Mini Project Reporton

Sliding Mode Control of Powered Orthosissubmitted by

BISWAJIT DEBNATH(M140201EE)

in partial fulfillment of the requirementsfor the award of the degree of

Master of Technologyin

Electrical Engineering(Instrumentation and Control Systems)

Under the guidance ofDr. S.J. Mija

Department of Electrical EngineeringNational Institute of Technology, Calicut

CalicutKerala 673 601

May 2015

Contents

1 INTRODUCTION 21.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 LITERATURE SURVEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 ORGANIZATION OF REPORT . . . . . . . . . . . . . . . . . . . . . . . 4

2 LAB HELICOPTER SYSTEM 52.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 SYSTEM MODELLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 CONTROLLER DESIGN 83.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 NOMINAL CONTROLLER DESIGN . . . . . . . . . . . . . . . . . . . . . 83.3 ROBUST COMPENSATOR DESIGN . . . . . . . . . . . . . . . . . . . . 93.4 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 RESULTS AND DISCUSSION 104.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 CONCLUSIONS 14

REFERENCES 15

2

List of Figures

1.1 Boeing HC-1B Chinook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Lab Helicopter system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1 Control Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Control Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Control Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1 Elevation Response with State F/B Controller . . . . . . . . . . . . . . . . 104.2 Travel Response with State F/B Controller . . . . . . . . . . . . . . . . . . 114.3 Elevation Tracking for Square wave Ref Input w/o disturbance . . . . . . . 124.4 Travel Tracking for Square wave ref Input w/o disturbance . . . . . . . . . 124.5 Elevation Response for square wave ref.input with disturbance . . . . . . . 134.6 Travel Response for square wave ref. input with disturbance . . . . . . . . 13

1

Chapter 1

INTRODUCTION

1.1 INTRODUCTIONHelicopters, either man controlled or an autonomous unmanned vehicle system have

gained much attention these days due to their versatile functions like, remote areal ex-ploration, remote sensing, imaging and surveillance etc. Considerable attention has beenattracted to the analysis and control of helicopters due to their potential military andcivil applications as well as scientific significance.

The attitude and position controller is an important part of a helicopter and its designproblem attracts much attention of the researchers. High levels of agility, manoeuvrability,and the capability of operating in degraded visual environments and adverse weatherconditions are the current trends of helicopter control design. Helicopter flight controlsystems should make these performance requirements achievable by improving trackingperformance and disturbance rejection capabilities. Robustness is one of the critical issues,which must be considered in the control system design for high-performance autonomoushelicopter. In general, most control designs are based on linearised helicopter dynamicsusing the widely adopted concept of stability derivatives. However, in recent years thereis considerable research related to helicopter flight control based on non-linear dynamicrepresentations.

The system considered in this report is a lab helicopter with 3-degrees of freedom. Thesystem has similar dynamics with the Boeing Chinook HC-1B Tandem helicopters, whichare used for rescue operations in military.

Figure 1.1: Boeing HC-1B Chinook

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A linear time-invariant robust control design is mentioned here, which deals with the po-sition control problem under aggressive manoeuvres and time-varying wind disturbances.This controller consists of two parts, a nominal state feedback controller and a robustcompensator. The nominal part is applied to obtain desired tracking for the nominallinear system while the robust compensator is added to restrain the effects of parame-ter perturbations, non-linear uncertainties and external disturbances because of the windgusts. The nominal dynamical performances can be specified by the optimal controllerand the robust compensator can restrain the effects of uncertainties.

The attitude control problems involving the elevation and pitch angles were investigatedfor the 3-DOF helicopter and full state feedback control schemes were applied. However,in position control problems, the elevation and travel angles are required to track theirdesired reference values and the pitch angle is considered as the inner dynamics. Becauseof MIMO system with serious inter-axis couplings, a full state feedback control methodcan not be used. Since some states cannot be measured directly, these states are estimatedby indirectly by investigating pitch angle.

