Exact and Approximate Modeling of Linear Systems

Markovsky, Ivan; Willems, Jan C.; Huffel, Sabine Van; Moor, Bart De

doi:10.1137/1.9780898718263

Cited by 109 publications

(116 citation statements)

References 0 publications

Supporting

Mentioning

116

Contrasting

Order By: Relevance

“…The data-driven simulation problem is defined in , where its applications in subspace system identification Van Overschee and De Moor (1996);Markovsky et al (2006) are presented. Its fundamental role in data-driven control is shown in Rapisarda (2007, 2008).…”

Section: Closed-loop Data-driven Simulationmentioning

confidence: 99%

An algorithm for closed-loop data-driven simulation

Markovsky

2009

IFAC Proceedings Volumes

Self Cite

View full text Add to dashboard Cite

Closed-loop data-driven simulation refers to the problem of constructing trajectories of a closed-loop system directly from data of the plant and a representation of the controller. Conditions under which the problem has a solution are given and an algorithm for computing the solution is presented. The problem formulation and its solution are in the spirit of the deterministic identification algorithms, i.e., in the theoretical analysis of the method, the data is assumed exact (noise free).Keywords: Simulation, system identification, behaviors. PRELIMINARIES AND NOTATIONWe use the behavioural language, see Willems (1986). A discrete-time dynamical system with w manifest variables is a subset of the signal space (R w ) N , i.e., the set of functions from the set of natural numbers N to R w . We assume that the manifest variables w have a given input/output partitionWe consider linear, time-invariant, and finite dimensional plants and controllers. A kernel representation R(σ )w = 0, were σ is the backwards shift operator σ w(t) := w(t + 1), is parameterized by the polynomial matrix R ∈ R p×(m+p) [z], and an image representation w = M(σ )g is parameterized by the polynomial matrix M ∈ R (m+p)×m [z].The Hankel matrix with t block rows, composed of the signal w ∈ (R w ) T is denoted byThe banded upper-triangular Toeplitz matrix with t blockcolumns, related to the polynomial r ∈ R 1×r [z], deg(r) =: n is denoted by CLOSED-LOOP DATA-DRIVEN SIMULATIONThe data-driven simulation problem is defined in , where its applications in subspace system identification Van Overschee and De Moor (1996);Markovsky et al. (2006) are presented. Its fundamental role in data-driven control is shown in Rapisarda (2007, 2008). This paper further develops the concept of data-driven simulation to closed-loop data-driven simulation, defined as follows. Problem 1. (Closed-loop data-driven simulation). Givenfind the set of responses w r of the closed-loop system B C to the reference signal r r . Solution and computational algorithmFor given w d , C , r r , we aim to compute the signals w r , such that (r r , w r ) ∈ B C ⇐⇒ w r ∈ B (r r , w r ) ∈ CAssumption that B is controllable, there is M ∈ R w×m [z], such that B = image M(σ ) . Consider a kernel representation of the controller C = { (r, w) | R(σ ) col(r, w) = 0 }.

show abstract

Section: Closed-loop Data-driven Simulationmentioning

confidence: 99%

An algorithm for closed-loop data-driven simulation

Markovsky

2009

IFAC Proceedings Volumes

Self Cite

View full text Add to dashboard Cite

show abstract

“…Section 4 deals with the inverse problem of modeling the forward and backward solution space of a system, and Section 5 combines the results of the previous sections into a single algorithm. Lastly, in Section 6, following the behavioral framework of Antoulas and Willems (1993), Markovsky et al (2006), Willems (1986;, or Zerz (2008a;2008b;2011), we study the conditions under which the constructed system is most powerful. Section 7 concludes the paper.…”

Section: Introductionmentioning

confidence: 99%

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

Moysis

Karampetakis

2017

International Journal of Applied Mathematics and Computer Science

View full text Add to dashboard Cite

This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ)β(k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ). This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ)β(k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.

show abstract

“…Due to the linear constraint, problem (4) is a structured total least-squares (STLS) problem [6,Ch. 4].…”

Section: Introductionmentioning

confidence: 99%

A Recursive Restricted Total Least-Squares Algorithm

Rhode

Usevich

Markovsky

et al. 2014

IEEE Trans. Signal Process.

Self Cite

View full text Add to dashboard Cite

Abstract-We show that the generalized total least squares (GTLS) problem with a singular noise covariance matrix is equivalent to the restricted total least squares (RTLS) problem and propose a recursive method for its numerical solution. The method is based on the generalized inverse iteration. The estimation error covariance matrix and the estimated augmented correction are also characterized and computed recursively. The algorithm is cheap to compute and is suitable for online implementation. Simulation results in least squares (LS), data least squares (DLS), total least squares (TLS), and RTLS noise scenarios show fast convergence of the parameter estimates to their optimal values obtained by corresponding batch algorithms.

show abstract

Exact and Approximate Modeling of Linear Systems

Cited by 109 publications

References 0 publications

An algorithm for closed-loop data-driven simulation

An algorithm for closed-loop data-driven simulation

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

A Recursive Restricted Total Least-Squares Algorithm

Contact Info

Product

Resources

About