Computation of Structural Decomposition for Linear ...



Computation of Structural Decomposition for Linear Singular Systems

MINGHUA HE and BEN M. CHEN

Department of Electrical and Computer Engineering

The National University of Singapore

10 Kent Ridge Crescents, Singapore 117576

REPUBLIC OF SINGAPORE

Abstract: We present in this paper computation algorithms for the structural decomposition of general linear multivariable singular systems. Such kind of decomposition has a distinct feature of capturing and displaying all the structural properties, such as the finite and infinite zero structures, invertibility structures and redundant dynamics of the given system. The computation will make it a powerful and convenient tool in solving control problems for singular systems.

Keywords: Computation; MATLAB programs; singular systems; structure decomposition; structure properties.

1. Introduction

Consider the following linear singular system

[pic] (1)

where [pic]. It is also alternatively called a descriptor system, an implicit system or a generalized linear system in literature. Linear singular system has attracted researchers for more than three decades since many real systems, such as electrical power systems, can be naturally modeled as singular systems. Among the issues discussed for linear singular system [7], system structure and equivalence is an essential topic. Campbell [1] presented an effective structural decomposition method and got the corresponding equivalent system, while Verghese et al. [11] defined a strong system equivalence using a trivial augmentation and deflation technique. On the other hand, structural invariants also received much attention in literature. Further, Misra et al. [10] and Liu et al. [9] have presented their algorithms to compute the invariant structural indices of singular systems. More recently, He and Chen [5] and He et al [6] have developed a structural decomposition method for single-input single-output and multivariable linear singular systems respectively. Such a structural decomposition can not only give the invariant structural indices but also explicitly display the structural properties, such as the finite and infinite zero dynamics, invertibility structures and redundant dynamics of the given systems. And it is expected to be a powerful tool in solving system and control problems as its counterpart in nonsingular linear system [3].

This paper focuses on the computation algorithms of the structural decomposition. First, to make this paper more self-contained, we briefly describe the structural decomposition theorem in Section 2. And in Section 3, the main MATLAB computation programs are presented while several numerical examples will be included in Section 4. Finally, a conclusion will be drawn in Section 5.

2. Structural Decomposition Theorem

and Its Properties

We first summarize the structural decomposition of general multivariable singular systems in compact matrix form. And its essential properties will also be given in brief in this section.

Theorem 2.1 Consider the general multivariable linear singular system [pic] in (1). Then, there exist nonsingular state and output transformations [pic] and [pic], and a nonsingular transformation [pic], as well as an m(m input transformation [pic], whose inverse has all its elements being some polynomials of [pic] (i.e., its inverse contains various differentiation operators), which together give a structural decomposition of [pic] and display explicitly its structural properties.

This structural decomposition can be described in the following equation form,

[pic] (2)

and

[pic]

[pic](3)

and for each [pic],

[pic](4)

[pic]

The structural decomposition can also be expressed in the following compact form.

[pic]

[pic]

[pic]

(5)

[pic]

[pic]

where

[pic] (6)

and

[pic] (7)

The equation form of this theorem and detail proof can be found in He, Chen and Lin [6]. Here, we briefly introduce the essential properties of this structural decomposition.

Property 2.1 The given system [pic] in (1) is stabilizable if and only if [pic] is stabilizable, and it is detectable if and only if [pic] is detectable, where [pic], (8)

and

[pic]. (9)

Property 2.2 The invariant zeros of the given system [pic] are the eigenvalues of [pic]. The normal rank of [pic] is equal to [pic]. Here [pic] is the dimension of [pic].

Property 2.3 The given system [pic] has [pic]infinite zero of order 0. And its infinite zero structure (of order greater than 0) is given by

[pic] , (10)

that is, for each [pic], [pic] has an infinite zero of order [pic], respectively.

Property 2.4 The given system [pic] is right invertible if and only if [pic]and hence [pic]are non-existent, is left invertible if and only if [pic]and hence [pic]are non-existent, and is invertible if and only if both [pic] and [pic] are non-existent.

The properties show that our structural decomposition can explicitly display the structure properties of the given singular system, and hence it is expected to be a powerful tool in solving singular system and control problems.

3. MATLAB Computation Programs

for the Structural Decomposition

As mentioned before, a detailed constructive decomposition algorithm will not be given here due to the limit pages. And in this section, we will give brief descriptions of the main functions for the computation. The computation programs introduced are all in MATLAB codes.

