# Popov form and the explicit equations of inverse systems/Popovi kuju ja poordsusteemi ilmutatud vorrandid.

1. INTRODUCTION

In  the problem of right inversion is addressed for nonlinear control systems, described by the set of input--output (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  the IA is presented in terms of differential one-forms, exactly like in  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 . 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.

2. PRELIMINARIES

2.1.I/o equations

Consider a discrete-time multi-input multi-output nonlinear system, described by the explicit set of higher-order difference equations, relating the inputs [u.sub.k], k = 1,..., m, the outputs y.sub.i,i = 1,..., p, and a finite number of their time shifts:

[mathematical expression not reproducible] (1)

where j = 1,...,p, l = 1,...,m, and [[phi].sub.i] are real meromorphic functions. The word 'explicit' means that the variable [y.sub.i](t + [n.sub.i]) does not appear on the right-hand side of the ith equation, i.e. [n.sub.ii] < [n.sub.i].It is assumed that system (1) is strictly causal, i.e. [s.sub.il] < [n.sub.i]. The functions [[phi].sub.i] are defined on an open and dense subset of [R.sup.(n+1)(p+m)], whereas n = max[n.sub.i].

Definition 1. The set of i/o equations (1) is said to be in the strong Popov form with respect to the output if (a) [n.sub.1] [less than or equal to] [n.sub.2] [less than or equal to][??][less than or equal to] [n.sub.p];

(b) for all [[phi].sub.i],i = 1,...,p the following conditions hold:

(i) [n.sub.ik] < n.sub.i if k [less than or equal to] i;

(ii) [n.sub.ik] [less than or equal to] [n.sub.i] if k > i;

(iii) [n.sub.ki] < [n.sub.i] if k [not equal to] i.

Compared with the definition of the strong Popov form for implicit equations in , we have made in Definition 1 technical simplification [j.sub.i] = i for the explicit equation (1). With this additional assumption condition (v) in the strong Popov form (see ) 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.sub.[SIGMA]] with system (1). If (1) involves any denominators, then these denominators have to be included in [S.sub.[SIGMA]] together with their shifts and powers. The typically infinite set [S.sub.[SIGMA]] can be generated by a finite generator set [??]. The set [??] generates [S.sub.[SIGMA]] if each element of [S.sub.[SIGMA]] can be obtained from a finite number of elements of [??] 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.sub.[SIGMA]] = {1} and only in this case [S.sub.[SIGMA]] is a finite set (i.e. [S.sub.[SIGMA]] = [??]).

Hereinafter we use the notation [zeta] for a variable [zeta](t), [[zeta].sup.[k]] for its k-step time shift [zeta](t + k),k[member of] Z. In such notation (1) takes the form

[mathematical expression not reproducible] (2)

where l = 1,...,m and j = 1,...p.

We also consider the equations in the form

[mathematical expression not reproducible] (3)

where l = 1,..., m, j = 1,..., p, and [micro] = min(m, p). Together with [[SIGMA].sub.u] the multiplicative set [S.sub.[SIGMA]] is considered.

Definition 2. I/o equations (3) are said to be in the strong Popov form with respect to the input if

(a) [[sigma].sub.1] [less than or equal to] [[sigma].sub.2] [less than or equal to] [less than or equal to] [[sigma].sub.[micro]];

(b) for all [X.sub.k],k=1,...,[micro] the following conditions hold:

(i) [a.sub.kl] < [[sigma].sub.k] if l [less than or equal to] k;

(ii) [[sigma].sub.kl] [less than or equal to] [[sigma].sub.k] if l > k;

(iii) [[sigma].sub.lk] < [[sigma].sub.k] if l [not equal to] k.

In Definition 2, like in Definition 1, we have made a technical simplification for the explicit equations that in the kth equation of [[SIGMA].sub.u] the variable [u.sub.k] appears with the highest shift.

