Robust H State-Feedback Control Design for Fuzzy Singularly Perturbed Systems with Markovian Jumps: An LMI Approach

Transcription

1 Robust H State-Feedback Control Design for Fuzzy Singularly Perturbed Systems with Markovian Jumps: An LMI Approach 75 Robust H State-Feedback Control Design for Fuzzy Singularly Perturbed Systems with Markovian Jumps: An LMI Approach Wudhichai Assawinchaichote and Sing Kiong Nguang Non-members ABSTRACT This paper examines the problem of designing a robust H state-feedback controller for a class of uncertain Markovian jump nonlinear singularly perturbed systems described by a Takagi-Sugeno (TS fuzzy model with Markovian jumps Based on a linear matrix inequality (LMI approach LMI-based sufficient conditions for the uncertain Markovian jump nonlinear singularly perturbed systems to have an H performance are derived The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard but also to nonstandard nonlinear singularly perturbed systems A numerical example is provided to illustrate the design developed in this paper Keywords: H control Singularly perturbed systems Markovian Jumps Fuzzy systems INTRODUCTION Markovian jump systems sometimes called hybrid systems with a state vector consists of two components; ie the state (differential equation and the mode (Markov process The Markovian jump system changes abruptly from one mode to another mode caused by some phenomenon such as environmental disturbances changing subsystem interconnections and fast variations in the operating point of the system plant etc The switching between modes is governed by a Markov process with the discrete and finite state space Over the past few decades the Markovian jump systems have been extensively studied by many researchers; see This is due to the fact that jumping systems have been a subject of the great practical importance For the past three decades singularly perturbed systems have been intensively studied by many researchers; see 7 Singularly perturbed systems also known as multiple time-scale dynamic systems SP4R37: Manuscript received on August 5 5 ; revised on June 9 5 The Department of Electronic and Telecommnunication Engineering King Mongkut s University of Technology Thonburi 9 Prachautits Rd Bangkok 4 Thailand wudhichaiasa@kmuttacth The Department of Electrical and Computer Engineering The University of Auckland Private Bag 99 Auckland New Zealand normally occur due to the presence of small parasitic parameters typically small time constants masses etc In state space such systems are commonly modelled using the mathematical framework of singular perturbations with a small parameter say determining the degree of separation between the slow and fast modes of the system However it is necessary to note that it is possible to solve the singularly perturbed systems without separating between slow and fast mode subsystems But the requirement is that the parasitic parameters must be large enough In the case of having very small parasitic parameters which normally occur in the description of various physical phenomena a popular approach adopted to handle these systems is based on the socalled reduction technique According to this technique the fast variables are replaced by their steady states obtained with frozen slow variables and controls and the slow dynamics is approximated by the corresponding reduced order system This time-scale is asymptotic that is exact in the limit as the ratio of the speeds of the slow versus the fast dynamics tends to zero In the last few years the research on singularly perturbed systems in the H sense has been highly recognized in control area due to the great practical importance H -optimal control of singularly perturbed linear systems under either perfect state measurements or imperfect state measurements has been investigated via differential game theoretics approach Although many researchers have studied the H control design