Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeB10.4
Nonlinear Adaptive Control Design for Non-minimum Phase Hypersonic Vehicle Models with Minimal Control Authority Lisa Fiorentini∗
Andrea Serrani
The Ohio State University Columbus, OH 43210, USA
The Ohio State University Columbus, OH 43210, USA
Abstract— The design of a nonlinear robust controller for a non-minimum phase model of the longitudinal dynamics of an air-breathing hypersonic vehicle is presented in this work. When flight-path angle is selected as a regulated output and the elevator is the only control surface available for the pitch dynamics, longitudinal models of air-breathing hypersonic vehicle dynamics exhibit unstable zero-dynamics. The approach proposed in this paper uses a combination of small-gain arguments and adaptive control techniques for the design of a statefeedback controller that achieves asymptotic tracking of velocity and flight-path angle reference trajectories in spite of model uncertainties. The method reposes upon a suitable redefinition of the internal dynamics of a control-oriented model of the vehicle dynamics and uses a time-scale separation between the controlled variables to manage the peaking phenomenon occurring in the system. Simulation results on a full nonlinear vehicle model illustrate the effectiveness of the methodology.
I. I NTRODUCTION Among many challenges encountered when designing flight control systems for air-breathing hypersonic vehicles (HSVs), one of the most severe is the presence of exponentially unstable zero-dynamics when longitudinal velocity and flight-path angle are selected as regulated output. As discussed in [1], a non-minimum phase behavior arises as a consequence of elevator-to-lift coupling. Early work on nonlinear design for longitudinal models of HSV dynamics ignore this non-minimum phase effect altogether [2]–[4]. In [5], approximate feedback linearization was applied to the model of Bolender and Doman [6] by strategically ignoring the elevator-to-lift coupling. A combination of backstepping and adaptive dynamic inversion was adopted in [7] for the design of a dynamic state-feedback controller that used both elevator and canard as control surfaces. Although beneficial for controllability, the presence of a canard is problematic for the vehicle structure, as this control surface must withstand a significant thermal stress at hypersonic speed. Therefore, it is of interest to address the case in which the elevator is the only aerodynamic surface available for controlling the vehicle attitude. The difficulties encountered in this new setup are not limited to uncompensated non-minimum phase behavior: in [7], the canard input provided a supplementary stabilizing action and ∗ Corresponding author, Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Ave, Columbus, OH. Email:
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
was used to enforce the desired trim condition in steadystate, whereas in the new scenario integral augmentation is required to reconstruct the desired equilibrium. In [8] we took a first step in this direction by presenting a controller that uses a combination of high-gain and low-amplitude feedback. The control strategy relied to a certain extent on exact linearization, as a consequence, the important issue of robustness with respect to parameter model uncertainties was not addressed. Here, we present a new control strategy that does not require exact linearization, but employes adaptive dynamic inversion. The approach is based upon a suitable redefinition of the internal dynamics of the system and makes use of a gain-dependent change of coordinates which, by enforcing a time-scale separation between the controlled variables, allows one to manage the peaking phenomenon occurring in the system. Stable adaptation ensures robustness with respect to uncertainty on the model parameters, whereas small-gain arguments are employed in the stability analysis. The proposed approach yields a guaranteed domain of attraction for given ranges of parameter variations. The rigid-body control-oriented model considered in [7] is used for controller design and stability analysis, while, the full nonlinear model in [6], which includes structural flexibility, is employed for closed-loop simulations. The paper is organized as follows: Section II introduces the vehicle model and the control objective. In Section III, an analysis on the system zero-dynamics is presented; the control design is developed in section IV, and the stability analysis is discussed in Section V. Finally, simulation results are shown in Section VI, while final remarks are given in Section VII. II. V EHICLE M ODEL AND P ROBLEM F ORMULATION The longitudinal model of the rigid-body vehicle dynamics considered in this study is written as follows [6] T cos _ − D − g sin a V˙ = m L + T sin _ g − cos a a˙ = mV V e˙ = Q M . (1) Q˙ = Iyy This model comprises four rigid-body state variables x = [V, a , e , Q]T and two control inputs u = [\, b e ]T which af-
