PHÂN TÍCH ỔN ĐỊNH BỀN VỮNG CỦA HỆ THỐNG ĐIỀU KHIỂN H-INFINITY CHO CÁC ĐỘNG CƠ MỘT CHIỀU KÍCH TỪ ĐỘC LẬP

Finally, in order to ensure that the closed-loop system is stable when the excited field is under variations, the authors apply the well- known structure singular value to evaluat[r]

ROBUSTNESS STABILITY ANALYSIS FOR H-INFINITY CONTROL OF

SEPARATELY EXCITED DC MOTORS

Nguyen Thi Mai Huong * , Nguyen Tien Hung

University of Technology - TNU

ABSTRACT

This paper is dealt with the problem of robustness analysis for a separately excited DC motor control system with field weakening utilizations For sake of simplicity, we propose a linear control design for the DC motor in which the excited field is considered to be unchanged instead

of using nonlinear control technique for the combination of armature voltage and field current regulations Then the authors design a linear controller design that guarantees stability of the controlled system against motor’s parameter uncertainties Finally, in order to ensure that the closed-loop system is stable when the excited field is under variations, the authors apply the well-known structure singular value to evaluate the stability of the controlled system when the armature resistance, inductance, motor constant and excited field are changed in the same time The research results will be demonstrated in the Matlab/Simulink environment

Key words: Separately excited DC machine; field control; linear robust control;structure

singular value, robustness stability

INTRODUCTION*

Separately excited DC motor machines

(SEDCMs) arestill utilized in many

applications since they own capability to be

simply and effectively controlled over a wide

rangeof the rotor speed, especially for field

weakening utilizations[ HYPERLINK \l

"Gop89" 1 , HYPERLINK \l "ZZL03" 2

, HYPERLINK \l "RHa94" 3 ,

HYPERLINK \l "MHN96" 4 ]

In the normal operation, the field current is

fixed at a maximum value and it can be

viewed as at a constant value Under the

circumstances, the SEDCM can be described

by linear differential equations and linear

control techniques can be applied to the

system However, in the field weakening

region, when the variation of the field

currenthas to be taken into account, the

system turns to benonlinear because of a

product of field flux and armaturecurrent as

well as a product of field current and

rotorspeed[ HYPERLINK \l "Ngu17" 5 ]

*

In the literature, several strategies have been proposed to control the speed of a

SEDCM.In3]}, a multi-input multi-output

(MIMO) controller wasdesigned for a SEDCM using an on-line linearizationalgorithm in which the applied armature and the fieldvoltage are driven simultaneously An input-output linearizationtechnique based on canceling the nonlinearities in theSEDCM model and finding a direct relationship betweenthe motor output and input quantities is proposed in [ HYPERLINK \l "MHN96" 4 ] In 6]}, an



H controller is designed in order to ensure that the performance of the controlled system

is maintained with respect to the changes of motor parameters in specified ranges

In this paper we employed the famous tool known as the structure singular value in order

to investigate the stability of the controlled system with an Hcontroller design for SEDCMs The controller design is followed

by the spirit of the idea in [ HYPERLINK \l

"VuN14" 6 ] where armature resistance, inductance and motor constant are considered

to be varied with time while the field current is

kept constant Then the robustness stability of

the closed-loop controlled system is tested for

the case in which the motor’s parameters and

the excited field are changed in the same time

ROBUST CONTROLLER DESIGN AND

ANALYSIS



H } control of linear time-invariant

systems

The following discussion in this section is

mainly based on [6]

A standard setup for Hcontrol is

presentedin Fig 1, where w represents the

generalized disturbances, z the controlled

variable, u the control input and y the

measuement output, while P is a linear

time-invariant system described as

 

& p

x Ax B w Bu

z C x d w Eu

y Cx Fw

(1)

Fig 1 The interconnection of the system

The goal in Hcontrol is to find a stabilizing

linear time-invariant (LTI) controller Kthat

minimizes the Hnorm of the closed-loop

system| | l( , P K ) | |, where l( ,P K)is

lower linear fractional transformation of P

and K, which is nothing but the closed-loop

transfer functionw  zin Fig 1

In order to achieve certain desired shapes of

the closed-loop transfer functions, such as

dictated by requirements on bandwidth,

weights are introduced and we consider

minimizing the H-norm of | | l( , P K % ) | |,

where l( , P K % ) is the closed-loop transfer

function w %  z %in Fig 2, W z and Wware real-rational proper weighting functions with suitable band-pass and characteristics

