Solution Manual for Modern Control Systems 12th Edition by Dorf

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C H A P T E R

1

Introduction to Control Systems

There are, in general, no unique solutions to the following exercises and problems. Other equally valid block diagrams may be submitted by the student.

Exercises E1.1

A microprocessor controlled laser system: Controller 

Desired power  output

Error 

-

Micro-      processor 

Currenti(t)

Laser 

Power  Sensor 

power 

A driver controlled cruise control system: Controller

Desired speed

Power  out

Measurement

Measured

E1.2

Process

Process

Foot Fo ot ped pedal al

-

Driver

Car and Engine

Actual auto speed

Measurement

Visual Visu al indica indication tion of speed speed

E1.3

Speedometer

Although the princi Although principle ple of conservation conservation of momen momentum tum explains much much of  the process of fly-casting, there does not exist a comprehensive scientific explanation of how a fly-fisher uses the small backward and forward motion of the fly rod to cast an almost weightless fly lure long distances (the 1

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2

CHAPTER 1

Introduction Introduc tion to Contro Controll Syste Systems ms

current world-record is 236 ft). The fly lure is attached to a short invisible leader about 15-ft long, which is in turn attached to a longer and thicker Dacron line. The objective is cast the fly lure to a distant spot with deadeye accuracy so that the thicker part of the line touches the water first and then the fly gently settles on the water just as an insect might. Fly-fisher Desired position of  the fly

Controller

-

Wind disturbance

Mind and body of the fly-fisher

Process

Rod, line, and cast

Actual

position of the fly

Measurement

Visual indication of the position of the fly

E1.4

Vision of the fly-fisher

An autofocus autofocus camera camera control control system: system: One-way trip time for the beam

Conversion factor (speed of light or   sound)

K1 Beam Emitter/ Receiver Beam return

Distance to subject

Subject Lens focusing motor

Lens

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3

Exercises E1.5

Tack acking ing a sailboat as the wind shifts shifts::

Error 

Desired ailboat irection

-

Controller 

 Actuators

Sailor

Rudder and sail adjustment

Wind

Process

Sailboat

Actual sailboat direction

Measurement Measured sailboat direction

Gyro compass

E1.6

An automated highway highway control control system merging two two lanes of traffic: Controller 

Error 

Desired gap

-

Embedded computer

 Actuators

Brakes, gas or steering

Process

Active vehicle

Actual gap

Measurement Measured gap

Radar

E1.7

Using the speedometer, speedometer, the driv driver er calculates calculates the differ difference ence between between the measured speed and the desired speed. The driver throotle knob or the brakes as necessary to adjust the speed. If the current speed is not too much over the desired speed, the driver may let friction and gravity slow the motorcycle down. Controller 

Desired speed

Error 

-

Driver

 Actuators

 Throttle or brakes

Measurement Visual indication of speed

Speedometer

Process

Motorcycle

Actual motorcycle speed

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4

CHAPTER 1 E1.8

Introduction Introduc tion to Contro Controll Syste Systems ms

Human biofeedback biofeedback control system: Controller 

Desired body temp

Process

Hypothalumus

-

Message to blood vessels

Actual body temp

Human body

Measurement Visual indication of body temperature

E1.9

 TV display

Body sensor

E-enabled aircraft with ground-based flight path control: Corrections to the flight path

Desired Flight Path

-

Controller

Aircraft

Gc(s)

G(s)

Flight Path Health Parameters

Meteorological data

Location and speed

Optimal flight path

Ground-Based Computer Network 

Desired Flight Path

E1.10

Specified Flight Trajectory

Optimal flight path Health Parameters

Meteorological data

Corrections to the flight path

Gc(s)

G(s)

Controller

Aircraft

Location and speed

Flight Path

Unmanned Unmann ed aerial aerial ve vehic hicle le use used d for crop mon monito itorin ringg in an aut autono onomou mouss mode: Trajectory error 

