Passivity and passification for a class of singularly perturbednonlinear systems via neural networks

Shiping Wen, Zhigang Zeng, and Tingwen Huang

Abstract—This paper is concerned with the problem ofpassivity and passification for a class of singularly perturbednonlinear systems (SPNS) via neural network. By constructinga proper functional as well as the linear matrix inequalities(LMIs) technique, some novel sufficient conditions are derivedto make SPNS passive. The allowable perturbation bound 𝜉∗

can be determined via certain algebra inequalities, the proposedcontroller based on neural network will make SPNS passive forall 𝜉 ∈ (0, 𝜉∗). Finally, a numerical example is given to illustratethe theoretical results.

Keywords: Passivity and passification; SPNS; Neural net-work

I. INTRODUCTION

In many practical systems, the presence of small masses,moments of inertia, and resistances gives rise to two-time-scale (singularly perturbed) systems. The singularly per-turbed systems represented by slow and fast subsystems havebeen studied by many researchers [1]-[3]. Using the singu-larly perturbation method, the stability of a high dimensionalsystem can be analyzed based on the lower order subsystems,slow and fast subsystems. Recently, several works haveconsidered stability analysis and stabilization of singularperturbation systems with the stability bound 𝜉∗ [4]-[5].

There are many studies about linear singularly perturbedsystems [6]-[8], and SPNS [9]-[14]. On the other hand, thepassivity and passification problems for practical systemshave been attracting great attention. The passivity theoryplays an important role in circuits, networks, systems andcontrol. It provides a useful way to deal with delay systems[15], fuzzy systems [16], networked control systems [17],hybrid systems [18], etc.

In the past decades, neural networks have been extensivelystudied [19]-[26], and successfully applied in many areassuch as combinatorial optimization, signal processing, imageprocessing and pattern recognition. And adaptive controlhave been found in many literatures [27]-[29]. However, forthe neural network based passivity analysis and passificationfor SPNS is very few. Therefore, the purpose of this paperis to shorten such a gap.

Manuscript received December 15, 2011. The work is supported by theNatural Science Foundation of China under Grants 60974021 and 61125303,the 973 Program of China under Grant 2011CB710606, the Fund forDistinguished Young Scholars of Hubei Province under Grant 2010CDA081.

Shiping Wen and Zhigang Zeng are with the Department of ControlScience and Engineering, Huazhong University of Science and Tech-nology, and Key Laboratory of Image Processing and Intelligent Con-trol of Education Ministry of China, Wuhan, Hubei, 430074, China(Email:zgzeng527@126.com).

Tingwen Huang is with Texas A & M University at Qatar, Doha 5825,Qatar (Email: tingwen.huang@qatar.tamu.edu).

Motivated by the above discussion, in this paper, we in-vestigate the problems of passivity and passification of SPNSvia neural networks. The main contributions of this papercan be summarized as follows: (i) the passivity analysis isfirst extended to the SPNS; (ii) a novel Lyapunov functionalcombined with the matrix analysis techniques is developedto obtain sufficient conditions under which the closed-loopsystem is globally passive in the sense of expectation. Thesesufficient conditions are given in the form of LMIs that canbe solved numerically.

The rest of the paper is organized as follows. In SectionII, the system studied in this paper is proposed and somepreliminaries are given. In Section III, we will address theneural network adaptive controller design scheme in detail.Passivity analysis of the neural network adaptive controllerand the passification bounds of the singularly perturbationparameter are also carried out in this section. In SectionIV, an illustrative example is constructed to demonstratedthe effectiveness and usefulness of the acquired results andfinally, conclusions are drawn in Section V.

Notation. The notation used through the paper is fairlystandard. ℕ is the set of natural numbers and ℕ

+ standsfor the set of nonnegative integers; ℝ

𝑛 and ℝ𝑛×𝑚 denote,

respectively, the 𝑛 dimensional Euclidean space and the set ofall 𝑛×𝑚 real matrices. The notation 𝑃 > 0(≥ 0) means that𝑃 is positive definite(positive semi-definite). In symmetricblock matrices or complex matrix expressions, we use anasterisk (∗) to represent a term that is induced by symmetryand diag{⋅ ⋅ ⋅ } stands for a block-diagonal matrix. Matrices,if their dimensions are not explicitly stated, are assumed tobe compatible for algebraic operations. 𝔼{𝑥} stands for theexpectation of stochastic variable 𝑥. 𝑙2[0,+∞) is the spaceof square integrable vectors. The notation ∣∣.∣∣ stands for theusual 𝑙2 norm while ∣.∣ refers to the Euclidean vector norm.Sometimes, when no confusion would arise, the dimensionsof a function or a matrix will be omitted for convenience.