1405
WeB10.4 TABLE I
¯ _ , \) M(
A DMISSIBLE RANGES FOR STATE , INPUT, AND VARIABLES OF INTEREST Var V a e Q \ be _ h q¯
Vehicle Velocity Flight-Path Angle (FPA) Pitch Angle Pitch Rate Fuel-to-air Ratio Elevator Deflection Angle-of-Attack, _ = e − a Vehicle Altitude, h˙ = V sin a Dynamic Pressure, q¯ =
lV 2 2
Min Value 7500 ft/s −3 deg −5 deg −10 deg/s 0.05 −20 deg −5 deg 85000 ft
Max Value 11000 ft/s 3 deg 5 deg 10 deg/s 1.5 20 deg 5 deg 135000 ft
182.5 psf
2200 psf
h−h
l
Air Density, l =
0 l0 e− hs
7e-6
slugs/ft3
_ (_ ) := C_ 2 _ 2 + C_ _ + C0 , and C¯Mb := where CM M M M b b −CM /CL > 0. System (3) has an equilibrium at (e , Q) = (e ⋆ , 0) and \ = \⋆ , where the constant values e ⋆ , \⋆ ⋆ are determined by the ⋆trim condition⋆ at⋆ V = V , namely ⋆ ⋆ ⋆ ¯ T (e , \ ) cos e − D e = 0 , M(e , \ ) = 0 . Since it can be verified that for the considered flight envelope ¯ e , \)/, e |e =e ⋆ ,\=\⋆ > 0, it follows that the equilibrium , M( (e , Q) = (e ⋆ , 0) of the pitch dynamics (3) is a hyperbolic saddle. As a result, any attempt to apply a standard dynamic inversion algorithm results in unstable internal dynamics.
8e-5 slugs/ft3
fect (1) through thrust, T , pitching moment, M, and lift, L. The output to be controlled is selected as y = [V, a ] T . For the purpose of control design and stability analysis, the following approximations have been considered [5]: T (_ , \)
M(_ , be ) L(_ , be ) D(_ )
3
2
≈ qS ¯ [(CT\_ _ 3 + CT\_ _ 2 + CT\_ _ + CT\ )\ +(CT3 _ 3 + CT2 _ 2 + CT1 _ + CT0 )] _2 2 0 _ b _ + CM _ + CM + CM be ≅ zT T + q¯c¯ S CM ≅ qS ¯ CL_ _ + CL0 + CLb be 2 ≅ qS ¯ CD_ _ 2 + CD_ _ + CD0 . (2)
The design model retains the dominant features of the higher fidelity model of [6], including the non-minimum phase behavior. It is assumed that all the model coefficients are subject to uncertainty. The vector of all uncertain parameters is denoted by P ∈ R p and it is assumed that P ∈ U P where UP is a compact convex set that includes the nominal value P 0 of P and represents the admissible range of variation of P. The goal pursued in this study is to design a dynamic state-feedback controller to steer the state of system (1) from a given compact set of initial conditions, x 0 ∈ U0 , to a desired trim condition x ⋆ = [V ⋆ , 0, e ⋆ , 0]T , along smooth exogenous reference trajectories y ref (t) = [Vref (t), aref (t)]T robustly with respect to the considered model parameter uncertainty. Clearly, limt→' Vref (t) = V ⋆ and, to have level flights, limt→' aref (t) = 0. Table I determines the considered flight envelope of the vehicle. Herein, we denote with A ⊂ R10 the admissible range for all variables in Table I. It should be noted that, once a desired trim condition V ⋆ is reached, e ⋆ cannot be determined a priori due to parameter uncertainty. III. Z ERO - DYNAMICS OF THE M ODEL The system (1) has vector relative degree r = [1, 1] with respect to the regulated output y, hence a 2-dimensional zero-dynamics with respect to e = [V − V ⋆ , a ]T . Applying the decoupling control input b e⋆ (_ , \, a ) := −CL_ (_ )/CLb + [mg cos a − T (_ , \) sin _ ]/(qSC ¯ Lb ) where CL_ (_ ) := CL_ _ + CL0 , and choosing the initial condition a (0) = 0, one obtains the zero dynamics of the system (1) with respect to a
e˙
=
Q,
¯ e , \) Iyy Q˙ = M(
b mg cos a C¯M _ b _ ¯ = q¯ S c¯ CM (_ ) + CMCL (_ ) − qS ¯ + zT + c¯ C¯Mb sin _ T (_ , \)
IV. C ONTROL D ESIGN The control philosophy for the velocity subsystem will essentially remain the same as in [7]. For the rotational dynamics (a , e , Q), a change of coordinates is applied to the FPA dynamics to redefine the internal dynamics of the system. The system is augmented with an integrator to reconstruct the desired trim condition e ⋆ , and a commanded reference ecmd for the pitch angle is chosen to stabilize the resulting internal dynamics. Finally, the pitch angle is controlled through the pitch moment by means of b e . A. Translational dynamics Let V˜ = V − Vref , then in the new coordinates mV˙˜ = T cos _ − D − mg sin a − mV˙ref .
