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Neural computation for robust approximate pole assignment
Neurocomputing 25 (1999) 191}211
Neural computation for robust approximate pole assignment Daniel W.C. Ho *, James Lam, Jinhua Xu , Hei Ka Tam Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Hong Kong Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong Received 5 January 1998; accepted 15 December 1998
Abstract This paper provides an approach for output feedback robust approximate pole assignment. It is formulated as an unconstrained optimization problem and solved via the gradient #ow approach which is ideally suited for neural computing implementation. A schematic circuit architecture of the neural network is suggested. Simulation results are used to demonstrate the e!ectiveness of the proposed method. 1999 Elsevier Science B.V. All rights reserved. Keywords: Output feedback; Robustness; Approximate pole assignment; Neural networks; Gradient #ow
1. Introduction Consider a linear time-invariant multivariable system described by x "Ax#Bu,
(1)
y"Cx,
(2)
where x31L, u31K and y31N represent the state, input, and output vectors, respectively, A31L"L, B31L"K, and C31N"L are constant matrices. As usual, it is assumed that B and C are of full column and row rank, respectively. By applying a constant
* Corresponding author. 0925-2312/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 9 9 ) 0 0 0 5 7 - 0
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output feedback law u"Ky to (1) and (2), K31K"N, the closed-loop system is given by x "(A#BKC)x. It was established that under the conditions of (A, B) completely controllable and (C, A) completely observable and that m#p'n [15] or mp'n [24], there exist feedback matrices K for almost any set of self-conjugate complex numbers +j , j ,2, j , as the desired poles of the closed-loop system. The problem of "nding L one such K is referred to as the output feedback exact pole assignment (OFEPA). In the special case when C"I , the n;n identity matrix, we have state feedback exact L pole assignment (SFEPA). The solution of the output feedback pole assignment problem is, in general, nonunique for a multivariable system. The freedom on the choice of K may be exploited for optimizing other design objectives. One of these objectives is to maximize the robustness of the assigned poles in face of perturbation. The state feedback robust exact pole assignment (SFREPA) problem was formulated by Kautsky et al. [13] in terms of minimizing some conditioning measures of the closed-loop state matrix. More recently, Byers and Nash [2] tackled this problem using Newton's method and truncated-Newton method. While state feedback exact pole assignment was considered by Kautsky and many others, it is commonly recognized that in many applications, the poles assigned are not required to be exactly the same as those speci"ed. This is because the closed-loop system with poles approximately close to the desired ones will possess similar desired behavior [6]. Up to this stage, output feedback robust exact pole assignment (OFREPA) remains a di$cult open problem to be solved. A more tractable variation is to relax the exact pole assignment constraint. This leads to the idea of output feedback robust approximate pole assignment (OFRAPA) which allows inexactness in the assignment of poles [6]. The extra freedom on the choice of K is then exploited to further improving the closed-loop robustness against perturbation. There is an extensive literature published on various kinds of exact and robust pole assignment problems (see [31] for a survey). For OFRAPA, Chu [6] presented an approach using rank-one update algorithms while Oh et al. [19] used the Lagrange multiplier method with built-in regional constraints. Chu's method relies on a suitable choice of a large number of weights, their e!ects are not entirely transparent to the designer. The algorithm of Oh et al., on the other hand, does not converge globally and is sensitive to the starting point for some illconditioned problems. The present work is motivated by recent applications of neural optimization. In [8], Chua and Lin presented a general form of an analog circuit to solve nonlinear programming problems, and in [9], they presented a circuit implementation that is capable of solving various programming problems without computation. Tank and Hop"eld [23] introduced a linear programming neural network and they showed that the energy function of the network was monotonically nonincreasing with time. Kennedy and Chua [14] proposed a neural network circuit implementation that
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could be used to solve a class of nonlinear programming problems. Chiu et al. [5] examined the Hop"eld-like networks utilizing the fact that they are gradient systems. Lillo et al. [17] used neural networks to solve constrained optimization problems. The network circuit was shown to be a gradient system that can be viewed as a penalty method approximation of the original constrained optimization problems. In [33], Zak et al. classi"ed the neural network models for linear programming into three classes, and they compared each class in terms of model complexity, complexity of individual neuron and accuracy of the solutions. Recently, neural networks are applied to solve linear algebra and matrix algebra problems which were formulated as optimization problems [3,4,25}28]. In [29], Wang and Wu applied the neural network approach to compute the state feedback gain of SFEPA by solving the Sylvester equation, and in [30], they used multilayer recurrent neural networks to compute the state feedback gain matrix with the minimum Frobenious norm. However, the robustness of the closed-loop systems has not been taken into account in their work. In [16], Lam and Yan proposed the gradient #ow approach to solve the SFREPA problem by minimizing the Frobenious condition number of the closed-loop eigenvector matrix. Due to the presence of matrix inverse operation in the di!erential equations and the need of "nding solutions of matrix equations, the formulation of SFREPA in [16] does not allow a straightforward implementation in the form of neural networks. In this paper, we proposed a method which inherits the properties of the gradient #ow approach of [16] and extends it to (i) the output feedback case, (ii) cover the case of inexact pole assignment, and (iii) formulate the optimization in such a manner that the solution of the ODE's admits a simple neural network realization. Furthermore, the present approach (iv) allows the overlapping of open-loop and closed-loop poles which is not allowed in [16] and in many standard pole assignment algorithms, and (v) incorporate eigenstructure robustness which is not considered in [29,30]. Schematic circuit diagrams for constructing the neural networks are given. In contrast with the method by Lam et al. [16], our approach requires neither matrix inversion nor solutions of matrix equations. The idea of this paper has been modi"ed to some mechanical systems in [12]. In addition to these nice features, the user needs only to handle a few weights in the guaranteed global convergent optimization process. In the next section, preliminary results on robust pole assignment will be provided. In Section 3, the formulation of the OFRAPA are described in detail. A gradient #ow based algorithm is given and the neural network implementation is also presented. Moreover, the neural complexity of the network will be studied. The simulation results of three examples are shown in Section 4. Finally, conclusions are given in Section 5.
2. Preliminary results To have a self-contained development, we provide some preliminary results on the existence and construction of an output feedback gain for exact pole assignment (see
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for example [13]). Results on the measures of robustness will also be brie#y reviewed. 2.1. Measure of approximate pole assignment We say that a matrix is real pseudo-diagonal if it is a real block-diagonal matrix containing 1;1-blocks for real eigenvalues and 2;2-blocks of the form
p
u
!u
p
for complex conjugate eigenvalue pairs p$uj. Denote spec(M) as the set of eigenvalues of the matrix M. Given a real pseudo-diagonal matrix K31L"L with spec(K)"+j , j ,2, j , and ¹"X\, there exists K satisfying L (A#BKC)¹"¹K, (3) X(A#BKC)"KX
(4)
if and only if ¹ and X satisfy ;2(A¹!¹K)"0, (XA!KX)< "0, ¹X"I,
(5) (6) (7)
where
B"[; ; ]
Z
0
,
(8)
<2 (9) C"[Z 0] ! <2 are the QR factorizations of B and C2, respectively. The matrices [; ; ], [< < ] are orthogonal, matrices Z and Z are nonsingular. Here, unless K is diagonal, ¹ is ! not a closed-loop eigenvector matrix, but there exists a unitary matrix = such that ¹= is an eigenvector matrix. Furthermore, if K exists then it is given by K"Z\;2(¹KX!A)< Z\ (10) ! A K is said to be assignable if there exists a matrix pair (¹, X) to satisfy (5)}(7) simultaneously. Correspondingly, the pair (¹, X) is said to be admissible for such K. It is important to realize that the set of all admissible (¹, X) is nonempty for a given K if and only if K is assignable. In the case of approximate pole assignment, Eqs. (5)}(7) are not required to be satis"ed exactly. By de"ning the errors E " : ;2(A¹!¹K), 2 E " : (XA!KX)< , 6
(11) (12)
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E " : ¹X!I, 26 E " : X¹!I 62 and with K given by (10), we have
195
(13) (14)
