Adaptive Failure Compensation for Aircraft Tracking Control Using Engine Differential Based Model

Adaptive Failure Compensation for Aircraft Tracking Control Using

Engine Differential Based Model

Yu Liu , Xidong Tang and Gang Tao Suresh M. Joshi Department of Electrical and Computer Engineering Mail Stop 308 University of Virginia NASA Langley Research Center Charlottesville, VA 22903 Hampton, VA 23681 Abstract — An aircraft model that incorporates independently patterns. An algorithm based on multiple model adaptive re- adjustable engine throttles and ailerons is employed to develop configuration control approach was presented and illustrated an adaptive control scheme in the presence of actuator failures.

by simulation results of the F/A-18 aircraft during carrier This model captures the key features of aircraft flight dynamics landing. In [12], an F-16 fighter aircraft subject to asymmet- when in the engine differential mode. Based on this model an ric actuator failure was discussed, including system modeling adaptive feedback control scheme for asymptotic state tracking is developed and applied to a transport aircraft model in the and control system design. The problem was formulated as presence of two types of failures during operation, rudder a nonlinear disturbance rejection problem in the presence failure and aileron failure. Simulation results are presented to of actuator failures and simulation results using an F-16 demonstrate the adaptive failure compensation scheme.

aircraft model were discussed. In [6], fault-tolerant control system design against stuck actuators was investigated using Keywords : Actuator failures, adaptive compensation, aircraft an iterative learning observer that provides information of the flight control, engine differentials, tracking.

system state estimates and fault compensation transients. The performance of the controller design was evaluated using an I. I NTRODUCTION F-8 aircraft model.

Effective compensation of control component failures is In this paper, we present a failure compensation scheme crucial for aircraft flight safety. Considerable research has based on an adaptive control approach that can utilize the focused on the design of control systems that can provide remaining (functioning) controls to achieve desired perfor- safe performance when failures occur. In [5], an emergency mance in the presence of uncertain system failures. To com- flight control system that can utilize engine thrusts to ma- pensate for aircraft failures such as rudder failure or engine neuver an aircraft was developed and tested on an MD- malfunction, asymmetric engine thrusts may be inevitably 11 airplane. In [7], a propulsion controlled aircraft design needed [10]. For the design of such control schemes, an by H-infinity model matching was introduced. In [1], an aircraft model with independently adjustable engine thrusts indirect adaptive LQ controller was developed for aircraft is necessary. In [8], we derived such an aircraft model and control, which is able to implicitly reconfigure the control used it to develop an adaptive failure compensation control law using on-line estimates of the changed aircraft dynamics, scheme using engine differentials for state regulation . In this so that the failures in the pitch control channel or the paper, we shall use this aircraft model to develop an adaptive horizontal stabilizer can be accommodated. In [2], several failure compensation control scheme for state tracking , and multivariable adaptive control algorithms for flight control apply the scheme to a transport aircraft model. We shall reconfiguration were presented with a failure characterized consider two simulation cases representing realistic scenarios by a locked left horizontal tail surface. An adaptive controller in which the rudder or an aileron are stuck at unknown was used to compensate this failure. In [13], a direct adaptive constant values at unknown time instants.

reconfigurable flight control algorithm was presented. An on- The paper is organized as follows. In Section 2, we line adaptive neural network was applied to regulate the error describe an engine differential based aircraft flight dynamic between the plant model and the actual aircraft, and appli- model. In Section 3, we develop an adaptive compensation cation of this control approach to a tailless advanced fighter scheme that is able to handle uncertain actuator failures and aircraft was demonstrated. In [9], an algorithm for aircraft guarantee asymptotic state tracking. In Section 4, we apply failure detection and compensation was presented, which this compensation scheme to a transport aircraft model and incorporated multiple model adaptive estimation methods.

present simulation results to illustrate the effectiveness of the In this approach failures are detected by a bank of parallel scheme.

