Eigenstructure assignment in a class of second-order ... - Springer Link

Journal of Control Theory and Applications 3 (2006) 302–308

Eigenstructure assignment in a class of second-order dynamic systems Guosheng WANG 1 , Qiang LV 1 , Guangren DUAN 2 (1.Department of Control Engineering, Academy of Armored Force Engineering, Beijing 100072, China; 2.Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin Heilongjiang 150001, China)

Abstract: Eigenstructure assignment using the proportional-plus-derivative feedback controller in a class of secondorder dynamic system is investigated. Simple, general, complete parametric expressions for both the closed-loop eigenvector matrix and the feedback gains are established based on two simple Smith form reductions. The approach utilizes directly the original system data and involves manipulations only on n−dimensional matrices. Furthermore, it reveals all the degrees of freedom which can be further utilized to achieve additional system specifications. An example shows the effect of the proposed approach. Keywords: Second-order dynamic systems; Eigenstructure assignment; Proportional-plus-derivative control; Parametric solutions

1

Introduction

Second-order dynamic systems capture the dynamic behaviour of many natural phenomena and have found wide applications in vibration and structure analysis. Most of the results in second-order dynamic systems are focused on stabilization (for e.g. [1∼3]) and pole assignment [4∼9], Merovitch et al. [4], Bhaya and Desoer [5], Joshi [6] considered pole assignment in second-order dynamic systems through the independent modal space control technique, while Juang and Maghmi [7,8] adopted the first-order approach. Chu and Datta [9] proposed two new methods for pole assignment in second-order dynamic systems, which modified some existing results (for e.g. [7,8]). Eigenstructure assignment in linear control systems design has attracted much attention for the last decades (see [10∼23]). Recently, Duan and Liu [23] has considered eigenstructure assignment using proportional plus derivative state feedback controller in second-order dynamic systems possessing the following form:

verted into system (1) when the coefficient matrix E is nonsingular, this conversion involves matrix inverse manipulation which may produce numerically unreliable results, and the rounding error caused in this very first step by this conversion is consequently carried over to the final results through all the subsequent steps; b) Dealing with system (2) in its original form is not only preferable from the numerical reliability point of view, but also has the advantage that the developed approach can be easily generalized into the case that the coefficient matrix E is singular, in which case the system (2) is really a general second-order descriptor system.

(2)

Based on the controllability of the matrix triple (E, A, B), an extremely simple complete parametric approach for eigenstructure assignment in second-order dynamic system (2) via proportional plus derivative feedback is proposed. Simple, complete parametric expressions for both the closed-loop eigenvectors and the feedback gains are given. The obtained expressions for both the closed-loop eigenvectors and the feedback gains contain a group of parameter vectors which represent the degrees of freedom existing in the eigenstructure assignment design. Besides this group of parameter vectors, the closed-loop eigenvalues may also be treated as a part of the design freedom. These design parameters can be properly chosen to produce a closed-loop system with some desired system specifications.

where E , A , B and C are matrices of appropriate dimensions. This generalization is not trivial because of the following two aspects: a) Although system (2) can be con-

This paper includes 6 sections. Section 2 gives the formulation of the eigenstructure assignment problem for secondorder dynamic systems (2), while Section 3 presents some

q¨ − A q˙ − Cq = Bu.

(1)

This paper is a generalization of Duan and Liu [23]. It considers eigenstructure assignment via proportional-plusderivative co-ordinate control in second-order dynamic systems of the following form: E q¨ − A q˙ − Cq = Bu,

Received 24 January 2005; revised 21 March 2006.

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3 Preliminaries

preliminary results. The general parametric approach to the eigenstructure assignment problem is developed in Section 4. Section 5 gives an illustrative example. Concluding remarks follow in Section 6.

Definition 2 if and only if

2

If (E, A, B) is controllable, there clearly exist two unimodular matrices P (s) and Q(s) satisfying

Problem formulation

Consider a second-order dynamic system (2), where q ∈ R is the generalized co-ordinate vector; u ∈ Rr is the input vector; and E, A , B , C are matrices of appropriate dimensions. Let       I 0 0 I 0    , A = , B = (3) E = 0E CA B n

and x=

  q q˙

rank[A − sE B] = n, ∀s ∈ C.

