# Integrated Guidance and Control Design of Rolling-Guided Projectile Based on Adaptive Fuzzy Control with Multiple Constraints

Authors: Shang Jiang; Fuqing Tian; Shiyan Sun
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In the terminal guidance section of large caliber naval gun-guided projectile while striking nearshore maneuvering target, an integrated guidance and control (IGC) method based on an adaptive fuzzy and block dynamic surface sliding mode (AFCBDSM) was proposed with multiple constraints, including the impact angle, control limitations, and limited measurement of the line of sight (LOS) angle rate. The strict feedback cascade model of rolling-guided projectile IGC in space was constructed, and the extended state observer (ESO) was used to estimate the LOS angle rate and uncertain disturbances inside and outside the system, such as target maneuvering, model errors, and wind. A nonsingular terminal sliding mode (NTSM) was designed to zero the LOS angle tracking errors and LOS angle rate in finite time, with the adaptive exponential reaching law. The cascade system was effectively stabilized by the block dynamic surface sliding mode, which prevented differential explosions. To compensate for the saturated nonlinearity of canard control constraints, an adaptive Nussbaum gain function was adopted. The switching chatter of the block dynamic surface sliding mode was reduced through adaptive fuzzy control. Proven by Lyapunov theory, the LOS angle tracking error and LOS angle rate were convergent in finite time, the closed-loop system was uniformly ultimately bounded (UUB), and the system states could be made arbitrarily small at the steady state. Hardware-in-the-loop simulation (HILS) experiments showed that the AFCBDSM provided the guided projectile with good guidance performance while striking targets with different maneuvering forms.


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## Body

## 1. Introduction

In recent years, the rapid development of high-tech and advanced naval warfare theory has required naval gun weapons to be capable of continuous naval surface fire support and accurate striking of maritime and coastal targets. The naval gun-guided projectile, rolling at low speeds during flight, not only possesses a higher firing speed, more ammunition carrying capacity, and superior cost effectiveness than missiles, but it also has a longer range and higher accuracy compared with traditional ammunition, thereby providing reliable naval surface fire supports for amphibious forces [1].Terminal guidance and control play a core role in achieving accurate strikes of naval gun-guided projectile. As attack and defense equipment systems are upgraded, the relative movement speed between the projectile and target increases, which makes the frequency of the guidance loop close to that of the control loop. Consequently, the frequency assumption of spectral separation might be invalid [2]. Due to the limited space of guided projectile control modules, it is necessary for guidance and control systems to share sensors, such as gyroscopes and accelerometers, to promote economic efficiency and reliability. Furthermore, to achieve a better damage effect, the multiple constraints of actual combat should be fully considered, such as the impact angle [3], nonlinear saturation of canard deflection [4], and limited measurements of the LOS angle rate [5]. The traditional time-scale separation design method that merely meets the miss-distance constraint has difficulty meeting the aforementioned requirements, which has attracted experts, scholars, and engineers to the IGC design method with multiple constraints.IGC, first proposed by Williams et al. [6], utilizes the coupling relationship between the guidance and control system to construct a direct connection through aerodynamic angles and form a cascade closed-loop system. Based on the relative motion between the projectile and target, the control law of canard deflection can be directly calculated by IGC algorithm. Scholars have subsequently achieved results in IGC, combining it with robust control [7, 8], dynamic surface control [9, 10], adaptive control [7–9], sliding mode control (SMC) [11–14], fuzzy control [15], and other modern control theories. Yang et al. [7] proposed a robust IGC design method for guided projectile based on a SMC observer. Nevertheless, its high performance depended on precise measurements of the LOS angle rate, which was too demanding for rolling-guided projectile.Backstepping control requires calculating high-order derivatives of virtual control laws, which easily leads to differential expansion. Therefore, Seyedipour et al. [9] designed a dynamic surface and adopted a low-pass filter to avoid differential expansion, which simplified the design process by guaranteeing the guidance performance and system stability. Combining it with ESO, Shao and Wang [10] designed a dynamic surface backstepping controller. The model error, target maneuvering, and other uncertain disturbances were observed by the ESO, which significantly reduced the switching range of the dynamic surface. The ESO, initially proposed by Han [16], is a feasible scheme to determine the time-varying nonlinear IGC problem with uncertain disturbances of which a simple algorithm can accurately observe the system states and internal and external disturbances without a precise model of the research object. As the sole actuator of the guided projectile, a canard usually exhibits deflection saturation phenomenon, which allows the available overload to be less than the required overload and easily leads to the system loss of control or even instability. Hence, Wen et al. [8] combined an adaptive Nussbaum gain function with the dynamic surface sliding mode, which effectively solved the nonlinear saturation of the actuator and stabilized the IGC cascade system.Possessing strong robustness against instability factors, such as system parameter perturbations and external disturbances, SMC has been widely applied to the IGC design. Shtessel and Tournes [11] regarded the normal acceleration generated by target maneuver as a disturbance and designed the IGC method based on the high order sliding mode, which is robust against uncertainty disturbances of the target maneuver. To meet the terminal impact angle constraint, Wu and Yang [12] defined the impact angle as the angle between the projectile velocity and horizontal plane and proposed an effective IGC design scheme. However, it was pointed out that it is more universal to define the impact angle as the angle between the projectile velocity and target velocity at the moment of impact [13]. Furthermore, combining a block dynamic surface with ESO, an IGC method was proposed by Guo and Liang [14], which allowed the nonrolling near-space interceptor to obtain better guidance performance. It is well known that chatter is a difficult and urgent problem in SMC, which can be partially improved by adopting a continuous saturation function. Resulting from the difficulty of determining switching gains, the effect of buffeting reduction must be further improved. Therefore, Ran et al. [15] designed an adaptive fuzzy system using the LOS angle rate and distance between the projectile and target, which not only effectively weakened chattering but it also improved the robustness of the whole system.The previous research on IGC basically focused on nonrolling vehicles in a plane or space with a single constraint. However, there have been few IGC designs for rolling aircraft. A continuous roll character significantly enhances the couple relationship between the pitch and yaw channel of naval gun-guided projectiles, and multiple constraints introduce challenges to the finite-time convergence and stability of the system. With the above constraints and disturbances comprehensively considered in this paper, the AFCBDSM is proposed. It was proven using Lyapunov theory that the LOS angle tracking errors and LOS angle rate could converge in finite time, the closed-loop system was UUB, and the system states could be made arbitrarily small at the steady state. The main innovations were as follows: (1) with multiple constraints and various disturbances comprehensively considered, the IGC strict feedback cascade model for rolling naval gun-guided projectile was constructed, (2) the switching chatter of the block dynamic surface sliding mode was effectively weakened through adaptive fuzzy control, (3) the finite-time convergence of the LOS angle-tracking errors, LOS angle rate, and UUB of the closed-loop system were strictly proven, and (4) the effectiveness and feasibility of the AFCBDSM were verified by the designed HILS, which could satisfy multiple constraints.


