Recent developments on nonmodal and partial modal approaches for control of vibration

Applied Numerical Mathematics 30 (1999) 41–52

Recent developments on nonmodal and partial modal approaches for control of vibration Biswa Nath Datta 1 Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA

Abstract In this paper, we present a brief overview of some of the recent nonmodal and partial modal algorithms for feedback stabilization and eigenvalue assignment problems arising in control of vibration modeled by systems of matrix second-order differential equations. For a complete up-to-date account, see Datta (1999).  1999 Published by Elsevier Science B.V. and IMACS. All rights reserved.

1. Introduction In this paper we consider numerical methods for several important control problems modeled by a system of second-order differential equations: M q(t) ¨ + D q(t) ˙ + Kq(t) = 0.

(1.1)

Many important vibrating structures ranging from machine tools to bridges, aircrafts, high rise buildings, space crafts are modeled by systems of differential equations of the above type. It is then required to control unwanted vibrations by applying suitable control forces. Let B be the input (control) matrix and u be the control vector. We then have the second-order control system of the following type: M q(t) ¨ + D q(t) ˙ + Kq(t) = Bu.

(1.2)

In most vibration applications, the matrices M, K and D are known, respectively, as the mass, stiffness and damping matrices. The mass matrix is also known as the inertia matrix, because it arises from the inertial forces in the system. The damping and stiffness matrices arise, respectively, from dissipative forces proportional to the velocity and the elastic forces proportional to the displacement. In most of the applications the mass, stiffness, and damping matrices are assumed to be symmetric; and, furthermore, M is assumed to be positive definite. Symmetric positive definite, positive semidefinite, 1 E-mail: [email protected].

0168-9274/99/$19.00  1999 Published by Elsevier Science B.V. and IMACS. All rights reserved. PII: S 0 1 6 8 - 9 2 7 4 ( 9 8 ) 0 0 0 8 3 - X

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negative definite and negative semidefinite matrices will be denoted respectively by > 0, > 0, < 0, 6 0. Thus, M = M T = mass or inertia matrix (M > 0), K = K T = stiffness matrix, D = D T = damping matrix. Further assumptions on the stiffness and damping matrices will be made, as the situations demand. If the damping matrix D is absent, the system is called an undamped system; otherwise, it is a damped system. An obvious approach for solving a control problem associated with (1.2) is to construct a first-order realization of the second-order system, and then apply one of the standard well-established first-order techniques for that problem. For an account of these techniques, see [7]. The standard first-order realization of the second-order control system (1.2) is 

x(t) ˙ =

0 −M −1 K





I 0 x(t) + −1 −1 −M D M B

The system (1.2) can also be rewritten in the form 

−K 0

0 M





z˙ (t) =



−K z(t) + −D

0 −K



0 B



ub(t).

(1.3)



ub(t).

(1.4)

As we see, the standard first-order realization requires the inversion of the mass matrix M; thus, if this matrix is ill-conditioned, its inverse, and therefore the state matrix of the first-order system, will not be computed correctly. The other first-order realizations such as (1.4) do not require the inversion of M, but it gives rise to a descriptor control system of the type b(t). E x˙ = Ax + Bb u

The numerical methods for control problems of descriptor-control systems are not well-developed. The matrix E here can be ill-conditioned too. Furthermore, none of these realizations respect the exploitable properties such as definiteness, sparsity, etc., of the data matrices M, K and D. An approach, widely known in the engineering literatures, is the Independent Modal Space Control (IMSC) approach (see [14,20]). As the name suggests, the IMSC approach aims at decomposing the problem into n independent problems by using the modes (eigenvectors) of the associated quadratic eigenvalue problem:


P (λ)x = λ2 M + λD + K x = 0. This eigenvalue problem, being nonlinear and nonsymmetric, is difficult to solve in a numerically effective way. Indeed, numerically viable methods for computing the eigenvalues and eigenvectors of the quadratic eigenvalue problem P (λ)x = 0, especially for large and sparse matrices, are virtually nonexistent. The state-of-the-art techniques can compute only a few extremal eigenvalues of the pencil [21]. On the other hand, there are many practical situations such as the design of large-space structures [1, 2,15], control of power systems, etc., which give rise to very large-scale problems. The IMSC approach and the approach through first-order realization are not practical for such large and sparse problems.

