suppressing chaotic/hyperchaotic …flyingv.ucsd.edu/mostafa/papers/dscc2017_5006.pdf · sion time...

Proceedings of the ASME 2017 Dynamic Systems and Control ConferenceDSCC 2017

October 11-13, 2017, Tysons Corner, Virginia, USA

DSCC2017-5006

SUPPRESSING CHAOTIC/HYPERCHAOTIC DYNAMICS OF SMART VALVESNETWORK USING DECENTRALIZED AND CENTRALIZED SCHEMES

Peiman Naseradinmousavi and Mostafa BagheriDynamic Systems and Control LaboratoryDepartment of Mechanical Engineering

San Diego State University, San Diego, USA

Hashem AshrafiuonDirector of Center for Nonlinear Dynamics and Control

Department of Mechanical EngineeringVillanova University, Villanova, USA

Marcello CanovaDepartment of Mechanical & Aerospace Engineering

Ohio State University, Columbus, USA

David B. SegalaNaval Undersea Warfare Center

1176 Howell Street, Newport, USA

ABSTRACTIn this effort, we present a comprehensive comparative study

of decentralized and centralized adaptive schemes to control theso-called “Smart Valves” network employed in many applica-tions including, but not limited to, Municipal Piping Systems andoil and gas fields. The network being considered here typicallyincludes scores of coupled solenoid actuated butterfly valves. Wehere examine the multiphysics network of two interconnected ac-tuated sets. The network undergoes the coupled chaotic andhyperchaotic dynamics subject to some initial conditions andcritical parameters. The control schemes’ trade-offs are thor-oughly investigated with respect to robustness, computationalcost, and practical feasibility of control inputs in the presence ofstrong nonlinear interconnections and harmful chaotic and hy-perchaotic responses.

1 IntroductionThe decentralized and centralized control of large-scale net-

works have received considerable attention, particularly in thepresence of coupled harmful responses. The coupled danger-ous dynamical responses, including chaotic and hyperchaotic(with larger domain of attraction) ones, can be transmitted amongneighbor sets. Consequently, robust and practically feasible con-trol schemes are needed to be implemented in driving such large-

scale systems.

The system being studied here is the so-called “SmartValves” network containing scores of interconnected actuatedbutterfly valves coupled in series. This multiphysics sys-tem broadly deals with several aspects of electro-magneto-mechanical-fluid units. We particularly analyze two sets of cou-pled bi-directional solenoid actuated butterfly valves. The net-work plays an important role in proper and efficient performanceof many critical infrastructures which include, but are not lim-ited to, the US Navy, oil and gas fields, petrochemical plants,and more importantly, Municipal Piping Systems. The first uti-lizes a distributed flow control system for cooling purposes andtherefore, the failure of such a crucial unit would expectedly im-pose considerable costs of restoration and operation. A robustand practically feasible control scheme is hence required to miti-gate the effects of the harmful dynamic responses in the presenceof uncertainties involved with such a large-scale network.

We have previously reported broad analytical and experi-mental studies [1–13] for both an isolated actuator-valve arrange-ment and a network of two interconnected solenoid actuated but-terfly valves operating in series. Novel third-order (nondimen-sional) and sixth-order analytical models of the single and twosets of solenoid actuated butterfly valves were derived, respec-tively, dealing with the coupled nonlinear magnetic, hydrody-namic, and bearing torques. The transient chaotic and crisis,

1 Copyright c© 2017 by ASME

and coupled chaotic and hyperchaotic dynamics of both the sin-gle and interconnected sets were captured, respectively, by ex-posing the system to some critical parameters and initial con-ditions. The constrained isolated and then coupled design andoperational optimization problems were solved using the stabil-ity constraints imposed by the dynamic analysis. Finding spe-cific research work to capture and control the chaotic and hy-perchaotic dynamics of such a multiphysics network, using boththe centralized/decentralized schemes, is somewhat difficult al-though some efforts have been reported for similar case studies.Many efforts have been addressed in [14–28] to thoroughly in-vestigate the decentralized and centralized schemes for a varietyof electromechanical systems.

We here briefly represent the interconnected analyticalmodel of two sets (for completeness) along with the criticalinitial conditions and parameters resulted in the dangerous re-sponses. Both the centralized (coupled) and decentralized adap-tation and control laws will be formulated with respect to the in-terconnected dynamics of the system; in the presence of chaoticand hyperchaotic responses. The results will be thoroughly dis-cussed to address the practical feasibility and robustness of boththe adaptive methods, for vanishing the chaotic and hyperchaoticdynamics, in addition to their computational costs.

k1 bd1

Plunger

Coil

Coil

Yoke

k2 bd2

Plunger

Coil

Coil

Yoke

α1 α2Pin Poutqin qout

qv

(a)

R1

Pin PoutP1 P2

R2

α1 α2

L1 L2

RL1 Rcon RL2

Dv1 Dv2

Pcon1 Pcon2

1 2

(b)

FIGURE 1. (a) A schematic configuration of two bi-directionalsolenoid actuated butterfly valves subject to the sudden contraction; (b)A coupled model of two butterfly valves in series without actuation

2 Mathematical Modeling

The system, which is being considered here, is two bi-directional solenoid actuated butterfly valves operating in series.The system undergoes a sudden pipe contraction as shown in Fig1. The plungers are connected to the valves’ stems through therack and pinion arrangements yielding kinematic constraints.

