Adaptive heading tracking control of surface vehicles with unknown control directions and full state constraints

Neurocomputing 359 (2019) 517–525

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Brief papers

Adaptive heading tracking control of surface vehicles with unknown control directions and full state constraints Zhenyu Gao a, Ge Guo a,b,c,∗ a

School of Control Engineering, Northeastern University Qinhuangdao, Qinhungdao 066004, China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China c Department of Automation, Dalian Maritime University, Dalian 116026, China b

a r t i c l e

i n f o

Article history: Received 4 February 2019 Revised 29 April 2019 Accepted 11 June 2019 Available online 19 July 2019 Communicated by Shaocheng Tong Keywords: Heading tracking control Nussbaum gain control Command filter Barrier Lyapunov functions Adaptive control

a b s t r a c t This paper investigates the heading tracking control of surface vehicles in the presence of unknown control directions and full state constraints. To achieve the control objective, the neural networks and command filter-based backstepping technique are utilized to construct an adaptive controller. The Nussbaum gain technique is employed to handle the problem of unknown control directions, and all states of the system are guaranteed to remain within their constraints based on the Barrier Lyapunov functions (BLFs). The proposed control protocol can guarantee that all signals of the closed-loop system are bounded and all states of the system are ensured to remain in the predefined compact sets. The simulation results are provided to illustrate the effectiveness of the proposed control scheme.

1. Introduction Over the past few decades, considerable attention has been paid to the control issues of surface vehicles (SVs), especially heading tracking control, due to its importance for operation safety and economy. Many advanced control techniques have been used to improve the steering performance of SVs in tracking the desired heading. Some outstanding examples include neural network adaptive control [1], fuzzy control [2], robust control [3], finite-time control [4,5], sliding control [6], fuzzy neural network control [7], model predictive control [8], etc. Most existing literatures are based on a common assumption that the control direction of the vehicle is known in advance. However, in some situations, the control direction is unknown, making the control design problem rather complicated. Despite the practical importance, this problem has not been fully addressed in the heading tracking control of surface vehicles. Nussbaum gain function as an effective tool to handle the unknown control directions was first given in [9]. Based on this technique, various control problems were studied by many researchers. In [10], a Nussbaumtype course keeping controller was given for ships with unknown control direction. These results cannot be directly applied to the ∗ Corresponding author at: School of Control Engineering, Northeastern University Qinhungdao, Qinhungdao, 066004, China. E-mail address: [email protected] (G. Guo).

https://doi.org/10.1016/j.neucom.2019.06.091 0925-2312/© 2019 Published by Elsevier B.V.

© 2019 Published by Elsevier B.V.

heading tracking control problem of surface vehicles with rudder dynamics. It is worth noting that, as in all practical engineering systems, some or all SVs outputs or states need to keep in some compact sets due to system specifications, performance or safety requirements. Any violation of the constraints may lead to misbehavior of the transient response, performance degradation, or even instability of SVs. Recently, the barrier Lyapunov function approach to nonlinear control problems subject to output/state constraints give new hints to us [11–13]. In [11], a symmetric Barrier Lyapunov function based control scheme was proposed for trajectory tracking of surface vehicles with output constraint. Recently, in [12,13], a range constrained formation tracking control problem of surface vehicles was investigated by using Barrier Lyapunov Functions.The aforementioned works mainly focus on vehicle systems with output or partial state constraints. This paper is concerned with a heading tracking control problem of surface vehicles subject to full state constraints and unknown control directions. To the best of our knowledge, such a problem has not been investigated yet. Generally, many practice plants are subject to the effect of the uncertainties. As the universal approximations, the neural network [14,15] or the fuzzy logic system [16,17] have been involved in the adaptive tracking control strategies to approximate the uncertainties. In practice, the dynamics of vehicle is influenced by the speed and the load and the instantly changing ocean conditions. Therefore, there evidently exist parameter uncertainties in the vehicle

