H∞ performance state estimation of delayed static neural networks based on an improved proportional-integral estimator

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H∞ performance state estimation of delayed static neural networks based on an improved proportional-integral estimator Guoqiang Tan, Zhanshan Wang∗, Cong Li College of Information Science and Engineering, Northeastern University, Shenyang 110819, China

a r t i c l e

i n f o

Article history: Received 7 June 2019 Revised 22 August 2019 Accepted 3 November 2019 Available online xxx Keywords: H∞ performance State estimation Time-varying delay Static neural networks Proportional-integral estimator with exponential gain term

a b s t r a c t In this paper, an improved proportional-integral (PI) estimator is presented to analyze the problem of H∞ performance state estimation of static neural networks with disturbance. An exponential gain term is added to the PI estimator, which leads to the convenience of analysis and design. In order to guarantee the H∞ performance state estimation, a less conservative delay-dependent criterion is derived by using an improved reciprocally convex inequality. Finally, simulation results are given to verify the advantage of the presented approach. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Static neural networks (SNNs) have been applied in a large number of fields [1–5], such as static imagine processing, associative memory, classification, and combinatorial optimization. For the study of SNNs, one of the research directions is state estimation problem. Thus, under the presence of disturbance, how to improve state estimation accuracy is a difficult problem, and the relevant research results are given in [6–10]. In [7], a Luenberger estimator was used to study H∞ state estimation problem of delayed SNNs, and a less conservative criterion was derived. In [8], the problem of H∞ performance state estimation for delayed SNNs was studied by an improved Luenberger estimator called Arcak-type estimator, and much better performance was achieved. In practice, time delay is inevitable during the applications of engineering [11–20], tremendous efforts have been devoted to study delayed SNNs. Among these references, the state estimation of SNNs is usually based on the Luenberger estimator. The characteristic of the Luenberger estimator is that the output error is used as a proportional gain term. Then, in order to design a suitable proportional gain term, many useful mathematical inequalities were employed to improve state estimation accuracy [21–25]. If the integral terms of output error are involved in the Luenberger estimator, the famous PI estimator is derived. The PI estimator has been used in control community [26–28]. In the analysis and design of PI estimator, how to design gain parameters of the PI estimator to improve estimation accuracy is one of the key problems. In [29], the method of dissipative analysis was used to deal with such problem. However, owing to limitation of the condition of dissipative

∗

Corresponding author. E-mail addresses: [email protected] (G. Tan), [email protected] (Z. Wang), [email protected] (C. Li).

https://doi.org/10.1016/j.amc.2019.124908 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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analysis, it will be difficult to derive a less conservative delay-dependent criterion when the PI estimator is directly used to estimate neuron states. Based on the above analysis, an improved PI estimator is presented in this paper, that is, an exponential gain term is involved in the integral terms of output error of the PI estimator. Meanwhile, an improved reciprocally convex inequality is presented, which leads to a less conservative delay-dependent criterion and much better performance can be achieved. The main contributions of this paper are given as follows: 1. An improved PI estimator with exponential gain term is presented for delayed SNNs, which can improve the state estimation accuracy and facilitate the analysis procedure of PI estimator. 2. An improved reciprocally convex inequality with three slack matrix variables is presented, which leads to a less conservative delay-dependent criterion to design the improved PI estimator. The rest of this paper is organized as follows. In Section 2, the problem of H∞ state estimation for delayed SNNs with disturbance is described, and an improved PI estimator and an improved reciprocally convex inequality are presented. In Section 3, a less conservative delay-dependent criterion is derived based on the improved reciprocally convex inequality. In Section 4, a numerical example is provided to illustrate the effectiveness of the proposed method. In Section 5, the conclusion is drawn. Notations: The notations used in this paper are standard. Rn represents the n-dimensional Euclidean space and Rn×m stands for the set of all n × m real matrices. The block-diagonal matrix can be denoted by diag{}. The identity matrix can be denoted by I with appropriate dimension. The space of square-integrable vector functions over [0, ∞) can be denoted by L2 [0, ∞). R > 0 (R < 0) signifies that R is a real positive definite (negative definite) symmetric matrix. AT and A−1 denote the transpose and inverse of a matrix A. ‘∗ ’ denotes the symmetric term of a symmetric block matrices. The notation Sn+ represents the set of positive definite symmetric matrices of Rn×n . 2. Problem description and preliminaries The delayed static neural network with disturbance is described by:

⎧ x˙ (t ) = −Ax(t ) + f (g(t )) + B1 ω (t ) ⎪ ⎪ ⎨g(t ) = W x(t − k(t )) + J y(t ) = C0 x(t ) + C1 x(t − k(t )) + B2 ω (t ) ⎪ ⎪ ⎩z(t ) = L0 x(t ) + L1 x(t − k(t )) x(t ) = ψ (t ), t ∈ [−d, 0]

(1)

where x(t ) ∈ Rn is the neuron state vector, y ∈ Rm is the network measurement vector, z(t ) ∈ R p is the linear combination of state vectors to be estimated, ω (t ) ∈ Rq is the disturbance vector belonging to L2 [0, ∞). A is a diagonal and positive definite matrix, W is a delayed connection weight matrix, B1 , B2 , C0 , C1 , L0 , and L1 are real matrices with appropriate dimensions. f (x ) = [ f1 (x1 ), f2 (x2 ), . . . , fn (xn )]T is the continuous neuron activation function, J = [J1 , J2 , . . . , Jn ]T is an external input vector, k(t) is a time-varying delay, ψ (t) is a initial condition defined on [−d, 0]. Assumption 1. The neural activation function fi ( · ) satisfies

0≤

f i ( h1 ) − f i ( h2 ) ≤ λi , h1 − h2

h1 = h2 ∈ R, i = 1 , 2 , . . . , n

(2)

where λi is a known nonnegative scalar. Assumption 2. For real scalars d > 0 and k1 , k2 ∈ R, k(t) satisfies

0 ≤ k(t ) ≤ d,

−∞ < k1 ≤ k˙ (t ) ≤ k2 < ∞

(3)

In this paper, an improved PI estimator is designed as follows:

⎧˙ xˆ(t ) = −Axˆ(t ) + f (gˆ(t )) + K1 (y(t ) − yˆ(t )) + K2 e−at ξ (t ) ⎪ ⎪ −at ⎪ ⎪ ⎨gˆ(t ) = W xˆ(t − k(t )) + J + K3 (y(t ) − yˆ(t )) + K4 e ξ (t ) ˙ξ (t ) = y(t ) − yˆ(t ) yˆ(t ) = C0 xˆ(t ) + C1 xˆ(t − k(t )) ⎪ ⎪ ⎪ ⎪ ⎩zˆ(t ) = L0 xˆ(t ) + L1 xˆ(t − k(t )) xˆ(t ) = 0, t ∈ [−d, 0]

(4)

where xˆ(t ) ∈ Rn , zˆ(t ) ∈ R p , ξ (t ) ∈ Rm , a is a constant satisfying a > 0. K1 , K2 , K3 , and K4 are the gain matrices to be computed. Remark 1. The H∞ state estimation problem has been studied for (1), and the corresponding state estimation results are given based on the Luenberger estimator [25]. We continue to study the state estimation problem of (1), and improve the state estimation accuracy by using an improved PI estimator. It is shown [25] that the Luenberger estimator is used for state estimation, that is, K2 = K4 = 0 in (4). The PI estimator takes the form of K1 (y(t ) − yˆ(t )) + K2 ξ (t ), that is, a = 0 in (4). Please cite this article as: G. Tan, Z. Wang and C. Li, H∞ performance state estimation of delayed static neural networks based on an improved proportional-integral estimator, Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124908

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3

However, for the improved PI estimator, an exponential gain term is taken into account. It is obvious that the improved PI estimator (4) includes the Luenberger estimator [25] and the PI estimator [26–28] as special cases. Remark 2. The advantage of adding an exponential gain term e−at to the improved PI estimator (4) is that the dynamic response performance is improved, then the analysis and design of the improved PI estimator is facilitated. Meanwhile, the advantage of the improved PI estimator is that the convergence rate of estimation error system can be adjusted, then the conservatism of the improved PI estimator based on the improved reciprocally convex inequality is reduced. Define υ (t ) = x(t ) − xˆ(t ), z˜(t ) = z(t ) − zˆ(t ), g˜(t ) = g(t ) − gˆ(t ), ξ¯ (t ) = e−at ξ (t ), then

g˜(t ) = g(t ) − gˆ(t ) = − K3C0 υ (t ) + (W − K3C1 )υ (t − k(t )) − K4 ξ¯ (t ) − K3 B2 ω (t ). Then, from (1) and (4), the error system can be obtained as follows:

⎧ υ˙ (t ) = −(A + K1C0 )υ (t ) − K1C1 υ (t − k(t )) + f¯(g˜(t )) ⎪ ⎨ + (B1 − K1 B2 )ω (t ) − K2 ξ¯ (t ) ˙ (t ) = C0 υ (t ) + C1 υ (t − k(t )) + B2 ω (t ) ⎪ ξ ⎩ z˜(t ) = L0 υ (t ) + L1 υ (t − k(t ))

(5)

where f¯ (g˜(t )) = f (g(t )) − f (gˆ(t )). Clearly, for each nonnegative scalar λi (i = 1, . . . , n ), f¯i (g˜i (t )) satisfies

f¯i (g˜i (t ))[ f¯i (g˜i (t )) − λi g˜i (t )] ≤ 0.

