Active Disturbance Rejection Fuzzy Controller for Roll Stabilization of Autonomous Underwater Vehicle under Wave Disturbance

Considering the case of autonomous underwater vehicle navigating with low speed near water surface, a newmethod for designing of roll motion controller is proposed in order to restrain wave disturbance effectively and improve roll stabilizing performance under different sea conditions. Active disturbance rejection fuzzy control is applied, which is based on nonlinear motion model of autonomous underwater vehicle and the principle of zero-speed fin stabilizer. Extended state observer is used for estimation of roll motion state and unknownwave disturbance.Wavemoment is counteracted by introducing compensation term into the roll control law which is founded on nonlinear feedback. Fuzzy reasoning is used for parameter adjustment of the controller online. Simulation experiments on roll motion are conducted under different sea conditions, and the results show better robustness improved by active disturbance rejection fuzzy controller of autonomous underwater vehicle navigating near water surface.


Introduction
AUV (Autonomous Underwater Vehicle) rolls severely if it is navigating near water surface and wave disturbance has an obvious effect on its motion attitude.Violent roll motion often discontinues normal working of AUV and even endangers crew or important equipment [1].So, it is necessary to study an effective control pattern for solution to the problem of AUV motion attitude control.Moreover, traditional fin stabilizer is hard to generate enough lift when AUV is navigating with low speed [2].Consequently roll motion is very difficult to control at the moment.Then a new pattern of fin stabilizer working is required to realize effective roll control in low speed navigation.Maritime Research Institute Netherlands, KoopNautic Holland, and Quantum Controls Ltd. have ever cooperated in the research on zerospeed fin stabilizer system [3].Harbin Engineering University designs and tests zero-speed fin stabilizers [4] since 2005.
Considering the characteristic of AUV motion with low speed near water surface, roll attitude is controlled by zerospeed fin stabilizer which is based on Weis-Fogh device.In this paper, ADRFC (Active Disturbance Rejection Fuzzy Controller) is designed because of its strong capacity for self-adaptation.Compared to traditional PID which is based on the precise linear model of controlled system, ADRFC is more applicable to nonlinear system with time-variant parameters or incomplete model structure.From this point of view, ADRFC reduces the requirement on model accuracy in the premise of ensuring control performance.The obvious advantage of ADRFC lies in its robustness.But PID is only applicable to a certain stable condition and requires a combination of adaptive strategy to improve its robustness.
In the design of ADRFC, ESO (Extended State Observer) is used for estimation of system unmodeled dynamics and unknown wave disturbance, which are controlled effectively through compensation method [5].If AUV motion model is not accurate and sea conditions are varying constantly, the method is very fit for designing AUV roll stabilizing control.
Because AUV roll motion state estimated by ESO is required for nonlinear feedback control, gains of state feedback are obtained through fuzzy reasoning method.Then, control law for the closed loop system is designed and self-adjustment of control parameters is realized online.Reference [2] discussed how lift force generated by zero-speed fin stabilizer based on Weis-Fogh device changes with  (angle between the two wings of fin) in nonperfect fluid.From lift force model in [2], it is easily found that lift force not only has relations to , but also varies with  (angular rate of fin wings flapping) and ω (angular acceleration of fin wings flapping).In order to realize lift force control accurately and effectively, a secondorder Riccati differential equation about ω is required to be solved in nature.But, the differential equation is very complicated and hard to solve in analytical mathematics.Thus, approximate linearization method is used in this paper to calculate lift force; namely, the hypothesis is made that ω is varying while  and  keep invariant in a very small sampling time interval [, +1],  is calculated by integrating ω in [,  + 1] when sampling time is  + 1, and similarly  is calculated by integrating  in [,  + 1].In approximate linearization method,  control is replaced by ω control.As a result, lift force calculation becomes more accurate, and design of lift force control is simplified.

