Suppressing Chaotic and Hyperchaotic Dynamics of Smart Valves NetworkUsing A Centralized Adaptive Approach

Peiman Naseradinmousavi1, Hashem Ashrafiuon2, and Mostafa Bagheri3

Abstract— Catastrophic chaotic and hyperchaotic dynamicalbehavior have been experimentally observed in the so-called“Smart Valves” network, given certain critical parameters andinitial conditions. The centralized network-based control ofthese coupled systems may effectively mitigate the harmfuldynamics of the valve-actuator configuration which can bepotentially caused by a remote set and would gradually affectthe whole network. In this work, we address the centralizedcontrol of two bi-directional solenoid actuated butterfly valvesdynamically coupled in series subject to the chaotic andhyperchaotic dynamics. An interconnected adaptive schemeis developed and examined to vanish both the chaotic andhyperchaotic dynamics and return the coupled network to itssafe domain of operation.

I. INTRODUCTION

Dangerous dynamical behaviors of multidisciplinary sys-tems, in particular the electromechanical ones, need to becontrolled in order to avoid the expected failure of the large-scale network. The so-called “Smart Valves” network, con-taining many interconnected electro-magneto-mechanical-fluid components, plays an important role in proper andefficient performance of many critical infrastructures whichinclude, but are not limited to, municipal piping systems,oil and gas fields, petrochemical plants, and the US Navy.A robust control scheme is hence required to mitigate theeffects of the harmful dynamic responses in the presence ofuncertainties involved with such a large-scale network.

We have reported broad analytical and experimental stud-ies [1]–[15] for both an isolated actuator-valve arrangementand a network of two interconnected solenoid actuatedbutterfly valves operating in series. Finding specific researchwork to capture and control the chaotic and hyperchaoticdynamics of such a multiphysics network is somewhat dif-ficult although some efforts have been reported for similarcase studies. Wan and Jian [16] have studied gear dynamicswith turbulent journal bearings mounted hybrid squeeze filmdamper. In order to avoid the nonsynchronous chaotic vibra-tions, they utilized an increased proportional gain Kp = 0.1to control this system. It was shown that the pinion trajectorywill leave chaotic motion to periodic motion in the steadystate under control action. Morel et al. [17] proposed a new

1Peiman Naseradinmousavi is with the Department of MechanicalEngineering, San Diego State University, San Diego, CA 92115, USAhttp://peimannm.sdsu.edu

2Hashem Ashrafiuon is with the Department of MechanicalEngineering, Villanova University, Villanova, PA 19085, USAhttp:/homepage.villanova.edu/hashem.ashrafiuon/

3Mostafa Bagheri is with the Department of Mechanical and AerospaceEngineering, San Diego State University and University of California SanDiego, USA. http://peimannm.sdsu.edu/members.html

nonlinear feedback, which induces chaos and which wasable at the same time to achieve a low spectral emissionand to maintain a small ripple in the output. The design ofthis new and simple controller was based on the proprietythat chaotified nonlinear systems present many independentchaotic attractors of small dimensions. Chen and Liu [18] in-vestigated chaos control of fractional-order energy demand-supply system by two different control strategies: a linearfeedback control and an adaptive switching control strategyvia a single control input. Some other efforts related to thechaos control can be found in [19]–[22].We here brieflyrepresent the interconnected analytical model of two sets (forcompleteness) along with the critical initial conditions andparameters resulted in the dangerous responses. The coupledadaptation and control laws will be formulated with respectto the interconnected dynamics of the system. The resultswill be thoroughly discussed to address the robustness ofadaptive scheme for vanishing the chaotic and hyperchaoticdynamics.