## 2. Model Establishment

### 2.1. Motion and Mechanical Model

The relative motion relationship between the projectile and target in space is shown in Figure1, where Px0y0z0, Px2y2z2, Px6y6z6, Tx7y7z7, and Tx8y8z8 represent the projectile reference coordinate (NP), projectile trajectory coordinate (PV), LOS coordinate (QS), target reference coordinate (NT), and target trajectory coordinate (TV), respectively. Furthermore, P, T, R, θQ, and ψQ denote the projectile, target, distance between the projectile and target, LOS inclination angle, and LOS azimuth angle, respectively. θP, ψP, vP, θT, ψT, and vT denote the trajectory inclination angle, trajectory azimuth angle, and speed of the guided projectile and target, respectively.Figure 1
Relative motion relationship between the projectile and target in space.In addition,θ, ψ, γ, α, β, δz, and δy represent the pitch angle, yaw angle, roll angle, attack angle, sideslip angle, pitch canard angle, and yaw canard angle, respectively. Because α, β, δz, and δy change periodically with the roll of the projectile, it is necessary to establish nonrolling coordinates [17], including quasi-projectile coordinates Px4y4z4 and quasi-velocity coordinates Px5y5z5. α∗, β∗, δzeq, and δyeq represent the quasi attack angle, quasi sideslip angle, equal pitch canard angle, and equal yaw canard angle, respectively, which are utilized to calculate forces and moments of the rolling-guided projectile. The transformation relations are given by(1)α∗β∗=cosγsinγ−sinγcosγαβ,δzeqδyeq=cosγsinγ−sinγcosγδzδy.Assumption 1 (see [7]).
The guided projectile can be regarded as a rigid body, and the target can be regarded as a particle.R, θQ, ψQ, θP, ψP, vP, θ, ψ, and γ can be measured easily. P and T only possess acceleration in the normal direction of the respective velocity, which always satisfies vP>vT.
The transformation matrix fromPx2y2z2 to Px6y6z6 is as follows:(2)CQSPV=ςQSPV11ςQSPV12ςQSPV13ςQSPV21ςQSPV22ςQSPV23ςQSPV31ςQSPV32ςQSPV33.
The relative motion relationship between the projectile and target [18] is given by(3)θ¨Qψ¨Q=−2R˙Rθ˙Q−ψ˙Q2sinθQcosθQ−aPy6R+aTy6R︸dθT−2R˙Rψ˙Q+2ψ˙Qθ˙QtanθQ+aPz6r−aTz6r︸dψT,where dθT and dψT are unknown disturbances caused by the target maneuver and aPx6,aPy6,aPz6T and aTx6,aTy6,aTz6T are components of the projectile acceleration aP and target acceleration aT in Px6y6z6, respectively. r is the projection of R on the yaw plane. The components of aP in Px2y2z2 and the components of aT in Tx8y8z8 are defined as 0,aPy2,aPz2T and 0,aTy8,aTz8T, respectively. The transformation relation for aP from Px2y2z2 to Px6y6z6 can be obtained as follows:(4)aPy6aPz6=aPy2aPz2+ςQSPV22−1aPy2+ςQSPV23aPz2︸daPyςQSPV32aPy2+ςQSPV33−1aPz2︸daPz,where daPy and daPz are coordinate transformation errors. The combined external force FP acting on the projectile mainly consists of a gravitational force G, lift force Ry, Magnus force Rz, equal operating force Fδeq, and wind force Fw. The components of FP in the Py2 and Pz2 axes are the normal force FPy2 and lateral force FPz2, respectively. The external moment M mainly consists of a static moment Mz, equatorial damping moment Mzz, Magnus moment My, equal operating moment Mδeq, and wind moment Mw. The components of M in the Pz4 and Py4 axes are the pitch moment Mz4 and yaw moment My4, respectively. Their formulas can be written as follows:(5)FPy2=−mgcosθP︸Gy2+QScy′α∗︸Ryy2−QSdγ˙cz″vPβ∗︸Rzy2+QScyδzsatmδzeq︸Fδeqy2+QScy′Δαx+Δαy−QSdγ˙cz″vPΔβz︸Fwy2+dFPy2,FPz2=−QScy′β∗︸Ryz2−QSdγ˙cz″vPα∗︸Rzz2−QScyδzsatmδyeq︸Fδeqz2−QScy′Δβz−QSdγ˙cz″vPΔαx+Δαy︸Fwz2+dFPz2,Mz4=QSlmz′α∗︸Mzz4−QSldmzz′vPθ˙︸Mzzz4−QSldγ˙my″vPβ∗︸Myz4+QSlccyδzsatmδzeq︸Mδeqz4+QSlmz′Δαx+Δαy−QSldγ˙my″vPΔβz︸Mwz4+dMz4,My4=QSlmz′β∗︸Mzy4−QSldmzz′cosθvPψ˙︸Mzzy4+QSldγ˙my″vPα∗︸Myy4+QSlccyδzsatmδzeq︸Mδeqy4+QSlmz′Δβz+QSldγ˙my″vPΔαx+Δαy︸Mwy4+dMy4,where θ˙ and ψ˙ could be obtained through sensors of the projectile, such as microinertial navigation device and m, Q, S, d, l, and lc are the mass, dynamic pressure, reference area, reference diameter, reference length, and distance from the canard to the pressure center of the guided projectile, respectively. cy′, cz″, and cyδz represent the lift coefficient derivative, Magnus force coefficient joint partial derivative, and canard lift coefficient derivative, respectively. mz′, mzz″, and my″ are the static moment coefficient derivative, equatorial damping moment coefficient derivative, and Magnus moment coefficient joint partial derivative, respectively. dFPy2, dFPz2, dMPz4, and dMPy4 represent unknown model errors of the force and moment. Δαx and Δαy are the additional attack angle caused by the vertical wind wx0 and plumb wind wy0, respectively, and Δβz is the additional sideslip angle from the cross wind wz0. These are related as follows:(6)Δαx=−arctanwx0sinθPvP,Δαy=arctanwy0cosθPvP,Δβz=−arctanwz0cosψPvP,where satmδzeq is a continuous nondifferentiable saturation function defined as follows:(7)satmδzeq=δzeq,δzeq≤δcmax,δcmaxsignδzeq,δzeq>δcmax,where δcmax is the maximum canard deflection angle and satmδyeq has the same form. For the convenience of the IGC design, the following reasonable assumption is made.Assumption 2 (see [12]).
F
P
y
2 and FPz2 are mainly generated by α∗ and β∗, respectively. The forces generated by δzeq and δyeq account for a small proportion of the lift force, which can be regarded as bounded uncertain disturbances.
The kinetics and kinematic equations of a projectile body rotating around mass center yield the following:(8)θ¨=Mz4Jz4+Jx4−Jy4Jz4ψ˙2sinθcosθ︸dMz4ψ˙+Jx4Jz4ψ˙γ˙cosθ︸dMz4γ˙,ψ¨=My4Jy4cosθ+Jy4+Jz4−Jx4Jy4ψ˙θ˙tanθ︸dMy4θ˙−Jx4θ˙γ˙Jy4cosθ︸dMy4γ˙.
Considering the delay characteristic of the canard as a first-order inertial link with time constantτc, the IGC model of a guided projectile can be obtained as follows:(9)θ¨Qψ¨Q=−2R˙Rθ˙Q−ψ˙Q2sinθQcosθQ+gcosθPR−2R˙Rψ˙Q+2ψ˙Qθ˙QtanθQ︸f2+−QScy′mR00−QScy′mr︸a2α∗β∗+−Rzy2+Fδy2+Fwy2+dFPy2mR−daPyR+dθTRzz2+Fδz2+Fwz2+dFPz2mr+daPzr+dψT︸d2,α˙∗β˙∗=−QScy′mvPα∗+QSdγ˙cz″mvP2β∗+gcosθPvP−QScy′mvPcosθPβ∗−QSdγ˙cz″mvP2cosθPα∗︸f3+1001︸a3θ˙ψ˙+−Fδy2+Fwy2+dFPy2mvPFδz2+Fwz2+dFPz2mvPcosθP︸d3,θ¨ψ˙=QSlmz′Jz4α∗−QSldmzz′Jz4vPθ˙−QSldγ˙my″Jz4vPβ∗+Jx4−Jy4Jz4ψ˙2sinθcosθ+Jx4Jz4ψ˙γ˙cosθQSlmz′Jy4cosθβ∗−QSldmzz′Jy4vPψ˙+QSldγ˙my″Jy4vPcosθα∗+Jy4+Jz4−Jx4Jy4ψ˙θ˙tanθ−Jx4θ˙γ˙Jy4cosθ︸f4+QSlccyδzJz400QSlccyδzJy4cosθ︸a4satmδzeqsatmδyeq+Mwz4+dMz4Jz4Mwy4+dMy4Jy4cosθ︸d4,δ˙zeqδ˙yeq=−1τcδzeqδzeq︸f5+1τc1001︸bδzeqcδyeqc+d51d52︸d5.


### 2.2. Nearshore Maneuvering Target Model

A maritime and coastal maneuvering target can be approximately expressed by the following first-order term:(10)a˙Ty8a˙Tz8=aTy8c−aTy8τTy8aTz8c−aTz8τTz8,where τTy8 and τTz8 are time constants and aTy8c and aTz8c are normal and lateral acceleration commands of the target, respectively, which are unknown and bounded.


### 2.3. Impact Angle Model

In the pitch plane, the impact angleϑE is defined as the angle between vP and vT at the final impact moment. Zeroing the relative normal velocity of the projectile and target leads to [3](11)θQf=θTf−arctansinϑEvT/vP−cosϑE,where θQf and θTf are the final LOS inclination angle and final target trajectory inclination angle, respectively, which can be acquired by unmanned reconnaissance. For any specified ϑE, there exists a unique θQf corresponding to it, and the impact angle constraint can be converted into a LOS angle constraint.


### 2.4. System States Space

The system state variables, input variables, and output variables are defined as follows:(12)x1=θQ−θQfψQ−ψQf,x2=θ˙Qψ˙Q,x3=α∗β∗,x4=θ˙ψ˙,x5=δzeqδyeq,u=δzeqcδzeqc,y=x1.For the convenience of the IGC design, a continuous differentiable hyperbolic tangent function vectorgx5 is introduced to describe the canard deflection saturation:(13)gx5j=δcmaxex5j/δcmax−e−x5j/δcmaxex5j/δcmax+e−x5j/δcmax,j=1,2.The derivative function vector ofgx5 is g˙x5=g˙x51,g˙x52T and(14)g˙x5j=4ex5j/δcmax+e−x5j/δcmax2.Moreover, the IGC strict feedback state space of a continuous rolling-guided projectile can be obtained as follows:(15)x˙1=a1x2,x˙2=f2+a2x3+d2,x˙3=f3+a3x4+d3,x˙4=f4+a4gx5+d4,x˙5=f5+bu+d5,a1=1001,where d4=d40+a4satmx5−gx5. For the convenience of the IGC design, the following reasonable assumption is made.Assumption 3 (see [13]).
The disturbancesdij i=2,3,4,5;j=1,2 and their first derivative are bounded, i.e., there are unknown normal numbers Dij and D˙ij that satisfy dij≤Dij and d˙ij≤D˙ij, respectively.Remark 1.
According to the definition ofd4, one could be obtained that d4 is a continuous function in its definition domain and there is no discontinuous point. As result of the left derivative is not equal to the right derivative at x5j=±δcmax, and d4 is only nonderivative at x5j=±δcmax in its definition domain. However, d4 is derivable in its definition domain except these two points, and the derivative is bounded.


