Periodical report

postdoc researcher Dr. Aleksey Balabanov

Ръководител проф. д.т.н. Тодор Стоилов

27.05.2015

2

Brief introduction about myself Crimea, Sevastopol

Senior Lecturer on department of Technical cybernetics

Automatic control system,

Algorithms and data structures,

Components and structure of control systems

Sevastopol National Technical University

3

The control problem considers

the global management of the

traffic. The green in one direction

corresponds to red to other

direction.

Cooperative movement of vehicles in a platoon is a way

- to increase traffic capacity

- to maintain suitable distance and keep safe movement

- to reduce fuel consumption

- …

My planed work here

The research during the AComIn project will constitute a feasibility study of the ideas laid down in optimization and intelligent control area. The

main aim of the research will be design, modeling, test and simulation of the control algorithms, which will be applied for large scale and

complex systems. The targeted systems for control and optimization will be from transport domain.

The following two general classes of transport systems will be under consideration

cooperative car-following in a tightly spaced platoon intersection green time duration control

4

My so far achievements in the frame of ACOMIN I have processed a contemporary overview of the problems (my library contains 57 books, 150 articles)

I have performed some numerical explorations and repeated someone results (towards cooperative

car-following and intersection control)

I have learnt SmartLab environments (AIMSUN software)

I have prepared three theses to science conferences. The thesis topics are

Fast decentralized optimal control algorithm on the basis of Bass’ relation for vehicles in a platoon1 (accepted)

Применение метода резольвенты при поиске решений некоторых частных видов матричных алгебраических

уравнений Риккати2 (published)

H бесконечность оптимизация ведомого транспортного средства в колонне2 (published)

I have prepared a byproduct conference thesis

Применение процедур метода резольвенты при проверке устойчивости матрицы2 (published)

I have prepared an article with topic

An issue of multiple solution search of the linear-quadratic optimization problem for not completely controllable

dynamic system (is searched where to publish)

Conferences:

1. The 7th Balkan Conference in Informatics. September 2-4, 2015. Craiova, Romania

2. Intelligent Systems, Control and Mechatronics – 2015. May 13-15, 2015. Sevastopol, Sevastopol State University

5

Other activities

- workshop in the frame of COST Committee Core Group meeting “Satellite Positioning

Performance Assessment for Road Transport“, Sofia, Best Western EXPO hotel, 10th

March 2015

- Poster “Urban traffic management”, Doors Open Days in IICT – BAS, Sofia, Bulgaria,

April 17-18, 2015.

- interview for annual ACOMIN project video report

6

Working with SmartLab environment

7

8

The function editor allows the user to define cost functions using the Python language. Several types of functions are

presented in Aimsun

9

Reinforce of decentralized linear-quadratic optimal control of vehicles in a platoon. The issue of control for not completely

controllable dynamic system. Cooperative movement of vehicles in a platoon is a way

- to increase traffic capacity

- to maintain suitable distance and keep safe movement

- to reduce fuel consumption, est.

Main challenges:

- the platoon model is a large scale dynamic system

- some vehicle parameters are not precisely known and we need on-board implementation of

control law synthesis

Tasks:

- the platoon should maintain assigned constant velocity plv and safe inter-vehicle distance des

id ;

10

DECENTRALIZED CONTROL SCHEME AND OVERLAPPING STRUCTURE

Linearized model of N vehicles platoon are described as

(1)

11

PROBLEM FORMULATION

Constant time headway distance policy des v x

i i i id k v k , (2)

Distance deviation i id d – 1 1

des v x

i i i i i i id x x l k v k . (3)

The leader vehicle dynamic

0

1 1 1 1 1 1 1( ) ( ) ( ), (0)t t t x A x B u x x , (4)

where 1 1

1

0 1

0

A , 1 1

1

0

B , 1 1 1

Tv a x , 1 1

desa u . (5)

12

Model of second vehicle (first subsystem) and its control

0

2 2 2 2 2 2 1 2 2( ) ( ) ( ) ( ), (0)t t t t x A x B u B u x x , (6)

where

1

2

2 2

1

2

0 1 0 0 0

0 0 0 0

1 0 0 1

0 0 0 0 1

0 0 0 0

vk

A , 2

1

2

0

0

0

0

B ,

1

2

2

0

0

0

0

B , (7)

2 1 1 2 2 2

Tv a d v a x , 2 2

desa u , 1 1 1 1( , )f v a u . (8)

2 2 1 1 2 2 2( , , , , )f v a d v a u – ? (9)

Control of third vehicle

3 3 2 2 2 3 3 3( , , , , , )f d v a d v a u , (10)

or

3 3 1 1 2 2 2 3 3 3( , , , , , , , )f v a d v a d v a u . (11)

13

The problem is control synthesis for not completely controlled system

11 2,11 ,1 1

2 221 22 2

( ) ( )( )

( ) ( )

n mn nt tdt

t tdt

0A 0x xu

x xA A B, (12)

where 1

1

nRx is a state vector that connected to preceding vehicle(s),

2

2

nRx is a state vector that connected to current vehicle,

11A is a Hurwitz matrix.

