Research on the Reliability of a Two-Robot Security System with Early Warning Function

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Introduction
With the progress and development of the times, robots are becoming more and more widely used.Robots are mechanical devices that automatically perform work, replacing or assisting human work, such as manufacturing, construction, or hazardous work.Terefore, the stability and reliability of the robot security system has become a hot issue.Early warning function refers to the ability to provide early warning and timely issuance of alarm signals before a system or component malfunctions, allowing sufcient time to predict various impending disasters and reduce economic losses caused to humans by disasters.Te reliability [1] of a system means the ability or property of the system to accomplish the specifed functions under the specifed time and the specifed using conditions.It is also one of the important characteristics of a repairable system.Te robot security system with early warning function can accurately detect the abnormal condition of the parts of the robot working system according to the early warning signal and carry out advanced control, continue to work, or carry out maintenance, so as to reduce the economic losses.In order to ensure the security and reliability of the system and avoid the occurrence of unexpected accidents, the early warning function is introduced into the repairable robot security system in this paper.Based on the abstract Cauchy problem theory, the system reliability is analyzed.
Te repairable system is an important system in the research of reliability mathematics, and reliability is one of the important contents in the research of the repairable system.Many scholars and researchers at home and abroad have done a lot of research on this kind of system, and achieved fruitful results.Some of the literatures adopt the method of qualitative analysis; for example, Wang et al. used linear operator semigroup theory in literatures [2][3][4][5][6][7][8][9] to study the semigroup properties of the main operator of repairable systems composed of faulty components and repairmen and discussed the well-posedness of the system model solution by using functional analysis methods.In literatures [10][11][12][13][14][15], Kamranfar et al. studied the stability of repairable systems.On the basis of discussing the asymptotic stability of the solutions of a repairable system with two diferent components connected in parallel, Wang et al. studied the controllability of the system in the zero state and the criterion of using the controllability of the system in literatures [16][17][18][19][20][21][22][23][24], and obtained the corresponding optimal control element of the system by using the minimization sequence, thus solving the optimal control problem of the system solution.In other literatures, the reliability index of the system is obtained by quantitative analysis.For example, Marco et al. obtained the reliability index of the system in the literatures [25][26][27][28][29][30][31][32] by using the Laplace transformation inversion method, MATLAB mathematical software calculation, linear diferential equation with constant coefcients satisfying initial conditions, and so on.In addition, some literatures combine the quantitative analysis method with computer technology to perform the numerical calculation and simulation of the repairable system.For example, Gao et al. studied a repairable system with early warning function in literature [33], utilizing that when the risk coefcient α ⟶ ∞, the warning system approximates a new model with weak solution-the model of a nonearly warning system; the corresponding conclusions were drawn and numerical simulation examples were given by computer, which greatly enriches the theory and practice of the repairable system with an early warning function; Zhou et al. studied the reliability of the system in literatures [34][35][36][37] and simulated the graphs of transient and steady-state reliability of the system by using Maple software.In summary, in the existing literatures and materials both domestically and internationally, the method of combining qualitative analysis with quantitative analysis is less efective than that of using qualitative or quantitative analysis alone.In this article, a kind of two-robot security system with an early warning function is taken as the research object, and attempts are made to combine qualitative analysis with quantitative analysis to study the reliability of the system.
Te main contents of this article are the modeling of the robot safety system, the well-posedness and controllability of the system solution, the reliability index of the system, and the numerical experiment of the reliability theory results.However, with the development and application of system reliability, when the system models become more complex, new methods and theories are needed to guide them, which is an important development problem of system reliability research in the future.
Te reliability of repairable system has achieved certain achievements in both theory and application.Nowadays, in the 21st century, the research on reliability has been elevated both domestically and internationally to the high level of understanding of saving resources and energy.Products (or equipment) are developing in the direction of gradually improving reliability and gradually reducing maintenance time, maintenance personnel, and maintenance costs.At the same time, reliability technology is the result of the joint development of multiple felds and technologies.It belongs to a comprehensive basic industry, and the current development trend is moving towards comprehensiveness, usability, automation, informatization, virtualization, and intelligence.Tus, higher economic benefts and stronger competitiveness can be achieved.

