Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time

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Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time Yanhui Li∗, Peng Bo, Ji Qi College of Electrical and Information Engineering, Northeast Petroleum University, Daqing, Heilongjiang 163318, PR China Received 28 January 2019; received in revised form 28 July 2019; accepted 10 September 2019 Available online xxx

Abstract This paper investigates the problem of robust H∞ fixed-order filtering for a class of linear parametervarying (LPV) switched delay systems under asynchronous switching that the system parameter matrices and the time delays are dependent on the real-time measured parameters. The so-called asynchronous switching means that there are time delays between the switching of filters and the switching of system modes. By constructing the parameter-dependent and mode-dependent Lyapunov-Krasovskii functional which is allowed to increase during the running time of active subsystem with the mismatched filter, and using the mode-dependent average dwell time (MDADT) switching method, the sufficient conditions for exponential stability and satisfying a novel weighted H∞ criterion are derived. As there exist couplings between Lyapunov-Krasovskii functional matrices and system parameter matrices, we utilize slack matrices to decouple them. Based on the above results, a suitable weighted H∞ fixed-order filter can be obtained in the form of the parameter linear matrix inequalities (PLMIs). By virtue of approximate basis function and gridding technique, the design of weighted H∞ fixed-order filter can be transformed into the solution of the finite dimensional LMIs. Finally, a numerical example is presented to verify both the effectiveness and the low conservatism of the parameter-dependent and mode-dependent fixed-order filtering method proposed in this paper. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author. E-mail address: [email protected] (Y. Li).

https://doi.org/10.1016/j.jfranklin.2019.09.016 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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1. Introduction Since the general system models of continuous systems or discrete systems are not precise enough to describe complex engineering and social systems, this kinds of models are also difficult to achieve satisfactory effects in terms of filtering or control. Switched systems, however, consist of a set of subsystems (continuous or discrete) and a switching signal that schedule the switching between them, which can provide a more flexible framework for modeling complex systems [1]. The research on switched systems has already attracted considerable attention of scholars, and they find that switched systems have a wide-ranging applied background in communication systems, flight and air traffic control systems and power electronics systems [2–4]. Due to the switched systems have been applied in various aspects, the basic issues of stability analysis are widely worked out [5,6]. The common Lyapunov method can not achieve satisfactory stability under arbitrary switching, multiple-Lyapunov method, that is, mode-dependent Lyapunov method under constrained switching signals shows greater flexibility [7]. The most common constrained switching method include dwell time (DT), average dwell time (ADT) and MDADT. Among the above three constrained switching methods, only the MDADT switching is dependent on the system modes for avoiding stay longer on a specific mode which has quite poor performance, and it also shows that the obtained results by the MDADT switching method can further reduce conservatism [8,9]. The state estimation problem has been studied widely for all kinds of dynamic systems. Such as both the Kalman filtering and the robust filtering are introduced to cope with the issue of state estimation [10]. The well known Kalman filtering theory, however, requires that the systems are described by an accurate mathematical model with known dynamic characteristics, and the noise inputs belong to strict Gaussian process, these restraints are difficult to satisfy the demands of actual systems. Therefore, the H∞ filtering method has been introduced to deal with the problem, and many significant results about switched systems have been reported [6,11,12]. For examples, robust H∞ filtering for uncertain discrete-time switched singular systems has been investigated in [13], and robust finite-time boundedness of H∞ filtering for switched systems has also been addressed in [14]. Obviously, the aforementioned works mainly focus on the linear switched systems which have been studied in a large number of literatures [8,15]. The LPV systems are a special form of time-varying systems, which can be employed to describe the time-varying characteristics of many systems [16]. Recently, the exploration on the LPV switched systems has also received an amount of attention [17,18]. The robust LPV current control problem for switched reluctance motor systems is studied in [19]. In [20], it aims to investigate the problem of H∞ output tracking control for a class of switched LPV systems, which have been applied on an aero-engine model. It is also worth noting that the time delays exist in many practical systems, which are always the sources of poor performance, instability and oscillation. Obviously, it is also important to study effect of time delays on switched systems. One of the most essential time delays is the state delay, which has gained a lot of research attention on switched systems [21,22]. Another important time delay is the switching signal delay, which is caused by filter that takes time to identify the active subsystem, thus there may be a time lag between switching of the filter and the switching of corresponding subsystem, and results in asynchronous switching and deteriorates performance of switched systems [23]. Most research on switched delay systems only consider either state delay or switching signal delay. Therefore, it is more meaningful to study the switched systems with two kinds of time delays than ones with only a class of delays [24]. Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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On the other hand, many results have been obtained for the LPV switched delay systems, but the asynchronous filtering problems based on MDADT switching are seldom reported. As pointed out in [16,25,26], filters are designed for the LPV switched systems with the ADT switching, neither the MDADT switching nor the reduced-order filtering problem is considered. It can be seen that the design of asynchronous H∞ fixed-order filter for the LPV switched delay systems with MDADT has not been fully investigated. Therefore, the exploration on the issue is also challenging and profound significance. This paper investigates the asynchronous weighted H∞ fixed-order filtering problem for the LPV switched delay systems with MDADT, the sufficient conditions of exponentially stable with H∞ performance have been derived. The main contributions of this paper are as follows. • We consider that the asynchronous switching between the filter and the switched systems is inevitable in actual systems, and the parameter-dependent state delay is also very common in many kinds of engineering systems. They are all the sources of instability, undesirable performance and even make the switched systems uncontrollable. Therefore, not only the asynchronous behaviours, but also the synchronous behaviours are considered for the filtering problem of the LPV switched systems with parameter-dependent state delay. Under the unified framework, the asynchronous H∞ fixed-order filter design method of LPV switched systems is given, which not only ensures that the filter error system is exponential stability but also meets the weighted H∞ noise attenuation level. • Since the MDADT switching method focuses on the performance of each subsystem instead of all subsystems simultaneously, that is, there is not mutual affection or constraint among subsystems. Based on the above consideration, in order to further relax the conservativeness resulted from the nonlinearity and the common Lyapunov-Krasovskii functional, we utilize the MDADT switching method, and construct an appropriate parameter-dependent and mode-dependent Lyapunov-Krasovskii functional, which is allowed to increase with bounded rate during the asynchronous interval. We know that the results have less conservatism than the assumption of the asynchronous LPV switched delay systems are stable all the time, then the weighted H∞ stability criterion with less conservatism is derived. • Given that the LPV systems are more useful to deal with smooth nonlinear problems than the linear time-invariant (LTI) ones. Therefore, motivated by the LPV switched filtering method, in order to highlight the advantages of the LPV fixed-order filter, thus the corresponding results of LTI fixed-order filter have been developed. By comparison, the results of the LPV fixed-order filter have less conservatism than the results of LTI one. Therefore, it is clear that the proposed filtering method is very important in both theoretical and practical aspects. The rest of this paper is organized as follows. In Section 2, problem formulation, a lemma and three definitions are given. The weighted H∞ performance analysis for the LPV switched delay systems is provided in Section 3. In Section 4, the asynchronous weighted H∞ fixedorder filter design for the LPV switched delay systems is presented. Two numerical examples are provided to show the effectiveness of the proposed method in Section 5. Finally, the conclusion is summarized in Section 6. Notation 1. In this paper, the used notations are fairly standard. n is n-dimensional Euclidean space, and subscript  T is the matrix transposition. Respectively, I represents the identity matrix and 0 represents the zero matrix with appropriate dimensions. L2 [t0 , ∞) is the space Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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of square-integrable  vector functions over [t0 , ∞), and for ω(t) ∈ L2 [t0 , ∞), its norm is given ∞ T by ω(t )2 = t0 ω (t )ω(t )dt , ∀t ≥ t0 ; diag{. . .} stands for a block-diagonal matrix. The ‘∗’ in a matrix stands for the transposed element in the symmetric position. 2. Problem formulation Consider the following LPV switched systems with time-varying delays: ⎧ x˙(t ) = Aσ (t ) (ρ(t ))x(t ) + Adσ (t ) (ρ(t ))x(t − d (ρ(t ))) + Bσ (t ) (ρ(t ))ω(t ) ⎪ ⎪ ⎨y(t ) = C (ρ(t ))x(t ) + C σ (t ) dσ (t ) (ρ(t ))x(t − d (ρ(t ))) + Dσ (t ) (ρ(t ))ω(t ) z(t ) = E (ρ(t )) x(t ) ⎪ σ (t ) ⎪ ⎩ x(t ) = ϕ(t ), ∀t ∈ [−d˜, t0 ],

