Simple Adaptive Delta Operator Aircraft Flight Control for Accommodation of Loss of Control Effectiveness

A new proof for stability of delta operator simple adaptive control is presented in terms of a set of Linear Matrix Inequalities (LMIs). The paper shows how to design a feedforward gain to satisfy the LMIs over a polytope of loss of control effectiveness failures. The MATLAB Robust Control Toolbox is used to find the feedforward gain with the smallest norm that satisfies the LMIs. Examples are presented of the F/A-18 aircraft and the Innovative Control Effectors (ICE) tailless aircraft that show the design of a feedforward gain for a loss of control effectiveness in any one control effector. The designs use a fixed eigenstructure assignment controller for an inner loop augmented with the simple adaptive controller. Simulations of both aircraft include simultaneous loss of control effectiveness failure and lateral wind gust. Simulation results for the F/A-18 aircraft show that the adaptive controller achieves almost perfect tracking whereas the nonadaptive controller cannot achieve a coordinated turn when an aileron failure occurs. The ICE tailless aircraft uses sideslip, washed-out stability axis yaw rate, and stability axis roll rate feedback for both the inner loop eigenstructure assignment controller and the simple adaptive controller. However, the adaptive controller also uses bank angle feedback. Simulation results for the ICE tailless aircraft show that the adaptive controller achieves almost perfect tracking whereas the nonadaptive controller diverges when an all moving tip failure occurs.


Introduction
Aircraft flight control systems are designed with extensive redundancy to ensure a low probability of failure. tive controller. Both loops use sideslip, washed-out yaw rate, and roll rate feedback sampled at 200 Hz. Simulation results show that the adaptive controller achieves almost perfect tracking whereas the nonadaptive controller cannot achieve a coordinated turn when an aileron failure occurs. A second example is presented of the ICE tailless aircraft [20] that shows the design of a feedforward gain for a loss of control effectiveness in any one control effector of either 50% elevon, 50% all moving tips, or 50% yaw thrust vectoring. Here both the inner loop eigenstructure assignment controller and the adaptive controller use sideslip, washed-out stability axis yaw rate, and stability axis roll rate feedback. However, the adaptive controller also uses bank angle feedback. Both loops use a sampling rate of 1 kHz. Simulation results show the adaptive controller achieves almost perfect tracking whereas the nonadaptive controller diverges when an all moving tip failure occurs.
A preliminary version of this paper [21] was presented at the AIAA Guidance, Navigation and Control Conference. This revised version includes 1) an extended explanation of the feedforward gain design method, 2) an extended discussion of almost strictly positive real and its relationship to minimum phase for delta operator systems, 3) the addition of a lateral wind gust to the ICE aircraft example, and 4) new time responses that are consistent for both examples with a control effectiveness failure at 5 sec and a lateral wind gust at 10 sec with a duration of 10 sec. The addition of a lateral wind gust to the ICE aircraft resulted in a more difficult problem. This problem was solved with the novel idea of adding bank angle feedback to the adaptive controller, but not the inner loop eigenstructure assignment controller, in order to achieve excellent tracking during a simultaneous loss of control effectiveness failure and lateral wind gust.  The unified state space model proposed by Middleton and Goodwin [17] is valid for both the discrete and continuous time cases simultaneously. This unified model, which is assumed to be a minimal realization, is described by [ ( ) a p y t is the output to be controlled and where D is square and nonsingular. We remark that in aug-menting the plant Equation (3) to obtain Equation (4) we are not physically modifying the system, instead we are just defining a metasystem that will allow us to use the simple adaptive control SAC methodology.

Problem Statement
The control objective is to design an adaptive control signal ( ) p u t such that all signals in the closed loop system are bounded and the augmented plant output ( ) a p y t tracks the output of a reference model given by [21]: We remark that the order of the plant may be much greater than the order of the reference model. That is, . m p n n 

