Robust fuzzy stabilization of hybrid discrete delay T–S systems

Accepted Manuscript

Robust Fuzzy Stabilization of Hybrid Discrete Delay T–S Systems Magdi S. Mahmoud PII: DOI: Reference:

S0016-0032(17)30494-5 10.1016/j.jfranklin.2017.09.025 FI 3161

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

13 January 2017 22 August 2017 26 September 2017

Please cite this article as: Magdi S. Mahmoud, Robust Fuzzy Stabilization of Hybrid Discrete Delay T–S Systems, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.09.025

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Robust Fuzzy Stabilization of Hybrid Discrete Delay T–S Systems Magdi S. Mahmoud

Abstract

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Novel results on robust fuzzy stabilization are developed hereafter for discrete hybrid systems in the (T–S) fuzzy framework. The systems are subject to fractional parametric uncertainties and bounded time-delays. Linear matrix inequalities-based conditions are derived for stability analysis. By optimizing an appropriate performance measure, both stabilizing gains and the switching signal are generated. The analytical findings are validated on a typical water-quality system application to demonstrate the efficacy of the developed techniques.

Keyword Fuzzy discrete systems, Bounded delays, Hybrid fuzzy stabilization, Linear fractional uncertainty, LMIs.

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I. I NTRODUCTION

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The utilization of (T–S) fuzzy framework [1], [2] signifies a universal candidate for dealing with the stability and stabilization problems of dynamic nonlinear systems. It turns out that these problems have been received an increasing interest in the last decade and consequently a rich body of literature has appeared [3]–[8] using a single quadratic Lyapunov function (SQLF). A basis dependent Lyapunov-Krasovskii function was introduced [9] to guarantee that fuzzy discrete delay systems is robustly stable. The work of [10] concerns the H∞ control design for type 2-fuzzy systems examined under dynamic output feedback while casting the uncertainties in terms of bounds on the membership functions. This leads to an improved hybrid stabilizing controller under which closedloop stability was guaranteed. For multivariable nonlinear systems having nonstrict-feedback form, adaptive fuzzy control problem was treated in [11]. A variable partition approach is presented to determine an adaptive state feedback that guarantees the system stabilizability. For stochastic T–S fuzzy systems with Markovian switching and random delays [12], a state feedback controller preserving reliable mixed H∞ and passivity-based criterion was developed to ensure stability of the Markovian switching stochastic fuzzy system in presence random delays and actuator faults. In [13], the problem of designing reliable filter for a class of discrete-time T-S fuzzy time-delay systems was treated under strict dissipativity performance. The problem of designing adaptive fuzzy controller was investigated in [14] for nonlinear uncertain hybrid systems in strict-feedback form. The controller was based on output feedback to deal with hybrid systems containing unknown nonlinearities and dead-zone. Output feedback design for fuzzy uncertain discrete systems is developed in [15] and model-based robust networked control scheme is presented in [16]. A decentralized delay-dependent filter is constructed in [17] to guarantee the asymptotic stability with prescribed H∞ performance for nonlinear fuzzy interconnected delay systems. In [18]–[20] results on adaptive control of fuzzy systems are provided. Stabilization techniques are given in [21]–[22] for a class of nonlinear hybrid systems. Some recent results on fuzzy stochastic systems were reported in [23], [24], [25] based on sliding mode control and in [26], [27] using output feedback control. On a parallel development, stability and control issues in hybrid systems have commanded the interest of several researchers during the last decade where basic concepts and results can found in [28]–[40] and their references. Adopting arbitrary switching, results are reported in [31] employing hybrid Lyapunov functionals. A fundamental property of a hybrid system is the ability to handle dynamical systems with unstable constituents through the construction of suitable switching signals [28], [30]. There has been two main approaches for control design: • The first approach incorporates single Lyapunov function, and • The second approach adopts several Lyapunov functions. Manuscript MsM-KFUPM-SwitchedFuzzy.tex M. S. Mahmoud is with the Systems Engineering Department, KFUPM, P. O. Box 5067, Dhahran 31261, Saudi Arabia, e-mail: [email protected].

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Naturally, the second approach is more flexible [34], [35]. In this paper, we target hybrid discrete delay systems (HDDS), which have numerous applications including networked control systems and power systems [30]. Theoretical studies were recently pursued for HDD systems in [38]–[47] where alternative methods of design were presented. Our unique departure in embedding HDDS into fuzzy configuration to allow systematically treating nonlinear systems. On a different research front, the design problem of output feedback stabilization with H∞ performance for continuous-time TS fuzzy systems was studied in [48], [49], [50] using pole placement constraints. Motivated by the foregoing results, it appears that the problem designing hybrid fuzzy stabilization for nonlinear discrete fuzzy delay systems has not fully considered. Therefore, the major objective of this paper is to fill out this gap. It investigates the design problem of robust fuzzy stabilizing control for discrete HDDS represented by fuzzy models and subject to parametric uncertainties. The design goal is to reach global exponential stabilization of the closed-loop fuzzy system. Therefore, the work entails focusing on the joint design of two feedback signals: control signal and switching signal. The main contributions of our work are: 1) Development of a novel model set–up of a wide class of nonlinear discrete fuzzy systems incorporating unknown-but-bounded delays and fractional parametric uncertainties. 2) Establishing new results for robust control and deriving previous results on full parametric uncertainties as special cases. 3) Design a robust hybrid state-feedback fuzzy control strategy and a switching signal that ensure the global asymptotic stability of the origin and minimizing an appropriate quadratic criteria. The design is based on an improved fuzzy-basis-dependent Lyapunov–Krasovskii functional. 4) Extending the results to delay-dependent hybrid fuzzy control design. 5) Demonstrating the effectiveness of the analytical results on a typical system application in water-quality control. P t t Notations: Let Rn denote Euclidean space with vector-norm ||α||22 = ∞ k=0 α (k)α(k). We use S , to mean −1 the matrix transpose and S to mean the matrix inverse of square matrix S . For a symmetrical matrix S, the expression S > 0 accounts for the positive definiteness of S . I is the unit matrix with arbitrary dimension. Unless otherwise specified, compatibility of matrix operations prevails. The symbol • represents a term that is induced by symmetry in block symmetric matrices. II. P ROBLEM D ESCRIPTION

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The multi-controller configuration depicted in Fig. 1 is by far one of the fundamental modeling facets of hybrid systems [37]. Bearing this in mind, we consider a nonlinear hybrid discrete delay fuzzy system subject to parametric uncertainties and represented in state-space by a (T-S) model of the type: F or j = 1, 2, ..., N IF θj1 is M`j1 and ... and θjg is M`jg , T HEN ` x(k + 1) = A`ξ(x) x(k) + Dξ(x) x(k − dξ(x) ) ` ` +Bξ(x) u(k) + Mξ(x) q(k) + Γ`ξ (x)w(k)

` ` p(k) = Eξ(x) x(k) + Hξ(x) x(k − dξ(x) ) + L`ξ(x) q(k) ` z(k) = Cξ(x) x(t) + G`ξ(x) x(k − dξ(x) )

q(k) = ∆ p(k), ∆ ∈ ∆

(1) (2) (3)

where x(k) ∈ Rn , u(k) ∈ Rm , w(k) ∈ Rv , z(k) ∈ Rv are the state, control input, exogenous input and observed output vectors, respectively and the function ξ(x) : Rn → N = {1, ..., N } is a switching rule. The vectors p(k) ∈ Rs , q(k) ∈ Rr are internal variables. For all j ∈ N, ` = 1, 2, ..., rj , A`j ∈ Rn×n , Dj` ∈ Rn×n , Mj` ∈ Rn×r , Cj` ∈ Rv×n , G`j ∈ Rv×n , Ej` ∈ Rs×n , Hj` ∈ Rs×n , L`j ∈ Rs×r and Γ`j ∈ Rn×v , are known matrices with rj being the number of fuzzy rules in subsystem j and {M`j1 ...M`jg } are the fuzzy sets. The interval delay dξ(x) (k) is expressed as

