EE 4314: Control SystemsLectures: Tue/Thu, 2:00-3:20, NH 202 Instructor: Indika Wijayasinghe, Ph.D.Office hours: Tue/Thu 9:00 am – 11:00 am, NH 250, or by appointment.

Course TAs: Raghavendra Sriram, Fahad Mirza

Course info: http://www.uta.edu/ee/ngs/ee4314_control/

Grading policy: 5 Homework – 20%

6 Labs – 20% Midterm I (in-class) – 20%

Midterm II (take-home) – 20%Final (in-class) – 20%

Grading criteria: on curve based on class average

Syllabus• Assignments:

– Homework contains both written and/or computer simulations using MATLAB. Submit code to the TA’s if it is part of the assignments.

– Lab sessions are scheduled in advance, bi-weekly, so that the TA’s can be in the lab (NH 148). While the lab session is carried out in a group, the Lab report is your own individual assignment.

– Examinations: Three exams (two midterms, one final), in class or take home.

– In rare circumstances (medical emergencies, for instance) exams may be retaken and assignments can be resubmitted without penalty.

– Missed deadlines for take-home exams and homework: Maximum grade drops 15% per late day (every 24 hours late).

Honor Code

• Academic Dishonesty will not be tolerated. All homework and exams are individual assignments. Discussing homework assignments with your classmates is encouraged, but the turned-in work must be yours. Discussing exams with classmates is not allowed. Your take-home exams and homework will be carefully scrutinized to ensure a fair grade for everyone.

• Random quizzes on turned-in work: Every student will be required to answer quizzes in person during the semester for homework and take home exam. You will receive invitations to stop by during office hours. Credit for turned in work may be rescinded for lack of familiarity with your submissions.

• Attendance and Drop Policy: Attendance is not mandatory but highly encouraged. If you skip classes, you will find the homework and exams much more difficult. Assignments, lecture notes, and other materials are going to be posted, however, due to the pace of the lectures, copying someone else's notes may be an unreliable way of making up an absence. You are responsible for all material covered in class regardless of absences.

Textbooks & Description• Textbook:

– G.F. Franklin, J.D. Powell, A. Emami-Naeni, Feedback Control of Dynamic– Systems, 7th Ed., Pearson Education, 2014, ISBN 978-0-13-349659-8

• Other materials (on library reserve)– K. Ogata, Modern Control Engineering, 5-th ed, 2010, Pearson Prentice Hall ISBN13:9780136156734,

ISBN10:0136156738– Student Edition of MATLAB Version 5 for Windows by Mathworks, Mathworks Staff, MathWorks Inc.– O. Beucher, M. Weeks, Introduction to Matlab & Simulink, A project approach, 3-rd ed., Infinity Science

Press, 2006, ISBN: 978-1-934015-04-9– B.W. Dickinson, Systems: Analysis, Design and Computation, Prentice Hall, 1991, ISBN: 0-13-338047-5.– R.C. Dorf, R.H. Bishop, Modern Control Systems, 10th ed., Pearson Prentice Hall, 2005, ISBN: 0-13-

145733-0• Catalog description:

– Catalog description: EE 4314. CONTROL SYSTEMS (3-0) Analyses of closed loop systems using frequency response, root locus, and state variable techniques. System design based on analytic and computer methods.

– This is an introductory control systems course. It presents a broad overview of control techniques for continuous and discrete linear systems, and focuses on fundamentals such as modeling and identification of systems in frequency and state-space domains, stability analysis, graphical and analytical controller design methods.

– The course material is divided between several areas:• Control Systems: classification, modeling, and identification• Basics of Feedback: performance and stability• Control Design Methods: frequency domain, state-space• Programming exercises using MATLAB and Simulink• Laboratory experiments

Tentative Course SchedulePart 1: Introduction to systems & system modeling• Week 1 - January 20, 22, Lectures 1, 2

• Introduction to feedback control systems and brief history • History of Feedback Control • Review of basics: Matrix and vector algebra, complex numbers, integrals and series,

Differential equations and linear systems • Notes for quick review

• Week 2 - January 27, 29, Lectures 3, 4 • Lecture 3: Dynamic Models of Mechanical systems. • Lecture 4: Dynamic Models of Mechanical systems • Homework #1 handed out on January 29 • Lab #1: Matlab and Simulink Hands on Lab at NH 148

• Week 3 - Feb 3, 5, Lectures 5, 6 • Lecture 5: Dynamic Models of Circuits • Lecture 6: Dynamic Models of Electromechanical Systems

• Week 4 - February 10, 12, Lectures 7, 8 • Lecture 7: Dynamic Models of DC motors, thermal and fluid system • Lecture 8: System identification of first and second order systems • Homework #1 due February 12, Homework #2 is posted • Lab #2: Identification of a DC Motor Transfer Function