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)

Fig. 1. (a) A schematic configuration of two bi-directional solenoid actuatedbutterfly valves subject to the sudden contraction; (b) A coupled model oftwo butterfly valves in series without actuation

II. MATHEMATICAL MODELING

The system, which is being considered here, is two bi-directional solenoid actuated butterfly valves operating inseries. The system undergoes a sudden pipe contractionas shown in Fig 1. The plungers are connected to thevalves’ stems through the rack and pinion arrangementsyielding kinematic constraints. We have previously derivedthe interconnected analytical model of two sets operating inseries [2]–[9] and briefly represent here for completeness.As can be observed in Fig. 1(b), the valves are modeled aschanging resistors:

Rni(αi) =ei

(piα3i + qiα2

i + oiαi + γi)2, (i = 1, 2) (1)

where, Rn1 and Rn2 indicate the resistances of the upstreamand downstream valves, respectively, and 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 =

2018 Annual American Control Conference (ACC)June 27–29, 2018. Wisconsin Center, Milwaukee, USA

978-1-5386-5428-6/$31.00 ©2018 AACC 1671

−695.1, and γ2 = 807.57 for two different valves’ diameters.Also the flow between the valves in addition to the suddencontraction are modeled as constant resistors based on theHagen-Poiseuille and Borda-Carnot formulas:

RLi =128µfLiπD4

vi

, (i = 1, 2) (2)

Rcon =8Kcon

π2D4v2

(3)

where, Kcon = 0.5(1−β2)

√sin(θ2

), β indicates the ratio of

minor and major diameters(Dv2Dv1

), θ is the angle of approach

(the pipe contraction angle), µf stands for the fluid dynamicviscosity, Dv1 and Dv2 are the upstream and downstreamvalves’ diameters, respectively, L1 and L2 indicate the pipelengths before and after contraction, and RL1 and RL2 arethe constant resistances. Therefore, two valves operating inseries can be modeled as a set of five resistors leading us toderive mathematical expressions of the pressures after andbefore the upstream and downstream valves, respectively, asfollows.

P1 =Rn2Pin +Rn1Pout +Rn1(RL1 +RL2 +Rconqv)qv

(Rn1 +Rn2)(4)

P2 =Rn2Pin +Rn1Pout −Rn2(RL1 +RL2 +Rconqv)qv

(Rn1 +Rn2)(5)

where qv is the volumetric flow rate. These interconnectedpressures were used in developing both the coupled hydro-dynamic and bearing torques [2], [3]:

Th1 = (a1α1eb1α1

1.1

− c1ed1α1)(Pin − P1)

= (a1α1eb1α1

1.1

− c1ed1α1) ×

e1(p1α

31+q1α

21+o1α1+γ1)2∑2

i=1ei

(piα3i+qiα

2i+oiαi+γi)2

× (Pin − Pout − (RL1 +RL2 +Rconqv)qv) (6)

Th2 = (a′1α2eb′1α

1.12 − c′1e

d′1α2)(P2 − Pout)

= (a′1α2eb′1α2

1.1

− c′1ed′1α2) ×

e2(p2α

32+q2α

22+o2α2+γ2)2∑2

i=1ei

(piα3i+qiα

2i+oiαi+γi)2

× (Pin − Pout − (RL1 +RL2 +Rconqv)qv) (7)Tb1 =C1∆P1(Rn1, Rn2, RL1, RL2, Rcon) (8)Tb2 =C2∆P2(Rn1, Rn2, RL1, RL2, Rcon) (9)

where, a1 = 0.4249, a′1 = 0.1022, b1 = −18.52, b′1 =−17.0795, c1 = −7.823 × 10−4, c′1 = −2 × 10−4,d1 = −1.084, d′1 = −1.0973, C1 = C2 = 0.5AdµDs,∆P1 = Pin − P1, ∆P2 = P2 − Pout, and Pin and Poutare the given inlet and outlet pressures, respectively. Notethat Ds is the stem diameter of the valve and µ standsfor the friction coefficient of the bearing area. We havepreviously established that the hydrodynamic torque acts as ahelping load pushing the valve to be closed and is typicallyeffective for when the valve angle is lower than 60◦ [3],[5]; the effective range was experimentally examined [5]confirming the helping behavior of the hydrodynamic torqueby presenting positive values. The bearing torque, due toits friction-based nature, always acts as a resisting load.Based on the analytical formulas addressed above, the sixth-order interconnected dynamic equations of two bi-directional

(a) (b)

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

solenoid actuated butterfly valves were developed as follows:

z1 = z2 (10)

z2 =1

J1

[r1C21N

21 z

23

2(C11 + C21(gm1 − r1z1))2− bd1z2 − k1z1

+

(Pin−Pout−(RL1+RL2+Rconqv)qv)e1(p1z31+q1z

21+o1z1+γ1)

