1.Introduction

Dissimilar Redundant Actuation Systems(DRAS),composed of Hydraulic Actuator(HA)and Electro-Hydrostatic Actuator(EHA),have been increasingly gaining consideration for the application in the large commercial aircraft due to their ability to avoid possible common mode failures.1–3DRAS has the advantage of fast response and high reliability associated with HA actuator,while avoiding common mode failure by introducing redundant EHA.4The research on DRAS system has become the mainstream of the large aircraft development programs.The latest Airbus aircraft,A380 and A350,have adopted DRAS system of 2H/2E type.5

The HA systems commonly experience gradual faults caused by oil leakage and constriction of oil flow which ultimately lead to the system performance degradation.Although these faults may have limited initial side-effect,over the long life-time they can result in increasingly adverse effects on the actuation system and impact the overall system performance.Therefore,measures must be taken to maintain certain level of acceptable system performance under the gradually increasing system fault conditions.The Fault-Tolerant Control(FTC)techniques6–8are commonly used to remedy the faults and maintain the aircraft system safety at a high level,and could also be applied to designing the FTC mechanism for the actuation system with gradual faults.

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Design of FTC system requires that the system parameters be known,which is difficult to realize in real application.9Researchers have proposed various control methods for uncertain systems operating under very specific working conditions and for short period of time,specifically,for the cases when gradual faults do not have significant effect on the system performance.However,there are few control methods for systems with varying and uncertain parameters over long period of time,which prompted many researchers to focus on the system identification techniques10–13commonly applied to systems under normal operating conditions and over a specific time period.However,when system faults occur,the corresponding system parameters would drift,which would increase the identification errors of the parameter values.When the system suffers multiple faults of different severity levels,the system parameters would present uncertainty.Therefore,traditional approaches addressing system identification are inadequate for effective control design for the systems with increasing errors of parameters.This is especially evident in the case of a system with changing parameters caused by the gradual faults of the imprecise system model.The robust control techniques,14,15adaptive design techniques16,17and the Linear Parameter Varying(LPV)control design methods18can deal with the system uncertainty to some degree,but they exhibit limitations when applied to systems with multiple gradual faults which occur over a long working period.

Design of an effective FTC mechanism for DRAS experiencing gradual faults,is a complex problem which involves not only system identification over a long period of time but also design of the fault-tolerant controller.This paper proposes a Fault Mode Probability Factor(FMPF)based FTC strategy for multiple fault modes of DRAS over a long working period.Since the identification errors would increase due to the multiple gradual faults,the cycle of sampling method is used to record the multiple fault modes and provide the historical statistics as the fault information.Applying the proposed FMPF with expectation operator,the historical statistics can be used to estimate current degree and values of multiple faults.Finally,the amended system model based on FMPF can be obtained and used to design the fault-tolerant controller.Based on the amended system model,the control gain can be determined using Linear Quadratic Regulator(LQR).19–21To balance the system model precision and the computation time,a Moving Window(MW)method is used to determine the amount of applied historical statistics,following a certain window size,at the end of the most recent sampling period,new system parameter estimation results are imported into the applied historical statistics while the oldest set is removed,and then the chosen historical statistics are processed by using the FMPF to update the system parameters.The updated system parameters are then used to modify the control law to compensate for the effects of the changed gradual faults.In this paper,the Monte Carlo method is used to provide simulation data for different periods.Several case studies of DRAS,subjected to multiple faults,are performed to analyze the effectiveness of this method and associated design approach.

The main contribution of this paper is the FMPF based FTC strategy for multiple fault modes of DRAS under long term working conditions,where the control gain of the faulttolerant controller is updated with each sampled data set.This approach allows the system performance to be maintained within reasonable range even under changed working conditions caused by the gradual faults.Compared with the existing FTC design methods,the proposed FMPF approach applies the expectation operator on the historical statistics resulting in a novel way to comprehensively utilize the system statistical information,where law of large numbers and central limit theorem can be used to provide theoretical foundations,and meanwhile,the MW method balances the system model precision and the computation time.

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Notation:Throughout this paper,the superscript T speci fies matrix transposition,Δ speci fies parameter error.The symbol P stands for probability and E expectation operator.Re{λ(·)}is the form of eigenvalue real part.w(t)∈ L2[0，∞)is a quadratic differential function.max(·)and min(·)express maximum value and minimum value respectively.

