Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 530162, 9 pages http://dx.doi.org/10.1155/2013/530162
Research Article Adaptive Neural Network Control with Backstepping for Surface Ships with Input Dead-Zone Guoqing Xia, Xingchao Shao, Ang Zhao, and Huiyong Wu College of Automation, Harbin Engineering University, 150001 Harbin, China Correspondence should be addressed to Xingchao Shao;
[email protected] Received 17 June 2013; Revised 27 August 2013; Accepted 28 August 2013 Academic Editor: Jyh-Horng Chou Copyright Β© 2013 Guoqing Xia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper addresses the problem of adaptive neural network controller with backstepping technique for fully actuated surface vessels with input dead-zone. The combination of approximation-based adaptive technique and neural network system is used for approximating the nonlinear function of the ship plant. Through backstepping and Lyapunov theory synthesis, an indirect adaptive network controller is derived for dynamic positioning ships without dead-zone property. In order to improve the control effect, a dead-zone compensator is derived using fuzzy logic technique to handle the dead-zone nonlinearity. The main advantage of the proposed controller is that it can be designed without explicit knowledge about the ship motion model, and dead-zone nonlinearity is well compensated. A set of simulations is carried out to verify the performance of the proposed controller.
1. Introduction Ship moving in the sea performs strong nonlinear and coupled property. It is difficult for researchers to obtain the ship motion model. Most modern nonlinear control theories, such as sliding mode control, backstepping technique, and feedback linearization, are mostly based on the knowledge about system models. For these control strategies, the controller cannot be derived without explicit knowledge about the system model. Meanwhile, dead-zone characteristic is quite common in actuators, that is, rudders or propellers. But few works consider the influences on control performance brought by the dead-zone characteristic. Fuzzy logic system and neural network are commonly used for approximating the unknown terms or uncertainties in the systems [1β6]. An adaptive fuzzy decentralized output-feedback control problem was discussed for a class of nonlinear large-scale systems, and fuzzy logic systems were employed to approximate the unknown nonlinear functions in the control design in [1]. Zhu and Li proposed a stable decentralized adaptive fuzzy sliding mode control scheme for reconfigurable modular manipulators to satisfy the concept of modular software in [2], and a first-order TakagiSugeno fuzzy logic system was introduced to approximate the unknown dynamics of subsystem by using adaptive
algorithm. The fuzzy basis function was used to approximate an unknown nonlinear function according to some adaptive laws in [3], and then the state observer was designed for estimating the states of the plant, upon which an adaptive fuzzy sliding mode controller was investigated. In [4], a radial basis function (RBF) network were employed to estimate and compensate the uncertainties of ship dynamics and disturbances in controller design. In [5], an adaptive neural network control scheme for robot manipulators with actuator nonlinearities was presented, and the RBF network was introduced to emulate the unknown parameters. An adaptive RBF neural network controller was adopted to learn the unknown upper bound of model uncertainties and external disturbances in [6]. The approach taken in this paper tries to overcome the necessity for shipβs mathematical models by using an adaptive RBF network control algorithm to estimate the unknown nonlinear functions. Meanwhile, adaptive RBF network can improve the robustness of the control systems. In real applications, the dead-zone characteristic is quite common in actuators. Sometimes the dead-zone nonlinearity affects the control system performance. Generally, the dead-zone parameters are unknown and time-variant, which cause challenging problem for the control system. Dead-zone commonly affects all practical systems, such as
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Mathematical Problems in Engineering
The main advantage of the proposed method is that the controller can be designed without explicit knowledge about the ship motion model, and control input saturation and dead-zone nonlinearity are well compensated. The rest of this paper is organized as follows. In Section 2, a model of fully actuated surface vessels with dead-zone is established. Section 3 contains the design of adaptive RBFNN controller with backstepping, the auxiliary design system is introduced to handle the input saturation, and an FLS is
XE
π (x, y)
Y
(i) To the best of our knowledge, it is the first time in the literature that both input saturation and deadzone are considered during controller design for fully actuated surface ships motion control problems. (ii) For handling the limit of propellers, the auxiliary design system is introduced to analyze the input saturation, and the effect of dead-zone nonlinearity is compensated by using fuzzy logic system (FLS). (iii) The adaptive method and radial basis function neural network (RBFNN) system are combined to estimate the unknown nonlinear functions of the ship plant. (iv) Under the assumptions of the inexistence of the input saturation and the dead-zone characteristic in actuator, an adaptive RBFNN controller using backstepping technique for fully actuated surface ships is derived without knowing the ship motion model.
X
mechanics and electronics, so the study on dead-zone has drawn much interest in the control community for a long time [7β12]. To solve the problem brought by the deadzone, adaptive dead-zone inverses were proposed in [7β 9]. Reference [7] built a continuous dead-zone inverse for linear system with unmeasurable dead-zone outputs, while asymptotical adaptive cancellation of an unknown deadzone was achieved analytically under the condition that the output of a dead-zone was measurable in [8]. For a general nonlinear actuator dead-zone of unknown width, [9] presented a compensation scheme using neural network. In [11], adaptive control with adaptive dead-zone inverse has been introduced by giving a matching condition to the reference model. By utilizing the integral-type Lyapunov function and introducing an adaptive compensation for the upper bound of the optimal approximation error and the dead-zone disturbance, a robust adaptive neural controller for a single input single output nonlinear system was derived in [12]. By utilizing a description of a dead-zone feature, an adaptive law was used to estimate the properties of the deadzone model intuitively and mathematically, without constructing a dead-zone inverse in [13]. Departing from existing approximate adaptive dead-zone compensations, [14] used indirect parameter estimation algorithms along with online condition monitoring to obtain an accurate estimation of the unknown dead-zone when certain relaxed persistentexcitation conditions are satisfiedβa theoretical result that cannot be achieved with the existing methods. This paper proposes the work on adaptive neural network control with backstepping technique for fully actuated surface vessels with constraint in the actuator. The main contributions of this paper are as follows.
