lectures on system theory

8/8/2019 Lectures on System Theory

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Constantin MARIN

LECTURES ON

SYSTEM THEORY

2008


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PREFACE.

In recent years systems theory has assumed an increasingly important role

in the development and implementation of methods and algorithms not only for

technical purposes but also for a broader range of economical, biological and

social fields.

The success common key is the notion of system and the system oriented

thinking of those involved in such applications.This is a student textbook mainly dedicated to determine a such a form of

thinking to the future graduates, to achieve a minimal satisfactory understanding

of systems theory fundamentals.

The material presented here has been developed from lectures given to

the second study year students from Computer Science Specialisation at

University of Craiova.

Knowledge of algebra, differential equations, integral calculus, complex

variable functions constitutes prerequisites for the book. The illustrative

examples included in the text are limited to electrical circuits to fit the technicalbackground on electrical engineering from the second study year.

The book is written with the students in the mind, trying to offer a

coherent development of the subjects with many and detailed explanations. The

study of the book has to be accomplished with the practical exercises published

in our problem book | 19|.

It is hoped that the book will be found useful by other students as well as

by industrial engineers who are concerned with control systems.


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CONTENTS

1. DESCRIPTION AND GENERAL PROPERTIES OF SYSTEMS.

1.1. Introduction 1

1.2. Abstract Systems; Oriented Systems; Examples. 2Example 1.2.1. DC Electrical Motor. 3

Example 1.2.2. Simple Electrical Circuit. 5

Example 1.2.3. Simple Mechanical System. 8

Example 1.2.4. Forms of the Common Abstract Base. 9

1.3. Inputs; Outputs; Input-Output Relations. 11

1.3.1. Inputs; Outputs. 11

1.3.2. Input-Output Relations. 12

Example 1.3.1. Double RC Electrical Circuit. 13

Example 1.3.2. Manufacturing Point as a Discrete Time System. 17

Example 1.3.3. RS-memory Relay as a Logic System. 18

Example 1.3.4. Black-box Toy as a Two States Dynamical System. 20

1.4. System State Concept; Dynamical Systems. 22

1.4.1. General aspects. 22

Example 1.4.1. Pure Time Delay Element. 24

1.4.2. State Variable Definition. 26

Example 1.4.2. Properties of the i-is-s relation. 28

1.4.3. Trajectories in State Space. 29

Example 1.4.3. State Trajectories of a Second Order System. 30

1.5. Examples of Dynamical Systems. 33

1.5.1. Differential Systems with Lumped Parameters. 33

1.5.2. Time Delay Systems (Dead-Time Systems). 35

Example 1.5.2.1. Time Delay Electronic Device. 35

1.5.3. Discrete-Time Systems. 36

1.5.4. Other Types of Systems. 36

1.6. General Properties of Dynamical Systems. 37

1.6.1. Equivalence Property. 37

Example 1.6.1. Electrical RLC Circuit. 38

1.6.2. Decomposition Property. 40

1.6.3. Linearity Property. 40

Example 1.6.2. Example of Nonlinear System. 41

1.6.4. Time Invariance Property. 41

1.6.5. Controllability Property. 41

1.6.6. Observability Property. 41

1.6.7. Stability Property. 41

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2. LINEAR TIME INVARIANT DIFFERENTIAL SYSTEMS (LTI).

2.1. Input-Output Description of SISO LTI. 42

Example 2.1.1. Proper System Described by Differential Equation. 46

2.2. State Space Description of SISO LTI. 49

2.3. Input-Output Description of MIMO LTI. 512.4. Response of Linear Time Invariant Systems. 54

2.4.1. Expression of the State Vector and Output Vector in s-domain. 54

2.4.2. Time Response of LTI from Zero Time Moment. 55

2.4.3. Properties of Transition Matrix. 56

2.4.4. Transition Matrix Evaluation. 57

2.4.5. Time Response of LTI from Nonzero Time Moment . 58

3. SYSTEM CONNECTIONS.

3.1. Connection Problem Statement. 60

3.1.1. Continuous Time Nonlinear System (CNS). 60

3.1.2. Linear Time Invariant Continuous System (LTIC). 60

3.1.3. Discrete Time Nonlinear System (DNS). 60

3.1.4. Linear Time Invariant Discrete System (LTID). 60

3.2. Serial Connection. 64

3.2.1. Serial Connection of two Subsystems. 64

3.2.2. Serial Connection of two Continuous Time Nonlinear Systems (CNS). 65

3.2.3. Serial Connection of two LTIC. Complete Representation. 663.2.4. Serial Connection of two LTIC. Input-Output Representation. 66

3.2.5. The controllability and observability of the serial connection. 67

3.2.5.1. State Diagrams Representation. 68

3.2.5.2. Controllability and Observability of Serial Connection. 71

3.2.5.3. Observability Property Underlined as the Possibility to Determine

theInitial State if the Output and the Input are Known. 73

3.2.5.4. Time Domain Free Response Interpretation for

an Unobservable System. 75

3.2.6. Systems Stabilisation by Serial Connection. 763.2.7. Steady State Serial Connection of Two Systems. 80

3.2.8. Serial Connection of Several Subsystems. 81

3.3. Parallel Connection. 82

3.4. Feedback Connection. 83

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4. GRAPHICAL REPRESENTATION AND REDUCTION OF SYSTEMS.

4.1. Principle Diagrams and Block Diagrams. 84

4.1.1. Principle Diagrams. 84

4.1.2. Block Diagrams. 84

Example 4.1.1. Block Diagram of an Algebraical Relation. 85Example 4.1.2. Variable's Directions in Principle Diagrams and

Block Diagrams. 87

Example 4.1.3. Block Diagram of an Integrator. 89

4.1.3. State Diagrams Represented by Block Diagrams. 89

4.2. Systems Reduction Using Block Diagrams. 92

4.2.1. Systems Reduction Problem. 92

4.2.2. Analytical Reduction. 92

4.2.3. Systems Reduction Through Block Diagrams Transformations. 93

4.2.3.1. Elementary Transformations on Block Diagrams. 93Example 4.2.1. Representations of a Multi Inputs Summing Element. 96

4.2.3.2. Transformations of a Block Diagram Area by Analytical

Equivalence. 96

4.2.3.3. Algorithm for the Reduction of Complicated Block Diagrams. 96

Example 4.2.2. Reduction of a Multivariable System. 98

4.3 Signal Flow Graphs Method (SFG). 106

4.3.1. Signal Flow Graphs Fundamentals. 106

4.3.2. Signal Flow Graphs Algebra. 107

Example 4.3.1. SFGs of one Algebraic Equation. 110Example 4.3.2. SFG of two Algebraic Equations. 111

4.3.3. Construction of Signal Flow Graphs. 113

4.3.3.1. Construction of SFG Starting from a System of Linear Algebraic

Equations. 113

Example 4.3.3. SFG of three Algebraic Equations. 114

4.3.3.2. Construction of SFG Starting from a Block Diagram. 115

Example 4.3.4. SFG of a Multivariable System. 115

4.4. Systems Reduction Using State Flow Graphs. 116

4.4.1. SFG Reduction by Elementary Transformations. 1174.4.1.1. Elimination of a Self-loop. 117

4.4.1.2. Elimination of a Node. 118

4.4.1.3. Algorithm for SFG Reduction by Elementary Transformations. 120

4.4.2. SFG Reduction by Mason's General Formula. 121

Example 4.4.1. Reduction by Mason's Formula of a Multivariable System. 123

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5. SYSTEMS REALISATION BY STATE EQUATIONS.

5.1. Problem Statement. 125

5.1.1. Controllability Criterion. 126

5.1.2. Observability Criterion. 1265.2. First Type I-D Canonical Form. 127

Example 5.2.1. First Type I-D Canonical Form of a Second Order

System. 130

5.3. Second Type D-I Canonical Form. 132

5.4. Jordan Canonical Form. 134

5.5State Equations Realisation Starting from the Block Diagram 137

6. FREQUENCY DOMAIN SYSTEMS ANALYSIS.6.1. Experimental Frequency Characteristics. 139

6.2. Relations Between Experimental Frequency Characteristics and

Transfer Function Attributes. 142

6.3. Logarithmic Frequency Characteristics. 145

6.3.1. Definition of Logarithmic Characteristics. 145

6.3.2. Asymptotic Approximations of Frequency Characteristic. 146

6.3.2.1. Asymptotic Approximations of Magnitude Frequency

Characteristic for a First Degree Complex Variable Polynomial. 146

3.2.2.2. Asymptotic Approximations of Phase Frequency Characteristicfor a First Degree Complex Variable Polynomial. 148

6.4. Elementary Frequency Characteristics. 150

6.4.1. Proportional Element. 150

6.4.2. Integral Type Element. 151

6.4.3. First Degree Polynomial Element. 152

6.4.4. Second Degree Polynomial Element with Complex Roots. 153

6.4.5. Aperiodic Element. Transfer Function with one Real Pole. 158

6.4.6. Oscillatory element. Transfer Function with two Complex Poles. 159

6.5. Frequency Characteristics for Series Connection of Systems.

162

6.5.1. General Aspects. 162

Example 6.5.1.1. Types of Factorisations. 164

6.5.2. Bode Diagrams Construction Procedures. 165

6.5.2.1. Bode Diagram Construction by Components. 165

Example 6.5.2.1. Examples of Bode Diagram Construction by

Components. 165

6.5.2.2. Directly Bode Diagram Construction. 168

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7. SYSTEMS STABILITY.

