ECE 301 Introduction to System Theory

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ECE 301 Introduction to System Theory

• Robert S. Lynch
• ITE Room 125
• Phone [phone], Fax [phone]
• Email lynchrs_at_npt.nuwc.navy.mil
• Office Hours Can arrange at UCONN by appointment
  
• http//www.engr.uconn.edu/ece/

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• Reading Assignment 1, 2.1 - 2.3
• Optional Reading Brogan 1 and 3
• Problem Set 1 Ch. 2 2, 5, 8, 9, 10, 12 due next
  class
• Today
• Introduction
• Motivation ? Course Overview
• Math. Descriptions of Systems Review
• Classification of Systems
• Linear Systems
• LTI Systems
• Next Time Sections 2.3 - 2.7, 3.1 - 3.3

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1. INTRODUCTION

• 1.1 Motivation
• What is a "system"?
• A physical process or a mathematical model of a
  physical process that relates a set of input
  signals to yield another set of output signals

• Examples Cars, circuits, bank accounts, stock
  markets
• Two general categories of signals/systems
• Continuous-time signals/systems

• Examples Signals in cars and circuits
• Described by differential eqs., e.g., dy/dt
  ay(t) bu(t)

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• Signals themselves could be discontinuous
• Discrete-time signals/systems

• Examples Money in a bank account, quarterly
  profit
• No derivative exists
• Signals described by difference equations, e.g.,
  yk1 ayk buk
• They are quite similar, and shall be treated in
  parallel
• What is "System Theory"?
• Understanding the physical system under
  consideration
• Describing the system mathematically
• Analyzing the properties
• Controlling it to meet certain criteria

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• Why are you in the class?
• Foundation for most control/communication courses
• Included in MS and Ph.D. examinations
• What are the prerequisites?
• ECE 202 Signals and Systems Working knowledge of
  
• Laplace transform
• z-transform
• Differential equations
• Linear algebra, and
• Modeling of electrical, mechanical, and financial
  systems

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Example A simple electric circuit

• Understanding the components and
  interconnections
• What are the components? How to model them?

• Interconnections
• KVL Voltage across a loop 0
• KCL Current to a node 0

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• An integral-differential or differential equation
• Input-output description or external description
• Analyzing the properties/responses
• For example, find the output i(t) given u(t) and
  IC.
• For now, shall use Laplace transform (Effective
  for LTI systems)

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Laplace Transform, A Quick Review

• Key Properties
• Linearity a1f1(t) a2f2(t) ? a1F1(s) a2F2(s)
• Derivative theorem f'(t) ? s?F(s) - f(0-)
  f'(-1)(t) ? F(s)/s
• Converting linear constant coefficient
  differential equations into algebraic equations
• Differentiation in the frequency domain t?f(t) ?
  (-1)F'(s)
• Convolution h(t)?f(t) ? H(s)?F(s)
• Time and frequency shifting f(t-t0)u(t-t0) ?
  e-st0 F(s) es0t f(t) ? F(s - s0)

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• Time and frequency scaling f(at) ? 1/a F(s/a)
  for a gt 0
• Initial Value Theorem f(0) lims?? sF(s)
• Final Value Theorem f(?) lims?0 sF(s) if all
  the poles of sF(s) have strictly negative real
  parts
• Example (Continued)

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• An algebraic equation as opposed to an
  integral-differential equation. Solution

• What can we say about it?
• It has two components, one caused by input, and
  the other by IC
• How about the voltage across the capacitor?

• What is the system's transfer function?

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• Assume that the ICs are zero, then

• Frequency domain analysis
• How to obtain the response in time domain?

• Suppose that L C 1, R 2, v0 i0 0, and
  u(t) U(t) (unit step function). Then

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• Does this make sense for the circuit?

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• Laplace transform is not effective for time
  varying systems
• Example. Solve y'(t) t y(t) f(t). How?
• Take Laplace transform and recall that t?f(t) ?
  (-1)F'(s)
• (sY(s) - y(0-)) (- Y'(s)) F(s)
• It is still a differential equation, not an
  algebraic equation
• The use of Laplace transform is restricted to LTI
  systems

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v(t)

• State-Space Description
• What are the state variables?
• Voltage across C and current through L

• What is the state equation?

