# Design of Particle Swarm Optimization Adaptive Sliding Mode Controller Based on an Extended State Observer for the Longitudinal Motion of a Supercavitating Vehicle with Input Saturation

Authors: Guang Yang; Faxing Lu; Ling Wu; Junfei Xu
Journal: Journal of Sensors
 (2023)

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## Abstract

The aim of this paper is to present a novel sliding mode control scheme for the supercavitating vehicle trajectory tracking problem that subjects to external disturbances and actuator saturation with two symmetric elevators and a cavitator as actuators by analytical methods and computer simulations. Firstly, the nonlinear and highly coupled dynamic and kinematic models of a supercavitating vehicle are presented in a comprehensive way by taking the cavity memory effect and time-variant planing force into consideration. The PSO algorithm is employed to optimize the control parameters for achieving better and more practical tracking performance by minimizing the objective function. A second-order extended state observer (ESO) is utilized to estimate the unknown external and state-dependent disturbances and compensates for control inputs. In addition, an antiwindup compensator is adopted to cope with actuator saturation. Finally, the proposed control scheme is employed for complex trajectory tracking of a supercavitating vehicle under various conditions by conducting comparative numerical simulations. Rigorous theoretical analysis and simulation results indicate that the proposed control scheme can achieve satisfactory tracking performance and have a good capability of robustness.


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## Body

## 1. Introduction

The development of supercavitation technology can improve the velocity of underwater vehicles considerably, which has broad prospects in both military and civilian fields. A successful and famous application is a supercavitating torpedo-“Shkval,” developed by Russia arousing the attention of military powers subsequently. The supercavitating vehicle is enveloped by supercavity while sailing, resulting in isolation from ambient water so that the resistance decreases by an order of magnitude and the high speed can be realized in excess of 75 m/s. However, a vehicle enveloped by supercavity will lose the balance between buoyancy and gravity because of the reduced wet surface which is different from the conventional UUV completely wrapped by water. Without applying external controllers, the supercavitating vehicle may sail unstably and descend constantly. Meanwhile, nonlinear planing force will be caused by its collision with the supercavity wall. The unique hydrodynamic characteristics, highly coupled nonlinearities, and the cavity memory effect have brought challenges to the controller design for supercavitating vehicle motion.In recent years, various efforts have been conducted to study supercavitating vehicles. In the aspect of supercavitating vehicle modeling, the dynamic model contains 12 state variables and 6 degrees of freedom. Dzielski and Kurdila [1] proposed a dynamic benchmark model in the dive-plane which is widely adopted in the subsequent research [2–6], but the dynamic model in the reference [1] did not consider the cavity memory effect and the fin efficiency was set to a constant. Such simplifications more or less exist in the subsequent literatures. Besides, different coordinate system definitions result in different dynamic models of a supercavitating vehicle [7–9]. Therefore, the dynamic model of a supercavitating vehicle is reconstructed in this paper, considering the cavity memory effect and time-varying fin efficiency comprehensively.The control of a supercavitating vehicle has aroused much academic attention worldwide. Dzielski [7] proposed the linear feedback control using the cavitator deflection as a control input, but their research did not further delve into the uncertainty and nonlinearity of the supercavitating vehicle model. Shao et al. [10] designed a switching controller based on the existence of planing force following the principles of the linear quadratic regulator (LQR), in a bid to eliminate the generation of planing force. Vanek et al. [2, 11] transformed a supercavitating vehicle nonlinear model into a linear parameter varying (LPV) model, which could improve the performance of controller under the coupling effect of control equation. Fan et al. [12] adopted the adaptive fuzzy sliding mode control for the depth control as an effective solution for the disturbance and nonlinear terms of a supercavitating vehicle. Li et al. [13] applied PID control method to the control for a supercavitating vehicle. Qiang et al. [14] transformed the original system into a cascade connection of two subsystems by using backstepping and designed a state feedback controller with the circle criterion method based on the nonlinear bounded conditions. However, the memory effect on supercavitating vehicles was not considered. Xiong and Bao [15] studied the dynamic characteristics of the closed-loop control for a supercavitating vehicle, and realized the sailing stability of a vehicle by adopting the variable gain in the feedback control law. Han et al. [16] designed a robust variable-gain H2/H∞ controller subject to multiobject constraints, which could satisfactorily track the trajectory. Considering the uncertainties and external disturbances of the model, Li et al. 8] and Wang et al. [17] employed a radial basis function (RBF) neural network as the compensation for unknown terms and put forward the corresponding control strategy. Evidently, the trajectory tracking control of a supercavitating vehicle is still a hot topic.The dynamics of supercavitating vehicles are actually a multi-input and multioutput (MIMO) system with highly coupled nonlinearities. In another word, the cavitator and fins need to cooperate to regulate the motion of a supercavitating vehicle with six degrees of freedom (DOF). For a supercavitating vehicle, the trajectory following and stability in longitudinal motion are fundamental to task execution. Moreover, longitudinal motion control and lateral motion control have something in common. Therefore, this paper is mainly about the longitudinal motion control. In reality, dynamics uncertainty and the external disturbances induced by ocean currents are huge obstacles to the trajectory tracking accuracy which may cause instability. Many researchers have proposed a diversity of methods to address this kind of control problem in terms of AUV, USV, and spacecraft [18–20] other than supercavitating vehicles. In addition, the actuators of a supercavitating vehicle unlikely provide unlimited control inputs, and the limitation of physical input saturation always leads to the undesirable inaccuracy [21, 22]. Consequently, enhancing the ability of tackling the actuator saturation can boost the practicality of the control scheme. Since the stability and performance for the closed-loop system largely depend on the selection of control parameters involved in control scheme design [23, 24], thereby, PSO algorithm is introduced to adjust the control parameters by minimizing the designed objective function.Motivated by the previous works and afore-mentioned considerations, this paper presents a novel PSO-based sliding mode control scheme to address the trajectory tracking problem of a supercavitating vehicle in presence of actuator saturation and time-varying unknown disturbances. The main contributions are summarized as follows:(1)
According to the unique hydrodynamic characteristics of a supercavitating vehicle, the highly coupled nonlinear dynamic and kinematic models are rebuilt considering the nonlinear planing force and time-variant fin efficiency induced by the centerline derivation between the supercavitating vehicle and cavity, which takes the cavity memory effect into consideration(2)
Compared with the previous control studies about the supercavitating vehicle motion, the control parameters of the controller are determined manually. In this paper, the control parameters are obtained by the PSO for the controller to achieve desired tracking performance by minimizing the designed objective function(3)
The sliding mode control scheme is deeply integrated with a second-order ESO and an antiwindup compensator [25–27] for practical purposes. The ESO is introduced to estimate the external disturbances induced by the ocean current and the state-dependent disturbances. The antiwindup compensator is employed to attenuate the effect of actuator saturationThe remainder of this paper is organized as follows. Sections2 and 3 present the nonlinear dynamic and kinematic models of a supercavitating vehicle and the state space in the longitudinal plane, respectively. In Section 4, the kinematic control law is generated from the resulting error kinematics. Then, the sliding mode control scheme is adopted to design the dynamic control law in the presence of external disturbances and input saturation. The derivative process and theoretical proof of the proposed controller are presented. Section 5 validates the effectiveness and robustness of this proposed control scheme through plenty of numerical simulations. Finally, the conclusion of this research is drawn.


## 2. Model of the Supercavitating Vehicle

### 2.1. Configuration and Parameters of the Supercavitating Vehicle

In this paper, a common supercavitating vehicle model, which has been widely adopted in many kinds of literature [28, 29], is used with a disc cavitator mounted at the nose. Supercavity forms by the artificial ventilation from the cavitator which also works as an actuator. At 1.35 m from the cavitator, cruciform fins are placed. The model parameters are listed in Table 1, and the structure of the supercavitating vehicle is illustrated in Figure 1.Table 1
Model parameters of supercavitating vehicle.
SymbolDescriptionValuesLbVehicle length1.8 mRbVehicle radius0.0508 mRnCavitator radius0.0191 mρsSeawater density1020 kg/m3ρbVehicle density2040 kg/m3LfRudder height0.07 mSfElevator area0.0011 m2Figure 1
Model of the supercavitating vehicle.
(a)
3D view(b)
Lateral view


### 2.2. Definition of Coordinate Systems

To study the longitudinal motion of a supercavitating vehicle, only elevators are taken into account in this paper. Six coordinate systems are defined shown in Figure2, including body-fixed reference frame oxbybzb, translation reference frame o1x1y1z1 obtained by shifting the origin of earth-fixed reference frame oexeyeze to the center of gravity, velocity reference frame oxvyvzv, cavitator reference frame oxcyczc rotating about zc-axis, and elevator reference frame ofxfyfzf. All coordinate systems satisfy the right-hand rule. The origin of oxbybzb is fixed at the center of the disc cavitator with u,v,w and p,q,r representing the linear and angular velocity vector, respectively. V,0,0 is the velocity vector in oxvyvzv. x,y,z and θ,ψ,φ are the position and attitude of a vehicle in oexeyeze, respectively. Moreover, δθ and δh denote the cavitator deflection angle and elevator deflection angle, respectively, which are positive in the upward direction. From the point view of simplification, the six-DOF model of the supercavitating vehicle is decoupled. In the uncoupled longitudinal subsystem, roll motion is ignored because of great inertia resulting from its high speed [30]. Therefore, the longitudinal motion of a vehicle can be described by u,v,r. In this paper, it is assumed that the forward speed u remains constant.Figure 2
Coordinate system schematic diagram.


