Simulation on Stern-Rudder Independent Control for Submarine Motion in a Vertical Plane

BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 15, No 6 Special Iue on Logitic, Informatic and Service Science Sofia 2015 Print ISSN: 1311-9702; Online ISSN: 1314-4081 DOI: 10.1515/cait-2015-0078 Simulation on Stern-Rudder Independent Control for Submarine Motion in a Vertical Plane Xiong Ying, Hu Jun Jie South-Central Univerity for Nationalitie, Wuhan, 430074 China Email: 304074461@qq.com Abtract: To independently control the depth and trim of the ubmarine, a fuzzy control method on tern-rudder i propoed, and the advantage of a ingle ternrudder in low noie control i elaborated. Taking into account the character of the ingle tern-rudder, a fuzzy controller i deigned. The imulation reult have hown that the algorithm ha very good control preciion, which decreae the rudder and radiated noie greatly. Keyword: Submarine, motion in a vertical plane, fuzzy control. 1. Introduction The ubmarine pace motion in water can be decompoed into two plane motion, namely the weak maneuvering of ubmarine moving in the horizontal plane with hip movement above water a a main reearch direction to maintain and change, which doe not involve depth change; and a ubmarine in vertical plane movement, which mainly tudie the pitch and depth keeping and change without involving heading change. The vertical plane movement i an important apect which i different from a hip on the urface, a well a an important guarantee for the ubmarine to make full ue of it tactical performance. Therefore, it i very important to tudy the law of vertical urface movement of the ubmarine [1-5]. A we all know, the reduction of the ubmarine noie can efficiently increae the detection range of a onar, and uing double rudder ubmarine noie controlling 178

ha greater effect than uing ingle tern rudder control. Since the onar i located in the ubmarine bow, in low noie ailing condition, keeping zero for a hell around the rudder can reduce the diturbance to the urrounding fluid of the peripheral rudder hull, which eliminate plahing of the hydrodynamic noie of rudder movement. At the ame time around the hell actuator, a cloed tate mechanical noie i alo found, o the pipeline of the hydraulic hock noie and the component friction noie are removed. Therefore, ingle tern rudder control can achieve a local noie reduction effect, thereby increaing the detection range of the onar to a certain extent. In addition, conidering the ubmarine maneuvering afety, a general ubmarine tern rudder i in the wake of the propeller and rudder tern in the propeller, o the rudder effect i ignificantly enhanced. Beide, the ubmarine tern rudder i far away from the centre of confining the rudder hull of the ubmarine, o compared with the peripheral rudder hull, the advantage of the tern rudder i the arm. Uually the ratio of the length of the tern rudder arm to the confining hell rudder arm i about 1:3, due to the three time longer length of the tern rudder compared to the confining rudder hull in the ame area. The trong control ability of the tern rudder lead to eaier trim or retore the trim. In addition, the tern rudder i equipped with larger area and exhibition ide than the ailplane, o it can provide more lift power. Thu, the tern rudder control ability i tronger than the encloed hell rudder. Thi i the reaon for uing a ingle tern rudder control rather than a ingle round hell rudder control. To um up, the ingle tern rudder control ha great advantage in ubmarine low noie operation; it can reduce to ome extent the ubmarine elf-noie and radiation noie and improve the ubmarine tealth and combat performance. Thi paper adopt the cheme of ingle tern rudder control to further reduce the ubmarine noie. 2. The deign of a fuzzy control cheme In the cae of ingle ubmarine tern rudder control, the pitch and depth control of a ubmarine are taken into account. In the deign of a controller there are four control input: depth deviation, deviation change rate, trim of the deviation and it change rate, which i likely to give rie to a complex fuzzy controller tructure, large rule bae, affecting the olution efficiency of fuzzy nature. When the ubmarine i in depth, generally the firt thing i to control the vertical tilt, when the vertical tilt i in command, and then according to the current depth of the deviation to control the depth of the pecified depth. Therefore, thi paper will ue two fuzzy controller, a controller manipulating the tern rudder to control the pitch, and another controller to control the depth through tern rudder manipulation. At the tarting tage of the depth motor, when the directive trim i reached, the control i witched to the depth controller, and through the depth bia the depth control i executed to a further intruction. Since the ubmarine ha the right to make up a torque, at the ame time the torque will be pulled to a zero longitudinal tilt. It tructure i hown in Fig. 1. 179

