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Discrete Robust Disturbances Absorbing Control: Applications of Traffic Flow Control
IFAC DECOM-TT 2004 Automatic Systems for Building the Infrastructure in Developing Countries October 3 - 5, 2004 Bansko, Bulgaria Copyright © IFAC AUTOMATIC SYSTEMS FOR BUILDING THE INFRASTRUCTURE IN DEVELOPING COUNTRIES Bansko, Bulgaria, 2004
DISCRETE ROBUST DISTURBANCES
ABSORBING
CONTROL: APPLICATIONS OF TRAFFIC FLOW CONTROL
Nina Nikolova
1754, Sofia, Kliment Ohridski 1, Technical University, block 9, Department ANP e-mail: [email protected]
Abstract: In this paper a proof is given for the applicability of the method for synthesis of discrete-time robust disturbance-absorbing systems with internal model. Copyright © 2004 IFAC
Keywords: robust control, internal model control, discrete systems, disturbance
1. INTRODUCTION
The robust disturbance-absorbing systems with internal model are known in the literature (Morari and Zafiriou 1990; Nikolova 2004; Nikolov 2003) and they combine the effectiveness of robust systems with internal model and disturbance-absorbing systems, as is the method of the free parameter and the balance equation with a partial disturbance absorbing for its syntheses.
2. PURPOSE
The purpose of the present design is to prove the applicability of the method for discrete-time robust control systems synthesis. The task is to design a discrete-time robust control system with and internal model and with partial disturbance-absorbing using data from a numerical example after solving the differential equation of the controller, and to apply it to a programmable controller and to compare the results with the continuous-time cases.
3. INITIAL CONDITIONS
The structure of the robust control system with an internal model with disturbance-absorbing is shown in Fig. 1. It differs from the robust control system with an internal model in the disturbances absorbing filter A das p added in the structure of the basic control-
ler R M p . The absorber and the basic controller
form the controller with disturbances absorbing fea p . ture R DAS M The method of the free parameter and the balance equation of the partial disturbances absorbing solve the problem (Morari and Zafiriou 1990; Nikolova 2004) for the analytical synthesis of processes in case of a prior uncertainty using the criteria (1): 1. robust stability and robust performance of the system for a prior given set of 3 according the requirements (1a); 2. Local Quality Criteria (1b) - process with minimal mean square error H ; 3. minimal Euclidian norm (1c) for the particles of the balance equation for disturbances absorbing on the controlled value; 4. equilibrium for the stability reserve (1c) (optimal modulus),
and in the case of the following initial conditions: a priori given G * , 3 , V , time series for the error of the system in working conditions where u - control signal (fig.1); the matrixes E , G and C (3) reflect dependence on vector y of vector u , [ is disturbance, u - control vector, the system state vector x (3) correspond to the systems with absorbing of disturbances; Q D [ is the model of the generalized disturbance [ with wave structure.