Roll Stabilizing Principle of AUV with Low Speed
Schematic diagram of AUV roll stabilizing system is shown in Figure 1.Motion attitude of AUV with low speed is controlled through actuation of system controller.Wings of fin stabilizer actively flap around the fin axis with high frequency in sea water.Lift on the wing surface is generated under driving of servo system.Lift righting moment counteracts wave moment effectively, and then roll motion amplitude is reduced [6].Relevant parameters of fin stabilizer are shown as follows: span length is 0.25 m, chord length is 0.5 m, and AUV navigating speed is 1.832 m/s.According to traditional roll stabilizing theory, lift force is calculated from where  is sea water density,   is projected area of fin Considering special working pattern of zero-speed fin stabilizer, force analysis during normal working of fin commits to category of unsteady flow problem [7].When zerospeed fin stabilizer flaps in perfect or nonperfect fluid, lift generated on the fin can be analyzed by using potential theory and vortex action theory instead of fix wing theory [8].Through relevant deduction, lift model of zero-speed fin stabilizer can be described as In (2),  is span length,  is sea water density,   is coefficient of drag force,  is proportion factor, 2 is chord length,  is distance from fin axis to midpoint of chord length,  is angular rate of fin wings flapping,  is additional moment of inertia,  is distance from fin axis to the point where force on additional mass acts, and  1 and  2 are both constants.
Because the problem discussed in this paper is attitude control of AUV with low navigating speed, it is necessary to consider additional effect of water flow on lift force while sea water flows through fin surface with relative flow speed.The additional lift is in relation to navigating speed, and it is timevariant; namely, additional lift can be denoted as Δ lift (, ).Thus, lift model of fin stabilizer when AUV is navigating with low speed can be given by  lift =  zero + Δ lift (, ) . ( If navigating speed is 1.832 m/s, the value of Δ lift (, ) is much less than that of  zero .Ratio of additional lift to total lift is 3%-4%.So, (4) can be approximately accepted in simulations:

Calculation of Wave Moment Near Water Surface
Practical AUV figure can be simplified and approximately described as combination object composed of a cylinder in the middle and two half spheres on both symmetrical sides as shown in Figure 2.
AUV figure is expressed as the following mathematical equations: where AUV total length  is 5.3 m, half sphere radius  is 0.5 m, and (, , ) is coordinate of a random point on AUV surface.Through analysis of force which is acted on AUV, wave moment encountered when AUV is navigating near water surface can be calculated.Because velocity and acceleration at each point on AUV surface are different and sea water passes by each point on AUV surface with different velocity and acceleration, wave moment near water surface can be calculated through integration method.From ( 5), coordinate distance between (, , ) and (  ,   ,   ), which are two random points on AUV surface, can be expressed as because model of long crested wave can be expressed as where  is wave amplitude,   is wave frequency, and wave phase   is a random variable subject to uniform distribution in 0 ∼ 2.Velocity and acceleration of wave at the point (, , ) can be obtained and given by In ( 8), ℎ  is depth from the center of water particle motion trajectory to water surface, and   is wave number.Thus, wave moment is expressed as where   is coefficient of drag force,  is sea water density, and   is coefficient of additional mass.  is 1.0 for cylinder moving towards water flow, and 0.5 for sphere. and  are vertical velocity and acceleration at the point (, , ), respectively.Equation ( 9) is expanded, and then expression for numerical calculation of wave moment is given by

Design of AUV ADRFC System. ADRFC is generally composed of ESO and NF (nonlinear feedback)
. Schematic diagram of AUV ADRFC system is shown in Figure 3.
In Figure 3,   and   are set values of roll angle and roll angular rate, respectively.ESO outputs, namely,  1 ,  2 , and  3 are estimated values of , , and  wave , respectively. 1 is error of the desired variable and the real variable for roll angle. 2 is error of the desired variable and the real variable for roll angular rate. 1 and  2 are coefficients of NF.  is control input and  is coefficient of control input. 0 is control input when wave moment is not compensated.

ESO Design.
ESO is mainly used for estimating state and disturbance.Convergence and estimation error of ESO can be analyzed by applying SSR (Self Stable Region) theory and piecewise smooth Lyapunov function [11][12][13][14][15][16].Here, basic principle of ESO is not stated in detail.ESO theory is directly applied for design of roll motion state observer.Equation ( 12) is substituted into (18), and then (19) is obtained through further simplification: where  is estimation error of roll angle,  1 ,  2 , and  3 are ESO coefficients; expression for  and fal(,   , ) ( = 1, 2, 3) are given by In (26),   is exponential factor, and  is filter factor.  and  should satisfy the following relations: 0 <   < 1,  > 0.