The General Tracking Problem
We summarize the general tracking problem for a known plant. These results have appeared in Kaufman, Barkana, and Sobel [6] and Barkana [5]. Let the input command ( ) m u t be the output of an unknown command generating systems of the form , such that, if the augmented plant could reach and move along them, its output would perfectly track the output of the reference model. That is, the ideal trajectories are targets that the augmented plant tries to reach or at least be close to, in order to have bounded tracking errors. Mathematically, where the ideal trajectories are defined as ( ) ( ) ( ) 11 12 p m m x t X x t X u t * = + (10) and the ideal control signal is defined as from Equation (10) into Equation (9), and using ( ) m u t from Equation (8), gives a condition for the existence of the ideal target trajectories: or 11 12 0 Therefore, the ideal control in Equation (11) and the ideal augmented plant in Equation (14) Thus, since the reference model is stable, we only require that ( ) is an auxiliary input signal. The control in Equation (17), however, requires calculations of 11 X and 12 X and also explicit knowledge of the system dynamics. As an alternative, the direct adaptive control algorithm known as Simple Adaptive Control (SAC) is used to calculate the gains which enable the plant to get bounded tracking errors. Note that SAC only requires that the plant outputs be available for measurement.

Adaptive Control Algorithm
The unified form of the adaptive algorithm is as follows The adaptive gains are concatenated into matrix ( ) The concatenated gain m p e t y t y t = − (22) where ( ) ( ) ( ) in continuous time in discrete time and T and T are time invariant weighting matrices. The σ term in Equation (20) was originally proposed by Ioannou and Kokotovic [22] and it is used to avoid divergence of the integral gains in the presence of bounded disturbances.

Almost Strictly Positive Real and Minimum-Phase Concepts
The following development shows sufficient conditions for a system to be ASPR in the delta domain.
Lemma 2: The unified system described by the minimal realization in Equation (4), with ( ) 0 d t = , is SPR if and only if there exists a positive-definite symmetric matrix P that satisfies the following LMI Proof: The result follows trivially from Proposition 4 in Collins, Haddad, Chellaboina, and Song [23] and the observation that as 0 ∆ → , Equation (24) approaches the continuous time result given by Lemma 4 in Kottenstette and Antsaklis [24].
Substituting Equation (27) into Equation (4), with ( ) 0 A. Cano, K. Sobel Using Lemma 1 we have that Equation (29) is SPR (or Equation (4) is ASPR) if and only if there exists a positive-definite symmetric matrix P such that Now we derive necessary conditions for the unified system to be minimum-phase (MP). The zero dynamics are obtained from ( ) a p y t of Equation (4) and are given by Substituting Equation (31) into the first equation of Equation (4), with ( ) 0 d t = , gives the zero-dynamics equation . If the unified system in Equation (4) is MP then z A must be asymptotically stable. That is, Equation (4) is MP if there exist a positive-definite symmetric matrix P such that or, equivalently, all the eigenvalues of z A must reside inside the circle of radius 1 ∆ centered at 1 γ = − ∆ in the complex plane. We are now in a position to state and prove the following lemma. (4), with D nonsingular and ( ) 0 d t = , is MP, then it is ASPR. Proof: Assume that the unified model in Equation (4) is MP and D is nonsingular. We want to show that there exists a stabilizing, positive-definite symmetric gain e K sufficiently large that leads to a closed loop system that is SPR. To this end, consider the control signal of Equation (25) that leads to a closed loop system in Equation (28), which, using Equation (26), can be rewritten as

Lemma 3: If the unified system in Equation
Applying the left-hand side of Lemma 1 to Equation (35), and assuming a positive-definite symmetric matrix P, we have that Showing that L < 0 would imply that Equation (35) is SPR and, by noting that that Equation (34) is also SPR or, equivalently, that Equation (4) is ASPR, as desired. Therefore we must show that for e K sufficiently large, 0 L < . To this end we use Schur's complement lemma and note that 0 First note that 0 Λ < follows from the assumption that Equation (4) is MP, since the zeros of the closed loop system with a constant output feedback gain e K are identical to the zeros of the open loop system [26]. Next we show that ( ) is nondefinite, these two terms are bounded and hence we can establish that, for a sufficiently large positive-defi-nite gain e K , the following inequality can be satisfied Thus we will have 0 Φ < , which is a necessary condition for L to be negative-definite. Now consider A s e K becomes more positive-definite, A and p C approach a limiting bounded matrix and hence Ψ is bounded. Let Furthermore, since Q is also bounded, we can similarly establish that, for e K sufficiently large, the following inequality will hold ( ) Note that as we make e K more positive-definite, the left-hand side of Equation (37) approaches a limiting bounded matrix while the right-hand side becomes more positive-definite so that the inequality can be satisfied. Hence

( )
Sch 0 Λ < for e K sufficiently large. This completes the proof. 

