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H_/H<sub>∞</sub> Sensor Fault Detection and Isolation of Uncertain Time-Delay Systems
H_/H Sensor Fault Detection and Isolation of Uncertain Time-Delay Systems
Journal of Electrical Engineering and Technology. 2014. Jan, 9(1): 313-323
Copyright © 2014, The Korean Institute of Electrical Engineers
  • Received : July 08, 2013
  • Accepted : September 25, 2013
  • Published : January 01, 2014
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About the Authors
Sung Chul Jee
Dept. of Electronic Engineering, Inha University, Korea. (jeesch@ inha.edu)
Ho Jae Lee
Corresponding Author: Dept. of Electronic Engineering, Inha University, Korea. (mylchi@inha.ac.kr)
Do Wan Kim
Dept. of Electrical Engineering, Hanbat National University, Korea. (dowankim@hanbat.ac.kr)

Abstract
Sensor fault detection and isolation problems subject to H _/ H , performance are concerned for linear time-invariant systems with time delay in a state and parametric uncertainties. To that end, a model-based observer bank approach is pursued. The design conditions for both continuous- and discrete-time cases are formulated in terms of matrix inequalities, which are then converted to the problems solvable via an algorithm involving convex optimization.
Keywords
1. Introduction
Fault detection and isolation (FDI) for dynamic systems holds a major field in the modern control theory academia, due to the ever increasing demand from industry for higher safety and reliability standards [1 - 3] . A great deal of research works on FDI has been released from different points of view, among which a successful one is a model-based observer approach using effective design tools that have been accumulated in the field over decades. See [2] and good references therein.
When dealing with single-output systems, isolating a fault on a sensor is identical to detecting one. On the other hand, it is more complicated in multi-output systems, because constructing a non-interactive map (diagonal transfer matrix for the linear time-invariant (LTI) case) from faults to residuals is particularly challenging, even if the system is LTI [4] . A simple yet effective alternative could be a scheme via an observer bank, where as many observers as sensors are involved [5] . The residual generated by each observer in the bank should be as sensitive to the faults on all sensors except each one and as robust against disturbance, as possible, in a reasonable (for example, an H _ and an H [6 - 10] ) sense. Then the fault can be isolated based on a suitable voting scheme [5 , 11] . However, time delay and uncertainty which are widely believed to exist in physical model configurations [12] could be as the major obstacles against satisfactory FDI.
Paper [5] introduces an application of the observer bank-based FDI method to load-frequency control problem without deliberating time delay, uncertainty, nor any performance index. In [11] , a robust FDI for a robot manipulator is developed, where the residual sensitivity to the fault is considered in the H sense. A solution to FDI is presented in the frame of H model-matching in [13] . However, these referred works do not consider the time delay. Besides, although a fault detector, rather than an isolator, is designed with delays and/or uncertainties in [1 , 14 - 16] , available literatures are relatively few for the FDI of LTI systems with the time delay and the uncertainty in the H _/ H criterion. This motivates our present study.
In this paper, we concern design conditions of the H _/ H sensor FDI observer banks for continuous- and discrete-time LTI systems subject to the time delay in the state and the parametric uncertainty. The bank is composed of the sensors’ number of observers. Both the observer gain and the residual gain (that is not necessarily symmetric or triangular due to additional manipulation such as the matrix square root or a Cholesky factorization) are taken into account as the design variables. Sufficient conditions to find the gains are developed in terms of matrix inequality. An algorithm involving a convex optimization is presented based on the cone complementary linearization technique [12] .
We follow standard notations: A = AT ≺ 0 is a negative definite matrix. ‖ x ‖ stands for a Euclidean norm while ‖ x 2 means the 2 norm. Symbol * denotes a transposed element in a symmetric position. Ellipsis He{ S } := S + ST is used for simplicity. For any vector y ∈ ℝ m and matrix C ∈ ℝ m×n , we define as follows:
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2. Preliminaries
Consider the following uncertain time-delay LTI system:
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where x ∈ ℝ n is the state; y ∈ ℝ m is the sensor output; d ∈ [0, du ], du ∈ ℝ >0 , is the known time-varying delay; and w ∈ ℝ l and f ∈ ℝ m belonging 2 to are the disturbance and the sensor fault, respectively. Matrices Δ A , Δ B , and Δ C represent the parametric uncertainties that satisfy the following assumption:
Assumption 1 : Δ A , Δ B , and Δ C are the real valued matrix functions fulfilling
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where ■ means no restriction on that entry, F 1 , F 2 , E 1 , and E 2 are known constant real matrices of appropriate dimensions and 𝝨 is time-varying real matrix satisfying 𝝨 T 𝝨≼ I
Without loss of generality, assume the observability of ( A , C ). We introduce an m -observer bank in which the p th member should detect all faults on all sensors excluding the p th one. Such a requirement is realized if the p th observer takes the following form
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where p ∈ ℐ M := {1, 2,…, m },
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∈ ℝ n is the estimated state; ŷ ∈ ℝ m is the observer output; r ∈ ℝ m−1 is the residual; and L and H are the observer and the residual gains to be designed, respectively.
