Control System Modeling

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UNIT 1

CONTROL SYSTEM MODELING

1. Introduction

Systems are sets of components, physical or otherwise, which are

connected in such a manner as to form and act as entire units. Control is the

effort to make systems act as desired. A process is the action of a system oralternatively, a system in action.

Humans have created control systems as technical innovations to

enhance the quality and comfort of their lives. Human engineered control

systems are part of automation, which is a feature of our modern life. They are

applied in several aspects of our daily life- in heating and air conditioning tocontrol our living environment and in many of our household appliances. They

significantly relieve us from the burden of operation of complex systems and

processes and enable us to achieve control with desired precision. Control

systems enable accurate positioning and control of machine tools in metal

cutting operations and automate manufacturing processes. They automaticallyguide and control space vehicles, aircraft, large sea going vessels, and high-

speed ground transportation systems. Modern automation of a plant involves

components such as sensors, instruments, computers and application of

techniques of data processing and control. The principles and techniques ofautomatic control may be applied in a wide variety of systems in order toenhance the quality of their performance.

Control systems are not human inventions; they have naturally evolvedin the earths living system. The action of automatic control regulates the

conditions necessary for life in almost all living things. They possess sensing

and controlling systems and counter disturbances. An automatic temperature

control system, for example, makes it possible to maintain the temperature of

the human body constant at the right value despite varying ambient conditions.

The human body is a very sophisticated biochemical processing plant in which

the consumed food is processed and glands automatically release the required

quantities of chemical substances as and when necessary in the process. The

stability of the human body and its ability to move as desired are due to some

very effective motion control systems. A bird in flight, a fish swimming inwater or an animal on the run- all are under the influence of some very

efficient control systems that have evolved in them.

The field of automatic control is very well developed. The established

techniques in this field can be applied to the control of a wide range of systems- engineering systems such as machines and complex plants, natural systems


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such as biological and ecological systems, and non-physical systems such as

economic and sociological systems following the understanding of thesimilarity of the underlying problems.

Understanding a system for its properties is prerequisite to the creationof a control system for it. Before attempting to control a system, it is essentialto know how it generally behaves and responds to external stimuli. Such an

understanding is possible with the help of a model. The process of developinga model is known as modeling.

Physical systems are modeled by applying the phenomenological laws

that govern their behavior. For example, mechanical systems are described by

Newtons laws and electrical systems by Ohms, Faradays and Lenzs laws.These laws form the basis for the constitutive properties of the elements in asystem.

Consider an automobile's cruise control, which is a device designed to

maintain a constant vehicle speed; the desiredor reference speed, provided bythe driver. The system in this case is the vehicle. The system output is the

vehicle speed, and the control variable is the engine's throttle position whichinfluences engine torque output.

OPEN LOOP & CLOSED LOOP:

A primitive way to implement cruise control is simply to lock the

throttle position when the driver engages cruise control. However, on

mountain terrain, the vehicle will slow down going uphill and accelerate going

downhill. In fact, any parameter different than what was assumed at designtime will translate into a proportional error in the output velocity, including

exact mass of the vehicle, wind resistance, and tire pressure. This type of

controller is called an open-loop controller because there is no direct

connection between the output of the system (the vehicle's speed) and theactual conditions encountered; that is to say, the system does not and can not

compensate for unexpected forces.

In a closed-loop control system, a sensor monitors the output (the

vehicle's speed) and feeds the data to a computer which continuously adjusts

the control input (the throttle) as necessary to keep the control error to a

minimum (that is, to maintain the desired speed). Feedback on how the system

is actually performing allows the controller (vehicle's on board computer) to

dynamically compensate for disturbances to the system, such as changes inslope of the ground or wind speed. An ideal feedback control system cancels
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out all errors, effectively mitigating the effects of any forces that may or may

not arise during operation and producing a response in the system thatperfectly matches the user's wishes.