YE
Figure 1: Definition of the earth-fixed ππΈ ππΈ ππΈ and the body-fixed πππ reference frames.
utilized to approximate the nonlinearity of the dead-zone in actuators. Then a set of simulations is taken in Section 4 to verify the control effect of the proposed method. Finally, conclusions are made in Section 5.
2. Model of Surface Vessels 2.1. Ship Motion Model for Surface Vessels. For the horizontal motion of a fully actuated surface vessel, the kinematics and dynamics models can be described by (1), for more details see [15]. πΜ = J (π) ^, M^Μ + C (^) ^ + D (^) ^ = π + RΞ€ (π) b,
(1)
where π = [π₯, π¦, π]Ξ€ , ^ = [π’, V, π]Ξ€ , and π₯, π¦ denote the coordinates of the vessel in the earth-fixed frame ππΈ ππΈ ππΈ (see Figure 1), and π is the heading angle of the ship. π’, V, and π express the velocities in surge, sway, and yaw, respectively in the body-fixed reference frame πππ, π = [ππ₯ , ππ¦ , ππ ]Ξ€ represents the control inputs, that is, forces and moments produced by the propellers, M stands for the mass and inertia matrix, C(^) denotes the Coriolis-centripetal matrix, D(^) is the damping matrix, and b is a bias term representing slowly varying environmental forces and moments caused by the wind, second-order waves, and currents. J(π) is a state dependent transformation matrix which can be written as cos π β sin π 0 J (π) = J (π) = [ sin π cos π 0] . 0 1] [ 0
(2)
Note that the matrix J(π) is nonsingular for all π β R3 and Jβ1 (π) = JΞ€ (π); for example, J(π)JΞ€ (π) = I. The system model can be rewritten as [16] Mπ (π) πΜ + Cπ (^, π) πΜ + Dπ (^, π) πΜ = JβΞ€ (π) π + b,
(3)
Mathematical Problems in Engineering
3
where βΞ€
Mπ (π) = J
β1
(π) MJ (π) ,
Cπ (^,π) = JβΞ€ (π) [C (^) β MJβ1 (π) JΜ (π)] Jβ1 (π) ,
(4)
z1 = π β ππ .
Dπ (^,π) = JβΞ€ (π) D (^) Jβ1 (π) .
(5)
(i) Mπ (π) is a positive definite matrix, and it is bounded, which means that there exists a constant π0 > 0, such that 0 < Mπ (π) β€ π0 I, where I is a 3rd-order identity matrix. (ii) The matrix Mπ (π) and Cπ (^,π) have the following property: the matrix MΜ π (π) β 2Cπ (^,π) is skewsymmetric and for all x β R3 the following relation is satisfied: β^, π β R3 .
(6)
2.2. Model for Controller Design. In order to discuss conveniently and simplify the controller design, state transformation is introduced here to obtain a new form of the system model. Let x1 = π and x2 = πΜ then the system model (1) can be written as; xΜ1 = x2 , xΜ2 =
Mβ1 π
(π) u β
Mβ1 π
(π) Cπ (^,π) x2
β1 βMβ1 π (π) Dπ (^,π) x2 + Mπ (π) b,
zΜ1 = πΜ β πΜ π = xΜ1 β πΜ π = x2 β πΜ π .
(7) (8)
where u = JβΞ€ (π)π = J(π)π.
3. Control Strategy An adaptive fuzzy logic control using backstepping technique is designed for fully actuated surface vessels with dead-zone in actuator. To compensate the dead-zone nonlinearity, a fuzzy logic system is introduced. 3.1. Backstepping Design without Dead-Zone Step 1. Define the desired position and heading vector as ππ , and a perfect guidance system is introduced to generate the
(10)
Choose x2 as virtual control and defined as π1 β x 2 = z 2 + πΌ 1 ,
where π0π is the πth control input of the designed control law π0 . Assume that the system parameters are unknown and bounded and the following properties are satisfied.
xΞ€ (MΜ π (π) β 2Cπ (^,π)) x = 0,
(9)
Taking the time derivation of (9) yields
Considering the presence of input saturation constraints on π, we have ππ min β€ ππ β€ ππ max (π = π₯, π¦, π), where ππ min and ππ max are the known lower limit and upper limit of input saturation constraints. Thus, the control input is defined as , π0π > ππ max , π { { π max ππ = {π0π , ππ min β€ π0π β€ ππ max , { {ππ min , π0π < ππ min ,
tracking target ππ ; and its first- and second-order derivation πΜ π and πΜ π are also generated. Then the control objective is to design a controller to make the ship track the target ππ . Define the tracking error as
(11)
where z2 is the second error vector, and πΌ1 is the stable function which is defined later. Substitute (11) into (10), we obtain zΜ1 = x2 β πΜ π = z2 + πΌ1 β πΜ π .
(12)
Choose the stable function πΌ1 as πΌ1 = βπ1 z1 + πΜ π ,
(13)
where π1 is a positive definite diagonal matrix. The Lyapunov function is chosen for the z1 -subsystem as 1 π1 = zΞ€1 z1 . 2
(14)
Then the time derivation of (14) is π1Μ = zΞ€1 zΜ1 = zΞ€1 (z2 + πΌ1 β πΜ π ) = βzΞ€1 π1 z1 + zΞ€1 z2 .
(15)
If z2 = 0, then π1Μ β€ 0 which implies that the z1 -subsystem is stable. Step 2. From (11) we can get z2 = x2 β πΌ1 .