A#BKC"A#; Z Z\;2(¹KX!A)< Z\Z <2 ! ! "A#; ;2(¹KX!A)< <2 "A#(I!; ;2)(¹KX!A)(I!< <2) "¹KX!; ;2(¹KX!A)!; ;2(¹KX!A)< <2 (15) "¹KX!(¹KX!A)< <2!; ;2(¹KX!A)< <2. (16) The reason behind the use of both E and E is that, as long as they are nonzero, the 26 62 smallness of either group does not imply the smallness of the other. For good approximate pole assignment, both groups are required to be su$ciently small. By applying Bauer}Fike theorem [32] to (15), (16), and realizing that ¹\E ¹"XE X\"E , we know that if jK is an eigenvalue of A#BKC, then 26 26 62 there is an eigenvalue j of K, such that G "jK !j "4min(i (¹), i (X)) G $ $ ;+(i (¹)#E # #i (X)#E # )#K# ##X\# #E # $ 26 $ $ 62 $ $ $ 6 $ ##¹\# #E # ,. (17) $ 2 $ For su$ciently small #E # , #E # , #E # and #E # , the eigenvalues of A#BKC 2 $ 6 $ 26 $ 62 $ will be close to the desired poles. 2.2. Measure of robustness There are various ways to measure the conditioning of A#BKC. Let M denote either ¹ or X. Two common measures are the spectral condition number given by i (M)"#M# #M\# and the Frobenius condition number given by
(18)
i (M)"#M# #M\# . (19) $ $ $ The latter was used in [2,16,19] for optimization. The Bauer}Fike theorem gives that the spectral variation of the closed-loop state matrix A#BKC due to an unstructured perturbation D is bounded by #M# #M\# #D# for any matrix norm # ) #. Therefore i (M) and i (M) provide meaningful measures on the sensitivity of the $ closed-loop eigenvalues due to unstructured perturbation in A#BKC. The two measures are related by 14n\i (M)4i (M)4i (M). $ $
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The condition numbers achieve their respective minima when the eigensystem is perfectly conditioned, that is when M is orthogonal. However, this is not usually attained as ¹ and X have to satisfy, respectively, (5) and (6). The main advantage of using i (M) lies in the di!erentiability with respect to M. Hence, we will adopt this as $ a measure of the closed-loop conditioning in this presentation. For any c31, cO0, we have i (M)"i (cM). This implies that i (M) has a singu$ $ $ lar Hessian and the minimization may not converge to a solution. To eliminate such a scaling problem, we minimize an alternative measure as in [2,16] given by U(M) " : #M###M\# . (20) $ $ It can be shown that for any admissible ¹ or X, M minimizes U(M) will also minimize i (M) (see [16]). At the minimum point MH, we have i (MH)"U(MH). The $ $ OFRAPA problem considered here is to minimize U(M) over all (¹, X) for an assignable K. When ¹X"I, we have U(¹)"U(X)"#¹###X# . $ $ 3. Problem formulation, neural computation and its network implementation 3.1. Problem formulation In the present OFRAPA problem, we consider both (¹, X) as an unconstrained optimization variable. Based on the penalty function method, the OFRAPA problem is formulated as the following minimization problem: min t(¹, X), 2 6 where
(21)
t(¹, X) " : #¹###X##j(S(E = )#S(= E )) $ $ 2 2 6 6 #o(S(E )#S(E )), (22) 26 62 j, o are large positive real numbers called penalty parameters, = , = are positive 2 6 diagonal matrices for controlling the pole assignment accuracy of each pole. For M"[m ]31KP"KA, GH KP KA S(M) " : p(m ), GH G H where p is some strictly nonnegative convex function such that p(0)"0. The penalty function S is then a nonnegative function, S(M)"0 if and only if M"0. This means the penalty function terms S(E = ), S(= E ), S(E ) and S(E ) in (22) are equal 2 2 6 6 26 62 to zero if and only if the error matrices E , E , E and E are equal to zero 2 6 26 62 respectively. With p denoting the derivative of p, we de"ne *S(M) D(M) " : "[p (m )] GH *M
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Table 1 Summary of some common functions of p( ) )
p ()) p ()) p ())
p( ) )
p ( ) )
( ) ) 2b ln(cosh( ) /b)), b'0 2( ) )arctan( ) )!2 ln (1#( ) )
2( ) ) 2 tanh ( ) /b) 2 arctan( ) )
as the matrix derivative of S(M) with respect to M. Three commonly used p( ) ) and their derivatives are summarized in Table 1. In particular, for the "rst case, we have S(M)"#M#. $ Notice that the unconstrained ¹ and X are not necessarily admissible for the given K. However, if it turns out that K is assignable, the penalty parameters j and o together then determine the accuracy of the approximation of the desired poles. For j<1 and o<1, it is expected that the approximation becomes more accurate and A#BKC has eigenvalues close to those of K. 3.2. Neural computation As in [16], the minimization process is formulated in terms of a gradient #ow. In their approach, the minimization problem (21) is recast into an associated system of "rst-order di!erential equations where the limiting solution is sought. In their work, however, the gradient #ow does not admit a simple neural network realization due to the term ¹\ which is not instantly available for generating the gradient unless we have an explicit form of ¹\ in terms of the elements of ¹. There is no such problem in the present case since ¹\ is approximated by the matrix X. With ¹"[t ], X"[x ], the gradient #ow associated with (21) is given by GH GH *t d¹ "!k , *¹ dt
¹(0)"¹ ,
(23)
dX *t "!k , dt *X
X(0)"X ,
(24)
where *W "2¹#jA2; D(E = )= !j; D(E = )= K2 2 2 2 2 2 2 *¹ #oD(E )X2#oX2D(E ), 26 62
(25)
*W "2X#j= D(= E )<2A2!jK2= D(= E )<2 6 6 6 6 6 6 *X #o¹2D(E #oD(E )¹2 26 62
(26)
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and k'0 is some positive constant and ¹ and X are the initial conditions. It is further noticed that W(¹, X)3C is coercive (a continuous function f (v) de"ned for all v31L is said to be coercive if lim f (v)"#R). This implies that W(¹, X) has ,T, global minimizers which can be found among the critical points of W(¹, X) [20]. Moreover, W(¹, X) has compact sublevel sets and thus the solution (¹(t), X(t)) is de"ned on [0, R) and converges to (¹H, XH) corresponding to a minimum point of W [11]. With P, Q31L"L, de"ne 1P, Q2 " : trace(P2Q). Then
dW *W d¹ *W dX " , # , dt *¹ dt *X dt "!k trace "!k
*W 2 *W *¹ *¹
*W *W # *¹ *X $ $
#trace
*W 2 *W *X *X
40. This guarantees that W(¹(t), X(t)) decreases strictly monotonically in time provided that (¹(0), X(0)) is not a critical point. Normally, the appropriate choice of the scaling constant k is based on experimentation and familiarity with this class of optimization problems. In fact, the value k can theoretically be set arbitrarily large in the gradient #ow. Generally speaking, the larger the value of k the faster the speed of convergence. Since the ordinary di!erential equations to be solved are autonomous, the e!ect of k may be viewed as a scaling parameter on the time-axis. On the other hand, the magnitude of the penalty parameters j, o are not in general dependent upon the dimension of the problem. They are mainly used to provide a relative emphasis between conditioning and accuracy of pole assignment. 3.3. Neural network implementation Although there are many standard numerical routines to solve the ODEs (23) and (24), the process of obtaining a solution can be very time consuming. As parallel distributed processing models, neural networks are composed of many massively connected simple neurons operating concurrently. The inherently parallel and distributed processing nature in neural computation o!ers the neural networks approach computational advantages over the existing sequential algorithms. For such computational neural networks, the solution of high-order nonlinear matrix di!erential equations is not a formidable task, but rather a natural one [11]. The schematic diagram of the neural networks for computing matrix K is shown in Fig. 1, it gives an overall picture of a three-layer networks. The system of di!erential Equations (23) and (24) is implemented directly by the "rst two layers, detailed circuit diagrams are shown in Figs. 2 and 3. Each layer consists of an array of neurons. The "rst layer computes matrices D(E = ), D(= E ), D(E ) and D(E ), as shown in 2 2 6 6 26 62
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199
Fig. 1. The schematic diagram of the three-layer neural networks.
Fig. 2. There are (n!m);n neurons for computing D(E = ), n;(n!p) for 2 2 D(= E ), n;n for D(E ) and n;n for D(E ). An activation function is incorpor6 6 26 62 ated in each neuron of the "rst layer. The second layer, as shown in Fig. 3, computes matrices ¹ and X. There are n;n neurons for computing ¹ and X, respectively. An integration operation is performed in each neuron of the second layer. The outputs of the second layer are fed back to the "rst layer and itself. The neural computation for robust approximate pole assignment feedback gain matrix K according to (10) is completed with a third layer consisting of m;p neurons shown in Fig. 4. The overall neural computation process may be interpreted as having an equivalent closed-loop feedback con"guration if the output of the third layer is applied to the system as depicted in Fig. 5. The choice of the convex function p as described in Section 3.1 determines the corresponding activation function p (the derivative of p) in the "rst layer, which can be linear (p "p ) or nonlinear (p "p or p ). Here p , p and p are respectively, the linear function, the hyperbolic tangent function and the arctangent function. They are used widely in the literature as the activation functions of neural networks. Nonlinear bounded activation function can improve the quality of the continuous-valued solution in the presence of spiky noise.
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Fig. 2. The circuit of the "rst layer neuron.
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Fig. 3. The circuit of the second layer neuron.
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Fig. 4. The circuit of the third layer neuron.
Fig. 5. Block diagram of the recurrent neural network.
To measure the resources for an implementation, we consider the neuron complexity (size and operations) of the network which is de"ned as the number of multiplications/divisions ( ) and additions/substractions ( ) performed by the neuron (see [33]). Table 2 shows the number of neurons (C) of each stage of the computation and the neuron complexity. It should be noted that when n is large, all the neurons have both and operations of order less than 3n (most of them are around 2n).
