General Periodic Cruise Guidance Optimization for Hypersonic Vehicles

Abstract

Periodic cruise has the potential to improve the fuel efficiency of a hypersonic vehicle. However, the optimization of periodic cruise is very difficult and can only be performed inefficiently through trial-and-error due to the parameterized form. In this paper, we systematically optimized the hypersonic periodic cruise using the pseudo-spectral method (PSM) scheme. We specify the main variables as the given forms of periodic functions and parameterize the periodic guidance. The characteristic parameters can then be considered as augmented states to generate augmented dynamics. Therefore, periodic cruise optimization can be directly obtained by using GPOPS (Gauss Pseudo-spectral OPtimization Software). The numerical results demonstrate the effectiveness of the proposed method. The approach in this case study can be generalized to solve similar trajectory optimization problems that can be parameterized in a unified manner.

1. Introduction

Nowadays, hypersonic vehicles are attractive in many countries [1,2,3]. In addition to aerodynamic hypersonic gliders, powered hypersonic vehicles primarily operate at constant velocities at specific altitudes because this strategy can be easily implemented in practice. The cruising altitude and velocity were determined according to the principle of fuel efficiency maximization to achieve a long flight range. In other words, the flight range, with respect to the fuel consumption, should be as large as possible. In the future, there is a need to further improve the fuel efficiency of the cruising phase to make room for useful payloads and reduce the weight and volume of the launch vehicle. Therefore, a more advantageous flight mode should be developed instead of the traditional steady-state cruise, to increase the efficiency-cost of fuel consumption. This kind of research is particularly important for inter-continental hypersonic vehicles.

The conventional steady-state cruise is the simplest paradigm in the sense of optimality, which is an achievable 2 degree-of-freedom (DOF) trajectory guidance law in terms of altitude and velocity at the expense of a certain flight range. The theoretically optimal trajectory is an infinite DOF curve, which cannot be implemented in practice. In addition to the steady-state cruise, a periodic cruise of more than 2 DOF may improve the fuel efficiency in hypersonic flight. Periodic control is a repetition of the same control actions in each cycle, and the results of optimal periodic control (OPC) have appeared intermittently since the 1950s. Early OPC germinated from the chemical engineering process to achieve economic benefits [4,5,6,7,8,9]. Since then, once it was properly developed, the OPC concept was suitable for handling a much wider variety of dynamical systems. Besides chemical processes, cruise control of automobiles and aircrafts is a typical application of periodic control during long-period normal operations because of the economic reasons [10,11,12,13,14,15,16,17,18,19,20]. According to [11,12,13,14,15,16,17,18,19,20], a flight vehicle cruising at non-constant altitudes and velocities with proper power switches may save the fuel consumption to some extent, or it can fly at a longer range. For example, the thrust can be switched off during the descending phase in a cycle such that the gravitational force can be fully utilized to reduce the velocity loss, while the thrust can be switched on again at a certain time in the ascending phase to compensate for the velocity loss. This conversion between the potential energy and kinetic energy has the possibility of enhancing the fuel efficiency. This is quite attractive in the aerospace industry.