(4)
By introducing the vector of uncertain parameters 1 , the regressor ^1 and the input matrix B 1 respectively as 1 = 3 2 2 S [CT\_ ,CT\_ ,CT\_ ,CT\ , CT3 , CT2 , CT1 , CT0 ,CD_ ,CD_ ,CD0 , m/S]T , ^1 (x, yref ) = [01×4 , −q¯_ 3 cos _ , −q¯_ 2 cos _ , −q¯_ cos _ , −qcos ¯ _ , q¯_ 2 , q¯_ , q, ¯ g sin a + V˙ref ]T andB1(x) = q¯ cos _ [_ 3 , 2 T _ , _ , 1, 01×8 , ] Eq. (4) can be written in the linearly parameterized form mV˙˜ = 1T B1 \ − ^1 . For the sake of simplicity, the arguments of functions or vectors will be omitted when no confusion arises. Let ˆ1 be a vector of estimates of 1 , ˜1 = ˆ1 − 1 and O1 be the compact set in which e1 is assumed to range, obtained by letting the entries of e1 vary within the parameter set P. The control law for the equivalence air ratio is chosen as \ =
1 − kV V˜ + ^T1 ˆ1 T ˆ 1 B1
(5)
where kV > 0 is a gain parameter. Consequently, the update law for the parameter estimate ˆ1 is chosen as (6) ˙ˆ1 = ˙˜1 = Proj V˜ K1 B1 \ − ^1 ˆ1 ∈ O1
where K1 ∈ R12×12 is a symmetric positive definite matrix and Projˆ1 ∈ O1 is a smooth parameter projection [9]. As a result the velocity error dynamics can be written as
(3)
1406
V˙˜ = −kV V˜ + ˜1T (^1 − B1 \) .
(7)
WeB10.4 3.5
B. Rotational dynamics Let a˜ = a − aref , then the rotational dynamics read as L + T sin _ g − cos a − a˙ref = mV V ˙ e = Q M Q˙ = . Iyy
Vref = 7500 ft/s
= 11500 ft/s
ref
_* [deg] 1
0.5
0
(9)
1 Iyy CLb >0 b Vref c¯ mCM
b C µ¯ 2 1 z 1 T L − tan _ ⋆ := µ2 (Vref , _ ⋆ ) = ⋆ b Vref cC cos _ V ¯ M ref
µ1 (Vref ) = −
and _ ⋆ (m, l ,Vref ) is a state and reference-dependent variable that will be defined in the sequel. Since µ 1 and µ2 are unknown due to parametric uncertainty, the state j 2 is not available for feedback. Applying (9), one obtains f1 + R2 j2 + (µ˙ 1 − µ1 R2 )Q + ( f2 − µ2 R2 )V˜ −a˙ref − µ2V˙ref + R2 aref (10)
where f 1 (x, u, yref ), f2 (x, u, yref ) and R2 (x) are continuous functions not reported here for reasons of space limitations. Consider the algebraic equation in the variable _ ⋆
b _ _2 CL CM CLb CM 2 _ ⋆2 _ _ ⋆ ¯ ¯ µ _ µ − C − + + − C C 2 D 2 D L _ b b CM CM (11)
describing the trim condition for _ . Since µ¯ 2 (_ ⋆ ) depends linearly on tan _ ⋆ , for small angle approximations (tan _ ≈ _ ⋆ ) (11) is a cubic equation in _ ⋆ . Applying Cardano’s method it is possible to verify that (11) has one real root and a pair of complex conjugate roots. Denote the real root by _ ⋆ = _ ⋆ (m, l ,Vref ). Since f 1 |_ =_ ⋆ = 0, by continuity and
0
1
2
3
4
5
6
7
l [slugs/ft3]
Fig. 1.