1.2 LITERATURE SURVEYMany approaches have been done to reduce the effects of the wind gusts. The influences

of external wind were measured by sensors of the wind velocity and a feedforward con-troller was applied. Earlier to counter the effects of the wind disturbances, the wind gustvelocity is measured with a rotary vane anemometer, and an optimal control approachwas applied. Furthermore, it was shown that the wind disturbances could be addressedwithout the measuring equipments of the wind velocity and could be reduced by the adap-tive output regulation method or the disturbance observation control technique. Windgusts were considered as uncertainties in and an active disturbance rejection controllerwas designed to restrain the effects of the wind gusts. Many previous experimental worksfocused on the stabilization of the helicopters in hovering or near hovering conditions inthe presence of the persistent wind gusts with a fixed velocity, while further investiga-tion to obtain better tracking performances for aggressive manoeuvres under time-varyingwind disturbances remains challenging

A non-linear controller was proposed with a non-linear model predictive control and anon-linear disturbance observer to estimate the bounded lumped disturbances involvingvariations of the helicopter dynamics and external influences introduced by wind turbu-lences. However, further investigation to design a decentralized and linear time-invariantcontroller to restrain the effects of the lumped disturbances with only uniform finite normbounds were not fully discussed here. A standard H- controller can also be appliedto achieve robust flight control. However, reducing the influences of model uncertainties,non-linear dynamics and external wind disturbances cannot be guaranteed in the wholefrequency range by this control approach.

One feature of this control method compared with previous approaches on trajectorytracking control problem under wind disturbances is that the wind velocity does notneed to be measured. Another feature is that the desired dynamical and steady-statetracking performances can be obtained even in aggressive mission under the influencesof time-varying wind gusts. The applications of the proposed robust control method can

be extended to a conventional 6-DOF helicopter, whose dynamics shares some specialfeatures of the laboratory helicopter such as the non-linearities, parametric uncertaintiesand external wind disturbances involved. Its robust controller can be designed in a similarway with a nominal controller and a robust compensator.

1.3 OBJECTIVESThe objective is to develop a robust position controller for 3-DoF lab helicopter system

to improve the system performance under the effect of the external wind disturbances.And to develop a tracking controller using state feedback approach, which helps the systemto trace the desired reference trajectory.

1.4 ORGANIZATION OF REPORTThe report is organised as follows. In Chapter 2, lab helicopter system theory is pre-

sented and its approximated linear model is derived. In Chapter 3, The design procedureof controller is mentioned. And results and conclusion over the designed control are givenin Chapter 4 and 5 respectively.

Chapter 2

LAB HELICOPTER SYSTEM

2.1 INTRODUCTIONthe laboratory helicopter has 3-DOF. That are elevation, pitch and travel. The heli-

copter frame, with two motors called front motor and back motor installed at its two ends,is suspended from an instrumented joint mounted at one end of a long arm. The frameis free to pitch around this arm and the arm is free to elevate and travel. The motorsare installed with propellers and a positive voltage to either motor results in an positiveelevation torque of the helicopter frame. Front motor applied with positive voltage causesa positive pitch torque whereas back motor with positive voltage causes a negative pitchtorque. A travel torque of the helicopter frame can be produced if the pitch angle isnon-zero. An attitude controller is aimed at tracking references of the elevation and pitchchannels, whereas a position controller is focused on trajectory tracking for the elevationand travel angles. The system is shown by fig 2.1.

Figure 2.1: Lab Helicopter system

2.2 SYSTEM MODELLINGThe modelling provides the input-output relationship of the system in the mathematical

form of a ordinary differential equations. Model equations are useful to understand thesystem state dynamics and behaviour of the system. For lab helicopter system a whitebox modelling approach is used, i.e. input-output relations are obtained by using thephysical laws like force-energy balance, mass balance and total energy conservation of thesystem. To develop the model for system certain assumptions are made and followingterms are defined.