Mathematical Model
A robot is a complex system including mechanical, electronic, electrical, hydraulic, pneumatic, computer, and other types of components and control software.It is relatively complicated to study its reliability and safety [15,16].In order to consider this issue more clearly, this article assumes that the robot security system consists of two robots、security devices and a repairman [17].Assuming that the entire system does not consider the repair and replacement process and starts to run at the time t � 0. At the time t � 0, the two robots and the safety device are brand new, the system starts to operate normally, and the maintenance personnel goes on holiday.If the system fails, the repairman immediately terminates the vacation and repairs the faulty system immediately.If N(t) represents the state at the moment t, the system has the following situations: Te state-transfer chart of the system is shown in Figure 1.
In order to facilitate the modeling and model analysis, the following general assumptions can be made according to the state transition diagram of the repairable system: (1) Various faults are independent of each other in the statistical sense (2) Only when both robots fail (or the safety device fails due to conventional reasons), the whole system is in a fault state (3) Te failure rate of the system is constant, and the repair rate after system failure is nonconstant (4) Te normal working time of the robot security system follows the exponential distribution function F � 1 − e − λt , t ≥ 0, and λ > 0 (5) Te repair time of faulty parts of robot system follows the general distribution function G � 1 − e − μ i (x)t , t ≥ 0, μ i (x) > 0, and i � 1, 2 (6) Te two robots are exactly the same and repaired as new Since the time distribution of transitions between states of the system does not completely obey the negative exponential distribution, it can be known from the above hypothesis that N(t) represents the state at the moment t, so N(t), t ⩾ 0 { } is not a Markov process.But we can make it a high-dimensional Markov process by using the method of supplementary variable, assuming that the fault system is in 2 Journal of Mathematics a state of maintenance, the supplementary variable X i (t)(i � 2, 3) represents the maintenance time from the beginning of maintenance to the present, and let ) is a continuous-time twodimensional Markov process; that is, t at any time, given the concrete values of N(t) and X i (t), then the probability law of the process N(t), X i (t) | t ⩾ 0  (i � 2, 3) after time t has nothing to do with the history of the process before time t.Let P i (t)(i � 0, 1, c) denote the probability that the system is in state i at time t, P i (t, x)(i � 2, 3) represents the probability density of the time t that the system is in state i and the faulty part has been repaired x, i.e., Here, it should be noted that although P i (t, x)(i � 2, 3) is only defned as 0⩽x < t, but for the sake of discussion, P i (t, x)(i � 2, 3) can be defned according to the actual physical background of the system and P i (t, x)(i � 2, 3) is extended on x > t, that is, supplementary defnition P i (t, x) � 0, x > t, i � 2, 3.At the same time, the system is out of state i(i � 2, 3) of the risk function; that is, the fx quotiety of the faulty component in the system in the state i(i � 2, 3) can be defned as follows by the conditional probability [21,22]: And from the actual physical meaning of μ i (x), the following reasonable assumptions can be made: Te following is a discussion of the system state transition after ∆t time.Terefore, let λ represents the damage rate of the running system robot caused by its own reasons, λ ci indicates the normal failure rate of the system in the state i(i � 0, 1), λ s indicates the human fault quotiety of the operating system, and α represents the damage quotiety of hot standby robot, µ indicates the constant fx quotiety of the running robot, µ c indicates the constant fx quotiety of the running system, and P i (t) indicates the probability that the system is in state i at the moment t(i � 0, 1, c), P i (t, x) represents the probability that the system is in the state i and the repaired time x at the moment t, (t, x) in [0, ∞] × [0, ∞].μ i (x) represents the fx quotiety when the system is in state i and the repaired time x.Ten, it is deduced from the formula of total probability and the properties of Markov process (for convenience, it is assumed that ∆x is the same as ∆t): P 0 (t + ∆t) � P (at t when the system is in state 0, ∆t the system remains in state 0) + P (at t when the system is in state 1, ∆t the system is fxed to state 0) + P (at t when the system is in state c, ∆t the system is fxed to state 0) + P (at t when the system is in state 2, ∆t the system is fxed to state 0) + P (at t when the system is in state 3, ∆t the system is fxed to state 0): From formula (5) and the defnition of derivative, Journal of Mathematics 3 P 1 (t + ∆t) � P (at t the system is in state 1, the faulty robot in ∆t has not been repaired, and the other robot has not failed, and the system in ∆t has not left state 1) + P (at t the system is in state 0, and one robot in ∆t has failed): From formula (7) and the defnition of derivative, P c (t + ∆t) � P (at t when the system is in state c, ∆t the system routine fault has not been repaired, and the system has not left the state c) + P (at t when the system is in state 0, ∆t the system routine fault) + P (at t when the system is in state 1, ∆t the system routine fault): From formula (9) and the defnition of derivative, 4 Journal of Mathematics Similarly, there are P i (t + ∆t, x + ∆t) � P (at t the system is in state i, and the repair time of the faulty components is x, ∆t the system has not left state i), � P (at t the system is in state i, and the repair time of the faulty components is x) × P (∆t the failure components are not repaired yet), Derived from formula (11) and the defnition of partial derivative, Te boundary conditions and initial conditions of the system are discussed below [20].
Because P 2 (t, 0) represents the probability that the system just enters the state 2 at t, that is, the probability that the system just leaves the state 1 at t, i.e., From formula (13) and the defnition of derivative, In addition, P 3 (t, 0) represents the probability that the system just enters state 3 at t, that is, the probability that the system just leaves state 0 or state 1 at t, i.e., From formula (15) and the defnition of derivative Assuming that time t � 0, both parts are good, that is, the initial condition is At this time, from probability analysis formulas ( 5) to (17) [18,19], the integro-diferential equations of the mathematical model of the two-robot security system with early warning function can be described in the Figure 1 are