• A set of first-order differential equations

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• It describes the behaviors inside the system by
  using the state variables v(t) and i(t)
• How to describe the output?

• The output equation
• Combined with the state equation, we have the
  state-space description or internal description
• How to analyze the system?
• Can also use Laplace transform

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as expected
? Quantitative analysis
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• What else can be said about this system?
• Is the system controllable? observable? stable?
  
• These are "qualitative analysis" as opposed to
  the previous "quantitative analysis"
• Analysis is one of our major emphases
• What happens if the performance of a system is
  not satisfactory?
• Design How to realize a system, adjust system
  parameters (e.g., the resistance R), or design
  feedback control to meet certain specifications

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• Design is our final goal
• System realization
• State feedback and state estimators
• Pole placement and model matching
• Introduction to optimal control
• The focus will be on linear systems, and
  nonlinear systems will be covered occasionally,
  mostly on stability

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1.2 Course Overview

• Textbooks
• Chi-Tsong Chen, Linear System Theory and Design,
  3rd Edition, Oxford University Press, 1999
  (Required)
• William L. Brogan, Modern Control Theory, 3rd
  edition, Prentice Hall, 1991 (Optional)
• Web Sites
• ECE Course Page http//www.engr.uconn.edu/ece/
  for lecture notes after lectures and solutions

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• Goals To provide a thorough understanding about
  systems theory and multivariable system design
• Tentative Outline (13 lectures)
• Introduction (0.5 W)
• Modeling How to model a physical system (1 W)
• The fundamentals of linear algebra (3.5 W)
• Analysis
• Quantitative How to derive response for a given
  input (1W)
• Qualitative How to analyze controllability,
  observability, and stability (2 W)

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• Design
• How to realize a system given its mathematical
  description (1 W)
• How to design a control law so that system
  response satisfies certain criteria (1 W)
• How to design an observer to estimate the state
  of the system (1 W)
• Pole Placement and Model Matching (1 W)
• How to design optimal control laws (1 W)
• Shall treat continuous-time and discrete-time
  systems in parallel

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• How to maximize your grade?
• Grading
• Classroom Participation 5
• Homework Assignments 20
• Review Paper Presentation 5
• Mid Term 22
• Term Project 22
• Final Examination 27
• Total 101
• ? Prepare, participate, review, put all
  together, and stay on top

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• General Rules
• Homework can be done individually or in teams of
  two. Partners can be dynamic.
• Homework should be clear, concise, and complete
• Late assignments will be discounted 10 a day, up
  to 5 days
• Homework solutions will be provided on the ECE
  web a week after the due date

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• Starting 9/25, each lecture will be divided into
  two parts. The first 2.5 hours will be used to
  present course materials, and the remaining 0.5
  hour will be used for students to present reviews
  of recent papers
• Paper reviews should be based on relevant and
  recent (2000 and up) journal articles
• Mid term We will find a three-hour slot
• Term projects can be done individually or in
  teams of two on relevant system-related topics or
  applications based on at least two recent papers.
  
• Proposals are due on Monday October 2
• Numerical implementation and testing are required

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• Presentations are scheduled on Monday Dec. 4, and
  final reports are due on Friday, Dec. 8
• Final Examination We will find a three-hour slot
  
• Comments and discussions are encouraged in class,
  after class, via phone/email, or by appointment.
• NO CHEATING!

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2. Mathematical Descriptions of Systems(Review,
or another viewpoint)

• Classification of Systems
• Linear Systems
• Linear time invariant (LTI) Systems

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2.1 Classification of Systems

• Basic assumption When an input signal is applied
  to the system, a unique output is obtained
• Q. How do we classify systems?
• Number of inputs/outputs with/without memory
  causality dimensionality linearity time
  invariance
• The number of inputs and outputs
• When p q 1, it is called a single-input
  single-output (SISO) system
• When p gt 1 and q gt 1, it is called a multi-input
  multi-output (MIMO) system

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• Memoryless vs. with Memory
• If y(t) depends on u(t) only, the system is said
  to be memoryless, otherwise, it has memory
• An example of a memoryless system?

A purely resistive circuit

• An example of a system with memory?

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• i(t) depends on i(t0) and u(?) for t0 ? ? ? t,
  not just u(t)
• A system with memory
• Causality No output before an input is applied

• A system is causal or non-anticipatory if y(t0)
  depends only on u(t) for t ? t0 and is
  independent of u(t) for t gt t0
• Is the circuit discussed last time causal?

