1 Introduction
In this paper we study second order discretetime descriptor systems of the form
(1) 
We will also discuss the initial value problem of the associated singular difference equation (SiDE)
(2) 
together with some given initial conditions
(3) 
Here the solution/state , the inhomogeneity , the input , where , and for each . Three matrix sequences , , take values in , and takes values in . We notice that all the results in this paper also can be carried over to the complex case and they can also be easily extended to systems of higher order. However, for sake of simplicity and because this is the most important case in practice, we restrict ourselves to the case of real and second order systems.
The SiDE (2), on one hand, can be considered as the resulting equation obtained by finite difference or discretization of some continuoustime DAEs or constrained PDEs. On the other hand, there are also many models/applications in reallife, which lead to SiDEs, for example Leotief economic models, biological backward Leslie model, etc, see e.g. Aga00 ; Ela13 ; Kel01 ; Lue79 .
While both DAEs and SiDEs of first order have been wellstudied from both theoretical and numerical points of view, the same maturity has not been reached for higher order systems. In the classical literature for regular difference equations, e.g. Aga00 ; Ela13 ; Kel01 , usually new variables are introduced to represent some chosen derivatives of the state variable such that a high order system can be reformulated as a first order one. Unfortunately for singular systems, this approach may induce some substantial disadvantages. As have been fully discussed in LosM08 ; MehS06 for continuoustime systems, these disadvantages include: (1st) increase the index of the singular system, and therefore the complexity of a numerical method to solve it; (2nd) increase the computational effort due to the bigger size of a new system; (3rd) affect the controllability/observability of the corresponding descriptor system since there exist situations where a new system is uncontrollable while the original one is. Therefore, the algebraic approach, which treats the system directly without reformulating it, has been presented in LosM08 ; MehS06 ; Wun06 ; Wun08 in order to overcome the disadvantages mentioned above. Nevertheless, even for second order SiDEs, this method has not yet been considered.
Another motivation of this work comes from recent research on the stability analysis of high order discretetime systems with timedependent coefficients LinNT16 ; MehT15 . In these works, systems are supposed to be given in either strangenessfree form or linear statespace form. This, however, is not always the case in applications, and hence, a reformulation procedure would be required.
Therefore, the main aim of this article is to set up a comparable framework for second order SiDEs and for discretetime descriptor systems as well. It is worth marking that the algebraic method proposed in LosM08 ; MehS06 is applicable theoretically but not numerically due to two reasons: (1st) The condensed forms of the matrix coefficients are really big and complicated; (2nd) The system’s transformations are not orthogonal, and hence, not numerically stable. In this work, we will modify this method to make it more concise and also computable in a stable way.
The outline of this paper is as follows. After giving some auxiliary results in Section 2, in Sections 3 and 4 we consecutively introduce index reduction procedures for SiDEs and for descriptor systems.
A desired strangenessfree form and a constructive algorithm to get it will be presented in Theorem 3.8 and Algorithm 1 (Section 3).
A resulting system from this algorithm allows us to fully analyze structural properties such as existence and uniqueness of a solution, consistency and hidden constraints, etc.
For descriptor systems, where feedback also takes part in the regularization/solution procedure, besides the strangenessfree form presented in Theorem 4.7, regularization via first order feedback is discussed in Theorem 4.4.
In order to get stable numerical solutions of these systems, in Section 5 we study the difference array approach in Algorithm 2 and Theorem 5.6 aiming at bringing out the strangenessfree form of a given system.
Finally, we finish with some conclusions.
2 Preliminaries
In the following example we demonstrate some difficulties that may arise in the analysis of second order SiDEs.
Example 2.1.
Consider the following second order descriptor system, motivated from Example 2, MehS06 .
(4) 
Clearly, from the second equation , we can shift the time forward to obtain
Inserting these into the first equation of (4), we find out the hidden constraint
Consequently, we deduce the following system, which possess a unique solution
Let in this new system, we obtain a constraint that must obey. This example showed us some important facts. Firstly, one can use some shift operators and rowmanipulation (Gaussian eliminations) to derive hidden constraints. Secondly, a solution only exists if initial conditions and an input fulfill certain consistency conditions. Finally, in this example the solution depends on the future input. This property is called noncausality and cannot happen in the case of regular difference equations.
For matrices , , the pair is said to have no hidden redundancy if
Otherwise, is said to have hidden redundancy
. The geometrical meaning of this concept is that the intersection space
contains only the zerovector
. Here, for any given matrix , by we denote its transpose. We denote by (resp., ) the real vector space spanned by the rows of (resp., rows of ).Lemma 2.2.
(HaM12 ) Consider full row rank matrices , , and assume that for none of the matrix pairs has a hidden redundancy. Then has full row rank.
Lemma 2.3 below will be very useful later for our analysis, in order to remove hidden redundancy in the coefficients of (2).
Lemma 2.3.
Consider two matrix sequences , which take values in and , respectively. Furthermore, assume that they satisfy the constant rank assumptions
Then there exists a matrix sequence in such that the following conditions hold.

