23 PARAMETRIC EIGENSTRUCTURE ASSIGNMENT FOR DESCRIPTOR SYSTEMS VIA PROPORTIONAL

23

Parametric Eigenstructure Assignment for Descriptor Systems via Proportional plus Derivative State Feedback

© ICROS, KIEE and Springer 2011

__________

Manuscript received August 13, 2009; revised February 22, 2010 and July 9, 2010; accepted July 21, 2010. Recommended by Editor Jae Weon Choi. This work was supported by the Chinese National Natural Science Foundation under Grant No. 10671046.

Biao Zhang is with the Department of Mathematics, Harbin Institute of Technology, Harbin, P. R. China (e-mail: zhangb@ hit.edu.cn).

1. INTRODUCTION

In this paper, we consider the control of the following linear descriptor system

(1)

with proportional plus derivative (PD) state feedback

(2)

where in (1), are respectively, the state vector and the input vector; and are real matrices with where in (2), are respectively, the proportional (P) state feedback gain matrix and the derivative (D) state feedback gain matrix.

It is well known from classical control theory that derivative feedback is very essential for improving the stability and the performance of a control system (see e.g., [1-3]). In [1], PD controllers are employed to provide anticipatory action for overshoot reduction in the responses. In [2] and [3], PD controllers are used to achieve decoupling of linear systems. For descriptor systems, derivative feedback is even more important since it can alter many properties of a descriptor system, which a pure proportional state feedback can not. Because of this reason, the use of PD control law in descriptor systems has been intensively and widely studied since the early 1980s [3-17].

Eigenstructure assignment in descriptor linear systems is a very important problem in descriptor systems theory and has been studied during the past three decades. In particular, the topic of eigenstructure assignment for descriptor systems via PD state feedback has been studied by a number of researchers [13-17]. Chen and Chang [13] and Jin [14] considered the problem of eigenstructure assignment in descriptor systems using the special case of PD state feedback where i.e., the constant-ratio proportional plus derivative (CRPD) state feedback, while Duan and Patton [15], Owens and Askarpour [16], and Wang and Lin [17] considered the problem of eigenstructure assignment in descriptor systems via PD state feedback. The work in [13] depends on the properties of the standard form descriptor systems, but the computed gain matrix and assigned eigenvectors are in the original coordinates, and no transformations are needed. However, the result requires the solution of (sE−A)⁻¹ or and thus requires more computational work and may subject to numerical problems. The works in [14] and [15] are based on the right coprime factorization of the open-loop system. The work in [14] removes the restriction that the closed-loop finite eigenvalues are different from the open-loop eigenvalues, which was required in [13], while the work in [15] removes the restriction required in [13] that the assigned closed-loop finite eigenvalues are different from the open-loop eigenvalues and also releases the open-loop regularity assumption required in both [13] and [14]. However, the results in [14] and [15] require right coprime matrix polynomials to be determined, and thus are not desirable to use in high dimension cases because the determination of right coprime matrix polynomials is computationally expensive and not in general numerically reliable [18]. The works in [16] and [17] are based on solving a recursive eigenvector chains of the matrix pair Unlike [14] and [15], the works in [16] and [17] do not require right coprime matrix polynomials to be determined, and thus overcome the defect of [14] and [15]. However, the results in [16] and [17] contain a series of iterative computations with parameter vectors involved, and thus are complex and need more computational work [19,20]. Moreover, all the reported works for eigenstructure assignment by PD state feedback except the work [17] cannot assign infinite eigenvalues to the closed-loop system.

This paper considers eigenstructure assignment in the descriptor system (1) via PD state feedback (2). We relate the problem to the following recursive equations

(3)

where with of full row-rank are known matrices; are to be determined. Based on a presented simple complete explicit parametric solution to (3), a simple complete parametric approach for eigenstructure assignment in the descriptor system (1) via PD state feedback (2) is proposed. The proposed approach possesses the following features: 1) it does not impose any condition on the closed-loop eigenvalues, simultaneously assigns arbitrary n finite and infinite eigenvalues to the closed-loop system and guarantees the closed-loop regularity; 2) it is very simple and needs less computational work; 3) it gives general complete parametric expressions for the closed-loop eigenvectors, and the P and D state feedback gain matrices.

2. FORMULATION OF THE PROBLEM

Assume that the descriptor system (1) is complete controllable (C-controllable), i.e., system (1) satisfies the following C-controllability assumption.