Tentative Course SchedulePart 2: Feedback and Its Time Domain Analysis• Week 5 - Feb 17, 19, Lectures 9, 10

• Lecture 9:Stability and Mason's rule • Lecture 10: Introduction of state space model • Lab #1 report due February 19

• Week 6 - Feb 24, 26, Lectures 11, 12 • Lecture 11: State space model • Lecture 12: First analysis of feedback system • Homework #2 due February 26, Homework #3 handed out • Lab #3: Identification of a State-Space Model

• Week 7 - Mar 3, 5, Lectures 13, Midterm exam I • Midterm study guide • In-class Midterm I on March 5, covers: system modeling, basic feedback, and state-

space methods. • Lab #2 report due March 3 in class

• Week 8 - Mar 9 - 13, Spring break • Week 9 - Mar 17, 19, Lecture 14, Lecture 15

• Lecture 14: PID control • Lecture 15: Response of State-space equation • Homework #3 due on March 19

Tentative Course SchedulePart 3: Feedback and Design Methods in Frequency Domain• Week 10 - Mar 24, 26, Lectures 16, 17

• Lecture 16: Full-state feedback, Ackerman's Formula • Lecture 17: Optimal control - Linear Quadratic Regulation. • Lab #4: Speed Motor Control using PID • Lab #3 report due March 26

• Week 11 - Mar 31, Apr 2, Lectures 18, 19 • Lecture 18: Root-locus design 1 • Lecture 19: Root-locus design 2 • Homework #4 handed out on Apr 2

• Week 12 - Apr 7, 9, Lectures 20, 21 • Lecture 20: Root-locus design 3 • Lecture 21: Root-locus design 4 • Lab #5: LQR control of Mass-Spring-Damper System

• Week 13 - Apr 14, 16, Lectures 22, 23 • Midterm II (Take Home), posted April 16, due April 23, covers frequency domain

techniques • Lecture 22: Bode plot, PM, GM • Lecture 23: Lead-Lag compensator • Homework #4 due April 16, Homework #5 handed out • Lab #4 report due April 14 in class

Tentative Course SchedulePart 4: Digital Control• Week 14 - Apr 21, 23, Lectures 24, 25

• Lecture 24: Nyquist plot • Lecture 25: Nyquist stability criterion • Lab #6: Digital Control

• Week 15 - Apr 28, 30, Lectures 26, 27 • Lecture 26: Digital Control: Z-transform • Lecture 27: Digital Control: Controller design

• Week 16 - May 5, 7, Course recap and exam preparation • Lecture 28: Digital Control • Exam preparation and final study guide • Homework #5 due May 5 at class • Lab #5 report due May 5 at class

• Week 17 - May 12 • Final exam (in-class) (comprehensive) in class, no calculator • Time: 2:00 pm - 4:30 pm • Bring a 6-page, double-sided cheat sheet, handwriting only • Lab #6 report due May 12

Course Objectives

Students should be familiar with the following topics:1. Modeling of physical dynamic systems2. Block diagrams3. Specifications of feedback system performance4. Steady-state performance of feedback systems5. Stability of feedback systems6. Root-locus method of feedback system design7. Frequency-response methods8. Nyquist’s criterion of feedback loop stability9. Design using classical compensators10. State variable feedback

Textbook Reading and Review• Course Refresher:

– Math: complex numbers, matrix algebra, vectors and trigonometry, differential equations.

– Programming: MATLAB & Simulink– EE 3317 (Linear Systems), 3318 (Discrete Signals and Systems)

• For weeks 1 & 2– Read Chapter 1, Appendix A (Laplace Transformation) of Textbook– Read History of Feedback Control by Frank Lewis

• http://arri.uta.edu/acs/history.htm

• Purpose of weekly assigned textbook readings– To solidify concepts– To go through additional examples– To expose yourselves to different perspectives– Reading is required. Problems or questions on exams might cover reading

material not covered in class.

Signals and Systems

• Signal:– A set of data or information – Examples: audio, video, image, sonar, radar, etc.– It provides information on the status of a physical

system. – Any time dependent physical quantity

• System:– Object that processes a set of signals (input) to

produce another set of signals (outputs).– Examples:

• Hardware: Physical components such as electrical, mechanical, or hydraulic systems

• Software: Algorithm that computes an output from input signals

?x(t)

u(t) y(t)

Signal Classification

– Continuous Time vs. Discrete Time• Telephone line

signals, Neuron synapse potentials

• Stock Market, GPS signals

– Analog vs. Digital• Radio Frequency (RF)

waves, battery power• Computer signals,

HDTV images

Analog, continuous time Digital, continuous time

Analog, discrete time Digital, discrete time

Signal Classification

– Deterministic vs. Random• Predictable: FM Radio Signals• Non-predictable: Background

Noise Speech Signals– Periodic vs. Aperiodic

• Sine wave• Sum of sine waves with non-

rational frequency ratio

System Classification

– Linear vs. Nonlinear• Linear systems have the property of

superposition– If U →Y, U1 →Y1, U2 →Y2 then

» U1+U2 → Y1+Y2» A*U →A*Y

• Nonlinear systems do not have this property, and the I/O map is represented by a nonlinear mapping.