### 2.3. Dynamic Model

Inoxbybzb, the dynamic equation of a supercavitating vehicle is created on the basis of Newton-Euler equations. The mass distribution of a supercavitating vehicle is symmetric about the oxbyb plane; that is, the inertial tensor Izx and Izy are both zero. To give the dynamic equations and the forthcoming simplification of a supercavitating vehicle, the following assumptions are made.Assumption 1.
The added mass is ignored.Assumption 2.
The mass change of a vehicle arising from the loss of fuel for sailing is ignored.Above all, the dynamic model in the longitudinal plane can be derived as(1)mv̇+mxgṙ+mru=Fcby+Gbby+Ffby+Fpby,mxgv̇+Izzṙ+mxgur=Mgbz+Mfbz+Mpbz,Fcby=−12CxρsSnV2cosδθ+αsinδθ,Gbby=−mgcosθ,Ffby=CxρsSfsnVf2sinδh+αfcosδh,Mgbz=−xgGbby,Mfbz=xfFbby,Cx=Cx01+σ.The variablesFcby, Gbby, Ffby, and Fpby represent the components of the cavitator hydrodynamic force, gravity, elevator hydrodynamic force, and planing force expressed in the yb-axis direction, respectively. Mgbz, Mfbz, and Mpbz denote the pitch moments generated byGbby, Ffby, and Fpby separately. Cx is the inflow drag coefficient of the cavitator. Cx0 is the drag coefficient at zero cavitation number, which is equal to 0.83 according to Reference [31]. Sn and Sf indicate the area of the cavitator and the elevator, respectively. s is the similarity ratio of hydrodynamic coefficients between the cavitator and the elevator; in this paper, set s=1. xg and xf represent xb-axis coordinates of center of gravity and elevator pivot separately, Vf is the resultant velocity at position [xf, 0, 0]; n is the elevator efficiency which will be demonstrated below. The supercavity model needs to be constructed since Fpby and n are the physical variables in regard to the relative position relationship between a vehicle and supercavity.


### 2.4. Cavity Model

In this paper, the supercavity model adopts the principle of independence of cavity section expansion put forth by Loginovich [32]. The cavity expansion rule is related to the velocity and size of the cavitator, cavitation number, and resistance at the time of passing the cross section, which is known as the cavity memory effect [33].
(2)MaximumcavityradiusRc=RnCxkσ,CavitylengthLc2Rn=1σCx01+σln1σ,where k=1+50σ/1+56.2σ and cavitation number σ=0.03.It is assumed that the cavity is symmetric about its maximum section. Then, a semiempirical closed-form solution for cavity shape [34] is presented as
(3)Rx=Rc1−1−Rn2Rc21−2xLc2/η,0≤x≤Lc2.


### 2.5. Interaction of the Supercavitating Vehicle and Supercavity Surface

The hydrodynamic force acting on elevators is related to the immersion of elevators into the cavity wall as shown in Figure3. The elevator efficiency n is the ratio of the immersion length and fin span Lf. Define n as
(4)n=Rb+Lf−Rcf2−Δy2Lf,where Δy is the deviation between the vehicle centerline and cavity centerline along the y-axis direction; Rcf is the radius of the supercavity.Figure 3
Sketch of elevator immersion depth.Planing force at the vehicle stern is a nonlinear force generated by the protrusion of the vehicle stern across supercavity surface when the vehicle sails. For this reason, planing force is a function of immersion depthhp and immersion angle αp at the rear shown in Figure 4. In Reference [35], a water tunnel experiment was conducted and verified that the Hassan model was closer to the experimental results. Consequently, the Hassan model is selected to calculate planing force. According to cavity memory effect, the current cavity section at the rear is actually generated by the previous motion of a cavitator at the moment of passing the section. The cavity expands in the radial direction perpendicular to the velocity direction. Thus, planing force of a supercavitating vehicle is a nonlinear force related to current time and past time.Figure 4
Position relationship between the vehicle and the supercavity.The origin ofoxbybzb at the time t and t+τ (time delay τ=Lb/u) is at the position xbt,ybt and xbt+τ,ybt+τ, respectively, in oexeyeze. The cavity centerline angle with respect to the horizontal direction xe-axis can be expressed as θcτ=θt−αt.As shown in Figure4, the immersion depth and immersion angle at the time t+τ can be expressed as
(5)hp=yt+τ−θt+τ×Lb−yt−Rcp+Rb,contactwithupperboundary,yt−yt+τ+θt+τ×Lb−Rcp+Rb,contactwithlowerboundary,αp=θt+τ−θct+ṘcV,contactwithupperboundary,θt+τ−θct−ṘcV,contactwithlowerboundary,αt=−atanvtu,where αt is the vehicle angle of attack at time t. Rcp is the cavity radius at the rear. Ṙc is the supercavity contraction ratio which is used as an augmentation to αp [36].
(6)Ṙc=−20/170.821+σ/σ.1/2V1−4.5σ/1+σK123/17K21.92/σ−3,K1=LbRn1.92σ−3−1−1,K2=1−1−4.5σ1+σK140/171/2.Based on the Hassan model, planing force can be formulated as(7)Fp=ρπV2Rb2sinαpt,τcosαpt,τ1−Rcp−Rbhpt,τ+Rcp−Rb2Rb+hpt,τRb+2hpt,τ.


### 2.6. Kinematic Model

By utilizing the transformation betweenoxbybzb and oexeyeze, the kinematic model of a supercavitating vehicle which is similar with the conventional AUV [37] is obtained as
(8)ẋ=ucosθ−vsinθ,ẏ=usinθ+vcosθ,θ̇=r.


## 2.1. Configuration and Parameters of the Supercavitating Vehicle

In this paper, a common supercavitating vehicle model, which has been widely adopted in many kinds of literature [28, 29], is used with a disc cavitator mounted at the nose. Supercavity forms by the artificial ventilation from the cavitator which also works as an actuator. At 1.35 m from the cavitator, cruciform fins are placed. The model parameters are listed in Table 1, and the structure of the supercavitating vehicle is illustrated in Figure 1.Table 1
Model parameters of supercavitating vehicle.
SymbolDescriptionValuesLbVehicle length1.8 mRbVehicle radius0.0508 mRnCavitator radius0.0191 mρsSeawater density1020 kg/m3ρbVehicle density2040 kg/m3LfRudder height0.07 mSfElevator area0.0011 m2Figure 1
Model of the supercavitating vehicle.
(a)
3D view(b)
Lateral view


## 2.2. Definition of Coordinate Systems

To study the longitudinal motion of a supercavitating vehicle, only elevators are taken into account in this paper. Six coordinate systems are defined shown in Figure2, including body-fixed reference frame oxbybzb, translation reference frame o1x1y1z1 obtained by shifting the origin of earth-fixed reference frame oexeyeze to the center of gravity, velocity reference frame oxvyvzv, cavitator reference frame oxcyczc rotating about zc-axis, and elevator reference frame ofxfyfzf. All coordinate systems satisfy the right-hand rule. The origin of oxbybzb is fixed at the center of the disc cavitator with u,v,w and p,q,r representing the linear and angular velocity vector, respectively. V,0,0 is the velocity vector in oxvyvzv. x,y,z and θ,ψ,φ are the position and attitude of a vehicle in oexeyeze, respectively. Moreover, δθ and δh denote the cavitator deflection angle and elevator deflection angle, respectively, which are positive in the upward direction. From the point view of simplification, the six-DOF model of the supercavitating vehicle is decoupled. In the uncoupled longitudinal subsystem, roll motion is ignored because of great inertia resulting from its high speed [30]. Therefore, the longitudinal motion of a vehicle can be described by u,v,r. In this paper, it is assumed that the forward speed u remains constant.Figure 2
Coordinate system schematic diagram.


## 2.3. Dynamic Model

Inoxbybzb, the dynamic equation of a supercavitating vehicle is created on the basis of Newton-Euler equations. The mass distribution of a supercavitating vehicle is symmetric about the oxbyb plane; that is, the inertial tensor Izx and Izy are both zero. To give the dynamic equations and the forthcoming simplification of a supercavitating vehicle, the following assumptions are made.Assumption 1.
The added mass is ignored.Assumption 2.
The mass change of a vehicle arising from the loss of fuel for sailing is ignored.Above all, the dynamic model in the longitudinal plane can be derived as(1)mv̇+mxgṙ+mru=Fcby+Gbby+Ffby+Fpby,mxgv̇+Izzṙ+mxgur=Mgbz+Mfbz+Mpbz,Fcby=−12CxρsSnV2cosδθ+αsinδθ,Gbby=−mgcosθ,Ffby=CxρsSfsnVf2sinδh+αfcosδh,Mgbz=−xgGbby,Mfbz=xfFbby,Cx=Cx01+σ.The variablesFcby, Gbby, Ffby, and Fpby represent the components of the cavitator hydrodynamic force, gravity, elevator hydrodynamic force, and planing force expressed in the yb-axis direction, respectively. Mgbz, Mfbz, and Mpbz denote the pitch moments generated byGbby, Ffby, and Fpby separately. Cx is the inflow drag coefficient of the cavitator. Cx0 is the drag coefficient at zero cavitation number, which is equal to 0.83 according to Reference [31]. Sn and Sf indicate the area of the cavitator and the elevator, respectively. s is the similarity ratio of hydrodynamic coefficients between the cavitator and the elevator; in this paper, set s=1. xg and xf represent xb-axis coordinates of center of gravity and elevator pivot separately, Vf is the resultant velocity at position [xf, 0, 0]; n is the elevator efficiency which will be demonstrated below. The supercavity model needs to be constructed since Fpby and n are the physical variables in regard to the relative position relationship between a vehicle and supercavity.


## 2.4. Cavity Model

In this paper, the supercavity model adopts the principle of independence of cavity section expansion put forth by Loginovich [32]. The cavity expansion rule is related to the velocity and size of the cavitator, cavitation number, and resistance at the time of passing the cross section, which is known as the cavity memory effect [33].
(2)MaximumcavityradiusRc=RnCxkσ,CavitylengthLc2Rn=1σCx01+σln1σ,where k=1+50σ/1+56.2σ and cavitation number σ=0.03.It is assumed that the cavity is symmetric about its maximum section. Then, a semiempirical closed-form solution for cavity shape [34] is presented as
(3)Rx=Rc1−1−Rn2Rc21−2xLc2/η,0≤x≤Lc2.


