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Robust Regulation of Metro Lines Using Time-Variant Control Law
Copyright © IFAC Robust Control Design Milan, Italy, 2003
ELSEVIER
IFAC PUBUCATIONS www.elsevier comllocale/ifac
ROBUST REGULATION OF METRO LINES USING TIME-VARIANT CONTROL LAW
Winderson O. Assis • Basilio E. A. Milani ••
• Escola de Engenharia Maud, Instituto Maud de Tecnologia, Prara Maud I, 09580-900 Sao Caetano do SuI SP, Brazil. •• Faculdode de Engenharia EIetrica e de Computarao, Universidade Estadual de Campinas, CP' 6101, 13081-970 Campinas SP, Brazil.
Abstract: This paper presents a new formulation for robust traffic regulation of metro lines. The approach uses a linear time-variant state feedback control law computed in real-time. The formulation is based on the stability analysis using the eigenvalues of the feedback system and consider the traffic model uncertainties and the constraints on its state and control variables. It is assumed the constraints on model variables and uncertain parameter domain defined by convex compact polyhedra. The simplicity and computational efficiency of this formulation makes it applicable to real-time regulation of nowadays metro lines and presents better performance than the obtained using already known robust regulation approaches. Copyright © 2003 [FAC Keywords: Robust regulation, Metro lines, Linear programming, Optimization, Stability analysis.
Milani and Correa (1997) proposed a fonnulation based on linear programming for robust constrained regulation of metro lines, considering explicitly the constraints on the state and control variables, the parameter uncertainties and random operational disturbances. The constraints on model variables and uncertain parameter domain were defined by convex compact polyhedra. Exploring structural properties of the traffic model, the method gives a linear time-invariant control law, with bidiagonal structure, obtained solving independent linear programming problems, related one to one with the platforms of the line.
1. INTRODUcnON
The traffic of trains in high-frequency metro lines is known to be naturally unstable. Consider, for example, a train delayed with respect to the nominal time schedule of the line (Cury et al., 1980), (Assis and Milani, 2(02). Due to this delay, its time interval relative to the preceding train is increased, more passengers will be waiting at the platforms to get on the train, resulting in increased delay (Campion et al., 1985). Thus, traffic control is necessary to keep the departure time at the stations, the time interval between successive trains as close as possible to their nominal values. The control actions consist of instructions given by the controller to the trains at the stations, increasing or decreasing their speed during the running time to the next station and/or their waiting time at the platform. The control is naturally constrained by the speed range allowed to the trains, minimum waiting time at the stations and traffic security rules. The time interval between successive trains is also constrained by the traffic security rules and the maximum train occupancy.
This paper presents a new fonnulation for robust traffic regulation of metro lines using a possibly nonlinear time-variant state feedback control law computed in real time. The formulation is based on eigenvalues assignement of the closed-loop matrix.
2. TRAFFIC MODELING
=
Consider n trains (i I,···, n) in an open metro line with N platforms (k = 1,2"", N) (figure I).
275
where Yj. Uj and Vj are N dimensional vectors representing the state, the control and the external disturbances of the system, respectively; C is the vector of uncertain parameters; A(C) and B(C) are given by:
Defining Yi(k) as the deviation of the actual departure time of train i in platform k with respect to the nominal time schedule, Campion et al. (1985) proposed the basic equation for the traffic of trains: 2
N
N-I
-e(l) l-e(1)
0--0-55-0-0
A(C) ~
0
-e(2)
l-e(2)
l-e(2)
0
0
sentido do rifego
Fig. I. Open metro line with N platforms
1
l-e(1)
(1- c(k+ 1))Yi(k+ I)
0
B(C) ~
=Yi(k)-
-c(k+ I)Yi_l(k+ I) +ui(k+ I) +vi(k+ I)
(1)
0
where: 0 :5 c(k) :5 I is related to passengers demand, inherently uncertain; vi(k) is the random disturbance; ui(k) is the control action applied to train i between the platforms k - I and k in order to increase (ui(k) > 0) or to decrease (ui(k) < 0) the running time.
0
I
0
,I 0
(5)
-e(N) I l-e(N) l-e(N) 0 I
l-e(2)
0
. ,-u
(6)
3. CONSTRAINED ROBUST REGULATION PROBLEM (CRRP) This section presents the constrained robust regulation approach proposed in (Milani and Correa, 1997). Consider an open metro line with N platforms and equations (4)-(6). Consider also the uncertain parameters C, the disturbances Vj and the control Uj, restricted to the polyhedrons:
Throughout this paper: for two real matrices nxm, A = (ai,j) and B = (bi,j) , A :5 B is equivalent to ai,j :5 bi,j for all i,j such that I :5 i :5 n and I :5 j :5 m. A ~ I are equivalent to ai,j ~ 0, ai,j ~ I, respectively. IAI (lai,jD·
=
a) Stations Sequential Model (SSM)
:5 C :5 cV -dv :5 Vj :5 d v -dll :5 Uj :5 dll CL
The equations (I) can be put in matrix form as the following station sequential model (SSM), suitable for stability analysis (Campion et al.,1985):
YK+I = AK(YK + UK+I
+ VK+d
where CL, cV E 9tN ,O < CL :5 cV and dll E 9tN ,dll ~ O.
(2)
YK~ [YI(k) Y2(k)
Yn(k)f; Yk= 1,2,oo·,N
UK~ [uI(k) u2(k)
un(k)f;Yk= 1,2,oo·,N
VK~ [vI(k) v2(k)
vn(k)f; Yk= 1,2, .. ·,N
~ [I ~(~~i)l) I-eg+ I) ::~ o
0
~
-dh
]
(3)
The real time model (RTM) (Campion et al.,1985), presents a state-space formulation suitable for control design:
Yj ~ [Yj_I(I) Yj-2(2) Uj ~ [uj(l) uj_I(2)
Vj~ [vAI) vj_I(2) C ~ [c(l) c(2)
00'
00'
00.
