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Tracking Control of Join-Free Timed Continuous Petri Net Systems
Copyright 0Analysis and Design of Hybrid Systems Alghero. Italy 2006
PUBLICATIONS
TRACKING CONTROL OF JOIN-FREE TIMED CONTINUOUS PETRI NET SYSTEMS Jing Xu, Laura Recalde, Manuel Silva
Instituto de Investigacidn de Ingenieria d e Aragdn (I3A) Departamento d e Informa'tica e Ingenieria de Sistemas, Centro Polite'cnico Superior, Uniuersidad d e Zaragoza, Email:(~xu,lrecalde,silva}Qunizar.es
Abstract: A new low-and-high gain algorithm is presented for tracking control of timed coiitiiiuous Petri Net (coiitPN) systems working urider infinite servers semantics. The iiihereiit properties of timed coritPN determine that the control signals must be iioii-negative aiid upper bounded by functions of system states. In the proposed control approach, LQ theory is first utilized t o design a low-gain controller such that the control signals satisfy the input coiistraiiits. Based on the low-gain cont,roller, a higli-gain t,erm is fiirt,lier added to improve the syst,em transient performance. In order t o guarantee global tracking corivergeiice aiid smoot'hness of control signals, in our work, a new mixed t'racking t'rajectory (state st,ep and ramp) is ut,ilized instead of a pure st,ep reference signal. Copyright 02006 IFAC
1. INTRODTICTION
For cont,rol of timed contPN systems, we will begin from Join-Free(JF) timed coritPN systems with step-tracking cont,rol target. For a JF t,imed contPN syst,em, a linear differential equat,ion describes its behavior, but it' is subject' to certain input constraints, i.e. the control signal must be non-negat'ive and upper bounded by a funct'ion of system st'ates. 'I'liese constraints result in a hybrid system, more precisely a piecewise linear system. 'I'he main challenge in our work is t,o develop control laws under these special input const'raints so that global tracking coiivergeiice can be ensured.
Petri Nets (PNs) are powerful mathematical tools with appealing graphical representations for tlie modeling, analysis and synthesis of Discrete Even Systems (DESs). However, like in all DESs, discrete PN systems suffer from tlie so called state explosion problem. One possible way t o partially tackle this problem is t o fluidify the discrete P N models. The resiilt'ing continuous Petri Net' (cant'PN) systems have the potential for the applications of more analytical t,echniqiies which were originally developed for coiitiiiuous and hybrid systems. Fiirt'liermore, analogous to discrete P N systems, time has been introduced t o coritPN systems, which leads t'o timed cont'PN systems. Steady state control is studied in (Mahulea et al.: 2005); assuming all transitions can be controlled: it can be solved by meaiis of a LPP. Nevertheless, dynamic control of timed coiitPN systems needs further investigations aiid great improvements.
'I'he input' const'raints can be t'reated as input saturat,ions. As input, sat,urat,ion is one of the common phenomena encountered in cont,rol systerns: hitherto lots of works have been done. In (Wredenhage aiid Bklanger, 1994), a kind of piecewise-linear control law was derived. However, such a design method will lead t o a lowgain controller. To improve tlie control performance: a low-and-high gain approach was given in (Saberi et al., 1996). Recent'ly, several nonlinear control met'liods were fiirt'lier present'ed (Lin et
30
have n and rn elements, respectively. 'I'he marking m belongs t o R+n7 where R+ is the set of 11011negative real numbers, and bot,h P r e and Post are of the size n x r n . For w t P U T7the set of its input' and output' nodes are denot'ed as *w,and 'w*. respectively. N is JF (or reiidez-vous free) iff i t E T, I*tl = 1. If m is reachable froin mo through a sequence u E R+m, tlie state equation is m = mo C . u , where C = Post - P r e is the token flow mat,rix and u is t,he firing count vector.
al.: 199X), which inaiiily focus 011 the high-gain design in tlie low-aiid-high gain algoritliins aiinirig t o achieve better traiisieiit control performance. It should be pointed out that coiriirioii assumptions in all these works are that the lower saturation bounds are negat'ive constants and the upper saturation bounds are positive constants. However, in JF t,imed contPN syst,ems,t,he lower sat,uration bound is zero and the upper sat'iiration bound depends on syst'em st'at'es. Therefore, the existing control st'rat'egies for systems wit'h input' sat'urations cannot be applied direct'ly.
+
2.2 Tarried Continuous Petri Net Systems
A timed contPN system can be represented as ( N ,A, mo), where X [ t ] > 0 is t'he int'ernal firing rate of transition t . The state equation has an explicit dependence 011 time m(7) = mo CU(T), where 7 is time. Deriving with respect t o it: m(7) = C&(-r)is obtained. Define f ( 7 ) = & ( T ) > wliicli denote flows of transitions. The state equation is m(7) = Cf(7). For notation simplicity, 7 will be omitted in the rest of tlie paper. For the definition of flow f 7different semantics have been iiitroduced and t'he most import'ant ones are infinite servers and finite servers. Infinite servers semantics will be considered in t,liis work, which usually provides a much bet'ter approximation of discret'e behaviors. Under infinite server semant,ics,f is t,he product of X[t] and tlie iiistaritaiieous enabling of tlie transition, i.e. f[t] = X[t] . enab(t,m) = X[t]. 1riiii~~.~{m[23]/Pre[23, t]}. As in JF nets any traiisitioii has only one input place, the flow can be expressed as f = @ . m7 where @ t R t m x n aiid @[t,p]= X[t]/Pre[p7t] if p = * t , @[t,p]= 0 otherwise. Moreover, each row of @ has only one
In t,his paper, a new low-and-high gain approach will be proposed for st'ep-tracking cont'rol of JF timed contPN syst,ems. 'I'he presenkd algorithm can ensure global asympt'ot'ical convergence in presence of t,he input const,raint,sexisting in JF timed contPN syst,ems.According t,o init,ialmarking: desired marking arid net structure, a new reference t'rajectory is const'ruct'ed.Based 011 t'he new tracking target, a low-gain controller is designed according t o LQ theory so that the control signals are within the required regions. Siiice the upper bounds of the control inputs depend 011 the system states, tlie design method in (Wredeiihage arid Bdanger: 1994) fails t o work. By coinbiriiiig the new t,racking traject,ory design wit,h LQ met,hod, a novel design scheme for the low-gain part is given in our work. Analogous to t'he works of (Saberi et al., 1996), a high-gain part' is fiirt'lier added to make bet't'er use of t'lie available cont'rol authority, and consequently faster syst'em response can be obt,ained. Not,e that, in t,he high-gain term, the control vect,or of the low-gain part is maint,ained, wliicli makes possible the analysis of system stability. Rigorous proof is provided t o guarantee the globa 1 asyrriptot ica 1 convergence.
