IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 11, NOVEMBER 2000 3087 MIMO System Identification: State-Space and Subspace Approximations versus Transfer Function and Instrumental Variables Petre Stoica, Fellow, IEEE, and Magnus Jansson, Member, IEEE Abstract—The identification of multi-input multi-output (MIMO) linear systems has recently received a new impetus with the introduction of the state-space (SS) approach based on subspace approximations. This approach has immediately gained popularity, owing to the fact that it avoids the use of canonical forms, requires the determination of only one structural param- eter, and has been empirically shown to yield MIMO models with good accuracy in many cases. However, the SS approach suffers from several drawbacks: There is no well-established rule tied to this approach for determining the structural parameter, and, perhaps more important, the SS parameter estimates depend on the data in a rather complicated way, which renders almost futile any attempt to analyze and optimize the performance of the estimator. In this paper, we consider a transfer function (TF) approach based on instrumental variables (IV), as an alternative to the SS approach. We use the simplest canonical TF parameterization in which the denominator is equal to a scalar polynomial times the identity matrix. The analysis and optimization of the statistical accuracy of the TF approach is straightforward. Additionally, a simple test tailored to this approach is devised for estimating the single structural parameter needed. A simulation study, in which we compare the performances of the SS and the TF approaches, shows that the latter can provide more accurate models than the former at a lower computational cost. Index Terms—Correlation, covariance analysis, identification, multivariable systems, parameter estimation, state space methods, statistics, time domain analysis, transfer function matrices. I. INTRODUCTION T HE STATE-SPACE (SS) approach to multi-input multi-output (MIMO) system identification (see, e.g., [1]–[3]) does not require the use of a canonical parameterization for the SS equations, and hence, it avoids the daunting task of structural index determination. Additionally, it was empirically shown to yield estimated models with good accuracy in many cases. These two facts have led to a quick acceptance of the Manuscript received July 12, 1999; revised August 1, 2000. This work was supported in part by the Swedish Foundation for Strategic Research (SSF), the Swedish Foundation for International Cooperation in Research and Higher Ed- ucation (STINT), and the Foundation BLANCEFLOR Boncompagni-Ludovisi, née Bildt. The associate editor coordinating the review of this paper and ap- proving it for publication was Prof. Philippe Loubaton. P. Stoica is with the Department of Systems and Control, Information Tech- nology, Uppsala University, Uppsala, Sweden. M. Jansson is with the Department of Signals, Sensors, and Systems, Signal Processing, Royal Institute of Technology, Stockholm, Sweden (e-mail: mag- nusj@s3.kth.se). Publisher Item Identifier S 1053-587X(00)09296-5. SS without too much concern for comparison with alternative approaches. However, the SS is not free of drawbacks, as we explain shortly, and hence, better approaches may exist. In this paper, the focus is on output error models for which the main problem is the estimation of the system transfer func- tion (from the input to the output). We can solve this problem by using the SS approach as well as by estimating the transfer func- tion (TF) parameters directly by means of an IV method (the latter approach is simply referred to as the TF in what follows). The TF approach of this paper is based on a simple canonical model in which the denominator of the transfer function is set equal to a scalar polynomial times the identity matrix. Similarly to the SS approach, the aforementioned TF model has only one structural parameter that has to be determined from the avail- able data. The SS approach is not statistically optimal in any known sense. Attempts to optimize the statistical accuracy of this ap- proach have had only a limited success, owing to the compli- cated mapping from the data to the estimated transfer function (see, e.g., [4], where the focus was on pole estimates only). 1 For the TF approach proposed in this paper, on the other hand, the parameter estimates depend on the data in a fairly simple manner. Consequently, the analysis and optimization of the sta- tistical accuracy of this approach is straightforward. An additional problem associated with the SS approach is the lack of a well-established rule for estimating the structural pa- rameter (that is, the dimension of the state vector). Rank tests [5] appear to be the most natural choice for such a task as they rely on quantities that are computed in the SS approach (hence, rank tests are structural parameter estimation rules tied to the SS approach). However, a satisfactory approach to rank determina- tion has not yet crystallized. By contrast, we present a simple -based test that can be used to estimate the structural param- eter of the transfer function model used in the TF approach. In fairness to SS, we should note that this approach extends in a relatively straightforward way to the case in which a noise model also needs to be estimated. The TF approach cannot be used for such a purpose in a direct manner (it can only be used in an indirect manner in which the noise is modeled using the estimated residuals of the TF output error equation). 1 We note in passing that the underlying state-space equation estimated in the SS approach wanders, in a realization dependent manner, in the set of simi- larity-equivalent state-space equations. Hence, the performance of this approach should be judged in terms of quantities that are invariant to similarity transfor- mations, such as the transfer function matrix. 1053–587X/00$10.00 © 2000 IEEE