Tong LI, Zhenyu JIANG, Huo YANG, Cheng HU, Shifeng ZHANG,*

a Unmanned Systems Research Center, National Innovation Institute of Defense Technology, Beijing 100071, China

b College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China

KEYWORDS

Abstract Aircraft undergoing actuator failures into under-actuation have been seldom studied in literature. Aiming at addressing actuator failures of Total Loss of Effectiveness (TLOE) as well as Partial Loss of Effectiveness(PLOE)resulting in different system actuations,reconfigurable Fault-Tolerant Control (FTC) is proposed for supersonic wingless missiles under actuation redundancy.The under-actuated system of TLOE failure patterns is solved by transformation to cascade systems through a‘shape variable’.Meanwhile,actuator TLOE faults of different unknown failure patterns from proper actuation to under-actuation are accommodated by a reconfigurable adaptive law on a multiple-model basis.The backstepping technique with the Extended State Observer(ESO)method adopted as a basic strategy is applied to an established symmetric coupled missile system with actuator PLOE faults, modeling errors, and external disturbances. Additionally, the nonlinear saturation characteristics of actuators are settled by an auxiliary system with the Nussbaum function technique. The stability of the control system is analyzed and proven through Lyapunov theory.Numerical simulations are implemented in the presences of aerodynamic uncertainties,gust disturbance,and actuator failures.Results demonstrate the effectiveness of the proposed method with satisfactory tracking performance and actuator fault tolerance capacity.

1. Introduction

Supersonic missiles offer one of the most important national defenses in modern wars with high maneuverability and penetration capability. However, actuator failures resulting from large hinge moments on fin servos continue to present a substantial challenge. In general, actuator failures may cause severe performance deterioration of control systems, and sometimes lead to system instability or even catastrophic accidents.Partial Loss of Effectiveness(PLOE)and Total Loss of Effectiveness(TLOE)faults are the most common and typical actuator failures experienced by supersonic missiles.1,2Accommodation of such failures becomes an extremely vital and significant topic,especially for life-critical missile systems.The redundancy widely employed in the control system design of missiles becomes a double-edged sword when facing PLOE and TLOE faults. On one hand, the redundancy introduces additional uncertainty in actuation, resulting in exactly different behaviors of control systems.In particular,missiles suffering from actuator complete failures may transfer from overactuation to under-actuation, which will be a main topic discussed in this study.On the other hand,the existence of redundancy offers possibilities on a solution to PLOE and TLOE faults, especially for under-actuation patterns, which certainly improves the survivability of control systems.This approach is mainly focused on supersonic missiles with PLOE and TLOE faults of actuators under actuation redundancy.

Growing demand for safety and reliability in modern control systems has drawn a broad range of interest and research attention on Fault-Tolerant Control (FTC) for actuator failures2. The actuator failure accommodation control problem can be addressed in two categories:passive and active.Passive approaches3–9are mainly based on robust control theory with fixed conservative controllers. Nevertheless, with more uncertainty and complexity attached on actuator failures, passive methods always fail to satisfy the performance requirements for high robustness and reliability. Remarkable approaches have been conducted for active control strategies.Such efforts lead to great development of considerable reconfigurable FTC research on accommodation of actuator failures, including multiple-model-based designs,10–13eigenstructure assignment,14,15neural networks/fuzzy logic-based designs,16–18model predictive control,19,20sliding mode control,21–23adaptive control,1,12,13,24–31observer estimation-based designs,32–34etc.Compensation for actuator PLOE faults can be well addressed by the above approaches. Among these FTC methods, adaptive control and observer estimation-based control are promising approaches to achieve asymptotic tracking and stabilization as well as an effective reconfigurable scheme under TLOE faults.In the approaches of Refs.26–28,stabilizing nonlinear systems under unknown actuator failures is realized through backstepping techniques with adaptive failure pattern selectors. Incorporated with system structural information,adaptive control can achieve desired system performance through only one controller with a parameter adaption mechanism to accommodate actuator failures. On the other hand,observer estimation-based control is mostly studied by research in the Extended State Observer(ESO)and the Disturbance Observer (DOB). The ESO is a core functional part in Active Disturbance Rejection Control (ADRC) which was fully established by Han35,36in the 2000s, while the DOB was firstly proposed in the 1980s by Ohishi et al.to observe and compensate for the load torque.37In Ref.34,actuator failures of a Bank-to-Turn(BTT)missile were compensated for by an ESO designed with dynamic surface control.The uncertainties of actuator failures accompanied with modeling errors and external disturbances are regarded as a ‘total disturbance’ to be compensated for in the input channel in the ESO method.The conflict between performance specifications and robustness to actuator failures and disturbances can be independently resolved by separation of direct control and observer estimation.

