- Email: [email protected]
Global finite-time heading control of surface vehicles
Author’s Accepted Manuscript Global Finite–Time Heading Control of Surface Vehicles Ning Wang, Shuailin Lv, Zhongzhong Liu
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)01582-9 http://dx.doi.org/10.1016/j.neucom.2015.10.106 NEUCOM16284
To appear in: Neurocomputing Received date: 26 September 2015 Revised date: 19 October 2015 Accepted date: 28 October 2015 Cite this article as: Ning Wang, Shuailin Lv and Zhongzhong Liu, Global Finite– Time Heading Control of Surface Vehicles, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.10.106 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Global Finite–Time Heading Control of Surface Vehicles Ning Wang, Shuailin Lv and Zhongzhong Liu Marine Engineering College, Dalian Maritime University, Dalian 116026, P. R. China E-mail: [email protected]
Abstract In this paper, a global finite-time heading control (GFHC) scheme for surface vehicles is proposed. The salient features of the GFHC scheme are triplefold: 1) A discontinuous control law is proposed to guarantee the finite-time stability of the entire closed-loop heading control system. 2) The finite-time convergence leads to accurate heading control and remarkable disturbance rejection. 3) Furthermore, it reveals that the proposed GFHC scheme treats asymptotic heading controllers as special cases. Simulation studies and comprehensive comparisons on various scenarios demonstrate the effectiveness and superiority of the proposed GFHC scheme. Keywords: Heading control, global finite-time stability, surface vehicles. 1. Introduction In the community of ship automation, course keeping plays a significant role in the entire control system, and is directly related to the operation, economy, safety and effectiveness of the ship control system. The ship heading control problem has been attracting much attention from various researchers. In order to realize the position and heading control of ships and oil rigs, Fossen and Perez [1] applied the Kalman filter to estimate unmeasurable Preprint submitted to Neurocomputing
October 31, 2015
states and thereby realizing output-feedback based position and heading control of ships and oil rigs. However, the foregoing approach usually requires nice compensation for low-frequency disturbances. An accurate and economic optimization method for ship heading control system was proposed in [2], whereby simulation results showed that the optimization algorithm was reliable. However, it inevitably suffers from fine tuning parameters of a PID controller. A nonlinear controller based on the backstepping technique and the sliding mode control for an air cushion vehicle was proposed in [3]. Note that the steering dynamics and/or uncertainties have not been fully addressed. Recently, intelligent approaches via fuzzy logic systems [4], neural networks [5] and fuzzy neural networks [6, 7], etc., have been intensively studied and applied to tracking control of surface vehicles. Unlike traditional fuzzy/neural control approaches which require predefined structure for an approximator, these self-organizing fuzzy neural network (SOFNN) based adaptive (robust) control schemes [4-7] are able to achieve remarkable performance in terms of both tracking and approximation, since the structure of the online fuzzy/neural approximator is persistently evolving and is updated according to tracking accuracy, in addition to parameter identification. Similar idea has been applied to flight vehicles [8-10], whereby system uncertainties and disturbances can be handled. However, only asymptotic convergence can be obtained in the previous schemes. By virtue of finite-time stability of homogeneous systems [11], non-smooth continuous control approaches have been developed rapidly and have been applied to finite-time controller synthesis. In [12], a finite-time controller for robot manipulators was designed via both state feedback and dynamic out-
2
put feedback approaches. Recently, a finite-time control scheme via output feedback for tracking autonomous underwater vehicles has been proposed in [13]. The finite-time control problem of multiple manipulators [14] has also been addressed by employing the homogeneous theory which allowed unmodeled dynamics to be handled. The aforementioned finite-time control schemes achieved superior control performance in terms of both convergence rate and disturbance rejection ability. In order to improve the maneuverability and reduce the consumption of fuel, it is desired to maintain the surface vehicle in a scheduled line course. Two important performance indexes pertaining to nonlinear control systems are the convergence rate and the robustness to disturbances, which are also crucial in the course keeping of surface vehicles. In this context, the finitetime control method with fast convergence and strong disturbance rejection ability is highly required to solve the course keeping problem of surface vehicles. In this paper, a global finite-time heading control (GFHC) scheme is proposed for the heading control of surface vehicles. In the GFHC scheme, a discontinuous control law is employed to realize the finite-time stability of the entire closed-loop heading control system. In this context, faster convergence and stronger disturbance rejection ability can be achieved for the ship heading control. The rest of this paper is organized as follows. Section 2 formulates preliminaries about finite-time stability, homogeneity and the problem statement of heading control. The heading dynamics of a surface vehicle is addressed in Section 3. The GFHC scheme together with corresponding stability analysis
3
are presented in Section 4. Simulation studies and comprehensive comparisons are conducted in Section 5. Conclusions are drawn in Section 6. 2. Preliminaries and Problem Statement 2.1. Preliminaries Consider the system x ∈ Rn
x˙ = f (x),
(1)
with x = xe being the equilibrium. We recall fundamental definitions as follows: Definition 1 (Asymptotic Stability [15]). The equilibrium xe = 0 of system (1) is globally asymptotically stable if there exits a function V (x) satisfying (i) V (0) = 0. (ii) V (x) > 0, ∀x = 0, and V (x) is radically unbounded. (iii) V˙ (x) ≤ 0. (iv) V˙ (x) does not vanish identically along any trajectory in Rn , other than the null solution x = 0. Definition 2 (Homogeneity [16]). Let f (x) = (f1 (x), ..., fn (x))T be a continuous vector field. f (x) is homogeneous of degree k ∈ R with respect to the dilation (r1 , ..., rn ), if, for any given (r1 , ..., rn ) ∈ Rn as well as ε > 0, fi (εr1 x1 , ..., εrn xn ) = εk+ri fi (x), 4
i = 1, ..., n, ∀x ∈ Rn .
