H∞ performance for delayed singular nonlinear Markovian jump systems with unknown transition rates via adaptive control method

Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

H∞ performance for delayed singular nonlinear Markovian jump systems with unknown transition rates via adaptive control method ∗

Guowei Yang a,d , Binghua Kao b , Ju H. Park c , , Yonggui Kao c ,

∗

a

Jiangsu Key Laboratory of Auditing Information Engineering, School of Information Engineering, Nanjing Audit University, Nanjing 211815, PR China b Mathematics Science College, Inner Mongolia Normal University, Hohhot, 010022, PR China c Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea d School of Electronic Information, Qingdao University, Qingdao 266071, PR China

article

info

Article history: Received 23 August 2017 Accepted 1 February 2019 Available online xxxx Keywords: Uncertain singular Markovian jump system Adaptive control method H∞ performance Unknown TRs

a b s t r a c t We work on the investigation of the H∞ performance for uncertain stochastic singular Markovian jump systems with both nonlinear external disturbance and general unknown transition rates (TRs) in this paper. There exist uncertainties in both parameters and TRs. Every transition rate could be totally unavailable or only its estimated value is available. With a generalized H∞ disturbance attenuation level γ , some robust controllers are designed such that the delayed singular Markovian jump system is stochastically admissible. The novel controllers designed in this paper consist of an adaptive controller and a linear controller. Moreover, some sufficient linear matrices inequalities (LMIs) conditions are provided according to Lyapunov stability theory and Itoˆ ’s differential formula. Finally, an example is presented to demonstrate our main results. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Singular systems have attracted extensive interests because of their extensive applications in a lot of applications like power systems, aerospace engineering, social economic systems, chemical processes, biological systems, network analysis, aircraft control systems, electrical networks, solar thermal central receivers, robotic manipulator systems and so on [1–5]. On the other hand, driven by continuous-time Markov chains, Markovian jump systems (MJSs) are very prevalent in modeling practical systems. In the parameters and structures of these practical systems, some phenomena such as abrupt environment disturbances, changing subsystem interconnections and random component failures or repairs may lead to random abrupt changes [6–13]. Therefore, lots of works have focused on stability analysis, filtering problems and control design for singular Markovian jump systems (SMJSs) [14–21], which assume the full information of the TRs. The authors in [17] considered robust normalization and guaranteed cost control for a class of uncertain singular Markovian jump systems via hybrid impulsive control. The authors in [18] discussed stabilization of descriptor Markovian jump systems with partially unknown transition probabilities. However, the authors did not consider time delay in these two papers. In fact, time delay occurs in many practical systems and may result in the poor performance of the systems, therefore, it is necessary for us to consider the time-delay systems. ∗ Corresponding authors. E-mail addresses: [email protected] (J.H. Park), [email protected] (Y. Kao). https://doi.org/10.1016/j.nahs.2019.02.003 1751-570X/© 2019 Elsevier Ltd. All rights reserved.

34

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

Besides, all physical system models are subject to uncertainties, due to the external disturbance and modeling errors [22–26]. The uncertainties of SMJSs in this paper is considered to exist not only in parameters but also in TRs. In the jumping process, however, it is often hard to truly estimate the TRs that determines the MJS behaviors in a large part. So numerous researchers have had much attention in the synthesis and analysis for MJSs with uncertain TRs. Now there have been many results about three types of descriptions with uncertain TRs. The first type is Bounded Uncertain TRs (BUTRs), where the upper bounds and lower bounds of every TR must be known although the precise value may not be known, such as [27,28]. The second type is Partly Unknown TRs (PUTRs) [29–33], where every TR is completely available or totally unavailable. It is worth mentioning that the above two types are too restrictive to model actual situations due to the precise estimate difficulties for every TR in practice. In this paper, we consider the third type called General Unknown Transition Rates (GUTRs) [9,34,35], where every TR could be totally unknown or only its estimate is known. It is obvious that the BUTR models and PUTR models can be regarded as special cases of GUTR models which have more applications in the last few years. As a control method, adaptive control must be adapted to a controlled system with initially uncertain parameters or initially [36]. Besides, it has an extensive use in lots of fields, such as robot manipulators, power systems, software quality improvement, solar energy collector systems, aerospace applications and so on [37]. Recently, Hu et al. [38] investigated adaptive sliding mode tracking control problem for a flexible air-breathing hypersonic vehicle. Salarieh and Alasty [39] considered adaptive synchronization for two uncertain chaotic systems. Wang et al. [40] discussed H∞ problem for delayed uncertain stochastic nonlinear Markovian jump systems based on adaptive control method. Zhao et al. [41] considered adaptive tracking control issues for uncertain switched nonlinear systems. Li et al. [42] probed adaptive sliding mode control for interval type-2 fuzzy systems. In this work, we aim to propose class of adaptive controller for SMJSs with GUTRs, which has not been greatly investigated and is challenging in many practical applications. For reasons discussed above, this paper concentrates on H∞ control for uncertain stochastic nonlinear singular Markovian jump systems with GUTRs, nonlinear external disturbance and time-varying delay through adaptive control method. We will design a controller with a linear part and an adaptive part to obtain sufficient conditions of the H∞ stochastically admissible for uncertain stochastic nonlinear singular Markovian jump systems with GUTRs. Section 2 describes the system model and introduces some prior knowledge. Section 3 provides a new class of robust controller that consists of an adaptive controller and a linear controller via Lyapunov theory and Itoˆ ’s differential formula, guaranteeing the admissibility of the considered system with the desired disturbance attenuation level. Section 4 uses a numerical example to display the effectiveness of the given methods. At last, Section 5 gives the conclusions. 2. Problem statement and preliminaries We consider the following uncertain stochastic nonlinear singular Markovian jump systems (USNSMJSs) with GUTRs: Edx(t)

{

z(t)

= + =

[(A(r(t)) + ∆A(r(t)))x(t) + (Aτ (r(t)) + ∆Aτ (r(t)))x(t − τ (t)) + f (x, x(t − τ (t)), v, r(t)) (B(r(t)) + ∆B(r(t)))u(r(t))]dt + σ (t , x(t), x(t − τ (t)), r(t))dω(t), C (r(t))x(t)

(1)

where x(t) ∈ Rn and z(t) ∈ Rp denote the system state and control output, respectively; the time-varying delay is described by τ (t), x(t) = ψ (t) ∈ C ([−h¯ , 0], R) is the initial condition, where h¯ = supt ≥0 τ (t) and C ([−h¯ , 0], R) stands for the set of all continuous functions from [−h¯ , 0] to R; ω(t) = (ω1 (t), . . . , ωn (t))T , defined on a complete probability space (Ω , F , {Ft }t ≥0 , P), is a n dimensional Brownian motion with a natural filtration {Ft }t ≥0 . σ : R+ × Rn × Rn → Rn×n is named the noise intensity function matrices. r(t)(t ≥ 0) is a right-continuous Markov chain on a probability space (Ω , F , {Ft }t ≥0 , P), which takes values in a finite state space N = {1, 2, . . . , s} with generator Π = {πij } and is given by

{ P {r(t + ∆) = j|r(t) = i} =

πij ∆ + o(∆), if i ̸= j,

(2)

1 + πii ∆ + o(∆), if i = j,

o(∆)

where ∆ > 0, lim∆→0 ∆∑ = 0, and πij ≥ 0, (i, j ∈ N, j ̸ = i), indicates the TR from mode i at time t to mode j at s time t + ∆, and πii = − j=1,j̸=i πij , for each i ∈ N. u(r(t)) ∈ Rm shows the control input and A(r(t)), Aτ (r(t)), B(r(t)) denote the feedback connection weight matrices, the delayed feedback weight matrices and the input feedback connection weight matrices, respectively. Moreover, in the system parameters, ∆A(r(t)), ∆Aτ (r(t)), ∆B(r(t)) represent the uncertainties. f (x, x(t − τ (t)), v, r(t)) is the continuous nonlinear function of the mode r(t) and v (t) ∈ L2 [0, ∞) is the unknown external disturbance. For every well-posed initial condition, we suppose the right-hand side of system (1) is continuous such that the uniqueness and existence of the solution can be ensured. For every possible value of r(t) = i ∈ N, let A(r(t)) = Ai , Aτ (r(t)) = Aτ i , B(r(t)) = Bi , ∆A(r(t)) = ∆Ai , ∆Aτ (r(t)) = ∆Aτ i , ∆B(r(t)) = ∆Bi where Ai , Aτ i , Bi are known constant matrices and Ai , ∆Aτ i , ∆Bi are uncertain matrices with appropriate dimensions. Besides, we have u(r(t)) = ui (t). Assume that the mode TR matrix Π ≜ (πij ) is generally uncertain. For example, we write the TR matrix for system (1) with s operation modes as follows:

⎡ πˆ 11 + ∆11 ? ⎢ ⎢ .. ⎣ . ?