(16)
Differencing (16) with respect to time yields zΜ2 = xΜ2 β πΌΜ 1 .
(17)
Substitute (8) into (17), then (17) becomes β1 β1 β1 zΜ2 = Mβ1 π u β Mπ Cπ x2 β Mπ Dπ x2 + Mπ b β πΌΜ 1 .
(18)
Note that, we omit the variables of coefficient matrices for denoting conveniently. For convenience of constraint effect analysis of the input saturation, the following auxiliary design system is given by σ΅¨σ΅¨ Ξ€ σ΅¨σ΅¨ σ΅¨z Ξuσ΅¨ + 0.5ΞuΞ€ Ξu { {βK π β σ΅¨σ΅¨ 2 σ΅¨σ΅¨ π + Ξu, βπβ β₯ π, 1 Μπ = (19) βπβ2 { { βπβ < π, {0, where Ξu = u β u0 = J(π)(π β π0 ), K1 = KΞ€1 > 0, π is a small positive design parameter, and π β R3 is the state
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of the auxiliary design system. Control command π0 will be designed later. For z2 -subsystem, the Lyapunov function is chosen as 1 1 π2 = π1 + πΞ€ π + zΞ€2 Mπ z2 . 2 2
Du (u)
(20) βdβ 0
Take the time derivation of (20), which yields
d+
u
1 π2Μ =π1Μ + zΞ€2 Mπ zΜ2 + πΞ€ πΜ + zΞ€2 MΜ π z2 2 1 = βzΞ€1 π1 z1 + zΞ€1 z2 + zΞ€2 Mπ (xΜ2 β πΌΜ 1 ) + zΞ€2 MΜ π z2 + πΞ€ πΜ 2
Figure 2: Unsymmetrical dead-zone nonlinearity.
β1 = βzΞ€1 π1 z1 + zΞ€1 z2 + zΞ€2 Mπ (Mβ1 π u β Mπ Cπ x2
3.2. Dead-Zone Compensator Design. The unsymmetrical dead-zone nonlinearity is shown as Figure 2, and it can be described as
β1 βMβ1 π Dπ x2 + Mπ bβπΌΜ 1 )
1 + zΞ€2 MΜ π z2 + πΞ€ πΜ 2 β€ βzΞ€1 π1 z1 + zΞ€1 z2 + zΞ€2 (u β Cπ x2 β Dπ x2 + b β Mπ πΌΜ 1 ) 1 1 σ΅¨ σ΅¨ + zΞ€2 MΜ π z2 β πΞ€ (K1 β I3 ) π β σ΅¨σ΅¨σ΅¨σ΅¨zΞ€2 Ξπσ΅¨σ΅¨σ΅¨σ΅¨ . 2 2
(21)
According to the skew-symmetric property of the matrix Μ Mπ (π)β2Cπ (^,π) in (6), we have (1/2)zΞ€2 MΜ π z2 = zΞ€2 Cπ (^,π)z2 . Substituting it into (21) yields π2Μ β€ βzΞ€1 π1 z1 + zΞ€1 z2 + zΞ€2 Γ (u + Cπ z2 β Cπ x2 β Dπ x2 + b β Mπ πΌΜ 1 )
(22)
1 σ΅¨ σ΅¨ β πΞ€ (K1 β I3 ) π β σ΅¨σ΅¨σ΅¨σ΅¨zΞ€2 Ξπσ΅¨σ΅¨σ΅¨σ΅¨ . 2
π’ + πβ , { { π = π·π’ (π’) = {0, { {π’ β π+ ,
π = π’ β satπ (π’) , βπ , π’ < βπβ , { { β satπ (π’) = {π’, βπβ β€ π’ < π+ , { π , π’ β₯ π+ . { +
(28)
where ππ = [ππ+ , ππβ ]Ξ€ , and let D = diag{π1 , . . . , ππ }, it becomes (23)
Let f = βCπ πΌ1 β Dπ x2 β Mπ πΌΜ 1 , and (23) can be written as
π = u β satπ· (u) .
(24)
Then the ideal control law can be chosen as
if (π€π is positive) , then (π’π = π€π + πΜπ+ ) ,
where π2 is a positive definite diagonal matrix, Μf is a nonlinear function designed later, and k is a robust term for disturbance and estimated error.
(30)
Ξ€
Μ = [πΜ , πΜ ] is estimate of dead-zone d = [π , π ]Ξ€ . and d π π+ πβ π π+ πβ After compensating the dead-zone, the control input π’π is π’π = π€π + π€πΉπ ,
(25)
(29)
According to the characteristic of dead-zone, the rules for compensation are designed as
if (π€π is negative) , then (π’π = π€π + πΜπβ ) ,
π2Μ β€ βzΞ€1 π1 z1 + zΞ€1 z2 + zΞ€2 (u + f + b)
uβ0 = βπ2 (z2 β π) β z1 β Μf + k,
(27)
For multiple input multiple output system such as dynamics positioning ships, control input of the πth control loop is
π2Μ β€ βzΞ€1 π1 z1 + zΞ€1 z2 + zΞ€2 (u β Cπ πΌ1 β Dπ x2 β Mπ πΌΜ 1 )
1 σ΅¨ σ΅¨ β πΞ€ (K1 β I3 ) π β σ΅¨σ΅¨σ΅¨σ΅¨zΞ€2 Ξπσ΅¨σ΅¨σ΅¨σ΅¨ . 2
(26)
where π’ is the control input before the dead-zone, π is the output of dead-zone model, and πβ , π+ are unknown positive constant. Then the dead-zone model can be denoted as
ππ = π· (π’π ) = π’π β satππ (π’) ,
Invoking (11) and (22) becomes
1 σ΅¨ σ΅¨ + zΞ€2 b β πΞ€ (K1 β I3 ) π β σ΅¨σ΅¨σ΅¨σ΅¨zΞ€2 Ξπσ΅¨σ΅¨σ΅¨σ΅¨ . 2
π’ < βπβ , βπβ β€ π’ < π+ , π’ β₯ π+ ,
(31)
where π€πΉπ is a compensation term which is determined by if (π€π β π+ (π€π )) , then (π€πΉπ = πΜπ+ ) if (π€π β πβ (π€π )) , then (π€πΉπ = βπΜπβ ) ,
(32)
Mathematical Problems in Engineering
5
Μ = where π+ (β
), πβ (β
) are the membership degree of π€π and d π Ξ€ Ξ€ Μ Μ [ππ+ , ππβ ] . Let wπΉ = [π€πΉ1 , . . . , π€πΉπ ] ; then the control action after compensation is Μ Ξ€ π (w) , u = w + wπΉ = w + D
(33)
h1 x1
x2 .. .