II. PRELIMINARIES

In this paper, we aim to consider the following SPNS:⎧⎨⎩

�̇�(𝑡) = 𝑓1(𝑥(𝑡), 𝑧(𝑡)) + 𝑔1(𝑥(𝑡), 𝑧(𝑡))𝑢(𝑡)+ 𝐸1𝜔(𝑡),

𝜉�̇�(𝑡) = 𝑓2(𝑥(𝑡), 𝑧(𝑡)) + 𝑔2(𝑥(𝑡), 𝑧(𝑡))𝑢(𝑡)+ 𝐸2𝜔(𝑡),

𝑦1(𝑡) = 𝐺11𝑥(𝑡) +𝐺12𝜔(𝑡),𝑦2(𝑡) = 𝐺21𝑧(𝑡) +𝐺22𝜔(𝑡),

(1)

978-1-4673-1490-9/12/$31.00 ©2012 IEEE

WCCI 2012 IEEE World Congress on Computational Intelligence June, 10-15, 2012 - Brisbane, Australia IJCNN

3187

where 𝑥(𝑡) =[𝑥1(𝑡) 𝑥2(𝑡) ⋅ ⋅ ⋅ , 𝑥𝑚(𝑡)

]𝑇 ∈ ℝ𝑚×1

and 𝑧(𝑡) =[𝑧1(𝑡) 𝑧2(𝑡) ⋅ ⋅ ⋅ , 𝑧𝑛(𝑡)

]𝑇 ∈ ℝ𝑛×1 denote

the state vectors, 𝑦1(𝑡) ∈ ℝ𝑚×1 is the output of 𝑥(𝑡),

𝑦2(𝑡) ∈ ℝ𝑛×1 is the output of 𝑧(𝑡), 𝑓1(𝑥(𝑡), 𝑧(𝑡)) ∈

ℝ𝑚×1, 𝑓2(𝑥(𝑡), 𝑧(𝑡)) ∈ ℝ

𝑛×1, 𝑔1(𝑥(𝑡), 𝑧(𝑡)) ∈ ℝ𝑚×𝑝,

𝑔2(𝑥(𝑡), 𝑧(𝑡)) ∈ ℝ𝑛×𝑝 are nonlinear functions, 𝑢(𝑡) ∈ ℝ

𝑝×1

is the control input, 𝜔(𝑡) ∈ ℝ𝑞×1 is the disturbance,

which is assumed to an arbitrary signal in 𝑙2[0,∞). 𝐺11 ∈ℝ

𝑚×𝑚, 𝐺21 ∈ ℝ𝑛×𝑛, 𝐸1, 𝐺12 ∈ ℝ

𝑚×𝑞, 𝐸2, 𝐺22 ∈ ℝ𝑛×𝑞

are constant matrices, and 𝜉 is the singular perturbationparameter. While

𝑓1(𝑥(𝑡), 𝑧(𝑡)) = 𝐴11𝑥(𝑡) +𝐴12𝑧(𝑡) + Δ𝑓11(𝑡)𝑥(𝑡)

+ Δ𝑓12(𝑡)𝑧(𝑡),

𝑓2(𝑥(𝑡), 𝑧(𝑡)) = 𝐴21𝑥(𝑡) +𝐴22𝑧(𝑡) + Δ𝑓21(𝑡)𝑥(𝑡)

+ Δ𝑓22(𝑡)𝑧(𝑡),

𝑔1(𝑥(𝑡), 𝑧(𝑡)) = 𝐵1𝑔(𝑡), 𝑔2(𝑥(𝑡), 𝑧(𝑡)) = 𝐵2𝑔(𝑡),

𝐴11 ∈ ℝ𝑚×𝑚, 𝐴12ℝ

𝑚×𝑛, 𝐴21ℝ𝑛×𝑚, 𝐴22 ∈ ℝ

𝑚×𝑛, Δ𝑓11 ∈ℝ

𝑚×𝑚, Δ𝑓12 ∈ ℝ𝑚×𝑛, Δ𝑓21 ∈ ℝ

𝑛×𝑚, Δ𝑓22 ∈ ℝ𝑛×𝑛,

𝐵1 ∈ ℝ𝑚×𝑝, 𝐵2 ∈ ℝ

𝑛×𝑝, 𝑔(𝑡) ∈ ℝ𝑝×𝑝, then SPNS (1) can

be rearranged as the following form,⎧⎨⎩

Δ(𝜉)�̇�(𝑡) = 𝐴𝑋(𝑡) + Δ𝑓(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)+𝐵𝑔(𝑥(𝑡), 𝑧(𝑡))𝑢(𝑡) + 𝐸𝜔(𝑡),

