2.2 Adaptive Robust Fault Tolerant Controller (ARFTC)
2.2.5 Simulation Results and Discussion
(L2 bound of the output error increases), which is undesirable. Hence, the superiority of the proposed scheme in offering better transient response is well clarified from the stability analysis leading to the inferences as discussed above.
Table 2.1: Aircraft model parameters Terminologies for the aircraft model α The angle of attack (AoA)
VT The velocity or true air speed θ The pitch angle
q The pitch rate
δe1, δe2 The elevator deflections of a two-piece augmented elevator m The mass of the aircraft
Iy The moment of inertia along the bodyy ρ The air density
S The wing area
¯
c The mean chord
Pni Thrust produced by theith engine φs The horizontal stabilizer deflection angle
¯
q The velocity head
Table 2.2: Values of aircraft parameters in simulation Aircraft parameters in simulation
m= 3×105kg CDm2 = 0.0348 CLb0= 0.00615 Iy= 4.5278×107kg CDm1 = 4.45×10−5 CMb3= 2.39
¯
c= 8.324m CDm0 = 0.00992 CMb2=−1.46 S= 511m2 CL2=−8.25×10−6 CMb1=−0.00032 ρ= 0.59kg/m3 CL1= 2.44×10−3 CMb0= 0.12 zeng1=zeng4 = 0.94m CL0= 0.1839 CM2 = 2.492×10−5 zeng2=zeng3 = 2.53m CLb2= 5.15 CM1 =−6.64×10−3 CDm3= 3.27 CLb1= 0.00121 CM0 =−1
P=41631 N φs= 0.0128 dCdφM
s =−2.8374
dCL dδe1 =dCL
dδe2 =CL2VT2+CL1VT +CL0
dCM
dδe1 =dCM
dδe2 =CM2VT2+CM1VT +CM0 (2.64) The terminologies used for the model in (2.57)-(2.58) are illustrated in Table 2.1. The aircraft pa- rameters used in the simulation are set based on the data sheet in [111] and are reproduced in Table 2.2. The state variables are chosen as [x1, x2, x3, x4] = [α, VT, θ, q], respectively. Longitudi- nal control is performed through four elevator segments and by thrust from the four engines. Pitch angle control is achieved mainly through the four piece elevator segments. Under normal operation, the inboard and outboard elevators move together and hence a two piece elevator can be formed with an outer and an inner elevator segment. This is done for mere simplicity in modeling and control design for the aircraft. The elevator deflection angles of such a two piece augmented eleva- tor is designated as δe1 and δe2 and chosen as the two actuators u1 and u2. Moreover, the term Pn (N ewton) collectively defines the thrust generated by the four engines of the aircraft, that is, Pn1 = Pn2 = Pn3 =Pn4 = Pn. Furthermore, φs represents the horizontal stabilizer deflection angle.
The trim point is given as αtrim = 0.0162 rad,qtrim= 0 rad/s,VTtrim = 230m/s,θtrim = 0.0162 rad, δetrim= 0,φstrim= 0.0128rad,Pntrim= 41631N. The magnitude of the unknown control coefficients are given as b1 =b2 = 2.77×10−5.
The control objective is to design an actuator fault tolerant control strategy to control the elevator deflection angles u1 and u2 such that the pitch angle θ given by x3 tracks a reference signal yr gen- erated from a reference system P
d even if the actuators driving the elevators lose their effectiveness partially or experience an uncertain stuck failure. As explained in Section 2, there exists a diffeo- morphism, which transforms the system P
p to P
c in strict feedback form and P
z denoting the internal dynamics of the system. The diffeomorphism for the aircraft model (2.57)-(2.58) is given by T(x) = [η1 η2 ξ3 ξ4]T = [T1(x) T2(x) x3 x4]T yielding P
c as, ξ˙3=ξ4
ξ˙4=a(ξ, η) + Xm j=1
bjβj(ξ, η)(KjuHj + ¯uF j) (2.65)
Accordingly, the transformations T1(x) and T2(x) are found as solutions to the partial differential equations given below.
∂T1
∂x1g1(x) +∂T1
∂x2g2(x) +∂T1
∂x3g3(x) + ∂T1
∂x4g4(x) = 0 (2.66)
∂T2
∂x1g1(x) +∂T2
∂x2g2(x) +∂T2
∂x3g3(x) + ∂T2
∂x4g4(x) = 0 (2.67) where, g1, g2, g3, g4 are nonlinear smooth actuation functions in x2 described as,
g1(x) g2(x) g3(x) g4(x)
=
−0.000502483x2(0.00321 + 0.0000426x2−1.44×10−7x22) 0
0
0.0000277133x22(−0.0176−0.000116x2+ 4.35×10−7x22)
Thereafter solving the partial differential equations (PDEs) in (2.66) and (2.67) yields,
η1=T1(x) =x1x2−6.6586261x4 (2.68)
η2=T2(x) =x2 (2.69)
Hence, the unforced zero dynamics can be expressed as η˙ = ̺(η1, η2,0,0) with ξ3 = ξ4 = 0 as the input. It is now shown that the zero dynamics is locally input to state stable (ISS) [109]. The equi- librium point ofη˙ =̺(η1, η2,0,0) is found to beηe= [0 3806.14]T. The stability of the unforced zero dynamics at equilibrium is investigated by checking the eigen values of ∂̺(η,0)/∂η at ηe. The eigen values are found to be−9.17485and −1.00182, and sinceℜ(λ(∂̺(η,0)/∂η|ηe)) <0, it is proved that the unforced zero dynamics are locally asymptotically stable.