Compared to the application of OPC, the theoretical analysis of OPC is quite difficult. In the theoretical research, the main focuses are the sufficient and necessary conditions for OPC [21,22]. The sufficient condition aims to give a criterion for improving the optimal steady-state control through cyclic operations [23]. The existing sufficient conditions are based on the Hamilton-Jacobi theory [24]. The state-of-the-art sufficient condition is the so-called π-test in the frequency domain [25,26,27]. However, the sufficient condition is only a guarantee of existence, which can slightly improve the steady-state performance with slight perturbation, without considering optimality. We must resort to the necessary conditions for finding the optimal periodic solution. The necessary conditions are almost unexceptionally based on the maximum principle [28], which provides a unique basis for the subsequent optimization of characteristic parameters with the help of Hamiltonian functions by using gradient-based methods, including direct and indirect schemes [29]. The indirect philosophy analytically derives the necessary condition and then obtains the optimal solution. These methods use the calculus of variations to establish a set of first-order optimality equations. However, it is quite difficult to solve these equations because such a two-point boundary value problem (TPBVP) is well known for its difficulty. So far, the shooting method [30] is one of the few methods that can handle TPBVP by iteratively correcting the initial values through the terminal discrepancy. Nevertheless, the shooting method is too sensitive to the choice of the initial guess. Therefore, successful application of the shooting method heavily relies on a refined understanding of the plant and an accurate initial guess. Besides, the existing experiments showed that the convergence behavior of the first-order optimization schemes is poor, relative to a class of aircraft cruise problems [30]. In short, there is currently no effective indirect numerical method to obtain the optimal periodic solution. Due to the complicated calculation process and special expertise requirement, the above-mentioned methods lack easy utility, in general, for fast design and evaluation. To date, there is no professional software that can handle periodic optimization problems in a unified framework. Therefore, we must apply existing and mature optimization software to deal with this problem.

In recent years, direct methods, especially pseudo-spectral methods (PSM), have become increasingly popular in the preliminary design phase of the aerospace industry with the development of computers and the corresponding optimization software, GPOPS (Gauss Pseudo-spectral OPtimization Software), which is based on some effective nonlinear programming methods, such as SQP (Sequential Quadratic Programming) [31,32,33,34,35]. GPOPS has been successfully applied to many trajectory optimization problems. Although PSM is widely used in trajectory optimization problems because of its maturity and module standardization, there are no reports on its application in the periodic cruise optimization to our best knowledge, except [36] which partitioned a cycle into four segments to impose different kinds of constraints in a tiresome manner. The main difficulty is that the parameterized periodic trajectory is beyond the framework of PSM, which is based on the polynomial fitting in terms of Gauss nodes to handle infinite-DOF curve optimization problems with independent variables to be optimized. In other words, PSM cannot be directly applied to optimize trajectory with a specified form that implies strong interdependence of the variables. At present, it is hard to find other means, other than PSM, to solve this problem. Based on the current popularity and acceptance of GPOPS, the use of PSM to optimize periodic cruise trajectories can provide a general, convenient, and effective methodology for practitioners. Since the periodic cruise cannot be embedded directly into the GPOPS software, the problem is how to achieve this flexibly. As a matter of fact, many trajectory optimization problems are parameterized to facilitate implementation. We hope for a general solving method for this wide range of problems, by using the GPOPS rather than a particular solution.

In this paper, the Gauss pseudospectral method (GPM) is used to obtain the optimal periodic trajectory of the HL-20 hypersonic aircraft in a general manner, different to the method in [36]. The periodic form is firstly specified, and then the characteristic parameters are considered as augmented states. The originally physical states plus these augmented states establish an augmented system. The periodic form can be added as a path constraint embedded in the GPOPS, with acceptable accuracy. Mathematical simulations are performed to verify the proposed approach.

The remaining parts of the paper are organized as follows. The preliminary knowledge is presented in Section 2. The solving strategy is offered in detail in Section 3. The simulation results are presented in Section 4. The conclusions are given in Section 5.

2. Preliminary Preparation

2.1. Mathematical Model

In this paper, the HL-20 hypersonic vehicle model is employed [15] as a benchmark. In the trajectory optimization and guidance design phase, we assume that the attitude control can be ideally realized; in other words, there is no control error and the hypersonic vehicle can be considered as a mass point. For the sake of brevity, the spherical model of the Earth is ignored without affecting the essence of the optimization algorithm. The dynamic model for the mass point can be described as

$$\{\begin{array}{l}\frac{dh}{dt}=M\mathrm{a}\sin \gamma \\ \frac{dM}{dt}=\frac{T\cos \alpha -D-m\mathrm{g}\sin \gamma }{m\mathrm{a}}\\ \frac{d\gamma }{dt}=\frac{T\sin \alpha +L}{mM\mathrm{a}}+\cos \gamma \left(\frac{M\mathrm{a}}{{\mathrm{R}}_{\mathrm{e}}+h}-\frac{\mathrm{g}}{M\mathrm{a}}\right)\\ \frac{dm}{dt}=-\frac{T}{\mathrm{g}{I}_{\mathrm{sp}}}\\ \frac{dr}{dt}=M\mathrm{a}\cos \gamma \left(\frac{{\mathrm{R}}_{\mathrm{e}}}{{\mathrm{R}}_{\mathrm{e}}+h}\right)\end{array},$$