? ?

.. . πˆ s2 + ∆s2

πˆ 13 + ∆13 πˆ 23 + ∆23 .. .

··· ··· .. .

?

···

?

⎤

πˆ 2s + ∆2s ⎥ ⎥ .. ⎦ . πˆ ss + ∆ss

(3)

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

35

where πˆ ij and ∆ij ∈ [−δij , δij ](δij ≥ 0) represent the estimate value and the estimate error of the uncertain TR πij respectively, and πˆ ij as well as δij are known. ‘‘?’’ is the complete unknown TR indicating the unknown estimate value πˆ ij and estimate i i error bound. For all i ∈ N, the set U i means U i = Uki ∪ Uuk with Uki ≜ {j : The estimate value of πij is known for j ∈ N}, Uuk ≜

{j : The estimate value of πij is unknown for j ∈ N}. Moreover, if Uki ̸= Ø, it could be described as Uki = {ki1 , ki2 , . . . , kimi }, where kimi ∈ N+ denotes the mth bound-known element with the index kimi in the ith row of matrix Π .

Remark 1. It should be noticed that either the BUTR or PUTR models are less general than the above description about uncertain TRs. The following two uncertain models are rewritten to explain: BUTR model(see [27,28]):

⎡ πˆ 11 + ∆11 ⎢πˆ 21 + ∆21 ⎢ .. ⎣ .

πˆ 12 + ∆12 πˆ 22 + ∆22 .. .

⎤ πˆ 1s + ∆1s πˆ 2s + ∆2s ⎥ ⎥ .. ⎦ . πˆ s1 + ∆s1 πˆ s2 + ∆s2 · · · πˆ ss + ∆ss ∑s ∑s with πˆ ij − δij ≥ 0(∀j ∈ N, j ̸ = i), πˆ ii = − j=1,j̸=i πˆ ij , and δii = − j=1,j̸=i δij . PUTR model(see [29–32]): ⎡ ⎤ π11 ? π13 · · · ? ? π23 · · · π2s ⎥ ⎢ ? ⎢ . . .. .. ⎥ .. ⎣ . ⎦ .. . . . . ? πs2 ? · · · πss ··· ··· .. .

(4)

(5)

Obviously, the GUTR model (3) will reduce to the BUTR model (5) when Uki ̸ = Ø ∀i ∈ N, and reduce to the PUTR model (5) if δij = 0, ∀i ∈ N, ∀j ∈ Uki . Obviously, the BUTR or PUTR models are less general than GUTR model (3), which means it is more practicable. It is assumed that the known estimate values of the TRs are defined as follows. Assumption 1. If Uki = N, then πˆ ij − δij ≥ 0, (∀j ∈ N, j ̸ = i), πˆ ii = −

∑s

j=1,j̸ =i

πˆ ij ≤ 0, and δii =

Assumption 2. If Uki ̸ = N, and i ∈ Uki , then πˆ ij − δij ≥ 0, (∀j ∈ Uki , j ̸ = i), πˆ ii + δii ≤ 0, and

∑

∑s

j∈Uki

j=1,j̸ =i

δij > 0;

πˆ ij ≤ 0;

Assumption 3. If Uki ̸ = N and i ∈ / Uki , then πˆ ij − δij ≥ 0, (∀j ∈ Uki ). Note that the above three assumptions ∑are the direct result from the features of TRs, which means they are reasonable (e.g. πij ≥ 0(∀i, j ∈ / N, i ̸= j) and πii = − sj=1,j̸=i πij ). The following assumptions deal with uncertainties, nonlinear disturbances, noise intensity function matrices and timevarying delays: Assumption 4. ∆Ai and ∆Aτ i are unknown time-varying uncertainties and mismatch norm-bounded with

∆Ai = Hi Fi (t)Ei , ∆Aτ i = Hτ i Fτ i (t)Eτ i , where HI , Hτ i , Ei and Eτ i are known real constant matrices with proper dimensions. Fi (t) and Fτ i (t) are unknown matrix and satisfy FiT (t)Fi (t) ≤ I and FτTi (t)Fτ i (t) ≤ I for every mode i ∈ N. ∆Bi is unknown time-varying uncertainties but match norm-bounded with

∆Bi = Bi ∆(i, t)E¯ i , ∥∆(i, t)E¯ i ∥ ≤ ψi < 1, t ≥ 0, where Ei is a known real constant matrix with proper dimensions. For every i ∈ N, ψi is a non-negative known constant. ∆(i, t) is an unknown matrix. And the nonlinear disturbance f (x, x(t − τ (t)), v, i) is satisfied with the so-called matching condition for all x and v : f (x, x(t − τ (t)), v, i) = Bi f1 (x, x(t − τ (t)), v, t), ∥f1 ∥ ≤ l1 ∥x∥ + l2 ∥x(t − τ (t))∥ + l3 ∥v∥ where l1 , l2 , l3 are positive constants being unknown or known, and what is more, the time-varying delay satisfies:

τ (t) ≤ h¯ , τ˙ (t) ≤ h < 1 where h¯ and h are constant numbers. The noise intensity function matrices σ : R+ × Rn × Rn → Rn×n is locally Lipschitz continuous which satisfies the linear growth condition [6]. Moreover, σ meets the following condition: trace[σ T (t , x, x(t − τ (t), i))σ (t , x, x(t − τ (t), i))] ≤ ∥Mi x(t)∥ + ∥Mhi x(t − τ (t))∥.

36

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

Remark 2. In Assumption 4, the Limiting condition of the delay derivative τ˙ (t) ≤ h < 1 is too restrictive, which can be relaxed. Actually, a lot of papers now have no longer asked the delay derivative less than 1 [8]. In some real systems, however, time delays often change slowly. So the constraint of the delay derivative τ˙ (t) ≤ h < 1 still has great significance. In this paper, we can also use the similar method in [8] to do with the relaxed condition. Definition 1. (i) The pair (E , A(rt )) is said to be regular, if det(sE − A(rt )) is not identically zero for each mode rt = i ∈ N. (ii) The pair (E , A(rt )) is said to be impulsive free, if it is regular and deg(det(sE − A(rt ))) = rank(E). (iii) The autonomous jump time-delay system (1) with v (t) = 0 and u(·) = 0 is said to be stochastically stable, if for all finite ψ (t) ∈ Rn defined on [−h¯ , 0] and initial mode r0 ∈ N, there exists an M > 0 satisfying T

∫

xT (t)x(t)dt | ψ, r0 ≤ xT (0)Mx(0).

lim E

T →∞

0

System (1) with v (t) = 0 and u(·) = 0 is said to be stochastically admissible if it is regular, impulse free and stochastically stable. Definition 2. If there exists a linear state feedback as follows: u(t) = K (rt )x(t) where K (rt ) is a gain controller for each rt ∈ S so that the closed-loop system is stochastically admissible for every initial condition (r0 , ψ (·)), then system (1) is said to be stabilizable. Definition 3. Given a real number γ > 0, system (1) is said to possess the γ disturbance attenuation property if for all v (t) ∈ L2 [0, ∞), v ̸= 0, the system (1) is stochastically admissible and the response z : [0, ∞] → Rp under zero initial condition, i.e., ψ = 0, satisfies ∫ ∞ ∫ ∞ E v T (t)v (t)dt z T (t)z(t)dt ≤ γ 2 0

0

or, equivalently, ∞

∫

(z T (t)z(t) − γ 2 v T (t)v (t))dt } ≤ 0, x(0) = 0.

J = E{

(6)

0

∫∞

Remark 3. Let Tz v denote the system from the exogenous input v (t) to the controlled output z(t) and ∥z ∥2 = (E{

∫∞ 1 1 dt }) 2 , ∥v∥2 = { 0 v T (t)v (t)dt } 2 , then the H∞ norm of Tz v is ∥Tz v ∥∞ = supv (t)∈L2 [0,∞)

∥z ∥2 . ∥v∥2

0

z T (t)z(t)

Hence, (6) implies ∥Tz v ∥∞ ≤ γ .

