On the transformation of a nonlinear discrete-time input – output system into the strong row-reduced form

The paper addresses the problem of the transformation of nonlinear discrete-time systems, described by implicit higherorder difference equations, into the strong row-reduced form. The motivating example illustrates the phenomenon that sometimes equations in the row-reduced form may contain higher-order shifts of output variables than the corresponding row degrees. This means that, in general, linear transformations of equations are not enough for transforming equations into the strong row-reduced form. Therefore, in this paper we study the possibility of using local nonlinear transformations to reduce the order of a system. A constructive (up to the solution of a system of partial differential equations) step-by-step algorithm is provided. It is followed by several illustrative examples.


INTRODUCTION
The transformation of a set of higher-order nonlinear input-output (i/o) equations into a row-reduced form is an important problem in control theory for several reasons.First, the row-reduced form may be seen as an intermediate step towards a so-called doubly-reduced (i.e., both row-and column-reduced) or Popov form (see [1,2,6,7]).Thus, an algorithm for transforming arbitrary system equations into the row-reduced form is necessary to obtain the double-reduced or Popov form.Second, this form can serve as a good starting point for the application of realization procedures (see, for example, [4] and the references therein).The realization problem is a fundamental research topic in nonlinear control theory, which studies the possibility of transforming a set of higher-order i/o difference equations into a classical state-space form.Moreover, the sum of row degrees of the system in the row-reduced form defines the order of the realization, i.e., the number of state variables.In addition, the form under study also shows explicitly when not all of the inputs are free (independent) variables, or, when the system is not right invertible since certain functions of outputs are not affected by controls.
The problem has been studied by various authors, both in continuous-time [10] and discrete-time cases [3,5].The results of this paper can be understood as an extension of those presented in [3,5,9].
In [3] a specific version of a leading coefficient matrix was assumed with m 1 = • • • = m p = 0 (see Eq. (4) below).A particular solution, based on linear i/o equivalence transformations, was proposed in [5].In such a case the original and transformed equations are related to each other through the transformation over the field of meromorphic functions in system variables by applying to the original set of equations an operator defined in terms of a unimodular polynomial matrix whose indeterminate may be interpreted as a forward-shift operator.The transformation in [5] was found from the variational (i.e., globally linearized) system description, presented in terms of the polynomial matrices and differentials of system variables.This (polynomial) description was used to calculate a unimodular matrix by means of elementary matrix operations.Finally, the unimodular transformation matrix was applied to the original system of equations to find its row-reduced form.This approach works well in many situations.However, there exist numerous examples when this method leads to equations of row-reduced form, which contain higher-order shifts of output variables than their row degrees.The row-reducedness property as defined in [5] is actually a property of linearized equations (differential one-forms describing the linearized equations) and not the property of equations themselves.In particular, the set of i/o equations is called row-reduced if and only if their linearization is row-reduced.However, in general, the row-reducedness property cannot be easily translated back to the original system equations.This fact motivated us to introduce a new stronger definition of a row-reduced system based on row orders (see Definitions 4 and 6).In [9] local nonlinear transformations were used to transform a continuous-time system to the row-reduced form.Note that the continuous-time case is different from its discrete counterpart, since the verification whether a system is in the row-reduced form or not is done in a slightly different manner (see Example 1 below).In addition, the results for the continuous-time case from [9] rely on a special case of the rank theorem and are valid locally under certain constant rank assumptions.Finally, it should be mentioned that in this paper we work with system equations and not with linearized description as in [5].
Based on the reasons mentioned above, the main goals of this paper are: to present a new definition of the strong row-reducedness property of a system and to specify a larger class of local nonlinear i/o equivalence transformations.For the first purpose we combine two ideas from [3] and [5].Moreover, we adopt some ideas from [9] to achieve the second goal.Of course, even though the existence of such nonlinear transformations can be proven, their computation may be a difficult task.
The paper is organized as follows.Section 2 recalls the main notions and definitions regarding the i/o equivalence and row-reduced form.It is followed by a motivating example that illustrates the difficulties one may face when applying the results of [5].Section 3 is devoted to local nonlinear i/o equivalence transformations and presents also the algorithm allowing one to transform a set of i/o equations into the strong row-reduced form.A number of illustrative examples are given in Section 4. Concluding remarks are drawn in the final section.

