Nonlinear Aaptive Ship Course Tracking Control Base on Backstepping an Nussbaum Gain Jialu Du, Chen Guo Abstract A nonlinear aaptive controller combining aaptive Backstepping algorithm with Nussbaum gain technique is propose for ship course tracking steering without a priori knowlege about the sign of control coefficient. It is theoretically prove that the presente aaptive controller guarantees that all signals in the uncertain nonlinear ship motion system is uniform boune an the output of the controlle uncertain nonlinear ship motion system asymptotically tracks the output of the ship course reference moel. The results of simulation for two ships steering moels show that the esigne controller can be properly aopte to the ship course tracking with goo compatibility. I. INTRODUCTION SHIPHANDLING, in general, is a complicate control problem. The ship course control irectly influences the maneuverability, economy an security of ship navigation an the combat capacity of warship []. Since the application of PID control law to ship navigation control in 9 s, the navigation controls have been highly emphasize ue to the navigation safety, energy-saving an lowering working intensity of the crew. Many avance control algorithms, such as moel reference aaptive control [], neural network aaptive control [3], [4], have been trie to apply to ship steering since the 98s. So far, Nomoto linear moel has been wiely use in most ship motion control esigns. However, uner some steering conitions like a course-changing operation, it is necessary to consier the hyroynamic nonlinearities. Hence, the ship control system moel becomes nonlinear. For a certain moel, a state feeback linearizing control law can be esigne accoring to reference [5], while feeback linearization with This work was supporte in part by the Ministry of Communications of P.R.C. uner grant (95-5-5-3), in part by the Founation of China Scholarship Council uner grant ([999]53), an in part by the Founation of The State Key Lab. of Intelligent Technology an Systems, Tsinghua University, China, uner grant (7). Jialu Du is with the School of Automation an Electrical Engineering, Dalian Maritime University, Dalian, P.R. China (phone: 4-469583; e-mail: ul66@63.com). Chen Guo is with Lab.of Simulation an Control of Navigation Systems, Dalian Maritime University, Dalian, P.R. China (e-mail: guocl@yahoo.com.cn). saturating an slew rate limiting actuators is iscusse by Tzeng et al. [6]. But it oes not possess the robustness to the changes of parameter an moel, because feeback linearization metho requires that system parameters an structure shoul be accurately known. It is obvious that the mathematical moel of ship steering system is uncertain when varying of spee, loaing conition an environment are consiere. Now, the stuy an evelopment of nonlinear aaptive autopilot with robustness is a hot research topic in the fiel of ship motion control. A robust aaptive nonlinear control algorithm was presente for ship steering autopilot with both parametric uncertainty an unknown boun of input isturbance base on proection approach by using Lyapunov stability theory in [7]. However, the esign proceure propose in [7] requires a priori knowlege of the sign of unknown control coefficient. Backstepping approach is a new algorithm evelope in recent years, which can be use to esign the aaptive controller of a large class of nonlinear control system with unknown constant parameter [8]. It is one of the most efficient ways in solving the control problem of uncertain nonlinear systems. This paper, uner the conition that the nonlinearity of ship steering moel is consiere an the assumption that the parameters of the moel are uncertain an the control irection is unknown, proposes a new aaptive control algorithm for ship course nonlinear system by incorporating the technique of Nussbaum-type function [9] into Backstepping esign. The problem that the parameters are uncertain an the sign of control coefficient is unknown in course-controlling system is solve by the algorithm presente in this paper. The asymptotic stability of ship movement course error system is achieve. The goal of course-changing aaptive tracking control of vessels is realize. Finally, simulation stuies on two vessels mathematic moel are performe to show the feasibility an robustness of the propose control algorithm. II. MATHEMADICAL MODLES OF SHIP STEERING The linear Nomoto moel has been wiely accepte in esigning ship course controller []. T r& r = Kδ ()
