Phu Du Hong Bui ,Sm-Sng You ,Hwn-Seong Kim ,Sng-Do Lee

a Department of Mechanical Engineering,The University of Tulsa,Tulsa,OK 74104,United States

b Division of Mechanical Engineering,Korea Maritime and Ocean University,Busan 49112,Republic of Korea

c Division of Logistics Engineering,Korea Maritime and Ocean University,Busan 49112,Republic of Korea

d Division of Navigation and Information System,Mokpo National Maritime University,Mokpo,Jeollanam-do 58628,Republic of Korea

Keywords:Vehicle dynamics Planing force Vertical maneuver H-infinity control Anti-windup Actuator saturation

ABSTRACT The control synthesis of a high-speed supercavitating vehicle (HSSV) faces many difficulties such as the stability,control,and maneuvering with dynamical uncertainties due to parameter perturbations,external disturbances,unmodeled dynamics,measurement noises,and actuator constraints.Inspired by the HSSV dynamical analysis,this paper proposes the H∞ (i.e."H-infinity") robust control synthesis to generate a robust low-order controller,which is intended for real implementations to ensure active control actions.Particularly,the presented control scheme includes a feedback component and an anti-windup compensator.The anti-windup synthesis is to provide system stability under actuator saturations.Extensive simulations show that the designed controller provides good performances with high robustness for vertical plane manoeuver while effectively eliminating planning forces,exogenous disturbances and noises as well as overcoming cavitator saturations.

1.Introduction

Conventional underwater vehicles often have limited speeds due to considerable friction drag acting on their hull skins.Their speeds can be dramatically increased under a condition called supercavitation with the vehicle configuration.When underwater vehicles such as torpedoes or submarines move at a certain high speed,fluid pressure drops below water’s vapor pressure,and gaseous cavities can be formed.If a single cavity size is maintained big enough to encompass the whole vehicle body,the friction drag is minimized,and the vehicle can move at a very high speed [1].Recently,several HSSVs have been designed and serving mostly military communities in some countries such as Russia,Germany,the United States,and Iran.In the military,the cavitating torpedo has been dominating over the fully wetted ones thanks to its extremely high speed,which can be more than 100 m/s.This kind of moving technique is still in its early stage and will bring many advantages if it is broadly applied to other underwater vehicle types.Currently,some navy prototypes can transport small groups of personnel at a speed of more than 15 m/s [2,3],and the goal of 50 m/s is to be reached in the near future.

The main actuators of HSSV include the cavitator mounted on the nose and the fins in the aft part of the vehicle.The cavitator is designed to generate and maintain the cavity.Together with the fins,it manipulates the vehicle orientation and stability [4].A typical configuration of the proposed HSSV is given in Fig.1,where the earth-fixed frame is denoted as (OE-XE-ZE) and the body-fixed frame is (Ob-x-z).At very high speeds,the cavitator contacts water at a certain deflection angle,then generates supercavitating phenomenon providing lift force.The lift forces provided by the cavitator and control surface of the elevators and immersion planing force share the responsibility of the equilibrium problem [5].Depending on the shape of the cavitator with the size and immersion of the control surfaces,the vehicle body may be inherently stable or unstable [4].

The vehicle control is considerably challenging due to the plant complexity in cavity dynamics,nonlinear forces,parametric uncertainties,disturbances,as well as actuator constraints [6].The cavity dynamics are very complex with dynamical coupling between internal terms.Even a slight variation in the cavitation number,cavity radius,or any other parameters determining the cavity shape will cause strong influences on the vehicle dynamical behaviors.Furthermore,most dynamics that describe the cavity use assumptions or simplifications.Some other uncertainties include the vehicle mass and inertia due to the fuel consumption mass.When there is so little contact with water,the balance between the hydrodynamic forces and moments breaks down,making HSSVs control more difficult [4].The nonlinearity exists in the vehicle under the form of the planing behavior.Planing occurs when the vehicle aft end pierces the supercavity,leading to high amplitude forces that push the vehicle aft end toward the center of the supercavity bubble [7].These forces can cause instabilities or oscillations known as the tail-slap phenomenon.The actuators in the HSSV system also have their saturation.The actuator saturation will lead to the windup phenomenon when the integral term in the controller accumulates a significant error during the rise and eventually resulting in excessive overshooting.The practice control of HSSVs requires considering actuator saturations in case of sudden steering to avoid windup,possibly leading to instability and actuator damages.