The pitch axis is a line perpendicular to the length of body frame of helicopter at centerof gravity of the body frame and the pitch angle is the rotation of system about the pitchaxis. An elevation axis is the line parallel to length of body at the base frame and thetravel axis is the vertical line perpendicular to the elevation axis. The system modellingconventions are

The Helicopter is in horizontal position, when elevation angle is zero. Travel angle increases positively, when body rotates in counter-clock wise direction.

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Pitch angle is positive, when thrust due to front motor is higher than that of backmotor - nose above horizon position. There exist a mechanical limit for pitch angle.

trajectories and their derivatives are piecewise uniformly bounded The external wind disturbances and their derivatives are bounded.

The mathematical equations of the rotational motions of the 3-DOF helicopter can bederived by the EulerLagrange formula. In order to incorporate parameter uncertaintiesand external wind disturbances into the helicopter model, the dynamical model of thislab helicopter can be described as

(t) = a1(t) + a2sin(t) + bcos(t)(Vf (t) + Vb(t)) + w(t) (2.1)(t) = a1(t) + a2sin(t) + b(Vf (t) Vb(t) + w(t) (2.2)(t) = a1 + a2sin(t)(Vf (t) + Vb(t)) + a2Vopsin(t) + w(t) (2.3)

where (t) is the elevation angle, (t) is the pitch angle, (t) is the travel angle, Vf (t)and Vb(t) are the control voltages of the front motor, and the back motor, respectively,wi(t) (i = , , ) are the additional forces acting on the helicopter from the externalwind gusts, Vop is a positive constant, ai1 (i = , , ) are the damping coefficients forthe three angles, bi (i = , , ) and a2 are the voltage-to-torque scaling factors in eachchannel and ai2 (i = , , ) positive coefficients.

The parameters can be split into two parts, one is nominal parts; denoted by superscriptN and other is uncertain parts represented by . They are given below by equation (2.4)and (2.5) also control inputs ui(t) are defined, are given by equation (2.6) and (2.7).

aij = aNij + aij (2.4)bi = bNi + bi (2.5)

u(t) = Vf (t) + Vb(t) (2.6)u(t) = Vf (t) Vb(t) (2.7)

The helicopter model given by equations (2.1), (2.2) and (2.3) can be rewritten as lin-earised model considering the qi(t) as equivalent disturbances occurring in the each chan-nel.

(t) = aN1(t) + aN2(t) + bN u(t) + q(t) (2.8)(t) = aN1(t) + aN2(t) + bN u(t) + q(t) (2.9)(t) = aN1 + bN (t) + q1(t) (2.10)

By observing the above equations, it is clear that the pitch and travel are coupled. whereaschange in pitch or travel value will not effect the dynamics of elevation channel. Thecoupled channels are made separate as given below.Differentiating the equation no (2.10) twice, we get

...(t) = aN1(t) + bN (t) + q1(t) (2.11).... (t) = aN1

...(t) + bN (t) + q1(t) (2.12)

Substituting the (t) value from equation (2.9) in the equation (2.12).... (t) = aN1

...(t) + bN [aN1(t) + aN2(t) + bN u(t) + q(t)] + q1(t) (2.13)....

(t) = aN1...(t) + aN1bN (t) + bN aN2(t) + bN bN u(t) + bN q(t) + q1(t) (2.14)

From equation (2.11), substitute the bN (t) in equation (2.14).... (t) = aN1

...(t) + aN1[

...(t) aN1(t) q1(t)] + bN aN2(t) + bN bN u(t) + bN q(t) + q1(t)(2.15)....

(t) = aN1...(t) + aN1

...(t) aN1aN1(t) aN1 q1(t) + aN2bN (t) + bN bN u(t) + bN q(t) + q1(t)(2.16)

From equation (2.10), substitute the bN (t) value in the equation (2.16) and rearrangingthe parameters we get the equation as.

