Nonlinear Adaptive Flight Control Using Backstepping ... - AIAA ARC
JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 24, No. 4, July– August 2001
Nonlinear Adaptive Flight Control Using Backstepping and Neural Networks Controller Taeyoung Lee¤ and Youdan Kim† Seoul National University, Seoul 151-742, Republic of Korea A nonlinear adaptive ight control system is proposed using a backstepping and neural networks controller. The backstepping controller is used to stabilize all state variables simultaneously without the two-timescale assumption that separates the fast dynamics, involving the angular rates of the aircraft, from the slow dynamics, which includes angle of attack, sideslip angle, and bank angle. It is assumed that the aerodynamic coef cients include uncertainty, and an adaptive controller based on neural networks is used to compensate for the effect of the aerodynamic modeling error. Under mild assumptions on the aerodynamic uncertainties and nonlinearities, it is shown by the Lyapunov stability theorem that the tracking errors and the weights of neural networks exponentially converge to a compact set. Finally, nonlinear six-degree-of-freedom simulation results for an F-16 aircraft model are presented to demonstrate the effectiveness of the proposed control law.
Nomenclature F I L ; M; N p; q; r qN T V ®; ¯ ±e ; ±a ; ±r Á; µ ; Ã
F
= = = = = = = = = =
search, the gain of the inner-loop controller is set much larger than that of the outer-loop controller, and it is assumed that the aircraft dynamics satis es this property. This, therefore, does not guarantee closed-loop stability. Schumacher and Khargonekar analyzed theoretically the stability of the ight control system with the two-timescale separation assumption.4 Using the Lyapunov theory, they determined the minimal inner-loop gain guaranteeing closed-loop stability. However, this approach is so complicated and conservativethat the calculated value of the minimal inner-loop gain is too large to be applied in the ight controller. It may excite unmodeled dynamics or saturate the control inputs and, therefore, cause robustness problems. Another dif culty related to the applicationof feedbacklinearization to a ight control system is that a complete and accurate aircraft dynamic model including aerodynamic coef cients is required. It is dif cult to identify accurately aerodynamic coef cients because they are nonlinear functions of several physical variables. Gain scheduling with a linear H1 design is a traditional approach to overcome this problem; however, it can guarantee the desired performance only when conditions comprise a small perturbation and slow variance. Neural networks have been proposed recently as an adaptive controller for nonlinearsystems.5;6 By the use of their universalapproximation capability,the adaptive controller based on neural networks can be designed without signi cant prior knowledge of the system dynamics. In ight control problems, the applications of adaptive neural networks can be found in Refs. 7 and 8. Single-layer neural networks are used to compensate for unknown dynamics7 and inversion error.8 This paper proposes a backstepping and neural networks controller for a nonlinear ight dynamic system and analyzes the stability of the proposedcontrol system using the Lyapunovtheory.The controller is designed by using the backstepping approach9 with the assumption that all aerodynamic coef cients are fully understood. The proposed method also intermediately uses the fast states p; q, and r as control inputs. However, it considersthe transientresponses of the fast states and does not require the two-timescale assumption. Therefore, it is not necessary to make the controller gain impracticably large to guaranteethe closed-loopstability because the timescale separation assumption is not used in the design or analysis. The effects of the modeling error in some aerodynamic coef cients are also considered and are compensated for by multilayer neural networks.The parametersof the neural networks are adjusted to offset the term generatedby the modeling error. The adaptivecontroller based on multilayer neural networks is an extension of the work described in Ref. 6. In that paper, only the state variables can be used as input for the neural networks. This paper generalizes it by adding a robust control term, which allows the use of neural network inputs that do not belong to the states.The main contributionof
aerodynamic force about the body- xed frame moment of inertia aerodynamic rolling, pitching, yawing moment roll, pitch, yaw rate about the body- xed frame dynamic pressure thrust velocity angle of attack, sideslip angle elevator, aileron, rudder angle roll, pitch, yaw angle
I.
Introduction
EEDBACK linearization is a theoretically established and widely used method in controlling nonlinear systems. Exact input– output linearization is often used to control speci c output variable sets in nonlinear ight control problems.1 Unfortunately, this direct application of feedback linearization requires the second or third derivatives of uncertain aerodynamic coef cients and does not guarantee internal stability for nonminimum phase systems. Another approach to designing ight control laws with feedback linearization is to separate the ight dynamics into fast and slow dynamics by using timescale properties.2;3 This method allows the controller design process to be performed without state transformation, because each separated subsystem is square: The number of control inputs is equal to the number of states. The design process of this method can be divided into two steps. In the outer loop, the controller for the slow states ®; ¯, and Á is designedto facilitatetracking of the given commands by assuming that the fast states p; q, and r are control inputs, which achieve their commanded values instantaneously. With the slow states controller designed in the outer loop, a separated inner-loop controller is designed to make the fast states p; q, and r follow the outer loop’s control input trajectories using the real control inputs: aileron, rudder, and elevator. This method can be justi ed only if there is suf cient timescale separation between the inner- and outer-loop dynamics because the fast states p; q, and r are used as control inputs in the outer-loopsystem. Hence, the states p; q, and r in the inner loop should be much faster than the states ®; ¯, and Á in the outer loop. The stability of this timescale separation approach may be analyzed by the singular perturbation theory. However, in most nonlinear ight control reReceived 29 November 1999; revision received 1 July 2000; accepted for c 2000 by the American Institute publication 2 October 2000. Copyright ° of Aeronautics and Astronautics, Inc. All rights reserved. ¤ Graduate Student, Department of Aerospace Engineering. † Associate Professor, Department of Aerospace Engineering; ydkim @snu.ac.kr. Senior Member AIAA. 675
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LEE AND KIM
II.
2
qP D I5 pr ¡ I6 . p ¡ r / C I7 M
(5)
rP D ¡I2 qr C I8 pq C I4 L C I9 N
(6)
I1 D ¡
I7 D (1)
(2)
1 ; Iy
I8 D
Ix z ; I x Iz ¡ Ix2z
Ix I x Iz ¡ I x2z
¤
¤
C .br =2V /C lr .®/ qN Sb
Substituting the aerodynamic coef cients into the ight dynamic equations yields
3 2
¡.sin ® =cos ¯/C x q .®/cN C .cos ® =cos ¯/C z q .®/cN ¡cos ® sin ¯C x q .®/cN ¡ sin ® sin ¯C z q .®/cN 0
¡.sin ® =cos ¯/C x ±e .®/ C .cos ® =cos ¯/C z ±e .®; ¯/ ½VS 4 ¡cos ® sin ¯C x ±e .®/ ¡ sin ® sin ¯C z ±e .®; ¯/ C 2m 0
0 cos ¯C y±a .¯/ 0
1 0 sin Á tan µ
32 3
¡sin ® tan ¯ p ¡cos ® 54 q 5 r cos Á tan µ
32 3
p 0 cos ¯C yr .®/b54 q 5 r 0
32 3
0 ±e cos ¯C y±r .¯/5 4±a 5 0 ±r
3
.1=cos ¯/.sin ® sin µ C cos ® cos Á cos µ / g C 4cos ® sin ¯ sin µ C cos ¯ cos µ sin Á ¡ sin ® sin ¯ cos Á cos µ 5 V 0
2 3
Ix z Iy
£
2
2
I9 D
I6 D
L D Cl .®; ¯/ C C l±a .®; ¯/±a C C l±r .®; ¯/±r C .bp=2V /Cl p .®/
(3)
2
Iz ¡ I x ; Iy
Ix .I x ¡ I y / C I xz2 ; I x Iz ¡ I x2z
®P ¡.sin ® =cos ¯/[T C C x .®/qN S] C .cos ® =cos ¯/C z .®/qN S ¡cos ® tan ¯ 4¯P 5 D 1 4¡cos® sin ¯[T C C x .®/qN S] C cos ¯C y .¯/qN S ¡ sin ® sin ¯C z .®; ¯/qN S 5C 4 sin ® mV 0 1 ÁP 0 ½S 4 C cos ¯C y p .®/b 4m 0
I5 D
£
g sin ® sin ¯ Fz C [cos ® sin ¯ sin µ C cos ¯ cos µ sin Á mV V
2
Ix z .Ix ¡ I y C Iz / I x Iz ¡ I x2z
Fx D C x .®/ C C x ±e .®/±e C .cq N =2V /C x q .®/ qN S
cos ® sin ¯ cos ¯ Fy ¯P D sin ®p ¡ cos ®r ¡ [T C Fx ] C mV mV
2 3
I4 D
I2 D
De nitionsof state and controlvariables,forces,and moments in the precedingequationsare describedin the Nomenclature.Itis assumed that the aerodynamicforcesand moments are expressedas functions of angle of attack, sideslip angle, angular rates, and control surface de ection.11 For example, Fx and L are expressed as follows:
sin ® ®P D ¡cos ® tan ¯ p C q ¡ sin ® tan ¯r ¡ [T C Fx ] mV cos ¯
¡ sin ® sin ¯ cos Á cos µ ]
I z .I z ¡ I y / C Ix2z ; Ix Iz ¡ Ix2z
Iz ; 2 I x Iz ¡ I xz
I3 D
C g[¡cos ® cos ¯ sin µ C sin ¯ sin Á cos µ
¡
(8)
sin Áq C cos Ár (9) ÃP D cos µ where the moments of inertia Ii ; i D 1; 2; : : : ; 9, are de ned as follows:
Problem De nition
g cos ® Fz C [sin ® sin µ C cos ® cos Á cos µ ] m V cos ¯ V cos ¯
(7)
µP D cos Áq ¡ sin Ár
cos ® cos ¯ sin ¯ sin ® cos ¯ VP D [T C Fx ] C [Fy ] C [Fz ] m m m
C
2
ÁP D p C tan µ .sin Áq C cos Ár /
This paper presents a controller for a nonlinear aircraft. The task of the controller is to track the commands of ®; ¯, and Á when aerodynamic model uncertainties exist. The body- xed axes, nonlinearequationsof motion for an aircraft over a at Earth are given by10
C sin ® cos ¯ cos Á cos µ ]
(4)
pP D I2 pq C I1 qr C I3 L C I4 N
this paper is that the stability of nonlinear ight dynamics is proven mathematically without unrealistic restrictions. This paper is organized as follows. A nonlinear ight model is described in Sec. II. In Sec. III, the backstepping controller is designed when modeling error does not exist. The neural networks adaptive controller is then designed in Sec. IV when aerodynamic modeling error is present. Finally, a numerical simulation of a sixdegree-of-freedom F-16 aircraft model is performed to verify the effectiveness of the proposed algorithm in Sec. V.
2
3
2
(10)
3
I2 pq C I1 qr pP I3 Cl .®; ¯/qN Sb C I4 C n .®; ¯/qN Sb 4qP 5 D 4 I5 pr ¡ I6 . p 2 ¡ r 2 /5 C 4 5 I7 C m .®/qN S cN rP I4 Cl .®; ¯/qN Sb C I9 C n .®; ¯/qN Sb ¡I2 qr C I8 pq
2 ½VS 4 C 4
2
I3 C l p .®/b C I4 C n p .®/b 0 I4 C l p .®/b C I9 C n p .®/b 0
C qN S 4 I7 C m ±e .®/cN 0
0 I7 C m q .®/cN 0
32 3
I3 C lr .®/b C I4 C nr .®/b p 5 4q 5 0 I4 C lr .®/b C I9 C nr .®/b r
I3 Cl ±a .®; ¯/b C I4 C n±a .®; ¯/b 0 I4 Cl ±a .®; ¯/b C I9 C n±a .®; ¯/b
32 3
I3 Cl ±r .®; ¯/b C I4 C n±r .®; ¯/b ±e 5 4±a 5 0 I4 Cl ±r .®; ¯/b C I9 C n±r .®; ¯/b ±r
(11)
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LEE AND KIM
µ ¶
µ
µP 0 D 0 ÃP
cos Á sin Á =cos µ
¶
2 3
p ¡sin Á 4q 5 cos Á =cos µ r
(12)
Lemma 1 can be proved by a suf cient condition that the set of rows of g1 is linearly independent. According to Assumption 3 and Lemma 1, the size of g1¡1 and g1a is bounded by some positive constants cg1a ; cg¡1 :
xP1 D f1 .®; ¯/ C g1 .®; ¯; Á; µ /x 2 C g1a .®; ¯/x 2 (13)
C h 1 .®; ¯/u C f 1g .®; ¯; Á; µ / xP2 D f2 .®; ¯; p; q; r/ C f 2a .®; ¯/x 2 C g2 .®; ¯/u
(14)
xP3 D f3 .Á; µ /x 2
(15)
where f; g, and h represent the terms of Eqs. (10– 12) in order. The preceding equations are mainly used for the controller design and stability analysis processes of the following sections.
