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Nonlinear gain-scheduled flight controller design via stable manifold method
Aerospace Science and Technology 80 (2018) 301–308
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Nonlinear gain-scheduled flight controller design via stable manifold method Anh Tuan Tran a,∗ , Noboru Sakamoto b , Koichi Mori c a b c
Department of Mechanical Systems Engineering, Nagoya University, Japan Department of Mechatronics, Faculty of Science and Engineering, Nanzan University, Japan Department of Aerospace Engineering, Nagoya University, Japan
a r t i c l e
i n f o
Article history: Received 31 January 2018 Received in revised form 20 May 2018 Accepted 3 July 2018 Available online 10 July 2018 Keywords: Nonlinear optimal control Stable manifold Flight controller Stabilization/Control augmentation system F-18 HARV
a b s t r a c t This research presents a nonlinear gain-scheduled flight controller design method via a stable manifold theory in order to handle the nonlinearities of the F-18 High Alpha Research Vehicle due to the change in the aerodynamic characteristics at different angles of attack and the airspeed variation. The designed longitudinal flight control system consists of a nonlinear gain-scheduled stabilization augmentation system which is designed using the stable manifold method, and a linear gain-scheduled control augmentation system which consists of proportional and integral gains. The nonlinear longitudinal flight controller is verified in a 6 degree-of-freedom simulator. © 2018 Elsevier Masson SAS. All rights reserved.
1. Introduction One of the desired characteristics of modern fighter or acrobatic aircraft is the capability to fly at high angles of attack or post-stall conditions. However, the aerodynamic characteristics become extremely complex and highly nonlinear when the angle of attack exceeds a stall angle. At that moment, many aerodynamic coefficients of the aircraft dramatically change their values even with a small increment of the angle of attack, i.e., the aircraft experiences adverse phenomena, such as significant lift and pitching control losses, asymmetric stall, and auto-rotation, which generally cause linear controllers to be inadequate and inefficient [1,2]. A traditional approach to take into account the nonlinearities of the aircraft in the flight controller design process is to use gain-scheduling technique. In this approach, a number of operating points for specified flight conditions are defined, and then the mathematical model of the aircraft is linearized at each flight condition. A family of linear controller candidates for those flight conditions is constructed, then a gain-scheduled controller is obtained by blending controller candidates to cover the entire flight envelope [3–9]. The advantage of this approach is that it is simple and practical. However, it requires many operating points to
*
Corresponding author. E-mail addresses: [email protected] (A.T. Tran), [email protected] (N. Sakamoto), [email protected] (K. Mori). https://doi.org/10.1016/j.ast.2018.07.002 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.
capture the nonlinearity of the aircraft, especially in the post-stall region. There are also other approaches which can consider the changes of aerodynamic characteristics in order to enable the aircraft to fly at high angles of attack or to perform super-maneuvers. It appears that nonlinear dynamic inversion and sliding mode control are two of the most applied methods to design nonlinear flight controllers recently. The nonlinear dynamic inversion approach [10–14] directly takes into account the nonlinearities within the entire flight envelope. Therefore, it can control the aircraft in the linear and post-stall regions by using a single controller. However, this method requires a precise knowledge of the aircraft model and is sensitive to modeling errors and parameter uncertainties. This sensitivity may be critical since it is not easy to obtain the exact model of the aircraft in practice. The sliding mode control approach [15–18] can also handle post-stall flights as well as perform super-maneuvers like Herbst and Cobra maneuvers. It is a robust nonlinear control design approach which can deal with modeling errors and parameter uncertainties efficiently. However, it experiences an undesirable oscillation phenomenon, called chattering, in implementation due to the discontinuous control action of the method. Some alternative approaches which can handle nonlinearities of the aircraft are, for instance, nonlinear optimal control [19–21], back-stepping [22–25], and neural networks [26–29]. This paper presents a flight controller design method for the longitudinal motion control of the F-18 High Alpha Research Vehicle (HARV) [30,31]. Although the HARV is equipped with thrust
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vectoring mechanism, only aerodynamic control surfaces are used to control the aircraft in this research. The flight controller structure considered in this research is similar to the conventional one which consists of a stabilization augmentation system (SAS) that uses proportional gains to feedback attitude rates, and a control augmentation system (CAS) that uses proportional and integral gains to feedback attitude angle errors. However, in this research, we take the angle of attack of the aircraft as a state variable in the SAS design to deal with the changes in the aerodynamic characteristics at high alpha angles. Moreover, since the variation of the aerodynamic characteristics is highly nonlinear, a nonlinear SAS is designed via the stable manifold method [32–36]. Additionally, since the dynamics of the aircraft also depend on the airspeed, the gain-scheduling technique is applied to deal with the change of airspeed in flight. The traditional gain-scheduling technique usually requires many operating points corresponding to specified flight conditions in order to capture the nonlinearities of the aircraft. Then, a linear controller candidate is designed for each flight condition. In this research, a set of sparse operating points is considered. For that reason, the nonlinearities of the aerodynamic characteristics between adjacent operating points are strong, which corrupt the control performance of the linear controller. The proposed nonlinear controller, on the other hand, does take into account those nonlinearities in the controller design process. Therefore, it can achieve better control performance compared with the linear one. A partial result of this research is briefly discussed in [37]. In this paper, the controller design method, result and discussion will be given in details. The outline of this paper is as follows: In Section 2, a nonlinear mathematical model of the longitudinal motion of the aircraft is presented. Section 3 describes the structure of the gain-scheduled controller considered in this research. In Sections 4 and 5, the gain-scheduled SAS and CAS are designed. The simulation results and discussions are shown in Section 6. Section 7 concludes this paper. 2. Mathematical model This section describes the nonlinear equations of motion (EOMs) of the HARV. In this research, we focus on the motion control in the longitudinal direction. The perturbations in the lateral–directional motion are kept at small values via a feedback controller. Therefore, the lateral–directional motion is omitted due to its weak impact on the longitudinal variables. For that reason, this paper only consider the longitudinal motion model of the aircraft. Since the purpose is to control the aircraft using only aerodynamic control surfaces, the thrust vectoring mechanism is unused. Hence, the direction of the thrust is fixed along the body x-axis. The general nonlinear EOMs for the longitudinal motion are reported in [38] (page 115) and are summarized as below
α˙ = q + q˙ =
M I yy
1 mV T
(− L − F T sin α + mg D cos(θ − α )), (1)
,
θ˙ = q, where α is the angle of attack, q is the pitch angular velocity, m is the mass of aircraft, V T is the total velocity, L is the lift force, F T is the thrust, g D is the gravitational constant, I y y is the pitching moment of inertia, and M is the pitch moment. In these equations, the definitions of the lift force L and the pitch moment M are
L=
1 2
1
ρ V T2 SC L , M = ρ V T2 S c¯ C M , 2
(2)
Fig. 1. Experimental data of C L 0 (α , δe ) at Mach 0.6 and its approximation which is expressed in (3).
where ρ is the air density, S is the reference area, c¯ is the mean aerodynamic chord. C L and C M are the total lift and moment coefficients. In general, those coefficients depend on numerous parameters such as the altitude h, aircraft velocity V T , angle of attack α , etc. In this research, we assume that around an operating point, the changes of the aerodynamic coefficients with respect to the altitude and velocity are small and can be neglected. Therefore, they can be expressed as functions of the angle of attack α and elevator deflection δe as below
c¯ ˙ , C Lq (α )q + C L α˙ (α )α 2V T c¯ ˙ . = C M 0 (α , δe ) + C Mq (α )q + C M α˙ (α )α 2V T
C L = C L 0 (α , δe ) + CM
In the above equations, C ∗∗ denotes the aerodynamics derivative coefficients which are reported in [30]. Fig. 1 illustrates the values of C L 0 (α , δe ) at the altitude h = 15000 ft and Mach 0.6. It can be seen from this figure that the stall angle is around 35 deg. When the angle of attack exceeds the stall angle, the aerodynamic coefficients dramatically change, and the aircraft is beyond the linear region. In order to design the flight controller for the HARV, the aerodynamic data of the aircraft are expressed in high order polynomials which are constructed by using a polynomial fitting method [33]. For example, the approximated polynomial function of C L 0 (α , δe ) is shown below
C L 0 (α , δe ) = 0.1262 + 0.1094α − 0.0022α 2
+ 1.0893E − 05α 3 + 0.0131δe − 0.0002α δe + 4.6787E − 07α 2 δe .