° ° ° a !! ° ° ° ° ° b !! ° ® ° c !! ° ° ° ° ° d !! ° ° °¯
° °° ® K " m ° ° e y0 ¯°
3
2
H § ¨ ¨ ¨ ¨ ¨ ¨ ©
§ ¨ ¨ ©
c
2
^
jZ A
[]
½ ° G * jZ °° 1 "m Z K sup K "m 1 ; ,Z ¾ f ° Z 0 0 ° K Z " sup e y 1 ; e y 1 , m f Z ¿° f f 2 2 1 H 2 t dt ; min H 2 min H jZ d Z R 2S f 0 E ud F [ mi n , u u r u d
³
A k opt G [] jZ SA
[]
jZ A
Q
c jZ
[]
A
§
O ¨¨ 1 W O p D
©
G jZ
CLASS
S
CLASS
S
y C x E u
· 1 ¸ ¸ p ¹ WF O p 1
1
WI O
u
r
d
· ¸ ¸ ¹
· d 0 , Z t 0.1 Z [] ¸ ¸ ¹
1 1 R k , jZ G [] jZ
R
`
S
2
A
R M p k
d "m Z
³
S
§ ¨ 20 l o g 10 ¨ ©
G jZ G* jZ
' G:
· ¸ ¸ ¸ ¸ ¸ ¸ ¹
½ ° ° ° ° ° ° ° ° ° ¾ ° ° ° ° ° ° ° ° °¿
(1)
(2)
F [
(3)
p
(4)
A
DAS
p
A
k
-1 2
Q
A opt
k
jZ
R M
RMM
S
R M
G jZ
jZ A
[]
S
RMM
S
G
Q jZ 2
A
S
[]
(5)
jZ
[]
A
[]
S
° ° ° ® ° ° °® : °° ¯¯
k
0.1 Z
[]
A opt
ª § « 1 ¨ R ¨ «¬ ©
§ ; ¨Z ¨ ©
[]
M
d dZ
j:
G
1
1 R
M
k
j:
[]
jZ
G
[]
Q
2
· ¸ ¸ ¹
A opt
j:
d dZ
jZ
º » »¼
1
d1
(6) ) y RMM H 0
[]
jZ
· ½° 0 ¸¾ ¸° ¹¿
y0 p
Q p
R M p Q p
A p
G* p
R DAS M p
u p
[ p y p G p
Fig. 1. Robust Disturbances Absorbing system
The algorithm of the method is as follows: 1. analytical design of the robust controller R M according with (2), using the known nominal model G ; 2. choose (using the typical waveforms presented in the first column of Table 1 for the a prior known trend of the error for the robust control system with an internal model), of the functional bases f i t
^
presented in column 2 of the same Table 1;
`
3. defining the state model Q D [ of the generalized disturbance with wave structure presented in column 3 of Table 1 and the chosen basis; 4. analytical design of the absorber A as in structure (fig.1) of the system and according (4), where the corresponded function Q 2-1 p is designed as in the end column of the Table 1. The optimal value k
A
(4) of the gain k of the absorber A is determined according (5) or (6).
Table 1. Typical waveforms, functional bases, state model of the generalized disturbance with wave structure and absorber
^ f t `
[ t
Q D [
i
1 2
p
c1 c 2 t
[( t)
100
Q
80 60 40
^1 , t `
d 2[ d t2
X t
1 T p2
^1 , t , t `
d 3[ d t3
X t
1 T p3
^1 ,t ,t ,t `
d 4[ d t4
X t
1 T p4
^1 , e `
d 2[ d[ D d t2 dt
20 0 0 -20
1
2
-40
[
-60
[
-80
[
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
[
-100
[( t)
100
80
c1 c 2 t c 3 t 2
60
40
2
20
0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
-20
-40 -60
-80 -100
[( t)
100
80
c1 c 2 t c 3 t 2 c 4 t 3
60
40
2
20
0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
8
1
2
-20
3
-40
-60
-80 -100
c1 c 2 e D t
[( t )
100 80
60 40
D t
20 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
-20 -40
1 T2 p 2 T1 p 2
X t
-60
-80 -100
c1 c 2 e D t c 3 e E t c 4 e J t
[( t )
100
80 60 40
20 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
^1 ,e
D t
,e
E t
,e
J t
d 4[ d 3[ D E J 3 4 dt dt
`
20
D E DJ E J
-20 -40 -60
-80
d 2[ d[ D EJ d t2 dt
1 T4 p 4 T3 p 3 T2 p 2 T1 p
X t
-100
° a 0 y > k n T0 @ a 1 y > k n 1 T0 @ a n y k T0 ® b 0 y 0 > k m T0 @ b 1 y 0 > k m 1 T0 @ b m y 0 k T0 °¯
(7)
W
y p y0 p
p
p
V jZ ,
p
bm p
m
bm1 p
an p
n
a n1 p
Re p j Im p ; Re p
m 1 n 1
b0 p a0 p
0
(8)
0
V , Im p Z , j
1
(9)
W
z
y z y0 z
E 0 z m E 1 z m 1 E m D 0 z n D 1 z n 1 D n
(10)
z
z
e
T0 p
e
T0V
Re z j Im z ; Re z
cos Z T j e e
T0V
T0V
sin Z T
c o s Z T , Im z
e
T0V
sin Z T
(11)
z
e
T0 p
;
p
T 0 1 l n z
(12)
The differential equation, which describes the control system, is as in (7) ( a 0 y a n and b 0 y b m - coeffi-
(Madjarov 1973) is performed for the designed robust system. The initial conditions and the results from the analytical synthesis are presented in Table 2. The results are modeled and simulated.
