8/2/2019 L10_OptAdv

1/12

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

8/2/2019 L10_OptAdv

2/12

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.

8/2/2019 L10_OptAdv

3/12

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=

+

8/2/2019 L10_OptAdv

4/12

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

8/2/2019 L10_OptAdv

5/12

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.

8/2/2019 L10_OptAdv

6/12

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+

8/2/2019 L10_OptAdv

7/12

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+

8/2/2019 L10_OptAdv

8/12

Normalized step responses for exact ramp input tracking

based on ITAE criterion, are as given below.

Exact Ramp Tracking Design

8/2/2019 L10_OptAdv

9/12

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

= =

+ +

= =

+ +

8/2/2019 L10_OptAdv

10/12

The above method can be extended to higher order open

loop transfer functions. Following tables show this.

Open Loop Gain Based Optimality

8/2/2019 L10_OptAdv

11/12

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=

+ + +

8/2/2019 L10_OptAdv

12/12

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.