Well-Posedness of the System Solution
Since the repairable robot system with early warning function contains both integral and diferential, it is difcult to directly deal with it.Terefore, it is necessary to perform the necessary conversion before the reliability analysis [23].
In order to facilitate the subsequent discussion of the adaptability of the system solution, the system equations ( 18)-( 23) are translated into an abstract Cauchy problem in Banach space [24,25].For this defnition, make Take state space among which P � (P 0 , P 1 , P c , P 2 (x), P 3 (x)).At this point, it can be proved (X, ‖ • ‖) is a Banach space.Te following operators A and B and their domains D(A), D(B) are defned as follows: is a strictly succession function and meets P(0) � P 0 , P 1 , P c , P 2 (0), P 3 (0)  � P 0 , P 1 , P c , λP 1 , λ s And for { } of the operator B is defned as follows: Terefore, the system equations ( 18)-( 23) can be rewritten as an abstract Cauchy problem in Banach space: Several lemmas are given below, which play a signifcant part in describing the semigroup characteristics of system operators.

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So, from equations ( 29) and (30) and Fubini theorem, we have
Proof.Suppose P i (t) ∈ C ∞ 0 [0, ∞) and there is a constant C i , so that for any t ∈ [0, C i ], there are P i (t) � 0, (i � 2, 3).Construct a set L � P(t) � P 0 , P 1 , P c , P 2 (t), P 3 (t) ), and there is a constant C i > 0, makes for arbitrary t ∈ 0, C i  , always have At this time, using the knowledge of functional analysis, it is not difcult to verify that L is dense in Banach space X.Terefore, to verify that D(A) is dense in X, just prove D(A) is dense in L. For this, take P � (P 0 , P 1 , P c , P 2 (t), P 3 (t)) ∈ L; then, for any i(i � 2, 3), there is a constant , there is always P i (t) � 0, (i � 2, 3).Let f s (0) � P 0 , P 1 , P c , f s 2 (0), f s 3 (0)  � P 0 , P 1 , P c , λP 1 , λ s P 0 + λ s P 1 , f s (t) � P 0 , P 1 , P c , f s 2 (t), f s 3 (t)  � P 0 , P 1 , P c , P 2 (t), P 3 (t) . ( Among them . At this point, it is easy to verify f s (t) ∈ D(A), and