of linear singularly perturbed systems for many years the H control design of nonlinear singularly perturbed systems remains as an open research area This is due to in general nonlinear singularly perturbed systems can not be decomposed into slow and fast subsystems Recently a great amount of effort has been made on the design of fuzzy H for a class of nonlinear systems which can be represented by a Takagi-Sugeno (TS fuzzy model; see 8 Recent studies 8 show that a fuzzy model can be used to approximate global behaviors of a highly complex nonlinear system In this fuzzy model local dynamics in different state space regions are represented by local linear systems The overall model of the system is obtained by blending of these linear models through nonlinear fuzzy membership functions Unlike conventional

2 76 ECTI TRANSACTIONS ON ELECTRICAL ENG ELECTRONICS AND COMMUNICATIONS VOL3 NO AUGUST 5 modelling which uses a single model to describe the global behavior of a system fuzzy modelling is essentially a multi-model approach in which simple submodels (linear models are combined to describe the global behavior of the system Employing the existing fuzzy results 8 on the singularly perturbed system one ends up with a family of ill-conditioned linear matrix inequalities resulting from the interaction of slow and fast dynamic modes In general ill-conditioned linear matrix inequalities are very difficult to solve What we intend to do in this paper is to design a robust H fuzzy state-feedback controller for a class of uncertain nonlinear singularly perturbed systems with Markovian jumps First we approximate this class of uncertain nonlinear singularly perturbed systems with Markovian jumps by a Takagi-Sugeno fuzzy model with Markovian jumps Then based on an LMI approach we develop a technique for designing a robust H fuzzy state-feedback controller such that the L -gain of the mapping from the exogenous input noise to the regulated output is less than a prescribed value To alleviate the ill-conditioned linear matrix inequalities resulting from the interaction of slow and fast dynamic modes these ill-conditioned LMIs are decomposed into -independent LMIs and -dependent LMIs The -independent LMIs are not ill-conditionedandthe-dependent LMIs tend to zero when approaches to zero If is sufficiently small the original ill-conditioned LMIs are solvable if and only if the -independent LMIs are solvable The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard but also to nonstandard singularly perturbed systems This paper is organized as follows In Section system descriptions and definition are presented In Section 3 based on an LMI approach we develop a technique for designing a robust H fuzzy statefeedback controller such that the L -gainofthemapping from the exogenous input noise to the regulated output is less than a prescribed value for the system described in Section The validity of this approach is demonstrated by an example from a literature in Section 4 Finally conclusions are given in Section 5 SYSTEM DESCRIPTIONS AND DEFINI- TIONS The class of nonlinear uncertain singularly perturbed system with Markovian jumps under consideration is described by the following TS fuzzy model with Markovian jumps: E ẋ(t = r i= µ i(ν(t A i (η(t + A i (η(tx(t +B i (η(t + B i (η(tw(t +B i (η(t + B i (η(tu(t x( = z(t = r i= µ i(ν(t C i (η(t + C i (η(tx(t +D i (η(t + D i (η(tu(t ( I E = >isthe singular perturbation parameter ν(t = ν (t ν ϑ (t is I the premise variable that may depend on states in many cases µ i (ν(t denote the normalized timevarying fuzzy weighting functions for each rule ϑ is the number of fuzzy sets x(t R n is the state vector u(t R m is the input w(t R p is the disturbance which belongs to L z(t R s is the controlled output the matrix functions A i (η(t B i (η(t B i (η(t C i (η(t D i (η(t A i (η(t B i (η(t B i (η(t C i (η(t and D i (η(t are of appropriate dimensions {η(t} is a