Mohamed Benmensour

Sema Djennane-Bousmaha

Abdou Boucekkine
email: abdou.boucekkine@univ-rennes1.fr

New Cyclic Oligothiophenes And Their Co-Oligomers Derivatives Of Molecular Engineering Interest

Keywords: Macrocyclic conducting polymers, polythiophene, polybithiophene, bifuran, bipyrrole, DFT 1 1

The electronic properties of several series of bridged oligomers and their cooligomers derivatives of five-membered biheterocycles (bithiophene, bipyrrole, bifuran bridged by CO or CS group) in their linear and cyclic structures (planar rings and crowns), have been studied using Density Functional Theory (DFT) calculations. The stability of ring forms of bridged oligo(6) biheterocycles is evidenced. Moreover, when replacing the CO bridging group by the CS one, in all oligomers the HOMO-LUMO energy gap is reduced, due mainly to the strongest stabilization of the LUMO. This energy gap is further reduced with the introduction of bridged bithiophene or bridged bifuran in the cyclic chain of bridged bipyrrole, thus suggesting an improvement of the conductivity properties of such species. The CS bridged co-oligo(3)(bithiophenebipyrrole) and CS bridged co-oligo(3) (bifuran-bipyrrole) seems to be promising candidates for molecular engineering applications.

INTRODUCTION 2

Polyheterocycles, such as polypyrrole, polythiophene, polyaniline and their derivatives, have attracted considerable attention since they can lead to conducting polymer materials. Their conductivity after adequate doping, their good mechanical property and their stability made these compounds candidates for applications in microelectronics and also in various domains of nanotechnology [START_REF]Handbook of Conducting Polymers[END_REF][START_REF] Roncali | [END_REF]. New materials based on functionalized heterocycles are actively researched. Experimentally, the synthesis of polyheterocyclic species after functionalization of the monomer or after fusing or bridging monomer units [START_REF] Roncali | [END_REF][3][4] constitute a way to access to new polyheterocyclic polymers with enhanced properties. For instance, different polyalkylthiophenes [5] and polybithiophenes exhibiting a low electronic band gap (1.10-1.20 eV) have been obtained by fusing a benzene and a thiophene ring or by fixing electro-attracting groups on bithiophenes bridged by a substituted sp 2 carbon atom [6].

Fusing of different heterocyclic monomers by electrochemical copolymerization was also developed [7][8][9][10][11][12][13][14][15] in the case of oxidation potentials close to each other. Thus, different copolymers were synthesized electrochemically: pyrrole with bithiophene [9] pyrrole with furan [10] pyrrole with methyl-pyrrole [11,12], pyrrole with thiophene [13], pyrrole with N-phenyl substituted pyrrole [14]. The electrochemical copolymerization of furan with 3-methylthiophene [15] and with 3-chlorothiophene [16] was also performed: the measurement of the spectro-electrochemical properties of the obtained copolymers shows that they are intermediate between those of their corresponding homopolymers. Moreover the band gap of polyfuran is reduced by incorporation of substituted thiophene units in the polymeric chain. It is also worth noting the synthesis and characterization of new light-emitting copolymers containing 3,4-dialkoxythiophenes [17], and of fluorescent polyheteroaromatic oxadiazoles with alternating electron-donating thiophene and electron-withdrawing oxadiazole units [18] . To our knowledge, the idea of bridging electron-donating and electron-withdrawing groups along a conjugated chain has been initiated by Havinga et al. [19].

The design and synthesis of conjugated copolymers with donor-acceptor architectures for potential applications in microelectronic devices, photovoltaic cells, electrochromics, [20][21][22] are in continuous development as well as the synthesis and characterization of low-bandgap materials based on benzodithiophene and benzodifuran units Moreover, since the discovery of fullerenes and carbon nanotubes [23], great interest is focused on systems exhibiting cyclic or tubular structures for which interesting optical, electronic, conducting properties are expected. Mishra et al. [24] describe the synthesis, characterization and properties of mixed macrocycles based on thiophenes and other unsaturated units. Other works highlight interesting electronic and optoelectronic properties of such species [25][26][27][28][29][30] and of cyclo(n)pyrrole [31]. On the theoretical side, various results have been obtained; in 2012, Shuang Huang et al., using DFT/ MPWB95/6-31G* computations studied the ring size effect on properties of macrocyclic thiophene and pyrrole derivatives [32a]. Then, these authors carried out a ZINDO computational study of the electronic structures, UV-Vis spectra and static second-order nonlinear optical susceptibilities of macrocyclic (trithiophene-diacetylene)n (n=2,5) that showed that macrocyclic thiophene derivatives compounds potentially exhibit large static second-order nonlinear optical susceptibilities [32b]. A previous work focused on 2D-macrocyclo(8)thiophenes [33] which brought to light that there exists a noticeable interaction between π-orbitals of individual macrocycles in tubular aggregates leading to a decrease of the band gap with the number of units in neutral aggregates. Moreover, oxidized 2D macrocyclic oligothiophenes are able to form tubular aggregates because of π-π stacking between macrocycles. In the same way, Manuel Garcia et al. [34] investigated theoretically the geometrical and electronic structures as well as electronic properties of Donor-Acceptor Tubular Nanoaggregates of Cyclic Oligocyclo(8)thiophenes and cyclo(8)(3,4dicyanothiophene); they concluded that donoracceptor tubular aggregates could be a promising new class of photovoltaic materials due to a small and adjustable band gap, strong charge separation in the excited state, and high possible mobility of the charge carriers due to their low relaxation energies. Otherwise, cyclic Oligothiophenes [35a] have been studied theoretically at the B3LYP/6-31G(d) level of theory; the results show that HOMO-LUMO gap of cyclooligothiophene is lower than those of corresponding linear oligothiophene. Thus, cyclic oligoheterocycles seem to be promising candidates for applications in electronics, optoelectronics, photovoltaic cells [35b,c].

For our part, we plan to study new co-oligomers of brigded five-membered heterocycles in their linear structure and macrocyclic structures that we expect to be interesting for technological applications. To our knowledge, the sole reported calculations on this kind of species are the ones of Bakhshi and col. [36] who studied the electronic properties and conduction properties of copolymers based on furan and thiophene. The latter studies considered only linear structures, the calculations using ab-initio Hartree-Fock crystal orbital method. In a previous work [37], we applied standard B3LYP/6-31G(d) computations to study the structure and electronic properties of linear oligomers based on bithiophene bridged by a sp 2 carbon bearing a chalcogen atom (O, S, Se and Te), and of their oligobifuran and oligobipyrrole analogues. An important reduction of the HOMO-LUMO gap was observed for the whole series of biheterocyclic compounds when substituting the extracyclic oxygen atom by a chalcogen atom of higher atomic number. In the present work, several novel series of cooligo five-membered heterocycles, where every biheterocycle (bithiophene, bipyrrole and bifuran) is bridged by CO or by CS groups, will be considered. The most stable combination of these heterocyclic species, considering different linear and cyclic architectures, will be first studied; then we shall investigate if their electronic and optoelectronic properties are adequate for potential technological applications. Our preliminary design will be based on standard DFT computations.

METHOD OF CALCULATION AND SYSTEMS UNDER CONSIDERATION

Density Functional Theory (DFT) [38] calculations have been carried out using the B3LYP functional [39] and the standard 6-31G(d) basis set. Another double zeta basis set, namely the LANL2DZ one, also supplemented by polarization functions on heavy atoms, has also been used, because 6-31G (d) is not available for heavy elements. The DFT calculations were carried out using the Gaussian 09 package [START_REF] Gaussian | Gaussian Inc[END_REF]. The geometry optimizations were followed by vibration frequencies calculations. The calculations have been carried out without any constraint of symmetry.

The structures of the considered unbridged and bridged oligo(6)biheterocycles and cooligo(3) biheterocycles (X and X'=NH,O,S and Y=S,O) in linear and cyclic chains, are given in Figure 1. We have considered the energetically favoured αα ' linkage for all linear and cyclic chains. The linear unbridged oligo (6) biheterocyclic model that we considered, is based on a unit constituted by two planar rings, αα' linked in such a way that the X heteroatoms are in trans position (Fig 1a). The same linkage was considered between two bridged biheterocycles in linear bridged oligo (6) biheterocycles and closed bridged oligo (6) biheterocycles ( Figures 1b,1b',1c,1c').The planar cyclic unbridged and bridged oligo (6) biheterocyclic and co-oligo(3) biheterocyclic models we have considered, are constituted by two units in such a way that the heteroatoms are in a cis position (Figures 1a',1b',1c'). These cyclic planar structures have been our starting point for the geometry optimization of each of the closed oligomers. We found that the optimized final structures could either remain in a planar conformation (a ', b', c') or take a tubular conformation (a ", b", c"). The a" one is a tubular structure where the X heteroatoms are in trans position whereas b" and c" refer to different relative positions of the bridged biheterocyclic units. The structural and electronic properties of each studied oligomer are calculated considering the most stable optimized conformation. The used abbreviation for linear and cyclic unbrigded oligo( 6 For the sake of consistency, the values n = 6 for 1a, 1a', 1a'', 1b, 1b'and 1b'' and n = 3 for 1c,1c' and 1c'', n being the number of units in the oligomeric chains, have been chosen.

Y X X X X X X X X X X X X Linear structures planar ring structures crown ring structures X X X X X X X X X X X X X X X X H H H H 6 a : L6(H-H) a' :C6(H-H) cisoid a'' : C6(H-H) transoid Y X X
(HYH) C6(HYH) X' X X' X X X' X' X Y Y Y Y X X X' X' Y Y X' X' Y X X Y X' X' Y X X Y X X Y X' X' Y X' X' Y X X Y X' X' Y X x Y X X Y X' X' Y X X X X X X X X Y Y Y Y X X X X Y Y 7
3. RESULTS AND DISCUSSION 3-1. UNBRIDGED LINEAR AND CYCLIC OF OLIGO(6) BITHIOPHENE, OF OLIGO(6) BIPYRROLE, AND OF OLIGO(6) BIFURAN First, in order to assess the validity of our approach, the structural and electronic properties of well known oligomers of bithiophene, bipyrrole and bifuran, and more specifically oligo (6)biheterocycles in linear (open) and cyclic (ring) chains, have been studied. The optimized values of relevant geometrical parameters are reported in table 1 as well as the drawings of the obtained cyclic structures. All structures have been checked to be ground state ones considering their normal modes of vibration (no imaginary frequency). The optimization of closed structures of cyclic oligo(6)bifuran and cyclic oligo(6)bipyrrole, starting from a planar cisoid ring ( Figure 1a'), led to cyclic transoid conformers of tubular (crown) forms (Fig. 1a'') (dihedral angles O10-C9-C4-O3 equal to 143° and N11-C10-C4-N3 equal to 129°) whereas the optimized geometries of cyclic oligo(6)bi thiophene remain cisoid planar (dihedral angles equal to 0.0 or ~180.°). The optimized structures (Table 1) in the three cases exhibit only small bond lengths alternation: for example for C6(F-F), the distances C6-C5, C5-C4 and C4-C9 are respectively equal to ca. 1.43 Å, 1.38Å and 1.44 Å. Regarding the unbridged linear biheterocycles (Figure 1a), we note that all open structures remain trans planar.