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0 < αj ≤ dξ(x) (k) ≤ βj ,

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j∈N

where the constants αj > 0, βj > 0 are known. It is crucial to observe that we could eliminate the variables (p, q) Lj 6= 0if so desired. The initial condition vector x(s) = κ(s), s ∈ [−βj , −βj+1 , .., 0] where κ(s) represents a prescribed differentiable vector-valued function. The uncertain matrix ∆ is an element of the set ∆ represented by ∆ = {∆ ∈ Rr×s : ||∆||∞ ≤ 1}

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Fig. 1. Hybrid delay fuzzy system

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Remark 1: The model (1)-(3) is a generalized version of the work reported in [43]–[45]. The results developed hereafter intended to generalize the previous results to nonlinear discrete-time systems by adopting the notion of fuzzy framework. Following the work of [3], [4] and utilizing the standard fuzzy inference method, that is, a singleton fuzzifier, minimum fuzzy inference, and central-average defuzzifier, model (1)-(3) can be cast into:  `  ` Aξ(x) x(k) + Dξ(x) x(k − dξ(x) ) Pr x(k + 1) = ω ` (θq (k))  ``=1  Pr ` ` ` `=1 ω (θq (k)) Bξ(x) u(k) + Mξ(x) q(k) + Γξ (x)w(k) Pr + ` `=1 ω (θq (k)) r X   ` ` ` = µ` (θq (k)) A`ξ(x) x(k) + Dξ(x) x(k − dξ(x) ) + Bξ(x) u(k) + Mξ(x) q(k) + Γ`ξ (x)w(k) Pr

` (θ

q (k))

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`=1 ω

`=1

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= Aξ(x) (µ)x(k) + Dξ(x) (µ)x(k − dξ(x) ) + Bξ(x) (µ)u(k) + Mξ(x) (µ)q(k) + Γξ (x)(µ)w(k)  `  Pr ` ` ` `=1 ω (θq (k)) Eξ(x) x(k) + Hξ(x) x(k − dξ(x) ) + Lξ(x) q(k) Pr p(k) = ` `=1 ω (θq (t)) r X  `  ` = µ` (θq (k)) Eξ(x) x(k) + Hξ(x) x(k − dξ(x) ) + L`ξ(x) q(k)

(6)

`=1

= Eξ(x) (µ)x(k) + Hξ(x) (µ)x(k − dξ(x) ) + Lξ(x) (µ)q(k)

(7)

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 `  Cξ(x) x(t) + G`ξ(x) x(k − dξ(x) ) Pr z(k) = ` `=1 ω (θq (t)) r X  `  µ` (θq (k)) Cξ(x) x(t) + G`ξ(x) x(k − dξ(x) ) = Pr

`=1 ω

` (θ

q (k))

`=1

= Cξ(x) (µ)x(t) + Gξ(x) (µ)x(k − dξ(x) )

q(k) = ∆ p(k), ∆ ∈ ∆ `

µ (θq (k)) =

Y ω ` (θq (k)) Pr j ` (M`q (θq (k))) , ωj` (θq (k)) = ω (θ (k)) q `=1

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where

(8) (9)

(10)

q=1

and M`q (θq (k)) is the grade of membership of (θq (k)) in M`q . It is not difficult to see that r Y X

M`q (θq (k)) ≥ 0,

(M`q (θq (k))) > 0, ∀ k

`=1 q=1

r X

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Hence, for all k, we have

µ`j (θqj (k)) ≥ 0, ` = 1, 2, ..., r;

µ` (θq (k)) = 1

(11)

`=1

Remark 2: The work of [32], [40] and their references essentially dealt with delayless non-fuzzy systems. Given a set of N state-feedback gain matrices K : {K1 , ..., KN }, the approach addressed seeks a switching function ξ(x) : Rn → K such that the feedback hybrid control u(k) = Kξ (x(k)) x(k)

(12)

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guarantees the global asymptotic stabilization of the closed loop system. In this work, we build on this result and extend it further to nonlinear fuzzy systems. Remark 3: For wide class of time-delay systems, the controller could be designed to be delay-dependent or delayindependent such that the closed-loop system satisfies prescribed stability criteria. In the literature, there are several reported efforts on both types. In this paper, we selected the controller to be delay-independent (corresponding to static state feedback strategy) and subsequently provided a complete math analysis to achieve the target design. For the time being, we consider that the matrices K : {K1 , ..., KN } are made available at the disposal of the designer before proceeding to deciding on the stabilizing switching function ξ(.). Subsequent improvements will be achieved in the stabilization phase. Remark 4: It is important to observe that in [22], the authors adopt the T–S fuzzy modeling approach to study the problem of switching stabilization for a class of hybrid continuous nonlinear systems, with possibly unstable subsystems, based on average dwell time (ADT) switching policy. In the same spirit, the work reported in [48] focuses on continuous fuzzy nonlinear plants with unknown parameters to design a switching stabilizing controller. Our work in this paper improves upon [22] in advocating a hybrid-controller configuration of Fig. 1 for a class of hybrid discrete fuzzy nonlinear time-delay systems with parametric uncertainties (1)-(3) and jointly determining the feedback gains and the switching signal while minimizing a suitable guaranteed cost. III. ROBUST H YBRID S TABILIZATION

Our goal now is to establish procedure of finding a stabilizing switching signal ξ(x) which ensures the asymptotic stabilization at origin globally and achieves an optimal bounding of the measure J(ξ) given by J(K, ξ) = max

∆∈∆

To accomplish our goal, we define

N X

t zm (k)zm (k)

m=1

dsj = (βj − αj ), δx(k) = x(k + 1) − x(k)

(13)

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Let L(x(k)) = minj∈N Vj (x(k)) expresses a Lyapunov-Krasovskii functional (LKF) candidate where k−1 X

Vj (µ) = xt (k)Pj (µ)x(k) +

xt (j)Qj (µ)x(j) +

k−1 X

xt (j)Sj (µ)x(j) +

+

X

k−1 X

xt (m)Qj (µ)x(m)

j=−βj +1 m=k+j

j=k−αj −αj −1

X

k−1 X

t

δx (m)Raj (µ)δx(m) +

j=−βj m=k+j

−1 X

k−1 X

δxt (m)Rcj (µ)δx(m)

(14)

j=−βj m=k+j

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+

xt (j)Zj (µ)x(j)

j=k−αj

j=k−dj (k) −αj

k−1 X

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where 0 < Pj (µ), 0 < Qj (µ), 0 < Zj (µ), 0 < Raj (µ), 0 < Rcj (µ), 0 < Sj (µ) are fuzzy-basis-dependent matrices. A method to achieve our goal is fulfilled by the following theorem: Theorem 1: Given Vj (µ) in (14). For all j ∈ N, we assume the bounding factors αj > 0, αj > 0 are specified. There exist fuzzy-basis-dependent matrices Pj (µ), Qj (µ), Zj (µ), Sj (µ), Raj (µ), Rcj (µ), Θaj (µ), Θcj (µ), Ψaj (µ), Ψcj (µ), Φaj (µ) and Φcj (µ) such that the following inequalities are satisfied for j ∈ N   Ωj (µ) Λj (µ) Ξj (µ) −Υj (µ) 0  < 0 Πj (µ) =  • (15) • • −I   Πoj (µ) Πaj (µ) Ψaj (µ) −Φaj (µ) Ejt (µ)Lj (µ)  • −Πcj (µ) Ψcj (µ) −Φcj (µ) Hjt (µ)Lj (µ)      Ωj (µ) =  • • −Zj (µ) 0 0    • • • −Sj (µ) 0 t • • • • Lj (µ)Lj (µ) − I    t Λaj (µ) Λcj (µ) , Ξj (µ) = Cj (µ) Gj (µ) 0 0 0 Λj (µ) = 0 0 p p  p  d Φ (µ) d Ψ (µ) β Θ (µ) Atj (µ)Pj (µ) p sj aj p sj aj p j aj  dsj Φcj (µ) dsj Ψcj (µ) βj Θcj (µ) Djt (µ)Pj (µ)   Λaj (µ) =   0 0 0 Mjt (µ)Pj (µ)  0 0 0 0 p   p t t dp βp sj (Aj (µ) − I) Raj (µ) j (Aj (µ) − I) Rcj (µ)   dsj Djt (µ)Raj (µ) βj Djt (µ)Rcj (µ)  Λcj (µ) =    0 0 0 0   Υj (µ) = diag Raj (µ) + Rcj (µ) Raj (µ) Rcj (µ) Pj (µ) Raj (µ) Rcj (µ) , Πoj (µ) = −Pj (µ) + (ds + 1)Qj (µ) + Zj (µ) + Sj (µ) + Θaj (µ) + Θtaj (µ) + Ejt (µ)Ej (µ),