Stability Analysis
Theorem 1: If the unified delta plant to be controlled is ASPR with the adaptive scheme consisting of the augmented plant, the SAC control law and its gain adaptation formula, together with 1  The next theorem due to Belkharraz and Sobel [16] describes a sufficient condition for the boundedness of the Lyapunov functions at the failure instants.
The Lyapunov functions at the failure instants given by

Robust Simple Adaptive Control Tracking
We now extend the results of Theorem 1 to the case where the matrices , are known to reside within a given convex hull of matrices, also known as a matrix polytope. The development here is similar to the work of Ben-Yamin, Yaesh, and Shaked [15] for shift operator systems. However, the results here for delta operator systems are explicitly in terms of the sampling period ∆ . Let i Ω be the set of the matrices such that each i Ω belongs to the polytope defined as: where the k Ω 's in Equation (39) represent the vertices of the polytope. Alternatively, Equation (38) can be described as ( ) Co Ω the adaptive scheme consisting of the augmented plant, the SAC control law, and its gain adaptation formula, create bounded gains and state signals.

Feedforward Gain Design
We propose a method which uses the MATLABLMI toolbox and the Optimization toolbox to design the feedforward gain D. Given the strictly-proper plant in Equation (3), which may not be inherently ASPR, we seek a gain D to augment the system and obtain a proper plant in Equation (4) which is ASPR. This will enable us to use SAC to generate an adaptive control signal ( ) p u t such that all signals in the closed loop system are bounded and the augmented plant output ( ) a p y t tracks the output of a reference model. It follows from Lemma 3 that if the plant is MP, with D nonsingular, then it is ASPR. Thus we use the MP property, which can be easily verified using Equation (33), to obtain a gain D with the smallest norm possible so that We reiterate that in augmenting the plant we are not physically modifying the system, instead we are just realizing a metasystem that is ASPR and which will allow the use of SAC.
We use a convex matrix polytope whose vertices, defined as LMIs in MATLAB, represent the unfailed plant and several failed plants which are augmented with the same feedforward gain D that makes each of them MP. Once the vertices of the polytope are MP, then all the possible plants within the polytope are also MP. Note that when D is not specified, Equation (33) is no longer an LMI but a bilinear matrix inequality (BMI) in variables 1 D − and P. On the other hand, when D is given, Equation (33) is an LMI in the variable P and is only feasible when there exists a 0 P > that satisfies it. Thus our approach consists of using an optimization routine which iteratively specifies and substitutes a gain D into Equation (33), and simultaneously minimizes the Frobenius norm of D and checks the feasibility of the LMI constraints. We minimize the Frobenius norm of D by using the fmincon function from the MATLAB Optimization toolbox [27] which finds the minimum of a multivariate function with nonlinear constraints.
In this paper we consider a single failure in any one control effector. Suppose that the plant has m control effectors. When considering a single failure in any one control effector we define m polytopes with two vertices each; one vertex for the unfailed plant and the other for the failed plant. For the m polytopes we define each of the 2m vertices as an instance of the LMI in Equation (33) using the MATLAB LMI Control Toolbox [28]. We can, however, define only m vertices for each control effector failure plus a shared vertex for the unfailed plant for a total of 1 m + LMIs. Note that although we are allowed to define only 1 m + LMIs in our computer program, we still retain the notion that only the unfailed plant and any other vertex representing a failed plant form the required convex polytope. Since P has to be positive-definite, we need to define an additional LMI that will guarantee that 0 P > for a total of 2 m + LMIs. In order to avoid the ambiguity that results from marginal infeasibility of the LMI constraints when P is close to zero, this additional LMI in our program is defined as 3 10 P I − > , instead of 0 P > . This will guarantee that P is strictly positive definite. Note that this does not affect the LMI constraints since each vertex is homogeneous in P. The definition of the LMI constraints is the first step of the design process shown in the flowchart in Figure 1.
Next, to initialize the optimization, a gain 0 D is obtained using the randn function from MATLAB which returns a square matrix of pseudo-random numbers drawn from a normal distribution with a variance of unity. We use fmincon to find a D with a small Frobenius norm which is constrained to satisfy an LMI set that represents the m polytopes described above. We will perform a specified number of optimization runs with a certain number of iterations each. At each iteration, a D is substituted into the set of 2 m + LMIs which is then solved for P. We remark that P is the same for every LMI in the set. The feasibility of the LMI is monitored by the parameter tmin which must be strictly negative in order to guarantee the feasibility of our LMI set for a given D.
It is possible for an optimization run to reach the maximum number of iterations before converging to a final gain, or to actually converge to a feedforward gain which we refer to as con D . In the former case, as shown in the flowchart in Figure 1, we check if the maximum number of optimization runs has been reached before obtaining another initial condition from the random number generator to start a new optimization run. In the latter case, however, we check if the norm of con D is smaller than the norm of the initial gain. That is, if  It is important to note that by considering an LMI set consisting of a single polytope corresponding to a failure in one control effector, and using the D obtained from the optimization, we can increase the percentage of loss of control effectiveness in steps of 0.1 and check if the feasibility of the LMI set is maintained for additional percentage failure. Depending on the type of failure, the D may or may not allow more loss of control effectiveness than the amount that was initially defined for each failure.
We remark that when searching for a D for plants with a single failure in any one control effector, the LMI set in design process can be defined to include only one polytope at a time. This would require, however, finding a different D for each type of effector failure and so we would be forced to first identify the failure in order to use the appropriate feedforward gain. Furthermore, no claims are made about the convergence rate and optimality of the proposed feedforward design process.