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Our above discussions lead us to state that numerically viable algorithms for control problems of second-order systems ideally should be such that: (i) they are capable of exploiting the properties of the coefficient matrices such as sparsity, definiteness, etc., and (ii) only a minimal knowledge of the eigenvalues and eigenvectors (frequencies and modes) of the associated quadratic eigenvalue problem is needed (possibly only a small number of eigenvalues and eigenvectors that can be computed using the existing techniques or can be measured in a vibration laboratory). Some work to this effect has been done recently. Nonmodal and partial modal approaches (approaches that do not either require knowledge of eigenvalues and eigenvectors at all or only require partial knowledge), have been developed for feedback stabilization [10], for the partial pole placement problems [8,9], and for the robust pole placement problems [4]. We will describe some of these recent work here. For notational convenience, we will sometimes denote q(t) by q, x(t) by x, q(t) ˙ by q, ˙ q(t) ¨ by q, ¨ etc. 2. Feedback-control problems for second-order model Suppose a control force u is applied to a structure modeled by the second-order system (1.1). The controlled system then becomes (1.2): M x(t) ¨ + D x(t) ˙ + Kx(t) = Bu(t), where u(t) is the input (control) vector, B is the n × p control matrix. Suppose that the state variables are known. Also, assume that the matrix M is symmetric and positive definite, and the matrices K, and D are symmetric and positive semidefinite. We want to modify the behavior of the controlled system by choosing the input vector u appropriately. Choose ˙ u = F1 x(t) + F2 x(t). The closed-loop system then becomes ˙ + (K − BF1 )x(t) = 0. M x(t) ¨ + (D − BF2 )x(t) In order to stabilize the above system the feedback matrices F1 and F2 should be chosen appropriately so that the system is asymptotically stable. The pencil P (λ) = λ2 M + λD + K is asymptotically stable if and only if all its 2n eigenvalues have negative real parts. Feedback stabilization problem for a second-order control system. Given the n × n symmetric positive semidefinite quadratic pencil P (λ) = λ2 M + λD + K and the matrix B of order n × m, find matrices F1 and F2 such that the closed-loop pencil Pc (λ) = λ2 M + λ(D − BF2 ) + (K − BF1 ) has all its 2n eigenvalues negative real parts. Next, we formulate the partial eigenvalue assignment problem. Suppose that the second-order control system M x¨ + D x˙ + Kx = Bu is unstable, but only p eigenvalues are not in the left half plane (p < 2n). Then the problem is to choose the feedback matrices F1 and F2 such that the p unstable eigenvalues are replaced by suitable arbitrarily

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chosen numbers µ1 , . . . , µp ; while leaving the (2n − p) stable eigenvalues unchanged. This gives rise to the partial eigenvalue assignment problem. Partial eigenvalue assignment problem for a second-order control system. Given the n × n symmetric positive semidefinite quadratic pencil P (λ) = λ2 M + λD + K and the matrix B of order n × m, find the matrices F1 and F2 such that p eigenvalues of the closed-loop pencil Pc (λ) = λ2 M + λ(D − BF2 ) + (K − BF1 ) can be arbitrarily chosen, and the other 2n − p eigenvalues remain the same as those of the pencil P (λ). The single-input partial eigenvalue assignment for a second-order control system In particular, in the single-input case the problem is to find the vectors f and g such that the pencil


