# Discrete Sliding Mode Control for Hypersonic Cruise Missile

Authors: Yong Hua Fan; Peng Peng Yan; Fan Wang; Hong Yang Xu
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

A discrete variable structure control (DVSC) with sliding vector is presented to track the velocity and acceleration command for a hypersonic cruise missile. In the design an integrator is augmented to ensure the tracking with zero steady-state errors. Furthermore the sliding surface of acceleration is designed using the error of acceleration and acceleration rate to avoid the singularity of control matrix. A proper power rate reaching law is utilized in this proposal; therefore the state trajectory from any initial point can be driven into the sliding surface. Besides, in order to validate the robustness of controller, the unmolded dynamic and parameter disturbance of the missile are considered. Through simulation the proposed controller demonstrates good performance in tracking velocity and acceleration command.


---

## Body

## 1. Introduction

Recently hypersonic vehicle has attracted attention because of its potential military and civil application [1–7]. With the breakthrough of the scramjet and heat safeguard, the hypersonic vehicle can be developed into cruise missile to realize global strike. The control system of hypersonic cruise missile exhibits high performance of tracking both velocity command and guidance command. Moreover the model of hypersonic missile has unmolded dynamic and parameter disturbance. Thus it is a challenge for hypersonic missile autopilot design [8].The exploration of the various design methods for hypersonic aircraft has been an active field. Matthew [9] reported an adaptive linear quadratic (ALQ) altitude and velocity tracking control algorithm for the longitudinal model of a generic air-breathing hypersonic flight vehicle. A nonlinear controller for an air-breathing hypersonic vehicle was designed by Fiorentini et al. [10]. The sliding variable structure control also shows its advantage in robustness to the disturbance [11–14]. Using this method, Xu et al. [4] presented an adaptive sliding mode control for hypersonic aircraft with nonlinear and model uncertainty. A high-order sliding mode control was investigated by Yang and Tian [11] and then a sliding variable structure controller is designed for a reentry hypersonic missile. Yu and Yuri [12] used high-order sliding surface to design a hypersonic missile controller with the trajectory restriction. In case there is uncertainty, neural network based hypersonic flight control research could be found in literatures [14–17].Although these literatures have satisfactory effect for a continuous plant, the computer on board always works discontinuously, and thus controller on the basis of continuous system cannot offer the same performance by a digital computer with a certain sampling interval. This can cause the system instability [13, 18–20].This paper presents a design method of hypersonic cruise missile autopilot using discrete variable structure control. The main task of autopilot is to track commands of the acceleration and velocity simultaneously. So the multiple sliding surfaces are designed (for both subsystems of acceleration and velocity). To track with zero steady errors, an integrator is introduced into them. In this paper, chattering is reduced by a proper design of power rate reaching law. The unmolded dynamics and uncertainties are also included to examine the robustness of the discrete variable structure controller.This paper is organized as follows. The dynamics and discrete model of hypersonic cruise missile are given in Section2. And the design of discrete variable structure controller is presented in Section 3. Then we conduct simulations and give the results to demonstrate the reasonability of our design. Finally, we make a conclusion in Section 5.


## 2. Model of Hypersonic Cruise Missile

### 2.1. Description of Hypersonic Cruise Missile

The configuration of hypersonic cruise missile with scramjet is shown as in Figure1.Figure 1
Configuration of hypersonic cruise missile.The cowl of scramjet is arranged in the mandible of hypersonic cruise missile. The fore body is unsymmetrical, where the guidance system and warhead are located. And the lower fore body is the inlet of engine. Two elevator-ailerons and two rudders are used to control pith, roll, and yaw movement. The thrust is adjusted by throttle. The longitudinal model of the cruise missile considered in this paper is(1)V˙=Tcos⁡α-Dm-gsin⁡γ,γ˙=L+Tsin⁡αmV-gcos⁡γV,α=θ-γ,q˙=MyyIyy,T=12ρV2SCT,D=12ρV2SCD,L=12ρV2SCL,Myy=12ρV2Sc-CMα+CMδe+CMq,where the expressions of aerodynamic coefficients are denoted by CL, CD, and CM, respectively, and CT denotes thrust coefficient.


