Continuity of Formal Power Series Products in Nonlinear Control Theory

The interconnection of simple input-output systems to form more complex and useful systems is commonplace in science and engineering. When each component is a nonlinear dynamical system, a weighted infinite sum of iterated integrals known as a Chen–Fliess series, \(F_c\), provides a convenient way to represent its local behavior [15‐17, 35, 43]. When this series converges on some set of admissible inputs, \(F_c\) defines a so called Fliess operator. It is uniquely specified by a formal power series c in \(\mathbb {K}\langle \langle X \rangle \rangle \), known as its generating series, where X is a finite set of indeterminates, and \(\mathbb {K}\) is a suitable field. An interconnection of two Chen–Fliess series \(F_c\) and \(F_d\), represented by \(F_c\Box F_d\), induces a corresponding algebra \((\mathbb {K}\langle \langle X \rangle \rangle ,\Box ^\prime )\) so that \(F_c\Box F_d=F_{c\Box ^\prime d}\) [13‐15, 25, 29]. Algebras defined in this manner provide computational frameworks for explicitly computing the generating series of interconnected systems for the purposes of analysis and design, especially in the field of nonlinear control theory. Historically, the coefficients of c have been determined by direct calculations using state space models derived from physical laws and other first principles [33, 38]. But with the growth of adaptive control and new types of learning based technologies, there is increasing interest in estimating these coefficients using real-time data and numerical methods from the field of system identification [23, 39]. Assuming that a given sequence of estimates asymptotically approaches its true value as more data are collected, a difficult problem in its own right, there is a fundamental question regarding continuity.

Consider a sequence of generating series \(c_i\), \(i\ge 1\) known to produce a sequence of corresponding Fliess operators \(F_{c_i}\), \(i\ge 1\). If \(c_i\rightarrow c\) in some manner, is it also true that \(F_{c_i}\), \(i\ge 1\) converges to a well-defined Fliess operator \(F_c\), i.e., does the limit point c ensure a convergent Chen–Fliess series? The answer, of course, depends directly on the ambient sets and the assumed topologies. For example, in [9, 44] the claim is shown to be false on the subset of locally convergent series in \(\mathbb {K}\langle \langle X \rangle \rangle \), (i.e., a set of generating series under which their corresponding Fliess operators are known to converge for sufficiently small arguments in an \(L^p\) norm sense.) endowed with the ultrametric topology and where the operator space has an \(L^p\) type topology. As the ultrametric topology mirrors the algebra but provides almost no information on the analytic behavior of the series, this outcome is not surprising. On the other hand, in [9] the claim is shown to be true when the ultrametric topology is replaced with a certain Banach topology on a subspace. Ultimately, the question boils down to identifying topological vectors spaces contained in \(\mathbb {K}\langle \langle X \rangle \rangle \) which ensure that every limit point is a generating series with a well-defined Fliess operator in some sense.

The main goal of this paper is to address a natural follow-up question: Suppose \(c_i,d_i\in \mathbb {K}\langle \langle X \rangle \rangle \), \(i\ge 1\) are two sequences of generating series converging to c and d, respectively, such that \(F_{c_i}\) and \(F_{d_i}\), \(i\ge 1\) are well defined Fliess operators as are their limits. Assuming that \(c\Box ^\prime d\) has the same convergence properties as c and d (this theory is well understood, see [30, 42, 44]), under which conditions does \(c_i\Box ^\prime d_i\mapsto c\Box ^\prime d\)? Is it even possible to identify infinite-dimensional spaces with respect to which the products are smooth or even analytic?

Three formal power series products will be considered: the shuffle product, which models a type of parallel connection [15]; a composition product modeling series connections [13, 14, 29]; and a group product for a transformation group known to model dynamic output feedback, a central object of study in control theory [25]. In addition, the continuity of the shuffle inverse will be addressed. (A preliminary version of this analysis was presented in [40].) The shuffle group appears in the context of feedback linearization [26, 27]. In each case continuity will be considered in both the Fréchet and Silva topologies. In addition, analyticity of these products will be characterized. It should be noted that the Fréchet topology was used in [44] to show that the shuffle and composition products preserve a type of global convergence. Continuity issues in this setting are beyond the scope of the present paper. However, the Fréchet topology is employed as a natural (locally convex) topology on the space of all power series. Convergence in this topology does not preserve growth bounds. Thus, it is necessary to endow the space of locally convergent series with the finer Silva topology.