Pc (λ) = λ2 D + λ D − bf T + K − bg T




(2.1)

has the first p eigenvalues arbitrarily chosen, while the rest remain the same as those of the open-loop pencil P (λ). Finally, we state the robust eigenvalue assignment problem. As is well known (see [7]), even when a numerically stable algorithm is used to solve an eigenvalue assignment problem, the eigenvalues of the computed closed-loop pencil may be very different from the eigenvalues that we need to assign. An analysis on the conditioning of the eigenvalue assignment problem (see [7]) shows that the conditioning of the eigenvector matrix of the closed-loop matrix is a major contributing factor to this effect. Thus, in practice, it may not be enough to just assign the eigenvalues, but the eigenvectors of the closed-loop pencil should be chosen in such a way that they are as well-conditioned as possible (robust). This leads to the robust eigenvalue assignment problem. Robust eigenvalue assignment problem for a second-order control system. Given the n × n symmetric positive semidefinite quadratic pencil P (λ) = λ2 M + λD + K, find the matrices F1 and F2 such that the closed-loop pencil Pc (λ) = λ2 M + λ(D + BF2 ) + (K + BF1 )

(2.2)

has a desired set of eigenvalues and the associated eigenvectors are as well-conditioned as possible (robust). We next describe some recent algorithms for solving these problems. The feedback stabilization algorithm (Algorithm 3.1) is nonmodal—it does not require any knowledge of eigenvalues and eigenvectors of the open-loop pencil. The algorithm is due to Datta and Rincón [10]. The algorithms for the partial eigenvalue assignment problems are partial modal: they require only a partial knowledge of the eigenvalues and the eigenvectors. Specifically, only the small number of eigenvalues that are required to be reassigned and the corresponding eigenvectors are needed. These small number of eigenvalues can either be computed or be measured in a vibration laboratory, given the physical parameters of the system. The single-input partial eigenvalue assignment algorithm (Algorithm 4.1) is due to Datta et al. [9]. We also describe an algorithm (Algorithm 5.1) for robust eigenvalue assignment problem. This algorithm is due to Chu and Datta [4].

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An additional advantage of these algorithms is that they work exclusively with the data matrices M, D and K of the second-order pencil, thus allowing the exploitation of structural properties such as symmetry, definiteness, bandness, sparsity, etc., which occur frequently in practical applications.

3. Nonmodal solution of the feedback stabilization problem Here we present a nonmodal solution of the feedback stabilization problem. Consider the closed-loop pencil Pc (λ) = λ2 M + λ(D − BF2 ) + (K − BF1 ) = λ2 M + λD 0 + K 0 , where D 0 = D − BF2 ,

K 0 = K − BF1 .

The well-known classical stability criterion for a quadratic pencil due to Rayleigh states that if the data matrices M, K and D of the pencil P (λ) = λ2 M + λD + K are all symmetric and positive definite, then the pencil P (λ) is asymptotically stable. From the Rayleigh criterion of stability, we know that the Pc (λ) will have all its eigenvalues with negative real parts if F1 and F2 are chosen so that K 0 and D 0 are symmetric positive definite. Assume that the system is controllable. Then it can be seen, using the eigenvector criteria of controllability, that this will happen if F1 and F2 are chosen as F1 = −C1 B T ,

F2 = −C2 B T ,

where C1 and C2 are arbitrary m × m symmetric positive definite matrices (see Lancaster [19]). An alternative way to see this is to use the expression for the real parts of the eigenvalues of the pencil Pc (λ). Let (λc , xc ) be an eigenpair of the pencil Pc (λ). Let λc = α + iβ. Then it has been shown in [10] that −|λc |2 Dx0 c −|λc |2 (D + BC2 B T )xc = , α= |λc |2 Mxc + Kx0 c |λc |2 Mxc + (K + BC1 B T )xc where Axc = xcT Axc . Since the system is controllable, we must have xcT B 6= 0, which implies that xcT BC1 B T xc = (BC1 B T )xc > 0. Similarly, xcT BC2 B T xc > 0. Since M = M T > 0, K = K T > 0 and D = D T > 0, then we have D + BC2 B T




xc




= xcT D + BC2 B T xc > 0

and





K + BC1 B T xc = xcT K + BC1 B T xc > 0. Thus α < 0. The above discussions lead to the following theorem and the algorithm. Theorem 3.1 (Existence of stabilizing feedback solutions). Given the n × n symmetric positive semidefinite pencil P (λ) = λ2 M + λD + K (M = M T > 0, K = K T > 0, D = D T > 0), and the input

