Assume that there is a nonlinear system:

x ˙ ( t ) = f ( x ( t ) ) ,         x ( 0 ) = 0 ,         f ( 0 ) = 0 (1)

where x = [ x 1 , x 2 , ⋯ , x n ] T . f ( x ) is a continuous nonlinear function.

Definition If system f ( x ) satisfies the following points, the equilibrium x = 0 of System (1) is fixed-time stable:

System f ( x ) it has global finite time stability and T ( x ) converges to the origin within a finite convergence time;

There is a stable time equation T ( x ) , Guaranteeing T ( x ) ≤ T max , where T max is a constant.

Lemma 1 If there is a Lyapunov function V ( x ) satisfied V ˙ ( x ) ≤ − ( α V ( x ) p + β V ( x ) q ) k , it shows that the origin of the system is stable at a fixed-time, which means that V ( x ) can converge to V ( x ) = 0 in a fixed time based on any initial value. The convergence time boundary is:

T ≤ 1 α ( 1 − p ) + 1 β ( q − 1 ) (2)

where α , β , p , q are positive constant, and 0 < p < 1 , q > 1 .

Lemma 2 For any nonnegative real constants [ x 1 , x 2 , ⋯ , x n ] ∈ R , the following

inequality we can holds: ( ∑ i = 1 n | x i | ) υ ≤ ∑ i = 1 n | x i | υ , where υ ∈ R + , υ ∈ ( 0 , 1 ] .

If υ ∈ R + , υ > 1 , and ( ∑ i = 1 n | x i | ) υ ≤ n υ − 1 ∑ i = 1 n | x i | υ . (3)

Lemma 3 For arbitrary positive real constants a, b, and c and positive real constants p and q satisfying 1 / p + 1 / q = 1 , the following inequality holds:

a b ≤ c p a p p + c − q b q q (4)

The ship’s motions are described in two right-hand coordinate frames as shown in Figure 1. The earth-fixed frame indicated by O X 0 Y 0 Z 0 is an inertial frame and the ship-fixed frame indicated by AXYZ is a non-inertial frame. The origin O of the earth-fixed frame can be chosen as any point on the earth’s surface. The axis O X 0 is directed to the north, the axis O Y 0 is directed to the east, and the axis O Z 0 points towards the center of the earth. When the ship is port-starboard symmetric, the origin A of the ship-fixed frame is located at the gravity center of the ship. The axis AX is directed from aft to fore, the axis AY is directed to starboard, and the axis AZ is directed from top to bottom. The planes X 0 Y 0 and XY are parallel to the still water surface.

The mathematical model that describes the ship motions in DP mode is

η ˙ = R ( ψ ) ν M ν ˙ + C ( ν ) ν + D ( ν ) ν = τ + b (5)

η = [ x , y , ψ ] T represents the ship position vector in the north-east-down frame; ν = [ u , v , r ] T represents the ship velocity vector in the ship-fixed frame; τ = [ τ 1 , τ 2 , τ 3 ] T and b = [ b 1 , b 2 , b 3 ] T respectively represent propeller thrust and external environmental force. R ( ψ ) is a transformation matrix between coordinate systems. M ( v ) is mass matrix, C ( v ) is Coriolis centripetal matrix. D ( v ) is damping coefficient matrix and the following equalities hold

M ( v ) = [ m − X u 0 0 0 m − Y v m x G − Y r 0 m x G − N v I z − N τ ] (6)

C ( v ) = [ 0 0 − m ( x G + v ) + Y v v + Y r r ] 0 0 m u − X u u m ( x G r + v ) − Y v v − Y r r − m u + X u u 0 ]

D ( v ) = [ − X u 0 0 0 − Y v − Y r 0 − N v − N r ]

Define dynamic positioning formula:

z = R ( ψ ) v (7)

where z = [ z 1 , z 2 , z 3 ] T .

So the formula can become:

{ η ˙ = z z ˙ = R ( ψ ) M − 1 τ + δ η ˙ d = z d δ = R ( ψ ) s ( r ) v − R ( ψ ) M − 1 C ( v ) v − R ( ψ ) M − 1 D ( v ) v + R ( ψ ) M − 1 b (8)

where δ denotes unknown concentrated interference, and η d = [ x d , y d , ψ d ] denotes the desired position.

Assume that the concentrated interference δ is continuously differentiable, that satisfies ‖ δ ˙ ‖ ≤ H n , H n is Upper limit constant.

