Control_Theory_Unit_1_31_October_2012.docx

7/27/2019 Control_Theory_Unit_1_31_October_2012.docx

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CONTROL THEORYAPPLICATION TO FLIGHT CONTROL SYSTEMS

B. Tech., Aeronautical Engineering

Fourth Year, First Semester

JNTUH

2009-2013 to 2012-2016

UNIT I

CONTROL SYSTEMSMODELING,

PERFORMANCE TIME, FREQUENCY AND S-DOMAIN DESCRIPTION

Note:

This unit is repeated four times citing examples from two text books Kuo and two editions of Ogata, three

different prints -2000, 2005, 2007.

Depending on the book you follow, you can choose the particular part, but cross references have been

inevitable.

Letter refers to Author, first number refers to Section and the next number refers to page no.

(The number occasionally refers to Example no.)

YYechout, S&LStevens & Lewis, O3- OgataThird edition,

O4(5)OgataFourth edition, 2005, O4(7)OgataFourth edition, 2007

NNelson KuoB.C. Kuo[More details at the end]

Part1 of 4.

For the students who are primarily referring to B.C. Kuo:

Dynamic systems, principal constituents, input, output, process: Kuo Sec 4-3, pp138-145,

S1-1-2, pp 2-8,

Block diagram representation: S 1-1-3, 1-1-4, 1-2, 1-2-1, 1-2-2 pp 9-13

Inputs

control input, noise: S1-2-4, 1-2-4, pp 14-15Also read Section 1-3, 1-3-1,1-3-2 pp16-19

Discrete data system on page 17 pertains to Digital control system.

[Read S 3-11, pp 106-117 before reading digital systems in Unit VIII]

Tracking Performance: Section 10-5O3, pp 686

Sensitivity:- - Sensitivity of the gain of the overall system is defined in Sec 1-2-3, pp 13-14


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Function of control as regulation (hold), tracking (command)

[See notes of Unit V, reproduced below]

Autopilotspurpose, functioning Y - 9.2 - 444

Inputsfour cases are discussed.Holdkeeping the output constant,

commandinput, (Command here may be interpreted as a particular value)

trackfollow a given input profile

Pitch altitude hold - Y9.2.1446

Altitude hold - flying at a constant height Y9.2.2447

Bank angle holdkeeping roll angle constant Y9.2.3449

Heading holdmaintaining a constant yaw and the required bank angleY 9.2.4450

[The above topics are given from page 327 onwards in S&L]

Velocity holdS&Lpage 334-335

The Mach-hold autopilot is chiefly used on commercial passenger jets during climb and

descent. During a climb the throttles may be set at a fairly high power lever, and feedback of

Mach number to the elevator will be used to achieve a constant Mach climb(Ex 4.6-3 on

page 335 of S&L)Figure on page 335 of Ref S&L or fig 9.25 on page 448 of Y may be

modified to show velocity hold Block Diagram, by replacing h by v. Aircraft Transfer

function may be replaced by a general TF and sensor is for measuring v.

Aircraft Transfer functions are given in Block diagrams given in Ref Y.

Maneuver autopilotY9.2444n is load factor coming on aircraft during manoeuvre.

Pitch rate feedbackY Ex 9.5page 459

Block diagrams of Control Systems: S3-3-1, pp 84 -88

Gain formula: S 3-9, pp99-100.

Compare with Rules for Block diagram algebra in Ogata Section 3-3, page 68.

Control in every day life, pervasiveness of control in nature, engineering and societal

system, the importance of study of control system etc. - Examples in initial pages of all

control system books refer to these.

Robust control: S10-10, pp778-779Robustness is insensitivity to external disturbance and

parameter variation. Also, insensitivity to noise.

Modelling of dynamical systems by differential equations: See spring-mass system etc.,

on page 16 (of the first set) of Notes of Prof. G.V.S.S. Sastry, Wg. Cdr. (Rtd.).

KuoSection 4, pp 134- 160. Ex 4-4, pp151-154, explains spring-mass model. Ex 4-6,

pp155-158, explains a two-spring system.

Stochastic control: See Page 8 (of the second set) of Notes of Prof. G.V.S.S. Sastry, Wg.

Cdr. (Rtd.).


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Linear time-invariant systems and linear time-varying systems:

[See Ogata, S3-1, pp 58-59 of O3.]