Next it will be shown that the output feedback transformation group is a locally convex Lie group (see [37] for a survey on (infinite-dimensional) Lie theory). This result builds on the development of a pre-Lie algebra presented in [12, 18]. Lie groups have a long history in feedback control theory originating with the work of Brockett in [7]. More recent applications in this context have appeared in [27, 28], albeit only in the formal case where an explicit differential structure is not specified. The present work will provide a means to fill this gap. Finally, the regularity of these Lie groups is investigated. Roughly speaking, regularity of a Lie group asks for the existence and smooth parameter dependence of certain ordinary differential equations on the Lie group. Note that since the Lie groups at hand are not modeled on Banach spaces, the usual theory for existence and uniqueness of ordinary differential equations does not apply. However, it is shown that the Fréchet Lie groups are regular. While some progress on the regularity problem is made for the Silva Lie groups, their regularity largely remains an open problem that the authors plan to pursue in future work.

Throughout this paper let \(\mathbb {K}\in \{\mathbb {R},{\mathbb C}\}\), namely either the field of real numbers \(\mathbb {R}\) or the field of complex numbers \({\mathbb C}\). It will be essential to admit complex coefficients in order to discuss analyticity of mappings on infinite-dimensional spaces. Note that the continuity results are unaffected by this choice. Refer to Appendix A for more information regarding calculus on infinite-dimensional spaces and “Appendix B” for a summary of the primary notation used in this paper.

In this section, the continuity of the various products modeling system interconnections described in the previous section is proved. The main goal is to establish continuity on spaces of locally convergent series. In [9], the authors described the space of locally convergent Chen–Fliess series as a locally convex space carrying a Silva space topology. That construction is summarized first, and then the continuity results are presented.

Fix \(M>0\) and definefor each \(c\in \mathbb {K}^{\ell } \langle \langle X \rangle \rangle \). The set of all c with \(\left\| c\right\| _{\ell _{\infty ,M}}<\infty \) is denoted by \(\ell _{\infty ,M}(X^*,\mathbb {K}^\ell )\). It is straightforward to check that \(\ell _{\infty ,M}(X^*,\mathbb {K}^\ell )\) is a vector subspace of \(\mathbb {K}^{\ell } \langle \langle X \rangle \rangle \) and that the assignment \(\left\| \cdot \right\| _{\ell _{\infty ,M}}\) is a norm on \(\ell _{\infty ,M}(X^*,\mathbb {K}^\ell )\). This space is a Banach space as it is isometrically isomorphic to the Banach space of all bounded functions \(\ell _\infty (X^*,\mathbb {K}^\ell ):= \{c :X^* \rightarrow \mathbb {K}^\ell : \sup _{\eta }\left| (c,\eta ) \right| <\infty \}\). The Banach space of generating series bounded with respect to the constant M obviously does not capture all locally convergent series. Indeed for larger M one obtains series which converge only on a smaller disc. To capture all locally convergent series in one space, it is necessary to pass to the limit of these Banach spaces as described next.

One can show that the inclusion mappings in this sequence are compact operators, hence the resulting space is a Silva space [6, 10]. Since the sequence \(M_k = k\), \(k\in \mathbb N\) is cofinal, one can always find an \(M\in \mathbb N\) for which \(\left\| c\right\| _{\ell _{\infty ,M}}<\infty \). Thus, one could equivalently work only with \(M\in \mathbb N\). Though the Silva space topology is more complicated than the Banach spaces from which it was built, some of its properties make it very amenable for the applications considered here. The most important properties are summarized in the next lemma. Refer to [45] for proofs and more information about Silva spaces.

Perhaps the most striking property of the Silva topology is that one can address continuity and differentiability questions in the Banach spaces from which the Silva space is built. This will be demonstrated in the next section addressing the continuity of formal power series products.

In this section, it is proved that the formal power series products and inverse presented in the previous sections are not only continuous but also analytic. Note that on the infinite-dimensional spaces involved, both complex and real analyticity make sense, cf. Appendix A. For real analyticity, one needs only to identify the complexification of the spaces \({\mathbb R}\langle \langle X \rangle \rangle \) and \({\mathbb R}_{LC}\langle \langle X \rangle \rangle \).