(1)

where x(t) ∈ n denotes the system state vector; ω(t) ∈ q denotes the noise signal vector which is assumed to belong to L2 [t0 , ∞); y(t) ∈ m is the measured output vector; z(t) ∈ p is the signal to be estimated; ϕ(t) is a given initial vector function on the segment [−d˜, t0 ]. σ (t): [t0 , ∞) is the constant switching signal function, which takes the value in the finite set  = {1, 2, . . . , N }, where t0 = 0 is initial instant and N > 1 is the number of subsystems. When σ (t ) = i ∈ , the system parameter matrices Aσ (t) (·), Adσ (t) (·), Bσ (t) (·), Cσ (t) (·), Cdσ (t) (·), Dσ (t) (·) and Eσ (t) (·) associated with mode i can be denoted by Ai (·), Adi (·), Bi (·), Ci (·), Cdi (·), Di (·) and Ei (·), respectively. The system parameter matrices and the parameter-dependent delays d(·) are assumed to be dependent on time-varying parameter vector ρ(t), and the delays d(·) satisfy 0 < d (·) ≤ d˜ < ∞, ∀t ≥ 0, for all i ∈ . The parameter vector ρ(t ) = [ρ1 (t ), ρ2 (t ), . . . , ρs (t )] satisfies ρl (t ) ∈ [ρ l , ρ¯l ],and the parameter variation rate τl (t ) = ρ˙l (t ) ∈ [τ l , τ¯l ], ∀l ∈ {1, . . . , s} to be measurable in real time. For convenience, the time-varying parameters ρ(t), ρ l (t) and τ l (t) denoted by ρ, ρ l and τ l , respectively. The switching signal is time-dependent, that is σ (t ) : {(t0 , σ (t0 )), (t1 , σ (t1 )), . . . , (tk , σ (tk ))}, k ∈ 1, 2, . . . , where tk denotes the kth switching instant. We consider a v-order filter (when v = n for a full-order filter, and when 1 ≤ v < n, for reduced-order filter) described by the following form:  x˙ f (t ) = Aσ f (t ) (ρ)x f (t ) + Bσ f (t ) (ρ)y(t ) (2) z f (t ) = Cσ f (t ) (ρ)x f (t ), where x f (t ) ∈ v is the filter state vector, zf (t) ∈ p is the filter output vector, and Aσ f (t ) (·), Bσ f (t ) (·), Cσ f (t ) (·) are the parameter matrices to be determined later. σ f (t) denotes the practical switching signal of the filter, during asynchronous switching, for convenience, it can be written as σ f (t ) = {(t0 , σ f (t0 )), (t1 + 1 , σ f (t1 + 1 )), . . . , (tk + k , σ f (tk + k ))}. Wherein, σ f (t0 ) = σ (t0 ), σ f (tk + k ) = σ (tk ), k < inf k≥1 (tk+1 − tk ), k > 0(k < 0) represent the period that the switching instants of the filter lag behind (or exceed) those of systems. Denote filtering error e(t ) = z(t ) − z f (t ), augmented state vector x˜(t ) = [x T (t ) x Tf (t )]T and K = [I 0]. When t ∈ [t0 , t1 ) ∪ [tk + k , tk+1 ), k = 1, 2, . . . , that is to say, during the matched period, for each value σ (t ) = i ∈ , σ f (t ) = i ∈ , then we can obtain the following filtering error system: ⎧ ˙˜ ) = A˜ i (ρ)x˜(t ) + A˜ di (ρ)K x˜(t − d (ρ)) + B˜ i (ρ)ω(t ) ⎨x(t 1  : e(t ) = C˜i (ρ)x˜(t ) (3) ⎩ x˜(t ) = [ϕ T (t ) 0]T , ∀t ∈ [−d˜, t0 ] Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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And when t ∈ [tk , tk + k ), k = 1, 2, . . . , that is to say, during the mismatched period, thus, σ (t ) = i ∈ , σ f (t ) = j ∈ , for any 0 < j < i ∈ , we can obtain the following filtering error system: ⎧ ˙˜ ) = A˜ i j (ρ)x˜(t ) + A˜ di j (ρ)K x˜(t − d (ρ)) + B˜ i j (ρ)ω(t ) ⎨x(t 2 : e(t ) = C˜i j (ρ)x˜(t ) (4) ⎩ T T ˜ x˜(t ) = [ϕ (t ) 0] , ∀t ∈ [−d , t0 ] Therein, denotes the whole filtering error system that contains 1 and 2 , and



0 Ai (ρ) Bi (ρ) , B˜ i (ρ) = , A˜ i (ρ) = B fi (ρ)Ci (ρ) A fi (ρ) B fi (ρ)Di (ρ)

 Adi (ρ) , C˜i (ρ) = Ei (ρ) −C fi (ρ) A˜ di (ρ) = B fi (ρ)Cdi (ρ)



0 Bi (ρ) ˜ , Bi j (ρ) = , A f j (ρ) B f j (ρ)Di (ρ)

 Adi (ρ) , C˜i j (ρ) = Ei (ρ) −C f j (ρ) A˜ di j (ρ) = B f j (ρ)Cdi (ρ) A˜ i j (ρ) =



Ai (ρ) B f j (ρ)Ci (ρ)

In this paper, we aim to design an asynchronous switching fixed-order filter for systems (1), let the filtering error system  be exponentially stable with weighted H∞ performance level γ . Furthermore, we give the following lemma and definitions that will play key roles in our derivation. Lemma 1 [27]. If there exist functions ϰ(t) and v(t ) satisfying (t ˙ ) ≤ −λ(t ) + kv(t ), so the following inequality:  t (t ) ≤ e−λ(t−t0 ) (t0 ) + k t0 e−λ(t−τ ) v(τ ) dτ holds, where λ > 0, k > 0, t ≥ t0 . Definition 1 [28]. Under the switching signal σ (t), if there exist constants ζ > 0, ϱ > 0, and the trajectory of the filtering error system  satisfies x˜(t ) ≤ ζ x˜(t0 )e−(t−t0 ) , ∀t ≥ t0 , where x(t0 ) = sup−d˜≤θ≤t0 ϕ(θ ), thus filtering error system  is said to be exponentially stable. Definition 2 [8]. For a switching signal σ (t) and any T ≥ t ≥ 0, let Nσ i (T , t ) be the switching number of the ith subsystem activated over the interval [t, T ], and Ti (T , t ) denotes the total running time of the ith subsystem over the interval [T , t], ∀i ∈ . We say that the σ (t) has a mode-dependent average dwell time τ ai , if there exist positive numbers N0i (we call N0i as the mode-dependent chatter bounds here. As commonly used in the literature, for convenience, we choose N0 = 0 in this paper), and τ ai such that Nσ i (T , t ) ≤ N0i + Ti (T , t )/τai , where ∀T ≥ t ≥ 0 holds. Definition 3. For scalars ησ (t) > 0 and γ > 0, the filtering error system  has a prescribed exponential weighted H∞ performance γ under the switching signal σ (t), that the following conditions are satisfied: (1) The filtering error system  is exponentially stable when the ω(t ) = 0; (2) Under zero initial condition ϕ(t ) = 0, ∀t ∈ [−d˜, t0 ], the following inequality holds for all nonzero ω(t) ∈ L2 [t0 , ∞): Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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t

− e ησ (t ) (t − t0 ) eT (s)e(s)ds ≤ γ 2

t0

t

ωT (s)ω(s)ds.

t0

3. Weighted H∞ performance analysis of LPV switched delay systems Before proceeding further, we need to make some notes. For the asynchronous filtering error system , we are interested in finding a LPV filter that the specific form is presented in Eq. (2). Although instability or other worse performance may occur when the filter and LPV switched delay systems (1) run during the mismatched period, the filtering error system  to be exponentially stable with the anticipated weighted H∞ performance constraint by using the asynchronous filter. The following Theorem 1 gives a sufficient condition to achieve this aim. Theorem 1. Consider the filtering error system . Let 0 < α i < 1, β i ≥ 0, μ1 > 1, μ2 > 1, γ > 0 be given scalars. If there exist continuously differentiable matrices 0 < Pi (ρ) ∈ (n+v)×(n+v) , 0 < Pi j (ρ) ∈ (n+v)×(n+v) , and matrices 0 < Qi ∈ n × n , 0 < Qij ∈ n × n , 0 < Ri ∈ n × n , 0 < Rij ∈ n × n for any 0 < j < i ∈ , make the following inequalities hold: ⎡ ⎤ 1 Pi (ρ)A˜ di (ρ) Pi (ρ)B˜ i (ρ) d˜A˜ Ti (ρ)K T Ri C˜iT (ρ) ⎢∗ 2 0 d˜A˜ Tdi (ρ)K T Ri 0 ⎥ ⎢ ⎥ 2 T T ⎢∗ (5) ∗ −γ I d˜B˜ i (ρ)K Ri 0 ⎥ ⎢ ⎥<0 ⎣∗ ∗ ∗ −d˜Ri 0 ⎦ ∗ ∗ ∗ ∗ −I ⎡

3 ⎢∗ ⎢ ⎢ ⎢∗ ⎢ ⎣∗ ∗

Pi j (ρ)A˜ di j (ρ) 4 ∗ ∗ ∗

Pi (ρ) ≤ μ1 Pi j (ρ),

Pi j (ρ)B˜ i j (ρ) 0 −γ 2 I ∗ ∗ Pi j (ρ) ≤ μ2 Pj (ρ)

d˜A˜ Ti j (ρ)K T Ri j d˜A˜ Tdi j (ρ)K T Ri j d˜B˜ iTj (ρ)K T Ri j −d˜Ri j ∗

⎤ C˜iTj (ρ) 0 ⎥ ⎥ ⎥ 0 ⎥<0 ⎥ 0 ⎦ −I

0< j
Qi ≤ μ1 Qi j , Qi j ≤ μ2 Q j ; Ri ≤ μ1 Ri j , Ri j ≤ μ2 R j ,

0< j
(6)