is applied to the first equation of (8), where
0 CLb CM 2mg = 0 − µ¯ 2 (_ ⋆ )CD0 − 2 b l SVref CM
V
1.5
(8)
It can be shown that, as a consequence of the “sector boundedness” property proved in [7], the equilibrium a˜ = a˜⋆ is locally asymptotically stable. This observation suggests to consider y¯ = [V, e ]T as the output of system (1). System (1) has relative degree 2 with respect to the “output” e and has a 1-dimensional LAS internal dynamics. To compute the internal dynamics of the system (1) with respect to y, ¯ the following change of coordinates
+ CL0 −
= 10500 ft/s
2
L(−a˜⋆ , be⋆ ) − T (−a˜⋆ , \⋆ ) sin a˜⋆ − mg cos a˜⋆ = 0 T (−a˜⋆ , \⋆ ) cos a˜⋆ − D(−a˜⋆) − mg sin a˜⋆ = 0 C_ (_ ) zT T (−a˜⋆ , \⋆ ) . be⋆ = − M b − b CM q¯c¯ SCM
=
= 8500 ft/s
V
ref
2.5
When (e , Q) = (0, 0) the FPA error dynamics in system (8) has an equilibrium at a˜ = a˜∗ = 0 defined by the trim condition a˙ref = 0, \ = \⋆ , and be = be⋆ , where
j˙2
ref
Vref = 9500 ft/s
a˙˜
j2 = a˜ + µ1 Q + µ2V˜
V
3
8 ï5
x 10
Plots of _ ⋆ (l ) for different values of Vref and m = 147 slug/ft.
using the same arguments in Ref. [7], it is possible to show that the function f 1 satisfies a “sector boundedness” property. Specifically, one can write f 1 := R1 (x, u, yref )(_ − _ ⋆ ) where R1 satisfies h 1 ≤ R1 (x, u, yref ) ≤ h 1 for all (x, u, yref ) in the feasible flight envelope defined by Table I, for some positive constants h 1 , h 1 . To obtain the desired equilibrium a˜⋆ = 0 it is necessary to enforce j 2⋆ = 0. If _ ⋆ were know, this could be accomplished by imposing e = _ ⋆ . However, due to parameter uncertainty, _ ⋆ is unknown and hence the system is augmented with an integrator, whose dynamics are given by j˙1 = a˜ = j2 − µ1 Q − µ2V˜ . The augmented rotational dynamics have now a 2-dimensional zero-dynamics with respect to e which no longer have a LAS equilibrium. As a result, a new output e˜ = e + ecmd is considered, where e cmd must be designed to stabilize the equilibrium of the zerodynamics using the states available for feedback. To stabilize the integrator dynamics and shift the equilibrium to j 2⋆ = 0, define j˜2 = j2 + k1 j1 + _ ⋆ , j˜1 = j1 + _ ⋆ /k1 , where k1 > 0 is a design parameter. As a consequence _˙ ⋆ j˙˜1 = −k1 j˜1 + j˜2 − µ1 Q − µ2V˜ + (12) k1 j˙˜2 = R1 (_ −_ ⋆ )+(R2 + k1 )j˜2 − k1 (R2 + k1 )j˜1 − a˙ref − µ2V˙ref +[µ˙ 1 − µ1 (R2 + k1)]Q + [ f2 − µ2 (R2 + k1 )]V˜ + R2 aref + _˙ ⋆ Using the expressions of l and h˙ reported in Table I and applying small angle approximation, one obtains
_˙ ⋆ =
, _⋆ ˙ , _⋆ , _⋆ ˙ , _⋆ l l˙ ≈ Va . Vref + Vref − , Vref ,l , Vref , l hS
Plots of _ ⋆ versus l have been obtained considering nominal values of the parameters and different values of V ref and m. For reasons of space limitation, only the case m = 147 slug/ft is shown in Figure 1 (changing the value of m only results in a vertical translation of these curves.) The graphical analysis show that , _ ⋆ /, l is strictly negative for all feasible values of Vref and l . By continuity, there exists h 3 > 0 such that R3 (x) := −
