(166b) Adaptive Design of Rotated Inputs for Model Order Determination in Subspace Identification

Correct model order determination is important for building a multivariable dynamic model that can be effectively used in controller design. Various kinds of inputs can be used for a model identification experiment, with pseudo random binary signals (PRBS) being the most common. However, PRBS inputs may be inadequate to clearly determine the correct order of a multivariable model, especially in the case of an ill-conditioned system. In such a case, the model order may be over- or under-estimated, leading to subsequent identification of a poor model. Misra and Nikolaou 2003) showed that input design based on random rotated inputs generates experimental data that can clearly reveal the correct model order in case of an ill-conditioned system. Inspiration for the rotated inputs design was based on a related approach suggested for identification of multivariable models that satisfy the integral controllability property (Koung and Macgregor1994, Darby and Nikolaou2009). The concept of input rotation stems from singular value decomposition of the original model structure and subsequent definition of transformed inputs resulting from premultiplication of the original inputs by a rotation (i.e. orthonormal) matrix.

However, the rotated inputs approach produces the correct model order when the rotation angles are close to the actual rotation angles. Since, the actual gain matrix always unknown before the design of experiment, large deviations from the actual rotation angle may produce input designs that generate data leading to incorrect model order (Micchi and Pannocchia2008). In the current work, it is demonstrated, via numerical simulation, that by adaptively changing the rotation angles, the angles converge towards their actual values, which leads to correct model order. The computer-simulated experiment started with rotation angles deliberately chosen far from their actual values, and the angles were adaptively changed until they converged. The model order of the system from the identification experiment with the adaptively rotated input design was found to be same as the actual system order.

More specifically, the adaptive design method was implemented on a 2x2 system that has been extensively studied in literature (Skogestad and Morari 1987) as well as on a 5x5 model of an FCC unit (Darby 2008). For the 2x2 system the model order was determined correctly as 2 by adaptively changing the rotation angle. By contrast, rotated inputs with the rotation angle 00 (far from the actual rotation angle, 400) without adaptation led to incorrect model order 1. PRBS inputs design also resulted in incorrect model order of 1 over the same time allocated for identification. Figure 1 shows a comparison of all three input design methods for a 2x2 system. This figure shows the singular values of a projection matrix that appears in subspace identification and whose nonzero singular values are as many as the system order. For a 5x5 system MATLAB system identification toolbox was used to select the model order for various input designs (Figure 2). The actual order was estimated as 15 for this 5x5 system. Adaptive rotated input design determines the model order very close to actual order, whereas, other input designs significantly underestimate the order.

The preceding results provide justification for using the proposed adaptive input design for multivariable system identification. A theoretical analysis of the observed behavior will follow in a future publication.

1(a)

1(b)

1(c)

Figure 1: Order of the 2x2 system, as indicated by singular values of the projection matrix, for a) PRBS inputs design, b) rotated inputs design with wrong rotation angle and c) adaptively rotated inputs design. With adaptively rotated inputs the model order determination is very clear since, the difference in singular values is very large. For rotated inputs with the wrong rotation angle, the second singular value is so small that it looks close to noise thus resulting in incorrect model order.

2(b)

2(c)

Figure 2: Order of the 5x5 system, as indicated by singular values of the projection matrix, for a) PRBS inputs design, b) rotated inputs design with wrong rotation angle and c) adaptively rotated inputs design.

References

Darby, M. L. (2008). Studies of online optimization methods for experimental test design and state estimation. PhD, University of Houston.‌

Koung, C. W. and J. F. Macgregor (1994). "Identification for Robust Multivariable Control - the Design of Experiments." Automatica 30(10): 1541-1554.

Micchi, A. and G. Pannocchia (2008). "Comparison of input signals in subspace identification of multivariable ill-conditioned systems." Journal of Process Control 18(6): 582-593.‌

Misra, P. and M. Nikolaou (2003). "Input design for model order determination in subspace identification." Aiche Journal 49(8): 2124-2132.

Skogestad, S. and M. Morari (1987). "Control Configuration Selection For Distillation-Columns." Aiche Journal 33(10): 1620-1635.