H∞ Control of Network Control System for Singular Plant

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Information Technology and Control 2018/1/47 140

H∞ Control of Network

Control System for

Singular Plant

ITC 1/47

Journal of Information Technology and Control

Vol. 47 / No. 1 / 2018 pp. 140-150

DOI 10.5755/j01.itc.47.1.19982 © Kaunas University of Technology

H∞ Control of Network Control System for Singular Plant

Received 2018/01/19 Accepted after revision 2018/02/05

http://dx.doi.org/10.5755/j01.itc.47.1.19982

Wei Wei

School of Computer Science and Engineering, Xi’an University of Technology, Xi’an 710048, China; Qilu University of Technology (Shandong Academy of Sciences); Shandong provincial Key Laboratory of Computer Network; e-mail: [email protected]

Xunli Fan

School of Information Science & Technology, Northwest University, Xi’an 710127, China; e-mail: [email protected]

Marcin Woźniak

Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland; e-mail: [email protected]

Houbing Song

Department of Electrical, Computer, Software, and Systems Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114-3900, USA; e-mail: [email protected]

Wei-Li

The Center for Distributed and High Performance Computing, School of Information Technologies, The University of Sydney, Sydney, Australia; e-mail: [email protected]

Ye Li

School of Computer Science and Engineering, Xi’an University of Technology, Xi’an 710048, China; Qilu University of Technology (Shandong Academy of Sciences); Shandong provincial Key Laboratory of Computer Network

Peiyi Shen

National school of Software, Xidian University, Xi’an 710071, P.R.China, China; e-mail: [email protected]

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141 Information Technology and Control 2018/1/47

This paper investigates H∞ control method for a class of Singular Network Control Systems (SNCS) based on

singular plant. Considering the network delay, external disturbance, impulse behavior and structural

instabil-ity of singular plant, the H∞ control of SNCS with state feedback and dynamic output feedback are investigated

respectively by approach of Linear Matrix Inequality (LMI). The existence of the H∞ control law, the solving of

the H∞ control law and the disturbance degree are discussed in the following sections of the paper. Simulation

results illustrate the effectiveness and feasibility of the given approach.

KEYWORDS:Singular Network Controls, H∞ control, Linear Matrix Inequality, Network delay.

1. Introduction

Network Control System (NCS) is a distributed and a real-time feedback control system where the sys-tem node situated at different geographical position exchanges state information and control informa-tion with the controller through a communicainforma-tion network [20]. Network bandwidth and restraint of communication mechanism such as network delay and data packet loss exist typically in network com-munication channel, which makes NCS loses invari-ability, integrality, causality and certainty [18], and due to this fact the study of NCS is more complicated and challenging.

The traditional control theories and methods are not suitable for NCS, which makes rapid development over the past few years. Since the end of the last centu-ry, the research of NCS experiences the process from simple to complex, from single to comprehensive and from special to general. A large number of results have been reported, for instance, system complexity analy-sis [23], quantized dynamic output feedback control [3], observer-based controller design [25], state

esti-mation and stabilization [6], H∞ control method [24],

fault-tolerant control [4], guaranteed cost control [8], co-design [10].

The results in the existing literature are focused on linear system. However, the study of SNCS based on singular system has not been addressed intensively. The dynamics of singular system is quite different from normal linear system and have many character-istics such as no causality, no solution, no uniqueness and structure instability, etc. [22]. In fact, the research on SNCS is still in the primary stage, and the existing results are limited to system modeling, stability anal-ysis and control method [1-2, 5, 11-12, 14-17].

This paper aims to study the stabilization and H∞

con-trol method for a class of SNCS subject to the double characteristics of a singular systems and NCS.

Net-work delay, input disturbance of limited energy, and impulse behavior are taken into consideration. The

H∞ control method of SNCS with state feedback and

dynamic output feedback are presented respectively

by means of LMI. The existence H∞ control law, H∞

control law approach and disturbance attenuation degree in different feedback are presented. Finally, a simulation is given to illustrate the effectiveness of the proposed method [7, 9, 13, 19, 21].