Μ ,...,d Μ } and π(w) is described as Μ = diag{d where D 1 π Ξ€
π (w) = [π+ (π€1 ) , βπβ (π€1 ) , . . . , π+ (π€π ) , βπβ (π€π )] . (34)
h2
Ξ£
.. .
xn hm
Figure 3: The structure of RBFNN.
Then membership of π€π is designed as π+ (π€π ) = {
0, π€π < 0, 1, π€π β₯ 0,
1, π€π < 0, πβ (π€π ) = { 0, π€π β₯ 0.
(35)
According to Theorem 1 in [17], the control input after the dead-zone is Μ Ξ€ πΏ, Μ Ξ€ π (w) + D π=wβD
(36)
Μ =DβD Μ = diag{πΜ1 , . . . , πΜ1 } and βπΏβ β€ βπ. where D With the control input chosen as (25) and w = π½β1 (π)uβ0 , the time derivation of π2 becomes π2Μ β€ βzΞ€1 π1 z1 + zΞ€1 z2 + zΞ€2 [ β π2 (z2 β π) β z1 β Μf + f + b Μ Ξ€ π (w) + Jβ1 D Μ Ξ€ πΏ] + k β Jβ1 D 1 σ΅¨ σ΅¨ β πΞ€ (K1 β I3 ) π β σ΅¨σ΅¨σ΅¨σ΅¨zΞ€1 Ξπσ΅¨σ΅¨σ΅¨σ΅¨ 2 β€
βzΞ€1 π1 z1 +
zΞ€2
β
zΞ€2 π2
σ΅©2 σ΅©σ΅© σ΅©X β cπ σ΅©σ΅©σ΅© βπ = exp (β σ΅© ), 2ππ2
π = 1, 2, . . . , π,
(40)
where cπ = [ππ1 , ππ2 , . . . , πππ ]Ξ€ is the central vector of the ith node and ππ > 0 is the basis width parameter of the πth node. Define the weight vector of the RBF neural network as W = [π€1 , π€2 , . . . , π€π ]Ξ€ ; then the output of the RBFNN can be expressed as follows: π¦ = HΞ€ W.
(z2 β π)
(41)
In order to approximate the nonlinear function f which is relative to the ship modeled parameters and the system states, the elements of f are, respectively, estimated by corresponding RBFNN as follows:
Μ Ξ€ π (w) + Jβ1 D Μ Ξ€ πΏ) (f β Μf + k + b β Jβ1 D
1 β πΞ€ (K1 β I3 ) π. 2
The structure of RBFNN is shown in Figure 3. RBFNN is a forward network with three layers: input layer, implicit layer, and output layer. The mapping from input to output is nonlinear, while it is linear from implicit layer to output. RBFNN can approximate nonlinear function locally, so its learning rate is fast and the local minimization problem can be avoided. Assume there are π inputs and π implicits nodes in the RBF network. Let X = [π₯1 , π₯2 , . . . , π₯π ]Ξ€ be the input vector; then denote the radial basis vector as H = [β1 , β2 , . . . , βπ ]Ξ€ , where βπ is Gaussian function defined as
(37)
We can choose the robust term and adaptive law of deadzone width as
πΜπ (x) = Hπ (x)Ξ€ Wπ ,
(π = 1, 2, 3) .
(42)
Then define the Gaussian basis function vector π as HΞ€1 0 0 π1 W1 Μf (x, π) = [π2 ] = [ 0 HΞ€ 0 ] [W2 ] = π(x)Ξ€ π, (43) 2 Ξ€ [π3 ] [ 0 0 H3 ] [W3 ]
k (π‘) = βbπ sgn (z2 ) ,
(38)
Μ ΜΜ = βΞπ (w) zΞ€ Jβ1 β 1 πD, D 2 2
(39)
where πΞ€ (x) = diag{HΞ€1 , HΞ€2 , HΞ€3 } and π = [WΞ€1 , WΞ€2 , WΞ€3 ] .
where π > 0, Ξ > 0 are diagonal matrices and 0 β€ βbβ β€ bπ.
Assumption 1. The output of RBFNN Μf(x, π) is continuous.
3.3. Adaptive RBFNN Controller Design. From the description of f mentioned above, we can see that it contains the knowledge about the system model. In order to design a control law without explicit knowledge about the ship motion model, an adaptive RBFNN is introduced to approximate the nonlinear function f. And here Μf is the adaptive neural network system used to approximate f.
Assumption 2. There exist an ideal approximation Μf(x, πβ ), such that for any small positive constant π0 σ΅© σ΅© (44) max σ΅©σ΅©σ΅©σ΅©f (x) β Μf (x, πβ )σ΅©σ΅©σ΅©σ΅© β€ π0 ,
Ξ€
where πβ is the optimal weight vector of the RBFNN for approximating f(x).
