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Linear adaptive actuator failure compensation for wing rock motion control
Accepted Manuscript Linear adaptive actuator failure compensation for wing rock motion control
Sabri Boulouma, Salim Labiod, Hamid Boubertakh
PII: DOI: Reference:
S1270-9638(16)30989-0 http://dx.doi.org/10.1016/j.ast.2017.03.025 AESCTE 3964
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Aerospace Science and Technology
Received date: Revised date: Accepted date:
2 November 2016 15 February 2017 17 March 2017
Please cite this article in press as: S. Boulouma et al., Linear adaptive actuator failure compensation for wing rock motion control, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.025
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Linear adaptive actuator failure compensation for wing rock motion control Sabri Bouloumaa,* Salim Labiodb and Hamid Boubertakhc LAJ, Faculty of Science and Technology, University of Jijel, Ouled Aissa, BP. 98, 18000, Jijel, Algeria. a [email protected], b [email protected], c [email protected] Abstract In this paper, a novel adaptive actuator failure compensation control strategy is developed for wing rock motion control in the presence of both system and actuator failure uncertainties. The proposed strategy can compensate for both total and partial loss of effectiveness. A proportional actuation scheme of redundant aileron segments is used. This allows bringing the faulty multi-input single-output nonlinear system into an equivalent perturbed single-input single-output system. Afterward, an adaptive actuator failure compensation control scheme is developed around a linear approximation of an ideal feedback linearization controller. A failure estimation and compensation term is appended to this controller to account for possible actuator failures. Closed-loop stability and tracking performance are proved based on Lyapunov theory and a piecewise analysis is also introduced to show that stability properties hold despite the presence of parameter jumps caused by abrupt actuator failures. Simulation results on a small scale wind tunnel based wing rock model with redundant actuators show the effectiveness of the proposed adaptive control strategy. Keywords: Wing rock; Adaptive control; Actuator failures; Redundant actuators; Nonlinear systems. 1.
Introduction
As a result of the increasing demands for performance, precision and higher maneuverability, control systems has become more complex. Thus, failures have become more common, especially in aircraft and spacecraft control systems [1,2]. Failures can take place at different locations within the control loop. They can appear at the plant itself, at the sensors or at the actuators as illustrated in Fig. 1. Actuator failures may lead to severe performance deterioration or even system instability. They can cause catastrophic accidents involving human lives and millions worth equipment if they are not handled properly using appropriate control designs [3]. Recently, actuator failure compensation has become an active area of research that has undergone remarkable progress in both theory and practice. Various actuator failure compensation control designs have been proposed in the research literature. These techniques generally fall into two categories, the passive approaches and the active approaches [3–5]. In the passive approaches, a fixed controller is designed by considering a set of predefined actuator failures. These approaches are generally based on the robust control theory and aim to optimize system’s performance under actuator failures using linear quadratic (LQ) control designs and H ∞ [6,7] or using convex optimization based on linear matrix inequalities (LMI) [8]. These techniques are limited in that they deal only with a restricted set of failures [4]. On the other hand, active actuator failure compensation approaches are designed to take action at the time of failure occurrence. Among these approaches one can find multiple model based designs [9–11], fault diagnosis and isolation (FDI) designs based residual generation or state observers [3,12], fuzzy logic and artificial neural networks techniques [13,14], and adaptive control based designs [2,5,15–27]. Adaptive control provides adaptation mechanisms to adjust controllers to cope with parametric, structural and external disturbances. The adaptive control approach has been extensively used to design controllers for systems with actuator failures. Many classical linear and nonlinear adaptive control techniques have been extended and customized to compensate for actuator failures [5]. A recent overview on adaptive actuator failure compensation was also presented in [15]. In [16,17], the authors proposed adaptive control schemes for linear redundant systems with lock in place failure types for state and output tracking. In [18,19], the authors proposed adaptive control schemes for multivariable nonlinear systems with unknown actuator failures using feedback linearization techniques. In [20,21], the authors proposed an adaptive backstepping control schemes for a class of parametric strict feedback systems with unknown actuator failures. In these works, system parameters as assumed to be known or partially known, i.e. defined as a product of known nonlinear functions with unknown
Corresponding author. E-mail address : [email protected]
parameters while the actuator failures are supposed unknown, in terms of time value and pattern. To the authors’ best knowledge, the case of unknown nonlinear systems with unknown actuator failures has been rarely considered except in some few works. For instance, in [22,24–26], an adaptive actuator failure compensation for uncertain nonlinear systems with uncertain actuator failures is designed based on the adaptive fuzzy and neural control techniques. In these works, the idea was to bring the unknown system with unknown actuator failures to an ordinary system with unknown parameters via an adequate actuation scheme, then existing adaptive fuzzy and neural control techniques can be applied to the system with consideration of possible parameter jumps caused by abrupt faults [21,24]. These techniques are effective but do not provide any information or indication on the failures, which is sometimes required. Besides the computational burden of such fuzzy and neural techniques may limit their practical implementation. Fig. 1. Actuator, plant and sensor faults From the system’s architecture viewpoint, one way to increase system’s resilience is through actuation redundancy, i.e. the use of multiple actuators that can fulfill the same task. These actuators can be similar in structure and task such as segmented rudder in an aircraft or can be different in structure but can fulfill the same task. In an aircraft, for instance, differential engine thrust can compensate for a failed rudder [5]. However, from the controller design viewpoint, actuator redundancy brings more challenges, as the controller should drive the actuators and ensure cooperation between them such that in case one or more actuators fail, the remaining healthy actuators react consequently to compensate for the effect of the failed actuators. This means that the designed controllers should manage this redundancy in an effective way. Wing rock is an oscillatory rolling motion of an aircraft which occurs at high angles of attack. This oscillatory motion can be seen in high-speed civil transport and combat aircraft. The main aerodynamic parameters of wing rock are (i) angle of attack (ii) angle of sweep (iii) leading edge extensions (iv) slender forebody. Therefore, aircrafts that are susceptible to the wing rock phenomenon are those containing these parameters: such as aircraft with highly swept wings operating with leading edge extensions. The motion is characterized by an increase in amplitude up to a limit cycle as shown in Fig. 2, the final state is usually stable and defined by large roll oscillations [28–31]. The control of wing rock is a relevant issue as high-speed civil transport and combat aircraft can encounter it in their flight envelope, seriously compromising handling qualities and maneuvering capabilities. Many nonlinear and adaptive control schemes have been proposed for wing rock suppression [28,30–32]. These designs are efficient regarding their ability to suppress wing rock oscillations and their robustness against system’s uncertainties. However, the aforementioned works dealt only with the failurefree case. To the authors’ knowledge, except for the work given in [5], where an adaptive backstepping actuator failure control design for wind rock motion has been proposed, the case of actuator failure compensation for wing rock system has not been dealt with. In this paper, a redundant wing rock model is considered, the nonlinear model is based on parameter identification of wind tunnel experimental data [29]. In addition, it is assumed that three actuators deliver the control input to the system. These actuators are susceptible to fail during flight. An integrated, model-free linear adaptive controller with a feedforward actuator failure estimation and compensation term is developed to suppress wing rock oscillations. The proposed controller deals with both system uncertainties and actuator failures. It is simple in terms of the computational burden, compared to other works based on fuzzy logic and neural networks such as in [21,22,24,25,33,34]. Besides, this scheme does not require fault detection and isolation (FDI) module, which makes it more effective for real-time implementation, as the FDI module requires some latent time for fault detection, which may affect real-time constraints. In the same time, this controller gives an online estimation of the failure as opposed to previous adaptive approximation based techniques [21,24], where actuator failures are blindly compensated without having any information on their nature or time of occurrence. This lack of information makes them limited in case of failures needing immediate intervention. To summarize, the main contributions of this work can be listed as follows: (i) a new controller that can estimate the failures and compensate for their effects is proposed. (ii) the class of actuator failures is extended to time varying patterns compared to works in [5,21]. (iii) Stability and tracking requirements are proved rigorously while taking into account parameter jumps caused by abrupt actuator failures. (iv) the application to the augmented wing rock model, where system parameters are assumed unknown, this problem is rarely considered in the literature and brings a new challenge to the design of flexible and reliable aircraft. The remaining of the paper is organized as follows, in section 2, the nonlinear model of a redundant wing rock system is described and the control problem is formulated. In section 3 the control design steps are outlined along with stability and tracking proofs. In
section 4 simulation results are presented and discussed. Conclusions and suggestions for future works are presented is section 5. 2.