continuous-time discrete-state Markov process taking values in a finite set S = { } with transition probability matrix Pr = {P ık (t} given by P ık (t = Pr(η(t + =k η(t =ı { λık +O( if ı k = +λ ıı +O( if ı = k ( > and lim O( = Hereλ ık is the transition rate from mode ı (system operating mode to mode k (ı k and λ ıı = s k=k ı λ ık (3 For the convenience of notations we let µ i = µ i (ν(t η = η(t and any matrix M(µ ı = M(µ η = ı The matrix functions A i (η B i (η B i (η C i (η and D i (η represent the timevarying uncertainties in the system and satisfy the following assumption Assumption : A i (η =F (x(tηth i (η B i (η =F (x(tηth i (η B i (η =F (x(tηth 3i (η C i (η =F (x(tηth 4i (η and D i (η =F (x(tηth 5i (η H ji (η j = 5 are known matrices which characterize the structure of the uncertainties Furthermore there exists a positive function ρ(η such that the following inequality holds: F (x(tηt ρ(η (4

3 Robust H State-Feedback Control Design for Fuzzy Singularly Perturbed Systems with Markovian Jumps: An LMI Approach 77 We recall the following definition Definition : Suppose γ is a given positive number A system of the form ( is said to have the L -gain less than or equal to γ if Tf E {z T (tz(t γ w T (tw(t} dt x( = (5 E stands for the mathematical expectation for all T f and all w(t L T f Note that for the symmetric block matrices we use ( as an ellipsis for terms that are induced by symmetry 3 ROBUST H FUZZY STATE-FEEDBACK CONTROL DESIGN This section provides the LMI-based solutions to the problem of designing a robust H fuzzy statefeedback controller that guarantees the L -gainofthe mapping from the exogenous input noise to the regulated output to be less than some prescribed value First we consider the following H fuzzy statefeedback which is inferred as the weighted average of the local models of the form: u(t = µ j K j x(t (6 j= Then we describe the problem under our study as follows Problem Formulation: Given a prescribed H performance γ> design a robust H fuzzy statefeedback controller of the form (6 such that the inequality (5 holds Before presenting our first main result the following lemma is needed Lemma : Consider the system ( Given a prescribed H performance γ> for ı = sif there exist matrices P =P T positive constants δ and matrices Y j j = r such that the following -dependent linear matrix inequalities hold: P > (7 Ψ ii (ı < i = r (8 Ψ ij (ı +Ψ ji (ı < i < j r (9 Ψ ij (ı = Φ ij (ı ( T ( T ( T R B T i γr ( T ( T Υ ij (ı γr ( T Z T (ı P(ı ( Φ ij (ı =A i E P +E P A T i +B i Y j +Yj T B T i +λ ıı E P ( Υ ij (ı = C i E P + D i Y j ( R =diag{δiiδii} (3 ( Z(ı = λı E P λ ı(ı+ E { P(ı = diag E E λ ı(ı E P (4 P λ ıs E P P ( E P (ı } P (ı + E P (s (5 with B i = I I I B i C i = γρh T i ℵρH4 T i D i = ℵ = ℵC T i ℵρH5 T i T T γρh3 T i ℵD T i I + ρ H T i H j then the inequality (5 holds Furthermore a suitable choice of the fuzzy controller is u(t = µ j K j x(t (6 j= K j =Y j (P E (7 Proof: The closed-loop state space form of the fuzzy system model ( with the controller (6 is given by E ẋ(t = µ i µ j (A i +B i K j x(t + A i + B i K j x(t +B i + B i w(t x( = (8 or in a more compact form E ẋ(t= r ( r µ iµ j A i +B i K j x(t + B i R w(t x( = (9 B i = I I I B i ( F (x(tıth i x(t w(t=r F (x(tıth i w(t F (x(tıth 3i K j x(t ( w(t

4 78 ECTI TRANSACTIONS ON ELECTRICAL ENG ELECTRONICS AND COMMUNICATIONS VOL3 NO AUGUST 5 Consider a Lyapunov functional candidate as follows: V (x(tı=γx T (tq x(t ı S ( Note that Q is constant for each ı For this choice we have V (ı =andv (x(tı only when x(t Now let consider the weak infinitesimal operator of the joint process {(x(tıt } whichis the stochastic analog of the deterministic derivative {(x(tıt } is a Markov process with infinitesimal operator given by 5 V (x(tı=γẋ T (tq x(t+γx T (tq ẋ(t s +γx T (t λ ık Q (kx(t = k= µ i µ j (γx T (tq (A i +B i K j x(t +γx T (ta i +B i K j T Q x(t +γx T (tq B i R w(t +γ w T (tr B T i Q x(t s +γx T (t λ ık Q (kx(t (3 k= Adding and subtracting ℵ z T (tz(t +γ r r r r i= j= m= n= µ iµ j µ m µ n w T (tr w(t to and from (3 we get V (x(tı= ℵ z T (tz(t +γ µ i µ j µ m µ n w T (tr w(t m= n= +ℵ z T (tz(t T x(t +γ µ i µ j µ m µ n w(t m= n= T A i +B i K j Q +Q A i +B i K j + ( T s k= λ ıkq (k R B T i Q γr x(t (4 w(t Now let us consider the following terms: γ m= n= µ i µ j µ m µ n w T (tr w(t ρ γ µ i µ j µ m µ n x T (t δ m= n= { } H T i H m +KjH T 3 T i H 3m K n x(t +ℵ γ w T (tw(t (5 and ℵ z T (tz(t ℵ m= n= µ i µ j µ m µ n x T (t { C i +D i K j T C m +D m K n +ρ H 4i +H 5i K j T } H 4m +H 5m K n x(t (6 ℵ = Hence γ ( +ρ r i= r j= HT i H j m= n= µ i µ j µ m µ n w T (tr w(t +ℵ z T (tz(t µ i µ j µ m µ n (x T (t m= n= Ci + D T i K j R Cm + D m K n x(t + ℵ γ w T (tw(t C i = D i = γρh T i ℵρH T 4 i ℵC T i ℵρH5 T i Substituting (7 into (4 we have γρh T 3 i ℵD T i T T (7 V (x(tı ℵ z T (tz(t+γ ℵ w T (tw(t +γ µ i µ j µ m µ n m= n= x(t w(t T Φ ijmn(ı x(t w(t Φ ijmn (ı = (A i +B i K j T Q +Q A i +B i K j + γ Ci + D T i K j ( T R Cm + D m K n + s k= λ ıkq (k R B T i Q γr (8

5 Robust H State-Feedback Control Design for Fuzzy Singularly Perturbed Systems with Markovian Jumps: An LMI Approach 79 Using the fact m= n= µ i µ j µ m µ n Mij T N mn µ i µ j Mij T M ij +N ij Nij T we can rewrite (8 as follows: V (x(tı ℵ z T (tz(t+γ ℵ w T (tw(t T x(t +γ µ i µ j Φ w(t ij(ı x(t w(t = ℵ z T (tz(t+γ ℵ w T (tw(t T +γ µ x(t x(t i Φ w(t ii(ı w(t i= T ( x(t +γ µ i µ j Φ w(t ij(ı +Φ ji(ı i= i<j x(t (9 w(t Φ ij (ı = (A i +B i K j T Q +Q (A i +B i K j + γ Ci + D T i K j R Ci + D i K j + s k= λ ıkq (k R B T i Q Pre and post multiplying (3 by ( P Ξ = I with P =Q we obtain ΞΦ ij (ı Ξ = P A T i +Y j T BT i +A i P +B i Y j + γ Ci P + D T i Y j R Ci P + D i Y j + s k= λ ıkp P (kp R B T i ( T γr ( T γr Note that (3 is the Schur complement of Ψ ij (ı defined in ( Using (8-(9 we learn that (3 Φ ii (ı < i= r (3 Φ ij (ı +Φ ji (ı < i<j r (33 (3 Following from (9 (3 and (33 we know that V (x(tı < ℵ z T (tz(t+γ ℵ w T (tw(t (34 Applying the operator E T f ( dt on both sides of (34 we obtain E Tf V (x(tıdt Tf < E ( ℵ z T (tz(t+γ ℵ w T (tw(tdt From the Dynkin s formula it follows that E Tf V (x(tıdt = EV (x(t f ı(t f Substitute (36 into (35 yields (35 EV (x(ı((36 Tf < E ( ℵ z T (tz(t+γ ℵ w T (tw(tdt EV (x(t f ı(t f + EV (x(ı( Using (34 and the fact that V (x( = ı( = and V (x(t f ı(t f > we have E Tf { } z T (tz(t γ w T (tw(t dt < (37 Hence the inequality (5 holds This completes the proofoflemma Remark : The linear matrix inequalities given in Lemma becomes ill-conditioned when is sufficiently small which is always the case for the singularly perturbed system In general these illconditioned linear matrix inequalities are very difficult to solve Thus to alleviate these ill-conditioned linear matrix inequalities we have the following theorem which does not depend on Now we are in the position to present our first result Theorem : Consider the system ( Given a prescribed H performance γ> for ı = sif there exist matrices P positive constants δ and matrices Y j j = r such that the following -independent linear matrix inequalities hold: EP+P D > (38 Ψ ii < i = r (39 Ψ ij +Ψ ji < i < j r (4 EP = P T E P D = DP T E =

6 8 ECTI TRANSACTIONS ON ELECTRICAL ENG ELECTRONICS AND COMMUNICATIONS VOL3 NO AUGUST 5 ( I Ψ ij = ( D = I Φ ij ( T ( T ( T R B T i γr ( T ( T Υ ij γr ( T Z T P (4 Φ ij = A i P +P T A T i +B i Y j +Yj T B T i +λ ıı P (4 Υ ij = C i P + D i Y j (43 R = diag{δiiδii} (44 ( Z = λı P λ ı(ı P λ ı(ı+ P λıs P (45 P = diag{ P ( P (ı P (ı + P (s } (46 P = P +PT (47 with B i = I I I B i C i = γρh T i ℵρH4 T i ℵC T i D i = ℵρH5 T i ℵ = I + ρ γρh T 3 i ℵD T i T H T i H j then there exists a sufficiently small ˆ >suchthat the inequality (5 holds for ( ˆ Furthermore a suitable choice of the fuzzy controller is u(t = T µ j K j x(t (48 i= K j =Y j (P (49 Proof: Suppose there exists a matrix P such that the inequality (38 holds then P is of the following form: ( P P = P T (5 P 3 with P =P T > andp 3 =P3 T > Let P =E (P + P (5 with P =( P (5 Substituting (5 and (5 into (5 we have ( P P P = P T (53 P 3 Clearly P =P T and there exists a sufficiently small ˆ such that for ( ˆ P > Using the matrix inversion lemma we learn that P = P +M E (54 M = P P ( I + P P P Substituting (5 and (54 into ( we obtain Ψ ij +ψ ij (55 the -independent linear matrix Ψ ij isdefined in (4 and the -dependent linear matrix is ψ ij = A i P + P T A T i +B i Y j +Y T j B T i ( T ( T ( T +λ ıı ˆ P ( T ( T ( C i P T + D ( T i Y j Z T P (56 λı with Z =( ˆ P λı(ı ˆ P λı(ı+ ˆ P λıs ˆ P P = diag{ ˆ P ( ˆ P } (ı ˆ P (ı + ˆ P (s ˆ P = P + P T and Y j = K j M Note that the - dependent linear matrix tends to zero when approaches zero Employing (38-(4 and knowing the fact that for any given negative definite matrix W there exists an > suchthatw + I < one can show that there exists a sufficiently small ˆ > such that for ( ˆ (8 and (9 hold Since (7-(9 hold using Lemma the inequality (5 holds for ( ˆ 4 ILLUSTRATIVE EXAMPLE Consider a modified series dc motor model based on 3 as shown in Fig which is governed by the following difference equations: J d ω(t dt = K m L f ĩ (t (D + D ω(t L dĩ(t dt = Rĩ(t K m L f ĩ(t ω(t+ṽ (t (57

7 Robust H State-Feedback Control Design for Fuzzy Singularly Perturbed Systems with Markovian Jumps: An LMI Approach 8 ω(t = ω(t ω ref (t is the deviation of the actual angular velocity from the desired angular velocity ĩ(t =i(t i ref (t is the deviation of the actual current from the desired current Ṽ (t = V (t V ref (t is the deviation of the actual input voltage from the desired input voltage J is the moment of inertia K m is the torque/back emf constant D is the viscous friction coefficient and R a R f L a and L f are the armature resistance the field winding resistance the armature inductance and the field winding inductance respectively with R = R f + R a and L = L f + L a Note that in a typical seriesconnected dc motor the condition L f L a holds When one obtains a series-connected dc motor we have i(t =i a (t =i f (t Now let us assume that J J The parameters for the system are given as R = Ω L f =5 H D =5 N m/rad/s and K m = N m/a Substituting the parameters into (58 we get ẋ (t 5 5 = J J x (t x (t ẋ (t 5x (t x (t + w(t+ u(t J J x (t x (t x (t x (t z(t = + u(t x (t (59 x(t =x T (t x T (t T is the state variables w(t =w T (t w T (t T is the disturbance input u(t is the controlled input and z(t is the controlled output The control objective is to control the state variable x (t for the range x (t N N For the sake of simplicity we will use as few rules as possible Note that Fig shows the plot of the membership function represented by M (x (t = x (t+n N N Fig: A modified series dc motor equivalent circuit Giving x (t = ω(t x (t =ĩ(t andu(t =Ṽ (t (57 becomes ẋ (t ẋ (t = D (J+ J K m L f x (t + K ml f x (t x (t (J+ J x (t R u(t (58 and M (x (t = x (t N N N Knowing that x (t N N the nonlinear system = L represents a small parasitic parameter Assume that the system is aggregated into 3 modes as shown in Table : Table : System Terminology Mode ı Moment of Inertia J ± J (kg m Small 5 ±% Normal 5 ±% 3 Large 5 ±% The transition probability matrix that relates the three operation modes is given as follows: P ık = Fig: Membership functions for the two fuzzy set (59 can be approximated by the following TS fuzzy model E ẋ(t = µ i A i + A i x(t i= +B i w(t+b i u(t x( = z(t = i= µ i C i x(t+d i u(t µ i is the normalized time-varying fuzzy weighting functions for each rule i = x(t = x (t x (t

