CN109521676B - Self-adaptive sliding mode fault-tolerant control method of probability distribution time-lag system - Google Patents

Self-adaptive sliding mode fault-tolerant control method of probability distribution time-lag system Download PDF

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CN109521676B
CN109521676B CN201811582376.2A CN201811582376A CN109521676B CN 109521676 B CN109521676 B CN 109521676B CN 201811582376 A CN201811582376 A CN 201811582376A CN 109521676 B CN109521676 B CN 109521676B
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胡军
张红旭
李泽昊
关馨郁
李佳兴
徐沈阳
武晗
高萍萍
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Harbin University of Science and Technology
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Abstract

A self-adaptive sliding mode fault-tolerant control method of a probability distribution time-lag system. Belonging to the field of modular fault-tolerant control. The existing sliding mode control method has the problems that the system performance is influenced by the fact that the uncertainty, the probability distribution time lag, the actuator fault and the unknown upper bound of the external disturbance cannot be processed simultaneously. Establishing a dynamic model of a control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound; designing a sliding mode surface of the established dynamic model of the control system; calculating the corresponding sliding mode of the sliding mode surface of the dynamic model; obtaining a judging condition for ensuring the performance of the sliding mode by utilizing the sliding mode and through the Lyapunov stability theorem; obtaining a gain matrix according to the obtained discrimination condition; and designing an adaptive law to carry out sliding mode control according to the gain matrix. The invention can ensure the stable control under the condition that the system performance is influenced by the uncertainty, the probability distribution time lag, the actuator fault and the unknown upper bound of the external disturbance.

Description

Self-adaptive sliding mode fault-tolerant control method of probability distribution time-lag system
Technical Field
The invention relates to the field of control of an uncertain system with probability distribution time lag, in particular to a self-adaptive sliding mode fault-tolerant control method of the uncertain system with probability distribution time lag.
Background
The model fault-tolerant control is an important research branch in the control field and is widely applied to the science and technology industries such as nuclear industry, aerospace, robots and the like. As the degree of automation of modern control systems becomes more and more complex and with increasing operating times, the internal components of the plant inevitably age, fail or even fail. Once a fault occurs, the system performance is inevitably seriously affected, which in turn causes serious economic loss and even endangers personal safety. Therefore, how to improve the safety, reliability and stability of the system becomes a hot problem which needs to be solved urgently in the field of control science. The corresponding sliding mode fault tolerance control technology is developed. Due to the design of the sliding mode, the parameters of the object and the disturbance, the variable structure control has stronger robustness. In addition, when external disturbance and parameter information are unknown, the traditional sliding mode control method is difficult to meet the performance requirements of the system, and the adaptive sliding mode control technology ensures good operation of the system performance by designing adaptive law to continuously adjust the parameter information, so that the method is widely concerned by scholars.
Disclosure of Invention
The invention aims to solve the problem that the existing sliding mode control method cannot simultaneously process system uncertainty, probability distribution time lag, actuator faults and unknown upper bound external disturbance influence system performance, and provides a self-adaptive sliding mode fault-tolerant control method of a probability distribution time lag system.
A self-adaptive sliding mode fault-tolerant control method of a probability distribution time-lag system comprises the following specific processes:
establishing a dynamic model of a control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound;
step two, designing a sliding mode surface of the dynamic model of the control system which is established in the step one and has system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound;
step three, calculating a corresponding sliding mode according to the sliding mode surface of the dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound, which is designed in the step two;
step four, obtaining a judging condition for ensuring the performance of the sliding mode by utilizing the sliding mode obtained in the step three and through the Lyapunov stability theorem;
step five, obtaining a gain matrix according to the judgment condition obtained in the step four;
and step six, designing a self-adaptive law according to the gain matrix obtained in the step five, and realizing sliding mode control of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound.
The invention has the beneficial effects that:
the method solves the problem that the traditional sliding mode control method cannot simultaneously process system uncertainty, probability distribution time lag, actuator faults and unknown external disturbance upper bound, thereby influencing the overall performance of the system. The invention simultaneously considers that the values of the time lag information of the system in different sections have different probabilities and the actuator of the system has a fault pair control system H∞The influence of the performance is realized by selecting a proper Lyapunov function, and effective information of time lag is fully utilized. Compared with the self-adaptive sliding mode fault-tolerant control method of the existing uncertain system, the self-adaptive sliding mode fault-tolerant control method can simultaneously process the uncertainty of the system, the probability distribution time lag, the fault of an actuator and the unknown upper bound of external disturbance, a sliding mode control method based on the linear matrix inequality solution is used, the purpose of parameter perturbation resistance is achieved, and the self-adaptive sliding mode fault-tolerant control method is suitable for the self-adaptive sliding mode fault-tolerant control of the probability distribution time lag system.