BLOCK DIAGRAM REDUCTION TECHNIQUE

Rule:1

Rule: 2 (Associative and Commutative Properties)

Rule: 3 (Distributive Property)

Rule: 4 (Blocks in Parallel)


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Rule: 5 (Positive Feedback Loop)

Rule: 6 (Negative Feedback loop)


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Equivalent: 1


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Equivalent: 2

Examples of Block Diagram Reduction

SIGNAL-FLOW GRAPH

A signal-flow graph (SFG) is a special type of block diagram[1]

and

directed graphconstrained by rigid mathematical rules, that is a graphical

means of showing the relations among the variables of a set of linear algebraic

relations. Nodes represent variables, and are joined by branches that haveassigned directions (indicated by arrows) and gains. A signal can transmit only

in the direction of the arrow.
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UNIT-2

TIME RESPONSE ANALYSIS

TIME-DOMAIN RESPONSE ANALYSISThe time-domain approach is a unified method for analyzing and designingsystems modeled by either modern or classical approach. Time-domain

response analysis functions below can be used to study the output behavior of

a linear time-invariant system driven by system inputs and initial states.

The step or impulse response plots the system response with respect to time

when the system input is a step or impulse function, respectively. The initial

response plots the system response to the initial state x0 with zero-input. The

simulation response plots the system response to any input and initial statespecified by the user. The initial state of zero is assumed by default.

Function Description

Step response

Plot step response of a system in time domain.

Impulse response

Plot impulse response of a system in time

domain.

Initial response

Plot time domain initial response of a system

represented in state space.

Simulation response Simulate system response to an arbitrary input.

STEADY STATE ERROR

Control systems are used to control some physical variable. That variable

may be a temperature somewhere, the attitude of an aircraft or a frequency in a

communication system. Whatever the variable, it is important to control thevariable accurately.

If you are designing a control system, how accurately the system

performs is important. If it is desired to have the variable under control take

on a particular value, you will want the variable to get as close to the desired

value as possible. Certainly, you will want to measure how accurately you cancontrol the variable. Beyond that you will want to be able to predict howaccurately you can control the variable.

To be able to measure and predict accuracy in a control system, astandard measure of performance is widely used. That measure of
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performance is steady state error - SSE - and steady state error is a concept thatassumes the following:

The system under test is stimulated with some standard input.Typically, the test input is a step function of time, but it can also be aramp or other polynomial kinds of inputs.

The system comes to a steady state, and the difference between the inputand the output is measured.

The difference between the input - the desired response - and the output- the actual response is referred to as the error.

Goals For This Lesson

Given our statements above, it should be clear what you are about in thislesson. Here are your goals.

Given a linear feedback control system,

Be able to compute the SSE for standard inputs, particularly step input

signals.Be able to compute the gain that will produce a prescribed level of SSE in

the system.

Be able to specify the SSE in a system with integral control.

In this lesson, we will examine steady state error - SSE - in closed loopcontrol systems. The closed loop system we will examine is shown below.

The system to be controlled has a transfer function G(s). There is a sensor with a transfer function Ks. There is a controller with a transfer function Kp(s) - which may be a

constant gain.

What Is SSE?

We need a precise definition of SSE if we are going to be able to predict a

value for SSE in a closed loop control system. Next, we'll look at a closedloop system and determine precisely what is meant by SSE.

In this lesson, we will examine steady state error - SSE - in closed loopcontrol systems. The closed loop system we will examine is shown below.


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The system to be controlled has a transfer function G(s). There is a sensor with a transfer function Ks. There is a controller with a transfer function Kp(s).

o A controller like this, where the control effort to the plant isproportional to the error, is called a proportional controller.

In our system, we note the following:

The input is often the desired output. In other words, the input is whatwe want the output to be. If the input is a step, then we want the outputto settle out to that value.

The output is measured with a sensor. Often the gain of the sensor isone. That is especially true in computer controlled systems where the

output value - an analog signal - is converted into a digitalrepresentation, and the processing - to generate the error, E(s) - takesplace inside a computer.

The signal, E(s), is referred to as the error signal.

The error signal is the difference between the desired input and themeasured input.

The error signal is a measure of how well the system is performing atany instant. When the error signal is large, the measured output doesnot match the desired output very well.

A step input is often used as a test input for several reasons.

A step input is really a request for the output to change to a new,constant value.

If the system is well behaved, the output will settle out to a constant,steady state value.

The difference between the measured constant output and the inputconstitutes a steady state error, or SSE.

