Least squares pole assignment by memoryless output feedback

SYSTIJB & CONTIIOL ELSEVIER

Systems & Control Letters 29 (1996) 31-42

Least squares pole assignment by memoryless output feedback 1 D a n - c h i J i a n g * , J.B. M o o r e Department of Systems Enoineerin9, Australia National University, Canberra, ACT 0200, Australia Received 8 January 1996; revised 10 April 1996

Abstract

In this paper, a pole assignment problem for a linear time-invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed. The cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum. This approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters. The relation between the modified cost functions and the original pole assignment problem is revealed. The proposed approach is compared with other existing approaches and illustrated by numerical results.

I. Introduction

The pole placement problem of linear systems has been an important issue for decades. Compared to the pole placement via state feedback or dynamic feedback, the same problem via output feedback is more complex. Different approaches from linear system theory, combinatorics, complex function theory and algebraic geometry are used to explore this problem and yet the results are not complete. See the survey paper [1] for some details. There are some recent works concerning system pole assignment by memoryless output feedback. For example, see [4, 5, 8, 9, I 1, 12] and references cited therein. For a linear time-invariant control system with n states, m inputs and l outputs, it is known that a necessary condition under which system poles can be assigned freely by output feedback is ml>~n. This condition is generically sufficient if the inequality is strictly satisfied. See [12] for details. In [8, 11] this result is understood in a simpler way. In particular, a computational procedure is developed in [8] to compute an output feedback gain based on a generalised degenerate output feedback gain. Along this line a sufficient rank condition and the associated computational scheme is reported in [14] based on a linearization technique. As pointed in [15], the computational result might be practically infeasible to implement. In [4, 9], another approach is introduced to consider this problem as a least squares optimisation problem. There some cost functions are devised to force the closed-loop system poles as close to the target system poles as possible in a least square sense. This approach has several advantages. First, it converts the exact

* Corresponding author. Supported by the funding of the activities of the Cooperative Research Centre for Robust and Adaptive Systems by the Australian Government under the Cooperative Research Centres Program. ] A part of this paper was submitted to 1995 student paper contest of IEEE ACT Section by the first author and awarded the 1st prize in the postgraduate category. 0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PII S01 67-69 11 ( 9 6 ) 0 0 0 4 4 - 8

32

D.-C. Jiang, J.B. Moore~Systems & Control Letters 29 (1996) 31-42

pole assignment problem into an optimisation problem which can be solved numerically by many mature software packages. Second, even though exact pole assignment by output feedback may not be feasible, it can always provide a reasonable alternative which is optimal with respect to the least squares of the difference between system poles and target poles. Third, it may exclude complex mathematical computation on a Grassmannian manifold. Furthermore, the corresponding eigenstructure is obtained automatically if the pole assignment problem is exactly solved. However, the cost function used in [4] has no compact sublevel sets. In fact, some numerical simulations have been conducted by us which shows that a trajectory of a negative gradient system does not converge to an equilibrium point. To remove this drawback, [9] restricts the system to be symmetric so that the state transformation matrices belong to a compact manifold. Work remains to be done to cope with the drawback in the general case. The sum of determinants of the return difference of the closed-loop system at all poles is used as a cost function in [5]. This cost function is transformed on a Grassmannian manifold to a sum of n items. Each item is a product of two functions. One is a positive function of a feedback gain matrix and the target poles, and another is a determinant of a Hermitian projection matrix related to target poles plus another constant matrix. Then, an auxiliary cost function is constructed by replacing the positive functions with constant parameters. For this auxiliary cost function, sublevel sets are compact and hence the existence of global minimum and convergence of gradient system are guaranteed. In this paper, a bound condition on the state transformation matrix and its inverse matrix is imposed. This condition is always satisfied in practical settings. In fact, if a global minimum exists, the corresponding state transformation matrix and its inverse matrix can be bounded by some constant real matrix. Solely on the bases or by further removing some redundancy, two types of modifications are made to the cost function. The modified cost functions have the required properties such as the existence of global minimum, convergence of gradient flow, etc. Furthermore, the exact solution if it exists can also obtained as the global minimum of the modified cost functions. For the pole assignment problem, another important issue is the robustness against parameter uncertainty or disturbance. Tens of papers are published to address the problem by static state feedback. See [2, 7]. One of the critical techniques is that the eigenvectors corresponding to each closed-loop eigenvalue form a linear subspace and therefore can be easily parametrized. But this technique is not available for the case of memoryless output feedback. In this paper, the proposed approach can also be adjusted to accommodate the robust pole assignment problem by memoryless output feedback.