7.1. Problem Statement. 170

7.2. Algebraical Stability Criteria. 171

7.2.1. Necessary Condition for Stability. 1717.2.2. Fundamental Stability Criterion. 171

7.2.3. Hurwitz Stability Criterion. ` 171

7.2.4. Routh Stability Criterion. 173

7.2.4.1. Routh Table. 173

7.2.4.2. Special Cases in Routh Table. 174

Example 7.2.1. Stability Analysis of a Feedback System. 176

7.3. Frequency Stability Criteria. 177

7.3.1. Nyquist Stability Criterion. 177

7.3.2. Frequency Quality Indicators. 1787.3.3. Frequency Characteristics of Time Delay Systems. 180

8. DISCRETE TIME SYSTEMS.

8.1. Z - Transformation. 181

8.1.1. Direct Z-Transformation. 181

8.1.1.1. Fundamental Formula. 181

8.1.1.2. Formula by Residues. 182

8.1.2. Inverse Z-Transformation. 183

8.1.2.1. Fundamental Formula. 1838.1.2.2. Partial Fraction Expansion Method. 185

8.1.2.3. Power Series Method. 185

8.1.3. Theorems of the Z-Transformation. 186

8.1.3.1. Linearity Theorem. 186

8.1.3.2. Real Time Delay Theorem. 186

8.1.3.3. Real Time Shifting in Advance Theorem. 186

8.1.3.4. Initial Value Theorem. 186

8.1.3.5. Final Value Theorem. 186

8.1.3.6. Complex Shifting Theorem. 1878.1.3.8. Partial Derivative Theorem. 188

8.2. Pure Discrete Time Systems (DTS). 190

8.2.1. Introduction ; Example. 190

Example 8.2.1. First Order DTS Implementation. 190

8.2.2. Input Output Description of Pure Discrete Time Systems. 193

Example 8.2.2.1. Improper First Order DTS. 193

Example 8.2.2.2. Proper Second Order DTS. 193

8.2.3. State Space Description of Discrete Time Systems. 195

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9. SAMPLED DATA SYSTEMS.

9.1. Computer Controlled Systems. 1979.2. Mathematical Model of the Sampling Process. 2049.2.1. Time-Domain Description of the Sampling Process. 204

9.2.2. Complex Domain Description of the Sampling Process. 2059.2.3. Shannon Sampling Theorem. 2079.3. Sampled Data Systems Modelling. 2099.3.1. Continuous Time Systems Response to Sampled Input Signals. 2099.3.2. Sampler - Zero Order Holder (SH). 2119.3.3. Continuous Time System Connected to a SH. 2129.3.4. Mathematical Model of a Computer Controlled System. 2139.3.5. Complex Domain Description of Sampled Data Systems. 215

10. FREQUENCY CHARACTERISTICS FOR DISCRETE TIMESY STEMS10.1.Frequency CharacteristicsDefinition. 217

10.2. Relations Between Frequency CharacteristicsandAttributes of Z-Transfer Functions. 218

10.2.1. Frequency Characteristics of LTI Discrete Time Systems. 21810.2.2. Frequency Characteristics of First Order Sliding Average Filter. 22010.2.3. Frequency Characteristics of m-Order Sliding Weighted Filter. 22110.3. Discrete Fourier Transform (DFT). 222

11. DISCRETIZATION OF CONTINUOUS TIME SYSTEMS.11.1. Introduction. 224

11.2. Direct Methods of Discretization. 22511.2.1. Approximation of the Derivation Operator.

Example 11.2.1. LTI Discrete Model Obtained by Direct Methods. 22511.2.2. Approximation of the Integral Operator. 22511.2.3. Tustin's Substitution. 22611.2.4. Other Direct Methods of Discretization. 227

11.3. LTI Systems Discretization Using State Space Equations. 22811.3.1. Analytical Relations. 228

11.3.2. Numerical Methods for Discretized Matrices Evaluation. 230

12. DISCRETE TIME SYSTEMS STABILITY.12.1. Stability Problem Statement. 231

Example 12.1.1. Study of the Internal and External Stability. 23212.2. Stability Criteria for Discrete Time Systems. 23412.2.1. Necessary Stability Conditions. 23412.2.2. Schur-Kohn Stability Criterion. 23412.2.3. Jury Stability Criterion. 23412.2.4. Periodicity Bands and Mappings Between Complex Planes. 23512.2.5. Discrete Equivalent Routh Criterion in the "w" plane. 238

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1. DESCRIPTION AND GENERAL PROPERTIES OF SYSTEMS.

1.1. Introduction

The systems theory or systems science is a discipline well defined, whosegoal is the behaviour study of different types and forms of systems within a

unitary framework of notions, performed on a common abstract base.

In such a context the systems theory is a set of general methods,

techniques and special algorithms to solve problems as analysis, synthesis,

control, identification, optimisation irrespective whether the system to which they

are applied is electrical, mechanical, chemical, economical, social, artistic,

military one.

In systems theory it is the mathematical form of a system which is

important and not its physical aspect or its application field.There are several definitions for the notion of system, each of them

aspiring to be as general as possible. In common usage the word " system " is a

rather nebulous one. We can mention the Webster's definition:

" A system is a set of physical objects or abstract entities united

(connected or related) by different forms of interactions or interdependencies as

to form an entirety or a whole ".

Numerous examples of systems according to this definition can be

intuitively done: our planetary system, the car steering system , a system of

algebraical or differential equations, the economical system of a country.A peculiar category of systems is expressed by so called "physical

systems" whose definition comes from thermodynamics:

"A system is a part (a fragment) of the universe for which one inside and

one outside can be delimited from behavioural point of view".

Later on several examples will be done according to this definition.

The system theory is the basement of the control science which deals with

all the conscious activities performed inside a system to accomplish a goal under

the conditions of the external systems influence. In the control science three main

branches can be pointed out: Automatic control, Cybernetics, Informatics.The automatic control or just automatic, as a branch of the control science deals

with automatic control systems.

An automatic control system or just a control system, is a set of objects

interconnected in such a structure able to perform, to elaborate, command and

control decisions based on information got with its own resources.

There are also many other meanings for the notion of control systems.

Automation means all the activities to put to practice automatic control

systems.

1. DESCRIPTION AND GENERAL PROPERTIES OF SYSTEMS. 1.1. Introduction.

.

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1.2. Abstract Systems; Oriented Systems; Examples.

Any physical system (or physical object), as an element of the real world,

is a part (a piece) of a more general context. It is not an isolated one, its

interactions with the outside are performed by exchanges of information, energy,material. These exchanges alter its environments that cause modifications in time

and space of some of its specific (characteristic) variables.

In Fig. 1.1. , such a representation is realised.

Information

Energy

Material

INSIDE

OUTSIDE

inputs outputs

u1 y1u2 y2

yrup

u y

System

descriptor

Figure no. 1.2.1. Figure no. 1.2.2

The physical system (object) interactions with the outside are realised

throughout some signals so called terminal variables. In the systems theory, the

mathematical relations between terminal variables are important. These

mathematical relations define the mathematical model of the physical system.

By an abstract system one can understand the mathematical model of a

physical system or the result of a synthesis procedure.

A causal system, feasible system or realisable system, is an abstract

system for which a physical model can be obtained in such a way that its

mathematical model is precisely that abstract system.

An oriented system is a system (physical or abstract) whose terminal

variables are split, based on causalities principles, in two categories:

input variables and output variables.

Input variables (or just "inputs" ) represent the causes by which the outside

affects (inform) the inside.

Output variables (or just "outputs" ) represent the effects of the external

and internal causes by which "the inside" affects ( influences or inform) the

outside. The output variables do not affect the input variables. This is the

directional property of a system: the outputs are influenced by the inputs but not

vice versa.

When an abstract system is defined starting from a physical system

(object), first of all the outputs are defined (selected). The outputs represent

those physical object attributes (qualities) which an interest exists for, taking into

consideration the goal the abstract system is defined for.

The inputs of this abstract system are all the external causes that affect the

above chosen outputs.

1. DESCRIPTION AND GENERAL PROPERTIES OF SYSTEMS. 1.2. Abstract Systems; Oriented Systems; Examples.

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Practically are kept only those inputs that have a significant influence (into

a defined precision level context) on the chosen outputs.

Defining the inputs and outputs we are defining that border which

expresses the inside and the outside from behavioural point of view.

Usually an input is denoted by u, a scalar if there is only one input, or by acolumn vector u=[u1 u2 .. up]

T if there are p input variables.

The output usually is denoted by y if there is only one output or by a

column vector y=[y1 y2 .. yr]T if there are r output variables.

Scalars and vectors are written with the same fonts, the difference between

them, if necessary, is explicitly mentioned.

An oriented system can be graphically represented in a block diagram as it

is depicted in Fig. 1.2.2. by a rectangle which usually contains a system

descriptor which can be a description of or the name of the system or a symbol

for the mathematical model, for its identification. Inputs are represented byarrows directed to the rectangle and outputs by arrows directed from the

rectangle.

Generally there are three main graphical representations of systems:

1. The physical diagram or construction diagram. This can be a picture of a

physical object or a diagram illustrating how the object is built or has to be build.

2. The principle diagram or functional diagram, is a graphical representation of a

physical system using norms and symbols specific to the field to which the

physical system belongs, represented in a such way to understand the functioning

(behaviour) of that system.3. The block diagram is a graphical representation of the mathematical relations

between the variables by which the behaviour of the system is described. Mainly

the block diagram illustrates the abstract system. The representation is performed

by using rectangles or flow graphs.

Example 1.2.1.DC Electrical Motor.

Let us consider a DC electrical motor with independent excitation voltage.

As a physical object it has several attributes: colour, weight, rotor voltage(armature voltage), excitation voltage (field voltage), cost price, etc. It can be

represented by a picture as in Fig. 1.2.3. This is a physical diagram, any skilled

people understand that it is about a DC motor, can identify it, and nothing else.