P (s)[A − sE B]Q(s) = [ 0 I ], ∀s ∈ C. Partition Q(s) as follows   Q11 (s) Q12 (s) Q(s) = , Q11 ∈ Rn×r . Q21 (s) Q22 (s)

,

(4)





E x˙ = A x + B u.

(5)

If the feedback controller

  u = K0 q + K1 q˙ = Kx, K = K0 K1 ∈ Rr×2n (6)

is applied to the system (2) or (5), the following closed-loop system can be obtained: E  x˙ = Ac x,

(7)

where Ac = A + B  K.

(8)

Definition 1 Suppose that s ∈ C and A , E are matrices of appropriate dimensions, s is called a generalized eigenvalue of the matrix pair (E, A) if and only if Av = sEv has a nonzero solution v, while v is called a generalized eigenvector of the matrix pair (E, A). For simplicity, we only consider that (E  , Ac ) possesses distinct generalized eigenvalues si , i = 1 , 2 , · · · , 2n . Put Λ = diag[s1 s2 · · · s2n ],

(9)

where the matrix E is singular, and denote the generalized right eigenvector matrix by V , then det(V ) = 0 and, by definition, (A + B  K)V = E  V Λ.

(10)

Thus the eigenstructure assignment problem to be solved in this paper for the second-order dynamic system can be described as follows: Problem ESA (Eigenstructure assignment) Given matrices, E, A, C ∈ Rn×n , B ∈ Rn×r and a set of distinct self-conjugate complex numbers si , i = 1 , 2 , · · · , 2n , find matrices K ∈ Rr×2n and V ∈ C2n×2n , with det(V ) = 0, satisfying equation (10), where the matrices E  , A andB  are given as in (3).

(11)

(12)

(13)

We can then give a general solution to AV + BW = EV Λ + R,

then (2) can be written in the following first-order descriptor linear system: 

The matrix triple (E, A, B) is controllable

(14)

where E, A , B are given as before, R ∈ Rn×p and Λ = diag[s1 s2 · · · sp ], while V ∈ C

n×p

and W ∈ C

r×p

(15)

are to be determined.

Lemma 1 Let the matrix triple (E, A, B) be controllable, then all the solutions to (14) can be written by columns as     vi fi = Q(si ) , i = 1 , 2, · · · , p (16) wi P (si )ri or, equivalently, as  vi = Q12 (si )P (si )ri + Q11 (si )fi , wi = Q22 (si )P (si )ri + Q21 (si )fi ,

(17)

where fi ∈ Cr , i = 1 , 2 , · · · , p , are arbitrary parameter vectors. Proof It can be observed that the matrix equation (14) can be equivalently converted into    v  i = ri , i = 1 , 2, · · · , p. (18) A − si E B wi From (12) and (16), we can obtain    v  i A − si E B wi     fi = A − si E B Q(si ) P (si )ri     f i = P −1 (si )P (si ) A − si E B Q(si ) P (si )ri     fi = P −1 (si ) 0 I = ri . P (si )ri Therefore, the vectors given by (16) satisfy equations in (18). Now let’s show that the vectors vi and wi satisfying equation (18) can be expressed in the form of (16). Pre-multiplying by P (si ) both sides of (18), and using

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(12), yields 

Let



 vi 0 I Q−1 (si ) wi 

fi



= P (si )ri , i = 1, 2, · · · , p. (19) 

−1

=Q

ei

unimodular matrices H(s) and L(s) satisfying



(si )

vi wi

 , i = 1 , 2, · · · , p,

(20)

H(s)[Q12 (s)P (s)C + sIn − Q11 (s)]L(s) = [ 0 In ]. Partition the matrix L(s) satisfying (24) as   L11 (s) L12 (s) , L11 (s) ∈ Rn×r . L(s) = L21 (s) L22 (s)

(24)

(25)

Lemma 3 Suppose that (24) holds for some pair of the unimodular matrices H(s) and L(s). Then all the vectors y and z satisfying

then (19) becomes     f i = P (si )ri , i = 1 , 2, · · · , p, 0I ei

[Q12 (s)P (s)C + sI] y − Q11 (s)z = 0

(26)

are given as

which produces ei = P (si )ri , i = 1 , 2 , · · · , p.

y = L11 (s)g, z = L21 (s)g

(21)

with g ∈ C being an arbitrary parameter vector.