(1)

where $h$ is the altitude (m); $M$ is the Mach number; $\gamma$ is the elevation angle (rad); $m$ is the mass of the flight vehicle (kg); $r$ is the range (m); $\mathrm{a}$ is the normal sound speed and assumes the constant of 340 m/s for convenience; $\alpha$ is the angle of attack (rad); $T$ is the thrust (N); $D$ is the drag (N); $L$ is the lift (N); $g\approx 9.81 {\mathrm{m}/\mathrm{s}}^2$ is the gravitational constant; $R_e=6378 \mathrm{km}$ is the radius of the Earth; and $I_{\mathrm{sp}}$ is the specific impulse of the booster. The detail of this model can be seen in [15].

According to the definition of the periodic cruise, the period refers to the flight range rather than the flight time. Therefore, we can reformulate Equation (1) as

$$\{\begin{array}{l}{h}^{\prime }=\frac{dh}{dr}=\tan \gamma \left(1+\frac{h}{{\mathrm{R}}_{\mathrm{e}}}\right)\\ M^{\prime }=\frac{dM}{dr}=\frac{T\cos \alpha -D-m\mathrm{g}\sin \gamma }{M{\mathrm{a}}^{2}m\cos \gamma }\left(1+\frac{h}{{\mathrm{R}}_{\mathrm{e}}}\right)\\ {\gamma }^{\prime }=\frac{d\gamma }{dr}=\left(\frac{L+T\sin \alpha -m\mathrm{g}\cos \gamma }{m{M}^2{\mathrm{a}}^2\cos \gamma }+\frac{1}{{\mathrm{R}}_{\mathrm{e}}+h}\right)\left(1+\frac{h}{{\mathrm{R}}_{\mathrm{e}}}\right)\\ m^{\prime }=\frac{dm}{dr}=-\frac{T}{\mathrm{g}{I}_{\mathrm{sp}}M\mathrm{a}\cos \gamma }\left(1+\frac{h}{{\mathrm{R}}_{\mathrm{e}}}\right)\end{array},$$

(2)

which is the foundation for later optimization.

2.2. Problem Formulation

In this paper, it is necessary to find an optimal periodic cruising trajectory at an initial altitude to minimize the fuel consumption per unit of flight range. In short, we hope to seek a cost-effective trajectory.

The complete issue can be formulated as a standard nonlinear optimal control problem. Considering the dynamic Equation (2), the average fuel consumption or the cost function can be defined as [15]

$$J=\frac{1}{r_{\mathrm{c}}}{\displaystyle \underset{0}{\overset{r_{\mathrm{c}}}{\int }}\frac{T}{g{I}_{\mathrm{sp}}M\mathrm{a}\cos \gamma }}\left(1+\frac{h}{{\mathrm{R}}_{\mathrm{e}}}\right)dr,$$

(3)

for a specified range of $r_{\mathrm{c}}$ or the range of one cycle. The control constraints include the angle of attack and the opening degree of the engine, $s\left(0\le s\le 1\right)$. The process constraints include the altitude ($h$), the Mach number ($M$), and the elevation angle ($\gamma$), which should be within their reasonable ranges, respectively. The terminal constraints are

$$\{\begin{array}{l}h\left(r_c\right)=h\left(0\right)\\ M\left(r_c\right)=M\left(0\right)\\ \gamma \left(r_c\right)=\gamma \left(0\right)\end{array},$$

