## 3 Internal Model Control  
IMC uses the plant model as a component of the controller [18]. The IMC has a number of benefits compared with the traditional PI (proportional-integral) control. First of all, the closed-loop stability is guaranteed merely by selecting a stable controller if an accurate plant model can be  
Figure 1: Digital filtering system with filtered-X least mean square (FX-LMS) algorithm. obtained. Furthermore, the controller design is relatively straightforward and closed-loop performance properties, such as the rising time, settling time, and time constant, are directly correlated with the control parameters. This makes online tuning of the controller very convenient. The IMC structure is a modification of a classical feedback structure, with the process model applied definitely as an internal part of the controller [19]. The general structure of an IMC system is shown in Figure 2.  
The purpose of designing an IMC controller is to force the output to respond to a set point change with desired manners and counter the effects of disturbances. If the controller gain is the inverse of the model gain, and the controller is tuned to assure stability, the process output \(y\) will eventually reach and maintain the set point \(y_{s}\) in the absence of new disturbances. Unfortunately, a model is never perfect. In addition, if the model has any dynamics at all, rather than just a gain, no controller can perfectly invert the process model. A controller can, however, come very close to inverting the process model. Just how close it can come is the subject of the IMC design problem, in order to expect perfect models. When no zeros are near the imaginary axis or in the right half of the s-plane, the inverse of the model is stable and not overly oscillatory. If the form of the process is defined as Equation (3), where \(N(s)\) and \(D(s)\) are polynomials of \(s\), the internal model controller can be chosen easily as follows. \[p(s)=\frac{N(s)}{D(s)} \tag{3}\]  
\[q(s)=\widetilde{q}(s)f(s)=\frac{D(s)}{N(s)(\lambda s+1)^{r}} \tag{4}\]  
In this form, the filter \(f(s)\) has two parameters, \(\lambda\) and \(r\). The relative order \(r\) is the order of the denominator minus the order of the numerator and it is selected to be large enough for \(q(s)\), in order to be proper. The filter time constant \(\lambda\) is selected in order to avoid excessive noise amplification. The main advantage is that closed-loop stability is assured simply by choosing a stable internal model controller. In addition, it has systematic and relatively straightforward design methods, with advantages such as internal stability, stability robustness, and performance robustness. The structure is also intuitive, and the closed-loop performance characteristics are directly related to the controller parameters, making online tuning of the IMC very convenient. Consider the following model for an arbitrary plant \(T(s)\) with second-order linear system dynamics \(P(s)\) and time-varying delay term \(D(s)\), as follows:  
\[T(s)=P(s)\cdot D(s)=\frac{a{\omega_{n}}^{2}}{s^{2}+2\zeta{\omega_{n}}s+{\omega _{n}}^{2}}\cdot e^{-\tau s} \tag{5}\]  
Here, \(a\), \(\omega_{n}\), \(\zeta\), and \(\tau\) are the proportional gain, natural frequency, damping ratio, and time delay, respectively. As IMC is a linear controller that requires a linear system model, the exponential function in the model needs to be linearized. For this purpose, (0, 1) the Pade approximation is applied so as to  
Figure 2: Internal model control (IMC) system. convert the exponential function into a rational function, such that the following, and then the actuator model, becomes linearized, as follows:  
\[e^{-\tau s}\approx\frac{1}{\tau s+1}=\widetilde{D}(s) \tag{6}\]  
\[\widetilde{T}(s)=\frac{Y}{V}(s)\cdot\widetilde{D}(s)=\frac{a{\omega_{n}}^{2}}{s ^{2}+2\zeta\omega_{n}s+{\omega_{n}}^{2}}\cdot\frac{1}{\tau s+1} \tag{7}\]  
With this model, an internal model controller can be designed.