Beom-Soo Kim,¹ Il-Joo Shim,² Myo-Taeg Lim,³ and Young-Joong Kim³

Abstract

An efficient computational method is presented for state space analysis of singular systems via Haar wavelets. Singular systems are those in which dynamics are governed by a combination of algebraic and differential equations. The corresponding differential-algebraic matrix equation is converted to a generalized Sylvester matrix equation by using Haar wavelet basis. First, an explicit expression for the inverse of the Haar matrix is presented. Then, using it, we propose a combined preorder and postorder traversal algorithm to solve the generalized Sylvester matrix equation. Finally, the efficiency of the proposed method is discussed by a numerical example.

1. Introduction

Wavelets are mathematical functions that cut up data into different frequency components and then study each component with a resolution matched to its scale. Wavelets are now being applied in many areas of science and engineering [1–4]. Much attention has been focused on the use of wavelet transforms to investigate dynamic systems. This is due to the powerful ability of wavelet transforms to decompose time series in time-frequency domain and wavelet basis functions. Chen and Hsiao [3, 4] derived a Haar operational matrix for integration and solved the lumped and distributed parameter systems by constructing operational matrices of various order. The main characteristic of this technique is that it converts a differential equation into an algebraic one with the result that the solution procedures are greatly reduced and simplified. This approach gives new insight into the use of the Haar wavelet method.

Singular systems (also referred to as descriptor or semistate systems) arise more naturally than do state-variable descriptions in the analysis of many sorts of systems. Examples occur in electrical networks, neural networks, control systems, chemical systems, economic systems, and so on (see [5, 6] and references therein). These systems are governed by a mixture of differential and algebraic equations. The complex nature of singular systems causes many difficulties in the analytical and numerical treatment of such systems.

Recently, Haar wavelet technique was applied to state analysis and observer design of singular systems [7]. This approach replaces the state function and the forcing function by the truncated Haar series, respectively. Then the state trajectories are obtained by solving a generalized Sylvester matrix equation. But there exists a trade-off between the resolution of the wavelets and the computation time. The accuracy of the solution can be achieved by increasing the resolution level, but this requires more computation time and very large memory.

In this paper, an efficient computational method is presented for state space analysis of singular systems via Haar wavelets. First, an explicit expression for the inverse of the Haar matrix is presented. This inverse matrix also has a recursive structure. By using this matrix, we propose a combined preorder and postorder traversal algorithm. Then, the full-order generalized Sylvester matrix equation should be solved in terms of the solutions of simple linear matrix equations. Finally, the efficiency of the proposed method is discussed by a numerical example.

2. Kronecker Product

Let and be and matrices, respectively. The Kronecker product of the matrices, denoted by , is defined as (2.1) The operator transforms a matrix of size to a vector of size by stacking the columns of . Some properties of the Kronecker product are given below [8]: (2.2)

3. Haar Wavelets and Their Properties

Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet that generates orthogonal bases of . The simplest and most basic of the wavelet systems is the Haar wavelet which is a group of square waves with magnitudes of in certain intervals and zeros elsewhere [9]. The scaling function and mother wavelet are defined by, respectively, (3.1) Then, all the other basis functions are obtained by dilation and translation of the mother wavelet as follows: (3.2) where , integer is a dilation parameter, integer is a shift parameter, and the intervals are given by , , and . Since the support of the Haar wavelet is , any square integrable function can be written as an infinite linear combination of Haar functions (3.3) where the Haar coefficients are determined by (3.4) where denotes the inner product. In practical applications, Haar series are truncated to terms, that is, (3.5) where Haar functions coefficient vector and Haar functions vector are defined as and .

Integrals of the Haar functions with respect to variable form ramp and triangular waveforms standing with uniform slope, respectively, at the positions of the corresponding rectangular functions. The group of these integrals can be expressed as follows: (3.6) Then, the Haar matrix is defined as (3.7) where .