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steering dynamics. For the heading tracking of surface vehicle in the presence of unmodeled dynamics, adaptive control technique has become the most widely used solution [10,14,23–25]. In [14], an adaptive course-tracking control scheme was presented based on the radial basis function neural network (RBFNN) and Lyapunov stability theory. In [10], an adaptive robust control law is proposed for the course tracking problem of vehicles subject to external disturbances and input saturation. In this paper, an adaptive heading tracking control method is studied for surface vehicles with unknown control directions and full state constraints. The main contributions of this paper are summarized as follows: (i) Nussbaum gain technique is employed to solve the problem of unknown control directions; (ii) By employing BLFs and neural networks (NNs), an adaptive control law is constructed to guarantee all states of the vehicle remain in a constrained set while ensuring all the signals of the heading control system globally uniform ultimate bounded. (iii) The compensatorbased command filter technique is introduced to mitigate the explosion of complexity problem due to consecutive differentiations involved in the control design procedure, making the resulted scheme simple and easy to implement in engineering practice. Finally, by using the Lyapunov stability analysis, it can be proved that all the signals of the closed-loop systems are bounded, all the state constraints are never violated and the tracking errors are within the small neighborhood of the origin. This paper is organized as follows. Section 2 is the problem formulation and preliminaries. Section 3 presents the controller design and stability analysis. Section 4 is the simulation studies. Some conclusions are drawn in Section 5. 2. Problem formulation and preliminaries

Consider the following Norrbin nonlinear model of surface vehicles for heading tracking controller design [4]:

(1)

where ψ ∈ R is the heading angle, δ ∈ R is the actual control of rudder angle, T ∈ R is the time constant, K ∈ R is the gain constant, and α ∈ R is the Norrbin coefficient. In order to get better maneuverability, it is more reasonable to take the characteristics of rudder in account, whose dynamics can be described as [18]:

δ˙ = −

1 K δ + E δE TE TE

(2)

where δE ∈ R is the command rudder angle, KE ∈ R is the control gain, and TE ∈ R is the time constant of the rudder. The nonlinear steering motion equations (1)–(2) can be transformed into the following nonlinear system in strict-feedback form:

ψ˙ = r r˙ = φ0 (r ) + θ0 δ δ˙ = φ1 (δ ) + θ1 δE

Remark 1. The model parameters K, T, α of the vehicle steering are related with the depth of water, the speed and the load of the vehicle, they are difficult to be accurately determined. Besides, the parameters of rudder are easily affected by its own characteristics and external environment. Hence, Assumption 1 is reasonable. Remark 2. In general, the absolute maximum allowed output rudder angle does not exceed 35◦ . Assumption 2 [25]. The sign of control coefficients θi (i = 0, 1 ) are unknown. Remark 3. Taking θ 0 as an example, if the line-movement of the vehicle is stable, T > 0 in (1), whereas if the line-movement of the vehicle is not stable, T < 0 in (1) [19]. As such, the assumption that the sign of control coefficients θi , (i = 0, 1 ) in (3) are unknown is more practical. Assumption 3 [11]. The desired heading ψ d is a smooth bounded signal with bounded time derivatives ψ˙ d . Here, ψ d is assumed to satisfy −ψ ≤ ψd ≤ ψ¯ d and |ψ d | ≤ B0 , |ψ˙ d | ≤ B1 where ψ d , ψ¯ d , B0 , d B1 are positive constants, and max{ψ , ψ¯ } ≤ B0 ≤ kc1 . d

d

2.2. Preliminaries In this subsection, we review some useful lemmas which will serve as the basis of the coming controller design and performance analysis. Lemma 1 [20]. There exist the positive constants kbi , and the variables zi satisfy the interval Ωz = {z ∈ R||zi | < kbi , i = 1, 2, . . . n}. For the interval Ωz , the following inequality holds:

2.1. Problem formulation

T ψ¨ + ψ˙ + α ψ˙ 3 = K δ

Assumption 1 [10]. The nonlinear functions φ i and constants θi , (i = 0, 1 ) are unknown in the design, the model parameters K, T, α and the rudder parameters KE , TE are unknown constants, too.

(3)

where φ0 (r ) = − T1 r − αT r 3 , θ0 = KT , φ1 (δ ) = − T1 δ, and θ1 = TE E E with θi , (i = 0, 1 ) are constants. In this study, all the states are constrained in the compact sets, i.e., |ψ | < kc1 , |δ | < kc2 , |δ E | < kc3 , where kci , (i = 1, 2, 3 ) are known positive constants. The objective of this paper is to design an adaptive controller for the surface vehicle so that the actual heading ψ tracks the desired heading ψ d , while guaranteeing the boundedness of all the signals without violating the state constrains. The following assumptions will be needed in later sections. K

log

k2bi k2bi

−

zi2

≤

zi2 k2bi

− zi2

(4)

Definition 1 [9]. Any continuous even function N(ζ ) is called a Nussbaum-type when there are the following properties:

 1 s N ( ζ )d ζ = +∞, s→∞ s 0  1 s lim inf N (ζ )dζ = −∞. s→∞ s 0 lim sup

(5)