III. Controller Design and Stability Analysis When designing a ight control system with the two-timescale assumption, the inner-loop controller is designed to control the fast states x2 using the control input u, where the desired values of the fast states x 2d are given by the outer loop. In the outer loop, the controller is designed to control the slow states x 1 using the fast states x 2 as control inputs. The inner-loop controller neglects the transient responses of the fast states x 2 . It assumes that the fast states track their commanded values instantaneously and that the control surface de ection has no effect on the outer-loop dynamics. The structure of the two-timescale controller is shown in Fig. 1. In each feedback loop, control laws x2d and u are designed separately. In this paper, the backstepping method is used to design a controller.9 The backstepping design procedure can be viewed as the two-timescale approach because the fast states x 2 are used as control inputs for the slow states x 1 intermediately. However, this methodologyconsidersthe transientresponsesof the fast states and, therefore,does not require the timescale separationassumption.The following assumptionsare used in the design and analysisprocesses. Assumption 1: The desired trajectories x1d D [® d ; ¯ d ; Á d ]T are bounded as
®£ ¤® ® x d ; xP d ; xR d ® · cd 1
1
where cd 2 R is a known positive constant and k ¢ k denotes the 2-norm of a vector or a matrix. Assumption 2: The total velocity and the dynamic pressure are constant. VP D 0;
qPN D 0
(18)
jµ j · µm < ¼ =2
where µm 2 R is a positive constant. The following lemma is used in the design and analysis processes. Lemma 1: There exist positive constants ®m ; ¯ m , and µm 2 R such that g1 .®; ¯; Á; µ / is invertible for all Á 2 R and all ®; ¯, and µ 2 R satisfying j®j · ®m ; j¯j · ¯m , and jµ j · µm .
Fig. 1
(20)
Assumption 5: The following inequalities are satis ed for constants cg1a and cg¡1 in Eqs. (19) and (20): 1
(21)
cg1a cg ¡1 < 1 1
Note that g1a is composed of the aerodynamic coef cient terms related to the angular rate and multiplied by a very small quantity ½ =m, and therefore, the magnitude of g1a is very small. Also note that the norm of cg¡1 is mainly in uenced by the pitch angle µ , 1 and therefore Assumption 5 is closely related to the maximum pitch angle in Lemma 1. For the aerodynamic model considered in this study, the numerical value of cg1a cg ¡1 is less than 0:13. 1 Assumption 6: There exist positive constants®m and ¯m 2 R such that g2 .®; ¯/ is invertible for all ® and ¯ 2 R satisfying j®j · ®m and j¯j · ¯ m . Assumption 7: The control surface de ection has no effects on the aerodynamic force component: h 1 .®; ¯/ D 0
(22)
z 1 D x1 ¡ x1d
(23)
Note that g2 in Eq. (14) represents the input matrix of the control u to the dynamics of the angular rates p; q; and r. Also, h 1 in Eq. (13) represents the aerodynamic force component caused by the control surface de ection. Because the control surfaces of the aircraft are designedto control each axes’ angular rate of aircraft independently, the input matrix g2 is invertiblefor all cases, and the magnitude of h 1 is very small compared to other aerodynamic terms in the dynamic equation. Therefore, it can be assumed that g2 is invertible, and h 1 D 0. Numerical studies for the aerodynamic model considered in this study also show that g2 is always invertible, and the size of h 1 is negligible. Let us introducethe error state variablesz 1 and z 2 2 R3 as follows: z 2 D x2 ¡
x2d
(24)
where x1d and x 2d are the desiredtrajectoriesof x1 and x2 , respectively. Note that x 1d is given by command signals and x2d will be de ned later. Using Eqs. (13) and (14), and Assumption 7, the dynamic equations of the error states are given as follows:
(17)
Assumption 3: There exist positive constants®m and ¯m 2 R such that the magnitudes and derivatives of f 1 ; f 2 ; f 2a ; g1a , and g2 are bounded for all ® and ¯ 2 R satisfying j®j · ®m and j¯j · ¯m . Assumption 4: The magnitude of µ is bounded as
(19)
1a
(16)
1
1
® ® ®g1 .®; ¯; Á; µ /¡1 ® · c ¡1 g1 ® ® ®g1a .®; ¯/® · cg
Let us de ne the states x 1 ; x 2 2 R3 ; x3 2 R2 , and the control inputs u 2 R3 as follows: x1 D [®; ¯; Á]T , x2 D [p; q; r]T , x 3 D [µ; Ã ]T , and u D [±e ; ±a ; ±r ]T . With the choices of x1 ; x 2 ; x 3 , and u, the ight dynamic Eqs. (10– 12) can be rearranged as
zP 1 D xP1 ¡ xP1d D f 1 C g1 x 2 C g1a x 2 C f 1g ¡ xP1d zP 2 D xP2 ¡
xP2d
D f 2 C f 2a x 2 C g2 u ¡
xP2d
(25) (26)
Theorem 1: Consider the system in Eqs. (25) and (26), where the control input u is de ned in Eq. (27). Then, the solutions of the system are locally uniformly ultimately bounded:
£
u D g2¡1 ¡k 2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ A
¤
(27)
where and A 2 R are de ned in Eqs. (28) and (29) and k 1 and k 2 2 R are positive design parameters: x2d
3£1
£
x 2d D g1¡1 ¡k1 z 1 ¡ f 1 ¡ f 1g C xP 1d
Structure of the two-timescale controller.
¤
(28)
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LEE AND KIM
¤ @ x2d £ f1 C g1 x2 C g1a x 2 C f 1g @ x1
A D f2 C f 2a x2 ¡ ¡
VP1 · ¡2¹1 V1 C
£ ¤ @ x 2d f 3 x 2 ¡ g1¡1 k 1 xP1d C xR1d x @ 3
(29)
Furthermore, the bound of the tracking error may be kept as small as desired by adjusting the design parameters. Proof: By Assumptions 1 and 3, the following inequalities are satis ed for certain positive constants c f1 ; c f 1g , and cxP d : 1
(30)
k f 1 .®; ¯/k · c f 1
® ® ® f 1g .®; ¯; Á; µ /® · c f 1g ® ® ®xP d ® · c d
(31)
2
h
g 1¡1
i
(33)
k1 kz 1 k C c f 1 C c f 1g C cxP d 1
Let us consider the Lyapunov function candidate, V1 D 12 z 1T z 1 C 12 z 2T z 2
(34)
The derivative of the Lyapunov function V1 along the trajectoriesof Eqs. (25) and (26) is given by @ V1 @ V1 (35) VP1 D zP 1 C zP 2 @z 1 @ z2 Substitution of Eqs. (25) and (28) into Eq. (35) yields the following equation:
¤ @ V1 £ ¤ @ V1 @ V1 £ VP1 D ¡k 1 z 1 C g1a x2 C g1 x2 ¡ x2d C zP 2 @z 1 @ z1 @ z2 D ¡k1 kz 1 k2 C z 1T g1a x 2 C z T1 g1 z 2 C z 2T zP 2 D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T1 g1a z 2 C z 1T g1 z 2 C z 2T zP 2 (36) Substituting Eqs. (26) and (29) into Eq. (36) yields
¶
@ x2d @xd @xd @xd xP 1 ¡ 2 xP3 ¡ 2d xP1d ¡ 2d xR1d @ x1 @ x3 @ x1 @ xP 1
(37)
£
D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T2 g1Ta z 1 C g1T z 1 C A C g2 u
¤
(38)
Because f 1 in Eq. (28) is dependenton V , =@ V / VP must be included in Eq. (37); however, it is neglected by using Assumption 2 in this study. Substituting Eq. (27) into Eq. (38) and using Eqs. (20) and (33) give the following equations: the term ¡z 2T .@ x2d
VP1 D ¡k1 kz 1 k2 ¡ k 2 kz 2 k2 C z T1 g1a x 2d · ¡k1 kz 1 k2 ¡ k 2 kz 2 k2
®
® ®
¡
¢
1
¢
µ
· ¡
k1 k1 .1 ¡ c1 /kz 1 k2 ¡ .1 ¡ c1 / kz 1 k 2 2 C
¼ c21 c22 2 2 kz 1 k C kz 2 k · D1 D z 1 ; z 1 2 R 2¹1 k1 .1 ¡ c1 / »
(42)
Because c1 and c2 do not depend on k1 and k2 , the size of the set D1 can be made arbitrarily small by adjusting the design parameters k1 and k 2 . Theorem 1 represents the design procedure of the controller to track the ®; ¯, and Á commands, and also shows that the tracking error of the control system converges to a compact set whose size is adjustable by the design parameters. Because the timescale separation assumption is not used in this study, it is not necessary to make the control gain large to guarantee closed-loop stability.
IV.
Neural Networks Adaptive Controller Design
In the preceding section, the backstepping controller is designed with the assumption that the aerodynamic characteristics are fully known. Because aerodynamic coef cients are highly nonlinear and dependent on lots of physical variables, it is very dif cult to identify them exactly. The difference between the mathematical model and the real system may cause performance degradation. To overcome this drawback, multilayer neural networks are used in this study. The weights of the neural networks are adjusted to compensate for the effect of the modeling error. The adaptive controller based on multilayer neural networks is an extension of the work described in Ref. 6. In this paper, it is generalized such that the variables that do not belong to the states can be used as neural networks’ inputs. A.
Effect of Modeling Error
In this paper, the modeling error in the body- xed angular rates dynamics is considered. The identi ed values of f 2 ; f 2a , and g2 in Eq. (14) are de ned as fO2 ; fO2a , and gO2 , respectively.Then, the control input is expressed as follows: uO D gO 2¡1 ¡k2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ AO
(43)
£ ¤ @ x 2d f 3 x 2 ¡ g1¡1 k 1 xP1d C xR1d @ x3 Substituting Eqs. (43) and (44) into Eq. (38) gives ¡
(44)
£
VP1 D ¡k1 kz 1 k2 C z 1T g1a x2d C z T2 g1Ta z 1 C g1T z 1 C A C g2 uO C g2 u ¡ g2 u D ¡k1 kz 1 k2 ¡ k 2 kz 2 k2 C z 1T g1a x 2d 4
C z 2T g2 [uO ¡ u] D ¡k 1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d ¡ z 2T 11
O represents the term caused by the modeling where 11 ´ g2 [u ¡ u] error, that is, the modeling error of f 2 ; f2a , and g2 adds the term z 2T 11 to the derivative of the Lyapunov function. B.