## 2.1. Motion and Mechanical Model

The relative motion relationship between the projectile and target in space is shown in Figure1, where Px0y0z0, Px2y2z2, Px6y6z6, Tx7y7z7, and Tx8y8z8 represent the projectile reference coordinate (NP), projectile trajectory coordinate (PV), LOS coordinate (QS), target reference coordinate (NT), and target trajectory coordinate (TV), respectively. Furthermore, P, T, R, θQ, and ψQ denote the projectile, target, distance between the projectile and target, LOS inclination angle, and LOS azimuth angle, respectively. θP, ψP, vP, θT, ψT, and vT denote the trajectory inclination angle, trajectory azimuth angle, and speed of the guided projectile and target, respectively.Figure 1
Relative motion relationship between the projectile and target in space.In addition,θ, ψ, γ, α, β, δz, and δy represent the pitch angle, yaw angle, roll angle, attack angle, sideslip angle, pitch canard angle, and yaw canard angle, respectively. Because α, β, δz, and δy change periodically with the roll of the projectile, it is necessary to establish nonrolling coordinates [17], including quasi-projectile coordinates Px4y4z4 and quasi-velocity coordinates Px5y5z5. α∗, β∗, δzeq, and δyeq represent the quasi attack angle, quasi sideslip angle, equal pitch canard angle, and equal yaw canard angle, respectively, which are utilized to calculate forces and moments of the rolling-guided projectile. The transformation relations are given by(1)α∗β∗=cosγsinγ−sinγcosγαβ,δzeqδyeq=cosγsinγ−sinγcosγδzδy.Assumption 1 (see [7]).
The guided projectile can be regarded as a rigid body, and the target can be regarded as a particle.R, θQ, ψQ, θP, ψP, vP, θ, ψ, and γ can be measured easily. P and T only possess acceleration in the normal direction of the respective velocity, which always satisfies vP>vT.
The transformation matrix fromPx2y2z2 to Px6y6z6 is as follows:(2)CQSPV=ςQSPV11ςQSPV12ςQSPV13ςQSPV21ςQSPV22ςQSPV23ςQSPV31ςQSPV32ςQSPV33.
The relative motion relationship between the projectile and target [18] is given by(3)θ¨Qψ¨Q=−2R˙Rθ˙Q−ψ˙Q2sinθQcosθQ−aPy6R+aTy6R︸dθT−2R˙Rψ˙Q+2ψ˙Qθ˙QtanθQ+aPz6r−aTz6r︸dψT,where dθT and dψT are unknown disturbances caused by the target maneuver and aPx6,aPy6,aPz6T and aTx6,aTy6,aTz6T are components of the projectile acceleration aP and target acceleration aT in Px6y6z6, respectively. r is the projection of R on the yaw plane. The components of aP in Px2y2z2 and the components of aT in Tx8y8z8 are defined as 0,aPy2,aPz2T and 0,aTy8,aTz8T, respectively. The transformation relation for aP from Px2y2z2 to Px6y6z6 can be obtained as follows:(4)aPy6aPz6=aPy2aPz2+ςQSPV22−1aPy2+ςQSPV23aPz2︸daPyςQSPV32aPy2+ςQSPV33−1aPz2︸daPz,where daPy and daPz are coordinate transformation errors. The combined external force FP acting on the projectile mainly consists of a gravitational force G, lift force Ry, Magnus force Rz, equal operating force Fδeq, and wind force Fw. The components of FP in the Py2 and Pz2 axes are the normal force FPy2 and lateral force FPz2, respectively. The external moment M mainly consists of a static moment Mz, equatorial damping moment Mzz, Magnus moment My, equal operating moment Mδeq, and wind moment Mw. The components of M in the Pz4 and Py4 axes are the pitch moment Mz4 and yaw moment My4, respectively. Their formulas can be written as follows:(5)FPy2=−mgcosθP︸Gy2+QScy′α∗︸Ryy2−QSdγ˙cz″vPβ∗︸Rzy2+QScyδzsatmδzeq︸Fδeqy2+QScy′Δαx+Δαy−QSdγ˙cz″vPΔβz︸Fwy2+dFPy2,FPz2=−QScy′β∗︸Ryz2−QSdγ˙cz″vPα∗︸Rzz2−QScyδzsatmδyeq︸Fδeqz2−QScy′Δβz−QSdγ˙cz″vPΔαx+Δαy︸Fwz2+dFPz2,Mz4=QSlmz′α∗︸Mzz4−QSldmzz′vPθ˙︸Mzzz4−QSldγ˙my″vPβ∗︸Myz4+QSlccyδzsatmδzeq︸Mδeqz4+QSlmz′Δαx+Δαy−QSldγ˙my″vPΔβz︸Mwz4+dMz4,My4=QSlmz′β∗︸Mzy4−QSldmzz′cosθvPψ˙︸Mzzy4+QSldγ˙my″vPα∗︸Myy4+QSlccyδzsatmδzeq︸Mδeqy4+QSlmz′Δβz+QSldγ˙my″vPΔαx+Δαy︸Mwy4+dMy4,where θ˙ and ψ˙ could be obtained through sensors of the projectile, such as microinertial navigation device and m, Q, S, d, l, and lc are the mass, dynamic pressure, reference area, reference diameter, reference length, and distance from the canard to the pressure center of the guided projectile, respectively. cy′, cz″, and cyδz represent the lift coefficient derivative, Magnus force coefficient joint partial derivative, and canard lift coefficient derivative, respectively. mz′, mzz″, and my″ are the static moment coefficient derivative, equatorial damping moment coefficient derivative, and Magnus moment coefficient joint partial derivative, respectively. dFPy2, dFPz2, dMPz4, and dMPy4 represent unknown model errors of the force and moment. Δαx and Δαy are the additional attack angle caused by the vertical wind wx0 and plumb wind wy0, respectively, and Δβz is the additional sideslip angle from the cross wind wz0. These are related as follows:(6)Δαx=−arctanwx0sinθPvP,Δαy=arctanwy0cosθPvP,Δβz=−arctanwz0cosψPvP,where satmδzeq is a continuous nondifferentiable saturation function defined as follows:(7)satmδzeq=δzeq,δzeq≤δcmax,δcmaxsignδzeq,δzeq>δcmax,where δcmax is the maximum canard deflection angle and satmδyeq has the same form. For the convenience of the IGC design, the following reasonable assumption is made.Assumption 2 (see [12]).
F
P
y
2 and FPz2 are mainly generated by α∗ and β∗, respectively. The forces generated by δzeq and δyeq account for a small proportion of the lift force, which can be regarded as bounded uncertain disturbances.
The kinetics and kinematic equations of a projectile body rotating around mass center yield the following:(8)θ¨=Mz4Jz4+Jx4−Jy4Jz4ψ˙2sinθcosθ︸dMz4ψ˙+Jx4Jz4ψ˙γ˙cosθ︸dMz4γ˙,ψ¨=My4Jy4cosθ+Jy4+Jz4−Jx4Jy4ψ˙θ˙tanθ︸dMy4θ˙−Jx4θ˙γ˙Jy4cosθ︸dMy4γ˙.
Considering the delay characteristic of the canard as a first-order inertial link with time constantτc, the IGC model of a guided projectile can be obtained as follows:(9)θ¨Qψ¨Q=−2R˙Rθ˙Q−ψ˙Q2sinθQcosθQ+gcosθPR−2R˙Rψ˙Q+2ψ˙Qθ˙QtanθQ︸f2+−QScy′mR00−QScy′mr︸a2α∗β∗+−Rzy2+Fδy2+Fwy2+dFPy2mR−daPyR+dθTRzz2+Fδz2+Fwz2+dFPz2mr+daPzr+dψT︸d2,α˙∗β˙∗=−QScy′mvPα∗+QSdγ˙cz″mvP2β∗+gcosθPvP−QScy′mvPcosθPβ∗−QSdγ˙cz″mvP2cosθPα∗︸f3+1001︸a3θ˙ψ˙+−Fδy2+Fwy2+dFPy2mvPFδz2+Fwz2+dFPz2mvPcosθP︸d3,θ¨ψ˙=QSlmz′Jz4α∗−QSldmzz′Jz4vPθ˙−QSldγ˙my″Jz4vPβ∗+Jx4−Jy4Jz4ψ˙2sinθcosθ+Jx4Jz4ψ˙γ˙cosθQSlmz′Jy4cosθβ∗−QSldmzz′Jy4vPψ˙+QSldγ˙my″Jy4vPcosθα∗+Jy4+Jz4−Jx4Jy4ψ˙θ˙tanθ−Jx4θ˙γ˙Jy4cosθ︸f4+QSlccyδzJz400QSlccyδzJy4cosθ︸a4satmδzeqsatmδyeq+Mwz4+dMz4Jz4Mwy4+dMy4Jy4cosθ︸d4,δ˙zeqδ˙yeq=−1τcδzeqδzeq︸f5+1τc1001︸bδzeqcδyeqc+d51d52︸d5.


## 2.2. Nearshore Maneuvering Target Model

A maritime and coastal maneuvering target can be approximately expressed by the following first-order term:(10)a˙Ty8a˙Tz8=aTy8c−aTy8τTy8aTz8c−aTz8τTz8,where τTy8 and τTz8 are time constants and aTy8c and aTz8c are normal and lateral acceleration commands of the target, respectively, which are unknown and bounded.


## 2.3. Impact Angle Model

In the pitch plane, the impact angleϑE is defined as the angle between vP and vT at the final impact moment. Zeroing the relative normal velocity of the projectile and target leads to [3](11)θQf=θTf−arctansinϑEvT/vP−cosϑE,where θQf and θTf are the final LOS inclination angle and final target trajectory inclination angle, respectively, which can be acquired by unmanned reconnaissance. For any specified ϑE, there exists a unique θQf corresponding to it, and the impact angle constraint can be converted into a LOS angle constraint.


## 2.4. System States Space

The system state variables, input variables, and output variables are defined as follows:(12)x1=θQ−θQfψQ−ψQf,x2=θ˙Qψ˙Q,x3=α∗β∗,x4=θ˙ψ˙,x5=δzeqδyeq,u=δzeqcδzeqc,y=x1.For the convenience of the IGC design, a continuous differentiable hyperbolic tangent function vectorgx5 is introduced to describe the canard deflection saturation:(13)gx5j=δcmaxex5j/δcmax−e−x5j/δcmaxex5j/δcmax+e−x5j/δcmax,j=1,2.The derivative function vector ofgx5 is g˙x5=g˙x51,g˙x52T and(14)g˙x5j=4ex5j/δcmax+e−x5j/δcmax2.Moreover, the IGC strict feedback state space of a continuous rolling-guided projectile can be obtained as follows:(15)x˙1=a1x2,x˙2=f2+a2x3+d2,x˙3=f3+a3x4+d3,x˙4=f4+a4gx5+d4,x˙5=f5+bu+d5,a1=1001,where d4=d40+a4satmx5−gx5. For the convenience of the IGC design, the following reasonable assumption is made.Assumption 3 (see [13]).
The disturbancesdij i=2,3,4,5;j=1,2 and their first derivative are bounded, i.e., there are unknown normal numbers Dij and D˙ij that satisfy dij≤Dij and d˙ij≤D˙ij, respectively.Remark 1.
According to the definition ofd4, one could be obtained that d4 is a continuous function in its definition domain and there is no discontinuous point. As result of the left derivative is not equal to the right derivative at x5j=±δcmax, and d4 is only nonderivative at x5j=±δcmax in its definition domain. However, d4 is derivable in its definition domain except these two points, and the derivative is bounded.