(2)
Let V (x) : Rn → R be a continuous scalar function, if ∀ε > 0, ∃δ > 0 and dilation (r1 , ..., rn ) ∈ Rn , ri > 0, i = 1, ..., n, thus V (εr1 x1 , ..., εxn xn ) = εσ V (x),
i = 1, ..., n, ∀x ∈ Rn .
(3)
V (x) is a homogeneous function of degree σ with respect to dilation (r1 , ..., rn ). An important lemma about finite-time stability is ready to be given here. Lemma 1 (Finite-time Stability [12]). System (1) is global finite-time stable if it is globally asymptotically stable and is homogeneous with a negative degree k < 0. Throughout this paper, the foregoing definitions and Lemma 1 help to derive the main result. 2.2. Problem Statement The main task during the automatic navigation in the sea is to provide crews a safe working environment, comfortable living environment as well as to short the voyage cycle, reduce fuel consumption and save voyage. In this context, a precise and stable course keeping and changing plays an important role in this process. Hence, our objective is to design a heading controller such that the actual heading ψ can converge to the desired heading ψd with fast convergence and little overshoot which are actually trapped into a dilemma in traditional asymptotic control methods. Moreover, due to model uncertainties and unknown disturbances imposed on a surface vehicle, a finite-time control method is highly desired to realize remarkable performance of disturbance rejection pertaining to the complex heading dynamics of a surface vehicle. 5
3. Heading Dynamics Consider the Nomoto equation for steering dynamics of a surface vehicle as follows [17]:
K(1 + T3 s) ψ(s) = δ(s) s(1 + T1 s)(1 + T2 s)
(4)
where T1 T2 and T3 are time constants, and K is the system gain. It can further be written as the first-order Nomoto model at low frequency given by T ψ¨ + ψ˙ = Kδ
(5)
where T = T1 + T2 − T3 . Remark 1. System (5) is the linear model for heading dynamics of a surface vehicle. For most ships, system (5) properly describes the dynamic behavior for the case where only little rudder angle and low rudder operating frequency are considered. However, the linear dynamics in system (5) cannot address well the heading behavior when the surface vehicle changes its course with large rudder angle. Actually, system (5) is only the linearization of the ship heading dynamics. However, such a linear model can only be valid for the case where both little rudder angle and low rudder frequency are considered. In this ˙ is usually employed to address the complex context, a nonlinear term H(ψ) heading dynamics with large maneuvers. Hence, the nonlinear heading model can be written as follows: ˙ = Kδ T ψ¨ + H(ψ)
6
(6)
where ˙ = α0 + α1 ψ˙ + α2 ψ˙ 2 + α3 ψ˙ 3 H(ψ)
(7)
with Norrbin coefficients αi > 0(i = 0, 1, 2, 3) and rudder reflection angle δ. 4. Controller Design and Stability Analysis In this section, the GFHC scheme via state feedback for surface vehicles is proposed by employing the homogeneous theory. Moreover, the finite-time stability of the entire closed-loop heading control system can be guaranteed by using the Lyapunov approach. 4.1. Controller Design The heading dynamics in (6) can be rewritten as follows: ⎧ ⎪ ⎨ x˙ 1 = x2
(8)
⎪ ⎩ x˙ 2 = 1 (Kδ − α0 − α1 x2 − α2 x22 − α3 x32 ) T where x1 = ψ and x2 = ψ˙ are measurable states.
The desired heading is denoted as ψd . Incorporating the feedback linearization into the finite-time controller design, we design the GFHC scheme as follows: u=−
T k1 T k2 |x1 − ψd |β1 sgn(x1 − ψd ) − |x2 |β2 sgn(x2 ) + φ(x2 ) K K
where φ(x2 ) =
α0 α1 α2 α3 + x2 + x22 + x32 K K K K
with positive constants k1 > 0, k2 > 0, 0 < β1 < 1, and β2 =
(9)
(10) 2β1 . 1+β1
It should
be noted that the term φ(x2 ) is used to feedback linearize the nonlinearity pertaining to the heading dynamics in (8). 7