2. Output feedback optimal least squares pole assigmnent Consider linear time-invariant systems with output £c(t) -~ Ax(t) + Bu(t),

( 1)

y ( t ) = Cx(t),

(2)

where x ( t ) , y ( t ) , u ( t ) are the system state, the output and the input with dimension n, l,m, respectively. Here A,B, C are constant matrices with appropriate dimensions. Without loss of generality, assume that B, C are of full column and row rank, respectively. Given a set of n complex numbers in which those pure complex numbers are in conjugate pairs, the pole assignment problem by memoryless output feedback is to find an output feedback u = F y such that the closed-loop system poles coincides with the set of complex numbers. 2.1. Least squares pole assignment

Given a real n x n matrix F, the least squares optimal assignment problem by memoryless output feedback is posed as [4]:

D.-C JianO, J.R Moorel Systems & Control Letters 29 (1996) 31-42

33

Problem 1. A pair (T,F), where T is a state transformation matrix and F is a memoryless output feedback gain matrix, is said to optimally (least squares) assign the closed-loop system poles to those of the matrix F if it minimises the following cost function:

J(F,F) := IIT(A + BFC)T -1 - 1, I[2 ,

(3)

where the matrix norm is Frobenius norm. Problem 1 has already been analysed in [4]. All transformed closed-loop systems (T(A + BFC)T -1, TB, CT - l ) form a smooth manifold which is an orbit of Lie group GL(n,R x R taxI, where GL(n,R) is the general linear group containing all nonsingular n x n matrices with matrix product operation. The group action of R m×t is a matrix summation operation. With respect to the normal Riemannian metric [4] the gradient is computed as

gradJ(T,F) = - ( [ T ( A + BFC)T -1 - F, T-T(A + BFC)TT]T, BTTT(T(A + BFC)T -1 - F)T-TcT), where T -T := (T -1

)T.

(4)

However, the cost function J(T,F) has no compact sublevel sets. It is easily seen that

J(T,F) = J(2T, F) for any nonzero constant real number 2, and one can always construct a sequence in the sublevel set 6e(e) := {(T,F) E GL(n, ~) x Rmxt: J ( T , F ) < e } , say, (kTo, Fo), k = 1,2 ..... for any (To,Fo) E 5e(e) which is not convergent. If the difference of eigenvalues 11T(A +BFC)T -1 -1" HF is considered as a function o f t he triple (A(T,F),B(T,F),C(T,F)) which is the orbit of the Lie group GL(n, R) x R mxl, i.e.,

J(A,B,C)

= I I A - rl12,

then, J(A,B, C) does not have compact sublevel sets either, because B(2T, F ) = 2TB ~ +oo as 2 ~ +c~. Therefore, the technique used to prove Theorem 3.2 in Ch. 5 of [4] is not applicable in this case. If we consider the negative gradient system of J(T,F) defined as (~/',P) = - g r a d J ( T , F ) ,

(5)

Only the following weak results hold Lemma 1. For the system (5),

(1) any trajectory has no finite escape time; (2) along any trajectory o f ( 5 ) the essential upper bound O( t ) of gradient norm on [0,oo) converges to zero, where U(t) is given by U(t) := i n f { g E •: IlgradJ(T(t),F(t)){lF
II(T(t)'F(t))ll2

=

fJogradJ(T(s),F(s)) fo t

d S 2 F f<.O t IlgradJ(T(s),F(s))ll 2 ds fOt ds

t dJ

= -t

~s ds = (J(T(O),F(O))

which establishes the claim (1).

