What are and why are sine and cosine modulated integrals used?

Question

I have found the definition of the following formulas in a paper regarding active vibration control, where they are called sine and cosine modulated integrals.

$y$ is measurement signal with a strong periodic component of frequency $N\Omega$

$$y^{(i)}_{Nc}=\frac{2}{T}\int^{T}_{0}y^{(i)}(ϕ)\cos(Nϕ)dϕ$$

$$y^{(i)}_{Ns}=\frac{2}{T}\int^{T}_{0}y^{(i)}(ϕ)\sin(Nϕ)dϕ$$

where $\phi=\Omega t$.

From these the vector $y_N^{(i)}$ is defined as $y_N^{(i)} = \begin{bmatrix} y_{Nc}^{(1)}\\ y_{Ns}^{(1)}\\\vdots\end{bmatrix}$

The same is done for the control input(s) $u$. Then a quadratic cost function to be minimized at each step is defined using these newly introduced signals in this way:

$J(k) = y^T_NQy_N+u^T_NRu_N $ where $Q$ and $R$ are just two weighing matrices.

Are they introduced to consider the sine and cosine components at the considered frequency of the signal?

Can someone explain their meaning and why are they introduced?

Another question: suppose I have the value $y_N$, how can I invert the relationship to get $y^{(i)}$?

Here is the link for the paper.

Answer 1

As someone made me notice in the comments, the integrals are just the coefficients of the Fourier Series for that frequency component (when the series is expressed with the cosine and sine parts separately).

The integral is performed on a period much bigger then the period corresponding to the frequency of the disturbance so that a better averaged is obtained.

To get back the signal in the time domain it is sufficient to multiply the two components by the cosine and sine respectively. Of course in this way the signal has only one frequency component.