plex variable, y 0 p is the input signal, y p is the controlled signal.
The differential equation, which is appropriate for controllers programming is:
cients). The continuous time transfer W p function corresponded to (7) is (8), where p (9) is a com-
y 0 k T0
The generalized equation (10) for the discrete time transfer function (Madjarov 1973) of the dynamic system W z , as a discrete analog of (8) is defined by using discrete operators or their approximations (8), where: z (11) - complex variable, corresponded to the variable p and the sampling time T 0 according (12).
k 1 T 0.53 y k 2 T 0.311 y k 3 T 0.011 y k 4 T (13) 0.019 u k T 0.015 u k 1 T 0.022 u k 2 T 0.012u k 3 T 0.006 u k 4 T 1.73 y 0
0
0
0
0
0
0
0
0
0
0
0
0
5. CONCLUTION
4. PROBLEM SOLVE
Applying the known method of the free parameter and the balance equation with a partial disturbances absorbing for syntheses (Morari and Zafiriou 1990; Nikolova 2004) on a numerical example, a robust control system with an internal model and disturbances absorbing is designed. The proposed method is applied for the synthesis of a control system of a plant-model for traffic flow process (Nikolov E. 2003), are shown in Tabl.2. Using the Tustin operator in case of sampling time T 0 1 s a z-transformation
The present work describes the tasks for the synthesis of robust control systems with an internal model and disturbance-absorbing filter. A new method for the synthesis of discrete-time robust control system of the presented class is shown. It differs from the known methods in its applicability for direct controller programming in distributed control systems. The results are compared with the continuous systems and the analysis confirms the feasibility of the new method.
Table 2. The initial conditions and the results from the analytical synthesis of controllers and plantmodels for traffic flow process
G p
1 , 25
10 p 1
p p
2 2
3 p3 3 p3
R
M
p
0 ,16088
1 § ¨ 1 0.909 p p 2 5 p 1 ¨© 8p
0 ,05952 z 0 ,05952 G z z 0 , 9048
R
M
z
0 ,16
1 · ¸ ¸ 0.998 p 1 ¹
1 ,8866
0 ,12 z 2 0 , 25 z 0 ,12 z 2 0 , 4 z 0 , 33
0 ,07692 z 0 ,1538 z 1 z 0 ,1538 z 0 ,07692 2
2
u
§ 0 , 33 z 0 , 33 0 ,61 z 0 ,61 u¨ ¨ z 0 , 33 z 0 , 33 ©
0 ,020 z 0 ,041 z 0 ,02 z 1 ,33 z 0 ,33 2
2
· ¸ ¸ ¹
ROBUST DISTURBANCES-ABSORBING SYSTEMS WITH INTERNAL MODEL
DISKRET ROBUST DISTURBANCESABSORBING SYSTEMS WITH INTERNAL MODEL T0=1 s, Tustin
Fig. 2. The results from the simulations of a continuous control system and a discrete control system of a plant-model for traffic flow process REFERENCES
Morari M., E. Zafiriou (1990), Robust Process Control, p. 479, Prentice Hall, Englewood Cliffs, NJ Nikolova N. (2004), Robust Control Systems Research, Ph.D. Thesis, p. 40, Technical University, Sofia
Nikolov E. (2003), Applied Methods for Process Control - I, p. 358, Technical University Sofia, ISBN 954-438-334-4 Madjarov N. (1973), Theory of Automatic Control. Impulse Systems - III, p. 333, Technica, Sofia
DISCRETE ROBUST DISTURBANCES
ABSORBING
CONTROL: APPLICATIONS OF TRAFFIC FLOW CONTROL
Nina Nikolova
1754, Sofia, Kliment Ohridski 1, Technical University, block 9, Department ANP e-mail: [email protected]
Abstract: In this paper a proof is given for the applicability of the method for synthesis of discrete-time robust disturbance-absorbing systems with internal model. Copyright © 2004 IFAC
Keywords: robust control, internal model control, discrete systems, disturbance
1. INTRODUCTION
The robust disturbance-absorbing systems with internal model are known in the literature (Morari and Zafiriou 1990; Nikolova 2004; Nikolov 2003) and they combine the effectiveness of robust systems with internal model and disturbance-absorbing systems, as is the method of the free parameter and the balance equation with a partial disturbance absorbing for its syntheses.