8 8 ECTI TRANSACTIONS ON ELECTRICAL ENG ELECTRONICS AND COMMUNICATIONS VOL3 NO AUGUST 5 A = F (x(tıth E = A =F (x(tıth N A ( = 5N N A ( = 5N N A ( = 5N N A ( = 5N N A (3 = 5N N A (3 = 5N B =B = B =B = C =C = and D =D = with F (x(tıt Then we have 5 5 H = J J N and H = 5 J 5 J N In this simulation we select N = 3 andn =3 Using the LMI optimization algorithm and Theorem with =5 γ = and δ( = δ( = δ(3 = we obtain the results given in Fig 3 Fig 4 and Fig 5 Remark : Employing results given in 8 and Matlab LMI solver 8 it is easy to realize that when <5 for the state-feedback control design LMIs become ill-conditioned and Matlab LMI solver yields an error message Rank Deficient However the state-feedback fuzzy controller proposed in this paper guarantee that the inequality (5 holds for the system (59 Fig 3 shows the result of the changing between modes during the simulation with the initial mode at mode and =5 The disturbance input signal w(t which was used during simulation is given in Fig 4 The ratio of the regulated output energy to the disturbance input noise energy obtained by using the H fuzzy controller is depicted in Fig 5 The ratio of the regulated output energy to the disturbance input noise energy tends to a constant value which is about 94 So γ = 94 = 97 which is less than the prescribed value Finally Table shows the performance index γ for different values of Mode Time (sec Fig3: The result of the changing between modes during the simulation with the initial mode at mode The disturbance input w(t 5 5 Ratio of the regulated output energy to the disturbance energy Time (sec Fig4: The disturbance input w(t Time (sec Fig5: The ratio of the regulated ( output energy to Tf z the disturbance noise energy T (tz(tdt T f with w T (tw(tdt =5

9 Robust H State-Feedback Control Design for Fuzzy Singularly Perturbed Systems with Markovian Jumps: An LMI Approach 83 The performance index γ for different val- Table : ues of The performance index γ State-feedback control design > 5 CONCLUSION This paper has investigated the problem of designing a robust H state-feedback controller for a class of uncertainty Markovian jump nonlinear singularly perturbed systems that guarantees the L - gain from an exogenous input to a regulated output to be less or equal to a prescribed value First we approximate this class of uncertain Markovian jump nonlinear singularly perturbed systems by a class of uncertain Takagi-Sugeno fuzzy models with Markovian jumps Then based on an LMI approach LMIbased sufficient conditions for the uncertain Markovian jump nonlinear singularly perturbed systems to have an H performance are derived The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard but also to nonstandard nonlinear singularly perturbed systems An illustrative example is used to illustrate the effectiveness of the proposed design techniques References HJKushnerStochastic Stability and Control New York: Academic Press 967 E B Dynkin Markov Process Berlin: Springer- Verlag W M Wonham On a matrix Riccati equation of stochastic control SIAM J Contr vol 6 pp X Feng K A Loparo Y Ji and H J Chizeck Stochastic stability properties of jump linear system IEEE Tran Automat Contr vol 37 pp C E de Souza and M D Fragoso H control for linear systems with Markovian jumping parameters Control-Theory and Advanced Tech vol 9 pp E K Boukas and Z K Liu Suboptimal design of regulators for jump linear system with timemultiplied quadratic cost IEEE Tran Automat Contr vol 46 pp E K Boukas and H Yang Exponential stabilizability of stochastic systems with Markovian jump parameters Automatica vol 35 pp M A Rami and L Ei Ghaoui H statefeedback control of jump linear systems Proc Conf Decision and Contr pp PShiandEKBoukas H control for Markovian jumping linear system with parametric uncertainty J of Opt Theory and App vol 95 pp K Benjelloun E K Boukas and O L V Costa H control for linear time delay with Markovian jumping parameters J of Opt Theory and App vol 5 pp E K Boukas Z K Liu and G X Liu Delaydependent robust stability and H control of jump linear systems with time-delay Int J of Contr V Dragan P Shi and E K Boukas Control of singularly perturbed system with Markovian jump parameters: An H approach Automatica vol 35 pp Z Pan and T Basar H optimal control for singularly perturbed systems Part