Research on Sliding Mode Method about Three-Dimensional Integrated Guidance and Control for Airto-Ground Missile

1.Henan University of Science and Technology – School of Information Engineering – Luoyang/Henan – China. 2.Henan University of Science and Technology – Henan Key Laboratory of Robot and Intelligent Systems – Luoyang/Henan – China. *Correspondence author: lymjw@163.com Received: Dec. 1, 2017 | Accepted: Mar. 5, 2018 Section Editor: Luiz Martins-Filho ABSTRACT: Based on adaptive sliding mode-control and back-stepping design method, an integrated guidance and control method with less calculation is proposed, which is designed for air-to-ground missile during the terminal course in threedimensional space. The model of the control system with nonlinear and coupling is simplifi ed, then the integrated guidance and control model in pitch and yaw channel is established. The coupling terms and modeling error between channels is considered as unknown bounded disturbance. An extended state observer is developed to estimate and compensate the unknown disturbance. In the design process, the block dynamic surface method is adopted, and the fi rst order low pass fi lter is introduced to avoid the problem of differential explosion present in the traditional back-stepping design method during the process of differentiating virtual control variable. The Lyapunov stability theory is used to prove the stability of the system. Finally, in the case of nominal and positive and negative perturbations of model parameters, the simulation experiments are carried outto verify the effectiveness of the proposed IGC algorithm.


INTRODUCTION
Th e traditional design method of the missile guidance and control system is based on the idea of separation; the system is divided into fast loop (control loop) and slow loop (guidance loop).First the guidance loop is designed and the desired overload is obtained.Aft er, in order to track the desired overload, the control loop is designed.During the terminal course, with the distance between the missile and the target becoming closer, the frequency of the guidance loop becomes faster, the coupling between the two loops becomes larger, thence it is diffi cult to design each subsystem separately, which always leads to a larger miss distance.At the moment, in order to achieve comprehensive index requirements, such as the angle changes slowly, the miss distance is as small as possible and the trajectory of the missile is relatively smooth.
It is necessary to exploit the synergistic relationship of guidance and control system.Hence such a design approach usually leads to excessive design iterations.As a result, the idea of IGC method is not to distinguish the guidance loop and control loop, taking into account the coupling relationship between the guidance loop and control loop as a whole (Shima et al. 2006;

MOTION MODEL
The three-dimensional motion model of air-to-ground missile has features with complex nonlinear, strong coupling and parameter uncertain, which makes the design of the control system difficult.The model needed to be simplified in the actual design; generally the missile's three-dimensional guidance problem is divided into pitch plane and yaw plane.In this paper, we make the following assumptions about the integration model of air-to-ground missile:

•
The rolling channel of missile is relatively stable, ignoring the missile's rolling motion.

•
For one channel, the effect of coupling with the rest of the channels is considered as unknown bounded uncertainties.

•
In the terminal course, the missile has no thrust, the speed of the missile and the target do not change.Based on the above assumptions and considering Hou (2011), the pitch and yaw channel IGC model of the missile is established as follows.In Fig. 1, M and T are the position of the missile and the target, where V mp and V tp are the speed of the missile and the target in the pitch plane, the θ and θ tp respectively represent the flight path angle of the missile and target, the normal acceleration of the missile and the target in the pitch plane are respectively represented by a m and a tp , q p is the azimuth angle of the line-of-sight, R p is the distance of missile and target in the pitch plane.Define the angle q p , angle θ tp and angle θ above the baseline for positive and vice versa (Eq.1).In Fig. 1, M and T are the position of the missile and the target, where Vmp and Vtp are t speed of the missile and the target in the pitch plane, the θ and θtp respectively represent the flig path angle of the missile and target, the normal acceleration of the missile and the target in the pit plane are respectively represented by am and atp, qp is the azimuth angle of the line-of-sight, Rp the distance of missile and target in the pitch plane.Define the angle qp, angle θtp and angle θ abo the baseline for positive and vice versa (Eq.1). (1 where Vqp is the line of sight rate, perpendicular to the line of missile-target.Equation 2 can obtained: (2) where, ∆qp is an unknown disturbance.( where Vqp is the line of sight rate, perpendicular to the line of missile-target.Equation 2 can b obtained: (2) where, ∆qp is an unknown disturbance.