Πaj (µ) = −Θaj (µ) + Θtcj (µ) + Φaj (µ) − Ψaj (µ) + Ejt (µ)Hj (µ)

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t Πcj (µ) = Qj (µ) + Θcj (µ) + Θtcj (µ) − Φcj (µ) − Φtcj (µ) + Ψcj (µ) + Ψtcj (µ) − Hj(µ) Hj (µ)

(16)

then the switching signal ξ(x) = arg minj∈N Vj (µ) guarantees the global exponential stability of system (1)-(3) and ∞ X J(ξ) = max z t (m)z(m) < min ˆj sup ||x(m)||2 (17) ∆∈∆

ˆj

=

m=0

j∈N

m∈[−αj ,0]

max{λM (Pj (µ)), aj (µ), cj (µ)}

aj (µ) = (βj − αj + 1)λM (Qj (µ)) + λM (Zj (µ)) + λM (Sj (µ)) + 2αj λM (Raj (µ) + Rcj (µ)) cj (µ) = 2βj λM (Raj (µ) + Rcj )(µ)

Proof 1: Define the indices and the associated switching signal I((µ)) = {j ∈ N : Vj ((µ)) = L((µ))}, ξ(x(k)) = j, k ≥ 0 f or some j ∈ N

(18)

(19) (20)

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Straightforward computation of the first-difference of δVj (µ) = Vj (µ, k + 1) − Vj (µ, k) given the trajectories of (1) with w(k) ≡ 0 yield: δVj (µ) = Vj (µ, k + 1) − Vj (µ, k)

= [Aj x(k) + Dj x(k − dj (k) + Mj q(k)]t Pj (µ)[Aj x(k) + Dj x(k − dj (k) + Mj q(k)] + xt (k)Qj (µ)x(k) − xt (k − dj (k))Qj (µ)x(k − dj (k)) + xt (k)Zj (µ)x(k) k−αj

X

j=k−βj +1

xt (j)Qj (µ)x(j) − xt (k)Pj (µ)x(k) + xt (k)Sj (µ)x(k) − xt (k − βj )Sj (µ)x(k − βj )

+ (βj − αj )xt (k)Qj (µ)x(k) − xt (k − αj )Zj (µ)x(k − αj )

+ (βj − αj )δxt (k)Raj (µ)δx(k) + αj δxt (k)Rcj (µ)δx(k) −

k−αj −1

X

j=k−βj

δxt (j)Raj (µ)δx(j) −

Introduce the identities

k−1 X

δxt (j)Rcj (µ)δx(j)

j=k−βj

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[2xt (k)Ψaj (µ) + 2xt (k − dj (k))Ψcj (µ)][x(k − αj ) − x(k − dj (k)) − [2xt (k)Θaj (µ) + 2xt (k − dj (k))Θcj (µ)][x(k) − x(k − dj (k)) − t

t

k−αj −1

X

(21)

δx(j)] = 0

j=k−dj (k)

k−1 X

δx(j)] = 0

j=k−dj (k)

[2x (k)Φaj (µ) + 2x (k − dj (k))Φcj (µ)][x(k − d(k)) − x(k − βj ) −

k−dj (k)−1

X

δx(j)] = 0

(22)

j=k−βj

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for some fuzzy-basis-dependent matrices Θaj (µ), Φaj (µ), Ψaj (µ), Θcj (µ), Φcj (µ), Ψcj (µ). Proceeding further, we consider (1)–(3) under ∆t ∆ ≤ I with β(k) = [xt (k) xt (k − d(k)) q(k)]t . Now, we cast t p (k)p(k) − q t (k)q(k) ≥ 0 in the form  Ejt (µ)Ej (µ) Ejt (µ)Hj (µ) Ejt (µ)Lj (µ) • Hjt (µ)Hj (µ) Hjt (µ)Lj (µ)  β(t) ≥ 0 β t (k)  t • • Lj (µ)Lj (µ) − I

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

(23)

By algebraic manipulations, we get from (21)-(23)

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b j (µ) ξ(k) δVj (µ) ≤ ξ t (k) Ω b j (µ) = Ωj (µ) + Ξt1 (µ)(Raj (µ) + Rcj (µ))−1 Ξ1 (µ) + Ξt2 (µ)R−1 (µ)Ξ2 (µ) + Ξt3 (µ)Rcj (µ)Ξ3 (µ), Ω aj   Ωoj (µ) Ωaj (µ) Ψaj (µ) −Φaj (µ) Ejt (µ)Lj (µ)  • −Ωcj (µ) Ψcj (µ) −Φcj (µ) Hjt (µ)Lj (µ)     , Ωj (µ) =  • • −Zj (µ) 0 0    • • • −Sj (µ) 0 • • • • Ltj (µ)Lj (µ) − I  t  x (k) xt (k − dj (k)) xt (k − αj ) xt (k − βj ) q t (k) ξ t (k) =

(24)

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p

 dsj Φtcj (µ) 0 0 , p  p  dsj Ψtaj (µ) dsj Ψtcj (µ) 0 0 , Ξ2 (µ) = p   p βj Θtaj (µ) βj Θtcj (µ) 0 0 Ξ3 (µ) =

Ξ1 (µ) =

dsj Φtaj (µ)

Ωoj (µ) = Atj (µ)Pj (µ)Aj (µ) − Pj (µ) + (dsj + 1)Qj (µ) + Zj (µ) + Sj (µ) + Θaj (µ) + Θtaj (µ) + dsj (Aj (µ) − I)t Raj (µ)(Aj (µ) − I) + βj (Aj (µ) − I)t Rcj (µ)(Aj (µ) − I) + Ejt (µ)Ej (µ)

Ωcj (µ) = −Djt (µ)Pj (µ)Dj (µ) − dsj Djt (µ)Raj (µ)Dj (µ) − dsj Djt (µ)Rcj (µ)Dj (µ) + Qj (µ)

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+ Θcj (µ) + Θtcj (µ) − Φcj (µ) − Φtcj (µ) + Ψcj (µ) + Ψtcj (µ) − Hjt (µ)Hj (µ),

Ωaj (µ) = Atj (µ)Pj (µ)Dj (µ) + dsj (Aj (µ) − I)t Raj (µ)Dj (µ) + dsj (Aj (µ) − I)t Rcj (µ)Dj (µ) + Ejt (µ)Hj (µ)

− Θaj (µ) + Θtcj (µ) + Φaj (µ) − Ψaj (µ)

(25)

b j (µ) < 0 thereby entailing δVj (µ) < 0, ξ(k) 6= 0. When the By Schur complements, it follows from (24) that Ω switching signal ξ(x) = j, j ∈ N, it is easy to see that δL(µ, x(k)) = L(µ, x(k + 1)) − L(µ, x(k)) < min(δV` (µ, x(k)))

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`∈N

(26)