𝑌 (𝑡) = 𝐺1𝑋(𝑡) +𝐺2𝜔(𝑡),(2)

where Δ(𝜉) =

[𝐼𝑚 00 𝜉𝐼𝑛

], 𝑋(𝑡) =

[𝑥𝑇 (𝑡) 𝑧𝑇 (𝑡)

]𝑇,

𝑌 (𝑡) =[𝑦𝑇1 (𝑡) 𝑦𝑇2 (𝑡)

]𝑇, 𝐴,𝐵, and 𝐸,𝐺1, 𝐺2 are

known constant matrices with 𝐴 ∈ ℝ(𝑛+𝑚)(𝑛+𝑚), 𝐵 ∈

ℝ(𝑛+𝑚)𝑝 and 𝐸 ∈ ℝ

(𝑛+𝑚)𝑞 which are defined as 𝐴 =[𝐴11 𝐴12

𝐴21 𝐴22

], 𝐵 =

[𝐵𝑇

1 𝐵𝑇2

], 𝐸 =

[𝐸𝑇

1 𝐸𝑇2

],

𝐺1 = diag{𝐺11, 𝐺21}, 𝐺2 = diag{𝐺12, 𝐺22}. And𝑔(𝑥(𝑡), 𝑧(𝑡)) ∈ ℝ

𝑝×𝑝 is a known matrix function andthe determinant of 𝑔(𝑥(𝑡), 𝑧(𝑡)) is not equal to zero,Δ𝑓(𝑥(𝑡), 𝑧(𝑡)) ∈ ℝ

(𝑛+𝑚)(𝑛+𝑚) is a matrix function withnonlinear term of some state valuables in every element. Inthis paper, the function Δ𝑓(𝑥(𝑡), 𝑧(𝑡)) is a continuously anddifferentiable matrix function which can be approximated bya neural network as the following,

Δ𝑓(𝑥(𝑡), 𝑧(𝑡)) = 𝐵𝑊𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)) + 𝜑(𝑥(𝑡), 𝑧(𝑡)) (3)

where 𝑊 ∈ ℝ𝑝×𝑟 denotes optimal unknown constant

weighting matrix, 𝜙(𝑥(𝑡), 𝑧(𝑡)) ∈ ℝ𝑟(𝑛+𝑚) is a given basis

function such that each component of 𝜙(𝑥(𝑡), 𝑧(𝑡)) takesvalues between 0 and 1. 𝜑(𝑥(𝑡), 𝑧(𝑡)) ∈ ℝ

(𝑛+𝑚)(𝑛+𝑚) is anapproximation error. To this end, the following assumptionis given

Assumption 1. ∣∣𝜑(𝑥(𝑡), 𝑧(𝑡))𝑇𝜑(𝑥(𝑡), 𝑧(𝑡))∣∣ ≤ 𝜎−1, where𝜎−1 is a real constant.

Before formulating the main problem, we first give thefollowing definition.

Definition 1. System (2) is said to be passive if there existsa scalar 𝛽 > 0 such that

2𝔼

{∫ 𝑡

0

𝜔𝑇 (𝑠)𝑌 (𝑠)𝑑𝑠

}≥ −𝛽𝔼

{∫ 𝑡

0

𝜔𝑇 (𝑠)𝜔(𝑠)𝑑𝑠

}(4)

for all 𝑡 > 0 under zero initial condition.

Remark 1. The concept of system passivity is related tothe output signal and the external input signal. The physicalmeaning of a passive system is that the increment of thenonlinear system energy from 𝑠 = 0 to 𝑠 = 𝑡 is alwaysless than or equal to the supply from the external energy.This means that the motion of a passive system is alwaysaccompanied by energy dissipation.

To obtain our results, the following Lemmas will be em-ployed.

Lemma 1. (Schur Complement). Given constant matrices𝑆1, 𝑆2 and 𝑆3, where 𝑆1 = 𝑆𝑇

1 and 𝑆2 = 𝑆𝑇2 . Then 𝑆1 +

𝑆𝑇3 𝑆

−12 𝑆3 < 0 if and only if

[𝑆1 𝑆𝑇

3

𝑆3 −𝑆2

]< 0, or

[ −𝑆2 𝑆3

𝑆𝑇3 𝑆1

]< 0.

