CN109901395B - Self-adaptive fault-tolerant control method of asynchronous system - Google Patents

Abstract

The invention relates to a self-adaptive fault-tolerant control method of an asynchronous system. The method comprises the steps of describing an asynchronous phenomenon between a system mode and a controller mode by using a hidden Markov model, designing a corresponding adaptive law to estimate parameters of upper and lower bounds of faults and disturbance of an actuator, and designing a state feedback controller to automatically compensate the influences of the faults and the disturbance based on an estimated result. Compared with a fault detection and identification mechanism, the method has the advantages of independence on an online estimation value, high resource utilization rate, quicker response and the like, and the control system aimed at by the method has universality, so that the method has a wide application background.

Description

Self-adaptive fault-tolerant control method of asynchronous system

Technical Field

The invention relates to an asynchronous self-adaptive fault-tolerant control method applied to a Markov jump system with actuator faults and unknown disturbance, and belongs to the field of fault-tolerant control.

Background

Today, as industrial technologies are rapidly developed, requirements for various performance indexes of a system are improved, the structure of the system is more complex, and the design difficulty of a controller is greatly increased. In some switching systems, often, due to time delay, when a controller is still in the previous mode after the mode of the system is switched, a mismatch behavior exists between the controller and the system; or in some markov jump systems, the modality of the controller cannot be synchronized with the modality of the actual system due to inaccurate information of the system modality that may be provided. The two asynchronous situations are ubiquitous in the actual production process, so the method has great application value in the application field of the actual control system.

The production process of the complex process is usually in extreme environments such as high temperature, high pressure or low temperature vacuum, and the like, and when improper operation or a control system fails, risks such as production interruption, explosion, toxic gas leakage and the like may be caused. With the increasing trend of the industrial process towards upsizing and complicating, and the emergence of large-scale and high-level integrated automation systems, the requirement for control quality is increasingly highlighted, and the attention on fault detection, fault diagnosis and fault-tolerant control systems is promoted. Fault tolerant control is a fault tolerant technique for a system, which means that when one or more critical components in the system fail, the system can continue to operate safely and stably. The method has the significance of ensuring that the system can still stably run when a fault occurs and has acceptable performance indexes. Therefore, the research on fault-tolerant control is very realistic.

The existing production equipment becomes increasingly complex, various faults are likely to occur in the operation process, the performance index of the system is reduced and the production efficiency is influenced, the equipment is damaged and even the life safety of operators is threatened, the negative influence caused by the faults cannot be effectively compensated by the traditional feedback control, the rapid development of fault-tolerant control is promoted, the structure or parameters of the system can be changed by unpredictable faults, and the control method for weakening, inhibiting and compensating or even eliminating the influence of the faults by automatically correcting relevant parameters on line is rapidly developed. Due to the difference of the design method of the controller and the difference of the design method of the adaptive law, a plurality of adaptive control methods can be formed. In practice, the controlled object often has uncertainty, that is, the mathematical model of the controlled object is not completely known, or the characteristics of the controlled object change when the parameters of the model are time-varying or disturbed. The main reasons for forming the uncertainty of the controlled object are as follows: due to the complexity of modern industrial devices and processes, it is difficult to know exactly its dynamic characteristics purely by means of mechanistic analysis; the characteristics of the controlled object have uncertainty due to the influence of the external uncertainty environment of the controlled object; the characteristics of the controlled object can change during the operation. According to the controlled object with uncertainty, how to design a controller can automatically adjust the parameters of the controller according to the uncertainty of the controlled object, so that the output of the controlled object tracks the reference input, and the tracking error between the output of the controlled object and the reference input meets the requirement, namely the task of adaptive control.

Disclosure of Invention

In order to improve the stability of a control system and the dynamic performance index of the system and enable the system to have smaller conservatism while considering the robustness of the system, the invention designs an indirect self-adaptive controller, estimates the failure rate and the upper and lower limits of disturbance by designing a corresponding self-adaptive law, takes the estimated value as a parameter to reduce the negative influence caused by the disturbance and fault and maintain the stability and the performance index of the system, so that the system can still keep a considerable operating state when the disturbance and the actuator fault exist.

