Popov form and the explicit equations of inverse systems

The paper addresses the invertibility problem for discrete-time nonlinear control systems, described by the input–output equations. The necessary and sufficient conditions for the existence of left and right inverse systems are given. The explicit equations of inverse systems are found by transforming the system equations into the strong Popov form with respect to inputs. The results are obtained under the assumption that the equations are transformable into the strong Popov form using linear equivalence transformations over the field of meromorphic functions.


INTRODUCTION
In [7] the problem of right inversion is addressed for nonlinear control systems, described by the set of inputoutput (i/o) difference equations. The solution of the problem, that is necessary and sufficient conditions of right invertibility, is given based on the inversion algorithm (IA), extended for this class of systems. The IA is traditionally expressed in a form involving the implicit function theorem (IFT). However, in [7] the IA is presented in terms of differential one-forms, exactly like in [5] for nonlinear systems, described by the state equations. Therefore, the algorithm does not use the IFT. This form of the IA is certainly efficient for checking invertibility. To find the explicit equations of the right inverse, one has to integrate the set of one-forms, obtained at the last step of the IA, which may be a difficult task.
In this paper an alternative approach is suggested, based on the strong Popov form with respect to the control variables of the set of i/o equations. One can easily find the explicit equations of the inverse system when the set of original equations will be transformed into the strong Popov form with respect to the control variable. Note that our results are obtained under the assumption that the equations are transformable into the strong Popov form using linear equivalence transformations, see [9]. Our results address also the left inversion problem, not studied so far for this class of systems to the authors' knowledge. The new approach is computationally more efficient and transparent, though both approaches result, in principle, in the same equations of the inverse system 1 . As shown via the example, our results agree with those developed for nonlinear systems, described in terms of the state equations. However, our results are more general since not all nonlinear i/o equations are realizable in the state space form. According to our knowledge, the approach is also new for linear systems.

I/o equations
Consider a discrete-time multi-input multi-output nonlinear system, described by the explicit set of higherorder difference equations, relating the inputs u k , k = 1, . . . , m, the outputs y i , i = 1, . . . , p, and a finite number of their time shifts: y j (t), . . . , y j (t + n i j ), u ι (t), . . . , u ι (t + s iι ) where j = 1, . . . , p, ι = 1, . . . , m, and ϕ i are real meromorphic functions. The word 'explicit' means that the variable y i (t + n i ) does not appear on the right-hand side of the ith equation, i.e. n ii < n i . It is assumed that system (1) is strictly causal, i.e. s iι < n i . The functions ϕ i are defined on an open and dense subset of R (n+1)(p+m) , whereas n = max n i . (1) is said to be in the strong Popov form with respect to the output if (a) n 1 n 2 · · · n p ; (b) for all ϕ i , i = 1, . . . , p the following conditions hold:

Definition 1. The set of i/o equations
Compared with the definition of the strong Popov form for implicit equations in [2], we have made in Definition 1 technical simplification j i = i for the explicit equation (1). With this additional assumption condition (v) in the strong Popov form (see [2]) is always satisfied. This assumption allows us to avoid double indices and does not bring along any restrictions since this is always doable by renumbering the output coordinates 2 , see also Remark 3. Assumption 1. System (1) is assumed to be in the strong Popov form with respect to output.
We associate a multiplicative set S Σ with system (1). If (1) involves any denominators, then these denominators have to be included in S Σ together with their shifts and powers. The typically infinite set S Σ can be generated by a finite generator set S 0 Σ . The set S 0 Σ generates S Σ if each element of S Σ can be obtained from a finite number of elements of S 0 Σ by applying a finite number of multiplications and backward and forward shifts to these elements. If (1) does not include any denominators, then we set S Σ = {1} and only in this case S Σ is a finite set (i.e. S Σ = S 0 Σ ). Hereinafter we use the notation ζ for a variable ζ (t), ζ [k] for its k-step time shift ζ (t + k), k ∈ Z. In such notation (1) takes the form where ι = 1, . . . , m and j = 1, . . . p. 1 It has to be mentioned that both the inversion algorithm and transformation of equations into the strong Popov form allow some freedom in their choices, not affecting the invertibility property but possibly the inverse system equations if m ̸ = p. 2 Note that renumbering the output coordinates is not a row operation on globally linearized system equations.
Definition 2. I/o equations (3) are said to be in the strong Popov form with respect to the input if (a) σ 1 σ 2 · · · σ µ ; (b) for all χ k , k = 1, . . . , µ the following conditions hold: In Definition 2, like in Definition 1, we have made a technical simplification for the explicit equations that in the kth equation of Σ u the variable u k appears with the highest shift.
With system Σ, described by equations (2), we associate a vector function Φ : The system Σ defines the inversive difference field Q Σ with the shift operator δ Σ . In particular, the shift of y is defined by the right-hand side of equation (2), see more in [2] 3 . Each element of Q Σ is the image of a meromorphic function under the map e Σ . Basically the map e Σ allows us to exclude (or include) the zeros, defined by equations (2), from (into) the elements of the field Q Σ , and in this way to find the simplest representatives of the functions in Q Σ . See [2] for a precise definition and Example 1 below.