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matrix B of order m × n, of full rank, there exists an one-parameter family of feedback matrices F1 and F2 such that the closed-loop pencil P c(λ) = λ2 M + λ(D − BF2 ) + (K − BF1 ) is asymptotically stable; provided that the system is controllable. An explicit expression for F1 and F2 is given by F1 = −C1 B T ,

F2 = −C2 B T ,

where C1 and C2 are arbitrary m × m symmetric positive definite matrices. Remark. In [10], C1 and C2 were chosen to be C1 = C2 = (B T M −1 B)−1 . The above generalization is due to Lancaster (1997), who also pointed out that the positive semidefinite condition of D can be relaxed to Re(D) > 0, where Re(D) = 12 (D + D), if D is complex. Since D is assumed to be real here, this generalization does not apply. For a feedback stabilization algorithm for a gyroscopic system, see [13]. Algorithm 3.1. Nonmodal stabilizing algorithm Inputs: M = M T > 0, K = K T > 0, D = D T > 0, B the m × n control matrix. Outputs: Feedback matrices F1 and F2 such that the pencil Pc (λ) = λ2 M + λ(D − BF2 ) + (K − BF1 ) is asymptotically stable. Step 1. Choose two arbitrary m × m symmetric positive definite matrices C1 and C2 . Step 2. Form F1 = −C1 B T ,

F2 = −C2 B T .

4. A partial modal solution for the partial eigenvalue assignment problem (the single-input case) Let (X, Λ) be the eigenvector–eigenvalue matrix-pair of the quadratic pencil P (λ) = λ2 M + λD + K. Then MXΛ2 + DXΛ + KX = 0. Partition X and Λ in the form X = (X1 , X2 ),

Λ = diag(Λ1 , Λ2 ),

where X1 ∈ C n×p and Λ1 = diag(λ1 , . . . , λp ). Define the vectors f and g by f = MX1 Λ1 β

and

g = −KX1 β,

where β is an arbitrary p-vector. We then show that with this choice of f and g, the (2n − p) eigenvalues λp+1 , . . . , λ2n of the closedloop pencil


Pc (λ) = λ2 M + λ D − bf T + K − bg T




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remain unchanged by feedbacks; that is, they are the same as those of the open-loop pencil: Pc (λ) = λ2 M + λD + K. In terms of the eigenvalue and eigenvector matrices, this amounts to proving that





MX2 Λ22 + D − bf T X2 Λ2 + K − bg T X2 = 0. To prove this, we consider the eigen-decomposition equation again: MXΛ2 + DXΛ + KX = 0. From this, we obtain







MX2 Λ22 + D − bf T X2 Λ2 + K − bg T X2 = MX2 Λ2 + DX2 Λ2 + KX2 − bβ T Λ1 X1T MX2 Λ2 − X1T KX2







= −bβ T Λ1 X1T MX2 Λ2 − X1T KX2 , because MX2 Λ2 + DX2 Λ2 + KX2 = 0. Furthermore, Λ1 X1T MX2 Λ2 − X1T KX2 = 0, because the left hand side is the p × (2n − p) top right (and therefore zero) block of a diagonal matrix, by virtue of the following orthogonality relation (proved in [9]): ΛXT MXΛ − XT KX = D1 , where D1 is a diagonal matrix, provided that the eigenvalues of the pencil P (λ) are all distinct. Thus