Lemma 2 [21]. Let V(t) ≤ 0 and ζ (t) be smooth functions defined on [0, tf ], and N(ζ (t)) be smooth Nussbaum-type function. If the following inequality holds:

V (t ) ≤ c1 + e−c2 t



t 0

[ϕ N (ζ (τ )) + 1]ζ˙ (τ )dτ ,

(6)

where c1 , c2 and ϕ are positive constants. Then, V(t), ζ (t) and t ˙ 0 N (ζ (τ ) )ζ (τ )d τ must be bounded on [0, tf ]. In this paper, an even Nussbaum function is chosen as N (ζ ) = ζ 2 cos(ζ ). In addition, there are some Nussbaum-type  functions  2 2 that are common, such as ζ sin (ζ ) and exp ζ cos π2ζ . 2.3. Radial basis function neural networks Due to the property of universal approximation capabilities over a compact set, the radial basis function neural networks (RBFNNs) have been widely used in the control design for uncertain nonlinear systems. In this paper, the unknown continuous function φ (x) defined on a compact set NN will be approximated by using the RBFNNs. Namely, φ (x ) = W T S(x ), where x ∈ NN ⊂ Rn denotes the input vector, W = [w1 , w2 , . . . , wl ]T denotes the weight

Z. Gao and G. Guo / Neurocomputing 359 (2019) 517–525

vector with l > 1 being the number of NN nodes, and S(x ) = T [s1 (x ), s2 (x ), . . . , sl (x )] denotes the basis function vector with si (x) being selected as the Gaussian function, which have the form



si (x ) = exp

− ( x − μi ) T ( x − μi )



ηi2

(7)

where μi = [μi1 , μi2 , . . . , μin ]T denote the center of the receptive field and ηi is the width of the Gaussian function. It has been proved that, provided sufficient number of NN nodes l, WT S(x) can approximate any continuous φ (x ) : Rq → R over a compact set x → Rq to any desired accuracy as

φ (x ) = W ∗T S(x ) +  (x )



W ∗ := arg minn W ∈R

  T   sup ϕ (x ) − W S(x ) .

x∈ NN

with z1,1 = αr (t ) and α˙ r (t ) = z1,2 are the output of the filter, wr > 0, fr ∈ (0, 1] are the design parameters, and the initial values are chosen as z1,1 (0 ) = αr (0 ) and z1,2 (0 ) = 0. The command filter may have errors if not appropriately designed and calibrated, which will in turn affect the performance. Here we introduce a compensating signal mechanism to deal with this issue. To do this, define the filtering error ω1 = αr − αrd . Then, the filtered compensating signal ξ r is generated by the following system:

ξ˙r = −κ1 ξr + ω1 , ξr (0 ) = 0

Remark 4. The RBFNNs-based approximation, which is used here, can be replaced by other unknown continuous function approximations such as fuzzy logic systems (FLSs), wavelet NNs, and multilayer NNs.

e1 k2b1 − e21

V˙ 1 =

e2 − k1 e1 + ψ˙ d −

=−

k1 e21 k2b1 − e21

−

−κ1 ξr2 + ξr ω1 .

(17)

Based on Young’s inequality, we have

e21 1 e1 e2 1 ≤  + e22 , 2 2 k2 − e2 2 2 − e1 1 b1

k2b1

κ1 e1 ξr k2b1

−

e21

κ1

e21 κ1 2 + ξ ,  2 k2 − e2 2 2 r 1 b1

≤

1 2 1 2 ξ + ω1 . 2 r 2

In this section, a heading tracking control law is proposed for surface vehicle incorporating the Nussbaum function and the Barrier Lyapunov function into the adaptive command-filter technique. The whole controller design process is given as follow: Step 1: Define the heading tracking error e1 as follows:

In the light of (18), one has

V˙ 1 ≤ −

(10)

+

(11)

≤−

k1 e21 k2b1

Consider the Barrier Lyapunov function candidate V1 as follows:

V1 =

1 1 log 2 + ξr2 , 2 2 kb1 − e21

(12)

V˙ 1 =

αr

e1 = −k1 e1 + ψ˙ d − + κ1 ξr , 2 k2b1 − e21

(13)

e2

1

e2

1

1 1 +  + e22 2 k2 − e2 2 2 k2 − e2 2 2 1 1 b1 b1

e21 κ1 2 1 1 + ξ − κ1 ξr2 + ξr2 + ω1  2 k2 − e2 2 2 r 2 2 1 b1 k1 e21

k2b1

− e21

V˙ 1 ≤ −k1V1 +

+

1 2 1 2 e + ω . 2 2 2 1

(19)