The goal is to construct numerical efficient approach that will be appropriate for vehicle on-board

computer implementation.

14

Linear quadratic (LQ) optimization for system (12)

System

0( ) ( ) ( ), (0) , , n mt t t R R x Ax Bu x x x u . (13)

Performance index

T T

0

( ) ( ) ( ) ( ) ( ) minJ t t t t dt

u x Px u Ru , (14)

,A B, 0, 0T T P P R R are known matrix.

( ) ( )t tu Gx , (15)

1 T G R B X, (16)

X is a stabilizing solution (SS) of algebraic Riccati equation (ARE)

,

T

n n A X XA P XQX 0 , 1 TQ BR B . (17)

* A A BG A QX (18)

15

Hamilton matrix 2 2n n

TR

A QH

P A (19)

2( ) det( )ns s I H (20)

1

2( ) ( )ns s Θ I H (21)

if ( ) H , then ( ) H , (22)

BASS’ RELATION

2 ,( ) ,n

n nq

IH 0

X (23)

1

1 1 0 2( ) n n

n nq q q q

H H H H I (24)

0

( )n

k

k

k

q s q s

(25)

( ) ( 1) ( ) ( )ns q s q s (26)

RESOLVENT METHOD

2 ,

n

n n

IU 0

X (27)

1

( )2

C

s dsj

U Θ (28)

16

Special structure of algebraic Riccati equation

,

T

n n A X XA P XQX 0 , (29)

1 211 ,

21 22

n n

A 0A

A A, 1 ,

2

n m

0B

B, 11 21

21 22

T

P PP

P P,

1 1 1 2

2 1

, ,

, 22

n n n n

n n

0 0Q

0 Q, 1

22 2 2

TQ B R B , (30)

1 1 2 1 2 2 2

11 21 22 2, , ,n n n n n n n m

R R R R

A A A B ,

1 1 2 1 2 2

11 21 22, , ,n n n n n n

R R R

P P P 22Q 2 2n nR

11 21

21 22

T

X XX

X X, 1 1 2 1 2 2

11 21 22, ,n n n n n n

R R R

X X X . (31)

2 222 22 22 22 22 22 22 22 ,

T

n n A X X A P X Q X 0 , (32)

1 211 ,

*

21 22 21 22 22 22

n n

A 0A

A Q X A Q X, (33)

1

2 21 22

T G R B X X . (34)

17

Bass’ approach steps:

1) finding characteristic polynomial (20) of the Hamiltonian matrix (19);

2) performing factorization (26);

3) finding matrix series of matrix (24);

4) getting solution of the linear equation (23).

1 2 1 1 1 2

2 1

2 1

11 , , ,

21 22 , 22

11 21 11 21

21 22 , 22

n n n n n n

n n

T T T

T

n n

A 0 0 0

A A 0 QH

P P A A

P P 0 A

. (35)

18

22 22

22 22

T

A QH

P A,

22( ) det ns s I H (36)

2( ) ( 1) ( ) ( )n

s g s g s ; (37)

1 , 0,1,..., 1k k k n H H H , 11 12

21 22

k k k

k

k k

H HH H

H H, 1 1 2

2 1 2

,0

,

n n n

n n n

I 0H

0 I (38)

11 11 12

1 22 22

12 11 12

1 22 22

21 21 22

1 22 22

22 21 22

1 22 22

,

,

,

,

0,1,..., 1,

k k k

T

k k k

k k k

T

k k k

k n

H H A H P

H H Q H A

H H A H P

H H Q H A

(39)

11 12

21 22

( )g

V VV H

V V1 2

V V , (40)

2 22 22 1 2 ,n n V X V 0 . (41)

19

according to (33) ( ) ( ) ( )q s c s g s , (42)

1 11( ) det nc s s I A (43)

2 22 22 22( ) det( )ng s s I A Q X (44)

( ) ( ) ( )q c gH H H . (45)

( ), ( )c gc g V H V H , (46)