, _⋆ l V > h3 > 0 , l hS
(13)
for all admissible flight conditions. As a result , _⋆ ˙ _˙ ⋆ = Vref + R3 j˜2 − k1 j˜1 − µ1 Q − µ2V˜ + aref . (14) , Vref
1407
WeB10.4 Since _ − _ ⋆ = e˜ + ecmd − j˜2 + k1 j1 + µ1 Q + µ2V˜ − aref , the reference command for e is chosen as e cmd = −k1 j1 + aref . Finally, looking at system (12), it can be noticed that both Q and e˜ affect the (j˜1 , j˜2 )-dynamics. This system is not in one of the standard forms [10] which assume that only one state of the chain of integrators perturbs the zero-dynamics, and a peaking phenomenon [9] may occur in the system. To preserve the stability of the interconnection of the internal dynamics with the (e˜ , Q)-dynamics, a gain-dependent change of coordinates will be applied to enforce a time-scale separation between the two subsystems. Let r 1 = k12 j˜1 , r2 = k1 j˜2 . ˜ ˜ k2 is a positive gain, 2 = Define Qb = Q +\_k32 e , where 2 ¯ M , zT CT , zT CT\_ , zT CT\_ , zT CT\ , zT CT3 , zT CT2 + S/Iyy cC 0 T , ^ (x, u) = −q¯ 0, _ 3 \, _ 2 , z C1 + cC _ , z C0 + cC cC ¯ M ¯ M ¯ M T T T T 2 T T _ 2 \, _ \, \, _ 3 , _ 2 , _ , 1 , and B2 (q) ¯ = q, ¯ 01×8 . Let ˆ2 be a vector of estimates of 2 with estimation error ˜2 = ˆ2 − 2 , and let O2 be the compact set in which e 2 is assumed to range, obtained by letting the entries of e 2 vary within the parameter set P. By choosing 1 T ˆ ^2 2 − (k2 + kQ )Q˜ + K22 e˜ − k1 k2 a˜ + k2 a˙ref be = T ˆ2 B2 where kQ > 0 is a gain parameter, the augmented rotational dynamics assume the following form r˙ 1 = k1 +R3 −r1 + r2 + k1(k2 µ1 e˜ − µ1 Q˜ − µ2V˜ ) + k1 d1 r˙ 2 = −(R1 − R2 − R3 − k1 )r2 − (R2 + R3 + k1 )r1 + k1 d2 +k1 {R1 − k2 µ˙ 1 + µ1 (R1 − R2 − R3 − k1 ) } e˜ +k1 µ˙ 1 + µ1 (R1 − R2 − R3 − k1 ) Q˜ +k1 f2 + µ2 (R1 − R2 − R3 − k1) V˜ ˜ a˙ − k µ V˜ e˙˜ = −k (1−k µ )e˜ − r + r +(1−k µ )Q− 2
1 1
1
2
1 1
Q˙˜ = −kQ Q˜ + ˜2T (^2 − B2 be ) .
ref
2 2
(15)
where d1 and d2 are vanishing disturbances. The update law for the parameter estimate ˆ2 is chosen as (16) ˆ˙2 = Proj Q˜ K2 B2 be − ^2 ˆ2 ∈ O2
where K2 ∈ R9×9 is a symmetric positive definite matrix.
V. C LOSED - LOOP S YSTEM S TABILITY A NALYSIS Looking at the closed-loop dynamics, defined by (6)-(7) and (15)-(16), it is possible to notice that the adaptive control strategy adopted enforces a “nearly cascade” structure in ˜ only the system as r1 , r2 and e˜ affect the (V˜ ,Q)-dynamics through the regressors ^ 1 and ^2 . Using the Lyapunov func˜ ˜ T −1 ˜ tion candidate W0 = 1/2 (V˜ 2 + Q˜ 2 + ˜1T K−1 1 1 + 2 K2 2 ) , it is straightforward to prove that, as long as the trajectories of the overall system are defined, V˜ and Q˜ remain bounded and converge asymptotically to zero while the parameter estimates remain bounded. As a consequence, the next and final step in the stability analysis is to show that no finite escape time occurs in the system and limt→' (r1 (t), r2 (t), e˜ (t)) = (0, 0, 0) . To this aim, the L ' bounds1 of the trajectories 1 In
the following, we use the notation adopted in [11].
