We examine when differentially flat nonlinear control systems with more than two inputs can be rendered static feedback linearizable by prolongations of suitably chosen inputs after applying a static input transformation. Building on the structure of the time derivatives of a flat output, we derive sufficient conditions that guarantee such prolongations yield a static feedback linearizable system. In the two-input case, prior work established precise links between relative degrees, the highest derivative orders occurring in a flat parameterization, and the minimal dimension of a linearizing dynamic extension, leading to necessary and sufficient criteria for systems that become static feedback linearizable after at most two prolongations of such suitably chosen inputs. This work extends this analysis to systems with more than two inputs and derives particular results for the three-input case.
A central objective in nonlinear control theory is to characterize systems that can be transformed into canonical forms such as chains of integrators using suitable state and input transformations. In this context, differential flatness [1] is a fundamental concept: A nonlinear control system
with n state variables x and m control inputs u is called differentially flat, or simply flat, if there exists an m-tuple of differentially independent functions y = φ(x, u, u (1) , . . . , u (ν) ),
where u (ν) denotes the ν-th time derivative of u, such that (x, u) = F (y, y (1) , . . . , y (r) ).
This means both x and u can be expressed via the socalled flat parameterization (3) in terms of the flat output (2) and finitely many of its time derivatives. This property enables systematic solutions to generally non-trivial tasks such as trajectory planning and tracking [2]. In flatness-based tracking control design, one typically applies an endogenous feedback that exactly linearizes the system such that the closed-loop input-output dynamics reduce to m integrator chains y (lj ) j without a general solution, as highlighted, e.g., by [3]- [7].
A well-known subclass of flat systems with a comprehensive solution [8] are static feedback linearizable (SFL) systems, i.e., systems that can be exactly linearized by a static state feedback u = α(x, w). 1 The corresponding flat output is referred to as a linearizing output.
In general, flat systems admit exact linearization by endogenous dynamic feedback-roughly speaking, an endogenous feedback involves only system variables, i.e., states, inputs, and time derivatives of inputs-which can be interpreted as static feedback linearization of a suitable dynamic extension. A well-studied class of endogenous dynamic feedback laws arises from original input prolongations, i.e., dynamic extensions obtained by finitely many derivatives of the original control inputs. Flat systems that become SFL after a finite number of such input prolongations have been investigated in depth, see [9]- [12]. Necessary and sufficient conditions for this property were recently established in [13]. Taking a more general perspective, one may ask whether a system can be rendered SFL by prolongations of selected components of a suitably transformed input ū = ϕ(x, u). We refer to this problem simply as static feedback linearizability via prolongations.
The minimal order of a dynamic extension required to render a system SFL with φ as a linearizing output is called the differential difference of φ, denoted by d diff (φ). The differential difference of a system, simply denoted by d diff , is the smallest differential difference among all possible flat outputs. As shown in [5], every flat system with d diff = 1 becomes SFL after a one-fold prolongation of a suitably chosen input. Control-affine systems with differential difference of one have been fully characterized with respect to flatness in [5]. 2Previous work shows that every (x, u)-flat system with two inputs becomes SFL after d diff -fold prolongations of a suitably chosen control input [14], directly linking the state dimension, relative degrees, highest derivative orders in the flat parameterization, and the differential difference. By additionally establishing procedures to identify such suitable input transformations without explicitly computing a flat output, it has been shown that the gap to SFL systems can be systematically reduced. Building on these results, necessary and sufficient conditions have been established for flatness of two-input systems with d diff ≤ 2 [4], [15]. Motivated by these developments, we study static feedback linearizability via prolongations for systems with multiple, that is, more than two, inputs.
Our contribution is two-fold:
A) We derive sufficient conditions under which flat multiinput systems are SFL via a minimal number of prolongations. B) For (x, u)-flat three-input systems, we describe the structure of the time derivatives of the flat-output components under suitable input transformations and we characterize SFL via minimal prolongations in terms of the relative degrees, the highest derivative orders occurring in the flat parameterization and the differential difference.
Our paper is structured as follows: Section II introduces notation and terminology followed by Section III recalling key results for SFL of two-input systems. Our main results accompanied by illustrating examples are presented in Section IV followed by Section V giving a conclusion and an outlook for further research. Detailed proofs are presented in the appendix.
This work utilizes tensor notation and the Einstein summation convention. Consider an n-dimensional smooth manifold X with local coordinates x = (x 1 , . . . , x n ) and an m-tuple of functions h = (h 1 , . . . , h m ) : X → R m . By ∂ x h, we denote the m × n Jacobian matrix, and ∂ x i h j represents the partial derivative of h j with respect to x i . The notation dh represents the
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