Non-commutative polynomials
The field Q Σ and the shift operator δ Σ induce the ring of non-commutative polynomials in a variable Z over Q Σ , denoted by Q Σ [Z; δ Σ ]. The multiplication is defined by the linear extension of the following rules: Z · a := (δ Σ a)Z and a · Z := aZ, where a ∈ Q Σ and δ Σ a means δ Σ evaluated at a (so for example (aZ µ ) · (bZ ν ) = a(δ  Definition 4. [9,11] A polynomial matrix W ∈ Q Σ [Z, δ Σ ] p×q with non-zero rows is called row-reduced if its leading row coefficient matrix L(W ) has full row rank 4 over the field Q Σ . If W contains zero rows, then W is called row-reduced if its submatrix consisting of non-zero rows is row-reduced. Definition 5. [11] Matrix W ∈ Q Σ [Z, δ Σ ] p×q is in the Popov form if W is row-reduced with the rows sorted with respect to their degrees (σ 1 · · · σ p ) and for all non-zero rows w i• there is a column index j i (called the pivot index) such that 3 In [2] the notations Q Φ S and δ Φ were used respectively for Q Σ and δ Σ . 4 The matrix W ∈ Q Σ [Z, δ Σ ] p×q is said to have full row rank if rankW = min(p, q).

Linearized i/o equations
Our goal is to represent system (2) in terms of polynomials from Q Σ [Z; δ Σ ]. For that purpose we apply the differential operation d to equations (2) to obtain j and Z β du ι := du [β ] ι allows us to rewrite (4) as where Example 1. Consider the system taken from [9]: By (5) and (6) the matrix Next we define the equivalence classP. To find one of the simplest representatives of the class, we take into account system equations (7) in matrix P and obtain In examples we often make computations using representatives of elements from classes. By abuse of notation we then writeP =P, whereP is the equivalence class andP is the representative of this class.
Let us define the action of the ring Q Σ [Z; δ Σ ] on the field Q Σ by the linear extension of the formula Z s a := δ s Σ a, where a ∈ Q Σ . Note that Z a = δ Σ a, as the action of the polynomial on the element of the field, is different from Za = (δ Σ a)Z as a product of polynomials Z and a in the ring of polynomials. Assume that the unimodular matrix U transforms the matrixQ into the Popov form. The matrix U ∈ Q Σ [Z, δ Σ ] p×p , applied as an operator to system equations (2), i.e. U Φ, is called the linear equivalence transformation.
Assumption 2. It is assumed that system (1) (or equivalently, system (2)) can be transformed into the strong Popov form (3) with respect to input u using linear equivalence transformations.

