# Parameters Optimization-Based Tracking Control for Unmanned Surface Vehicles

Authors: Renqiang Wang; Huaran Yan; Qinrong Li; Yubo Deng; Yutong Jin
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, a type of tracking controller on the basis of parameters optimization was proposed for unmanned surface vehicles (USVs). Taking into account the unique nonlinear and large inertia characteristics of USVs, an iterative sliding mode control (ISMC) was adopted to construct the controller including the USVs’ main engine speed controller to determine the longitudinal velocity and the steering controller to control the lateral displacement. In designing, the hyperbolic tangent function with the saturation characteristic is introduced to design the output feedback control law of nonlinear iterative sliding mode. Then, the differential evolution algorithm (DEA) is applied to construct the parameters optimization system for acquiring the optimal parameters of the proposed controller, and the control quality with adaptive ability and robustness of the optimized controller is achieved. It is verified by computer experiments that the optimized controller realizes the tracking control for USV under interference; meanwhile, compared with the iterative sliding mode controller, the control performance of the controller is better and the robustness of that is stronger.


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## Body

## 1. Introduction

In recent years, the development of information, computers, artificial intelligence, and other technologies has greatly promoted the process of ship intelligence, making it possible to realize unmanned smart ships. Smart ships are the inevitable trend of future ship development and have good application requirements and development prospects. Unmanned surface vessels (USVs) are one of the ways to move towards unmanned ships. Among them, autonomous navigation technology including ship course tracking control and trajectory tracking control is one of the core technologies of unmanned ship motion control and has become a research hotspot in the academic world. The engineering requirements put forward higher requirements for the accuracy of USV control, studying the motion control of underactuated USV having important theoretical significance and practical value. However, the unique underactuated characteristics of the USV and the changeable external environment seriously affect its operating performance.The application prospect of ships’ trajectory tracking control in practical engineering is promoted by increasing of research on trajectory tracking control. The simulation of tracking the preset ships’ trajectory on the actual model of 1 : 70 was realized on the basis of the output feedback method proposed by Michiel [1]. Zhao [2] combined sliding mode control and reinforcement learning algorithms and used an adaptive integral sliding mode controller to suppress chattering, but this method can only be used for tensor product models. He [3] combined the robust adaptive learning control method and the sliding mode control method to complete the trajectory tracking and directly used the integral sliding mode surface to suppress chattering. However, the disadvantage of [1–3] is that they are all in the ideal state, i.e., no disturbance or just continuous disturbance. In order to improve the dynamic disturbance immunity of the control algorithm, the trajectory tracking under the time-varying disturbance was achieved based on the backstepping method proposed by Yang [4]. Besides, the model predictive control was proposed by Annamalai [5] to control the sudden disturbance of the ship, and the effectiveness was verified by simulation experiments.Sliding mode control method was applied to the tracking control for ships or USVs with the uncertainty. An adaptive terminal sliding mode tracking control with nonsingular integral-type second order was proposed by Saleh Mobayen [6] for dynamic nonlinear system with uncertainties. And the method is applied to the thruster system, so that the chattering of the control output is lower and the amplitude is smaller. The proportional integral derivative (PID) control mode is incorporated into the sliding surface designing, and an adaptive PID-SMC technique is constructed to realize tracking control of UAVs with disturbances [7]. The radial basis function (RBF) neural network [8] online compensation is integrated into the sliding mode control designing, which solves the control input limitation problem of the USV motion system. At the same time, the RBF neural network also reduces the chattering [9] of control system designed by user. Literature [10] uses model predictive control to adapt to the environment online to control the sudden disturbance of the ship, with good results. The application of sliding mode control to ship trajectory tracking has similar research results. The auxiliary compensation system constructed by RBF is directly used to solve the input saturation problem of ship, and the desired effect that control input jumps out of the saturation limit area was achieved [11]. In literature [12], in view of the uncertainty of model parameters and the time-varying interference of wind, wave, and current, a sliding mode control method was adopted to construct the controller which tracks the virtual ship’s reference trajectory. In order to obtain a better control effect, the literature [13] used coordinate changes to transform the system into a chain system, designed an exponential control law, designed a sliding mode controller based on a nonlinear model, and achieved good results in the simulation process. In literature [14], a method to robust controllers with model uncertainty and different types of disturbances was proposed by using sliding mode control technology, so that the scheme of waypoint tracking for underactuated autonomous underwater vehicles was developed.Regarding engineering requirements, the output and velocity constraints of USVs should be considered to improve safety for USVs. In literature [15], by introducing a barrier Lyapunov function into the line-of-sight guidance, the specific transient tracking performance in terms of position error is guaranteed. A novel constrained yaw rate controller is proposed to ensure time-varying yaw rate constraint satisfaction, in which the yaw rate barrier is required to vary with the speed of the hovercraft. The neuroadaptive control for complicated underactuated systems with simultaneous output and velocity constraints [16] was proposed by Yang Tong. Different output constraint-related auxiliary functions are constructed in the Lyapunov function candidate to generate nonlinear displacement/angle-limited terms to control all state variables. Then, the elaborately designed velocity constraint-related terms are directly introduced into the presented controller, so that both actuated and unactuated velocity constraints are ensured.On the mathematical model of dynamic motion of USVs, based on the response model and the Fossen model, there is a big gap between the control inputs including control forces and torques, obtained by the above control method and the actual propulsion input of USVs. Especially for the underactuated USVs with single propeller and single rudder, the control inputs should be obtained in the form of rudder angle and main engine speed [17]. For this, the control inputs in the form of rudder angle and main engine speed were achieved for trajectory tracking control of underactuated USVs [18], under the nonlinear separation model proposed by mathematical model group (MMG) [19]. Similarly, Jia Heming [20] adopted nonlinear iterative sliding mode methods to control the course and track tracking of surface ships and unmanned underwater vehicles, respectively, and achieved good control effects. On the one hand, this method avoids the linearity of the model; on the other hand, it also avoids the uncertainty in USVs’ model and uncertain interference items. However, the parameters of the controller are fixed and make the controller not adaptive. The adaptive recursive sliding mode control strategy [21] with minimum parameter was proposed for nonlinear system with time varying, which expands the limited output range.Nowadays, intelligent control technology is developing rapidly. Moreover, some intelligent control methods, such as neural network [20] and fuzzy algorithm [22], have been used in the design of ship motion tracking control system. In literature [22], Shen proposed a fuzzy and shaped sliding mode flight control method with interesting brain evaluation. The mathematical model of the “Wenzhuhai” bulk carrier was used for control simulation. The results proved that the designed control performance is good and has strong robustness. In literature [23], Zhao Shunli proposed an adaptive sliding mode control method with dynamic surface based on neural network, and simulations show that the ship can quickly stabilize on a certain specified heading. In literature [24], Wang Hao proposed a cooperative path tracking algorithm with dynamic surface based on neural network, which realizes path tracking under interference conditions. In literature [25], Shen Zhipeng introduced a fuzzy system to optimize the predictable sliding mode parameters, designed a flight control that mimics the sliding mode control with sail navigation aids, and achieved good control effects. The particle swarm algorithm (PSA) with swarm optimization performance was used to optimize multiparameters in the iterative sliding mode control of underactuated ships, and the precise control of trajectory tracking is realized [26]. In literature [27], the adaptive sliding mode control with RBF neural network optimized by improved genetic optimization algorithm was designed for ships’ heading tracking control, and computer experiments verified the method is effective for steering system. In literature [28], the ship heading control system is designed by using the adaptive sliding mode control method combined with RBF neural network, which effectively solves the fast heading tracking under the condition of limited control input. It is concluded that in the sliding mode control, the control parameters have a significant effect on the adaptability and robustness of the controller, and the use of intelligent methods to optimize the parameters of the controller can make the control system achieve better accuracy. In addition, as far as the current intelligent optimization algorithm is concerned, under the same circumstances, the differential evolution algorithm (DEA) [29] has been proven to be the fastest evolutionary algorithm compared with genetic algorithm (GA) [30] and particle swarm algorithm (PSA) [26].Inspired by observations, a type of tracking controller on the basis of parameters optimization was proposed for unmanned surface vehicles (USVs) in paper. Taking into account the unique nonlinear and large inertia characteristics of USVs, an iterative sliding mode control (ISMC) was adopted to construct the controller including the USVs’ main engine speed controller to determine the longitudinal velocity and the steering system controller to control the lateral displacement. In designing, the hyperbolic tangent function with the saturation characteristic is introduced to design the nonlinear iterative sliding mode output feedback control law. Then, the differential evolution algorithm (DEA) is applied to construct the parameters optimization system for acquiring the optimal parameters of the proposed controller. With a certain type of USV model, it is obtained by computer experiments that the optimized controller proposed in paper realizes the tracking control for USV under interference; meanwhile, compared with the iterative sliding mode controller, its control performance is better, the control chattering is smaller, and the robustness is stronger.The merit of the paper is to optimize the previous tracking controller. Specifically, it includes (1) introducing the hyperbolic tangent function with saturation characteristics into the iterative sliding mode design. On the one hand, it can effectively overcome the chattering problem of the control input objectively existing in the sliding mode control design, and, on the other hand, it can also effectively avoid the hassle of getting control inputs into saturation. (2) Using the differential evolution algorithm (DEA) with swarm optimization performance to optimize various control parameters in the iterative sliding mode control design online, the optimization coupling problem of multiple parameters is solved at one time, and the tedious process of repeated trial and error is avoided. It greatly improves the adaptive ability and robustness of the system and conforms to the design and development concept of unmanned boats.At the same time, it is necessary to solve the speed limitation problem of USV trajectory tracking control. In response to this problem, this paper designs an iterative sliding mode control algorithm based on the speed model of the diesel engine for USV and then guarantees that the speed of the USV meets the set speed range.


