高阶非线性系统输入时延跟踪控制：新误差补偿方法

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"这篇论文提出了一种针对带有输入时延的高阶非线性系统的新型误差补偿跟踪控制策略。在自适应神经网络控制框架下，该方法旨在解决由于非线性、输入延迟以及传统跟踪误差所带来的问题，如输出曲线滞后、大峰值误差和初始误差导致的稳定性问题。通过构建预测和补偿项、辅助信号以及阻尼项三个误差补偿项，论文作者尝试克服高阶非线性对控制精度的影响，并增强系统性能。然而，高阶不等式的放大可能会削弱误差补偿方案的效果。" 基于标题和描述，本文关注的核心知识点如下： 1. **高阶非线性系统**：这种系统具有多个相互作用的动态变量，其行为不能简单地用线性模型来描述。它们通常出现在复杂的工程领域，如航空航天、电力系统和机械工程。 2. **输入时延**：系统中的输入信号到达执行器并产生效果之间存在时间延迟，这可能会影响控制性能，导致稳定性问题和跟踪精度下降。 3. **自适应神经网络控制**：这是一种智能控制策略，利用神经网络的非线性映射能力和自我学习特性来适应系统参数的变化和不确定性，以实现精确控制。 4. **误差补偿方法**：为了改善控制系统性能，通过预测和补偿误差来减少跟踪误差，提高系统的跟踪精度。文章提出的新型误差补偿控制策略包括预测和补偿项、辅助信号以及阻尼项。 5. **控制器设计**：设计一个能够有效处理高阶非线性和输入时延的控制器是关键，目标是确保系统稳定且能准确跟踪期望的轨迹。 6. **跟踪误差问题**：传统的跟踪误差控制面临输出曲线滞后、峰值误差大以及初始误差可能导致系统不稳定的问题。文章提出的补偿方法旨在解决这些问题。 7. **不等式放大与控制精度**：高阶非线性系统的特性可能导致不等式的放大，从而降低控制精度。论文探讨了如何在这种情况下保持控制策略的有效性。这篇研究聚焦于在复杂环境中，如何利用自适应神经网络控制和创新的误差补偿技术，优化带有输入时延的高阶非线性系统的跟踪控制性能，同时克服由高阶非线性引起的控制难题。

SHI et al.: NOVEL ERROR-COMPENSATION CONTROL FOR A CLASS OF HIGH-ORDER NONLINEAR SYSTEMS 4079

where Z =[Z

,...,Z

]

∈ 

⊂ R

is the input vector,

W =[w

,...,w

]

∈ R

is the weight vector, l denotes the

NN node number, and (Z) =[φ

(Z),...,φ

(Z)]

∈ R

with φ

(Z) being chosen as the commonly used Gaussian

functions as follows:

(Z) = exp



−(Z − μ

)

(Z − μ

)



, i = 1, 2, ...., l (4)

where μ

=[μ

,μ

,...,μ

]

is the center of the receptive

ﬁeld and ζ is the width of the Gaussian functions.

As proven in [32], by choosing enough neural nodes,

the neural networks (3) can approximate any continuous

function over a compact set 

⊂ R

with arbitrary accuracy

f (Z) = W

∗T

(Z) + δ(Z), ∀Z ∈ 

(5)

where W

∗

is the ideal constant weight vector, and δ(Z) is the

approximation error satisfying |δ(Z)|≤ε with ε>0being

an unknown constant.

For Gaussian RBFNNs, an upper bound on the norm of the

basis function vector (Z) isgiveninLemma3.

Lemma3[41]:Consider the Gaussian RBF networks (3)

and (4). Let ρ = (1/2) min

i=j

μ

− μ

, then an upper

bound of (Z) is obtained as

(Z)≤

∞



k=0

3q(k + 2)

q−1

−2ρ

/ζ

:= 

where  is a positive constant, k = 0, 1,...,+∞, ζ is the

width of the Gaussian functions, and q is the dimension of

input Z.

Remark 1: It has been proven in [41] and [42] that  is

a limited value and independent of NN input Z and node

number l.

III. M

AIN RESULTS

To quantify the control objective, the tracking error denoted

by r ∈ Risdeﬁnedas

r = y − y

. (6)

To tackle these difﬁculties that the phenomenon that output

curve always lags behind the desired trajectory, the instability

caused by the initial value of the modiﬁed tracking error,

and some big peak tracking errors, two compensation terms

˙y

and e

and an auxiliary signal ς are introduced to modify

the tracking error r. The modiﬁed tracking error denoted by

is deﬁned as

= r − e

˙y

− ς − e

. (7)

Next, the deﬁnitions of e

˙y

, ς,ande

will be given in

Sections III-A–III-C.

A. Tracking Lag Compensation

From [29] and [39], it can be ﬁnd that good tracking

performances have been obtained by their simulation results,

and however, there exist signiﬁcant lags between their output

curves and desired trajectories, respectively. If the variation

tendency of the desired trajectory can be obtained in advance,

the lag phenomenon will be eliminated by a precompensation

method. As is known, the time derivative of the desired trajec-

tory can be used to predict its variation tendency. Therefore,

we deﬁne the tracking lag compensation (TLC) term e

˙y

= ι ˙y

(8)

where ι is a positive design parameter.

B. Peak Tracking Error Compensation

Although the tracking lag phenomenon is compensated

in Section III-A, some big peak errors still exist in the

tracking error curve. To weaken these big peak tracking errors,

an auxiliary system is constructed as

˙ς =−κς − hr,ς(0) = 0(9)

where ς ∈ R denotes the auxiliary signal and is used as peak

tracking error-compensation (PTEC) term, κ>0andh > 0

are design parameters satisfying κ +h >(h

/2) +(1/2),and

r = r −r

denotes the difference between the actual tracking

error r and the desired tracking error r

deﬁned as

⎧

⎪

⎨

⎪

⎩

max

, r > r

max

r, −r

max

≤ r ≤ r

max

−r

max

, r < −r

max

where r

max

> 0 is the upper bound of the desired tracking

error to be designed.

Remark 2: Since there is no aprioriknowledge of the

derivative of r, the tracking error cannot be modiﬁed by r.

It can be observed from (9) that ς represents a ﬁltered version

of r and tends to zero if |r|≤r

max

,andfor|r| > r

max

, ς

becomes nonzero. From the view of the solution of (9), ς is a

convolution of an exponential term with r and converges to

zero if and only if r becomes zero. Hence, by introducing

ς to (7), we can weaken these big peak tracking errors.

Remark 3: The value of r

max

decides whether compensation

is required for the actual tracking error. To be illustrated, with

the help of the auxiliary system (9) and auxiliary signal ς,

all the error values r satisfying |r| > r

max

will be weak-

ened signiﬁcantly but would not be narrowed to the range

of [−r

max

, r

max

C. Initial Tracking Error Compensation

To deal with the difﬁculty that our considered closed-loop

system is sensitive to the initial value of the modiﬁed tracking

error e

, an initial tracking error-compensation (ITEC) term

denoted by e

is constructed as

= e

−λt

(0) − e

˙y

(0)) = β

−λt

where λ is a positive design parameter, β

= z

(0) − e

˙y

(0),

and z

(0) = x

(0).

Remark 4: From (7), it can be ﬁnd that if the initial

values of TLC term e

˙y

and state x

(0) are nonzero, then,

there may exist a big initial value e

(0). In the simulation

experiment, the nonzero initial error e

(0) can easily lead to

the instability of the closed-loop system, and the experiment

can be implemented only for the case that e

(0) is in a quite

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