T x o
q q tp p cos(q ) cos(q ) sin(q ) sin(q ) In this paper the following dynamic model of missile in pitch channel is adopted (Eq.(Shima et al. 2006). ( where α is the angle of attack, ωz is the a is the pitch rate, m is the mass of the missile, G is gravity of the missile, q is dynamic pressure, S is the feature area of the missile, L is the refere length of the missile, Iz is the moment of inertia around the pitch axis, δz is the elevator deflection where V qp is the line of sight rate, perpendicular to the line of missile-target.Equation 2 can be obtained: where, ∆q p is an unknown disturbance. In this paper the following dynamic model of missile in pitch channel is adopted (Eq. 3) (Shima et al. 2006).
where α is the angle of attack, ω z is the a is the pitch rate, m is the mass of the missile, G is the gravity of the missile, q is dynamic pressure, S is the feature area of the missile, L is the reference length of the missile, I z is the moment of inertia around the pitch axis, δ z is the elevator deflection, ϑ is the pitch angle, θ is the flight path angle, C y α is the rise lift coefficient corresponding to The normal acceleration of the missile in pitch channel is as follows (Eq.4): missile attack angle, mz ωz , mz α and mz δz respectively represent the pitch moment coefficient corresponding to ωz, α and δz.The normal acceleration of the missile in pitch channel is as follows (Eq.4): (4) According to the above analysis, the IGC model of the missile can be obtained as follows (Eq.5): (5)

cos( )
According to the above analysis, the IGC model of the missile can be obtained as follow (Eq.5): (5)
The IGC model in pitch channel can be simplified as follows (Eq.6): where:

Motion model in yaw
Similarly, the yaw channel IGC model is established according to the relative motion in the yaw channel and the motion equation of the missile body.
The normal acceleration of the missile in yaw channel can be described as follows (Eq.7): (7) The IGC model in yaw channel is established as follows (Eq.8): The IGC model in pitch channel can be simplified as follows (Eq.6): (6) where:

Motion model in yaw
Similarly, the yaw channel IGC model is established according to the relative motion in the yaw channel and the motion equation of the missile body.
The IGC model in pitch channel can be simplified as follows (Eq.6): (6) where:

Motion model in yaw
Similarly, the yaw channel IGC model is established according to the relative motion in th yaw channel and the motion equation of the missile body.
The IGC model in pitch channel can be simplified as follows (Eq.6): where:

Motion model in yaw
Similarly, the yaw channel IGC model is established according to the relative motion in the yaw channel and the motion equation of the missile body.
The normal acceleration of the missile in yaw channel can be described as follows (Eq.7): (7) The IGC model in yaw channel is established as follows (Eq.8): The IGC model in pitch channel can be simplified as follows (Eq.6): where:

Motion model in yaw
Similarly, the yaw channel IGC model is established according to the relative motion in the yaw channel and the motion equation of the missile body.
The normal acceleration of the missile in yaw channel can be described as follows (Eq.7): (7) The IGC model in yaw channel is established as follows (Eq.8): (5) According to the above analysis, the IGC model of the missile can be obtained as follows (Eq.5): where ∆ 1, ∆ 2, ∆ 3 are unknown bounded uncertainties.
The IGC model in pitch channel can be simplified as follows (Eq.6): where:

MOTION MODEL IN YAW
Similarly, the yaw channel IGC model is established according to the relative motion in the yaw channel and the motion equation of the missile body.
The normal acceleration of the missile in yaw channel can be described as follows (Eq.7): The IGC model in yaw channel is established as follows (Eq.8): xx/xx 05/15 where ∆ 4 , ∆ 5 , ∆ 6 are unknown bounded uncertainties, R y is the projection of distance between missile and target in yaw plane, v y is the projection of missile speed V in the yaw plane, β is the sideslip angle of the missile, ω y is yaw rate, I y is the moment of inertia around the yaw axis, δ y is the rudder deflection, C z β is the lateral force coefficient which corresponding to the missile sideslip angle β, m y β , m y ωy , and m y δy respectively represent the yaw moment coefficient corresponding to β, ω y and δ y .
The IGC model in yaw channel can be rewritten as follows (Eq.9): sideslip angle of the missile, ωy is yaw rate, Iy is the moment of inertia around the yaw axis, δy is the rudder deflection, Cz β is the lateral force coefficient which corresponding to the missile sideslip angle β, my β , my ωy , and my δy respectively represent the yaw moment coefficient corresponding to β, ωy and δy.

