Robust Pole Assignment in Descriptor Linear Systems ... - Science Direct

European Journal of Control (2002)8:136±149 # 2002 EUCA

Robust Pole Assignment in Descriptor Linear Systems via State Feedback* Guang-Ren Duan1,2 N.K. Nichols3,y and Guo-Ping Liu4,z 1

School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Stranmillis Road, Belfast BT9 5AH, UK; Center for Control Theory and Guidance Technology, Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, P. R. China; 3Department of Mathematics, The University of Reading, P. O. Box 220, Reading RG6 6AF, UK; 4School of Mechanical, Materials, Manufacturing Eng. and Management, University of Nottingham, University Park, Nottingham NG7 2RD, UK 2

The problem of eigenvalue assignment with minimum sensitivity in multivariable descriptor linear systems via state feedback is considered. Based on the perturbation theory of generalized eigenvalues of matrix pairs, the sensitivity measures of the closed-loop finite eigenvalues are established in terms of the closed-loop normalized right and left eigenvectors. By combining these measures with a recently proposed general parametric eigenstructure assignment result for descriptor linear systems via state feedback, the robust pole assignment problem is converted into an independent minimization problem. The optimality of the obtained solution to the robust pole assignment problem is totally dependent on the solution to the independent minimization problem. The closed-loop eigenvalues are also taken as a part of the design parameters and are optimized, together with the other degrees of freedom, within certain desired regions on the complex plane. The approach takes numerical stability into consideration and also gives good robustness for the closed-loop regularity. An example is worked out with multiple solutions, both the indices and the numerical robustness test demonstrate the effect of the proposed approach.



The original version of the paper has been presented at the 2001 European Control Conference. E-mail: [email protected] z E-mail: [email protected]. Correspondence and offprint requests to: G.-R. Duan, School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Stranmills Road, Belfast BT9 5AH, UK. Tel.: ‡44(0)1232 27 4123; Fax: ‡44(0)1232 661729. E-mail: [email protected], [email protected] y

Keywords: Descriptor Linear Systems; Eigenstructure Assignment; Eigenvalue Sensitivities; Minimization; Parameter Perturbations; State Feedback

1. Introduction It has been known since the early 1960s that certain degrees of freedom exist in pole assignment for multivariable linear systems. More specifically, the state feedback gain which assigns a set of closed-loop eigenvalues is generally non-unique. Such a fact can be well revealed by a series of eigenstructure assignment approaches (Fahmy and O'Reilly 1983,1988,1989, O'Reilly and Fahmy 1985, Fletcher et al. 1986, Lewis and Ozcaldira 1989, Duan 1992a, 1993b, 1994, 1995, 1998, 1999, Duan and Patton 1997, 1998, Liu and Patton 1998). The degrees of freedom in a pole assignment design can be further utilized to achieve additional system performances. Such an idea gives rise to many design problems, and one of them is the so-called robust pole assignment problem which aims at selecting the design freedom such that the closedloop eigenvalues are as insensitive as possible to perturbations in the components of the closed-loop system coefficient matrices. As is well-known, the stability as well as the transient response of a linear system are mainly determined by the eigenvalues, or Received January 31, 2001; Accepted in revised form December 18, 2001; Recommended by S. Weiland and S. Morse.

Robust Pole Assignment in Descriptor Systems

poles, of the system. For a linear system with certain eigenvalues very sensitive to perturbations in certain elements of the system coefficient matrices, a very small perturbation in those elements of the system coefficient matrices may significantly affect the dynamical property of the system, or even make the system unstable. Therefore, the stability as well as the dynamical property of a linear system with smaller eigenvalue sensitivities are more robust in the sense that they are less affected by perturbations in the system coefficient matrices. Due to such an importance, the problem of robust pole assignment has been intensively studied in the last two decades. However, most of the results are obtained for the case of conventional linear systems (e.g. Kautsky et al. 1985, Sun 1987, Kautsky and Nichols 1990, Owens and O'Reilly 1989, Duan 1992b, 1993a, Lam and Yan 1995). For the case of descriptor systems, this problem has not yet received much attention. Three pieces of pioneering work in this aspect are due to Kautsky and Nichols (1986), Kautsky et al. (1989) and Syrmos and Lewis (1992). Kautsky and Nichols (1986) and Kautsky et al. (1989) extend their earlier well-known techniques in Kautsky et al. (1985), developed for conventional linear systems, to the case of descriptor systems, and lay a special emphasis on the closed-loop regularity. Syrmos and Lewis (1992) proposed a robustness theory for the generalized spectrum of descriptor linear systems, and presented a compact theory for the robust eigenvalue assignment problem in descriptor linear systems using the concept of chordal metric. Very recently, Duan and Patton (1999) studied robust pole assignment in descriptor linear systems via proportional plus partial derivative state feedback. Due to the capacity of the derivative feedback, their work concentrates on the case that the closed-loop system possesses n ( ˆ the system order) finite closed-loop eigenvalues. Robust pole assignment is closely related with eigenstructure assignment because the closed-loop eigenvalue sensitivity measures, which are essential for robust pole assignment designs, are determined by the closed-loop eigenvectors. Eigenstructure assignment in a degenerate descriptor linear system is much more complicated than that in a conventional linear system because of the following: (a) In a descriptor linear system, the number of ®nite eigenvalues which can be assigned to the closed-loop is not greater than rank (E). Thus, in®nite closed-loop eigenvalues exist when the matrix E is de®cient. (b) For eigenstructure assignment in a conventional system, the feedback gain has a unique