4.2. Stability Analysis It is essential that the GFHC law can make the actual heading angle converge to the desired heading angle in a finite time. The corresponding stability analysis is stated here. ˙ = (ψd , 0) of system (8) together with Theorem 1. The equilibrium (ψ, ψ) the GFHC law (9) is global finite-time stable. ˙ Then the closed-loop system of the Proof. Let x3 = ψ − ψd , x4 = x˙ 3 = ψ. ship heading dynamic model (8) under the GFHC law (9) is as follows: ⎧ ⎨ x˙ 3 = x4 ⎩ x˙ = −k |x |β1 sgn(x ) − k |x |β2 sgn(x ) 4 1 3 3 2 4 4 Consider the following Lyapunov function x3 1 |μ|β1 sgn(μ)dμ + x24 V (x3 , x4 ) = k1 2 0
(11)
(12)
The derivative of V (x3 , x4 ) along the dynamics (11) is obtained as follows: V˙ (x3 , x4 ) =k1 x4 |x3 |β1 sgn(x3 ) + x4 (−k1 |x3 |β1 sgn(x3 ) − k2 |x4 |β2 sgn(x4 )) = − k2 x4 |x4 |β2 sgn(x4 )
(13)
= − k2 |x4 |β2 +1 which is negative semi-definite with V˙ (x3 , x4 ) = 0 if and only if x4 = 0. In this context, we have x4 = 0 and x˙ 4 = 0, if V˙ (x3 , x4 ) is identical to zero. Combining with system (11), we get −k1 |x3 |β1 sgn(x3 ) − k2 |x4 |β2 sgn(x4 ) = 0 8
(14)
which implies x3 = 0. It follows that V˙ (x3 , x4 ) does not vanish identically along any solution other than (x3 , x4 ) = (0, 0). The result satisfies Definition 1. To prove system (11) is global finite-time stable, according to Lemma 1, we need to validate a negative homogeneous degree σ < 0. To be specific, system (11) can be rewritten as follows: ⎧ 1+β 1+β ⎨ f1 (ε1 x3 , ε 2 1 x4 ) = ε 2 1 f1 (x3 , x4 ) β1 −1 1+β 1 ⎩ f (ε1 x , ε 1+β + 21 2 x ) = ε 2 f2 (x3 , x4 ) 2 3 4
where 0 < β1 < 1, β2 = negative, i.e., σ =
β1 −1 2
2β1 , 1+β1
(15)
and the corresponding homogeneous degree is
1 < 0 with respect to the dilation (r1 , r2 ) = (1, 1+β ). 2
It follows from Lemma 1 that the equilibrium of system (8) together with the GFHC law (9) is global finite-time stable. This concludes the proof. Remark 2. When β1 increases to power one, the closed-loop system composed of the Norrbin ship model (8) and the GFHC controller (9) becomes globally asymptotically stable. The closed-loop system is as follows: ⎧ ⎨ x˙ 3 = x˙ 4 ⎩ x˙ = −k |x | sgn(x ) − k |x | sgn(x ) 4 1 3 3 2 4 4
(16)
where the asymptotic control can be derived from backstepping technique. Remark 3. It can be seen from (13) that the state x4 converge to the origin in a finite time, and the convergence rate within a small vicinity of the origin is much larger than that of an asymptotic control scheme, since the power β2 can be less than one rather than identically unity pertaining to the asymptotic case in (16). Together with (11), faster convergence for x3 can steadily be obtained. 9
In simulation studies, we will show that the GFHC scheme can perform better behavior than the asymptotic heading controller. 5. Simulation Studies In order to demonstrate the effectiveness of the proposed GFHC scheme, we conduct simulation studies and comprehensive comparisons on a surface vehicle in [18], the main parameters are T = 78.41s, K = 0.21s−1 , α0 = 0, α1 = 2.2386, α2 = 0, α3 = 1988.4. The GFHC controller parameters are chose as k1 = 490, k2 = 338, and β1 = 1/7 and β1 = 1/9 respectively. The simulation results are shown in Figs. 1-4. It should be noted that the GFHC scheme would degrade to a conventional backstepping based heading control method if we set β1 = 1. In this context, comparisons with backstepping based approaches have also been made in the sequel. As a consequence, the superiority of the proposed GFHC scheme can be clearly validated.
10
14 −.ψ
d
−β1=1/7
12
:β =1 1
ψ/degree
10
8
6
4
2
0
0
1
2
3
4
5
t/s
Figure 1: Heading curves with β1 = 17 , x(0) = [0, 0]T .
30 −.ψ
d
−β1=1/9 :β =1
ψ/degree
25
1
20
15
10
5
0
1
2
3
4
t/s
Figure 2: Heading curves with β1 = 19 , x(0) = [30, 0]T .
11
5
35 −.ψ
d
−β1=1/9
30
:β =1 1
ψ/degree
25
20
15
10
5
0
0
2
4
6
8
10
12
14
t/s
Figure 3: Heading curves with β1 = 19 , x(0) = [0, 0]T . 35 −.ψ
d
−β1=1/7
30
:β1=1
ψ/degree
25
20
15
10
5
0
2
4
6
8
10
12
t/s
Figure 4: Heading curves with β1 = 17 , x(0) = [30, 0]T .