Assumption 1: for all and

If the PD feedback control law (2) is applied to (1), a closed-loop system is obtained in the form

(4)

with

(5)

Let where are a group of distinct self-conjugate complex numbers and be the set of eigenvalues of the matrix pair and denote the algebraic and geometric multiplicity of by m_i and q_i, respectively, then there are q_i chains of generalized eigenvectors of associated with Denote the lengths of those q_i chains by then the following relations hold:

(6)

(7)

Let the right eigenvector chains of the matrix pair associated with finite eigenvalue be denoted by Then they satisfy

(8)

Let then is the zero eigenvalue of the matrix pair (A_c, E_c). Denote the right eigenvector chains of the matrix pair (A_c, E_c) associated with by Then we have the following equations by definition

(9)

Now the problem of eigenstructure assignment (EA) via PD feedback controller (2) for the descriptor system (1) can be stated as follows: Determine a pair of real matrices and a group of vectors such that the following three requirements are simultaneously satisfied.

1) all the equations in (8) and (9) hold;

2) vectors are linearly independent;

3) the matrix pair (E_c, A_c) is regular, i.e., −A_c) is not identically zero.

3. GENERAL SOLUTION OF EQUATION (3)

Using matrix elementary transformation and in view of the assumption that L is of full row-rank, a pair of matrices and can be obtained such that

(10)

Partition the matrix Q as follows:

(11)

From (10) and (11), we obtain

(12)

Denote

(13)

Then the general solution of (3) is given in the following theorem.

Theorem 1: All solutions of (3) are given by

(14)

where are a group of arbitrarily chosen free parameter vectors; are determined by (10)-(13).

Proof: First, let us show that the vectors given by (14) satisfy the equations in (3). Using (12) and (13), we have

which holds for all Therefore, the vectors given by (14) satisfy the equations in (3).

Next, let us show that the solution (14) is complete, that is, it contains the maximum degrees of freedom, and therefore forms a complete parametric solution to (3).

It is obvious that the maximum degrees of freedom involved in the general solution to the group of equations in (3) is l(t − s), while the solution (14) happen to contain l(t − s) parameters. Thus, we need only to show that these parameters involved in the solution (14) all contributes to the vectors z_k independently.

Let

and

then the equations in (3) can be written in the compact matrix form

Since the matrix Q is nonsingular, the matrix H₁ (=Q₂) is of full column-rank. Thus, the matrix H is of full column-rank. Therefore, all elements in f contribute independently to With this we complete the proof. 

4. SOLUTION OF THE PROBLEM EA

Let

(15)

Denote

(16)

(17)

Then the equations in (8) can be equivalently written as

(18)

Applying matrix elementary transformations to the matrices and in view of Assumption 1, we obtain two sets of matrices and satisfying

(19)

Denote

(20)

It can be easily verify that the following holds

(21)

Partition the matrix as follows:

(22)

Denote

(23)

where Partition the matrix as follows:

(24)

By Theorem 1, the general complete parametric expressions for the closed-loop eigenvectors associated with the finite closed-loop eigenvalues, together with the corresponding vectors are obtained as

(25)

where are a group of arbitrarily chosen free parameter vectors; are determined by (19)-(24).

Let

(26)

Denote

(27)

(28)

Then the equations in (9) can be equivalently written as

(29)

Applying matrix elementary transformation to the matrix [E B] and in view of Assumption 1, we obtain a pair of matrices and satisfying

(30)

Denote

(31)

It can be easily verify that the following holds

(32)

Partition the matrix Q_∞ as follows:

(33)

Denote

(34)

where   Partition the matrix as follows:

(35)

By Theorem 1, the general complete parametric expressions for the closed-loop eigenvectors associated with the infinite closed-loop eigenvalues, together with the corresponding vectors are obtained as

(36)

where are a group of arbitrarily chosen free parameter vectors; are de-termined by (30)-(35).

Define the matrix V_f as follows:

Similarly, columns of matrices W_f and S_f are also composed of i =1, Then the equations in (15) can be equivalently written in the unified matrix forms

(37)

Again, define the matrix V_∞ as

and similarly matrices W_∞ and S_∞ have the same form. Then the equations in (26) can be equivalently written in the unified matrix forms

(38)

By combining (37) and (38) we have

(39)

In order that real matrices K₁ and K₂ to be solved from (39), we choose for a real finite eigenvalue whereas for a complex conjugate pair of finite eigenvalues From (25), it is easy to see that this condition can be equivalently converted into the following constraint on the group of parameter vectors

Constraint 1: for a real finite eigenvalue whereas for a complex conjugate pair of finite eigenvalues

To ensure the requirement 2) in the problem EA, we need to supply the following constraint on the group of parameter vectors

Constraint 2:

When this constraint is met, the gain matrices K₁ and K₂ are given by

(40)

Let

and

Then all the equations in (8) and (9) can be written respectively in the following matrix forms

(41)

(42)

Concerning the regularity of system (4), we have the following lemma.

Lemma 1: Let V_f, V_∞ and K₁, K₂ be matrices satisfying equations (41) and (42) and Constraint 2. Then the matrices K₁ and K₂ make the matrix pencil regular if and only if

(43)

Proof: Using (41) and (42), we obtain

(44)

Therefore, we have

(45)

Since and is not identically zero, it follows from (45) that is not identically zero if and only if condition (43) holds. 