(3)
Note that only two sets of data corresponding to two positions of the elevator are provided in [30]. Therefore, the aerodynamic data of C L 0 (α , δe ) at the other elevator angles are estimated using (3). Fig. 1 shows that the approximated data of C L 0 (α , δe ), which are calculated from (3), and the experimental data, which are described in [30], are matching. Substituting the approximated functions of the aerodynamic derivative coefficients into (1), eliminating the high order terms of δe (δek , k = 2, 3, . . . ) since they are small, and assuming that θ − α is insignificant, which makes cos(θ − α ) ≈ 1, ones obtain
α˙ = f 1 (α , q) + g1 (α , q)δe , q˙ = f 2 (α , q) + g 2 (α , q)δe ,
θ˙ = q,
(4)
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Table 1 Nominal flight conditions. Parameter
Cond. 1
Cond. 2
Cond. 3
Unit
h VT
15000 300 0.2246 0.2246 −0.0619 9203
15000 450 0.0733 0.0733 −0.0132 7969
15000 650 0.0293 0.0293 0.0030 9188
ft ft/s rad rad rad lbs
α θ δe T a
Cond. is short for Condition.
4.1. Problem formulation Fig. 2. Control block diagram.
where f 1 (α , q), f 2 (α , q), g 1 (α , q) and g 2 (α , q) are appropriately calculated. Note that, in (4), the velocity V T and the thrust F T take their nominal values and are constant around a specific flight condition. 3. Gain-scheduled flight controller structure This section describes the flight controller structure of the HARV for the longitudinal motion control. As mentioned in Section 1, the longitudinal flight controller of the HARV in this research has similar structure to the conventional S/CAS with a modification that the angle of attack of the aircraft is feed-backed in the SAS in order to deal with the variation in the aerodynamic characteristics with respect to the angle of attack. The block diagram of the flight controller structure is shown in Fig. 2. In this research, two cases of the SAS are considered: the first one is the linear SAS which contains only proportional gains to feedback the information of the angle of attack and pitch rate; the second one is the nonlinear SAS which is designed to deal with the nonlinearities due to the changes in the aerodynamic characteristics when the HARV is at the high angle of attack. As for the CAS design, since the nonlinearities are taken into account in the SAS, only PI controllers are used in both cases. As can be seen in (1), the EOMs of the aircraft also depend on the velocity V T . When the aircraft operates in a broad operating region with different flight conditions, it is necessary to consider the effect of the velocity since its variation may affect the stability of the closed-loop stability and the control performance. In other words, a controller designed for a given velocity may become unstable, or the control performance may deteriorate when the velocity alters. A conventional approach to deal with the alternation of the velocity is to use the gain-scheduling technique. In the gain-scheduling approach, the aircraft is trimmed at different flight conditions, and the nonlinear EOMs (1) are linearized at those conditions. Then, a family of controller candidates is designed using those linear EOMs. Finally, the gain-scheduled controller to cover all flight conditions is obtained by interpolating the candidates at two adjacent flight conditions. This research considers three flight conditions which correspond to different velocity values, namely 300 ft/s, 450 ft/s and 600 ft/s, around the altitude h of 15000 ft. The aircraft is trimmed at those flight conditions, and the results are shown in Table 1. By substituting the nominal parameters at trimmed flight conditions (Table 1) into (4), both linear and nonlinear EOMs of the aircraft corresponding to each flight condition can be derived. 4. Nonlinear and linear gain-scheduled SAS In this section, a nonlinear optimal SAS based on the stable manifold method is first designed, then a linear optimal SAS via the LQR method using the same cost function is considered.
The nonlinear EOMs of the aircraft at each flight condition are derived, and the corresponding controller is designed. In order to design the SAS, only the first two equations of (4) are used. Those equations are rewritten in the following form
x˙ = f (x) + g (x)u , T where x = α q , f (x) = f 1 (α , q) T g 1 (α , q) g 2 (α , q) , u = δe . Let us denote A=
∂ f (x) , ∂ x x=0
(5) f 2 (α , q)
T
,
g (x) =
B = g (0).
The optimal control problem is to find a control law u such that the closed-loop system of (5) is asymptotically stable and the following cost function is minimized
∞ J = (x T Q x + u T Ru )dt , 0
where Q ∈ R2×2 and R ∈ R are weighting matrices. 4.2. Derivation of the Hamiltonian system The pre-Hamiltonian H D [39] for the optimal control problem of (5) is written as
H D (x, p , u ) = p T ( f (x) + g (x)u ) + x T Q x + u T Ru ,
(6)
where p is the co-state. The minimizing vector u¯ of (6) is obtained by solving
∂ H D (x, p , u ) = 0, ∂u which gives the result
1 u¯ (x, p ) = − R −1 g (x) T p . 2
(7)
A Hamilton–Jacobi equation (HJE) is obtained by substituting u¯ (x, p ) into the pre-Hamiltonian H D (6)
H (x, p ) = p T f (x) −
where p = ∂ V /∂ x1
1 4
p T g (x) R −1 g (x) T p + x T Q x = 0,
∂ V /∂ x2
T
(8)
and V is an unknown function.
T
can be determined by solving Next, if p = ∂ V /∂ x1 ∂ V /∂ x2 the HJE, the optimal control input u¯ (x, p ) is attained from (7). To accomplish this goal, the stable manifold method [32–34] is utilized. The author of [32] proved that the stabilizing solution of the HJE (8) is on the stable manifold of the Hamiltonian system
x˙ =
∂H (x, p ), ∂p
p˙ = −
∂H (x, p ). ∂x
(9)
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The above Hamiltonian system can be written in the following form in which the linear and nonlinear parts are separated
A x˙ = p˙ −2Q
−R B −AT
x N x (x, p ) + , p N p (x, p )
(10)
where R B = 12 B R −1 B T , and N x (x, p ), N p (x, p ) are higher order terms. A linear coordinate transformation using an appropriate matrix T ∈ R2n×2n is taken [32]
x p
= T −1
x , p
T=
I P
S PS + I
(11)
and S is the solution of the Lyapunov equation
In the new coordinates (x , p ), the linear part of (10) is diagonalized
=
F 0
0 −F T
n (x , p ) x + x , n p p (x , p )
(12)
A − R B P is Hurwitz, and nx (x , p ), np (x , p )
where the matrix F = are higher order terms.
4.3. Controller construction By using the stable manifold method reviewed in the Appendix, solution curves on the stable manifold S of (9) are calculated, then the stable manifold S can be approximated. Since S is locally canonically surjective onto x-space, one can represent S as p = π (x) with π (x) ∈ R2 is an appropriate function. It can be
T
written that p = π (x) = π1 (x) π2 (x) , where π1 (x) ∈ R and π2 (x) ∈ R are the elements of π (x). Later in section 4.4, some solution curves for this problem will be shown in Fig. 3. By calculating many solution curves that cover the domain of interest, they will shape the stable manifold S of (9), and then π (x) can be approximated. There are several ways to obtain π (x). One way is to use a polynomial fitting algorithm to approximate π (x) with polynomials [33]. Another way is to use an interpolation method to represent the relationship between x and p [40]. In this research, the second approach is chosen. Finally, the optimal control input u¯ (x, p ) is
1 u¯ (x, π (x)) = − R −1 g (x) T π (x). 2
(13)
The LQR controller of the optimal control problem of (5) is
1 u 0 = − R −1 B T P x. 2
(14)
Note that (14) is the linearization of (13) at the origin, which is
B = g (0),
∂ π (x) ∂x
x=0
300 450 600
7.19 6.87 9.17
Kα
−48.21 −18.92 −20.61
K
Optimal
Kp
Ki
Kp
Ki
[−50, 0] [−40, 0] [−40, 0]
[−50, 0] [−40, 0] [−40, 0]
−50 −40 −40
−20 −10 −10
In order to cover a broad operating region, a family of controllers at different flight conditions is designed; then a gainscheduled controller is constructed. In Sections 4.1–4.3, the design methods of the nonlinear and linear controllers are introduced. The same weighting matrices Q and R are chosen to design the nonlinear and linear controllers at each flight condition as follows
( A − R B P ) S + S ( A − R B P )T = R B .
Kq
4.4. Gain-scheduling algorithm
A T P + P A − P R −1 B T P + Q = 0,
x˙ p˙
V (ft/s)
,
where P is the stabilizing solution of the algebraic Riccati equation
Table 2 Designed S/CAS gains.
= P.
(15)
The LQR controller (14) can be written as
u 0 = K α α + K q q, where K α and K q are the proportional gains which are calculated as K α K q = − 12 R −1 B T P .