Schur Formula and Schur Complement Lemma Let

Let _A_∈_Rr×r_,_B_∈_Rr×(n−r)_,_C_∈_R(n−r)×r_,_D_∈_R(n−r)×(n−r)_，_and

_A

_{be an invertible matrix. The Schur formula has the}

following three forms：

(i) �𝐴𝐴𝐴𝐴_{0 D}₋_CA𝐵𝐵𝐵𝐵−1_B�=�_{−𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴}𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟)−1 _{𝐼𝐼𝐼𝐼} 0

(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� �𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷�, (ii) �𝐴𝐴𝐴𝐴_{𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷 − 𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴}0 −1_{𝐵𝐵𝐵𝐵�}=�𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵_{𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷�} �𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) −𝐴𝐴𝐴𝐴

−1_{𝐵𝐵𝐵𝐵} 0 𝐼𝐼𝐼𝐼(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)�,

, and A be an invertible matrix. The Schur formula has the following three forms:

(i)

Let

A

_∈

R

r×r

,

B

_∈

R

r×(n−r)

,

C

_∈

R

(n−r)×r

,

D

_∈

R

(n−r)×(n−r)

，and

A

be an invertible matrix. The Schur formula has the following three forms：

(i) �𝐴𝐴𝐴𝐴_{0 D}₋_CA𝐵𝐵𝐵𝐵 −1_B�=�

𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) 0

−𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴−1 _{𝐼𝐼𝐼𝐼}

(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� �𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷�,

(ii) �𝐴𝐴𝐴𝐴_{𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷 − 𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴}0 −1_{𝐵𝐵𝐵𝐵�}=�𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵_{𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷�} �𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) −𝐴𝐴𝐴𝐴 −1_{𝐵𝐵𝐵𝐵} 0 𝐼𝐼𝐼𝐼(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟) �, (ii)

Let

A

_∈

R

r×r

,

B

_∈

R

r×(n−r)

,

C

_∈

R

(n−r)×r

,

D

_∈

R

(n−r)×(n−r)

，and

A

be an invertible matrix. The Schur formula has the following three forms：

(i) �𝐴𝐴𝐴𝐴_{0 D}₋_CA𝐵𝐵𝐵𝐵 −1_B�=�

𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) 0 −𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴−1 _{𝐼𝐼𝐼𝐼}

(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� �𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷�, (ii) �𝐴𝐴𝐴𝐴_{𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷 − 𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴}0 −1_{𝐵𝐵𝐵𝐵�}=�𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵_{𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷�} �𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) −𝐴𝐴𝐴𝐴

−1_{𝐵𝐵𝐵𝐵}

0 𝐼𝐼𝐼𝐼(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)�, (iii)

(iii) �𝐴𝐴𝐴𝐴₀ _{𝐷𝐷𝐷𝐷 − 𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴}0 −1_{𝐵𝐵𝐵𝐵�}=�

(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� �𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷� �

𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) −𝐴𝐴𝐴𝐴−1𝐵𝐵𝐵𝐵 0 𝐼𝐼𝐼𝐼(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟) �,

in which, e.g., 𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) denotes identity matrix of the size 𝑟𝑟𝑟𝑟×𝑟𝑟𝑟𝑟. The other identity matrices are of sizes that fit the Schur

lemma.

Schur Complement Lemma:

Let Q∈𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟×𝑟𝑟𝑟𝑟_{, S}_{∈ 𝑅𝑅𝑅𝑅}𝑟𝑟𝑟𝑟×(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)_{, R}_{∈ 𝑅𝑅𝑅𝑅}(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)×(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)_and_{� 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆}

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}�<0. Then if and only if

(i) R<0

，_Q-_{𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅}−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇<0

or

(ii) Q<0

，R-_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{𝑄𝑄𝑄𝑄}−1_{𝑆𝑆𝑆𝑆}<0.

Proof：Since the contract transform does not change the matrix positive definition, we first prove that _{� 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆}

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is

equivalent to R<0，Q-_{𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅}−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_.

If Q<0, through the Schur formula, we have the following: �𝐼𝐼𝐼𝐼(₀𝑟𝑟𝑟𝑟) −𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅_{𝐼𝐼𝐼𝐼} −1

(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 𝑅𝑅𝑅𝑅� �

−𝑅𝑅𝑅𝑅−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇 _{𝐼𝐼𝐼𝐼}

(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)�=�

Q− 𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇 ₀

0 𝑅𝑅𝑅𝑅�.

Correspondingly, � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is equivalent to R<0，Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇< 0.

[End of Proof]

Notes: Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{is the Schur complement of Q.}

PROBLEM DESCRIPTION

The singular sample of the network control system is described as presented in [1].

Figure 1 General Structure of SNCS

In Figure 1, u, w and z are control input, measurement state or measurement output, input external disturbance and

expectation output, and τ is network-induced delay. The aim of positioning is to guarantee stable running of the system

independently of any external disturbances so that the expected output of the system is not affected.

