Create basis functions for a gain that varies as a periodic function of one scheduling variable.

shapefcn = fourierBasis(2);

shapefcn is a handle to a function of one variable that returns an array of four values corresponding to the first two harmonics of a periodic function on x = [–1,1]:

Use shapefcn as an input argument to tunableSurface to define a gain surface of the form:

The variable x is a normalized version of the scheduling variable for your tunable surface. Because the basis functions created by fourierBasis act on normalized variables, your gain-scheduled system must use design points whose endpoint values delineate exactly one period. For example, suppose you use the following design points:

alpha = [-7,-4,-1,2,5];
domain = struct('alpha',alpha);
K = tunableSurface('K',0,domain,shapefcn);

In normalizing the domain, the software assumes that the gain surface, K, is periodic in alpha such that K(-7) = K(5).

Create a two-dimensional Fourier basis for periodic functions of x and y on the domain $${\left[-1,1\right]}^N$$. The basis functions should go up to the third harmonic in both the x and y dimensions.

F2D = fourierBasis(3,2);

This function is the outer product of two vectors:

x = fourierBasis(3);
y = fourierBasis(3);

Equivalently, you can obtain the outer product using ndBasis.

F = fourierBasis(3);
F2D = ndBasis(F,F);

The values in the vector returned by F include cross-terms such as $$\sin \left(\pi x\right)\cos \left(\pi y\right)$$ and $$\sin \left(3\pi x\right)\cos \left(2\pi y\right)$$.