However, the various kinds of FTC methods mentioned above are specifically studied for proper actuation cases,where the number of failure-free actuators is no less than the number of system outputs. Thus, despite that actuator PLOE and TLOE faults resulting in properly actuated systems can be effectively solved by adaptive control or observer estimationbased control, under-actuation of control systems resulting from actuator complete failures remains a tough task. Traditional under-actuated control methods based on energy strategy38are not suitable for complex systems of missiles. Olfati-Saber39showed a prominent solution for under-actuated nonlinear mechanic systems which could be transformed to cascade systems through a ‘shape variable’ derived from a Lagrangian inertia matrix. The application of a ‘shape variable’ to under-actuated systems has the following advantages:a ‘shape function’ can be simply obtained to mold the related‘shape variable’as an intermediate variable in the cascade system; the backstepping method can be directly applied to the cascade system. Consequently, the ‘shape variable’ strategy provides an effective solution for under-actuated systems with TLOE faults. Another issue lies on the saturation characteristic of actuators. Physical input saturation of actuators always exists in aircraft control systems,especially for under-actuated systems. Constraints on the magnitude of actuators may lead to undesirable inaccuracy or even instability, if the saturation problem is not properly handled. In Refs.34,40, the issue was addressed by a smooth hyperbolic tangent function to approximate the saturation where an auxiliary system based on Nussbaum function technique was applied to deal with the derivative. Results effectively verified the feasibility of this technique.

In light of this, reconfigurable FTC for a supersonic wingless missile under actuation redundancy with actuator PLOE and TLOE faults is proposed in this study. The underactuated system of TLOE failure patterns is solved by transformation to cascade systems based on a ‘shape variable’. Meanwhile, actuator TLOE faults of different unknown failure patterns from proper actuation to under-actuation are accommodated by a reconfigurable adaptive law on a multiple-model basis. The backstepping technique is applied to the nonlinear symmetric coupled system with the ESO method adopted to compensate for actuator PLOE faults, modeling errors, and external disturbances. Furthermore, the nonlinear saturation characteristics of actuators are settled by an auxiliary system with the Nussbaum function technique. Aerodynamic uncertainties,gust disturbance,and different PLOE and TLOE failure patterns are considered in numerical simulation to validate the effectiveness of the proposed method. Main contributions of this study are concluded as follows: (A) proposing a solution for different system actuation resulting from actuator TLOE faults of unknown failure patterns based on reconfigurable FTC;(B)establishing symmetric cascade coupling models under different TLOE failure patterns for wingless aircraft;(C)resolving the problems of accommodation of actuator failures, disturbance rejection, and input saturation in a synchronous manner.

The remainder of this paper is organized as follows. In Section 2,the symmetric coupling model of the missile is established, and actuator PLOE and TLOE failure patterns are introduced. In Section 3, a reconfigurable FTC design is derived in detail with stability analysis.In Section 4,numerical simulations are performed and analyzed to demonstrate the effectiveness of the proposed method. Finally in Section 5,conclusions are drawn.

2. Problem formulation

In this section, the model of a supersonic wingless missile is established. The failure patterns of actuators will be introduced after.

2.1. Modeling of missile dynamics

To begin with, the following assumptions should be made before the establishment of a missile dynamic model.

Assumption 1.Theparametersofcenter-of-massdynamicsare treatedasconstants(frozencoefficients),includingmass(m),speed(V),accelerationofgravity(g),Machnumber(Ma),andairdensity(ρ).Thelong-termdynamicsandgravitycanbe neglectedinthemodeling.