For singular plant, its state response contains not only the exponential term as normal systems, but also the pulse term and input derivative item, which will make the whole system has a pulse behavior. The pulse behavior decreases not (iii) �𝐴𝐴𝐴𝐴₀ _{𝐷𝐷𝐷𝐷 − 𝐶𝐶𝐶𝐶𝐴𝐴𝐴𝐴}0 −1_{𝐵𝐵𝐵𝐵�}=�

(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� �𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷� �

𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) −𝐴𝐴𝐴𝐴−1𝐵𝐵𝐵𝐵 0 𝐼𝐼𝐼𝐼(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟) �,

in which, e.g., 𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) denotes identity matrix of the size 𝑟𝑟𝑟𝑟×𝑟𝑟𝑟𝑟. The other identity matrices are of sizes that fit the Schur

lemma.

Let Q∈𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟×𝑟𝑟𝑟𝑟_{, S}_{∈ 𝑅𝑅𝑅𝑅}𝑟𝑟𝑟𝑟×(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)_{, R}_{∈ 𝑅𝑅𝑅𝑅}(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)×(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)_and_{� 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆}

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}�<0. Then if and only if

(i) R<0

，_Q-_{𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅}−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇<0

or

(ii) Q<0

，R-_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{𝑄𝑄𝑄𝑄}−1_{𝑆𝑆𝑆𝑆}<0.

Proof：Since the contract transform does not change the matrix positive definition, we first prove that _{� 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆}

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is

equivalent to R<0，Q-_{𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅}−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_.

If Q<0, through the Schur formula, we have the following: �𝐼𝐼𝐼𝐼(₀𝑟𝑟𝑟𝑟) −𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅_{𝐼𝐼𝐼𝐼} −1

Correspondingly, � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is equivalent to R<0，Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇< 0.

[End of Proof]

Notes: Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{is the Schur complement of Q.}

For singular plant, its state response contains not only the exponential term as normal systems, but also the pulse term and input derivative item, which will make the whole system has a pulse behavior. The pulse behavior decreases not

in which, e.g., I(r)denotes identity matrix of the size

r × r. The other identity matrices are of sizes that fit the Schur lemma.

Let Q∈ Rr × r_{, S}_∈_Rr × (n–r) _{, R}_∈_R (n–r) × (n–r) _and_{� 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆} 𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}�<0.

(i)

R<0，Q-_{𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅}−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_<0

or

(ii)

Q<0，_{R-𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{𝑄𝑄𝑄𝑄}−1_{𝑆𝑆𝑆𝑆<0.}

Proof：Since the contract transform does not change the matrix positive definition, we first prove that _{� 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆} 𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is

equivalent to R<0，Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇.

IfQ<0, through the Schur formula, we have the following: �𝐼𝐼𝐼𝐼(₀𝑟𝑟𝑟𝑟) −𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅_{𝐼𝐼𝐼𝐼(} −1

𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 𝑅𝑅𝑅𝑅� �

𝐼𝐼𝐼𝐼(𝑟𝑟𝑟𝑟) 0 −𝑅𝑅𝑅𝑅−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇 _{𝐼𝐼𝐼𝐼}

Correspondingly, � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is equivalent to R<0，Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 < 0.

[End of Proof]

Notes: Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{is the}_{Schur complement}_{of Q.}

For singular plant, its state response contains not only the exponential term as normal systems, but also the pulse term

and input derivative item, which will make the whole system has a pulse behavior. The pulse behavior decreases not

Then if and only if (i) R<0, Q-SR–1_ST_<0

or

(3)

Information Technology and Control 2018/1/47 142

Proof: Since the contract transform does not change the matrix positive definition, we first prove that � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆_{𝑆𝑆𝑆𝑆}_{𝑇𝑇𝑇𝑇} _{𝑅𝑅𝑅𝑅}�<0.

(i)

or

(ii)

Q<0，_{R-𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{𝑄𝑄𝑄𝑄}−1_{𝑆𝑆𝑆𝑆<0.}

IfQ<0, through the Schur formula, we have the following: �𝐼𝐼𝐼𝐼(₀𝑟𝑟𝑟𝑟) −𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅_{𝐼𝐼𝐼𝐼(} −1

𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟)� � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 𝑅𝑅𝑅𝑅� �

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is equivalent to R<0，Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇< 0.

[End of Proof]

is equivalent to R<0, Q-SR–1_ST_.

If Q<0, through the Schur formula, we have the fol-lowing:

�𝐼𝐼𝐼𝐼(₀𝑟𝑟𝑟𝑟) −𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅_{𝐼𝐼𝐼𝐼} −1

Correspondingly, � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆_{𝑆𝑆𝑆𝑆}_{𝑇𝑇𝑇𝑇} _{𝑅𝑅𝑅𝑅}� <0 is equivalent to R<0，Q-_{𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅}−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇 _{< 0.}

[End of Proof]

In Figure 1, u, w and z are control input, measurement state or measurement output, input external disturbance and expectation output, and τ is network-induced delay. The aim of positioning is to guarantee stable running of the system independently of any external disturbances so that the expected output of the system is not affected.

