Dynamic output feedback controller design for affine T–S fuzzy systems with quantized measurements

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ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements Huimin Wang a, Guang-Hong Yang a,b,n a b

College of Information Science and Engineering, Northeastern University, Shenyang 110819, PR China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, PR China

ar t ic l e i nf o

a b s t r a c t

Article history: Received 14 December 2015 Received in revised form 4 May 2016 Accepted 7 June 2016 This paper was recommended for publication by Prof. A.B. Rad.

This paper investigates the problem of dynamic output feedback control for afﬁne T–S fuzzy systems with quantized measurements. By using the S-procedure, the unmatched regions between the plant and the controller caused by quantization errors are considered in control synthesis. A novel piecewise dynamic output feedback control design is presented, which reduces the worst case peak output due to quantization errors and guarantees an H 1 -norm bound constraint. In contrast to the existing results, the derived design condition leads to better steady-state performance and less computational burden. Numerical examples are given to show the superiority and effectiveness of the new method. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Afﬁne T–S fuzzy systems Dynamic output feedback control Quantized measurement S-procedure

1. Introduction Since a large number of information processing units need to be applied in modern engineering control systems, signal quantization problem has been becoming an interesting and active research topic. Much attention has been paid to the topic from the engineering control ﬁeld, such as [1–8] and the references therein. Especially, in [2], the authors investigated the quadratic stabilization problem for discrete-time single-input–single-output (SISO) linear time-invariant systems with quantized feedback. It has been proved in [2] that the quantizer needs to be logarithmic for a quadratically stabilizable system, and the coarsest quantization density is given explicitly in terms of the system's unstable poles. Furthermore, the authors in [3] generalized the work in [2] to multi-input–multi-output (MIMO) systems, and solved the control problems by employing a simple sector bound to model the quantization error. The approaches mentioned above provide effective solutions for quantized feedback control problems of linear systems. However, since almost all realistic physical processes exhibit nonlinear dynamics, the research on nonlinear systems has gained lots of interest in past few years. In these studies, fuzzy logic is a powerful tool for the control of complex systems, some signiﬁcant work has been made for fuzzy control of nonlinear systems [9,10]. Especially, an important approach to nonlinear system feedback control is to model the considered nonlinear system as a T–S fuzzy system [11], which has attracted more and more attention [12–20]. Since signal quantization has a signiﬁcant impact on the performance, many results have been obtained for the problems of stability analysis and controller design for T–S fuzzy systems with signal quantization [21–26]. Most of the existing results are derived based on the PDC strategy, i.e., the membership functions of the fuzzy controller/ﬁlter are the same as the ones of the fuzzy plant, such as in [23–25]. One more advantage of sharing the same membership functions in the fuzzy plant and fuzzy controller/ﬁlter is that more powerful relaxation conditions can be employed, such as the techniques proposed in [13,14]. However, when the fuzzy plant has uncertain membership functions, the above cited PDC controller/ﬁlter design methods cannot be employed. In [27–29], the authors proposed novel controller/ ﬁlter design methods based on the non-PDC strategy, i.e., the membership functions of the fuzzy controller/ﬁlter are different from the ones of the fuzzy plant. Although these methods can solve the controller/ﬁlter design problem for the fuzzy systems with uncertain membership functions, the computational burden is heavy for implementing. n

Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang, 110819, PR China. Tel.: þ 086 024 83681939. E-mail addresses: [email protected] (H. Wang), [email protected] (G.-H. Yang).

http://dx.doi.org/10.1016/j.isatra.2016.06.007 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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2

The motivation of this paper is to investigate the output quantized feedback control problem for afﬁne T–S fuzzy systems with normbounded uncertainties, and reduce the conservatism of existing results. First, with the structural information encoded in the fuzzy rules, the output-space is partitioned into operating and interpolation regions. Then, a piecewise dynamic output feedback controller is designed. Due to the existence of the quantization errors between system outputs and quantized outputs, the plant and the controller do not always evolve in the same region at the same time, especially within the neighborhoods of region boundaries. By using the S-procedure, the quantization error information is fully considered, and an H 1 piecewise dynamic output feedback controller design method is derived in the formulation of LMIs. Finally, two examples are given to illustrate the effectiveness and merit of the proposed method. The main contributions of this paper are summarized as follows: (1) The afﬁne T–S fuzzy model is exploited to describe the considered nonlinear system, which has fewer rules and more powerful function approximation capabilities than the linear T–S fuzzy model discussed in [21–25]. (2) The inﬂuence of quantization on premise variables is considered, and the unmatched regions caused by quantization errors are considered in control synthesis. Compared with the results without considering the inﬂuence of quantization on premise variables, the proposed control scheme can guarantee better steady performance when premise variables of fuzzy systems contain quantization signals. Note that, the fuzzy ﬁlter design problem for afﬁne T–S fuzzy systems with measurement errors has been addressed in [28]. However, for the output quantized feedback control problem of afﬁne T–S fuzzy systems, the proposed method in [28] cannot be directly used. (3) A novel piecewise dynamic output feedback controller design method is derived in the form of LMIs, which loosens the restrictive constraint on the input matrices imposed in [26]. This paper is organized as follows: following the introduction, the afﬁne T–S fuzzy model description and the problem statement are presented in Section 2. In Section 3, the output quantized feedback control problem is addressed. Two examples are given in Section 4 to show the effectiveness and merit of the proposed method. Finally, conclusions are drawn in Section 5.