### 2.2. Discrete Model of Cruise Missile

Letx=[θ,γ,α,V,q]T be the state vector, and u=[δt,δe]T is the control vector. The longitudinal model of the cruise missile can be linearized: (2)
X
˙
t
=
A
X
t
+
B
U
t
,
(3)
X
=
θ
,
γ
,
α
,
V
,
q
T
,
(4)
U
=
δ
t
,
δ
e
T
=
U
1
,
U
2
T
,
(5)
A
=
0
0
0
0
1
0
V
cos
⁡
γ
0
sin
⁡
γ
0
0
g
sin
⁡
γ
V
T
+
L
α
m
V
T
V
α
+
L
V
m
V
0
0
-
g
sin
⁡
γ
V
-
T
+
L
α
m
V
-
T
V
α
+
L
V
m
V
1
0
M
y
y
γ
I
y
y
M
y
y
α
I
y
y
M
y
y
V
I
y
y
M
y
y
q
I
y
y
,
(6)
B
=
0
0
0
0
T
δ
t
m
0
0
0
L
δ
e
m
V
-
L
δ
e
m
V
0
M
y
y
δ
e
I
y
y
T
,where θ, γ, α, V, and q represent velocity, pitch angle, flight trajectory angle, angle of attack, and pitch rate, respectively. δe is the angle of elevator and δt is the control of throttle setting.AssumingTs is the sampling interval of onboard computer and the plant is integrated in a sampling interval, the discrete state equations are(7)Xk+1=dA·Xk+dB·Uk,(8)Xk=θkγkαkVkqkT,(9)dA=eATs,(10)dB=∫0TseAtdtB,Uk=δtkδekT.


## 2.1. Description of Hypersonic Cruise Missile

The configuration of hypersonic cruise missile with scramjet is shown as in Figure1.Figure 1
Configuration of hypersonic cruise missile.The cowl of scramjet is arranged in the mandible of hypersonic cruise missile. The fore body is unsymmetrical, where the guidance system and warhead are located. And the lower fore body is the inlet of engine. Two elevator-ailerons and two rudders are used to control pith, roll, and yaw movement. The thrust is adjusted by throttle. The longitudinal model of the cruise missile considered in this paper is(1)V˙=Tcos⁡α-Dm-gsin⁡γ,γ˙=L+Tsin⁡αmV-gcos⁡γV,α=θ-γ,q˙=MyyIyy,T=12ρV2SCT,D=12ρV2SCD,L=12ρV2SCL,Myy=12ρV2Sc-CMα+CMδe+CMq,where the expressions of aerodynamic coefficients are denoted by CL, CD, and CM, respectively, and CT denotes thrust coefficient.


## 2.2. Discrete Model of Cruise Missile

Letx=[θ,γ,α,V,q]T be the state vector, and u=[δt,δe]T is the control vector. The longitudinal model of the cruise missile can be linearized: (2)
X
˙
t
=
A
X
t
+
B
U
t
,
(3)
X
=
θ
,
γ
,
α
,
V
,
q
T
,
(4)
U
=
δ
t
,
δ
e
T
=
U
1
,
U
2
T
,
(5)
A
=
0
0
0
0
1
0
V
cos
⁡
γ
0
sin
⁡
γ
0
0
g
sin
⁡
γ
V
T
+
L
α
m
V
T
V
α
+
L
V
m
V
0
0
-
g
sin
⁡
γ
V
-
T
+
L
α
m
V
-
T
V
α
+
L
V
m
V
1
0
M
y
y
γ
I
y
y
M
y
y
α
I
y
y
M
y
y
V
I
y
y
M
y
y
q
I
y
y
,
(6)
B
=
0
0
0
0
T
δ
t
m
0
0
0
L
δ
e
m
V
-
L
δ
e
m
V
0
M
y
y
δ
e
I
y
y
T
,where θ, γ, α, V, and q represent velocity, pitch angle, flight trajectory angle, angle of attack, and pitch rate, respectively. δe is the angle of elevator and δt is the control of throttle setting.AssumingTs is the sampling interval of onboard computer and the plant is integrated in a sampling interval, the discrete state equations are(7)Xk+1=dA·Xk+dB·Uk,(8)Xk=θkγkαkVkqkT,(9)dA=eATs,(10)dB=∫0TseAtdtB,Uk=δtkδekT.