RIGHT INVERSE SYSTEM
In this section it is assumed that p m. Consider the set of i/o equations in the strong Popov form (2), satisfying Assumptions 1 and 2 together with the associated set S Σ . Denote by u a control sequence {u(t),t 0} and by y the output sequences {y(t),t 0}. We also consider the system where ι = 1, . . . , m, j = 1, . . . , p in the strong Popov form with respect to u = [u 1 , . . . , u m ] T . Note that the systems Λ and Σ u have similar structure and both are related to the original system Σ. However, they are introduced for different purposes and do not have to coincide. In particular, we assume that p m for Λ. Together with Λ we consider a multiplicative set S Λ . Let S be the smallest multiplicative set containing S Σ and S Λ .
• The pair (y, u) is acceptable with respect to the multiplicative set S (shortly S-acceptable) if for any ψ ∈ S and any t 0, ψ(y(t), . . . , Remark 1. If S is finitely generated, then the set of S-acceptable pairs (y, u) is generic in the following sense. For every k 0 the set of finite sequences (y(0), . . . , y(k), u(0), . . . , u(k)), obtained from an acceptable pair (y, u), is open and dense in some subset of R (k+1)(p+m) . This follows from the fact that non-acceptable pairs (y, u) satisfy a finite number of analytic equations. For a similar reason for an acceptable y there is a generic set of sequences u such that (y, u) is an acceptable pair. Definition 6. System Λ is a right inverse of Σ if for any S-acceptable y there exists u such that ( y, u) is acceptable and ( y, u) solves Λ and after substituting u = u to Σ and setting y i (k) =ỹ i (k), k = 0, . . . , n i − 1, we get the solution y of Σ, satisfying y i (k) =ỹ i (k), k n i . The right inverse of Σ is denoted by Σ −1 R . Proposition 2. For an S-acceptable sequence y there are infinitely many solutions u of Λ such that ( y, u) is S-acceptable. They correspond to a generic set of initial values u k (l), k = 1, . . . , p; l = 0, . . . , σ k − 1 and parameters u κ (l), κ = p + 1, . . . , m, l 0.
From the above, if the desired trajectory for the system Σ is fed into the right inverse system Σ −1 R , then the outputs of the right inverse generate the inputs u, resulting in y at the output of Σ; see Fig. 1. Proof. Necessity. The proof is by contradiction. Assume that (2) is right invertible but, contrary to the claim, ρ(Q) < p. If ρ(Q) < p, then by Proposition 1 and Lemma 1, the matrix Q = UQ in the Popov form has at least one zero row. Thus globally linearized system equations (5) involve relation between differentials dy 1 , . . . , dy p , solely, not involving any of inputs u 1 , . . . , u m . Such a system is not right invertible by Definitions 6 and 7, since one cannot guarantee that y i (k) =ỹ i (k), k n i .
Sufficiency. Assume that ρ(Q) = p and show that then system (2) is right invertible. According to Definitions 6 and 7, this is so when one can provide the rules for computing the input sequence {ũ(t),t 0} such that y(t) =ỹ(t) for t 0. Following [2,9], one can find the matrix Q := UQ in the Popov form with no zero rows. Under the assumptions of the theorem, by Lemma 1, also ρ( Q) = p. The application of the linear equivalence transformation U to equations (2) yields the system in the strong Popov form (3) with respect to the inputs. We show that (3) together with the multiplicative set S Σ u is really the right inverse of (2). Thus we set S to be the smallest multiplicative set containing S Σ and S Σ u . For the sake of transparency the remaining part of the proof is presented for the multi-input single-output case, where p = 1 and m = 2. The simplification does not change the idea of the proof. Let Σ be given by where ξ = (y, y [1] , . . . , y [n−1] , u 1 , . . . , u ). Assume that 5 s 1 s 2 . Due to Assumption 2 one can transform (9) via linear equivalence transformation to where ψ = α ϕ for e Σ (α) ∈ Q Σ . We will show that (10) is the right inverse of Σ. Let u 1 be the solution of for S-acceptable y, some initial values u 1 (0), . . . , u 1 (s 1 − 1), and someũ 2 (k), k 0, such that (ỹ,ũ 1 ,ũ 2 ) is Sacceptable (by Proposition 2). So (ỹ,ũ 1 , Let y be the solution of (9) for this ( u 1 , u 2 ) and initial conditions y(ℓ) =ỹ(ℓ) for ℓ = 0, . . . , n − 1. So On the other hand, one can transform (11) tõ via multiplying by α −1 . From (12) and (13) one gets y [n] =ỹ [n] and consequently, y [k] =ỹ [k] for k n.
Remark 2. Although the strong Popov form itself is unique, the right inverse system is not necessarily unique. Namely, if m > p, the equations of inverse are parametrized by m − p inputs that can be chosen freely. Expressing u [1] 1 , u [1] 3 from (20) yields alternative equations of the right inverse system, parametrized by u 2 and its shifts u [1] 1 = −y [2] 1 − u [1] 2 y 2 , u [1] 3 = −(u [1] 2 + y [3] 2 )/y 1 , satisfying Definition 6. However, the equations are not in the strong Popov form with respect to inputs, since condition (i) of Definition 2 is not satisfied for the second equation.
Remark 3. Sometimes the existence of an inverse system depends on the choice of variables. For instance, the system y [3] = (u [2] 1 ) 2 +u [2] 2 cannot be transformed into the explicit form with respect to u 1 (required by the Popov form), using the linear transformations, because for that the nonlinear transformation is necessary. However, one can find the inverse system by transforming the equations into the explicit form with respect to variable u 2 via linear transformation, obtaining u [2] 2 = y [3] − (u [2] 1 ) 2 . Observe that the latter system is not in the strong Popov form according to Definition 2, because it does not match with the system description (3) where from the ith equation the variable u i is expressed. Relaxing the assumption j i = i, made in this paper, reveals that the system u [2] 2 = y [3] − (u [2] 1 ) 2 satisfies the conditions of the strong Popov form, as defined in [2].
Example 4. The goal of this Example is to demonstrate that indices s iι in Σ are not the same as the indices in the inverse system; they change. For instance, given the system in the strong Popov form with respect to outputs Σ : y together with the set S Σ = {1}, its right inverse is ) .
The comparison of maximal input shifts in the original and inverse systems reveals that shifts in the inverse are lower than or equal to those appearing in the original system. Indeed, for (24) the indices To find the explicit equations of the right inverse for a system described by i/o equations there is no need to realize the equations in the state space form. However, our approach is consistent with the earlier results for state equations. The example below demonstrates that the diagram in Fig. 2 commutes. (20) in the strong Popov form with respect to outputs. Following the approach in this paper, we transform (20) into the Popov form with respect to inputs u 1 and u 2 , obtaining (21).