2. System description and problem statement 2.1. System description By choosing system outputs as premise variables, the following uncertain afﬁne T–S fuzzy dynamic model is used to represent a class of complex nonlinear systems 8 r X > > _ ¼ hl ðyðtÞÞððAl þ ΔAl ðtÞÞxðtÞ þ Bl uðtÞ þ Gl wðtÞ þ al þ Δal ðtÞÞ xðtÞ > > > > l¼1 > > > > r X < yðtÞ ¼ hl ðyðtÞÞC l xðtÞ ð1Þ > l¼1 > > > > r > X > > > hl ðyðtÞÞDl xðtÞ > : zðtÞ ¼ l¼1

where r is the number of inference rules; xðtÞ A Rn is the system state vector; uðtÞ A Rm is the control input vector; wðtÞ A Rp is the dis T turbance input which belongs to L2 ½0; 1Þ; yðtÞ ¼ y1 ðtÞ y2 ðtÞ ⋯ ys ðtÞ is the system output vector, which is chosen as the premise variable; P hl ðyðtÞÞ is said to be normalized membership function with satisfying hl ðyðtÞÞ Z 0, rl ¼ 1 hl ðyðtÞÞ ¼ 1; zðtÞ A Rr is the controlled output vector; Al ; Bl ; Gl ; al ; C l ; Dl are constant matrices, and al is an afﬁne term included in each local subsystem, which can approximate the original nonlinear system more accurately [30]; ðΔAl ðtÞ; Δal ðtÞÞ denotes the uncertainty terms of the lth local model, which is assumed to be in the form of ð2Þ ΔAl ðtÞ Δal ðtÞ ¼ W l Δl ðtÞ½U l1 U l2 where Wl, Ul1 and Ul2 are known real constant matrices, and Δl ðtÞΔl ðtÞ r I. In order to facilitate control synthesis for the afﬁne fuzzy system (1), similar to [31], the membership functions of the fuzzy propositions are assumed to be trapezoidal, and the output-space is partitioned into operating regions and interpolation regions. In each operating region, there exists some l such that hl ðyðtÞÞ ¼ 1, and all other membership functions equal to zero and the dynamic of the _ ¼ ðAl þ ΔAl ðtÞÞxðtÞ þ Bl uðtÞ þ Gl wðtÞ þ al þ Δal ðtÞ. In between operating regions, there are interpolation regions where system is given by xðtÞ 0 o hl ðyðtÞÞ o 1, and the system dynamics are given by a convex combination of several afﬁne models. Here, fSi gi A F DRs denotes a polyhedral partition of the outputs, and F denotes the set of region indexes. For each region Si, the set K(i) contains the indexes for the system matrices used in the interpolation within that region. In addition, we let F 0 D F be the set of indexes for regions that contain origin and F 1 D F be the set of indexes of the regions that do not contain the origin for short. Hence, in each region, the global system in (1) can be expressed with a blending of m A KðiÞ subsystems 8 b b b b b b _ > > < xðtÞ ¼ ðA i þ ΔA i ðtÞÞxðtÞ þ B i uðtÞ þ G i wðtÞ þ a i þ Δa i ðtÞ b i xðtÞ ð3Þ yðtÞ ¼ C > > : zðtÞ ¼ D b i xðtÞ; yðtÞ A S i : T

where b ¼ A i

X

hm ðyðtÞÞAm ;

m A KðiÞ

bi ¼ a

X

hm ðyðtÞÞam ;

X

hm ðyðtÞÞΔAm ðtÞ;

m A KðiÞ

X

Δabi ðtÞ ¼

m A KðiÞ

bi ¼ B

X

ΔAb i ðtÞ ¼

hm ðyðtÞÞΔam ðtÞ;

m A KðiÞ

hm ðyðtÞÞBm ;

m A KðiÞ

bi ¼ G

X

hm ðyðtÞÞGm ;

m A KðiÞ

Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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Fig. 1. Logarithmic quantizer (shown for positive input values only).

bi ¼ C

X

hm ðyðtÞÞC m ;

m A KðiÞ

X

bi ¼ D

hm ðyðtÞÞDm ;

m A KðiÞ

X

hm ðyðtÞÞ 4 0;

hm ðyðtÞÞ ¼ 1:

m A KðiÞ

Similar to [32], for yðtÞ A S i , the quadratic inequality is constructed as follows, which will be used in control synthesis for the S-procedure: b T T iq C b i xðtÞ þ 2uT C b F iq ðyðtÞÞ ¼ yT ðtÞT iq yðtÞ þ 2uTiq yðtÞ þ viq ¼ xT ðtÞC i iq i xðtÞ þviq r0;

q ¼ 1; 2; …; s:

ð4Þ

Remark 1. The uncertain afﬁne T–S fuzzy system (1) can be used to represent a class of nonlinear systems considering plant modeling error and nonlinear uncertainty structure for each individual rule. The similar nonlinear modeling via uncertain fuzzy systems can also be seen in the literatures, e.g., [26,28,29]. To avoid unnecessarily complicated notations, in this paper, we only consider the uncertainty terms appearing in the matrices Al and al. Nevertheless, the approaches that are to be developed in this paper can be easily extended to the case when the uncertainty terms also exist in the matrices Gl, Dl. 2.2. Problem statement In this paper, we consider the dynamic output feedback control using quantized outputs, that is T yq ðtÞ ¼ Q½yðtÞ ¼ Q½y1 ðtÞ Q½y2 ðtÞ ⋯ Q½ys ðtÞ

ð5Þ

where Q½ is chosen as a symmetric, static, and time-invariant logarithmic quantizer. For each output channel, the associated set of quantization levels is expressed as n o ð6Þ V j ¼ 7 vij : vij ¼ ρij v0j ; i ¼ 7 1; 72; … [ f 7v0j g [ f0g; 0 o ρj o 1; 1 r j rs: where ρj A ð0; 1Þ denotes the quantization density. A small ρj corresponds to a coarse quantizer, while a large ρj means a dense quantizer. Each of the quantization level vij corresponds to a segment of the measured output yj(t) such that the quantizer maps the whole segment of the measured output yj(t) to this quantization level. In addition, these segments form a partition of R, i.e., they are disjoint, and their union equals to R. For the jth output channel, the associated logarithmic quantizer Q½ is depicted in Fig. 1 and deﬁned as follows, with 1ρ δj ¼ 1 þ ρjj : 8 ρij v0j ρij v0j > i > if 1 þ > ρj v0j δj oyj ðtÞ r 1 δj ; < Q½yj ðtÞ ¼ 0 ð7Þ if yj ðtÞ ¼ 0; > > > : Q½ y ðtÞ if y ðtÞ o 0: j

j

It is shown from Fig. 1 that, the quantization error between yj(t) and Q½yj ðtÞ can be expressed as δj yj ðtÞ r Q½yj ðtÞ yj ðtÞ r δj yj ðtÞ;