where r is the yaw rate, δ is ruer angle, T is time constant, K is ruer gain. This is mainly attribute to its simple structure an relative easiness in obtaining the moel parameters from stanar trial ata of a ship. Despite its popularity, the Nomoto moel is only vali for small ruer angles an low frequencies of ruer action. In orer to better escribe the ship steering ynamic behavior so that steering equation is also vali for rapi an large ruer angles, a nonlinear characteristic for r is ae in equation (). The following nonlinear moel is suggeste in this paper. 3 T r& r αr = Kδ () where α is calle Norrbin coefficient. The value of α can be etermine via a spiral test []. The moel parameters vary significantly with operating conitions such as the forwar spee. Uner these varying operating conitions, it is teious an ifficult to etermine properly the parameters of the controller. So we assume that moel parameters T, K, α are unknown constant parameters in the esign. Noting the heaing angle ψ, we have ψ& = r (3) We select state variable as x = ψ, x = r, u is = δ control variable, then equations () an (3) are transforme into the followings: x & = x (4.) = θ u θ ϕ, ( x ) = x& (4.) y = x (4.3) where θ = T, θ = α T, θ = K T, ϕ, = x, ϕ 3, = x. The heaing angle ψ is the output of ship course control system. The moel in (4) will be use for our propose autopilot esign. Eviently, this is a matching uncertain nonlinear system of single input-single output (SISO) in which nonlinear function is known. When the parameter θ is known, e.g., θ =, it is ust in parametric strict-feeback form [8]. T > when a ship is line movement stable, whereas T <. In this paper, we evelop an aaptive tracking control esign scheme of ship course changing controller which oes not require a priori knowlege about the sign of control coefficient θ. y The control obective of this paper is to make the output of the system (4) asymptotically track a esire time-variant reference traectory ψ, while keeping all the close-loop signals boune. First, we make the following assumptions. Assumption : x = ψ, x = r are both measurable. Assumption : The smooth reference traectory ψ an its first erivatives ψ &, & are known an boune. ψ III. DESIGN OF SHIP NONLINEAR ADAPTIVE CONTROLLER During course-changing operation it is esirable to specify the ynamics of the esire heaing instea of using a constant reference signal. One simple way to o this is by applying moel reference techniques. The obective of a course-changing operation is to perform the operation quickly with minimum overshoot. The ieal performance can be given by the reference moel (5) [] such as ω n ψ = ψ r (5) s ξω s ω where the input r n ψ of the reference moel is the course-changing heaing comman, the output ψ of the reference moel is the esire smooth course-changing signal. ξ an ω are the parameters that escribe the n close-loop system response characteristics. In this paper, we choose ω =.5ra s, ξ =. 8. n / The reference moel (5) can be interprete as a pre-filter for the commane heaing. The pre-filter ensures that numerical ifficulties associate with large step inputs are avoie. A secon-orer moel is sufficient to generate the esire smooth course signal ψ with known an boune erivatives ψ &, &. ψ The aaptive controller of course tracking of the system (4) is esigne as follows. Our esign consists of steps. The esigns of both the control law an the aaptive laws are base on the change of coorinates z = ψ ψ = x ψ (6.) z = x φ ( z, ψ& ) (6.) where the function φ is referre to as intermeiate control function which will be esigne later using an appropriate Lyapunov function V, while z is ust course tracking error. At the secon step, the actual control u appears an the esign is complete. Step : Let us stuy the following subsystem of (4) x & = (7) x In light of (6.), equation (7) becomes z& = x ψ& (8) where is taken as a virtual control input. x Consier a Lyapunov function caniate V V = z (9) n