System modeling and problem formulation
In this section, the state space model of a redundant wing rock system is derived and the wing rock phenomenon is described. Afterward, the actuator failure compensation control problem is formulated. 2.1. Wing rock dynamic model In most works, the study of wing rock motion is based on wind tunnel experimental investigations on 80° swept delta wing models. The experimental geometries have simple triangular geometries. These models present stable limit cycles and correctly reproduce the dominant effect of primary wing vortices. the mathematical models of 80° swept slender wings have been developed based on parameter identification of experimental data [29,30]. In fact, these wing-rock mathematical models are not complete roll dynamics for practical control design because aerodynamics at high angles of attack (AOA) is very complex. Some unmodeled uncertainties such as different configurations and sizes should be considered in these models. Furthermore, the disturbance is also an important factor since aircraft operate in uncertain environments. For this work, we consider the wing rock system studied in [29,30]. The non-dimensional differential equation (single degree of freedom roll dynamics) describing the free motion of the roll angle φ is given by
φ (t ) + aˆ0φ (t ) + aˆ1φ (t ) + aˆ2 φ (t ) φ (t ) + aˆ3φ 3 (t ) + aˆ4φ 2 (t ) φ (t ) = 0
(1)
where the coefficients aˆ0 , aˆ2 ,, aˆ4 are parameters related to the aircraft’s operating conditions (i.e. angle of attack, Reynold number, and wing characteristics). Introducing the reference time t s = b 2V , with V is the air speed and b is the wing span, equation (1) can be written as follows [30]
φ (t ) +
aˆ0 aˆ aˆ aˆ φ (t ) + 1 φ (t ) + aˆ2 φ (t ) φ (t ) + 23 φ 3 (t ) + 4 φ 2 (t ) φ (t ) = 0 ts ts t s2 ts
(2)
The wing rock model exhibits oscillation for a large span of angles of attack. To illustrate this situation, let us consider an analytical wing rock model based on parameter identification of wind tunnel based experimental data. The model has a wing span b = 0.169m , root chord c r = 0.479m , and sweep Λ = 80° , airspeed
V = 30m/s ( Re = 950000 ) , and angle of attack α = 32.5° , the corresponding coefficients are given as
aˆ0 = 0.00723 , aˆ1 = −0.03104 , aˆ2 = 0.53884 , aˆ3 = 0.00623 , aˆ4 = 0.04189 [30]. The free to roll motion for an initial roll angle φ ( 0 ) = 1° and roll rate φ (t ) = 0 deg/s in the first case and φ ( 0 ) = 10° , φ (t ) = 0 deg/s in the second case are shown in Fig. 2. The free motion starts to oscillate and settles at a limit cycle. Thus it is the task of the controller to suppress these oscillations. Fig. 2. Free to roll motion of the wing rock for different release angles Now, including ts in the aˆi coefficients, the controlled wing rock model equation with dimensional derivatives can be written as follows
φ (t ) + a0φ (t ) + a1φ (t ) + a2 φ (t ) φ (t ) + a3φ 3 (t ) + a4φ 2 (t ) φ (t ) = u
(3)
with a0 = aˆ0 t s2 , a1 = aˆ1 t s , a2 = aˆ2 , a3 = aˆ3 t s2 , a4 = aˆ4 t s . The equivalent control input u is designed to ensure wing rock oscillation suppression, and it is related to the controlling torque T by the following equation
u = −T I xx
(4)
where I xx is the inertia of the model, and T is the controlling torque. In this paper, it is assumed that the controlling torque is provided by three aileron segments as follows [5] T =
1 ρV 2Sb (Cl da1δ a1 + Cl da 2δ a 2 + Cl da 3δ a 3 ) 2
(5)
where ρ is the air density, S is the wing surface and Cl dai , i = 1, 2,3 are the derivatives of the roll moment coefficient with respect to the aileron’s deflection angles δai . T
Denoting x = [ x 1 , x 2 ] = ª¬φ , φ º¼ and u = [u 1 , u 2 , u 3 ] = [δ a1 , δ a 2 , δ a 3 ] , the state space model of the wing rock system with augmented actuation system can be written as follows T
T
T
x1 = x2 x2 = − a0 x1 − a1 x2 − a2 x2 x2 − a3 x12 − a4 x12 x2 + b1u1 + b2 u2 + b3u3
(6)
with bi = ρV 2SbCl dai 2I xx , i = 1, 2,3 . The state space model (6) can be put into the following compact form
x1 = x 2
(7)
x 2 = f ( x ) + g T ( x ) u with f ( x ) = −a0 x 1 − a1x 2 − a2 x 2 x 1 − a3 x 12 − a4 x 12 x 2 and g ( x ) = [b1 , b 2 , b 3 ]
T
. Taking y (t ) = φ (t ) , (7)
becomes
y = f ( x ) + g T ( x ) u
(8)
2.2. Actuator failure model The actuator failures considered in this work fall into two types; the first type is total loss of effectiveness i.e. at time instant t j , some inputs are no longer influenced by the issued control actions, this is represented by
u j ( t ) = u j (t ) , t ≥ t j
(9)
For some unknown t j ≥ 0 and j = 1, 2,3 , where
u j (t ) = u j 0 + ¦ l =j 1u jl δ jl (t ) n
(10)
for unknown scalar constants u jl and known scalar signals δ jl (t ) , n j is the number of δ jl (t ) in the j th actuator failure pattern. A special case of actuator failures is when u jl = 0, l = 1, , n j , this characterizes the lock in place failure types, i.e. at the time t j , the j th actuator is stuck at the position u j 0 . This type is common in practical systems such as flight control segments which may be stuck at unknown positions during flight. Remark 1: Notice that the failure value u j (t ) is unknown, (10) represents a possible approximation
(decomposition) of u j (t ) according to basis functions such as Fourier series decomposition, wavelet
decomposition, Laguerre polynomials, etc. However, in this decomposition, the basis functions are obviously known whereas the decomposition (approximation) coefficients are unknown since u j (t ) is assumed unknown. For further analysis, the actuator failure signals are expressed as
u j (t ) = β Tj ϖ j (t ) T
where β j = ªu j 0 , u j 1 , , u jn j º ∈ ¬ ¼
n j +1
(11) T
, and ϖ j (t ) = ª1, δ j 1 (t ) , , δ jn j (t ) º ∈ ¬ ¼
n j +1
.
The second type of actuator failures commonly encountered in practical control systems is partial loss of effectiveness or actuator fading. In this case, the input is still partially influenced by the control action. Mathematically, this can be described by
u j (t ) = ρ j (t )v j (t ) , 0 < ρ j (t ) ≤ 1, t ≥ t 0
(12)
In the presence of type (9) and/or type (12) failures, the control vector applied to the system will be given by
u (t ) = ρ (t )( I − σ )v (t ) + σ u (t )
(13)
where ρ (t ) = diag { ρ1 , ρ2 , ρ3 } , σ = diag {σ 1 , σ 2 , σ 3 } , with σ j = 1 if the jth actuator fails as (9) and σ j = 0 otherwise. The control objective is to design an actuator failure estimation and compensation control scheme for the wing rock system (1) so that, in addition to system model uncertainties, in the presence of unknown type (9) and/or (12) actuator failures (in terms of pattern, time, and value), the output of the system tracks a reference signal y d (t ) with known derivatives up to order n, while keeping all closed-loop system signals bounded. In other words, the roll angle and roll rate should be forced to track predefined trajectories. For actuator failure compensation control design, the following assumption is imposed on the system [5]: Assumption 1: The wing rock system (8) is designed so that, for any p 2 actuators failing as (9), 0 ≤ p 2 ≤ 2 and any p1 actuators failing as (12), 0 ≤ p 2 ≤ 3 , the remaining effective part of the actuators can still achieve the control objective with acceptable performance levels (the system remains controllable). Remark 2: Assumption 1 assumes that all actuators are ideal, i.e. even with one remaining actuator; which is partially effective (worst case), the system can be controlled with this actuator alone. Remark 3: Assumption 1 implies that there are actuators that can perform the same task, i.e. there exists a kind of actuation redundancy within the system, such that if one or more actuators fail as (9), the other healthy actuators must compensate for its effect on the system. It is worth to note here that redundancy is inherent to the system, in other words, it is part of the problem and not part of the controller design task. Our aim is thus to design a controller that manages this redundancy in an efficient way. Regarding this redundant structure of the system, one can think of an equal or proportional actuation scheme [5,21]. For the latter case, the control signal vector is given by
v (t ) = b ( x )v 0 (t )
(14)
where v 0 (t ) is a scalar control signal to be designed and b ( x ) = ª¬b1 ( x ) , b 2 ( x ) , b3 ( x ) º¼
T
is a vector of
coefficients describing the contribution of each actuator. Remark 4: Such an actuation scheme is reasonable, for instance, segmented actuators in an aircraft (aileron or rudder) can be controlled by one control signal which is distributed among the different control segments. Remark 5: From the controller design point of view, the actuation scheme (14) simplifies considerably the controller design task. Indeed, this scheme brings the controller design problem for a faulty MISO system to that of a SISO system with the only difference that system functions may not be continuous due to actuator failures. This will be discussed later in section 3. 3.
Linear adaptive actuator failure compensation controller design
3.1. Preliminary analysis With actuation scheme (14), in the presence of type (9) and/or (12) actuator failures, (8) becomes: y ( 2) = f ( x ) + g T ( x ) σ u + g T ( x ) ρ ( I − σ )v (t )
Using (11) and (14), (15) can be written as
(15)
y ( 2 ) = f ( x ) + g T ( x ) ρ ( I − σ ) b ( x )v 0 + ¦ j =1 g j ( x ) σ j β Tj ϖ j (t ) 3
(16)
Let us denote g ( x ) = g T ( x ) ρ (t )( I − σ ) b ( x ) and κ *j = g j ( x ) σ j β j g ( x ) which is a time varying parameter, then (16) simplifies to y ( 2 ) = f ( x ) + g ( x )v 0 + g ( x ) ¦ j =1 κ *jT ϖ j (t ) 3
(17)
Equation (17) can be thought of as being a simple SISO nonlinear system with a disturbance like term for which a control signal v 0 (t ) is sought to force the output y (t ) to track the reference signal y d (t ) and ensure closedloop system stability. Let us define the output tracking error e (t ) as follows
e (t ) = y d ( t ) − y ( t )
(18)
The corresponding filtered tracking error s (t ) is given by
s (t ) = e (t ) + λe (t ) , λ >0
(19)
In (19), s (t ) = 0 represents a linear differential equation with constant coefficients whose solution implies that the tracking error and its derivative converge to zero [35]. Henceforth, the control objective will be the design of a controller that brings s (t ) to zero. Moreover, if s (t ) is bounded, the tracking error and its derivative will be also bounded, more precisely, if one has s (t ) ≤ Φ where Φ is a positive constant, it can be concluded that: e (t ) ≤ Φ λ and e (t ) ≤ 2Φ [35].
The time derivative of s (t ) along system trajectories is given by s = v − f ( x ) − g ( x )v 0 − g ( x ) ¦ j =1 κ *jT ϖ j (t ) 3
(20)
with v = y d(2) + λe . In the case where both the system parameters and actuator failures are known, f ( x ) , g ( x ) and κ *j , j = 1, 2,3 will be also known. The control objectives can be met through the following ideal control law [36,37] v 0* =
1 ( −f ( x ) + v + ks + k 0 tanh ( s ε 0 ) ) − ¦ 3j =1κ *jT ϖ j (t ) g (x )
(21)
where k > 0 , k 0 > 0 and ε 0 is a small positive constant. In fact, with the ideal control (21), the time derivative of
s (t ) along system trajectories simplifies to s = −ks − k 0 tanh ( s ε 0 )
(22)
From (22), it can be easily checked that s (t ) and consequently, e (t ) and e (t ) are bounded and converge to zero as time t tends to infinity. The ideal control law (21) can split into two terms as: v 0* = v *p + v f* , where v *p and v f* are given by v *p =
1 ( −f ( x ) + v + ks + k 0 tanh ( s ε 0 ) ) g (x )
(23)
v f* = ¦ j =1v f , j = − ¦ j =1 κ *T j ϖ j (t ) 3
3
(24)
The term v *p characterizes the control law that accounts for effective actuators (partially or completely effective) while the term v f* compensates for actuators that have totally lost their effectiveness. The problem is that f ( x ) , g ( x ) and the coefficients κ *j , j = 1, 2,3 are unknown. Therefore, the control law (21) cannot be implemented. In the following, an adaptive version of the controller (21) is proposed. 3.2. Adaptive controller design In this subsection, an adaptive version of the ideal controller (21) is developed. For this, the plant controller is approximated by a linear controller while the parameters κ *j of v f* are estimated online. Assume that the term
v *p can be approximated as follows [37] v p = θ T E (t )
(25)
where E (t ) = [1, e , e ] is the error vector, and θ = [θ1 , θ 2 , θ 3 ] . Let us denote θ * the optimal value of θ , then T
T
v 0* = θ *T E (t ) − ¦ j =1 κ *jT ϖ j (t ) + ε ( E ) 3
(26)
where ε ( E ) is the optimal approximation error for the term v p . The adaptive version of (26) is v 0 = θ T E (t ) − ¦ j =1 κ Tj ϖ j (t ) 3
(27)
Before proceeding, the following assumption on the approximation error ε ( E ) is imposed.