Substituting (21) into (20), we can get (16).

Proof Rewriting (26) as

Definition 3 The descriptor linear system (5) is controllable if the matrix triple (E  , A , B  ) is controllable, that is, 





rank[A − sE B ] = 2n, ∀s ∈ C.

(22)

Lemma 2 Let the matrix triple (E, A, B) be controllable, then system (5) is also controllable if and only if rank[Q12 (s)P (s)C +sI

(27)

r

[Q12 (s)P (s)C + sI − Q11 (s)]

The above matrix can easily be transformed, by elementary transformations, into the following form   Q12 (s)P (s)C + sI −Q11 (s) 0 . 0 0 I

From this we can conclude that (22) holds if and only if (23) is valid. Note that (23) is equivalent to the existence of a pair of

=0

H(s) [Q12 (s)P (s)C + sI − Q11 (s)]   y −1 × L(s)L (s) = 0, z which reduces, in view of (24), to     y = 0. 0 In L−1 (s) z Letting

  g

−1

=L

h

(28)

(s)

  y z

(29)

(30)

(31)

and substituting (31) to (30), we can obtain h = 0. Thus it can be obtained from (31) that     y g = L(s) , z 0 which is equivalent to (27).

4

Solution to problem ESA

4.1

Case of undetermined closed-loop eigenvalues

Denote



W = KV, V =

Thus we have rank[A − sE  B  ]   Q12 (s)P (s)C + sI −Q11 (s) 0 = rank . 0 0 I

z

and pre-multiplying by H(s) both sides of (28), gives

−Q11 (s)] = n, ∀s ∈ C. (23)

Proof Using (12) and the structure of matrices E  ,A and B  , we can obtain       I 0 I 0    A − sE B 0 P (s) 0 Q(s)     I 0 −sI I 0 I 0 = C A − sE B 0 Q(s) 0 P (s)    −sI I 0 I 0 = P (s)C P (s)[A − sE B] 0 Q(s)   −sI [ I 0 ]Q(s) = P (s)C P (s)[A − sE B]Q(s)   −sI Q11 (s) Q12 (s) = . P (s)C 0 I

  y

V1 V2

 , V1 , V2 ∈ Cn×2n .

(32)

Then (10) can be decomposed into V2 = V1 Λ

(33)

AV2 + BW = EV2 Λ − CV1 .

(34)

and Applying Lemma 1 to (34), obtains  v2i = −Q12 (si )P (si )Cv1i + Q11 (si )fi , wi = −Q22 (si )P (si )Cv1i + Q21 (si )fi ,

(35)

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G. WANG et al. / Journal of Control Theory and Applications 3 (2006) 302–308

where i = 1, 2, · · · , 2n. Clearly, (33) can be written in its vector form: v2i = si v1i , i = 1, 2, · · · , 2n.

(36)

Combining (36) and (35), yields [Q12 (si )P (si )C + si In ] v1i − Q11 (si )fi = 0, (37) where i = 1, 2, · · · , 2n, which is clearly in the form of (26). Applying Lemma 3 to (37), gets v1i = L11 (si )gi , i = 1, 2, · · · , 2n,

(38)

fi = L21 (si )gi , i = 1, 2, · · · , 2n,

(39)

and r

where gi ∈ C , i = 1 , 2 , · · · , 2n are a group of parametric vectors. Substituting (38) into (36), produces v2i = si L11 (si )gi , i = 1 , 2 , · · · , 2n.