## 3. AFCBDSM Design

The purpose of the AFCBDSM design is to generate an appropriate control lawu for equation (14) that can allow x1 and x2 to converge to any small neighborhood of zero in finite time and guarantee the closed-loop system is UUB [19], and the system states converge to a small neighborhood of the origin, with limited measurements of x2, the control saturation of x5, and the unknown bounded disturbances di.


### 3.1. ESO Design

To acquire an accurate observation value ofd2, the following third-order ESO is designed, with the observational variables defined as zx1=zx11,zx12T, zx2=zx21,zx22T, and zd2=zd21,zd22T:(16)e21=zx1−x1,e22=zx2−x2,e23=zd2−d2,z˙x1=zx2−β21e21,z˙x2=f2+a2x3+zd2−β22fale21,σ21,η21,z˙d2=−β23fale21,σ22,η22,where β21=β211100β2122, σ21=σ211,σ212T, η21=η211,η212T, β21jj>0, 0<σ21j<1, 0<η21j<1, and fale21,σ21,η21 is the function vector fale211,σ211,η211,fale212,σ212,η212T [16].In a previous report [3], the stability of the second- and third-order ESO designed in cascade systems were deduced and proven. By choosing appropriate parameters values, in particular, letting β23jj be much larger than β22jj and β21jj, the ESO given by equation (16) can observe d2 accurately and quickly. Similarly, for observing di, the observational variables are defined as zxi=zxi1,zxi2T and zdi=zdi1,zdi2T and the second order ESO is designed as follows:(17)ei1=zxi−xi,ei2=zdi−di,z˙x3=f3+a3x4+zd3−β31e31,z˙x4=f4+a4g+zd4−β41e41,z˙x5=f5+bu+zd5−β51e51,z˙di=−βi2falei1,σi,ηi,i=3,4,5.


### 3.2. Nonsingular Terminal Sliding Mode Design

Lemma 1 (see [14]).
Considering the following system:(18)x˙=fx,t,f0,t=0,x∈Rn,where x∈Rn is the system state vector, gx,t: U0×R⟶Rn is a nonlinear continuous function defined on U0×R, and U0∈Rn is an open neighborhood of the origin xt=0. Assuming Vx is a C1 smooth positive definite function defined on U∈Rn and V˙x+CVAx is negative semidefinite on U∈Rn for C∈R+ and A∈0,1, there exists an area U0 such that any Vx that starts from U0∈Rn can reach Vx≡0 in finite time. Furthermore, if Tr is the time needed to reach Vx≡0, then Tr≤V1−Ax0/C1−A, where Vx0 is the initial value of Vx.
A kind of NTSM is used to construct the sliding mode vectors:(19)s2=s21s22=x1+μ1x21δ1signx21μ2x22δ2signx22,where μj>0 and 1<δj<2. The time derivative of equation (19) can be obtained as follows:(20)s˙2=x2+μ1δ1x21δ1−100μ2δ2x22δ2−1f2+a2x3+d2.
To ensure good dynamic properties in the sliding mode approach process, the sliding mode adaptive approach law is designed as follows:(21)s˙2=−k2signs2−R˙Rc2s2,where k2=k21100k222, e21i≤k2ii, c2=c21100c222, c2jj>0, and signs2 is a symbolic function vector signs21,signs22T. Combining equations (20) and (21) and removing the singular factors x211−δ1/μ1δ1 and x221−δ2/μ2δ2 yields,(22)x¯3=−a2−1x212−δ1signx21μ1δ1x222−δ2signx22μ2δ2+f2+zd2+k2signs2+R˙Rc2s2.Theorem 1.
For the subsystem composed of the first two equations of equation (15), adopting the ESO given by equation (16) and the control law given by equation (22), the system state variables x1 and x2 will converge to any small neighborhood of zero in finite time.Proof.
A Lyapunov function is chosen asV2=s2Ts2/2, and the time derivative of V2 can be obtained:(23)V˙2=s2Ts˙2=∑j=12s2js˙2j=∑j=12s2jx2j+μjδjx2jδj−1f2j+a2jjx¯3j+d2j=∑j=12μjδjx2jδj−1s2jd2j−zd2j−k2jjsigns2j−R˙Rc2jjs2j=−∑j=12μjδjx2jδj−1R˙Rc2jjs2j2−e21js2j−k2jjs2j≤−∑j=12k2jj−e21js2j≤−mink211−e211,k222−e212⋅22V21/2.
According to Lemma1, s2 can converge to 0 in finite time while x2≠0, which is also satisfied if x2=0 [20]. Thus, the following can be obtained:(24)x˙1j=−μj−1/δjx1j1/δjsignx1j.
A Lyapunov function is chosen asVx1j=x1j2/2, and the time derivative of Vx1j can be obtained:(25)V˙x1j=−μj−1/δjx1j1/δj+1=−21+δj/2δjμj−1/δjVx1j1+δj/2δj.
According to Lemma1, one can obtain x1, and x2 can converge to any small neighborhood of 0 in finite time. The proof ends.


### 3.3. Block Dynamic Surface Sliding Mode Design

s
2 is defined as the second block dynamic surface sliding mode, and the third, fourth, and fifth block dynamic surface sliding modes are defined as follows:(26)s3=x3−x3d,s4=x4−x4d,s5=g−gd,where g is the abbreviated notation for the function gx5. Virtual control laws x¯4 and g¯ can be obtained [3] as follows:(27)x¯4=−f3+zd3−x˙3d+k3signs3+c3s3,g¯=−a4−1f4−x˙4d+zd4+k4signs4+c4s4.To avoid direct differentiation with respect tox¯3, x¯4, and g¯, the filtered virtual control laws x3d, x4d, and gd are obtained using the following first-order filter:(28)τ3x˙3d+x3d=x¯3,x3d0=x¯30,τ4x˙4d+x4d=x¯4,x4d0=x¯40,τ5g˙d+gd=g¯,gd0=g¯0,where τ3, τ4, and τ5 are positive constants.Definition 1 (see [21]).
If the continuous functionNχ satisfies the following property limt⟶∞sup1/t∫0tNχdχ=∞ and limt⟶∞sup1/t∫0tNχdχ=−∞, then Nχ is a Nussbaum function.
For handling the nonlinear saturation of the canard deflection effectively, control laws are designed as follows:(29)u¯=−g˙f5+g˙d−k5signs5−c5s5,u=b−1Nχu¯−zd5,where ki and ci i=3,4,5 are positive parameters whose form and scope are similar to k2 and c2, respectively. According to Definition 1, the following Nussbaum function is implemented:(30)Nχ=eχ12cosχ100eχ22cosχ2,where χ1 and χ2 are adaptive variables of the Nussbaum function with the following adaptive law:(31)χ˙=χ˙1,χ˙2T=γχ1u¯1s51,γχ2u¯2s52T,where γχ1 and γχ2 are positive constants.


### 3.4. Adaptive Fuzzy System Design

Resulting from the time variability and nonlinearity of uncertain disturbances, it is difficult to determine appropriate switching gains, which causes serious chatter of the sliding mode. Possessing the universal approximation ability, the adaptive fuzzy system, composed of product inference, singleton fuzzification, center-average defuzzification, and Gauss membership functions [22], will be used to approximate observation error of disturbance. Thus, the high frequency chatter of the control law could be effectively weakened.The input variable vector is defined asw=w1,…,wnT∈Ω, then there are ∏j=1npj fuzzy rules in common as follows:(32)Rulel1,…,ln:ifw1isA1l1,…,wnisAnlnthenΞwκisBl1,…,ln,ln=1,…,pn,where Anln are fuzzy sets associated with the fuzzy membership functions, and Ξwκ is the output variable as follows:(33)Ξwκ=κTξw=∑l1=1p1⋯∑ln=1pnκl1,…,ln∏i=1nϕAiliwi∑l1=1p1⋯∑ln=1pn∏i=1nϕAiliwi,where κ=κ1,…,1,…,κp1,…,pnT is an adaptive fuzzy parameter vector in the dimension of ∏i=1npi, ϕAnlnwn is the Gauss membership function, ξw=ξ1,…,1w,…,ξp1,…,pnwT is the fuzzy basis vector in the dimension of ∏i=1npi, and the element is as follows:(34)ξl1,…,lnw=∏i=1nϕAiliwi∑l1=1p1⋯∑ln=1pn∏i=1nϕAiliwi.Lemma 2 ([see22]).
For a continuous functionΓ defined in a closed set Ω⊆Rn and any precision ε, there must exist an adaptive fuzzy system Ξwκ composed of equations (33) and (34) such that it satisfies supw∈ΩΓ−Ξwκ≤ε.
To guarantee the finite time convergence ofx1 and x2, the adaptive fuzzy systems are designed for the virtual control law given by equation (27) and the control law given by equation (29). The optimal adaptive fuzzy parameter approximation vectors are defined as follows:(35)κij∗=argminsupwij∈Ω−ei2j−κijTξijwij,κ5j∗=argminsupw5j∈Ω−g˙jje52j−κ5jTξ5jw5j,where wij=sij,s˙ijTi=3,4,5. According to Lemma 2, the following inequalities hold for any given small constant εij:(36)supwij∈Ω−ei2j−κij∗Tξijwij≤εij,i=3,4;j=1,2,supw5j∈Ω−g˙jje52j−κ5j∗Tξ5jw5j≤ε5j,j=1,2.
The approximation errors of the fuzzy adaptive parameter vectors were defined asκ˜ij=κij∗−κij, of which the adaptive approach laws are as follows:(37)κ˙ij=λijξijwijsij−κij,i=3,4,5;j=1,2,where λij>0. From equations (27) and (29), the virtual control laws and control law are, respectively, modified as follows:(38)x¯4=−f3+zd3−x˙3d+Ξ3w3κ3+c3s3,g¯=−a4−1f4−x˙4d+zd4+Ξ4w4κ4+c4s4,u¯=−g˙f5+g˙d−Ξ5w5κ5−c5s5,where Ξiwiκi=Ξi1wi1κi1,Ξi2wi2κi2T.


