Observer-based control design of semi-Markovian jump systems with uncertain probability intensities and mode-transition-dependent sojourn-time distribution

Applied Mathematics and Computation 372 (2020) 124968

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Observer-based control design of semi-Markovian jump systems with uncertain probability intensities and mode-transition-dependent sojourn-time distribution Khanh Hieu Nguyen, Sung Hyun Kim∗ School of Electrical Engineering, University of Ulsan, Daehak-ro 93, Nam-Gu, Ulsan 680-749, South Korea

a r t i c l e

i n f o

Article history: Received 17 June 2019 Revised 1 November 2019 Accepted 8 December 2019

Keywords: Semi-Markovian jump systems Probability distribution and intensity Observer-based control Relaxation process

a b s t r a c t This paper deals with the problem of H∞ observer-based control for a class of continuoustime semi-Markovian jump systems (S-MJSs) with more detailed observational information. First, to explore the impact of uncertain probability intensities, a mathematical analysis is accomplished, from which some useful inequality conditions on the sum of transition rates (TRs) are obtained. Further, to come up with more accurate bounds of TRs, the mode-transition-dependent probability distribution of sojourn time is imposed on the mechanism of forming TRs. Lastly, by devising a compatible relaxation process that can embrace all the conditions found in our derivation, the resultant observer-based stabilization conditions are formulated in terms of linear matrix inequalities. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Over the past several decades, rapidly growing attention has been witnessed in the research of Markovian jump systems (MJSs) that are recognized as one of the most successful models in representing a class of stochastic hybrid systems subject to random abrupt changes such as component faults or failures, sudden environmental changes, and changing subsystem interconnections (refer to [1–3], and references therein). For instance, considerable efforts have been contributed to the studies on MJSs to deal with various kind of practical systems such as fault-tolerant systems [4–6], power systems [7,8], actuator saturation [9–11], communication systems [12,13], and networked control systems [14–16]. Recently, to overcome the limitation of homogeneous MJSs with exponential distributions for the sojourn-time (representing the interval between two consecutive jumps), a great deal of attention has been drawn to the study of semi-MJSs (S-MJSs) because such systems can ultimately offer the possibility to represent a continuous Markov process with nonexponential sojourn-time distributions. As prominent achievements, Hou et al. [17] studied the problem of stochastic stability for linear systems with phase-type semi-Markovian jump parameters. After that, Huang and Shi [18,19] relaxed the probability distribution of sojourn time from exponential to Weibull distribution, and then developed a sojourn-time partition technique that can reduce the conservatism of stability conditions. Furthermore, to achieve less conservative stabilization conditions of S-MJSs, Nguyen et al. [20] focused on imposing the impact of uncertain probability intensities in their derivation, and Kim [21] proposed the use of the mode-transition-dependent sojourn-time distributions. Recently, Zhang et al. [22] proposed the exponentially modulated periodic distributions of sojourn time to address the stability analysis and control synthesis problems for S-MJSs. ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (S.H. Kim).

https://doi.org/10.1016/j.amc.2019.124968 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

2

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

However, a remarkable thing is that most of the above results are only applicable to the state-feedback control design of S-MJSs under the assumption that full state is accessible. In practice, as is well known, the control synthesis problem is often essentially confronted with a critical situation where all system states are not fully measurable. Thus, to alleviate such concerns successfully, it is of crucial importance to recover the states since they provide a lot of information in calculating the corresponding control input. In this sense, the problem of designing a mode-dependent observer-based controller for S-MJSs also deserves special attention. Among the recent approaches regarding the observer-based control design issue, Liu et al. [23] addressed the problem of sliding mode control for networked control systems with semi-Markovian switching and random measurement, and Chen et al. [24] proposed the observer-based sensor fault-tolerant control for S-MJSs. Further, Jiang et al. [25] investigated the problem of observerbased fuzzy sliding mode control for finite-time synthesis of S-MJSs with immeasurable premise variables. Most recently, to deal with the periodic reference tracking problem, Ma et al. [26] proposed the observer-based H∞ modified repetitive controller for S-MJSs with logarithmically quantized output measurements. However, it is worth noticing that, in [24], the probability distribution of sojourn-time is established regardless of mode transitions, and in [23,25,26], no description on the probability distribution of sojourn-time is given during defining the transition rates. Moreover, since the sojourn-time is not regarded as a random variable therein, the bounds of transition rates can be occasionally infinite, as discussed in [27]. Especially, the probability intensity from one mode to the other has to be estimated and analyzed statistically, for instance, via the relative frequency distribution. Hence, when forming the mode transition rates from the estimated probability intensities, it is necessary to take into account the presence of either their norm-bounded or polytopic uncertainties caused by the estimate errors. Nevertheless, to the best of our knowledge, there has been almost no research directly attempting to explore the impact of such uncertain probability intensities, especially while covering the problem of observer-based control design for S-MJSs with mode-transition-dependent distributions of the sojourn time. To take a step forward from the above concerns, this paper proposes a method that can exploit both uncertain probability intensities and mode-transition-dependent sojourn-time distribution while addressing the problem of H∞ observer-based control for S-MJSs. Overall, the main contributions of this paper can be highlighted as follows. • By fully describing the process of deriving transition rates (TRs) from the probability distribution of the sojourn-time, this paper offers a theoretical framework to incorporate the impact of uncertain probability intensities into S-MJSs. A remarkable feature is that this framework leads to some useful inequality conditions on the sum of TRs, as well as their boundary conditions, which ultimately has a beneficial effect on tightly reshaping the allowable region of TRs. Further, to alleviate the numerical stiffness arising from a specific case where the upper bounds of TRs become infinite, the involved sojourn-time herein is consistently interpreted as a random variable while deriving a set of stochastic stabilization conditions. • Different from [23–26], this paper provides a possible way to impose the mode-transition-dependent probability distribution of the sojourn time in our derivation, which plays an important role in forming more accurate bounds of TRs. In other words, constructing more realistic TR-boundary conditions is allowed on basis of this imposition, which essentially affects deriving less conservative stabilization conditions through the proposed relaxation process. Moreover, in this paper, the probability distribution of the sojourn time is established to be of a general form, including Weibull distribution, such that a broad range of application systems can be covered. • As is well-known, the stability and stabilization problem of S-MJSs generally demands the relaxation process to obtain a finite number of LMIs from uncertain TR-dependent matrix inequalities. In the same vein, this paper also proposes a compatible relaxation method that can embrace all the derived inequality conditions and strict boundary conditions in the underlying stabilization conditions. Further, to constitute a trade-off between computational complexity and performance quality, this paper offers an alternative set of observer-based stabilization conditions for S-MJSs. The validity of the proposed method is demonstrated through illustrative examples. The rest of the paper is organized as follows. In Section 2, we present the mathematical of semi-MJSs and preliminaries results. Section 3 gives main results of this paper. The validity of the proposed method is demonstrated through illustrative examples in Section 4. Finally, the paper is concluded in Section 5. Notations: The notations X ≥ Y and X > Y mean that X − Y is positive semi-definite and positive definite, respectively. In symmetric block matrices, (∗ ) is used as an ellipsis for terms induced by symmetry. The triplet notation (, F, P ) denotes a probability space, where , F, and P represent the sample space, the algebra of events, and the probability measure defined on F, respectively. E{ · } denotes the mathematical expectation; diag( · ) stands for a diagonal matrix with diagonal entries; col(v1 , v2 , . . . , vn ) = [vT1 vT2 . . . vTn ]T for scalar or vector vi ;  denotes the Kronecker product; S1 ࢨS2 indicates the set of elements in the set S1 , but not in the set S2 ; λmax ( · ) denotes the maximum eigenvalue of the argument; He{Q} is used to represent Q + QT for any square matrix Q; and  L2 [0, ∞ ) denotes the space of square integrable vector functions of a given dimension over [0, ∞). For S = {1, 2, . . . , s}, Qi D = diag(Q1 , Q2 , . . . , Qs ), i∈S