(7)

(8)

and MDADT holds for τai > τai∗ =

ln μ1 μ2 ηi

T − (t0 , t ) βi + ηi ≥ , 0 < ηi < αi ; + T (t0 , t ) αi − ηi

(9)

(10)

The filtering error system is exponentially stable, where the T − (t0 , t ) denotes the total matched period during [t0 , t] and the T + (t0 , t ) denotes the total mismatched period during [t0 , t], wherein: Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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 s   ∂Pi (ρ) τl + αi Pi (ρ) + K T Qi K ; ∂ρ l l=1   s   ∂d (ρ) ˜ Qi ; 2 = −e−αi d 1 − τl ∂ρl l=1  s   ∂Pi j (ρ) T ˜ ˜ τl − βi Pi j (ρ) + K T Qi j K ; 3 = Ai j (ρ)Pi j (ρ) + Pi j (ρ)Ai j (ρ) + ∂ρ l l=1   s   ∂d (ρ) τl Qi j ; 4 = − 1 − ∂ρl l=1 1 = A˜ Ti (ρ)Pi (ρ) + Pi (ρ)A˜ i (ρ) +

Proof. When t ∈ [t0 , t1 ) ∪ [tk + k , tk+1 ), k = 1, 2, . . . , for the ith subsystem, we choose the parameter-dependent Lyapunov-Krasovskii functional as Vi (x˜t , ρ) = V1i (x˜t , ρ) + V2i (x˜t , ρ) + V3i (x˜t , ρ)

(11)

wherein V1i (x˜t , ρ) = x˜T (t )Pi (ρ)x˜(t )

t V2i (x˜t , ρ) = eαi (s−t ) x˜T (s)K T Qi K x˜(s)ds t−d (ρ) t t

V3i (x˜t , ρ) =

t−d˜ s

T eαi (θ−t ) x˙˜ (θ )K T Ri K x˙˜(θ )d θ d s

Take time derivative of Vi (x˜t , ρ) which partly depends on ρ, and define ψ (t ) = [x˜T (t ) x˜T (t − d (ρ))K T ωT (t )] hence we can obtain the results:  V˙i (x˜t , ρ) ≤ x˜T (t ) A˜ Ti (ρ)Pi (ρ) + Pi (ρ)A˜ i (ρ) +

  s   ∂Pi (ρ) T τl + αi Pi (ρ) + K Qi K x˜(t ) ∂ρl l=1

+ 2x˜T (t )Pi (ρ)A˜ di (ρ)K x˜(t − d (ρ)) + 2x˜T (t )Pi (ρ)B˜ i (ρ)ω(t ) − αiVi (x˜t , ρ)   s   ∂d (ρ) −αi d˜ −e 1− x˜T (t − d (ρ))K T Qi K x˜(t − d (ρ)) τl ∂ρ l l=1  T  T + d˜ψ (t ) A˜ i (ρ) A˜ di (ρ) B˜ i (ρ) K T Ri K A˜ i (ρ) A˜ di (ρ) B˜ i (ρ) ψ (t ) ≤ ψ T (t )1 ψ (t ) − αiVi (x˜t , ρ) + γ 2 ωT (t )ω(t ) wherein ⎡

1 1 = ⎣ ∗ ∗

Pi (ρ)A˜ di (ρ) 2 ∗

⎤ ⎡ T ⎤ ⎡ T ⎤T A˜ i (ρ) A˜ i (ρ) Pi (ρ)B˜ i (ρ) ⎦ + d˜⎣A˜ T (ρ)⎦K T Ri K ⎣A˜ T (ρ)⎦ 0 di di 2 −γ I B˜ iT (ρ) B˜ iT (ρ)

For all ψ(t) = 0 and ω(t ) = 0, according to the inequality (5), it follows that V˙i (x˜t , ρ) ≤ −αiVi (x˜t , ρ)

t ∈ [t0 , t1 ) ∪ [tk + k , tk+1 ), k = 1, 2, . . . ,

(12)

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Let t1 , t2 , . . . , tk denote the switching instants of σ (t) over the interval [t0 , t], from the inequality (12), and yields:  t ∈ [t0 , t1 ) V (t ) ≤ Vi (t0 )e−αi (t−t0 ) V (t ) = i (13) Vi (t ) ≤ Vi (tk + k )e−αi (t−tk −k ) t ∈ [tk + k , tk+1 ), k = 1, 2, . . . , When t ∈ [tk , tk + k ), k = 1, 2, . . . , Lyapunov-Krasovskii functional:

considering

the

following

parameter-dependent

Vi j (x˜t , ρ) = V1i j (x˜t , ρ) + V2i j (x˜t , ρ) + V3i j (x˜t , ρ)

(14)

wherein: V1i j (x˜t , ρ) = x˜T (t )Pi j (ρ)x˜(t )

t V2i j (x˜t , ρ) = e−βi (s−t ) x˜T (s)K T Qi j K x˜(s)ds t−d (ρ) t t

V3i j (x˜t , ρ) =

t−d˜ s

T e−βi (θ−t ) x˙˜ (θ )K T Ri j K x˙˜(θ )d θ d s

Take the time derivative of Vi j (x˜t , ρ), which partly depends on ρ, so obviously, we can get the following results:    s   ∂P (ρ) i j τl −βi Pi j (ρ) + K T Qi j K x˜(t ) V˙i j (x˜t , ρ) ≤ x˜T (t ) A˜ Ti j (ρ)Pi j (ρ) + Pi j (ρ)A˜ i j (ρ) + ∂ρ l l=1 + 2x˜T (t )Pi j (ρ)A˜ di j (ρ)K x˜(t − d (ρ)) + 2x˜T (t )Pi j (ρ)B˜ i j (ρ)ω(t ) + βiVi j (x˜t , ρ)   s   ∂d (ρ) τl − 1− x˜T (t − d (ρ))K T Qi j K x˜(t − d (ρ)) ∂ρ l l=1  T  T + d˜ψ (t ) A˜ i j (ρ) A˜ di j (ρ) B˜ i j (ρ) K T Ri j K A˜ i j (ρ) A˜ di j (ρ) B˜ i j (ρ) ψ (t ) ≤ ψ T (t )2 ψ (t ) + βiVi (x˜t , ρ) + γ 2 ωT (t )ω(t ) wherein ⎡

3 2 = ⎣ ∗ ∗

Pi j (ρ)A˜ di j (ρ) 4 ∗

⎡ ⎤ ⎡ ⎤T ⎤ A˜ Ti j (ρ) A˜ Ti j (ρ) Pi j (ρ)B˜ i j (ρ) ⎥ T ⎢ ⎥ ⎦ + d˜⎢ 0 ⎣A˜ Tdi j (ρ)⎦K Ri j K ⎣A˜ Tdi j (ρ)⎦ −γ 2 I B˜ iTj (ρ) B˜ iTj (ρ)

For all ψ(t) = 0 and ω(t ) = 0, according to the inequality (6), we can conclude V˙i j (x˜t , ρ) ≤ βiVi j (x˜t , ρ),

t ∈ [tk , tk + k ), k = 1, 2, . . .

(15)

From the inequality (13) and the inequality (15). Therefore the Lyapunov-Krasovskii statisfies: ⎧ ⎨Vi (t ) ≤ Vi (t0 )e−αi (t−t0 ) , V (t ) = Vi j (t ) ≤ Vi j (tk )eβi (t−tk ) , ⎩ Vi (t ) ≤ Vi (tk + k )e−αi (t−tk −k ) ,

t ∈ [t0 , t1 ), t ∈ [tk , tk + k ), k = 1, 2, . . . , t ∈ [tk + k , tk+1 ), k = 1, 2, 3, . . . ,

(16)

If μ1 > 1, μ2 > 1, according to the inequalities (7) and (8), it holds that Vi (t) ≤ μ1 Vij (t), Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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Vij (t) ≤ μ2 Vj (t), ∀i, j ∈ θ . Therefore, when t ∈ [tk , tk + k ), k = 1, 2, . . . , we have the relationship Nσ f (t ) (t0 , t ) = Nσ (t ) (t0 , t ) − 1, and Lyapunov functional satisfies: Vσ (tk )σ f (tk−1 +k−1 ) (t ) ≤ Vσ (tk )σ f (tk−1 +k−1 ) (tk )eβσ (t ) (t−tk ) ≤ μ2Vσ (tk−1 ) (tk− )eβσ (t ) (t−tk ) ≤ μ2Vσ (tk−1 ) (tk−1 + k−1 )eβσ (t ) (t−tk )−ασ (t ) (tk −(tk−1 +k−1 )) ≤ μ2 μ1Vσ (tk−1 )σ f (tk−2 +k−2 ) (tk−1 )eβσ (t ) (t−tk +k−1 )−ασ (t ) (tk −tk−1 −k−1 ) N

≤ μ2 σ (t )

(tk−2 ,t )

Nσ f (t ) (tk−2 +k−2 ,t )

Vσ (tk −2) (tk−1 )eβσ (t ) (t−tk +k−1 )−ασ (t ) (tk −tk−1 −k−1 )

μ1

(t ,t )

Nσ

(t ) (t0 ,t )