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correspondence relation with the closed-loop eigenvector matrix. This fact no longer holds for the descriptor system case. There exists some extra degree of freedom in the solution of the feedback gain of a descriptor system (Duan 1998, 1999). (c) Regularity problem rises with eigenstructure assignment in descriptor linear systems. When system parameter perturbation exists, closedloop regularity should also maintain certain robustness. In this paper, the eigenstructure assignment approach in descriptor linear systems via state feedback, recently proposed by Duan (1998), is adopted. This approach assigns the maximum number of finite closed-loop eigenvalues, guarantees closed-loop regularity, and provides the complete parametric expressions for both the closed-loop eigenvectors and the state feedback gain. The design freedom provided by this approach is composed of three parts, namely, the finite closed-loop eigenvalues, the group of parameter vectors found in the solution of the closed-loop eigenvectors, and the parameter matrix found in the solution of the state feedback gain. By using this eigenstructure assignment result and the closed-loop eigenvalues sensitivity measures obtained based on a result of the perturbation theory of generalized eigenvalues proposed by Stewart (1975), robust pole assignment in descriptor linear systems via state feedback is converted into a minimization problem. Two methods are proposed for solving this minimization problem. Both methods take numerical stability into consideration by either avoiding solution of matrix inverses or by minimizing the condition numbers of the matrices whose inverses are involved. Due to the advantages of the eigenstructure assignment approach used, the approach proposed for the robust pole assignment problem possesses the following several features. (a) The procedures for solution of the proposed robust pole assignment problem are in a sequential order, and no ``going back'' procedures are needed. (b) The ®nite closed-loop eigenvalues are also included in the design parameters and are optimized within certain desired ®elds on the complex plane, thus a closed-loop system with better robustness and desired transient performance can be obtained. (c) The optimality of the solution to the whole robust pole assignment problem is solely dependent on the optimality of the solution to the minimization problem converted.

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(d) As consequences of the numerical stability procedures taken, the closed-loop system also possesses better regularity robustness. The paper is divided into six sections. In the next section, the problem of robust pole assignment in descriptor linear systems via state feedback is formulated. Section 3 provides two aspects of preliminaries, one is the derivation of the closed-loop eigenvalue sensitivity measures, the other is the statement of a special case of the eigenstructure assignment result proposed by Duan (1998). Solution to the problem of robust pole assignment is considered in Section 4. An example of order 5 is examined in Section 5, and multiple solutions are derived and compared and tested as well. Concluding remarks follow in Section 6.

2. Formulation of the Problem Consider the following descriptor linear system E x ˆ Ax ‡ Bu,

…2:1†

where  represents the differential operator d/dt, in the continuous-time case, or the one step forward operator q (qx(k) ˆ x(k ‡ 1)), in the discrete-time system case. x 2 Rn, u 2 Rr are, respectively, the state descriptor vector and the input vector; E, A and B are matrices of proper dimensions and satisfy rank(E) ˆ m, rank(B) ˆ r, and the following controllability assumption: Assumption A1. Rank[sE

A B] ˆ n, for all s 2 C.

When the following state feedback controller u ˆ Kx,

K 2 Rrn

…2:2†

is applied to system (2.1), the closed-loop system is obtained in the following form: E x ˆ Ac x,

…2:3a†

where Ac ˆ A ‡ BK:

…2:3b†

The eigenvalues of the closed-loop system (2.3) are determined by the relative eigenvalues of the matrix pencil [Ac E] (Wilkinson 1965). It is well-known that the matrix pencil [Ac E] possesses indefinite relative eigenvalues when rank(E) < n, and has, when it is regular, at most m ˆ rank(E) finite relative eigenvalues. Therefore, the maximum number of finite relative eigenvalues can be assigned to the closed-loop

system (2.3) is m ˆ rank(E). Further, in view of the fact that non-defective matrix pair [Ac E] possesses relative eigenvalues with less sensitivities to the matrix parameter perturbations (Kautsky et al. 1989), the closed-loop finite eigenvalues are restricted to be a set of m distinct, but self-conjugate complex numbers. The problem of robust pole assignment to be solved in this paper can be stated as follows. Problem RPA. Given system (2.1) satisfying Assumption A1, and a set of regions i, i ˆ 1, 2, . . . , m on the complex plane, which are symmetric about the real axis, seek a state feedback controller in the form of (2.2), such that the following requirements are met: 1. The closed-loop system (2.3) is regular, that is, the closed-loop matrix pencil det[sE (A ‡ BK)] is not identically zero. 2. The closed-loop system (2.3) has m distinct ®nite relative eigenvalues si, i ˆ 1, 2, . . . , m satisfying si 2 i, i ˆ 1, 2, . . . , m. 3. The ®nite closed-loop eigenvalues si, i ˆ 1, 2, . . . , m, are as insensitive as possible to parameter perturbations in the closed-loop system matrices E and Ac. Remark 2.1. The requirement si 2 i, i ˆ 1, 2, . . . , n, in the above problem represents the requirement on the closed-loop stability and performance property. For a real closed-loop eigenvalue si, the region i may be chosen to be an interval [ai bi]. For a pair of complex eigenvalues si and sl, the regions i and l may often be chosen as

i ˆ fsi ˆ i ‡ l jji 2 ‰ai bi Š, l 2 ‰al bl Šg

…2:4a†

and

l ˆ fsl ˆ i

l jji 2 ‰ai bi Š, l 2 ‰al bl Šg:

