# Asymptotic Stability of Delay-Controlled Nonlinear Stochastic Systems with Actuator Failures

Authors: N. Zhou; R. H. Huan
Journal: Shock and Vibration
 (2017)

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## Abstract

The problem of asymptotic stability of delay-controlled nonlinear stochastic systems with actuator failures is investigated in this paper. Such a system is formulated as a continuous-discrete hybrid system based on the random switch model of failure-prone actuator. Time delay control force is converted into delay-free one by randomly periodic characteristic of the system. Using limit theorem and stochastic averaging, an approximate formula for the largest Lyapunov exponent of the original system is then derived, from which necessary and sufficient conditions for asymptotic stability are obtained. The validity and utility of the proposed procedure are demonstrated by using a stochastically driven nonlinear two-degree system with time delay feedback and actuator failure.


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## Body

## 1. Introduction

Modern technological systems rely on sophisticated control systems to meet increased performance requirements. In such systems, control components (e.g., actuator or sensor) are often subjected to random failures [1, 2]. It is known that control component failures can cause performance deterioration of control systems or even system instability leading to catastrophic accidents. Therefore, stability analysis for systems with control component failures has been an important and challenging research problem.In practical applications, actuators are very important in transferring the controller output to the structure, and actuator failures are the typical faults appearing in control systems. In the last few decades, issues concerning stability and control problem of systems with actuator failures have been examined in the literatures. Mahmoud et al. [3] analyzed the stochastic stability of fault tolerant control systems in the presence of noise using the Lyapunov function approach. Gundes [4] studied the stability of the MIMO feedback system under possible failures in either the sensor-connections or actuator-connections. Aberkane et al. [5] dealt with the problem of robust H∞ control of active fault tolerant control systems with Markovian parameters via dynamic output feedback. A state feedback output tracking adaptive control scheme which is developed for plants with actuator failures is proposed by Tao et al. [6]. Seo and Kim [7] designed a robust and reliable H∞ control for linear uncertain systems with actuator failures. The previous studies focused mainly on the linear systems or the one-dimensional nonlinear system. Furthermore, time delay is assumed to be ignorable. However, the phenomenon of time delay is common in control systems, which is a source of instability [8–10]. Thus, development of methodology for the stability analysis of delay-controlled nonlinear systems with actuator failures is thus much deserving.In the present research, asymptotic stability of delay-controlled nonlinear systems with actuator failures will be investigated. The organization of this paper is as follows. In the next section, the random switch model of failure-prone actuator is established. Section3 converts time delay control force into the delay-free one using randomly periodic characteristic of the system. Reduced averaged equations are obtained in Section 4. Section 5 gives the necessary and sufficient conditions for asymptotic stability of the system. An illustrated example is presented in Section 6, and conclusion follows thereafter.


## 2. Formulation of Problem

### 2.1. Model of Possible Failure Actuator

The behavior of a possible failure actuator in a control system can be described as(1)uat=κut,where u(t) is commanding signal to the actuator and ua is the actual output of the possible failure actuator. κ∈0,1 is the failure factor which expressed changed actuator characteristics. The case κ=0 corresponds to complete failure of the actuator and therefore no output; κ=1 refers to a functional actuator which is working normally. Therefore, in a real control system, the actuator may have two states: a “functional” state and a “failure” state. The “functional” actuator may change to failure actuate due to occasional breakdowns and the failure actuator can also be recovered during a finite time interval due to either efficient mobile maintenance or full redundancy. The random switches between these two states occur as a continuous-time Markov process [3, 5, 11]. Then, (1) can be rewritten in the following Markov jump form:(2)uat=κstut,where s(t) is a Markov process, which points to state of the actuator, and it takes discrete values from a finite set S={1,2}. The transition rate from the functional state to failure state is assumed to be λ1 and the backward transition rate is λ2. Then, the transition rate matrix Λ of s(t) can be specified by [12](3)Λ=-λ1λ1λ2-λ2.Denote byμ(u) the stationary or equilibrium probability that s(t)=u(u=1,2). Then μ(u) satisfies the following equations [13]:(4)∑u=12μuλui=0,∑u=12μu=1,i=1,2.μ(u) will be used in the next section.