Describing an asynchronous phenomenon between a system mode and a controller mode by using a hidden Markov model, and selecting a continuous system with unchanged linear time as a model of a controlled object, wherein the specific model is as follows:

wherein x (t) e RⁿIs a system state matrix, u^A(t)∈R^mFor a control input matrix from an actuator to a controlled object, ω (t) ∈ R^qFor external interference input, a (r (t)), B (r (t)), and W (r (t)) are system parameter matrices of appropriate dimensions, and r (t) is a markov process for time-continuous values over a finite set S ═ 1,2,3, …, S.

Transition probability matrix of system mode P ═ { pi ═ pi_ijIs defined as:

wherein，

Pi when i ≠ j_ij≥0，

π_ijThe change rate of the system jump is shown, and N is the total number of the subsystems.

Controller transition probability matrix Ω ═ μ_iφIs defined as:

Pr{σ(t)＝φ|r(t)＝i}＝u_iφ (3)

wherein

u_iφAnd M is the total number of the controllers, wherein M is the probability of the jump of the controllers.

And (2) considering that only partial failure faults of the actuator occur, wherein the form is as follows:

u^A(t)＝ρu(t) (4)

where ρ ═ diag (ρ)₁,ρ₂,…,ρ_h) Is the failure rate of the actuator and partial failure of the actuator

When rho_hWhen the value is 1, the h-th actuator operates normally; when rho_hWhen the h-th actuator is completely failed, the situation is shown as 0; when 0 is present<ρ_h<1 the h-th actuator is in a partial failure state.

The new system state equation can be rewritten as follows:

x(t)＝A_ix(t)+B_iρu(t)+W_iw(t) (5)

step (3) designing an adaptive law to estimate unknown partial failure rate rho and the upper and lower bounds of unknown disturbance

The specific design is as follows:

wherein

And

is an estimate of the upper and lower bounds of the disturbance,

is an estimate of the partial failure rate.

σ＝[max{x^TP_icol₁(B_i)},max{x^TP_icol₂(B_i)}，…，max{x^TP_icol_m(B_i)}]，

ζ_k＝[max{F_i}_1k,max{F_i}_2k,…,max{F_i}_mk]^T，

Wherein col₂(B_i) Representation matrix B_iσ is the matrix x^TP_iB_iMaximum element matrix, ζ, after maximum value of each column_kAccording to a matrix F_iAnd taking the most valued vector after the most valued vector. S is a matrix with disturbance estimates, and a reduction is mainly performed for the following transformations.

Where ρ ═ diag_h[ρ_h],

w＝[w ₁,w ₂,…,w _q]^TDue to ρ_h,

wIs an unknown constant, the error system can be written in the form:

and (4) designing a self-adaptive robust state feedback fault-tolerant controller to ensure that the closed-loop system is gradually stable. The controller is designed as follows:

wherein K_1φ,K_2φ,K_3φRespectively, the gain of the controller is the gain of the controller,

respectively, an estimated value of the failure rate and estimated values of the upper and lower bounds of the disturbance. Tau is_w＝diag{τ_w1,τ_w2,…,τ_w3},

The system equation can be written as follows by equations (5) and (9):

step (5) for the fault tolerant control system described in equation (10), the proposed controller gain K of the adaptive feedback controller is applied_2φ,K_3φThe design is as follows:

k is designed for any rho by a method of linear matrix inequality_1φ,P_i：

Wherein X_i>0，X_iIs an arbitrary positive definite matrix to obtain

The invention has the beneficial effects that: the invention designs an indirect adaptive controller, estimates the failure rate and the upper and lower bounds of disturbance by designing a corresponding adaptive law, takes the estimated value as a parameter to reduce the negative influence caused by the disturbance and the fault and maintain the stability and the performance index of the system, so that the system can still keep a considerable running state when the disturbance and the actuator fault exist.

Description of the drawings:

FIG. 1: a system state of the open loop control system;

FIG. 2: the system state of the closed-loop system with actuator failure;

FIG. 3: introducing a system state aiming at an actuator fault controller by a closed-loop system;

FIG. 4: the system state of the mechanical arm open-loop control system;

FIG. 5: the system state of the closed-loop system of the mechanical arm is in the fault state of the actuator;

FIG. 6: the closed loop system of the mechanical arm introduces a system state for the actuator fault controller.