A differential equation is linear if the coefficients are constants or functions only of the

independent variable. If the coefficients are constants, then they are time-invariant. A morerigorous definition is given below. Dynamic systems that are composed of linear time-

invariant lumped-parameter components may be described by linear time-invariant (constant

coefficient) differential equations. Such systems are called linear time-invariant (or linear

constant-coefficient) systems.

Systems that are represented by differential equations whose coefficients are functions of

time are called linear time-varying systems.

Linear and non-linear systems:

In linear systems, the coefficients of derivatives like

are constants or functions

only of the independent variable. Also, functions like sin or cos would not be present.

Non-linear differential equations are those whose solutions do not obey the law of

superposition.

Definition: A linear system is a system which has the property that if:

(a) an input x1(t) produces an output y1(t), and

(b) an input x2(t) produces an output y2(t), then

(c) an input c1 x1(t) + c2 x2(t) produces an output c1 y1(t) + c2 y2(t) for all pairs of inputs x1(t)

and x2(t) and all pairs of constants c1 and c2.

First order, Second order and higher order systems:

Second order differential equation is of the form

,

while first order differential equation is of the form

See Ogata, S4-2, pp 136-139 of O3.

Second order system: Earlier referred - KuoSection 4, pp 134-160. S4-4, pp151-15,

explaining spring-mass model, and Ex 4-6, pp155-158, explaining a two spring systemare

second order systems.Higher order systemsUnit 6, 7you study nth order systems. The differential equation

contains the terms like. See Eq 3-3, p 79. KuoSection 2-3, pp25-28.

SISOSingle input single output systems

MIMOMultiple input multiple output systemsS 3-2-3, p81referred to as multi-

variable system. Fig 3-5 - Block diagram on p87.


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Linearization of non-linear systemsCertain approximations have to be made to make the

coefficients constants at for a limited interval.

Time-invariant linear systemsThe differential equation is linear and the coefficients are

constants and not functions of the independent variable or time.

Control System performancetime domain description output response to control

inputs: As referred to earlier, KuoSection 4, pp 134- 160. Ex 4-4, pp151-154, explains

spring-mass model. The displacement as a function of time is given. Analysis without

referring to Laplace Transforms is given in OgataEx 3-2, pp73-74. Here it is solved in

state-space model. In the class it was solved as a solution of the second order differential

equation. All of these are examples of time domain description, where system variables are

obtained as functions of time, when acted on by an external force. The external force may be

impulse, step or ramp (or any other function).

Characteristic parameters, relation to system parametersexplained in the examples shown

aboveSee the response curve given on page 29 of the first set of notes given by Prof. Sastry

overshoot, steady state error, time constant, rise time, tmax etc.

See Kuo S7-4, pp385-389.

Indicial functions:

The time response in lift and moment to step changes in angle of attack and pitch rate are

termed indicial functions. [NACA R 1188, by Murray Tobak].

Transient Response of a second order system: Kuo, S7-5 pp387-388

Review Fourier transformsSee your old notes of Mathematics.

Laplace Transforms: Kuo S2-4, 2-4-2 pp28-30.

Application to differential equations: [] () () [

]

() ()()

[

] () ()() , where, ()

() See tables in Ogata or any other text book.

Laplace Transform: Kuo S2-6, pp 41-43

Transfer function: Kuo, S3-2, S3.3 pp78-88

Terminology on page 85

GH is loop transfer function. Ogata and many others call it Open-loop transfer function

OLTF

M is used only by some authors. is CLTFclosed loop transfer function.Read Kuo, S3-10-1, 3-10-2, 3-10-3, pp 102-105 to arrive at state and output equations

mentioned in Unit VI.

Kuo - Section 3-11 for Digital system of Unit VIII.


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Poles and zeros are defined in Kuo S2-2-4, 2-2-5, pp24-25. See Ex 2-6 on page 38.

Characteristic equation: Kuo S3-2-2, page 80. 1+GH =0 is the characteristic equation of a

closed loop control system with negative feedback.

Kuo, S3-2-2, page 80: The characteristic equation of a linear system is defined as the equation

obtained by setting the denominator polynomial of the transfer function to zero.

Frequency and damping ratio of dominant poles: The lowest frequency pole is called

the dominant pole because it dominates the effect of all of the higher frequency poles [from

Wikipedia]

Relation of transfer function to impulse response: I guess this means system response for

impulse input.