We have previously derived the interconnected analyticalmodel of two sets operating in series [1–4] and briefly representhere for completeness. Some simplifying assumptions have beenmade to develop the analytical formulas of the coupled sets. Thefirst one is to assume a negligible magnetic “diffusion time” withrespect to a nominal operation time (40s). Note that the diffu-sion time has an inverse relationship with the amount of currentapplied. The second assumption is the existence of laminar flow.We have carried out experimental work for an isolated set to val-idate the assumption of laminar flow. The total flow loads, in-cluding hydrodynamic (Th) and bearing (Tb) torques, were mea-sured experimentally. An acceptable consistency was observedbetween the analytical and experimental approaches [29].

Based on the analytical formulas addressed in [1–4],the sixth-order interconnected dynamic equations of two bi-directional solenoid actuated butterfly valves were developed asfollows:

z1 = z2 (1)

z2 =1J1

[r1C21N2

1 z23

2(C11 +C21(gm1− r1z1))2 −bd1z2− k1z1

+

(Pin−Pout−(RL1+RL2+Rconqv)qv)e1(p1z3

1+q1z21+o1z1+γ1)2

∑i=1,4ei

(piz3i +qiz2

i +oizi+γi)2

×[(a1z1eb1z1

1.1 − c1ed1z1)−C1× tanh(Kz2)]]

(2)

z3 =(V1−R1z3)(C11 +C21(gm1− r1z1))

N21

−

r1C21z3z2

(C11 +C21(gm1− r1z1))(3)

z4 = z5 (4)

z5 =1J2

[r2C22N2

2 z26

2(C12 +C22(gm2− r2z4))2 −bd2z5− k2z4

+

(Pin−Pout−(RL1+RL2+Rconqv)qv)e2(p2z3

4+q2z24+o2z4+γ2)2

∑i=1,4ei

(piz3i +qiz2

i +oizi+γi)2

×[(a′1z4eb′1z4

1.1 − c′1ed′1z4)−C2× tanh(Kz5)]]

(5)

z6 =(V2−R2z6)(C12 +C22(gm2− r2z4))

N22

−

r2C22z5z6

(C12 +C22(gm2− r2z4))(6)


where, bdi indicates the equivalent torsional damping, ki is the

0 5 10 15 20 25−80

−60

−40

−20

0

20

40

60

80

αi(degree)

αi

(

rad s

)

α1

α2

(a)

0 5 10 15 20 25 30−150

−100

−50

0

50

100

150

αi(degree)

αi

(

rad s

)

α1

α2

(b)

FIGURE 2. (a) The coupled sets’ phase portraits for Initial1; (b) Thecoupled sets’ phase portraits for Initial2

equivalent torsional stiffness, Vi stands for the supply voltage, riindicates the radius of the pinion, C1 and C2 are the reluctancesof the magnetic path without air gap and that of the air gap, re-spectively, Ni stands for the number of coils, gmi is the nominalair gap, Ji indicates the polar moment of inertia of the valve’sdisk, and Ri is the electrical resistance of coil. z1 = α1, z2 = α1,and z3 = i1 indicate the upstream valve’s rotation angle, angu-lar velocity, and actuator current, respectively. z4 = α2, z5 = α2,and z6 = i2 stand for the downstream valve’s rotation angle, an-gular velocity, and actuator current, respectively. The networkparameters are listed in Table 1.

Note that we could capture, for the first time, the coupledchaotic and hyperchaotic dynamics of the interconnected sets [1,3] by examining the critical values of bdi = µi = 1×10−7 for twodifferent initial conditions of Initial1 = [20(deg) 0 0 20(deg) 0 0]and Initial2 = [2(deg) 0 0 2(deg) 0 0], respectively. Shown inFigs. 2(a) and 2(b) are the chaotic and hyperchaotic dynamics ofthe coupled actuated valves, respectively. Some powerful toolsof the nonlinear analysis, including the Lyapunov exponents andPoincare map [30], were used in distinguishing among the natureof harmful responses, as shown in Figs. 3(a)-3(f). One and twopositive Lyapunov exponents along with irregular Poincare mapsconfirmed the chaotic and hyperchaotic dynamics of the actuatedvalves, respectively. Such dangerous responses need to be van-ished using a nonlinear control scheme, due to the nonlinear andcoupled nature of the network, in order to return the intercon-nected sets to their stable domains. The operationally optimizedvalves’ motions, on the other hand, are utilized in the nonlinearcontrol scheme as desirable trajectories. Based on the inevitableunknown parameters of such a coupled network, the nonlinearmodel-based adaptive scheme looks as an effective approach tobe employed in stabilizing the system subject to the chaotic andhyperchaotic dynamics.