Fig 2 The weighted interconnection of the system

Sub-optimal H control

Let us now consider a generalized plant P where weights are incorporated already as follows

0

    

& p

(2)

If the linear time-invariant controller K is expressed as

  

    

&c c c c

A B

the closed-loop system l( ,P K) admits the following state-space description:





    



    

  

 

&

w

where



(5)

The H control problem is to find an LTI controller which renders stable and such that





holds true [7], where  0 is a given

number that specifies the performance level

This is the so-called sub-optimal H

problem



H controller synthesis

Using the bounded real lemma for (6), the

matrix is stable and (6) is satisfied if and

only if the LMI

0 0

0

1 0

0

0 0





    

p

T

I

I

holds for some f 0 Unfortunately this

inequality is not affine in and in the

controller parameters which are appearing in

the description of , , , However,

a by now standard procedure allows to

eliminate the controller parameters from these

conditions, which in turn leads to convex

constraints in the matrices XandY that

appear in the partitioning of

1

,



   

according to that of in (5) One then

arrives at the following synthesis LMIs for



H -design [8]:

0

0

0









f

p

p

(7)

where X and Y are basis matrices for

the subspaces

respectively

After having obtained X and Y that satisfy (7) for some level  , the controller parameters can be reconstructed by using the projection lemma [9] This procedure for



H -synthesis is implemented in the robust control toolbox [10]

Standard setup for robustness analysis

Let us denote m n

c as the set of all causal linear operators that mapLn2[0, )  into

2[0, ) 

m

L Recall that, roughly, an operator is causal if the past output (at any current time)

is not affected by any modification of the future of the input signal

Let us now consider the standard setup for stability analysis as given in



 p p

M is a known causal linear time-invariant operator and   p p

c

is a general causal operator For some set of uncertainties Δ  c p p we say that M is robustly stable againstΔif the feedback interconnection of M and Δ in Fig.3 is

well-posed (I M   has a causal inverse) and stable (( I M   )1 has finite energy-gain)for all  Δ

Structured singular value analysis

The structured singular value, denoted by ,

is apowerful tool for robustness analysis against lineartime-invariant (LTI) structured uncertainties [11,12,13]

Fig 3 The standard setup for robust stability

analysis

For the standard set up as depicted in Fig 3and in view of the fact that we are considering parametric uncertainties, we define the structure of theuncertainty block in

the scope of robustness analysis for the

SEDCM control as

 diag(1 1, , ) : , 1, , 

Δ

u

where I r i is an r ri i identitymatrix The

size of the uncertainty  Δis measured in

terms ofthe largest singular value as ( )

Since is diagonal, let us recall that

1, ,

   i u i

From the Nyquist theorem it is well-known

that, for a stable system M in Fig.3,the

feedback interconnection is well-posed and

stable if and only if

det(I M j ( ) )      0,  { }, Δ (8)

Definition 1.The structured singular

valueΔ( M ) of the complex matrixMwith

respect to the structure setΔ is defined as the

smallest

( )

  that renders the matrix I   M

singular, i.e

( )

inf ( ) : , det( ) 0







Δ M

andΔ( M )  0if there is no  Δsuch that

det(I   M ) 0

In view of (8), well-posedness and robust

stability of theinterconnection in Fig.3 is

henceguaranteed for all Δ with

1

{ }





  

In the literature [14], it is shown that exact

calculation of thestructured singular value is a

very hard problem.Fortunately, lower and

upper bounds can be computed efficiently

with the-tools in Matlab[15]



ROBUSTNESS EVALUATION

SEDCM model

The electrical behavior of the SEDCM can be expressedby the following equation:







&

m

m m

m

where

1 2

2

,

120 0











   

   











a

m

m

L

x

K J

1

1 0

0 1 1

0

0

20

    



a a

L m

v L

B

J

T K

a

e is the back-electromotive force (EMF)of the motor, Te is the electrical torque, TLis the load torque, va is the terminal voltage,

a

R is thearmature resistance, Lais the armatureinductance, K mis the motor constant, ia is the armature current,nis the motor speed, J isthe inertial torque of the motor, is the field flux, respectively

