Application of H-Infinity Control on Boost Converter

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In the following sections, the design procedures for application of. H-Infinity Control on .... Bode plot of the non-ideal boost converter is depicted in Fig. 3, and the ...
Application of H-Infinity Control on Boost Converter Prepared by Omar Abdel-Rahim Ph. D Student, Electrical and Electronic Engineering Department, Utsunomiya University 147102H Submitted to Prof. Mitsu Hirata

Introduction During

the

past

decade,

there

have

been

many

theoretical

advances in the field of robust control, and, in particular, h-infinity control.

A

problem

breakthrough

in

feasible

1988

algorithms

in

led

to

for

of

weighting outputs input

input

functions and

reflect

disturbances

solution

the

finding

of the key steps in the formulation

the

of

the

development

h-infinity

h-infinity of

control

computationally

optimal

controllers.

One

h-infinity control design approach is

and

are

output

utilized

the

spatial

and

the

weighting

functions.

normalize

the

to and

frequency

performance

the

These

inputs

dependency

and

of

specifications

the

of

the

output variables [1]-[4]. The

weighting

performance

(i.e.,

time

function

should

tracking

performance

overshoot.

In

frequency

gain,

one

would

function

have

the

expect,

selected

response) low

and

general,

is

the

more

at

gain

limiting

low at

larger the

the

overshoot the

reflect

characteristics.

gain

high

to

the

The

sensitivity

frequencies

high

is

overshoot

for

frequencies

magnitude

of

limited. is

desired

good

to

limit

the

high

However,

accomplished

as by

adding more damping at the expense of response speed, which sets up a design tradeoff between overshoot and response speed. Pulse-Width-Modulation nonlinear average describes

and model,

(PWM)

time-variant we

systems.

obtain

approximately

the

a

dc–dc After

linear behavior

converters linearization

time-invariant of

the

are of

the

model

that

system

for

frequencies up to half of the switching frequency. The tasks of the controller in dc–dc converters are to [5]: 1) Assure stability of the closed-loop system; 2)

Minimize

sensitivity

to

load

changes,

i.e.,

reduce

the

output

impedance; 3) Attenuate input–output transmission (low audio susceptibility). Traditional

designs

systems

are

linearization).

of

feedback

based A

on

major

loops

in

dc–dc

frequency-domain

problem

is

switched-mode

analysis

(after

however,

when

encountered,

the transfer function of the power stage (from the control input to the

output

cases RHPZ

voltage)

(e.g.,

boost

severely

has or

a

fly

restricts

right-half-plane back

the

Zero

converters),

closed-loop

(RHPZ).

the

band

In

presence

width

that

such

of

the

can

be

obtained by the classical frequency domain approach, [5]-[7]. H-∞ Infinity Control: SOME GENERAL FACTS For a stable and proper transfer function G(s), the  norm is defined as

‖()‖ = Where

 

()  

the

maximum



singular

is

(1) value

of

a

complex

matrix A and is defined as

() =  (∗ ) Note

that

maximum

∗ is

value

the of



conjugate eigenvalues.



transpose If

G(s)



(2)

of

A,

is

a

Output (SISO) system, (1) takes on a simpler form as

and Single

 Input

is

the

Single

  |()|

‖()‖ = i.e.,



the

norm





corresponds

to

the

maximum

Furthermore, the following inequality

‖()‖ < 1 ⟺ | ()| < 1, ∀ Obtained

from

(3)

is

commonly



used

substituted for the  norm condition.

(3) of

gain

G(s).

(4)

when

the

gain

condition

is

Using (4), the robust stability condition can be represented by the

 norm condition of

‖& '‖ < 1









H-Infinity Control for Boost DC-DC Converter

In

the

following

H-Infinity details.

sections,

Control

Design

on

procedure



(5)

the

design

procedures

for

application

of

dc-dc

boost

converter

are

described

in

and

design

equations

all

are

based

on

the IEEE Transaction in [8]. Most of the results in this report are based on the design equations developed in [8]. A. THE BOOST CONVERTER

As converter

a

design

operated

example, in

we

took

continuous

a

typical

conduction

low-power

mode,

shown

boost

in

Fig.

1, which transforms a nominal 12-V input voltage into 24 V at its output.