(6)

It’s obvious that, for any real matrix T = diag{t1 , t2 , . . . , tn } > 0, (6) can be rewritten as the following form

2 f¯T (g˜(t ))T [ f¯(g˜(t )) − g˜(t )] ≤ 0

(7)

where  = diag{λ1 , λ2 , . . . , λn }. The problem of H∞ state estimation is given as follows. Given a parameter γ > 0, then the error system (5) with ω (t ) = 0 is globally asymptotically stable when the improved PI estimator is properly designed. In addition, the following inequality holds if the zero-initial conditions are satisfied:



∞ 0

z˜T (t )z˜(t )dt < γ 2



∞

0

ωT (t )ω (t )dt.

(8)

Lemma 1. Let Ri ∈ Rm×m (Ri = RTi > 0 ), ζi ∈ Rm (i = 1, 2 ), real scalar σi ∈ [0, 1](i = 1, 2 ), and real scalars α > 0, β > 0 satisfying α + β = 1. If there exist real symmetric matrices M ∈ Rm×m , N ∈ Rm×m , and any real matrix X ∈ Rm×m such that

 



R1 − M XT

X ≥ 0, R 2 − σ2 M

(9)



R 1 − σ1 N XT

X ≥0 R2 − N

(10)

the following inequality holds:

1

α

ζ1T R1 ζ1 +

1

β

ζ2T R2 ζ2 ≥ ζ1T [R1 + β M + σ1 β 2 N/α ]ζ1 + 2ζ1T X ζ2 + ζ2T [R2 + α N + σ2 α 2 M/β ]ζ2 .

(11)

Proof. It is noted that the inequality (11) can be rewritten as the following form

    ζ1 T ζ1

≥0 ζ2 ζ2

where

=

β

α R1 − β M −

−X T

σ1 β 2 α N



−X α R − α N − σ2 α 2 M . 2 β β

Thus, the inequality (11) holds if we can prove that ≥ 0. Now, define H = diag{−[α /β ]1/2 I, [β /α ]1/2 I}, one has



H H = α T

R1 − M XT





X R −σ N +β 1 T 1 R 2 − σ2 M X



X . R2 − N

Noting (9) and (10), one can easily get that HT H ≥ 0. Thus, ≥ 0.



Remark 3. It is worth noting that Lemma 1 is an extension of the reciprocally convex inequality (16) in [25]. It’s easy to get that the (16) in [25] is a special case of (11) when σ1 = σ2 = 0. It is obvious that (11) is much less conservative than (16) in [25]. Please cite this article as: G. Tan, Z. Wang and C. Li, H∞ performance state estimation of delayed static neural networks based on an improved proportional-integral estimator, Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124908

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3. Main results In this section, we present a delay-dependent criterion to design the improved PI estimator. Theorem 1. Given γ > 0, d > 0, a > 0, ε 1 > 0, ε 2 > 0, and k1 , k2 satisfying (3), the H∞ state estimation problem is solvable if 2n n n 3n×3n , real there exist real symmetric matrices P1 ∈ S7+n , P2 ∈ Sm + , Q1 , Q2 ∈ S+ , R, Z ∈ S+ , T = diag{t1 , t2 , . . . , tn } ∈ S+ , M, N ∈ R n ×n n ×m 3 n ×3 n matrices G1 ∈ R , Y1 , Y2 , Y3 , Y4 ∈ R , and X ∈ R such that

⎡ ( 0 , k ) | 1 k∈{k1 ,k2 } ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

⎡ (d, k )| 1 k∈{k1 ,k2 } ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

T2

−2T ∗ ∗ ∗

T2

−2T ∗ ∗ ∗

3 5

3 5

−aP2 ∗ ∗

−aP2 ∗ ∗

χ6T ⎤ 0⎥ 0 ⎥<0 ⎦

4 6 7 −γ 2 I ∗

χ6T ⎤ 0⎥ 0 ⎥<0 ⎦

4 6 7 −γ 2 I

(12)