, , ,

is orthogonal, and ,

has full row rank, and the pair has no hidden redundancy.
Proof.
Since the proof is essentially the same as in the continuoustime case, we refer the interested readers to the proof of Lemma 2.7, HaMS14 . ∎
Remark 2.4.
i) In the special case, where has full row rank and the pair has no hidden redundancy, we will adapt the notation of an empty matrix and take
, , .
ii) Furthermore, we notice, that whenever the smallest singular value of
and the largest one do not differ very much in size, then we can stably compute the matrix . Both matrices and will play the key role in our index reduction procedure presented in the next section.For any given matrix , by
we denote an orthogonal matrix whose columns span the left null space of
. By we denote an orthogonal matrix whose columns span the vector space . From basic linear algebra, we have the following lemma.Lemma 2.5.
The matrix is nonsingular, the matrix has full row rank, and the following identity holds
Proof.
A simple proof can be found, for example, in GolV96 . ∎
3 Strangenessindex of second order SiDEs
In this section, we study the solvability analysis of the second order SiDE (2) and that of its corresponding IVP (2)–(3). Many regularization procedures and their associated index notions have been proposed for first order systems, see the survey Meh13 and the references therein. Nevertheless, for high order systems, only the strangenessindex has been proposed in the continuoustime case in MehS06 ; Wun08 . Thus, it is our purpose to construct a comparable regularization and index concept for discretetime system (2).
Let
we call the behavior matrix sequence of system (2). Thus, (2) can be rewritten as
(5) 
Clearly, by scaling (2) with a pointwise nonsingular matrix sequence in , we obtain a new system
(6) 
without changing the solution space. This motivates the following definition.
Definition 3.1.
Lemma 3.2.
Consider the behavior matrix sequence of system (2). Then for all , we have that
(7) 
where the matrices , , have full row rank. Here, the numbers , , , are rowsizes of the block rows of . Furthermore, these numbers are invariant under left equivalent transformations. Thus, we can call them the local characteristic invariants of the SiDE (2).
Proof.
Analogous to the continuoustime case, we will apply an algebraic approach (see Bru09 ; MehS06 ), which aims to reformulate (2) into a socalled strangenessfree form, as stated in the following definition.
Definition 3.3.
In order to perform an algebraic approach, an additional assumption below is usually needed.
Assumption 3.4.
Assume that the local characteristic invariants , , become global, i.e., they are constant for all . Furthermore, assume that two matrix sequences and have constant rank for all .
Remark 3.5.
Remark 3.6.
In (8), the quantities , , and are dimensions of the second order dynamics part, the first order dynamics part, and the algebraic (zero order) part, respectively. Furthermore,
is exactly the degree of freedoms.
Let us call the number the upper rank of system (2). Clearly, is invariant under left equivalence transformations. Rewrite (5) block rowwise, we obtain the following system for all .
(10a)  
(10b)  
(10c)  
(10d) 
Since the matrices , , have full row rank, the number of scalar difference equations of order (resp. , and ) in (2) is exactly (resp. and ), while is the number of redundant equations. Now we are able to define the shiftforward operator , which acts on some or whole equations of system (10). This operator maps each equation of system (10) at the time instant to the equation itself at the time , for example
(11) 
Clearly, under Assumption 3.4, this shift operator can be applied to equations of system (10). In order to reveal all hidden constraints of (10) we propose the idea that for each , we use equations of order less than to reduce the number of scalar equations of order . This task will be performed as follows. Firstly, by applying Lemma 2.3 to two matrix pairs and , we obtain matrix sequences , , and , , of appropriate sizes such that for all , the following conditions hold true.

For , the matrices are orthogonal.

The following identities hold true.
(12a) (12b) 
Both matrix pairs , have no hidden redundancy.
Lemma 3.7.
Proof.
The proof is not too difficult but rather lengthy and technical, so we leave it to A. ∎
Consider system (13), we see that the upper rank of the behavior matrix is
In conclusion, after performing a socalled index reduction step, which passes from (10) to (13), we have reduced the upper rank at least by . Continue in this fashion until , we obtain the following algorithm.
Input: The SiDE (2) and its behavior form (5).
Output: A strangenessfree SiDE of the form (8) and the strangenessindex .
After each index reduction step the upper rank has been decreased at least by , so Algorithm 1 terminates after a finite number of iterations, which will be called the strangenessindex of the SiDE (2).
Theorem 3.8.
Consider the SiDE (2) and assume that Assumption 3.4 is satisfied for any and any considered within the loop, such that the strangenessindex is welldefined by Algorithm 1. Then the SiDE (2) has the same solution set as the strangenessfree SiDE
(14) 
where the matrix has full row rank for all , and the functions and consist of the components of (at most).
Proof.
To illustrate Algorithm 1, we consider the following example.
Example 3.9.
Given a parameter , we consider the second order SiDE
(15) 
for all . Fortunately, the behavior matrix
is already in the block diagonal form, so we do not need to perform Step 2 in Algorithm 1. Furthermore, all constant rank conditions required in Assumption 3.4 are satisfied. We observe that
By directly verifying, we see that the matrix pair has hidden redundancy, while the pair does not. Now we choose , , , . Notice that the fact is nonempty leads to the appearance of . Furthermore, the resulting system (13) reads
(16) 
Since the leading coefficient matrix associated with becomes zero, so for notational convenience we do not write this term.
Go back to Step 3, we see that the following two cases may happen.
i) If , then Algorithm 1 terminates here, and the strangenessindex is . The number of timeshift appear in the inhomogeneity in the strangenessfree formulation (16) is .
ii) If , then the matrix pair have hidden redundancy. Now we choose , , . The resulting system (13) now reads
(17) 
Algorithm 1 terminates here, and the strangenessindex is . However, the number of timeshifts appearing in the inhomogeneity in the strangenessfree formulation (3.9) remains .
Corollary 3.10.
Under the assumption of Theorem 3.8, the following statements hold true.
Another direct consequence of Theorem 5.3 is that we can obtain an inherent regular difference equation as follows.
Corollary 3.11.
Remark 3.12.
Unlike the procedures in Bru09 ; LosM08 ; MehS06 , we do not change the variable . This approach permits us to simplify significantly the condensed forms in these references. We emphasize that as in (3.5), we only require five constant rank conditions within one step of index reduction, instead of seven as in MehS06 . Therefore, by this way the domain of application for SiDEs (and also for DAEs in the continuoustime case) will be enlarged. This approach is also useful for the control analysis of the descriptor system (1), as will be seen in the next section.
Remark 3.13.
i) Within one loop of Algorithm 1, for each
, we have used four Singular Value Decompositions (SVDs) to remove the hidden redundancies in two matrix pairs. The total cost depends on the problems itself, i. e., depending on sizes of the matrix pairs which applied SVDs. Nevertheless, it does not exceed
.ii) Unfortunately, since , , are not orthogonal, in general Algorithm 1 could not be stably implemented. For the numerical solution to the IVP (2)(3), we will consider a suitable numerical scheme in Section 5.
4 Regularization of second order descriptor systems
Based on the index reduction procedure for SiDEs in Section 3, in this section we construct the strangenessindex concept for the descriptor system (1). The solvability analysis for first order descriptor systems with variable coefficients have been carefully discussed in ByeKM97 ; KunMR01 ; Rat97 . Nevertheless, for second order descriptor systems, this problem has been rarely considered. We refer the interested readers to LosM08 ; Wun08 for continuoustime systems.
It is well known that in regularization procedures of continuoustime systems, one should avoid differentiating equations that involve an input function, due to the fact that it may not be differentiable. We will also keep this spirit, and hence, will not shift any equation that involve an input function, since it may destroy the causality of the considered system, as in Example 2.1. Instead of it, we will also incorporate proportional state and first order feedback within each index reduction step of the regularization procedure, as will be seen later. Now let us present two auxiliary lemmas, which will be very useful later.
Lemma 4.1.
Given four matrices , , in and in . Let us consider the following matrices whose columns span orthogonal bases of the associated vector spaces
Then the following assertions hold true.

The matrices , , and are orthogonal.

The matrices , , , , and have full row rank.

Moreover, there exists an orthogonal matrix such that
(19) where the matrices , , , , have full row rank.
Proof.
The first two claims followed directly from Lemma 2.5. To prove the third claim, we construct the desired matrix as follows
Comments
There are no comments yet.