In this experiment, we will use a device that has one heater and one fan. We will model both the heather and the fan using an optimization technique. Thus, we will lean more toward mathematical modeling than physical modeling to ensure no redundancies in the model’s parameters. As we know, it is difficult to perform consistent parameter optimization if the parameters are redundant. The obtained results will be susceptible to the given initial guesses.
The proposed model is as follows:
$$ \begin{equation} \dot{y}=\frac{K u(t-\theta)-y(t)}{\tau} \end{equation} $$
where $y(t)$ is the heater temperature and $u(t)$ is the applied PWM input to the heater. This model is the time-domain version of the previous FOPTD model. We select time-domain modeling for intuitiveness to know how the fan will fit into the equation.
Let us rewrite Eq. 1 such that it is clear the physical meaning of each term in that equation.
$$ \begin{equation} \dot{y}(t)=\underbrace{\frac{K}{\tau}u(t-\theta)}{\text{External input}}-\underbrace{\frac{1}{\tau}y(t)}{\text{Convection}} \end{equation} $$
Let us take $y(t)=T(t)-T_0$ where $T(t)$ is the temperature of the heater and $T_{\infty}$ is the ambient (room) temperature.
$$ \begin{equation} \underbrace{\dot{T}(t)}{\text{Storage}}=\underbrace{\frac{K}{\tau}u(t-\theta)}{\text{Generation}}-\underbrace{\frac{1}{\tau}[T(t)-T_{\infty}]}_{\text{Dissipation}} \end{equation} $$
Eq. 2 physically makes more sense since this is actually an energy balance equation that we learn in thermodynamics class: heat stored = heat generated + heat dissipated. Now the question is where we should fit in the fan parameter.
The way heat is dissipated is through natural convection and the rate for the convection is given by $1/\tau$. Intuitively, we can say that turning on the fan will increase the convection rate while turning off the fan will reduce the convection rate. This forced convection rate is obviously a function of the fan’s rotational speed ($\omega$). Now, we can update our model as follows:
$$ \begin{equation} \underbrace{\dot{T}(t)}{\text{Storage}}=\underbrace{\frac{K}{\tau}u(t-\theta)}{\text{Generation}}-\underbrace{\bigg(\underbrace{\frac{1}{\tau}}{\text{Natural}}+\underbrace{L\omega(t)}{\text{Forced}}\bigg)[T(t)-T_{\infty}]}_{\text{Dissipation}} \end{equation} $$
where $L$ is the forced heat transfer coefficient that relates the fan's rotational speed to the resulting forced convection rate. As we can see in Eq. 4, we now have a system with two inputs and one output. However, the second input manipulates one model parameter linearly. We call such a system as an affine linear parameter varying (affine LPV) system. Notice the model in Eq. 4 has the following general form.
$$ \begin{equation} \dot{y}(t)=A(\omega)y(t)+Bu(t-\theta) \end{equation} $$
$A$ is not constant. It depends affinely on $\omega$. However, it is a unique LPV system since the varying parameter also acts as another exogenous input.
By taking:
$$ \begin{equation} \dot{T}(t)=\frac{T(k)-T(k-1)}{\Delta t} \end{equation} $$
we can discretize Eq. 4 as follows.