Aerospace Science and Technology 13 (2009) 105–113 www.elsevier.com/locate/aescte
Adaptive backstepping-based flight control system using integral filters Chao-Yong Li ∗,1 , Wu-Xing Jing 2 , Chang-Sheng Gao 3 Department of Aerospace Engineering, Harbin Institute of Technology, 150001, China Received 10 September 2007; received in revised form 28 April 2008; accepted 16 May 2008 Available online 22 May 2008
Abstract A backstepping control design procedure for uncertain nonlinear flight control system expressible in parameter-strict feedback form is presented in this paper. The proposed backstepping procedure, in association with sliding model control technique, exploits the possibility of avoiding, under certain suitable assumptions, the overparameterization problem existing in the classical backstepping process. In particular, a sliding-model-based integral filter is introduced to facilitate the development of the derivation of the virtual inputs, thus reducing the computational load with regard to the standard backstepping procedure. Moreover, in simulations, the control parameters in the resulted controller are optimally tuned using a genetic algorithm so as to show the full potential of the proposed control system. © 2008 Elsevier Masson SAS. All rights reserved. Keywords: Backstepping control; Flight control; Integral filters; Genetic algorithm
1. Introduction In the design of flight control system, it is a common practice to linearized the dynamics model for various flight conditions [1,20], then the linear design techniques are applied to schedule the control parameters obtained above as functions of flight conditions, such as altitude, Mach number, etc. However, as soon as the intrinsic limitations of feedback linearization control and, in particular, the impossibility to deal with systems with parameter uncertainties became apparent, much effort was used by many researchers in order to develop adaptive controllers for nonlinear flight control problem. Particularly, in the designing of adaptive autopilots, the control parameters change through an internal process to adapt to the flight envelope, as well as to meet control requirements [1,17]. With the development of the recursive adaptive control scheme named backstepping [9], which became a key result for many researchers involved with adaptive control problems. Indeed, the backstepping philosophy makes the design of the * Corresponding author.
E-mail address:
[email protected] (C.-Y. Li). 1 Ph.D Student, Department of Aerospace Engineering, HIT, China. 2 Professor, Department of Aerospace Engineering, HIT, China. 3 Assistant professor, Department of Aerospace Engineering, HIT, China.
1270-9638/$ – see front matter © 2008 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2008.05.002
feedback control strategy systematic. In particular, it consists of a step-by-step coordinate transformation interlaced with the determination of a virtual Lyapunov-based control signal and a suitable tuning function involving the parameters, producing, as a result, at the last step, the actual control input and parameter update law. Varies applications of the adaptive backstepping control techniques demonstrate its superiority over classical controllers [2,3,8,11,14–16,19,21–24,26]. Especially in the flight control problem, in which case, unlike the traditional control philosophy, it could guarantee the stability and tracking performance in three channels simultaneously. However, despite of its advantage in multi-variable control problems, the main drawback associated with this technique is its computational complexity (i.e., overparameterization), which, naturally, increases with the system order. The desire of overcoming such drawback, as well as that of strengthening the robustness of the resulted controller are among the reasons why the possibility of combining the basic adaptive backstepping procedure with other techniques born in the robust control area [7,10,18], or, in particular, with sliding model control has been investigated [27]. With regard to the flight control problem, Harkegärd [3] developed a backstepping controller that is globally stabilizing around the small AOA. Kim [10] introduced a robust nonlinear missile pitch autopilot using adaptive backstepping procedure.