…2:4b†

3. Preliminaries This section states a result on eigenstructure assignment in descriptor linear systems via state feedback, and presents, in terms of the closed-loop eigenstructure, the sensitivity measures of the finite eigenvalues of the closed-loop system (2.3). 3.1. Closed-Loop Eigenstructure Assignment Duan (1998) has given a full treatment on eigenstructure assignment in descriptor linear systems via state feedback. Here, as only necessary to solution of the proposed Problem RPA, a special result in Duan (1998), for the case that the finite closed-loop

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eigenvalues are restricted to be distinct from each other, is stated. For the closed-loop system (2.3), the right and left normal eigenvector matrices associated with the infinite eigenvalues, denoted by V1 2 Rn…n m† and T1 2 Rn…n m† , are defined respectively by EV1 ˆ 0,

VT1 V1 ˆ In

m

…3:1†

and

and for an arbitrary matrix Q such that the matrix [V1 Q] forms an orthogonal matrix, the matrix (QTV) is nonsingular, and the corresponding feedback gain K is given by K ˆ W…QT V† 1 QT ‡ W1 VT1 ‰I V…QT V† 1 QT Š with W ˆ ‰D…s1 †f1 D…s2 †f2


D…sm †fm Š

TT1 E ˆ 0,

TT1 T1 ˆ In

m:

…3:2†

It is shown in Duan (1996) that, when Assumption A1 is satisfied, there exist a pair of real coefficient, right coprime polynomial matrices N(s) and D(s) of dimensions n  r and r  r, respectively, satisfying the following relation …A

sE†N…s† ‡ BD…s† ˆ 0:

…3:3†

Lemma 3.1. (Duan 1998). Given system (2.1) satisfying Assumption A1. Let V1 2 Rn…n m† and T1 2 Rn…n m† be two matrices satisfying (3.1) and (3.2), N(s) and D(s) be a pair of right coprime matrix polynomials which are of dimensions n  r and r  r, respectively, and satisfy (3.3). Then (1) There exist a group of distinct, self-conjugate complex numbers si, i ˆ 1, 2, . . . , m, a matrix V 2 Cnm, and a real matrix K 2 Rrn, such that det[sE (A ‡ BK)] is not identically zero, and …A ‡ BK†V ˆ EV

…3:4†

holds for  ˆ diag‰s1 s2


sm Š

…3:5†

if and only if there exists (a) a group of parameter vectors fi, i ˆ 1, 2, . . . , m, satisfying sl . Constraint C1: fi ˆ fl if si ˆ  Constraint C2: det…PT E‰N…s1 †f1 N…s2 †f2


for some P 2 Rn  m. N…sm †fm Š† 6ˆ 0 (b) a parameter matrix W1 2 Rr…n m† satisfying Constraint C3:det…TT1 AV1 ‡ TT1 BW1 † 6ˆ 0 (2) When the above conditions (a) and (b) are met, the matrix V is given by V ˆ ‰N…s1 †f1 N…s2 †f2


N…sm †fm Š

…3:6†

…3:7†

…3:8†

where in (3.6) and (3.8), fi, i ˆ 1, 2, . . . ,m, is a group of design parameter vectors satisfying Constraints C1 and C2, and in (3.7), W1 is a real parameter matrix satisfying Constraint C3. 3.2. Closed-Loop Eigenvalue Sensitivity Measures In order to solve the robust pole assignment problem formulated in Section 2, proper sensitivity measures for the closed-loop eigenvalues need to be established. Let M, N 2 Rn  n,  be a simple finite relative eigenvalue of the non-defective matrix pair [M N]. Then, the right and left eigenvectors x and y of the matrix pair [M N] associated with eigenvalue  are defined by Mx ˆ Nx,

yT M ˆ yT N:

…3:9†

Moreover, this pair of right and left eigenvectors x and y are said to be normalized (Stewart 1975) if they satisfy yT Nx ˆ 1,

yT Mx ˆ :

…3:10†

Definition 3.1. Let M, N 2 Rnn,  be a simple finite relative eigenvalue of the nondefective matrix pair [M N ], and x and y are a pair of normalized right and left eigenvectors of the matrix pair [M N ] associated with the eigenvalue , then the following quantity c…† ˆ

kyk2 kxk2

…1 ‡ jj2 †1=2

…3:11†

is called the condition number corresponding to the eigenvalue . Based on the above concept, the following perturbation result of generalized eigenvalue problem of matrix pairs can be stated. Proposition 3.1. (Stewart 1975). Let M, N 2 Rnn,  be a simple finite relative eigenvalue of the nondefective matrix pencil [M N]. Then perturbations of order O()

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in the components of matrices M and N cause perturbations of order O["c()] in the eigenvalue , here c() is the condition number corresponding to the eigenvalue .

numbers corresponding to the closed-loop eigenvalues, which serve as the closed-loop eigenvalue sensitivity measures, can be given.

The above proposition indicates that the condition number c() given by (3.11) well serves a measure for the sensitivity of the eigenvalue  with respect to perturbations in the matrices M and N. Similar to the concept of normalized right and left eigenvectors, normalized right and left eigenvector matrices can also be defined. Take the closed-loop system (2.3), for example, its right eigenvector matrix V is defined by (3.4), and its left eigenvector matrix T is defined by

Lemma 3.2. Let the conditions in Proposition 3.2 hold. Then the condition numbers corresponding to the closed-loop finite eigenvalues si, i ˆ 1,2, . . . , m, are given as follows:

TT …A ‡ BK† ˆ TT E:

…3:12†

A pair of right and left eigenvector matrices V and T of the closed-loop system (2.3) is said to be normalized if they further satisfy TT EV ˆ I and

TT Ac V ˆ :