INPUT-OUTPUT EQUIVALENCE AND ROW-REDUCED FORM
Consider a nonlinear discrete-time multi-input multi-output (MIMO) control system, described by the set of implicit higher-order i/o difference equations ϕ i (y(t), y(t + 1), . . . ,y(t + n), u(t), u(t + 1), . . . ,u(t + n)) = 0, i = 1, . . . ,p, ( where t ∈ Z, u(t) ∈ R m is a vector of input variables, y(t) ∈ R p is a vector of output variables, and ϕ i is a meromorphic function.Sometimes, to simplify the exposition, the abridged notations are used.In particular, if a time-dependent variable is denoted as ξ (t), then ξ [k] (t) stands for the kth-step forward time shift ξ (t + k) and ξ [−l] (t) for the lth-step backward time shift ξ (t − l) with k, l ∈ Z + .Furthermore, we may leave the time argument t to make the notation even more compact, i.e., ξ := ξ (t).
Recall briefly the algebraic formalism from [5] that is used in this paper.Let A be the ring of analytic functions in a finite number of variables from the sets Y = {y i := y i and u [0] j := u j .For the function F, depending on variables from Y and U , the forward-shift operator σ : A → A is defined as follows: , y, y [1] , . . ., u [−1] , u, u [1] , . . .
Then, σ (y , and σ −1 (y for i = 1, . . ., p, j = 1, . . ., m, and k, l ∈ Z.Note that A is a difference ring with the shift operator, being an automorphism. Let S be a multiplicative subset of the ring A , meaning that 1 ∈ S , 0 ̸ ∈ S and if α ∈ S and β ∈ S , then αβ ∈ S .Assume that S is invariant with respect to both σ and σ −1 .Then, A S := S −1 A = {α/β | α ∈ A and β ∈ S } defines the localization of the ring A with respect to S .Observe that A S is an inversive difference ring with the shift operator σ given by σ (α/β ) := σ (α)/σ (β ) and S may be interpreted as a subset of A S due to the natural injection α → α/1.
Let Φ = {ϕ 1 , . . ., ϕ p } be a finite subset of A S .Note that Φ may be interpreted as a system of implicit i/o equations.Let I S := ⟨Φ⟩ S be the smallest ideal of A S that contains all forward and backward shifts of ϕ i , i.e., I S is generated by {σ k (ϕ i ) | i = 1, . . ., p, k ∈ Z}.Note that I S is a difference ideal, since it is closed with respect to all shifts of ϕ i .Observe that Φ may be considered as a subset of S −1 A for some other multiplicative set S .For that reason we put S in the notation of the ideal I S .
Assumption 2. I S is proper, i.e., different from the entire ring.
Properness of the ideal I S is equivalent to the condition S ∩ I S = / 0. In particular, numerators of ϕ i do not belong to S .
Observe that S is constructed for system (1).However, when applying equivalence transformations with Eqs (1), S may have to be extended to S by including possible expressions that do not equal zero, restricting in this way the domain of definition.When we start, some functions ϕ i in (1) may have denominators that, together with their forward/backward shifts and powers, should be included in the set S .If the functions are analytic, one may set S := {1}, meaning that S −1 A = A .Of course, additional denominators that show up in the row-reduction should also be included in S together with their shifts and powers.That is, we extend our initial S by adding an infinite number of elements.The infinite S can be briefly described by its generator S 0 .The set S 0 generates S if each element of S can be obtained from a finite number of elements of S 0 by applying a finite number of multiplications and backward/forward shifts to these elements.
Let A S /I S be the quotient ring.It consists of cosets φ = φ + I S for φ ∈ A S .We define addition and multiplication in this new ring by φ + ψ := φ + ψ and φ • ψ := φ • ψ.These definitions do not depend on the choice of a representative in a coset.Since I S is a prime ideal, A S /I S is an integral ring.Now we can redefine σ on A S /I S (denoted by σ Φ to indicate the dependence on Φ) as σ Φ ( φ) = σ (φ).The operator σ Φ is well defined and bijective, so σ −1 Φ is well defined on A S /I S .Let Q Φ S denote the field of fractions of the ring A S /I S .Since σ Φ can be naturally extended to the field of fractions, Q Φ S is an inversive difference field.Definition 1.The sequence of pairs {(u(t), y(t)),t ≥ 0} is called a solution of (1) if, for any t ≥ 0, u(t) and y(t) satisfy the equations ϕ i (y(t), . . . ,y(t + n), u(t), . . . ,u(t + n)) = 0, i = 1, . . . ,p. Definition 2. Two systems of the form (1) are called i/o equivalent if their solutions coincide.Definition 3.An i/o equivalence transformation for system (1) is an invertible transformation of the system equations to another set of equations of the form (1), being i/o equivalent with the original system equations.Definition 4. The row orders 1 T , in the output y, denoted by µ i , are the largest integers such that in (1) for i = 1, . . ., p and some j ∈ {1, . . ., p}.
In other words, µ i is the highest forward-shift of the output component 2 , appearing nontrivially in ϕ i .Next, we set µ := (µ 1 , . . . ,µ p ) and define the p × p-dimensional matrix M µ as the matrix with the (i, j)th element given by ∂ ϕ i /∂ y for i, j = 1, . . ., p.
In the continuous-time case the matrix M µ is enough to verify whether the original system is in the row-reduced form or not (see [10]).However, in the case of discrete-time systems one has to multiply M µ by certain diagonal matrix (as defined below) from the left (see the explanation in Example 1).Let N µ = max µ i and m = (m 1 , . . . ,m p ) with m i = N µ − µ i for i = 1, . . . ,p. Define σ 0 := id A S and is called the leading coefficient matrix of system (1).(5) in which the second equation is a forward shifted version of the first.Obviously, we do not want to call this set of equations to be in a row-reduced form.Find, according to Definition 4, the row orders of system (5) as µ = (1, 2).Then, by ( 3) and rank A S M µ = 2 indicating that Eqs (5) are independent.However, for (5), N µ = max{µ 1 , µ 2 } = 2 and m = (1, 0).Compute, according to (4), Observe that rank A S L µ = 1 as expected. 1The notion of row degrees is kept for the indices ρ i , defined in [5], as the largest integers such that ∂ ϕ i /∂ y Observe that the indices µ i are greater than or equal to ρ i . 2 If ϕ i does not depend on y or ∂ ϕ i /∂ y [µ i ] j does not exist, we set µ i = −1.Recall that µ i = 0 corresponds to the case when ϕ i depends on y j only and not on its shifts.