(4)

to ensure a complete cycle. However, it is not appropriate to take $r_{\mathrm{c}}$ as the integral upper limit when $r_{\mathrm{c}}$ is not a constant. Then, we define another equivalent cost function as

$$J=\frac{m_{\mathrm{i}}-m_{\mathrm{t}}}{r_{\mathrm{c}}},$$

(5)

where $m_{\mathrm{i}}$ and $m_{\mathrm{t}}$ are the initial and terminal mass of one cycle, respectively. Note that Equations (3) and (5) are equivalent, as they are two representations of the average fuel consumption relative to the flight range in one cycle. The static form of Equation (5) is easier for an optimizer to solve. In this way, the problem of using Equation (3) can be avoided, and the computational complexity can be reduced.

2.3. Brief Introduction of GPM

Consider dynamic equation [31,32,33,34,35]:

$$\dot{x}=f\left(x\left(t\right),u\left(t\right),t\right),$$

(6)

where $x\left(t\right)\in {\mathrm{R}}^n$ and $u\left(t\right)\in {\mathrm{R}}^m$ are state and control variables, respectively, and the continuous function $f$ is defined on ${\mathrm{R}}^n\times {\mathrm{R}}^m\times \mathrm{R}\to {\mathrm{R}}^n$.

When the cost function, Equation (5), is only in terms of terminal states as in this periodic optimization paradigm, the traditional optimal control problem can be formulated in a unified form as minimization of

$$J=\mathsf{\Phi }\left[x\left(t_0\right),t_0,x\left(t_f\right),t_f\right],$$

(7)

subject to the boundary equality constraint of

$$\varphi \left(x\left(t_0\right),t_0,x\left(t_f\right),t_f\right)=0,$$

(8)

and the process inequality constraint of

$$C\left(x\left(t\right),u\left(t\right),t\right)\le 0,$$

(9)

where $\mathsf{\Phi }$, $\varphi$, and $C$ are continuous functions defined on ${\mathrm{R}}^n\times \mathrm{R}\times {\mathrm{R}}^n\times \mathrm{R}\to \mathrm{R}$, ${\mathrm{R}}^n\times \mathrm{R}\times {\mathrm{R}}^n\times \mathrm{R}\to {\mathrm{R}}^q$, and ${\mathrm{R}}^n\times {\mathrm{R}}^m\times \mathrm{R}\to {\mathrm{R}}^r$, respectively; $t_0$ and $t_f$ are the initial and final time instants of the optimization interval. The detailed description of GPM can be found in [31,32,33,34,35]. In GPM, the original optimal control problem can be approximated as static nonlinear programming (NLP). The essence of GPM is to replace the original infinite-dimension static NLP by eliminating differential and integral equations. There are many effective methods to solve NLP, and SQP is a famous one that is widely used because of its reliability. Nowadays, reliable software, SNOPT (Sparse Nonlinear OPTimizer, Stanford University, Stanford, CA, USA), is available to solve SQP problems in a unified framework.

3. Augmented States Based Periodic Cruise Optimization

Due to the mathematical form constraint of periodic cruising trajectory, GPM cannot be directly applied. To solve this problem, we formulate the periodic functional constraint as a path constraint by treating the characteristic parameters as augmented states. Then, we augment the original system with additional path constraints embodying the periodic property.

The periodic altitude can be in the following form:

$$h=h_a\cos \omega r+h_b,$$

(10)

where $h_a$, $h_b$, and $\omega$ are the characteristic parameters to be optimized. Considering the first sub-Equation (2):

$$\frac{dh}{dr}=\tan \gamma \left(1+\frac{h}{{\mathrm{R}}_{\mathrm{e}}}\right),$$

(11)

As $h\ll {\mathrm{R}}_{\mathrm{e}}$ up to more than two orders of magnitudes, we can approximate Equation (11) as:

$$\frac{dh}{dr}=\tan \gamma ,$$

(12)

Combining Equations (10) and (12) yields:

$$\gamma =-{\tan }^{-1}\left(h_a\omega \sin \omega r\right),$$

(13)