+
a
lion-zero element. V j E AT = { 1:2: . . . r n } : the lion-zero eleineiit for tlie j-tli row of @ is denoted a as 4j,i where i E N = {1,2;..,n}. Therefore, V j E A{, we have fj = q5j>irni,where fj is t'he jt h element of f arid rrii is the i-th eleineiit of m. The evolut,ion of the marking can be writ,ten as follows: ~
The paper is organized as follows. Sect,ion 2 introduces the required concepts of coiitPN systems arid timed coritPN systems. The control for JF timed coiitPN systems is forinulated in Section 3 . How to construct the new tracking trajectory is outlined in Section 4. Section 5 focuses on the development of cont'rol laws and t'he analysis of global asyrriptotical convergence property. An illustrative example is given in Section 6. Finally, Section 7 concludes the paper.
m=C.f=A.m
(1)
where A = C . @. Finally the definition of conservativeness aiid some related properties for a JF timed coritPN are listed as follows.
Definition 1. A P N is conseruative iff 3y > 0, such t,liat y.C = 0. Any left non-negat,ive anniiller of matrix C, i.e. y, is called P-semiflow. A Psemiflow is minimal if 1 is the great,est common divisor of its elements, aiid its support does riot strictly contain the support of other P-semiflow.
2. CON'I'INTJOTJS PETRI NET SYSTEMS
2.1 Untrmed Contznuous Petrz Net Systems A contPN system can be described as (N,mo), where N = ( P , T , P r e , P o s t )specifies the net structure (P and T are disjoint (finite) sets of places and transitions, and P r e and Post are incidence matrices), and mo is the initial marking. N is assumed t o be connected. while P aiid T
Property 1. (Teruel et al., 1997) Let A' be a JF net.
N is strongly connected, a) N has at most one rriiiiiinal P-semiflow:
1.1 If
31
Assumption 1. V i E N , mo,%> 0 and md,%> 0.
b) N is conservative iff it has one P-semiflow. 1.2 If N is conservative, it is coiisisterit iff it is strongly connected.
Remark 1. If we coiisider optiinal steady states in practical manufacture syst’ems,generally md > 0 is valid. On t,he ot,lier hand, if some element,s of mo are zeros, as md > 0, a firing sequence can always be found such t’liat mo[a > m and m > 0 (Recalde et al., 1999). Then the control algorit’lim proposed in our work can be further applied.
Property 2. Let A’ be a conservat’ive and strongly connected JF t,imed contPN. The matrix A for A’ defined in (1) has only one zero-eigenvalue. Moreover, the rerriaiiiiiig eigeiivalues of A are with negative real part’s.
4. DESIGN O F NEW TRACKING REFERENCE
Proof: As N is conservative arid strongly coiinected, from Property 1.2, N is consistent,. From the Proposition 1 in ( J i m h e z et al., 2005), for the conservative and consistent, J F PN A’, t,he mat,rix A has only one zero-valued eigenvalue. On the other hand, as A is a Metzler matrix aiid the eigenvalue of zero is the (Frobenius) dominant, eigeiivalue (Jiinhez et al., 2005), tlie rerriairiirig eigenvalues of A are wit’li negative real part’s. W
To ensure global convergence arid sinootliiiess of the control signal, a step target md is replaced by tlie following refereiice trajectory m,(T).
a
where m,(r) E R+n, m,o = m,(0) and h > 0 determines the time when m,(7) = md. Here we choose m,o = mo 6(md mo), where 0 5 6 < 1 is a parameter t o be designed. The parameter h is chosen such that, for given m,o and md, valid control actions u,o E R+m aiid uTh-E Rtm exist for the following equations.
3. PROBLEM FORMULATION
+
We will restrict our research t o coiiservative aiid strongly connect,ed JF t,imed contPN systems and assume all tlie markings are observable. For concise expression, “timed cont,PN” will be writ,ten as “contPN”. Like all t,he ot,her systems, cont,rol act’ions can also be introduced t’o modify aut,onomous evolution of P N systems. The possible control actioii that can be applied t o PN systems is to slow down t,lieir unforced firing flows. Hence, tlie forced flows of controlled traiisitioiis becoine f-u, where u is t,hecontrol signal and must satisfy 0 5 u 5 f . Corisideriiig (l),a JF coritPN system with a control actioii caii be described as follows:
a
m = C ( @ m - u) = A m -
Bu,
-
(5)
Proposztzon 1. For given mo aiid reachable
md,
6
aiid h can always be fourid such that (5) aiid (6) are valid. Moreover, as 6 increase. smaller h can be chosen.
(2)
a
where m E R+n, A E Rnxn, B = C E Rnxm and u E R+m. It should be n o k d t,liat the constraints oii input u lead tlie closed system t o be a piecewise linear system.
Proof: Given mo and reachable m d , a 2 0 can always be fouiid so that md = mo B a . Thus.
+
md -
Our control object’ive is t’o construct cont’rollaws such that m arid u respectively converge t o desired values: md and ud. ‘Yo sat,isfy reachabilit,y, md inust fulfill that y . md = y . mo where y E Rn is the basis of P-semiflows. For ud, 0 5 ud 5 @md inust be satisfied, i.e. V j E M 7 0 5 U d , j 5 & , i r r i d > i where U d , j aiid md,i are tlie j-tli aiid i-th elements of ud aiid md respectively. Moreover, as md is constaiit: according t o ( 2 ) : tlie desired coiitrol input, must, be a solut,ionof the following equat,ion. 0 = Amd
-
Bud.
m,o
=B(l-
&)a.
(7)
Substitutiiig (7) iiito (5) aiid (6) yields
B@m,o - Bu,o
=
1 B ( l - 6)a-, h 1
B@md B u , ~ -= B(1 -
-
6)~:.
h
Obviously, u,o = @m,o (1 6)ai and u,h- = @md - (1 - 6 ) a i are solutions for the above equations. From 0 5 u , ~5 @m,o. we have: -
(3)
In our work, the following assumption is made for mo and md.
32
-
From 0 5 u,h- 5 @md,we can obtain that 1
O I U - < -
1
h-1-6
where uly is the low-gain part, uhg is the highgain term arid u,(r) is tlie reference control input defined as follows:
@md.
Therefore, tlie filial constraints for h are: U
Smd
1
-@md h -< rniii{ @mo+ 1-6'1-6
0 5 -
(11)
where u,o arid
u,h-
are given in (12) aiid (13)
respectively. Moreover, V d t Rm. s a t ( d ) [sat(dl), . . . , sat(dm)lT and V j E A l , .sat(d3) is
where, for al E R+n and a2 E Rtn, t,he i-t'li element of min(a1, a2} is defined as min(al,i, a2.i}. From Assumption 1, all t,he element,s of mo and md are strictly positive. Hence: rniii{@mo =md, 6 &@md} is positive. Considering u 2 0, for a given 0 5 6 < 1: a sufficiently large h > 0 can always be fourid such that (11) is valid. Moreover, for a chosen u 7as 6 increase: smaller h can be obtained.
a
=
+ Define e = m,(T)
Proposition 1 clearly shows t,hat, t,he inpiit' constraints result in tlie coiistraiiits for h and: for a chosen u , smaller h can be realized by increasing 6. Hence: t o obtain faster response, in (4), m,o is designed instead of using mo direct,ly. However, larger 6 will lead t o larger initial error, wliicli may destroy t'he t'racking convergence. Hence, 6 and h should be properly chosen such t'liat bot'li tracking convergence and possible fast, response can be obtained.