TABLE 1. The system parameters

ρ 1000 kgm3 v 3 m

s

J1,2 0.104×10−1(kg.m2) N2 3300

N1 3300 C11,22 1.56×106(H−1)

Dv1 0.2032(m) Dv2 0.127(m)

Ds1,s2 0.01(m) Pout 2(kPa)

k1,2 60(N.m−1) C21,22 6.32×108(H−1)

L1 2(m) L2 1(m)

r1,2 0.05(m) θ 90◦

Pin 256(kPa) gm1,m2 0.1(m)

µ f 0.018 (Kg.m−1.s−1) λ1,2 1

n1,2 10 bdi = µi 1×10−7

ε1,2 5×10−3 e1 7.2×105

p1 461.9 q1 -405.4

o1 -1831 γ1 2207

e2 4.51×105 p2 161.84

q2 -110.53 o2 -695.1

γ2 807.57

3 Control and Adaptation Laws

3.1 Adaptive Centralized Approach

The nonlinear model-based adaptive control method [31] isused in stabilizing the unstable system in order to track the de-sired trajectories [1,3] defined based on the critical initial condi-tions as follows:

αdi =π

3tanh(10−4t3)+

π

9, Initial1 (7)

αdi =π

3tanh(10−4t3)+

π

90, Initial2 (8)

The so-called “S-Shaped” trajectories are highly energy-efficient[3, 9] and yield smooth dynamic responses avoiding the repeat-edly observed dangerous phenomenon of “Water Hammering”.The coupled dynamic Eqs. 2 and 5 can be rewritten as the fol-lowing:

Jiαi +bdiαi + kiαi =riC2iN2

i i2i2(C1i +C2i(gmi− riαi))2


0 1000 2000 3000 4000 5000 6000−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (s)

Lj

L1=−0.0072

L2=−0.0102

L3=−0.0101

L4=−0.0113

L5=−21.569

L6=+0.1535

(a)

30 40 50 60 70 80 90

0.05

0.1

0.15

0.2

0.25

0.3

θ

Lj

(b)

47

48

49

50

51

52

53

54

α1

α1

(c)

61.5

62

62.5

63

63.5

64

64.5

α2

α2

(d)

51

51.5

52

52.5

α1

α1

(e)

64.4

64.5

64.6

64.7

64.8

64.9

65

α2

α2

(f)

FIGURE 3. (a) The Lyapunov exponents for Initial1; (b) The positiveLyapunov exponents for Initial2 vs. different approach angles (θ ); (c)The poincare map for Initial1 of the upstream set; (d) The poincare mapfor Initial1 of the downstream set; (e) The Poincare map for Initial2 ofthe upstream set; (f) The poincare map for Initial2 of the downstreamset

+A1Rni

∑2i=1 Rni

[T ′hi−T ′bi tanh(Kαi)], (i = 1,2) (9)

where, A1 = (Pin − Pout − (RL1 + RL2 + Rconqv)qv), T ′h1 =

a1α1eb1α11.1 − c1ed1α1 , T ′h2 = a′1α2eb′1α2

1.1 − c′1ed′1α2 , and T ′bi =0.5AdiµiDsi. Assuming,

Mi =2Ji(C1i +C2i(gmi− riαi))

2

riC2iN2i

,Bi =2bdi(C1i +C2i(gmi− riαi))

2

riC2iN2i

Ki =2ki(C1i +C2i(gmi− riαi))

2

riC2iN2i

,Coi =2(C1i +C2i(gmi− riαi))

2

riC2iN2i

Eq. 9 can be rewritten as follows.

Miαi +Biαi +Kiαi = ui

+A1CoiRni

∑2i=1 Rni

[T ′hi−T ′bi tanh(Kαi)], (i = 1,2) (10)

We define the valves’ tracking errors and their first and secondtime derivatives as the following:

ei = αdi−αi, ei = αdi− αi, ei = αdi− αi, (i = 1,2)

This yields,

Miei = Miαdi−Miαi = Miαdi +Biαi +Kiαi

− ui−A1CoiRni

∑2i=1 Rni

[T ′hi−T ′bi tanh(Kαi)], (i = 1,2) (11)

The combined tracking errors [31] and their first time derivativesare as follows:

si = ei +λiei, si = ei +λiei, (i = 1,2)

where λ ’s are strictly positive numbers listed in Table 1. Premul-tiplying by Mi and substituting from Eq. 11, we have,

Misi = Miαdi +Biαi +Kiαi−ui +Miλiei

− A1CoiRni

∑2i=1 Rni

[T ′hi−T ′bi tanh(Kαi)], (i = 1,2) (12)

Based on the interconnected dynamics of the network, we chosethe following quadratic Lyapunov function candidate:

V =12

[2

∑i=1

(sTi Misi + Θ

Ti Γ−1i Θi)

](13)

where Γi is a symmetric positive definite matrix and Θi is thesystem’s lumped parameter estimation error (Θi = Θi− Θi). Dif-ferentiating the Lyapunov function (Eq. 13) yields,

V =2

∑i=1

(sTi Misi +

12

sTi Misi− Θ

Ti Γ−1i

˙Θi) (14)


By defining the regression vectors as the following:

WiΘi = Miαdi +Biαi +Kiαi−A1CoiRni

∑ Rni

× [T ′hi−T ′bi tanh(Kαi)]+Miλiei +12

Misi, (i = 1,2) (15)

The V can be easily rewritten as follows.