The invention adjusts the influence of the upper bound of unknown parameters on the system performance by constructing an adaptive lawThe stability method of the aid Lyapunov function ensures that the designed adaptive sliding mode control rate can still quickly converge the system state track to the sliding mode surface when the system is subjected to actuator faults and the upper bound of external disturbance is unknown, and further ensures that the sliding mode is in the H state∞Is randomly and gradually stable in the sense of.
The invention is not influenced by the condition that the actuator is in failure and stuck faults under the condition that the system time-lag information and the located interval have a certain probability relation, and ensures the control performance of the system.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a state trace diagram for an open loop system;
FIG. 3 is a state trace diagram for a closed loop system;
FIG. 4 is a state trace of the slip form face;
FIG. 5 is an adaptive variable
Figure BDA0001918217430000021
A trajectory diagram of (a);
FIG. 6 is an adaptive variable
Figure BDA0001918217430000022
A trajectory diagram of (a);
FIG. 7 is an adaptive variable
Figure BDA0001918217430000023
A trajectory diagram of (a);
FIG. 8 is a plot of a controlled output response versus an external disturbance response;
Detailed Description
The first embodiment is as follows:
in this embodiment, a method for adaptive sliding mode fault-tolerant control of a probabilistic distributed time lag system includes:
establishing a dynamic model of a control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound;
step two, designing a sliding mode surface of the dynamic model of the control system which is established in the step one and has system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound;
step three, calculating a corresponding sliding mode according to the sliding mode surface of the dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound, which is designed in the step two;
step four, obtaining a judging condition for ensuring the performance of the sliding mode by utilizing the sliding mode obtained in the step three and through the Lyapunov stability theorem;
step five, designing a solving algorithm according to the judgment conditions obtained in the step four, and solving a gain matrix;
and step six, designing a self-adaptive law according to the gain matrix obtained in the step five, and realizing sliding mode control of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound.
The second embodiment is as follows:
different from the first specific embodiment, in the first step, the state space form of the established dynamic model of the control system with unknown upper bound of system uncertainty, probability distribution time lag, actuator fault and external disturbance is as follows:
Figure BDA0001918217430000031
in the formula (I), the compound is shown in the specification,
Figure BDA0001918217430000032
is a system state variable at the time t,
Figure BDA0001918217430000033
the derivative of the system state x (t) at time t, x (t- τ (t)) is the system state variable at time t- τ (t),
Figure BDA0001918217430000034
is an external disturbance at time t;
Figure BDA0001918217430000035
actual control input of the control system at time t; z (t) is the controlled output of the system at time t; tau (t) is formed by [0, tau ∈2]For a bounded time-varying time lag, τ2Is the upper bound of τ (t); a is a system matrix of the control system, AτIs a system time lag matrix of the control system, B is an input matrix of the control system, D1For the disturbance matrix of the control system, C for the controlled output matrix of the control system, D2For controlling a controlled disturbance matrix of the system, Aτ，B，C，D1，D2Are all known; Δ A represents a system uncertainty matrix, and satisfies Δ A ═ EFH, where E represents a left-end metric matrix, H represents a right-end metric matrix, E and H are known quantities and are used to characterize a norm-bounded uncertainty matrix, F is an unknown quantity, and F satisfies FTF is less than or equal to I which is an identity matrix; w (t) e L2[0, + ∞) is the energy-bounded external disturbance for the time t control, L2[0, + ∞) represents an energy bounded space on [0, + ∞); phi (t) is a known initial condition of the control system;
Figure BDA0001918217430000036
a euclidean space representing the n dimensions,
Figure BDA0001918217430000041
euclidean sum of m dimensions
Figure BDA0001918217430000042
A Euclidean space representing the w dimension;
(1) said actual control input uF(t) obeying the following fault model:
Figure BDA0001918217430000043
