An algorithm for closed-loop data-driven simulation

Abstract

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 }.