# Gliding-Guided Projectile Attitude Tracking Controller Design Based on Improved Adaptive Twisting Sliding Mode Algorithm

Authors: Wenguang Zhang; Wenjun Yi
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

The finite-time attitude tracking control for gliding-guided projectile with unmatched and matched disturbance is investigated. An adaptive variable observer is used to provide estimation for the unmeasured state which contains unmatched disturbance. Then, an improved adaptive twisting sliding mode algorithm is proposed to compensate for the matched disturbance dynamically with better transient quality. Finally, a proof of the finite-time convergence of the closed-loop system under the disturbance observer and the adaptive twisting sliding mode-based controller is derived using the Lyapunov technique. This attitude tracking control scheme does not require any information on the bounds of uncertainties. Simulation results demonstrate that the proposed method which is able to acquire the minimum possible values of the control gains guaranteeing the finite-time convergence performs well in chattering attenuation and tracking precision.


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## Body

## 1. Introduction

The most significant difference between gliding-guided projectile (GGP) and traditional projectile is that the former one is equipped with wing assembly. On the one hand, the wing assembly makes the projectile be capable of gliding, which enables the gliding-guided projectile to strike the target from a greater distance. On the other hand, the wing assembly provides the projectile with the ability of controlling its flight path, which means that the gliding-guided projectile can achieve precise strikes [1, 2]. Due to these advantages, the gliding-guided projectile has become the current research hotspot. It should be noted that the main way to ensure the stability of a projectile’s airframe is a high body roll-axis spin rate, which is a significant difference from traditional missiles, and is the source of nonlinear coupling between the yaw and pitch channel of GGP. Therefore, the design of GGP, especially the design of the attitude controller, is a challenging task. This is due to the fact that the GGP model suffers from high nonlinearity and strong coupling characteristics. In addition, a GGP is vulnerable to various disturbances during flight, such as the aerodynamic parameter perturbation, the unmodeled dynamics, and strong and sudden wind gusts which change greatly along the latitude and longitude [3].To compensate for uncertainties and disturbances suffered by the flight control system, a lot of works have been carried out. In [4], the authors propose a pitch attitude and lateral decoupling control based on active disturbance rejection control for aircraft. To obtain robustness, they utilize an extended state observer (ESO). In [5], the finite-time attitude control for a reusable launch vehicle with unmatched disturbance is investigated, and they propose an adaptive multivariable disturbance compensation method to estimate the disturbance. In [6], the adaptive control on the longitudinal dynamics of a hypersonic flight vehicle in the presence of wind effects is investigated. The unknown disturbance due to wind is estimated by neural networks. In [7], they propose a dual closed-loop control framework for attitude control of a quadrotor. In the dual-loop framework they proposed, active disturbance rejection control and proportional-derivative control are used in the inner and outer loops, respectively. To cancel the effect of dynamic disturbance, they use an ESO. In [8], they utilize the prescribed performance control technique to tackle the issue of attitude control of hypersonic flight vehicles (HFVs), which largely improves the transient characteristics of HFVs. In [9], the authors propose an intelligent flight control scheme based on reinforcement learning technique. Besides, many other control methods, such as fuzzy control, neural network control, backstepping control, and model following control, have been applied on this issue [10–20]. It should be noted that various feedback linearizations have been applied to this issue [21–23]. The aims of such methods are to cancel the nonlinear dynamics via mathematical transformation so as to control the system in a linear manner. It is able to provide a full envelope nonlinear flight control system. However, the design techniques need much information about the system; otherwise, the output will not be desired.Higher-order sliding mode (HOSM) control is another popular method to deal with an uncertain system. A twisting controller (TC) is historically the first 2-SMC algorithm [24] that drives the output and derivative of the system to the origin in finite time in the presence of disturbance whose boundary is known. If the boundary of the disturbance is unknown, it is necessary to design adaptive twisting sliding mode control (ATSMC) methods to satisfy the condition for sliding mode to exist. One efficient scheme is designing algorithms which are able to adjust dynamically the control gains. In [25], this kind of adaptive twisting sliding mode control is integrated into attitude tracking controller for the reentry reusable launch vehicle. In [26], the twisting sliding mode control law is modified with a proposed gain adaptation strategy to improve the attitude tracking performance of quadrotor unmanned aerial vehicles. In [27], the Lyapunov theory is used to design the ATSMC. However, the aforementioned methods fail to search the minimum possible value of the control gain. As we know, the amplitude of chattering which hinders the applications of the sliding mode control is proportional to the magnitude of discontinuous control.In [28], an adaptive algorithm is designed via equivalent control. Their method is able to search the minimum possible value of control. In their work, the gains can be adjusted adaptively to compensate for the disturbances with lower values without knowledge of the bounds of the disturbances. However, this method may lead to large chattering when the values of the gains are close to the ideal ones. This is due to the fact that the changing rate of the gains still keeps a large value in such circumstances.Motivated by the aforementioned work, we propose a robust attitude tracking scheme for GGP with chattering attenuation in this paper. The error dynamics of the attitude tracking system suffers from both matched and unmatched disturbances, where the bounds of disturbances are not known. In order to estimate the unmeasured state influenced by unmatched disturbance, an adaptive observer is utilized. Then, an improved equivalent control-based adaptive twisting sliding mode control (IECATSMC) algorithm for [28] is designed and adopted by the controller to compensate for the matched disturbance dynamically. The main contributions of this paper can be summarized as follows:
(i)
The improved adaption algorithm is able to adaptively adjust the changing rates of the gains on a larger scale. Thus, the transient characteristics of the adaptation strategy in [28] are improved. Since the proposed control scheme is able to acquire the minimum possible values of the control gains, it performs better in chattering attenuation than the fixed gain TSMC and the existing ATSMC, which can be seen from the simulation results given in the latter(ii)
A high-performance disturbance estimator proposed in [29] is used in this paper to reconstruct the unmeasurable system state, which extends the application of this estimator. More importantly, by doing so, the finite-time convergence property of the closed-loop system is guaranteed(iii)
A proof of the finite-time convergence of the closed-loop system under the observer and the IECATSMC-based controller is derived using the Lyapunov stability theoryThe paper is organized as follows: the GGP attitude kinematics model is given in Section2; then, error dynamics of attitude tracking system is derived. To estimate the unmeasured state in error dynamics, an adaptive observer is utilized in Section 3. The improved adaptive twisting algorithm which is able to search the minimum possible values of the control gains is investigated for the attitude control in Section 4, and a proof of the finite-time convergence for the overall system is derived. The simulation results are provided in Section 5. Finally, concluding remarks are summarized in Section 6.Notations. Throughout the paper the following notations will be used. For a given vector x=x1,⋯,xi,⋯,xnT∈R, denote ∥x∥=xTx. For a given scalar variable y, denote ∣y∣ as the absolute value of y.