⎡



Qi j

L

i, j∈S

0

⎢ ⎢Q21 ⎣ ..

=⎢

. Q s1

··· 0 .. . ···

··· .. . .. . Qs(s−1)

⎤

0 .. ⎥ .⎥

  ⎥, Q i i ∈ S ⎦

0 0

⎡

⎤

Q1 ⎢Q 2 ⎥ = ⎢ . ⎥, ⎣ .. ⎦ Qs

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

3

Fig. 1. Description of semi-Markov process {φ (t), t ≥ 0}.

where Qi and Qi j denote real submatrices with appropriate dimensions. 2. System description and preliminaries Let us consider the following continuous-time system:



x˙ (t ) = A(φ (t ))x(t ) + B(φ (t ))u(t ) + E (φ (t ))w(t ), y(t ) = C (φ (t ))x(t ) + D(φ (t ))w(t ), z(t ) = G(φ (t ))x(t ) + H (φ (t ))u(t ),

(1)

where x(t ) ∈ Rnx , u(t ) ∈ Rnu , y(t ) ∈ Rny , z(t ) ∈ Rnz , w(t ) ∈ Rnw , and φ (t ) ∈ Nφ = {1, 2, . . . , nφ } denote the state, the control input, the measurement output, the performance output, the disturbance input belonging to L2 [0, ∞ ), and the system operation mode, respectively; and A(φ (t ) = i ) = Ai , B(φ (t ) = i ) = Bi , C (φ (t ) = i ) = Ci , D(φ (t ) = i ) = Di , E (φ (t ) = i ) = Ei , G(φ (t ) = i ) = Gi , and H (φ (t ) = i ) = Hi are known system matrices with appropriate dimensions. Especially, {φ (t), t ≥ 0} is characterized by a semi-Markov process that contains (i) the kth transition instant tk , for k ∈ N+ , and (ii) a Markov chain {k ∈ Nφ , k ∈ N+ } that satisfies the following transition probability:

q˜i j = qi j + qi j = Pr(k+1 = j | k = i ),

∀ j = i,

where qij denotes the estimated transition probability from mode i to mode j, and qi j ∈ [δ i j , δ i j ] denotes the estimate error (or the uncertainty of qij ). If δ i j = δ i j = 0, then q˜i j = qi j (completely known). Otherwise, q˜i j ∈ [qi j , qi j ], where qi j = qi j + δ i j and qi j = qi j + δ i j . Furthermore, the semi-Markov process can be said to be essentially associated with the Markov renewal process {(k , tk ), k ∈ N+ } such that φ (t ) = k , ∀tk ≤ t < tk+1 (refer to [28,29]). For detailed analysis, let Tk = tk − tk−1 > 0 be the sojourn time of the system mode between (k − 1 )th and kth jumps (see Fig. 1). Then, it can be established that



φ (t + δ ) = j | φ (t ) = i

 Pr h + δ ≥ Tk+1 , k+1 = j | h < Tk+1 , k = i ,  = Pr h + δ < Tk+1 | h < Tk+1 , k = i ,

Pr

if j = i, if j = i,

where t = tk + h (i.e., h = t − tk ) and δ denotes a time interval. In addition, let the cumulative distribution function of h be given as follows:

Fi j (h ) = Pr{h ≥ Tk+1 | k+1 = j, k = i}, Fi (h ) = Pr{h ≥ Tk+1 | k = i} =



(2)

q˜i j Fi j (h ).

(3)

j∈Nφ \{i}

Then, based on the probability density function fi j (h ) = lim (Fi j (h + δ ) − Fi j (h ))/δ, the transition rate π ij (h) is formulated as δ →0

follows:

πi j (h ) = q˜i j · ηi j (h ), ηi j (h ) = πii (h ) = −



f i j (h ) , 1 − Fi (h )

∀ j = i,

(4)

q˜i j ηi j (h ).

(5)

j∈Nφ \{i}

Especially, in (3), it is worth noticing that the uncertainty of q˜i j ∈ [qi j , qi j ] entails the lower and upper bounds of Fi (h),

i.e., Fi (h ) ∈ [F i (h ), F i (h )], which definitely influence the value of ηi j (h ) ∈ [η (h ), ηi j (h )] by (4). Thus, it is also assured that ij

4

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968









E ηi j (h ) = E pη (h ) + (1 − p)ηi j (h ) = pη + (1 − p)ηi j ∈ [η , ηi j ], for 0 ≤ p ≤ 1, where η = E[η (h )] = ij ij ij ij ij ∞ and ηi j = E[ηi j (h )] = 0 ηi j (h ) fi j (h )dh. Hence, the expectation of π ij (h) satisfies

∞ 0

ηi j (h ) fi j (h )dh

    π˜ i j = E πi j (h ) ∈ qi j ηi j , qi j ηi j , ∀ j = i,

(6)

     π˜ ii = E πii (h ) = − q˜i j E ηi j (h )