.... (t) = (aN1 + aN1)

...(t) + (aN2 aN1aN1)(t) aN2aN1(t) + bN bN u + q(t)(2.17)

q(t) = q1(t) aN1 q1(t) + bN q(t) aN2q1(t) (2.18)

Thus equation (2.17) gives the decoupled model for travel channel of helicopter system andequation (2.18) gives the equivalent disturbance function occurring in the travel system.

2.3 SUMMARYThe lab helicopter is a non-linear, coupled MIMO system. The system is having three

degrees of freedom and the model of the system is obtained by applying Euler-Lagrangeformula over each degree. The obtained model have non-linearity and have interactionwith the other states (pitch to travel) of the system. This model is further linearised andinteracting states are separated by decoupling the system.

The position control is developed by considering the linearised-decoupled model, whichis given by equation (2.17) and (2.8) for travel and elevation channel respectively. Thedetails of controller design are explained in chapter 3.

Chapter 3

CONTROLLER DESIGN

3.1 INTRODUCTIONThe control block structure for 3-Dof lab helicopter system is shown in fig 4.3. The

robust controller is designed by applying the static linear feedback control for the nominallinear system and signal compensation for uncertainties. The control inputs ui(t) (i = ,) have two parts: the nominal control inputs uNi (t) (i = , ) and the control inputsvi(t) (i = , ) based on robust compensation technique. Thus, the control inputs havethe following forms.

ui(t) = uNi (t) + vi(t) (3.1)

Figure 3.1: Control Block Diagram

3.2 NOMINAL CONTROLLER DESIGNTo design the nominal state feedback control, first we neglect the uncertainty qi(t)

where (i = , ). The control law is given by the equation given below.

uNi (t) = (KiXi(t) + ri(t))/cN (3.2)

where Ki are chosen such that AiH = Ai + BiKi (i = , ) are Hurwitz matrices. ki(i = 1, 2, 3) and kj ( j = 1, 2, . . . , 5) can be determined by the pole placement methodbased on the steady-state error, overshoot and settling time requirements for the nominallinear system. The system will take form

Xi(t) = AiHXi(t) +Bi(cNi vi(t) + qi(t)) (3.3)

However, for the travel channel, (t) and (t) cannot be obtained by the measurements.Therefore as full state feedback nominal controller cannot be used here. These states areestimated by sensing the pitch dynamics of the system hence, redesign the nominal controlinput uN (t) as

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3.3 ROBUST COMPENSATOR DESIGNIn order to restrain the influences of uncertainties, robust compensator are added. Therobust compensating signals vi(s) (i = , ) are produced as

vi(s) = Fi(s)qi(s)/cNi (3.4)

Figure 3.2: Control Block Diagram

Figure 3.3: Control Block Diagram

3.4 SUMMARYPosition control design for a lab helicopter system is stated in this chapter. The con-

troller has two parts, the state feedback controller and a robust compensator. Statefeedback controller is considered as nominal controller, it is required for smooth trackingof the reference values and Robust compensator is designed based on H- control syn-thesis method. The effectiveness of these controllers over performance of the system isvalidated by doing MATLAB simulation. The results are discussed in Chapter 4.

Chapter 4

RESULTS AND DISCUSSION

4.1 INTRODUCTION

4.2 RESULTSThe effectiveness of the controller is analysed by doing the simulation using the Simulink

tool in MATLAB. The output responses, which are obtained by developing a MATLABcode for elevation and travel system is shown in fig 4.1 and fig 4.2. These responses arewith respect to the reference value as 1 degree unit input and without considering thedisturbance occurring in the system.

Figure 4.1: Elevation Response with State F/B Controller

Figure 4.2: Travel Response with State F/B Controller

The developed SimuLink models for elevation channel and travel channel are shown inthe fig 3.2 and fig 3.3 of previous Chapter 3. The model structure is exactly similar tothe control block diagram explained in previous Chapter 3.