## 2. Gliding-Guided Projectile Attitude Kinematics Model

In actual flight, the speed of the GGP is variable due to the varying thrust and drag. However, if we study the dynamics of the projectile which is gliding along a very short flight path, we can safely think that the flight speed and rotational speed of the projectile are constant. Besides, we can also neglect the variation of the dynamic pressure and Mach number as well as other physical parameters. Then, under the assumption that the attack angle and side slip angle are small, the attitude kinematics model of the gliding-guided projectile can be constructed as [30, 31]
(1)θ̇=α1α−α2β,−φ̇cosθ=α1β−α2α,ϑ¨=b1α−b2β+b4δz−α3δy−b3ϑ̇+b5cosϑ.ψ−b5ϑ̇,ψ¨cosϑ=b1β+b2α+b4δy−α3δz−b3cosϑ.ψ̇−b5ϑ̇,where ai and bi denote kinetic coefficients, where a1=QS/mvCLα+CLδ, a2=QS/mvCμ′′γ̇d/v, a3=QS/mvCLδ, b1=QS/Emz′+mσ′, b2=QSlγ̇D/Evmy′′, b3=QSlD/Evmzz′, b4=QSl/Emσ′, and b5=Cγ̇/E.By accepting that the attitude change speed is more faster than the change speed of the velocity vector direction of the projectile center of mass on the short flight path [30], and without loss of generity, we assume that ϑ̇≈α̇, φ̇≈β̇. From (2) and the assumption of α=ϑ−θ, β=ψ−φcosθ, and cosθ≈cosϑ, we obtain
(2)α̇=ϑ̇−a1α+a2β,β̇=ψ̇−φ̇cosϑ+ψ−φϑ̇sinϑ,dϑ−θdt2=−k1α̇−k2β̇+k3α−k4β+k5δz−k6δy,dψ−φcosθdt2=−k1β̇−k2α̇+k3β+k4α+k5δy−k6δz,where k1=b3+a1, k2=b5+a2, k3=b1+b3a1+b5a2, k4=b2−b3a2−b5a1, k5=b4, and k6=a3.Assume that the steering gear system of the projectile is designed as a second-order one. For the convenience of study, we only consider the steady output of equivalent rudder deflection [32]:
(3)δzδy=kskrcosγdsinγd−sinγdcosγdδzcδyc,where
(4)ks=11−Ts2γ̇22+2μsTsγ̇2,kr=τdγ̇+arccos1−Ts2γ̇21−Ts2γ̇22+2μsTsγ̇2,where δyc and δzc stand for equivalent rudder deflect angle control commands for pitching channel and yaw channel, respectively; ks is the gain of canard; Ts denotes the time constant of steer gear system; μs denotes the system damping ratio; τd represents delay time;, and γd represents delay phase angle.Taking matched and unmatched disturbances into consideration, from ((1)) and ((2)), we get
(5)α̇=ϑ̇−a1α+a2β+d1,β̇=ψ̇−φ̇cosϑ+ψ−φϑ̇sinϑ+d2dϑ−θdt2=f1+Δf1+A1ΘT+B1+ΔB1U+ḋ1,dψ−φcosθdt2=f2+Δf2+A2ΘT+B2+ΔB2U+ḋ2,where
(6)Θ=αβT,(7)U=δyc,δzcT,(8)f1=−k1α̇−k2β̇+k3α−k4β,(9)f2=−k1β̇−k2α̇+k3β+k4α,(10)A1=k3−k4T,(11)A2=k4k3T,(12)B1=κ6κ5T,(13)B2=κ5−κ6T,(14)κ5=k5kskrcosγd−k6kskrsinγd,(15)κ6=k5kskrsinγd+k6kskrcosγd,besides, Δf1, Δf2, ΔB1, and ΔB2 are uncertainties induced by parameter variation and model simplification; d1,d2 stand for external unmeasurable disturbances.The object is to design a control inputU=δyc,δzcT so as to drive the attitude angle vector Θ to follow the control command Θc=αc,βcT in spite of the internal and external disturbances.Thus, defining the attitude tracking error aseΘ=Θ−Θc, using (1) and recalling the relationships α=ϑ−θ and β=ψ−φcosθ, the error dynamics for (5) can be calculated as
(16)ėΘ=zΘ,żΘ=F+AΘ+BU−Θ¨c+Δ,where
(17)zΘ=ϑ̇−a1α+a2βψ̇−φ̇cosϑ+ψ−φϑ̇sinϑ−Θ̇c+Δ1,(18)Δ1=d1d2T,(19)F=f1f2T(20)A=A1A2T,(21)B=B1B2T,(22)Δ=Δf1+ΔU1Δf2+ΔU2+Δ̇1,where the deviations ΔU1=ΔB1U,ΔU2=ΔB2U induced by the uncertainties in B1 and B2 can be seen as internal disturbances.The following assumption adopted from [29] is needed to the existences of causal control schemes for system (5).Assumption 1.
The disturbancesΔf1, Δf2, ΔB1, ΔB2, d1, and d2 and the first-order derivatives of them are bounded, but the boundaries are unknown.