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• An example of a non-causal system?
• y(t) u(t 2)

• Can you truly build a physical system like this?
  
• What is an example of a non-causal system in
  practice?
• If you can invent such a system, let me know. We
  will be rich through Connecticut Lottery

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• The Concept of State
• The state of a system at t0 is the information at
  t0 that, together with ut0,?), uniquely
  determines the behavior of the system for t ? t0
• The number of state variables the number of ICs
  needed to solve the problem
• For an LRC circuit, the number of state variables
  the number of C the number of L (except for
  degenerated cases)
• A natural way to choose state variables as what
  we have done earlier vc and iL
• Is this the unique way to choose state variables?

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• Any invertible transformation of the above can
  serve as a state, e.g.,

• Although the number of state variables 2, there
  are infinite numbers of representations
• Order of dimension of a system The number of
  state variables
• If the dimension is a finite number ? Finite
  dimensional (or lumped) system
• Otherwise, an infinite dimensional (or
  distributed) system

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• Q. Give an example of an infinite dimensional
  system

A delay line

• Given u(t) for t ? 0, what information is needed
  to know y(t) for t ? 0?

• We need an infinite amount of information ? An
  infinite dimensional system

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• Today
• Introduction
• Motivation
• Course Overview
• Mathmatical Descriptions of Systems Review
• Classification of Systems
• Linear Systems
• LTI Systems

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2.2 Linear Systems

• Linearity
• Double the efforts double the outcome?
• Suppose we have the following state-input-output
  pairs

• What would be the output of

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• If it is true Additivity
• How about

• If it is true Homogeneity
• Combined together to have

• If it is true Superposition or linearity
  property
• A system with such a property a Linear System

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• Are R, L, and C linear elements?

• Yes (differentiation is a linear operation)

Affine
Nonlinear
Linear

• Also, KVL and KCL are linear constraints. When
  put together, we have a linear system

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Response of a Linear System

• The additivity property implies that

• Response zero-input response zero-state
  response

• How to obtain the response of a linear system to
  a given u(t) with zero IC?
• Use the linearity property. How?

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• Let ??(t-ti) be a pulse at time ti with width ?
  and height 1/?

Area 1

• Let the system response to ??(t-ti) at time t be
  g?(t, ti)
• Then what?
• A general input u(t) can be approximated as a sum
  of such pulses
• The response y(t) would then be the sum of such
  responses based on linearity

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• What is ??(t-ti) in the limit as ? ?0?

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• Thus far, we have used linearity
• What if the system is causal?
• g(t,?) Response at t from a unit impulse at ?

• A system is said to be relaxed at t0 if the
  initial state at t0 is 0
• In this case, y(t) for t ? t0 is caused
  exclusively by u(t) for t ? t0

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State-Space Description

• How about for a system with p inputs and q
  outputs?
• Have to analyze the relationship for input/output
  pairs

gij(t,?) The impulse response between the jth
input and ith output

• A linear system can be described by

• The derivation of solutions will be done later

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2.3 Linear Time-Invariant (LTI) Systems

• Time Invariance The characteristics of a system
  do not change over time
• What are some of the LTI examples? Time-varying
  examples?
• What happens for an LTI system if u(t) is delayed
  by T? Have to watch out ICs

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• If the initial state is also shifted to time t0
  T, then the two responses should be the same,
  only shifted by T

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• What happens to the unit impulse response when
  the system is LTI?

• Only the difference between t and ? matters
• What happens to y(t)?

Convolution integral
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Transfer-Function Matrix

• The above convolution leads to the use of Laplace
  transform
• This then converts the differential equations
  into algebraic equations for easy solutions as
  below

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• Transfer function, the Laplace
  transform of the unit impulse response
• For a MIMO system, we have

Transfer-function matrix, or transfer matrix
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• Today
• Introduction
• Motivation
• Course Overview
• Mathematical Descriptions of Systems Review
• Classification of Systems
• Linear Systems
• LTI Systems
• Next Time 2.3 - 2.7, 3.1 - 3.2.
• Linearization Examples Discrete-time systems
• Linear Algebra
• Basis, representation, and orthonormalization