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Control & InstrumentationLab Assignment

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Student Name

Student ID Number

Instructor (professor)

Institutional affiliation

Location(state, country)

Date of submission

TABLE OF CONTENTS

INTRODUCTION...........................................................................................................................1

AIMS OF THE EXPERIMENT......................................................................................................3

ROTARY INVERTED PENDULUM SYSTEM MODEL.............................................................3

ROTARY INVERTED PENDULUM SIMULINK MODEL.........................................................7

CONTROLLER DESIGN...............................................................................................................8

DISCUSSION..................................................................................................................................8

CONCLUSION & FUTURE WORKS...........................................................................................8

REFERENCES................................................................................................................................9

LIST OF FIGURES

Figure 1 ROTPEN rotary inverted pendulum on LABVIEW [source: Quanser].........................................3

Figure 2 Free body diagram of the rotary inverted pendulum......................................................................4

Figure 3ROTPEN Matlab Simulink Model-Overview................................................................................9

Figure 4 Step response ROTPEN Matlab Simulink Model Output............................................................12

Figure 5 Bode Diagram to show magnitude and phase of ROTPEN Pendulum........................................13

Figure 6 Open loop system Simulation Model Block................................................................................14

Figure 7 Root Locus for Transfer Function 1............................................................................................15

Figure 8 Root Locus for transfer function 2...............................................................................................16

Figure 9 Pole location for a closed loop system........................................................................................18

Figure 10 State feedback controller for a closed loop ROTPEN pendulum...............................................18

Figure 11 The ROTPEN controller for the Pendulum link........................................................................19

Figure 12 System Simulation Balance Control Response for the closed loop............................................20

Figure 13 Results form Simulation of the ROTPEN model.......................................................................21

Figure 14 ROTPEN Model........................................................................................................................22

Figure 15 State Feedback controller segment............................................................................................22

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INTRODUCTION...........................................................................................................................1

AIMS OF THE EXPERIMENT......................................................................................................3

ROTARY INVERTED PENDULUM SYSTEM MODEL.............................................................3

ROTARY INVERTED PENDULUM SIMULINK MODEL.........................................................7

CONTROLLER DESIGN...............................................................................................................8

DISCUSSION..................................................................................................................................8

CONCLUSION & FUTURE WORKS...........................................................................................8

REFERENCES................................................................................................................................9

LIST OF FIGURES

Figure 1 ROTPEN rotary inverted pendulum on LABVIEW [source: Quanser].........................................3

Figure 2 Free body diagram of the rotary inverted pendulum......................................................................4

Figure 3ROTPEN Matlab Simulink Model-Overview................................................................................9

Figure 4 Step response ROTPEN Matlab Simulink Model Output............................................................12

Figure 5 Bode Diagram to show magnitude and phase of ROTPEN Pendulum........................................13

Figure 6 Open loop system Simulation Model Block................................................................................14

Figure 7 Root Locus for Transfer Function 1............................................................................................15

Figure 8 Root Locus for transfer function 2...............................................................................................16

Figure 9 Pole location for a closed loop system........................................................................................18

Figure 10 State feedback controller for a closed loop ROTPEN pendulum...............................................18

Figure 11 The ROTPEN controller for the Pendulum link........................................................................19

Figure 12 System Simulation Balance Control Response for the closed loop............................................20

Figure 13 Results form Simulation of the ROTPEN model.......................................................................21

Figure 14 ROTPEN Model........................................................................................................................22

Figure 15 State Feedback controller segment............................................................................................22

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INTRODUCTION

The rotary inverted pendulum is a nonlinear system whose initial state is unstable hence

the need for control. There are different methods implemented in the control of the rotary

inverted pendulum. In the analysis, the velocity of the pendulum center of mass, is considered for

a system displaced in the angle, a, and in the x-direction [1]-[5].The system has a motor that

moves the cart along a straight track with the pendulum attached to the cart using a pin joint. The

axis of rotation of the pendulum link is considered to be horizontal and it is perpendicular to the

cart’s direction of motion. The input of the system is the force that is applied to the cart through

the motor. The horizontal link is coupled such that it links directly or by connecting to a gearing

of the motor shaft and the rotary motion [6]. The QNET rotary inverted pendulum kit is

comprised of the dc motor, L-shaped arm, and pendulum, and two optical encoders, Elvis II

board. The motor that runs the pendulum is mounted on a metallic chamber. The pendulum is

suspended on a horizontal axis at the end of the arm. The pendulum and arm angle are measured

by two separate encoders [7]. The control variable is the input voltage to the pulse width

modulated amplifier that controls the motor.

Figure1ROTPEN rotary inverted pendulum on LABVIEW [source: Quanser]

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The rotary inverted pendulum is a nonlinear system whose initial state is unstable hence

the need for control. There are different methods implemented in the control of the rotary

inverted pendulum. In the analysis, the velocity of the pendulum center of mass, is considered for

a system displaced in the angle, a, and in the x-direction [1]-[5].The system has a motor that

moves the cart along a straight track with the pendulum attached to the cart using a pin joint. The

axis of rotation of the pendulum link is considered to be horizontal and it is perpendicular to the

cart’s direction of motion. The input of the system is the force that is applied to the cart through

the motor. The horizontal link is coupled such that it links directly or by connecting to a gearing

of the motor shaft and the rotary motion [6]. The QNET rotary inverted pendulum kit is

comprised of the dc motor, L-shaped arm, and pendulum, and two optical encoders, Elvis II

board. The motor that runs the pendulum is mounted on a metallic chamber. The pendulum is

suspended on a horizontal axis at the end of the arm. The pendulum and arm angle are measured

by two separate encoders [7]. The control variable is the input voltage to the pulse width

modulated amplifier that controls the motor.

