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Linear System Theory

Wonhee KimLecture 1

March 7, 2017

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Overview

I Course Information

I Prerequisites

I Course Outline

I What is Control Engineering?

I Examples of Control Systems

I Structure of Control Systems

I Linear Systems

I Nonlinear Systems

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Course Information

I Instructor: Wonhee Kim

I Email/Office: [email protected]/Room 310-439

I Course Website: wonheekim.wordpress.com

I Textbook: Chi-Tsong Chen, Linear System Theory, 4th edition,Oxford University Press, available at the UNIST bookstore

I Grading Policy: HW(20%), Midterm(40%), Final Exam (40%)

I Check the course website for updates

I Original lecture note supported by Prof. Jun Moon (UNIST)

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Prerequisites

I linear algebra

I signals and systems: Laplace transform (z-transform)

I differential and difference equations

I some familiarity of classical control theory (Bode plot, root locus,Nyquist, PID, etc) and programming skills (e.g. MATLAB) would behelpful

I We will cover the required mathematical skills during the lectures

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Tentative Course Topics

I linear algebraI vector space, norm, inner product, column and null spaces,

eigenvalues, basis, rank, similarity transformation, Jordan form, etc.

I ordinary differential equations (and difference equations)I state space representation, solutions, matrix exponential, etc

I stability

I controllability, observability, decomposition

I linear system designI state-feedback, pole-placement, observer design

I optimal control (if time permits)

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What is Control Engineering?

I Control Engineering (Wikipedia): Control engineering is theengineering discipline that applies control theory to design systemswith desired behaviors.

I Control Theory (Wikipedia): Control theory is an interdisciplinarybranch of engineering and mathematics that deals with the behaviorof dynamical systems with inputs, and how their behavior ismodified by feedback.

I In this course, we will study mathematical control theory

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Examples of Control Systems

I Aircraft, Vehicles, Defense, Circuits, Communication, PowerSystems, Social Networks, Economics, etc

I Desired behavior: safety, speed, position, price, power, etc.

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Examples of Control SystemsI Segway: https://www.youtube.com/watch?v=rmlg5QkusFQ

(control of an inverted pendulum)

I HDD: control of spindle and actuator (sensor) motors

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https://www.youtube.com/watch?v=rmlg5QkusFQ

Structure of Control Systems

Plant SensorActuator +-

Controller

d

+

n

u

r

y

y

I BlockI Plant: system that needs to be controlled (ODE or difference

equations) ⇒ motor, aircraft, pendulum, etcI Actuator/sensorI Controller: controller that controls the plant

I SignalI Input (r): reference signalI Output (y): sensor signalI Noise (n) / Disturbance (d): unwanted signal (need to reduce their

effect)I Control (u): control signal (we need to design)

I Structure: Open loop / Feedback9 / 22

Continuous-Time Linear Time-Varying (LTV) System

x(t) =dx(t)

dt= A(t)x(t) + B(t)u(t) + D(t)d(t)

y(t) = C (t)x(t) + E (t)u(t) + F (t)n(t)

I t ≥ 0: timeI x ∈ Rn: stateI u ∈ Rm: controlI d ∈ Rl : disturbanceI y ∈ Rp: (sensor) outputI n ∈ Rq: noiseI A(t): n × n system matrixI B(t): n ×m input matrixI D(t): n × l disturbance matrixI C (t): p × n output matrixI E (t): p ×m feedthrough matrixI F (t): p × q noise matrix

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Continuous-Time Linear Time-Invariant (LTI) System

x(t) =dx(t)

dt= Ax(t) + Bu(t) + Dd(t)

y(t) = Cx(t) + Eu(t) + Fn(t)

I t ≥ 0: timeI x ∈ Rn: stateI u ∈ Rm: controlI d ∈ Rl : disturbanceI y ∈ Rp: (sensor) outputI n ∈ Rq: noiseI A: n × n system matrixI B: n ×m input matrixI D: n × l disturbance matrixI C : p × n output matrixI E : p ×m feedthrough matrixI F : p × q noise matrix

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LTV and LTI SystemsI LTV system: constants are time-varying

