Research Article | Open Access
Integrated Guidance and Control for Hypersonic Morphing Missile Based on Variable Span Auxiliary Control
The morphing aircraft can improve the flight performance of hypersonic vehicles by satisfying the flight requirements of large airspace and large velocity field. In this paper, for the hypersonic variable span missile, the dynamic model and aerodynamic model are established on the variable span characteristic. The adaptive dynamic surface control back-stepping method is used to establish the integrated guidance and control (IGC) with terminal angular constraint in the dive phase of the hypersonic variable span missile. The span variety is used to assist the lift control to achieve fast and stable control for the centroid motion. The simulation results demonstrate that the feasibility and robustness of the IGC method of the hypersonic variable span missile is better than the invariable span.
In the dive phase, problems such as large Mach number variation range and large overload for a hypersonic vehicle are obvious. The translational and rotational aircraft dynamics are characterized by being fast time-varying, being nonlinear, having a strong coupling, and being uncertain . The traditional aircraft guidance and the control system mainly separate the control and guidance subsystems, without utilizing the coupling information between subsystems based on engineering experience or singular perturbation theory. The integrated guidance control (IGC) system can make full use of the coupling information between the subsystems to improve the performance of the entire system. For the attitude control of hypersonic vehicles, Wang et al. proposed a hierarchical predictive control method  and developed a rapid generation method of multitarget entry trajectory  that could solve the problem of effective attitude controller and reentry guidance for the general hypersonic vehicle model. Hou and Duan proposed an integrated design method for guidance and control based on an adaptive dynamic surface which avoids the differential explosion problem of the traditional back-stepping method . Wang et al. designed a method of the guidance control system based on banking turning control and dynamic surface control, and the uncertain term was estimated by the state observer and compensated by the control law . Zhao et al. proposed a method of IGC for hypersonic vehicles that satisfies the terminal angular constraint based on the full-coupling model . Then, Zhao et al. studied a method of IGC based on L2 gain interference consistency, which simplifies the structure and calculation of the IGC method . Wang et al. studied the guidance control problem of hypersonic vehicles with composite constraints in the dive phase. The integrated model based on the sliding mode dynamic surface can realize the three-dimensional angle constraint of landing velocity .
For a hypersonic waverider aircraft, the intelligent morphing technology can effectively solve the problem that aerodynamic performance deteriorates sharply when deviating from the design state [9, 10]. When intelligent morphing technology is applied to the flying missile for the battlefield environment and combat mission changes, the accuracy, combat effectiveness, and cost-effectiveness ratio can greatly improve . In 2004, Jae-Sung et al. studied the aerodynamic and aeroelastic properties of a variable span cruise missile similar to the shape of the Tomahawk and showed that the symmetrical increase in the span of the two wings can effectively reduce the drag and greatly improve the flight range . In the modeling of morphing aircraft, Seigler et al. established a dynamic model of large-scale morphing flight for flight control problems and discussed various control methods . Yang et al. developed dynamic modeling and dynamic response analysis of the variable sweep and variable span aircraft and proved that the combination of a suitable variant mode and variant velocity can reduce the workload of the flight control system . Yue et al. analyzed the longitudinal multibody dynamics of the Z-wing morphing aircraft and showed that the variant of aerodynamic characteristics is the main factor affecting the dynamic characteristics of the aircraft during the folding process . A joint simulation method based on Missile DATCOM and MATLAB proposed by Zhang Fair is applied to the study of variable wing dynamic characteristics of typical axisymmetric missiles . Nobleheart et al. designed a time-varying system for a single-neural network adaptive controller for morphing aircraft . In terms of the control of the morphing aircraft, Dong et al. studied the smooth-switching linear variable parameter (LPV) control problem for a class of variable span aircraft to ensure the stability and robustness of the aircraft system . Gandhi et al. proposed a control scheme that used model identification to realize morphing control and keep the aircraft stable . In terms of the ballistic design of the morphing missile, Huang et al. used the Radau pseudospectral method (RPM) to optimize the gliding trajectory of the variable sweep aircraft with the target of largest range . Wei-Ming et al. designed a trajectory-attitude dual-loop adaptive sliding mode controller for the variable sweep missile to ensure the stability of the ballistic tracking to the optimal scheme when the wing is scheduled to be actuated .