Q1 =
0.1 0 , R 1 = 0.3, 0 1000
Q2 =
0.1 0 , R 2 = 1.5, 0 1000
0.1 0 , R 3 = 1.5, Q3 = 0 1000 where Q i and R i (i = 1, 2, 3) are the weighting matrices corresponding to the flight condition i. The resultant linear gains K α and K q for each flight condition are shown in Table 2. As mentioned in Section 4.3, the data interpolation method is used to construct the nonlinear controller. To this end, a twodimensional look-up table of π (x) is built for each flight condition. It has 49 × 49 grid data points and covers the domain of interest (α , q) of −10 deg 70 deg × −70 deg/s 70 deg/s . For visualization, the data representing the stable manifolds π (x) = π1 (x) π2 (x) T corresponding to three flight conditions are illustrated in Fig. 3. It is shown in Fig. 3 that the stable manifolds corresponding to the flight conditions are different due to the changes in the mathematical models. In the figures of the stable manifold corresponding to the flight condition 1, three trajectories start from different initial conditions of angle of attack α and pitch rate q and converge to the origin which corresponds to the equilibrium point. One can observe the disparities between the stable manifold π (x) (meshed surface) and its approximation at the origin (plain surface) which is calculated by using (15). Those disparities become significant when the values of α and q go far from the origin (trim condition). They represent how strong the nonlinearities of the aircraft are at different states of α and q. Also, by looking at these figures, ones can predict the states of the aircraft at which the control performance of the linear controller might degrade. Next, the gain-scheduled controller u GS is designed. A practical way is to linearly interpolate nearby controller candidates. The interpolation rule is shown below
u GS = (1 − η)u i + ηu i +1 , where i and i + 1 are two adjacent operating points such that V i V T V i +1 , u i and u i +1 are the corresponding linear (or nonlinear) controllers at those operating points, and
η=
VT − Vi V i +1 − V i
.
The calculation of η can be interpreted that: if V T = V i , the controller u i will be used; if V T = V i +1 , the controller u i +1 will be used; if V i < V T < V i +1 , both u i and u i +1 will be used for calculating the control input.
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305
Fig. 3. The components π1 (x) (meshed surface, left-hand side column), π2 (x) (meshed surface, right-hand side column) of the stable manifolds S and their linear approximations obtained from the stabilizing solution of the algebraic Riccati equation (plain surface) at the flight conditions 1, 2, 3, respectively. In the first row, which corresponds to the flight condition 1, three lines show the examples of stabilization trajectories starting from different initial conditions.
5. Gain-scheduled CAS This section presents the controller design method of the gainscheduled CAS. Based on the closed-loop system including the SAS, the CAS is designed. Similar to the SAS design, a family of CAS for three flight conditions is designed, then the gain-scheduled controller is constructed. In the previous section, the change in the aerodynamic characteristics was already taken into account in the SAS design. Therefore, only linear CAS is considered in this section. The CAS consists of proportional and integral gains (PI controller). In order to design the CAS, the linearization of the SAS closed-loop system is used. Note that the linearization of the nonlinear optimal controller design via stable manifold method (13) coincides with the LQR one (14). Hence, the linearized closed-loop SAS and the resultant CAS in both cases of the nonlinear SAS and the linear SAS are identical. The CAS design method is very similar to the one presented in [7,8]. An admissible set K of the proportional gain K p and the
integral gain K i is determined. This set should be appropriately chosen in order to avoid actuator magnitude saturation. Next, the admissible set K is gridded, and the gain K p and K i at each node will be evaluated using the cost function
T f e 2 dt ,
J CAS =
(16)
0
where T f is a large value, says 100. This cost function can be calculated in MATLAB. Then, the optimal gains K p and K i are the ones such that when tracking a step reference input, the cost function (16) is minimized. Table 2 shows the designed optimal gains for each flight condition. Finally, the gain-scheduled CAS is obtained using the method similar to the one used in the gain-scheduled SAS design. In addition, a simple PI controller for the thrust control is designed to regulate the velocity of the HARV
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Fig. 4. Responses of the velocity, angle of attack α , pitch rate q, pitch angle θ when using only one candidate S/CAS controller at the operating point V T = 450 (ft/s). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
t δT = K V p ( V com − V T ) + K V i
( V com − V T )dt , 0
where V com is the reference velocity, K V p = 300, K V i = 20. 6. Simulation result and discussion The designed controllers are verified in a 6 degree-of-freedom nonlinear simulator. The aerodynamic characteristics of the HARV obtained from wind tunnel tests are reported in [30], which correspond to the altitude of 15,000 ft and Mach number range between 0.3 and 0.9. Interpolation is required to obtain aerodynamic data of nearby flight conditions. Moreover, the actuator models of the control surfaces and thrust [30] are taken into account in the simulation. Note that the elevator has the rate limit of 40 deg/s and position limits of −24 deg and 10.5 deg, where the positive direction is trailing-edge down [30,41]. In Sections 4 and 5, two types of gain-scheduled controllers are designed. The first one is the nonlinear gain-scheduled controller which contains the nonlinear optimal SAS, and the second one is the linear gain-scheduled controller which has the LQR SAS. Recall that both of them have the same CAS. In this section, simulations are conducted to evaluate the effectiveness of the gain-scheduling technique as well as to compare the control performances of the nonlinear gain-scheduled controller and the linear gain-scheduled one. 6.1. Linear and nonlinear controllers without gain-scheduling technique Figs. 4 and 5 show the simulation results of the nonlinear and linear controllers in which only one S/CAS candidate corresponding to the flight condition 2 is used to control the pitch angle of the HARV. Although the nominal velocity of the flight condition 2 is V T = 450 ft/s, in this simulation, the velocity command V com is set to be 400 ft/s to verify the robustness of the designed controller. The aircraft is controlled to follow the pitch angle command θcom .
Fig. 5. Responses of the elevator when using only one candidate S/CAS controller at the operating point V T = 450 (ft/s).
The simulation results show that the nonlinear controller has a faster convergence rate than the linear one. The main discrepancy in the control performances of the linear and nonlinear controllers is at the time when the HARV is at the high angle of attack. At this moment, the HARV is far from the linear region; and the aerodynamic characteristics vary drastically in accordance with the angle of attack α . In the design process of the nonlinear controller, this variation in the aerodynamic characteristics is appropriately taken into account. Conversely, it is not considered in the linear controller. As a consequence, it can be seen that the nonlinear controller has better tracking performance than the linear one. For the rest of the time, since the aircraft is at low angle of attack, the discrepancy between the nonlinear and linear controllers is insignificant. The cost functions (16) of the linear controller ( J linear ) and the nonlinear one ( J nonlinear ) are calculated. The results are
J linear = 6346,
J nonlinear = 4644,
J nonlinear J linear
= 73%.
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Fig. 6. Responses of the velocity, angle of attack
307
α , pitch rate q, pitch angle θ when using the gain-scheduled controllers.
6.2. Linear and nonlinear gain-scheduled controllers Figs. 6 and 7 show the simulation results in which the nonlinear gain-scheduled controller and the linear gain-scheduled one are evaluated. As mentioned before, the EOMs of the aircraft depend on the changes in the aerodynamic characteristics at different angles of attack as well as the variation of airspeed. These simulations analyze the capabilities of the two aforementioned gainscheduled controllers to handle such nonlinearities. Also, the same pitch angle command pattern θcom as the one in the previous simulation is given to compare with the controllers that do not use the gain-scheduling technique in Figs. 4 and 5. It can be seen that the nonlinear gain-scheduled controller has a better overall tracking performance than the linear gainscheduled one. However, the discrepancy between these controllers is not as remarkable as it is in the previous simulation (Figs. 4 and 5). The reason is that in this case, multiple aircraft models at different flight conditions are considered. Therefore, the aerodynamic characteristics and airspeed variation during the flight are taken into account in the design processes of both linear and nonlinear gain-scheduled controllers. As a result, the linear gain-scheduled one can achieve a similar control performance as the nonlinear gain-scheduled one does within the considered flight conditions. The cost functions (16) are calculated. The results are
J linear = 5244,
J nonlinear = 4340,
J nonlinear J linear
= 83%.
In the comparison between the nonlinear and linear gainscheduled controllers (Figs. 6 and 7) with the ones without gain scheduling technique (Figs. 4 and 5), the overall tracking performance and the cost function are improved thanks to the implementation of the gain scheduling technique. For the nonlinear controller that uses only one SAS/CAS candidate of the flight condition 2 (Figs. 4 and 5), although it considers the change in the aerodynamic characteristics in the SAS design, the variation of airspeed is not taken into account. Therefore, the implementation of
Fig. 7. Responses of the elevator when using the gain-scheduled controllers.
the gain scheduling technique (Figs. 6 and 7) further improves its tracking performance. 7. Conclusion This research evaluates the benefits of the gain-scheduling technique and the nonlinear control approach via the stable manifold method in flight controller design. The simulation results show that by taking into account the nonlinear aerodynamic characteristics of the aircraft in the controller design process, the nonlinear controller can perform better in the nonlinear domain, i.e., at high angle of attack, compared with the linear one. When the gainscheduling technique is employed, it improves the control performance of both linear and nonlinear ones. However, the nonlinear controller is still better than the linear one in term of reducing tracking error by a remarkable amount. Conflict of interest statement There is no conflict of interest.