Assumption 2: It is assumed that for all actuator failures, the error ε ( E ) in (26) is bounded as follows
ε 2 ( E ) ≤ ε 0 s 2 + ε1
(28)
where ε 0 and ε1 are positive constants. In the following, update laws for the controller parameters θ and κ j , j = 1, 2,3 are designed. Let us define the following control errors ev p = v *p − θ T E (t ) = θT E (t ) + ε ( E )
(29)
ev f , j = −κTj ϖ j (t ) , j = 1, 2,3
(30)
with θ = θ * − θ and κ j = κ *j − κ j , j = 1, 2,3 . The prediction errors ev p , ev f , j , j = 1, 2,3 respectively measure the discrepancy between the unknown functions v *p , v f* , j , j = 1, 2,3 and the control inputs v p , v f , j , j = 1, 2,3 . By adding and subtracting g ( x )v *p to (20) and using (21) we obtain
(
s = − ks − k 0 tanh ( s ε 0 ) + g ( x ) ev p + ¦ j =1ev f , j 3
)
(31)
Now, consider the following quadratic cost function of the control error J (θ , κ j ) =
(
1 3 g ( x ) ev p + ¦ j =1ev f , j 2
)
2
=
(
1 3 g ( x ) v *p − θ T E (t ) − ¦ j =1 κ j ϖ j (t ) 2
)
2
(32)
Using the gradient descent method, the update laws for θ and κ j that minimize this cost function are given by
θ = −η1∇θ J (θ , κ )
(33)
κ j = −η 2 ∇κ J (θ , κ ) , j = 1, 2,3
(34)
j
(
where η1 and η2 are positive constants. From (32), we have ∇θ J (θ , κ ) = − g ( x ) E (t ) ev p + ¦ j =1ev f , j
(
∇κ j J (θ , κ ) = g ( x )ϖ j (t ) ev p + ¦ j =1ev f , j 3
) hence (33) and (34) can be written respectively as (
θ = η1 g ( x ) E (t ) ev + ¦ j =1ev
(
3
p
κ j = −η 2 g ( x )ϖ j (t ) ev + ¦ j =1ev
(
In (35) and (36) the term g ( x ) ev p + ¦ j =1ev f , j 3
(
3
p
f ,j
f ,j
)
) , j = 1, 2, 3
3
),
(35) (36)
) is not available. However, from (31), it can be pulled out as )
g ( x ) ev p + ¦ j =1ev f , j = s + ks + k 0 tanh ( s ε 0 ) 3
(37)
Substituting (37) into (35) and (36) yields
θ = η1 E (t ) ( s + ks + k 0 tanh ( s ε 0 ) )
(38)
κ j = −η 2ϖ j (t ) ( s + ks + k 0 tanh ( s ε 0 ) ) , j = 1, 2, 3
(39)
To ensure the boundedness of the parameters vector θ and to improve the robustness of the adaptive laws (38) and (39) in the presence of approximation errors, a σ − modification term is introduced as follows [38]
θ = η1 E (t ) ( s + ks + k 0 tanh ( s ε 0 ) ) − η1σθ
(40)
κ j = −η 2ϖ j (t ) ( s + ks + k 0 tanh ( s ε 0 ) ) − η 2σκ j , j = 1, 2, 3
(41)
where σ is a small positive constant. The adaptive laws are modified so that the derivative of the Lyapunov function used to analyze this adaptive law becomes negative in the space of the parameter estimates when these parameters exceed certain bound [38]. 3.3. Stability and tracking analysis Suppose that one or more actuators fail at time instants t i , i = 1,..., m 0 , and at time intervals t ∈ [t i , t i +1 ) there are p1 ,0 ≤ p1 < 3 actuators fail as (9) and p 2 , 0 ≤ p 2 ≤ 3 fail as (12). In order to analyze the tracking error convergence and the stability of the closed-loop system, let us consider the following Lyapunov-like function defined in a piecewise fashion over the time intervals t ∈ [t i , t i +1 ) with fixed failure pattern Vi =
1 2 1 T 1 s + θ θ+ 2 2η1 2η 2
3
¦ κ
T j
j =1
The time derivative of V i over the time interval t ∈ [t i , t i +1 ) is given as
κ j
(42)
1 1 1 Vi = ss − θT θ + θT θ* −
η1
η1
η2
3
¦ κ
T j
j =1
κ j +
3
1
η2
¦ κ
κ *j
T j
j =1
(43)
Using (31), (40) and (41), (43) can be written as follows
(
)
(
Vi = s − ks − k 0 tanh ( s ε 0 ) + g ( x ) ev 0 − θT Eg ( x ) ev 0 − σθ +
1 T * 1 θ θ + ¦ κTj ϖ j (t ) g ( x ) ev 0 + σκ j + 3
η1
j =1
(
)
3
η2
¦ κ
)
(44)
κ *j
T j
j =1
where ev 0 = ev p + ¦ j =1ev f , j . By virtue of (29) and (30), (43) can be re-written as follows 3
(
)
Vi = − ks 2 − k 0 s tanh ( s ε 0 ) + sg ( x ) ev 0 − ev p − ε ( E ) g ( x ) ev 0 1 1 + σθT θ + θT θ* − ¦ ev f , j g ( x ) ev 0 + σ ¦ κTj κ j +
η1
3
3
j =1
j =1
3
η2
¦ κ
T j
j =1
(45)
κ *j
Given that ev 0 = ev p + ¦ j =1ev f , j , (45) can be written as 3
Vi = −ks 2 − k 0 s tanh ( s ε 0 ) + sg ( x ) ev 0 − g ( x ) ev20 + ε ( E ) g ( x ) ev 0 + σθT θ +
1 T * θ θ +σ
η1
3
¦ κ
T j
κj +
j =1
1
η2
3
¦ κ
T j
j =1
(46)
κ *j
By virtue of assumption 2 and using the following inequalities
σθT θ ≤ −
σ 2 σ * 2 σ θ + θ , σκTj κ j ≤ − κ j 2
2
2
2
2 1 T * σ 2 1 1 σ θ θ ≤ θ + 2 θ* , κTj κ *j ≤ κ j η1 4 η2 4 ση1
ε ( E ) g ( x ) ev ≤ 0
2
+ +
σ
2
κ *j
2 1
ση
2 2
κ *j
(47) 2
(48)
1 1 g ( x ) ev20 + g ( x ) ε 2 ( E ) ≤ g ( x ) ev20 + g ( x ) ( ε 0 s 2 + ε1 ) 4 4
sg ( x ) ev 0 ≤
(49)
1 g ( x ) ev20 + g ( x ) s 2 4
(50)
Equation (46) can be bounded as follows 2 1 σ σ 3 Vi ≤ − g ( x ) ev20 − k 0 s tanh ( s ε 0 ) − ( k − g ( x )(1 + ε 0 ) ) s 2 − θ − ¦ j =1 κ j 2 4 4 σ * 2 1 * 2 σ 1 3 3 * 2 * 2 + θ + θ + ¦ j =1 κ j + 2 ¦ j =1 κ j + g ( x ) ε1 2 ση12 2 ση 2
2
(51)
Since the parameter vectors θ * , κ *j , j = 1, 2,3 and their derivatives θ* , κ *j , j = 1, 2,3 along with g ( x ) are assumed bounded for a fixed failure pattern. A positive bound ψ i can be defined for each fixed failure pattern as follows
§σ * θ ©2
ψ i = sup ¨ t ∈[t i ,t i +1 )
Then (51) simplifies to
2
+
1
ση12
2
θ* +
σ 2
¦
3 j =1
2
κ *j +
1
ση22
¦
3 j =1
κ *j
2
· + g ( x ) ε1 ¸ ¹
(52)
2 1 σ σ 3 Vi ≤ − g ( x ) ev20 − k 0 s tanh ( s ε 0 ) − ( k − (1 + ε 0 ) g ( x ) ) s 2 − θ − ¦ j =1 κ j 2 4 4
Assuming that the design parameter k
2
+ψ i
(53)
is chosen such that k > (1 + ε 0 ) δ1 , δ1 > g ( x ) , and define
γ = min ( 2 × ( k − (1 + ε 0 )δ1 ) , 0.5ση1 , 0.5ση 2 ) , then (53) can be further simplified as 1 γ γ 2 γ θ − Vi ≤ − g ( x ) ev20 − k 0 s tanh ( s ε 0 ) − s 2 − 2 2 2η1 2η 2
¦
3 j =1
κ j
2
+ψ i
(54)
From the definition of V i in (42), we can write Vi ≤ −γV i + ψ i
(55)
Now we can prove the following theorem that shows the boundedness of all variables in the closed loop system. Theorem 1: Consider the wing rock system (8) subject to type (9) and (12) actuator failures, using the actuation scheme (14), the adaptive controller (27) with parameter update laws (40) and (41) guarantees that the closedloop system is uniformly ultimately bounded (UUB) stable and the output tracking error converges to a small neighborhood of the origin. Proof:
Equation (55) implies that, over each time interval t ∈ [t i , t i +1 ) , where the failure pattern is fixed, given that
V i ≥ ψ i γ , we have Vi < 0 . By integrating (55) over a fixed failure pattern interval t ∈ [t i , t i +1 ) , we obtain V i (t ) ≤V i (t i+ ) e −γ (t − ti ) +
ψi γ
(56)
Now, let us denote V (t ) the extension of V i over the whole time domain. Due to actuator failures
θ, κ j , j = 1, 2,3 and consequently V (t ) will exhibit finite jumps at each time instant t i , i = 1, 2, , N , with t N +1 = ∞ , i.e. there are no further actuator failures after t N . Let us also denote ΔV i the jump on V (t ) caused by jumps on θ, κ j , j = 1, 2,3 at time t i . t i− and t i+ are the time instants just before and after the occurrence of failure respectively. Hence, starting from t1 we can write V (t 1+ ) = V (t 1− ) + ΔV 1
(57)
And from (56) we have
V (t 1− ) ≤V (t 0 ) e
ψ1 γ
(58)
ψ1 + ΔV 1 γ
(59)
− γ (t1 −t 0 )
+
From (57) and (58) we can write
V (t 1+ ) ≤V (t 0 ) e
− γ (t1 −t 0 )
+
Likewise, we proceed for t 2 ,, t N , we end up by the following inequality
ψi N + ¦ ΔV i i =1 γ i =1 N
V (t N+ ) ≤V (t 0 ) + ¦
(60)
Now, after t N there are no further failures. By integrating (55) over the time interval t ∈ [t N , ∞ ) , we can write
V (t ) ≤V (t N+ ) e
− γ ( t −t N
)
+
ψ N +1 γ
(61)
By substituting (60) into (61) we obtain N N ψ ψ § · − γ t −t V (t ) ≤ ¨V (t 0 ) + ¦ i + ¦ ΔV i ¸ e ( N ) + N +1 γ γ i =1 i =1 © ¹
(62)
It can be concluded that s (t ) , θ (t ) , κ j (t ) , j = 1, 2,3 and consequently u (t ) are bounded. Regarding (42) and (62), we conclude that, after time t N , where no further actuator failure will occur, we have N N 2ψ N +1 ψ § · −0.5γ (t −t N ) + s (t ) ≤ 2 ¨V (t 0 ) + ¦ i + ¦ ΔV i ¸e γ γ i =1 i =1 © ¹
(63)
Equation (63) implies that V (t ) is exponentially bounded and converges to a residual set defined by: s (t ) ≤ 2ψ N +1 γ . This means that e and e converge to residual sets defined as e (t ) ≤ 2ψ N +1 γ λ ,
e (t ) ≤ 2 2ψ N +1 γ . These sets can be made smaller by an adequate choice of the design parameters λ , σ ,η0 .
4.