-Amd
Bu,(T) = Am,o
U
-
P
(md
-
'T
1 m,o)h
1 -m,o)l - Aw-0 = -(md
E [h', co)
-
+ [Amd
+ (md
-
(md
l r
-
m,o)-]h h
1 m,o)- + A m r ( r ) . h
(16)
According t o (3) arid (16), tlie following is valid:
@ niin{m,.rr.m,i} m,o, we have 0 < @ niin m,,,m 0 aiid a < @ inin{ mo,md}. Consequently, u,o arid B u,h- can be rewrit,t,enas follows:
-
+ Ae + Bu
-
@ Inin{m,u,m~t} . From tlie definition of 1-8
@md
m. From (2) and (4),we have
From t,he definit,ion of u,('T) and considering ( 5 ) and (6), for 'T E [O, h-I, it, can be derived t'liat'
To simplify the design procedure, in the proposed algorithm, S will be designed first,. It,s design will be given in next Section. Now let us discuss how to calculate h for a given 6. According to (8) and (lo), the constraints for h can be rewritt'en as
a h -<
-
+ Am,(r)> r t [O: X]
Bu,(T) =
Amd:
r t [h+>co)
Substituting (14) into (15) and considering the above results. (15) can be rewritten as e
= Ae
+ Bsat(ulg + u h g ) .
(17)
CoritPNs with at least one P-serniflow are 11011controllable: according t o the classical linear control t,heory (Mahulea et al., 2005). Hence, a t'ransformation mat'rix H E RnX" is const'ruct'ed to separat,e t'he syst'em states into controllable and iioii-controllable parts. The first row of H is a basis of P-semiflow (here only one vect'or; Property 1.1) aiid the reinairiiiig rows are completed with elementary vect'ors such that, H is full rank. Define e = He. Tlieii (17) becomes
(13)
Remark 2. For given mo and md, u is designed first. Based on the chosen u , the minimum P such that a < @ min(m0, md} is chosen. 0 5. TRACKING CONTROL O F JF CONTPN
e
SYS'I'E1CIS
= Ae
+ B.sat(uLg+ Uhg)
(18)
where A = HAH-' and B = HB. According to the definitions of A, B and H and considering y . C = 0 , it can be derived that the first rows of both A and B are zeros, which leads t o 61 = 0.
The control signal u is designed as follows:
33
5.2 Design of the parameters 6 and y
From the definition of e and H, we have el(0) = y ( r n d , l -rn1(0)). As ymd = ym(O),el(0) = 0 can be obtained. Therefore, YT E [0,00), el(^) = 0. Moreover, the controllable part of (18) is
6,
= Ace,
+ Bcsat(ulg + u h g ) ,
a
Define e(W,p) = {e, : eTWe, 5 p } , where p = eT(O)We,(O).6 and y are designed off-line such t’liat Ye, E e(W,p) and Y j E A{, -kje, 2 -c1.j arid -k$e, 5 c a ; j , where cl;j is the j - t h eleineiit of
(19)
a
c1, c2,j =
a [ e 2 , .. . , en]TE RnP1, A, E R ( n - l ) x ( n - l ) where e, =
ulg
and
row of K 7 k$ = kj - q$,isi arid of S. Note that c ~2 ,0. ~
Uhg
7
(eFQe,
+ yuTRu)d.r,
is the i-th row
Proposition 2. Let ( N :A: mo) be a conservative arid strongly coiiiiected JF coiitPN system. For given Q and R,6 and y can always be found such that Ve, E t(W,p) arid V j E AT, -kje, 2 - q j and -kie, 5 c2,j.
designed t o minimize the following quadratic perforrriaiice criterion
=
si
is t’he j-t’h
The following proposition implies the existence of tlie solutions of 6 arid y.
ulg is
J(e,(O))
&,irnd,j- Ud,j}, kj
a
arid B, t R ( n - l ) x m . Froin tlie definition of e, we have e = H-le. As el = 0, e can be rewritten as Se, where S E Rnx(npl) is Hpl without the first column.
5.1 Deszgn of
L T
Inin{$-,
(20)
0
Proof: Same as tlie design of contains two cases.
where Q E R(npl)x(npl) is a diagonal positive definite matrix, R = ding(r1, . . . , r,) is positive definite and the y > 0 is a parameter t o be designed,
Case 1. cl
ulg, tlie
proof also
>0
Y j E M , the maximinn values of 1 k,e,l and I kie.1 subjected to eTWe, 5 p are as follows (Wredenhage and Belanger, 1994): -
-
Define c1 = rriiii{u,o. U,h-. u d } . Obviously. c1 2 0. The design of ulg is classified into two cases. Case 1. cl
>0
The low gain controller is ulg = -Ke,, where K = LRplBTW and W can be found from the Y following Riccati equation,
To satisfy the design requirements, we need to prove that. V,j E A I ,
Case 2. cl have zero-elements Based 011 (25) and (26) and considering the definitions of p, k, and k i , 6, and y, can be calculated for every J E A I . Therefore, 6 = min,t~l{6,} and y = rriaxJt~~{yJ}.VJ t AI. tlie existeiice of yJ arid 6, can be disscused according t o two cases:
To clearly explain tlie basic idea, assume oiily one element of c1, i.e. cl,, ( z E A I ) , is zero. However, if c1 have several zero-elements. the design of ulg can be derived analogously. In this Case, W is calculated froin tlie following Riccati equation.
+
WAC AFW
-
LW(B,
7 Rpl(B, - AB,)TW + Q
-
AB,)
= 0,
In t’hiscase, let’ 6, = 0 and yj can be any positive value. (22)
B.c2,j > 0
where AB, E R ( n - l ) x m , the z-th column of AB, is same as tlie z-th coluinri of B, arid all the remaining columns of AB, are 0. As A, is stable (from Property 2), the solution of W always exists. The low gain controller is ulg = -Ke,, where K = ‘Rpl(Bc AB,)TW. From the 7definition of AB,, it can be derived that the zt h row of K , i.e. k,, is 0. Hence. AB,K = 0.
From t,he definition of kj and ki, it, can be derived that any given y,kj arid ki are finite constant matrices, i.e. irrelated with 6. Therefore, V j t A’ f7 both kjWplkT arid kiWplk$Tare constants. On the other hand, smaller 6, will lead t o smaller initial error which further results in smaller ec(0) arid p. Therefore, as both q j arid c ~are, strictly ~ posit,ive constant,s, a posit,ive 63,which is small enough, can always be found such that (25) and (26) are valid.
-
For tlie high-gain term. in both Case 1 arid Case 2, Uhg = -IBTWe, where I is a positive constant.
34
Case 2.
c1
have zero-elements
II.
Assume el., = 0. 6 aiid y are designed off-line such that Ye, E e(W,p),-kje, 2 -rl.j ( Y j E {l;..,z l , z l,...,rn}) and -k$e, 5 c2;j ( Y j E M ) . According t'o the result' in Case 1 and considering the strictly posit'iveness of el.? ( V j t { 1 : . . . , z - l , z + 1 , . . . 7 r n } ) 7tlie existence of 6 and y can also be guaranteed. On the ot'lier hand, as k, = 0, -kje, = 0. Consequently, -k,e, 2 -el,, is always valid. Therefore, 6 and y caii always be fouiid such that Ve, E t(W,p) arid V j t AT, -kje, 2 -cl,j and -kge, 5 c2.j.