The

current

source

switching in

the

frequency

is

240

output

to

simulate

is

(changes

in

the

output

current).

inductor,

switch,

diode,

and

The

capacitor

kHz.

changes

series are

The

role in

resistance all

of

the

the

load

of

the

shown.

The

controller imposes the duty cycle), which is the ratio between the

time '*+ when the switch is closed and the period'. Applying the

well-known a

state-space

averaging

that,- = ,. = ,/ = 0,

moment

procedure

[8]

we

the

find

and

assuming

following

for

transfer

function from the duty cycle to the output voltage 1234 .

() =

6 ( 5234 789)6

Recall

5:; -
?6 234 A(7BC H

=78



)

and

N*LM

see

and

that

and O

the



denote

fixed

denote

above

(6) values

deviations

transfer

at

from

function

an this

exhibits

Right Hand Pole or Zero (RHPZ) 6 Q5:;

PB = -5 6 234









If we do not neglect ,- , ,. STO ,/ function

discussed

above







(7)

then the formula of the transfer

becomes

more

complicated,

of

course,

but it retains the same features. Transfer function from the control input O to the output voltage N*LM were found to be:

1234 .

() =

8 .77V(BWXWY )(8WZWZ )  6 BW[77BXZ







(8)

Input-to-output voltage transfer function of boost converter is described as follow:

G^_ (S) =

b c (d) a f b e (d) b a g(d)h

=

ZZV(BWXWY )

 6 BW[77BXZ





(9)

Output current-to- output voltage transfer function of boost converter is described as follow: P* (j) =



b (S) V o p Ino (S) db rSs=0

= −.12rj+45460srj+4100s j +4311j+5200000

(10)

Figure 1: DC-DC Boost Converter

B. H-∞ Infinity Control for the boost converter

ω{

v in

PO

iout

W

vout

z

d

C

Wd

y

Figure 2. The control of the converter as a standard H∞ control problem.

In the diagram, depicted in Fig. 2, v* is the converter, w is

the controller to be designed, and x is a stable weight function.

The vector  contains the perturbations (NK+ andy*LM ), the output z is

the

weighted

measurement

error

outputs

of

signal, the

the

vector

(N*LM andNK+ ),

plant

{

contains

and

the

the

control

input | is the duty cycleO. We denote by '}~ the transfer function

from w to z (in closed loop, i.e., with the controller connected to the

plant).

controller

The C,

standard

which

problem

stabilizes

of

the

H∞

control

is

plant

(i.e.,

the

to

find

a

closed-loop

system

is

stable)

weight

function

frequencies

(it

perturbations the

in

weight

that

the

bigger

the

minimization

the

is

can

frequencies

the W.

weighted

where

more

and

at

Thus, output

of

the

for

the

different

By

adjusting

example,

worse

way

around.

Note

input–output

voltage-

both

design

objective

impedance.

and

W

weight

function

can

be

combined

to

The

state-space

equations

of

the

averaged

and

which is a system with inputs &, | and outputsz, {.

better

of

impedance,

our

of

expense

other

the

The

presence

disturbing).

the

output

of'}~ .

norm

importance

obtain,

'}~ are

of

function

H∞

relative

frequencies

and

the

frequencies,

signal we

attenuation

of

the

for

error

lower higher

weight

minimizes

expresses

components

disturbance by

at at

the

is

also

function,

performance attenuation

x

and

The

multiplied includes

converter

form

v*

the

plant

P,

linearized

plant

v*

in our design example are

€ =   + 7‚  + Z‚ |

N*LM = w  + )7‚  + )Z‚ |

(11)

Where q is the state of the converter (the inductance current and

capacitor

voltage).

The

matrices

appearing

in

equations are

−4208 −2283 „ 2086 −103.1 4975 228.3 7‚h ƒ „ 0 −4535 119540 Z‚ = ƒ „ −5370  = ƒ

w = …0.046 1† )7‚ = …0

−0.1†

(12) (13) (14) (15) (16)

the

state

)Z‚ = …−0.118†

(17)

For our design example, we chose the weight function

x () =

BZ∗‡K∗[X BZ∗‡K∗X

(18)

By combining v* with a realization of W, we obtain the plant P, which is of order three as follow:

w ()

−24300000(j − 102. 10Y )(j + 12140)(j + 4120) 1.87. 107[ =ˆ ‰ (j − 4.47. 107W )(j + 3140)(j + 45460) j − 4.47. 107W (19) The

raw

extremely

w has

controller big

positive

an

pole,

unstable

which

cannot

pole—actually be

realized

by

an any

means. (The closed-loop system would be stable, even if we could

w realize and use it.) To overcome this problem, the unstable pole was approximated by a constant Š

8



8Š 

, For|j| ≪ |S|.

