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本文档为一篇研究论文，标题为《Event-Driven Nonlinear Discounted Optimal Regulation Involving a Power System Application》，涉及的主要内容是通过神经网络近似架构，在事件驱动的自适应批评框架下处理非线性折扣最优调控问题。文章的主旨是采用改进的学习算法，使得通过训练神经网络得到事件驱动的最优控制律，同时文章还包含了稳定性保证和仿真说明。此外，不需要在实施过程中采用初始稳定控制策略，且闭环系统被构建为一个脉冲模型。通过使用Lyapunov方法来解决相关稳定性问题，并且通过仿真研究，包括对电力系统的应用，来验证本文设计方法的有效性。从文档内容中可以提取出以下知识点： 1. 非线性最优调控：研究在电力系统中，如何进行非线性最优调控，以达到优化系统性能的目的。这通常涉及到复杂的数学问题，如汉密尔顿-雅可比-贝尔曼方程求解。 2. 事件驱动的自适应批评框架（Event-Driven Adaptive Critic Framework）：这种框架是为了优化控制律而设计的，特别是在遇到系统状态变化等“事件”时，能够进行即时的控制调整。 3. 神经网络近似架构（Neural Network Approximation Architecture）：由于“维数的诅咒”问题，即随着系统维数的增加，计算最优控制策略变得非常困难。利用神经网络的逼近能力，可以对非线性最优调控问题进行迭代计算。 4. 改进的学习算法（Improved Learning Algorithm）：为了获得事件驱动的最优控制律，文中提出了一种改进的学习算法，使系统能够在训练神经网络的过程中进行自我调整。 5. 稳定性保证（Stability Guarantee）：研究中特别强调了系统的稳定性问题，并通过Lyapunov方法来保证系统的稳定性。 6. 电力系统应用（Power System Application）：文档中提到的仿真研究包括了电力系统应用，说明了该理论和技术在实际电力系统调控中的适用性和有效性。 ***apunov方法（Lyapunov Approach）：在闭环系统稳定性分析中，Lyapunov方法提供了一种理论框架，用以分析和证明控制系统的稳定性。 8. 脉冲模型（Impulsive Model）：闭环系统被描述为脉冲模型，意味着系统行为在某些特定时刻会经历急剧的变化。 9. Hamilton–Jacobi–Bellman方程（Hamilton–Jacobi–Bellman Equation）：这是动态规划中用于解决最优控制问题的基本方程，尤其适用于非线性系统的最优调控问题。 10. 国家自然科学基金、北京自然科学基金、美国国家科学基金会等的资金支持（NSFC Funding）：研究得到了多个国家和机构的资助，表明其研究的重要性以及被学术界认可的程度。以上知识点反映了在电力系统应用中事件驱动非线性折扣最优调控的重要性和应用前景，同时也展示了神经网络和动态规划在复杂系统控制中应用的深度与广度。研究者们通过采用先进的算法和理论，为解决实际工程问题提供了新的思路和方法。

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 64, NO. 10, OCTOBER 2017 8177

Event-Driven Nonlinear Discounted Optimal

Regulation Involving a Power System Application

Ding Wang, Member, IEEE, Haibo He, Senior Member, IEEE, Xiangnan Zhong, Student Member, IEEE,

and Derong Liu, Fellow, IEEE

Abstract—By employing neural network approximation

architecture, the nonlinear discounted optimal regulation

is handled under event-driven adaptive critic framework.

The main idea lies in adopting an improved learning algo-

rithm, so that the event-driven discounted optimal control

law can be derived via training a neural network. The stabil-

ity guarantee and simulation illustration are also included.

It is highlighted that the initial stabilizing control policy is

not required during the implementation process with the

combined learning rule. Moreover, the closed-loop system

is formulated as an impulsive model. Then, the related sta-

bility issue is addressed by using the Lyapunov approach.

The simulation studies, including an application to a power

system, are also conducted to verify the effectiveness of

the present design method.

Index Terms—Adaptive/approximate dynamic program-

ming, approximation, discount factor, event-driven control,

neural networks, optimal regulation, power system.