The time erivative of V along the solution of (8) V& = z x ψ& ) () ( Let the intermeiate control function φ be φ ( z, ψ& c z ψ& () ) = where the esign constant c > will be chosen later. A irect substitution of x = z into () by using () yiels The unesire effects of φ V & () = cz zz z & on V will be ealt with at the next step. Step In light of (6.), (8) an (), the erivative of is z& = θ u θ ϕ = =, c ( c z z & ψ ) = θ u θ ϕ ϕ (3), where ϕ c z c z & ψ,3 = ( z, z, ψ& & ).,3 z is a smooth function of In orer to cope with the unknown sign of control coefficient θ, the Nussbaum gain technique which was originally propose in [9] is employe in this paper. A function N( ) is calle a Nussbaum-type function [9] if it has the following properties: lim sup s N( k) k = (4.) s s s lim inf N( k) k = (4.) s s Commonly use Nussbaum functions inclue: k cos( k), k sin( k), an exp( k )cos(( π / ) k). In this paper, an even Nussbaum function k cos( k) is exploite. We now give the following lemma regaring to the property of Nussbaum gain which is use in the controller esign. The proof can be foun in []. Lemma : Let V ( ) an k( ) be smooth functions efine on [, t ) with V (t), t, t ), N( ) be f [ f an even smooth Nussbaum-type function, an θ be a nonzero constant. If the following inequality hols: t V ( t) ( θ N( k( τ )) ) k& ( τ ) τ C t [, t f ) (5) where C represents some suitable constant, then V (t), k(t) an [, t f ). t ( θ N( k( τ )) ) k& ( τ ) τ must be boune in Let the control input be esigne as the following actual aaptive control with u = N( k)( c z N ( k) = k z ϕ cos( k),3,3 = = k& = c ϕ z ˆ θ ϕ (8) where c is a positive esign constant an N( ) is an even smooth Nussbaum-type function. ˆ θ,, are the Let the parameter aaptation laws be ˆ& θ = ϕ z V & = V& = c, V z ( θ u θ ϕ = & ( ˆ θ ) ˆ θ θ = θ = c z zz zn( k)( = θ ϕ, z z zz c z ϕ ϕ,3 c z z,,3 ϕ, 3 ) = = ( θ N( k) )( c z ϕ,3z ˆ & ( θ )( ˆ θ θ ϕ, z ) ˆ θ ϕ, ) (6), z (7) parameter estimates of the unknown parameters θ,., (9) The istinguishing feature in (6) - (9) is that,, are containe in the upate law of k(t), which is the argument of the Nussbaum-type gain N(k) []. Consier the Lyapunov function caniate V = V z ( ˆ θ ) = θ () The time erivative of V along the solution of (3), an (6) - (9) is = ˆ θ ϕ, ) ( ˆ ) ˆ& θ θ θ = ˆ θ ϕ, z )
= ( c ) z ( c z 4 ( θ N ( k) ) k& ) ( z z ) ( c ) z ( c a ) z ( θ N ( k) ) k& 4a () So far the esign proceure is complete. Due to the smoothness of the aaptive control (6), the resulting close-loop system (4), (6)-(9) amits a solution efine on its maximum interval of existence [, t ). Using lemma f to () when c c >, we conclue that V ( t), k(t) 4 an ( θ N( k(τ )) ) k( & τ τ, hence, z, z, an t N( k) ) are boune. Furthermore, z, z is square integrable accoring to (), all on [, t f ). In turn φ an the original state x, x are also boune on [ t ). Therefore, no finite-time escape phenomenon may occur an t =. Thus, applying Barbalat s lemma, we f conclue that lim z ( t ) = lim z ( t). Since t = ψ ψ ψ t = z, y = ψ, the actual heaing of ship asymptotically tracks the esire changing reference heaing. The above facts prove the following theorem. Theorem. Suppose the feeback control scheme escribe by (6)-(9) is applie to the uncertain strictfeeback ship motion nonlinear system (4) with completely unknown control coefficient θ, then all signals in the resulting close-loop aaptive system are ultimately boune. Furthermore, the close-loop error system z & = cz z z& = θ u = θ ϕ, ϕ achieves uniform asymptotic stability. The asymptotic track of ship course follows from y = ψ ψ.,3 IV. SIMULATION STUDIES In this section, two simulation examples are presente, which valiate control law (6) - (9). In the simulation, let the input ψ of the reference moel be the square wave r signal whose perio is 4s an magnitue is 3. Example. The ship ynamics parameters of equation (4) use in the simulation stuy are T = s, K =. 3 /s, α =.3 s. These values are obtaine from ientification results of a frigate at a spee of m/s [6]. We choose the control esign parameters as c =. 3,, f c = 8. The initial values are selecte as k ( ) =.6 * π, x ˆ ˆ () = x ) = θ () = θ (). The simulation ( = results of a course-changing operation where a heaing change of 6 is involve are plotte in Fig. to Fig. 4. Example. The test be is a small ship with length of 45m. The motion of the ship escribe by the moel of equation (4) has the following set of ynamics parameters at a forwar spee of 5 m/s: T = 3s, K =. 