## 3. Design of Variable Structure Controller

The flight control system of hypersonic cruise missile is designed to track the commands of velocity and acceleration. Define the error of tracking as(11)ayk-ayck⟶0,Vk-Vck⟶0.Let(12)E=ay+a˙y.a
y is longitudinal acceleration, a˙y is the rate of longitudinal acceleration, and the tracking error can be written as(13)Ek-Eck⟶0,Vk-Vck⟶0.The structure of the discrete control system of the hypersonic cruise missile is shown as in Figure2.Figure 2
Structure of the discrete control system.


### 3.1. Design of Discrete Sliding Surface

Selecting multiple sliding surfaces,(14)Sk+1=S1k+1S2k+1=00.If the acceleration and velocity subsystem are chosen directly as the sliding surface,(15)S1k+1=ayk+1-ayk+c1ayk-ayck,(16)S2k+1=Vk+1-Vk+c2Vk-Vck,where c1 and c2 are the unknown parameters.The acceleration can be written as(17)ay=Vγ˙.Differentiating the acceleration expression, (17) yields(18)a˙y=Vγ¨=Vθ¨-α¨=Vq˙-α¨.Substitutingq˙ and α˙ in terms of the sate equation (2) into (18) yields(19)a˙y=-VA33α˙.Discretizing (19) yields(20)ayk+1=-VA33αk+1,ayk=-VA33αk.Substituting (20) and V(k+1) in terms of the sate equation (7) into (15) and (16), respectively, yields (21)Sk+1=SXXk+SYUk+SZ.To be simple,dA(2,1) is written as dA21; the other similar denotation is the same. Therefore the matrixes in (21) are(22)SX=-VA33dA31-VA33dA32-VA33dA33+c1-1-VA33dA34-VA33dA35dA41dA42dA43dA44+c2-1dA45,SY=-VA33dB31-VA33dB32dB41dB42,SZ=-c1ayck-c2Vck.FordB31=dB32=0, the matrix SY-1 will not exist. Then, U(k) cannot be achieved. Therefore the sliding surface cannot be chosen as (15). As long as the differential of state variable includes Δq˙, the matrix SY is invertible. From the derivative of (19), the second-order derivative of acceleration is (23)a¨y=-VA33A33α˙+q˙.In the right-hand side of (23), Δq˙ is included and the matrix SY is invertible; the control law U(k) can be obtained.Thereby, the form of control variable is as follows:(24)E=ay+a˙y.Differentiating (24), we have(25)E˙=a˙y+a¨y=-VA33A33+1α˙+q˙.In terms of discretization (25) is(26)Ek=-VA33A33+1αk+qk,(27)Ek+1-Ek=-VA33A33+1·αk+1-αk+qk+1-qk.Substituting the discrete state equations α(k+1) and q(k+1) into (27) yields(28)Ek+1-Ek=SE1Xk+SE2Uk.The elements of matrix SE2 are not all zeros.Choosing the sliding surface of the acceleration subsystem,(29)S1k+1=bEk+1-Ek+cEk-Eck,where b and c are designed parameters.To enable the system for tracking the command with zero steady-state errors, a discrete integrator is added in the sliding surface:(30)S1k+1=aS1k+TsbEk+1-Ek+cEk-Eck,where a is an unknown coefficient to enhance the performance of integrator. Let a=a1, Tsb=1, and Tsc=c1; rearranging the equation yields(31)S1k+1=a1S1k+Ek+1-Ek+c1Ek-Eck,(32)S1k+1-S1k=Ek+1-Ek+c1Ek-Eck-1-a1S1k.With solution of (31) and (32) for E(k+1)-E(k) and E(k) can be obtained. Substituting E(k+1)-E(k) and E(k) into (32), we have(33)S1k+1-S1k=SX1Xk+SY1Uk+SZ1,where, SX1∈R1×6, SY1∈R1×2, with SX1, SY1, and SZ1 given by(34)SX1=-A33VA33+1dA31+dA51A33+1dA32+dA52A33+1dA33+dA53+c1-1A44+1A33+1dA34+dA54A33+1dA35+dA55+c1-1T,SY1=-A33VA33+1dB31+dB51A33+1dB32+dB52T,SZ1=-c1Eck-1-a1S1k.For the velocity subsystem, the sliding surface can be chosen as(35)S2k+1=a2S2k+Vk+1-Vk+c2Vk-Vck,(36)S2k+1-S2k=Vk+1-Vk+c2Vk-Vck-1-a2S2k.Substituting the discrete state equation V(k+1) into (36) yields(37)S2k+1-S2k=SX2Xk+SY2Uk+SZ2,where(38)SX2=dA41dA42dA43dA44+c2-1dA45,SY2=dB41dB42,SZ2=-c2Vck-1-a2S2k.Solving (33) and (37), we have(39)S1k+1-S1kS2k+1-S2k=SXXk+SYUk+SZ,where SX=SX1SX2, SY=SY1SY2, SZ=SZ1SZ2.