-J(r(t),F(t)))t<~J(T(O),F(O))t

34

D.-C. Jiang, J.B. Moore~Systems & Control Letters 29 (1996) 31 42

By noticing that

J(T(t),F(t)) =-

IlgradJ(T(s),F(s))ll2F ds

is decreasing and positive the claim (2) is established.

[]

In fact, the numerical computation of some examples conducted by us shows that the gradient along a trajectory may diverge. For details of the computation results of one example, see Section 4. Thus it motivates us to propose modified approaches in the following subsection.

2.2. Modified least squares pole assignment problems In Section 2.1 it is observed that there exists a redundancy in the cost function J(T,F), i.e., J ( T , F ) = J(2T, F). One may expect that by removing all redundancy in the cost function the local minimum can be unique and hence the global minimum. However, it is already known that the output feedback gain matrix which achieves the closed-loop system pole assignment is not unique. Example 5.2 in [1] shows that there are four output feedback gain matrices that assign the closed system poles to the open-loop system poles. Furthermore, these output feedback gain matrices are not proportional to each other. If there is a global minimum of the cost function J(T,F), the global minimum can always be achieved by a state transformation matrix T* which is of unit Frobenius norm and its inverse norm is bounded by some constant positive real number M. Therefore it is reasonable to minimise the cost function J ( T , F ) in a set that the state transformation matrices are of unit Frobenius norm and the norm of their inverse matrices is bounded, or in a set that the norm of both state transformation matrices and their inverse matrices is bounded. In the following, accordingly two types of modification are made to the minimisation problem. First consider the set of all state transformation matrices with unit Frobenius norm denoted as OffF : = { T E GL(n,~): HTIIF = 1}.

It is known [4] that the GL(n, ~) × R'nxt is an orbit of itself considered as a Lie group by the following Lie group action: fl : (GL(n, •) x ~,nxl) x (GL(n, ~) x Rtaxi) ---+(GL(n, R) x R mxt) ((7~,fi), (T,F)) ~ (TT, F + F) This set is a smooth manifold with the manifold structure induced by Lie group action. Its tangent space at (T,F) can be calculated as

T(T,F)(GL(n,~) × ~m×l) = {(XT, L) : X E ~nxn,L E ~mxl}. q/F is a subset of GL(n, R) and has the following properties. Lemma 2. ~llF is a submanifold in GL(n, ~). Its tangent space at T is calculated as TT(~[F) = { X T : X

E ~nxn and tr(TTTX) = 0}.

Proof. Since d(IITIIF) = d(tr(TTT)) is a cotangent vector field which is nonzero everywhere on GL(n,R), the co-distribution spanned by d(H TIIF) is one-dimensional. One can choose ~1 = tr(TrT) as one component of a coordinate ~ on GL(n, ~). Therefore, °gF = {~ E GL(n, ~): ~l = 1} is an (n 2 - 1)-dimensional embedded submanifold of GL(n, R). The tangent space of °//e at T is calculated as

TT(°I[F)

=

{d(IITI[F)} ± = {XT: X E ~nxn and tr(TTTX) -----0}.

The proof is complete.

[]

D.-C Jiang, J.B. MoorelSystems & Control Letters 29 (1996) 31-42

35

To obtain a better convergence property of the cost function, a bound condition can be imposed on IIT-111F, where F stands for Frobenius, for the following reasons. First, it is already known that if IlT-~llr is large, the closed-loop system poles are very sensitive to system parameter disturbances. The detail will be revealed in the next section or referred to [2, 7]. Second, for any (T,F) that minimises the cost function, IIT-~II~ is always bounded by some appropriate constant. Then, the following problem with an adjustable constant M > 0 can be proposed to accommodate the system optimal pole assignment instead of Problem 1. Problem 2.