1 1

1

1

1 1 0 2

n nc

n nc c c

V H H H I , 2 2c n nR V , (47)

2 2

2

1

1 1 0 2

n ng

n ng g g

V H H H I , 2 2g n nR V . (48)

1 2 1 1 1 2

2 1

2 1

11

, , ,

21 22 24

,

31 32 33 34

41 42 44

,

k n n n n n n

k k n n kk

k k k k

k k n n k

H 0 0 0

H H 0 HH

H H H H

H H 0 H

. (49)

22 11 42 21 24 21 44 22, , , , 1,2,...,k k k k k k k k k H H H H H H H H (50)

11 33

11 11, , 1,2,...kk T

k k k H A H A . (51)

20

1 2 1 1 1 2

2 1

2 1

11 , , ,

21 22 , 24

31 32 33 34

41 42 , 44

n n n n n n

n n

n n

V 0 0 0

V V 0 VV

V V V V

V V 0 V

, (52)

1 2 1 1 1 2

2 1

2 1

11 , , ,

21 22 , 24

31 32 33 34

41 42 , 44

c

n n n n n n

c c c

n nc

c c c c

c c c

n n

V 0 0 0

V V 0 VV

V V V V

V V 0 V

,

1 2 1 1 1 2

2 1

2 1

11 , , ,

21 22 , 24

31 32 33 34

41 42 , 44

g

n n n n n n

g g g

n ng

g g g g

g g g

n n

V 0 0 0

V V 0 VV

V V V V

V V 0 V

, (53)

22 11 24 12 42 21 44 22, , ,g g g g V V V V V V V V . (54)

11 11 1, 1( )c

n nc V A 0 , 11 11 11 1, 1

c g

n n V V V 0 (55)

21

2 1

1 2 2

2 1

, 24 22

21

33 34 32 ( 2 ),

22

, 44 42

n nT

n n n

n n

0 V VX

V V V 0X

0 V V

. (56)

1 233 21 32 34 22 ,

T

n n V X V V X 0 , (57)

32 32 11 33 32 34 21

34 32 12 33 34 34 22

33 33 33

,

,

.

c c g c

c c g c

c g

V V V V V V V

V V V V V V V

V V V

(58)

32 33 34 32 33 34, , , , ,c c c g g gV V V V V V (59)

32 32 33 34

1 22 21 22

34 32 33 34

1 22 21 22

33 33 T

1 22 1 2

,

,

, 0,1, ,max( , ) 1,

T

k k k k

T T

k k k k

k k k n n

H H A H P H P

H H Q H A H A

H H A

(60)

1 2

32

0 ,n nH 0 , 1 2

34

0 ,n nH 0 , 2

33

0 nH I .

22

To resolvent method

1

2 1 0 2

0

1,

2

n

n n k k n

k

U I H Ψ Ψ I , (61)

21

2 ( )

r

r

C

sds

j s

, 0,1,..., 1r n , (62)

2

1

2n U I HS , (63)

1

( )2

C

x dsj

S Ω

, 2x s , S=1

1

0

nk

n k

k

G , (64)

0 1 1 0...m m m m , 0 1 , 0,1,..., 1m n , 2( ) ( )x s

s x

, (65)

1 2

2 1 2 1 2 2

1 2

11 12 13 ,

, 22 , ,2

31 41 11 ,

41 42 12 22

n n

n n n n n n

T T

n n

T T

G G G 0

0 G 0 0G H

G G G 0

G G G G

, (66)

23

2

1

11 11 12 13 21 31 22 22( ) ( ), ( ) ( ), ( ) ( ), ( ) ns s s s s s s s

Θ Θ Θ Θ Θ Θ Θ I A (67)

11 11 12 13 21 31, , U U U U U U ,

222 22

1( )

2n

C

s dsj

U Θ 0

. (68)

( )xΩ 1

2nx

I G

1 2

2 1 2 1 2 2

1 2

11 12 13 ,

, 22 , ,

31 41 11 ,

41 42 12 22

( ) ( ) ( )

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

n n

n n n n n n

T T

n n

T T

x x x

x

x x x

x x x x

Ω Ω Ω 0

0 Ω 0 0

Ω Ω Ω 0

Ω Ω Ω Ω

, (69)

1 2

2 1 2 2 2 1 2 2

1 1 2

2

11 12 12 ,

, , , ,

21 41 11 ,

41 42 12

n n

n n n n n n n n

T T

n n n

T

n

U U U 0

0 0 0 0U

U U I U 0

U U U I

, (70)

12 11 12 41

T X X U U . (71)

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FLOPS EVALUATION

Flop is one of the following operations , , , /, …

Supposed for simplicity:

1. all steps are performed with the same algorithms;

2. Hamilton matrix is not sparse;

3. Case 1 2n n ;

4. Relative profit.

the volume of reduced computations:

BASS’ RELATION RESOLVENT METHOD

on step 1 – up to 4 times profit;

on step 2 – up to 6,6 times profit;

on step 3 – up to 8,5 times profit;

on step 4 – up to 4,2 times profit.

on step 1 – up to 4 times profit;

on step 2 – up to 3,7 times profit;

on step 3 – up to 4,5 times profit;

on step 4 – up to 6,2 times profit.