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Appendix A. Review of stable manifold method In order to calculate the stable manifold of (12), one defines the sequences xk (t , ξ ) and pk (t , ξ )
xk +1 = e F t ξ +
t
e F (t −s) nx xk (s), pk (s) ds,
0
pk +1 = −
∞
(A.1)
e − F (t −s) np T
xk (s), p s (s) ds,
t
for k = 1, 2, . . . , and the initial conditions are
x0 = e F t ξ,
p 0 = 0,
where ξ ∈ Rn is arbitrary. Theorem 1 (Sakamoto and van de Schaft, 2008 [32]). For sufficiently small |ξ |, the sequences xk (t , ξ ) and pk (t , ξ ) are convergent to zero, that is, xk (t , ξ ), pk (t , ξ ) → 0 as t → ∞ for all k = 0, 1, 2, . . . Furthermore, xk (t , ξ ) and pk (t , ξ ) are uniformly convergent to a solution of (12) on [0, ∞) as k → ∞. Let x (t , ξ ) and p (t , ξ ) be the limit of xk (t , ξ ) and yk (t , ξ ), respectively. Then, x (t , ξ ) and y (t , ξ ) are the solution on the stable manifold of (12), that is, x (t , ξ ), p (t , ξ ) → 0 as t → ∞. Therefore, by using the sequence (A.1), one can calculate flows on the stable manifold of (12). The flows on the stable manifold S of the Hamiltonian system (9) in the original coordinates are given by
x(t , ξ ) =T p (t , ξ )
x (t , ξ ) . p (t , ξ )
References [1] J.R. Chambers, S.B. Grafton, Aerodynamic characteristics of airplanes at high angle of attack, Tech. Rep. NASA-TM-74097, NASA Langley Research Center, Hampton, Virginia, 1977. [2] J.R. Chambers, High-angle-of-attack aerodynamics: lessons learned, in: 4th Applied Aerodynamics Conference, 1986. [3] J. Ackermann, Multi-model approaches to robust control system design, in: Uncertainty and Control, 1985, pp. 108–130. [4] W.J. Rugh, J.S. Shamma, Research on gain scheduling, Automatica 36 (10) (2000) 1401–1425. [5] R.A. Nichols, R.T. Reichert, W.J. Rugh, Gain scheduling for H-infinity controllers: a flight control example, IEEE Trans. Control Syst. Technol. 1 (2) (1993) 69–79. [6] J. Wang, N. Sundararajan, A nonlinear flight controller design for aircraft, Control Eng. Pract. 3 (6) (1995) 813–825. [7] M. Sato, K. Muraoka, Flight controller design and demonstration of quad-tiltwing unmanned aerial vehicle, J. Guid. Control Dyn. 38 (6) (2015) 1071–1082. [8] A.T. Tran, N. Sakamoto, M. Sato, K. Muraoka, Control augmentation system design for quad-tilt-wing unmanned aerial vehicle via robust output regulation method, IEEE Trans. Aerosp. Electron. Syst. 53 (1) (2017) 357–369. [9] D.J. Leith, W.E. Leithead, Gain-scheduled control: relaxing slow variation requirements by velocity-based design, J. Guid. Control Dyn. 23 (6) (2000) 988–1000. [10] G.J. Balas, Flight control law design: an industry perspective, Eur. J. Control 9 (2–3) (2003) 207–226. [11] R.J. Adams, J.M. Buffington, S.S. Banda, Design of nonlinear control laws for high-angle-of-attack flight, J. Guid. Control Dyn. 17 (4) (1994) 737–746. [12] S.A. Snell, D.F. Enns, W.L. Garrard Jr., Nonlinear inversion flight control for a supermaneuverable aircraft, J. Guid. Control Dyn. 15 (4) (1992) 976–984.
[13] W.C. Reigelsperger, S.S. Banda, Nonlinear simulation of a modified F-16 with full-envelope control laws, Control Eng. Pract. 6 (3) (1998) 309–320. [14] D. Bugajski, D. Enns, Nonlinear control law with application to high angle-ofattack flight, J. Guid. Control Dyn. 15 (3) (1992) 761–767. [15] S.N. Singh, M.L. Steinberg, A. Page, Nonlinear adaptive and sliding mode flight path control of F/A-18 model, IEEE Trans. Aerosp. Electron. Syst. 39 (4) (2003) 1250–1262. [16] S. Seshagiri, E. Promtun, Sliding mode control of F-16 longitudinal dynamics, in: American Control Conference, 2008, pp. 1770–1775. [17] E.M. Jafarov, R. Tasaltin, Robust sliding-mode control for the uncertain MIMO aircraft model F-18, IEEE Trans. Aerosp. Electron. Syst. 36 (4) (2000) 1127–1141. [18] B.K. Mukherjee, M. Sinha, Extreme aircraft maneuver under sudden lateral CG movement: modeling and control, Aerosp. Sci. Technol. 68 (2017) 11–25. [19] W.L. Garrard, J.M. Jordan, Design of nonlinear automatic flight control systems, Automatica 13 (5) (1977) 497–505. [20] A.T. Tran, N. Sakamoto, Y. Kikuchi, K. Mori, Pilot induced oscillation suppression controller design via nonlinear optimal output regulation method, Aerosp. Sci. Technol. 68 (2017) 278–286. [21] R. Cory, R. Tedrake, Experiments in fixed-wing UAV perching, in: AIAA Guidance, Navigation and Control Conference and Exhibit, August 2008. [22] L. Sonneveldt, Q. Chu, J. Mulder, et al., Nonlinear flight control design using constrained adaptive backstepping, J. Guid. Control Dyn. 30 (2) (2007) 322–336. [23] T. Lee, Y. Kim, Nonlinear adaptive flight control using backstepping and neural networks controller, J. Guid. Control Dyn. 24 (4) (2001) 675–682. [24] J. Farrell, M. Sharma, M. Polycarpou, et al., Backstepping-based flight control with adaptive function approximation, J. Guid. Control Dyn. 28 (6) (2005) 1089–1102. [25] J. Cayero, B. Morcego, F. Nejjari, Modelling and adaptive backstepping control for TX-1570 UAV path tracking, Aerosp. Sci. Technol. 39 (2014) 342–351. [26] B.S. Kim, A.J. Calise, Nonlinear flight control using neural networks, J. Guid. Control Dyn. 20 (1) (1997) 26–33. [27] A.J. Calise, R.T. Rysdyk, Nonlinear adaptive flight control using neural networks, IEEE Control Syst. Mag. 18 (6) (1998) 14–25. [28] K. Rokhsaz, J. Steck, Use of neural networks in control of high-alpha maneuvers, J. Guid. Control Dyn. 16 (5) (1993) 934–939. [29] D.I. Ignatyev, A.N. Khrabrov, Neural network modeling of unsteady aerodynamic characteristics at high angles of attack, Aerosp. Sci. Technol. 41 (2015) 106–115. [30] J. Cao, J. Frederick Garrett, E. Hoffman, H. Stalford, Analytical Aerodynamic Model of a High Alpha Research Vehicle Wind-Tunnel Model, Tech. Rep. NASACR-187469, Georgia Institute of Technology, 1990. [31] D. Enns, D. Bugajski, R. Hendrick, G. Stein, Dynamic inversion: an evolving methodology for flight control design, Int. J. Control 59 (1) (1994) 71–91. [32] N. Sakamoto, A.J. van der Schaft, Analytical approximation methods for the stabilizing solution of the Hamilton–Jacobi equation, IEEE Trans. Autom. Control 53 (10) (2008) 2335–2350. [33] N. Sakamoto, Case studies on the application of the stable manifold approach for nonlinear optimal control design, Automatica 49 (2) (2013) 568–576. [34] N. Sakamoto, A recent progress in the optimal control design under various constraints, in: 7th IFAC Conference on Manufacturing Modelling, Management, and Control, June 2013, pp. 1506–1511. [35] N. Sakamoto, Analysis of the Hamilton–Jacobi equation in nonlinear control theory by symplectic geometry, SIAM J. Control Optim. 40 (6) (2002) 1924–1937. [36] A.T. Tran, N. Sakamoto, A general framework for constrained optimal control based on stable manifold method, in: IEEE 56th Annual Conference on Decision and Control, CDC, December 2016. [37] A.T. Tran, N. Sakamoto, Application of stable manifold method to aircraft longitudinal flight controller design, in: The 60th Japan Joint Automatic Control Conference, Tokyo, Japan, November 2017. [38] B.L. Stevens, F.L. Lewis, Aircraft Control and Simulation, 2nd edition, John Wiley & Sons, 2003. [39] A.J. van der Schaft, L 2 -gain and passivity techniques in nonlinear control, vol. 2, Springer, 2000. [40] A.T. Tran, S. Suzuki, N. Sakamoto, Nonlinear optimal control design considering a class of system constraints with validation on a magnetic levitation system, IEEE Control Syst. Lett. 1 (2) (2017) 418–423. [41] K.W. Iliff, K.S.C. Wang, Extraction of Lateral–Directional Stability and Control Derivatives for the Basic F-18 Aircraft at High Angles of Attack, NASA Dryden Flight Research Center, Edwards, CA United States, Tech. Rep. NASA-TM-4786, 1997.