in the formula, ρ is unknownA time-varying diagonal matrix representing an actuator failure factor, rho satisfying rho e { rho ∈ [)1,ρ2,…,ρL}；
Figure BDA0001918217430000044
A factor indicative of a failure of the actuator,
Figure BDA0001918217430000045
satisfy the requirement of
Figure BDA0001918217430000046
L is the number of fault modes; for each j e {1,2, …, L }, there is
Figure BDA0001918217430000047
Further for each i e {1,2, …, m }, there is
Figure BDA0001918217430000048
Satisfy the requirement of
Figure BDA0001918217430000049
Figure BDA00019182174300000410
Satisfy the requirement of
Figure BDA00019182174300000411
Or
Figure BDA00019182174300000412
Where ρ isiAnd
Figure BDA00019182174300000413
are all known constants, piA lower bound on the failure factor representing the ith component of the actuator,
Figure BDA00019182174300000414
representing the upper bound of the failure factor of the ith component of the actuator; u. off(t) represents an actuator stuck-at fault,
Figure BDA00019182174300000415
the actuator stuck fault comprises the following conditions:
Figure BDA00019182174300000416
(2) for a bounded time-varying time lag τ (t) e [0, τ2]In the presence of a τ1Satisfy 0 ≦ τ1＜τ2Then there is τ (t) e [0, τ)1]Or τ (t) ∈ (τ)1,τ2](ii) a Introducing Bernouli variable alpha (t) to characterize tau (t) epsilon [0, tau1]And τ (t) ∈ (τ)1,τ2]In both cases, α (t) ═ 1 indicates τ (t) ∈ [0, τ1]Where α (t) ═ 0 denotes τ (t) ∈ (τ)1,τ2]And τ (t) satisfies the following probability distribution:
Figure BDA00019182174300000421
Figure BDA00019182174300000417
where Prob { α (t) ═ 1} is a probability when α (t) ═ 1, and E { α (t) } is an expectation of a random variable α (t);
on the other hand, two functions are introduced
Figure BDA00019182174300000418
And
Figure BDA00019182174300000419
respectively satisfy:
Figure BDA00019182174300000420
the system (1) is then represented as:
Figure BDA0001918217430000051
(3) for the actuator stuck fault uf(t) and external perturbations w (t), ufEach variable u of (t)f,i(t) all have an unknown upper bound, each variable w of w (t)p(t) all have an unknown upper bound, i.e.
Figure BDA0001918217430000052
Figure BDA0001918217430000053
And
Figure BDA0001918217430000054
all unknown fixed constants, i ═ 1,2, …, m, p ═ 1,2, …, w; the input matrix B is decomposed into B ═ B1B2，
Figure BDA0001918217430000055
And
Figure BDA0001918217430000056
the ranks of the N-ary-type organic matter are all l, and l is less than or equal to m; m is the dimension of the actuator, and w is the dimension of the external disturbance;
let us say for an arbitrary ρ e { ρ1,ρ2,…,ρL}, there are rank (B)2ρ)＝rank(B2) And (A, B) is controllable.
The third concrete implementation mode:
different from the second specific embodiment, in the second step, the sliding mode surface is designed for the dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound, which is established in the first step, and the formula of the designed sliding mode surface is as follows:
Figure BDA0001918217430000057
in the formula (I), the compound is shown in the specification,
Figure BDA0001918217430000058
is a matrix of sliding mode coefficients,
Figure BDA0001918217430000059
l is a free matrix; and has GB1I ═ I; k is a gain matrix to be designed; s (x (t)) is the sliding mode function at time t.
4. The adaptive sliding mode fault-tolerant control method of the probability distribution time-lag system according to claim 3, characterized in that: in the third step, according to the sliding mode surface of the dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound, which is designed in the second step, the process of calculating the corresponding sliding mode is specifically as follows:
when the state track of the system runs on the sliding mode surface, the system has
Figure BDA00019182174300000511
An equation of the sliding mode is obtained by using the equation (4), and is shown in the equation (5):
Figure BDA00019182174300000510
5. the adaptive sliding mode fault-tolerant control method of the probability distribution time-lag system according to claim 4, characterized in that: in the fourth step, the process of obtaining the judgment condition for ensuring the performance of the sliding mode by using the sliding mode obtained in the third step and through the lisinoprofen stability theorem specifically comprises the following steps:
if the controlled output z (t) and the external disturbance w (t) in the control system satisfy:
Figure BDA0001918217430000061
the control system has H∞Performance index, where γ represents a given interference suppression level and E is the mathematical expectation;
