lst - spring 2015 - lecture - week 1-2

EE-503 Linear System Theory Spring 2015

Lecture-1: Feb. 06, 2015

Chapter 1:INTRODUCTION

1.1 INTRODUCTION

Study and Design of Physical SystemsEmpirical MethodsAnalytical Methods

Analytical Study of Physical Systems Modeling Development of Mathematical Descriptions Analysis Design

Distinction between Physical Systems and ModelsPhysical SystemModelMathematical equations using physical laws

AnalysisQuantitative: Response of the system excited by certain inputs.Qualitative: General properties of systems (stability, controllability,

observability)Design techniques often evolve from this study.

If the Response of a system is unsatisfactory, the system must be modified;By adjusting system parameters.By introducing compensators.

Success is based on the selection of a model.

Most difficult and important task is the selection of a model close enough to a physical system and yet simple enough to be studies analytically.

1.2 OVERVIEW

The systems to be studied in this course are limited to linear systems.Every linear system can be expressed by

Linear Systems (Continuous-Time)

Linear and Time-Invariant Systems(Continuous-Time)

Input-output or external description

Internal descriptionState-space equations

Linear System Theory – Spring-2015 1

Transfer Function Approach (input-output or external description)LTI systems with zero initial conditions

y¿

( s )=g¿

( s )u¿

( s )

this textbook uses circumflex denotes the Laplace Transform of the variable

Y(s)=G(s)U(s)

bold-case letters are used to represent matrix variables

Note: SISO Systems: Transfer Function MIMO Systems: Transfer Matrix

1. The transfer function of a system is a mathematical model in that it is an operational method of expressing the differential equation that relates the output variable to the input variable.

2. The transfer function is a property of a system itself, independent of the magnitude and nature of the input or driving function.

3. The transfer function includes the units necessary to relate the input to the output; however, it does not provide any information concerning the physical structure of the system. (The transfer functions of many physically different systems can be identical.)

4. If the transfer function of a system is known, the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system.

5. If the transfer function of a system is unknown, it may be established experimentally by introducing known inputs and studying the output of the system. Once established, a transfer function gives a full description of the dynamic characteristics of the system, as distinct from its physical description.

Modern Control Theory multiple-input, multiple-output. linear or nonlinear systems. time invariant or time varying

systems. Systems are modeled in the form

of state-space equations.

Modern control theory is essentially a time-domain approach and frequency domain approach in certain cases.

Conventional Control Theory only linear time-invariant (LTI), single-

input single-output (SISO) systems. Transfer Matrix for MIMO.

Non-linear systems are addressed using linearization.

Systems are modeled in the form of Transfer Function.

Conventional control theory is a complex frequency-domain approach.


t0 t

Chapter 2:MATHEMATICAL DESCRIPTION OF SYSTEMS

2.1 INTRODUCTION

Continuous-Time SystemsA system is continuous-time if it accepts continuous-time signals as its input and generates continuous-time signal as its output.

Discrete-Time SystemsA system is discrete-time if it accepts discrete-time signals as its input and generates discrete-time signal as its output.

2.1.1 Causality and Lumpedness Memoryless system: output y(t0) depends only on the input u(t) at t0.

Example: resistive circuit.

Causal or nonanticipatory system: output y(t0) depends on past input u(t), t<t0, and current input u(t0) but not on the future input, u(t), t>t0

Noncausal or anticipatory system: the current output also depends on future input. System can anticipate what will be applied in the future.

No physical system can anticipate the future input. Therefore, every physical system is causal.

Definition 2.1 The state x(t0) of a system at t0 is the information at t0 that, together with the input u(t), for t t0, determines uniquely the output y(t) for all t t0.

If we know the state at t0, there is no more need to know the input u(t) applied before t0 in determining the output y(t) after t0.


Output is partly excited by the initial state x(t0) and partly by the input u(t) applied at and after t0. It means that there is no need to know the input applied before t0.

Lumped System: if its number of state variables is finite or its state is a finite vector. A system is called distributed system if its state has infinitely many state variables (for example, a transmission line).

Network shown below is a lumped system: its state consists of three states x1, x2 and x3.

2.2 LINEAR SYSTEMS

Additivity and Homogeneity Zero-Input Response Zero-State Response Input-output description (external description) State-space equations (internal description)

x(t) = Ax(t) + Bu(t) y(t) = Dx(t) + Du(t)

n: states p: inputs q: outputs

A: nn B: np C: qn D: qp


Irrational function of s. Unit-delay, as shown below, is a distributed system, infinitely many state

variables.