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Aerospace Science and Technology www.elsevier.com/locate/aescte
Nonlinear gain-scheduled flight controller design via stable manifold method Anh Tuan Tran a,∗ , Noboru Sakamoto b , Koichi Mori c a b c
Department of Mechanical Systems Engineering, Nagoya University, Japan Department of Mechatronics, Faculty of Science and Engineering, Nanzan University, Japan Department of Aerospace Engineering, Nagoya University, Japan
a r t i c l e
i n f o
Article history: Received 31 January 2018 Received in revised form 20 May 2018 Accepted 3 July 2018 Available online 10 July 2018 Keywords: Nonlinear optimal control Stable manifold Flight controller Stabilization/Control augmentation system F-18 HARV
a b s t r a c t This research presents a nonlinear gain-scheduled flight controller design method via a stable manifold theory in order to handle the nonlinearities of the F-18 High Alpha Research Vehicle due to the change in the aerodynamic characteristics at different angles of attack and the airspeed variation. The designed longitudinal flight control system consists of a nonlinear gain-scheduled stabilization augmentation system which is designed using the stable manifold method, and a linear gain-scheduled control augmentation system which consists of proportional and integral gains. The nonlinear longitudinal flight controller is verified in a 6 degree-of-freedom simulator. © 2018 Elsevier Masson SAS. All rights reserved.
1. Introduction One of the desired characteristics of modern fighter or acrobatic aircraft is the capability to fly at high angles of attack or post-stall conditions. However, the aerodynamic characteristics become extremely complex and highly nonlinear when the angle of attack exceeds a stall angle. At that moment, many aerodynamic coefficients of the aircraft dramatically change their values even with a small increment of the angle of attack, i.e., the aircraft experiences adverse phenomena, such as significant lift and pitching control losses, asymmetric stall, and auto-rotation, which generally cause linear controllers to be inadequate and inefficient [1,2]. A traditional approach to take into account the nonlinearities of the aircraft in the flight controller design process is to use gain-scheduling technique. In this approach, a number of operating points for specified flight conditions are defined, and then the mathematical model of the aircraft is linearized at each flight condition. A family of linear controller candidates for those flight conditions is constructed, then a gain-scheduled controller is obtained by blending controller candidates to cover the entire flight envelope [3–9]. The advantage of this approach is that it is simple and practical. However, it requires many operating points to
*
Corresponding author. E-mail addresses: [email protected] (A.T. Tran), [email protected] (N. Sakamoto), [email protected] (K. Mori). https://doi.org/10.1016/j.ast.2018.07.002 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.
capture the nonlinearity of the aircraft, especially in the post-stall region. There are also other approaches which can consider the changes of aerodynamic characteristics in order to enable the aircraft to fly at high angles of attack or to perform super-maneuvers. It appears that nonlinear dynamic inversion and sliding mode control are two of the most applied methods to design nonlinear flight controllers recently. The nonlinear dynamic inversion approach [10–14] directly takes into account the nonlinearities within the entire flight envelope. Therefore, it can control the aircraft in the linear and post-stall regions by using a single controller. However, this method requires a precise knowledge of the aircraft model and is sensitive to modeling errors and parameter uncertainties. This sensitivity may be critical since it is not easy to obtain the exact model of the aircraft in practice. The sliding mode control approach [15–18] can also handle post-stall flights as well as perform super-maneuvers like Herbst and Cobra maneuvers. It is a robust nonlinear control design approach which can deal with modeling errors and parameter uncertainties efficiently. However, it experiences an undesirable oscillation phenomenon, called chattering, in implementation due to the discontinuous control action of the method. Some alternative approaches which can handle nonlinearities of the aircraft are, for instance, nonlinear optimal control [19–21], back-stepping [22–25], and neural networks [26–29]. This paper presents a flight controller design method for the longitudinal motion control of the F-18 High Alpha Research Vehicle (HARV) [30,31]. Although the HARV is equipped with thrust
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vectoring mechanism, only aerodynamic control surfaces are used to control the aircraft in this research. The flight controller structure considered in this research is similar to the conventional one which consists of a stabilization augmentation system (SAS) that uses proportional gains to feedback attitude rates, and a control augmentation system (CAS) that uses proportional and integral gains to feedback attitude angle errors. However, in this research, we take the angle of attack of the aircraft as a state variable in the SAS design to deal with the changes in the aerodynamic characteristics at high alpha angles. Moreover, since the variation of the aerodynamic characteristics is highly nonlinear, a nonlinear SAS is designed via the stable manifold method [32–36]. Additionally, since the dynamics of the aircraft also depend on the airspeed, the gain-scheduling technique is applied to deal with the change of airspeed in flight. The traditional gain-scheduling technique usually requires many operating points corresponding to specified flight conditions in order to capture the nonlinearities of the aircraft. Then, a linear controller candidate is designed for each flight condition. In this research, a set of sparse operating points is considered. For that reason, the nonlinearities of the aerodynamic characteristics between adjacent operating points are strong, which corrupt the control performance of the linear controller. The proposed nonlinear controller, on the other hand, does take into account those nonlinearities in the controller design process. Therefore, it can achieve better control performance compared with the linear one. A partial result of this research is briefly discussed in [37]. In this paper, the controller design method, result and discussion will be given in details. The outline of this paper is as follows: In Section 2, a nonlinear mathematical model of the longitudinal motion of the aircraft is presented. Section 3 describes the structure of the gain-scheduled controller considered in this research. In Sections 4 and 5, the gain-scheduled SAS and CAS are designed. The simulation results and discussions are shown in Section 6. Section 7 concludes this paper. 2. Mathematical model This section describes the nonlinear equations of motion (EOMs) of the HARV. In this research, we focus on the motion control in the longitudinal direction. The perturbations in the lateral–directional motion are kept at small values via a feedback controller. Therefore, the lateral–directional motion is omitted due to its weak impact on the longitudinal variables. For that reason, this paper only consider the longitudinal motion model of the aircraft. Since the purpose is to control the aircraft using only aerodynamic control surfaces, the thrust vectoring mechanism is unused. Hence, the direction of the thrust is fixed along the body x-axis. The general nonlinear EOMs for the longitudinal motion are reported in [38] (page 115) and are summarized as below
α˙ = q + q˙ =
M I yy
1 mV T
(− L − F T sin α + mg D cos(θ − α )), (1)
,
θ˙ = q, where α is the angle of attack, q is the pitch angular velocity, m is the mass of aircraft, V T is the total velocity, L is the lift force, F T is the thrust, g D is the gravitational constant, I y y is the pitching moment of inertia, and M is the pitch moment. In these equations, the definitions of the lift force L and the pitch moment M are
L=
1 2
1
ρ V T2 SC L , M = ρ V T2 S c¯ C M , 2
(2)
Fig. 1. Experimental data of C L 0 (α , δe ) at Mach 0.6 and its approximation which is expressed in (3).
where ρ is the air density, S is the reference area, c¯ is the mean aerodynamic chord. C L and C M are the total lift and moment coefficients. In general, those coefficients depend on numerous parameters such as the altitude h, aircraft velocity V T , angle of attack α , etc. In this research, we assume that around an operating point, the changes of the aerodynamic coefficients with respect to the altitude and velocity are small and can be neglected. Therefore, they can be expressed as functions of the angle of attack α and elevator deflection δe as below
c¯ ˙ , C Lq (α )q + C L α˙ (α )α 2V T c¯ ˙ . = C M 0 (α , δe ) + C Mq (α )q + C M α˙ (α )α 2V T
C L = C L 0 (α , δe ) + CM
In the above equations, C ∗∗ denotes the aerodynamics derivative coefficients which are reported in [30]. Fig. 1 illustrates the values of C L 0 (α , δe ) at the altitude h = 15000 ft and Mach 0.6. It can be seen from this figure that the stall angle is around 35 deg. When the angle of attack exceeds the stall angle, the aerodynamic coefficients dramatically change, and the aircraft is beyond the linear region. In order to design the flight controller for the HARV, the aerodynamic data of the aircraft are expressed in high order polynomials which are constructed by using a polynomial fitting method [33]. For example, the approximated polynomial function of C L 0 (α , δe ) is shown below
C L 0 (α , δe ) = 0.1262 + 0.1094α − 0.0022α 2
+ 1.0893E − 05α 3 + 0.0131δe − 0.0002α δe + 4.6787E − 07α 2 δe .
