Integrated Guidance and Control of Interceptor Missile Based on Asymmetric Barrier Lyapunov Function
In this study, a novel integrated guidance and control (IGC) algorithm based on an IGC method and the asymmetric barrier Lyapunov function is designed; this algorithm is designed for the interceptor missile which uses a direct-force/aerodynamic-force control scheme. First, by considering the coupling between the pitch and the yaw channels of the interceptor missile, an IGC model of these channels is established, and a time-varying gain extended state observer (TVGESO) is designed to estimate unknown interferences in the model. Second, by considering the system output constraint problem, an asymmetric barrier Lyapunov function and a dynamic surface sliding-mode control method are employed to design the control law of the pitch and yaw channels to obtain the desired control moments. Finally, in light of redundancy in such actuators as aerodynamic rudders and jet devices, a dynamic control allocation algorithm is designed to assign the desired control moments to the actuators. Moreover, the results of simulations show that the IGC algorithm based on the asymmetric barrier Lyapunov function for the interceptor missile allows the outputs to meet the constraints and improves the stability of the control system of the interceptor missile.
The interceptor missile plays an important role in modern anti-missile systems. Given the rapid development of hypersonic aircraft, demands on accurate guidance and control of the interceptor missile has become increasingly stringent. To cope with high-speed aircraft with a strong penetration capability, the interceptor missile generally adopts a direct-force/aerodynamic-force control scheme that can accelerate the speed of command response. The guidance and control system of the interceptor missile is a highly dynamic and multivariate system of strong coupling, rapid temporal change, and uncertainty, thereby making it critical to the examination of the guidance and control of interceptor missile at high speed with strict control requirements.
Integrated guidance and control (IGC) refers to using the relative interceptor–target motion information and the dynamic information of the interceptor missile to generate a control force that drives it to strike the target. IGC allows for a rational allocation of interceptor missile control capability to not only maintain the interceptor missile’s flight attitude but also improve the precision of its guidance [1, 2]. The authors in [3, 4] explored the IGC control law by using high-order sliding-mode control methods and backstepping control algorithms. On the basis of backstepping control algorithms, the authors in [5, 6] considered saturation factors and introduced dynamic models of actuators models to design algorithms for longitudinal and anti-saturation IGC for aircraft. An adaptive sliding-mode control method was used to implement the design of an IGC system on the pitch plane to configure missiles to attack ground targets , but the problem of target mobility was not considered. Based on a backstepping method, the missile control loop was regarded as a second-order element in the study , which introduced a first-order integral filter to estimate the derivative of the input to the virtual control and design the control law. Based on a three-dimensional (3D) ICG model, a robust adaptive backstepping method was implemented to design an IGC algorithm for a missile . Methods of IGC design have been widely used in guidance and control systems of aircraft , missiles [11–13], and unmanned aerial vehicles [14, 15]. In recent years, researchers have combined this method with modern control theories, such as dynamic surface control , optimal control , and predictive control , to generate methods of IGC design for aircrafts. However, most prevalent IGC design methods do not consider the coupling relationship between the pitch and the yaw channels. Moreover, to improve the stability of the guidance process, constraints concerning the angular velocities of the line of sight during the guidance of the interceptor missile should be considered.
Nonlinearity commonly exists in the IGC system of the interceptor missile. In recent years, with practical engineering problems’ increasing demand on the control performance, there has been considerable progress in the development of nonlinear control theory, especially in adaptive control , neural network control , fuzzy control , etc. All of these have laid a solid foundation for the in-depth research of nonlinear control theory. However, in actual engineering applications, the use of nonlinear systems is always subject to input, output, and state constraints, among others, and violation of these constraints can result in the control system’s downgraded performance. Therefore, it has now become an important research direction, when constructing the control system to consider the effect of these constraints and to properly handle them in the controller design process. The authors in  proposed an adaptive neural network constrained control algorithm for single-input/single-output nonlinear stochastic switching systems; this algorithm constructed traditional Lyapunov function to handle constraint control, which achieved good results. As barrier Lyapunov function does not require an exact solution of the system, the constrained control method based on barrier Lyapunov function has been widely used in state constraint and output constraint problems in recent years. Barrier Lyapunov function is a special type of continuous function, unlike traditional Lyapunov function which is radially unbounded, in barrier Lyapunov function, when the parameters approach the limit value, the function value will tend to infinity to ensure that the control system satisfies the constraints . Barrier Lyapunov function can be utilized to satisfy the constraints of both symmetric and asymmetric constraint controls, even when the constraint is a time-varying asymmetric one. According to the authors in , barrier Lyapunov function has been used to solve constraint problems in a hybrid PDE-ODE system that describes a nonuniform gantry crane system. The authors in  proposed an adaptive fuzzy neural network control method using impedance learning for a constrained robot system based on barrier Lyapunov function. According to the authors in [26, 27], barrier Lyapunov functions have been used to solve constraint problems in nonlinear and uncertain systems and to expand the definitions of the constraints. Methods based on these functions can effectively solve problems of symmetrically and asymmetrically constrained control. The authors in [28, 29] have expanded output control constraints to include time-varying outputs while relaxing the limitations on the initial values of control systems. By combining barrier Lyapunov functions with dynamic surface control technologies, some studies [30, 31] have proposed barrier Lyapunov function-based methods suitable for constrained dynamic surface control to solve the computational inflation problem caused by backstepping control. Researchers subsequently applied this method to brushless DC motors , plane braking systems , and hypersonic aircraft [34, 35] to achieve satisfactory results in terms of constrained control. However, few studies have investigated the application of this method to interceptor missile control. Design methods based on barrier Lyapunov functions are advantageous because they can solve the output constraint problem of interceptor missile guidance control systems and improve their stability.
