A normal form for two-input forward-flat nonlinear discrete-time systems

We show that every forward-flat nonlinear discrete-time system with two inputs can be transformed into a structurally flat normal form by state- and input transformations. This normal form has a triangular structure and allows to read off the flat output, as well as a systematic construction of the parameterisation of all system variables by the flat output and its forward-shifts. For flat continuous-time systems, no comparable normal form exists.


Introduction
In the 1990s, the concept of flatness has been introduced by Fliess, Lévine, Martin and Rouchon for nonlinear continuous-time systems (see e.g. Fliess et al., 1992Fliess et al., , 1995Fliess et al., , 1999. Flat continuous-time systems have the characteristic feature that all system variables can be parametrised by a flat output and its time derivatives. They form an extension of the class of static feedback linearisable systems and can be linearised by endogenous dynamic feedback. Their popularity stems from the fact that a lot of physical systems possess the property of flatness and that the knowledge of a flat output allows an elegant solution to motion planning problems and a systematic design of tracking controllers. For nonlinear discrete-time systems, flatness can be defined analogously to the continuous-time case. The main difference is that time derivatives have to be replaced by forward-shifts. Like in the continuoustime case, flat systems form an extension of static feedback linearisable systems. The problem of static feedback linearisation for discrete-time systems is already solved, see Grizzle (1986), Jakubczyk (1987) and Aranda-Bricaire et al. (1996). An important difference to the continuous-time case is the existence of discrete-time systems that can be linearised by dynamic feedbacks which are not contained within CONTACT Johannes Diwold johannes.diwold@jku.at the class of endogenous dynamic feedbacks as proposed in Aranda-Bricaire and Moog (2008). The corresponding linearising output then depends not only on forward-shifts but also on backward-shifts of system variables. A recent definition of flatness proposed in Guillot and Millérioux (2020) includes also backwardshifts of the input variables in the flat output. However, within this contribution, similar to Kaldmäe and Kotta (2013), Kolar, Kaldmäe, et al. (2016) we restrict ourselves to forward-shifts in the flat output and therefore use the term forward-flatness. The property of forward-flatness is equivalent to linearisability by an endogenous dynamic feedback as proposed in Aranda-Bricaire and Moog (2008). In general, the analysis of flat systems can be divided into two separate tasks. First, we are interested in checking whether a system is flat or not in order to clarify if flatness based control strategies can be applied in principle. In Grizzle (1986) and Nijmeijer and van der Schaft (1990), an efficient test for static feedback linearisable systems, which is based on the computation of certain distributions, can be found. As we have shown in , this test can be generalised to the class of forwardflat systems. The test is based on the results of . It should be noted that for continuous-time flat systems no comparable test is available so far. Second, in order to use the flatness property for control strategies, the knowledge of a flat output as well as the corresponding parameterisation of all system variables is necessary. For this purpose, the use of structurally flat normal forms (see e.g. Schöberl, 2014;Schöberl & Schlacher, 2014) has turned out to be helpful. Structurally flat normal forms allow to read off the flat output, as well as a systematic construction of the parameterisation of all system variables. The most famous example for such a normal form is the Brunovsky normal form. However, a transformation to Brunovsky normal form is possible if and only if the system is static feedback linearisable. In , we have shown that every forward-flat system can be transformed into a structurally flat implicit normal form. The main feature of this normal form is that the equations depend on the system variables in a triangular manner. The reason for the implicit character of this normal form is that the required coordinate transformations are possibly more general than the usual state-and input transformations. In the present contribution, in order to preserve an explicit state representation we restrict ourselves to state-and input transformations. We introduce a structurally flat explicit system representation, and show that for every forward-flat system with two inputs the existence of a transformation into such a representation is guaranteed. Hence, for two-input forward-flat systems we also use the term explicit triangular normal form. We want to emphasise that for flat continuous-time systems no comparable normal form exists.
The paper is organised as follows: In Section 2 we recapitulate the concept of flatness in terms of forward-shifts and the corresponding test according to . In Section 3 we discuss certain coordinate transformations, which will be useful later on. In Section 4 we introduce a structurally flat explicit triangular form. Then, we prove that every two-input forward-flat discrete-time system can be transformed into such a representation by successive state-and input transformations. Finally, in Section 5, we illustrate our results by an example.