MX2 Λ22 + D − bf T X2 Λ2 + K − bg T X2 = 0. We summarize the above discussion in the form of a theorem below. Theorem 4.1 (Invariance of unassigned spectrum by feedback). Given the quadratic pencil P (λ) = λ2 M + λD + K with M = M T > 0, D = D T , K = K T , an (n × 1)-vector b, and the first p eigenpair (Λ1 , X1 ), define f = MX1 Λ1 β,

g = −KX1 β,

where β is an arbitrary (p × 1)-vector. Then for any arbitrary choice of β, the last (2n − p) eigenvalues of the closed-loop


Pc (λ) = λ2 M + D − bf T + K − bg T




remain the same as those of the open-loop pencil P (λ). Remark. From Theorem 4.1 we see that there is a p-parameters family of feedback vectors that can be used to change some eigenvalues in the closed-loop pencil while keeping the remaining unchanged. 4.1. Choosing β In order to use pole assignment problem, we need to choose β which p Theorem 4.1 to solve the partial p will move λj j =1 of the pencil P (λ) to {µj }j =1 in Pc (λ), if that is possible. If there is such a vector β, then there exist an eigenvector matrix Y ∈ C n×p : Y = (y1 , y2 , . . . , yp ),

yj 6= 0, j = 1, 2, . . . , m,

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and a matrix C = diag(µ1 , µ2 , . . . , µp ) which are such that





MY C 2 + D − bf T Y C + K − bg T Y = 0. Substituting for f, g and rearranging, we have




MY C 2 + DY C + KY = bβ T Λ1 X1T MY D − X1T KY = bβ T Z1T = bcT , where Z1 = CY T MX1 Λ1 − Y T KX1 and c = Z1 β is a vector that will depend on the scaling chosen for the eigenvectors in Y . To obtain Y , we can solve for each of the eigenvectors yi using the equations


µ2j M + µj D + K yj = b,

j = 1, 2, . . . , p.

This corresponds to choosing the vector c = (1, 1, . . . , 1)T . So, having computed the eigenvectors, we could solve the p × p square linear system Z1 β = (1, 1, . . . , 1)T for β, and hence determine the vectors f , g. The above discussions lead us to formulate the following algorithm for the solution of the partial eigenvalue assignment problem. Algorithm 4.1. The single-input partial eigenvalue assignment algorithm Inputs: M = M T > 0, K = K T , D = D T , b an n-vector, Λ01 = diag{µ1 , . . . , µp }, closed under complex conjugation. Assumption: Numbers µ1 , . . . , µp ; λ1 , λ2 , . . . , λ2n are all distinct, where λ1 , λ2 , . . . , λ2n are the eigenvalues of p(λ). Outputs: The feedback vectors f and g such that the spectrum of the closed-loop pencil


Pc (λ) = λ2 M + λ D − bf T + K − bg T




is {µ1 , . . . , µp ; λp+1 , . . . , λ2n}, where λp+1 , . . . , λ2n are the last 2n − p eigenvalues of the open-loop pencil P (λ). Step 1. Obtain the first p eigenvalues λ1 , . . . , λp of the pencil p(λ) = λ2 M + λD + K, that need to be reassigned and the corresponding eigenvectors x1 , . . . , xp . Form Λ1 = diag(λ1 , . . . , λp ),

X1 = (x1 , . . . , xp ).

Step 2. Solve for y1 , . . . , yp :


µ2j M + µj D + K yj = b,

j = 1, . . . , p.

Step 3. Form Z1 = Λ01 Y T MX1 Λ1 − Y T KX1 , where Y = (y1 , . . . , yp ), and Λ01 = diag (µ1 , . . . , µp ). Step 4. Solve for β: Z1 β = (1, 1, . . . , 1)T . Step 5. Form f = MX1 Λ1 β,

g = −KX1 β.