Neural Networks Structure
Given an input x0nn 2 R N1 , the three-layer neural networks as shown in Fig. 2 has an output ynn 2 R N3 as
c12 c22 ¡ k 2 kz 2 k2 2k1 .1 ¡ c1 /
k1 c12 c22 .1 ¡ c1 /kz 1 k2 ¡ k2 kz 2 k2 C 2 2k1 .1 ¡ c1 /
¤
¤ @xd £ AO D fO2 C fO2a x2 ¡ 2 f 1 C g1 x2 C g1a x 2 C f 1g @ x1
1
1
¶2
(41)
Note that ¹1 is positive because c1 is less than 1 by Assumption 5. Equation (40) implies that VP1 < 0 when V1 > c21 c22 =[4¹1 k1 .1 ¡ c1 /]. Therefore, the error states z 1 and z 2 are bounded and converge exponentially to the residual set D1 :
¡
C c f 1g C c xP d kz 1 k ¡ k2 kz 2 k2 D ¡k1 .1 ¡ c1 /kz 1 k2 C c1 c2 kz 1 k
c1 c2 ¡ k 1 .1 ¡ c1 /
1
¹1 D min[.k 1 =2/.1 ¡ c1 /; k 2 ]
¤
C ® g1a ®kz 1 k® x2d ® · ¡k1 1 ¡ cg1a cg¡1 kz 1 k2 C cg1a cg¡1 c f 1
¡ k 2 kz 2 k2 D ¡
c2 D c f 1 C c f 1g C cxP d
1
£
VP1 D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T1 g1a z 2 C z 1T g1 z 2 C z 2T [A C g2 u]
®
c1 D cg1a cg¡1 ;
µ
VP1 D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T1 g1a z 2 C z 1T g1 z 2 C z 2T f2 C f 2a x2 C g2 u ¡
where the constants c1 ; c2 , and ¹1 are de ned as follows:
(32)
The bound of kx 2d k can be computed using Eqs. (19), (30), (31), and (32) as
® d® ®x ® · c
(40)
3
xP1
1
c12 c22 2k 1 .1 ¡ c1 /
(39)
ynni D
N2 X
"
Á
N1 X
wi j ¾ j D1
kD1
!
v j k x0nnk
#
C µv j Cµwi ;
i D 1; 2; : : : ; N 3 (45)
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LEE AND KIM
where
d¾ ¾ .Oz / D dz z D zO 0
and w 2 R3 is de ned as
¡
¢
¡
¢
w.t / D ¡ WQ T ¾ 0 VO T xnn V T xnn ¡ W T O VQ T xnn ¡ ².xnn / (51)
Fig. 2
Furthermore, kwk satis es the following inequality for some positive constants Ci ; i D 1; 2; 3; 4:
Structure of three-layer neural networks.
Q F C C 3 k Zk Q F kx 1 k C C 4 k ZQ k F kx 2 k kwk · C 1 C C 2 k Zk
where v jk is the weight connectingthe rst layer to the second layer, wi j is the weight connecting the second layer to the third layer, µ is a bias, and N i is the number of neurons in the i th layer. The sigmoid activation function ¾ .¢/ is de ned by ¡z
(46)
¾ .z/ D 1=.1 C e /
The neural networks input– output mapping equation (45) can be expressed in matrix form as follows:
¡
ynn D W T ¾ V T xnn
¢
(47)
where W 2 R N2 C 1 £ N3 ; V 2 R N1 C 1 £ N2 ; xnn 2 R N1 C 1 , and ¾:R N2 7! R N2 C 1 are de ned as follows:
2
µ w1
w11 w21 :: :
6 W D 4µw2 T
:: :
w12 w22 :: :
3
¢¢¢ ¢ ¢ ¢7 5; :: :
2
µ v1
6 V D 4µv2 T
:: :
h xnn D 1;
0 xnn ; 1
0 xnn ;:::; 2
£
0 xnn N1
v11 v21 :: :
v12 v22 :: :
3
¢¢¢ ¢ ¢ ¢7 5 :: :
iT
¡
¾.z/ D 1; ¾ .z 1 /; ¾ .z 2 /; : : : ; ¾ z N2
¡
¢
¢¤T
kV k F · V M
¡
¢
¢
¡
¢
(53)
The Taylor series expansion of ¾ in Eq. (53) may be written as
¡ ¢ ¡ ¢ d¾ ¡ ¢ VQ T xnn C O VQ T xnn (54) ¾ V T xnn D ¾ VO T xnn C dz z D VO T xnn By the use of Eqs. (53) and (54), the hidden layer output error equation can be written as follows:
¡
¢
¡
¾Q D ¾ 0 VO T xnn VQ T xnn C O VQ T xnn
¢
(55)
Then, the error of the output layer is expressed as follows:
¡
¢
¡
¢
¡
¢
WO T ¾ VO T xnn ¡ 1 D WO T ¾ VO T xnn ¡ W T ¾ V T xnn ¡ ².xnn /
¡
¢
£ ¡
¢
¡
D ¡ WQ T ¾ VO T xnn ¡ W T ¾ V T xnn ¡ ¾ VO T xnn
¢¤
¡ ².xnn /
(56)
¢
¡
(49)
(50)
¢
¡
¢
WO T ¾ VO T xnn ¡ 1D ¡ WQ T ¾ VO T xnn ¡ WQ T ¾ 0 VO T xnn VQ T xnn
¡
¢
¡
¡ WO T ¾ 0 VO T xnn VQ T xnn ¡ W T O VQ T xnn ¡ ².xnn /
£ ¡
¢
¡
¢
¡
¢
D ¡ WQ T ¾ VO T xnn ¡ ¾ 0 VO T xnn VO T xnn ¡ WO T ¾ 0 VO T xnn VQ T xnn C w
(48)
where W M and V M 2 R are known positive constants and k ¢ k F denotes the Frobenious norm of a matrix. The ideal weight matrices W and V satisfying Eq. (48) cannot be determined in anticipation because we have no information on the error term 1. Instead, the estimated values of the ideal weights, WO and VO , are used in the controller, and they are adjusted by the adaptive laws. Consequently, there exists an effect caused by the difference between the ideal weights W and V , and the estimated weights WO and VO . This effect is stated in the following lemma. Lemma 2: Let us de ne the weight estimation errors as WQ D W ¡ WO ; VQ D V ¡ VO , and Z D diag[W; V ]. Given a neural networks’ input xnn , the output error is expressed as ¡ ¢ £ ¡ ¢ ¡ ¢ ¤ WO T ¾ VO T xnn ¡ 1 D ¡ WQ T ¾ VO T xnn ¡ ¾ 0 VO T xnn VO T xnn ¡ WO T ¾ 0 VO T xnn VQ T xnn C w
¡
¾Q D ¾ ¡ ¾O D ¾ V T xnn ¡ ¾ VO T xnn
¡
where ².xnn / is the approximationerror satisfyingk².xnn /k · ² N for all xnn in some input space. The following assumptions are used in the design and analysis of the adaptive control law presented in this paper. Assumption 8: Here, x0nn D [x 1d ; x1 ; x2 ]T is de ned as the input vector of the neural networks, and this input satis es Eq. (48) for some ² N . Assumption 9: The ideal weights are bounded in the sense that kW k F · W M ;
Proof: The output error of the hidden layer for the input xnn is de ned as
Substituting Eq. (55) and W D WQ C WO and VQ D V ¡ VO into Eq. (56) yields
Multilayer neural networks can approximate a nonlinear function to any desired accuracy. This is known as the universal approximation capability.12;13 That is, for a continuous function 1:R N1 7! R N3 and an arbitrary constant ² N > 0, there exist an integer N2 , the number of neurons in the hidden layer, and ideal constant weight matrices W 2 R N 2 C 1 £ N3 and V 2 R N1 C 1 £ N2 such that 1 D W T ¾ V T xnn C ².xnn /
(52)
¢
¤ (57)
Therefore, Eq. (50) has been proved. When the sigmoid function ¾ .¢/ and its derivative d¾ .z/=dz are bounded by some constants and Assumption 8 is used, it can be shown that the high-order term O in Eq. (54) satis es the following inequality:
® ¡ ¢® ® ¡ ¢® ® ¡ ¢® ®O VQ T xnn ® · ®¾ V T xnn ® C ®¾ VO T xnn ® ® ¡
¢
®
C ®¾ 0 VO T xnn VQ T xnn ® · c C c C ck VQ k F kxnn k · c C ck VQ k F C ckx 1 kkVQ k F C ckx 2 kk VQ k F
(58)
where c is a generic symbol used to denote any nite constant. Note that the property of kAx k · k Ak F kxk is used in the preceding equations. Finally, Eq. (52) is proved by using Eqs. (51) and (58) and hboxAssumption 8 as follows: kw.t /k · kWQ k F cVM .c C ckx1 k C ckx 2 k/ C W M .c C ckVQ k F Q F C ckx 1 kk VQ k F C ckx2 kkVQ k F / C ² N · C 1 C C 2 k Zk C C 3 k ZQ k F kx 1 k C C4 k ZQ k F kx2 k
(59)
This lemma shows that the neural networks output error caused by the weight estimation error can be expressed as Eq. (50) and that the size of the term w in Eq. (50) is bounded by Eq. (52).
680 C.
LEE AND KIM
Design of the Adaptive Controller
The followinglemma de nes a robustterm v and shows a property for the stability analysis. Lemma 3: Consider w 2 R3 de ned as in Eq. (51) and v 2 R3 ; » , and kv 2 R de ned as vD¡
z2» » kz 2 k» C ²
(60)
» D kv .Z M C k ZO k F /.kx 1 k C kx 2 k/
(61)
kv ¸ maxfC3 ; C 4 g
(62)
1
2
2
1a
1
¤ ¤
PQ ] C .1=° / tr [ VQ T VPQ ] ¡ g2 u C .1=°w / tr [ WQ T W v
z 2T .w C v/ · kz 2 k[C 1 C C 2 k ZQ k F ] C ²
(63)
Proof: With the choice of v in Eq. (60), z 2T .w C v/ is expressed
where is the ideal control input when the exact aerodynamic model is available and is expressed as follows:
£
¤
¡
C v/ D
z T2 w
.kz 2 k» /2 ¡ kz 2 k» C ²
(64)
.kz 2 k» /2 C C 4 k ZQ k F kx 2 k] ¡ kz 2 k» C ²
(65)
Furthermore, from the de nition ZQ D Z ¡ ZO and Eqs. (61) and (62), we have Q F kx1 k C C 4 k ZQ k F kx 2 k · kv kZ ¡ ZO k F .kx 1 k C kx 2 k/ · » C 3 k Zk
(66) When Eqs. (65) and (66) are used, Eq. (63) can be proved as follows:
C C 2 k ZQ k F ] C
.kz 2 k» /2 D kz 2 k[C 1 kz 2 k» C ²
kz 2 k» ² · jz 2 k[C 1 C C2 k ZQ k F ] C ² kz 2 k» C ²
(67)
PQ ] C .1=° / tr [ VQ T VPQ ] Cz 2T v C .1=°w / tr [ WQ T W v
£
¤
¡
¢
(68)
where AO and v are de ned as in Eqs. (44) and (60), respectively. WO and VO are computed by the following adaptive laws:
£ ¡ ¢ ¡ ¢ ¤ P WO D ¡°w ¾ VO T xnn z T2 ¡ ¾ 0 VO T xnn VO T xnn z T2 ¡ · °w WO h ¡ 0
P VO D ¡°v xnn ¾ VO T xnn
¢T
WO z 2
iT
¡ ·°v VO
(69) (70)
where ·; °w , and °v 2 R are some positive design parameters. Furthermore,the bound of the tracking error and the neuralnetworks parameter estimation error may be kept as small as desiredby adjusting the design parameters. Proof: Let us consider the Lyapunov function candidate, V2 D 12 z 1T z 1 C 12 z T2 z 2 C .1=2°w / tr [ WQ T WQ ] C .1=2°v / tr [ VQ T VQ ]
(71)
¢
(74)
¤
where 12 is de ned as g2 [u ¡ u] and represents the term caused by the modeling error. Substituting Eqs.(50), (69), and (70) into the preceding equation gives the following equation.