EXTENDED STATE OBSERVER
The uncertainties of the system with nonlinear can be estimated by the extended state observer and compensate in the control system by feedback.In this paper, coupling between pitch and yaw channel, system parameters perturbation and modeling error are seen as unknown bounded uncertainties, which are estimated by extended state observers and treated as extend states.(Xia et al. 2011) The design principle of the extended state observer is illustrated by taking the first closed-loop subsystem of Eq. 6 as an example. (10) In this system, d1 is unknown uncertainty.The system (Eq.10) can be described as follows where: , x5 = β, x6 = ωy, uy = δy, , , d5 = Δ5, d6 = Δ6, , .

EXTENDED STATE OBSERVER
The uncertainties of the system with nonlinear can be estimated by the extended state observer and compensate in the control system by feedback.In this paper, coupling between pitch and yaw channel, system parameters perturbation and modeling error are seen as unknown bounded uncertainties, which are estimated by extended state observers and treated as extend states.(Xia et al. 2011) The design principle of the extended state observer is illustrated by taking the first closed-loop subsystem of Eq. 6 as an example. (10) In this system, d1 is unknown uncertainty.The system (Eq.10) can be described as follows where x1d is the extend state of d1, g1(t) is the differential of extend state.The second-order extend state observer is designed as follows (Eq.12): (12) where Z11 and Z12 respectively represent the estimated values of state value x1 and disturbance value d1, E11 is state value error, γ11 and γ12 are the gain of ESO, µ1 and δ1 are the parameters of the ESO.
where x1d is the extend state of d1, g1(t) is the differential of extend state.The second-order exte state observer is designed as follows (Eq.12): (12) where Z11 and Z12 respectively represent the estimated values of state value x1 and disturbance val d1, E11 is state value error, γ11 and γ12 are the gain of ESO, µ1 and δ1 are the parameters of the ES The function fal is defined as follows (Eq.13): (13) where x1d is the extend state of d1, g1(t) is the differential of extend state.The second-order extend state observer is designed as follows (Eq.12): (12) where Z11 and Z12 respectively represent the estimated values of state value x1 and disturbance value d1, E11 is state value error, γ11 and γ12 are the gain of ESO, µ1 and δ1 are the parameters of the ESO.
The function fal is defined as follows (Eq.13): (13) The observation error of the ESO is defined as follows (Eq.14): rudder deflection, Cz β is the lateral force coefficient which corresponding to the missile sideslip angle β, my β , my ωy , and my δy respectively represent the yaw moment coefficient corresponding to β, ωy and δy.

EXTENDED STATE OBSERVER
The uncertainties of the system with nonlinear can be estimated by the extended state observer and compensate in the control system by feedback.In this paper, coupling between pitch and yaw channel, system parameters perturbation and modeling error are seen as unknown bounded uncertainties, which are estimated by extended state observers and treated as extend states.(Xia et al. 2011) The design principle of the extended state observer is illustrated by taking the first closed-loop subsystem of Eq. 6 as an example. (10) In this system, d1 is unknown uncertainty.The system (Eq.10) can be described as follows ( ) rudder deflection, Cz β is the lateral force coefficient which corresponding to the missile sideslip angle β, my β , my ωy , and my δy respectively represent the yaw moment coefficient corresponding to β, ωy and δy.

EXTENDED STATE OBSERVER
The uncertainties of the system with nonlinear can be estimated by the extended state observer and compensate in the control system by feedback.In this paper, coupling between pitch and yaw channel, system parameters perturbation and modeling error are seen as unknown bounded uncertainties, which are estimated by extended state observers and treated as extend states.(Xia et al. 2011) The design principle of the extended state observer is illustrated by taking the first closed-loop subsystem of Eq. 6 as an example. (10) In this system, d1 is unknown uncertainty.The system (Eq.10) can be described as follows ( ) where:

EXTENDED STATE OBSERVER
The uncertainties of the system with nonlinear can be estimated by the extended state observer and compensate in the control system by feedback.In this paper, coupling between pitch and yaw channel, system parameters perturbation and modeling error are seen as unknown bounded uncertainties, which are estimated by extended state observers and treated as extend states.(Xia et al. 2011) The design principle of the extended state observer is illustrated by taking the first closed-loop subsystem of Eq. 6 as an example.
In this system, d 1 is unknown uncertainty.The system (Eq.10) can be described as follows (Eq.11): where x 1d is the extend state of d 1 , g 1 (t) is the differential of extend state.The second-order extend state observer is designed as follows (Eq.12): where Z 11 and Z 12 respectively represent the estimated values of state value x 1 and disturbance value d 1 , E 11 is state value error, γ 11 and γ 12 are the gain of ESO, μ 1 and δ 1 are the parameters of the ESO.The function fal is defined as follows (Eq.13): The observation error of the ESO is defined as follows (Eq.14): ( 14) The dynamic equation of observation error can be described as follows (Eq.15): The observation error of the ESO is defined as follows (Eq.14): ( 14) The dynamic equation of observation error can be described as follows (Eq.15): (15)  , , ) According to a large number of numerical simulations experience, the selection of the observer parameters obey the following rules.For nonlinear function fal (E11, µ1, δ1), choosing µ1 = 1/2 n-1 , δ1 = h, where n is the order of the observer, h is the integration step time of the numerical simulations.The selection of parameters γ11 and γ12 is related to h, usually the value of γ are chosen as follows: γ11 = 1/h, γ11 = 1/3h 2 .Based on the above design principle of ESO, ESO is designed for the remaining five subsystems of the closed loop system (Eq.6) and the closed loop system (Eq.9), and the state value and the disturbance value are estimated.Finally, we get the estimate values Zi1, Zi2 respectively about system state value, Ei1, Ei2 is the observer error.The parameters can be designed as γi1, γi2, µi, δi, i = (1,2,3,4,5,6).

IGC controller design in pitch
Step 1 Considering the first subsystem of Eq. 6 (Eq.16): (16) Define x1c as the command signal of the first subsystem in Eq. 6.In order to achieve the purpose of guidance, zero the line of sight angular velocity, that is .Define the subsystem tracking error as follows (Eq.17).
Then the dynamic error of the equation can be described as follows (Eq.18): (18) The estimated value Z12 of disturbance d1 can be obtained by ESO, then, according to the ides of back-stepping design, the following virtual control value is designed (Eq.19): Then the dynamic error of the equation can be described as follows (Eq.18): (18) The estimated value Z12 of disturbance d1 can be obtained by ESO, then, according to the ides of back-stepping design, the following virtual control value is designed (Eq.19): Then the dynamic error of the equation can be described as follows (Eq.18): (18) The estimated value Z12 of disturbance d1 can be obtained by ESO, then, according to the ides of back-stepping design, the following virtual control value is designed (Eq.19): The parameter k1 is a positive constant that will be designed in controller.
Step 2 Considering the second subsystem of Eq. 6 (Eq.20): The observation error of the ESO is defined as follows (Eq.14): The dynamic equation of observation error can be described as follows (Eq.15): The observer error is determined by the observer parameter γ 11 , γ 12 , μ 1 , δ 1 .By selecting the appropriate parameters, the observation values of the ESO x 1 and d 1 can converge to a small neighborhood of zero within a finite time.
According to a large number of numerical simulations experience, the selection of the observer parameters obey the following rules.For nonlinear function fal(E 11 , μ 1 , δ 1 ), choosing μ 1 = 1/2 n-1 , δ 1 = h, where n is the order of the observer, h is the integration step time of the numerical simulations.The selection of parameters γ 11 and γ 12 is related to h, usually the value of γ are chosen as follows: Based on the above design principle of ESO, ESO is designed for the remaining five subsystems of the closed loop system (Eq.6) and the closed loop system (Eq.9), and the state value and the disturbance value are estimated.Finally, we get the estimate values Z i1 , Z i2 respectively about system state value, E i1 , E i2 is the observer error.The parameters can be designed as 2,3,4,5,6).

IGC CONTROLLER DESIGN IN PITCH
Step 1 Considering the first subsystem of Eq. 6 (Eq.16): Define x 1c as the command signal of the first subsystem in Eq. 6.In order to achieve the purpose of guidance, zero the line of sight angular velocity, that is x 1 → 0. Define the subsystem tracking error as follows (Eq.17).
Then the dynamic error of the equation can be described as follows (Eq.18): The estimated value Z 12 of disturbance d 1 can be obtained by ESO, then, according to the ides of back-stepping design, the following virtual control value is designed (Eq.19): The parameter k 1 is a positive constant that will be designed in controller.
Step 2 Considering the second subsystem of Eq. 6 (Eq.20): Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.21): Step 2 Considering the second subsystem of Eq. 6 (Eq.20): (20) Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.21): (21) The dynamic error equation can be written as follows (Eq.22): Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.21): (21) The dynamic error equation can be written as follows (Eq.22): In the process of back-stepping design (Hwang and Tahk 2006), the derivative of the virtual control value will lead to differential explosion.In order to overcome this shortcoming, the first-order low-pass filter is designed by dynamic surface method, and the output of the filter is selected as the estimation of the input x2c.The first-order low-pass filter is designed as follows (Eq.