Now, considering LMI (15) and applying the Schur-complement for ξ(x(k)) = j and using inequality (23), it follows that δL(µ, x(k)) < −z t (k)z(k). This holds true since Vm (µ, x(k)) ≥ Vj (µ, x(k)) = L(µ, x(k)) for all m ∈ N and all j ∈ I(x). This emphasizes the stabilizing feature of the switching rule ξ(x). Moreover, it follows from (15)-(16) ∀ ∆ ∈ ∆ that δL(µ, x(k)) < −z t (k)z(k). From the standard properties of (24), we have δV (µ, x(k))) ≤ λM (Ωj (µ)) ||x(k)||22

(27)

λM (Pj (µ))||x(k)||22

+ aj (µ)

k−1 X

j=k−βj

ED

V (µ) ≤

M

and considering (14), it is readily evident, given aj , cj from (19)–(20), that ||x(j)||22

k−1 X

+ cj (µ)

j=k−βj

Selecting T > βj + 1 and ϕ > 1, it is not difficult to prove that T−1 X

k−1 X

ϕ

||x(j)||22

PT

k=0 j=k−βj

k

T−1 X

k−1 X

≤ βj ϕ

From (27)–(29), little algebra leads to

||x(j)||22

sup j∈[−βj ,0]

ϕk ||x(j + 1)||22 ≤ βj ϕβj

CE

k=0 j=k−βj

βj

T X j=1

βj

+ βj ϕ

T−1 X j=1

ϕj ||x(j)||22 + βj ϕβj

||x(j + 1)||22

(28)

ϕj ||x(j)||22 sup

j∈[−βj ,0]

||x(j)||22

(29)

AC

ϕk+1 V (µ, k + 1) − ϕk V (µ, k) ≤ [ϕk (ϕ − 1)λM (Pj (µ)) + ϕk+1 λM (Ωj )]||x(k)||2

+ aj ϕk (ϕ − 1)

k−1 X

j=k−αj

||x(j)||22 + cj ϕk (ϕ − 1)

k−1 X

j=k−βj

||x(j + 1)||22

which when summed up over the interval [k = 0 → k = J − 1], it yields ϕT j V (µ, T) − V (µ, 0) ≤ [(ϕ − 1)λM (Pj (µ)) + ϕλM (Ωj )] + aj (µ)(ϕ − 1) + cj (µ)(ϕ − 1)

T−1 X

j−1 X

j=0 m=j−βj

T−1 X

j−1 X

j=0 m=j−βj

J−1 X j=0

(30)

ϕj ||x(j)||2

ϕj ||x(m)||22 ϕj ||x(m + 1)||22

(31)

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8

Proceeding, it can be easily inferred from (28), given ˆj (µ) in (18), that V (µ, T) ≥ λm (Pj (µ))||x(T)||22 , V (µ, 0) ≤ ˆj (µ)

sup

j∈[−βj ,0]

||x(j)||22

(32)

By considering (29)-(32), we obtain ϕT V (µ, T) ≤ V (µ, 0) + [(ϕ − 1)λM (Pj (µ)) + ϕλM (Ωj (µ)) J−1 X + (aj (µ) + cj (µ))(ϕ − 1)βj ϕαj ] ϕj ||x(j)||22 + (aj (µ) + cj (µ))(ϕ − 1)βj ϕ

In the light of (32), (33) simplifies into ||x(T)||2 ≤

CR IP T

j=0

αj

sup

j∈[−dM ,0]

||x(j)||22

1 1 β [M + (aj + cj )(ϕ∗ − 1)βj ϕ∗ j ]( )J sup ||x(j)||2 λm ϕ∗ j∈[−βj ,0]

(33)

(34)

AC

CE

PT

ED

M

AN US

subject to selecting ϕ∗ satisfying the algebraic relation (ϕ∗ −1)λM (Pj (µ))+ϕ∗ λM (Ωoj (µ))+(aj (µ)+cj (µ))(ϕ∗ − β 1)αj ϕ∗ j = 0. Obviously, this guarantees the exponential stability of hybrid system (1) with the performance measure J(ξ) bounded by (17) and therefore the proof is completed.  Remark 5: Observe that the switching rule ξ(x) = arg minj∈N Vj (x) belongs to the min-projection strategy [39] which means that the selected the switching rule ξ as ”subsystem variable” corresponds to the smallest projection of linear vector field Vj (x)x on the vector x. The intuition behind this strategy is that the system should become stable if such selection is made possible. Recalling (10)–(11), it is quite clear that the role of the membership functions is a ”normalization” of the systems dynamics to enable the fuzzy approximation process. In this regard, there is no direct relationship between the switching rules and the membership functions. Remark 6: The global asymptotic stabilization of the feedback-controlled fuzzy system is guaranteed by the switching rule ξ(x) = arg minj∈N Vj (x) whenever the LMIs of Theorem 1 achieve feasibility. Also provided is the optimal performance measure J(ξ). It is significant to note that when LMI (15) is feasible ∀ j ∈ N, it immediately implies the validity of the switching signal held constant ξ(k) = j ∈ N, ∀k ≥ 0. This is a novel result in the context of hybrid fuzzy systems and hence it is interpreted as the stability of system (1)-(3) is preserved under constant switching. It follows that ||z||22 /||q||22 < 1 and it is further emphasized that the procedure of proving Theorem 1 embodies the fact that the stability on the boundary of adjacent regions is implied by (26) and the hybrid discrete-time system passes through all modes of operation leading to the computation of the upper bound on the cost function. Remark 7: With reference to the work in [10], [15]–[17], [20], the delay term dξ(x) (k) in the present work is characterized by (4) and it is time-varying in response to the switching signal ξ(x), within known bounds αj , βj , j ∈ N. For stability analysis, we constructed a proper LKF in (14) which includes delay-independent and delay-dependent terms as well as fuzzy-basis-dependent matrices. In this regard, our work provides efficient method for handling delays in the context of hybrid (switched) fuzzy systems. A. Special case

Consider the following uncertain hybrid fuzzy discrete-time system: x(k + 1) = Aξ(x) (µ)x(k) + Bξ(x) (µ)u(k) + Mξ(x) (µ)q(k) + Γξ (x)(µ)w(k) p(k) = Eξ(x) (µ)x(k) + Lξ(x) (µ)q(k)

(35) (36)

z(k) = Cξ(x) (µ)x(t) q(k) = ∆p(k), ∆ ∈ ∆

(37)

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Corollary 1: Let the following LMIs   ˆ j (µ) Λ ˆ aj (µ) Cj (µ) Ω Πj (µ) =  • −Pj (µ) 0  < 0 • • −I    t  t −Pj (µ) + Ej (µ)Ej (µ) Ejt (µ)Lj (µ) Aj (µ)Pj (µ) ˆ ˆ Ωj (µ) = Λaj (µ) = • Ltj (µ)Lj (µ) − I Mjt (µ)Pj (µ)

(38) (39)

J(ξ) = max

∆∈∆

∞ X

CR IP T

have feasible solution Pj , j ∈ N. Hence, the switching signal ξ(x) = arg minj∈N Vj (µ) with Vj (µ) = xt (k)Pj (µ)x(k) guarantees the exponential stability of the hybrid fuzzy system (35)-(37) and ensure that z t (m)z(m) < min λM (Pj (µ))||x(k)||2 j∈N

m=0

(40)

Proof 2: It follows from Theorem 1 by selecting Qj (µ) ≡ 0, Zj (µ) ≡ 0, Sj (µ) ≡ 0, Raj (µ) ≡ 0, Rcj (µ) ≡ 0, Θaj (µ) ≡ 0, Θcj (µ) ≡ 0, Ψaj (µ) ≡ 0, Ψcj (µ) ≡ 0, Φaj (µ) ≡ 0, Φcj (µ) ≡ 0. IV. D ELAY-D EPENDENT H YBRID S TABILIZATION

AN US

In the sequel, we target a fuzzy stabilization, possessing the robustness and delay-dependence properties, for the system x(k + 1) = Aj (µ)x(k) + Dj (µ)x(k − dj (k)) + Mj (µ)q(k) + Γj (µ)w(k) + Bj (µ)u(t)