UAV UA V

Controller 

-

Gc( s  s))

Flight Trajectory

G( s  s))

Sensor  Location with respect to the ground

Map Correlation Algorithm

Ground  photo

Camera

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5

Exercises E1.11

An inverted inverted pendulum control control system using an optical encoder encoder to measure the angle of the pendulum and a motor producing a control torque:  Actuator 

Voltage

Error 

Desired angle

-

Controller 

 

Process

Torque

Motor 

Pendulum

 Angle

Measurement

Measured angle

E1.12

In the video game, the player player can serve serve as both the controller and the sensor. The objective of the game might be to drive a car along a prescribed path. The player controls the car trajectory using the joystick using the visual queues from the game displayed on the computer monitor. Controller 

Desired game objective

Optical encoder 

Error 

-

Player 

 Actuator 

Joystick

Measurement

Player  (eyesight, tactile, etc.)

Process

Video game

Game objective

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6

CHAPTER 1

Introduction Introduc tion to Contro Controll Syste Systems ms

Problems P1.1

Desired temperature set by the driver

An automobile interior interior cabin temperature temperature control system system block diagram:

Error 

-

Controller 

Process

 Thermostat and air conditioning unit

Automobile cabin

Automobile cabin temperature

Measurement Measured temperature

P1.2

 Temperature  Temp erature sensor

A human operator controlled valve valve system: Controller 

Process

Error *

Desired fluid output *

-

 Tank 

Valve

Fluid output

Measurement Visual indication of fluid output *

Meter * = operator functions

P1.3

A chemical chemical composition composition control control block diagram: diagram: Controller 

Process

Error Desired chemical composition

-

Mixer tube

Valve

Measurement Measured Mea sured chemical composition

Infrared analyzer

Chemical composition

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7

Problems P1.4

A nuclear nuclear reactor reactor control control block diagram: diagram: Controller 

Process

Error Desired power pow er level

Reactor and rods

Motor and amplifier

-

Output power level

Measurement Measured chemical

Ionization chamber

composition

P1.5

A light seeking seeking control control system system to track the sun: sun:

Measurement

Light source

Dual Photocells

P1.6

Controller 

Ligh intensity

 Trajectory Planner

Desired carriage position

Controller 

-

Motor, carriage, and gears

K 

Photocell carriage position

If you assume that increasing increasing worker’s worker’s wages wages results in incre increased ased prices, prices, then by delaying or falsifying cost-of-living data you could reduce or eliminate the pressure to increase worker’s wages, thus stabilizing prices. This would work only if there were no other factors forcing the cost-of-living up. Government price and wage economic guidelines would take the place of additional “controllers” in the block diagram, as shown in the block diagram. Controller 

Process Market-based prices

Initial wages

Process

Motor inputs

Error

-

Industry

Government price guidelines

Controller 

Wage increases

Government wage guidelines

Cost-of-living

K 1

Prices

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8

CHAPTER 1 P1.7

Introduction Introduc tion to Contro Controll Syste Systems ms

Assume that the can Assume cannon non fires initiall initially y at exa exactl ctly y 5:0 5:000 p.m. p.m... We ha hav ve a positive feedback system. Denote by ∆t ∆t  the time lost per day, and the net time error by E  by  E T  T  . Then the follwoing relationships hold: ∆t  = 4/3 min. min. + 3 min. min.  = 13 13/ /3 min. min. and E T  13/ /3 min./ min./day day . T  = 12 days × 13 Therefore, the net time error after 15 days is min. E T  T  = 52 min.