1 rj r s:

ð8Þ

Further, we have yq ðtÞ ¼ Q½yðtÞ ¼ ðI þ Π ΛðtÞÞyðtÞ

ð9Þ

where Π ≔diagfδ1 ; δ2 ; …; δs g, Λ ðtÞΛðtÞ r I. T

Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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Fig. 2. Dynamic output feedback control of afﬁne T–S fuzzy systems with quantized measurements.

Fig. 3. Regions of y(t). (For interpretation of the references to color in this ﬁgure caption, the reader is referred to the web version of this paper.)

The control problem of fuzzy dynamic systems with output quantization is shown in Fig. 2, and the dynamic output feedback controller is designed to have the following form: 8 b b_ > > < x ðtÞ ¼ AFi x ðtÞ þ BFi yq ðtÞ þaFi ; b ð10Þ uðtÞ ¼ C Fi x ðtÞ þ σ Fi ; > > : zF ðtÞ ¼ DFi b x ðtÞ; yq ðtÞ A S i where b x ðtÞ A Rn is the estimated state vector; AFi ; BFi, aFi, CFi,

σFi, DFi are the controller gains to be determined.

Remark 2. Due to the existence of the quantization errors between system outputs y(t) and quantized outputs yq(t), the plant and the controller do not always evolve in the same region at the same time, especially within the neighbourhoods of region boundaries. An example of membership functions and the associated partitioning for an afﬁne fuzzy system with a single system output is shown in Fig. 3. For this example, when the controller lies in the boundary of region S 1 (the red part), the plant maybe lies in its neighbourhood S 2 (the green part). For S i , S ni denotes the set of the cells adjacent to S i , and ni denotes the corresponding set of indexes of these cells. For yq ðtÞ A S i , S ðni ;δi Þ denotes the regions in S ni that y(t) may lie, which can be concluded according to the quantization error bound in (8). Further, S ki denotes the set of S ðni ;δi Þ ⋃S i . Applying the controller (10) to the system (3), the augmented system is described as 8 < ξ_ ðtÞ ¼ Aeik ξðtÞ þGek wðtÞ þ aeik i i i ð11Þ : zðtÞ ¼ Deiki ξðtÞ; for yq ðtÞ A S i ; yðtÞ A S ki : where ξðtÞ ¼ ½xT ðtÞ xF ðtÞT T , zðtÞ ¼ zðtÞ zF ðtÞ, and 2 3 " # b þ ΔA b ðtÞ b k C Fi A b B ki ki G i ki 4 5 ; Aeiki ¼ ; Geki ¼ b BFi ðI þ ΠΛðtÞÞC AFi 0 ki " aeiki ¼

b k σ Fi bki ðtÞ þ B bki þ Δa a i aFi

# ;

h i b k DFi : Deiki ¼ D i

Based on the discussion in Remark 2, the augmented system described in (11) can be classiﬁed into the following two situations. Situation 1: (_ ξ ðtÞ ¼ Aei ξðtÞ þ Gei wðtÞ þaei ð12Þ zðtÞ ¼ Dei ξðtÞ; for yq ðtÞ A S i ; yðtÞ A S i :

Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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Situation 2: 8 < ξ_ ðtÞ ¼ Aein ξðtÞ þ Gen wðtÞ þ aein i i i : zðtÞ ¼ Deini ξðtÞ;

for yq ðtÞ A S i ;

5

ð13Þ

yðtÞ A S ðni ;δi Þ :

where "

b i C Fi B

#

"

b i σ Fi bi ðtÞ þ B bi þ Δa a

#

"

bi G

#

; aei ¼ ; Gei ¼ ; bi aFi BFi ðI þ ΠΛðtÞÞC AFi 0 2 3 " # b n ðtÞ b n þ ΔA b n C Fi b B A b b i i i 5; aein ¼ a ni þ Δa ni ðtÞ þ B ni σ Fi ; ¼4 i bn aFi BFi ðI þ ΠΛðtÞÞC AFi i

Aei ¼

Aeini

b þ ΔA b ðtÞ A i i

" Geni ¼

bn G i 0

# ;

h i b i DFi ; Dei ¼ D

h i b n DFi : Deini ¼ D i

In this paper, we will design a piecewise dynamic output feedback controller in the form of (10), such that

The augmented system (11) is asymptotically stable when wðtÞ 0. The L2-norm of the operator from the disturbance w(t) to the estimated output error zðtÞ is less than γ under zero initial conditions Z

1 0

z T ðtÞzðtÞdt r γ 2

Z

1

wT ðtÞwðtÞdt

ð14Þ

0

for all nonzero wðtÞ A L2 ½0; 1Þ.