### 3.2. Design of Discrete Variable Structure Controller

Choose the power rate reaching law as(40)S˙=-εsgn⁡S-δS.The discrete form of (40) is (41)Sk+1-Sk=-TsδSk-εTssgn⁡Sk,where Ts, ε>0, and δ>0 denote sampling interval, reaching rate, and exponent of reaching rate, respectively. The monotone reaching condition of the discrete power rate reaching law is(42)Tsδ<1.Therefore the reaching conditions of acceleration subsystem and velocity subsystem are(43)S1k+1-S1k=-Tsδ1S1k-ε1Tssgn⁡S1k,S2k+1-S2k=-Tsδ2S2k-ε2Tssgn⁡S2k.Rearranging (43) yields(44)S1k+1-S1kS2k+1-S2k=-Snd1Sk-Snd2sgn⁡Sk,where(45)Snd1=Tsδ100Tsδ2,Snd2=ε1Ts00ε2Ts.With the solutions of (39) and (44), the control law in terms of vector is(46)Uk=-SY-1SXXk+SZ+Snd1Sk+Snd2sgn⁡Sk,where δt(k)=U1(k), δe(k)=U2(k).


## 3.1. Design of Discrete Sliding Surface

Selecting multiple sliding surfaces,(14)Sk+1=S1k+1S2k+1=00.If the acceleration and velocity subsystem are chosen directly as the sliding surface,(15)S1k+1=ayk+1-ayk+c1ayk-ayck,(16)S2k+1=Vk+1-Vk+c2Vk-Vck,where c1 and c2 are the unknown parameters.The acceleration can be written as(17)ay=Vγ˙.Differentiating the acceleration expression, (17) yields(18)a˙y=Vγ¨=Vθ¨-α¨=Vq˙-α¨.Substitutingq˙ and α˙ in terms of the sate equation (2) into (18) yields(19)a˙y=-VA33α˙.Discretizing (19) yields(20)ayk+1=-VA33αk+1,ayk=-VA33αk.Substituting (20) and V(k+1) in terms of the sate equation (7) into (15) and (16), respectively, yields (21)Sk+1=SXXk+SYUk+SZ.To be simple,dA(2,1) is written as dA21; the other similar denotation is the same. Therefore the matrixes in (21) are(22)SX=-VA33dA31-VA33dA32-VA33dA33+c1-1-VA33dA34-VA33dA35dA41dA42dA43dA44+c2-1dA45,SY=-VA33dB31-VA33dB32dB41dB42,SZ=-c1ayck-c2Vck.FordB31=dB32=0, the matrix SY-1 will not exist. Then, U(k) cannot be achieved. Therefore the sliding surface cannot be chosen as (15). As long as the differential of state variable includes Δq˙, the matrix SY is invertible. From the derivative of (19), the second-order derivative of acceleration is (23)a¨y=-VA33A33α˙+q˙.In the right-hand side of (23), Δq˙ is included and the matrix SY is invertible; the control law U(k) can be obtained.Thereby, the form of control variable is as follows:(24)E=ay+a˙y.Differentiating (24), we have(25)E˙=a˙y+a¨y=-VA33A33+1α˙+q˙.In terms