This can provide a practical guidance law for a hypersonic vehicle to achieve periodic cruise; otherwise, it is difficult to implement accurate tracking of the altitude, as shown in Equation (10), due to the slow altitude response [37,38,39,40]. According to Equation (10), the cycle range is

$$r_c=2\pi /\omega ;$$

(14)

To sum up, three characteristic parameters describe a periodic solution, which should be optimized. We specify these constant parameters as augmented states as

$$\{\begin{array}{l}\frac{d{h}_a}{dr}=0\\ \frac{d{h}_b}{dr}=0\\ \frac{d\omega }{dr}=0\end{array},$$

(15)

Combining the original four states, we obtain seven-state augmented dynamics. Subsequently, we perform the optimization for this system. We may include Equation (10) as a path constraint as

$$\left|h-\left(h_a\cos \omega r+h_b\right)\right|<h_{\epsilon },$$

(16)

where $h_{\epsilon }$ is an acceptable tolerance for numerical computation. In this way, the path constraints can be naturally embedded in the GPOPS.

4. Numerical Examples and Analysis

In this section, numerical examples are provided to illustrate the effectiveness of the proposed method. We attempt to investigate the optimality and accuracy. Optimality is the performance improvement compared with the traditional optimal steady-state cruise. Here, the performance is only the average fuel consumption. The comparison results can explain the practical significance of the periodic cruise and the proposed method. The accuracy investigation can reveal the compatibility of the results.

The limits of the control variables, $\alpha$ and $s$, are determined according to the original data benchmark, as illustrated in

Table 1. Limits of control variables.

Variable	Lower Limit	Upper Limit
$\alpha$ (deg)	5	20
$s$	0	1

.

4.1. Optimal Velocities of Steady-State Cruise at Different Altitudes

First, the optimal velocities of steady state cruise at different altitudes should be obtained, laying the foundation for further periodic optimization and comparison. The optimal velocity can be obtained through trajectory optimization by nulling the altitude rate, the Mach number rate, and the elevation angle rate in (2), respectively. The Matlab (Mathworks, Natick, MA, USA) function of fmincon is invoked to conduct this mission, wherein fmincon finds the constrained minimum of a function of several variables. In this method, we convert all dynamic equations into algebraic equations in terms of $M$, $\alpha$, and $s$ in a steady situation for a specified altitude, $h$. Here, $M$ is the state variable, while $\alpha$ and $s$ are two control variables, which have their respective ranges to be determined by the user. The cost function is the fuel consumption rate with respect to these three variables. The optimization results for OSS cruise in the altitude range [42 km, 45 km] are shown in

Table 2. Steady-state cruise optimization results.

Altitude (km)	Mach Number	Fuel Consumption (kg/km)
45	14.536	1.566
44	13.980	1.564
43	13.413	1.569
42	12.830	1.580

. Compared with the optimal steady-state result in [15] of 1.555 kg/km, these results are slightly higher because of the numerical calculation discrepancies from different software. The optimal cruising velocity is larger at a higher altitude, in line with our empirical knowledge.

4.2. Optimization by Using Augmented States

In this part, we seek the optimal periodic cruising solutions by using the proposed method at the specified altitudes, as shown in Table 2. We select the altitude error tolerance, $h_{\epsilon }$, as 80 m by trial-and-error to ensure the fitting accuracy and convergence simultaneously.

To facilitate the optimization process, we specify the initial altitude, $h_i$ (also the terminal altitude, $h_t$), as the maximum altitude of each cycle, $h_{\max }$. Thus, we have $h_i=h_t=h_{max}$ and ${\gamma }_i={\gamma }_t=0$. According to the definition of a periodic cruise, the initial Mach number, $M_i$, equals the terminal Mach number, $M_t$. In the following study, all optimization procedures consistently use these constraints, as shown in

Table 3. Limits of state and path constraints.

Variable	Value
$h_{\max }$ (km)	$h_0$
$h_{\min }$ (km)	$h_0-10$
${\gamma }_{\max }$ (deg)	5
$M_{\min }$	$M_0-1$
$M_{\max }$	$M_0+1$
$m_{\min }$ (kg)	80,000

, for the sake of fairness.

