# Active Vibration Control Study of Harmonic Excitation for Voigt-Kelvin System
## 3 Analysis of Control Parameters
In the case where the function \(g_{i}(P_{r},\omega_{r},\varphi_{r},t_{r})\), \(i=1\ldots 3\) is not canceled due to the restrictions of the control system (permissible maximum force control, speed control, etc.) \(P_{r},\omega_{r},\varphi_{r}\), \(t_{r}\) parameters can be searched for the minimum value of function \(g_{i}(P_{r},\omega_{r},\varphi_{r},t_{r})\). In this case, the \(t_{r}\) parameter can be regarded as the time taken to reach the  
Figure 6: Blok diagram of closed-loop control. minimum \(g_{i,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})\). Therefore, the problem becomes a complex problem when identifying the extremely real functions of several real variables. If we know the finite intervals of control parameters \(P_{r}\in\left[P_{r}^{\min};P_{r}^{\max}\right]\), \(\omega_{r}\in\left[\omega_{r}^{\min};\omega_{r}^{\max}\right]\), \(\varphi_{r}\in\left[\varphi_{r}^{\min};\varphi_{r}^{\max}\right]\) and \(t_{r}\in\left[t_{r}^{\min};t_{r}^{\max}\right]\), we can determine a multitude of solutions. To begin with, we can determine the critical points of a system consisting of the following equalities:  
\[\frac{\partial g_{i}}{\partial P_{r}}=0;\ \ \frac{\partial g_{i}}{\partial \omega_{r}}=0;\ \ \frac{\partial g_{i}}{\partial\varphi_{r}}=0;\ \ \frac{\partial g_{i}}{\partial t_{r}}=0 \tag{34}\]  
The solutions of the \(P_{r},\omega_{r},\varphi_{r},t_{r}\) variables can be determined for each combination of control parameter values of finite fields, as defined above according to the capacity of an electrodynamic shaker using complex numerical methods such as the [PERSON] or Duhamel method [9; 26]. To simplify the problem, taking into account the response system if external excitation produces a disruptive force, \(F_{T}\) has the form \(X_{2}(t)\cos[\omega_{2}(t)+\theta_{2}]\), as obtained from Equation (28). It can be observed that the effect of this force is more important than the overall movement of the mass \(m\) and the problem of disturbing forces due to the displacement amplitude tending toward zero, i.e., when \(X_{2}(t)\to 0\), can be studied  
Due to the complexity of the functions \(g_{i,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})\), instead of the [PERSON] method or Duhamel finite integrals, a simple numerical method was preferred, using a minimum of an array in four sizes (4D--matrix) in the following form:  
\[g_{i,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})=\left[P_{r}^{\min};P_{r}^{\max} \right]x\left[\omega_{r}^{\min};\omega_{r}^{\max}\right]x\left[\varphi_{r}^{ \min};\varphi_{r}^{\max}\right]x\left[t_{r}^{\min};t_{r}^{\max}\right] \tag{35}\]  
The steps described in this chapter represent a clear detailing of the _method for identifying the minimum function_, which is based on the 4D matrix denoted as \(g_{i,\min}\), as mentioned in the previous Equation (35). This method depends on the parameters obtained by this calculation method, the necessary optimal adjustment parameters \(P_{r},\omega_{r},\varphi_{r}\), and the duration of time \(t_{r}\) in which vibration control is obtained. The necessary adjustment parameters represent the necessary technical characteristics of the electrodynamic shaker (ED) for this control method to be efficient and possible.