Minimise

)(T,F)

subjectto

T E qlF,

= IIr(A + B F C ) T -1 -

IIT-III2~
and

rl12

(6)

F E ~ m×t.

Another type of problem can also be posed similarly, however, with redundancy. Problem 3.

Minimise

J(T,F) IIT(A + SFC)r -~ - vii2

subject to

TeGL(n,R),

=

IITII2..
and

F E

Nmxl.

(7)

These are constrained optimisation problems. They can be converted into unconstrained optimisation problems by introducing additional penalty items in the cost functions, i.e., the cost functions can be defined as

JI(T,F) := J(T,F) + p{IM-II T-11121-M+ IIT-1112}2

(8)

J 2 ( T , F ) := J(T,F) + Pl {IM -

(9)

and

IIr 1121-g+ IIT 112}2 + p2{lM-II T-' 1121-g+ IIT-' 112}2,

where p, Pl, P2 are tuning parameters. The Frobenius norm is used here instead of the 2-norm for two reasons. One reason is that

IITII=~ IITIIF~ nllTIl=. A similar inequality is also available for T -1. Another reason is that the Frobenius norm is smooth, easier to calculate, and in harmony with the first term in the cost functions. Now the following lemma holds:

Lemma 3. The cost functions Ji( T, F ), i = 1,2 have compact sublevel sets in ql F × R mx t or GL(n, R) × Rmxt, respectively, i.e., f o r any e > O, the sets

6el(e) := {(T,F) E ~ F × ~m×l :JI(T,F)<<.e} and

5:2(e) := {(T,F) E GL(n, R) x ~mxl :J2(T,F)<~e} are compact.

Proof. For any sequence {(Tn,Fn)},,~=l in the sublevel set L~°I, since qgF is bounded, it follows that there exist a convergent subsequence of {Tn}n~l. Without loss of generality, we assume

lim T~ = T*. n ---* ¢ X )

D.-C. Jian#, J.B. Moore~Systems & Control Letters 29 (1996) 31-42

36

Since +

it is clear that T* is the limit of Tn and invertible. Therefore, there exist two positive numbers M1,N1 > O, such that for any n > N1,

IlZnllg<~M~,

IlZnlllr ~
Since for any m × l matrix Y,

IIYIl=~
min{m, l}ll YII2,

it follows that

liEn liE <~min{m, l} II(BTB)- 1BTBEnCCT(ccT)-IlI2 ~< min{m, I}II(BTB)-IBTI[2IIcT(ccT)-I II2IIT~-I(Tn(A + BFnC)T~ -1 - r)Tn - A + T~-IFTnlI2

<~min{m,I}I[(BTB)-1BTIIzIIcT(ccT)-I]I2{IIT~-lYl(Tn,En)TnlI2 + [IT~-IFT, II2 + [IAI]2},

(10)

where B is of full column rank and C is of full row rank by assumption of the system model (1),(2). Therefore, Fn is also hounded. Hence there exists a convergent subsequence of {Fn}n°°__l. Now it follows from the continuity of Frobenius norm that if {(Tn,Fn)}~l ~ (T*,F*), then

(T*,E*)

•

~(~).

Similarly, one can prove that 5e2(e) is compact.

[]

By the compactness of sublevel sets, the existence of a global minimum of the corresponding cost function is guaranteed. In order to obtain the gradient of cost function JI(T,F), we need to equip the manifold q/r x R m×t with an appropriate Riemannian metric. Given (T,F) E q/e × R"xt, let trl denote the following map: (71 : ~ n x n ~

~,

X H al(X) = tr(TTTX). For any X E ~n×n, let X = X ° + X ±, where X ° E ker(al),

X -[ E [ker(al)]±.