With system [SIGMA], described by equations (2), we associate a vector function [??] where [??]. The system [SIGMA] defines the inversive difference field [??] with the shift operator [[delta].sub.[SIGMA]]. In particular, the shift of [??] is defined by the right-hand side of equation (2), see more in [.sup.3]. Each element of [??] is the image of a meromorphic function under the map [e.sub.[SIGMA]]. Basically the map [e.sub.[SIGMA]] allows us to exclude (or include) the zeros, defined by equations (2), from (into) the elements of the field [??], and in this way to find the simplest representatives of the functions in [??]. See  for a precise definition and Example 1 below.

2.2. Non-commutative polynomials

The field [??] and the shift operator [[delta].sub.[SIGMA]] induce the ring of non-commutative polynomials in a variable Z over [??], denoted by [??]. The multiplication is defined by the linear extension of the following rules:

[mathematical expression not reproducible]

where a [member of] [??] and [[delta].sub.[SIGMA]]a means [[delta].sub.[SIGMA]] evaluated at a (so for example (a[Z.sup.[micro]]) x (b[Z.sup.v]) = a([??]b)[Z.sup.[micro]+v]).

Let [??] be the set of p x q-dimensional matrices, whose entries are polynomials in [??][Z, [[delta].sub.[SIGMA]]]. Let us denote the ith row of the matrix W [member of] [??] by [??] For the non-zero row [??] we define its degree deg [??] = [[sigma].sub.i] as the exponent of the highest power in Z present in [??] for i = 1,..., p. If [??] = 0, we define [[sigma].sub.i] = -[infinity]. The vector of the row degrees is denoted by [sigma] := [[[sigma].sub.1],..., [[sigma].sub.p]]. The degree of the matrix W is defined as degW:=max{[[sigma].sub.1],...,[[sigma].sub.p]}.LetN = deg W, e = [1,...,1],andM=[[m.sub.1],...,[m.sub.p]]:=N x e-[sigma]. By [Z.SUP.M] we denote the diagonal p x p matrix with the diagonal elements [Z.sup.m1],...,[Z.sup.mp].

Definition 3. The matrix L(W) such that [Z.SUP.M]W = L(W)[Z.SUP.N] + lower degree terms is called the leading row coefficient matrix of W.

Definition 4. [9,11] A polynomial matrix W [member of] [??] with non-zero rows is called row-reduced if its leading row coefficient matrix L(W) has full row [rank.sup.4] over the field [??]. IfW contains zero rows, then W is called row-reduced if its submatrix consisting of non-zero rows is row-reduced.

Definition 5.  Matrix W [member of] [??] is in the Popov form if W is row-reduced with the rows sorted with respect to their degrees ([[sigma].sub.1] [less than or equal to]xxx [less than or equal to] [[sigma].sub.p]) and for all non-zero rows [??] there is a column index [j.sub.i] (called the pivot index) such that

(i) [??].sub. is monic;

(ii) deg [w.sub.ik] < deg [??] if k < [j.sub.i];

(iii) deg [w.sub.ik] [less than or equal to] deg [??] if k [greater than or equal to] [j.sub.i];

(iv) deg [??] < deg [??] if k [not equal to] i;

(v) if deg [??] = deg [??] and i < k, then [j.sub.i] < [j.sub.k] (if the degrees of the rows are equal, then the pivot indices are increasing).

Proposition 1.  For any matrix W [member of] [??] there exists a unimodular matrix U [member of] [??] such that U x Wis in the Popov form.

2.3. Linearized i/o equations

Our goal is to represent system (2) in terms of polynomials from [??][Z; [[delta].sub.[SIGMA]]]. For that purpose we apply the differential operation d to equations (2) to obtain

[mathematical expression not reproducible] (4)

for i = 1,...,p. The polynomial variable Z is interpreted as the shift operator. Defining [Z.sup.[alpha]]d[y.sub.j] := [??] and [Z.sup.[beta]] d[u.sub.l] := [??] allows us to rewrite (4) as

P(Z)dy + Q(Z)du = 0, (5)

where P [member of] [??] and Q [member of] [??] are polynomial matrices, whose elements [p.sub.ij],[q.sub.it] [member of] [??][Z; [[delta].sub.[SIGMA]]] are

[mathematical expression not reproducible] (6)