+ k,e,).
(31)
2 - 7 ~ , , j + - 7 ~ ~ >+j kje, 5 0. (32)
-kje,
On t'he other hand,
*
+uhg,j
5 -ur,j + -lbEjWe, 5 -ur,j
ulg,j
5 + kje,. uhg,j
-ur>j - u l g , j
(33)
From (32) arid (33): it can be derived that bTjWe, > 0. Herice: froin (31),eTWb,,jvj < 0.
Theorem 1. Let ( N ,A, mo) be a conservative and strongly coiiriected JF coritPN system. For aiiy mo > 0 aiid aiiy reachable md > 0 (Assunipfzon 1 ) aiid u d , control law (14) with tlie parameters 6 aiid y designed in Proposzfzon 2 caii eiisure the global asyinptotical coiivergeiice of both tlie syst,em markings and t,he cont,rol signals.
111. uig.j
+ uhg,j 2 & , m i
-
ur.j
Similarly to t'he proof in II, eTWb,,jvj be derived.
I
< 0 can
From I. 11 aiid III, V < -eTQe,. Hence, t(W7p ) is a positively iiivariarit region. As ec(0) t t(W74): e , ( ~t) t(W,p ) for all T 2 0. Therefore, since V < -eTQec, e,, e and e asymptot,ically converge t o zero. Furthermore, the coiivergerice of e, leads to the convergence of u t'o ~ ~ ( 7 ) .
Proof:
>0
Assume e, E e(W,p ) , where e(W,p ) is defined in Proposition 2. Proposition 2 iinplies that 6 arid y can always be found so t'liat', Ye, E e(W,p ) and Y j E A I , -kje, 2 -cl,j and -k$e, 5 c2,j. Define V = eTWe,, where W is obtained from (21). Hence:
Case 2.
c1
have zero-elements
The convergence analysis is qiiit,e similarly t,o that in Case 1. From (19),the error dynamics can be rewritten as follows:
Q, = [A,
v = e,.T w e , + eFwQ,.
= eTWb,,,(-u,,,
As -kje, 2 -cl,j, coiisideririg the defiiiitioris of alld u T ( T ) , we have
5.3 Asymptotical convergence analysis
Case 1. c1
+ uhg,j 5 -ur,j
eTWb,,,u,
+
-
7Llg.j
-
(B,
-
AB,)K]e,
+ B,[.snf(uig+ u h g )
'
+Ke,]
(27)
On the ot'her hand, (19)can be rewrit't'en as
ulg,j
As AB,K
-
=
AB,Ke,.
(34)
0, (34) becomes
+ 7 L h g ; j is not' saturat'ed. Hence,
eFW6,,juj = eFWb,,j(-kje, - lbzjWe, = -l(e,TWb,,j)2 5 0.
11122 eleineiitary vectors, H is chosen as H
+ kje,) (30)
01000 0 0 1 00 00010
= -
35
00001
Fig. 1. Timed Join-Free Net System. Fig. 3. Control signals. 7. CONCLUSION
E-
--
2
EC
The main concern of our work is t o construct proper control laws for step-trackiiig of timed coritPN systems in presence of the existing staterelated input constraints. To guarantee global corivergeiice aiid smoothness of states arid control signals. the design inetliod for a step-rainp tracking trajectory has been outlined. With tlie new tracking target, a novel low-and-high gain control method has been further proposed.
*EE-
Tlme 5
-
8 Tlme ( 5 )
Fig. 2. Convergence of markings. REFERENCES Assume an init,ial marking as mo = [ 2 , 3 , 5 , 6 ,9IT a.nd a. desired marking m d = [3,5,2,7,8IT.To maximize t,he flows of t,he steady st,at,e,t,he desired filial control input u d = [l:3 , 0 75, 6IT.
.Jimi.nez, E., J . Julvez, L. Recalde and M. Silva (2005). On cont,rollabilit,yof t,imed continuous Petri net syskms: the join free case. In: Proceedings of the 44th IEEE CDC and E 2UO5. Seville, Spain. Liii, Z., M. Pachter aiid S. Baiida (1998). Toward improvement, of tracking performance - nonlinear feedback for linear systems. International Journal of Control 70(1), 1-11. Mahiilea, C., A. Ramirez, L. Recalde and M. Silva (2005). St'eady state control, zero valued poles and token conservation laws in continuous net' syst'ems. In: Proceedings of the International Workshop on Control of Hybrid and Discrete Event Systems. Miami, TJSA. Recalde: L.: E. 'I'eruel and M. Silva (1999). Autonomous coiitiriuous P/T systems. In: Application and Theory of Petri Nets 1999 ( S . Donatelli and J . Kleijn, Eds.). Vol. 1639 of Lecture Notes in Computer Science. Springer. pp. 107-126. Saberi: A,, Z. Liri arid A. R. Tee1 (1996). Coiitrol of linear system with saturating actuat'ors. IEEE li-ansactions on Automatic Control 41(3): 368-378. Teruel, E., J . M. Coloin arid M. Silva (1997). Choice-free Petri nets: A model for deterrninistic coiicurreiit systems with bulk services and arrivals. IEEE li-ansactions on Systems, Man, and Cybernetics 27(1), 73-83. Wredenhage, G. F. arid P. R. Bklanger (1994). Piecewise-linear LQ control for systems with input constraints. Automatica 30(3), 403416.
+
Based 011 mo and md7 n = [1:0,2: 1,2IT a [ l ,1: 1,1,1IT where a 2 0. Here we randomly choose a = 1: heiice n = [a71 , 3 7 2 7 3 ] TCoii. sideririg n7 mo and m d , /? = 1.5 is the miniinurri value such that 5 @miii{mo7md}. According to (12) and (13), it' can be derived t'liat U,O = [0.6667,2.3333,3,4.6667,7IT and U,h- = [1.6667,4.3333,0,5.6667,6IT. Hence, C1 = [0.6667: 2.3333,O: 4.6667, 6IT. As c1 has one zeroelement, ulg is calciilat'ed based on (22). Let 6 = 0.2 aiid y = 5. It is easy t o verify that V j E {1,2,3,4,5}, bot'li (25) and (26) are valid. Then, h = p(1 S ) = 1.2. For simplicity, choose Q = I d X 4 and R = I s X 5 for the low-gain design. For t,he high-gain t,erm, any positive 1 can guarantee the t,rackingconvergence. Generally, smaller 1 will lead t o slower system response. However, due t'o the existence of input sat'uration, when 1 is sufficiently large, the system responses have little difference by further increasing 1. In our simulation: we choose 1 = 10.
5
~
The siinulatioii results are sliowri in Figures 2 and 3 . Figure 2 illiistrat,es the convergence of t,he markings urider tlie designed control law. Figure 3 shows the control signals u arid tlie state-related upper bound, i.e. q$,imi. ( Note the net structure determines that V j E {1,2: 3: 4,5}, i = j.) It can be seen that 0 5 uj 5 q$,irni and the firial control signals converge t o tlie desired ones.