P1.8

The student-teac student-teacher her learning process:  

Controller 

Lectures

Error Desired knowledge

-

Process

 Teacher

Knowledge

Student

Measurement

Exams

Measured knowledge

P1.9

A human human arm control control system: system:  

Controller  u Desired arm location

e

 y

Process

 s Brain

 Nerve signals

 z  Measurement

Visual indication of  arm location

Pressure Eyes and  pressure receptors

Arm & muscles

d 

Arm location

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9

Problems P1.10

An aircraft aircraft flight path path control control system system using GPS: Controller 

Desired flight path from air traffic controllers

 Actuators

Computer Auto-pilot

Error 

-

Process

Ailerons, elevators, rudder, and engine power

Flight path

Aircraft

Measurement Measured flight path

P1.11

Global Positionin Positioning g System

The accurac accuracy y of the clock is depe depende ndent nt upon a con consta stant nt flow from the orifice; the flow is dependent upon the height of the water in the float tank. The height of the water is controlled by the float. The control system controls only the height of the water. Any errors due to enlargement of  the orifice or evaporation of the water in the lower tank is not accounted for. The control system can be seen as: Controller 

Desired height of the water in float tank 

P1.12

Process

Flow from upper tank to float tank 

Float level

-

Actual height

Assume that the turret Assume turret and fantai fantaill are at 90 , if  θ  θ w  =  θ F  -90 . The fantail operates on the error signal θ signal  θ w -  θ T  , and as the fantail turns, it drives the turret to turn. ◦

◦

y

Wind

qW = Wind angle qF   = Fantail angle q T  = Turret angle

Controller 

* 

qW qW *    

qF q T

 Turret

x

-

Process  Torque

Error

Fantail

Fantail

Gears & turret

q T

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10

CHAPTER 1 P1.13

Introduction Introduc tion to Contro Controll Syste Systems ms

This scheme scheme assumes assumes the person p erson adjusts adjusts the hot water for temperature temperature control, and then adjusts the cold water for flow rate control. Controller 

Error 

Desired water temperature

Process

Hot water system

Valve adjust

-

Hot water

Actual water temperature and flow rate Desired water flow rate

Cold water system

Valve adjust

-

Cold water

Measurement

Measured water flow Measured water temperature

P1.14

Human: visual and touch

If the rewards rewards in a specific trade is greater greater than the average average reward, reward, there is a positive influx of workers, since q (t) =  f 1 (c(t) − r (t)) )).. If an influx of workers occurs, then reward in specific trade decreases, since c(t) = −f 2 (q (t)) )).. Controller 

Average rewards r (t )

P1.15

Error 

-

 f  1(c(t )-r  )-r (t )) ))

Process q(t )

Total of  rewards c(t )

A computer computer controlled controlled fuel injection injection system: system: Controller 

Desired Fuel Pressure

)) - f  2(q(t ))

-

Process

Electronic Control Unit Measurement Measured fuel pressure

Fuel Pressure Sensor

High Pressure Fuel Supply Pump and Electronic Fuel Injectors

Fuel Pressure

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11

Problems

With the onset of a fev With fever, er, the body thermos thermostat tat is turned turned up. The bo dy adjusts by shivering and less blood flows to the skin surface. Aspirin acts to lowers the thermal set-point in the brain.

P1.16

Controller 

Desired temperature or set-point from body thermostat in the brain

Process

Adjustments within the body

-

Body temperature

Body

Measurement Measured body temperature

Internal sensor

P1.17

Hitting a baseball is arguably one of the most difficult feats in all of sports. Given that pitchers may throw the ball at speeds of 90 mph (or higher!), batters have only about 0.1 second to make the decision to swing—with bat speeds aproaching 90 mph. The key to hitting a baseball a long distance is to make contact with the ball with a high bat velocity. This is more important than the bat’s weight, which is usually around 33 ounces (compared to Ty Cobb’s bat which was 41 ounces!). Since the pitcher can throw a variety of pitches (fast ball, curve ball, slider, etc.), a batter must decide if the ball is going to enter the strike zone and if possible, decide the type of pitch. The batter uses his/her vision as the sensor in the feedback loop. A high degree of eye-hand coordination is key to success—that is, an accurate feedback control system.

