Envelope-constrained H∞ filtering for nonlinear systems with quantization effects: The finite horizon case

Automatica 93 (2018) 527–534

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Envelope-constrained H∞ filtering for nonlinear systems with quantization effects: The finite horizon case✩ Lifeng Ma a , Zidong Wang b,c, *, Qing-Long Han d , Hak-Keung Lam e a

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, UK d School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia e Department of Informatics, School of Natural & Mathematical Science, King’s College London, WC2R 2LS, UK b c

article

info

Article history: Received 1 October 2016 Received in revised form 20 September 2017 Accepted 1 February 2018

Keywords: Nonlinear systems H∞ filtering Envelope constraint Quantization effects Finite-horizon filtering

a b s t r a c t This paper is concerned with the envelope-constrained H∞ filtering problem for a class of discrete nonlinear stochastic systems subject to quantization effects over a finite horizon. The system under investigation involves both deterministic and stochastic nonlinearities. The stochastic nonlinearity described by statistical means is quite general that includes several well-studied nonlinearities as its special cases. The output measurements are quantized by a logarithmic quantizer. Two performance indices, namely, the finite-horizon H∞ specification and the envelope constraint criterion, are proposed to quantify the transient dynamics of the filtering errors over the specified time interval. The aim of the proposed problem is to construct a filter such that both the prespecified H∞ requirement and the envelope constraint are guaranteed simultaneously over a finite horizon. By resorting to the recursive matrix inequality approach, sufficient conditions are established for the existence of the desired filters. A numerical example is finally proposed to demonstrate the effectiveness of the developed filtering scheme. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Due to the significance in control and signal processing, the nonlinear filtering problem has been attracting constant research interest in the past several decades. A number of approaches have been developed to deal with the filtering problem for nonlinear stochastic systems, among which some of the most widely used include but are not limited to Bayes filtering, particle filtering, extended Kalman filtering (EKF) and unscented Kalman filtering (UKF). The Bayes filter aims to, in a recursive fashion, estimate the hidden state by using the available measurements and the process model (Garcia, Hausotte, & Amthor, 2013). Based on the Bayesian theory in combination with the concept of sequential importance sampling, particle filtering is particularly useful in coping with nonlinear and/or non-Gaussian problems (Djuric et al., 2003). However, the high computational complexity largely hinders the ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Xing Zheng under the direction of Editor Torsten Söderström. Corresponding author at: College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China. E-mail addresses: [email protected], [email protected] (L. Ma), [email protected] (Z. Wang), [email protected] (Q. Han), [email protected], [email protected] (H. Lam).

*

https://doi.org/10.1016/j.automatica.2018.03.038 0005-1098/© 2018 Elsevier Ltd. All rights reserved.

utilization of particle filters. Another recursive filter that should be mentioned is the celebrated Kalman filter (Kalman, 1960), which is in fact a linear version of Bayes filter for systems subject to Gaussian noises. As for nonlinear stochastic Gaussian systems, several invariants based on Kalman filters have been developed among which the most widely applied are EKF and UKF. EKF provides an approximation of an optimal estimate by linearizing the nonlinear system at the state estimates, which has found wide applications in both theoretical research and engineering practice (Einicke & White, 1999). However, it is no longer applicable when the process/measurement models are highly nonlinear, which gives rise to the so-called unscented Kalman filtering. The UKF uses a deterministic sampling technique known as the unscented transform to pick a minimal set of sample points around the mean value and could give more accurate estimates than EKF especially for those highly nonlinear systems (Sarkka, 2007). The past several decades have seen a surge of research interest on the H∞ filtering problems for nonlinear systems and several effective approaches have been exploited to deal with filtering problems with the requested disturbance attenuation level, see e.g. Dong, Wang, Ding, and Gao (2016), Einicke and White (1999), Shaked and Berman (1995), Shen and Deng (1997), Shen, Wang, Shu, and Wei (2010) and Takaba and Katayama (1996). On another research frontier, networked control systems (NCSs) have attracted

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L. Ma et al. / Automatica 93 (2018) 527–534

much attention owing to their clear application insights in a wide range of areas (Zhang, Han, & Jia, 2015; Zhang, Han, & Zhang, 2017). It has been recognized that, in the context of NCSs, the quantization effects stemming from analog-to-digital conversion processes are ubiquitous, which would probably lead to the deterioration of the system performance. In the NCS research, there are mainly two types of quantization models, namely, the uniform quantization (Tsai & Song, 2009) and the logarithmic quantization (Fu & Xie, 2005). In particular, a sector-bound technique has been presented in Fu and Xie (2005) that is capable of coping with the logarithmic quantization issues conveniently, and such an elegant paradigm has then been quickly followed in the area, see e.g. Liu, Liu, and Alsaadi (2016) and Shen et al. (2010). The envelope-constrained filtering (ECF) algorithm has been stirring some research interest in the past few decades. The main idea of ECF algorithm is to confine the output of the filtering error (stimulated by a specified input) into a prescribed envelope. Such an envelope is determined by the desired output and tolerance band. The ECF technique has found successful applications in a variety of engineering branches ranging from signal processing to digital communications (Cantoni, Vo, & Teo, 2001; Ding, Wang, Shen, & Dong, 2015). Up to now, several methodologies have been utilized in the literature to deal with the envelope-constrained filtering problems, see, e.g. Tan, Soh, and Xie (2000) and Zang, Cantoni, and Teo (1999). It should be pointed out that almost all the results relevant to ECF have been concerned with the linear timeinvariant systems. When it comes to general nonlinear time-varying systems, the corresponding envelope-constrained filtering problem has not been thoroughly investigated yet and this motivates us to shorten such a gap in the current study. It is, therefore, the main purpose of this paper to deal with the identified challenges by launching a major study on the so-called envelope-constrained H∞ filtering problem. The rest of this paper is organized as follows. Section 2 formulates the envelope-constrained H∞ filtering problem for discretetime nonlinear system subject to quantization effects. The main results are presented in Section 3 where sufficient conditions for solvability of the addressed filtering problem are given in terms of recursive linear matrix inequalities (RLMIs). Section 4 gives a numerical example and Section 5 outlines our conclusion.