3. Output quantized feedback controller design 3.1. Performance analysis Firstly, we will provide an H 1 performance analysis for the corresponding augmented system (11) described as follows. Lemma 1. For a prescribed H 1 performance index γ 4 0, if there exist a matrix P ¼ P T 4 0, and a set of scalars τiq 40, τini q 4 0 (i A F, q ¼ 1; 2; …; s), such that " T # Aei P þ PAei þ DTei Dei PGei o0 ð15Þ GTei P γ2 I for i A F 0 , and 2

s X T T τiq X T1 Cb i T iq Cb i X 1 þ DTei Dei 6 Aei P þ PAei 6 q¼1 6 6 6 GTei P 6 6 s X 6 4 τiq uTiq Cb i X 1 aTei P

PGei

Paei

s X q¼1

γ I 2

0

q¼1

for i A F 1 , and 2 s X T T τini q X T1 Cb ni T ini q Cb ni X 1 þ DTeini Deini 6 Aeini P þ PAeini 6 q¼1 6 6 6 GTeni P 6 6 s X 6 4 τini q uTini q Cb ni X 1 aTeini P

3

T

τiq X T1 Cb i uiq 7

7 7 7 7o0 7 7 7 5

0 s X

τiq viq

ð16Þ

q¼1

s X

PGeni Pa eini

q¼1

γ2 I 0

q¼1

T

3

τini q X T1 Cb ni uini q 7

s X

0

τini q vini q

7 7 7 7o0 7 7 5

ð17Þ

q¼1

for i A F. The matrices T ini q , uini q , vini q are constructed according to the region yðtÞ A S ðni ;δi Þ . Then, the augmented system (11) is asymptotically stable with a given H 1 norm bound γ. Proof.. See the part I of Appendix.□ 3.2. Piecewise dynamic output feedback controller design Due to the S-procedure, it is difﬁcult to convert the non-convex conditions in Lemma 1 to convex ones by the equivalent transformation. To obtain convex controller design conditions, the following assumption is considered: Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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6

Assumption 1. Assume that Bl (l ¼ 1; 2; …; r) are of full column rank, and let invertible matrices Tl be with T l Bl ¼

I mm 0

:

ð18Þ

Remark 3. Assumption 1 is commonly used in the literatures, e.g., [26,33]. For each Bl, the corresponding Tl generally is not unique. A special Tl can be obtained by " Tl ¼ and

P

ðBTl Bl Þ 1 BTl P

#

is a matrix composed of the rows which are mutually independent and perpendicular to the columns of Bl.

To facilitate controller design, the following lemmas are presented. Lemma 2 ([34]). Let matrices M ¼ M T ; S; N; and ΔðtÞ be real matrices of appropriate dimensions, the inequality M þ SΔðtÞN þ N T Δ ðtÞST o0 T

ð19Þ

holds for all Δ ðtÞΔðtÞ rI if and only if for some positive scalar ϵ 40 such that T

"

M þ ϵN T N ST

#

S ϵI

o0

Lemma 3 ([35]). Let

ð20Þ

Φ be a symmetric matrix, P be a positive-deﬁnite matrix. The following statements are equivalent:

(a) There exists a matrix P 4 0, such that Φ þ PA þAT P o 0. (b) For a given constant α 4 0, there exist matrices F, P 40 such that "

#

Φ 2α P P þ ðA þ αIÞT F o0: F FT P þF T ðA þ αIÞ

ð21Þ

Based on Lemmas 1–3, the following theorem can be used to solve the H1 dynamic output feedback controller design problem. h i 4 0, Theorem 1. For a prescribed H 1 performance index γ 4 0 and positive scalars αi, βi, γi, if there exist matrices P ¼ P T ¼ PPT11 PP 12 22 12 h i F i111111 0 F i1111 ¼ F i111121 F i111122 , Fi1122, Fi211, Fi212, Fi22, Fi23, Fi311, Fi312, Fi32, Fi33, A Fi , B Fi , a Fi , C Fi , σ Fi , DFi, and positive scalars τiq, τini q , ϵim such that 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

2αi P 11 þ ϵim ðU Tm1 U m1 þ C Tm C m Þ 2α

n

2αi P 22

T i P 12

n

n

n

n

n

n

n

n

n

n

n

n

n 7

ðγ 2 þ2αi ÞI

n

n

n

n

n

Ξ im55 Ξ im65

n

n

n

n

Ξ im66

n

n

n

n

F i211

F i212

F Ti22 F i22

n

n

n

0

ϵim I

n

n

0 0

0 0

ϵim I 0

I

0

0

Ξ im51 Ξ im61

Ξ im52 Ξ im62

0

0

Ξ im53 Ξ im63 I þ αi Y Ti22

0

0

0

0 Dm

3

n

0 DFi

T T W Tm T Tm Y i1111 β i W m T m Y i1111

γiΠ

0 0

T

T B Fi

Π

T

T B Fi

0

0

7 7

n 7 7

7 7 7 7o0 7 7 7 7 7 7 7 5

n 7

n

ð22Þ

for i A F 0 , m A KðiÞ, and 2

Ξ im11

6 2α P T i 12 6 6 6 0 6 6 Ξ im41 6 6 6 Ξ im51 6 6 Ξ 6 im61 6 6 0 6 6 0 6 6 6 0 6 6 6 0 4 Dm

n

2αi P 22 0

n

n

n

n

ðγ 2 þ 2αi ÞI

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n 7

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

Y i22 Y Ti22

n

n

n

n

Y i32 Y Ti23

F Ti33 F i33

n

n

n

0

0

ϵim I

n

n

n

n 7 7

0

0

Ξ im52 Ξ im62 0

Ξ im53 Ξ im63 I þ αi Y Ti22 αi Y Ti23

Ξ im44 Ξ im54 Ξ im64 αi Y Ti32 I þ αi Y Ti33

F i311

F i312

0

0

0

W Tm T Tm Y i1111

βi W Tm T Tm Y i1111

0

0

0

γ i Π T B Fi

Π T B Fi

0

0

0

ϵim I

n

DFi

0

0

0

0

0

0

0

0

I

0

3

n

n

n

Ξ im55 Ξ im65

n

Ξ im66

F i211

F i212

T

T

7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 5 ð23Þ

Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

H. Wang, G.-H. Yang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

for i A F 1 , m A KðiÞ, and 2 0

Ξ im11

6 6 2αi P T12 6 6 0 6 6 6 Ξ 0im41 6 6 6 Ξ im51 6 6 Ξ im61 6 6 0 6 6 6 0 6 6 6 0 6 6 6 0 4 Dm