23):
( where τ1 is the filter time constant.Define the filter error b1 of virtual control value x2c as follows (Eq.24): (24) Then, the error dynamic equation can be described as follows (Eq.25): (25) In the process of back-stepping design (Hwang and Tahk 2006), the derivative of the virtual control value will lead to differential explosion.In order to overcome this shortcoming, the first-order low-pass filter is designed by dynamic surface method, and the output of the filter is selected as the estimation of the input x2c.The first-order low-pass filter is designed as follows (Eq.

23):
( where τ1 is the filter time constant.Define the filter error b1 of virtual control value x2c as follows (Eq.24): (24) Then, the error dynamic equation can be described as follows (Eq.25): (25) The estimated value Z12 of disturbance d2 can be obtained by ESO, then, according to the idea of back-stepping method, the following virtual control value is designed (Eq.26): In the process of back-stepping design (Hwang and Tahk 2006), the derivative of the virtual control value will lead to differential explosion.In order to overcome this shortcoming, the first-order low-pass filter is designed by dynamic surface method, and the output of the filter is selected as the estimation of the input x2c.The first-order low-pass filter is designed as follows (Eq.

23):
( where τ1 is the filter time constant.Define the filter error b1 of virtual control value x2c as follows (Eq.24): (24) Then, the error dynamic equation can be described as follows (Eq.25): (25) The estimated value Z12 of disturbance d2 can be obtained by ESO, then, according to the idea of back-stepping method, the following virtual control value is designed (Eq.26): first-order low-pass filter is designed by dynamic surface method, and the output of the filter is selected as the estimation of the input x2c.The first-order low-pass filter is designed as follows (Eq.

23):
( where τ1 is the filter time constant.Define the filter error b1 of virtual control value x2c as follows (Eq.24): (24) Then, the error dynamic equation can be described as follows (Eq.25): (25) The estimated value Z12 of disturbance d2 can be obtained by ESO, then, according to the idea of back-stepping method, the following virtual control value is designed (Eq.26): The parameter k1 is a positive constant that will be designed in controller.
Step 3 Considering the third subsystem of Eq. 6 (Eq.27): (27) Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.28): The parameter k1 is a positive constant that will be designed in controller.
Step 3 Considering the third subsystem of Eq. 6 (Eq.27): (27) Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.28): (28) In order to eliminate the tracking error between x3c and x3, slide-mode control method is The parameter k1 is a positive constant that will be designed in controller.
Step 3 Considering the third subsystem of Eq. 6 (Eq.27): (27) Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.28): (28) In order to eliminate the tracking error between x3c and x3, slide-mode control method is adopted (Shtessel and Tournes 2009); define the sliding mode manifold as follows (Eq.29): The parameter k1 is a positive constant that will be designed in controller.
Step 3 Considering the third subsystem of Eq. 6 (Eq.27): (27) Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.28): (28) In order to eliminate the tracking error between x3c and x3, slide-mode control method is adopted (Shtessel and Tournes 2009); define the sliding mode manifold as follows (Eq.29): (29) The differential of S1 can be obtained as follows (Eq.30): (25) The dynamic error equation can be written as follows (Eq.22): In the process of back-stepping design (Hwang and Tahk 2006), the derivative of the virtual control value will lead to differential explosion.In order to overcome this shortcoming, the first-order low-pass filter is designed by dynamic surface method, and the output of the filterx 2c is selected as the estimation of the input x 2c .The first-order low-pass filter is designed as follows (Eq.23): where τ 1 is the filter time constant.Define the filter error b 1 of virtual control value x 2c as follows (Eq.24): Then, the error dynamic equation can be described as follows (Eq.25): The estimated value Z 12 of disturbance d 2 can be obtained by ESO, then, according to the idea of back-stepping method, the following virtual control value is designed (Eq.26): The parameter k 1 is a positive constant that will be designed in controller.
Step 3 Considering the third subsystem of Eq. 6 (Eq.27): Define the tracking error of the second subsystem in Eq. 6 as follows (Eq.28): In order to eliminate the tracking error between x 3c and x 3 , slide-mode control method is adopted (Shtessel and Tournes 2009); define the sliding mode manifold as follows (Eq.29):

xx/xx 08/15
The differential of S 1 can be obtained as follows (Eq.30): (29) The differential of S1 can be obtained as follows (Eq.30): (30) Similarly, according to the design method of Step 2, the first-order low-pass filter can be described as follows (Eq.31): where τ2 is the filter time constant.Define filtering error of virtual control value x3c as follows (Eq.