Consider the following nonlinear system defined on the horizon [0, N ]:

⎧ ⎨ xk+1 = f (xk ) + g(xk ) + Bk wk yk = h(xk ) + Dk vk ⎩ z k = Lk x k

(1)

where xk ∈ Rnx , yk ∈ Rny and zk ∈ Rnz represent, respectively, the system state, the measurement output and the signal to be estimated. wk ∈ l2 ([0, N ]; Rnw ) and vk ∈ l2 ([0, N ]; Rnv ) are the disturbance inputs. Bk , Dk and Lk are known time-varying matrices with appropriate dimensions. The deterministic nonlinearities f (xk ) and h(xk ) are known and analytic everywhere over the finite horizon [0, N ]. On the other hand, the stochastic nonlinearity g(xk ) is assumed to have the following first moment for all xk :

E{g(xk )|xk } = 0

(2)

with the covariance given by

E{g(xk )g T (xj )|xk } = 0, k ̸ = j E{g(xk )g (xk )|xk } =

q ∑ l=1

[ σ (·) ≜ σ1 (·)

σ2 (·)

σny (·)

···

]

which is symmetric, i.e., σj (−y) = −σj (y) (j = 1, 2, . . . , ny ). The quantizer is assumed to be logarithmic type and the process of the quantization is described by

[ σ (yk ) = σ1 (y(1) k )

σ2 (y(2) k )

···

(n )

σny (yk y )

]T

ϱ ϱ

T l,k l,k

(j)

(j)

(j)

(j)

Uj = ±µ ˆi , µ ˆ i = χji µ ˆ 0 , i = 0, ±1, ±2, . . . ∪ 0 ,

{

}

(j)

0 < χj < 1, µ ˆ 0 > 0.

)

{ }

(5)

where χj (j = 1, 2, . . . , ny ) is the quantization density. Each of the quantization level corresponds to a segment such that the quantizer maps the whole segment to this quantization level. According to Fu and Xie (2005), the associated quantizer is defined as follows:

σ (y(j) k ) =

⎧ 1 + χj (j) 1 + χj (j) ⎪ ⎪ µ ˆ (j) , µ ˆ i ≤ y(j) µ ˆi ⎪ k ≤ ⎨ i 2 2χj 0, ⎪ ⎪ ⎪ ⎩ − σ (−y(j) k ),

(j)

yk = 0

(6)

(j)

yk < 0.

Consequently, it can be easily seen from the above definition (6) that the following inequality holds:

(

σ (yk ) − G1 yk

) )T ( σ (yk ) − G2 yk ≤ 0

(7)

where G1 ≜ diagny {2χj /(1 + χj )} and G2 ≜ diagny {2/(1 + χj )}. Since 0 < χj < 1, it is obvious that 0 ≤ G1 < I ≤ G2 . Then, σ (yk ) can be decomposed as follows:

σ (yk ) = G1 yk + ϕ (yk )

(8)

where ϕ (yk ) is a nonlinear vector-valued function which, from (7), satisfies (9)

with G being defined as G ≜ G2 − G1 . Definition 1 ( Durieu, Walter, & Polyak, 2001). A bounded ellipsoid E (c , P , n) of Rn with a nonempty interior in the mean square sense can be defined by E (c , P , n) ≜ {x ∈ Rn : E{(x − c)T P −1 (x − c)} ≤ 1}

where c ∈ Rn is the center of E (c , P , n) and P > 0 is a positive definite matrix. In this paper, the filter to be designed is of the following form: xˆ k+1 = Fk xˆ k + Hk σ (yk ),

xˆ 0 = 0.