(14)

z˙ 1,1 = z1,2 (15)

1 2 e + 0 2 2

(20)

with 0 = 12 ω12 . Step 2: Taking the time derivative of e2 , one obtains:

e˙ 2 = r˙ − α˙ rd

  φ0 (r ) + θ0 e3 + αδd − α˙ rd

(21)

with e3 = δ − αδd . Select the following Barrier Lyapunov function candidate V2 as:

V2 =

where k1 , γ 1 and κ 1 are positive constants with γ1 ≥ κ1 + 1 and κ 1 ≥ 1. To generate the stabilizing function αrd and its derivative α˙ rd , the nominal stabilizing function is then passed through the following command filter

z˙ 1,2 = −2 fr wr z1,2 − w2r (z1,1 − αr ).

γ1

Based on Lemma 1, if |e1 | ≤ kb1 , we have the following inequal-

=

with e2 = r − αrd . To stabilize (13), the virtual input α r is chosen as:

γ1

−

ity

where kb1 = kc1 − B0 , and ξ r is the compensating signal which will be designed later. Taking the time derivation of V1 , one obtains

e1 e˙ 1 + ξ˙r ξr k2b1 − e21  e1  = 2 e2 + αrd − ψ˙ d + ξ˙r ξr 2 kb1 − e1

−

e21

(18)

κ1

and taking the time derivative of e1 yields:

k2b1



e21 e1 e2 κ1 e1 ξr + 2 + 2  2 k2 − e2 2 kb1 − e21 kb1 − e21 1 b1

ξr ω 1 ≤

e˙ 1 = r − ψ˙ d

e1 − ψ˙ d + κ1 ξr 2 k2b1 − e21

γ1

3. Controller design and stability analysis

e1 = ψ − ψd

γ1

+ξr (−κ1 ξr + ω1 )

(9)

Since the activation function S(x) is bounded, there exists a positive constant S∗ ∈ R such that S(x) ≤ S∗ .

(16)

Substituting (14) into (13), we can obtain

(8)

where W∗ is the ideal unknown constant vector and  (x) is the unknown approximation error and bounded over the compact set satisfying | (x)| ≤  ∗ with  ∗ > 0. The ideal weigh vector W∗ is defined as

519

k2 1 1 1 T −1 ˜ Γ W ˜ 0, log 2 b2 2 + ξδ2 + W 2 2 2 0 0 kb2 − e2

(22)

˜0 = W ˆ 0 − W0 , kb2 ≥ 0 where Γ0 = Γ0T > 0 is the design parameter, W is the design constant, and ξ δ is the compensating signal designed later. Taking the time derivative of V2 , one has

V˙ 2 = =

e2 e˙ 2 ˜ 0 Γ −1W ˆ˙ 0 + ξ˙δ ξδ +W 0 − e22

k2b2 e2



   φ0 (r ) + θ0 e3 + αδd − α˙ rd k2b2 − e22

˜ 0 Γ −1W ˆ˙ 0 + ξ˙δ ξδ . +W 0

(23)

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Z. Gao and G. Guo / Neurocomputing 359 (2019) 517–525

Design the virtual controller α δ as

αδd = N (ζδ )αδ ,

(24)

−

e22 e2  ( r ) 1 1 ≤  +  ∗2 , 2 2 2 k2 − e2 2 2 kb2 − e2 2 b2

with

αδ = k2 e2 − α˙ rd + Wˆ 0 S(r ) +

γ2

e2 − κ2 ξδ , 2 k2b2 − e22

(25)

e2 αδ , k2b2 − e22

(26)

where k2 , γ 2 and κ 2 are positive constants with γ2 ≥ κ2 + 1 and κ 2 ≥ 1. Similar to Step 1, in order to get αδd and α˙ δd , the nominal stabilizing function is then passed through the following command filter

z˙ 2,1 = z2,2

ξ˙δ = −κ2 ξδ + ω2 , ξδ (0 ) = 0

(28)

According to (24), (23) can be rewrite as:

V˙ 2 =

θ0 e2 N (ζδ )αδ k2b2 − e22



(29)

+

γ2 2



e22 k2b2 − e22

κ2 ξδ e2 k2b2 − e22

2

γ2 2



k2b2 − e22

+

θ0 e 2 e 3 k2b2 − e22

e2

+



α

e2

ˆ 0S ˙ rd − W k2b2 − e22



k2b2 − e22 −1 ˆ˙ 0 W0

k2 e22 k2b2

−

e22



Y

φ0 (r ) − α˙ rd



− κ2 ξδ + ξδ ω2 .