In table 2, we give the HOMO-LUMO energy gap values computed for the most stable structures; they are compared to those already published (Table 2, column Theo.). The experimental values of the polymer energy gaps are also given for comparison (Table 2, column Exp.). In this table the following letters, T, F and P indicate respectively thiophene, furan and pyrrole oligomers. As expected, we notice that the values of the energy gap decrease with the number n of linked units [37]. For n=6 we note that the sole energy gap value obtained for the linear structure species a, b and c (Figure 1) are in good agreement with the gap measured experimentally for polythiophene (2.3-2.0 eV), polyfuran (2.7 eV) and polypyrrole (3.0 eV) .The calculated values for linear and cyclic cisoid oligothiophene, i.e. Considering our results obtained using the two basis sets we note that the LANL2DZ(d) gaps are, as expected, very close to the 6-31G (d) values. The differences are lower by about 0.05 eV for oligo (6) bithiophene and oligo ( 6) bifuran (linear and cyclic oligomers) and by about 0.5 eV for oligo (6) bipyrrole (linear and cyclic oligomers). The obtained agreement with experiment gives us confidence that the energy band gaps estimations, for the unknown systems which are studied in this paper, are reliable. Our remaining computations will be carried out using the B3LYP functional and the LANL2DZ(d) basis set. 234, the HOMO energy obtained after bridging remains relatively constant independently of the CO or CS moiety for both the linear and cyclic structures. On the contrary, an important decrease of the energy of the LUMO for all compounds after such substitution can be noticed. This result can be easily rationalized considering the frontier MOs of the unbridged and the bridged cyclo oligo (6) biheterocycles which are displayed on figures 2-1, 2-2 , 2-3. Indeed, these species exhibit practically the same HOMO. In contrast, the LUMO depends on the nature of the bridging group thus explaining the observed energy lowering.

3-3. DFT CALCULATIONS ON LINEAR AND CYCLIC STRUCTURES OF CO-OLIGO (3) BRIDGED BITHIOPHENE, BIPYRROLE AND BIFURAN DERIVATIVES

The study is now extended to linear and ring structures of co-oligo (3) bridged bithiophene, bipyrrole and bifuran derivatives, noted respectively L3((HYH)(H'YH')) and C3((HYH) (H'YH')) where H ≠ H ' are T, F, P for respectively thiophene, furan and pyrrole; Y is O or S atom of the bridging group CY.

The co-oligomers are obtained by considering the periodical sequence of two different bridged biheterocycles (HYH) and (H'YH'). All cases are considered. The purpose of this study is to investigate the effect of the substitution of a bridged biheterocycle (HYH) by another one (H'YH') aiming to find a suitable combination allowing potential applications.

The computed values at the B3LYP/LANL2DZ(d) level of theory are reported in Table 6 and 7 respectively for the linear and cyclic forms; the results for the reference oligomers are also given.

It can be noticed for the linear chains (Table 6) that (i) the energy gap values obtained for the bridged co-oligomers are all lower than those of their corresponding reference oligomers. This suggests that the properties of conductivity of co-oligomers could be higher than that of corresponding oligomers; (ii)when the bridging group is CO, the substitution of bridged bipyrrole by bridged bifuran or by bridged bithiophene does not affect the energy gap: the value obtained for all co-oligomers is practically the same (between 1.66 and 1.68 eV).

(iii) on the contrary, we notice that this effect is significant when the bridging group is CS: the energy gap of the oligo(6) bipyrrole L6 (PYP) passes from 1.58 eV to 1.14 eV (L3((PYP) (TYT))) and 1.02 eV (L3( PYP)(FYF))) when a bridged bipyrrole is replaced, respectively, by a bridged bithiophene and bridged bifuran.

In Figure 3, we display the frontier MO diagrams of these species. We notice that: (i) the In Figure 3 We considered the bridged oligo(6)bithiophene, the bridged oligo(6)bifuran, and the bridged oligo(6)bipyrrole, as well as the bridged co-oligo(3)(bithiophene-bipyrrole) and bridged cooligo(3)(bithiophene-bifuran) obtained replacing three bridged bithiophene by respectively three bridged bipyrrole or by three bridged bifuran considering different linking modes (leading to a linear or ring structure). The bridged co-oligo(3)(bifuran-bipyrrole) resulting from the substitution of three bridged bifuran by respectively three bridged bipyrrole as also been studied. Interestingly, it appears that for the linear chains, a significant decrease of the gap value after replacement of bridged biheterocycle by another can be obtained; the gap values obtained for the bridged co-oligomers are all lower than those of the corresponding reference oligomers. This effect is significant when the bridging group is CS. The energy gap lowering is due to the energy decrease of the LUMO of the considered species. This is well rationalized considering the frontier MO diagrams of the studied species.

For the ring-shaped chains, the cooligomerization of cyclo bridged oligo (6) The values obtained for the cyclo bridged co-oligo(3)(bifuran-bipyrrole) are slightly lower than those of cyclo bridged oligo(6)bifuran .

Our calculations suggest that the properties of conductivity of bridged linear and cyclic cooligomer should be better than that of corresponding basis bridged oligomers. Among them, the cyclo bridged co-oligo(3)(bithiophene-bipyrrole) and the cyclo bridged co-oligo(3) (bifuran-bipyrrole) seem to be good candidates for potential technological applications.

  [20a] on cyclopentadithiophene (CPDT) [20b], or based on cyclopentathiophene (CPT) as donors and diazole derivatives or triazole derivatives as acceptors [20c].

  )biheterocycle with 12 rings is 6(H-H) where H = (T, F, P) for respectively thiophene, furan and pyrrole; the L or C prefix indicates if the structure is respectively linear or cyclic. The denominations for the bridged linear (b) and cyclic (b',b") bridged oligo (6)biheterocycle are respectively L6(HYH) and C6(HYH) where Y= ( O or S). For the co-oligomers (c), (c') and (c'') the denomination is L3(HYH)(H'YH') and C3(HYH)(H'YH') where H ≠ H' = (pyrrole or thiophene or furan) and Y = ( O, S ).

Fig. 1 .

 1 Fig. 1. Unbridged and bridged oligo(6)biheterocycles and cooligo(3) biheterocycles (X and X'=NH,O,S and Y=S,CO) in linear and cyclic chains.

  c : L3(HYH)(H'YH') c' : C3(HYH)(H'YH') c'' : C3(HYH)(H'YH') b : L6(HYH) b' :C6(HYH) b'' : C6(HYH) C6

  Table1. B3LYP/ LAN2DZ(d) optimized geometry of cyclic oligo(6)bithiophene (C6(T-T), of cyclic oligo(6)bifuran (C6(F-F)) and of cyclic oligo(6)bipyrrole (C6(P-P)) n1, n2, n3,n4: 9-10(-179.) 8-9-10-11(0.) 8-9-10-15(-179.)

  HOMO of L3 (TST) and of L3(PSP) are π MOs fully delocalized over the rings and do not involve the CS group. (ii) The LUMO of L3 (TST) exhibits a strong contribution of the three bridging groups CS and of the sulfur atoms of the rings. On the contrary, the LUMO of L3( PSP) which is of higher energy (-2.73 eV vs. -3.54 eV for L3(TST)) is mostly localized on one central CS group. (iii) the π* LUMO of the co-oligomers is localized on the TST moieties with an almost zero contribution of the PSP fragments; this results first to a LUMO energy of the co-oligomer similar to L3(TST) ca. -3.55 eV, and secondly to a significant lowering of the LUMO energy of L3(TSTPSP) relatively to L3(PSP); (iv) The smallest gap value (1.02 eV) is obtained for the combination L3 (PSPFSF) due to stabilization of the LUMO after introduction of the FSF sequence within the oligo (6) bridged bipyrrole.

  -2 and 3-3 are given respectively the frontier MO diagrams of the cyclo CS bridged co-oligo(3)(bithiophene-bipyrrole), noted C3((TST)(PSP)) and cyclo CS bridged cooligo(3)(bifuran-bipyrrole) noted C3((FSF)(PSP)). The frontier MO diagrams of cyclo oligo(6)biheterocycles noted C3((TST)(TST)), C3((PSP)(PSP)), C3((FSF)(FSF)) which are the basic oligomers before co-oligomerization are also given. Orbital analysis for C3((TST) (PSP)) shows that: (i) the HOMO of C3 ((TST)(TST)), of C3 ((TST)(FSF)) and of C3 ((PSP) (PSP)) are π MOs fully delocalized over the rings and do not involve the CS group. (ii) the LUMO of C3 ((TST)(TST)) is a π* MO fully delocalized over the rings and does not involve the CS group (iii) the LUMO of C3 ((PSP)(PSP)) exhibits a strong contribution of the six bridging groups CS (iv) the HOMO of C3 ((TST)(PSP)), is a π MO fully delocalized over the rings which does not involve neither the CS group nor heteroatoms of ring. The π* LUMO of the co-oligomers expands mainly over three TST moieties with an almost zero contribution of the PSP fragments; this results in a significant lowering of the LUMO energy of C3((TST)(PSP)) relatively to C3((PSP)(PSP)); (iv) the gap value (1.17 eV) obtained for the combination C3 ((TST)(PSP)) is closer to the C3 ((TST)(TST ) one, i.e. 0.80 eV rather than the C3 ((PSP)(PSP)) values equal to 2.05 eV. 4. CONCLUSION A series of unbridged and bridged oligomers of five-membered heterocycles (bithiophene, bipyrrole, bifuran bridged by CO or by CS group, exhibiting a linear or ring-shaped macrocyclic structure has been studied by means of standard B3LYP/LANL2DZ(d) DFT computations. In particular, the effects of bridging and co-oligomer linking on the structural and electronic properties of the oligo (6) bridged biheterocycles, have been investigated. Different ring structures can be obtained. The optimization of closed structures starting from planar cyclic cisoid oligo(6)bifuran and cyclic oligo(6)bipyrrole leads to cyclic transoïd conformers corresponding to tubular (crown) forms whereas the optimized structure of cyclic oligo(6)bithiophene remain cisoid planar. The full optimizations of the unbridged linear biheterocycles, starting from a planar geometry, show that all open structures remain trans planar. The ring structures exhibit lower HOMO-LUMO gaps than the linear ones.

  bipyrrole with cyclo bridged oligo(6)bifuran or cyclo bridged oligo(6)bithiophene leads to a decrease of the computed energy gap of the cyclo bridged oligo(6)bipyrrole. The values obtained for the cyclo bridged co-oligo(3)(bithiophene-bipyrrole) are intermediate between those of the cyclo bridged oligo(6)bipyrrole and cyclo bridged oligo(6)bithiophene regardless of bridging group.