to ensure the above-mentioned H∞The performance index is established, the Lyapunov is stableThe qualitative theory is as follows:
E{L V(ξ(t))}+z(t)Tz(t)-γ2wT(t)w(t)＜0
wherein:
Figure BDA0001918217430000062
Figure BDA0001918217430000063
Figure BDA0001918217430000064
Figure BDA0001918217430000065
Figure BDA0001918217430000066
Figure BDA00019182174300000611
M＝[0 Aτ 0 -A τ 0 0 0]，
ξT(t)＝[xT(t)xT(t-τ1(t))xT(t-τ1)xT(t-τ2(t))xT(t-τ2)wT(t)yT(t)]，
A＝A+BK，
Figure BDA0001918217430000067
E1＝(I-B1G)E，
Figure BDA0001918217430000068
in the formula, V (xi (t)) is t timeThe Lyapunov function of (g), L V (ξ (t)) is the weak infinitesimal operator of V (ξ (t)); x is the number ofT(t) is the transpose of x (t), yT(t) is the transpose of y (t), ξT(t) is the transpose of ξ (t), MTIs the transpose of M;
from the Lyapunov stability theory described above, system H is obtained∞The conditions for discriminating the progressive stabilization in the sense are:
Figure BDA0001918217430000069
wherein the content of the first and second substances,
Figure BDA00019182174300000610
is symmetrical;
Figure BDA0001918217430000071
Figure BDA0001918217430000072
Figure BDA0001918217430000073
Figure BDA0001918217430000074
Figure BDA0001918217430000075
Figure BDA0001918217430000076
Figure BDA0001918217430000077
Figure BDA0001918217430000078
Figure BDA0001918217430000079
Figure BDA00019182174300000710
Figure BDA00019182174300000711
Figure BDA00019182174300000712
Figure BDA00019182174300000713
Figure BDA00019182174300000714
Figure BDA00019182174300000715
Figure BDA00019182174300000716
Figure BDA00019182174300000717
Figure BDA00019182174300000718
Figure BDA00019182174300000719
Figure BDA00019182174300000720
Figure BDA00019182174300000721
Figure BDA00019182174300000722
Figure BDA00019182174300000723
Figure BDA0001918217430000081
Figure BDA0001918217430000082
Figure BDA0001918217430000083
Figure BDA0001918217430000084
Figure BDA0001918217430000085
Figure BDA0001918217430000086
Figure BDA0001918217430000087
Figure BDA0001918217430000088
Figure BDA0001918217430000089
Figure BDA00019182174300000810
Φ＝STAT+XTBT
wherein denotes a symmetric portion of the block matrix; omega11Is omega q1 st row and 1 st column of the block matrix, omega12Is omegaqRow 1 and column 2 block matrices,
Figure BDA00019182174300000811
is omegaqRow 1, column 3, block matrix, omega14Is omegaqRow 1, column 4, block matrix, omega15Is omegaqRow 1, column 5, block matrix, omega16Is omegaqRow 1, column 6, block matrix, omega22Is omegaqRow 2 and column 2 block matrix, omega33Is omegaqRow 3 and column 3 of the block matrix, omega44Is omegaqRow 4 and column 4 of the block matrix, omega55Is omegaqRow 5 and column 5 block matrices,
Figure BDA00019182174300000812
is a matrix omega11The block matrix of the r row and the s column;
Figure BDA00019182174300000813
is a matrix
Figure BDA00019182174300000814
The (j) th block matrix of (a),
Figure BDA00019182174300000815
is a matrix
Figure BDA00019182174300000816
The transpose of (a) is performed,
Figure BDA00019182174300000817
is a matrix N54Is transposed, STIs a transpose of the matrix S, X being the matrix XTTransposing; matrix array
Figure BDA00019182174300000818
S is a positive definite symmetric matrix to be solved;
Figure BDA00019182174300000819
N54and X is the free matrix to be solved; rho1,ρ3,ρ5Given a constant greater than zero.
The sixth specific implementation mode:
different from the fifth specific implementation manner, in the sixth step, according to the discrimination condition obtained in the fourth step, a solving algorithm is designed to obtain the gain matrix K, and the specific process is as follows:
step six: initialization parameter ρ1＞0，ρ2＞0，ρ3＞0，ε1,0＞0，ε2,0＞0，
Figure BDA00019182174300000820
Figure BDA0001918217430000091
And the maximum number of iterations NαAnd Nβ(ii) a Let k be 0 and k be equal to 0,kk＝0；
step six and two: solving for
Figure BDA0001918217430000092
The following linear matrix inequality with optimization is satisfied:
Figure BDA0001918217430000093
Figure BDA0001918217430000094
wherein the content of the first and second substances,
Figure BDA0001918217430000095
if κ ≈ 4, a feasible solution is found, and a gain matrix K ═ XS may be found-1(ii) a Otherwise, executing the next step;
step six and three: adding 1 to the k value, if the k value after adding 1 is less than NαIf yes, executing step six and step two; otherwise, executing the next step;
step six and four: reinitializing parameter ρ1＞0，ρ2＞0，ρ3＞0，ε1,0＞0，ε2,0＞0，
Figure BDA0001918217430000096
Figure BDA0001918217430000097
And adding 1 to the value of kk;
if the value of kk plus 1 is less than NβMaking the value of k added with 1 be zero, and executing the step six to two; otherwise, exiting and finding no feasible solution.