Ma d e i nCr a i o v a

E P T i p M CC 3

Cr

UrUe Ir

Ur

Ue

Cr

S1

ext

Ur

Ue

Cr S2ext

int

Ir

Figure no. 1.2.3. Figure no. 1.2.4. Figure no. 1.2.5.


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We can look at the motor as to an oriented object from the systems theory

point of view. In a such way we shall define the inside and the outside .

1. Suppose we are interested about the angular speed of the motor axle. Thiswill be the output of the oriented system we are defining now. The inputs to this

oriented system are all the causes that affect the selected output , into anaccepted level of precision. To do this, knowledge on electrical engineering is

necessary. The inputs are: rotor voltage Ur, excitation voltage Ue, resistant

torque Cr, external temperature ext. Now the oriented system related to the DCmotor having the angular speed as output, into the agreed level of precision isdepicted in Fig. 1.2.4. The mathematical relations between and Ur, Ue, Cr,ext are denoted by S1 which expresses the abstract system. This abstractsystem is the mathematical model of the physical oriented object

(or system) as defined above.

2. Suppose now we are interested about two attributes of the above DC motor:

the rotor current Ir and the internal temperature int. These two variables areselected as outputs. The inputs are the same Ur, Ue, Cr, ext . The resultedoriented system is depicted in Fig. 1.2.5. The abstract system for this case is

denoted by S2.

Any one can understand that S1 S2 even they are related to the samephysical object so a conclusion can be drawn: For one physical object (system)

different abstract systems can be attached depending on what we are looking for.

Example 1.2.2.Simple Electrical Circuit.

Let we consider an electrical circuit represented by a principle diagram as

it is depicted in Fig. 1.2.6.

ii =0R

iC0

24 6 8

volts

Controlled voltage generator

u1 =xuC u2=y

Zi=

K2

iC

Voltage amplifieru= y= u2

1S

1S

Tx + x = K uy = K x

.1

2

Figure no. 1.2.6. Figure no. 1.2.7.From the above principle diagram we can understand how this physical

object behaves.

Suppose we are interested about the amplifier voltage u2 only, so it will be

selected as output and the notation y=u2 is utilised and marked on the arrow

outgoing from the rectangle as in Fig. 1.2.7. Under a common usage of this

circuit we accept that the unique cause which affects the voltage u2 is the knob

position in the voltage generator so the input is u2 and is denoted by u= andmarked on the incoming arrow to the rectangle as depicted in Fig. 1.2.7.


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The abstract system attached to this oriented physical object, denoted by

S1, is expressed by the mathematical relations between y=u2 and u= . Withelementary knowledge of electrical engineering one can write:

x=uC; iC=C ; -u1+ Ri + x=0; u1=K1=K1u; i=iC; y=K2x; T=RC; x

(1.2.1)S1 : T

x +x = K1u

y = K2x

In (1.2.1) the abstract system is expressed by so called "state equations".

Here the variable x at the time moment t , x(t), represents the state of the system

in that time moment t. The first equation is the proper state equation and the

second is called "output relation".

The same mathematical model S1 can be expressed by a single relation: a

differential equation in y and u as

(1.2.2)S1 : T

y +y = K1K2uwhich can be presented as an input-output relation(1.2.3)S1 : R(u, y) = 0 where R(u,y) = T

y +y K1K2u

The three above forms of an abstract system representations are called

implicit forms or representation by equations. The time evolution of the system

variables are solutions of these equations starting in a time moment t0 with given

initial conditions for t t0 .The time evolution of the capacitor voltage x(t) can be obtained integrating

(1.2.1) for t t0 and x(t0)=x0 or just from the system analysis as,

. (1.2.4)x(t) = ett0

T x0 + K1T

t0

t

etT u()d

We can observe that the value of x at a time moment t , denoted x(t), depends on

four entities:

1. The current (present) time variable t in which the value of x is expressed.

2. The initial time moment t0 from which the evolution is considered.

3. An initial value x0 which is just the value of x(t) for t=t0.

This is called the initial state.

4. All the input values u() on the time interval [t0

,t],called observation interval,

are expressed by the so called input segment where,u[t0,t]. (1.2.5)u[t0,t] = {(, u()/ [t0, t]}

Putting into evidence these four entities any relation as (1.2.4) is written in

a concentrate form as

(1.2.6)x(t) = (t, t0, x0, u[t0,t])called the input-initial state-state relation, on short i-is-s relation.

Also by substituting (1.2.4) into the output relation from (1.2.1) we get the

output time evolution expression,

(1.2.7)y(t) = K2ett0

T x0 + K1K2T t0

t

etT u()d


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which also depends on the four above mentioned entities and can be considered

in a concentrate form as

. (1.2.8)y(t) = (t, t0, x0, u[t0,t])This is called input-initial state-output relation on short i-is-o relation.

The time variation of the input is expressed by a functionu: TU, t u(t) (1.2.9)

so the input segment is the graph of a restriction on the observation intervalu[t0,t][t0,t] of the function u. In our case the set U of the input values can be for

example the interval [0,10] volts.

Someone who manages the physical object represented by the principle

diagram knows that there are some restrictions on the time evolution shape of the

function u. For example could be admitted piecewise continuous functions or

continuous and derivative functions only.

We shall denote by the set of admissible inputs,={u / u: TU , admitted to be applied to a system)} (1.2.10)

Our system S1 is well defined specifying three elements: the set and thetwo relations and ,

S1={, , } (1.2.11)This is a so called explicit form of abstract systems representation or the

representation by solutions.

An explicit form can be presented in complex domain applying the Laplace

transform to the differential equation, if this is a linear with time-constant

coefficients one, as in (1.2.2):

L{T

y(t) +y(t)} = L{K1K2u(t)} T[sY(s) y(0)} + Y(s) = K1K2U(s)

(1.2.12)Y(s) = K1K2Ts + 1

U(s) + TTs + 1

y(0)

We can see that the differential equation has been transformed to an algebraical

equation simpler for manipulations. But as in any Laplace transforms, the initial

values are stipulated for t=0 not t=t0 as we considered. This can be easily

overtaken considering that the time variable t in (1.2.12) is t-t0 from (1.2.2). With

this in our mind the inverse Laplace transform of (1.2.12) will give us the relation

(1.2.7) where y(0)=K2x(0)=K2x0 and tt-t0.From (1.2.12) we can denote

(1.2.13)H(s) = K1K2Ts + 1

which is the so called the transfer function of the system. The transfer function

generally can be defined as the ratio between the Laplace transform of the output

Y(s) and the Laplace transform of the input U(s) into zero initial conditions,

. (1.2.14)H(s) =Y(s)U(s) y(0)=0

The system S1 can be represented by the transfer function H(s) also.


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Sometimes an explicit form is obtained using the so called integral-

differential operators. Denoting by D=d/dt the differential operator then the

differential equation (1.2.2) is expressed as TDy(t) + y(t)=K1K2u(t)

from where formally one obtain,

(1.2.15)y(t) = K1K2TD + 1u(t) y(t) = S(D)u(t) where S(D) = K1K2

TD + 1so the system S1 can be represented by the integral-differential operator S(D).

Now suppose that in the other context we are interested in the current i of

the physical object represented by the principle diagram from Fig. 1.2.6. The

output is now y(t)=i(t) and the input, considering the same experimental

conditions, is u(t)=(t) also. This oriented system is represented in Fig. 1.2.8.

u= y= i2S 2S

.Tx + x = K u1

y = 1

Rx

R

1K+ u

Figure no. 1.2.8.

The mathematical model of this oriented system is now the abstract system

S2 represented, for example, by the state equations as in Fig. 1.2.8.

Of course any form of representation can be used as discussed above on

the system S1. Because S1 S2 we can withdraw again the conclusion:"For the same physical object it is possible to define different abstract

systems depending on the goal ".

Example 1.2.3.Simple Mechanical System.

Let us consider a mechanical system whose principle diagram is

represented in Fig. 1.2.9.

A

B

KVKPx

y

f

SpringDampingsystem

1S

1S

Tx + x = K uy = K x

.1

2

u= f y

Figure no. 1.2.9. Figure no. 1.2.10.

If a force f is applied to the point A of the main arm, whose shifting with

respect to a reference position is expressed by the variable x, then the point B of

the secondary arm has a shift expressed by the variable y. To the movement

determined by f, the spring develops a resistant force proportional to x , by a

factor KP and the damper one proportional to the derivative of x by KV..

Suppose we are interested on the point B shifting only so the variable y is

selected as to be the output of the oriented system which is defining. Under

common experimental conditions, the unique cause changing y is the force fwhich is the input denoted as u=f.


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The oriented system with above defined input and output is represented in

Fig.1.2.10. where by S1 itis denoted a descriptor of the abstract system.Writing

the force equilibrium equations we get

.KPx + KVx= f ; y = K2x

Dividing the first equation by KP and denoting T=KV/KP ; K1=1/KP ; u=fwe get the mathematical model as state equations,

(1.2.16)S1 :

Tx +x = K1u

y = K2xThis set of equations expresses the abstract object of the mechanical

system. Formally S1 from (1.2.16) is identical with S1 from (1.2.1) of the previous

example. Even if we have different physical objects they are characterised (for

above chosen outputs) by the same abstract system. This abstract system is a

common base for different physical systems. Any development we have done for

the electrical system, relations (1.2.2)--(1.2.15), is available for the mechanical

system too. These constitute the unitary framework of notions we mentioned in

the systems theory definition.