For (X1T, F1),(X2T, F2) E Tr(°gp) x Rre×t, the Riemannian metric can be defined as

(((X~T,F~),(X2T, F2))) := 2 t r ( ( x ( ) T x ~ ) + 2tr(F~F2). Now we have the following theorem. Theorem 1. For the cost function Jl(T,F), the following results hold: (1) Its gradient is calculated as grad Jl I(r,F) = (,~I([T(A + B F C ) T -l - F,T-T(A + B F C ) T T] - 2p(lM -

IIT-'II~

- M + lIT -~ ]I~.)T-TT-1 )T, BTTT[T(A + BEC)T -l - F]T-TCT),

I (1 1)

D.-C. Jiang, J.R MoorelSystems & ControlLetters 29 (1996) 31-42

37

where [.4,B] is the matrix Lie bracket of any square matrix A, B defined as [A,8] = A~

- ~A.

3~1 is the projection operator from •nxn to [ker(ol)] -k. It can be calculated by the following formula: vcc(3ol(X)) = tr(TTTX)v "TTT" IITTT II2 ect ).

(12)

(2) The equilibrium point set 81 is 81 = { ( T , F ) : ~ I ( [ T ( A + S F C ) T -~ - r , T - T ( A + B F C ) T T] - 2 p ( l M - lit -~ II~ I

- M + I I r - l l l ~ ) r - T r - 1 ) r = 0, eTrT[T(A + e F t ) T - ' - r]T-TC T = 0}. (3) The negative gradient system is defined as

(T,F) = -grad Jl [~r,r). Along its trajectory starting from any point (To, Fo) E q/F x R s×~, the cost function decreases strictly at any nonequilibrium point. (4) Along any trajectory, the gradient converges to a zero matrix. (5) Any trajectory converges to a connected subset of 81. The proof is standard. Interested readers may refer to the technique in [4, p. 159]. It is omitted to save space. Remark 1. If (T1,F1) exactly assigns system poles, then, there exists a positive number M* > 0 such that for any M>.M*, J1(T,F)~JI(TbFI) = 0. In fact, one can choose M* = IIrFXll~. Similarly, let the Riemannian metric of GL(n, R) x ~mxl be defined as

(((3(1T,F1 ), (X2T, F2))) :-- 2 tr((X1 )Tx2) -~- 2 tr(FITF2). We have the following theorem. Theorem 2. For the cost function J2(T,F), the following results hold: (1) Its gradient is calculated as grad J2 I(T,F)----(([T(A + BFC)T -1 - F, T-T(A + BFC)T T] + 2pl(lM-II T I1~-I

+ IITII2 - M ) r r T - 2p2(IM-IIT-111,~ I + IIT-111~-+M)T-TT-~) T, BTTT(T(A + BFC)T -1 - F)T-TC T)

(13)

(2) The equilibrium point set 82 is

82 = {(T,F) : ([T(A + BFC)T-' - F , T-T(A + BFC)T T] + 2pI(IM-II T II~l + IITII,~ - g ) T T T - 2p2([M- IIT-111~ I+IIT-111,~ +

M)T-TT-1) =

0,

BTTT(T(A + BFC)T -1 - F)T-TC T = 0}. (3) The negative gradient system is defined as (f',fi') : -grad J2 I(r,F) •

Along its trajectory starting from any point (To,Fo ) E GL(n, R) x R taxI, the cost function decreases strictly at any nonequilibrium point.

38

D.-C. JianO, J.B. Moore~Systems & Control Letters 29 (1996) 31-42

(4) Along any trajectory, the gradient converges to a zero matrix. (5) Any trajectory converges to a connected subset o f g2.

Remark 2. These two cost functions J I ( T , F ) and J2(T,F) may not have the same equilibrium point. However, it can be shown by simple computation that (1) if (T1,F1) = arg min J1, then, min J2 ~ 1. (2) if (T2,F2)=arg min J2, then, min .71 <~JI(T2,F2) where Jl is the same as Jl but parameter M is chosen as 2M.