(10)
(11) where dy E 9tN ,dy > O. This corresponds to assume that (11) is positively invariant with respect to system (4) which is equivalent to require that:
(4)
f Uj-N+I (N) f
00.
:5 Yj+l - Yj :5 dh
In order to guarantee the stability in the sense of Lyapunov of the traffic of trains in the metro line, the states Yj, along all trajectories of system (4) will be considered restricted to the polyhedron:
... l-e(k+l)
=A(C)Yj + B(C)[Uj + Vj]
< I, dv E 9tN ,dv ~ 0
where dh E 9tN ,dh ~ O.
b) Real Time Model (RTM)
Yj+1
(8)
(9)
Train occupancy requirements and security rules impose limits on the admissible variation of the time interval between two successive trains (Yj+1 - Yj), which will be considered restricted to the polyhedron:
AK = (CK+I)-I where CK+l is the following bidiagonal 9tnxn matrix:
CK+I
(7)
Yj-N(N)
(12) must be satisfied for all Yj in (11).
Vj-N+I(N)f
Consider system (4)-(6), constraints (7)-(12) and a linear time-invariant state feedback control law:
c(N))
276
(13)
Definition 3.1: Constrained Robust Regulation Problem (CRRP): find a matrix F E 9t NxN such that for all
(17) equation (16) can be rewritten as
C (7), Vj (8), Yj (12), j ~ 0, the following constraints are jointly satisfied (Milani and Correa, 1997):
= [A(C) +B(C)F]Yj+B(C)Vj
Yj +!
(18)
(14)
=
- fydy ::5 Yj+1 ::5 fydy
::5 Uj ::5 f IIdll -fhdh ::5 Yj+1 - Yj ::5 fhdh -dv ::5 Vj ::5 d v 0::5 f y ,fll ,fh::5 IN - f
where f
y, f
11 ,
IIdll
Proposition 4.1: The time-variant system GXf(k)Yx+ I AXf(k)Yx, where c(k+ 1) is known, is asymptoticaly stable if the following constraints are satisfied:
=diag(Yyl "'YyN) Fx
=diag(Yhl "'YhN)
0 ···0 0]
~ ~l f~2 ::: ~ ~ fk
(15)
f ll =diag(YIII"'YIIN)
fh
(19)
YX+l = (Gx;AXf)Yx
rh are diagonal matrices: fy
where Fx and Gx are defined as control matrices with tenns fkj,i e gkj,r Defining AXf(k) = AX(IN +Fx) and GXf IN - Gx, after some algebraic manipulations, one has:
(20)
[
0 0 ···Ofkn
In (Milani and Correa, 1997) it is showed that the solution of the CRRP can be obtained solving N reduced order independent linear programming problems, related one to one to the platfonns of the metro line.
~ ~]
o
0
...
0 0 ... ,k,,_1 0
(21)
4. STABILITY ANALYSIS Consider the SSM (2)-(3) with v;(k)
fk; + 1 < A . k, - 1- c(k+ I) - k,
(22)
o::5 Akj < Am ::5 1
(23)
-A . <
=0:
c(k+l)
YX+I
=AxYx + AxUx+1 I
gkj
(16)
o
!-c(k+ I) -c(.l:+ I) (I-c(k+ Ill2
I-c(k+ 1) -c(k+ 1) (I-c(k+ 1))2
(c(k+ I» (I-c(k+ 1))3
(-I)"-I(c(k+ 1))_-3 (I-c(k+I»- 2
0 0
0 0
0
0
Vk=O,I,···,N-l
(platforms)
Vi= 1,2,···,n
(trains)
where Am is an scalar that represents the desired stability degree. Proof: It can be verified, if the constraints (22) and (23) are satisfied, substituting in (21) gives:
(-I)n-I(c(k+ 1})n-1 (-l)n(c(k+ I))n-2 (I-c(k+ I))n (I-c(k+ 1))_-1
0 0 1 I-c(k+I)
(24)
= I-c(k+ I)
-c(k+ 1) I (I-c(k+ 1»2 l-c(k+I)
Matrix Ax is lower triangular with eigenvalues equal to 1/(1 - c(k+ 1». Since 0::5 c(k+ I) < I, the eigenvalues ofAx are outside the unity circle, showing the unstable behavior of traffic in the metro line in the abscence of control. Moreover, since the system not stationary, all the eigenvalues inside the unity circle is not sufficient to assure asymptotic stability. Considering the state feedback control law:
Substituting (26) in (27):
277
=
I., +1 I-c(k+ I)
o
ft l + 1
(1-c(Hl}U) 1-c(H I) 1-c(H I}U
o
1-c(k+I)
(26) 1.. + 1
o
o
o
o
o I'l + 1
l-c(k+l)
It is easy to verify that G'K;AKj is a diagonal matrix Ik·+ 1 ·th I WI e ements l-c(k+I)'
:::
It can be verified in (24):
...
'v'k=O,I,···,N-I
f.,+1 )(I-C(k+l}U) ( 1-c(H I) 1-c(H I}U
!k;+l I-
c(k+ I) - 2:5 fk; :5 -c(k+ I);
(34)
Substituting (36) and the control (33) in G'K;AKf (35): (28)
'v'i= 1,2,···,n; 'v'k=O,I,···,N-I
Substituting in (28), it can be verified that all elements of G'K;AKj presents magnitudes less than I, which concludes the proof. 0
A., (1-c(H I)u) 1-c(HI} "" c(H I}u "" c(H 1)(I-c(H I}u} 1-c(H I} (l-c(H 1}}2
Proposition 4.1 assumes parameters c(k) perfectly known. For uncertain parameters c(k). the following proposition gives sufficient robust stability conditions.