£ ¡
VP2 D ¡k1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d C z 2T ¡ WQ T ¾ VO T xnn
¡
¢
¤
©
£ ¡
¢
¡
©
£ ¡
C tr VQ T xnn ¾ 0 VO T xnn
¡
¢
¡
¢
¢T
¤T
WO z 2
T
C · VO
¢
ª
¤ª
ª¢¢
(75)
T
When the propertyof the trace, tr[yx ] D x y, is used, the preceding equation is simpli ed as follows: O C z T .w C v/ VP2 D ¡k 1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d C · tr[ ZQ T Z] 2
(76)
O D tr[ ZQ T Z ] ¡ tr[ ZQ T Z] Q · k ZQ k F Z M ¡ Using the property tr[ ZQ T Z] 2 Q k Zk F and Eqs. (39) and (63), we have
£ k1 c12 c22 VP2 · ¡ .1 ¡ c1 /kz 1 k2 ¡ k2 kz 2 k2 C C · k ZQ k F Z M 2 2k1 .1 ¡ c1 / ¤
Q 2 C kz 2 k[C 1 C C 2 k ZQ k F ] C ² D ¡ ¡ k Zk F
µ
The designprocessof the adaptivecontrollerand stabilityanalysis are stated in the following theorem. Theorem 2: Consider the system Eqs. (25) and (26), where the controlinputu is de ned in Eq. (68)andthe adaptivelaws are de ned in Eqs. (69) and (70). Then, the trajectories of the system as well as the neural networks’ weights are locally uniformly ultimately bounded: u D gO2¡1 ¡k 2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ AO C WO T ¾ VO T xnn C v
¡
C tr WQ T ¾ VO T xnn z 2T ¡ ¾ 0 VO T xnn VO T xnn z 2T C · WO
z 2T .w C v/ · kz 2 k[C 1 C C 2 k ZQ k F C C 3 k ZQ k F kx 1 k
(73)
VP2 D ¡k1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d ¡ z 2T 12 C z 2T WO T ¾ VO T xnn
¡ ¾ 0 VO T xnn VO T xnn ¡ WO T ¾ 0 VO T xnn VQ T xnn C w C v
Substituting Eq. (52) into the preceding equation gives
z 2T .w C v/ · kz 2 k[C 1 C C 2 k ZQ k F C » ] ¡
¢
u ¤ D g2¡1 ¡ k2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ A C WO T ¾ VO T xnn C v
©
z T2 .w
(72)
u¤
With the de nition of u ¤ in Eq. (73), Eq. (72) becomes
where ² is a positive constant. Then, the following inequality is satis ed:
as
When Eq. (38) is used the derivative of the Lyapunov function V2 along the trajectories of Eqs. (25) and (26) with the adaptive laws, Eqs. (69) and (70), are computed as £ VP2 D ¡k1 kz 1 k2 C z T g1a x d C z T g T z 1 C g T z 1 C A C g2 u C g2 u ¤
¡
k2 k2 C1 kz 2 k2 ¡ kz 2 k ¡ k2 2 2
C C 2 kz 2 kk ZQ k F C
¶2 ¡
k1 .1 ¡ c1 /kz 1 k2 2
· Q 2 · k Z k F ¡ [k ZQ k F ¡ Z M ]2 2 2
c12 c22 C2 · Z 2M C 1 C²C 2k 1 .1 ¡ c1 / 2k2 2
(77)
With the de nition C5 D [c12 c22 =2k1 .1 ¡ c1 /] C .C 12 =2k 2 / C .· Z 2M =2/ C ², the preceding equation becomes k1 k2 · VP2 · ¡ .1 ¡ c1 /kz 1 k2 ¡ kz 2 k2 C C 2 kz 2 kk ZQ k F ¡ k ZQ k2F 2 2 2 C C5 D ¡
µ
k1 k2 · Q 2 .1 ¡ c1 /kz 1 k2 ¡ kz 2 k2 ¡ k Zk F 2 4 4
1 kz 2 k ¡ 2 k ZQ k F
¶T
2
k2 6 2 4 ¡C 2
3
µ
¶
¡C 2 7 kz 2 k C C5 · 5 k Zk Q F 2
(78)
If k2 and · are chosen to satisfy k2 · ¡ 4C 22 > 0, the last matrix in the preceding equation becomes positive de nite. Therefore, the following inequality is satis ed: VP2 · ¡.k 1 =2/.1 ¡ c1 /kz 1 k2 ¡ .k 2 =4/kz 2 k2 ¡ .· =4/k ZQ k2F C C 5
(79)
For a positive constant ¹2 satisfying 0 < ¹2 < minf.k1 =2/.1 ¡ c1 /; k 2 =4; · =4 minf°w ; °v gg, nally, the following equation is obtained for the derivative of the chosen Lyapunov candidate function:
681
LEE AND KIM
VP2 · ¡2¹2 V2 C C 5
(80)
Equation (80) implies that VP2 < 0 when V2 > C 5 =2¹2 . Hence, the error states z 1 and z 2 and the weight estimation error ZQ are bounded and converge exponentially to the residual set D2 :
The F-16 aircraft model is used in this paper,11 and the following command values of ®; ¯, and Á are applied to the aircraftin a steadystate level ight of V D 500 ft/s, h D 10,000 ft: ®d D 2:659;
©
®d D 10;
kz 1 k2 C kz 2 k2 D2 D z 1 ; z 2 2 R3 ; ZQ F 2 R N1 C N2 C 2 £ N2 C N3 C .1=maxf°w ; °v g/k ZQ k2F · C 5 =¹2
Because c1 ; c2 ; C 1 ; Z M , and ² are independentof k 1 and k 2 , the size of the set D2 can be made arbitrarily small by adjusting the design parameters k1 and k2 . Theorem 2 represents the design procedure of the adaptive controller to track the ®; ¯, and Á commands when the modeling errors exist. This shows that, if the controlleris applied, the tracking errors and the parameter estimation error of the neural networks converge to a compact set and also shows that the size of the set is adjustable by tuning the design parameters. The universal approximation theorem only guarantees the existence of the ideal weight and the ideal number of the hidden layer neurons. In this paper, the weight of neural networks are adjusted by the adaptive laws; however, the size of hidden layer neuron N 2 is xed. It also affects the approximation capacity of the neural networks. If too small a value of N 2 is chosen, the neural networks may not compensate for the effect of the modeling error properly. Therefore, the value of N 2 should be chosen carefully in considerationof the complexity and the size of the modeling error.
V.
Numerical Simulation
In Sec. III, the design methodology of the ight controller to track the ®; ¯, and Á commands is proposed when full knowledge of the aerodynamic characteristics is available, and in Sec. IV, the adaptive controller based on neural networks is designed to eliminate the effect caused by the aerodynamic modeling error. This section presents numerical simulation results for each controller to demonstrate the performance of the proposed nonlinear control laws.
Fig. 3
®d D ¡ 2;
ª
¯d D 0; ¯d D 0; ¯d D 0;
Ád D 0 deg;
0·t ·1s
Ád D 50 deg;
1 · t · 10 s
Ád D 0 deg;
10 · t · 20 s
To obtain differentiable commands satisfying Assumption 1, the third-order linear command lter is used. The aerodynamic modeling error is made arbitrarily, and the average error for each coef cient is listed in Table 1. The controller design parameters are chosen as follows: k1 D 3, k2 D 8, · D 0:2, k v D 0:153, ² D 0:001, Z M D 0:6142, and °w D °v D 30. Figure 3 represents the simulation results of the backstepping controller described in Sec. III when the exact aerodynamic model is available. The solid line represents the simulation result of the backsteppingcontroller proposed in Sec. II, and the dotted line represents the command signal. As expected, the proposed backstepping controller makes the ®; ¯, and Á follow the command values in a satisfactory way. Figure 4 shows the simulation results when the modeling error exists. The solid lines represent the simulation result of the adaptivecontroller based on neural networks in Sec. IV, and the dashed lines represent the result of the controller in Sec. III. The dotted lines represent the command signals. It is shown that the system output of the adaptive controller tracks the command quite well, even if large modeling uncertaintyexists. It can be said that the performance of the system is not degraded in the case of the neural networks adaptive controller. Table 1
Average modeling errors of the aerodynamic coef cients, %
Coef cients
Error
Coef cients
Error
Coef cients
Error
Cl Cl p C lr Cl±a Cl±r
79.6 16.0 148.9 141.8 69.8
Cm Cm q Cm ±e —— ——
207 77.6 146.2 —— ——
Cn Cn p Cnr Cn ±a Cn ±r
180.1 86.7 94.9 48.3 228.5
Time response of x1 and u (without modeling error): backstepping controller in Sec. III —— , and command¢ ¢ ¢ ¢ .
682
Fig. 4
¢ ¢ ¢ ¢ .
LEE AND KIM
Time response of x1 and u (with modeling error): adaptive controller in Sec. IV —— , backstepping controller in Sec. III - - - -, and command
VI.
Conclusions
A controllerfor a six-degree-of-freedom nonlinear ight model is proposed,and its stability is analyzedby using the Lyapunovtheory. The backstepping controller is used to track the ®; ¯, and Á commands with the assumption that the aerodynamic characteristicsare fully understood. It is shown that, if the controller is applied, the tracking error exponentiallyconverges to a compact set and the size of the set can be made arbitrarilysmall by tuning the design parameters. It is not necessary to make the controller gain excessively large to guarantee stability because the timescale separation assumption is not used. An adaptive controller based on neural networks is used to compensate for the effects of the aerodynamic modeling errors. The neural networks’ parameters are adjusted to offset the error term. The closed-loop stability of the error states and the parameters of the neural networks are examined by the Lyapunov theory, and it is shown that the error states and the parameter estimation errors exponentially converge to a compact set whose size is adjustable by the design parameters. Finally, a nonlinear simulation of an aircraft maneuver is performed to demonstrate the performance of the proposed control laws.
Acknowledgment This research was supported by the Korea Science and Engineering Foundation Grant 1999-2-305-004-3.
References 1
Lane, S. H., and Stengel, R. F., “Flight Control Design Using Non-Linear Inverse Dynamics,” Automatica , Vol. 24, No. 4, 1988, pp. 471– 483. 2 Menon, P., Badgett, M., and Walker, R., “Nonlinear Flight Test Trajec-
tory Controllers for Aircraft,” Journal of Guidance, Control, and Dynamics, Vol. 10, No. 1, 1987, pp. 67 – 72. 3 Snell, S. A., Enns, D. F., and Garrard, W. L., Jr., “Nonlinear Inversion Flight Control for a Supermaneuverable Aircraft,” Journal of Guidance, Control, and Dynamics, Vol. 15, No. 4, 1992, pp. 976– 984. 4 Schumacher, C., and Khargonekar, P. P., “Stability Analysis of a Missile Control System with a Dynamic Inversion Controller,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 3, 1998, pp. 508– 515. 5 Farrell, J. A., “Stability and Approximator Convergence in Nonparametric Nonlinear Adaptive Control,” IEEE Transactions on Neural Networks, Vol. 9, No. 5, 1998, pp. 1008– 1020. 6 Lewis, F. L., Yesildirek, A., and Liu, K., “Multilayer Neural-Net Robot Controller with Guaranteed Tracking Performance,” IEEE Transactions on Neural Networks, Vol. 7, No. 2, 1996, pp. 388– 399. 7 Singh, S. N., Yim, W., and Wells, W. R., “Direct Adaptive and Neural Control of Wing-Rock Motion of Slender Delta Wings,” Journal of Guidance, Control, and Dynamics, Vol. 18, No. 1, 1995, pp. 25 – 30. 8 Kim, B. S., and Calise, A. J., “Nonlinear Flight Control Using Neural Networks,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 1, 1997, pp. 26– 33. 9 Krsti´ c, M., Kanellakopoulos, I., and Kokotovi´c, P., Nonlinear and Adaptive Control Design, Wiley, New York, 1995, Chap. 2. 10 Stevens, B. L., and Lewis, F. L., Aircraft Control and Simulation, Wiley, New York, 1992, Chap. 2. 11 Morelli, E. A., “Global Nonlinear Parametric Modeling with Application to F-16 Aerodynamics,” Proceedings of the 1998 American Control Conference, IEEE Publications, Piscataway, NJ, 1998, pp. 997– 1001. 12 Hornik, K., Stinchcombe, M., and White, H., “Multilayer Feedforward Networks are Universal Approximators,” Neural Networks, Vol. 2, No. 5, 1989, pp. 359– 366. 13 Funahashi, K., “On the Approximate Realization of Continuous Mapping by Neural Networks,” Neural Networks, Vol. 2, No. 3, 1989, pp. 183– 192.