Kalman filters and a reconfiguration algorithm is used to redistribute control commands to the non-failed surfaces.

II. E NGINE D IFFERENTIAL B ASED M ODEL In [3], a new parametrization for the modeling of control effector failures in flight control applications was proposed, As described in [8], a nonlinear aircraft dynamic model including lock in place, hard over and loss of effectiveness in body-axis coordinate system which incorporates engine (2) (2) (1) (4) (1) differentials can be described by the force equations where A and B are zero matrices, A , A , B and (4) B are of the same forms as in the literature [4], and the m ( ˙ u + qw − rv ) = X − mg sin θ + ( T + T ) cos ǫ (II.1) L R matrices     m ( ˙ v + ru − pw ) = Y + mg cos θ sin φ (II.2) 0 0 0 0 0 0 0 ′′ ′′ ¯ ¯ ¯ ¯   0 T − T  T T 0 0  u w δ δ  t tr  m ( ˙ w + pv − qu ) = Z + mg cos θ cos φ − ( T + T ) sin ǫ (II.3) l L R   (3) (3) ′ ′ ¯ ¯  ′′′ ′′′  ¯ ¯ A =  T T 0 0  , B = 0 T − T u w  δ δ  t tr   l   and moment equations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I ˙ p + I ˙ r +( I − I ) qr + I qp = L + l ( T − T ) sin ǫ (II.4) x xz z y xz L R (II.13) 2 2 represent the effect of engine thrust differentials, that is, if I ˙ q + ( I − I ) pr + I ( r − p ) = M (II.5) y x z xz the left and right engine thrusts are equal, these matrices are I ˙ r + I ˙ p + ( I − I ) qp − I qr = N + l ( T − T ) cos ǫ z xz y x xz L R zero. See [8] for details of this model.

(II.6) We note that this aircraft model is different from standard where m is the mass of the aircraft. u , v and w are the body- models used in most of the literature [4] that assume equal axis components of the velocity of the center of mass. p , q engine thrusts and aileron angles. This engine differential and r are the body-axis components of the angular velocity based model in which the two engine thrusts and the ailerons of the aircraft. X , Y and Z are the body-axis aerodynamic are taken into account separately captures the essential forces about the center of mass. L , M and N are the body- dynamics of the aircraft in the engine differential mode, axis aerodynamic torques about the center of mass. θ and φ and is capable of coping with some actuator failures such as are the Euler pitch and roll angles of the aircraft. ǫ represents rudder failures or engine failure, which cannot be achieved the angle between thrust and body x -axis. I are the moments i without using engine differentials. Therefore it is desirable (or products) of inertia in body axes. g is the gravitational to develop an adaptive control scheme for aircraft actuator force per unit mass. T and T are the left and right engine L R failure compensation using engine differentials.

thrusts, and l is the distance between engines and x – z plane.

By applying the linearization procedure around the equi- III. A DAPTIVE F AILURE C OMPENSATION librium point of interest, we can obtain the linearized aircraft In this section, we shall first formulate an actuator failure model with engine differentials. For this purpose, the state compensation problem for linear systems, and then develop and control vectors of the linearized model are an adaptive failure compensation scheme for closed-loop T x = [ u w q θ v r p φ ψ ] (II.7) stability and asymptotic tracking of the system state variables T in the presence of certain actuator failures.

U = [ δ δ δ δ δ δ ] (II.8) e t t a a r l r l r where the notation “ δ ” has been dropped from δx and δU A. Problem Formulation for simplicity of presentation. Thus ( u , v , w ) represent the Consider the linear time-invariant system velocity perturbations along each axis and ( p , q , r ) are the angular velocity perturbations about each axis. ( θ , φ , ψ ) are n m ˙ x = Ax + Bu, x ∈ R , u ∈ R , (III.1) the pitch, roll and yaw angle perturbations, and δ , δ , δ , e a a l r T δ are the deflection perturbations of the elevator, the left whose actuators u = [ u , u , . . . , u ] may fail during r 1 2 m and right ailerons and the rudder. δ and δ are the left and system operation. A typical failure model is t t l r right throttle perturbations.

