Stochastic Control-Oriented Modeling of Flexible Air-Breathing Hypersonic Vehicle

The flexible dynamics of commonly used air-breathing hypersonic vehicle model are not tractable for control design and the inevitable stochastic perturbations are usually neglected. Aiming at these deficiencies, reduced flexible dynamics are deducted in this paper and a stochastic control-oriented vehicle model is established accordingly. The responses of the original system to the deterministic and the stochastic part of the generalized force, which is treated as the input of the flexible dynamic system, are analyzed. After that, the simplified flexible dynamics is deduced to approximate the responses.The reduced flexible dynamics, which are tractable for control design since they greatly reduce the complexity of the original dynamics, are comprised of a simple function of the determined generalized force and an Ornstein-Uhlenbeck colored noise. Finally, the longitudinal dynamics in parametric strict feedback form are obtained by substituting the reduced flexible dynamics into the original model. The applicability of the simplified flexible dynamics is validated through the numerical simulations.


Introduction
Air-breathing hypersonic vehicle (AHV) is a reliable and cost-effective technology for access to space and providing rapid global response capability; thus considerable efforts have been made to the further development and design of AHV [1][2][3][4].Notwithstanding the success of NASA's X-43A and X-51A experimental vehicle and the applying of advanced control strategies [5][6][7][8], the design of guidance and control systems for hypersonic vehicles is still an open problem [9,10].
The modeling of the AHV dynamics is a challenging task due to various reasons [11,12].First, the strong coupling between propulsive and aerodynamic forces results from the location of the scramjet engine.Second, the flexible effects are caused by the length and slender geometry.In addition, significant uncertainties and stochastic noises affect the vehicle, as a result of the variability of its characteristics with the flight conditions, fuel consumption, thermal effects, and measurement noises of sensors [11,13,14].
Many efforts have been made to develop simulation models for AHV to match the real-world physics.The main results focus on the longitudinal dynamics, since the AHV rarely takes transverse maneuvering limited by the scramjet engine [4,13,14].The first attempt to build an analytical AHV model was made by Chavez and Schmidt [15].After that, Clark et al. employed computational fluid dynamics (CFD) and analytical analysis to model the longitudinal dynamics [16].Then a nonlinear, physics-based model of the longitudinal dynamics for AHV called the first-principle model (FPM) was proposed by Bolender and Doman, which captures a number of complex interactions between the propulsion system, aerodynamics, and structural dynamics [11].To overcome the analytical intractability of FPM, a simplified control-oriented model (COM) was obtained by Parker et al. [17].Considering that the flexible effects are coupled into the pitch rate equation in FPM and COM, Williams et al. modified the flexible dynamics and got the decoupled COM (DCOM) using the assumed-modes method [18].Both the COM and DCOM have been widely used in the controller design [19][20][21][22][23] since it retains the relevant dynamical features of the FPM and offers the advantage of being analytically tractable.The subsequent dynamic models of AHV are mostly deduced from COM and DCOM [24,25].However, DCOM is still quite complex for controller design since the flexible dynamic equations are second-order systems, which is hard to substitute into the rigid dynamic and there are no well-developed methods to directly get the controller.Hence the existing controller developed based on DCOM makes use of feedback from the rigid states only and keeps the flexible states "frozen" at nominal trim condition, meaning that the flexible dynamics are ignored during the controller design [23,26,27].In recent years, Krylov subspace methods have been employed to compute reduced-order models of high-order linear time invariant systems [28,29], which have been used for reduced-order modeling of large-scale dynamical systems [30].The reducedorder modeling for the flexible dynamics of AHV has not been researched.Another significant drawback is that none of the existing AHV models considers the inevitable stochastic perturbations caused by the airflow, operational error of aerodynamic control surfaces, and measurement noise of sensors.
By analyzing the above-mentioned deficiencies, this paper modified the longitudinal dynamics of DCOM proposed in [18] and simplified the flexible dynamics.Meanwhile, the stochastic perturbations are taken into consideration during the simplification in order to establish a stochastic decoupled control-oriented AHV model (SDCOM).The reduced flexible dynamics greatly reduce the complexity of the original dynamics.The proposed stochastic AHV model is derived from physics-based model and it is tractable for stochastic control design.The rest of this paper is organized as follows: the longitudinal dynamics of DCOM are briefly introduced in Section 2. The responses of flexible dynamics to deterministic signal and Gaussian white noise signal are analyzed.Then the flexible dynamics are simplified in Section 3. The longitudinal dynamics of SDCOM are established in Section 4. The numerical simulations are given in Section 5 to illustrate the validity of the simplified system.