Up until now,many control strategies have been designed to enhance the maneuvering ability of supercavitating vehicles.They include LQG [6],linear parameter-varying control [8],model predictive control [9],robust control andμcontrol [10],nonlinear control and sliding mode control [11],and LTI output-feedback control [12].The control method seems plentiful,but relatively few works have accounted for actuator constraints,especially the robust control with such constraints.On the other side,some primary controllers are simple and easy for application purposes but lack robustness while robustH∞ andμ-controllers offer robustness,but their high orders are barriers for them to be deployed in present electronic devices.

Fig.1.Schematic diagram of a typical HSSV model.

In order to solve the above restrictions,this paper proposes a robust control synthesis with a fixed structure and low order guaranteeing robust performance and real implementations.The controller also includes a compensator to account for cavitator saturation to self-equip the anti-windup function.Unlike some convention anti-windup gains,the current compensator is simultaneously synthesized with the structured feedback controller via nonsmooth multi-objectiveH∞ optimization.It is known that optimization problems are common in many disciplines and various domains [13,14,15]but mostly they are single boundary value problems [16].In this paper,the optimization problem is solved to satisfy two boundaries,one for system stability and the other for saturation compensation.The simulation result shows that the proposed controller can do good maneuvering with a fast response.It also shows the ability to attenuate the external disturbance and provides robustness under the form of fault-tolerance design with the nominal,minimum and maximum models respecting the allowed percentages of uncertainties in the vehicle speed,cavitation number and lift coefficient.The dynamic performance of the proposed controller provides potentials in the implementation of an active robust structured controller for new generations of HSSVs.

2.Vehicle model and dynamical analysis

2.1.Mathematical model

There have been several different models of HSSVs which are in general highly complex,nonlinear,and coupled systems.Aiming at the design of the robust vehicle controller,this paper focuses on the dynamic model proposed by Dzielski and Kurdila [17],which has four states to describe the dive-plane dynamics.This model is well defined and is suitable for control design purposes.Note that in this study,the forward speed is assumed to be constant and defined along the vehicle axis.The state-space representation in the body-fixed frame that describes the dynamical behavior of the HSSV is defined as in Eq.(1).It can be seen that besides the portions of a regular state-space equation,it also contains two other parts including the contributions of gravity and planing force.The four state variables of the vehicle model include the vertical positionz,the vertical speedw,the pitch angleθ,and the pitch rateq.Two actuation inputs include the elevator deflection angleδeand the cavitator deflection angleδc:

where the system parameters are given as follows:

whereRcis the cavity radius andthe cavity radius contraction rate,Rthe supercavitating body radius,Rncavitator radius,Fpthe planing force,h’the immersion depth,αthe angle of attack,Lthe vehicle length,nthe fin effectiveness ratio with respect to the cavitator,mthe density ratio of the body to water,gthe gravity acceleration,Cx0the cavitator lift force coefficient,andσthe cavity number.For dynamical analysis and control synthesis,the model parameters are described in Table 1.

Table 1 Vehicle model parameters.

Fig.2.Characteristics of the normalised planing force.

2.2.Vehicle dynamical behavior

The planing force is a noncontinuous force that only exists if the vertical speed of the HSSV reaches a certain threshold (see Fig.2).This hydrodynamic force is one of the fundamental nonlinear factors in this underwater vehicle model [17].It is assumed that the planing force only depends on the vertical velocity and is a monotonically increasing function of the immersionh’and attack angleαwhich finally depends on vertical speedw.Fig.2 shows the relation between the normalized planing force and vertical speedw.It can be observed that the planing force only occurs if the magnitude of the vertical velocity is greater thanwo,where the vehicle aft end pierces the supercavity.This force has a tendency of pushing the vehicle aft end toward the center of the supercavity bubble.When alternatively happening in two opposite directions,the force will cause the tail-slap phenomenon,which will lead to instability without a proper controller.