Assumption 2.Attitudeparametersincludingflightpathangle(θ),headingangle(σ),angleofattack(α),andsideslipangle(β)canbetakenassmallvalueswhichsatisfy:sinA≈A,cosA≈1, sinAsinB≈0,whereAandBbelongtotheEuler anglesmentionedabove.

Consider the missile traveling from Mach 3.1 at an altitude of 497 m when the first-stage solid engine ends boosting with separation and the second-stage solid ramjet starts working.The trajectory dynamics based on the assumptions above can be described by the following equations34:

whereP,Y,Z, and γ respectively denote the thrust, lift force,side force, and roll angle. The attitude dynamic equations of the missile are given by41

where ωx1, ωy1, and ωz1are the body axis roll, yaw, and pitch angular velocities, respectively.Jx1,Jy1, andJz1are the roll,yaw, and pitch moments of inertia, respectively.Mx1,My1,andMz1are the moments in roll, yaw, and pitch channels of the body axis, respectively.

Remark 1.Assumption 1andAssumption 2aremadefor convenienceofcontrolsystemdesign,andmeanwhileare reasonableinengineeringpractice.Theapproximationsunder theassumptionswillresultinacceptableun-modeleddynamics,whichintendstobecompensatedforthroughanobserver estimationschemeincontrolsystemdesign.

The actuators of the missile are 4 tail fins with deflection servos, which are set as an ‘X’ configuration shown in Fig.1.The figure is at a view from the tail to the head of the missile,where the arrows represent the force directions when deflections are positive. The nonlinear saturation characteristics of the fin deflection are considered as the following form:

where δk, δuk, and δmkrepresent the fin deflection, control input, and maximum deflection, respectively.

The aerodynamic data is based on numerical calculation and wind tunnel experiments. Aiming at facilitating problem formulation and control system design, high-dimensional and nonlinear aerodynamics are expressed in the traditional linear and decoupled forms in Eq. (4). Modeling errors resulting from approximations will be treated as disturbances injected to the missile dynamics.

whereSmandlkare respectively the characteristic area and length. ω-x1=ωx1lk/V, ω-y1=ωy1lk/V, and ω-z1=ωz1lk/Vare respectively the roll, yaw, and pitch non-dimensional angular velocities.q=ρV2/2 is the dynamic pressure. The symbols with superscripts are partial derivatives of aerodynamic coefficients.

Since the missile has a symmetrical configuration without wings, the relationship between aerodynamic coefficients and moments of inertia can be obtained as follows:

Substituting Eqs. (4) and (5) into Eqs. (1) and (2), the trajectory dynamics and attitude dynamics of the missile can be restated by the following equations:

Fig.1 Configuration of tail fins.

Remark 2.Themissiledynamicmodelestablishedabove becomesacascadesystem,whichissimilartothemodelof BTTmissiles.Eachofthethreechannelshasastrictfeedback form,whichshowsanintuitiveideafortheapplicationof backsteppingmethods.Furthermore,thesymmetricformsofthe pitchandyawchannelsofferoperationalsolutionstoFTC systemsundersevereractuatorfailures.

2.2. Actuator failure patterns

In this study, PLOE and TLOE faults of the fin servos are mainly focused. Since the efficiency loss faults of fin servos in aerospace engineering are always unrepeatable and unrecoverable, the failure number is finite during a flight. The system input under PLOE and TLOE faults can be derived as12,13

where δekis the expected control deflection,δfkis a failure constant, and σkdenotes the actuator failure pattern indicator as

Approaches on the condition that 0 ＜σk＜1 (PLOE faults) have been well conducted, where actuator failures can be regarded as compensable disturbances and uncertainties.However,problems when complete failures of excess actuators occur as σk=0 have rarely been discussed. A system may transfer from an over-actuated system to an under-actuated system under TLOE faults of actuators,where the methodologies adopted in control system design will be quite different.

