l10_optadv

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    Optimal Control Options

    Binomial Standard Form Based Control

    Butterworth Standard Form Based Control

    ITAE Standard Form for Zero Ramp Error

    Open Loop Parameter Based Control

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    Other Optimality Options

    The ITAE based optimal design has good peak

    overshoot, but not so good settling time. Particularly,

    when the systems are of higher order.Also, there can be situations where, ramp input error

    also needs to be driven to zero.

    Thus, there are additional techniques for generating

    optimal control designs that improve upon the basic

    ITAE optimal design.

    In cases where open loop time constants are fixed, we

    can obtain optimal close loop performance by adjusting

    open loop gain.

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    Binomial expansion based characteristic equations assume

    that all modes are critically damped, thus, we have

    The corresponding polynomials are given below. Theseactually provide a slow 1st order type response.

    Binomial Expansion Based Design

    2 2 3 2 2 3

    1 0 2 0 0 3 0 0 0

    4 3 2 2 3 4

    4 0 0 0 0

    5 4 2 3 3 2 4 5

    5 0 0 0 0 0

    6 5 2 4 3 3 4 2 5 6

    6 0 0 0 0 0 0

    ( ) : ; ( ) : 2 ; (s):s 3 3

    ( ) : 4 s 6 4 ;

    ( ) : 5 s 10 10 5

    ( ) : 6 s 15 20 15 6

    s s a s s a s a a s a s a

    s s a a s a s a

    s s a a s a s a s a

    s s a a s a s a s a s a

    + + + + + +

    + + + +

    + + + + +

    + + + + + +

    ( )0

    0

    ( )Ensures exact step input tracking

    ( )

    n

    n

    aC s

    R s s a=

    +

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    Butterworth form aims to place all close loop poles

    uniformly on a circle in left half of s-plane, with its

    centre at the origin of the s plane.

    The corresponding polynomials are given below.

    This means that lower bandwidth modes decay faster.

    Butterworth Form Based Design

    2 2 3 2 2 31 0 2 0 0 3 0 0 0

    4 3 2 2 3 4

    4 0 0 0 0

    5 4 2 3 3 2 4 5

    5 0 0 0 0 0

    6 5 2 4 3 3 4 2 5

    6 0 0 0 0 0

    ( ) : ; ( ) : 1.4 ; (s):s 2 2

    ( ) : 2.6 s 3.4 2.6 ;

    ( ) : 3.24 s 5.24 5.24 3.24

    ( ) : 3.86 s 7.46 9.13 7.46 3.86

    s s a s s a s a a s a s a

    s s a a s a s a

    s s a a s a s a s a

    s s a a s a s a s a s

    + + + + + +

    + + + +

    + + + + +

    + + + + + +6

    0a

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    The corresponding step responses are;

    Butterworth Form Based Response

    It is seen that

    Butterworth form

    ensures nearly thesame settling time,

    but the peak overshoot

    increases with increasein the system order.

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    Settling time based optimal designs are for a 5% ripple

    and applicable characteristic polynomials are given below.

    Settling Time Based Design

    2 2 3 2 2 3

    1 0 2 0 0 3 0 0 0

    4 3 2 2 3 4

    4 0 0 0 0

    5 4 2 3 3 2 4 5

    5 0 0 0 0 0

    6 5 2 4 3 3 4 2

    6 0 0 0 0

    ( ) : ; ( ) : 1.4 ; (s):s 1.55 2.10( ) : 1.60 s 3.15 245 ;

    ( ) : 1.575 s 4.05 4.10 3.24

    ( ) : 1.45 s 5.10 5.30 6.25

    s s a s s a s a a s a s as s a a s a s a

    s s a a s a s a s a

    s s a a s a s a s

    + + + + + +

    + + + +

    + + + + +

    + + + + +5 6

    0 03.425a s a+

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    Consider an nth order control ratio as given below.

    In this case the close loop transfer function exactly tracks

    both the step and the ramp inputs. The applicable ITAEbased characteristic polynomials are,

    Zero Ramp Error Based Optimality

    1 0

    n 1

    1 1 0

    ( )

    ( ) sn

    n

    a s aC s

    R s a s a s a

    +=

    + + + +

    2 2 3 2 2 3

    2 0 0 3 0 0 0

    4 3 2 2 3 4

    4 0 0 0 0

    5 4 2 3 3 2 4 5

    5 0 0 0 0 0

    6 5 2 4 3 3 4 2 5

    6 0 0 0 0 0

    ( ) : 3.2 ; (s):s 1.75 3.25

    ( ) : 2.41 s 4.93 5.14 ;

    ( ) : 2.19 s 6.50 6.30 5.24

    ( ) : 6.12 s 13.42 17.16 14.14 6.76

    s s a s a a s a s a

    s s a a s a s a

    s s a a s a s a s a

    s s a a s a s a s a s

    + + + + +

    + + + +

    + + + + +

    + + + + +6

    0a+

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    Normalized step responses for exact ramp input tracking

    based on ITAE criterion, are as given below.

    Exact Ramp Tracking Design

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    Consider open loop transfer function and corresponding

    unity feedback control ratio, as given below.

    It can be shown that for above transfer function, ITAEerror is a minimum for K > 1.25 & remains constant.

    Thus, we can choose K = 1.25 in this case and for a given

    T, we can determine the gain K.

    Open Loop Gain Based Optimality

    2

    ' '

    '' '2 ' '

    ( )( ) ; ;

    S(1+Ts) ( ) Ts

    ( ) ;( ) s

    K C s K G s

    R s s K

    C s K K KT R s s K

    = =

    + +

    = =

    + +

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    The above method can be extended to higher order open

    loop transfer functions. Following tables show this.

    Open Loop Gain Based Optimality

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    Consider the following open loop system.

    For each of the following methods, obtain the optimal

    cascade controller and discuss its important features.

    (1) Binomial Standard Form

    (2) Butterworth Form

    (3) Settling Time Form

    (4) ITAE Form for Zero Ramp Error

    (5) ITAE based Optimal Gain Form

    Optimal Control Example

    6

    G(s) ;( 1)( 2)( 3)s s s s=

    + + +

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    Optimal Control Summary

    There can be many different ways in which the optimumperformance can be defined.

    It is also possible to achieve tracking requirements, while

    optimizing the transient performance.

    In general, no method is superior to the other as the

    quality of solution depends on the system order.