## 2.5. Interaction of the Supercavitating Vehicle and Supercavity Surface

The hydrodynamic force acting on elevators is related to the immersion of elevators into the cavity wall as shown in Figure3. The elevator efficiency n is the ratio of the immersion length and fin span Lf. Define n as
(4)n=Rb+Lf−Rcf2−Δy2Lf,where Δy is the deviation between the vehicle centerline and cavity centerline along the y-axis direction; Rcf is the radius of the supercavity.Figure 3
Sketch of elevator immersion depth.Planing force at the vehicle stern is a nonlinear force generated by the protrusion of the vehicle stern across supercavity surface when the vehicle sails. For this reason, planing force is a function of immersion depthhp and immersion angle αp at the rear shown in Figure 4. In Reference [35], a water tunnel experiment was conducted and verified that the Hassan model was closer to the experimental results. Consequently, the Hassan model is selected to calculate planing force. According to cavity memory effect, the current cavity section at the rear is actually generated by the previous motion of a cavitator at the moment of passing the section. The cavity expands in the radial direction perpendicular to the velocity direction. Thus, planing force of a supercavitating vehicle is a nonlinear force related to current time and past time.Figure 4
Position relationship between the vehicle and the supercavity.The origin ofoxbybzb at the time t and t+τ (time delay τ=Lb/u) is at the position xbt,ybt and xbt+τ,ybt+τ, respectively, in oexeyeze. The cavity centerline angle with respect to the horizontal direction xe-axis can be expressed as θcτ=θt−αt.As shown in Figure4, the immersion depth and immersion angle at the time t+τ can be expressed as
(5)hp=yt+τ−θt+τ×Lb−yt−Rcp+Rb,contactwithupperboundary,yt−yt+τ+θt+τ×Lb−Rcp+Rb,contactwithlowerboundary,αp=θt+τ−θct+ṘcV,contactwithupperboundary,θt+τ−θct−ṘcV,contactwithlowerboundary,αt=−atanvtu,where αt is the vehicle angle of attack at time t. Rcp is the cavity radius at the rear. Ṙc is the supercavity contraction ratio which is used as an augmentation to αp [36].
(6)Ṙc=−20/170.821+σ/σ.1/2V1−4.5σ/1+σK123/17K21.92/σ−3,K1=LbRn1.92σ−3−1−1,K2=1−1−4.5σ1+σK140/171/2.Based on the Hassan model, planing force can be formulated as(7)Fp=ρπV2Rb2sinαpt,τcosαpt,τ1−Rcp−Rbhpt,τ+Rcp−Rb2Rb+hpt,τRb+2hpt,τ.


## 2.6. Kinematic Model

By utilizing the transformation betweenoxbybzb and oexeyeze, the kinematic model of a supercavitating vehicle which is similar with the conventional AUV [37] is obtained as
(8)ẋ=ucosθ−vsinθ,ẏ=usinθ+vcosθ,θ̇=r.


## 3. Construction and Verification of the Supercavitating Vehicle State Space in the Longitudinal Plane

The dynamic and kinematic models need to be linearized in order to create the longitudinal plane state space of a supercavitating vehicle. The resultant velocity at the location of elevatorsVf is approximately equal to the resultant velocity of the vehicle V. The velocity angle of attack at elevators is αf=−v+xf.r/V. Thus, along with some additional small angle assumptions, the state space of a supercavitating vehicle can be expressed as
(9)ẏθ̇v̇ṙ=Ayθvr+Bδθδh+C+DFp,where
(10)A=0uI20002,2M−1A0,B=02,2M−1B0,C=02,1M−1C0,D=02,1M−1D0,M=mmxgmxgIz,A0=−ρsCxSfsnu−mu+ρsCxSfsnuxf−ρsCxSfsnuxf−muxg+ρsCxSfsnuxf2,B0=−12ρsCxSnu2ρsCxSfsnu20ρsCxSfsnu2xf,C0=−mg−mgxg,D0=1xp.In order to verify whether the linearization is reasonable, set the angular velocity of the vehicle001, and the angle of attack, cavitator deflection angle, and elevator deflection angle vary within the range −π/18,π/18. The initial and linearized hydrodynamic forces acting on the cavitator and the elevator are calculated as shown in Figure 5. The maximum deviation is 6.6% and 3.9%, respectively. Additionally, the deviation is positively correlated with the angle of attack, cavitator deflection, and elevator deflection. In the middle of the angle range, the initial and linearized hydrodynamic forces basically coincide with each other. Therefore, the linearization of kinematic and dynamic models is validated.Figure 5
Comparison of nominal and approximate hydrodynamic forces.
(a)
Comparison between nominal and approximate values of hydrodynamic force on the cavitator(b)
Comparison between nominal and approximate values of hydrodynamic force on the elevatorSimulation is carried out to analyze the open-loop characteristics of the supercavitating vehicle. It is assumed that the initial state of the supercavitating vehicle isxyθvr=00000, and there is no control input. As shown in Figure 6, the uncontrolled state variables of the supercavitating vehicle constantly deviate from its initial state; in other words, the vehicle goes deeper ceaselessly and can not remain at the desired depth. At the beginning of voyage, the vehicle is encapsulated by the cavity. The head of the vehicle first deflects upwards under the effect of gravity, which generates the positive pitch rate. Meanwhile, the vertical velocity increases and the depth decreases. For this reason, contact with the lower cavity wall generates planing force, which generates the restoring force and torque. This is repeated until the vehicle loses stability completely. Therefore, a controller should be designated for the stability and motion control of a supercavitating vehicle.Figure 6
Open loop characteristics of supercavitating vehicle with delay effect.


## 4. Motion Controller Design

### 4.1. Scheme of the Proposed Control Method

As is known, sliding mode control has been widely applied to the trajectory tracking problem in the presence of external disturbances and actuator saturation for its good capability of robustness. As shown in Figure7, a successive control structure is designed. The cascaded controller is divided into two subsystems, including kinematic control and dynamic control. Kinematic control law is designed to obtain the virtual desired vertical velocity and pitch rate based on the Lyapunov stability theory in the first subsystem. In the second subsystem, a sliding mode control method is utilized to obtain the control efforts δθ and δh in the presence of highly coupled nonlinearities, external disturbances, and actuator saturation. Dynamic controller adjusts the actuators to regulate the actual vertical velocity and pitch rate converging to the kinematic control law in finite time. In order to achieve more precise and practical control performance, the PSO algorithm is introduced to the control method for obtaining the optimal parameters. An ESO is proposed to estimate the external disturbances in the dynamic model of the system and the control efforts are compensated accordingly. In practical application, input saturation must be taken into consideration because the control law may exceed the maximum output of actuators. For attenuating the influence of the actuator saturation, the antiwindup compensator is introduced to the control scheme. Based on the control scheme proposed above, the motion control for a supercavitating vehicle can achieve good performances, which has good robustness, high accuracy in the presence of coupled nonlinearities, and external disturbances.Figure 7
The proposed control scheme.


### 4.2. Design of the Motion Controller

#### 4.2.1. Design of Kinematic and Dynamic Control Law

According to the criterion for controllability, the system is controllable withn∈0,1 and rankP=4 with P=BABA2BA3B.Based on the dynamic model presented in Section1, the control inputs from the cavitator and elevators act directly on the vertical velocity and pitch rate, which indirectly affect the depth and pitch angle. For better understanding, the state space of a supercavitating vehicle is rewritten into
(11)Ẋ1=A1X1+X2,Ẋ2=A2X2+B2U+C2+D2Fp+dt,where
(12)X1=yθ,X2=vr,U=δθδh,A1=0u00,A2=M−1A0, B2=M−1B0, C2=M−1C0, D2=M−1D0, and dt=dvdrT denotes the time-varying external disturbances induced by ocean currents.The desired position vector ofX1 is X1,d. The virtual desired states of X2 is X2,d. Accordingly, the tracking errors can be described as E1=X1−X1,d and E2=X2−X2,d, where E1=eyeθT and E2=everT.To derive the kinematic control law, define the first Lyapunov function candidate as(13)V1=12E1TE1.And its derivative can be presented as(14)V̇1=E1TĖ1=E1TA1X1+E2+X2,d−Ẋ1,d.Given a desired path and the trajectory tracking errors, design the virtual kinematic control law to generate the desired vertical velocityvd and desired pitch rate rd as follows:
(15)X2,d=Ẋ1,d−C1E1−A1X1,where
(16)C1=c100c2is a positive definite diagonal matrix that needs to be designed.By designing a reasonable dynamic control law to makeE2 converge to zero, then V̇1<0, which means trajectory tracking errors can converge to an arbitrarily small neighborhood around zero.Given the highly coupled dynamic model with nonlinearities, select the linear sliding mode surface with regard toE1 and E2 to design the dynamic control law to obtain the effective control input δθ and δh.
(17)S=KE+1E2,where
(18)K=k100k2is a positive definite diagonal matrix that needs to be designed.Combine equations (14)–(17).
(19)Ė1=−K+C1E1+S.The accessibility of the sliding mode condition guarantees that any original state in space can reach the sliding mode surface in finite time. When the system reaches the sliding mode surfaceS=0, then the sliding mode state equation of E1 can be obtained as
(20)Ė1=−K+C1E1.Make sure−K−C1 satisfies the Hurwitz matrix, so that E1 can converge to zero in finite time. Set
(21)−K−C1=λ100λ2,λ1,λ2<0.Define the second Lyapunov function candidate asV2=V1+1/2STS, differentiating V2, and utilizing equations (11), (15), and (17) yields
(22)V̇2=E1TĖ1+STK+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+B2U+C2+D2Fp+dt−X¨1,d+A1Ẋ1.Accordingly,(23)Ṡ=K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+B2U+C2+D2Fp+dt−X¨1,d+A1Ẋ1.In order to improve the convergence performance, an improved reaching law and boundary layer is introduced in the control method. The introduction of terms enhances the antichattering effect. Additionally, the saturation function can regulate the reaching speed. Outside the boundary layer, dynamic system approaches quickly, and approaches smoothly inside the boundary layer.The reaching law is chosen as(24)Ṡ=−εS2satS−HS,where
(25)S=s1s2T,ε>0,H=h100h2is a positive definite diagonal matrix.The definition of saturation function is given as(26)satsi=sgnsi,abssi>Δ,siΔ,abssi≤Δ,i=1,2.Adopting the backstepping techniques, the control input of a supercavitating vehicle can be derived as(27)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp+dt−X¨1,d+A1Ẋ1+εS2satS+HS.For analyzing the stability, substitute equation (27) into equation (22); then, the derivative V̇2 can be obtained
(28)V̇2=E1TE2−E1TC1E1−εS2STsatS−STHS.Proof.
By rearranging equation (28), the first three terms of V̇2 can be expanded as follows:
(29)E1TE2−E1TC1E1−STHS=−c1+h1k12ey2−2h1k1−1eyev−h1ev2−c2+h2k22eθ2−2h2k2−1ereθ−h2er2=−eyevQ1eyev−eθerQ2eθer,where
(30)Q1=c1+h1k12h1k1−12h1k1−12h1,Q2=c2+h2k22h2k2−12h2k2−12h2.Just make sure bothQ1 and Q2 are positive definite matrixes by determining the parameters properly; E1TE2−E1TC1E1−STHS<0 can be guaranteed.In addition,(31)−εS2STsatS=−εS2s1+s2,abss1&abss2>Δ,−εΔS4,abss1&abss2≤Δ,−εS2s1+s22Δ,abss1>Δ&abss2≤Δ,−εS2s12Δ+s2,abss1≤Δ&abss2>Δ.To sum up,V̇2 is negatively definite. According to the Lyapunov stability theory, it can be guaranteed that E1 and E2 can converge to a small neighborhood around zero in finite time.Furthermore, when the close-loop system is subjected to the disturbances, the severe chattering of control inputs appears due to the chattering of the sliding mode surface. In order to attenuate the effect of the chattering, the reaching law is improved as(32)Ṡ=−HS−DtanhSε,where tanhS/ε is a continuous and smooth hyperbolic tangent function compared with the saturation function which has sharp corners.


