helipresosmc_icet

Problem Defintion Mathematical Modelling Open Loop Analysis Nonlinear Analysis SMC of Helicopter CCP Control CAS Control Vertical Takeoff Questions

Robust Altitude Tracking of a Helicopter usingSliding Mode Control Structure

Yasir Awais Butt &Aamer Iqbal Bhatti

Muhammad Ali Jinnah University, Islamabad, PakistanControl and Signal Processing Research Group

Control and Signal Processing Research Group Yasir Awais Butt & Aamer Iqbal Bhatti


Outline

1 Problem Defintion

2 Mathematical Modelling

3 Open Loop Analysis

4 Nonlinear Analysis

5 SMC of Helicopter

6 CCP Control

7 CAS Control

8 Vertical Takeoff

9 Questions



Introduction and Problem Definition



Motivation



Motivation

• Rotory wing flight is fascinating• Challenging platform for nonlinear analysis• Development of practical application based on theoreticalresults is possible

• Growing interest and increased use of small model helicoptersfor applications that do not warrant use of a full scalehelicopter or where piloted flight is dangerous

• Extensive work on Sliding Mode by CASPR on Automotiveapplication

• Multiagent / formation flight of helicopters is current researchproblem being investigated by many universities



Problem Definition

• Autonomous flight was second major objective to be achieved• Nonlinear analysis of complete helicopter model is difficult dueto model complexity

• Reduce 6 DOF model to 1 DOF model to analyze the heavedynamics

• Design and analyse performance of sliding mode controller forhelicopter heave dyanmics



Mathematical Modelling



Thrust Generating Mechanism

Figure : Rotor feathering motion and its role in manueverability ofhelicopter



Helicopter Heave Dynamics Model

Ref:J. Kaloust C.Ham Z.Qu, Nonlinear autopilot control design fora 2 DOF helicopter model, lEEE Proceedings of Control TheoryApplication, Vol. 144, No.6, November 1997, pp 612-616

x1 = x2

x2 = a0 +a1x2 +a2x22 + (a3 +a4x4−

√a5 +a6x4)x2

3

x3 = a7 +a8x3 + [a9 sin(x4) +a10]x23 +u1

x4 = x5

x5 = a11 +a12x4 +a13x23 sin(x4) +a14x4 +u2

x = [ z z Ω θc θc ]T

u = [ u1 u2]T

Where u1 is the throttle input and u2 is the collective pitch input



Open Loop Analysis



Open Loop Response Initial Condition Alone



With Constant Throttle Input



Nonlinear Analysis and Design



Lyapunov Stability Analysis

Designing u1and u2:-

u1 = (−x1x2

x4− a2

x4x32 −Γx2x4−a7−a9sin(x5)x2

4 )

u2 = (−x5−a11−a12x5−a13x24 sin(x5))

V now becomes: -

V = k1x21 +a1x

22 +a8x

24 +a14x

26



Control Lyapunov Function (Regulator Problem)

Figure : Response of System with Control Lyapunov Function and ZeroRef Input



Sliding Mode Control of HeaveDynamics of Helicopter



Sliding Mode Tracking Dynamics

• Consider x = x−xd where x is the original state and xd is thedesired state.

• Sliding surface is given as: -

s = x1 + x2 + · · ·+ xns = c1(x1−xd) + c2(x2−x2d) + · · ·+ (xn−xnd)s = c1(x1− xd) + c2(x2− x2d) + · · ·+ (xn− xnd)

Ref:Applied Nonlinear Control Chap 7 - Slotine



Sliding Mode Tracking Dynamics

Figure : Sliding Surfaces Comparison



Peculiarities - Tracking Dynamics for Helicopter

• Don’t need to track all the states• For hovering only vertical position is to be tracked• Vertical velocity at hovering is zero• Rotor angular speed and collective pitch is to be variable toreject disturbances

• Two inputs, therefore two sliding surfaces



Methodology

• Hovering control with constant collective pitch (CCP)• Hovering control with constant rotor angular speed (CAV)• Vertical takeoff and hovering using multiple sliding surfaces