r1 (t), r2 (t) and e˜ (t) will be first analyzed separately. Then, it will be shown that the (r 1 , r2 )-system is input-to-statestable (ISS) with no restriction on the inputs ( e˜ , z), and that the e˜ -system is ISS with no restriction on the inputs (r , z), ˜ d1 , d2 , a˙ref ]T . Finally, where r = [ r1 , r2 ]T and z = [ V˜ , Q, the interconnection of the r -system and the e˜ -system will be analyzed. By introducing suitable functions b 1 , b2 , b3 and b4 , the r and e˜ -dynamics can be written as r˙ 1 = k1 + R3 − r1 + r2 + k1 k2 µ1 e˜ + k1 b1 z (17) r˙ 2 = −(R1 − R2 − R3 − k1 )r2 − (R2 + R3 + k1 )r1 +k1 k2 b2 e˜ + k1b3 z (18) ˙ ˜ ˜ (19) e = −k2 (1 − k1 µ1 )e − r1 + r2 + b4z . Define µ1M = b¯ 1
=
b¯ 2
=
b¯ 3
=
max
(x,u,yref )∈A
max
(x,u,yref )∈A
max
(x,u,yref )∈A
max
(x,u,yref )∈A
µ1 , µ2M =
max
(x,u,yref )∈A
| µ2 |
1 |µ2 |, µ1 , R3 ¯h1 + |µ˙ 1 | + µ1 |R1 − R2 − R3 | + 1 | f2 | + |µ2| |R1 − R2 − R3 | + 1 , |µ˙ 1 | + µ1 |R1 − R2 − R3 | + 1 , 1 ,
and let ¡ be a design parameter such that 0 < ¡ < 1. The following propositions hold. Proposition 5.1: For all k 1 , k2 > 0 system (17) is ISS without restriction and on the inputs (r 2 , e˜ , z). Moreover, the state r1 satisfies the infinity bound 6k1 b¯ 1 r2 ' 2k1 k2 µ1M ˜ , e ' , z' . r1 ' < max |r10 |, 1−¡ ¡ ¡
Proof: Let us consider the Lyapunov function candidate W1 (r1 ) = 12 r12 . Since for (x, u, y ref ) ∈ A , |b1 | < b¯ 1 , the derivative of W1 along trajectories of system (17) satisfies ˙ W1 < − k1 + R3 |r1 | (1 − ¡ )|r1 | − |r2| ¡ |r1 | ¡ |r1 | M ˜ ¯ + − k1 k2 µ1 |e | + − 3k1b1 |z| 2 2 and therefore using (13) it follows that |r2 | 2k1 k2 µ1M ˜ 6k1 b¯ 1 |r1 | > max , |e |, |z| ⇒ W˙ 1 < 0 . 1−¡ ¡ ¡ As a result, W1 is an ISS Lyapunov function for system (17) and hence (see [11, Lemma 3.3]) the infinity bound for r 1 is proved to hold. Proposition 5.2: There exists k 1a > 0 such that, for all k1 < k1a and k2 > 0, system (18) is ISS without restriction on the inputs (r 1 , e˜ , z). Moreover, for k 2 > 1 the state r2 satisfies the asymptotic bound c1 (k1 )c2 (k1 ) r2 ' < max |r20 | , r1 ' , 1−¡ 2k1 k2 b¯ 2 c¯1 ˜ 6k1 b¯ 3 c¯1 e ' , z' ¡ ¡ where c1 (k1 ) and c2 (k1 ) are two positive functions of k 1 and c¯1 is a positive constant .
1408
WeB10.4 0.04
Proof: Using the same arguments used to conclude that 0 < h 1 ≤ R1 ≤ h 1 , it is also possible to show [7] that there exists a constant R S such that min
(x,u,yref )∈A
{R1 − R2 − R3 } > 0 .
Vref = 9000, l = 4.8eï005, \ = 1.3, m = 147.9 V
ref
(20)
Since a stronger condition on the function R 1 will be required in the sequel, a graphical proof that (20) holds will be provided later in the paper. Let k 1a = min 1, R2S and select k1 < k1a and k2 > 1. As a result 1 c1 (k1 ) := max < c1 (k1a ) =: c¯1 R1 − R2 − R3 − k 1 (x,u,yref )∈A c2 (k1 ) :=
max
(x,u,yref )∈A
Vref = 8000, l = 4.2eï005, \ = 0.5, m = 93.57
0.02 f(_+_*) ï 2 |R2 + R3| _
R S :=
Vref = 7500, l = 5.5eï005, \ = 0.8, m = 190.2 0.03
= 10000, l = 5eï005, \ = 0.7, m = 100.1
Vref = 11000, l = 2eï005, \ = 1, m = 126.1
0.01
0
ï0.01
ï0.02
ï0.03 ï5
Fig. 2.
ï4
ï3
r22
Consider the Lyapunov function candidate W 2 (r2 ) = whose derivative along trajectories of (18) satisfies 1−¡ ˙ 2 < − | r2 | |r2 | − c2(k1 )|r1 | W c1 (k1 ) ¡ ¡ + |r2 | − k1k2 b¯ 2 |e˜ | + |r2 − 3k1 b¯ 3 |z| . 2c¯1 2c¯1 Therefore,
0 _ [deg]
1
2
3
4
5
Plot of the function f (_ + _⋆ ) − 2|R2 + R3 |_ .