#### 4.2.2. Control Parameter Optimization Based on PSO

As aforementioned, the proposed sliding mode control contains various parameters to be designed, which can be transformed into a multiobjective optimization problem. The objective function with constraints is presented as follows:(33)minJ=maxδθ+maxδh+∫0Tζey+eθdts.t.ci,ki,hi∈0,upperbound,i=1,2hici+ki−14>0.From a practical point of view, the fitness function consists of maximum of absolute values of control inputs and kinematic control law tracking errors in order to achieve minimum energy consumption and tracking errors. According to Lyapunov theorem, as long as the parameter constraints are satisfied, the trajectory errors will converge to zero; thereby, greater importance should be attached to the maximum deflection angle of the cavitator and elevators; then set weight coefficientζ=0.5 to balance control inputs and kinematic control law tracking errors. The PSO iterative process is shown as Figure 8, and related equations are listed as follows:
(34)Vi+1j=ωVij+n1r1pBestij−Xij+n2r2gBesti−Xij,Xi+1j=Xij+Vi+1j,where ω is inertia weight factor, n1 and n2 denote cognitive factors, r1 and r2 denote random numbers satisfying a standard normal distribution, Vij represents the velocity of particle Xij in swarm i, pBestij is the best between particle Xi−1j and Xij, and gBesti is the best particle in swarm i.Figure 8
The PSO algorithm process.


#### 4.2.3. Design of Extended State Observer

Time-varying external disturbances that arise from the unexpected environment are an obstacle to accurate control, since it brings great difficulties to precise modeling and controller design. In this subsection, an extended state observer is proposed to estimate unknown external disturbances in dynamic model [38, 39] and achieve disturbance attenuation via effective compensation.To facilitate the control design, make an assumption that the time-variant external disturbancedt is bounded, satisfying dv≤ρ1, ḋv≤ρ2, dr≤ϖ1, and ḋr≤ϖ2. The further simplified model of dynamic model can be obtained
(35)Ẋ2=fX2+dt,ḋt=ht,where fX2 is the modeled part. ht is the derivative of the unknown external disturbances dt which is regarded as an extended state. A second-order extended state observer is utilized.
(36)e1=Z1−X2,Ż1=Z2+fX2−β1e1,Ż2=−β2fale1,α,δ,dt=Z2+ξ,fale,α,δ=eδα−1,e≤δ,eαsigne,e>δ,where Z1 is the estimation of the vertical velocity and pitch rate, Z2 is the estimation of disturbances, and e1 is the error of state variables. The observer gains β1 andβ2 are used to regulate the convergence speed of the observer. ξ=ξvξrT is the approximation error of the ESO. And ξ is bounded, satisfying.
(37)ξv≤ψ1,ξr≤ψ2,where ψ1 and ψ2 are both positive and unknown.After making compensation for disturbances, the sliding mode control law is modified as(38)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp−X¨1,d+A1Ẋ1+Z2+εS2satS+HS+ψ^signS.With the adaptation lawsψ^̇i=σisi,i=1,2.
(39)ψ^=ψ^100ψ^2,ψ^1, and ψ^2 denote the estimations of ψ1 and ψ2. σ1 and σ2 are positive constants. sign. denotes the unit sign function.Define the third Lyapunov function candidate [40] as
(40)V3=V2+12σ1ψ~12+12σ2ψ~22,where ψ~1=ψ1−ψ^1 and ψ~2=ψ2−ψ^2 denote the estimation errors.Substituting equation (38) into the derivative of V3, then V̇3 can be expressed as
(41)V̇3=E1TE2−E1TC1E1−εS2STsatS−STHS+ξvs1−ψ^1s1−ψ~1s1+ξrs2−ψ^2s2−ψ~2s2≤E1TE2−E1TC1E1−εS2STsatS−STHS.Therefore,V̇3 is negative definite, which satisfies the Lyapunov stability theory. In other words, dynamic and kinematic errors can converge uniformly and asymptotically.


#### 4.2.4. Design of an Antiwindup Compensator

Due to the physical saturation of actuators, an antiwindup compensator is introduced to the proposed control scheme to attenuate the effects of saturation. The actual cavitator deflection angleuθ and elevator deflection angle uh can be given as, respectively,
(42)uθ=satδθ=mθsignδθ,δθ≥mθ,δθ,δθ<mθ,uh=satδh=mhsignδh,δh≥mh,δh,δh<mh,where mθ and mh are the upper limits of cavitator and elevator saturation constraints, respectively.In order to attenuate the influence caused by input saturations, the antiwindup compensator [26, 27] can be designed as
(43)Ṗ=−QP−STB2ΔU+1/2ΔUTΔUP2P+ΔU,P≥μ,0,P<μ,where P=pθphT is the auxiliary variable, ΔU=ΔδθΔδhT, Δδθ=uθ−δθ, and Δδh=uh−δh.
(44)Q=q100q2is a parameter matrix to be designed.After application of this modification, the control input will be as follows:(45)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp−X¨1,d+A1Ẋ1+Z2+εS2satS+HS−P+ψ^signSDefine the fourth Lyapunov function candidate as(46)V4=V3+12PTP.Considering the input saturation, the dynamic model and derivative of sliding mode surface should be rewritten as follows:(47)Ẋ2=A2X2+B2U+ΔU+C2+D2Fp+dt,Ṡ=K+C1Ė1+A2E2+Ẋ1d−C1E1−A1X1+B2U+B2ΔU+C2+D2Fp+dt−X¨1d+A1Ẋ1.Substituting equations (45) into the derivative of V4, V̇4 can be given as
(48)V̇4≤E1TE2−E1TC1E1−εS2STsatS−STHS+STHP−PTQP−12ΔUTΔU+PTΔU.According to equation (48), verification of some terms is just necessary.
(49)−STHS+STHP−PTQP−12ΔUTΔU+PTΔU=−h1s12−h2s22+s1h1pθ+s2h2ph−q1pθ2−q2ph2−12Δδθ2−12Δδh2+pθΔδθ+phΔδh.In fact, some inequalities hold as follows:(50)s1pθ≤s122γ1+γ1pθ22,s2ph≤s222γ2+γ2ph22,pθΔδθ≤pθ22γ3+γ3Δδθ22,phΔδh≤ph22γ4+γhΔδh22,whereγ1, γ2, γ3, and γ4 are design parameters.Substituting equation (50) into equation (49), one can be obtained
(51)−STHS+STHP−PTQP−12ΔUTΔU+PTΔU≤−h1−h12γ1s12−h2−h22γ2s22+h1γ12−q1+12γ3pθ2+h2γ22−q2+12γ4ph2−12−γ32Δδθ2−12−γ42Δδh2.In conclusion,V̇4≤0 can be guaranteed by setting proper design parameters which satisfy the following conditions:
(52)γi≥12,i=1,2,γi≤1,i=3,4,h1γ12−q1+12γ3<0,h2γ22−q2+12γ4<0.According to Barbalat’s Lemma, the close-loop control method is asymptotically stable, and the tracking errors can converge to a small neighborhood around zero in finite time.


## 4.1. Scheme of the Proposed Control Method

As is known, sliding mode control has been widely applied to the trajectory tracking problem in the presence of external disturbances and actuator saturation for its good capability of robustness. As shown in Figure7, a successive control structure is designed. The cascaded controller is divided into two subsystems, including kinematic control and dynamic control. Kinematic control law is designed to obtain the virtual desired vertical velocity and pitch rate based on the Lyapunov stability theory in the first subsystem. In the second subsystem, a sliding mode control method is utilized to obtain the control efforts δθ and δh in the presence of highly coupled nonlinearities, external disturbances, and actuator saturation. Dynamic controller adjusts the actuators to regulate the actual vertical velocity and pitch rate converging to the kinematic control law in finite time. In order to achieve more precise and practical control performance, the PSO algorithm is introduced to the control method for obtaining the optimal parameters. An ESO is proposed to estimate the external disturbances in the dynamic model of the system and the control efforts are compensated accordingly. In practical application, input saturation must be taken into consideration because the control law may exceed the maximum output of actuators. For attenuating the influence of the actuator saturation, the antiwindup compensator is introduced to the control scheme. Based on the control scheme proposed above, the motion control for a supercavitating vehicle can achieve good performances, which has good robustness, high accuracy in the presence of coupled nonlinearities, and external disturbances.Figure 7
The proposed control scheme.