In other words, γ disturbance attenuation implies γ suboptimal H∞ control. We intend to propose the controller ui (t) in this work so that the H∞ disturbance attenuation performance of system (1) with (3) can be achieved regarding the system initial conditions as: ∞

∫

(z T (t)z(t) − γ 2 v T (t)v (t))dt } ≤ 0,

E{

(7)

0

with γ is a given attenuation level. For any bounded input v (t) ̸ = 0, if LV ≤ γ 2 v T (t)v (t) − z T (t)z(t),

(8)

system (1) with (3) is stochastically admissible and (7) is fulfilled based on the analysis in book [6]. Thus, system (1) with (3) is said to be H∞ stochastically admissible with the disturbance attenuation γ . Lemma 1 ([43]). For matrices R = RT , H, E and Q = Q T > 0 with appropriate dimensions, then R + HF (t)E + E T F T (t)H T < 0 satisfying F T (t)F (t) ≤ I, if and only if there is a positive constant ε > 0 so that R + ε −1 HQ −1 H T + ε E T QE < 0. Lemma 2 ([1]). Let C ∈ Rn×n be a positive definite matrices and B ∈ Rn×m , then for any A ∈ Rm×m

( F =

A B

BT C

)

>0

if and only if A > BT C −1 B. 3. Main results We plan to propose an adaptive controller ui (t) in this work so that the system (1) with (3) is H∞ stochastically admissible. Firstly, we assume that the upper bounds l1 , l2 and l3 are completely known, then Theorem 1 is obtained to give a sufficient condition for the H∞ stable of USNSMJSs (1) with GUTRs. Moreover, we give two corollaries from Theorem 1 with no

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

37

uncertain terms or no Markovian jump mode, respectively. Secondly, we assume that the upper bounds l1 , l2 and l3 are unknown, then Theorem 2 is also obtained to give a sufficient condition for the H∞ stable of USNSMJSs (1) with GUTRs. Moreover, we give two corollaries from Theorem 2 with non-measured state variable x(t) or no Markovian jump mode, respectively. Firstly, the upper bounds l1 , l2 and l3 are assumed known and can be used in the linear controller. Then the following Theorem 1 and Corollaries 1–2 are obtained. Theorem 1. The system (1) with (3) is H∞ stochastically admissible, if there is a set of positive definite matrices Pi = PiT > 0, Q > 0, R > 0 and a series of positive numbers εi1 , εi2 , ρi such that: E T Pi = PiT E ≥ 0, Case I: i ∈ / that

Uki

and

(9)

= { ,...,

Uki

ki1

Γ11

⎛

⎜ ATτ i Pi E ⎜ ⎜ HiT Pi E ⎜ ⎜ H T Pi E Γ1 = ⎜ ⎜ E T (P τi i − P )E l ⎜ k1 ⎜ .. ⎜ ⎝ . E T (Pki − Pl )E m

kimi

n× n

}, there is a series of positive definite matrices Tij ∈ R

(i ∈ / Uk , j ∈ Uk ) and l ∈ Uuk such

E T Pi Aτ i

E T Pi Hi

E T Pi Hτ i

E T (Pki − Pl )E

···

E T (Pki − Pl )E

Φi,2

0

0 0 0

−εi1 I

0 0

0 0

−1 −εi2 I

.. .

.. .

0

0

0

··· ··· ··· ··· .. . ···

0 0 0 0

.. .

0 0 0 −Tiki

−1

.. .

1

0

i

i

⎞

mi

1

0

i

.. . −Tikim

i

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ < 0. ⎟ ⎟ ⎟ ⎟ ⎠

(10)

i

and E T P i E ≤ ρi I

(11)

hold with the given constant ηˆ 1 > 0, where Γ11 = Ξ +

ρ

T i Mi Mi

CiT Ci 1I i i Uk Uuk

Ω11

⎛

⎜ ATτ i Pi E ⎜ ⎜ HiT Pi E ⎜ ⎜ H T Pi E Ω1 = ⎜ ⎜ E T (P τi i − P )E l ⎜ k1 ⎜ .. ⎜ ⎝ . E T (Pki − Pl )E m

T

(Pj − Pl )E +

∑

δij2

j∈Uki 4

where Ω11 = Ξ +

∑

j∈Uki

−1 T

−1 T Ei Ei + Tij , Ξ = E T Pi Ai + ATi Pi E + Q +εi1

E T Pi Aτ i

E T Pi Hi

Pi H τ i

E T (Pki − Pl )E

···

E T (Pki − Pl )E

Φi,2

0

0 0 0

−εi1 I

0 0

0 0

−1 −εi2 I

.. .

.. .

0

0

0

··· ··· ··· ··· .. . ···

0 0 0 0

.. .

0 0 0 −Riki l

−1

0

i

,

πˆ

ij E j∈Uki T M M i hi hi

+ dR, and Φi,2 = −(1 − h)Q + ρ + εi2 Ehi Eτ i + ηˆ 1 I. i ) ̸= ∅ and Uki = {ki1 , . . . , kimi }, there exists a set of positive definite matrices Rij l ∈ Rn×n (i, j ∈ Uki , l ∈ Uuk

+ ηˆ + ,

Case II: i ∈ such that

∑

1

.. .

0

1

⎞

mi

.. . −Rikim l

⎟ ⎟ ⎟ ⎟ ⎟ ⎟<0 ⎟ ⎟ ⎟ ⎟ ⎠

i

πˆ ij E T (Pj − Pl )E +

∑

δij2

j∈Uki 4

Rijl .

= ∅, there exists a set of positive definite matrices Wij ∈ Rn×n (i, j ∈ Uki ) such that ⎛ ⎞ Υ11 E T Pi Aτ i E T Pi Hi Υ11 E T (P1 − Pi )E · · · E T (Ps − Pi )E T Φi,2 0 0 0 ··· 0 ⎜ A τ i Pi E ⎟ ⎜ ⎟ −1 0 −εi1 I 0 0 ··· 0 ⎜ HiT Pi E ⎟ ⎜ ⎟ −1 T H P E 0 0 −ε I 0 · · · 0 ⎜ ⎟ < 0. i τi i2 Υ1 = ⎜ ⎟ T 0 0 0 −Wi1 ··· 0 ⎜E (P1 − Pi )E ⎟ ⎜ ⎟ .. .. .. .. .. .. .. ⎝ ⎠ . . . . . . . T E (Ps − Pi )E 0 0 0 0 ··· −Wis

Case III: i ∈

Uki

i Uuk

where

Υ11 = Ξ +

s ∑

(12)

( πˆ ij E (Pj − Pi )E + T

j=1,j̸ =i

δij2 4

) Wij

.

According to Assumptions 1–4, we design the controller via a switching function ξi (x) = BTi Pi x(t) as ui (x, t) = uiL (t) + uia (t),

(13)

38

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51 l2 +l2

θˆ

i where the linear controller uiL = − 2(11−ψ 2)ηˆ ξi (x) and the adaptive controller uia = − 2(1−ψ ξ (x) with: ) i i

i

1

θ˙ˆi = −ηˆ 2 (θˆi − θ¯ ) + ∥ξi ∥2 , in which θ¯ =

l23 γ2

(14)

> 0 and γ > 0, ηˆ 2 > 0 is an adjustable scalar.

Proof. We consider a type of Lyapunov–Krasovskii function as follows: V (x, r(t)) =

4 ∑

Vm (x, r(t))

(15)

m=1

∫t

∫0 ∫t

where V1 (x, r(t)) = xT (t)E T P(r(t))Ex(t) , V2 (x, r(t)) = t −τ (t) xT (s)Qx(s)ds, V3 (x, r(t)) = 12 θ˜ 2 , and V4 (x, r(t)) = −d t +θ xT (s) Rx(s)dsdθ , in which P(r(t)) = Pi , (r(t) = i) is a symmetric positive constant matrices, Q is a positive constant matrices, and θ˜i = θˆi − θ¯ . One can get the following equation along the trajectories of system (1) s

∑ ∂V ⏐ ∂V πij V (x(t), t , i) + x˙ T (t) ⏐γ =i + t ∂t ∂x

LV (x, i) =

j=1

s

= xT (t)(E T Pi Ai + ATi Pi E +

∑

πij E T Pj E + Q )x(t) + 2xT (t)E T Pi ∆Ai x(t)

j=1

+ 2x (t)E Pi Aτ i x(t − τ (t)) − (1 − τ˙ (t))xT (t − τ (t))Qx(t − τ (t)) T

T

+ 2xT (t)E T Pi ∆Aτ i x(t − τ (t)) + 2xT (t)E T Pi Bi (uiL (t) + uia (t)) + 2xT (t)Pi E T ∆Bi (uiL (t) + uia (t)) + 2xT (t)Pi E T f (x, x(t − τ (t)), v, t) + trace[σ T (t , x, x(t − τ (t), i))E T Pi E σ (t , x, x(t − τ (t), i))] ∫ t n ∑ 1 xT (s)Rx(s)ds + πij θ˜ 2 + θ˜i θ˙˜i + dxT (t)Rx(t) − t −d

+

s ∑

πij

t

∫

xT (s)Qx(s)ds + t −τ (t)

j=1

2

j=1 s ∑

πij

∫

0

∫

xT (s)Rx(s)dsdθ. t +θ

−d

j=1

t

Here, note that s ∑

πij

∫

πij

∫

j=1

0

−d

j=1 s ∑

xT (s)Qx(s)ds =

t −τ (t)

j=1 s ∑

t

∫

⎛

t

x (s)Rx(s)dsdθ = ⎝ T

t +θ

πij θ˜ 2 = ⎝ 2

πij

t

)(∫

xT (s)Qx(s)ds

s ∑

⎞ πij ⎠

j=1 s ∑ j=1

)

= 0,

t −τ (t)

j=1

⎛ 1

(∑ s

0

∫

t

∫

−d

xT (s)Rx(s)dsdθ = 0, t +θ

⎞ 1

πij ⎠ θ˜ 2 = 0. 2

According to Assumption 4 and Lemma 1, we can obtain 2xT (t)E T Pi ∆Ai x(t) = 2xT (t)E T Pi Hi Fi (t)Ei x(t) −1 T ≤ εi1 xT (t)E T Pi Hi HiT Pi Ex(t) + εi1 x (t)EiT Ei x(t),

and 2xT (t)E T Pi ∆Aτ i x(t − τ (t)) = 2xT (t)E T Pi Hτ i Fτ i (t)Ehi x(t − τ (t)) −1 T T ≤ εi2 xT (t)E T Pi Hτ i HτTi Pi Ex(t) + εi2 x (t − τ (t))Ehi Ehi x(t − τ (t)).