32):
(32) Then, the error dynamic equation can be described as follows (Eq.33): (33) The estimate value Z12 of disturbance d2 can be obtained by ESO, and then the Eq. 30 can be rewritten as follows (Eq.34): (34) In order to make the tracking error quickly converge to zero, proximity method with adaptive ability is selected, which differential as follows (Eq.35): (35) where τ2 is the filter time constant.Define filtering error of virtual control value x3c as follows (Eq.

32):
(32) Then, the error dynamic equation can be described as follows (Eq.33): (33) The estimate value Z12 of disturbance d2 can be obtained by ESO, and then the Eq. 30 can be rewritten as follows (Eq.34): (34) In order to make the tracking error quickly converge to zero, proximity method with adaptive ability is selected, which differential as follows (Eq.35): (35) where k1 > 0, σ1 > 0.
where τ2 is the filter time constant.Define filtering error of virtual control value x3c as follows (Eq.

32):
(32) Then, the error dynamic equation can be described as follows (Eq.33): (33) The estimate value Z12 of disturbance d2 can be obtained by ESO, and then the Eq. 30 can be rewritten as follows (Eq.34): (34) In order to make the tracking error quickly converge to zero, proximity method with adaptive ability is selected, which differential as follows (Eq.35): (35) where k1 > 0, σ1 > 0.
Then, the error dynamic equation can be described as follows (Eq.33): (33) The estimate value Z12 of disturbance d2 can be obtained by ESO, and then the Eq. 30 can be rewritten as follows (Eq.34): (34) In order to make the tracking error quickly converge to zero, proximity method with adaptive ability is selected, which differential as follows (Eq.35): (35) where k1 > 0, σ1 > 0.
Then, the error dynamic equation can be described as follows (Eq.33): (33) The estimate value Z12 of disturbance d2 can be obtained by ESO, and then the Eq. 30 can be rewritten as follows (Eq.34): (34) In order to make the tracking error quickly converge to zero, proximity method with adaptive ability is selected, which differential as follows (Eq.35): (35) where k1 > 0, σ1 > 0.
With the change of the distance between the missile and the target Rp, the rata that switching function move to the sliding mode manifold is adjusted adaptively, so as to achieve the effect o weakening the buffeting (Eq.36): (36) where r1 > 0.
Define σ1 as the estimate error (Eq.37): (37) Thus the control value δy is obtained as (Eq.38): With the change of the distance between the missile and the target Rp, the rata that switching function move to the sliding mode manifold is adjusted adaptively, so as to achieve the effect of weakening the buffeting (Eq.36): (36) where r1 > 0.
Define σ1 as the estimate error (Eq.37): (37) Thus the control value δy is obtained as (Eq.38): With the change of the distance between the missile and the target Rp, the rata that switching function move to the sliding mode manifold is adjusted adaptively, so as to achieve the effect of weakening the buffeting (Eq.36): (36) where r1 > 0.
Define σ1 as the estimate error (Eq.37): (37) Thus the control value δy is obtained as (Eq.38): (38) Finally, we get the IGC controller based on the ESO and back-stepping sliding mode.
Equation 38 is the final control equation.
Similarly, according to the design method of Step 2, the first-order low-pass filter can be described as follows (Eq.31): where τ 2 is the filter time constant.Define filtering error of virtual control value x 3c as follows (Eq.32): Then, the error dynamic equation can be described as follows (Eq.33): The estimate value Z 12 of disturbance d 2 can be obtained by ESO, and then the Eq. 30 can be rewritten as follows (Eq.34): In order to make the tracking error quickly converge to zero, proximity method with adaptive ability is selected, which differential as follows (Eq.35): where k 1 > 0, σ 1 > 0.
With the change of the distance between the missile and the target R p , the rata that switching function move to the sliding mode manifold is adjusted adaptively, so as to achieve the effect of weakening the buffeting (Eq.36): where r 1 > 0.
Define σ 1 as the estimate error (Eq.37): Thus the control value δ y is obtained as (Eq.38): Finally, we get the IGC controller based on the ESO and back-stepping sliding mode.Equation 38 is the final control equation.