Except extreme or uncontrollable cases (only one or none of the actuators works), TLOE failure patterns in the form{σ1, σ2, σ3, σ4} are given by

(1) None of the actuators has a complete failure (overactuated):{1, 1, 1, 1}-r1;

(2) One actuator totally fails (fully-actuated):{1, 1, 1, 0}-r2,{1, 1, 0, 1}-r3, {1, 0, 1, 1}-r4, {0, 1, 1, 1}-r5;

(3) Two actuators totally fail (under-actuated):{1, 1, 0,0}-r6, {1, 0, 1, 0}-r7, {1, 0, 0, 1}-r8, {0, 1, 1, 0}-r9, {0,1, 0, 1}-r10, {0, 0, 1, 1}-r11,

where rj（j=1， 2， ...， 11）denote numbered complete failure patterns.

The situations in cases(1)and(2)with proper actuation can be addressed through regular control methodologies. Nevertheless, the system under case (3) will be under-actuated with one control freedom lost, which may lead to unstable zerodynamics through traditional feedback linearization. To this end, reconfigurable under-actuated control becomes necessary in control system design.

In this study,the objectives of the control system design can be expressed as follows:

(1) to provide satisfactory tracking performance with the given trajectory guidance.

(2) to maintain the performance under different TLOE failure patterns.

(3) to achieve robustness to disturbances and uncertainties including modeling errors, input saturation and PLOE faults of actuators, external disturbances, etc.

3. Fault-tolerant and reconfigurable control system design and analysis

This section will introduce reconfigurable FTC with the ESO method based on the control objectives.

3.1. Proper actuation control strategy

The missile system is properly actuated under actuator TLOE failure patternsr1-r5.The actuators have redundancy on the control of three channels when the system is over- or fullyactuated. Consequently, the standard backstepping method can be applied to these normal cases.

where

It can be verified that the relative degrees of the outputs corresponding to the three channels are equal to system orders.Consider a standard diffeomorphism for the system states with Lie derivatives ξrj=Tc（x）(j=1， 2， ... ， 5), where

The system dynamics are given by

where

and Grjis the set of serial numbers of the actuators without TLOE faults under patternrj.

Given the tracking command

the standard backstepping procedure is defined as

The ESO method is applied to each of the three channels to compensate for modeling errors and external disturbances,which is described by

Define the input actuation matrix as

which is nonsingular under certain failure patterns of properly actuated cases. The inputs of fin deflections are distributed by

whereA+represents the generalized inverse of matrixA.

3.2. Under-actuation control strategy

The missile system is under-actuated with two inputs under actuator TLOE failure patternsr6-r11. Direct application of backstepping will result in zero dynamics. Thus, modeling of the system has to be reconsidered to match the case. Firstly,‘shape variables’39should be introduced before the modeling.

Definition 1.Letqj∈GwhereGisacyclicgroup.Decompose theconfigurationmanifoldasQ=Qx×G.TheactionofGonQ isamappingΦ:Q×G→Q.Forafixj∈｛1， 2，... ，n｝,let p∈Gandq∈Q,anddefineΦ（q，p）=（q1，q2， ... ，qj+p， ... ，qn）.Formally,qjiscalledanexternalvariable,if theLagrangeinertiamatrixofsystemswithkineticsymmetry satisfiesM（q）=M（Φ（q，p））.LetQx=G1×G2×...×Gn representtheconfigurationmanifoldofallexternalvariables whereGiisaone-dimensionalcyclicgroup.ThenQ=Qx×Qs,whereQsiscalledtheshapespace,i.e.,theconfiguration manifoldoftheshapevariables.

According to Definition 1, ‘shape variables’ are the only elements that appear in the inertia matrix of a mechanical system. It has been proven39that there exists a global diffeomorphism obtained from the Lagrangian of a system that transforms the dynamics of an under-actuated system with kinetic symmetry into a reduced-order system with a new configuration vector and a well-defined Lagrangian function parameterized by ‘shape variables’ which satisfy the unforced Euler-Lagrange equations. In other words, ‘shape variables’are configuration variables that can eliminate the coupling effect for under-actuated systems with kinetic symmetry via a global change of coordinates that decouples the dynamics of‘external variables’ and ‘shape variables’. High-order underactuated systems can be reduced and normalized through generalized momentums conjugate to‘external or shape variables’together with their integrals.42To simplify the understanding of‘shape variables’,control system design under actuator failure patternr6is shown to explain its application to modeling.

