GUO Hang ,WANG Zheng ,FU Bin ,CHEN Kang ,FU Wenxing ,and YAN Jie

1.Research Center for Unmanned System Strategy Development,Northwestern Polytechnical University,Xi’an 710072,China;2.Unmanned System Research Institute,Northwestern Polytechnical University,Xi’an 710072,China

Abstract: An impact angle constrained fuzzy adaptive fault tolerant integrated guidance and control method for Ski-to-Turn(STT) missiles subject to unsteady aerodynamics and multiple disturbances is proposed.Unsteady aerodynamics appears when flight vehicles are in a transonic state or confronted with unstable airflow.Meanwhile,actuator failures and multisource model uncertainties are introduced.However,the boundaries of these multisource uncertainties are assumed unknown.The target is assumed to execute high maneuver movement which is unknown to the missile.Furthermore,impact angle constraint puts forward higher requirements for the interception accuracy of the integrated guidance and control (IGC) method.The impact angle constraint and the precise interception are established as the object of the IGC method.Then,the boundaries of the lumped disturbances are estimated,and several fuzzy logic systems are introduced to compensate the unknown nonlinearities and uncertainties.Next,a series of adaptive laws are developed so that the undesirable effects arising from unsteady aerodynamics,actuator failures and unknown uncertainties could be suppressed.Consequently,an impact angle constrained fuzzy adaptive fault tolerant IGC method with three loops is constructed and a perfect hit-to-kill interception with specified impact angle can be implemented.Eventually,the numerical simulations are conducted to verify the effectiveness and superiority of the proposed method.

Keywords: integrated guidance and control (IGC),impact angle constraint,unsteady aerodynamics,fault tolerant control (FTC),actuator failures.

1.Introduction

The guidance and control system have a significant influence on the flight performance of the vehicles.Conventionally,guidance and control loops are designed separately and combined by the guidance commands on normal acceleration or flight attitude [1,2].It has been verified from engineering applications that separated guidance and control systems can be applied to extensive scenarios.However,the transition delay between guidance and control systems usually leads to large miss distance and cannot fully meet some specific expectations,such as the terminal impact angle constraint.As a result,the integrated guidance and control (IGC) method is developed as a mainstream scheme which considers the coupling relation between the two subsystems directly and is able to improve the interception accuracy on the basis of reducing design cost [3-6].

The IGC method was originally proposed by Williams et al.in 1983 [7],and as the IGC dynamics took many engineering factors into account,such as actuator failures,model uncertainties,numerous modern control approaches have been adopted to enhance the capacity of the original IGC,e.g.,linear optimal control theory [8,9],sliding mode control [10-18],model predictive control [19,20],back-stepping control [21-23],feedback linearization[24,25],and the state-dependent Riccatti equation(SDRE) [26].

With regard to the IGC method with impact angle constraint,a great deal of research has been conducted in the past decades.In [3],in the presence of input saturation and actuator failures,a three-dimensional integrated guidance and control law with impact angle constraint was developed,using the dynamic surface control and extended state observer (ESO) techniques.A partial IGC method by the sliding mode control technique was developed and terminal impact angle constraints in both azimuth and elevation could be satisfied with high interception accuracy [6].By combining the sliding mode control method with the nonlinear extended disturbance observer technique,an IGC law was proposed to satisfy the impact angle constraint and the enhancement on the rapidity and performance of the guidance and control system was confirmed by numerical simulations in [13].The linear quadratic regular (LQR) method was utilized to construct an IGC law with impact angle constraint on the basis of linearized flight dynamics,and the better interception performance of the IGC law than separated guidance and control (SGC) was verified in [27].An impact angle constrained IGC scheme was put forward with the non-singular terminal sliding mode control (NTSMC)technique for missile to intercept a maneuverable target in [28],and the ESO was established to cope with the model parameter uncertainties and the target maneuvering.In [29],a three-dimensional IGC scheme was presented for a Ski-to-Turn (STT) interceptor,considering impact angle and actuator saturation constraints,and strong robustness of the proposed IGC was guaranteed by the combination of sliding mode control and finite-time super-twisting ESO (STESO).