Nonlinear Adaptive Flight Control Using Backstepping and Neural Networks Controller Taeyoung Lee¤ and Youdan Kim† Seoul National University, Seoul 151-742, Republic of Korea A nonlinear adaptive ight control system is proposed using a backstepping and neural networks controller. The backstepping controller is used to stabilize all state variables simultaneously without the two-timescale assumption that separates the fast dynamics, involving the angular rates of the aircraft, from the slow dynamics, which includes angle of attack, sideslip angle, and bank angle. It is assumed that the aerodynamic coef cients include uncertainty, and an adaptive controller based on neural networks is used to compensate for the effect of the aerodynamic modeling error. Under mild assumptions on the aerodynamic uncertainties and nonlinearities, it is shown by the Lyapunov stability theorem that the tracking errors and the weights of neural networks exponentially converge to a compact set. Finally, nonlinear six-degree-of-freedom simulation results for an F-16 aircraft model are presented to demonstrate the effectiveness of the proposed control law.
Nomenclature F I L ; M; N p; q; r qN T V ®; ¯ ±e ; ±a ; ±r Á; µ ; Ã
F
= = = = = = = = = =
search, the gain of the inner-loop controller is set much larger than that of the outer-loop controller, and it is assumed that the aircraft dynamics satis es this property. This, therefore, does not guarantee closed-loop stability. Schumacher and Khargonekar analyzed theoretically the stability of the ight control system with the two-timescale separation assumption.4 Using the Lyapunov theory, they determined the minimal inner-loop gain guaranteeing closed-loop stability. However, this approach is so complicated and conservativethat the calculated value of the minimal inner-loop gain is too large to be applied in the ight controller. It may excite unmodeled dynamics or saturate the control inputs and, therefore, cause robustness problems. Another dif culty related to the applicationof feedbacklinearization to a ight control system is that a complete and accurate aircraft dynamic model including aerodynamic coef cients is required. It is dif cult to identify accurately aerodynamic coef cients because they are nonlinear functions of several physical variables. Gain scheduling with a linear H1 design is a traditional approach to overcome this problem; however, it can guarantee the desired performance only when conditions comprise a small perturbation and slow variance. Neural networks have been proposed recently as an adaptive controller for nonlinearsystems.5;6 By the use of their universalapproximation capability,the adaptive controller based on neural networks can be designed without signi cant prior knowledge of the system dynamics. In ight control problems, the applications of adaptive neural networks can be found in Refs. 7 and 8. Single-layer neural networks are used to compensate for unknown dynamics7 and inversion error.8 This paper proposes a backstepping and neural networks controller for a nonlinear ight dynamic system and analyzes the stability of the proposedcontrol system using the Lyapunovtheory.The controller is designed by using the backstepping approach9 with the assumption that all aerodynamic coef cients are fully understood. The proposed method also intermediately uses the fast states p; q, and r as control inputs. However, it considersthe transientresponses of the fast states and does not require the two-timescale assumption. Therefore, it is not necessary to make the controller gain impracticably large to guaranteethe closed-loopstability because the timescale separation assumption is not used in the design or analysis. The effects of the modeling error in some aerodynamic coef cients are also considered and are compensated for by multilayer neural networks.The parametersof the neural networks are adjusted to offset the term generatedby the modeling error. The adaptivecontroller based on multilayer neural networks is an extension of the work described in Ref. 6. In that paper, only the state variables can be used as input for the neural networks. This paper generalizes it by adding a robust control term, which allows the use of neural network inputs that do not belong to the states.The main contributionof
aerodynamic force about the body- xed frame moment of inertia aerodynamic rolling, pitching, yawing moment roll, pitch, yaw rate about the body- xed frame dynamic pressure thrust velocity angle of attack, sideslip angle elevator, aileron, rudder angle roll, pitch, yaw angle
I.
Introduction
EEDBACK linearization is a theoretically established and widely used method in controlling nonlinear systems. Exact input– output linearization is often used to control speci c output variable sets in nonlinear ight control problems.1 Unfortunately, this direct application of feedback linearization requires the second or third derivatives of uncertain aerodynamic coef cients and does not guarantee internal stability for nonminimum phase systems. Another approach to designing ight control laws with feedback linearization is to separate the ight dynamics into fast and slow dynamics by using timescale properties.2;3 This method allows the controller design process to be performed without state transformation, because each separated subsystem is square: The number of control inputs is equal to the number of states. The design process of this method can be divided into two steps. In the outer loop, the controller for the slow states ®; ¯, and Á is designedto facilitatetracking of the given commands by assuming that the fast states p; q, and r are control inputs, which achieve their commanded values instantaneously. With the slow states controller designed in the outer loop, a separated inner-loop controller is designed to make the fast states p; q, and r follow the outer loop’s control input trajectories using the real control inputs: aileron, rudder, and elevator. This method can be justi ed only if there is suf cient timescale separation between the inner- and outer-loop dynamics because the fast states p; q, and r are used as control inputs in the outer-loopsystem. Hence, the states p; q, and r in the inner loop should be much faster than the states ®; ¯, and Á in the outer loop. The stability of this timescale separation approach may be analyzed by the singular perturbation theory. However, in most nonlinear ight control reReceived 29 November 1999; revision received 1 July 2000; accepted for c 2000 by the American Institute publication 2 October 2000. Copyright ° of Aeronautics and Astronautics, Inc. All rights reserved. ¤ Graduate Student, Department of Aerospace Engineering. † Associate Professor, Department of Aerospace Engineering; ydkim @snu.ac.kr. Senior Member AIAA. 675
676
LEE AND KIM
II.
2
qP D I5 pr ¡ I6 . p ¡ r / C I7 M
(5)
rP D ¡I2 qr C I8 pq C I4 L C I9 N
(6)
I1 D ¡
I7 D (1)
(2)
1 ; Iy
I8 D
Ix z ; I x Iz ¡ Ix2z
Ix I x Iz ¡ I x2z
¤
¤
C .br =2V /C lr .®/ qN Sb
Substituting the aerodynamic coef cients into the ight dynamic equations yields
3 2
¡.sin ® =cos ¯/C x q .®/cN C .cos ® =cos ¯/C z q .®/cN ¡cos ® sin ¯C x q .®/cN ¡ sin ® sin ¯C z q .®/cN 0
¡.sin ® =cos ¯/C x ±e .®/ C .cos ® =cos ¯/C z ±e .®; ¯/ ½VS 4 ¡cos ® sin ¯C x ±e .®/ ¡ sin ® sin ¯C z ±e .®; ¯/ C 2m 0
0 cos ¯C y±a .¯/ 0
1 0 sin Á tan µ
32 3
¡sin ® tan ¯ p ¡cos ® 54 q 5 r cos Á tan µ
32 3
p 0 cos ¯C yr .®/b54 q 5 r 0
32 3
0 ±e cos ¯C y±r .¯/5 4±a 5 0 ±r
3
.1=cos ¯/.sin ® sin µ C cos ® cos Á cos µ / g C 4cos ® sin ¯ sin µ C cos ¯ cos µ sin Á ¡ sin ® sin ¯ cos Á cos µ 5 V 0
2 3
Ix z Iy
£
2
2
I9 D
I6 D
L D Cl .®; ¯/ C C l±a .®; ¯/±a C C l±r .®; ¯/±r C .bp=2V /Cl p .®/
(3)
2
Iz ¡ I x ; Iy
Ix .I x ¡ I y / C I xz2 ; I x Iz ¡ I x2z
®P ¡.sin ® =cos ¯/[T C C x .®/qN S] C .cos ® =cos ¯/C z .®/qN S ¡cos ® tan ¯ 4¯P 5 D 1 4¡cos® sin ¯[T C C x .®/qN S] C cos ¯C y .¯/qN S ¡ sin ® sin ¯C z .®; ¯/qN S 5C 4 sin ® mV 0 1 ÁP 0 ½S 4 C cos ¯C y p .®/b 4m 0
I5 D
£
g sin ® sin ¯ Fz C [cos ® sin ¯ sin µ C cos ¯ cos µ sin Á mV V
2
Ix z .Ix ¡ I y C Iz / I x Iz ¡ I x2z
Fx D C x .®/ C C x ±e .®/±e C .cq N =2V /C x q .®/ qN S
cos ® sin ¯ cos ¯ Fy ¯P D sin ®p ¡ cos ®r ¡ [T C Fx ] C mV mV
2 3
I4 D
I2 D
De nitionsof state and controlvariables,forces,and moments in the precedingequationsare describedin the Nomenclature.Itis assumed that the aerodynamicforcesand moments are expressedas functions of angle of attack, sideslip angle, angular rates, and control surface de ection.11 For example, Fx and L are expressed as follows:
sin ® ®P D ¡cos ® tan ¯ p C q ¡ sin ® tan ¯r ¡ [T C Fx ] mV cos ¯
¡ sin ® sin ¯ cos Á cos µ ]
I z .I z ¡ I y / C Ix2z ; Ix Iz ¡ Ix2z
Iz ; 2 I x Iz ¡ I xz
I3 D
C g[¡cos ® cos ¯ sin µ C sin ¯ sin Á cos µ
¡
(8)
sin Áq C cos Ár (9) ÃP D cos µ where the moments of inertia Ii ; i D 1; 2; : : : ; 9, are de ned as follows:
Problem De nition
g cos ® Fz C [sin ® sin µ C cos ® cos Á cos µ ] m V cos ¯ V cos ¯
(7)
µP D cos Áq ¡ sin Ár
cos ® cos ¯ sin ¯ sin ® cos ¯ VP D [T C Fx ] C [Fy ] C [Fz ] m m m
C
2
ÁP D p C tan µ .sin Áq C cos Ár /
This paper presents a controller for a nonlinear aircraft. The task of the controller is to track the commands of ®; ¯, and Á when aerodynamic model uncertainties exist. The body- xed axes, nonlinearequationsof motion for an aircraft over a at Earth are given by10
C sin ® cos ¯ cos Á cos µ ]
(4)
pP D I2 pq C I1 qr C I3 L C I4 N
this paper is that the stability of nonlinear ight dynamics is proven mathematically without unrealistic restrictions. This paper is organized as follows. A nonlinear ight model is described in Sec. II. In Sec. III, the backstepping controller is designed when modeling error does not exist. The neural networks adaptive controller is then designed in Sec. IV when aerodynamic modeling error is present. Finally, a numerical simulation of a sixdegree-of-freedom F-16 aircraft model is performed to verify the effectiveness of the proposed algorithm in Sec. V.
2
3
2
(10)
3
I2 pq C I1 qr pP I3 Cl .®; ¯/qN Sb C I4 C n .®; ¯/qN Sb 4qP 5 D 4 I5 pr ¡ I6 . p 2 ¡ r 2 /5 C 4 5 I7 C m .®/qN S cN rP I4 Cl .®; ¯/qN Sb C I9 C n .®; ¯/qN Sb ¡I2 qr C I8 pq
2 ½VS 4 C 4
2
I3 C l p .®/b C I4 C n p .®/b 0 I4 C l p .®/b C I9 C n p .®/b 0
C qN S 4 I7 C m ±e .®/cN 0
0 I7 C m q .®/cN 0
32 3
I3 C lr .®/b C I4 C nr .®/b p 5 4q 5 0 I4 C lr .®/b C I9 C nr .®/b r
I3 Cl ±a .®; ¯/b C I4 C n±a .®; ¯/b 0 I4 Cl ±a .®; ¯/b C I9 C n±a .®; ¯/b
32 3
I3 Cl ±r .®; ¯/b C I4 C n±r .®; ¯/b ±e 5 4±a 5 0 I4 Cl ±r .®; ¯/b C I9 C n±r .®; ¯/b ±r
(11)
677
LEE AND KIM
µ ¶
µ
µP 0 D 0 ÃP
cos Á sin Á =cos µ
¶
2 3
p ¡sin Á 4q 5 cos Á =cos µ r
(12)
Lemma 1 can be proved by a suf cient condition that the set of rows of g1 is linearly independent. According to Assumption 3 and Lemma 1, the size of g1¡1 and g1a is bounded by some positive constants cg1a ; cg¡1 :
xP1 D f1 .®; ¯/ C g1 .®; ¯; Á; µ /x 2 C g1a .®; ¯/x 2 (13)
C h 1 .®; ¯/u C f 1g .®; ¯; Á; µ / xP2 D f2 .®; ¯; p; q; r/ C f 2a .®; ¯/x 2 C g2 .®; ¯/u
(14)
xP3 D f3 .Á; µ /x 2
(15)
where f; g, and h represent the terms of Eqs. (10– 12) in order. The preceding equations are mainly used for the controller design and stability analysis processes of the following sections.