�𝐼𝐼𝐼𝐼(₀𝑟𝑟𝑟𝑟) −𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅_{𝐼𝐼𝐼𝐼} −1

Correspondingly, � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆_{𝑆𝑆𝑆𝑆}_{𝑇𝑇𝑇𝑇} _{𝑅𝑅𝑅𝑅}� <0 is equivalent to R<0，Q-_{𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅}−1_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇 _{< 0.}

[End of Proof]

In Figure 1, u, w and z are control input, measurement state or measurement output, input external disturbance and expectation output, and τ is network-induced delay. The aim of positioning is to guarantee stable running of the system independently of any external disturbances so that the expected output of the system is not affected.

Correspondingly, _{� 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆}

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}�<0.

(i)

or

(ii)

Q<0，R-_{𝑆𝑆𝑆𝑆}𝑇𝑇𝑇𝑇_{𝑄𝑄𝑄𝑄}−1_{𝑆𝑆𝑆𝑆}_<0.

IfQ<0, through the Schur formula, we have the following: �𝐼𝐼𝐼𝐼(₀𝑟𝑟𝑟𝑟) −𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅_{𝐼𝐼𝐼𝐼} −1

𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇 _{𝑅𝑅𝑅𝑅}� <0 is equivalent to R<0，Q-𝑆𝑆𝑆𝑆𝑅𝑅𝑅𝑅−1𝑆𝑆𝑆𝑆𝑇𝑇𝑇𝑇< 0.

[End of Proof]

is equivalent to R<0, Q-SR–1_ST _<0.

[End of Proof]

Notes: Q- SR–1_ST _{is the Schur complement of Q.}

Problem Description

In Figure 1, u, w and z are control input, measurement state or measurement output, input external distur-bance and expectation output, and τ is network-in-duced delay. The aim of positioning is to guarantee stable running of the system independently of any ex-ternal disturbances so that the expected output of the system is not affected.

Figure 1

General Structure of SNCS

For singular plant, its state response contains not only the exponential term as normal systems, but also the pulse term and input derivative item, which will make the whole system has a pulse behavior. The pulse be-havior decreases not only the system performance and even leads to the unstable state, which is a fatal destructiveness to the system. For network communi-cation, due to the limited network bandwidth and the restraint of communication mechanism, the network communication obtains uncertainty and complexity.

The presented model shows a singular system state:

          + = + = + − + = ) ( ) ( ) ( ) ( ) ( ) ( )) ( ( ) ( ) ( & 2 2 1 1 0 t w H t x C t z t w H t X C t y t w H l t Bu t Ax t Ex

, (1)

where x(t)∈𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛_,_u(t)_{∈𝑅𝑅𝑅𝑅}𝑚𝑚𝑚𝑚_,_y(t)_{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{and z(t)} _{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{are state vector, control input vector, output vector and expectation} output vector, respectively. A∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{, B}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑚𝑚𝑚𝑚_and_{𝐶𝐶𝐶𝐶}

1,𝐶𝐶𝐶𝐶2∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑙𝑙𝑙𝑙are constant matrices, E∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛is a singular matrix; w(t) is finite energy external disturbance, and 𝐻𝐻𝐻𝐻0,𝐻𝐻𝐻𝐻1,𝐻𝐻𝐻𝐻2are constant matrices.

When the singular plant is regular and impulse free, the equation (1) can be equivalently transformed as:

1 1 1 1 1

2 2 2 2

11 1 12 2 1

21 1 22 2 2

& ( ) ( ) ( ) ( ) & ( ) ( ) ( 1) ( )

. ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x t A x t B u t l W w t

x t x t B t W w t

y t C x t C x t H w t

z t C x t C x t H w t

= + − +   ₌ ₊ _{− +}   ₌ ₊ ₊   ₌ ₊ ₊ 

The state feedback control is

[

]

_      = ) ( ) ( ) ( 2 1 2 1 K _xx k_k

K k

u .

Let 𝑥𝑥𝑥𝑥�=[ x1𝑇𝑇𝑇𝑇(𝑘𝑘𝑘𝑘) 𝑢𝑢𝑢𝑢𝑇𝑇𝑇𝑇(𝑘𝑘𝑘𝑘 −1)]T , the state feedback SNCS close-loop model is

𝑥𝑥𝑥𝑥�(k+1)=�(𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑+𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾1) (𝐵𝐵𝐵𝐵11(𝐿𝐿𝐿𝐿)− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2

𝐾𝐾𝐾𝐾1 − 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2 � 𝑥𝑥𝑥𝑥�+�𝑊𝑊𝑊𝑊

0− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2 𝑊𝑊𝑊𝑊2 − 𝐾𝐾𝐾𝐾2𝑊𝑊𝑊𝑊 �w(k).

The dynamic output feedback controller is

�𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘+ 1) =𝐴𝐴𝐴𝐴𝑐𝑐𝑐𝑐𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘) +𝐵𝐵𝐵𝐵𝑐𝑐𝑐𝑐𝑦𝑦𝑦𝑦(𝑘𝑘𝑘𝑘) 𝑢𝑢𝑢𝑢(𝑘𝑘𝑘𝑘) =𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘) .

Let 𝑥𝑥𝑥𝑥̅=[ x1𝑇𝑇𝑇𝑇 x𝑐𝑐𝑐𝑐𝑇𝑇𝑇𝑇 𝑢𝑢𝑢𝑢𝑇𝑇𝑇𝑇]T, the SNCS close-loop model is as follows:

) ( 0 ) ( 0 0 ) ( ) ( ) 1

( 1 12 2

0 2 12 11 10 k w W BcC H B W k x C B C B A B l b C l B A k x _c c c c c c d           − +           − =

+ . (2)

Whether the system uses state feedback and output feedback or not, the SNCS close-loop system model is a linear

normal system depending on time delay τ.

H∞CONTROL

1. State feedback H∞ control

Theorem 1: If there exist positive definite matrices 𝑆𝑆𝑆𝑆̅, 𝑅𝑅𝑅𝑅�, such that

(1)

where x(t) ∈ Rn

, u(t) ∈ Rm, , y(t) ∈ Rl and z(t) ∈ Rl are state vector, control input vector, output vector and

expectation output vector, respectively. A ∈ Rn×n_{, B}

∈Rn×m_{and C}

1, C2∈Rn×l are constant matrices, E ∈

Rn×n_{is a singular matrix; w(t) is finite energy external}

disturbance, and H0, H1, H2_{are constant matrices.}

          + = + = + − + = ) ( ) ( ) ( ) ( ) ( ) ( )) ( ( ) ( ) ( & 2 2 1 1 0 t w H t x C t z t w H t X C t y t w H l t Bu t Ax t Ex

, (1)

where x(t)∈𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛_,_{u(t)∈𝑅𝑅𝑅𝑅}𝑚𝑚𝑚𝑚_,_{y(t)∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{and z(t)} _{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{are state vector, control input vector, output vector and expectation}

output vector, respectively. A∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{, B}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑚𝑚𝑚𝑚_and_{𝐶𝐶𝐶𝐶}

1,𝐶𝐶𝐶𝐶2∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑙𝑙𝑙𝑙are constant matrices, E∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛is a singular matrix; w(t)

is finite energy external disturbance, and 𝐻𝐻𝐻𝐻0,𝐻𝐻𝐻𝐻1,𝐻𝐻𝐻𝐻2are constant matrices.

1 1 1 1 1

2 2 2 2

11 1 12 2 1

21 1 22 2 2

& ( ) ( ) ( ) ( ) & ( ) ( ) ( 1) ( )

. ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

= + − +   ₌ ₊ _{− +}   ₌ ₊ ₊   ₌ ₊ ₊ 

[

]

_      = ) ( ) ( ) ( 2 1 2

1 _x _k

k x K K k u .

Let 𝑥𝑥𝑥𝑥�=[ x1𝑇𝑇𝑇𝑇(𝑘𝑘𝑘𝑘) 𝑢𝑢𝑢𝑢𝑇𝑇𝑇𝑇(𝑘𝑘𝑘𝑘 −1)]T, the state feedback SNCS close-loop model is

𝑥𝑥𝑥𝑥�(k+1)=�(𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑+𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾1) (𝐵𝐵𝐵𝐵11(𝐿𝐿𝐿𝐿)− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2

𝐾𝐾𝐾𝐾1 − 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2 � 𝑥𝑥𝑥𝑥�+�𝑊𝑊𝑊𝑊

0− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2 𝑊𝑊𝑊𝑊2

− 𝐾𝐾𝐾𝐾2𝑊𝑊𝑊𝑊 �w(k).