Based on Young’s inequality, one has

θ0 e 2 e 3

e22 1 1 ≤  + θ02 e23 ,  2 2 2 2 2 kb2 − e2 k2b2 − e22

κ2 e2 ξδ κ2 κ2 2 ≤ + ξ , 2 k2 − e2 2 2 δ k2b2 − e22 2 b2 e22

(33)

−

e22

−

α1 W˜ 0 2 2

+

α1 W0 2 2

we have

k2b2

k2 −e22 b2

≤

e22

(34)

in the interval |e2 | < kb2 , then,

k2 −e22 b2

1 V˙ 2 ≤ −k2V2 + 1 + θ0 N (ζδ )ζ˙ δ + ζ˙ δ + θ02 e23 , 2

(35)

α W 2

(36)

The following Barrier Lyapunov function V3 is chosen as:

k2 1 1 T −1 ˜ Γ W ˜1 log 2 b3 2 + W 2 2 1 1 kb3 − e3

(37)

e3 e˙ 3 ˜ 1 Γ −1W ˆ˙ 1 +W 1 k2b3 − e23 e3



φ1 (δ ) + θ1 u − α˙ δd



k2b3 − e23

˜ 1 Γ −1W ˆ˙ 1 . +W 1

(38)

Here, the actual control input is designed as:

u = N (ζu )αu ,

(39)

with

αu = k3 e3 − α˙ δd + Wˆ 1 S(δ ),



(40)

and

e22

ζ˙u =

θ0 e 2 e 3 κ2 ξδ e2 2 + k2 − e2 + k2 − e2 2 2 b2 b2 2

k2 e22 k2b2

It is a fact that log

=



˜ 0 S (r ) +  (r ) e2 W −

.

1 1 1 +θ0 N (ζδ )ζ˙ δ + ζ˙ δ + θ02 e23 + ω22 +  ∗2 . 2 2 2

V˙ 3 =

k2b2 − e22

k2b2

2

˜1 = W ˆ 1 − W1 , and where Γ1 = Γ1T > 0 is the design parameter, W kb3 ≥ 0 is the design constant. Taking the time derivative of V3 yields:

(r )



−

2

W0 2

+

and in the light of (30)–(33), we obtain

V3 =

˜ 0 Γ −1W ˆ˙ 0 − κ2 ξ 2 + ξδ ω2 +W δ 0

e22

˜ 0Γ +W

k2 e22

+

θ0 N (ζδ )ζ˙δ + ζ˙δ − −

2

b2

θ0 N (ζδ )ζ˙δ + ζ˙δ − −

=

φ0 (r ) − α˙ rd

k2b2 − e22

˜ 0 Γ −1W ˆ˙ 0 + ξδ − κ2 ξδ + ω2 +W 0 =



θee θ0 N (ζδ )ζ˙δ + ζ˙δ + 20 2 32 + kb2 − e2 k2b2 − e22   ˆ 0 S(r ) + γ2 2 e2 2 − κ2 ξδ e2 k2 e2 − α˙ rd + W 2 k −e −

(32)

e˙ 3 = δ˙ − α˙ δd .

Substituting (25) and (26) into (29), we have

V˙ 2 =

W˜ 0 2

α W˜ 2

˜ 0 Γ −1W ˆ˙ 0 + ξ˙δ ξδ . +W 0

e2



where 1 = − 1 2 0 + 1 2 0 + 12 ω22 + 12  ∗2 . Step 3: Taking the time derivative of e3 , one obtains:

 d

e2 φ0 (r ) − α˙ r θ0 e 2 e 3 + k2b2 − e22 k2b2 − e22

+

e2 φ0 (r ) − α1Wˆ 0 k2b2 − e22

˜ 0T W ˆ0 ≤ − −W

(27)

with z2,1 = αr (t ) and α˙ r (t ) = z2,2 are the output of the filter, wδ > 0, fδ ∈ (0, 1] are the design parameters, and the initial values are chosen as z2,1 (0 ) = αr (0 ) and z2,2 (0 ) = 0. Define the filtering error ω2 = αδ − αδd , and the filter compensating signal ξ δ is generated by the following system:



with α 1 is a positive design parameter. Considering the following facts by completion of squares:

V˙ 2 ≤ −

z˙ 2,2 = −2 fδ wδ z2,2 − w2δ (z2,1 − αr ).