[[delta].sub.ij] is the Kronecker symbol, and dy = [[d[y.sub.1],...,d[y.sub.p]].SUP.T], du = [[d[u.sub.1],...,d[u.sub.m]].SUP.T]. Equation (5) describes the (globally) linearized system, associated with equations (2). For P and Q we define the equivalence classes P := [e.sub.[SIGMA]](P), where [e.sub.[SIGMA]][(P).sub.ij] := [e.sub.[SIGMA]]([p.sub.ij]), Q := [e.sub.[SIGMA]](Q), where [e.sub.[SIGMA]][(Q).sub.it] := [e.sub.[SIGMA]]([q.sub.it]). The following example explains the action of operator [e.sub.[SIGMA]].

Example 1. Consider the system taken from :

[mathematical expression not reproducible] (7)

By (5) and (6) the matrix

[mathematical expression not reproducible]

Next we define the equivalence class P. To find one of the simplest representatives of the class, we take into account system equations (7) in matrix P and obtain

[mathematical expression not reproducible]

In examples we often make computations using representatives of elements from classes. By abuse of notation we then write P = P, where P is the equivalence class and P is the representative of this class.

Let us define the action of the ring [??][Z;[[delta].sub.[SIGMA]]] on the field [??] by the linear extension of the formula [Z.sup.s] [??] a:=[??], where a [member of] [??]. Note that Z [??] a = [[delta].sub.[SIGMA]]a, as the action of the polynomial on the element of the field, is different from Za = ([[delta].sub.[SIGMA]]a)Z as a product of polynomials Z and a in the ring of polynomials. Assume that the unimodular matrix U transforms the matrix Q into the Popov form. The matrix U [member of] [??], applied as an operator to system equations (2), i.e. U [??] [PHI], 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.

3. RIGHT INVERSE SYSTEM

In this section it is assumed that p [less than or equal to] 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.sub.[SIGMA]]. Denote by u a control sequence {u(t),t [greater than or equal to] 0} and by y the output sequences {y(t),t [greater than or equal to] 0}. We also consider the system

[mathematical expression not reproducible] (8)

where l = 1,...,m, j = 1,...,p in the strong Popov form with respect to u=[[[u.sub.1],...,[u.sub.m]].SUP.T]. Note that the systems A and [[SIGMA].sub.u] have similar structure and both are related to the original system [SIGMA]. However, they are introduced for different purposes and do not have to coincide. In particular, we assume that p [less than or equal to] m for [LAMBDA]. Together with [LAMBDA] we consider a multiplicative set [S.sub.[LAMBDA]]. Let S be the smallest multiplicative set containing [S.sub.[SIGMA]] and [S.sub.[LAMBDA]].

* The pair (y, u) is acceptable with respect to the multiplicative set S (shortly S-acceptable) if for any [psi] [member of] S andanyt [greater than or equal to] 0, [psi](y(t),...,y(t + [p.sub.[psi]]), u (t),..., u (t + [m.sub.[psi]])) [not equal to] 0.

* The sequence y is S-acceptable if there is u such that (y, u) is an S-acceptable pair.

Remark 1. If S is finitely generated, then the set of S-acceptable pairs (y, u) is generic in the following sense. For every [greater than or equal to] 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.SUP.(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 A is a right inverse of [SIGMA] if for any S-acceptable y there exists u such that (y,u) is acceptable and (y,u) solves A and after substituting u = u to [SIGMA] and setting y.sub.i(k) =[y.sub.i] (k), k = 0,...,[n.sub.i]-1, we get the solution y of [SIGMA], satisfyingyi(k) = [y.sub.i](k),k [greater than or equal to] n. The right inverse of [SIGMA] is denoted by [??].

Proposition 2. For an S-acceptable sequence y there are infinitely many solutions u of A such that (y,u) is S-acceptable. They correspond to a generic set of initial values [u.sub.k](l), k = 1,...,p; l = 0,..., [[sigma].sub.k] - 1 and parameters [u.sub.[kappa](l), [kappa] = p + 1,...,m,l [greater than or equal to] 0.