) ( 0 ) ( 0 0 ) ( ) ( ) 1

( 1 12 2

0 2 12 11 10 k w W BcC H B W k x C B C B A B l b C l B A k x c c c c c c d           − +           − =

+ . (2)

H∞CONTROL

Theorem 1: If there exist positive definite matrices 𝑆𝑆𝑆𝑆̅, 𝑅𝑅𝑅𝑅�, such that

The state feedback control is           + = + = + − + = ) ( ) ( ) ( ) ( ) ( ) ( )) ( ( ) ( ) ( & 2 2 1 1 0 t w H t x C t z t w H t X C t y t w H l t Bu t Ax t Ex

, (1)

where x(t)∈𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛_,_u(t)_{∈𝑅𝑅𝑅𝑅}𝑚𝑚𝑚𝑚_,_y(t)_{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{and z(t)} _{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{are state vector, control input vector, output vector and expectation}

output vector, respectively. A∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{, B}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑚𝑚𝑚𝑚_and_{𝐶𝐶𝐶𝐶}

1 1 1 1 1

2 2 2 2

11 1 12 2 1

21 1 22 2 2

& ( ) ( ) ( ) ( ) & ( ) ( ) ( 1) ( )

. ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

= + − +   ₌ ₊ _{− +}   ₌ ₊ ₊   ₌ ₊ ₊ 

[

]

_      = ) ( ) ( ) ( 2 1 2

1 _x _k

k x K K k u .

𝑥𝑥𝑥𝑥�(k+1)=�(𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑+𝐵𝐵𝐵𝐵10_{𝐾𝐾𝐾𝐾1}(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾1) (𝐵𝐵𝐵𝐵11(𝐿𝐿𝐿𝐿)₋− 𝐵𝐵𝐵𝐵10_{𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2}(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2� 𝑥𝑥𝑥𝑥�+�𝑊𝑊𝑊𝑊0− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2 𝑊𝑊𝑊𝑊2

Let 𝑥𝑥𝑥𝑥̅=[ x1𝑇𝑇𝑇𝑇 x𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐 𝑢𝑢𝑢𝑢𝑇𝑇𝑇𝑇]T, the SNCS close-loop model is as follows:

) ( 0 ) ( 0 0 ) ( ) ( ) 1

( 1 12 2

0 2 12 11 10 k w W BcC H B W k x C B C B A B l b C l B A k x c c c c c c d           − +           − =

+ . (2)

H∞CONTROL

Let

only the system performance and even leads to the unstable state, which is a fatal destructiveness to the system. For

network communication, due to the limited network bandwidth and the restraint of communication mechanism, the

network communication obtains uncertainty and complexity.





















+

=

+

=

+

−

+

=

)

(

)

(

)

(

)

(

)

(

)

(

))

(

)

(

)

(

&

2 2 1 1 0

t

w

H

t

x

C

t

z

t

w

H

t

X

C

t

y

t

w

H

l

t

Bu

t

Ax

t

E

x

, (1)

where x(t)∈𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛_,_u(t)_{∈𝑅𝑅𝑅𝑅}𝑚𝑚𝑚𝑚_,_y(t)_{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{and z(t)} _{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{are state vector, control input vector, output vector and expectation} output vector, respectively. A∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{, B}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑚𝑚𝑚𝑚_and_{𝐶𝐶𝐶𝐶1,}_{𝐶𝐶𝐶𝐶2}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑙𝑙𝑙𝑙_{are constant matrices, E}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{is a singular matrix; w(t)}

1 1 1 1 1

2 2 2 2

11 1 12 2 1

21 1 22 2 2

& ( )

( )

(

)

( )

& ( )

( )

( 1)

( )

.

( )

x

t

A x t B u t l W w t

x

t

x t B t

W w t

y t

C x t C x t H w t

z t

C x t C x t H w t

=

+

− +





₌

₊

_{− +}





₌

₊



₌

₊



[

]

_











=

)

(

)

(

)

(

2 1 2

1

_x

_k

k

x

K

k

u

.

𝑥𝑥𝑥𝑥�(k+1)=�(𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑+𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) _{𝐾𝐾𝐾𝐾} 𝐾𝐾𝐾𝐾1) (𝐵𝐵𝐵𝐵11(𝐿𝐿𝐿𝐿)− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2

1 − 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2 � 𝑥𝑥𝑥𝑥�+�𝑊𝑊𝑊𝑊

0− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2 𝑊𝑊𝑊𝑊2

�𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘+ 1) =_{𝑢𝑢𝑢𝑢(𝑘𝑘𝑘𝑘) =}𝐴𝐴𝐴𝐴𝑐𝑐𝑐𝑐𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐_{𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐}(𝑘𝑘𝑘𝑘) +_{(𝑘𝑘𝑘𝑘)}𝐵𝐵𝐵𝐵𝑐𝑐𝑐𝑐𝑦𝑦𝑦𝑦(𝑘𝑘𝑘𝑘).