I. INTRODUCTION

HERE exists a severe difﬁculty called the “curse of dimen-

sionality” when solving the Hamilton–Jacobi–Bellman

equation during nonlinear optimal regulation design. Hence,

a series of iterative methods have been developed to tackle

the optimal control problems. Among them, neural networks

have served as an important function approximation archi-

tecture to perform the iterative calculation. In fact, neural

Manuscript received December 19, 2016; revised February 12, 2017

and March 18, 2017; accepted April 4, 2017. Date of publication April 27,

2017; date of c urrent version September 11, 2017. This work was sup-

ported in part by the National Natural Science Foundation of China under

Grant U1501251, Grant 61533017, Grant 61233001, Grant 51529701,

and Grant 61520106009, in part by the Beijing Natural Science Foun-

dation under Grant 4162065, in part by the U.S. National Science Foun-

dation under Grant ECCS 1053717, and in part by the Early Career

Development Award of SKLMCCS. (Corresponding author: Haibo He.)

D. Wang is with The State Key Laboratory of Management and Con-

trol for Complex Systems, Institute of Automation, Chinese Academy of

Sciences, Beijing 100190, China, and also with the School of Computer

and Control Engineering, University of Chinese Academy of S ciences,

Beijing 100049, China. This research was conducted when D. Wang

was a Visiting Scholar at the Department of Electrical, Computer, and

Biomedical Engineering, University of Rhode Island, Kingston, RI 02881

USA (e-mail: ding.wang@ia.ac.cn).

H. He and X. Zhong are with the Department of Electrical, Computer

and Biomedical Engineering, University of Rhode Island, Kingston, RI

02881 USA (e-mail: he@ele.uri.edu; xzhong@ele.uri.edu).

D. Liu is with the School of Automation, Guangdong University of

Technology, Guangzhou 510006, China (e-mail: derongliu@foxmail.

com).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TIE.2017.2698377

networks are widely adopted to perform learning and ap-

proximation abilities for practical feedback stabilization prob-

lems [1]–[5], especially for optimal control design [5]. Re-

cently, a signiﬁcant breakthrough has been made in adaptive

boundary control design and stability analysis for inﬁnite di-

mensional systems with constraints [2], [3]. Based on neural

networks, adaptive/approximate dynamic programming [6], [7]

is regarded as a key technique to design optimal controllers

adaptively and forward-in-time. In the last decade, extensive

developments of adaptive/approximate dynamic programming

have been achieved in aspect of optimal control for discrete-time

systems [8]–[12] and continuous-time systems [13]–[17]. For

example, Modares and Lewis [14] studied the linear quadratic

tracking control with discounted cost function of partially un-

known continuous-time systems based on the reinforcement

learning technique. One also has observed the extensions of

regulation design to various intelligent control strategies for

complex systems, such as decentralized control [18], consen-

sus control [19], H

∞

control [20], and tracking control [21].

However, the aforementioned results are mainly obtained based

on time-driven design, which would inevitably cause frequent

adjustments of the actuator state and might result in serious en-

ergy consumption. Thus, conducting time/event transformation

to achieve event-driven design has become a new avenue for the

feedback control community.

With the rapid development of network-based systems, more

and more control loops are closed through some communication

media. The increasing interest in saving the computational load

of networked control systems results in an extensive attention

with respect to the event-triggering mechanism. Under the gen-

eral framework of event-driven approaches, the actuators are

updated only when certain conditions are violated to guarantee

both stability performance and control efﬁciency of the target

plants. Recently, the event-driven adaptive critic control has pro-

vided a new channel to implement nonlinear intelligent optimal

stabilization [22]–[26]. For example, Sahoo et al. [22] presented

a novel approximation-based event-triggered control scheme

of multi-input multioutput unknown nonlinear continuous-time

systems in afﬁne form. Vamvoudakis et al. [23] proposed an

event-driven tracking control algorithm for nonlinear systems

with an inﬁnite horizon discounted cost function. Under the

new framework, the designed controller is only updated when

an event is triggered, thereby reducing the computational burden

of two processes including neural network learning and adaptive

optimal control.

See http://www.ieee.org/publications

standards/publications/rights/index.html for more information.

8178 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 64, NO. 10, OCTOBER 2017

Nevertheless, to the best of our knowledge, the existing work

of event-based adaptive critic designs is mainly conducted

for optimal regulation [24] and robust H

∞

control [25], [26]

with undiscounted cost function. Besides, an initial stabilizing

control policy is a common requirement of the literature

[23]–[26], which narrows the application of event-based design

to a certain extent. Motivated by the above consideration and

the results of [27]–[29], in this paper, the authors aim at using

the neural network approximator to derive the event-based

nonlinear discounted optimal control law without depending

on the special initial condition. In summary, it improves the

previous results [13], [15]–[17], [22]–[26], [28] in terms of

introducing the discounted cost function and reinforcing the

learning mechanism, under an event-based environment. The

rest of this paper is organized as follows. A brief description of

the nonlinear discounted optimal control problem is provided

in Section II. Neural network approximation for event-based

nonlinear discounted optimal control is developed with sta-

bility analysis in Section III. The application studies and the

concluding remarks are presented in Sections IV and V.

