Normalization and stabilization of neutral descriptor hybrid systems based on P-D feedback control

Normalization and stabilization of neutral descriptor hybrid systems based on P-D feedback control

Journal Pre-proof

Normalization and stabilization of neutral descriptor hybrid systems based on P-D feedback control Guangming Zhuang, Jianwei Xia, Wei Sun, Qian Ma, Zhen Wang, Yanqian Wang PII: DOI: Reference:

S0016-0032(19)30752-5 https://doi.org/10.1016/j.jfranklin.2019.10.020 FI 4218

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

25 May 2019 31 August 2019 18 October 2019

Please cite this article as: Guangming Zhuang, Jianwei Xia, Wei Sun, Qian Ma, Zhen Wang, Yanqian Wang, Normalization and stabilization of neutral descriptor hybrid systems based on P-D feedback control, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.10.020

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

Normalization and stabilization of neutral descriptor hybrid systems based on P-D feedback control Guangming Zhuanga , Jianwei Xiaa,b,† , Wei Suna, , Qian Mac , Zhen Wangd , Yanqian Wange a b

School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, P. R. China c

d

School of Mathematical Sciences, Liaocheng University, Liaocheng Shandong 252059, PR China

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China

College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China e

School of Information and Control Engineering, Qingdao University of Technology, Qingdao, Shandong, 266520, China

E-mail: [email protected]; [email protected]; [email protected]; [email protected]; wangzhen [email protected]; [email protected]

Abstract This paper is concerned with the problem of normalization and stabilization of neutral descriptor hybrid systems with neutral and time-varying state delays. By designing mode-dependent delayed proportionalderivative state feedback and mode-dependent delayed proportional-derivative output injection controllers, respectively, the stabilization problem of neutral descriptor hybrid systems is transformed into the stability problem of normal neutral hybrid systems via normalization technique. Sufficient conditions are provided in terms of linear matrix inequalities by constructing mode-dependent and delay-dependent LyapunovKrasovskii functional. Simulation examples including a partial element equivalent circuit system are employed to verify the usefulness and validity of the controllers design method. Keywords: Descriptor hybrid systems, Normalization, Delayed P-D feedback control, Neutral delays, Time-varying delays

1

Introduction Descriptor systems are also known as singular systems, constrained systems, differential-algebraic systems,

degenerate systems, generalized state space systems, etc., [1, 2]. This kind of systems are formulated by a set of coupled algebraic and differential equations, which include information not only on dynamic constraints but also on static constraints. Therefore, descriptor systems can model various kinds of practical systems, †

Corresponding author. E-mail address: [email protected]

1

such as electrical networks, power systems, flexible robots, social economic systems, aerospace engineering, and so on [3, 4]. Besides the finite dynamic modes, there exist infinite dynamic modes and infinite non-dynamic modes in descriptor systems, and the infinite dynamic modes can result in destructive impulse. Thus, the admissibility analysis including regularity, non-impulsiveness (causality in discrete case), stability and control synthesis for descriptor systems becomes more complicated than that of normal systems (also known as state-space systems, regular systems). The study for descriptor systems not only has great theoretical importance, but also has practical significance [5–7]. Over the past decades, a great deal of attention has been paid to the admissibility analysis and control synthesis of descriptor systems, and various kinds of important results including normalization have been published successively [8–11]. The so-called normalization of descriptor systems is to design appropriate proportional-derivative (P-D) feedback controller such that the corresponding closed-loop system becomes normal system, enabling the mature theory and application of normal systems be extended to descriptor systems [1]. The advantage of normalization is that the infinite poles of descriptor systems can be assigned to finite via the P-D feedback control, and the impulse behaviors are eliminated in the closed-loop state response [4, 9]. On the other hand, hybrid systems are an important kind of dynamical systems, which contain two classes of states with continuous and discrete values, respectively [12–15].The hybrid systems driven by Markovian process (also known as Markovian switching systems or Markovian jump systems) can model many practical systems, which often undergo abrupt changes in their parameters or/and structures due to sudden external disturbances, interconnections changes, component repairs/failures , etc., [16–20]. Therefore, such hybrid systems have received much attention during the past decades; see, for instance, [21–26]. When descriptor systems encounter various sudden changes in parameters and/or structures, which results in the descriptor hybrid systems (DHSs) [2, 3]. Recently, a large number of influential achievements such as admissibility analysis, feedback control, sliding mode control, normalization, passification, filtering, etc., have been reported in [27–32] and the references therein. It should be mentioned that the aforesaid references involved just the retarded time delays (the system states suffered time delays). Noting that time delays are inevitable in various kinds of practical system, which often result in system performance degradation or even seriously damage to system stability [33–40]. Besides the retarded type time delays, neutral type time delays often exist in many real systems. The so-called neutral type time delay refers to the time delays involved in state derivative [41–43]. Reference [43] addressed that partial element equivalent circuit (PEEC) can be modeled as a neutral delay system. When neutral type time delays appear in descriptor systems, the neutral descriptor systems are formed [44]. Very recently, some attention has been paid to neutral descriptor systems; see, [45] and the references therein. Noting that feedback control can eliminate the deviation between the system state/output and the expected 2

behavior and obtain the expected system performance based on feedback principle [46–55]. To the best of the authors’ knowledge, so far, there is little literature about feedback control for neutral descriptor hybrid systems (NDHSs), much less the case of time-varying delays in states. The key challenges arise from the fact that the researchers not only need to design feedback controller, but also ensure that the closed-loop NDHSs can satisfy the admissibility, which includes regularity, non-impulsiveness/causality and stability. Fortunately, descriptor systems normalization technique can realize the aim of admissibility of NDHSs via proportional-derivative state feedback (PDSF) control and proportional-derivative output injection (PDOI) control [56]. This motivates the current investigation. Considering that the feedback controller depends not only on the current state but also on the past state due to the time lag between the state observation and the feedback control reaching [31, 57]. Thus, the delay feedback control is necessary and more realistic for dynamic systems. In this paper, we will investigate the problem of normalization and stabilization of NDHSs with neutral and time-varying state delays. By designing mode-dependent delayed proportional-derivative state feedback (DPDSF) and mode-dependent delayed proportional-derivative output injection (DPDOI) controllers, respectively, the stabilization problem of NDHSs is transformed into the stability problem of normal neutral hybrid systems (NHSs) via normalization technique. Sufficient conditions will be provided in terms of linear matrix inequalities (LMIs) by constructing mode-dependent and delay-dependent Lyapunov-Krasovskii functional. Simulation examples including a partial element equivalent circuit (PEEC) system will be employed to verify the usefulness and validity of the controllers design method. The main contributions of this paper are outlined below: (1) mode-dependent delayed proportional-derivative state feedback (DPDSF) and mode-dependent delayed proportional-derivative output injection (DPDOI) controllers are designed effectively for NDHSs; (2) the neutral and time-varying delays are considered simultaneously, and the method can be employed to deal with time invariant neutral descriptor hybrid systems; (3) the stabilization problem of NDHSs is transformed into the stability problem of normal neutral hybrid systems via normalization technique. Notation. E {·} is expectation operator, sym(A) represents A + AT , (∗) means a symmetry term in matrices.