The Decoupled Control-Oriented Model
A sketch of the AHV geometry is shown in Figure 1 and the detailed structure parameters can be seen in [11,17,18].
The DCOM was developed by Williams et al. [18] on the bases of FPM proposed by Bolender and Doman [11]; the longitudinal dynamics are derived using Lagrange's equations and compressible flow theory.The assumed-modes method [31] is employed to establish the flexible dynamic equations.Unlike COM, the flexible modes are orthogonal to the rigid body modes; therefore the interaction between rigid and flexible dynamics occurs only through the aerodynamic forces.The scramjet engine model is adopted from Chavez and Schmidt [15].The longitudinal dynamics of DCOM written in the stability axis coordinate system are where where  = 0.5 ( The parameters and coefficients can be found in [32].It is worth noting that all of the coefficients of the DCOM are subject to uncertainty.The role of the canard deflection is to adaptively decouple lift from elevator commands, so the following relationship holds to eliminate the nonminimum phase behavior [33]:

The Simplification of Flexible Dynamics
It can be seen from the DCOM model given in Section 2 that the flexible dynamic equations are It is obvious that the flexible dynamics are second-order linear systems and the flexible states   are the responses of the system to the input signal.Notice that the value of   is inevitably affected by stochastic perturbations caused by the airflow, operational errors, and measurement noises, we separate   into two kinds of signals: the deterministic signal    and the stochastic signal    .Then the responses of the flexible dynamics to   can be accordingly separated into two parts:    and    , which represent the responses to    and    , respectively.Since the first three flexible states   have the similar expression and decoupled with each other, hereafter we take  1 , for example, to explain the simplification process.

Response to Deterministic Signal.
In this section, the input signal of system ( 5) is considered as the deterministic steady signal and ( 5) can be rewritten as The above equation can be expressed as where The analytical solution of ( 8) can be deducted using linear system theory as follows [34]: We can use inverse Laplace transform tables to compute  A : where Define the following notations: and we have Substitute ( 14) into (10) to obtain the analytical solution of   1 : From ( 15), we can conclude that   1 converges exponentially to   1 / 2 1 ; the rate of convergence is determined by  1  1 .It is worth noting that the results above are got under the assumption that   1 is steady, which is surely not accurate.However, from the results got by Oppenheimer et al. [35] and Bolender and Doman [11], the natural frequencies of the rigid body states range from 0.09 rad/s to 4 rad/s, which are far less than the natural frequencies of the flexible states, which range from 15 rad/s to 120 rad/s [18].So the main factors on   1 change far slower than the rate of convergence of   1 .From the engineering practical view, it is reasonable to assume that    1 is steady within a small time interval.