Lemma 2. It is known that for any positive constant 𝜌 andany matrices 𝑊 and 𝑉 ,

𝑊𝑇𝑉 + 𝑉 𝑇𝑊 ≤ 𝜌𝑊𝑇𝑊 + 𝜌−1𝑉 𝑇𝑉. (5)

The objective of the following section is how to designthe neural network controller such that SPNS (2) is passivity.Define the following matrices, 𝑃𝜉 = 𝑃 + 𝜉𝑇 and 𝑃𝜉 > 0,

where 𝑃 =

[𝑃11 0𝑃𝑇12 𝑃22

], 𝑇 =

[0 𝑃12

0 0

], 𝑃11 = 𝑃𝑇

11 >

0 and 𝑃22 = 𝑃𝑇22 > 0. Furthermore, we assume 𝑃−1 = 𝑄.

III. MAIN RESULTS

In this section, we will present a sufficient condition interms of LMIs, under which SPNS (2) is passive.

Theorem 1. If there exist positive-definite matrices 𝑄,𝑅,and real matrix 𝑀 , positive constants 𝜎, 𝛼, 𝜀, such that⎡

⎣ Λ 𝑄𝑇 𝑄𝑇

∗ −𝜎𝐼 0∗ ∗ −𝛼𝐼

⎤⎦ < 0, (6)

[𝛼𝐺1𝐺

𝑇1 − 2𝐺2 − 𝛽𝐼 𝐸𝑇

∗ −𝜀𝐼]< 0, (7)

then when the neural network adaptive controller and theadaptive law is defined as follows:

𝑢(𝑡) = 𝑔(𝑥(𝑡), 𝑧(𝑡))−1[𝐾𝑋(𝑡) − �̄�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)

],

˙̄𝑊 = 𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝐵, (8)3188

�̂� = �̄� − 𝑊 , 𝐾 = 𝑀𝑃 , SPNS (2) is globally passivefor all 𝜉 ∈ (0, 𝜉∗), where 𝜉∗ is a positive solution of thefollowing inequality:

𝜇2𝜉2 + 𝜇1𝜉 − 𝜇0 < 0, (9)

Λ = 𝑄𝑇𝐴𝑇 +𝐴𝑄+𝑀𝑇𝐵𝑇 +𝐵𝑀 + 𝜀𝐼 +𝑅,

𝜇2 = ∣∣𝜀𝑇𝑇𝑇 ∣∣,𝜇1 = ∣∣(𝐴+𝐵𝐾)𝑇𝑇 + 𝑇𝑇 (𝐴+𝐵𝐾) + (𝑇𝑇𝑃 + 𝑃𝑇𝑇 )𝜀∣∣,𝜇0 = ∣∣𝜆min(𝑃

𝑇𝑅𝑃 )∣∣.

Proof. Choose functional 𝑉 (.) ∈ 𝜈 : ℝ𝑛 × ℝ+ → ℝ

+ to be

𝑉 (𝑡) = 𝑋𝑇 (𝑡)𝑃𝑇𝜉 Δ(𝜉)𝑋(𝑡) + tr(�̂�𝑇 �̂� ). (10)

Let 𝒜 be the weak infinitesimal generator, then

𝒜𝑉 (𝑡) = 2𝑋𝑇 (𝑡)𝑃𝑇𝜉 Δ(𝜉)�̇�(𝑡) + 2𝑡𝑟(�̂�𝑇 ˙̂

𝑊 ),

= 𝑋𝑇 (𝑡)[(𝐴+𝐵𝐾)𝑇𝑃𝜉 + 𝑃𝑇

𝜉 (𝐴+𝐵𝐾)]𝑋(𝑡)

+𝑋𝑇 (𝑡)[(𝑓(𝑥(𝑡), 𝑧(𝑡)) −𝐵�̄�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))𝑇𝑃𝜉

+ 𝑃𝑇𝜉 (𝑓(𝑥(𝑡), 𝑧(𝑡)) −𝐵�̄�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))

]𝑋(𝑡)

+𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝐸𝜔(𝑡) + 𝜔𝑇 (𝑡)𝐸𝑇𝑃𝜉𝑋(𝑡)

+ tr(𝐵𝑇𝑃𝜉𝑋(𝑡)𝑋𝑇 (𝑡)𝜙𝑇 (𝑥(𝑡), 𝑧(𝑡))�̂� )

+ tr(�̂�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝐵)

= 𝑋𝑇 (𝑡)[(𝐴+𝐵𝐾)𝑇𝑃𝜉 + 𝑃𝑇

𝜉 (𝐴+𝐵𝐾)]𝑋(𝑡)

+𝑋𝑇 (𝑡)[(𝑓(𝑥(𝑡), 𝑧(𝑡)) −𝐵𝑊𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))𝑇𝑃𝜉

+ 𝑃𝑇𝜉 (𝑓(𝑥(𝑡), 𝑧(𝑡)) −𝐵𝑊𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))

]𝑋(𝑡)

+𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝐸𝜔(𝑡) + 𝜔𝑇 (𝑡)𝐸𝑇𝑃𝜉𝑋(𝑡)