Using Assumption 4 and the definition of uiL (t), we derive 2xT (t)E T Pi Bi uiL (t) + 2xT (t)E T Pi ∆Bi uiL (t)

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

39

≤ 2xT (t)E T Pi Bi uiL (t) + 2∥xT (t)E T Pi Bi ∥∥∆i (t)Ei ∥∥uiL (t)∥ l2 + l22 ψi (l21 + l22 ) ≤− 1 ∥ξi ∥2 + ∥ξi ∥2 2(1 − ψi )ηˆ 1 2(1 − ψi )ηˆ 1 l2 + l22 ∥ξi ∥2 . ≤ 1 ηˆ 1

(16)

By the definition of uia (t) with the adaptive law (14) and Assumption 4 one can get that 2xT (t)E T Pi Bi uia (t) + 2xT (t)E T Pi ∆Bi uia (t)

=−

θˆi 1 − ψi

xT (t)E T Pi Bi BTi Pi Ex(t) +

θˆi 1 − ψi

∥xT (t)E T Pi Bi ∥∥∆(t)E¯ i ∥∥BTi Pi Ex(t)∥

ψi θˆi θˆi ∥ξi ∥2 + ∥ξi ∥2 1 − ψi 1 − ψi = −θˆi ∥ξi ∥2 . ≤−

(17)

In view of Assumption 4, it follows 2xT (t)E T Pi Bi f1 (x, x(t − τ (t)), v, t) ≤ 2∥xT (t)E T Pi Bi ∥(l1 ∥x∥ + l2 ∥x(t − τ (t))∥ + l3 ∥v∥)

¯ i ∥2 + γ 2 v T (t)v (t). ≤ 2l1 ∥ξi ∥∥x∥ + 2l2 ∥ξi ∥∥x(t − τ (t))∥ + θ∥ξ

(18)

Again, employing adaptive law (14) and (16)–(18), we can derive

˙

2xT (t)E T Pi (Bi + ∆Bi )uiL (t) + 2xT (t)E T Pi (Bi + ∆Bi )uia (t) + 2xT (t)E T Pi f (x, x(t − τ (t)), v, i) + θ˜i θ˜i

≤−

l21 + l22

ηˆ 1

¯ i ∥2 ∥ξi ∥2 − θˆi ∥ξi ∥2 + 2l1 ∥ξi ∥∥x∥ + 2l2 ∥ξi ∥∥x(t − τ (t))∥ + θ∥ξ

+ γ 2 v T (t)v (t) + (θˆi − θ¯ )[−ηˆ 2 (θˆi − θ¯ ) + ∥ξi ∥2 ] ≤ ηˆ 1 xT (t)x(t) + ηˆ 1 xT (t − τ (t))x(t − τ (t)) + γ 2 v T (t)v (t). By (11) and Assumption 4, it follows that trace[σ T (t , x, x(t − τ (t), i))E T Pi E σ (t , x, x(t − τ (t), i))]

≤ ρi xT (t)MiT Mi x(t) + ρi xT (t − τ (t))MhiT Mhi x(t − τ (t)). Then, we obtain LV (x, i) ≤ xT (t)(E T Pi Ai + ATi Pi E + Q + ρi MiT Mi −1 T + εi1 E T Pi Hi HiT Pi E + εi2 E T Pi Hτ i HhiT Pi E + εi1 Ei Ei

+ ηˆ 1 I + CiT Ci +

n ∑

πij E T Pj E + dR)x(t) + 2xT (t)E T Pi Aτ i x(t − τ (t))

j=1

−1 T + x (t − τ (t))(−(1 − h)Q + ρi MhiT Mhi + εi2 Ehi Eτ i + ηˆ 1 I)x(t − τ (t)) T

− xT (t)CiT Ci x(t) + γ 2 v T (t)v (t). According to the definition of a GUTR matrix in (3), we now discuss the above inequality under Assumptions 1–3 in three cases. Case I: i ∈ / Uki . It should be noted that in this case

∑

i l ∈ Uuk , l ̸= j such that E T Pl E − E T Pj E ≥ 0. We define

i ,j̸ =i j∈Uuk

⎛ ⎞ ∑ ⎜∑ ⎟ Φi ≜ Ξ + E T ⎝ πij Pj + πii Pi + πij Pj ⎠ E j∈Uki

i ,j̸ =i j∈Uuk

πij = −πii −

∑

j∈Uki

πij and πij ≥ 0, and there must be an

(19)

40

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

then we have

∑

Φi = Ξ +

∑

πij E T Pj E + πii E T Pi E +

j∈Uki

πij E T Pj E

i ,j ̸ =i j∈Uuk

⎛

⎞

∑ ⎟ ⎜ πij ⎠ Pl E πij E T Pj E + πii E T Pi E + E T ⎝−πii −

≤Ξ+

∑

=Ξ+

∑

j∈Uki

j∈Uki

πij E T (Pj − Pl )E

(20)

j∈Uki

=Ξ+

∑

(πˆ ij + ∆ij )E T (Pj − Pl )E

j∈Uki

=Ξ+

∑

πˆ ij E T (Pj − Pl )E +

∑

j∈Uki

∆ij E T (Pj − Pl )E .

j∈Uki

Besides, due to Lemma 1, one can get

∑

∆ij E T (Pj − Pl )E

j∈Uki

=

∑ [1 2

j∈Uki

∆ij E T (Pj − Pl )E +

∑ δij2

1 2

∆ij E T (Pj − Pl )E

[

≤

j∈Uki

4

] (21)

] Tij + E T (Pj − Pl )ETij−1 E T (Pj − Pl )E .

From (19)–(21), we have

Φi ≤ Ξ +

∑

πˆ ij E T (Pj − Pl )E +

j∈Uki

∑ δij2 j∈Uki

4

∑[

E T (Pj − Pl )ETij−1 E T (Pj − Pl )E .

]

(22)

E T (Pj − Pl )ETij−1 E T (Pj − Pl )E < 0.

(23)

Tij +

j∈Uki

Hence, Φi < 0 holds if

Ξ+

∑

πˆ ij E T (Pj − Pl )E +

j∈Uki

∑ δij2 j∈Uki

4

Tij +

∑[

]

j∈Uki

If there exist Pi , Q , R, εi1 , εi2 , ρi , Tij such that the LMIs (9)–(11) hold, then by Lemma 2, (23) holds. It can be seen LV (x) ≤ γ 2 v T (t)v (t) − z T (t)z(t).

It follows from (8) that system (1) is stochastically admissible at zero with H∞ disturbance level γ . i i i Case II: i ∈ Uki and Uuk ̸= Ø, there must be an l ∈ Uuk such that E T P(l)E − E T P(j)E ≥ 0, for ∀j ∈ Uuk . It holds that

⎛

⎞ ∑ ∑ ⎜ ⎟ Θi ≜ Ξ + E T ⎝ πij Pj + πij Pj ⎠ E j∈Uki

i j∈Uuk

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

41

Then

⎞ ∑ ∑ ⎟ ⎜ πij Pj + πij Pl ⎠ E Θi ≤ Ξ + E T ⎝ ⎛

j∈Uki

∑

=Ξ+

πij E T Pj E −

j∈Uki

i j∈Uuk

∑

πij E T Pl E

j∈Uki

=Ξ+

∑

=Ξ+

∑

(πˆ ij + ∆ij )E T (Pj − Pl )E

j∈Uki

πˆ ij E T (Pj − Pl )E +

j∈Uki

∑

∆ij E T (Pj − Pl )E .

j∈Uki

Moreover, because of Lemma 2, we can obtain that

∑

∆ij E T (Pj − Pl )E

j∈Uki

=

∑ [1 2

j∈Uki

∑ δ [

≤

j∈Uki

∆ij E T (Pj − Pl )E +

2 ij

4

1 2

∆ij E T (Pj − Pl )E

] (24)

] 1 T Rijl + E T (Pj − Pl )ER− ijl E (Pj − Pl )E .