(10)

Denote ek ≜ xk − xˆ k and z˜k ≜ zk − zˆk . Subtracting (10) from (1) and taking (8) into account, we obtain the following filtering error system:

⎧ ⎪ ⎨ ek+1 = f (xk ) + g(xk ) + Bk wk( −)Fk xˆ k − Hk G1 h(xk ) − Hk ϕ yk − Hk G1 Dk vk ⎪ ⎩ z˜ = L e . k k k By defining

xTk Υl,k xk

(

(4)

where yk (j = 1, 2, . . . , ny ) denotes the jth entry of the vector yk . For each σ (·), the set of quantization levels is described by

( ) ϕ T (yk ) ϕ (yk ) − Gyk ≤ 0

2. Problem formulation

T

where ϱl,k and Υl,k ≥ 0 (l = 1, 2, . . . , q) are, respectively, known column vectors and matrices with compatible dimensions. In this paper, the quantization effects are taken into consideration. Denote the quantizer as

(3)

Φk ≜

∂ f (x) ⏐⏐ ∂ h(x) ⏐⏐ , Ψk ≜ , ⏐ ⏐ ∂ x x=ˆxk ∂ x x=ˆxk

(11)

L. Ma et al. / Automatica 93 (2018) 527–534

and utilizing the Taylor series expansion formula, we linearize the nonlinear functions f (xk ) and h(xk ) around the state estimate xˆ k as follows: f (xk ) = f (xˆ k ) + Φk (xk − xˆ k ) + L1 ∆1 (xk − xˆ k )

(12)

h(xk ) = h(xˆ k ) + Ψk (xk − xˆ k ) + L2 ∆2 (xk − xˆ k )

(13)

n ×n

n ×n

where L1 ∈ R x l1 and L2 ∈ R y l2 are known scaling matrices, n ×n n ×n and ∆1 ∈ R l1 x and ∆2 ∈ R l2 x are unknown matrices such that ∥∆1 ∥ ≤ 1 and ∥∆2 ∥ ≤ 1. Taking (12) and (13) into consideration, we reformulate the filtering error system (11) as

⎧ ek+1 = f (xˆ k ) − Fk xˆ k − Hk G1 h(xˆ k ) ⎪ ⎪ ⎨ + (Φk + L1 ∆1 − Hk G1 Ψk − Hk G1 L2 ∆2 )ek ⎪ + Ek ξk − Hk ϕ (yk ) + g(xk ) ⎪ ⎩ z˜k = Lk ek

(14)

Theorem 1. Given γ > 0, Ξ > 0 and {Fk , Hk }0≤k≤N . For system (14), the H∞ performance index is guaranteed if there exist a sequence of ¯ ≤ 0 (Ξ¯ = positive definite matrices {Pk }0≤k≤N +1 with P0 − γ 2 Ξ diag{0, Ξ , 0}), and a sequence of positive scalars {εk }0≤k≤N such that

[ Ωk − εk

(15)

nz −i

i−1

This paper aims to design filter (10) such that the following requirements are achieved simultaneously: (R1) (H∞ specification) Given γ > 0 and Ξ > 0, the output z˜k of the filtering error system (14) satisfies

1,

k=0

0,

1≤k≤N

{ } Λi (ψk − βk ) ≤ E Λi z˜k◦ ≤ Λi (ψk + βk )

Π11

(18)

3. Main results 3.1. H∞ requirement

Hk ≜ 0

[

Lk ≜ 0

]T

0 , Γ ≜ 0

HkT

− Lk

[

[

0

I

]T

I ,

EkT Pk+1 Hk ⎦ ,

⎥

(21)

Next, for presentation simplicity, we denote L¯ 1 ≜ 0

[

LT1

[

0 , I˜ ≜ I

]

I≜ 0

Inl

L˜ 1 ≜ 0

0

[

]T

LT1 , L¯ 2 ≜ 0

1

[

]T −LT2 GT1 HkT 0 , ] 0 0 0 ,

]T

L¯ T1 ,

0

]T

L¯ T2 , L˜ ≜ ϵ1,k L˜ 1

ϵ2,k L˜ 2 J ≜ diag{ϵ1,k Inl , ϵ2,k Inl , ϵ1,k Inx , ϵ2,k Inx }, 1 2 ⎡ L˜ 2 ≜ 0

[

0

0

[

I˜ T

0

I˜ T ,

]

0

Φk − Hk G1 Ψk Φk

⎤

0⎦ ,

Theorem 2. Given γ > 0, Ξ > 0 and {Fk , Hk }0≤k≤N be given. For filtering error system (14), the H∞ performance index is guaranteed if there exist a sequence of positive definite matrices {Pk }0≤k≤N +1 with ¯ ≤ 0, sequences of positive scalars {εk , ϵ1,k , ϵ2,k }0≤k≤N , and P0 − γ 2 Ξ sequences of positive scalars {αl,k }0≤k≤N (l = 1, 2, . . . , q) such that

ϱlT,k G T −Pk−+11

] ≤ 0,

[ ˜k Π ∗

L˜

−J

]

≤ 0.

(22)

Proof. See Appendix B.

I ,

]

0 ,

]

3.2. Envelope constraint

f˜k ≜ f (xˆ k ) − Fk xˆ k − Hk G1 h(xˆ k ),

˜ k ≜ Φk + L1 ∆1 − Hk G1 Ψk − Hk G1 L2 ∆2 , Φ ⎡ ⎤ [ 1 0 0 0 ⎢ ˜ ⎥ ˜k Ak ≜ ⎣ fk Φ 0⎦ , Ek ≜ Bk Bk f (xˆ k ) Φk + L1 ∆1 0

Lemma 1. Given {Fk , Hk }0≤k≤N . If there exist a sequence of positive definite matrices {Qk }0≤k≤N +1 , sequences of positive scalars {τ1,k , τ2,k , τ3,k , τ4,k , τ5,k }0≤k≤N satisfying the following recursive linear matrix inequality:

0 −Hk Dk . 0

]

[ −Mk Σk

Accordingly, from (1) and (14), we have:

{

EkT Pk+1 Ek − γ 2 Inξ

⎤

Proof. See Appendix A.