7

3

n

n

n

n

n

2αi P 22

n

n

n

n

n

n

n

n

ðγ 2 þ 2αi ÞI

n

n

n

n

n

n

n

n

n

n

n

n 7

0 0

0

Ξ im52 Ξ im62

Ξ im53 Ξ im63 I þ αi Y Ti22 αi Y Ti23

0 0 0

Ξ Ξ im54 Ξ im64 αi Y Ti32 I þ αi Y Ti33

0

0 DFi

0 im44

0

0 0

n 7 7

n

n

Ξ im55 Ξ im65

n

n

n

n

n

n

Ξ im66

n

n

n

n

n

n

n

n

n

n

F i211

F i212

Y i22 Y Ti22

n

n

n

n

F i311 W Tm T Tm Y i1111 T T B Fi i

F i312

Y i32 Y Ti23

F Ti33 F i33

n

n

n

0

0

ϵim I

n

n

0

0

0

ϵim I

n

0

0

0

0

I

γΠ

0 0

n

β

T T i W m T m Y i1111 T T B Fi

Π

0

0

7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 5 ð24Þ

for i A F, m A Kðni Þ, where

Ξ im11 ¼

s X q¼1

Ξ im41 ¼

s X

s X

τini q C Tm T ini q C m 2αi P 11 þ ϵim ðU Tm1 U m1 þ C Tm C m Þ;

q¼1

τiq uTiq C m þ ϵim U Tm2 U m1 ; Ξ 0im41 ¼

q¼1

Ξ 0im44 ¼

s X

τiq C Tm T iq C m 2αi P 11 þ ϵim ðU Tm1 U m1 þ C Tm C m Þ; Ξ 0im11 ¼ s X q¼1

τini q uTini q C m þ ϵim U Tm2 U m1 ; Ξ im44 ¼

s X

τiq viq þ ϵim U Tm2 U m2 2αi I;

q¼1

"

τini q vini q þ ϵim U Tm2 U m2 2αi I; Ξ im51 ¼ P 11 þ Y Ti1111 T m Am þ γ i B Fi C m þ αi Y Ti1111 T m ; Ξ im52 ¼ P 12 þ

q¼1

Ξ im53 ¼ Y Ti1111 T m Gm þ αi Y Ti211 ; Ξ im54 ¼ Y Ti1111 T m am þ

σ Fi 0

þ γ i a Fi þ αi Y Ti311 ; Ξ im55 ¼ T Tm Y i1111 Y Ti1111 T m ; "

Ξ im61 ¼ P T12 þ βi Y Ti1111 T m Am þ B Fi C m þ αi βi Y Ti1111 T m ; Ξ im62 ¼ P 22 þ βi Ξ im64 ¼ βi Y Ti1111 T m am þ βi

σ Fi 0

# C Fi þ γ i A Fi þ αi γ i Y Ti1122 ; 0

C Fi 0

# þ A Fi þ αi Y Ti1122 ; Ξ im63 ¼ βi Y Ti1111 T m Gm þ αi Y Ti212 ;

þa Fi þ αi Y Ti312 ; Ξ im65 ¼ γ i Y i1122 βi Y Ti1111 T m ; Ξ i66 ¼ Y i1122 Y Ti1122 :

Then, the augmented system (11) is asymptotically stable with a given H 1 norm bound γ, and the controller gains are given by AFi ¼ ðY Ti1122 Þ 1 A Fi ;

BFi ¼ ðY Ti1122 Þ 1 B Fi ;

C Fi ¼ ðY Ti111111 Þ 1 C Fi ;

ðY Ti1122 Þ 1 A Fi ;

BFi ¼ ðY Ti1122 Þ 1 B Fi ;

C Fi ¼ ðY Ti111111 Þ 1 C Fi ;

AFi ¼

aFi ¼ ðY Ti1122 Þ 1 a Fi ;

σ Fi ¼ ðY Ti111111 Þ 1 σ Fi ;

DFi ;

for i A F 1 :

DFi ; for i A F 0 ;

ð25Þ

Proof.. See the part II of Appendix.□ Remark 4. In [23–25], the authors discussed the controller/ﬁlter design problem for T–S fuzzy systems with quantized measurements. It should be pointed out that all of the above mentioned results are obtained based on the PDC strategy, and the inﬂuence of quantized measurements on premise variables is not considered. In contrast, by using the S-procedure, Theorem 1 considers the unmatched regions caused by the quantization errors between system outputs and quantized outputs in the controller design. Furthermore, to obtain convex controller design conditions, the slack variables Fi are introduced to eliminate the couplings between the system matrices Am(Bm, or am) and the Lyapunov matrix P. The relaxation of the Lyapunov matrix leads to less restrictive synthesis conditions than [33] where the Lyapunov matrix is constrained as a diagonal structure. Similar to [23–25], if the inﬂuence of quantized measurements on premise variables is not considered, based on Theorem 1, the controller design conditions are obtained as follows. h i Corollary 1. For a prescribed H 1 performance index γ 4 0 and positive scalars αi, βi, γi, if there exist a symmetric matrix P ¼ PP T11 PP12 4 0, 22 h i 12 F i111111 0 appropriate dimension matrices A Fi , B Fi , a Fi , C Fi , σ Fi , DFi, F i1111 ¼ F i111121 F i111122 , Fi1122, Fi211, Fi212, Fi22, Fi23, Fi311, Fi312, Fi32, Fi33, and positive scalars

τiq, ϵim such that the inequalities (22) and (23) hold for i A F, m A KðiÞ. Then, the augmented system (11) is asymptotically stable with a given H1 norm bound γ, and the controller gains are given in (27).