## 2. Problem Description

For the USV equipped with one propeller and one rudder, the driving sources to realize trajectory tracking control are rudder angle and main engine speed, while the driving sources of Fossen model [31] for trajectory tracking control are forces and torques. The MMG model overcomes this shortcoming. So, this paper adopts the MMG model [20, 21] to design the trajectory tracking controller for USVs. The MMG model is expressed as(1)u˙=XH+XP+XR+Xwind+Xwave+m+myvrm+mxv˙=YH+YP+YR+Ywind+Ywave−m+mxurm+myr˙=NH+NP+NR+Nwind+NwaveIz+Jzx˙=ucosφ−vsinφ+uccosφcy˙=vcosφ+usinφ+ucsinφcφ˙=r,where XH, YH, and NH represent longitudinal force, lateral force, and rotational moment, respectively; XP, YP, and NP represent the longitudinal force, lateral force, and rotational moment of propeller, respectively; XR, YR, and NR stand for longitudinal force, lateral force, and rotational moment of rudder, respectively; Xwind, Ywind, and Nwind represent longitudinal force, lateral force, and rotational moment, respectively; Xwave, Ywave, and Nwave represent longitudinal force, lateral force, and rotational moment, respectively; m represents USV’s mass; mx and my represent additional USV’s mass in the longitudinal and transverse directions, respectively; Iz and Jz represent the moment of inertia of the USV, respectively; u stands for USV’s longitudinal speed; uc stands for current speed; v stands for USV’s lateral speed; r stands for rotation speed; x and y stand for USV’s longitudinal and lateral position; φ stands for USV’s heading; and φc stands for current direction.Figure1 is a schematic diagram of the tracking deviation of the USV trajectory. Oxd,yd is the coordinate of the USV’s actual trajectory; Gx,y is the coordinate of the USV’s current trajectory. The distance between these two points is ρD. ψr is the expected USV’s heading and ψ is the USV’s heading. xe and ye stand for the deviation of longitudinal and lateral of trajectory.Figure 1
Trajectory tracking diagram for ships.The expected point of the trajectory points to the actual point and the direction angle isθr. So, (2) is obtained from Figure 1.(2)ψr=arctany˙dx˙dθr=arctany−ydx−xdρD=y−yd2+x−xd2xe=ρD⋅cosθr−ψrye=ρD⋅sinθr−ψrψe=ψ−ψr.In the actual maneuvering process, according to the characteristics of USV navigation, its speed (including longitudinal and lateral speed), acceleration, and external disturbances are usually smooth and bounded; the longitudinal velocity is greater than the flow velocity, and both are much greater than the lateral velocity; that is,u>uc?v.In order to make the track deviationye⟶0, parameters optimization based tracking control for USVs is designed.