Linear fractional transformation representation ofSEDCMs

Let Ra,La,Kmare uncertain parameters,

we can represent them as follows

0 0 0







L L K

p

R

(11)

where Ra0, La0, Km0are the nominal values of motor parameters; pr, pl, pmand

,

1   , 1

  r l m  represent the variations of the system parameters, respectively

The model of the SEDCM with these

uncertainties is shown in Fig.4

Fig 4 The uncertain model of the SEDCM

This model can be rearranged to theM 

configuration with matrix M G  m given by

1

2

    

&

&

1 4 44 2 4 4 43

m

G

L a

(12)

in which





m

A

1

2

0





m

B

p

, 0 2

2

0

0



a L B

,

 

2  0 1

0 1

0 0

0 0

0



a

m m

R C

K K

,

11



12

0

T

a

D

21 0 0 0 0 , 22  0 0

 m

 

 

 

 

 

 

l r m n i

u u w u

u ’

0 0 0

0 0 0









l r m

m m

,

 

 

 

 

 

 

l r m n i

y y z y

y .

With the LFT representation of the SEDCM model we can now derive a standard control structure for the synthesis of an H

-controller as depicted in Fig.5 Here, Gmis the linear time-invariant part of the plant,

m is the uncertainty block as given in (13),

in

K is the H controller that is to be designed In this configuration, n ref is the reference input, v ais the controller output,

nis the controlled output

The system representation with additional field flux uncertainty

In order to employ the -tools for robustness analysis of the controlled system with respect to the machine uncertainties including the field flux variation, we first pull the uncertainties out

of the plant to get the standardM  configuration as shown in

Fig.3

Fig 5 Closed-loop control of SEDCM

Let    0 pf f In combination with

(11) weinfer the model of the SEDCM with

uncertainties as shown in Fig.6

Fig 6 The model of the SEDCM

with filed flux uncertainty

Similarly as above, the model in Fig 6can be

rearranged to the M  configuration with

matrix M given by

1

2

&

&

1 4 4 44 2 4 4 4 43

M

L a

in which







m

A

K

,

1

1

2 2 0

0 0





   

f

l

p

p

B

0 2

2

1 0 0





a

L B

,

2

1 0

0 1

0

0 1

0

1 0

0 0

0 0

0 0

0







a

m m

m m

R

K C

K

K K

,

1

11

1

0







f

l

m

m

p

p

D

p

p

,

12

0

1



T

a

D

L

12

0

1



T

a

D

L

,

21

0 0 0 0 0 0

0 0 0 0 0 0

22

0 0

0 0

D

and the  matrix is given by













l r m m m m

Now we can easily close the loop with the



H controller at a nominal value of the field

flux and get the standard setup for  analysis

as depicted in Fig.7, where M is the transfer

function from wto z

ANALYSIS RESULTS

The uncertainty structure fits into the

framework of the structure singular value

analysis Since M is known to be stable and

normalization 1    r, l, m, f 1, robust

stability is guaranteed if the structured

singular value satisfiesΔ(M j r( )) 1 for

all  { }

Fig 7 The standard MM ¡ ¢¡ ¢ configuration

for ¹¹ analysis

In order to guarantee operation of the

controlled system, the designed controller is

expected to maintain stability when the

armature resistance, inductance and motor

constant vary around 50% of their nominal

values withparameters of a SEDCM presented

in Appendix A

The chosen weighting functions for



H controller synthesis are as follows

45 0.15





s s

W

0.1 1.15





t

s s W

(14)

Frequency response

Fig.8 and Fig 9 show the frequency

responses of the controlled system with the



H controller and the inverse of the weighting functions (see equations (14)) We

can see in Fig 8 and Fig 9the relevant magnitude plots of the complementary sensitivity and sensitivity functions of the closed-loop system with the performance requirements achieved by W t and W s In

Fig.8, the solid curve shows the response of

the output nwith respect to the reference inputsn ref The inverse of the weighting functions W tare depicted by the dashed line

Similarly, in Fig.9, the solid curve shows the

response of the controlled errorwith respect to the reference inputsn ref The inverse of the weighting functions W sare depicted by the dashed line