Detailed Description

The invention will be described in further detail below with reference to an embodiment shown in the drawings.

The invention provides an asynchronous self-adaptive fault-tolerant control method applied to a Markov jump system with actuator faults and unknown disturbance, which comprises the following steps:

(1) the method comprises the following steps of describing an asynchronous phenomenon between a system mode and a controller mode by using a hidden Markov model, selecting a continuous system with unchanged linear time as a model of a controlled object, and specifically:

wherein x (t) e RⁿIs a system state matrix, u^A(t)∈R^mFor a control input matrix from an actuator to a controlled object, ω (t) ∈ R^qFor external interference input, a (r (t)), B (r (t)), and W (r (t)) are system parameter matrices of appropriate dimensions, and r (t) is a markov process for time-continuous values over a finite set S ═ 1,2,3, …, S.

Hopping probability matrix pi of system mode ═ pi_ijIs defined as:

wherein the content of the first and second substances,

pi when i ≠ j_ij30，

π_ijThe change rate of the system jump is shown, and N is the total number of the subsystems.

Controller transition probability matrix Ω ═ μ_iφIs defined as:

Pr{σ(t)＝φ|r(t)＝i}＝u_iφ (3)

wherein

u_iφAnd M is the total number of the controllers, wherein M is the probability of the jump of the controllers.

For ease of understanding, step 1 is now explained as follows: generally speaking, the modeling of the system needs to have certain universality, the system model is a typical fault and disturbance state equation, and the used symbols are general default symbols, so that ambiguity can be well reduced.

(2) Considering that the actuator only has partial failure fault, the form is as follows:

u^A(t)＝ρu(t) (4)

where ρ ═ diag (ρ)₁,ρ₂,…,ρ_h) Is the failure rate of the actuator and partial failure of the actuator

When rho_hWhen the value is 1, the h-th actuator operates normally; when rho_hWhen the h-th actuator is completely failed, the situation is shown as 0; when 0 is present<ρ_h<1 the h-th actuator is in a partial failure state.

The new system state equation can be rewritten as follows:

x(t)＝A_ix(t)+B_iρu(t)+W_iw(t) (5)

for ease of understanding, step 2 is now explained as follows: the following assumptions must first be made:

assume that 1: all states of the system are measurable at any time;

assume 2: { A_i,B_iRho for any fault model rho e { rho ∈¹…ρ^LAll controllable;

assume that 3: for the system (2) there is a matrix function F, let W_i＝B_iF_i(purpose is disturbance and control matching);

assume 4: any fault model rho epsilon { rho ∈ }¹…ρ^L}，rank[B_iρ]＝rank[B_i]。

The above 4 assumptions are necessary to proceed to the next step.

(3) Designing an adaptation law to estimate the unknown fractional failure rate p and the upper and lower bounds of the unknown disturbance

The specific design is as follows:

wherein

And

is an estimate of the upper and lower bounds of the disturbance,

is an estimate of the partial failure rate.

σ＝[max{x^TP_icol₁(B_i)},max{x^TP_icol₂(B_i)}，…，max{x^TP_icol_m(B_i)}]，

ζ_k＝[max{F_i}_1k,max{F_i}_2k,…,max{F_i}_mk]^T，

Wherein col₂(B_i) Representation matrix B_iσ is the matrix x^TP_iB_iMaximum element matrix, ζ, after maximum value of each column_kAccording to a matrix F_iAnd taking the most valued vector after the most valued vector. S is oneThe matrix with the disturbance estimates is subjected to a reduction mainly for the subsequent transformations.

Where ρ ═ diag_h[ρ_h],

w＝[w ₁,w ₂,…,w _q]^TDue to ρ_h,

wIs an unknown constant, the error system can be written in the form:

for ease of understanding, step 3 is now explained as follows: the corresponding adaptive laws are designed to better estimate the failure rate and the upper and lower bounds of the disturbance, but the actual values of the failure rate and the disturbance do not need to be accurately estimated. This is where the present invention distinguishes the method of separation from normal fault diagnosis, and the present invention does not rely on accurate estimates.