## 3. Design Tracking Controller of USV Motion

### 3.1. Theorem

In order to facilitate the design of the controller, the theorem is described as follows:Consider a first-order nonlinear state space model,(3)y=fx,u,t.where x is the state variable, u is the input, and y is the output.In order to facilitate the subsequent iterative sliding mode design and solution, the total assumptions need to be considered in the process of designing the USV trajectory controller in this paper.Assumptions 1.
(see [32]).(1)
fx,u,t in (3) is a continuous differentiable function.(2)
The gain of the control input is unknown, and it is assumed that∂f/∂u>0.Theorem 1.
The hyperbolic tangent function with saturation characteristic is adopted to construct the nonlinear feedback control law of iterative sliding mode.(4)u˙=−kpy−εtanhy,where kp>0 and ε>0.The state function is defined asx˙=gu, and gu represents continuous function with boundness. The uniform asymptotic stabilization of system (3) can be guaranteed under the control law (4).Proof.
First, (5) is derived by taking the derivative of (4).(5)y˙=∂f∂xx˙+∂f∂uu˙.
The Lyapunov function is constructed as follows:(6)Vy=12y2.
Then, (6) is referred by taking the derivative of(7)V˙y=12y2=yy˙=y∂f∂xx˙+∂f∂uu˙=∂f∂xguy+∂f∂u−kpy−εtanhyy=∂f∂u−kpy−εtanhy+∂f∂xguy≤−∂f∂ukpy+εtanhy−∂f∂xguy.
Further, considering∂f/∂u>0, the following inequality is obtained with constants kp and ε.(8)∂f∂ukpy+εtanhy−∂f∂xgu>0.
Furthermore, the following results can be obtained.(9)V˙y<0/.
Therefore, the theorem is proved under the assumptions mentioned above.


### 3.2. Design ISMC Controller

According to the conceptual analysis of exponential reaching law, the gain term of the sliding mode surface function is to adjust the rise time, and its value is as large as possible; the gain term of the hyperbolic tangent function is used to adjust switching amplitude, and its value should be as small as possible to weaken the degree of switching. The hyperbolic tangent function is used instead of the sign function to further weaken the system chattering caused by sliding mode surface switching. The gain term of the variable in the hyperbolic tangent function can adjust the amplitude, which promotes the smooth function and weakens the switching.The lateral component of the ship’s trajectory is realized by manipulating the rudder angle. The steering effect produced by steering is restricted by the ship’s longitudinal speed. At the same time, the longitudinal component of the ship’s trajectory is realized by manipulating the speed of the main engine. Therefore, in order to achieve trajectory tracking control, it is necessary to complete the lateral deviation control and the longitudinal deviation control at the same time.


#### 3.2.1. Design of Iterative Sliding Mode Controller for Longitudinal Deviation

The longitudinal deviationxe is controlled by changing the speed n of diesel engine, and the feedback control of nonlinear iterative sliding mode is designed as follows:(10)σ11xe=k11tanhk12xe+x˙e,σ12σ11=k13tanhk14σ11+σ˙11,n˙σ12=−k15σ12−k16tanhσ12,where tanhx is strictly bounded; k11, k12, k13, k14, k15, and k16 are design parameters.It is obtained by (10) that when σ12⟶0, there are σ11⟶0 and xe⟶0. So, the control goal becomes a stabilization control problem of σ12.After expandingσ12 in (10), it can be referred that(11)σ12σ11=k13tanhk13σ11+k11ddttanhk12xe+x˙e=k13tanhk13σ11+k11k12x˙ecosh2k12xe+x˙e.Considering the system model in (2), ignoring variables that have nothing to do with the speed, it is obtained that(12)∂σ12∂n=∂x¨e∂n=∂∂nu˙cosψe−v˙sinψe.When the actual ship is navigating normally, the lateral thrust generated by the propeller can be ignored (except for low-speed reverse). Therefore, (12) can be approximated as(13)∂σ12∂n≈∂∂nu˙cosψe=∂XP∂ncosψe/m+mx.The deviation between the heading and the track direction when USV is in normal navigating is generally no more than 90° and the relationship between propeller thrust and speed is strictly monotonic. Therefore, it can be inferred that(14)∂σ12∂n>0.According to (14), it can be seen that since the hyperbolic tangent and hyperbolic cotangent functions are strictly bounded, the design trajectory is sufficiently smooth; if the system is controllable, k11, k12, k13, k14∈R and ideal speed n∗t∈−nmax,nmax exist to guarantee σ12⟶0.According to the theorem, when the system and external interference are sufficiently smooth, the sliding mode surface feedback control lawn˙σ12=−k15σ12−k16tanhσ12 can make σ12 asymptotically stable. Therefore, the stability of the longitudinal trajectory tracking error xe is proved, that is, the deviation converges.