≤ · · · ≤ μ2 σ (t ) 0 μ1 f Vσ (t0 ) (t0 )eβσ (t ) (t−tk +k−1 + · · · +1 )−ασt [(tk −tk−1 −k−1 )+ · · · +(t2 −t1 −1 )+(t1 −t0 )] Nσ (t0 ) (t0 ,t ) ≤ μ−1 Vσ (t ) (t0 )eβσ (t ) (t−tk +k−1 + · · · +1 )−ασ (t ) [(tk −tk−1 −k−1 )+ · · · +(t2 −t1 −1 )+(t1 −t0 )] (17) 1 (μ2 μ1 ) N

It follows from the inequality (10) that: βσ (t ) T + (t0 , t ) − ασ (t ) T − (t0 , t ) ≤ −ησ (t ) (t − t0 )

(18)

From the inequalities (17) and (18), yields: t−t0

τai V (t ) ≤ μ−1 Vσ (t0 ) (t0 )e−ησ (t ) (t−t0 ) = μ−1 1 (μ2 μ1 ) 1 Vσ (t0 ) (t0 )e

−(ησ (t ) −

lnμ1 μ2 τai

)(t−t0 )

(19)

When t ∈ [t0 , t1 ) ∪ [tk + k , tk+1 ), k = 1, 2, . . . , it is known that Nσ f (t ) (t0 , t ) = Nσ (t ) (t0 , t ), yields: V (t ) = Vσ (tk ) (t ) ≤ μ1Vσ (tk )σ f (tk−1 +k−1 ) (t ) ≤ Vσ (t0 ) (t0 )e

−(ησ (t ) −

lnμ1 μ2 τai

)(t−t0 )

(20)

Let a = mini, j∈,i = j λmin (Pδ(tk ) (ρ), Pδ(tk )δ f (tk−1 +k−1 ) (ρ)), b = maxi, j∈,i = j λmax (Pδ(tk ) (ρ) + ˜2 d˜λmax Qδ(tk ) + d λmax Rδ(tk ) , Pδ(tk )δ f (tk−1 +k−1 ) (ρ) + d˜λmax Qδ(tk )δ f (tk−1 +k−1 ) + 2 d˜2 λ R ), 2 max δ(tk )δ f (tk−1 +k−1 )

 x˜(t ) ≤

we can get:

lnμ μ b − 1 (η − 1 2 )(t−t0 ) x˜(t0 )e 2 σ (t ) τai a

So we can conclude that the filtering error system  is exponentially stable. Denoting (t ) = eT (t )e(t ) − γ 2 ωT (t )ω(t ), according to the above conditions, we can get ⎧ ⎨

⎡ T ⎤⎫ C˜i (ρ)C˜i (ρ) 0 0 ⎬ J1 = (t ) + V˙i (ξt , ρ) + αiVi (ξt , ρ) = ψ T (t ) 1 + ⎣ 0 0 0⎦ ψ (t ) ⎩ ⎭ 0 0 0 ⎧ ⎡ T ⎤⎫ C˜i j (ρ)C˜i j (ρ) 0 0 ⎬ ⎨ J2 = (t ) + V˙i j (ξt , ρ) − βiVi j (ξt , ρ) = ψ T (t ) 2 + ⎣ 0 0 0⎦ ψ (t ) ⎩ ⎭ 0 0 0 From the inequalities (5) and (6) and the Lemma 1, we know J1 < 0, J2 < 0. Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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⎧ t ⎪ Vi (t ) ≤ Vi (t0 )e−αi (t−t0 ) − t0 e−ηi (t−s) (s)ds; ⎪ ⎪ t ⎨ Vi j (t ) ≤ Vi j (tk )eβi (t−tk ) − tk e−ηi (t−s) (s)ds; V (t ) = ⎪Vi (t ) ≤ Vi (tk + k )e−αi (t−(tk +k )) ⎪ t ⎪ ⎩ − tk +k e−ηi (t−s) (s)ds;

t ∈ [t0 , t1 ), t ∈ [tk , tk + k ), k = 1, 2, . . . , t ∈ [tk + k , tk+1 ), k = 1, 2, 3, . . . , (21)

When t ∈ [tk , tk + k ), k = 1, 2, . . . , and considering the relationship Nσ f (t ) (t0 , t ) = Nσ (t ) (t0 , t ) − 1, it follows that

t V (t ) ≤ Vσ (tk )σ f (tk−1 +k−1 ) (tk )eβσ (t ) (t−tk ) − e−ησ (t ) (t−s) (s)ds tk

≤ μ2Vσ (tk−1 ) (tk−1 + k−1 )eβσ (t ) (t−tk )−ασ (t ) (tk −(tk−1 +k−1 ))

t

tk −ησ (t ) (t−s) − μ2 e (s)ds − e−ησ (t ) (t−s) (s)ds tk−1 +k−1

tk

≤ μ2 μ1Vσ (tk−1 )σ f (tk−2 +k−2 ) (tk−1 )eβσ (t ) (t−tk +k−1 )−ασ (t ) (tk −tk−1 −k−1 )

tk−1 +k−1 − μ2 μ1 e−ησ (t ) (t−s) (s)ds

− μ2

≤

tk−1 tk

e−ησ (t ) (t−s) (s)ds −

t

e−ησ (t ) (t−s) (s)ds

tk−1 +k−1 tk Nσ (t ) (tk−2 ,t ) Nσ f (t ) (tk−2 +k−2 ,t ) μ2 μ1 Vσ (tk −2) (tk−1 )eβσ (t ) (t−tk +k−1 )−ασ (t ) (tk −tk−1 −k−1 )

t N (s,t ) Nσ (t ) (s,t ) −ησ (t ) (t−s) − μ2 σ (t ) μ1 f e (s)ds tk−1

≤ ···

(t ,t )

Nσ

(t ) (t0 ,t )

Vσ (t0 ) (t0 )eβσ (t ) (t−tk +k−1 + · · · +1 )−ασt [(tk −tk−1 −k−1 )+ · · · +(t2 −t1 −1 )+(t1 −t0 )] ≤ μ2 σ (t ) 0 μ1 f

t − eNσ (t ) (s,t )lnμ2 +Nσ f (t ) (s,t )lnμ1 −ησ (t ) (t−s) (s)ds (22) N

t0

Under zero initial condition, that is x˜(t0 ) = 0, the inequality (22) implies:

t 0≤− eNσ (t ) (s,t )lnμ2 +Nσ f (t ) (s,t )lnμ1 −ησ (t ) (t−s) (s)ds

(23)

t0

The inequality (23) is equivalent to the follow inequality

t eNσ (t ) (s,t )lnμ2 +Nσ f (t ) (s,t )lnμ1 −ησ (t ) (t−s) eT (s)e(s)ds t0

≤ γ2

t

eNσ (t ) (s,t )lnμ2 +Nσ f (t ) (s,t )lnμ1 −ησ (t ) (t−s) ωT (s)ω(s)ds

(24)

t0

Multiplying both sides of the inequality (24) by e−Nσ (t ) (t0 ,t )lnμ2 −Nσ f (t ) (t0 ,t )lnμ1 yields:

t eNσ (t ) (s,t0 )lnμ2 +Nσ f (t ) (s,t0 )lnμ1 −ησ (t ) (t−s) eT (s)e(s)ds t0

≤γ

2

t

eNσ (t ) (s,t0 )lnμ2 +Nσ f (t ) (s,t0 )lnμ1 −ησ (t ) (t−s) ωT (s)ω(s)ds

(25)

t0

Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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where Nσ (t ) (s, t0 )lnμ2 + Nσ f (t ) (s, t0 )lnμ1 − ησ (t ) (t − s) ≤ −ησ (t ) (t − t0 ) + lnμ1 . So we can get the following inequality:

t

e

−ησ (t ) (t−t0 ) T

e (s)e(s)ds ≤ γ

t

2

t0

e

−ησ (t ) (t−t0 )

ω (s)ω(s)ds ≤ γ T

t

2

t0

ωT (s)ω(s)ds

(26)

t0

When t ∈ [t0 , t1 ) ∪ [tk + k , tk+1 ), k = 1, 2, . . . , we have the relationship Nσ f (t ) (t0 , t ) = Nσ (t ) (t0 , t ), and it follows that Lyapunov functional satisfies: V (t ) = Vσ (tk ) (t ) ≤ μ1Vσ (tk )σ f (tk−1 +k−1 ) (t ) Nσ f (t ) (t0 ,t )

≤ μ1

N

μ2 σ (t )

(t0 ,t )