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Nomenclature α Angle-of-attack β Sideslip angle γ Roll angle ϑ Pitch angle Y Atmospheric lift Z Atmospheric side force P Thrust v Speed ρ Air density q Dynamics pressure = ρv 2 /2 s Reference area l Reference length m Mass ωx , ωy , ωz Body rates in the body frame
Lee [11] applied neural networks technique to compensate for the aerodynamic modeling error in the design procedure. Lian [14] studied its application in the reentry vehicle attitude control systems. Farrell [2] studied the command filtered backstepping approach in controlling UAV (unmanned aerial vehicles). Sonneveldt [22] examined the application of constrained backstepping approach in aircraft control problem. Singh and Steinberg [19,23] developed a backstepping adaptive control law with increased robustness through the incorporation of integrated error in the Lyapunov function. Hu [7] introduced a robust adaptive autopilot for BTT block control model using robust control function and nonlinear tracking differentiator. Ju [8] proposed a longitudinal backstepping flight control system, which is valid for all flight envelope. Furthermore, problem of differentiating a composite reference trajectory (i.e., virtual input) at each step is another worth noting drawback associated with the backstepping procedure. Although, the analytic determination is possible, it becomes tedious as the number of iterations of the backstepping approach increases and when parameter adaptation is involved. An effective way to deal with such problem is to use the command/integral filters [2], which has been well addressed in developing a control technique, similar to backstepping schemes, named adaptive dynamics surface control [21,26]. Moreover, Lu [15,16] studied the first/second order integral filters from an inverse Taylor series viewpoint. Stotsky [24] examined the use of sliding mode filters to simply the backstepping control method. The results in their paper show that the integral filters can guarantee a good performance in backstepping procedure. However, the application of integral filters in the flight control system has not been fully addressed in the open literature, especially incorporating with sliding model control technique. This paper focuses mainly on the development of an adaptive backstepping-based flight control system, taking into account the unknown uncertainties/disturbance. It differs from prior work in three main aspects. First, the sliding model control technique is introduced in the backstepping procedure so as to develop an easy-implemented controller, as well as to avoid
δx , δy , δz Deflection angles in the body frame Ix , Iy , Iz Moments of inertia of missile’s body m∗x , m∗y , m∗z Atmospheric moment coefficients AOA Angle-of-attack BTT Bank-to-turn STT Skid-to-turn xˆ Estimated value of the vector x x˜ Estimated error of the vector x Subscript x, y, z v b
Along x, y, and z axes, respectively In speed frame In body frame
the possibility of the overparameterization problem. Second, a second-order sliding mode integral filter is introduced to facilitate the development of the derivation of the virtual control input with uncertainty terms included. Third, a genetic algorithm (GA) is utilized to optimize the selection of the fixed control parameters, so as to show the full potential of the proposed control system. 2. Formulation of the missile dynamics The missile model used in this paper derives from the nonlinear model of a generic STT missile. In this model, the dynamics equations of AOA, sideslip angle are developed in the speed frame [12], while the dynamics equation of the roll angle is developed in the body frame [12]. For details, please refer to Appendix A. Moreover, the above-mentioned three angles are used as the state variables of the system, as well as the body rates, whose dynamics equations are developed in the body frame. Note that, although the AOA and sideslip angle could not be measured directly via onboard sensor, it could be estimated reasonable accurate and they are directly related to the gyros rates, and also using these angles as state variable is a common experience in the development and analysis of missile flight control system. First, defining the states x 1 , x 2 ∈ R 3 , and the control input u as follows: x 1 = [α β γ ]T ,
x 2 = [ωx ωy ωz ]T ,
u = [δx δy δz ]T
(1)
With certain assumptions [14,20], the missile dynamics equations can be simply expressed in state-space form as follows: x˙1 = f 1 (x 1 ) + A(x 1 )x 2 + B(x 1 )u
(2)
x˙2 = f 2 (x 1 , x 2 ) + Cu
(3)
The definitions of all the matrices in (2) and (3) can be found in Appendix A. Without loss of generality, the parameter uncertainty, input uncertainty, and function uncertainty are taken into account in the proposed control system. Then, the dynamics equations (2) and (3) can be rewritten in abbreviate form as
C.-Y. Li et al. / Aerospace Science and Technology 13 (2009) 105–113
x˙1 = f 1 + f 1 + Ax 2 + Bu + Bu + d 1
(4)
x˙2 = f 2 + f 2 + Cu + Cu + d 2
(5)
where f i denotes the parameter uncertainty, which is owing to the time variations of the atmospheric coefficients system, B and C are the input uncertainties, d i denotes the function uncertainty, which is due to the modeling errors and external disturbances, i = 1, 2. 3. Nonlinear controller and stability analysis The development of flight control has been dominated by many classical control schemes, while this tradition has produced many reliable and effective control systems, recent years have seen a growing interest in the applications of robust, nonlinear, adaptive, and backstepping control theories in this field. In this section, the adaptive backstepping procedure and the sliding model control techniques are introduced so as to develop a nonlinear flight control system, whose function is to track the commanded angles with an acceptable accuracy, and stabilize the body rate simultaneously. The following assumptions are used in the design and analysis procedure.