Managing with specific methods the abstract system, expressed in one of

the form (1.2.2) -- (1.2.15) , some results are obtained. These results equally can

be applied both to the electrical system and the mechanical system. Of course in

the first case, for example, x is the capacitor voltage but in the second case the

meaning of x is the paint A shifting.

Such a study is called model based study. We can say that the mechanical

system is a physical model for an electric system and vice versa because they are

related to the same abstract system.

Example 1.2.4.Forms of the Common Abstract Base.

The goal of this example is to manipulate the abstract system (1.2.1) or

equivalently (1.2.16) from mathematical point of view using one element of the

common abstract base, the Laplace transform on short LT. Finally the solutions

(1.2.4) and (1.2.7) will be obtained.

Now we write (1.2.1) putting into evidence the time variable t as

T +x(t)=K1u(t), tt0 , x(t0)=x0 (1.2.17)x. (t)y(t)=K2x(t) (1.2.18)

The main problem is to get the expression of x(t) because y(t) is obtained

by a simple substitution. As we know the one side Laplace transform always

uses as initial time t0=0 and we have to obtain (1.2.4) depending on any t0. It is

admitted that all the functions restrictions for t0 are original functions.Shall we denote by

X(s)=L{x(t)} , U(s)=L{u(t)}

the Laplace transforms of x(t) and u(t) respectively.


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We remember that the Laplace transform of a time derivative function,

admitted to be an original function is,

L{ (t)}=sX(s) - x(0+)x.

where x(0+)=x(t) .t=0+ = limt0 ; t >0

x(t)

If x(t) is a continuous function for t=0 we can simpler write x(0) against to

x(0+). However we can prove that the state of a differential system driven by

bounded inputs is always a continuous function.

So, applying the LT to (1.2.17) we obtain

T[sX(s)-x(0)]+X(s)=K1U(s)

X(s)= U(s) + x(0) (1.2.19)K1

Ts + 1T

Ts + 1which gives us the expression of the state in complex domain but with initial state

in t=0.

We remember now the convolution product theorem of LT:If F1(s) = L{f1(t)} and F2(s) = L{f2(t)} then,

F1(s)F2(s) = L{ } = L{ }0t

f1(t )f2()d 0t

f1()f2(t )d

and in the inverse form,

L-1{F1(s)F2(s)} = = (1.2.20)0t

f1(t )f2()d 0t

f1()f2(t )dThe inverse LT of (1.2.19) is

x(t)=L-1{ } + L-1{ }x(0). (1.2.21)K1

Ts + 1 U(s) T

Ts + 1

We know from tables thatL-1 , for t 0 (1.2.22)T

Ts + 1= e

t

T

L-1 , for t 0 (1.2.23)K1Ts + 1

= K1T

et

T

Identifying now by

andF1(s) =K1

Ts + 1 f1(t) =

K1T

et

T f1(t ) =K1T

etT

F2(s) = U(s) f2(t) = u(t) f2() = u()

taking into consideration (1.2.20) applied to (1.2.21) after the substitution of(1.2.22), we have,

which is written asx(t) = 0t[K1

Te

tT ]u()d + e

t

T x(0), t 0

. (1.2.24)x(t) = et

T x(0) + K1T 0

t

e t

T u()d= (t,0,x(0), u[0,t]), t 0

This is the state evolution starting at the initial time moment t=0, from the

initial state x(0) and has the form of the input-initial state-state relation (i-is-s).

For t = t0 from (1.2.24) we obtain,


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x(t0)=e (1.2.25)

t0

T x(0) + K1T 0

t0e

t0T u()d

= (t0,0 ,x(0), u[0,t 0])

Substituting x(0) from (1.2.25)

x(0)=e ,t0

T x(t0) K1T 0

t0e

T u()d

into (1.2.24) we obtain

x(t)=e +t

Te

t0

T x(t0) K1T 0

t0e

T u()d

K1T 0

t0e

tT u()d + K1

T t0t

etT u()d

x(t)=e (1.2.26)tt0

T x(t0) +K1T t0

te

tT u()d

= (t, t0, x(t0), u[t0,t])

which is just (1.2.4), if we are taking into consideration that x(t0) = x0.

Now from (1.2.24), (1.2.25) and (1.2.26) we observe that

(1.2.27)x(t) = (t, 0, x(0), u[0,t]) (t, t0, (t0,0 ,x(0), u[0,t0]), u[t0,t])

x(t0)which is the so called the state transition property of the i-is-s relation.

According to this property, the state at any time moment t, x(t), as the

result of the evolution from an initial state x(0) at the time moment t=0 with an

input is the same as the state obtained in the evolution of the system startingu[0,t]at any intermediate time moment t0 from an initial state x(t0) with an input ifu[t0,t]the intermediate state x(t0) is the result of the evolution from the same initial state

x(0) at the time t=0 with the input . It has to be precised thatu[0,t0](1.2.28)u[0,t] = u[0,t0] u[t0,t]

Two conclusions can be drawn from this example:1. Any intermediate state is an initial state for the future evolution.

2. An initial state x0 at a time moment t0 contains all the essential

information from the past evolution able to assure the future evolution if the

input is given starting from that time moment t0.

To obtain the relation (1.2.7), we have only to substitute x=y/K2 in

(1.2.26) getting, if we denote x(t0)=x0

(1.2.29)y

(t

) =K

2e

tT0T x

0 +

K1K2

T t0

te

tT u

()d

= (t, t

0, x

(t

0), u

[t0,t])This is an input-initial state-output relation (i-is-o).


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1.3. Inputs; Outputs; Input-Output Relations.

1.3.1. Inputs; Outputs.

The time variable is denoted by letter t for the so called continuous-time

systems and by letter k for the so called discrete-time systems.The time domain T , or observation domain, is the domain of functions

describing the time evolution of variables. For continuous-time systems T Rand for discrete-time systems T Z . Sometimes the letter t is utilised as timevariable also for discrete-time systems thinking that t Z .

The input variable is the function

u :TU; tu(t), (1.3.1)where U is the set of input values (or the set of all inputs).

Usually if there are p inputs expressed by real numbers.U Rp

The set of admissible inputs , is the set of functions u allowed to beapplied to an oriented system.

The input segment on a time interval [t0,t1] called observation intervalis the graphic of the function u on this time interval, :u[t0,t1]

u ={(t,u(t)), t1]} (1.3.2)[t0,t1] t [t0,

When we are saying that to a system an input is applied on a time-interval

[t0, t1], we have to understand that the input variable changes in time according to

the given graphic ,that is according to the input segment. Sometimes foru[t0,t1]

easier writing we understand by u , depending on context, the following:u - a function as (1.3.1).

u - a segment as (1.3.2) on an understood observation interval.[t0,t1]u(t) -the law of correspondence of the function (1.3.1).

Also u(t) can be seen as the law of correspondence of the function (1.3.1)

or the value of this function in a specific time moment denoted t.

All these conveniences will be encountered for all other variables in this

textbook.

The output variable is the function

y :TY; ty(t), (1.3.3)where Y is the set of output values (or the set of all outputs). Usually ifY R r

there are r outputs expressed by real numbers.

We denote by the set of possible outputs that is the set of all functions ythat are expected to be got from a physical system if inputs that belong to areapplied.

The input-output pair. If an input is applied to a physical system theu[t0,t1]output time response is expressed by the output segment , wherey[t0,t1]

={(t,y(t)), (1.3.4)y[t0,t1] t [t0, t1]}that means to an input corresponds an output.

1. DESCRIPTION AND GENERAL PROPERTIES OF SYSTEMS. 1.3. Inputs; Outputs; Input-Output Relations.

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The pair of segments

(1.3.5)[u[t0,t1] ; y[t0,t1]] = (u;y)observed to a physical system is called an input-output pair.

It is possible that for the same input another output segment tou[t0,t 1] y[t0,t1]a

be obtained, that means also the pair is an input-output[u[t0,t 1] ; y[t0,t 1]a ] = (u;ya)pair of that system as depicted in Fig. 1.3.0.

0 t

u u ,[ ]t0 t1

[

]

t0 t1

0 t

y y,[ ]t0 t1

[

[

]

]

,[ ]t0 t1ya

t0 t1

Figure no. 1.3.0.

In the example of the electrical device from Ex. 1.2.2.

or of the mechanical system from Ex. 1.2.3. , the solution of

the differential equation (1.2.2) for t t0 and x(t0)=x0 is

y(t) = ett0

T x0 +K1K2

T t0

t

etT u()d = (t, t0, x0, u[t0,t])

For the same input u[t0,t1] , the output depends also on

the value x0=x(t0) which is the voltage across the capacitor

terminals C in Ex. 1.2.2. or the arm position (point A ) in

Ex. 1.2.3. at the time moment t0.

1.3.2. Input-Output Relations.

The totality of input-output pairs that describe the behaviour of a physical

object is just the abstract system. Instead of specific list of input time functions

and their corresponding output time functions, the abstract system is usually

characterised as a class of all time functions that obey a set of mathematical

equations. This is in accord once with the scientific method of hypothesising an

equation and then checking to see that the physical object behaves in a manner

similar to that predicted by the equation /2/.

Practically an abstract system is expressed by the so called input-output

relation which can be a differential or difference equation, graph, table or

functional diagram.

A relation implicitly expressed by R(u,y)=0 or explicitly expressed by an

operator S, y=S{u}, is an input-output relation for an oriented system if:

1. Any input-output pair observed to the system is checking this relation.

2. Any pair (u,y) which is checking this relation is an input-output pair of that

oriented system.

We have to mention that by the operatorial notation y=S{u} or just y=Su,

we understand that the operator S is applied to the input (function) u and as a

result, the output (function) y is obtained.

