Optimal Ascent Guidance for Air-Breathing Launch Vehicle Based on Optimal Trajectory Correction

An optimal guidance algorithm for air-breathing launch vehicle is proposed based on optimal trajectory correction. The optimal trajectory correction problem is a nonlinear optimal feedback control problem with state inequality constraints which results in a nonlinear and nondifferentiable two-point boundary value problem (TPBVP). It is difficult to solve TPBVP on-board. To reduce the on-board calculation cost, the proposed guidance algorithm corrects the reference trajectory in every guidance cycle to satisfy the optimality condition of the optimal feedback control problem. By linearizing the optimality condition, the linear TPBVP is obtained for the optimal trajectory correction. The solution of the linear TPBVP is obtained by solving linear equations through the Simpson rule. Considering the solution of the linear TPBVP as the searching direction for the correction values, the updating step size is generated by linear search. Smooth approximation is applied to the inequality constraints for the nondifferentiable Hamiltonian. The sufficient condition for the global convergence of the algorithm is given in this paper. Finally, simulation results show the effectiveness of the proposed algorithm.


Introduction
The development of space technology has given rise to the expectation that launchers will become low cost and fully reusable.The launch vehicle with hypersonic air-breathing propulsion is considered to reduce the cost of payloads taken to the Earth's orbit.The air-breathing launch has inherent features that make it a candidate for future space transportation [1].The impulse of air-breathing propulsion, which is approximately 3000 s [2], is significantly higher than that of a rocket (360 s).Air-breathing propulsion brings high impulse as well as strong nonlinear, coupling aerodynamic and thrust.
The traditional design category for ascent guidance is to drive an optimal nominal trajectory off-board.The guidance problem is then transformed into a tracking problem for the designed optimal nominal trajectory.The design of a trajectory could be formulated as a global optimization task [3].The methods for numerical optimization of continuous dynamic systems could be termed "Hamiltonian" (indirect method) and "Transcription" (direct method) [4].
For the optimization problem with inequality constraints, smoothing approximation was considered in [5,6].A filled function approach for nonsmooth constrained global optimization was presented in [7].Linear-quadratic optimization was implemented to optimal control in [8].Direct and indirect optimization methods were implemented to the offboard trajectory optimization problem in previous literature.The numerical algorithms of trajectory optimization for vehicles were summarized and systematically analyzed in [9,10].A new concept of pseudocontrol sets to solve optimal control problems was proposed in [11].This approach reduces the calculation cost by combining large-scale linear programming algorithms with discretization of the continuous system dynamics on small segments.An algorithm for multiobjective optimization was presented in [12].Intelligent algorithms can also be used for trajectory optimization problems.The particle swarm optimization (PSO) method was implemented to the space trajectory optimization in [13].The simulation results showed the effective of PSO in finding the optimal solution to the space trajectory optimization problems, with great numerical accuracy.Approximate numerical methods of optimization were presented for multiorbit noncoplanar orbit transfers of low-thrust spacecraft in [14].
Many control methods were implemented to trajectory tracking guidance and control problem [15].In the previous literature, many researchers focused on the robust control method [16,17].It has obtained successful application in the industry [18,19].For the networked control system, a Hinfinity step tracking control method was presented in [20].An adaptive fuzzy robust control for a class of nonlinear systems was proposed in [21].An adaptive guidance law and off-board trajectory optimization for air-breathing launch vehicle were addressed in [22].In that paper, the optimal control problem was solved using SQP method.And the adaptive guidance law was developed using a feedback loop based on a second-order rate controller for angle of attack.A robust state feedback guidance law was generated in real time using the indirect Legendre pseudospectral feedback method in [23].In that paper, the guidance problem was converted into a trajectory state regulation problem which is a linear time varying system.