Measures of Performance (MOP): measures derived from the dimensional parameters (bothphysical and structural) and measure attributes of system behavior. MOPs quantify the set of

selected parameters. Examples include sensor detection probability, sensor probability of

false alarm, and probability of correct identification.

Measures of Effectiveness (MOE): measure of how a system performs its functions within its

environment. An MOE is generally an aggregation of MOPs. Examples include

survivability, probability of raid annihilation, and weapon system effectiveness.

[Download from InternetEstablishing System Measures of Effectiveness by John M.

Green.]

Partial Fraction decomposition of transfer function: Kuo S 2-5, pp 34-37

Frequency domain description:

Kuo, Appendix A, pp864-

Eq A-1 Magnitude of G(j) and phase of G(j) are defined. Please note this is not G(s), but

the specific case, where the real part is zero.

Polar plotA plot of the magnitude versus phase in the polar coordinates, as varies fro m

zero to infinity.Bode plotA plot of the magnitude in decibels |()| versus in semi-logcoordinates

Section A1Polar plot, Section A2 Bode Plot.

The polar plot is often called Nyquist Plot.

Polar plot is for any G. Nyquist plot is for GH, [loop transfer function or OLTF]

Frequency Transfer function: I suspect it refers to G(j)

Corner frequencies, resonant frequencies, peak gainPlease study Bode plots.Resonant frequency is the frequency at which the peak resonance M, (Mr),


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occurs. [Kuo, S9-1-2, p 544.] Some books use Mp as the symbol.

The resonant frequency is the frequency at which the magnitude ofM(j) is the highest.

In general the Mrshould be between 1.1 and 1.5.

Characteristics of a system are stability, bandwidth, overall gain, disturbance andsensitivity.

Bandwidth: Bandwidth BW is the frequency at which |()| drops to 70.7 percent or 3 dBdown from its zero-frequency value. {Kuo}

In general, the bandwidth of a control system gives an indication of the transient response

properties in the time domain. Kuo, p544.

The frequency range , in which the magnitude of the closed loop does not dropminus3dB is called the bandwidth of the system. {Ogata}. The bandwidth indicates the

frequency where the gain starts to fall off from its low-frequency value. Thus, the bandwidth

indicates how well the system will track an input sinusoid. The rise time and the bandwidth

are inversely proportional to each other.

Another definition: BW is that range of frequencies of the input over which the system will

respond satisfactorily.

Second order system is dealt with in Kuo, S 9-2, pp544- 549.

First order system is given in Ogata S 4-2, pp136-139 of O3.

END OF Part 1.-.-.-.-.-.-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-..-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-.-

Part 2 of 4.

For students who are primarily referring to Ogata, Third Edition:

Dynamic systems, principal constituents, input, output, process: SeeOgata, S1-2, pp 2-6

of Third edition [O(3)].

Block diagram representation: Ogata S3-3, pp63-67 of O3Rules for block diagram algebrapage 68 of O3

Inputscontrol input, noise: Control input is r(t) and R(s), NoiseBlock diagrams in

Ogata, Section 10-3, pp 680-681 and 684-685 of O3 have noise as additional input N(s). This

is usually measurement error of the output (referred to as sensor error).

Discrete data system on {page 17 of Kuo} pertains to Digital control system.

[Read Kuo. S 3-11, pp 106-117 before reading digital systems in Unit VIII]

Tracking Performance: Ogata, Section 10-5see page 686 and then page 685 of O3


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()(), where Y(s) and R(s) are output and input respectively. Tracking error is thedifference between the desired output and achieved output. To keep it small, must be close to unity over a wide range of frequency.

Sensitivity: (i) - Sensitivity of the gain of the overall system is defined in Kuo, Sec 1-2-3,pp 13-14

Ogata Section 10-5pp 686-687 of O3

(ii) Sddegree of disturbance rejection

(iii) Sensitivity to modeling errors




Inputsfour cases are discussed.

Holdkeeping the output constant,





Bank angle holdkeeping roll angle constant Y9.2.3449Heading holdmaintaining a constant yaw and the required bank angleY 9.2.4450





Mach number to the elevator will be used to achieve a constant Mach climb (Ex 4.6-3 on







Block diagrams of Control Systems: Ogata, Examples 3-1, 3-2, pp 69-70 and pp73-75 of

O3. [These are in addition to those cited earlier.]





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Robust control: Ogata, S10-5, pp 685-686 of O3Robustness is insensitivity to external

disturbance and parameter variation. Also, insensitivity to noise.



