## 4 Formulation of Multi-Objective Design Problem  
The conceptual scheme in Figure 4 represents a general structure for a formulation of a constrained multi-objective optimization problem within the framework of Mechatronic Concurrent Design that can be expressed in the form  
\[ \begin{array}{rcl}\underset{\mathbf{p}\in\mathbb{R}^{q}}{min}&\mathbf{\Phi}(\mathbf{x}, \mathbf{P},t)&=&[\Phi_{1},\Phi_{2},\ldots,\Phi_{m}]^{T}\end{array}  \tag{1}\]  
with  
\[\Phi_{i} = \int_{t_{0}}^{t_{f}}F_{i}(\mathbf{x},\mathbf{P},t)\,dt, i=1,2,\ldots,m \tag{2}\] \[\mathbf{x} =[x_{1},x_{2},\ldots,x_{n}]^{T} \mathbf{x}\in\mathbb{R}^{n}\] (3) \[\mathbf{P} =[p_{1},p_{2},\ldots,p_{q}]^{T} \mathbf{P}\in\mathbb{R}^{q} \tag{4}\]  
subject to  
\[\mathbf{G}(\mathbf{x},\mathbf{P},t)\leq 0 \mathbf{G}\in\mathbb{R}^{r} \tag{5}\] \[\mathbf{H}(\mathbf{x},\mathbf{P},t)=0 \mathbf{H}\in\mathbb{R}^{s}\qquad(s<q)\] (6) \[p_{j}^{L}\leq p_{j}\leq p_{j}^{U} j=1,2,\ldots,q \tag{7}\]  
and by the system characteristics  
\[\dot{\mathbf{x}} =f(\mathbf{x}(\mathbf{u},\mathbf{P},t),\mathbf{u}(\mathbf{P},t),\mathbf{P},t) \tag{8}\] \[\mathbf{x}(t_{0}) =\mathbf{x}_{0} \tag{9}\]  
where \(\mathbf{\Phi}\) is the objective function vector and its components; \(\Phi_{m}\) is the \(m\)th objective function of a selected criterion; \(t\) is the time variable; \(t_{0}\) is the initial time and \(t_{f}\) is the final time of the operation range to be considered for a system evaluation; \(F_{i}\) is a function of a performance criterion in terms of equations of motion, parameters and time variable; \(\mathbf{x}\) is the state-space vector of a dynamic model of a mechatronic system under design with \(n\) state variables; \(\mathbf{P}\) is the design vector with \(q\) design parameters; \(\mathbf{G}\) is the vector of inequality constraints with \(r\) elements; \(\mathbf{H}\) is the vector of equality constraints with \(s\) elements; \(p_{j}^{L}\) and \(p_{j}^{U}\) are the lower and upper bounds of the \(j\)th design parameter; \(\dot{\mathbf{x}}\) is the gradient state-space vector; \(\mathbf{x}_{0}\) is the initial condition vector of the dynamic model; and \(\mathbf{u}\) is the signal control vector. Figure 5 illustrates a representation of the mathematical formulation with Equations (1  
and the search space, also called as the objective space. A design vector \(\mathbf{P}\) contains all the design parameters such as dimensions, masses, control parameters, and operation measurements. The parameter space, labeled as A in Figure 5 is defined by the bounds of design variables in Equation (7). The feasible space labeled as B in Figure 5, corresponds to all \(\mathbf{P}\) candidate vectors that satisfy constraints described in Equations (5)-(9), which represent strict design requirements. Inequality constraints \(\mathbf{G}\) in Equation (5) delimits larger regions in the feasible space, and some examples of this type of constraints in a merchantonic design are associated with kinematic characteristics, working operation regions, and limiting conditions. Equality constraints \(\mathbf{H}\) in Equation (6) describe strict requirements and can represent for example mobility conditions in a design problem. The system's characteristics are described with a dynamic model in Equation (8) and its initial conditions in Equation (9) which also constrains the optimization problem. The state variables of this dynamic model represent, for example, position, velocity, acceleration, electrical current, or voltage characteristics. The dynamic model is affected by design parameters and inputs for actuators where the signal control vector \(\mathbf{u}\) is a function in terms of design parameters. In this case, those parameters represent the gains of a selected control algorithm.