F/A-18 Aircraft and Reference Model
Consider the linearized lateral dynamics of the F/A-18 aircraft described in [12]. The rigid body states are lateral velocity v (ft/s), yaw rate r (deg/s), roll rate p (deg/s), and bank angle φ (deg from [16] shown in Table 1, which was designed using eigenstructure assignment for the unfailed aircraft. This constant output feedback gain will be placed around both the aircraft and the reference model. Thus, the adaptive algorithm will control the closed loop aircraft. The block diagram of the adaptive control system is shown in Figure 2.
Barkana, Rusnak, and Weiss [19] have shown that a constant parallel feedforward gain D can be implemented as part of the adaptive controller. Therefore, nothing is added in parallel with the aircraft in the implementation of the adaptive controller. However, the gain D does create an algebraic loop. Barkana As shown in Figure 3. The equivalence between Figure 2 and Figure 3 is shown in detail in [19]. Therefore, Table 1. Eigenstructure assignment gain for the F/A-18 aircraft from Belkharraz and Sobel [16].
and m p is the magnitude of the roll rate pulse in deg/sec.

Bounded Input Disturbance
In this example we consider a bounded input disturbance in the form of a lateral gust ( ) g v t , which is described in [16] and given by ( )

Feedforward Gain for the F/A-18 Aircraft
For the F/A-18 aircraft there are a set of five LMIs. These include 1) an LMI representing the unfailed aircraft, 2) an LMI for positive definite P, and 3) three LMIs for the three failure polytopes. Each of the three failure polytopes has two vertices with one vertex for the unfailed aircraft and a second vertex for the aircraft with one control effector failure. So the three failure LMIs represent a) the aircraft with a failure in the trailing edge flaps, b) the aircraft with a failure in the ailerons, and c) the aircraft with a rudder failure. Belkharraz and Sobel [16] used an 80% effectiveness failure in any one control effector, and so we choose each of the failed vertices for the F/A-18 aircraft to be defined with an 80% loss of control effectiveness. Using our proposed method with a sampling rate of 200 Hz, we found the feedforward gain matrix D shown in Table 2 that has a Frobenius norm of 0.0889. This feedforward gain was obtained by choosing the D with minimum Frobenius norm from among those D matrices with positive definite P. Out of the 300 optimization runs, 16 converged to a feasible feedforward gain; the maximum Frobenius norm was 0.8457. Once the gain D was found, we considered each of the three two-vertex polytopes that include the unfailed aircraft and the aircraft with one control effector failure. By modifying our LMI program to include only three LMI's (one for the unfailed aircraft, one for the aircraft with one failure, and one for the 3 10 P I − > constraint) we search for a positive definite P with the same feedforward matrix obtained in the previous optimization. Clearly, when the effectiveness failure is at most 80%, a feasible solution to the new system of LMI's is guaranteed to exist. However, if we keep increasing the effectiveness failure in steps of 0.1 we find that the single failed F/A-18 aircraft remains minimum-phase for a 92% trailing edge flap failure, a 99% aileron failure, and a 80% rudder failure.