V =2

∑i=1

[sT

i [WiΘi−ui]− ΘTi Γ−1i

˙Θi

](16)

The appropriate control inputs are hence chosen as the following:

ui =WiΘi +nisi, (i = 1,2) (17)

where,

WiΘi = Miαdi + Biαi + Kiαi−A1CoiRni

∑ Rni

× [T ′hi−T ′bi tanh(Kαi)]+ Miλiei +12

ˆMisi, (i = 1,2) (18)

We can easily develop the regression (Wi ∈ ℜ1×17) (and subse-quently estimated lumped parameter vectors (Θi ∈ ℜ17×1)) asfollows.

Wi =[α

2i αdi,αiαdi, αdi,α

2i αi,αiαi, αi,α

3i ,α

2i ,αi,

A1Rni

∑ Rni[T ′hi−T ′bi tanh(Kαi)]α

2i ,

A1Rni

∑ Rni[T ′hi−T ′bi tanh(Kαi)]αi,

A1Rni

∑ Rni[T ′hi−T ′bi tanh(Kαi)],α

2i ei,αiei, ei,αisi,si],(i = 1,2)(19)

Substituting the control inputs into Eq. 16 gives,

V =2

∑i=1

sTi [WiΘi−WiΘi︸︷︷︸

WiΘi

−nisi]− ΘTi Γ−1i

˙Θi

=2

∑i=1

sTi WiΘi− sT

i nisi− ΘTi Γ−1i

˙Θi

=2

∑i=1

[sTi Wi−Γ

−1i

˙Θ

Ti ]Θi− sT

i nisi (20)

which leads us to develop the following parameters’ adaptationlaws:

˙Θi = ΓiW T

i si (21)

Substituting Eq. 21 into Eq. 20 yields,

V =2

∑i=1−sT

i nisi ≤ 0 (22)

Based on Eq. 22, we need to prove V → 0 as t → ∞ revealingsi→ 0 when t→ ∞, or simply:

V → 0⇒ si→ 0

Since V is positive, Barbalat’s lemma [31] confirms that V ap-proaches zero if it is uniformly continuous and its time derivativeV is bounded:

V is bounded⇒ V → 0⇒ si→ 0

We can easily derive V as follows.

V =−22

∑i=1

sTi nisi (23)

Eq. 23 implies,

si and si are bounded⇒ V is bounded⇒ V → 0⇒ si→ 0

Note that V is bounded due to V ≥ 0 and V ≤ 0 indicating thatsi and Θi are also bounded. This would in turn reveals that αi,αi, αdi, αdi, αdi (si = f (αi, αi, αdi,αdi)), and Θi are bounded.Combining Eqs. 12, 15, and 17 gives,

Misi +

[ni +

12

Mi

]si =WiΘi, (i = 1,2) (24)

Note that the bounded αi and αi result in bounded Mi and Miyielding bounded si due to the bounded si, Wi, and Θi. Thebounded si and si result in the bounded V and one can easily con-clude that V and si→ 0 as t → ∞ [31]. This obviously indicatesthat ei and ei tend to zero as t→∞. We hence can guarantee boththe global stability of the coupled network (the boundedness of


Time(s)0 10 20 30 40

0.0542

0.0544

0.0546

0.0548Θ1

Θ18

Time(s)0 10 20 30 40

-0.2228

-0.2226

-0.2224

-0.2222

-0.222Θ2

Θ19

Time(s)0 10 20 30 40

0.228

0.2285

0.229

Θ3

Θ20

Time(s)0 10 20 30 40

×10-3

-10

-5

0

5Θ4

Θ21

(a)

Time(s)0 10 20 30 40

-0.02

-0.015

-0.01

-0.005

0Θ5

Θ22

Time(s)0 10 20 30 40

-0.06

-0.04

-0.02

0

0.02Θ6

Θ23

Time(s)0 10 20 30 40

313.34

313.36

313.38

313.4

313.42

Θ7

Θ24

Time(s)0 10 20 30 40

-1284.52

-1284.5

-1284.48

-1284.46

Θ8

Θ25

(b)

FIGURE 4. The centralized scheme’s parameter estimation for Θ1 toΘ8 of the upstream set and Θ18 to Θ25 of the downstream set.

αi, αi, and Θi) and convergence of the tracking errors (ei).

3.2 Adaptive Decentralized ApproachThe decentralized scheme addressed in [32, 33] is used

in suppressing the chaotic/hyperchaotic dynamics of the twostrongly interconnected sets. We represent the formulations, thedirect decentralized adaptation and control laws, for complete-ness although the detailed stability proof can be easily foundin [34]. A large-scale network containing N interconnected sets,here N = 2 (Eq. 10), can be generally written in the form of:

xi = Fi(x1, ...,xN)+gi(xi)ui (25)yi = hi(x1, ...,xN) (26)

where, xi = [xi,1,xi,2, ...,xi,mi ]T is the state vector of ith set, x =

[xT1 ,x

T2 , ...,x

TN ]

T indicates the full state of the whole network, andFi(.), gi(.), and h(.) are smooth functions. ui and yi stand for thedecentralized input and output of the ith set, respectively. Theoutput dynamics of the ith set, by having strong relative degreeof ni, can be written as follows.