36
PUBLICATIONS
TRACKING CONTROL OF JOIN-FREE TIMED CONTINUOUS PETRI NET SYSTEMS Jing Xu, Laura Recalde, Manuel Silva
Instituto de Investigacidn de Ingenieria d e Aragdn (I3A) Departamento d e Informa'tica e Ingenieria de Sistemas, Centro Polite'cnico Superior, Uniuersidad d e Zaragoza, Email:(~xu,lrecalde,silva}Qunizar.es
Abstract: A new low-and-high gain algorithm is presented for tracking control of timed coiitiiiuous Petri Net (coiitPN) systems working urider infinite servers semantics. The iiihereiit properties of timed coritPN determine that the control signals must be iioii-negative aiid upper bounded by functions of system states. In the proposed control approach, LQ theory is first utilized t o design a low-gain controller such that the control signals satisfy the input coiistraiiits. Based on the low-gain cont,roller, a higli-gain t,erm is fiirt,lier added to improve the syst,em transient performance. In order t o guarantee global tracking corivergeiice aiid smoot'hness of control signals, in our work, a new mixed t'racking t'rajectory (state st,ep and ramp) is ut,ilized instead of a pure st,ep reference signal. Copyright 02006 IFAC
1. INTRODTICTION
For cont,rol of timed contPN systems, we will begin from Join-Free(JF) timed coritPN systems with step-tracking cont,rol target. For a JF t,imed contPN syst,em, a linear differential equat,ion describes its behavior, but it' is subject' to certain input constraints, i.e. the control signal must be non-negat'ive and upper bounded by a funct'ion of system st'ates. 'I'liese constraints result in a hybrid system, more precisely a piecewise linear system. 'I'he main challenge in our work is t,o develop control laws under these special input const'raints so that global tracking coiivergeiice can be ensured.
Petri Nets (PNs) are powerful mathematical tools with appealing graphical representations for tlie modeling, analysis and synthesis of Discrete Even Systems (DESs). However, like in all DESs, discrete PN systems suffer from tlie so called state explosion problem. One possible way t o partially tackle this problem is t o fluidify the discrete P N models. The resiilt'ing continuous Petri Net' (cant'PN) systems have the potential for the applications of more analytical t,echniqiies which were originally developed for coiitiiiuous and hybrid systems. Fiirt'liermore, analogous to discrete P N systems, time has been introduced t o coritPN systems, which leads t'o timed cont'PN systems. Steady state control is studied in (Mahulea et al.: 2005); assuming all transitions can be controlled: it can be solved by meaiis of a LPP. Nevertheless, dynamic control of timed coiitPN systems needs further investigations aiid great improvements.
'I'he input' const'raints can be t'reated as input saturat,ions. As input, sat,urat,ion is one of the common phenomena encountered in cont,rol systerns: hitherto lots of works have been done. In (Wredenhage aiid Bklanger, 1994), a kind of piecewise-linear control law was derived. However, such a design method will lead t o a lowgain controller. To improve tlie control performance: a low-and-high gain approach was given in (Saberi et al., 1996). Recent'ly, several nonlinear control met'liods were fiirt'lier present'ed (Lin et
30
have n and rn elements, respectively. 'I'he marking m belongs t o R+n7 where R+ is the set of 11011negative real numbers, and bot,h P r e and Post are of the size n x r n . For w t P U T7the set of its input' and output' nodes are denot'ed as *w,and 'w*. respectively. N is JF (or reiidez-vous free) iff i t E T, I*tl = 1. If m is reachable froin mo through a sequence u E R+m, tlie state equation is m = mo C . u , where C = Post - P r e is the token flow mat,rix and u is t,he firing count vector.
al.: 199X), which inaiiily focus 011 the high-gain design in tlie low-aiid-high gain algoritliins aiinirig t o achieve better traiisieiit control performance. It should be pointed out that coiriirioii assumptions in all these works are that the lower saturation bounds are negat'ive constants and the upper saturation bounds are positive constants. However, in JF t,imed contPN syst,ems,t,he lower sat,uration bound is zero and the upper sat'iiration bound depends on syst'em st'at'es. Therefore, the existing control st'rat'egies for systems wit'h input' sat'urations cannot be applied direct'ly.
+
2.2 Tarried Continuous Petri Net Systems
A timed contPN system can be represented as ( N ,A, mo), where X [ t ] > 0 is t'he int'ernal firing rate of transition t . The state equation has an explicit dependence 011 time m(7) = mo CU(T), where 7 is time. Deriving with respect t o it: m(7) = C&(-r)is obtained. Define f ( 7 ) = & ( T ) > wliicli denote flows of transitions. The state equation is m(7) = Cf(7). For notation simplicity, 7 will be omitted in the rest of tlie paper. For the definition of flow f 7different semantics have been iiitroduced and t'he most import'ant ones are infinite servers and finite servers. Infinite servers semantics will be considered in t,liis work, which usually provides a much bet'ter approximation of discret'e behaviors. Under infinite server semant,ics,f is t,he product of X[t] and tlie iiistaritaiieous enabling of tlie transition, i.e. f[t] = X[t] . enab(t,m) = X[t]. 1riiii~~.~{m[23]/Pre[23, t]}. As in JF nets any traiisitioii has only one input place, the flow can be expressed as f = @ . m7 where @ t R t m x n aiid @[t,p]= X[t]/Pre[p7t] if p = * t , @[t,p]= 0 otherwise. Moreover, each row of @ has only one
In t,his paper, a new low-and-high gain approach will be proposed for st'ep-tracking cont'rol of JF timed contPN syst,ems. 'I'he presenkd algorithm can ensure global asympt'ot'ical convergence in presence of t,he input const,raint,sexisting in JF timed contPN syst,ems.According t,o init,ialmarking: desired marking arid net structure, a new reference t'rajectory is const'ruct'ed.Based 011 t'he new tracking target, a low-gain controller is designed according t o LQ theory so that the control signals are within the required regions. Siiice the upper bounds of the control inputs depend 011 the system states, tlie design method in (Wredeiihage arid Bdanger: 1994) fails t o work. By coinbiriiiig the new t,racking traject,ory design wit,h LQ met,hod, a novel design scheme for the low-gain part is given in our work. Analogous to t'he works of (Saberi et al., 1996), a high-gain part' is fiirt'lier added to make bet't'er use of t'lie available cont'rol authority, and consequently faster syst'em response can be obt,ained. Not,e that, in t,he high-gain term, the control vect,or of the low-gain part is maint,ained, wliicli makes possible the analysis of system stability. Rigorous proof is provided t o guarantee the globa 1 asyrriptot ica 1 convergence.