 ∈ where Qi =

  q˜i j

j∈Nφ \{i}

j∈Nφ \{i}



− max Qi

q˜i j ηi j , − min Qi

j∈Nφ \{i}



 q˜i j ηi j ,

(7)

j∈Nφ \{i}

   q˜i j ∈ [q , qi j ],  j∈N \{i} q˜i j = 1 . That is, π˜ i j is bounded as π˜ i j ∈ [π i j , π i j ], for all j ∈ Nφ . ij φ

Remark 1. Different from [23–26], this paper offers the possibility to express more accurate boundary conditions on timevarying transition rates by taking into account the following two factors: (i) the mode-transition-dependent distribution of sojourn time, i.e., Fij (h), and (ii) the uncertain probability intensities q˜i j ∈ [qi j , qi j ]. Now, let us consider the following observer-based control law:

˙

x˜(t ) = A(φ (t ))x˜(t ) + B(φ (t ))u(t ) − L(φ (t ))(y(t ) − y˜(t )), y˜(t ) = C (φ (t ))x˜(t ), u(t ) = F (φ (t ))x˜(t ),

(8)

where x˜(t ) ∈ Rnx and y˜(t ) ∈ Rny denote the estimated state and the estimated output, respectively; L(φ (t)) and F(φ (t)) denote the filter gain and the control gain to be designed later, respectively. Then, for φ (t ) = i, the resultant closed-loop system for (1) and (8) is given as follows:

x¯˙ (t ) = A¯ i x¯ (t ) + E¯i w(t ), z(t ) = G¯ i x¯ (t ),

(9)

where x¯ (t ) = col(x(t ), e(t )), e(t ) = x(t ) − x˜(t ),









Ai +Bi Fi A¯ i = 0

−Bi Fi Ei , E¯i = , Ai +LiCi Ei +Li Di

G¯ i = Gi +Hi Fi

−Hi Fi .





Definition 1 ([30,31]). System (9) with w(t) ≡ 0 is stochastically stable if its solution is such that, for any initial condition x0 and φ 0 ,



T

lim E

T →∞

0

   ||x¯ (t )|| dt x¯0 , φ0 < ∞. 2

(10)

Definition 2 ([32,33]). System (9) is stochastically stable with γ -disturbance attenuation if system (9) is stochastically stable in the absence of disturbances and, for all 0 = w(t ) ∈ L2 [0, ∞ ), the response z(t) satisfies



T

lim E

T →∞

0

  ||z(τ )||22 − γ 2 ||w(τ )||22 dτ  x¯0 = 0, φ0 < 0.

(11)

To sum up, the main purpose of this paper is to design an observer-based controller for S-MJSs (9) with uncertain transition rates and mode transition-dependent sojourn-time distributions in the H∞ sense, especially by proposing an efficient relaxation scheme for parameterized linear matrix inequalities (PLMIs) with uncertain transition rates governed by (6) and (7). Lemma 1 ([34]). For given matrices Q, X, and Y = Y T < 0 with appropriate dimensions, then following inequality holds:

X Y X T ≤ −He{QX T } − QY −1 Q T . 3. Control design The following lemma provides the stochastic stability and H∞ performance condition for (9), formulated in terms of PLMIs. Lemma 2. Suppose that there exit Pi > 0, for all i ∈ Nφ , and a positive scalar γ such that



0 > L(π˜ i j ) =

i + G¯ Ti G¯ i (∗ )



Pi E¯i , −γ 2 I

(12)

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

where i =

 j∈Nφ

5

π˜ i j Pj + He{Pi A¯ i }. Then closed-loop system (9) is stochastically stable with γ -disturbance attenuation.

Proof. Let us consider the following Lyapunov function:

V (t, φ (t )) = x¯T (t )P (φ (t ))x¯ (t ),

(13)

where P(φ (t ) = i ) = Pi > 0. Then, the weak infinitesimal operator ∇ of the stochastic process {x¯ (t ), φ (t )} acting on V(t, φ (t)) is given by

  1 ∇V (t, φ (t )) = lim E V (t + δ, φ (t + δ ) = j | φ (t ) = i ) − V (t, φ (t ) = i ) δ →0 δ

  1 T T T = lim E p(h, δ ) · x¯ (t + δ )Pj x¯ (t + δ ) + p¯ (h, δ ) · x¯ (t + δ )Pi x¯ (t + δ ) − x¯ (t )Pi x¯ (t ) , δ →0 δ j∈N \{i}

(14)

φ

where the probabilities p(h, δ ) and p¯ (h, δ ) are given as



p(h, δ ) = Pr h + δ ≥ Tk+1 , k+1 = j | h < Tk+1 , k = i = Pr{k+1 = j | k = i} ·



Pr{h + δ ≥ Tk+1 , h < Tk+1 | k+1 = j, k = i} Pr{h < Tk+1 | k = i}

Fi j (h + δ ) − Fi j (h ) , 1 − Fi (h )

= q˜i j ·

(15)



p¯ (h, δ ) = Pr h + δ < Tk+1 , k+1 = i | h < Tk+1 , k = i



Pr{h + δ < Tk+1 | k = i} 1 − Fi (h + δ ) = . Pr{h < Tk+1 | k = i} 1 − Fi (h )

=

(16)

Further, by (15) and (16), it can be found that

f i j (h ) = πi j ( h ) , 1 − Fi (h )  1 1 j∈Nφ \{i} q˜i j (Fi j (h ) − Fi j (h + δ )) • lim ( p¯ (h, δ ) − 1 ) = lim 1 − Fi (h ) δ →0 δ δ →0 δ  f i j (h ) =− q˜i j = πii (h ), 1 − Fi (h ) • lim

1

δ →0 δ

• lim

δ →0

p(h, δ ) = q˜i j

1

δ

j∈Nφ \{i}



x¯T (t + δ )Pi x¯ (t + δ ) − x¯T (t )Pi x¯ (t ) = 2x¯T (t )Pi x¯˙ (t ).