)

(

0 )

(

0

0 )

(

)

(

)

1 (

1 12 2

0 2 12 11 10

k

w

W

BcC

H

B

W

k

x

C

B

C

B

A

B

l

b

C

l

B

A

k

x

c c c c c c d













−

+













−

=

+

. (2)

Whether the system uses state feedback and output feedback or not, the SNCS close-loop system model is a linear normal system depending on time delay τ.

H∞CONTROL

1. State feedbackH∞ control

only the system performance and even leads to the unstable state, which is a fatal destructiveness to the system. For network communication, due to the limited network bandwidth and the restraint of communication mechanism, the network communication obtains uncertainty and complexity.





















+

=

+

=

+

−

+

=

)

(

)

(

)

(

)

(

)

(

)

(

))

(

)

(

)

(

&

2 2 1 1 0

t

w

H

t

x

C

t

z

t

w

H

t

X

C

t

y

t

w

H

l

t

Bu

t

Ax

t

E

x

, (1)

where x(t)∈𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛_,u(t)_{∈𝑅𝑅𝑅𝑅}𝑚𝑚𝑚𝑚_,_y(t)_{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{and z(t)}_{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{are state vector, control input vector, output vector and expectation}

output vector, respectively. A∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{, B}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑚𝑚𝑚𝑚_and_{𝐶𝐶𝐶𝐶1}_,_{𝐶𝐶𝐶𝐶2}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑙𝑙𝑙𝑙_{are constant matrices, E}_{∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{is a singular matrix; w(t)}

1 1 1 1 1

2 2 2 2

11 1 12 2 1

21 1 22 2 2

& ( )

( )

(

)

( )

& ( )

( )

( 1)

( )

.

( )

x

t

A x t B u t l W w t

x

t

x t B t

W w t

y t

C x t C x t H w t

z t

C x t C x t H w t

=

+

− +





₌

₊

_{− +}





₌

₊



₌

₊



[

]

_











=

)

(

)

(

)

(

2 1 2

1

_x

_k

k

x

K

k

u

.

Let 𝑥𝑥𝑥𝑥�=[ x1𝑇𝑇𝑇𝑇(𝑘𝑘𝑘𝑘) 𝑢𝑢𝑢𝑢𝑇𝑇𝑇𝑇(𝑘𝑘𝑘𝑘 −1)]T , the state feedback SNCS close-loop model is

𝑥𝑥𝑥𝑥�(k+1)=�(𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑+𝐵𝐵𝐵𝐵10_{𝐾𝐾𝐾𝐾1}(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾1) (𝐵𝐵𝐵𝐵11(𝐿𝐿𝐿𝐿)₋− 𝐵𝐵𝐵𝐵10_{𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2}(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2� 𝑥𝑥𝑥𝑥�+�𝑊𝑊𝑊𝑊0− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2 𝑊𝑊𝑊𝑊2

)

(

0 )

(

0

0 )

(

)

(

)

1 (

1 12 2

0 2 12 11 10

k

w

W

BcC

H

B

W

k

x

C

B

C

B

A

B

l

b

C

l

B

A

k

x

c c c c c c d













−

+













−

=

+

. (2)

H∞CONTROL

Theorem 1: If there exist positive definite matrices 𝑆𝑆𝑆𝑆̅, 𝑅𝑅𝑅𝑅�, such that , the state feedback SNCS

close-loop model is

only the system performance and even leads to the unstable state, which is a fatal destructiveness to the system. For

network communication, due to the limited network bandwidth and the restraint of communication mechanism, the

network communication obtains uncertainty and complexity.





















+

=

+

=

+

−

+

=

)

(

)

(

)

(

)

(

)

(

)

(

))

(

)

(

)

(

&

2 2 1 1 0

t

w

H

t

x

C

t

z

t

w

H

t

X

C

t

y

t

w

H

l

t

Bu

t

Ax

t

E

_x

, (1)

where x(t)∈𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛_,_u(t)_{∈𝑅𝑅𝑅𝑅}𝑚𝑚𝑚𝑚_,_y(t)_{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{and z(t)} _{∈ 𝑅𝑅𝑅𝑅}𝑙𝑙𝑙𝑙 _{are state vector, control input vector, output vector and expectation} output vector, respectively. A∈ 𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛×𝑛𝑛𝑛𝑛_{, B∈ 𝑅𝑅𝑅𝑅}𝑛𝑛𝑛𝑛×𝑚𝑚𝑚𝑚_{and𝐶𝐶𝐶𝐶}

1 1 1 1 1

2 2 2 2

11 1 12 2 1

21 1 22 2 2

& ( )

( )

(

)

( )

& ( )

( )

( 1)

( )

_.