For example if in the differential equation from Ex. 1.2.2.,T x

+x

=K

1u

we substitute x=y/K2, we obtain,


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, (1.3.6)Ty +y = K1K2u R(u, y) = 0 R(u, y) = T

y +y K1K2u

which is an implicit input-output relation.

By denoting , the time derivative operator, we obtainD = ddt

Dy =y

TDy(t) + y(t) K1K2u(t) = 0 (TD + 1)y(t) K1K2u(t) = 0 (1.3.7)y(t) = K1K2

TD + 1u(t) y(t) = Su(t) , S = K1K2

TD + 1Here is an explicit input-output relation by any(t) = Su(t)

integral-differential operator S . This relation is expressed in time domain but it

can be expressed in every domain if one to one correspondence exists.

For example we can express in s-complex domain applying to (1.3.6) the

Laplace transform,

Y(s)= (1.3.8)K1K2

Ts + 1

U(s) + T

Ts + 1

x(0)

from where an operator H(s) called transfer function is defined,

H(s)= . (1.3.9)Y(s)U(s) x(0)=0

= K1K2Ts + 1

The relation between the Laplace transformation of the output Y(s) and the

Laplace transformation of the input U(s) which determined that output into zero

initial conditions

Y(s)=H(s)U(s) (1.3.10)

is another form of explicit input-output relation.

Example 1.3.1.Double RC Electrical Circuit.

Let us consider an electrical network obtained by physical series

connections of two simple R-C circuits whose principle diagram is represented in

Fig. 1.3.1.

C 1

1A 1i

1A' 1B'

1B

1C

1R

2A' 2B'

2A 2i 2B2R

2Cu =x

C 2u =xu

Ci1

Ci 2

y

i=0

1 2

u yS

Figure no. 1.3.1. Figure no. 1.3.1.

Suppose that the second circuit runs empty and the first is controlled by

the voltage u across the terminals A1,,A1' , and we are interested in the voltage y

across the terminals B2,B2'.

Because the output y is defined, under common conditions only the voltage

u affects this selected output so the oriented system is specified as depicted in

Fig. 1.3.2.

The abstract system denoted by S will be defined establishing themathematical relations between u and y.


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To do this first we observe from the principle diagram that there are 8

variables as time functions involved : u, i1, iC1, uC1=x1, iC2, uC2=x2, i2 and y.

The other variables R1, R2, C1, C2 are constants in time and represent the

circuit parameters , any skilled people in electrical engineering understand on the

spot.Because u is a cause (an input) it is a free variable and we have to look for

7 independent equations. These equations can be written using the Kirckoff's

theorems and Ohm's law:

1. 2. 3. 4.ic1 = i1 i2 ic2 = i2 ic1 = C1x.

1 ic2 = C2x.

2

5. 6. 7. y=x2i1 =1

R1(x1 + u) i2 = 1

R2(x1 x2)

We can observe that two variables x1 and x2 appear with their first order

derivative so eliminating all the intermediate variables a relation between u and y

will be obtained as a second order differential equation. But first shall we keepthe variables x1 and x2 and their derivative.

Denoting by T1=R1C1 and T2=R2C2 the two time constants, after some

substitutions we obtain,

(1.3.11)T1x.

1 = (1 +R1R2

)x1 +R1R2

x2 + u

(1.3.12)T2x.

2 = x1 x2y=x2 (1.3.13)

which after dividing by T1, T2 respectively, they take the final form

(1.3.14)x.

1 = 1

T1 (1 +R1R2 )x1 +

1T1

R1R2 x

2 +1

T1 u(t)S: (1.3.15)x

.2 = 1

T2x1 1

T2x2

y=x2 (1.3.16)

The equations (1.3.14), (1.3.15), (1.3.16) are called state equations

related to that oriented system and they constitute the abstract system S in state

equations form. We can rewrite these equations into a matrix form,

S: (1.3.17)x.

= Ax + bu y=cTx + du (1.3.18)

where:

; ; ; d = 0 , (1.3.19)A =

1T1

(1 + R1R2

) 1T1

R1

R21

T2 1

T2

b =

1

T1

0

c =

0

1

and generally they are called:

A the system matrix

b the command vector

c the output vectord the direct input-output connection factor .


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The input-output relation R(u.y)=0 , as mentioned before, can be

expressed as a single differential equation in u and y. To do this we can use for

example (1.3.14), (1.3.15), (1.3.16) or simpler (1.3.11), (1.3.12), (1.3.13).

Substituting x2 from (13) in (12) and multiplying by T1 we obtain

.T1T2y.

= T1x1 T1y T1 x1 = T1T2y.

+ T1y ()Applying the first derivative to (*) and substituting from (11) andT1x

.1

x1 = T2y.

+ yfrom (*) we.. obtain,

T1T2y + T1y.

= (1 + R1R2

)(T2y.

+ y) + R1R2

y + u

which finally goes to

. (1.3.20)T1T2y + [T1 + (1 +R1R2

)T2]y.

+ y = u

This is a differential equation expressing the mathematical model (theabstract system) of the oriented system. It can be presented as an i-o relation

(1.3.21)R(u, y) = 0 where R(u, y) = T1T2y + [T1 + (1 +R1R2

)T2]y.

+ y u

If we denoted by we can express the i-o relations into an explicit formd

dt= D

y(t)= (1.3.22)1

T1T2D2 + [T1 + T2(1 +

R1

R2)]D + 1

u(t) y(t) = S(D)u(t)

where S(D) is an integral-differential operator.

For simplicity shall we consider the following values for parameters:

R1=R ; R2=2R ; C1=C ; C2=C/2 T1=T2=T=RCso the differential equation (1.3.20) becomes

. (1.3.23)T2y + 2.5T y.

+ y = uWe can express the i-o relation into a complex form by using the Laplace

transform. Applying the Laplace transform to (1.3.23) we get

Y(s)=L{y(t)} ; U(s)=L {u(t)} T2[s2Y(s) sy(0+) y

.(0+)] + 2,5T[sY(s) y(0+)] + Y(s) = U(s)

Y(s) = 1

T

2

s

2

+ 2,5Ts + 1

U(s) +T2s + 2,5T

T

2

s

2

+ 2,5Ts + 1

y(0+) + T2

T

2

s

2

+ 2, 5Ts + 1

y.(0+)

(1.3.24)

We denote by L(s) the characteristic polynomial

L(s) = T2s2+2,5Ts+1 (1.3.25)

so the output in complex domain is,

(1.3.26)Y(s) = 1L(s)

U(s) +T2s + 2,5T

L(s)y(0+) + T

2

L(s)y.(0+)

As we can see, the Laplace transform of the output Y(s), depends on the

Laplace transform of the input U(s) and on two initial conditions : y(0+) , the

value of the output, and , the value of the time derivative of the outputy.

(0+

)both on the time-moment 0+.


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Let H(s) be

(1.3.27)H(s) = 1L(s)

=Y(s)U(s) zero initial condition

where H(s) is called the transfer function of the system.

The transfer function of a system is the ratio between the Laplacetransform of the output and the Laplace transform of the input which determines

that output into zero initial conditions if and only if this ratio is not depending on

the form of the input.

By using the inverse Laplace transformwe can obtain the time answer (the

output response) of this system. The characteristic equation

L(s) = T2s2+2,5Ts+1 = 0 has the roots,

s1,2= (1.3.28)(5

2T 1

225T2 16T2 )/2T2

so the characteristic polynomial is presented as

L(s) = T2(s-1)(s-2) with 1= -1/2T ; 2= -2/T . (1.3.29)One way to calculate the inverse Laplace transform is to use the partial

fraction development of rational functions from Y(s) as in (1.3.26)

H(s) = 1T2(s1)(s2)

; H(s) = As1

+ Bs2

A = 23T

; B = 23T

T2s+2,5T

T2(s1)(s2)= A1

s1+ B1

s2 A1 = 43 ; B1 =

1

3

T2

T2(s1)(s2)= A2s1 +

B2

s2 A2 = 2T3 ; B2 =

2T

3

Y(s) =2

3T 1

s1 1

s2 U(s) +1

3 4

s1 1

s2 y(0) +2T

3 1

s1 1

s2 y.

(0)L-1 L-11

s1= e1t = 1(t) 1s2 = e

2t = 2(t)

L-1 L-11

s1U(s) = 0

t 1(t )u()d 1s2U(s) = 0t 2(t )u()d

y(t) = 13

[41(t) 2(t)] y(0) + 2T3

[1(t) 2(t)] y.(0) + 2

3T 0t

[1(t ) 2(t )]u()d

where (1.3.30)

(1.3.31)1(t) = e1t = et

2T

(1.3.32)2(t) = e2t = et

T/2

By using the same procedure like to the first order R-C circuit presented in

Ex. 1.2.2 we can express this time relation depending on the initial time t 0 as:

y(t) =1

3[41(t t0) 2(t t0)]y(t0) + 2T3 [1(t t0) 2(t t0)]y

.(t0) +

(1.3.33)+ 23T t0

t [1(t ) 2(t )]u()dAs we can see the general response by output depends on: t, t0, the initial

state x0 , and the input , where the state vector is defined asu[t0,t](1.3.34)x1(t0) = y(t0) ; x2(t0) = y

.(t0) x(t0) = [x1(t0) x2(t0)]T = x0

The relation (1.3.33) is an input-initial state-output relation of the form(1.3.35)y(t) = (t, t0, x0, u[t0,t])


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Example 1.3.2. Manufacturing Point as a Discrete Time Ststem.