However, the accuracy of the trajectory tracking method is low with the disturbance and the modeling error.It lacks the autonomy and adaptability to cope with the nonnominal vehicle and mission conditions needed for future reusable launch vehicles [24].To improve the performance and the accuracy of the guidance, the optimal guidance method generates the prospective trajectory by fast trajectory optimization on-board based on current flight state.This guidance method has become a research focus with the advance in on-board computation capability.Ping Lu used the indirect method to pose the trajectory optimization problem as a nonlinear two-point boundary value problem (TPBVP) in [25,26].From this model, the optimal thrust vector that satisfied the optimality condition was derived.A finite difference method was employed to solve the nonlinear TPBVP through numerical calculations.Similarly, a fast trajectory optimization for hypersonic air-breathing vehicles was presented in [27].The indirect method was implemented for ascent trajectory optimization on-board considering the features of the air-breathing vehicle.Ping Lu's work is significant for the optimal ascent guidance.However, the TPBVP is nonlinear and nondifferentiable for the air breathing launch vehicle.It is difficult to solve the nonlinear and nondifferentiable TPBVP on-board in every guidance cycle.Using the direct method, a new guidance concept based on nonlinear programming (NLP) method was proposed in [24].NLP-based guidance concepts appear advantageous over conventional methods because the on-board guidance algorithm allows a single algorithm to be implemented for different vehicles and missions.The optimal control problem was parameterized into a nonlinear programming problem that was solved by the gradient projection algorithm.In [28], the reference trajectory was updated for disturbance by an onboard algorithm that satisfied the real-time requirement.A new real-time guidance method derived from the optimality condition was proposed in [29].In that paper, the simple guidance parameters were updated in real time.In [30], a guidance method was presented for online launcher ascent trajectory updating based on neural networks.In the paper, the utilization of a neural network approximation was used online during the ascent flight, with a training process performed off-line.
In this paper, we present an optimal guidance algorithm for air breathing launch vehicle based on optimal trajectory correction to reduce the on-board calculation cost.Considering the current vehicle state as the initial condition, the optimal trajectory correction problem is referred to as a nonlinear optimal control problem with state inequality constraints.For the real-time requirement of the on-board algorithm, the linear TPBVP is obtained for optimal trajectory correction by linearizing the optimality condition in this paper.The Simpson rule is applied to transform the linear TPBVP into linear equations.Considering the solution of the linear TPBVP as the searching direction for the correction values, the updating step size is generated by linear search.Smooth approximation is applied to the inequality constraints for the nondifferentiable Hamiltonian.The sufficient condition for the global convergence of the algorithm is given in this paper.Finally, simulation results for different cases of the modeling error show the effectiveness of the proposed algorithm.In summation, the main contributions of this paper are given as follows.
(1) We reduce the on-board calculation cost of the guidance method a lot.Comparing with the methods in previous literatures which solves nonlinear equations in every guidance cycle, the proposed guidance method in this paper solves linear equations once in every guidance cycle only.
(2) We obtain the relationship between the global convergence and the guidance cycle of the online algorithm.
The sufficient condition of the global convergence of this algorithm is given.
The remainder of this paper is organized as follows.Section 2 presents the state normalized-energy differential equations of motion of the vehicle and the optimal control problem for optimal ascent guidance.Section 3 provides details on the optimal guidance algorithm.Section 4 presents an analysis of the global convergence of the proposed algorithm.Section 5 discusses the proposed differentiable approximation for Hamiltonian.Section 6 presents the simulations for the Generic Hypersonic Vehicle (GHV) model and scramjet engine.Section 7 presents the conclusion.

Problem Formulation
2.1.Ascent Dynamic.The formulation of motion of the airbreathing vehicle in the longitudinal plane is presented in this section.As shown in Figure 1, the thrust , gravitation , and the aerodynamic lift force  and drag force  are acting on the vehicle.The angle of attack  is the angle between the body axis and the vector of the velocity V.The flight path angle  is the angle between the vector of the velocity V and the ground plane.The motion of the vehicle in the longitudinal plane can be described as follows:

Center of mass
where , ,  sp , and  are the altitude, weight of the vehicle, impulse, and gravity acceleration, respectively.The lift force  and the drag force  are given as follows: where  and  ref are the air density of the current altitude and the reference area respectively,   and   are the lift coefficient and the drag coefficient, respectively, which are the nonlinear functions of the angle of attack  and Mach number   .For the air-breathing engine, the thrust is given by where  is the nonlinear function of , ,   , and the throttle command .
In [25], the equations were normalized to reduce the numerical calculation error.Variable substitutions are applied as follows: where  0 and  0 are Earth radius and the ground gravity acceleration, respectively.Considering the thrust  and aerodynamic forces as a composition of forces, we define the normalized accelerations   and   as follows: We define The normalized equations of motion are obtained from (1), ( 4), (5), and (6) as follows: The mission of ascent is to send the vehicle to the required final state which is denoted as (  ,   ,   ).Define normalized-energy  as follows: From ( 8) we obtain The integration interval [ 0 ,   ] is fixed.

Optimal Feedback Control Problem for Ascent Guidance.
In this section, the optimal feedback control problem for optimal ascent guidance is addressed.This problem differs from trajectory optimization off-board in that the initial state of the optimal control problem is obtained from the navigation system.We denote the state and the guidance command of the trajectory as With the feedback vehicle state   obtained from navigation system, the optimal ascent guidance algorithm is used to generate new reference trajectory [() ()]  and output () from the following optimal control problem: where   is the normalized energy for current vehicle state   .

Optimal Ascent Guidance Algorithm
In [25,26], the optimal control problem was solved on-board in every guidance cycle.However, it is difficult to solve the nonlinear and nondifferentiable TPBVP on-board for airbreathing launch vehicles.In this section, we propose an optimal guidance algorithm.This guidance algorithm updates the reference trajectory [ ]  in every guidance cycle to deal with the unknown disturbance.For the current state   of the th guidance cycle, the optimal feedback control problem is transformed into a linear TPBVP problem using optimality condition linear approximation.The searching direction of the correction values   is obtained by solving the linear TPBVP.With the searching direction, the new reference trajectory is generated by linear search.For the th guidance cycle, the trajectory update is performed as follows: The initial reference trajectory [ 0 ()  0 ()]  is derived from the off-board from trajectory optimization by direct method.The flow chart of the algorithm is shown in Figure 2. Figure 3 shows the time sequence of algorithm.

Linear Approximation for the Optimality Condition.
In this section, we linearize the optimality condition for optimality trajectory updating.The optimal control problem (11) without inequality constraints is given as follows: The Hamiltonian of ( 13) is given as follows: where  is the costate vector.The state equations, costate equations, and optimality condition for the optimal control problem are given by ẋ =  (, ) , The initial and transversality conditions are given by We denote the trajectory correction variables as follows: The first-order Taylor expansion of (15) The following equations are established: Substituting (20) This equation is a linear TPBVP about the variables [Δ Δ ]  .The linear TPBVP is solved using the Simpson rules in Section 6.

Backtracking Line Search for the
Step Size.In this section, the step size   is determined by backtracking line search considering the solution of the linear TPBVP as the search direction.The penalty function for problem (13) where  is the penalty parameter satisfied  > max ||.We denote where [Δ Δ ]  is the solution of the linear TPBVP (21).
The minimizer of penalty function   (, ) corresponds to the solution of problem (11).Backtracking line search was used for large-scale nonlinear optimization problems in [31].The step size   is determined by backtracking line search that satisfies where  1 ∈ (0, 1),  2 ∈ (0, 1), and  +1 (  ,   ) is the variational of the functional  +1 () caused by variational   of  on   .Considering   as the solution of ( 21),  +1 (  ,   ) ≤ 0 is established.The new state and guidance command of the reference trajectory are generated as follows:

The Global Convergence of the Algorithm
In [32,33], the convergence of the off-board algorithm was analyzed.But the relationship between the global convergence and the guidance cycle  has not been discussed for onboard optimization in the previous literatures.In this section, we state and prove the global convergence of the on-board algorithm proposed in the preceding section.The solution sequence [    ]  generated by the algorithm will converge to the solution of the feedback optimal control problem.Unlike off-board trajectory optimization, the initial condition of the feedback optimal control problem depends on the current state of the vehicle in real time.For the th guidance cycle, the optimal feedback control problem is described as (13).The penalty function   for this feedback optimal control problem is given by (22).For the subsequent guidance cycle, the optimal feedback guidance problem is given by where Δ is the guidance cycle and  +1 is the new state of vehicle from navigation system at the time   + Δ.The penalty function for optimal problem (24) is given by We define the entire penalty function as follows: p (, ) =   (, ) The entire penalty function p (, ) includes the penalty function on the whole integration interval [ 0 ,   ].
Proof.From ( 22), ( 28), (29) and we can obtain that From (32), we obtain lim Thus, the solution sequence [    ]  generated by the algorithm will converge to the solution of the optimal control problem.

Smooth Approximation for Inequality Constraints
The nondifferentiable TPBVP from optimal control problem (11) with the inequality constraints brings difficulties for the proposed algorithm, which can only solve a smooth problem.Smooth approximations for nondifferentiable optimization problems have been studied in [5,6,34].In this section we introduce the smooth approximation method for the inequality constraints.If the inequality constraint function is about the state and the guidance command the Hamiltonian for the optimal control problem (11) with inequality constraints (39) is given by  =  (, ) +    (, ) +  max { (, ) , 0} .
However, the Hamiltonian (40) is a nondifferentiable function, which brings difficulty for the proposed algorithm.
For the smooth Hamiltonian approximation, a differentiable function is substituted for max{(, ), 0}.We denote the nondifferentiable function as follows: where () is the step function Considering (, ) as the approximation function of (), the differentiable approximation to max{0, } is obtained as To assess the capability of the proposed algorithm to deal with disturbance, the aerodynamic coefficients bias and the initial state error in the simulations are given by following cases: (1) Δ  = −0.1 , Δ  = +0.1 .
(2) Δ  = −0.1 , Δ  = +0.1 , Δ ini = +500 m.The simulation results of the optimal guidance algorithm are shown in Figures 4,5,6,7, and 8.The altitude, velocity, flight path angle, angle of attack, and dynamic pressure are, respectively, shown in Figures 4 to 8. As expected, the final state is achieved under aerodynamic bias and initial state error.Figure 8 shows that the state inequality constraint is satisfied.The dynamic pressure is less than 140000 Pa for different cases.
Table 2 shows the terminal error and the fuel cost of the optimal guidance algorithm.The terminal accuracy is high for different cases.Moreover, the fuel cost of the optimal guidance algorithm is lower than that of the tracking guidance algorithm.A greater initial state error will result in more fuel savings.In summary, with lower fuel cost the results of the proposed optimal guidance algorithm satisfies all the equality and inequality constraints.The simulation results show the great potential for the final flight experimental test.

Conclusion
In this paper, we present an optimal guidance algorithm for air-breathing launch vehicle based on optimal trajectory correction.The proposed guidance algorithm corrects the reference trajectory in every guidance cycle to satisfy the optimality condition of the optimal feedback control problem.By linearizing the optimality condition, the linear TPBVP is obtained for optimal trajectory correction.The solution of the linear TPBVP is derived using the Simpson rule.The new trajectory is generated by a linear search for the step size of the solution.Smooth Hamiltonian approximation is implemented to the inequality constraints.The sufficiency condition for the global convergence of the guidance algorithm is given in this paper.
Finally, simulations for 5 different cases of modeling and initial state error are presented.Compared with the tracking guidance method, the simulation results prove the low fuel cost and high precision of the proposed optimal guidance method.

Figure 1 :
Figure 1: The forces acting on the vehicle.

Figure 2 :
Figure 2: Flow chart of the optimal ascent guidance algorithm.

Figure 4 :Figure 5 :
Figure 4: The altitude of the simulation results.

Table 1 :
Initial and required final states of the vehicle.The simulations end once the final normalized energy   is reached.The optimal guidance algorithm updates the trajectory every guidance cycle, which is 0.3 s.The initial and required final states of the air-breathing vehicle are shown in Table1.The number of the discrete nodes  = 10.The dynamic pressure inequality constraint is given by