As locally convex spaces, the complexification of \({\mathbb R}\langle \langle X \rangle \rangle \) is \({\mathbb C}\langle \langle X \rangle \rangle \). This is clear on the level of vector spaces, and for the topology simply note that as topological vector spaces \({\mathbb C}\langle \langle X \rangle \rangle = {\mathbb R}\langle \langle X \rangle \rangle \oplus i{\mathbb R}\langle \langle X \rangle \rangle \). Similarly, the complexification of the Silva space \({\mathbb R}_{LC}\langle \langle X \rangle \rangle \) is \({\mathbb C}_{LC}\langle \langle X \rangle \rangle \). Again this is clear on the level of vector spaces but more complicated on the level of the vector space topologies. However, also the vector space topologies coincide as it is easy to see that for every \(M >0\) the Banach space \(\ell _{\infty ,M}(X^*,{\mathbb C}^\ell )\) is the complexification of \(\ell _{\infty ,M}(X^*,\mathbb {R}^\ell )\), and the inductive limit of a sequence of compact operators between Banach spaces commutes with the formation of complexifications [32,  Theorem 3.4].

Having identified the complexification of the infinite-dimensional spaces, observe that the shuffle product, the composition product and the shuffle inverse are all well defined on both the complexification and on the real space. Hence, if it can be proved that these mappings are holomorphic on the complexification, then real analyticity is obtained for the corresponding mappings on the real space. Before continuing with the shuffle product and the shuffle inverse, it is helpful to recall a special type of locally convex algebra.

It is well known that the unit group of a CIA is an infinite-dimensional Lie group. Before stating the next result, recall some standard notation. For a differentiable map \(f :M \rightarrow N\) between manifolds, one denotes by \(Tf :TM \rightarrow TN\) the tangent map. The tangent map generalizes the differential of a map, i.e., locally in charts or on open subsets of a vector space, the tangent Tf is given as (f, df), where df is the differential. With this notation in place, recall the following standard notions from infinite-dimensional Lie theory stated below.

Every Banach Lie group is \(C^0\)-regular (cf. [37]). Several important results in infinite-dimensional Lie theory are only available for regular Lie groups. For example, the interplay between Lie algebra and Lie group hinges on regularity as this property guarantees existence of a smooth Lie group exponential function. Moreover, if one wants to lift morphisms of Lie algebras to the Lie group by integration, this requires the group to be regular, cf. [34].

A Lie group structure on the group \((\delta + \mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle ,\circ ,\delta )\) is presented in this section. This group is known to have an associated graded and connected Hopf algebra \((H,\mu ,\Delta )\) as described in Sect. 2.2 and more completely in [25,  Section 3]. This structure will play an important role in the proof of the Lie group property. Note, however, that \((\delta +\mathbb {K}^m\langle \langle X \rangle \rangle ,\circ ,\delta )\) is not the character group of said Hopf algebra, and thus the Lie theory for such groups from [5, 10] is not directly applicable. The main claim, as stated below, is established from first principles.

The argument for the Lie group structure on subsets of locally convergent series can be adapted almost verbatim to the case where no convergence of the series is assumed.

While the Fréchet  Lie group \(\delta + \mathbb {K}^m\langle \langle X \rangle \rangle \) is much simpler (topologically speaking) than the Silva group \(\delta +\mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle \), it supplies a useful template for the Lie theoretic arguments considered next, namely identifying the Lie algebra and proving that a Lie group is regular in the sense of Milnor. The first goal is to establish these properties for the simpler FréchetLie group. Subsequently, it is shown that these results then imply corresponding properties for the Silva Lie group. However, it is first necessary to introduce a new structure which will yield a convenient description of the Lie bracket. This structure is the so called pre-Lie product, which was developed in [18] for the case where \(m=1\) and generalized in [12,  Section 3.2] for the case where \(m\ge 1\).

This product can be viewed as the linear part of the group product, that is,where \({\mathcal O}(c,d^2)\) denotes all terms depending linearly on c and on higher powers of d. One can show that the pre-Lie product preserves the length of words in the sense that \((\eta \lhd \xi ,\nu )=0\) when \(\left| \eta \right| +\left| \xi \right| \ne \left| \nu \right| \). Therefore, the product is well defined as it is locally finite. Moreover, defining \(d[0]=0\), the recursive formulas reduce to a single expression