## 3. Finite-Time Adaptive Observer forzΘ

As we see from (6), the error state contains unknown disturbance Δ1; therefore, it cannot be measured directly. To addresss this problem, an adaptive finite-time variable observer is used for (5) in this section.Assumption 2.
Suppose that the disturbance∥Δ∥≤C and ∥Δ̇∥≤C′, where C and C′ are unknown.For system (5), the controller based on twisting algorithm can be designed as
(23)U=1B−F−AΘ−Θ¨c−r1eΘ∥eΘ∥−r2zΘ∥zΘ∥,where r1,r2>0 are control gains. It should be noted that the contoller (16) is impractical because of the existence of lumped disturbance Δ, whose upper bound is unknown. Meanwhile, as mentioned before, the error state zΘ is not measurable, so an adaptive finite-time observer is needed. Inspired by [29, 33], an adaptive finite-time observer for (16) is formulated as
(24)ϖ̇1=−kΓ1sΓ∥sΓ∥1/2+ϖ2,ϖ̇2=−kΓ2sΓ∥sΓ∥,where the auxiliary sliding variable
(25)sΓ=ϖ1−eΘ,and the adaptive gains kΓ1 and kΓ2 are
(26)k̇Γ1=ω1ι12,if∥sΓ∥>σT,0,if∥sΓ∥≤σT,kΓ2=ε1kΓ1,where σT represents the arbitrary small threshold value, and ω1, ι1, and ε1 are positive constants. It can be concluded that zΘ can be estimated by the term ϖ2 in finite time regardless of the unknown bounded uncertainty Δ [29].


## 4. Main Results

Ifr1 and r2 in (16) satisfy r1>2C and r2>C, the system represented by (5) is stable [34]. However, to make traditional twisting sliding mode control strategies meet the demand is a challenging thing because the upper bound of disturbance is often unknown especially in the case of gliding-guided projectile. Therefore, adaptive methods which can adjust the values of control gains have been proposed [27]. It should be noted that such methods usually cause large gains, which may result in an undesirable aggressive chattering phenomenon. In this section, an improved adaptive twisting sliding mode control method is developed to search the minimum possible values of control with better transient quality.Usingz^Θ estimated in (17), the actual adaptive twisting controller is then designed as
(27)U=1B−F−AΘ−Θ¨c−r1eΘ∥eΘ∥−r2z^Θ∥z^Θ∥,where the estimation z^Θ=ϖ2=zΘ+e with e represents the estimation error of zΘ. Substituting (25) into (5), the closed-loop system is changed into
(28)ėΘ=z^Θ−e,żΘ=−r1eΘ∥eΘ∥−r2z^Θ∥z^Θ∥+Δ^,where Δ^=ė+Δ.