P1.18

Define the follo Define following wing variables variables:: p   = output pressure, f s   = spring force = Kx Kx,, f d   = diaphragm force = Ap Ap,, and f v   = valve force = f s - f d . The motion of the valve is described by y¨ = f v /m /m where  where m  is the valve mass. The output pressure is proportional to the valve displacement, thus  p =  p  = cy  cy ,  where  where c  c  is the constant of proportionality.

Constant of  proportionality

Spring

Screw displacement  x(t )

 K 

 f  s

-

Valve position

 f  v

Valve

c

 y

Diaphragm area

 f  d

 A

Output pressure  p(t )

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12

CHAPTER 1 P1.19

Introduction Introduc tion to Contro Controll Syste Systems ms

A control control system to keep keep a car at a given relative relative position position offset from a lead car:

 Throttle

Position of follower

Follower car

Actuator

-

Controller

u

Relative position

-

Position of lead

Lead car

Fuel throttle (fuel)

Video camera & processing algorithms

Reference photo

Desired relative position

P1.20

A control control system for a high-performanc high-performancee car with an adjustable wing: wing:

Desired road adhesion

-

Computer

Process

 Actuator 

Controller 

Adjustable wing

Road conditions

Race Car

Road adhesion

Measurement

Measured road adhesion

P1.21

 K 

 Tire internal strain gauges

A control control system for a twin-lift twin-lift helicopter helicopter system: system: Measurement Measured separation

Radar

distance

Desired separation distance

-

Controller 

Separation distance

Pilot Desired altitude

Process

Helicopter Altitude

Measurement Measured altitude

Altimeter

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13

Problems P1.22

The desired desired building deflection deflection would not necessaril necessarily y be b e zero. Rather it would be prescribed so that the building is allowed moderate movement up to a point, and then active control is applied if the movement is larger than some predetermined amount. Process Controller 

Desired deflection

Hydraulic stiffeners

-

Building

Deflection

Measurement

Measured deflection

P1.23

Strain gauges Strain on truss structure

 K 

The human-lik human-likee face of the robot might have have micro-actuator micro-actuatorss place placed d at strategic strat egic point p ointss on the inte interior rior of the malleable facial structure. structure. Cooperative control of the micro-actuators would then enable the robot to achieve various facial expressions. Controller 

Process

Error Desired actuator position

-

Voltage

Electromechanical actuator

Amplifier

Actuator position

Measurement

Measured position

P1.24

Position sensor

We mig might ht envisi envision on a sen sensor sor embedde embedded d in a “gu “gutte tter” r” at the base of the windshield which measures water levels—higher water levels corresponds to higher intensity rain. This information would be used to modulate the wiper blade speed. Process

Controller 

Desired wiper speed

Wiper blade and motor

Electronic Control Unit

-

Measurement

 K 

Measured water level

Water depth sensor

Wiper blade speed

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14

CHAPTER 1

Introduction Introduc tion to Contro Controll Syste Systems ms

A feedback feedback control control syste system m for the space traffic control: control:

P1.25

Controller 

Desired orbit position

Error 

-

Control law

 Actuator  Jet commands

Process Applied forces

Reaction control jets

Satellite

Actual

orbit position

Measurement Measured orbit position

Radar or GPS

P1.26

Earth-based Earthbased control control of a microrove microroverr to point the camera: camera: Microrover Camera position command

Receiver/  Transmitter  Transmitt er

Controller

G(s)

G  c(s)

Rover position

C   a   m   e  r    Measured camera a   M     p  o   position e  a   s   i   t   i    s  u   o  n   r   e   d     c   o   c   a   m   m   m   e  r    a  n   a   d      p  o   s  i   t    i   o   n  

P1.27

Desired Charge Level

Camera

Camera Position

Sensor

Control Cont rol of a methanol methanol fuel fuel cell: cell:

-

Controller

Recharging System

Gc(s)

GR(s)

 

Methanol water solution

G(s) Sensor

Measured charge level

Fuel Cell

H(s)

Charge Level

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15

Advanced Advan ced Probl Problems ems

Advanc Adv anced ed Pro Proble blems ms AP1.1

Control Con trol of a robotic microsurgic microsurgical al device: device:

Microsurgical robotic manipulator 

Controller  Desired End-effector  Position

-

 

Gc( s  s))

G( s  s))

End-effector  Position

Sensor 

 H ( s  s))

AP1.2

An advanced advanced wind energy energy system viewed viewed as a mechatronic mechatronic system: system: AERODYNAMIC DESIGN STRUCTURAL DESIGN OF THE TOWER ELECTRICAL AND POWER SYSTEMS

SENSORS Rotor rotational sensor Wind speed speed and direction sensor sensor ACTUATORS Motors for manipulatiing the propeller pitch

Physical System Modeling

CONTROL SYSTEM DESIGN AND ANALYSIS ELECTRICAL SYSTEM DESIGN AND ANALYSIS POWER GENERATION AND STORAGE

Sensors and Actuators WIND ENERGY SYSTEM

Software and Data Acquisition

CONTROLLER ALGORITHMS DATA ACQUISTION: WIND SPEED AND DIRECTION ROTOR ANGULAR SPEED PROPELLOR PITCH ANGLE

AP1.3

Signals and Systems

Computers and Logic Systems

COMPUTER EQUIPMENT FOR CONTROLLING THE SYSTEM SAFETY MONITORING SYSTEMS

The aut automa omatic tic par parall allel el par parkin kingg sys system tem mig might ht use mu multi ltiple ple ult ultras rasoun ound d sensor sen sorss to mea measur suree dis distan tances ces to the par park ked aut automo omobil biles es and the cur curb. b. The sensor measurements would be processed by an on-board computer to determine the steering wheel, accelerator, and brake inputs to avoid collision and to properly align the vehicle in the desired space.

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16

CHAPTER 1

Introduction Introduc tion to Contro Controll Syste Systems ms

Even though the sensors may accurately measure the distance between the two parked vehicles, there will be a problem if the available space is not big enough to accommodate the parking car. Controller 

Desired automobile  position

Error 

On-board computer 

-

Actuators

 

Steering wheel, accelerator,, and accelerator  brake

Process

Actual automobile  position

Automobile

Measurement

Position of automobile relative to parked cars and curb

Ultrasound

There are various various control control methods that can be considered, including placing the controller in the feedforward loop (as in Figure 1.3). The adaptive optics block diagram below shows the controller in the feedback loop, as an alternative control system architecture.

AP1.4

Process

Astronomical object Uncompensated image

 Astronomical telescope mirror 

Compensated image

Measurement

Wavefront reconstructor 

Wavefront corrector 

Wavefront sensor 

 Actuator & controller 

AP1.5

Desired floor 

Error

-

The control control system might might have an inner loop for controlling controlling the acceleracceleration and an outer loop to reach the desired floor level precisely.

Controller #2

Outer Loop

Desired acceleration

Error

-

Controller #1

Elevator  motor, cables, etc.

Inner Loop Measured acceleration

Acceleration Measurement

Elevator 

 

Floor 

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17

Advanced Advan ced Probl Problems ems

An obstacle avoidanc avoidancee con control trol system would keep the robotic vacuum cleaner from colliding with furniture but it would not necessarily put the vacuum cleaner on an optimal path to reach the entire floor. This would require another sensor to measure position in the room, a digital map of  the room layout, and a control system in the outer loop.

AP1.6

Process Desired distance from obstacles

Error

-

Controller 

Measured distance from obstacle

Motors, wheels, etc.

