Two parametric approaches for eigenstructure ... - Springer Link

Journal of Control Theory and Applications 1 (2003) 59 - 64

Two parametric approaches for eigenstructure assignment in second-order linear systems Guangren DUAN (Center for Control Theory and Guidance Technology, Harbin Institute of Technology, P. O. Box 416, Harbin Heilon~iang 150001, China)

Abstract: This paper considers eigenstrucmre assignment in second-order hnear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through estabhshing two general parametric solutions to this type of man'ix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-,loop eigenvector matrices. The first one mainly depends on a series of singular value decomposifous, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-'loop eigenvalues to be set undetemlined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Kaeywords: Second-order hnear systems; Eigenstructure assignment; Proportional pins derivative feedback; Parametric solutions; Singular value decomposition; Right factofization

1

Introduction

The second-order linear systems capture the dynamic behavior of many natural phenomena, and have found applications in many fields, such as vibration and structural analysis, spacecraft control and robotics control, and hence have attracted a great deal of attention ([ 1 - 12] ). In this paper, we consider the control of the following second-order dynamical linear system MX + Dx + Kx = B u , (1.1) where x E F~n and u E R" are the state vector and the control vector, respectively; M, D, K ~ R n• and B ~ R ~x' are the system coefficient matrices. In certain applications, the matrices M, D and K are called the mass matrix, the structural damping matrix and the stiffness matrix, respectively. These coefficient matrices satisfy the following assumption. A s s u m p t i o n 1 det ( M ) # 0, rank ( B ) = r. In terms of the control of the second-order linear system ( 1.1 ) , most of the results are focused on stabilization (for e.g. [2] and [ 3 ] ) and pole assignment ( [ 4 - 9 ] ) . Furthermore, many theoretical results for second-order systems have been developed via the corresponding extended firstorder state-space model 2 = Aez + Beu, (1.2) where -

M-1K

-

M-1D

'

M

"

(1.3) Therefore, these results eventually involve manipulations on 2n dimensional matrices Ae and Be. Eigenstructure assignment is a very important problem in

linear control systems design, and has been one of the foci of attention in the last two decades (see[ 13, 14] and the references therein). However, for eigenstmcture assignemnt in second-order linear systems, there have not been many results ( [ 10 - 12 ] ). Reference [ 10 ] considers eigenstructure assignment in a special class of second-order linear systems using inverse eigenvalue methods. Reference [ 11 ] proposes an algorithm for eigenstmcture assignment in the second-order linear system ( 1.1 ) , with M and K being symmetric positive definite, and D being symmetric positive semidefinite. This algorithm is attractive because it utilizes only the original system data M, D and K. Very recently, eigenstructure assignment in the second-order linear system ( 1 . 1 ) using a proportional-plus-derivative feedback controller is considered in [ 12]. Simple, general, and complete parametric expressions in direct dosed forms for both the closed-loop eigenvector matrix and the feedback gains are established. As in [ 11 ], the approach utilises directly the original system data M, D and K, and involves manipulations on only n-dimensional matrices. However, the approach has the disadvantage that it requires the controllability of the matrix pair ( D , B ) , which is not satisfied in some applications. This paper considers eigenstmcture assignment in the second-order linear system ( 1.1 ) via proportional plus derivative coordinate control. The only requirements on the system coefficient matrices are that M is nonsingular and B has flail coltmm rank. Based on a series of singular value decompositions and the right factorization of the system, two complete parametric approaches are proposed. Simple and complete parametric expressions for both the closed-loop eigenvector matrices and the feedback gains are established.

This work was supported in part by the Chinese Outstanding Youth Foundafon (No .69504002).