Weighting Matrices for the F/A-18 Aircraft
We now describe our selection process for the weighting matrices for the F/A-18 aircraft using a computer simulation with the reference model input in Equation (42), but without the lateral gust in Equation (44). In order to simplify the approach, we first let T and T be diagonal matrices and also let T T = . We then take our first set of candidates to be 11

T T I = =
. A computer simulation for this candidate shows no acceptable tracking of the reference model for the first 20 seconds of the simulation and so it is rejected. We then choose to make the weights for the a y e 's (the first three entries in T and T) larger. That is, we choose our second set of candidates as ing everywhere except at 2 t = sec where there is an unacceptable jump which results from an unrealistic deflection rate in the control signals, and so it is rejected. We now let T T ≠ and recall that T is the weighting matrix for the proportional part of the adaptive algorithm. Therefore, we make an effort to have the entries of T be smaller than those of T. This is because we want to avoid having any jumps from appearing in the simulation. To this end we choose our third set of candidates to be ( )

Simulation Results for the F/A-18 Aircraft
We perform non-adaptive simulations with the fixed gain controller  Figure 4, where the black line corresponds to the reference model, the red line corresponds to a 92% trailing edge flap failure, the green line corresponds to a 99% aileron failure, and the blue line corresponds to a 80% rudder failure. Observe the unacceptable tracking performance in sideslip angle β , yaw rate r, and roll rate p for each surface failure. Furthermore, the coordinated turn is not achieved when an aileron failure occurs. Recall that here we feed back the washed out yaw rate wo r (deg/s), but we plot the true yaw rate r (deg/s). Finally, we perform adaptive simulations of the F/A-18 to accommodate the same surface failures and input disturbance using the proposed adaptive controller with feedforward matrix D as given in Table 2. The weighting matrices used in the simulation are the same as those obtained above for the unfailed adaptive response. Here we let 0.002 σ = . The adaptive time responses are shown on the right side of Figure 4. The adaptive control surface deflections are rate limited. Observe the almost perfect tracking in sideslip angle β , yaw rate r, and roll rate p.

Tailless Aircraft and Reference Model
We now consider the linearized dynamics of the Innovative Control Effectors (ICE) aircraft which was described in Nieto-Wire and Sobel [20]. The state variables are velocity T V (ft/s), angle of attack α (rad), pitch angle θ (rad), pitch rate q (rad/s), engine power level, sideslip angle β (rad), bank angle φ (rad), roll rate p (rad/s), and yaw rate r (rad/s). The control effectors are throttle th δ (0-1), symmetric pitch flap pflap δ  B , and p C are given in [20]. The continuous time model for the ICE aircraft is converted to the delta model by using the c2del function from the MATLAB Delta Toolbox with a sampling rate of 1 kHz. We use the method proposed in [20] to compute the eigenstructure assignment gain ICE eig dK for the unfailed aircraft which is shown in Table 3. We assign the desired dutch roll damping ratio ζ , natural frequency n ω , and roll subsidence eigenvalues as 0.707 ζ = where ( ) c p t is the pilot roll rate command given in Equation (43). The 4/3 gain in ( ) m u t has been added to the pilot stick for the purpose of achieving zero steady-state error to a s p command.

Feedforward Gain for the Tailless Aircraft
In this example we consider loss of control effectiveness failures in any one control effector. Here we add bank angle feedback in the implementation of the adaptive controller only. We linearly map the five lateral control effectors into four, and use left and right elevons, the all moving tips, and yaw thrust vectoring to yield a total of four control surfaces. That is, we map left and right all moving tips into a single control signal as: . This is done because our adaptive algorithm requires that the number of inputs equal the number of outputs. We also require that the failures be symmetric; otherwise cross coupling effects between the lateral and longitudinal axes must be considered. For the tailless aircraft we define a set of five LMIs. These include 1) an LMI representing the unfailed aircraft, 2) an LMI for positive definite P, and 3) three LMIs for the three failure polytopes. Each of the three failure polytopes has two vertices with one vertex for the unfailed aircraft and a second vertex for the aircraft with one control effector failure. So the three failure LMIs represent a) the aircraft with a failure in the elevons, b) the aircraft with a failure in the all moving tips, and c) the aircraft with a yaw thrust vectoring failure. Since we do not know in advance how much loss of control effectiveness can be effectively accommodated by the adaptive controller, we choose each of the failed vertices for the tailless aircraft to be defined with a 50% loss of control effectiveness. Using our proposed method with a sampling rate of 1 kHz, we found the feedforward gain matrix D shown in Table 4 that has a Frobenius norm of 0.0043. This feedforward gain was obtained by choosing the D with minimum Frobenius norm from among those D matrices with positive definite P using 300 optimization runs. Out of the 300 optimization runs, 29 converged to a feasible feedforward gain; the maximum Frobenius norm was 4.3351.
In this case increasing the percentage of loss of control effectiveness failure for each polytope individually, with the D obtained from the optimization, does not yield feasible LMIs beyond 50%.