Forward-flatness of discrete-time systems
Throughout this contribution we consider discretetime nonlinear systems in explicit state representation of the form with dim(x) = n, dim(u) = m and smooth functions f i (x, u). Geometrically, the system (1) can be interpreted as a map f from a manifold X × U with coordinates (x, u) to a manifold X + with coordinates x + . Furthermore, we assume that the system meets rank (∂ (x,u) f ) = n, which is a necessary condition for accessibility and consequently also for flatness. Apart from this, we assume that the system possesses no redundant inputs, i.e. rank(∂ u f ) = m, and define forward-flatness according to .
Definition 2.1: A system (1) is said to be forwardflat around an equilibrium (x 0 , u 0 ), if there exists an m-tuple of functions where u [α] denotes the α-th forward-shift of u, such that the n + m coordinate functions x and u can be expressed locally by y and forward-shifts of y up to some finite order, i.e. 1 The m-tuple (2) is called a forward-flat output.
The test for forward-flatness, as stated in , is based on the construction of sequences of nested distributions on X × U and X + . The construction makes use of the system equations (1) and the map π : X × U → X + defined by as well as their tangent maps f * and π * . In the following, we call a distribution D on X × U projectable (with respect to the map f ), if its pushforward f * (D) induces a well-defined distribution on X + . Whether a distribution is projectable or not can be checked by the use of adapted coordinates on X × U. Further details can be found in .

Algorithm 2.1:
Step k = 0: Define the distribution Then compute the largest subdistribution which is projectable with respect to the map f of (1). The distribution D 0 is involutive 2 and its pushforward is a well-defined involutive distribution on X + .
Step k ≥ 1: Compute and the largest subdistribution which is projectable with respect to the map f of (1). The distribution D k is involutive and its pushforward The procedure according to Algorithm 2.1 yields a unique nested sequence of projectable and involutive distributions on X × U and a unique nested sequence of involutive distributions Whether a system is forward-flat or not can now be checked by use of the following theorem. For the proof we refer to . The test for forward-flatness contains the test for static feedback linearisability (see Nijmeijer & van der Schaft, 1990) as a special case. The only difference is that the distributions (5), according to Algorithm 2.1, are defined as the largest projectable subdistributions D k ⊂ E k , while in the static feedback linearisable case these distributions coincide, i.e. D k = E k .

Theorem 2.2:
Since all distributions E k must be completely projectable, the test for static feedback linearisability is more restrictive. For a system that meets Theorem 2.1, a single step where D k = E k can be interpreted as a defect in the test of static feedback linearisability, as we will demonstrate by the following example.
Example 2.1: For the system we obtain the sequence of distributions The computations can be found in some detail in the appendix. Despite the fact that E 1 is not completely projectable, i.e. D 1 = E 1 , the distribution Steps with D k = E k may occur several times through the algorithm. However, like in Example 2.1, for the last distribution the relation Dk −1 = Ek −1 holds. Since we will use this relation in Section 4, we establish the following lemma.
Thus, Ek −1 is projectable and according to the definition of Dk −1 we have Dk −1 = Ek −1 .