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Table 2 Neuron Complexity (size and operations) Neurons
C
Layer 1 D(E = ) 2 2 GH D(= E ) 6 6 GH D(E ) ) 26 GH D(E ) ) 62 GH
n(n!m) n(n!p) n n
n(2n#1)#1 n(2n#1)#1 n n
n(2n!1)#1 n(2n!1)#1 n n
Layer 2 ¹ GH X GH
n n
3n(n!m)#4n!m#5 3n(n!p)#4n!p#5
2n(n!m)#2n 2n(n!p)#2n
Layer 3 K GH
mp
2n(n#1)
2n!1
4. Simulation results When the optimal solution, (¹H, XH), of (21) is obtained, the optimal feedback gain matrix KH is computed using (10). To compute the resulting spectral condition number i of the eigenvector matrix of AH"A#BKHC, we "rst compute an A eigenvector matrix < of AH. Suppose K has distinct eigenvalues, then the required A eigenvector matrix <"< D can be obtained by "nding a scaling matrix D"diag(d , d ,2, d ) with d '0 such that i (<) is a minimum via standard optim L G ization under a generalized eigenvalue formulation [1]. In the following, we shall simulate, using conventional serial/digital computers, the behavior of the neural networks implementation of (23) and (24) via three examples. The computer simulation program is written in Fortran running on Vax 8800 calling the IMSL routines. The storage requirement is around 26 arrays of size n each, where n is the state-dimension of the problem. Possible improvements may arise from the implementation using purpose-built hardware, for example, involving a hybrid computer. The implementation details based on CMOS technology may be found in Cichocki and Unbehauen [4]. Nevertheless, this is out of the scope of the present work. Example 1. Consider a 5-state, 3-input, 3-output pilot plant evaporator model [17,18,20] given by
0
0
0 !0.041
!0.0034 0 0.0013
0
0
0
A" 0
0
!1.1471 0
0
0
0
!0.0036 0
0
0
0.094
0.0057
0 !0.0510
,
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B"
!1
0
0
0
0
0
0
0
0.948
0.916
!1
0
!0.598
0
0
0 0 0 0 1
, C" 1 0 0 0 0 0 0 0 1 0
with open-loop eigenvalues at 0, 0, !0.041, !0.051 and !1.1471. The nominal closed-loop poles are !0.0572, !0.1898, !0.2159, !0.5220 and !1.1461. By following the gradient #ow formulation proposed in Section 3 with W(¹, X) as the potential function to be minimized, "ve sets of results using di!erent values of j and o are obtained with p"p and = "= "I . Our results are then compared to 2 6 other published methods in Table 3 (see [18]), in which a metric called optimal matching distance [22] is employed to measure the pole assignment accuracy. The optimal matching distance M is de"ned as
M(S, spec(A )) " : min A L
max "jK !j " , LG G G
where S"+j ,2, j , and spec(A )"+jK ,2, jK , are the set of nominal and assigned L A L poles respectively with n denoting all permutations of +1, 2,2, n,. Notice that, apart from the Oh et al. method, their approaches are only for exact pole assignment, and there is no robustness consideration in both Minimis's method [18] and Roppenecker and O'Reilly's method [21]. We emphasize that the method due to Oh et al. allows regional pole assignment and may achieve a small condition number as shown in Table 3. However, this is associated with an increase in pole assignment error. In this example, our Tests 3}5 show clearly the tradeo! between the reduction of condition number and the smallness of pole assignment error when the condition number is small. In Test 1, j and o are chosen to be large values to ensure pole assignment accuracy. In Test 2, an improvement on the closed-loop eigenstructure robustness is obtained at the expense of relaxing the exactness of the locations of the desired poles. More emphasis is paid on pole insensitivity in Tests 3}5. A good compromise between the tradeo! of condition number reduction and pole assignment accuracy is obtained in Test 3 (see also Fig. 6(a)}(f)) while Test 5 gives the smallest i among all results, although the assigned spectrum inevitably departs from the desired one to some extent. The trajectories of the real and imaginary parts of the poles in Test 3 over the "rst three seconds are shown in Fig. 6(g) and (h). The dotted lines in Fig. 6(g) correspond to the desired pole values. It can be observed that the poles moved quickly toward these values. There are two poles eventually settled at values close to the desired values. The trajectories of the poles are real except over the period from t"10\ to 10\ where a complex pole pair is formed as a result of the collision of two real eigenvalues (see Fig. 6(h)).
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205
Table 3 Comparison of results for Example 1 i
Closed-loop spectrum
M
Neural network Test 1 (j, o"10, k"10\)
88.8
+!0.0571, !0.1899, !0.2159, !0.5220, !1.1461,
0.0001
Neural network Test 2 (j, o"10, k"10\)
23.0
+!0.0418, !0.1939, !0.2160, !0.5204, !1.1452,
0.0154
Neural network Test 3 (j, o"10, k"1)
14.8
+!0.0424, !0.1382, !0.2158, !0.5087, !1.1451,
0.0516
Neural network Test 4 (j, o"3.5;10, k"1)
10.8
+!0.0426, !0.0846, !0.2157, !0.5096, !1.1452,
0.1052
Neural network Test 5 (j"10, o"10, k"1)
9.5
+!0.0414, !0.1637, !0.2157, !0.0247, !1.1470,
0.3063
Miminis's method
341.0
+!0.0572, !0.1898, !0.2159, !0.5219, !1.1461,
0.0001
Oh, Gu and Spurgeon's method (exact,
91.8
+!0.0572, !0.1898, !0.2159, !0.5219, !1.1461,
0.0001
Oh, Gu and Spurgeon's method (regional)
10.2
+!0.0412, !0.0803, !1.1384, !1.1484, !2.0000,
0.9225
Roppenecker and O'Reilly's method
348.5
+!0.0572, !0.1898, !0.2159, !0.5220, !1.1461,
0.0001
Example 2. Now, consider a 4-state, 2-input, 3-output system [7,10,19] given by
A"
0
1 0 0
0 0
1
1 0 0
1 0
!1 0 0 0 0
0 0 0
,
B"
0 0 0 1
,
1 0 0 0
C" 0 0 1 0 0 0 0 1
with open-loop eignvalues at 0, 0, 1.618 and !0.618. The nominal closed-loop poles are !1, !2, !3 and !4. For the sake of assigning the set of closed-loop poles very accurately, we use = "= "I , j"o"10 and k"10\. Our results are 2 6 compared with others in Table 4 (see [19]). It is clear that when we set j and o to such huge values, the closed-loop spectrum can be assigned nearly exactly with i kept small enough as well.
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Fig. 6. Simulation Results of Example 1.
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207
Table 4 Comparison of results for Example 2
Neural network Tests 1, 2 (p"p , p ) Neural network Test 3 (p"p , b"10) Neural network Test 4 (p"p ,b"10) Neural network Test 5 (p"p ,b"10) Chu, Nichols and Kautsky's method Fletcher's method Oh, Gu and Spurgeon's method (exact)
i
Closed-loop spectrum
499.2 499.2 499.3 532.6 778.2 884.8 499.2
+!1.00000, !2.00000, !3.00000, !4.00000, +!1.00000, !2.00001, !2.99999, !4.00000, +!0.99992, !2.00072, !2.99891, !4.00015, +!0.99202, !2.07947, !2.88429, !4.01404, +!0.973, !2.226, !2.918, !3.800, +!1.0, !2.0, !3.0, !4.0, +!1.00000, !2.00000, !2.99997, !4.00001,
Example 3. Lastly, we consider a 2-input, 5-state distillation column model [3,4,24] given by
!0.1094
0.0628
0
0
0
1.306
!2.132
0.9807
0
0
0
1.595
!3.149
1.547
0
0
0.0355
2.632
!4.257
1.855
0
0.00227
0
0.1636
!0.1625
A"
0
0
0.0638
0
B" 0.0838 !0.1396 0.1004 !0.2060 0.0063 !0.0128
,
The eigenvalues are at !0.077324, !0.014232, !0.89531, !2.8408 and !5.9822. They are required to be reassigned into !1$j, !0.2, !0.5 and !1 via a state feedback gain matrix K and we have
K"
!1
1
!1 !1
0
0
0
0
0
0
0
0
!0.2
0
0
0
0
0
!0.5
0
0
0
0
0
!1
.