∗

For the brevity of later presentation, we denote

]T

ATk Pk+1 Hk

∗ ∗ HkT Pk+1 Hk ≜ diag{0, 0, −Inx , 0}, [ ] ˜k . ≜ −Gh(xˆ k ) −GΨk 0 −GD

[ −αl,k

xTk , ϕk ≜ ϕ (yk ), G ≜ 0

ATk Pk+1 Ek

(17)

where the sequences of vectors {ψk }0≤k≤N and {βk }0≤k≤N represent, respectively, the desired output and the tolerance band.

eTk

(20)

f (xˆ k ) 0 ⎤ ⎡ q ∑ Γ T Υl,k Γ αl,k − Pk + LTk Lk 0 0⎥ ⎢ ⎥ ⎢ ⎥, ˜ k ≜ ⎢ l=1 Ω ⎥ ⎢ ∗ −γ 2 Inξ 0⎦ ⎣ ∗ ∗ 0 [ ] ˜k [ ] −εk Πk + Ω A¯kT ˜ . A¯k ≜ A¯k Ek Hk , Πk ≜ A¯k −Pk−+11

under the zero-initial condition, the corresponding output z˜k◦ of filtering error system (14) satisfies

[ ηk ≜ 1 [

≤0

A¯k ≜ ⎣f (xˆ k ) − Fk xˆ k − Hk G1 h(xˆ k )

k=1

{

2Inz − GL2 LT2 G

(16)

for any nonzero ξk ̸ = 0. (R2) (Envelope constraint) Given the following input

ξk◦ =

Π21

1

N N ∑ ∑ { } E ∥˜zk ∥2 ≤ γ 2 ∥ξk ∥2 + γ 2 eT0 Ξ e0 k=1

T Π21

⎡ ¯ k + LTk Lk Ω ⎢ ∗ Ωk ≜ ⎣

[

Λi ≜ [ 0 · · 0 1 0 · · 0 ].  ·  ·

]

Π11

where

Π21

where ξk ≜ [wkT vkT ]T and Ek ≜ [Bk − Hk G1 Dk ]. Before giving the main objective of this paper, for the brevity of later presentation, we denote

529

ηk+1 = Ak ηk + Ek ξk + G g(Γ ηk ) + Hk ϕk , z˜k = Lk ηk .

(19)

] ΣkT ≤0 −Qk+1

(23)

where

[ ( ) Ψ˜ 21 ≜ −G h(xˆ k ) + D˜ k ξk◦

−GΨk Θk

0

] −GL2 ,

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L. Ma et al. / Automatica 93 (2018) 527–534

{ Ψ˜ k ≜ diag

0

Ψ˜ 21

Ψ˜ 21

2Iny

[

] T

[

{ Λ ˜ (11) k ˜ k ≜ diag Λ ˜ (21) Λ k ˜ (11) Λ ≜ −ˆxTk k

q ∑

⎡ −(Λi βk )2 + λk ⎢ 0 ⎣ −Λi ψk

} ,0 ,

] ˜ (12) Λ k

}

, 0, 0, 0, Inx ,

˜ (22) Λ k

0

−λk Iϑ Λi Lk Θk

−Λi ψk

⎤

⎥ ΘkT LTk ΛTi ⎦ ≤ 0 −1

(28)

where the parameter Pk+1 is updated recursively according to Pk+1 = −1 Pk+1 , and the parameter Θk is computed iteratively by decomposing Qk such that Qk = Θk ΘkT .

ϱl,k ϱlT,k Υl,k xˆ k ,

l=1

˜ (12) Λ ≜ −ˆxTk k

q ∑

l=1 q ΘkT

˜ (21) Λ ≜− k

Proof. The proof follows directly from Theorems 1–3 and is therefore omitted here.

ϱl,k ϱlT,k Υl,k Θk ,

∑

Remark 1. So far, we have discussed the envelope-constrained H∞ filtering problem for general nonlinear system subject to quantization effects. Within the established theoretical framework, our results can be extended to certain filtering problems with other performance requirements/constraints such as security performance (Ma, Wang, Han, & Lam, 2017), ellipsoidal constraints (Ma, Wang, & Lam, 2017), consensus performance (Ma, Wang, Han, & Liu, 2017), communication cost (Zhang, Wang, Liu, Ding, & Alsaadi, 2017) and dissipativity (Xu, Lu, Peng, Xie, & Xue, 2017).