Coordinate transformations
For static feedback linearisable systems, the sequences of distributions (5) and (6) can be straightened out simultaneously by suitable state transformations. As shown in Nijmeijer and van der Schaft (1990), in such coordinates the system (1) exhibits a triangular form. The primary objective of the present paper is to prove that every forward-flat system with two inputs allows a similar explicit 3 triangular representation. The key tool is again to straighten out the sequences of distributions (5) and (6) simultaneously (in Example 2.1 we already have such special coordinates). However, now we will need in general both state-and input transformations. For systems with more than two inputs, there is no guarantee that an explicit triangular representation exists at all.
Before we state our main results in Section 4, we discuss certain state-and input transformations which will be useful for straightening out the sequences of distributions. In order to preserve an explicit system representation like (1), within this contribution we restrict ourselves to state-and input transformationŝ where both x and x + are transformed equally. The transformed system is given bŷ where −1 (x,û) denotes the inverse of (9). Like in the static feedback linearisable case, the first step in achieving a triangular representation is to straighten out the sequence (6). Since (6) is a nested sequence of involutive distributions on X + , by an extension of the Frobenius theorem there exists a state transformation simultaneously. The system in new coordinates reads as 4x and meets for k = 1, . . . ,k. Additionally, from the definition of E k according to (4) and (12), it follows automatically that E k is also straightened out and reads as for k = 0, . . . ,k − 1. The D-sequence, in contrast, is only straightened out automatically if the system possesses the stronger property of static feedback linearisability, since then D k = E k . Thus, for finding coordinates that straighten out both sequences of distributions (5) and (6), the transformation (11) alone is not sufficient. Therefore, we need an additional transformation that straightens out the D-sequence while the -sequence remains straightened out. In the following, we introduce transformations that meet the latter condition.
Lemma 3.1: State-and input transformations of the formx preserve the structure (12) of the sequence of distributions (6), i.e.
The proof follows from the triangular structure of the state transformation of (16). The input transformation does not affect the -sequence.
For systems with two inputs the distributions of the D-sequence have a very special structure. Since we will deal with two-input systems in Section 4.2, we state the following important lemma.

Lemma 3.2:
Consider an n-dimensional manifold Z with coordinates ζ = (ζ 1 , . . . , ζ n ) and an involutive distribution for some i, j ≥ k. There exists a transformation of the coordinate ζ j such that The proof can be found in the appendix. For twoinput systems we will encounter distributions D k of the form (17) on the manifold X × U, and straighten them out by state-or input transformations of the type (18). These transformations will also exhibit the structurepreserving form (16) with respect to the -sequence.

Explicit triangular form
In Nijmeijer and van der Schaft (1990) it is shown how a static feedback linearisable system can be transformed into Brunovsky normal form. In the first step, a state transformation is performed that straightens out the sequences of distributions simultaneously. This yields an explicit triangular system representation which can be interpreted as a composition of smaller subsystems. With respect to the inputs of these subsystems, there may occur redundancies. Following Nijmeijer and van der Schaft (1990), further state-and input transformations are successively performed in order to obtain the Brunovsky normal form. The above mentioned redundancies appear if the chains of the Brunovsky normal form have different lengths.
For forward-flat systems that are not static feedback linearisable, a transformation to Brunovsky normal form is not possible. Thus, we introduce a more general structurally flat explicit triangular form that can be obtained by straightening out the D-andsequences by suitable state-and input transformations. For two-input systems, we prove that such a transformation which straightens out both sequences of distributions simultaneously always exists. Subsequently, similar to the static feedback linearisable case, redundant inputs of the subsystems can be eliminated by further structure-preserving state-and input transformations. For the resulting system representation we use the term explicit triangular normal form. It allows to read off a forward-flat output and the corresponding parameterising map, according to Definition 2.1, in a systematic way.

Explicit triangular form for multi-input systems
In the following we present an explicit triangular representation for forward-flat systems. Note, we do not refer to it as a normal form, since for forward-flat systems with an arbitrary number of inputs the existence of such coordinates is not guaranteed in general.