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Theorem 4.2 (The partial pole-placement theorem). The vectors f and g constructed by the above algorithm are such that the spectrum of


Pc (λ) = λ2 M + λ D − bf T + K − bg T




is {µ1 , . . . , µp ; λp+1 , . . . , λ2n }, where {λ1 , . . . , λp ; λp+1 , . . . , λ2n } is the spectrum of P (λ) = λ2 M + λD + K. 4.2. Explicit expression for β It can be shown [9] that if the components βj of the vector β are chosen as βj =

m 1 µj − λj Y µi − λj , T λj λ − λj b xj i=1 i

j = 1, 2, . . . , p,

i6=j

then the closed-loop pencil Pc (λ) has the spectrum {µ1 , . . . , µp ; λp+1 , . . . , λ2n }. 4.3. Important features of the algorithm (1) The algorithm needs only a partial knowledge of the eigenvalues and the eigenvectors. (2) The knowledge of the damping matrix D is not needed in computing the vectors f and g. (3) There is no spill-over; that is, the eigenvalues that are not required to change remain unchanged. Note. We emphasize that feature (2) is really an asset for practical applications, because, in practical design, one does not, in general, have an explicit knowledge of the damping matrix, but one can measure a few required eigenvalues and eigenvectors explicitly. Remark. (1) The paper [9] contains several theoretical results on the quadratic pencil P (λ) = λ2 M + λD + K, such as, the generalized Rayleigh quotient, orthogonality relations of the eigenvectors, etc., which are of independent interest. (2) The paper [8] contains a partial modal algorithm for the multi-input eigenvalue assignment problem of the quadratic pencil P (λ). The algorithm reassigns a subset of eigenvalues while maintaining the stability of the remaining eigenvalues.

5. Solution of the robust eigenvalue assignment problem for a second-order system In this section, we present an algorithm for the robust eigenvalue assignment problem that can be implemented by working on the matrices M, D and K exclusively. This algorithm is a modification of an algorithm by Juang and Maghami [17] which is an adaptation to the second-order model of the well-known first order algorithm for the robust eigenvalue assignment by Kautsky et al. [18]. For another new robust eigenvalue assignment algorithm, see [4].

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5.1. Development of the method First n eigenvalues λ1 , . . . , λn and their corresponding eigenvectors x1 , . . . , xn of the closed-loop system (2.2) satisfy 



λ2k M + λk (D + BF2 ) + (K + BF1 ) xk = 0,

with k = 1, . . . , n. The above eigenvalue equation can be rearranged to Γk φk = 0 with   
 xk Γk ≡ λ2k M + λk D + K , B , φk ≡ . (λk F2 + F1 )xk T Select φkT = (φ k , φbkT ), with φ k ∈ C n and φbk ∈ C m from the null space of Γk . The feedback matrices F1 and F2 then satisfy   Φ e b e [F1 , F2 ]ψ = Φ, ψ ≡ , ΦΛ

b ≡ [φb1 , . . . , φbn ] and Λ ≡ diag(λ1 , . . . , λn ). Considering the real and complex with Φ ≡ [φ 1 , . . . , φ n ], Φ parts, the above equation can be written in real form     ΦF ΦR b b e , [F1 , F2 ]Ψ = ΦF , ΦR , Ψ ≡ Φ R ΛF + Φ F ΛR Φ R ΛR − Φ F ΛF where the subscripts (·)R and (·)F indicate respectively the real and imaginary parts. Unfortunately, Ψe is of dimension 2n × n and consequently Ψ has at most rank n and is thus singular. Note that the complex equation involving Ψe and the real equation involving Ψ are equivalent. (The difference in the number of equations is a mirage, as some of the real equations are redundant, due to the relationship between the real and complex parts of the eigenvalue equation for a real matrix.) A trivial but revealing observation is that Ψ has n zero columns when the poles to be assigned are all real. Note also that although the equation involving Ψe is complex, a real solution is guaranteed because complex conjugate poles are assigned. Note that Ψe forms n columns of S1 (see Step 2 of Algorithm 5.1), which is always nonsingular with appropriately chosen eigenvectors. The immediate implication is that a method similar to the Juang– Maghami method will work if 2n poles are assigned. The null-spaces defining the eigenvectors (Step 2) can be obtained by pre-multiplying by QT2 (from Step 1).