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Mathematical Problems in Engineering Denote eπ as the approximate error of the ideal RBFNN: eπ = f (x) β Μf (x, πβ ) .
(45)
According to zΞ€2
Μ Ξ€ π (w) β Jβ1 D Μ Ξ€ πΏ) = zΞ€ Jβ1 D Μ Ξ€ (π (w) β πΏ) (Jβ1 D 2 Μ Ξ€ (π (w) β πΏ) zΞ€ Jβ1 ) , = tr (D 2 (53)
According to the approximate capability of RBFNN, it is easy to obtain that eπ is bounded. Denote the bound as σ΅© σ΅© eπ0 = sup σ΅©σ΅©σ΅©σ΅©f (x) β Μf (x, πβ )σ΅©σ΅©σ΅©σ΅© ,
(46)
where Μf(x, πβ ) = π(x)Ξ€ πβ . f(x) is bounded, so πβ is bounded and βπβ βπΉ β€ πmax . Let πΜ = πβ β π, and the adaptive law is chosen as Ξ€
πΜ π = ππ (π§2π ππ Ξ€ (x)) β 2ππ ππ ,
(π = 1, 2, 3) ,
(47)
where ππ > 0 and ππ > 0 are designed constant and π§2π is the πth element of z2 . Let πΎ β R3πΓ3π , π
β R3πΓ3π , and define matrices as πΎ = diag {π1 Iπ , π2 Iπ , π3 Iπ } , π
= diag {π1 Iπ , π2 Iπ , π3 Iπ } ,
(48)
where Iπ is a πth order identity matrix. Then the adaptive law (47) becomes Ξ€ πΜ = πΎ(zΞ€2 πΞ€ (x)) β 2π
π.
(49)
3.4. Stability Analysis. Now we can choose the total Lyapunov function as 1 1 Ξ€ Μ . Μ Ξ€ Ξβ1 D) π = π2 + πΜ πΎβ1 πΜ + tr (D 2 2
(50)
Ξ€ β€ βzΞ€1 π1 z1 β zΞ€2 π2 (z2 β π) + zΞ€2 (f β π(x)Ξ€ π) β πΜ πΎβ1 πΜ
Μ Ξ€ ΞD) ΜΜ Μ Ξ€ π (w) β Jβ1 D Μ Ξ€ πΏ) + tr (D β zΞ€2 (Jβ1 D
(54)
Μ Ξ€ (βπ (w) zΞ€ Jβ1 + πΏzΞ€ Jβ1 + Ξβ1 D)) ΜΜ . = tr (D 2 2 Μ =DβD Μ we have = βD; ΜΜ then From D Μ Ξ€ (βπ (w) zΞ€ Jβ1 + πΏzΞ€ Jβ1 β Ξβ1 D)) ΜΜ π = tr (D 2 2 Μ Μ Ξ€ (πΏzΞ€ Jβ1 + 1 Ξβ1 (D β D))) = tr (D 2 2
(55)
1 σ΅© Μ σ΅©σ΅© ππ σ΅© σ΅© σ΅© Μ σ΅©σ΅©σ΅© Μ , Μ Ξ€ Ξβ1 D) σ΅©σ΅© β β€ βπ σ΅©σ΅©σ΅©z2 σ΅©σ΅©σ΅© σ΅©σ΅©σ΅©σ΅©D π· σ΅©σ΅©σ΅©D tr (D σ΅©σ΅© + 2ππ π σ΅© σ΅© 2 where ππ = πmin (π), ππ = πmin (Ξ) and π·π > βDβ, βπΏβ β€ βπ, so (51) yields σ΅© σ΅© σ΅© σ΅© πΜ β€ βzΞ€1 π1 z1 β zΞ€2 (π2 β π) z2 + σ΅©σ΅©σ΅©σ΅©zΞ€2 σ΅©σ΅©σ΅©σ΅© β
σ΅©σ΅©σ΅©σ΅©f β π(x)Ξ€ πβ σ΅©σ΅©σ΅©σ΅© Ξ€ + π β πΜ πΎβ1 πΜ + zΞ€2 π(x)Ξ€ πΜ + zΞ€2 (b β bπ sgn (z2 ))
1 β πΞ€ (K1 β I3 β πβ1 πΞ€2 π2 ) π 2 1 σ΅© σ΅©2 1 β€ βzΞ€1 π1 z1 β zΞ€2 (π2 β π) z2 + σ΅©σ΅©σ΅©σ΅©zΞ€2 σ΅©σ΅©σ΅©σ΅© + π0 2 + π 2 2
(56)
1 β πΞ€ (K1 β I3 β πβ1 πΞ€2 π2 ) π. 2 Substitute adaptive law (39) into (56) and we can get 1 πΜ β€ βzΞ€1 π1 z1 β zΞ€2 (π2 β π) z2 β πΞ€ (K1 β I3 β πβ1 πΞ€2 π2 ) π 2 Ξ€ Ξ€ 1 + π2 β πΜ [(zΞ€2 πΞ€ (x)Ξ€ ) β πΎβ1 (πΎ(Z2 πΞ€ (x)) β 2π
π)] 2
1 + zΞ€2 (b + k) β πΞ€ (K1 β I3 ) π 2 β€ βzΞ€1 π1 z1 β zΞ€2 π2 (z2 β π) + zΞ€2 (f β π(x)Ξ€ πβ + b + k) Ξ€ Μ Ξ€ (π (w) β πΏ) β πΜ πΎβ1 πΜ + zΞ€2 π(x)Ξ€ (πβ β π) β zΞ€2 Jβ1 D
(51)
1 σ΅© σ΅©2 + π + σ΅©σ΅©σ΅©σ΅©zΞ€2 σ΅©σ΅©σ΅©σ΅© 2 1 = βzΞ€1 π1 z1 β zΞ€2 (π2 β π) z2 + zΞ€2 z2 + π2 2 Ξ€ 1 + 2πΜ πΎβ1 π
π + π β πΞ€ (K1 β I3 β πβ1 πΞ€2 π2 ) π 2
1 1 Ξ€ = βzΞ€1 π1 z1 β zΞ€2 (π2 β I3 ) z2 + π2 + 2πβ πΎβ1 π
π 2 2
It is obviously that
where π β R3Γ3 and π > 0.