I LTI system: constants are time-invariant

I LTI system can be converted into the transfer function via theLaplace (or z) transformation

I LTV and LTI systems: first-order ODE (first-order recursiveequation)

I also called state equation or state system

I state x : position, velocity, acceleration, etc, which capture thebehavior of the system

I scalar (one-dimensional) u and y : single-input-single-output (SISO)system

I In this course, we consider continuous-time LTV and LTI systemswhen D = E = F = 0 (system without disturbance, noise andfeedthrough terms)

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Discrete-Time LTV and LTI Systems

I discrete-time LTV system

x(t + 1) = A(t)x(t) + B(t)u(t) + D(t)d(t)

y(t) = C (t)x(t) + E (t)u(t) + F (t)n(t)

I discrete-Time LTI System

x(t + 1) = Ax(t) + Bu(t) + Dd(t)

y(t) = Cx(t) + Eu(t) + Fn(t)

I t ∈ Z+ = {0, 1, 2, ...}

I difference equation (first-order recursive equation)

I x , y , u are sequences

I sampled system: x(t) := x(tT ) (T : sampling period)

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Nonlinear Systems

I continuous-time nonlinear system

x(t) = f (t, x(t), u(t), d(t)), y(t) = g(t, x(t), u(t), n(t))

I discrete-time nonlinear system

x(t + 1) = f (t, x(t), u(t), d(t)), y(t) = g(t, x(t), u(t), n(t))

I Example: x(t) = x2(t), x(t) = cos(t)

I Linear system can be obtained by linearization of a nonlinear system⇒ next class

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Why Study Linear Systems?

I Linear system is a special case of nonlinear systems

I Why do we study linear systems?

I If you do not understand linear systems, you cannot understandnonlinear systems

I Nonlinear systemI Existence of solution?I Hard to analyze its dynamic behaviorI Hard to see its input/output characteristics

I Linear systemI Solution always existsI System characteristics depend on coefficients of the systemI Computationally inexpensiveI Easy to implement (real-time systems)I Linear algebra is the most effective toolI Many applications can be represented by linear systems (circuits,

aircraft, missile, communication, traffic, guidance, economics)

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Continuous-Time LTI System

I continuous-time SISO-LTI system

x(t) = Ax(t), x(0) = x0, (A ∈ R, A 6= 0)

I This is a continuous-time SISO autonomous system (no input, u)

I Solution

x(t) = eAtx0

I The behavior of x(t) is determined by the value of A

I A > 0: |x(t)| → ∞ as t →∞ for all x0 ∈ R (unstable)

I A < 0: |x(t)| → 0 as t →∞ for all x0 ∈ R (stable)

I The scalar A determines the speed of convergence or divergence of|x(t)| ( ⇒ system characteristics depend on coefficients of thesystem)

I What if A is a matrix?

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I two-dimensional diagonal continuous-time LTI system

x(t) = Ax(t), x(0) = x0, A =

(a 00 −a

), a 6= 0

I Solution

x(t) =

(eat

e−at

)x0

I What is limt→∞ ‖x(t)‖ ?

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I continuous-time SISO-LTI system with control u

x(t) = Ax(t) + Bu(t), x(0) = 0

y(t) = Cx(t) (A,B,C ∈ R, A < 0)

I Problem: find appropriate u so that y = h ∈ RI Naive approach: consider static input and output (u, x , y constant)

x(t) = 0 = Ax + Bu, y = h = Cx

I Then (why?)

u(t) = − A

CBh

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I time-response plot when A = −1, B = C = 1, h = 0.5

I MATLAB Simulink: control system toolbox

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I time-response plot when A = −1, B = C = 1, h = 0.5

I MATLAB Simulink: control system toolbox

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I This is one simple approach to design u

I There are may ways of designing control u to achieve the desiredcontrol performance

I In this course, we will study some design techniques of u for LTIsystems

I We will not cover classical control theory

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Next Class

I classical and modern control theory

I LTI systems

I linearization

I modeling

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linear system theory - wordpress.com · why study linear systems? i linear system is a special case...

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