+ tr(𝐵𝑇𝑃𝜉𝑋(𝑡)𝑋𝑇 (𝑡)𝜙𝑇 (𝑥(𝑡), 𝑧(𝑡))�̂� )

+ tr(�̂�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝐵)

−𝑋𝑇 (𝑡)[(𝐵�̂�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))𝑇𝑃𝜉

+ 𝑃𝑇𝜉 𝐵�̂�

𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))]𝑋(𝑡)

≤ 𝑋𝑇 (𝑡)[(𝐴+𝐵𝐾)𝑇𝑃𝜉 + 𝑃𝑇

𝜉 (𝐴+𝐵𝐾)]𝑋(𝑡)

+𝑋𝑇 (𝑡)[(𝑓(𝑥(𝑡), 𝑧(𝑡)) −𝐵𝑊𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))𝑇𝑃𝜉

+ 𝑃𝑇𝜉 (𝑓(𝑥(𝑡), 𝑧(𝑡)) −𝐵𝑊𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))

]𝑋(𝑡)

+ 𝜀𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝑃𝜉𝑋(𝑡) + 𝜀−1𝜔𝑇 (𝑡)𝐸𝑇𝐸𝜔(𝑡)

+ tr(𝐵𝑇𝑃𝜉𝑋(𝑡)𝑋𝑇 (𝑡)𝜙𝑇 (𝑥(𝑡), 𝑧(𝑡))�̂� )

+ tr(�̂�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝐵)

−𝑋𝑇 (𝑡)[(𝐵�̂�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡)))𝑇𝑃𝜉

+ 𝑃𝑇𝜉 𝐵�̂�

𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))]𝑋(𝑡)

As tr[Π1Π2] = tr[Π2Π1] for any matrices Π1 and Π2 with

appropriate dimensions, then

𝑋𝑇 (𝑡)𝜙𝑇 (𝑥(𝑡), 𝑧(𝑡))�̂�𝐵𝑇𝑃𝜉𝑋(𝑡)

= 𝑡𝑟(𝑋𝑇 (𝑡)𝜙𝑇 (𝑥(𝑡), 𝑧(𝑡))�̂�𝐵𝑇𝑃𝜉𝑋(𝑡)

)= tr

(𝐵𝑇𝑃𝜉𝑋(𝑡)𝑋𝑇 (𝑡)𝜙𝑇 (𝑥(𝑡), 𝑧(𝑡))�̂�

),

𝑋𝑇 (𝑡)𝑃𝑇𝜉 𝐵�̂�

𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)

= tr(𝑋𝑇 (𝑡)𝑃𝑇

𝜉 𝐵�̂�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)

)= tr

(�̂�𝑇𝜙(𝑥(𝑡), 𝑧(𝑡))𝑋(𝑡)𝑋𝑇 (𝑡)𝑃𝑇

𝜉 𝐵).

Combined with Assumption 1,

𝒜𝑉 (𝑡) ≤ 𝑋𝑇 (𝑡){[

(𝐴+𝐵𝐾)𝑇𝑃 + 𝑃𝑇 (𝐴+𝐵𝐾)

+ 𝜀𝑃𝑇𝑃 + 𝜎−1𝐼]+ 𝜉

[(𝐴+𝐵𝐾)𝑇𝑇

+ 𝑇𝑇 (𝐴+𝐵𝐾) + 𝜀(𝑇𝑇𝑃 + 𝑃𝑇𝑇 )]

+ 𝜉2[𝜀𝑇𝑇𝑇

]}𝑋(𝑡)

+ 𝜀−1𝜔𝑇 (𝑡)𝐸𝑇𝐸𝜔(𝑡). (11)

Consider the following index:

𝐽(𝑡) = 𝒜𝑉 (𝑡) − 2𝜔𝑇 (𝑡)𝑌 (𝑡) − 𝛽𝜔𝑇 (𝑡)𝜔(𝑡)

≤ 𝒜𝑉 (𝑡) + 𝛼−1𝑋𝑇 (𝑡)𝑋(𝑡)

+ 𝜔𝑇 (𝑡)(𝛼𝐺1𝐺

𝑇1 − 2𝐺2 − 𝛽𝐼

)𝜔(𝑡)

≤ 𝑋𝑇 (𝑡){[

(𝐴+𝐵𝐾)𝑇𝑃 + 𝑃𝑇 (𝐴+𝐵𝐾)

+ 𝜀𝑃𝑇𝑃 + 𝜎−1𝐼 + 𝛼−1𝐼]

+ 𝜉[(𝐴+𝐵𝐾)𝑇𝑇 + 𝑇𝑇 (𝐴+𝐵𝐾)