(41)

p(k) = Ej (µ)x(k) + Hj (µ)x(k − dj (k)) + Lj (µ)q(k) + Tj (µ)u(k)

(42)

q(k) = ∆p(k), ∆ ∈ ∆

(43)

z(k) = Cj (µ)x(t) + Gj (µ)x(k − dj (k)) + Fj (µ)u(t)

M

using a fuzzy version of the state-feedback hybrid control (12) along with the performance measure J(K, ξ) = max

ED

∆∈∆

N X

zjt (k)zj (k)

(44)

j=1

where ∀ j = 1, ..., N, the sequences w(k) = ej δ(k), zj (k) are input-output of the feedback-controlled fuzzy system

PT

x(k + 1) = Aˆj (µ)x(k) + Dj (µ)x(k − dj (k)) + Mj (µ)q(k) + Γj (µ)w(k) ˆj (µ)x(k) + Hj (µ)x(k − dj (k)) + Lj (µ)q(k) p(k) = E z(k) = Cˆj (µ)x(t) + Gj (µ)x(k − dj (k))

(46) (47) (48)

AC

CE

q(k) = ∆p(k), ∆ ∈ ∆ ˆ ˆj (µ) = Ej (µ) + Tj (µ)Kj (µ) Aj (µ) = Aj (µ) + Bj (µ)Kj (µ), E Cˆj (µ) = Cj (µ) + Fj (µ)Kj (µ)

(45)

with ej being the jth column of identity matrix. It is readily evident that the exponential stability property is attained when the inequality   ˆ j (µ) Ξ ˆ j (µ) ˆ j (µ) Λ Ω ˆ j (µ) Πj (µ) =  • (49) −Υ 0 <0 • • −I ˆ j (µ), .., Υ ˆ j (µ), respectively correspond to Ωj (µ), .., Υj (µ) with Aˆj (µ), E ˆj (µ), Cˆj (µ) replacing is feasible where Ω Aj (µ), Ej (µ), Cj (µ). The desired result is provided by the following theorem:

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Theorem 2: Let αj > 0, αj > 0 for all j ∈ N be specified. The fuzzy-basis-dependent matrices Xj (µ) > b j (µ) > 0, b bj (µ) > 0, Θ b aj (µ), Θ b cj (µ), Φ b aj (µ), Ψ b aj (µ), Ψ b aj (µ), Φ b cj (µ), 0, Yj (µ), Q Sj (µ) > 0, Z b b Raj (µ), Rcj (µ), j ∈ N satisfy the following LMIs for all j ∈ N  b b j (µ) Ξ b j (µ) Xj (µ)E t (µ)  Ωj (µ) Λ j t  •  b − Υ (µ) 0 X (µ)H j j  j (µ)  < 0 (50)  •  • −I 0 • • • −I 

CR IP T

 b oj (µ) Π b aj (µ) Ψ b aj (µ) −Φ b aj (µ) Xj (µ)E t (µ)Lj (µ) Π j   b cj (µ) Ψ b cj (µ) −Φ b cj (µ) • −Π Hjt (µ)Lj (µ)     b b Ωj (µ) =  , • • −Zj (µ) 0 0   b   • • • −Sj (µ) 0 • • • • Ltj (µ)Lj (µ) − I b j (µ) + Z bj (µ) + b b oj (µ) = −Xj (µ) + (ds + 1)Q b aj (µ) + Θ b taj (µ), Π Sj (µ) + Θ b aj (µ) = −Θ b aj (µ) + Θ b tcj (µ) + Φ b aj (µ) − Ψ b aj (µ), Π

b j (µ) + Θ b cj (µ) = Q b cj (µ) + Θ b tcj (µ) − Φ b cj (µ) − Φ b tcj (µ) + Ψ b cj (µ) + Ψ b tcj (µ), Π   b aj (µ) Λ b cj1 (µ) Λ b cj2 (µ) Λ b j (µ) = Λ 0 0 0 h i b aj1 (µ) = b aj1 (µ) Λ b aj2 (µ) Λ Λ p   p b aj (µ) b aj (µ) dsj Φ dsj Ψ p  p b b cj (µ)  dsj Ψ b aj (µ) =  dsj Φcj (µ)  Λ  0 Xj (µ)Mjt (µ)  0 0  √  b aj (µ) Xj (µ)At (µ) + Yt (µ)B t (µ) αj Θ j j j  √ b   αj Θcj (µ) Xj (µ)Djt (µ) b aj2 (µ) =  Λ     0 Xj (µ)Mjt (µ) 0 0  p  t t dsj (Xj (µ)Aj (µ) + Yj (µ)Bjt (µ) − I) p   dsj Djt (µ) b cj1 (µ) =   Λ   0 0   p t βj (Xj (µ)Aj (µ) + Yjt (µ)Bjt (µ) − I) p   βj Djt (µ) b cj2 (µ) =   Λ   0 0 h i   b j (µ) = diag Υ b j (µ) = Cj (µ)Xj (µ) Gj (µ) 0 0 0 t , b j1 (µ) Υ b j2 (µ) , Ξ Υ h i b j1 (µ) = diag R b aj (µ) + R b cj (µ) R b aj (µ) R b cj (µ) Xj (µ) , Υ h i b j2 (µ) = diag 2I − R b aj (µ) 2I − R b cj (µ) Υ

AC

CE

PT

ED

M

AN US

where

(51)

The switching signal ξ(x) = arg minj∈N Vj (µ) with Vj (µ) in (14) and the hybrid matrix gain Kj (µ) = Yj (µ)X−1 j (µ) guarantee the global asymptotic stability of the fuzzy system (1)-(3) along with J(K, ξ) < min T r(Γto (µ)X−1 j (µ)Γo (µ)) j∈N

(52)

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Proof 3: Consider system (45)-(48) together with inequality (49). Invoking the congruent transformation T (µ) = [Xj (µ), Xj (µ), Xj (µ), Xj (µ), I, Xj (µ), Xj (µ), Xj (µ), Xj (µ), Xj (µ), Xj (µ), I], Xj (µ) = P−1 j (µ)

along with the change of variables b j (µ) = Xj (µ)Wj (µ)Xj (µ), Q b j (µ) = Xj (µ)Qj (µ)Xj (µ) Yj (µ) = Kj (µ)Xj (µ), W bj (µ) = Xj (µ)Zj (µ)Xj (µ), b b aj (µ) = Xj (µ)Θaj (µ)Xj (µ), Z Sj (µ) = Xj (µ)Sj (µ)Xj (µ), Θ b cj (µ) = Xj (µ)Θcj (µ)Xj (µ), Φ b aj (µ) = Xj (µ)Φaj (µ)Xj (µ), Ψ b aj (µ) = Xj (µ)Ψaj (µ)Xj (µ), Θ

CR IP T

b aj (µ) = Xj (µ)Ψaj (µ)Xj (µ), Φ b cj (µ) = Xj (µ)Φcj (µ)Xj (µ), R baj (µ) = Xj (µ)Raj (µ)Xj (µ), Ψ bcj (µ) = Xj (µ)Rcj (µ)Xj (µ) R

and Schur complement operations, LMI (49) will be converted into LMI (50) subject to (51). Proceeding further, the output zj (k) of system (45)-(48) with zero initial condition and input w(k) = ej δ(k) can be determined from the same system with zero input and initial condition xo = Γo ej , j = 1, ..N . Thus it follows from (44) that J(K1 , ..., KN ξ) ≤ max ||zjt (k)zj (k)||22 ∆∈∆

J(K, ξ) <

AN US

According to Theorem 1, the above expression is cast into N X j=1

By the trace properties, we reach

min(Γo ej )t Pj (Γo ej ) j∈N

J(K, ξ) < min T r(Γto X−1 j Γo ) j∈N



M

Hence, the proof is completed. Consider the system

x(k + 1) = Aj (µ)x(k) + Mj (µ)q(k) + Γj (µ)w(k) + Bj (µ)u(t)