In view of the above analysis, to aim at the interceptor missile which uses a direct-force/aerodynamic-force control scheme, and to consider the coupling relationship between the pitch and the yaw channels as well as the constraints on the system’s output, this study proposes an IGC algorithm based on the asymmetric barrier Lyapunov function. First, a time-varying gain extended state observer (TVGESO) is designed to estimate interferences in the system. Second, an asymmetric barrier Lyapunov function and a dynamic surface sliding-mode control method, respectively, are used to design control laws for the interceptor missile to obtain the desired moments. Finally, a dynamic control allocation algorithm is designed to allocate the desired control moments. The results of simulations show that the proposed algorithm enables the outputs to meet the constraints and improves the stability of the interceptor missile control systems.
2. IGC Model of Interceptor Missile
The relationship of relative motion between the interceptor missile and its target in 3D space is shown in Figure 1.
In the figure, refers to an inertial coordinate system and refers to a line-of-sight coordinate system, respectively; and refer to the interceptor missile and the target, respectively; and are the vertical and horizontal angles of the line of sight with the interceptor missile and the target, respectively, and denotes the relative distance between the interceptor missile and the target. The model of the relative motion of the interceptor missile and target is as follows: where and denote the vertical and horizontal angular velocities of the line of sight with the interceptor missile and the target, respectively; and denote the longitudinal and lateral accelerations of the interceptor missile, respectively; and and denote the longitudinal and lateral accelerations of the target, respectively.
The interceptor missile uses a direct-force/aerodynamic-force control scheme. Assuming that the direct force is adjustable and continuous, the angles of deflection of the rudder equivalent to the direct force in the pitch and the yaw channels are, respectively, defined as where and denote the thrust generated by the jet device, and is the maximum steady-state thrust of the jet devices.
In light of the coupling relationship between the pitch and yaw channels of the interceptor missile, its dynamic model is expressed as follows: where is the reference area of the interceptor missile; is the dynamic pressure; is the speed of the interceptor missile; is its reference length; is the average distance between the jet device and its center of mass; is the mass of the interceptor missile; and denote the attack angle and the sideslip angle, respectively; and denote the angular velocities of the pitch and the yaw, respectively; and denote the angles of deflection of the aerodynamic rudder; and denote the amplification factors of moment (used to describe the effect of the mutual interference between lateral jets and incoming flow on the aerodynamic moment of the interceptor missile); , , , and refer to the disturbances and uncertain interferences at each link of the system; and refer to the moments of inertia; , , , , , , , and refer to the relevant aerodynamic forces and coefficients of moment; and and refer to the vertical and lateral overloads of the interceptor missile, respectively.
Assuming that the angle of the line of sight of the interceptor missile in the terminal guidance stage changes slightly and that the angle of line of sight and direction of velocity of the interceptor missile are relatively small, let and . According to Equations (1)–(4), by defining , , , , , , , and , one can have the nonlinear IGC model in the pitch channel for the interceptor missile: where , , , , , , , , and , with representing the moment generated jointly by both the aerodynamic rudders and the jet devices in the pitch channels.
Similarly, the nonlinear IGC model for the interceptor missile in its yaw channel is as follows: where , , , , , , , , and , with representing the moment generated jointly by both the aerodynamic rudders and the jet devices in the yaw channel.
3. Design of TVGESO
A TVGESO can estimate nonlinear uncertainties in the model of the system and feed the estimates back into the control system for compensation. To eliminate the effects of unknown uncertain interferences , , , , , and in the system models in Equations (5) and (6) on the control system of the interceptor missile, a TVGESO is designed to estimate these interferences.
