This is a set of examples of how operations with ellipsoids and polytopes can be used to assess model quality and modelling uncertainty. If the model has been identified already and parameter uncertainty is given in the form of confidence ellipsoids or polytope hard-bounds, then geometric methods can be used to assess the effect of the parameter uncertainty on a particular application of the model. If the model has not been produced yet, then also reliable parameter bounding models can be computed beside statistical and probabilistic models. The models based on "hard bounds" can be conservative and can be used to evaluate the possibility of guaranteed performance in applications. If the parameter bounding models produced are overly conservative then either further experiments can be performed to calculate less conservative parameter feasibility sets, or the same data can be analyzed with improved analysis of noise and modelling errors.
The aim of the examples below is to clarify some of the hottest issues in modelling, which are subject to much discussion in applications, because they are concerned with model uncertainty. The rest of this text contains a short description of the demos which can be run under MATLAB. The following four demos can be accessed via FTP as indicated below:
Many practical systems are basically linear in their operating range and allow for a physical model to be set up for them. Bounds on measurement accuracy of outputs are usually known. Actuators can also provide control inputs with their accuracy known. All this often allows for setting up a model which is linear in its parameters but non-linear in its I/O relationship. A general form of such a model with k inputs, m outputs and altogether
m*d linear parameters
y1(t) = param11*f11(y1(t-1),...,ym(t-n), u1(t-1),...,uk(t-n)) + ... + param1d*f1d(y1(t-1),...,ym(t-n), u1(t-1),...,uk(t-n)) + e1(t)
y2(t) = param21*f21(y1(t-1),...,ym(t-n), u1(t-1),...,uk(t-n)) + ... + param2d*f2d(y1(t-1),...,ym(t-n), u1(t-1),...,uk(t-n)) + e2(t)
: : : :
ym(t) = paramm1*fm1(y1(t-1),...,ym(t-n), u1(t-1),...,uk(t-n)) + ... + parammd*fmd(y1(t-1),...,ym(t-n), u1(t-1),...,uk(t-n)) + em(t) .
Assuming that bounds are known on the error terms ei(t), i=1,...,m , this example illustrates the computation of polytope parameter sets on the basis of a given sequence of I/O measurements. The measurements are taken from an electromechanical system with some non-linearities. The advantage of computing a model based on a parameter set is that it naturally provides an uncertainty description of the model. In statistical estimation of model parameters the uncertainty is usually based on the inverse of the Fisher information matrix to compute the covariance matrix of the parameter estimates. The covariance matrix obtained this way has been proved to be an unreliable assessment of model accuracy, as it is shown in various publications on robust identification.
This demo can be accessed using the FTP facility http://sun1.bham.ac.uk/veress/ftparchive/parbound.m .
On-line tracking of parameter changes is fairly straightforward if parameter-set models are used which represent the feasible set of plant parameters at each time instant. Using sets reflects the fact that we don't have infinite amount of information on our plant, which the determination of a single parameter vector would require. Allowing for changes in plant parameters according to some bounded set means that the model set has to be enlarged according to the sum of the old set-model and the set of possible parameter increments. Collecting new I/O measurements at the same time allows to reduce the size of the set-model . Thus the two opposing operations of repeatedly increasing and reducing the parameter set allow for parameter tracking.
This idea is illustrated on a third-order FIR model in an m-file accessible via the FTP facility http://sun1.bham.ac.uk/veress/ftparchive/parbound.m .
The is and m-file which can be run on MATLAB 4 and above to demonstrate how sensitive pole-placement control is to small inaccuracies in the coefficients of SISO ARX models. The demo is asking for a plant transfer function to be specified by typing in the zeros, the poles and the steady-state gain. The asymptotic covariance matrix of the least-squares estimates is computed by assuming unit variance white-noise input to test the plant. To obtain the normal matrix of a confidence ellipsoid of 0.95 probability, the covariance is multiplied by 1.65n-1/2 . This confidence ellipsoid is examined for the possibility of pole-zero cancellation by computing its orthogonal projections parallel to hyperplanes which represent lower order models in the parameter space, and displayed on the same two-dimensional plane. If the area swept over by the projected family of ellipsis contains the origin, then the confidence ellipsoid contains a lower order model obtainable by pole-zero cancellation. This implies that controllability of the associated state-space model is lost and the poles of the closed-loop system might not be possible to guarantee to be kept in a prescribed region.
This m-file allows to experiment with models close to pole-zero cancellation. The important observation is that pole- zero cancellation occurs in general all too often because of the sensitivity of ARX models.
The FTP facility http://sun1.bham.ac.uk/veress/ftparchive/estimate.m can be used to copy the freeware demo m-file.
Based on techniques similar to the one in 1. above, this routine can be used to test whether the confidence ellipsoid associated with an estimated ARX model can be reduced by pole-zero cancellation to a lower order model. . This confidence ellipsoid is examined for the possibility of pole-zero cancellation by computing its orthogonal projections parallel to hyperplanes which represent lower order models in the parameter space, and displayed on the same two-dimensional plane. If the area swept over by the projected family of ellipsis contains the origin, then the confidence ellipsoid contains a lower order model obtainable by pole-zero cancellation. This routine is available using the FTP facility http://sun1.bham.ac.uk/veress/ftparchive/ordtest.m