– Examples: Diode, Dry Friction, Robot Arm at High Speeds.

– Memoryless vs. Dynamical• A memoryless system is represented by a

static (non-time dependent) I/O map: Y=f(U). – Example: Amplifier – Y=A*U, A- amplification

factor.

• A dynamical system is represented by a time-dependent I/O map, usually a differential equation:

– Example: dY/dt=A*u, Integrator with Gain A.

0

0)sin(

2

2

2

2

L

g

dt

d

L

g

dt

d Exact Equation, nonlinear

Approximation around vertical equilibrium, linear

System Classification

– Time-Invariant vs. Time Varying• Time-invariant system parameters do not change over time. Example: pendulum,

low power circuit, robots.• Time-varying systems perform differently over time. Example: human body during

exercise, rocket.

– Stable vs. Unstable• For a stable system, the output to bounded inputs is also bounded. Example:

pendulum at bottom equilibrium• For an unstable system, the ouput diverges to infinity or to values causing

permanent damage. Example: Inverted pendulum.

stable unstable

System Modeling

• Building mathematical models based on observed data, or other insight for the system.– Parametric models (analytical): ODE, PDE– Non-parametric models: ex: graphical models

- plots, or look-up tables.

Types of Models• White (clear or glass) Box Model

– derived from first principles laws: physical, chemical, biological, economical, etc.

– Examples: RLC circuits, MSD mechanical models (electromechanical system models).

• Black Box Model– model is based solely from measured data– No or very little prior knowledge is used.– Example: regression (data fit)

• Gray Box Model – combination of the two– Determination of the model structure relies on prior knowledge

while the model parameters are mainly determined by measurement data

White Box Systems: Electrical

• Defined by Electro-Magnetic Laws of Physics: Ohm’s Law, Kirchoff’s Laws, Maxwell’s Equations

• Example: Resistor, Capacitor, Inductor

u

Riu

i

C

ui

L

RLC Circuit as a System

Kirchoff’s Voltage Law (KVL):

u1

L

C

R

uu3

u2RLCq(t)

u(t) i(t)

White Box Systems: Mechanical

Newton’s Law:

M

K

BF MSD

x(t)

F(t) x(t)

Mechanical-Electrical Equivalance:

F (force) ~V (voltage)x (displacement) ~ q (charge)M (mass) ~ L (inductance)B (damping) ~ R (resistance)1/K (compliance) ~ C (capacitance)

White Box vs Black Box Models

White Box Models Black-Box Models

Information Source First Principle Experimentation

Advantages Good ExtrapolationGood understandingHigh reliability, scalability

Short time to developLittle domain expertise requiredWorks for not well understood systems

Disadvantages Time consuming and detailed domain expertise required

Not scalable, data restricts accuracy, no system understanding

Application Areas Planning, Construction, Design, Analysis, Simple Systems

Complex processesExisting systems

This course deals with both white and black models which are linear

Linear System

• Why study continuous linear analysis of signals and systems when many systems are nonlinear in practice?– Basis for digital signals and systems– Many dynamical systems are nonlinear but some

techniques for analysis of nonlinear systems are based on linear methods

– Methods for linear systems often work reasonably well, for nonlinear systems as well

– If you don’t understand linear dynamical systems you certainly can’t understand nonlinear systems

LTI Models

• State space form of linear time varying dynamical system

dx/dt= A(t)x(t) + B(t)u(t),

y(t) = C(t)x(t) + D(t)u(t)• where:

– x(t) = state vector (n-vector)– u(t) = control vector (m-vector)– y(t) = output vector (p-vector)– A(t) = nxn system matrix, B(t) = nxm input matrix– C(t) = pxn output matrix, D(t) = pxm matrix

Linear Systems in Practice

• Most linear systems encountered are time-invariant: A, B, C, D are constant, i.e., don’t depend on time– Examples: second-order electromechanical systems with

constant coefficients (mass, spring stiffness, etc)

• when there is no input u (hence, no B or D) system is called autonomous– Examples: filters, uncontrolled systems

• when u(t) and y(t) are scalar, system is called single-input, single-output (SISO)