### 4.1. Improved Adaptive Twisting Sliding Mode Control Algorithm Design

To simplify the adaptive controller design, the ideal estimate ofzΘ will be exploited, and the following representation for (26) can be considered:
(29)ẋ1=x2,ẋ2=−h1tx1∥x1∥−h2tx2∥x2∥+f,where the state variables x1,x2∈Rn, and the control gains h1=Lt,h2=0.5Lt with Lt to be designed. f represents lumped uncertainty, which satisfies ∥f∥≤D0 and ∥ḟ∥≤D1, where the constants D0 and D1 are unknown.Before we proceed, the following lemma is needed to be introduced.Lemma 3.
Considering the system defined as (29), if the gain L satisfies L>2D0, then the states x1 and x2 will converge to origin in finite time.
The proof of Lemma3 is given in Appendix A.Thus, the problem to be solved in the following part becomes to design a gain adaptive algorithm, which ensures the conditionL>2D0 holds without any knowledge about the disturbance f.Theorem 4.
For the uncertain systems (27), the dynamics of the control gains are designed as h1=Lt,h2=0.5Lt, where Lt is defined as
(30)L̇=eρξ−ζηLsignξ−ML−L+++Mμ−L+,z+≔1ifz≥0,0ifz<0,where ρ, ζ, η, M, and L+ are positive constants, and they satisfy M>eρ∣ξ∣−ζηL+, L>Dp0. ξ satisfies
(31)ξ=∥x1∥x1∥+0.5x2∥x2∥eq∥−ε,where −eq is called the equivalent control [23] function, which is an average value ranging from (-1, 1). ε is a positive constant and satisfies ε∈0,0.5. Then, the states xii=1,2 will converge to origin in finite time.Proof.
The analysis can be divided into three phases.Phase 1.
the gainL satisfies L<2D0.The system (28) is on reaching phase, so we have
(32)∥x1∥x1∥−0.5x2∥x2∥eq∥>∥x1∥x1∥∥−0.5∥x2∥x2∥∥=0.5.Recall thatε∈0,0.5, thus
(33)ξ=0.5−ε>0,so the value of L will keep increasing until it is large enough, which means ∫t0t1eρ∣ξ∣−ζηLsignξ−ML−L+++Mμ−L+dt>2D0 can be achieved.AssumingL<l+, we can have
(34)L̇≥ηL+Mμ−L+≥ηL,here, eρ∣ξ∣−ζ≥1 is used. Therefore, by solving the inequation L≤2D0, we can obtain t1≥ln∣2D0−Lt0+eηt0∣, which means the condition L≤2D0 can be achieved at most after ln∣2D0−Lt0+eηt0∣.Phase 2.
the gainL satisfies L>2D0.According to Lemma3, the states x1 and x2 will converge to the origin in finite time. Furthermore, the convergence time t2 satisfies t2≤t0+t1+16λ¯max3/4PV01/4t1/E¯, where the definitions of P and E¯ can be seen in Appendix A.Phase 3.
the system enters the sliding mode.First, Let us assume thatL∈μ,L+, which means ∥f∥/ε>μ, and the time derivative of signx1+0.5signx2eq exist. A Lyapunov function candidate is defined as Vξ=1/2ξ2. Since system (27) has entered the sliding mode in phase 3, the following equation holds
(35)∥ueq∥=L∥x1∥x1∥+0.5x2∥x2∥eq∥=∥f∥,where ueq=Lx1/∥x1∥+0.5x2/∥x2∥eq is defined as equivalent control input. Taking the time derivative of Vξ, assuming that Lt∈μ,L+, which means that ∥f∥/ε>μ and using (28), (29), and (33) yield
(36)V̇0ξ=ξ̇ξ=ξddt∥x1∥x1∥+0.5x2∥x2∥eq=ξddt∥f∥L=−∥f∥ξL−2L̇+ξL−1ddt∥f∥=−∥f∥ξL−2eρ∣ξ∣−ζηLsignξ−M−L−L+++Mμ−L++ξL−1fTḟ∥f∥−1≤−εeρ∣ξ∣−ζημ∣ξ∣L−1η∣ξ∣+∣ξ∣L−1D1=−∣ξ∣L−1εeρ∣ξ∣−ζημ−D1Ifη>D1/εμ∣eρ∣ξ∣−ζ, we will have
(37)V̇ξ≤−2εeρξ−ζημ−D1L+Vξ.Therefore,Vξ=0 is achievable, at least after tf=L+/εeρ∣ξ∣−ζημ−D1∣ξt2∣. After the adaption process for L is over t>tf, we obtain
(38)∥x1∥x1∥+0.5x2∥x2∥eq∥=∥f∥L=ε,which means L=∥f∥/ε. Another case is that ∥f∥/ε>μ, then L increases until Lt=μ, and it will be maintained at that level. The theorem is proven.Ifε is very close to 0.5, the minimal possible gain will be found for the current value of disturbance. As a result, the amplitude of chattering will be reduced.Remark 5.
The main feature of the improved adaptive algorithm is that it can adaptively adjust the changing rate of the gainL, which is realized by a regulatory factor eρ∣ξ∣−ζ. The factor enables the changing rate of L to be larger or smaller than η. In general, when ∣ξ∣ becomes larger than ζ, which means L is far away from its ideal value, eρ∣ξ∣−ζ can also increase and become larger than 1. Hence, L will tend more quickly toward its ideal value. On the other hand, once ∣ξ∣ enters a small region Cξ=∣ξ∣≤ζ, which means L has been already very close to its ideal value, eρ∣ξ∣−ζ will become smaller than 1. As a result, the amplitude of gain oscillation will be reduced so that the condition L>2∥Δ∥ can be safely ensured.Remark 6.
The functionx1/∥x1∥+0.5x2/∥x2∥eq can be derived by low pass filter
(39)τż1+z1=x1∥x1∥+0.5x2∥x2∥,z1=0,0T,with a small time constant τ>0, respectively. The output z1 satisfies
(40)∥z1−x1∥x1∥+0.5x2∥x2∥eq∥≤Oτ→τ→00.