60

G. DUAN / Journal of Control Theory and Applications 1 (2003) 59- 64

These expressions contain a group of parameter vectors that and represent the design degrees of freedom, which can be V' = VA. (2.8) properly further chosen to produce a closed-,loop system Proof Since with some desired system specifications. As a result, the 0 I _M-l( second approach, which uses the fight factorization of the system, is a natural generalization of the parametric approach in [ 16] proposed for first-order state-space systems. _M-l( K - B F o ) V - M - J ( D - B F 1 ) V' ' With this approach, besides the group of paranleter vec- and tors, the closed-,loop eigenvalues may also be treated as part = of the design freedom since they appear directly in the exV'A ' pressions of the eigenvector matrix and the feedback gains, the equation ( 2 . 6 ) is clearly equivalent to the relation and hence are not necessarily chosen a pr/or/. (2.8) and The paper is composed of five sections. Section 2 for- M-I(KBFo)V- M-~(DBF1)V' = V'A. mulates the eigenstmcture assigm'nent problem for second(2.9) order linear systems and relates it to the problem of solving Further, substituting ( 2 . 8 ) into ( 2 . 9 ) yields equation a type of second-order Sylvester matrix equations. Section (2.7). [] 3 proposes two complete parametric solutions to the type of The above lemma states that the Jordan matrix of Aec is second-order Sylvester matrix equations. Based on the soluA if and only if there exists V C C n• satisfying ( 2 . 9 ) , tions proposed in Section 3, two parametric methods are and in this case the corresponding eigenvector matrix of Aec presented in Section 4 for the formulated eigenstmcture asis given by signment problem. An illustrative example is given in Section 5. V~ = VA " (2.10)

v,

]

[

2

Problem formulation

For the second-order dynamical system (1. 1 ), by choosing the following control law u = Fox + FiSt, Fo, F1 E ]~fxn, (21 1) we obtain the closed-loop system as follows: My + ( D - BFj)Sc + ( K - B F o ) x --- 0. (2.2) Note that det ( M ) # 0, the above system ( 2 . 2 ) can be written in the first-order state-space form = a~cz (2.3) with

[ Aec =

,

0 - M-I( K-

BFo)

-

M-I(D

] -

BF1)

(2.4) Recall the fact that a nondefective matrix possesses eigenvalues which are less sensitive to the parameter perturbations in the matrix, we here require the closed-loop matrix Ae~ to be nondefective, that is, the Jordan form of the matrix Aec possesses a diagonal form:

With the above understanding, the problem of eigenstructure assigrnnent in the second-order dynamical system (1. 1 ) via the proportional plus derivative feedback law (2.1) can be stated as follows. Problem ESA(Eigenstmcture assignment) Given system ( 1.1 ) satisfying Assumption 1, and the matrix A = diag ( S l , S Z , ' " , s z n ) , w i t h s i , i = 1 , 2 , " ' , 2 n , beinga group of self-conjugate complex numbers (not necessarily distinct), find a general parametric form for the matrices F0, Fl ~ R rx" and V ~ ~nx2n such that the matrix equation (2.7) and the condition V (2.11) det[ VA ] # 0 are satisfied. Let I V = F~VA + F o V = [Fo

FI][VA.

v] (2.12)

and then (2.7) becomes

MVA 2 + DVA + K V = B W . (2.13) A = diag( 31, $ 2 , " " , S2n), (2.5) Clearly, equation (2.13) becomes the type of generalized where si, i = 1 , 2 , " ' , 2 n , are clearly the eigenvalues of Sylvester matrix equation investigated in [ 1 5 - 17 ] when the matrix Aec. M = 0. In view of this fact, we call the equation (2.13) L e m m a 1 Let Aec and A be given by ( 2 . 4 ) and the second-order generalized Sylvester matrix equation. ( 2 . 5 ) , respectively. Then there exist matrices V, V' E It follows that, once a pair of matrices V and IV that satC ~x2'~ satisfying isfy the second-order generalized Sylvester matrix equation (2.13) and condition (2.11 ) are obtained, a pair of control gain matrices can be easily obtained from ( 2 . 1 2 ) as if and only if follows: MVA 2 + ( D -

BF1) VA + ( K -

BFo) V = 0

(2.7)