STABILITY ANALYSIS IN PITCH
Lemma 1: For a Lyapunov Function x 2c :[0, ∞)∈R, the solution of the inequality equation V .
According to lemma 1 and Eq.50, the following equation should be established (Eq.51): (51) To sump up, according to Eq. 41 the third closed loop subsystem is proved to be stable based on Lyapunov stability theory.By designing the control law, the state tracking error e3 can be converged to a neighborhood of zero, which can ensure that the first and the second subsystems of the system (Eq.6) are semi-globally unanimous bounded, thus, the tracking error of the system converges, and the global stability of the closed-loop system is obtained.

IGC controller design in yaw
Similarly to the pitch channel, the controller in yaw channel can be designed.
By designing an appropriate control law makes e3 converge to zero neighborhoods.In order to satisfy the form of Theorem 1, Eq. 49 can be expressed as follows (Eq.50): (50) Define κ and c as follows: , .
According to lemma 1 and Eq.50, the following equation should be established (Eq.51): (51) To sump up, according to Eq. 41 the third closed loop subsystem is proved to be stable based on Lyapunov stability theory.By designing the control law, the state tracking error e3 can be converged to a neighborhood of zero, which can ensure that the first and the second subsystems of the system (Eq.6) are semi-globally unanimous bounded, thus, the tracking error of the system converges, and the global stability of the closed-loop system is obtained.

IGC controller design in yaw
Similarly to the pitch channel, the controller in yaw channel can be designed.

Stability analysis in yaw
According to the stability analysis method of the system (Eq.6), the closed-loop system (Eq.9) can be proved to be globally stable.
By designing an appropriate control law makes e3 converge to zero neighborhoods.In order to satisfy the form of Theorem 1, Eq. 49 can be expressed as follows (Eq.50): (50) Define κ and c as follows: , .
According to lemma 1 and Eq.50, the following equation should be established (Eq.51): (51) To sump up, according to Eq. 41 the third closed loop subsystem is proved to be stable based on Lyapunov stability theory.By designing the control law, the state tracking error e3 can be converged to a neighborhood of zero, which can ensure that the first and the second subsystems of the system (Eq.6) are semi-globally unanimous bounded, thus, the tracking error of the system converges, and the global stability of the closed-loop system is obtained.

IGC controller design in yaw
Similarly to the pitch channel, the controller in yaw channel can be designed.

Stability analysis in yaw
According to the stability analysis method of the system (Eq.6), the closed-loop system (Eq.9) can be proved to be globally stable.
By designing an appropriate control law makes e3 converge to zero neighborhoods.In order to satisfy the form of Theorem 1, Eq. 49 can be expressed as follows (Eq.50): (50) Define κ and c as follows: , .
According to lemma 1 and Eq.50, the following equation should be established (Eq.51): (51) To sump up, according to Eq. 41 the third closed loop subsystem is proved to be stable based on Lyapunov stability theory.By designing the control law, the state tracking error e3 can be converged to a neighborhood of zero, which can ensure that the first and the second subsystems of the system (Eq.6) are semi-globally unanimous bounded, thus, the tracking error of the system converges, and the global stability of the closed-loop system is obtained.

IGC controller design in yaw
Similarly to the pitch channel, the controller in yaw channel can be designed.

Stability analysis in yaw
According to the stability analysis method of the system (Eq.6), the closed-loop system (Eq.9) can be proved to be globally stable.

NUMERICAL SIMULATIONS
Combining Eq. 46 and Eq.48, the following results can be obtained (Eq.49): By designing an appropriate control law makes e 3 converge to zero neighborhoods.In order to satisfy the form of Theorem 1, Eq. 49 can be expressed as follows (Eq.50): Define κ and c as follows: According to lemma 1 and Eq.50, the following equation should be established (Eq.51): To sump up, according to Eq. 41 the third closed loop subsystem is proved to be stable based on Lyapunov stability theory.By designing the control law, the state tracking error e 3 can be converged to a neighborhood of zero, which can ensure that the first and the second subsystems of the system (Eq.6) are semi-globally unanimous bounded, thus, the tracking error of the system converges, and the global stability of the closed-loop system is obtained.

