On robust stabilization of Markovian jumpsystems with uncertain switching probabilities

Transcription

1 Automatca 41 (2005) Bref Paper On robust stablzaton of Markovan jumpsystems wth uncertan swtchng probabltes Junln Xong a James Lam a Hujun Gao b Danel W.C. Ho c a Department of Mechancal Engneerng Unversty of Hong Kong Hong Kong b Space Control and Inertal Technology Research Center Harbn Insttute of Technology Harbn Chna c Department of Mathematcs Cty Unversty of Hong Kong Hong Kong Receved 20 Aprl 2004; receved n revsed form 6 September 2004; accepted 2 December 2004 Abstract Ths bref paper s concerned wth the robust stablzaton problem for a class of Markovan jump lnear systems wth uncertan swtchng probabltes. The uncertan Markovan jump system under consderaton nvolves parameter uncertantes both n the system matrces and n the mode transton rate matrx. Frst a new crteron for testng the robust stablty of such systems s establshed n terms of lnear matrx nequaltes. Then a suffcent condton s proposed for the desgn of robust state-feedback controllers. A globally convergent algorthm nvolvng convex optmzaton s also presented to help construct such controllers effectvely. Fnally a numercal smulaton s used to llustrate the developed theory Elsever Ltd. All rghts reserved. Keywords: Lnear matrx nequaltes; Markovan parameters; Robust stablty; Uncertan systems 1. Introducton Markovan jumplnear systems (MJLSs) have been ntensvely studed over the past decade. The reason s manly that MJLS s a sutable mathematcal model to represent a class of dynamc systems subject to random abrupt varatons n ther structures and has many applcatons such as target trackng problems manufactory processes and fault-tolerant systems (Marton 1990). From a mathematcal pont of vew MJLSs can be regarded as a specal class of stochastc systems wth system matrces changed randomly at dscrete tme nstances governed by a Markov process and reman lnear tme nvarant between the random jumps. MJLSs also Ths paper was not presented at any IFAC meetng. Ths paper was recommended for publcaton n revsed form by Assocate Edtor T. Chen under the drecton of Edtor I. Petersen. Correspondng author. Tel.: ; fax: E-mal addresses: jlxong@hkusua.hku.hk (J. Xong) james.lam@hku.hk (J. Lam) hjgao@ht.edu.cn (H. Gao) madanel@ctyu.edu.hk (D.W.C. Ho). belong to the category of hybrd systems wth fnte operaton modes. Each operaton mode corresponds to some dynamc system and the mode transtons from one to another are governed by a Markov process as well. The analyss and synthess problems have been extensvely studed such as the controllablty and observablty (J & Chzeck 1990) stablty and stablzaton (J & Chzeck 1990; Feng Loparo J & Chzeck 1992; El Ghaou & Ram 1996; Boukas Sh & Benjelloun 1999; Mao 2002) H 2 control (Costa Val & Geromel 1999; do Val Geromel & Goncalves 2002) H control (de Faras Geromel do Val & Costa 2000) flterng (Xu Chen & Lam 2003) and model reducton (Zhang Huang & Lam 2003). In partcular for lnear contnuoustme MJLSs wth uncertantes only n system matrces the robust stablty property can be tested by checkng the exstence of the soluton to a set of coupled lnear matrx nequaltes (LMIs) (Sh Boukas & Agarwal 1999). Unfortunately almost all of the work done on robust control of MJLSs s bult upon the assumpton that swtchng probabltes are known precsely a pror. However n /$ - see front matter 2005 Elsever Ltd. All rghts reserved. do: /j.automatca

2 898 J. Xong et al. / Automatca 41 (2005) practce only estmated values of mode transton rates are avalable and estmaton errors referred to as swtchng probablty uncertantes may also lead to nstablty or at least degraded performance of a system as the uncertantes n system matrces do. Ths pont s demonstrated by the numercal smulaton presented n ths paper. Therefore further work s needed to tackle the more realstc stuaton wth uncertan swtchng probabltes. In the lterature two dfferent types of descrptons about uncertan swtchng probabltes have been consdered n the context of robust stablzaton. The frst one s the polytopc descrpton where the mode transton rate matrx s assumed to be n a convex hull wth known vertces (El Ghaou & Ram 1996; Costa et al. 1999). The other type s descrbed n an element-wse