Also, we have

Θi ≤ Ξ +

∑

πˆ ij E (Pj − Pl )E + T

j∈Uki

[ ∑ δij2 4

j∈Uki

] Rijl + E (Pj − Pl )ERijl E (Pj − Pl )E . T

−1 T

(25)

Hence, Θi < 0 holds if

Ξ+

∑

πˆ ij E T (Pj − Pl )E +

j∈Uki

∑ δij2 4

j∈Uki

Rijl +

∑[

1 T E T (Pj − Pl )ER− ijl E (Pj − Pl )E < 0.

]

(26)

j∈Uki

If there exist Pi , Q , R, εi1 , εi2 , ρi , Rijl such that the LMIs (9) and (12) hold, thus by Lemma 2, (26) holds. It can be seen that LV (x) ≤ γ 2 v T (t)v (t) − z T (t)z(t)

It follows from (8) that system (1) is stochastically admissible at zero with H∞ disturbance level γ . i Case III: i ∈ Uki and Uuk = Ø. Similarly, it holds that s ∑

Υi ≜ Ξ +

πij E T Pj E + πii E T Pi E

j=1,j̸ =i s ∑

=Ξ+

πij E T (Pj − Pi )E

j=1,j̸ =i s ∑

=Ξ+

(πˆ ij + ∆ij )E T (Pj − Pi )E

j=1,j̸ =i s ∑

≤Ξ+

s ∑ 1 [ δij2 Wij + E T (Pj − Pi )EWij−1 E T (Pj − Pi )E ].

πˆ ij E T (Pj − Pi )E +

j=1,j̸ =i

j=1,j̸ =i

4

Hence, Υi < 0 holds if

Ξ+

s ∑ j=1,j̸ =i

πˆ ij E T (Pj − Pl )E +

s ∑ δij2 j=1,j̸ =i

4

Wij +

s ∑ [

E T (Pj − Pl )EWij−1 E T (Pj − Pl )E < 0.

]

(27)

j=1,j̸ =i

If there exist Pi , Q , R, Wij , εi1 , εi2 , ρi such that the LMIs (9) and (13) holds, then by Lemma 2, (27) holds. It can be seen that LV (x) ≤ γ 2 v T (t)v (t) − z T (t)z(t).

42

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

It follows from (8) that system (1) with (3) is stochastically admissible at zero with H∞ disturbance level γ .

■

Remark 4. Note that we can choose the disturbance attenuation level on the basis of the demand for practical applications. Once the disturbance attenuation level is given, we can propose the adaptive controller connected with the real level. In view of this, the H∞ control problems studied in our work has a wider range of applications than the existing papers. When ∆Ai = 0, ∆Aτ i = 0 and ∆Bi = 0, namely there is no uncertain terms in the system parameters, it is obvious to see the following corollary from Theorem 1: Corollary 1. Assume that ∆Ai = 0, ∆Aτ i = 0 and ∆Bi = 0 in system (1). Thus system (1) with (3) is H∞ stochastically admissible, if there is a series of positive definite matrices Pi = PiT > 0, Q > 0, R > 0 and a set of positive constants ρi , ηˆ 1 , so that: E T Pi = PiT E ≥ 0, Case I: i ∈ / that

Uki

and

Uki

(28)

= { ,..., ki1

kimi

(i ∈ / Uk , j ∈ Uk ) and l ∈ Uuk such i

i

n×n

}, there exist a set of positive definite matrices Tij ∈ R

i

˜1 Γ ˜11 Γ

⎛

E T PAτ i

E T (Pki − Pl )E

···

E T (Pki − Pl )E

0

··· ··· .. . ···

0 0

−(1 − h)Q + ρi MhiT Mhi + ηˆ 1 I

⎜ A T Pi E ⎜ ⎜ E T (P τi i − P )E l k1 =⎜ ⎜ .. ⎜ ⎝ . E T (Pki − Pl )E m

0

.. .

−Tiki 1 .. .

0

0

⎞

mi

1

⎟ ⎟ ⎟ ⎟ < 0. ⎟ ⎟ ⎠

.. . −Tikim

i

(29)

i

and E T P i E ≤ ρi I where

˜11 = Ξ ˜+ Γ

∑

πˆ ij E T (Pj − Pl )E +

j∈Uki

∑ δij2 4

j∈Uki

Tij ,

˜ = E T Pi Ai + ATi Pi E + Q + ρi MiT Mi + ηˆ 1 I + CiT Ci + dR. Ξ i i ) Case II: i ∈ Uki , Uuk ̸= ∅ and Uki = {ki1 , . . . , kimi }, there exists a series of positive definite matrices Rij l ∈ Rn×n (i, j ∈ Uki , l ∈ Uuk such that

˜11 Ω

⎛

⎜ AT Pi E ⎜ ⎜ E T (P τi i − P )E l k1 ˜1 = ⎜ Ω ⎜ .. ⎜ ⎝ . E T (Pki − Pl )E m

E T PAτ i

E T (Pki − Pl )E

···

E T (Pki − Pl )E

−(1 − h)Q + ρi MhiT Mhi + ηˆ 1 I

0 −Riki l

··· ··· .. . ···

0 0

0

.. .

1

.. .

0

1

0

⎞

mi

⎟ ⎟ ⎟ ⎟ < 0. ⎟ ⎟ ⎠

.. . −Rikim l

(30)

i

i

where

˜11 = Ξ ˜+ Ω

∑

πˆ ij E T (Pj − Pl )E +

j∈Uki

∑ δij2 4

j∈Uki

Rijl .

i Case III: i ∈ Uki , Uuk = ∅, there is a series of positive definite matrices Wij ∈ Rn×n (i, j ∈ Uki ) such that

⎛

˜11 Υ T

⎜ Aτ i Pi E ⎜E T (P − P )E 1 i ˜ Υ1 = ⎜ ⎜ .. ⎝ . E T (Ps − Pi )E

E T PAτ i −(1 − h)Q + ρi MhiT Mhi + ηˆ 1 I 0

.. .

0

where

˜11 = Ξ ˜+ Υ

s ∑ j=1,j̸ =i

( πˆ ij E (Pj − Pi )E + T

δij2 4

) Wij

.

E T (P1 − Pi )E 0 −Wi1

.. .

··· ··· ··· .. .

0

···

E T (Ps − Pi )E 0 ⎟ ⎟ 0 ⎟ < 0. ⎟

⎞

.. .

−Wis

⎠

(31)

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

43

Through a switching function ξi (x) = BTi Pi x(t), we choose the controller ui (x, t) = uiL (t) + uia (t) according to Assumptions 1– 4, where the linear controller uiL = −

−ηˆ 2 (θˆi − θ¯ ) + ∥ξi ∥2 , in which θ¯ =

l23

γ2

l21 +l22 2ηˆ 1

ˆ ξi (x) and the adaptive controller uia = − θ2i ξi (x) with the adaptive law θˆ˙i =

> 0 and ηˆ 2 > 0 is an adjustable constant.

One can have the following corollary if there is no Markovian jump mode in system (1): Corollary 2. Assume that there is no Markovian jump mode in system (1), that is A(r(t)) = A and so on. If there are positive constants ε1 , ε2 , ρ and positive definite matrices P = P T => 0, Q > 0, R > 0 so that: E T P = P T E ≥ 0,

⎛

Φ1

(32)

E T PAτ

0 0

Φ2

T

⎜ Aτ PE ⎜ 0 ⎜ ⎝H T PE

Φ3

0 0 0

HhT PE

0 0

E T PH 0 0 −ε1−1 I 0

E T PHh 0 ⎟ ⎟ 0 ⎟ < 0, 0 ⎠ −ε2−1 I

⎞

(33)

and E T PE ≤ ρ I

(34)

where

Φ1 = E T PA + AT PE + Q + ε1−1 E T E + ρ M T M + ηˆ 1 I + C T C + dR and

Φ2 = −(1 − h)Q + ε2−1 EhT Eh + ρ MhT Mh + ηˆ 1 I . Based on Assumptions 3–4 and the function ξ (x) = BT Px(t), choose the controller as u(x, t) = uL (t) + ua (t), where the linear l2 +l2

˙

θ ξ (x) with the adaptive law θˆ = −ηˆ 2 (θˆ − θ¯ ) + ∥ξ (x)∥2 , controller uL = − 2(11−ψ2)ηˆ ξ (x) and the adaptive controller ua = − 2(1−ψ )

in which θ¯ =

l23 γ2

ˆ

1

> 0 and ηˆ 2 > 0 is an adjustable constant. Thus system (1) is H∞ stochastically admissible.