Simulation results and discussion
To assess the effectiveness and performance of the proposed actuator failure compensation control strategy, a comparative simulation is conducted on the analytical wind rock model. This model is based on parameter identification of wind tunnel experimental data [29,30]. The triangular wing shape configuration is considered with a wing span b = 0.169m , root chord c r = 0.479m , sweep Λ = 80° , airspeed V = 30 m/s(Re = 950000) and angle of attack α = 32.5° . The other parameters of the model are computed from the physical variables of the wing, the parameters ai are computed from aˆi and t s . Thus we obtain a0 = 922.6568 , a1 = −11.0201 ,
a2 = 0.5388 , a3 = −785.2666 , a4 = 14.8722 , for the parameters bi they are computed using the equation, bi = ρV 2SbCl dai 2I xx , i = 1, 2,3 which gives b1 = −149.1759, b2 = −111.8819, b3 = −111.8819 .
The control
objective is to suppress the roll angle and rate oscillations, i.e. the output should track the reference y d (t ) = 0° . First, a proportional damper is considered. It is commonly used in aircraft stability augmentation systems (SAS), besides it has a very simple form, the segments deflection of the damper are proportional to the roll rate [30]. The three inputs are then given as: δ ai = k i t s φ , i = 1, 2, 3 . In this controller, the ailerons deflection angles are considered proportional to the roll rate δ ai = k i t s φ , where t s is introduced for dimensional reasons [30]. Taking
k i = 10, i = 1, 2,3 , the simulation is carried out with and without failures. Second, the model based adaptive actuator failure compensation strategy proposed in [5] is considered The control strategy uses a separate actuation scheme among actuators. It is primarily designed to compensate for the very particular case of lock in place (stuck) actuator failures. The control law aims to track a desired output generated from a given reference model. The entries of the control vector are given by: v j = (1 g j ( x
) ) π j ω (t ) , j
T
= 1, 2, 3 , where ω (t ) = ª¬ x T , y d − f ( x ) , g T ( x ) º¼
, and π j is a parameter vector
whose update law is defined as follows: π j = −e PB m Γ j ω (t ) , j = 1, 2,3 , for more on this strategy refer to [5]. T
The different design parameters are chosen as follows: the reference model is
( A m , B ,C )
with:
A m = [ 0 1, −16 −8] , B = [ 0 1] ,C = [1 0 ] , Γ j = 50 I 2 , P = [ 25 20 , 20 20 ] , the initial values of the T
parameter vectors π j , j = 1, 2,3 are chosen all zero.
Third, we consider the application of the proposed linear adaptive actuator failure compensation controller. The design parameters of the controller are chosen as: λ = 5,η1 = η2 = 20, ε 0 = 0.005, σ = 0.1, k = 20, k 0 = 10 . The initial values of the controller parameters θ and κ j , j = 1, 2,3 are taken zero. The actuation proportional gains are chosen b1 ( x ) = 2, b2 ( x ) = 2, b3 ( x ) = 1 . Consider first he failure free case, the simulation is carried out for 10s, which seems to be sufficient regarding the fast dynamics of the wind tunnel based model campared to real one. Fig. 3 shows the roll angle and roll rate evolution for the three controllers, i.e, the proportional roll damper, the model based adaptive controller and the proposed model-free adaptive controller, Fig. 4 shows the different control signals. Simulaiton results show that the three controllers can suppress the oscillations when all the actuators are healthy. From the simulaiton results, it can be shown that for the chosen design parameters, the proposed controller ensures a fast transient response compared to the adaptive compensation controller and the proportional roll damper, which presents some transient oscillation. The control efforts for the controllers are shown in Fig. 4. It can be seen that the proposed control shows less peaks compared to the proportional damper and the adaptive controller.
Fig. 3. Roll angle and roll rate (no failures) Fig. 4. Control signals and parameters (no failures)
Now, in order to simulate the largest possible failure cases, we consider the following failure scenarios: Failure scenario 1: With initial release angle and rate φ ( 0 ) = 1°, φ ( 0 ) = 0°/s respectively, assume that the first segment remains intact i.e. u1 ( t ) = v1 ( t ) . At time t = 3 s , the second segment is only 50% effective, i.e. u 2 ( t ) = 0.5 v 2 ( t ) . At the time
t = 6 s , the third segment is stuck at position u 3 ( t ) = u 3 = 1 5 .The roll angle and
roll rate time histories are shown in Fig. 5, the control signals and parameters are shown in Fig. 6. It can be seen that, the three controllers compensate the partial loss of effectiveness. However, when the third actuator is locked in place at time 6s, the proportional damper is unable to compensate for the failure and a roll angle offset appears. This can be checked from the control signal, we see that the control signals from the damper remained passive and did not react to the failure. On the other hand, we see that both the model-based and the proposed linear adaptive controllers were able to compensate the lock in place failure, and the controls were reconfigured to account for the failures.
Fig. 5. Roll angle and roll rate for the three controllers (scenario 1) Fig. 6. Control vectors and parameters (scenario 1) Failure scenario 2: With initial release angle and rate
φ ( 0) = 1°,φ ( 0) = 0°/s
respectively, assume that the first
segment remains intact i.e. u1 (t ) = v 1 (t ) . At time t = 3s , the second segment is only 60% effective, i.e. u 2 ( t ) = 0.6 v 2 ( t ) . At the time
t = 5 s , the third segment becomes uncontrollable and starts oscillating according
to the pattern u 3 (t ) = 10 − 30sin ( 5t ) + 20 cos ( 5t ) . The simulation results are shown in Figs. 7 and 8. The roll angle and roll rate curves show that, neither the proportional damper nor the adaptive controller can compensate for the time varying failure at time t = 5 s , oscillations in both cases are present. However, the proposed controller was effective in dealing with the failures, the roll angle and rate where kept very close to zero. Fig. 7. Roll angle and roll rate for the three controllers (scenario 2) Fig. 8. Control signals and parameters (scenario 2)
Failure scenario 3: With initial release angle and rate
φ ( 0) = 1°,φ( 0) = 0°/s respectively, we assume that the
first segment remains intact, i.e. u1 ( t ) = v1 ( t ) . At time t = 3s , the third segment is stuck at a value u 3 ( t ) = 10 . At time t = 6s , the second segment is only 30% effective, i.e. u 2 ( t ) = 0.3v 2 ( t ) , and at time t = 8s , the third segment recovers from its failure, i.e. u 3 ( t ) = v 3 ( t ) . Simulation results are shown in Figs. 9 and 10, it can be seen that when an actuator recovers from a failure it will be exploited automatically by the control scheme to control the system except for the proportional damper that is passive to such faults, the controller do not react. Fig. 9. Roll angle and roll rate (scenario 3) Fig. 10. Control signals and parameters (scenario 3) Failure scenario 4: Consider the worst-case failure scenario, with initial release angle and rate
φ ( 0) =10°,φ( 0) = 0°/s
respectively. Assume that starting from t = 5s , the first segment is only 70% effective.
At time instant t = 4 s the second segment locks at a value u 2 (t ) = 20 and then recovers at time t = 8s . For the third segment, we assume that at time t > 3s , it starts oscillating according to the pattern u 3 ( t ) = 1 − 30 sin ( 5 t ) + 20 cos ( 5 t ) . The simulation results are shown in Figs. 11 and 12, it can be seen that the
proportional damper was unable to compensate for actuator failures and stronger roll and rate oscillations appear. When the second actuator recovers from the failure, oscillations of smaller amplitude persist. The adaptive model based controller can barely keep the roll angle at zero in the presence of such failures, while the proposed controller shows superiority in dealing with such failures, the roll rate and angle were regulated around the origin. Once again, we see that the control effort is intelligently redistributed among healthy actuators if an actuator fails. Besides, if an actuator recovers from a failure, it is automatically exploited to control the system. To summarize, the proportional damper is passive to actuator failures. It can compensate for the partial loss of effectiveness seamlessly, but it is unable to react against the total loss of effectiveness. Roll offsets and roll oscillations were present which despite the smaller value compared to the free to roll case they can be annoying to the pilot and limit the performance of such controller. Another drawback of this controller is that it depends on the wing span b and the air speed V . This may limit its practical implementation if b and V were uncertain due to measurement errors or due to changes of the wingspan due to partial damage of the wing [39]. On the other hand, the adaptive model based controller given in [5] can compensate for partial loss of effectiveness and lock in place failures, but it cannot compensate for a time varying failure. Besides, it is a model based design, i.e. exact model of the system is needed to implement the controller. This is difficult to obtain in real flight as the wing rock motion is a complex multi-degrees of freedom system with uncertainties and perturbations. Faced to these limitations, the proposed strategy holds as a good remedy, it can compensate for the largest cases of actuator failures including the time-varying failures, and it does not need exact system model to be implemented which makes it more suitable for real flight applications. Fig. 11. Roll angle and roll rate (scenario 4) Fig. 12. Control signals and parameters (scenario 4) Remark 6: An attractive feature of the proposed controller is that it is a model-free design. That is, it does not depend upon exact model of the system. This is interesting given that the wind rock system studied in this paper is based on wind tunnel experiments, where a 10s simulation time interval was adequate to study the dynamics and response of the system. In real flight condition the time dynamics are very large of the order of 1000 s [28,31]. Besides, the real phenomenon is a multi-degrees of freedom phenomenon, it is also subject to the effect of aerodynamic asymmetries, coupling and wing-body vortex interactions which make it more complex and hard to model. Even with the present model with the identified parameters, there is still an error margin in the parameters a i or bi which can be affected by changing physical parameters during flight. Given that little
information on the system is required and only the system functions f ( x ) and g ( x ) are different from the real
flight models, the proposed control design will be applicable for real flight operation. Another feature of the proposed design is that it is versatile and not limited to the present case of wind tunnel based wing rock system, in fact, any high order nonlinear system having the form of equation (17) and satisfying the stated assumptions can be effectively controlled via the proposed actuator failure compensation design. 5.