It helps to get a feel for how things go. So, below we'll examine a system

that has a step input and a steady state error. That system is the same block

diagram we considered above.


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For the example system, the controlled system - often referred to as the plant -is a first order system with a transfer function:

G(s) = Gdc/(st + 1)

We will consider a system with the following parameters.

Gdc = 1 t = 1 Ks = 1. Kp can be set to various values in the range of 0 to 10, The input is always 1.

Here is a simulation you can run to check how this works. In this

simulation, the system being controlled (the plant) and the sensor have theparameters shwon above. You can adjust the gain up or down by 5% using the

"arrow" buttons at bottom right. You can also enter your own gain in the text

box, then click the red button to see the response for the gain you enter. Theactual open loop gain is shown in the text box above the red button.

UNIT3

FREQUENCY RESPONSE ANALYSIS

BODE PLOT

A Bode plot is a graph of the logarithm of the transfer function of a linear,

time-invariant system versus frequency, plotted with a log-frequency axis, to

show the system's frequency response. It is usually a combination of a Bode

magnitude plot (usually expressed as dB ofgain) and a Bode phase plot (thephase is the imaginary part of the complex logarithm of the complex transferfunction).

The magnitude axis of the Bode plot is usually expressed as decibels ofpower,

that is by the 20 log rule: 20 times the common logarithm of the amplitudegain. With the magnitude gain being logarithmic, Bode plots make
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multiplication of magnitudes a simple matter of adding distances on the graph(in decibels), since

A Bode phase plot is a graph of phase versus frequency, also plotted on a log-

frequency axis, usually used in conjunction with the magnitude plot, toevaluate how much a signal will be phase-shifted. For example a signaldescribed by:Asin(t) may be attenuated but also phase-shifted. If the system

attenuates it by a factorxand phase shifts it by the signal out of the system

will be (A/x) sin(t). The phase shift is generally a function of

frequency.

Phase can also be added directly from the graphical values, a fact that is

mathematically clear when phase is seen as the imaginary part of the complexlogarithm of a complex gain.

In Figure 1(a), the Bode plots are shown for the one-pole highpass filterfunction:

wherefis the frequency in Hz, and f1 is the pole position in Hz,f1 = 100 Hz in

the figure. Using the rules for complex numbers, the magnitude of thisfunction is

while the phase is:

Care must be taken that the inverse tangent is set up to return degrees, not

radians. On the Bode magnitude plot, decibels are used, and the plottedmagnitude is:

In Figure 1(b), the Bode plots are shown for the one-pole lowpass filter

function:

Also shown in Figure 1(a) and 1(b) are the straight-line approximations to theBode plots that are used in hand analysis, and described later.

The magnitude and phase Bode plots can seldom be changed independently of

each otherchanging the amplitude response of the system will most likelychange the phase characteristics and vice versa. For minimum-phase systems

the phase and amplitude characteristics can be obtained from each other withthe use of the Hilbert transform.

If the transfer function is a rational function with real poles and zeros, then the

Bode plot can be approximated with straight lines. These asymptotic
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approximations are called straight line Bode plots or uncorrected Bode plots

and are useful because they can be drawn by hand following a few simplerules. Simple plots can even be predicted without drawing them.

The approximation can be taken further by correcting the value at each cutofffrequency. The plot is then called a corrected Bode plot.

POLAR PLOT

The polar plot is created using the Radar chart.

NYQUIST PLOT

Polar graphsThere are a number of polar graph options for studying control systems

including the nyquist, inverse polar plot and the nichols plot. The nyquist

open loop polar plot indicates the degree of stability, and the adjustments

required and provides stability information for systems containing timedelays. Polar plots are not used exclusively because,without powerful

computing facilities, they can be difficult to generate at a detailed level and

they do not directly yield frequency values.

The Nyquist plot is obtained by simply plotting a locus of imaginary(G(j ))

versus Real(G(j )) at the full range of frequencies from ( - to + ) It isvery easy to produce nyquist plots by hand or by using proprietary software


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packages such as Matlab. Links below show how bode and nyquist plots can

be produced using Excel and using Mathcad. The plots below have been

produced in minutes using Mathcad..

The nyquist plot fundamentals are shown below....