Algorithm 5.1. The robust eigenvalue assignment algorithm Inputs: M = M T > 0, K = K T , D = D T , {λ11 , . . . , λ1n , λ21 , . . . , λ2n } 2n eigenvalues to be assigned. Outputs: F1 , F2 the feedback matrices such that the closed-loop pencil Pc (λ) + λ2 M + λ(D + BF2 ) + (K + BF1 ) has the eigenvalues {λ11 , λ12, . . . , λ1n; λ21 , λ22 , . . . , λ2n}, and the eigenvectors are as well-conditioned as possible. Step 1 (Initiation). Let the prescribed closed-loop poles be {λ1k } and {λ2k }, k = 1, . . . , n, and Λi ≡ diag{λi1 , . . . , λin }, i = 1, 2. Form the QR decomposition of B, so that   RB = Q1 RB , Q = [Q1 , Q2 ], B = [Q1 , Q2 ] 0

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with RB being upper triangular and nonsingular, and Q being orthogonal. Step 2 (Eigenvector selection). For i = 1, 2 and k = 1, . . . , n, select xik from the null spaces (using QR decompositions) defined by


QT2 λ2ik M + λik D + K xik = 0, so that with X1i = [xi1 , . . . , xin ] (i = 1, 2), X1 = [X11 , X12 ] and T1 = Λ1 ⊕ Λ2 , so as to minimize the condition number of     X1 X11 X12 = . S1 = X1 T1 X11 Λ1 X12 Λ2 Step 3 (Feedback matrices calculation). Solve the well-conditioned system RB [F2 , F1 ]S1 = −QT1 [R1 , R2 ], for the feedback matrices F1 and F2 , with Ri = MX1i Λ2i + DX1i Λi + KX1i

(i = 1, 2).

Remark. (1) The eigenvector selection in Step 2 can be done in a similar fashion as in [18]. Note that from the assumed controllability property of the system, we can deduce easily that the dimensions of the null spaces defined in Step 2 are exactly p. As a result, we can only assign eigenvalues of multiplicity less than or equal to p, otherwise a more involved procedure involving defective (thus ill-conditioned) eigenvalues and principal vectors will be required. On the other hand, open-loop eigenvalues can be re-assigned with different eigenvectors. As indicated in Step 2, a possibility is to select xik so that the condition number of S1 , is optimized. (2) In [16], it was indicated that only n eigenvalues are assignable, but we have seen above that it is possible to assign all the 2n eigenvalues while minimizing the condition number of the eigenvector matrix. References [1] M.J. Balas, Trends in large space structure control theory: fondest hopes, wildest dreams, IEEE Trans. Automat. Control 27 (1982) 522–535. [2] A. Bhaya and C. Desoer, On the design of large flexible space structures (LFSS), IEEE Trans. Automat. Control 30 (1985) 1118–1120. [3] E.K. Chu, Approximate pole assignment, Internat. J. Control 59 (1993) 471–484. [4] E.K. Chu and B.N. Datta, Numerically robust pole assignment for second-order systems, Internat. J. Control 64 (1996) 1113–1127. [5] B.N. Datta, Linear and numerical linear algebra in control theory: some research problems, Linear Algebra Appl. 197/198 (1994) 755–790. [6] B.N. Datta, Numerical Linear Algebra and Applications (Brooks/Cole, 1995). [7] B.N. Datta, Numerical Methods for Linear Control Systems Design and Analysis (1999). [8] B.N. Datta, S. Elhay and Y. Ram, An algorithm for the partial multi-input pole assignment of a second-order control system, in: Proc. IEEE Conference on Decision Control (1996) 2025–2029. [9] B.N. Datta, S. Elhay and Y. Ram, Orthogonality and partial pole assignment for the symmetric definite quadratic pencil, Linear Algebra Appl. 257 (1997) 29–48.

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