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Assumption 3. The magnitude of the actual/virtual control input, as well as their derivations, during the adaptive backstepping procedure is bounded due to its physical limitation. Moreover, the desired commanded angles, or in other words, the output of the guidance system (i.e., AOA, sideslip angle), are continuous with respect to time. Assumption 4. There exists positive constant αm such that A is invertible for all α ∈ R 3 satisfying that |α| αm . Moreover, the disturbance C is relatively small in magnitude compared with C, such that it does not affect the invertibility property of C, which means (C + Ωmin ) is invertible. Before we start, the error variables of the designed flight control system are introduced as follows: z1 = x 1 − x 1d
(12)
z2 = x 2 − x 2d
(13)
Assumption 1. The control surface deflections have no effect on the aerodynamic force components, or this effect could be ignored in practical perspective, that is
where x 1d and x 2d are the desired values of x 1 and x 2 , respectively. x 1d is comprised of the commanded AOA, sideslip angle, and the roll angle, and x 2d will be determined later. In particular, for the STT missile, the bank angle γv is usually assumed to be zero throughout the engagement. Therefore, in this case, the commanded roll angle could be determined via solving the following geometric equation:
(B + B)u ≈ 0
cos α sin β sin ϑ − sin α sin β cos ϑ cos γcmd
(6)
The function of the control surfaces is to control each axis’ angular rate independently. It should be noted that this is a general assumption in solving such problems [10,11,14]. Assumption 2. The parameter and input uncertain terms in (4) and (5) is partially known and can be simply represented as
+ cos β cos ϑ sin γcmd = 0
(14)
where γcmd is the desired commanded roll angle. From (11) and (12), we have z˙1 = x˙1 − x˙1d = Ax 2 + f 1 + ϕ1 θ˜1 + ϕ1 θˆ1 + d 1 − x˙ 1d
(15)
f 1 = θ1 ϕ1
(7)
We further assume that x 2d is the virtual input to (15), and the desired virtual control is
f 2 = θ2 ϕ2
(8)
x 2d = −A−1 (c1 z1 + f 1 + ϕ1 θˆ1 + η1 − x˙ 1d )
(16)
where θi and ϕi are known smooth function and unknown constant matrices, respectively, i = 1, 2. Moreover, the function uncertainty is assumed to be unknown. However, the upper boundary of its magnitude is known as
where c1 is a designed positive diagonal matrix. η1 is a robust term designed to cancel the function uncertainty, and
d i ψi
As stated previously, with the inclusion of the uncertainty or disturbance in the virtual input (16), it is difficult in finding its derivatives because the signal may not be practically differentiable due to noises and/or disturbances, and the problem of overparameterization will occur with the increase of steps as well. In view of this, a second-order sliding model integral filter is presented in this paper so as to eliminate the analytic computation of x˙2d , which will be used as reference in the backstepping procedure. It is worth stressing that the proposed filter works also for the high-order backstepping procedures, just using the output of the (i − 1)th filter as the input to the ith filter, i = 1, 2, . . . , n. The proposed integral filters are presented as follows:
(9)
We further assume that the input uncertainty satisfies the wellknown triangle function [11,14,18] Ωmin C Ωmax
(10)
where Ωmin and Ωmax are known positive define diagonal matrices. With above assumptions, the proposed missile dynamics equations can be rendered to a parameter-strict feedback form with unmatched uncertainties, results x˙ 1 = f 1 (x 1 ) + A(x 1 )x 2 + θ1 ϕ1 + d 1 (11) x˙ 1 = f 2 (x 1 , x 2 ) + C(x 1 , x 2 )u + θ2 ϕ2 + d 2