y(ni)i =

l

∑w=1

ζi,w fi,w(xi)+ηiui +∆i(x1, ...,xN),(i = 1,2) (27)

where, y(ni)i indicates the nith time derivative of yi, and ζi,w and

ηi are unknown parameters which will be decentrally estimated.Note that there is no assumption on sign of ηi (ηi 6= 0) and∆i(x1, ...,xN) ≤ εi (εi > 0) stands for the effects of the other in-terconnected sets. As defined for the centralized scheme, thetracking error signal is ei = rdi︸︷︷︸

αdi

− yi︸︷︷︸αi

,(i = 1,2). The aim is

to design a decentralized adaptive scheme for each set such thatthe outputs yi(i = 1,2) track the desirable trajectories (Eqs. 7 &8) with respect to the existing strong nonlinear interconnectionsamong the two sets by using only local measurements.

The error vector of the ith set is written as ei =

[ei, ei, ...,e(ni−1)i ]T leading to the following time derivative:

ei = [ei,e(2)i , ...,e(ni)

i ]T ,(i = 1,2) (28)

We can easily write the error dynamics as follows:

e(ni)i = r(ni)

di − y(ni)i ,(i = 1,2) (29)

Substituting Eq. 27 into Eq. 29 yields,

e(ni)i = r(ni)

di −5

∑w=1

ζi,w fi,w(xi)−ηiui−∆i,(i = 1,2) (30)

where, fi,1 = tanh(Kαi), f1,2 = α1eb1α1.11 , fi,3 = αi, fi,4 =

αi, f1,5 = ed1α1 , f2,2 =α2eb′1α1.12 , f2,5 = ed′1α2 , and the correspond-

ing ζi,w and ηi are estimated through adaptation laws; ζi,1 =0.5AdiµiDsi,ζ1,2 = a1,ζi,3 = ki,ζi,4 = bdi,ζ1,5 = c1,ζ2,2 = a′1,and ζ2,5 = c′1.

The tracking error of the ith set needs to followe(ni)

i + ai,ni−1e(ni−1)i + ...+ ai,0ei = 0 such that Li(s) = s(ni) +

ai,ni−1s(ni−1) + ...+ ai,0 is Hurwitz. The decentralized control


signal of the ith set is calculated as follows:

ui =νi

ηi(31)

where,

νi =(

ai,0ei +ai,1ei +ai,2e(2)i + ...+ai,ni−1e(ni−1)i

)−

5

∑w=1

ζi,w fi,w(xi)+ r(ni)di

+ εisign(eTi PibT

i ), i = (1,2) (32)

Substituting Eq. 31 into Eq. 30, with respect to ηiηi

= 1− ηiηi

Time(s)0 10 20 30 40

1316.18

1316.2

1316.22

1316.24

1316.26

Θ9

Θ26

Time(s)0 10 20 30 40

-5.5

-5

-4.5

Θ10

Θ27

Time(s)0 10 20 30 40

20.5

21

21.5

22

22.5

Θ11

Θ28

Time(s)0 10 20 30 40

-24

-23

-22

-21

-20

Θ12

Θ29

FIGURE 5. The centralized scheme’s parameter estimation for Θ9 toΘ12 of the upstream set and Θ26 to Θ29 of the downstream set.

where ηi = ηi−ηi, the error dynamics becomes,

e(ni)i = r(ni)

di −5

∑w=1

ζi,w fi,w(xi)−νi +ηi

ηiνi−∆i,(i = 1,2)(33)

Combining Eqs. 32 and 33 yields,

e(ni)i =

5

∑w=1

ζi,w fi,w(xi)− εisign(eTi PibT

i )

−(

ai,0ei +ai,1ei +ai,2e(2)i + ...+ai,ni−1e(ni−1)i

)

+ηi

ηiνi−∆i,(i = 1,2) (34)

where ζi,w = ζi,w−ζi,w. The error dynamics of the ith set, by sub-stituting Eq. 34 into Eq. 28, can be developed as the followingmatrix form:

ei = Aiei +bi

(5

∑w=1

ζi,w fi,w(xi)− εisign(eTi PibT

i )+ηi

ηiνi

− ∆i) ,(i = 1,2) (35)

The Hurwitz matrix Ai and vector bi for the two interconnectedsets subject to the chaotic and hyperchaotic dynamics are chosenas follows.

Ai =

0 1− 1︸︷︷︸

ai,0

−7×107︸︷︷︸ai,1

bi = [0 1]T (36)

The Hurwitz Ai leads to a unique positive definite Pi to be ob-tained through the Lyapunov equation:

ATi Pi +PiAi =−Qi (37)

where Qi is a positive definite matrix in which we select as fol-lows (for the two coupled sets):

Qi =

[qi 00 qi

]×103 (38)

where qi=0.1 and qi=0.8 for the chaotic and hyperchaotic re-sponses, respectively. The following decentralized adaptationlaws [32, 33] are used in estimating the unknown parameters:

˙ζ i,w = −γζ i fi,weT

i PibTi (39)

˙ηi = −γη ieT

i PibTi νi

ηi,(i = 1,2) (40)

where γζ i = γη i = 0.1. The decentralized control and adapta-tion laws developed through Eqs. 31, 39, and 40 guarantee [34]asymptotic convergence of the tracking errors to zero and alsoboundedness of the closed-loop network. We have implementedthe decentralized formulations in MATLAB to be compared withthe centralized scheme with respect to the computational cost,


feasibility, and robustness issues.