Proof. Follows from Remark 1.

Definition 7. System (2) is said to be right invertible if its right inverse system [??] exists in the sense of Definition 6.

From the above, if the desired trajectory for the system [??] is fed into the right inverse system [??], then the outputs of the right inverse generate the inputs u, resulting in y at the output of [SIGMA]; see Fig. 1.

Definition 8.  The rank of the matrix W [member of] [??], denoted as [rho](W), is defined to be equal to the maximum number of [??]-linearly independent rows of W.

Lemma 1. (, Lemma 3.7) Multiplying the matrix W [member of] [??] by the unimodular matrix U [member of] [??] from the left does not change its rank, i.e. [rho](W)=[rho](UW).

Theorem 1. Let p[less than or equal to]m. Under Assumption 2 system (2) is right invertible iff [??] = p.

Proof. Necessity. The proof is by contradiction. Assume that (2) is right invertible but, contrary to the claim, [??] < p. If [??] < p, then by Proposition 1 and Lemma 1, the matrix [??] in the Popov form has at least one zero row. Thus globally linearized system equations (5) involve relation between differentials [??],...,[??], solely, not involving any of inputs [u.sub.1],...,[u.sub.m]. Such a system is not right invertible by Definitions 6 and 7, since one cannot guarantee that [??].

Sufficiency. Assume that [??] 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 [??] such that [??] for t [greater than or equal to] 0. Following [2,9], one can find the matrix [??] in the Popov form with no zero rows. Under the assumptions of the theorem, by Lemma 1, also [??]. 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 [??] is really the right inverse of (2). Thus we set S to be the smallest multiplicative set containing [S.sub.[summation]] and [??]. 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 [summation] be given by

[mathematical expression not reproducible] (9)

where [??]. Assume that (5) [s.sub.1] [greater than or equal to] [s.sub.2]. Due to Assumption 2 one can transform (9) via linear equivalence transformation to

[mathematical expression not reproducible] (10)

where [??]. We will show that (10) is the right inverse of [summation]. Let [??] be the solution of

[mathematical expression not reproducible]

for S-acceptable [??], some initial values [u.sub.1](0),...,[u.sub.1]([S.sub.1] - 1), and some [??], such that ([??]) is Sacceptable (by Proposition2). So ([??]), where [??] satisfies

[mathematical expression not reproducible] (11)

Let y be the solution of (9) for this ([??]) and initial conditions [??] for l = 0,...,n-1. So

[mathematical expression not reproducible] (12)

On the other hand, one can transform (11) to

[mathematical expression not reproducible] (13)

via multiplying by [[alpha].sup.-1]. From (12) and (13) one gets [??] and consequently, [??]

Example2. Consider the system

[mathematical expression not reproducible] (14)

Since equation (14) does not contain denominators, the set [S.sub.[summation]] = {1}. The explicit equation of the right inverse system is

[mathematical expression not reproducible] (15)

and the set [??]. Assume that the reference sequence [??] is given such that

[mathematical expression not reproducible] (16)

Now one can find the input sequence [??] as follows. First, [??] can be chosen arbitrarily. By (15),

[mathematical expression not reproducible] (17)

We show that substituting [??] into equations (14) yields, by (16), y(t,y(0),y(1), [??],...,[??] = y(t), t [greater than or equal to] 2. First, note that y(0) = y(0), y(1) = y(1). Applying u(t) = u(t) in (14) yields for t = 2: y(2) = u(0) y (0) y (1) +u(1). Due to (16) and (17), y(2) = u(0)y(0)y(1) +y(2)-u(0)y(0)y(1) =y(2). Taking t = 1 and u(t)=u(t)fort = 1,2 in (14), we get

y(3) = u(1) y (1) y (2) + u(2). (18)

Substituting u(2) from (17) into (18) yields

y(3) = u(1)y(1)y(2) +y(3) - u(1)y(1) y(2). (19)

Rewrite (19) as y(3) = u(1) [y(1)y(2) -y(1)y(2)] +y(3). Note that y(1) = y(1) due to initial conditions, and on the previous step we have shown that y(2) = y(2). Thus the equalities y(1)y(2) -y(1)y(2) = 0 and y(3) = y(3) hold. In a similar manner, one can show that y(t) = y(t) for t [greater than or equal to] 4. Consequently, system (14) is right invertible by Definition 7.