Fig. 1. Single tern rudder fuzzy control tructure 3. The depth deign of a fuzzy controller The depth fuzzy controller will chooe a two-dimenional fuzzy controller [6-11]. The fuzzy controller ha two controller input, the depth deviation Δ ξ = ξz ξ, and depth deviation change rateγ = dδ ξ dt. The depth deviation rate of the change i a differential ignal, but the depth of the ubmarine deviation change rate cannot be directly obtained by the enor. And a the depth i quantified by a deviation differential digital, we alo need a differentiator to get the micro component. In traditional conditioning theory, the cholar have adopted the method of direct differential to get the depth and trim of the firt-order micro component, which i alo one of the commonly ued proceing method. But with the claic differentiator, differential ignal contaminated by random ignal amplifier noie pollution will occur and caue the burr and jitter of the differential ignal. Through mathematical deduction, we have obtained that the output ignal yt () i compoed of a differential ignal and magnified 1/T noie ignal of time. The time contant T i very mall, o the noie amplification i very eriou, which ometime can completely ubmerge the differential ignal vt &(), hence the claical differential link everely amplifie the effect of noie. Therefore, in the proce of differential treatment we need to avoid the tate, in which the variable differential i producing other variable. For thee reaon, thi paper will ue a new type of a tracking differentiator. The teepet nonlinear tracking differentiator can overcome the noie amplification effect of econd and higher order differential link. The teepet nonlinear tracking differentiator algorithm formula can be preented a follow: x& 1 = x2. x& 2 = ε 180

The output equation i y = u& = x, including ε = fthan( x1 ux, 2, rh, 0). In the formula u denote the input ignal, the depth i ξ ; u& the depth of the tracking differentiator output differential ignal. ξ, and d = rh0, d0 = h0d, y = x1 u+ h0x2, 2 a0 = d + 8 r y, ( a0 d) x2 + ign( y ), y > d0, fthan( x 2 1 u, x2, r, h0) = a = y x2 +, y d0, h rign( a), a > d, fthan = a r, a d, d h0 = Kh. The depth of the ignal a an input, the claic differentiator and differential ignal of the teepet nonlinear tracking differentiator can be obtained repectively. Fig. 2. Two kind of differentiator ignal comparion A it can be een from Fig. 2, the claical differentiator of the differential ignal tremble i obviou and even ditorting. The teepet nonlinear tracking differentiator out of the differential ignal i very mooth, a well a the deign of trimming and heading of the controller, o we can ue the teepet nonlinear tracking differential device to track the longitudinal tilt and heading of the differential ignal. Thu, we can improve the control effect of the fuzzy controller to a large extent. 181

3.1. Theory and baic theory of domain The depth of the deviation theory domain for Δ ξ = ξz ξ to [10, 10], the depth deviation rate i γ = dδ ξ dt, the univere i [2, 2], and the control theory of tern rudder angle domain δ [6, 6]. According to the actual ituation, chooe the depth deviation control ytem of the baic theory of the domain a [100, 100], the depth deviation rate of the baic theory of domain [2, 2], and the tern rudder angle of the baic theory of the domain [30, 30]. 3.2. Quantization factor and caling factor In determinitic theory and baic theory of domain, the quantization factor and the caling factor can be determined a follow: The depth of the deviation quantization factor K e2 = 10 /100 = 0.1. The depth of the deviation change rate quantization factor K ec2 = 0.5. The output of the caling factor K w2 = 30 / 6 = 5. 3.3. Fuzzy language variable and determination of the memberhip function The input and output language variable of the fuzzy controller i et to five, repectively {(NB), (NS), (ZE), (PS), (PB)}. According to the fuzzy language, the variable can be obtained from depth deviation, deviation rate and tern rudder angle. The memberhip function of the fuzzy ubet i hown in Fig 3, 4 and 5. Fig. 3. Memberhip function of the depth of the deviation Δ ξ Fig. 4. Memberhip function of the depth of the deviation change rate γ 182

3.4. Determination of the fuzzy rule Fig. 5. Memberhip function of the tern rudder angle To obtain the fuzzy rule of the depth of the controller require ubmarine control engineering theory knowledge, a well a expert advice and practical operational experience from the older generation. For the depth of a motor during the ubmarine manipulation, the teering peronnel experience i very important. In general, the deign of the fuzzy rule will be baed on the actual operation proce. In the proce of the depth of the motor, we can refer to the tep repone of a typical fuzzy rule being pecified. δ Fig. 6. Depth of the ubmarine maneuver control proce of a typical tep repone Fig. 6 decribe the typical tep repone of the depth of ubmarine maneuver. At the beginning, near point a, the depth of the deviation Δ ξ i very large, and the depth of the deviation rate of change γ i very mall. In order to eliminate the depth deviation at thi moment, we need a large rudder angle ignal in fat variable depth to minimize the deviation ubmarine. Hence, near to the point a, the rule i: if the deviation Δξ i PB and depth γ i ZE or NS, the tern rudder angle δ i NB (negative). Near point b in Fig. 6, the depth of the deviation Δ ξ i very mall, and the deviation change rate γ i large. To prevent the ytem overhoot and ocillation at thi moment, we need a control ignal of PS in order to avoid exceive overhoot 183