Control with Constant CollectivePitch



Simplified Model

• Assumption• Collective Pitch x4 = 0.122rad• Pitch Rate x5 = 0

• Reduced Problem - Hovering by throttle control

x1 = x2x2 = f2 = a0 +a1x2 +a2x

22 + (a3 +a4x4−

√a5 +a6x4)x2

3x3 = f3 +u1 = a7 +a8x3 + [a9 sinx4 +a10]x2

3 +u1



Sliding Surface Design

Sliding Surface

s = x3 + c2x2 + c1x1

s = x3 + c2x2 + c1 ˙x1 = x3 + c2x2 + c1x1

Equivalent control

ueq =−f3− c2f2− c1x2−Ksign(s)



Reachability

Lyapunov Method

V =12s2

V = ss =−K | s |



Finite Time Convergence

Global Convergence

Perruquetti, W.; Barbot, J.P. (2002). Sliding Mode Control inEngineering.If

V ≤−µ√V

α

where µ > 0 and 0< α ≤ 1, then the system is globally finite timeconvergent.In this system:-

V = ss =−K | s |=−√2Ksign(s)

√V

Hence global convergence condition is not satisfied.



Finite Time ConvergenceSarah SPURGEON, “Finite time convergent control using terminalsliding mode”, Journal of Control Theory and Applications (2004)Suppose s(0) = x3 + c2x2 + c1x1 > 0, the system converges to thesliding surface in finite time, if, ∃T > 0 such that s(T ) = 0.

x3 =−Ksign(s)−2

∑i=1

ci fi

Integrating both sides.

∫ t

0dx3 =

∫ t

0(−Ksign(s)−

2

∑i=1

ci fi )dt

x3(t) =−Kt−2

∑i=1

cixi (t) +2

∑i=1

cixi (0) + x3(0)



Finite Time ConvergenceNow x3(t)≤ 0 ensures that trajectories do reach the sliding surface.Lets calculate time to reach surface: -

t ≥ x3(0)

K−

2

∑i=1

cixi (t) +2

∑i=1

cixi (0)

Finite Time Convergence Condition

x3(0) > K2

∑i=1

cixi (t)−K2

∑i=1

cixi (0)

Thus ∃T > 0 for which trajectories converge to sliding surface iffabove condition is satisfied. Result can be generalized for xn for thetrivial sliding surface used ordinarily.



Matched Disturbance RejectionA miniature helicopter is particularly susceptible to disturbancesuch as wind gusts, weight changes etc that may enter the systemin affine or non-affine manner. Consider an affine disturbance γ forconstant rotor angular speed.

x1 = f1 = x2

x2 = f2 = a0 +a1x2 +a2x22 + (a3 +a4α−

√a5 +a6α)x2

3

x3 = f3 +u1 = a7 +a8x3 + [a9 sinx4 +a10]x23 +u1 + γ (1)

We calculate the bound of this disturbance through Lyapunovapproach. Consider again Lyapunov function (12).

V =−K |s|+ γs



Hovering at 100m



Hovering at 1000m



Hovering at 100m - Collective Pitch 0.22rad



Matched Disturbance



Tracking Time Varying TrajectorySine wave tracking, Amplitude 20, Frequency 5rad/sec, Offset 100



Tracking Time Varying TrajectorySine wave tracking, Amplitude 5, Frequency 0.5rad/sec, Offset 100



Tracking Time Varying TrajectorySine wave tracking, Amplitude 5, Frequency 5Hz, Offset 100



Control with Constant AngularSpeed



Simplified Model

• Assumption• Main rotor angular velocity x3 = 100rad/sec• Main rotor angular acceleration x3 = 0

• Reduced Problem - Hovering by collective pitch control

x1 = x2x2 = f2 = a0 +a1x2 +a2x

22 + (a3 +a4x4−

√a5 +a6x4)x2

3x4 = x5x5 = f5 +u2 = a11 +a12x4 +a13x

23 sin(x4) +a14x4 +u2



Sliding Surface Design

• Sliding Surface

s = x5 + c4x4 + c2x2 + c1x1

s = x5 + c4x4 + c2x2 + c1 ˙x1 = x5 + c4x4 + c2x2 + c1x1

• Equivalent control

ueq =−f5− c2f2− c4x5− c1x2−Ksign(s)



Hovering at 100m via Collective Pitch Control



Control Effort



Vertical Takeoff



Helicopter Takeoff Theory



Helicopter Takeoff Strategy



Takeoff and Hovering



Rotor Angular Speed and Control Effort



Questions


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