ˆ ,

ϱ ϱ

T l,k l,k Υl,k xk

l=1 q

˜ (22) Λ ≜ −ΘkT k

∑

ϱl,k ϱlT,k Υl,k Θk ,

l=1

˜ k + diag{1 − τ1,k , τ1,k Iϑ Mk ≜ τ4,k Ψ˜ k + τ5,k Λ − (τ2,k + τ3,k )ΘkT Θk , τ2,k Inl1 , τ3,k Inl2 , 0, 0}, [ ] Σk ≜ Σk(11) Σk(12) L1 −Hk G1 L2 −Hk Inx , (11)

≜ f (xˆ k ) − Fk xˆ k − Hk G1 h(xˆ k ) + Ek ξk◦ ,

(12)

≜ (Φk − Hk G1 Ψk )Θk ,

Σk Σk

4. Numerical example

with Θk being a factorization of Qk (i.e., Qk = Θk ΘkT ), then xk ∈ E (xˆ k , Qk , nx ) holds for all k ∈ [0, N ]. In other words, the state estimation error ek is confined in the ellipsoid E (0, Qk , nx ) at each time step k over the finite horizon [0, N ]. Proof. See Appendix C. Theorem 3. Given {Fk , Hk }0≤k≤N , {ψk }0≤k≤N and {βk }0≤k≤N . The envelope constraint defined in (18) is achieved if there exist a sequence of positive definite matrices {Qk }0≤k≤N +1 and sequences of positive scalars {λk , τ1,k , τ2,k , τ3,k , τ4,k , τ5,k }0≤k≤N satisfying

[ −Mk Σk

] ΣkT ≤ 0, −Qk+1

⎡ −(Λi βk )2 + λk ⎢ 0 ⎣ −Λi ψk

(24)

−Λi ψk

0

−λk Iϑ Λi Lk Θk

⎤

⎥ ΘkT LTk ΛTi ⎦ ≤ 0 −1

(25)

where Θk is a factorization of Qk (i.e., Qk = Θk ΘkT ). Proof. See Appendix D. 3.3. Filter design Theorem 4. Given γ > 0, Ξ > 0, {ψk }0≤k≤N and {βk }0≤k≤N . The output estimation error z˜k satisfies simultaneously the prespecified H∞ specification and envelope constraint if, under the initial condition {P0 ≤ γ 2 Ξ¯ , Q0 ≥ 0}, there exist sequences of positive definite matrices {Pk , Qk }0≤k≤N +1 , sequences of real-valued matrices {Fk , Hk }0≤k≤N , sequences of positive scalars {εk , ϵ1,k , ϵ2,k , λk , τ1,k , τ2,k , τ3,k , τ4,k , τ5,k }0≤k≤N , sequences of positive scalars {αl,k }0≤k≤N (l = 1, 2, . . . , q) satisfying the following set of recursive linear matrix inequalities

[

[

−αl,k ∗ −Mk Σk

] ϱlT,k G T ≤ 0, −Pk+1 ] ΣkT ≤ 0, −Qk+1

[ ˜k Π ∗

L˜

−J

]

≤ 0,

(26)

(27)

Consider the following univariate non-stationary growth model (UNGM) investigated in Gordon, Salmond, and Smith (1993): ⎧ xk ⎪ ⎪ xk+1 = 0.5xk + 25 ⎪ 1 + x2k ⎪ ⎪

+ 8 cos(1.2 ∗ (k + 1)) + g(xk ) + wk

⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

yk =

x2k

+ vk

20 zk = 0.2xk .

It is worth mentioning that in Gordon et al. (1993), wk and vk are assumed to obey Gaussian distribution with zero mean and known variances. However, in practical engineering, apart from the stochastic noises, sometimes systems might be corrupted by other different kinds of disturbances such as the energy bounded disturbances investigated in this paper. Therefore, in this paper, we assume wk = 0.5e−0.2k sin(k) and vk = 5 cos(2k)/(k + 1), which, obviously, are two energy bounded disturbance sequences. On the other hand, note that in Gordon et al. (1993), the item g(xk ) does not exist. However, in many cases, the system may contaminate with the stochastic nonlinearities owing to a variety of reasons such as random failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. Thus, in order to better reflect the engineering reality and present a comprehensive model, we take into account the stochastic nonlinearity g(xk ) with the form of g(xk ) = 0.06 sign[xk ]xk ςk where ςk is a Gaussian white sequence with unitary covariance. Then it can be easily checked that g(xk ) satisfies

E{g(xk )|xk } = 0, E{g(xk )g T (xk )|xk } = 0.0036xTk xk . In this example, the parameters of the logarithmic quantizer

σ (·) are taken as µ ˆ 0 = 3 and χ = 0.6. Then, it can be obtained that G1 = 0.75 and G2 = 1.25. To study the H∞ performance, in the simulation, choose γ = 1.5 and Ξ = 0.5. Set the initial values by P0 = diag{0, 0.5, 0}, Q0 = I2 and xˆ 0 = 0. According to Theorem 4, the presented

time-varying LMIs can be solved recursively by utilizing Matlab software. The simulation results are presented in Figs. 1–2. Fig. 1 depicts the trajectories of the state xk and its estimate. It can be observed

L. Ma et al. / Automatica 93 (2018) 527–534

531

errors over the specified horizon. By means of recursive matrix inequalities approach, sufficient conditions of the existence of the desired filter have been established guaranteeing simultaneously the pre-specified H∞ criterion and the envelope constraint. An illustrative example has been proposed to show the effectiveness and applicability of the presented filtering scheme. Acknowledgments

Fig. 1. The state xk and its estimate.