## 4.2. Design of the Motion Controller

### 4.2.1. Design of Kinematic and Dynamic Control Law

According to the criterion for controllability, the system is controllable withn∈0,1 and rankP=4 with P=BABA2BA3B.Based on the dynamic model presented in Section1, the control inputs from the cavitator and elevators act directly on the vertical velocity and pitch rate, which indirectly affect the depth and pitch angle. For better understanding, the state space of a supercavitating vehicle is rewritten into
(11)Ẋ1=A1X1+X2,Ẋ2=A2X2+B2U+C2+D2Fp+dt,where
(12)X1=yθ,X2=vr,U=δθδh,A1=0u00,A2=M−1A0, B2=M−1B0, C2=M−1C0, D2=M−1D0, and dt=dvdrT denotes the time-varying external disturbances induced by ocean currents.The desired position vector ofX1 is X1,d. The virtual desired states of X2 is X2,d. Accordingly, the tracking errors can be described as E1=X1−X1,d and E2=X2−X2,d, where E1=eyeθT and E2=everT.To derive the kinematic control law, define the first Lyapunov function candidate as(13)V1=12E1TE1.And its derivative can be presented as(14)V̇1=E1TĖ1=E1TA1X1+E2+X2,d−Ẋ1,d.Given a desired path and the trajectory tracking errors, design the virtual kinematic control law to generate the desired vertical velocityvd and desired pitch rate rd as follows:
(15)X2,d=Ẋ1,d−C1E1−A1X1,where
(16)C1=c100c2is a positive definite diagonal matrix that needs to be designed.By designing a reasonable dynamic control law to makeE2 converge to zero, then V̇1<0, which means trajectory tracking errors can converge to an arbitrarily small neighborhood around zero.Given the highly coupled dynamic model with nonlinearities, select the linear sliding mode surface with regard toE1 and E2 to design the dynamic control law to obtain the effective control input δθ and δh.
(17)S=KE+1E2,where
(18)K=k100k2is a positive definite diagonal matrix that needs to be designed.Combine equations (14)–(17).
(19)Ė1=−K+C1E1+S.The accessibility of the sliding mode condition guarantees that any original state in space can reach the sliding mode surface in finite time. When the system reaches the sliding mode surfaceS=0, then the sliding mode state equation of E1 can be obtained as
(20)Ė1=−K+C1E1.Make sure−K−C1 satisfies the Hurwitz matrix, so that E1 can converge to zero in finite time. Set
(21)−K−C1=λ100λ2,λ1,λ2<0.Define the second Lyapunov function candidate asV2=V1+1/2STS, differentiating V2, and utilizing equations (11), (15), and (17) yields
(22)V̇2=E1TĖ1+STK+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+B2U+C2+D2Fp+dt−X¨1,d+A1Ẋ1.Accordingly,(23)Ṡ=K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+B2U+C2+D2Fp+dt−X¨1,d+A1Ẋ1.In order to improve the convergence performance, an improved reaching law and boundary layer is introduced in the control method. The introduction of terms enhances the antichattering effect. Additionally, the saturation function can regulate the reaching speed. Outside the boundary layer, dynamic system approaches quickly, and approaches smoothly inside the boundary layer.The reaching law is chosen as(24)Ṡ=−εS2satS−HS,where
(25)S=s1s2T,ε>0,H=h100h2is a positive definite diagonal matrix.The definition of saturation function is given as(26)satsi=sgnsi,abssi>Δ,siΔ,abssi≤Δ,i=1,2.Adopting the backstepping techniques, the control input of a supercavitating vehicle can be derived as(27)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp+dt−X¨1,d+A1Ẋ1+εS2satS+HS.For analyzing the stability, substitute equation (27) into equation (22); then, the derivative V̇2 can be obtained
(28)V̇2=E1TE2−E1TC1E1−εS2STsatS−STHS.Proof.
By rearranging equation (28), the first three terms of V̇2 can be expanded as follows:
(29)E1TE2−E1TC1E1−STHS=−c1+h1k12ey2−2h1k1−1eyev−h1ev2−c2+h2k22eθ2−2h2k2−1ereθ−h2er2=−eyevQ1eyev−eθerQ2eθer,where
(30)Q1=c1+h1k12h1k1−12h1k1−12h1,Q2=c2+h2k22h2k2−12h2k2−12h2.Just make sure bothQ1 and Q2 are positive definite matrixes by determining the parameters properly; E1TE2−E1TC1E1−STHS<0 can be guaranteed.In addition,(31)−εS2STsatS=−εS2s1+s2,abss1&abss2>Δ,−εΔS4,abss1&abss2≤Δ,−εS2s1+s22Δ,abss1>Δ&abss2≤Δ,−εS2s12Δ+s2,abss1≤Δ&abss2>Δ.To sum up,V̇2 is negatively definite. According to the Lyapunov stability theory, it can be guaranteed that E1 and E2 can converge to a small neighborhood around zero in finite time.Furthermore, when the close-loop system is subjected to the disturbances, the severe chattering of control inputs appears due to the chattering of the sliding mode surface. In order to attenuate the effect of the chattering, the reaching law is improved as(32)Ṡ=−HS−DtanhSε,where tanhS/ε is a continuous and smooth hyperbolic tangent function compared with the saturation function which has sharp corners.


### 4.2.2. Control Parameter Optimization Based on PSO

As aforementioned, the proposed sliding mode control contains various parameters to be designed, which can be transformed into a multiobjective optimization problem. The objective function with constraints is presented as follows:(33)minJ=maxδθ+maxδh+∫0Tζey+eθdts.t.ci,ki,hi∈0,upperbound,i=1,2hici+ki−14>0.From a practical point of view, the fitness function consists of maximum of absolute values of control inputs and kinematic control law tracking errors in order to achieve minimum energy consumption and tracking errors. According to Lyapunov theorem, as long as the parameter constraints are satisfied, the trajectory errors will converge to zero; thereby, greater importance should be attached to the maximum deflection angle of the cavitator and elevators; then set weight coefficientζ=0.5 to balance control inputs and kinematic control law tracking errors. The PSO iterative process is shown as Figure 8, and related equations are listed as follows:
(34)Vi+1j=ωVij+n1r1pBestij−Xij+n2r2gBesti−Xij,Xi+1j=Xij+Vi+1j,where ω is inertia weight factor, n1 and n2 denote cognitive factors, r1 and r2 denote random numbers satisfying a standard normal distribution, Vij represents the velocity of particle Xij in swarm i, pBestij is the best between particle Xi−1j and Xij, and gBesti is the best particle in swarm i.Figure 8
The PSO algorithm process.


### 4.2.3. Design of Extended State Observer

Time-varying external disturbances that arise from the unexpected environment are an obstacle to accurate control, since it brings great difficulties to precise modeling and controller design. In this subsection, an extended state observer is proposed to estimate unknown external disturbances in dynamic model [38, 39] and achieve disturbance attenuation via effective compensation.To facilitate the control design, make an assumption that the time-variant external disturbancedt is bounded, satisfying dv≤ρ1, ḋv≤ρ2, dr≤ϖ1, and ḋr≤ϖ2. The further simplified model of dynamic model can be obtained
(35)Ẋ2=fX2+dt,ḋt=ht,where fX2 is the modeled part. ht is the derivative of the unknown external disturbances dt which is regarded as an extended state. A second-order extended state observer is utilized.
(36)e1=Z1−X2,Ż1=Z2+fX2−β1e1,Ż2=−β2fale1,α,δ,dt=Z2+ξ,fale,α,δ=eδα−1,e≤δ,eαsigne,e>δ,where Z1 is the estimation of the vertical velocity and pitch rate, Z2 is the estimation of disturbances, and e1 is the error of state variables. The observer gains β1 andβ2 are used to regulate the convergence speed of the observer. ξ=ξvξrT is the approximation error of the ESO. And ξ is bounded, satisfying.
(37)ξv≤ψ1,ξr≤ψ2,where ψ1 and ψ2 are both positive and unknown.After making compensation for disturbances, the sliding mode control law is modified as(38)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp−X¨1,d+A1Ẋ1+Z2+εS2satS+HS+ψ^signS.With the adaptation lawsψ^̇i=σisi,i=1,2.
(39)ψ^=ψ^100ψ^2,ψ^1, and ψ^2 denote the estimations of ψ1 and ψ2. σ1 and σ2 are positive constants. sign. denotes the unit sign function.Define the third Lyapunov function candidate [40] as
(40)V3=V2+12σ1ψ~12+12σ2ψ~22,where ψ~1=ψ1−ψ^1 and ψ~2=ψ2−ψ^2 denote the estimation errors.Substituting equation (38) into the derivative of V3, then V̇3 can be expressed as
(41)V̇3=E1TE2−E1TC1E1−εS2STsatS−STHS+ξvs1−ψ^1s1−ψ~1s1+ξrs2−ψ^2s2−ψ~2s2≤E1TE2−E1TC1E1−εS2STsatS−STHS.Therefore,V̇3 is negative definite, which satisfies the Lyapunov stability theory. In other words, dynamic and kinematic errors can converge uniformly and asymptotically.


### 4.2.4. Design of an Antiwindup Compensator

Due to the physical saturation of actuators, an antiwindup compensator is introduced to the proposed control scheme to attenuate the effects of saturation. The actual cavitator deflection angleuθ and elevator deflection angle uh can be given as, respectively,
(42)uθ=satδθ=mθsignδθ,δθ≥mθ,δθ,δθ<mθ,uh=satδh=mhsignδh,δh≥mh,δh,δh<mh,where mθ and mh are the upper limits of cavitator and elevator saturation constraints, respectively.In order to attenuate the influence caused by input saturations, the antiwindup compensator [26, 27] can be designed as
(43)Ṗ=−QP−STB2ΔU+1/2ΔUTΔUP2P+ΔU,P≥μ,0,P<μ,where P=pθphT is the auxiliary variable, ΔU=ΔδθΔδhT, Δδθ=uθ−δθ, and Δδh=uh−δh.
(44)Q=q100q2is a parameter matrix to be designed.After application of this modification, the control input will be as follows:(45)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp−X¨1,d+A1Ẋ1+Z2+εS2satS+HS−P+ψ^signSDefine the fourth Lyapunov function candidate as(46)V4=V3+12PTP.Considering the input saturation, the dynamic model and derivative of sliding mode surface should be rewritten as follows:(47)Ẋ2=A2X2+B2U+ΔU+C2+D2Fp+dt,Ṡ=K+C1Ė1+A2E2+Ẋ1d−C1E1−A1X1+B2U+B2ΔU+C2+D2Fp+dt−X¨1d+A1Ẋ1.Substituting equations (45) into the derivative of V4, V̇4 can be given as
(48)V̇4≤E1TE2−E1TC1E1−εS2STsatS−STHS+STHP−PTQP−12ΔUTΔU+PTΔU.According to equation (48), verification of some terms is just necessary.
(49)−STHS+STHP−PTQP−12ΔUTΔU+PTΔU=−h1s12−h2s22+s1h1pθ+s2h2ph−q1pθ2−q2ph2−12Δδθ2−12Δδh2+pθΔδθ+phΔδh.In fact, some inequalities hold as follows:(50)s1pθ≤s122γ1+γ1pθ22,s2ph≤s222γ2+γ2ph22,pθΔδθ≤pθ22γ3+γ3Δδθ22,phΔδh≤ph22γ4+γhΔδh22,whereγ1, γ2, γ3, and γ4 are design parameters.Substituting equation (50) into equation (49), one can be obtained
(51)−STHS+STHP−PTQP−12ΔUTΔU+PTΔU≤−h1−h12γ1s12−h2−h22γ2s22+h1γ12−q1+12γ3pθ2+h2γ22−q2+12γ4ph2−12−γ32Δδθ2−12−γ42Δδh2.In conclusion,V̇4≤0 can be guaranteed by setting proper design parameters which satisfy the following conditions:
(52)γi≥12,i=1,2,γi≤1,i=3,4,h1γ12−q1+12γ3<0,h2γ22−q2+12γ4<0.According to Barbalat’s Lemma, the close-loop control method is asymptotically stable, and the tracking errors can converge to a small neighborhood around zero in finite time.