• when input & output signal dimensions are more than one, MIMO (Multi-Input-Multi-Output)– Example: Aircraft – MIMO

Linear System Description in Frequency Domain

Block Diagrams

• Block Diagram Model: – Helps understand flow of information (signals) through a complex

system– Helps visualize I/O dependencies– Elements of block diagram:

• Lines: Signals• Blocks: Systems• Summing junctions• Pick-off points

Transfer Function Summer/Difference Pick-off point

H(s)U(s) Y(s) +

+

U2

U1 U1+U2 U U

U

Block Diagram: Simplification Rules

Block Diagram: Reduction Rules

Block Diagram: Reduction Rules

Automatic Control

• Control: process of making a system variable converge to a reference value– Tracking control (servo): reference value = changing– Regulation control: reference value = constant

(stabilization)

• Open Loop vs. closed loop control

ControllerK(s)

PlantG(s)

+

-

Sensor GainH(s)

++

ControllerK(s)

PlantG(s)

r

r

y y

- No output measurement

- Known system

- No disturbance

Feedback Control

• Role of feedback:– Reduce sensitivity to system parameters (robustness)– Disturbance rejection– Track desired inputs with reduced steady state errors,

overshoot, rise time, settling time (performance)• Systematic approach to analysis and design

– Select controller based on desired characteristics• Predict system response to some input

– Speed of response (e.g., adjust to workload changes)• Approaches to assessing stability

Feedback System Block Diagram• Temperature control system

– Control variable: temperature– Initial set temp=55F, At time=6, set temp=65F

Feedback System Block Diagram

• Process: house• Actuator: furnace• Sensor: Thermostat

• Controller: computes control input• Actuator: a device that influences the

controlled variable of the process• Disturbance: heat loss (unknown,

undesired)

Key Transfer Functions

)()()(

)(

)(

)(

)(

)(21 sGsG

sE

sU

sU

sY

sE

sY :eedforwardF

)()()()(

)( :Loop 21 sHsGsGsE

sB

)()()(1

)()(

)(

)( :

21

21

sHsGsG

sGsG

sR

sY

Feedback

PlantControllerS)(sU )(sY)(sR )(sE

Transducer

)(sB

+

–

)(1 sG )(2 sG

)(sH

Reference

Basic Control Actions: u(t)

)(

)()()(:control Derivative

)(

)()()(:control Integral

)(

)()()(:control alProportion

0

sKsE

sUte

dt

dKtu

s

K

sE

sUdtteKtu

KsE

sUteKtu

dd

it

i

pp

Summary of Basic Control• Proportional control

– Multiply e(t) by a constant

• PI control– Multiply e(t) and its integral by separate constants– Avoids bias for step

• PD control– Multiply e(t) and its derivative by separate constants– Adjust more rapidly to changes

• PID control– Multiply e(t), its derivative and its integral by separate constants– Reduce bias and react quickly

)()( teKtu p

dtteKteKtut

ip 0

)()()(

)()()( tedt

dKteKtu dp

)()()()(0

tedt

dKdtteKteKtu d

t

ip

Feedback System Block Diagrams

• Automobile Cruise Control

disturbance

Input

Output

Brief History of Feedback Control

• The key developments in the history of mankind that affected the progress of feedback control were:

• 1. The preoccupation of the Greeks and Arabs with keeping accurate track of time. This represents a period from about 300 BC to about 1200 AD. (Primitive period of AC)

• 2. The Industrial Revolution in Europe, and its roots that can be traced back into the 1600's. (Primitive period of AC)

• 3. The beginning of mass communication and the First and Second World Wars. (1910 to 1945). (Classical Period of AC)

• 4. The beginning of the space/computer age in 1957. (Modern Period of AC).

Primitive Period of AC

Float Valve for tank level regulators Drebbel incubator furnace control (1620)

(antiquity)

Primitive Period of AC

James Watt

Fly-Ball Governor

For regulating steam

engine speed

(late 1700’s)

Classical Period of AC

• Most of the advances were done in Frequency Domain.• Stability Analysis: Maxwell, Routh, Hurwitz, Lyapunov (before

1900).

• Electronic Feedback Amplifiers with Gain for long distance communications (Black, 1927) – Stability analysis in frequency domain using Nyquist criterion

(1932), Bode Plots (1945).

• PID controller (Callender, 1936) – servomechanism control

• Root Locus (Evans, 1948) – aircraft control

Modern Period of AC

• Time domain analysis (state-space)• Bellmann, Kalman: linear systems (1960)• Pontryagin: Nonlinear systems (1960) – IFAC• Optimal controls• H-infinity control (Doyle, Francis, 1980’s) – loop shaping

(in frequency domain).• MATLAB (1980’s to present) has implemented math

behind most control methods.