Proposition 4.2: The time-variant system G KI( k) YK+ 1 = AKj(k)YK, uncertain c(k) (7) is asymptoticaly stable if the constraints (22), (23), (25) and the following ones are satisfied: -A < fk;+ 1
(29)
-A < fk;+1
(30)
c(k+ 1)u Kk;= l-c(k+I)U
o "'l(l-c(H l)u} I c(HI}
,-"-1..,,.,]
(31)
(35)
• c(H I)
(I-c(k+ I)U)(I-c(k+ I)L)n-1 c(k+ I)U -c(k+ I)L
~-..:....,..,-:...:-:-~-:-:...:...--.--..'--.:..._+
It can be verified that if c(k+ I) = c(k+ I)U (28) is obtained. Nevertheless, in the worst case, c(k + I) c(k + I)L and Ak; = Am in (25) for all i 1,2,· .. ,n, (37) becomes:
+(I-c(k+ I)Lt- 2 + (1- c(k+ Ilt- 3 c(k+ I)L+ L n-2 +(c(k+ I))
(33)
(27)
Using the worst case in (25):
+... + (I-c(k+ I)L)(c(k+ I)Lt-
]
Considering the worst case for Ak; in (31) and (32) when (c(k) = c(k)u):
Akj{c(k+ I) -I) -1:5 A; :5 Ak;(I-c(k+ 1))-1
'v'i=I,2,···,n;
~
3
=
+
(I-c(k+ l)L)n
< Am(c(k+ I)U -c(k+ I)L)
'v'k=O,I, .. ·,N-I 'v'i= 1,2,· .. ,n
(32) G'K;AKj
(platforms)
= "':~~~:~r}
(trains)
I.",c(H ')u I.",c(H I)L(I-c(H flu} l-c(HI}L (I-c(HI}L)l
Proof: Equation (27) can be rewritten as:
[
(_I)' 1.",(c(H I)Lr-lc(H I}u +(_I}"-I 1.",(c(H I}L}'-'(I-c(H I)u) (I-c(k+ I}Lr-' (1-c(H I)L).
ft, + I ( 1- «H I)u ) 1-c(H I) 1-c(H 1)0 "1 (f.\ + I) _ c(H 1)(/'1 + I)) (I-C(k+ I}u ) ( .-c(H I} ('-c(k+ I}}l l-c(H I)U
o "'l(l-c(k+ I}u) l-c(k+ I}L
278
=
(l-c(k+ I)U)(I-c(k+ I)L)n-1 c(k+ I)U -c(k+ I)L + +(1- c(k+ I)Lr- 2 + (I - c(k+ I)L)n-3 c(k+ I)L+ + ... + I - c(k+ I)L(c(k+ I)Lr- 3 +
Now, considering the worst case for YK in the SSM (21) and analysing matrix (38) it can be noted that the system is asymptoticaly stable if the sum of the absolute value of the terms related to train n is less than 1. After some algebraic manipulations it can be verified that it is equivalent to satisfy the constraint (34), which concludes the proof. 0
(l-c(k+ I)L)n Ln 2 +(c(k+ I) ) - < Am(c(k+ I)U -c(k+ I)L) Note at platform k = 1 the control is constant given by c(k)u. The designer can adjust the stability degree by Am, but respecting the conditions in (42). Nevertheless it is necessary to verify that, depending on the vector YK, for lower values in Am, the problem can be unfeasible, due the saturation of the control. So, it is necessary to satisfy the control bounds defined by:
It can be verified that (21) considering (22) - (26) can be rewritten as:
[~
I-~+I)
YK+I =
0
10,+1 1-c(Hl) ...
0
0
0
" 1..
]
-Yukduk :5lk,k_IYi(k-1)+c(k)uyi_ l (k) :5Yukduk
YK (37)
0:5 Yuk :5 I
... I-«HI)
5. REAL-TIME ROBUST REGULATION (RTRR)
Analysing SSM (2), it can be verified that:
The real-time robust regulation of metro lines (RTRR) can be formulated as independent reduced order linear programming problems related one to one with platforms of the line, initiating at the 2 nd platform. In the first platform the control is constant II ,I = c(k)u. Thus the state Yi( I) can be obtained directly by:
1 Ik; ( ) Yi(k+1)= l_c(k+I)Yi(k)+ 1-c(k+l/i k (38) Note that if the control is:
ui(k+ I) = !k;Yi(k) +c(k+ I)Yi_l(k+ 1)
(39)
Yi(I)= this equation is equivalent to the basic equation for the traffic of trains described in (I) assuming vi(k) = O. This also can be obtained from the state feedback equation proposed (17). Assuming FK and GK obtained by the constraints (22) - (26), SSM (2) and the matrix AK (16) it is easy to obtain (41). Analysing (14) and the formulation proposed by Correa and Milani (1997), it can be verified that the terms !k; for i = 1,2"", n described in SSM (41) are equivalent to the terms Ik,k-I presented in RTM (4). Similarly, if Ik,k = c(k + I) in RTM, without uncertainties, the SSM will have the control ui(k+ I) described by (41). Thus the sufficient condition for the RTM system to be asymptoticaly stable corresponds to:
Yj+l F=
= [A (C) +B(C)F]Yj +B(C)Vj
I
For the others (N - I) platforms, the following formulation is proposed: P.L. k; k = 2 : N:
Ik,k-I + I Ik,k - c(k) Yi(k)= I-c(k) Yi(k-1)+ I-c(k) Yi-I(k)+ I
+ I_c(k)dvk
(40)
!k,k-l + I
Vk=2,3,···,N
Ik,k =c(k)u
Vk= 1,2,···,N
0:5 Ak,k-I
o :5 Ak,k-I < Am :5
< Am :5 I
(43) I
-Yhkdhk :5 Yi(k) - Yi-I (k) :5 Yhkdhk (l-c(k)U)(I-c(k)L)N-1 c(k)U - c(k)L + +(1- c(k)L)N-2 + (1- c(k)L)N-3 c (k)L +
Vk = 2,3,···,N
-Ak,k-I:5 l-c(k)L :5 Ak,k-I
-Yykdyk :5 Yi(k) :5 Yykdyk -Yukduk :5 Ik,k-IYi(k - I) + c(k)uYi _ 1(k) :5 Yukduk Ik,k-I + I -Ak,k-I:5 1- c(k) :5 Ak,k-I Ik,k = c(k)U
0 0··· IN-I,N-I 0 0 0··· IN,N-I IN,N
Ik,k-I + I -Ak,k-I :5 1- c(k)U :5 Ak,k-I
11,1 - c(l) 1 l-c(l) Yi-I(I)+I_c(I)Vi(l) (42)
where Vie I) is the random disturbance and Yi-I (I) is known by the initial condition.