3

2

Problem Formulation and Preliminaries Given complete probability space (Ω, F, P ) with the usual filtration {Ft }t≥0 , we consider the underlying

NDHS with time-varying delays:    E (rt ) x˙ (t) − N (rt ) x˙ (t − τ )    ˜ (rt ) u (t, rt ) , = A (rt ) x (t) + Ad (rt ) x (t − τ (t)) + B      x(t) = φ(t), ∀ t ∈ [−τ, 0],

(1)

where x(t) ∈ Rn is the system state, φ(t) is the initial state, u(·) ∈ Rm is the control input. The matrix

E (rt ) ∈ Rn×n is singular and rank(E (rt )) = r < n. {rt } is a right continuous Markovian process, which

takes values in a finite set S = {1, 2, · · · , N }. P {rt+∆t

  πij ∆t + o (∆t) , = j |rt = i } =  1 + π ∆t + o (∆t) , ii

is known as transition probability, ∆t > 0, lim

∆t→0

o(∆t) ∆t

i 6= j

(2)

i=j

= 0. πij is the well-known transition rate, Π = [πij ]N ×N

is called transition rate matrix, where πij ≥ 0 for i 6= j, and πii = −

N X

πij .

(3)

τ˙ (t) ≤ µ,

(4)

j6=i,j=1

τ (t) is the time-varying state delay, which satisfies 0 ≤ τ (t) ≤ τ < ∞, where τ and µ are constant scalars. The main aim of this paper is to design mode-dependent delayed proportional-derivative feedback (DPDF) controller (5) such that the NDHS (1) can be normalized and stabilized. ˜ (rt ) u(t, rt ) = B(rt )Kp (rt )x (t) + C(rt )Kl (rt )x (t − τ (t)) − D(rt )Kd (rt )x˙ (t) , B h i ˜ (rt ) = B (r ) C (r ) −D (r ) . where B t t t

(5)

Our main measure is to implement mode-dependent delayed proportional-derivative state feedback and

output injection control for the NDHSs with time-varying delays. The delayed proportional-derivative state feedback (DPDSF) control is to design B(rt ), C(rt ), D(rt ) when Kp (rt ), Kl (rt ) and Kd (rt ) are given, while the delayed proportional-derivative output injection (DPDOI) control is to design Kp (rt ), Kl (rt ), Kd (rt ) when B(rt ), C(rt ) and D(rt ) are given; see, e.g., [56, 57] and the references therein. Definition 1 ([1, 4]) The NDHS (1) is called normalizable if there exists a mode-dependent DPDF controller (5) such that | E(rt ) + D(rt )Kd (rt ) |6= 0 and the corresponding closed-loop system [E(rt ) + D(rt )Kd (rt )] x˙ (t) − N (rt ) x˙ (t − τ ) = [A(rt ) + B(rt )Kp (rt )] x (t) + [Ad (rt ) + C(rt )Kl (rt )] x (t − τ (t)) 4

(6)

is a normal neutral hybrid system (NHS). Then, the closed-loop system (6) can be rewritten as: x˙ (t) − [E(rt ) + D(rt )Kd (rt )]−1 N (rt ) x˙ (t − τ ) = [E(rt ) + D(rt )Kd (rt )]−1 [A(rt ) + B(rt )Kp (rt )] x (t)

(7)

+ [E(rt ) + D(rt )Kd (rt )]−1 [Ad (rt ) + C(rt )Kl (rt )] x (t − τ (t)) . Definition 2 ([2]) The NHS (1) and (7) are said to be stochastically stable, respectively, if there exists a scalar M(φ(·), r0 ) > 0 such that lim E

T →∞

Z

T

0

 |x (t)| dt x(s) = φ(s), r0 , s ∈ [−τ, 0] 2

(8)

≤ M(φ(·), r0 ).

For simplicity, in the sequel, for all rt = i ∈ S, matrix E (rt ) is replaced by Ei , A (rt ) is represented by Ai , Ad (rt ) is depicted by Adi , and so on. Remark 1 Noting that a suitable DPDF controller can ensure the regular closed-loop system (6) or (7) is impulse-free. According to ([1, 4]), the necessary and sufficient condition for normalization of descriptor   I . Therefore, in this paper, we always system (1) is rank(Ei Di ) = n due to Ei + Di Kdi = (Ei Di )  Kdi assume rank(Ei Di ) = n so as to ensure descriptor system (1) can be normalized. 

Lemma 1 ([15]) For a positive scaler τ and matrices V, U satisfying  [−τ, 0] → Rn satisfies the following integral, then Z t −τ χ˙ T (s)V χ(s)ds ˙ ≤ ψ T (t)Πψ(t),

V U ∗

V



 ≥ 0, if function χ(t ˙ + ·) :

t−τ

where iT , χT (t) χT (t − τ (t)) χT (t − τ )   −V V −U U     Π =  ∗ −2V + U + U T V − U  .   ∗ ∗ −V

ψ(t) =

3

h

Main Results

Theorem 1 The NHS (7) is stochastically stable, if for every i ∈ S, there exist matrices Q > 0, R > 0, W > 0, V ≥ 0, Pˆi , U such that

 

V

U

∗

V 5



 > 0,

(9)



Pˆi [Ei + Di Kdi ] > 0,

Ψ Ψ2i U Pˆi Ni  1i   ∗ Ψ3i V − U 0 T Ψi + Υi ΞΥi =    ∗ ∗ −Q − V 0  ∗ ∗ ∗ −W −1 [Ei + Di Kdi ] Ni < 1,

where

(10) 

    + ΥTi ΞΥi < 0,   

(11)

(12)

Ψ1i = sym(Pˆi (Ai + Bi Kpi )) + Q + R − V N P + πij [Pˆj (Ej + Dj Kdj )], j=1

Ψ2i = Pˆi (Adi + Ci Kli ) + V − U ,

Ψ3i = −2V + U + U T + (µ − 1)R, Ξ = W+ τ 2 V , h iT −1 [E + D K ] (A + B K ) i i i i pi di  h iT   [Ei + Di Kdi ]−1 (Adi + Ci Kli )  ΥTi =   0   h iT [Ei + Di Kdi ]−1 Ni



    .   