Shall we consider a working point (manufacturing point). In this

manufacturing point the work is made by a robot which manufactures

semiproducts. We suppose that it works in a daily cycle. Suppose that in the day

k the working point is supplied with uk semiproducts from which only apercentage ofk is transformed in finite products. The working point has a storehouse which contains in the day k , xkfinite products. Suppose that daily (may be

in the day k) a percentage of (1-k) from the stock is delivered to other sections.Also at day k the production is reported as being yk, a percentage k from thestock. We want to see what are the evolution of the stock and the report of the

stock for any day. It can be observed that the time variable is a discrete one, the

integer number k .

u yk kS

Figure no. 1.3.3.

This working point can be interpreted as an

oriented system having the daily report yk as theoutput. The input is the daily rate of supply uk so the

oriented system is defined as in Fig. 1.3.3. where the

input and the output are string of numbers.

The mathematical model is determined from the above description. If we

denote by xk+1 the stock for the next day, it is composed from the left stock

xk-(1-k)xk=kxk and the new stockkuk,xk+1=kxk+kuk (1.3.36)

yk=kxk (1.3.37)This is the abstract system S of the working point looked upon as anoriented system. These are difference equations expressing a discrete-time

system.

We can determine the solution of this system of equations by using a

general method, but in this case we shall proceed to difference equation

integration step by step.

The day p+1: xp+1=pxp+pup | k-1...p+1The day p+2: xp+2=p+1xp+1+p+1up+1 | k-1...p. . . . . . . . . . . . . . . . . . . . . . . . . . . . .The day k-2 : xk-2=k-3xk-3+k-3 uk-3 | 12The day k-1 : xk-1=k-2xk-2+k-2uk-2 | k-1The day k : xk=k-1xk-1+k-1uk-1 | 1

Denoting by the discrete-time transition matrix (in this example it is a(k, p)scalar),

; (k,k)=1 (1.3.38)(k, p) = j=p

k1j = k1k2...p+1p

and adding the above set of relations each multiplied by the factor written on the

right side, we obtain


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(1.3.39)xk = (k, p)xp + j=p

k1[(k, j + 1)juj]

yk = kxk (1.3.40)We observe that (1.3.39) is an input-initial state-state relation of the form,

(1.3.41)xk = (k, k0, xk0 , u[k0,k1])where xk is the state on the current time k (in our case the day index), x k0 is the

initial state on the initial time moment p=kk0.

The output evolution is

(1.3.42)yk = k(k, p)xp + j=p

k1[k(k, j + 1)juj]

which is an input-initial state-output relation of the form,

(1.3.43)yk = (k, k0, xk0 , u[k0,k1])

Example 1.3.3. RS-memory Relay as a Logic System.

Let us consider a physical object represented by a principle diagram as in

Fig. 1.3.4. This is a logic circuit performing an RS-memoryrelay based.

x

x

xa

b

c

d

L

y

+ E

i

Relay

SB

RB

Figure no. 1.3.4.

try

s tt

S

Figure no. 1.3.5.

=0xt0 =1xt0

S0 S1

S0 S1S=

Figure no. 1.3.6.

Here SB represents a normal opened button (the set-button) and RB a

normal closed button (the reset-button).

By normal it is understood "not pressed".When the button SB is pushed the current can run through the terminals

a-b and when the button RB is pushed the current can not run through the

terminals c-d. By x are denoted the open-normal contacts of the relay.

The normal status (or state) of the relay is considered that when the current

through the coil is zero. L is a lamp which lights when the relay is activated. The

functioning of this circuit can be explained in words:

If the button RB is free, pushing on the button SB a current i run activating

the relay whose contact x will shortcut the terminals a-b and the lamp is turned

on. Even if the button SB is released the lamp keeps on lighting.


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If the button RB is pushed, the current i is cancelled and the relay becomes

relaxed, the lamp L is turned off.

The variables encountered in this description SB, RB, i, x, y are associated

with the variables st, rt, it, xt, yt called logical variables .

They represent the truth values, at the time moment t, of the propositions:st: "The button SB is pushed" "Through terminals a-b the current can run".rt: "The button RB is pushed" "Through terminals c-d the current can not run"it: "

xt:"The relay is activated" "The relay normal opened contacts are connected".yt: "The lamp lights".

These logical variables can take only two values denoted usually by the

symbols 0 and 1 on a set B={0;1} which represents false and true.

The set B is organised as a Boolean algebra. In a Boolean algebra three

binary fundamental operations are defined: conjunction " ", disjunction " "and negation " ".

Suppose we are interested about the lamp status so the output is y(t)=yt.

This selected output depends on the status of buttons SB, RB only (it is supposed

that the supply voltage E is continuously applied) as external causes, so the input

is the vector u(t) = [st rt ]T.

An oriented system is defined now as depicted in Fig. 1.3.5.

The mathematical relations between u(t) and y(t), defining the abstract

system S are expressed as logical equations.

The value of logical variable it is given byit= . (1.3.44)(s t x t) rt

Because of the mechanical inertia, the status of the relay changes after a

small time interval finite or ideally 0 to the value of it,. (1.3.45)xt+ = it

Of course the status of the lamp equals to the status of the relay, so

yt=xt (1.3.46)

Substituting (1.3.44) into (1.3.45) together with (1.3.46) we obtain the

abstract system S as,

(1.3.47)xt+ = (s t xt) r tS:

yt=xt (1.3.48)To determine the output of this system beside the two inputs st; rt we

need another piece of information (the value of xt , the state of the relay: 1 - if

the relay is activated and 0 - if the relay is not activated.


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It does not matter how we shall denote these information A (or on) for 1

and B )or off) for 0. If we know that the state of the relay is A we can determine

the output evolution if we know the inputs.

If we are doing experiments with this physical system on time interval

[t0,t1] for any t0, t1, a set S of input-output pairs (u,y)= can be(u[t0,t1], y[t0,t1])observed,

, (1.3.49)S = {(u[t0,t 1], y[t0,t 1])/ observed , t0, t1 R, u[t0,t 1] }The set S can describe the abstract system. It can be split into two subsets

depending whether a pair (u,y) is obtained having equal to0 or 1:xt0S0={(u,y)S / if =0}={(u,y)S / if =A} ={(u,y)S / if =on} (1.3.50)xt0 xt0 xt0S1={(u,y)S / if =1}={(u,y)S / if =B} ={(u,y)S / if =off} (1.3.51)xt0 xt0 xt0

It can be proved that

S0

S

1

= S ; S0

S

1 = , (1.3.52)

as depicted in Fig. 1.3.6.

Also inside of any subset Si the input uniquely determines the output

(1.3.53)(u, ya) Si, (u, yb) Si ya yb i = 0; 1From this we understand that the initial state is a label which parametrize

the subsets Si S as (1.3.53).

Example 1.3.4. Black-box Toy as a Two States Dynamical System.

Let us consider a black-box, as a toy, someone received. The box is

covered, nothing can be seen of what it contains inside , but it has a controllablevoltage supplier with a voltage meter u(t) across the terminals A-B and a voltage

meter y(t) across the terminals C-D, as depicted in Fig. 1.3.7.

V y(t)R

R R

X

VVoltageSupplayer

u(t)

Figure no. 1.3.7.

A

B

C

D

Playing with this black-box, we register the evolution of the inputu(t) and

of the resulted output y(t) . We are surprised that sometimes we get the

input-output pair

, (1.3.54)(u[t0,t1], y[t0,t1] =1

2u[t0,t1]) y() =

1

2u()

but other times we get the input-output pair

. (1.3.55)(u[t0,t1], y[t0,t1] =2

3u[t0,t1]) y() =

2

3u()

Doing all the experiments possible we have a collection of input-output

pairs which constitute a set S as (1.3.49).


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This collection S will determine the behaviour of the black-box. For the

same input applied u(t) we can get two outputs or . Oury(t) = 12u(t) y(t) = 2

3u(t)

set of input-output pairs can be split into two subsets: S0

if they correspond to

(1.3.54) and S1

if they correspond to (1.3.55).

Of course S0 S1 = S ; S0 S1 = .If someone gave us an input we would not be able to say what theu[t0,t1]

output is because we have no idea from which subset, S0 or S1,, to select the right

pair. Some information is missing to us.

Suppose that the box cover face has been broken so we can have a look

inside the box as in Fig. 1.3.7.

Now we can understand why the two sets of input-output pairs (1.3.54),

(1.3.55) were obtained.

The box can be in two states depending of the switch status: opened or

closed. We can define the box state by a variable x which takes two valuesnominated as: {off ; on} or {0 ; 1} or {A ; B}.

Now the subset S0can be equivalently labelled by one of the marks: "off",

"0", "A" and the subset S1

by : "on", "1", "B" respectively.

It does not matter how the position of the switch is denoted (labelled). The

switch position will determine the state of the black-box.

The state is equivalently expressed by one of the variables:

x {off;on} or x {0 ; 1} or x {A; B}If someone gives us an input and an additional piece of informationu[t0,t 1]

formulated as: "the state is on " that means x=on or as "the state is B" that means

etc. we can uniquely determine the output selecting it from thex = B y(t) = 23u(t)

subset S1.

With this example our intention is to point out that the system state

appears as a way of parametrising the input-output pair subsets inside which one

input uniquely determines one output only.

Also we wanted to point out that the state can be presented in different

forms, the same input-output behaviour can have different state descriptions.

In this example the state of the system can not be changed by the input,

such a system being called state uncontrollable.


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1.4. System State Concept; Dynamical Systems.

1.4.1. General aspects.

As we saw in the above examples, to determine univocally the output,

beside the input, some initial conditions have to be known.For example in the case of simple RC circuit we have to know the voltage

across the capacitor, for the mechanical system the initial position of the arm, in

the case of double RC circuit the voltages across the two capacitors, for the

manufacturing point the initial value of the stock, for the relay based circuit the

initial status of the relay.