However,the above-mentioned research did not take the model nonlinearities and uncertainties as well as actuator failures into consideration sufficiently and simultaneously.That is,some dominating factors were simplified,and the consequent model errors were regarded as disturbances.To deal with actuator and sensor failures occurring in engineering applications pervasively,fault tolerant control (FTC) has attracted many research interests[30-34].For the fuzzy adaptive FTC problem,a tremendous amount of research focused on the active FTC control methods,which adaptively possessed reconfigurable structures and adjustable control parameters [35-48].In detail,in [40],an adaptive fuzzy FTC approach had been utilized for digital communication networks by the sliding mode control method.In addition,fuzzy adaptive FTC schemes had been widely studied and enhanced by adaptively adjusting control schemes to compensate model uncertainties and failures in an efficient way[41-44].A novel fault tolerant IGC structure was constructed for a class of STT missiles subjected to rapidly changing actuator failures and coupled multisource uncertainties in [49] and satisfactory hit-to-kill interception performance could be obtained.To this date,there are plenty of research results on the active FTC method and FTC method begins to combine with IGC scheme to tackle with multisource uncertainties,actuator failures,and model nonlinearities simultaneously.

As far as the authors could be concerned,unsteady aerodynamics has not been considered sufficiently and concretely in the common IGC problems,which in most cases is regarded as unknown disturbances and tackled with common robust techniques.Nevertheless,unsteady aerodynamics exerts a significant impact on flight control system design of multiple vehicles and control surface effectiveness tends to be reduced [50-56].In the transonic flight regime,linear and nonlinear aeroelastic analyses of rolling maneuvers are performed by transonic small disturbance theory and the effects arising from unsteady aerodynamics are presented based on the predictions of aerodynamics coefficients [50].In [51],the longitudinal and lateral/directional stability and control characteristics of F/A-18E at transonic state were analyzed and it turned out that the flow on the F/A-18E at transonic speeds was massively separated and unsteady.Similar results about the characteristics of the unsteady aerodynamics were drawn for X-31 [52] and F-16XL[53].A precise mathematical model for the unsteady aerodynamics of transonic cruise aircraft was formulated from the pitch oscillations point of view,and the prediction of dynamic behavior were in perfect agreement with the experimental data even in high angle of attacks [54].Recently,the combined experimental and computational study of the transonic features was adopted to obtain deeper insights on the phenomenon of unsteady flows and shock oscillations and related suggestions on the design of vehicle structure to suppress the adverse effect of unsteady aerodynamics passively were proposed in[55,56].In summary,unsteady aerodynamics has become a challenge during the IGC design process of transonic aircraft.Unfortunately,there is a lack of research on the fuzzy adaptive fault tolerant IGC method for certain vehicles with unsteady aerodynamics.Therefore,we are motivated to devote to this frontier research.

In this paper,an impact angle constrained fuzzy adaptive IGC method is proposed for the STT missiles subject to unsteady aerodynamics and multiple disturbances.The main challenge originates from implementing hit-tokill interception with certain impact angle in the face of complicated coupling relations among model uncertainties,actuator failures and unsteady aerodynamics.The above-mentioned multiple disturbances bring about great difficulties without any doubt to the fuzzy adaptive IGC structure design.Based on the accurate description on the dominating disturbances in the IGC problem,the planar mathematical model of IGC is formulated as a nonlinear strict-feedback system which involves complex and multiple uncertainties.The target is assumed to execute high maneuver movement during interception process,which is unknown to the missile.Besides,the unsteady aerodynamics is modeled according to [54].As a result,several reciprocal adaptive laws are derived to tackle with the multiple disturbance concurrently and to achieve a hit-tokill interception with specific impact angle.In contrast with the existing literature,the main contributions of the proposed method in this paper can be concluded in the following aspects:

(i) A novel IGC scheme for the STT missiles with unsteady aerodynamics and multiple disturbances is proposed,which can achieve the hit-to-kill interception with expected impact angle.

(ii) Unsteady aerodynamics is introduced from the point of view of engineering applications and is precisely described.Together with model uncertainties and actuator failures,all dominating factors are dealt with simultaneously in the sense of robustness and adaptation.

(iii) The intrinsic coupling effects arising from impact angle constraint,unsteady aerodynamics as well as target maneuver are compensated by a series of reciprocal adaptive laws and fuzzy logic systems.Besides,the boundaries of the target’s maneuverability,nonlinear uncertainties as well as actuator failures,are not required in advance.

The structure of this paper is organized as follows: The mathematical model of the fuzzy adaptive fault tolerant IGC problem as well as relevant preliminaries are stated in Section 2.The adaptive fuzzy fault tolerant IGC approach is developed in detail in Section 3.Numerical simulations are conducted,and the performance of the proposed method is evaluated in Section 4,followed by the eventual conclusions in Section 5.

