Answer Happy

Consider the following state-space model: 0 1.132 0 0 0 0 -0.1712 0 -0.12 1 -0.0538 0 -1 0.0705 0 -1.1013 x = 0 0 1 0 0 0 0 0.0485 0 -0.8556 4.419 0 -1.665 -0 -0.2909 0 1.0532 -0.6859 1.575 0 -0.0732¹ [1 0 0 0 0 y = 0 1 0 0 0 x. Lo 0 1 0 This model constitutes a linearized form of the equations of an aircraft motion in the vertical plane when small perturbations about a flight condition are considered. The elements of the state vector x = (X₁ X2 X3 X4 X5)" are as follows: x₁:height error relative to ground [m] X₂: forward speed [m/s] x3: pitch angle [degrees] X4: rate of change of pitch angle [degrees/s] x5: vertical speed [m/s], and the elements of the input vector u = (u₁ U₂ U₂) ¹ are ₁: spoiler angle [degrees] U₂: forward acceleration due to engine thrust [m/s²] u3: elevator angle [degrees] Vertical Stabilizer Control Yaw Winglet Decrease Drag Rudder Change Yaw Horizontal Stabilizer Control Pitch Trims Leading Edge Slats Increase Lift Elevator Change Pitch Empennage x+ Wing Generate Lift 0 0 น Aileron Change Roll Spoiler Air breaks/ Speed Breaks Trailing Edge Flaps Increase Lift and Drag

The objective of the design is to ensure that the response of these three output variables to step changes are fast, non-oscillatory, non-overshooting, acceptable in steady-state and non-interacting (if possible). a) Is the system controllable? (2 points) b) Is the system observable? (2 points) c) Is the system output controllable? (4 points) d) Find the eigenvalues of the system. (2 points) e) Find the transfer function matrix of the system. (2 points) f) Design a full-order observer. Plot actual and estimated states. (20 points) g) Design a reduced-order observer based on eigenvalue assignment. (20 points) h) Design a reduced-order observer based on Sylvester equation. (20 points) i) Design a state feedback controller to place the eigenvalues of the system to desired locations. (16 points) j) Close the loop (output feedback) so, such that the system response is as close as possible to the design requirements you have decided. (12 points)