自主水下航行器的迭代自适应动态规划轨迹跟踪控制算法
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本文档标题为《基于迭代自适应动态规划的自主水下航行器轨迹跟踪控制》(Nonlinear Trajectory-Tracking Control for Autonomous Underwater Vehicle Based on Iterative Adaptive Dynamic Programming)，发表在《智能与模糊系统》(Journal of Intelligent & Fuzzy Systems)第37卷(2019年)的4205-4215页，DOI为10.3233/JIFS-190294，由IOSPress出版。研究者是厦门大学航空航天工程学院的高峰、刘丽君和郑宇教授。
文章的核心内容围绕自主水下航行器的最优轨迹跟踪控制问题展开。作者提出了一种迭代自适应动态编程算法，通过结合两种迭代过程，即i-iteration（迭代控制策略）和j-iteration（迭代价值函数），来解决这一复杂问题。i-iteration的目标是获取逐次优化的轨迹跟踪控制律，而j-iteration则在i-iteration过程中负责计算相应的价值函数。这种方法将原始的最优跟踪控制问题转化为一个最优调节问题，通过系统转换实现。
算法的关键在于采用策略迭代和值迭代的交互策略，这是一种自适应动态规划的策略，旨在找到最优化的控制策略和状态反馈函数，从而确保系统的稳定性和效率。神经网络技术被应用到算法的实现中，以处理非线性系统的复杂性，并提高算法的适应性和泛化能力。
作者详细分析了算法的收敛性和最优性特性，证明了该方法在解决自主水下航行器轨迹跟踪控制问题上的有效性。通过模拟例证，展示了迭代自适应动态编程算法在实际应用场景中的优良性能，能够有效地追踪预设的轨迹，同时保持系统的鲁棒性和稳定性。
这篇论文为自主水下航行器的轨迹跟踪控制提供了一种创新且高效的解决方案，对于推进无人水下技术的发展以及实时控制系统的优化设计具有重要意义。通过阅读和理解这份文档，读者可以深入理解自适应动态规划在复杂控制系统中的应用，以及如何通过迭代策略提高控制性能。
G. Che et al. / Nonlinear trajectory-tracking control for AUV base on ADP 4207
J(η) =
J
1
(η)0
3×3
0
3×3
J
2
(η)
(2)
J
1
(η) =
⎛
⎜
⎝
cos ψ cos θ
11
12
sin ψ cos θ
21
22
− sin θ cos θ sin φ cos θ cos φ
⎞
⎟
⎠
(3)
J
2
(η) =
⎛
⎜
⎝
1 tan θ sin φ tan θ sin φ
0 cos φ − sin φ
0 sin φ sec θ cos φ sec θ
⎞
⎟
⎠
(4)
where
11
= cos ψ sin θ sin φ − sin ψ cos φ;
12
= cos ψ sin θ cos φ + sin ψ sin φ;
21
= sin ψ sin θ sin φ + cos ψ cos φ;
22
=
sin ψ sin θ cos φ − cos ψ sin φ.
The dynamic model system is established via laws
of Newton:
[M]
˙
ξ + [C(ξ)]ξ + [D(ξ)]ξ + g(η) = τ (5)
where [M] ∈
6×6
is the inertia matrix and its inverse
[M]
−1
;[C(ξ)] ∈
6×6
is the Coriolis and centripetal
matrix; [D(ξ)] ∈
6×6
is the damping matrix; g(η) ∈
6
is the gravity and buoyancy forces vector; τ ∈
6
is the generalized thrust force vector.
The equation (5) can be transformed as follow:
˙
ξ = [M]
−1
([C(ξ)] − [D(ξ)]ξ − g(η) + τ) (6)
2.2. Problem formulation
We suppose that the sample time is very short.
According to equation (6), the discrete-time dynamic
system is described as follows:
ξ(k + 1) = f (ξ(k)) + ι(ξ(k))u(ξ(k)) (7)
where u(ξ(k))is the system control input. For opti-
mal tracking control problem, the control objective
is to find an optimal control u
∗
(ξ(k)), so as to make
equation (7) track the desired trajectory ξ
d
(k). For
simplicity, u(ξ(k)) is replaced by u(k).
The tracking error is defined as:
e(k) = ξ(k) − ξ
d
(8)
The control input error is described as :
ν(k) = u(k) − u
d
(k) (9)
u
d
(k) = ι
−1
(ξ
d
(k))(ξ
d
(k + 1) − f (ξ
d
(k)) − [M]
−1
g)
(10)
where u
d
(k) is the expected control input and
introduced for analytical purpose. By substituting
equations (8), (9) and (10), the new system is obtained
as follows:
e(k + 1) = f (e(k) + ξ
d
(k)) + ι(e(k) + ξ
d
(k))
ι
−1
(ξ
d
(k))(ξ
d
(k + 1) − f (ξ(k)))
−ξ
d
(k + 1) + ι(e(k) + ξ
d
(k))ν(k)
(11)
where u
d
(k) and u
d
(k + 1) are desired control vec-
tors.
The equation (11) can be represented as:
e(k + 1) = F (e(k),ν(k)) (12)
where e(k) is the sate vector and ι(k) is the control
vector.
According to Bellman’s optimality principle, the
control performance index function
J
∗
(e(k)) = min
ν(k)
∞
i=k
γ
i−k
U(e(i),ν(i)) (13)
where U(e(k),ι(k)) = e
T
(k)Qe(k) + ν
T
(k)Rν(k)is
the utility function,γ is the discount factor with 0 <
γ ≤ 1; and Q and R are symmetric and positive-
define matrices.
In other words,J
∗
(e(k)) satisfies the discrete-time
HJB equation. Therefore, the optimal control law can
be expressed as:
υ
∗
(k) = arg min
ν(k)
{U(e(k),ν(k)) + γJ
∗
(e(k + 1))}
(14)
Bellman principle yields a backwards-in-time pro-
cedure for solving the optimal control problem,
because that must know the optimal policy at time
k + 1 according to equation (14) that used to deter-
mine the optimal policy at time k. However, it
always causes that it is often computationally unten-
able to run true dynamic programming due to
the backward numerical process required for its
solutions. In order to overcome this difficulty, approx-
imate the performance index function is proposed.
sectionTrajectory-tracking control based on ADP
algorithm
2.3. Derivative of policy iterative ADP algorithm
In this section, the policy iterative ADP algorithm
is presented, and the value function and control law
are updated by iterations. First of all, we start with an
initial admissible control law
ˆ
ν
0
(k), and let
ˆ
J
0
(e(k +
1)) satisfy the HJB equation:
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