Proof. Defining {xt = x(t + θ), −τ ≤ θ ≤ 0}, then we obtain a new Markovian process {(xt , rt ), t ≥ τ } with initial state (r0 , φ(·)). For the NHS (7), we select P (rt ) > 0 and the following Lyapunov-Krasovskii functional candidate: V (xt , rt ) = V1 (xt , rt ) + V2 (xt , rt ) + V3 (xt , rt ) + V4 (xt , rt ) + V5 (xt , rt ) ,

(13)

where V1 (xt , rt ) = xT (t) P (rt )x (t), Rt V2 (xt , rt ) = t−τ xT (s) Qx (s)ds, Rt V3 (xt , rt ) = t−τ (t) xT (s) Rx (s)ds, Rt V4 (xt , rt ) = t−τ x˙ T (s) W x˙ (s)ds, R0 Rt V5 (xt , rt ) = τ −τ t+θ x˙ T (s) V x˙ (s)dsdθ.

Let A be the weak infinitesimal generator of the Markovian process {(xt , rt ), t ≥ τ }, A acting on V (·) by AV (xt , rt = i) = lim

h→0

E[V (xt+h ,rt+h )|xt ,rt =i]−V (xt ,rt =i) , h

6

(14)

then we have AV1 (xt , rt = i) = 2xT (t) Pi [Ei + Di Kdi ]−1 [(Ai + Bi Kpi ) × x (t) + (Adi + Ci Kli ) x (t − τ (t)) + Ni x˙ (t − τ )] + xT (t)

N X

(15)

πij Pj x (t) ,

j=1

AV2 (xt , rt = i) = xT (t) Qx (t) − xT (t − τ ) Qx (t − τ ) ,

(16)

AV3 (xt , rt = i) ≤ xT (t) Rx (t) − (1 − µ)xT (t − τ (t)) R

(17)

× x (t − τ (t)) , AV4 (xt , rt = i) = x˙ T (t) W x˙ (t) − x˙ T (t − τ ) W x˙ (t − τ ) , Z t 2 T x˙ T (s) V x˙ (s) ds. AV5 (xt , rt = i) = τ x˙ (t) V x˙ (t) − τ

(18) (19)

t−τ

By Lemma 1, we get

−τ

Z

ζ(t) =

h

where

t

t−τ



¯ x˙ T (s) V x˙ (s) ds ≤ ζ T (t)Πζ(t),

xT (t) −V

 ¯ = Π  ∗  ∗

xT (t

− τ (t))

V −U

−2V + U + U T ∗

Connecting (15)-(20), and let Pi [Ei + Di Kdi ]−1 = Pˆi , we get

xT (t

− τ)

U

iT

(20)

,



  V − U .  −V

¯ < 0, AV (xt , rt = i) ≤ ζ¯T (t)Ψi ζ(t) where ¯ = ζ(t)

h

xT (t) xT (t − τ (t)) xT (t − τ ) x˙ T (t − τ )

Then, there exist positive scaler %, for every x(t) 6= 0, have

(21) iT

.

AV (xt , rt = i) ≤ −%xT (t)x(t).

(22)

By Dynkin’s formula, we have that E{V (xt1 , rt1 )} − E{V (x0 , r0 )} ≤ −%E{

Z

E{V (xT , rT )} − E{V (xt1 , rt1 )} ≤ −%E{ then we have E{

Z

0

T

t1

xT (s)x(s)ds},

0

Z

T

xT (s)x(s)ds},

t1

xT (s)x(s)ds} ≤ %−1 E{V (x0 , r0 )}.

By Definition 1, the NHS (7) is stochastically stable. 7

(23) 2

Remark 2 According to [41], the norm of neutral parameter matrix need to be strictly less than 1 so as to ensure the stability of normal neutral system. Therefore, [Ei + Di Kdi ]−1 Ni < 1 is a necessary constrained

condition in this paper.

Remark 3 It should be pointed out that the term xT (t)

N P

πij Pj x (t) is the sum about j, which is introduced

j=1

due to the conditional expectation: E [V1 (xt+h , rt+h ) |xt , rt = i] in the weak infinitesimal generator: AV (xt , rt = i) = lim

h→0

E[V (xt+h ,rt+h )|xt ,rt =i]−V (xt ,rt =i) . h

Now, we are in a position to design the mode-dependent DPDSF controller (5) to realize the normalization and stabilization of the NDHS (1). Theorem 2 Consider the NDHS (1) with DPDSF controller (5), the closed-loop NHS (7) is stochastically stable, if for every i ∈ S, there exist matrices Pˆi , Q > 0, R > 0, W > 0, V ≥ 0, U , Xi , Yi , Zi such that (9) and the following LMIs are satisfied. Pˆi Ei + Zi Kdi > 0,

 

      Φi =       

where

NiT PˆiT

−I



(24)

I − sym(Pˆi Ei + Zi Kdi )

∗

Φ1i Φ2i

Pˆi Ni Φ4i

U

∗

Φ3i

V −U

0

Φ5i

∗

∗

−Q − V

0

0

∗

∗

∗

−W

Φ6i

∗

∗

∗

∗

Φ7i

∗

∗

∗

∗

∗

Φ1i = sym(Pˆi Ai + Xi Kpi ) + Q + R − V N P + πij [Pˆj Ej + Zj Kdj ], j=1

Φ2i = Pˆi Adi + Yi Kli + V − U ,

Φ3i = −2V + U + U T + (µ − 1)R,

Φ4i = τ (Pˆi Ai + Xi Kpi )T ,

Φ5i = τ (Pˆi Adi + Yi Kli )T , Φ6i = τ (Pˆi Ni )T , Φ7i = V − sym(Pˆi Ei + Zi Kdi ),

Φ8i = (Pˆi Ai + Xi Kpi )T ,

8



 < 0, Φ8i

(25) 

  Φ9i    0   < 0,  Φ10i    0   Φ11i

(26)

Φ9i = (Pˆi Adi + Yi Kli )T , Φ10i = (Pˆi Ni )T , Φ11i = W − sym(Pˆi Ei + Zi Kdi ). Then, the desired mode-dependent DPDSF controller (5) can be obtained by Bi = Pˆi−1 Xi , Proof. Noting that

Ci = Pˆi−1 Yi ,

Di = Pˆi−1 Zi .

−1 [Ei + Di Kdi ] Ni < 1 ⇔

NiT [Ei + Di Kdi ]−T [Ei + Di Kdi ]−1 Ni < I. By Schur complement, (28) can be transformed into   −I NiT PˆiT   < 0. ∗ −Pˆi [Ei + Di Kdi ] [Ei + Di Kdi ]T PˆiT

Due to

 

−I ∗



NiT PˆiT

<0

I − sym(Pˆi [Ei + Di Kdi ])

(27)

(28)

(29)

(30)

ensures (29), then let Pˆi Di = Zi in (30), we get (25), which implies (12). By Schur complement, we find that (11) is equivalent to  Ψ τ ΥTi ΥTi  i   ∗ −V −1 0  ∗ ∗ −W −1



   < 0. 