The seventh embodiment:
different from the sixth specific embodiment, in the sixth step, an adaptive law is designed according to the gain matrix K obtained in the fifth step, so as to implement a sliding mode control process of a control system with system uncertainty, probability distribution time lag, actuator failure, and unknown external disturbance upper bound, specifically:
the following adaptive laws are designed:
Figure BDA0001918217430000098
Figure BDA0001918217430000099
Figure BDA00019182174300000910
wherein F (t) is γ0isT(x(t))B2iKix(t)，γ0i，γ1i，γ2gGiven a gain constant greater than zero; u. offi0，wg0，
Figure BDA0001918217430000101
Is a given initial value; b is2iIs a matrix B2The ith column; d1gIs a matrix D1Column g of (1); kiIs the ith row of the matrix K;
according to the designed adaptive law, the following controllers are designed:
u(t)＝Kx(t)+u1(t)
wherein the content of the first and second substances,
Figure BDA0001918217430000102
Figure BDA0001918217430000103
in this equation, oa is a given real number greater than zero,
Figure BDA0001918217430000104
the following examples were used to demonstrate the beneficial effects of the present invention:
example 1:
consider a practical example of a rocket fairing structure. The launch vehicle payload fairing is a protective shroud that protects the air from wind pressure, high temperatures, and structural vibrations and dampens fairing vibrations. State variable x (t) ═ x1(t)x2(t)x3(t)x4(t)]T。x1(t) denotes the structural displacement, x2(t) is the velocity vector, (x)3(t),x4(t)) is the intra-cavity pressure vector. Due to the variability of the external environment, the launch vehicle payload fairing must be affected by external disturbances such as airflow, temperature, etc. Considering the above example, under the control of a controller designed, the rocket fairing system can still maintain a desired level of interference suppression when affected by a variety of factors. The specific selection of all parameters involved in the system is as follows.
System parameters:
Figure BDA0001918217430000105
Figure BDA0001918217430000106
H＝[0 1 1 0]，
F＝0.1，
Figure BDA0001918217430000111
τ1＝0.1，
τ2＝0.3，
Figure BDA0001918217430000112
ρ＝diag{ρ 1,ρ 2,…,ρ m}＝diag{0.65,0.7,0.7}，
Figure BDA0001918217430000113
γ0i＝0.01，
γ1i＝0.01，γ2g＝0.01。
then it can be obtained
Figure BDA0001918217430000114
Further select
Figure BDA0001918217430000115
By the proposed algorithm, the corresponding matrix can be obtained as
Figure BDA0001918217430000116
Figure BDA0001918217430000117
Further, gain matrix is obtained
Figure BDA0001918217430000118
Initial value x of given system parameter0＝[-0.5 0.3 1 -1]T，
Figure BDA0001918217430000119
Figure BDA00019182174300001110
In order to avoid the oscillation situation of the system state, sign (s (x)) in the simulation
Figure BDA00019182174300001111
Instead, let d equal 0.005, the simulation results are shown in fig. 2-8.
The effect of the constructed sliding mode controller is as follows:
FIG. 2 is a state trace diagram for an open loop system; FIG. 3 is a state trace diagram for a closed loop system; FIG. 4 is a state trace of the slip form face; FIG. 5 is an adaptive variable
Figure BDA00019182174300001112
A trajectory diagram of (a); FIG. 6 is an adaptive variable
Figure BDA00019182174300001113
A trajectory diagram of (a); FIG. 7 is an adaptive variable
Figure BDA0001918217430000121
A trajectory diagram of (a); FIG. 8 is a graphical depiction of the response of a controlled output versus the response of an external disturbance.
As can be seen from fig. 2-3, the system conditions are unstable in the open loop case and stable in the closed loop case. 3-8, the adaptive sliding mode control method can effectively inhibit external disturbance, and when the variable of the actuator is subjected to a locking fault, although the system state and the sliding mode surface function have transient fluctuation, the state track can still be rapidly converged to the sliding mode surface, so that a stable state is achieved.
