Global approximation based adaptive RBF neural network control for supercavitating vehicles

LI Yang,LIU Mingyong,ZHANG Xiaojian,and PENG Xingguang

School of Marine Science and Technology,Northwestern Polytechnical University,Xi’an 710072,China

Abstract:A global approximation based adaptive radial basis function(RBF)neural network control strategy is proposed for the trajectory tracking control of supercavitating vehicles(SV).A nominal model is built firstly with the unknown disturbance.Next,the control scheme is established consisting of a computed torque controller(CTC)for the practical vehicle and an RBF neural network controller to estimate model error between the practical vehicle and the nominal model.The network weights are adapted by employing a Lyapunov-based design.Then it is shown by the Lyapunov theory that the trajectory tracking errors asymptotically converge to a small neighborhood of zero.The control performance of the proposed controller is illustrated by simulation.

Keywords:radial basis function(RBF)neural network,computed torque controller(CTC),adaptive control,supercavitating vehicle(SV).

1.Introduction

The supercavitating vehicle(SV)has gained a lot of attention from the military domain in recent years,as it helps the vehicle to have low drag and achieve high speed using encapsulation by cavitation inside a liquid[1-5].Most studies with respect to the control of SV are to establish a mode,and then design the controller based on it[3,4].However,the supercavitation breaks the conventional balance of the underwater vehicle,the forces on the vehicle are quite challenging to balance due to the lack of buoyancy,which also causes additional challenges for the attitude and orbit control[6].In the last decades,most studies focus on the planning force,which is a complex nonlinear term related to the underwater depth in which the vehicle locates as well as the angle at which the vehicle travels.Mathematical models and control strategies investigating the SV have been presented in numerous scientific literatures.A longitudinal dynamics model was developed in[1],and the switching controllers were also simultaneously presented.The influence of the cavitation structure on the dynamics model with respect to the vehicle was proposed in[2].A benchmark model,proposed in[3]for the pitch-plane dynamics of an SV,was widely used to design the controller for the sake of simplification.Consider the fin effectiveness issue,Mao et al.[4]designed an adaptive law based on the back stepping technique.Several nonlinear control approaches such as sliding model control and quasi-linear parameter varying(LPV)control for the dive plane dynamics of an SV were investigated in[7],in which a high-gain observer was also designed to estimate the immeasurable state.The impact of the different cavitation numbers and Froude numbers on cavitation was presented in[8].Hardware in loop simulation for the control of the SV was made in[9].Furthermore,a neural network based adaptive control was designed to deal with the modeling uncertainty in[10].By using computational fluid dynamics(CFD),the supercavitating flows through a typical SV model with different cavitation numbers and angles of attack was simulated in[11].

However,it is hard to establish an accurate model,actually.The model we obtained is the so called nominal model and the deviation between the nominal model and the practical vehicle is often ignored. The SV in double cavity state is what we study in this paper,and it is not only a transition phase,but also an independent state in sailing.The existing research has not considered the uncertainty while modeling the system.In the control problem of the vehicle in this state,modeling uncertainty is an important issue that impacts the control performance.Hence,a neural network(NN)-based adaptive control is necessary for highly uncertain nonlinear systems[12-14].

The NN plays an important role in machine vision,deep learning and controller design[12-14].The application of controller design focuses on the approximation process,which has superior performance on approximating to any function[15-19].For a class of unknown discrete-time nonlinear systems,an NN strategy is first constructed to identify the unknown controlled system,and then facilitate the implementation of the iterative algorithm[18].An NN-based adaptive observer and a sliding mode adaptive output feedback controller were designed in[19],in which the trajectory tracking problem was solved in the presence of uncertainties and disturbances.In[20],a new adaptive NN control scheme was proposed,where the objective for the robot end effector is specified as a dynamic region.To deal with the system uncertainties,radial basis function(RBF)NN was used to approximate the unknown model of biped robots in[21].Based on the strict-positive-real Lyapunov stability theory,an adaptive output feed back NN controller was designed for a single input single output(SISO)non-affine nonlinear system in[22].In addition,various backstepping design schemes have been reported,which combine the backstepping technique with adaptive NNs[15,23,24].

The rest of this paper is organized as follows.In Section 2,a dynamic model of the SV is built.In Section 3,the proposed method is introduced with respect to the adaptive RBF NN and the Lyapunov theorem.The simulations are carried out in Section 4.Finally,this paper is concluded in Section 5.

2.SV nominal model

2.1 Preparation for the modeling

Most of the system parameters of SV in double cavity state,as shown in Fig.1,come from benchmark high-speed SV model in[3].Consider the cavitator and the direct lateral force equipment on the aft body as actuators,and there is no tail rudders design for the whole body.The longitudinal dynamics of an SV is expressed in a body- fixed reference frame whose origin is at the center of gravity,the x-axis pointing forward along the vehicle axis of the center,the y-axis to the larboard and the z-axis pointing down[4],which can be seen in Fig.2.