III. Controller Design and Stability Analysis When designing a ight control system with the two-timescale assumption, the inner-loop controller is designed to control the fast states x2 using the control input u, where the desired values of the fast states x 2d are given by the outer loop. In the outer loop, the controller is designed to control the slow states x 1 using the fast states x 2 as control inputs. The inner-loop controller neglects the transient responses of the fast states x 2 . It assumes that the fast states track their commanded values instantaneously and that the control surface de ection has no effect on the outer-loop dynamics. The structure of the two-timescale controller is shown in Fig. 1. In each feedback loop, control laws x2d and u are designed separately. In this paper, the backstepping method is used to design a controller.9 The backstepping design procedure can be viewed as the two-timescale approach because the fast states x 2 are used as control inputs for the slow states x 1 intermediately. However, this methodologyconsidersthe transientresponsesof the fast states and, therefore,does not require the timescale separationassumption.The following assumptionsare used in the design and analysisprocesses. Assumption 1: The desired trajectories x1d D [® d ; ¯ d ; Á d ]T are bounded as
®£ ¤® ® x d ; xP d ; xR d ® · cd 1
1
where cd 2 R is a known positive constant and k ¢ k denotes the 2-norm of a vector or a matrix. Assumption 2: The total velocity and the dynamic pressure are constant. VP D 0;
qPN D 0
(18)
jµ j · µm < ¼ =2
where µm 2 R is a positive constant. The following lemma is used in the design and analysis processes. Lemma 1: There exist positive constants ®m ; ¯ m , and µm 2 R such that g1 .®; ¯; Á; µ / is invertible for all Á 2 R and all ®; ¯, and µ 2 R satisfying j®j · ®m ; j¯j · ¯m , and jµ j · µm .
Fig. 1
(20)
Assumption 5: The following inequalities are satis ed for constants cg1a and cg¡1 in Eqs. (19) and (20): 1
(21)
cg1a cg ¡1 < 1 1
Note that g1a is composed of the aerodynamic coef cient terms related to the angular rate and multiplied by a very small quantity ½ =m, and therefore, the magnitude of g1a is very small. Also note that the norm of cg¡1 is mainly in uenced by the pitch angle µ , 1 and therefore Assumption 5 is closely related to the maximum pitch angle in Lemma 1. For the aerodynamic model considered in this study, the numerical value of cg1a cg ¡1 is less than 0:13. 1 Assumption 6: There exist positive constants®m and ¯m 2 R such that g2 .®; ¯/ is invertible for all ® and ¯ 2 R satisfying j®j · ®m and j¯j · ¯ m . Assumption 7: The control surface de ection has no effects on the aerodynamic force component: h 1 .®; ¯/ D 0
(22)
z 1 D x1 ¡ x1d
(23)
Note that g2 in Eq. (14) represents the input matrix of the control u to the dynamics of the angular rates p; q; and r. Also, h 1 in Eq. (13) represents the aerodynamic force component caused by the control surface de ection. Because the control surfaces of the aircraft are designedto control each axes’ angular rate of aircraft independently, the input matrix g2 is invertiblefor all cases, and the magnitude of h 1 is very small compared to other aerodynamic terms in the dynamic equation. Therefore, it can be assumed that g2 is invertible, and h 1 D 0. Numerical studies for the aerodynamic model considered in this study also show that g2 is always invertible, and the size of h 1 is negligible. Let us introducethe error state variablesz 1 and z 2 2 R3 as follows: z 2 D x2 ¡
x2d
(24)
where x1d and x 2d are the desiredtrajectoriesof x1 and x2 , respectively. Note that x 1d is given by command signals and x2d will be de ned later. Using Eqs. (13) and (14), and Assumption 7, the dynamic equations of the error states are given as follows:
(17)
Assumption 3: There exist positive constants®m and ¯m 2 R such that the magnitudes and derivatives of f 1 ; f 2 ; f 2a ; g1a , and g2 are bounded for all ® and ¯ 2 R satisfying j®j · ®m and j¯j · ¯m . Assumption 4: The magnitude of µ is bounded as
(19)
1a
(16)
1
1
® ® ®g1 .®; ¯; Á; µ /¡1 ® · c ¡1 g1 ® ® ®g1a .®; ¯/® · cg
Let us de ne the states x 1 ; x 2 2 R3 ; x3 2 R2 , and the control inputs u 2 R3 as follows: x1 D [®; ¯; Á]T , x2 D [p; q; r]T , x 3 D [µ; Ã ]T , and u D [±e ; ±a ; ±r ]T . With the choices of x1 ; x 2 ; x 3 , and u, the ight dynamic Eqs. (10– 12) can be rearranged as
zP 1 D xP1 ¡ xP1d D f 1 C g1 x 2 C g1a x 2 C f 1g ¡ xP1d zP 2 D xP2 ¡
xP2d
D f 2 C f 2a x 2 C g2 u ¡
xP2d
(25) (26)
Theorem 1: Consider the system in Eqs. (25) and (26), where the control input u is de ned in Eq. (27). Then, the solutions of the system are locally uniformly ultimately bounded:
£
u D g2¡1 ¡k 2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ A
¤
(27)
where and A 2 R are de ned in Eqs. (28) and (29) and k 1 and k 2 2 R are positive design parameters: x2d
3£1
£
x 2d D g1¡1 ¡k1 z 1 ¡ f 1 ¡ f 1g C xP 1d
Structure of the two-timescale controller.
¤
(28)
678
LEE AND KIM
¤ @ x2d £ f1 C g1 x2 C g1a x 2 C f 1g @ x1
A D f2 C f 2a x2 ¡ ¡
VP1 · ¡2¹1 V1 C
£ ¤ @ x 2d f 3 x 2 ¡ g1¡1 k 1 xP1d C xR1d x @ 3
(29)
Furthermore, the bound of the tracking error may be kept as small as desired by adjusting the design parameters. Proof: By Assumptions 1 and 3, the following inequalities are satis ed for certain positive constants c f1 ; c f 1g , and cxP d : 1
(30)
k f 1 .®; ¯/k · c f 1
® ® ® f 1g .®; ¯; Á; µ /® · c f 1g ® ® ®xP d ® · c d
(31)
2
h
g 1¡1
i
(33)
k1 kz 1 k C c f 1 C c f 1g C cxP d 1
Let us consider the Lyapunov function candidate, V1 D 12 z 1T z 1 C 12 z 2T z 2
(34)
The derivative of the Lyapunov function V1 along the trajectoriesof Eqs. (25) and (26) is given by @ V1 @ V1 (35) VP1 D zP 1 C zP 2 @z 1 @ z2 Substitution of Eqs. (25) and (28) into Eq. (35) yields the following equation:
¤ @ V1 £ ¤ @ V1 @ V1 £ VP1 D ¡k 1 z 1 C g1a x2 C g1 x2 ¡ x2d C zP 2 @z 1 @ z1 @ z2 D ¡k1 kz 1 k2 C z 1T g1a x 2 C z T1 g1 z 2 C z 2T zP 2 D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T1 g1a z 2 C z 1T g1 z 2 C z 2T zP 2 (36) Substituting Eqs. (26) and (29) into Eq. (36) yields
¶
@ x2d @xd @xd @xd xP 1 ¡ 2 xP3 ¡ 2d xP1d ¡ 2d xR1d @ x1 @ x3 @ x1 @ xP 1
(37)
£
D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T2 g1Ta z 1 C g1T z 1 C A C g2 u
¤
(38)
Because f 1 in Eq. (28) is dependenton V , =@ V / VP must be included in Eq. (37); however, it is neglected by using Assumption 2 in this study. Substituting Eq. (27) into Eq. (38) and using Eqs. (20) and (33) give the following equations: the term ¡z 2T .@ x2d
VP1 D ¡k1 kz 1 k2 ¡ k 2 kz 2 k2 C z T1 g1a x 2d · ¡k1 kz 1 k2 ¡ k 2 kz 2 k2
®
® ®
¡
¢
1
¢
µ
· ¡
k1 k1 .1 ¡ c1 /kz 1 k2 ¡ .1 ¡ c1 / kz 1 k 2 2 C
¼ c21 c22 2 2 kz 1 k C kz 2 k · D1 D z 1 ; z 1 2 R 2¹1 k1 .1 ¡ c1 / »
(42)
Because c1 and c2 do not depend on k1 and k2 , the size of the set D1 can be made arbitrarily small by adjusting the design parameters k1 and k 2 . Theorem 1 represents the design procedure of the controller to track the ®; ¯, and Á commands, and also shows that the tracking error of the control system converges to a compact set whose size is adjustable by the design parameters. Because the timescale separation assumption is not used in this study, it is not necessary to make the control gain large to guarantee closed-loop stability.
IV.
Neural Networks Adaptive Controller Design
In the preceding section, the backstepping controller is designed with the assumption that the aerodynamic characteristics are fully known. Because aerodynamic coef cients are highly nonlinear and dependent on lots of physical variables, it is very dif cult to identify them exactly. The difference between the mathematical model and the real system may cause performance degradation. To overcome this drawback, multilayer neural networks are used in this study. The weights of the neural networks are adjusted to compensate for the effect of the modeling error. The adaptive controller based on multilayer neural networks is an extension of the work described in Ref. 6. In this paper, it is generalized such that the variables that do not belong to the states can be used as neural networks’ inputs. A.
Effect of Modeling Error
In this paper, the modeling error in the body- xed angular rates dynamics is considered. The identi ed values of f 2 ; f 2a , and g2 in Eq. (14) are de ned as fO2 ; fO2a , and gO2 , respectively.Then, the control input is expressed as follows: uO D gO 2¡1 ¡k2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ AO
(43)
£ ¤ @ x 2d f 3 x 2 ¡ g1¡1 k 1 xP1d C xR1d @ x3 Substituting Eqs. (43) and (44) into Eq. (38) gives ¡
(44)
£
VP1 D ¡k1 kz 1 k2 C z 1T g1a x2d C z T2 g1Ta z 1 C g1T z 1 C A C g2 uO C g2 u ¡ g2 u D ¡k1 kz 1 k2 ¡ k 2 kz 2 k2 C z 1T g1a x 2d 4
C z 2T g2 [uO ¡ u] D ¡k 1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d ¡ z 2T 11
O represents the term caused by the modeling where 11 ´ g2 [u ¡ u] error, that is, the modeling error of f 2 ; f2a , and g2 adds the term z 2T 11 to the derivative of the Lyapunov function. B.