(40)

Theorem 1 Let the matrix triple (E, A, B) and system (5) be both controllable, then all the solutions to Problem ESA are given by   L11 (s1 )g1 L11 (s2 )g2 · · · L11 (s2n )g2n (41) V= s1 L11 (s1 )g1 s2 L11 (s2 )g2 · · · s2n L(s2n )g2n and K = W V −1 ,

(42)

where W is a matrix with columns given by wi = [Q21 (si )L21 (si )−Q22 (si )P (si )CL11 (si )]gi (43) and gi ∈ Cr , i = 1 , 2 , · · · , 2n, are parameter vectors satisfying the following constraints: Constraint C1 gi = g¯j iff si = s¯j , Constraint C2 det(V ) = 0. Based on the above theorem, a procedure for eigenstructure assignment in second-order dynamic system (2) can be given as follows. Algorithm ESA1 Step 1 Solve the pair of unimodular matrices P (s) and Q(s) satisfying (12), and partition matrix Q(s) as in (13). Step 2 Solve the pair of unimodular matrices H(s) and L(s) satisfying (24), and partition matrix L(s) as in (25). Step 3 Establish the general parametric forms for matrices V and W according to (41) and (43), respectively. Step 4 Find a group of parameter vectors gi ∈ Cr , i = 1, 2, · · · , 2n, satisfying Constraints C1 and C2. Step 5 Calculate matrices V and W based on their expressions in Step 3 and the parameters obtained in Step 4. Step 6 Calculate the gain matrix K according to (42). 4.2

Case of prescribed closed-loop eigenvalues

When the closed-loop eigenvalues si , i = 1, 2, · · · , 2n are prescribed, the solutions (16) and (17) to the generalized Sylvester matrix equation (14) are actually dependent

on the constant matrices: ˜ i = Q(si ), i = 1, 2, · · · , 2n, P˜i = P (si ), Q

(44)

which satisfy the following: ˜ i = [0 I], i = 1, 2, · · · , 2n. P˜i [A − si E B]Q

(45)

Therefore, we can, instead of seeking the polynomial matrices P (s) and Q(s) satisfying the relation (12), find these constant matrices directly so as to avoid polynomial matrix manipulations. The constant matrices Pi = P (si ), Qi = Q(si ), i = 1, 2, · · · , 2n can be easily obtained by singular value decompositions. In fact, by applying singular value decomposition to the matrix [A − si E B], we obtain two orthogonal matrices Pi and Qi satisfying the following equations: Pi [A − si E B]Qi = [0 Σi ], i = 1, · · · , 2n,

(46)

where Σi , i = 1, 2, · · · , 2n are diagonal matrices with positive diagonal elements. By rearranging (46) in the following form: Σi−1 Pi [A − si E B]Qi = [0 I], i = 1, 2, · · · , 2n, (47) ˜ i , i = 1, 2, · · · , 2n we obtain the constant matrices P˜i and Q satisfying (45) as ˜ i = Qi , i = 1, 2, · · · , 2n. P˜i = Σ −1 Pi , Q

(48)

Partitioning the matrix Qi , i = 1, 2, · · · , 2n as follows:   Qi11 Qi12 , Qi11 ∈ Rn×r . (49) Qi = Qi21 Qi22 Corollary 1 Let the matrix triple (E, A, B) be controllable, and the diagonal matrix Λ be defined as in (15) with si , i = 1 , 2 , · · · , p known. Then all the solutions to the matrix equation (14) can be given by columns as     vi fi = Qi , i = 1, 2, · · · , p (50) wi Σi−1 Pi ri or, equivalently, as  vi = Qi12 Σi−1 Pi ri + Qi11 fi

wi = Qi22 Σi−1 Pi ri + Qi21 fi

, i = 1, 2, · · · , p, (51)

wherefi ∈ Cr , i = 1 , 2 , · · · , p are a group of arbitrary parameter vectors. By applying singular value decomposition to the matrix [Qi12 Pi C + si I − Qi11 ], we obtain two orthogonal matrices Hi and Li satisfying the following equation: Hi [Qi12 Σi−1 Pi C + si I − Qi11 ]Li = [0 Ξi ], i = 1, · · · , 2n,

(52)

where Ξi , i = 1, 2, · · · , 2n are diagonal matrices with positive diagonal elements. Note that (52) can be arranged in the following form for i = 1, 2, · · · , 2n: Ξi−1 Hi [Qi12 Σi−1 Pi C + si I − Qi11 ]Li = [0 I]. (53) Matrices H(si ) and L(si ), i = 1, 2, · · · , 2n in Lemma 3 can thus be substituted by the matrices Ξi−1 Hi and Li ,

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i = 1, 2, · · · , 2n. Further partition the matrix Li as   Li11 Li12 , Li11 ∈ Rn×r . Li = Li21 Li22

⎡ ⎢ C=⎢ ⎣

(54)

⎤

−10

5

⎡

0

⎤ 1

⎥ ⎢ ⎢ 20 ⎥ ⎦ , B = ⎣0 20 −20 0

0

5 −25

⎥ 0⎥ ⎦.