way. In ths case the elements of the mode transton rate matrx are measured n practce and error bounds are gven at the same tme (Benjelloun Boukas & Sh 1997; Boukas et al. 1999). In many stuatons the element-wse uncertanty descrpton can be more convenent as well as natural. On the other hand the element-wse descrpton can be formulated nto an equvalent polytopc descrpton but the total number of vertex matrces of the convex hull wll be extremely large when the total number of system modes s greater than three. In ths paper we consder the element-wse uncertantes n the mode transton rate matrx and propose a new crteron for robust stablty and a new approach for the robust stablzaton of the uncertan MJLSs. It s mportant to pont out that the feasble soluton set to the robust stablzaton problem n ths paper s not convex. Smlar condtons also appear n the H control problem for lnear tme-nvarant systems usng reduced-order output-feedback controllers (El Ghaou Oustry & Ram 1997; Lebfrtz 2001) and the robust H control problem for lnear tme-delay systems (Lee Moon Kwon & Park 2004). Numercally t s dffcult to solve such non-convex problem drectly. However the cone complementarty lnearzaton algorthm (El Ghaou et al. 1997) and the sequental lnear programmng matrx method (SLPMM) (Lebfrtz 2001) have been developed for ths purpose based on the LMI machnery. Consequently such problems can be solved systematcally and effectvely. Ths paper consders the robust stablzaton problem for MJLSs wth uncertan swtchng probabltes. The am s to desgn a robust state-feedback controller such that the closed-loopsystem s quadratcally mean square stable over all admssble uncertantes both n system matrces and n the mode transton rate matrx. The analyss problem can be tackled n terms of the solvablty of a set of coupled LMIs and the assocated synthess problem can be dealt wth usng the SLPMM algorthm. A numercal example s used to demonstrate the usefulness and applcablty of the developed theory. Notaton: The notatons n ths paper are standard. R n and R m n denote respectvely the n-dmensonal Eucldean space and the set of all m n real matrces. R + and S n n denote the set of all postve real numbers and the set of all n n real symmetrc postve defnte matrces respectvely. The notaton X Y (respectvely X>Y) where X and Y are real symmetrc matrces means that X Y s postve sem-defnte (respectvely postve-defnte). I s the dentty matrx wth compatble dmensons. The superscrpt T denotes the transpose for vectors or matrces and trace( ) stands for the trace of a square matrx. 2 refers to the Eucldean norm for vectors. Moreover let (Ω F P) be a complete probablty space. E( ) stands for the mathematcal expectaton operator. 2. Problem statement Consder the followng MJLSs defned on a complete probablty space (Ω F P): ẋ(t) = Â(ˆr(t))x(t) + ˆB(ˆr(t))u(t) t 0 (1) where x(t) R n s the system state u(t) R m s the control nput. The mode jumpng process {ˆr(t)t 0} s a contnuous-tme dscrete-state homogeneous Markov process on the probablty space takes values n a fnte state space S {1 2...s} and has the mode transton probabltes { ˆπj δt + o(δt) f j = Pr(ˆr(t + δt) = j ˆr(t) = ) = 1 +ˆπ δt + o(δt) f j = where δt >0 lm δt 0 o(δt)/δt = 0 and ˆπ j 0 ( j Sj = ) denotes the swtchng rate from mode to mode j and ˆπ s j = ˆπ j for all S. In ths paper both system matrces  Â(ˆr(t) = ) ˆB ˆB(ˆr(t)= ) S and the mode transton rate matrx ˆΠ (ˆπ j ) are not precsely known a pror but belong to the followng admssble uncertanty domans respectvely: D a {A + E F H a : F T F I for all S} D b {B + E F H b : F T F I for all S} D π {Π + ΔΠ : Δπ j ε j ε j 0 for all j Sj = } (2a) (2b) (2c) where matrces A B E H a H b and Π (π j ) are known constant real matrces of approprate dmensons whle F and ΔΠ (Δπ j ) denote the uncertantes n the system matrces and the mode transton rate matrx respectvely. For all j Sj = π j ( 0) denotes the estmated value of ˆπ j and the error between them s referred as to Δπ j whch can take any value n ε j ε j ; For all S π s j = π j and Δπ s j = Δπ j. Let x(t; x 0 ˆr 0 ) be the trajectory of the system state of (1) from ntal system state x 0 x(0) R n and ntal operaton mode ˆr 0 ˆr(0) S. We have the followng defntons and proposton throughout the paper.