12
14
The actual (solid line) and the desired heading (dash-doted line) are shown in Figs. 1-2, from which we can see that the proposed GFHC controller can track the desired heading with faster convergence rate than the backstepping based asymptotic controller (doted line) which coincides with the statement in Remark 3. In addition, the control system has no overshoot, and the tracking errors can be rendered to zero in a finite time. When the surface vehicle change its course, we can see that the actual states are able to track the desired ones with rapid transient response and high steady-state accuracy, the corresponding simulation results are shown in Figs. 3-4. In this context, by virtue of the fast convergence rate pertaining to the GFHC, the proposed GFHC scheme can track the desired heading with rapid transient response, no overshoot, and the tracking errors can be rendered to zero in a finite time. Moreover, the GFHC scheme is superior to the traditional asymptotic heading controller. 6. Conclusions In this paper, a novel discontinuous control scheme termed global finitetime heading control (GFHC) for a surface vehicle has been proposed. By virtue of the finite-time stability, the discontinuous controller is realized by combining LaSalle’s invariant theorem and negative degree of homogeneity. As a consequence, the proposed GFHC scheme renders asymptotic heading controllers as special cases. Moreover, the finite-time stability of the entire heading control system has been proven by employing the Lyapunov approach. 13
Acknowledgments The authors would like to thank the Editor-in-Chief, Associate Editor and anonymous referees for their invaluable comments and suggestions. This work is supported by the National Natural Science Foundation of P. R. China (under Grants 51009017 and 51379002), Applied Basic Research Funds from Ministry of Transport of P. R. China (under Grant 2012-329-225-060), China Postdoctoral Science Foundation (under Grant 2012M520629). References [1] T.I. Fossen, T. Perez, Kalman filtering for positioning and heading control of ships and offshore rigs, IEEE Trans. Control Syst. 29(6)(2009) 32–46. [2] S.M. Yuan, Y. Yao, H.S. Du, J.H. Cheng, Research on optimization technology of the ship heading control performance, in: Proceedings of IEEE International Conference on Mechatronics and Automation, 2013, pp. 1610–1614. [3] J. Li, D.C. Song, Z.Y. Liu, Sliding mode control of ACV heading tracking based on backstepping, in: Proceedings of IEEE International Conference on Mechatronics and Automation, 2013, pp. 360–364. [4] N. Wang, M.J. Er, J.C. Sun, Y.C. Liu, Adaptive robust online constructive fuzzy control of a complex surface vehicle System, IEEE Trans. Cybern. DOI: 10.1109/TCYB.2015.2451116. [5] N. Wang, J.C. Sun, M.J. Er, Y.C. Liu, A novel extreme learning con-
14
trol framework of unmanned surface vehicles, IEEE Trans. Cybern. DOI: 10.1109/TCYB.2015.2423635. [6] N. Wang, M.J. Er, M. Han, Dynamic tanker steering control using generalized ellipsoidal-basis-function-based fuzzy neural networks, IEEE Trans. Fuzzy Syst. 23(5)(2015) 1414–1427. [7] N. Wang, M.J. Er, Self-constructing adaptive robust fuzzy neural tracking control of surface vehicles with uncertainties and unknown disturbances, IEEE Trans. Control Syst. Technol. 23(3)(2015) 991–1002. [8] B. Xu, C.G. Yang, Y.P. Pan, Global neural dynamic surface tracking control of strict-feedback systems with application to hypersonic flight vehicle, IEEE Trans. Neural Netw. Learn. Syst. 26(10)(2015) 2563–2575. [9] B. Xu, Z.K. Shi, C.G. Yang, F.C. Sun, Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form, IEEE Trans. Cybern. 44(12)(2014) 2626–2634. [10] B. Xu, C.G. Yang, Z.K. Shi, Reinforcement learning output feedback nn control using deterministic learning technique, IEEE Trans. Neural Netw. Learn. Syst. 25(3)(2014) 635–641. [11] S.P. Bhat, D.S. Bernstein, Finite-time stability of homogeneous systems, in: Proceedings of American Control Conference, 1997, pp. 2513–2514. [12] Y.G. Hong, Y.S. Xu, H. Jie, Finite-time control for robot manipulators, Syst. Control Lett. 46(2002) 243–253.
15
[13] S.H. Li, X.Y. Wang, L.J. Zhang, Finite-time output feedback tracking control for autonomous underwater vehicles, IEEE J. Ocean Eng. 40(3)(2015) 727–751. [14] B. Zhang, Y.M. Jia, J.P. Du, J. Zhang, Finite-time consensus control for multiple manipulators with unmodeled dynamics, in: Proceedings of American Control Conference, 2013, pp. 5380–5385. [15] H.J. Marquez, Nonlinear Control Systems: Analysis and Design, John Wiley & Sons, Inc. Hoboken, New Jersey, 2003. [16] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Syst. Control Lett. 19(6)(1992) 467–473. [17] P. Yan, J.D. Han, Z.W. Wu, Nonlinear backstepping design of ship steering controller: using unscented kalman filter to estimate the uncertain parameters, in: Proceedings of IEEE International Conference on Automation and Logistics, 2007, pp. 126–131. [18] F.C. Meng, N. Wang, B.J. Dai, Y.C. Liu, Indirect fuzzy adaptive heading control of surface ships, in: Proceedings of the 33rd Chinese Control Conference, 2014, pp. 4576–4580.
16
Biography
Ning Wang received his B. Eng. degree in Marine Engineering and the Ph.D. degree in control theory and engineering from the Dalian Maritime University (DMU), Dalian, China in 2004 and 2009, respectively. From September 2008 to September 2009, he was financially supported by China Scholarship Council (CSC) to work as a joint-training Ph.D. student at the Nanyang Technological University (NTU), Singapore. In the light of his significant research at NTU, he received the Excellent Government-funded Scholars and Students Award in 2009. He is currently an Associate Professor with the Marine Engineering College, Dalian Maritime University (DMU), Dalian 116026, China. Dr. Wang received the Nomination Award of Liaoning Province Excellent Doctoral Dissertation, the DMU Excellent Doctoral Dissertation Award and the DMU Outstanding Ph.D. Student Award in 2010, respectively. He also won the Liaoning Province Award for Technological Invention and the honour of Liaoning BaiQianWan Talents, Liaoning Excellent Talents, Science and Technology Talents the Ministry of Transport of the P. R. China, Youth Science and Technology Award of China Institute of Navigation, and Dalian Leading Talents. His research interests include fuzzy logic systems, artificial neural networks, machine learning, nonlinear control, self-organizing fuzzy neural modeling and control, ship intelligent control, and dynamic ship navigational 17
safety assessment. He currently serves as an Associate Editor of the Neurocomputing.
Shuailin Lv received his B. Eng. degree in Marine Electronic and Electrical Engineering and B. E. degree in International Economics and Trade from the Dalian Maritime University (DMU), Dalian, China in 2015. He is currently pursuing his Master degree at DMU, Dalian 116026. His research interests include sliding mode control , ship motion control, and finite-time control.
Zhongzhong Liu received his B. Eng. degree in Marine Electronic and Electrical Engineering from the Dalian Maritime University (DMU), Dalian, China in 2015. He is currently pursuing his Master degree at DMU, Dalian 116026. His research interests include underactuated surface vessel, adaptive modeling and control.