The present invention is capable of other embodiments and its several details are capable of modifications in various obvious respects, all without departing from the spirit and scope of the present invention.

Claims (1)

1. A self-adaptive sliding mode fault-tolerant control method of a probability distribution time-lag system is characterized by comprising the following steps: the method comprises the following specific processes:
establishing a dynamic model of a control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound;
step two, designing a sliding mode surface of the dynamic model of the control system which is established in the step one and has system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound;
step three, calculating a corresponding sliding mode according to the sliding mode surface of the dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound, which is designed in the step two;
step four, obtaining a judging condition for ensuring the performance of the sliding mode by utilizing the sliding mode obtained in the step three and through the Lyapunov stability theorem;
step five, obtaining a gain matrix according to the judgment condition obtained in the step four;
designing a self-adaptive law according to the gain matrix obtained in the step five, and realizing sliding mode control on a control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound;
in the first step, the state space form of the established dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound is as follows:
Figure FDA0003242004200000011
in the formula (I), the compound is shown in the specification,
Figure FDA0003242004200000012
is a system state variable at the time t,
Figure FDA0003242004200000013
the derivative of the system state x (t) at time t, x (t- τ (t)) is the system state variable at time t- τ (t),
Figure FDA0003242004200000014
is an external disturbance at time t;
Figure FDA0003242004200000015
actual control input of the control system at time t; z (t) is the controlled output of the system at time t; tau (t) is formed by [0, tau ∈2]For a bounded time-varying time lag, τ2Is the upper bound of τ (t); a is a system matrix of the control system, AτIs a system time lag matrix of the control system, B is an input matrix of the control system, D1For the disturbance matrix of the control system, C is the control systemSystematic controlled output matrix, D2For controlling a controlled disturbance matrix of the system, Aτ，B，C，D1，D2Are all known; Δ A represents a system uncertainty matrix, and satisfies Δ A ═ EFH, where E represents a left-end metric matrix, H represents a right-end metric matrix, E and H are known quantities and are used to characterize a norm-bounded uncertainty matrix, F is an unknown quantity, and F satisfies FTF is less than or equal to I which is an identity matrix;
Figure FDA0003242004200000016
is an energy-bounded external disturbance of the control at time t,
Figure FDA0003242004200000017
represents an energy bounded space on [0, + ∞); phi (t) is a known initial condition of the control system;
Figure FDA0003242004200000018
a euclidean space representing the n dimensions,
Figure FDA0003242004200000019
euclidean sum of m dimensions
Figure FDA00032420042000000110
A Euclidean space representing the w dimension;
(1) said actual control input uF(t) obeying the following fault model:
Figure FDA00032420042000000111
in the formula, rho is an unknown time-varying diagonal matrix and represents an actuator failure factor, and rho satisfies rho epsilon { rho [ ]1,ρ2,…,ρL}；
Figure FDA0003242004200000021
Indicates the actuator isThe factor of the barrier is the factor of the barrier,
Figure FDA0003242004200000022
satisfy the requirement of
Figure FDA0003242004200000023
L is the number of fault modes; for each j e {1,2, …, L }, there is
Figure FDA0003242004200000024
Further for each i e {1,2, …, m }, there is
Figure FDA0003242004200000025
Satisfy the requirement of
Figure FDA0003242004200000026
Figure FDA0003242004200000027
Satisfy the requirement of
Figure FDA0003242004200000028
Or
Figure FDA0003242004200000029
Wherein the content of the first and second substances,ρ iand
Figure FDA00032420042000000210
are all constant values which are known to be,ρ ia lower bound on the failure factor representing the ith component of the actuator,
Figure FDA00032420042000000211
representing the upper bound of the failure factor of the ith component of the actuator; u. off(t) represents an actuator stuck-at fault,
Figure FDA00032420042000000212
the actuator stuck fault comprises the following conditions:
Figure FDA00032420042000000213
(2) for a bounded time-varying time lag τ (t) e [0, τ2]In the presence of a τ1Satisfy 0 ≦ τ1＜τ2Then there is τ (t) e [0, τ)1]Or τ (t) ∈ (τ)1,τ2](ii) a Introducing Bernouli variable alpha (t) to characterize tau (t) epsilon [0, tau1]And τ (t) ∈ (τ)1,τ2]In both cases, α (t) ═ 1 indicates τ (t) ∈ [0, τ1]Where α (t) ═ 0 denotes τ (t) ∈ (τ)1,τ2]And τ (t) satisfies the following probability distribution:
Figure FDA00032420042000000214
Figure FDA00032420042000000215
wherein Prob { α (t) ═ 1} is a probability that α (t) ═ 1,
Figure FDA00032420042000000216
is the expectation of a random variable α (t);
introducing two functions
Figure FDA00032420042000000217
And
Figure FDA00032420042000000218
respectively satisfy:
Figure FDA00032420042000000219
the system (1) is then represented as:
Figure FDA00032420042000000220
(3) for the actuator stuck fault uf(t) and external perturbations w (t), ufEach variable u of (t)f,i(t) all have an unknown upper bound, each variable w of w (t)p(t) all have an unknown upper bound, i.e.