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Te above mentioned formula indicates that D(A) is dense in L, so D(A) is dense in X.
Proof.We split the proof this theorem into three steps.
Te frst step is to verify that A + B creates C 0 hemigroup T(t).
It can be verifed by the demi-closed operator theorem that A is a closed linear operator; from the defnition of B operator, B is a bounded linear operator, and From the above mentioned proof and the Hille-Yosida generation theorem, we know that the operator A creates a C 0 hemigroup, and then from the perturbation theorem of the semigroup, the operator A + B creates a C 0 hemigroup, denoted as T(t).
Te second step is to verify that A + B creates a hemigroup T(t) that is a positive C 0 semigroup.
In fact, when y i (i � 0, 1, c, 2, 3) is a nonnegative vector, P is a nonnegative vector, so (cI − A) − 1 is a positive operator.At the same time, it is known from the meaning of B that B is also a positive operator.Hence, exists and is bounded because Te third step is to prove that A + B creates positive C 0 compressed hemigroup T(t).
In fact, whatever P � (P 0 , P Among them Ten, A + B is a difusion operator.From the above mentioned proof and Pillips theorem, A + B creates a positive contraction C 0 hemigroup T(t).Terefore, by the uniqueness theorem of a C 0 hemigroup, T(t) is a positive contraction C 0 hemigroup.Te well-posedness of the system solution is discussed below; because the system equations ( 18)-( 23) are complex equations composed of diferential, partial diferential and integral, it is difcult to solve them directly.Terefore, in order to discuss the existence and uniqueness of the 10 Journal of Mathematics nonnegative solution of the system, system (18)-( 23) is converted into the form of convolution Volterra integral equation.To this end, notation is introduced: For convenience,  P i (t, x),  P i (t),  μ i (x) is still represented by P i (t, x), P i (t), μ i (x) in the following system equations.P i (t, x) can be obtained from the partial diferential equations (21): First, substituting (42) into (18), then make t − x � τ as a variable to obtain Since the initial condition P 0 (0) � 1, s − τ � v, P 0 (t) can be obtained as follows: Among them, Similarly, P 1 (t) and P c (t) can be solved by ( 19) and ( 20), respectively: Among them, Substituting formulas ( 44) and (46) into formula (22), respectively: 12

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Put expressions (44), ( 46), ( 47), (49), and (50) together to form the convolution Volterra integral equations: And write the Volterra integral equation in vector form Among them, According to the above mentioned equation, the following theorem holds.

□ Theorem 4. Te existence and uniqueness of nonnegative solutions for the system (18)-(23) are a necessary and sufcient condition for the existence and uniqueness of nonnegative solutions for the Volterra integral equation (52).
For any T > 0, from the expressions of f(t) and k(t − τ), each component f i (t)(i � 0, 1, c, 2, 3) of f(t) and each k i (t − τ)(i � 2, 3) of k(t − τ) are nonnegative bounded functions, according to documents [36,37] the following theorem holds.

and is unique on C[0, T].
To sum up, the following theorem holds.

Theorem 6. Te system (18)-(23) has a unique nonnegative solution on C[0, T].
Te main results of this paper and the well-posedness conclusions of the system solution are given below.

□
Theorem 8 (see [30]).0 is the simple eigenvalue of the operator A + B.
Proof.Discuss the equation (A + B)P � 0, that is, Solve the above equations to get Assume P 0 > 0, from the system of equations ( 57) and (58), It can be seen that, , so P � (P 0 , P 1 , P c , P 2 (x), P 3 (x)) is the eigenvector of 0 eigenvalue corresponding to the operator