  1.99-2.19 eV are very close to the corresponding experimental value, i.e. 2.0-2.3 eV. We did not find necessary to run calculations for higher linear oligomers in order to get a better estimate of the energy band gap by extrapolation. Such a work has already been done by other authors using similar DFT techniques. Our values are similar to the published results of Salztner et al. [41a] who employed the B3P86-30% functional and CEP-31G(d) and LANL2DZ basis sets. It is worth noting that the different DFT functionals, B3LYP for our part and B3P86-30% which has been used by Saltzner, give similar energy gaps [41b].

Table 2

 2

	. B3LYP/LANL2DZ(d) energy gaps (in eV) of oligo (n) biheterocycles (linear and
	cyclic oligomers)
	Compounds	Basis set					Theo.	Exp.
	Ln(H-H)	6-31G*		LANL2DZ(d) [39]	[2, 3, 6]
	n	1 a	2 a	3 a	6	6	3	°°Ln(T-T)
		4.23 3.08 2.50 2.23	2.19	2.30 b 2.21 c	2.3-2.0
	Cn(T-T)	-	-	-1.95	1.99
	cisoid
	Ln(F-F)	4.67 4.09 2.81 2.60	2.56
	Cn(F-F)	-	-	-2.82	2.86	2.67	2.7
	transoid
	Ln(P-P)	4.90 4.27 3.38 3.45	2.98
	Cn(P-P)				3.53	3.77	3.16	3.0
	transoid
	a:Values from a previous work [37]
	b: CEP-31G(d) basis set [41]
	c: B3P86-30% functional and LANL2DZ basis set [41]

Table 5

 5 LUMO gap in all cases. This effect is more pronounced for the cyclic structures than for the corresponding open structures. For example the energy gap of cyclic unbridged oligo(6) bithiophene i.e. 1.99 eV, decreases to 0.84 eV and 0.80 eV after bridging respectively by the CO and CS groups. The energy gaps of cyclic CO bridged dithiophene derivatives (0.84 eV) are lower than those of cyclic CO bridged difuran derivatives ( 1.56 eV) which are themselves lower than those of cyclic CO bridged bipyrrole derivatives (2.76 eV). As pointed out in our previous work[38] on linear structures, we similarly obtain for the cyclic structures a more important reduction of the energy gap with the CS bridging group than with the CO one. The gap values are respectively 0.80 eV, 1.06 eV, 2.05 eV for CS bridged dithiophene derivatives, cyclic CS bridged difuran derivatives, and cyclic CS bridged bipyrrole derivatives. Besides, as it can be seen in Tables

	. B3LYP/LANL2DZ(d) computed energies (eV) of linear and cyclic bridged oligo (6)
	bipyrrole derivatives (Figure 1)
	bridged	B3LYP/ LANL2DZ(d)	Exp
	oligo(6)bipyrrole a				[2, 3,6]
		-ε HOMO	-ε LUMO Gap	Gap pol
	L6(P-P)	4.04	1.06	2.98 3.0
	L6(POP)	4.64	2.26	2.38
	L6(PSP)	4.60	2.91	1.69
	C6(P-P)trans	4.51	0.74	3.77
	C6(POP)	5.07	2.31	2.76
	C6(PSP)	5.05	3.00	2.05
	a L6(P-P): linear unbridged oligo(6) bipyrrole; C6(P-P): cyclic unbridged oligo (6) bipyrrole.
	L6(PYP): linear bridged oligo(6) bipyrrole; C6(PYP): cyclic bridged oligo (6) bipyrrole with Y= ( O
	or S).
	It is not possible to compare our results to those obtained by Saltzner and Coll. [41] for
	similar species, because they only studied the case n = 1 (corresponding to 2 heterocyclic
	rings). Obviously the HOMO-LUMO gaps that we computed for high oligomers (n=6) are
	much lower than those of Saltzner.

It is worth noting that substituting by O or S the sp 2 carbon atom of the bridge reduces the 14 HOMO-

Table 6

 6

	. B3LYP/ LANL2DZ(d) computed energies (eV) of the brigded oligo(6) and co-
	oligo(3) bithiophene, bipyrrole and bifuran derivatives (linear structures)
	compounds	Y	CO		CS
		-ε HOMO 4.59		4.54
	L6 (PYP)	-ε LUMO 2.35 gap	2.24	2.96	1.58
		-ε HOMO 5.50		5.33
	L6(FYF)	-ε LUMO 3.48 gap	2.02	3.86	1.47
		-ε HOMO 5.44		5.31
	L6(TYT)	-ε LUMO 3.61 gap 1.83 (1.93) 2	3.89	1.42
		-ε HOMO 4.64		4.56
	L3( PYP)(FYF))	-ε LUMO 2.98 gap	1.66	3.54	1.02
		-ε HOMO 4.75		4.69
	L3((PYP)(TYT))	-ε LUMO 3.09 gap	1.66	3.55	1.14
		-ε HOMO 5.00		4.89
	L3((FYF)(TYT))	-ε LUMO 3.32 gap	1.68	3.66	1.23
	From the results of Table 7 for the cyclic brigded oligo(6) and co-oligo(3) bithiophene,
	bipyrrole and bifuran derivatives, it can be seen that: (i) substituting in the sequence [(FYF)
	(FYF)] a FYF fragment by a TYT fragment results in a small change of the value of the
	energy gap, i.e. from 1.56 eV to 1.58 (when Y = CO) and 1.06 to 1.16 eV (where Y = CS) in
	the C6 (FYF) case; (ii) the gap values obtained for C3 ((TYT) (FYF)) co-oligomer are higher
	than those of the same unit oligomers C6 (TYT) and C6 (FYF); (iii) substitution in the
	sequence [(PYP) (PYP)] of a bridged bipyrrole PYP unit by a bridged bifuran FYF decreases
	the energy gap from 2.76 eV (C6 (PYP) value) to 1.52 eV (when Y = CO), and from 2.05 to

Acknowledgements

GENCI-IDRIS-CINES (Grants Nos. 2012-080649 and 2013-080649) are acknowledged for a grant of computational time.


M Sigurani
email: manuela.sigurani@uni-bayreuth.de

C Stöcker
email: stoecker@atp.rub.de

L Grüne

J Lunze
email: lunze@atp.rub.de

Experimental evaluation of two complementary decentralized event-based control methods

Keywords: Event-based control, Ultimate boundedness, Networked control

Event-based control aims at the reduction of the feedback communication effort among the sensors, controllers and actuators in control loops. The feedback communication is invoked by some well-defined triggering condition. This paper presents a new method for the decentralized event-based control of physically interconnected systems and shows its experimental evaluation.

The novel method is based on two complementary approaches, called the global and the local approach, which jointly ensure the ultimate boundedness of the closed-loop system. The global approach steers the state of each subsystem into a target region, whereas the local approach makes the state remain in this set in spite of exogenous disturbances and the effect of the interconnections to other subsystems. This event-based control method is applied to a continuous flow process to show its practical implementation and to evaluate the analytical results on the basis of experiments.

system, Interconnected systems, Thermofluid Process

Introduction

In event-based control the communication among the components of a control system is restricted to time instants at which the exchange of current information is necessary to ensure a desired behavior of the closed-loop system. Its triggering scheme contrasts with the current practice, where the controller is implemented on digital hardware and the control task is executed periodically (sampled-data control). As the main reason for using this kind of implementation, the analysis and design of sampled-data control loops can be based on a well-established theory. However, the periodic sampling, computing and updating of actuator signals is carried out whether required or not, which can lead to a waste of communication resources. To use these communication resources more efficiently, event-based control has been proposed as an alternative to periodic control.

u 1 (t) u N (t) x 1 (t)
x N (t) This paper investigates decentralized event-based control of interconnected systems Σ i (i = 1, . . . , N ) (Fig. 1). The event-based controller for each subsystem Σ i consists of an event generator E i and a control input generator C i which communicate over a network only at certain event times.

d 1 (t) d N (t) . . . Network Σ 1 Σ N Σ E 1 C 1 E N C N
The aim of this paper is twofold: First, it is presented how two approaches to decentralized event-based control can be combined to jointly accomplish ultimate boundedness of the overall control system. These two approaches, which are subsequently specified as the global approach and the local approach to decentralized event-based control, are based on methods which have been recently published by [START_REF] Grüne | Numerical ISS controller design via a dynamic game approach[END_REF] and Stöcker et al. (2013), respectively. The global approach drives the state of each subsystem into a target region in finite time where the local approach takes over and keeps the state within this region in spite of exogenous disturbances or interconnections to other systems. The second aim of this paper is to demonstrate how these complementary approaches are applied in practice to a continuous flow process. The analytical results derived in the first part of this paper are evaluated by experiments.

