1. Consider the control system shown in the block diagram

where K₁(s) = K₁ (τ_ns+1) / s; K₂(s) = K_q (s² + ω²n) / (s² +2ζ_ns+ ω_n²) and the UAV dynamics G(s) are described using the model x^• = Ax+ Bu , y= Cx , u = δ , y = [A, q]^T , with G(s)= C(sI - A)^-1B .

a) Derive a state space model for the controller (A_c,B_c,C_c,D_c), withas the input to the controller and δ as the output from the controller.

b) Describe what effect the transfer function K₁ (s) has on the system response.

c) Describe what effect the transfer function K₂(s) has on the system response.

2. Consider the following aircraft dynamics.

a) Derive the transfer functions for A_z / δ and q / δ.

b) Using the data in the above table, compute the poles and zeros for these transfer functions at α = 6 deg and α = 16 deg.

c) What is the primary difference between the dynamics at α = 6 deg and α = 16 deg? What parameter in the model determines this difference?

3. Consider the following two linear time invariant systems

Find a similarity transformation relating system a) with system b). If no transformation exists, explain why.

4. a) Transform this system to diagonal form:

b) True or false

T F Only the zeroes of the transfer function matrix change when transforming to diagonal form. T F The above (A,B,C) is a minimal realization.

5 Consider the following notch filter

How many states are required to implement this filter? Derive the state space matrices for this transfer function: (A,BC,D)