18
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)01582-9 http://dx.doi.org/10.1016/j.neucom.2015.10.106 NEUCOM16284
To appear in: Neurocomputing Received date: 26 September 2015 Revised date: 19 October 2015 Accepted date: 28 October 2015 Cite this article as: Ning Wang, Shuailin Lv and Zhongzhong Liu, Global Finite– Time Heading Control of Surface Vehicles, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.10.106 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Global Finite–Time Heading Control of Surface Vehicles Ning Wang, Shuailin Lv and Zhongzhong Liu Marine Engineering College, Dalian Maritime University, Dalian 116026, P. R. China E-mail: [email protected]
Abstract In this paper, a global finite-time heading control (GFHC) scheme for surface vehicles is proposed. The salient features of the GFHC scheme are triplefold: 1) A discontinuous control law is proposed to guarantee the finite-time stability of the entire closed-loop heading control system. 2) The finite-time convergence leads to accurate heading control and remarkable disturbance rejection. 3) Furthermore, it reveals that the proposed GFHC scheme treats asymptotic heading controllers as special cases. Simulation studies and comprehensive comparisons on various scenarios demonstrate the effectiveness and superiority of the proposed GFHC scheme. Keywords: Heading control, global finite-time stability, surface vehicles. 1. Introduction In the community of ship automation, course keeping plays a significant role in the entire control system, and is directly related to the operation, economy, safety and effectiveness of the ship control system. The ship heading control problem has been attracting much attention from various researchers. In order to realize the position and heading control of ships and oil rigs, Fossen and Perez [1] applied the Kalman filter to estimate unmeasurable Preprint submitted to Neurocomputing
October 31, 2015
states and thereby realizing output-feedback based position and heading control of ships and oil rigs. However, the foregoing approach usually requires nice compensation for low-frequency disturbances. An accurate and economic optimization method for ship heading control system was proposed in [2], whereby simulation results showed that the optimization algorithm was reliable. However, it inevitably suffers from fine tuning parameters of a PID controller. A nonlinear controller based on the backstepping technique and the sliding mode control for an air cushion vehicle was proposed in [3]. Note that the steering dynamics and/or uncertainties have not been fully addressed. Recently, intelligent approaches via fuzzy logic systems [4], neural networks [5] and fuzzy neural networks [6, 7], etc., have been intensively studied and applied to tracking control of surface vehicles. Unlike traditional fuzzy/neural control approaches which require predefined structure for an approximator, these self-organizing fuzzy neural network (SOFNN) based adaptive (robust) control schemes [4-7] are able to achieve remarkable performance in terms of both tracking and approximation, since the structure of the online fuzzy/neural approximator is persistently evolving and is updated according to tracking accuracy, in addition to parameter identification. Similar idea has been applied to flight vehicles [8-10], whereby system uncertainties and disturbances can be handled. However, only asymptotic convergence can be obtained in the previous schemes. By virtue of finite-time stability of homogeneous systems [11], non-smooth continuous control approaches have been developed rapidly and have been applied to finite-time controller synthesis. In [12], a finite-time controller for robot manipulators was designed via both state feedback and dynamic out-
2
put feedback approaches. Recently, a finite-time control scheme via output feedback for tracking autonomous underwater vehicles has been proposed in [13]. The finite-time control problem of multiple manipulators [14] has also been addressed by employing the homogeneous theory which allowed unmodeled dynamics to be handled. The aforementioned finite-time control schemes achieved superior control performance in terms of both convergence rate and disturbance rejection ability. In order to improve the maneuverability and reduce the consumption of fuel, it is desired to maintain the surface vehicle in a scheduled line course. Two important performance indexes pertaining to nonlinear control systems are the convergence rate and the robustness to disturbances, which are also crucial in the course keeping of surface vehicles. In this context, the finitetime control method with fast convergence and strong disturbance rejection ability is highly required to solve the course keeping problem of surface vehicles. In this paper, a global finite-time heading control (GFHC) scheme is proposed for the heading control of surface vehicles. In the GFHC scheme, a discontinuous control law is employed to realize the finite-time stability of the entire closed-loop heading control system. In this context, faster convergence and stronger disturbance rejection ability can be achieved for the ship heading control. The rest of this paper is organized as follows. Section 2 formulates preliminaries about finite-time stability, homogeneity and the problem statement of heading control. The heading dynamics of a surface vehicle is addressed in Section 3. The GFHC scheme together with corresponding stability analysis
3
are presented in Section 4. Simulation studies and comprehensive comparisons are conducted in Section 5. Conclusions are drawn in Section 6. 2. Preliminaries and Problem Statement 2.1. Preliminaries Consider the system x ∈ Rn
x˙ = f (x),
(1)
with x = xe being the equilibrium. We recall fundamental definitions as follows: Definition 1 (Asymptotic Stability [15]). The equilibrium xe = 0 of system (1) is globally asymptotically stable if there exits a function V (x) satisfying (i) V (0) = 0. (ii) V (x) > 0, ∀x = 0, and V (x) is radically unbounded. (iii) V˙ (x) ≤ 0. (iv) V˙ (x) does not vanish identically along any trajectory in Rn , other than the null solution x = 0. Definition 2 (Homogeneity [16]). Let f (x) = (f1 (x), ..., fn (x))T be a continuous vector field. f (x) is homogeneous of degree k ∈ R with respect to the dilation (r1 , ..., rn ), if, for any given (r1 , ..., rn ) ∈ Rn as well as ε > 0, fi (εr1 x1 , ..., εrn xn ) = εk+ri fi (x), 4
i = 1, ..., n, ∀x ∈ Rn .