0 −I

(13)

0 −I

∗

where

1 (k(t ), k˙ (t )) = 11 (k(t ), k˙ (t )) − 12 (k(t )) 11 (k(t ), k˙ (t )) T = ψ1T P1 ψ2 + ψ2T P1 ψ1 − T χ0 − χ0T  + d2 α10 (R + Z )α10

+ χ1T Q1 χ1 + (1 − k˙ (t ))χ3T (Q2 − Q1 )χ3 − χ2T Q2 χ2

12 (k(t )) = χ4T {1T [(Rˆ + (1 − δ )M + (1 − δ )2 N]1 + 22T Z˜ 2 }χ4 + χ5T [22T Z˜ 2 + 1T (Rˆ + δ N + δ 2 M )1 ]χ5 + χ4T 1T X 1 χ5 + χ5T 1T X T 1 χ4

2 = G1  + [−Y3C0 α1 + (T W − Y3C1 )α2 ]   3 = −T Y2 + α1T C0T P2 + α2T C1T P2 , 4 = T GT1 B1 − Y1 B2 5 = −Y4 , 6 = −Y3 B2 , 7 = P2 B2 ,  = α10 + ε1 α1 + ε2 α2   αi = 0n×(i−1)n In×n 0n×(10−i)n , i = 1, . . . , n   χ0 = GT1 A + Y1C0 α1 + Y1C1 α2 + GT1 α10  T T T     χ1 = α10 , α1 , χ2 = α9T , α3T T , χ3 = α8T , α2T T     χ4 = α2T , α1T , α5T , α7T T , χ5 = α3T , α2T , α4T , α6T T , χ6 = L0 α1 + L1 α2   ψ1 = α1T , α2T , α3T , (d − k(t ))α4T , k(t )α5T , (d − k(t ))α6T , k(t )α7T T  T ψ2 = α10 , (1 − k˙ (t ))α8T , α9T , (1 − k˙ (t ))α2T − α3T , α1T − (1 − k˙ (t ))α2T ,  (1 − k˙ (t ))α4T + k˙ (t )α6T − α3T , α5T − (1 − k˙ (t ))α2T − k˙ (t )α7T T 1 = [11 12 13 14 ], 2 = [21 22 23 24 ] 11 = [−I, I, −I]T , 12 = [I, I, I]T , 13 = [0, −2I, −6I]T , 14 = [0, 0, 12I]T 21 = [0, 0]T , 22 = [I, I]T , 23 = [−2I, −4I]T , 24 = [0, 6I]T with Zˆ = diag{Z, 3Z, 5Z }, Z˜ = diag{2Z, 4Z }, Rˆ = diag{R, 3R, 5R}, δ = k(t )/d. Furthermore, the improved PI estimator gain matrices Ki (i = 1, 2, 3, 4 ) are designed as follows: −1 Ki = G−T Yi (i = 3, 4 ). 1 Yi (i = 1, 2 ), Ki = T

(14)

Proof. Choose the following Lyapunov functional candidate

V (t ) = π1T (t )P1 π1 (t ) + e−at ξ T (t )P2 ξ (t ) +



t

t −k(t )

π2T (s )Q1 π2 (s )ds

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 + +

t −k(t )

t−d  t t−d

π2T (s )Q2 π2 (s )ds + d





0

t

t+θ

−d

υ˙ T (s )Rυ˙ (s )dsdθ

(d − t + s )2 υ˙ T (s )Z υ˙ (s )ds

(15)

where π1 (t ) = [υ T (t ), υ T (t − k(t )), υ T (t − d ), (d − k(t ))ν1T (t ), k(t )ν2T (t ), (d − k(t ))ν3T (t ), k(t )ν4T (t )]T . (t )]T , and

ν1 (t ) =

5

1 d − k(t )



t −k(t )

t−d

υ (s )ds, ν2 (t ) =



1 k(t )



t

t −k(t )

π2 (t ) = [υ˙ T (t ), υ T

υ (s )ds

t −k(t ) 1 (t − k(t ) − s )υ (s )ds 2 (d − k(t )) t−d  t 1 ν4 (t ) = 2 (t − s )υ (s )ds. k (t ) t −k(t )

ν3 (t ) =

Taking the time-derivative of V(t) with respect to t along the trajectories of the error system (5), one has

V˙ (t ) ≤ 2π1T (t )P1 π˙ 1 (t ) − aξ¯ T (t )P2 ξ¯ (t ) + 2ξ¯ T (t )P2 ξ˙ (t ) + π2T (t )Q1 π2 (t ) − π2T (t − d )Q2 π2 (t − d ) + (1 − k˙ (t ))π2T (t − k(t ))(Q2 − Q1 )π2 (t − k(t )) + d2 υ˙ T (t )(R + Z )υ˙ (t ) + ϕ1 (t ) + 2ϕ2 (t ) where



ϕ1 (t ) = −d ϕ2 (t ) = −

t

t−d



t

t−d

υ˙ T (s )Rυ˙ (s )ds

(16)

(17)

(d − t + s )υ˙ T (s )Z υ˙ (s )ds.