Vσ (t0 ) (t0 )e−ησ (t ) (t−s) −

t

eNσ f (t ) (s,t )lnμ2 +Nσ f (t ) (s,t )lnμ1 −ησ (t ) (t −s) (s)ds

t0

(27) Under zero initial condition, that is x˜(t0 ) = 0, the above inequality (27) implies:

t

eNσ (t ) (s,t0 )lnμ2 +Nσ f (t ) (s,t0 )lnμ1 −ησ (t ) (t−s) eT (s)e(s)ds

t0

≤γ

2

t

eNσ (t ) (s,t0 )lnμ2 +Nσ f (t ) (s,t0 )lnμ1 −ησ (t ) (t−s) ωT (s)ω(s)ds

(28)

t0

where Nσ (t ) (s, t0 )lnμ2 + Nσ f (t ) (s, t0 )lnμ1 − ησ (t ) (t − s) ≤ −ησ (t ) (t − t0 ). So we can get the following inequality:

t

e−ησ (t ) (t−t0 ) eT (s)e(s)ds ≤ γ 2

t0

t

e−ησ (t ) (t−t0 ) ωT (s)ω(s)ds ≤ γ 2

t0

t

ωT (s)ω(s)ds

(29)

t0

It can be concluded from the inequalities (26) and (29) that:

t

e

−ησ (t ) (t−t0 ) T

e (s)e(s)ds ≤ γ

t0

2

t

ωT (s)ω(s)ds

(30)

t0

The proof is completed.  Remark 1. The Theorem 1 provides a sufficient condition for exponentially stable with a weighted H∞ performance for the filtering error system  under MDADT. By resetting αi = α, βi = β, ∀i ∈ θ , we know that the ADT switching is a special case of MDADT switching. αi βi αβ The CMDADT is the MDADT switching signal with α i , β i . And the CADT is the ADT switching αβ αi βi signal with α, β (in this case αi = α, βi = β), then there are CADT ∈ CMDADT . The γ ADT and the γ MDADT are the H∞ performance of the filtering error system  under two switching methods, and the inequality γ ADT ≤ γ MDADT is known from Lu et al. [18]. Remark 2. In actual operation, the inequalities (9) and (10) are difficult to apply directly. Thus, the max is known to be a positive scalar that represents the maximum period of the filter switching instants lag behind those systems. We can present a simple condition of the mode-dependent average dwell time, that is, the inequality (9) can be reduced to the following condition: Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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τai > τai∗ = max



   lnμ1 μ2 βi + ηi , + 1 max , 0 < ηi < αi ηi αi − ηi

(31)

The proof of the inequality (31) can be seen in [23]. Remark 3. As there exist couplings between the system parameter matrices and the Lyanpunov-Krasovskii functional matrices, which are a parameter nonlinear problem. Next, we can get a more convenient result from the Theorem 2. Theorem 2. Consider the filtering error system . For given scalars 0 < α i < 1, β i ≥ 0, μ1 > 1, μ2 > 1, γ > 0, if there exist continuously differentiable matrices 0 < Pi (ρ) ∈ R(n+v)×(n+v) , 0 < Pi j (ρ) ∈ R(n+v)×(n+v) and matrices 0 < Qi ∈ Rn × n , 0 < Qij ∈ Rn × n ; 0 < Ri ∈ Rn × n , 0 < Rij ∈ Rn × n , Ui , for any 0 < j < i ∈  satisfying the inequalities (7) and (8) and: ⎡ −Ui − UiT ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗

UiT A˜ i (ρ) + Pi (ρ)

UiT A˜ di (ρ)

UiT B˜ i (ρ )

UiT

d˜K T Ri

1

0

s   −αi d˜ τl ∂P∂ρi (ρl ) −e Qi 1 −

0

0

0

0

0

0

∗

∗

−γ I

0

0

∗

∗

∗

−Pi (ρ )

∗

∗

∗

∗

0 −d˜Ri

∗

∗

∗

∗

∗

∗



l=1

2

0

⎤

⎥ C˜iT (ρ )⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ <0 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −I

(32) ⎡ −Ui − UiT ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗

UiT A˜ i j (ρ) + Pi j (ρ)

UiT A˜ di j (ρ)

UiT B˜ i j (ρ )

UiT

d˜K T Ri j

2

0

s   ∂Pi j (ρ ) −Qi j 1 − τl ∂ρ l

0

0

0

0

0

0

∗

∗

−γ 2 I

0

0

∗

∗

∗

−Pi j (ρ )

∗

∗

∗

∗

−d˜Ri j

∗

∗

∗

∗

∗

∗

l=1

0

0

⎤

⎥ C˜iTj (ρ )⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ <0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦ −I

(33) Then the filtering error system  is exponentially stable with a guaranteed weighted H∞ performance attenuation level γ for the switching signal with MDADT satisfying the inequalities (9) and (10). where:  s   ∂Pi (ρ) τl + K T Qi K ; 1 = (αi − 1)Pi (ρ) + ∂ρ l l=1  s   ∂Pi j (ρ) τl + K T Qi j K ; 2 = (−βi − 1)Pi j (ρ) + ∂ρ l l=1 Proof. The inequality (32) is equivalent to: ϒi + EiT UiT Fi + FiT Ui Ei < 0 Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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wherein: ⎡ ⎤ 0 Pi (ρ) 0 0 0 0 0 ⎢∗ 1 0 0 0 C˜iT (ρ)⎥ 0s 

⎢ ⎥ ⎢ ⎥  ∂Pi (ρ) −αi d˜ ⎢∗ τl ∂ρl 0 ∗ −e Qi 1 − 0 0 0 ⎥ ⎢ ⎥ l=1 ⎥ ϒi = ⎢ 2 ⎢∗ ∗ ∗ −γ I 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ ∗ ∗ −Pi (ρ) 0 0 ⎥ ⎢ ⎥ ⎣∗ ∗ ∗ ∗ ∗ −d˜Ri 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ −I   Ei = I 0 0 0 0 0 0 ; Fi = −I A˜ i (ρ) A˜ di (ρ) B˜ i (ρ) I 0 0 ; where the null spaces of matrices Ei and Fi are ℵEi and ℵFi , respectively, that is



0 F˜i ; ℵEi = (n+v)×r ; ℵFi = Ir×r Ir×r  where r = 4n + 2v + q; F˜i = A˜ i (ρ) A˜ di (ρ) B˜ i (ρ) I 0 0 . Applying the Projection Lemma from [16], we conclude ⎡ ⎤ 1 0 0 0 C˜iT (ρ) 0s 

⎢ ⎥  ˜ ⎢∗ τl ∂P∂ρi (ρ) 0 −e−αi d Qi 1 − 0 0 0 ⎥ ⎢ ⎥ l l=1 ⎢ ⎥ 2 ℵTEi ϒi ℵEi = ⎢ ∗ −γ I 0 0 0 ⎥ ⎢∗ ⎥<0 ⎢∗ ⎥ ∗ ∗ −P (ρ) 0 0 i ⎢ ⎥ ⎣∗ ˜ ∗ ∗ ∗ −d Ri 0 ⎦ ∗ ∗ ∗ ∗ ∗ −I (34) ⎡

1 − Pi (ρ) ⎢ ∗ ⎢ ⎢ ∗ ⎢ ℵTFi ϒi ℵFi =⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Pi (ρ)A˜ di (ρ) 2 ∗ ∗ ∗ ∗

Pi (ρ)Bi (ρ) 0 −γ 2 I ∗ ∗ ∗

Pi (ρ) 0 0 −Pi (ρ) ∗ ∗

d˜A˜ Ti (ρ)K T Ri d˜A˜ Tdi (ρ)K T Ri d˜B˜ iT (ρ)K T Ri d˜K T Ri −d˜Ri ∗

⎤ C˜iT (ρ) 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥<0 0 ⎥ ⎥ 0 ⎦ −I (35)

By means of the Schur complement lemma in the inequality (35), we can obtain the inequality (5) in the Theorem 1. Meanwhile, the inequality (34) is part of the inequality (32). Similarly, the inequality (6) is also decoupled by the above method. Thus, the filtering error system is exponentially stable with a guaranteed weighted H∞ noise attenuation level γ .  4. Asynchronous weighted H∞ fixed-order filter design for LPV switched delay systems Next, we will design the weighted H∞ fixed-order filter for the LPV switched delay systems (1) based on the Theorem 2. Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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Theorem 3. Consider the filtering error system  and MDADT switching logic satisfying the inequality (9). For given scalars 0 < α i < 1, β i ≥ 0, μ1 > 1, μ2 > 1, γ > 0, the filtering error system  is exponentially stable with a guaranteed weighted H∞ performance attenuation level γ , if and only if, there exist continuously differentiable matrices 0 < P˜1i (ρ) ∈ Rn×n , 0 < P˜1 j (ρ) ∈ Rn×n ; P˜2i (ρ) ∈ Rn×v , P˜2 j (ρ) ∈ Rn×v , P˜3i (ρ) ∈ Rv×v , P˜3 j (ρ) ∈ Rv×v , 0 < P˜1i j (ρ) ∈ Rn×n , P˜2i j (ρ) ∈ Rn×v , 0 < P˜3i j (ρ) ∈ Rv×v , matrices 0 < Qi ∈ Rn × n , n×n n×n n×n n×n n×n 0 < Qj ∈ R , 0 < Qij ∈ R , 0 < Ri ∈ R , 0 < Rj ∈ R , 0 < Rij ∈ R and suitable dimensional matrices Si , Mi , Wi , A˜ f i (ρ), B˜ f i (ρ), C˜ f i (ρ), A˜ f i j (ρ), B˜ f i j (ρ), C˜ f i j (ρ) for any 0 < j < i ∈  satisfying the inequality (8) and ⎡ −Si − SiT ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗

−H T Mi − H T WiT −Wi − WiT ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