2∑i=1,4

ei(piz3i+qiz

2i+oizi+γi)

2

×

[(a1z1e

b1z11.1

− c1ed1z1)− C1× tanh(Kz2)

]](11)

z3 =(V1 −R1z3)(C11 + C21(gm1 − r1z1))

N21

−

r1C21z3z2(C11 + C21(gm1 − r1z1))

(12)

z4 = z5 (13)

z5 =1

J2

[r2C22N

22 z

26

2(C12 + C22(gm2 − r2z4))2− bd2z5 − k2z4

+

(Pin−Pout−(RL1+RL2+Rconqv)qv)e2(p2z34+q2z

24+o2z4+γ2)

2∑i=1,4

ei(piz3i+qiz

2i+oizi+γi)

2

×

[(a′1z4e

b′1z41.1

− c′1ed′1z4)− C2× tanh(Kz5)

]](14)

z6 =(V2 −R2z6)(C12 + C22(gm2 − r2z4))

N22

−

r2C22z5z6(C12 + C22(gm2 − r2z4))

(15)

where, bdi indicates the equivalent torsional damping, kiis the equivalent torsional stiffness, Vi stands for the supplyvoltage, ri indicates the radius of the pinion, C1 and C2 arethe reluctances of the magnetic path without air gap and thatof the air gap, respectively, Ni stands for the number of coils,gmi is the nominal air gap, Ji indicates the polar moment ofinertia of the valve’s disk, and Ri is the electrical resistanceof coil. z1 = α1, z2 = α1, and z3 = i1 indicate the upstreamvalve’s rotation angle, angular velocity, and actuator current,respectively. z4 = α2, z5 = α2, and z6 = i2 stand forthe downstream valve’s rotation angle, angular velocity, andactuator current, respectively. The network parameters arelisted in Table I.

Note that we could capture, for the first time, the cou-pled chaotic and hyperchaotic dynamics of the intercon-nected sets [2] by examining the critical values of bdi =

1672

TABLE ITHE SYSTEM PARAMETERS

ρ 1000 kgm3 v 3ms

J1,2 0.104× 10−1(kg.m2) N2 3300N1 3300 C11,22 1.56× 106

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

L1 2(m) L2 1(m)r1,2 0.05(m) θ 90◦

Pin 256(kPa) gm1,m2 0.1(m)µf 0.018 (Kg.m−1.s−1) λ1,2 1n1,2 10 bdi = µi 1× 10−7

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)

Fig. 3. (a) The Lyapunov exponents for Initial1; (b) The positive Lyapunovexponents for Initial2 vs. different approach angles (θ)

µi = 1 × 10−7 for two different initial conditions ofInitial1 = [20(deg) 0 0 20(deg) 0 0] and Initial2 =[2(deg) 0 0 2(deg) 0 0], respectively. Shown in Figs. 2(a)and 2(b) are the chaotic and hyperchaotic dynamics of thecoupled actuated valves, respectively; the red and blue linesindicate the angular velocities vs. rotation angles of theupstream and downstream valves, respectively. Some power-ful tools of the nonlinear analysis, including the Lyapunovexponents and Poincaré map, were used in distinguishingamong the nature of harmful responses, as shown in Figs.3(a) & 3(b). One and two positive Lyapunov exponents alongwith irregular Poincaré maps confirmed [2] the chaotic andhyperchaotic dynamics of the actuated valves, respectively.Such dangerous responses need to be vanished using anonlinear control scheme, due to the nonlinear and couplednature of the network, in order to return the interconnectedsets to their stable domains. The operationally optimizedvalves’ motions, on the other hand, are utilized in thenonlinear control scheme as desirable trajectories. Basedon the inevitable unknown parameters of such a couplednetwork, the nonlinear model-based adaptive scheme looksas an effective approach to be employed in stabilizing thesystem subject to the chaotic and hyperchaotic dynamics.