…3:13†

It is clear that the corresponding columns vi and ti of a pair of normalized right and left eigenvector matrices V and T are also a pair of normalized right and left eigenvectors. Proposition 3.2. (Duan 1998). Let the conditions (a) and (b) in Lemma 3.1 hold, and matrices V and K be given according to the second conclusion of Lemma 3.1. Then (1) for an arbitrary matrix P 2 Rnm such that [T1 P] forms an orthogonal matrix, the matrix (PTEV) is nonsingular, and (2) any left eigenvector matrix T of the closed-loop system (2.3), associated with the ®nite closed-loop eigenvalues, which forms a normalized pair with a right eigenvector matrix V, is given by T ˆ ‰I

T1 1T …AV1 ‡ BW1 †T ŠP…PT Ev†

T

,

…3:14† where 1 ˆ

TT1 AV1

‡

TT1 BW1

P 2 Rn  m is a matrix such that [T1 P] forms an orthogonal matrix. Furthermore, this matrix T is invariant with different choices of the matrix P. Based on Lemma 3.1 and the above two propositions, the following result about the condition

ci ˆ

kti k2
kN…si †fi k2 …1 ‡ jsi j2 †1=2

,

i ˆ 1, 2; . . . , n,

…3:15†

where ti is the ith column of the matrix T given in Proposition 3.2. All the other variables are as defined in Lemma 3.1. Let max(M) and min(M) respectively denote the maximum and minimum singular values of the matrix M. To finish this section, we finally reveal a property of the matrix (PT EV) involved in the expression (3.14) for the eigenvector matrix T. Lemma 3.3. Let matrices T1, E and V be described as above. Then, the condition number cond…PT EV† ˆ

max …PT EV† min …PT EV†

…3:16†

is invariant to different choices of such matrix P that [T1 P] forms an orthogonal matrix. Proof. In view of the definition of singular values of matrices, there holds 2i …PT EV† ˆ i …PT EVVT ET P†,

i ˆ 1, 2, . . . m

where i(M) and i(M) represent the ith singular value and eigenvalue of matrix M, respectively. Therefore, in order to show that cond(PTEV) is invariant to matrix P, it suffices only to show that (PTEVVTETP) is invariant to matrix P. Utilizing (3.2) and the fact that [P T1] forms an orthogonal matrix, yields ‰P

T1 Š 1 EVVT ET ‰P T

T1 Š

ˆ ‰P T1 Š EVVT ET ‰P T1 Š " # PT ˆ EVVT ET ‰P T1 Š TT1  T  P EVVT ET P 0 ˆ : 0 0

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Since similarity transformation does not change matrix eigenvalues, the above relation gives …EVVT ET † ˆ …PT EVVT ET P† [ f0g: This clearly shows that (PTEVVTETP) is invariant of matrix P. &

4. Solution to Problem RPA It follows from Lemma 3.1 that the design freedom existing in the closed-loop eigenstructure assignment consists of the following three parts:

 Establish the parametric expressions of Constraints C2 and C3.  Solve a matrix P 2 Rnm such that [P T1] forms an orthogonal matrix, or equivalently, ®nd a matrix P 2 Rnm satisfying TT1 P ˆ 0,

PT P ˆ Im :

…4:2†

 Solve a matrix Q 2 Rn  m such that [Q V1] forms an orthogonal matrix, or equivalently, ®nd a matrix Q 2 Rn  m satisfying VT1 Q ˆ 0,

QT Q ˆ Im :

…4:3†

1. the ®nite closed-loop eigenvalues si, i ˆ 1, 2, . . . , m; 2. the group of parameter vectors fi, i ˆ 1, 2 , . . . , m, satisfying Constraints C1 and C2; and 3. the parameter matrix W1 satisfying Constraint C3.

 Specify a group of closed-loop eigenvalues location regions i, i ˆ 1, 2, . . . ,m, according to the closedloop stability and performance requirements.

It follows from the well-known pole assignment result for descriptor linear systems that under Assumption A1 and the following condition (Kautsky et al. 1989)

4.2. Step II ± Optimization

rank…‰B E AV1 Š† ˆ n or rank…‰B E ‡ AV1 VT1 Š† ˆ n

…4:1†

all the above three parts of design parameters exist, and in most cases, some or even all of these three parts of parameters are not unique. With any choice of these three parts of design parameters, the matrices K and V given according to the second conclusion of Lemma 3.1 result in a closed-loop system (2.3), which is regular, has m finite closed-loop relative eigenvalues si, i ˆ 1, 2, . . . , m, and takes the matrix V as its right eigenvector matrix. To further solve Problem RPA, a set of such design parameters need to be sought, which minimizes the closed-loop eigenvalue sensitivity measures given in (3.15). With the help of Lemma 3.1, solution to Problem RPA can be performed in three steps as explored in the following. 4.1. Step I ± Preparation The purpose of this step is to get all the variables ready to be used in the next two steps. These include the following:  Solve the coprime polynomial matrices N(s) and D(s) satisfying the right coprime factorization (3.3).  Solve the two matrices T1 and V1 satisfying (3.1) and (3.2).  Establish the parametric expressions of matrices V and W according to (3.6) and (3.8).