ΜΜ Μ Ξ€ (βπ (w) + πΏ) zΞ€ Jβ1 ) + tr (D Μ Ξ€ Ξβ1 D) = tr (D 2
+ πΜ [(zΞ€2 πΞ€ (x)Ξ€ ) β πΎβ1 π]Μ
Ξ€ Μ Μ Ξ€ ΞD) ΜΜ πΜ β€ π2Μ + πΜ πΎβ1 πΜ + tr (D
zΞ€2 π2 π β€ πzΞ€2 z2 + πβ1 πΞ€ πΞ€2 π2 π,
ΜΜ Μ Ξ€ Ξβ1 D) Μ Ξ€ π (w) β Jβ1 D Μ Ξ€ πΏ) + tr (D π β βzΞ€2 (Jβ1 D
Ξ€
Differentiate (50) with respect to time as
Μ Ξ€ ΞD) ΜΜ β πΞ€ (K1 β 1 I3 ) π. + tr (D 2
we can obtain
(52)
1 β 2πΞ€ πΎβ1 π
π + π β πΞ€ (K1 β I3 β πβ1 πΞ€2 π2 ) π. 2
(57)
Mathematical Problems in Engineering
7
If π
and πΎ are positive definite diagonal matrices, then Ξ€ (π β π) πΎβ1 π
(πβ β π) β₯ 0. So the following inequality is satisfied: β
β Ξ€ β1
Ξ€ β1
Ξ€ β1
β Ξ€ β1
β
2π πΎ π
π β 2π πΎ π
π β€ βπ πΎ π
π + π πΎ π
π .
(58)
According to 0 < Mπ (π) β€ π0 I, that is, βMβ1 π β€ β(1/π0 ); then Ξ€ 1 1 πΜ β€ βπ 1π zΞ€1 z1 β π 2π zΞ€2 Mπ z2 β ππ πΞ€ π β π
π πΜ πΎβ1 πΜ π0 2
1 Ξ€ + 2πβ πΎβ1 π
πβ + π2 + π. 2
Substituting (58) into (57) yields 1 1 Ξ€ πΜ β€ βzΞ€1 π1 z1 β zΞ€2 (π2 β I3 β π) z2 + π2 + πβ πΎβ1 π
πβ 2 2
Define π0 = min{2π 1π , (2/π0 )π 2π , π
π , ππ , ππ }, then
1 β πΞ€ πΎβ1 π
π + π β πΞ€ (K1 β I3 β πβ1 πΞ€2 π2 ) π 2 =
βzΞ€1 π1 z1
β
zΞ€2
Ξ€ 1 Μ Ξ€ Ξβ1 D)) Μ πΜ β€ β π0 (zΞ€1 z1 + zΞ€2 Mπ z2 + πΞ€ π + πΜ πΎβ1 πΜ + tr (D 2
1 (π2 β I3 β π) z2 β πΞ€ πΎβ1 π
π 2
1 σ΅©σ΅© Μ σ΅©σ΅© σ΅© Μ σ΅©σ΅©2 σ΅© σ΅© σ΅© Μ σ΅©σ΅©σ΅© σ΅©σ΅© β ππσ΅©σ΅©σ΅©D σ΅© + π2 + βπ σ΅©σ΅©σ΅©z2 σ΅©σ΅©σ΅© (σ΅©σ΅©σ΅©σ΅©D σ΅©σ΅© + πππ·π σ΅©σ΅©σ΅©D σ΅© σ΅© σ΅©σ΅© ) 2
1 Ξ€ β π πΎ π
π + π2 + 2πβ πΎβ1 π
πβ + π 2 β Ξ€ β1
β
1 β π (K1 β I3 β πβ1 πΞ€2 π2 ) π. 2
Ξ€
+ 2πβ πΎβ1 π
πβ
Ξ€
(59)
Ξ€
Due to (πβ + π) πΎβ1 π
(πβ + π) β₯ 0, we obtain Ξ€
Ξ€
βπβ πΎβ1 π
π β πΞ€ πΎβ1 π
πβ β€ πΞ€ πΎβ1 π
π + πβ πΎβ1 π
πβ .
(60)
1 σ΅© Μ σ΅©σ΅© σ΅©σ΅© π· σ΅©σ΅©σ΅©D 2ππ π σ΅© σ΅©
= βπ0 π + πππ, (66)
Ξ€ Ξ€ πΜ πΎβ1 π
πΜ = (πβ β π) πΎβ1 π
(πβ βπ) β Ξ€ β1
= π πΎ π
πβ + πΞ€ πΎβ1 π
π β 2π πΎ π
π
(61)
Ξ€
β€ 2πβ πΎβ1 π
πβ + 2πΞ€ πΎβ1 π
π, which implies that 1 Ξ€ Ξ€ Μ βπβ πΎβ1 π
πβ β πΞ€ πΎβ1 π
π β€ β πΜ πΎβ1 π
π. 2 Now (59) becomes
1 Ξ€ + 2πβ πΎβ1 π
πβ β πΞ€ (K1 β I3 β πβ1 πΞ€2 π2 ) π + π. 2
Ξ€ Μ + where πππ = 2πβ πΎβ1 π
πβ + (1/2)π2 + βπβz2 ββDβ Μ (1/2ππ )π·πβDβ > 0. So we can obtain that π and all the signals of the closeloop system are bounded [18].