η1 =
z1 ψ12 zT1 z1
(17)
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(λˆ 1 − x 2d ) ξ1 (λˆ 1 − x 2d ) λˆ˙ 1 = − − τ1 λˆ 1 − x 2d + ζ1 λˆ 2 − λˆ˙ 1 ξ2 (λˆ 2 − λˆ˙ 1 ) − λˆ˙ 2 = − τ2 λˆ − λˆ˙ + ζ 2
1
(18) (19)
2
where τi is the time constant of the filter, ξi and ζi are the designed constants, i = 1, 2. Obviously, with ξi assumed to be zero, the proposed filters are reduced to a classical integral filters. It should be pointed out that, with the inclusion of the sliding model control component, the fast convergence of the estimation error produced by the proposed integral filters is guaranteed, which will be analytically studied during the stability analysis. Similar integral filters associated with different control schemes can also be found in varies applications [15,24,25], and the performance demonstrates their feasibility within the backstepping procedure. Furthermore, the candidate Lyapunov function for the first system of (11) is designed in the form V 1 = zT1 z1 /2 + θ˜1T Ξ1 θ˜1 /2 + (λˆ 1 − x 2d )T (λˆ 1 − x 2d )/2
(20)
where Ξ1 is the designed positive definite matrix. Obviously, the third term in (20) is used to stabilize the estimation error of the proposed filters. Consequently, evaluating its time derivative along the solutions of the system (15), results V˙ 1 = zT1 z˙ 1 + θ˜1T Ξ1 θ˙˜ 1 + (λˆ 1 − x 2d )T (λˆ˙ 1 − x˙ 2d )
(21)
Substituting (15) and (16) into (21), we have V˙ 1 −zT1 c1 z1 + zT1 Az2 + θ˜1T Ξ1−1 (Ξ1 ϕ1T z1 − θ˙ˆ 1 ) + (λˆ − x )T (λˆ˙ − x˙ ) 1
2d
1
2d
(22)
(23)
Then, (22) becomes V˙ 1 −zT1 c1 z1 + zT1 Az2 + (λˆ 1 − x 2d )T (λˆ˙ 1 − x˙ 2d )
(24)
Furthermore, substituting (18) into (24), we have V˙ 1 −zT1 c1 z1 + zT1 Az2 ξ1 λˆ 1 − x 2d − (λˆ 1 − x 2d )T − x˙ 2d λˆ 1 − x 2d + ζ1
(25)
Recalling Assumption 3 and assuming that the derivative of the virtual input x 2d is bounded with a known vector 1 , that is x˙ 2d x˙ 2d max = 1
(26)
Moreover, the parameter ξ1 can be designed as ξ1 = ς1 1 , where ς1 > 1. Hence, we have V˙ 1 −zT1 c1 z1 + zT1 Az2 − (λˆ 1 − x 2d )T 1
ς1 λˆ 1 − x 2d −1 λˆ 1 − x 2d + ζ1
λˆ 1 − x 2d >
ζ1 ς1 − 1
(27)
(28)
With the preceding condition, the system will be bounded stable at the origin (i.e., z1 = 0, z2 = 0), and also, with such condition, the actual estimation error of the proposed filter can be guaranteed within a compact set determined in the form: λˆ 1 − x 2d
ζ1 ς1 − 1
(29)
Obviously, the estimation error of the filter can be adjusted sufficiently small by choosing ζ1 appropriately, and with the inclusion of the sliding mode control component, (29) can be arrived in finite time. Next, from (13), we have z˙ 2 = x˙ 2 − x˙ 2d = f 2 + Cu + ϕ2 θˆ2 + ϕ2 θ˜2 − x˙ 2d
(30)
The candidate Lyapunov function in this case is defined as V 2 = V 1 + zT2 z2 /2 + θ˜2T Ξ2 θ˜2 /2 + (λˆ 2 − λˆ˙ 1 )T (λˆ 2 − λˆ˙ 1 )/2
(31)
Similarly, choosing the parameter update law for θ˙ˆ 2 as θ˙ˆ 2 = Ξ2 ϕ2T z2
The parameter update law for θˆ in (22) can then be designed as θ˙ˆ 1 = Ξ1 ϕ1T z1
Apparently, if λˆ 1 − x 2d = 0 and the following relation is satisfied, the time derivative of the Lyapunov function will be rendered to negative
(32)
Consequently, the parameter ξ2 can be designed as ξ2 = ς2 2 , where ς2 > 1. We have V˙ 2 −zT1 c1 z1 + zT1 Az2 + zT2 f 2 + (C + C)u + ϕ2 θˆ2 − λˆ 2 ς2 λˆ 2 − λˆ˙ 1 ˙ T ˆ ˆ − (λ2 − λ1 ) 2 −1 λˆ 2 − λˆ˙ 1 + ζ2
(33)
where λˆ˙ 1 max = 2 . Then, the control input for the proposed flight control system can be designed as: u = −C −1 [c2 z2 + Az1 + f 2 + ϕ2 θˆ2 − λˆ 2 ] + uvsc
(34)
where uvsc is the nonlinear sliding mode control component. Substituting (34) into (33), we have V˙ 2 −zT1 c1 z1 + zT2 −c2 z2 + (C + C)uvsc + Cueq ς2 λˆ 2 − λˆ˙ 1 ) − (λˆ 2 − λˆ˙ 1 )T 2 −1 (35) λˆ 2 − λˆ˙ 1 ) + ζ2 where ueq = −C −1 [c2 z2 + Az1 + f 2 + ϕ2 θˆ2 − λˆ 2 ].