Event-based control is a useful means to reduce the communication while accomplishing a desired control performance which has been shown in several simulation and experimental studies, e.g. by [START_REF] Hendricks | Problems in event based engine control[END_REF]; [START_REF] Heemels | Asynchronous measurement and control: A case study on motor synchronization[END_REF]; [START_REF] Kwon | Event-based modelling and control for the burnthrough point in sintering processes[END_REF]; [START_REF] Yook | Trading computation for bandwidth: reducing communication in distributed control systems using state estimators[END_REF]; [START_REF] Sandee | Analysis and experimental validation of a sensorbased event-driven controller[END_REF]; [START_REF] Henningsson | Comparison of lti and event-based control for a moving cart with quantized position measurement[END_REF]; [START_REF] Lehmann | Extension and experimental evaluation of an event-based state-feedback approach[END_REF]; [START_REF] Trimpe | An experimental demonstration of a distributed and event-based state estimation algorithm[END_REF]. Besides the investigation of the practical applicability, a lot of effort has been spend on developing a profound theory on event-based control starting with the works of [START_REF] Arzén | A simple event-based PID controller[END_REF]; [START_REF] Åström | Comparison of periodic and event-based sampling for first-order stochastic systems[END_REF] and has been continued in recent years, e.g. by [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]; [START_REF] Henningsson | Sporadic event-based control of first-order linear stochastic systems[END_REF]; [START_REF] Gawthrop | Event-driven intermittent control[END_REF]; Lunze & Lehmann (2010); [START_REF] Yu | Event-triggered real-time scheduling for stabilization of passive and output feedback passive systems[END_REF]; [START_REF] Donkers | Output-based eventtriggered control with guaranteed L ∞ -gain and improved decentralised event-triggering[END_REF]; [START_REF] Wang | On event design in event-triggered feedback systems[END_REF]; [START_REF] Molin | On the optimality of certainty equivalence for event-triggered control systems[END_REF]. Most of the literature on event-based control is concerned with stabilization. The problem of rendering the system asymptotically or exponentially stable using event-based feedback has been studied, among others, by [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]; [START_REF] Mazo | Decentralized event-triggered control over wireless sensor/actuator networks[END_REF]; De [START_REF] De Persis | On a small-gain approach to distributed event-triggered control[END_REF]; [START_REF] Garcia | Model-based event-triggered control with time-varying network delays[END_REF]; [START_REF] Wang | On event design in event-triggered feedback systems[END_REF]. Except for the works of Wang & Lemmon (2011a) and Stöcker & Lunze (2013), the price for asymptotic stability is usually a more frequent communication the cl! oser the state converges to the desired target point. Moreover, the eventbased control approaches which achieve asymptotic stability do not tolerate model uncertainties or exogenous disturbances. ¿From a practical perspective it is more preferable to steer the system state into a target region, rather than a point, and maintain it there, which is known in literature as ultimate boundedness (see e.g. [START_REF] Khalil | Nonlinear Systems[END_REF]). Eventbased control approaches which aim at this property have been presented by Lunze & Lehmann (2010); [START_REF] Grüne | Two complementary approaches to event-based control[END_REF]; [START_REF] Lehmann | Event-Based State-Feedback Control[END_REF]; [START_REF] Tallapragada | On event triggered trajectory tracking for control affine nonlinear systems[END_REF]; [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF]; [START_REF] Donkers | Output-based eventtriggered control with guaranteed L ∞ -gain and improved decentralised event-triggering[END_REF] where the last two references proposed approaches that render the control system L p -stable or L ∞ -stable, respectively. According to the definition of ultimate boundedness [START_REF] Khalil | Nonlinear Systems[END_REF]) the state must enter the target set within some finite time T and remain there for all t ≥ T . However, in none of the publications that study ultimate boundedness of event-based control systems [START_REF] Grüne | Two complementary approaches to event-based control[END_REF]; [START_REF] Lehmann | Event-Based State-Feedback Control[END_REF]; [START_REF] Tallapragada | On event triggered trajectory tracking for control affine nonlinear systems[END_REF]), the time T (or a bound on it) is derived. In contrast, this paper gives an upper bound on the time T in which the state reaches the target set.

While early works on event-based control mainly have been focused on single-loop systems, some recent publications deal with decentralized control [START_REF] Mazo | Decentralized event-triggered control over wireless sensor/actuator networks[END_REF]; Stöcker et al. (2013); [START_REF] Garcia | Decentralized model-based eventtriggered control of networked systems[END_REF] or distributed control [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF]; De Persis et al. ( 2011)).

In the existing literature the plant is considered to be exclusively described by either nonlinear dynamics [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]; [START_REF] Mazo | Decentralized event-triggered control over wireless sensor/actuator networks[END_REF] 2009)) or by linear dynamics (Lunze & Lehmann (2010); [START_REF] Donkers | Output-based eventtriggered control with guaranteed L ∞ -gain and improved decentralised event-triggering[END_REF]). The approach presented in this paper uses both a nonlinear model for the global approach and a linearized model for the local approach.

In this way, this paper follows an idea that has been published by [START_REF] Grüne | Two complementary approaches to event-based control[END_REF] for single-loop systems and extends it to decentralized control.

Based on the separation of the control problem (ultimate boundedness of the closed-loop system) into two problems, namely 1. steering the state from an initial state into a target region and 2. keeping the state in the target region in spite of exogenous disturbances or interconnections with other subsystems, the global and the local approach are tailored to the respective task. Hence, the combination of both approaches leads to a new method for the decentralized stabilization and disturbance attenuation of interconnected systems.

Although the global and the local approach use different models (nonlinear vs. linear) and follow different ideas (stabilization vs. disturbance attenuation), they are similar in the sense that event-based controllers for the isolated subsystems are designed and their robustness with respect to the interactions of the subsystems are proved by small-gain theorems.

Outline of the paper. The general control problem is formally stated in Sec. 2.

The global approach and the local approach are explained in Sections 3 and 4, respectively. The combination of both control methods is described in Sec. 5.

Sections 6 and 7 introduce the continuous flow process, shows the implementation of the decentralized event-based controllers in MATLAB/Simulink and gives the experimental results and their discussions.

Nomenclature. IR and IR + denote the set of real numbers and nonnegative real numbers, respectively, and B 1 (0) denotes the closed unit ball. For a scalar s, |s| denotes the absolute value. For a vector v ∈ IR n or a matrix M ∈ IR m×n the |•|-operator applies to every element. A comparison between two vectors v, ṽ ∈ IR n holds element-wise, like v ≤ ṽ amounts to v i ≤ ṽi for all i = 1, . . . , n, where v i and ṽi denote the i-th element of the vectors v and ṽ, respectively. For a real square matrix M ∈ IR n×n , λ P (M ) denotes its Perron root (largest eigenvalue of M ). A block diagonal matrix with the matrices A i for i = 1, . . . , N on the main diagonal is represented by A = diag (A i ). M + denotes the pseudoinverse of a matrix M ∈ IR m×n that is determines according to

M + = (M ⊤ M ) -1 M ⊤ ,
where M ⊤ ∈ IR n×m represents the transpose of M . The asterisk * symbolizes the convolution-operator, e.g.

G * u = t 0 G(t -τ )u(τ )dτ.
For a signal x : IR + → IR n , the limit from above at time t ∈ IR + is denotes by x(t + ) = lim s↓t x(s). We introduce the following sets of comparison functions: K = {γ : R + → R + , γ is continuous, strictly increasing, and γ(0) = 0},

K ∞ = {γ ∈ K : γ is unbounded} and KL = {β : R + × R + → R + : β(•, t) ∈
K and for each fixed s ≥ 0 it is decreasing to zero as t → ∞}.

Problem formulation

Structure of the event-based control system

The structure of the event-based control system investigated in this paper is illustrated in Fig. 1. The overall plant Σ is composed of N physically interconnected subsystems Σ i , (i = 1, . . . , N ). The subsystem Σ i is controlled by an event-based controller consisting of the control input generator C i and the event generator E i . The event-based control method that is proposed in this paper is decentralized in two senses:

1. The event times t k at which a feedback communication is invoked are determined by the event generators E i based on locally available information only and 2. the control input generator C i produces the control input for subsystem Σ i using only information that is received at the event times from the event generator E i .

The latter implies the decentralized communication structure as shown in Fig. 1, where the information is transmitted from the event generator E i to the corresponding control input generator C i , but to no other component.

Note that this communication structure opens the possibility to realize the feedback communication of the respective control loops over various networks. A communication from E i to C i is only invoked at the event times t k , which is expressed by the dashed line in Fig. 1. The solid lines represent continuous information links. In this paper, the information transmission over the communication network is assumed to occur instantaneously and without delays or packet losses.

Plant model

The overall plant is composed of N subsystems and is described by the nonlinear state-space model Σ :

           ẋ1 (t) = f 1 (x 1 (t), . . . , x N (t), u 1 (t), d 1 (t)) . . . ẋN (t) = f N (x 1 (t), . . . , x N (t), u N (t), d N (t)) (1)
with initial conditions x i (0) = x 0i for all i ∈ N := {1, . . . , N }. In (1),

x i ∈ X i ⊂ IR n i , u i ∈ U i ⊂ IR m i and d i ∈ D i ⊂ IR w i
are the state, the control input and the disturbance of the i-th subsystem denoted by Σ i . In the following,

D i := d i | |d i | ≤ di , (2)
where the vector di element-wise denotes the maximum magnitude of the disturbance vector d i (t). The state, the control input and the disturbance of the overall plant Σ are represented by

x(t) = x ⊤ 1 (t), . . . , x ⊤ N (t) ⊤ u(t) = u ⊤ 1 (t), . . . , u ⊤ N (t) ⊤ d(t) = d ⊤ 1 (t), . . . , d ⊤ N (t) ⊤ .
The full state space, set of controls and set of disturbances are denoted by

X = X 1 × . . . × X N , U = U 1 × . . . × U N and D = D 1 × . . . × D N , respectively.

Control aim

The control aim is formulated in terms of ultimate boundedness:

Definition 1 (Ultimate boundedness). The solution x(t) of (1) is called ultimately bounded (UB) to the set A if for each initial condition x 0 ∈ X there exists a time T (x 0 ) > 0 such that

x(t) ∈ A, ∀ t ≥ T (x 0 ) (3)
holds for all admissible disturbances d(t) ∈ D. The system (1) is said to be ultimately bounded if its state x(t) is UB.

