Autopilot Analysis and EP Scheme for the Twin Otter under Iced Conditions.

Transcription

1 Autopilot Analysis and EP Scheme for the Twin Otter under Iced Conditions. Vikrant Sharma University of Illinois 4-46

2 Objectives Investigate the autopilot behavior under iced conditions. Develop an envelope protection scheme for the autopilot system to operate under iced conditions. 4-47

3 Overview of the talk Pitch attitude hold (PAH) autopilot (A/P) structure Stability analysis of the PAH A/P Reference value limits through Linear Matrix Inequalities analysis Envelope Protection scheme Work in progress 4-48

4 Block Diagram for PAH K i /s θ ref + - K θ integrator Actuator Dynamics δ e A/C Dynamics θ q K q PAH 4-49

5 Affine Model of Closed Loop PAH The closed loop model is affinely dependent on the icing parameter η, i.e. x & = A(η ) A is a function of η and can be written in the form: A(η) = A 0 + η(t)a 1 Where η, with = [0, η max ] x 4-50

6 Quadratic Stability The PAH A/P system given by x & = A(η) x is stable iff K=K T > 0 s.t. A(η) T K + KA(η) < 0 For all η [0,η max ] PROBLEM: This places an infinite number of constraints on the symmetric matrix K 4-51

7 Quadratic Stability SOLUTION Since the system is affinely dependent on η the quadratic stability condition is the same as the following TWO LMIs A(η=0) T K + KA(η=0) < γi and A(η=η max ) T K + KA(η=η max ) < γi where γ is a negative scalar and η(t) [0, η max ] The above can be solved to obtain minimum possible values of γ using LMILAB in MATLAB 4-52

8 Linear Matrix Inequalities (LMI s) What are they? An expression of the form: F(x) = F 0 + x 1 F x m F m > 0 where X = (x 1,..,x m ) is a vector of decision variables F 0,.., F m are real symmetric matrices, Inequality > 0 means positive definite 4-53

9 Results 4-54

10 Conclusion The linear analysis show that the Pitch Attitude hold A/P will maintain stability under icing There is a small degradation in the guaranteed stability level 4-55

11 Simulation results Comparison of the Pitch Hold PID controller response at V = 60 m/s to a pitch up by 6 degrees. 4-56

12 Envelope Protection for PAH Autopilot PROBLEM: Insure α ( t) <α max ( η( t)) For all time APPROACH: Keep θ ref (t) low enough so that the above is insured 4-57

13 Limits on the Reference Pitch Using LMI Techniques Can find smallest γ s.t. if θ ref < 1/γ then α( t) Values obtained using α max =18.1 degrees for the clean and degrees for iced case and δ emax = 26 degrees δ e < α ( t) < δ max e max α max = Clean Case Velocity (m/s) Maximum ref allowed (deg) Velocity (m/s) Iced Case (η=0.1) Maximum allowed (deg) θ (t) θ ref (t)

14 An Alternate Envelope Protection Scheme Earlier LMI technique might be conservative. Look at step pilot inputs Look at steady state response of the angle of attack Step Response Envelope Protection System Steady State Estimation LIMIT DETECTION Estimate value of limited parameter in steady state LIMIT AVOIDANCE Find control value that causes the limited parameter to reach envelope limit in steady state 4-59

15 Table of limits on θ ref (Clean Case) from SS and LMI approach (α max =18.1 deg) Steady State LMI approach Velocity (m/s) Maximum step θ ref allowed (deg) Velocity (m/s) Maximum θ ref allowed (deg)

16 PAH A/P with EP Module Pilot input (θ ref ) Command limiter Limited θ ref Closed Loop PAH dynamics x, Aircraft States θ ref limits Reference Command limit Calculation α stall Limit Calculation η EP MODULE 4-61

17 Data Generation Data is generated by issuing a range of reference pitch commands at different flying conditions V trim θ ref η Closed loop PAH dynamics α ss Steady state angle of attack values corresponding to trim state values of V, η and θ ref are recorded 4-62

18 EP Module Coding Scheme EVERY 5 SECONDS Treat the state reached as a trim state Use the data generated to obtain max maximum allowable θ ref ( θ ref ) at that state α ss = f ( V, η, θ max ref ) α stall ( η) Compare θ ref at the current point with max the θ ref value and pitch down if necessary 4-63

19 A scenario A/C trimmed at V = 50 m/s with η=0.2 at H=2300m. A pitch up command of 4 degrees issued. 4-64

20 Another Scenario A pitch up command of 4 degrees with V=60m/s is issued and ice starts to build and grows from η=0 at t=0 to η=0.2 at t=50 s. 4-65

21 Continued 4-66

22 Summary The pitch command inputs need to be reduced in case of icing to stay within the prescribed limit The EP module works fine with varying stall angle limits with icing 4-67

23 Work in Progress Test the steady state based envelope protection module in the simulator Complete study on alternative PAH A/P design 4-68