## 3.1. ESO Design

To acquire an accurate observation value ofd2, the following third-order ESO is designed, with the observational variables defined as zx1=zx11,zx12T, zx2=zx21,zx22T, and zd2=zd21,zd22T:(16)e21=zx1−x1,e22=zx2−x2,e23=zd2−d2,z˙x1=zx2−β21e21,z˙x2=f2+a2x3+zd2−β22fale21,σ21,η21,z˙d2=−β23fale21,σ22,η22,where β21=β211100β2122, σ21=σ211,σ212T, η21=η211,η212T, β21jj>0, 0<σ21j<1, 0<η21j<1, and fale21,σ21,η21 is the function vector fale211,σ211,η211,fale212,σ212,η212T [16].In a previous report [3], the stability of the second- and third-order ESO designed in cascade systems were deduced and proven. By choosing appropriate parameters values, in particular, letting β23jj be much larger than β22jj and β21jj, the ESO given by equation (16) can observe d2 accurately and quickly. Similarly, for observing di, the observational variables are defined as zxi=zxi1,zxi2T and zdi=zdi1,zdi2T and the second order ESO is designed as follows:(17)ei1=zxi−xi,ei2=zdi−di,z˙x3=f3+a3x4+zd3−β31e31,z˙x4=f4+a4g+zd4−β41e41,z˙x5=f5+bu+zd5−β51e51,z˙di=−βi2falei1,σi,ηi,i=3,4,5.


## 3.2. Nonsingular Terminal Sliding Mode Design

Lemma 1 (see [14]).
Considering the following system:(18)x˙=fx,t,f0,t=0,x∈Rn,where x∈Rn is the system state vector, gx,t: U0×R⟶Rn is a nonlinear continuous function defined on U0×R, and U0∈Rn is an open neighborhood of the origin xt=0. Assuming Vx is a C1 smooth positive definite function defined on U∈Rn and V˙x+CVAx is negative semidefinite on U∈Rn for C∈R+ and A∈0,1, there exists an area U0 such that any Vx that starts from U0∈Rn can reach Vx≡0 in finite time. Furthermore, if Tr is the time needed to reach Vx≡0, then Tr≤V1−Ax0/C1−A, where Vx0 is the initial value of Vx.
A kind of NTSM is used to construct the sliding mode vectors:(19)s2=s21s22=x1+μ1x21δ1signx21μ2x22δ2signx22,where μj>0 and 1<δj<2. The time derivative of equation (19) can be obtained as follows:(20)s˙2=x2+μ1δ1x21δ1−100μ2δ2x22δ2−1f2+a2x3+d2.
To ensure good dynamic properties in the sliding mode approach process, the sliding mode adaptive approach law is designed as follows:(21)s˙2=−k2signs2−R˙Rc2s2,where k2=k21100k222, e21i≤k2ii, c2=c21100c222, c2jj>0, and signs2 is a symbolic function vector signs21,signs22T. Combining equations (20) and (21) and removing the singular factors x211−δ1/μ1δ1 and x221−δ2/μ2δ2 yields,(22)x¯3=−a2−1x212−δ1signx21μ1δ1x222−δ2signx22μ2δ2+f2+zd2+k2signs2+R˙Rc2s2.Theorem 1.
For the subsystem composed of the first two equations of equation (15), adopting the ESO given by equation (16) and the control law given by equation (22), the system state variables x1 and x2 will converge to any small neighborhood of zero in finite time.Proof.
A Lyapunov function is chosen asV2=s2Ts2/2, and the time derivative of V2 can be obtained:(23)V˙2=s2Ts˙2=∑j=12s2js˙2j=∑j=12s2jx2j+μjδjx2jδj−1f2j+a2jjx¯3j+d2j=∑j=12μjδjx2jδj−1s2jd2j−zd2j−k2jjsigns2j−R˙Rc2jjs2j=−∑j=12μjδjx2jδj−1R˙Rc2jjs2j2−e21js2j−k2jjs2j≤−∑j=12k2jj−e21js2j≤−mink211−e211,k222−e212⋅22V21/2.
According to Lemma1, s2 can converge to 0 in finite time while x2≠0, which is also satisfied if x2=0 [20]. Thus, the following can be obtained:(24)x˙1j=−μj−1/δjx1j1/δjsignx1j.
A Lyapunov function is chosen asVx1j=x1j2/2, and the time derivative of Vx1j can be obtained:(25)V˙x1j=−μj−1/δjx1j1/δj+1=−21+δj/2δjμj−1/δjVx1j1+δj/2δj.
According to Lemma1, one can obtain x1, and x2 can converge to any small neighborhood of 0 in finite time. The proof ends.


## 3.3. Block Dynamic Surface Sliding Mode Design

s
2 is defined as the second block dynamic surface sliding mode, and the third, fourth, and fifth block dynamic surface sliding modes are defined as follows:(26)s3=x3−x3d,s4=x4−x4d,s5=g−gd,where g is the abbreviated notation for the function gx5. Virtual control laws x¯4 and g¯ can be obtained [3] as follows:(27)x¯4=−f3+zd3−x˙3d+k3signs3+c3s3,g¯=−a4−1f4−x˙4d+zd4+k4signs4+c4s4.To avoid direct differentiation with respect tox¯3, x¯4, and g¯, the filtered virtual control laws x3d, x4d, and gd are obtained using the following first-order filter:(28)τ3x˙3d+x3d=x¯3,x3d0=x¯30,τ4x˙4d+x4d=x¯4,x4d0=x¯40,τ5g˙d+gd=g¯,gd0=g¯0,where τ3, τ4, and τ5 are positive constants.Definition 1 (see [21]).
If the continuous functionNχ satisfies the following property limt⟶∞sup1/t∫0tNχdχ=∞ and limt⟶∞sup1/t∫0tNχdχ=−∞, then Nχ is a Nussbaum function.
For handling the nonlinear saturation of the canard deflection effectively, control laws are designed as follows:(29)u¯=−g˙f5+g˙d−k5signs5−c5s5,u=b−1Nχu¯−zd5,where ki and ci i=3,4,5 are positive parameters whose form and scope are similar to k2 and c2, respectively. According to Definition 1, the following Nussbaum function is implemented:(30)Nχ=eχ12cosχ100eχ22cosχ2,where χ1 and χ2 are adaptive variables of the Nussbaum function with the following adaptive law:(31)χ˙=χ˙1,χ˙2T=γχ1u¯1s51,γχ2u¯2s52T,where γχ1 and γχ2 are positive constants.


## 3.4. Adaptive Fuzzy System Design

Resulting from the time variability and nonlinearity of uncertain disturbances, it is difficult to determine appropriate switching gains, which causes serious chatter of the sliding mode. Possessing the universal approximation ability, the adaptive fuzzy system, composed of product inference, singleton fuzzification, center-average defuzzification, and Gauss membership functions [22], will be used to approximate observation error of disturbance. Thus, the high frequency chatter of the control law could be effectively weakened.The input variable vector is defined asw=w1,…,wnT∈Ω, then there are ∏j=1npj fuzzy rules in common as follows:(32)Rulel1,…,ln:ifw1isA1l1,…,wnisAnlnthenΞwκisBl1,…,ln,ln=1,…,pn,where Anln are fuzzy sets associated with the fuzzy membership functions, and Ξwκ is the output variable as follows:(33)Ξwκ=κTξw=∑l1=1p1⋯∑ln=1pnκl1,…,ln∏i=1nϕAiliwi∑l1=1p1⋯∑ln=1pn∏i=1nϕAiliwi,where κ=κ1,…,1,…,κp1,…,pnT is an adaptive fuzzy parameter vector in the dimension of ∏i=1npi, ϕAnlnwn is the Gauss membership function, ξw=ξ1,…,1w,…,ξp1,…,pnwT is the fuzzy basis vector in the dimension of ∏i=1npi, and the element is as follows:(34)ξl1,…,lnw=∏i=1nϕAiliwi∑l1=1p1⋯∑ln=1pn∏i=1nϕAiliwi.Lemma 2 ([see22]).
For a continuous functionΓ defined in a closed set Ω⊆Rn and any precision ε, there must exist an adaptive fuzzy system Ξwκ composed of equations (33) and (34) such that it satisfies supw∈ΩΓ−Ξwκ≤ε.
To guarantee the finite time convergence ofx1 and x2, the adaptive fuzzy systems are designed for the virtual control law given by equation (27) and the control law given by equation (29). The optimal adaptive fuzzy parameter approximation vectors are defined as follows:(35)κij∗=argminsupwij∈Ω−ei2j−κijTξijwij,κ5j∗=argminsupw5j∈Ω−g˙jje52j−κ5jTξ5jw5j,where wij=sij,s˙ijTi=3,4,5. According to Lemma 2, the following inequalities hold for any given small constant εij:(36)supwij∈Ω−ei2j−κij∗Tξijwij≤εij,i=3,4;j=1,2,supw5j∈Ω−g˙jje52j−κ5j∗Tξ5jw5j≤ε5j,j=1,2.
The approximation errors of the fuzzy adaptive parameter vectors were defined asκ˜ij=κij∗−κij, of which the adaptive approach laws are as follows:(37)κ˙ij=λijξijwijsij−κij,i=3,4,5;j=1,2,where λij>0. From equations (27) and (29), the virtual control laws and control law are, respectively, modified as follows:(38)x¯4=−f3+zd3−x˙3d+Ξ3w3κ3+c3s3,g¯=−a4−1f4−x˙4d+zd4+Ξ4w4κ4+c4s4,u¯=−g˙f5+g˙d−Ξ5w5κ5−c5s5,where Ξiwiκi=Ξi1wi1κi1,Ξi2wi2κi2T.