+
a
lion-zero element. V j E AT = { 1:2: . . . r n } : the lion-zero eleineiit for tlie j-tli row of @ is denoted a as 4j,i where i E N = {1,2;..,n}. Therefore, V j E A{, we have fj = q5j>irni,where fj is t'he jt h element of f arid rrii is the i-th eleineiit of m. The evolut,ion of the marking can be writ,ten as follows: ~
The paper is organized as follows. Sect,ion 2 introduces the required concepts of coiitPN systems arid timed coritPN systems. The control for JF timed coiitPN systems is forinulated in Section 3 . How to construct the new tracking trajectory is outlined in Section 4. Section 5 focuses on the development of cont'rol laws and t'he analysis of global asyrriptotical convergence property. An illustrative example is given in Section 6. Finally, Section 7 concludes the paper.
m=C.f=A.m
(1)
where A = C . @. Finally the definition of conservativeness aiid some related properties for a JF timed coritPN are listed as follows.
Definition 1. A P N is conseruative iff 3y > 0, such t,liat y.C = 0. Any left non-negat,ive anniiller of matrix C, i.e. y, is called P-semiflow. A Psemiflow is minimal if 1 is the great,est common divisor of its elements, aiid its support does riot strictly contain the support of other P-semiflow.
2. CON'I'INTJOTJS PETRI NET SYSTEMS
2.1 Untrmed Contznuous Petrz Net Systems A contPN system can be described as (N,mo), where N = ( P , T , P r e , P o s t )specifies the net structure (P and T are disjoint (finite) sets of places and transitions, and P r e and Post are incidence matrices), and mo is the initial marking. N is assumed t o be connected. while P aiid T
Property 1. (Teruel et al., 1997) Let A' be a JF net.
N is strongly connected, a) N has at most one rriiiiiinal P-semiflow:
1.1 If
31
Assumption 1. V i E N , mo,%> 0 and md,%> 0.
b) N is conservative iff it has one P-semiflow. 1.2 If N is conservative, it is coiisisterit iff it is strongly connected.
Remark 1. If we coiisider optiinal steady states in practical manufacture syst’ems,generally md > 0 is valid. On t,he ot,lier hand, if some element,s of mo are zeros, as md > 0, a firing sequence can always be found such t’liat mo[a > m and m > 0 (Recalde et al., 1999). Then the control algorit’lim proposed in our work can be further applied.
Property 2. Let A’ be a conservat’ive and strongly connected JF t,imed contPN. The matrix A for A’ defined in (1) has only one zero-eigenvalue. Moreover, the rerriaiiiiiig eigeiivalues of A are with negative real part’s.
4. DESIGN O F NEW TRACKING REFERENCE
Proof: As N is conservative arid strongly coiinected, from Property 1.2, N is consistent,. From the Proposition 1 in ( J i m h e z et al., 2005), for the conservative and consistent, J F PN A’, t,he mat,rix A has only one zero-valued eigenvalue. On the other hand, as A is a Metzler matrix aiid the eigenvalue of zero is the (Frobenius) dominant, eigeiivalue (Jiinhez et al., 2005), tlie rerriairiirig eigenvalues of A are wit’li negative real part’s. W
To ensure global convergence arid sinootliiiess of the control signal, a step target md is replaced by tlie following refereiice trajectory m,(T).
a
where m,(r) E R+n, m,o = m,(0) and h > 0 determines the time when m,(7) = md. Here we choose m,o = mo 6(md mo), where 0 5 6 < 1 is a parameter t o be designed. The parameter h is chosen such that, for given m,o and md, valid control actions u,o E R+m aiid uTh-E Rtm exist for the following equations.
3. PROBLEM FORMULATION
+
We will restrict our research t o coiiservative aiid strongly connect,ed JF t,imed contPN systems and assume all tlie markings are observable. For concise expression, “timed cont,PN” will be writ,ten as “contPN”. Like all t,he ot,her systems, cont,rol act’ions can also be introduced t’o modify aut,onomous evolution of P N systems. The possible control actioii that can be applied t o PN systems is to slow down t,lieir unforced firing flows. Hence, tlie forced flows of controlled traiisitioiis becoine f-u, where u is t,hecontrol signal and must satisfy 0 5 u 5 f . Corisideriiig (l),a JF coritPN system with a control actioii caii be described as follows:
a
m = C ( @ m - u) = A m -
Bu,
-
(5)
Proposztzon 1. For given mo aiid reachable
md,
6
aiid h can always be fourid such that (5) aiid (6) are valid. Moreover, as 6 increase. smaller h can be chosen.
(2)
a
where m E R+n, A E Rnxn, B = C E Rnxm and u E R+m. It should be n o k d t,liat the constraints oii input u lead tlie closed system t o be a piecewise linear system.
Proof: Given mo and reachable m d , a 2 0 can always be fouiid so that md = mo B a . Thus.
+
md -
Our control object’ive is t’o construct cont’rollaws such that m arid u respectively converge t o desired values: md and ud. ‘Yo sat,isfy reachabilit,y, md inust fulfill that y . md = y . mo where y E Rn is the basis of P-semiflows. For ud, 0 5 ud 5 @md inust be satisfied, i.e. V j E M 7 0 5 U d , j 5 & , i r r i d > i where U d , j aiid md,i are tlie j-tli aiid i-th elements of ud aiid md respectively. Moreover, as md is constaiit: according t o ( 2 ) : tlie desired coiitrol input, must, be a solut,ionof the following equat,ion. 0 = Amd
-
Bud.
m,o
=B(l-
&)a.
(7)
Substitutiiig (7) iiito (5) aiid (6) yields
B@m,o - Bu,o
=
1 B ( l - 6)a-, h 1
B@md B u , ~ -= B(1 -
-
6)~:.
h
Obviously, u,o = @m,o (1 6)ai and u,h- = @md - (1 - 6 ) a i are solutions for the above equations. From 0 5 u , ~5 @m,o. we have: -
(3)
In our work, the following assumption is made for mo and md.
32
-
From 0 5 u,h- 5 @md,we can obtain that 1
O I U - < -
1
h-1-6
where uly is the low-gain part, uhg is the highgain term arid u,(r) is tlie reference control input defined as follows:
@md.
Therefore, tlie filial constraints for h are: U
Smd
1
-@md h -< rniii{ @mo+ 1-6'1-6
0 5 -
(11)
where u,o arid
u,h-
are given in (12) aiid (13)
respectively. Moreover, V d t Rm. s a t ( d ) [sat(dl), . . . , sat(dm)lT and V j E A l , .sat(d3) is
where, for al E R+n and a2 E Rtn, t,he i-t'li element of min(a1, a2} is defined as min(al,i, a2.i}. From Assumption 1, all t,he element,s of mo and md are strictly positive. Hence: rniii{@mo =md, 6 &@md} is positive. Considering u 2 0, for a given 0 5 6 < 1: a sufficiently large h > 0 can always be fourid such that (11) is valid. Moreover, for a chosen u 7as 6 increase: smaller h can be obtained.
a
=
+ Define e = m,(T)
Proposition 1 clearly shows t,hat, t,he inpiit' constraints result in tlie coiistraiiits for h and: for a chosen u , smaller h can be realized by increasing 6. Hence: t o obtain faster response, in (4), m,o is designed instead of using mo direct,ly. However, larger 6 will lead t o larger initial error, wliicli may destroy t'he t'racking convergence. Hence, 6 and h should be properly chosen such t'liat bot'li tracking convergence and possible fast, response can be obtained.