Fig.1 Schematic diagram of an SV

The main difference between the proposed double cavity SV(DCSV)in this paper and the previous SV model[1-4,6,7,11]is the forces that play on the aft body.In the previous SV model,the forces on the aft body are mainly planning force and thrust,while the tail of the DCSV model has buoyancy Fb,thrust FT,direct lateral force Fv,hydraulic resistance Ffand tail differential pressure ΔF.The forces on the DCSV are shown in Fig.3,whose parameters are based on the literature from Dzielski and Kurdila[3]as presented in Table 1.

Fig.3 Forces on the DCSV

Table 1 System parameters of the DCSV

The differences forces cause the discrepancy on the dynamic between these two kinds of models.The forces on the DCSV include the fluid lift forceand resistanceon the cavitator which is on the nose of the vehicle and gravity Fg.By taking advantage of momentum equation and angular momentum theorem,the DCSV can be described as follows:

where u,v,w are the linear velocities in three directions,p,q,r are the angular velocities in roll,pitch and yaw,Jx,Jy,Jzare the rotational inertia around three coordinate axes.m is the mass of the vehicle,Mc=Fclc,otherwise,Fwand Mw=Fwlwis the resultant force and the moment of force acting on the wetted body,respectively.Here lc=17/28L,lw=-11/28L.

2.2 Cavity model

The cavity starts from the cavitator,and expands until it reaches the maximum radius and then closes in the aft body or encapsulates the whole vehicle,and the cavitator continuously makes the cavity while the vehicle is moving.The parameter that represents the characteristics of the cavity is the cavitation number σc.

where p∞is the static pressure,pcis the pressure inside the cavity.The cavitation number refers to the tendency of cavitation.When the vehicle sailing in a definite range of depth,p∞is constant,as well as the vehicle velocity.Therefore,the cavitation number σcis in relation to the pressure inside pccontrolled only by volumetric rate of the gas Qimeasured in m3/s.

where Fris the Froude number,h is the depth of the vehicle.When the speed V is constant,it means Fris invariable.According to Savchenko[5],the profile of cavitation depends on the gas supply rate input,which can be utilized by a semi-empirical formula:

wherer(xc)is the cavity radius at xc,which is the distance from the cavity curve origin.cdis the drag coefficient.Fig.4 shows the varying cavity profile for the Savchenko model when the cavitation number is σ=0.05,0.1,0.2.Here,kr=r(xc)/r,kL=lcl/L,lclis the length of the cavity.

Fig.4 Cavity profiles according to the varying cavitation numbers

2.3 Cavitator model

The cavitator is a crucial part of the SV,since it is the source of the cavity and the main control approach by changing the angle to generate forces and moments.The force on the cavitator are illustrated in Fig.5.

Fig.5 Cavitator

The hydrodynamic forces acting on the cavitator include a forcealong Xeaxis and another forcealong Zeaxis.

where α is the angle of attack,δc is the cavitator deflection angle,ρ is the fluid density,Scis the cross sectional area of the cavitator.Lcis the lift force of the fluid acting on the cavitator,and Dcis hydrodynamicdrag.

2.4 Gravity

Gravityacts at the center of the vehicle gravity,and ignores changes in the mass of the fuel,and when expressed in the body frame,it is given by

2.5 Forces acting on the wetted body

In this DCSV model,there is no planning force but buoyancy Fb,hydraulic drag Ffand differential pressure ΔFpacting on the wetted body,as is illustrated in Fig.6.

Fig.6 Forces acting on the wetted body

The wetted area and volume of the aft body can be calculated by the shape of the cavity.The buoyancy and its moment can be written as

where vwetis the volume of the wetted body.

The drag on the aft body is mainly the fluid viscous drag,and it is calculated by using Hughes line for the frictional coefficient[10].

where Cfis the friction coefficient.Ffacts in the negative direction of the vehicle forward speed.Reis the Reynolds number.

The differential pressure on the tail is caused by different angles and the cross-sectional area between the tail and water when the vehicle is maneuvering.It can be written as

where Sfis the maximum cross-sectional area of the stressed surface,clfis the lift coefficient.μas an efficiency coefficient is introduced for the length of the cavity cannot be measured when maneuvering.

2.6 Dynamic model of SV

In this study,we assume that the velocity is constant,ignoring the shape memory effort of the cavity.Hence,we can obtain the dynamical equation of the longitudinal plane:

where Jyis the moment of inertia in the y-axis direction.

We choose depth z,pitching angle θ,longitudinal velocity w and pitching angle velocity q as state variables,and cavitator angle δcand direct lateral force Fvas control inputs.Consider the kinematical equations presented in[1-3],the dynamical model of DCSV can be presented as

To simplify the form,the model can be expressed as

3.Adaptive RBF neural network controller design

3.1 Model treatment

Even after the model of DCSV has been obtained, some parameters are still difficult to acquire,such as the length of cavity.Hence,a model like(14)is called a nominal model,where there exists a number of differences from the actual vehicle.In order to facilitate the design of the RBF NN controller,model(14)can be rewritten as follows:

where M(ϕ)=B-1,C(ϕ)=-B-1A,G(ϕ)=-B-1F,τ is the control input and d is the unknown bounded disturbance.