## 4. System Stability Analysis

Lemma 3 ([see23]).
LetVt and χt be smooth functions, which are defined on 0,tf with Vt≥0. ∀t∈0,tf, and N⋅ is a Nussbaum gain function. If the following inequality holds, then Vt and χt must be bounded on 0,tf:(39)Vt≤V0e−Ct+MC1−e−Ct+e−Ctγχ∫0tχ˙ξNχ−1eCτdτ,where C>0 and M>0. The errors of the virtual control laws are defined as follows:(40)yi=xid−x¯i,yg=gd−g¯,i=3,4.
Differentiatingxid and gd yields(41)x˙id=−yiτi,g˙d=−ygτ5.
Through some simple simplifications, we obtain the following:(42)y˙i=−yiτi−x¯˙i,y˙g=−ygτ5−g¯˙.
According to a previous report [24], there are positive real numbers Mij and Mgj such that the inequalities x¯˙ij≤Mij and g¯˙j≤Mgj hold. Through some simple calculations, we obtain(43)xi=si+yi+x¯i,g=s5+yg+g¯.
We obtain the following Lyapunov function of the whole system state:(44)V=12∑i=25siTsi+12∑i=34yiTyi+12ygTyg+12∑j=12∑i=351λijκ˜ijTκ˜ij.Theorem 2.
For system (15), which satisfies Assumptions 1–3, the closed-loop system is UUB, and the system states could be made arbitrarily small at the steady state, with the application of the ESO given by equations (16) and (17), the block dynamic surface sliding mode given by equations (18) and (26), the control law given by equations (22) and (37), and the appropriate parameter selections.Proof.
For convenience, taking the third dynamic surfaces31 as an example, the simplification process of equalities and inequalities related to s31 is deduced as follows. From equations (27) and (28), one can obtain(45)s31s˙31=s31x˙31−x˙3d1=s31f31+a311s41+y41+x¯41+d31−x˙3d1=s31s41+y41+s31d31−zd31−κ31Tξ31w31−c31s31≤−c31−1s312+s4122+y4122+s31−e321−κ31Tξ31w31+κ31∗Tξ31w31−κ31∗Tξ31w31≤−c31−1s312+s4122+y4122+s31−e321−κ31∗Tξ31w31+s31κ˜31Tξ31w31≤−c31−32s312+s4122+y4122+ε3122+κ˜31Tξ31w31s31.
Differentiatingκ˜31Tκ˜31/2λ31 yields(46)κ˜31Tκ˜˙31λ31=κ˜31T−λ31ξ31w31s31+κ31λ31=−κ˜31Tξ31w31s31+κ˜31Tκ31∗−κ˜31λ31≤−κ˜31Tξ31w31s31−κ˜31Tκ˜312λ31+κ31∗Tκ31∗2λ31.
From equations (40) and (41), we have(47)y31y˙31=y31−x¯˙31−y31τ3=−y312τ3−y31x¯˙31≤−y312τ3+y312m312ρ2+ρ24.
Being similar tos31, equalities and inequalities related to other dynamic surface could be obtained with no more detailed description. The derivative of equation (44) is as follows:(48)V˙=∑j=12∑i=25sijs˙ij+∑i=34yijy˙ij+ygjy˙gj+∑i=351λijκ˜ijTκ˜˙ij=∑j=12s2jx2j+μjδjx2jδj−1f2j+a2jjs3j+y3j+x¯3j+d2j+s3jf3j+a3jjs4j+y4j+x¯4j+d3j−x˙3dj+s4jf4j+a4jjs5j+ygj+g¯j+d4j−x˙4dj+s5jg˙jjf5j+bjuj+d5j−g˙dj+∑i=34yij−x¯˙ij−yijτi+ygj−g˙j−ygjτ5−∑i=35κ˜ijTξijwijsij+∑i=351λijκ˜ijTκij∗−κ˜ij≤∑j=12μjδjx2jδj−1a2jjs2js3j+y3j+μjδjx2jδj−1s2jd2j−zd2j−k2jjsigns2j−R˙Rc2jjs2j+s3js4j+y4j+s3jd3j−zd3j−κ3jTξ3jw3j−c3jjs3j+a4jjs4js5j+ygj+s4jd4j−zd4j−κ4jTξ4jw4j−c4jjs4j+s5jg˙jjNjjχj−1u¯j+g˙jjd5j−zd5j−κ5jTξ5jw5j−c5jjs5j+∑i=34yij−yijτi+Mij+ygj−ygjτ5+Mgj−∑i=35κ˜ijTξijwijsij−12∑i=351λijκ˜ijTκ˜ij+12∑i=351λijκij∗Tκij∗≤∑j=12μjδjx2jδj−1a2jjs2j2+s3j22+y3j22−μjδjx2jδj−1R˙Rc2jjs2j2+s3j2+s4j22+y4j22−c3jjs3j2+s3j22+ε3j22+a4jjs4j2+s5j22+ygj22−c4jjs4j2+s4j22+ε4j22−c5jjs5j2+s5j22+ε5j22+s5jg˙jjNjjχj−1u¯j+∑i=34Mij2ρ2−1τiyij2+Mgj2ρ2−1τ5ygj2−12∑i=351λijκ˜ijTκ˜ij+32ρ2+12∑j=12∑i=351λijκij∗Tκij∗≤∑j=12−μjδjx2jδj−1R˙Rc2jj−a2jj︸m1s2j2−c3jj−μjδjx2jδj−1a2jj2−32︸m2s3j2−c4jj−a4jj−1︸m3s4j2−c5jj−a4jj2−12︸m4s5j2−1τ3−M3j2ρ2−μjδjx2jδj−1a2jj2︸m5y3j2−1τ4−M4j2ρ2−12︸m6y4j2−1τ5−Mgj2ρ2−a4jj2︸m7ygj2−12∑i=351λij︸m8κ˜ijTκ˜ij+∑j=12χ˙jγχjg˙jjNjjχj−1+32ρ2+12∑j=12∑i=35εij2+12∑j=12∑i=351λijκij∗Tκij∗︸ζ.
The appropriate parameters are selected to meet the following requirements, which can guarantee the system stability:(49)c2jj≥ℓ+a2jjRR˙,c3jj≥ℓ+μjδjx2jδj−1a2jj2+32,c4jj≥ℓ+a4jj+1,c5jj≥ℓ+a4jj2+12,1τ3≥ℓ+m3j2ρ2+μjδjx2jδj−1a2jj2,1τ4≥ℓ+m4j2ρ2+12,1τ5≥ℓ+mgj2ρ2+a4jj2,where ℓ is an arbitrary positive constant. Letting the positive constant ϖ=minmii=1,…,8, equation (48) can be simplified as follows:(50)V˙≤−2ϖV+ζ+∑j=12χ˙jγχjg˙jjNjjχj−1.
Supposeϖ>ζ+∑j=12χ˙j/γχjg˙jjNjjχj−1/2b, then it could be obtained that V˙<0 on V=b, which means that V≤b is an invariant set, and if Vtr≤b, then Vt≤b holds for all t≥tr. By the comparison principle [23], the following can be concluded from equation (50):(51)0≤Vt≤V0e−2ϖt+ζ2ϖ1−e−2ϖt+e−2ϖt∑j=121γχj∫0tχ˙jg˙jjNjjχj−1e2ϖτdτ.
According to Lemma 3, one could obtain thatV is bounded by ζ/2ϖ and the closed-loop system is UUB. Furthermore, ζ is a constant independent of ϖ, so the upper bound ζ/2ϖ could be made arbitrarily small through tuning the design parameters, of which the purpose is for ζ small enough and ϖ big enough. Thus, the system states could be made arbitrarily small at the steady state. The proof ends.Similar to the existing literatures [25], Theorem 2 shows the existence of the control algorithm to ensure the stability of the closed-loop system but does not provide a quantitative criterion on how to select the design parameters. In fact, it is difficult to give such a criterion. By far, the design parameters can only be selected by trial and error.