u ( t ) = ¯ u , t ≥ t , i ∈ { 1 , 2 , . . . , m } , (III.2) In our study, we consider a steady-state rectilinear wings- i i i level flight condition as the equilibrium point. For this where t is the unknown failure time instant and ¯ u is the i i steady-state flight condition, the derivatives of all states, the unknown failure constant [11]. An example of such actuator angular velocity components ( p , q , r ) and the roll angle φ at failures is when an aircraft control surface (such as the rudder the equilibrium point are all zero, that is, or an aileron) is stuck at some unknown fixed position at an ˙ ˙ ˙ [ ˙ u ˙ w ˙ q θ ˙ v ˙ r ˙ p φ ψ ] = 0 (II.9) unknown time instant.

x , U o o The control objective is to design an adaptive state feed- p = q = r = φ = ψ = v = 0 , (II.10) o o o o o o back control signal to be applied to the actuators in u , where x and U are determined as o o to ensure closed-loop signal boundedness, and asymptotic T tracking: lim ( x ( t ) − x ( t )) = 0 , where x ( t ) is a desired t →∞ d d x = [ u w 0 θ 0 0 0 0 0 ] , o o o o state trajectory, in the presence of unknown actuator failures.

T U = [ δ δ δ δ δ δ ] . (II.11) o eo t t alo aro ro lo ro B. Adaptive Compensator Designs By applying the linearization around this equilibrium point, we can obtain the linearized aircraft model as In the presence of actuator failures, u ( t ) can be expressed [ ] [ ] (1) (2) (1) (2) as A A B B 4 × 4 4 × 5 4 × 3 4 × 3 ˙ x = x + U, (II.12) (3) (4) (3) (4) u ( t ) = v ( t ) + σ (¯ u − v ( t )) , (III.3) A A B B 5 × 4 5 × 5 5 × 3 5 × 3 m m × 1 where v ( t ) ∈ R is the applied control input vector, ¯ u = R , are the parameters updated from the adaptive laws T [¯ u , ¯ u , . . . , ¯ u ] is the failure vector, and σ represents the 1 2 m ˙ T ˆ failure pattern and is defined as K = − Γ xe P b , i = 1 , 2 , . . . , m (III.10) i i i T ˙ ˆ κ = − γ r e P b , i = 1 , 2 , . . . , m (III.11) i i d i σ = diag { σ , σ , . . . , σ } (III.4) 1 2 m ˙ T ˆ θ = − λ e P b , i = 1 , 2 , . . . , m, (III.12) i i i with σ = 1 if the i th actuator has failed, that is, u = ¯ u , i i i T T and σ = 0 otherwise. The failures are assumed to occur i where Γ = Γ > 0 , γ = γ > 0 , λ > 0 , b is the i th i i i i i i T instantaneously, i.e., σ are piecewise constant functions of i column of B , i = 1 , 2 , . . . , m , and P = P > 0 satisfying m time. There are 2 possible combinations of actuator states (III.7). Γ , γ , and λ denote the design parameters for the i i i m (each actuator is either normal or failed), and therefore 2 − 1 adaptive laws. This adaptive actuator failure compensation ¯ possible failure patterns that constitute a set denoted by Σ .

scheme has the following desired properties: The system (III.1) can then be rewritten as Theorem 3.1: The control law (III.9), updated from (III.10)–(III.12) and applied to the system (III.1) subject ˙ x ( t ) = Ax ( t ) + B ( I − σ ) v ( t ) + Bσ ¯ u. (III.5) to the actuator failures (III.2) under Assumption 3.1, en- sures that all closed-loop system signals are bounded and For our adaptive control design for actuator failure compen- lim ( x ( t ) − x ( t )) = 0 , for any failure pattern σ ∈ Σ sation, the following assumption is needed: t →∞ d with uncertain parameters.

