Generating fuzzy controllers for ship steering

Generating Fuzzy Controllers for Ship Steering Juan Contreras Department of Naval Engineering Escuela Naval Almirante Padilla Cartagena, Colombia [email protected] Abstract— In this paper we present a new method to generate fuzzy controller from training data. The antecedent partition uses triangular sets with 0.5 interpolations avoiding the presence of complex overlapping that happens in other methods. Singleton consequents are employed and least square method is used to adjust the consequents. This approach is not a hybrid system and does not employ other techniques, like neural network or genetic algorithm. The applicability of the proposed approach is demonstrated by application for controlling the directional heading of a cargo ship Keywords: fuzzy identification; fuzzy control; ship steering dynamics; ship control I. INTRODUCTION A ship is characterized by a nonlinear function describing the static maneuvering characteristics. The ship course control influences the maneuverability, economy and security of ship navigation and the combat capacity of warship. So far, Nomoto linear model [1] has been widely used in most ship motion control designs. However, the linear Nomoto model is only valid if the ship make small deviations from a straight line path (δ<5°) and for low frequencies of rudder action. Under some steering conditions like a course-changing operation, it is necessary to consider the hydrodynamic nonlinearities. Hence, the ship control system model becomes nonlinear [2]. Over the past twenty years a number of new methods were developed for designing controllers to control non-linear dynamic systems [3]. Navigation in low speed regimes is difficult due to manoeuvring problems connected with a relatively big mass of the ship and a limited dimension of the rudder, which must be significantly deflected to obtain the required change in the ship’s course [4], [5]. The construction of fuzzy controllers involves the selection and tuning of many parameters as the shape and distribution of ownership functions of input variables, rule base, logic operators used, shape and distribution of consequents, etc. The great number of parameters needed to obtain a fuzzy model has obstructed the development of a unique technique for modeling, especially in the case of fuzzy identification from experimental input and output data. One of the first proposals to automatically design a fuzzy system from data was the table look-up écheme [6]. However, when the number of inputs and membership functions are huge, the number of fuzzy rules increases exponentially [7]. Sugeno and Yasukawa [8] proposed a methodology to identify fuzzy model parameters using singleton consequents, but it requires many rules and presents a poor description capacity. This paper presents a new approach for the development of fuzzy controllers from data. The approach has been used in system identification [9], [10]. The methodology used in this paper to get the fuzzy controller from input and output data is presented in two phases: At the first, the inference error method is presented and then the fuzzy identification algorithm to generate a fuzzy model; at the second phase, the method is applied to a set of simulated data for obtaining a fuzzy controller for steering a ship on the course. II. FUZZY IDENTIFICATION APPROACH A. Inference Error A fuzzy rule: “if u is A, then y is B ”, where u and y represent two numeric variables, and A ⊂ U and B ⊂ Y, are two fuzzy input and output sets respectively, defined at the universes U and Y, is equivalent to the equation: u A ( u ) ≤ u B ( y) (1) The inference error ε, is given by 0 u A (u ) ≤ u B ( y) ⎧ ∈≈ ⎨ ⎩u A (u ) − u B ( y ) … u A (u ) > u B ( y ) (2) There are an infinite number of solutions which satisfy (2). Hence, we will look for a unique solution: a fuzzy rule of the kind “If u is A, then y is B” with a null inference error, must fulfill the condition u A ( u ) = u B ( y) (3) In this approach, the consequent u B ( y) will be a singleton. It is because fuzzy models with singleton consequents can be easily understood and adjusted [11]. 978-1-61284-968-3/11/$26.00 ©2011 IEEE If the system has n inputs, it must be represented by rules of the kind “If u1 is A1 and u2 is A2 and …and um is Am, then y is B”, and the generated system must fulfill the condition ((u A1 ( x k ) ∧ (u A 2 ( x k ) ∧ ... ∧ (u A m ( x k )) = u B ( y k ) (4) C. Fuzzy Identification Algorithm Given a collection of experimental input and output data {xk, yk}, k =1, ..., N, where xk is the p-dimensional input array x 1k , x k2 , … , x kn and y yk is the one-dimensional output array, the algorithm is defined by the following steps: 1. where ∧ represents a t-norm, or an aggregation operator, of fuzzy logic { } (i ) x ∈ ℜ are input arrays and y are output scalars. See Fig. 1. (i ) k B. Fuzzy Model Structure 