[7.:: f~2 g:.: g g o o
(41)
+1
+... + (I -
c(k)L)(c(k)L)N-3 + LN-2 (l-c(k)L)N +(c(k)) < Am(c(k)U -c(k)L)
Vk = 2,3,···,N
279
.
where z ~ 0, q ~ 0 and p ~ 0 are scalars used for relative weighting of Ak,k-lo Yyk, Yh! and Yut; Am can be considered as variable in the problem or adjusted by the designer to obtain the desired stability degree; c(k)U is the upper bound constant related to the passenger demand; dut, dyk and dh! are the limits imposed to the control, state and time interval between two consecutive trains; v;(k) is the random disturbance. The delays (states y;(k-1) and Y;_I(k) are known or estimated on-line nearly simultaneously when the trains arrive at the platforms. The formulation have to consider the worst case of uncertainties and random perturbance. Thus, (45) must be feasible for c(k) = c(k)U and c(k) = c(k)L, and similarly for dvk = d~ and dvk =d~k'
. . "
.. I .
. I. . I
r
11
8..
0
-0
'1111
-.
1111
• Ptat1'onns
"
I
~
"
Q
1.
1
I
" ,
,
--. Plat1'onns •
4.
11111 Ptat1'onns
"
Fig. 2. Constrained Robust Regulation (CRRP)
.. I ~
IlIil : I
]: I
6. NUMERICAL EXAMPLE
=
10 plataforms and Consider a metro line with N passenger demand parameters bounded by:
~
!--.
~tl
~.
111111 .. ..
1111111
-", Ptatt'onns " '., Ptatt'onns " 4. Ptatt'onns •
cV =[.200
.210 .250 .200 .120 .ISO .250 .170 .ISO .125]
Fig. 3. Real-Tune Robust Regulation
cc- = [.180 .189 .225 .ISO .108 .135 .225 .153 .162 .112]
7. CONCLUSION
Also consider the state vector Yj, the interval (Yj+1 Yj), the control Vj and the disturbance Vj bounded by:
dy = [30 30 30 30 30 30 30 30 30 30] dh
dll
= [60 60 60 60 60 60 60 60 60 60] = [20 20 20 20 20 20 20 20 2020] d v =[2222222222]
A new approach for robust regulation of metro lines with nonlinear time-variant state feedback control law was presented. The control law is computed in realtime, assuring asymptotic stability in the presence of traffic model uncertainties and constraints on its state and control variables. The constraints on model variables, the disturbances and uncertain parameter domain were defined by convex compact polyhedra. A numerical example ilustrates the efficiency of the proposed approach which presents better regulation performance than known robust regulation approach using time-invariant feedback control law. The computational efficiency of this approach makes it applicable to real-time regulation of nowadays metro lines.
(44)
(45) (46)
(47)
To i1ustrate the performance obtained using the formulation RTRR comparing with the time-invariant control law proposed in CRRP (Milani and Correa, 1997), are presented a set of 100 simulated trajectories for the delay (state Yj(k), interval (yj(k) - Yj_1 (k)) and control (uj(k)) for a train j along the platforms, starting from the initial condition:
Yo= [30 -300 -30300030 -30
8. REFERENCES
of
Assis, W.O. e Milani, B.E.A. (2002) "Generation of Optimal Schedules for Metro Lines Using Model Predictive Control" - International Federation of Automatic Control - 15th IFAC World Congress, Barcelona, Spain. Campion, G., Van Breusegem, v., Pinson, P. and Bastin, G. (1985) "Traffic Regulation of an Underground Railway Transportation System By State Feedback" - Optimal Control Application & Methods, vol.6, pp. 385-402. Cury, lE., Gomide, EA. and Mendes, MJ. (1980) "A Methodology for Generation of Optimal Schedules for an Undeground Railway System" - IEEE Transactions on Automatic Control, vol. 25, n.2. Milani, REA e Correa, S.S. (1997) "Decentralized Robust Regulation of Metro Lines." Proc. American Control Conference, pp. 218, 219.
The simulations consider random C and Vj' Figure 1 presents the results obtained using the formulation CRRP and considering the weighting parameters in the performance criterium: p 1; q
=
= !.
Figure 2 presents the results obtained using the real0,3 and time robust regulation, considering Am p= 1 weighting parameters: z = 1; q =
!;
=
Comparing the results of the RTRR, where the feedback control matrix F is time-variant, with the results presented in CRRP, the better performance of the realtime regulation approach is evident. It can be verified the proposed approach uses the maximum possible control to eliminate the delays and intervals deviations.