(3)
Note that only two sets of data corresponding to two positions of the elevator are provided in [30]. Therefore, the aerodynamic data of C L 0 (α , δe ) at the other elevator angles are estimated using (3). Fig. 1 shows that the approximated data of C L 0 (α , δe ), which are calculated from (3), and the experimental data, which are described in [30], are matching. Substituting the approximated functions of the aerodynamic derivative coefficients into (1), eliminating the high order terms of δe (δek , k = 2, 3, . . . ) since they are small, and assuming that θ − α is insignificant, which makes cos(θ − α ) ≈ 1, ones obtain
α˙ = f 1 (α , q) + g1 (α , q)δe , q˙ = f 2 (α , q) + g 2 (α , q)δe ,
θ˙ = q,
(4)
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Table 1 Nominal flight conditions. Parameter
Cond. 1
Cond. 2
Cond. 3
Unit
h VT
15000 300 0.2246 0.2246 −0.0619 9203
15000 450 0.0733 0.0733 −0.0132 7969
15000 650 0.0293 0.0293 0.0030 9188
ft ft/s rad rad rad lbs
α θ δe T a
Cond. is short for Condition.
4.1. Problem formulation Fig. 2. Control block diagram.
where f 1 (α , q), f 2 (α , q), g 1 (α , q) and g 2 (α , q) are appropriately calculated. Note that, in (4), the velocity V T and the thrust F T take their nominal values and are constant around a specific flight condition. 3. Gain-scheduled flight controller structure This section describes the flight controller structure of the HARV for the longitudinal motion control. As mentioned in Section 1, the longitudinal flight controller of the HARV in this research has similar structure to the conventional S/CAS with a modification that the angle of attack of the aircraft is feed-backed in the SAS in order to deal with the variation in the aerodynamic characteristics with respect to the angle of attack. The block diagram of the flight controller structure is shown in Fig. 2. In this research, two cases of the SAS are considered: the first one is the linear SAS which contains only proportional gains to feedback the information of the angle of attack and pitch rate; the second one is the nonlinear SAS which is designed to deal with the nonlinearities due to the changes in the aerodynamic characteristics when the HARV is at the high angle of attack. As for the CAS design, since the nonlinearities are taken into account in the SAS, only PI controllers are used in both cases. As can be seen in (1), the EOMs of the aircraft also depend on the velocity V T . When the aircraft operates in a broad operating region with different flight conditions, it is necessary to consider the effect of the velocity since its variation may affect the stability of the closed-loop stability and the control performance. In other words, a controller designed for a given velocity may become unstable, or the control performance may deteriorate when the velocity alters. A conventional approach to deal with the alternation of the velocity is to use the gain-scheduling technique. In the gain-scheduling approach, the aircraft is trimmed at different flight conditions, and the nonlinear EOMs (1) are linearized at those conditions. Then, a family of controller candidates is designed using those linear EOMs. Finally, the gain-scheduled controller to cover all flight conditions is obtained by interpolating the candidates at two adjacent flight conditions. This research considers three flight conditions which correspond to different velocity values, namely 300 ft/s, 450 ft/s and 600 ft/s, around the altitude h of 15000 ft. The aircraft is trimmed at those flight conditions, and the results are shown in Table 1. By substituting the nominal parameters at trimmed flight conditions (Table 1) into (4), both linear and nonlinear EOMs of the aircraft corresponding to each flight condition can be derived. 4. Nonlinear and linear gain-scheduled SAS In this section, a nonlinear optimal SAS based on the stable manifold method is first designed, then a linear optimal SAS via the LQR method using the same cost function is considered.
The nonlinear EOMs of the aircraft at each flight condition are derived, and the corresponding controller is designed. In order to design the SAS, only the first two equations of (4) are used. Those equations are rewritten in the following form
x˙ = f (x) + g (x)u , T where x = α q , f (x) = f 1 (α , q) T g 1 (α , q) g 2 (α , q) , u = δe . Let us denote A=
∂ f (x) , ∂ x x=0
(5) f 2 (α , q)
T
,
g (x) =
B = g (0).
The optimal control problem is to find a control law u such that the closed-loop system of (5) is asymptotically stable and the following cost function is minimized
∞ J = (x T Q x + u T Ru )dt , 0
where Q ∈ R2×2 and R ∈ R are weighting matrices. 4.2. Derivation of the Hamiltonian system The pre-Hamiltonian H D [39] for the optimal control problem of (5) is written as
H D (x, p , u ) = p T ( f (x) + g (x)u ) + x T Q x + u T Ru ,
(6)
where p is the co-state. The minimizing vector u¯ of (6) is obtained by solving
∂ H D (x, p , u ) = 0, ∂u which gives the result
1 u¯ (x, p ) = − R −1 g (x) T p . 2
(7)
A Hamilton–Jacobi equation (HJE) is obtained by substituting u¯ (x, p ) into the pre-Hamiltonian H D (6)
H (x, p ) = p T f (x) −
where p = ∂ V /∂ x1
1 4
p T g (x) R −1 g (x) T p + x T Q x = 0,
∂ V /∂ x2
T
(8)
and V is an unknown function.
T
can be determined by solving Next, if p = ∂ V /∂ x1 ∂ V /∂ x2 the HJE, the optimal control input u¯ (x, p ) is attained from (7). To accomplish this goal, the stable manifold method [32–34] is utilized. The author of [32] proved that the stabilizing solution of the HJE (8) is on the stable manifold of the Hamiltonian system
x˙ =
∂H (x, p ), ∂p
p˙ = −
∂H (x, p ). ∂x
(9)
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The above Hamiltonian system can be written in the following form in which the linear and nonlinear parts are separated
A x˙ = p˙ −2Q
−R B −AT
x N x (x, p ) + , p N p (x, p )
(10)
where R B = 12 B R −1 B T , and N x (x, p ), N p (x, p ) are higher order terms. A linear coordinate transformation using an appropriate matrix T ∈ R2n×2n is taken [32]
x p
= T −1
x , p
T=
I P
S PS + I
(11)
and S is the solution of the Lyapunov equation
In the new coordinates (x , p ), the linear part of (10) is diagonalized
=
F 0
0 −F T
n (x , p ) x + x , n p p (x , p )
(12)
A − R B P is Hurwitz, and nx (x , p ), np (x , p )
where the matrix F = are higher order terms.
4.3. Controller construction By using the stable manifold method reviewed in the Appendix, solution curves on the stable manifold S of (9) are calculated, then the stable manifold S can be approximated. Since S is locally canonically surjective onto x-space, one can represent S as p = π (x) with π (x) ∈ R2 is an appropriate function. It can be
T
written that p = π (x) = π1 (x) π2 (x) , where π1 (x) ∈ R and π2 (x) ∈ R are the elements of π (x). Later in section 4.4, some solution curves for this problem will be shown in Fig. 3. By calculating many solution curves that cover the domain of interest, they will shape the stable manifold S of (9), and then π (x) can be approximated. There are several ways to obtain π (x). One way is to use a polynomial fitting algorithm to approximate π (x) with polynomials [33]. Another way is to use an interpolation method to represent the relationship between x and p [40]. In this research, the second approach is chosen. Finally, the optimal control input u¯ (x, p ) is
1 u¯ (x, π (x)) = − R −1 g (x) T π (x). 2
(13)
The LQR controller of the optimal control problem of (5) is
1 u 0 = − R −1 B T P x. 2
(14)
Note that (14) is the linearization of (13) at the origin, which is
B = g (0),
∂ π (x) ∂x
x=0
300 450 600
7.19 6.87 9.17
Kα
−48.21 −18.92 −20.61
K
Optimal
Kp
Ki
Kp
Ki
[−50, 0] [−40, 0] [−40, 0]
[−50, 0] [−40, 0] [−40, 0]
−50 −40 −40
−20 −10 −10
In order to cover a broad operating region, a family of controllers at different flight conditions is designed; then a gainscheduled controller is constructed. In Sections 4.1–4.3, the design methods of the nonlinear and linear controllers are introduced. The same weighting matrices Q and R are chosen to design the nonlinear and linear controllers at each flight condition as follows
( A − R B P ) S + S ( A − R B P )T = R B .
Kq
4.4. Gain-scheduling algorithm
A T P + P A − P R −1 B T P + Q = 0,
x˙ p˙
V (ft/s)
,
where P is the stabilizing solution of the algebraic Riccati equation
Table 2 Designed S/CAS gains.
= P.
(15)
The LQR controller (14) can be written as
u 0 = K α α + K q q, where K α and K q are the proportional gains which are calculated as K α K q = − 12 R −1 B T P .
