Jose A. Ramos

A note on the cross Riccatian and related properties for symmetric stochastic realizations

Recent results concerning cross Grammians for both SISO and symmetric MIMO systems are extended for stochastic systems. The extension is based on a cross Riccatian matrix and several results from stochastic realization theory. It is shown that the cross Riccatian can be obtained from the solution to a cross Riccati equation carrying properties from both a forward and a backwards innovations representation. Other symmtry properties similar to those for the cross Grammian are introduced, along with connections to balanced stochastic realizations.

Mathematical modeling, system identification, and controller design of a two tank system

In this paper we present a case study involving mathematical modeling, system identification, and controller design of a two tank fluid level system. The case study is motivated by a realistic application of a two tank problem. We address some fundamental control oriented issues such as physical plant design and identification, transformation from discrete-time to continuous-time, and finally the controller design. We also introduce a novel physical system identification algorithm consisting of subspace identification, followed by a similarity transformation computation to extract the physical parameters of the system. The controller design is done by pole placement.

A 2-D realization theory for Markov chains

The dynamics of time-homogeneous Markov chain models is studied from a state-space modeling point of view. It is shown that a Markov chain model can be embedded in a 2-D realization theory where Markov parameters correspond to higher-order transition probabilities. The implication of formulating a Markov chain model in this state-space domain is that many equivalent representations may exist, some of which may have better robustness properties. A modified Hankel approximation algorithm is presented which exactly matches all the Markov parameters. The algorithm is an extension of the 2-D harmonic retrieval algorithm of D.V.B. Rao et al. (1984).<<ETX>>

A Subspace Approach for Identifying Bilinear Systems with Deterministic Inputs

In this paper we introduce an identification algorithm for MIMO bilinear systems subject to deterministic inputs. The new algorithm is based on an expanding dimensions concept, leading to a rectangular, dimension varying, linear system. In this framework the observability, controllability, and Markov parameters are similar to those of a time-varying system. The fact that the system is time invariant, leads to an equaivaleet linear deterministic subspace algorithm. Provided a rank condition is satisfied, the algorithm will produce unbiased parameter estimates. This rank condition can be guaranteed to hold if the ratio of the number of outputs to the number of inputs is larger than the system order. This is due to the typical exponential blow-out in the dimensions of the Hankel data matrices of bilinear systems, in particular for deterministic inputs since part of the input subspace cannot be projected out. Other algorithms in the literature, based on Walsh functions, require that the number of outputs is at least equal to the system order. For ease of notation and clarification, the algorithm is presented as an intersection based subspace algorithm. Numerical results show that the algorithm reproduces the system parameters very well, provided the rank condition is satisfied. When the rank condition is not satisfied, the algorithm will return biased parameter estimates, which is a typical bottleneck of bilinear system identification algorithms for deterministic inputs.

A note on a cross Riccatian and related properties for symmetric stochastic realizations

Recent results concerning cross Grammians for both SISO and symmetric MIMO systems are extended for stochastic systems. The extension is based on a cross Riccatian matrix and several results from stochastic realization theory. It is shown that the cross Riccatian can be obtained from the solution to a cross Riccati equation carrying properties from both a forward and a backwards innovations representation. Other symmtry properties similar to those for the cross Grammian are introduced, along with connections to balanced stochastic realizations.

Detection of abnormalities in the signal averaged electrocardiogram: a subspace system identification approach

This paper addresses the detection and classification of low amplitude signals within the QRS complex of the signal-averaged electrocardiogram. The raw data is used to fit a state-space model using the N4SID algorithm and the residual from the model are then used for detection. The fundamental assumption behind the state-space model is that the residuals are a white noise process. Therefore, anything that cannot be modeled with the state-space model will show up in the residuals as flow amplitude signal+noise. Compared to typical residuals, the low amplitude signal behaves as influential observations and can be treated as outliers. Diagnostic tests and analysis on the residuals will then lead to detection and classification of abnormalities in the intra-QRS complex. Residual analysis in this paper includes whiteness and Gaussian tests, statistical process control, and the use of a tracking signal. The end result is a tool to aid the physician in diagnosing the heart condition of a patient.

A total least squares approach for data reduction of longterm ECG recordings

A major problem associated with longterm ECG recordings is the enormous volume of data they contain and the requirement of an efficient procedure for its archival in reduced form is highly desirable. The storage of multiple recordings poses limitations, especially when they are to be used at a later time for applications involving high resolution mode, such as the contextual analysis of ECG. This paper presents an application of the singular value decomposition (SVD) within a total least squares (TLS) framework for data reduction of longterm ECG recordings. We present a combined formulation of denoising and data reduction via a TLS approach. Following beat delineation, archival of the ECG beat is accomplished using the reduced parameter set obtained using the TLS approximation in the discrete cosine transform (DCT) domain. Casting the transform domain ECG signal into a structured form using only the significant DCT coefficients resulted in a substantial reduction of the computational complexity involved in estimating the model parameters. With the reduced parameter set obtained using the proposed TLS approach, it was possible to archive multiple recordings of ambulatory Holter ECG data in a personal computer with only a limited storage capacity.

Canonical correlation analysis of the deterministic realization problem

Publisher Summary Recently, a fairly simple state-space system identification algorithm has been introduced, which uses I/O data directly. This new algorithm falls in the class of subspace methods and resembles a well known Markov parameter based realization algorithm. The heart of the subspace algorithm is the computation of a state vector sequence from the intersection of two subspaces. Once this state vector sequence is obtained, the system matrices can be easily obtained by solving an over-determined linear system of equations. It has been shown that this new subspace algorithm is equivalent to a canonical correlation analysis on certain data matrices. This chapter applies the classical theory of canonical correlation analysis to the deterministic identification problem. It is shown that the first n pairs of canonical eigenvectors contain all the information necessary to compute the state vector sequence in the original algorithm. The chapter shows the connections with forward-backwards models, a duality property common to canonical correlation based identification algorithms. This is useful for computing balanced realizations (an optional step). It is shown that, unlike the stochastic realization problem, certain dual covariance constraints have to be satisfied in order for the state sequence to be admissible thereby forcing the optimal solution to be of a canonical correlation analysis type. Algorithms that do not satisfy all the constraints are only approximate. The chapter concludes with derivation of deterministic realization algorithms using ideas from their stochastic realization counterparts.

Canonical Variate Modeling and Approximation of Linear Systems

This paper treats balanced realization problems within the framework of canonical variate analysis. By applying the concepts of statistical studies, duality diagrams, and the RV-coefficient to deterministic systems, it is shown how both the identification (Hankel approach) and transformation (Grammian approach) based balanced realization problems lead to dual interpretations. It is further shown that both approaches lead to a minimum distance problem (equivalently maximum RV-coefficient) between certain observability and controllability properties of a linear system. The motivation for this optimization problem follows from the singular value decomposition, the orthogonal procrustes problem, and the RV-coefficient. The solution has the format of a generalized singular value decomposition.