280
ELSEVIER
IFAC PUBUCATIONS www.elsevier comllocale/ifac
ROBUST REGULATION OF METRO LINES USING TIME-VARIANT CONTROL LAW
Winderson O. Assis • Basilio E. A. Milani ••
• Escola de Engenharia Maud, Instituto Maud de Tecnologia, Prara Maud I, 09580-900 Sao Caetano do SuI SP, Brazil. •• Faculdode de Engenharia EIetrica e de Computarao, Universidade Estadual de Campinas, CP' 6101, 13081-970 Campinas SP, Brazil.
Abstract: This paper presents a new formulation for robust traffic regulation of metro lines. The approach uses a linear time-variant state feedback control law computed in real-time. The formulation is based on the stability analysis using the eigenvalues of the feedback system and consider the traffic model uncertainties and the constraints on its state and control variables. It is assumed the constraints on model variables and uncertain parameter domain defined by convex compact polyhedra. The simplicity and computational efficiency of this formulation makes it applicable to real-time regulation of nowadays metro lines and presents better performance than the obtained using already known robust regulation approaches. Copyright © 2003 [FAC Keywords: Robust regulation, Metro lines, Linear programming, Optimization, Stability analysis.
Milani and Correa (1997) proposed a fonnulation based on linear programming for robust constrained regulation of metro lines, considering explicitly the constraints on the state and control variables, the parameter uncertainties and random operational disturbances. The constraints on model variables and uncertain parameter domain were defined by convex compact polyhedra. Exploring structural properties of the traffic model, the method gives a linear time-invariant control law, with bidiagonal structure, obtained solving independent linear programming problems, related one to one with the platforms of the line.
1. INTRODUcnON
The traffic of trains in high-frequency metro lines is known to be naturally unstable. Consider, for example, a train delayed with respect to the nominal time schedule of the line (Cury et al., 1980), (Assis and Milani, 2(02). Due to this delay, its time interval relative to the preceding train is increased, more passengers will be waiting at the platforms to get on the train, resulting in increased delay (Campion et al., 1985). Thus, traffic control is necessary to keep the departure time at the stations, the time interval between successive trains as close as possible to their nominal values. The control actions consist of instructions given by the controller to the trains at the stations, increasing or decreasing their speed during the running time to the next station and/or their waiting time at the platform. The control is naturally constrained by the speed range allowed to the trains, minimum waiting time at the stations and traffic security rules. The time interval between successive trains is also constrained by the traffic security rules and the maximum train occupancy.
This paper presents a new fonnulation for robust traffic regulation of metro lines using a possibly nonlinear time-variant state feedback control law computed in real time. The formulation is based on eigenvalues assignement of the closed-loop matrix.
2. TRAFFIC MODELING
=
Consider n trains (i I,···, n) in an open metro line with N platforms (k = 1,2"", N) (figure I).
275
where Yj. Uj and Vj are N dimensional vectors representing the state, the control and the external disturbances of the system, respectively; C is the vector of uncertain parameters; A(C) and B(C) are given by:
Defining Yi(k) as the deviation of the actual departure time of train i in platform k with respect to the nominal time schedule, Campion et al. (1985) proposed the basic equation for the traffic of trains: 2
N
N-I
-e(l) l-e(1)
0--0-55-0-0
A(C) ~
0
-e(2)
l-e(2)
l-e(2)
0
0
sentido do rifego
Fig. I. Open metro line with N platforms
1
l-e(1)
(1- c(k+ 1))Yi(k+ I)
0
B(C) ~
=Yi(k)-
-c(k+ I)Yi_l(k+ I) +ui(k+ I) +vi(k+ I)
(1)
0
where: 0 :5 c(k) :5 I is related to passengers demand, inherently uncertain; vi(k) is the random disturbance; ui(k) is the control action applied to train i between the platforms k - I and k in order to increase (ui(k) > 0) or to decrease (ui(k) < 0) the running time.
0
I
0
,I 0
(5)
-e(N) I l-e(N) l-e(N) 0 I
l-e(2)
0
. ,-u
(6)
3. CONSTRAINED ROBUST REGULATION PROBLEM (CRRP) This section presents the constrained robust regulation approach proposed in (Milani and Correa, 1997). Consider an open metro line with N platforms and equations (4)-(6). Consider also the uncertain parameters C, the disturbances Vj and the control Uj, restricted to the polyhedrons:
Throughout this paper: for two real matrices nxm, A = (ai,j) and B = (bi,j) , A :5 B is equivalent to ai,j :5 bi,j for all i,j such that I :5 i :5 n and I :5 j :5 m. A ~ I are equivalent to ai,j ~ 0, ai,j ~ I, respectively. IAI (lai,jD·
=
a) Stations Sequential Model (SSM)
:5 C :5 cV -dv :5 Vj :5 d v -dll :5 Uj :5 dll CL
The equations (I) can be put in matrix form as the following station sequential model (SSM), suitable for stability analysis (Campion et al.,1985):
YK+I = AK(YK + UK+I
+ VK+d
where CL, cV E 9tN ,O < CL :5 cV and dll E 9tN ,dll ~ O.
(2)
YK~ [YI(k) Y2(k)
Yn(k)f; Yk= 1,2,oo·,N
UK~ [uI(k) u2(k)
un(k)f;Yk= 1,2,oo·,N
VK~ [vI(k) v2(k)
vn(k)f; Yk= 1,2, .. ·,N
~ [I ~(~~i)l) I-eg+ I) ::~ o
0
~
-dh
]
(3)
The real time model (RTM) (Campion et al.,1985), presents a state-space formulation suitable for control design:
Yj ~ [Yj_I(I) Yj-2(2) Uj ~ [uj(l) uj_I(2)
Vj~ [vAI) vj_I(2) C ~ [c(l) c(2)
00'
00'
00.
