L₂-optimal identification of MIMO errors-in-variables models from the v-gap geometrical interpretation

Abstract

An L 2 -optimal identification method is extended to cope with MIMO errors-in-variables (EIV) model estimation based on a geometrical interpretation for the v-gap metric. The L 2 -optimal approximate models are composed of system and noise models and characterised by a normalised right graph symbol (NRGS) and its complementary inner factor (CIF), respectively. This metric can be evaluated as the supreme of sine values of the maximal principal angles between NRGS frequency responses of two concerned models. In order to make full use of the angular cosine formula for complex vectors to reduce computational loads, a CIF of the NRGS of the perturbed model is introduced and thus, the system parameter optimisation can be efficiently solved by sequential quadratic programming methods. With the estimated system model, the associated noise model can be built by right multiplication of an inner matrix. Finally, a simulation example demonstrates the effectiveness of the proposed identification method.

A convex optimization algorithm for frequency‐domain identification in the v‐gap metric

“…Because of the vast difference in computational complexity between these two LMI problems, the computational efficiency of the FPLS algorithm is much higher than that of the SLMI algorithm. Moreover, superior to the SQP algorithm proposed in [19], the FPLS algorithm needs no selection of appropriate initial values for parameter initiation. The computational efficiency of these three algorithms will be further compared with a simulation example given in Section 4.…”

“…As for the parametrization of the approximate model P . /, the generalized orthonormal basis functions are applied in the following way [19,30]:…”

“…In order to use the v-gap metric as an optimization criterion to estimate model parameters, Date and Vinnicombe [17,18] proposed a sequential linear matrix inequality (SLMI) algorithm, but its computational complexity climbs up in the presence of amounts of frequency-response data. Geng et al [19][20][21][22] adopted this algorithm to study the errors-in-variables (EIV) model identification. Therefore, developing an even more efficient algorithm for minimization of the v-gap metric is of great importance for its widespread applications in control and identification [23][24][25][26][27].…”

“…In order to estimate MIMO EIV models, Geng et al [19] explored a geometrical interpretation of the v-gap metric, and the associated optimization problem can be solved by the sequential quadratic programming (SQP) algorithm, in which the inappropriate initiation can lead to local convergence. For the sake of further improving on the parameter optimization efficiency, a convex optimization algorithm is proposed in this paper to identify a SISO open-loop system from noisy frequency-domain measurements.…”

“…With this interpretation, the resulting parameter optimization can be efficiently solved by a sequence of feasibility problems with a decreasing line search. Different from the NRGS identification in [19], the proposed algorithm directly estimates an open-loop model of the underlying system. Moreover, an NRGS associated with the estimated open-loop model can be easily constructed from some specified procedure and is free from consideration of additional constraints, such as coprimeness and inner property.…”