γ13 γ23 γ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗

γ14 γ24 γ34 γ44 ∗ ∗ ∗ ∗ ∗ ∗

γ15 γ25 0 0 γ55 ∗ ∗ ∗ ∗ ∗

γ16 γ26 0 0 0 −γ 2 I ∗ ∗ ∗ ∗

SiT MiT H 0 0 0 0 −P˜1i (ρ ) ∗ ∗ ∗

H T WiT WiT 0 0 0 0 −P˜2i (ρ ) −P˜3i (ρ ) ∗ ∗

d˜Ri 0 0 0 0 0 0 0 −d˜Ri ∗

⎤ 0 ⎥ 0 ⎥ ⎥ T ˜ Ei (ρ ) ⎥ ⎥ −C˜ Tf i (ρ )⎥ ⎥ ⎥ 0 ⎥ ⎥<0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦ −I

(36)

⎡ −Si − SiT ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗

−H T Mi − H T WiT −Wi − WiT ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



γ13  γ23  γ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗



γ14  γ24  γ34  γ44 ∗ ∗ ∗ ∗ ∗ ∗



γ15  γ25 0 0  γ55 ∗ ∗ ∗ ∗ ∗



γ16  γ26 0 0 0 −γ 2 I ∗ ∗ ∗ ∗

SiT MiT H 0 0 0 0 −P˜1i j (ρ ) ∗ ∗ ∗

H T WiT WiT 0 0 0 0 −P˜2i j (ρ ) −P˜3i j (ρ ) ∗ ∗

d˜Ri j 0 0 0 0 0 0 0 −d˜Ri j ∗

⎤ 0 ⎥ 0 ⎥ ⎥ T ˜ Ei (ρ ) ⎥ ⎥ −C˜ Tf i j (ρ )⎥ ⎥ ⎥ ⎥ 0 ⎥< 0 ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎦ 0 −I

(37)

P˜1i (ρ) ∗

P˜2i (ρ) P˜ (ρ) ≤ μ1 1i j ˜ P3i (ρ) ∗ ≤ μ2

P˜1 j (ρ) ∗

P˜2i j (ρ) P˜ (ρ) ; 1i j ˜ P3i j (ρ) ∗

P˜2i j (ρ) P˜3i j (ρ)

P˜2 j (ρ) ; ∀0 < j < i ∈ ; P˜3 j (ρ)



(38)

Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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 where H = Iv×v γ13 γ14 γ15 γ16 γ23 γ24 γ25 γ26 γ33 γ34 γ44 γ55

15

0v×(n−v) and

= SiT Ai (ρ) + H T B˜ f i (ρ)Ci (ρ) + P˜1i (ρ); = H T A˜ f i (ρ) + P˜2i (ρ); = SiT Adi (ρ) + H T B˜ f i (ρ)Cdi (ρ); = SiT Bi (ρ) + H T B˜ f i (ρ)Di (ρ); = MiT H Ai (ρ) + B˜ f i (ρ)Cdi (ρ) + P˜2iT (ρ); = A˜ f i (ρ) + P˜3i (ρ); = MiT H Adi (ρ) + B˜ f i (ρ)Cdi (ρ); = MiT H Bi (ρ) + B˜ f i (ρ)Di (ρ); s   ˜1i (ρ) τl ∂ P∂ρ + Qi ; = (αi − 1)P˜1i (ρ) + l l=1  s  ˜2i (ρ) τl ∂ P∂ρ ; = (αi − 1)P˜2i (ρ) + l l=1  s  ˜3i (ρ) τl ∂ P∂ρ ; = (αi − 1)P˜3i (ρ) + l l=1

s   ˜ τl ∂d∂ρ(ρ) = −e−αi d Qi 1 − l l=1



γ13  γ14  γ15  γ16  γ23  γ24  γ25  γ26 

γ33 

γ34 

γ44 

γ55

= SiT Ai (ρ) + H T B˜ f i j (ρ)Ci (ρ) + P˜1i j (ρ); = H T A˜ f i j (ρ) + P˜2i j (ρ); = SiT Adi (ρ) + H T B˜ f i j (ρ)Cdi (ρ); = SiT Bi (ρ) + H T B˜ f i j (ρ)Di (ρ); = MiT H Ai (ρ) + B˜ f i j (ρ)Ci (ρ) + P˜2i j (ρ); = A˜ f i j (ρ) + P˜3i j (ρ); = MiT H Adi (ρ) + B˜ f i j (ρ)Cdi (ρ); = MiT H Bi (ρ) + B˜ f i j (ρ)Di (ρ); s   ∂ P˜1i j (ρ) τl ∂ρ + Qi j ; = (−βi −1)P˜1i j (ρ)+ l l=1  s  ∂ P˜2i j (ρ) τl ∂ρ ; = (−βi − 1)P˜2i j (ρ) + l l=1  s  ∂ P˜3i j (ρ) τl ∂ρ ; = (−βi − 1)P˜3i j (ρ) + l l=1

s   τl ∂d∂ρ(ρ) = −Qi j 1 − l l=1

Moreover, the admissible filter can be constructed by following equations: Suppose the filter with the synchronous behaviours:

−T



A fi (ρ) B fi (ρ) Wi 0 A˜ fi (ρ) B˜ fi (ρ) = ∗ C fi (ρ) ∗ I ∗ C˜ fi (ρ)

(39)

Suppose the filter with the asynchronous behaviours:

−T



A f j (ρ) B f j (ρ) Wi 0 A˜ fi j (ρ) B˜ fi j (ρ) = ∗ C f j (ρ) ∗ I ∗ C˜ fi j (ρ) Proof. Let the matrices Pi (ρ), Pij (ρ) and Ui be partitioned as



P (ρ) P2i (ρ) P (ρ) P2i j (ρ) U1i , Pi j (ρ) = 1i j , Ui = Pi (ρ) = 1i ∗ P3i (ρ) ∗ P3i j (ρ) U3i H

(40)

H T U2i U4i



Perform congruence transformations to the inequalities (32) and (33) by matrix diag{Ji , Ji , I, I, Ji , I, I}, respectively. And define the following matrices as





I 0 P˜1i (ρ) P˜2i (ρ) P˜1i j (ρ) P˜2i j (ρ) T T , Ji Pi (ρ)Ji = , J Pi j (ρ)Ji = , Ji = 0 U4i−1U3i ∗ P˜3i (ρ) i ∗ P˜3i j (ρ) Si = U1i , Mi = U2iU4i−1U3i , Wi = U3Ti U4i−T U3i , and A˜ fi (ρ) ∗ A˜ f j (ρ) ∗

T B˜ fi (ρ) U = 3i ∗ C˜ fi (ρ)

T B˜ f j (ρ) U = 3i ˜ ∗ C f j (ρ)



A fi (ρ) B fi (ρ) U4i−1U3i 0 ∗ C fi (ρ) ∗ I



−1

0 A fi j (ρ) B fi j (ρ) U4i U3i 0 I ∗ C fi j (ρ) ∗ I

0 I



So, we can deduce the inequalities (36)–(38). Based on the Eqs. (39) and (40), the U3i and U4i are necessary matrices to construct the filter parameter matrices. Therefore considering Wi = Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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U3Ti U4i−T U3i , the transfer function of the filter with synchronous behaviours from y(t) to zf (t) can be expressed as Gz f y = C fi (ρ)(sI − A fi (ρ))−1 B fi (ρ) and then we can get G˜ z f i y = C˜ fi (ρ)(sI − Wi−T A˜ fi (ρ))−1 B˜ fi (ρ), the transfer function of the filter with asynchronous behaviours from y(t) to zf (t) can be expressed as G˜ z f i j y = C˜ fi j (ρ)(sI − Wi−T A˜ fi j (ρ))−1 B˜ fi j (ρ). The proof is completed.  Remark 4. Because of the above result depends on the parameter ρ, the filter design is corresponding to PLMIs. We utilize approximate basis function and gridding technique, the filter design can be transformed into solution problem of finite number of LMIs. Corollary 1. Furthermore, considering the LPV switched delay systems (1) and let 0 < α i < 1, β i ≥ 0, μ1 > 1, μ2 > 1 be given scalars. We can obtain a weighted H∞ performance γ that ensure exponential stability of the filtering error system  by constructing and solving the following convex optimization problem: min γ subject to the inequalities (8), (36)–(38) and the parameters of admissible filter can be obtained by the Eqs. (39) and (40). 5. Numerical example Consider the LPV switched delay systems (1) which consist of two subsystems: Subsystem 1: ⎡

⎤ −6.8 + 0.1ρ1 + 0.1ρ2 0 1.1 0 ⎢ ⎥ 0 −4.8 + 0.1ρ1 + 0.1ρ2 2.8 0 ⎢ ⎥ A1 (ρ ) = ⎢ ⎥ ⎣ ⎦ −2.2 0 −1.9 + 0.1ρ1 + 0.1ρ2 0 0 1.1 0 −3.7 + 0.1ρ1 + 0.1ρ2 ⎡ ⎤ 0.18 + 0.1ρ1 + 0.1ρ2 0 −0.1 0 ⎢ ⎥ 0 −0.2 + 0.1ρ1 + 0.1ρ2 0 0 ⎢ ⎥ Ad1 (ρ ) = ⎢ ⎥ ⎣ ⎦ 0 0 0.1 + 0.1ρ1 + 0.1ρ2 0 0.1 0 0 0.2 + 0.1ρ1 + 0.1ρ2 ! "T ! " B1 (ρ ) = 0.8 0.9 + 0.2ρ2 1.2 1.1 + 0.1ρ1 , C1 (ρ ) = 1.8 2.8 + 0.1ρ1 4.7 3.2 + 0.2ρ2 ! " Cd1 (ρ ) = 0.3 0.1ρ1 −0.3 0.1ρ2 , D1 (ρ ) = 1 + 0.1ρ1 + 0.1ρ2 , ! " E1 (ρ ) = 1.8 3.1 + 0.1ρ1 4.9 3.1 + 0.1ρ2