Proof: Let k1 < k1a and k2 > 1; then, since Propositions 5.1 and 5.2 hold, the result follows [11] for 1 c¯1 c¯2 , c3 = max 1, 1−¡ 1−¡ M 2 µ1 2b¯ 2 c¯1 2c¯1 c¯2 µ1M , , c4 = max ¡ ¡ (1 − ¡ ) ¡ (1 − ¡ ) ¯ 6b1 6b¯ 3c¯1 6c¯1 c¯2 b¯ 1 , c5 = max , ¡ ¡ (1 − ¡ ) ¡ (1 − ¡ ) if the following small-gain condition is satisfied
2k1 k2 b¯ 2 c¯1 ˜ 6k1 b¯ 3 c¯1 c1 (k1 )c2 (k1 ) |r2 | > max |r1 |, |e |, |z| 1−¡ ¡ ¡
c1 (k1 )c2 (k1 ) < 1. (1 − ¡ )2
implies W˙ 2 < 0. As a result, W2 is an ISS Lyapunov function for system (18) and hence the claim follows. Proposition 5.3: There exists k 1b > 0 such that, for all k1 < k1b and k2 > 0, system (19) is ISS without restriction on the inputs (r , z). Moreover 6 z' 4 r ' , . (21) e˜ ' < max |e˜0 |, k2 (1 − k1 µ1M ) k2 (1 − k1 µ1M ) Proof: Consider the Lyapunov function candidate W3 (r1 ) = 12 e˜ 2 and let k1 < k1b := 1/ max µ1M , µ2M . Then, in the feasible flight envelope |b 4 | < 1, and the derivative of W3 along trajectories of system (19) satisfies W˙ 3 < −|e˜ | k2 (1 − k1 µ1M )|e˜ | − 2|r | − 3|z| . Since
ï1
{|R2 + R3| + k1 } < c2 (k1a ) =: c¯2 . 1 2
|e˜ | > max
ï2
4 |r | 6 |z| , M k2 (1 − k1 µ1 ) k2 (1 − k1 µ1M )
˙3 < 0 ⇒ W
W3 is an ISS Lyapunov function for system (19). The infinity bound for e˜ (t) is therefore proved. Proposition 5.4: There exists k 1c > 0 such that, for all k1 < k1c and k2 > 0 the system given by the interconnection of systems (17) and (18) is ISS without restriction on the inputs (e˜ , z). Moreover, for k 2 > 1 the combined state r satisfies the infinity bound (22) r ' < max c3 |r0 | , k1 k2 c4 e˜ ' , k1 c5 z' . where c3 , c4 and c5 are positive constants .
(23)
Since both c1 (k1 ) and c2 (k1 ) are continuous and decreasing functions of k 1 , if we can show that c 1 (0)c2 (0) < 1, then by continuity, there exist k 1∗ > 0 and ¡ ∗ , (0 < ¡ ∗ < 1), so that (23) is satisfied for any ¡ < ¡ ∗ and k1 < k1∗ . As a result, to prove (23) it is sufficient to show that max
(x,u,yref )∈A
{R1 − 2|R2 + R3 |} > 0
which, recalling that f 1 := R1 · (_ − _ ⋆ ), is equivalent to prove that >0 _ >0 =0 _ =0 . (24) f (_ + _ ⋆ ) − 2|R2 + R3 |_ 1 the state r and e˜ satisfy the infinity bounds r ' < max c3 |r0 | , k1 k2 c4 |e˜0 | , k1 c6 z' . 6z' 4c3 |r0 | , e˜ ' < max |e˜0 |, (25) k2 (1 − k1 µ1M ) k2 (1 − k1 µ1M ) where c6 is a positive constants .
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(b) Flight Path Angle Tracking and Tracking Error
(a) Velocity Tracking and Tracking Error Fig. 3.
Simulation Results.