## 5. Experimental Results and Discussions

To verify the effectiveness and feasibility of the AFCBDSM, the control module of a naval gun-guided projectile was selected as the control object and a HILS system was built, as shown in Figure2, which mainly included a host computer, three-axis turntable, industrial personal computer (IPC), and control module, including a microcontrol unit (MCU) and dual channel canard. The main performance indexes of the canard and turntable are shown in Tables 1 and 2, respectively, which meet the requirements of the HILS.Figure 2
Hardware-in-the-loop simulation system.Table 1
Main indexes of canard.
Index
Value
Output torque (N·m)
≥3
Max canard angle (deg)
±18
Response frequency (Hz)
≥6
Baud rate (bps)
115200
Max error of canard deflection (deg)
±0.1Table 2
Main indexes of turntable.
Index
Value
Pitch angle (deg)
±80
Yaw angle (deg)
±40
Roll angle (deg)
Continuous
Angle accuracy (deg)
±0.1
Pitch rate (deg·s−1)
≥10
Yaw rate (deg·s−1)
≥10
Roll rate (deg·s−1)
≥3600
Overload mass (kg)
≤15The core component of the HILS system was the STM32F429IGT6 MCU, which configured the RS232 bus communication, interruption, direct memory access (DMA), and analog-to-digital converter (ADC) according to the program compiled in Keil. The host computer controlled the states of the hardware-in-the-loop simulation system by sending control instructions to the control module, such as the initial, start, stop, and recovery instructions. After starting the simulation, the MCU uses a Kalman filter to process the canard deflection feedback collected by the ADC in real time and obtained attitude feedback from the IPC. These feedback parameters and the fourth order Runge–Kutta method were used to iterate the differential equations with a 10 ms step, which was composed of six-degree-of-freedom equations of the projectile, three-degree-of-freedom equations of the target, the relative motion equations, the Kalman filter, and the AFCBDSM. The corresponding parameters were solved and sent back to the host computer. Meanwhile, the MCU sent the canard deflection and attitude commands to the dual channel canard and IPC, respectively. Based on the commands from the MCU, the canard drove its wings to deflect and the IPC drove the turntable to rotate. The simulation stopped until the distance between the projectile and target was less than the lethal radius. To ensure the efficient operation of the MCU and real-time communication between the devices, DMA was applied to the links with large amount of data.The parameters of the projectile, the parameters of the motion and system, and the parameters of the guidance and control were shown in Tables3–5, respectively. In order to reflect AFCBDSM, it does not need x2, zx2 used in equations (22) and (37). In addition, the radius of the seeker blind area was set as 50 m, where the control law of the canard deflection was unchanged and the projectile relied on inertial flight. The initial values of the adaptive fuzzy system parameters vectors κij were 0, and the following Gauss fuzzy membership function was adopted:(52)ϕaijlijsij=e−10sij+1−lij−1/22,i=3,4,5;j=1,2;lij=1,…,5,ϕaijlijs˙ij=e−5s˙ij+2−lij−1/22,i=3,4,5;j=1,2;lij=1,…,9.Table 3
Parameters of projectile.
Parameter
Value
m (kg)
43.500
l (m)
1.492
l
c (m)
0.217
d (m)
0.130
δ
c
max (deg)
15
τ
c
0.1
c
y
′
23.130
c
z
″
0.337
c
y
δ
z
4.925
m
z
′
−7.035
m
z
z
′
11.444
m
y
″
−0.337Table 4
Parameters of motion and system.
Parameter
Value
Initial position ofP (m)
(0, 4000, 0)
Initial position ofT (m)
(3000, 0, 300)
InitialθP, ψP (deg)
(−10, 0)
InitialθT, ψT (deg)
(0, 0)
θ
Q
f, ψQf (deg)
(−65, −10)
d
F
P
y
2, dFPz2 (N)
5
sin
t
d
M
P
z
4, dMPy4 (N·m)
5
cos
t
d
5
1, d52 (deg·s−1)
sin
t
τ
T
y
8
0.01
τ
T
z
8
0.01
v
P (m·s−1)
357
v
T (m·s−1)
30
w
x
0 (m·s−1)
5
w
y
0 (m·s−1)
5
w
z
0 (m·s−1)
5Table 5
Parameters of guidance and control.
Item
Value
β
21
11
8
σ
21
1
0.02
δ
1
1.5
τ
3
0.01
β
21
22
8
σ
21
2
0.01
δ
2
1.7
τ
4
0.01
β
22
11
80
η
21
1
0.1
k
2
11
0.3
τ
5
0.01
β
22
22
80
η
21
2
0.025
k
2
22
0.5
λ
3
1
50
β
23
11
550
μ
1
12
c
2
11
3
λ
3
2
60
β
23
22
550
μ
2
15
c
2
22
5To strike targets with sinusoidal and square wave maneuvers, the HILS and digital simulation of the AFCBDSM were carried out, respectively. For comparison, the adaptive dynamic surface control (ADSC) [23] and conventional guidance and control (G&C) design method [25] were simulated numerically under the same conditions. G&C used the PID controller and proportional navigation guidance (PNG) with navigation constant of 4. To reduce chatter of control law, ADSC used the following continuous saturation function instead of a symbolic function:(53)sat⋅=⋅⋅+δ,δ>0,where δ is a small positive constant.


### 5.1. Operating Condition 1: Sinusoidal Maneuvering Target

The acceleration commands for a sinusoidal maneuvering target were set as follows:(54)aTy8c=3sintm⋅s−2,aTz8c=−3sintm⋅s−2.The experimental results and curves are, respectively, shown in Figure3 and Table 6, where AFCBDSM-D, AFCBDSM-H, ADSC-D, and G&C-D represent digital simulations of AFCBDSM, HILS of AFCBDSM, digital simulation of ADSC, and digital simulation of G&C, respectively.Figure 3
Experimental curves of operating condition 1. (a) Trajectory of projectile and target. (b) Normal overload. (c) Quasi-attack angle. (d) Equal pitch canard angle. (e) Pitch canard angle. (f) LOS inclination angle rate. (g) Pitch angle. (h)zd21 and disturbance d21. (i) zd41 and disturbance d41.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table 6
Experimental results of operating condition 1.
Method
θ
Q error (deg)
ψ
Q error (deg)
Miss distance (m)
Hit time (s)
AFCBDSM-D
0.09
0.07
0.15
14.87
AFCBDSM-H
0.13
0.12
0.18
14.90
ADSC-D
0.27
0.24
0.31
15.12
G&C-D
0.38
0.35
0.53
15.34As shown in Figure3(a), the trajectory of the projectile and target illustrated that AFCBDSM, ADSC, and G&C could both make the guided projectile hit the sinusoidal maneuvering target. As shown in Table 6, one can obtain that the trajectory of the AFCBDSM was flatter than ADSC and G&C, which further improved the miss distance, hit time, and LOS angle tracking errors. This showed that the adaptive fuzzy control effectively improved the terminal guidance performance. The change in the normal overload can be analyzed from Figure 3(b). Because ADSC only used a continuous saturation function with constant coefficients, the chatter of the dynamic surface sliding mode was larger and the peak value and range of overload were significantly larger than that of the AFCBDSM, which made the variation of the control instructions smoother and converge near the end because of the adaptive fuzzy system and block dynamic surface, which is more in line with the actual operational needs.As shown in Figure3(c), the variation of α∗ was basically consistent with normal overload, which verifies the rationality of Assumption 2. Significantly affected by the channel coupling and winds during the terminal guidance process, ADSC and G&C mutated in the initial stage and even diverged in the final stage, which was not conducive to stable flight. Nevertheless, under the joint control of the ESO and adaptive fuzzy systems, the AFCBDSM reduced the peak value and variation range of α∗.As shown in Figure3(d), by introducing the adaptive Nussbaum function, the AFCBDSM effectively handled the nonlinear saturation and protected canard better, so as to avoid failing to reach the tactical index due to canard deflection saturation.Based on Figure3(e), the pitch channel of the canard could meet requirements of the actual canard deflection in the switching frequency and amplitude. Resulting from the adaptive adjustment of fuzzy parameters, the AFCBDSM further ameliorated the range and mutation of δz, unlike ADSC and G&C.The change ofθ˙Q is shown in Figure 3(f). Under the control of the AFCBDSM, θ˙Q maintained a stable convergent state after 8 s, which not only indicated x1 and x2 were able to converge to zero in finite time, but also verified the correctness of the finite-time convergent property of s2 given in Theorem 1.Figure3(g) shows that the change in θ was smooth and continuous during the guidance process. It also illustrated that the pitch channel of turntable exhibited good servo performance. Under the control of adaptive fuzzy systems and block dynamic surface sliding modes, the AFCBDSM exhibited not only a faster convergence rate than ADSC and G&C, but also smaller overshoot.Figures3(f), 3(h), and 3(i) revealed the good observational properties and robustness of the designed ESO, which could quickly provide accurate observations of θ˙Q and uncertain disturbances for effectively hitting a sinusoidal maneuvering target. Additionally, it could reduce the hardware design requirement of the guided projectile.


### 5.2. Operating Condition 2: Square Wave Maneuvering Target

The acceleration commands of a square wave maneuvering target were set as follows:(55)aTy8c=3signsintm⋅s−2,aTz8c=−3signsintm⋅s−2.The experimental results and curves were shown in Figure4 and Table 7, respectively.Figure 4
Experimental curves of operating condition 2. (a) Trajectory of projectile and target. (b) Lateral overload. (c) Quasi sideslip angle. (d) Equal yaw canard angle. (e) Yaw canard angle. (f) LOS azimuth angle rate. (g) Yaw angle. (h)zd22 and disturbance d22. (i) zd42 and disturbance d42.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table 7
Experimental results of operating condition 2.
Method
θ
Q error (deg)
ψ
Q error (deg)
Miss distance (m)
Hit time (s)
AFCBDSM-D
0.11
0.09
0.17
14.89
AFCBDSM-H
0.14
0.12
0.20
14.93
ADSC-D
0.32
0.28
0.34
15.17
G&C -D
0.41
0.37
0.58
15.41As shown in Figure4(a), both the AFCBDSM, ADSC, and G&C could be utilized in the terminal guidance stage of the guided projectile while attacking a square wave maneuvering target. And from Table 7, the AFCBDSM optimized miss distance, hit time, and LOS angle tracking errors, which was conducive to fine adjustments of the guided projectile in the terminal guidance process.The trend of lateral overload is demonstrated in Figure4(b). With fixed gain parameters, ADSC relied only on the dynamic surface sliding mode to stabilize the system, which caused the initial value to change greatly, the peak value to become large, and the terminal convergence rate to be slow. However, the AFCBDSM effectively weakened the chatter of the block dynamic surface sliding modes by the adaptive adjustment of the fuzzy parameters and the terminal convergence rate was faster, which demonstrates the robustness of the system.According to Figure4(c), the change in β∗ was basically consistent with lateral overload, which confirms the correctness of Assumption 2. Compared with ADSC and G&C, the projectile had a smaller peak and slower change of β∗ under the control of the AFCBDSM through the ESO and adaptive fuzzy systems, which could adapt to various unknown disturbances.As can be seen from Figure4(d), ADSC and G&C easily exhibited saturation of the canard deflection, owing to it did not account for the nonlinear saturation of the canard deflection. Moreover, under the internal and external disturbances of the system, δyeq varied widely and rapidly, which usually wastes precious energy and can even cause irreversible damage to canard. However, the AFCBDSM could avoid this, which verified the effectiveness of the adaptive Nussbaum gain function for handling control saturation of the actuator.As shown in Figure4(e), the yaw channel of the canard could meet the requirements of the actual canard deflection in the switching frequency and amplitude. Unlike ADSC and G&C, the AFCBDSM further optimized the range and mutation of δy.From the analysis of Figure4(f), the channel coupling effect acting on the yaw channel was greater than that acting on the pitch channel, which resulted in a slower convergence rate of ψ˙Q than that of θ˙Q. ADSC and G&C were more affected by disturbances due to the ineffective processing of disturbances and chatter, and it even failed to reach convergence at hitting time. Whereas, with the AFCBDSM, ψQ and ψ˙Q converged to zero after 14 s under the control of the designed ESO and adaptive fuzzy systems, which indicated s2 could converge in finite time.Because of the influence of channel coupling, the ESO could not completely observe the uncertain disturbanced42 initially. However, it subsequently entered a continuous smooth state quickly. From another perspective, the yaw channel of the turntable possessed good tracking characteristics.Figures4(f), 4(h), and 4(i) showed that the designed ESO possessed good observational properties and robustness. Even with fast variations and a wide range of disturbances, the ESO could observe the LOS azimuth angle rate and disturbances quickly and accurately, which provided necessary feedback information.