[F0

F1] = W[VA] -1

(2.14)

G. DUAN / Journal of Control Theoryand Applications 1 (2003) 59 - 64 Therefore, to solve Problem ESA, the key step is to find a solution to the following problem. P r o b l e m SSE (Second-order Sylvester equation ) Given the matrices M , D , K E R'~• E R n• satisfying Assumption 1, and a diagonal matrix A = d i a g ( s l , s 2 , ' " , S q ) E C q• (2.15) find a parameterization for all the matrices V E C "• and W E C ~• satisfying the second-order Sylvester matrix equation ( 2 . 1 3 ) . It should be noted that the number of the columns of the matrices V, W and A in the above Problem SSE are changed into q because it makes the Problem SSE more general.

3

Solution to problem SSE Denote V = W =

IV 1

/32

[w 1

"'"

w2

(3.1)

Vq],

"-"

Wq],

(s2iM + siD + K ) v i = Bw i, i = 1 , 2 , ' " , q . (3.3) 3.1 Case of p r e s c r i b e d si, i = 1 , 2 , ' " , q. The equations in ( 3 . 3 ) can be further written in the following foma

Wi

= 0, i = 1 , 2 , - . . , q ,

where Hi = [sZM + siD + K

- g],

(3.4)

i = 1,2,-'" , q . (3.5)

This states that [ vii

~

ker Hi, i =

1,2,'",q.

(3.6)

Wi

The following algorithm produces two sets of constant matrices Ni and Oi , i = 1 , 2 , " ' , q, to be used in the representation of the solution to the matrix equation ( 2 . 1 3 ) . A l g o r i t h m l(Solving Ni and Di, i = 1 , 2 , ' " , q) S t e p 1 Through applying SVD to the matrix Hi, i = 1 , 2 , ' " , q , obtain two sets of matrices Pi ~ C nxn and Qi E C(n+r)• = 1,2,'",q, satisfying 9p.jiiQ

[diag

i

IIi "'[Nil Di = 0, i = 1 , 2 , " ' , q .

0

for ker Hi. The above deduction clearly yields the following result. T h e o r e m 1 Let ni, i = 1 , 2 , ' " , q, be defined by ( 3 . 8 ) ,and N i E R n• and Di ~ ~;~r• i = 1 , 2 , " ' , q, be obtained via Algorithm 1. Then all the matrices V and W satisfying the second-order Sylvester matrix equation ( 2 . 1 3 ) can be pararneterized by columns as follows

wi

B], i = 1,2,-",q.

(3.8) Obtain the matrices Ni E R nx(n+r-nl) and Di E R rx("+r-~) , i = 1,2, " " , q, by partitioning the matrix Qi as follows:

,q,

Di

where f/ E C"+r-nl, i = 1,2, " " , q, are a set of arbitrary parameter vectors. ,~ Definition 1 The second-order dynamical system ( 1.1 ) is called controllable if"and only if" the corresponding extended first-order state-space representation ( 1 . 2 ) and ( 1 . 3 ) is controllable. Regarding the controllabiliW of system ( 1.1 ) , we have the following basic result. L e m m a 2 The second-order dynamical system ( 1.1 ) is controllable if and only if rank [ sZ M + sD + K B ] = n , V s E C. (3.12) Proof By the well-known PBH criterion, we need only to show that condition ( 3 . 1 2 ) is equivalent to rank [ A e - sI2,~ Be] = 2 a , V s E C, where Ae and Be are given by ( 1 . 3 ) . Since rank [ Ae - sI2, Be ] [ -sI, I~ 0 ] = rank - M - I K - M - 1 D - sin M - 1 B

= raok [ s,. K

[

0 ' (3.7)

i = 1,2,...,q, where 6 i > 0, i = 1 , 2 , " ' , hi, are the singular values of Hi, and ni = rank [s2iM + siD + K

(3.10)