The fixed-time state observer obtains accurate information of speed and unknown disturbance in a comprehensive manner, so the observer can be designed as:

{ η ^ ˙ = z ^ − k 1 s i g α 1 ( η − η ^ ) − c 1 s i g β 1 ( η − η ^ ) z ^ ˙ = R ( ψ ) M − 1 τ + δ + k 2 s i g α 2 ( η − η ^ ) + c 2 s i g β 2 ( η − η ^ ) δ ^ ˙ = k 3 s i g α 3 ( η − η ^ ) + c 3 s i g β 3 ( η − η ^ ) + γ s i g n ( η − η ^ ) (9)

where η ^ , z ^ , δ ^ are estimated value, parameter α i = i α − ( i − 1 ) , β i = i β − ( i − 1 ) , i = 1 , 2 , 3 . Where α ∈ ( 1 − ε 1 , 1 ) , β ∈ ( 1 , 1 + ε 2 ) and ε 1 , ε 2 are sufficiently small positive real number, and H n < γ .

The state observer gain is Hurwitz matrix:

A = [ − k 1 1 0 − k 2 0 1 − k 3 0 0 ] (10)

k 1 > 0 , k 2 > 0 , k 3 > 0

The observation error of the observer converges to the origin within a fixed time, and convergence time is:

T i ≤ 1 k i , 1 1 1 − α i + 1 k i , 2 1 1 − β i (11)

The resulting estimation error e ˜ 1 = η − η ^ , e ˜ 2 = z − z ^ and e ˜ 3 = δ − δ ^ . Following equations hold:

{ e ˜ ˙ 1 = e ˜ 2 − k 1 s i g α 1 ( η − η ^ ) − c 1 s i g β 1 ( η − η ^ ) e ˜ ˙ 2 = e ˜ 3 − k 2 s i g α 2 ( η − η ^ ) − c 2 s i g β 2 ( η − η ^ ) e ˜ ˙ 3 = δ ˙ − k 3 s i g α 3 ( η − η ^ ) − c 3 s i g β 3 ( η − η ^ ) − γ s i g n ( η − η ^ ) (12)

If there are positive definite matrices P 1 and Q 1 make:

A 1 T P 1 + P 1 A 1 = − Q 1 (13)

where A 1 is Hurwitz matrix:

A 1 = [ − k 1 1 0 − k 2 0 1 − k 3 0 0 ] (14)

According to Lemma 1, we can conclude that the error converges to zero within a fixed time, and the boundary time is:

T f = λ max ( 2 − α i ) ( P 1 ) λ min ( Q 1 ) ( 1 − α i ) + λ min ( P 1 ) λ min ( Q 2 ) ( β i − 1 ) ρ (15)

where ρ is positive and ρ ≤ λ min ( P 1 )

When e i = 0 happen after T f , e i will remain zero. Then you can get e 1 = e ˙ 1 = 0 , e 2 = e 3 = 0 . In other words, for all values of t ≥ T 1 , there is a upper limit time T 1 .

e ˙ 3 = δ ˙ − γ s i g n ( η − η ^ ) = 0,         t ≥ T 1 (16)

In reality, it is impossible to strictly implement e 3 = 0 due to defects such as noise, sampling steps, delay and interference. Therefore, a small range of convergence region e 3 | τ is proposed, Allow to reach the convergence area e 3 in a short time.

T d = e 3 | τ γ − H n (17)

Therefore, the upper bound of the convergence time T u = T f + T d of the fixed-time extended state observer is obtained, completing the proof of the theorem.

Using backstepping technology to solve the system status tracking problem, the design is as follows:

Make s 1 = η − η d , therefore

s 1 = η ˙ − η ˙ d = z − η ˙ d (18)

Designing the virtual control rate:

z * = − k 1 s i g ( s 1 ) p − c 1 s i g ( s 1 ) q + z ^ (19)

where p, q, k 1 and c 1 are designing parameters.

We can get

s 2 = z − z * . (20)

Because s ˙ 2 = R ( ψ ) M − 1 τ + δ − z ˙ * , we can design control input:

τ = ( − k 2 s i g ( s 2 ) p − c 2 s i g ( s 2 ) q + z ˙ * − s 1 − δ ⌢ ) R ( ψ ) − 1 M (21)

where k 2 , c 2 designing parameters.