Assumption 3.1: ( A, B ) is known and stabilizable, and − 1 T ¯ Proof : For Q = Q + P BR B P , using (III.7), we obtain there exists a non-empty set Σ of “recoverable” failures such ¯ that rank[ B ( I − σ )] = rank[ B ] ∀ σ ∈ Σ . Σ is a subset of Σ .

T ¯ P ( A + BK ) + ( A + BK ) P = − Q < 0 . (III.13) Remark 3.1: This assumption characterizes the built-in redundancy needed for failure compensation as well as all the Suppose that at time t there are p < m actuator failures, failure patterns that can be accommodated. This condition is that is, u ( t ) = ¯ u , i = i , i , . . . , i , { i , i , . . . , i } ⊂ i i 1 2 p 1 2 p needed for the existence of a (fixed) failure compensation { 1 , 2 , . . . , m } , and that actuator failures happen at time controller that can achieve the desired performance when instants t , with t < t , k = 1 , 2 , . . . , N .

k k k +1 the system and failure parameters are known . The adaptive From the condition of Assumption 3.1: rank[ B ( I − σ )] = control design for unknown failure parameters is developed rank[ B ] , ∀ σ ∈ Σ , it follows that for each σ ∈ Σ , there exist based on the same condition. For instance, when the aircraft m × n m × m r constant matrices K ∈ R and κ ∈ R such that σ σ rudder fails during flight, this condition can still be satisfied so that the aircraft can be controlled by the remaining B ( I − σ ) K = BK, B ( I − σ ) κ = Bκ. (III.14) σ σ actuators and the failure can be accommodated (which is demonstrated in the simulation in Section 4.2). 2 Therefore, for each σ , there are constant K satisfying σ For asymptotic tracking, we first present a desired nominal T T T ¯ design for the system (III.1) without any actuator failures. P [ A + B ( I − σ ) K ] + [ A + ( I − σ ) K B ] P = − Q < 0 , σ σ The nonadaptive nominal controller is (III.15) and κ satisfying B ( I − σ ) κ r ( t ) = Bκr ( t ) . In addition σ σ d d u ( t ) = Kx ( t ) + κr ( t ) , (III.6) T m d there exists constant θ = [ θ , θ , . . . , θ ] ∈ R , where θ , 1 2 m i i 6 = i , i , . . . , i , are solutions of the following equation − 1 T m × n 1 2 p where K = − R B P ∈ R is an optimal LQ gain ∑ ∑ with P satisfying the Riccati equation b θ = − b ¯ u , (III.16) i i j j T − 1 T i 6 = i ,i ,...,i j = i ,i ,...,i A P + P A − P BR B P + Q = 0 (III.7) 1 2 p 1 2 p T and θ = 0 , for i = i , i , . . . , i , such that B ( I − σ ) θ = for some chosen n × n matrix Q = Q > 0 and m × m matrix i 1 2 p T m r Bσ ¯ u .