The case of state feedback is considered as a special case of output feedback with C"I . In this case, we use the convex function p"p and k"1. Di!erent penalty parameters as well as weighting matrices = and = are used and the results are 2 6
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D.W.C. Ho et al. / Neurocomputing 25 (1999) 191}211 Table 5 Comparison of results for Example 3
Neural network Test 1 (= "= "I, j, o"3000) 2 6 Neural network Test 2 (= "= "I, j, o"500) 2 6 Neural network Test 3 (= "= "W, j, o"3000) 2 6 Lam and Yan's method Byers and Nash's method Kautsky, Nichols and van Dooren's method 1
i
#K#
32.1 30.9 31.9 33.6 33.1 39.4
361.6 328.0 331.0 337.4 354.9 311.5
Table 6 Corresponding percentage errors of the eigenvalues for Example 3
Neural network Test 1 Neural network Test 2 Neural network Test 3
Closed-loop spectrum
Percentage errors
+!1.05$0.99j, !0.19, !0.49, !1.00, +!1.25$0.92j, !0.15, !0.44, !0.98, +!1.14$0.96j, !0.17, !0.48, !0.98,
+3.25, 7.01, 1.95, 0.41, +18.36, 23.00, 11.98, 1.61, +10.55, 12.76, 3.70, 1.58,
summarized in Table 5. In all the cases, our i are smaller than those obtained by the others. The assigned closed-loop spectrum and the corresponding pole percentage errors for the three sets of results are shown in Table 6. Again, in the "rst case, the penalty parameters j and o are chosen to be large values to ensure the pole assignment accuracy. In the second case, improvement on the closed-loop eigenstructure robustness is well expected by choosing smaller penalty parameters. However, one consequence associated with such action is that the set of assigned poles becomes relatively less accurate. In fact, the percentage errors for the poles (except !0.98) increase a lot when j and o decreases from 3000 to 500. Hence in the third case, we try to use weighting matrices = and = which are set to be 2 6
W"
1.6
0
0
0
0
0
1.6
0
0
0
0
0
1.6
0
0
0
0
0
0
0
0
1.3 0 0
1
to re"ne the accuracy of individual assigned poles. From Table 5, we can observe the e!ect of such action is that percentage errors for the assigned poles !1.14$0.96j, !0.17 and !0.48 are made smaller signi"cantly. Remark. For di!erent initial conditions with "xed set of j, o and k in all the examples, although it is found that ¹, X, and K do not always converge to the same values, the
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objective function values achieved do not vary signi"cantly. Di!erent initial conditions have small e!ect on the resulting condition number and the quality of the accuracy of the poles. The computational time for all the simulations are reasonably fast, for instance, they are all within 10 s for 2n"50 optimization variables in the "rst example.
5. Conclusions In this paper, we have presented a neural network approach to compute the output feedback gain for robust approximate pole assignment. The gradient #ow formulation allows the optimization process to admit a simple recurrent neural network realization. The e!ectiveness of the neural computation process is demonstrated by numerical simulations.
Acknowledgements This work is supported by City University research grant. The authors would like to thank the reviewers' constructive comments.
References [1] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, SIAM Studies in Applied Mathematics, 1994. [2] R. Byers, S.G. Nash, Approaches to robust pole assignment, Int. J. Control 49 (1) (1989) 97}117. [3] A. Cichocki, R. Unbehauen, Neural Networks for Optimization and Signal Processing, Wiley, England, 1993. [4] A. Cichocki, R. Unbehauen, Simpli"ed neural networks for solving linear least squares and total least squares problems in real time, IEEE Trans. Neural Networks 5 (6) (1994) 910}923. [5] C. Chiu, C.Y. Maa, M.A. Shanblatt, Energy function analysis of dynamic programming neural networks, IEEE Trans. Neural Networks 2 (4) (1991) 418}426. [6] E.K. Chu, Approximate pole assignment,, Int. J. Control 58 (2) (1993) 471}484. [7] E.K.-W. Chu, N.K. Nichols, J. Kautsky, Robust pole assignment by output feedback, Proceedings of 4th IMA Conference on Control Theory, 1984, pp. 137}146. [8] L.O. Chua, G.N. Lin, Nonlinear optimization with constraints: a cookbook approach, Int. J. Circuit Theory Appl. 11 (2) (1983) 141}159. [9] L.O. Chua, G.N. Lin, Nonlinear programming without computation, IEEE Trans. Circuits Systems CAS-31 (2) (1984) 182}188. [10] L.R. Fletcher, An intermediate algorithm for pole placement by output feedback in linear multivariable control system, Int. J. Control 31 (1980) 1121}1136. [11] U. Helmke, J.B. Moore, Optimization and Dynamical Systems, Springer, Berlin, 1994. [12] D.W.C. Ho, J. Lam, J. Xu, Robust Approximate Pole Assignment for Second-Order Systems: Neural Network Computation, Journal of Guidance, Control and Dynamics 21 (6) (1998) 923}929. [13] J. Kautsky, N.K. Nichols, P. van Dooren, Robust pole placement in linear state feedback, Int. J. Control 41 (5) (1995) 1129}1155. [14] M.P. Kennedy, L.O. Chua, Neural networks for nonlinear programming, IEEE Trans. Circuits Systems 35 (5) (1988) 554}562.
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[15] H. Kimura, A further result on the problem of pole assignment by output feedback, IEEE Trans. Automat. Control 25 (3) (1977) 458}463. [16] J. Lam, W.Y. Yan, A gradient #ow approach to robust pole assignment problem, Int. J. Nonlinear Robust Control 5 (1995) 175}185. [17] W.E. Lillo, M.H. Loh, S. Hui, S.H. Zak, On solving constrained optimization problems with Neural networks: a penalty method approach, IEEE Trans. Neural Networks 4 (6) (1993) 931}940. [18] G. Miminis, Using de#ation in the pole assignment problem with output feedback, Proceedings of 3rd International Conference on Aerospace Computational Control, 1989. [19] M. Oh, D.W. Gu, S.K. Spurgeon, Robust pole assignment in a speci"ed region using output feedback, Optim. Control Appl. Meth. 14 (1993) 57}66. [20] A.L. Peressini, F.E. Sullivan, J.J. Uhl Jr., The Mathematics of Nonlinear Programming, Springer, New York, 1988. [21] G. Roppenecker, J. O'Reilly, Parametric output feedback controller design, Automatica 25 (1989) 259}265. [22] G.W. Stewart, Introduction to Matrix Computations, Academic Press, New York, 1973. [23] D.W. Tank, J.J. Hop"eld, Simple &Neural' optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit, IEEE Trans. Circuits Systems CAS 33 (5) (1986) 533}541. [24] X. Wang, Pole placement by static output feedback, J. Math. Systems Estimation Control 2 (2) (1992) 205}218. [25] J. Wang, Recurrent neural networks for solving linear matrix equations, Comput. Math. Appl. 26 (9) (1993) 23}34. [26] L.X. Wang, J.M. Mendel, Three-dimensional structured networks for matrix equation solving, IEEE Trans. Comput. 40 (12) (1991) 1337}1346. [27] L.X. Wang, J.M. Mendel, Parallel structured networks for solving a wide variety of matrix algebra problems, J. Parallel Distributed Comput. 14 (1992) 236}247. [28] J. Wang, G. Wu, Recurrent neural networks for LU decomposition and cholesky factorization, Math. Comput. Modelling 18 (6) (1993) 1}8. [29] J. Wang, G. Wu, Recurrent neural networks for synthesizing linear control systems via pole placment, Int. J. Systems Sci. 26 (12) (1995) 2369}2382. [30] J. Wang, G. Wu, A multilayer recurrent neural networks for on-line synthesis of minimum-norm linear feedback control systems via pole placement, Automatica 32 (3) (1996) 435}442. [31] B.A. White, Eigenstructure assignment: a survey, Proc. Inst. Mech. Engrs. 209 (1995) 1}11. [32] J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. [33] S.H. Zak, V. Upatising, S. Hui, Solving linear programming problems with neural networks: a comparative study, IEEE Trans. Neural Networks 6 (1) (1995) 94}104.
Dr. Daniel W.C. Ho received a "rst class B.Sc., M.Sc. and Ph.D. degrees in mathematics from the University of Salford in 1980, 1982 and 1986, respectively. From 1985 to 1988, Dr Ho was a Research Fellow in Industrial Control Unit, University of Strathclyde. In 1989, he joined the Department of Mathematics, City University of Hong Kong, where he is currently an Associate Professor. His research interests include H control theory, robust pole assignment problem, adaptive neural wavelet identi"cation. Dr. James Lam received a "rst class B.Sc. degree in mechanical engineering from the University of Manchester in 1983. He then obtained the M.Phil. and Ph.D. degrees in the area of control engineering from the University of Cambridge in 1985 and 1988, respectively. Dr. Lam has held faculty positions at the now City University of Hong Kong and the University of Melbourne. He is now an Associate Professor in the Department of Mechanical Engineering, the University of Hong Kong, and is holding a Concurrent Professorship. Ms. Jinhua Xu received her B.S. and M.Eng. degrees in control engineering from Xi'an Jiaotong University, China, in 1998 and 1991, respectively. From 1991 to 1995, she worked in Dept. of Automatic Control Engineering, Xi'an Jiaotong University. Since 1995, she worked as a research assistant and then was admitted as a Ph.D. student in City University of Hong Kong. Her research studies are about wavelet neural networks, system identi"cation and adaptive control.
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Mr. Patrick Tam received his B.Eng.(Hons.) degree in Mechanical Engineering from the University of Hong Kong in 1994. Since graduation, he has been conducting research studies in the "eld of control engineering at the Department of Mechanical Engineering, the University of Hong Kong. His Ph.D. thesis focuses on the use of optimization methods in solving various robust pole assignment problems in control system design.