+ 𝜀(𝑇𝑇𝑃 + 𝑃𝑇𝑇 )]+ 𝜉2

[𝜀𝑇𝑇𝑇

]}𝑋(𝑡)

+ 𝜔𝑇 (𝑡)[𝜀−1𝐸𝑇𝐸 + 𝛼𝐺1𝐺

𝑇1

− 2𝐺2 − 𝛽𝐼]𝜔(𝑡)

≤ 𝑋𝑇 (𝑡){𝑃𝑇

[𝑄𝑇 (𝐴+𝐵𝐾)𝑇 + (𝐴+𝐵𝐾)𝑄

+ 𝜀𝐼 +𝑄𝑇𝜎−1𝑄+𝑄𝑇𝛼−1𝑄]𝑃

+ 𝜉∣∣(𝐴+𝐵𝐾)𝑇𝑇 + 𝑇𝑇 (𝐴+𝐵𝐾)

+ 𝜀(𝑇𝑇𝑃 + 𝑃𝑇𝑇 )∣∣ + 𝜉2∣∣𝜀𝑇𝑇𝑇 ∣∣}𝑋(𝑡)

+ 𝜔𝑇 (𝑡)[𝜀−1𝐸𝑇𝐸 + 𝛼𝐺1𝐺

𝑇1

− 2𝐺2 − 𝛽𝐼]𝜔(𝑡)

≤ 𝑋𝑇 (𝑡){𝑃𝑇

[𝑄𝑇𝐴𝑇 +𝐴𝑄+𝑀𝑇𝐵𝑇 +𝐵𝑀

+ 𝜀𝐼 +𝑄𝑇𝜎−1𝑄+𝑄𝑇𝛼−1𝑄]𝑃

+ 𝜉∣∣(𝐴+𝐵𝐾)𝑇𝑇 + 𝑇𝑇 (𝐴+𝐵𝐾)

+ 𝜀(𝑇𝑇𝑃 + 𝑃𝑇𝑇 )∣∣ + 𝜉2∣∣𝜀𝑇𝑇𝑇 ∣∣}𝑋(𝑡)

+ 𝜔𝑇 (𝑡)[𝜀−1𝐸𝑇𝐸 + 𝛼𝐺1𝐺

𝑇1

3189

− 2𝐺2 − 𝛽𝐼]𝜔(𝑡). (12)

If

𝑄𝑇𝐴𝑇 +𝐴𝑄+𝑀𝑇𝐵𝑇 +𝐵𝑀 + 𝜀𝐼

+𝑄𝑇𝜎−1𝑄+𝑄𝑇𝛼−1𝑄 < −𝑅,then

𝐽(𝑡) < 𝑋𝑇 (𝑡)[− 𝑃𝑇𝑅𝑃 + 𝜉𝜇1𝐼 + 𝜉2𝜇2𝐼

]𝑋(𝑡)

+ 𝜔𝑇 (𝑡)[𝜀−1𝐸𝑇𝐸 + 𝛼𝐺1𝐺

𝑇1

− 2𝐺2 − 𝛽𝐼]𝜔(𝑡)

≤ 𝑋𝑇 (𝑡)[− 𝜇1𝐼 + 𝜉𝜇1𝐼 + 𝜉2𝜇2𝐼

]𝑋(𝑡)

+ 𝜔𝑇 (𝑡)[𝜀−1𝐸𝑇𝐸 + 𝛼𝐺1𝐺

𝑇1

− 2𝐺2 − 𝛽𝐼]𝜔(𝑡). (13)

From condition (6), it is easy to obtain

𝒜𝑉 (𝑡) − 2𝜔𝑇 (𝑡)𝑌 (𝑡) − 𝛽𝜔𝑇 (𝑡)𝜔(𝑡) < 0, (14)

which means

2𝔼

{∫ 𝑡

0

𝜔𝑇 (𝑠)𝑌 (𝑠)𝑑𝑠

}

≥ 𝔼

{𝑉 (𝑡) − 𝛽

∫ 𝑡

0

𝜔𝑇 (𝑠)𝜔(𝑠)𝑑𝑠

}

≥ −𝛽𝔼{∫ 𝑡

0

𝜔𝑇 (𝑠)𝜔(𝑠)𝑑𝑠

}. (15)

Therefore, SPNS (2) is globally passive.

Remark 2. If the solution of (6) has 𝑄 = diag{𝑄11, 𝑄22},and 𝑃 = 𝑄−1, then 𝑃12 = 0, therefore 𝑇 = 0. According to(9), we can get the stability bound 𝜉∗ → ∞.