## 4.2.1. Design of Kinematic and Dynamic Control Law

According to the criterion for controllability, the system is controllable withn∈0,1 and rankP=4 with P=BABA2BA3B.Based on the dynamic model presented in Section1, the control inputs from the cavitator and elevators act directly on the vertical velocity and pitch rate, which indirectly affect the depth and pitch angle. For better understanding, the state space of a supercavitating vehicle is rewritten into
(11)Ẋ1=A1X1+X2,Ẋ2=A2X2+B2U+C2+D2Fp+dt,where
(12)X1=yθ,X2=vr,U=δθδh,A1=0u00,A2=M−1A0, B2=M−1B0, C2=M−1C0, D2=M−1D0, and dt=dvdrT denotes the time-varying external disturbances induced by ocean currents.The desired position vector ofX1 is X1,d. The virtual desired states of X2 is X2,d. Accordingly, the tracking errors can be described as E1=X1−X1,d and E2=X2−X2,d, where E1=eyeθT and E2=everT.To derive the kinematic control law, define the first Lyapunov function candidate as(13)V1=12E1TE1.And its derivative can be presented as(14)V̇1=E1TĖ1=E1TA1X1+E2+X2,d−Ẋ1,d.Given a desired path and the trajectory tracking errors, design the virtual kinematic control law to generate the desired vertical velocityvd and desired pitch rate rd as follows:
(15)X2,d=Ẋ1,d−C1E1−A1X1,where
(16)C1=c100c2is a positive definite diagonal matrix that needs to be designed.By designing a reasonable dynamic control law to makeE2 converge to zero, then V̇1<0, which means trajectory tracking errors can converge to an arbitrarily small neighborhood around zero.Given the highly coupled dynamic model with nonlinearities, select the linear sliding mode surface with regard toE1 and E2 to design the dynamic control law to obtain the effective control input δθ and δh.
(17)S=KE+1E2,where
(18)K=k100k2is a positive definite diagonal matrix that needs to be designed.Combine equations (14)–(17).
(19)Ė1=−K+C1E1+S.The accessibility of the sliding mode condition guarantees that any original state in space can reach the sliding mode surface in finite time. When the system reaches the sliding mode surfaceS=0, then the sliding mode state equation of E1 can be obtained as
(20)Ė1=−K+C1E1.Make sure−K−C1 satisfies the Hurwitz matrix, so that E1 can converge to zero in finite time. Set
(21)−K−C1=λ100λ2,λ1,λ2<0.Define the second Lyapunov function candidate asV2=V1+1/2STS, differentiating V2, and utilizing equations (11), (15), and (17) yields
(22)V̇2=E1TĖ1+STK+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+B2U+C2+D2Fp+dt−X¨1,d+A1Ẋ1.Accordingly,(23)Ṡ=K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+B2U+C2+D2Fp+dt−X¨1,d+A1Ẋ1.In order to improve the convergence performance, an improved reaching law and boundary layer is introduced in the control method. The introduction of terms enhances the antichattering effect. Additionally, the saturation function can regulate the reaching speed. Outside the boundary layer, dynamic system approaches quickly, and approaches smoothly inside the boundary layer.The reaching law is chosen as(24)Ṡ=−εS2satS−HS,where
(25)S=s1s2T,ε>0,H=h100h2is a positive definite diagonal matrix.The definition of saturation function is given as(26)satsi=sgnsi,abssi>Δ,siΔ,abssi≤Δ,i=1,2.Adopting the backstepping techniques, the control input of a supercavitating vehicle can be derived as(27)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp+dt−X¨1,d+A1Ẋ1+εS2satS+HS.For analyzing the stability, substitute equation (27) into equation (22); then, the derivative V̇2 can be obtained
(28)V̇2=E1TE2−E1TC1E1−εS2STsatS−STHS.Proof.
By rearranging equation (28), the first three terms of V̇2 can be expanded as follows:
(29)E1TE2−E1TC1E1−STHS=−c1+h1k12ey2−2h1k1−1eyev−h1ev2−c2+h2k22eθ2−2h2k2−1ereθ−h2er2=−eyevQ1eyev−eθerQ2eθer,where
(30)Q1=c1+h1k12h1k1−12h1k1−12h1,Q2=c2+h2k22h2k2−12h2k2−12h2.Just make sure bothQ1 and Q2 are positive definite matrixes by determining the parameters properly; E1TE2−E1TC1E1−STHS<0 can be guaranteed.In addition,(31)−εS2STsatS=−εS2s1+s2,abss1&abss2>Δ,−εΔS4,abss1&abss2≤Δ,−εS2s1+s22Δ,abss1>Δ&abss2≤Δ,−εS2s12Δ+s2,abss1≤Δ&abss2>Δ.To sum up,V̇2 is negatively definite. According to the Lyapunov stability theory, it can be guaranteed that E1 and E2 can converge to a small neighborhood around zero in finite time.Furthermore, when the close-loop system is subjected to the disturbances, the severe chattering of control inputs appears due to the chattering of the sliding mode surface. In order to attenuate the effect of the chattering, the reaching law is improved as(32)Ṡ=−HS−DtanhSε,where tanhS/ε is a continuous and smooth hyperbolic tangent function compared with the saturation function which has sharp corners.


## 4.2.2. Control Parameter Optimization Based on PSO

As aforementioned, the proposed sliding mode control contains various parameters to be designed, which can be transformed into a multiobjective optimization problem. The objective function with constraints is presented as follows:(33)minJ=maxδθ+maxδh+∫0Tζey+eθdts.t.ci,ki,hi∈0,upperbound,i=1,2hici+ki−14>0.From a practical point of view, the fitness function consists of maximum of absolute values of control inputs and kinematic control law tracking errors in order to achieve minimum energy consumption and tracking errors. According to Lyapunov theorem, as long as the parameter constraints are satisfied, the trajectory errors will converge to zero; thereby, greater importance should be attached to the maximum deflection angle of the cavitator and elevators; then set weight coefficientζ=0.5 to balance control inputs and kinematic control law tracking errors. The PSO iterative process is shown as Figure 8, and related equations are listed as follows:
(34)Vi+1j=ωVij+n1r1pBestij−Xij+n2r2gBesti−Xij,Xi+1j=Xij+Vi+1j,where ω is inertia weight factor, n1 and n2 denote cognitive factors, r1 and r2 denote random numbers satisfying a standard normal distribution, Vij represents the velocity of particle Xij in swarm i, pBestij is the best between particle Xi−1j and Xij, and gBesti is the best particle in swarm i.Figure 8
The PSO algorithm process.


## 4.2.3. Design of Extended State Observer

Time-varying external disturbances that arise from the unexpected environment are an obstacle to accurate control, since it brings great difficulties to precise modeling and controller design. In this subsection, an extended state observer is proposed to estimate unknown external disturbances in dynamic model [38, 39] and achieve disturbance attenuation via effective compensation.To facilitate the control design, make an assumption that the time-variant external disturbancedt is bounded, satisfying dv≤ρ1, ḋv≤ρ2, dr≤ϖ1, and ḋr≤ϖ2. The further simplified model of dynamic model can be obtained
(35)Ẋ2=fX2+dt,ḋt=ht,where fX2 is the modeled part. ht is the derivative of the unknown external disturbances dt which is regarded as an extended state. A second-order extended state observer is utilized.
(36)e1=Z1−X2,Ż1=Z2+fX2−β1e1,Ż2=−β2fale1,α,δ,dt=Z2+ξ,fale,α,δ=eδα−1,e≤δ,eαsigne,e>δ,where Z1 is the estimation of the vertical velocity and pitch rate, Z2 is the estimation of disturbances, and e1 is the error of state variables. The observer gains β1 andβ2 are used to regulate the convergence speed of the observer. ξ=ξvξrT is the approximation error of the ESO. And ξ is bounded, satisfying.
(37)ξv≤ψ1,ξr≤ψ2,where ψ1 and ψ2 are both positive and unknown.After making compensation for disturbances, the sliding mode control law is modified as(38)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp−X¨1,d+A1Ẋ1+Z2+εS2satS+HS+ψ^signS.With the adaptation lawsψ^̇i=σisi,i=1,2.
(39)ψ^=ψ^100ψ^2,ψ^1, and ψ^2 denote the estimations of ψ1 and ψ2. σ1 and σ2 are positive constants. sign. denotes the unit sign function.Define the third Lyapunov function candidate [40] as
(40)V3=V2+12σ1ψ~12+12σ2ψ~22,where ψ~1=ψ1−ψ^1 and ψ~2=ψ2−ψ^2 denote the estimation errors.Substituting equation (38) into the derivative of V3, then V̇3 can be expressed as
(41)V̇3=E1TE2−E1TC1E1−εS2STsatS−STHS+ξvs1−ψ^1s1−ψ~1s1+ξrs2−ψ^2s2−ψ~2s2≤E1TE2−E1TC1E1−εS2STsatS−STHS.Therefore,V̇3 is negative definite, which satisfies the Lyapunov stability theory. In other words, dynamic and kinematic errors can converge uniformly and asymptotically.