The system can be summarized as follows:

{ s ˙ 1 = − k 1 s i g ( s 1 ) p − c 1 s i g ( s 1 ) q + s 2 + z ⌢ − η ˙ d s ˙ 2 = − k 2 s i g ( s 2 ) p − c 2 s i g ( s 2 ) q + δ − s 1 − δ ⌢ (22)

As shown in the structural design shown in Figure 2, the impact of unknown interference on the ship is clearly reflected.

System Lyapunov function V = 1 2 s 1 2 + 1 2 s 2 2 .

We have:

V ˙ = s 1 s ˙ 1 + s 2 s ˙ 2       = − ( k 1 | s 1 | p + 1 + c 1 | s 1 | q + 1 ) − ( k 2 | s 2 | p + 1 + c 2 | s 2 | q + 1 ) + s 1 ( z ⌢ − η ˙ d ) + s 2 ( δ − δ ⌢ )       ≤ − ( k 1 | s 1 | p + 1 + c 1 | s 1 | q + 1 ) − ( k 2 | s 2 | p + 1 + c 2 | s 2 | q + 1 ) + | s 1 | | z ⌢ − η ˙ d | + | s 2 | | δ − δ ⌢ | (23)

According to the lemma 3:

{ | s 1 | | z ⌢ − η ˙ d | ≤ l p + 1 s 1 p + 1 p + 1 + l − ( q + 1 ) | z ⌢ − η ˙ d | q + 1 q + 1 | s 2 | | δ − δ ⌢ | ≤ h p + 1 s 2 p + 1 p + 1 + h − ( q + 1 ) | δ − δ ⌢ | q + 1 q + 1 (24)

where H > 0 Represents the upper limit of | z ˙ * | .

From the above formula:

V ˙ ≤ − ( k 1 − l p + 1 p + 1 ) | s 1 | p + 1 − c 1 | s 1 | q + 1 − ( k 2 − h p + 1 p + 1 ) | s 2 | p + 1 − c 2 | s 2 | q + 1         + l − ( q + 1 ) q + 1 | z ⌢ − η ˙ d | q + 1 + h − ( q + 1 ) q + 1 | δ − δ ⌢ | q + 1 (25)

Choosing design parameters such that

k 1 > l p + 1 p + 1 , c 1 > 0 , k 2 > h p + 1 p + 1 , c 2 > 0 . (26)

According to the lemma 4:

V ˙ ≤ − ζ 1 ( | s 1 | p + 1 + | s 2 | p + 1 ) − ζ 2 ( | s 1 | q + 1 + | s 2 | q + 1 ) + ζ 3       = − ζ 1 ( | s 1 2 | p + 1 2 + | s 2 2 | p + 1 2 ) − ζ 2 ( | s 1 2 | q + 1 2 + | s 2 2 | q + 1 2 ) + ζ 3       ≤ − 3 1 − p 2 V p + 1 2 − 3 1 − q 2 V q + 1 2 + ζ 3 (27)

where

{ ζ 1 = min { ( k 1 − l p + 1 p + 1 ) , ( k 2 − h p + 1 p + 1 ) } ζ 2 = min { c 1 , c 2 } ζ 3 = l − ( q + 1 ) q + 1 | z ⌢ − η ˙ d | q + 1 + h − ( q + 1 ) q + 1 | δ − δ ⌢ | q + 1 (28)

Because the global scope of | z ⌢ − η ˙ d | and | δ − δ ⌢ | are bounded, ζ 3 is bounded, so:

V ˙ ≤ − 3 1 − p 2 V p + 1 2 − 3 1 − q 2 V q + 1 2 + ζ 3 (29)

For all t ≥ max { T 1 , T 2 } cases, the limit range of the system can be calculated:

− 3 1 − p 2 V p + 1 2 − 3 1 − q 2 V q + 1 2 + ζ 3 = 0 (30)

So we can get the equations:

{ − 3 1 − p 2 V p + 1 2 + ζ 3 2 = 0 − 3 1 − q 2 V q + 1 2 + ζ 3 2 = 0 (31)

We have:

lim t → ∞ V ( t ) ≤ 2 min { ( ζ 3 2 × 3 1 − p 2 ) 2 1 − p , ( ζ 3 2 × 3 1 − q 2 ) 2 1 − q } (32)

T 3 ≤ 1 3 1 − p 2 ( p − 1 2 ) + 1 3 1 − q 2 ( q − 1 2 )

Because of the fixed-time observer, all state errors are limited and the formulas show that | z ⌢ − η ˙ d | and | δ − δ ⌢ | can converge to zero in finite times T 1 and T 2 . Therefore, the final settling time of the system is T = max { T 1 , T 2 , T 3 } .