R = R > 0 , and the reference input r ( t ) ∈ R and d m × m r κ ∈ R are chosen for some desired system trajectory. From (III.5), (III.8), and III.9, we have With this nominal controller, the closed-loop system is ˆ ˆ ˙ e ( t ) = Ax ( t ) + B ( I − σ )( Kx ( t ) + ˆ κr ( t ) + θ ) + Bσ ¯ u d ˙ x ( t ) = ( A + BK ) x ( t )+ Bκr ( t ) , based on which, we define d the desired state trajectory x ( t ) from the reference system − ( A + BK ) x ( t ) − Bκr ( t ) d d d = Ax ( t )+ B ( I − σ )( K x ( t )+ κ r ( t )+ θ )+ Bσ ¯ u σ σ d ˙ x ( t ) = ( A + BK ) x ( t ) + Bκr ( t ) . (III.8) d d d ˆ − ( A + BK ) x ( t ) − Bκr ( t )+ B ( I − σ )[( K − K ) x ( t ) d d σ Define the tracking error e ( t ) = x ( t ) − x ( t ) . As the new d ˆ +(ˆ κ − κ ) r ( t ) + ( θ − θ )] (III.17) σ d adaptive failure compensation scheme for asymptotic state tracking, the feedback control law is With equations (III.14) and (III.16), the dynamic equation for tracking error can be simplified as ˆ ˆ v ( t ) = Kx ( t ) + ˆ κr ( t ) + θ, (III.9) d ˆ T m × n ˙ e ( t ) = ( A + BK ) e ( t ) + B ( I − σ )[( K − K ) x ( t ) ˆ ˆ ˆ ˆ σ where K = [ K , K , . . . , K ] ∈ R , ˆ κ = 1 2 m T m × m T r ˆ ˆ ˆ ˆ ˆ [ˆ κ , ˆ κ , . . . , ˆ κ ] ∈ R , and θ = [ θ , θ , . . . , θ ] ∈ +(ˆ κ − κ ) r ( t ) + ( θ − θ )] . (III.18) 1 2 m 1 2 m σ d 2 ∞ ∞ ∞ ˆ With the adaptive laws (III.10)–(III.12), a Lyapunov func- we can conclude that e ∈ L ∩ L , K ∈ L , ˆ κ ∈ L , ∞ ˆ tion candidate can be chosen as and θ ∈ L . For the nominal design (III.6), A + BK is ∑ ∞ asymptotically stable such that x ( t ) ∈ L with a bounded d T T − 1 ˆ ˆ V = e P e + ( K − K ) Γ ( K − K ) i i i i ∞ i reference input r ( t ) . Hence we conclude that x ( t ) ∈ L , d i 6 = i ,i ,...,i 1 2 p ∞ ∞ so does v ( t ) . Furthermore, since ˙ x ( t ) ∈ L and ˙ x ∈ L , ∑ ∑ d T − 1 − 1 2 ˆ 2 + (ˆ κ − κ ) γ (ˆ κ − κ ) + λ ( θ − θ ) i i i i i i given that e ( t ) ∈ L , we also have lim e ( t ) = 0 . 2 i i t →∞ i 6 = i ,i ,...,i i 6 = i ,i ,...,i 1 2 p 1 2 p The physical meaning of state tracking for the aircraft dynamic model linearized at an equilibrium point ( x , U ) is o o for each time interval ( t , t ) , k = 0 , 1 , . . . , N , with t = k k +1 0 that the operation of the aircraft follows a desired trajectory 0 and t = ∞ , where K is the i th row of K and κ is N +1 i σ i the i th row of κ . The time-derivative of V in each ( t , t ) σ k k +1 in a neighborhood of the equilibrium