Weighting Matrices for the Tailless Aircraft
An approach similar to that described for obtaining the weights for the adaptive algorithm in the FA-18 aircraft example yields the weights

Simulation Results for the Tailless Aircraft
We now perform computer simulations using the ICE model for different control effector failures. Consider the roll rate pilot command ( ) c p t is given by Equation (43). We first perform non-adaptive simulations with the fixed gain controller  Figure 5, where the black line corresponds to the reference model time response, the red line corresponds to a 50% elevon failure, the green line corresponds to a 50% all moving tip failure, and the blue line corresponds to a 50% yaw thrust vectoring failure.
Observe the poor tracking performance in sideslip angle β , stability axis yaw rate s r , and stability axis roll rate s p for each surface failure. Recall that we feed back the washed out stability axis yaw rate ( ) s wo r (deg/s) but here we plot the stability axis yaw rate s r (deg/s). Next we perform adaptive simulations of the ICE aircraft to accommodate the surface failures and input disturbance using the proposed adaptive controller with feedforward matrix D as given in Table 4. We initialize the adaptive gains as 0, 0, , which corresponds to initializing the failed plant with the eigenstructure assignment feedback which was designed for the unfailed aircraft. Here we combine the five lateral control signals from ( ) m u t into four signals that go into the adaptive controls. The adaptive algorithm yields four adaptive control signals which are then mapped back into five control signals for the tailless aircraft. The amount of failure and weighting matrices used in the adaptive simulation are the same as those used in the non-adaptive simulation. Here we let 0.002 σ = . The adaptive time histories are shown on the right side of Figure 5. The adaptive control surface deflections are rate limited. Observe the almost perfect tracking performance of the adaptive controller in sideslip angle β , stability axis yaw rate s r , and stability axis roll rate s p   Figure 6, where the black line corresponds to the reference model time response, the red line corresponds to a 50% elevon failure, the green line corresponds to a 50% all moving tip failure, and the blue line corresponds to a 50% yaw thrust vectoring failure. By comparing the left sides of Figure 5 and Figure 6, we can clearly see that the fixed controller performance deteriorates considerably due to the disturbance. Observe how the fixed controller, on the left side of Figure 6, starts diverging once the wind gust occurs, as can be seen in the bank angle and yaw rate outputs, and does not recover. Compare this to the response of the adaptive controller shown in the right side of Figure 6 which exhibits almost perfect tracking and is able to successfully accommodate considerable loss of control effectiveness failures even in the presence of a bounded lateral wind gust disturbance.

Conclusion
A new proof that yields a sufficient condition for stability in the delta domain for simple adaptive control in terms of a linear matrix inequality has been presented. We have shown how to compute a feedforward gain D Figure 6. Tailless Aircraft at 1 kHz. Failures at t = 5 sec: 50% in any one control effector. 5 fps lateral wind gust disturbance at t = 10 sec with duration of 10 sec. that makes the augmented plant minimum-phase, and thus ASPR, by defining an LMI set that represents a convex control effector failure polytope. The approach consists of minimizing the Frobenius norm of D subject to LMI constraints. The designs used a fixed eigenstructure assignment controller for an inner loop augmented with the simple adaptive controller. The adaptive algorithm and the proposed method to compute the feedforward gain have been applied to both an F/A-18 aircraft and a tailless aircraft with lateral wind gust disturbances. A feedforward gain was designed for an F/A-18 aircraft for a loss of control effectiveness in any one control effector of 92% trailing edge flap, 99% aileron, or 80% rudder. Furthermore, a feedforward gain was designed for a tailless aircraft for a loss of control effectiveness in any one control effector of 50% elevon, 50% all moving tips, or 50% yaw thrust vectoring. Computer simulations for both aircraft with a failure in any one control effector under lateral gust conditions exhibited almost perfect tracking with the adaptive algorithm whereas the nonadaptive F/A-18 controller could not achieve a coordinated turn when an aileron failure occurred and the nonadaptive tailless aircraft controller diverged when an all moving tip failure occurred. , .