III. CONTROL AND ADAPTATION LAWSThe nonlinear model-based adaptive control method [23],

[24] is used in stabilizing the unstable system in order totrack the desired trajectories [2], [4] defined based on thecritical initial conditions as follows:

αdi =π

3tanh(10−4t3) +

π

9, Initial1 (16)

αdi =π

3tanh(10−4t3) +

π

90, Initial2 (17)

The so-called “S-Shaped” trajectories are highly energy-efficient [4] and yield smooth dynamic responses avoidingthe repeatedly observed dangerous phenomenon of “WaterHammering”. The coupled dynamic Eqs. 11 and 14 can berewritten as the following:

Jiαi + bdiαi + kiαi =riC2iN

2i i

2i

2(C1i + C2i(gmi − riαi))2

+A1Rni∑2i=1Rni

[T ′hi − T ′bi tanh(Kαi)], (i = 1, 2) (18)

where, A1 = (Pin − Pout − (RL1 + RL2 + Rconqv)qv),T ′h1 = a1α1e

b1α11.1−c1ed1α1 , T ′h2 = a′1α2e

b′1α21.1−c′1ed

′1α2 ,

T ′b1 = C1, and T ′b2 = C2. Assuming,

Mi =2Ji(Ci1 + C2i(gmi − riαi))

2

riC2iN2i

, Bi =2bdi(Ci1 + C2i(gmi − riαi))

2

riC2iN2i

Ki =2ki(Ci1 + C2i(gmi − riαi))

2

riC2iN2i

, Coi =2(Ci1 + C2i(gmi − riαi))

2

riC2iN2i

Eq. 18 can be rewritten as follows.

Miαi +Biαi +Kiαi = ui

+A1CoiRni∑2

i=1Rni[T ′hi − T ′bi tanh(Kαi)], (i = 1, 2) (19)

We define the valves’ tracking errors and their first andsecond time 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∑2

i=1Rni[T ′hi − T ′bi tanh(Kαi)], (i = 1, 2)(20)

The combined tracking errors [23], [24] and their first timederivatives are as follows:

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

where λ’s are strictly positive numbers listed in Table I.Premultiplying by Mi and substituting from Eq. 20, we have,

Misi = Miαdi +Biαi +Kiαi

− ui −A1CoiRni∑2

i=1 Rni[T ′hi − T ′bi tanh(Kαi)] +Miλiei (21)

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

V =1

2

2∑i=1

(sTi Misi + ΘTi Γ−1i Θi)

(22)

where Γi is a symmetric positive definite matrix and Θi is thesystem’s lumped parameter estimation error (Θi = Θi−Θi).Differentiating the Lyapunov function (Eq. 22) yields,

V =

2∑i=1

(sTi Misi +1

2sTi Misi − ΘT

i Γ−1i

˙Θi)

=

2∑i=1

sTi[Miαdi +Biαi +Kiαi − ui −

A1CoiRni∑Rni

1673

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)

Fig. 4. The parameter estimation for Θ1 to Θ8 of the upstream set andΘ18 to Θ25 of the downstream set.

× [T ′hi − T ′bi tanh(Kαi)] +Miλiei +1

2Misi

]− ΘT

i Γ−1i

˙Θi

](23)

By defining the regression vectors as the following:

WiΘi = Miαdi +Biαi +Kiαi −A1CoiRni∑

Rni

× [T ′hi − T ′bi tanh(Kαi)] +Miλiei +1

2Misi, (i = 1, 2) (24)

The V can be easily rewritten as follows.

V =

2∑i=1

[sTi [WiΘi − ui]− ΘT

i Γ−1i˙Θi

](25)

The appropriate control inputs are hence chosen as thefollowing:

ui = WiΘi + nisi, (i = 1, 2) (26)

where,

WiΘi = Miαdi + Biαi + Kiαi −A1CoiRni∑

Rni

× [T ′hi − T ′bi tanh(Kαi)] + Miλiei +1

2ˆiMsi, (i = 1, 2)(27)

We can easily develop both the regression (Wi ∈ <1×17)and estimated lumped parameter vectors (Θi ∈ <17×1).Substituting the control inputs into Eq. 25 gives,

V =

2∑i=1

sTi [WiΘi −WiΘi︸ ︷︷ ︸WiΘi

−nisi] − ΘTi Γ−1

i˙Θi

=

2∑i=1

sTi WiΘi − sTi nisi − ΘTi Γ−1

i˙Θi

=

2∑i=1

[sTi Wi − Γ−1i

˙ΘTi ]Θi − sTi nisi (28)

which leads us to develop the following parameters’ adapta-tion laws:

˙Θi = ΓiW

Ti si (29)

Substituting Eq. 29 into Eq. 28 yields,

V =

2∑i=1

−sTi nisi ≤ 0 (30)

Based on Eq. 30, by Lasalle-Yoshizawa theorem, we getthat si → 0 as t → ∞. This implies by the input-to-statestability property of the system ei+λiei = si, that ei and ei

0 5 10 15 20 25 30 35 40Time(s)

-40

-30

-20

-10

0

10

20

i i(A

)

Upstream ActuatorDownstream Actuator

(a)

0 5 10 15 20 25 30 35 40−60

−40

−20

0

20

40

60

80

T ime(s)

Tmag(N.m

)

Upstream ActuatorDownstream Actuator

(b)

Fig. 5. (a) The control inputs; (b) The magnetic torques

tend to zero as t→∞. We hence guarantee both the globalstability of the coupled network (the boundedness of αi, αi,and Θi) and convergence of the tracking errors (ei).

IV. RESULTS

The values of ni used in the coupled control inputs (Eq.26) are listed in Table I and Γi = diag[10]17×17. Fig.4 presents the sample estimation process of the unknownparameters (Θ1–Θ8) & (Θ18–Θ25) for both the upstreamand downstream sets subject to the Initial1 revealing theparameters convergence within the nominal operation time of40(s); the Initial1 yielded the coupled chaotic dynamics. Notethat the initial values of Θ1–Θ34 used in the adaptation lawsare 90% of their nominal values. Such initial values are in-tentionally selected to yield meaningful estimated parameterswith 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 parameters timely converge and therefore, we expect toobserve stable operations of both the coupled actuated valves.Note that this approach, based on the “sufficient richness”condition [23], [24], would not exactly estimate the unknownparameters such that it expectedly yields values to allow thedesired task to be carried out.

The estimated parameters, based on Eq. 26, would helpgenerate powerful control inputs, the applied currents of thebi-directional solenoid actuators (ii), to vanish the chaoticdynamics of the interconnected sets (Fig. 2(a)) and thendrive the coupled valves to track the desirable trajectories,which we addressed earlier (Eq. 16). Shown in Fig. 5(a) arethe control inputs for both the upstream and downstreamactuators. As expected, the currents consist of two phases asshown in Fig. 5(a). The first phase, with oscillatory negativevalues of the currents, suppresses the coupled chaotic dynam-ics due to the Initial1 resulting in downward/slightly upwardmotions of the plungers, which would consequently avoid thesudden jumps of the valves. During the second phase of thecontrol process, the currents gradually take positive valuesindicating that the plungers move upward and therefore, thevalves rotate toward the desirable trajectories.

It is of a great interest to observe that the control inputof the downstream set is remarkably higher than that of theupstream one, in particular for the first phase of the controlprocess. The physical interpretation of such higher values ofthe control input used in the downstream set can be explainedthrough the effects of the flow loads acting on the valve, in

1674

0 5 10 15 20 25 30 35 4020

30

40

50

60

70

80

T ime(s)

αi(degree)

α1

α2

Desired Trajectory

(a)

0 5 10 15 20 25 30 35 40−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

T ime(s)

ei(rad)

Upstream ValveDownstream Valve

(b)

10−1

100

101

−8

−6

−4

−2

0

2

4

6

T ime(s)

s i(t)

Upstream ValveDownstream Valve

(c)

Fig. 6. (a) The valves’ rotation angles; (b) The error signals; (c) Thecombined tracking error signals

particular the hydrodynamic torque:

Th2Th1

∝(Dv2

Dv1

)3

×(cv1cv2

)2

(31)

Tb2Tb1

∝(Dv2cv1Dv1cv2

)2

(32)

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

3i +qiα

2i +oiαi+γi);

we have provided the values of pi, qi, oi, and γi in Section2. We have previously reported [2]–[9] that a smaller pipediameter yields both the higher hydrodynamic and bearingtorques due to the higher coefficient of the upstream valvethan that of the downstream one (Eqs. 31 and 32). Fromanother aspect, the hydrodynamic torque is a helping load[2]–[9] to close the symmetric valve whereas the bearingone is a resistance (friction-based) torque for the valve’soperation. The downstream set with a smaller pipe diameter,subject to the chaotic dynamics of the Initial1, undoubtedlyneeds more suppressing control input to mitigate the desta-bilizer effects of the higher hydrodynamic torque acting onthe valve. For the second phase of the control process, thehigher resistance bearing torque acting on the downstreamset inevitably demands slightly higher control input to pushthe valve to the desirable trajectory.