Infrared sensors

Robotic vacuum cleaner 

Distance from obstacles

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18

CHAPTER 1

Introduction Introduc tion to Contro Controll Syste Systems ms

Desi De sign gn Pr Prob oble lems ms CDP1.1

The machine tool with the movable table in a feedback control configuration: Controller 

Error 

Desired position  x

Amplifier

-

 Actuator 

Process

Machine tool with table

Positioning motor

Actual position  x

Measurement

Position sensor

Measured position

DP1.1

Use the stereo stereo system and amplifiers amplifiers to cancel out the noise by emitting emitting signalss 180 out of phase with the noise. signal ◦

Process

Controller  Noise signal Desired noise = 0

Shift phase by 180 deg

-

Machine tool with table

Positioning motor

Noise in cabin

Measurement

Microphone

DP1.2

Desired peed f auto et by river

1/ K   K 

An automobile cruise control control system: Controller 

Desired shaft speed

-

Electric motor

Process

Automobile and engine

Valve

Measurement

Measured shaft speed

Shaft speed sensor

Drive shaf t speed

 K 

Actual speed of auto

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19

Design Desig n Probl Problems ems DP1.3

An automoted automoted cow cow milking system: system: Measurement Cow location

Vision system   Actuator

Controller 

Motor and gears

-

Desired cup location

Process

Location of cup

Robot arm and cup gripper

Cow and milker

Milk 

Measurement

Vision system

Measured cup location

DP1.4

A feedback feedback control control system system for a robot welder: welder: Controller 

Error 

Desired position

-

Process

Computer and amplifier

Voltage

Motor and arm

Weld top position

Measurement

Vision camera

Measured position

DP1.5

A control control system for one wheel of a traction control control system: system: Antislip controller

Engine torque

+

-

Wheel dynamics

+

-

Wheel speed

Sensor +

Actual slip

1/ Rw

Vehicle dynamics

Brake torque

+

Vehicle speed

Antiskid controller

Rw = Radius of wheel

Sensor

Measured slip

© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

20

CHAPTER 1 DP1.6

Introduction Introduc tion to Contro Controll Syste Systems ms

A vibration vibration damping system for the Hubble Space Space Telesco Telescope: pe: Controller 

Desired  jitter = 0

Error 

-

Computer

 Actuators

Gyro and reaction wheels

Process Signal to cancel the jitter

Spacecraft dynamics

Jitter of  vibration

Measurement

Measurement of 0.05 Hz jitter

DP1.7

A control control system system for for a nanorobot: nanorobot: Controller 

Desired nanorobot position

Rate gyro sensor

Error 

-

Biocomputer 

 Actuators

  Plane surfaces and propellers

Process

Nanorobot

 Actual nanorobot position

Measurement

External beacons

Many concepts from underwater robotics can be applied to nanorobotics within the bloodstream. For example, plane surfaces and propellers can provide the required actuation with screw drives providing the propulsion. The nanorobots can use signals from beacons located outside the skin as sensors to determine their position. The nanorobots use energy from the chemical reaction of oxygen and glucose available in the human body. The control system requires a bio-computer–an innovation that is not yet available. For further reading, see A. Cavalcanti, L. Rosen, L. C. Kretly, M. Rosenfeld, and S. Einav, “Nanorobotic Challenges n Biomedical Application, Design, and Control,”   IEEE ICECS Intl Con Conf. f. on Ele Electr ctroni onics, cs, Cir Circuit cuits  s  and Systems , Tel-Aviv, Israel, December 2004. DP1.8

The feedback feedback control system might might use gyros and/or accelerome accelerometers ters to measure angle change and assuming the HTV was originally in the vertical position, the feedback would retain the vertical position using commands to motors and other actuators that produce produced d torques and could move move the HTV forward and backward.

© 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

21

Design Desig n Probl Problems ems

Process Desired angle from vertical (0o)

Error

-

Controller 

Measured angle from vertical

Motors, wheels, etc.

Gyros & accelerometers

HTV

Angle from vertical