The purpose of this step is to obtain the design parameters W1, si and fi, i ˆ 1, 2, . . . , m, with which the solution to Problem RPA can be obtained with the help of Lemma 3.1. Since Constraints C2 and C3 guarantee the closed-loop regularity (Duan 1998), the problem to be solved in this step is to seek these design parameters W1, si and fi, i ˆ 1, 2, . . . , m, satisfying Constraints C1±C3 and si 2 i, i ˆ 1, 2, . . . , m, and at the same time, minimizing J0 …W1 , si , fi , i ˆ 1, 2, . . . , m† ˆ

m X iˆ1

i c2i ,

…4:4†

where ci, i ˆ 1, 2, . . . , m, are the closed-loop eigenvalue sensitivity measures defined by (3.15), i, i ˆ 1, 2, . . . , m, are a group of positive weighting factors. Obviously, this problem can be solved by directly applying some optimization algorithm to the following minimization problem: minimize J0 …W1 , si , fi , i ˆ 1, 2, . . . , m† fsi ;fi ;W1 g

s:t: si 2 i ,

i ˆ 1, 2, . . . , m

Constraints C1 C3: …4:5† However, since the index J0 contains two matrix inverses, namely, 11 and (PTEV) 1, measures for numerical stability must be taken. It follows from Proposition 3.2 that, when Constraint C2 is met, the matrix (PTEV), with the matrix P satisfying (4.2), is nonsingular. This states

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that Constraint C2 is equivalent to existence of a matrix P satisfying (4.2) such that the matrix (PTEV) is nonsingular. Due to this fact, in the following we make the convention that the matrix P in Constraint C2 satisfies (4.2). In order to derive a numerically stable solution to the above optimization problem (4.5), we need either to avoid solution of the matrix inverses and 11 and (PTEV) 1, or, according to matrix inverse perturbation theory, to have the following indices:

J1 ˆ 11 2 1 2

…4:6†

2

2

…4:7†

to be sufficiently small. Since the closed-loop regularity is ensured by Constraints C2 and C3 (Duan 1998), it is important to note that the closed-loop system also possesses robust regularity when indices J1 and J2 are small. The matrix P satisfying (4.2) is called the companion matrix in Duan (1998) and is introduced to reduce the order of the matrix whose inverse is involved in solution of the left closed-loop eigenvector matrix T. This matrix P does not affect the value of index J0 although it appears in the expression of this index (Duan 1998). Notice that J2 ˆ cond(PTEV), it follows from Lemma 3.3 that this matrix P does not affect index J2 either. Hence, there is no need to optimize the matrix P in minimizing indices J0 and J2. It follows from the above reasoning that the most straightforward way of solving the robust pole assignment problem numerical efficiently is to seek all the parameters W1, si, fi, i ˆ 1, 2, . . . , m, simultaneously by solving the following minimization: minimizefJ0 ‡ 1 J1 ‡ 2 J2 g fsi ; fi ;W1 g

s:t: Constraints C1 C3 si 2 i ,

1. Solve minimize J1 …W1 † fW1 g

s:t: Constraint C3:

…4:9†

2. Compute 11 based on the obtained W1. 3. Solve

and



J2 ˆ PT EV …PT EV† 1

us to first optimize W1 by minimizing J1. With the optimal W1, index J1 is small and hence a good solution can be obtained for 11 . We can then proceed to minimize a joint index of J0 and J2 through optimizing parameters si and fi, i ˆ 1, 2, . . . , m. The procedures can be given as follows:

…4:8†

i ˆ 1, 2, . . . , m,

where 1 and 2 are two positive scalar weighting factors. However, since this method optimizes all the parameters in one single optimization problem, it gives small robustness indices theoretically, while has the drawback of being more complicated and requiring a large computation load. In the following of this section, two simpler methods are presented. Method A. It is clear that the matrix 1 has relation with only the parameter matrix W1. This fact allows

minimizefJ0 ‡ 2 J2 g fsi , fi g s:t: Constraints C1 and C2 si 2 i ,

…4:10†

i ˆ 1, 2, . . . , m:

The advantage of above Method A is that it obviously has a much less computation load. The disadvantage of this method is that the part of the degree of freedom, W1, is all used to improve the conditioning to solution of 11 , while no contribution is given to minimization of the eigenvalue sensitivity measures. Method B. This method seeks parameters W1, si and fi, i ˆ 1, 2, . . . , m, simultaneously by solving the following optimization: minimizefJ0 ‡ 1 J1 g fsi ;fi ;W1 g

s:t: Constraints C1 C3 si 2 i ,

…4:11†

i ˆ 1, 2, . . . , m:

The index in the above optimization problem (4.11) does not involve J2. The key point is that J2 does not need to be minimized when index J0 is, in this method, computed in each iteration of optimization using the following algorithm which avoids computation of the inverse of matrix (PTEV). Algorithm 4.1 1. Solve 1 and  ˆ ‰I

T1 1T …AV1 ‡ BW1 †T ŠP

…4:12†

with the matrix P taken to be the one obtained in Step I. 2. Solve, using some numerically stable algorithm for linear equations (e.g. the QR algorithm), the

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matrix T from T…PT EV†T ˆ :

…4:13†

3. Compute ci ˆ

kti k2 kN…si †fi k2 …1 ‡ jsi j2 †1=2

,

i ˆ 1, 2, . . . , m,

…4:14†

where ti is the ith column of the matrix T obtained in Step II. 4. Compute J0 according to (4.4). Remark 4.1. Combining the ideas of Methods A and B, Method A can be further simplified. As a matter of fact, the optimization problem (4.10) in the third step of Method A can be substituted by the following one: minimize J0 fsi , fi , i ˆ 1, 2, . . . , mg fsi ; fi g

s:t: Constraints C1 and C2 si 2 i , i ˆ 1, 2, . . . , m, where the index J0 is computed using Algorithm 4.1 in each iteration of optimization. The specific minimization problems (4.10) and (4.11) involved in the above methods can be solved using some standard optimization packages. We have found that the matlab command constr is simple and effective for solving all these minimization problems. Generally speaking, it is hard to derive specific optimization algorithms for these minimization problems, which guarantee convergence and optimality. 4.3. Step III ± Solution of K When the parameters W1, si and fi, i ˆ 1, 2, . . . , m, are obtained, the matrices V and W can then be derived. Therefore, the state feedback gain matrix K can be readily calculated from (3.7). However, formula (3.7) involves a matrix inverse (QTV) 1 which might give a poor solution of K. To overcome this problem, Duan (1998) has proposed the following second general expression for K: K ˆ W0 QT ‡ W1 VT1 ,