2.IGC problem formulation and preliminaries

2.1 Mathematical model of IGC problem

The relative kinematics between an STT missile and a target in pitch plane is illustrated in Fig.1,whereMandTdenote the missile and the target respectively.Rrepresents the relative distance andλrepresents the line-ofsight (LOS) from the missile to the target.VM,θMandaMare the missile’s velocity,flight path angle,and normal acceleration.VT,θT,andaTare the target’s velocity,flight path angle,and normal acceleration.In addition,the parametersR,aM,VM,aT,VTare introduced for mathematical calculations,which represent the magnitude of the vectorsR,aM,VM,aT,VTrespectively.

Fig.1 Relative kinematics between an STT missile and a target in pitch plane

From Fig.1,the equations of relative kinematics between the missile and the target [27] can be obtained as follows:

Differentiating the second equation in (1) yields

The longitudinal kinematics of both the missile and the target [26] can be expressed by the following equations:

where (xM,yM) are the missile’s planar coordinates inxandydirections respectively,(xT,yT) are the target’s planar coordinates inxandydirections respectively,Dandmdenote the missile’s drag force and mass respectively,andgrepresents the gravity acceleration.The normal accelerationaMof the missile can be calculated by

whereLis the missile’s lift force.According to the flight dynamics of the missile,taking the unsteady aerodynamics into consideration [53],the lift and drag force can be given by

In the terms of engineering applications,the actuator failures commonly occur during flight missions,which induces undesirable performance degradation of the guidance and control systems.FTC is of vital importance to construct a robust and adaptive flight system,which is an important part of our research.In this paper,the actuator failures are introduced as follows:

where χ(t) represents the scaling factor of the actuators gain failures,u(t) represents the control input of the actuator,anddδ(t) represents the deviation faults of the actuator.Here,we assume thatdδ(t) is bounded but unknown for the missile and χ(t) takes value in the interval (0,1],which implies that the actuator does not encounter total loss of effectiveness during interception process.

2.2 IGC design objective

Denote λ*as the expected constant angle of LOS.Because the hit-to-kill interception can be achieved when LOS angle rate λ˙ can be sustained as zero [16],the design objective of fault tolerant IGC problem is summarized as

Lete=λ-λ*and then the design objective in (9) can be rewritten as

Next,we introduce a new variable σ and it satisfies

wherek＞0 is a constant.According to the sliding mode control theory,the design objective (10) can be achieved if

in finite time and the convergence rate depends on the parameterk.

Consequently,the design objective of the fault tolerant IGC problem is to construct an effective fault tolerant IGC scheme to guarantee that σ can converge to 0 in finite time in the face of unsteady aerodynamics and multiple disturbances concurrently.

Remark 1Here,multiple disturbances arise from actuator failures,the uncertainties of model nonlinearities and the measurement noises,which are extensively encountered in engineering applications.In addition,to implement the precise interception with impact angle constraint,both the aerodynamics of the missile and high maneuver of the target have significant effect and further bring about great difficulties to developing the fault tolerant IGC method.

2.3 Fuzzy logic systems and lemmas

With regard to the unknown model nonlinearities in the IGC problem,we resort to fuzzy logic systems (FLSs) to compensate them.Taking vectorx=[x1,x2,···,xn]Tas the input,the logic rules of an FLS [45] can be expressed as follows:

The bound of the approximation error and the universal approximation property are analyzed in Lemma 1.

Lemma 1[47] Given any continuously differentiable functionf(x1,x2,···,xn) defined onU=[a1,b1]×[a2,b2]×···×[an,bn],there exists an FLSy(x1,x2,···,xn)established by (13) with bounded derivative such thatf(x)-y(x)|＜ε,where ε is an arbitrarily small positive constant.

Furthermore,the following lemma is presented which will be used in the subsequent IGC design process.

Lemma 2[46] For any ε ＞0 ands∈R,we have

where κ is a constant and κ=0.278 5.

3.Fuzzy adaptive fault tolerant IGC with impact angle constraint

3.1 Structure of fuzzy adaptive fault tolerant IGC scheme

In this subsection,an IGC scheme is constructed with a three-loop integrated configuration as depicted in Fig.2.By designing the outer guidance loop,the virtual normal acceleration commandis achieved to ensure that the LOS angle λ can converge to the expected LOS angleλ*while the LOS angle rate λ˙ can converge to 0 at the same time.By means of middle maneuver loop,the virtual pitch angular rate commandis obtained to maintain that the actual normal accelerationis able to track the commandin a rapid and precise way under the condition that the target’s unknown maneuver is compensated and the unwanted effects from unsteady aerodynamics and model nonlinearities are suppressed.In addition,in the inner attitude loop,the control inputu(t) of the elevator is derived to track the pitch angular rate commandin the face of actuator failures,unsteady aerodynamics and model uncertainties from nonlinear characters.