(31)

Implementing congruent transformation to (31) by diag{I PiT PiT }, we get that 

τ ΥTi PiT

ΥTi PiT

Ψ  i   ∗ −Pi V −1 PiT  ∗ ∗

0 −Pi W −1 PiT



   < 0. 

(32)

Noting that −Pi V −1 PiT ≤ V − Pi − PiT , −Pi W −1 PiT ≤ W − Pi − PiT , the inequality (32) can be ensured by (33):



Ψi

   ∗  ∗

τ ΥTi PiT

ΥTi PiT

V − Pi − PiT

0 W − Pi − PiT

∗



   < 0. 

(33)

Applying Pi [Ei + Di Kdi ]−1 = Pˆi , Pˆi Di = Zi , and let Pˆi Bi = Xi , Pˆi Ci = Yi , we can easily finish the proof of Theorem 2. The detailed proof is omitted here.

2

9

Remark 4 Noting that output injection is the dual of state feedback[56], it is naturally to consider output injection control for the NDHSs (1). However, based on Theorem 1 and Theorem 2, DPDOI controller design for the NDHSs (1) is not easily to be implemented. Therefore, in the sequel, we will design mode-dependent DPDOI controller when τ˙ (t) ≤ µ < 1. Theorem 3 The NHS (7) is stochastically stable, if for every i ∈ S, there exist matrices Q > 0, R > 0, W > 0, Hi > 0, Pˇi such that (12) and the following matrix inequalities are satisfied. Pˇi [Ei + Di Kdi ] > 0,

where

πij Pˇj (Ej  ˆ Ψ  1i   ∗ ˆ i + ΥT ΞΥ ˆ i= Ψ i   ∗  ∗

+ Dj Kdj ) < Hj , j 6= i  ˆ 2i Ψ 0 Pˇi Ni   ˆ 3i Ψ 0 0  ˆ i < 0,  + ΥTi ΞΥ  ∗ −τ Q 0   ∗ ∗ −W

(34) (35)

(36)

ˆ 1i = sym(Pˇi (Ai + Bi Kpi )) + τ Q + R + P Hj Ψ j6=i

+πii Pˇi (Ei + Di Kdi ),

ˆ 2i = Pˇi (Adi + Ci Kli ), Ψ ˆ 3i = (µ − 1)R, Ξ ˆ = W, Ψ  h iT −1 [E + D K ] (A + B K ) i di i i pi  h i iT   [Ei + Di Kdi ]−1 (Adi + Ci Kli )  ΥTi =   0   h iT [Ei + Di Kdi ]−1 Ni



    .   

Proof. Selecting the following Lyapunov-Krasovskii functional candidate: V (xt , rt ) = V1 (xt , rt ) + V2 (xt , rt ) + V3 (xt , rt ) + V4 (xt , rt ) ,

(37)

where V1 (xt , rt ) = xT (t) P (rt )x (t), Rt V2 (xt , rt ) = τ t−τ xT (s) Qx (s)ds, Rt V3 (xt , rt ) = t−τ (t) xT (s) Rx (s)ds, Rt V4 (xt , rt ) = t−τ x˙ T (s) W x˙ (s)ds.

Let Pi [Ei + Di Kdi ]−1 = Pˇi , similar to Theorem 1, we can prove that NHS (7) is stochastically stable. The

detailed proof is omitted here.

2 10

Theorem 4 Consider the NDHS (1) with DPDOI controller (5), the closed-loop NHS (7) is stochastically ˆ > 0, R ˆ > 0, W ˆ > 0, H ˆ i > 0, X ¯ i , Y¯i , Z¯i such that the stable, if for every i ∈ S, there exist matrices P¯i , Q following LMIs are satisfied.



where

Ei P¯iT + Di Z¯i > 0,

(38)

ˆ j − P¯j − P¯ T < 0, j 6= i, πij (Ej P¯jT + Dj Z¯j ) + H j   −I NiT 0      ∗ −sym(Ei P¯iT + Di Z¯i ) P¯i  < 0,   ∗ ∗ −I

(39)

ˆ ˆ ¯T  Φ1i Φ2i 0 Ni Pi  ˆ 3i 0  ∗ Φ 0   ˆ 4i  ∗ ∗ Φ 0    ∗ ˆ 5i ∗ ∗ Φ   ˆi =  ∗ Φ ∗ ∗ ∗    ∗ ∗ ∗ ∗    ∗ ∗ ∗ ∗    ∗ ∗ ∗  ∗  ∗ ∗ ∗ ∗

√ ¯T τ Pi

P¯i

(40)

P˜i

ˆ 6i Φ

0 0

0

0

0

ˆ 7i Φ

0

0

0

0

0

0

0

0

ˆ 8i Φ

0

ˆ −Q

0

0

0

0

∗

ˆ −R

0

0

0

∗

∗

˜i −H

0

0

∗

∗

∗

ˆ 9i Φ

P¯i

∗

∗

∗

∗

ˆ −W



            < 0,           

(41)

ˆ 1i = sym(Ai P¯ T + Bi X ¯ i ) + πii (Ei P¯ T + Di Z¯i ), Φ i i ˆ 2i = Adi P¯ T + Ci Y¯i , Φ i ˆ 3i = (1 − µ)(R ˆ − P¯i − P¯ T ), Φ i ˆ 4i = τ (Q ˆ − P¯i − P¯ T ), Φ i ˆ 5i = W ˆ − P¯i − P¯ T , Φ i

ˆ 6i = P¯i AT + X ¯ T BT , Φ i i i ˆ 7i = P¯i AT + Y¯ T C T , Φ i i di ˆ 8i = P¯i N T , Φ i ˆ 9i = −sym(Ei P¯ T + Di Z¯i ), Φ i   P˜i = P¯i P¯i · · · P¯i 1×(N −1) , n o ˆ i+1 · · · H ˆN . ˜ i = diag H ˆ1 H ˆ2 · · · H ˆ i−1 H H

Then, the desired DPDOI controller (5) can be acquired by ¯ i P¯ −T , Kpi = X i

Kli = Y¯i P¯i−T ,

Kdi = Z¯i P¯i−T .

(42)

Proof. Similar to the proof of Theorem 2, (40) ensures (12) based on Schur complement and congruent transformation technique as Pˇi−1 = P¯i . 11

By Schur complement, (36) is equivalent to  ˇi =  Ψ

ˆi Ψ

ΥTi

∗

−W −1



 < 0.