2. PURPOSE
The purpose of the present design is to prove the applicability of the method for discrete-time robust control systems synthesis. The task is to design a discrete-time robust control system with and internal model and with partial disturbance-absorbing using data from a numerical example after solving the differential equation of the controller, and to apply it to a programmable controller and to compare the results with the continuous-time cases.
3. INITIAL CONDITIONS
The structure of the robust control system with an internal model with disturbance-absorbing is shown in Fig. 1. It differs from the robust control system with an internal model in the disturbances absorbing filter A das p added in the structure of the basic control-
ler R M p . The absorber and the basic controller
form the controller with disturbances absorbing fea p . ture R DAS M The method of the free parameter and the balance equation of the partial disturbances absorbing solve the problem (Morari and Zafiriou 1990; Nikolova 2004) for the analytical synthesis of processes in case of a prior uncertainty using the criteria (1): 1. robust stability and robust performance of the system for a prior given set of 3 according the requirements (1a); 2. Local Quality Criteria (1b) - process with minimal mean square error H ; 3. minimal Euclidian norm (1c) for the particles of the balance equation for disturbances absorbing on the controlled value; 4. equilibrium for the stability reserve (1c) (optimal modulus),
and in the case of the following initial conditions: a priori given G * , 3 , V , time series for the error of the system in working conditions where u - control signal (fig.1); the matrixes E , G and C (3) reflect dependence on vector y of vector u , [ is disturbance, u - control vector, the system state vector x (3) correspond to the systems with absorbing of disturbances; Q D [ is the model of the generalized disturbance [ with wave structure.
° ° ° a !! ° ° ° ° ° b !! ° ® ° c !! ° ° ° ° ° d !! ° ° °¯
° °° ® K " m ° ° e y0 ¯°
3
2
H § ¨ ¨ ¨ ¨ ¨ ¨ ©
§ ¨ ¨ ©
c
2
^
jZ A
[]
½ ° G * jZ °° 1 "m Z K sup K "m 1 ; ,Z ¾ f ° Z 0 0 ° K Z " sup e y 1 ; e y 1 , m f Z ¿° f f 2 2 1 H 2 t dt ; min H 2 min H jZ d Z R 2S f 0 E ud F [ mi n , u u r u d
³
A k opt G [] jZ SA
[]
jZ A
Q
c jZ
[]
A
§
O ¨¨ 1 W O p D
©
G jZ
CLASS
S
CLASS
S
y C x E u
· 1 ¸ ¸ p ¹ WF O p 1
1
WI O
u
r
d
· ¸ ¸ ¹
· d 0 , Z t 0.1 Z [] ¸ ¸ ¹
1 1 R k , jZ G [] jZ
R
`
S
2
A
R M p k
d "m Z
³
S
§ ¨ 20 l o g 10 ¨ ©
G jZ G* jZ
' G:
· ¸ ¸ ¸ ¸ ¸ ¸ ¹
½ ° ° ° ° ° ° ° ° ° ¾ ° ° ° ° ° ° ° ° °¿
(1)
(2)
F [
(3)
p
(4)
A
DAS
p
A
k
-1 2
Q
A opt
k
jZ
R M
RMM
S
R M
G jZ
jZ A
[]
S
RMM
S
G
Q jZ 2
A
S
[]
(5)
jZ
[]
A
[]
S
° ° ° ® ° ° °® : °° ¯¯
k
0.1 Z
[]
A opt
ª § « 1 ¨ R ¨ «¬ ©
§ ; ¨Z ¨ ©
[]
M
d dZ
j:
G
1
1 R
M
k
j:
[]
jZ
G
[]
Q
2
· ¸ ¸ ¹
A opt
j:
d dZ
jZ
º » »¼
1
d1
(6) ) y RMM H 0
[]
jZ
· ½° 0 ¸¾ ¸° ¹¿
y0 p
Q p
R M p Q p
A p
G* p
R DAS M p
u p
[ p y p G p
Fig. 1. Robust Disturbances Absorbing system
The algorithm of the method is as follows: 1. analytical design of the robust controller R M according with (2), using the known nominal model G ; 2. choose (using the typical waveforms presented in the first column of Table 1 for the a prior known trend of the error for the robust control system with an internal model), of the functional bases f i t
^
presented in column 2 of the same Table 1;
`
3. defining the state model Q D [ of the generalized disturbance with wave structure presented in column 3 of Table 1 and the chosen basis; 4. analytical design of the absorber A as in structure (fig.1) of the system and according (4), where the corresponded function Q 2-1 p is designed as in the end column of the Table 1. The optimal value k
A
(4) of the gain k of the absorber A is determined according (5) or (6).
Table 1. Typical waveforms, functional bases, state model of the generalized disturbance with wave structure and absorber
^ f t `
[ t
Q D [
i
1 2
p
c1 c 2 t
[( t)
100
Q
80 60 40
^1 , t `
d 2[ d t2
X t
1 T p2
^1 , t , t `
d 3[ d t3
X t
1 T p3
^1 ,t ,t ,t `
d 4[ d t4
X t
1 T p4
^1 , e `
d 2[ d[ D d t2 dt
20 0 0 -20
1
2
-40
[
-60
[
-80
[
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
[
-100
[( t)
100
80
c1 c 2 t c 3 t 2
60
40
2
20
0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
-20
-40 -60
-80 -100
[( t)
100
80
c1 c 2 t c 3 t 2 c 4 t 3
60
40
2
20
0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
8
1
2
-20
3
-40
-60
-80 -100
c1 c 2 e D t
[( t )
100 80
60 40
D t
20 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
-20 -40
1 T2 p 2 T1 p 2
X t
-60
-80 -100
c1 c 2 e D t c 3 e E t c 4 e J t
[( t )
100
80 60 40
20 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
^1 ,e
D t
,e
E t
,e
J t
d 4[ d 3[ D E J 3 4 dt dt
`
20
D E DJ E J
-20 -40 -60
-80
d 2[ d[ D EJ d t2 dt
1 T4 p 4 T3 p 3 T2 p 2 T1 p
X t
-100
° a 0 y > k n T0 @ a 1 y > k n 1 T0 @ a n y k T0 ® b 0 y 0 > k m T0 @ b 1 y 0 > k m 1 T0 @ b m y 0 k T0 °¯
(7)
W
y p y0 p
p
p
V jZ ,
p
bm p
m
bm1 p
an p
n
a n1 p
Re p j Im p ; Re p
m 1 n 1
b0 p a0 p
0
(8)
0
V , Im p Z , j
1
(9)
W
z
y z y0 z
E 0 z m E 1 z m 1 E m D 0 z n D 1 z n 1 D n
(10)
z
z
e
T0 p
e
T0V
Re z j Im z ; Re z
cos Z T j e e
T0V
T0V
sin Z T
c o s Z T , Im z
e
T0V
sin Z T
(11)
z
e
T0 p
;
p
T 0 1 l n z
(12)
The differential equation, which describes the control system, is as in (7) ( a 0 y a n and b 0 y b m - coeffi-
(Madjarov 1973) is performed for the designed robust system. The initial conditions and the results from the analytical synthesis are presented in Table 2. The results are modeled and simulated.
plex variable, y 0 p is the input signal, y p is the controlled signal.