Response to Stochastic Signal.
In the practical flight process, the stochastic perturbations caused by the turbulent flow and measurement errors of sensors are modeled as zeromean Gaussian white noise [36].Assume that the stochastic part of th generalized force    is a zero-mean Gaussian white noise   () and   () ∼ (0,  2   ()), where () is the Dirac's delta function.Denote the standard Gaussian white noise as () ∼ (0, ()); then we have Assume that the response of flexible dynamic system to the stochastic input  1 () is   1 , so the following equation holds: According to stochastic process theory [37],   1 is a colored noise because the Gaussian white noise is the input of the second-order dynamic system.This colored noise is not tractable for stochastic control design, so it is necessary to simplify the system.In practical application, the Ornstein-Uhlenbeck colored noise, which is a response of a first-order dynamic system to Gaussian white noise, is commonly used and has received a great amount of research [38,39].The ultimate purpose of this section is to construct an Ornstein-Uhlenbeck process, which has distribution characteristics similar to   1 .Let  > 0 and  = const.; the following Itô stochastic differential equation is called the Langevin equation, where () is 1-dimensional standard Brownian motion.The solution stochastic process  1 () ( ≥ 0) is called the Ornstein-Uhlenbeck process, which is a colored noise.It is obvious that the Ornstein-Uhlenbeck process is the response of a first-order system to Gaussian white noise signal.
Let the input noise signal () be a constant  to explore the solution of systems ( 17) and ( 18); the analytical solution of ( 17) can be got as the following according to the solving process in Section 3.1: where   10 , η  10 are initial values of   1 and η  1 ;  11 ,  12 are given in (13).
Then consider the following differential equation: It is easy to give the analytical solution of (20) as follows [40]: The task is to find proper parameters  and  to achieve that the stochastic processes  1 () and   1 have the similar distribution.By comparing (19) and ( 21), the following equations can be deduced to guarantee the same convergent rate and the convergent steady value: So we have From the above points, we can conclude that the solution process  1 () of the following Langevin equation has the distribution characteristics similar to the stochastic process   1 .Synthesize the results in Sections 3.1 and 3.2; the reduced flexible generalized coordinates   in the longitudinal dynamics (1) can be obtained as where   are determined by the following differential equations: Expression (25), which greatly reduces the complexity of the former flexible states dynamics, is comprised of a simple function of the determined generalized force and an Ornstein-Uhlenbeck colored noise.It is tractable for control design since there are many well-developed methods to handle the Ornstein-Uhlenbeck process [37].

The Longitudinal Dynamics of Stochastic AHV Model
Considering that some flight parameters of AHV can also suffer from stochastic perturbations, the stochastic perturbations of , ,  are modeled as zero-mean Gaussian white noises, which are denoted as   ,   ,   , respectively [36].The high-order terms and the cross-terms are omitted for convenience.Substitute expression (25) into the AHV longitudinal dynamics (1) to get the following nonlinear system in parametric strict feedback form: where Δ * ( * = , , , ) represents the uncertainties including parameter perturbations of the curve-fitted coefficients, the high-order terms and cross-terms omitted during the deduction, and so on;  = [ 1 ,  2 ,  3 ]  ; other functions and symbols in (27) are introduced as follows. ( where (2) The Flight-Path Angle  Equation where (3) The Angle of Attack  Equation ) , where It can be seen from the above equations that expression (27) is not in the strict feedback form since the term sin  is included in the function   , and the control input   enters functions   ,   , and   through the term   .However, there are some reasons why we can still allege that the system ( 27) is suitable for backstepping control.First, the natural frequencies of the -equation is larger than -equation, so that the term sin  can be treated as constant.Second, the aerodynamic control surfaces have significant inertial delay [41], meaning that   can be seen as constant when calculating the generalized force   .Third, it is assumed that all model parameters are subject to uncertainty up to 40%, and the necessary significations are considered acceptable in order to design guidance and control systems for AHV [23].
To sum up, the proposed stochastic AHV model possesses many advantages: first, the control design is more convenient since the strong coupling between the flexible states and the rigid body states and the control variables are canceled; second, the distribution of the colored noise is known and the system is in parametric strict feedback form, so the control can be easily designed based on a combination of backstepping and stochastic system theory.

Numerical Simulations
The reduced form (25) of the flexible generalized coordinate   has been tested in simulation to validate the correctness of our results obtained in the previous section.The first three structure natural frequencies and damping ratios are set as the following according to literatures [26,35]: 19.7437, 47.7851, 94.8203]  rad/s  = [0.7,0.6, 0.5]  . (36)