Motivating example
Let us study the motivating example that illustrates difficulties one may face when applying the approach proposed in [5].Recall that the elements of the field Q Φ S are not fractions of functions but abstract fractions (equivalence classes of functions) since the construction of Q Φ S is based on the quotient ring A S /I S .In the following example we use the (simplest) representatives of these equivalence classes.
Example 2. Consider the set of i/o equations and perform the calculations according to the approach from [5].Since there are no denominators in (6), we set S := {1}.Next, one can find the row degrees for system (6) as ρ = {ρ 1 , ρ 2 } = {3, 2}.Then we have to reorder equations T with respect to the row degrees starting from the lowest that can be done by means of multiplication by the permutation matrix ] .
Note that we have to extend the set S as S := {1, σ k (cos β [1] ) | k ∈ Z} and σ k (cos β [1] ) ̸ ∈ I S .Now the transformation matrix U(z) can be found as Finally, compute the row-reduced form of the system by applying the transformation operator U(z), denoted by the symbol , to functions in the original system description as follows: where the application of z to a function is defined as z ξ = σ (ξ ).It is easy to observe that the second (transformed) function still depends on y Observe that on the level of linearized system equations (in terms of the one-forms) the transformation U(z) results in the row-reduced form.In fact, the multiplication of the linearized system description ] T by the transformation matrix U(z) from the left yields One may also observe that the leading coefficient matrix of P(z) has the full rank, i.e., However, according to Definition 6, the transformed system ( 7) is not in the strong row-reduced form since and therefore, rank A S L µ = 1.Note that the difference stems from the difference between the row degrees and the row orders: the row orders µ i , defined by (4), are either greater than or equal to the row degrees ρ i .This comes from the fact that in computation of ρ i we take the values of the elements from I S equal to zero.Note that, according to the definition of row degrees, we have to find such ρ i for which the derivative of a function does not belong to the ideal, i.e., ∂ ϕ i /∂ y j ̸ ∈ I S .For example, the partial derivatives of (8) with respect to y [2] 1 and y [3] 1 are −1 and − tan 2 (β [1] ), respectively.Since tan(β [1] ) = sin(β [1] )/ cos(β [1] ) and sin(β [1] ) ∈ I S (the shifted version of the second equation in ( 6)), for Eq. ( 8) we have µ 2 = 3 and ρ 2 = 2.To conclude, this example points to the fact that sometimes the linear transformations from [5] cannot transform the system equations into the strong row-reduced form.

NONLINEAR INPUT-OUTPUT EQUIVALENCE TRANSFORMATIONS
The following lemma has been proved in [9, Lemma 6.2] and corrected in [8] in the smooth (C ∞ ) case.It will be used to construct nonlinear equivalence transformations.Lemma 1.Let f 1 , . . ., f p be analytic functions on an analytic manifold Ω and depending analytically on the parameter ξ ∈ R k for some k such that the dimension of the codistribution span{d f 1 , . . ., d f p } is constant.Suppose there exist analytic functions λ 2 , . . . ,λ p on Ω depending on ξ such that