SD.m

This is the main function, that is, structural decomposition function for general linear singular systems. The function transforms the given singular system [pic] into its structural decomposition form [pic], which can explicitly display all the structural properties, such as the finite and infinite zero structures, invertibility structures and even redundant dynamics of the given system.

sys_hat.m

This function separates two kinds of redundant states from the original system. One kind of redundant states [pic]are static and identical zero all the time, whereas the other redundant states [pic] are linear combination of appropriate order of system input's derivatives. Such states are associated with the so called impulse modes, which are introduced by the derivatives of the system input.

pre_decom.m

This one is to perform a fast-slow decomposition (see e.g., [4] for details) for the given singular system. With two constant transform matrices [pic] and [pic], it transforms the given singular system into two subsystems, one is nonsingular and the other is singular. The decomposition can be characterized as the following transformations,

[pic]

[pic], (11)

where [pic] is a nilpotent matrix.

ctr_cf.m

The function transform a matrix pair [pic] into its control canonical form as follows,

[pic] (12)

where [pic] is completely controllable while [pic] is totally uncontrollable.

bdc_cf.m

This function decomposes a complete controllable pair [pic] into a special block controllability canonical form [2], in which every submatrix block corresponds to a distinct input channel. The decomposition process can be described as follows,

[pic] (13)

where [pic] are Jordan blocks with zero eigenvalue and

[pic]. (14)

kronecker.m

The function transform the given system's system matrix [pic] to its Kronecker Canonical Form with two constant transform matrices [pic] and [pic],

[pic]

(15)

Here every block of the diagonal entries in [pic] is associated the distinct structure index.

SCB.m

This is the function of structural decomposition for linear nonsingular system. The function was developed by Lin and Chen [8], and it decomposes a given linear system [pic] and explicitly displays its structural properties. The function SD.m is its natural extension to linear singular systems.

r_jordan.m

This function transforms a real matrix [pic] to its Jordan canonical form.

The functions introduced here are only some main procedures, and there are still many other functions needed in our computation. But due to the limit of page, we can not introduce every function here. However, this omission will not affect our illustration of computation process in the following section.

4. Some Numerical Examples

To illustrate the computation of our structural decomposition algorithm, we present in this section two numerical examples, one is of single-input and single-out linear singular system while the other is of multi-input and multi-output case.

Let us first look at the following single-input and single-output system,

[pic]

[pic] (16)

the statement

[pic]returns

[pic] (17)

[pic] , (18)

[pic], (19)

and

[pic](20)

[pic]

And finally this decomposition result can be verified by the following operation. The statement

[pic] returns

[pic] (21)

From [pic], the Kronecker Canonical Form of the given system, we can see clearly that the structure indices are the same as our computation results.

Now we look at the following multi-input multi-output linear singular system,

[pic] (22)

Then its structural decomposition form is in the results of the following statement,

[pic]And the results is

[pic]

[pic] (23)

[pic] (24)

And

[pic]

(25)

And the corresponding structure indices are

[pic] (26)

Again, this result can be verified by the following computation,

[pic],

and its computation result is:

[pic]

[pic] (27)

[pic]

Thus, with these two numerical examples, we illustrate the computation process of our structural decomposition algorithm. It can be seen that the MATLAB functions are effective in computing the structural decomposition.

5. Conclusions

We have presented in this paper MATLAB computation functions for the structural decomposition technique for general linear singular systems. The structural decomposition has a distinct feature of explicitly capturing and displaying the structure properties, which make it a powerful tool in solving system and control problems as its counterpart in nonsingular systems. The numerical examples showed that our computation programs are effective in giving a singular system’s structural decomposition form. They thus enhance the structural decomposition’s role as a powerful tool in solving practical problems.

References:

[1] S. L. Campbell, Singular System of Differential

Equations II, Pitman, New York, 1982.

[2] B. M. Chen, Robust and [pic] Control,

Springer, London, 2000.

[3] B. M. Chen, A. Saberi, P. Sannuti and Y.

Shamash, Construction and parameterization of

all static and dynamic [pic]-optimal state

feedback solutions, optimal fixed modes and

fixed decoupling zeros, IEEE Transactions on

Automatic Control, Vol. 38, 1993, pp.248-261.

[4] L. Dai, Singular control systems, Springer-

Verlag, Berlin, 1989.

[5] M. He and B. M. Chen, Structural

decomposition of linear singular systems: The

single-input and single-output case, Systems

and Control Letters, Vol. 47, No. 4, 2002,

pp.325-332.

[6] M. He, B. M. Chen and Z. Lin, Structural

decomposition of general multivariable linear

singular systems, Submitted to publish.

[7] F. L. Lewis, A survey of linear singular systems,

Circuits, Systems, and Signal Processing, Vol.

5, No. 1, 1986, pp.3-36.

[8] Z. Lin and B. M. Chen, Linear systems and

control toolbox, Technical Report, Department

of Electrical and computer engineering,

University of Virginia, USA, 2000

[9] X. Liu, B. M. Chen and Z. Lin, Computation of

structural invariants of singular linear systems,

Proceedings of the 2002 Information, Decision

and Control Symposium, Adelaide, Australia,

2002, pp.35-40.

[10] P. Misra, P. V. Dooren and A. Varga,

Computation of structural invariants of

generalized state-space systems, Automatica,

Vol 30, 1994, pp. 1921-1936.

[11] G. C. Verghese, B. C. Levy and T. Kailath, A

generalized state-space for singular systems,

IEEE Transactions on Automatic Control, Vol.

26, No. 4, 1981, pp.811-831.

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