4. Case Study (62)
1 1 Ξ€ 1 πΜ β€ βzΞ€1 π1 z1 β zΞ€2 (π2 β I3 β π) z2 β πΜ πΎβ1 π
πΜ + π2 2 2 2
(63)
Choose appropriate π2 , π, K1 such that π2 β (1/2)I3 β π and K1 β (1/2)I3 β πβ1 πΞ€2 π2 are positive definite, and π 1π , π 2π , π
π and ππ are, respectively, the minimum eigenvalue of π1 , π2 β (1/2)I3 β π, π
and K1 β (1/2)I3 β πβ1 πΞ€2 π2 . Then (63) becomes Ξ€ 1 πΜ β€ βπ 1π zΞ€1 z1 β π 2π zΞ€2 z2 β π
π πΜ πΎβ1 πΜ 2 1 Ξ€ β ππ πΞ€ π + 2πβ πΎβ1 π
πβ + π2 + π 2 1 ΜΞ€ β1 Μ Ξ€ = βπ 1π zΞ€1 z1 β π 2π zΞ€2 Mβ1 π Mπ z2 β ππ π π + π πΎ π
π 2 1 Ξ€ + 2πβ πΎβ1 π
πβ + π2 + π. 2
1 Ξ€ σ΅© σ΅© σ΅© Μ σ΅©σ΅©σ΅© = βπ0 π + 2πβ πΎβ1 π
πβ + π2 + βπ σ΅©σ΅©σ΅©z2 σ΅©σ΅©σ΅© σ΅©σ΅©σ΅©σ΅©D σ΅©σ΅© 2 +
Then
β Ξ€ β1
(65)
(64)
A computer simulation has been used to evaluate the performance of the adaptive neural network control with backstepping for fully actuated surface ships with dead-zone in actuator. The ship system used for simulation is described as follows: 0 0 9.1948 Γ 107 M=[ 0 9.1948 Γ 107 1.2607 Γ 109 ] , 0 1.2607 Γ 109 1.0724 Γ 1011 ] [ Dπ = [ [
1.5073 Γ 106 0 0 0 8.1687 Γ 106 β1.3180 Γ 108 ] , 0 β1.3180 Γ 108 1.2568 Γ 1011 ]
Dπ (^) = β diag {π|π’|π’ |π’| , π|V|V |V| , π|π|π |π|} , (67) where π|π’|π’ |π’|, π|V|V |V|, π|π|π |π| are hydrodynamic coefficients, π|π’|π’ = β2.9766 β
104 , π|V|V = β8.0922 β
104 , and π|π|π = β1.2228 β
1012 . Note that the elements of matrices are the nominal values of the ship plant. The bound of dead-zone is set as πβ = 0.5 Γ 105 ,
π+ = 0.6 Γ 105 .
(68)
8
Mathematical Problems in Engineering 0.5 xe (m)
xe (m)
0.05 0 β0.05 0
50
100
200
250
β0.5
300
xe (m) 0
50
100
200
250
250
300
0
50
100
150
200
250
300
200
250
300
Time (s) 1
0
0
50
100
150
200
250
0 β1
300
Time (s)
0
50
100
150 Time (s)
Figure 4: Tracking error of the position and heading.
Figure 6: Tracking error of the position and heading with PID controller.
Γ105 1
Γ105 2 0
50
100
Γ104 5
150 t (s)
200
250
300
πx (N)
0
0
0
Γ10 2
β5 50
100
Γ106 5
150 t (s)
200
250
300
50
100
150 t (s)
200
250
300
b = [0.25 Γ 105 sin (0.1π‘) , 0.25 Γ 105 sin (0.1π‘) , (69)
Ξ€
And parameters of the ship model vary from 0.5 to 1.5 times the size of its nominal value. The limit of control input is set as Ξ€
|π| β€ [0.2 Γ 106 , 0.15 Γ 106 , 0.15 Γ 108 ] .
100
150
200
250
300
150 t (s)
200
250
300
150
200
250
300
t (s)
50
100
50
100
7
0 0
t (s)
To verify the robustness of the controller and the control effect, the unpredictable disturbances and parameter uncertainties are introduced. The environmental disturbances acting on the ship can be treated together as
0.25 Γ 106 sin (0.1π‘)] .
0
Γ10 1
β1
Figure 5: Control inputs in surge, sway, and yaw.
50 5
0 β2
0 0
πy (N)
t (s) 0
0 β2
ππ (Nm)
πx (N)
200
β0.2
300
πe (deg)
πe (deg)
150
β0.1
πy (N)
150
0
Time (s)
0.1
ππ (Nm)
100
0.2
β0.1
β5
50
Time (s)
0
β1
0
Time (s)
0.1 xe (m)
150
0
(70)
Figure 7: Control inputs in surge, sway, and yaw with PID controller.