-Amd
Bu,(T) = Am,o
U
-
P
(md
-
'T
1 m,o)h
1 -m,o)l - Aw-0 = -(md
E [h', co)
-
+ [Amd
+ (md
-
(md
l r
-
m,o)-]h h
1 m,o)- + A m r ( r ) . h
(16)
According t o (3) arid (16), tlie following is valid:
@ niin{m,.rr.m,i} m,o, we have 0 < @ niin m,,,m
-
+ Ae + Bu
-
@ Inin{m,u,m~t} . From tlie definition of 1-8
@md
m. From (2) and (4),we have
From t,he definit,ion of u,('T) and considering ( 5 ) and (6), for 'T E [O, h-I, it, can be derived t'liat'
To simplify the design procedure, in the proposed algorithm, S will be designed first,. It,s design will be given in next Section. Now let us discuss how to calculate h for a given 6. According to (8) and (lo), the constraints for h can be rewritt'en as
a h -<
-
+ Am,(r)> r t [O: X]
Bu,(T) =
Amd:
r t [h+>co)
Substituting (14) into (15) and considering the above results. (15) can be rewritten as e
= Ae
+ Bsat(ulg + u h g ) .
(17)
CoritPNs with at least one P-serniflow are 11011controllable: according t o the classical linear control t,heory (Mahulea et al., 2005). Hence, a t'ransformation mat'rix H E RnX" is const'ruct'ed to separat,e t'he syst'em states into controllable and iioii-controllable parts. The first row of H is a basis of P-semiflow (here only one vect'or; Property 1.1) aiid the reinairiiiig rows are completed with elementary vect'ors such that, H is full rank. Define e = He. Tlieii (17) becomes
(13)
Remark 2. For given mo and md, u is designed first. Based on the chosen u , the minimum P such that a < @ min(m0, md} is chosen. 0 5. TRACKING CONTROL O F JF CONTPN
e
SYS'I'E1CIS
= Ae
+ B.sat(uLg+ Uhg)
(18)
where A = HAH-' and B = HB. According to the definitions of A, B and H and considering y . C = 0 , it can be derived that the first rows of both A and B are zeros, which leads t o 61 = 0.
The control signal u is designed as follows:
33
5.2 Design of the parameters 6 and y
From the definition of e and H, we have el(0) = y ( r n d , l -rn1(0)). As ymd = ym(O),el(0) = 0 can be obtained. Therefore, YT E [0,00), el(^) = 0. Moreover, the controllable part of (18) is
6,
= Ace,
+ Bcsat(ulg + u h g ) ,
a
Define e(W,p) = {e, : eTWe, 5 p } , where p = eT(O)We,(O).6 and y are designed off-line such t’liat Ye, E e(W,p) and Y j E A{, -kje, 2 -c1.j arid -k$e, 5 c a ; j , where cl;j is the j - t h eleineiit of
(19)
a
c1, c2,j =
a [ e 2 , .. . , en]TE RnP1, A, E R ( n - l ) x ( n - l ) where e, =
ulg
and
row of K 7 k$ = kj - q$,isi arid of S. Note that c ~2 ,0. ~
Uhg
7
(eFQe,
+ yuTRu)d.r,
is the i-th row
Proposition 2. Let ( N :A: mo) be a conservative arid strongly coiiiiected JF coiitPN system. For given Q and R,6 and y can always be found such that Ve, E t(W,p) arid V j E AT, -kje, 2 - q j and -kie, 5 c2,j.
designed t o minimize the following quadratic perforrriaiice criterion
=
si
is t’he j-t’h
The following proposition implies the existence of tlie solutions of 6 arid y.
ulg is
J(e,(O))
&,irnd,j- Ud,j}, kj
a
arid B, t R ( n - l ) x m . Froin tlie definition of e, we have e = H-le. As el = 0, e can be rewritten as Se, where S E Rnx(npl) is Hpl without the first column.
5.1 Deszgn of
L T
Inin{$-,
(20)
0
Proof: Same as tlie design of contains two cases.
where Q E R(npl)x(npl) is a diagonal positive definite matrix, R = ding(r1, . . . , r,) is positive definite and the y > 0 is a parameter t o be designed,
Case 1. cl
ulg, tlie
proof also
>0
Y j E M , the maximinn values of 1 k,e,l and I kie.1 subjected to eTWe, 5 p are as follows (Wredenhage and Belanger, 1994): -
-
Define c1 = rriiii{u,o. U,h-. u d } . Obviously. c1 2 0. The design of ulg is classified into two cases. Case 1. cl
>0
The low gain controller is ulg = -Ke,, where K = LRplBTW and W can be found from the Y following Riccati equation,
To satisfy the design requirements, we need to prove that. V,j E A I ,
Case 2. cl have zero-elements Based 011 (25) and (26) and considering the definitions of p, k, and k i , 6, and y, can be calculated for every J E A I . Therefore, 6 = min,t~l{6,} and y = rriaxJt~~{yJ}.VJ t AI. tlie existeiice of yJ arid 6, can be disscused according t o two cases:
To clearly explain tlie basic idea, assume oiily one element of c1, i.e. cl,, ( z E A I ) , is zero. However, if c1 have several zero-elements. the design of ulg can be derived analogously. In this Case, W is calculated froin tlie following Riccati equation.
+
WAC AFW
-
LW(B,
7 Rpl(B, - AB,)TW + Q
-
AB,)
= 0,
In t’hiscase, let’ 6, = 0 and yj can be any positive value. (22)
B.c2,j > 0
where AB, E R ( n - l ) x m , the z-th column of AB, is same as tlie z-th coluinri of B, arid all the remaining columns of AB, are 0. As A, is stable (from Property 2), the solution of W always exists. The low gain controller is ulg = -Ke,, where K = ‘Rpl(Bc AB,)TW. From the 7definition of AB,, it can be derived that the zt h row of K , i.e. k,, is 0. Hence. AB,K = 0.
From t,he definition of kj and ki, it, can be derived that any given y,kj arid ki are finite constant matrices, i.e. irrelated with 6. Therefore, V j t A’ f7 both kjWplkT arid kiWplk$Tare constants. On the other hand, smaller 6, will lead t o smaller initial error which further results in smaller ec(0) arid p. Therefore, as both q j arid c ~are, strictly ~ posit,ive constant,s, a posit,ive 63,which is small enough, can always be found such that (25) and (26) are valid.
-
For tlie high-gain term. in both Case 1 arid Case 2, Uhg = -IBTWe, where I is a positive constant.
34
Case 2.
c1
have zero-elements
II.
Assume el., = 0. 6 aiid y are designed off-line such that Ye, E e(W,p),-kje, 2 -rl.j ( Y j E {l;..,z l , z l,...,rn}) and -k$e, 5 c2;j ( Y j E M ) . According t'o the result' in Case 1 and considering the strictly posit'iveness of el.? ( V j t { 1 : . . . , z - l , z + 1 , . . . 7 r n } ) 7tlie existence of 6 and y can also be guaranteed. On the ot'lier hand, as k, = 0, -kje, = 0. Consequently, -k,e, 2 -el,, is always valid. Therefore, 6 and y caii always be fouiid such that Ve, E t(W,p) arid V j t AT, -kje, 2 -cl,j and -kge, 5 c2.j.
