Palindromic polynomials, time-reversible systems, and conserved quantities

The roots of palindromic and antipalindromic polynomials appear in pairs (s,1/s). A polynomial with such roots is antipalindromic if and only if in addition, it has a root at 1 of an odd multiplicity. The result has applications in system theory: 1) any kernel representation of a discrete-time, time-reversible, scalar, autonomous LTI system is either palindromic or antipalindromic. (Similar statement holds for systems with inputs.) 2) LTI systems with palindromic or antipalindromic kernel representations have nontrivial conserved quantities

On the linear quadratic data-driven control

The classical approach for solving control problems is model based: first a model representation is derived from given data of the plant and then a control law is synthesized using the model and the control specifications. We present an alternative approach that circumvents the explicit identification of a model representation. The considered control problem is finite horizon linear quadratic tracking. The results are derived assuming exact data and the optimal trajectory is constructed off-line

An adapted version of the element-wise weighted total least squares method for applications in chemometrics

The Maximum Likelihood PCA (MLPCA) method has been devised in chemometrics as a generalization of the well-known PCA method in order to derive consistent estimators in the presence of errors with known error distribution. For similar reasons, the Total Least Squares (TLS) method has been generalized in the field of computational mathematics and engineering to maintain consistency of the parameter estimates in linear models with measurement errors of known distribution. In a previous paper [M. Schuermans, I. Markovsky, P.D. Wentzell, S. Van Huffel, On the equivalance between total least squares and maximum likelihood PCA, Anal. Chim. Acta, 544 (2005), 254–267], the tight equivalences between MLPCA and Element-wise Weighted TLS (EW-TLS) have been explored. The purpose of this paper is to adapt the EW-TLS method in order to make it useful for problems in chemometrics. We will present a computationally efficient algorithm and compare this algorithm with the standard EW-TLS algorithm and the MLPCA algorithm in computation time and convergence behaviour on chemical data

On errors-in-variables estimation with unknown noise variance ratio

We propose an estimation method for an errors-in-variables model with unknown input and output noise variances. The main assumption that allows identifiability of the model is clustering of the data into two clusters that are distinct in a certain specified sense. We show an application of the proposed method for system identification

Algorithms for deterministic balanced subspace identification

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”

A Recursive Restricted Total Least-squares Algorithm

International audienceWe show that the generalized total least squares (GTLS) problem with a singular noise covariance matrix is equivalent to the restricted total least squares (RTLS) problem and propose a recursive method for its numerical solution. The method is based on the generalized inverse iteration. The estimation error covariance matrix and the estimated augmented correction are also characterized and computed recursively. The algorithm is cheap to compute and is suitable for online implementation. Simulation results in least squares (LS), data least squares (DLS), total least squares (TLS), and RTLS noise scenarios show fast convergence of the parameter estimates to their optimal values obtained by corresponding batch algorithms