(2)
Let V (x) : Rn → R be a continuous scalar function, if ∀ε > 0, ∃δ > 0 and dilation (r1 , ..., rn ) ∈ Rn , ri > 0, i = 1, ..., n, thus V (εr1 x1 , ..., εxn xn ) = εσ V (x),
i = 1, ..., n, ∀x ∈ Rn .
(3)
V (x) is a homogeneous function of degree σ with respect to dilation (r1 , ..., rn ). An important lemma about finite-time stability is ready to be given here. Lemma 1 (Finite-time Stability [12]). System (1) is global finite-time stable if it is globally asymptotically stable and is homogeneous with a negative degree k < 0. Throughout this paper, the foregoing definitions and Lemma 1 help to derive the main result. 2.2. Problem Statement The main task during the automatic navigation in the sea is to provide crews a safe working environment, comfortable living environment as well as to short the voyage cycle, reduce fuel consumption and save voyage. In this context, a precise and stable course keeping and changing plays an important role in this process. Hence, our objective is to design a heading controller such that the actual heading ψ can converge to the desired heading ψd with fast convergence and little overshoot which are actually trapped into a dilemma in traditional asymptotic control methods. Moreover, due to model uncertainties and unknown disturbances imposed on a surface vehicle, a finite-time control method is highly desired to realize remarkable performance of disturbance rejection pertaining to the complex heading dynamics of a surface vehicle. 5
3. Heading Dynamics Consider the Nomoto equation for steering dynamics of a surface vehicle as follows [17]:
K(1 + T3 s) ψ(s) = δ(s) s(1 + T1 s)(1 + T2 s)
(4)
where T1 T2 and T3 are time constants, and K is the system gain. It can further be written as the first-order Nomoto model at low frequency given by T ψ¨ + ψ˙ = Kδ
(5)
where T = T1 + T2 − T3 . Remark 1. System (5) is the linear model for heading dynamics of a surface vehicle. For most ships, system (5) properly describes the dynamic behavior for the case where only little rudder angle and low rudder operating frequency are considered. However, the linear dynamics in system (5) cannot address well the heading behavior when the surface vehicle changes its course with large rudder angle. Actually, system (5) is only the linearization of the ship heading dynamics. However, such a linear model can only be valid for the case where both little rudder angle and low rudder frequency are considered. In this ˙ is usually employed to address the complex context, a nonlinear term H(ψ) heading dynamics with large maneuvers. Hence, the nonlinear heading model can be written as follows: ˙ = Kδ T ψ¨ + H(ψ)
6
(6)
where ˙ = α0 + α1 ψ˙ + α2 ψ˙ 2 + α3 ψ˙ 3 H(ψ)
(7)
with Norrbin coefficients αi > 0(i = 0, 1, 2, 3) and rudder reflection angle δ. 4. Controller Design and Stability Analysis In this section, the GFHC scheme via state feedback for surface vehicles is proposed by employing the homogeneous theory. Moreover, the finite-time stability of the entire closed-loop heading control system can be guaranteed by using the Lyapunov approach. 4.1. Controller Design The heading dynamics in (6) can be rewritten as follows: ⎧ ⎪ ⎨ x˙ 1 = x2
(8)
⎪ ⎩ x˙ 2 = 1 (Kδ − α0 − α1 x2 − α2 x22 − α3 x32 ) T where x1 = ψ and x2 = ψ˙ are measurable states.
The desired heading is denoted as ψd . Incorporating the feedback linearization into the finite-time controller design, we design the GFHC scheme as follows: u=−
T k1 T k2 |x1 − ψd |β1 sgn(x1 − ψd ) − |x2 |β2 sgn(x2 ) + φ(x2 ) K K
where φ(x2 ) =
α0 α1 α2 α3 + x2 + x22 + x32 K K K K
with positive constants k1 > 0, k2 > 0, 0 < β1 < 1, and β2 =
(9)
(10) 2β1 . 1+β1
It should
be noted that the term φ(x2 ) is used to feedback linearize the nonlinearity pertaining to the heading dynamics in (8). 7
4.2. Stability Analysis It is essential that the GFHC law can make the actual heading angle converge to the desired heading angle in a finite time. The corresponding stability analysis is stated here. ˙ = (ψd , 0) of system (8) together with Theorem 1. The equilibrium (ψ, ψ) the GFHC law (9) is global finite-time stable. ˙ Then the closed-loop system of the Proof. Let x3 = ψ − ψd , x4 = x˙ 3 = ψ. ship heading dynamic model (8) under the GFHC law (9) is as follows: ⎧ ⎨ x˙ 3 = x4 ⎩ x˙ = −k |x |β1 sgn(x ) − k |x |β2 sgn(x ) 4 1 3 3 2 4 4 Consider the following Lyapunov function x3 1 |μ|β1 sgn(μ)dμ + x24 V (x3 , x4 ) = k1 2 0
(11)
(12)
The derivative of V (x3 , x4 ) along the dynamics (11) is obtained as follows: V˙ (x3 , x4 ) =k1 x4 |x3 |β1 sgn(x3 ) + x4 (−k1 |x3 |β1 sgn(x3 ) − k2 |x4 |β2 sgn(x4 )) = − k2 x4 |x4 |β2 sgn(x4 )
(13)
= − k2 |x4 |β2 +1 which is negative semi-definite with V˙ (x3 , x4 ) = 0 if and only if x4 = 0. In this context, we have x4 = 0 and x˙ 4 = 0, if V˙ (x3 , x4 ) is identical to zero. Combining with system (11), we get −k1 |x3 |β1 sgn(x3 ) − k2 |x4 |β2 sgn(x4 ) = 0 8
(14)
