# Semi-Active System of Vehicle Vibration Damping
## 4 Model of the Control System  
In the system of the damper control, signals are processed that represent the physical quantities characterizing the vehicle state. In the subsequent discussion, denotations of physical quantities and corresponding signals are the same. The central part of the considered control system is a programmable controller, where the signal vector \(\tau^{\text{opt}}\in\mathbb{R}^{4}\) is determined, corresponding to the optimal vector of the friction forces \(T^{\text{opt}}\) defined by the relationship in Formula (15). Fundamentals applied to formulate this relationship are provided by criterial function \(\mathcal{K}\), defining the influence of the friction forces on the intensity of the vehicle vibration. In this work, the intensity of the vehicle vibration will be estimated on the basis of vector \(\text{A}\in\mathbb{R}^{3}\) defining the acceleration of the vehicle body:  
\[\text{A}=-\text{M}^{-1}(\text{P}+\text{HT}), \tag{30}\]  
where vector of forces \(\text{P}\in\mathbb{R}^{3}\) described with the Formula (26),  
M--matrix of the body inertia. \[\text{M}\in\mathbb{R}^{3\;\text{x}\;3},\;\text{M}:=\text{diag}\Big{(}\text{m},\text{J}_{\text{x}},\text{J}_{\text{y}}\Big{)}, \tag{31}\]  
Formula (30) determines the relationship between the vectors of friction forces \(\text{T}\in\mathbb{R}^{4}\) and accelerations of the vehicle body \(\text{A}\in\mathbb{R}^{3}\). On the basis of the above, the first criterial function is determined, which sets the norm for the acceleration vector \(\text{A}\in\mathbb{R}^{3}\) according to the following formula:  
\[\mathcal{K}^{(1)}(\text{T}):=\frac{1}{2}\text{A}^{\text{T}}\text{A}, \tag{32}\]  
Hence, after taking into account Formula (30), the following is obtained:  
\[\mathcal{K}^{(1)}(\text{T})=\frac{1}{2}(\text{P}+\text{HT})^{\text{T}}\text{ M}^{-2}(\text{P}+\text{HT}), \tag{33}\]  
and after transformations, there are:  
\[\mathcal{K}^{(1)}(\text{T})=\frac{1}{2}\text{TDT}+\text{d}^{\text{T}}\text{T} +\frac{1}{2}\text{P}^{\text{T}}\text{M}^{-2}\text{P}, \tag{34}\]  
if  
\[\text{D}\in\mathbb{R}^{4\times 4},\;\text{D}:=\text{H}^{\text{T}}\text{M}^{-2} \text{H}, \tag{35}\]  
\[\text{d}\in\mathbb{R}^{4},\;\text{d}:=\text{H}^{\text{T}}\text{M}^{-2}\text{P}. \tag{36}\]  
The optimization task is formulated in the following way: vector \(\text{T}\in\Omega(\text{V})\) needs to be determined for which the value of function \(\mathcal{K}^{(1)}\) is the smallest in the set of vectors \(\Omega(\text{V})\) described with Formula (9). Taking into account that function \(\mathcal{K}^{(1)}\) and set \(\Omega(\text{V})\) are convex allows for concluding that the above task has a solution; because matrix \(\text{D}\) is not strictly convex--which ensues from Formulae (25) and (35)--the solution to this task may be ambiguous. This means that in set \(\Omega(\text{V})\), there exists a subset of vectors for which the values of function \(\mathcal{K}^{(1)}\) are the same and equal to the minimum. Thus, the optimizing issue considered herein comes down to determining a subset of vectors \(\text{T}\), as mentioned before, and selecting one of those \(T^{\text{opt}}\). This task can be rendered as follows:  
\[\text{T}^{\text{opt}}\in\text{Arg}\min_{\text{T}\in\Omega(\text{V})}\mathcal{ K}^{1}(\text{T})\, \tag{37}\]  
For the vector of forces \(T^{\text{opt}}\in\Omega(\text{V})\) so determined, the norm of the vector of body accelerations, denoted with Formula (32), reaches the minimum. It needs to be noted that vector \(\text{T}^{\text{opt}}\), selected in such a way, changes along the vector \(\text{U},\;\text{V}\in\mathbb{R}^{3}\) because the force vector \(\text{P}\in\mathbb{R}^{3}\) (Formula (26)) depends on them and the set \(\Omega\) (Formula (8)). This means, that relationship (37) is non-stationary because vectors \(\text{U},\;\text{V}\) change over time. The method of determining solution (37) can be found further in this work.