Consider that the system Σ as in (1) and a target set A = A 1 × . . . × A N are given, where A i ⊂ X i denotes the target set for the subsystem Σ i . The aim of the first part of this paper is to develop a method for the design of decentralized event-based state-feedback controllers K i : X i → U i that render the closed-loop system ultimately bounded to the set A.

A global and a local approach

The proposed event-based control approach follows an idea that has been presented by [START_REF] Grüne | Two complementary approaches to event-based control[END_REF] for single-loop event-based control systems.

As illustrated in Fig. 2, the problem of finding the decentralized event-based controllers K i that render the closed-loop system ultimately bounded is subdivided into a global problem and a local problem, which are solved by two complementary approaches:

1. The global event-based control approach drives the state x i (t) of each subsystem Σ i from the initial state x 0i into the target set A i , while taking possible constraints on the states or on the control inputs into account.

2. Local approach -Linear control. For the local approach, the target set A is considered as a bounded surrounding of the operating point x in which the system Σ is described by a linearized model with sufficient accuracy. The control input generator C i determines the control input u i (t) using a linear model Σ si of the subsystem Σ i with a continuous state-feedback controller.

The event generator E i monitors the deviation between the actual subsystem state x i (t) and the state x si (t) of the model Σ si used for the generation of the control input u i (t) and it triggers an event whenever this deviation exceeds a defined threshold.

Switching from the global to the local approach. The control input generator C i as well as the event generator E i include the respective components which are designed according to the global and the local approach. C i and E i work as specified by the global approach as long as x i (t) / ∈ A i . T i denotes the time instant at which x i (T i ) enters the set A i . At time t = T i , both C i and E i switch their functionality and proceed with working according to the local approach and, hence, the state x i (t) is maintained within the set A i for all t ≥ T i .

The global approach

Main idea

In this section the problem of optimally controlling a nonlinear interconnected system to a desired target set A by means of a quantized state-feedback law is considered. We utilize the structure of the interconnected system by separately deriving a feedback law for each subsystem Σ i .

For the calculation of an event-based robustly stabilizing feedback law for each subsystem we use a set oriented approach which has been previously proposed for systems without perturbations in [START_REF] Grüne | An algorithm for event-based optimal feedback control[END_REF], [START_REF] Grüne | Two complementary approaches to event-based control[END_REF]. As this approach yields only practical stability, we will utilize practical versions of all stability properties involved. For simplicity of exposition, throughout this section we assume that the target set A contains a neighborhood of the origin.

We sample the nonlinear state-space model ( 1) to obtain a discrete-time representation of the continuous-time sampled-data system Σ :

x(k + 1) =            x 1 (k + 1) = f 1 (x 1 (k), . . . , x N (k), u 1 (k), d 1 (k)) . . . x N (k + 1) = f N (x 1 (k), . . . , x N (k), u N (k), d N (k)) (4) = f (x(k), u(k), d(k)),
k = 0, 1, . . ., with initial conditions x i (0) = x 0i for all i = 1, . . . , N , where

x i ∈ X i ⊂ IR n i , u i ∈ U i ⊂ IR m i and d i ∈ D i ⊂ IR w i . Infinite sequences
of control and perturbation are denoted by u = (u(0), u(1), . . .) ∈ U N 0 and

d = (d(0), d(1), . . .) ∈ D N 0 .

Small gain approach

We want to construct a decentralized static state feedback controller u P , such that the closed loop system

x(k + 1) = f (x(k), u P (x(k)), d(k)) =: h(x(k), d(k)) (5)
is rendered input-to-state practically stable in the following sense Definition 2. System (5) is called input-to-state practically stable (ISpS)

with respect to δ, ∆ d ∈ R ≥0 on a set Y ⊂ X if there exist β ∈ KL and γ ∈ K,
such that the solutions of the system satisfy

x(k) ≤ max{β( x 0 , k), γ( d ∞ ), δ}, (6)
for all x 0 ∈ Y, all d ∈ D N 0 with d ∞ ≤ ∆ d and all k ∈ N 0 .
Since we assumed that the target set A contains a neighborhood of the origin, this property ensures ultimated boundedness w.r.t. A provided δ and d are sufficiently small. If (6) holds with γ = 0 the system is called uniformly practically asymptotically stable.

The idea we pursue for the decentralized design is to derive an ISpScontroller for each subsystem Σ i , where the input from the other subsystems is considered as an additional perturbation. Then, stability of the overall system can be ensured via a small-gain argument. The central tool for this purpose are ISpS Lyapunov functions since ISpS can be characterized through them. Here we provide the definition of an ISpS Lyapunov function for the subsystems.

Definition 3. A continuous function V i : X i → R ≥0 is called ISpS Lya- punov function for the i-th subsystem Σ i of (4) on a sublevel set Y i = {x i ∈ X i | V i (x i ) ≤ ℓ i } for some ℓ i > 0 with respect to a Lyapunov target set A V i ⊂ Y i , if there exist functions α i , α i ∈ K ∞ , α i , µ ij ∈ K ∪ {0}, µ i ∈ K, a value w ∈ R >0 such that for all x ∈ Y i \A V
i the inequalities and implications

α i ( x i ) ≤ V i (x i ) ≤ α i ( x i ) (7)
and

V i (x i ) ≥ max{ max j {µ ij ( x j ∞ )}, µ i ( d ∞ )}} ⇒ V i (x i (k + 1)) -V i (x i (k)) ≤ -α i (|x i |) (8)
hold for all w ∈ V with w ≤ w.

The functions µ ij and µ i are called Lyapunov gains. By definition we set µ ii ≡ 0. Note that the Lyapunov target A V will in general be smaller than the target set A since no invariance of A V can be guaranteed, for details see the discussion after Theorem 1.

From small-gain theorems (see [START_REF] Dashkovskiy | Small gain theorems for large scale systems and construction of ISS Lyapunov functions[END_REF]) it is known that, if the gains µ ij are sufficiently small, a Lyapunov function V for the overall system exists. While this result is currently only proved for continuous time systems, our experiments indicate that it also holds in an event based setting.

The following theorem states that the existence of an ISpS Lyapunov function implies that the system is ISpS. This is fairly straightforward if all data is continuous. However, since in our approach the ISpS-Lyapunov function and the resulting feedback for the subsystems may not be continuous, we introduce the following conditions, cf. (Grüne & Sigurani, 2013, Assumptions 7 and 8).

Assumption 1. The map f : X ×U ×V → R n in (4) is uniformly continuous in w = 0 in the following sense: there exists a γ d ∈ K ∞ , such that for all x ∈ X , u ∈ U and d ∈ D f (x, u, d) -f (x, u, 0) ≤ γ d ( d ). ( 9
)
Assumption 2. The map h : X × D → R n in (5) is bounded for d = 0 on A in the following sense: there exists γ x ∈ K ∞ such that for all sufficiently small sets A ⊂ X with 0 ∈ A and each x ∈ A we have

f (x, u P (x), 0) ≤ γ x ( x ). ( 10
)
Now we can state the relation between the existence of an ISpS Lyapunov function and the ISpS property of the system, cf. (Grüne & Sigurani, 2013, Theorem 10).

Theorem 1. Consider system (5) satisfying Assumptions 1 and 2 and assume that the system admits an ISpS Lyapunov function V . Then the system

is ISpS on Y, with ∆ d → ∞ as ℓ → ∞ and δ → 0 as sup x∈A V x → 0, provided δ ≤ α -1 (ℓ) holds.
The relation between the size of the Lyapunov target A V and the practical stability parameter δ is made more precise in [START_REF] Grüne | Numerical ISS controller design via a dynamic game approach[END_REF], Theorem 10). However, for our purpose the decisive property is that δ → 0 as sup x∈A V x → 0. Since the target A contains a neighborhood of the origin, this convergence implies ultimate boundedness w.r.t. A for vanishing perturbation if δ and thus A V are sufficiently small. In practice, given A one can determine an appropriate A V by numerical simulations.

Conversion to a robust stabilization problem

The goal of this section is to design an ISpS-controller for each subsystem Σ i . Recall that we consider the influence of the other states x j , j = i ∈ 1, . . . , N, as perturbations and thus extend the perturbation of Σ i to

v i = (x j . . . x N -1 d i ) ∈ V i = X j × . . . × X N -1 × D i , j = i ∈ 1, .
. . , N . Since this section only deals with a single subsystem, for simplicity we will omit the indices i in the sequel, i.e., we write the subsystem as

x(k + 1) = f (x(k), u(k), v(k)). (11)
In order to design an ISpS controller for (11), we follow the approach in [START_REF] Grüne | Numerical ISS controller design via a dynamic game approach[END_REF], which in turn is based on ideas from [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF]. This approach uses that system (5) is ISpS if and only if it is practically robustly stable. Practical robust stability means that there exists e : X × V → V and η ∈ K ∞ such that the system

x(k + 1) = f (x(k), u P (x(k)), e(x(k), ṽ(k))) (12)
with V = B 1 (0) is uniformly practically asymptotically stable, where e is such that for each v ∈ V with v ≤ η( x ) there exists ṽ ∈ V with e(x, ṽ) = v.

Solution of the robust stabilization problem

In order to construct a controller rendering the system uniformly practically asymptotically stable we employ the dynamic game approach from [START_REF] Grüne | Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property[END_REF] which in turn relies on ideas from [START_REF] Grüne | Global optimal control of perturbed systems[END_REF], [START_REF] Junge | A set oriented approach to global optimal control[END_REF]. This approach introduces a discretization of the state space X using a finite partition P of boxes P . Below, we describe the discrete time version of this approach. Its event-based extension, as introduced and discussed in [START_REF] Grüne | An algorithm for event-based optimal feedback control[END_REF] and [START_REF] Grüne | Two complementary approaches to event-based control[END_REF], is straightforward using the observation that the control value only changes when the state moves from one partition element P to another. Hence, an event is triggered whenever the state leaves a partition element. This event based implementation is used in the experiments in Section 7.

Let a target set A V be given which contains a neighborhood of the origin.