ESO Coefficient
Figure 4: ESO coefficient self-adjustment with neurons.
In Figure 4, upper neuron and lower neuron are used for adjustment of  1 and  2 , respectively.Adopted control algorithm is expressed as In (27),  is current time, learning speed  1 > 0,   () is upper neuron weight, and   1 is coefficient of upper neuron gain.If  3 () = ż1 − ż1 (), upper neuron input is given by where set value ż1 = 0, and ż1 () is roll angular rate estimated by ESO.Similarly, control algorithm of lower neuron is expressed as If  4 () = ż2 − ż2 (), lower neuron input is given by Set value ż2 = 0, ż2 () is roll angular acceleration estimated by ESO, learning speed  2 > 0,   () is lower neuron weight, and   2 is coefficient of lower neuron gain.Calculation unit 1 and calculation unit 2 in Figure 4 can be described as ( 22) and (23).Thus, algorithm of ESO coefficient self-adjustment with neurons is composed of (21)-( 23) and ( 27)-(30).

NF Design.
In Figure 3, NF is used for constituting control law.NF algorithm can be expressed as In (33),  4 and  5 are exponential factors and  1 and  2 are filter factors.Compensation term in (34), namely, ( 3 + )/, embodies the nature of ADRFC.When the system is affected by unknown wave disturbance, controller simultaneously gives estimation and compensation.Equation (34) is expression for angular acceleration calculation of fin stabilizer wings flapping; namely, control input  can be

Simulation Results
Total length of AUV is 5. Statistics of roll stabilizing performances under different sea conditions is shown in Table 1.Roll stabilizing performance of ADRFC is defined as the following expression: Roll stabilizing performance = (standard deviation of roll angle without roll control − standard deviation of roll angle with ADRFC)/(standard deviation of roll angle without roll control).(2) Simulation results when significant wave height is 1.2 m and encountering angle is 90 deg are given in Figures 11-13.Roll stabilizing performance of PID is shown in Figure 12.Compared to Figure 13, roll stabilizing performance in Figure 12 is not satisfied.The method proposed in this paper is obviously superior to traditional PID.This result proves that PID is only applicable to precise linear model and ADRFC is more applicable to nonlinear system.The same conclusion can be drawn from Figures 14-16 where significant wave height is 1.5 m and encountering angle is 90 deg.

Conclusions
(3) Fin stabilizer based on Weis-Fogh device is used for roll stabilizing because lift righting moment generated by common fin stabilizer is too small to counteract wave moment when AUV is navigating with low speed.Roll stabilizing performance of fin stabilizer based on Weis-Fogh device is favorable as shown in Table 1.
Because the precise model of AUV roll motion is required in the design of traditional PID, and its robustness is not strong enough for variant sea conditions, roll stabilizing performance of traditional PID which is designed for a certain sea condition deteriorates obviously with sea conditions varying.
Furthermore, AUV will be endangered when violent roll motion appears.In this paper, precise model of AUV roll motion is not required in the design of ADRFC.Some uncertain factors (e.g., wave disturbance) are not necessarily calculated accurately according to sea condition forecast.
3 m, AUV height is 0.5 m, AUV width is 1 m, navigating depth is 10.5 m, and navigating speed is 1.832 m/s.Set values of roll angle and roll angular rate are both 0; namely,   =   = 0. Propeller rotational rate is 52.359 rad/s.Relevant parameters of zero-speed fin stabilizer are given as follows: span length is 0.25 m, chord length is 0.5 m, and distance form fin axis to midpoint of chord length is 0.125 m.Exponential factors are given as follows:  1 = 0.5,  2 =  4 =  5 = 0.25,  3 = 0.125.Filter factors are given as  =  1 =  2 = 0.4.Initial values of  1 ,  2 , and  3 are set as 15, 20, and 30, respectively.Parameters of the two self-adaptive neurons are given as follows: gain coefficients are both 0.5, and learning speeds are both 2; namely,   1 =   2 = 0.5,  1 =  2 = 2. Significant wave height and encountering angle are denoted by   and , respectively.

( 1 )
Simulation results shown in Figures 5-10 demonstrate that ADRFC embodies favorable robustness and satisfactory performance of roll stabilizing when significant wave height is 1 m and encountering angle varies from 45 deg to 135 deg.There is no instability phenomenon in AUV roll motion.
,   is coefficient of lift force, and  is AUV navigating speed.

Table 1 :
Statistics of roll stabilizing performance.