A list of notations used in the paper is given. R represents

the set of all real numbers. R

is the Euclidean space of all

n-dimensional real vectors. R

n×m

is the space of all n × m real

matrices. ·denotes the vector 2-norm of a vector in R

the matrix 2-norm of a matrix in R

n×m

. I

represents the n ×

n identity matrix. λ

max

(·) and λ

min

(·) calculate the maximal

and minimal eigenvalues of a matrix, respectively. Let Ω be a

compact subset of R

and A (Ω) be the set of admissible control

policies (deﬁned in [13]) on Ω. N = {0, 1, 2,...} denotes the

set of all nonnegative integers. In addition, the superscript “T”

is used to represent the transpose operation and ∇(·)  ∂(·)/∂x

is employed to denote the gradient operator.

II. P

ROBLEM STATEMENT

Dynamical systems with input-afﬁne form are common in the

literature and engineering. Consider a class of continuous-time

afﬁne nonlinear systems described by

˙x(t)=f(x(t)) + g(x(t))u(t) (1)

where x(t) ∈ Ω ⊂ R

is the state vector, u(t) ∈ Ω

⊂ R

the control vector, and f(·) and g(·) are differentiable in their

arguments with f(0) = 0. Let the initial state at t =0be x(0) =

and x =0be the equilibrium point of the controlled plant.

In general, the system (1) is assumed to be controllable. In

the optimal regulation design, it is always required to derive a

state feedback control law u(x) with respect to a speciﬁed cost

function. Choose two positive deﬁnite matrices Q ∈ R

n×n

and

R ∈ R

m ×m

and let

U(x(τ),u(τ)) = x

(τ)Qx(τ )+u

(τ)Ru(τ)

represent the utility function. Then, similar to [14], [23], and

[29], one can deﬁne the inﬁnite horizon discounted cost function

J(x(t),u)=



∞

−γ (τ −t)

U(x(τ),u(τ))dτ (2)

where γ>0 is a discount factor. Note that with the discount

factor, one can modulate the convergence speed of the regulation

design and reduce the value of the optimal cost. For simplicity,

the cost J(x(t),u) is written as J(x(t)) or J(x) in the sequel.

What one shall concern is the cost function starting from t =0,

which is denoted as J(x(0)) = J(x

For an admissible control policy u ∈ A (Ω), if the related

cost function (2) is continuously differentiable, one can derive

a limit-based expression with three parts

lim

T →0



t+ T

−γ (τ −t)

U(x(τ),u(τ))dτ



 

Part 1

+ lim

T →0

J(x(t + T )) − J(x(t))



 

Part 2

+ lim

T →0



−γT

− 1



J(x(t + T ))



 

Part 3

=0.

Then, one derives the inﬁnitesimal version called the nonlinear

Lyapunov equation as follows:

0=U(x, u)



 

Part 1

+(∇J(x))

[f(x)+g(x)u]



 

Part 2

− γJ(x)





Part 3

with J(0) = 0. Deﬁne the Hamiltonian of system (1) as

H(x, u, ∇J(x)) = U (x, u)+(∇J(x))

[f(x)

+ g(x)u] − γJ(x).

If one recalls the Bellman’s well-known optimality principle,

then the optimal cost function

∗

(x) = min

u∈A (Ω)



∞

−γ (τ −t)

U(x(τ),u(τ))dτ

satisﬁes the so-called Hamilton–Jacobi–Bellman equation

min

H(x, u, ∇J

∗

(x)) = 0.

Using the results of [13] and [25], the optimal control law is

∗

(x)=−

−1

(x)∇J

∗

(x). (3)

Considering the optimal controller (3), the Hamilton–Jacobi–

Bellman equation turns to be the form

0= U(x, u

∗

)+(∇J

∗

(x))

[f(x)+g(x)u

∗

] − γJ

∗

(x)

= x

Qx −

(∇J

∗

(x))

g(x)R

−1

(x)∇J

∗

(x)

+(∇J

∗

(x))

f(x) − γJ

∗

(x),J

∗

(0) = 0. (4)

Note that (4) is known as the classical time-based Hamilton–

Jacobi–Bellman equation, that is, H(x, u

∗

, ∇J

∗

(x)) = 0, which

is difﬁcult to address in theory. This fact inspires us to devise an

approximate control strategy to overcome the difﬁculty. Next,

we investigate the event-driven adaptive critic design.

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