Note that A V i is the target set for the state of the i-th subsystem. We select a stage cost g : X × U → R which penalizes the distance to the origin and define the accumulated cost as

J(x 0 , u, v) := k(A V ,x 0 ,u,v) k=0 g(x(k, x 0 , u, v), u(k)),
where

k(A V , x 0 , u, v) denotes the smallest k for which x(k, x 0 , u, v) ∈ A V holds.
The computation of V is performed by a graph theoretic approximation of the dynamics of the system on the partition P. Since the model includes both control and perturbation the resulting graph takes the form of a hypergraph. The computation of V can then be carried out by solving a generalized min-max shortest path problem on this hypergraph. This yields an approximation V P of V which is constant on each element P of the partition P. An approximation u P of the optimal controller is obtained through the dynamic programming principle

u P (x) := argmin u∈U g(x, u) + sup x ′ ∈f (x,u,W ) V P (x ′ ) . ( 13
)
This controller is defined on the stabilizable set w.r.t. V P given by S P := {x ∈ X | V P (x) < ∞}. For details we refer to [START_REF] Grüne | Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property[END_REF]. Note that neither V P nor u P are continuous, hence we introduced Assumptions 1 and 2.

The following theorem in (Grüne & Sigurani, 2013, Theorem 12) summarizes the derivation outlined in this section.

Theorem 2. Consider the subsystem (11) satisfying Assumptions 1 and 2 and let the function V P denote the approximate optimal value function constructed according to the algorithm presented in [START_REF] Grüne | Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property[END_REF] on a

given partition P and target A V for system (12). Denote the corresponding feedback by u P . Let ℓ ≤ max s∈S P V P (s) and let α, α ∈ K ∞ denote functions such that (7) holds on Y = {x ∈ X | V P (x) ≤ ℓ}.

Then the system is ISpS on Y, with ∆ d → ∞ as ℓ → ∞ and δ → 0 as

sup x∈A V x → 0, provided δ ≤ α -1 (ℓ) holds.
We can thus use the proposed algorithm to compute the feedback u i,P for the subsystem (11) for i = 1, . . . .N . According to Theorem 2, the feedback u P (x) = (u 1,P (x 1 ), . . . , u N,P (x N )) will then also render system (5) ISpS, provided the gains are sufficiently small.

The local approach

This section presents a decentralized event-based state-feedback approach which, once the state x i (t) of subsystem Σ i has entered the target set

A i at time T i , ensures that x i (t) ∈ A i holds for t ≥ T i in spite of disturbances d i (t)
and of the interconnections to the remaining subsystems. For the overall system, the relation x(t) ∈ A ⇔ x i (t) ∈ A i , for all i ∈ N holds (cf. Fig. 3).

Models

The target set A i is a surrounding of the operating point xi of Σ i , in which Σ i is described for t ≥ T i by the linear state-space model

Σ i :            ẋi (t) = A i x i (t) + B i u i (t) + E i d i (t) + E si s i (t) x i (T i ) = x Ti z i (t) = C zi x i (t), (14)
where s i ∈ IR p i and z i ∈ IR q i denote the coupling input and coupling output, respectively. Σ i is interconnected with the remaining subsystems according to the relation

s i (t) = N j=1 L ij z j (t), (15)
where L ii = 0 holds for all i ∈ N by assumption. The control input generator C i and the event generator E i for Σ i are designed by using the method of Lunze & Lehmann (2010), which is applied here for the isolated subsystems

ẋi (t) = A i x i (t) + B i u i (t) + E i d i (t), x i (T i ) = x Ti , i ∈ N (16)
and leads to the components explained in the next sections.

Components of the event-based control loops

Control input generators C i . In the interval t ∈ [t k i , t k i +1 ), the control input generators C i are represented by the model

C i :            Σ si :    ẋsi (t) = Āi x si (t) + E i di (t k i ) x si (t + k i ) = x i (t k i ) u i (t) = -K i x si (t) (17)
for i ∈ N , where x si ∈ IR n i denotes the state and di (t k i ) is a disturbance estimate. This event-based control approach works with any disturbance estimation method that yields bounded estimates di (t k i ), e. g. with the trivial estimation di (t k i ) ≡ 0 for all k i ∈ IN 0 or with the more sophisticated disturbance estimate presented in Stöcker et al. (2013). In ( 17), the state-feedback gain K i is chosen such that the matrix

Āi := A i -B i K i is Hurwitz.
Event generators E i . Like the control input generators, the event generators include the model Σ si defined in (17). In order to determine the event times

t k i (k i = 1, 2, . . .), E i monitors the difference state x ∆i (t) := x i (t) -x si (t)
and triggers an event, whenever the condition

|x ∆i (t)| -ēi ∞ = 0 (18)
is satisfied, where ēi ∈ IR n i + denotes the event threshold vector. Hence, E i continuously measures the subsystem state x i (t) and it determines the event times t k i using the model

E i :            Σ si :    ẋsi (t) = Āi x si (t) + E i di (t k i ) x si (t + k i ) = x i (t k i ) t k i := min t > t k i -1 |x ∆i (t)| -ēi ∞ = 0 . ( 19
)
At the event time t k i , E i transmits the current subsystem state x i (t k i ) to C i and this information is used in both components to reset the state x si of the model Σ si , which implies x ∆i (t

+ k i ) = 0 for all event times t k i , k i ∈ IN 0 .

Stability analysis

The control input generator C i and the event generator E i are designed under the assumption of vanishing interconnections (s i (t) = 0 for all i ∈ N )

and with the aim to ensure the stability of the isolated event-based control loops. This section presents a condition on the interconnection matrix L for which the stability of the isolated event-based control loops implies the stability of the overall control system.

The following theorem summarizes a stability test which has been derived

by Stöcker et al. (2013) using the comparison principle (see [START_REF] Lunze | Feedback Control of Large-Scale Systems[END_REF]).

Theorem 3 (Stöcker et al. (2013)). The overall event-based control system that consists of the interconnected subsystems ( 14), ( 15) and the decentralized event-based controllers (17), ( 19) is ultimately bounded if the condition

λ P ∞ 0 Ḡxs (t) L Cz dt < 1 (20)
is satisfied with

Ḡxs (t) = diag e Āi t E si , (21)
Cz = diag (|C zi |) . (22)
The matrix

L =         0 |L 12 | . . . |L 1N | |L 21 | 0 . . . |L 2N | . . . . . . . . . . . . |L N 1 | |L N 2 | . . . 0         (23)
represents a bound on the interconnections among the subsystems.

The stability condition ( 20) is a small-gain condition requiring the interconnection among the subsystems to be sufficiently weak. Hence, (20) can be used to find a bound on the interconnection up to which the stability of the overall system is guaranteed.

The following result explicitely states a region B in which the state x(t) of the overall system is maintained by the decentralized event-based controller ( 17), ( 19).

Theorem 4 (Stöcker et al. (2013)). Consider the interconnected subsystems ( 14), ( 15) together with the decentralized event-based controllers (17), ( 19) and assume that the condition (20) is satisfied. The set

B := x = x ⊤ 1 . . . x ⊤ N ⊤ ∈ IR n (|x 1 | . . . |x N |) ⊤ ≤ b(ε, δ) (24)
is positive invariant for the overall control system (14), ( 15), ( 17), ( 19), with the ultimate bound

b(ε, δ) = I n - ∞ 0 Ḡxs (t) L Cz dt -1 (ε + δ) (25)
where

ε = ∞ 0 diag e Āi t B i K i dt • ē⊤ 1 . . . ē⊤ N ⊤ , (26) δ
= ∞ 0 diag e Āi t E i dt • d⊤ 1 . . . d⊤ N ⊤ ( 27
)
and the matrices Ḡxs (t), Cz and L are given in (21)-( 23).

The inverse matrix in (25) exists if the condition (20) is satisfied. Theorem 4 shows that the size of the set B depends upon the disturbance magnitudes di and the event thresholds ēi and that it can be adjusted by appropriately setting the event thresholds ēi for all i ∈ N .

Combination of the global and the local approach

This section explains how the global and the local approach are merged in order to obtain an event-based controller that renders the overall system ultimately bounded. For a practical application it is reasonable to combine both approaches, as they are complementary in the following ways:

• For steering the state into the target region, the controller of the global approach must be designed on the basis of the nonlinear plant model ( 1) which appropriately describes the system behavior within the overall state space and includes possible constraints regarding the set of states and control inputs. Once the state enters the target region, it is sufficient to only consider the system behavior in the neighborhood of the operating point by means of a linearized model ( 14), ( 15), as done by the controller of the local approach.

• The quantized state [x i (t)] provides enough information for the controller of the global approach to drive the state towards the target region. Moreover, the quantization reduces the amount of data for the feedback communication. But for keeping the state within the target region and counteracting exogenous disturbances and interconnections to other systems, the controller of the local approach requires more accurate feedback information about the state x i (t).

The overall event-based controller combines the two approaches by including in each component the functionalities of both the global and local approach.

Besides the previously defined triggering conditions, all event generators E i also includes the logic that induces the switching from the global to the local approach in both components. The switching time for the event-based controller for subsystem Σ i is given by

T i := inf t ≥ 0 | x i (t) ∈ A i .
At this time, E i switches the functionality from the global to the local approach and transmits a respective command to the corresponding control input generator C i . The local approach keeps the state x i (t) within the target set A i for all t ≥ T i . Note that the decentralized event-based controllers decide locally at which time they switch from the global to the local approach and, thus, the switching occurs asynchronously in time (Fig. 3).

x 0 x 1 x 2 A X A 1 A 2 T 1 T 2
Figure 3: Switching from the global to the local approach 6. Application to a continuous flow process

Hardware description

The proposed event-based control is tested and evaluated on the pilot plant at the Institute of Automation and Computer Control at Ruhr-University Bochum, Germany (Fig. 4). The plant includes four cylindrical storage tanks, three batch reactors and a buffer tank which are connected over a complex pipe system and it is constructed with standard industrial components including more than 70 sensors and 80 actuators.

Process description

The experimental setup is illustrated in Fig. 6. The main components are the two reactors B and S in which continuous flow processes shall be realized.

Reactor B is connected to the storage tank T 1 from where the inflow can be controlled by means of the valve angle u T1 . Via the pump PB a part of the outflow is pumped out into the buffer tank TW (and is not used further in the process) while the remaining outflow is conducted to the reactor S. The temperature ϑ B (t) of the water in reactor B is influenced by the cooling unit (CU) using the input u CU or by the heating rods that are driven by the signal d H . The inflow from the storage tank T 3 to the reactor S can be adjusted by means of the opening angle u T3 . Reactor S is additionally fed by the fresh water supply (FW) from where the inflow is set by means of the valve angle d F . Equivalently to reactor B, the outflow of reactor S is s! plit and one part is conveyed via the pump PS to TW and the other part is pumped to the reactor B. The temperature ϑ TS (t) of the liquid in reactor S can be increased by the heating rods that are controlled by the signal u H .