Then the nonlinear behavior of the HSSV is analyzed to offer a vision of the system stability.Fig.3 illustrates the phase portrait of the two primary states;the vertical speedwand the pitching rateq.The phase plane trajectories show that the system is an unstable focus around the equilibrium point.The spiral diverges away from the origin.Through this analysis,it strongly indicates that the system needs an active controller to keep the vehicle stable while producing precise maneuvering.

3.Structured H∞ anti-windup control via the nonsmooth optimization

Nonsmooth multi-objective optimization has been developed over the last decade.This technique has proved its abilities to optimize different control structures varying from simple gains,PIDs,to free-order transfer functions or state-space models.It is also able to do anti-windup control with magnitude and rate saturation.The control structure is to maintain robust performance while minimizing the adverse effects of saturation and protecting the actuators [18].This optimization algorithm will be applied in this paper to produce a robust controller that provides an anti-windup function and low order structure to be ready for real-time implementation.

Fig.3.Phase portrait of the HSSV model.

3.1.Anti-windup control structure

Consider the linear HSSV systemG(s)to be controlled with the magnitude and rate saturation in its actuators,as illustrated in Fig.4.

In this configuration,the anti-windup signalw=us-uc,so that the anti-windup action becomes active only when saturation occurs.The control signalucwill include an anti-windup compensatorJ(s)and a feedback component(s)as follows:

It is noted that the augmented controller(s)is defined with additional inputv,comparing to the conventional controllerK(s).The saturated control signalusis verified by|us|≤SMand≤SRwhereSMandSRare the saturation thresholds for magnitude and rate,respectively.

3.2.Anti-windup design with H ∞ control framework

In order to recast the anti-windup design intoH∞framework,following the formalism from [19],the nonlinear operatorΦis replaced by a function of nonlinear uncertaintyΔΦ=I-Φwhich is null in the nominal case.From the block diagram in Fig.4,the general control scheme has been proposed for the HSSV as in Fig.5.

In this control framework,r(s)is the reference model to be tracked,dthe exogenous disturbance,Wpthe low-pass filter to weight the error between the plant output and the reference trajectory,WΔthe high-pass filter to penalize high-frequency dynamics in the anti-windup actions.The linear time-invariant (LTI) operatorT(s)is computed such as:

whereΩJandΩKare the designed blocks satisfying the following standard upper linear fractional transformation (LFT):

Note that for two matricesMandNof appropriate dimensions,the upper and lower LFT are defined asFu(M,N):=M22+M21N(IM11N)-1M12andFl(M,N):=M11+M12N(I-M22N)-1M21,respectively,where the matricesMijare partitions ofM.

Rearranging the feedback control system in Fig.5 leads to the general control structure using LFT,as illustrated in Fig.6.

Using the small-gain theorem,the control problem which is to preserve system stability despite saturations turns out to be minimizing theH∞norm of the transfer function fromwΔtozΔ.The idea can be justified by the following multi-objectiveH∞control design problem [18]:

wherePnom(s)is the nominal augmented plant from (r,d,wK) to(zr,zu,zK);Prob(s)describes the robust plant that includes the nominal plant and the anti-windup related signals from (wΔ,wJ) to(zΔ,zJ);the parameters (c1andc2) emphasize the satisfaction of system’s performances.The bi-objective problem in Eq.(20) can be recast into the following nonsmooth program,which findsJ(s)andK(s)such that:

where

ThisH∞anti-windup problem is a nonsmooth optimization problem proposed in reference [20].In this procedure,the feedback controllerK(s)and the anti-windup compensatorJ(s)are simultaneously synthesized,guaranteeing the robustness of the HSSV.It is important to notice that the stability of the controller is satisfied if a value less than one is achieved for the program in Eq.(21).It is also known that there are lots of dynamic uncertainties in the HSSV model.They can be the cause of the varying hydrodynamic configurations,drag force,planing force,adding mass,cavity number,cavity radius,system coefficient,or from the unmodeled dynamics,etc.The nominal HSSV model is only a typicalG(s).The real operating model should be any in a set of the possible plant.In order to improve the robustness level,the perturbed systemversus additional uncertainties affecting the nominal plant is defined by:

The system designers have extended the proposed strategy for a multi-model case by defining {G1(s),...,Gn(s)},a family of the possible plant,including worse cases deduced from Eq.(23).The initial optimization problem now can be extended for the family set as [19]:

Finally,from the above information,the robust controller has been synthesised through some trial-and-errors with an optimal value of 0.95 that can guarantee the closed-loop stability.