Conclusion
A model-free adaptive controller design was developed for wing rock motion control in the presence of system uncertainties and uncertain actuator failures. The control strategy was built around a self-tunning controller, which is based on the prediction error, and a failure compensation controller that accounts for actuator failures. This alleviates the need for a fault detection and isolation algorithm (FDI) which makes it more efficient and suitable for real time implementation. The proposed controller requires minimum information about the system and the failures. This makes it more attractive as real flight conditions are subject to uncertainties and modeling errors. A rigorous proof of tracking and stability requirements was also presented. The simulation results show the feasibility of the proposed control strategy and its superiority compared to a proportional roll damper and an established model based design, which cannot deal with such a large set of actuator failures. As a future work, we propose the extension of the study to multivariable systems. Besides, we can also focus on seeking more relaxed design conditions and enlarging the set of compensable actuator failures. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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List of figures
Fig. 1. Actuator, plant and sensor faults
Fig. 2. Free to roll motion of the wing rock for different release angles
Fig. 3. Roll angle and roll rate (no failures)
Fig. 4. Control signals and parameters (no failures)
Fig. 5. Roll angle and roll rate for the three controllers (scenario 1)
Fig. 6. Control vectors and parameters (scenario 1)
Fig. 7. Roll angle and roll rate for the three controllers (scenario 2)
Fig. 8. Control signals and parameters (scenario 2)
Fig. 9. Roll angle and roll rate (scenario 3)
Fig. 10. Control signals and parameters (scenario 3)
Fig. 11. Roll angle and roll rate (scenario 4)
Fig. 12. Control signals and parameters (scenario 4)
Sabri Boulouma, Salim Labiod, Hamid Boubertakh
PII: DOI: Reference:
S1270-9638(16)30989-0 http://dx.doi.org/10.1016/j.ast.2017.03.025 AESCTE 3964
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
2 November 2016 15 February 2017 17 March 2017
Please cite this article in press as: S. Boulouma et al., Linear adaptive actuator failure compensation for wing rock motion control, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.025
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Linear adaptive actuator failure compensation for wing rock motion control Sabri Bouloumaa,* Salim Labiodb and Hamid Boubertakhc LAJ, Faculty of Science and Technology, University of Jijel, Ouled Aissa, BP. 98, 18000, Jijel, Algeria. a [email protected], b [email protected], c [email protected] Abstract In this paper, a novel adaptive actuator failure compensation control strategy is developed for wing rock motion control in the presence of both system and actuator failure uncertainties. The proposed strategy can compensate for both total and partial loss of effectiveness. A proportional actuation scheme of redundant aileron segments is used. This allows bringing the faulty multi-input single-output nonlinear system into an equivalent perturbed single-input single-output system. Afterward, an adaptive actuator failure compensation control scheme is developed around a linear approximation of an ideal feedback linearization controller. A failure estimation and compensation term is appended to this controller to account for possible actuator failures. Closed-loop stability and tracking performance are proved based on Lyapunov theory and a piecewise analysis is also introduced to show that stability properties hold despite the presence of parameter jumps caused by abrupt actuator failures. Simulation results on a small scale wind tunnel based wing rock model with redundant actuators show the effectiveness of the proposed adaptive control strategy. Keywords: Wing rock; Adaptive control; Actuator failures; Redundant actuators; Nonlinear systems. 1.
Introduction
As a result of the increasing demands for performance, precision and higher maneuverability, control systems has become more complex. Thus, failures have become more common, especially in aircraft and spacecraft control systems [1,2]. Failures can take place at different locations within the control loop. They can appear at the plant itself, at the sensors or at the actuators as illustrated in Fig. 1. Actuator failures may lead to severe performance deterioration or even system instability. They can cause catastrophic accidents involving human lives and millions worth equipment if they are not handled properly using appropriate control designs [3]. Recently, actuator failure compensation has become an active area of research that has undergone remarkable progress in both theory and practice. Various actuator failure compensation control designs have been proposed in the research literature. These techniques generally fall into two categories, the passive approaches and the active approaches [3–5]. In the passive approaches, a fixed controller is designed by considering a set of predefined actuator failures. These approaches are generally based on the robust control theory and aim to optimize system’s performance under actuator failures using linear quadratic (LQ) control designs and H ∞ [6,7] or using convex optimization based on linear matrix inequalities (LMI) [8]. These techniques are limited in that they deal only with a restricted set of failures [4]. On the other hand, active actuator failure compensation approaches are designed to take action at the time of failure occurrence. Among these approaches one can find multiple model based designs [9–11], fault diagnosis and isolation (FDI) designs based residual generation or state observers [3,12], fuzzy logic and artificial neural networks techniques [13,14], and adaptive control based designs [2,5,15–27]. Adaptive control provides adaptation mechanisms to adjust controllers to cope with parametric, structural and external disturbances. The adaptive control approach has been extensively used to design controllers for systems with actuator failures. Many classical linear and nonlinear adaptive control techniques have been extended and customized to compensate for actuator failures [5]. A recent overview on adaptive actuator failure compensation was also presented in [15]. In [16,17], the authors proposed adaptive control schemes for linear redundant systems with lock in place failure types for state and output tracking. In [18,19], the authors proposed adaptive control schemes for multivariable nonlinear systems with unknown actuator failures using feedback linearization techniques. In [20,21], the authors proposed an adaptive backstepping control schemes for a class of parametric strict feedback systems with unknown actuator failures. In these works, system parameters as assumed to be known or partially known, i.e. defined as a product of known nonlinear functions with unknown
Corresponding author. E-mail address : [email protected]
parameters while the actuator failures are supposed unknown, in terms of time value and pattern. To the authors’ best knowledge, the case of unknown nonlinear systems with unknown actuator failures has been rarely considered except in some few works. For instance, in [22,24–26], an adaptive actuator failure compensation for uncertain nonlinear systems with uncertain actuator failures is designed based on the adaptive fuzzy and neural control techniques. In these works, the idea was to bring the unknown system with unknown actuator failures to an ordinary system with unknown parameters via an adequate actuation scheme, then existing adaptive fuzzy and neural control techniques can be applied to the system with consideration of possible parameter jumps caused by abrupt faults [21,24]. These techniques are effective but do not provide any information or indication on the failures, which is sometimes required. Besides the computational burden of such fuzzy and neural techniques may limit their practical implementation. Fig. 1. Actuator, plant and sensor faults From the system’s architecture viewpoint, one way to increase system’s resilience is through actuation redundancy, i.e. the use of multiple actuators that can fulfill the same task. These actuators can be similar in structure and task such as segmented rudder in an aircraft or can be different in structure but can fulfill the same task. In an aircraft, for instance, differential engine thrust can compensate for a failed rudder [5]. However, from the controller design viewpoint, actuator redundancy brings more challenges, as the controller should drive the actuators and ensure cooperation between them such that in case one or more actuators fail, the remaining healthy actuators react consequently to compensate for the effect of the failed actuators. This means that the designed controllers should manage this redundancy in an effective way. Wing rock is an oscillatory rolling motion of an aircraft which occurs at high angles of attack. This oscillatory motion can be seen in high-speed civil transport and combat aircraft. The main aerodynamic parameters of wing rock are (i) angle of attack (ii) angle of sweep (iii) leading edge extensions (iv) slender forebody. Therefore, aircrafts that are susceptible to the wing rock phenomenon are those containing these parameters: such as aircraft with highly swept wings operating with leading edge extensions. The motion is characterized by an increase in amplitude up to a limit cycle as shown in Fig. 2, the final state is usually stable and defined by large roll oscillations [28–31]. The control of wing rock is a relevant issue as high-speed civil transport and combat aircraft can encounter it in their flight envelope, seriously compromising handling qualities and maneuvering capabilities. Many nonlinear and adaptive control schemes have been proposed for wing rock suppression [28,30–32]. These designs are efficient regarding their ability to suppress wing rock oscillations and their robustness against system’s uncertainties. However, the aforementioned works dealt only with the failurefree case. To the authors’ knowledge, except for the work given in [5], where an adaptive backstepping actuator failure control design for wind rock motion has been proposed, the case of actuator failure compensation for wing rock system has not been dealt with. In this paper, a redundant wing rock model is considered, the nonlinear model is based on parameter identification of wind tunnel experimental data [29]. In addition, it is assumed that three actuators deliver the control input to the system. These actuators are susceptible to fail during flight. An integrated, model-free linear adaptive controller with a feedforward actuator failure estimation and compensation term is developed to suppress wing rock oscillations. The proposed controller deals with both system uncertainties and actuator failures. It is simple in terms of the computational burden, compared to other works based on fuzzy logic and neural networks such as in [21,22,24,25,33,34]. Besides, this scheme does not require fault detection and isolation (FDI) module, which makes it more effective for real-time implementation, as the FDI module requires some latent time for fault detection, which may affect real-time constraints. In the same time, this controller gives an online estimation of the failure as opposed to previous adaptive approximation based techniques [21,24], where actuator failures are blindly compensated without having any information on their nature or time of occurrence. This lack of information makes them limited in case of failures needing immediate intervention. To summarize, the main contributions of this work can be listed as follows: (i) a new controller that can estimate the failures and compensate for their effects is proposed. (ii) the class of actuator failures is extended to time varying patterns compared to works in [5,21]. (iii) Stability and tracking requirements are proved rigorously while taking into account parameter jumps caused by abrupt actuator failures. (iv) the application to the augmented wing rock model, where system parameters are assumed unknown, this problem is rarely considered in the literature and brings a new challenge to the design of flexible and reliable aircraft. The remaining of the paper is organized as follows, in section 2, the nonlinear model of a redundant wing rock system is described and the control problem is formulated. In section 3 the control design steps are outlined along with stability and tracking proofs. In
section 4 simulation results are presented and discussed. Conclusions and suggestions for future works are presented is section 5. 2.
