Algorithms for deterministic balanced subspace identification

Abstract

New algorithms for identification of a balanced state space representation are proposed. They are based on a procedure for the estimation of impulse response and sequential zero input responses directly from data. The proposed algorithms are more efficient than the existing alternatives that compute the whole Hankel matrix of Markov parameters. It is shown that the computations can be performed on Hankel matrices of the input-output data of various dimensions. By choosing wider matrices, we need persistency of excitation of smaller order. Moreover, this leads to computational savings and improved statistical accuracy when the data is noisy. Using a finite amount of input-output data, the existing algorithms compute finite time balanced representation and the identified models have a lower bound on the distance to an exact balanced representation. The proposed algorithm can approximate arbitrarily closely an exact balanced representation. Moreover, the finite time balancing parameter can be selected automatically by monitoring the decay of the impulse response. We show what is the optimal in terms of minimal identifiability condition partition of the data into "past" and "future".

Economic model predictive control of nonlinear process systems using empirical models

“…Some of the various subspace system identification algorithms in the literature include multi-variable output error state-space algorithm (MOESP) [24,41,28,33], the Canonical Variate Algorithm (CVA) [44], and numerical algorithms for subspace statespace system identification (N4SID) [27]. Identifying the deterministic part of a MIMO state-space model using SMI methods has proven to be successful in the context of industrial settings [41,43,33,32,31]. Combining subspace methods with MPC has also been considered (see, for instance, [30] and the references contained therein).…”

System-theoretic Interpretation of the Mode Sensing Hypothesis

“…Hence, the state covariance matrix R D lim t!1 R.t / is the stochastic counterpart of the deterministic controllability Gramian W c , albeit additionally weighted by the input noise covariance matrix V , as can be seen by comparing (47) with (40) and (48) with (44). The additional presence of the noise covariance matrix V in (47) and (48) is necessary, because V expresses the stochastic character of input w in (45) and was thus absent in (8) for deterministic input u.…”

“…Generalisation to the nonlinear case represents a much greater challenge for at least three reasons: (i) the nonlinear theory of balanced realisations is much less well developed than in the LTI/LTV case; (ii) operationally effective tools (system identification) are much less well developed; (iii) it is not immediately obvious whether nonlinear models from (i) may be expected to capture the essential nonlinearity of insect flight dynamics and control. Finally, although we defer discussion of subspace identification methods to a later paper, the framework that we have outlined is operationally effective for inferring the proposed models from experimental data [22,23,43,44,45]. A short bibliography is provided overleaf as a guide to some of the wider relevant literature.…”

Data-driven simulation and control

“…Typically T ini and n(B) are small compared to the length T of the given trajectory, however, T r might be big. In (Markovsky et al 2005a) a refinement of the data-driven simulation algorithm is presented (in the special cases of impulse response and free response computation) that requires persistency of excitation of order 1 + T ini + n(B).…”