The differential equation, which is appropriate for controllers programming is:
cients). The continuous time transfer W p function corresponded to (7) is (8), where p (9) is a com-
y 0 k T0
The generalized equation (10) for the discrete time transfer function (Madjarov 1973) of the dynamic system W z , as a discrete analog of (8) is defined by using discrete operators or their approximations (8), where: z (11) - complex variable, corresponded to the variable p and the sampling time T 0 according (12).
k 1 T 0.53 y k 2 T 0.311 y k 3 T 0.011 y k 4 T (13) 0.019 u k T 0.015 u k 1 T 0.022 u k 2 T 0.012u k 3 T 0.006 u k 4 T 1.73 y 0
0
0
0
0
0
0
0
0
0
0
0
0
5. CONCLUTION
4. PROBLEM SOLVE
Applying the known method of the free parameter and the balance equation with a partial disturbances absorbing for syntheses (Morari and Zafiriou 1990; Nikolova 2004) on a numerical example, a robust control system with an internal model and disturbances absorbing is designed. The proposed method is applied for the synthesis of a control system of a plant-model for traffic flow process (Nikolov E. 2003), are shown in Tabl.2. Using the Tustin operator in case of sampling time T 0 1 s a z-transformation
The present work describes the tasks for the synthesis of robust control systems with an internal model and disturbance-absorbing filter. A new method for the synthesis of discrete-time robust control system of the presented class is shown. It differs from the known methods in its applicability for direct controller programming in distributed control systems. The results are compared with the continuous systems and the analysis confirms the feasibility of the new method.
Table 2. The initial conditions and the results from the analytical synthesis of controllers and plantmodels for traffic flow process
G p
1 , 25
10 p 1
p p
2 2
3 p3 3 p3
R
M
p
0 ,16088
1 § ¨ 1 0.909 p p 2 5 p 1 ¨© 8p
0 ,05952 z 0 ,05952 G z z 0 , 9048
R
M
z
0 ,16
1 · ¸ ¸ 0.998 p 1 ¹
1 ,8866
0 ,12 z 2 0 , 25 z 0 ,12 z 2 0 , 4 z 0 , 33
0 ,07692 z 0 ,1538 z 1 z 0 ,1538 z 0 ,07692 2
2
u
§ 0 , 33 z 0 , 33 0 ,61 z 0 ,61 u¨ ¨ z 0 , 33 z 0 , 33 ©
0 ,020 z 0 ,041 z 0 ,02 z 1 ,33 z 0 ,33 2
2
· ¸ ¸ ¹
ROBUST DISTURBANCES-ABSORBING SYSTEMS WITH INTERNAL MODEL
DISKRET ROBUST DISTURBANCESABSORBING SYSTEMS WITH INTERNAL MODEL T0=1 s, Tustin
Fig. 2. The results from the simulations of a continuous control system and a discrete control system of a plant-model for traffic flow process REFERENCES
Morari M., E. Zafiriou (1990), Robust Process Control, p. 479, Prentice Hall, Englewood Cliffs, NJ Nikolova N. (2004), Robust Control Systems Research, Ph.D. Thesis, p. 40, Technical University, Sofia
Nikolov E. (2003), Applied Methods for Process Control - I, p. 358, Technical University Sofia, ISBN 954-438-334-4 Madjarov N. (1973), Theory of Automatic Control. Impulse Systems - III, p. 333, Technica, Sofia