ROBUSTNESS STABILITY ANALYSIS FOR H-INFINITY CONTROL OF SEPARATELY EXCITED DC MOTORS

(1)

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

(2)

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, uu the control input and y the measuement output, while P is a linear time- invariant system described as

  

 

& _p

p 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_andW_ware 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

p p

x A B B x

z C D E w

y C F u

(2) If the linear time-invariant controller K is expressed as

 

   

  

   

    

&_c _c _c _c

c c

A B

x x

C D

u y (3)

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

 

    

    

  

 

&

z w (4)

where

   

   

   

     

c c p c

c c c

p c c p c

A BD C BC B BD F B C A B F C ED EC D ED F

(5)

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





 

 

 

 (6)

(3)

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

0 0

0 0 0 0

0 0

0 1

0

0 0



 

    

    

     

    

    

p

I T I

I I 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

* *

    

 ^T   ^T 

X U Y V

U V

according to that of in (5). One then arrives at the following synthesis LMIs for

H-design [8]:

0



 

 

 

  

 

  

  

 

 

  

 

  

  

 

 

f

p

T T

p p

T T T

X p p X

p p

T T

p p

T T T

Y p p Y

p p

Y I I X XA A X XB C

B X I D

C D I

AY YA B YC

B I D

C Y D I

(7)

where _X and _Y are basis matrices for

   

ker

C F

0 , ker

B^T

0

E^T

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 _c^{m n}^ as the set of all causal linear operators that mapLⁿ₂

[0, )

 into

2

[0, )



Lm . 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 Fig.3whereM _^{p p}^ is a known causal linear time-invariant operator and   _c^{p p}^ 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

(4)

the scope of robustness analysis for the SEDCM control as



diag(1 1,..., ) : , 1,...,



    

Δ Ir u rIu i i u

where I_r_i is an r r_i _i identitymatrix. The size of the uncertainty Δis measured in terms ofthe largest singular value as ( ) . Since is diagonal, let us recall that

( ) max 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.



¹



( )

inf ( ) : , det( ) 0

^Δ M ^       

Δ I M

and



_Δ

(

M

)



0

if 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

{ }

( ) sup ( ( ))

   



  

 

   ^Δ M j  (9) 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].

HCONTROL OF SEDCMSAND

ROBUSTNESS EVALUATION SEDCM model

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





&

m m

m

m m m

m

x A x B u

y C x ⁽¹⁰⁾

where

1 2

2

,

12 0

0



   

 

 





   

   

  

 





a m

a a a

m

R K

x i

L

x n

x

A L

K J 1

1 0

, , ,

0 1

1

0

20 0

 

       

 

 

 

 

a a

m m

m L

L v

C u

B

J K T

e

ais the back-electromotive force (EMF)of the motor,

T

_e is the electrical torque,

T

_Lis the load torque,

v

_a is the terminal voltage,

R

a is thearmature resistance,

L

_ais the armatureinductance, K_mis the motor constant,

i

_a is the armature current,

n

is the motor speed, J isthe inertial torque of the motor,



is the field flux, respectively.

Linear fractional transformation representation ofSEDCMs

Let

R

_a,

L

_a,

K

_mare uncertain parameters, we can represent them as follows

0 0

0



 

a a r r

a a l l

m m m m

R p

L L K

p

K p

R

(11)

where

R

_a₀,

L

_a₀,

K

_m₀are the nominal values of motor parameters;

p

_r,

p

_l,

p

_mand

,

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.

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Fig. 4. The uncertain model of the SEDCM This model can be rearranged to theM configuration with matrix M G _m given by

1 1

2 2

1 2

1 11 12

2 21 22

1 2

   

   

 

   

 

    

    

    

   

   

&

1 4 44 2 4 4 43

m

l l

r r

n n

i i

G

L a

x x

y u

A B B

y u

C D D

y u

C D D

y u

y T

y v

(12)

in which

0 0 1 0 0

2 0

0



 

 

   

a a m a

m

R L K L

A K

,

0 1 0

1

2

0

0 0 0



 

 

   

l r a m a

m

p p L p L

B p

,

0 2

2

0

 0

 

   La

B

,

 

2 

0 1

C

,

0 0 1 0 0

0 1

0 0

0



 

 

 

 

 

 

a a m a

a

m m

R L K L

C R

K K

,

0 1 0

11

0 0

0 0 0 0



 

 

 

  

 

 

r a m a

p L p L

D

,

12

0

0 0 0 0

0 0 0

 

  

 

T

a

D L

,

   

21 0 0 0 0 , 22  0 0

D D

.

 _m

w z (13)

  

  

  

l r

m n

i

u w u

u u

’

0 0 0



 

 

  

 

 

l r

m m

m

,

  

  

  

l r

m n

i

y z y

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, G_mis the linear time-invariant part of the plant,

_m is the uncertainty block as given in (13), Kin_{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

(6)

Let    ₀ p_{f 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 1

2 2

1 2

1 11 12

2 21 22

1 2

  

   

   

    

    

    

    

 

   

   

&

1 4 4 44 2 4 4 4 43

l l

r r

n n

i i

f f

M

g g

L a

x x

y u

A B B

y u

C D D

y u

C D D

y u

y T

y v

in which

0 1 0

0 0

2 0 0 0





   

 

  

  

 

a m

a a

m

R K

L L

A

K

,

1

0 0 0

1

2 2 0

0 0

0 0 0 0



 



 

   

 

  

  

 

r f m

l a a a

f m

p p p

p L L L

B

p p

2 0

2

0 1

 0



 

 

  

 

 

La

B

,

2

1 0 0 1

  

  

 

C

,

0 1 0

0 0

0 1 0

0

1 0

0 0

0





   

 

  

 

 

 

a m

a a

a

m m

R K

L L

R

K C

K

K K

,

1

0 0 0

11

1

0

0 0

0 0 0 0 0 0

0 0 0 0 0





 

   

 

  

 

  

 

r f m

l a a a

m

p p p

p L L L

D

p

,

12

0

0 0 0 0 0 0

0 1 0 0 0 0



 

 

  

 

T

a

D

L

12

0

0 0 0 0 0 0

0 1 0 0 0 0



 

 

  

 

 

T

a

D

L

,

21

0 0 0 0 0 0 0 0 0 0 0 0

  

  

 

D

,

22

0 0 0 0

  

  

 

D

and the  matrix is given by

0 0 0 0 0



 

 

   

 

 

l r

m m

(7)

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

due to the

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 W s

(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.

9

the 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.

(8)

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

l a

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 R_a ^0.076^

Armature inductance L_a ^0.00157H

Field resistance R_f ³¹⁰

Field inductance L_f ^232.5H

Field-armature mutual inductance L_af

3.32H

Total inertia J ₁₀kg m

.

²

Viscous friction coefficient B_m ^0.32N.m.s

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REFERENCES

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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

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