The precise values of the parameters M(ϕ),C(ϕ)and G(ϕ)are difficult to acquire due to measuring errors,environment disturbance,etc.Therefore,we describe the dynamic system using the nominal model that is denoted by M0(ϕ),C0(ϕ)and G0(ϕ).Besides,the differences between the nominal model and the practical vehicle are represented by ΔM(ϕ), ΔC(ϕ)and ΔG(ϕ).These variables satisfy the following relationship:

Now(15)can be expressed as

Then

Based on the computed torque controller(CTC),we can obtain the control law:

where Kpis a positive definite matrix and ϕdis a desired trajectories vector.

where e=ϕ-ϕd,which is defined as the trajectory tracking error vector.It is obvious that errors would tend to zero if proportional gain Kpis chosen in the favorable situation.

3.2 Controller design

The RBF NN to approximate a continuous function f(x)can be expressed as

where x∈Rnis the input vector,W ∈Rmis the weight matrix,?(x)is the approximate error.φ(x)=[φ1(x),φ2(x),...,φn(x)]T,with φi(x)being the RBF function,the Gaussian RBF function is

where ci,biare the center and width of the ith neuron.

Assumption 1 Assume there exists an optimal weight vector W∗,given an arbitrary small positive constant ζ,so that the approximation error η of systems satisfies

Substituting control law(19)into(15)yields

then

37.After having drunk five or six glasses: The implication is that the king s good will has been increased with an abundance of alcohol while his critical thinking has been adversely affected. He fails to be cynical of the perfect circumstances.Return to place in story.

Hence,(25)can be rewritten as

Theorem 1 Consider the DCSV dynamic model which is described by(16)and(18),based on the CTC technique,the system can achieve the tracking control under the closed-loop control laws proposed as(19).

Proof Consider the Lyapunov function candidate as

where γ＞0,and P is a positive definite symmetric matrix,and satisfies the Lyapunov equation as

then,(30)can be written as

Noting that

then

where h＞0.

Substituting(34)into(32),we obtain

According to the characteristics of norm F,

hence

then

where ηsupis the upper bound of η,Wmaxis the maximal element of the ideal weight,λmin(Q)and λmax(P)denote the minimum eigenvalue of matrix Q and the maximum eigenvalue of matrix P.

or

Then the roundedness of the closed-loop system can be obtained as

or

Hence,it can satisfy the system stability via adjusting the parameter matrices as Q,P,etc. ?

4.Simulation

In this section,simulations of path tracking control for the DCSV are conducted to demonstrate the performance of our algorithms.Here we mainly consider the depth and pitch angle tracking,where the reference signals are set as

Some other parameters are chosen as h=0.1,γ=80,Q=10I4×4,the designed parameter in CTC is picked as Kp=100I4×4,and the initial state of the system is[zθwq]T=[00.0120.1]T.For RBF NN,the parameters of Gaussian function ciand biare chosen as[-2 -1 0 1 2]and 8,and the initial weight value is 0.1.The structure of the NN is designed as two cases:2-5-2,which means it has 2 inputs,5 hidden nodes and 2 outputs and 2-12-2,which means it has 2 inputs,12 hidden nodes and 2 outputs.The states responses are illustrated in Fig.7 and Fig.8.Fig.7 illustrates the desired and actual trajectories of the depth z and pitch angle θ,and nice tracking performance is obtained,otherwise,the overshoot amount is within the acceptable range.The performance of 12 hidden-layer nodes is better than that of 5 nodes.Fig.8 shows the actual trajectories of the longitudinal velocity w and pitching angle velocity q,and it can be seen that two state variables are stable within certain ranges basically after 0.3 s.

Fig.7 Desired trajectory and actual trajectory of z and θ

Fig.8 Actual trajectory of w and q

From Fig.9 we can see that two control inputs are bounded.Furthermore,the cavitator angle is always positive,and the direct lateral force is almost opposite,since the forces on DCSV have Fg＞Fb,the control inputs need to overcome the gravity to be stable.Fig.10 indicates that we can guarantee the tracking errors ez,eθconverge to a small neighborhood.Moreover,the more hidden-layer nodes the smaller tracking errors can we obtain.

Fig.10 Trajectory tracking errors of z and θ

5.Conclusions

In this paper,a dynamical nominal modeling of DCSV as well as the tracking control using neural network based adaptive control is presented.RBFs are used to approximate the errors between the nominal model and the actual vehicle,whose weight parameters are adjusted based on Lyapunov stability theorem.Simulation results carried out on the nonlinear system demonstrate the effectiveness of the proposed approaches.From the simulation results,we can see that the tracking control can ensure the stability of the closed-loop system for both depth and pitching angle tracking control of the DCSV as well as the errors can converge to a certain range.