point. When we apply is this adaptive failure compensation scheme to aircraft flight T T T T ˙ V = e [ P ( A + BK ) + ( A + K B ) P ] e control, we want the aircraft to maintain the desired trajec- T tory that was originally set for the nominal case of no failure, ˆ ˆ +2 e P B ( I − σ )[( K − K ) x ( t )+(ˆ κ − κ ) r ( t )+( θ − θ )] σ σ d ∑ ∑ even if unknown actuator failures occur. Theorem 3.1 gives a T − 1 ˙ T − 1 ˆ ˆ ˙ +2 ( K − K ) Γ K + 2 (ˆ κ − κ ) γ ˆ κ i i i i i i i i solution to the problem of state tracking. The stabilizability i 6 = i ,i ,...,i i 6 = i ,i ,...,i 1 2 p 1 2 p and rank condition in Assumption 3.1 characterizes the ∑ − 1 ˙ ˆ ˆ +2 λ ( θ − θ ) θ system redundancy condition needed for actuator failure i i i i i 6 = i ,i ,...,i 1 2 p compensation. As shown in next section, it is satisfied for T T T T the rudder or aileron failure case and the system state is = e [ P ( A + BK ) + ( A + K B ) P ] e T able to track the desired trajectory asymptotically, which ˆ ˆ +2 e P B ( I − σ )[( K − K ) x ( t )+(ˆ κ − κ ) r ( t )+( θ − θ )] σ σ d ∑ ∑ implies that the aircraft can maintain the desired performance T T T T ˆ − 2 ( K − K ) xe P b − 2 (ˆ κ − κ ) r e P b i i i i i d i under normal as well as failure conditions. Assumption i 6 = i ,i ,...,i i 6 = i ,i ,...,i 1 2 p 1 2 p 3.1 also requires the knowledge of A and B . However, it ∑ T ˆ − 2 ( θ − θ ) e P b , i i i may be noted that, if the actual system matrix given by i 6 = i ,i ,...,i p 1 2 A = A + δA (where δA is the parameter error) is such that p T ( A + BK ) P + P ( A + BK ) < 0 , the asymptotic tracking p p For the considered actuator failure pattern, that is, u ( t ) = i and signal boundedness of Theorem 3.1 will still hold. (This ¯ u , σ = 1 , i = i , i , . . . , i , using the fact that B ( I − i i 1 2 p ∑ T ˆ ˆ condition basically implies that the nominal LQ regulator σ )( K − K ) = b ( K − K ) and the commu- σ i i i i 6 = i ,i ,...,i 1 2 p used for generating the desired trajectory is designed to be tativity property of the matrix trace operator, i.e., T r ( XY ) = robust to parameter uncertainties). Further research is needed T r ( Y X ) , the following equalities hold: in order to investigate robustness of the adaptive scheme to ∑ T T T ˆ ˆ e P B ( I − σ )( K − K ) x ( t ) = ( K − K ) xe P b , σ i i i model errors, and to relax the requirement of knowledge of i 6 = i ,i ,...,i 1 2 p A and B .