Thus, from (14), it follows that

∇V (t, φ (t ))

1

= lim

δ

δ →0

E



p(h, δ )x¯T (t + δ )Pj x¯ (t + δ )

j∈Nφ \{i}



+ ( p¯ (h, δ ) − 1 )x¯T (t + δ )Pi x¯ (t + δ ) + x¯T (t + δ )Pi x¯ (t + δ ) − x¯T (t )Pi x¯ (t ) ,





=E

 πi j (h )x¯ (t )Pj x¯ (t ) + 2x¯ (t )Pi x¯˙ (t ) T

T

j∈Nφ

=



j∈Nφ

 π˜ i j x¯T (t )Pj x¯ (t ) + 2x¯T (t )Pi A¯ i x¯ (t ) + E¯i w(t ) 

i = ζ (t ) (∗ ) T

Pi E¯i 0







x¯ (t ) ζ (t ), ζ (t ) := . w(t )

(17)

(i) Stability condition: From (17), let us first consider the case where w(t) ≡ 0. Then, ∇ V (t, φ (t )) = x¯T (t ) i x¯ (t ). Accordingly, by the generalized Dynkin’s formula, we have

6

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968



   E{V (t, φ (t ))} − V (0, φ0 ) = E ∇V (t )dt x¯0 , φ0 0

 T   ≤ max(λmax ( i )) · E ||x¯ (t )||22 dt  x¯0 , φ0 , T

i∈N φ

(18)

0

which leads to



T

E 0

||x¯ (t )||

2 2 dt

   x¯0 , φ0 ≤ −

V ( 0 , φ0 ) . max(λmax ( i )) i∈N φ

Therefore, since (12) implies i < 0 (i.e., maxi∈Nφ (λmax ( i )) < 0), it is guaranteed that



T

lim E

T →∞

0

||x¯ (t )||

2 2 dt

   x¯0 , φ0 < ∞.

That is, condition (12) ensures that closed-loop system (9) with w(t) ≡ 0 is stochastically stable, as mentioned in Definition 1. (ii) H∞ performance: For the overall system (9), we have

∇V (t, φ (t )) + ||z(t )||22 − γ 2 ||w(t )||22 = ζ T (t )L(π˜ i j )ζ (t ).

(19)

Thus, by (12), it is ensured that

||z(t )||22 − γ 2 ||w(t )||22 < −∇V (t, φ (t )). Accordingly, for x¯0 = 0, it holds that



T

E 0

 ||z(τ )||22 − γ 2 ||w(τ )||22 dτ < −E{V (T , φ (T ))} < 0.

Therefore, by Definition 2, the proof is completed.



The following lemma provides the stochastic H∞ observer-based stabilization condition for (9), formulated in terms of PLMIs. Lemma 3. Suppose that there exit matrices 0 < P¯1i ∈ Rnx ×nx , 0 < P2i ∈ Rnx ×nx , L¯ i ∈ Rnx ×ny , and F¯i ∈ Rnu ×nx ; and a scalar γ > 0 such that the following conditions hold: for all i ∈ Nφ ,

⎡

−I ⎢ (∗ )

⎢ ⎢ 0 > ⎢ (∗ ) ⎢0 ⎣0 0



Ri j 0≤ (∗ )

Gc Ac

(∗ ) (∗ ) 0 0

−Hi F¯i −Bi F¯i −2μP¯1i (∗ ) (∗ ) (∗ )

0 Ei

0 0

−Q1i −He{Q2i } 0 (∗ )

μI

0 0

⎤ ⎥



⎥ Q1i ⎥ +  π˜ hT Ri j − P¯1i ⎥ ij Q2i ⎥ 0 j∈Nφ \{i} ⎦ Eo −γ 2 I

0 Ao (∗ )



0 h, P2 j − P2i

(20)



P¯1i , ∀ j ∈ Nφ P¯1 j

\ {i},

(21)

where









Ac = He Ai P¯1i + Bi F¯i , Ao = He P2i Ai + L¯ iCi ,



Gc = Gi P¯1i + Hi F¯i , Eo = P2i Ei + L¯ i Di , h =

0 0

I 0

0 0

0 0

0 I



0 . 0

Then closed-loop system (9) is stochastically stable with γ -disturbance attenuation, and the observer-based control gains are designed as follows: Li = P2i −1 L¯ i and Fi = F¯i P¯1−1 . i

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968







7



Proof. To begin with, let us take Pi = diag P1i , P2i and Pi = diag P1i , P1i . Then, the coupling terms in (12) are represented as follows:







• He Pi A¯ i = He

P1i 0

0 P2i

 = He Pi

 = Pi



0

• Pi E¯i = Pi

P¯1i





−Bi Fi Ai + LiCi





−Hi F¯i Pi = Gc

Ei

P¯1i P2i Ei + L¯ i Di



P1i Eo



(∗ )



   = Pi Ei = Pi Ei , ¯

 −Bi F¯i , Gi = Gc P¯1i AoP¯1i

Ac

 

−Hi F¯i Pi = Gi Pi ,

where P¯1i = P1−1 , F¯i = Fi P¯1i , L¯ i = P¯2i Li , i

Ai =



−Bi F¯i Pi P2i Ai + L¯ iCi P¯1i

−Bi F¯i Pi = Pi Ai Pi , ¯ P1i AoP¯1i

Ac (∗ )



Ai + Bi Fi 0

Ai P¯1i + Bi F¯i

• G¯ i = Gi P¯1i + Hi F¯i









−Hi F¯i , Ei =



Ei

.

P¯1i Eo

Further, by the Schur complement, (12) is rearranged as follows:

⎡

0 > ⎣ (∗ ) 0

0 Pi E¯i ⎦ ⇔ 0 > −γ 2 I

i (∗ )

where i = Pi Ai Pi +

⎡

⎤

G¯ i

−I



0 > ⎣ (∗ )

Ai + P¯ i

−I (∗ ) 0

Gi Pi

i (∗ )



0 Pi Ei , −γ 2 I

(22)

π˜ i j Pj . Thus, performing a congruence transformation (22) with diag(I, P−1 , I ) offers i

j∈Nφ

−I



Gi j∈Nφ

π˜ i j Pj P¯ i

⎤

Ei

⎦,

(23)

−γ 2 I

(∗ )

0

0

where P¯ i = P−1 . i    Meanwhile, noting that π˜ ii = − j∈N \{i} π˜ i j , it is available that ˜ i j Pj = j∈N \{i} π˜ i j (Pj − Pi ), and condition j∈Nφ π φ φ (23) becomes

⎡

−I ⎢ (∗ ) 0>⎣ (∗ ) 0 where

!

0 Ei ⎥ ⎦, P¯1i Eo −γ 2 I

(24)

"



(2, 2 ) = Ac + P¯1i

⎤

−Hi F¯i −Bi F¯i ( 3, 3 ) (∗ )

Gc ( 2, 2 ) (∗ ) (∗ )

π˜ i j (P1 j − P1i ) P¯1i ,

j∈Nφ \{i}

!