Remark 5. The results given in this paper are obtained with the assumption that the input matrices Bl s are of full column rank. As stated in [35], if there exist invertible matrices T l ; l ¼ 1; 2; …r, such that I nc nc 0 ; for l ¼ 1; 2; …r: T l Bl ¼ 0 0 where nc o m, which means that the input matrices are not full column rank, and then a novel piecewise dynamic output feedback Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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8

controller is designed with the form: 8 b > x ðtÞ þ BFi yq ðtÞ þaFi ; x_ ðtÞ ¼ AFi b > < uðtÞ ¼ C~ Fi b x ðtÞ þ σ~ Fi ; > > : z ðtÞ ¼ D x F Fi bðtÞ; yq ðtÞ A S i h nc i h nc i C σ where C~ Fi ¼ 0Fi , σ~ Fi ¼ 0Fi . The corresponding dynamic output feedback control synthesis method is proposed as follows.

ð26Þ

h i 4 0, Corollary 2. For a prescribed H 1 performance index γ 4 0 and positive scalars αi, βi, γi, if there exist a symmetric matrix P ¼ PP T11 PP 12 h i 12 22 0 , F , F , F , F , F , F , F , F , appropriate dimension matrices A Fi , B Fi , a Fi , C Fi A Rnc n , σ Fi A Rnc 1 , DFi, F i1111 ¼ FF i111111 , F i1122 i211 i212 i22 i23 i311 i312 i32 i33 and i111121 F i111122

positive scalars τiq, τini q , ϵim such that the inequalities (22)–(24) hold for iA F, m A KðiÞ. Then, the augmented system (11) is asymptotically stable with a given H 1 norm bound γ, and the controller gains are given by AFi ¼ ðY Ti1122 Þ 1 A Fi ;

BFi ¼ ðY Ti1122 Þ 1 B Fi ;

C nFic ¼ ðY Ti111111 Þ 1 C Fi ;

AFi ¼ ðY Ti1122 Þ 1 A Fi ; aFi ¼ ðY Ti1122 Þ 1 a Fi ;

BFi ¼ ðY Ti1122 Þ 1 B Fi ; T nc 1 Fi ; Fi ¼ ðY i111111 Þ

C nFic

σ

σ

DFi ; for i A F 0 ;

¼ ðY Ti111111 Þ 1 C Fi ;

DFi ;

for i A F 1 :

ð27Þ

Remark 6. Note that the conditions given in Theorem 1, Corollaries 1 and 2 involve some tuning parameters: αi, βi, γi. When these parameters are given, the conditions become convex and can be solved with the LMI Toolbox. As stated in [26,36,37], a simple way to address the tuning issue is the trial-and-error method, and another way is to apply some numerical optimization search algorithms under a set of initial scaling parameters (such as genetic algorithm or the program fminsearch in the optimization toolbox of MATLAB). Remark 7. In [26], the authors considered the output feedback control problem for afﬁne fuzzy systems with quantized measurements. The transitions of the state variables and their estimations could happen among all regions, and the controller design result is under heavy computations. In addition, the results are under the assumption that the input matrix B is a common one, but the convexifying techniques are inefﬁcient for the systems with uncommon input matrices Bl. In this paper, the system outputs are chosen as premise variables of the fuzzy plant, and a piecewise dynamic output feedback controller is designed with the quantized outputs. Inspired by [35], with slack variables Fi, the proposed controller design method can be applied to the systems with uncommon input matrices Bl. Denote n0 as the number of operating regimes, n1 as the number of interpolation regimes. Table 1 is given to show the number of needed LMI constraints on different methods. It can be seen from Table 1 that, the proposed method needs less LMI constraints than [26].

4. Simulation examples 4.1. Example 1 In this subsection, the following afﬁne T–S fuzzy system is used to show the effectiveness and merit of the proposed control design method. Plant Rule Rl: IF y1 ðtÞ is Fl, then 8 _ > < xðtÞ ¼ Al xðtÞ þ Bl uðtÞ þ al þ Gl wðtÞ yðtÞ ¼ C l xðtÞ > : zðtÞ ¼ D xðtÞ; l A f1; 2; 3g l

where

2

3 0:3057 0:5371 1 6 7 0:4755 5; A2 ¼ A3 ¼ 4 0:0414 1:2015 0:1103 0 1:5312 2 3 0:2 1 0 0 6 7 ; Dl ½2 1 0; Gl 4 0:1 5; C l 0 1 0:5 0

2

3 2 1:2560 0 6 7 6 a2 ¼ a3 ¼ 4 0:5627 5; A1 ¼ 4 0:1217 0:0145 0

5:0523

1:2154

2:1210

0:5038

a

1:2554 þ b

3 7 5;

and the membership functions Fl are shown in Fig. 4. The parameters a (0 ra r5) and b (0 rb r 1) in A1 are assigned values in a prescribed grid to check the feasibility of the associated control problem under different approaches. For comparison, the input matrix Bl is assumed as the following two cases: Case 1: Case 2:

The input matrix Bl is a common one with Bl ½1 0:5 0T . The input matrix Bl is an uncommon one with B1 ¼ ½2 0:5 0T , B2 ¼ B3 ¼ ½1 0:5 0T .

Choosing the disturbance input as wðtÞ ¼ 3 sin ð2t þ 1Þ, and the quantization density ρl 0:6. For Case 1, Fig. 5 is given to show the stabilization regions obtained respectively using the control design method of [26] and the method proposed in Theorem 1 of this paper. Δ Table 1 LMI numbers under different methods. Methods

The number of LMIs

Theorem 1 in this paper Theorem 3.1 in [26]

5n0 þ 4n1 n20 þ 4n0 n1 þ 4n21

Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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Fig. 4. Membership functions in Example 1.

▵

Fig. 5. Stabilization regions by respectively the methods in [26] ( ) and this paper (

▵; ○) under Case 1.