(43)

ˇ i by diag{I Pi } and its transpose, respectively, we get Left- and right-multiplying Ψ   T T ˆ Ψ Υi Pi ˜i =  i  < 0, Ψ −1 T ∗ −Pi W Pi where

Pi = Pˇi [Ei + Di Kdi ] ,

ΥTi PiT



[Ai + Bi Kpi ]T PˇiT

   [Adi + Ci Kli ]T PˇiT =   0  NiT PˇiT

(44)



   .   

Left- and right-multiplying Pˇi [Ei + Di Kdi ] by Pˇi−1 and its transpose, respectively, we get (38). Left- and right-multiplying πij Pˇj (Ej + Dj Kdj ) − Hj , by Pˇj−1 and its transpose, respectively, we have (39). Left- and ˜ i by diag{Pˇ −1 Pˇ −1 Pˇ −1 Pˇ −1 Pˇ −1 } and its transpose, respectively, we can acquire right-multiplying Ψ i i i i i −Pˇi−1 Pi W −1 PiT Pˇi−T

= − [Ei + Di Kdi ] Pˇi−T PˇiT W −1 Pˇi Pˇi−1 [Ei + Di Kdi ]T

(45)

≤ Pˇi−1 W Pˇi−T − sym(Ei P¯iT + Di Z¯i ).

ˆ ¯ i , Kli P¯ T = Y¯i , Kdi P¯ T = Z¯i , R−1 = R, By Schur complement, we get (41) with Pˇi−1 = P¯i , Kpi P¯iT = X i i ˆ W −1 = W ˆ , H −1 = H. ˆ Meanwhile, the desired DPDOI controller (5) can be realized by (42). Q−1 = Q,

4

Illustrative Examples

Example 1 Consider the NDHS (1) with three modes and the following parameters:         1 0 0 2 0 0 3 0 0 3 1 −1.5                 E1 =  0 1 0  , E2 =  0 1 0  , E3 =  0 1 0  , N1 =  2 −0.3 3 ,         0 0 0 0 0 0 0 0 0 1 4 −4       5 2 −1.6 2.5 −0.5 0.2 4 3 −1.4             N3 =  4 −0.2 A1 =  0.3 N2 =  3 −0.4 4 , 2.6 0.2  , 5 ,       −0.1 0.3 −2.5 2 3 −3 4 2 −5       1.5 −1.5 0.3 2.5 −0.4 0.4 0.03 −0.2 0.1             , A = , A = A2 =  0.4  0.5  0.2 2.7 0.3  2.8 0.4  0.05 0.04  , 3 d1       −0.2 0.4 −2.8 −0.4 0.2 −2.7 −0.04 0.1 0.2       0.2 −0.12 0.21 0.1 −0.2 0.15 0.21 0.33 0.15             Ad2 =  0.03 , A = , K =     0.2 −0.04 0.1 0.25 −0.3 0.02 0.12 −0.13  , p1 d3       −0.2 0.2 0.1] −0.07 0.14 0.15 −0.12 0.14 0.23 12

2



0.23

0.33

0.14





0.22

0.32

0.16





−1.1

0.3

2.5



            Kp2 =  0.03 0.13 −0.12  , Kp3 =  0.04 0.14 −0.11  , Kl1 =  1.1 0.1 −2.1  ,       −0.13 0.15 0.25 1.2 −1.5 2.2 −0.11 0.13 0.24       −0.1 0.2 0.3 −1.3 0.1 2.4 −1.2 0.4 2.6             , K = Kl2 =  1.3 , K =     0.2 0.1 0.2  , 1.2 0.3 −2.3 0.2 −2.2 d1 l3       −0.2 0.1 0.4 1.4 −1.4 2.1 1.6 −1.3 2.3       −1.7 1.2 0.5 −0.3 0.2 0.5 −0.2 0.1 0.4             Π =  0.5 −1.8 1.3  . Kd3 =  0.3 0.1 0.4  , Kd2 =  0.1 0.2 0.3  ,       0.7 0.5 −1.2 −0.2 0.4 0.3 −0.3 0.2 0.2 3.5 modes

Markovian jump modes

3

2.5

2

1.5

1

0.5

0

500

1000

1500

Time t

2000

2500

3000

State of closed−loop NHS via DPDSF control

Figure 1: Markovian jump modes.

X(:,1) X(:,2) X(:,3)

1.5

1

0.5

0

−0.5

−1

−1.5

0

500

1000

1500

Time t

2000

2500

3000

Figure 2: The state x(t) of the closed-loop NHS (7) via DPDSF control. (1) DPDSF control. Assume r0 = 3, Markovian jump mode rt ∈ {1, 2, 3} changes as in Fig.1. Solving LMIs (9), (24)-(26) in Theorem 2 when µ = 1.7, τ = 5.5, we get the desired PDSF controller parameters such

13

that



−0.0141 −0.0074

  [E1 + D1 Kd1 ]−1 (A1 + B1 Kp1 ) =  0.0084  −0.0011  −0.0184   −1 [E2 + D2 Kd2 ] (A2 + B2 Kp2 ) =  0.0036  −0.0023  −0.0025   [E3 + D3 Kd3 ]−1 (A3 + B3 Kp3 ) =  −0.0063  −0.0019  −0.0159   [E1 + D1 Kd1 ]−1 (Ad1 + C1 Kl1 ) =  0.0095  −0.0013  −0.0206   [E2 + D2 Kd2 ]−1 (Ad2 + C2 Kl2 ) =  0.0041  −0.0026  −0.0028   −1 [E3 + D3 Kd3 ] (Ad3 + C3 Kl3 ) =  −0.0069  −0.0021

−0.0141 −0.0042 −0.0040 −0.0150 0.0070 −0.0029 −0.0155 0.0076 −0.0084 −0.0159 −0.0048 −0.0045 −0.0168 0.0079 −0.0032 −0.0171 0.0084

| [E1 + D1 Kd1 ]−1 N1 | = 10−14 × 1.8597 < 1,

0.0042



  0.0011  ,  −0.0182  −0.0014   −0.0072  ,  −0.0160  0.0059   0.0051  ,  −0.0039  0.0048   0.0012  ,  −0.0205  −0.0015   −0.0081  ,  −0.0179  0.0065   0.0057  ,  −0.0042

| [E2 + D2 Kd2 ]−1 N2 | = 10−14 × 9.0752 < 1, | [E3 + D3 Kd3 ]−1 N3 | = 10−14 × 2.3062 < 1. Let the initial condition be x (0) = [ −1.2 1.5 0.8 ]T , Fig.2 shows the state x(t) of the closed-loop NHS (7) via DPDSF control under the conditions of Theorem 2. (2) DPDOI control.  2.1   B1 =  0.2  −1.2  −1.1   C1 =  1.1  1.2  −1.3   C3 =  1.2  1.6