3 J. Xong et al. / Automatca 41 (2005) Defnton 1 (de Faras et al. 2000). The nomnal Markovan jumpsystem of (1) (wth u(t) 0) s sad to be mean square stable f lm t E ( x(t; x 0 ˆr 0 ) 2 2 ) = 0 for any ntal condtons x 0 R n and ˆr 0 S. Proposton 2 (de Faras et al. 2000). The nomnal Markovan jump system of (1) (wth u(t) 0) s mean square stable f and only f the coupled LMIs A T P + P A + π j P j < 0 for all S (3) are feasble for a set of matrces {P : P S n n S}. Defnton 3. The uncertan Markovan jumpsystem (1) (wth u(t) 0) s sad to be quadratcally mean square stable f there exsts a set of matrces {P : P S n n S} such that  T P + P  + ˆπ j P j < 0 for all S (4) hold over all admssble uncertanty domans n (2). 3. Robust stablty analyss The goal of ths secton s to developa new analyss result of robust stablty for system (1) wth uncertanty domans (2). Here we establsh a new crteron for testng the robust stablty property n terms of LMIs. Theorem 4. Uncertan Markovan jump system (1) (wth u(t) 0) s quadratcally mean square stable f there exst sets: {P : P S n n S} {λ : λ R + S} and {λ j : λ j R + j Sj = } such that Q P E M E TP λ I 0 < 0 for all S (5) M T 0 Λ where Q = A T P + P A + + λ H T a H a π j P j j = λ j ε 2 j I M = P P 1 P P 1 P P +1 P P s Λ = dag(λ 1 I...λ ( 1) Iλ (+1) I...λ s I). Proof. Consder the uncertan system (1) wth u(t) 0 and the admssble uncertanty domans (2) we have  = A + E F H a and ˆπ j = π j + Δπ j for all j S. Accordng to Defnton 3 the uncertan Markovan jumpsystem (1) s quadratcally mean square stable f there exsts a set of matrces {P : P S n n S} such that (A + E F H a ) T P + P (A + E F H a ) + (π j + Δπ j )P j < 0 for all S. Ths nequalty can be further wrtten as A T P + P A + π j P j + Ha T F T ET P + P E F H a + j = 1 2 Δπ j (P j P ) Δπ j (P j P ) < 0. The above nequalty holds for all F TF I and Δπ j ε j f there exst real numbers λ R + λ j R + ( j Sj = ) such that A T P + P A + π j P j + λ Ha T H a + 1 P E E T λ P + j = λj 4 ε2 j I + 1 (P j P ) 2 < 0 (6) λ j whch s equvalent to nequalty (5) n vew of Schur complement equvalence. Remark 5. Although the uncertanty doman (2c) can be formulated nto a fx polytope (El Ghaou & Ram 1996; Costa et al. 1999) by ntroducng L 2 s(s 1) vertex matrces the test for the robust stablty of the uncertan system (1) needs to check the solvablty of a (L + 1)sn (L + 1)sn (compared to (s + 1)sn (s + 1)sn here) lnear matrx nequalty system wth respect to (n(n + 1)/2)s (compared to (n(n + 1)/2)s + s(s 1) here) scalar varables based on Theorem 3.3 of El Ghaou and Ram (1996). For example consder the case n = 2 s = 4 to test the robust stablty we can translate the uncertanty doman (2c) nto a fx polytopc descrpton by ntroducng 4096 vertex matrces then we need to test the solvablty of a (compared to here) lnear matrx nequalty system wth respect to 12 (compared to 24 here) scalar varables. On the other hand the lterature (Benjelloun et al. 1997; Boukas et al. 1999) have consdered the robust stablty problem wth a smlar uncertanty doman to (2c) where they bounded Δπ j by the upper bound ε j for all j S n every LMI (Theorem 2.2 of Benjelloun et al. (1997) Theorem 3 of Boukas et al. (1999)). Ths approach may be more conservatve than ours n many cases based on numercal experences. 4. Robust stablty synthess Ths secton deals wth the robust stablzaton problem for MJLSs wth uncertan swtchng probabltes. We am