Let e := x
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and
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. The residual is then generated by the following augmented error dynamics
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where
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and
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For excellent FDI, (2) is desired to be designed so that the effect of
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to r is encouraged and the effect of w to r is attenuated. In connection with this, we recall the following performance measures:
Definition 1 : For a map from
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to r in (3), the H _ performance is defined by [9]
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For a map from w to r , the H performance is defined by
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Definition 2 : Let the evaluation function be
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and the threshold function be
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where TW ∈ ℝ >0 is the constant time window. In cases where it is necessary to display explicitly the association of J r and J th with the residual from the p th observer, we write J rp and J th p , respectively. Define an FDI logic as Table 1 .
FDI logic
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FDI logic
Problem 1 : Given the fault sensitivity level β ∈ ℝ >0 and the disturbance attenuation level 𝛾 ∈ ℝ >0 , find L and H in (2) so as to satisfy
  • (C1) (2) is asymptotically stable whenf= 0 andw= 0;
  • (C2)_>βwhenw= 0 with the initial conditionx(0) =(0) = 0;
  • (C3)∞<γwhenf= 0 withx(0) =(0) = 0.
3. Main Results
We recall the following lemmas, before proceeding further.
Lemma 1 : Given any matrices X = XT ≻ 0 , and M and vectors a (·) and b (·) defined on an interval 𝛀 , the following inequality holds [17] :
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Lemma 2 : For any compatible matrices S = ST ≺ 0, F , and E , the following equivalence holds:
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for some ε ∈ ℝ >0 , where 𝝨 T 𝝨 ≼ I .
Lemma 3 : For any compatible matrices Q , R ,
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, and Ȓ , the following inequality holds:
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Proof: It is readily proved as follows:
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Now, we summarize the main result on Problem 1.
Theorem 1 ((β, 𝛾)-H_H FDI): Given β , 𝛾 ∈ ℝ >0 (2) has (β, 𝛾)-H_H FDI performance and is asymptotically stable, if there exist , P = PT := blockdiag { P 1 , P 2 } ≻ 0, Q = QT ≻0, X = XT ≻ 0, Z = ZT ≻ 0, Ĥ = ĤT ≻ 0,
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, W , and Y , such that
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where
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Then the gains are given by ( L , H ) = ( P 2 −1 N , Ĥ −1 W ).
Proof: Denote
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and
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. Define the following positive definite function
V ( z ( t α ), α ∈ [0, du ]) := V 1 + V 2 + V 3
where
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Using the basic calculus fact
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(3) is rewritten as
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Letting
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and b ( α ) := Pz ( α ) , and utilizing Lemma 1, we further compute
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Replacing Y = X M P and assuming
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yield
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From (9), if
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= 0, it holds
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Integrating the Hamilton—Jacobi—Bellman (H—J—B) inequality
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from 0 to ∞, the following relation holds
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Indeed (10) is necessity for (6) because for all ( z , z ( t d ), w ) ∈ ℝ 2n \ {0} × ℝ 2n \ {0} × ℝ l
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where we have used Schur complement, the congruence transformation, and Lemma 2. This also implies the asymptotic stability of (2) ((C1)). The proof of (7) ⇒ (C2) is along a similar line to the proof of (6) ⇒ (C3). Next, letting w = 0 in (9) allows one to manipulate
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Considering the following H—J—B inequality
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it is easy to see (12) is a sufficiency for
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Using (11), we know for all ( z , z ( t d ),
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)∈ ℝ 2n \{0}×ℝ 2n \{0}×ℝ m−1 , that
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where we have used Schur complement, the congruence transformation, and Lemmas 2 and 3. ■
Algorithm 1 Iterative algorithm
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Remark 1: The matrix inequalities in Theorem 1 are not linear due to the terms ĤĤ and − PZ −1 P . It provokes a non-convex feasibility problem that is generally difficult to solve. One may simply attempt to find a feasible solution set through a tractable convex optimization algorithm after recovering their convexity. For instance, we could set Ĥ = I and P = Z , which however, may produce a quite conservative design result.
