# Balanced input excitation for identification of ill-conditioned nxn systems with n>2

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Publication Details

List of Authors: Ramkrishna Ghosh, Kurt E. Häggblom, Jari M. Böling

Editors: S. Skogestad

Publisher: Norwegian University of Science and Technology (NTNU)

Place: Trondheim

Publication year: 2015

Book title: Proceedings of the 19th Nordic Process Control Workshop

Start page: 30

End page: 30

Number of pages: 1

Abstract

Typically, 75% of the cost associated with an advanced control project goes into model development (Gevers 2005). Hence, efficient modeling and system identification techniques suited for industrial use and tailored for control design applications are crucial (Hjalmarsson2005). For an ill-conditioned system, the identification input design is sensitive to the distribution of the singular values of the system gain matrix. The construction of input excitations in smart way is crucial for a successful identification. Excitations can introduce one direction at a time or all directions simultaneously. Consider a singular value decomposition of the steady-state gain matrix, i.e. *G(0) = WΣV*^{T}where *V* and *W* are unitary matrices and *Σ* is a diagonal matrix of singular values, *σ*_{1} ≥ σ_{2} ≥ ... σ_{n} ≥ 0. An input *u*^{i} = V_{i} σ_{i}^{-1} will then produce the output *y*^{i} = V_{i}, with *||y*^{i}|| = 1. To properly excite all directions *i = 1,2,..., n*, we need to apply inputs u^{i} that vary between *u*^{i}_{-} = -σ_{i}^{-1}V_{i} and *u*^{i}_{+} = +σ_{i}^{-1}V_{i} (Koung and MacGregor, 1993). This can be achieved by any kind of input signals (step sequence, PRBS or multi-sinusoidal). Furthermore, this excitations can be added one direction at a time or simultaneously.

The effectiveness of various input excitation methods for ill-conditioned MIMO systems is studied. Three types of perturbations are used: step, PRBS (pseudo-random binary sequence) and multi-sine inputs. The methods are evaluated for a distillation column stripper system with four-inputs and four-outputs described by a transfer function model. Various plant-friendliness measures (e.g. crest factor, PIPS, peak factor etc.) are considered in input excitation design. The relation between the phase and the crest factor of a multisine is quite complicated and cost function contains number of local minima. A plant friendly multisine signal is generated through Polya’s best approximation of Chebyshev norm and crest factor optimization (Guillaume et al. 1991, Pintelon and Schoukens 2004). A Guillaume phase modified zippered signal was found to be the best option among the studied excitation signals. The distribution of the output signal over the gain directions is good for this signal. This is in agreement with a previous study of a 2 × 2 system (Rivera et al. 2009). Projections of the outputs from all experiments onto the high- and low-gain are compared. This projections are obtained by using the *W*-matrix from the SVD of gain matrix *G(0) = WΣV*^{T} and work as a measure for how well a certain direction is excited. As expected, all the experiments that not specifically excite the low-gain direction do excite the low-gain direction much less than the high-gain direction