The proposed method was carried out utilizing the initial altitudes and Mach numbers as shown in Table 2, and the corresponding results are shown in

Table 4. Periodic cruise optimization results.

Altitude (km)	Mach Number	Fuel Consumption (kg/km)	Cycle Range (km)	Improvement in Fuel Consumption (%)
45	14.536	1.5275	877	2.46
44	13.980	1.5298	849	2.19
43	13.413	1.5373	807	2.02
42	12.830	1.5492	786	1.95

To be concrete, we take 45 km and 43 km optimization results as two examples.

. It can be seen that periodic cruise can improve fuel efficiency compared with the corresponding optimal steady-state cruise, which is more pronounced at higher altitudes. Compared with the results of [15,36] of around 3% improvement, 2% fuel consumption improvement was achieved. Note that the unique advantage of the proposed method is its unified dealing with parameterized trajectory optimization with little programming work. In [15], the solving software NCONF (Visual Numerics, Houston, TX, USA) is rarely used nowadays and is hard to find; and in [36], the segmented optimization results have poor convergence and oversensitivity to parameter specifications. Therefore, we seek a general solving method for practitioners rather than a particular solution here. It should be also noted that the improvement degree is about 2%, like the results of 3% in [15,36], which is not quite as robust to the uncertainties in the model. In other words, this result may be worse than that of the optimal steady state when there are uncertainties. In fact, the trajectory optimization is static, and the robustness is beyond the scope of this paper.

For the 45 km case, the optimal periodic altitude is

$$h=4595\cdot \cos (0.0072\cdot r_{\mathrm{km}})+40699,$$

(17)

As shown in Figure 1, it can be seen that the discrete altitude nodes almost match the cosine curve. The Mach number is shown in Figure 2. The elevation angle is shown in Figure 3, wherein the line curve is obtained by using Equation (13) with the corresponding characteristic parameters. The fitting curve matches the optimal elevation angle and is relatively satisfactory, which can provide a proper guidance command in practice. The angle of attack is shown in Figure 4. The thrust throttle is shown in Figure 5, and we can see that the engine is turned on at around 500 km. It is clear that the thrust is switched off during the descending phase, such that the gravitational force can be fully utilized to reduce the drag effect; while the thrust is switched on again in the ascending phase to compensate for the velocity loss. In this way, the fuel efficiency can be improved further. According to this result, the engine can operate at only two modes, on or off, with $s=1$ or $s=0$, respectively. This fact is preferred due to its simplicity.

Next, we investigate the 43 km case for further validation. The optimal periodic altitude can be obtained as

$$h=3660\cdot \cos (0.0078\cdot r_{\mathrm{km}})+39148,$$

(18)

The states are shown in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, respectively. Similar observations can be obtained with respect to the previous example. In Figure 6 and Figure 8, periodic altitude and elevation angle approximations with relatively satisfactory precision can offer two-level approximate references for the guidance, which is preferred by practitioners. According to Figure 1 and Figure 6, we can see that the vibration amplitude is smaller for the lower initial altitude. Figure 10 also shows an off-on engine mode as already mentioned above.

5. Conclusions

Based on the flexible use of GPM, this paper attempted to solve the optimization problem of the periodic cruising trajectory of hypersonic vehicles. The characteristic parameters of a cycle were considered as augmented states, and the difficulty of handling the periodic constraints could be avoided. The optimal solution for a periodic cruise can be systematically obtained without relying on specific models and aerodynamic characteristics. Numerical simulations were provided to illustrate the effectiveness of the proposed methods. The optimization results revealed that the periodic cruise consumes fuel more efficiently than the steady-state cruise. This work can provide a novel use of GPM to solve the parameterized guidance problem, which is not easy to optimize although it is required by practitioners. In this way, GPM can serve as a useful tool for a wider range of applications in optimal trajectory design that have parameterized forms.