### 4.2. Stability Analysis for the Closed-Loop System

In this subsection, the stability of the gliding guided projectile closed-loop system (5) is analyzed.Theorem 7.
For the attitude system (5), the adaptive twisting controller is designed as (25), with the adaptive gains defined as r1=Lt,r2=0.5Lt, where Lt is obtained from Theorem 4; then, the states eΘ,zΘ will converge to the origin in finite time.
The proof of Theorem7 is given in Appendix B.In order to illustrate the entire control architecture in this paper, the schematic diagram is shown in Figure1.Figure 1
Schematic diagram of the proposed control strategy.


## 4.1. Improved Adaptive Twisting Sliding Mode Control Algorithm Design

To simplify the adaptive controller design, the ideal estimate ofzΘ will be exploited, and the following representation for (26) can be considered:
(29)ẋ1=x2,ẋ2=−h1tx1∥x1∥−h2tx2∥x2∥+f,where the state variables x1,x2∈Rn, and the control gains h1=Lt,h2=0.5Lt with Lt to be designed. f represents lumped uncertainty, which satisfies ∥f∥≤D0 and ∥ḟ∥≤D1, where the constants D0 and D1 are unknown.Before we proceed, the following lemma is needed to be introduced.Lemma 3.
Considering the system defined as (29), if the gain L satisfies L>2D0, then the states x1 and x2 will converge to origin in finite time.
The proof of Lemma3 is given in Appendix A.Thus, the problem to be solved in the following part becomes to design a gain adaptive algorithm, which ensures the conditionL>2D0 holds without any knowledge about the disturbance f.Theorem 4.
For the uncertain systems (27), the dynamics of the control gains are designed as h1=Lt,h2=0.5Lt, where Lt is defined as
(30)L̇=eρξ−ζηLsignξ−ML−L+++Mμ−L+,z+≔1ifz≥0,0ifz<0,where ρ, ζ, η, M, and L+ are positive constants, and they satisfy M>eρ∣ξ∣−ζηL+, L>Dp0. ξ satisfies
(31)ξ=∥x1∥x1∥+0.5x2∥x2∥eq∥−ε,where −eq is called the equivalent control [23] function, which is an average value ranging from (-1, 1). ε is a positive constant and satisfies ε∈0,0.5. Then, the states xii=1,2 will converge to origin in finite time.Proof.
The analysis can be divided into three phases.Phase 1.
the gainL satisfies L<2D0.The system (28) is on reaching phase, so we have
(32)∥x1∥x1∥−0.5x2∥x2∥eq∥>∥x1∥x1∥∥−0.5∥x2∥x2∥∥=0.5.Recall thatε∈0,0.5, thus
(33)ξ=0.5−ε>0,so the value of L will keep increasing until it is large enough, which means ∫t0t1eρ∣ξ∣−ζηLsignξ−ML−L+++Mμ−L+dt>2D0 can be achieved.AssumingL<l+, we can have
(34)L̇≥ηL+Mμ−L+≥ηL,here, eρ∣ξ∣−ζ≥1 is used. Therefore, by solving the inequation L≤2D0, we can obtain t1≥ln∣2D0−Lt0+eηt0∣, which means the condition L≤2D0 can be achieved at most after ln∣2D0−Lt0+eηt0∣.Phase 2.
the gainL satisfies L>2D0.According to Lemma3, the states x1 and x2 will converge to the origin in finite time. Furthermore, the convergence time t2 satisfies t2≤t0+t1+16λ¯max3/4PV01/4t1/E¯, where the definitions of P and E¯ can be seen in Appendix A.Phase 3.
the system enters the sliding mode.First, Let us assume thatL∈μ,L+, which means ∥f∥/ε>μ, and the time derivative of signx1+0.5signx2eq exist. A Lyapunov function candidate is defined as Vξ=1/2ξ2. Since system (27) has entered the sliding mode in phase 3, the following equation holds
(35)∥ueq∥=L∥x1∥x1∥+0.5x2∥x2∥eq∥=∥f∥,where ueq=Lx1/∥x1∥+0.5x2/∥x2∥eq is defined as equivalent control input. Taking the time derivative of Vξ, assuming that Lt∈μ,L+, which means that ∥f∥/ε>μ and using (28), (29), and (33) yield
(36)V̇0ξ=ξ̇ξ=ξddt∥x1∥x1∥+0.5x2∥x2∥eq=ξddt∥f∥L=−∥f∥ξL−2L̇+ξL−1ddt∥f∥=−∥f∥ξL−2eρ∣ξ∣−ζηLsignξ−M−L−L+++Mμ−L++ξL−1fTḟ∥f∥−1≤−εeρ∣ξ∣−ζημ∣ξ∣L−1η∣ξ∣+∣ξ∣L−1D1=−∣ξ∣L−1εeρ∣ξ∣−ζημ−D1Ifη>D1/εμ∣eρ∣ξ∣−ζ, we will have
(37)V̇ξ≤−2εeρξ−ζημ−D1L+Vξ.Therefore,Vξ=0 is achievable, at least after tf=L+/εeρ∣ξ∣−ζημ−D1∣ξt2∣. After the adaption process for L is over t>tf, we obtain
(38)∥x1∥x1∥+0.5x2∥x2∥eq∥=∥f∥L=ε,which means L=∥f∥/ε. Another case is that ∥f∥/ε>μ, then L increases until Lt=μ, and it will be maintained at that level. The theorem is proven.Ifε is very close to 0.5, the minimal possible gain will be found for the current value of disturbance. As a result, the amplitude of chattering will be reduced.Remark 5.
The main feature of the improved adaptive algorithm is that it can adaptively adjust the changing rate of the gainL, which is realized by a regulatory factor eρ∣ξ∣−ζ. The factor enables the changing rate of L to be larger or smaller than η. In general, when ∣ξ∣ becomes larger than ζ, which means L is far away from its ideal value, eρ∣ξ∣−ζ can also increase and become larger than 1. Hence, L will tend more quickly toward its ideal value. On the other hand, once ∣ξ∣ enters a small region Cξ=∣ξ∣≤ζ, which means L has been already very close to its ideal value, eρ∣ξ∣−ζ will become smaller than 1. As a result, the amplitude of gain oscillation will be reduced so that the condition L>2∥Δ∥ can be safely ensured.Remark 6.
The functionx1/∥x1∥+0.5x2/∥x2∥eq can be derived by low pass filter
(39)τż1+z1=x1∥x1∥+0.5x2∥x2∥,z1=0,0T,with a small time constant τ>0, respectively. The output z1 satisfies
(40)∥z1−x1∥x1∥+0.5x2∥x2∥eq∥≤Oτ→τ→00.


