1. 1 Project seminar on - By:- Tejaswinee Darure

2. Contents: 1] Aircraft (Boing-737) modeling a] Longitudinal Dynamics b] State space representation of system c] Response for each state and output 2] Pole placement technique 3] Linear quadratic regulator design 4] Kalman filter design + LQG 5] Loop transfer recovery 6] Conclusion 2

3. 3 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 Longitudinal Dynamics

4. 4 u’ w ’ q ’ θ ’ = x u x w x q x θ z u z w z q z θ m u m w m q m θ 0 0 1 0 uwqθuwqθ + x η x τ z η z τ m η m τ 0 0 ητητ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 uwqθuwqθ θγ θγ State space representation for longitudinal stability + =

5. 5 ∫BC A u x xˆ Y A = -0.8 -0.0006 -13.2 0 0 -0.014 -16.64 -32.2 1 -0.0001 -1.65 0 1 0 0 0 B = -19 -2.5 -0.66 -0.5 -0.16 -0.6 0 0 C = 0 0 0 1 0 0 -1 1 State space representation for longitudinal stability

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7. 7 Pole placement technique :- B A C∫ -K y u x xˆxˆ K = 0.4764 -0.0026 -0.1567 -1.0735 -3.8432 0.0213 1.3051 -2.8896 Where desired poles will be at- P = -1.1212+3.44664j -1.1212-3.44664j -0.0058+0.0264j -0.0058-0.0264j

8. 8  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)=A CL x(t) where A CL =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. A CL 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.

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10. 10 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, f J(t,t f ) = Let, quadratic cost function be, f J(t,t f ) =  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.

11. 11  If A- B*K is stable, u= -K*x be control law where K is optimal gain matrix then let us define, X T (Q+K T RK)X =  P is positive definite real symmetric matrix. After taking derivative and comparing, We get, -(Q+K T RK)= (A-B*K) T P-P(A-B*K) i.e. matrix P should satisfies above equation. By solving this we obtain, K = R -1 B T P Therefore control law is, u = -K x(t) = - R -1 B T P >>[K,P,e] = lqr(A,B,Q,R) where, e = Eigen values of A CL

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13. 13 Another example A = 0.4158 1.025 -0.00267 -0.0001106 -0.08021 0 -5.5 -0.8302 -0.06549 -0.0039 -5.115 0.809 0 0 0 1 0 0 -1040 -78.35 -34.83 -0.6214 -865.6 -631 0 0 0 0 -75 0 0 0 0 0 0 -100 B = 0 0 0 0 75 0 0 100 C= -1419 -146.43 -40.2 -0.9412 -1285 -564.66 0 1 0 0 0 0 D= 0 0 0 0 Q=I R=I

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17. 17  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. Conclusion:

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