2.System description

The DRAS system is composed of one HA system and one EHA system as shown in Fig.1.In the normal operating condition,only HA drives the control surface while EHA is in the follower mode,i.e.backup mode.This type of active/passive(A/P)operating mode,known as HA/EPmode,is the most common operating mode since HA system has better performance than EHA system.Consequently,the proposed FTC strategy is developed for this operating mode and leaves the EA/HPoperating mode as an alternate FTC strategy.

2.1.Modeling of HA/EPmode under normal operating condition

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This work was co-supported by the National Natural Science Foundation of China(Nos.51620105010,51675019 and 51575019),the National Basic Research Program of China(No.2014CB046402),the Fundamental Research Funds for the Central Universities of China(YWF-17-BJ-Y-105)and the"111"Project of China.

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Fig.1 Schematic of dual DRAS system.

where Bh,Beand Bdare damping coefficients of hydraulic cylinders of HA and EHA systems,and the control surface;mh,meand mdare masses of hydraulic cylinder pistons of HA and EHA systems and the control surface;Ahis hydraulic cylinder area of HA;Vhis the hydraulic cylinder total volume of HA;Ehis bulk modulus;Kce=(Kc+Chl),Kcis coefficient of flow-pressure,and Chlis leakage coefficient of hydraulic cylinder of HA;Kqis coefficient of flow change;Kvis proportionality coefficient of servo valve;τvis the servo valve time constant.

2.2.Modeling of HA/EPmode with multiple gradual faults

According to the published research,24,25the gradual faults of HA,which cause the parameters drift,can be classi fied in the following five types:(A)the change of servo valve time constant τvdue to the servo valve blockage;(B)the change of servo valve gain Kvdue to the servo valve leakage;(C)the change of bulk modulus Ehdue to the air entrapment in oil;(D)the change of damping coefficient Bhdue to the increasing motion damping;(E)the change of leakage coefficient Chldue to the hydraulic cylinder leakage.These types of gradual faults may be difficult to detect but would result in degradation of the system performance.The uncertainty modulation matrices are used to model these faults in DRAS system and to describe the parameter uncertainty caused by these faults.Assuming that the system does not experience sensor faults,the DRAS model of the HA/EPmode with gradual faults can be represented in the following form:

where ΔA and ΔB are deviations in the original state and input matrices caused by parameter change due to the system faults.The system matrices and the fault modulation matrices can be expressed as follows:

Proof.See Appendix A.

3.Amended model based on fault mode probability factor

Since gradual faults have stochastic characteristics,they can be described using probabilistic approach.As the gradual faults converge to a particular state,that state can be regarded as the expectation of the final gradual fault changed state and could be used to describe the gradual fault process and design appropriate FTC strategy.

The effects of all the gradual faults have been modeled as uncertainty matrices ΔA and ΔB which can be regarded as random variables(ΔA(t)，ΔB(t))due to the gradual change of DRAS faults.The matrices(ΔATi，ΔBTi)are updated after every sampling period T followed by the parameter estimation procedure,where Tiis the ith sampling period.According to Law of Large Numbers and the gradual change of DRAS faults,the matrix class sequence of gradual fault parameters will converge to the expected value(ΔAE，ΔBE),

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Since the matrix class sequence(ΔATi，ΔBTi)can be obtained from periodically sampled historical statistics,and since they may vary only with the sampling period,therefore,if there are N sets of the sequence (ΔAT1，ΔBT1) →(ΔATN，ΔBTN),they can be regarded as independent with the same distribution.Since the matrix class sequence is historical statistics of gradual faults at different periods,the sequence can have limited non-zero matrix variance σ2.Therefore,according to the central limit theorem,the error with the expectation(ΔAE，ΔBE)can be described as

If there is sufficient amount of historical statistics,namely,for sufficiently large sampling number N,the error distribution can be described as

where α is the confidence coefficient.This approach provides a way to modify the system model under gradual fault conditions,which is affected by the number of the historical sampling sets.

3.1.Moving window based determination principle for historical statistics

Eq.(6)shows that as historical statistics set number N becomes larger,the probability of the error between the matrix class sequence and their expectation will be close to the normal distribution,namely,increasing historical statistics will result in increased accuracy of expectation.However,two factors need to be considered:(A)although a large number of applied historical statistics within reasonable time range can be used to estimate a more accurate set of system parameters,they also increase the computation time,and(B)very old historical statistics can have negative effect when used to estimate the current system parameters.To balance the system model precision with required computation time,a MW method is introduced to reduce the negative effect of the old historical statistics.