Proof: Let k1 < k1d and k2 > 1, where k1d = min k1b , k1c , 1/(4c4 + µ1M ), 1/(2µ1M ), 3/(2c5 ) ; then, since Propositions 5.3 and 5.4 hold, the small-gain condition 4k 1 c4 /(1 − k1 µ1M ) < 1 is satisfied for k1 < k1d , the result follows [11] for c 6 = max{12c4, c5 }. Proposition 5.5 proves that the internal variables of the system remain bounded and therefore no finite escape time occurs in the system. Moreover, since d a = 0 and V˜ a = ˜ a = 0, then we can conclude [11] that r a = 0 and 0, Q ˜ e a = 0. As a result the control objective is achieved.
vehicle dynamics which employs only the elevator as aerodynamic control surface. The main focus of the paper is on counteracting the exponentially non-minimum phase behavior of the rigid-body FPA dynamics. The method reposes upon the redefinition of the internal dynamics of the systems and upon a gain-dependent change of coordinates which enforces a time-scale separation between the controlled variables. Model uncertainties are dealt with by adaptive control and small-gain arguments are employed for stability analysis. Simulation results validate the proposed methodology. R EFERENCES
VI. SIMULATIONS To validate the controller derived in the previous section, simulations have been performed on the vehicle model implemented in SIMULINK . For reasons of space limitation, only one representative case study will be presented here, which is the second case study reported in [8]. The vehicle is not trimmed at t = 0, h(0) = 86000 ft and V (0) = 7850 ft/s. The velocity reference trajectory is generated to let the vehicle reach the desired final trim condition V ∗ = 10500 ft/s. The flight path angle reference trajectory is generated filtering a step s(t) such that, s(t) = 5 for 0 < t < 20 and s(t) = 0 for t ≥ 20, with a first-order pre-filter with natural frequency t f = 0.02 rad/s and damping factor c f = 0.95. The controller gains have been chosen as kV = 100, kQ = 10, k1 = 0.5, k2 = 4, K1 = 0.1 × I12×12 and K2 = 0.1 × I9×9 . Comparing Fig. 3(a)-(b) with the corresponding ones in [8], it is possible to see that the velocity tracking errors are comparable, while the tracking error for the FPA here is one order of magnitude smaller than the one obtained in [8]. This is due to the fact that in [8] exact dynamic inversion was applied to the FPAdynamics while the different methodology presented here makes use of adaptive control to take care of parametric uncertainty. VII. C ONCLUSIONS
[1] M. A. Bolender and D. B. Doman, “Flight path angle dynamics of air-breathing hypersonic vehicles,” AIAA Paper 2006-6692, 2006. [2] C. Marrison and R. Stengel, “Design of robust control systems for a hypersonic aircraft,” Journal of Guidance, Control, and Dynamics, vol. 21, no. 1, pp. 58–63, 1998. [3] Q. Wang and R. Stengel, “Robust nonlinear control of a hypersonic aircraft,” Journal of Guidance, Control, and Dynamics, vol. 23, no. 4, pp. 577–85, 2000. [4] H. Xu, M. Mirmirani, and P. Ioannou, “Adaptive sliding mode control design for a hypersonic flight vehicle,” Journal of Guidance, Control, and Dynamics, vol. 27, no. 5, pp. 829–38, 2004. [5] J. T. Parker, A. Serrani, S. Yurkovich, M. A. Bolender, and D. B. Doman, “Control-oriented modeling of an air-breathing hypersonic vehicle,” Journal of Guidance, Control, and Dynamics, vol. 30, no. 3, pp. 856–869, 2007. [6] M. A. Bolender and D. B. Doman, “A nonlinear longitudinal dynamical model of an air-breathing hypersonic vehicle,” Journal of Spacecraft and Rockets, vol. 44, no. 2, pp. 374–387, 2007. [7] L. Fiorentini, A. Serrani, M. Bolender, and D. Doman, “Nonlinear robust adaptive control of flexible air-breathing hypersonic vehicles,” Journal of Guidance, Control and Dynamics, vol. 32, no. 2, MarchApril 2009. [8] L. Fiorentini, A. Serrani, M. A. Bolender, and D. B. Doman, “Nonlinear control of non-minimum phase hypersonic vehicle models,” in Proceedings of the 2009 American Control Conference, St. Louis, MO, 2009. [9] H. K. Khalil, Nonlinear systems. Upper Saddle River, N.J.: Prentice Hall, 2002. [10] A. Isidori, Nonlinear Control Systems. Berlin, Germany: SpringerVerlag, 1995, pp 293-312. [11] A. R. Teel, “A nonlinear small gain theorem for the analysis of control systems with saturation,” IEEE Transactions on Automatic Control, vol. 41, no. 9, pp. 1256–70, 1996.
In this paper, we have presented a methodology for robust adaptive nonlinear control of the longitudinal hypersonic
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