Such profiles of the control inputs for both the sets areexpected to be observed for the driving magnetic torques(forces) as nonlinear functions of the control inputs in addi-tion to the valves’ rotation angles/plungers’ displacements.Fig. 5(b) presents the driving magnetic torques of both thecoupled sets in which the two phases of the control processcan be distinguished as we discussed for the currents. Theoscillatory negative values of the magnetic torques suppressthe chaotic dynamics along with mitigating the effects ofthe hydrodynamic torques. The positive magnetic torques(forces) move the plungers upward and subsequently, thevalves move toward the desirable trajectories. The higheramount of the driving magnetic torque of the downstreamset, for the second phase of the control process, looks logicalto overcome the higher resistance bearing torque than thatof the upstream one.

Time(s)10-3 10-2 10-1 100 101

i i(A

)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

39.5 40-150

-100

-50

0

50

Upstream ActuatorDownstream Actuator

(a)

10−3

10−2

10−1

100

101

−1200

−1000

−800

−600

−400

−200

0

200

T ime(s)

Tmag(N.m

)

Upstream ActuatorDownstream Actuator

39.5 40−1000

−500

0

500

(b)

Fig. 7. (a) The control inputs; (b) The magnetic torques

Shown in Fig. 6(a) are the upstream and downstreamvalves’ rotation angles indicating that both the sets track thedesirable trajectories (Eq. 16) by applying the control inputswhich expectedly vanish the coupled chaotic dynamics dueto the Initial1. Both the error (ei) and combined trackingerror (si) signals shown in Figs. 6(b) and 6(c), respectively,converge to zero revealing that the valves’ angles (αi) tend tothe desirable trajectories (αdi) within the nominal operationtime.

Time(s)0 5 10 15 20 25 30 35 40

αi(degree)

0

10

20

30

40

50

60

70α

1

α2

(a)

10−3

10−2

10−1

100

101

−0.01

−0.005

0

0.005

0.01

0.015

0.02

T ime(s)

e i(rad)

101.599

101.602

−5

0

5

10

15

20x 10

−3

Upstream ValveDownstream Valve

(b)

10−3

10−2

10−1

100

101

−15

−10

−5

0

5

10

15

T ime(s)

s i(t)

Upstream ValveDownstream Valve

(c)

Fig. 8. (a) The valves’ rotation angles; (b) The error signals; (c) Thecombined tracking error signals

Figs. 7(a) and 7(b) present the control inputs and drivingmagnetic torques, respectively, used in vanishing the coupledhyperchaotic dynamics caused by the Initial2 (Fig. 2(b)). Asexpected, the hyperchaotic dynamics of both the sets with thelarger domains of attractions, which we have thoroughly ad-dressed in [2], [4], would require significantly higher valuesof the control inputs to be vanished than those of the chaoticones. The considerable control inputs expectedly result inthe higher driving magnetic torques than the ones used inthe network subject to the chaotic dynamics (Fig. 7(b)).The two phases of the control process, which we discussedfor the chaotic case, can be observed for the hyperhcaoticone such that the oscillatory negative control inputs/drivingmagnetic torques suppress the hyperchaotic dynamics. Notethat the green boxes shown in Figs. 7(a) and 7(b) reveal theincremental values of the control inputs/torques to rotate thevalves to the desirable trajectories.

Shown in Fig. 8(a) are both the upstream and downstreamvalves’ rotation angles revealing that the valves’ motionstend to the desirable trajectories (Eq. 17). Figs. 8(b) and8(c) present the convergence of both the error and combinedtracking error signals to zero, respectively. Consequently,it is straightforward to conclude that the adaptation and

1675

control laws guarantee both the global stability of the couplednetwork and convergence of the tracking errors in which weanalytically discussed in Section 3.