I: Perfect state measurements Automatica vol 9 pp Z Pan and T Basar H optimal control for singularly perturbed systems Part II: Imperfect state measurements IEEE Trans Automat Contr vol 39 pp E Fridman State-feedback H control of nonlinear singularly perturbed systems Int J Robust Nonlinear Contr vol 6 pp P Shi and V Dragan Asymptotic H control of singularly perturbed system with parametric uncertainties IEEE Trans Automat Contr vol 44 pp P V Kokotovic H K Khalil and J O Reilly Singular Perturbation Methods in Control: Analysis and Design London: Academic Press S K Nguang and P Shi H fuzzy output feedback control design for nonlinear systems: An LMI approach Proc IEEE Conf Decision and Contr pp Z X Han and G Feng State-feedback H controller design of fuzzy dynamic system using LMI techniques Fuzzy-IEEE 98 pp B S Chen C S Tseng and Y Y He Mixed H /H fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach IEEE Trans Fuzzy Syst vol 8 pp K Tanaka T Ikeda and H O Wang Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stability H control theory and linear martix inequality IEEE Trans Fuzzy Syst vol 4 pp H O Wang K Tanaka and M F Griffin An approach to fuzzy control of nonlinear systems: Stability and design issues IEEE Trans Fuzzy Syst vol 4 no pp D P de Farias J C Geromel J B R do Val

10 84 ECTI TRANSACTIONS ON ELECTRICAL ENG ELECTRONICS AND COMMUNICATIONS VOL3 NO AUGUST 5 and O L V Costa Output feedback control of Markov jump linear systems in continuous-time IEEE Trans Automat Contr vol 45 pp S K Nguang and P Shi H fuzzy output feedback control design for nonlinear systems: An LMI approach IEEE Trans Fuzzy Syst vol pp S K Nguang Robust nonlinear H output feedback control IEEE Trans Automat Contr vol 4 pp W Assawinchaichote and S K Nguang Fuzzy control design for singularly perturbed nonlinear systems: An LMI approach ICAIET (Kota Kinabalu Malaysia pp W Assawinchaichote and S K Nguang Fuzzy observer-based controller design for singularly perturbed nonlinear systems: An LMI approach Proc IEEE Conf Decision and Contr (Las Vegas pp P Gahinet A Nemirovski AJ Laub and M Chilali LMI Control Toolbox For Use with MATLAB MA: The MathWorks Inc S Boyd L El Ghaoui E Feron and V Balakrishnan Linear Matrix Inequalities in Systems and Control Theory vol 5 Philadelphia: SIAM S Mehta and J Chiasson Nonlinear control of a series dc motor: Theory and experiment IEEE Trans Ind Electron vol 45 pp Sing Kiong Nguang graduated (with first class honours from the Department of Electrical and Computer Engineering of the University of Newcastle Australia in 99 and received the PhD degree from the same university in 995He is currently holding a senior lectureship in the Department of Electrical and Electronic Engineering of the University of Auckland New Zealand He has over 5 journal papers and over 3 conference papers/presentations on nonlinear control design nonlinear H- infinity control systems nonlinear time-delay systems nonlinear sampled-data systems biomedical systems modelling fuzzy modelling and control biological systems modelling and control and food and bioproduct processing He is currently serving as Associate Editor for IEEE Control System Society Conference Editor Board and is a senior member of IEEE Wudhichai Assawinchaichote received the BEng (Hons degree in Electronic Engineering from Assumption University Bangkok Thailand in 994; the MS degree in Electrical Engineering from the Pennsylvania State University (Main Campus PA USA in 997 and the PhD degree with the Department of Electrical and Computer Engineering from the University of Auckland New Zealand (-4 He is currently working as a lecturer in the Department of Electronic and Telecommunication Engineering at King Mongkut s University of Technology Thonburi Bangkok Thailand His research interests include fuzzy control robust control and filtering Markovian jump systems and singularly perturbed systems He is a member of the IEEE