ED

p(k) = Ej (µ)x(k) + Lj (µ)q(k) + Tj (µ)u(k)

(53) (54)

z(k) = Cj (µ)x(t) + Fj (µ)u(t) q(k) = ∆p(k), ∆ ∈ ∆

(55)

AC

CE

PT

under the hybrid control (12) and together with the performance measure (44). We have the following result Corollary 2: The fuzzy-basis-dependent matrices Xj (µ) > 0, Yj (µ) satisfy the following LMIs for j ∈ N   e j (µ) Λ e j (µ) Xj (µ)C t (µ) Xj (µ)E t (µ) Ω j j  • −Xj 0 Xj (µ)Hjt (µ)  <0  (56)  •  • −I 0 • • • −I   t −Xj (µ) Xj (µ)Ej (µ)Lj (µ) e j (µ) = Ω , • Ltj (µ)Lj (µ) − I   Xj (µ)Atj (µ) + Yjt (µ)Bjt (µ) e j (µ) =   Xj (µ)Djt (µ) Λ (57) Xj (µ)Mjt (µ) The switching rule ξ(x) = arg minj∈N Vj (µ) with Vj (µ) = xt (k)Pj (µ)x(k) and the hybrid matrix gain Kj (µ) = Yj (µ)X−1 j (µ) ensure global asymptotic stability of the system (1)-(3) and J(K, ξ) < min T r(Γto (µ)X−1 j (µ)Γo (µ)) j∈N

(58)

Proof 4: It can be easily deduced from Theorem 2 by selecting Qj (µ) ≡ 0, Zj (µ) ≡ 0, Sj (µ) ≡ 0, Raj (µ) ≡ 0, Rcj (µ) ≡ 0, Θaj (µ) ≡ 0, Θcj (µ) ≡ 0, Ψaj (µ) ≡ 0, Ψcj (µ) ≡ 0, Φaj (µ) ≡ 0, Φcj (µ) ≡ 0.

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12

V. S IMULATION E XAMPLES In this section, we present two examples to illustrate the effectiveness of the developed theoretical developments A. Water pollution control

`=1

z1 (k) =

2 X

µ`1

  ` ` C1 x1 (k) + G1 x1 (k − d1 )

µ`2

  ` ` ` ` ` A2 x2 (k) + D2 x2 (k − d2 ) + B2 u2 (k) + M2 q2 (k) + Γ2 w2 (k)

`=1

p2 (t) = z2 (t) =

`=1 3 X

`=1 3 X

µ`1

  ` ` ` E2 x2 (k) + H2 x2 (k − d2 ) + L2 q2 (k)   ` ` C2 x2 (k) + G2 x2 (k − d2 )

ED

x2 (k + 1) =

3 X

  ` ` ` E1 x1 (k) + H1 x1 (k − d1 ) + L1 q1 (k)

M

p1 (k) =

µ`1

AN US

`=1

2 X

CR IP T

The purpose of this example is to illustrate the application of Theorem 2. From the extensive studies on the River Nile [51], [52], we consider a portion of length about 50 km subjected to severe dumps of industrial waste. For the purpose of constructing water pollution model, we subdivided that portion along the fresh-water stream into four consecutive-reaches of lengths 10 km, 10 km, 20 km, 20 km. With the available data values [51], [52], a four-reach water pollution model of the type (1)–(3) is constructed. The model of the first two reaches (subsystems) represents two aggregate bio-strata of river pollutant: Algae and Ammonia products while the corresponding model of the third and fourth reaches represents three aggregate bio-strata of river pollutant: Algae, Heavy ammonia products and Light ammonia products. The standard unit of measurements is milligram per liter (mg/L). The overall dynamic model is described by:   2 X ` ` ` ` ` ` µ1 A1 x1 (k) + D1 x1 (k − d1 ) + B1 u1 (k) + M1 q1 (k) + Γ1 w1 (k) x1 (k + 1) =

µ`2

  ` ` ` ` ` A3 x3 (k) + D3 x3 (k − d3 ) + B3 u3 (k) + M3 q3 (k) + Γ3 w3 (k)

PT

`=1

`=1 2 X

CE

x3 (k + 1) =

2 X

p3 (k) =

`=1

AC

z3 (k) =

x4 (k + 1) =

2 X

µ`3

  µ`1 E3` x3 (k) + H3` x3 (k − d3 ) + L`3 q3 (k) µ`1

  ` ` C3 x3 (k) + G3 x3 (k − d3 )

µ`4

  ` ` ` ` ` A4 x4 (k) + D4 x3 (k − d4 ) + B4 u4 (k) + M4 q4 (k) + Γ4 w4 (k)

`=1

3 X `=1

p4 (k) =

3 X

µ`1

`=1

z4 (k) =

3 X `=1

µ`1

  ` ` ` E4 x4 (k) + H4 x3 (k − d4 ) + L4 q4 (k)   ` ` C4 x4 (k) + G4 x4 (k − d4 )

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13

where x1 (k) = [xt11 (k) xt21 (k)], x2 (k) = [xt12 (k) xt22 (k)], x3 (k) = [xt13 (k) xt23 (k) xt33 (k)] and x4 (k) = [xt14 (k) xt24 (k) xt34 (k)], representing the state vectors of subsystem (reach) 1, 2, 3 and 4, respectively. The associated matrices are given by: Reach model 1

:



:



C12 H11 Reach model 2 A12

=

C21 = A22

=

C22 = A32

=

C23 = B21 = H21 = :













D31 =

AC

C31 = A23 =

C32 = B31 = H31 =

−0.1 −0.05



,

−0.05 −0.07



,

−0.03 −0.04



,

     −0.7 1.2 0 0.4 1.3  −0.3 −0.4 −0.1  , Γ13 =  0.2  , M31 =  2.4  , −0.5 0.6 −0.1 0.3 1.5       −0.06 −0.01 0.02 2.3 −0.04 −0.03 −0.02  −0.05 −0.03 0.02  , M32 =  1.7  , D32 =  −0.05 −0.06 −0.04  , 1.8 −0.01 0.06 −0.05 0.01 −0.02 −0.04       1 1 0.2 0.1 0.3 , E3 = 0 0.9 0.1 , G3 = 1.8 0 1.1 ,     −0.4 1.4 0.1 0.1  −0.2 −0.5 0.2  , Γ23 =  0.2  , L13 = [0.6], L23 = [1.7] 0 −0.2 −0.5 0.3       2 0.3 0 0.2 , E3 = 0 0.8 0.2 , G23 = 0.9 0 0.1 ,     1 2 2 , B3 = , α3 = 5, β3 = 25, 5 3     0.1 0 −0.02 , H32 = −0.2 0.03 0

CE

A13 =



PT

Reach model 3



      0 2.2 0.1 −0.1 0.2 1 1 1 , D2 = , M2 = , Γ2 = −0.04 0.8 0.5 0.3 −0.4      0.3 0 , E21 = 0 0.8 , G12 = 1.6 0 , L12 = [1.8],       −0.06 1 0.2 −0.2 0.1 2 2 2 , D2 = , M2 = , Γ2 = 2.8 −0.02 0.8 0.6 −0.3      2 2 2 0.5 0 , E2 = 0 0.6 , G2 = 1.2 0 , L2 = [1.4],       1 −0.08 0.1 −0.2 0 3 3 3 , D2 = , M2 = , Γ2 = 0 2.8 0.3 1 −0.1      3 3 3 0.4 0 , E2 = 0 0.4 , G2 = 1.9 0 , L2 = [1.1],      2 4 3 , B23 = , α2 = 5, β2 = 30, , B22 = 4 2 3      0.3 −0.01 , H22 = −0.1 0.01 , H23 = −0.3 0.04