WT1 V:

1. Solve the matrices V and W according to (3.6) and (3.8) respectively based on the parameters W1, si and fi, i ˆ 1, 2, . . . , m obtained in Step II. 2. Solve the matrix from the linear matrix equation (4.17) using some numerically stable algorithm for solution of linear equations (e.g. the QR algorithm). Here the matrix Q is obtained in Step I. 3. Calculate the matrix K according to (4.16). To end up with this section, let us further make some remarks about the above approach. Remark 4.2. For a real eigenvalue sk ˆ k, we may choose a real corresponding parameter vector fk ˆ gk. For a pair of conjugate eigenvalues sl and si ˆ sl ˆ i ‡ l j, we may choose the corresponding parameter vectors as fl and fi ˆ fl ˆ gi ‡ gl j, with all i, gi being real. Then, Constraint C1 holds automatically. Remark 4.3. For a real eigenvalue sk ˆ k, we may choose the corresponding region k ˆ [ak bk]. For a pair of conjugate eigenvalues sl and si ˆ sl ˆ i ‡ l j, we may choose the corresponding regions i and l as in (2.4). In this way, the constraint si 2 i, i ˆ 1, 2, . . . , m, involved in all the optimizations (4.8), (4.10), (4.11) and (4.15), may be expressed in the following form with the set of new parameters i, and gi, i ˆ 1, 2, . . . , n: a i   i  bi ,

i ˆ 1, 2, . . . , m:

Remark 4.4. Because of the completeness of the eigenstructure assignment approach used, the optimality of the solution to the robust pole assignment problem obtained using the above proposed approach is totally dependent on that of the solution to the optimization problem formulated.

…4:16†

where W0 is a matrix of proper dimension satisfying the following linear equation: W0 …QT V† ˆ W

The matrix Q is an arbitrary matrix satisfying (4.3). Based on (4.16) and (4.17), the following procedures can be given for solving the feedback gain matrix K, which does not involve solution of matrix inverses:

…4:17†

5. The example Consider a system in the form of (2.1) with the following matrix parameters (Kautsky et al. 1989,

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Chu et al. 1997, Duan and Patton 1998): 2

0 0

0 0

6 6 6 Eˆ6 6 0:82 0 6 0 4 0 0 0 2 0 1:1 6 0 0 6 6 Aˆ6 6 1:23 0 6 0 4 0 0

2

0 6 T B ˆ 40 0

0

3 1:72 0 7 0 07 7 0 0 07 7, 7 0 0 05 0 0 1 3 0 0 0 7 1:56 0 07 7 0 1:98 0 7 7 7 0 0 05 1:01 0 0

0 0

1:55 0 0 1:07 0

The right and left eigenvector matrix V1 and T1 associated with the infinite closed-loop eigenvalue are easily obtained as

0

0 0

3 0 7 2:5 5:

1:11

0



 0 1 0 0 0 T , V1 ˆ 0 0 1 0 0   0 1 0 0 0 T T1 ˆ : 0 0 0 1 0 Let 2

2

By a method given by Duan (1996), the following accurate solution to the right coprime factorization (3.3) is obtained: 0

xij 2 C, i, j ˆ 1, 2, 3

then

5.1. New Solutions

167:5085

3

6 7 fi ˆ 4 xi2 5, xi3

Obviously, n ˆ 5, and rank(E) ˆ rank(B) ˆ m ˆ r ˆ 3. Furthermore, it is easy to verify that Assumption A1 is satisfied. Duan and Patton (1998) and Kautsky et al. (1989) both considered assignment of the closed-loop finite eigenvalues s1 ˆ 0.5, s2 ˆ 1 and s3 ˆ 2. By minimizing the spectral norm of the feedback gain matrix, Duan and Patton (1998) obtained a gain matrix for this problem, which will be referred as Solution 1 in the following. By using their proposed robust pole assignment algorithm Kautsky et al. (1989) obtained two solutions for the problem, which will be referred as Solution 2 (the first one) and Solution 3 (the second one) in the following.

2

xi1

0

3

6 7 0 288:11462s 0 7 6 6 7 N…s† ˆ 6 843:975 155s 7 6 476:625 317:75s 7 7 6 0 184:25935 0 5 4 0 0 156:55 2 3 479:7 ‡ 319:8s 849:42 156s 6 7 D…s† ˆ 4 192:5565 128:371s 340:9659 0 5: 0 0 0

6 6 vi ˆ 6 6 4

3 167:5085xi1 7 288:11462si xi2 7 …476:625 ‡ 317:75si †xi1 843:975xi2 ‡ 155si xi3 7 7, 5 184:25935xi2 156:55xi3 i ˆ 1, 2, 3, 2

3 …479:7 ‡ 319:8si †xi1 ‡ 849:42xi2 156si xi3 7 …192:5565 ‡ 128:371si †xi1 340:9659xi2 5, 0

6 wi ˆ 4

i ˆ 1, 2, 3:

Put 2

x11 X ˆ 4 x12 x13

x21 x22 x23

3 x31 x32 5, x33

2

W1

z11 ˆ 4 z21 z31

3 z12 z22 5: z32

It can be verified that Constraints C2 and C3 are given, respectively, by
ˆ det…X† 6ˆ 0 and 1:55…z11 z32

z12 z31 †

1:56z31 6ˆ 0:

The two matrices satisfying (4.2) and (4.3) can be taken as 2

1 60 6 Pˆ6 60 40 0

0 0 1 0 0

3 0 07 7 07 7, 05 1

2

1 60 6 Qˆ6 60 40 0

0 0 0 1 0

3 0 07 7 07 7: 05 1

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Robust Pole Assignment in Descriptor Systems

By using Method B and the MATLAB command constr, the following three solutions to the problem are obtained:

Table 3 lists the actual closed-loop eigenvalues s0i , i ˆ 1±3, corresponding to solutions 1±6. Clearly, the closed-loop eigenvalues have drifted from

Solution 4 2

0:6004413 K ˆ 4 0:1362303 0:3946905

2:5163467 0:4244683 0:8416625

3 0:5665919 0:2794157 5: 0:7175041

0:8210607 0:2644560 1:7602383

0:5976893 0:3382794 1:9280632

0:7191178 0:0937929 1:0235496

0:8361169 0:4291316 3:8611859

3 4:0755356 0:3868824 0:2136272 1:0920495 5: 0:8956908 0:6551404

1:9869080 0:0189260 0:6932277

0:6850267 0:3300014 4:7998693

3:8410919 2:6108797 0:7104634

Solution 5 2

0:1090470 K ˆ 4 0:2170617 0:6832669 Solution 6 2

0:8630988 K ˆ 4 0:8204082 2:1824224

The regions of the closed-loop eigenvalues corresponding to the above Solutions 4±6 are respectively taken as

1 ˆ 0:5, 2 ˆ 1, 3 ˆ 2,

1 ˆ ‰ 1 0:5Š, 2 ˆ ‰ 2 1Š, 3 ˆ ‰ 3

2Š

and 2Š,

1 ˆ ‰ 5

2;3 ˆ f2  3 j 2 2 ‰ 2

1Š, 3 2 ‰ 3 3Šg:

The obtained corresponding optimizing parameters are shown in Table 1. 5.2. Analysis of Solutions Table 2 gives, for each solution, the closed-loop eigenvalue sensitivity measures ci, i ˆ 1±3, and the spectral norm of the condition number vector c ˆ ‰ c1

c2

c3 Š:

…5:7†

The magnitude of each of the feedback gain K are also shown in Table 2. It can be seen from this table that inclusion of closed-loop eigenvalues into the design parameters indeed further improves the robustness. Such a fact can be easily understood from the fact that the vi's are functions of the powers of the closed-loop eigenvalues.

3 2:5839856 0:8757252 5: 0:9687696

their nominal values. This is caused by the computation error and the truncation in the digits of the entries in the matrix K. The amount of drift for each solution is also shown in Table 3 by the number  defined by 8 !1=2 3 > X > 2 0 > > …si s1 † , for solutions 1 5 < iˆ1 ˆ  1=2 > > 0 2 0 2 0 2 > …s s † ‡ …Im…s s †† ‡ …Re…s s †† , > 1 2 2 1 2 2 : for solution 6. It can be observed from the values of  in this table and the values of kck2 in Table 2 that the result comes out almost consistent with the theory that for solutions which have smaller eigenvalue sensitivity measures, the closed-loop eigenvalue drifts caused by the truncation error in K are also small. To further demonstrate the robustness of the solutions, a numerical test is carried out, in which the following perturbations to matrices E, A and B are considered:
Ek …1, 4† ˆ 0:001k
randn…size…1††,
Ek …3, 1† ˆ 0:001k
randn…size…1††,
Ak ˆ 0:001k
randn…size…A††,
Bk ˆ 0:001k
randn…size…B††,

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Table 1. Optimal parameters corresponding to solutions 4±6. Parameters

Solution 4

s1 s2 s3 x11 x12 x13 x21 x22 x23 x31 x32 x33 z11 z21 z31 z12 z22 z32

Solution 5

0.5 1 2 0.158762 0.0001742 0.8137357 0.0737054 0.0000021 0.3482712 0.0821066 0.0002352 0.2958685 0.8210607 0.2644560 1.7602383 0.5976893 0.3382794 1.9280632

Solution 6

0.5 1.8370639 3 3.1631727 1.5987402 0.1477861 0.4391780 1.0839968 0.0592666 2.6718713 2.8465096 0.0417510 1.9025805 0.0018031 0.1516478 0.7372848 0.4187045 4.1402561

2 1‡3i 1 3i 5.6061441 4.7658532 9.3304736i 4.7658532‡9.3304736i 1.7258848 2.7088541‡2.7620399i 2.7088541 2.7620399i 3.4649445 11.3770959 7.5767253i 11.3770959‡7.5767253i 1.9869080 0.0189260 0.6932277 0.6850267 0.3300014 4.7998693

Table 2. Robustness measures and magnitudes of solutions. Solutions

c1

c2

c3

kck2

kKk2

kKkF

1 2 3 4 5 6

137.4587 2.7060 3.3533 0.9401 0.7797 0.6666

316.4962 2.1727 1.3020 1.1863 0.7830 0.5997

146.5296 1.6482 1.6572 0.8470 0.8079 0.5997

374.8809 3.8418 3.9606 1.7345 1.3689 1.0787

0.0720 0.7327 1.7806 2.8810 4.2468 5.4540

0.0969 0.8180 1.9618 4.0867 6.0927 8.0427

Table 3. Finite closed-loop eigenvalues. Solutions 1 2 3 4 5 6

s01 0.4943537 0.4995358 0.5000145 0.5000000 1.3488797 2.0000000

s02

s03



1.0057941 1.9973683 1.0005676 1.9998846 1.0000098 2.0000127 1.0000000 1.9999999 2.4769677 4.2505349 0.999999933.0000000i

where randn is a command in MATLAB. Randn (size(M)) generates a random matrix of the same size as matrix M, whose components are chosen from a normal distribution with zero and variance 1. Applying each solution to the perturbed system, corresponding to each k, we can obtain the finite eigenvalues si (k), i ˆ 1±3, of the perturbed closed-loop system.

