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Journal Article•DOI•

A note on persistency of excitation

Jan C. Willems1, Paolo Rapisarda2, Ivan Markovsky1, Bart De Moor1•
Katholieke Universiteit Leuven1, Maastricht University2
01 Apr 2005-Systems & Control Letters (Elsevier Science)-Vol. 54, Iss: 4, pp 325-329
TL;DR: It is proved that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior.
About: This article is published in Systems & Control Letters.The article was published on 2005-04-01 and is currently open access. It has received 615 citations till now. The article focuses on the topics: Matrix (mathematics) & Linear independence.

Summary (1 min read)

Jump to: [1. Introduction] – [2. Linear time-invariant systems] – [3. Sequences with spanning windows] and [4. Comments and corollaries]

1. Introduction

  • The authors examine consequences of persistency of excitation using the behavioral language.
  • The problem studied may be posed as follows.

2. Linear time-invariant systems

  • The authors denote the class of systems =(N, Rw,B) satisfying (i)–(iii) by Lw.
  • The kernel representation associated with a givenB ∈ L(B) is the smallest possible lag over all kernel representations ofB.
  • In fact, L(B) is also the smallest such thatN B generates the moduleNB.
  • These integers are all readily computable from a kernel representation, and certainly from an input/state/output representation ofB (see [10, Section 7]).

3. Sequences with spanning windows

  • The following is the main result of the paper.
  • The inclusion leftkernel(HL(w̃)) ⊇ NLB is obvious.

4. Comments and corollaries

  • In other words, the authors have to assume a ‘deeper’ persistency of excitation oñu than the width of the windows of(ũ, ỹ) which are considered.
  • So, in particular, ifL l(B), and under persistency of excitation of orderL+n(B), HL(w̃) has full row rank.
  • The observed system signal then completely specifies the laws of the system.
  • Special cases that were studied are:u white noise[2] andu periodic [9, Theorem 2].

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Citations
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Journal Article•DOI•
Overview of total least-squares methods

[...]

Ivan Markovsky1, Sabine Van Huffel2•
University of Southampton1, Katholieke Universiteit Leuven2
01 Oct 2007-Signal Processing
TL;DR: It is explained how special structure of the weight matrix and the data matrix can be exploited for efficient cost function and first derivative computation that allows to obtain computationally efficient solution methods.

745 citations

Cites background from "A note on persistency of excitation..."

  • ...Moreover the solution is unique whenever A is of full column rank, which can be translated to a persistency of excitation condition on a, see [79]....

    [...]

Subspace model identification-Part 1. The output-error state-space model identification class of algorithms

[...]

M. Verhaegen
01 Jan 1992
TL;DR: Two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data are presented: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set of equations.
Abstract: In this paper, we present two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data. The algorithms have a number of common features. They are classified as one of the subspace model identification schemes, in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input-output data. This structure is then exploited in the calculation of a realization. Another common feature is their algorithmic organization: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set (or sets) of equations. The schemes assume that the underlying system has an output-error structure and that a measurable input sequence is available. The latter characteristic indicates that both schemes are versions of the MIMO Output-Error State Space model identification (MOESP) approach. The first algorithm is denoted in particular as the (elementary MOESP scheme)...

660 citations

Journal Article•DOI•
System Identification Methods for (Operational) Modal Analysis: Review and Comparison

[...]

Edwin Reynders1•
Katholieke Universiteit Leuven1
09 Feb 2012-Archives of Computational Methods in Engineering
TL;DR: In this article, the authors extensively review operational modal analysis approaches and related system identification methods and compare them in an extensive Monte Carlo simulation study, and then compare the results with the results obtained in an experimental setting.
Abstract: Operational modal analysis deals with the estimation of modal parameters from vibration data obtained in operational rather than laboratory conditions. This paper extensively reviews operational modal analysis approaches and related system identification methods. First, the mathematical models employed in identification are related to the equations of motion, and their modal structure is revealed. Then, strategies that are common to the vast majority of identification algorithms are discussed before detailing some powerful algorithms. The extraction and validation of modal parameter estimates and their uncertainties from the identified system models is discussed as well. Finally, different modal analysis approaches and algorithms are compared in an extensive Monte Carlo simulation study.

481 citations

Proceedings Article•DOI•
Data-Enabled Predictive Control: In the Shallows of the DeePC

[...]

Jeremy Coulson1, John Lygeros1, Florian Dörfler1•
ETH Zurich1
25 Jun 2019
TL;DR: In this paper, a data-enabled predictive control (DeePC) algorithm is presented that computes optimal and safe control policies using real-time feedback driving the unknown system along a desired trajectory while satisfying system constraints.
Abstract: We consider the problem of optimal trajectory tracking for unknown systems. A novel data-enabled predictive control (DeePC) algorithm is presented that computes optimal and safe control policies using real-time feedback driving the unknown system along a desired trajectory while satisfying system constraints. Using a finite number of data samples from the unknown system, our proposed algorithm uses a behavioural systems theory approach to learn a non-parametric system model used to predict future trajectories. The DeePC algorithm is shown to be equivalent to the classical and widely adopted Model Predictive Control (MPC) algorithm in the case of deterministic linear time-invariant systems. In the case of nonlinear stochastic systems, we propose regularizations to the DeePC algorithm. Simulations are provided to illustrate performance and compare the algorithm with other methods.

411 citations

Journal Article•DOI•
Data-Driven Model Predictive Control With Stability and Robustness Guarantees

[...]