Most of the above researches were aimed at the IGC system of the hypersonic vehicle and the stability control of the morphing aircraft. There were few reports on the control research of the hypersonic morphing missile and how to use the span variant to assist the ballistic control. The research in this paper is aimed at the IGC for a class of hypersonic variable span missiles. Based on the adaptive dynamic surface method, an IGC method in the dive phase with terminal angular constraint is designed and the span variant is skillfully applied to the auxiliary control of the IGC to enhance the control accuracy and reduce the workload of the control system. The variable span auxiliary control strategy was hardly studied for hypersonic missiles before. The results of simulation of the new method proposed in the paper show that the total flight time and the miss distance of the variable span are less than invariable span shown. The variety of span could adjust the lift-to-drag ratio of the missile so that it is easier to realize the terminal angular constraint, and the variable span reduces the workload for the attitude control system. At the same time, the robustness of a variable span missile is stronger than that of an invariable span missile.
In Section 2 in the paper, the geometric model, aerodynamic model, and 6 DOF motion model of the hypersonic deformation missile are established. In Section 3, the IGC model in the dive phase of the hypersonic morphing missile is established and the variable extension control method adapts to the IGC method. Section 4 displays the numerical simulation results and analyses of the IGC method. Section 5 is the conclusion for the paper.
2. The Motion Model of Hypersonic Variable Span Missile
2.1. The Shape and Parameters of Hypersonic Variable Span Missile
The outline of the shape of the variable-span hypersonic missile selected in this paper is shown in Figure 1.
The shape parameters and morphing modes are shown in Figure 2. The lift of the morphing missile was provided by a pair of horizontal front wings, and the four-rear wing was a -shaped layout. The span variant of the wing is synchronously changed on both sides of the wings without differential changes. The synchronous or differential rotation of the four tails controls the rotational motion of the missile.
(1) Geometric Parameters of the Missile. is the length of the warhead. is the length of the total missile. is the diameter of the main body and bottom of the warhead
(2) Variable Span Geometry. is the distance from the front end of the wing root to the nose of the warhead. is the chord length of the wing root. is the chord length of the wing tip. is the span of the wing. is the sweep angle of the leading edge of the wing
(3) Geometric Parameters of the Tails. is the distance from the front end of the tail root to the nose of warhead. is the chord length of the tail root. is the chord length of the tail tip. is the span of the tail, and is the sweep angle of the leading edge of the tail
is the span of the invariant wing, and is the maximum span of the variable span wing. When is assumed as the span of the wing in a variable span state, the morphing rate of the wing span is defined as
The missile characteristic parameters are shown: in Table 1, the mass of the missile is , and the mass of the wing is and and m2. The moments of inertia are , , and . The reference area is . The longitudinal reference length is , and the lateral–directional reference length is .
2.2. Hypersonic Variable Span Missile Dynamic Model
2.2.1. Centroid Dynamic Model
Due to the small range of the dive phase, the Earth’s rotation can be ignored. The centroid motion equation of the variable span missile in the ballistic coordinate frame is 
In Eq. (2), is the acceleration vector in the ballistic coordinate frame. is the flight velocity, is the flight path angle, and is the heading angle. In Eq. (2), the component of gravitational acceleration in the ballistic coordinate frame is 
In Eq. (3), is the gravity constant of the Earth, is the radius of the Earth, and is the mode of the Earth’s radial diameter , which is expressed in the launch coordinate frame as . In Eq. (3), the expression of is 
In Eq. (4), and are the transformation matrixes of the velocity coordinate frame to the ballistic coordinate frame and the body coordinate frame, respectively. is the additional force due to the span variety and the expression is 
In Eq. (5), denotes the two sides of the wing, respectively. is the angular rates of the missile’s rational movement, and is the position vector of the wing mass center relative to the missile mass center. The expression of in the body coordinate frame is 
The wing is trapezoidal, and the span of the wing varies in the plane of the body coordinate frame. Based on the geometric knowledge, could be expressed in the body coordinate frame as
According to Eq. (7), the first and second derivative of with time is
In Eq. (9), is the dynamic pressure. is the atmospheric density which can be calculated according to the standard atmospheric model. are the lift coefficient, the drag coefficient, and the lateral force coefficient, respectively. is the reference area when . The missile reference area varies during the morphing process. However, for the convenience of calculations, Eq. (11) is regarded as invariant and the influence of the reference area variant caused by the span variant is classified into the equivalent aerodynamic coefficient as
The equivalent aerodynamic coefficient calculation model is in Section 2.3.
2.2.2. Rotational Dynamic Model
According to the literature, the rotational motion model of the variable span missile can be
In Eq. (13), is the transformation matrix of the launch coordinate frame to the body coordinate frame, and the other components of each variable are expressed in the body coordinate frame.
In Eq. (11), are the roll, yaw, and pitch moment, respectively, of the missile as
In (14), are the roll, yaw, and pitch moment coefficients, respectively. Similar to Eq. (11), the variant in the reference area caused by the span variant is converted into the equivalent aerodynamic torque coefficient as
The equivalent aerodynamic moment coefficient calculation model is in Section 2.3.
2.3. Aerodynamic Model of Hypersonic Variable Span Missile
Due to the lack of models and aerodynamic research of hypersonic variable span missiles, the mature hypersonic morphing aircraft is unavailable. This paper uses the Missile DATCOM software of the US Air Force Laboratory (AFL) to estimate the variable span missile aerodynamic model. The software is used to calculate the aerodynamic data under different combinations of morphing rate, Mach number, attack angle, side slip angle, and roll, yaw, and pitch fin deflection angles. Then, the model is identified by the least squares method. The aerodynamic model of the variable span missile is
In Eq. (16), the subscript of each coefficient represents the variable contained in the coefficient matrix. According to the model identified by the obtained aerodynamic data, the structural form of each variable in Eq. (17) is
In Eq. (17), is the coefficient matrix of different aerodynamic coefficients () relative to different variables (). is the coefficient matrix of relative to the morphing rate. For the designing of the IGC method, the aerodynamic load model needs to be processed as
In Eq. (18), . is the partial derivative vector of the lift coefficient for the attack angle. The uncertainty includes the influence of pitch fin deflections on the lift. ,, and are the partial derivatives of the roll moment coefficient for the roll fin deflections, the yaw moment coefficient for the yaw fin deflections, and the pitch moment coefficient for the pitch fin deflections, respectively. The uncertainty includes the influence of the quadratic pitch fin deflections on the pitching moment.
3. IGC Method in the Dive Phase with Variable Span Auxiliary Control
3.1. IGC Model in the Dive Phase
In this paper, the IGC design of the dive phase of the hypersonic variable span missile is studied. The fixed point on the ground is selected as the target, and the IGC model in the dive phase with the terminal angular constraint is established.
3.1.1. Relative Motion of Variable Span Missile and Target
As shown in Figure 3, the East–North–Up (ENU) coordinate frame is formed by a plane tangential to the Earth’s surface and is attached to the target. The East axis is labeled , the North axis is labeled , and the up axis is labeled . Define the line-of-sight (LOS) coordinate frame so that its origin is on the target where the -axis points to the vehicle, the -axis is in the horizontal plane of the target, and the -axis is determined using the right-hand rule. is the angle between the direction of velocity and the LOS, and is the longitudinal velocity azimuth in the dive plane. is the distance between the center of mass of the variable span missile and the target point. and are the angles of elevation and azimuth of LOS.
The relative dynamic model of the variable span missile and the target point by the line-of-sight angles are expressed as 
The relationship between the components in the LOS coordinate frame and the trajectory coordinate frame of the missile acceleration vector can be expressed as  where are the elements in the conversion matrix of the trajectory coordinate frame to the LOS coordinate frame. represents the row, and represents the column.
To study the effect of the span variant on the IGC method, the hypersonic missile adopts the BTT control strategy in the dive phase to show the influence of the span variant more clearly. Therefore, the lateral force can be neglected. The influence of the lateral force in Equation (2) is regarded as the uncertainty term. Substituting Eq. (18) into Eq. (2) yields  where the uncertain items and can be expressed as 
For missiles and fixed targets on the ground, the relative motion between them can be expressed as  then where is the terminal angle. Because of at the terminal time, it can be obtained as
In Eq. (26),
3.1.2. Rotational Motion Model
The rotational motion model by the attack, sideslip, and bank angle is 
Then, the rotational motion model that is available for control can be denoted as
In Eq. (29),
3.1.3. Processing of the Rotational Dynamic Model
When ignoring the Earth’s rotation, the launch coordinate frame is equivalent to the ground inertial system. By substituting Eq. (18) into Eq. (11), the simplified state equations with the input of fin deflections can be denoted as 
Then, the rotational dynamic model that is available for control can be denoted as
The variant rate of the three-axis moment of inertia caused by the variable span is difficult to calculate accurately, so it is regarded as an uncertainty in the Eq. (36).
The method of solved by will be introduced in Section 2.2.
3.2. The IGC Method in the Dive Phase with Variable Span Control
The IGC model in the dive phase guidance control with terminal angular constraint is shown in Eq. (37). The adaptive dynamic surface back-stepping control method is designed to make the system stable and make the output as close as possible to zero and make the system have strong robustness for uncertain factors.
3.2.1. Design of the Adaptive Dynamic Surface Back-Stepping IGC Method
Therefore, let us assume to make nonsingularity as in the whole dive phase. And is the bounded closed set in , so is a tight set.
The design steps of the adaptive dynamic surface back-stepping IGC method are as follows .
Step 1. Design the dynamic surface 
where is the angular error coefficient, of which the magnitude determines the weight of the terminal angular error in the dynamic surface.
The rate of change of the dynamic surface is designed according to the exponential approach law. where and are positive gain constants, and are the gain-of-saturation function, and and are the boundary layer thickness. is a saturation function which defined as  When the missile is far from the target point and is large, the rate of approaching law in (42) will be slow so that the missile has a small overload in the initial stage. When the missile approaches the target point and is small, the rate of approaching law in (42) increases so that does not diverge to improve the accuracy of hitting. The selection of the saturation function can effectively eliminate the chattering of the first virtual control variable.
Deriving the dynamic surface and then combining (42) can get the first virtual control input as In Eq. (44), the result of deriving the terminal angle is In Eq. (44), , , and .
According to , it can be translated as According to the aerodynamic model of the variable span missile, the structural form of the lift coefficient caused by the attack angle is It can be solved as In Eq. (48), take as the main factor affecting for lift, and is the previous moment attack angle.
Then, In order to exert the advantage of the variable span hypersonic missile, the morphing rate and attack angle of the wing are coordinated controls. The roadmap of the IGC method for a hypersonic morphing missile based on variable span auxiliary control is shown in Figure 4, while the IGC method with invariable span is shown in Figure 5.
The principle of coordinated control shown in Figure 4 is to translate the change of the lift coefficient into a change of the attack angle and the morphing rate, so that the span variant can withstand the change demand of the lift command and reduce the change demand of the attack angle control. While in the traditional IGC method with invariable span, the change demand of the lift command is only reached by the attack angle. It would improve the stability and velocity of missile IGC mission. That is, Solve Eq. (50) to get In Eq. (51), take as the main factor affecting for lift; is the previous moment value of each variable.
According to the model of the aerodynamic coefficient of the missile, where is the coefficient matrix of with respect to the morphing rate, and the expression is Let us substitute Eq. (53) into (3.35) and deriving on both sides to get Combining Eq. (51) and Eq. (54), the variation law of the morphing rate can be obtained as In order to prevent differential explosion, it will pass through the first-order filter as
Step 2. Design the dynamic surface
where , is the output value of the following first-order filter as
where is the time constant of the filter.
Design the second virtual control input as where , , and is an estimate of the upper bound of the uncertainty , which can be obtained by the following adaptive law where and are positive constants.
Step 3. Design the dynamic surface
Similarly, is the output value of the following first-order filter as
where is the time constant of the filter. The control input of the system is designed as
where and is an estimate of the upper bound of the uncertainty , which can be obtained by the following adaptive law as
where and are positive constants.
In summary, the adaptive dynamic surface back-stepping IGC method for variable span control of the system (3.20) can be expressed as
3.2.2. Stability Analysis
The stability of the closed-loop system obtained using the control law in Eq. (65) is analyzed below.
Define the filter error and , respectively, as
Then, the derivative of the filtering error is
We define the upper bound estimation error of the norm as 
Then, its derivative is
can be defined as
The derivative of is
The derivative of is
The derivative of is
In summary, the entire closed-loop system with the control law can be composed of the dynamic surface vector, the filtered error vector, and the upper bound estimation error of the uncertainty as
The Lyapunov function is defined as where , , and .
The derivatives of the Lyapunov function (80) with respect to time are
According to Young’s inequality, we can get 
Since is a diagonal matrix, the following inequality is established. where is the minimum value of the diagonal elements in each gain matrix . Since the uncertainty satisfies the following inequality, is a positive constant. So, the following inequality is established : and