By defining , , and , one obtains
Considering the system in Equation (5) and Equation (7), one can design the following TVGESO to estimate acceleration of the target: where ; and , respectively, are the estimated values of and ; and are the estimated errors; and and are the time-varying gain coefficients designed for the state observer. They are defined as and , respectively. Function is defined as where is the adaptive coefficient and is greater than zero. As indicated in the literature , appropriate values of the coefficient can ensure that the error system of the TVGESO is stable for a limited time.
Similarly, by estimating interferences and of the angle-of-attack loop and the pitch angular velocity loop, respectively, in Equation (5), one can obtain the following: where and , the interferences and are estimated as and , respectively, and the estimation errors for them are denoted by and , respectively.
4. Design of the Dynamic Surface Sliding-Mode Control Law Based on Asymmetric Barrier Lyapunov Function
Let us define as an open region containing the origin and the barrier Lyapunov function as a scalar function defined in for the system . It also has the following characteristics: (1) smooth and positive definite, (2) has a first-order continuous partial derivative at each point in , (3) tends to infinity when approaches the edge of , and (4) satisfies the expression for if , where .
Assumption 2. For any , there exist constants and that satisfy and , with their derivatives satisfying , , , and .
Assumption 3. For any 和 , there exist functions and as well as positive constants and satisfying and , such that they make the system track command , and its time derivative satisfies and as well as for . There exists a continuous set satisfying .
Given that the IGC model of interceptor missile is a mismatching and uncertain system, to enable the guidance and control system to accurately pursue the target, and not violating the constraints on the control system, the control law allows using a dynamic surface sliding mode algorithm based on the asymmetric barrier Lyapunov function. It can enable the control system to pursue the target highly precisely, meanwhile ensuring that the closed-loop system is consistent and ultimately bounded, and the tracking error converges to a small set.
4.1. Design of Control Law for Pitch Channel
For the system in Equation (5), define as the system’s track command signal. (1)Define the first dynamic error surface:
Taking the derivative of , one obtains the error dynamic equation:
Because the backstepping method does not have a perfect solution to the expansion of items and the problems caused by the expansion of items in the derivation process of the virtual control, this shortcoming is particularly prominent in the higher-order system. By using the dynamic surface control method and using the first-order filter to calculate the derivative of the virtual control, the expansion of the differential items can be eliminated and the controller and parameters can be designed simply . Introduce virtual control variables and . To avoid complicated calculations of the expansion of the number of items during the derivation of the virtual control variables, the virtual control variable before filtering is passed through a first-order low-pass filter to become the virtual control variable :
In the above expression, is the filter’s time constant, , and the filtering error is defined as .
Considering boundary layer errors of the dynamic surface, one can construct the following asymmetric barrier Lyapunov function: where , , represents a natural logarithm, and and represent output constraints.
Given the independence characteristic of the output constraints and , the tracking error constraints and can be designed independently. When constraints and are constant, and , the output constrained control can be extended to a static asymmetric constraint, whereas the output constraint becomes a symmetric constraint when . This means that the initial output can be changed depending on the setting of the constraint. It is evident that the asymmetric barrier Lyapunov function relaxes the constraint on the initial condition of the output.
As shown by Equation (16), the expression simplifies to if and only if and simultaneously. Therefore, is a positive-definite function in the range . Moreover, given , is a piecewise continuously differentiable function in the ranges and . Therefore, is a valid Lyapunov function that can ensure that the system’s output error is constrained in the ranges and .
Substituting the estimated values of TVGESO in Equation (9), one can design the virtual control variable for the first dynamic surface as shown in Equation (18): where , , and , and the control coefficient is defined as
According to Equation (15), the derivative of the virtual control variable for the error surface after filtering is (2)Define the second dynamic error surface as
By taking the derivation of , one can obtain the equation for the dynamic error as follows:
Introduce two virtual control variables and . By passing the virtual control variable through a first-order low-pass filter, one obtains the virtual control variable :
In Equation (23), denotes the filter time constant, , and the filtering error is defined as .
Construct the following Lyapunov function:
According to Equation (23), the derivative of the virtual control variable for the error surface after filtering is (3)Define the third dynamic error surface:
By taking the derivative of , one can obtain the equation of error dynamic as follows:
Construct the following Lyapunov function:
4.2. Stability Analysis of Control Law for Pitch Channel
where and are functions. Suppose and . If the inequality, is valid in the domain , and and are positive constants, then is still in the set for .