#### 3.2.2. Design of Iterative Sliding Mode Controller for Lateral Deviation

In order to construct the first-order function between sliding mode surface and rudder angleδ, a fourth-order nonlinear iterative sliding mode is designed, which is specifically divided into the following steps.The first step is to design the nonlinear sliding mode of the track deviation as(15)σ21ye=k21tanhk20ye+y˙e,where k20∼k21∈R+.It is obtained that whenσ21⟶0, y˙e⟶−k21tanhk0ye. Therefore, the control of tracking is transformed into the control of σ21.In the second step, considering the large inertia characteristics of the ship, in order to prevent excessive turn rate, the following sliding mode with integral is designed:(16)σ22σ21,φe=ψe+k22∫tanhσ21dt,where k22∈R+.Obviously, whenσ22⟶0, ψe⟶−k22∫tanhσ21dt and ψ˙e⟶−k22tanhσ21.Considering that the longitudinal and lateral components of the rudder force are much smaller than the torque and thatu≫v, it is proved that when σ22⟶0, σ21⟶0 [19]. So, the control of tracking is transformed into the control of σ22.In the third step, combined with the strictly bounded characteristics of the hyperbolic tangent function, the following sliding mode is designed in order to construct the control rudder angleδ,(17)σ23σ22=k23tanhσ22+σ˙22σ24σ23=k24tanhσ23+σ˙23,where k23∼k24∈R+.According to the relationship between the sliding surfaces in (17), the control goal is further transformed from σ22 to σ24.Next, deriving the monotonic function betweenσ24 and δ, the theorem is used to design the feedback control law.Equation (18) is obtained by expanding σ21.(18)σ21=k21tanhk20ye+vcosψe+usinψe+ucsinψc,σ21 is revisited as the output y of system (3), and σ21 is subjected to ψe. So, ψe is regarded as u in (4).Further, it is referred that(19)∂σ21∂φe=ucosψe−vsinψe.The deviation between the heading and the track direction when USV is in normal navigating is generally no more than 90°. So,cosψe>0, sinψe>0. Furthermore, u?v. Thus, ∂σ21/∂φe>0.Therefore,u is designed as(20)u˙=ψ˙e=−k22tanhσ1.According to the theorem,∂σ21/∂φe>0 in (19) achieves asymptotic stability, and σ21⟶0. Thence, ye is asymptotically stable, too.Next,σ24⟶0 is realized by a type of control law which is constructed as (21) so that ye is guaranteed to converge stably.(21)δ˙=−k25σ24−εtanhσ24.where k25∈R+, ε∈R+.Equation (22) is acquired by expanding σ24.(22)σ24σ23=k24tanσ23+k23r+k22tanhσ21coshσ222+k22k20k21y˙e/coshk20ye2+y¨e/coshσ212+NH+NP+NR+Nwind+NwaveIz+Jz.It is considered that, in (22), NR, XR, and YR are related to δ, so(23)∂σ24∂δ=∂∂δNR/Iz+Jz+∂∂δk22y¨e/coshσ212=∂∂δNR/Iz+Jz+k22∂XR∂δsinψ/m+mx/coshσ212+k22∂YR∂δcosψ/m+my/coshσ212.where the calculation of NR is shown in the following:(24)NR=xR+aHxHFNcosδ,where FR is the rudder force and FR>0, xR is the longitudinal coordinate of the lateral point of action on the rudder, aH is correction factor, and xH is the distance from the lateral force to the center of gravity; moreover, xR, aH, and xH are all scalars; δ is the rudder angle of the rudder, and cosδ>0 while δ∈−35°,35°.So,∂NR/∂δ>0 is established.Considering the actual steering,XR and YR are all smaller than NR. With the boundedness of trigonometric function and hyperbolic trigonometric functions, (25) holds under k22.(25)∂σ24∂δ>0.Forasmuch, from (21), (25), and the theorem, σ24 converges to zero asymptotically and ye shows asymptotic convergence too. Therefore, through the design of the above-mentioned control system, the tracking of the planned path of the USV is realized.


### 3.3. Parameters Optimization Based on DEA

It can be known from the USV’s mathematical model of maneuvering motion that the USV’s roll and external disturbances will cause frequent chattering of the maneuvering control system. Therefore, the design parameters of the controller must be dynamically adjusted to weaken and overcome this chattering.To enhance the effectiveness and service life of control system, the parametersk11∼k16,k20∼k25 of the designed controller are optimized based on DEA, and the adaptive ability and robustness of the system are improved together. Next, an online optimization system is designed based on DEA as in Figure 2, which is integrated into the trajectory tracking control system designed with ISMC method for USV.Figure 2
The optimization trajectory tracking control system for USVs.It is known that solving the optimization problem based on DEA is to determine the final optimal gene combination by solving the extreme value of the evaluation index, which is the optimal control parameter combination. Combining with the optimization system design idea of Figure2, the evaluation index function is constructed:(26)J=1N∑n=1Nλ1⋅ye2+λ2⋅ψe2+λ3⋅δ˜2,ye≤D,where J is evaluation index function, ye is trajectory tracking deviation, ψe is heading deviation, δ˜ is rudder angle chattering, N is the total number of iteration, and λ1∼λ3 stand for coefficients. Simultaneously, the sizes of λ1∼λ3 are set generally on the basis of environmental conditions.δ˜ in (26) is calculated [21] by (27)δ˜=M−M¯M¯,where M¯ is the preset value of rudder angle. M is the cumulative sum of the absolute values through n iterations. If M increases, δ˜ increases; conversely, if M decreases, δ˜ decreases.M in (27) is calculated by(28)Mt=∑t=0nρklδt−lT−δt−l−1T,where T is simulation period, δt−lT and δt−l−1T are the rudder angle values of different periods, n is the number of accumulation, and ρkl is defined as(29)ρkl=0,δklT−δkl−1T⋅δkl−1T−δkl−2T≥0,1,δklT−δkl−1T⋅δkl−1T−δkl−2T<0,If chattering occurs, the trend of the output will change together. The chattering will be acquired in the last cycle by (28).The process of optimizing the parametersk11∼k16,k20∼k25 of the designed controller is shown in Figure 3. The optimized architecture consists of two parts, DEA module and Simulink module. The two parts interact through individual genes (parameters of ISMC controller) and evaluation index function J. Among them, the DEA module assigns parameters to individuals by receiving evaluation indicators; the Simulink module solves the evaluation indicators based on the USB motion model via acquiring the individuals of DEA corresponding to the parameters. The end point of the optimization process is to obtain the smallest evaluation index function.Figure 3
Optimization process of control parameters.The solution process of Simulink module can be found in this section, which is the design unit of the control system, and the solution process of the DEA module is as follows.Step 1.
The gene population representing the design parameters of the controller is initialized, and an initial genetic individualxi0 is randomly generated.(30)xi,00=randixU−xL+xL,where xU and xL represent the upper and lower limits of the individual; randi is 0,1.Step 2.
The initial gene population is assigned to parametersk11∼k16,k20∼k25 which are transmitted to the Simulink module for performing the track tracking operation. Then, J is calculated and the termination condition is judged. If it is met, end; otherwise, go to Step 3.Step 3.
Mutation Operation. Three individuals (xp1,j, xp2,j, and xp3,j) are selected randomly, which are different from xi,j in j-th generation population, to perform mutation operation for generating a variant gene individual hi,jt+1.(31)hi,jt+1=xp1,jt+Fxp2,jt−xp3,jt,where p1∼p3 are numbers, i≠p1≠p2≠p3; G is evolutionary algebra, j<G; F⋅ is a factor.Step 4.
Cross Operation. To enhance diversity of gene population, new individual vit+1 with variant individual hi,jt+1 and evolution individual xi,jt is generated.(32)vi,jt+1=hi,jt+1,randi≤CR,xi,jt,randi>CR,where CR is probability, and CR∈0,1.Step 5.
Next-generation population individuals are generated on the basis of the greedy strategy.xit+1 is chosen by f⋅, and next generation particles are generated; then, return to Step 2, and the mentioned-above evolution operation is repeated until G reaches the maximum evolution algebra.(33)xit+1=vit+1,fvit+1≤fxitxit,fvit+1>fxit,where f⋅=1/J+1.


## 3.1. Theorem

In order to facilitate the design of the controller, the theorem is described as follows:Consider a first-order nonlinear state space model,(3)y=fx,u,t.where x is the state variable, u is the input, and y is the output.In order to facilitate the subsequent iterative sliding mode design and solution, the total assumptions need to be considered in the process of designing the USV trajectory controller in this paper.Assumptions 1.
(see [32]).(1)
fx,u,t in (3) is a continuous differentiable function.(2)
The gain of the control input is unknown, and it is assumed that∂f/∂u>0.Theorem 1.
The hyperbolic tangent function with saturation characteristic is adopted to construct the nonlinear feedback control law of iterative sliding mode.(4)u˙=−kpy−εtanhy,where kp>0 and ε>0.The state function is defined asx˙=gu, and gu represents continuous function with boundness. The uniform asymptotic stabilization of system (3) can be guaranteed under the control law (4).Proof.
First, (5) is derived by taking the derivative of (4).(5)y˙=∂f∂xx˙+∂f∂uu˙.
The Lyapunov function is constructed as follows:(6)Vy=12y2.
Then, (6) is referred by taking the derivative of(7)V˙y=12y2=yy˙=y∂f∂xx˙+∂f∂uu˙=∂f∂xguy+∂f∂u−kpy−εtanhyy=∂f∂u−kpy−εtanhy+∂f∂xguy≤−∂f∂ukpy+εtanhy−∂f∂xguy.
Further, considering∂f/∂u>0, the following inequality is obtained with constants kp and ε.(8)∂f∂ukpy+εtanhy−∂f∂xgu>0.
Furthermore, the following results can be obtained.(9)V˙y<0/.
Therefore, the theorem is proved under the assumptions mentioned above.


## 3.2. Design ISMC Controller

According to the conceptual analysis of exponential reaching law, the gain term of the sliding mode surface function is to adjust the rise time, and its value is as large as possible; the gain term of the hyperbolic tangent function is used to adjust switching amplitude, and its value should be as small as possible to weaken the degree of switching. The hyperbolic tangent function is used instead of the sign function to further weaken the system chattering caused by sliding mode surface switching. The gain term of the variable in the hyperbolic tangent function can adjust the amplitude, which promotes the smooth function and weakens the switching.The lateral component of the ship’s trajectory is realized by manipulating the rudder angle. The steering effect produced by steering is restricted by the ship’s longitudinal speed. At the same time, the longitudinal component of the ship’s trajectory is realized by manipulating the speed of the main engine. Therefore, in order to achieve trajectory tracking control, it is necessary to complete the lateral deviation control and the longitudinal deviation control at the same time.


### 3.2.1. Design of Iterative Sliding Mode Controller for Longitudinal Deviation

The longitudinal deviationxe is controlled by changing the speed n of diesel engine, and the feedback control of nonlinear iterative sliding mode is designed as follows:(10)σ11xe=k11tanhk12xe+x˙e,σ12σ11=k13tanhk14σ11+σ˙11,n˙σ12=−k15σ12−k16tanhσ12,where tanhx is strictly bounded; k11, k12, k13, k14, k15, and k16 are design parameters.It is obtained by (10) that when σ12⟶0, there are σ11⟶0 and xe⟶0. So, the control goal becomes a stabilization control problem of σ12.After expandingσ12 in (10), it can be referred that(11)σ12σ11=k13tanhk13σ11+k11ddttanhk12xe+x˙e=k13tanhk13σ11+k11k12x˙ecosh2k12xe+x˙e.Considering the system model in (2), ignoring variables that have nothing to do with the speed, it is obtained that(12)∂σ12∂n=∂x¨e∂n=∂∂nu˙cosψe−v˙sinψe.When the actual ship is navigating normally, the lateral thrust generated by the propeller can be ignored (except for low-speed reverse). Therefore, (12) can be approximated as(13)∂σ12∂n≈∂∂nu˙cosψe=∂XP∂ncosψe/m+mx.The deviation between the heading and the track direction when USV is in normal navigating is generally no more than 90° and the relationship between propeller thrust and speed is strictly monotonic. Therefore, it can be inferred that(14)∂σ12∂n>0.According to (14), it can be seen that since the hyperbolic tangent and hyperbolic cotangent functions are strictly bounded, the design trajectory is sufficiently smooth; if the system is controllable, k11, k12, k13, k14∈R and ideal speed n∗t∈−nmax,nmax exist to guarantee σ12⟶0.According to the theorem, when the system and external interference are sufficiently smooth, the sliding mode surface feedback control lawn˙σ12=−k15σ12−k16tanhσ12 can make σ12 asymptotically stable. Therefore, the stability of the longitudinal trajectory tracking error xe is proved, that is, the deviation converges.