Q1 =
0.1 0 , R 1 = 0.3, 0 1000
Q2 =
0.1 0 , R 2 = 1.5, 0 1000
0.1 0 , R 3 = 1.5, Q3 = 0 1000 where Q i and R i (i = 1, 2, 3) are the weighting matrices corresponding to the flight condition i. The resultant linear gains K α and K q for each flight condition are shown in Table 2. As mentioned in Section 4.3, the data interpolation method is used to construct the nonlinear controller. To this end, a twodimensional look-up table of π (x) is built for each flight condition. It has 49 × 49 grid data points and covers the domain of interest (α , q) of −10 deg 70 deg × −70 deg/s 70 deg/s . For visualization, the data representing the stable manifolds π (x) = π1 (x) π2 (x) T corresponding to three flight conditions are illustrated in Fig. 3. It is shown in Fig. 3 that the stable manifolds corresponding to the flight conditions are different due to the changes in the mathematical models. In the figures of the stable manifold corresponding to the flight condition 1, three trajectories start from different initial conditions of angle of attack α and pitch rate q and converge to the origin which corresponds to the equilibrium point. One can observe the disparities between the stable manifold π (x) (meshed surface) and its approximation at the origin (plain surface) which is calculated by using (15). Those disparities become significant when the values of α and q go far from the origin (trim condition). They represent how strong the nonlinearities of the aircraft are at different states of α and q. Also, by looking at these figures, ones can predict the states of the aircraft at which the control performance of the linear controller might degrade. Next, the gain-scheduled controller u GS is designed. A practical way is to linearly interpolate nearby controller candidates. The interpolation rule is shown below
u GS = (1 − η)u i + ηu i +1 , where i and i + 1 are two adjacent operating points such that V i V T V i +1 , u i and u i +1 are the corresponding linear (or nonlinear) controllers at those operating points, and
η=
VT − Vi V i +1 − V i
.
The calculation of η can be interpreted that: if V T = V i , the controller u i will be used; if V T = V i +1 , the controller u i +1 will be used; if V i < V T < V i +1 , both u i and u i +1 will be used for calculating the control input.
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305
Fig. 3. The components π1 (x) (meshed surface, left-hand side column), π2 (x) (meshed surface, right-hand side column) of the stable manifolds S and their linear approximations obtained from the stabilizing solution of the algebraic Riccati equation (plain surface) at the flight conditions 1, 2, 3, respectively. In the first row, which corresponds to the flight condition 1, three lines show the examples of stabilization trajectories starting from different initial conditions.
5. Gain-scheduled CAS This section presents the controller design method of the gainscheduled CAS. Based on the closed-loop system including the SAS, the CAS is designed. Similar to the SAS design, a family of CAS for three flight conditions is designed, then the gain-scheduled controller is constructed. In the previous section, the change in the aerodynamic characteristics was already taken into account in the SAS design. Therefore, only linear CAS is considered in this section. The CAS consists of proportional and integral gains (PI controller). In order to design the CAS, the linearization of the SAS closed-loop system is used. Note that the linearization of the nonlinear optimal controller design via stable manifold method (13) coincides with the LQR one (14). Hence, the linearized closed-loop SAS and the resultant CAS in both cases of the nonlinear SAS and the linear SAS are identical. The CAS design method is very similar to the one presented in [7,8]. An admissible set K of the proportional gain K p and the
integral gain K i is determined. This set should be appropriately chosen in order to avoid actuator magnitude saturation. Next, the admissible set K is gridded, and the gain K p and K i at each node will be evaluated using the cost function
T f e 2 dt ,
J CAS =
(16)
0
where T f is a large value, says 100. This cost function can be calculated in MATLAB. Then, the optimal gains K p and K i are the ones such that when tracking a step reference input, the cost function (16) is minimized. Table 2 shows the designed optimal gains for each flight condition. Finally, the gain-scheduled CAS is obtained using the method similar to the one used in the gain-scheduled SAS design. In addition, a simple PI controller for the thrust control is designed to regulate the velocity of the HARV
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Fig. 4. Responses of the velocity, angle of attack α , pitch rate q, pitch angle θ when using only one candidate S/CAS controller at the operating point V T = 450 (ft/s). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
t δT = K V p ( V com − V T ) + K V i
( V com − V T )dt , 0
where V com is the reference velocity, K V p = 300, K V i = 20. 6. Simulation result and discussion The designed controllers are verified in a 6 degree-of-freedom nonlinear simulator. The aerodynamic characteristics of the HARV obtained from wind tunnel tests are reported in [30], which correspond to the altitude of 15,000 ft and Mach number range between 0.3 and 0.9. Interpolation is required to obtain aerodynamic data of nearby flight conditions. Moreover, the actuator models of the control surfaces and thrust [30] are taken into account in the simulation. Note that the elevator has the rate limit of 40 deg/s and position limits of −24 deg and 10.5 deg, where the positive direction is trailing-edge down [30,41]. In Sections 4 and 5, two types of gain-scheduled controllers are designed. The first one is the nonlinear gain-scheduled controller which contains the nonlinear optimal SAS, and the second one is the linear gain-scheduled controller which has the LQR SAS. Recall that both of them have the same CAS. In this section, simulations are conducted to evaluate the effectiveness of the gain-scheduling technique as well as to compare the control performances of the nonlinear gain-scheduled controller and the linear gain-scheduled one. 6.1. Linear and nonlinear controllers without gain-scheduling technique Figs. 4 and 5 show the simulation results of the nonlinear and linear controllers in which only one S/CAS candidate corresponding to the flight condition 2 is used to control the pitch angle of the HARV. Although the nominal velocity of the flight condition 2 is V T = 450 ft/s, in this simulation, the velocity command V com is set to be 400 ft/s to verify the robustness of the designed controller. The aircraft is controlled to follow the pitch angle command θcom .
Fig. 5. Responses of the elevator when using only one candidate S/CAS controller at the operating point V T = 450 (ft/s).
The simulation results show that the nonlinear controller has a faster convergence rate than the linear one. The main discrepancy in the control performances of the linear and nonlinear controllers is at the time when the HARV is at the high angle of attack. At this moment, the HARV is far from the linear region; and the aerodynamic characteristics vary drastically in accordance with the angle of attack α . In the design process of the nonlinear controller, this variation in the aerodynamic characteristics is appropriately taken into account. Conversely, it is not considered in the linear controller. As a consequence, it can be seen that the nonlinear controller has better tracking performance than the linear one. For the rest of the time, since the aircraft is at low angle of attack, the discrepancy between the nonlinear and linear controllers is insignificant. The cost functions (16) of the linear controller ( J linear ) and the nonlinear one ( J nonlinear ) are calculated. The results are
J linear = 6346,
J nonlinear = 4644,
J nonlinear J linear
= 73%.
A.T. Tran et al. / Aerospace Science and Technology 80 (2018) 301–308
Fig. 6. Responses of the velocity, angle of attack
307
α , pitch rate q, pitch angle θ when using the gain-scheduled controllers.
6.2. Linear and nonlinear gain-scheduled controllers Figs. 6 and 7 show the simulation results in which the nonlinear gain-scheduled controller and the linear gain-scheduled one are evaluated. As mentioned before, the EOMs of the aircraft depend on the changes in the aerodynamic characteristics at different angles of attack as well as the variation of airspeed. These simulations analyze the capabilities of the two aforementioned gainscheduled controllers to handle such nonlinearities. Also, the same pitch angle command pattern θcom as the one in the previous simulation is given to compare with the controllers that do not use the gain-scheduling technique in Figs. 4 and 5. It can be seen that the nonlinear gain-scheduled controller has a better overall tracking performance than the linear gainscheduled one. However, the discrepancy between these controllers is not as remarkable as it is in the previous simulation (Figs. 4 and 5). The reason is that in this case, multiple aircraft models at different flight conditions are considered. Therefore, the aerodynamic characteristics and airspeed variation during the flight are taken into account in the design processes of both linear and nonlinear gain-scheduled controllers. As a result, the linear gain-scheduled one can achieve a similar control performance as the nonlinear gain-scheduled one does within the considered flight conditions. The cost functions (16) are calculated. The results are
J linear = 5244,
J nonlinear = 4340,
J nonlinear J linear
= 83%.
In the comparison between the nonlinear and linear gainscheduled controllers (Figs. 6 and 7) with the ones without gain scheduling technique (Figs. 4 and 5), the overall tracking performance and the cost function are improved thanks to the implementation of the gain scheduling technique. For the nonlinear controller that uses only one SAS/CAS candidate of the flight condition 2 (Figs. 4 and 5), although it considers the change in the aerodynamic characteristics in the SAS design, the variation of airspeed is not taken into account. Therefore, the implementation of
Fig. 7. Responses of the elevator when using the gain-scheduled controllers.
the gain scheduling technique (Figs. 6 and 7) further improves its tracking performance. 7. Conclusion This research evaluates the benefits of the gain-scheduling technique and the nonlinear control approach via the stable manifold method in flight controller design. The simulation results show that by taking into account the nonlinear aerodynamic characteristics of the aircraft in the controller design process, the nonlinear controller can perform better in the nonlinear domain, i.e., at high angle of attack, compared with the linear one. When the gainscheduling technique is employed, it improves the control performance of both linear and nonlinear ones. However, the nonlinear controller is still better than the linear one in term of reducing tracking error by a remarkable amount. Conflict of interest statement There is no conflict of interest.