Velocity Feedback control

Basic Rules for constructing Nyquist plots

In control systems a transfer function to be assessed is often of the form

This transfer function is modified for frequency response analysis by replacingthe s with j


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Assuming the function is proper and n > m..he Nyquist plot will have the

following characteristics. Crude plots to be may be produced relatively easily

using these characteristics.

1.

Asymptotic behavior.. For n - m > 1, the Nyquist plot approaches theorigin at an asymptotic angle of -(n-m) /2...

2. Assuming G(s) = K(s)/s k. For k poles at zero, the Nyquist plot comes infrom infinity at an angle of -(n-m) /2

3. In a system with no OL zeros, the plot of G(j) will decreasemonotonically as rises above the level of the largest imaginary part ofthe poles; This will also be true for large enough even in the presence

of zeros.(Ref plot 1 below).

4. The plot will cross the imaginary axis when Real G(j) =0 and willcross the real axis when Imaginary G(j) = 0, ( for crossing of thenegative real axis use Arg- G(j)= )

Relative stability assessments Using the Nyquist Plot

As identified in the page on frequency response Frequency response The

nyquist plots are based on using open loop performance to test for closed loop

stability. The system will be unstable if the locus has unity value at a phase

crossover of 180o

( ).

Two relative stability indicators "Gain Margin" and "Phase Margin" may be

determined from the suitable Nyquist Plots. The degree of gain margin is

indicated as the amount the gain is less than unity when the plot crosses the

180o

axis (Phase crossover). The phase margin is the angle the phase is less

than 180o

when the gain is unity. The values are generally identified by use of

Bode plots

The phase margin is clearly illustrated below
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NyquistStabilityCriterion:In the nyquist plots below the area covered to the right of the locus(shaded) is

the Right Hand Plane (RHP)

A closed loop control system is absolutely stable if the roots of thecharacteristic equation have negative real parts. This means the poles of the

closed loop transfer function, or the zeros of the denominatior ( 1 + GH(s)) ofthe closed loop tranfer function, must lie in the (LHP). The nyquist stability

criterion establishes the number of zeros of (1 + GH(s) in the RHP directly

from the Nyquist stability plot of GH(s) as indicated below.

The closed loop control system whose open loop transfer function is GH(s) is

stable only if..

N = -Po 0


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where

1) P o = the number of G(s) poles in the RHP 0

2) N = total number of CW encirclements of the (-1,0) in the G(s) plane.

If N > 0 the number of zeros (Z o) in the RHP is determined by Z o = N + P o

If N 0 the (-1,0) point is not enclosed by the nyquist plot.If N and P 0 then the system is absolutely stable only if N = 0. That is if andonly if the (-1,0) point does not lie in the shaded region..

Considering the LH plot above of 1/s(s+1). The (-1,0) point is not in the RHP

therefore N 0 (N= 1). The poles of GH are at s= 0 and s = +1 . S=

+1 is in the RHP and therefore P o = 1.N - P o Indicating that this system is unstable..

There are Z o = N + P o zeros of 1+GH in the RHP.

Nyquist Plots A number of typical nyquist plots are shown below to illustrate

the various shapes.

Plot 1..... 1 /(s + 2)

1 /(s + 2)


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Note that G(i0) = 0.5 and as increases to the plot approaches zero along

the negative locus.G(j) moves from 0 to 0.5 as - to 0

G(j ) = 0

The asymptotic angle approaching 0 is = -90o

and

Plot 2.....1 /(s2

+ 2s + 2)

1 /(s2

+ 2s + 2)

The zero-frequency behaviour is:G(j0) =0.5

G(j ) = 0The asymptotic angle is = -180

o

Plot 3.....s(s+1) /(s3

+ 5.s2

+3.s + 4 )

s(s+1) /(s3

+ 5.s2

+3.s +4 )

Plot 4.....(s+1) /[(s+2)(s+3)]


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((s+1) /[(s+2)(s+3)]

Plot 5.....1 /s(s-1).. an unstable regime

UNIT 4

STABILITY ANALYSIS

Routh-Hurwitz Criterion

The RouthHurwitz stability criterion is a necessary (and frequently

sufficient) method to establish the stability of a single-input, single-output(SISO), linear time invariant (LTI) control system. More generally, given a

polynomial, some calculations using only the coefficients of that polynomial

can lead to the conclusion that it is not stable. For the discrete case, see theJury test equivalent.