(10)
(11) where dy E 9tN ,dy > O. This corresponds to assume that (11) is positively invariant with respect to system (4) which is equivalent to require that:
(4)
f Uj-N+I (N) f
00.
:5 Yj+l - Yj :5 dh
In order to guarantee the stability in the sense of Lyapunov of the traffic of trains in the metro line, the states Yj, along all trajectories of system (4) will be considered restricted to the polyhedron:
... l-e(k+l)
=A(C)Yj + B(C)[Uj + Vj]
< I, dv E 9tN ,dv ~ 0
where dh E 9tN ,dh ~ O.
b) Real Time Model (RTM)
Yj+1
(8)
(9)
Train occupancy requirements and security rules impose limits on the admissible variation of the time interval between two successive trains (Yj+1 - Yj), which will be considered restricted to the polyhedron:
AK = (CK+I)-I where CK+l is the following bidiagonal 9tnxn matrix:
CK+I
(7)
Yj-N(N)
(12) must be satisfied for all Yj in (11).
Vj-N+I(N)f
Consider system (4)-(6), constraints (7)-(12) and a linear time-invariant state feedback control law:
c(N))
276
(13)
Definition 3.1: Constrained Robust Regulation Problem (CRRP): find a matrix F E 9t NxN such that for all
(17) equation (16) can be rewritten as
C (7), Vj (8), Yj (12), j ~ 0, the following constraints are jointly satisfied (Milani and Correa, 1997):
= [A(C) +B(C)F]Yj+B(C)Vj
Yj +!
(18)
(14)
=
- fydy ::5 Yj+1 ::5 fydy
::5 Uj ::5 f IIdll -fhdh ::5 Yj+1 - Yj ::5 fhdh -dv ::5 Vj ::5 d v 0::5 f y ,fll ,fh::5 IN - f
where f
y, f
11 ,
IIdll
Proposition 4.1: The time-variant system GXf(k)Yx+ I AXf(k)Yx, where c(k+ 1) is known, is asymptoticaly stable if the following constraints are satisfied:
=diag(Yyl "'YyN) Fx
=diag(Yhl "'YhN)
0 ···0 0]
~ ~l f~2 ::: ~ ~ fk
(15)
f ll =diag(YIII"'YIIN)
fh
(19)
YX+l = (Gx;AXf)Yx
rh are diagonal matrices: fy
where Fx and Gx are defined as control matrices with tenns fkj,i e gkj,r Defining AXf(k) = AX(IN +Fx) and GXf IN - Gx, after some algebraic manipulations, one has:
(20)
[
0 0 ···Ofkn
In (Milani and Correa, 1997) it is showed that the solution of the CRRP can be obtained solving N reduced order independent linear programming problems, related one to one to the platfonns of the metro line.
~ ~]
o
0
...
0 0 ... ,k,,_1 0
(21)
4. STABILITY ANALYSIS Consider the SSM (2)-(3) with v;(k)
fk; + 1 < A . k, - 1- c(k+ I) - k,
(22)
o::5 Akj < Am ::5 1
(23)
-A . <
=0:
c(k+l)
YX+I
=AxYx + AxUx+1 I
gkj
(16)
o
!-c(k+ I) -c(.l:+ I) (I-c(k+ Ill2
I-c(k+ 1) -c(k+ 1) (I-c(k+ 1))2
(c(k+ I» (I-c(k+ 1))3
(-I)"-I(c(k+ 1))_-3 (I-c(k+I»- 2
0 0
0 0
0
0
Vk=O,I,···,N-l
(platforms)
Vi= 1,2,···,n
(trains)
where Am is an scalar that represents the desired stability degree. Proof: It can be verified, if the constraints (22) and (23) are satisfied, substituting in (21) gives:
(-I)n-I(c(k+ 1})n-1 (-l)n(c(k+ I))n-2 (I-c(k+ I))n (I-c(k+ 1))_-1
0 0 1 I-c(k+I)
(24)
= I-c(k+ I)
-c(k+ 1) I (I-c(k+ 1»2 l-c(k+I)
Matrix Ax is lower triangular with eigenvalues equal to 1/(1 - c(k+ 1». Since 0::5 c(k+ I) < I, the eigenvalues ofAx are outside the unity circle, showing the unstable behavior of traffic in the metro line in the abscence of control. Moreover, since the system not stationary, all the eigenvalues inside the unity circle is not sufficient to assure asymptotic stability. Considering the state feedback control law:
Substituting (26) in (27):
277
=
I., +1 I-c(k+ I)
o
ft l + 1
(1-c(Hl}U) 1-c(H I) 1-c(H I}U
o
1-c(k+I)
(26) 1.. + 1
o
o
o
o
o I'l + 1
l-c(k+l)
It is easy to verify that G'K;AKj is a diagonal matrix Ik·+ 1 ·th I WI e ements l-c(k+I)'
:::
It can be verified in (24):
...
'v'k=O,I,···,N-I
f.,+1 )(I-C(k+l}U) ( 1-c(H I) 1-c(H I}U
!k;+l I-
c(k+ I) - 2:5 fk; :5 -c(k+ I);
(34)
Substituting (36) and the control (33) in G'K;AKf (35): (28)
'v'i= 1,2,···,n; 'v'k=O,I,···,N-I
Substituting in (28), it can be verified that all elements of G'K;AKj presents magnitudes less than I, which concludes the proof. 0
A., (1-c(H I)u) 1-c(HI} "" c(H I}u "" c(H 1)(I-c(H I}u} 1-c(H I} (l-c(H 1}}2
Proposition 4.1 assumes parameters c(k) perfectly known. For uncertain parameters c(k). the following proposition gives sufficient robust stability conditions.