Example 3. Consider the system with 3 inputs and 2 outputs:

[mathematical expression not reproducible] (20)

with the set [S.sub.[summation]] = {1}. After rewriting system (20) in explicit form (2), one obtains the indices [n.sub.1]=2, [n.sub.11] = -[infinity], [n.sub.12] = 0, [s.sub.11] = 1, [s.sub.12] = 1, [s.sub.13] = -[infinity]

[n.sub.2] = 3, [n.sub.21] =0, [n.sub.22] = -[infinity], [s.sub.21] = -[infinity], [s.sub.22] = 1, [s.sub.23] = 1.

By Definition 1 the system is in the strong Popov form (with respect to outputs), but not in the strong Popov form with respect to inputs, since condition (iii) of Definition 2 is not fulfilled. For system (20) the matrix

[mathematical expression not reproducible]

and so the rank [rho](Q) = 2 = p. Due to Theorem 1 the right inverse system exists for (20). The application of Algorithm 1 from  to Q yields the transformation matrix

[mathematical expression not reproducible]

Applying the operator U to [[[phi].sub.1], [[phi].sub.2]].sup.T] yields

[mathematical expression not reproducible]

The indices of the transformed equations (after rewriting the system in the explicit form) are

[[sigma].sub.1] = 1, [[sigma].sub.11] = -[infinity], [[sigma].sub.12] = -[infinity], [[sigma].sub.13] = 1, [[sigma].sub.2] =1, [[sigma].sub.21] = -[infinity], [[sigma].sub.22] = -[infinity], [[sigma].sub.23] = 1;

conditions (i)-(iii) of Definition 2 are fulfilled. Therefore [??] can be expressed from [[phi].sub.1] = 0, [[phi].sub.2] = 0 as

[mathematical expression not reproducible] (21)

where [u.sub.3] is considered as a free parameter. System (21) is in the strong Popov form with respect to input.

Let us show that equations (21) allow us to compute the sequence {u(t), t > 0}, required in Definition 6. Assume that the reference sequence {([y.sub.1](t), [y.sub.2](t)),t [greater than or equal to] 0} is given and

[mathematical expression not reproducible] (22)

To compute the input sequence {([u.sub.1](t),[u.sub.2](t)),t [greater than or equal to] 0}, we choose the sequence {[u.sub.3](t),t [greater than or equal to] 0} arbitrarily and compute by (21) and (22):

[u.sub.1](1) =-[y.sub.1](2) + [u.sub.3](1)[y.sub.1](0)[y.sub.2](0)+[y.sub.2](0)[y.sub.2](3), [u.sub.2](1) =-[u.sub.3](1)[y.sub.1](0)-[y.sub.2](3), [u.sub.1](2) =-[y.sub.1](3) + [u.sub.3](2)[y.sub.1](1)[y.sub.2](1)+[y.sub.2](1)[y.sub.2](4), [u.sub.2](2) =-[u.sub.3](2)[y.sub.1](1)-[y.sub.2](4), (23)

...

We show next that substituting [u.sub.1](t), [u.sub.2](t), and [u.sub.3] (t), t [greater than or equal to] 0 into equations (20) yields, using (22),

[y.sub.i](t,[y.sub.1](0),[y.sub.1](1),[y.sub.2](0),[y.sub.2](1),[y.sub.2](2),u(0),...,u(t-1))= [y.sub.i] (t), t[greater than or equal to]0,i = 1, 2.

Replacing [u.sub.1](t) and [u.sub.2](t) respectively by [u.sub.1](t) and [u.sub.2](t) from (23) yields [y.sub.1](2) = -[u.sub.1](1) - [u.sub.2](1)[y.sub.2](0), [y.sub.2](3) = -[u.sub.2](1) - [u.sub.3](1)[y.sub.1](0). Due to (22),

[mathematical expression not reproducible]

In a similar manner one can show that [y.sub.1](t) =[y.sub.1](t),t [greater than or equal to] 3 and [y.sub.2](t) = [y.sub.2](t),t [greater than or equal to] 4. Consequently, system (20) is right invertible by Definition 7.