### 2.2. Nonlinear Stochastic Systems with Actuator Failures

Consider ann degree-of-freedom (DOF) controlled stochastically driven nonlinear system with n actuators marked as Ω=1,2,…,n. The actuators in the system are distinguished in two groups with regard to potential failures. One group is the set of actuators that are susceptible to failure, denoted by Ωf. The other is a set of actuators that are robust to faults eventually, by Ωn=Ω-Ωf. The motion of the system is usually governed by n second-order stochastic differential equations in the generalized displacements. These second-order equations can always be recast as 2n first-order equations in the Hamiltonian formulation(5a)q˙i=∂H~′∂pi,(5b)p˙i=-∂H~′∂qi-εc~ij′q,p∂H~′∂pj+εκistuiqτ,pτ+ε1/2fikq,pWkt,where i,j=1,2,…,N, k=1,2,…,m, and qi and pi are the generalized displacements and momenta, respectively; q=(q1,…,qn)T and p=(p1,…,pn)T. ε is a small parameter. H~′=H~′(q,p) is the Hamiltonian. -εc~ij′(q,p)∂H~/∂pj and ε1/2fik(q,p)Wk(t) represent the weak damping and pure parametric random excitations, respectively. Wk(t) are independent Gaussian white noises with zero means and correction functions E[Wk(t)Wl(t+τ)]=2Dklδ(τ). qτ=q(t-τ) and pτ=p(t-τ) are system’s states with time delay. κi[s(t)] are the two-state failure factors(6)κi1=1,i∈Ωn,0,i∈Ωf,κi2=1,i∈Ω.Note that, by modeling the transition between two states of the failure-prone actuator as a Markov jump process, the stochastic stability of system (5a) and (5b) can be studied in the framework of Markov jump hybrid systems.


## 2.1. Model of Possible Failure Actuator

The behavior of a possible failure actuator in a control system can be described as(1)uat=κut,where u(t) is commanding signal to the actuator and ua is the actual output of the possible failure actuator. κ∈0,1 is the failure factor which expressed changed actuator characteristics. The case κ=0 corresponds to complete failure of the actuator and therefore no output; κ=1 refers to a functional actuator which is working normally. Therefore, in a real control system, the actuator may have two states: a “functional” state and a “failure” state. The “functional” actuator may change to failure actuate due to occasional breakdowns and the failure actuator can also be recovered during a finite time interval due to either efficient mobile maintenance or full redundancy. The random switches between these two states occur as a continuous-time Markov process [3, 5, 11]. Then, (1) can be rewritten in the following Markov jump form:(2)uat=κstut,where s(t) is a Markov process, which points to state of the actuator, and it takes discrete values from a finite set S={1,2}. The transition rate from the functional state to failure state is assumed to be λ1 and the backward transition rate is λ2. Then, the transition rate matrix Λ of s(t) can be specified by [12](3)Λ=-λ1λ1λ2-λ2.Denote byμ(u) the stationary or equilibrium probability that s(t)=u(u=1,2). Then μ(u) satisfies the following equations [13]:(4)∑u=12μuλui=0,∑u=12μu=1,i=1,2.μ(u) will be used in the next section.


