Department of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010

Size: px

Start display at page:

Download "Department of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010"

Myrtle Baldwin
5 years ago
Views:

1 Department of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010 Problem 1: Control of Short Period Dynamics Consider the following linear model for the short-period dynamics of an aircraft: [ ] [ ] ẋ = x + δ e where x := [ Angle of Attack α (rad), Pitch Rate q (rad/sec) ] T and δ e is the elevator deflection (rad). Show that the transfer function from δ e to α is based on the state-space equations. G(s) = 65 s 2 + 8s + 65

2 Problem 1a: Bode Plot Sketch the Bode plot for G Magnitude Frequency (rad/sec) Phase (deg) Frequency (rad/sec) Figure 1: Bode Plot

3 Problem 1b: PD Control Design Consider the feedback loop in the figure shown below. Design a Proportional-Derivative control law, K(s) = k d s+k p so that the closed loop meets the following requirements: State state error due to a unit step command α cmd is 0.1. Closed loop poles have damping 0.7 d α cmd e K(s) u G(s) α Figure 2: α Tracking Feedback Loop

4 Problem 1c: Step Response Sketch the response of α due to a unit step command for the feedback system with the controller designed in part b.

5 Problem 1d: Bending Mode Suppose that the aircraft has a bending mode that was neglected during the control design. A model of the aircraft with the bending mode is G b (s) := G(s)H(s) where G(s) was given in part a and the bending mode is modeled as: H(s) := s 2 + 2s Sketch a Bode plot for the loop transfer function G b (s)k(s) using the controller designed in part b. What is the impact of the bending mode? How would you design a controller (not necessarily a PD controller) to meet the design specifications in part b and also address any issues that arise due to this bending mode? You don t need to perform a detailed design. You only need to describe the steps you would pursue in redesigning the controller Magnitude Frequency (rad/sec) Phase (deg) Frequency (rad/sec) Figure 3: Bode Plot

6 Problem 1e: Sensor Noise Let s return to the aircraft model without the bending mode, G(s), and consider the feedback loop below with additional input n to model noise in the α sensor. K(s) is the PD control law designed in part b. Compute the transfer function from AOA sensor noise to elevator command. Suppose alpha signal is very noisy. 1. How would you redesign the control law to deal with the sensor noise? 2. Suppose you had the option of using a pitch rate sensor in addition to the α sensor. What control architectures might you consider to use both sensors? 3. What factors would impact your choice between the design options in 1) and 2)? d α cmd e K(s) u G(s) α n Figure 4: α Tracking Feedback Loop with Sensor Noise

7 Problem 2: Linearization Level flight trajectories of a conventional aircraft in the horizontal plane can be reasonably described by the following four-state two-control nonlinear dynamic model V = u 1 βv 2 w 1 (1) Ψ = g V u 2 + w 2 (2) ẋ = V sin Ψ + w 3 (3) ẏ = V cos Ψ + w 4 (4) where V is the normalized airspeed, Ψ is the heading angle measured clockwise from the North, (x,y) are (East, North) coordinates, (u 1,u 2 ) correspond to thrust and bank angle, and w k s are disturbances. For convenience, it is assumed that β = 0.1. The state and control variables in vector forms are x = V Ψ x y, u = [ u1 u 2 ] (5) Determine the trim conditions of the system around the nominal conditions of V 0 = 1,Ψ 0 = 90 o. Linearize this set of equations around the trim trajectory of x(t) = t,y(t) = 0 assuming w k = 0. This trim trajectory corresponds to V 0 = 1,Ψ 0 = 90 o. Call linearized states and controls as δv,δψ,δx,δy,δu 1,δu 2, where δv = V V 0, etc. Express the linearized system in a state-space form and determines the corresponding A and B matrices. Assuming [δx,δy] is the output vector, obtain the C and D matrices.

8 Problem 3. Linear System as a Predictive Model We are now going to use the above linearized system model as the basis to predict future system behaviors. In particular in using the linear system to predict future states, the initial state values are estimated from measurements and in general contain errors δx 0 = δˆx 0 + z 0 = δ ˆV δ ˆΨ δˆx δŷ + z V z Ψ z x z y (7) This error z 0 can affect the model s ability to accurately predict the future behavior of the aircraft. Analyze the sensitivity of the predicted linearized system response δx(t) with respect to initial state errors z 0 δx(t) z 0 In this analysis, let s assume that both disturbances and controls are zero. What is the effect of an initial airspeed error (z V ) on the future position along the East δx(t)? What is the effect of an initial heading error (z Ψ ) on the future position along the East δx(t)?

Mech 6091 Flight Control System Course Project. Team Member: Bai, Jing Cui, Yi Wang, Xiaoli

Mech 6091 Flight Control System Course Project Team Member: Bai, Jing Cui, Yi Wang, Xiaoli Outline 1. Linearization of Nonlinear F-16 Model 2. Longitudinal SAS and Autopilot Design 3. Lateral SAS and Autopilot