## 4.2.4. Design of an Antiwindup Compensator

Due to the physical saturation of actuators, an antiwindup compensator is introduced to the proposed control scheme to attenuate the effects of saturation. The actual cavitator deflection angleuθ and elevator deflection angle uh can be given as, respectively,
(42)uθ=satδθ=mθsignδθ,δθ≥mθ,δθ,δθ<mθ,uh=satδh=mhsignδh,δh≥mh,δh,δh<mh,where mθ and mh are the upper limits of cavitator and elevator saturation constraints, respectively.In order to attenuate the influence caused by input saturations, the antiwindup compensator [26, 27] can be designed as
(43)Ṗ=−QP−STB2ΔU+1/2ΔUTΔUP2P+ΔU,P≥μ,0,P<μ,where P=pθphT is the auxiliary variable, ΔU=ΔδθΔδhT, Δδθ=uθ−δθ, and Δδh=uh−δh.
(44)Q=q100q2is a parameter matrix to be designed.After application of this modification, the control input will be as follows:(45)U=−B2−1K+C1Ė1+A2E2+Ẋ1,d−C1E1−A1X1+C2+D2Fp−X¨1,d+A1Ẋ1+Z2+εS2satS+HS−P+ψ^signSDefine the fourth Lyapunov function candidate as(46)V4=V3+12PTP.Considering the input saturation, the dynamic model and derivative of sliding mode surface should be rewritten as follows:(47)Ẋ2=A2X2+B2U+ΔU+C2+D2Fp+dt,Ṡ=K+C1Ė1+A2E2+Ẋ1d−C1E1−A1X1+B2U+B2ΔU+C2+D2Fp+dt−X¨1d+A1Ẋ1.Substituting equations (45) into the derivative of V4, V̇4 can be given as
(48)V̇4≤E1TE2−E1TC1E1−εS2STsatS−STHS+STHP−PTQP−12ΔUTΔU+PTΔU.According to equation (48), verification of some terms is just necessary.
(49)−STHS+STHP−PTQP−12ΔUTΔU+PTΔU=−h1s12−h2s22+s1h1pθ+s2h2ph−q1pθ2−q2ph2−12Δδθ2−12Δδh2+pθΔδθ+phΔδh.In fact, some inequalities hold as follows:(50)s1pθ≤s122γ1+γ1pθ22,s2ph≤s222γ2+γ2ph22,pθΔδθ≤pθ22γ3+γ3Δδθ22,phΔδh≤ph22γ4+γhΔδh22,whereγ1, γ2, γ3, and γ4 are design parameters.Substituting equation (50) into equation (49), one can be obtained
(51)−STHS+STHP−PTQP−12ΔUTΔU+PTΔU≤−h1−h12γ1s12−h2−h22γ2s22+h1γ12−q1+12γ3pθ2+h2γ22−q2+12γ4ph2−12−γ32Δδθ2−12−γ42Δδh2.In conclusion,V̇4≤0 can be guaranteed by setting proper design parameters which satisfy the following conditions:
(52)γi≥12,i=1,2,γi≤1,i=3,4,h1γ12−q1+12γ3<0,h2γ22−q2+12γ4<0.According to Barbalat’s Lemma, the close-loop control method is asymptotically stable, and the tracking errors can converge to a small neighborhood around zero in finite time.


## 5. Numerical Simulation and Verification

In order to verify whether the proposed control method is reasonable and effective, three numerical simulation cases are conducted. Case 1 is presented to illustrate the effectiveness of proposed SMC based PSO under different initial conditions without external disturbances. Case 2 is utilized to demonstrate the ability of the antiwindup compensator and the performance of ESO under the external disturbances adopting the optimal control parameters. Case 3 is a curvilinear trajectory tracking considering external disturbances and input saturation to further verify the trajectory tracking performance and robustness of the proposed control scheme.


### 5.1. Case 1

In case 1, the desired path is chosen as a rectilinear path at a fixed depth, and the state variables arexyθvr=75t30000. In order to verify the adaptability and tracking performance of SMC based on PSO, two initial conditions are chosen listed as follows, which represent the typical task scenarios. The PSO parameters are selected in Table 2. In order to illustrate the ability of the PSO algorithm, choose two sets of control parameters (P1 and P2) listed in Table 3 as the comparison of the optimal parameters. The PSO results are shown in Table 4.Table 2
The PSO parameters.
NameInertia weightωAcceleration factorn1Acceleration factorn2Value range(c1,c2)Value range(h1,h2,k1,k2)Value range(V)Value0.60.50.5(0, 30](0, 30][-1, 1]Table 3
Two sets of SMC parameters.
c1c2h1h2k1k2P1111111P2303030303030Table 4
The optimal SMC parameters.
Conditionc1c2h1h2k1k2I10.074616.23013.710660.650490.005342210.1064II4.2789917.00194.679681.10520.20375810.9177Condition I: The vehicle is ascending, the initial state isxyθvr=0200.100.Condition II: The vehicle exceeds the desired depth, the initial state isxyθvr=040−0.100.As presented in Figures9 and 10, the supercavitating vehicle can reach and move along the desired trajectory under all conditions. Though the trajectory tracking errors under both conditions with P2 are driven to zero immediately, the maximal deflection angles of the cavitator and elevators reach 28.8 rad and 61.6 rad, respectively. It is because the larger the control parameters h1 and h2 are, the faster the system can converge to zero. Obviously, this set of control parameters is undesirable. In terms of the fitness values, the ones with optimal parameters decrease by nearly 50% under both conditions compared with those with P1 shown in Figure 11. And the vehicle can reach the desired depth with optimal control parameters under both conditions within nearly 2 s compared with those with P1. Besides, tracking velocities with optimal control parameters under both conditions are stable and smooth depicted in Figure 12. In order to obtain the minimum objective function value, the control inputs with optimal control parameters slightly exceed the actuator saturation in a short period during the whole time domain shown in Figure 13. Time histories of planing force and elevator efficiency can be viewed from Figures 14 and 15, which reflect the position relationship between the vehicle and cavity. The vehicle is completely encapsulated by supercavity from nearly 0.8 s under both conditions with optimal control parameters. And it can be observed that there is a shift between the complete-encapsulated supercavitating vehicle centerline and supercavity centerline which leads to elevator efficiency change after 0.8 s under both conditions.Figure 9
Trajectory tracking in case 1.Figure 10
Position and orientation tracking errors in case 1.Figure 11
Comparison of fitness values under different conditions.Figure 12
Tracking velocities in case 1.Figure 13
Control inputs in case 1.Figure 14
Planing forces in case 1.Figure 15
Elevator efficiency in case 1.According to the simulation result of case 1, in the absence of external disturbances and input saturation, the PSO-based controller is proved to have good tracking performance under different initial conditions considering the cavity memory effect and nonlinear planing force.


### 5.2. Case 2

In order to verify the functionality of the proposed control scheme, it is assumed that(53)ht=6costWGN,where WGN is the white Gaussian noise with the frequency of 5 Hz and the amplitude of 0.2. Set condition I in the previous subsection as the initial state and the desired trajectory as well. The corresponding optimal control parameters are selected. Parameters of the antiwindup compensator are chosen as γ1=γ2=γ3=γ4=1, q1=3, and q2=1.The parameters D and ε of the improved reaching law in equation (32) are selected as 40 and 0.02, respectively. Besides, the parameters of ESO are given as β1=5, β2=10, δ=0.1, α=0.22, σ1=0.5, and σ2=0.1. The cavitator and elevator saturation are characterized by mθ=30° and mh=45°, respectively.The simulation results are shown in Figures16–25. It is seen that the both controllers adopting saturation function (sat) and hyperbolic tangent function (tanh), respectively, in the reaching laws with ESO perform better than that without ESO. In Figures 16 and 17, both trajectory tracking errors can both converge to zero with ESO while those without ESO generate periodical oscillation around the desired trajectory. In Figure 18, tracking velocities with sat and tanh both perform slighter chattering and faster convergence rate with ESO than those without ESO. Moreover, the vertical velocity without ESO cannot even be stable. Due to the existence of external disturbances and input constraints, the actuators need to change control efforts periodically during the whole time domain shown in Figure 19. By contrast, the reaching law adopting tanh can effectively suppresses the chattering of the control inputs, which is more practical. It is mainly because the nonlinearity in the reaching law adopting sat causes severe chattering of the sliding mode surfaces, and the smooth hyperbolic tangent function can attenuate the chattering. In addition, time history of planing force and elevator efficiency can also illustrate that the proposed control scheme with ESO and the antiwindup compensator achieve better performance than that with the only antiwindup compensator in Figures 20 and 21. The estimation values of external disturbances are presented in Figures 22 and 23. Based on the estimation, the compensations for δθ and δh are shown in Figures 24 and 25. It is obvious that external disturbances can be well estimated by the proposed ESO. Therefore, the effectiveness of ESO and the antiwindup compensator is validated.Figure 16
Trajectory tracking in case 2.Figure 17
Position and orientation tracking errors in case 2.Figure 18
Tracking velocities in case 2.Figure 19
Control inputs in case 2.Figure 20
Planing forces in case 2.Figure 21
Elevator efficiency in case 2.Figure 22
Disturbancev estimation in case 2.Figure 23
Disturbancer estimation in case 2.Figure 24
Compensation forδθ.Figure 25
Compensation forδh.


