control system modeling
TRANSCRIPT
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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.