### 3.2.2. Design of Iterative Sliding Mode Controller for Lateral Deviation

In order to construct the first-order function between sliding mode surface and rudder angleδ, a fourth-order nonlinear iterative sliding mode is designed, which is specifically divided into the following steps.The first step is to design the nonlinear sliding mode of the track deviation as(15)σ21ye=k21tanhk20ye+y˙e,where k20∼k21∈R+.It is obtained that whenσ21⟶0, y˙e⟶−k21tanhk0ye. Therefore, the control of tracking is transformed into the control of σ21.In the second step, considering the large inertia characteristics of the ship, in order to prevent excessive turn rate, the following sliding mode with integral is designed:(16)σ22σ21,φe=ψe+k22∫tanhσ21dt,where k22∈R+.Obviously, whenσ22⟶0, ψe⟶−k22∫tanhσ21dt and ψ˙e⟶−k22tanhσ21.Considering that the longitudinal and lateral components of the rudder force are much smaller than the torque and thatu≫v, it is proved that when σ22⟶0, σ21⟶0 [19]. So, the control of tracking is transformed into the control of σ22.In the third step, combined with the strictly bounded characteristics of the hyperbolic tangent function, the following sliding mode is designed in order to construct the control rudder angleδ,(17)σ23σ22=k23tanhσ22+σ˙22σ24σ23=k24tanhσ23+σ˙23,where k23∼k24∈R+.According to the relationship between the sliding surfaces in (17), the control goal is further transformed from σ22 to σ24.Next, deriving the monotonic function betweenσ24 and δ, the theorem is used to design the feedback control law.Equation (18) is obtained by expanding σ21.(18)σ21=k21tanhk20ye+vcosψe+usinψe+ucsinψc,σ21 is revisited as the output y of system (3), and σ21 is subjected to ψe. So, ψe is regarded as u in (4).Further, it is referred that(19)∂σ21∂φe=ucosψe−vsinψe.The deviation between the heading and the track direction when USV is in normal navigating is generally no more than 90°. So,cosψe>0, sinψe>0. Furthermore, u?v. Thus, ∂σ21/∂φe>0.Therefore,u is designed as(20)u˙=ψ˙e=−k22tanhσ1.According to the theorem,∂σ21/∂φe>0 in (19) achieves asymptotic stability, and σ21⟶0. Thence, ye is asymptotically stable, too.Next,σ24⟶0 is realized by a type of control law which is constructed as (21) so that ye is guaranteed to converge stably.(21)δ˙=−k25σ24−εtanhσ24.where k25∈R+, ε∈R+.Equation (22) is acquired by expanding σ24.(22)σ24σ23=k24tanσ23+k23r+k22tanhσ21coshσ222+k22k20k21y˙e/coshk20ye2+y¨e/coshσ212+NH+NP+NR+Nwind+NwaveIz+Jz.It is considered that, in (22), NR, XR, and YR are related to δ, so(23)∂σ24∂δ=∂∂δNR/Iz+Jz+∂∂δk22y¨e/coshσ212=∂∂δNR/Iz+Jz+k22∂XR∂δsinψ/m+mx/coshσ212+k22∂YR∂δcosψ/m+my/coshσ212.where the calculation of NR is shown in the following:(24)NR=xR+aHxHFNcosδ,where FR is the rudder force and FR>0, xR is the longitudinal coordinate of the lateral point of action on the rudder, aH is correction factor, and xH is the distance from the lateral force to the center of gravity; moreover, xR, aH, and xH are all scalars; δ is the rudder angle of the rudder, and cosδ>0 while δ∈−35°,35°.So,∂NR/∂δ>0 is established.Considering the actual steering,XR and YR are all smaller than NR. With the boundedness of trigonometric function and hyperbolic trigonometric functions, (25) holds under k22.(25)∂σ24∂δ>0.Forasmuch, from (21), (25), and the theorem, σ24 converges to zero asymptotically and ye shows asymptotic convergence too. Therefore, through the design of the above-mentioned control system, the tracking of the planned path of the USV is realized.


## 3.2.1. Design of Iterative Sliding Mode Controller for Longitudinal Deviation

The longitudinal deviationxe is controlled by changing the speed n of diesel engine, and the feedback control of nonlinear iterative sliding mode is designed as follows:(10)σ11xe=k11tanhk12xe+x˙e,σ12σ11=k13tanhk14σ11+σ˙11,n˙σ12=−k15σ12−k16tanhσ12,where tanhx is strictly bounded; k11, k12, k13, k14, k15, and k16 are design parameters.It is obtained by (10) that when σ12⟶0, there are σ11⟶0 and xe⟶0. So, the control goal becomes a stabilization control problem of σ12.After expandingσ12 in (10), it can be referred that(11)σ12σ11=k13tanhk13σ11+k11ddttanhk12xe+x˙e=k13tanhk13σ11+k11k12x˙ecosh2k12xe+x˙e.Considering the system model in (2), ignoring variables that have nothing to do with the speed, it is obtained that(12)∂σ12∂n=∂x¨e∂n=∂∂nu˙cosψe−v˙sinψe.When the actual ship is navigating normally, the lateral thrust generated by the propeller can be ignored (except for low-speed reverse). Therefore, (12) can be approximated as(13)∂σ12∂n≈∂∂nu˙cosψe=∂XP∂ncosψe/m+mx.The deviation between the heading and the track direction when USV is in normal navigating is generally no more than 90° and the relationship between propeller thrust and speed is strictly monotonic. Therefore, it can be inferred that(14)∂σ12∂n>0.According to (14), it can be seen that since the hyperbolic tangent and hyperbolic cotangent functions are strictly bounded, the design trajectory is sufficiently smooth; if the system is controllable, k11, k12, k13, k14∈R and ideal speed n∗t∈−nmax,nmax exist to guarantee σ12⟶0.According to the theorem, when the system and external interference are sufficiently smooth, the sliding mode surface feedback control lawn˙σ12=−k15σ12−k16tanhσ12 can make σ12 asymptotically stable. Therefore, the stability of the longitudinal trajectory tracking error xe is proved, that is, the deviation converges.