Ogata, Ex 3-2, pp 73-74 of O3explains spring-mass model. Example in the sub-section

Mechanical System, pp 83-85 of O3 explains a slightly different model, where the spring-

mass system is on a cart.

If you can read the electrical examples in the next few pages, it is good. But, before that read

page 67 of O3, procedures for drawing block diagrams.


Cdr. (Rtd.).


Ogata, Section 3-1, pp 58-59 of O3.


independent variable. It is time-invariant, if the independent variable is time and the

coefficients are constant. A more rigorous definition for a linear system is given below.

Dynamic systems that are composed of linear time-invariant lumped-parameter components

may be described by linear time-invariant (constant coefficient) differential equations. Such

systems are called linear time-invariant (or linear constant-coefficient) systems.Systems that are represented by differential equations whose coefficients are functions of






superposition.

Definition: A linear system is a system which has the property that if:


(b) an input x2(t) produces an output y2(t), then(c) an input c1 x1(t) + c2 x2(t) produces an output c1 y1(t) + c2 y2(t) for all pairs of inputs x1(t)




,


See Ogata, S4-2, page 136-139 of O3.


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Second order system: Earlier referred example explaining spring-mass model is a second

order system.

Higher order systemsUnit 6, 7you study nth order systems. The differential equation


See Ogata, S4-3, pp 150-154 of O3. If you can study Section 4-3, pp 141-150 of O3, it willbe helpful.


MIMOMultiple input multiple output systemsKuo, S 3-2-3, p81referred to as multi-

variable system. Kuo, Fig 3-5 - Block diagram on p87. Ogata, Section 3-4, page 70 of O3.


coefficients constants at for a limited interval. Ogata, S3-10, pp 100-102 of O3.


constants and not functions of the independent variable, time.



spring-mass model. The displacement as a function of time is given.

Analysis without referring to Laplace Transforms is given in OgataEx 3-2, pp73-74 of O3.

Here it is solved in state-space model. In the class it was solved as a solution of the second

order differential equation. All of these are examples of time domain description, where

system variables are obtained as functions of time, when acted on by an external force. The

external force may be impulse, step or ramp (or any other function).

Characteristic parameters, relation to system parametersexplained in the examples referred

to aboveSee the response curve given on page 29 of the first set of notes given by Prof.

G.V.S.S. Sastryovershoot, steady state error, time constant, rise time, tmax etc.

Also, see S4-3, Sub-section Definitions of transient response specifications, page 150 of

O3.

Indicial function:The time response in lift and moment to step changes in angle of attack and pitch rate are


Transient Response of a second order system: Ogata, Section 4-3, page 150 of O3.


Laplace Transforms: Application to differential equations: [] () ()

[ ] () ()()


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[

] () ()() , where, ()

()

See tables in Ogata or any other text book.

Laplace Transform: Ogata, S 2-3, pp 17- 23 of O3.

Transfer function: S3-2, S3-3, page 60 and pp 65-66 of O3

Terminology is given clearly by Kuo, S3-3-1, page 85


OLTF




Poles and zeros: Ogata, S2-2, page 16 of O3Rigorous definition is given in the text.

[Hint: Roots of the numerator equated to zero are zeros, and roots of the denominator equated

to zero are poles, but definition in the text has to be used.]

Characteristic equation: The equation 1+GH =0 is the characteristic equation of a closed

loop control system with negative feedback.

In the state-space system, | | is the characteristic polynomial of G(s). See Ogata, S3-4, at theend of the Sub-section Correlation between transfer function and state-space equations, page 75 of

O3.

Kuo, S3-2-2, page 80: The characteristic equation of a linear system is defined as the equationobtained by setting the denominator polynomial of the transfer function to zero.



Wikipedia]


impulse input.

Measures of Performance (MOP): measures derived from the dimensional parameters (bothphysical and structural) and measure attributes of system behavior. MOPs quantify the set of







Green.]


11/23

Partial Fraction decomposition of transfer function: Ogata, S2-5, pp37-38 of O3.


Ogata, S8-1, Subsection - Steady state output to sinusoidal inputs (Pp 471-472 of O3:Magnitude of G(j) is given as amplitude ratio, and phase of G(j) is defined as the phase

shift of the output sinusoid with respect to the input sinusoid. Please note this is not G(s), but

the specific case, where the real part is zero. Some call this simply angle of G(j).