AN US

A21

M

C11

ED

B11

CR IP T

       −0.3 0.1 0.2 1 −0.05 0 1 1 1 = , Γ1 = , M1 = , D1 = , −0.4 −0.2 0.3 2.8 −0.02 −0.03       5 4 = , B12 = , E11 = 0.1 0.05 , α1 = 4, β1 = 10 2 1       0.5 0.01 , E11 = 0.1 0.6 , G11 = 1.2 0 , L11 = [0.4], =         −0.9 −0.5 0.5 2 −0.08 −0.04 2 2 2 = , Γ1 = , M1 = , D1 = , −0.3 −1.2 0.2 1.2 0 −0.1       0.7 0.02 , E12 = 0.2 0.9 , G21 = 1.5 0 , L21 = [1.2], =     0.1 −0.02 , H12 = −0.2 0.03 =

A11

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A14 = C41 = A24 = C42 = D41 =

A34 = C43 = D41 = α4 = H42 =



     −0.7 1.2 0.5 0.4 1.3  0.3 −0.4 0.3  , Γ14 =  0.2  , M41 =  2.4  , −0.1 0.6 −0.5 0.3 0.5       1 1 0.2 0 0 , E4 = 0.1 0 0.9 , G4 = 1.8 0 0 , L14 = [0.6]       −0.4 1.4 0.2 0.1 2.3  −0.2 −0.5 0  , Γ24 =  0.2  , M42 =  1.7  , 0 −0.3 −0.7 0.1 1.5       2 2 0.3 0 0 , E4 = 0 0.2 0.8 , G4 = 0.9 0 0 , L24 = [0.8]     −0.06 −0.01 0.01 −0.04 −0.04 0.01  −0.05 −0.03 0.02  , D42 =  −0.05 −0.06 0.02  , 0.01 0.05 −0.02 −0.01 0.03 −0.07       −0.5 1.3 0.1 0.2 1.3  −0.5 −0.4 0  , Γ34 =  0.1  , M43 =  2.2  , 0 −0.1 −0.8 0.2 1.8       3 3 0.5 0 0.1 , E4 = 0 0.3 0.6 , G4 = 0.8 0 0 , L34 = [0.8]         0.07 −0.01 0 5 3 4 3 2 1  0.02 −0.03  0 , , B4 = , B4 = , B4 = 3 5 4 0.05 0 0.03   10, β4 = 35, H41 = 0.2 0 −0.04 ,     0.1 0.1 −0.02 , H43 = −0.2 0.03 0.05

CR IP T

:

AN US

Reach model 4

14

For simulation of the premise variables, we suppose the function µkj (θk ) is as follows: 1 1 + 1.5exp(− x211 ) µ12 (x11 ) = 1 − µ11 (x11 ) 1 µ21 (x21 ) = 1 + 1.3 exp(−x21 ) 1 µ22 (x21 ) = 1 + 1.7 exp(−x21 ) 2 µ2 (x21 ) = 1 − µ21 (x21 ) − µ22 (x21 ) 1 µ31 (x31 ) = 1 + exp(−1.4x31 ) µ32 (x31 ) = 1 − µ31 (x31 ) 1 µ41 (x21 ) = 1 + 1.8 exp(−x41 ) 1 µ42 (x21 ) = 1 + 1.2 exp(−x41 ) 4 µ2 (x21 ) = 1 − µ41 (x21 ) − µ42 (x41 )

(59)

µ12 (θ1 ) =

(60)

ED

µ21 (θ2 ) =

M

µ11 (θ1 ) = µ11 (x11 ) =

PT

µ22 (θ2 ) = µ23 (θ2 ) =

AC

CE

µ31 (θ3 ) =

µ32 (θ3 ) = µ41 (θ4 ) = µ42 (θ4 ) = µ43 (θ4 ) =

(61) (62) (63) (64) (65) (66) (67) (68)

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Therefore, the T–S fuzzy models of four isolated reaches are of the following form: Reach 1 : Rule1 :

If x11 (k) is (59) then

x1 (k + 1) = A11 x1 (k) + D11 x1 (k − d1 ) + B11 u1 (k) + M11 q1 (k) + Γ11 w1 (k) p1 (k) = E11 x1 (k) + H11 x1 (k − d1 ) + L11 q1 (k)

Rule2 :

z1 (k) = C11 x1 (k) + G11 x1 (k − d1 ) If x11 (k) is (60) then

y1 (t) = E12 x1 (t) + H12 x1 (k − d1 ) + L21 q1 (k) z1 (t) = C12 x1 (t) + G21 x1 (k − d1 ) Reach 2 :

Rule1 :

If x21 (k) is (61) then

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x1 (k + 1) = A21 x1 (k) + D12 x1 (k − d1 ) + B12 u1 (k) + M12 q1 (k) + Γ21 w1 (t)

x2 (k + 1) = A12 x2 (k) + D21 x1 (k − d2 ) + B21 u2 (t) + M21 q2 (k) + Γ12 w2 (t)

Rule2 :

AN US

p2 (k) = E21 x2 (k) + H21 x1 (k − d2 ) + L12 q2 (k) z2 (k) = C21 x2 (k) + G12 x1 (k − d2 ) If x11 (k) is (62) then

x2 (k + 1) = A22 x2 (k) + D22 x1 (k − d2 ) + B22 u2 (k) + M22 q2 (k) + Γ22 w2 (k) p2 (k) = E22 x1 (k) + H22 x1 (k − d2 ) + L22 q2 (k)

z2 (k) = C22 x1 (k) + G22 x1 (k − d2 ) If x21 (kt) is (63) then

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Rule3 :

x2 (k + 1) = A32 x2 (k) + D23 x1 (k − d2 ) + B23 u2 (k) + M23 q2 (k) + Γ32 w2 (k) p2 (k) = E23 x1 (k) + H23 x1 (k − d2 ) + L32 q2 (k)

ED

z2 (k) = C23 x1 (k) + G32 x1 (k − d2 ) Reach 3 :

If x13 (k) is (64) then

PT

Rule1 :

x3 (k + 1) = A13 x3 (k) + D31 x3 (k − d3 ) + B31 u3 (k) + M31 q3 (k) + Γ13 w3 (k) p3 (k) = E31 x3 (k) + H31 x1 (k − d3 ) + L13 q3 (k)

CE

z3 (k) = C31 x3 (k) + g31 x1 (k − d3 )

AC

Rule2 :

If x13 (t) is (65) then

x3 (k + 1) = A23 x3 (t) + D32 x3 (k − d3 ) + B32 u3 (k) + +M32 q3 (k) + Γ23 w3 (k) p3 (p) = E32 x3 (k) + H32 x3 (k − d3 ) + L23 q3 (k) z3 (p) = C32 x3 (k) + G23 x3 (k − d3 )

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Subsystem Closed loop 1

10

x1 x2

5

States

0

−5

−15

0

50

100

150 Time

200

Fig. 2. Closed-loop response of subsystem (reach) 1

Reach 4 : If x14 (k) is (66) then

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Rule1 :

250

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−10

x4 (k + 1) = A14 x4 (t) + D41 x4 (t − d4 ) + B41 u4 (t) + M41 q4 (k) + Γ14 w4 (t) p4 (k) = E41 x3 (t) + H41 x4 (t − d4 ) + L12 q4 (k)

Rule2 :

z4 (k) = C41 x3 (t) + G14 x4 (t − d4 ) If x14 (t) is (67) then

x3 (k + 1) = A24 x4 (t) + D42 x4 (t − d4 ) + B42 u4 (t) + M42 q4 (k) + Γ24 w4 (t)

Rule3 :

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p3 (k) = E42 x4 (t) + H42 x4 (t − d4 ) + L14 q4 (k) z3 (k) = C42 x4 (t) + G24 x4 (t − d4 ) If x14 (t) is (68) then

ED

x4 (k + 1) = A34 x4 (t) + D43 x4 (t − d4 ) + B43 u4 (t) + M43 q4 (k) + Γ34 w4 (t) p3 (k) = E43 x4 (t) + H43 x4 (t − d4 ) + M43 q4 (k) z3 (k) = C43 x4 (t) + G34 x4 (t − d4 )

CE

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To illustrate the effectiveness of our quantized fuzzy control design, we employ the MATLAB-LMI solver and it is found that the feasible solution of Theorem 2 yields the following gain matrices and performance levels:     −0.4023 0.5102 , K2 = 0.3715 −0.1671 , K1 =     −0.8522 0.3361 −1.0034 , K4 = −1.3291 0.0732 −0.4428 K3 =

AC

Typical simulation results are shown in Figs 2-5 for the closed-loop response of the algae (first state) and ammonia products (second state) trajectories due to input disturbance wj (k) = 0.5exp(−0.1k)sin(0.01πk) starting from the initial state conditions:     9.2 −10 , Reach 2 = 9.9 −11.2 , Reach 1 =     −1 2.6 3.5 , Reach 4 = −1 4 2 Reach 3 =

From these results, the smooth behavior of the state trajectories for the four subsystem is recorded. This illuminates the fact that the developed control algorithm has been able to regulate the water quality constituents and clear the stream in quite short time. The ensuing profiles of control input (in-reach water speed) are depicted in Figs. 6-9.