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 [??] from (20) yields alternative equations of the right inverse system, parametrized by u.sub.2 and its shifts

[mathematical expression not reproducible]

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 [??] cannot be transformed into the explicit form with respect to u.sub.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.sub.2] via linear transformation, obtaining [??]. 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.sub.i] is expressed. Relaxing the assumption [j.sub.i] = i, made in this paper, reveals that the system [??] satisfies the conditions of the strong Popov form, as defined in .

Example 4. The goal of this Example is to demonstrate that indices [s.sub.il] in [summation] 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

[mathematical expression not reproducible] (24)

together with the set [S.sub.[summation]] = {1}, its right inverse is

[mathematical expression not reproducible] (25)

with [??]. The transformation matrix 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

[s.sub.11]=1, [s.sub.12] = 0,

[s.sub.21]=3, [s.sub.22] = 0,

while for inverse system (25)

[[sigma].sub.1] = 1, [[sigma].sub.11] = -[infinity], [[sigma].sub.12]=0,

[[sigma].sub.2] = 2, [[sigma].sub.21] = -[infinity], [[sigma].sub.22] = 0.

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.

Example 5. (Continuation of Example 3) Consider system (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.sub.1 and u.sub.2, obtaining (21).

An alternative way is to start by transforming i/o equations (20) into the state space form as

[mathematical expression not reproducible] (26)

where [??], (see ). Applying the inversion algorithm from  allows us to find the equations of the right inverse system of (26) as

[mathematical expression not reproducible] (27)

The order of inverse system (27) can be reduced if we take into account that [x.sub.1]=[y.sub.1],[x.sub.2]=[y.sub.2], and [??], meaning that the first three equations in (27) are just identities (see more in , p. 81):

[mathematical expression not reproducible] (28)

where [[eta].sub.1] = [x.sub.4] = ([u.sub.1] + [y.sup..sub.1])/[u.sub.2] and [[eta].sub.2] = [x.sub.5] = ([u.sub.2] + [y.sup..sub.2)/[u.sub.3]. Eliminating state variables from (28) yields exactly i/o equations (21), obtained directly from (20).

Example 6. Consider Example 5.2 from :

[mathematical expression not reproducible] (29)

and the set [S.sub.[summation]] = {1}. Transforming the system into the strong Popov form with respect to inputs yields the following right inverse system:

[mathematical expression not reproducible] (30)

and the set [S.sub.[LAMBDA]] = {1}. The application of the IA from  to system (29) results also in (30).

4. LEFT INVERSE SYSTEM

In this section we assume that p [greater than or equal to] m. Let us consider the following system:

[mathematical expression not reproducible] (31)

where l = 1,...,m, j = 1,...,p, together with the multiplicative set [S.sub.[GAMMA]]. Let S be the smallest multiplicative set containing [S.sub.[[summation]] and [S.sub.[GAMMA]].

Definition 9. System [DAMMA] is a left inverse of [summation] if for any S-acceptable u there exists y such that (y,u) is S-acceptable and solves [summation] and after substituting y = y to [GAMMA] and setting [u.sub.k](l) = [u.sub.k](l), l = 0,..., [[sigma].sub.k] - 1, we get solution u of [GAMMA], satisfying [u.sub.k](l) = [u.sub.k](l), l [greater than or equal to] [[sigma].sub.k]. Then the left inverse of [summation] is denoted by [[summation].sup.-1.sub.L].

Definition 10. System [summation] is left invertible if there exists a left inverse of [summation] in the sense of Definition 9.

If system (2) is left invertible, then it is possible to reconstruct uniquely the input u from the knowledge of the observed output sequence y.

Theorem 2. Let p[greater than or equal to]m. Under Assumption 2 system (2) is left invertible iff [rho](Q) = m.