C.-Y. Li et al. / Aerospace Science and Technology 13 (2009) 105–113
Therefore, V˙ 2 −zT1 c1 z1 + zT2 −c2 z2 + C + C uvsc + Cueq ς2 λˆ 2 − λˆ˙ 1 ) ˙ T ˆ ˆ − (λ2 − λ1 ) 2 −1 (36) λˆ 2 − λˆ˙ 1 ) + ζ2 Therefore, the sliding model control component (i.e., uvsc ) of the control input is designed as uvsc = (C + Ωmin )−1 h¯ with −Ωmax ueq h¯ = −Ωmin ueq
(37)
z2 0 z2 < 0
(38)
Hence, in this case V˙ 2 −zT1 c1 z1 − zT2 c2 z2 − (λˆ 2 − λˆ˙ 1 )T 2
ς2 λˆ 2 − λˆ˙ 1 ) −1 λˆ − λˆ˙ ) + ζ 2
1
(39)
2
In the same way as (27), in order to render V˙ 2 < 0, we must have ζ2 (40) λˆ 2 − λˆ˙ 1 > ς2 − 1 Consequently, with such controller, the estimation error of the filter can be guaranteed within the set determined in the form λˆ 2 − λˆ˙ 1
ζ2 ς2 − 1
(41)
Therefore, the proposed control system is overall asymptotically stable in its origin (z1 = z2 = 0), and the estimated errors of the filters are all bounded and converge exponentially to a predetermined set. Also, since the included designed parameters do not depend on each other, the size of the set can be made sufficiently small by adjusting the corresponding parameters ζi (i = 1, 2) appropriately. It is worth stressing that the designed controller, (34), consists of two parts: one is linear (ueq ) control component, and the other is nonlinear (uvsc ) control component, which is used to cancel the nonlinearities caused by the uncertainties. Otherwise, the performance of the control system will be significantly compromised. Moreover, the resultant control system, compared with other work in this subject, seems to be simple with enhanced robustness in the unexpected uncertainties of the system. 4. Simulation results and discussions In order to demonstrate the performance of the proposed flight control system, simulations are presented in this section. Note that the output of the DG (differential geometric) guidance system is utilized as the reference/commanded guidance angles [4,12,13], which are continuous and differentiable with respect to time. Moreover, the target’s AOA are engaging a sinusoidal maneuver during the homing guidance of the proposed engagement. Also, the initial conditions of the engagement are given as follows:
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• Initial coordinate of the missile (m): [0 0 0] • Initial velocity of the missile (m/s): [0 0 0] • Launch azimuth angle (deg): 44.6; Launch elevation angle (deg): 71.8 • Initial coordinate of the target (m): [50000 50000 50000] • Initial velocity of the target (m/s): [−1000 −1000 −1000] • Initial mass of the missile (kg): 1000; Thrust (N ): 65000; Burn time (s): 15; Impulse: 250 • Moment of inertia (kg m2 ): Ix = 20, Iy = 120, Iz = 120 • Reference area (m2 ): 0.2; Reference length (m): 2 • Simulation step (s): 0.01; Time constant of the guidance system (s): 0.3 In particular, the air density is looked up from the Standard Atmospheric Model using the current flight altitude [20]. The needed moment coefficients and atmospheric coefficients are also determined through a looked up table using the current flight status (i.e., Mach number, altitude).Moreover, without loss of generality, in all the simulations, the uncertainty/disturbance terms (i.e., parameter/input/function uncertainty) in the dynamics equations are randomly selected within 15% of their nominal values in all the simulations. Furthermore, in order to show the full potential of the proposed control system, the controller parameters (i.e., C i , Ξi , i , ςi , ζi , τi , ξi , i = 1, 2) are optimally chosen using a genetic algorithm (GA) [5,6]. In which case, the cost function subjected to be minimized is