Figure 4.3: Elevation Tracking for Square wave Ref Input w/o disturbance

Elevation system output response for a square wave reference input of period 30 sec andamplitude of 30 degrees with uniform disturbance is shown in fig 4.5. and Travel systemresponse for a square wave reference input of 30 sec period and 30 degree of magnitudewith uniform time varying disturbance is shown in the fig 4.6.

4.3 DISCUSSION

4.4 SUMMARYAt first the system performance is obtained by using state feedback controller only fora unit reference input value. And the state behaviour is discussed. The tracking of set

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Figure 4.4: Travel Tracking for Square wave ref Input w/o disturbance

Figure 4.5: Elevation Response for square wave ref.input with disturbance

point value is further analysed, using a state feedback controller alone and by applying asquare wave reference input to a system. Also a system performance using State feedbackcontroller and Robust compensator is discussed. The conclusions over the results arepresented in Chapter 5.

Figure 4.6: Travel Response for square wave ref. input with disturbance

Chapter 5

CONCLUSIONS

Using the robust compensator and state feedback controller a robust position controlis achieved for a 3DoF lab helicopter under the effect of the disturbances, where the statefeedback controller is responsible for tracking errors to the neighbourhood of the originin closed loop system and robust compensator provides the output robustness against thewind disturbances.

Robustness of the system, depends upon the effective tuning of the robust compensator.The parameters of the compensator are selected such that, it will suppress the unnecessaryfrequency region where the singular value of the system is high. This means that thecompensator will reduce the effect of the high frequency disturbance on the output of thesystem.

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REFERENCES

[1] Hao Liu, XiafuWang and Yisheng Zhong, "Robust position control of a lab helicopterunder wind disturbances,"IET Control Theory Appl., 2014, Vol. 8, Iss. 15, pp. 1555-1565.

[2] Bo Zheng and Yisheng Zhong, Member, IEEE, "Robust Attitude Regulation of a 3-DOF Helicopter Benchmark: Theory and Experiments,"IEEE Trans. on IndustrialElectronics, vol. 58, no. 2, February 2011.

[3] Hao Liu, Geng Lu, and Yisheng Zhong, "Robust LQR Attitude Control of a 3-DOFLaboratory Helicopter for Aggressive Maneuvers,"IEEE Trans. on Industrial Electron-ics, vol. 60, no. 10, October 2013.

[4] Thomas Kiefer, Knut Graichen, and Andreas Kugi, "Trajectory Tracking of a 3DOFLaboratory Helicopter Under Input and State Constraints,"IEEE Trans. on ControlSystems Technology, vol. 18, no. 4, July 2010.

[5] Ioannis A. Raptis, Kimon P. Valavanis, Senior Member, IEEE, and Wilfrido A.Moreno, Member, IEEE, "A Novel Nonlinear Backstepping Controller Design for Heli-copters Using the Rotation Matrix,"IEEE Trans. on Control Systems Technology, vol.19, no. 2, March 2011.

[6] Franois Lonard, Adnan Martini, and Gabriel Abba, Member, IEEE, "Robust Nonlin-ear Controls of Model-Scale Helicopters Under Lateral and Vertical Wind Gusts,"IEEETrans. on Control Systems Technology, vol. 20, no. 1, January 2012.

[7] Gareth D. Padfield, "Helicopter Flight Dynamics,"Second Edition, Blackwell Publish-ing, North America 2007.

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INTRODUCTIONINTRODUCTIONLITERATURE SURVEYOBJECTIVESORGANIZATION OF REPORT

LAB HELICOPTER SYSTEMINTRODUCTIONSYSTEM MODELLINGSUMMARY

CONTROLLER DESIGNINTRODUCTIONNOMINAL CONTROLLER DESIGNROBUST COMPENSATOR DESIGNSUMMARY

RESULTS AND DISCUSSIONINTRODUCTIONRESULTSDISCUSSIONSUMMARY

CONCLUSIONSREFERENCES