1) Membership Functions The universe partitioning of the input variables in the learning process will be done with normalized triangular sets with specific overlapping of 0.5. The triangular membership functions allow the reconstruction of the linguistic value at the same numeric value after a defuzzyfication method has been applied [12]. In addition, the overlapping in 0.5 assures that the supports of the fuzzy sets are different. The fuzzy sets generated by the output variable will be a singleton Organization of the N pair set of input – output data with i = 1...N ; k = 1,..., p ,, where x k( i ) , y ( i ) 2. p Determination of universe ranges of each variable, according to maximum and minimum values of associated data xk− , xk+ , y − , y + . [ ] [ ] 2) Distribution of the Membership Functions The triangular fuzzy sets of input variables will be distributed symmetrically at each respective universe 3) Operators For combining the antecedents OWA operators will be used [13]. 4) Inference Method L f ( x (i ) ) = ∑y j Fig. 1. Data organization 3. Distribution of triangular membership functions over each universe. As a general condition the vortex with ownership value one (modal value) falls at the middle of the region covered by the membership function while the other two vortexes, with membership values equal to zero, fall in the middle of the two neighboring regions. See Fig 2 4. Calculate the position of the modal values from the input variable(s), according to ( ) m j x (i ) j =1 L (5) ∑ m j (x (i ) ) j =1 where m j ( x (i ) ) = u j A1 ( x1(i ) ).u j A2 ( x 2(i ) ).….u j An ( xn(i ) ) (6) if u ( ) Akn ysk( n ) = j is the output grade of the jth rule of a Sugeno fuzzy system, y is the singleton value corresponding to rule j. Because we use fuzzy partition with normalized triangular sets with specific output space of data x evaluation of the k-th input variable at the n-th set of the corresponding partition. The output value corresponding to this projection is given by the value of the i-th position of output array y. See Fig. 3. i j j =1 Then, (5) can be expressed as ∑ y m (x( ) ) j i j f ( x (i ) ) = y[i ] (i ) ∑ m (x( ) ) = number of input variables. L (8) end where ysk(n ) corresponds to the projection over the L overlapping of 0.5, ( x k(i ) ) = 1 j =1 p when p is the number of input variables. (7) 5. Rule determination. Initially, the number of rules are equal to the number of sets of each input variable multiplied by the number of variables; in other words, n × k . The membership function associated with a consequent will be the antecedent of this rule. Antecedents of rules with the same consequent are merged using OWA operator, thus reducing the number of rules as seen in Fig. 4. 6. Model validation described by (7) using the inference rules with sufficient excitation. This problem can be solved using recursive least square (RLS). The algorithm RLS looks like [14] method Fig. 2. Triangular sum-1 partition Fig. 4. Generating rules θ ( k + 1) = θ (k ) + γ (k )[ y (k + 1) − Wk +1θ (k )] where { Wk = w1k , w2k ,… wLk γ (k ) = Parameters adjust, relocating the output singletons using the least squares method. Equation (7) can be expressed in the form: ( ) f x (i ) = L ∑y j (9) ( ) w j x (i ) = ( ) m j x (i ) = wij p (10) Output values can be represented as Y = Wθ + E , that, in matrix form is given by ⎡ y1 ⎤ ⎡ w11 ⎢ 2⎥ ⎢ 2 ⎢ y ⎥ = ⎢ w1 ⎢ ⎥ ⎢ ⎢ L⎥ ⎢ n ⎢⎣ y ⎥⎦ ⎢⎣ w1 Y … w1L ⎤ ⎡ y ⎤ ⎡ e1 ⎤ ⎥⎢ 2⎥ ⎢ ⎥ … wL2 ⎥ ⎢ y ⎥ ⎢e2 ⎥ + ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ … wLn ⎥⎦ ⎢⎣ y L ⎥⎦ ⎣en ⎦ 1 w12 w22 w2n W θ (11) (15) (16) (17) with the initial value P(0) obtained in step 4 of this algorithm. 8. w j ( x (i ) ) j =1 where P( k )Wk +1 Wk +1 P (k )WkT+1 + 1 P (k + 1) = [1 − γ ( k )Wk +1 ]P (k ) Fig. 3. Generating consequents 7. } (14) Finish if either the square error measure MSE is not greater than a previously established value or the number of membership functions is more than 9. In any other case, increment by 1 the number of sets in the input variable (the number of partition member) and turn back to step 3. III. NONLINEAR MODEL OF SHIP STEERING A. Example I The linear Nomoto model has been widely accepted in designing ship course controller [2] •• • T ψ (t ) + ψ (t ) = Kδ (t ) (18) E where E is the approximation error, which should be minimized. Using the quadratic error norm, we have: W TY θ = T = (W TW )−1W T Y (12) W W The solution is valid if and only if (13) rank (W T W ) −1 = dim(θ ) Equation (13) implies that all rules have to receive enough excitation during training but in practice it is not always guaranteed so the application of this method will generate catastrophic results for those rules with low excitation and a significant bias in the where ψ (t) is the yaw angle, δ is ruder angle, T is time constant, K is rudder gain. However, the linear Nomoto model is only valid if the ship make small deviations