## 4.2. Stability Analysis for the Closed-Loop System

In this subsection, the stability of the gliding guided projectile closed-loop system (5) is analyzed.Theorem 7.
For the attitude system (5), the adaptive twisting controller is designed as (25), with the adaptive gains defined as r1=Lt,r2=0.5Lt, where Lt is obtained from Theorem 4; then, the states eΘ,zΘ will converge to the origin in finite time.
The proof of Theorem7 is given in Appendix B.In order to illustrate the entire control architecture in this paper, the schematic diagram is shown in Figure1.Figure 1
Schematic diagram of the proposed control strategy.


## 5. Simulation and Results

To verify the effectiveness of the adaptive algorithm-based controller, the attitude motion of a certain type of GGP is taken as the simulation project. The parameters of the GGP are shown in Table1. The initial values for the plant are x0=0.160−0.10470T. Reference commands are taken as αc=cos0.5πt+10 deg, βc=0 deg.Table 1
Parameters for a gliding-guided projectile.
b1/s−2b2/s−2b3/s−2b4/s−2b5/s−2−129.400.29190.5191.886a1/s−1a2/s−1a3/s−1γ̇r/sτ/s0.28700.136100.015ksTs/sμsδmax/oδ̇max/rad/s11/1500.720200To obtain the equivalent control information, we use a low-pass filter formulated as (37) in the simulation, and the parameter is set as τ=0.015. Simulations are performed in MATLAB/simulink environment, where sample time is set as 0.001 s, and fourth-order Runge-Kutta method is chosen as the numerical integration method.