uT W u u + eT W e e + abs(eT )W s t (42) J= where W u , W e , and W s are the weighting matrices, e = [z1 z2 ]T is the error state. Genetic algorithms search the solution space of a function through the use of simulated evolution, i.e., the survival of the first strategy. In general, the fittest individuals of any population tend to reproduce and survive to the next generation, thus improving successive generations. However, inferior individuals can, by chance, survive and also reproduce. Genetic algorithms have been shown to solve linear and nonlinear problems by exploring all regions of the state space and exponentially exploiting promising areas through mutation, crossover, and selection operations applied to individuals in the population. Thus, the main advantage of using Genetic algorithms are that they do not get trapped in local minimal, and they can use any cost function that can be computed in a reasonable amount of time. As indicated in Fig. 1, as expected, the proposed flight control system could guarantee the achieved angles track the commanded angles effectively, even in the presence of the uncertainties. Moreover, the tracking error achieves its maximum at the beginning of the engagement, and stays at a negligible range at the rest. Furthermore, Obviously that the commanded roll angle is relatively small compared with the guidance angles, which indicates the guidance system is reasonable in the sense of stabilizing the missile’s body. The time history of the Euler angles is presented in Fig. 2, the time history of the body rates is proposed in Fig. 3. Appar-
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Fig. 1. Time history of commanded and achieved angles.
Fig. 2. Time history of Euler angles.
ently, either the roll angle or the roll rate is small in magnitude, which means the proposed control system could stabilize the body, while guarantees the tracking of the commanded angles. Also, as mentioned before, the maneuver is mainly caused by the AOA, this is the reason of why the ϑ and ωz are relatively bigger than the other angles. Time history of the deflection angles in the body frame is shown in Fig. 4. It is clear that the performance advantage of the proposed controller doesn’t come at the expense of the control effort. Moreover, the magnitude of δx and δy is much smaller compared with δz . Also, the deflection angles δz has a similar
response to ωz , both, in general, in a reverse-sinusoidal type, which indicates the main function of the actuators is to stabilize the body rates. 5. Conclusion An adaptive autopilot for a nonlinear six-degree-of-freedom missile dynamics is introduced in this paper, using backstepping and sliding model control techniques. Simulation results indicate that the resultant control system works well and effectively in missile flight control system.
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Fig. 3. Time history of body rates.
Fig. 4. Time history of deflection angles.
During the backstepping procedures, the sliding mode control technique is introduced, compensating for the disturbing terms and simplify the design of the control system in the last step. Also, the integral filters, incorporating with the sliding mode control component, are presented so as to facilitate the computation of the derivations of the virtual inputs during the standard backstepping procedure. The results demonstrate that the estimation error of the proposed filters is acceptable, and with the inclusion of the sliding mode control component, the convergence time of the filters is fast.