Neural computation for robust approximate pole assignment Daniel W.C. Ho *, James Lam, Jinhua Xu , Hei Ka Tam Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Hong Kong Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong Received 5 January 1998; accepted 15 December 1998
Abstract This paper provides an approach for output feedback robust approximate pole assignment. It is formulated as an unconstrained optimization problem and solved via the gradient #ow approach which is ideally suited for neural computing implementation. A schematic circuit architecture of the neural network is suggested. Simulation results are used to demonstrate the e!ectiveness of the proposed method. 1999 Elsevier Science B.V. All rights reserved. Keywords: Output feedback; Robustness; Approximate pole assignment; Neural networks; Gradient #ow
1. Introduction Consider a linear time-invariant multivariable system described by x "Ax#Bu,
(1)
y"Cx,
(2)
where x31L, u31K and y31N represent the state, input, and output vectors, respectively, A31L"L, B31L"K, and C31N"L are constant matrices. As usual, it is assumed that B and C are of full column and row rank, respectively. By applying a constant
* Corresponding author. 0925-2312/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 9 9 ) 0 0 0 5 7 - 0
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output feedback law u"Ky to (1) and (2), K31K"N, the closed-loop system is given by x "(A#BKC)x. It was established that under the conditions of (A, B) completely controllable and (C, A) completely observable and that m#p'n [15] or mp'n [24], there exist feedback matrices K for almost any set of self-conjugate complex numbers +j , j ,2, j , as the desired poles of the closed-loop system. The problem of "nding L one such K is referred to as the output feedback exact pole assignment (OFEPA). In the special case when C"I , the n;n identity matrix, we have state feedback exact L pole assignment (SFEPA). The solution of the output feedback pole assignment problem is, in general, nonunique for a multivariable system. The freedom on the choice of K may be exploited for optimizing other design objectives. One of these objectives is to maximize the robustness of the assigned poles in face of perturbation. The state feedback robust exact pole assignment (SFREPA) problem was formulated by Kautsky et al. [13] in terms of minimizing some conditioning measures of the closed-loop state matrix. More recently, Byers and Nash [2] tackled this problem using Newton's method and truncated-Newton method. While state feedback exact pole assignment was considered by Kautsky and many others, it is commonly recognized that in many applications, the poles assigned are not required to be exactly the same as those speci"ed. This is because the closed-loop system with poles approximately close to the desired ones will possess similar desired behavior [6]. Up to this stage, output feedback robust exact pole assignment (OFREPA) remains a di$cult open problem to be solved. A more tractable variation is to relax the exact pole assignment constraint. This leads to the idea of output feedback robust approximate pole assignment (OFRAPA) which allows inexactness in the assignment of poles [6]. The extra freedom on the choice of K is then exploited to further improving the closed-loop robustness against perturbation. There is an extensive literature published on various kinds of exact and robust pole assignment problems (see [31] for a survey). For OFRAPA, Chu [6] presented an approach using rank-one update algorithms while Oh et al. [19] used the Lagrange multiplier method with built-in regional constraints. Chu's method relies on a suitable choice of a large number of weights, their e!ects are not entirely transparent to the designer. The algorithm of Oh et al., on the other hand, does not converge globally and is sensitive to the starting point for some illconditioned problems. The present work is motivated by recent applications of neural optimization. In [8], Chua and Lin presented a general form of an analog circuit to solve nonlinear programming problems, and in [9], they presented a circuit implementation that is capable of solving various programming problems without computation. Tank and Hop"eld [23] introduced a linear programming neural network and they showed that the energy function of the network was monotonically nonincreasing with time. Kennedy and Chua [14] proposed a neural network circuit implementation that
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could be used to solve a class of nonlinear programming problems. Chiu et al. [5] examined the Hop"eld-like networks utilizing the fact that they are gradient systems. Lillo et al. [17] used neural networks to solve constrained optimization problems. The network circuit was shown to be a gradient system that can be viewed as a penalty method approximation of the original constrained optimization problems. In [33], Zak et al. classi"ed the neural network models for linear programming into three classes, and they compared each class in terms of model complexity, complexity of individual neuron and accuracy of the solutions. Recently, neural networks are applied to solve linear algebra and matrix algebra problems which were formulated as optimization problems [3,4,25}28]. In [29], Wang and Wu applied the neural network approach to compute the state feedback gain of SFEPA by solving the Sylvester equation, and in [30], they used multilayer recurrent neural networks to compute the state feedback gain matrix with the minimum Frobenious norm. However, the robustness of the closed-loop systems has not been taken into account in their work. In [16], Lam and Yan proposed the gradient #ow approach to solve the SFREPA problem by minimizing the Frobenious condition number of the closed-loop eigenvector matrix. Due to the presence of matrix inverse operation in the di!erential equations and the need of "nding solutions of matrix equations, the formulation of SFREPA in [16] does not allow a straightforward implementation in the form of neural networks. In this paper, we proposed a method which inherits the properties of the gradient #ow approach of [16] and extends it to (i) the output feedback case, (ii) cover the case of inexact pole assignment, and (iii) formulate the optimization in such a manner that the solution of the ODE's admits a simple neural network realization. Furthermore, the present approach (iv) allows the overlapping of open-loop and closed-loop poles which is not allowed in [16] and in many standard pole assignment algorithms, and (v) incorporate eigenstructure robustness which is not considered in [29,30]. Schematic circuit diagrams for constructing the neural networks are given. In contrast with the method by Lam et al. [16], our approach requires neither matrix inversion nor solutions of matrix equations. The idea of this paper has been modi"ed to some mechanical systems in [12]. In addition to these nice features, the user needs only to handle a few weights in the guaranteed global convergent optimization process. In the next section, preliminary results on robust pole assignment will be provided. In Section 3, the formulation of the OFRAPA are described in detail. A gradient #ow based algorithm is given and the neural network implementation is also presented. Moreover, the neural complexity of the network will be studied. The simulation results of three examples are shown in Section 4. Finally, conclusions are given in Section 5.
2. Preliminary results To have a self-contained development, we provide some preliminary results on the existence and construction of an output feedback gain for exact pole assignment (see
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for example [13]). Results on the measures of robustness will also be brie#y reviewed. 2.1. Measure of approximate pole assignment We say that a matrix is real pseudo-diagonal if it is a real block-diagonal matrix containing 1;1-blocks for real eigenvalues and 2;2-blocks of the form
p
u
!u
p
for complex conjugate eigenvalue pairs p$uj. Denote spec(M) as the set of eigenvalues of the matrix M. Given a real pseudo-diagonal matrix K31L"L with spec(K)"+j , j ,2, j , and ¹"X\, there exists K satisfying L (A#BKC)¹"¹K, (3) X(A#BKC)"KX
(4)
if and only if ¹ and X satisfy ;2(A¹!¹K)"0, (XA!KX)< "0, ¹X"I,
(5) (6) (7)
where
B"[; ; ]
Z
0
,
(8)
<2 (9) C"[Z 0] ! <2 are the QR factorizations of B and C2, respectively. The matrices [; ; ], [< < ] are orthogonal, matrices Z and Z are nonsingular. Here, unless K is diagonal, ¹ is ! not a closed-loop eigenvector matrix, but there exists a unitary matrix = such that ¹= is an eigenvector matrix. Furthermore, if K exists then it is given by K"Z\;2(¹KX!A)< Z\ (10) ! A K is said to be assignable if there exists a matrix pair (¹, X) to satisfy (5)}(7) simultaneously. Correspondingly, the pair (¹, X) is said to be admissible for such K. It is important to realize that the set of all admissible (¹, X) is nonempty for a given K if and only if K is assignable. In the case of approximate pole assignment, Eqs. (5)}(7) are not required to be satis"ed exactly. By de"ning the errors E " : ;2(A¹!¹K), 2 E " : (XA!KX)< , 6
(11) (12)
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E " : ¹X!I, 26 E " : X¹!I 62 and with K given by (10), we have
195
(13) (14)