The two reactors are coupled by the flow from reactor B to reactor S and vice versa, where the coupling strength can be adjusted by means of the valve angles u BS and u SB . The ratio of the volume that is used for the coupling of the systems and the outflow to TW is set by the valve angles u BW and u SW .

Both reactor B and reactor S are equipped with sensors that continuously measure the level and the temperature of the contents.

E 1 E 2 C 1 C 2 FW TW TW CU PB PS T T L L Reactor B Reactor S T 1 T 3 u T3 u CU u T1 u H d H d F u BS u SB u BW u SW l B ϑ B l S ϑ S Ethernet

Plant model

In the following, the behavior of the level and the temperature in reactor B and reactor S are considered as subsystem Σ 1 and Σ 2 , respectively. Hence, the states of the subsystems are represented by x 1 (t) = l B (t) ϑ B (t)

⊤ and

x 2 (t) = l S (t) ϑ S (t) ⊤ . The continuous flow process is represented by the nonlinear state-space model

lB (t) = A -1 B q 1B (u T1 (t)) + q SB (l S (t), u SB ) -q BW (l B (t), u BW ) -q BS (l B (t), u BS ) (28a) θB (t) = (A B l B (t)) -1 q 1B (u T1 (t))(ϑ 1 -ϑ B (t)) + q SB (l S (t), u SB )(ϑ S (t) -ϑ B (t)) + q C (u CU (t))(ϑ C -ϑ B (t)) + H B d H (t) (28b) lS (t) = A -1 S q 3S (u T3 (t)) + q BS (l B (t), u BS ) -q SW (l S (t), u SW ) -q SB (l S (t), u SB ) + q FS (d F (t)) (28c) θS (t) = (A S l S (t)) -1 q 3S (u T3 (t))(ϑ 3 -ϑ S (t)) + q BS (l B (t), u BS )(ϑ B (t) -ϑ S (t)) + q FS (d F (t))(ϑ F -ϑ S (t)) + H S u H (t) . (28d)
Here,

q 1B (u T1 (t)) = 1.61 × 10 -4 • u T1 (t) (29a) q 3S (u T3 (t)) = 1.81 × 10 -4 • u T3 (t) (29b)
denote the flows from the storage tanks T 1 and T 3 to the reactors B and S, respectively.

q C (u CU (t)) = 0.97 × 10 -4 • u CU (t) (29c)
is the flow of the coolant and

q BS (l B (t), u BS ) = K BS (u BS ) 2gl B (t) (29d) K BS (u BS ) = 10 -4 •      1.02 • u BS , 0 ≤ u BS ≤ 0.1 2.13 • u BS -0.11, 0.1 < u BS ≤ 1 q SB (l S (t), u SB ) = K SB (u SB ) 2gl S (t) (29e)
K SB (u SB ) = 10 -4 •      0.90 • u SB , 0 ≤ u SB ≤ 0.1 1.68 • u SB -0.08, 0.1 < u SB ≤ 1
denote the flows from reactor B to reactor S and vice versa with the specific valve parameters K BS and K SB (m 3 /m). Finally,

q BW (l B (t), u BW ) = K BW (u TB ) 2gl B (t) (29f)
K BW (u BW ) = 10 -4 •      0.96 • u TB , 0 ≤ u BW ≤ 0.1 2.01 • u TB -0.10, 0.1 < u BW ≤ 1 q SW (l S (t), u SW ) = K SW (u SW ) 2gl S (t) (29g) K SW (u SW ) = 10 -4 •      0.79 • u SW , 0 ≤ u SW ≤ 0.1 1.42 • u SW -0.06, 0.1 < u SW ≤ 1
denote flows of volume from the reactors B and S into the buffer reactor TW with the specific valve parameters K BW and K SW (m 3 /m). All flows have the unit m 3 /s. All parameters are listed in Table 1.

Due to technical limitations the subsystem states x 1 = (l B , ϑ B ) ⊤ and

H B 4.8 × 10 -3 m 3 K/s
The control inputs u 1 = (u T1 , u CU ) ⊤ and u 2 = (u T3 , u H ) ⊤ are limited to

the set U = U 1 × U 2 with U 1 = [0; 1] × [0; 1] , U 2 = [0; 1] × [0; 1] . (31)
Note that the components which are used for the control are highlighted in gray in Fig. 6. The disturbance characteristics are accomplished by means of the heating with disturbance input d 1 (t) = d H (t) in reactor B and the additional water inflow in reactor S that is set by the valve angle d 2 (t) = d F (t). The disturbances are considered to be bounded to

d 1 ∈ D 1 = [0; 0.1] , d 2 ∈ D 2 = [0; 0.25] . (32)

Specification of the control aim

The state x(t) of the overall system shall be steered from a given initial state x 0 ∈ X into the target region

A = A 1 × A 2 with A 1 = [0.3; 0.36] m × [291.7; 297.7] K (33a) A 2 = [0.31; 0.37] m × [297.2; 303.2] K (33b) around the operating point x1 =   lB θB   =   0.33 m 294.7 K   , x2 =   lS θS   =   0.34 m 300.2 K   (34)
and maintained in A for all time in spite of the influence given in (32) and interconnections. The interconnections among both subsystems are set by the valve angles u BS and u SB which are fixed to

u BS = 0.19, u SB = 0.22 (35)
throughout the experiments. Moreover, the choice

u BW = 0.21, u SW = 0.29 (36)
defines the outflow from the reactors B and S to the buffer tank TW.

Decentralized event-based controller resulting from the global approach

The global approach calculates a decentralized event-based controller for each subsystem Σ 1 and Σ 2 , utilizing the algorithm described in Section 3.

For our experiment we run the algorithm without external disturbances, i.e., we set d H = d F = 0 and consider only the state of the other subsystem as disturbance by setting v 1 = (l S θ S ) and v 2 = (l B θ B ). The functions

e 1 (x 1 , ṽ1 ) =   0.34 + 1.28(l B -0.33) 2 ṽ11 300.2 + 1053.4(l B -0.33) 2 + 0.631338(θ B -294.7) 2 ṽ12   (37)
and

e 2 (x 2 , ṽ2 ) =   0.33 + 0.750312(l S -0.34) 2 ṽ21 294.7 + 1239.4(l S -0.34) 2 + 1.49094(θ S -300.2) 2 ṽ12   (38)
are used to convert the problem into a robust stabilization problem, thus obtaining system (12) with ṽ1 = (ṽ 11 ṽ12 ) T ∈ [-1; 1] 2 and ṽ2 = (ṽ 21 ṽ22 ) T ∈ [-1; 1] 2 . The additive terms in e 1 and e 2 are necessary because in contrast to Section 3 the operating point in our experiment is not located at the origin.

Note that both subsystems have a cascaded (or triangular) structure and it has turned out beneficial to choose the e i to reflect this structure, i.e., the first components of the e i are independent of θ B or θ S , respectively.

For obtaining the discrete time system (4) we use a sampling time of 2 seconds. For the discretization of each of the state spaces X 1 and X 2 from (30) we use a partition P of 8 × 8 equally sized rectangular elements. The target set A V consists of the partition element containing the operating point (34), i.e., A V 1 = [0.33; 0.3475]×[290.3375; 295.025] and A V 2 = [0.33; 0.3475]× [296.9; 300.65], i.e., we choose a smaller set than A in (33). The stage costs are chosen as

g 1 (x 1 , u 1 ) = 1 0.0196 (l B -0.33) 2 + 1 1406.25 (θ B -294.7) 2 (39)
and

g 2 (x 2 , u 2 ) = 1 0.0196 (l S -0.34) 2 + 1 1406.25 (θ S -300.2) 2 . ( 40
)
For constructing the hypergraph we also need to discretize the control and perturbation input sets. We accomplish this by choosing the disturbances ṽi ∈ {-1; 0; 1} 2 and discretizing the control inputs for Σ 1 by 9×5 equidistant values and for Σ 2 by 9 × 4 equidistant values.

As mentioned in Section 3.4, an event is triggered whenever the state leaves a partition element. In order to algorithmically check whether this happens, we need to define a maximum number R of steps after which an element P is considered as not left, cf. the discussion after Definition 2.1 in [START_REF] Grüne | An algorithm for event-based optimal feedback control[END_REF]. Here we choose R = 600. In order to increase the efficiency of the algorithm we moreover use one step of past information for the controller design, for details see (Grüne & Müller, 2009, Section III) or (Grüne et al., 2010, Section 3.4).

The resulting approximated optimal value functions V 1 of Σ 1 (left) and V 2 of Σ 2 (right) depending on the initial value are depicted in Figure 7. One notes that the values of V 2 are much higher than those of V 1 because the cooling unit in reactor B is much slower than the heating unit in reactor S.

This effect is also visible when looking at the maximal time needed to reach the target set from a given partition element which is shown in Figure 8.

Decentralized event-based controller resulting from the local approach

The local approach uses a linearized model of the continuous flow process.

Therefore, the nonlinear system (28), ( 29

L 12 =   1 0 0 1   , L 21 =   1 0 0 1   .
The control input generators C 1 and C 2 determine the control inputs according to the model ( 17) using the feedback gains which implies ultimate boundedness of the overall closed-loop system. Moreover, the stability condition (20) can be used to show that the event-based control system is robustly stable with respect to changes of the coupling strength. The evaluation of the condition (20) for the interconnection

K 1 =   10
L = κ   O L 12 L 21 O   , (42)
in dependence upon a scaling factor κ ∈ IR + yields that the overall system is UB for 0 ≤ κ < κ crit = 2.6.

In order to satisfy the control aim (3) with the target set A = A 1 × A 2 as defined in (33), the event threshold vectors of the event generators E 1 and for i = 1, 2. For the practical realization, the inequality (45) substitutes the equality (18) as due to the implementation on digital hardware the event condition is checked periodically with sampling period T s = 0.3 s which generally only detects the exceeding of the condition. Nevertheless, in relation to the time constants of the process the sampling is fast and, thus, the error that is introduced by the sampling is negligible. A more detailed analysis of event-based control with discrete-time sampling is given in Grüne et al.