4 Results

T ime(s)0 10 20 30 40

×10-3

-1

0

1

2

3

ζ1,1ζ

2,1

(a)

T ime(s)0 10 20 30 40

×10-5

-3

-2

-1

0

ζ1,2ζ

2,2

(b)

T ime(s)0 10 20 30 40

-0.1

-0.05

0

0.05

ζ1,3ζ

2,3

(c)

T ime(s)0 10 20 30 40

×10-3

-1

0

1

2

3

ζ1,4ζ

2,4

(d)

T ime(s)0 10 20 30 40

-0.01

0

0.01

0.02

0.03

0.04

ζ1,5ζ

2,5

(e)

T ime(s)0 10 20 30 40

10

15

20

25η

1η

2

(f)

FIGURE 6. The decentralized scheme’s parameter estimation for ζi,w

and ηi of both the upstream and downstream sets.

The values of ni used in the coupled centralized control in-puts (Eq. 17) are listed in Table 1 and Γi = diag[10]17×17. Thevalues of εi utilized in developing the decentralized control in-puts (Eq. 31) are given in Table 1 and selected to mitigatethe effects of strong nonlinear interconnections among the setswhich is inevitably a fundamental requirement of the decentral-ized scheme. For the centralized method, Figs. 4 and 5 presentthe estimation process of the unknown parameters (Θ1–Θ29) forboth the upstream and downstream sets subject to the Initial1 re-vealing the parameters convergence within the nominal operationtime of 40(s); the Initial1 yielded the coupled chaotic dynam-ics. The Θ1–Θ12 and Θ18–Θ29 indicate the sample parametersof the upstream and downstream sets, respectively. Note that theinitial values of the parameters used in the centralized adapta-tion laws are 90% of their nominal values. Such initial values

0 10 20 30 40−40

−20

0

20

T ime(s)

ii(A

)

Upstream Actuator

Downstream Actuator

(a)

T ime(s)0 10 20 30 40

i i(A

)

-40

-20

0

20

40Upstream ActuatorDownstream Actuator

(b)

0 10 20 30 40−100

−50

0

50

100

T ime(s)

Tmag(N

.m)

Upstream Actuator

Downstream Actuator

(c)

T ime(s)0 10 20 30 40

Tmag(N

.m)

-100

0

100

200Upstream ActuatorDownstream Actuator

(d)

FIGURE 7. (a) The centralized control inputs; (b) The decentralizedcontrol inputs; (c) The centralized magnetic torques; (d) The decentral-ized magnetic torques.

are intentionally selected to yield meaningful estimated parame-ters with respect to the electro-magneto-mechanical-fluid natureof the network. It is of a great interest to observe that, despitethe dominant chaotic dynamics resulted from the Initial1, the pa-rameters timely convergence and therefore, we expect to observestable operations of both the coupled actuated valves. Note thatboth the centralized and decentralized approaches, based on the“sufficient richness” condition [31], would not exactly estimatethe unknown parameters such that the schemes expectedly yieldvalues to allow the desired task to be carried out.

Figs. 6(a) to 6(f) present the decentralized scheme’s estima-tion process for ζi,w and ηi of both the upstream and downstreamsets. Note that some researchers have reported inefficiency [35]and lack of robustness of the decentralized schemes in the pres-ence of strong nonlinear interconnections which are expected tobe observed for the network subject to the chaotic/hyperchaoticdynamics. The estimated parameters validate the lack of robust-ness of the decentralized scheme in comparison with the cen-tralized one. Few abrupt converging process of the parameterswould potentially magnify the shortcomings of the decentralizedscheme due to the significant effects of strong nonlinear inter-connections between two sets. We hence expect to observe un-desirable control inputs.

For the centralized scheme, the estimated parameters, basedon Eq. 17, would help generate powerful control inputs, the ap-plied currents of the bi-directional solenoid actuators (ii), to van-ish the chaotic dynamics of the interconnected sets (Fig. 2(a))and then drive the coupled valves to track the desirable trajecto-ries, which we addressed earlier (Eq. 7). Shown in Fig. 7(a)are the control inputs for both the upstream and downstream


actuators. As expected, the currents consist of two phases asshown in Fig. 7(a). The first phase, with oscillatory negativevalues of the currents, suppresses the coupled chaotic dynamicsdue to the Initial1 resulting in downward/slightly upward mo-tions of the plungers, which would consequently avoid the sud-den jumps of the valves. During the second phase of the controlprocess, the currents gradually take positive values indicatingthat the plungers move upward and therefore, the valves rotatetoward the desirable trajectories. It is of a great interest to ob-serve that the control input of the downstream set is remarkablyhigher than that of the upstream one, in particular for the firstphase of the control process. The physical interpretation of suchhigher values of the control input used in the downstream set canbe explained through the effects of the flow loads acting on thevalve, in particular the hydrodynamic torque:

Th2

Th1∝

(Dv2

Dv1

)3

×(

cv1

cv2

)2

(41)

Tb2

Tb1∝

(Dv2cv1

Dv1cv2

)2

(42)

where cv1 and cv2 are the upstream and downstream valves’ co-efficients, respectively (cvi(αi) = piα

3i + qiα

2i + oiαi + γi). We

have previously reported [1–3, 9, 10, 12] that a smaller pipe di-ameter yields both the higher hydrodynamic and bearing torquesdue to the higher coefficient of the upstream valve than that ofthe downstream one (Eqs. 41 and 42). From another aspect, thehydrodynamic torque is a helping load [1–6, 8–13] to close thesymmetric valve whereas the bearing one is a resistance (friction-based) torque for the valve’s operation. The downstream setwith a smaller pipe diameter, subject to the chaotic dynamics ofthe Initial1, undoubtedly needs more suppressing control inputto mitigate the destabilizer effects of the higher hydrodynamictorque acting on the valve. For the second phase of the con-trol process, the higher resistance bearing torque acting on thedownstream set inevitably demands slightly higher control inputto push the valve to the desirable trajectory.

Such profiles of the control inputs of the centralized methodfor both the sets are expected to be observed for the drivingmagnetic torques (forces) as nonlinear functions of the controlinputs in addition to the valves’ rotation angles/plungers’ dis-placements. Fig. 7(c) presents the centralized driving magnetictorques of both the coupled sets in which the two phases of thecontrol process can be distinguished as we discussed for the cur-rents. The oscillatory negative values of the magnetic torquessuppress the chaotic dynamics along with mitigating the effectsof the hydrodynamic torques. The positive magnetic torques(forces) move the plungers upward and subsequently, the valvesmove toward the desirable trajectories. The higher amount ofthe driving magnetic torque of the downstream set, for the sec-

0 10 20 30 4020

40

60

80

Time(s)

αi(degree)

α1

α2

Desired Trajectory

(a)

0 10 20 30 4020

40

60

80

Time(s)

αi(degree)

α1

α2

Desired Trajectory

(b)

0 10 20 30 40−0.03

−0.02

−0.01

0

0.01

0.02

Time(s)

ei(rad)

Upstream ValveDownstream Valve

(c)

T ime(s)0 10 20 30 40

ei(rad)

×10-3

-2

0

2

4Upstream ValveDownstream Valve

(d)

100−10

−5

0

5

10

Time(s)si(t)


(e)

FIGURE 8. (a) The centralized valves’ rotation angles; (b) The de-centralized valves’ rotation angles; (c) The centralized error signals; (d)The decentralized error signals; (e) The centralized combined trackingerror signals.

ond phase of the control process, looks logical to overcome thehigher resistance bearing torque than that of the upstream one.

Despite the centralized control inputs/magnetic torques, thedecentralized scheme leads to the so-called “Bang-Bang” con-trol inputs and magnetic torques shown in Figs. 7(b) and 7(d),respectively. Obviously, the implementation of the Bang-Bangcontrol inputs for the network subject to the chaotic dynamicscan be questionable and hence would probably not be feasible.As expected, such control inputs can result in failures of the ac-tuation units and gradually damage the whole network. Notethat the decentralized scheme yields slightly higher values of theBang-Bang control inputs and magnetic torques than those of thecentralized ones.

Shown in Fig. 8(a) are the upstream and downstream valves’rotation angles subject to the centralized scheme indicating thatboth the sets track the desirable trajectories (Eq. 7) by applyingthe control inputs which expectedly vanish the coupled chaoticdynamics due to the Initial1. Both the error (ei) and combined


10−2

100−200

−150

−100

−50

0

50

Time(s)

i i(A

)

101.596

101.602

−150

−100

−50

0

50

Upstream ActuatorDownstream Actuator

(a)

T ime(s)0 10 20 30 40

i i(A

)

-150

-100

-50

0

50


(b)

10−2

100−1500

−1000

−500

0

500

Time(s)

Tmag(N

.m)


39.5 40−1000

−500

0

500

(c)

T ime(s)0 10 20 30 40

Tmag(N

.m)

-800

-600

-400

-200

0

200


(d)

FIGURE 9. (a) The centralized control inputs; (b) The decentralizedcontrol inputs; (c) The centralized magnetic torques; (d) The decentral-ized magnetic torques.

tracking error (si) signals shown in Figs. 8(c) and 8(e), respec-tively, converge to zero revealing that the valves’ angles (αi) tendto the desirable trajectories (αdi) within the nominal operationtime.