In what follows, we alternatively solve the problem to obtain a better result based on the cone complementary linearization technique [12] . Introduce V and S such that ĤĤ V and PZ −1 P S or
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Let (6) ' and (7) ' be (6) and (7) with the respective replacements of ĤĤ and − PZ −1 P by V and S . Then we know that (6) ' ,(7) ' ,(8)⇒(6)−(8). Defining new variables ( M, K, U, J, R ) := ( V −1 , Ĥ −1 , S −1 , P −1 , Z −1 ), (13) is represented as
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Then, the problem concerned above is converted to the following nonlinear minimization one involving LMIs:
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Algorithm 1 summarizes an iterative LMI approach to MP 1.
4. Paralleling to Discrete-Time Case
In this section, we discuss the FDI problem in the discrete-time domain. Consider the following discrete-time uncertain time-delay LTI system:
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where wk , fk l 2 , and dk ∈ [0, du ] ⊂
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The m - observer bank whose p th observer is in the form of
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is desired to be designed to possess the ( β , 𝛾 )- H _/ H performance in the l 2 -norm sense. The augmented error dynamics is written as
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where
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and
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.
Lemma 4: For any compatible matrices X = XT ≻ 0, Y , N , Z = ZT ≻ 0, following inequality is satisfied [12] :
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where
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Theorem 2: Given β , 𝛾 ∈ ℝ >0 , (17) has ( β , 𝛾)- H _/ H FDI performance and is asymptotically stable, if there exist P :=blockdiag{ P 1 , P 2 } = PT ≻ 0, Q = QT ≻ 0, X = XT ≻ 0, Ĥ = ĤT ≻ 0, Z = ZT ≻ 0,
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, W , and Y , such that
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and (19), where
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Then the gains are given by
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.
Proof: Define the positive definite function V k := V 1k + V 2k + V 3k + V 4k , where
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and
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where Δ zh := zh +1 zh . Since
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, one rewrite (18) as
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By assigning a := zk , b := Δ zh and
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in Lemma 4, we obtain
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Moreover, we have
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From the above relations, Δ Vk is majorized by
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In case of
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is written as
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Summing up the discrete-time H—J—B inequality
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along (18) from 0 to ∞ yields
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Inequality (21) guarantees this 𝛾- H performance in the l 2 -norm sense, because for all ( zk , zk dk , wk ) ∈ ℝ 2n \ {0} × ℝ 2n \ {0} × ℝ l one can derive
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where we have used Schur complement, the congruence transformation, and Lemma 2. This also implies the asymptotic stability of (17). Finally, it is not difficult to prove that (21) is the sufficiency for the β - H performance in the l 2 -norm sense when one considers (22) with wk = 0 to induce the following inductions
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where we have used Schur complement, the congruence transformation, Lemma 2, and Lemma 3 with the assignments
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, R := H ,
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,
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therein.
Though the design condition in Theorem 2 is casted in a nonlinear form, it can be efficiently solved by convex optimization algorithms in a manner similar to Algorithm 1 through the following conversion.
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5. Example
Consider the state-space data for (1) borrowed from [15]
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and du = 1. We further suppose by Assumption 1 that Δ A , Δ B , and Δ C and are decomposed as
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Since y ∈ ℝ 3 , a three-observer bank is employed. With a slight abuse of notations, in the sequel, the subscript p ∈ { 1, 2, 3 } denotes each observer in the bank. First, we attempt to find the gains by fixing Ĥp = I and Pp = Zp in Theorem 1, which convexifies the inequalities. However, as is concerned, it turns out to fail in finding any feasible solutions, even though we do our best in adjusting βp ’s, 𝛾p ’s, and εp ’s. On the other hand, for the given ( β 1 , β 2 , β 3 )=(1.1, 1.2, 1.6), and 𝛾p = 1, and , εp = 0.1, p ∈ { 1, 2, 3 }, we succeed in finding the following gains by Algorithm 1 in 8 iterations for the first and the second observer, respectively, and 16 iterations for the third one:
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For simulation, we introduce a delay d ∈ ℝ ≽0 and a disturbance w 2 randomly varying within [0, du ] and (−0.5, 0.5) , respectively. A fault f 2 forms the transient pattern as
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for 50 s . Set Twp = 3 , then the threshold is generally calculated as
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under the zero-initial condition for both the system and the observer bank.