3. Robust output feedback poles assigmnent If system parameters are not precisely known or subject to disturbances, the closed-loop system sensitivity should be taken into consideration in the system pole assignment problem. This is the so-called robust pole assignment problem. There are many papers devoted to address this problem in a state feedback context. For example, see [2, 7]. In the state feedback settings, eigenvectors of the closed-loop system matrix form some linear subspace [7]. On this basis many algorithms can be designed to minimise the 2-norm or other norm of the state transformation matrix, which is a matrix consisting of the closed-loop eigenveetors. If only output feedback can be used, those eigenvectors no longer form linear spaces. Therefore, techniques as used in [2, 7] are not available to the robust output feedback pole assignment problem. In this section, two optimisation problems will be proposed in the same way as in Section 2 to handle the problem by output feedback. Let (A,B, C) denote the nominal system parameters and (AA, AB, AC) denote parameter disturbances. Then,

II T(A

+ B F C ) T -1 - T(A + A A + (B + A B ) F ( C + AC))T -1112

=l[ T { A A + A B F C + B F A C + A B F A C } T -1 ll2

11TIIFIIT -~ IIF {ll~xAII2 + IIFII2(II~xBII2IICII2+ IIBIIzlIACII2 + It~xBII211ACII2)}. Therefore, it is sensible to accommodate the objective of insensitivity to parameter disturbance and pole assignment jointly by penalising T -1 and F of the following cost functions in minimisation problems: J,1( T,F) = k, ][T(A + B F C ) T -1 -

r ll~F+k2 IIT -1 [1~ +k~ IIFII2F,

(14)

where (T,F) E q/F × R mxt, and kl,k2,k3~O are tuning parameters and

Jr2(T,F) = kl IIT(A + B F C ) T -I - rll~ +k2 IITII~ +k3 IIT-1112 +k+llF I1~,

(15)

where T E GL(n, ]~) and F E ~mxt. The following results can be obtained by the same techniques used in Section 2. Lemma 4. Cost functions J,~, i = 1,2 have compact sublevel sets. Therefore, the global minima o f cost functions can be reached in any nonempty sublevel sets.

Theorem 3. For cost function Jri, i = 1,2, the following results hold: (1) Their gradients with respect to the corresponding Riemannian metrics are calculated as

gradJrl [(T,F) ----- ( ~ l ( k l [ T ( A + B F C ) T -1 - F, T-T(A + B F C ) T T] - k2 T - T T -1 )T, BTTT[T(A + B F C ) T -~ - F ] T - T C r + k3F)

(16)

D.-C. dianO, £B. Moore/Systems & Control Letters 29 (1996) 31-42 grad Jr2 I(r,F) = (kl [T(A + BFC)T -l - F, T-T(A + BFC)T T] +

k2TT T -

39

k3T-TT -l)T,

klBTTT[T(A + BFC)T -~ - FIT-TC T + k4F).

(17)

(2) The equilibrium point set ~1 and #~2 are 8rl = {(T,F) : ~l(kl [T(A + BFC)T -l - F, T-T(A + BFC)T T] - k2T-TT -1) = O, [T(A + B F C ) T -1 - F I T - T + k 3 F = 0},

t~r2 = {(T,F) : kl[T(A + BFC)T -1 - F, T-T(A + BFC)T T] + k2TT T - k3T-TT -1) = O, klBTTT[T(A + BFC)T -1 - F]T-TC T + k4F = 0}. (3) The negative gradient systems are defined as ('P,F) = -gradJrl ](T,F), (T,/~) = -grad Jr2 I~r,v). Along their trajectories startino from any point (To, Fo) E q/F x R taxi or (To,Fo) E GL(n, R) x R mxl, the cost functions decrease strictly at any nonequilibrium point. (4) Along any trajectory, the gradients converge to a zero matrix. (5) Any trajectory of gradient system of Jri converges to a connected subset of 8,~,i = 1,2. To justify these results, we have the following lemma. Lemma 5. I f the system poles can be exactly assigned by output feedback, then, by fixing tuning parameters k2,k3 in J~l or k2,ka,k4 in J~2 and letting kl ~ +oo, the global minimal point of Jr1 or d~2 has a subsequence that converges to a point ( T*, F*) which minimises the cost function dr l or J~2 while exactly assi#ning system poles. Proof. If k2,k3 are fixed, the cost function JrI(T,F) is nondecreasing with respect to kl. For any ( T , F ) that exactly assigns system poles, min