It is of a great interest to observe that the decentralized ap-proach also pushes the valves to track the desirable trajectories(Figs. 8(b) and 8(d)) with even considerably smaller tracking er-rors. The tracking errors of the decentralized method are signif-icantly smaller than those of the centralized one. Consequently,we deal with an important trade-off between the smaller trackingerrors and Bang-Bang control inputs of the decentralized methodin the presence of strong nonlinear interconnections and chaoticdynamics. This trade-off can be expected to become more crucialby adding more coupled sets. Another trade-off is computationalcost of the centralized and decentralized schemes. Note that thecomputational time of the decentralized method, for only twosets, is almost one-sixtieth of the centralized one. It is straightfor-ward to conclude that the computational cost of the decentralizedscheme would be significantly lower than that of the centralizedone by adding more interconnected sets subject to the chaoticdynamics. Therefore, the centralized adaptive method, by yield-ing smoother/practically feasible control inputs/magnetic torquesand better robustness, is an effective approach in comparisonwith the decentralized one (even with smaller tracking error andcomputation time) to control the network of actuated butterflyvalves subject to the chaotic dynamics.

Figs. 9(a) and 9(c) present the centralized control inputs anddriving magnetic torques, respectively, used in vanishing the cou-

0 10 20 30 400

20

40

60

80

T ime(s)

αi(degree)

α1

α2

Desired Trajectory

(a)

T ime(s)0 10 20 30 40

αi(degree)

0

20

40

60

80

α1

α2

Desired Trajecctory

(b)

10−2

100

−0.01

0

0.01

0.02

T ime(s)

ei(rad)

101.599

101.602

−5

0

5

10

15

20x 10

−3


(c)

T ime(s)0 10 20 30 40

ei(rad)

-0.8

-0.6

-0.4

-0.2

0


(d)

10−3

10−2

10−1

100

101

−15

−10

−5

0

5

10

15

T ime(s)

si(t)


(e)

FIGURE 10. (a) The centralized valves’ rotation angles; (b) The de-centralized valves’ rotation angles; (c) The centralized error signals; (d)The decentralized error signals; (e) The centralized combined trackingerror signals.

pled hyperchaotic dynamics caused by the Initial2 (Fig. 2(b)).As expected, the hyperchaotic dynamics of both the sets withthe larger domains of attractions, which we have thoroughly ad-dressed in [1, 3], would require significantly higher values of thecontrol inputs to be vanished than those of the chaotic ones.

The considerable control inputs expectedly result in thehigher driving magnetic torques than the ones used in the net-work subject to the chaotic dynamics (Fig. 9(c)). The two phasesof the control process, which we discussed for the chaotic case,can be observed for the hyperchaotic one such that the oscilla-tory negative control inputs/driving magnetic torques suppressthe hyperchaotic dynamics. Note that the green boxes shown inFigs. 9(a) and 9(c) reveal the incremental values of the controlinputs/torques to rotate the valves to the desirable trajectories.

The Bang-Bang control inputs/magnetic torques, which wediscussed for the chaotic response, are expected to be observedfor the network subject to the hyperchaotic dynamics. Shownin Figs. 9(b) and 9(d) expectedly reveal higher Bang-Bang con-


trol inputs/magnetic torques for the hyperchaotic responses thanthose of the chaotic ones; again due to the larger domains of at-tractions of both the sets.

Shown in Fig. 10(a) are both the upstream and downstreamvalves’ rotation angles revealing that the valves’ motions tend tothe desirable trajectories (Eq. 8) subject to the centralized con-troller. Figs. 10(c) and 10(e) present the convergence of both theerror and combined tracking error signals to zero, respectively.Consequently, it is straightforward to conclude that the central-ized adaptation and control laws guarantee both the global sta-bility of the coupled network and convergence of the tracking er-rors in which we analytically discussed in Section 3. Figs. 10(b)and 10(d) reveal that both the valves, by utilizing the decentral-ized controller, experience considerably higher tracking errorsand, again for the hyperchaotic responses, the implementation ofBang-Bang control inputs can be potentially harmful.

5 Conclusions and Future WorkIn this effort, we presented the novel sixth-order inter-

connected dynamic equations of the two coupled bi-directionalsolenoid actuated butterfly valves. The network underwent theharmful chaotic and hyperchaotic dynamics subject to some ini-tial conditions and critical parameters of the equivalent viscousdamping and the friction coefficient of bearing area. We thenthoroughly studied both the centralized and decentralized adap-tive schemes for different aspects of robustness, computationalcost, and practically feasible control inputs. The issues of ro-bustness and practical feasibility of control efforts expectedlybecome more critical in the presence of strong nonlinear inter-connections among sets and harmful dynamical responses.

It was shown that the decentralized adaptive scheme yieldssignificantly smaller tracking errors in comparison with the cen-tralized one (for the chaotic dynamics) although the latter showsa much better performance by revealing more robust estimationprocess and feasible control inputs. The decentralized scheme,in the presence of chaotic/hyperchaotic dynamics, has the short-comings of abrupt estimation process and, more importantly,Bang-Bang control inputs which would expectedly result in thefailures of the coupled actuation units and gradually the wholenetwork. Another trade-off is the computation cost of both theschemes. The computation time of the decentralized scheme, forthis particular problem subject to the chaotic/hyperchaotic dy-namics, is at least one-sixtieth of the centralized one and weexpect to spend much lower computation time by adding morecoupled agents. In summary, the decentralized controller needsto be utilized carefully, in particular for the systems subject tothe interconnected chaotic and hyperchaotic dynamics.

We currently focus on developing the interconnected modelof a network of n actuated valves operating in series.

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