0

1

Corollary 2 Let Pi , Qi11 , Qi12 and Σi be given as previous. Then all the vectors y and z satisfying   i −1 (55) Q12 Σi Pi C + si I y − Qi11 z = 0

1) Matrices P (s) and Q(s) satisfying (12) can be obtained as

are given as

and y=

Li11 g,

z=

Li21 g

(56)

with g ∈ Cr being an arbitrary parameter vector. Theorem 2 Let the matrix triple (E, A, B) and the descriptor linear system (5) be both controllable, and the desired closed-loop eigenvalues si , i = 1 , 2 , · · · , 2n be given a priori. Then all the solutions to Problem ESA are given by   L111 g1 L211 g2 · · · L2n 11 g2n (57) V = s1 L111 g1 s2 L211 g2 · · · s2n L2n 11 g2n and K = W V −1 ,

(58)

where W is a matrix with columns given by wi =

[Qi21 Li21

−

Qi22 Σi−1 Pi CLi11 ]gi ,

i = 1 , 2 , · · · , 2n

(59)

r

and gi ∈ C , i = 1 , 2 , · · · , 2n, are a group of parameter vectors satisfying the same constraints C1 and C2 as in Theorem 1. Based on the above theorem, an algorithm for solution to Problem ESA in the case of prescribed closed-loop eigenvalues can be given as follows. Algorithm ESA2 Step 1 Solve the matrices Pi , Qi and Σi satisfying (46), and partition Qi , i = 1, 2, · · · , 2n as (49). Step 2 Solve the matrices Hi , Li and Ξi satisfying (52), and partition Li , i = 1 , 2 , · · · , 2n as (54). Step 3 Establish the general parametric forms for matrices V and W according to (57) and (59), respectively. Step 4 Find parameter vectors gi ∈ Cr , i = 1, 2, · · · , 2n satisfying Constraints C1 and C2. Step 5 Calculate matrices V and W based on their expressions in Step 3 and the parameters obtained in Step 4. Step 6 Calculate the gain matrix K according to (58).

5

An illustrative example

Consider a second-order dynamic system (2) with the following parameters: ⎡ ⎤ ⎡ ⎤ 1 0 0 −2.5 0.5 0 ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ E=⎢ ⎣ 0 1 0 ⎦ , A = ⎣ 0.5 −2.5 2 ⎦ , 0 0 −1 0 2 −2

P (s) = diag[ 1 2 1 ] ⎡

−4

2s + 5

0

1

⎢ ⎢ 1 0 0 0 ⎢ ⎢ Q(s) = ⎢ 0 1 0 0 ⎢ ⎢ 2(s+2.5)2 −0.5 −4s−10 1 s + 2.5 ⎣ −2 2−s 0 0

0

⎤

⎥ 0⎥ ⎥ ⎥ 0⎥ . ⎥ 0⎥ ⎦ 1

2) Note that   Q12 (s)P (s)C + sI −Q11 (s) ⎡ ⎤ 10 + s −50 40 −(2s + 5) 4 ⎢ ⎥ =⎢ s 0 −1 0 ⎥ ⎣ 0 ⎦. 0 0 s 0 −1 Matrices H(s) and L(s) satisfying (23) are obtained as ⎡ ⎤ −0.005 0.005(2s + 5) −0.02 ⎢ ⎥ H(s) = ⎢ −1 0 ⎥ ⎣ 0 ⎦, 0 0 −1 ⎡ ⎤ 2s2 + 5s + 50 −4 2s − 15 0 0 ⎢ ⎥ ⎢ s + 10 0 1 0 0⎥ ⎢ ⎥ ⎢ ⎥ L(s) = ⎢ 0 1 0 0 0⎥ . ⎢ ⎥ ⎢ s(s + 10) 0 s 1 0⎥ ⎣ ⎦ 0 s 0 0 1 3) Let

 gi =

αi



βi

, i = 1, · · · , 6.