Theorem 4.1: Assume there exists a state-and input transformationx
that straightens out the sequences (5) and (6) Here z k denotes a selection of components of (û,x 1 , . . . , In such coordinates, the system (1) has the triangular form . . .
and meets The proof can be found in the appendix. Hereinafter, we state the intention of defining z k . Since the -sequence is straightened out, likewise is E k = span{∂x k , . . . , ∂x 1 , ∂û}. Furthermore, by assumption the D-sequence is also straightened out, and due to D k ⊂ E k it follows that D k might not contain all components of ∂x k , . . . , ∂x 1 , ∂û. Therefore, we introduce the variable z k , which acts as placeholder and describes states and/or inputs of (û,x 1 , . . . ,x k ) so that D k reads as (22). It is important to mention that the variables z 0 , . . . , zk −1 contain all inputs and states exceptxk (see the proof in the appendix). Note, the system (23) is still in an explicit state representation (1), as we can always replace z k by the corresponding states and inputs. We clarify the definition of z k using the system of Example 2.1.
Example 4.1: Consider the system (8) of Example 2.1. In this example, both the D-and the -sequence are already straightened out. Thus, the coordinate transformation of Theorem 4.1 is just a renaminĝ According to Theorem 4.1 we define such that the D-sequence of distributions reads as and the system follows as The system has the structure of (23), and a forward-flat output is given by y = (x 1 3 , x 2 3 ).
It is important to emphasise that for systems with m > 2 inputs the existence of a state-and input transformation (20) that straightens out both sequences (5) and (6) simultaneously is not guaranteed. Thus, an explicit triangular form (23) does not necessarily exist. However, at least a transformation into an implicit triangular form as discussed in  is always possible.

Explicit triangular form for two-input systems
In the following we restrict ourselves to forward-flat systems with two inputs and state our main result.

Theorem 4.2: A two-input forward-flat system (1) is locally transformable into an explicit triangular representation (23).
According to Theorem 4.1, we must show that there exists a state-and input transformation (20) which straightens out the sequences of distributions (5) and (6) simultaneously. We start with the system representation (13), where the -sequence has already been straightened out by a suitable state transformation. Next, we want to straighten out the D-sequence step by step, starting with D 0 . For this purpose, we can exploit the fact that for systems with two inputs the dimension of these distributions grows in every step by either one or two. In the first case, the distribution D k is of the form (17) and can be straightened out by a transformation (18), whereas in the latter case the distribution D k is already straightened out and no transformation is required.
The following algorithm straightens out the Dsequence step by step with state-and input transformations that preserve the structure of the -sequence according to Lemma 3.1. In every step k, after performing the transformation the corresponding states or inputs are renamed by z k and z k,c . As mentioned before, z k is just a selection of states and inputs so that whereas z k,c denotes the complementary state or input so that To keep the successive transformations readable, after each step k we return from the hat notation for the transformed variables again to the bar notation. However, for the final system representation after the last step we use the hat notation.

Algorithm 4.1:
Step k = 0, we distinguish between the two cases: (a) If the entire input distribution is projectable, i.e. D 0 = E 0 , then there is no need for an input transformation because E 0 is already straightened out. We define z 0 = (u 1 , u 2 ) and z 0,c is empty.
up to a renumbering of the components of u. According to Lemma 3.2, there exists an input transformationû 1 = u 1 (x, u) such that D 0 = span{∂ u 2 }. We define z 0 = u 2 and z 0,c =û 1 .
Finally, the distributions are given by Step k = 1, . . . ,k − 1, we repeat the procedure with the distribution It can be shown that the dimensions of z k−1,c andx k meet Thus, in every step we must distinguish three cases: (a) If the entire distribution E k is projectable, i.e. D k = E k , then there is no need for a transformation because E k is already straightened out. We define z k = (x k , z k−1,c ) and z k,c is empty.
up to a renumbering of the components ofx k . According to Lemma 3.2, there exists a state transformationx We define z k =x 2 k and z k,c =x 1 k . (c) If D k = E k and dim(x k ) = 1, then necessarily also dim(z k−1,c ) = 1. Otherwise, we would have D k = E k and case (a) would apply. Thus, According to Lemma 3.2, there exists a transformationẑ such that D k = span{∂ z 0 , . . . , ∂ z k−1 , ∂x k }. Since zk −1,c could represent both an input or state variable of the system, the transformation is either an input-or a state transformation. We define z k =x k and z k,c =ẑ k−1,c .
Finally, the distributions are given by Within the algorithm, only state transformations (27) and input-or state transformations (28) are performed. Since both are of the structure-preserving form (16), the resulting entire transformation law is also of the form (16) and the -sequence remains straightened out. Thus, after the last step of the algorithm, both sequences of distributions are straightened out according to (21) and (22) in Theorem 4.1.