Fin deflections δ1and δ2are available without TLOE faults under patternr6. Let

The original nonlinear system is described by

where

1 as follows:

The system dynamics are then given by

where

The application of the backstepping method to tracking ofqHcandqlcis defined similarly as

The ESO method is adopted to compensate for disturbances and uncertainties in both channels, which is described by

Thus, the decoupled control inputs can be obtained from

The control law under failure patternris eventually given by

From the above, κj(j=1， 2， ... ， 11) is not continuous and has the following characteristics:

Direct adaptive control is applied to the estimation of the failure pattern selector based on the above characteristics with the following adaptive laws:

whereA+denotes the generalized inverse of matrixA, ^κ is the estimation of κ=[κ1， κ2， ... ， κ11]T, Λ=diag｛Λ1， Λ2， ... ，Λ11｝ is the adaptive gain matrix of estimation, and

is the generalized input matrix where

It is noteworthy that the last row vector of the matrix in the adaptive law in Eq. (41) is based on the third condition in Eq. (41). This is aimed at the avoidance of a singular matrix problem to realize pseudo-inverse of the matrix, which is also functionally similar to the projection operator.

Considering the saturation in Eq.(40),a direct application of the backstepping method to the system is not available.Therefore,a bounded saturation function is adopted as follows:

Define the fin deflection input error as

Remark 5.Thereconfigurationschemeisbasedonthefailure patternselectortoswitchcontrolstrategiesunderdifferent failurepatterns,whichissimilartomultiple-modelmethodology.TheadaptivelawisaimedatcancelingtherelatedtermsinthedeflectionsissolvedbyanauxiliarysystemwiththeNussbaum functiontechnique,makingdirectapplicationofbackstepping controlstrategiesavailable.Similarly,commandfilterscanalso beappliedtodifferentialcalculation.

3.4. Stability analysis

The stability analysis is based on the Lyapunov function.Consider the Lyapunov function candidate as

The following assumption and lemma36,40should be proposed before the analysis.

Lemma 1.χ（·）andN（χ）respectivelydenoteasmoothfunction definedon（ti，tj）andaNussbaumgainfunction.Thefollowing functionVε（·）isboundedon（ti，tj）:

whereK＞0 and ν ＞0 are constants, and ε is a positive variable.

The time derivative of Eq. (47) is derived as

Lyapunovfunctiondiscussedinnextsectiontorealizethe stabilityoftheestimation.Inaddition,thesaturationofthefinwhererpdenotes the actual failure pattern experienced by the missile actuators, and

Take the direct integral of Eq. (53) as

In accordance with Lemma 1,the Lyapunov functionVTis bounded from Eq.(54).Meanwhile,the bound can be reduced by increasingKthrough related parameters of control system design.However,as a function of time,VTwill jump to several bounded values with the variation of κ when an actuator failure pattern changes, which leadsVTto be piecewise continuous. SinceVTis differentiable and decreases in time intervals between jumping points,VTis bounded and convergent under a finite number of actuator failures. Therefore, the system is globally stable based on the Lyapunov stability analysis above.

4. Numerical simulation

In this section, 6-DOF numerical simulation is performed on the nonlinear missile system to validate the capability of the designed control system. The missile intends to lower the height at first to realize sea skimming at a fixed altitude of 10 m and then conduct an ‘S’ shape of lateral maneuver with a maximum distance of 500 m. The missile trajectory starts at the activation of the second-stage solid ramjet with initial states of velocityV0=1065.936 m/s, altitudeH0=496.944 m, and flight path angle θ0=3.490°. According to the designed trajectory,the shutdown time of the first-stage solid engine is chosen as the characteristic point for control system design.Table 1 presents the nominal coefficients of missile dynamics and specifications of the control system. The aerodynamic data is based on 7-D lookup tables of angle of attack, sideslip angle, Mach number, and 4 fin deflections.