(4) An adaptive robust state feedback fault-tolerant controller is designed to ensure that a closed-loop system is gradually stable. The controller is designed as follows:

wherein K_1φ,K_2φ,K_3φRespectively, the gain of the controller is the gain of the controller,

respectively, an estimated value of the failure rate and estimated values of the upper and lower bounds of the disturbance. Tau is_w＝diag{τ_w1,τ_w2,…,τ_w3},

The system equation can be written as follows by equations (5) and (9):

step (5) for the fault tolerant control system described in equation (10), the proposed controller gain K of the adaptive feedback controller is applied_2φ,K_3φThe design is as follows:

k is designed for any rho by a method of linear matrix inequality_1φ,P_i：

Wherein X_i>0，X_iIs an arbitrary positive definite matrix to obtain

The following conclusions can be drawn to illustrate that the closed loop system is consistently and ultimately stable.

Conclusion 1: aiming at the closed-loop fault-tolerant control system (5) meeting 4 assumptions, if a positive definite matrix P exists, the condition feedback controller (9) and the adaptive law (6) are introduced to ensure that

I.e. the closed loop system is asymptotically stable.

Validity verification is performed using a simple example:

for equation of state

And selecting the parameters. The parameters of three modes are selected as follows:

the hopping probability of the system mode is as follows:

the hopping probability of the controller is as follows:

where the system is open loop unstable.

Before 60 seconds, the perturbation w (t) [ -0.5,0.2sin (5t), -0.3]^TAnd the actuator is not failed, i.e. ρ ═ diag [1,1 ═ d](ii) a At 60 seconds, the system has partial failure of the actuator, i.e. p ═ diag [0.27,0.45 ═ d]The controller is introduced for the partially failed part up to 130 seconds. The state of the open loop system is shown in fig. 1, and it can be seen that the state of the system is divergent, so the open loop system is an unstable system. When the system is operating without failure, the system state is as shown in FIG. 2. As can be seen from the figure, when the controller is introduced into the open loop system, the state of the closed loop system converges to zero. The state of the closed loop system with a fault is shown in fig. 3, and it is obvious that after a period of time from the fault, the state begins to diverge, but under the action of the controller, the state finally converges to zero. Thus, this example can be said to beObviously, the controller designed by the invention can well compensate the influence of actuator faults and interference on asynchronous phenomena of a system mode and a controller mode.

(7) Consider a single link robotic arm system for validation as follows:

θ (t) is the angular position of the robotic arm; u (t) is a control input, d₁(t),d₂(t) is the external disturbance, M is the mass of the payload, g is the gravitational acceleration, L is the length of the robot arm, J_iIs the moment of inertia and D is the viscous friction. The parameters g-9.8, L-0.4 and D-2.

Definition of

Then (13) can be written as follows:

the parameters are selected as follows: j. the design is a square₁＝1.0,J₂＝1.2,J₃＝2.0,J₄1.5, the system transition matrix and the controller hopping matrix are as follows:

the initial conditions were:

the partial failure parameter of the controller is rho 0.45, and the external disturbance is respectively: d₁＝e^-1sin(t),d₂＝e^-1sin(t)。

The gain of the controller can be obtained according to theorem 1. In the simulation process, the fault state of the system is considered, and in the first 50 seconds of the operation of the system, the disturbed operation of the system has no actuator fault, which means that rho is 1, and the actuator has partial fault from 50 seconds, and is described as rho being 0.45.

The state of the open-loop system is shown in fig. 4, and it can be seen that the state of the open-loop system converges but does not converge to 0, which indicates that the open-loop system is only a stable system, not asymptotically stable. Therefore, in order to improve the performance of the entire system, it is necessary to introduce a controller. First consider a system operating without a fault, the system state is shown in fig. 5. As can be seen, when the controller is introduced into an open loop system, the state of the closed loop system converges to zero. On the basis, partial faults of the actuator after the system operates for several seconds are considered. The final result is shown in fig. 6, and it is clear that under the control of the designed controller, the trend of the state starts to diverge after a certain time of failure occurrence, but finally tends to 0. Therefore, the numerical calculation example based on the single-link robot arm system also shows that the designed controller can well compensate the influence of actuator faults and interference on the Markov jump system with asynchronous phenomenon, and the effectiveness and the superiority of the invention are also reflected.