Regularity of the Fréchet Lie group \(\delta + \mathbb {K}\langle \langle X \rangle \rangle \) is investigated next. For a curve \(\gamma _\delta (t) = (\delta + \gamma (t)) \in \delta + \mathbb {K}\langle \langle X \rangle \rangle \) consider the Lie type differential equationwhere \(c :[0,1] \rightarrow \mathbb {K}\langle \langle X \rangle \rangle \) is a continuous curve. For every \(\eta \in X^*\) observe that \((\gamma _\delta (t),\eta ) = (\gamma (t),\eta )\). Now since the coordinate functions are continuous and linear, a differential equation is obtained for every word \(\eta \in X^*\):The computations in Example 5.1 have been used above, and the products of elements in \(\mathbb {K}^m\) are taken as componentwise products. Note now that the sum in (17) only appears if \(|\eta |_{x_0} \ne 0\). Hence, if a word does not contain the letter \(x_0\), then the differential equation (17) reduces toSince \((c(t),\eta )\) is a continuous \(\mathbb {K}^m\)-valued curve, one can solve the above equation for all \(t \in [0,1]\). Now if \(\eta \) is a word with \(|\eta |_{x_0} \ne 0\), observe that all elements in (17) appearing as coefficients of evaluations of \(\gamma \) are continuous \(\mathbb {K}^m\)-valued curves of the form \((c(t),\eta )\) orIt is now proved via induction on the length of the words that equation (17) admits a solution on [0, 1] for every word. Note first that for any word without an \(x_0\) (such as the empty word, which is the only length zero element), the statement follows directly from the integral equation (18). If \(|\eta | = n >1\) assume that the statement is true for all words of lower length. If \(|\eta |_{x_0} =0\), the statement follows again from (18). To obtain solutions for the words of length n containing \(x_0\), pick an enumeration \((\eta _i)_{i \in I_n}\) of words of length n. Using the enumeration and (19), defineThen, (17) together with the observation that \((\eta \lhd c , \eta )=0\) give rise to the following inhomogeneous system of linear differential equations on \((\mathbb {K}^m)^{|I_n|}\):Now by the induction hypothesis the inhomogeneity \({\varvec{b}}_n\) in (20) is already completely determined by the previous computations. Furthermore, the coefficient matrix \(C_n\) is determined by c and thus continuous in t. Hence, one can solve the system (20) and obtain a solution on [0, 1] (via the usual solution theory for linear differential equations on finite-dimensional spaces). This completes the induction, and thus, one can iteratively solve the inhomogeneous linear system (20) for every \(n \in \mathbb {N}_0\) with a unique solution on [0, 1]. Following [11,  §6] (cf. also [2]), the solution to the Lie type equation (16) is the solution to the infinite system of differential equations (18) andThe earlier discussion has shown that this system is lower diagonal, i.e., the right-hand side of the equation in degree n depends only on the solutions up to degree n. One can now solve the differential equation on the Fréchet space by adapting the argument in [11,  p. 79-80]: Lower diagonal systems can be solved iteratively component-by-component, if each solution exists on a time interval \([0, \varepsilon ]\) for some fixed \(\varepsilon > 0\). Choosing \(\varepsilon =1\), observe that the Lie type equation (16) admits a unique global solution which can be computed iteratively. Thus, the following result is evident.

Observe that one can leverage the regularity of the Fréchet Lie group in the investigation of the regularity for the Silva Lie group \(\delta +\mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle \). The inclusion \(\iota :\delta + \mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle \rightarrow \delta + \mathbb {K}^m\langle \langle X \rangle \rangle \) is a Lie group morphism which relates the solutions of the evolution equation on the Silva and the Fréchet  Lie groups. Indeed, [21, 1.16] shows that for a continuous curve \(c :[0,1] \rightarrow \mathbf {L}(\delta +\mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle )=\mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle \) a solution to the evolution equation (16) in \(\delta + \mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle \) must satisfywhere c is interpreted canonically as a curve into \(\mathbf {L}(\delta +\mathbb {K}^m\langle \langle X \rangle \rangle )=\mathbb {K}^m\langle \langle X \rangle \rangle \) via the natural inclusion. Hence, the Silva Lie group will be \(C^0\)-semiregular if and only if it can be proved that the solutions to the evolution equation on the Fréchet Lie group are bounded when the curve c is bounded. Unfortunately, at present it is not obvious how to bound these solutions to the evolution equation, which leads to the following.

Open problem: Is the Silva Lie group \(\delta + \mathbb {K}^{m}_{LC}\langle \langle X \rangle \rangle \) \(C^0\)-semiregular?