Integration of the Haar function vector can be written as (3.8) where is the -square operational matrix of integration which satisfies the following recursive formula [3]: (3.9) where is an -square zero matrix. The Haar matrix also has the following recursive formula [3]: (3.10) Particularly, it was proven that the following relationship holds [3]: (3.11) where and . This diagonal matrix also can be represented in the recursive form (3.12) where , and is called a resolution scale or level.

We present the following lemma which will be used to decompose the generalized Sylvester matrix equation.

Lemma 3.1. Let be a Haar matrix defined in (3.10). Then, its inverse matrix has the following recursive form: (3.13)

Proof. We assume that has the following recursive structure: (3.14) where are constants to be determined. Now, we multiply and : (3.15) Then, using the property of , we obtain (3.16) Thus, satisfy  .

4. Singular Linear System

Consider a linear continuous-time singular system described by (4.1) where denotes the vector of state variables, denotes the vector of manipulated inputs, , are matrices, is generally singular, and is a matrix. Without loss of generality, we assume that and (4.1) is regular, that is, . Regularity means that the solution is uniquely determined by the given initial value and input .

If the input function vector is square integrable in the interval , then it can be represented in a Haar function basis as (4.2) where is a Haar coefficient matrix and can be obtained by the method described in Section 3. Likewise, is expanded in Haar function basis (4.3) where is the unknown matrix to be determined. From the definition of the Haar function, the initial state can be represented as follows: (4.4) Integrating (4.3) from to , we have (4.5) Integrating (4.1) and using (3.8) and (4.4), after canceling , we obtain (4.6) where we define . Thus, the differential matrix equation (4.1) has been transformed to a generalized Sylvester matrix equation that must be solved for . Equation (4.6) can be solved by using Kronecker product as in [6] (4.7) where is a unit matrix. Equation (4.7) can be solved by LU factorization. However, the coefficient matrix has dimension , making this approach impractical except for small systems. There are other methods for solving the Sylvester matrix equation (4.6), for example, the Bartels-Stewart algorithm, Krylov subspace method, and matrix sign function method (see [10] and references therein). In [3, 11], recursive algorithms were derived to solve the equations of type for linear systems. It should be noted that the algorithm in [3] is not applicable to a generalized Sylvester matrix equation (4.6), since E is a singular matrix.

4.1. Decomposition and Recursive Binary Tree

Under the assumption that is a nonsingular matrix, (4.6) can be written as the following Sylvester equation: (4.8) To decompose (4.8), we split and by columns: (4.9) where , , with . Here denotes the matrix that is decomposed at level with . Then, we obtain the following reduced-order matrix equations: (4.10)(4.11) Since is a singular matrix, is also singular. Thus, we postmultiply by both sides of (4.11) to express in terms of (4.12) Substituting (4.12) into (4.10) yields (4.13) Therefore, the original problem is decomposed into a reduced-order generalized Sylvester matrix equation (4.13) and a matrix algebraic equation (4.12). Again postmultiplying by both sides of (4.13), we have(4.14) In (4.14), we define (4.15) Then, the matrix is an upper triangular matrix and has the following recursive form: (4.16) where denotes -square matrix with all elements being 1 (see Appendix ).

Substituting (4.16) into (4.14) and splitting and the right-hand side of (4.14) by columns yields (4.17) where (4.18) Thus, (4.17) is decomposed into two matrix equations with dependent and independent subsystems. (4.19)(4.20) In (4.19) and (4.20), we first solve for and then after updating the right-hand side of (4.20) with respect to , solve for . Since (4.19) and (4.20) have the same form as (4.17) and is still an upper triangular matrix, they can be decomposed into two subsystems in which the dimension has been reduced by half, respectively. Therefore, we recursively decompose each equation into two equations until no further decomposition is possible in which all are column vectors. This procedure constructs the binary tree as shown in Figure 1.

Figure 1: Binary tree for resolution scale .