(18)

Now, we begin to tackle the integral terms ϕi (t ), (i = 1, 2 ). By applying the second-order Bessel-Legendre inequality [30], one has

ϕ1 (t ) = − d ≤−



t −k(t )

t−d

υ˙ T (s )Rυ˙ (s )ds − d



t

t −k(t )

υ˙ T (s )Rυ˙ (s )ds

1 1 ρ T (t )1T Rˆ1 ρ2 (t ) − ρ1T (t )1T Rˆ1 ρ1 (t ) 1−δ 2 δ

and

ϕ2 (t ) = −



t −k(t )

t−d

(d − t + s )υ˙ T (s )Z υ˙ (s )ds −

− (d − k(t ))



t

t −k(t )



t

t −k(t )

(19)

(k(t ) − t + s )υ˙ T (s )Z υ˙ (s )ds

υ˙ T (s )Z υ˙ (s )ds

≤ − ρ2T (t )2T Z˜ 2 ρ2 (t ) − ρ1T (t )2T Z˜ 2 ρ1 (t ) − where

1 δ

−1

 ρ1T (t )1T Zˆ 1 ρ1 (t )

(20)

   t  t 1 1 ρ1 (t ) = υ (t − k(t )), υ (t ), υ (s )ds, 2 (t − s )υ (s )ds k(t ) t −k(t ) k (t ) t −k(t )  ρ2 (t ) = υ (t − d ), υ (t − k(t )),

1 d − k(t )



t −k(t )

t−d

υ (s )ds,

1 (d − k(t ))2



t −k(t )

t−d

 (t − k(t ) − s )υ (s )ds .

Then, one has

ϕ1 (t ) + 2ϕ2 (t ) ≤ F (δ ) − 2ρ1T (t )(2T Z˜ 2 − 1T Zˆ 1 )ρ1 (t ) − 2ρ2T (t )2T Z˜ 2 ρ2 (t )

(21)

where

F (δ ) = −

1

δ

ρ1T (t )1T (Rˆ + 2Zˆ )1 ρ1 (t ) −

1 ρ T (t )1T Rˆ1 ρ2 (t ). 1−δ 2

(22)

By applying Lemma 1 with ξ1 = 1 ρ1 (t ), ξ2 = 1 ρ2 (t ), R1 = Rˆ + 2Zˆ , R2 = Rˆ, α = δ, σ1 = δ, σ2 = 1 − δ, it is true that

F (δ ) ≤ − ρ1T (t )1T [Rˆ + 2Zˆ + (1 − δ )M + (1 − δ )2 N]1 ρ1 (t ) − ρ2T (t )1T [Rˆ + δ N + δ 2 M]1 ρ2 (t ) − 2ρ1T (t )1T X 1 ρ2 (t ).

(23)

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Then, from (21)–(23), one has

ϕ1 (t ) + 2ϕ2 (t ) ≤ −ηT (t )12 (k(t ))η (t ). [υ T (t ), υ T (t

Letting η (t ) = equation in (5), one has

− k(t )), υ T (t

(24)

− d ), ν1T (t ), ν2T (t ), ν3T (t ), ν4T (t ), υ˙ T (t

− k(t )), υ˙ T (t

− d ), υ˙ T (t )]T ,

χ¯ 0 η (t ) + f¯(g˜(t )) + (B1 − K1 B2 )ω (t ) − K2 ξ¯ (t ) = 0

and noting the first

(25)

where χ¯ 0 = −(A + K1C0 )α1 − K1C1 α2 − α10 , and it is also true that

2ηT (t )T GT1 [χ¯ 0 η (t ) + f¯ (g˜(t )) + (B1 − K1 B2 )ω (t ) − K2 ξ¯ (t )] = 0.

(26)

Substituting K1 = G−T Y1 into (26) yields 1

¯ − 2ηT (t )T χ0 η (t ) + 2ηT (t )T GT1 [ f¯ (g˜(t )) + (B1 − G−T 1 Y1 B2 )ω (t ) − K2 ξ (t )] = 0.