This indicates that the columns of [ N Dii l f o r m a s e t o f b a s i s

(~Yl,a2,"',Gni) 01

=

Step 2

]

[* Nil, i = 1,2,..-,q. (3.9) Qi = . Di As a resttlt of ( 3 . 7 ) and ( 3 . 9 ) , the matrices Ni E R n• andDi ~ R rx('~+r-ni), i = 1 , 2 , ' " , q , obtained through the above Algorithm I satisfy

(3.2)

then, in view of ( 2 . 1 5 ) , we can convert the second-order Sylvester matrix equation ( 2 . 1 3 ) into the following column fOrlT1

Hi

61

= rank

0]

'n -D-

sM

B

0]

- D - sM

B

0 - K-

Ds - s2M

= n + rank [s2M + sD + K B ] , the conclusion is readily reached. [] Based on the above lemma, the following corollary of Theorem 1 can be immediately derived. Corollary 1 Let system ( 1.1 ) be controllable, and A be given by ( 2 . 1 5 ) , then the degrees of freedom existing in the general solution to the second-order Sylvester matrix equation ( 2 . 1 3 ) is qr"

G. DUAN / Journal of Control Theoryand Applications 1 (2003) 59 - 64

62

P r o o f Due to the controllability of system ( 1 . 1 ) , we have from Lemma 2 n i = rt, i = 1 , 2 , " ' , q . Thus the conclusion immediately follows from Theorem 1. [] 3 . 2 C a s e of u n d e t e r m i n e d s~, i = 1 , 2 , - . . , q. By performing the right factorization of

P ( s ) [ s 2 M + sD + K - B ] Q ( s ) = [I= 0]. S t e p 2 Obtain the pair of polynomial matrices N ( s ) ~n • [ S ] and D ( s ) E R r• [ s ] by partitioning the unimodular matrix Q ( s ) as follows:

G(s) = (s2M + sD + K ) - 1 B ,

D(s)

we can obtain a pair of polynomial matrices N ( s ) an• and D ( s ) 6 arXr[s] satisfying

6

( s ; M + sD + K ) - I B = N ( s ) D - l ( s ) . ( 3 . 1 3 ) T h e o r e m 2 Let the system ( 1 . 1 ) be controllable, a n d N ( s ) ~ R~Xr[S] and D ( s ) 6 R'Xr[S] satisfy the right factorization ( 3 . 1 3 ) . Then (1) The matrices V and W given by ( 3 . 1 ) , ( 3 . 2 ) a n d wi

D ( s i ) l fi, i = 1 , 2 , ' " , q

(3.14)

satisfy the second-order Sylvester matrix equation ( 2 . 1 3 ) for allj~ E C~,i = 1 , 2 , ' " , q . (2) When rank

[ N( si)]

D(s~)l r, i = 1 , 2 , " - , q

(3.15)

hold, ( 3 . 1 4 ) gives all the solutionsto Problem SSE. P r o o f It follows from ( 3 . 1 3 ) that

{ (s2iM + siO + g ) N ( s i ) - BO(si) = O,

"

It is worth pointing out that the pair of polynomial matrices N ( s ) ~ ~:~nxr[$] and O ( s ) ~ Rrxr[8] satisfying the right factorization ( 3 . 1 3 ) obtained from the above Algorithm 2 are right coprime since

N(s) l

rank [ D ( s ) J = r, V s ~ C. This condition implies the condition ( 3 . 1 5 ) which ensures the completeness of the solution ( 3 . 1 4 ) . To finish this section, let us finally give a remark on the extension of the result. R e m a r k 1 The main results in this section can be easily extended into the case that the matrix A is a general Jordan form. In fact, when A is replaced with the following Jordan matrixJ _- Blockdiag ( J1, J 2 , ' " , Jv) 6 C qxq , with

si Ji =

1

.

8i

"..".