Fig 8 Output response with reference input

It is clear from Fig 8 and Fig 9 that the

sensitivity and complementary sensitivity functions are below the inverse of the performance weighting functions The bandwidths corresponding to the channels



ref

n nis about 2 10 2rad/s

Fig 9 Reference input with error

Time response

Fig 10 shows the time responses of the

controlled system for a step input The solid

lineshows the response of the outputn with

respect to the reference inputn ref As it can

be seen from the figure, the controlled output

follows the reference input in about 0.03s

This indicates a fast dynamic of the controlled

system with the Hcontroller

Fig 10 Step response of the output with respect to

the reference input

Robustness test

Robust stability of the controlled system up to

50% uncertainty in the armature resistance,

50% uncertainty in the armature inductance,

50% uncertainty in the motor constant and

50% uncertainty in the field flux is

investigated by setting p r 0.5R a,

0.5



p L , p m 0.5K m, andp f 0.5

The frequency responses of the upper bound

(the solid line) and lower bound (the dashed

line) of the structured singular value over

the frequency interval [0,1000] are shown in

Fig.11 The maximum value of  is

about0.72 10 2 which means that the

controlled system remains stable as long as

the deviations of R a, L a, K m, and from

their nominal values obey the above bounds

Fig 11 Robust stability analysis with

CONCLUSION This paper shows a design of an



H controller for a speed control loop of SEDCMs in which the armature resistance, inductance, and motor constant are considered

to be uncertainties at frozen value of the field flux In order to ensure that the designed controller guarantees the performance achievement even when the field flux is decreased below its nominal value in the field weakening region, the well-known structure singular value analysis is employed to test robustness of the closed-loop controlled system The analysis results shown that the performance of the closed-loop system has been maintained when the armature resistance, inductance, motor constant and excited field are changed in the same time ACKNOWLEDGEMENTS

The authors thank the Thai Nguyen University of Technology (TNUT) for financial support for our research

APPENDIX A

DC MACHINE PARAMETERS Armature resistance Ra 0.076 Armature inductance La 0.00157H

Field resistance R f 310 Field inductance L f 232.5H Field-armature mutual

inductance L af

3.32H

.

kg m

Viscous friction coefficient Bm 0.32N.m.s

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TÓM TẮT

PHÂN TÍCH ỔN ĐỊNH BỀN VỮNG CỦA HỆ THỐNG ĐIỀU KHIỂN

H-INFINITY CHO CÁC ĐỘNG CƠ MỘT CHIỀU KÍCH TỪ ĐỘC LẬP

Nguyễn Thị Mai Hương * , Nguyễn Tiến Hưng

Trường Đại học Kỹ thuật công nghiệp – ĐH Thái Nguyên

Bài báo này giải quyết vấn đềphân tích ổn định bền vững của hệ thống điều khiển tốc độ động cơ một chiều kích từ độc lập có điều chỉnh từ thông Để đơn giản, các tác giả đề xuất coi từ thông kích từ là một tham số không thay đổi và như vậy có thể sử dụng mô hình tuyến tính của động cơ trong thiết kế bộ điều khiển thay vì sử dụng phương pháp thiết kế bộ điều khiển phi tuyến khi kết hợp điều khiển điện áp phần ứng và từ thông kích từ Từ đó, các tác giả đã thiết kế một bộ điều khiển Htuyến tính đảm bảo tính ổn định của hệ thống chống lại sự thay đổi của các tham số bất định của động cơ Cuối cùng, để đảm bảo hệ thống kín có thể làm việc ổn định khi thay đổi từ thông kích từ, các tác giả đã sử dụng phương pháp phân tích giá trị suy biến cấu trúc để đánh giá tính ổn định của hệ thống kín khi cả điện trở, điện kháng phần ứng, hằng số động cơ và từ thông kích từ thay đổi cùng một lúc Các kết quả nghiên cứu sẽ được thể hiện trong môi trường Matlab/Simulink

Từ khóa: Động cơ điện một chiều kích từ độc lập;điều chỉnh từ thông; điều khiển bền vững tuyến

tính;phân tích giá trị suy biến; ổn định bền vững

Ngày nhận bài: 22/8/2018; Ngày phản biện: 16/9/2018; Ngày duyệt đăng: 12/10/2018

*