## 5.1. Operating Condition 1: Sinusoidal Maneuvering Target

The acceleration commands for a sinusoidal maneuvering target were set as follows:(54)aTy8c=3sintm⋅s−2,aTz8c=−3sintm⋅s−2.The experimental results and curves are, respectively, shown in Figure3 and Table 6, where AFCBDSM-D, AFCBDSM-H, ADSC-D, and G&C-D represent digital simulations of AFCBDSM, HILS of AFCBDSM, digital simulation of ADSC, and digital simulation of G&C, respectively.Figure 3
Experimental curves of operating condition 1. (a) Trajectory of projectile and target. (b) Normal overload. (c) Quasi-attack angle. (d) Equal pitch canard angle. (e) Pitch canard angle. (f) LOS inclination angle rate. (g) Pitch angle. (h)zd21 and disturbance d21. (i) zd41 and disturbance d41.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table 6
Experimental results of operating condition 1.
Method
θ
Q error (deg)
ψ
Q error (deg)
Miss distance (m)
Hit time (s)
AFCBDSM-D
0.09
0.07
0.15
14.87
AFCBDSM-H
0.13
0.12
0.18
14.90
ADSC-D
0.27
0.24
0.31
15.12
G&C-D
0.38
0.35
0.53
15.34As shown in Figure3(a), the trajectory of the projectile and target illustrated that AFCBDSM, ADSC, and G&C could both make the guided projectile hit the sinusoidal maneuvering target. As shown in Table 6, one can obtain that the trajectory of the AFCBDSM was flatter than ADSC and G&C, which further improved the miss distance, hit time, and LOS angle tracking errors. This showed that the adaptive fuzzy control effectively improved the terminal guidance performance. The change in the normal overload can be analyzed from Figure 3(b). Because ADSC only used a continuous saturation function with constant coefficients, the chatter of the dynamic surface sliding mode was larger and the peak value and range of overload were significantly larger than that of the AFCBDSM, which made the variation of the control instructions smoother and converge near the end because of the adaptive fuzzy system and block dynamic surface, which is more in line with the actual operational needs.As shown in Figure3(c), the variation of α∗ was basically consistent with normal overload, which verifies the rationality of Assumption 2. Significantly affected by the channel coupling and winds during the terminal guidance process, ADSC and G&C mutated in the initial stage and even diverged in the final stage, which was not conducive to stable flight. Nevertheless, under the joint control of the ESO and adaptive fuzzy systems, the AFCBDSM reduced the peak value and variation range of α∗.As shown in Figure3(d), by introducing the adaptive Nussbaum function, the AFCBDSM effectively handled the nonlinear saturation and protected canard better, so as to avoid failing to reach the tactical index due to canard deflection saturation.Based on Figure3(e), the pitch channel of the canard could meet requirements of the actual canard deflection in the switching frequency and amplitude. Resulting from the adaptive adjustment of fuzzy parameters, the AFCBDSM further ameliorated the range and mutation of δz, unlike ADSC and G&C.The change ofθ˙Q is shown in Figure 3(f). Under the control of the AFCBDSM, θ˙Q maintained a stable convergent state after 8 s, which not only indicated x1 and x2 were able to converge to zero in finite time, but also verified the correctness of the finite-time convergent property of s2 given in Theorem 1.Figure3(g) shows that the change in θ was smooth and continuous during the guidance process. It also illustrated that the pitch channel of turntable exhibited good servo performance. Under the control of adaptive fuzzy systems and block dynamic surface sliding modes, the AFCBDSM exhibited not only a faster convergence rate than ADSC and G&C, but also smaller overshoot.Figures3(f), 3(h), and 3(i) revealed the good observational properties and robustness of the designed ESO, which could quickly provide accurate observations of θ˙Q and uncertain disturbances for effectively hitting a sinusoidal maneuvering target. Additionally, it could reduce the hardware design requirement of the guided projectile.


## 5.2. Operating Condition 2: Square Wave Maneuvering Target

The acceleration commands of a square wave maneuvering target were set as follows:(55)aTy8c=3signsintm⋅s−2,aTz8c=−3signsintm⋅s−2.The experimental results and curves were shown in Figure4 and Table 7, respectively.Figure 4
Experimental curves of operating condition 2. (a) Trajectory of projectile and target. (b) Lateral overload. (c) Quasi sideslip angle. (d) Equal yaw canard angle. (e) Yaw canard angle. (f) LOS azimuth angle rate. (g) Yaw angle. (h)zd22 and disturbance d22. (i) zd42 and disturbance d42.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table 7
Experimental results of operating condition 2.
Method
θ
Q error (deg)
ψ
Q error (deg)
Miss distance (m)
Hit time (s)
AFCBDSM-D
0.11
0.09
0.17
14.89
AFCBDSM-H
0.14
0.12
0.20
14.93
ADSC-D
0.32
0.28
0.34
15.17
G&C -D
0.41
0.37
0.58
15.41As shown in Figure4(a), both the AFCBDSM, ADSC, and G&C could be utilized in the terminal guidance stage of the guided projectile while attacking a square wave maneuvering target. And from Table 7, the AFCBDSM optimized miss distance, hit time, and LOS angle tracking errors, which was conducive to fine adjustments of the guided projectile in the terminal guidance process.The trend of lateral overload is demonstrated in Figure4(b). With fixed gain parameters, ADSC relied only on the dynamic surface sliding mode to stabilize the system, which caused the initial value to change greatly, the peak value to become large, and the terminal convergence rate to be slow. However, the AFCBDSM effectively weakened the chatter of the block dynamic surface sliding modes by the adaptive adjustment of the fuzzy parameters and the terminal convergence rate was faster, which demonstrates the robustness of the system.According to Figure4(c), the change in β∗ was basically consistent with lateral overload, which confirms the correctness of Assumption 2. Compared with ADSC and G&C, the projectile had a smaller peak and slower change of β∗ under the control of the AFCBDSM through the ESO and adaptive fuzzy systems, which could adapt to various unknown disturbances.As can be seen from Figure4(d), ADSC and G&C easily exhibited saturation of the canard deflection, owing to it did not account for the nonlinear saturation of the canard deflection. Moreover, under the internal and external disturbances of the system, δyeq varied widely and rapidly, which usually wastes precious energy and can even cause irreversible damage to canard. However, the AFCBDSM could avoid this, which verified the effectiveness of the adaptive Nussbaum gain function for handling control saturation of the actuator.As shown in Figure4(e), the yaw channel of the canard could meet the requirements of the actual canard deflection in the switching frequency and amplitude. Unlike ADSC and G&C, the AFCBDSM further optimized the range and mutation of δy.From the analysis of Figure4(f), the channel coupling effect acting on the yaw channel was greater than that acting on the pitch channel, which resulted in a slower convergence rate of ψ˙Q than that of θ˙Q. ADSC and G&C were more affected by disturbances due to the ineffective processing of disturbances and chatter, and it even failed to reach convergence at hitting time. Whereas, with the AFCBDSM, ψQ and ψ˙Q converged to zero after 14 s under the control of the designed ESO and adaptive fuzzy systems, which indicated s2 could converge in finite time.Because of the influence of channel coupling, the ESO could not completely observe the uncertain disturbanced42 initially. However, it subsequently entered a continuous smooth state quickly. From another perspective, the yaw channel of the turntable possessed good tracking characteristics.Figures4(f), 4(h), and 4(i) showed that the designed ESO possessed good observational properties and robustness. Even with fast variations and a wide range of disturbances, the ESO could observe the LOS azimuth angle rate and disturbances quickly and accurately, which provided necessary feedback information.


## 6. Conclusion

Aiming at the terminal guidance stage for hitting coastal maneuvering targets, the AFCBDSM was designed for large caliber naval gun-guided projectiles. It was theoretically proven and demonstrated with HILS experiments that the AFCBDSM exhibited well-terminal guidance performances while hitting a target with different maneuvering forms. The results are highlighted as follows:(1)
With multiple constraints and various disturbances comprehensively considered, the IGC strict feedback cascade model for rolling naval gun-guided projectile was constructed(2)
The switching chatter of the block dynamic surface sliding mode was effectively weakened by the adaptive fuzzy system(3)
The finite time convergence of LOS angle tracking errors, LOS angle rate, and UUB of the closed-loop system were strictly proven(4)
The effectiveness and feasibility of the AFCBDSM were validated through the designed HILS system, which possesses certain theoretical and engineering values



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*Source: 6309462-2019-12-24.xml*