which implies x3 = 0. It follows that V˙ (x3 , x4 ) does not vanish identically along any solution other than (x3 , x4 ) = (0, 0). The result satisfies Definition 1. To prove system (11) is global finite-time stable, according to Lemma 1, we need to validate a negative homogeneous degree σ < 0. To be specific, system (11) can be rewritten as follows: ⎧ 1+β 1+β ⎨ f1 (ε1 x3 , ε 2 1 x4 ) = ε 2 1 f1 (x3 , x4 ) β1 −1 1+β 1 ⎩ f (ε1 x , ε 1+β + 21 2 x ) = ε 2 f2 (x3 , x4 ) 2 3 4
where 0 < β1 < 1, β2 = negative, i.e., σ =
β1 −1 2
2β1 , 1+β1
(15)
and the corresponding homogeneous degree is
1 < 0 with respect to the dilation (r1 , r2 ) = (1, 1+β ). 2
It follows from Lemma 1 that the equilibrium of system (8) together with the GFHC law (9) is global finite-time stable. This concludes the proof. Remark 2. When β1 increases to power one, the closed-loop system composed of the Norrbin ship model (8) and the GFHC controller (9) becomes globally asymptotically stable. The closed-loop system is as follows: ⎧ ⎨ x˙ 3 = x˙ 4 ⎩ x˙ = −k |x | sgn(x ) − k |x | sgn(x ) 4 1 3 3 2 4 4
(16)
where the asymptotic control can be derived from backstepping technique. Remark 3. It can be seen from (13) that the state x4 converge to the origin in a finite time, and the convergence rate within a small vicinity of the origin is much larger than that of an asymptotic control scheme, since the power β2 can be less than one rather than identically unity pertaining to the asymptotic case in (16). Together with (11), faster convergence for x3 can steadily be obtained. 9
In simulation studies, we will show that the GFHC scheme can perform better behavior than the asymptotic heading controller. 5. Simulation Studies In order to demonstrate the effectiveness of the proposed GFHC scheme, we conduct simulation studies and comprehensive comparisons on a surface vehicle in [18], the main parameters are T = 78.41s, K = 0.21s−1 , α0 = 0, α1 = 2.2386, α2 = 0, α3 = 1988.4. The GFHC controller parameters are chose as k1 = 490, k2 = 338, and β1 = 1/7 and β1 = 1/9 respectively. The simulation results are shown in Figs. 1-4. It should be noted that the GFHC scheme would degrade to a conventional backstepping based heading control method if we set β1 = 1. In this context, comparisons with backstepping based approaches have also been made in the sequel. As a consequence, the superiority of the proposed GFHC scheme can be clearly validated.
10
14 −.ψ
d
−β1=1/7
12
:β =1 1
ψ/degree
10
8
6
4
2
0
0
1
2
3
4
5
t/s
Figure 1: Heading curves with β1 = 17 , x(0) = [0, 0]T .
30 −.ψ
d
−β1=1/9 :β =1
ψ/degree
25
1
20
15
10
5
0
1
2
3
4
t/s
Figure 2: Heading curves with β1 = 19 , x(0) = [30, 0]T .
11
5
35 −.ψ
d
−β1=1/9
30
:β =1 1
ψ/degree
25
20
15
10
5
0
0
2
4
6
8
10
12
14
t/s
Figure 3: Heading curves with β1 = 19 , x(0) = [0, 0]T . 35 −.ψ
d
−β1=1/7
30
:β1=1
ψ/degree
25
20
15
10
5
0
2
4
6
8
10
12
t/s
Figure 4: Heading curves with β1 = 17 , x(0) = [30, 0]T .
12
14
The actual (solid line) and the desired heading (dash-doted line) are shown in Figs. 1-2, from which we can see that the proposed GFHC controller can track the desired heading with faster convergence rate than the backstepping based asymptotic controller (doted line) which coincides with the statement in Remark 3. In addition, the control system has no overshoot, and the tracking errors can be rendered to zero in a finite time. When the surface vehicle change its course, we can see that the actual states are able to track the desired ones with rapid transient response and high steady-state accuracy, the corresponding simulation results are shown in Figs. 3-4. In this context, by virtue of the fast convergence rate pertaining to the GFHC, the proposed GFHC scheme can track the desired heading with rapid transient response, no overshoot, and the tracking errors can be rendered to zero in a finite time. Moreover, the GFHC scheme is superior to the traditional asymptotic heading controller. 6. Conclusions In this paper, a novel discontinuous control scheme termed global finitetime heading control (GFHC) for a surface vehicle has been proposed. By virtue of the finite-time stability, the discontinuous controller is realized by combining LaSalle’s invariant theorem and negative degree of homogeneity. As a consequence, the proposed GFHC scheme renders asymptotic heading controllers as special cases. Moreover, the finite-time stability of the entire heading control system has been proven by employing the Lyapunov approach. 13
Acknowledgments The authors would like to thank the Editor-in-Chief, Associate Editor and anonymous referees for their invaluable comments and suggestions. This work is supported by the National Natural Science Foundation of P. R. China (under Grants 51009017 and 51379002), Applied Basic Research Funds from Ministry of Transport of P. R. China (under Grant 2012-329-225-060), China Postdoctoral Science Foundation (under Grant 2012M520629). References [1] T.I. Fossen, T. Perez, Kalman filtering for positioning and heading control of ships and offshore rigs, IEEE Trans. Control Syst. 29(6)(2009) 32–46. [2] S.M. Yuan, Y. Yao, H.S. Du, J.H. Cheng, Research on optimization technology of the ship heading control performance, in: Proceedings of IEEE International Conference on Mechatronics and Automation, 2013, pp. 1610–1614. [3] J. Li, D.C. Song, Z.Y. Liu, Sliding mode control of ACV heading tracking based on backstepping, in: Proceedings of IEEE International Conference on Mechatronics and Automation, 2013, pp. 360–364. [4] N. Wang, M.J. Er, J.C. Sun, Y.C. Liu, Adaptive robust online constructive fuzzy control of a complex surface vehicle System, IEEE Trans. Cybern. DOI: 10.1109/TCYB.2015.2451116. [5] N. Wang, J.C. Sun, M.J. Er, Y.C. Liu, A novel extreme learning con-