## 3.2.2. Design of Iterative Sliding Mode Controller for Lateral Deviation

In order to construct the first-order function between sliding mode surface and rudder angleδ, a fourth-order nonlinear iterative sliding mode is designed, which is specifically divided into the following steps.The first step is to design the nonlinear sliding mode of the track deviation as(15)σ21ye=k21tanhk20ye+y˙e,where k20∼k21∈R+.It is obtained that whenσ21⟶0, y˙e⟶−k21tanhk0ye. Therefore, the control of tracking is transformed into the control of σ21.In the second step, considering the large inertia characteristics of the ship, in order to prevent excessive turn rate, the following sliding mode with integral is designed:(16)σ22σ21,φe=ψe+k22∫tanhσ21dt,where k22∈R+.Obviously, whenσ22⟶0, ψe⟶−k22∫tanhσ21dt and ψ˙e⟶−k22tanhσ21.Considering that the longitudinal and lateral components of the rudder force are much smaller than the torque and thatu≫v, it is proved that when σ22⟶0, σ21⟶0 [19]. So, the control of tracking is transformed into the control of σ22.In the third step, combined with the strictly bounded characteristics of the hyperbolic tangent function, the following sliding mode is designed in order to construct the control rudder angleδ,(17)σ23σ22=k23tanhσ22+σ˙22σ24σ23=k24tanhσ23+σ˙23,where k23∼k24∈R+.According to the relationship between the sliding surfaces in (17), the control goal is further transformed from σ22 to σ24.Next, deriving the monotonic function betweenσ24 and δ, the theorem is used to design the feedback control law.Equation (18) is obtained by expanding σ21.(18)σ21=k21tanhk20ye+vcosψe+usinψe+ucsinψc,σ21 is revisited as the output y of system (3), and σ21 is subjected to ψe. So, ψe is regarded as u in (4).Further, it is referred that(19)∂σ21∂φe=ucosψe−vsinψe.The deviation between the heading and the track direction when USV is in normal navigating is generally no more than 90°. So,cosψe>0, sinψe>0. Furthermore, u?v. Thus, ∂σ21/∂φe>0.Therefore,u is designed as(20)u˙=ψ˙e=−k22tanhσ1.According to the theorem,∂σ21/∂φe>0 in (19) achieves asymptotic stability, and σ21⟶0. Thence, ye is asymptotically stable, too.Next,σ24⟶0 is realized by a type of control law which is constructed as (21) so that ye is guaranteed to converge stably.(21)δ˙=−k25σ24−εtanhσ24.where k25∈R+, ε∈R+.Equation (22) is acquired by expanding σ24.(22)σ24σ23=k24tanσ23+k23r+k22tanhσ21coshσ222+k22k20k21y˙e/coshk20ye2+y¨e/coshσ212+NH+NP+NR+Nwind+NwaveIz+Jz.It is considered that, in (22), NR, XR, and YR are related to δ, so(23)∂σ24∂δ=∂∂δNR/Iz+Jz+∂∂δk22y¨e/coshσ212=∂∂δNR/Iz+Jz+k22∂XR∂δsinψ/m+mx/coshσ212+k22∂YR∂δcosψ/m+my/coshσ212.where the calculation of NR is shown in the following:(24)NR=xR+aHxHFNcosδ,where FR is the rudder force and FR>0, xR is the longitudinal coordinate of the lateral point of action on the rudder, aH is correction factor, and xH is the distance from the lateral force to the center of gravity; moreover, xR, aH, and xH are all scalars; δ is the rudder angle of the rudder, and cosδ>0 while δ∈−35°,35°.So,∂NR/∂δ>0 is established.Considering the actual steering,XR and YR are all smaller than NR. With the boundedness of trigonometric function and hyperbolic trigonometric functions, (25) holds under k22.(25)∂σ24∂δ>0.Forasmuch, from (21), (25), and the theorem, σ24 converges to zero asymptotically and ye shows asymptotic convergence too. Therefore, through the design of the above-mentioned control system, the tracking of the planned path of the USV is realized.


## 3.3. Parameters Optimization Based on DEA

It can be known from the USV’s mathematical model of maneuvering motion that the USV’s roll and external disturbances will cause frequent chattering of the maneuvering control system. Therefore, the design parameters of the controller must be dynamically adjusted to weaken and overcome this chattering.To enhance the effectiveness and service life of control system, the parametersk11∼k16,k20∼k25 of the designed controller are optimized based on DEA, and the adaptive ability and robustness of the system are improved together. Next, an online optimization system is designed based on DEA as in Figure 2, which is integrated into the trajectory tracking control system designed with ISMC method for USV.Figure 2
The optimization trajectory tracking control system for USVs.It is known that solving the optimization problem based on DEA is to determine the final optimal gene combination by solving the extreme value of the evaluation index, which is the optimal control parameter combination. Combining with the optimization system design idea of Figure2, the evaluation index function is constructed:(26)J=1N∑n=1Nλ1⋅ye2+λ2⋅ψe2+λ3⋅δ˜2,ye≤D,where J is evaluation index function, ye is trajectory tracking deviation, ψe is heading deviation, δ˜ is rudder angle chattering, N is the total number of iteration, and λ1∼λ3 stand for coefficients. Simultaneously, the sizes of λ1∼λ3 are set generally on the basis of environmental conditions.δ˜ in (26) is calculated [21] by (27)δ˜=M−M¯M¯,where M¯ is the preset value of rudder angle. M is the cumulative sum of the absolute values through n iterations. If M increases, δ˜ increases; conversely, if M decreases, δ˜ decreases.M in (27) is calculated by(28)Mt=∑t=0nρklδt−lT−δt−l−1T,where T is simulation period, δt−lT and δt−l−1T are the rudder angle values of different periods, n is the number of accumulation, and ρkl is defined as(29)ρkl=0,δklT−δkl−1T⋅δkl−1T−δkl−2T≥0,1,δklT−δkl−1T⋅δkl−1T−δkl−2T<0,If chattering occurs, the trend of the output will change together. The chattering will be acquired in the last cycle by (28).The process of optimizing the parametersk11∼k16,k20∼k25 of the designed controller is shown in Figure 3. The optimized architecture consists of two parts, DEA module and Simulink module. The two parts interact through individual genes (parameters of ISMC controller) and evaluation index function J. Among them, the DEA module assigns parameters to individuals by receiving evaluation indicators; the Simulink module solves the evaluation indicators based on the USB motion model via acquiring the individuals of DEA corresponding to the parameters. The end point of the optimization process is to obtain the smallest evaluation index function.Figure 3
Optimization process of control parameters.The solution process of Simulink module can be found in this section, which is the design unit of the control system, and the solution process of the DEA module is as follows.Step 1.
The gene population representing the design parameters of the controller is initialized, and an initial genetic individualxi0 is randomly generated.(30)xi,00=randixU−xL+xL,where xU and xL represent the upper and lower limits of the individual; randi is 0,1.Step 2.
The initial gene population is assigned to parametersk11∼k16,k20∼k25 which are transmitted to the Simulink module for performing the track tracking operation. Then, J is calculated and the termination condition is judged. If it is met, end; otherwise, go to Step 3.Step 3.
Mutation Operation. Three individuals (xp1,j, xp2,j, and xp3,j) are selected randomly, which are different from xi,j in j-th generation population, to perform mutation operation for generating a variant gene individual hi,jt+1.(31)hi,jt+1=xp1,jt+Fxp2,jt−xp3,jt,where p1∼p3 are numbers, i≠p1≠p2≠p3; G is evolutionary algebra, j<G; F⋅ is a factor.Step 4.
Cross Operation. To enhance diversity of gene population, new individual vit+1 with variant individual hi,jt+1 and evolution individual xi,jt is generated.(32)vi,jt+1=hi,jt+1,randi≤CR,xi,jt,randi>CR,where CR is probability, and CR∈0,1.Step 5.
Next-generation population individuals are generated on the basis of the greedy strategy.xit+1 is chosen by f⋅, and next generation particles are generated; then, return to Step 2, and the mentioned-above evolution operation is repeated until G reaches the maximum evolution algebra.(33)xit+1=vit+1,fvit+1≤fxitxit,fvit+1>fxit,where f⋅=1/J+1.