By using (25) and (36)-(38), condition (43), which ensures the closed-loop regularity, can be turned into the following constraint also on the group of parameter vectors as follows.

Constraint 3:

To summarize, we have the following general result for the solution to the problem EA.

Theorem 2: The problem EA has solutions if and only if there exists a group of parameter vectors satisfying Con-straints 1-3. When this condition is met, the group of closed-loop eigenvectors (or the closed-loop eigenvector matrix is given by (25) and (36), and the feedback gain matrices K₁, K₂ are given by (40) with Constraints 1-3 satisfied.

Remark 1: The above approach for eigenstructure assignment in descriptor systems by PD state feedback has several advantages over the approaches of previously published works in the following.

1) The approach does not impose any condition on the closed-loop eigenvalues, and simultaneously assigns arbitrary n finite and infinite eigenvalues to the closed-loop system. Consequently, the approach automatically removes the restriction required in [13] that the assigned closed-loop finite eigenvalues are different from the open-loop eigenvalues and the restriction required in [13-16] that all the assigned closed-loop eigenvalues are finite. The approach also releases the open-loop regularity assumption required in [13] and [14].

2) The approach involves mainly matrix elementary transformations (19) and (30). Unlike [13], it does not require the solution of or to be determined. Unlike [14] and [15], it does not require right coprime matrix polynomials to be determined. Unlike [16] and [17], it does not contain iterative computations. Thus, comparing with the approaches in [13-17], the approach is much simpler and needs less computational work.

3) Unlike [16] and [17], the approach gives the direct, explicit parametric solution to the problem EA. It is known that a direct, explicit solution usually provides much convenience in some system design problems [19].

5. EXAMPLES

Two examples are given in this section. The first one is to demonstrate the effect of the proposed approach; the second one is an application of the proposed approach to model refinement.

Example 1: Consider a system of the form (1) with the following matrix parameters [13,15]

where It is easy to verify that Assumption 1 is satisfied.

In the following, we consider the assignment of the following closed-loop eigenstructure:

In this case, from (19)-(24), we have

From (30)-(35), we have

Therefore the closed-loop eigenvectors are given by

and the corresponding vectors are given by

and

By specially choosing

which satisfy Constraints 1-3, we obtain

Then, according to (40), the P and D state feedback gain matrices are given by

Example 2 (Model refinement [21]): Given a second-order dynamical model of the form

(46)

where M is the positive definite mass matrix, D is the positive definite (semidefinite) damping matrix, K is the positive definite (semidefinite) stiffness matrix, H is the disturbance input influence matrix, x is a n l vector of displacements, and f is a l l vector of disturbances to the system. Find position, velocity, and acceleration gain matrix that reassign a desired subset of the eigenvalues of the model, along with partial mode shapes.

It is known [21] that the dynamics of the refined system may be written as

(47)

where and are symmetric matrices satisfying and > 0 respectively. Clearly, the first-order descriptor representation of this system is

(48)

where

We consider the case n =4. The mass, damping, and stiffness matrices are, respectively, chosen as [22]

The eigenvalues of this system are

It is desired that assigning the eigenvalues to their target values while ensuring that the remaining eigenvalues …, 6, and the corresponding eigenvectors …, 6, remain invariant (a property known as the no spill-over phenomenon [23]). Let

From (19)-(25), the general parametric solution of the eigenvector and the corresponding vectors of the extended first-order system (48) associated with are given by

where and S₇ are deter-mined by (19)-(24). Assume that

(49)

Then the gain matrices K₁ and K₂ are given by

Since we have

with

(50)

Assume that

(51)

We focus on finding low gain matrices and Then we may define an objective as

where are positive scalars representing the weighting factors, represents the spectral norm. Therefore the model refinement problem can be converted into the following minimization problem

Taking and using the Matlab command fmincon, the solution to this minimization problem is obtained as

With this group of parameters, we have

6. CONCLUSIONS

This paper deals with eigenstructure assignment in descriptor systems via PD state feedback. Based on a simple complete explicit parametric solution to a group of recursive equations, a parametric approach for eigenstrucure assignment in descriptor systems via PD state feedback is proposed. The proposed approach possesses the following features: 1) it does not impose any condition on the closed-loop eigenvalues, simultaneously assigns arbitrary n finite and infinite eigenvalues to the closed-loop system and guarantees the closed-loop regularity; 2) it is simple and needs less computational work; 3) it gives general complete parametric expressions for the closed-loop eigenvectors, and the P and D state feedback gain matrices.

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B iao Zhang received his B.S. degree in Mathematics from Peking University, Beijing, an M.S. degree in Applied Mathematics, and a Ph.D. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, in 1984, 1989, and 2007, respectively. He is currently an associate professor in the Department of Mathematics at Harbin Institute of Technology, Harbin. His research interests include eigenstructure assignment, robust control, and descriptor systems.