The criterion establishes a systematic way to show that the linearizedequations of motion of a system have only stable solutions exp(pt), that is
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where all p have negative real parts. It can be performed using eitherpolynomial divisions or determinant calculus.

Using Euclid's algorithm

The criterion is related to RouthHurwitz theorem. Indeed, from the

statement of that theorem, we have where:

p is the number of roots of the polynomial f(z) located in the left half-plane;

q is the number of roots of the polynomial f(z) located in the right half-plane (let us remind ourselves that fis supposed to have no roots lying

on the imaginary line);

w(x) is the number of variations of the generalized Sturm chain obtainedfrom P0(y) and P1(y) (by successive Euclidean divisions) where f(iy) =P0(y) + iP1(y) for a realy.

By the fundamental theorem of algebra, each polynomial of degree n must

have n roots in the complex plane (i.e., for anfwith no roots on the imaginary

line, p+q=n). Thus, we have the condition that f is a (Hurwitz) stablepolynomial if and only ifp-q=n (the proofis given below). Using the Routh

Hurwitz theorem, we can replace the condition on p and q by a condition on

the generalized Sturm chain, which will give in turn a condition on the

coefficients off.

Using matrices

Letf(z) be a complex polynomial. The process is as follows:

1. Compute the polynomials P0(y) and P1(y) such thatf(iy) = P0(y) + iP1(y)wherey is a real number.

2. Compute the Sylvester matrix associated to P0(y) and P1(y).3. Rearrange each row in such a way that an odd row and the following

one have the same number of leading zeros.4. Compute each principal minor of that matrix.5. If at least one of the minors is negative (or zero), then the polynomial f

is not stable.

ROOT LOCUS TECHNIQUES:

Introduction

A root loci plot is simply a plot of the s zero values and the s poles on a

graph with real and imaginary coordinates. The root locus is a curve of the
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location of the poles of a transfer function as some parameter (generally the

gain K)isvaried.The locus of the roots of the characteristic equation of the

closed loop system as the gain varies from zero to infinity gives the name of

the method. Such a plot shows clearly the contribution of each open loop pole

or zero to the locations of the closed loop poles.

This method is very powerful graphical technique for investigating the

effects of the variation of a system parameter on the locations of the closed

loop poles. General rules for constructing the root locus exist and if the

designer follows them, sketching of the root loci becomes a simple matter.

The closed loop poles are the roots of the characteristic equation of the system.From the design viewpoint, in some systems simple gain adjustment can

move the closed loop poles to the desired locations. Root loci are completed

to select the best parameter value for stability.

A normal interpretation of improving stability is when the real part of a

pole is further left of the imaginary axis.

Open and Closed Loop Transfer Functions

A control system is often developed into an equation as shown below

D(s) = (s -p 1).(s -p 2).. (s -p n) is the characteristic equation for the system...

Normally n > m.

F(s) = 0 when s = z 1,z 2... z n..These values of s are called zeros

F(s) = infinity when s = p 1, p 2....p n...These values of s are called poles..Below is shown a root loci plot with a zero of -2 and a pole of (-2 + 2 j ). In

practice one complex pole /zero always comes with a second one mirrored

around the real axis


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The Transfer function F(s) can also be written in polar form using

vectors(modulus-argument.

The complex numbers in polar form have the following elementaryproperties..

|z 1 .z 2 |= |z 1 |.|z 2 |..&..|z 1 / z 2 | = |z 1 | / |z 2 |

A typical feedback system is shown below

The open-loop transfer function between the forcing input R(s) and themeasured output Y1(s) =

T1(s) = K.G(s)H(s)

The closed-loop transfer function =

K is the value of the open loop gain>

1+ KG(s)H(s) is the characteristic equation

A closed loop pole (when T(s) = infinity) must satisfy


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K.G(s).H(s) = -1

This is can be interpreted using vectors

If G(s) = Q(s)/P(s) and H(s) = W(s)/V(s) Then the characteristic

equation for the open loop transfer function =

P(s).V(s)= 0

The corresponding characteristic equation for the closed loop system =

P(s).V(s) + K.Q(s)W(s) = 0

Unit 5

Sampled Data control systems

A type of digital control system in which one or more of the input oroutput signals is a continuous, or analog, signal that has been sampled. There

are two aspects of a sampled signal: sampling in time and quantization in

amplitude. Sampling refers to the process of converting an analog signal from

a continuously valued range of amplitude values to one of a finite set ofpossible numerical values. This sampling typically occurs at a regular

sampling rate, but for some applications the sampling may be aperiodic or

random.