(31)

Construct the adaptive law as

ˆ˙ 0 = Γ0 W

and

ζ˙δ =

1 2 1 2 ξ + ω2 . 2 δ 2

ξδ ω 2 ≤

e3 αu . k2b3 − e23

(41)

Using (39), (38) can be rewrite as



(30)



e3 φ1 (δ ) − α˙ δd e3 θ1 N (δu )αu ˜ 1 Γ −1W ˆ˙ 1 V˙ 3 = + +W 1 k2b3 − e23 k2b3 − e23 Substituting (40) and (41) into (42), we have



V˙ 3 = θ1 N (ζu )ζ˙ u + ζ˙ u − +

e3



 d

φ1 (δ ) − α˙ δ k2b3 − e23

ˆ 1 S (δ ) e3 k3 e3 − α˙ δd + W k2b3 − e23 ˜ 1 Γ −1W ˆ˙ 1 +W 1



(42)

Z. Gao and G. Guo / Neurocomputing 359 (2019) 517–525

521

35 30 25

15 10

r

and

d

[deg]

20

5 0 -5 -10 0

50

100

150

200

250

300

350

400

450

500

Time [s] Fig. 1. The reference heading ψ r and desired heading ψ d of the surface vehicle.

= θ1 N (ζu )ζ˙ u + ζ˙ u − ˜ 1Γ +W



k3 e23 k2b3

−

e23

−

˜ 1 S (δ ) +  (δ ) e3 W k2b3

−1 ˆ˙ 1 W1

−



Proof. Consider the following Lyapunov function candidate V for the whole closed-loop system as:

e23 (43)

=

Construct the adaptive law as

ˆ˙ 1 = Γ1 W



e3 φ1 (δ ) − α2Wˆ 1 k2b3 − e23



V˙ 3 ≤ −

k2b3 − e23

where 2 = −

+ 2 + θ1 N (ζu )ζ˙ u + ζ˙ u

α2 W˜ 1 2 2

It is a fact that log we have

+

α2 W1 2

k2b3

k2 −e23 b3

2

≤

e23

V˙ 3 ≤ −k3V3 + 2 + θ1 N (ζu )ζ˙ u + ζ˙ u .

k2 1 1 T −1 1 1 ˜ Γ W ˜ 1 + ξr2 + ξ 2 log 2 b3 2 + W 2 2 1 1 2 2 δ kb3 − e3

(47)

Taking the time derivative of V yields:

V˙ ≤ −k1 log

in the interval |e3 | < kb3 . Then,

k2b1 k2b1 − e21

− k2 log

k2b2 k2b2 − e22

− k3 log

k2b3 k2b3 − e23

1 1 + e22 + θ0 e23 + θ0 N (ζδ )ζ˙ δ + ζ˙ δ + θ1 N (ζu )ζ˙ u + ζ˙ u 2 2 +0 + 1 + 2

(45)

+ 12  ∗2 .

k2 −e23 b3

k2 k2 1 1 1 T −1 ˜ Γ W ˜0 log 2 b1 2 + log 2 b2 2 + W 2 2 2 0 0 kb1 − e1 kb2 − e2 +

(44)

with α 2 is a positive design parameter. Similar to (19), (34), we obtain

k3 e23

V = V1 + V2 + V3

(48)

According to Lemma 1, (48) can be rewrite as:

V˙ ≤ −ρV + C + θ0 N (ζδ )ζ˙ δ + ζ˙ δ + θ1 N (ζu )ζ˙ u + ζ˙ u

(49)

where

(46)

The main contents in this section are summarized as the following theorem. Theorem 1. For the nonlinear surface vehicle steering motion model (3) under Assumptions 1–3, by designing the actual control input (39) with virtual control laws (14), (25), command filters (15), (27), and adaptive laws (32), (44), the proposed control protocol can ensure that: (1) the signals of the closed-loop system are bounded. (2) all the state constraints of the system are never violated. (3) the tracking error signal e1 will remain within the compact set  defined by  = {e1 | −  ≤ e1 ≤ } where  will be defined later on.

ρ = min {k1 , k2 , k3 , κ1 , κ2 , λmin (Γ0 ), λmin (Γ1 )} and

C=

1 2 1 e + θ0 e23 + θ0 N (ζδ )ζ˙ δ + ζ˙ δ + θ1 N (ζu )ζ˙ u 2 2 2 +ζ˙ u + 0 + 1 + 2 .