System modeling and problem formulation
In this section, the state space model of a redundant wing rock system is derived and the wing rock phenomenon is described. Afterward, the actuator failure compensation control problem is formulated. 2.1. Wing rock dynamic model In most works, the study of wing rock motion is based on wind tunnel experimental investigations on 80° swept delta wing models. The experimental geometries have simple triangular geometries. These models present stable limit cycles and correctly reproduce the dominant effect of primary wing vortices. the mathematical models of 80° swept slender wings have been developed based on parameter identification of experimental data [29,30]. In fact, these wing-rock mathematical models are not complete roll dynamics for practical control design because aerodynamics at high angles of attack (AOA) is very complex. Some unmodeled uncertainties such as different configurations and sizes should be considered in these models. Furthermore, the disturbance is also an important factor since aircraft operate in uncertain environments. For this work, we consider the wing rock system studied in [29,30]. The non-dimensional differential equation (single degree of freedom roll dynamics) describing the free motion of the roll angle φ is given by
φ (t ) + aˆ0φ (t ) + aˆ1φ (t ) + aˆ2 φ (t ) φ (t ) + aˆ3φ 3 (t ) + aˆ4φ 2 (t ) φ (t ) = 0
(1)
where the coefficients aˆ0 , aˆ2 ,, aˆ4 are parameters related to the aircraft’s operating conditions (i.e. angle of attack, Reynold number, and wing characteristics). Introducing the reference time t s = b 2V , with V is the air speed and b is the wing span, equation (1) can be written as follows [30]
φ (t ) +
aˆ0 aˆ aˆ aˆ φ (t ) + 1 φ (t ) + aˆ2 φ (t ) φ (t ) + 23 φ 3 (t ) + 4 φ 2 (t ) φ (t ) = 0 ts ts t s2 ts
(2)
The wing rock model exhibits oscillation for a large span of angles of attack. To illustrate this situation, let us consider an analytical wing rock model based on parameter identification of wind tunnel based experimental data. The model has a wing span b = 0.169m , root chord c r = 0.479m , and sweep Λ = 80° , airspeed
V = 30m/s ( Re = 950000 ) , and angle of attack α = 32.5° , the corresponding coefficients are given as
aˆ0 = 0.00723 , aˆ1 = −0.03104 , aˆ2 = 0.53884 , aˆ3 = 0.00623 , aˆ4 = 0.04189 [30]. The free to roll motion for an initial roll angle φ ( 0 ) = 1° and roll rate φ (t ) = 0 deg/s in the first case and φ ( 0 ) = 10° , φ (t ) = 0 deg/s in the second case are shown in Fig. 2. The free motion starts to oscillate and settles at a limit cycle. Thus it is the task of the controller to suppress these oscillations. Fig. 2. Free to roll motion of the wing rock for different release angles Now, including ts in the aˆi coefficients, the controlled wing rock model equation with dimensional derivatives can be written as follows
φ (t ) + a0φ (t ) + a1φ (t ) + a2 φ (t ) φ (t ) + a3φ 3 (t ) + a4φ 2 (t ) φ (t ) = u
(3)
with a0 = aˆ0 t s2 , a1 = aˆ1 t s , a2 = aˆ2 , a3 = aˆ3 t s2 , a4 = aˆ4 t s . The equivalent control input u is designed to ensure wing rock oscillation suppression, and it is related to the controlling torque T by the following equation
u = −T I xx
(4)
where I xx is the inertia of the model, and T is the controlling torque. In this paper, it is assumed that the controlling torque is provided by three aileron segments as follows [5] T =
1 ρV 2Sb (Cl da1δ a1 + Cl da 2δ a 2 + Cl da 3δ a 3 ) 2
(5)
where ρ is the air density, S is the wing surface and Cl dai , i = 1, 2,3 are the derivatives of the roll moment coefficient with respect to the aileron’s deflection angles δai . T
Denoting x = [ x 1 , x 2 ] = ª¬φ , φ º¼ and u = [u 1 , u 2 , u 3 ] = [δ a1 , δ a 2 , δ a 3 ] , the state space model of the wing rock system with augmented actuation system can be written as follows T
T
T
x1 = x2 x2 = − a0 x1 − a1 x2 − a2 x2 x2 − a3 x12 − a4 x12 x2 + b1u1 + b2 u2 + b3u3
(6)
with bi = ρV 2SbCl dai 2I xx , i = 1, 2,3 . The state space model (6) can be put into the following compact form
x1 = x 2
(7)
x 2 = f ( x ) + g T ( x ) u with f ( x ) = −a0 x 1 − a1x 2 − a2 x 2 x 1 − a3 x 12 − a4 x 12 x 2 and g ( x ) = [b1 , b 2 , b 3 ]
T
. Taking y (t ) = φ (t ) , (7)
becomes
y = f ( x ) + g T ( x ) u
(8)
2.2. Actuator failure model The actuator failures considered in this work fall into two types; the first type is total loss of effectiveness i.e. at time instant t j , some inputs are no longer influenced by the issued control actions, this is represented by
u j ( t ) = u j (t ) , t ≥ t j
(9)
For some unknown t j ≥ 0 and j = 1, 2,3 , where
u j (t ) = u j 0 + ¦ l =j 1u jl δ jl (t ) n
(10)
for unknown scalar constants u jl and known scalar signals δ jl (t ) , n j is the number of δ jl (t ) in the j th actuator failure pattern. A special case of actuator failures is when u jl = 0, l = 1, , n j , this characterizes the lock in place failure types, i.e. at the time t j , the j th actuator is stuck at the position u j 0 . This type is common in practical systems such as flight control segments which may be stuck at unknown positions during flight. Remark 1: Notice that the failure value u j (t ) is unknown, (10) represents a possible approximation
(decomposition) of u j (t ) according to basis functions such as Fourier series decomposition, wavelet
decomposition, Laguerre polynomials, etc. However, in this decomposition, the basis functions are obviously known whereas the decomposition (approximation) coefficients are unknown since u j (t ) is assumed unknown. For further analysis, the actuator failure signals are expressed as
u j (t ) = β Tj ϖ j (t ) T
where β j = ªu j 0 , u j 1 , , u jn j º ∈ ¬ ¼
n j +1
(11) T
, and ϖ j (t ) = ª1, δ j 1 (t ) , , δ jn j (t ) º ∈ ¬ ¼
n j +1
.
The second type of actuator failures commonly encountered in practical control systems is partial loss of effectiveness or actuator fading. In this case, the input is still partially influenced by the control action. Mathematically, this can be described by
u j (t ) = ρ j (t )v j (t ) , 0 < ρ j (t ) ≤ 1, t ≥ t 0
(12)
In the presence of type (9) and/or type (12) failures, the control vector applied to the system will be given by
u (t ) = ρ (t )( I − σ )v (t ) + σ u (t )
(13)
where ρ (t ) = diag { ρ1 , ρ2 , ρ3 } , σ = diag {σ 1 , σ 2 , σ 3 } , with σ j = 1 if the jth actuator fails as (9) and σ j = 0 otherwise. The control objective is to design an actuator failure estimation and compensation control scheme for the wing rock system (1) so that, in addition to system model uncertainties, in the presence of unknown type (9) and/or (12) actuator failures (in terms of pattern, time, and value), the output of the system tracks a reference signal y d (t ) with known derivatives up to order n, while keeping all closed-loop system signals bounded. In other words, the roll angle and roll rate should be forced to track predefined trajectories. For actuator failure compensation control design, the following assumption is imposed on the system [5]: Assumption 1: The wing rock system (8) is designed so that, for any p 2 actuators failing as (9), 0 ≤ p 2 ≤ 2 and any p1 actuators failing as (12), 0 ≤ p 2 ≤ 3 , the remaining effective part of the actuators can still achieve the control objective with acceptable performance levels (the system remains controllable). Remark 2: Assumption 1 assumes that all actuators are ideal, i.e. even with one remaining actuator; which is partially effective (worst case), the system can be controlled with this actuator alone. Remark 3: Assumption 1 implies that there are actuators that can perform the same task, i.e. there exists a kind of actuation redundancy within the system, such that if one or more actuators fail as (9), the other healthy actuators must compensate for its effect on the system. It is worth to note here that redundancy is inherent to the system, in other words, it is part of the problem and not part of the controller design task. Our aim is thus to design a controller that manages this redundancy in an efficient way. Regarding this redundant structure of the system, one can think of an equal or proportional actuation scheme [5,21]. For the latter case, the control signal vector is given by
v (t ) = b ( x )v 0 (t )
(14)
where v 0 (t ) is a scalar control signal to be designed and b ( x ) = ª¬b1 ( x ) , b 2 ( x ) , b3 ( x ) º¼
T
is a vector of
coefficients describing the contribution of each actuator. Remark 4: Such an actuation scheme is reasonable, for instance, segmented actuators in an aircraft (aileron or rudder) can be controlled by one control signal which is distributed among the different control segments. Remark 5: From the controller design point of view, the actuation scheme (14) simplifies considerably the controller design task. Indeed, this scheme brings the controller design problem for a faulty MISO system to that of a SISO system with the only difference that system functions may not be continuous due to actuator failures. This will be discussed later in section 3. 3.
Linear adaptive actuator failure compensation controller design
3.1. Preliminary analysis With actuation scheme (14), in the presence of type (9) and/or (12) actuator failures, (8) becomes: y ( 2) = f ( x ) + g T ( x ) σ u + g T ( x ) ρ ( I − σ )v (t )
Using (11) and (14), (15) can be written as
(15)
y ( 2 ) = f ( x ) + g T ( x ) ρ ( I − σ ) b ( x )v 0 + ¦ j =1 g j ( x ) σ j β Tj ϖ j (t ) 3
(16)
Let us denote g ( x ) = g T ( x ) ρ (t )( I − σ ) b ( x ) and κ *j = g j ( x ) σ j β j g ( x ) which is a time varying parameter, then (16) simplifies to y ( 2 ) = f ( x ) + g ( x )v 0 + g ( x ) ¦ j =1 κ *jT ϖ j (t ) 3
(17)
Equation (17) can be thought of as being a simple SISO nonlinear system with a disturbance like term for which a control signal v 0 (t ) is sought to force the output y (t ) to track the reference signal y d (t ) and ensure closedloop system stability. Let us define the output tracking error e (t ) as follows
e (t ) = y d ( t ) − y ( t )
(18)
The corresponding filtered tracking error s (t ) is given by
s (t ) = e (t ) + λe (t ) , λ >0
(19)
In (19), s (t ) = 0 represents a linear differential equation with constant coefficients whose solution implies that the tracking error and its derivative converge to zero [35]. Henceforth, the control objective will be the design of a controller that brings s (t ) to zero. Moreover, if s (t ) is bounded, the tracking error and its derivative will be also bounded, more precisely, if one has s (t ) ≤ Φ where Φ is a positive constant, it can be concluded that: e (t ) ≤ Φ λ and e (t ) ≤ 2Φ [35].
The time derivative of s (t ) along system trajectories is given by s = v − f ( x ) − g ( x )v 0 − g ( x ) ¦ j =1 κ *jT ϖ j (t ) 3
(20)
with v = y d(2) + λe . In the case where both the system parameters and actuator failures are known, f ( x ) , g ( x ) and κ *j , j = 1, 2,3 will be also known. The control objectives can be met through the following ideal control law [36,37] v 0* =
1 ( −f ( x ) + v + ks + k 0 tanh ( s ε 0 ) ) − ¦ 3j =1κ *jT ϖ j (t ) g (x )
(21)
where k > 0 , k 0 > 0 and ε 0 is a small positive constant. In fact, with the ideal control (21), the time derivative of
s (t ) along system trajectories simplifies to s = −ks − k 0 tanh ( s ε 0 )
(22)
From (22), it can be easily checked that s (t ) and consequently, e (t ) and e (t ) are bounded and converge to zero as time t tends to infinity. The ideal control law (21) can split into two terms as: v 0* = v *p + v f* , where v *p and v f* are given by v *p =
1 ( −f ( x ) + v + ks + k 0 tanh ( s ε 0 ) ) g (x )
(23)
v f* = ¦ j =1v f , j = − ¦ j =1 κ *T j ϖ j (t ) 3
3
(24)
The term v *p characterizes the control law that accounts for effective actuators (partially or completely effective) while the term v f* compensates for actuators that have totally lost their effectiveness. The problem is that f ( x ) , g ( x ) and the coefficients κ *j , j = 1, 2,3 are unknown. Therefore, the control law (21) cannot be implemented. In the following, an adaptive version of the controller (21) is proposed. 3.2. Adaptive controller design In this subsection, an adaptive version of the ideal controller (21) is developed. For this, the plant controller is approximated by a linear controller while the parameters κ *j of v f* are estimated online. Assume that the term
v *p can be approximated as follows [37] v p = θ T E (t )
(25)
where E (t ) = [1, e , e ] is the error vector, and θ = [θ1 , θ 2 , θ 3 ] . Let us denote θ * the optimal value of θ , then T
T
v 0* = θ *T E (t ) − ¦ j =1 κ *jT ϖ j (t ) + ε ( E ) 3
(26)
where ε ( E ) is the optimal approximation error for the term v p . The adaptive version of (26) is v 0 = θ T E (t ) − ¦ j =1 κ Tj ϖ j (t ) 3
(27)
Before proceeding, the following assumption on the approximation error ε ( E ) is imposed.