Polar plotA plot of the magnitude versus phase in the polar coordinates, as varies from

zero to infinity. Ogata, S8-4, pp 504-505 of O3.

Bode plotA plot of the magnitude in decibels |()| versus in semi-logcoordinates. Bode diagrams are explained in Ogata, S8-2, pp 473-483 of O3. The General

procedure for plotting Bode diagrams is given at the end of the section, pp 483-484 of O3.

Bodepronounced Bodee, Boday (and as per internet, Bodah, probably in his mother

tongue.]




Corner frequencies, resonant frequencies, peak gainPlease study Bode plots.Resonant frequency is the frequency at which the peak resonance M, (Mr),

occurs. [Kuo, S9-1-2, p 544. Ogata, S8-2, page 483 of O3] Some books use Mp as the

symbol.



Characteristics of a system are stability, bandwidth, overall gain, disturbance and

sensitivity.



properties in the time domain. [See Kuo, p 544, Ogata, S8-9, pp 554-555 of O3]






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Second order system is dealt with in Ogata S 4-2, pp136-139 of O3.

First order system is given in Ogata, S4-3, pp 150-154 of O3.

End of Part 2.

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Part 3 of 4.

For students who are primarily referring to Ogata, Fourth Edition printed

in 2005 [O4(5)]:

Dynamic systems, principal constituents, input, output, process: Ogata, S1-2, pp 2-6 of

O4(5).

Block diagram representation: Ogata S3-3, pp 58-63 of O4(5)

Rules for block diagram algebrapage 68 of O3 {not seen in O4(5)}


Ogata, Section 10-4, pp 701-705 of O4(5) have noise as additional input N(s). This is usually

measurement error of the output (referred to as sensor error).



Tracking Performance: Ogata, Section 10-5see page 686 and then S10-4, page 685 of O3

{I could not see the relevant part in O4}


Sensitivity: (i)

- Sensitivity of the gain of the overall system is defined in Kuo, Sec 1-2-3,pp 13-14






Autopilots

purpose, functioning Y - 9.2 - 444


13/23





Pitch altitude hold - Y9.2.1446Altitude hold - flying at a constant height Y9.2.2447







Mach number to the elevator will be used to achieve a constant Mach climb (Ex 4.6 -3 on







Block diagrams of Control Systems: Ogata, Examples 3-2, 3-3, pp 69-70 and pp73-75 of

O4(5). [These are in addition to those cited earlier.]

Control in every day life, pervasiveness of control in nature, engineering and societalsystem, the importance of study of control system etc. - Examples in initial pages of all




[Please see Ogata, Third Edition]



Ogata, Ex 3-3, pp 73-74 of O4(5) explains spring-mass model. Example in the sub-section

Mechanical Systems, Ex. 3-8 pp 81-82 of O4(5) explains a slightly different model, where

the spring-mass` system is on a cart.




Cdr. (Rtd.).



14/23

Ogata, Section 3-1, page 54 of O4(5).



coefficients are constant. A more rigorous definition for a linear system is given below.Dynamic systems that are composed of linear time-invariant lumped-parameter components


systems are called linear time-invariant (or linear constant-coefficient) systems.







superposition.Definition: A linear system is a system which has the property that if:







,

while first order differential equation is of the form See Ogata, S5-2, page 221-224 of O4(5).


order system.



See Ogata, S5-3, pp 230-231 of O4(5). If you can study Section 5-3, pp224-230 of O4(5), it

will be helpful.



variable system. Kuo, Fig 3-5 - Block diagram on p87. Ogata, Section 3-4, page 70, O4(5).


coefficients constants at for a limited interval. Ogata, S3-10, pp 112-114 of O4(5).




15/23




Analysis without referring to Laplace Transforms is given in OgataEx 3-3, pp73-74 of

O4(5). Here it is solved in state-space model. In the class it was solved as a solution of thesecond order differential equation. All of these are examples of time domain description,

where system variables are obtained as functions of time, when acted on by an external force.

The external force may be impulse, step or ramp (or any other function).




Also, see S5-3, Sub-section Definitions of transient response specifications, pp 229-231 of

O4(5).

Indicial function:



Transient Response of a second order system: Ogata, Section 5-3, page 230 of O4(5).


Laplace Transforms: Application to differential equations: [] () () [

]

() ()()

[

] () ()() , where, ()

()


Laplace Transform: Ogata, S 2-3, pp 13-20 of O4(5).