## 2.2. Nonlinear Stochastic Systems with Actuator Failures

Consider ann degree-of-freedom (DOF) controlled stochastically driven nonlinear system with n actuators marked as Ω=1,2,…,n. The actuators in the system are distinguished in two groups with regard to potential failures. One group is the set of actuators that are susceptible to failure, denoted by Ωf. The other is a set of actuators that are robust to faults eventually, by Ωn=Ω-Ωf. The motion of the system is usually governed by n second-order stochastic differential equations in the generalized displacements. These second-order equations can always be recast as 2n first-order equations in the Hamiltonian formulation(5a)q˙i=∂H~′∂pi,(5b)p˙i=-∂H~′∂qi-εc~ij′q,p∂H~′∂pj+εκistuiqτ,pτ+ε1/2fikq,pWkt,where i,j=1,2,…,N, k=1,2,…,m, and qi and pi are the generalized displacements and momenta, respectively; q=(q1,…,qn)T and p=(p1,…,pn)T. ε is a small parameter. H~′=H~′(q,p) is the Hamiltonian. -εc~ij′(q,p)∂H~/∂pj and ε1/2fik(q,p)Wk(t) represent the weak damping and pure parametric random excitations, respectively. Wk(t) are independent Gaussian white noises with zero means and correction functions E[Wk(t)Wl(t+τ)]=2Dklδ(τ). qτ=q(t-τ) and pτ=p(t-τ) are system’s states with time delay. κi[s(t)] are the two-state failure factors(6)κi1=1,i∈Ωn,0,i∈Ωf,κi2=1,i∈Ω.Note that, by modeling the transition between two states of the failure-prone actuator as a Markov jump process, the stochastic stability of system (5a) and (5b) can be studied in the framework of Markov jump hybrid systems.


## 3. Randomly Periodic Solution Approximation

Equations (5a) and (5b) can be converted to the following Itô equations by adding Wong-Zakai correction terms:(7a)dqi=∂H~∂pidt,(7b)dpi=-∂H~∂qi-εc~ijq,p∂H~∂pj+εκistuiqt-τ,pt-τdt+ε1/2σikq,pdBkt,where H~ and c~ij(q,p) are the modified Hamiltonian and coefficients of damping, respectively; Bk(t) are standard Wiener processes such that σik(q,p)dBk(t)=fik(q,p)Wk(t)dt.Here, we only consider the case that the associated Hamiltonian system of system (7a) and (7b) is integrable [14]. Then, the Hamiltonian H~ has the form(8)H~q,p=∑i=1nH~iqi,pi,H~iqi,pi=12pi2+Uiqi,where H~1,H~2,…,H~n are n independent first integrals of the associated Hamiltonian system.For the case of small time delay, it is proved that the following approximate expressions hold [10]:(9)qit-τ≈cos⁡ωiτqit-sin⁡ωiτωipit,pit-τ≈cos⁡ωiτpit+ωisin⁡ωiτqit,where ωi is the average frequency.According to expression (9), the time delayed control forces κi[s(t)]uiq(t-τ),p(t-τ) can be expressed in terms of current state variables with time delay as a parameter; that is, κi[s(t)]uiq(t-τ),p(t-τ)=κi[s(t)]ui(q,p;τ). Then, splitting κi[s(t)]ui(q,p;τ) into conservative part and dissipative part and combining them with -∂H~/∂qi and -εc~ij(q,p)∂H~/∂pj, respectively, yield(10a)dqi=∂H′stτ∂pidt,(10b)dpi=-∂H′stτ∂qi-εcij′stq,p;τ∂H′stτ∂pjdt+ε1/2σikq,pdBkt,where H′st and cij′stare the Markov jump Hamiltonian and damping coefficients. Since the control forces are weak, it is reasonable to assume that the conservative part of the delayed control forces may change the Hamiltonian slightly but not change the integrability of the associated Hamiltonian system. Then, the resulting system (10a) and (10b) is still a quasi integrable Hamiltonian system.