LEFT INVERSE SYSTEM
In this section we assume that p m. Let us consider the following system: where ι = 1, . . . , m, j = 1, . . . , p, together with the multiplicative set S Γ . Let S be the smallest multiplicative set containing S Σ and S Γ . Proof. Sufficiency. The proof is, for transparency, presented for the single-input multi-output case where p = 2, m = 1, described by the equations together with the multiplicative set S Σ , where ξ = (y 1 , . . . , y 1] ). There exists, due to Assumption 2, a transformation operator U satisfying e Σ (U) ∈ Q Σ [Z, δ Σ ] 2×2 , which transforms equations (32) into the strong Popov form where ψ 1 (y 1 , . . . , y 2 ) = 0 and the second equation where ξ ′ = (y 1 , . . . , y , for some nonnegative integers ν 1 , ν 2 , υ 1 , υ 2 , σ s. We demonstrate that (33) together with the multiplicative set S Γ is really the left inverse of (32). In what follows we set S to be the smallest multiplicative set containing S Σ and S Γ .
From the above, if the outputs of the original system Σ are fed into the left inverse system Σ −1 L , the latter can reconstruct the inputs u of the original system on its output; see Fig. 3. Example 7. Consider the set of equations in the strong Popov form with respect to y 1 , y 2 , y 3 Σ : 1 u [2] 2 + u [3] 2 + y 1 y 2 (37) with the set S Σ = {1}. From (37) one obtains p = 3, m = 2. The matrix is row-reduced, since its leading coefficient matrix has rank 2. The application of Algorithm 1 from [2] tō Q gives the transformation matrix Applying the transformation U to system equations (37) yields 1 .

CONCLUSIONS
It was shown that transforming the system into the strong Popov with respect to inputs enables one to find the explicit equations of right and left inverse systems for the set of i/o equations, under the assumptions m p and p m, respectively.
Note that the linear equivalence transformations that construct the explicit equations of the inverse system are valid globally in the entire space besides a certain set S that consists of zeros of some functions. Therefore, the approach avoids using the IFT (yielding, in general, only local results) or integrating the set of 1-forms obtained by the IA (yielding the generic results that are valid almost everywhere). However, when one needs to apply nonlinear equivalence transformations, the inverse system is not necessarily defined globally. Moreover, such transformations are difficult to find, see [1], and it is unclear whether the approach, based on the strong Popov form, outperforms the earlier approach based on the IA.