Presented By:-
Tejaswinee Darure

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Longitudinal Dynamics
u(t) : axial velocity
w(t) : normal velocity
V(t) : velocity magnitude
α(t) : angle of attack
γ (t) : flight path angle
θ (t) : pitch angle
 x’(t) = Ax(t) + Bu(t) --------(1)
Consider , state feedback as -
u(t)= - K*x(t) --------(control law)
 This indicates that instantaneous states are given as feedback where K is a matrix of order 1*n called as state feedback matrix.
x’(t)=ACL x(t) where ACL=A-B*K --------(2)
 Hence stability and transient response of closed system is determined by the eigen values of matrix A-B*K.
 Depending on the selection of state feedback gain matrix K, the matrix A-B*K i.e. ACL can be made asymptotically stable.
 Thus system closed loop poles can be placed at arbitrary chosen locations by choosing appropriate state feedback matrix with the condition that system must be completely state controllable.
>>K= place(A,B, p)
where p will be desired pole locations.
Linear quadratic regulator for LTI system
 Optimal control problem is to find a control input u which causes the system to follow an optimal trajectory x(t) that minimizes performance criteria or cost function,
 Q and R are state and control weighing matrices and are always square and symmetric.
 J is always scalar quantity.
 Linear quadratic regulator (LQR) provides optimal control law for linear system by minimizing above quadratic cost function.
Conclusion:
 We found that it may be required to change plant’s characteristics by using a closed loop system, in which controller is designed to place the poles at desired location.
 Hence by selecting the controller gain matrix, K, we can place the close loop poles at desired location
 Optimal control allows us to directly formulate performance objectives of a control system.
 Linear quadratic regulator (LQR) provides optimal control law (-Kx) for linear system by minimizing quadratic cost function.