As polytopic uncertainty description can be used to char-acterize uncertain parameters, where the system matricesare supposed to contain partially unknown parameters andthey reside in a give polytope. An assumption is given afterTheorem 1 as follows: the matrices 𝐴,𝐵,𝐸,𝐺1, 𝐺2 containpartially unknown parameters. Assume that

Ω ≜ (𝐴,𝐵,𝐸,𝐺1, 𝐺2) ∈ ℛ,where ℛ is a given convex-bounded polyhedral domaindescribed by 𝑠 vertices

ℛ = {Ω(𝜆)∣Ω(𝜆) =𝑠∑

𝑖=1

𝜆𝑖Ω𝑖;𝑠∑

𝑖=1

𝜆𝑖 = 1, 𝜆𝑖 ≥ 0}, (16)

where Ω𝑖 = (𝐴𝑖, 𝐵𝑖, 𝐸𝑖, 𝐺1𝑖, 𝐺2𝑖) denote the vertices ofthe polytope. Since LMIs (6) and (7) in Theorem 1 areaffine in the system matrices, this theorem can therefore bedirectly used for the passivity and passification problem onthe basis of quadratic stability notion. Therefore, we presentthe following corollary without proof.

Fig. 1. An electronic circuit with parasitic capacitor and nonlinear resistor.

Corollary 1. Suppose system (2) contains polytopic un-certainty described in (16). There exists a neural networkadaptive controller (4) such that system (2) is globallypassive if there exist positive matrices 𝑄,𝑅, and real ma-trix 𝑀 , positive constants 𝜎, 𝛼, 𝜀 satisfy (6) and (7) for𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, where the matrices 𝐴,𝐵,𝐸,𝐺1, 𝐺2 aretaken with 𝐴𝑖, 𝐵𝑖, 𝐸𝑖, 𝐺1𝑖, 𝐺2𝑖, respectively.

IV. AN ILLUSTRATIVE EXAMPLE

In this section, we present an example to illustrate theeffectiveness of the proposed approach. Consider a nonlinearcircuit [10] illustrated in Fig. 1 with nonlinear resistor. The𝑉𝐶 − 𝐼𝑅 characteristics of the resistor is 𝐼𝑅 = 1

5𝑉3𝐶 − 1

5𝑉𝐶 .Applying the Kirchoff’s voltage and current laws, we canobtain the state equation as

𝐿𝐼𝐿(𝑡) = −𝐼𝐿𝑅− 𝑉𝐶 + 𝑎1𝑢+ 𝑑1(𝑡),

𝐶�̇�𝐶(𝑡) = 𝐼𝐿 − 1

5(𝑉 3

𝐶 − 𝑉𝐶) + 𝑎2𝑢+ 𝑑2(𝑡), (17)

where 𝜉 = 𝐶, 𝑎1 = 0.6, 𝑎2 = 0.8, 𝑅 = 1Ω and 𝐿 = 0.1𝐻 .Let 𝑥(𝑡) = 𝐿𝐼𝐿 and 𝑧(𝑡) = 𝑉𝐶 . 𝑑1(𝑡) and 𝑑2(𝑡) areexternal noises. Let 𝑑1(𝑡) = 0.4 cos(0.1𝜋𝑡) and 𝑑2(𝑡) =0.6 cos(0.2𝜋𝑡), then the state equation (17) can be rewrittenby the following SPNS:

�̇�(𝑡) = −10𝑥(𝑡) − 𝑧(𝑡) + 0.6𝑢(𝑡) + 𝑑1(𝑡),

𝜉�̇�(𝑡) = 10𝑥(𝑡) − 0.2(𝑧2(𝑡) − 1)𝑧(𝑡)

+ 0.8𝑢(𝑡) + 𝑑2(𝑡). (18)

And consider the output

𝑦1(𝑡) = 0.2𝑥(𝑡) + 0.2 cos(0.1𝜋𝑡),

𝑦2(𝑡) = 0.1𝑧(𝑡) + 0.1 cos(0.2𝜋𝑡),

then

𝐴 =

[ −10 −110 0.2

],Δ𝑓(𝑥(𝑡), 𝑧(𝑡)) =

[0 00 −0.2𝑧2(𝑡)

],

𝐵 =

[0.60.8

], 𝑔(𝑥(𝑡), 𝑧(𝑡)) = 1,

𝐸 =

[0.4 00 0.6

], 𝜔(𝑡) =

[cos(0.1𝜋𝑡)cos(0.2𝜋𝑡)

],

𝐺1 =

[0.2 00 0.1

], 𝐺2 =

[0.2 00 0.1

].