Proof. Sufficiency. The proof is, for transparency, presented for the single-input multi-output case where p = 2,m = 1, described by the equations

[mathematical expression not reproducible] (32)

together with the multiplicative set [S.sub.[summation], where [??]. There exists, due to Assumption 2, a transformation operator U satisfying [??], which transforms equations (32) into the strong Popov form

[mathematical expression not reproducible]

where [??] and the second equation

[mathematical expression not reproducible] (33)

where [??], for some nonnegative integers [[nu].sub.1],[[nu].sub.2],[v.sub.1],[v.sub.2],[sigma] [less than or equal to] s. We demonstrate that (33) together with the multiplicative set [S.sub.[GAMMA] is really the left inverse of (32). In what follows we set S to be the smallest multiplicative set containing [S.sub.[summation]] and [S.sub.[GAMMA]]. Take S-acceptable u and solve [summation] getting ([y.sub.1],[y.sub.2]). Then we obtain

[mathematical expression not reproducible]

From the linear equivalence transformation we get

[mathematical expression not reproducible] (34)

Let us solve (33), setting [y.sub.i] = [y.sub.i], i= 1,2, and u(k) = u(k) for k = 0,...,[sigma]-1. Then we get

[mathematical expression not reproducible] (35)

From (34) and (35) it follows that [u.sup.[[sigma]] = [u.sup.[[sigma]], as required and consequently, [u.sup.[k]] = [u.sup.[k]] for k [greater than or equal to] [sigma].

Necessity. Given the system [summation] and its left inverse [GAMMA], their linearized descriptions are respectively (5) and

d[GAMMA]: R(Z)du + S(Z)dy = 0 (36)

for some polynomial matrices [??] and [??]. From (36) we obtain du = -R[(Z).sup.-1]S(Z)dy. Let us substitute dy by -[P.sup.-1](Z)Q(Z)du from (5):du = -R[(Z).sup.-1]S(Z)(-[P.sup.-1](Z)Q(Z)du), resulting in I = [R.sup.-1]S[P.sup.-1]Q. This cannot be true unless the rank [rho](Q) = m. Note that the elements of [P.sup.-1] and [R.sup.-1] are from the quotient field of the Ore ring [??]. The both inverse matrices [P.sup.-1] and [R.sup.-1] exist, because [??] is the Ore ring, see .

From the above, if the outputs of the original system [summation] are fed into the left inverse system [[summation].sup.-1.sub.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.sub.1],[y.sub.2],[y.sub.3]

[mathematical expression not reproducible] (37)

with the set [S.sub.[summation]] = {1}. From (37) one obtains p = 3, m = 2. The matrix

[mathematical expression not reproducible]

is row-reduced, since its leading coefficient matrix has rank 2. The application of Algorithm 1 from  to Q gives the transformation matrix

[mathematical expression not reproducible]

and the set [S.sub.[GAMMA]] = {1}. Applying the transformation U to system equations (37) yields

[mathematical expression not reproducible] (38)

being in the strong Popov form with respect to [u.sub.1],[u.sub.2]. The explicit equations of the left inverse system are

[mathematical expression not reproducible] (39)

Alternatively, the left inverse system can be found via state equations, as shown in Fig. 2. For that purpose the following steps are necessary. First, the realization of original equations (37) is constructed:

[mathematical expression not reproducible] (40a)

[mathematical expression not reproducible] (40b)

Note that this is not always doable since not all i/o equations are realizable. Second, the left inverse system of state equations (40) is found. This can be done by expressing [u.sub.1] = [y.sup..sub.2] -[x.sub.2], [u.sub.2] = [x.sub.4]-[y.sub.1] from (40) and substituting [u.sub.1],[u.sub.2] into (40a):

[mathematical expression not reproducible] (41)

The third and last step is transforming the left inverse (41) into the form of i/o equations

[mathematical expression not reproducible] (42)

At first sight the first equations of (39) and (42) are different; moreover, the first equation of (39) depends on y.sub.3, whereas the first equation of (42) does not. However, note that (39) do not include the equation [[PSI].sub.1] = 0, being part of (38). We rewrite [[PSI].sub.1] = 0 as

[mathematical expression not reproducible] (43)

Shifting the first equation of (39) forward to obtain [??] and eliminating then [y.sup..sub.3] using (43) yield [y.sup..sub.1] = [y.sub.1] +[y.sup..sub.2], the first equation of (42). The second equations of (42) and (39) coincide.