While the device to be controlled is usually referred to as the plant,sampled-data control systems are also used to control processes. The term

plant refers to machines or mechanical devices which can usually be

mathematically modeled by an analysis of their kinematics, such as a roboticarm or an engine. A process refers to a system of operations such as a batch

reactor for the production of a particular chemical, or the operation of a

nation's economy. The output of the plant which is to be controlled is called

the controlled variable. A regulator is one type of sampled-data control system,

and its purpose is to maintain the controlled variable at a preset value (forexample, the robotic arm at a particular position, or an airplane turboprop

engine at a constant speed) or the process at a constant value (for example, the
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concentration of an acid, or the inflation rate of an economy). This input iscalled the reference or setpoint.

The second type of sampled-data control system is a servomechanism,

whose purpose is to make the controlled variable follow an input variable.Examples of servomechanisms are a robotic arm used to paint automobileswhich may be required to move through a predefined path in three-

dimensional space while holding the sprayer at varying angles, an automobile

engine which is expected to follow the input commands of the driver, a

chemical process that may require the pH of a batch process to change at a

specified rate, and an economy's growth rate which is to be changed byaltering the money supply. See also Analog-to-digital converter; Processcontrol; Regulator; Servomechanism.

The analog-to-digital converter changes the sampled signal into a binarynumber so that it can be used in calculations by the digital compensator. Since

a digital controller computes the control signal used to drive the plant, a

digital-to-analog converter must be used to change this binary number to an

analog voltage. The digital compensator in the typical sampled-data controlsystem takes the digitized values of the analog feedback signals and combines

them with the setpoint or desired trajectory signals to compute a digital control

signal, to actuate the plant through the digital-to-analog converter. A

compensator is used to modify the feedback signals in such a way that the

dynamic performance of the plant is improved relative to some performanceindex. See alsoControl systems; Digital computer; Digital control; Digital-to-analog converter.

CONTROLLABILITY

A system with internal state vectorx is called controllable if and only if

the system states can be changed by changing the system input.

A state x0 is controllable at time t0 if for some finite time t1 there exists

an input u(t) that transfers the state x(t) from x0 to the origin at time t1.

A system is called controllable at time t0 if every state x0 in the state-space is controllable.
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OBSERVABILITY

The state-variables of a system might not be able to be measured for any of

the following reasons:

1. The location of the particular state variable might not be physicallyaccessible (a capacitor or a spring, for instance).

2. There are no appropriate instruments to measure the state variable, orthe state-variable might be measured in units for which there does notexist any measurement device.

3. The state-variable is a derived "dummy" variable that has no physicalmeaning.

If things cannot be directly observed, for any of the reasons above, it can be

necessary to calculate or estimate the values of the internal state variables,using only the input/output relation of the system, and the output history of the

system from the starting time. In other words, we must ask whether or not it ispossible to determine what the inside of the system (the internal system states)

is like, by only observing the outside performance of the system (input and

output)? We can provide the following formal definition of mathematical

observability:

OBSERVABILITY

A system with an initial state,x(t0) is observable if and only if the valueof the initial state can be determined from the system output y(t) that has beenobserved through the time interval t0 < t< tf. If the initial state cannot be so

determined, the system is unobservable.

Complete Observability

A system is said to be completely observable if all the possible initial

states of the system can be observed. Systems that fail this criteria are said to

be unobservable.

A system state xi is unobservable at a given time ti if the zero-input

response of the system is zero for all time t. If a system is observable, then the

only state that produces a zero output for all time is the zero state. We can usethis concept to define the term state-observability.

STATE-OBSERVABILITY

A system is completely state-observable at time t0 or the pair (A, C) is

observable at t0 if the only state that is unobservable at t0 is the zero state x = 0.

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