(27)

Then, from (16)–(27), one can deduce

V˙ (t ) ≤ ηT (t )(k(t ), k˙ (t ))η (t ) + 2ηT (t )T GT1 [ f¯(g˜(t )) − K2 ξ¯ (t ) + (B1 − K1 B2 )ω (t )] − aξ¯ T (t )P2 ξ¯ (t ) + 2ξ¯ T (t )P2 ξ˙ (t ).

(28)

Now, from (7) and (28), one can get

z˜T (t )z˜(t ) − γ 2 ωT (t )ω (t ) + V˙ (t ) ≤ η1T (t ) η1 (t ) where

⎡ ⎢ =⎣

1 (k(t ), k˙ (t )) + χ6T χ6 ∗ ∗ ∗

T2

⎤ ¯4  ¯6 ⎥  ⎦ 7 2 −γ I

¯3  ¯5 

−2T ∗ ∗

(29)

−aP2 ∗

(30)

¯ 3 = −T GT K2 + α T C T P2 + α T C T P2  1 1 0 2 1 ¯ 4 = T GT (B1 − K1 B2 )  1 ¯ 5 = −T K4 ,  [ηT (t ),

and η1 (t ) = equivalent to

¯ 6 = −T K3 B2 

(31)

f¯T (g˜(t )), ξ¯ T (t ), ωT (t )]T . Furthermore, η1T (t ) η1 (t ) < 0 when  < 0. By Schur complement,  < 0 is

⎡ 1 (k(t ), k˙ (t )) ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

T2

−2T ∗ ∗ ∗

3 5

−aP2 ∗ ∗

4 6 7 −γ 2 I ∗

χ6T

⎤

0⎥ ⎥ 0 ⎥<0 0⎦ −I

(32)

¯ 3,  ¯ 4,  ¯ 5,  ¯ 6 by replacing Ki = G−T Yi (i = 1, 2 ), Ki = T −1Yi (i = 3, 4 ). Obviously, where 3 , 4 , 5 , 6 are obtained from  1 ˙ ˙ 1 (k(t ), k(t )) < 0 holds when k(t) ∈ [0, d] and k(t ) ∈ [k1 , k2 ] if and only if

1 ( 0 , k 1 ) < 0 , 1 (d, k1 ) < 0,

1 ( 0 , k 2 ) < 0 , 1 (d, k2 ) < 0.

(33)

Thus, (32) can be satisfied from (12) and (13). If zero-initial conditions are satisfied, it is easy to get that V (t )|t=0 = 0 and V(t) ≥ 0 (t > 0). Define

J (t ) =

∞ 0

[z˜T (t )z˜(t ) − γ 2 ωT (t )ω (t )]dt.

(34)

Then, from (29) and (32), one has

J (t ) ≤



 = That is,

∞ 0

∞ 0

0

∞

[z˜T (t )z˜(t ) − γ 2 ωT (t )ω (t )]dt + V (t )|t→∞ − V (t )|t=0 [z˜T (t )z˜(t ) − γ 2 ωT (t )ω (t ) + V˙ (t )]dt < 0.

z˜T (t )z˜(t )dt < γ 2

∞ 0

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ωT (t )ω (t )dt. This completes the proof. 

Remark 4. The significance of adding exponential gain term e−at to the improved PI estimator (4) is that, when dealing with e−at ξ T (t )P2 ξ (t ) of Lyapunov functional (15), the corresponding derivative is −ae−at ξ T (t )P2 ξ (t ) + 2e−at ξ T (t )P2 ξ˙ (t ), which is less than or equal to −ae−at ξ T (t )P2 ξ (t )e−at + 2e−at ξ T (t )P2 ξ˙ (t ). Therefore, the analysis and design of the improved Please cite this article as: G. Tan, Z. Wang and C. Li, H∞ performance state estimation of delayed static neural networks based on an improved proportional-integral estimator, Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124908