All this information defines the system state in the time moment from

which the input will affect the output.

The state of an abstract system is a collection of elements (the elements

can be numbers) which together with the input u(t) for all t t0 uniquelydetermines the output y(t) for all t t0 ..

In essence the state parametrizes the listing of input-output pairs.

The state is the answer to the question: "Given u(t) for t t0 and themathematical relationships between input and output (the abstract system), what

additional information is needed to completely specify y(t) for t t0 ? ".The system state at a time moment contains (includes) all the essential

information regarding the previous evolution to determine, starting with that time

moment, the output if the input is known.

A state variable denoted by the vector x(t) is the time function whose valueat any specified time is the state of the system at that time moment.

The state can be a set consisting of an infinity of numbers and in this case

the state variable is an infinite collection of time functions. However in most

cases considered, the state is a set of n numbers and correspondingly x(t) is a

n-vector function of time.

The state space, denoted by X, is the set of all x(t) values.

The state representation is not unique. There can be many different ways

of expressing the relationships of input to output. For example in the case of

black-box or of the logic circuit we can define the state as {on , off}, {A, B} and

so on. For the double RC circuit one state representation means the output value

y(t0) and the time derivative value of the output

The state of a system is related to a time moment. For example the state x 0at a time moment t0 is denoted by (x0 , t0 ) x(t0) .

=

The minimum number of state vector elements, for which the output can be

univocally determined, for a known (given) input, represents the system order or

the system dimension. Systems from Ex. 1.2.2. or Ex. 1.2.3. are of first order

1. DESCRIPTION AND GENERAL PROPERTIES OF SYSTEMS. 1.4. System State Concept; Dynamical Systems.

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because it is enough to know a single element x0 to determine the output

response as we can see in relation (1.2.7).

Also the discrete time system (1.3.36), (1.3.37) from Ex. 1.3.2. is of first

order because the response (1.3.42) can be uniquely determined if we know just

the number xp .But the abstract system (1.3.17), (1.3.18) or (1.3.20) from Ex. 1.3.1 is of

second order because, as we can see in relation (1.3.33) (considering for the sake

of convenience the particular case T1 = T2 =T the system dimension is not

affected), the output y(t) is uniquely determined by a given input if twou[t0,t]initial conditions y(t0) , are known as it is rewritten in (1.4.1)y

.(t0)

y(t) = 13[41(t t0) 2(t t0)]y(t0) + 2T3 [1(t t0) 2(t t0)]y

.(t0)+

(1.4.1)+ 23T t0

t [1(t )2 (t )]u()dThese initial conditions are the output value and the output time

derivative value at the time moment t0. These two values can be selected as

components of a vector, the state vector that means at thex0 =

x10

x20

=

y(t0)y.(t0)

time t0 the state is x0 . (t0, x0) = x(t0)If , for example, y(t0)=3V, =9V/sec. , the state at the time moment t0y

.(t0)

is expressed by the numerical vector so we can sayx0 =

3

9 =

3volts

9volts/sec .

that at the time moment t0 , the system is in the state [3 9]T. Because two

numbers are necessary to determine uniquely the output we can say that thissystem is a second order one.

The response from (1.3.33) can be arranged as:

y(t) = 1(t t0)

43y(t0) +

2T

3y.(t0)

+ 2(t t0)

13y(t0)

2T

3y.(t0)

+ 23T t0

t [..]u()d

(1.4.2)x10

x20

Denoting by

x10 = 4

3y(t0) + 2T3 y

.(t0)

x20

=

1

3y(t0)

2T

3 y

.

(t0)the output response can be written as(1.4.3)y(t) = 1(t t0)x1

0 + 2(t t0)x20 + 2

3T t0t [[1(t )2 (t )]]u()d

This output response can be univocally determined if the numbers and arex10

x20

known, that means they can constitute the components of the vector x0 =

x10

x20

Let us consider a concrete example for T=1sec. For

y(t0) = x10 = 3V; y

.(t0) = x2

0 = 9V/sec


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x10 = 4

33 + 21

39 = 10V; x2

0 = 133 21

39 = 7V/ sec;

In this form of the state vector definition we can say that at the time

moment t0 the system is in the state =[10 -7]T , that is different of the statex0

x0=[3 9]T but, applying the same input , we obtain the same output as itu[t0,t]

was obtained in the case of x0=[3 9]T .The following important conclusion can be pointed out:

The same input-output behaviour of an oriented system can be obtained by

defining the state vector in different ways.

If x is the state vector related to an oriented system and a square matrix T

is a non-singular one, detT0, then the vector is also a state vector for thex = Txsame oriented system. Both states x, above related will determine the samex

input-output behaviour.

In the above example, the two state relationships

x1 = 43x1 + 2T3 x2x2 = 13x1

2T

3x2

can be written in a matrix form

. (1.4.4)x = Tx,where T =

4

3

2T

3

13

2T3

,det T =

2T

3 0

For example in the case of logic circuit (Ex. 1.3.3.) or of black-box

(Ex.1.3.4.) we can define the state values as {on, off}, {A, B} and so on as we

discussed.

If the amount of the collection of numbers which define the state is a finite

one the state is defined as a column-vector: x=[x1 x2 . . xn]T.

The minimal number of the elements of this vector able uniquely to precise

(to determine) the output will define the system order or the system dimension.

When the amount of such a collection strictly necessary is infinite ( we can

say the vector x has an infinity number of elements) then the order of the system

is infinity or the system is infinite-dimensional.

Such a infinity-dimensional system is presented in the next example by the

pure time delay element.

Example 1.4.1. Pure Time Delay Element.

Let us consider a belt conveyor transporting dust fuel (dust coal for

example) utilised in a heating system represented by a principle diagram as

shown in Fig. 1.4.1. The belt moves with a speed v.

The thickness of the fuel is controlled by a mobile flap.

Suppose we are interested about the thickness in the point B at the end of

the belt expressed by the variable y(t) .


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This variable will be the output of the oriented system we are defining as

in Fig. 1.4.2. The input is the thickness realised on the flap position, point A, and

we shall denote it by the variable u(t).

The distance between points A and B is d . One piece of fuel passing from

A to B will take a period of time . = dv

u(t);y(t)

t

y(t)u(t)

t10

y(t1)t1u( )

t1

Figure no. 1.4.1.

y(t)=u(t- )u(t) y(t)Pure time delay

element

U(s) Y(s)se

Figure no. 1.4.2.

Conveyor belt

d

Controlled flapDust fuel (coal)

u(t)

speed v

ThicknessThicknessy(t)

A B

The input-output relation is expressed by the equation,

y(t) = u(t ) (1.4.5)We can read this relation as: The output at the time moment t equals to the

value the input u(t) had seconds ago. Such a dependence is illustrated in thediagram from Fig. 1.4.2. It is a so called a functional equation.

Now suppose an input is given. Can we determine the output y(t) foru[t0,t]

any t t0 ? What do we need in addition to do this ? Looking to the principlediagram from Fig. 1.4.1. or to the relation (1.4.5) we understand that in addition

to know all the thickness along the belt between the points A and B or in other

words all the values the input u(t) had during the time interval

[t0 , t0) .This collection of information constitutes the system state at the time

moment t0

and it will be denoted by x0

.

So the state at the time moment t0, (t0,x0 ) , denoted on short as

x0=x(t0)= is a set containing an infinite number of elementsxt0(1.4.6)x0 = xt0 = x(t0) = { u(), [t0 , t0) } = u[t0 , t0)

Because of that this system has the dimension of infinity.

At any time t the state is (t,x)=x(t) defined by

(1.4.7)x(t) = { u(), [t , t) } = u[t , t)All these intuitively observations may have a mathematical support

applying the Laplace transform to the input-output relation (1.4.5).


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We remember that

(1.4.8)L{u(t )} = L{u(t )1(t)} = es[U(s) +

0

u(t)estdt]

so the Laplace transform of the output is

(1.4.9)Y(s) = esU(s) + es 0

u(t)estdt = Y f(s) + Y l(s)where (1.4.10)Y f(s) = esU(s) yf(t) = (t ,0,0,u[0,t])

is the forced response which depends on the input u(t) only, of course here

depends on the Laplace transform U(s) which contains the input values u() forany 0. We paid for Laplace transform instrument simplicity with t0=0.

(1.4.11)Y l(s) = es

0

u(t)estdt

is the free response as the output response when the input is zero.

By zero input we must meanu(t) 0 t 0 U(s) 0s

from the convergence domain of U(s).

The free response depends on the initial state only (here at the initial time

moment t0=0 ) and as we can see from (1.4.11) it depends on all the values of

u() [ , 0) u[ , 0)so it looks naturally to choose the initial state as

.x0 = x(0) = u[ , 0)Now we can interpret the free response from (1.4.11) as

. (1.4.12)yl(t) = (t,0,x0, 0[0,t))From (1.4.10), (1.4.12) the general response ( the time image of (1.4.9) )

can be expressed as an input-initial state-output relation

. (1.4.13)y(t) = yf(t) + yl(t) = (t,0,0,u[0,t]) + (t,0,x0, 0 [0,t)) = (t,0,x0, u[0,t))

1.4.2. State Variable Definition.

The state variable is a function

x: TX , tx(t), (1.4.14)where X is the state space , which expresses the time evolution of the system

state. The state is not a constant (fixed) one.

It can be changed during the system time evolution, so the function x(t)

can be a non-constant one.

The graphic of this function on a time interval [t0, t1] , denoted by

(1.4.15)x[t0, t1] = {(t, x(t)), t [t0, t1] }is called the time state trajectory on the interval [t0, t1]. The state variable x(t)

is an explicit function of time but also depends implicitly on the starting time t0,

the initial state x(t0)=x0 and the input u() [t0,t].