Reliability of the System Solution
Te reliability of the system is one of the signifcant contents of the repairable system model.In the above system ( 18)-( 23), because the size of P 0 (t) indicates the probability that both robots are in good condition and can work normally at t � 0, the larger the P 0 (t) is, the closer the system is to working properly; therefore, the size of P 0 (t) determines the probability that the system will work properly and thus gets one of the important factors afecting the reliability of the system.So as to discuss the reliability of the system, several defnitions are given at frst.Defnition 9. P 0 (t) is called the transient reliability of system ( 18)-( 23).Defnition 10.If lim t ⟶ 0 P 0 (t) � P * 0 exists, then P * 0 is called the stable-state reliability of the system ( 18)-( 23).
Proof.Let the fault quotiety and fx quotiety be constants, i.e., Ten, the actual physical background of the system ( 18)-( 23) is  3 i�0 P i (t) + P c (t) � 1, which can be transformed into formula (18): A system of ordinary diferential equations can be converted from the upper system ( 18)-( 23 At this point, if Ten, the frst three equations of system ( 18)-( 23) can be converted into an abstract Cauchy problem: dP(t) dt � AP(t) + b, Te solution of the system (66) is obtained by using ordinary diferential equation theory and advanced algebra knowledge, including the following four steps: Step 1: Find all eigenvalues of matrix A Tat is, (r + 4κ + β)(r + 3κ + β)(r + β) � 0, then the eigenvalues of the matrix A are r Step 2: Seek e At , let Among them, q 1 (t) � e r 1 t , q 2 (t) �  t 0 e r 2 (t− s) q 1 (s)ds, q 3 (t) �  t 0 e r 3 (t− s) q 2 (s)ds.
Step 3: Find A − 1 .From the knowledge of linear algebra, it is easy to fnd the inverse matrix A − 1 of A, that is,

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Step 4: Finding the solution of the system (66), we have (73) Substitute the above formulas e At and A − 1 into the formula (73) and sort them out So, the transient reliability of the system (18)− ( 23) is Tus, by defnition of 10, the stable-state reliability of system (18)− ( 23) is From the above mentioned discussion, it is obvious that you can get P 0 (t) ≥ P * 0 , Terefore, according to the defnition of 11, the system ( 18)-( 23) is reliable.

Controllability of System Zero State
Using the method of functional analysis to fnd a control element μ * ∈ U, to study the controllability of P 0 (t), P 0 (t) in the system model can be transferred to the specifed state in a fnite time T(T > 0), and the allowable control set is selected as follows: and let η be the probability that the system reaches the desired state at fnite time T(T > 0) and fts e − 4κT < η < 1, then there is μ * ∈ U such that P 0 (T) � η.
Proof.By the formula of (75), Considering P 0 (T) as a function of the variable β, we derive dP 0 (T) and therefore Tus, we have Tis formula shows that P 0 (T) is a monotonically increasing function about the variable β, Note that lim β ⟶ 0 P 0 (T) � e − 4κT , lim β ⟶ ∞ P 0 (T) � 1. Terefore, for any η: e − 4κT < η < 1, according to the intermediate value theorem, there is β * ∈ U, such that P 0 (T) � η. (83) Te above mentioned proves that the zero state P 0 (T) of the system is controllable.
Teorem 14 indicates that the zero state P 0 (t) of the system is controllable, in other words, the initial state P 0 (t) of the system is controllable, but it does not mean that other states of the system are controllable, that is, the conclusion of controllability of the whole system is not necessarily derived from zero-state controllability, Similarly, to prove that the system is completely controllable, it is also necessary to prove other states of the system, such as P 1 (t), P c (t), P 2 (t, x), P 3 (t, x), etc. are controllable.