3190

0 0.5 1 1.5 2−6

−4

−2

0

2

4

6

t

x(t)

,z(t

)

x(t)z(t)

Fig. 2. States evolution 𝑥(𝑡), 𝑧(𝑡) of SPNS (18) with the neural networkadaptive controller (21) and the adaptive law (22) with 𝜎 = 0.9.

While 𝜎 = 0.9, then the desired solution can be determinedby (6) and (7) by the LMI Toolbox,

𝑀 =[ −1.2923 −17.2209

],

𝑃 =

[0.7973 0

0 0.6129

],

𝑅 =

[8.8561 −0.2280−0.2280 8.9891

],

𝐾 =[ −1.0304 −10.5547

],

𝛼 = 9.1601, 𝛽 = 9.1128,

𝜀 = 8.9226, 𝜉 → ∞.

If 𝜙(𝑥(𝑡), 𝑧(𝑡)) =[

11+𝑒−0.3𝑥(𝑡)

11+𝑒−0.7𝑧(𝑡)

], then the

neural network adaptive controller and the adaptive law areobtained as follows:

𝑢(𝑡) = −1.0304𝑥(𝑡) − 10.5547𝑧(𝑡) − 𝑥(𝑡)�̄�

1 + 𝑒−0.3𝑥(𝑡)

− 𝑧(𝑡)�̄�

1 + 𝑒−0.7𝑧(𝑡), (19)

˙̄𝑊 =( 𝑥(𝑡)

1 + 𝑒−0.3𝑥(𝑡)+

𝑧(𝑡)

1 + 𝑒−0.7𝑧(𝑡)

)×(0.4784𝑥(𝑡) + 0.4903𝑧(𝑡)

), (20)

where �̄� ∈ ℝ1×1. The proposed neural network adaptive

control (21) and adaptive law (22) is applied to the SPNS(18). Simulation results of the states 𝑥(𝑡), 𝑧(𝑡) and neuralnetwork adaptive controller 𝑢(𝑡), adaptive law �̄� are shownin Figs. 2-4 with 𝑥(0) = 6, 𝑧(0) = −5 and 𝜉 = 10.

While 𝜎 = 0.1, then the desired solution can be deter-mined by (6) and (7) by the LMI Toolbox,

𝑀 =[ −69.1965 −107.7597

],

𝑃 =

[0.7973 0

0 0.6129

],

𝑅 =

[46.5857 55.192355.1923 78.7812

],

0 0.5 1 1.5 2−10

0

10

20

30

40

50

t

u(t)

u(t)

Fig. 3. States evolution of the neural network adaptive controller (21) with𝜎 = 0.9.

0 0.5 1 1.5 22.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

t

w−

w−

Fig. 4. States evolution of the adaptive law (22) with 𝜎 = 0.9.

𝐾 =[ −55.1704 −66.0459

],

𝛼 = 106.6095, 𝛽 = 117.1773,

𝜀 = 9.9363, 𝜉 → ∞.

Then the neural network adaptive controller and the adaptivelaw are obtained as follows:

𝑢(𝑡) = −55.1704𝑥(𝑡) − 66.0459𝑧(𝑡) − 𝑥(𝑡)�̄�

1 + 𝑒−0.3𝑥(𝑡)

− 𝑧(𝑡)�̄�

1 + 𝑒−0.7𝑧(𝑡), (21)

˙̄𝑊 =( 𝑥(𝑡)

1 + 𝑒−0.3𝑥(𝑡)+

𝑧(𝑡)

1 + 𝑒−0.7𝑧(𝑡)

)×(0.4784𝑥(𝑡) + 0.4903𝑧(𝑡)

), (22)

Simulation results of the states 𝑥(𝑡), 𝑧(𝑡) and neural networkadaptive controller 𝑢(𝑡), adaptive law �̄� are shown in Figs.5-7 with the same initial values as above.

Remark 3. From the illustrative example, we can see that thelarger the approximation error is, the greater absolute valueof the control gain to make the system passivity is required.

V. CONCLUSIONS

In this paper, the passivity and passification have beenstudied for a class of SPNS via neural networks. A newfunctional has been used to design the neural network adap-tive controller, such that the closed-loop system is globally3191

0 0.5 1 1.5 2−6

−4

−2

0

2

4

6

8

t

x(t)

,z(t

)

x(t)z(t)

Fig. 5. States evolution 𝑥(𝑡), 𝑧(𝑡) of SPNS (18) with the neural networkadaptive controller (21) and the adaptive law (22) with 𝜎 = 0.1.

0 0.5 1 1.5 22.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

t

w−

w−

Fig. 6. States evolution of the neural network adaptive controller (21) with𝜎 = 0.1.

passive. And the controller parameters can be obtained bysolving certain LMIs. An illustrative example has been usedto show the effectiveness of the proposed method.

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