Exponential function e-s has infinite terms; hence it requires infinitely many states.

e−s=1+(−s )+(−s )2

2 !+

(−s )3

3 !+⋯

Rational function of sA lumped system has finite states and therefore the transfer function is a rational function of s.

Every rational function of s can be expressed as

G( s )=

N (s )D (s ) .

G(s) is proper if deg D(s) deg N(s) and G() = zero or nonzero constant.

G(s) is strictly proper if deg D(s) > deg N(s) and G() = 0.

G(s) is biproper if deg D(s) = deg N(s) and G() = nonzero constant.

G(s) is improper if deg D(s) < deg N(s) and |G()| = .


Example of a Transfer Matrix of a system with 2-inputs and 2-outputs.



State-space equations x(t) = Ax(t) + Bu(t) y(t) = Dx(t) + Du(t)

X(s) = [sI-A]-1 x(0) + [sI-A]-1 B U(s)

Y(s) = C[sI-A]-1 x(0) + C[sI-A]-1 BU(s) + DU(s)zero-input zero-state response

Transfer Function

with x(0)=0: G(s) = Y(s ) / U(s) = C[sI-A]-1 BU(s) + D

MATLAB [A, B, C, D] = tf2ss(num, den)[num, den] = ss2tf(A, B, C, D)step(num, den)sys = tf(num, den)step(sys)[y, t] = step(sys)[y, x, t] = step(A, B, C, D)

Laplace transform is not used in studying linear time-varying systems becauseL [A(t) x(t)] L [A(t)] L [x(t)]

2.3.1 Op-Amp Circuit Implementation

A system represented in state-space equations.

Draw simulation diagram from state equations

Simulation diagram


Op-Amp implementation

Note: With appropriate position of inverters, a system can be implemented with minimum number of Op-Amps.



2.4 LINEARIZATION

The principle of superposition does not apply to nonlinear systems.

Thus, for a nonlinear system the response to two inputs cannot be calculated by treating one input at a time and adding the results.

Although many physical relationships are often represented by linear equations, in most cases actual relationships are not quite linear.

In fact, a careful study of physical systems reveals that even so-called “linear systems” are really linear only in limited operating ranges.

In practice, many electromechanical systems, hydraulic systems, pneumatic systems, and so on, involve nonlinear relationships among the variables.

For example, the output of a component may saturate for large input signals. There may be a dead space that affects small signals. (The dead space of a component is a small range of input variations to which the component is insensitive.)

Linearization of Nonlinear Systems

In control engineering a normal operation of the system may be around an equilibrium point, and the signals may be considered small signals around the equilibrium. (It should be pointed out that there are many exceptions to such a case.)

However, if the system operates around an equilibrium point and if the signals involved are small signals, then it is possible to approximate the nonlinear system by a linear system.

Such a linear system is equivalent to the nonlinear system considered within a limited operating range. Such a linearized model (linear, time-invariant model) is very important in control engineering.

Linear Approximation of Nonlinear Mathematical ModelsTaylor series expansion

Input: x(t)Output: y(t)

Inputs: x1(t) & x2(t)

Output: y(t)______________________Linearize


2.5 Examples

Mechanical Systems: Spring-mass-damper systems

Characteristic of damper Characteristic of spring

Suspension system of an automobile

Input: xi, the vertical displacement due to road conditionOutput: xo, vertical displacement of vehicle’s body

Transfer function: Xo(s) / Xi(s) = ?

State equations: Assume x1= x0 (displacement) potential energy x2= d(x0)/dt (velocity) kinetic energy u = xi

y = x0


Example 2.7Example 2.8

Electrical Systems: RLC Networks

Transfer function Write differential equations using KVL (loop equations) and KCL (node

equations). Take Laplace transform of differential equations and find transfer function. OR Convert RLC network in s-domain (C 1/Cs, L Ls) and find transfer function

using loop/node equations.

Transfer function State variable: inductor current and capacitor voltage. Write state equations using loop and node equations. One state variable for capacitors in parallel, one state variable for inductors

in series.