The principle of the moving window is shown in Fig.2.Considering both of the previously mentioned factors and assuming that the optimal MW size is SMW=(N-1)T,the MW is changing from MW(1)to MW(N-1)(marked in green)before the MW reaches its desired length.During this period,increasing amount of historical statistics is used,and once the MW size achieves its optimal length(N-1)T,it will continue to maintain that length(marked in red).In the process,the old historical statistics will be removed from the applied set while the new statistics will be applied.Using the MW with its size of(N-1)T,there are always N set of historical statistics used to estimate the system model parameters.

Remark 1.In Eq.(7),the con fidence coefficient α is defined to measure the estimation accuracy of the system model parameters.Given a defined acceptable maximum value αmax,the determined optimal MW size SMWshould guarantee that α ≤ αmaxholds.Considering the computation time by de fining the computation time cost as t(cost),the SMWshould guarantee that t(cost)≤tmax.Therefore,MW size can be regarded as a multivariate function SMW=F({α|α ≤ αmax}，{t(cost)|t(cost)≤tmax}),which can be used to determine the optimal MW sizeby an optimization process.

3.2.Fault mode probability factor

In order to effectively utilize the historical statistics of the fault modes,the FMPF is proposed in this paper.The multiple faults occur at different levels and in different combinations,which can cause residual statistical information for the multiple faults.Choosing reasonable sampling period over the long working period,the gradual fault parameter change can be saved as historical statistics.The multiple faults can be represented by the parameter-dependent matrices of the state space form system as(ΔATi，ΔBTi).In all historical statistics,different matrix classes may be the same,namely,in different sampling periods:(ΔATi，ΔBTi)=(ΔATj，ΔBTj).In order to appropriately apply the historical statistics to design the FTC law,the FMPF method is first defined as follows:

Definition 1.FMPF.The probability of the multiple faults represents a combination of different faults at different degrees of fault severity,which is captured through statistical information.If there are N cycles of historical statistics,and some of the historical statistics are the same,the same historical statistics are captured in the same group and recorded with the number ncj,where cjis the jth category of the historical statistics.De fine the fault mode probability factor as

where ncjcan be used to describe the probability of the jth category of multiple faults(ΔATi，ΔBTi)cj.In addition,if there are total of k multiple categories of the historical statistics,all the FMPFs can be calculated as{Pc1，Pc2，···，Pck}.From all the calculated FMPFs,let Pmax=max{Pc1，Pc2，···，Pck}be the maximum probability factor,where in the case of Pcj=Pmax,the jth category multiple faults(ΔATi，ΔBTi)cjis the most common fault condition during the entire measured statistical time interval.

Fig.2 Moving window based applied historical statistics determination mechanism.

The proposed approach to deal with the system uncertainty is amending the uncertain system in state space form using FMPF.As a result,the uncertain matrices(ΔA，ΔB),which describe the multiple faults in the system,can be determined using FMPF and the corresponding historical statistics.Since all gradual faults converge to a specific state,the expectation operator,Eq.(5),can be applied to processing the statistics and amending the system model uncertainty as(ΔAE，ΔBE)=limt→∞E(ΔA，ΔB).Specifically,the expectation operator based on FMPF can be used in the following form:

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3.3.Amended model based on fault mode probability factor

Since the current system with multiple faults in uncertain form can be amended using the historical statistics,the amended model based on FMPF is given in the following section.

where k is the number of the multiple fault mode categories.The proposed FMPF of the historical statistics and the expectation operator can be used to amend DRAS system with multiple faults as

In this amended model,the system uncertain matrices(ΔA，ΔB)can be replaced by the determinant expectation of matricesbased on FMPF.Finally,the amended system model can be described as

The purpose of the FMPF based approach is to counteract the adverse effect of parameter uncertainty since it can reduce the parameter estimation errors by utilizing the historical statistics and expectation operator,and then to facilitate the FTC strategy design based on the amended system model.The theoretical foundation of the FMPF based approach is illustrated by the following lemma.