Transfer function: S3-2, S3-3, page 55 and pp 59-60 of O4(5).



OLTF




Poles and zeros: Ogata, S2-2, page 12 of O4(5). Rigorous definition is given in the text.




16/23




O4(5).





Wikipedia]


impulse input.

Measures of Performance (MOP): measures derived from the dimensional parameters (both

physical and structural) and measure attributes of system behavior. MOPs quantify the set of







Green.]

Partial Fraction decomposition of transfer function: Ogata, S2-5, pp 36-37 of O4(5).


Ogata, S8-1, SubsectionObtaining steady state output to sinusoidal inputs [pp 493-494 of

O4(5)]:

Magnitude of G(j) is given as amplitude ratio, and phase of G(j) is defined as the phase

shift of the output sinusoid with respect to the input sinusoid. Please note this is not G(s), butthe specific case, where the real part is zero. Some call this simply angle of G(j).


zero to infinity. Ogata, S8-4, pp 523-526 of O4(5).

Bode plotA plot of the magnitude in decibels |()| versus in semi-logcoordinates. Bode diagrams are explained in Ogata, S8-2, pp 497-507 of O4(5). The General

procedure for plotting Bode diagrams is given at the end of the section, page 507 of O4(5).

[Bodepronounced Bodee, Boday (and as per internet, Bodah, probably in his mother

tongue.]


17/23




Corner frequencies, resonant frequencies, peak gainPlease study Bode plots.

Resonant frequency is the frequency at which the peak resonance M, (Mr),

occurs. [Kuo, S9-1-2, p 544. Ogata, S8-2, page 507 of O4(5)] Some books use Mp as the

symbol.




sensitivity.



properties in the time domain. [See Kuo, p544, Ogata, S8-9, pp 572-573 of O4(5).]



indicates how well the system will track an input sinusoid. The rise time and the bandwidthare inversely proportional to each other.



Second order system is dealt with in Ogata S 5-3, pp 224-233 of O4(5).

First order system is given in Ogata, S5-2, pp 221-224 of O4(5).

End of Part 3.

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Part 4 of 4.

For students who are primarily referring to Ogata, Fourth Edition printed

in 2007 [O4(7)]:

Dynamic systems, principal constituents, input, output, process: Ogata, S1-2, pp 14-18 of

O4(7).Block diagram representation: Ogata S3-3, pp 70-75 of O4(7).


18/23

Rules for block diagram algebrapage 68 of O3 {not seen in O4(5) or O4(7)}


Ogata, Section 10-4, pp 713-717 of O4(7) have noise as additional input N(s). This is usually

measurement error of the output (referred to as sensor error).



Tracking Performance: Ogata, Section 10-5see page 686 and then S10-4, page 685 of O3

{I could not see the relevant part in O4}


Sensitivity: (i) - Sensitivity of the gain of the overall system is defined in Kuo, Sec 1-2-3,pp 13-14




[Please see Kuo and Ogata, Third Edition]
















Mach number to the elevator will be used to achieve a constant Mach climb (Ex 4.6 -3 on



function may be replaced by a general TF and sensor is for measuring v.Aircraft Transfer functions are given in Block diagrams given in Ref Y.


19/23



Block diagrams of Control Systems: Ogata, Examples 3-2, 3-3, pp 81-82 and pp 85-86 of

O4(7). [These are in addition to those cited earlier.]






[Please see Ogata, Third Edition]



Ogata, Ex 3-3, pp 85-86 of O4(5) explains spring-mass model. Example in the sub-section

Mechanical Systems, Ex. 3-8 pp 93-94 of O4(7) explains a slightly different model, where

the spring-mass` system is on a cart.



Stochastic control: See Page 8 (of the second set) of Notes of Prof. G.V.S.S. Sastry, Wg.Cdr. (Rtd.).


Ogata, Section 3-1, page 56 of O4(7).



coefficients are constant. A more rigorous definition for a linear system is given below.

Dynamic systems that are composed of linear time-invariant lumped-parameter components


systems are called linear time-invariant (or linear constant-coefficient) systems.







superposition.

Definition: A linear system is a system which has the property that if:(a) an input x1(t) produces an output y1(t), and


20/23




First order, Second order and higher order systems:Second order differential equation is of the form

,


See Ogata, S5-2, page 233-236 of O4(7).


order system.


contains the terms like

. See Eq 3-3, p 79. KuoSection 2-3, pp25-28.