Subsystem 2: ⎡ −7.1 + 0.2ρ1 + 0.2ρ2 ⎢ 0 ⎢ A2 (ρ ) = ⎢ ⎣ −2.1 0 ⎡ 0.19 + 0.2ρ1 + 0.2ρ2 ⎢ 0 ⎢ Ad2 (ρ ) = ⎢ ⎣ 0 0.2 ! B2 (ρ ) = 0.9 1 + 0.2ρ2 0.8

⎤ 0 ⎥ 0 ⎥ ⎥ ⎦ 0 −2.7 + 0.2ρ1 + 0.2ρ2 ⎤ 0 −0.1 0 ⎥ −0.2 + 0.2ρ1 + 0.2ρ2 0 0 ⎥ ⎥ ⎦ 0 0.1 + 0.2ρ1 + 0.2ρ2 0 0 0 0.32 + 0.2ρ1 + 0.2ρ2 "T ! " 1 + 0.2ρ1 , C2 (ρ ) = 2.1 2.9 + 0.2ρ1 3.2 5.2 + 0.2ρ2 0 −3.7 + 0.2ρ1 + 0.2ρ2 0 1

0.8 2.2 −1.9 + 0.2ρ1 + 0.2ρ2 0

Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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! Cd2 (ρ ) = 0.4 ! E2 (ρ ) = 2.1

0.2ρ1

−0.3

3.2 + 0.2ρ1

17

" 0.2ρ2 , D2 (ρ ) = 1 + 0.2ρ1 + 0.2ρ2 , " 4.1 5.2 + 0.2ρ2

Where ρ 1 (t)=sin(t), ρ2 (t ) = | cos(t )|, the parameters change within [−1,1] × [0,1], the approximate basis functions are chosen as f1 (ρ) = 1, f2 (ρ) = ρ1 (t ), f3 (ρ) = ρ2 (t ) and the parameter-dependent delay d (ρ) = 0.2ρ1 (t ) + 0.3. Take into account the full-order and reduced-order filtering problems for LPV switched delay systems (1), gridding the parameters space of ρ(t) with ten uniform grids, then by solving convex optimization problem of the Corollary 1, the filter parameters can be obtained: Case 1. The synchronous filter parameters under MDADT switching are as follows: The fourth-order filter parameters: ⎡ ⎤ −9.2405 −2.7378 −0.3282 −4.5083 ⎢−3.5190 −5.9665 −1.5669 −5.3072⎥ ⎥ A f1 (ρ) = ⎢ ⎣−6.0119 −5.6638 −2.3070 −7.1716⎦ −1.8203 −4.5520 −1.8855 −7.2008 ⎡ ⎤ 0.2944 0.1701 0.4988 0.0903 ⎢−0.1021 0.1420 0.6400 0.1111⎥ ⎥ρ +⎢ ⎣ 0.1598 0.1948 0.5050 0.2508⎦ 1 0.2469 0.0937 −0.2423 0.3652 ⎡ ⎤ 0.0915 0.0034 0.3862 0.1105 ⎢−0.0894 −0.1492 0.1748 −0.4021⎥ ⎥ρ +⎢ ⎣ 0.2671 0.1704 0.2775 0.2640 ⎦ 2 0.7185 0.7021 −0.0344 1.0802 ⎡ ⎤ −11.4628 −3.3650 −1.5544 −5.9499 ⎢ −1.0195 −8.1280 0.6546 −5.2774⎥ ⎥ A f2 (ρ) = ⎢ ⎣ −6.0617 −2.0830 −4.0577 −4.1219⎦ −1.5069 −1.5027 −2.7805 −6.2919 ⎡ ⎤ 0.2988 0.0993 0.3608 0.2492 ⎢ 0.1065 0.3140 0.3163 0.2230⎥ ⎥ρ +⎢ ⎣ 0.1336 0.0517 0.3136 0.2487⎦ 1 −0.0318 −0.1374 0.0882 0.2825 ⎡ ⎤ 0.7598 0.6376 0.4096 0.6103 ⎢ 0.5241 0.5680 −0.6092 0.2669 ⎥ ⎥ρ +⎢ ⎣ 0.5777 0.6822 0.4370 0.5475 ⎦ 2 −0.8626 −0.8330 0.8288 −0.4848  T B f1 (ρ) = −0.8547 −1.0484 −1.2930 −1.0654  T + 0.0789 0.0721 0.0878 0.0288 ρ1  T + 0.0583 −0.0279 0.0793 0.1716 ρ2  T B f2 (ρ) = −1.1156 −1.0054 −0.8511 −0.8094  T + 0.0770 0.0715 0.0580 0.0110 ρ1  T + 0.1643 0.0509 0.1582 −0.0863 ρ2 Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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 C f1 (ρ) = −3.1568 −4.2523 −1.4367 −5.0815  + −0.1333 −0.2837 0.3564 −0.0863 ρ1  + −0.3505 −0.1283 0.5898 −0.2823 ρ2  C f2 (ρ) = −2.8613 −4.0492 −2.2773 −5.1238  + −0.2083 −0.2217 0.2519 −0.0776 ρ1  + −0.2284 0.0000 0.2757 −0.2940 ρ2 The third-order filter parameters: ⎡ A f1 (ρ) =

A f2 (ρ) =

B f1 (ρ) = B f2 (ρ) = C f1 (ρ) = C f2 (ρ) =

⎤ ⎡ ⎤ 2.5186 −5.4526 −3.4891 0.2110 0.5549 0.1539 ⎣10.2573 −9.4521 −6.0022⎦ + ⎣−0.1379 0.9446 0.3750⎦ρ1 7.8801 −9.8826 −7.0383 0.3412 1.0285 0.2082 ⎡ ⎤ −0.3572 0.9248 0.1976 + ⎣−0.9039 1.0346 0.4102⎦ρ2 −0.7165 1.4849 0.4455 ⎡ ⎤ ⎡ ⎤ −4.3358 −3.6034 −3.6674 −0.2122 0.0196 −0.1749 ⎣ 10.7975 −7.9502 −5.2355⎦ + ⎣−0.6777 0.3339 −0.0796⎦ρ1 −7.2854 −2.1937 −4.1219 0.2330 −0.2030 −0.1363 ⎡ ⎤ 1.0085 −0.1964 0.4765 0.4275 −0.4314⎦ρ2 + ⎣−0.5522 1.6496 −0.4731 0.8943  T  T −0.5367 −0.7512 −0.9158 + 0.0766 0.1128 0.1317 ρ1  T + 0.0812 0.0731 0.1265 ρ2  T  T −0.4627 −0.4493 −0.4858 + −0.0147 0.0017 −0.0171 ρ1  T + 0.0478 −0.0280 0.0774 ρ2   9.2358 −10.1336 −6.8657 + 3.8305 −1.9276 −0.5669 ρ1  + 2.2767 −1.7530 0.9375 ρ2   10.4929 −11.5653 −9.6688 + 0.1954 −0.0817 −0.4737 ρ1  + −0.1453 −0.2745 0.2138 ρ2

Case 2. The asynchronous filter parameters under MDADT switching are as follows: The fourth-order filter parameters: ⎡

−4.7748 ⎢−3.2365 A f1 (ρ) = ⎢ ⎣−6.0995 −6.6760 ⎡ 0.2297 ⎢0.2694 +⎢ ⎣0.3180 0.1405

−2.9127 −5.9832 −5.7671 −3.9938

−0.8171 −1.0467 −1.7990 −0.8161

0.0909 0.2790 0.1909 −0.0255

0.0711 0.1360 0.1540 0.0865

⎤ −4.3044 −5.8193⎥ ⎥ −7.4654⎦ −6.9688 ⎤ 0.2044 0.3767⎥ ⎥ρ 0.3881⎦ 1 0.1791

Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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⎡