Assumption 2: It is assumed that for all actuator failures, the error ε ( E ) in (26) is bounded as follows
ε 2 ( E ) ≤ ε 0 s 2 + ε1
(28)
where ε 0 and ε1 are positive constants. In the following, update laws for the controller parameters θ and κ j , j = 1, 2,3 are designed. Let us define the following control errors ev p = v *p − θ T E (t ) = θT E (t ) + ε ( E )
(29)
ev f , j = −κTj ϖ j (t ) , j = 1, 2,3
(30)
with θ = θ * − θ and κ j = κ *j − κ j , j = 1, 2,3 . The prediction errors ev p , ev f , j , j = 1, 2,3 respectively measure the discrepancy between the unknown functions v *p , v f* , j , j = 1, 2,3 and the control inputs v p , v f , j , j = 1, 2,3 . By adding and subtracting g ( x )v *p to (20) and using (21) we obtain
(
s = − ks − k 0 tanh ( s ε 0 ) + g ( x ) ev p + ¦ j =1ev f , j 3
)
(31)
Now, consider the following quadratic cost function of the control error J (θ , κ j ) =
(
1 3 g ( x ) ev p + ¦ j =1ev f , j 2
)
2
=
(
1 3 g ( x ) v *p − θ T E (t ) − ¦ j =1 κ j ϖ j (t ) 2
)
2
(32)
Using the gradient descent method, the update laws for θ and κ j that minimize this cost function are given by
θ = −η1∇θ J (θ , κ )
(33)
κ j = −η 2 ∇κ J (θ , κ ) , j = 1, 2,3
(34)
j
(
where η1 and η2 are positive constants. From (32), we have ∇θ J (θ , κ ) = − g ( x ) E (t ) ev p + ¦ j =1ev f , j
(
∇κ j J (θ , κ ) = g ( x )ϖ j (t ) ev p + ¦ j =1ev f , j 3
) hence (33) and (34) can be written respectively as (
θ = η1 g ( x ) E (t ) ev + ¦ j =1ev
(
3
p
κ j = −η 2 g ( x )ϖ j (t ) ev + ¦ j =1ev
(
In (35) and (36) the term g ( x ) ev p + ¦ j =1ev f , j 3
(
3
p
f ,j
f ,j
)
) , j = 1, 2, 3
3
),
(35) (36)
) is not available. However, from (31), it can be pulled out as )
g ( x ) ev p + ¦ j =1ev f , j = s + ks + k 0 tanh ( s ε 0 ) 3
(37)
Substituting (37) into (35) and (36) yields
θ = η1 E (t ) ( s + ks + k 0 tanh ( s ε 0 ) )
(38)
κ j = −η 2ϖ j (t ) ( s + ks + k 0 tanh ( s ε 0 ) ) , j = 1, 2, 3
(39)
To ensure the boundedness of the parameters vector θ and to improve the robustness of the adaptive laws (38) and (39) in the presence of approximation errors, a σ − modification term is introduced as follows [38]
θ = η1 E (t ) ( s + ks + k 0 tanh ( s ε 0 ) ) − η1σθ
(40)
κ j = −η 2ϖ j (t ) ( s + ks + k 0 tanh ( s ε 0 ) ) − η 2σκ j , j = 1, 2, 3
(41)
where σ is a small positive constant. The adaptive laws are modified so that the derivative of the Lyapunov function used to analyze this adaptive law becomes negative in the space of the parameter estimates when these parameters exceed certain bound [38]. 3.3. Stability and tracking analysis Suppose that one or more actuators fail at time instants t i , i = 1,..., m 0 , and at time intervals t ∈ [t i , t i +1 ) there are p1 ,0 ≤ p1 < 3 actuators fail as (9) and p 2 , 0 ≤ p 2 ≤ 3 fail as (12). In order to analyze the tracking error convergence and the stability of the closed-loop system, let us consider the following Lyapunov-like function defined in a piecewise fashion over the time intervals t ∈ [t i , t i +1 ) with fixed failure pattern Vi =
1 2 1 T 1 s + θ θ+ 2 2η1 2η 2
3
¦ κ
T j
j =1
The time derivative of V i over the time interval t ∈ [t i , t i +1 ) is given as
κ j
(42)
1 1 1 Vi = ss − θT θ + θT θ* −
η1
η1
η2
3
¦ κ
T j
j =1
κ j +
3
1
η2
¦ κ
κ *j
T j
j =1
(43)
Using (31), (40) and (41), (43) can be written as follows
(
)
(
Vi = s − ks − k 0 tanh ( s ε 0 ) + g ( x ) ev 0 − θT Eg ( x ) ev 0 − σθ +
1 T * 1 θ θ + ¦ κTj ϖ j (t ) g ( x ) ev 0 + σκ j + 3
η1
j =1
(
)
3
η2
¦ κ
)
(44)
κ *j
T j
j =1
where ev 0 = ev p + ¦ j =1ev f , j . By virtue of (29) and (30), (43) can be re-written as follows 3
(
)
Vi = − ks 2 − k 0 s tanh ( s ε 0 ) + sg ( x ) ev 0 − ev p − ε ( E ) g ( x ) ev 0 1 1 + σθT θ + θT θ* − ¦ ev f , j g ( x ) ev 0 + σ ¦ κTj κ j +
η1
3
3
j =1
j =1
3
η2
¦ κ
T j
j =1
(45)
κ *j
Given that ev 0 = ev p + ¦ j =1ev f , j , (45) can be written as 3
Vi = −ks 2 − k 0 s tanh ( s ε 0 ) + sg ( x ) ev 0 − g ( x ) ev20 + ε ( E ) g ( x ) ev 0 + σθT θ +
1 T * θ θ +σ
η1
3
¦ κ
T j
κj +
j =1
1
η2
3
¦ κ
T j
j =1
(46)
κ *j
By virtue of assumption 2 and using the following inequalities
σθT θ ≤ −
σ 2 σ * 2 σ θ + θ , σκTj κ j ≤ − κ j 2
2
2
2
2 1 T * σ 2 1 1 σ θ θ ≤ θ + 2 θ* , κTj κ *j ≤ κ j η1 4 η2 4 ση1
ε ( E ) g ( x ) ev ≤ 0
2
+ +
σ
2
κ *j
2 1
ση
2 2
κ *j
(47) 2
(48)
1 1 g ( x ) ev20 + g ( x ) ε 2 ( E ) ≤ g ( x ) ev20 + g ( x ) ( ε 0 s 2 + ε1 ) 4 4
sg ( x ) ev 0 ≤
(49)
1 g ( x ) ev20 + g ( x ) s 2 4
(50)
Equation (46) can be bounded as follows 2 1 σ σ 3 Vi ≤ − g ( x ) ev20 − k 0 s tanh ( s ε 0 ) − ( k − g ( x )(1 + ε 0 ) ) s 2 − θ − ¦ j =1 κ j 2 4 4 σ * 2 1 * 2 σ 1 3 3 * 2 * 2 + θ + θ + ¦ j =1 κ j + 2 ¦ j =1 κ j + g ( x ) ε1 2 ση12 2 ση 2
2
(51)
Since the parameter vectors θ * , κ *j , j = 1, 2,3 and their derivatives θ* , κ *j , j = 1, 2,3 along with g ( x ) are assumed bounded for a fixed failure pattern. A positive bound ψ i can be defined for each fixed failure pattern as follows
§σ * θ ©2
ψ i = sup ¨ t ∈[t i ,t i +1 )
Then (51) simplifies to
2
+
1
ση12
2
θ* +
σ 2
¦
3 j =1
2
κ *j +
1
ση22
¦
3 j =1
κ *j
2
· + g ( x ) ε1 ¸ ¹
(52)
2 1 σ σ 3 Vi ≤ − g ( x ) ev20 − k 0 s tanh ( s ε 0 ) − ( k − (1 + ε 0 ) g ( x ) ) s 2 − θ − ¦ j =1 κ j 2 4 4
Assuming that the design parameter k
2
+ψ i
(53)
is chosen such that k > (1 + ε 0 ) δ1 , δ1 > g ( x ) , and define
γ = min ( 2 × ( k − (1 + ε 0 )δ1 ) , 0.5ση1 , 0.5ση 2 ) , then (53) can be further simplified as 1 γ γ 2 γ θ − Vi ≤ − g ( x ) ev20 − k 0 s tanh ( s ε 0 ) − s 2 − 2 2 2η1 2η 2
¦
3 j =1
κ j
2
+ψ i
(54)
From the definition of V i in (42), we can write Vi ≤ −γV i + ψ i
(55)
Now we can prove the following theorem that shows the boundedness of all variables in the closed loop system. Theorem 1: Consider the wing rock system (8) subject to type (9) and (12) actuator failures, using the actuation scheme (14), the adaptive controller (27) with parameter update laws (40) and (41) guarantees that the closedloop system is uniformly ultimately bounded (UUB) stable and the output tracking error converges to a small neighborhood of the origin. Proof:
Equation (55) implies that, over each time interval t ∈ [t i , t i +1 ) , where the failure pattern is fixed, given that
V i ≥ ψ i γ , we have Vi < 0 . By integrating (55) over a fixed failure pattern interval t ∈ [t i , t i +1 ) , we obtain V i (t ) ≤V i (t i+ ) e −γ (t − ti ) +
ψi γ
(56)
Now, let us denote V (t ) the extension of V i over the whole time domain. Due to actuator failures
θ, κ j , j = 1, 2,3 and consequently V (t ) will exhibit finite jumps at each time instant t i , i = 1, 2, , N , with t N +1 = ∞ , i.e. there are no further actuator failures after t N . Let us also denote ΔV i the jump on V (t ) caused by jumps on θ, κ j , j = 1, 2,3 at time t i . t i− and t i+ are the time instants just before and after the occurrence of failure respectively. Hence, starting from t1 we can write V (t 1+ ) = V (t 1− ) + ΔV 1
(57)
And from (56) we have
V (t 1− ) ≤V (t 0 ) e
ψ1 γ
(58)
ψ1 + ΔV 1 γ
(59)
− γ (t1 −t 0 )
+
From (57) and (58) we can write
V (t 1+ ) ≤V (t 0 ) e
− γ (t1 −t 0 )
+
Likewise, we proceed for t 2 ,, t N , we end up by the following inequality
ψi N + ¦ ΔV i i =1 γ i =1 N
V (t N+ ) ≤V (t 0 ) + ¦
(60)
Now, after t N there are no further failures. By integrating (55) over the time interval t ∈ [t N , ∞ ) , we can write
V (t ) ≤V (t N+ ) e
− γ ( t −t N
)
+
ψ N +1 γ
(61)
By substituting (60) into (61) we obtain N N ψ ψ § · − γ t −t V (t ) ≤ ¨V (t 0 ) + ¦ i + ¦ ΔV i ¸ e ( N ) + N +1 γ γ i =1 i =1 © ¹
(62)
It can be concluded that s (t ) , θ (t ) , κ j (t ) , j = 1, 2,3 and consequently u (t ) are bounded. Regarding (42) and (62), we conclude that, after time t N , where no further actuator failure will occur, we have N N 2ψ N +1 ψ § · −0.5γ (t −t N ) + s (t ) ≤ 2 ¨V (t 0 ) + ¦ i + ¦ ΔV i ¸e γ γ i =1 i =1 © ¹
(63)
Equation (63) implies that V (t ) is exponentially bounded and converges to a residual set defined by: s (t ) ≤ 2ψ N +1 γ . This means that e and e converge to residual sets defined as e (t ) ≤ 2ψ N +1 γ λ ,
e (t ) ≤ 2 2ψ N +1 γ . These sets can be made smaller by an adequate choice of the design parameters λ , σ ,η0 .
4.