See Ogata, S5-3, pp 242-243 of O4(7). If you can study Section 5-3, pp 236-241 of O4(7), it

will be helpful.



variable system. Kuo, Fig 3-5 - Block diagram on p87. Ogata, Section 3-4, page 82, O4(7).


coefficients constants at for a limited interval. Ogata, S3-10, pp 124-126 of O4(7). .






Analysis without referring to Laplace Transforms is given in OgataEx 3-3, pp 85-86 of

O4(7). Here it is solved in state-space model. In the class it was solved as a solution of the

second order differential equation. All of these are examples of time domain description,where system variables are obtained as functions of time, when acted on by an external force.

The external force may be impulse, step or ramp (or any other function).




Also, see S5-3, Sub-section Definitions of transient response specifications, pp 241-243 of

O4(7).

Indicial function:


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Transient Response of a second order system: Ogata, Section 5-3, page 240 of O4(7).


Laplace Transforms: Application to differential equations: [] () () [

]

() ()()

[

] () ()() , where, ()

()


Laplace Transform: Ogata, S 2-3, pp 25-30 of O4(7).

Transfer function: S3-2, S3-3, page 67 and page 72 of O4(7).



OLTF




Poles and zeros: Ogata, S2-2, page 24 of O4(7). Rigorous definition is given in the text.






O4(7).





Wikipedia]


impulse input.

Measures of Performance (MOP): measures derived from the dimensional parameters (both


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physical and structural) and measure attributes of system behavior. MOPs quantify the set of



Measures of Effectiveness (MOE): measure of how a system performs its functions within itsenvironment. An MOE is generally an aggregation of MOPs. Examples include



Green.]

Partial Fraction decomposition of transfer function: Ogata, S2-5, pp 44-45 of O4(7).


Ogata, S8-1, SubsectionObtaining steady state output to sinusoidal inputs [pp 505-506 of

O4(7)]:

Magnitude of G(j) is given as amplitude ratio, and phase of G(j) is defined as the phase

shift of the output sinusoid with respect to the input sinusoid. Please note this is not G(s), but

the specific case, where the real part is zero. Some call this simply angle of G(j).


zero to infinity. Ogata, S8-4, pp 535-538 of O4(7).

Bode plotA plot of the magnitude in decibels |()| versus in semi-logcoordinates. Bode diagrams are explained in Ogata, S8-2, pp 509-519 of O4(7). The Generalprocedure for plotting Bode diagrams is given at the end of the section, page 519 of O4(7).

[Bodepronounced Bodee, Boday (and as per internet, Bodah, probably in his mother

tongue.]




Corner frequencies, resonant frequencies, peak gainPlease study Bode plots.

Resonant frequency is the frequency at which the peak resonance M, (Mr),

occurs. [Kuo, S9-1-2, p 544. Ogata, S8-2, page 519 of O4(7)] Some books use Mp as the

symbol.




sensitivity.


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properties in the time domain. [See Kuo, p544, Ogata, S8-9, pp 584-585 of O4(7).]

The frequency range , in which the magnitude of the closed loop does not dropminus3dB is called the bandwidth of the system. {Ogata}. The bandwidth indicates thefrequency where the gain starts to fall off from its low-frequency value. Thus, the bandwidth





Second order system is dealt with in Ogata S 5-3, pp 236-245 of O4(7).

First order system is given in Ogata, S5-2, pp 233-236 of O4(7).

End of Part 4.

-.-.-.-.-.-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-..-.-.-.-.-.-.--.-.-.-.-.-.-.-.-.-.-.-

References:-

1. YYechout, Morris, Bossert and Hallgren, Introduction to Aircraft Flight Mechanics:

Performance, Static Stability, Dynamic Stability, and Classical Feedback Control, 2003.

2. S&LStevens & Lewis, Aircraft Control and Simulation, 2003.

3. O3Ogata, Modern Control Engineering, Fourth Edition, Second Impression, 2000.

4. O4(5) Ogata, Modern Control Engineering, Fourth Edition, Second Impression, 2005.

5. O4(7) Ogata, Modern Control Engineering, Fourth Edition, Second Impression, 2007.

6. NNelson, Flight Stability and Automatic Control, 2011.

7. KuoB.C. Kuo, Automatic Control Systems, 2003.

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