actual depth. So near to the point b, the rule i: if the depth of the deviation Δ ξ i NS or ZE and γ i NB, the tern rudder angle δ i PS. Near to point c and d the control behavior i imilar to the point near a and b, repectively. Thu, ome idea can be ummed up from the fuzzy rule of the fuzzy controller according to the value Δξ and the value γ of δ : when Δ ξ i bigger, we hould take a larger tern rudder angle, in order to eliminate the depth deviation quickly, which i dominated by the depth deviation Δ ξ at thi time in the controller. In term of the ize, uch a preventing a larger overhoot, the tern rudder angle Δ ξ mut be adequate, o a not to make a big overhoot. When Δ ξ i maller and γ i larger, the deviation i very mall, and the main tak i to prevent big overhoot or ocillation of the dominant, the tern rudder angle γ mut be the rudder. The timing of the compreive diplacement at the helm mut take an earlier action to proce the low and deep maneuver during the dynamic proce. Otherwie it will be too late to adjut the overhoot and the rudder angle. So thi cae will be baed on the imulation debugging repeatedly to find the appropriate rule. According to the depth of the ubmarine maneuver of a typical tep repone, the fuzzy rule bae on Δ ξ, γ and δ can be obtained. Table 1. The fuzzy rule bae on Δ ξ, γ andδ d NB NS ZE PS PB NB PB PM PM PS ZE NS PM PS PS ZE NS ZE PS ZE ZE NS NM PS PS ZE NS NS NM PB NS NM NM NM NB Uing MATLAB fuzzy toolbox, we can ee whether the deigned fuzzy controller performance i good or bad and ue the toolkit to output each linguitic variable fuzzy inference characteritic of the curved urface, a hown in Fig. 7. Fig. 7. ξ Δ and γ fuzzy reaoning, output characteritic curved urface 184

A it can be een from Fig. 7, the fuzzy reaoning output characteritic how urface mooth tranition of Δ ξ, γ and δ, with no mutation. Therefore, the control ytem of the fuzzy control rule i reaonable. 3.5. Fuzzy reaoning and olution of the fuzzy equation The fuzzy inference machine i the core of the fuzzy control ytem, according to the fuzzy input and fuzzy control rule. The fuzzy inference machine i to olve the fuzzy relation equation and obtain the fuzzy output. The fuzzy inference machine i mainly ued to control a ytem which ha a product inference engine. In thi paper we apply Mamdani minimum operation in the in-depth motor control ytem. The reult obtained by the fuzzy inference machine are fuzzy, and for actual control the fuzzy quantity i not clear, o one need to turn the fuzzy quantity into clear quantity. Thi i done in the theory of domain clear quantity within the cope of the cale tranform into actual control. Fuzzy control computed their tranformation within the cope of the clear quantity, accomplihed via the olution of a fuzzy equation. Thi paper adopt the fuzzy center olution method, o that the output of a fuzzy controller i a follow: M n M n * l l * l * y = y μi ( xi ) μi ( xi ), i= 1 i= 1 i= 1 i= 1 * where y i the output of the tern rudder angle, and y l i the center of the memberhip function of the deviation depth Δ ξ and deviation rate of change. The l * central point in the memberhip function μ i ( xi ) i the memberhip function of the deviation depth Δ ξ and deviation rate of change γ. 4. Trim of the fuzzy controller deign 4.1. Theory and baic theory of domain Trim deviation theory Δ θ = θz θ domain here i [ 2, 2], trim deviation rate κ = dδ θ dt of the theory of domain for [ 1, 1], and the control theory of tern rudder angle δ domain i [ 6, 6]. According to the actual ituation, chooe the trim deviation control ytem of the baic theory of the domain a [ 10, 10], trim the deviation rate of the baic theory of the domain a [ 1, 1], the tern rudder angle of baic theory of the domain a [ 30, 30]. 4.2. The quantization factor and caling factor In determinitic theory and baic theory of domain, the quantization factor and caling factor are determined a follow: 185

(1) Trim deviation quantitative factor K e2 = 2 /10 = 0.2. (2) The depth of the deviation rate of quantitative factor K ec2 = 1. (3) The output caling factor K w2 = 30 / 6 = 5. 4.3. Fuzzy language variable and determination of the memberhip function The input and output language variable of the fuzzy controller are et to even, repectively {(NB), (NM), (NS), (ZE), (PS), (PM), (PB)}. According to the fuzzy language variable, we can get the deviation rate and tern rudder angle, then trim the memberhip function of the fuzzy ubet a hown in Fig 8, 9 and 10. Fig. 8. Memberhip function of trim deviation Δ θ Fig. 9. Memberhip function of trim deviation rate κ Fig. 10. Memberhip function of the tern rudder angle δ 186