Lemma 1.The FMPF based amended model is more accurate than the uncertain model under the condition of the multiple gradual faults.Namely,using the same feedback matrix,the absolute values of the error between the real parts of eigenvalues for the FMPF based amended model and the actual one,and the error between the real parts of eigenvalues for the uncertain model and the actual one,hold up Eq.(A11).

where coefficients A and B matrices are defined in Eq.(2),whereas Δa22=-ΔBh/(mh+me+md)is damping fault factor, Δa32=-4ΔEhAh/Vhis the bulk modulus fault factor,Δa33=-[4ΔEh(Kc+ΔChl)]/Vhrepresents the bulk modulus and the internal leakage fault factor,Δa34=4ΔEhKq/Vhrepresents the bulk modulus fault factor,Δa44=-1/Δτvis the servo valve time constant fault factor,and Δb41= ΔKv/Δτvrepresents the servo valve block and leakage failure factor.

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Based on the amended model in Eq.(11),the controller of the FTC strategy is designed in the following section.

4.Design of FMPF based FTC strategy

The flowchart of FMPF based FTC approach is presented in Fig.3.In the initial stage of the actuation system,whether the gradual faults occur or not,a set of baseline control parameters should be determined.Since the gradual faults are being considered,all the relevant parameters of gradual faults should be preset and used to construct the uncertain matrices of the system model.Reasonable thresholds of parameters due to gradual faults within the FTC range are chosen.The gradual faults occur during a very long term,and in order to determine reasonable control parameters under gradual faults condition,a control parameter updating period is chosen.At every end of the period,the relevant parameters of gradual faults are estimated using system identification techniques.If all the parameters are within the corresponding thresholds,these parameters are then saved as a new set of the historical statistics.However,if some of the parameters deviate sharply from the corresponding thresholds,it can be conjectured that severe faults have occurred and that the current FTC law should be switched to another one.23In the case when the parameters move slightly outside the corresponding thresholds,the relevant control parameter matrices need to be appropriately adjusted.When the new set of gradual fault parameters is added to the historical statistics,the FMPF needs to be updated and the expectation parameter matrices of gradual faults need to be determined based on the new FMPF.Since the obtained system model,under gradual fault conditions,is determined by replacing the uncertain matrices with the expectation parameter matrices,the new FTC gain based on new determined system model can be solved and used to update the previous one.

Fig.3 Flowchart of FMPF based FTC mechanism.

The particular form of FTC for DRAS system with multiple gradual faults is introduced in the following sections.

It is assumed that the controller has fixed structure26,27such that:

(1)Under the multiple gradual faults condition,DRAS output can track the command order Souty(t)with respect to the reference signal r(t)without steady-state error,i.e.limt→∞e(t)=0,where e(t)=r(t)-Souty(t),and Sout=[1，0，0，0].

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(2)The optimized performance defined in Eq.(14)can be obtained through the designed controller.The corresponding augmented system can be described in the following form:

where is augmented state vector,is augmented output vector,andis augmented disturbance vector.The corresponding augmented matrices areThe controllability of this augmented system can be determined from its controllability verification matrix

In order to guarantee that the output can track the input signal,the controller is designed as per Ref.28by using the state and output error integration feedback as follows:

where K=[Ke，Kx]is the control gain matrix.The poles of this augmented system can be selected to have negative real parts,thus guaranteeing stability of the closed-loop system.

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The Linear-Quadratic(LQ)cost function is defined in the following form:

where Q and R are symmetric semi-positive-de finite and positive-de finite weighting matrices,respectively.The control law can be obtained by selecting the state weighting matrix

therefore, Error(M1)-min(Error(M2))=Re{λ(AE)-,where due to the expectation theory,at least the following conclusion can hold:

where P is a positive-definite matrix.The feedback control gain matrix K=[Ke，Kx]can be determined by using LQR,

Remark 2.The parameter γ represents the robust performance of the system and describes the quantitative relationship between the disturbance w and the system output y.The inequality ‖G(s)‖ ＜ γ means that the influence of the disturbance w on the system output y is limited in the gain level γ.

The controller with FTC function(Fig.4)is in the state and output error integration feedback form to realize the pole placement and guarantee desired performance in the presence of disturbance.The system uncertainty,due to multiple faults,is addressed by applying the matrix class sequence(ΔATi，ΔBTi)to obtain the matricesusing the proposed FMPF method.To minimize the cost function,Eq.(14),the weighting matrices can be chosen from LQR,and then the control gain matrix for the uncertain system can be solved.The condition for finding the fault to lerant controller gain is given by the following theorem.