A binary tree is a rooted tree in which each node has at most two children, designated as a left child and a right child. A full binary tree is a binary tree in which each node has exactly two children or none. A perfect (or complete) binary tree is a full binary tree in which all leaves have the same depth [12]. In Figure 1, the binary tree in the dotted box is a perfect binary tree of depth . An external node (or leaf node) is a node with no children. For instance, the nodes labeled 1, 9, 10, 11, 12, 13, 14, 15, and 16 in Figure 1 are external nodes.

Matrix equations corresponding to all external nodes of the perfect binary tree are classified into two types of equations described as follows: (4.21) Note that in equation (4.21), , . Thus, they become simple linear matrix equations as follows: (4.22)

4.2. Combined Preorder and Postorder Traversal Algorithm

Visiting all the nodes in a tree in some particular order is known as a tree traversal. A preorder traversal visits the root of a subtree, then the left and right subtrees recursively. A postorder traversal visits the left and right subtrees recursively, then the root node of the subtree [12]. For example, the preorder and postorder traversals of the binary tree shown in Figure 1 are as follows:

During the decomposition of (4.14), the right-hand side of the right child is split after updating it recursively as follows: (4.23) This splitting and updating sequence is a preorder traversal of the perfect binary tree from root node    . The unknown matrix is obtained by merging all column vectors . This sequence is a postorder traversal of the perfect binary tree from root node    . To update (4.23), we need which is obtained from the left child. Hence, to solve (4.22), it is necessary to update, split, and solve by using the following combined preorder and postorder traversal method.

The pseudocode of the proposed algorithm is as in Algorithm 1.

Algorithm 1

For example, at resolution scale , the proposed combined preorder and postorder traversal method is illustrated in Figure 2.

Figure 2: The combined preorder and postorder traversal for resolution scale .

In Figure 2, nodes 2, 3, 5, and 9 of preorder traversal are done at Step 1 and the remaining nodes are processed at Step 2. The computational efficiency of the proposed method is discussed in the next section.

5. An Illustrated Example

In this section, an example is presented to illustrate the proposed algorithm. We consider a singular linear system of (4.1) with (5.1) and . And we assume that is a unit step function. In the cases of and , the simulation results are depicted in Figures 3 and 4, respectively.

Figure 3: Case for resolution scale .

Figure 4: Case for resolution scale .

From these figures, it is clear that the solution accuracy is improved when the resolution scale is increased. However, it requires more computational time.

In (4.7), the LU factorization of involves flops. The cost of the proposed algorithm is the sum of the cost of WinSolver, , and the cost of WinTree, (see Appendix ). Since , the costs of WinSolver and can be rewritten as and , respectively. Thus, the total cost of the proposed algorithm is (5.2) Table 1 and Figure 5 show that the computational cost of the proposed algorithm is significantly less than the Kronecker product method, and that the flop counts are increasing rapidly with resolution scale. As the resolution scale grows, the flop counts of WinTree is increasing more rapidly than that of WinSolver since the sizes of matrices , and increase exponentially.

Table 1: Flop counts for various sizes of the matrix A and resolution scales.

Figure 5: Log plot of flop counts for the Kronecker product method and the proposed method.

6. Conclusions

An efficient computational method was presented for state space analysis of singular systems via Haar wavelets. The problem was formulated as a generalized Sylvester matrix equation. We presented an explicit expression for the inverse of the Haar matrix and a combined preorder and postorder traversal algorithm to solve the problem more effectively. The full-order generalized Sylvester matrix equation was solved in terms of the solutions of simple linear matrix equations by the proposed algorithm. The efficiency of the proposed method was demonstrated by a numerical example.

Appendices

A. Formula for

In this appendix, we derive a formula for . By using (3.13), (3.9), and (3.10), we can write (A.1) Since and , the above equation is rewritten as (A.2)

B. Flop Counts of the Combined Preorder and Postorder Traversal Algorithm

In this appendix, we show that the computational cost for the combined preorder and postorder traversal algorithm described in Section 4.2 can be obtained as follows:

(1) WinSolve

The total iteration number of “” is . Thus, WinSolve involves

(2) WinTree

(see Table 2).

Table 2

Therefore, the computational cost for WinTree can be calculated by (A.1)

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