Secondly, the upper bounds l1 , l2 and l3 cannot be used directly in the linear controller, since they are assumed to be unknown. For this reason, we propose an adaptive controller to ensure the system is H∞ controllable. Theorem 2. The system (1) with (3) is H∞ stochastically admissible, if there are a set of positive constants εi1 , εi2 , ρi and a set of positive definite matrices Pi = PiT > 0, Q > 0, R > 0 such that LMIs (9)–(13) hold. According to Assumptions 1–4, design the following controller as ui (x, t) = −

θˆ1 + θˆ2 θˆ3 ξi (x) − ξi (x), 2(1 − ψi )ηˆ 1 2(1 − ψi )

(35)

where the switching function ξi (x) = BTi Pi x(t) and the adaptive laws 1 θ˙ˆ1 = −τˆ1 (θˆ1 − θ¯1 ) + ∥ξi ∥2 , ηˆ 1 1 θ˙ˆ2 = −τˆ2 (θˆ2 − θ¯2 ) + ∥ξi ∥2 , ηˆ 1 θ˙ˆ3 = −τˆ3 (θˆ3 − θ¯3 ) + ∥ξi ∥2 in which θ¯1 = l21 , θ¯2 = l22 , θ¯3 =

l23

γ2

> 0 and τˆ1 > 0, τˆ2 > 0, τˆ3 > 0 are adjustable constants.

(36)

44

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

Proof. At first, we choose the following Lyapunov function candidate: 4 ∑

V (x, i) =

Vm (x, i)

m=1

∫t

where V1 (x, i) = xT (t)E T Pi Ex(t), V2 (x, i) =

θ˜ + θ˜22 + θ˜32 ), and V4 (x, i) = dsdθ in which Pi are positive constant matrices, θ˜1 = θˆ1 − θ¯1 , θ˜2 = θˆ2 − θ¯2 , and θ˜3 = θˆ3 − θ¯3 . t −τ (t)

xT (s)Qx(s)ds, V3 (x, i) =

1 ( 2 2 1

∫0 ∫t −d t +θ

xT (s)Rx(s)

Along the trajectories of system (1), one can obtain that s

LV (x, i) =

∑ ∂V ∂V ⏐ + x˙ T (t) ⏐γ =i + πij V (x, i) t ∂t ∂x j=1

= xT (t)(E T Pi Ai + ATi Pi E +

s ∑

πij E T Pj E + Q )x(t) + 2xT (t)E T Pi ∆Ai x(t)

j=1

+ 2xT (t)E T Pi Aτ i x(t − τ (t)) − (1 − τ˙ (t))xT (t − τ (t))Qx(t − τ (t)) + 2xT (t)E T Pi ∆Aτ i x(t − τ (t)) + 2xT (t)E T Pi Bi (uiL (t) + uia (t)) + 2xT (t)E T Pi ∆Aτ i x(t − τ (t)) + 2xT (t)E T Pi Bi ui (x, t) + 2xT (t)E T Pi ∆Bi ui (x, t) + 2xT (t)E T Pi f (x, x(t − τ (t)), v, t) + trace[σ T (t , x, x(t − τ (t), i))E T Pi E σ (t , x, x(t − τ (t), i))] ∫ t ˙ ˙ ˙ T ˜ ˜ ˜ ˜ ˜ ˜ xT (s)Rx(s)ds + θ1 θ1 + θ2 θ2 + θ3 θ3 + x (t)Rx(t)d − t −d s ∑

+

1

πij (θ˜ + θ˜ + θ˜ 2

j=1 s ∑

+

2 1

πij

2 3)

+

s ∑

πij

∫

t

∫

xT (s)Qx(s)ds t −τ (t)

j=1 0

∫

t

xT (s)Rx(s)dsdθ. t +θ

−d

j=1

2 2

From the relationship s ∑

πij

∫

T

x (s)Qx(s)ds =

(∑ s

t −τ (t)

j=1 s ∑

t

πij

∫

0

−d

j=1

x (s)Rx(s)dsdθ = ⎝ T

t +θ

1

πij (θ˜12 + θ˜22 + θ˜32 ) = ⎝

j=1

s ∑

2

= 0,

⎞ πij ⎠

∫

0

∫

−d

t

xT (s)Rx(s)dsdθ = 0, t +θ

⎞

s

∑

)

T

x (s)Qx(s)ds

j=1

⎛

s

∑

⎛

t t −τ (t)

j=1

t

∫

πij

)(∫

1

πij ⎠ (θ˜12 + θ˜22 + θ˜32 ) = 0,

j=1

2

and by using Assumption 4 and the description of ui (x, t) in (35), we obtain 2xT (t)E T Pi Bi ui (x, t) + 2xT (t)E T Pi ∆Bi ui (x, t) ≤ 2xT (t)E T Pi Bi ui (x, t) + 2∥xT (t)E T Pi Bi ∥∥∆(i, t)Ei ∥∥ui (x, t)∥ θˆ3 θˆ1 + θˆ2 ψi (θˆ1 + θˆ2 ) ψi θˆ3 2 2 2

≤−

≤−

(1 − ψi )ηˆ 1

∥ξi ∥ +

(1 − ψi )ηˆ 1

θˆ1 + θˆ2 ∥σi ∥2 − θˆ3 ∥ξi ∥2 . ηˆ 1

∥ξi ∥ −

1 − ψi

∥ξi ∥ +

1 − ψi

∥ξi ∥2

(37)

From (34), (18) and (36), we get

˙

˙

˙

2xT (t)E T Pi (Bi + ∆Bi )ui (t) + 2xT (t)E T Pi f (x, x(t − τ (t)), v, t) + θ˜1 θ˜1 + θ˜2 θ˜2 + θ˜3 θ˜3

≤−

θˆ1 + θˆ2 ¯ i ∥2 ∥ξi ∥2 − θˆ3 ∥ξi ∥2 + 2l1 ∥ξi ∥∥x∥ + 2l2 ∥ξi ∥∥x(t − τ (t))∥ + θ∥ξ ηˆ 1

+ γ 2 v T (t)v (t) + θ˜1 θ˙˜1 + θ˜2 θ˙˜2 + θ˜3 θ˙˜3

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

≤−

1

ηˆ 1

45

[(l1 ∥ξi ∥ − ηˆ 1 ∥x∥)2 + (l2 ∥ξi ∥ − ηˆ 1 ∥x(t − τ (t))∥)2 ] + ηˆ 1 ∥x∥2

+ ηˆ 1 ∥x(t − τ (t))∥2 + γ 2 ∥v∥2 − + (θˆ1 − θ¯1 )[−τˆ1 (θˆ1 − θ¯1 ) + + (θˆ2 − θ¯2 )[−τˆ2 (θˆ2 − θ¯2 ) +

1

ηˆ 1 1

ηˆ 1

θ¯1 θˆ1 ∥ξi ∥2 + ∥ξi ∥2 ηˆ 1 ηˆ 1

∥ξi ∥2 ] −

θˆ2 θ¯2 ∥ξi ∥2 + ∥ξi ∥2 ηˆ 1 ηˆ 1

∥ξi ∥2 ] − θˆ3 ∥ξi ∥2

+ θ¯3 ∥ξi ∥2 + (θˆ3 − θ¯3 )[−τˆ3 (θˆ3 − θ¯3 ) + ∥ξi ∥2 ] ≤ ηˆ 1 xT (t)x(t) + ηˆ 1 xT (t − τ (t))x(t − τ (t)) + γ 2 v T (t)v (t). Therefore, it follows that LV (x, i) ≤ xT (t)

(

E T Pi Ai + ATi Pi E + Q + ρi MiT Mi + εi1 E T Pi Hi HiT Pi E

−1 T Ei Ei + ηˆ 1 I + + εi2 E T Pi Hτ i HτTi Pi E + εi1

s ∑

) πij E T Pj E + dR x(t)

j=1

+x

T

(t)CiT Ci x(t)

+ 2x (t)E Pi Aτ i x(t − τ (t)) T

T

+ xT (t − τ (t))(−(1 − h)Q + ρi MhiT Mhi −1 T + εi2 Ehi Eτ i + ηˆ 1 I)x(t − τ (t)) − xT (t)CiT Ci x(t) + γ 2 v T (t)v (t).