8.5075E 7.4227E 2.1655E 9.0637E 7.3156E 9.5499E

3 4 5 8 8 8

For solutions 1±5, we have calculated the following values:

di ˆ

" 100 X kˆ1

‰Re…si …k†

si †Š

2

#1=2  100,

i ˆ 1 3:

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Robust Pole Assignment in Descriptor Systems

While for Solution 6, corresponding to which the normal closed-loop finite eigenvalue set contains complex ones, the di, i ˆ 1, 2, 3, are calculated by " d1 ˆ

100 X ‰s1 …k†

"

s1 Š2

"

dˆ

s2 †Š2 s2 †Š

2

3 X iˆ1

100,

kˆ1

100 X ‰Im…s2 …k† d3 ˆ

"

#1=2 

kˆ1

100 X d2 ˆ ‰Re…s2 …k†

The values of di, i ˆ 1, 2, 3, together with the value of

#1=2 

…5:8†

100, #1=2  100:

kˆ1

d2i

#1=2  3

…5:9†

are listed in Table 4. It is interesting to note from Tables 2 and 4 that again the results come out to be almost consistent with the theory, that is, for solution with a smaller eigenvalue sensitivity measure kck2, the corresponding overall drift magnitudes of the closed-loop eigenvalues measured by di and d are also smaller. In order to give a more intuitive picture, we have shown, in Fig. 1, the values of "

3 X …Re…si …k† d…k† ˆ

Table 4. Numerical robustness test results. d1

d2

d3

d

1 2 3 4 5 6

1.6667385 0.0259049 0.0323203 0.0194654 0.0139967 0.0125209

0.0773926 0.0238295 0.0177116 0.0080147 0.0108553 0.0119989

0.0738351 0.0193395 0.0278048 0.0071985 0.0064263 0.0111140

0.5567224 0.0133871 0.0153891 0.0074159 0.0062809 0.0068659

s1 Š2 ‡ ‰Re…s2 …k† s2 †Š2 1=2 ‡‰Im…s2 …k† s2 †Š2 =3, k ˆ 1 100

0.4

0.4

0.2

0.2

60

80

100

0

(c) 0.6

(d) 0.6

0.4

0.4

0.2

0.2

0 0

20

40

60

80

100

0

(e) 0.6

(f) 0.6

0.4

0.4

0.2

0.2

0 0

20

40

60

80

k ˆ 1 100

 d…k† ˆ ‰s1 …k†

(b) 0.6

40

3,

corresponding to Solutions 1±5, and the values of

(a) 0.6

20

#1=2 

iˆ1

Solutions

0 0

si ††

2

100

0

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

Fig. 1. Results of numerical robustness test: (a±f) Case of solutions 1±6.

148

for Solution 6. It is easily seen that the results of this robustness test are basically consistent with the results given by Tables 2±4. Remark 5.1. Since the perturbations in the open-loop system matrices are taken to be random matrices, each time we carry out this robustness test, a different table and a different figure are obtained. However, the relative relations among the values of Table 4, and the relative magnitudes of the curves in the subplot figures remain almost the same. Remark 5.2. It is well-known that, for conventional linear systems, small magnitude of the feedback gain applies lower closed-loop eigenvalue sensitivities. This fact no longer holds for descriptor linear systems. It can be seen from Table 2 that the Solution 1, which is obtained by minimizing the magnitude of K, possesses the smallest magnitude, but has the largest value of kck2. Tables 2 and 3 as well as Fig. 1 further show that this Solution 1 possesses really very poor robustness.

6. Concluding Remark This paper proposes a simple approach for robust pole assignment in descriptor linear systems via state feedback. The approach is based on a perturbation result for generalized eigenvalue problem of matrix pairs proposed by Stewart (1975) and a complete parametric eigenstructure assignment approach for descriptor linear systems via state feedback controllers, proposed by Duan (1998). Due to the features of the eigenstructure assignment result, the proposed approach has the following advantages: (a) The procedures for solution of the proposed robust pole assignment problem are in a sequential order, and no ``going back'' procedures are needed. (b) The ®nite closed-loop eigenvalues are included in the design parameters and are optimized within certain desired ®elds on the complex plane, thus a closed-loop system with better robustness and desired transient performance can be obtained. (c) The optimality of the solution to the whole robust pole assignment problem is solely dependent on the optimality of the solution to an independent minimization problem. (d) Measures of numerical stability have been taken, and as a consequence, the closed-loop system also possesses better regularity robustness. A detailed study of a 5 order example demonstrates the effect of the approach.

G.-R. Duan et al.

Acknowledgements This work was supported in part by the Outstanding Youth Foundation of the Chinese Ministry of Education and the Chinese National Outstanding Youth Science Foundation under Grant No. 69504002. The authors are grateful to the editors and reviewers for their helpful comments and suggestions.

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