4 900 J. Xong et al. / Automatca 41 (2005) to desgn a state-feedback controller such that the resultng closed-loopsystem s quadratcally mean square stable over all admssble uncertanty domans n (2). Consder the state-feedback control law u(t) = K(ˆr(t))x(t) (7) where K K(ˆr(t)= ) R m n ( S) s the controller to be determned. The closed-loopsystem s ẋ(t) ={A(ˆr(t)) + B(ˆr(t))K(ˆr(t)) + E(ˆr(t))F (ˆr(t)) H a (ˆr(t)) + H b (ˆr(t))K(ˆr(t))}x(t). (8) The followng result solves the robust stablzaton problem (RSP) for system (1) wth uncertan swtchng probabltes. Theorem 6. Consder uncertan Markovan jump system (1) there exsts a state-feedback control law (7) such that the closed-loop system (8) s quadratcally mean square stable f there exst sets of matrces P {P : P S n n S} X {X : X S n n S} V {V : V S n n S} Z {Z : Z S n n S} Y {Y : Y R m n S} Ξ {α : α R + S} Λ {λ j : λ j R + j Sj = } satsfyng the coupled LMIs Q 1 (H a X + H b Y ) T X H a X + H b Y α I 0 < 0 (9) X 0 Z Q2 M M T 0 Λ (10) wth equalty constrants P X = I V Z = I (11) for all S where Q 1 = (A X + B Y ) T + (A X + B Y ) + α E E T Q 2 = V + π j P j + 1 λ j εj 2 4 I j = M = P P 1 P P 1 P P +1 P P s Λ = dag(λ 1 I...λ ( 1) Iλ (+1) I...λ s I). In ths case controller (7) s gven by K = Y P S. Proof. Consder nequalty (6) let V Z 1 S n n for all S such that λj π j P j + 4 ε2 j I + 1 (P P j ) 2 V λ j j = whch s equvalent to (10) n vew of Schur complement equvalence. Replacng A and H a n (6) wth A + B K and H a + H b K respectvely yelds (A + B K ) T P + P (A + B K ) + 1 λ P E E T P + λ (H a + H b K ) T (H a + H b K ) + V < 0. Therefore the closed-loopsystem (8) s quadratcally mean square stable f the above nequalty holds for all S. Now pre- and post-multply both sdes of the above nequalty by P 1 Y K X α λ 1 and apply the changes of varables X P 1 wehave (A X + B Y ) T + (A X + B Y ) + α E E T + α 1 (H a X + H b Y ) T (H a X + H b Y ) + X V X < 0 whch s equvalent to (9) by Schur complement equvalence agan Computatonal method Defne the soluton set of Theorem 6 as X {(PXVZYΞ Λ) : (9) (10) (11) are satsfed}. Although the set X s not convex due to equalty constrants (11) SLPMM (Lebfrtz 2001) can be used to solve such non-convex problems. Frstly for computatonal purpose we ntroduce a suffcently small scalar β R + and replace Q 1 by Q 1 + βi for all S then nequalty (9) becomes Q 1 + βi (H a X + H b Y ) T X H a X + H b Y α I 0 0. (12) X 0 Z Secondly to fnd a soluton n X the equalty constrants (11) can be weakened to the sem-defnte programmng condtons P I V I 0 0 for all S (13) I I X Z Notce that the equalty constrants n (11) correspond to the boundares of the convex sets n (13). Fnally let X β {(PXVZYΞ Λ) : (10) (12) (13) are satsfed}. Now X β s a closed and convex set. The SLPMM algorthm can be employed to fnd a soluton of Theorem 6. The soluton of RSP s summarzed below. Algorthm RSP. For a gven precson δ R + let N be the maxmum number of teratons a suffcently small number β R + be gven and X β =. (1) Determne (P 0 X 0 V 0 Z 0 Y 0 Ξ 0 Λ 0 ) X β let k := 0. (2) Solve the followng convex optmzaton problem for the varables (PXVZYΞ Λ) X β : mn X β =1 trace(p X k + P k X + V Z k + V k Z ) where P k P k X k X k V k V k Z k Z k S. (3) Let T k := P L k := X U k := V R k := Z for all S.