A#BKC"A#; Z Z\;2(¹KX!A)< Z\Z <2 ! ! "A#; ;2(¹KX!A)< <2 "A#(I!; ;2)(¹KX!A)(I!< <2) "¹KX!; ;2(¹KX!A)!; ;2(¹KX!A)< <2 (15) "¹KX!(¹KX!A)< <2!; ;2(¹KX!A)< <2. (16) The reason behind the use of both E and E is that, as long as they are nonzero, the 26 62 smallness of either group does not imply the smallness of the other. For good approximate pole assignment, both groups are required to be su$ciently small. By applying Bauer}Fike theorem [32] to (15), (16), and realizing that ¹\E ¹"XE X\"E , we know that if jK is an eigenvalue of A#BKC, then 26 26 62 there is an eigenvalue j of K, such that G "jK !j "4min(i (¹), i (X)) G $ $ ;+(i (¹)#E # #i (X)#E # )#K# ##X\# #E # $ 26 $ $ 62 $ $ $ 6 $ ##¹\# #E # ,. (17) $ 2 $ For su$ciently small #E # , #E # , #E # and #E # , the eigenvalues of A#BKC 2 $ 6 $ 26 $ 62 $ will be close to the desired poles. 2.2. Measure of robustness There are various ways to measure the conditioning of A#BKC. Let M denote either ¹ or X. Two common measures are the spectral condition number given by i (M)"#M# #M\# and the Frobenius condition number given by
(18)
i (M)"#M# #M\# . (19) $ $ $ The latter was used in [2,16,19] for optimization. The Bauer}Fike theorem gives that the spectral variation of the closed-loop state matrix A#BKC due to an unstructured perturbation D is bounded by #M# #M\# #D# for any matrix norm # ) #. Therefore i (M) and i (M) provide meaningful measures on the sensitivity of the $ closed-loop eigenvalues due to unstructured perturbation in A#BKC. The two measures are related by 14n\i (M)4i (M)4i (M). $ $
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The condition numbers achieve their respective minima when the eigensystem is perfectly conditioned, that is when M is orthogonal. However, this is not usually attained as ¹ and X have to satisfy, respectively, (5) and (6). The main advantage of using i (M) lies in the di!erentiability with respect to M. Hence, we will adopt this as $ a measure of the closed-loop conditioning in this presentation. For any c31, cO0, we have i (M)"i (cM). This implies that i (M) has a singu$ $ $ lar Hessian and the minimization may not converge to a solution. To eliminate such a scaling problem, we minimize an alternative measure as in [2,16] given by U(M) " : #M###M\# . (20) $ $ It can be shown that for any admissible ¹ or X, M minimizes U(M) will also minimize i (M) (see [16]). At the minimum point MH, we have i (MH)"U(MH). The $ $ OFRAPA problem considered here is to minimize U(M) over all (¹, X) for an assignable K. When ¹X"I, we have U(¹)"U(X)"#¹###X# . $ $ 3. Problem formulation, neural computation and its network implementation 3.1. Problem formulation In the present OFRAPA problem, we consider both (¹, X) as an unconstrained optimization variable. Based on the penalty function method, the OFRAPA problem is formulated as the following minimization problem: min t(¹, X), 2 6 where
(21)
t(¹, X) " : #¹###X##j(S(E = )#S(= E )) $ $ 2 2 6 6 #o(S(E )#S(E )), (22) 26 62 j, o are large positive real numbers called penalty parameters, = , = are positive 2 6 diagonal matrices for controlling the pole assignment accuracy of each pole. For M"[m ]31KP"KA, GH KP KA S(M) " : p(m ), GH G H where p is some strictly nonnegative convex function such that p(0)"0. The penalty function S is then a nonnegative function, S(M)"0 if and only if M"0. This means the penalty function terms S(E = ), S(= E ), S(E ) and S(E ) in (22) are equal 2 2 6 6 26 62 to zero if and only if the error matrices E , E , E and E are equal to zero 2 6 26 62 respectively. With p denoting the derivative of p, we de"ne *S(M) D(M) " : "[p (m )] GH *M
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197
Table 1 Summary of some common functions of p( ) )
p ()) p ()) p ())
p( ) )
p ( ) )
( ) ) 2b ln(cosh( ) /b)), b'0 2( ) )arctan( ) )!2 ln (1#( ) )
2( ) ) 2 tanh ( ) /b) 2 arctan( ) )
as the matrix derivative of S(M) with respect to M. Three commonly used p( ) ) and their derivatives are summarized in Table 1. In particular, for the "rst case, we have S(M)"#M#. $ Notice that the unconstrained ¹ and X are not necessarily admissible for the given K. However, if it turns out that K is assignable, the penalty parameters j and o together then determine the accuracy of the approximation of the desired poles. For j<1 and o<1, it is expected that the approximation becomes more accurate and A#BKC has eigenvalues close to those of K. 3.2. Neural computation As in [16], the minimization process is formulated in terms of a gradient #ow. In their approach, the minimization problem (21) is recast into an associated system of "rst-order di!erential equations where the limiting solution is sought. In their work, however, the gradient #ow does not admit a simple neural network realization due to the term ¹\ which is not instantly available for generating the gradient unless we have an explicit form of ¹\ in terms of the elements of ¹. There is no such problem in the present case since ¹\ is approximated by the matrix X. With ¹"[t ], X"[x ], the gradient #ow associated with (21) is given by GH GH *t d¹ "!k , *¹ dt
¹(0)"¹ ,
(23)
dX *t "!k , dt *X
X(0)"X ,
(24)
where *W "2¹#jA2; D(E = )= !j; D(E = )= K2 2 2 2 2 2 2 *¹ #oD(E )X2#oX2D(E ), 26 62
(25)
*W "2X#j= D(= E )<2A2!jK2= D(= E )<2 6 6 6 6 6 6 *X #o¹2D(E #oD(E )¹2 26 62
(26)
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and k'0 is some positive constant and ¹ and X are the initial conditions. It is further noticed that W(¹, X)3C is coercive (a continuous function f (v) de"ned for all v31L is said to be coercive if lim f (v)"#R). This implies that W(¹, X) has ,T, global minimizers which can be found among the critical points of W(¹, X) [20]. Moreover, W(¹, X) has compact sublevel sets and thus the solution (¹(t), X(t)) is de"ned on [0, R) and converges to (¹H, XH) corresponding to a minimum point of W [11]. With P, Q31L"L, de"ne 1P, Q2 " : trace(P2Q). Then
dW *W d¹ *W dX " , # , dt *¹ dt *X dt "!k trace "!k
*W 2 *W *¹ *¹
*W *W # *¹ *X $ $
#trace
*W 2 *W *X *X
40. This guarantees that W(¹(t), X(t)) decreases strictly monotonically in time provided that (¹(0), X(0)) is not a critical point. Normally, the appropriate choice of the scaling constant k is based on experimentation and familiarity with this class of optimization problems. In fact, the value k can theoretically be set arbitrarily large in the gradient #ow. Generally speaking, the larger the value of k the faster the speed of convergence. Since the ordinary di!erential equations to be solved are autonomous, the e!ect of k may be viewed as a scaling parameter on the time-axis. On the other hand, the magnitude of the penalty parameters j, o are not in general dependent upon the dimension of the problem. They are mainly used to provide a relative emphasis between conditioning and accuracy of pole assignment. 3.3. Neural network implementation Although there are many standard numerical routines to solve the ODEs (23) and (24), the process of obtaining a solution can be very time consuming. As parallel distributed processing models, neural networks are composed of many massively connected simple neurons operating concurrently. The inherently parallel and distributed processing nature in neural computation o!ers the neural networks approach computational advantages over the existing sequential algorithms. For such computational neural networks, the solution of high-order nonlinear matrix di!erential equations is not a formidable task, but rather a natural one [11]. The schematic diagram of the neural networks for computing matrix K is shown in Fig. 1, it gives an overall picture of a three-layer networks. The system of di!erential Equations (23) and (24) is implemented directly by the "rst two layers, detailed circuit diagrams are shown in Figs. 2 and 3. Each layer consists of an array of neurons. The "rst layer computes matrices D(E = ), D(= E ), D(E ) and D(E ), as shown in 2 2 6 6 26 62
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199
Fig. 1. The schematic diagram of the three-layer neural networks.
Fig. 2. There are (n!m);n neurons for computing D(E = ), n;(n!p) for 2 2 D(= E ), n;n for D(E ) and n;n for D(E ). An activation function is incorpor6 6 26 62 ated in each neuron of the "rst layer. The second layer, as shown in Fig. 3, computes matrices ¹ and X. There are n;n neurons for computing ¹ and X, respectively. An integration operation is performed in each neuron of the second layer. The outputs of the second layer are fed back to the "rst layer and itself. The neural computation for robust approximate pole assignment feedback gain matrix K according to (10) is completed with a third layer consisting of m;p neurons shown in Fig. 4. The overall neural computation process may be interpreted as having an equivalent closed-loop feedback con"guration if the output of the third layer is applied to the system as depicted in Fig. 5. The choice of the convex function p as described in Section 3.1 determines the corresponding activation function p (the derivative of p) in the "rst layer, which can be linear (p "p ) or nonlinear (p "p or p ). Here p , p and p are respectively, the linear function, the hyperbolic tangent function and the arctangent function. They are used widely in the literature as the activation functions of neural networks. Nonlinear bounded activation function can improve the quality of the continuous-valued solution in the presence of spiky noise.
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Fig. 2. The circuit of the "rst layer neuron.
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Fig. 3. The circuit of the second layer neuron.
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Fig. 4. The circuit of the third layer neuron.
Fig. 5. Block diagram of the recurrent neural network.
To measure the resources for an implementation, we consider the neuron complexity (size and operations) of the network which is de"ned as the number of multiplications/divisions ( ) and additions/substractions ( ) performed by the neuron (see [33]). Table 2 shows the number of neurons (C) of each stage of the computation and the neuron complexity. It should be noted that when n is large, all the neurons have both and operations of order less than 3n (most of them are around 2n).