Let 3.3

1.5





    1.2 −1.3  , B2 =    1.4 2.3   0.3 2.5     0.1 −2.1  , C2 =    −1.5 2.2   0.1 2.4     0.3 −2.3  , D2 =    −1.3 2.3

2.3

3.3

0.3 −1.1 −1.2 1.3 1.4 −0.2 0.1 −0.3

1.4





2.2

    1.3 −1.2  , B3 =  0.4   −1.3 1.3 2.4   0.4 2.6 −0.1     0.2 −2.2  , D1 =  0.2   −1.4 2.1 −0.2   0.1 0.4 −0.3     0.2 0.3  , D3 =  0.3   0.2 0.2 −0.2

14

3.2

1.6



  1.4 −1.1  ,  1.5 2.5  0.2 0.3   0.1 0.2  ,  0.1 0.4  0.2 0.5   0.1 0.4  ,  0.4 0.3



−0.14

State of closed−loop NHS via DPDOI control

  Π =  0.05  0.07

0.09



0.05

  −0.13 0.08  .  0.05 −0.12 X(:,2) X(:,3) X(:,1)

1.5

1

0.5

0

−0.5

−1

−1.5

0

10

20

30

40

50

Time t

60

70

80

90

100

Figure 3: The state x(t) of the closed-loop NHS (7) via DPDOI control. Solving LMIs (38)-(41) in Theorem 4 when µ = 0.7, τ = 5.5, we get the desired DPDOI controller parameters: 

−2.6802

2.1394

4.5155





−2.7537

5.2902

1.8543



        Kp1 =  −1.6114 −4.7637 −1.6162  , Kp2 =  −2.7172 −8.6909 −1.6007  ,     −1.5002 6.7791 −3.9590 −1.8205 9.1466 −3.4606     −2.4206 3.6195 2.7777 −0.3200 0.0850 −0.2361         Kp3 =  −4.7304 −8.7753 −1.4598  , Kl1 =  −0.4304 0.2415 −0.2349  ,     −0.5115 8.5649 −4.6886 −0.0942 0.0766 −0.1206     −0.0988 −0.1776 −0.0865 −0.1612 −0.2051 0.1774         Kl2 =  −0.3311 −0.0652 −0.1591  , Kl3 =  −0.4249 −0.1814 0.3614  ,     −0.0682 −0.0238 −0.0884 −0.1072 −0.0193 0.0207     −9.8958 32.5947 −27.5996 −15.4676 40.9570 −14.7799         Kd1 =  88.5224 3.2722 −57.1673  , Kd2 =  −42.2613 29.3441 31.4621  ,     −33.5975 11.2824 43.5245 30.8851 12.0498 −13.1668   −15.0642 20.7475 −10.2852     Kd3 =  −30.3198 −4.0613 39.2955  .   21.6697 19.1198 −8.1760

15

Then, we have that 

−0.9616 −0.2210

  [E1 + D1 Kd1 ]−1 (A1 + B1 Kp1 ) =  −0.0963  −0.2969  −1.2297   −1 [E2 + D2 Kd2 ] (A2 + B2 Kp2 ) =  −0.1562  −0.3731  −1.1217   −1 [E3 + D3 Kd3 ] (A3 + B3 Kp3 ) =  −0.4586  −0.5598 

−0.9953 0.2041 −0.3803 −1.3047 0.2967 −0.4144 −1.1463 0.1708



0.2137

  0.3868  ,  −1.2063  −0.1272   0.1736  ,  −1.2461  0.2352   0.0869  ,  −1.2794 

0.0014 −0.0030 −0.0013     [E1 + D1 Kd1 ] (Ad1 + C1 Kl1 ) =  0.0003 0.0003 0.0007  ,   0.0012 0.0003 0.0003   0.0013 0.0005 0.0012     [E2 + D2 Kd2 ]−1 (Ad2 + C2 Kl2 ) =  −0.0011 0.0006 0.0008  ,   −0.0017 −0.0006 0.0014   0.0012 0.0001 0.0003     [E3 + D3 Kd3 ]−1 (Ad3 + C3 Kl3 ) =  0.0015 −0.0004 −0.0017  ,   −0.0012 0.0002 0.0008 −1

| [E1 + D1 Kd1 ]−1 N1 | = 0.4217 < 1,

| [E2 + D2 Kd2 ]−1 N2 | = 0.4818 < 1, | [E3 + D3 Kd3 ]−1 N3 | = 0.4969 < 1. Let the initial condition still be x (0) = [−1.2 1.5 0.8]T , Fig.3 depicts the state x(t) of the closed-loop NHS (7) via DPDOI control under the conditions of Theorem 4. Example 2 Consider the partial element equivalent circuit (PEEC) system originated from [43] with the control input u(t), and the detailed data are provided by [43]. Implementing the constraint of 8y(t) + 8z(t) =

16

u(t) in [43], then the PEEC system can be described by the NDHS (1) with the following parameters: 



1 0 0 0 0 0



−0.139 0.694 0.278 0 0 0

      E1 = E2 = E3 =       

  0 1 0 0 0 0    0 0 1 0 0 0  ,  0 0 0 0 0 0    0 0 0 0 0 0   0 0 0 0 0 0



    0 0.417 0 0 0   0.556      −0.278 0.556 0.139 0 0 0   , N1 = N2 = N3 =    0 0 0 0 0 0        0 0 0 0 0 0   0 0 0 0 0 0 

      3 A1 = 10 ×       

6

1

2

0

0

3

9

0

0

0

0



  0    1 2 6 0 0 0  ,  −8 0 0 −8 0 0    0 −8 0 0 −8 0   0 0 −8 0 0 −8   7 1 2 0 0 0     9 0 0 0 0   3       1 2 6 0 0 0 3  , A2 = 10 ×    −8 0 0 −8 0 0       0 −8 0 0 −8 0    0 0 −8 0 0 −8   8 1 2 0 0 0     9 0 0 0 0   3      1 2 6 0 0 0  3  , A3 = 10 ×     −8 0 0 −8 0 0      0 −8 0  0 −8 0   0 0 −8 0 0 −8

17



Ad1 = Ad2 = Ad3

State of closed−loop PEEC system via DPDOI control

B1 = B2 = B3  −0.17   Π =  0.07  0.1

0.1

−0.3 0 0 0

0



     −0.05 −0.05 −0.1 0 0 0       −0.05 −0.15 0 0 0 0  ,  =   0 0 0 0 0 0       0 0 0 0 0 0    0 0 0 0 0 0

= C1 = C2 = C3 = D1 = D2 = D3 = I6×6 ,  0.08 0.09   −0.19 0.12  .  0.06 −0.16 2 X(:,1) X(:,2) X(:,3) X(:,4) X(:,5) X(:,6)

1.5

1

0.5

0

−0.5

−1

−1.5

−2

0

20

40

60

80

100

120

140

160

180

200

Time t

Figure 4: The state x(t) of the closed-loop PEEC system via DPDOI control. Solving LMIs (38)-(41) in Theorem 4 when µ = 0.6, τ = 5.5, the desired DPDOI controller (5) can be realized, and let the initial condition be x (0) = [ −1.3 −0.5 −1.6 1.3 0.5 1.6 ]T , Fig.4 demonstrates the state x(t) of the closed-loop PEEC system via DPDOI control under the conditions of Theorem 4.

