real time adaptive PID controller based on RLS with unknown setpoint function

Question

I am looking to create a real-time adaptive PID controller using a recursive least squares method. I am trying to implement the method shown here:

https://www.hindawi.com/journals/ape/2014/507142/

I am stuck at the part where the authors express the output of the system in terms of the reference model and the input. They say that the output of the system should be equal to the reference output: $$y_m(t)=y(t)$$ Therefore, the output can be expressed as follows: $$M(s)r(t)=y(t)$$ My problem is that I don't understand how to use this information. I understand the basics of the RLS method, and the only part I am struggling with is the theory behind this multiplication. Why are they able to multiply a function of t by the model, which is a function of s, and obtain a function of t again? Why is it true that$$M(s)r(t)=y(t)\text{?}$$ Eventually, they use this to define the RLS equations: $$\mathbf{\hat\theta}(k)=\mathbf{\hat\theta}(k-1)+K(k)\left(u(k)-\phi^T(k)\mathbf{\hat\theta}(k-1)\right)$$ $$K(k)=\frac{P(k-1)\phi(k)}{1+\phi^T(k)P(k-1)\phi(k)}$$ In the above equations, $\mathbf{\hat\theta}(k)$ contains the PID controller gains that will be adjusted. The model multiplication I don't understand is contained inside $\phi(k)$. The paper defines: $$\phi=\phi_1(1-M(s))r(t)$$ $\phi_1$ simply contains the proportional, integral, and derivative transfer functions that make up a PID controller

To be clear, the only part I am having trouble with is the multiplication of a function of s by a function of t.

Answer 1

Right after $(6)$ it is stated:

... $M(s)$ is a given model reference transfer function which represents the ideal closed-loop dynamics.

So $M(s)$ is a transfer function. It is not entirely mathematically correct to write such a multiplication, however this notation is often used as short hand for

$$ M(s)\,\mathcal{L}\{r(t)\}(s) = \mathcal{L}\{y(t)\}(s), \tag{1} $$

with $\mathcal{L}\{x(t)\}(s)$ the Laplace transform of $x(t)$.

It can be noted that, if $M(s)$ is a ratio between polynomials in $s$ and is single input single output, $(1)$ can be written as

$$ \sum_{n=0}^N a_n\frac{d^n}{dt^n} r(t) = \sum_{n=0}^M b_n\frac{d^n}{dt^n} y(t), \tag{2} $$

with $a_n,b_n\in\mathbb{R}$.