4.4. Determination of the fuzzy rule According to the depth of the ubmarine maneuver of a typical tep repone, Δ θ, κ andδ of the fuzzy rule bae can be obtained; the rule are hown in Table 2. Table 2. Δ θ, κ and δ of the fuzzy rule bae d NB NM NS ZE PS PM PB NB PB PB PM PM PS PS ZE NM PB PM PM PS PS ZE ZE NS PM PM PS PS ZE NS NS ZE PS PS ZE ZE NS NS NM PS PS ZE ZE NS NS NM NM PM ZE NS NS NM NM NM NB PB NS NS NM NM NM NB NB Uing MATLAB fuzzy toolbox, we can ee each linguitic variable fuzzy inference characteritic of the curved urface, hown in Fig. 11. Fig. 11. Δ θ, κ and δ fuzzy reaoning, output characteritic curved urface A it can be een from Fig. 11, Δ θ, κ and δ of fuzzy reaoning, the output characteritic curved urface ha mooth tranition with no mutation. Accordingly, a trim controller of the fuzzy control rule i reaonable. 4.5. Fuzzy reaoning and olution of the fuzzy method Trim controller alo ue Mamdani minimum operation, method of fuzzy center fuzzy olution, the output of the controller: olution of the fuzzy method ue the fuzzy center olution, below i given the output of the controller: M n M n * l l * l * y = y μi ( xi ) μi ( xi ) i= 1 i= 1 i= 1 i= 1 * Where y i the output of the tern rudder angle; y l i the center of the memberhip function of the trim deviation Δ θ and trim deviation rate κ ; a the 187

l * central point in the memberhip function, μi ( xi ) the trim deviation Δ θ and trim deviation rate κ. are the memberhip function of 5. Sytem imulation In order to verify the control effect of the ingle tern rudder, in viewpoint of the ubmarine in water of infinite depth, width, tatic peed of 8 kn, hardware-in-theloop imulation i carried out for the motor depth. (1) The initial depth i 60 m, the target depth i 90 m, intruction trim 2, rudder peed 2.5 per 1, ampling time i 0.2, imulation time i 600. The imulation curve i hown in Fig 12 up to 14. 100 90 depth(m) 80 70 60 0 100 200 300 400 500 600 time() 0.5 0 Fig. 12. The depth of the curve pitch(deg) -0.5-1 188-1.5-2 0 100 200 300 400 500 600 time() Fig. 13. Shell rudder angle, tern rudder angle change curve A it can be een from the imulation, at depth of no overhoot, the mobile time i 263, there i no tatic difference after the teady tate. By the trim curve we can ee that at the beginning the motor depth in a trim controller i up to 2 intruction trim; under the action of 44 it reache the directive trim, the depth of the controller tart to work at thi time, the depth i 73.2 m, according to the current depth the depth of the deviation, the final depth of tability in 90 m depth are controlled. In the whole proce of the motor hell, the rudder angle i alway zero, the tern rudder i teering four time. Smooth and no ocillation phenomenon i noticed in the tranition proce. Therefore, we adopt the ingle tern rudder control in the motor proce, the twin rudder ignificantly reduce the teering input, o we can further reduce the radiated noie of a ubmarine in low noie condition. The noie to further reducing of the input of the bigget rudder angle will

retrict the tern rudder, the maximum angle of the tern rudder being 8, the rudder peed 2.5 per 1. (2) The peed i 6Kn, the initial depth i 60 m, the target depth i 90 m, the intruction of trim i 1.5, the tern rudder angle i the bigget one 10, the ampling time i 0.2, the imulation time i 800. 100 90 depth(m) 80 70 60 0 100 200 300 400 500 600 700 800 time() Fig. 14. The depth of the curve 1 pitch(deg) 0-1 -2 0 100 200 300 400 500 600 700 800 time() Fig. 15. The curve of trim Fig. 16. Shell rudder angle, tern rudder angle change curve From Fig 14 up to 16 we can ee that under the condition that the limit of the rudder depth i till not overhoot, the trim angle after reaching a directive trim angle of 1.5, under the action of the righting moment tabilize to zero. But in the cae of a low peed, the time of a ingle tern rudder maneuver i long, in 6 Kn depth correction under peed of 30 m, the maneuvering time i nearly 300, advere to the rapid mobility, but in order to reduce the noie under the appropriate acrifice mobility, it i worth. 189

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