In order to improve the stability of a control system and the dynamic performance index of the system and enable the system to have smaller conservatism while taking the robustness of the system into consideration, the invention designs an indirect self-adaptive controller, estimates the failure rate and the upper and lower bounds of disturbance by designing a corresponding self-adaptive law, takes an estimated value as a parameter to reduce the negative influence caused by the disturbance and the fault and maintain the stability and the performance index of the system; in addition, under the condition that the system mode is not matched with the controller mode, the invention further considers the condition that some characteristic information is extracted, and designs a corresponding self-adaptive controller according to the information to make up the problem of inaccurate information caused by mode mismatching, so that the system can still keep a considerable running state when disturbance, actuator failure and asynchronous phenomenon occur.

Claims (1)

1. The self-adaptive fault-tolerant control method of the asynchronous system is characterized by comprising the following steps:

describing an asynchronous phenomenon between a system mode and a controller mode by using a hidden Markov model, and selecting a continuous system with unchanged linear time as a model of a controlled object, wherein the specific model is as follows:

wherein x (t) e RⁿIs a system state matrix, u^A(t)∈R^mFor a control input matrix from an actuator to a controlled object, ω (t) ∈ R^qFor external interference input, a (r (t)), B (r (t)), and W (r (t)) are system parameter matrices of appropriate dimensions, and r (t) is a markov process for time-continuous values over a finite set S ═ 1,2,3, …, S };

hopping probability matrix pi of system mode ═ pi_ijIs defined as:

wherein the content of the first and second substances,

pi when i ≠ j_ij≥0，

π_ijThe change rate of the system jump is shown, and N is the total number of the subsystems;

controller transition probability matrix Ω ═ μ_iφIs defined as:

Pr{σ(t)＝φ|r(t)＝i}＝u_iφ (3)

wherein

u_iφThe probability of the jumping of the controller is shown, and M is the total number of the controllers;

and (2) considering that only partial failure faults of the actuator occur, wherein the form is as follows:

u^A(t)＝ρu(t) (4)

where ρ ═ diag (ρ)₁,ρ₂,…,ρ_h) Is the failure rate of the actuator and partial failure of the actuator

When rho_hWhen the value is 1, the h-th actuator operates normally; when rho_hWhen the h-th actuator is completely failed, the situation is shown as 0; when 0 is present<ρ_h<1, the h-th actuator is in a partial failure state;

the new system state equation is rewritten as follows:

x(t)＝A_ix(t)+B_iρu(t)+W_iw(t) (5)

step (3) designing an adaptive law to estimate unknown partial failure rate rho and the upper and lower bounds of unknown disturbance

w(ii) a The specific design is as follows:

wherein

And

is an estimate of the upper and lower bounds of the disturbance,

is an estimate of the partial failure rate;

σ＝[max{x^TP_icol₁(B_i)},max{x^TP_icol₂(B_i)}，…，max{x^TP_icol_m(B_i)}]，

ζ_k＝[max{F_i}_1k,max{F_i}_2k,…,max{F_i}_mk]^T，

wherein col₂(B_i) Representation matrix B_iσ is the matrix x^TP_iB_iMaximum element matrix, ζ, after maximum value of each column_kAccording to a matrix F_iTaking a most value vector after the most value; s is a matrix with disturbance estimation values, and simplification is mainly carried out on the subsequent transformation;

where ρ ═ diag_h[ρ_h],

w＝[w ₁,w ₂,…,w _q]^TDue to ρ_h,

wIs an unknown constant, the error system can be written in the form:

step (4), designing a self-adaptive robust state feedback fault-tolerant controller to ensure that a closed-loop system is gradually stable; the controller is designed as follows:

wherein K_1φ,K_2φ,K_3φRespectively, the gain of the controller is the gain of the controller,

respectively an estimated value of the failure rate and estimated values of the upper and lower bounds of disturbance; tau is_w＝diag{τ_w1,τ_w2,…,τ_w3},

The system equation is written as follows by equations (5) and (9):

step (5) for the fault tolerant control system described in equation (10), the proposed controller gain K of the adaptive feedback controller is applied_2φ,K_3φThe design is as follows:

k is designed for any rho by a method of linear matrix inequality_1φ,P_i：

Wherein X_i>0，X_iIs an arbitrary positive definite matrix to obtain

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