Remark 4.1:
In the last step k =k − 1, according to Lemma 2.1 the distribution Ek −1 is completely projectable and case (a) applies. Consequently, the last distribution meets Dk −1 = Ek −1 and zk −1,c is empty. Thus, it is ensured that the variables z 0 , . . . , zk −1 indeed contain all inputs and states exceptxk.

Explicit triangular normal form for two-input systems
The explicit triangular form (23) consists of thek subsystemsx . . .
with k = 1, . . . ,k. The parameterisation of the system variables of the system (23) by the forward-flat output can be obtained by determining step by step the parameterisation of the system variables of the subsystems (29), starting with the topmost subsystem If dim(x k ) = dim(z k−1 ) for all k = 1, . . . ,k, then due to the rank conditions (25) this is particularly simple, and y =xk with dim(xk) = m is a forward-flat output (see Example 4.1). By applying the implicit function theorem to the topmost subsystem (30) we immediately get the parameterisation of the variables zk −1 . Next, since zk −1 contains the state variablesxk −1 (see (24)), by applying the implicit function theorem to the equationŝ we get the parameterisation of the variables zk −2 . Continuing this procedure finally yields the parameterisation of all system variables by the forward-flat output y =xk. However, if dim(xk) < m, then for at least one k ∈ {2, . . . ,k} we have dim(x k ) < dim(z k−1 ), which means that the equationŝ cannot be solved for all components of z k−1 . In this case, the subsystem (29) has redundant inputs, and in addition toxk the forward-flat output has further components. The redundant inputs can be eliminated from the subsystem by suitable coordinate transformations of the structure-preserving form (16). The forward-flat output of the complete system (23) consists ofxk and the eliminated redundant inputs of all subsystems. For systems with two inputs there can occur exactly two cases. If dim(xk) = 2, then y =xk is a forwardflat output and none of the subsystems has redundant inputs. Otherwise, if dim(xk) = 1, then there is exactly one k ∈ {2, . . . ,k} with dim(x k ) = 1 < dim(z k−1 ) = 2. Thus, the corresponding subsystem (29) has one redundant input. By the use of the regular transformationz which is either an input-or state transformation but still of type (16), the subsystem reads aŝ . . .
and is independent of z 1 k−1 . The forward-flat output of the complete system is given by y = (xk, z 1 k−1 ). After the elimination of the possibly occurring redundant input of the subsystem (29) by the coordinate transformation (31), we refer to the resulting system representation as explicit triangular normal form for two-input systems.