14
trol framework of unmanned surface vehicles, IEEE Trans. Cybern. DOI: 10.1109/TCYB.2015.2423635. [6] N. Wang, M.J. Er, M. Han, Dynamic tanker steering control using generalized ellipsoidal-basis-function-based fuzzy neural networks, IEEE Trans. Fuzzy Syst. 23(5)(2015) 1414–1427. [7] N. Wang, M.J. Er, Self-constructing adaptive robust fuzzy neural tracking control of surface vehicles with uncertainties and unknown disturbances, IEEE Trans. Control Syst. Technol. 23(3)(2015) 991–1002. [8] B. Xu, C.G. Yang, Y.P. Pan, Global neural dynamic surface tracking control of strict-feedback systems with application to hypersonic flight vehicle, IEEE Trans. Neural Netw. Learn. Syst. 26(10)(2015) 2563–2575. [9] B. Xu, Z.K. Shi, C.G. Yang, F.C. Sun, Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form, IEEE Trans. Cybern. 44(12)(2014) 2626–2634. [10] B. Xu, C.G. Yang, Z.K. Shi, Reinforcement learning output feedback nn control using deterministic learning technique, IEEE Trans. Neural Netw. Learn. Syst. 25(3)(2014) 635–641. [11] S.P. Bhat, D.S. Bernstein, Finite-time stability of homogeneous systems, in: Proceedings of American Control Conference, 1997, pp. 2513–2514. [12] Y.G. Hong, Y.S. Xu, H. Jie, Finite-time control for robot manipulators, Syst. Control Lett. 46(2002) 243–253.
15
[13] S.H. Li, X.Y. Wang, L.J. Zhang, Finite-time output feedback tracking control for autonomous underwater vehicles, IEEE J. Ocean Eng. 40(3)(2015) 727–751. [14] B. Zhang, Y.M. Jia, J.P. Du, J. Zhang, Finite-time consensus control for multiple manipulators with unmodeled dynamics, in: Proceedings of American Control Conference, 2013, pp. 5380–5385. [15] H.J. Marquez, Nonlinear Control Systems: Analysis and Design, John Wiley & Sons, Inc. Hoboken, New Jersey, 2003. [16] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Syst. Control Lett. 19(6)(1992) 467–473. [17] P. Yan, J.D. Han, Z.W. Wu, Nonlinear backstepping design of ship steering controller: using unscented kalman filter to estimate the uncertain parameters, in: Proceedings of IEEE International Conference on Automation and Logistics, 2007, pp. 126–131. [18] F.C. Meng, N. Wang, B.J. Dai, Y.C. Liu, Indirect fuzzy adaptive heading control of surface ships, in: Proceedings of the 33rd Chinese Control Conference, 2014, pp. 4576–4580.
16
Biography
Ning Wang received his B. Eng. degree in Marine Engineering and the Ph.D. degree in control theory and engineering from the Dalian Maritime University (DMU), Dalian, China in 2004 and 2009, respectively. From September 2008 to September 2009, he was financially supported by China Scholarship Council (CSC) to work as a joint-training Ph.D. student at the Nanyang Technological University (NTU), Singapore. In the light of his significant research at NTU, he received the Excellent Government-funded Scholars and Students Award in 2009. He is currently an Associate Professor with the Marine Engineering College, Dalian Maritime University (DMU), Dalian 116026, China. Dr. Wang received the Nomination Award of Liaoning Province Excellent Doctoral Dissertation, the DMU Excellent Doctoral Dissertation Award and the DMU Outstanding Ph.D. Student Award in 2010, respectively. He also won the Liaoning Province Award for Technological Invention and the honour of Liaoning BaiQianWan Talents, Liaoning Excellent Talents, Science and Technology Talents the Ministry of Transport of the P. R. China, Youth Science and Technology Award of China Institute of Navigation, and Dalian Leading Talents. His research interests include fuzzy logic systems, artificial neural networks, machine learning, nonlinear control, self-organizing fuzzy neural modeling and control, ship intelligent control, and dynamic ship navigational 17
safety assessment. He currently serves as an Associate Editor of the Neurocomputing.
Shuailin Lv received his B. Eng. degree in Marine Electronic and Electrical Engineering and B. E. degree in International Economics and Trade from the Dalian Maritime University (DMU), Dalian, China in 2015. He is currently pursuing his Master degree at DMU, Dalian 116026. His research interests include sliding mode control , ship motion control, and finite-time control.
Zhongzhong Liu received his B. Eng. degree in Marine Electronic and Electrical Engineering from the Dalian Maritime University (DMU), Dalian, China in 2015. He is currently pursuing his Master degree at DMU, Dalian 116026. His research interests include underactuated surface vessel, adaptive modeling and control.
18