5 J. Xong et al. / Automatca 41 (2005) (4) If trace(t k Xk + P k Lk + U k Zk + V k Rk ) =1 2 trace(p k Xk + V k Zk ) < δ =1 then go to step(7) else go to step(5). (5) Compute θ 0 1 by solvng mn g(θ) θ 01 where g(θ) =1 +V k trace(p k + θ(t k P k )Xk + θ(lk Xk ) + θ(u k V k )Zk + θ(rk Zk )) (6) Let P k+1 := P k+θ (T k P k ) Xk+1 := X k+θ (L k X k k+1 ) V := V k+θ (U k V k ) Zk+1 := Z k+θ (R k Zk ) for all S and k := k + 1. If k<n then go to step(2) else go to step(7). (7) If trace(p kxk + V kzk ) = 2sn then a soluton s =1 found successfully else a soluton cannot be found. 5. Numercal example To llustrate the usefulness and flexblty of the theory developed n the paper we present a smulaton example. Attenton s focused on the desgn of a robust stablzng controller for a Markovan jumpsystem wth uncertan swtchng probabltes. Consder a uncertan Markovan jumpsystem (1) wth two operaton modes. The system data and the ntal condtons of (1) are as follows: A 1 = A 2 = B 1 = B = Π = x 0 = ˆr 1 0 = 1. The uncertantes n the mode transton rate matrx Π are such that Δπ 12 ε 12 wth ε 12 π 12 /2 Δπ 21 ε 21 wth ε 21 π 21 /2. The open-loop system s unstable. If no uncertantes exst a stablzng controller for the nomnal system can be obtaned by Theorem 6 of (Sh et al. 1999) as K 1 = K 2 = Applyng ths controller makes the resultng closed-loop system become mean square stable (see Fg. 1). However the tme (second) x Fg. 1. Closed-loopsystem wthout uncertantes tme (second) Fg. 2. Closed-loopsystem wth uncertantes. closed-loopsystem turns out to be unstable (see Fg. 2) f there exst swtchng probablty uncertantes say Δπ 12 = ε 12 and Δπ 21 = ε 21. Hence t s necessary to consder the uncertantes n Π when desgnng controller (7). Usng Algorthm RSP a robust controller (7) could be acheved such that the closed-loopsystem s robustly mean square stable over all admssble uncertantes Δπ 12 ε 12 and Δπ 21 ε 21 (see Fg. 3). To compute wth Algorthm RSP for ths example t s chosen that δ = N = 100 and β = The controller obtaned s K 1 = K 2 =

6 902 J. Xong et al. / Automatca 41 (2005) tme (second) 6. Concluson Fg. 3. Closed-loopsystem wth uncertantes. Ths paper has proposed a robust stablzaton method for a class of Markovan jumplnear systems wth uncertan swtchng probabltes. The robust stablty of such systems can be tested based on the feasblty of a set of coupled lnear matrx nequaltes. An algorthm nvolvng convex optmzaton was also proposed to construct a controller such that the uncertan system can be stablzed over all admssble uncertantes n the system matrces as well as n the mode transton rate matrx. A numercal example llustrated that the constructed controller could tolerate the uncertantes n the swtchng probabltes. Acknowledgements The work was partally supported by RGC HKU 7103/01P and RGC CtyU We thank the referees for ther careful readng of the paper. Ther suggestons have helped us to sgnfcantly mprove the presentaton of the paper. References Benjelloun K. Boukas E. K. & Sh P. (1997). Robust stablzablty of uncertan lnear systems wth Markovan jumpng parameters. In Proceedngs of the Amercan control conference. New Mexco USA (pp ). Boukas E. K. Sh P. & Benjelloun K. (1999). On stablzaton of uncertan lnear systems wth jumpparameters. Internatonal Journal of Control 72(9) Costa O. L. V. Val J. B. R. & Geromel J. C. (1999). Contnuoustme state-feedback H 2 -control of Markovan jumplnear system va convex analyss. Automatca de Faras D. P. Geromel J. C. do Val J. B. R. & Costa O. L. V. (2000). Output feedback control of Markov jump lnear systems n contnuous-tme. IEEE Transactons on Automatc Control 45(5) do Val J. B. R. Geromel J. C. & Goncalves A. P. C. (2002). The H 2 - control for jumplnear systems: cluster observatons of the Markov state. Automatca El Ghaou L. Oustry F. & Ram M. A. (1997). A cone complementarty lnearzaton algorthm for statc output-feedback and related problems. IEEE Transactons on Automatc Control 42(8) El Ghaou L. & Ram M. A. (1996). Robust state-feedback stablzaton of jumplnear systems va LMIs. Internatonal Journal of Robust and Nonlnear Control 6(9 10) Feng X. Loparo K. A. J Y. & Chzeck H. J. (1992). Stochastc stablty propertes of