## 4. Averaged Equations

According to the limit theorem [15], under the assumption of ergodicity of s(t), the solution of (10a) and (10b) converges in probability of the solution of the following weighted averaging equations as ε→0:(11a)dqi=∂Hτ∂pidt,(11b)dpi=-∂Hτ∂qi-ε∑u=1,2cij′uq,p;τ·μu∂Hτ∂pjdt+ε1/2σikq,pdBkt,where μ(u) is given in (4), and (12)Hτ=∑i=1nHiqi,pi;τ,Hiqi,pi;τ=∑u=1,2Hi′uqi,pi;τ·μu.It is noted from (12) that limiting theorem will not change the integrality of the quasi Hamiltonian system. Further assume that the associated Hamiltonian system is nonresonant. Then, by using the stochastic averaging method for the quasi integrable Hamiltonian systems [14], the following averaged Itô equations associated with (11a) and (11b) can be obtained: (13)dHi=miH;τdt+σ-ikH;τdBkt,where H=[H1,H2,…,Hn]T, and the drift coefficients mi(H;τ) and diffusion coefficients σ-ik(H;τ) are given by(14)miH;τ=1TH;τ∮-∑u=1,2crj′uq,p;τ·μu∂H∂pj∂Hi∂pr+12σrkσjk∂2Hi∂pr∂pj∏s=1n∂Hs∂ps-1dqs,bisH;τ=σ-ikσ-skH;τ=1TH;τ∮σjkσlk∂Hi∂pj∂Hs∂pl∏s=1n∂Hs∂ps-1dqs,where ∮·dqs represents an n-fold loop integral and(15)TH;τ=∮∏s=1n∂Hs∂ps-1dqs.After this approximation, the dimension of system is reduced significantly from 2n to n.


## 5. The Largest Lyapunov Exponent

Linearizing (13) at H=0 yields (16)dHi=miLH;τdt+σ-ikLH;τdBkt.Since (5a) and (5b) contain only parametric excitations, Q=P=0 is the trivial solution of (1) and H=0 is the trivial solution of (16). Then, the coefficients of (16) satisfy(17a)limH→0miLH;τ=0,limH→0σ-ikLH;τ=0,(17b)kmiLH;τ=miLkH;τ,kσ-ikLH;τ=σ-ikLkH;τ,k>0.Introduce the following new variables:(18a)ρ=12ln⁡H,(18b)αi=HiH,i=1,2,…,n.The Itô equations for ρ and αi are then obtained from (16) by using Itô differential rule as follows:(19a)dρ=Gα;τdt+Ξuα;τdBut,(19b)dαi=G-iα;τdt+Ξ-iuα;τdBut,where α=[α1,α2,…,αn]T and(20)Gα;τ=12∑i=1nmiLα;τ-14∑i,j=1n∑k=1mσ-ikLα;τσ-jkLα;τ,G-iα;τ=miLα;τ-αi∑s=1nmsLα;τ+12αi∑r,j=1n∑k=1mσ-rkLα;τσ-jkLα;τ-12∑j=1n∑k=1mσ-ikLα;τσ-jkLα;τ,Ξ-iuα;τ=σ-iuLα;τ-αi∑j=1nσ-juLα;τ.Define the Lyapunov exponent of system (16) as the asymptotic rate of exponential growth of the square-root of the Hamiltonian; that is,(21)Λ=limt→∞1tlog⁡H1/2.Integrating (19a) from 0 to t and then dividing it by t yield (22)12tlog⁡Ht=12tlog⁡H0+1t∫0tGα;τds+1t∫0tΞuα;τdBus.As t→∞, the first and the last terms in the right hand side of (22) vanish and, as a consequence,(23)Λ=limt→∞1t∫0tGα;τdτ.From (18b), it is seen that ∑i=1nαi=1, and only n-1 equations for αi in (19b) are independent. Without loss of generality, in the following, the first (n-1) equations for α1,α2,…,αn-1 are taken as independent equations. Let α′=[α1,α2,…,αn-1] be n-1 dimensional vector diffusion process. Suppose that α′ is ergodic over the interval 0≤αk≤1(k=1,2,…,n-1). Under this condition, the time averaging in (23) can be replaced by assembly averaging; that is,(24)Λ=EGα′;τ=∫Gα′;τpα′dα′,where p(α′) is the stationary probability density of vector process α′, which can be obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with first n-1 independent equations of (19b).For ann-dimensional (16), there exist n Lyapunov exponents. However, (24) has been proved to converge to the largest Lyapunov exponent Λmax [16]. Based on Oseledec ergodic theorem, a necessary and sufficient condition for asymptotic stability with probability one of the trivial solution of the system governed by (5a) and (5b) is approximately(25)Λmax=∫Gα′;τpα′dα′<0.