The controller is designed following the strategy discussed in Section 2.2.3 and the control law given by (2.20), with relative degree℘= 2. The initial state conditions are assumed to be[ 0.794 230 0.0162 0 ]T. The rate parameters in the adaptive law are chosen as γ = 0.5, ǫ1 = 0.5, ǫ2 = 0.6 and P = 0.5I2 and that in the virtual control law α1 is considered to bec1 = 30. The parameters in the nominal control lawwnom are selected asβ1 =40.5, β2 =30.5,ϑ1 = 35 andϑ2 = 34. The reference signal yrto be tracked
is generated from the reference system as in [4], given byGd(s) = s2 + 3s1 + 4 withydas the reference input, meaning that yr = L−1[Gd(s)]∗yd. The notations L−1[·] and ∗ denote the inverse Laplace transform and the convolution operator respectively. The signal yd is given as,
yd(t) =
0 0s≤t <5s 0.1 5s≤t <10s
0 10s≤t <15s
−0.1 15s≤t <20s 0 20s≤t <25s
(2.70)
This pattern of yd(t) in (2.70) is repeated for the next 25 s. The initial value of the adaptive pa- rameters ξˆand Γˆ are set as ξ(0) = [1 0]ˆ T and Γ(0) = 0. The functionsˆ β1 and β2 are given by, β1=β2=x22(−0.0088−0.000058x2 + 2.175×10−7x22).
Remark 2.5. The performance of the proposed methodology is compared with similar existing schemes.
For this purpose, an adaptive failure compensation method based on backstepping (BS) proposed in [32]
is considered. Also, an adaptive sliding mode control (ASMC) based failure control scheme has also been designed based on a partial actuator loss compensation method proposed in [4], for the sake of further comparison.
For the longitudinal aircraft model of Boeing 747-100/200, the basic controller proposed in [5] based on adaptive backstepping control (ABSC) is obtained considering similar notations as follows,
ujBS =−sgn(bj)1 bj
"
θˆ1
z1+c2z2+a0(ξ, η) + ˆΓa0(ξ, η)− ∂α1
∂ξ3ξ4+∂α1
∂y˙ry¨r
−y¨r
+βjθˆ2,j
# (2.71) θ˙ˆ1=γ1
"
z1+c1z1+a0(ξ, η) + ˆΓa0(ξ, η)− ∂α1
∂ξ3ξ4+∂α1
∂y˙ry¨r
−y¨r #
z2 (2.72)
θ˙ˆ2,j=γ2,jβjz2, j= 1,2 (2.73)
˙ˆΓ = 0.0005a0(ξ, η)z2 (2.74)
where, z1, z2 denote the error variables and α1 defines the virtual control as, z1 = ξ3 −yr, z2 = ξ4−α1−y˙r, α1 =−c1z1 The design parameters are chosen asc1 = 25, c2= 25, γ1= 1 and γ2,1= 0.1.
Herej = 1is only considered since case of total loss of only one actuator is possible for this particular illustration of aircraft control. Similarly, the adaptive sliding mode control (ASMC) methodology proposed in [4] assuming known control coefficients and modified to deal with partial loss as well as total actuator failure, is given as,
ujASM C = 1
bjβj(c( ˙yr−ξ4)−a0(ξ, η) + ¨yr+ ˆρsgn(σ) + ˆγ|ψ0|sgn(σ)) (2.75) ψ0 =c( ˙yr−ξ4)−a0(ξ, η) + ¨yr (2.76)
σ = ( ˙yr−ξ4) +c(yr−ξ3) (2.77)
˙ˆ
ρ=aρ|σ|, γ˙ˆ=aγ|ψ0||σ| (2.78)
where, c= 3, aρ= 0.3 and aγ = 5 are chosen values of the user defined parameters.
Simulation studies have been carried out considering a severe actuator failure scenario with modeling uncertainties. Transient performance is assessed on the basis of the integral time absolute error (ITAE) and settling time (ts) which are desired to be of low magnitude. A small value of ITAE ensures minimum overshoots and oscillations in the output under the action of the controller. The output tracking performance is evaluated in terms of the integral absolute error (IAE), which is widely used as a performance criterion in the design of efficient controllers. To evaluate the manipulated input usage, the total variation (TV) [113] is calculated as,
Pn
i=1|ui−ui+1|, which should have a small value. Energy of the control input is measured by using the 2-norm method.