5. 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[greater than or equal to]p and p[greater than or equal to]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 , and it is unclear whether the approach, based on the strong Popov form, outperforms the earlier approach based on the IA.

ACKNOWLEDGEMENTS

The work of Z. Bartosiewicz and M. Wyrwas was supported by grant No. S/WI/1/2016. The work of U. Kotta and M. Tonso was supported by the Estonian Research Council, personal research funding grant PUT481. The work of E. Pawluszewicz was supported by grant No. S/WM/1/2016. The publication costs of this article were covered by the Estonian Academy of Sciences.

REFERENCES

[1.] Bartosiewicz, Z., Belikov, J., Kotta, U., Tonso, M., and Wyrwas, M. On the transformation of a nonlinear discrete-time input-output system into the strong row-reduced form. Proc. Estonian Acad. Sci., 2016, 65, 220-236.

[2.] Bartosiewicz, Z., Kotta, U., Pawluszewicz, E., Tonso, M., and Wyrwas, M. The strong Popov form of nonlinear input-output equations. Proc. Estonian Acad. Sci., 2018, 67, 193-206.

[3.] Beckermann, B., Cheng, H., and Labahn, G. Fraction-free row reduction of matrices of Ore polynomials. J. Symb. Comput, 2006, 41, 513-543.

[4.] Cheng, H. Algorithms for Normal Forms of Matrices of Polynomials and Ore Polynomials. PhD thesis, University of Waterloo, Ontario, Canada, 2003.

[5.] Grizzle, J. W. A linear algebraic framework for the analysis of discrete-time nonlinear systems. SIAMJ. Control Optim., 1993, 31, 1026-1044.

[6.] Halas, M. An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica, 2008, 44, 1181-1190.

[7.] Kotta, U. Inversion algorithm for recursive nonlinear systems. Proc. Estonian Acad. Sci. Phys. Math., 1998, 47, 3-18.

[8.] Kotta, U. Inversion Method in the Discrete-Time Nonlinear Control Systems Synthesis Problems. Lecture Notes in Control and Information Science, Vol. 205. Springer-Verlag, London, 1995.

[9.] Kotta, U., Bartosiewicz, Z., Nomm, S., and Pawluszewicz, E. Linear input-output equivalence and row reducedness of discrete-time nonlinear systems. IEEE Trans. Autom. Control, 2011, 56, 1421-1426.

[10.] Kotta, U. and Tonso, M. Realization of discrete-time nonlinear input-output equations: polynomial approach. Automatica, 2012, 48, 255-262.

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Zbigniew Bartosiewicz (a), Ulle Kotta (b), Ewa Pawluszewicz(c), Maris Tonso (b*), and Malgorzata Wyrwas (a)

(a) Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15-351 Bialystok, Poland

(b) Department of Software Science, School of Information Technologies, Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia

(c) Department of Automatic Control and Robotics, Bialystok University of Technology, Wiejska 45A, 15-351 Bialystok, Poland

(*) Corresponding author, maris@cc.ioc.ee

Received 30 October 2017, accepted 24 May 2018, available online 17 October 2018

https://doi.org/10.3176/proc.2018.4.04

(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 [not equal to] p.

(2) Note that renumbering the output coordinates is not a row operation on globally linearized system equations.

(3) In  the notations [??] and [[delta].sub.[PHI]] were used respectively for [??] and [[delta].sub.[SIGMA]].

(4) The matrix W [member of] [??] is said to have full row rank if rank W = min (p,q).

(5) There is no loss of generality in assuming [s.sub.1] > [s.sub.2]. If [s.sub.1] < [s.sub.2], then renumbering [u.sub.1], [u.sub.2] allows us to bring the system into the necessary form.