## 6. Numerical Example

To demonstrate the application and validity of the proposed method, consider a stochastically excited nonlinear two-degree system with time delay feedback and actuator failure, governed by(26)q˙1=p1,p˙1=-ω12q1-β11p1-β22p2+κstut-τ+f11q1ξ1t+f12q22ξ2t,q˙2=p2,p˙2=-αq23-β21p1-β22p2+f21q1ξ1t+f22q22ξ2t,where ω1, α>0, and βij>0 are constants; ξi(t) is Gaussian white noise with zero mean and intensity 2D; κ[s(t)]u(t-τ) denotes the possible failure actuator with time delay. κ[s(t)] is the failure factor. The two-state Markov process s(t) denotes the state of the actuator: s(t)=1 is the “failure” state and s(t)=2 is the “functional” state. λ1 denotes the transition rate from the breakdown to the functional state and λ2 backward transition state. The transition matrix Λ=[λij] has the form of (3). The stationary distribution for s(t)=u is calculated from (4):(27)μu=∑i=1≠uλi∑i=1,2λi,u∈S=1,2.u(t-τ) can be specified as(28)ut-τ=-f1q1t-τ-f2p1t-τ.According to the approximate expression as (9), (26) becomes (29)q˙1=p1,p˙1=-∑u=1,2ω~1u2μuq1-∑u=1,2β~11uμup1-β22p2+f11q1ξ1t+f12q22ξ2t,q˙2=p2,p˙2=-αq23-β21p1-β22p2+f21q1ξ1t+f22q22ξ2t,where (30)ω~1u2=ω12+f1cos⁡ω1τ+f2ω1sin⁡ω1τκu,β~11u=β11+-f1sin⁡ω1τω1+f2cos⁡ω1τκu.Note that the Hamiltonian system associated with (29) is completely integrable. The associated Hamiltonian is(31)Hτ=∑i=1,2Hiqi,pi;τ,H1q1,p1;τ=12p12+12∑u=1,2ω~1u2·μuq12,H2q2,p2;τ=12p22+14αq22.After the limit averaging and stochastic averaging, the averaged Itô equation for system (29) can be derived as the form of (13). Linearizing the averaged Itô equation yields (32)dHi=miLH1,H2;τdt+σ-ikLH1,H2;τdBkt,where(33)m1LH1,H2=-∑u=1,2β~11uμuH1+f112D1∑u=1,2ω~1u2μuH1+η1f122D2H2,m2LH1,H2=-η2β22H2+f212D1∑u=1,2ω~1u2μuH1+η1f222D2H2,σ-1kLσ-1kLH1,H2=f112D1∑u=1,2ω~1u2μuH12,σ-2kLσ-2kLH1,H2=2η3f222D2H22,σ-1kLσ-2kLH1,H2=0,where(34)η1=8/9α1/2∫011-t4-1/2dt,η2=2∫011-t41/2dt∫011-t4-1/2dt,η3=8/α∫01t41-t41/2dt∫011-t4-1/2dt.Note that the drift and diffusion coefficients in (33) are homogeneous in Hi of order one. Introducing the transformation as the form in (18a) and (18b), the Itô equations for ρ and α1 are obtained (35)dρ=Gα1;τdt+Ξα1;τdBt,dα1=G-α1;τdt+Ξ-α1;τdBt,where(36)Gα1=μ1+f212D1∑u=1,2ω~1u2μuα1+μ2+f122D2η11-α1+Aα11-α14,G-α1=2μ1-μ2+A1-2α12α11-α1+f122D2η11-α12-f212D1∑u=1,2ω~1u2μuα12,Ξ-2α1=Aα121-α12,μ1=-12∑u=1,2β~11uμu+f112D1∑u=1,2ω~1u2μu,μ2=-12β22η2+12η1-η3f222D2A=f112D1∑u=1,2ω~1u2μu+2η3f222D2.Following (24), the largest Lyapunov exponent is then obtained(37)Λmax=∫01Gα1;τpα1dα1.The stationary probability densityp(α1) can be obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with Itô equation for α1 as(38)pα1=Cα11-α1α11-α14μ1-μ2/A×exp⁡-2Aη1f122D2α1+f212D11-α1∑u=1,2ω~1u2μu.Some numerical results are obtained as shown in Figures1–4 for system parameters(39)β11=0.01;β12=0.0;β21=0.0;β22=0.01;f11=1.0;f12=1.0;f21=1.0;f22=1.0;ω1=3.0;α=0.2;f1=0.04;f2=0.04;D1=0.01;D2=0.01,κ1=0.0;κ2=1.0.Figure 1
The largest Lyapunov exponentΛmax as function of transition rate λ1. The lines are obtained from numerical solution of (37) while the symbols are obtained from direct simulation of original system (26).Figure 2
The largest Lyapunov exponentΛmax as function of transition rate λ2. The lines are obtained from numerical solution of (37) while the symbols are obtained from direct simulation of original system (26).Figure 3