Example
Practical examples that are contained within the class of two-input forward-flat discrete-time systems, and thus possess an explicit triangular representation, are for instance the Euler-discretised model of an induction motor (e.g. in Chiasson, 1998;Schöberl, 2014) and the Euler-discretised simplified model of a car (e.g. in Kolar, 2017). However, for illustrational purposes we demonstrate our results with an academic example already discussed in . The system reads as and the sequences of distributions (5) and (6) are given by on X + . Following the procedure of Section 4, first we straighten out the -sequence by a state transformation of the form (11) withx 1 =x 1 1 ,x 2 = (x 1 2 ,x 2 2 ) and x 3 =x 1 3 . With the transformation }, and the system in new coordinates is given bȳ The D-sequence in new coordinates reads as Next, we use Algorithm 4.1 in order to straighten out the D-sequence and transform the system into the explicit triangular representation (23). Due to the fact that E 0 is not completely projectable, the case (b) applies and we need to perform an input transformationû 1 = u 1 + 2u 2 which yields D 0 = span{∂ u 2 }. We define z 0 = u 2 , z 0,c =û 1 and the distribution reads as In the second step, due to the fact that E 1 is completely projectable, case (a) applies. We just define z 1 = (x 1 1 , z 0,c ) = (x 1 1 ,û 1 ), and the distribution D 1 reads as Similarly, the last distribution E 2 is also completely projectable (cf. Lemma 2.1) and thus we have again case (a). We just define z 2 = (x 1 2 ,x 2 2 ), and the distribution D 2 reads as Consequently, with z 0 = (u 2 ), z 1 = (x 1 1 ,û 1 ), z 2 = (x 1 2 ,x 2 2 ) the system has the structure of (23) and reads aŝ x 1,+ 3 = z 1 2 + z 2 2 x 1,+ 2 =x 1 3 z 2 1 + z 1 1 x 2,+ 2 = z 2 1 x 1,+ 1 =x 1 3 + z 1 0 . (33) As mentioned before, the subsystems of (33) may still have redundant inputs. Indeed, because of dim(x 3 ) < dim(z 2 ), the inputs z 1 2 and z 2 2 of the topmost subsystem x 1,+ 3 = z 1 2 + z 2 2 are redundant. This redundancy can be eliminated by the final transformation Since z 2 2 represents a state variable, the Equation (34) defines a state transformation and can be rewritten aŝ Collecting all transformations we performed so far, we obtain the complete transformation which transforms the system (32) into the explicit triangular normal form with the forward-flat output y = (x 1 3 ,x 1 2 ). The parameterising map can now be constructed in a systematic way. From the first equation we immediately get the parameterisation ofx 2 2 . Inserting this parameterisation into the second and the third equation yields the parameterisation ofx 1 1 andû 1 . Finally, inserting the parameterisation ofx 1 1 into the last equation yields the parameterisation of u 2 . With the inverse of (35), the parameterisation of the original system variables x and u follows.

Conclusion
We have shown that every forward-flat nonlinear discrete-time system with two inputs can be transformed into a structurally flat explicit triangular normal form. In contrast to the implicit triangular form discussed in , this normal form is a state representation. The transformation is based on the sequences of distributions (5) and (6), that arise in the test for forward-flatness introduced in . If it is possible to straighten out both sequences of distributions simultaneously by state-and input transformations, then the transformed system has the triangular structure (23). For static feedback linearisable systems, even in the multiinput case with m > 2, this can always be achieved by a state transformation. For forward-flat systems that are not static feedback linearisable, in contrast, there is no guarantee that both sequences can be straightened out simultaneously, even if additionally input transformations are permitted. However, for forward-flat systems with two inputs, straightening out (5) and (6) by state-and input transformations is always possible. Thus, every forward-flat discrete-time system with two inputs can be transformed into an explicit triangular form.
It is important to emphasise that for flat continuoustime systems no comparable result exists. An obvious reason is that the explicit triangular form allows to read off a forward-flat output which depends only on the state variables. In contrast to continuous-time systems, it is shown in  that for forward-flat discrete-time systems such a flat output always exists. Notes 1. We use the same notation as in  and  with a multi-index R = (r 1 , . . . , r m ). 2. The largest projectable subdistribution D (with resp. to f ) of a distribution E is uniquely determined. Furthermore, if E is involutive, then D is also involutive (see . 3. By the term 'explicit' we refer to a state representation (1), in order to distinguish it from the implicit triangular representation discussed in . 4. Note that subsequently we will use the bar notation for system representations where the -sequence is already straightened out.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
The first author and the second author have been supported by the Austrian Science Fund (FWF) under grant number P 32151 and P 29964.