### 5.1. Comparison

In the simulation, severe disturbances are considered.Δf1, Δf2 are the 40% uncertainty in f1, f2, respectively. ΔB1 and ΔB2 are the 10% uncertainty in B1 and B2, respectively. The external disturbances are
(41)d1=9t−8cost2−14cost5,0≤t≤6,49t−8cost2−14cost5,t>6,d2=t+2πcosπt2−3πcosπt3,0≤t≤6,4t+2πcosπt2−3πcosπt3,t>6.For performance comparison, the ATSMC proposed in [34] and CTSMC proposed in [24] are also simulated under the same conditions. The gain adaptive schemes are formulated as
(i)
ATSMC [34]: the gain adaptive schemeL̇ is defined as(42)L̇=ϖ1/2γ11/γ1−2Lx1Tx1+∥x1∥x2Tx2/L∗−L3signV0−μ,ifL≥Lmin,χ,ifL>Lmin.The termV0 can be computed by
(43)V0=L2x1Tx1+γ1x1Tx2∥x1∥1/2+L∥x1∥x2Tx2+14∥x2∥4(ii)
CTSMC [24]: the gain adaptive scheme L̇ is defined as(44)L̇=0The parameters of the three methods are listed in Table2. Besides, to validate the effectiveness of the improved algorithm, the method named ECATSMC whose gain adaptive law is defined as [28]:
(45)L̇=ηLsignξ−ML−L+++Mμ−L+,is also simulated, and its control parameters are the same with those of IECATSMC.Table 2
Control parameters of the three algorithms.
MethodParametersIECATSMCρ=2, ζ=0.07, η=5.59, L+=50,μ=0.1, M=279.5279, ε=0.49, ω1=4.9, ι1=2.4, ε1=0.6, σT=10−3ATSMC [34]Lmin=32,ϖ1=4.9,γ1=1.1,μ1=0.0004,χ=40CTSMC [24]L=20Figure2 shows the curves of the estimation error of zΘ and the adaptive gains of the observer kΓ=kΓ1kΓ2T. From Figure 2(a), we can see that the estimation error can converge to the origin in a short time. Besides, it is shown in Figure 2(b) that the adaptive gains of the observer keep increasing when the estimation error is large; then, they remain constant.Figure 2
The curves of estimation errors and adaptive gains of the observer.
(a)(b)Figure3 denotes the tracking effect of yaw and pitch commands of the four algorithms. As we can see from Figure 3(a), they both can track the reference commands precisely. After the control begins, CTSMC responses fastest in tracking the commands, IEATSMC takes the second place, and ECATSMC takes the third place, ATSMC responds a little slower than the other three methods. What is more, we can also see from Figure 3(a) that IECATSMC and ECATSMC perform better in terms of tracking accuracy. CTSMC performs a little worse. The tracking effects for pitch command are shown in Figure 3(b). As we see in the picture, ECATSMC responds a little slower than the other three methods. Despite this, ECATSMC can track the command more precisely than CTSMC and ATSMC. Besides, we can also observe that the curve of pith angle under ATSMC deviates from the reference command curve around t=8 s. This is due to the fact that the intensive disturbance is imposed around that time. On the contrary, the tracking effects of the other three methods do not degrade.Figure 3
Tracking effect of yaw and pitch commands.
(a)(b)Figure4 shows the curves of the adaptive gains. We can see from the picture that the gains of IECATSMC and ECATSMC are slightly larger than 2∣Δ^∣ in almost the whole control process. It should be noted that this approximation can be easily adjusted by choosing the value of ε. We can also see that ATSMC can increase the gain to satisfy L>2∥Δ^∥ when 2∥Δ^∥ is larger than L. However, in other cases, the gain does not change. Besides, Figure 4 shows that the control gain L under CTSMC keep constant, although it satisfies L>2∥Δ^∥, its value is difficult to be set in practical applications given the upper boundary of disturbance is unknown.Figure 4
Adaptive gain curves.Figure5 shows the comparison between IECATSMC and ECATSMC and the curve of the exponential term. From Figure 5(a), we can see that at the beginning of the control process, the exponential term automatively increases to a large value to accelerate the changing rate of the gain L. Actually, when there is a sudden change occurring in ∥Δ^∥, the exponential term can always adjust its value in time. As a result, IECATSMC reponses faster than ECATSMC, which is demonstrated in Figure 5(b). What is more, when the disturbance changes gently, the exponential term can adjust its value around 1. The result is that IECATSMC has a smaller amplitude of oscillation than ECATSMC, which enables the former one to seldomly violate the condition of L>2∥Δ^∥.Figure 5
Comparison between IECATSMC and ECATSMC.
(a)(b)Figure6 denotes the curves of equivalent canard angle commands and equivalent canard angles. Because the results of IECATSMC and ECATSMC are very similar, here, we only show the results of the former one for a clear comparison with the other two methods. Because the values of equivalent canard angle commands are closely related with those of the control gains, we can see from Figures 6(a) and 6(b) that IECATSMC greatly attenuates the amplitude of chattering. This is also reflected in Figures 6(c) and 6(d).Figure 6
Equivalent canard angle commands.
(a)(b)(c)(d)Figure7 shows how the gain is adjusted in the three phases in a clearer manner. As we see from the picture, the gains r1=L and r2=0.5L continue to increase after control begins, and L>2∥Δ^∥ is satisfied after t>t1. The gains reach their maximum on t=t2. After that, the equivalent control input ∥ueq∥ equals the disturbance; thus, the gains keep decreasing until their curves follow the change of 2∥Δ^∥. Therefore, Figure 6 verifies the proof.Figure 7
Gains in three phases.


### 5.2. Monte Carlo Experiments

In order to further validate the robustness of the proposed control scheme, Monte Carlo experiments are performed 100 times. The external and internal disturbances used in this subsection are the same as that in Section5.1, and the standard derivations of the errors of the aerodynamic parameters are set to be 30% of the nominal values.It can be observed from Figure8 that the corresponding curves are kept within small envelops, and all of them can track the desired commands accurately.Figure 8
Tracking effect of yaw and pitch commands.
(a)(b)