The stability boundaries in the(β11,β22) plane for different transition rate λ1 when τ=0.3.Figure 4
The stability boundaries in the(β11,β22) plane for different transition rate λ1 when τ=0.6.Figure1 shows the largest Lyapunov exponent Λmax as function of transition rate λ1. It is seen that, for small delays, the largest Lyapunov exponent Λmax is a monotonous decreasing function of λ1. This is understandable: for small delay, the feedback control plays positive role in stability of system (26). As λ1 increases, the failure probability of the actuator decreases, and the stability of the system is enhanced as a result. However, for large delay, Λmax is a monotonous increasing function of λ1, as also shown in Figure 1. This implies that higher probability of actuator working in functional state deteriorates the performance of the dynamics. This is because that large delay feedback control brings negative damping to the system, as shown in (30), which has negative contribution to the stability of system. In Figure 2, the variation of Λmax with λ2 is displayed for different time delay. It is seen that as λ2 increases, Λmax increases for small delay while decreasing for large delay. The observation can be explained in a similar fashion as in Figure 1.In Figure3, the stability boundaries in the (β11,β22) plane for different transition rate λ1 in the small delay case are constructed by imposing Λmax=0. In small delay case, as λ1 is increased, the stability region of the system increases. Figure 4 shows the stability boundaries in (β11,β22) plane for different transition rate λ1 in the large delay case. Obviously, in large time delay case, large λ1 will reduce the stability region of the system.To verify the largest Lyapunov exponent calculated by using (37), the direct simulation of (26) is also obtained and shown in Figures 1 and 2 by using symbols ■. Observe that the symbols match closely with the solid lines, demonstrating the validity and accuracy of the proposed procedure.


## 7. Conclusions

In this paper, the asymptotic stability with probability of one of delay-controlled nonlinear systems with random actuator failures was investigated. Based on the random switch model of failure-prone actuator, the stability problem of such a system has been converted into the framework of Markov jump hybrid systems. Upon the limit principle and stochastic averaging, an approximate expression for the largest Lyapunov exponent of the original system was derived. Important results reported in the paper are summarized in the following statements:(1)
Actuator failure has significant effects on the asymptotic stability of the delay-controlled system. For small delay, high probability of actuator failure will degrade the stability while enhancing the stability for large delay.(2)
The validity and utility of the propose procedure have been demonstrated by using a delay-controlled nonlinear two-degree system with actuator failure. By comparison with direct system simulations, it has been observed that the procedure is fairly reliable in assessing asymptotic stability.



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