The failure scenario is as follows: 20% uncertainties in system dynamics and the actuator failure pattern is given as,
u1(t) =
( 0.3uH1(t), uH1(t),
t∈[20, ∞)
t∈[0, 20) , u2(t) =
( 2, uH2(t),
t∈[40, ∞)
t∈[0, 40) (2.79) The tracking performance of the proposed controller is compared with those of ASMC [4] and ABSC [5]
schemes in Figures 2.3(a)-(f). The tracking error obtained by using the proposed method and the control methodologies proposed in [4] and [5] are plotted in Figure 2.3(a). It is observed from Figure 2.3(a) that in comparison to [4] and [5], the system outputy = ξ3 (pitch angle θ) with the proposed scheme is capable of tracking the desired trajectory yr with marginal error and negligible overshoots in addition to a substantially low settling time. In contrast, the ASMC based scheme (2.75) exhibits a degraded transient performance with the output error swinging between −0.0125 and0.7×10−4 and never reaches an acceptable steady state value. Further, the tracking error obtained using the adaptive backstepping controller (ABSC) (2.71) never achieves a steady state and oscillates in the interval (- 0.001 , 0.002). The output tracking of the pitch angle θ using the proposed control strategy is shown in Figure 2.3(b) and the equivalent pitch rate q is shown in Figure 2.3(c). Figures 2.3(d)-(f) show the control signals utilizing the proposed method, ASMC and adaptive backstepping control (ABSC) respectively. The output tracking performance of each controller is measured in terms of ITAE and IAE. Comparison of tracking performances using the proposed scheme, ASMC [4] and backstepping [5]
can be found in Table 2.3. In Table 2.3, it is observed that the proposed control strategy yields an ITAE of 0.00947 in contrast to comparatively higher ITAE values of 0.481 and 0.523 using ASMC [4]
and ABSC [5] methods. Further, Table 2.3 shows that the proposed scheme yields an IAE of 0.00989 indicating small sustained oscillations in the output compared to significantly high IAE values of 0.044 and 0.025 in [4] and [5]. The total variation (TV) values for the inputs u1 and u2 in the proposed control scheme are measured to be 277.01 and 258.22 in comparison to quite high values in [4].
The time window between the onset of the fault or failure and its subsequent estimation is very
0 5 10 15 20 25 30 35 40 45 50
−0.08
−0.06
−0.04
−0.02 0 0.02
Time (s) Tracking error, z 1 = x 3 − y r
30 32 34 36 38 40 42 44 46 48
−12
−10
−8
−6
−4
−2 0 2x 10−3
(a)
0 10 20 30 40 50
−0.2
−0.1 0 0.1 0.2
Time (s)
Pitch angle, θ (rad)
(b)
0 10 20 30 40 50
−0.1
−0.05 0 0.05 0.1
Time (s)
Pitch rate q (rad/s)
(c)
0 10 20 30 40 50
−10
−5 0 5 10 15
Time (s) Control inputs u 1, u 2
(d)
0 10 20 30 40 50
−10 0 10 20 30 40
Time (s) Control inputs u 1, u 2
(e)
0 10 20 30 40 50
−10
−5 0 5 10 15
Time (s) Control inputs u 1, u 2
(f)
Figure 2.3: Plant response and control inputs under the considered fault scenario (2.79) using the proposed control scheme. (a) Output tracking error comparison of the proposed controller (red) with ASMC (blue) [4] and ABSC (green) [5]; (b) Pitch angle θ tracking using the proposed ARFTC; (c) Pitch rate q under the proposed ARFTC; (d) Control inputsu1 andu2corresponding to proposed ARFTC; (e) Control inputsu1 andu2 using ASMC [4]; (f) Control inputs u1 andu2 using ABSC procedure [5].
crucial from the point of FTC design and must be as small as possible. This time window can be made analogous to the settling time measure, signifying the time required by the overall control system to recover its performance after the failure occurs. The settling time ts has been measured at t = 40 s, when u1 experiences a partial loss of effectiveness and u2 undergoes a total loss. It is clear from the Figure 2.3 and Table 2.3 that using the proposed FTC scheme, the post fault overshoots and settling time could be substantially reduced. Moreover, the proposed FTC scheme spends lesser control effort than that in [4] with the faithful realization of its objective. The control energy spent by the proposed
controller is found to be almost at par with that of the basic controller (2.71) based on adaptive backstepping control (ABSC) [5]. However, the proposed design outperforms ABSC [5] and exhibits a superior post failure tracking performance. Besides, the scheme performs well in the presence of unknown control coefficientsbj compared to ASMC [4] and ABSC [5]. Further, the control input using ASMC in [4] is not smooth and experiences chattering which is undesired. However, the control signals using the proposed ARFTC strategy is continuous and smooth.
Table 2.3: Post failure tracking performance with 70% loss of effectiveness of u1 at t= 20s and stuck failure ofu2 at t= 40s
Performance
Criteria ARFTC ASMC[4] ABSC [5]
ITAE 0.00947 0.48141 0.52341
IAE 0.00989 0.04483 0.02505
Settling time ts 0.73 s - -
Control energy u1 472.99 525.38 474.14
u2 685.23 695.53 684.01
Total variation(TV) u1 277.01 1.31×103 103.61 u2 258.22 3.24×104 94.79