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PI estimator is facilitated. If the exponential gain term e−at is not added to estimator (4), the derivative of ξ T (t)P2 ξ (t) is ξ˙ T (t )P2 ξ (t ) + ξ T (t )P2 ξ˙ (t ), then, (12) and (13) will be infeasible, that is, the estimator gain matrices can not be designed. It further confirms the novelty of the improved PI estimator and the effectiveness of the proposed method. Remark 5. Note that this paper is mainly focused on the theoretical study on H∞ performance state estimation problem based on an improved PI estimator, has improved the estimation performance as those in [8,25]. For example, compared with the Lyapunov functionals proposed in some existing results [8,25], the advantage of the Lyapunov functional (15) lies in the adoption of the integral of output error. Meanwhile, an exponential gain term is added to the integral of output error, then the decay rate of the integral of output error is accelerated. Remark 6. Theorem 1 presents a delay-dependent criterion to guarantee H∞ performance state estimation of SNNs based on the improved PI estimator. It is clear from the proof that ϕ1 (t ) + 2ϕ2 (t ) are handled by the improved reciprocally convex inequality (11). The significance is that less conservative results can be obtained by Theorem 1 than [8,25], which can be illustrated by a numerical example. Remark 7. The major difficulties we have met in this paper are how to design the improved PI estimator and how to reduce the conservatism of the delay-dependent criterion. An exponential gain term e−at is added to the improved PI estimator, correspondingly, the analysis and design of the improved PI estimator is facilitated, which can be seen from the proof of Theorem 1. Meanwhile, it is known that the improved reciprocally convex inequality is less conservative than the traditional reciprocally convex inequality [21]. Therefore, a delay-dependent criterion based on the improved reciprocally convex inequality is derived, which is less conservative than some existing ones. In the sequel, for the same PI estimator, different mathematical processing methods can lead to different state estimation performance. In order to compare with the existing results, the reciprocally convex inequality in [25] is used to design the presented PI estimator, and the following criterion is given. Corollary 1. Given γ > 0, d > 0, a > 0, ε 1 > 0, ε 2 > 0, and k1 , k2 satisfying (3), the H∞ state estimation problem is solvable if 2n n n 3n×3n , real there exist real symmetric matrices P1 ∈ S7+n , P2 ∈ Sm + , Q1 , Q2 ∈ S+ , R, Z ∈ S+ , T = diag{t1 , t2 , . . . , tn } ∈ S+ , M, N ∈ R n ×n n ×m 3 n ×3 n matrices G1 ∈ R , Y1 , Y2 , Y3 , Y4 ∈ R , and X ∈ R such that

⎡˜ 1 (0, k )|k∈{k1 ,k2 } ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

⎡˜ 1 (d, k )|k∈{k1 ,k2 } ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

T2

−2T ∗ ∗ ∗

T2

−2T ∗ ∗ ∗

3 5

−aP2 ∗ ∗

3 5

−aP2 ∗ ∗

4 6 7 −γ 2 I

χ6T

4 6 7 −γ 2 I

χ6T

∗

∗

⎤

0⎥ 0⎥ ⎦<0 0 −I

(36)

⎤

0⎥ 0⎥ ⎦<0 0 −I

(37)

where

˜ 12 (k(t )) ˜ 1 (k(t ), k˙ (t )) =11 (k(t ), k˙ (t )) −  

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˜ 12 (k(t )) =χ T { T [(Rˆ + (1 − δ )M]1 + 2 T Z˜ 2 }χ4  4 1 2 + χ5T [22T Z˜ 2 + 1T (Rˆ + δ N )1 ]χ5 + χ4T 1T X 1 χ5 + χ5T 1T X T 1 χ4

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and the rest notations are the same as those defined in Theorem 1. Furthermore, the gain matrices of the improved PI estimator are designed as follows: −1 Ki = G−T Yi (i = 3, 4 ). 1 Yi (i = 1, 2 ), Ki = T

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Proof. Applying (16) in [25], (23) can be simplified as follows:

F (δ ) ≤ − ρ1T (t )1T [Rˆ + 2Zˆ + (1 − δ )M]1 ρ1 (t ) − ρ2T (t )1T [Rˆ + δ N]1 ρ2 (t ) − 2ρ1T (t )1T X 1 ρ2 (t ) and the rest of the proof is similar to Theorem 1, it is omitted.

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Remark 8. Actually, Corollary 1 is derived from the reciprocally convex inequality (16) in [25]. Because the Luenberger estimator in [25] is a special case of the improved PI estimator (4), the H∞ performance state estimation based on Corollary 1 will be better than that in [25]. It can be verified by a numerical example. Please cite this article as: G. Tan, Z. Wang and C. Li, H∞ performance state estimation of delayed static neural networks based on an improved proportional-integral estimator, Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124908

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G. Tan, Z. Wang and C. Li / Applied Mathematics and Computation xxx (xxxx) xxx Table 1 Obtained optimal H∞ performance index γmin via different λ.

