Controller and Observer - PowerPoint Presentation

Slide1

Controller and Observer Design

(Design

of Control System in State

Space)

(Design of Control System in State

Space by Pole placement)Slide2

References

Dr.

Radhakant

Padhi

,

Asstt

. Prof, IISC, Bangalore, through NPTEL

Modern Control Engineering by Katsuhiko Ogata, PHI Pvt. Ltd New Delhi Slide3

Pole Placement

Controller DesignSlide4

Pole Placement Technique

Poles of a control system (stable/unstable) can be place at desired location by pole placement technique. This is done to

Improve the performance of the system

Make the system stable

Increase the damping

Increase the response time

Etc

Slide5

Pole Placement Technique

Assumptions are

The system is completely state controllable

The sate variable are measureable and available for feedback

Control input (u) is unconstrained and single

Note:

For multi input system, the state feedback gain matrix is not unique Slide6

Pole Placement Technique

Objective:

The closed loop poles should lie

,…

. Which are their “desired locations”.

Difference from classical approach:

Not only the dominants poles, but “all poles” are forced to lie at specified desired locations.

In classical approach only dominants poles are placed at desired location

Necessary and Sufficient condition:

The system is completely state controllable

 Slide7

Philosophy of Pole placement control design

Let a system is represented by

---(1)

Put input u as

, put in equation (1)

K is called state feedback gain matrix (1xn)

and X is state vector

(nx1)

So KX will be scalar (=> single input)

---(2)

New closed loop state transition matrix

Its time response

---(3)

 Slide8

Philosophy of Pole placement control design …

Philosophy:

T

he matrix K is designed such a way that the two characterize equations are having same poles

A

B

u

Fig 1: Open loop Control system

X

+

+

A

B

u

Fig 2: Closed loop Control system

With u=-KX

-K

X

+

+Slide9

Placement control design (Controller Design)

There are three method:

Method 1:

Direct substitution method

(when order of system n≤3)

Method 2:

Bass-

Gura

Approach

Method 3:

Ackermann’s formula

Slide10

Controller Design by method 1:

Let the system is

steps are

Step 1: Check controllability of the system

Step 2:

Put u=-KX where

So

Step 3: Write characteristic equations of above new system

Step 4: Write Desired characteristic equation

Step 5: Compare above two characteristic equations and solve for k

1

, k

2

, k

3

by equating the power of s on both sides

 Slide11

Controller Design by method 2:

Let the system is

steps are

Step

1: Check controllability of the system

Step 2: Put u=-KX where

Step 3: Let the system is in first companion form (Controllable canonical form)

i.e

 Slide12

Controller Design by method 2…

Step 4: after putting the value of u in given system, now system will become

. So

---(4)

 Slide13

Controller Design by method 2…

Step 5:

---(5)Slide14

Controller Design by method 2…

Step 6: Comparing equations (4) & (5) we have Slide15

What if the system is not given in first companion form?

Answer is to convert it into

Companion Form

as follows

Define a transform

put the value of

Select the value of T such that

is in first companion form

Put T=MW

Where

is the controllability matrix

 Slide16

What if the system is not given in first companion form?...Slide17

Controller Design using Method 2: Bass-Gura

Approach

Step 1:

Check controllability of the system

Step 2:

Form the characteristic equation using matrix A

…

+

find

a

i

’s

Step 3:

find the transformation matrix T if system is not in

first

companion T=MW Step 4: Write the desired characteristic equation

Step 5:

The required state feedback matrix is

Note: Above approach is for any system (controllable canonical form or not). If system is in controllable canonical form put T=I (identity matrix)

 Slide18

Controller Design using Method 3:Ackermann’s Formula

Let

Desired characteristic equation

Caley

-Hamilton theorem states that every matrix A satisfies it own characteristic equation. So

For case n=3 consider the following identities

 Slide19

Controller Design using Method 3:Ackermann’s Formula

…

Multiplying the above identities with

respectively and adding them

---(6)

 Slide20

Controller Design using Method 3:Ackermann’s Formula …

From

Caley-Hemilton

theorem for

Also we have for A

Putting the values

&

of in equation (6)

0Slide21

Controller Design using Method 3:Ackermann’s Formula …

=>

Since

system is completely controllable inverse of the

controllability matrix

exists we obtain

=> ---(7)Slide22

Controller Design using Method 3:Ackermann’s Formula …

Pre multiplying both sides of the equation

(2)

with

[0 0 1]Slide23

Controller Design using Method 3:Ackermann’s Formula …

Hence

For an arbitrary positive integer

n

( number of states

)

Ackermann’s

formula

for the state feedback

gain matrix

K

is given

by are the coefficients of desired characteristic polynomial Slide24

Example

Example 1:

Consider the system defined by

where

By using the state feedback control u=-KX, it is desired to the closed loop poles at

and s=-10. Determine the sate feedback gain matrix K.

Solution:

First check the controllability of above system

 Slide25

Example ..

Controllability matrix

so rank of M =3. Hence system is completely state controllable.

Now we will solve this problem with previous three methods

 Slide26

Example ..

Method 1:

Direct substitution method

Put

u=-KX where

So

Write

characteristic equations of above new system

 Slide27

Example…

Desired Characteristic equation

comparing above two characteristic equations

k

1

= 199, k

2

= 55, k

3

= 8

So

 Slide28

Example

Method 2:

Characteristic equation of the given system

Comparing with

a

1

= 6, a

2

= 5, a

3

= 1

 Slide29

Example…

Desired Characteristic equation

Sate feedback gain matrix K is

Where T= I (identity matrix as system is in controllable canonical form)

 Slide30

Example…

Method 3: Ackermann’s Formula

 Slide31

Example…

So

 Slide32

Choice of closed loop poles:

Don’t choose the closed loop poles far away from the open loop poles, otherwise it will damage high control effort.

Don’t choose the closed loop poles very negative, otherwise the system will be fast reacting (

i.e

it will have a small time constant)

In frequency domain it will lead to large bandwidth and hence noise get amplified. Slide33

Controller for multi input system

The state feedback gain matrix (K

) becomes a matrix of

mxn

(Not vector of 1xn unlike single input system)

m = no of inputs and n = no of states

The state feedback gain matrix (K) is not uniqueSlide34

Summary wise

Define a linear combination of

control variables as new control

cariable

.

i.e

Figure Reference:

www.optisyn.com

 Slide35

Next: Observer Design

Thanks