jump lnear systems. IEEE Transactons on Automatc Control 37(1) J Y. & Chzeck H. J. (1990). Controllablty stablzablty and contnuous-tme Markovan jumplnear quadratc control. IEEE Transactons on Automatc Control 35(7) Lebfrtz F. (2001). An LMI-based algorthm for desgnng suboptmal statc H 2 /H output feedback controllers. SIAM Journal on Control and Optmzaton 39(6) Lee Y. S. Moon Y. S. Kwon W. H. & Park P. G. (2004). Delaydependent robust H control for uncertan systems wth a state-delay. Automatca Mao X. (2002). Exponental stablty of stochastc delay nterval systems wth Markovan swtchng. IEEE Transactons on Automatc Control 47(10) Marton M. (1990). Jump lnear systems n automatc control. New York: Marcel Dekker. Sh P. Boukas E. K. & Agarwal R. K. (1999). Kalman flterng for contnuous-tme uncertan systems wth Markovan jumpng parameters. IEEE Transactons on Automatc Control 44(8) Xu S. Chen T. & Lam J. (2003). Robust H flterng for uncertan Markovan jumpsystems wth mode-dependent tme delays. IEEE Transactons on Automatc Control 48(5) Zhang L. Huang B. & Lam J. (2003). H model reducton of Markovan jumplnear systems. Systems & Control Letters Junln Xong earned hs B.S. and M.S. degrees from Northeastern Unversty Shenyang Chna n 2000 and 2003 respectvely. He s now pursung hs Ph.D. degree n Mechancal Engneerng at Unversty of Hong Kong. Hs current research nterests nclude Markovan jumpsystems and networked control systems. James Lam receved a frst class B.Sc. degree n Mechancal Engneerng from the Unversty of Manchester n He was awarded the Ashbury Scholarshp the A.H. Gbson Prze and the H. Wrght Baker Prze for hs academc performance. From the Unversty of Cambrdge he obtaned the M.Phl. and Ph.D. degrees n the area of control engneerng n 1985 and 1988 respectvely. Hs postdoctoral research was carred out n the Australan Natonal Unversty between 1990 and Dr. Lam s a Scholar (1984) and Fellow (1990) of the Croucher Foundaton. Dr. Lam has held faculty postons at now the Cty Unversty of Hong Kong and the Unversty of Melbourne. He s now an Assocate Professor n the Department of Mechancal Engneerng the Unversty of Hong Kong and s holdng a Concurrent Professorshpat the Northeastern Unversty Guest Professorshpat the Huazhong Unversty of Scence and Technology Consultng Professorshpat the South Chna Unversty of Technology and Guest Professorshpof Shandong Unversty. Dr. Lam s a Chartered Mathematcan a Fellow of the Insttute of Mathematcs

7 J. Xong et al. / Automatca 41 (2005) and Its Applcatons (UK) a Senor Member of the Insttute of Electrcal & Electroncs Engneers (US) a Member of the Insttuton of Electrcal Engneers (UK). He s an Assocate Edtor of the Asan Journal of Control the Internatonal Journal of Appled Mathematcs and Computer Scence the Internatonal Journal of Systems Scence and the Conference Edtoral Board of IEEE Control System Socety. Hs research nterests nclude model reducton delay systems descrptor systems stochastc systems multdmensonal systems robust control and flterng fault detecton and relable control. Hujun Gao was born n Helongjang Provnce Chna n He receved the M.S. degree n Electrcal Engneerng from Shenyang Unversty of Technology Shengyang Chna n He s now pursung the Ph.D degree n control scence and engneerng from Harbn Insttute of Technology Harbn Chna. He served as a Research Assocate n the Department of Mechancal Engneerng Unversty of Hong Kong Hong Kong from November 2003 to September 2004 and joned Harbn Insttute of Technology n November Hs research nterests nclude model reducton robust control/flter theory tme-delay systems stochastc systems and two-dmensonal systems. Dr. Danel W.C. Ho receved a frst class B.Sc. M.Sc. and Ph.D. degrees n mathematcs from the Unversty of Salford (UK) n and 1986 respectvely. From 1985 to 1988 Dr. Ho was a Research Fellow n Industral Control Unt Unversty of Strathclyde (Glasgow Scotland). In 1989 he joned the Department of Mathematcs Cty Unversty of Hong Kong where he s currently an Assocate Professor. He s an Assocate Edtor of Asan Journal of Control. Hs research nterests nclude H control theory robust pole assgnment problem adaptve neural wavelet dentfcaton nonlnear control theory.