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10 x1 x2 5

0

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−15

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100

200

250

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Fig. 3. Closed-loop response of subsystem (reach) 2

150

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Fig. 4. Closed-loop response of subsystem (reach) 3

Fig. 5. Closed-loop response of subsystem (reach) 4

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Fig. 6. Control input profile to reach 1

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Fig. 7. Control input profile to reach 2

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Fig. 8. Control input profile to reach 3

Fig. 9. Control input profile to reach 4

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B. Example 2 For the purpose of illustrating the effect of quantization parameters, we consider an large-scale fuzzy system composed of two subsystems of the type (6)–(9) and having uniform quantizers with the following data:

A11 = B11

=

C11 = A21 = B12

=

C12 =

E21 N21

A22







   0.9 0.1 0 −0.1 0 0.2     =  0 0.5 −0.1  , D21 =  −0.2 −0.1 0.1  , Φ11 = 1 , Ψ11 = 0.5 , 0.1 0 0.4 −0.3 −0.2 −0.2     0.5 0.5   =  1.5  , Γ21 =  1  , G12 = −0.2 0.04 0.2 , 0.3 0.4       −0.02 0.01 0.1 , L12 = 0.03 0 0.02 , C21 = 0.6 1 0 , =     1 0 0 1 0 0 =  0 1 0  , M21 =  0 1 0  , 0 0 1 0 0 1       0.5 −0.2 0.04 0.2 0.9 0.1 0 =  0 0.5 −0.1  , D22 =  −0.4 −0.15 0  , B22 =  1.5  , 0.1 0 0.3 0.4 0.1 0 0.4       2 2 1 0.2 0.7 , E2 = −0.02 0.01 0.1 , L2 = 0.03 0.02 −0.01 , =     1 0 0 1 0 0 =  0 1 0  , M22 =  0 1 0  0 0 1 0 0 1

CE

G22



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B21



ED

A12

:



       0.8 0 −0.1 0 , D11 = , Φ11 = 0.5 , Ψ11 = 1 , 0.05 0.9 −0.2 −0.1        1 , G11 = 1 0.5 , L1 = 0.1 0.2 , E11 = 0.1 0.02 , 0.5        1 1 0 1 0 1 1 1 1 0.5 , Γ 1 = , M1 = , N1 = , 0.5 0 1 0 1        0.7 0 −0.3 0 , D12 = , Φ21 = 0.7 , Ψ21 = 0.8 , 0.03 0.6 −0.1 −0.2        2 , G21 = 0.4 1 , L1 = 0.4 0.3 , E12 = 0.3 0.02 , 0.6        0.5 1 0 1 0 2 2 2 0.6 1 , Γ1 = , M1 = , N1 = 2 0 1 0 1

PT

Subsystem 2



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:

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Subsystem 1

N22

AC

We let x1 (t) = [xt11 (t) xt21 (t)], x2 (t) = [xt12 (t) xt22 (t) xt32 (t)], representing the state vectors of subsystem 1 and 2, respectively. For simulation of the premise variables, we suppose the membership functions are as follows: 1 1 + 0.09[2 − x11 (t)]2 µ12 (x11 ) = 1 − µ11 (x11 ) 1 µ21 (x12 ) = 1 + 0.35[2 − x12 (t)]2 µ22 (x12 ) = 1 − µ21 (x12 ) µ11 (x11 ) =

(69) (70) (71) (72)

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Therefore, the T–S fuzzy models of two isolated subsystems are of the following form: Subsystem 1 : Rule1 :

If x11 (k) is (69) then

x1 (k + 1) = A11 x1 (k) + D11 x1 (k − d1 ) + B11 u1 (k) + c11 + Γ11 w1 (k) y1 (k) = C11 x1 (k) + E11 x1 (k − d1 ) + Φ11 w1 (k)

Rule2 :

z1 (k) = G11 x1 (k) + L11 x1 (k − d1 ) + Ψ11 w1 (k) If x11 (k) is (70) then

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x2 (k + 1) = A21 x1 (k) + D12 x1 (k − d1 ) + B12 u1 (k) + c21 + Γ21 w1 (k) y1 (k) = C12 x1 (k) + E12 x1 (t − d1 ) + Φ21 w1 (k)

z1 (k) = G21 x1 (k) + L21 x1 (k − d1 ) + Ψ21 w1 (k) Subsystem 2 :

Rule1 :

If x21 (k) is (71) then

x2 (k + 1) = A12 x2 (k) + D22 x2 (k − d2 ) + B21 u2 (k) + c12 + Γ12 w2 (k)

Rule2 :

AN US

y2 (t) = C21 x2 (k) + E21 x1 (k − d2 ) + Φ12 w2 (k) z2 (t) = G12 x2 (k) + L12 x1 (k − d2 ) + Ψ12 w2 (k) If x21 (k) is (72) then

x2 (k + 1) = A22 x2 (k) + D22 x1 (k − d2 ) + B21 u2 (k) + c22 + Γ22 w2 (t) y2 (k) = C22 x1 (k) + E22 x1 (k − d2 ) + Φ22 w2 (k)

z2 (k) = G22 x1 (k) + L22 x1 (k − d2 ) + Ψ22 w2 (k)

ED

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Using α1 = 10, β1 = 25, α2 = 15, β2 = 30, It is found by applying the MATLAB-LMI solver that the feasible solution of LMIs (50) of Theorem 2 is attained at     −0.6111 0.3472 , K2 = 0.7429 −0.6481 0.0564 K1 = The disturbance input is injected randomly with the initial state conditions:     1 0 , Subsystem 2 = 0 −0.3 0.1 , Subsystem 1 =

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The resulting state trajectories of both subsystems and the associated control input variations are plotted in Fig. 10. Smooth behavior is recorded. The variation in the switching signal pattern is recorded in Fig. 11. It is noted from the plotted graphs that the time span is different and the reason for this is to ensure smooth settling of the trajectories. VI. C ONCLUSIONS

AC

A robust stabilization scheme has been constructed for fuzzy hybrid nonlinear discrete systems in face of parametric uncertainties and unknown-but-bounded delays. The stabilization scheme has been emerged from statevariable feedback. The stability conditions of the feedback-controlled fuzzy system are LMIs-based and derived from an improved Lyapunov-Krasovskii functional. Both the switching signal and the stabilizing gains are jointly found from the optimization of a quadratic performance measure. The analytical findings have been validated on system applications to illustrate the effectiveness of the developed methodology. ACKNOWLEDGMENT The support of the Deanship of Scientific Research (DSR) at KFUPM through distinguished professorship project No. IN 141003 is hereby acknowledged.

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Fig. 10. State and control trajectories: (left top) Subsystem 1, (right top) Subsystem 2 and (bottom) Controls

Fig. 11. Switching signal pattern

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