] 1 6 Cpi•

i = 1,2,'",p,

i = 1,2,'",q, (3.16) Using (3.14) and ( 3 . 1 6 ) , we derive

(s2iM + siD + K)vi - Bwi

$i

following the development in [ 15,16], we can show that all matrices V and W satisfying the second-order Sylvester matrix equation ( 2 . 1 3 ) are given by

: [(S2i M + siO + K ) N ( s i) - B O ( s i ) ] f i v = [ v, v~ ... v~ ~ , =0, i = 1,2,'",q. Vi [ Vii Yi2 "'" Vipl 1, This states that the equations in ( 3 . 3 ) hold. Therefore, the and first conclusion of the theorem also hold. {w=[w, w2--- w~], It follows from Corollary 1 that, under the controllability of system ( 1.1 ) , the degrees of freedom existing in the Wi [ Wil wi2 "'" Wipi ]' general solution to the matrix equation ( 2 . 1 3 ) , with A with given by ( 2 . 1 5 ) , is qr, while in the solution ( 3 . 1 4 ) , the number of free parameters is just equal to qr. Further, it is D(si lllk, + ~D (~i) clear that all these parameters in the solution ( 3 . 1 4 ) have 1 [ N(k-l)(si) ] contributions when condition ( 3 . 1 5 ) holds. With this we (3.17) complete the proof. [] + (k - 1)! D ( k _ l ) ( s i ) J f 1 The right factorization ( 3 . 1 3 ) performs a fundamental k = 1,2,'",pi, i = 1,2,'",p, role in the solution ( 3 . 1 4 ) . When si,i = 1 , 2 , ' " , q , are where N ( s ) 6 R~• and O ( s ) E Rr• are a pair chosen to be different from the zeros of det ( s2M + sD + of polynomial matrices satisfying the right factorization K ) , we can take (3.13). N ( s ) = Adj (s2M + sD + K ) B , 4 Solution to problem ESA n(s) det (s2M + sD + K). For general numerical algorithms sohtng such right factorRegarding the solutions to Problem ESA, we have the izations, one can refer to [ 18, 19]. The following simple following two results which are based on the discussion in procedure can also be used. Section 2 and the results in Section 3. A l g o r i t h m 2(Right coprime factorization) Theorem 3 Let hi, i = 1 , 2 , - - - , 2 n , be given by S t e p 1 Under the K-controllability of system ( 1.1 ) , ( 3 . 8 ) , andNi ~ R ~• andDi E R r• i = find a pair of unimodular matrices P ( s ) and Q ( s ) , of ap- 1 , 2 , ' " , 2 n , be given by Algorithm 1. Then propriate dimensions, satisfying 1 ) Problem ESA has solutions if and only if there exist a

G. DUAN / Journal of Control Theory and Applications 1 (2003) 59 - 64 group of parametersj~ E C t~+r-ni , i = 1 , 2 , "" , 2 n , satisfying the following constraints: Constraint C I : f/ = ~ if si = ij. Constraint C2 a : det Vca # 0 with Vc~

[ Nl f l = tslNlfl

Nz f2 s2N2f2

"'"

N2,,f>, l

"'"

s2nN2nf2n ]"

(4.1) 2) W h e n the above condition is met, all the solutions to the Problem ESA are given by V = [Nlfl N 2 f 2 "'" N2nf2 "] (4.2) and [/70

F1] = [ O l f l

D2f2

D2nf2,,]Vc-1, (4.3) where j~ ~ cn+r-"~, i = 1 , 2 , "" , 2 n , are arbitrary parameter vectors satisfying Constraints C1 and C2a. Theorem 4 Let system ( 1 . 1 ) be controllable, and N ( s ) 6 Rn• s ] and D( s ) E R"• s ] be apair o f polynomial matrices satisfying the right factorization ( 3 . 1 3 ) . Then ( 1 ) Problem ESA has solutions if and only if there exist a group of parameters j~ E C r , i = 1 , 2 , " ' , 2 n , satisfying Constraint C1 and Constraint C2 b : det Vcb # 0 with

r u(.