This work was supported in part by the National Natural Science Foundation of China under Grants 61773209 and 61525305, the Research Fund for the Taishan Scholar Project of Shandong Province of China, the six talent peaks project in Jiangsu Province under Grant XYDXX-033, the Fundamental Research Funds for the Central Universities under Grant 30916011337, the Postdoctoral Science Foundation of China under Grant 2014M551598, International Postdoctoral Exchange Fellowship from the China Postdoctoral Council, and Alexander von Humboldt Foundation of Germany. Appendix A. Proof of Theorem 1 Proof. The proof can be conducted easily and is therefore omitted here due to the limitation of pages.

Appendix B. Proof of Theorem 2 Proof. Based on Theorem 1, we only need to prove that the matrix inequalities (22) imply the matrix inequality (20). According to Schur Complement Lemma (Boyd, Ghaoui, Feron, & Balakrishnan, 1994), the first inequality in (22) is true if and only if

ϱlT,k G T Pk+1 G ϱl,k ≤ αl,k .

(B.1)

By utilizing the property of matrix trace, we obtain tr G ϱl,k ϱlT,k G T Pk+1 ≤ αl,k .

[

˜◦

Fig. 2. The filtering error zk and its upper and lower bounds.

that all the estimates can track the true value with a satisfactory accuracy, which confirms that the proposed filtering algorithm performs quite well. In order to investigate the envelope constraint criterion, with the given input ξk◦ , the desired output ζk and the tolerance band βk are selected as ψk = −0.25 and βk = 1.95. The simulation result is shown in Fig. 2, where it can be clearly seen that the filtering error output is constrained by the pre-specified bounds. Therefore, it can be summarized that the envelope constraint can be achieved by using the exploited recursive filtering algorithm. 5. Conclusion In this paper, the envelope-constrained H∞ filtering problem has been discussed for a class of discrete time-varying nonlinear systems over a finite horizon. Both deterministic and stochastic nonlinearities are included in the system under consideration. The stochastic nonlinearity described by statistical means is quite general which could encompass certain frequently seen nonlinearities in the literature. An envelope-constrained performance index has been proposed to reflect the transient behavior of the filtering

]

(B.2)

Similarly, it follows from the Schur Complement Lemma (Boyd et al., 1994) that the second linear matrix inequality (22) holds if and only if

[ ˜k −εk Πk + Ω A¯k

A¯kT

]

−Pk−+11

˜ −1 L˜ T ≤ 0, + LJ

(B.3)

≤0

(B.4)

which is equivalent to

[ ˜k −εk Πk + Ω

AkT

−Pk−+11

Ak

]

where

[

Ak ≜ Ak

Ek

Hk .

]

It follows immediately from Schur Complement Lemma (Boyd et al., 1994) and (B.4) that

˜ k + AkT Pk+1 Ak ≤ 0. − εk Πk + Ω

(B.5)

Subsequently, taking (B.2) into consideration, we obtain

− εk Πk + Ωk ≤ 0.

(B.6)

Therefore, according to Theorem 1, we can conclude that the desired H∞ requirement is achieved. The proof is now complete.

532

L. Ma et al. / Automatica 93 (2018) 527–534

Appendix C. Proof of Lemma 1 Proof. Applying the input ξk◦ defined in (17) to the filtering error system (14), we obtain the one-step-ahead state estimation error as follows: ek+1 = f (xˆ k ) − Fk xˆ k − Hk G1 h(xˆ k ) + Ek ξk◦ − Hk ϕk

+ (Φk − Hk G1 Ψk )(xk − xˆ k ) + L1 ∆1 (xk − xˆ k ) − Hk G1 L2 ∆2 (xk − xˆ k ) + g(xk ).

{

}

where Q0 > 0 is any given matrix. Thus, when k = 0, x0 ∈ E (xˆ 0 , Q0 , nx ) is satisfied. Secondly, supposing that xk ∈ E (xˆ k , Qk , nx ) is true at k > 0, we shall demonstrate that the following holds:

{

}

(C.3)

{

}

q ∑

ϱl,k ϱlT,k Υl,k xˆ k + ϑkT ΘkT

+ xˆ Tk

ϱl,k ϱlT,k Υl,k Θk ϑk

l=1

q ∑

ϱl,k ϱlT,k Υl,k Θk ϑk + ϑkT ΘkT

q ∑

ϱl,k ϱlT,k Υl,k xˆ k ,

l=1

l=1

which is equivalent to E{ϖ ˜ ϖk } = 0. T k Λk

We are now in a position to demonstrate that the inequality (C.3) is true for the time instant k > 0. By means of Schur Complement Lemma (Boyd et al., 1994), the inequality (23) is true if and only if

− Mk + ΣkT Qk−+11 Σk ≤ 0,

(C.14)

which implies

{

(C.4)

Consequently, it follows from Ghaoui and Calafiore (2001) that there exists a vector ϑk ∈ Rnϑ (E{ϑkT ϑk } ≤ 1) such that

− τ1,k ϖkT U1 ϖk − τ2,k ϖkT U2 ϖk − τ3,k ϖkT U3 ϖk ˜ k ϖk ≤ 0. − τ4,k ϖkT Ψ˜ k ϖk − τ5,k ϖkT Λ

(C.5)

}

(C.15)

Accordingly, we know that

E ϖkT ΣkT Qk−+11 Σk ϖk ≤ 1

{

xk = xˆ k + Θk ϑk

q ∑

E ϖkT ΣkT Qk−+11 Σk ϖk − ϖkT diag{1, 0, 0, 0, 0, 0}ϖk

Since xk ∈ E (xˆ k , Qk , nx ) is true, we have

E (xk − xˆ k )T Qk−1 (xk − xˆ k ) ≤ 1.