from a straight line path (δ<5°) and for low frequencies of rudder action. In order to better describe the ship steering dynamic behavior so that steering equation is also valid for rapid and large rudder angles, the following nonlinear equation is suggested. ⎛ 1 1 ⎞ •• 1 ⎛• ⎞ + ⎟⎟ψ (t ) + H N ⎜ψ (t ) ⎟ = T1T2 ⎝ ⎠ ⎝ T1 T2 ⎠ ••• ψ (t ) + ⎜⎜ K T1T2 (19) • ⎛ ⎞ ⎜ T3 δ (t )+ δ (t ) ⎟ ⎝ ⎠ • δ max = 35[deg], 2.3 ≤ δ max ≤ 7[deg/ s] • where H N (ψ (t )) is a nonlinear function of the angular • • velocityψ (t ) . The function H N (ψ (t )) expresses the steadystate ••• set rudder angle, δz(t), while the output signal is the current rudder angle, δ(t). For the majority of ships the rudder angle and the speed of its change are kept within certain limits [17], where relation •• δ (t ) between • and ψ (t ) , when • ψ (t ) =ψ (t ) = δ (t ) = 0. It is usually required for the steering blade to move from one limiting position to another in a time shorter than 30 [s]. In this article it is assumed that the rate of rudder motion is approximately limited to ˙δmax = 6 [deg/s] until |δz − δ| ≤ 3 [deg], when the rudder operates in the linear region of the characteristic. The maximum rudder angle is δmax = 35 [deg]. For this assumption the steering gear dynamic characteristic was given by the following equation: The experiment performed to determine the shape of the • curve H N (ψ (t )) is called a “spiral test” and for the tanker examined in the present article it is approximated by the following function: • • ⎛• ⎞ H N ⎜ψ (t ) ⎟ = α ψ (t ) + β ψ (t ) ⎠ ⎝ 3 • K 1 δ (t ) = R δ z (t ) − δ (t ) TR TR (23) where TR = 156 [s] and KR = 96 [deg]. (20) The parameters α and β are real constants, such that α is always positive, and determined from the “spiral test”. They are chosen to be 1 in this paper: α = β = 1 in the model. The parameters K, T1, T2, T3 are defined as ⎛u⎞ K = K0 ⎜ ⎟ ⎝L⎠ Fig. 5. Block diagram of the steering gear system. (21) IV. ⎛L⎞ Ti = Ti 0 ⎜ ⎟, i = 1,2,3, ⎝u⎠ (22) where u is the longitudinal speed of the ship in [m/s], and L is the ship length in [m]. The mathematical model of the dynamic characteristics of the ship was taken from a model tanker described by [15] and modelled by a nonlinear thirdorder differential equation, referred to as Bech and Wenger’s model. RESULTS The nonlinear plant is guided by the difference equations (18) - (23). A simulation test was performed using Matlab/Simulink. Fig. 6 shows the results of simulation: desired yaw angle and real yaw angle. Fig.7 shows the rudder angle For the simulation the model parameters take on the following values [16]: K0 = -3.86, T10 = 5.66, T20 = 0.38, T30 = 0.89. The model parameters were determined for the speed u = 5 [m/s]. The length of the tanker is L = 161 [m]. The model of the dynamic characteristics of the ship was complemented by the model of the steering gear, as schematically shown in Fig. 5. The input signal passed to the steering gear comes from the autopilot and has the form of the Fig. 6. Desired yaw angle (continue) and real yaw angle (--) The array of singleton consequents is: ⎡− 34.375 ⎤ ⎢+ 34.375 ⎥ ⎢ ⎥ ⎢+ 0.229⎥ θ (k ) = ⎢ ⎥ ⎢− 0.226⎥ ⎢− 0.127⎥ ⎢ ⎥ ⎣⎢+ 0.128⎦⎥ Then, the output of the fuzzy control can be calculated using Y=W* θ(k), where Fig. 7. Input of the nonlinear ship steering process: rudder angle For the training process, past values of: rudder angle δz(k2), error ψe(k-1) and change in error ψc(k-1), between the desired yaw angle and the ship heading, are inputs of the fuzzy controller. The output is the present value of rudder angle δ(k) The structure of the fuzzy model obtained is showed in Fig. 8. It is not difficult to assign linguistic terms to each other membership functions. The labels S, and L denote, respectively, the linguistic terms “Small” and “Large”. The MSE achieved is 0.0175. With 2 input variables and 2 triangular membership functions for each input variable, 6 consequents (and 6 rules) are generated. (step 5 in fuzzy identification algorithm). The output of the obtained fuzzy control and the testing results are shown in Fig. 9. ⎡u S (δ Z (k − 2)) ⎢ ( ( − 2)) ⎢u L δ Z k ⎢u S (ψ e (k − 1)) W =⎢ ⎢u L (ψ e (k − 1)) ⎢u (ψ (k − 1)) ⎢ S C ⎣⎢u L (ψ C (k − 1)) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥ T Figure 9 shows a comparison between the real rudder angle and the output of the fuzzy control. Fig. 9. Real rudder angle (solid) and output of the fuzzy control (dotted). Fig. 8. Partitions of δz (k-2), ψe (k-1) and ψc (k-1) with 2 triangular membership functions V. CONCLUSION A new approach for the development of fuzzy controllers from experimental data was developed in this paper. The fuzzy identification algorithm proposed uses triangular membership function with 0.5 interpolations for antecedent partition avoiding the presence of complex overlapping that happens in other methods. 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