Numerical Simulation
According to Teorem 7, there is a unique nonnegative timedependent solution for the model of a repairable robot system with early warning function.In addition, the reliability of the two-robot security system with the function of early warning is discussed by using the theory and method of ordinary diferential equation, On this basis, the controllability of the zero state of the system is proved by using the method of functional analysis.Te following is a numerical simulation of the above results by using the numerical calculation method, which verifes the correctness of the Journal of Mathematics results of the system reliability theory [32].Assuming that the system failure rate and repair rate are constants κ, β, respectively, i.e., To do this, as shown by the above proof of reliability Teorem 13, the system is converted into an ordinary differential equation system (64), solve the linear equations, we have At this time, if κ � 0.3, β � 0.4, the numerical solution of system ( 18)-( 23) can be obtained by using Matlab mathematical software, and the result is shown in Figure 2.
It can be seen from Figure 2 that the system equations ( 18)-( 23) has a time-dependent asymptotically steady stablestate solution P * (x), which is consistent with the conclusion of the Teorem 12, at the same time, Teorem 12 indicates that the system equations ( 18)-( 23) is steady.It should be pointed out here that the importance of the Teorem 12 lies in that it not only proves the asymptotic stability [33] of the solution of the system equations ( 18)− ( 23), but also proves the existence of the stable-state solution P * (x) of the system related to the stable-state reliability P * 0 of the two-robot safety system with early warning function.
In addition, by selecting κ � 0.1, 0.2, 0.25, β � 0.25, 0.7, 0.95, respectively, the 3 groups of instantaneous reliability P 0 (t) and stable-state reliability P * 0 of the system equations ( 18)-( 23) can be obtained by using MATLAB software.For more intuitive comparison, put the charts of P 0 (t) and P * 0 together.Te result is shown in Figure 3: It can be seen from the Figure 3 that the transient reliability curves of the system equations ( 18)-( 23) are all above the stable-state reliability curves [34], when t⟶∞, P 0 (t)⟶P * 0 can be obtained by mathematical limit analysis, which is consistent with the reliability of the conclusion system equations ( 18)-( 23) of the Teorem 13.To some extent, the above results show that the numerical method can refect and depict that each state of the system tends to be stable with the change of time, and the results are in line with the actual situation of the robot security system.

Concluding Remarks
In this paper, the mathematical model of a two-robot safety system with an early warning function is studied, the semigroup characteristics of system operators are discussed by using linear operator semigroup theory, and the wellposedness of the solution of the system is proved.Under the assumption that the fault quotiety and fx quotiety of the system are constants, the early warning model equations are converted into ordinary diferential equations, the transient reliability and stable-state reliability of the system are obtained, and the reliability and zero-state controllability of the system are proved.Finally, the numerical solution of the ordinary diferential equation set of (64) is obtained by using Matlab mathematical software, and the graphs of the transient reliability and stable-state reliability of the system of equations ( 18)-( 23) are simulated, which shows that the results obtained by numerical calculation and numerical simulation are consistent with the proof of the above theory [35].
Te research on the repairable system model with an early warning function is mainly carried out by qualitative analysis, quantitative analysis, and the combination of qualitative analysis and quantitative analysis.Most of the existing literatures adopt qualitative analysis methods, but few use quantitative analysis methods.And the application of the qualitative analysis combined with the quantitative analysis method in the repairable system model is rarely reported in the repairable system model.First, the innovation of this paper is to combine the qualitative analysis with the quantitative analysis method in order to perfect and enrich the theory and method of repairable systems.Second, in the early warning system, when the warning prompt is invalid, the robot system model with an early warning function approaches to a repairable system model without an early warning function; the relationship between the early warning system and the nonearly warning system should be further studied, and the relationship between their steadystate solutions should be discussed.Finally, when studying the important indexes of system reliability, transient reliability, and steady-state reliability because of the relative complexity of the solution of the system (66) model, this paper uses the method of solving linear diferential equations with constant coefcients that satisfy initial conditions, which is computationally complex.Whether the solution 18 Journal of Mathematics process can be simplifed by the methods of Laplace transformation inversion, MATLAB mathematical software coding and compiling, and so on remains to be further studied.

Figure 1 :
Figure 1: State-transfer chart of the system.
For arbitrary, P ∈ D(A + B) have 〈(A + B)P, Q〉 � 0; thus, 0 is the simple eigenvalue of A + B.
One robot and safety device work, and one robot is in hot standby state State 1.One robot and safety device are working, and one robot is in a malfunction state State 2. Te state in which both robots are malfunctioning State 3. Te state of the system when the safety device fails State c. Te state of the normal faulty system