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Routine lsapprox

Polynomial approximation of tabulated data using linear least-squares.


This routine uses Linear Least-Squares to compute a multivariate polynomial approximation $f:\mathbb{R}^n\rightarrow\mathbb{R}^{n_1\times n_2}$ of a set of samples $\left\{y_k\in\mathbb{R}^{n_1\times n_2}, k\in [1, N]\right\}$ obtained for different values $\left\{x_k\in\mathbb{R}^n,k \in [1, N]\right\}$ of some explanatory variables $x\in\mathbb{R}^n$. The maximum degree of the polynomial function is set by the user. No maximum admissible error can be specified and a full polynomial expression is obtained, which minimizes the global relative error and the root-mean-square error between $\left\{f(x_k), k\in [1, N]\right\}$ and $\left\{y_k, k\in [1, N]\right\}$ for each entry (see errapprox for precise definitions).



Input arguments

The first four input arguments are mandatory:

X Values $\left\{x_k\in\mathbb{R}^n,k \in [1, N]\right\}$ of the explanatory variables $x$ ($n\times N$ array, where X(:,k) corresponds to $x_k$).
Y Samples $\left\{y_k\in\mathbb{R}^{n_1\times n_2}, k\in [1, N]\right\}$ to be approximated ($n_1\times n_2\times N$ array where Y(:,:,k) corresponds to $y_k$ in the general case, or possibly $1\times N$ array where Y(k) corresponds to $y_k$ if $n_1=n_2=1$).
names Names of the explanatory variables $x$ ($1\times n$ cell array of strings).
maxdeg Maximum degree of the approximating polynomial function $f$.

The fifth input argument options is an optional structured variable with fields:

maxexp Maximum exponent of each explanatory variable in the approximating polynomial function $f$ ($1\times n$ array). The default value is options.maxexp=maxdeg*ones(1,n).
Constraint to be exactly met (no least squares). options.Yeq is a $n_1\times n_2$ array and options.Xeq is a $n\times 1$ vector with associated values of the explanatory variables. The default values are options.Xeq=[] and options.Yeq=[].
trace Trace of execution (0=no, 1=text, 2=text+figures). The default value is options.trace=1.
viewpoint This option is applicable only if 3-D graphs are to be displayed (options.trace=2 and $n=2$). It represents the deviation with respect to the default viewpoint (see plotapprox). The default value is options.viewpoint=[0 0].

Output arguments

fdata Values $\left\{f(x_k)\in\mathbb{R}^{n_1\times n_2},k \in [1, N]\right\}$ of the approximating function $f$ (same size as Y).
flfr Linear fractional representation of the approximating function $f$ (GSS object if the GSS library is installed, LFR object otherwise if the LFR toolbox is installed).


If some values in X are much larger than 1, they must be scaled or normalized.


Drag coefficient of a generic fighter aircraft model:
load data_cx
Approximation on a rough grid:
options.viewpoint=[-100 0];

Validation on a fine grid:

See also



[1] C. Poussot-Vassal and C. Roos, "Generation of a reduced-order LPV/LFT model from a set of large-scale MIMO LTI flexible aircraft models", Control Engineering Practice, vol. 20, no. 9, pp.919-930, 2012.
[2] C. Roos, "Generation of LFRs for a flexible aircraft model", Optimization based clearance of flight control laws, Lecture Notes in Control and Information Sciences, vol. 416, pp. 59–77, Springer Verlag, 2011.