V. CONCLUSIONS

In this paper, we represented the interconnected sixth-order dynamic model of the network of two bi-directionalsolenoid actuated butterfly valves subject to the sudden con-traction. The network undergoes the coupled chaotic and hy-perchaotic dynamics for a set of initial conditions, the Initial1and Initial2, and critical parameters. The adaptation and con-trol laws were developed to vanish the chaotic/hyperchaoticdynamics and then push the dynamically coupled valves totrack the desirable trajectories.

For the initial conditions resulted in thechaotic/hyperchaotic dynamics, we revealed that thedownstream set required the higher control inputs/drivingmagnetic torques than those of the upstream one. Thetwo phases of the control process were also distinguishedto vanish the chaotic/hyperchaotic dynamics in additionto mitigating the harmful effects of the hydrodynamictorque (as a helping load) which expectedly magnifies theamplitude of dangerous stochastic oscillation. The firstphase by presenting the oscillatory negative control inputsremoved the chaotic/hyperchaotic dynamics. The controlinputs of the second phase gradually took the positive valuesto push the valves to the desirable trajectories. The controlinputs of the downstream set were shown to be higher thanthose of the upstream one, for the second phase of boththe chaotic and hyperchaotic cases, in order to overcomethe higher resistance bearing torque. We have previouslyestablished that the bearing torque of the downstream setwith a smaller pipe diameter is higher than that of theupstream one.

REFERENCES

[1] P. Naseradinmousavi, H. Ashrafiuon, and M. A. Ayoubi, “An adaptivecentralized approach to control chaotic and hyperchaotic dynamicsof smart valves network,” ASME Journal of Computational andNonlinear Dynamics, vol. 13, no. 1, pp. 011 002–011 002–11, 2018.

[2] P. Naseradinmousavi, D. B. Segala, and C. Nataraj, “Chaotic andhyperchaotic dynamics of smart valves system subject to a suddencontraction,” ASME Journal of Computational and Nonlinear Dynam-ics, vol. 11, no. 5, pp. 051 025–051 025–9, September 2016.

[3] P. Naseradinmousavi, M. Krstic, and C. Nataraj, “Design optimizationof dynamically coupled actuated butterfly valves subject to a suddencontraction,” ASME Journal of Mechanical Design, vol. 138, no. 4,pp. 041 402–041 402–11, April 2016.

[4] P. Naseradinmousavi, S. G. Machiani, M. A. Ayoubi, and C. Nataraj,“Coupled operational optimization of smart valve system subject todifferent approach angles of a pipe contraction,” Journal of Structuraland Multidisciplinary Optimization, vol. 55, no. 3, pp. 1001–1015,March 2017.

[5] P. Naseradinmousavi, “A novel nonlinear modeling and dynamicanalysis of solenoid actuated butterfly valves coupled in series,” ASMEJournal of Dynamic Systems, Measurement, and Control, vol. 137,no. 1, pp. 014 505–014 505–5, January 2015.

[6] P. Naseradinmousavi and C. Nataraj, “Optimal design of solenoidactuators driving butterfly valves,” ASME Journal of MechanicalDesign, vol. 135, no. 9, pp. 094 501–094 501–5, July 2013.

[7] ——, “Transient chaos and crisis phenomena in butterfly valves drivenby solenoid actuators,” Journal of Communications in NonlinearScience and Numerical Simulation, vol. 17, no. 11, pp. 4336–4345,November 2012.

[8] D. Lee, P. Naseradinmousavi, and C. Nataraj, “Nonlinear model-based adaptive control of a solenoid-valve system,” Journal of ControlScience and Engineering, vol. 2012, 2012.

[9] P. Naseradinmousavi and C. Nataraj, “Nonlinear mathematical model-ing of butterfly valves driven by solenoid actuators,” Journal of AppliedMathematical Modelling, vol. 35, no. 5, pp. 2324–2335, 2011.