Example 2.11

Example 2.12


Transformation of Mathematical Models with MATLAB

[A,B,C,D] = tf2ss(num,den) % transfer function to state-space

conversion

[num,den] = ss2tf(A,B,C,D) % state-space to transfer function

conversion

[num,den] = ss2tf(A,B,C,D,iu)% system with more than one inputs

RLC network of Figure 2.16

A = [-1/6 0 -1/3; 0 0 1; 1.2 -1/2 -1/2]; B = [1/6 1/3; 0 0; 0 0];C = [1 -1 -1; -0.5 0 0]; D = 0;

► Find transfer function matrix of the system using MAT LAB► Find the step response using MATLAB

-0.2

-0.1

0

0.1

0.2

0.3From: In(1)

To:

Out

(1)

0 20 40 60-1

-0.5

0

0.5

To:

Out

(2)

From: In(2)

0 20 40 60

Step Response

Time (seconds)

Am

plitu

de

Example 2.13


End of Lecture-3 Feb 13, 2015Lecture-4: Feb. 14, 2015

Lecture-4 cancelled due to PEC visit to Karachi. Makeup will be arranged later.


2.6 DISCRETE-TIME SYSTEMS

2.7 CONCLUDING REMARKS

2.8 PROBLEMS

Chapter-3 LINEAR ALGEBRA




Lecture-9: Mar. 06, 2015



Lecture-11: Mar13, 2015

Lecture-12 Mar. 14, 2015



Midterm Examination


Force b1>b2 b1 Viscous Static b2

Coulomb

Velocity


EE-503 Linear System Theory Spring 2015

Weeks Topics Readings

2Introduction to Linear Systems: LTI systems, Modeling in Continuous-Time and Discrete-Time, System Realization of continuous-time and discrete-time systems

Ch. 1, 2

2Linear Algebra: Basis, Representation and Orthonormlaization, Similarity Transformation, Diagonal Form and Jordan Form, Functions of a Square Matrix, Lyapnov Equation, Singular Value Decomposition

Ch. 3

2Solution of LTI State Equations, Discretization and Solution of Discrete-Time State Equations, Equivalent State Equations, Canonical Forms, Magnitude Scaling, State-Space Realization, Examples

Ch. 4

2Stability: Input-Output Stability of LTI Systems, Internal Stability, Lyapnov Theorem, Examples

Ch. 5

M I D T E R M EXAMINATION

2Controllability and Observability: Introduction to Controllability, Observability, Controllability Canonical Form, Observability Canonical Form, Examples

Ch. 6

2Canonical Decomposition, Conditions in Jordan-Form Equations, Discrete-Time State Equations, Examples

Ch. 6

2State Feedback and State Estimation: State Feedback, Regulation and Tracking, , State Estimation, Reduced Order State Estimators, Examples

Ch. 8

2Pole Placement, Compensator Equations using Classical Method, Poles Placement of Unity-Feedback Systems, Implementation of Compensators

Ch. 9

FINAL T E R M EXAMINATION

Textbook: C. T. Chen, “Linear System Theory and Design,” Oxford University Press, 1999.


C. T. Chen, “Linear System Theory and Design,” Oxford University Press, 1999.

Table of Contents

Chapter 1: INTRODUCTION Week

Lecture Date

1.1 Introduction 1 1 Feb. 09, 20151.2 OverviewChapter 2: MATHEMATICAL DESCRIPTION OF SYSTEMS2.1 Introduction 2.1.1 Causality and Lumpedness2.2 Linear Systems2.3 Linear Time-Invariant (LTI) Systems 2.3.1 Op-Amp Circuit Implementation2.4 Linearization 2 Feb. 10, 20152.5 Examples 2.5.1 RLC Networks2.6 Discrete-Time Systems 3 Feb. 15, 20152.7 Concluding RemarksChapter 3: LINEAR ALGEBRA3.1 Introduction3.2 Basis, Representation and Orthonormalization3.3 Linear Algebraic Equations3.4 Similarity Transformation3.5 Diagonal Form and Jordan Form3.6 Functions of Square Matrix3.7 Lyapunov Equation3.8 Some Useful Formulas3.9 Quadratic Form and Positive Definiteness3.10 Singular Value Decomposition3.11 Norms of MatricesCHAPTER 4: STATE-SPACE SOLUTIONS AND REALIZATIONS

CHAPTER 5: STABILITY

Chapter 6: CONTROLLABILITY AND OBSERVABILITY

Chapter 8: STATE-FEEDBACK AND STATE ESTIMATORS


lst - spring 2015 - lecture - week 1-2

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