Simulation results and discussion
To assess the effectiveness and performance of the proposed actuator failure compensation control strategy, a comparative simulation is conducted on the analytical wind rock model. This model is based on parameter identification of wind tunnel experimental data [29,30]. The triangular wing shape configuration is considered with a wing span b = 0.169m , root chord c r = 0.479m , sweep Λ = 80° , airspeed V = 30 m/s(Re = 950000) and angle of attack α = 32.5° . The other parameters of the model are computed from the physical variables of the wing, the parameters ai are computed from aˆi and t s . Thus we obtain a0 = 922.6568 , a1 = −11.0201 ,
a2 = 0.5388 , a3 = −785.2666 , a4 = 14.8722 , for the parameters bi they are computed using the equation, bi = ρV 2SbCl dai 2I xx , i = 1, 2,3 which gives b1 = −149.1759, b2 = −111.8819, b3 = −111.8819 .
The control
objective is to suppress the roll angle and rate oscillations, i.e. the output should track the reference y d (t ) = 0° . First, a proportional damper is considered. It is commonly used in aircraft stability augmentation systems (SAS), besides it has a very simple form, the segments deflection of the damper are proportional to the roll rate [30]. The three inputs are then given as: δ ai = k i t s φ , i = 1, 2, 3 . In this controller, the ailerons deflection angles are considered proportional to the roll rate δ ai = k i t s φ , where t s is introduced for dimensional reasons [30]. Taking
k i = 10, i = 1, 2,3 , the simulation is carried out with and without failures. Second, the model based adaptive actuator failure compensation strategy proposed in [5] is considered The control strategy uses a separate actuation scheme among actuators. It is primarily designed to compensate for the very particular case of lock in place (stuck) actuator failures. The control law aims to track a desired output generated from a given reference model. The entries of the control vector are given by: v j = (1 g j ( x
) ) π j ω (t ) , j
T
= 1, 2, 3 , where ω (t ) = ª¬ x T , y d − f ( x ) , g T ( x ) º¼
, and π j is a parameter vector
whose update law is defined as follows: π j = −e PB m Γ j ω (t ) , j = 1, 2,3 , for more on this strategy refer to [5]. T
The different design parameters are chosen as follows: the reference model is
( A m , B ,C )
with:
A m = [ 0 1, −16 −8] , B = [ 0 1] ,C = [1 0 ] , Γ j = 50 I 2 , P = [ 25 20 , 20 20 ] , the initial values of the T
parameter vectors π j , j = 1, 2,3 are chosen all zero.
Third, we consider the application of the proposed linear adaptive actuator failure compensation controller. The design parameters of the controller are chosen as: λ = 5,η1 = η2 = 20, ε 0 = 0.005, σ = 0.1, k = 20, k 0 = 10 . The initial values of the controller parameters θ and κ j , j = 1, 2,3 are taken zero. The actuation proportional gains are chosen b1 ( x ) = 2, b2 ( x ) = 2, b3 ( x ) = 1 . Consider first he failure free case, the simulation is carried out for 10s, which seems to be sufficient regarding the fast dynamics of the wind tunnel based model campared to real one. Fig. 3 shows the roll angle and roll rate evolution for the three controllers, i.e, the proportional roll damper, the model based adaptive controller and the proposed model-free adaptive controller, Fig. 4 shows the different control signals. Simulaiton results show that the three controllers can suppress the oscillations when all the actuators are healthy. From the simulaiton results, it can be shown that for the chosen design parameters, the proposed controller ensures a fast transient response compared to the adaptive compensation controller and the proportional roll damper, which presents some transient oscillation. The control efforts for the controllers are shown in Fig. 4. It can be seen that the proposed control shows less peaks compared to the proportional damper and the adaptive controller.
Fig. 3. Roll angle and roll rate (no failures) Fig. 4. Control signals and parameters (no failures)
Now, in order to simulate the largest possible failure cases, we consider the following failure scenarios: Failure scenario 1: With initial release angle and rate φ ( 0 ) = 1°, φ ( 0 ) = 0°/s respectively, assume that the first segment remains intact i.e. u1 ( t ) = v1 ( t ) . At time t = 3 s , the second segment is only 50% effective, i.e. u 2 ( t ) = 0.5 v 2 ( t ) . At the time
t = 6 s , the third segment is stuck at position u 3 ( t ) = u 3 = 1 5 .The roll angle and
roll rate time histories are shown in Fig. 5, the control signals and parameters are shown in Fig. 6. It can be seen that, the three controllers compensate the partial loss of effectiveness. However, when the third actuator is locked in place at time 6s, the proportional damper is unable to compensate for the failure and a roll angle offset appears. This can be checked from the control signal, we see that the control signals from the damper remained passive and did not react to the failure. On the other hand, we see that both the model-based and the proposed linear adaptive controllers were able to compensate the lock in place failure, and the controls were reconfigured to account for the failures.
Fig. 5. Roll angle and roll rate for the three controllers (scenario 1) Fig. 6. Control vectors and parameters (scenario 1) Failure scenario 2: With initial release angle and rate
φ ( 0) = 1°,φ ( 0) = 0°/s
respectively, assume that the first
segment remains intact i.e. u1 (t ) = v 1 (t ) . At time t = 3s , the second segment is only 60% effective, i.e. u 2 ( t ) = 0.6 v 2 ( t ) . At the time
t = 5 s , the third segment becomes uncontrollable and starts oscillating according
to the pattern u 3 (t ) = 10 − 30sin ( 5t ) + 20 cos ( 5t ) . The simulation results are shown in Figs. 7 and 8. The roll angle and roll rate curves show that, neither the proportional damper nor the adaptive controller can compensate for the time varying failure at time t = 5 s , oscillations in both cases are present. However, the proposed controller was effective in dealing with the failures, the roll angle and rate where kept very close to zero. Fig. 7. Roll angle and roll rate for the three controllers (scenario 2) Fig. 8. Control signals and parameters (scenario 2)
Failure scenario 3: With initial release angle and rate
φ ( 0) = 1°,φ( 0) = 0°/s respectively, we assume that the
first segment remains intact, i.e. u1 ( t ) = v1 ( t ) . At time t = 3s , the third segment is stuck at a value u 3 ( t ) = 10 . At time t = 6s , the second segment is only 30% effective, i.e. u 2 ( t ) = 0.3v 2 ( t ) , and at time t = 8s , the third segment recovers from its failure, i.e. u 3 ( t ) = v 3 ( t ) . Simulation results are shown in Figs. 9 and 10, it can be seen that when an actuator recovers from a failure it will be exploited automatically by the control scheme to control the system except for the proportional damper that is passive to such faults, the controller do not react. Fig. 9. Roll angle and roll rate (scenario 3) Fig. 10. Control signals and parameters (scenario 3) Failure scenario 4: Consider the worst-case failure scenario, with initial release angle and rate
φ ( 0) =10°,φ( 0) = 0°/s
respectively. Assume that starting from t = 5s , the first segment is only 70% effective.
At time instant t = 4 s the second segment locks at a value u 2 (t ) = 20 and then recovers at time t = 8s . For the third segment, we assume that at time t > 3s , it starts oscillating according to the pattern u 3 ( t ) = 1 − 30 sin ( 5 t ) + 20 cos ( 5 t ) . The simulation results are shown in Figs. 11 and 12, it can be seen that the
proportional damper was unable to compensate for actuator failures and stronger roll and rate oscillations appear. When the second actuator recovers from the failure, oscillations of smaller amplitude persist. The adaptive model based controller can barely keep the roll angle at zero in the presence of such failures, while the proposed controller shows superiority in dealing with such failures, the roll rate and angle were regulated around the origin. Once again, we see that the control effort is intelligently redistributed among healthy actuators if an actuator fails. Besides, if an actuator recovers from a failure, it is automatically exploited to control the system. To summarize, the proportional damper is passive to actuator failures. It can compensate for the partial loss of effectiveness seamlessly, but it is unable to react against the total loss of effectiveness. Roll offsets and roll oscillations were present which despite the smaller value compared to the free to roll case they can be annoying to the pilot and limit the performance of such controller. Another drawback of this controller is that it depends on the wing span b and the air speed V . This may limit its practical implementation if b and V were uncertain due to measurement errors or due to changes of the wingspan due to partial damage of the wing [39]. On the other hand, the adaptive model based controller given in [5] can compensate for partial loss of effectiveness and lock in place failures, but it cannot compensate for a time varying failure. Besides, it is a model based design, i.e. exact model of the system is needed to implement the controller. This is difficult to obtain in real flight as the wing rock motion is a complex multi-degrees of freedom system with uncertainties and perturbations. Faced to these limitations, the proposed strategy holds as a good remedy, it can compensate for the largest cases of actuator failures including the time-varying failures, and it does not need exact system model to be implemented which makes it more suitable for real flight applications. Fig. 11. Roll angle and roll rate (scenario 4) Fig. 12. Control signals and parameters (scenario 4) Remark 6: An attractive feature of the proposed controller is that it is a model-free design. That is, it does not depend upon exact model of the system. This is interesting given that the wind rock system studied in this paper is based on wind tunnel experiments, where a 10s simulation time interval was adequate to study the dynamics and response of the system. In real flight condition the time dynamics are very large of the order of 1000 s [28,31]. Besides, the real phenomenon is a multi-degrees of freedom phenomenon, it is also subject to the effect of aerodynamic asymmetries, coupling and wing-body vortex interactions which make it more complex and hard to model. Even with the present model with the identified parameters, there is still an error margin in the parameters a i or bi which can be affected by changing physical parameters during flight. Given that little
information on the system is required and only the system functions f ( x ) and g ( x ) are different from the real
flight models, the proposed control design will be applicable for real flight operation. Another feature of the proposed design is that it is versatile and not limited to the present case of wind tunnel based wing rock system, in fact, any high order nonlinear system having the form of equation (17) and satisfying the stated assumptions can be effectively controlled via the proposed actuator failure compensation design. 5.
Conclusion
A model-free adaptive controller design was developed for wing rock motion control in the presence of system uncertainties and uncertain actuator failures. The control strategy was built around a self-tunning controller, which is based on the prediction error, and a failure compensation controller that accounts for actuator failures. This alleviates the need for a fault detection and isolation algorithm (FDI) which makes it more efficient and suitable for real time implementation. The proposed controller requires minimum information about the system and the failures. This makes it more attractive as real flight conditions are subject to uncertainties and modeling errors. A rigorous proof of tracking and stability requirements was also presented. The simulation results show the feasibility of the proposed control strategy and its superiority compared to a proportional roll damper and an established model based design, which cannot deal with such a large set of actuator failures. As a future work, we propose the extension of the study to multivariable systems. Besides, we can also focus on seeking more relaxed design conditions and enlarging the set of compensable actuator failures. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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List of figures
Fig. 1. Actuator, plant and sensor faults
Fig. 2. Free to roll motion of the wing rock for different release angles
Fig. 3. Roll angle and roll rate (no failures)
Fig. 4. Control signals and parameters (no failures)
Fig. 5. Roll angle and roll rate for the three controllers (scenario 1)
Fig. 6. Control vectors and parameters (scenario 1)
Fig. 7. Roll angle and roll rate for the three controllers (scenario 2)
Fig. 8. Control signals and parameters (scenario 2)
Fig. 9. Roll angle and roll rate (scenario 3)
Fig. 10. Control signals and parameters (scenario 3)
Fig. 11. Roll angle and roll rate (scenario 4)
Fig. 12. Control signals and parameters (scenario 4)