TEOIIF,FE• mxt

J~,(T,F)<<.C, := k2 II~-1112 + k3 IIPII~

holds for any positive kl,k2,k3. Let {kl}i= i o¢1 be any sequence that tends to infinite, and the corresponding global minimal points of J~I(T,F) be denoted as (Ti,Fi). For any i > 0, (Ti,Fi) belongs to the sublevel set

oG°i(Cl) := {(T,F) E °RF × R mxl :drl(T,F)
,-~i- 1(C1) ~ o~'°1(C1),

from the compactness of ~al(C1) it follows that there exists a convergent subsequence of (Ti,Fi). Without loss of generality, assume that (Ti,Fi) ~ (T*,F*) E 6#l(Cl). By Jrl(Ti,Fi)<<.C1 for any i > 1 we know that lim [ITi(A + BFiC)T[-1 - FI]F = O.

i-"~ ¢x3

The optimality of (T*,F*) follows by noticing that ( T , F ) can be any point which exactly assigns system poles. The proof of the result corresponding to Jr2 follows in exactly the same way. []

D.-C. Jiang, £B. Moore~Systems & Control Letters 29 (1996) 31-42

40

4. Numerical examples

In the previous two sections, several cost functions and their gradient formulas are given to optimally assign system poles. A bound condition is imposed on the state transformation matrix and its inverse matrix to obtain a convergence property of cost function. If a cost index term related to a robustness against system parameter disturbances is introduced, the bound condition is not required to obtain the convergence of gradient flow. In this section, the proposed method is illustrated for the system that is considered in [1, Example 5.2]. Consider Example 5.2 in [1], where 1 1

2 3

-1 3 -1 -2

A=

-1

2 2

1

1 2 3 2 1 -3 -3 -2 -1 -1 1 3

1 3

1 1 1

-1 -1 --2 1 -3 2 1

B=

'

(ii) ,

C=

!ooooo) 1 0 0 0 0 . 0 1 0 0 0

In order to show the convergence property of gradient system, the optimal solution is obtained by directly calculating numerical solutions to these negative gradient systems. We use ODE45 command in Matlab to compute numerical solutions of several gradient systems. In Fig. 1, the function log(log(d(T,F)) and log(llgradJll 2) are plotted versus time t. The cost function is seen to be decreasing. However, one may observe that the algorithm is not stable and the norm of the gradient does not converge to zero. In Figs. 2 and 3, the gradient systems for indexes J1 and J2 are computed. The initial values of To,F0 are the same as that in Fig. 1, M = 10000, p = Pl = p 2 = 1. The costs log(log(J1)), grad J1, log(log(J2)), and grad J2 are plotted versus t. In these figures, not only cost functions are shown to be decreasing along the trajectories of gradient systems, but also Frobenius norms of gradients are shown to be converging to zero "exponentially". If the system is of high order, say, n states, m inputs and l outputs, the dimension of the gradient system is (n 2 + ml). Therefore, for high-order systems, it is computationally expensive to solve the group of ordinary differential equations. Newton's method involves computation of the inversion of Hessian which is also computationally expensive if the number of variables is large. Besides, the Hessian is not always invertible. One can use a truncated Newton method, a conjugate gradient method or a gradient method to keep computation and memory at each iterate low. These computational methods are addressed by many standard books. For example, see [3]. Details are not explicitly addressed in this report. The norm of gradient along the trajectory

The objective function along the trajectory 9.5

12

9.0

11

8.5

10

8.0

9

7.5

8

7.0 7

6.5

6

6.0

1

%

5.5

) i - L liiIL LJ~

4

5.0 4.5 0

)

i

i

i

1

2

3

4

|

5

I

)

6

7

0

1

2

3

Fig. 1. The computation results of the gradient system of J(T,F).