5

Conclusions This paper considered the problem of normalization and stabilization of NDHSs with neutral and time-

varying state delays. By designing mode-dependent DPDSF and mode-dependent DPDOI controllers, respectively, the stabilization problem of NDHSs has been transformed into the stability problem of normal NHSs via normalization technique. Sufficient conditions have been derived in terms of LMIs by constructing mode-dependent and delay-dependent Lyapunov-Krasovskii functional. Simulation examples including a PEEC system have been utilized to demonstrate the usefulness and validity of the proposed controllers design method.

18

Acknowledgment The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions, which greatly improved the quality and exposition of the paper. This work was supported in part by the National Natural Science Foundation of China under Grants 61773191, 61973148, 61573177, 61603170, 61773207, 61573008; the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant ZR2016JL025; Special Fund Plan for Local Science and Technology Development Lead by Central Authority; Undergraduate Education Reform Project of higher Education in Shandong Province under Grant M2018X047; Liaocheng University Education Reform Project Foundation under Grants G201811, 26322170267. Conflict of Interest: The authors declare that they have no conflict of interest.

References [1] L. Dai, Singular Control Systems, Springer, Berlin, 1989. [2] S. Xu, J. Lam, Robust Control and Filtering of Singular Systems, Springer, Berlin, 2006. [3] E. K. Boukas, Control of Singular Systems with Random Abrupt Changes, Springer, Berlin, 2008. [4] G. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, 2010. [5] A. Ech-charqy, M. Ouahi, E. H. Tissir, Delay-dependent robust stability criteria for singular time-delay systems by delay-partitioning approach, Int. J. Syst. Sci. 49(14) (2018) 2957-2967. [6] B. Sahereh, J. Aliakbar, K. S. Ali, Filtering for descriptor systems with strict LMI conditions, Automatica 80 (2017) 88-94. [7] J. R. Ch´avez-Fuentes, E. F. Costa, J. E. Mayta, M. H. Terra, Regularity and stability analysis of discrete-time Markov jump linear singular systems, Automatica 76 (2017) 32-40. [8] Z. Feng, W. Li, J. Lam, Dissipativity analysis for discrete singular systems with time-varying delay, ISA Trans. 64 (2016) 86-91. [9] C. Lin, Q. G. Wang, T. H. Lee, Robust normalization and stabilization of uncertain descriptor systems with normbounded perturbations, IEEE Trans. Autom. Control 50(4) (2005) 515-520. [10] G. Zhuang, Q. Ma, B. Zhang, S. Xu, J. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time-delays, Syst. Control Lett. 114 (2018) 1-10. [11] G. Zhuang, S. Xu, B. Zhang, H. Xu, Y. Chu, Robust H∞ deconvolution filtering for uncertain singular Markovian jump systems with time-varying delays, Int. J. Robust Nonlinear Control 26(12) (2016) 2564-2585. [12] Y. Chen, W. Zheng, Stability analysis and control for switched stochastic delayed systems, Int. J. Robust Nonlinear Control 26(2) (2016) 303-328.

19

[13] X. Zhao, P. Shi, Y. Yin, S. K. Nguang, New results on stability of slowly switched systems: a multiple discontinuous Lyapunov function approach, IEEE Trans. Autom. Control 62(7) (2018) 3502-3509. [14] S. Xu, J. Lam, X. Mao, Delay-dependent H∞ control and filtering for uncertain markovian jump systems with time-varying delays, IEEE Trans. Circuits Syst. Regul. Pap. 54(9) (2007) 2070-2077. [15] B. Zhang, W. Zheng, S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. Regul. Pap. 54(9) (2013) 1250-1263. [16] Z. G. Wu, P. Shi, Z. Shu, H. Su, R. Lu, Passivity-based asynchronous control for Markov jump systems, IEEE Trans. Autom. Control 62(4) (2017) 2020-2025. [17] S. Zhu, Q. L. Han, C. Zhang, L1 -stochastic stability and L1 -gain performance of positive Markov jump linear systems with time-delays: necessary and sufficient conditions, IEEE Trans. Autom. Control 62(7) (2017) 3634-3639. [18] G. M. Zhuang, J. Xia, W. Zhang, J. Zhao, Q. Sun, H. Zhang, State feedback control for stochastic Markovian jump delay systems based on LaSalle-type theorem, J. Franklin Inst. 355(5) (2018) 2179-2196. [19] H. Shen, Y. Wang, J. W. Xia, J. H. Park, Z. Wang, Fault-tolerant leader-following consensus for multi-agent systems subject to semi-Markov switching topologies: An event-triggered control scheme, Nonlinear Anal. Hybrid Syst 34 (2019) 92-107. [20] Z. Wang, L. Shen, J. Xia, H. Shen, J. Wang, Finite-time non-fragile l2 -l∞ control for jumping stochastic systems subject to input constraints via an event-triggered mechanism, J. Franklin Inst. 355(14) (2018) 6371-6389. [21] J. Xia, H. Gao, M. Liu, G. Zhuang, B. Zhang, Non-fragile finite-time extended dissipative control for a class of uncertain discrete time switched linear systems, J. Franklin Inst. 355(6) (2018) 3031-3049. [22] Z. Fei, S. Shi, Z. Wang, L. Wu, Quasi-time-dependent output control for discrete-time switched system with mode-dependent average dwell time, IEEE Trans. Autom. Control 63(8) (2018) 2647-2653. [23] G. Zhuang , S. Xu, J. W. Xia, Q. Ma, Z. Q. Zhang, Non-fragile delay feedback control for neutral stochastic Markovian jump systems with time-varying delays, Appl. Math. Comput. 355 (2019) 21-32. [24] X. Liang, J. Xia, G. Chen, H. Zhang, Z. Wang, Dissipativity- based sampled-data control for fuzzy Markovian jump systems, Appl. Math. Comput. 361(15) (2019) 552-564. [25] S. Zhu, Q. L. Han, C. Zhang, l1 -gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach, Automatica 50(8) (2014) 2098-2107. [26] S. Zhu, B. Wang, C. Zhang, Delay-dependent stochastic finite-time l1 -gain filtering for discrete-time positive Markov jump linear systems with time-delay, J. Franklin Inst. 354(15) (2017) 6894-6913. [27] G. Liu, J. H. Park, S. Xu, G. Zhuang, Robust non-fragile H∞ fault detection filter design for delayed singular Markovian jump systems with linear fractional parametric uncertainties, Nonlinear Anal. Hybrid Syst. 32 (2019) 65-78.