Theorem 1.Considering the closed-loop augmented system given by Eq.(12),for a given scalar γ＞ 0 in Remark 1 and for all nonzero disturbancesw(t)∈ L2[0，∞),choosing Q and R as weighting matrices of LQ index,and de fining symmetric positive de finite matrix P,if the LQR can find the P matrix,then the closed-loop system given in Eq.(12)has the upper bounds of performance index given in the following form:

and the gain matrix for the fault-tolerant controller of the closed-loop system can be determined as

Proof.See Appendix B.

5.Simulation analysis

5.1.Settings of fault scenarios

The system parameters used to verify the effectiveness of the proposed approach are the same as those used in Ref.22Typical multiple gradual faults of HA system are presented in Section 2.2.In order to demonstrate the advantage of the proposed FMPF based FTC approach,the Monte Carlo simulation method is used to simulate the historical statistics.Assuming that the MW size is SMW=30 and that it can satisfy the computation time cost requirement,the historical statistics are then processed using FMPF method and the expectation operator to obtain five different amended model results,which are designated as FMPF-1 to FMPF-5 and are shown in Table 1.

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Fig.4 Controller structure for system with multiple faults in uncertainty form.

Table 1 Historical statistics of multiple gradual fault modes.

5.2.Simulation results with FTC strategies

The first case(Fig.5)represents system response of an actuation system experiencing gradual faults to a step input with the displacement of 0.03 m.Two methods,based on the FTC approaches,are used to compare the system response.The first is FTC approach presented in Ref.29,which models the effects of gradual faults as system uncertain modes and applies robust control method to obtain the control parameters.The second is the FMPF based FTC approach.The method first estimates the unknown parameters using FMPF method on the historical statistics of the system experiencing gradual faults,and then determines the control parameters based on the estimated system.The simulation results indicate that system with FMPF based FTC approach has improved tracking performance(Fig.5(a)).The results(Fig.5(b))also indicate that using FMPF based FTC approach results in smaller tracking error.

Fig.5 Tracking response using original FTC and proposed FMPF based FTC when given square-wave order.

Fig.6 Tracking response using original FTC and proposed FMPF based FTC when given superimposed sine-wave order.

The same FTC approaches as the ones used in Fig.5 are used to compare the control effects when the sinusoidal input signal is applied to the actuation system experiencing gradual faults.The simulation results are shown in Fig.6,which indicate that the proposed controller has good tracking performance.Specifically,Fig.6(a)indicates that using FMPF based FTC approach results in close tracking of the input signal,where the amplitudes of the system response are very close to the ones of the given command,and the response time delay is also smaller than using the FTC approach without FMPF.Also,Fig.6(b)indicates that FMPF based FTC approach results in the system having smaller tracking error under the given command.

The effects of the historical statistics are analyzed by managing different amount of historical statistic using the proposed FMPF approach.Using the proposed FMPF method,five different models are designated as FMPF-1–5,where the model based on FMPF-1 is obtained based on the largest amount of historical statistic,whereas the FMPF-5 induced model is obtained using the least amount of historical statistic.

The simulation results,for the case when a step signal with displacement of 0.03[m]is given as the command input to the actuation system experiencing gradual faults,are shown in Fig.6.Using the same proposed FMPF based FTC approach,five models,based on different amount of historical statistic,are used to compare the control effects on system response.Fig.7(a)shows that system response has the settling rise time under the FMPF-1 based FTC approach,whereas FMPF-5 based FTC approach results in the slowest settling time.The results presented in Fig.7(b)indicate that FMPF-1 based FTC approach results in the lowest tracking error for the given command,whereas FMPF-5 based FTC approach results in the highest tracking error.Since FMPF-1 manages the largest amount of historical statistic and FMPF-5 manages the least amount of historical statistic,it illustrates that,within the coverage of MW size,more historical statistic used results in better control performance.

Fig.7 Tracking responses of when given square-wave order using different FMPF based FTC.

Fig.8 Tracking responses when given superimposed sine-wave order using different FMPF based FTC.