### 5.3. Case 3

In this subsection, a more complex scenario is conducted. The curvilinear trajectory is parameterized byx=75t, y=30+1.5sint, θ=−1/uẏ with severe initial condition xyθvr=0250.150.5. The external disturbances are time-variant and state-dependent listed as
(54)ht=sin0.3t,rcos0.3t.The input saturation is the same as that in case 2. PSO is conducted to obtain the optimal control parameters based on the same objective function, given asc1=1.04608, c2=1.29829, h1=3.02459, h2=27.3541, k1=16.8829, and k2=20.2797. The parameters of the antiwindup compensator are set as γ1=γ2=γ3=γ4=1, q1=3, and q2=15. And the parameters of ESO and the reaching law adopting tanh are the same as those in case 2. It can be viewed that under the proposed control scheme with sat and tanh in the reaching laws, the supercavitating vehicle can both converge to and move along the curvilinear trajectory with almost zero tracking errors shown from Figures 26 and 27. As shown in Figure 28, the actuator deflection angles briefly hit the bound of saturation in the initial tracking stage, since the tracking errors are relatively large. Furthermore, it can be observed planing force disappears rapidly and elevator efficiency implies the vehicle centerline coincides with the supercavity centerline within nearly 1.2 s presented as Figures 29 and 30. From Figures 31 and 32, ESO can all approximate the real values gradually although the external disturbances are related to the state variables. In terms of the chattering suppression, the reaching law adopting tanh also achieve better performance than that with sat. According to the simulation results above, it can be summarized that the effectiveness and performance of the proposed control scheme for the supercavitating vehicle in the task of rectilinear and curvilinear trajectory tracking are well verified through comparative numerical simulations.Figure 26
Trajectory tracking in case 3.Figure 27
Position and orientation tracking errors in case 3.Figure 28
Control inputs in case 3.Figure 29
Planing force in case 3.Figure 30
Elevator efficiency in case 3.Figure 31
Disturbancev estimation in case 3.Figure 32
Disturbancer estimation in case 3.


### 5.4. Computational Requirement

Three numerical simulation cases are carried out on the Matlab/Simulink platform, and the main configuration of the computer is 2.4GHz CPU and 32 GByte RAM. The maximum calculation cycle of three cases under different conditions is approximately 4.3 mins. Therefore, the proposed control scheme has good real-time performance.


## 5.1. Case 1

In case 1, the desired path is chosen as a rectilinear path at a fixed depth, and the state variables arexyθvr=75t30000. In order to verify the adaptability and tracking performance of SMC based on PSO, two initial conditions are chosen listed as follows, which represent the typical task scenarios. The PSO parameters are selected in Table 2. In order to illustrate the ability of the PSO algorithm, choose two sets of control parameters (P1 and P2) listed in Table 3 as the comparison of the optimal parameters. The PSO results are shown in Table 4.Table 2
The PSO parameters.
NameInertia weightωAcceleration factorn1Acceleration factorn2Value range(c1,c2)Value range(h1,h2,k1,k2)Value range(V)Value0.60.50.5(0, 30](0, 30][-1, 1]Table 3
Two sets of SMC parameters.
c1c2h1h2k1k2P1111111P2303030303030Table 4
The optimal SMC parameters.
Conditionc1c2h1h2k1k2I10.074616.23013.710660.650490.005342210.1064II4.2789917.00194.679681.10520.20375810.9177Condition I: The vehicle is ascending, the initial state isxyθvr=0200.100.Condition II: The vehicle exceeds the desired depth, the initial state isxyθvr=040−0.100.As presented in Figures9 and 10, the supercavitating vehicle can reach and move along the desired trajectory under all conditions. Though the trajectory tracking errors under both conditions with P2 are driven to zero immediately, the maximal deflection angles of the cavitator and elevators reach 28.8 rad and 61.6 rad, respectively. It is because the larger the control parameters h1 and h2 are, the faster the system can converge to zero. Obviously, this set of control parameters is undesirable. In terms of the fitness values, the ones with optimal parameters decrease by nearly 50% under both conditions compared with those with P1 shown in Figure 11. And the vehicle can reach the desired depth with optimal control parameters under both conditions within nearly 2 s compared with those with P1. Besides, tracking velocities with optimal control parameters under both conditions are stable and smooth depicted in Figure 12. In order to obtain the minimum objective function value, the control inputs with optimal control parameters slightly exceed the actuator saturation in a short period during the whole time domain shown in Figure 13. Time histories of planing force and elevator efficiency can be viewed from Figures 14 and 15, which reflect the position relationship between the vehicle and cavity. The vehicle is completely encapsulated by supercavity from nearly 0.8 s under both conditions with optimal control parameters. And it can be observed that there is a shift between the complete-encapsulated supercavitating vehicle centerline and supercavity centerline which leads to elevator efficiency change after 0.8 s under both conditions.Figure 9
Trajectory tracking in case 1.Figure 10
Position and orientation tracking errors in case 1.Figure 11
Comparison of fitness values under different conditions.Figure 12
Tracking velocities in case 1.Figure 13
Control inputs in case 1.Figure 14
Planing forces in case 1.Figure 15
Elevator efficiency in case 1.According to the simulation result of case 1, in the absence of external disturbances and input saturation, the PSO-based controller is proved to have good tracking performance under different initial conditions considering the cavity memory effect and nonlinear planing force.


## 5.2. Case 2

In order to verify the functionality of the proposed control scheme, it is assumed that(53)ht=6costWGN,where WGN is the white Gaussian noise with the frequency of 5 Hz and the amplitude of 0.2. Set condition I in the previous subsection as the initial state and the desired trajectory as well. The corresponding optimal control parameters are selected. Parameters of the antiwindup compensator are chosen as γ1=γ2=γ3=γ4=1, q1=3, and q2=1.The parameters D and ε of the improved reaching law in equation (32) are selected as 40 and 0.02, respectively. Besides, the parameters of ESO are given as β1=5, β2=10, δ=0.1, α=0.22, σ1=0.5, and σ2=0.1. The cavitator and elevator saturation are characterized by mθ=30° and mh=45°, respectively.The simulation results are shown in Figures16–25. It is seen that the both controllers adopting saturation function (sat) and hyperbolic tangent function (tanh), respectively, in the reaching laws with ESO perform better than that without ESO. In Figures 16 and 17, both trajectory tracking errors can both converge to zero with ESO while those without ESO generate periodical oscillation around the desired trajectory. In Figure 18, tracking velocities with sat and tanh both perform slighter chattering and faster convergence rate with ESO than those without ESO. Moreover, the vertical velocity without ESO cannot even be stable. Due to the existence of external disturbances and input constraints, the actuators need to change control efforts periodically during the whole time domain shown in Figure 19. By contrast, the reaching law adopting tanh can effectively suppresses the chattering of the control inputs, which is more practical. It is mainly because the nonlinearity in the reaching law adopting sat causes severe chattering of the sliding mode surfaces, and the smooth hyperbolic tangent function can attenuate the chattering. In addition, time history of planing force and elevator efficiency can also illustrate that the proposed control scheme with ESO and the antiwindup compensator achieve better performance than that with the only antiwindup compensator in Figures 20 and 21. The estimation values of external disturbances are presented in Figures 22 and 23. Based on the estimation, the compensations for δθ and δh are shown in Figures 24 and 25. It is obvious that external disturbances can be well estimated by the proposed ESO. Therefore, the effectiveness of ESO and the antiwindup compensator is validated.Figure 16
Trajectory tracking in case 2.Figure 17
Position and orientation tracking errors in case 2.Figure 18
Tracking velocities in case 2.Figure 19
Control inputs in case 2.Figure 20
Planing forces in case 2.Figure 21
Elevator efficiency in case 2.Figure 22
Disturbancev estimation in case 2.Figure 23
Disturbancer estimation in case 2.Figure 24
Compensation forδθ.Figure 25
Compensation forδh.


## 5.3. Case 3

In this subsection, a more complex scenario is conducted. The curvilinear trajectory is parameterized byx=75t, y=30+1.5sint, θ=−1/uẏ with severe initial condition xyθvr=0250.150.5. The external disturbances are time-variant and state-dependent listed as
(54)ht=sin0.3t,rcos0.3t.The input saturation is the same as that in case 2. PSO is conducted to obtain the optimal control parameters based on the same objective function, given asc1=1.04608, c2=1.29829, h1=3.02459, h2=27.3541, k1=16.8829, and k2=20.2797. The parameters of the antiwindup compensator are set as γ1=γ2=γ3=γ4=1, q1=3, and q2=15. And the parameters of ESO and the reaching law adopting tanh are the same as those in case 2. It can be viewed that under the proposed control scheme with sat and tanh in the reaching laws, the supercavitating vehicle can both converge to and move along the curvilinear trajectory with almost zero tracking errors shown from Figures 26 and 27. As shown in Figure 28, the actuator deflection angles briefly hit the bound of saturation in the initial tracking stage, since the tracking errors are relatively large. Furthermore, it can be observed planing force disappears rapidly and elevator efficiency implies the vehicle centerline coincides with the supercavity centerline within nearly 1.2 s presented as Figures 29 and 30. From Figures 31 and 32, ESO can all approximate the real values gradually although the external disturbances are related to the state variables. In terms of the chattering suppression, the reaching law adopting tanh also achieve better performance than that with sat. According to the simulation results above, it can be summarized that the effectiveness and performance of the proposed control scheme for the supercavitating vehicle in the task of rectilinear and curvilinear trajectory tracking are well verified through comparative numerical simulations.Figure 26
Trajectory tracking in case 3.Figure 27
Position and orientation tracking errors in case 3.Figure 28
Control inputs in case 3.Figure 29
Planing force in case 3.Figure 30
Elevator efficiency in case 3.Figure 31
Disturbancev estimation in case 3.Figure 32
Disturbancer estimation in case 3.


## 5.4. Computational Requirement

Three numerical simulation cases are carried out on the Matlab/Simulink platform, and the main configuration of the computer is 2.4GHz CPU and 32 GByte RAM. The maximum calculation cycle of three cases under different conditions is approximately 4.3 mins. Therefore, the proposed control scheme has good real-time performance.


## 6. Conclusion

In this paper, the control scheme of a supercavitating vehicle is proposed to address trajectory tracking problem in the presence of highly coupled nonlinearities, external disturbances, and input saturation. The dynamic and kinematic models of a supercavitating vehicle taking the cavity memory effect into consideration are formulated first. Subsequently, the virtual velocity control law is obtained based on the Lyapunov theorem to transform the trajectory tracking problem into a nonlinear control problem. Meanwhile, the dynamic controller combines the merits of SMC, PSO, and ESO, and the antiwindup compensator is introduced to design the dynamic control law, in which PSO is used to obtain the optimal control parameters, ESO estimates the external disturbances and compensates for control inputs, and auxiliary variables are introduced to handle input saturation by the antiwindup compensator. The stability of the closed-loop control system is guaranteed via the Lyapunov analysis. Moreover, the performance of the proposed control scheme is examined and validated by three simulation cases in terms of tracking accuracy, convergence rate, and system robustness. A future work will include the 6-DOF motion control for a supercavitating vehicle and the improvement of the control parameter optimization.



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*Source: 2938089-2023-02-22.xml*