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Appendix A. Review of stable manifold method In order to calculate the stable manifold of (12), one defines the sequences xk (t , ξ ) and pk (t , ξ )
xk +1 = e F t ξ +
t
e F (t −s) nx xk (s), pk (s) ds,
0
pk +1 = −
∞
(A.1)
e − F (t −s) np T
xk (s), p s (s) ds,
t
for k = 1, 2, . . . , and the initial conditions are
x0 = e F t ξ,
p 0 = 0,
where ξ ∈ Rn is arbitrary. Theorem 1 (Sakamoto and van de Schaft, 2008 [32]). For sufficiently small |ξ |, the sequences xk (t , ξ ) and pk (t , ξ ) are convergent to zero, that is, xk (t , ξ ), pk (t , ξ ) → 0 as t → ∞ for all k = 0, 1, 2, . . . Furthermore, xk (t , ξ ) and pk (t , ξ ) are uniformly convergent to a solution of (12) on [0, ∞) as k → ∞. Let x (t , ξ ) and p (t , ξ ) be the limit of xk (t , ξ ) and yk (t , ξ ), respectively. Then, x (t , ξ ) and y (t , ξ ) are the solution on the stable manifold of (12), that is, x (t , ξ ), p (t , ξ ) → 0 as t → ∞. Therefore, by using the sequence (A.1), one can calculate flows on the stable manifold of (12). The flows on the stable manifold S of the Hamiltonian system (9) in the original coordinates are given by
x(t , ξ ) =T p (t , ξ )
x (t , ξ ) . p (t , ξ )
References [1] J.R. Chambers, S.B. Grafton, Aerodynamic characteristics of airplanes at high angle of attack, Tech. Rep. NASA-TM-74097, NASA Langley Research Center, Hampton, Virginia, 1977. [2] J.R. Chambers, High-angle-of-attack aerodynamics: lessons learned, in: 4th Applied Aerodynamics Conference, 1986. [3] J. Ackermann, Multi-model approaches to robust control system design, in: Uncertainty and Control, 1985, pp. 108–130. [4] W.J. Rugh, J.S. Shamma, Research on gain scheduling, Automatica 36 (10) (2000) 1401–1425. [5] R.A. Nichols, R.T. Reichert, W.J. Rugh, Gain scheduling for H-infinity controllers: a flight control example, IEEE Trans. Control Syst. Technol. 1 (2) (1993) 69–79. [6] J. Wang, N. Sundararajan, A nonlinear flight controller design for aircraft, Control Eng. Pract. 3 (6) (1995) 813–825. [7] M. Sato, K. Muraoka, Flight controller design and demonstration of quad-tiltwing unmanned aerial vehicle, J. Guid. Control Dyn. 38 (6) (2015) 1071–1082. [8] A.T. Tran, N. Sakamoto, M. Sato, K. Muraoka, Control augmentation system design for quad-tilt-wing unmanned aerial vehicle via robust output regulation method, IEEE Trans. Aerosp. Electron. Syst. 53 (1) (2017) 357–369. [9] D.J. Leith, W.E. Leithead, Gain-scheduled control: relaxing slow variation requirements by velocity-based design, J. Guid. Control Dyn. 23 (6) (2000) 988–1000. [10] G.J. Balas, Flight control law design: an industry perspective, Eur. J. Control 9 (2–3) (2003) 207–226. [11] R.J. Adams, J.M. Buffington, S.S. Banda, Design of nonlinear control laws for high-angle-of-attack flight, J. Guid. Control Dyn. 17 (4) (1994) 737–746. [12] S.A. Snell, D.F. Enns, W.L. Garrard Jr., Nonlinear inversion flight control for a supermaneuverable aircraft, J. Guid. Control Dyn. 15 (4) (1992) 976–984.
[13] W.C. Reigelsperger, S.S. Banda, Nonlinear simulation of a modified F-16 with full-envelope control laws, Control Eng. Pract. 6 (3) (1998) 309–320. [14] D. Bugajski, D. Enns, Nonlinear control law with application to high angle-ofattack flight, J. Guid. Control Dyn. 15 (3) (1992) 761–767. [15] S.N. Singh, M.L. Steinberg, A. Page, Nonlinear adaptive and sliding mode flight path control of F/A-18 model, IEEE Trans. Aerosp. Electron. Syst. 39 (4) (2003) 1250–1262. [16] S. Seshagiri, E. Promtun, Sliding mode control of F-16 longitudinal dynamics, in: American Control Conference, 2008, pp. 1770–1775. [17] E.M. Jafarov, R. Tasaltin, Robust sliding-mode control for the uncertain MIMO aircraft model F-18, IEEE Trans. Aerosp. Electron. Syst. 36 (4) (2000) 1127–1141. [18] B.K. Mukherjee, M. Sinha, Extreme aircraft maneuver under sudden lateral CG movement: modeling and control, Aerosp. Sci. Technol. 68 (2017) 11–25. [19] W.L. Garrard, J.M. Jordan, Design of nonlinear automatic flight control systems, Automatica 13 (5) (1977) 497–505. [20] A.T. Tran, N. Sakamoto, Y. Kikuchi, K. Mori, Pilot induced oscillation suppression controller design via nonlinear optimal output regulation method, Aerosp. Sci. Technol. 68 (2017) 278–286. [21] R. Cory, R. Tedrake, Experiments in fixed-wing UAV perching, in: AIAA Guidance, Navigation and Control Conference and Exhibit, August 2008. [22] L. Sonneveldt, Q. Chu, J. Mulder, et al., Nonlinear flight control design using constrained adaptive backstepping, J. Guid. Control Dyn. 30 (2) (2007) 322–336. [23] T. Lee, Y. Kim, Nonlinear adaptive flight control using backstepping and neural networks controller, J. Guid. Control Dyn. 24 (4) (2001) 675–682. [24] J. Farrell, M. Sharma, M. Polycarpou, et al., Backstepping-based flight control with adaptive function approximation, J. Guid. Control Dyn. 28 (6) (2005) 1089–1102. [25] J. Cayero, B. Morcego, F. Nejjari, Modelling and adaptive backstepping control for TX-1570 UAV path tracking, Aerosp. Sci. Technol. 39 (2014) 342–351. [26] B.S. Kim, A.J. Calise, Nonlinear flight control using neural networks, J. Guid. Control Dyn. 20 (1) (1997) 26–33. [27] A.J. Calise, R.T. Rysdyk, Nonlinear adaptive flight control using neural networks, IEEE Control Syst. Mag. 18 (6) (1998) 14–25. [28] K. Rokhsaz, J. Steck, Use of neural networks in control of high-alpha maneuvers, J. Guid. Control Dyn. 16 (5) (1993) 934–939. [29] D.I. Ignatyev, A.N. Khrabrov, Neural network modeling of unsteady aerodynamic characteristics at high angles of attack, Aerosp. Sci. Technol. 41 (2015) 106–115. [30] J. Cao, J. Frederick Garrett, E. Hoffman, H. Stalford, Analytical Aerodynamic Model of a High Alpha Research Vehicle Wind-Tunnel Model, Tech. Rep. NASACR-187469, Georgia Institute of Technology, 1990. [31] D. Enns, D. Bugajski, R. Hendrick, G. Stein, Dynamic inversion: an evolving methodology for flight control design, Int. J. Control 59 (1) (1994) 71–91. [32] N. Sakamoto, A.J. van der Schaft, Analytical approximation methods for the stabilizing solution of the Hamilton–Jacobi equation, IEEE Trans. Autom. Control 53 (10) (2008) 2335–2350. [33] N. Sakamoto, Case studies on the application of the stable manifold approach for nonlinear optimal control design, Automatica 49 (2) (2013) 568–576. [34] N. Sakamoto, A recent progress in the optimal control design under various constraints, in: 7th IFAC Conference on Manufacturing Modelling, Management, and Control, June 2013, pp. 1506–1511. [35] N. Sakamoto, Analysis of the Hamilton–Jacobi equation in nonlinear control theory by symplectic geometry, SIAM J. Control Optim. 40 (6) (2002) 1924–1937. [36] A.T. Tran, N. Sakamoto, A general framework for constrained optimal control based on stable manifold method, in: IEEE 56th Annual Conference on Decision and Control, CDC, December 2016. [37] A.T. Tran, N. Sakamoto, Application of stable manifold method to aircraft longitudinal flight controller design, in: The 60th Japan Joint Automatic Control Conference, Tokyo, Japan, November 2017. [38] B.L. Stevens, F.L. Lewis, Aircraft Control and Simulation, 2nd edition, John Wiley & Sons, 2003. [39] A.J. van der Schaft, L 2 -gain and passivity techniques in nonlinear control, vol. 2, Springer, 2000. [40] A.T. Tran, S. Suzuki, N. Sakamoto, Nonlinear optimal control design considering a class of system constraints with validation on a magnetic levitation system, IEEE Control Syst. Lett. 1 (2) (2017) 418–423. [41] K.W. Iliff, K.S.C. Wang, Extraction of Lateral–Directional Stability and Control Derivatives for the Basic F-18 Aircraft at High Angles of Attack, NASA Dryden Flight Research Center, Edwards, CA United States, Tech. Rep. NASA-TM-4786, 1997.