Chapter 2Equations of MotionThe equations of motion constitute the basic building blocks of any system under study. These equations should be formulated as accurately as possible to model the desired system. δInaccuracies in the formulation of these equations could result in faulty system behavior that could be very difficult to understand. However, modern control systems are designed to compensate for model inaccuracies to some extent. It is very important to ensure that our model falls within this range. Errors could also enter the system during the calculation phase due to the precision and number of digits used to represent the values. The equations of motion for any airplane are nonlinear in nature. It is linearized by making certain assumptions. The hypotheses are as follows: The OX and OZ axes are the planes of symmetry. The polar moment of inertia J is zero in XY and YZ is equal to zero. The mass of the aircraft remains constant throughout the entire analysis. The plane is a rigid body. The Earth is an inertial reference. Formulation of the equations of motion: According to Newton's second law of motion, the vector sum of the total forces in a system is equal to the product of the mass (m) and the acceleration (a) of the system. Therefore, in this case, we have:∑ F= m (d Vt / dt ), where Vt = Aircraft Speed ​​(1.1)Since the mass m is constant. Therefore we have, F=m*(dVt/dt )+(d/dt) ∑( Vt+(dr/dt))* δm (1.2)F=m*(dVt/dt)+d^2/dt^2( ∑r*δm ) (1.3) r*δm = 0 since r is measured from the center of mass. Similarly the momentum can be found can be formulated as follows: δm=d/dt(δH)=d/dt(rxv ) δm (1.4)The velocity of this small element can be expressed as ...... half of the sheet ......= (A – B*K)x + Bv.We formulate a performance index J since there can be n number of optimal solutions to a problem. Once a performance index J has been formulated, we can call the solution that minimizes this performance index Optimal. This can be visualized as an energy function, since our ultimate goal is to reduce the energy consumption of the system. We define our performance index J as: J = ∫_0^∞▒xTQx + uTRu dtSubstituting the state variable Feedback control into this yields,J = ∫_0^∞▒xT(Q + KT RK ) x dtHere we have assumed that the input v(t) is equal to zero. This assumption is valid since we are only focusing on the internal stability of the system. In this performance index, both the states x(t) and the control input u(t) are weighted, so if J is small, neither x(t ) nor u(t) would be too large. If we can minimize J, then it is definitely finished.