The initial position and heading of the vessel is (0 m, 0 m, 0 deg), and the initial velocity is (0 m/s, 0 m/s, 0 rad/s) and the desired position is set as (2 m, 1 m, 5 deg). A smooth reference path is generated by a guidance system. The simulation results with proposed controller are shown in Figures 4 and 5, and the ones with PID controller are listed in Figures 6 and 7. The blue lines in Figures 4 and 5 are values without using dead-zone compensator, and the red ones are the results using it. Figure 4 shows that the dead-zone compensator improves the control performance and reduces the tracking errors. Figure 5 is the control inputs for both simulations with and without dead-zone compensator. From Figure 5, the control
Mathematical Problems in Engineering input is zero when the control command is in the deadzone, so its control effect is not excellent unless the deadzone is compensated well. The blue lines in Figures 6 and 7 are results without dead-zone in actuators, and the red ones are values with dead-zone characteristic. Comparing Figures 4 and 5 to Figures 6 and 7, the method proposed in this work ameliorates the control effect, and the tracking errors are 5β10 times smaller than the results using PID controller. Meanwhile, the forces and moments acting on the ship become smaller and smoother than the ones with PID method. So the control strategy designed in this work can protect the actuator from wear and reduce fuel consumption.
5. Conclusion In this paper, an adaptive neural network control using backstepping is derived for fully actuated surface ships with deadzone characteristics in actuator. A three degree of freedom model including disturbances has been established for ships. In order to overcome the difficulties brought by unknown model parameters, the adaptive RBFNN is introduced to approximate the unknown nonlinear functions needed in controller design. The dead-zone character is quite common in actuator for ships, and the existence of dead-zone affects the control effect of traditional controllers, especially when the dead-zone parameters are unknown. The fuzzy logic system is utilized here to handle this problem. Fuzzy logic technique can estimate the dead-zone parameters which are used for compensator design. It has been shown that the adaptive fuzzy control with the dead-zone compensator can drive the ship to the desired position with certain heading angles. The input saturation is overcome by adopting the auxiliary design system. Meanwhile, through the Lyapunov stable theory it is proved that the system is bounded for all states. The simulation results showed that the controller proposed in this work performs excellently for dynamic positioning ships.
References [1] S. Tong, Y. Li, and X. Jing, βAdaptive fuzzy decentralized dynamics surface control for nonlinear large-scale systems based on high-gain observer,β Information Sciences, vol. 235, pp. 287β307, 2013. [2] M. Zhu and Y. Li, βDecentralized adaptive fuzzy sliding mode control for reconfigurable modular manipulators,β International Journal of Robust and Nonlinear Control, vol. 20, no. 4, pp. 472β 488, 2010. [3] C.-C. Chiang and C.-H. Wu, βObserver-based adaptive fuzzy sliding mode control of uncertain multiple-input multipleoutput nonlinear systems,β in Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE β07), pp. 1β6, London, UK, July 2007. [4] J. L. Tao, Y. Yang, D. H. Wang, and C. Guo, βA robust adaptive neural networks controller for maritime dynamic positioning system,β Neurocomputing, vol. 110, no. 1, pp. 128β136, 2013. [5] J. Liu and Y. Lu, βAdaptive RBF neural network control of robot with actuator nonlinearities,β Journal of Control Theory and Applications, vol. 8, no. 2, pp. 249β256, 2010.
9 [6] J. Fei and H. Ding, βAdaptive sliding mode control of dynamic system using RBF neural network,β Nonlinear Dynamics, vol. 70, no. 2, pp. 1563β1573, 2012. [7] V. K. Deolia, S. Purwar, and T. N. Sharma, βBackstepping control of discrete-time nonlinear system under unknown dead-zone constraint,β in Proceedings of the International Conference on Communication Systems and Network Technologies (CSNT β11), pp. 344β349, Katra, Jammu and Kashmir, India, June 2011. [8] H. Cho and E.-W. Bai, βConvergence results for an adaptive dead zone inverse,β International Journal of Adaptive Control and Signal Processing, vol. 12, no. 5, pp. 451β466, 1998. Λ [9] R. R. SelmiΒ΄ c and F. L. Lewis, βDeadzone compensation in motion control systems using neural networks,β IEEE Transactions on Automatic Control, vol. 45, no. 4, pp. 602β613, 2000. [10] A. Taware and G. Tao, βAn adaptive dead-zone inverse controller for systems with sandwiched dead-zones,β International Journal of Control, vol. 76, no. 8, pp. 755β769, 2003. [11] X.-S. Wang, H. Hong, and C.-Y. Su, βModel reference adaptive control of continuous-time systems with an unknown input dead-zone,β IEEE Proceedings on Control Theory and Applications, vol. 150, no. 3, pp. 261β266, 2003. [12] T. Zhang and S. S. Ge, βRobust adaptive neural control of SISO nonlinear systems with unknown dead-zone and completely unknown control gain,β in Proceedings of the IEEE Conference on Control Applications, Computer-Aided Control Systems Design Symposium and International Symposium on Intelligent Control (ISIC β06), pp. 88β93, Munich, Germany, October 2006. [13] C.-C. Chiang, βAdaptive fuzzy tracking control for uncertain nonlinear time-delay systems with unknown dead-zone input,β Mathematical Problems in Engineering, vol. 2013, Article ID 363748, 13 pages, 2013. [14] C. Hu, B. Yao, and Q. Wang, βPerformance-oriented adaptive robust control of a class of nonlinear systems preceded by unknown dead zone with comparative experimental results,β IEEE/ASME Transactions on Mechatronics, vol. 18, no. 1, pp. 178β 189, 2013. [15] T. I. Fossen, Handbook of Marine Craft Hydrodynamics and Motion Control, John Wiley & Sons, Chichester, UK, 2011. [16] T. I. Fossen, Marine Control System, Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles, Trondheim, Norway, Marine Cybernetics, 2002. [17] J. K. Liu, Design and Matlab Simulation of Robot Control System, Tsinghua University Press, Beijing, China, 2012, (Chinese). [18] H. K. Khaill, Nonlinear System, Publishing House of Electronics Industry, Beijing, China, 2006.
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