Fig.2 Structure of the proposed three-loop fuzzy adaptive fault tolerant IGC scheme

3.2 Strict feedback state-space expression of IGC scheme

Combining (2) and (11) yields

Differentiating (4),the time derivative ofaMcan be given by

where Mach number Ma can be calculated asMa=VM/VstdandVstdis the standard velocity of sound.

Defining the state vectorx=[σ,aM,ωz]T,and combining (6),(15),and (16),the three-loop IGC model can be established as the following strict feedback state-space expression:

where Δfiand Δgi(i=1,2,3) are the multiple uncertainties arising from model nonlinearities and measurement errors.In addition,

Remark 2The dominating factor of the external disturbance termd1(t) is the unknown information of the maneuvering target,which can be tackled with adaptively in the outer guidance loop.In addition,the disturbance termd2(t) mainly results from unsteady aerodynamics and model nonlinearities,which can be coped with adaptively in the middle maneuver loop.In addition,the disturbance termd3(t) arises from the actuator failures and unsteady aerodynamics,which can be handled robustly and adaptively in the inner attitude loop.According to the practical experience,without any loss of generality,we assume that Δgi∈[-0.5,0.5](i=1,2,3).

3.3 Fuzzy adaptive fault tolerant IGC method design

With regard to the three-order strict feedback system by(17),a three-loop IGC scheme is constructed by three steps.

Step 1For the outer guidance loop,to begin with,a hyperbolic tangent function is utilized to deal with the rapidly changing disturbance termd1(t).Then,oriented to the IGC design objective,we introduce an integral type variables0=σ(τ)dτ to steer the variable σ(t) to 0 in a more effective way.With regard to the unknown nonlinear function Δf1,we formulate a fuzzy logic system [45]to compensate it,given by

As a result,the eventual control of the guidance loop can be derived,given by

where τy2is a positive constant.Then we introduce two new variables,expressed as

Then the dynamics ofs0,s1,andy2can be formulated as

For stability analyses of the proposed IGC method,consider the following Lyapunov function candidate:

Moreover,according to the Young’s inequality,the following inequalities can be derived:

For the stability analyses of the proposed IGC method,consider the following Lyapunov function candidate:

3.4 Stability analysis for the proposed IGC method

With regard to the fault tolerant IGC model (17),the transition control signals among three loops given by (21),(40),and (57),and the adaptive update laws given by(32),(48),and (62) are able to ensure that all the variables are uniformly bounded and the IGC objectiveσ converges to 0 asymptotically.

ProofThe Lyapunov function is chosen as

On the basis of the previous analysis,the time derivative ofLsatisfies

where

Then it is pretty clear that0 ≤L(t)≤max{L(0),εˉ/ω}for allt≥0.Eventually,all the variables are bounded during the whole interception.According to the Barbalat’s lemma,it can be further guaranteed that

Therefore,the objective of the proposed IGC method can be achieved and the proof is completed. □

Remark 3The apparent advantage of the proposed IGC method is that the unknown target’s maneuver,the unknown negative effects caused by the unsteady aerodynamics,as well as the unknown boundaries of the model nonlinearities and actuator failures are tackled with simultaneously.The highly accurate and adaptive estimations on the unknown boundaries of multiple disturbances ensure that the expected IGC performance with impact angle constraint can be accomplished.

4.Numerical simulations and analysis

In this section,numerical simulations are conducted to verify the effectiveness and superiority of the proposed IGC method.With the consideration of the target’s unknown maneuver,the unsteady aerodynamics and the multisource disturbance including model uncertainties and actuator faults,the availability of the proposed IGC method is verified firstly.Then,different cases on the IGC design objective,the target’s maneuver,the unsteady aerodynamics as well as the actuator failures are introduced to check the robustness and the adaptability of the proposed IGC method.Last but not least,regarding a class of IGC methods without the estimation of the unsteady aerodynamics (IGC-WEUA),and conventional SGC (CSGC) design method,the superiority of the proposed IGC method is confirmed.

The initial conditions of the numerical simulations are listed in Table 1.Besides,the aerodynamics coefficients of the missile are listed in Table 2.In addition,the structural parameters of the missile are given by Table 3.

Table 1 Initial conditions of the missile and the target

Table 2 Aerodynamics coefficients of the missile

Table 3 Structural parameters of the missile

4.1 Effectiveness and robustness verification

To verify the effectiveness and robustness of the proposed IGC method,three different cases on the IGC objective,unsteady aerodynamics,and actuator faults are introduced in the following numerical simulations,given by Table 4.The control gains and the initial values of the parameters in the proposed IGC method are selected,given by Table 5.The fuzzy membership functions are selected according to [47].