+ k,e,).
(31)
2 - 7 ~ , , j + - 7 ~ ~ >+j kje, 5 0. (32)
-kje,
On t'he other hand,
*
+uhg,j
5 -ur,j + -lbEjWe, 5 -ur,j
ulg,j
5 + kje,. uhg,j
-ur>j - u l g , j
(33)
From (32) arid (33): it can be derived that bTjWe, > 0. Herice: froin (31),eTWb,,jvj < 0.
Theorem 1. Let ( N ,A, mo) be a conservative and strongly coiiriected JF coritPN system. For aiiy mo > 0 aiid aiiy reachable md > 0 (Assunipfzon 1 ) aiid u d , control law (14) with tlie parameters 6 aiid y designed in Proposzfzon 2 caii eiisure the global asyinptotical coiivergeiice of both tlie syst,em markings and t,he cont,rol signals.
111. uig.j
+ uhg,j 2 & , m i
-
ur.j
Similarly to t'he proof in II, eTWb,,jvj be derived.
I
< 0 can
From I. 11 aiid III, V < -eTQe,. Hence, t(W7p ) is a positively iiivariarit region. As ec(0) t t(W74): e , ( ~t) t(W,p ) for all T 2 0. Therefore, since V < -eTQec, e,, e and e asymptot,ically converge t o zero. Furthermore, the coiivergerice of e, leads to the convergence of u t'o ~ ~ ( 7 ) .
Proof:
>0
Assume e, E e(W,p ) , where e(W,p ) is defined in Proposition 2. Proposition 2 iinplies that 6 arid y can always be found so t'liat', Ye, E e(W,p ) and Y j E A I , -kje, 2 -cl,j and -k$e, 5 c2,j. Define V = eTWe,, where W is obtained from (21). Hence:
Case 2.
c1
have zero-elements
The convergence analysis is qiiit,e similarly t,o that in Case 1. From (19),the error dynamics can be rewritten as follows:
Q, = [A,
v = e,.T w e , + eFwQ,.
= eTWb,,,(-u,,,
As -kje, 2 -cl,j, coiisideririg the defiiiitioris of alld u T ( T ) , we have
5.3 Asymptotical convergence analysis
Case 1. c1
+ uhg,j 5 -ur,j
eTWb,,,u,
+
-
7Llg.j
-
(B,
-
AB,)K]e,
+ B,[.snf(uig+ u h g )
'
+Ke,]
(27)
On the ot'her hand, (19)can be rewrit't'en as
ulg,j
As AB,K
-
=
AB,Ke,.
(34)
0, (34) becomes
+ 7 L h g ; j is not' saturat'ed. Hence,
eFW6,,juj = eFWb,,j(-kje, - lbzjWe, = -l(e,TWb,,j)2 5 0.
11122 eleineiitary vectors, H is chosen as H
+ kje,) (30)
01000 0 0 1 00 00010
= -
35
00001
Fig. 1. Timed Join-Free Net System. Fig. 3. Control signals. 7. CONCLUSION
E-
--
2
EC
The main concern of our work is t o construct proper control laws for step-trackiiig of timed coritPN systems in presence of the existing staterelated input constraints. To guarantee global corivergeiice aiid smoothness of states arid control signals. the design inetliod for a step-rainp tracking trajectory has been outlined. With tlie new tracking target, a novel low-and-high gain control method has been further proposed.
*EE-
Tlme 5
-
8 Tlme ( 5 )
Fig. 2. Convergence of markings. REFERENCES Assume an init,ial marking as mo = [ 2 , 3 , 5 , 6 ,9IT a.nd a. desired marking m d = [3,5,2,7,8IT.To maximize t,he flows of t,he steady st,at,e,t,he desired filial control input u d = [l:3 , 0 75, 6IT.
.Jimi.nez, E., J . Julvez, L. Recalde and M. Silva (2005). On cont,rollabilit,yof t,imed continuous Petri net syskms: the join free case. In: Proceedings of the 44th IEEE CDC and E 2UO5. Seville, Spain. Liii, Z., M. Pachter aiid S. Baiida (1998). Toward improvement, of tracking performance - nonlinear feedback for linear systems. International Journal of Control 70(1), 1-11. Mahiilea, C., A. Ramirez, L. Recalde and M. Silva (2005). St'eady state control, zero valued poles and token conservation laws in continuous net' syst'ems. In: Proceedings of the International Workshop on Control of Hybrid and Discrete Event Systems. Miami, TJSA. Recalde: L.: E. 'I'eruel and M. Silva (1999). Autonomous coiitiriuous P/T systems. In: Application and Theory of Petri Nets 1999 ( S . Donatelli and J . Kleijn, Eds.). Vol. 1639 of Lecture Notes in Computer Science. Springer. pp. 107-126. Saberi: A,, Z. Liri arid A. R. Tee1 (1996). Coiitrol of linear system with saturating actuat'ors. IEEE li-ansactions on Automatic Control 41(3): 368-378. Teruel, E., J . M. Coloin arid M. Silva (1997). Choice-free Petri nets: A model for deterrninistic coiicurreiit systems with bulk services and arrivals. IEEE li-ansactions on Systems, Man, and Cybernetics 27(1), 73-83. Wredenhage, G. F. arid P. R. Bklanger (1994). Piecewise-linear LQ control for systems with input constraints. Automatica 30(3), 403416.
+
Based 011 mo and md7 n = [1:0,2: 1,2IT a [ l ,1: 1,1,1IT where a 2 0. Here we randomly choose a = 1: heiice n = [a71 , 3 7 2 7 3 ] TCoii. sideririg n7 mo and m d , /? = 1.5 is the miniinurri value such that 5 @miii{mo7md}. According to (12) and (13), it' can be derived t'liat U,O = [0.6667,2.3333,3,4.6667,7IT and U,h- = [1.6667,4.3333,0,5.6667,6IT. Hence, C1 = [0.6667: 2.3333,O: 4.6667, 6IT. As c1 has one zeroelement, ulg is calciilat'ed based on (22). Let 6 = 0.2 aiid y = 5. It is easy t o verify that V j E {1,2,3,4,5}, bot'li (25) and (26) are valid. Then, h = p(1 S ) = 1.2. For simplicity, choose Q = I d X 4 and R = I s X 5 for the low-gain design. For t,he high-gain t,erm, any positive 1 can guarantee the t,rackingconvergence. Generally, smaller 1 will lead t o slower system response. However, due t'o the existence of input sat'uration, when 1 is sufficiently large, the system responses have little difference by further increasing 1. In our simulation: we choose 1 = 10.
5
~
The siinulatioii results are sliowri in Figures 2 and 3 . Figure 2 illiistrat,es the convergence of t,he markings urider tlie designed control law. Figure 3 shows the control signals u arid tlie state-related upper bound, i.e. q$,imi. ( Note the net structure determines that V j E {1,2: 3: 4,5}, i = j.) It can be seen that 0 5 uj 5 q$,irni and the firial control signals converge t o tlie desired ones.
36