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Table 2 Neuron Complexity (size and operations) Neurons
C
Layer 1 D(E = ) 2 2 GH D(= E ) 6 6 GH D(E ) ) 26 GH D(E ) ) 62 GH
n(n!m) n(n!p) n n
n(2n#1)#1 n(2n#1)#1 n n
n(2n!1)#1 n(2n!1)#1 n n
Layer 2 ¹ GH X GH
n n
3n(n!m)#4n!m#5 3n(n!p)#4n!p#5
2n(n!m)#2n 2n(n!p)#2n
Layer 3 K GH
mp
2n(n#1)
2n!1
4. Simulation results When the optimal solution, (¹H, XH), of (21) is obtained, the optimal feedback gain matrix KH is computed using (10). To compute the resulting spectral condition number i of the eigenvector matrix of AH"A#BKHC, we "rst compute an A eigenvector matrix < of AH. Suppose K has distinct eigenvalues, then the required A eigenvector matrix <"< D can be obtained by "nding a scaling matrix D"diag(d , d ,2, d ) with d '0 such that i (<) is a minimum via standard optim L G ization under a generalized eigenvalue formulation [1]. In the following, we shall simulate, using conventional serial/digital computers, the behavior of the neural networks implementation of (23) and (24) via three examples. The computer simulation program is written in Fortran running on Vax 8800 calling the IMSL routines. The storage requirement is around 26 arrays of size n each, where n is the state-dimension of the problem. Possible improvements may arise from the implementation using purpose-built hardware, for example, involving a hybrid computer. The implementation details based on CMOS technology may be found in Cichocki and Unbehauen [4]. Nevertheless, this is out of the scope of the present work. Example 1. Consider a 5-state, 3-input, 3-output pilot plant evaporator model [17,18,20] given by
0
0
0 !0.041
!0.0034 0 0.0013
0
0
0
A" 0
0
!1.1471 0
0
0
0
!0.0036 0
0
0
0.094
0.0057
0 !0.0510
,
204
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B"
!1
0
0
0
0
0
0
0
0.948
0.916
!1
0
!0.598
0
0
0 0 0 0 1
, C" 1 0 0 0 0 0 0 0 1 0
with open-loop eigenvalues at 0, 0, !0.041, !0.051 and !1.1471. The nominal closed-loop poles are !0.0572, !0.1898, !0.2159, !0.5220 and !1.1461. By following the gradient #ow formulation proposed in Section 3 with W(¹, X) as the potential function to be minimized, "ve sets of results using di!erent values of j and o are obtained with p"p and = "= "I . Our results are then compared to 2 6 other published methods in Table 3 (see [18]), in which a metric called optimal matching distance [22] is employed to measure the pole assignment accuracy. The optimal matching distance M is de"ned as
M(S, spec(A )) " : min A L
max "jK !j " , LG G G
where S"+j ,2, j , and spec(A )"+jK ,2, jK , are the set of nominal and assigned L A L poles respectively with n denoting all permutations of +1, 2,2, n,. Notice that, apart from the Oh et al. method, their approaches are only for exact pole assignment, and there is no robustness consideration in both Minimis's method [18] and Roppenecker and O'Reilly's method [21]. We emphasize that the method due to Oh et al. allows regional pole assignment and may achieve a small condition number as shown in Table 3. However, this is associated with an increase in pole assignment error. In this example, our Tests 3}5 show clearly the tradeo! between the reduction of condition number and the smallness of pole assignment error when the condition number is small. In Test 1, j and o are chosen to be large values to ensure pole assignment accuracy. In Test 2, an improvement on the closed-loop eigenstructure robustness is obtained at the expense of relaxing the exactness of the locations of the desired poles. More emphasis is paid on pole insensitivity in Tests 3}5. A good compromise between the tradeo! of condition number reduction and pole assignment accuracy is obtained in Test 3 (see also Fig. 6(a)}(f)) while Test 5 gives the smallest i among all results, although the assigned spectrum inevitably departs from the desired one to some extent. The trajectories of the real and imaginary parts of the poles in Test 3 over the "rst three seconds are shown in Fig. 6(g) and (h). The dotted lines in Fig. 6(g) correspond to the desired pole values. It can be observed that the poles moved quickly toward these values. There are two poles eventually settled at values close to the desired values. The trajectories of the poles are real except over the period from t"10\ to 10\ where a complex pole pair is formed as a result of the collision of two real eigenvalues (see Fig. 6(h)).
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Table 3 Comparison of results for Example 1 i
Closed-loop spectrum
M
Neural network Test 1 (j, o"10, k"10\)
88.8
+!0.0571, !0.1899, !0.2159, !0.5220, !1.1461,
0.0001
Neural network Test 2 (j, o"10, k"10\)
23.0
+!0.0418, !0.1939, !0.2160, !0.5204, !1.1452,
0.0154
Neural network Test 3 (j, o"10, k"1)
14.8
+!0.0424, !0.1382, !0.2158, !0.5087, !1.1451,
0.0516
Neural network Test 4 (j, o"3.5;10, k"1)
10.8
+!0.0426, !0.0846, !0.2157, !0.5096, !1.1452,
0.1052
Neural network Test 5 (j"10, o"10, k"1)
9.5
+!0.0414, !0.1637, !0.2157, !0.0247, !1.1470,
0.3063
Miminis's method
341.0
+!0.0572, !0.1898, !0.2159, !0.5219, !1.1461,
0.0001
Oh, Gu and Spurgeon's method (exact,
91.8
+!0.0572, !0.1898, !0.2159, !0.5219, !1.1461,
0.0001
Oh, Gu and Spurgeon's method (regional)
10.2
+!0.0412, !0.0803, !1.1384, !1.1484, !2.0000,
0.9225
Roppenecker and O'Reilly's method
348.5
+!0.0572, !0.1898, !0.2159, !0.5220, !1.1461,
0.0001
Example 2. Now, consider a 4-state, 2-input, 3-output system [7,10,19] given by
A"
0
1 0 0
0 0
1
1 0 0
1 0
!1 0 0 0 0
0 0 0
,
B"
0 0 0 1
,
1 0 0 0
C" 0 0 1 0 0 0 0 1
with open-loop eignvalues at 0, 0, 1.618 and !0.618. The nominal closed-loop poles are !1, !2, !3 and !4. For the sake of assigning the set of closed-loop poles very accurately, we use = "= "I , j"o"10 and k"10\. Our results are 2 6 compared with others in Table 4 (see [19]). It is clear that when we set j and o to such huge values, the closed-loop spectrum can be assigned nearly exactly with i kept small enough as well.
206
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Fig. 6. Simulation Results of Example 1.
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Table 4 Comparison of results for Example 2
Neural network Tests 1, 2 (p"p , p ) Neural network Test 3 (p"p , b"10) Neural network Test 4 (p"p ,b"10) Neural network Test 5 (p"p ,b"10) Chu, Nichols and Kautsky's method Fletcher's method Oh, Gu and Spurgeon's method (exact)
i
Closed-loop spectrum
499.2 499.2 499.3 532.6 778.2 884.8 499.2
+!1.00000, !2.00000, !3.00000, !4.00000, +!1.00000, !2.00001, !2.99999, !4.00000, +!0.99992, !2.00072, !2.99891, !4.00015, +!0.99202, !2.07947, !2.88429, !4.01404, +!0.973, !2.226, !2.918, !3.800, +!1.0, !2.0, !3.0, !4.0, +!1.00000, !2.00000, !2.99997, !4.00001,
Example 3. Lastly, we consider a 2-input, 5-state distillation column model [3,4,24] given by
!0.1094
0.0628
0
0
0
1.306
!2.132
0.9807
0
0
0
1.595
!3.149
1.547
0
0
0.0355
2.632
!4.257
1.855
0
0.00227
0
0.1636
!0.1625
A"
0
0
0.0638
0
B" 0.0838 !0.1396 0.1004 !0.2060 0.0063 !0.0128
,
The eigenvalues are at !0.077324, !0.014232, !0.89531, !2.8408 and !5.9822. They are required to be reassigned into !1$j, !0.2, !0.5 and !1 via a state feedback gain matrix K and we have
K"
!1
1
!1 !1
0
0
0
0
0
0
0
0
!0.2
0
0
0
0
0
!0.5
0
0
0
0
0
!1
.