From the table,the bandwidths of the tracking closed-loop dynamics in pitch,yaw,and roll channels are designed as 2,2,and 5 Hz, respectively. It is highlighted that the roll dynamics should be faster than those in pitch and yaw channels for thesake of stability.In addition,the bandwidths of ESOs in accordance with the tracking dynamics are designed as 15,15,and 5 times those of the closed-loop ones as 30, 30, and 25 Hz,respectively.Note that the closed-loop dynamics should be sufficiently fast over the trajectory guidance to promise tracking performance under actuator failures.

Table 1 Nominal coefficients of missile dynamics and control system specifications.

The simulation is carried out on MATLAB/SIMULINK software using the Runge-Kutta integral method with a fixed stepsize of 0.001 s. Four simulation scenarios are conducted for illustration.

Scenario 1.The nominal nonlinear missile dynamics is considered without external disturbances,aerodynamic uncertainties, or actuator faults.

Simulation results of Scenario 1 are shown in Figs. 2–6 whereHandlzrespectively represent the height and lateral distance. The guidance process is presented in Fig.2. A favored tracking performance is illustrated in Fig.3 with acceptable small delays. Uncertainties including modeling errors and un-modeled dynamics have been accurately estimated and compensated for by the ESOs. The variations of the attitudes in Fig.4 have reached the conditions of related assumptions and boundary constraints.The feasibility of the method under nonlinear dynamics is verified in Scenario 1.

Scenario 2.Aerodynamic uncertainties are included with values of +20% in forces and -20% in moments. PLOE faults of fin #1-fin #4 with σk=0.8 (k=1， 2， 3， 4) occur at 9 s when the missile is pulling up with the largest load required.

Fig.2 Trajectory of missile with reference in Scenario 1.

Fig.3 Results of tracking performance in Scenario 1.

Fig.4 Curves of related attitudes in Scenario 1.

Fig.5 Curves of angular velocities in Scenario 1.

Fig.6 Curves of fin deflections in Scenario 1.

Fig.7 Trajectory of missile with reference in Scenario 2.

Simulation results of Scenario 2 are shown in Figs.7–11.It can be seen in Fig.7 that the guidance error is reduced due to the positive uncertainty in aerodynamic forces. The control system has achieved a robust and satisfactory tracking performance under PLOE faults of actuators as shown in Fig.8. In addition,the transient process is quite smooth,since the influence caused by actuator failures is compensated for through the designed ESOs. In Fig.11, the fin deflections have a quick and small change corresponding to the PLOE faults imposed on actuators.It is worth mentioning that the disturbance rejection of ESOs has a fast response against actuator PLOE faults,assisting to achieve sufficient tracking accuracy.

Scenario 3.–20% uncertainty in aerodynamic forces and+20% uncertainty in aerodynamic moments are included.TLOE faults of fins #3 and #4 (patternr6) occurring at 9 s are taken into consideration. Additionally, a horizontal gust disturbance, modeled as a sinusoidal signal of a magnitude of 15 m/s and a frequency of 0.05 Hz at a direction of +45°from thexaxis of inertial coordinates, is introduced as an external disturbance.

Fig.8 Results of tracking performance in Scenario 2.

Fig.9 Curves of related attitudes in Scenario 2.

Fig.10 Curves of angular velocities in Scenario 2.

Fig.11 Curves of fin deflections in Scenario 2.

Fig.12 Trajectory of missile with reference in Scenario 3.