of discretization (25) is(26)Ek=-VA33A33+1αk+qk,(27)Ek+1-Ek=-VA33A33+1·αk+1-αk+qk+1-qk.Substituting the discrete state equations α(k+1) and q(k+1) into (27) yields(28)Ek+1-Ek=SE1Xk+SE2Uk.The elements of matrix SE2 are not all zeros.Choosing the sliding surface of the acceleration subsystem,(29)S1k+1=bEk+1-Ek+cEk-Eck,where b and c are designed parameters.To enable the system for tracking the command with zero steady-state errors, a discrete integrator is added in the sliding surface:(30)S1k+1=aS1k+TsbEk+1-Ek+cEk-Eck,where a is an unknown coefficient to enhance the performance of integrator. Let a=a1, Tsb=1, and Tsc=c1; rearranging the equation yields(31)S1k+1=a1S1k+Ek+1-Ek+c1Ek-Eck,(32)S1k+1-S1k=Ek+1-Ek+c1Ek-Eck-1-a1S1k.With solution of (31) and (32) for E(k+1)-E(k) and E(k) can be obtained. Substituting E(k+1)-E(k) and E(k) into (32), we have(33)S1k+1-S1k=SX1Xk+SY1Uk+SZ1,where, SX1∈R1×6, SY1∈R1×2, with SX1, SY1, and SZ1 given by(34)SX1=-A33VA33+1dA31+dA51A33+1dA32+dA52A33+1dA33+dA53+c1-1A44+1A33+1dA34+dA54A33+1dA35+dA55+c1-1T,SY1=-A33VA33+1dB31+dB51A33+1dB32+dB52T,SZ1=-c1Eck-1-a1S1k.For the velocity subsystem, the sliding surface can be chosen as(35)S2k+1=a2S2k+Vk+1-Vk+c2Vk-Vck,(36)S2k+1-S2k=Vk+1-Vk+c2Vk-Vck-1-a2S2k.Substituting the discrete state equation V(k+1) into (36) yields(37)S2k+1-S2k=SX2Xk+SY2Uk+SZ2,where(38)SX2=dA41dA42dA43dA44+c2-1dA45,SY2=dB41dB42,SZ2=-c2Vck-1-a2S2k.Solving (33) and (37), we have(39)S1k+1-S1kS2k+1-S2k=SXXk+SYUk+SZ,where SX=SX1SX2, SY=SY1SY2, SZ=SZ1SZ2.


## 3.2. Design of Discrete Variable Structure Controller

Choose the power rate reaching law as(40)S˙=-εsgn⁡S-δS.The discrete form of (40) is (41)Sk+1-Sk=-TsδSk-εTssgn⁡Sk,where Ts, ε>0, and δ>0 denote sampling interval, reaching rate, and exponent of reaching rate, respectively. The monotone reaching condition of the discrete power rate reaching law is(42)Tsδ<1.Therefore the reaching conditions of acceleration subsystem and velocity subsystem are(43)S1k+1-S1k=-Tsδ1S1k-ε1Tssgn⁡S1k,S2k+1-S2k=-Tsδ2S2k-ε2Tssgn⁡S2k.Rearranging (43) yields(44)S1k+1-S1kS2k+1-S2k=-Snd1Sk-Snd2sgn⁡Sk,where(45)Snd1=Tsδ100Tsδ2,Snd2=ε1Ts00ε2Ts.With the solutions of (39) and (44), the control law in terms of vector is(46)Uk=-SY-1SXXk+SZ+Snd1Sk+Snd2sgn⁡Sk,where δt(k)=U1(k), δe(k)=U2(k).