∑ T T T e P B ( I − σ )(ˆ κ − κ ) r ( t ) = (ˆ κ − κ ) r e P b , σ d i i d i IV. A PPLICATION TO F LIGHT C ONTROL i 6 = i ,i ,...,i 1 2 p ∑ In this section, we demonstrate application of the adaptive T T ˆ ˆ e P B ( I − σ )( θ − θ ) = ( θ − θ ) e P b .

i i i failure compensation technique to a transport aircraft by i 6 = i ,i ,...,i 1 2 p presenting some simulation results for trajectory tracking in the presence of unknown rudder and aileron failures. We So the time-derivative of V in each ( t , t ) is simplified k k +1 shall first describe the aircraft model used in simulation, and as then present the simulation results.

T ˙ ¯ V = − e Qe ≤ 0 , (III.19) A. Aircraft Model for Simulation Study where (III.13) is used for the last equality. It follows that e ∈ 2 ∞ ∞ ∞ For our simulation study, we use a transport aircraft model.

ˆ ˆ L ∩ L , and K ∈ L and θ ∈ L for i 6 = i , i , . . . , i , i i 1 2 p The airplane flies at a velocity of 774 ft/sec and an altitude where i , i , . . . , i are the indexes of failed actuators.

1 2 p of 40 kft. The linearized dynamic model is From (III.10)–(III.12), we have [ ] [ ] [ ] (1) (2) (1) (2) ˙ ˙ ˙ − 1 − 1 − 1 T A A B B ˆ ˆ ˆ 4 × 4 4 × 5 4 × 3 4 × 3 Γ K , Γ K , . . . , Γ K = − xe P B, 1 2 m ˙ x ( t ) = x ( t ) + U ( t ) (IV.1) 1 2 m (3) (4) (3) (4) A A B B [ ] 5 × 4 5 × 5 5 × 3 5 × 3 − 1 − 1 − 1 T ˙ ˙ ˙ γ ˆ κ , γ ˆ κ , . . . , γ ˆ κ = − r e P B, 1 2 m d 1 2 m (2) (2) (1) (4) (1) where A and B are zero matrices, and A , A , B [ ] (4) ˙ ˙ ˙ and B are of the same forms as in [8], and − 1 − 1 − 1 T ˆ ˆ ˆ λ θ , λ θ , . . . , λ θ = − e P B, (III.20) 1 2 m 1 2 m T x = [ u w q θ v r p φ ψ ] , ∞ ∞ ∞ ˆ ˆ T which implies that K ∈ L , ˆ κ ∈ L and θ ∈ L for i i i U = [ δ δ δ δ δ δ ] , e t t a a r l r l r     i = i , i , . . . , i , because B can be represented by a linear 1 2 p 0 0 0 0 0 0 0 0 . 001 0 . 001 0 0 0 0 . 8 − 0 . 7 combination of b , i 6 = i , i , . . . , i .

i 1 2 p     (3) (3) A = − 0 . 001 − 0 . 001 0 0 , B = 0 − 0 . 5 0 . 6     The function V is not continuous at t , k = 0 , 1 , . . . , N , k 0 0 0 0 0 0 0 and only has finite value jumps at those time instants. So 0 0 0 0 0 0 0 1.5 (3) (3) The non-zero terms in A and B represent the engine 1 0.5 thrust differential effect. The basic units used in this model −0.5 0 20 40 60 80 100 120 140 160 180 200 are ft, sec and crad (0.01 radian).

y−axis velocity x =v (solid) and desired x (dashed) (ft/sec) vs. time (sec) 5 d5 We consider two types of constant actuator failures: rudder 0.5 failure and aileron failure. The rudder failure is denoted as −0.5 −1 0 20 40 60 80 100 120 140 160 180 200 U ( t ) = U ( t ) t ≥ t , (IV.2) 6 6 f f roll angle x = φ (solid) and desired x (dashed) (deg) vs. time (sec) 8 d8 where t is the failure time instant. It represents the rudder f −0.5 stuck in its position at instant t , and cannot be moved. The f −1 aileron failure we consider is 0 20 40 60 80 100 120 140 160 180 200 yaw angle x = ψ (solid) and desired x (dashed) (deg) vs. time (sec) 9 d9 U ( t ) = 0 t ≥ t , (IV.3) 5 f Fig. 1. System states x = v , x = φ , x = ψ (Case I).

5 8 9 which indicates that at failure time instant t , the right f aileron angle drops to zero and is stuck from then on. These 0.5 failure patterns satisfy Assumption 3.1.

−0.5 0 20 40 60 80 100 120 140 160 180 200 B. Simulation Results left engine throttle δ (solid) and right engine throttle δ (dashed) (ft/sec ) vs. time (sec) tl tr In this subsection, we present the simulation results for 1 the asymptotic tracking of x ( t ) by x ( t ) to demonstrate d −1 −2 the performance of the system with the adaptive failure 0 20 40 60 80 100 120 140 160 180 200 left aileron δ (solid) and right aileron δ (dashed) (deg) vs. time (sec) compensation scheme described in Section 3.2. For our al ar simulation, the values of κ and r were chosen as r = 1 , 0.4 d d 0.2 and −0.2 −0.4 T 0 20 40 60 80 100 120 140 160 180 200 κ = [ 0 . 7344 0 . 6842 1 . 5974 − 0 . 5817 − 0 . 7853 1 ] rudder angle δ (deg) vs. time (sec) r that is, r = r ∈ R is a scalar, which leads to the final d d Fig. 2. Control signals: δ , δ , δ , δ , and δ (Case I).

tl tr al ar r values of the desired states: ∞ T x = [ 4 − 1 . 06 0 1 0 0 0 0 − 1 . 8 ] d immediately after its occurrence by the adaptive controller.