⎤ 0.1923 0.2223 0.1410 0.2985 ⎢0.1575 0.2206 0.1084 0.1911⎥ ⎥ +⎢ ⎣0.3482 0.3479 0.1946 0.4379⎦ρ2 0.2478 0.1768 0.1877 0.3256 ⎡ ⎤ −10.4003 −5.2765 −0.1947 −5.9415 ⎢ −0.6368 −10.3971 1.9896 −5.2031⎥ ⎥ A f2 (ρ) = ⎢ ⎣ −3.8890 −3.6070 −2.6897 −3.4928⎦ −3.3026 1.6828 −3.6509 −6.0801 ⎡ ⎤ 0.4240 0.3942 0.1366 0.4315 ⎢ 0.2044 0.3463 0.2461 0.3589 ⎥ ⎥ρ +⎢ ⎣ 0.3123 0.3970 0.2232 0.4477 ⎦ 1 −0.2351 −0.3454 0.0238 −0.1248 ⎡ ⎤ 0.4705 0.3220 0.1592 0.4746 ⎢−0.3881 −0.3522 −0.0071 −0.5569⎥ ⎥ρ +⎢ ⎣ 0.3772 0.3289 0.1873 0.4413 ⎦ 2 0.5652 0.4598 0.2263 0.6850  T B f1 (ρ) = −0.7328 −1.0426 −1.2855 −1.0945  T + 0.0422 0.0775 0.0760 0.0269 ρ1  T + 0.0644 0.0486 0.0977 0.0711 ρ2  T B f2 (ρ) = −1.1422 −1.0597 −0.7811 −0.6975  T + 0.1066 0.0869 0.1050 −0.0572 ρ1  T + 0.1080 −0.1009 0.0998 0.1446 ρ2  C f1 (ρ) = −3.6529 −4.1159 −1.2366 −5.1844  + −0.0276 −0.1512 0.0892 −0.0754 ρ1  + −0.0658 −0.1229 0.1779 −0.1416 ρ2  C f2 (ρ) = −3.6720 −3.9867 −1.2648 −4.8996  + −0.0868 −0.0927 0.1298 −0.1125 ρ1  + −0.0115 −0.2059 0.2259 −0.1346 ρ2 The third-order filter parameters: ⎡

4.1236 −5.2183 A f1 (ρ) = ⎣9.3050 −8.2919 9.1639 −9.0490 ⎡ 0.1635 0.0777 + ⎣0.1470 0.0629 0.1132 0.1657 ⎡ −5.1460 −2.9277 −8.3007 A f2 (ρ) = ⎣ 8.3596 −9.8242 −0.1011

⎤ ⎡ ⎤ −3.7369 −0.0783 0.0104 −0.0557 −6.2486⎦ + ⎣−0.2412 0.0990 0.0049 ⎦ρ1 −7.1128 −0.1842 0.0195 −0.0100 ⎤ 0.0670 0.1249⎦ρ2 0.1245 ⎤ ⎡ ⎤ −2.2577 −0.0568 −0.0232 0.1981 −2.8570⎦ + ⎣−0.0321 0.1678 −0.2343⎦ρ1 −4.5151 −0.2648 −0.1298 0.7203

Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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1.5 system output 4th−order filter output 3th−order filter output

z(t) and zf(t)

1

0.5

0 =6, a2=4

a1

3 2 1 0

−0.5

−1

0

0

20 20

40

60

40

60

80

100 80

100

t/sec Fig. 1. Output and estimate outputs under synchronous filtering with MDADT method.

⎡

B f1 (ρ) = B f2 (ρ) = C f1 (ρ) = C f2 (ρ) =

⎤ −0.2398 0.6234 0.0892 −0.1633 0.2583 ⎦ρ2 + ⎣ 0.1289 −0.6248 1.2582 −0.0735  T  T −0.3992 −0.5888 −0.6809 + −0.0057 −0.0020 −0.0071 ρ1  T + 0.0059 −0.0022 0.0056 ρ2  T  T −0.4475 −0.4711 −0.5245 + 0.0206 0.0042 0.0392 ρ1  T + 0.0598 −0.0131 0.1064 ρ2   10.6184 −10.7209 −7.9548 + 0.0343 −0.1408 −0.0279 ρ1  + 0.2241 −0.0487 0.0130 ρ2   6.6809 −8.1673 −5.8977 + 0.2929 −0.2430 −0.1399 ρ1  + 0.6002 −0.0941 −0.1998 ρ2

Let the initial conditions of the LPV switched delay systems (1) and the filter be x(0) = [1 − 1 1 − 1]T and x f (0) = [0 . . . 0]Tv×1 (v = 3 or 4), respectively, and the disturbance be ω(t ) = 0.5sin(0.2πt )e−1t . Simulation results are shown in the figures below, the Figs. 1 the 2 give out the trajectories of output and the estimate outputs for synchronous and asynchronous ∗ filtering with the minimal MDADT method which satisfying τa1 = 6 > τa1 = 5.5, τa2 = ∗ 4 > τa2 = 3.7062 by setting α1 = 0.3, α2 = 0.4; β1 = 0.25, β2 = 0.3; η1 = 0.25, η2 = 0.3, respectively. And the Figs. 3 and 4 give out the similar results under ADT method, respectively, and the ADT constraint satisfying τa = 6 > τa∗ = 5.5 by setting the parameters α = 0.3; β = 0.25; η = 0.25. The state responses of the filtering error system  under asynchronous switching are shown in the Fig. 5. By observing the Figs. 1–5, it is obvious that the figures show the effectiveness of the designed filter. Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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1.5 system output 4th−order filter output 3th−order filter output

z(t) and zf(t)

1

0.5

0 =6, a2=4

a1

3 2 1 0

−0.5

−1

0

0

20 20

40

60

40

80

60

100 80

100

t/sec Fig. 2. Output and estimate outputs under asynchronous filtering with MDADT method.

2 system output 4th−order filter output 3th−order filter output

1

z(t) and zf(t)

0 −1 −2 =4

a

−3

3 2 1 0

−4 −5

0

0

20 20

40

60

40

60

80

100 80

100

t/sec Fig. 3. Output and estimate outputs under synchronous filtering with ADT method.

Furthermore, in order to illustrate the parameter-dependent Lyapunov-Krasovskii functional method get less conservative results, thus, the results of parameter-independent functional method are also developed. The suboptimal γ obtained by different cases are given in the Table1. In order to provide concise expression, the parameter-dependent Lyapunov-Krasovskii Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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2 system output 4th−order filter output 3th−order filter output

1

z(t) and zf(t)

0 −1 −2 =4

a2

−3

3 2 1 0

−4 −5

0

0

20 20

40

60

40

80

60

100 80

100

t/sec Fig. 4. Output and estimate outputs under asynchronous filtering with ADT method.

a 1.5 x1 xf1

State1

1 0.5 0 −0.5 0

20

40

60

80

100

time(s)

b x

0.5

2

x State2

f2

0 −0.5 −1 0

20

40

60

80

100

time(s) Fig. 5. State responses of the filtering error system. Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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Table 1 The obtained suboptimal γ for different Lyapunov-Krasovskii functional under different switching methods. Switching schemes Switching signal parameters

MDADT switching μ1 = 1.9, μ2 = 1.6, β1 = 0.25, β2 = 0.3 α1 = 0.3, α2 = 0.4, η1 = 0.25, η2 = 0.3

ADT switching μ1 = 1.9, μ2 = 1.6 α = 0.3, β = 0.25, η = 0.25

Lyapunov-functional γ of 4th-order Synchronous Asynchronous γ of 3th-order Synchronous Asynchronous Switching signals

P-dependent P-independent 0.8044 1.0301 1.1160 1.2116 4.8881 6.8278 7.4642 7.8065 ∗ =max{4.4474, 5.5}=5.5 τa1 ∗ =max{3.7062, 3.5}=3.7062 τa2

P-dependent 0.7239 1.0014 4.0374 5.9895 τα∗ =max{4.4474,

P-independent 0.9112 1.0863 5.5599 6.2661 5.5}=5.5

Table 2 The obtained suboptimal γ for different Firter under different switching methods. Filter

LPV switched filter

Switching Schemes γ of 4th-order γ of 3th-order

Synchronous Asynchronous Synchronous Asynchronous

LTI switched filter

MDADT

ADT

MDADT

ADT

0.8044 1.1160 4.8881 7.4642

0.7239 1.0014 4.0374 5.9895

0.9176 1.1829 5.0103 7.5622

0.8278 1.0672 4.1443 6.0815

functional is represented by P-dependent and parameter-independent Lyapunov-Krasovskii functional is represented by P-inpendent. As shown in the Table 1, the parameter-dependent Lyapunov-Krasovskii functional has less conservativeness than the parameter-independent one, and the MDADT switching is better than the ADT, which is consistent with the Remark 1. Simultaneously, it also can be seen from the Table 1 that the performance level γ of the third-order filtering are more greater than the performance level γ of the full-order filtering. However, to design filter for some practical systems with the less stringent design accuracy, the reduced-order filter can greatly reduce the design difficulty and cost. According to the above results, to further highlight the superiority of the design method of the LPV switched filter, a LTI switched filter has also been designed. Notice that the weighted H∞ performance indexes for the LPV switched filter and the LTI switched filter are given in the Table 2, which has shown that the LPV filtering method gets less conservative results in the sense of the noise attenuation performance level. 6. Conclusion In this paper, the design problem of asynchronous weighted H∞ fixed-order filter for continuous-time LPV switched delay systems under MDADT is investigated. By constructing both the parameter-dependent and mode-dependent Lyapunov-Krasovskii functional which is allowed to increase during the running time of mismatched period with a limited increase rate, and the weighted H∞ performance for the system exponential stability is first established. By comparing with minimal ADT switching method, the MDADT switching is less rigid. Furthermore, the design of fixed-order filter is transformed into a finite-dimensional LMIs by using approximate basis function and gridding technique. Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016

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Please cite this article as: Y. Li, P. Bo and J. Qi, Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.016