Neural Networks Structure
Given an input x0nn 2 R N1 , the three-layer neural networks as shown in Fig. 2 has an output ynn 2 R N3 as
c12 c22 ¡ k 2 kz 2 k2 2k1 .1 ¡ c1 /
k1 c12 c22 .1 ¡ c1 /kz 1 k2 ¡ k2 kz 2 k2 C 2 2k1 .1 ¡ c1 /
¤
¤ @xd £ AO D fO2 C fO2a x2 ¡ 2 f 1 C g1 x2 C g1a x 2 C f 1g @ x1
1
1
¶2
(41)
Note that ¹1 is positive because c1 is less than 1 by Assumption 5. Equation (40) implies that VP1 < 0 when V1 > c21 c22 =[4¹1 k1 .1 ¡ c1 /]. Therefore, the error states z 1 and z 2 are bounded and converge exponentially to the residual set D1 :
¡
C c f 1g C c xP d kz 1 k ¡ k2 kz 2 k2 D ¡k1 .1 ¡ c1 /kz 1 k2 C c1 c2 kz 1 k
c1 c2 ¡ k 1 .1 ¡ c1 /
1
¹1 D min[.k 1 =2/.1 ¡ c1 /; k 2 ]
¤
C ® g1a ®kz 1 k® x2d ® · ¡k1 1 ¡ cg1a cg¡1 kz 1 k2 C cg1a cg¡1 c f 1
¡ k 2 kz 2 k2 D ¡
c2 D c f 1 C c f 1g C cxP d
1
£
VP1 D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T1 g1a z 2 C z 1T g1 z 2 C z 2T [A C g2 u]
®
c1 D cg1a cg¡1 ;
µ
VP1 D ¡k1 kz 1 k2 C z 1T g1a x 2d C z T1 g1a z 2 C z 1T g1 z 2 C z 2T f2 C f 2a x2 C g2 u ¡
where the constants c1 ; c2 , and ¹1 are de ned as follows:
(32)
The bound of kx 2d k can be computed using Eqs. (19), (30), (31), and (32) as
® d® ®x ® · c
(40)
3
xP1
1
c12 c22 2k 1 .1 ¡ c1 /
(39)
ynni D
N2 X
"
Á
N1 X
wi j ¾ j D1
kD1
!
v j k x0nnk
#
C µv j Cµwi ;
i D 1; 2; : : : ; N 3 (45)
679
LEE AND KIM
where
d¾ ¾ .Oz / D dz z D zO 0
and w 2 R3 is de ned as
¡
¢
¡
¢
w.t / D ¡ WQ T ¾ 0 VO T xnn V T xnn ¡ W T O VQ T xnn ¡ ².xnn / (51)
Fig. 2
Furthermore, kwk satis es the following inequality for some positive constants Ci ; i D 1; 2; 3; 4:
Structure of three-layer neural networks.
Q F C C 3 k Zk Q F kx 1 k C C 4 k ZQ k F kx 2 k kwk · C 1 C C 2 k Zk
where v jk is the weight connectingthe rst layer to the second layer, wi j is the weight connecting the second layer to the third layer, µ is a bias, and N i is the number of neurons in the i th layer. The sigmoid activation function ¾ .¢/ is de ned by ¡z
(46)
¾ .z/ D 1=.1 C e /
The neural networks input– output mapping equation (45) can be expressed in matrix form as follows:
¡
ynn D W T ¾ V T xnn
¢
(47)
where W 2 R N2 C 1 £ N3 ; V 2 R N1 C 1 £ N2 ; xnn 2 R N1 C 1 , and ¾:R N2 7! R N2 C 1 are de ned as follows:
2
µ w1
w11 w21 :: :
6 W D 4µw2 T
:: :
w12 w22 :: :
3
¢¢¢ ¢ ¢ ¢7 5; :: :
2
µ v1
6 V D 4µv2 T
:: :
h xnn D 1;
0 xnn ; 1
0 xnn ;:::; 2
£
0 xnn N1
v11 v21 :: :
v12 v22 :: :
3
¢¢¢ ¢ ¢ ¢7 5 :: :
iT
¡
¾.z/ D 1; ¾ .z 1 /; ¾ .z 2 /; : : : ; ¾ z N2
¡
¢
¢¤T
kV k F · V M
¡
¢
¢
¡
¢
(53)
The Taylor series expansion of ¾ in Eq. (53) may be written as
¡ ¢ ¡ ¢ d¾ ¡ ¢ VQ T xnn C O VQ T xnn (54) ¾ V T xnn D ¾ VO T xnn C dz z D VO T xnn By the use of Eqs. (53) and (54), the hidden layer output error equation can be written as follows:
¡
¢
¡
¾Q D ¾ 0 VO T xnn VQ T xnn C O VQ T xnn
¢
(55)
Then, the error of the output layer is expressed as follows:
¡
¢
¡
¢
¡
¢
WO T ¾ VO T xnn ¡ 1 D WO T ¾ VO T xnn ¡ W T ¾ V T xnn ¡ ².xnn /
¡
¢
£ ¡
¢
¡
D ¡ WQ T ¾ VO T xnn ¡ W T ¾ V T xnn ¡ ¾ VO T xnn
¢¤
¡ ².xnn /
(56)
¢
¡
(49)
(50)
¢
¡
¢
WO T ¾ VO T xnn ¡ 1D ¡ WQ T ¾ VO T xnn ¡ WQ T ¾ 0 VO T xnn VQ T xnn
¡
¢
¡
¡ WO T ¾ 0 VO T xnn VQ T xnn ¡ W T O VQ T xnn ¡ ².xnn /
£ ¡
¢
¡
¢
¡
¢
D ¡ WQ T ¾ VO T xnn ¡ ¾ 0 VO T xnn VO T xnn ¡ WO T ¾ 0 VO T xnn VQ T xnn C w
(48)
where W M and V M 2 R are known positive constants and k ¢ k F denotes the Frobenious norm of a matrix. The ideal weight matrices W and V satisfying Eq. (48) cannot be determined in anticipation because we have no information on the error term 1. Instead, the estimated values of the ideal weights, WO and VO , are used in the controller, and they are adjusted by the adaptive laws. Consequently, there exists an effect caused by the difference between the ideal weights W and V , and the estimated weights WO and VO . This effect is stated in the following lemma. Lemma 2: Let us de ne the weight estimation errors as WQ D W ¡ WO ; VQ D V ¡ VO , and Z D diag[W; V ]. Given a neural networks’ input xnn , the output error is expressed as ¡ ¢ £ ¡ ¢ ¡ ¢ ¤ WO T ¾ VO T xnn ¡ 1 D ¡ WQ T ¾ VO T xnn ¡ ¾ 0 VO T xnn VO T xnn ¡ WO T ¾ 0 VO T xnn VQ T xnn C w
¡
¾Q D ¾ ¡ ¾O D ¾ V T xnn ¡ ¾ VO T xnn
¡
where ².xnn / is the approximationerror satisfyingk².xnn /k · ² N for all xnn in some input space. The following assumptions are used in the design and analysis of the adaptive control law presented in this paper. Assumption 8: Here, x0nn D [x 1d ; x1 ; x2 ]T is de ned as the input vector of the neural networks, and this input satis es Eq. (48) for some ² N . Assumption 9: The ideal weights are bounded in the sense that kW k F · W M ;
Proof: The output error of the hidden layer for the input xnn is de ned as
Substituting Eq. (55) and W D WQ C WO and VQ D V ¡ VO into Eq. (56) yields
Multilayer neural networks can approximate a nonlinear function to any desired accuracy. This is known as the universal approximation capability.12;13 That is, for a continuous function 1:R N1 7! R N3 and an arbitrary constant ² N > 0, there exist an integer N2 , the number of neurons in the hidden layer, and ideal constant weight matrices W 2 R N 2 C 1 £ N3 and V 2 R N1 C 1 £ N2 such that 1 D W T ¾ V T xnn C ².xnn /
(52)
¢
¤ (57)
Therefore, Eq. (50) has been proved. When the sigmoid function ¾ .¢/ and its derivative d¾ .z/=dz are bounded by some constants and Assumption 8 is used, it can be shown that the high-order term O in Eq. (54) satis es the following inequality:
® ¡ ¢® ® ¡ ¢® ® ¡ ¢® ®O VQ T xnn ® · ®¾ V T xnn ® C ®¾ VO T xnn ® ® ¡
¢
®
C ®¾ 0 VO T xnn VQ T xnn ® · c C c C ck VQ k F kxnn k · c C ck VQ k F C ckx 1 kkVQ k F C ckx 2 kk VQ k F
(58)
where c is a generic symbol used to denote any nite constant. Note that the property of kAx k · k Ak F kxk is used in the preceding equations. Finally, Eq. (52) is proved by using Eqs. (51) and (58) and hboxAssumption 8 as follows: kw.t /k · kWQ k F cVM .c C ckx1 k C ckx 2 k/ C W M .c C ckVQ k F Q F C ckx 1 kk VQ k F C ckx2 kkVQ k F / C ² N · C 1 C C 2 k Zk C C 3 k ZQ k F kx 1 k C C4 k ZQ k F kx2 k
(59)
This lemma shows that the neural networks output error caused by the weight estimation error can be expressed as Eq. (50) and that the size of the term w in Eq. (50) is bounded by Eq. (52).
680 C.
LEE AND KIM
Design of the Adaptive Controller
The followinglemma de nes a robustterm v and shows a property for the stability analysis. Lemma 3: Consider w 2 R3 de ned as in Eq. (51) and v 2 R3 ; » , and kv 2 R de ned as vD¡
z2» » kz 2 k» C ²
(60)
» D kv .Z M C k ZO k F /.kx 1 k C kx 2 k/
(61)
kv ¸ maxfC3 ; C 4 g
(62)
1
2
2
1a
1
¤ ¤
PQ ] C .1=° / tr [ VQ T VPQ ] ¡ g2 u C .1=°w / tr [ WQ T W v
z 2T .w C v/ · kz 2 k[C 1 C C 2 k ZQ k F ] C ²
(63)
Proof: With the choice of v in Eq. (60), z 2T .w C v/ is expressed
where is the ideal control input when the exact aerodynamic model is available and is expressed as follows:
£
¤
¡
C v/ D
z T2 w
.kz 2 k» /2 ¡ kz 2 k» C ²
(64)
.kz 2 k» /2 C C 4 k ZQ k F kx 2 k] ¡ kz 2 k» C ²
(65)
Furthermore, from the de nition ZQ D Z ¡ ZO and Eqs. (61) and (62), we have Q F kx1 k C C 4 k ZQ k F kx 2 k · kv kZ ¡ ZO k F .kx 1 k C kx 2 k/ · » C 3 k Zk
(66) When Eqs. (65) and (66) are used, Eq. (63) can be proved as follows:
C C 2 k ZQ k F ] C
.kz 2 k» /2 D kz 2 k[C 1 kz 2 k» C ²
kz 2 k» ² · jz 2 k[C 1 C C2 k ZQ k F ] C ² kz 2 k» C ²
(67)
PQ ] C .1=° / tr [ VQ T VPQ ] Cz 2T v C .1=°w / tr [ WQ T W v
£
¤
¡
¢
(68)
where AO and v are de ned as in Eqs. (44) and (60), respectively. WO and VO are computed by the following adaptive laws:
£ ¡ ¢ ¡ ¢ ¤ P WO D ¡°w ¾ VO T xnn z T2 ¡ ¾ 0 VO T xnn VO T xnn z T2 ¡ · °w WO h ¡ 0
P VO D ¡°v xnn ¾ VO T xnn
¢T
WO z 2
iT
¡ ·°v VO
(69) (70)
where ·; °w , and °v 2 R are some positive design parameters. Furthermore,the bound of the tracking error and the neuralnetworks parameter estimation error may be kept as small as desiredby adjusting the design parameters. Proof: Let us consider the Lyapunov function candidate, V2 D 12 z 1T z 1 C 12 z T2 z 2 C .1=2°w / tr [ WQ T WQ ] C .1=2°v / tr [ VQ T VQ ]
(71)
¢
(74)
¤
where 12 is de ned as g2 [u ¡ u] and represents the term caused by the modeling error. Substituting Eqs.(50), (69), and (70) into the preceding equation gives the following equation.