Simulation results of Scenario 3 are shown in Figs. 12–16.Considering the trajectory guidance on a basis of time sequence, the guidance process has a large delay in Fig.12 under the negative uncertainty of aerodynamic forces. However, the control system still has a high accuracy of tracking performance, except vibrations during the reconfiguration at 9 s leading to the fluctuations of the trajectory guidance,which are shown in Fig.13. Unlike PLOE faults of actuators, the TLOE faults lead to under-actuation of the system, which has a significant impact on the tracking performance. It is worth pointing out that the roll angle in Fig.14 and the roll angular velocity in Fig.15 have swift responses against the TLOE faults of two actuators. Furthermore, the missile is maneuvering in a BTT-like manner.The roll angle finally converges to approximately 90°, which may result from the compensation for the gravity in the longitudinal plane for level flight during sea skimming. The solution of input saturation keeps the control law function normally with bounded responses of fin deflections shown in Fig.16. Simultaneously,the system is robust to the gust disturbance rejected by ESOs through fin deflections of sine wave shapes.

Fig.13 Results of tracking performance in Scenario 3.

Fig.14 Curves of related attitudes in Scenario 3.

Fig.15 Curves of angular velocities in Scenario 3.

Scenario 4.-20%uncertainties in both aerodynamic forces and aerodynamic moments along with the gust disturbance are involved.A TLOE fault of fin#4(patternr2)and a PLOE fault of fin#3 with σ3=0.8 occur at 9 s.Additionally,a TLOE fault of fin #2 and a PLOE fault of fin #1 with σ1=0.8 following the first failure(patternr7)occur at 12 s when the missile starts lateral maneuvering.

Fig.16 Curves of fin deflections in Scenario 3.

Fig.17 Trajectory of missile with reference in Scenario 4.

Fig.18 Results of tracking performance in Scenario 4.

Fig.19 Curves of related attitudes in Scenario 4.

Fig.20 Curves of angular velocities in Scenario 4.

Fig.21 Curves of fin deflections in Scenario 4.

Simulation results of Scenario 4 are shown in Fig.17–21.Similar to Scenario 3,there are conspicuous errors in the guidance process due to the loss of aerodynamic forces. The control system shows an excellent actuator fault tolerance capacity in this severe case. It can be seen from Fig.21 that the reconfiguration at 9 s for a fully-actuated system under the first TLOE fault is fast and smooth. In fact, the actuation redundancy design is originally aimed at overcoming the TLOE fault of one actuator which can be compensated for by the integral of errors distributed to the rest of actuators.The tracking accuracy in Fig.18 has nearly no change compared to that of Scenario 1,especially when the TLOE fault of one actuator occurs. Nevertheless, the TLOE faults of two actuators at 12 s still have a strong effect on the system responses, since the system transforms from fully-actuation to under-actuation.It should be highlighted that all the signals in Figs. 19 and 20 are bounded and stable under the actuator failures.Furthermore,the system presents strong robustness to the aerodynamic uncertainties, gust disturbance, actuator PLOE faults,and input saturation considered in the case.Similar to Scenario 3,the missile adopts a BTT strategy to maneuver after the complete failures of two actuators. Meanwhile,the roll angle finally converges to approximately 45° on account of the distribution and configuration of actuators shown in Fig.1.

The effectiveness and robustness of the designed control system have been validated by the simulation results.The proposed method is available to solve problems of an underactuated system resulting from TLOE faults of actuators. In addition, the designed control system shows prominent prospects in practical applications to dealing with distinct failure patterns of actuators, and provides an encouraging solution to FTC problems on a basis of multiple-model adaptive reconfiguration.

5. Conclusions

In this paper, a promoted solution of actuator PLOE and TLOE faults for supersonic missiles under actuation redundancy is studied and discussed. Considering the distinctive actuation conditions of a system under different failure patterns, a detailed description of reconfigurable FTC is introduced. The global stability of the system is proven with prudent deduction and analysis in Lyapunov theory. Numerical simulation results show a superb actuator fault tolerance capacity as well as a desired tracking performance of the developed control system in the presences of uncertain actuator failures, input saturation, disturbances, and uncertainties.Accompanied with fault tolerance, the robustness to external disturbances and un-modeled dynamics indicates prominent prospects for industrial applications, especially for highmaneuver and performance-critical aircraft. Future research intends to investigate an extension of the method to more general classes of nonlinear systems without symmetric structures.The relaxing condition of the requirement on full states availability can be viewed as a promising path for further study.

Acknowledgement

This work was supported by National Defense Science and Technology Commission of China.