λ

1.2

1.4

1.6

1.8

2.0

2.1

Th 1 [8] Prop 2 [25] Cor 1 Th 1

0.6536 0.3973 0.3965 0.3938

2.2387 0.5549 0.5533 0.5483

0.8289 0.8153 0.7967

1.7264 1.6392 1.5329

7.1543 6.6159 5.7365

27.489 21.486 15.755

Table 2 Obtained optimal H∞ performance index γmin via different d. d

0.6

0.8

1

1.2

1.4

1.6

Th 1 [8] Prop 2 [25] Cor 1 Th 1

0.4826 0.3750 0.3720 0.3710

1.0052 0.4760 0.4697 0.4665

0.6683 0.6648 0.6531

1.4055 1.1478 1.1023

5.6523 3.4825 3.0908

32.3475 23.1536

Fig. 1. Response of state estimation error υ (t ).

Remark 9. The numbers of decision variables of Theorem 1, [8,25] are 48.5n2 + (10.5 + 4m )n + (0.5 + 0.5m )m, 10n2 + (4 + 2m )n, and 39.5n2 + (7.5 + 2m )n, respectively. Although higher computational complexity is required, much better performance can be achieved by Theorem 1. 4. Numerical example In this section, an example is provided to show the effectiveness of the proposed method. Consider the system (1) with parameters from [8,25] as follows:

A = diag{1.06, 1.42, 0.88},



L0 =



1 1 0

0 0 −1

1 C0 = 0

0.5 −0.5

B1 = [0.2

0.2



0.5 1 ,W = 1



L1 = 0,

f (x ) = tanh(x )

−0.32 1.1 0.42

0.85 0.41 0.82





0 0 , C1 = 0.6 0 0.2]T , B2 = [0.4

1 0





−1.36 −0.5 −0.95

0.2 0.5

− 0.3]T .

Firstly, let  = λI (λ > 0 ), a = 2, d = 0.8, and k2 = −k1 = 0.6. For different values of λ with ε1 = 1.6, ε2 = 0.25, the corresponding optimal H∞ performance indices calculated by Theorem 1, Corollary 1 are listed in Table 1. Obviously, the obtained optimal H∞ performance index γmin grows as parameter λ increases. From Table 1, it is clear that both Theorem 1 and Corollary 1 provide a smaller optimal H∞ performance index γmin than those in [8, Th 1], [25, Prop 2]. Please cite this article as: G. Tan, Z. Wang and C. Li, H∞ performance state estimation of delayed static neural networks based on an improved proportional-integral estimator, Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124908

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Secondly, let  = diag{1.2, 1.3, 1.4}, a = 2, and k2 = −k1 = 0.5. For different upper bounds of d with ε1 = 1.6, ε2 = 0.25, the corresponding optimal H∞ performance indices calculated by Theorem 1, Corollary 1 are listed in Table 2. Clearly, the obtained optimal H∞ performance index γmin grows as parameter d increases. From Table 2, it is obvious that the results calculated by Theorem 1 and Corollary 1 are both smaller than those in [8, Th 1], [25, Prop 2]. Specifically, for d = 1.6, [8, Th 1], [25, Prop 2] cannot obtain any results for this example. Therefore, the results of Theorem 1 and Corollary 1 are better than those in [8, Th1], [25, Prop 2]. By applying Theorem 1, for d = 1.6, the optimal H∞ performance index γmin = 23.1536, k(t ) = 0.4 + 1.2cos(5t/6 ), and ω (t ) = e−t , the state estimation error υ (t ) is illustrated in Fig. 1. 5. Conclusion In this paper, the problem of H∞ performance state estimation of SNNs with time-varying delay has been studied. An improved PI estimator with exponential gain term is presented. In addition, an improved reciprocally convex inequality is presented to further reduce the conservatism of delay-dependent criterion. Finally, a numerical example is used to verify the effectiveness of the proposed method. In addition, how to apply the theoretical results to engineering applications is the future work. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (grant nos. 61973070, 61433004, and 61627809), the Liaoning Revitalization Talents Program (grant no. XLYC1802010), and in part by SAPI Fundamental Research Funds (grant no. 2018ZCX22). References [1] C.K. Ahn, P. Shi, L. Wu, Receding horizon stabilization and disturbance attenuation for neural networks with time-varying delay, IEEE Trans. Cybern. 45 (2015) 2680–2692. [2] Y. Liu, S.M. Lee, O.M. Kwon, J.H. Park, A study on h∞ state estimation of static neural networks with time-varying delays, Appl. Math. 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