= xˆ Tk

(C.1)

(C.2)

E (xk+1 − xˆ k+1 )T Qk−+11 (xk+1 − xˆ k+1 ) ≤ 1.

E{g T (xk )g(xk )}

l=1

In the following, we are going to prove the lemma by induction. Firstly, for k = 0, it can be known from x0 = 0 and xˆ 0 = 0 that:

E (x0 − xˆ 0 )T Q0−1 (x0 − xˆ 0 ) ≤ 1

Taking the statistical property of the stochastic nonlinearity g(xk ) into consideration, we obtain

}

(C.16)

which indicates that

where Θk is a factorization of Qk , i.e., Qk = Θk ΘkT . Substituting (C.5) into (C.1) yields

E (xk+1 − xˆ k+1 )T Qk−+11 (xk+1 − xˆ k+1 ) ≤ 1.

{

xk+1 − xˆ k+1

}

(C.17)

Thus, the induction is now accomplished and we conclude that xk ∈ E (xˆ k , Qk , nx ) holds for all k ∈ [0, N ]. The proof is now complete.

= f (xˆ k ) − Fk xˆ k − Hk G1 h(xˆ k ) + Ek ξk◦ + (Φk − Hk G1 Ψk )Θk ϑk + L1 ∆1 Θk ϑk − Hk G1 L2 ∆2 Θk ϑk − Hk ϕk + g(xk ).

(C.6)

Next, letting δ1,k ≜ ∆1 Θk ϑk , δ2,k ≜ ∆2 Θk ϑk and denoting a vector as

Appendix D. Proof of Theorem 3

we can rewrite the dynamics of state estimation error (C.6) as follows:

Proof. First, according to Lemma 1, we can know directly from (24) that the one-step-ahead estimation error ek belongs to the ellipsoid E (0, Qk , nx ), and therefore there exists a random vector ϑk (E{ϑkT ϑk } ≤ 1) such that

xk+1 − xˆ k+1 = Σk ϖk .

ek = Θk ϑk ,

[ ϖk ≜ 1

ϑkT

δ1T,k

δ2T,k

]T

g T (xk ) ,

ϕkT

(C.7)

On the other hand, the inequality E{ϑkT ϑk } ≤ 1 can be equivalently expressed in terms of vector ϖk as

E{ϖkT U1 ϖk } ≤ 0

(C.8)

where U1 ≜ diag{−1, Iϑ , 0, 0, 0, 0}. Noting δ1,k = ∆1 Θk ϑk and δ2,k = ∆2 Θk ϑk , we can infer from ∥∆1 ∥ ≤ 1 and ∥∆2 ∥ ≤ 1 that

δ1T,k δ1,k = ϑkT ΘkT ∆T1 ∆1 Θk ϑk ≤ ϑkT ΘkT Θk ϑk , δ δ

T 2,k 2,k

=ϑ

T T T k Θk ∆ 2 ∆ 2 Θk

ϑk ≤ ϑ

T T k Θk Θk

ϑk ,

(C.9) (C.10)

which can be rewritten by

ϖkT U2 ϖk ≤ 0, ϖ

T k U3

(C.11)

ϖk ≤ 0,

where U2 ≜ diag{0, − −ΘkT Θk , 0, Inl2 , 0, 0}.

(C.12)

ΘkT Θk

, Inl1 , 0, 0, 0} and U3 ≜ diag{0,

Likewise, we know from the inequality (9) that:

ϕkT ϕk − ϕkT Gyk ≤ 0 ⇐⇒ ϖkT Ψ˜ k ϖk ≤ 0.

(C.13)

Qk = Θk ΘkT .

(D.1)

Next, it is easy to see that the inequality (18) holds if and only if

( { ◦} )2 E Λi z˜k − Λi ψk ≤ (Λi βk )2 .

(D.2)

]T By defining a new vector ϖ ˜ k ≜ 1 E{ϑkT } , and taking into account the fact that E{ϑkT }E{ϑk } ≤ E{ϑkT ϑk } ≤ 1, we acquire ( { ◦} )2 E Λi z˜k − Λi ψk − (Λi βk )2 ( { } )2 ≤ E Λi z˜k◦ − Λi ψk − (Λi βk )2 + λk − λk E{ϑkT }E{ϑk } ] ([−(Λ β )2 + λ 0 i k k =ϖ ˜ kT 0 −λk Iϑ [ ] ]) −Λi ψk [ − Λ ψ Λ L Θ + ϖ ˜ k. (D.3) i k i k k ΘkT LTk ΛTi [

Consequently, by means of Schur Complement Lemma (Boyd et al., ( {1994), } it can be )2 readily known from the inequality (25) that E Λi z˜k◦ − Λi ψk − (Λi βk )2 ≤ 0 holds for all i ∈ {1, 2, . . . , nz }. Therefore, the envelope constraint defined in (18) is guaranteed at each time step k over the horizon [0, N ]. The proof is now complete.