£ ¡
VP2 D ¡k1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d C z 2T ¡ WQ T ¾ VO T xnn
¡
¢
¤
©
£ ¡
¢
¡
©
£ ¡
C tr VQ T xnn ¾ 0 VO T xnn
¡
¢
¡
¢
¢T
¤T
WO z 2
T
C · VO
¢
ª
¤ª
ª¢¢
(75)
T
When the propertyof the trace, tr[yx ] D x y, is used, the preceding equation is simpli ed as follows: O C z T .w C v/ VP2 D ¡k 1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d C · tr[ ZQ T Z] 2
(76)
O D tr[ ZQ T Z ] ¡ tr[ ZQ T Z] Q · k ZQ k F Z M ¡ Using the property tr[ ZQ T Z] 2 Q k Zk F and Eqs. (39) and (63), we have
£ k1 c12 c22 VP2 · ¡ .1 ¡ c1 /kz 1 k2 ¡ k2 kz 2 k2 C C · k ZQ k F Z M 2 2k1 .1 ¡ c1 / ¤
Q 2 C kz 2 k[C 1 C C 2 k ZQ k F ] C ² D ¡ ¡ k Zk F
µ
The designprocessof the adaptivecontrollerand stabilityanalysis are stated in the following theorem. Theorem 2: Consider the system Eqs. (25) and (26), where the controlinputu is de ned in Eq. (68)andthe adaptivelaws are de ned in Eqs. (69) and (70). Then, the trajectories of the system as well as the neural networks’ weights are locally uniformly ultimately bounded: u D gO2¡1 ¡k 2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ AO C WO T ¾ VO T xnn C v
¡
C tr WQ T ¾ VO T xnn z 2T ¡ ¾ 0 VO T xnn VO T xnn z 2T C · WO
z 2T .w C v/ · kz 2 k[C 1 C C 2 k ZQ k F C C 3 k ZQ k F kx 1 k
(73)
VP2 D ¡k1 kz 1 k2 ¡ k2 kz 2 k2 C z 1T g1a x 2d ¡ z 2T 12 C z 2T WO T ¾ VO T xnn
¡ ¾ 0 VO T xnn VO T xnn ¡ WO T ¾ 0 VO T xnn VQ T xnn C w C v
Substituting Eq. (52) into the preceding equation gives
z 2T .w C v/ · kz 2 k[C 1 C C 2 k ZQ k F C » ] ¡
¢
u ¤ D g2¡1 ¡ k2 z 2 ¡ g1Ta z 1 ¡ g1T z 1 ¡ A C WO T ¾ VO T xnn C v
©
z T2 .w
(72)
u¤
With the de nition of u ¤ in Eq. (73), Eq. (72) becomes
where ² is a positive constant. Then, the following inequality is satis ed:
as
When Eq. (38) is used the derivative of the Lyapunov function V2 along the trajectories of Eqs. (25) and (26) with the adaptive laws, Eqs. (69) and (70), are computed as £ VP2 D ¡k1 kz 1 k2 C z T g1a x d C z T g T z 1 C g T z 1 C A C g2 u C g2 u ¤
¡
k2 k2 C1 kz 2 k2 ¡ kz 2 k ¡ k2 2 2
C C 2 kz 2 kk ZQ k F C
¶2 ¡
k1 .1 ¡ c1 /kz 1 k2 2
· Q 2 · k Z k F ¡ [k ZQ k F ¡ Z M ]2 2 2
c12 c22 C2 · Z 2M C 1 C²C 2k 1 .1 ¡ c1 / 2k2 2
(77)
With the de nition C5 D [c12 c22 =2k1 .1 ¡ c1 /] C .C 12 =2k 2 / C .· Z 2M =2/ C ², the preceding equation becomes k1 k2 · VP2 · ¡ .1 ¡ c1 /kz 1 k2 ¡ kz 2 k2 C C 2 kz 2 kk ZQ k F ¡ k ZQ k2F 2 2 2 C C5 D ¡
µ
k1 k2 · Q 2 .1 ¡ c1 /kz 1 k2 ¡ kz 2 k2 ¡ k Zk F 2 4 4
1 kz 2 k ¡ 2 k ZQ k F
¶T
2
k2 6 2 4 ¡C 2
3
µ
¶
¡C 2 7 kz 2 k C C5 · 5 k Zk Q F 2
(78)
If k2 and · are chosen to satisfy k2 · ¡ 4C 22 > 0, the last matrix in the preceding equation becomes positive de nite. Therefore, the following inequality is satis ed: VP2 · ¡.k 1 =2/.1 ¡ c1 /kz 1 k2 ¡ .k 2 =4/kz 2 k2 ¡ .· =4/k ZQ k2F C C 5
(79)
For a positive constant ¹2 satisfying 0 < ¹2 < minf.k1 =2/.1 ¡ c1 /; k 2 =4; · =4 minf°w ; °v gg, nally, the following equation is obtained for the derivative of the chosen Lyapunov candidate function:
681
LEE AND KIM
VP2 · ¡2¹2 V2 C C 5
(80)
Equation (80) implies that VP2 < 0 when V2 > C 5 =2¹2 . Hence, the error states z 1 and z 2 and the weight estimation error ZQ are bounded and converge exponentially to the residual set D2 :
The F-16 aircraft model is used in this paper,11 and the following command values of ®; ¯, and Á are applied to the aircraftin a steadystate level ight of V D 500 ft/s, h D 10,000 ft: ®d D 2:659;
©
®d D 10;
kz 1 k2 C kz 2 k2 D2 D z 1 ; z 2 2 R3 ; ZQ F 2 R N1 C N2 C 2 £ N2 C N3 C .1=maxf°w ; °v g/k ZQ k2F · C 5 =¹2
Because c1 ; c2 ; C 1 ; Z M , and ² are independentof k 1 and k 2 , the size of the set D2 can be made arbitrarily small by adjusting the design parameters k1 and k2 . Theorem 2 represents the design procedure of the adaptive controller to track the ®; ¯, and Á commands when the modeling errors exist. This shows that, if the controlleris applied, the tracking errors and the parameter estimation error of the neural networks converge to a compact set and also shows that the size of the set is adjustable by tuning the design parameters. The universal approximation theorem only guarantees the existence of the ideal weight and the ideal number of the hidden layer neurons. In this paper, the weight of neural networks are adjusted by the adaptive laws; however, the size of hidden layer neuron N 2 is xed. It also affects the approximation capacity of the neural networks. If too small a value of N 2 is chosen, the neural networks may not compensate for the effect of the modeling error properly. Therefore, the value of N 2 should be chosen carefully in considerationof the complexity and the size of the modeling error.
V.
Numerical Simulation
In Sec. III, the design methodology of the ight controller to track the ®; ¯, and Á commands is proposed when full knowledge of the aerodynamic characteristics is available, and in Sec. IV, the adaptive controller based on neural networks is designed to eliminate the effect caused by the aerodynamic modeling error. This section presents numerical simulation results for each controller to demonstrate the performance of the proposed nonlinear control laws.
Fig. 3
®d D ¡ 2;
ª
¯d D 0; ¯d D 0; ¯d D 0;
Ád D 0 deg;
0·t ·1s
Ád D 50 deg;
1 · t · 10 s
Ád D 0 deg;
10 · t · 20 s
To obtain differentiable commands satisfying Assumption 1, the third-order linear command lter is used. The aerodynamic modeling error is made arbitrarily, and the average error for each coef cient is listed in Table 1. The controller design parameters are chosen as follows: k1 D 3, k2 D 8, · D 0:2, k v D 0:153, ² D 0:001, Z M D 0:6142, and °w D °v D 30. Figure 3 represents the simulation results of the backstepping controller described in Sec. III when the exact aerodynamic model is available. The solid line represents the simulation result of the backsteppingcontroller proposed in Sec. II, and the dotted line represents the command signal. As expected, the proposed backstepping controller makes the ®; ¯, and Á follow the command values in a satisfactory way. Figure 4 shows the simulation results when the modeling error exists. The solid lines represent the simulation result of the adaptivecontroller based on neural networks in Sec. IV, and the dashed lines represent the result of the controller in Sec. III. The dotted lines represent the command signals. It is shown that the system output of the adaptive controller tracks the command quite well, even if large modeling uncertaintyexists. It can be said that the performance of the system is not degraded in the case of the neural networks adaptive controller. Table 1
Average modeling errors of the aerodynamic coef cients, %
Coef cients
Error
Coef cients
Error
Coef cients
Error
Cl Cl p C lr Cl±a Cl±r
79.6 16.0 148.9 141.8 69.8
Cm Cm q Cm ±e —— ——
207 77.6 146.2 —— ——
Cn Cn p Cnr Cn ±a Cn ±r
180.1 86.7 94.9 48.3 228.5
Time response of x1 and u (without modeling error): backstepping controller in Sec. III —— , and command¢ ¢ ¢ ¢ .
682
Fig. 4
¢ ¢ ¢ ¢ .
LEE AND KIM
Time response of x1 and u (with modeling error): adaptive controller in Sec. IV —— , backstepping controller in Sec. III - - - -, and command
VI.
Conclusions
A controllerfor a six-degree-of-freedom nonlinear ight model is proposed,and its stability is analyzedby using the Lyapunovtheory. The backstepping controller is used to track the ®; ¯, and Á commands with the assumption that the aerodynamic characteristicsare fully understood. It is shown that, if the controller is applied, the tracking error exponentiallyconverges to a compact set and the size of the set can be made arbitrarilysmall by tuning the design parameters. It is not necessary to make the controller gain excessively large to guarantee stability because the timescale separation assumption is not used. An adaptive controller based on neural networks is used to compensate for the effects of the aerodynamic modeling errors. The neural networks’ parameters are adjusted to offset the error term. The closed-loop stability of the error states and the parameters of the neural networks are examined by the Lyapunov theory, and it is shown that the error states and the parameter estimation errors exponentially converge to a compact set whose size is adjustable by the design parameters. Finally, a nonlinear simulation of an aircraft maneuver is performed to demonstrate the performance of the proposed control laws.
Acknowledgment This research was supported by the Korea Science and Engineering Foundation Grant 1999-2-305-004-3.
References 1
Lane, S. H., and Stengel, R. F., “Flight Control Design Using Non-Linear Inverse Dynamics,” Automatica , Vol. 24, No. 4, 1988, pp. 471– 483. 2 Menon, P., Badgett, M., and Walker, R., “Nonlinear Flight Test Trajec-
tory Controllers for Aircraft,” Journal of Guidance, Control, and Dynamics, Vol. 10, No. 1, 1987, pp. 67 – 72. 3 Snell, S. A., Enns, D. F., and Garrard, W. L., Jr., “Nonlinear Inversion Flight Control for a Supermaneuverable Aircraft,” Journal of Guidance, Control, and Dynamics, Vol. 15, No. 4, 1992, pp. 976– 984. 4 Schumacher, C., and Khargonekar, P. P., “Stability Analysis of a Missile Control System with a Dynamic Inversion Controller,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 3, 1998, pp. 508– 515. 5 Farrell, J. A., “Stability and Approximator Convergence in Nonparametric Nonlinear Adaptive Control,” IEEE Transactions on Neural Networks, Vol. 9, No. 5, 1998, pp. 1008– 1020. 6 Lewis, F. L., Yesildirek, A., and Liu, K., “Multilayer Neural-Net Robot Controller with Guaranteed Tracking Performance,” IEEE Transactions on Neural Networks, Vol. 7, No. 2, 1996, pp. 388– 399. 7 Singh, S. N., Yim, W., and Wells, W. R., “Direct Adaptive and Neural Control of Wing-Rock Motion of Slender Delta Wings,” Journal of Guidance, Control, and Dynamics, Vol. 18, No. 1, 1995, pp. 25 – 30. 8 Kim, B. S., and Calise, A. J., “Nonlinear Flight Control Using Neural Networks,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 1, 1997, pp. 26– 33. 9 Krsti´ c, M., Kanellakopoulos, I., and Kokotovi´c, P., Nonlinear and Adaptive Control Design, Wiley, New York, 1995, Chap. 2. 10 Stevens, B. L., and Lewis, F. L., Aircraft Control and Simulation, Wiley, New York, 1992, Chap. 2. 11 Morelli, E. A., “Global Nonlinear Parametric Modeling with Application to F-16 Aerodynamics,” Proceedings of the 1998 American Control Conference, IEEE Publications, Piscataway, NJ, 1998, pp. 997– 1001. 12 Hornik, K., Stinchcombe, M., and White, H., “Multilayer Feedforward Networks are Universal Approximators,” Neural Networks, Vol. 2, No. 5, 1989, pp. 359– 366. 13 Funahashi, K., “On the Approximate Realization of Continuous Mapping by Neural Networks,” Neural Networks, Vol. 2, No. 3, 1989, pp. 183– 192.