It is easy to see that the system (1) is stochastically admissible with the H∞ disturbance level γ from a similar proof of Theorem 1. It is sometimes impossible to design the switching function by using x(t) in reality since the state variable x(t) is not always measurable. To solve this problem, we can use the output vector y(t) = D(r(t))x(t) which is measurable. Moreover, if there is G(r(t)) satisfying B(r(t))P(r(t)) = G(r(t))D(r(t)), the following corollary is obtained: Corollary 3. If there is a set of positive numbers εi1 , εi2 , ρi and a set of positive definite matrices Pi = PiT > 0, Q > 0, R > 0 such that (9)–(13) hold. Choose the controller according to Assumptions 3–4 and a switching function ϕi (x) = GTi y(t) as ui (t) = ˆ3 ˆ +θˆ2 − 2(1θ1−ψ ϕi (x) − 2(1θ−ψ ϕ (x), where θ˙ˆ1 = −τˆ1 (θˆ1 − θ¯1 ) + ηˆ1 ∥ϕi ∥2 , θ˙ˆ2 = −τˆ2 (θˆ2 − θ¯2 ) + ηˆ1 ∥ϕi ∥2 , θ˙ˆ3 = −τˆ3 (θˆ3 − θ¯3 ) + ∥ϕi ∥2 in )ηˆ ) i i

1

i

which θ¯1 = l21 , θ¯2 = l22 , θ¯3 = admissible.

l23

γ2

1

1

> 0 and τˆ1 > 0, τˆ2 > 0, τˆ3 > 0 are adjustable constants. Thus system (2) is H∞ stochastically

Similarly, one can obtain the following corollary if there is no Markovian jump mode in system (1): Corollary 4. If there are positive numbers ε1 , ε2 , ρ and positive definite matrices P = P T > 0, Q > 0, R > 0 such that (32), θˆ +θˆ

(33) and (34) hold. Choose the controller u(x, t) = − 2(11−ψ )2ηˆ ξ (x) −

ξ (x) = B Px(t) and the adaptive laws θ˙ˆ1 = −τˆ1 (θˆ1 − θ¯1 ) + T

which θ¯1 = l21 , θ¯2 = l22 , θ¯3 = admissible.

l23

γ2

1

1

ηˆ 1

θˆ3

ξ (x) according to Assumptions 3–4 and via a function ˙ ˆ ∥ξ ∥ , θ2 = −τˆ2 (θˆ2 − θ¯2 ) + ηˆ1 ∥ξ ∥2 , θ˙ˆ3 = −τˆ3 (θˆ3 − θ¯3 ) + ∥ξ ∥2 in 2(1−ψ )

2

1

> 0 and τˆ1 > 0, τˆ2 > 0, τˆ3 > 0 are adjustable constants. Then system (2) is H∞ stochastically

Remark 5. In Theorem 1, Corollaries 1 and 2, the upper bounds l1 , l2 , l3 are assumed known constants. In Theorem 2, Corollaries 3 and 4, the upper bounds l1 , l2 , l3 are assumed to be unknown in the linear controller and cannot be used directly. 4. A numerical example This section provides a numerical example to verify the effectiveness and feasibility of the designed methods. Assume that the TR matrix for system (1) with 4 operation modes is expressed as follows

Π=

[ −6.8 + ∆11 ? 3 + ∆31

4.5 + ∆12 −7.2 + ∆22 ?

2.3 + ∆13 ? , ?

]

with ∆ = 0.1, ∆11 ∈ [−0.1, 0.1], ∆12 ∈ [−0.06, 0.06], ∆13 ∈ [−0.04, 0.04], ∆22 ∈ [−0.05, 0.05], ∆31 ∈ [−0.02, 0.02].

46

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

And, the system parameters are considered as follows: 1 0 0

[ E=

0 1 0

]

[ −2.3

0 3 1

0 1

A3 =

7.5 0 0

0 0 , A1 = 0

0 −2.3 0

[

0 0 , A2 = 9.2

]

1 1 , ∆A1 = H1 F1 (t)E1 , H1 = 3 .2

]

F1 (t) = 0.25 cos(2t), E1 = 1.2

[

∆A3 = H3 F3 (t)E3 , H3 =

[

0

[ −4 0 0

Aτ 3 =

0 2 0

−1

0.56

0 ,

0 3.6 0.26

0 0 , 2.3

−3 −2 0.5

[

0 0 , ∆Aτ 1 = Hτ 1 Fτ 1 (t)Eh1 , Hτ 1 = 1

]

−0.09

∆Aτ 2 = Hτ 2 Fτ 2 (t)Eh2 , Hτ 2 =

[ ∆Aτ 3 = Hτ 3 Fτ 3 (t)Eh3 , Hτ 3 =

]

[ ] −0.25 0

,

−1

0.15 , Fτ 2 (t) = F2 (t),

]

0 [ −1.2 , Eh2 = 1.2 0.8

[

0.02

]

]

]

[

[

[ ] −0.01 0.21 , −0.65

0 ,

0 −1 , Aτ 2 = 0.29

Fτ 1 (t) = 0.1 sin(t), Eh1 = −0.06

Eh3 = −0.06

] −0.6 3.2 , −4

]

F3 (t) = 0.6 sin(t), E3 = 0.25

Aτ 1

−1 1.6 −2

0 0.02 , −0.15

[

1.2 0.26 0

2.3 0 −1.5

] −0.36 , ∆A2 = H2 F2 (t)E2 ,

0

[ ] −0.56 [ H2 = −0.2 , F2 (t) = 0.26e−t , E2 = 0.02 1.3

[ −4.56 −3 =

[

]

0

] −1 ,

0.15 0 , Fτ 3 (t) = 0.5 cos(0.2t), −0.65

]

0.3 ,

]

[ ] [ ] [ ] −2.6 −3.2 0.78 0 B1 = −7 , B2 = , B3 = −0.9 , 0.95 −1.5 1 f1 (x(t), x(t − τ (t)), v, t) = l1 cos(t)∥x(t)∥ + l2 ∥x(t − τ (t))∥ − l3 sin(0.25t), f (x(t), x(t − τ (t)), v, 1) = B1 f1 (x(t), x(t − τ (t)), v, t), f (x(t), x(t − τ (t)), v, 2) = B2 f1 (x(t), x(t − τ (t)), v, t), f (x(t), x(t − τ (t)), v, 3) = B3 f1 (x(t), x(t − τ (t)), v, t),

[ ] 0.1 , ∆B1 = B1 ∆(1, t)E¯ 1 , ∆(1, t) = 0.2sin(t) ] [ −2.3 E¯ 1 = , ψ1 = 0.82, ∆B2 = B2 ∆(2, t)E¯ 2 , −3.6 ∆(2, t) = cos(2t)

[

[ ] −0.65 ¯ sin(2t) , E2 = , 0.65 ]

ψ2 = 0.9192, ∆B3 = B3 ∆(3, t)E¯ 3 ,

[ ∆(3, t) = 0.2 cos(−t)

0.2 , E¯ 3 =

]

0.29 , ψ3 = 0.808, −4

[

]

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

47

σ (t , x(t), x(t − τ (t)), 1) = 0.3 sin(t)x(t) + 0.09 cos(2t)x(t − τ (t)), σ (t , x(t), x(t − τ (t)), 2) = −0.01 sin(0.5t)x(t − τ (t)), σ (t , x(t), x(t − τ (t)), 3) = 5.6 cos(0.2t)x(t), 5.3 C1 = 0

[

[ C3 =

−1 0 8.1 0

0 −1

0 −2.3 , C2 = −2.6 0

]

[ 0.9 , M1 = 5.6 −3

−1

Mh1 = 0.25

−0.09

[

]

0 1.6

]

M2 = −1.2

[

[

0 , 3

0.3 ,

]

−1

0 , M3 = M1 ,

]

0.18 , Mh2 = 0

]

−0.01

[

] −0.39 ,

Mh3 = Mh2 .

[

]T

1.6507 Q = 10 × −0.0218 −0.0004

−0.0218 1.6510 −0.0001

] −0.0004 −0.0001 , 1.3471

0.7572 R = 10 × 0.0001 −0.0001

0.0001 0.7576 −0.0005

] −0.0001 −0.0005 , 1.0228

Let the initial system state be x0 = −2 1 1.5 , the time-delay be τ (t) < h¯ = 1, h = 0.4. According to Theorem 1, we choose the positive scalars as ηˆ 1 = 4, ηˆ 2 = 0.6, γ = 1.8, k = 2, d = 0.6, ε11 = ε21 = ε31 = 0.25, ε12 = ε22 = ε32 = 8, ρ1 = 25, ρ2 = 86, ρ3 = 72. Firstly, we assume the upper bounds l1 = 8, l2 = 5, l3 = 2.5 are known constants to prove the effectiveness of Theorem 1. Through LMI toolbox of Matlab, we can successfully deal with LMIs (9)–(13) and derive the following appropriate matrices:

[

3

[

3

0.1394 −0.0497 0.1318

−0.0497 0.4702 0.0061

144.6755 0.4866 60.1195

0.4866 6.6000 −29.7327

[ P1 =

[ P2 =

1.7740 P3 = 10 × −0.9157 −1.5522

0.1318 0.0061 , 364.2052

]

60.1195 −29.7327 , 168.9002

]

−0.9157 9.6643 1.9372

] −1.5522 1.9372 , 2.5155

[

W12

2.0644 = 10 × 0.0013 −0.0000

0.0013 2.0648 0.0000

] −0.0000 0.0000 , 2.0666

W13

2.0609 = 10 × 0.0005 −0.0000

0.0005 2.0611 0.0000

] −0.0000 0.0000 , 2.0669

[

3

3

[

3

1.3845 6.1887 2.5768

6.1887 63.0705 −0.0838

2.5768 −0.0838 , 9.1771

3.8771 6.1020 14.3723

6.1020 10.2863 22.5585

14.3723 22.5585 . 57.7435

[ R223 =

[ T31 =

]

]

(38)

48

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

Fig. 1. System switching mode.