4

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The norm of gradient along the trajectory

The objective function along the trajectory 9.5

9000

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7000

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7.5 5000 7.0 4000

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)

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Fig. 2. The computation results of the gradient system of J1 (T, F).

The objective function along the trajectory

9.5 9.0

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Fig. 3. The computation results of the gradient system of J2(T,F).

5. Conclusions In this paper, several cost functions for optimally assigning the closed-loop system poles in a least squares sense have been constructed by penalising a state transformation matrix and its inverse matrix to be bounded or a state transformation matrix and its inverse matrix, and an output feedback gain matrix to be small. Gradients with respect to two types of Riemannian metric have been given and convergence properties of gradient systems and existence of global minima have been obtained. Numerical examples have also been given to illustrate the effectiveness of the proposed approach. Compared to existing works such as [4, 5, 9], the proposed approach is attractive in the following aspects. First, systems considered here are not restricted to a subclass of linear time-invariant control systems such as the class of symmetric systems. Second, the existence of global minimum and convergence of gradient system trajectories are guaranteed. Third, cost functions constructed are closely related to original pole assignment problem. In fact, if an exact solution of the output pole assignment problem is feasible, it can be approximated by the methods proposed in this paper. Fourth, insensitivity issues against system parameter disturbances can also be cast in the same framework.

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It is worth noting that only local minima are achieved here. Some computation methods can be applied to search for global minimum. But these methods are usually expensive in computation. It needs further research to find an economical method to search for a global optimal solution. References [1] C.I. Byrnes, Pole placement by output feedback, in: Three Decades of Mathematical Systems Theory, Lecture Notes in Control and Information Sciences, Vol. 135 (Springer, Berlin, 1989) 31-78. [2] R. Byers and S.G. Nash, Approaches to robust pole assignment, Internat. J. Control 45 (1989) 97-107. [3] J.E. Dennis and R.B. Sehnabel, Numerical Methods for Unconstrained Optimisation and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983). [4] U. Helmke and J.B. Moore, Optimisation and Dynamical Systems (Springer, London, 1991). [5] K. Hfiper and U. Helmke, Geometrical methods for pole assignment algorithms, Proc. 34th IEEE Conf. on Decision and Control New Orleans (1995). [6] D. Jiang and J.B. Moore, Least squares pole assignment by memorytess output feedback, presented at Mathematical Theory of Networks and Systems, 1996, St. Louis, MO. [7] J. Kautsky, N.K. Nichols and P. Van Dooren, Robust pole assignment in linear state feedback, Internat. J. Control 41 (1985) 1129-1155. [8] J. Leventides and N. Karcanias, Global asymptotic linearisation of the pole placement map: a closed-form solution for the constant output feedback problem, Automatica 31(9) (1995) 1303-1309. [9] R.E. Mahony and U. Helmke, System assignment and pole placement for symmetric realisations, J. Math. Systems Estim. Control 5 (1995) 267-270. [10] T.J. Owens and J. O'Reilly, Parametric static-feedback control with response insensitivity, lnternat. J. Control 45 (1987) 791-809. [11] J. Rosenthal, J.M. Schumacher and J.C. Willems, Generic eigenvalue assignment by memoryless real output feedback, Systems Control Lett. 26 (1995) 253-260. [12] X. Wang, Pole placement by static output feedback, J. Math. Systems Estim. Control 2(2) (1992) 205-218. [13] W.M. Wonham, On pole assignment in multi-input, controllable linear systems, IEEE Trans. Automat. Control AC-12(6) (1967) 660-665. [14] X. Wang, Grassmannian, Central projection, and output feedback pole assignment of linear systems, 1EEE Trans. Automat. Control AC-41(6) (1996) 786-794. [15] J. Rosenthal and X. Wang, Output feedback pole placement with dynamic compensators, IEEE Trans. Automat. Control AC-41(6) (1996) 830-843.