Response to Gaussian White
Noise.The stochastic noise intensity affecting the generalized force   is significant since the dynamic pressure of AHV during the flight is very large.For simplicity, we take the Gaussian white noise inputs as   () ∼ (0, 100 2 ()) to compare the responses of the original second-order flexible dynamics (17) and the simplified first-order dynamics (24).
Notice that    () and   () are both stochastic processes, which is a collection of random variables, so the characteristics of them must have been investigated from two aspects: one is the time characteristics of    () and   (); another one is the distribution properties of    () and   () at fixed time  fix .
For the time characteristics, we investigate the numerical characteristic functions of    () and   ().The expectations of    () and   () are denoted as   1.From these results, we can see that numerical characteristics of the stochastic process    () and   () are very close to each other; the reduced system has similar response to the Gaussian white noise to the original system.
As for the distribution properties of    () and   () at fixed time, 10000 Monte Carlo runs have been done and the deviations between the statistical distributions of    () and   () at time  fix = 10 s are given in Figure 3.
We can see that the statistical results of   ( fix ) at a given time are very close to    ( fix ).The statistical distribution comparison between    ( fix ) and   ( fix ) is given in Figure 4 to illustrate the distinctness.
The results show that the statistical distributions of   () at a given time are similar to    (), especially the first-order flexible generalized coordinate   1 (), which is the main factor   of the flexible states.From the above results, we can draw the conclusion that the reduced system (24) can be used to replace the flexible dynamics (17) for the Gaussian white noise inputs.

Response to Stochastic Time-Varying Signal Input.
The previous section only validates the response to the stochastic parts of the generalized force   .As discussed before, the generalized force   is comprised of two parts: the deterministic part    and the stochastic part    .In this section, the input signal is assumed to be a stochastic time-varying signal to meet the requirements of the reality.The generalized force   is chosen as the following based on the admissible ranges of the states and parameters.
From Figure 5 we can see that the reduced flexible system responses are basically the same as the original ones to the stochastic time-varying input signal.However, the original flexible states need to be calculated by a far more complicated flexible dynamics system.The applicability of the theoretical development has been illustrated through the simulations above.

Analysis of the Longitudinal Dynamics.
In this section, the behavior of open-loop longitudinal dynamics of SDCOM ( 27) is analyzed using geometric control technique.The models taken for comparison are FPM developed by Bolender and Doman [11], COM developed by Parker et al. [17], and DCOM proposed by Williams et al. [18].
The output of the models was selected as  = [, ]  and the control inputs are  = [  , ]  .First, the longitudinal dynamics were trimmed at  = 8 and ℎ = 85,000 ft.The trim condition is given in Table 2, where η  and η  are the flexible states of DCOM and SDCOM;   ,   , η  , η  represent the flexible states of FPM and COM.
The dynamics of AHV models have been linearized around the trim condition and the poles and zeros of the linearized systems are shown in Figure 6.
The actual pole and zero locations of the abovementioned models are given in Table 3.
Figures 6(a) and 6(b) show that both FPM and COM have 7 zeros and 9 poles and the locations of zeros and poles are very close.The two complex conjugate pairs correspond to the flexible dynamics, indicating that they are minimum phase.The pair of zeros that appears to be symmetric about the imaginary axis correspond to the  dynamics and one of them is nonminimum phase.The zero at the origin corresponds to the altitude state.In view of Figures 6(c) and 6(d), it is obvious that DCOM and SDCOM have 9 zeros and 11 poles.The three However, there still exist some limits of the proposed model: the coefficients of the model are subject to uncertainty up to 40%, which should be eliminated by real-word experiments; in the proposed model, only the stochastic elements in the flexible dynamics are considered, which can be improved by adding stochastic noises into the rigid body dynamics so that the model would be more realistic; another important work is to develop real-world examples.

Figure 1 :
Figure 1: The side view of X-43A.

Figure 2 :
Figure 2: The expectation and standard deviation error of    () and   ().
denotes the th flexible generalized coordinate, which is the proportion of the th flexible mode shape;   represents the th structure damping ratio;   is the th structure natural frequency;   is the th generalized force.
+    (, , ) 2is the dynamic pressure and  is the air density;   is the thrust-to-moment coupling coefficient;  is the mean aerodynamic chord;  denotes the reference area;  = [  ,   ]  represents elevator angular deflection and canard angular deflection;  represents the fuel equivalence ratio.The coefficients are 1) The Velocity  Equation

Table 3 :
Poles and zeros of the models.-oriented AHV model has been established.The flexible dynamic system of the traditional DCOM has been analyzed firstly by decomposing the generalized force into two parts.The simplified flexible dynamics have been deduced to approximate the responses to each part of the generalized force.Finally, the longitudinal dynamics of stochastic AHV model in parametric strict feedback form have been obtained by substituting the reduced form flexible dynamics into the original DCOM model.The applicability of the simplified flexible dynamics has been validated through numerical simulations. control