It should be pointed out that one of the advantages of using such integral filters is that the noise in control input is not directly propagated to its derivatives. It is easy to verify that are globally stable as long as input is Lebesgue measurable. Appendix A This appendix rigorously addresses the issue of the development of flight dynamics equations utilized in Section 2. Accord-
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⎡
ing to the definition of the speed frame, the missile’s velocity components in this frame can be expressed as: vx = v
vy = 0 vz = 0
(43)
Also, we have the following dynamics equation in the speed frame δv F + wv × v = (44) δt m where F represents the sum of all the external forces in the speed frame. Then, we have ⎧ ⎨ mv˙ = Fxv mωzv v = Fyv (45) ⎩ −mω v = F yv zv Also, according to the geometric relation between the speed frame and the body frame, we have w b = wv + α˙ + β˙
(46)
The preceding relation can be expressed in the speed frame as: ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 ωx 0 ωxv (47) C vb ⎣ ωy ⎦ = ⎣ ωyv ⎦ + C bv ⎣ 0 ⎦ + ⎣ β˙ ⎦ α˙ ωz b ωzv 0 where ⎡
cos β cos α C bv = ⎣ − cos β sin α sin β
sin α cos α 0
⎤ − sin β cos α sin β sin α ⎦ cos β
Substituting (47) into (45), we have ⎧ ωx cos β cos α − ωy cos β sin α + ωz sin β ⎪ ⎪ ⎪ ⎨ = ωxv + α˙ sin β ωx sin α + ωy cos α = ωyv β˙ ⎪ ⎪ ⎪ ⎩ −ωx sin β cos α + ωy sin β sin α + ωz cos β = ωzv + α˙ cos β
(48)
Also, from (45), we have Fyv Fzv ωzv = mv mv Therefore, (49) becomes ⎧ ⎨ α˙ = (−ωx sin β cos α + ωy sin β sin α + ωz cos β)/ cos β − Fyv /(mv) ⎩ β˙ = ω sin α + ω cos α + F /(mv) x y zv
(50)
(51)
In particular, the kinematics relation of the roll angle, in the body frame, could be expressed in the form: γ˙ = ωx − tan ϑ(ωy cos γ − ωz sin γ ) Therefore, the matrices used in system (2) are
Moreover, from the dynamics equations of the body rates in the body frame, we have ⎧ Ix ω˙ x − (Iy − Iz )ωy ωz ⎪ ⎪ ⎪ δ ⎪ ⎪ = (mδxx δx + mxy δy + mωx x ωx /vm )qSl ⎪ ⎪ ⎨ I ω˙ − (I − I )ω ω y y z x z x (53) δ ω β ⎪ = (my β + myy δy + my y ωy /vm )qSl ⎪ ⎪ ⎪ ⎪ Iz ω˙ z − (Ix − Iy )ωx ωy ⎪ ⎪ ⎩ δ ω = (mαz α + mzz δz + mz z ωz /vm )qSl Therefore, the matrices used in system (3) are ⎤ ⎡ δ 0 mδxx qSl/Ix mxy qSl/Ix ⎥ ⎢ δ C=⎣ ⎦ 0 0 myy qSl/Iy δ 0 0 mzz qSl/Iz f 2 (x 1 , x 2 ) ⎡ ⎤ [mωx x ωx qSl/vm − (Iz − Iy )ωy ωz ]/Ix ω = ⎣ [mβy βqSl + my y ωy qSl/vm − (Ix − Iz )ωx ωz ]/Iy ⎦ ω [mαz αqSl + mz z ωz qSl/vm − (Iy − Ix )ωx ωy ]/Iz References
Hence, the dynamics equations of AOA and sideslip angle could be derived in the form: ⎧ ⎨ α˙ = (−ωx sin β cos α + ωy sin β sin α + ωz cos β − ωzv )/ cos β (49) ⎩ β˙ = ω sin α + ω cos α − ω x y yv
ωyv = −
⎤ −Fyv /(mv) f 1 (x 1 ) = ⎣ Fzv /(mv) ⎦ 0 ⎤ ⎡ − cos α tan β sin α tan β 1 ⎦ sin α cos α 0 A(x 1 ) = ⎣ 1 − tan ϑ cos γv tan ϑ sin γv ⎡ ⎤ 0 0 −ρν 2 SCyδ /(2mv cos β) ⎦ B(x 1 ) = ⎣ 0 −ρν 2 SCzδ /(2mv) 0 0 0 0
(52)
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