Figure1ROTPEN rotary inverted pendulum on LABVIEW [source: Quanser]

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The rotary inverted pendulum is applied in the industries especially where items need to

be lifted from one point to another. The same system is implemented in robots and in the large

cranes used in the construction industry [8]. In some mall, the system is used to design the

Segway used to by security guards to monitor the premises. The ROTPEN kit in the 2 DOF

freely moves in two rotary directions.

AIMS OF THE EXPERIMENT

(a)To linearize a non-linear rotary inverted pendulum systems using Euler-Lagrange

equations of motion.

(b)To define the linear state-space representation of the rotary inverted pendulum system

(c)To develop a state-feedback or full feedback control to balance the pendulum in the

upright position using a pole placement.

(d)To simulate the open-loop and the closed loop systems of the ROTPEN inverted

pendulum so that the specifications are met and evaluating the performance.

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be lifted from one point to another. The same system is implemented in robots and in the large

cranes used in the construction industry [8]. In some mall, the system is used to design the

Segway used to by security guards to monitor the premises. The ROTPEN kit in the 2 DOF

freely moves in two rotary directions.

AIMS OF THE EXPERIMENT

(a)To linearize a non-linear rotary inverted pendulum systems using Euler-Lagrange

equations of motion.

(b)To define the linear state-space representation of the rotary inverted pendulum system

(c)To develop a state-feedback or full feedback control to balance the pendulum in the

upright position using a pole placement.

(d)To simulate the open-loop and the closed loop systems of the ROTPEN inverted

pendulum so that the specifications are met and evaluating the performance.

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ROTARY INVERTED PENDULUM SYSTEM MODEL

The system is modeled mathematically focusing on the rotational direction of the rotary

inverted pendulum arm. It is assumed that the system starts in the state of equilibrium and the

initial conditions are assumed to be zero [9]. The pendulum is set to move very few degrees

away from the vertical direction in order to satisfy the linear model.

Figure2Free body diagram of the rotary inverted pendulum

The velocity components of the rotary inverted pendulum in the x-direction are obtained

as,

VpenCOM=−Lcosα(̇α)^x−Lsinα(̇α)^y

Considering the rotating arm,

Varm=ṙθ

The x and y velocity components are expressed as,

vx=ṙθ−Lcosα(̇α)

vy=−Lsinα(̇α)

Obtaining the system dynamic equation based on the Euler-Lagrange Formulation,

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The system is modeled mathematically focusing on the rotational direction of the rotary

inverted pendulum arm. It is assumed that the system starts in the state of equilibrium and the

initial conditions are assumed to be zero [9]. The pendulum is set to move very few degrees

away from the vertical direction in order to satisfy the linear model.

Figure2Free body diagram of the rotary inverted pendulum

The velocity components of the rotary inverted pendulum in the x-direction are obtained

as,

VpenCOM=−Lcosα(̇α)^x−Lsinα(̇α)^y

Considering the rotating arm,

Varm=ṙθ

The x and y velocity components are expressed as,

vx=ṙθ−Lcosα(̇α)

vy=−Lsinα(̇α)

Obtaining the system dynamic equation based on the Euler-Lagrange Formulation,

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V=PEpend=mgh=mgLcosα

T=KEhub+KEVx

+KEVy

+KEpend

The moment of inertia for a rod is given based on the center of mass, such that,

Jcm=(1

12)MR2

The half of the pendulum length is described as L such that,

R=2L

Jcm=(1

12)M(2L)2=(1

3)ML2

The kinetic energy equation can be written as,

T=(1

2)Jeq̇θ2+(1

2)m(ṙθ−Lcosα(̇α))2+(1

2)m(−Lsinα(̇α))2+(1

2)Jcṁα2

Determining the KE function using the Lagrangian formulation, the equation results into,

L=T−V=(1

2)Jeq̇θ2+(2

3)mL2̇α2−mLrcosα(̇α)(̇θ)+(1

2)mr2̇θ2−mgLcosα

The two equations are used to obtain the angular components,

δ

δt(δL

δ̇θ)−δL

δθ=Toutput−Beq̇θ

δ

δt(δL

δ̇α)−δL

δα=0

Linearizing the equation atα=0,

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T=KEhub+KEVx

+KEVy

+KEpend

The moment of inertia for a rod is given based on the center of mass, such that,

Jcm=(1

12)MR2

The half of the pendulum length is described as L such that,

R=2L

Jcm=(1

12)M(2L)2=(1

3)ML2

The kinetic energy equation can be written as,

T=(1

2)Jeq̇θ2+(1

2)m(ṙθ−Lcosα(̇α))2+(1

2)m(−Lsinα(̇α))2+(1

2)Jcṁα2

Determining the KE function using the Lagrangian formulation, the equation results into,

L=T−V=(1

2)Jeq̇θ2+(2

3)mL2̇α2−mLrcosα(̇α)(̇θ)+(1

2)mr2̇θ2−mgLcosα

The two equations are used to obtain the angular components,

δ

δt(δL

δ̇θ)−δL

δθ=Toutput−Beq̇θ

δ

δt(δL

δ̇α)−δL

δα=0

Linearizing the equation atα=0,

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