A graph-oriented approach to address generically flat outputs in structured LTI discrete-time systems

Abstract

This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.

Introduction

This paper is concerned with flatness of discrete-time structured dynamical systems. Flatness of discrete-time systems is usually called difference flatness. It has been first reported in Sira-Ramirez and Agrawal (2004) and Fliess and Marquez (2000). It acts as the discrete-time counterpart of differential flatness, introduced in Fliess et al. (1995), that applies for continuous-time systems. Let us recall that for a flat continuous-time system, flatness gives a complete parametrization of all system variables (inputs and states) in terms of a finite number of independent variables and a finite number of their time derivatives. Those variables are called flat outputs. For a flat discrete-time system, the state variables as well as the input can be written as a function of the flat output and its backward/forward shifts. This being the case, flatness is interesting for both control and state reconstruction perspectives. For control purposes, the parametrization of the input in terms of outputs of the system provides in a straightforward manner a constructive way to design a feedforward control to track a prescribed trajectory of the plant output. The reader may consult Yong et al. (2015) or Chapter 5 in the book Sira-Ramirez and Agrawal (2004) for illustrative examples in the case of LTI discrete-time systems. As for state reconstruction, the parametrization of the state in terms of outputs of the system provides in a straightforward manner a constructive way to design an unknown input state observer. Such an issue has been discussed in Daafouz et al. (2006) in a general statement or for example in Shoukry et al. (2015) in the context of cybersecurity where the state reconstruction allows for detecting sensor attacks.

Most of the definitions, including the ones given in Sira-Ramirez and Agrawal (2004) and Yong et al. (2015) dealing with LTI systems, call for backward flatness or forward flatness, i.e., backward or forward shifts exclusively are involved in the expressions of the state and the input. However, more general definitions involving both backward and forward shifts have been recently proposed and motivated in Guillot and Millérioux (2020) and Diwold et al. (2022) for both linear and nonlinear systems. Difference flatness is motivated by the fact that some systems are intrinsically discrete (models of population growth, economy, biology, finance, discrete automata,…). Besides, it must be stressed that the property of flatness may not be preserved when a flat continuous-time system is discretized, even in the linear case. Hence, difference flatness for sampled-data systems should preferably be addressed directly within the discrete-time framework. Specific characterizations of flatness have been provided in the literature according to distinct classes of discrete-time systems as LTI systems (Sira-Ramirez and Agrawal, 2004, Yong et al., 2015), switched linear systems (Millérioux & Daafouz, 2009), LPV systems (Parriaux & Millérioux, 2013), or more general classes of nonlinear systems (Guillot and Millérioux, 2020, Kaldmäe and Kotta, 2013, Kolar, Kaldmäe, et al., 2016, Kolar, Schöberl, and Schlacher, 2016, Sato, 2012).

As it turns out, a general framework based on structural and graph-oriented approaches has never been proposed so far to deal with difference flatness of structured LTI systems. And yet, those approaches have been used with success over the years to characterize many structural properties of dynamical systems like controllability, observability (including with unknown inputs), and identifiability. The reader may refer to the survey Ramos et al. (2020) that gives an exhaustive overview of the works and applications of structural analysis from the seminal paper Dion et al. (2003) to most recent ones. We can also mention extension of results to other classes of systems like descriptor systems (Clark et al., 2017), bilinear systems (Boukhobza & Hamelin, 2007), switching systems (Boukhobza, 2012) or complex nonlinear networks (Kawano & Cao, 2019) to mention a few. An attempt to establish results on flatness had been proposed in Boukhobza and Millérioux (2016) but it was restricted to SISO systems and the approach was not suitable to tackle general LTI systems. Structural analysis allows to characterize properties independently of the exact values of the parameters and thus, to deal with systems of which the model equations are not known exactly. Furthermore, structural models usually involve equations derived from physical laws where the states are variables that get a physical meaning. Hence, structural properties are easily interpreted in terms of physical ones. In this respect, the applicability of the graph-oriented approaches is large and can also be efficient for sensor placements, reachability problems, reliability analysis, security in Cyber Physical Systems as in Dakil et al. (2015) and Gracy et al. (2020) but also in life sciences as biology (Liu & Linqiang, 2015) for example.

The aim of this paper is to propose a graph-oriented approach to address flatness for the class of structured LTI discrete-time systems. More specifically, necessary and sufficient graphical conditions for an output to be a structural flat output are given. These conditions can be checked by resorting to well-known algorithms, commonly used for finding successors and predecessors of vertex subsets, or for computing maximal linkings and essential vertices in a digraph. As a result, the proposed solution is simple to implement and has polynomial complexity.

The paper is organized as follows. Section 2 is devoted to the problem statement. The definitions of a difference flat output and a difference flat system are recalled and a preliminary result (Theorem 1) is established. It gives an algebraic characterization of a flat output in terms of invariant zeros. The result is quite general since it does not exclusively apply to structural systems. In Section 3, structured systems and the notion of structural flatness are introduced. Necessary background on graph-theoretic tools and recalls on digraph representation of LTI structured discrete-time systems are provided. In Section 4, the main result is established. It gives a necessary and sufficient condition (Theorem 2) for an output of an LTI system to be structurally flat. An equivalent characterization (Theorem 3) is also provided. In Section 5, the conditions are illustrated with some basic examples. Section 6 ends this paper with some concluding remarks and possible further work.

Standard notation: $I_k$, ($k\in \mathbb{N}$) stands for the $k$-dimensional identity matrix. For a vector $z$ of dimension $n$ ($n\in \mathbb{N}$), $z_i$ with $i\in \{1,\dots ,n\}$ denotes its $i$th component. For a $m\times l$-dimensional matrix $M$ (being $m$ and $l$ natural integers), $M(i,j)$ with $i\in \{1,\dots ,m\}$ and $j\in \{1,\dots ,l\}$ denotes the entry of $M$ located at row $i$ and column $j$.