Multiplying both sides of (49) by eσ t , one obtains





d (V˙ eσ t )/dt ≤ C + θ0 N (ζδ )ζ˙ δ + ζ˙ δ + θ1 N (ζu )ζ˙ u + ζ˙ u eσ t

(50)

Integrating (50) over [0, t], (50) further becomes





C −σ t C V (t ) ≤ V (0 ) − e + σ σ  t  + e −σ t θ0 N (ζδ )ζ˙δ + ζ˙δ + θ1 N (ζu )ζ˙u + ζ˙u eσ t dt 0

(51)

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Z. Gao and G. Guo / Neurocomputing 359 (2019) 517–525

35 ψd ψ

30

ψd and ψ [deg]

25 20 15 10 5 0 -5 -10 0

50

100

150

200

250

300

350

400

450

500

Time [s] Fig. 2. The desired heading ψ d and actual heading ψ of the surface vehicle.

15 e1 with constraint e1 without constraint

Heading error e1 [deg]

10

Δ -Δ

5 0 -5

15

0

10

-5

5

-10

0 100

-10

101

-15 400

102

401

402

-15 0

50

100

150

200

250

300

350

400

450

500

Time [s] Fig. 3. The heading error e1 of the surface vehicle.

In the light of Lemma 2, it is easy to obtain the boundedness of V(t) and ζi , (i= δ, μ ). Letting  be the upper bound of the integral t term e−σ t 0 θ0 N (ζδ )ζ˙ δ + ζ˙ δ + θ1 N (ζu )ζ˙ u + ζ˙ u eσ t d. According to the definition of V, the following inequality holds:

  k2 C −σ t 1 C log 2 b1 2 ≤ V (0 ) − e + + . 2 σ σ kb1 − e1

(52)

Then, one has

  12 C −σ t C |e1 | ≤ kb1 1 − e−2[V (0)− σ e + σ +] = 

(53)

From (49) and the optimal weight vector Wi , (i = 0, 1 ), we can ˜ i and Wi are bounded, know that the weight estimation errors W ˆ. and obtain the boundedness of the estimation weight vector W From x1 = e1 + ψd and |ψ d | ≤ B0 , one obtains that |x1 | ≤ |e1 | + |ψd | ≤ kb1 + B0 = kc1 . To verify that |r| ≤ kc2 , it needs to show that there is a constant α¯ r > 0 so that |αrd | ≤ α¯ r . In (14), αrd is a continuous function of e1 , ξ r and ψ˙ d . Since |e1 | ≤ kbi and ψ˙ d ≤ B1 , and α r is a continuous function, α r must be bounded. Then, α¯ r is existent and satisfying α¯ r + kb2 ≤ kc2 . Then, we have |r| ≤ |e2 | + |αrd | ≤ α¯ r + kb2 < kc2 . Likewise, we can in turn prove that |δ | ≤ kc3 . Thus,

the system state constraints are not violated. From (53), we can see, the tracking error e1 can converge to arbitrarily small compact set by selecting the design parameters appropriately. This completes the proof.  Remark 5. Compared with the conventional backstepping method in [23,24] and dynamic surface control (DSC) technique in [10,25], the compensator-based command filter method, in which the virtual control input passing a second-order filter, not only can avoid repeated differentiation of the virtual control input, but also can eliminate the filtering errors generated by DSC technique. 4. Simulation studies In this section, a simulation example is presented to validate the effectiveness of the designed command filter based adaptive heading tracking controller for surface vehicles. The dynamic parameters of vehicle steering model (1), and (2) are T = 0.332, K = 0.701, α = 0.001, TE = 2 and KE = 1 [22]. These values are obtained from identification results of a unmanned surface vehicle at a speed of U = 9m/s.

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523

αr [deg]

20 0 -20 0

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Time [s] Fig. 4. Virtual input α r , α δ and actual input δ E of the surface vehicle.

1

N (ζδ ) ζδ

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Time [s] Fig. 5. Nussbaum gain N(ζ δ ) and its argument ζ δ .

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N (ζu ) ζu

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Time [s] Fig. 6. Nussbaum gain N(ζ u ) and its argument ζ u .

350

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3.5

φˆ0 (r) φ0 (r)

3 2.5 2

φ0(r)

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Time [s] Fig. 7. The uncertain term φ 0 (r) and its estimate φˆ 0 (r ).

2.5 φˆ1 (δ) φ1 (δ)

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Time [s] Fig. 8. The uncertain term φ 1 (δ ) and its estimate φˆ 1 (δ ).