## 5.1. Comparison

In the simulation, severe disturbances are considered.Δf1, Δf2 are the 40% uncertainty in f1, f2, respectively. ΔB1 and ΔB2 are the 10% uncertainty in B1 and B2, respectively. The external disturbances are
(41)d1=9t−8cost2−14cost5,0≤t≤6,49t−8cost2−14cost5,t>6,d2=t+2πcosπt2−3πcosπt3,0≤t≤6,4t+2πcosπt2−3πcosπt3,t>6.For performance comparison, the ATSMC proposed in [34] and CTSMC proposed in [24] are also simulated under the same conditions. The gain adaptive schemes are formulated as
(i)
ATSMC [34]: the gain adaptive schemeL̇ is defined as(42)L̇=ϖ1/2γ11/γ1−2Lx1Tx1+∥x1∥x2Tx2/L∗−L3signV0−μ,ifL≥Lmin,χ,ifL>Lmin.The termV0 can be computed by
(43)V0=L2x1Tx1+γ1x1Tx2∥x1∥1/2+L∥x1∥x2Tx2+14∥x2∥4(ii)
CTSMC [24]: the gain adaptive scheme L̇ is defined as(44)L̇=0The parameters of the three methods are listed in Table2. Besides, to validate the effectiveness of the improved algorithm, the method named ECATSMC whose gain adaptive law is defined as [28]:
(45)L̇=ηLsignξ−ML−L+++Mμ−L+,is also simulated, and its control parameters are the same with those of IECATSMC.Table 2
Control parameters of the three algorithms.
MethodParametersIECATSMCρ=2, ζ=0.07, η=5.59, L+=50,μ=0.1, M=279.5279, ε=0.49, ω1=4.9, ι1=2.4, ε1=0.6, σT=10−3ATSMC [34]Lmin=32,ϖ1=4.9,γ1=1.1,μ1=0.0004,χ=40CTSMC [24]L=20Figure2 shows the curves of the estimation error of zΘ and the adaptive gains of the observer kΓ=kΓ1kΓ2T. From Figure 2(a), we can see that the estimation error can converge to the origin in a short time. Besides, it is shown in Figure 2(b) that the adaptive gains of the observer keep increasing when the estimation error is large; then, they remain constant.Figure 2
The curves of estimation errors and adaptive gains of the observer.
(a)(b)Figure3 denotes the tracking effect of yaw and pitch commands of the four algorithms. As we can see from Figure 3(a), they both can track the reference commands precisely. After the control begins, CTSMC responses fastest in tracking the commands, IEATSMC takes the second place, and ECATSMC takes the third place, ATSMC responds a little slower than the other three methods. What is more, we can also see from Figure 3(a) that IECATSMC and ECATSMC perform better in terms of tracking accuracy. CTSMC performs a little worse. The tracking effects for pitch command are shown in Figure 3(b). As we see in the picture, ECATSMC responds a little slower than the other three methods. Despite this, ECATSMC can track the command more precisely than CTSMC and ATSMC. Besides, we can also observe that the curve of pith angle under ATSMC deviates from the reference command curve around t=8 s. This is due to the fact that the intensive disturbance is imposed around that time. On the contrary, the tracking effects of the other three methods do not degrade.Figure 3
Tracking effect of yaw and pitch commands.
(a)(b)Figure4 shows the curves of the adaptive gains. We can see from the picture that the gains of IECATSMC and ECATSMC are slightly larger than 2∣Δ^∣ in almost the whole control process. It should be noted that this approximation can be easily adjusted by choosing the value of ε. We can also see that ATSMC can increase the gain to satisfy L>2∥Δ^∥ when 2∥Δ^∥ is larger than L. However, in other cases, the gain does not change. Besides, Figure 4 shows that the control gain L under CTSMC keep constant, although it satisfies L>2∥Δ^∥, its value is difficult to be set in practical applications given the upper boundary of disturbance is unknown.Figure 4
Adaptive gain curves.Figure5 shows the comparison between IECATSMC and ECATSMC and the curve of the exponential term. From Figure 5(a), we can see that at the beginning of the control process, the exponential term automatively increases to a large value to accelerate the changing rate of the gain L. Actually, when there is a sudden change occurring in ∥Δ^∥, the exponential term can always adjust its value in time. As a result, IECATSMC reponses faster than ECATSMC, which is demonstrated in Figure 5(b). What is more, when the disturbance changes gently, the exponential term can adjust its value around 1. The result is that IECATSMC has a smaller amplitude of oscillation than ECATSMC, which enables the former one to seldomly violate the condition of L>2∥Δ^∥.Figure 5
Comparison between IECATSMC and ECATSMC.
(a)(b)Figure6 denotes the curves of equivalent canard angle commands and equivalent canard angles. Because the results of IECATSMC and ECATSMC are very similar, here, we only show the results of the former one for a clear comparison with the other two methods. Because the values of equivalent canard angle commands are closely related with those of the control gains, we can see from Figures 6(a) and 6(b) that IECATSMC greatly attenuates the amplitude of chattering. This is also reflected in Figures 6(c) and 6(d).Figure 6
Equivalent canard angle commands.
(a)(b)(c)(d)Figure7 shows how the gain is adjusted in the three phases in a clearer manner. As we see from the picture, the gains r1=L and r2=0.5L continue to increase after control begins, and L>2∥Δ^∥ is satisfied after t>t1. The gains reach their maximum on t=t2. After that, the equivalent control input ∥ueq∥ equals the disturbance; thus, the gains keep decreasing until their curves follow the change of 2∥Δ^∥. Therefore, Figure 6 verifies the proof.Figure 7
Gains in three phases.


## 5.2. Monte Carlo Experiments

In order to further validate the robustness of the proposed control scheme, Monte Carlo experiments are performed 100 times. The external and internal disturbances used in this subsection are the same as that in Section5.1, and the standard derivations of the errors of the aerodynamic parameters are set to be 30% of the nominal values.It can be observed from Figure8 that the corresponding curves are kept within small envelops, and all of them can track the desired commands accurately.Figure 8
Tracking effect of yaw and pitch commands.
(a)(b)


## 6. Conclusion

In this paper, a robust attitude tracking control scheme is proposed for the gliding-guided projectile. The proposed control method does not require any information on the bounds of disturbances. Simulation results show that the proposed method is not only more efficient in chattering attenuation than existing algorithms but also more precise in tracking performance. Besides, in order to make the proposed attitude tracking controller be more robust, actuator-fault situation should be taken into consideration. Therefore, in our future work, we will extend the proposed control scheme with a fault-tolerant approach.



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*Source: 1480427-2020-10-31.xml*