Model error bounds for identification of stochastic models...

Details Model error bounds for identification of stochastic models... Model error bounds for identification of stochastic models...

1999-06-30

2001-08-21

Gordon, Paul P. (Department: 2121)

Data processing: generic control systems or specific application

Generic control system, apparatus or process

Optimization or adaptive control

C700S030000, C700S031000, C700S044000

Reexamination Certificate

active

06278898

ABSTRACT:

BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to the field of control system design, and more specifically, to model error bounds computation for system identification.
2. Background Information
Robust control system design entails designing a uncertainty-tolerant system such that the stability of the system is maintained for all perturbations which satisfy the uncertainty bounds. Uncertainty may take any forms such as noise or disturbance signals and transfer function modeling errors. Uncertainty must be quantified and its bounds must be found, in order to determine if the system will remain stable or to determine the worst-case behavior given the uncertainty. As such, it is desirable to know how large can the uncertainty be before instability occurs or before performance degrades beyond a bound. Various robust control designs have been developed such as H
∞
and &mgr;-synthesis designs. H
∞
robust control design enables stability performance and robustness properties of a system to be predicted with some certainty and it is developed to allow for modeling errors in system identification. The &mgr;-synthesis design method uses repeated iterations of an H
∞
design algorithm and invokes the structured singular value of the transfer function matrix of a system to test whether the design is robust.
Currently available software design methods for determining the model error bounds for system identification of stochastic models are however unsuitable (or incompatible) for use with robust control design methods. One major drawback of the identification methods that account for stochastic processes is that they provide model error bounds that are in terms of covariance of estimated parameters of the model structure or covariance of estimated transfer functions. However, because robust control design requires model error bounds to take the specific form known as “additive or multiplicative frequency-weighted singular value bounds”, the model error bounds in terms of covariance cannot be used directly in robust control design methods. There are several software design tools developed for robust control and system identification such as: Matlab® Control Module and System Identification Module, Matlab® &mgr;-synthesis Toolbox, Matlab® System Identification Toolbox (all available from Mathworks, Inc., Natick, Mass.), and MATRIXx® Robust Control Module and System Identification Module (available from Integrated Systems, Inc., Sunnyvale, Calif.). However, these current design tools do not allow computer automated or computer guided method to generate the model uncertainty from system identification in a mathematically compatible format usable in robust control design.
Moreover, for multi-input multi-output (MIMO) systems, the current design tools compute the self and cross variance of every pair of transfer function from the covariance of estimated parameters. These computations are impractical not only because the resulting pair-wise covariance cannot be directly used in robust control design, but also because these computations cannot be handled by ordinary workstations. Therefore, the available computing methods are laborious, time-consuming, and very expensive.
Additionally, when model order selection methods such as Akaike's Criterion, Modified Akaike's Criterion, or Rissanen's Minimum Descriptor Length Criteria are used with input-output data from closed-loop tests, biases in parameter estimates are unavoidable. These biases make model error bounds from covariance of estimated parameters incorrect. Thus, what is needed is a method for computing model error bounds for stochastic systems such that they are directly useable in H
∞
and &mgr;-synthesis robust control design methods.
SUMMARY OF THE INVENTION
A method for computing model error bounds for system identification of stochastic models is disclosed. The model error bounds take the form of additive frequency-weighted singular value bounds such that they are directly used in H
∞
and &mgr;-synthesis robust control design methods. The model error bounds are used in identification trajectory refinement and other control design cycles.
The method for computing the model uncertainty consists of the following steps: a closed-loop input-output trajectory is obtained by exciting the plant, and the input and output signals are thus collected; a high order ARX model estimate is selected and parameter estimates and residual signal are calculated; using the input signal and the calculated residual signal, the joint multivariate spectral density function is then computed; the noise sequence estimate and its spectral density function are also computed in order to further compute the covariance of multivariable transfer functions; and, using the covariance results, the estimate of the largest singular value of the additive uncertainty bound is then computed.
The largest singular value of the additive uncertainty bound is determined by performing a high number of simulations of the model uncertainty. Simulated values of the uncertainty are computed for a large data population, such that each candidate entry of simulated value lies on the 3-sigma ellipsoids defined by the covariance functions. For each simulated value of uncertainty, the maximum singular values are then determined. In order to determine the scalar uncertainty function needed for robust control design, the maximum over the population of the maximum singular values of uncertainty simulated values is then computed.
Additional features and benefits of the present invention will become apparent from the detailed description, figures, and claims set forth below.


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