[10] P. Naseradinmousavi, M. Bagheri, H. Ashrafiuon, M. Canova, andD. B. Segala, “Suppressing chaotic/hyperchaotic dynamics of smartvalves network using decentralized and centralized schemes,” in theASME 2017 Dynamic Systems and Control Conference, vol. 3, no.DSCC2017-5006, Tysons Corner, Virginia, USA, October 11-13 2017,p. V003T42A00.

[11] P. Naseradinmousavi, M. Krstic, M. Bagheri, and C. Nataraj, “Coupledchaotic and hyperchaotic dynamics of actuated butterfly valves oper-ating in series,” in The ASME 2016 Dynamic Systems and ControlConference, vol. 2, no. DSCC2016-9601, Minneapolis, Minnesota,USA, October 12-14 2016, p. V002T17A001.

[12] P. Naseradinmousavi, M. Bagheri, and C. Nataraj, “Coupled opera-tional optimization of smart valve system subject to different approachangles of a pipe contraction,” in The ASME 2016 Dynamic Systemsand Control Conference, vol. 1, no. DSCC2016-9627, Minneapolis,Minnesota, USA, October 12-14 2016, p. V001T02A001.

[13] P. Naseradinmousavi and C. Nataraj, “Design optimization of solenoidactuated butterfly valves dynamically coupled in series,” in Proceed-ings of the ASME 2015 Dynamic Systems and Control Conference,vol. 2, no. DSCC2015-9605, October 2015, p. V002T33A001.

[14] P. Naseradinmousavi, “Optimal design of solenoid actuated butterflyvalves dynamically coupled in series,” in Proceedings of the ASME2015 International Mechanical Engineering Congress & Exposition,vol. 4A: Dynamics, Vibration, and Control, no. IMECE2015-50094,Houston, November 13-19 2015, p. V04AT04A032.

[15] P. Naseradinmousavi and C. Nataraj, “A chaotic blue sky catastropheof butterfly valves driven by solenoid actuators,” in Proceedingsof the ASME 2011 International Mechanical Engineering Congress& Exposition, vol. 7: Dynamic Systems and Control; Mechatronicsand Intelligent Machines, Parts A and B, no. IMECE2011/62608,November (11-17) 2011, pp. 25–31.

[16] C.-W. Chang-Jian, “Gear dynamics analysis with turbulent journalbearings mounted hybrid squeeze film damper-chaos and active controlanalysis,” ASME Journal of Computational and Nonlinear Dynamics,vol. 10, no. 1, pp. 011 011–011 011–11, 2014.

[17] C. Morel, R. Vlad, and J.-Y. Morel, “Anticontrol of chaos reducesspectral emissions,” ASME Journal of Computational and NonlinearDynamics, vol. 3, no. 4, pp. 041 009–041 009–6, 2008.

[18] D. Chen and W. Liu, “Chaotic behavior and its control in a fractional-order energy demand-supply system,” ASME Journal of Computationaland Nonlinear Dynamics, vol. 11, no. 6, pp. 061 010–061 010–7, 2016.

[19] B. Wang, K. Shi, C. Zhang, and D. Zhu, “Fuzzy generalized predictivecontrol for nonlinear brushless direct current motor,” ASME Journal ofComputational and Nonlinear Dynamics, vol. 11, no. 4, pp. 041 004–041 004–7, 2015.

[20] R. Luo and Y. Zeng, “The control and synchronization of a class ofchaotic systems with output variable and external disturbance,” ASMEJournal of Computational and Nonlinear Dynamics, vol. 11, no. 5, pp.051 011–051 011–7, 2015.

[21] B. S. Reddy and A. Ghosal, “Asymptotic stability and chaotic motionsin trajectory following feedback controlled robots,” ASME Journal ofComputational and Nonlinear Dynamics, vol. 11, no. 5, pp. 051 012–051 012–11, 2015.

[22] P. Khamsuwan and S. Kuntanapreeda, “A linear matrix inequalityapproach to output feedback control of fractional-order unified chaoticsystems with one control input,” ASME Journal of Computational andNonlinear Dynamics, vol. 11, no. 5, pp. 051 021–051 021–7, 2016.

[23] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear andAdaptive Control Design. Wiley-Interscience, 1995.

[24] Slotine and Li, Applied Nonlinear Control. Prentice-Hall, 1991.

1676