The case of state feedback is considered as a special case of output feedback with C"I . In this case, we use the convex function p"p and k"1. Di!erent penalty parameters as well as weighting matrices = and = are used and the results are 2 6
208
D.W.C. Ho et al. / Neurocomputing 25 (1999) 191}211 Table 5 Comparison of results for Example 3
Neural network Test 1 (= "= "I, j, o"3000) 2 6 Neural network Test 2 (= "= "I, j, o"500) 2 6 Neural network Test 3 (= "= "W, j, o"3000) 2 6 Lam and Yan's method Byers and Nash's method Kautsky, Nichols and van Dooren's method 1
i
#K#
32.1 30.9 31.9 33.6 33.1 39.4
361.6 328.0 331.0 337.4 354.9 311.5
Table 6 Corresponding percentage errors of the eigenvalues for Example 3
Neural network Test 1 Neural network Test 2 Neural network Test 3
Closed-loop spectrum
Percentage errors
+!1.05$0.99j, !0.19, !0.49, !1.00, +!1.25$0.92j, !0.15, !0.44, !0.98, +!1.14$0.96j, !0.17, !0.48, !0.98,
+3.25, 7.01, 1.95, 0.41, +18.36, 23.00, 11.98, 1.61, +10.55, 12.76, 3.70, 1.58,
summarized in Table 5. In all the cases, our i are smaller than those obtained by the others. The assigned closed-loop spectrum and the corresponding pole percentage errors for the three sets of results are shown in Table 6. Again, in the "rst case, the penalty parameters j and o are chosen to be large values to ensure the pole assignment accuracy. In the second case, improvement on the closed-loop eigenstructure robustness is well expected by choosing smaller penalty parameters. However, one consequence associated with such action is that the set of assigned poles becomes relatively less accurate. In fact, the percentage errors for the poles (except !0.98) increase a lot when j and o decreases from 3000 to 500. Hence in the third case, we try to use weighting matrices = and = which are set to be 2 6
W"
1.6
0
0
0
0
0
1.6
0
0
0
0
0
1.6
0
0
0
0
0
0
0
0
1.3 0 0
1
to re"ne the accuracy of individual assigned poles. From Table 5, we can observe the e!ect of such action is that percentage errors for the assigned poles !1.14$0.96j, !0.17 and !0.48 are made smaller signi"cantly. Remark. For di!erent initial conditions with "xed set of j, o and k in all the examples, although it is found that ¹, X, and K do not always converge to the same values, the
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objective function values achieved do not vary signi"cantly. Di!erent initial conditions have small e!ect on the resulting condition number and the quality of the accuracy of the poles. The computational time for all the simulations are reasonably fast, for instance, they are all within 10 s for 2n"50 optimization variables in the "rst example.
5. Conclusions In this paper, we have presented a neural network approach to compute the output feedback gain for robust approximate pole assignment. The gradient #ow formulation allows the optimization process to admit a simple recurrent neural network realization. The e!ectiveness of the neural computation process is demonstrated by numerical simulations.
Acknowledgements This work is supported by City University research grant. The authors would like to thank the reviewers' constructive comments.
References [1] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, SIAM Studies in Applied Mathematics, 1994. [2] R. Byers, S.G. Nash, Approaches to robust pole assignment, Int. J. Control 49 (1) (1989) 97}117. [3] A. Cichocki, R. Unbehauen, Neural Networks for Optimization and Signal Processing, Wiley, England, 1993. [4] A. Cichocki, R. Unbehauen, Simpli"ed neural networks for solving linear least squares and total least squares problems in real time, IEEE Trans. Neural Networks 5 (6) (1994) 910}923. [5] C. Chiu, C.Y. Maa, M.A. Shanblatt, Energy function analysis of dynamic programming neural networks, IEEE Trans. Neural Networks 2 (4) (1991) 418}426. [6] E.K. Chu, Approximate pole assignment,, Int. J. Control 58 (2) (1993) 471}484. [7] E.K.-W. Chu, N.K. Nichols, J. Kautsky, Robust pole assignment by output feedback, Proceedings of 4th IMA Conference on Control Theory, 1984, pp. 137}146. [8] L.O. Chua, G.N. Lin, Nonlinear optimization with constraints: a cookbook approach, Int. J. Circuit Theory Appl. 11 (2) (1983) 141}159. [9] L.O. Chua, G.N. Lin, Nonlinear programming without computation, IEEE Trans. Circuits Systems CAS-31 (2) (1984) 182}188. [10] L.R. Fletcher, An intermediate algorithm for pole placement by output feedback in linear multivariable control system, Int. J. Control 31 (1980) 1121}1136. [11] U. Helmke, J.B. Moore, Optimization and Dynamical Systems, Springer, Berlin, 1994. [12] D.W.C. Ho, J. Lam, J. Xu, Robust Approximate Pole Assignment for Second-Order Systems: Neural Network Computation, Journal of Guidance, Control and Dynamics 21 (6) (1998) 923}929. [13] J. Kautsky, N.K. Nichols, P. van Dooren, Robust pole placement in linear state feedback, Int. J. Control 41 (5) (1995) 1129}1155. [14] M.P. Kennedy, L.O. Chua, Neural networks for nonlinear programming, IEEE Trans. Circuits Systems 35 (5) (1988) 554}562.
210
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[15] H. Kimura, A further result on the problem of pole assignment by output feedback, IEEE Trans. Automat. Control 25 (3) (1977) 458}463. [16] J. Lam, W.Y. Yan, A gradient #ow approach to robust pole assignment problem, Int. J. Nonlinear Robust Control 5 (1995) 175}185. [17] W.E. Lillo, M.H. Loh, S. Hui, S.H. Zak, On solving constrained optimization problems with Neural networks: a penalty method approach, IEEE Trans. Neural Networks 4 (6) (1993) 931}940. [18] G. Miminis, Using de#ation in the pole assignment problem with output feedback, Proceedings of 3rd International Conference on Aerospace Computational Control, 1989. [19] M. Oh, D.W. Gu, S.K. Spurgeon, Robust pole assignment in a speci"ed region using output feedback, Optim. Control Appl. Meth. 14 (1993) 57}66. [20] A.L. Peressini, F.E. Sullivan, J.J. Uhl Jr., The Mathematics of Nonlinear Programming, Springer, New York, 1988. [21] G. Roppenecker, J. O'Reilly, Parametric output feedback controller design, Automatica 25 (1989) 259}265. [22] G.W. Stewart, Introduction to Matrix Computations, Academic Press, New York, 1973. [23] D.W. Tank, J.J. Hop"eld, Simple &Neural' optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit, IEEE Trans. Circuits Systems CAS 33 (5) (1986) 533}541. [24] X. Wang, Pole placement by static output feedback, J. Math. Systems Estimation Control 2 (2) (1992) 205}218. [25] J. Wang, Recurrent neural networks for solving linear matrix equations, Comput. Math. Appl. 26 (9) (1993) 23}34. [26] L.X. Wang, J.M. Mendel, Three-dimensional structured networks for matrix equation solving, IEEE Trans. Comput. 40 (12) (1991) 1337}1346. [27] L.X. Wang, J.M. Mendel, Parallel structured networks for solving a wide variety of matrix algebra problems, J. Parallel Distributed Comput. 14 (1992) 236}247. [28] J. Wang, G. Wu, Recurrent neural networks for LU decomposition and cholesky factorization, Math. Comput. Modelling 18 (6) (1993) 1}8. [29] J. Wang, G. Wu, Recurrent neural networks for synthesizing linear control systems via pole placment, Int. J. Systems Sci. 26 (12) (1995) 2369}2382. [30] J. Wang, G. Wu, A multilayer recurrent neural networks for on-line synthesis of minimum-norm linear feedback control systems via pole placement, Automatica 32 (3) (1996) 435}442. [31] B.A. White, Eigenstructure assignment: a survey, Proc. Inst. Mech. Engrs. 209 (1995) 1}11. [32] J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. [33] S.H. Zak, V. Upatising, S. Hui, Solving linear programming problems with neural networks: a comparative study, IEEE Trans. Neural Networks 6 (1) (1995) 94}104.
Dr. Daniel W.C. Ho received a "rst class B.Sc., M.Sc. and Ph.D. degrees in mathematics from the University of Salford in 1980, 1982 and 1986, respectively. From 1985 to 1988, Dr Ho was a Research Fellow in Industrial Control Unit, University of Strathclyde. In 1989, he joined the Department of Mathematics, City University of Hong Kong, where he is currently an Associate Professor. His research interests include H control theory, robust pole assignment problem, adaptive neural wavelet identi"cation. Dr. James Lam received a "rst class B.Sc. degree in mechanical engineering from the University of Manchester in 1983. He then obtained the M.Phil. and Ph.D. degrees in the area of control engineering from the University of Cambridge in 1985 and 1988, respectively. Dr. Lam has held faculty positions at the now City University of Hong Kong and the University of Melbourne. He is now an Associate Professor in the Department of Mechanical Engineering, the University of Hong Kong, and is holding a Concurrent Professorship. Ms. Jinhua Xu received her B.S. and M.Eng. degrees in control engineering from Xi'an Jiaotong University, China, in 1998 and 1991, respectively. From 1991 to 1995, she worked in Dept. of Automatic Control Engineering, Xi'an Jiaotong University. Since 1995, she worked as a research assistant and then was admitted as a Ph.D. student in City University of Hong Kong. Her research studies are about wavelet neural networks, system identi"cation and adaptive control.
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Mr. Patrick Tam received his B.Eng.(Hons.) degree in Mechanical Engineering from the University of Hong Kong in 1994. Since graduation, he has been conducting research studies in the "eld of control engineering at the Department of Mechanical Engineering, the University of Hong Kong. His Ph.D. thesis focuses on the use of optimization methods in solving various robust pole assignment problems in control system design.