## 4. Simulations

Assuming the hypersonic cruise missile flight height isH0=25 Km, V0=6 Ma, the equilibrium condition can be given as(47)L0+T0sin⁡α0=mg,T0cos⁡α0=D0,Mz0=0.From solution of (47), the trim angle of attack α0=3.3420°, the trim angle of actuator δe0=-0.6257°, and the trim of throttle setting δt0=0.2418. At this equilibrium point, the coefficient matrixes of the longitudinal linearization model are(48)A=0000103.3×10-110.03649.3×10-700-3.3×10-11-0.0364-9.3×10-710-9.7489-21.317100000.31261.5×10-18-0.0584,B=08.7×10-5-8.7×10-56.8603000002.7378T.Suppose that the sample interval Ts=0.005 s; the state matrix in terms of discretization can be given:(49)dA=17.3×10-173.9×10-6-6×10-150.005011.8×10-44.6×10-94.6×10-701.1×10-100.9998-4.6×10-90.0050-0.0487-0.10661-2.7×10-405.86130.0016-3.6×10-120.9997,dB=-5.7×10-134.4×10-7-4.4×10-70.0343-3.4×10-103.4×10-52.1×10-93.4×10-5-1.2×10-60.0137T.Leta1=0.99, a2=0.98, c1=0.68, c2=0.3, δ1=0.815, δ2=0.4, ε1=0.005, and ε2=0.005; the control laws δt(k) and δe(k) can be obtained as in (40). The control system tracks a 2 g acceleration command and velocity command during cruising flight; the simulations are shown as in Figures 3–9.Figure 3
Response of acceleration.Figure 4
Response of velocity.Figure 5
Response of acceleration sliding surface.Figure 6
Response of velocity sliding surface.Figure 7
Response of throttle setting.Figure 8
Response of elevator.Figure 9
Response of angle of attack.The simulation results of discrete VSS controller law are shown as in Figures3–9. In Figures 3 and 4, the solid lines are the command and the dashed lines are the response of the control system. It is observed that the hypersonic missile can track both acceleration and velocity command. The controller also has no steady error. The sliding surfaces of acceleration and velocity subsystem are also reaching zero with the power reaching rate as in Figures 5 and 6. The steady response of angle of attack is 6.5 deg. with 2 g maneuver acceleration, and the elevator and throttle setting response are shown as in Figures 7 and 8. With the discrete VSS controller, perfect tracking is achieved.To validate the controller robustness to the model errors, the uncertainties of the model are considered asΔm≤0.03, ΔI≤0.05, ΔS≤0.03, ΔCD≤0.15, ΔCL≤0.10, ΔCm≤0.10, Δρ≤0.06, and ΔCT≤0.03. The Monte Carlo simulation results of 100 times are shown as in Figures 10–16.Figure 10
Results of acceleration.Figure 11
Results of velocity.Figure 12
Results of sliding surface of acceleration.Figure 13
Results of sliding surface of velocity.Figure 14
Results of elevator.Figure 15
Results of throttle setting.Figure 16
Results of angle of attack.It is shown that the controller has satisfactory performance to track the acceleration and velocity command with the model parameters uncertainties. For the errors of model mostly affect the acceleration of the missile, the sliding surface of acceleration has obvious response as shown in Figure12. However, the velocity subsystem is less affected by the parameters disturbance as shown in Figure 13. The elevator is −8~5 deg. with the uncertainties which is more great than result of the norm plant as shown in Figure 14.


## 5. Conclusion

In this paper a hypersonic cruise missile flight control system is designed using discrete variable structure controller to track the acceleration and velocity commands. An integrator is augmented to ensure that the tracking has no steady error. The multiple sliding surfaces are designed. And the control laws of acceleration and velocity subsystem are obtained. The uncertain missile plant with unmolded dynamic and parameter variations are considered. Simulation results demonstrate that the control system has good performance in tracking acceleration and velocity command and still has strong robustness to the uncertainties of model.



---

*Source: 2402794-2016-03-24.xml*