Table 2 Comparison of condition number under Case 1. Theorem 1 in this paper

Theorem 3.1 in [26]

23

49

indicates stabilization region by [26] and this paper; ○denotes parameter values that provide stabilization region according to this paper, but not according to [26]. It is apparent that our stabilization region is larger than that according to [26]. Furthermore, Table 2 shows the needed LMI constraints under the method in Theorem 1 of this paper and the method of [26]. For Case 2, due to the uncommon input matrix Bl, the approaches in [26] cannot be applied for this class of systems. For comparison, the method presented in Theorem 1 is compared with the result in Corollary 1, where the dynamic output feedback controller is obtained without considering the inﬂuence of quantized measurements on premise variables as in [23-25]. Fig. 6 is given to show the values of zðtÞ based on the proposed methods in Theorem 1 and Corollary 1 under a ¼0, b ¼1. It clearly shows that the controller derived in Theorem 1 can guarantee a better steady performance than the one in Corollary 1. 4.2. Example 2 Consider a highly nonlinear CSTR model [38,39], with constant liquid volume, the CSTR for an exothermic irreversible reaction A-B is described by the following dynamic model based on a component balance for reactant A and on an energy balance: 2 3 " # " # F ΔH E UA UA F V ðT fs x1 ðtÞÞ þ ρC p k0 exp Rx1 ðtÞ x2 ðtÞ V ρC p x1 ðtÞ 6 7 V ρC p _ ¼4 xðtÞ ð28Þ uðtÞ þ V wðtÞ 5þ F E 0 0 V ðC Af x2 ðtÞÞ k0 exp Rx ðtÞ x2 ðtÞ 1

where x1 ðtÞ is the reactor temperature, x2 ðtÞ is the concentration of A in the reactor, and uðtÞ is the temperature of the coolant stream. The Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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10 3 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

0

2

4

6

8

10

12

14

16

18

20

Fig. 6. The values of zðtÞ respectively by Theorem 1 and Corollary 1 under Case 2.

Table 3 Nominal operating conditions for the CSTR. Variable

Value

Variable

Value

F CAf

100 L/min 1 mol/L

E/R k0

8750 K

Tfs

350 K

UA

V ρ Cp ΔH

100 L 1000 g/L 0.239 J/gK

Tc CA T

5 104 J=minK 300 K 0.5 mol/L 350 K

7:2 1010 1=min

5 104 J=mol

objective is to control x1 ðtÞ by manipulating uðtÞ. The nominal operating conditions and plant parameters are given in Table 3, which corresponds to an unstable steady state. xo2 ¼ ½350; 0:5T is the equilibrium point of this system. The working range of state is x1 ðtÞ A ½300; 400. Next, we linearize the system around xo1 ¼ ½324:4; 0:875T , xo2 ¼ ½350; 0:5T , xo3 ¼ ½370:65; 0:2T at steady input uo ¼ 300 K. With the change of coordinates xðtÞ≔xðtÞ xo2 , uðtÞ≔uðtÞ uo , the following uncertain afﬁne T–S fuzzy model is designed to represent the nonlinear system (28): Plant Rule Rl: IF x1 ðtÞ is Fl, then 8 _ > < xðtÞ ¼ ðAl þ ΔAl ðtÞÞxðtÞ þBl uðtÞ þ al þ Δal ðtÞ þ Gl wðtÞ yðtÞ ¼ C l xðtÞ ð29Þ > : zðtÞ ¼ D xðtÞ; l A f1; 2; 3g l where

4:3790 209:1908 0:9757 29:0894 7:6367 842:2443 ; A2 ¼ ; A3 ¼ ; 0:0357 1:999 0:0101 1:1390 0:0513 5:0259 35:8862 94:9760 0 2:0921 1 a1 ¼ ; a2 ¼ ; a3 ¼ ; Bl ; Gl ; 0:1682 0:4488 0 0 0 0:5 0:03 C l ½1 0; Dl ½1 0; W 1 ¼ ; W2 ¼ W3 ¼ ; U l1 ½0:2 0:1; U l2 0:2; 0:1 0:05 A1 ¼

and the membership functions are shown in Fig. 7. The disturbance w(t) is assumed to be an unknown magnitude-bounded signal as shown in Fig. 8, whose magnitude is 1. The quantization density is chosen as ρl 0:8. The objective is to design a piecewise dynamic output feedback controller in the form of (10) such that the resulting closed-loop system is asymptotically stable with H 1 performance γ. Applying Theorem 1, we can get a piecewise dynamic output feedback controller while the scaling parameters are set as αi ¼ β i ¼ γ i 1. Due to the space limitation, the controller gains are omitted here. Simulations have been carried out to show the effectiveness of the obtained results. Fig. 9 is given to show the state responses of the corresponding closed-loop nonlinear system with the piecewise dynamic output feedback controller proposed in Theorem 1 under initial conditions xð0Þ ¼ ½0:5 0:5T . As shown in the picture, we can also know the effectiveness of the proposed piecewise dynamic output feedback control design. Moreover, the relations between the quantization density ρ and the minimal H 1 performance index γ under Theorem 1 are listed in Table 4, which shows that the more coarsely we quantize the measurement signal, the worse H 1 performance we would obtain. Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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11

Fig. 7. Membership functions in Example 2. 1.5

1

0.5

0

−0.5

−1

−1.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4

4.5

5

Fig. 8. Disturbance input w(t) in Example 2.

3

2.5 0.1

2

0.05 1.5

0 −0.05

1

−0.1 0.5

2

2.2

2.4

0

−0.5

−1

0

0.5

1

1.5

2

2.5

3

3.5

Fig. 9. State responses of the closed-loop system under ρ ¼ 0:8 in Example 2.