Three cases are simulated to investigate the validity of the designed bank.
i) In the fault-free-disturbance-activated case, the threshold is computed as J th p = 0.5 𝛾p = 0.5, p ∈ {1, 2, 3 }. As shown in Fig.1 , J r p , p ∈ { 1, 2, 3 }, does not exceed J th p for all t ∈ [0.50], from which one judge based on Table 1 that there does not occur any fault, as it really is. The evaluation
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=(0.8691,0.7275,0.9093)≺(𝛾 1 ,𝛾 2 ,𝛾 3 )=(1, 1, 1) confirms that the design goal — the 𝛾- H performance — is achieved.
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Residual evaluations when f = 0 but w2 \ {0}: Jrp (solid) Jthp and (dashed).
ii) Next, we simulate the fault-activated-disturbance-free case. In this case, the threshold is modified to
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Fig. 2 reveals the comparison
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Residual evaluations when f ∈ 2 \ {0} but : 0: Jrp (solid) and Jthp (dashed).
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By the FDI logic in Table 1 , the observer bank successfully declares that f 1 , f 2 , f 3 are isolated for t ∈ (5.97, 16.54), t ∈ (21.73, 30.70), t ∈ (36.54, 45.96), respectively, with a retard up to 2 s from the fault-arising time. The evaluated values
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=(1.4049, 1.5931, 2.1226) are greater than ( β 1 , β 2 , β 3 )=(1.1, 1.2, 1.6), entrywisely, which verifies the β - H _ performance of the observer bank.
iii) Now, we consider the fault-disturbance-activated case. The fault and the disturbance data applied in the previous runs are activated at the same time. Analyzing the result in Fig. 3 with the threshold
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Residual evaluation when f ≠ 0 and w ≠ 0: Jrp(solid) and Jthp (dashed).
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According to the FDI logic in Table 1 , we recognize that each sensor malfunctions for t ∈ (6.40, 16.10), t ∈ (22.06, 30.23), t ∈ (36.97, 45.48), in regular order. This means that the observer bank designed by the proposed method is robust against the disturbance, the uncertainties, and the time delay so that any misleading alarm is not issued, as should be expected.
iv) To highlight the benefit of the proposed method, we simulate a comparable scheme, Theorem 1 in [18] that does not consider the time-delay nor uncertainties. By this technique we compute
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and apply them to the fault-disturbance-activated case. As shown in Fig. 4 , the compared only isolates f 1 for t ∈ [4.87, 19.89) For the rest of the simulation-run time, any isolation is not declared but detection is kept alarmed although there exist no fault for t ∈ [30.35) and t ∈ [45,50), which less accurate than ours.
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Residual evaluation by Theorem 1 in [18] when f ≠ 0 and w ≠ 0: Jrp(solid) and Jthp (dashed).
6. Conclusions
In this paper, we presented the H _/ H FDI observer bank design techniques for uncertain time-delay LTI systems for both continuous- and discrete-time settings. Design conditions are developed in the format of LMIs using the cone complementary linearization algorithm. Simulation results convincingly demonstrated the effectiveness of the developed methodology.
Acknowledgements
This work was supported by INHA UNIVERSITY Research Grant.
BIO
Sung Chul Jee received B.S . and M.S. degrees from the Department of Electronic Engineering, Inha University, Incheon, Korea, in 2009 and 2011, respectively. Now, He is currently pursuing a Ph.D. degree at the same university. His research interests include fuzzy control systems, fault detection and isolation, and their applications.
Ho Jae Lee received B.S., M.S., and Ph. D. degrees from the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 1998, 2000, and 2004, respectively. In 2005, he was a Visiting Assistant Professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. Since 2006, he has been with the School of Electronic Engineering, Inha University, Incheon, Korea, where he is currently an Assistant Professor. His research interests include fuzzy control systems, hybrid dynamical systems, large-scale systems, and digital redesign.
Do Wan Kim received the B.S., M.S., and Ph.D. degrees from the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 2002, 2004, and 2007, respectively. He was with the Engineering Research Institute, Yonsei University, and a Visiting Scholar with the Department of Mechanical Engineering, University of California, Berkeley. In 2009, he was a Research Professor with the Department of Electrical and Electronic Engineering, Yonsei University. Since 2010, he has served on the faculty in the Department of Electrical Engineering, Hanbat National University, Daejeon, Korea. His current research interests include analysis and synthesis of nonlinear sampled-data control systems.
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