25

Bass’ approach applying for platoon control

22 2 2 2

1 2 3 4 1

0

( )i i i i i i iJ p d p v p a p v v dt

u u (72)

2( ) ( 1) ( ) ( )n

s g s g s , 3 2

2 1 0( )g s s g s g s g (73)

2

2

2

2 2 1 3

2

0 1

1 0 2

1 2 3 1 3 0 2 3

1

1

1

0

v v v

i i i

v

i

v

i

v

i

v v

i i

k k g k

k g

g k g g p

g k p

g g p

g k g k p g p g g p

V ,

2

2

2

2 2 1 3

2

0 1

1 0 2

1 2 3 1 3 0 2 3

1

1

1

0

v v v

i i i

v

i

v

i

v

i

v v

i i

k k g k

k g

g k g g p

g k p

g g p

g k g k p g p g g p

V (74)

2 22 22 1 2 ,n n V X V 0 (75)

26

1

1

1

3

1

2

0 1

0 0 1

v

ik

f f f

A , 3 2 3 2

3 1 1 2 1 2 1 0( ) v

is f s f k f s f sc s s cs c c (76)

1 4

2 1 1 4 1 2 4 2 4 1

1 3 4 2

32

4 4

0 0vc

i

f p

c p c p p f p c p k p

p f p c p p

V , 33 11

c Tc V A , 434

1

1 2

2 3

0 0

0

0 0

c

f

c f p

c f

V , (77)

434

1

2 1

2 3

0 0

0

0 0

g

f

f g p

g f

V ,

1 4

2 1 2 4 1 1 4 2 4 1

1 3 4 2

32

4 4

0 0vg

i

f p

g p f p p g p g p k p

p f p g p p

V , 33 11

g Tg V A (78)

Then by (58) we came to equation

1 233 21 32 34 22 ,

T

n n V X V V X 0 (79)

27

An issue of multiple solutions search of the linear-quadratic optimization problem for not

completely controllable dynamic system

Weight matrices ,P R are hard to choose

T T

0

( ) ( ) ( ) ( ) ( ) minJ t t t t dt

u x Px u Ru , (80)

RESOLVENT METHOD. The ratio of reduced flops

step First search of ARE SS Repeated search of ARE SS

1 on 4 times on 8 times

2 on 3,7 times on 6,9 times

3 on 4,5 times on 5,3 times

4 on 6,2 times on 6,2 times

Choose

weight matrices is the result

satisfactory?

Solve LQ problem end

yes

no

28

Modeling Results

0 5 10 15 20 25 30-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

time, [sec]

dis

tance

subsystem1

subsystem2

subsystem3

0 5 10 15 20 25 30 35-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

velo

city

time, [sec]

vehicle1

vehicle2

vehicle3

vehicle4

Figure 5. Spacing distance for N=4 Figure 6. Velocity for N=4

0 5 10 15 20 25 30-0.3

-0.2

-0.1

0

0.1

0.2

forc

e

time, [sec]

vehicle1

vehicle2

vehicle3

vehicle4

0 5 10 15 20 25 30 35

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

time, [sec]

dis

tance

subsystem1

subsystem2

subsystem3

subsystem4

subsystem5

subsystem6

subsystem7

Figure 7. Forces response, /i iF F (N=4) Figure 8. Spacing distance for N=8

29

Future activities, planned work and ideas

1. Continue explore exploration SmartLab environment toward the questions of how to:

- perform road net build in automated way;

- integrate own intersection control strategies.

2. Explore and implement new ideas toward intersection control strategies.

3. Connect two problems of cooperated vehicles move through intersection and

intersection control.

4. Consider a question of how to choose controllable or observable subspace for system

without preliminary exploration

5. Consider linear transformation for Bass’ and resolvent methods.

6. Explore numerical stability of resolvent methods.

7. Write an article about automated quality evaluation of control system.

30

Thank you for your kind attention

Your comments and questions, please

??