Lemma 1 (see ). For any two functions and , suppose and to be open sets. For system with as state, function is piecewise continuous with respect to , and satisfies the Lipschitz condition in . If there exist two functions and that are continuous and positively definite in their domains, respectively, the following two expressions are valid when or :
Theorem 1. For the closed-loop system Equation (5), if the virtual control variables and the control law satisfy Equations (18), (25), and (30), then for any set to which the initial condition belongs, there always exists a sufficiently large set that can prevent the system’s output constraints from being violated. Moreover, by selecting appropriate design parameters, it is possible to bound all closed-loop signals of the closed-loop system Equation (5) and make the output error converge in the neighborhood of the origin.
Proof 1. According to Equations (16), (24), and (29), one can construct a Lyapunov function that is a combination of asymmetric barrier Lyapunov functions and traditional Lyapunov functions as follows:
Define the estimated error of the TVGESO system to satisfy Equation (35): where , , and are positive constants.
It can be seen that the filtering errors are
By taking derivatives of and , one obtains the dynamic filtering errors:
It is clear that the variables in the system model and their derivatives are all bounded. If there exist continuous functions and that satisfy and , the variables and satisfy
For convenience of description, define the following parameters:
Then, Equation (48) can be rewritten as
In the set , one has
Then, Equation (50) can be further rewritten as
Define positive-definite matrix as
By selecting , where is a positive constant and denotes the minimum eigenvalue of matrix , one has
Define two sets and that satisfy and . From , one has
Therefore, set is an invariant set. According to Lemma 1, it can be seen that given , the following expression is valid for :
Therefore, it can be concluded that for any initial compact set defined by , there is always a sufficiently large compact set to make for .
Define parameters according to Equation (48) to satisfy the following rules:
Multiply both sides of Equation (54) by and arrange the terms to obtain the following expression:
Therefore, it is clear that by designing parameters , , , and and parameters and , it is possible to ensure that all closed-loop signals of the system are bounded. Opting to increase , , , and and decrease and can ensure that is sufficiently large to make the filtering errors and the error surface small enough to control accuracy. Q.E.D.
Remark 1. Theoretically, larger values of the designed parameters , , , and ; smaller ones of and ; the final boundary of the resulting error surfaces , , and ; and filtering errors and indicate an increase in the precision of control. However, in practice, very large values of , , , and , and very small ones of and can easily lead to input saturation in interceptor missile control systems, which triggers the saturation nonlinearity of the system such that the required overloads of the interceptor missile are beyond the available capacity, resulting in a reduction in the system’s control performance. Moreover, given the physical limitations of the low-pass filter, and should not be chosen to be arbitrarily small. Therefore, the parameters of the control algorithm should be properly selected in light of practical considerations.
4.3. Design of Control Law for Yaw Channel
Based on the design of the control law for the pitch channel, one can substitute , , and —the estimated interferences of the TVGESO—into the system of Equation (6) and define as the system’s track command sign for the following form of control law for the yaw channel: where , , , , , , , , , , , and . The parameters , , and are error surfaces; and are filter time constants, and ; and are virtual control variables of the system Equation (6); and and , respectively, are virtual control variables after filtering.
According to the stability analysis method in Equations (33)–(59), it can be proven that the closed-loop system of Equation (6) is stable and the system’s control precision can be achieved by designing appropriate parameters.
5. Dynamic Control Allocation Algorithm
Based on Equations (30) and (60), one can obtain the control moments jointly generated by the aerodynamic rudders and the jet devices using control inputs and as the desired control moments. Therefore, it is necessary to use a dynamic control allocation technique to distribute the desired control moments to the actuators.
Define virtual control moment as where , ,
, , , and .
Given that actuators are subject to physical constraints, such as structural and load-related constraints, the range of deflection and speed are limited. Define the control moment as in the feasible range . To achieve stable control performance, define the rate of change of the control moment to satisfy , where and are the minimum and maximum positional constraints of the actuators, respectively, with and being, respectively, the minimum and maximum speed constraints. Using to denote sampling time, one can rewrite the feasible range of the actuators as where and .
To achieve stable and smooth actuator trajectories to suppress the impacts of noise and interference on the controller, a dynamic control allocation algorithm is designed to solve the allocation problem between the aerodynamic rudders and the jet devices.
The dynamic control allocation problem can be expressed as the following hybrid optimization problem: where , , , , , and .
represents a norm defined as , denotes the desired control command of the actuators, denotes the minimum energy principle, denotes the minimum error principle for the desired control commands of the actuators, and denotes the minimum transition rate principle for the relative sampling time of the expected control commands.
Define diagonal matrices , , and with corresponding dimensions where
and with , , , , , , , , and the elements of the corresponding vectors .