"



(3, 3 ) = P¯1i AoP¯1i + P¯1i

π˜ i j (P2 j − P2i ) P¯1i

j∈Nφ \{i}

Especially, from (24), let us consider the following block matrix:



P¯1i AoP¯1i + P¯1i

 =

P¯1i 0

0 I





j∈Nφ \{i}

π˜ i j (P2 j − P2i ) P¯1i

Ao +

π˜ i j (P2 j − P2i ) (∗ )

j∈Nφ \{i}

Then, since (20) implies



Ao 0> (∗ )



 Eo + −γ 2 I



−γ 2 I

(∗ ) 

P¯1i Eo

j∈Nφ \{i}

 π˜ i j

P2 j − P2i 0



0 , 0

Eo −γ 2 I



P¯1i 0



0 . I

8

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

Lemma 1 allows that



( 3, 3 ) (∗ )

P¯1i Eo −γ 2 I

where



 ≤ −He



( 5, 5 ) = Ao +

μI

Q1i Q2i

0



P¯1i 0

0 I



 −

μI

Q1i Q2i

0



( 5, 5 ) (∗ )

Eo −γ 2 I

−1 

μI

Q1i Q2i

0

T ,

(25)

π˜ i j (P2 j − P2i ).

j∈Nφ \{i}

Thus, based on the inequality (25) and by the Schur complement, we can obtain the following sufficient condition for (24):

⎡

−I ⎢ (∗ ) ⎢ ⎢ (∗ ) 0>⎢ ⎢0 ⎣0 0

−Hi F¯i −Bi F¯i −2μP¯1i (∗ ) (∗ ) (∗ )

Gc ( 2, 2 ) (∗ ) (∗ ) 0 0

0 Ei −Q1i −He{Q2i } 0 (∗ )

⎤

0 0 μI 0 ( 5, 5 ) (∗ )

0 0 ⎥ ⎥ Q1i ⎥ ⎥. Q2i ⎥ Eo ⎦ −γ 2 I

(26)

Finally, since (21) guarantees Ri j ≥ P¯1i P1 j P¯1i , ∀j ∈ Nφ ࢨ{i}, it holds that



( 2, 2 ) = Ac +



π˜ i j (P¯1i P1 j P¯1i − P¯1i ) ≤ Ac +

j∈Nφ \{i}

π˜ i j (Ri j − P¯1i ).

j∈Nφ \{i}



Accordingly, it can be shown that condition (20) implies (26).

Remark 2. In general, since the observer-based stabilization control is formulated in terms of nonconvex terms, it is positively necessary to employ an LMI decoupling technique in order to handle such a nonconvexity (refer to [35,36]). In this sense, Lemma 3 also applies a decoupling technique to obtain the PLMI-based condition that guarantees the H∞ stochastic observer-based stabilization of (9). The following theorem provides the relaxed stochastic H∞ observer-based stabilization conditions for (9) with (6) and (7), formulated in term of LMIs. Theorem 1. For a prescribed μ > 0, suppose that there exit matrices 0 < P¯1i = P¯1Ti ∈ Rnx ×nx , 0 < P2i = P2Ti ∈ Rnx ×nx , Ri j = RTi j ∈



Rnx ×nx , L¯ i ∈ Rnx ×ny , F¯i ∈ Rnu ×nx , Q1i ∈ Rnx ×nw , Q2i ∈ Rnw ×nw , Xi j = a scalar γ > 0 such that the following conditions hold: for all i ∈ Nφ ,

 0 > Mi =  1 2

 0≤

i 

i j h

(∗ )



0

j∈Nφ \{i}



+

Xi(1) + Yi(1)

X12i j

X21i j

X22i j

(∗ )

Xi(2) + Yi(2)



X11i j

Xi(3) + Yi(3)



∈ R2nx ×2nx , Yi =

 ,

(27)

\ {i},

(28)

∀ j ∈ Nφ \ {i},

(29)

0 < He{Yi },

⎡

−I ⎢ (∗ ) ⎢ ⎢ i = ⎢(∗ ) ⎢0 ⎣0 0

(30) Gc Ac (∗ ) (∗ ) 0 0



R − P¯1i i j = i j 0

!

(1 )

Xi

=



Y12i ∈ R2nx ×2nx ; and Y22i



P¯1i , ∀ j ∈ Nφ P¯1 j

Ri j (∗ )

0 < He{Xi j },

where

Y11i Y21i

 j∈Nφ \{i}

−Hi F¯i −Bi F¯i −2μP¯1i (∗ ) (∗ ) (∗ )

0 Ei −Q1i −He{Q2i } 0 (∗ )

0 0 μI 0 Ao (∗ )

⎤

0 0 ⎥ ⎥ Q 1 i ⎥, ⎥ Q2i ⎥ Eo ⎦ −γ 2 I

(31)



0 , P2 j − P2i



(32)

−π i j π i j He h Xi j h T

" 



0 , h= 0

I 0

0 0

0 0

0 I



0 , 0

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968



Xi(2) = (π i j + π i j )Xi j h





Xi(3) =

j∈Nφ \{i} ,





Yi(1) = He −π ii π ii hT Yi h , Yi(2) = Yi(3) =





− He{Yi }



D j∈Nφ \{i}



+ He

 

−



− He Xi j

 π ii + π ii Yi h

− 2Yi

9

 D 

j∈Nφ \{i} ,

j∈Nφ \{i} ,



L

,

j,v∈Nφ \{i}



Ac = He Ai P¯1i + Bi F¯i , Ao = He P2i Ai + L¯ iCi , Gc = Gi P¯1i + Hi F¯i , Eo = P2i Ei + L¯ i Di . Then closed-loop system (9) is stochastically stable with γ -disturbance attenuation, and the observer-based control gains are designed as follows: Li = P2i −1 L¯ i and Fi = F¯i P¯1−1 . i Proof. Based on (31) and (32), condition (20) can be rearranged as follows:



0 > i +

π˜ i j hT i j h

j∈Nφ \{i}

⎡ i 1 # T   = i + i  h i j h j∈Nφ \{i} = T ⎣ 2 i j h j∈N \{i} φ 

(∗ ) 0

⎤ ⎦,

(33)

    # = π #  h T and  #i j j∈N \{i} ∈ Rm . And, in (33), the parameter matrix  can be relaxed through  i i φ

where T = I

(6) and (7). First, from (6), the following auxiliary condition is available under (29):



0≤



 π˜ i j − π i j π˜ i j − π i j hT He{−Xi j }h

j∈Nφ \{i}

   # T   #   #  h T π +π X h =  He{−Xi j } Dj∈Nφ \{i}  hT He{−π i j π i j Xi j }h + He  i h i h + i ij ij j∈Nφ \{i} ij  = T

(1 )

(∗ )

Xi

(2 )

Xi(3)

[−6 pt]Xi

j∈Nφ \{i}

 .

