the 4 states in the k step are stored in 1 to 4 columns of the k row of the matrix x, and x (k, [ 1:4) is written into]) Abbreviated as x (k,: in short).
In the step 4-4), the step of,
the operations carried out in the three cases are A, eta, K p 、K i 4 different matrices.
The method is used for estimating the state and the fault of the direct current motor driving simulation system, and compared with the state and fault estimation result of a traditional proportional-integral observer, and the method specifically comprises the following steps:
as shown in FIG. 3, the schematic diagram of the linear motor driving system, in which the control circuit includes circuit amplifiers and filters, etc., can be considered as a linear subsystem L 1 (ii) a The load can be regarded as a linear subsystem L 2 (ii) a The linear motor consists of a magnetic field winding, an armature winding, a bracket and the like, wherein coulomb friction exists; the linear motor is equivalent to a dead zone link and is represented by DZ; therefore, the whole linear motor driving system can be regarded as a dead zone sandwich system; the system is mainly used for a system requiring output of linear displacement and automatic control.
The simulation model of the linear motor driving system is as follows:
the input signal is u =20sin (0.8 t), the sampling time is 80s, the sampling period is 0.01s, and all the state and fault initial values are set to 0; then:
linear subsystem L 1 ：
Linear subsystem L 2 ：
Dead zone:
4 state variables x of linear motor drive system 11 、x 12 、x 21 、x 22 And the physical meanings represented by each are shown in Table 1.
TABLE 1 State variables of Linear Motor drive systems
Thus, with the state space equations and system parameters of the system, a corresponding matrix can be derived as follows.
C＝[0 0 0 00]And D = [0.04107 00 ]] T 。
The traditional observer dead zone link is a linear link, a proportional link is adopted for replacing the linear link, and the switching of the system between the linear link and the dead zone is omitted. Therefore, the conventional proportional-integral observer does not include a switching term, and the specific expression is as follows:
wherein
Two types of faults are simulated respectively:
the first type of fault, which is assumed to have a step fault at 10, represents a sudden fault in practical applications, a f ＝0.85，b f =1, when t is more than or equal to 0 and less than or equal to 10 f (t) =0, i.e. the first 10 seconds are non-faulty; when 10<When t is less than or equal to 80, u f (t) is 0.2, the sampling frequency is 100Hz, and K is selected according to the convergence condition of the switching proportional-integral observer pj ＝[0.1，0.1，0.1，0.1] T And K ij =1, characteristic value of Ae when j =1,3, 0.4480,0.8075,0.7975+0.0406i,0.7975-0.0406i,0.9494] T (ii) a When j =2, the characteristic value of Ae is [0.4500,0.8000,0.9500,0.8000+0.0316i,0.8000-0.0316i]] T Within the unit circle, i.e. whether the system is operating in the linear region or the dead zone, the characteristic value of Ae is within the unit circle.
By comparing fig. 4 and 5, 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 state estimation errors of the switching proportional-integral observer and the conventional observer are shown in fig. 6, the estimation error of the switching proportional-integral observer is smaller than that of the conventional proportional-integral observer from fig. 6, when a step fault occurs at 10s, the state estimation effects of the switching proportional-integral observer and the conventional observer are shown in fig. 4 and 5, the fault estimation effects of the switching proportional-integral observer and the conventional observer are shown in fig. 7, from fig. 7, the switching proportional-integral observer can accurately track the fault signal in time, but the conventional proportional-integral observer cannot track the fault signal at all; in summary, the switching proportional-integral observer has better performance than the conventional proportional-integral observer in terms of state estimation and fault estimation.
The second type of fault is assumed to be a sinusoidal signal with attenuated amplitude, representing a slowly varying fault in practical applications. The sampling frequency is also 100Hz, af =0.8 and bf =1, when 0 ≦ t ≦ 10, u f (t) =0, i.e. the first 10 seconds are not faulty; when 10<When t is less than or equal to 80, the fault is u f (t)＝e (-(t-30)/10) sin (pi (t-10)/10 + 1) +4, selection 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 (ii) a 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 attenuated amplitude, the state estimation effects of the switching proportional-integral observer and the conventional observer are as shown in fig. 8 and 9. From fig. 8-11, it can be seen that the switched proportional-integral observer is more accurate than the conventional observer model, and therefore, the switched proportional-integral observer is more effective than the conventional observer state estimation.
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,
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
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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