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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.
Linear System Theory – Spring-2015 2
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.
Linear System Theory – Spring-2015 3
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
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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()| = .
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Example of a Transfer Matrix of a system with 2-inputs and 2-outputs.
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Lecture-2: Feb. 07, 2015
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
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Op-Amp implementation
Note: With appropriate position of inverters, a system can be implemented with minimum number of Op-Amps.
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Lecture-3: Feb. 13, 2015
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
Linear System Theory – Spring-2015 9
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
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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
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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
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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.
Lecture-5: Feb. 20, 2015
2.6 DISCRETE-TIME SYSTEMS
2.7 CONCLUDING REMARKS
2.8 PROBLEMS
Chapter-3 LINEAR ALGEBRA
Lecture-6: Feb. 21, 2015
Lecture-7: Feb. 27, 2015
Lecture-8: Feb. 28, 2015
Lecture-9: Mar. 06, 2015
Lecture-10: Mar. 07, 2015
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Lecture-11: Mar13, 2015
Lecture-12 Mar. 14, 2015
Lecture-13: Mar. 20, 2015
Lecture-14: Mar. 21, 2015
Midterm Examination
Linear System Theory – Spring-2015 14
Force b1>b2 b1 Viscous Static b2
Coulomb
Velocity
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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.
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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
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