(34)

Further, from (7) and (30), it follows that

!

"!



0≤

π˜ i j + π ii

j∈Nφ \{i}



=



"

π˜ iv + π ii hT He{−Yi }h

v∈Nφ \{i}



π˜ i j π˜ iv hT He{−Yi }h + hT He{−π ii π iiYi }h +

j∈Nφ \{i} v∈Nφ \{i}



π˜ i j hT He{−(π ii + π ii )Yi }h.

(35)

j∈Nφ \{i}

That is, from the following separation:





π˜ i j π˜ iv hT He{−Yi }h

j∈Nφ \{i} v∈Nφ \{i}



# h =  i

⎡

He{−Yi }

T ⎢ ⎢ −2Yi ⎢ .. ⎣

(∗ ) ..

.

..

. ···

. −2Yi

··· .. . .. . −2Yi

condition (35) can be represented as follows:



# h 0≤  i

T 



He{−Yi }

D j∈Nφ \{i}

+ hT He{−π ii π iiYi }h + He



= T

Yi(1) Yi(2)

(∗ )

Yi(3)



.

+ He



(∗ ) .. .

⎤ ⎥ ⎥ # ⎥ i  h , ⎦

(∗ ) He{−Yi } − 2Yi



L j,v∈Nφ \{i}

  # h  i  

  #  h T − (π + π )Y h  i ii i ii

j∈Nφ \{i}

(36)

As a result, according to the S-procedure, combining (33) with (34) and (36) becomes 0 ≤ T Mi , which is ensured by (27). Eventually, (27) and (28) can be used as the relaxed conditions of (20) and (21) in Lemma 3. 

10

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

Fig. 2. Effect of the summation inequality (7).

Remark 3. As shown in Fig. 2, taking advantage of the summation inequality (7) allows to form a smaller region B than A that only denotes the boundary condition (6), i.e., B ≤ A. In other words, when relaxing the parameter matrix  in Theorem 1, the use of (7) opens a possibility to reduce the redundant region of π˜ i j , which plays an invaluable role in achieving less conservative stochastic stabilization conditions of S-MJSs with uncertain probability intensities. Since Theorem 1 tends to increase the number of scalar variables, this paper proposes the following corollary as an alternative to manage the trade-off between two conflicting requirements: performance quality and computational burden. Corollary 1. For a prescribed μ > 0, suppose that there exit matrices 0 < P¯1i = P¯1Ti ∈ Rnx ×nx , 0 < P2i = P2Ti ∈ Rnx ×nx , Ri j = RTi j ∈



Rnx ×nx , L¯ i ∈ Rnx ×ny , F¯i ∈ Rnu ×nx , Q1i ∈ Rnx ×nw , Q2i ∈ Rnw ×nw , Xi j = XiTj = the following conditions hold: for all i ∈ Nφ ,





0>

¯ i 

i j h





j∈Nφ \{i}

X12i j

(∗ )

X22i j



∈ R2nx ×2nx ; and a scalar γ > 0 such that



(∗ ) 

− Xi j

X11i j

,

D j∈Nφ \{i}

(37)



Ri j 0≤ (∗ )

P¯1i , ∀ j ∈ Nφ P¯1 j

¯ i = i + h

T

where





\ {i},

(38)

πi j (Ri j − P¯1i ) +

w2i j 4

w2i j 4

X11i j

πi j (P2 j − P2i ) +

(∗ )

j∈Nφ \{i}

 X12i j w2i j 4

h, X22i j

   i , ij , and h are defined in Theorem 1; πi j = 12 π i j + π i j , and wi j = 12 π i j − π i j . Then closed-loop system (9) is stochastically stable with γ -disturbance attenuation, and the observer-based control gains are designed as follows: Li = P2i −1 L¯ i and Fi = F¯i P¯1−1 . i

Proof. Using (31) and (32), condition (20) is represented as follows:

!

0 > i + h

"



T

π˜ i j i j h,

(39)

j∈Nφ \{i}

Further, letting π˜ i j = πi j + πi j , where πi j =

!

"



0 > i + hT

!

πi j i j h + hT

j∈Nφ \{i}

1 2 (π i j



+ π i j ) and |π ij | ≤ wij with wi j =

"

πi j i j h.

j∈Nφ \{i}

From Lemma 1, since it holds that

πi j i j = He

1 2

 w2 πi j i j ≤ i j Xi j + i j Xi−1 i j , j 4

the following condition implies (40):

!

0>i + hT

 j∈Nφ \{i}

$ %"  w2i j Xi j h+ hT i j Xi−1 i j h, πi j i j + j 4

j∈Nφ \{i}

1 2 (π i j

− π i j ), (39) becomes

(40)

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

11

Table 1 (Example 1) Comparison of H∞ performance levels for various μ and their computation complexity.

μ

1

2

3

4

5

6

NoV()

Theorem 1 Corollary 1

1.2841 1.3189

0.5406 0.5570

0.4346 0.4555

0.4134 0.4322

0.4141 0.4318

0.4193 0.4366

202 118

() NoV: number of scalar variables. Table 2 (Example 2) Comparison of H∞ performance levels for various μ and their computation complexity.

μ

3

4

5

10

15

20

NoV()

Theorem 1 Corollary 1

0.9564 0.9732

0.8522 0.8670

0.8142 0.8278

0.7737 0.7863

0.7675 0.7803

0.7659 0.7787

760 400

() NoV: number of scalar variables.

which is rearranged, by the Schur complement, as follows:

  0>

¯ i 

i j h



j∈Nφ \{i}

¯ i = i + h T ( where 





(∗ ) 

− Xi j

,

D j∈Nφ \{i}

j∈Nφ \{i} (πi j i j

w2i j

+

4

(41)

Xi j ))h.