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This functional dependency called input-initial state-state relation (i-is-s

relation) or just a trajectory (more precisely time-trajectory ), can be written as

(1.4.16)x(t) = (t, t0, x0, u[t0,t]) x0 = x(t0)A relation of the form (1.4.16) is an input-initial state-state relation

(i-is-s relation) and expresses the state evolution of a system if the following fourconditions are accomplished:

1. The uniqueness condition. For a given initial state x(t0)=x0 at time t0 and

a given well defined input the state trajectory is unique. This can beu[t0 , t]

expressed as: "A unique trajectory exists for all t > t0 , given(t, t0, x0, u[t0,t])the state x0 at time t0 and a real input u(), for t0 ".

2. The consistency condition . For t=t0 the relation (1.4.16) has to check

the condition :

. (1.4.17)x(t) t=t0 = x(t0) = (t0, t0, x0, u[t0,t0]) = x0Also

, (1.4.18) t1 t0 ,tt1,t>t1

lim (t, t1, x(t1), u[t1,t]) = x(t1)

that means a unique trajectory starts from each state.

3. The transition condition . Any intermediate state on a state trajectory is an

initial state for the future state evolution.

For any t2t0, one input takes the state x(t0) to x(t2),u[t0,t2]x(t2) = (t2, t0, x(t0), u[t0,t2])

but for any intermediate time t1, t0 t1 t2 , applying a subset of thatu[t0,t1] u[t0,t2]means(1.4.19)u[t0,t2] = u[t0,t1] u[t1,t 2]

we get the intermediate state x(t1)

x(t1) = (t1, t0, x(t0), u[t0,t1])which acting as an initial state from t1, will determine the same x(t2)

(1.4.20)x(t2) = (t2, t0, x(t0), u[t0,t2]) = (t2, t1, (t1, t0, x(t0) , u[t1,t 2]))

x(t1)

According to this property we can say that the input (or u) takes theu[t0,t]system from a state (t0, x0)=x(t0) to a state (t,x)=x(t) and if a state (t1,x1)=x(t1) is

on that trajectory, then the corresponding segment of the input will take the

system from x(t1) to x(t).

4. The causality condition . The state x(t) at any time t or the trajectories

do not depend on the future inputs u() for >t. This condition(t, t0, x0, u[t0,t])assures the causality of the abstract system which has to correspond to the

causality of the original physical oriented system.


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Example 1.4.2. Properties of the i-is-s relation.

Shall we consider the examples Ex. 1.2.2. and Ex. 1.2.3. for which the

abstract system is described by relations (1.2.1) or (1.2.16)

(1.4.21)S1 :

Tx +x = K1u

y = K2xThe voltage across the capacitor x , or the movement of the arm x , is the

state. Its time evolution is

(1.4.22)x(t) = ett0

T x0 +K1

T t0

t

etT u()d x(t) = (t, t0, x0, u[t0,t])

We can show that the relationship (1.4.22) accomplishes the four above

conditions, that means it is an i-is-s relation.

1. The uniqueness condition:

If and , the two state trajectories arex0 = x0 u[t0,t] = u[t0, t]

x (t) = ett0

T x0 +K1

T t0

t

etT u ()d

x (t) = ett0

T x0 +K1

T t0

t

etT u ()d

x (t) x (t) = ett0

T (x0 x0 )x0 +K1

T t0

t

etT [u () u ()]d 0 x (t) x (t)

=0 02. The consistency condition:

Substituting

t = t0 x(t0) = et0t0

T x0 +K1T

t0

t

etT u()d = x0 x(t0) = x0

3. The transition condition:

For t=t1, denoting x0=x(t0) , (1.4.22) becomes

x(t1) = et1t0

T x(t0) +K1T

t0

t1

e

t1

T u()d

and for t=t2 (1.4.22) is

.x(t2) = et2t0

T x(t0) +K1T

t0

t2

e

t2T u()d

Because and , we gete

t2t0T = e

t2t1T e

t1t0T

t0

t2

(..)d = t0

t1

(..)d + t1

t2

(..)d

x(t2) = et2t1

T et1t0

T x(t0) +K1T

t0

t1

et2t1

T et1

T u()d + K1T

t1

t2

et2

T u()d


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x(t2) = et2t1

T [et1t0

T x(t0) +K1T

t0

t1

e

t1

T u()d] + K1T

t1

t2

e

t2

T u()d

x(t1)

.x(t2) = et2t1

T x(t1) + K1T

t1

t2

et2

T u()d

4. The causality condition: Because in (1.4.22) u() is inside the integral from t0to t , x(t) is irrespective of u() >t .-------------------

Before we said, as a general statement, that the input affects the state and

the state influences the output. However there are systems to which inputs do

not influence the state or some components of the state vector.

Conversely there are systems to which outputs or some outputs are notinfluenced by the state. Such systems are called uncontrollable and unobservable

respectively, about which more will be analysed later on.

In Ex. 1.3.4., the black box system, the physical object is state

uncontrollable because no admitted input can make the switch to change its

position. If , for example, the wire to the output were broken then such a system

would be unobservable.

A state that is both uncontrollable and unobservable can not be detected

by any experiment and make no physical meaning.

1.4.3. Trajectories in State Space.

The input-initial state-state (i-is-s) relation

(1.4.23)x(t) = (t, t0, x0, u[t0,t])which expresses the time-trajectory of the state is an explicit function of time.

If the vector x is n-dimensional one there are n time-trajectories, one for

each component

. (1.4.24)xi(t) = i(t, t0, x0, u[t0,t]) , i = 1, ..., nThese time-trajectories can be plotted as t increases from t0, with t as an implicit

parameter, in n+1 dimensional space or as n separate plots xi(t), t t0 , i=1,..,n.Often this plot can be made by eliminating t from the solutions (1.4.24) of thestate equations, which is just a trajectory in state space.

If we denote xi=xi(t) , i=1,..,n , the i-is-s relation (1.4.24) is written as,

x1 = 1(t, t0, x0, u[t0,t])(1.4.25)xi = i(t, t0, x0, u[t0,t])

,xn = n(t, t0, x0, u[t0,t])and eliminating t from the n above relations we determine a trajectory in state

space, implicitly expressed as


29


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F(x1,x2,...,xn,t0,x(t0))=0 (1.4.26)

where it was supposed a given (known) input. Simpler expression are obtained if

the input is constant for any t.

If the state vector components are the output and its (n-1) time derivative

the state space is called phase space and the trajectory in phase space is calledphase trajectory.

The trajectory in state space can easier be obtained directly from state

equations, a system of first order differential equations. The plot can efficiently

be exploited for n=2 in state plane or phase plane.

For a given initial state (t0,x0) , we have denoted x0=x(t0), only one

trajectory is obtained. For different initial conditions a family of trajectories are

obtained called state portrait or phase portrait.

Because of the uniqueness condition accomplished by the i-is-s relation,

for a given input , one and only one trajectory passes through each point inu[t0,t]state space and exists for all finite tt0 . As a consequence of this, the statetrajectories do not cross one another.

Example 1.4.3.State Trajectories of a Second Order System.

Let we consider a simple second order system,

(1.4.27)

x.

1 = 1x1 + ux.

2 = 2x2 + u

x.

1(t) = 1x1(t) + u(t)x.

2(t) = 2x2(t) + u(t)

For simplicity let be u(t)0 .t 0 [t0,t] = {(, u()) = 0, [t0, t]}Under this hypothesis, the i-is-s is obtained by integrating the system (1.4.27),

(1.4.28)

x1(t) = e1(tt0)x1(t0) = 1(t, t0, x1(t0), 0[t0,t])x2(t) = e2(tt0)x2(t0) = 2(t, t0, x2(t0), 0[t0,t])

Supposing that we have 10 then the time-trajectories areplotted through the two components as in Fig. 1.4.3.

Eliminating the variable t from (1.4.28) we obtainx1

x1(t0) =e1(tt0) ;

x2

x2(t0) =e2(tt0)

[

x1

x1(t0)]

1

= [

x2

x2(t0)]

2

. (1.4.29)x2 = x20x1x10

2/1 F(x1, x2, x0) = 0

The same expression for (1.4.29) can be obtained directly from the

differential equation (1.4.27),

(1.4.30)

dx1

dt= 1x1

dx2

dt= 2x2

dx2dx1

= 21x2x1

x2 = x2(t0)x1

x1(t0)

2/1


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a'2a'1

a''1a''2

10x'

10x'' 1x (t)

2t0t 1t t

1


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For t0 = t , taking into consideration that u[t,t]=u(t) it becomes

y(t)=g(x(t), u(t), t) . (1.4.32)

This is an algebraical relation and is called also the output relation or

output equation.

By a dynamical system one can understand a set S of three elementsS = {, , } or S = {, , g} (1.4.33)

which are defined above, where:

the set of admissible inputs the input-initial state-state relation the input-initial state-output relationg the output relationThis is called the explicit form of a dynamical system , expressed by

relations (solutions) or trajectories.

Another form for dynamical system representation is the implicit form bystate equations

S = {, f, g} (1.4.34)whose solutions are the trajectories expressed by (1.4.33), where

f the vector function defining a set of equations: differential, difference,logical, functional.

The solution of the equations defined by f , for given initial state (t 0, x0) is

just the relation .For example, the first order system from Ex. 1.2.2. or Ex. 1.2.3. , can be

represented as:

The explicit form by relations (functions) or state trajectories is:

(1.4.35)S :

u

x(t) = ett0

T x(t0) +K1

T t0t

e t

T d

y(t) = K2x(t)

()()(g)

The implicit form by state equations is:

(1.4.36)S :

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