IGC CONTROLLER DESIGN IN YAW
Similarly to the pitch channel, the controller in yaw channel can be designed.

STABILITY ANALYSIS IN YAW
According to the stability analysis method of the system (Eq.6), the closed-loop system (Eq.9) can be proved to be globally stable.

NUMERICAL SIMULATIONS
The effectiveness of the IGC algorithm was verified by comparing with Liu's paper in three-dimensional (Liu et al. 2015).Simulation experiments are done using Matlab/Simulink.The target in ground with a constant speed is taken as the attack object, and the numerical simulation is carried out according to the following initial conditions (Table 1).
The initial speed of the missile is 400 m/s, its velocity component in the pitch plane is V p = V m cos(φ v ), and V y = V m cos(θ) is the velocity component of the missile in the yaw plane.The initial flight path angle θ 0 = -30°, heading angle φ v0 = 45°, the initial value of the attack angle α 0 , slide angle β 0 pitch angle rate ω z0 yaw angle rate ω y0 are all zero.The ground target moving The time constant of the filter in the dynamic surface method is designed as: τ 1 = τ 2 = τ 3 = τ 4 = 0.01.
Considering the random disturbances the missile suffered in the process of moving outside, ∆ i (t) = 0.3sin(3t), i = (1, 2, 3, 4, 5, 6), and some numerical simulation is carried out in the nominal case and the missile parameters were positive and negative.Due to space constraints in this paper, the following only shows the simulation results curves under the standard situation, and the simulation curves of the states and disturbances observed by ESO in pitch channel.
It can be seen from Fig. 2 that the ESO can make an accurate estimate of the uncertain disturbance (Fig. 3).In Liu's paper(2015) there was no estimate to the disturbance.
From Figs. 4 to 8, it can be seen that the changes of attack angle, sideslip angle, rudder deflection, elevator deflection are slow, deflection angles meet to the constraints condition, which no more than ±20°, therefore the result of the simulations satisfy the requirements of actual system.The angles of the missile state also change slowly in Liu's paper (2015).At the end of the simulation, Liu et al. (2015) encountered a big fluctuation.In this paper that problem was avoided.In order to verify the robustness of the designed control algorithm, we pull the atmospheric density, aerodynamic force coeffi cient, and aerodynamic moment coeffi cient moment of inertia with a positive and negative direction, then, the numerical simulations are carried out.

xx/xx 14/15
It can be seen in Table 2 that, in the case of aerodynamic coefficients and stability derivatives without pull, the missile miss distance within 1 m; in the case of negative or positive pull of the aerodynamic coefficients and stability derivatives, the missed distance can reach less than 3 m.In Liu's paper (2015) the simulation was done without pull, and its missing distance is smaller.The missile trajectory changes little with the pull, and the missile can accurately hit the target, which shows the robustness of the designed controller.

CONCLUSIONS
In this paper, an IGC model in pitch and yaw channel is designed for air-to-ground missile during the terminal course in three-dimensional space, considering the precise control of rolling channel.The back-stepping method and the adaptive proximity law sliding mode control method is adopted based on this model, moreover, the ESO is introduced to estimate the state value and disturbance value.The numerical simulations are carried out with the external disturbance and the pulled aerodynamic coefficients and stability derivatives.The numerical simulations results show that the designed controller has strong robustness and good guidance accuracy.In this paper, only the coupling of the pitch and yaw channels is taken into account in the modeling, and how to improve the controller in the case of the coupling of the rolling channel, which makes the application more general and will become the next research.
Taking the pitch channel as example, the vertical plane IGC model is established according to the relative motion relation of missile-target and the dynamic equation of the missile body.
of equation can be obtained as (Eq.43 (43 The third subsystem is closed-loop stable, according to the et al. 2016).Since the correlation variables and their derivation o thus, there is a nonnegative continuous function η1 satisfies the following equation is obtained (Eq.44): Then (Eqs.40 to 42): Set V = V1 + V2.Then (Eqs.40 to 42):

Figure 2 .
Figure 2. (a) Course of estimation of disturbance value (Pitch); (b) Course of estimation of disturbance value (Yaw).

Figure 3 .Figure 4 .
Figure 3. (a) Curves of estimation error of state value (Pitch); (b) Curves of estimation error of state value (Yaw).

Table 1 .
Initial state of missile and target.

Table 2 .
Instruction of parameter deviation.