Proposition 4.2: The time-variant system G KI( k) YK+ 1 = AKj(k)YK, uncertain c(k) (7) is asymptoticaly stable if the constraints (22), (23), (25) and the following ones are satisfied: -A < fk;+ 1
(29)
-A < fk;+1
(30)
c(k+ 1)u Kk;= l-c(k+I)U
o "'l(l-c(H l)u} I c(HI}
,-"-1..,,.,]
(31)
(35)
• c(H I)
(I-c(k+ I)U)(I-c(k+ I)L)n-1 c(k+ I)U -c(k+ I)L
~-..:....,..,-:...:-:-~-:-:...:...--.--..'--.:..._+
It can be verified that if c(k+ I) = c(k+ I)U (28) is obtained. Nevertheless, in the worst case, c(k + I) c(k + I)L and Ak; = Am in (25) for all i 1,2,· .. ,n, (37) becomes:
+(I-c(k+ I)Lt- 2 + (1- c(k+ Ilt- 3 c(k+ I)L+ L n-2 +(c(k+ I))
(33)
(27)
Using the worst case in (25):
+... + (I-c(k+ I)L)(c(k+ I)Lt-
]
Considering the worst case for Ak; in (31) and (32) when (c(k) = c(k)u):
Akj{c(k+ I) -I) -1:5 A; :5 Ak;(I-c(k+ 1))-1
'v'i=I,2,···,n;
~
3
=
+
(I-c(k+ l)L)n
< Am(c(k+ I)U -c(k+ I)L)
'v'k=O,I, .. ·,N-I 'v'i= 1,2,· .. ,n
(32) G'K;AKj
(platforms)
= "':~~~:~r}
(trains)
I.",c(H ')u I.",c(H I)L(I-c(H flu} l-c(HI}L (I-c(HI}L)l
Proof: Equation (27) can be rewritten as:
[
(_I)' 1.",(c(H I)Lr-lc(H I}u +(_I}"-I 1.",(c(H I}L}'-'(I-c(H I)u) (I-c(k+ I}Lr-' (1-c(H I)L).
ft, + I ( 1- «H I)u ) 1-c(H I) 1-c(H 1)0 "1 (f.\ + I) _ c(H 1)(/'1 + I)) (I-C(k+ I}u ) ( .-c(H I} ('-c(k+ I}}l l-c(H I)U
o "'l(l-c(k+ I}u) l-c(k+ I}L
278
=
(l-c(k+ I)U)(I-c(k+ I)L)n-1 c(k+ I)U -c(k+ I)L + +(1- c(k+ I)Lr- 2 + (I - c(k+ I)L)n-3 c(k+ I)L+ + ... + I - c(k+ I)L(c(k+ I)Lr- 3 +
Now, considering the worst case for YK in the SSM (21) and analysing matrix (38) it can be noted that the system is asymptoticaly stable if the sum of the absolute value of the terms related to train n is less than 1. After some algebraic manipulations it can be verified that it is equivalent to satisfy the constraint (34), which concludes the proof. 0
(l-c(k+ I)L)n Ln 2 +(c(k+ I) ) - < Am(c(k+ I)U -c(k+ I)L) Note at platform k = 1 the control is constant given by c(k)u. The designer can adjust the stability degree by Am, but respecting the conditions in (42). Nevertheless it is necessary to verify that, depending on the vector YK, for lower values in Am, the problem can be unfeasible, due the saturation of the control. So, it is necessary to satisfy the control bounds defined by:
It can be verified that (21) considering (22) - (26) can be rewritten as:
[~
I-~+I)
YK+I =
0
10,+1 1-c(Hl) ...
0
0
0
" 1..
]
-Yukduk :5lk,k_IYi(k-1)+c(k)uyi_ l (k) :5Yukduk
YK (37)
0:5 Yuk :5 I
... I-«HI)
5. REAL-TIME ROBUST REGULATION (RTRR)
Analysing SSM (2), it can be verified that:
The real-time robust regulation of metro lines (RTRR) can be formulated as independent reduced order linear programming problems related one to one with platforms of the line, initiating at the 2 nd platform. In the first platform the control is constant II ,I = c(k)u. Thus the state Yi( I) can be obtained directly by:
1 Ik; ( ) Yi(k+1)= l_c(k+I)Yi(k)+ 1-c(k+l/i k (38) Note that if the control is:
ui(k+ I) = !k;Yi(k) +c(k+ I)Yi_l(k+ 1)
(39)
Yi(I)= this equation is equivalent to the basic equation for the traffic of trains described in (I) assuming vi(k) = O. This also can be obtained from the state feedback equation proposed (17). Assuming FK and GK obtained by the constraints (22) - (26), SSM (2) and the matrix AK (16) it is easy to obtain (41). Analysing (14) and the formulation proposed by Correa and Milani (1997), it can be verified that the terms !k; for i = 1,2"", n described in SSM (41) are equivalent to the terms Ik,k-I presented in RTM (4). Similarly, if Ik,k = c(k + I) in RTM, without uncertainties, the SSM will have the control ui(k+ I) described by (41). Thus the sufficient condition for the RTM system to be asymptoticaly stable corresponds to:
Yj+l F=
= [A (C) +B(C)F]Yj +B(C)Vj
I
For the others (N - I) platforms, the following formulation is proposed: P.L. k; k = 2 : N:
Ik,k-I + I Ik,k - c(k) Yi(k)= I-c(k) Yi(k-1)+ I-c(k) Yi-I(k)+ I
+ I_c(k)dvk
(40)
!k,k-l + I
Vk=2,3,···,N
Ik,k =c(k)u
Vk= 1,2,···,N
0:5 Ak,k-I
o :5 Ak,k-I < Am :5
< Am :5 I
(43) I
-Yhkdhk :5 Yi(k) - Yi-I (k) :5 Yhkdhk (l-c(k)U)(I-c(k)L)N-1 c(k)U - c(k)L + +(1- c(k)L)N-2 + (1- c(k)L)N-3 c (k)L +
Vk = 2,3,···,N
-Ak,k-I:5 l-c(k)L :5 Ak,k-I
-Yykdyk :5 Yi(k) :5 Yykdyk -Yukduk :5 Ik,k-IYi(k - I) + c(k)uYi _ 1(k) :5 Yukduk Ik,k-I + I -Ak,k-I:5 1- c(k) :5 Ak,k-I Ik,k = c(k)U
0 0··· IN-I,N-I 0 0 0··· IN,N-I IN,N
Ik,k-I + I -Ak,k-I :5 1- c(k)U :5 Ak,k-I
11,1 - c(l) 1 l-c(l) Yi-I(I)+I_c(I)Vi(l) (42)
where Vie I) is the random disturbance and Yi-I (I) is known by the initial condition.