This trajectory represents a steady state flight condition (The other states can also converge to the desired trajectory in which the aircraft is climbing with a 4 ft/sec velocity after rudder failure, but are not shown in the figures due to perturbation along the x -axis, a -1.06 ft/sec velocity per- the limitation of space). The initial transients (before the turbation along the z -axis, a pitch angle of 1 crads (0.57 failure occurs) are because of the adaptive system response degrees), and a yaw angle of -1.8 crads (-1.0 degree). The when the system is first turned on with some arbitrary initial initial value of the state vector is zero, i.e., the airplane values of the adaptive control gains.

is in steady wings-level flight. The physical meaning of Case (II) . System performances with adaptive compensa- state tracking is that the aircraft flies from one steady state tion scheme and aileron failure (IV.3). The failure instant is flight condition to another steady state flight condition while t = 20 seconds and the parameter setting is the same as f closely following a reference state-trajectory. The choice Case (I). The results are shown in Figures 3 and 4.

of the reference trajectory (in particular, κ and r ) in this d paper was arbitrary, the main purpose being demonstration 1.5 of the adaptive scheme. For the controller design, we choose 0.5 Q = I and R = diag { 2 , 6 , 6 , 2 , 2 , 2 } . −0.5 0 50 100 150 200 250 300 y−axis velocity x =v (solid) and desired x (dashed) (ft/sec) vs. time (sec) For our simulation study, we examine two cases: (I) 5 d5 0.5 system responses with adaptive failure compensation with failure (IV.2) and (II) system responses with adaptive failure −0.5 −1 compensation with aileron failure (IV.3). 0 50 100 150 200 250 300 roll angle x = φ (solid) and desired x (dashed) (deg) vs. time (sec) 8 d8 Case (I) . System performances with adaptive compensation scheme and rudder failure (IV.2). The failure −0.5 instant is t = 5 seconds. Γ ( i = 1 , . . . , 6) are chosen f i −1 0 20 40 60 80 100 120 140 160 180 200 as [0 . 01 0 . 01 0 . 01 0 . 06 0 . 01 0 . 01 0 . 01 0 . 04 0 . 08] .

yaw angle x = ψ (solid) and desired x (dashed) (deg) vs. time (sec) 9 d9 γ and λ ( i = 1 , . . . , 6) are chosen as i i [0 . 01 0 . 05 0 . 05 0 . 02 0 . 02 0 . 01] . These design Fig. 3. System states x = v , x = φ , x = ψ (Case II).

5 8 9 parameters were chosen by trial and error. Some selected states and control signals are shown in Figures 1 and 2, In summary, in this study we simulated some typical which demonstrate how the rudder failure is accommodated aircraft motions for realistic rudder and aileron failure con- A CKNOWLEDGMENTS 0.5 −0.5 This research was partially supported by NASA Langley 0 20 40 60 80 100 120 140 160 180 200 left engine throttle δ (solid) and right engine throttle δ (dashed) (ft/sec ) vs. time (sec) Research Center under grant NCC-1-02006. The authors tl tr would like to thank the reviewers for their helpful comments.

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to handle actuator failures using functioning actuators that can be of types different from the failed actuators. An adaptive actuator failure compensation scheme was proposed, which guarantees closed-loop signal boundedness as well as asymptotic state tracking in the presence of unknown actuator failures occurring at unknown time instants. Sim- ulation results obtained for a large transport aircraft model indicate that the adaptive scheme can provide satisfactory performance in the presence of rudder or aileron failures, i.e., the functioning actuators automatically and seamlessly take over for the failed ones.

Several important and challenging issues need to be ad- dressed in future research. First, robustness of the adaptive scheme to model errors, and relaxation of the requirement of knowledge of the system matrices, need to be investigated.

In addition, the effects of actuator nonlinearities including output and rate saturation, as well as actuator dynamics, need to be addressed. Also, extensions of the adaptive scheme to model reference state and output tracking, as well as to nonlinear aircraft models, should be addressed.