( )

x

t

A x t B u t l W w t

x

t

x t B t

W w t

y t

C x t C x t H w t

z t

C x t C x t H w t

=

+

− +





₌

₊

_{− +}





₌

₊



₌

₊



[

]

_











=

)

(

)

(

)

(

2 1 2

1

_x

_k

k

x

K

k

u

.

𝑥𝑥𝑥𝑥�(k+1)=�(𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑+𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾1) (𝐵𝐵𝐵𝐵11(𝐿𝐿𝐿𝐿)− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2 𝐾𝐾𝐾𝐾1 − 𝐾𝐾𝐾𝐾2𝐵𝐵𝐵𝐵2 �

𝑥𝑥𝑥𝑥�+�𝑊𝑊𝑊𝑊0− 𝐵𝐵𝐵𝐵10(𝑙𝑙𝑙𝑙) 𝐾𝐾𝐾𝐾2 𝑊𝑊𝑊𝑊2 − 𝐾𝐾𝐾𝐾2𝑊𝑊𝑊𝑊 �w(k).

)

(

0 )

(

0

0 )

(

)

(

)

1 (

1 12 2

0 2 12 11 10

k

w

W

BcC

H

B

W

k

x

C

B

C

B

A

B

l

b

C

l

B

A

k

x

c c c c c c d













−

+













−

=

+

. (2)

H∞CONTROL

Theorem 1: If there exist positive definite matrices 𝑆𝑆𝑆𝑆̅, 𝑅𝑅𝑅𝑅�, such that The dynamic output feedback controller is

�𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘+ 1) =_{𝑢𝑢𝑢𝑢}₍_{𝑘𝑘𝑘𝑘}_{) =}𝐴𝐴𝐴𝐴𝑐𝑐𝑐𝑐_{𝐶𝐶𝐶𝐶}𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘) +𝐵𝐵𝐵𝐵𝑐𝑐𝑐𝑐𝑦𝑦𝑦𝑦(𝑘𝑘𝑘𝑘)

𝑐𝑐𝑐𝑐𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘) .

Let 𝑥𝑥𝑥𝑥̅=[ x1𝑇𝑇𝑇𝑇 x𝑇𝑇𝑇𝑇𝑐𝑐𝑐𝑐 𝑢𝑢𝑢𝑢𝑇𝑇𝑇𝑇]T, the SNCS close-loop model is as follows:

)

(

0 )

(

0

0 )

(

)

(

)

1 (

1 12 2

0 2 12 11 10

k

w

W

BcC

H

B

W

k

x

C

B

C

B

A

B

l

b

C

l

B

A

k

x

c c c c c c d













−

+













−

=

+

. (2)

H∞CONTROL

Theorem 1: If there exist positive definite matrices 𝑆𝑆𝑆𝑆̅, 𝑅𝑅𝑅𝑅�, such that Let

�𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘+ 1) =_{𝑢𝑢𝑢𝑢}₍_{𝑘𝑘𝑘𝑘}_{) =}𝐴𝐴𝐴𝐴𝑐𝑐𝑐𝑐_{𝐶𝐶𝐶𝐶}𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘) +𝐵𝐵𝐵𝐵𝑐𝑐𝑐𝑐𝑦𝑦𝑦𝑦(𝑘𝑘𝑘𝑘) 𝑐𝑐𝑐𝑐𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐(𝑘𝑘𝑘𝑘) .

Let 𝑥𝑥𝑥𝑥̅=[ x₁𝑇𝑇𝑇𝑇 _x 𝑐𝑐𝑐𝑐

𝑇𝑇𝑇𝑇 _{𝑢𝑢𝑢𝑢}𝑇𝑇𝑇𝑇_]T_{the SNCS close-loop model is as follows:}

)

(

0 )

(

0

0 )

(

)

(

)

1 (

1 12 2

0 2 12 11 10

k

w

W

BcC

H

B

W

k

x

C

B

C

B

A

B

l

b

C

l

B

A

k

x

c c c c c c d













−

+













−

=

+

. (2)

H∞CONTROL

Theorem 1: If there exist positive definite matrices 𝑆𝑆𝑆𝑆̅, 𝑅𝑅𝑅𝑅�, such that , the SNCS close-loop model is