From (41), the closed-loop eigenvectors are obtained as ⎤ ⎡ (2s2i + 5si + 50)αi − 4βi ⎥ ⎢ ⎥ ⎢ (si + 10)αi ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ β i ⎥ , i = 1, · · · , 6. ⎢ vi = ⎢ 2 ⎥ ⎢ (2si + 5si + 50)si αi − 4si βi ⎥ ⎥ ⎢ ⎥ ⎢ (si + 10)si αi ⎦ ⎣ si βi

Further, we have Q21 (s)L21 (s) − Q22 (s)P (s)CL11 (s)   T (s) −4s2 − 10s − 40 = , −2s2 − 40s − 200 2s + 20 − s2 where T (s) = 2s4 + 10s3 + 82s2 + 165s + 450.

G. WANG et al. / Journal of Control Theory and Applications 3 (2006) 302–308

Thus it follows from (43) that   T (si )αi − (4s2i + 10si + 40)βi . wi = −2(s2i + 20si + 100)αi + (2si + 20 − s2i )βi 4)∼6) Let the closed-loop eigenvalues be

307

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s1,2 = −1 ± i, s3,4 = −2 ± i, s5,6 = −3 ± i and the parameters be chosen as

[8] J. Juang, P. Maghami. Robust eigensystem assignment for state estimators using second-order models[J]. J. of Guidance, Control and

α1,2 = α5,6 = β3,4 = 1, α3,4 = β1,2 = β5,6 = 0. From (42), we can get     14 −29 36 −3 21 −6 , K1 = . K0 = 0 −20 25 0 −2 6 In the case that the parameters are chosen as

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we can get



K0 =  K1 =

0.6533

37.7333 −53.3867

1.6450 −12.775

14.42

−4.60

23.2933 −29.7333

−0.15

−2.56

4.4



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,  .

6 Conclusions This paper proposes a simple, complete parametric approach for eigenstructure assignment in a class of secondorder dynamic systems. Simple, complete parametric expressions for both the closed-loop eigenvector matrix and the feedback gains are established. It is shown that these expressions can be obtained as soon as two simple Smith form reductions are realized. With this approach, the closed-loop eigenvalues can also be set undetermined and utilized as a part of the design freedom. When the closed-loop eigenvalues are prescribed a priori, the two Smith form reductions can be replaced by two sets of singular value decompositions that are well regarded as numerically stale. References [1] F. Rincon. Feedback stabilization of second-order models[D]. USA: Northern Illinois University, 1992. [2] B. Datta, F. Rincon. Feedback stabilization of a second-order system: a nonmodal approach[J]. Linear Algebra and Its Applications, 1993, 188-189: 135 – 161. [3] D. Ho, H. Chan. Feedback Stabilization of Damped-gyroscopic Second-order Systems, MA-93-14[R]. City University, Hong Kong: Faculty of Science and Technology, 1993.

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Guosheng WANG was born in Hebei Province, P. R. China, in 1975. He received both the B. S. and M. S. degrees in 1999 and 2001, respectively, and the Ph. D. degree in Control Theory and Control Engineering from Harbin Institute of Technology in 2004. He is currently a lecture of Department of Control Engineering at Academy of Armored Force Engineering. His research interests include robust control, eigenstructure assignment, and second-order linear systems.

Qiang LV received the B. S. degree, the M. S and Ph. D. degrees in Control Systems Theory from Harbin Institute of Technology. He is currently a professor of Department of Control Engineering at Academy of Armored Force Engineering. His main research interests include robust control, armored force control and neural network control.

Guangren DUAN was born in Heilongjiang Province, P. R. China, in 1962. He received his B. S. degree in Applied Mathematics, and both his M. S. and PhD degrees in Control Systems Theory. Prof. Duan visited the University of Hull, UK, and the University of Sheffield, UK from December 1996 to October 1998,and worked at the Queen’s University of Belfast, UK from October 1998 to October 2002. Since August 2000, he has been elected Specially Employed Professor at Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese government. He is currently the Director of the Center for Control Systems and Guidance Technology at Harbin Institute of Technology. He is the author and co-author of over 400 publications. His main research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control and magnetic bearing control. Dr Duan is a Chartered Engineer in the UK, a Senior Member of IEEE and a Fellow of IEE.

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