## 4. Computer Simulation Experiment and Analysis

The USV model in [25] is applied to the computer simulation experiment of trajectory tracking control to verify the adaptive and robust performance of the designed controller. In computer experiment, initial position is (0, 0), speed of engine is 60 RPM, and speed is 15 knots. The center of the planned circular path to be tracked is (2000, 2000) and the radius is 2000 m; wind speed is 10 m/s and direction is 110°; current speed is 1.5 m/s and direction is 110°; wave height is 2 m and the encounter frequency is 0.5 Hz. Since the heading and speed of the USV change at any time, the vector relationship between the USV and the external disturbances also changes at any time. Therefore, the entire test can reflect the adaptability and robustness of the USV under the continuous change of external disturbances.During the USV’s circular trajectory tracking process, the ISMC controller and the DEA-optimized ISMC controller are simultaneously simulated for trajectory tracking.ε=0.001, M¯=0.1. k11∼k16 of ISMC controller are artificially preset as 10, 5, 15, 8, 26, and 5, and k20∼k25 of the ISMC controller are artificially preset as 17, 8, 7, 5, 16, and 5. k11∼k16,k20∼k25 of the DEA-optimized ISMC controller are obtained through the online optimization system, and G=50, CR=0.75, and F=0.25 of DEA.According to the above parameters settings, the two methods are simulated and compared; the description is the part of computer simulation results and analysis in next. First, the evaluation index function curve demonstrating the evolution process is shown in Figure4. Second, the comparative experiments are shown in Figures 5–12.Figure 4
The curve of evaluation index function.Figure 5
The curves of trajectory tracking.Figure 6
Error curve of lateral deviationye.Figure 7
Error curve of longitudinal deviationxe.Figure 8
Change curve of control rudder angleδ.Figure 9
Speed change curve of main enginen.Figure 10
Speed curve of USV.Figure 11
Change curve of parametersk11∼k16.Figure 12
Change curve of parametersk20∼k25.It is acquired from the trajectory tracking curve diagram of Figure5 and the variation curves of the lateral deviation ye and the longitudinal deviation xe shown in Figures 6 and 7 that both control methods can completely track the target trajectory. In Figure 6, the lateral deviation of the trajectory tracking control of the DEA-optimized ISMC controller is less than 40m, while the maximum position deviation of the trajectory tracking control of the ISMC controller is 188m, which is about 4.5 times of the former. In Figure 7, the longitudinal deviation of the trajectory tracking control of the DEA-optimized ISMC controller is less than 240 m, while the maximum position deviation of the trajectory tracking control of the ISMC controller is 310 m; it is larger. It is obtained that the DEA-optimized ISMC controller can track faster than the ISMC controller, and the yaw distance in the control process is smaller, which will save energy in practical applications.Figure8 is the control rudder angle change curve. It is found from Figure 8 that the maximum rudder angle of the USV using the DEA-optimized ISMC controller is only 12°, while using ISMC controller, it is about 18°; compared with the former, the applied rudder angle is increased by 50%. In addition, when the USV moves along a circular trajectory, the rudder angle output swing amplitude based on the DEA-optimized ISMC controller is less than 2°, while the rudder angle output swing amplitude based on the ISMC controller is about 5–6°, which is about 3 times that of the former. In engineering, the rudder angle swing amplitude is small and the wear and tear of the steering gear is small too, which is conducive to the working life of the steering gear. At this point, it is explained that the DEA-optimized ISMC controller is more suitable for the actual navigation of USV than the ISMC controller.Figure9 is the main engine speed change curve. It can be seen that after the introduction of the DEA optimization algorithm, the chattering changes are reduced correspondingly, and the effect of reducing the rudder angle buffeting is obvious. From the ship speed curve shown in Figure 10, it can be seen that the ship speeds of the two methods basically track the target speed. Due to the interference of sea conditions such as wind current, the actual ship speed fluctuates up and down 15kn, and the fluctuation based on the DEA optimization algorithm is small.Figures11 and 12 are two types of parameters adjustment curve based on DEA optimization algorithm. The parameters k11∼k16 in design of controller for longitudinal deviation and the parameters k20∼k25 in design of controller for lateral deviation are learned in real time through the DEA optimization system, which play a role in the adjustment process of chattering of rudder angle and speed of main engine, so as to enhance the effect of controller self-adaptation.Therefore, it can be concluded that the DEA-optimized ISMC controller has a better control effect than the ISMC controller. The former has better control performance; meanwhile, it is more in line with the actual USV project.In addition, the demonstration of the influence of model parameters on the adaptability and robustness of the controller can be verified according to Figures5 and 10. Referring to Figures 5 and 10, it can be seen that as the simulation progresses, the speed of the USV decreases with a maximum decrease of 50%; in this case, the designed optimized controller can still effectively track the predetermined circular trajectory. Therefore, it can be concluded that the designed control algorithm can still effectively track the preset trajectory when the speed parameters of the USV model change.


## 5. Conclusions

Taking into account that the driving source of Fossen model for trajectory tracking control is the three control variables of ship’s longitudinal thrust, lateral thrust, and turning moment, it is difficult to apply to engineering practice. The DEA-optimized ISMC controller is designed to realize the trajectory tracking control for USVs via MMG model, and it is guaranteed through the theorem proposed that all signals in the system are consistent and eventually bounded.Compared with ISMC controller, it is demonstrated that the proposed DEA-optimized ISMC controller has better control performance, smaller control chattering, and stronger robustness through computer simulation experiment. Thus, the adaptive and robust performance of the designed DEA-optimized ISMC controller is verified.



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*Source: 2242338-2022-03-18.xml*