Figure FDA00032420042000000221
Figure FDA00032420042000000222
And
Figure FDA00032420042000000223
all unknown fixed constants, i ═ 1,2, …, m, p ═ 1,2, …, w; the input matrix B is decomposed into B ═ B1B2，
Figure FDA00032420042000000224
And
Figure FDA00032420042000000225
the ranks of the N-ary-type organic matter are all l, and l is less than or equal to m; m is the dimension of the actuator, and w is the dimension of the external disturbance;
let us say for an arbitrary ρ e { ρ1,ρ2,…,ρL}, there are rank (B)2ρ)＝rank(B2) And (A, B) is controllable;
in the second step, the sliding mode surface is designed for the dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound established in the first step, and the designed sliding mode surface formula is as follows:
Figure FDA0003242004200000031
in the formula (I), the compound is shown in the specification,
Figure FDA0003242004200000032
is a matrix of sliding mode coefficients,
Figure FDA0003242004200000033
l is a free matrix; and has GB1I ═ I; k is a gain matrix to be designed; s (x (t)) is a sliding mode function at time t;
in the third step, according to the sliding mode surface of the dynamic model of the control system with system uncertainty, probability distribution time lag, actuator fault and unknown external disturbance upper bound, which is designed in the second step, the process of calculating the corresponding sliding mode is specifically as follows:
when the state track of the system runs on the sliding mode surface, the system has
Figure FDA0003242004200000034
An equation of the sliding mode is obtained by using the equation (4), and is shown in the equation (5):
Figure FDA0003242004200000035
in the fourth step, the process of obtaining the judgment condition for ensuring the performance of the sliding mode by using the sliding mode obtained in the third step and through the lisinoprofen stability theorem specifically comprises the following steps:
if the controlled output z (t) and the external disturbance w (t) in the control system satisfy:
Figure FDA0003242004200000036
the control system has H∞Performance index, where γ represents a given interference suppression level,
Figure FDA0003242004200000037
is a mathematical expectation;
to ensure the above-mentioned H∞The performance index is established, and the Lyapunov stability theory is：
Figure FDA0003242004200000038
Wherein:
Figure FDA0003242004200000039
V1(ξ(t))＝xT(t)Px(t)，
Figure FDA00032420042000000310
Figure FDA00032420042000000311
Figure FDA00032420042000000312
Figure FDA00032420042000000313
Figure FDA00032420042000000314
M＝[0 Aτ 0 -Aτ 0 0 0]，
ξT(t)＝[xT(t) xT(t-τ1(t)) xT(t-τ1) xT(t-τ2(t)) xT(t-τ2) wT(t) yT(t)]，
Figure FDA0003242004200000041
E1＝(I-B1G)E，
Figure FDA0003242004200000042
in the formula, V ([ xi ] (t)) is a Lyapunov function at time t,
Figure FDA0003242004200000043
a weak infinitesimal small operator of V (ξ (t)); x is the number ofT(t) is the transpose of x (t), yT(t) is the transpose of y (t), ξT(t) is the transpose of ξ (t), MTIs the transpose of M;
from the Lyapunov stability theory described above, system H is obtained∞The conditions for discriminating the progressive stabilization in the sense are:
Figure FDA0003242004200000044
wherein the content of the first and second substances,
Figure FDA0003242004200000045
is symmetrical;
Figure FDA0003242004200000046
Figure FDA0003242004200000047
Figure FDA0003242004200000048
Figure FDA0003242004200000049
Figure FDA00032420042000000410
Figure FDA00032420042000000411
Figure FDA00032420042000000412
Figure FDA00032420042000000413
Figure FDA00032420042000000414
Figure FDA00032420042000000415
Figure FDA00032420042000000416
Figure FDA00032420042000000417
Figure FDA0003242004200000051
Figure FDA0003242004200000052
Figure FDA0003242004200000053
Figure FDA0003242004200000054
Figure FDA0003242004200000055
Figure FDA0003242004200000056
Figure FDA0003242004200000057
Figure FDA0003242004200000058
Figure FDA0003242004200000059
Figure FDA00032420042000000510
Figure FDA00032420042000000511