(20)

A similar approximation was used to get rid of the very big zero lying these

in

the

right-half

approximations

controller

over

the

plane. do

Further

not

frequency

affect

range

of

investigation the

has

shown

performance

interest

(up

to

that

of

the

about

100

kHz).

The approximate  controller is

w() = ƒ

8X.XY(BW7Z )(B7Z7W ) (B[7W )(BWXWY )

− 0.0417„

(21)

We remark that the  controller C has a low dc gain compared to

the

voltage-mode

controller,

feed

forward

controller,

or

current-

mode

controller,

which

will

be

derived

later.

With

the

controller

C, the voltage-attenuation curve looks like that in Fig. 4. In the same

figure,

the

corresponding

curves

are

shown

for

two

other

possible choices of the weight function, namely x7 () =

xZ () =

BZ∗‡K∗[X BZ∗‡K∗X

(22)

BZ∗‡K∗[X BZ∗‡K∗X

(23)

The curves of the output impedance for the three weight functions, when

using

a

logarithmic

scale

for

the

impedance,

look

very

similar, so we omit them. Bode plot of the non-ideal boost converter is depicted in Fig. 3, and the bode plot of the designed controller is shown in Fig. 4. Bode

plot

of

the

closed

loop

system

with

different

weighting

factor are depicted in Fig. 5, 6 and 7, while Fig.8 shows the bode plot of the closed loop system without using any weighting factor. We

remark

compared

to

that the

the

h-infinity

voltage-mode

or current-mode controller

controller controller,

has feed

a

low

forward

dc

gain

controller,

Figure 3. Bode diagram of open loop boost converter.

Figure 4. Bode diagram of  controller.

Figure 5. Bode diagram of the closed loop system with weighting factor W.

Figure 6. Bode diagram of the closed loop system with weighting factor W1.

Figure 7. Bode diagram of the closed loop system with weighting factor W2.

Figure 8. Bode diagram of the closed loop system without any weighting factor.

References [1]-

Y.

flexible

Qian,

C.

structures

Hai, on

H. the

dong,” H-infinity

Tracking control

control method,

research “

2nd

of

high-order International

Conference

on

Advanced

Computer

Control

(ICACC),

F.

Novazzi

and

of

a

exchanger

27-29

March,

2010,

Pp. 111-115. [2]-F.

Delatore,

Infinity

model

bypasses

F.

Leonardi,

matching

“9th

IEEE

L.

control International

heat

Conference

J.

on

Cruz”

Multivariable

network

Control

(HEN)

and

Hwith

Automation

(ICCA), 19-21 Dec. 2011, Pp. 1108-1113. [3]- J. Mongkoltanata, D. Riu, X. Lepivert ” H infinity controller design for primary

frequency

European

control

Conference

on

of

energy

Power

storage

Electronics

in

islanding

and

MicroGrid

Applications

“15th

(EPE),

2-6

Sept. 2013, Pp. 1-11. [4]-

W.

infinity

Zhang, control

H. of

Wang ; X. linear

Xiaoming

systems

with

“Analytical time

formulas

delay

for

quasi-H

“American

Control

Conference, 2002. Proceedings of the 2002, Pp. 2233 – 2238, vol. 3. [5]- W. Erickson "Fundamental of Power electronics" second Edition. [6]- Cortes, P., Wilson, A., Rodriguez, J., Kouro, S. and Abu-Rub, H. “Model Predictive IEEE

Control

of

Transactions

on

Multilevel Industrial

Cascaded Electronics,

H-Bridge Vol.

57,

Control

of

Inverters, No.

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accepted August

at

2010,

2691 - 2699. [7]-

J.

Rodriguez,

P.

Cortes

“Predictive

Power

Converters

and

Electrical Drives” Wiley, IEEE. [8]- R. Naim, G. Weiss, and S. Yaakov “H∞ Control Applied to Boost Power Converters

“IEEE

12, NO. 4, JULY 1997.

TRANSACTIONS

ON

POWER

ELECTRONICS,

VOL.