It is clear that system (1) is H∞ controllable with the following adaptive controller

⎧ [ ] u1 (x, t) = −104 × 0.0007 −0.0195 2.1361 x(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − 2.7778θˆ [0.1107 −3.1564 345.6096] x(t), 1 ⎪  2 ⎪ ⎪  ⎪ [ ] ⎪ ˙θ = −0.6(θˆ − 1.9290) +    ⎪ ˆ ⎩ 1 1  0.1107 −3.1564 345.6096 x(t) . [ ] ⎧ u2 (x, t) = −104 × −7.6160 0.5926 −6.1371 x(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − 6.1881θˆ [−553.1409 43.0419 −445.7327] x(t), 2

⎪ ⎪  ⎪ ⎪ [ ⎪ ⎪ ⎩ θˆ˙2 = −0.6(θˆ1 − 1.9290) +  −553.1409 

43.0419

2  −445.7327 x(t)  .

[ ] ⎧ u3 (x, t) = −105 0.3799 −4.3312 −0.2542 x(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − 2604.1667θˆ [0.6556 −7.4749 −0.4387] x(t), 3 ⎪ ⎪  ⎪ ⎪ [ ⎪ ⎪ ⎩ θ˙ˆ3 = −0.6(θˆ1 − 1.9290) + 103 ×   0.6556 −7.4749

]

(39)

2  ] −0.4387 x(t)  .

Secondly, we assume the upper bounds l1 , l2 and l3 are all unknown to testify the feasibility of Theorem 2. Choose the constants τˆ1 = 175, τˆ2 = 225 and τˆ3 = 450. We also could solve the seem parameter matrices as those in (38) and (39), and by Theorem 2, the controller is designed as

[ ] ⎧ u1 (x, t) = −(θˆ1 + θˆ2 ) 0.0769 −2.1919 240.0066 x(t) ⎪ ⎪ ⎪ ⎪ ⎪ [ ] ⎪ ⎪ − θˆ3 0.3074 −8.7677 960.0266 x(t), ⎪ ⎪ ⎪ ⎪ ⎪  2 ⎪ ⎪ [  ⎪ ] ⎪ ⎪ θ˙ˆ1 = −175(θˆ1 − 64) + 0.25 0.1107 −3.1564 345.6096 x(t) , ⎨   ⎪  ⎪ ⎪ [ ⎪ ˙θˆ = −225(θˆ − 25) + 0.25 ⎪  0.1107 ⎪ ⎪ 2 2  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ [ ⎪ ⎪ ⎩ θ˙ˆ3 = −450(θˆ3 − 1.9290) +   0.1107

−3.1564

2  345.6096 x(t)  ,

−3.1564

345.6096 x(t)  .

]

]

2 

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

49

Fig. 2. Open-loop system state trajectories of xo1 , xo2 and xo3 .

⎧ u2 (x, t) ⎪ ⎪ ⎪ ⎪ ⎪ [ ] ⎪ ⎪ = −(θˆ1 + θˆ2 ) −855.7253 66.5871 −689.5617 x(t) ⎪ ⎪ ⎪ ⎪ ⎪ [ ] ⎪ ⎪ 3 ⎪ ⎪ − 10 × θˆ3 × −3.4229 0.2663 −2.7582 x(t), ⎪ ⎪ ⎪ ⎪  2 ⎪ ⎨˙ [  ]  , − 553 . 1409 43 . 0419 − 445 . 7327 θˆ1 = −175(θˆ1 − 64) + 0.25 x(t)   ⎪ ⎪ ⎪ ⎪ ⎪ 2  ⎪ ⎪  [ ] ⎪ ˙θ = −225(θˆ − 25) + 0.25 ⎪   ˆ ⎪ 2 2 ⎪  −553.1409 43.0419 −445.7327 x(t) , ⎪ ⎪ ⎪ ⎪ ⎪  2 ⎪ ⎪ [  ⎪ ] ⎪ ⎪ θ˙ˆ3 = −450(θˆ3 − 1.9290) +  −553.1409 43.0419 −445.7327 x(t) . ⎩   [ ] ⎧ u3 (x, t) = −103 × (θˆ1 + θˆ2 ) 0.4269 −4.8665 −0.2856 ⎪ ⎪ ⎪ ⎪ ⎪ [ ] ⎪ ⎪ − 104 × θˆ3 × 0.1707 −1.9466 −0.1142 , ⎪ ⎪ ⎪ ⎪ ⎪ 2  ⎪ ⎪  ⎪ [ ] ⎪ ˙θˆ = −175(θˆ − 64) + 250   ⎪ ⎨ 1 1  0.6556 −7.4749 −0.4387 x(t) ,

(40)

⎪  2 ⎪ ⎪ [  ] ⎪ ⎪ ⎪ θ˙ˆ2 = −225(θˆ2 − 25) + 250 0.6556 −7.4749 −0.4387 x(t) , ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  2 ⎪ ⎪ [  ] ⎪ ⎪ ⎩ θ˙ˆ3 = −450(θˆ3 − 1.9290) + 1000 0 . 6556 − 7 . 4749 − 0 . 4387 x(t)   . Lastly, we give the state trajectory curves to illustrate our methods. For Theorem 1, Figs. 1 and 2 shows the switching modes and the trajectories of system (1) without the controller, respectively. Fig. 3 indicates that system (1) is obviously asymptotically stochastically admissible under the controller (40). Remark 6. In fact, what is important is that the results in this work can be also applied to the MJSs with BUTRs or PUTRs. The BUTR methods, however, require the exactly known estimation of each rate and cannot be applicable to the GUTR model. Because the generally uncertain TR matrix can be partly known via replacing the uncertain TRs with unknown ones, PUTR methods can be used for GUTR model. On the other hand, we cannot utilize the information of the TRs’ estimates so that such methods are inescapably conservative for GUTRs.

50

G. Yang, B. Kao, J.H. Park et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 33–51

Fig. 3. Closed-loop system state trajectories of x1 , x2 and x3 by Theorem 1.

Remark 7. From [14–16], we know that MJS have many limitations in applications, since the jump time of a Markov chain is, in general, exponentially distributed, and the results obtained for the MJS are intrinsically conservative due to constant transition rates. Compared with the MJS, semi-Markovian jump systems (S-MJS) are characterized by a fixed matrix of transition probabilities and a matrix of sojourn time probability density functions. Due to their relaxed conditions on the probability distributions, S-MJS have much broader applications than the conventional MJS. Our works in this paper can be easily extended to S-MJS. Remark 8. The developed adaptive laws are provided to estimate certain parameters. The developed results still work if the parameters are time-varying. 5. Conclusion This work studied H∞ control for uncertain Markovian jump stochastic singular time-delay systems with general unknown TRs as well as nonlinear perturbations. Every TR in the considered system is totally unknown or only the estimation is known. According to Lyapunov theory and Itoˆ ’s differential formula, we design a class of robust controller with a linear part and an adaptive part. Firstly, we assume that the upper bounds are completely known, then a sufficient condition is given for the H∞ stochastically admissible of the given system. Moreover, two corollaries are given. Secondly, we assume that all the upper bounds are unknown, then a sufficient condition is also given for the H∞ stochastically admissible of the given system. And two corollaries are derived. The advantages and feasibility of the designed controllers are shown via a numerical example. We will extend our results to Markovian jump interval type-2 fuzzy systems and S-MJS in the future work. Acknowledgments This work of Y. Kao and G. Yang is supported by National Natural Science Foundation of China (61873071, 61772277), National Key R&D Program (2017YFC0804002), and the Natural Science Foundation of Jiangsu Province (BK20171494). Also, the work was of J.H. Park supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant number NRF-2017R1A2B2004671). Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] [2] [3] [4]

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