Figure FDA00032420042000000512
Figure FDA00032420042000000513
Figure FDA00032420042000000514
Figure FDA00032420042000000515
Figure FDA00032420042000000516
Figure FDA00032420042000000517
Figure FDA00032420042000000518
Figure FDA00032420042000000519
Figure FDA00032420042000000520
Figure FDA00032420042000000521
Φ＝STAT+XTBT
wherein denotes a symmetric portion of the block matrix; omega11Is omegaqRow 1, column 1 blocking momentsArray, omega12Is omegaqRow 1 and column 2 block matrices,
Figure FDA00032420042000000522
is omegaqRow 1, column 3, block matrix, omega14Is omegaqRow 1, column 4, block matrix, omega15Is omegaqRow 1, column 5, block matrix, omega16Is omegaqRow 1, column 6, block matrix, omega22Is omegaqRow 2 and column 2 block matrix, omega33Is omegaqRow 3 and column 3 of the block matrix, omega44Is omegaqRow 4 and column 4 of the block matrix, omega55Is omegaqRow 5 and column 5 block matrices,
Figure FDA00032420042000000523
is a matrix omega11The block matrix of the r row and the s column;
Figure FDA00032420042000000524
is a matrix
Figure FDA00032420042000000525
The (j) th block matrix of (a),
Figure FDA00032420042000000526
is a matrix
Figure FDA00032420042000000527
The transpose of (a) is performed,
Figure FDA00032420042000000528
is a matrix N54Is transposed, STIs a transpose of the matrix S, X being the matrix XTTransposing; matrix array
Figure FDA00032420042000000529
S is a positive definite symmetric matrix to be solved;
Figure FDA0003242004200000061
N54and X is the free matrix to be solved; rho1,ρ3,ρ5Given a constant greater than zero;
in the fifth step, a solving algorithm is designed according to the judgment conditions obtained in the fourth step, and a process of obtaining the gain matrix K is specifically as follows:
step five, first: initialization parameter ρ1＞0，ρ2＞0，ρ3＞0，ε1,0＞0，ε2,0＞0，
Figure FDA0003242004200000062
And maximum number of iterations
Figure FDA0003242004200000063
And
Figure FDA0003242004200000064
let k be 0, kk be 0;
step five two: solving for
Figure FDA0003242004200000065
a 1,2, …,4, h 1,2), satisfying the following linear matrix inequality with optimization:
Figure FDA0003242004200000066
Figure FDA0003242004200000067
wherein the content of the first and second substances,
Figure FDA0003242004200000068
if κ ≈ 4, a feasible solution is found, and a gain matrix K ═ XS may be found-1(ii) a Otherwise, executing the next step;
step five and step three: adding 1 to the k value, if the k value after adding 1 is less than
Figure FDA0003242004200000069
Executing the step six and two; otherwise, executing the next step;
step five and four: reinitializing parameter ρ1＞0，ρ2＞0，ρ3＞0，ε1,0＞0，ε2,0＞0，
Figure FDA00032420042000000610
And adding 1 to the value of kk;
if the value of kk plus 1 is less than
Figure FDA00032420042000000611
Making the value of k added with 1 be zero, and executing the step six to two; otherwise, quitting if no feasible solution is found;
in the sixth step, according to the gain matrix K obtained in the fifth step, a self-adaptation law is designed, and a sliding mode control process of a control system with system uncertainty, probability distribution time lag, actuator failure and unknown external disturbance upper bound is realized, specifically:
the following adaptive laws are designed:
Figure FDA00032420042000000612
Figure FDA00032420042000000613
Figure FDA0003242004200000071
in the formula (I), the compound is shown in the specification,
Figure FDA0003242004200000072
γ0i，γ1i，γ2ggiven a gain constant greater than zero; u. offi0，wg0，
Figure FDA0003242004200000073
Is a given initial value; b is2iIs a matrix B2The ith column; d1gIs a matrix D1Column g of (1); kiIs the ith row of the matrix K;
according to the designed adaptive law, the following controllers are designed:
u(t)＝Kx(t)+u1(t)
wherein the content of the first and second substances,
Figure FDA0003242004200000074
Figure FDA0003242004200000075
where e is a given real number greater than zero,
Figure FDA0003242004200000076
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