L. Ma et al. / Automatica 93 (2018) 527–534

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Lifeng Ma received the Ph.D. degree in Control Science and Engineering from Nanjing University of Science and Technology, Nanjing, China, in 2010. From August 2008 to February 2009, he was a Visiting Ph.D. Student in the Department of Information Systems and Computing, Brunel University London, U.K. From January 2010 to April 2010 and May 2011 to September 2011, he was a Research Associate in the Department of Mechanical Engineering, the University of Hong Kong. From March 2015 to March 2017, he was a Visiting Research Fellow at the King’s College London, U.K. He is currently a Professor in the School of Automation, Nanjing University of Science and Technology, Nanjing, China. His current research interests include nonlinear control and signal processing, variable structure control, distributed control and filtering, time-varying systems and multi-agent systems. He has published more than 20 papers in refereed international journals. He serves as an editor for Neurocomputing. He is a very active reviewer for many international journals.

Zidong Wang was born in Jiangsu, China, in 1966. He received the B.Sc. degree in mathematics in 1986 from Suzhou University, Suzhou, China, and the M.Sc. degree in applied mathematics in 1990 and the Ph.D. degree in electrical engineering in 1994, both from Nanjing University of Science and Technology, Nanjing, China. He is currently Professor of Dynamical Systems and Computing in the Department of Information Systems and Computing, Brunel University London, U.K. From 1990 to 2002, he held teaching and research appointments in universities in China, Germany and the UK. Prof. Wang’s research interests include dynamical systems, signal processing, bioinformatics, control theory and applications. He has published more than 300 papers in refereed international journals. He is a holder of the Alexander von Humboldt Research Fellowship of Germany, the JSPS Research Fellowship of Japan, William Mong Visiting Research Fellowship of Hong Kong. Prof. Wang serves (or has served) as the Editor-in-Chief for Neurocomputing, the Deputy Editor-in-Chief for International Journal of Systems Science, and an Associate Editor for 12 international journals, including IEEE Transactions on Automatic Control, IEEE Transactions on Control Systems Technology, IEEE Transactions on Neural Networks, IEEE Transactions on Signal Processing, and IEEE Transactions on Systems, Man, and Cybernetics—Part C. He is a Fellow of the IEEE, a Fellow of the Royal Statistical Society and a member of program committee for many international conferences.

Qing-Long Han received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Sc. and Ph.D. degrees in Control Engineering and Electrical Engineering from East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. From September 1997 to December 1998, he was a Post-doctoral Researcher Fellow with the Laboratoire d’Automatique et d’Informatique Industielle (now renamed as Laboratoire d’Informatique et d’Automatique pour les Systémes), École Supérieure d’Ing’enieurs de Poitiers (now renamed as École Nationale Supérieure d’Ingé nieurs de Poitiers), Université de Poitiers, France. From January 1999 to August 2001, he was a Research Assistant Professor with the Department of Mechanical and Industrial Engineering at Southern Illinois University at Edwardsville, USA. From September 2001 to December 2014, he was Laureate Professor, Associate Dean (Research and Innovation) with the Higher Education Division, and Founding Director of the Centre for Intelligent and Networked Systems at Central Queensland University, Australia. From December 2014 to May 2016, he was Deputy Dean (Research), with the Griffith Sciences, and Professor with the Griffith School of Engineering, Griffith University, Australia. In May 2016, he joined Swinburne University of Technology, Australia, where he is currently Pro Vice-Chancellor (Research Quality) and Distinguished Professor. His research interests include networked control systems, neural networks, time-delay systems, multiagent systems, and complex systems. Professor Han was appointed Chang Jiang (Yangtze River) Scholar Chair Professor by the Ministry of Education, China, in March 2010. He is one of The World’s Most Influential Scientific Minds: 2014, The World’s Most Influential Scientific Minds: 2015, and The World’s Most Influential Scientific Minds: 2016. He is a Highly Cited Researcher in Engineering according to Thomson Reuters. He is a Fellow of Institution of Engineers Australia.

534

L. Ma et al. / Automatica 93 (2018) 527–534 Hak-Keung Lam received the B.Eng. (Hons.) and Ph.D. degrees from Hong Kong Polytechnic University, Hong Kong, in 1995 and 2000, respectively.

and coauthored the monograph entitled Stability Analysis of Fuzzy-Model-Based Control Systems (Springer, 2011). His current research interests include intelligent control systems and computational intelligence.

From 2000 to 2005, he was a Post-Doctoral Fellow and Research Fellow with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University. He joined Kings College London, London, U.K., as a Lecturer, in 2005, where he is currently a Reader. He has coedited the books entitled Control of Chaotic Nonlinear Circuits (World Scientific, 2009) and Computational Intelligence and Its Applications (World Scientific, 2012),

Dr. Lam is an Associate Editor of the IEEE Transactions on Fuzzy Systems, the IEEE Transactions on Circuits and Systems II: Express Briefs, IET Control Theory and Applications, the International Journal of Fuzzy Systems, and Neurocomputing, and a Guest Editor and an Editorial Board Member for a number of international journals. He served as a Program Committee Member and an International Advisory Board Member for various international conferences, and a Reviewer for various books, international journals, and international conferences.