4.Simulation results and discussion

The efficiency of the technique was demonstrated through numerical simulations to illustrate the dynamical behavior of the vehicle system.Since the planing force only depends on the vertical speed,the simulations are carried for vertical maneuvering (or dive motion) of the underwater vehicle.Specifically,the responses of the pitch angle of the controlled system are going to be examined.It’s worth noting that the dive motion of the HSSV may be to keep its fixed depths,which are important for military or various missions.For the high-speed vehicle with rigorous maneuvering,the reference modelR(s)is selected in Eq.(25) as a second-order system with critical damping to gain fast responses with no oscillation and overshoots.

Fig.4.General structure for anti-windup compensator.

Fig.5.Completed control configuration.

The weighting functions have been selected after several trials as follows:

The parameter variations to check the control robustness is given in Table.1: the vehicle speed of 10%;the cavitator lift force coefficient of 20%;the cavity number of 10%.Then the design criterions of the closed-loop system are selected as follows:

·Rise time less than 0.5 s

·No overshoot with zero steady-state errors

·More than 50% of disturbance and noise rejected

·With overcoming parameter uncertainties

·With dealing with actuator constraints.

Based on the fundamentals described in Eqs.(17)-(24),the structured robustH∞ anti-windup controller has been synthesized for the vehicle.It consists of a feedback termK(s)and anti-windup compensatorJ(s).Note thatK(s)includes one second-order feedback function for the pitch angleθ,one gain for the vertical velocityw,and another gain for the feedback pitching rateq.Since the control actions in the elevator deflection usually are far from its saturation,the anti-windup term is only designed for the cavitator deflection angleδc.

In order to examine the effect of the controller on the parameter uncertainties,three models have been created as nominal(nom),minimal (min),and maximal (max) model relating to the dynamic models with the nom,max,and min parameter values,respectively.The three models represent all possible varying vehicle dynamics in the allowed range.The active controller has been synthesized so that its control effect spans all the models.The simulation is carried out to describe dive-plane dynamics.Hence,the vehicle maneuvering is to track the reference signal in longitudinal commands.

Fig.7 shows the transient responses of the models tracking with a reference of 1 rad in pitch angleθ.It can be observed that all models can reach the set point in a short time,with rising times less than 0.4 s.There are no overshoots,and steady-state errors go to zero within less than 1 s.Interestingly,the time responses of the three models are almost identical.Thus,the differences between them are unnoticeable.These behaviours demonstrate that the robust controller effectively deals with parametric uncertainties inhering in the HSSV dynamics.

Fig.6.LFT configuration of the proposed control scheme.

Fig.7.Transient responses of the perturbed multi-model set.

Since anti-windup is one of the designed targets,the next attention focuses on the control signals.Fig.8 shows the control actions for the elevator deflectionδe,the cavitator deflectionδcand its saturated signal if existed.For the sake of clarity,the signal illustration of each model is posed in a sub-figure.Assume that the saturated deflection of the cavitator is|δc|≤15o≈0.2617 rad.It can be observed from Figs.8 a and 8 b that the control signals are normal in the nom and max model.Remarkably,Fig.8 c shows that there is a saturation ofδcin the min model.It occurs when the control signalδclarger than 0.2617.The real control signal entering the system is saturatedδc.When saturation occurs,the compensatorJ(s)is activated and has an effect on the controller to avoid anti-windup.As can be seen in Fig.8 c,the transient responses of the min model are still perfect,satisfying all control criteria without overshoots.It proves that the anti-windup compensator can effectively cope with the cavitator saturation.Without the compensator,there will be an accumulation of errors when the saturation occurs,leading to dynamic overshoots,and then it will deteriorate the vehicle performance.Aiming at realizing the effect of the anti-windup compensator,the simulation of the min model is compared with a conventional PID controller.The simulation result shows that with the same saturation in Fig.8,the transient response of the PID controller in Fig.7 is deteriorated by a high overshoot,which does not satisfy the control criterion.