5 /s, α =. 4 s [4]. We aopt the same initial conitions an control esign parameters as the counterparts of example in the simulation. Fig. 5 to Fig. 8 epict the simulation results in this case. It can be seen from Fig. an Fig. 5 that the actual heaing angle ψ (soli-line) asymptotically tracks the output ψ (otte-line) of the reference moel as esire, an the control ruer angles (ashot-line) are smooth. In the Fig. to Fig. 4 an Fig. 6 to Fig. 8, the estimate parameters o not converge to the true value, the envelopes of their tracking curves are however convergent. They are boune as proven in Theorem. Therefore, the propose aaptive course autopilot is effective which tracks course changes for the ship motion system (4) with uncertain parameters an completely unknown control coefficient. Furthermore, the same initial conitions an control esign parameters are applie to the two ifferent ship steering moels use in the simulation stuies, so the robustness of the propose aaptive controller to the parameter changes is obviously given from Fig. to Fig. 8. V. CONCLUSION In orer to improve the performance of autopilot, this paper firstly establishes uncertain nonlinear mathematic moels of ship movement course. Base on that, a new nonlinear aaptive control esigning scheme is put forwar, which combines Backstepping algorithm with Nussbaum-type function without the nee of a priori knowlege about the sign of control gain. As to parameter uncertain nonlinear ship motion moels, the propose metho simultaneously esigns aaptive nonlinear ship course-tracking controller an parameter estimator which eals with the problems that the sign of control gain is unknown, an accomplishes aaptive tracking control of ship course. The control law of the aaptive algorithm is smooth. The numerical simulations illustrate that the feasibility of the esigne aaptive controller for ship course tracking system. θ
REFERENCES [] J. Q. Huang, Aaptive Control Theories an Its Rpplications in Ship systems, Beiing: National Defense Inustry Press, 99, pp. 68 75. [] J V. Amerongen, Aaptive steering of ships-a moel reference approach, Automatica, vol., pp. 3 4, January 984. [3] Y. S. Yang, Neural network aaptive control for ship automatic steering, International symposium of young investigators on information & computer & control, Beiing, China, 994, pp. 3 36. [4] M. A. Unar, D. J. Murray-smith, Automatic steering of ships using neural networks, Int. J. of Aaptive Control Signal Processing, vol. 3, pp. 3-8, June 999. [5] T. I. Fossen, High performance ship autopilot with wave filter, in Proceeings of the th International Ship Control Systems Symposium, Ottawa, Canaa, 993, pp..7.85. [6] C. Y. Tzeng, G. C. Goowub an S. Crisafulli, Feeback linearization of a ship steering autopilot with saturating an slew rate limiting actuator, Int. J. of Aaptive Control Signal Processing, vol. 3, pp. 3 3, February 999. [7] Y. S. Yang, Robust aaptive control algorithm applie to ship steering autopilot with uncertain nonlinear system, Shipbuiling of China, vol. 4, pp. 5, March. [8] M Krstic, I Kanellakopoulos, P V Kokotovic, Nonlinear an Aaptive Control Design, New York: Wiley, 995, ch.. [9] R. D. Nussbaum, Some remarks on the conecture in parameter aaptive control, Syst. Contr. Lett., vol. 3, pp. 43-46, November 983. [] X. L. Jia, Y. S. Yang, Ship Motion Mathematic Moel, Dalian: Dalian Maritime University Press, 998,ch. 6. [] T. I. Fossen, Guiance an Control of Ocean Vehicles, New York: Wiley, 994, pp.73-76. [] X. D. Ye, J. P. Jiang, Aaptive nonlinear esign without a priori knowlege of control irections, IEEE Trans. Automatic Control, vol. 43, pp. 67 6, November 998. Fig.. Aapting parameter:. Fig. 3. Aapting parameter:. Angle (eg) Fig.. Actual heaing angle ( ), reference heaing angle ( ), an control ruer angle ( ). N (k) k(t) Fig. 4. Nussbaum gain ( ) an its argument ( ).
Angle (eg) Fig. 5. Actual heaing angle ( ), reference heaing angle ( ), an control ruer angle ( ). N (k) k(t) Fig. 8. Nussbaum gain ( ) an its argument ( ). Fig. 6. Aapting parameter:. Fig. 7. Aapting parameter:.