[7.:: f~2 g:.: g g o o
(41)
+1
+... + (I -
c(k)L)(c(k)L)N-3 + LN-2 (l-c(k)L)N +(c(k)) < Am(c(k)U -c(k)L)
Vk = 2,3,···,N
279
.
where z ~ 0, q ~ 0 and p ~ 0 are scalars used for relative weighting of Ak,k-lo Yyk, Yh! and Yut; Am can be considered as variable in the problem or adjusted by the designer to obtain the desired stability degree; c(k)U is the upper bound constant related to the passenger demand; dut, dyk and dh! are the limits imposed to the control, state and time interval between two consecutive trains; v;(k) is the random disturbance. The delays (states y;(k-1) and Y;_I(k) are known or estimated on-line nearly simultaneously when the trains arrive at the platforms. The formulation have to consider the worst case of uncertainties and random perturbance. Thus, (45) must be feasible for c(k) = c(k)U and c(k) = c(k)L, and similarly for dvk = d~ and dvk =d~k'
. . "
.. I .
. I. . I
r
11
8..
0
-0
'1111
-.
1111
• Ptat1'onns
"
I
~
"
Q
1.
1
I
" ,
,
--. Plat1'onns •
4.
11111 Ptat1'onns
"
Fig. 2. Constrained Robust Regulation (CRRP)
.. I ~
IlIil : I
]: I
6. NUMERICAL EXAMPLE
=
10 plataforms and Consider a metro line with N passenger demand parameters bounded by:
~
!--.
~tl
~.
111111 .. ..
1111111
-", Ptatt'onns " '., Ptatt'onns " 4. Ptatt'onns •
cV =[.200
.210 .250 .200 .120 .ISO .250 .170 .ISO .125]
Fig. 3. Real-Tune Robust Regulation
cc- = [.180 .189 .225 .ISO .108 .135 .225 .153 .162 .112]
7. CONCLUSION
Also consider the state vector Yj, the interval (Yj+1 Yj), the control Vj and the disturbance Vj bounded by:
dy = [30 30 30 30 30 30 30 30 30 30] dh
dll
= [60 60 60 60 60 60 60 60 60 60] = [20 20 20 20 20 20 20 20 2020] d v =[2222222222]
A new approach for robust regulation of metro lines with nonlinear time-variant state feedback control law was presented. The control law is computed in realtime, assuring asymptotic stability in the presence of traffic model uncertainties and constraints on its state and control variables. The constraints on model variables, the disturbances and uncertain parameter domain were defined by convex compact polyhedra. A numerical example ilustrates the efficiency of the proposed approach which presents better regulation performance than known robust regulation approach using time-invariant feedback control law. The computational efficiency of this approach makes it applicable to real-time regulation of nowadays metro lines.
(44)
(45) (46)
(47)
To i1ustrate the performance obtained using the formulation RTRR comparing with the time-invariant control law proposed in CRRP (Milani and Correa, 1997), are presented a set of 100 simulated trajectories for the delay (state Yj(k), interval (yj(k) - Yj_1 (k)) and control (uj(k)) for a train j along the platforms, starting from the initial condition:
Yo= [30 -300 -30300030 -30
8. REFERENCES
of
Assis, W.O. e Milani, B.E.A. (2002) "Generation of Optimal Schedules for Metro Lines Using Model Predictive Control" - International Federation of Automatic Control - 15th IFAC World Congress, Barcelona, Spain. Campion, G., Van Breusegem, v., Pinson, P. and Bastin, G. (1985) "Traffic Regulation of an Underground Railway Transportation System By State Feedback" - Optimal Control Application & Methods, vol.6, pp. 385-402. Cury, lE., Gomide, EA. and Mendes, MJ. (1980) "A Methodology for Generation of Optimal Schedules for an Undeground Railway System" - IEEE Transactions on Automatic Control, vol. 25, n.2. Milani, REA e Correa, S.S. (1997) "Decentralized Robust Regulation of Metro Lines." Proc. American Control Conference, pp. 218, 219.
The simulations consider random C and Vj' Figure 1 presents the results obtained using the formulation CRRP and considering the weighting parameters in the performance criterium: p 1; q
=
= !.
Figure 2 presents the results obtained using the real0,3 and time robust regulation, considering Am p= 1 weighting parameters: z = 1; q =
!;
=
Comparing the results of the RTRR, where the feedback control matrix F is time-variant, with the results presented in CRRP, the better performance of the realtime regulation approach is evident. It can be verified the proposed approach uses the maximum possible control to eliminate the delays and intervals deviations.
280