Table 4 Three cases in numerical simulations

Table 5 Control gains and initial values of the parameters in the proposed IGC method

The condition for the numerical simulation to terminate can be shown as

Fig.3-Fig.8 exhibit the numerical simulations of the IGC problem in three cases.Specifically,the trajectories of the missile and the target of the IGC problem in three cases are presented in Fig.3.The flight states and elevator control inputs of the missile in three cases are shown in Fig.4.It is shown that the normal accelerationaMvariates between -10 m/s2and 40 m/s2,which is pretty acceptable for the precise interception against the highly maneuverable target.In Fig.5,the relative kinematics between the missile and the target,including the LOS λ,the rate of LOS,the relative distanceRand the approaching velocity,are illustrated.It is extremely obvious that the expected terminal LOS λ*can be guaranteed while desirable interception performance is achieved in all three cases in the simultaneous presence of unknown unsteady aerodynamics and multiple disturbances.As a result,the effectiveness and the robustness of the proposed IGC method can be verified.Meanwhile,the estimations of the adaptive parameters of the outer guidance loop are recorded in Fig.6.Then the estimations of the adaptive parameters of the middle maneuver loop are recorded in Fig.7.In addition,the estimations of the adaptive parameters of the inner attitude loop are recorded in Fig.8.Apparently,the estimations of the adaptive parameters of the proposed IGC method are converged or bounded during the whole engagement.In other words,the undesirable effects arising from the unknown unsteady aerodynamics and multiple disturbances are compensated adaptively.Thus,the proposed IGC method can be implemented in physical applications.

Fig.3 Trajectories of the missile and the target in three cases

Fig.4 Flight states and elevator control inputs of the missile in three cases

Fig.5 Relative kinematics between the missile and the target in three cases

Fig.6 Estimations of the adaptive parameters of the outer guidance loop in three cases

Fig.7 Estimations of the adaptive parameters of the middle maneuver loop in three cases

4.2 Superiority verification

To confirm the superiority of the proposed IGC method,numerical simulations are conducted,which introduce the IGC-WEUA method and the CSGC method at the same time.The three methods are applied in the same Case 1 and the simulation results are given by Fig.9 and Fig.10.

Fig.9 Trajectories of the missile and the target by three methods

Fig.10 Relative kinematics and elevator control input of the missile by three methods

The trajectories of the missile and the target by three methods are exhibited in Fig.9.The relative kinematics,including LOS λ and its rate λ˙,as well as the elevator control input δzare demonstrated in Fig.10.In Fig.9,the missile is capable of accurately engaging the highly maneuvering target by three methods.However,according to Fig.10,for the IGC-WEUA method and the CSGC method,the expected terminal LOS λ*cannot be achieved eventually,and the rate of LOS λ˙ are diverging rapidly during the terminal interception phase.Additionally,the elevator control input δzby the CSGC method is diverging at last,while that by the proposed IGC method and the IGC-WEUA method maintains within a reasonable extent during the whole interception process.Therefore,the superiority of the proposed IGC method can be validated consequently.With regard to the normal accelerationaM,it fluctuates apparently in the beginning and diverges at the engaging point by the IGC-WEUA method,while by the CSGC method,it oscillates during the whole process and diverges earliest among three methods.As for the proposed IGC method,the normal accelerationaMvariates slightly during the entire interception process and desirable engagement can be obtained.

5.Conclusions

In this paper,a fuzzy adaptive fault tolerant IGC method for a category of STT missiles maneuverable targets is proposed,and the expected terminal impact angle is introduced to the planar engagement scenario.Simultaneously,the undesirable effects induced by unsteady aerodynamics and multiple disturbances are considered in the form of the concrete mathematical model.To be specific,the actuator failures and multisource model uncertainties are treated as multiple disturbances.Several fuzzy logic systems are established to compensate the model nonlinearities and unknown uncertainties.With regard to the undesirable effects mentioned above,a fuzzy adaptive fault tolerant IGC scheme is proposed to deal with the impact angle constrained IGC problem.Firstly,a series of adaptive control gains and compensation terms are introduced,and then the undesirable effects and the unknown target’s maneuver can be coped with.Moreover,the reciprocal adaptive laws used to estimate the unknown control gains are established,which makes the proposed IGC method applicable.In addition,numerical simulations have been conducted to verify the effectiveness and the advantages of the proposed IGC method,which can be effectively applied to various engagement scenarios.