Fig.8.Control actions of the perturbed multi-models: a) nominal model;b) maximum model;c) minimum model.

Besides,one of the abilities to prove the robust performance of a controller is to deal with external disturbances.A simulation is carried in 10 s,and an exogenous disturbance is deliberately introduced at the fifth second as in Fig.9.The disturbance can be a current or any other external factor affecting the pitch angleθof the HSSV.It can be observed that the investigated models are slightly affected and well attenuated in general.There have been some variations in the pitch angles,only up to 20% of the magnitude.It confirms that 80% of the disturbance has been rapidly rejected.Whenever the disturbance changes its direction,the pitch angles are also reversed but eventually still keep their references.

As can be seen in Fig.10,the controller action is also validated through the control signals against the disturbance.The deflection angles have been controlled regarding the exogenous disturbance to drive the vehicle back to its desired trajectory.Both cavitator and elevators have been activated to stabilize the vehicle in less than 1 second.

As stated earlier,the nonlinear planing force occurs when the vertical velocity larger thanwo.In the above simulation,this force exists when the HSSV is tracking the set point under the interference of the disturbance.From Eq.(1),the planing force directly affects vehicle dynamics.It will lead to tail-slap and instability if uncontrolled properly.Fig.11 shows the magnitudes of the planing forces in three investigated model cases.They all disappear in short times,and it agrees with the transient responses in Fig.7.When the vehicle does not change its vertical direction and keeps the course,the vertical velocity goes to zero,and the planing force is eliminated.The elimination of the planing force sets the vehicle to the essential stable state,which guarantees that the controller successfully overcomes system uncertainty and nonlinearity.

Fig.9.Time responses of the perturbed multi-models due to disturbance input.

Fig.10.Control actions of the perturbed multi-models due to disturbance input: a)nominal model;b) maximum model;c) minimum model.

Fig.11.Elimination of nonlinear planing forces.

Fig.12.Time responses of the multi-models due to noise input.

The ability of noise reduction is simulated in Fig.12.A white noise at 100 Hz is introduced at the measured output.It can be observed that the multi-model set reacts to the noise with some variations,but with very low amplitudes compared with that of the noise.The parametric calculation shows that almost 80% of the noise input is eliminated.It indicates the robustness of the controlled system to the measured noise.

5.Conclusion

This paper deals with theH∞control synthesis with the structured anti-windup strategy to maintain robust stability with guaranteeing high performance for an uncertain HSSV system.An HSSV system presents some challenging dynamical problems in stability,control,and maneuvering due to the high speed,the varying uncertain dynamics,and the exogenous disturbances with measurement noises from underwater environments.In addition,the nonlinear analysis shows that the HSSV maneuvering basically requires robust controllers to cope with the uncertain dynamics.This work has solved three main control problems of the HSSV system,including the robustness against uncertainties and disturbances,the anti-windup approach and the low-order fixed control structure for real implementations.Notably,the controller components are simultaneously synthesized using a nonsmooth optimization technique to solve the multi-objectiveH∞problems.The simulation results show that the proposed control strategy can cope with the given level of parametric uncertainty by observing the system responses in the perturbed models.The controller can help eliminate 80% of external disturbances and noises as well as can avoid windup when saturations occur.In addition,the controlled vehicle can be robustly operated under external disturbance,noise,planing force,and actuator constraint.The remarkable feature is that the vehicle control system with robustness and performance can be easily implemented in the existent embedded systems as a result of a low-order fixed structure.Thus,the proposed scheme offers a potential control synthesis for HSSVs and new generations of high-speed underwater vehicles.

Declaration of Competing Interest

None.