20

[28] G. Liu, S. Xu, J. H. Park, G. Zhuang, Reliable exponential H∞ filtering for singular Markovian jump systems with time-varying delays and sensor failures, Int. J. Robust Nonlinear Control 28(14) (2018) 4230-4245. [29] Y. Ding, S. Zhong, S. Long, Asymptotic stability in probability of singular stochastic systems with Markovian switchings, Int. J. Robust Nonlinear Control 27(18) (2017) 4312-4322. [30] G. M. Zhuang, J. Xia, J. Feng, B. Zhang, J. Lu, Z. Wang, Admissibility analysis and stabilization for neutral descriptor hybrid systems with time-varying delays, Nonlinear Anal. Hybrid Syst. 33 (2019) 311-321. [31] R. Sakthivel, M. Joby, K. Mathiyalagan, S. Santra, Mixed H∞ and passive control for singular Markovian jump systems with time delays, J. Franklin Inst. 352(10) (2015) 4446-4466. [32] G. M. Zhuang, J. Xia, W. Zhang, W. Sun, Q. Sun, Normalization design for delayed singular Markovian jump systems based on system transformation technique, Int. J. Syst. Sci. 49(8) (2018) 1603-1614. [33] X. M. Zhang, Q. L. Han, J. Wang, Admissible delay upper bounds for global asymptotic stability of neural networks with time-varying delays, IEEE Trans. Neural Networks learning systems 99 (2018) 1-11. [34] X. M. Zhang, Q. L. Han, X. Ge, D. Ding, An overview of recent developments in Lyapunov-Krasovskii functionals and stability criteria for recurrent neural networks with time-varying delays, Neurocomputing 313 (2018) 392-401. [35] B. Liu, X. C. Jia, New absolute stability criteria for uncertain Lure systems with time-varying delays, J. Franklin Inst. 355(9) (2018) 4015-4031. [36] B. Liu, X. Ma, X. C. Jia, Further results on H∞ state estimation of static neural networks with time-varying delay, Neurocomputing 285 (2018) 133-140. [37] H. H. Lian, S. P. Xiao, Z. Wang, X. H. Zhang, H. Q. Xiao, Further results on sampled-data synchronization control for chaotic neural networks with actuator saturation, Neurocomputing 346 (2019) 30-37. [38] H. B. Zeng, X. G. Liu, W. Wang, S. P. Xiao, New results on stability analysis of systems with time-varying delays using a generalized free-matrix-based inequality, J. Franklin Inst. 356 (13) (2019) 7312-7321. [39] X. H. Chang, Z. M. Li, J. H. Park, Fuzzy generalized H2 filtering for nonlinear discrete-time systems with measurement quantization, IEEE Trans. Syst. Man Cybern. Syst. 48(12) (2018) 2419-2430. [40] X. Zhang, Q. Han, A. Seuret, F. Gouaisbaut, An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay, Automatica 84 (2017) 221226. [41] J. Hale, S. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [42] G. Zhuang, H. Zhang, J. Zhao, W. Sun, Q. Sun, Delay feedback controller design for neutral stochastic markovian jump systems, Journal of Liaocheng University (natural science edition) 31(4) (2018) 65-71. [43] D. Yue, Q. Han, A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model, IEEE Trans. Circuits Syst. Express Briefs 51(12) (2004) 685-689.

21

[44] E. Fridman, Stability of linear descriptor systems with delay: a Lyapunov-based approach, J Math. Anal. Appl. 273(1) (2002) 24-44. [45] Erol, H. E., ´Iftar, A. Stabilization of decentralized descriptor-type neutral time-delay systems by time-delay controllers. Automatica, 64, 262-269, 2016. [46] X. Zhang, Q. Han, Z. Wang, B. L. Zhang, Neuronal state estimation for neural networks with two additive timevarying delay components, IEEE Trans. cybernetics 47(10) (2017) 3184-3194. [47] D. Zhang, Q. Han, X. Zhang, Network-Based Modeling and Proportional-Integral Control for Direct-Drive-Wheel Systems in Wireless Network Environments, IEEE Trans. cybernetics doi: 10.1109/TCYB.2019.2924450, 2019. [48] W. Sun, S. Su, Y. Wu, J. Xia, V. T. Nguyen, Adaptive fuzzy control with high-order barrier Lyapunov functions for high-order uncertain nonlinear systems with full-state constraints, IEEE Trans. cybernetics doi: 10.1109/TCYB.2018.2890256, 2018. [49] W. Sun, J. Xia, G. Zhuang, X. Huang, H., Shen, Adaptive fuzzy asymptotically tracking control of full state constrained nonlinear system based on a novel Nussbaum-type function, J. Franklin Inst. 356(4) (2019) 1810-1827. [50] D. Zhang, Q. Han, X. Jia, Network-based output tracking control for T-S fuzzy systems using an event-triggered communication scheme, Fuzzy Sets Syst. 273(15) (2015) 26-48. [51] D. Zhang, Y. Yang, X. Jia, Q. Han, Investigating the positive effects of packet dropouts on network-based H∞ control for a class of linear systems, J. Franklin Inst. 353(14) (2016) 3343-3367. [52] J. Xia, J. Zhang, W. Sun, B. Zhang, Z. Wang, Finite-time adaptive fuzzy control for nonlinear systems with full state constraints, IEEE Trans. Syst. Man Cybern. Syst. 49 (7) (2019) 1541-1548. [53] J. Xia, J. Zhang, J. Feng, Z. Wang, G. Zhuang, Command filter-based adaptive fuzzy control for nonlinear systems with unknown control directions, IEEE Trans. Syst. Man Cybern. Syst. doi: 10.1109/ TSMC.2019.2911115, 2019. [54] X. Chang, R. Liu, Ju H. Park, A further study on output feedback H∞ control for discrete-time systems, IEEE Trans. Circuits Syst. Express Briefs doi: 10.1109/TCSII.2019.2904320, 2019. [55] X. Chang, R. Huang, H. Wang, L. Liu, Robust design strategy of quantized feedback control, IEEE Trans. Circuits Syst. Express Briefs doi: 10.1109/TCSII.2019.2922311, 2019. [56] A. J. Krener, A. Isidori, Linearization by output injection and nonlinear observers, Syst. Control Lett. 3(1) (1983) 47-52. [57] X. Mao, J. Lam, L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Syst. Control Lett. 57(11) (2008) 927-935.

22