The simulation results presented in Fig.8 are for the case when a sinusoidal signal is the command input to the actuation system experiencing gradual faults.The same FTC approach,i.e. five models based on different amount of historical statistic,is used to compare the control effects.The results presented in Fig.8(a)indicate that FMPF-1 based FTC approach results in the best tracking performance,whereas FMPF-5 based FTC approach results in the worst tracking performance.The simulation results presented in Fig.8(b)show that FMPF-1 based FTC approach results in the minimum tracking error,while FMPF-5 based FTC approach results in the maximum tracking error.It is apparent that,within the coverage of MW size,using more historical statistic results in better control performance.

6.Conclusions

A Fault Mode Probability Factor(FMPF)-based Fault-Tolerant Control(FTC)strategy for multiple gradual faults of Dissimilar Redundant Actuation System(DRAS)has been studied and evaluated.The typical gradual faults of DRAS system are modeled into the state space system in uncertainty form.The historical statistics of the fault modes are used to modify and correct the uncertain model using the proposed FMPF.A controller with fixed structure in state and output error integration feedback form is designed based on the modified model using FMPF.New updating method for the initial control gain under gradual fault condition is proposed.The simulation results indicate that the proposed FMPF based FTC control strategy is effective for the long term working condition of DRAS system experiencing gradual faults.The proposed strategy indicates that system response improves by increasing the amount of historical statistics.

In order to develop the strategy for a wider range of applications,the future work will address optimization of the MW size as well as applicability of the proposed FMPF based FTC strategy on the nonlinear systems.

Acknowledgements

where the state vector is defined as x(t)=[xh，˙xh，Ph，xv]T,xhand˙xhare the velocity and acceleration of the piston respectively,Phis the cylinder pressure,and xvis the servo valve displacement;u(t)is the system input to be designed;y(t)is the system output;w(t)is unknown disturbance.The state,input,output,and disturbance matrices are as follows23:

Appendix A.Proof of Lemma 1

Proof.The eigenvalue form of the uncertain system model,to transform the pair matrix(A+ΔA，B+ΔB)with Kx,can be obtained as

Let ρA∈ [-1，+1]and ρB∈ [-1，+1],express the change of parameters due to gradual faults,and then the following inequality can be obtained:

From Eq.(A2),it can be deduced that

The real parts of eigenvalues for the FMPF based amended model can be expressed as

The real parts of eigenvalues for the actual model can be expressed as

Then two errors can be obtained:One is the error between the real parts of eigenvalues for the FMPF based amended model and the actual one,whereas the other is the error between the real parts of eigenvalues for the uncertain model and the actual one.In order to prove the advantage of FMPF based FTC approach,it needs to be shown that the absolute value of the first error is less than the second one.

式中，i为仪器站编号，i=1、2、3。j为目标点编号，j=1、2、3，计算时，特别需要注意，当j-1=0时，用j=3替换；当j+1=4时，用j =1替换。k为1个仪器站上的观测序号，k=1、2、3、4。公式（5）～公式（17）中，i、j、k取值相同。

The error between the real parts of eigenvalues for the FMPF based amended model and the actual one can be determined as follows:

The model of HA/EPsystem,used in this work,has been previously developed and published by this research group.22The model is based on the assumption that the control surface is a rigid body of known mass and inertial moments.The forces acting on the system include the HA cylinder force,Fh,inertial and damping load of the EHA system,and aerodynamic force,FL.The state space representation of the system is given as follows:

De fine the absolute value of this error as

The error between the real parts of eigenvalues for the uncertain model and the actual one can be calculated as

为了验证本文方法的可行性，选取肤色，发色各异的人脸图像2000张，经过数据预处理后，放入到T-GAN模型中进行测试翻译。部分实验结果如图3所示，可以看出，本文算法在输入的各种肤色和发色手绘人脸图像上，根据人脸的特殊性，引入相应的文本描述，翻译结果能够达到一个较好的效果，接近真实图像，与文本描述内容也比较接近。

De fine the absolute value of this error as

Since the minimum of the error is the following form:

Q and input weighting matrix R,so as to minimize the total control cost function J.The optimal solution to minimize the total control cost can be obtained by solving the following Riccati equation:

Eq. (A11) deduces the conclusion that Error(M1)＜min(Error(M2)),so that

This proves Lemma 1.

Appendix B.Proof of Theorem 1

Proof.Since the control law form u(t)=Kxaug(t)=is used to stabilize the augmented system given in Eq.(12),using the method similar with the one used in Ref.26,this control law form is substituted into the LQ cost function form

This proves Theorem 1.

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