Table 4 Relation between the quantization density ρ and H 1 index γ in Example 2. ρ γ

0.9 0.0011

0.8 0.0024

0.7 0.0040

0.6 0.0060

0.5 0.0089

Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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5. Conclusion This paper investigates the problem of dynamic output feedback control for afﬁne T–S fuzzy systems with quantized measurements. First, system outputs are chosen as premise variables of the afﬁne fuzzy model. Then, with the structural information encoded in the fuzzy rules, the afﬁne fuzzy model is represented by multiple operating-regime-based models, and a piecewise dynamic output feedback controller is designed. The quantization errors between the system outputs and the quantized outputs are considered, as a result, the plant and the controller do not always evolve in the same region at the same time, especially within the neighborhoods of region boundaries. By using the S-procedure and introducing slack matrix variables, an H1 dynamic output feedback controller design method is obtained in the form of LMIs. Finally, two examples are used to illustrate the validity and effectiveness of the proposed method. Although this paper addresses the output quantized feedback problem for afﬁne T–S fuzzy systems, in practice, the control signal is usually quantized and sent to the plant. Therefore, extensions of the proposed method to the input quantized feedback control synthesis deserve further investigation.

Acknowledgements This work was supported in part by the Funds of National Science of China (Grant nos. 61273148, 61420106016, 61403070, 61273155, 61322312), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01), the Fundamental Research Funds for the Central Universities (Grant no. N140405006), and the Fok Ying Tung Education Foundation (Grant no. 141060).

Appendix A A.1. Part I–Proof of Lemma 1 Consider a candidate Lyapunov function as VðtÞ ¼ ξ ðtÞP ξðtÞ T

Here, without loss of generality, we just consider the proof for Situation 2, and the other can be obtained similarly. Based on the above Lyapunov function, we get T T T V_ ðtÞ þz T ðtÞzðtÞ γ 2 wT ðtÞwðtÞ ¼ ξ ðtÞðPAeini þ ATeini PÞξðtÞ þ 2ξ ðtÞðGeni wðtÞ þ aeini Þ þ ξ ðtÞDTeini Deini ξðtÞ γ 2 wT ðtÞwðtÞ 3 2 T T 2 3 2 3 ξðtÞ T 6 PAeini þAeini P þ Deini Deini PGeni Paeini 7 ξðtÞ 6 6 7 6 7 GTeni P γ2 I 0 7 ¼ 4 wðtÞ 5 6 74 wðtÞ 5 5 4 T 1 1 0 0 aein P i

Note that xðtÞ ¼ ½I 0ξðtÞ ¼ X 1 ξðtÞ. Then, considering the partition information of the region S ðni ;δi Þ , for τini q 4 0, we have 3 2 s s X X T T T T τ X C T C X 0 τ X C u ini q 1 ni ini q ni 1 ini q 1 ni ini q 72 2 3 6 3 7 ξðtÞ ξðtÞ T 6 q¼1 7 6 q¼1 76 6 7 6 7 0 0 0 74 wðtÞ 5 Z 0 4 wðtÞ 5 6 7 6 s s X X 7 6 1 4 τini q uTini q C ni X 1 0 τini q vini q 5 1 q¼1

q¼1

Then, with the S-procedure and the inequality (17), it follows: V_ ðtÞ þ z T ðtÞzðtÞ γ 2 wT ðtÞwðtÞ r 0 Integrating both sides of (30) yields the following: Z Z 1 ðV_ ðtÞ þ z T ðtÞzðtÞ γ 2 wT ðtÞwðtÞÞdt ¼ V ð1Þ Vð0Þ þ 0

ð30Þ

1 0

z T ðtÞzðtÞdt

Z

1 0

γ 2 wT ðtÞwðtÞdt r0

Considering the fact that xð0Þ ¼ 0 and Vð1Þ Z 0, we obtain the following: Z 1 Z 1 z T ðtÞzðtÞdt r γ 2 wT ðtÞwðtÞdt 0

0

then, the H 1 performance is fulﬁlled. If the disturbance wðtÞ ¼ 0, from (30), we have V_ ðtÞ o 0. Hence, the augmented system (11) is asymptotically stable. Thus, the proof is completed. Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i

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A.2. Part II–Proof of Theorem 1 Rewrite the inequality (17) as

Φini þ ΠΓ ini þ Γ Tini Π o 0 where

Φini

ð31Þ

2

s X τin q X T C T T in q C ni X 1 þDTeini Deini 6 6 q ¼ 1 i 1 ni i 6 6 0 ¼6 6 6 s X 6 4 τini q uTini q C ni

0

s X

q¼1

γ2I 0

q¼1

3

τini q X T1 C Tni uini q 7 0

s X

2

P

0

0

3

7 Π¼6 4 0 I 0 5; 0

0

I

2

Aeini

Γ ini ¼ 6 4 0 0

Geni 0 0

aeini

3

7 0 5: 0

q¼1

Then, by Lemma 3, the following inequality applies (31) 2 3 Φini 2αi Π Π þ ðΓ ini þ αi IÞT F i 4 5o0 Π þF Ti ðΓ ini þ αi IÞ F i F Ti where Let

τini q vini q

7 7 7 7; 7 7 7 5

ð32Þ

αi is a given constant, Fi is an introduced slack matrix. 2

F i1111 6 6 γ i F i1122 Fi ¼ 6 6 F 4 i211

βi F i1111 F i1122

0 0

F i212

F i22

3 0 7 0 7 7 F i23 7 5 F i33

F i311 F i312 F i32 P T and F i1111 ¼ k A Kðni Þ ϕk T k F i1111 , the function ( 1; k ¼ m; ϕk ¼ 0; else:

ϕk is deﬁned as

Then we have " # I mm ð33Þ T m Bm ¼ 0ðn mÞm h i 0 Let F i1111 ¼ FF i111111 , substitute (33) to (32), and apply Lemma 2 to (32), we can obtain the condition (24). Similarly, for the inequalities i111121 F i111122 (15) and (16), their sufﬁcient conditions (22) and (23) can be easily obtained.

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Please cite this article as: Wang H, Yang G-H. Dynamic output feedback controller design for afﬁne T–S fuzzy systems with quantized measurements. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.06.007i