E 2 are chosen to ē1 =   0.02 0.4   , ē2 =   0.02 0.4   , (43)
(2010).

Implementation of the event-based controllers

Each of the previously explained components of the decentralized eventbased controllers is implemented in MATLAB/Simulink using the concept of Level-2 MATLAB S-Functions. The implementations of the control input generator and event generator is independent of the index of the subsystem to which these components are assigned and, therefore, the index is omitted for the following explanations. The blocks Switch x and Switch trigger are used to route the signals of Global approach and Local approach where the criteria for passing the first input is select ≥ 1 in both switches. The block Global approach determines the integer value [x(t k )] that uniquely identifies the box in which the current state is. The signal trigger/t k is updated to the current event time t k whenever the state x(t) passes from one box to another (i. e. whenever an event is generated). The block Global approach detects the time instant t = T at which the state x(t) enters the target set A. At this time the block Global approach switches select from 1 to 0, indicating the activation of the block Local approach which then sets trigger/t k = -T. The transmission of the current state x(T ) and the time t k = -T to the control input generator signalizes the switching to the local approach by virtue of the negative event time. From then on, the block Local approach generates events according to the condition (45) and whenever this condition is satisfied it sets trigger/t k = t k .

Experimental evaluation

Description of the experiment

This section presents the results of an experiment where the state x(t) of the system (28), ( 29) is driven from the initial state

x 1 (0) =   l B (0) ϑ B (0)   =   0.40 317.2   , x 2 (0) =   l S (0) ϑ S (0)   =   0
A 1 A 2 X 1 X 2 x 1 (0) x 2 (0)
Figure 11: Trajectories of the state x 1 (t) and x 2 (t)

Experimental results

The behavior of the continuous flow process with decentralized eventbased control is illustrated in Figs. 11-14. Figure 11 gives an overview over the transition of the subsystem states x 1 (t) and x 2 (t) into the respective target regions. Once, the states enter the target regions, they are kept within these sets which shows that the control aim is fulfilled. Note that this aim is achieved despite model uncertainties which occur, since the model ( 28 compared to the transition of x 1 (t) to A 1 , which is due to the fact that x 2 (0) is much closer to A 2 as x 1 (0) is to A 1 . This is also reflected in the number of events triggered in both subsystems: In the reactor S only 5 events are triggered before the local approach is activated, whereas in reactor B, 48 events are generated before the target set A 1 is reached.

Figure 13 shows the disturbance rejection behavior of the continuous flow process with the local event-based control approach. The time intervals in which the disturbances d H (t) and d F (t) are active are highlighted in gray. The experiments show that in event-based control the feedback communication than 1150 s. In order to attenuate the disturbance d F (t) that affects reactor S in the gray highlighted interval, the feedback communication is induced more often, i. e. 10 events are generated. In total, only 13 events per 1602 s are triggered in reactor B and 26 events per 1897 s are triggered in reactor S.

Compared to a sampled-data control with a sampling period T s = 10 s (which is a typical choice for the considered continuous flow process), the feedback communication effort is considerably reduced by the event-based control.

Figure 14 provides a verification of the bounds (44) determined according to the analysis method in Theorem 4. It is shown that in both subsystems, the levels l B (t) and l S (t) exceed the calculated bounds, nevertheless, the maximum deviation between the levels and the respective bounds is less than 0.03 cm. From this investigation it can be concluded that the analysis method in Theorem 4 yields tight bounds for the considered class of systems, however, these bounds might not hold in the presence of model uncertainties.

Conclusion

This paper has presented a new method for the decentralized event-based control of physically interconnected systems and its practical application to a continuous flow process. The proposed control method is based on two approaches, referred to as global and local approach, that differ with respect to their control task for which they are specially designed. The global approach uses a nonlinear model of the plant and quantized state information in order to drive the state of each system into a target region. The state is kept within this target set in spite of exogenous disturbances and interconnections to other systems by the local approach, where a feedback is only invoked if the state deviates from a state estimate, determined using a linearized plant model, by some threshold. In this way, both approaches jointly achieve ultimate boundedness of the closed-loop system. The theoretical results have been evaluated on the basis of experimental results, obtained by the application of the control method to a continuous flow process. These results have mainly shown two facts: First, the decentralized event-based control method is robust with respect to model uncertainties and, second, the control aim is accomplished with considerably less feedback communication effort compared to the communication in sampled-data control.

Figure 1 :

 1 Figure 1: Structure of the event-based control system

Figure 5 Figure 4 :

 54 Figure 5 illustrates the automation concept for the pilot plant which is subdivided into three layers. On the top layer, the event-based control is implemented on an ordinary personal computer (PC). The functionalities of the control input generators C i and event generators E i are realized in MATLAB/Simulink executed with the sampling time T s = 0.3 s. As the sampling time T s is by a factor of more than 150 smaller than the time constants of the process, the control can be considered to be continuous. The PC is connected over a 100 Mbit/s Ethernet network with the programmable logic controllers (PLCs) on which subordinate controllers and several routines for the plant protection are implemented. On the field level (Sensor/actuator control) the actuator signals are applied and the sensor signals are sampled via the perepherials A to E that are connected over PROFIBUS DP with the PLCs.

Figure 5 :

 5 Figure 5: Automation concept for the pilot plant

Figure 6 :

 6 Figure 6: Experimental setup of the continuous flow process

  Heat coefficient of the heating in tank B H S 0.8 × 10 -3 m 3 K/s Heat coefficient of the heating in tank S ϑ 1 294.15 K Temperature of the fluid in tank T 1 ϑ 3 294.15 K Temperature of the fluid in tank T 3 ϑ C 282.65 K Temperature of the coolant ϑ F 294.15 K Temperature of the water supply x 2 = (l S , ϑ S ) ⊤ are restricted to the state space X = X 1 × X 2 with X 1 = [0.26; 0.40] m × [285.65; 323.15] K (30a) X 2 = [0.26; 0.40] m × [293.15; 323.15] K.

Figure 7 :Figure 8 :

 78 Figure 7: Value functions for Σ 1 (left) and Σ 2 (right) over state-space

Figure 9 :

 9 Figure 9: Simulink model of the event generator

TimeFigure 12 :

 12 Figure 12: Behavior of the global approach. In the upper two rows the trajectories of the levels l B (t), l S (t) and of the temperatures ϑ B (t), ϑ S (t) are shown for reactor B and S on the left-hand side or right-hand side, respectively. The next two rows depict the respective control inputs and the event time instant are marked by stems in the bottom figures.

Figure 14 :

 14 Figure 14: Verification of the ultimate boundedness analysis

  The local event-based control approach makes the set A i for each Σ i

	event generator E i . The control input generator C i is realized as a look-up
	table that can be computed offline.
	robustly positive invariant, i.e. once the state x i (t) enters A i it is kept
	within this set in spite of exogenous disturbances d i (t) and intercon-
	nections to other subsystems.
	x 2	A	X
	Global problem	Local problem
	x 0	State constraints	x 1
	Figure 2: The global and the local problem
	Global approach -Nonlinear control. Section 3 proposes an optimization-
	based method for the design of a decentralized event-based controller which
	accomplishes the transition of the state x(t) of the nonlinear system Σ given
	in (1) from an initial state x 0 to the target set A. The state-feedback law for
	subsystem Σ i uses quantized information [x i (t)] about the state x i (t) only.
	This quantization coarsely partitions the state space X i into a grid of boxes
	in each of which the control input u i (t) is held constant. The event-based
	character of the controller lies in updating the control input u i (t) only after
	the state x i (t) has crossed the boundary of a box, which is detected by the

Table 1

 1

			: Parameters
	Parameter Value	Meaning
	A B	0.07 m 2	Cross sectional area of tank B
	A S	0.07 m 2	Cross sectional area of tank S
	g	9.81 m/s 2	Gravitation constant

  which means that E 1 and E 2 trigger an event if either the level or the temper-Hence, the local approach ensures that the state x = (l B ϑ B l S ϑ S ) ⊤ is kept

	ature deviates by 2 cm or 0.4 K, respectively, from the corresponding model
	state. According to Eqs. (25)-(27), the choice (43) yields the ultimate bound
	b = (0.018 2.56 0.027 1.36) ⊤ .
	within the bounds
	l B (t) ∈ [0.312; 0.348], ϑ B (t) ∈ [292.1; 297.3]	(44a)
	l S (t) ∈ [0.313; 0.367], ϑ S (t) ∈ [298.8; 301.6]	(44b)
	for all t ≥ T (x 0 ) where the time T (x 0 ) is determined by the global approach.
	A comparison of the bounds (44) with the desired target set (33) shows that
	the decentralized event-based controllers with state-feedback gains (41) and
	the event thresholds (43) satisfy the control aim.
	The event generators E 1 and E 2 are implemented as in (19) with the
	event condition
	|x ∆i (t)| -ēi ∞ ≥ 0	(45)

  Figure 10: Simulink model of the control input generatorThe structure of the Simulink model of the control input generator is depicted in Fig.10. Similar to the model of the event generator, the main components are the blocks Global approach and Local approach. The first one determines the control input u(t) by selecting at the event times the appropriate entry from a look-up table which is linked to the integer [x(t k )]. In the block Local approach the information x(t k ) received at the event time t k is used to reinitialize the state of a model that is applied for the input generation. The decision whether the control input generated by the Global approach or by the Local approach shall be applied is implemented in the Embedded MATLAB Function Signal selector. third input of the block Switch u.

	[x(t_k)]	Global	u(t)
	x(t_k)	Local	u(t)

Acknowledgements

This work is supported by the German Research Foundation (DFG) within the priority program "Control Theory of Digitally Networked Dynamical Systems", grant numbers LU 462/26-2 and GR 1569/11-2 and by

is adapted to the current system behavior. In the time interval [103 s, 398 s], the state x 2 (t) is in the target set A 2 , whereas x 1 (t) is still outside of A 1 , which means that reactor S is considerably affected by reactor B via the interconnections. In this time interval, 9 events are generated in reactor S within less than 300 s, while in [398 s, 1550 s], where the coupling effect is small and the disturbance d F is not active, only 7 events are triggered in more