Remark 4. For given nφ , nx , nu , ny , and nw , Theorem 1 requires the following number of scalar variables: NoV = nφ (nφ − 1 )(4.5n2x + 0.5nx ) + 5n2x + nx + nx ny + nx nu + nx nw + n2w + 1. Meanwhile, for Corollary 1, the number of scalar variables de-

creases to NoV = (NoV of Theorem 1) −(2n2x − nx )nφ (nφ − 1 ) − 4n2x nφ . For more details, Tables 1 and 2 show the number of scalar variables for two methods. Remark 5. If other relaxation methods are developed in the future, Lemma 3 can be directly utilized in deriving the corresponding LMI-based stabilization conditions. Remark 6. As in [37–39], discussing the control synthesis problem for the discrete-time case is also a major issue in this area. However, it should be noted that, different from continuous-time Markov process, discrete-time Markov process is generally characterized by transition probabilities (TPs). Thus, when extending our approach to the case of discrete-time semi-MJSs, extra effort is required to consider the impacts arising from the presence of unknown TPs. 4. Illustrative examples Example 1: Let us consider the following S-MJS with three modes (Nφ = {1, 2, 3} ), used in [40]:



A1 =

 B1 =



−1.1 0.3



0.2 0.5 , A2 = 0.6 0.2





















0.4 −0.4 , A3 = −0.7 0.6

0.3 , 0.5



1.3 0.6 0.9 , B2 = , B3 = , 0.7 0.4 1.5







0.3 0.5 0.5 E1 = , E2 = , E3 = , 0.6 0.6 0.4





C1 = 1.3



0.6 , C2 = 1.1

D1 = 0.2, D2 = 0.5, D3 = −0.3,





G1 = 0.7



0.8 , G2 = 0.0











1.6 , C3 = 1.6

1 ,

1.0 , G3 = 0.8



0.5 ,

H1 = 0.4, H2 = 0.3, H3 = −0.3,

(42)

where the sojourn-time is supposed to be characterized by the Weibull distribution with the scale parameter α ij > 0 and the shape parameter β ij > 0 as follows: (α12 , β12 ) = (0.5, 2.0 ), (α13 , β13 ) = (1.0, 2.0 ), (α21 , β21 ) = (0.5, 1.5 ), (α23 , β23 ) = (1.5, 2.0 ), (α31 , β31 ) = (0.5, 1.5 ), and (α32 , β32 ) = (1.0, 2.0 ); and q˜i j ∈ [qi j , qi j ] are established as follows:

  qi j

i, j∈Nφ



=

× 0.3 0.7

0.2 × 0.2



0.6 0.5 , ×



  qi j

i, j∈Nφ

=

× 0.5 0.8

0.4 × 0.3



0.8 0.7 , ×

(43)

12

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

Fig. 3. (a)–(c) Cumulative distribution function Fij (h) and the corresponding F i (h ) and Fi (h), for all i; and (d)–(f) probability distribution function fij (h).

where × indicates an empty element with a null value. For more details, Fig. 3 shows the cumulative distribution function Fij (h) and probability distribution function fij (h), for i, j ∈ Nφ ࢨ{i}, where Fi (h) and F i (h ) denote the lower and upper bounds of Fi (h), obtained from (3) and (43). Accordingly, by (4), the lower and upper bounds of ηij are given as follows:

   × ηi j i, j∈Nφ = 1.3194

1.8761

   × ηi j i, j∈Nφ = 1.4799

2.0450



1.6738 × 4.3633

2.1529 1.6475 , ×

1.8587 × 5.6827

2.7595 2.2519 . ×

(44)



(45)

#i j as Eventually, the areas of qij ηij (h)fij (h) and qi j ηi j (h ) fi j (h ), depicted in Fig. 4, lead to the lower and upper bounds of π follows:





πij

πij



 i, j∈Nφ

−2.5794 0.3958 1.3132

=



 i, j∈Nφ

=

−1.9613 0.7399 1.6360



0.3348 −2.0203 0.8727

1.2917 0.8238 , −3.1363

0.7435 −1.4835 1.7048

2.2076 1.5763 . −2.3735

(46)



(47)

Based on such a set-up, Table 1 shows the minimum H∞ performance levels for several values of μ, obtained by Theorem 1 and Corollary 1. From Table 1, it can be found that Theorem 1 (that involves the summation inequality (7)) achieves better performance than Corollary 1 (that uses only the boundary condition (6)), but Corollary 1 is also an alternative well established from the viewpoint of decreasing the number of scalar variables. Meanwhile, for μ = 5.0, Theorem 1 offers the following observer-based control gains:

















F1 = −1.5491

−2.2629 , F2 = −2.0390

−2.7245 ,

F3 = −3.3225

−4.0240 , LT1 = −2.4654

−4.0649 ,

LT2 = −2.9861

−4.3316 , LT3 = −11.8580









−10.5980 .

(48)

Further, based on (48), Fig. 5(a) and (b) shows the state response of the closed-loop system with x0 = [0.5 1.0]T and x˜0 = [0.0 0.0]T , Fig. 5(c) shows the applied control input, and Fig. 5(d) plots the generated mode transition, where w(t ) = 10e−3t sin(0.015t ), for t ≥ 0. That is, from Fig. 5, it can be seen that (i) the estimated state x˜(t ) approaches the real state

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

13

Fig. 4. Lower and upper bounds of transition rates, that is, π i j (h ) and π ij (h).

Fig. 5. (a) state response of x1 (t) and x˜1 (t ), (b) state response of x2 (t) and x˜2 (t ), (c) control input u(t), and (d) mode evolution of φ (t).

x(t) closely as time increases, and (ii) all the states converge to zero, which illustrates the validity of (48). As a result, the performance improvement (shown in Table 1) and the state convergence (presented in Fig. 5) reveal the effectiveness of our approach for H∞ observer-based control synthesis of S-MJSs with uncertain probability intensities and mode-transitiondependent distribution functions of the sojourn time. Example 2: Let us consider the following aircraft dynamics described in terms of S-MJS model with three modes (Nφ = {1, 2, 3} ), (refer to [27]):

⎡

−0.0366 ⎢ 0.0482 Ai = ⎣ 0.1002 0

0.0271 −1.01 a1i 0

0.0188 0.0024 −0.707 1

⎤

−0.4555  −4.0208⎥ ⎦, Ei = 1 a2i 0

0

1

T

0 ,

14

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

 Bi =

0.4422 0.1761

 



0 1 , Gi = 1 0

Di =

0 1

T

−5.5200 4.4900

bi −7.5922

0 0



0 0

 , Ci =





0 0 , Hi = 0 1

0 0

0 0

1 0



0 , 1

1 0

(49)

with the parameters corresponding to airspeeds of 135, 60, and 170 knots:

⎡

a11

a12

⎣a21 b1

a13

⎤

⎡

⎤

a22

0.3681 a2 ⎦ = ⎣1.4200

0.0064 0.1198

0.5047 2.5460⎦,

b2

b3

3.5446

0.9775

5.1120

3

where x1 (t) is the horizontal velocity (knots), x2 (t) is the vertical velocity (knots), x3 (t) is the pitch rate (degrees per second), and x4 (t) is the pitch angle (degrees); and the cumulative distribution functions of the sojourn-time are supposed to be identical with those of Example 1; and q˜i j ∈ [qi j , qi j ] are established for simulation as follows:



  qi j

i, j∈Nφ

=

× 0.7 0.7



0.8 × 0.2

0.1 0.1 , ×



  qi j

i, j∈Nφ

=

× 0.9 0.8

0.9 × 0.3



0.2 0.3 , ×

(50)

where × indicates an empty element with a null value. In this example, the uncertain probability intensities (50) are modified from (43) such that the gap between the lower and upper bounds of transition rates can be increased and be worse, compared to Example 1. In other words, by (4), the lower and upper bounds of ηij are given as follows:

   × ηi j i, j∈Nφ = 1.7151

1.8761

   × ηi j i, j∈Nφ = 2.1395

2.0450



2.5180 × 4.3633

6.9491 3.5933 , ×

2.8501 × 5.6827

12.2131 9.5016 , ×

(51)



(52)

and the transition rate boundary matrices are given as follows:

   −4.7227 π i j i, j∈Nφ = 1.2005

0.6949 0.3593 , −3.1363

2.5651 −1.9029 1.7048

2.4426 2.8505 . −2.3735

1.3132

   −2.9611 π i j i, j∈Nφ = 1.9255

1.6360



2.0144 −4.3481 0.8727

(53)



(54)

Table 2 shows the minimum H∞ performance levels for several values of μ, obtained by Theorem 1 and Corollary 1. From Table 2, it can be found that Theorem 1 (in which the summation inequality (7) is imposed) achieves better performance than Corollary 1 (derived irrespective of (7)), as mentioned in Example 1. Meanwhile, for μ = 20, Theorem 1 offers the following observer-based control gains:



F1 =

 F2 =

 F3 =

−0.3208 1.0633

0.6509 0.7421

2.1726 , 1.2919

−3.2677 −1.7229

−0.0229 1.1515

0.9090 0.6574

2.7023 , 0.8468

−2.3056 −2.3325

−0.4361 0.8967

0.5515 0.7472

1.9516 , 1.5039

 LT1 =

 LT2 =



LT3



−2.6369 −2.1966







−47.6600 −1.0 0 0 0

−49.7362 −2 × 10−9

−13.9785 −1.0 0 0 0

13.9513 , 5 × 10−10

−78.2777 −1.0 0 0 0

−89.4056 2 × 10−11

−21.6523 −1.0 0 0 0

26.1801 , −2 × 10−11

−6.4677 −7 × 10−11

−6.7667 −1.0 0 0 0

−13.0971 = −1.0 0 0 0





1.2090 . 2 × 10−11

(55)

K.H. Nguyen and S.H. Kim / Applied Mathematics and Computation 372 (2020) 124968

15

Fig. 6. (a) state response of x1 (t) and x˜1 (t ), (b) state response of x2 (t) and x˜2 (t ), (c) state response of x3 (t) and x˜3 (t ), (d) state response of x4 (t) and x˜4 (t ), (e) control input u(t), and (f) mode evolution of φ (t).

Further, based on (55), Fig. 6(a)–(d) show the state response of the closed-loop system with x0 = [3.5 2.3 − 2.0 − 1.0]T and x˜0 = [0.0 0.0 0.0 0.0], Fig. 6(e) shows the applied control input to stabilize the closed-loop system, and Fig. 6(f) plots the generated mode transition, where the disturbance input are taken as w(t ) = 1/(1 + t 2 ), for t ≥ 0. That is, from Fig. 6, it can be seen that (i) the estimated state x˜(t ) approaches the real state x(t) closely as time increases and (ii) all the states converge to zero, which illustrates the validity of (55). As a result, from the simulation results of Table 2 and Fig. 6, we can demonstrate the applicability of the proposed method to the aircraft system, especially in the framework of H∞ observer-based control synthesis of S-MJSs with uncertain probability intensities and mode-transition-dependent distribution functions of the sojourn time. 5. Concluding remarks In this paper, we have proposed a method capable of designing an H∞ observer-based control for a class of continuoustime S-MJSs. To sum up, the main features of our method can be classified into three categories: (i) the impact of uncertain probability intensities has been deeply taken into account, (ii) more accurate bounds of TRs have been derived by using the mode transition-dependent distribution of the sojourn time, and (iii) based on a relaxation process, a set of observerbased stabilization conditions has been obtained in terms of LMIs. Lastly, our future work will be directed to developing an effective method of designing a sampled-data output-feedback control for continuous-time semi-MJSs while considering an asynchronous control mode. Acknowledgments This work was supported by the 2019 Research Fund of University of Ulsan (No. 2019-0354). References [1] [2] [3] [4] [5]

E.-K. Boukas, Stochastic Switching Systems: Analysis and Design, Springer, London, 2007. P. Shi, F. Li, A survey on Markovian jump systems: modeling and design, Int. J. Control, Auto. Syst. 13 (1) (2015) 1–16. Y. Kang, Y.-B. Zhao, P. Zhao, Stability Analysis of Markovian Jump Systems, Springer, London, 2017. F. Tao, Q. Zhao, Design of stochastic fault tolerant control for H2 performance, Int. J. Robust Nonlinear Control 17 (1) (2007) 1–24. J. Wang, K. Liang, X. Huang, Z. Wang, H. Shen, Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback, Appl. Math. Comput. 328 (2018) 247–262. [6] D. Zeng, Z. Pu, R. Zhang, S. Zhong, Y. Liu, G.-C. Wu, Stochastic reliable synchronization for coupled Markovian reaction-diffusion neural networks with actuator failures and generalized switching policies, Appl. Math. Comput. 357 (2019) 88–106. [7] K. Loparo, F. Abdel-Malek, A probabilistic approach to dynamic power system security, IEEE Trans. Circuits Syst. 37 (6) (1990) 787–798.

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