In this simulation, the reference heading of the vehicle is given as:

ψr =

⎧ 30, ⎪ ⎪ ⎨ 10,

−5,

⎪ ⎪ ⎩ 10,

30,

if 0 ≤ t < 100 if 100 ≤ t < 200 if 200 ≤ t < 300 if 300 ≤ t < 400 else

(54)

To obtain the smooth form of the reference signal ψ r , let ψ r pass through the following command filter:

ψ˙ d = rd r˙ d = −2 fψ wψ rd − w2ψ (ψd − ψr )

(55)

where ξ ψ and ωψ are the parameters of the command filter, which were used to describing the closed-loop response characteristics. Here, fψ = 0.8 and wψ = 50.

The parameter of the designed controller are k1 = 0.2, k2 = ˆ 0 (0 ) = W ˆ 1 ( 0 ) = 0, 2, k3 = 5, γ1 = 5, κ1 = 3, fr = 0.2, wr = 50, W Γ0 = diag(0.5 ), γ2 = 7, κ2 = 3, fδ = 0.8, α1 = 1.5, wδ = 50, Γ1 = diag(2 ) and α2 = 0.8. Choose the initial value of the state ψ (0 ) = r (0 ) = δ (0 ) = 0, which are constrained by |ψ | ≤ 40, |r| ≤ 30, and |δ | ≤ 30. The simulation results are shown in Figs. 1–8. Fig. 1 shows the reference course ψ r and the desired course ψ d which obtained by letting ψ r pass through the command filter (55). It can be seen ψ d is smoother, and more practical in practice. The desired heading and the actual heading of the surface vehicle are shown in Fig. 2, and the heading tracking error is shown in Fig. 3. From Figs. 2, 3 we can see that under the action of the proposed control law, the surface vehicle can track the desired course ψ d . In addition, from Fig. 4, it can be observed that the heading tracking error e1 never violet the constraint that given in (53). The designed virtual input α r , α δ and the actual input δ E are given in Fig. 4. In Figs. 5–8,

Z. Gao and G. Guo / Neurocomputing 359 (2019) 517–525

Nussbaum gain N(ζ ∗ ) and its argument ζ ∗ with ∗ = δ, u, the estimation φi , (i = 0, 1 ) are shown, which are all bounded as proved in Theorem 1. From these Figures, it can be demonstrated that the proposed adaptive controller can steer the surface vehicle with unknown control directions and full state constraints to track the desired course. 5. Conclusion In this paper, an adaptive controller has been developed for heading tracking control of surface vehicles in the presence of unknown control directions and full state constraints. To overcome the challenges, an adaptive tracking controller with Nussbaum gain technique is structured to counteract the lack of a priori knowledge of the sign of the control directions, the Barrier Lyapunov functions are employed to guarantee that states never violate the predetermined compact constraints. In addition, the command filters are incorporated into the backstepping control, which makes the control design simple and easy to implement. It can be proven that all the signals in the closed-loop systems are bounded, the tracking error fluctuates within the small sets around zero and the full state never violate their constraint sets based on Lyapunov analysis. The effectiveness of the presented method can be illustrated by a simulation example. For further investigations, it is of interest to consider the finite/fixed-time heading control of surface vehicles with unknown control directions and full state constraints. Declaration of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This work was supported by the National Natural Science Foundation of China under grant 61573077 and U1808205, and partially by National Key R&D Program of China under Grant 2017YFA070 030 0.

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Ge Guo received the B.S. and Ph.D. degrees from Northeastern University, Shenyang, China, in 1994 and 1998, respectively. In 1999, he joined the Lanzhou University of Technology, China, where he was the Director of the Institute of Intelligent Control and Robots and a Professor from 2004 to 2005. He has been with Dalian Maritime University, China, as a Professor with the Department of Automation. From 2009 to 2015, he held a visiting re search position at The Chinese University of Hong Kong, China. He is currently a Professor with Northeastern University. He has published over 100 journal papers within his areas of interest, which include intelligent transportation systems and cyber-physical systems. Dr. Guo is the Managing Editor of the International Journal of Systems, Control and Communications and an Associate Editor of several other journals like Information Sciences, the ACTA Automatica Sinica, and the IEEE Intelligent Transportation Systems Magazine. He was an honoree of the New Century Excellent Talents in University, Ministry of Education, China, in 2004, and a nominee for Gansu Top Ten Excellent Youths by the Gansu Provincial Government in 2005. He was the CAA Young Scientist Award Winner in 2017.