Then for fixed ξ there exists an analytic function F
The functions F ξ can be chosen in such a way that they depend analytically on ξ .
Proof.The proof in the analytic case is exactly the same as in the smooth case [9].
The function F ξ can be found by solving a certain system of partial differential equations.
If the rank of the matrix L µ of the new i/o system equals p, we have transformed the system equations into the i/o equivalent strong row-reduced form.Otherwise, we may repeat the above procedure.Note that at each step the sum of row degrees decreases, converging this way to some constant number greater than −p.After a finite number of steps we either obtain matrix L µ with rank p or obtain matrix L µ for which (possibly after permutation of the rows) the first p ′ rows are independent, while the last p − p ′′ rows are zero.In the latter case we obtain the i/o equations of the form where S is an extended multiplicative set obtained during the transformation procedure.Note that ϕ ⋆ p ′′ +1 , . . . ,ϕ ⋆ p depend only on input variables or are zeros.The above considerations give the proof of the following theorem.
Theorem 1.Consider a set of higher-order difference equations (1).Under Assumption 3 there exists a (local) equivalence transformation that allows one to transform the set of equations (1) into a strong rowreduced form, possibly together with some equations which are trivially satisfied, or define restrictions on input or output signals (20).
Using the theoretical considerations given above, we are ready to present an algorithm for transforming the set of i/o equations into the strong row-reduced form.
Step 0. Start of Algorithm.
Step 3. Reorder the elements in Φ with respect to the row orders starting from the lowest.This operation corresponds to the multiplication of the matrix Φ by a permutation matrix R from the left that can be obtained by (repeated) swapping of the ith and the jth rows of the identity matrix I p , resulting in a new matrix Φ.Then, reorder the elements of µ by multiplying it by the same permutation matrix from the right, i.e., µ = µR.
Step 6. Check whether rank A L µ = p.In case of an affirmative answer go to Step 9; otherwise, go to Step 7.
Step 7. Check whether Assumption 3 holds or not.If λ i+1 ̸ ∈ I S , solve Eq. ( 12) to find λ * υ and go to Step 8; otherwise it is not possible to complete the algorithm.
Step 8. From Lemma 1 and Proposition 1 it follows that there exists a function F ξ satisfying (17) and (18), which defines the transformation.Apply the obtained transformation F to Φ, resulting in the new system and proceed to Step 1.
Step 9.The system is in the strong row-reduced form.End of the algorithm.

EXAMPLES
Several illustrative examples are presented in this section.The first example shows that the approach proposed in this paper in some cases yields the same linear i/o equivalence transformation as the method from [5].The next two examples address the different aspects of the motivating example.The first of them shows how to calculate a local nonlinear transformation for system (6) from Example 2 which transforms equations into the strong row-reduced form.Recall that this is impossible using the linear transformation, since the equations obtained after the application of the method from [5] are in the row-reduced form (as expected), but not in the strong row-reduced form.In the next example we take these transformed equations (obtained after the application of the linear transformation) as a starting point and explain how to find a suitable nonlinear transformation.The final example is again intended to illustrate the applicability of a nonlinear transformation.The key moment here is that sometimes it is necessary to shift the elements of the leading coefficient matrix back to get the system of partial differential equations (17).Moreover, in this example the transformation depends on the parameter ξ .
Since ϕ 2 contains the denominators y 3 , we set S 0 = {1, y 2 , y 3 }.Then A S = S −1 A is a localization of the ring A with respect to the multiplicative subset S generated by S 0 .Let Φ = [ ϕ 1 ϕ 2 ϕ 3 ] T and compute, according to Definition 4, the row orders as µ = (µ 1 , µ 2 , µ 3 ) = (1, 3, 1).In order to permute the second and third elements of the vector Φ, it has to be multiplied by the permutation matrix as follows: Hence, we have µ = µR = (1, 1, 3).Set N µ := max µ i = 3, yielding m = (2, 2, 0).Using (3), find the matrix M µ as and the leading coefficient matrix L µ , according to (4), as follows: One can easily check that rank A S L µ = 2. Rows of the matrix L µ are linearly dependent.Moreover, the third row is a linear combination of the first and second rows, yielding It is easy to see that λ 3 = y [2] 2 y [2] 3 ̸ ∈ I S .Solving (22) with respect to λ k , for k = 1, 2, 3 and taking into account that γ = 0, we get λ * 1 = −y [2] 1 /y [2] 3 , λ * 2 = −u 2 /y [2] 2 , λ * 3 = 1.Note that the set S remains the same Finally, it is interesting to observe that when λ * υ , υ = 1, . . ., i do not depend on y [µ i+1 ] , then (17) yields the linear solution (see Remark 1).However, this solution does not necessarily coincide with the one obtained via the approach from [5], since we rely on a different definition of row-reducedness involving row orders.Though, this observation points to a link between approaches presented in this paper and that from [5], also, it raises a problem of finding a more general set of nonlinear i/o equivalence transformations, including transformations from [5] as a special case.This is the subject for future research.

Example 3 .
Consider the set of i/o equations ϕ