Julian Berberich1, Johannes Köhler1, Matthias A. Müller2, Frank Allgöwer1•
University of Stuttgart1, Leibniz University of Hanover2
18 Apr 2021-IEEE Transactions on Automatic Control
TL;DR: The presented results provide the first (theoretical) analysis of closed-loop properties, resulting from a simple, purely data-driven MPC scheme, including a slack variable with regularization in the cost.
Abstract: We propose a robust data-driven model predictive control (MPC) scheme to control linear time-invariant systems. The scheme uses an implicit model description based on behavioral systems theory and past measured trajectories. In particular, it does not require any prior identification step, but only an initially measured input–output trajectory as well as an upper bound on the order of the unknown system. First, we prove exponential stability of a nominal data-driven MPC scheme with terminal equality constraints in the case of no measurement noise. For bounded additive output measurement noise, we propose a robust modification of the scheme, including a slack variable with regularization in the cost. We prove that the application of this robust MPC scheme in a multistep fashion leads to practical exponential stability of the closed loop w.r.t. the noise level. The presented results provide the first (theoretical) analysis of closed-loop properties, resulting from a simple, purely data-driven MPC scheme.

381 citations

Cites background or methods from "A note on persistency of excitation..."

  • ...Moreover, the recent contributions [20]–[22] set up an MPC scheme based on [12], but no guarantees on recursive feasibility or closed-loop stability can be given, since neither terminal ingredients are included in the MPC scheme nor sufficient lower bounds on the prediction horizon are derived....

    [...]

  • ...An exposition of the main result of [12] in the classical state-space control framework and an extension to certain classes of nonlinear systems are provided in [16]....

    [...]

  • ...The result originates from behavioral systems theory [12], but we employ the formulation in the classical state-space control framework [16]....

    [...]

  • ...Moreover, since it relies on the data-driven system description from [12], the presented scheme is inherently an output-feedback MPC scheme and does not require online state measurements....

    [...]

  • ...Our approach relies on a result from behavioral systems theory, which shows that the Hankel matrix consisting of a previously measured input–output trajectory spans the vector space of all trajectories of an LTI system, given that the input component is persistently exciting [12]....

    [...]

References
More filters
Journal Article•DOI•
Paradigms and puzzles in the theory of dynamical systems

[...]

Jan C. Willems1•
University of Groningen1
01 Mar 1991-IEEE Transactions on Automatic Control
TL;DR: In this article, a self-contained exposition is given of an approach to mathematical models, in particular to the theory of dynamical systems, which leads to a new view of the notions of controllability and observability, and of the interconnection of systems.
Abstract: A self-contained exposition is given of an approach to mathematical models, in particular, to the theory of dynamical systems. The basic ingredients form a triptych, with the behavior of a system in the center, and behavioral equations with latent variables as side panels. The author discusses a variety of representation and parametrization problems, in particular, questions related to input/output and state models. The proposed concept of a dynamical system leads to a new view of the notions of controllability and observability, and of the interconnection of systems, in particular, to what constitutes a feedback control law. The final issue addressed is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. >

1,219 citations

Subspace model identification-Part 1. The output-error state-space model identification class of algorithms

[...]

M. Verhaegen
01 Jan 1992
TL;DR: Two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data are presented: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set of equations.
Abstract: In this paper, we present two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data. The algorithms have a number of common features. They are classified as one of the subspace model identification schemes, in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input-output data. This structure is then exploited in the calculation of a realization. Another common feature is their algorithmic organization: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set (or sets) of equations. The schemes assume that the underlying system has an output-error structure and that a measurable input sequence is available. The latter characteristic indicates that both schemes are versions of the MIMO Output-Error State Space model identification (MOESP) approach. The first algorithm is denoted in particular as the (elementary MOESP scheme)...

660 citations

Journal Article•DOI•
Subspace model identification Part 1. The output-error state-space model identification class of algorithms

[...]

Michel Verhaegen1, Patrick Dewilde1•
Delft University of Technology1
01 Nov 1992-International Journal of Control
TL;DR: In this paper, the authors present two algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data, which are classified as one of the subspace model identification schemes.
Abstract: In this paper, we present two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data. The algorithms have a number of common features. They are classified as one of the subspace model identification schemes, in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input-output data. This structure is then exploited in the calculation of a realization. Another common feature is their algorithmic organization: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set (or sets) of equations. The schemes assume that the underlying system has an output-error structure and that a measurable input sequence is available. The latter characteristic indicates that both schemes are versions of the MIMO Output-Error State Space model identification (MOESP) approach. The first algorithm is denoted in particular as the (elementary MOESP scheme)...

624 citations

Journal Article•DOI•
From time series to linear system-part I. Finite dimensional linear time invariant systems

[...]

Jan C. Willems1•
University of Groningen1
01 Sep 1986-Automatica
TL;DR: The structural indices of such systems are introduced and it is shown how an (AR) representation of a system having a given behaviour can be constructed.

530 citations

Journal Article•DOI•
From time series to linear system-Part II. Exact modelling

[...]

Jan C. Willems1•
University of Groningen1
01 Nov 1986-Automatica
TL;DR: First a mathematical vocabulary for discussing exact modelling is developed, and it is shown how the results of Part I guarantee the existence of a most powerful (AR) model for an observed time series.

251 citations

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Frequently Asked Questions (1)
Q1. What have the authors contributed in "A note on persistency of excitation" ?

The authors prove that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior.