nonlineardynamicmodel-basedadaptivecontrolof asolenoid ... · 2 journal of control science and...

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2012, Article ID 846458, 13 pagesdoi:10.1155/2012/846458

Research Article

Nonlinear Dynamic Model-Based Adaptive Control ofa Solenoid-Valve System

DongBin Lee, Peiman Naseradinmousavi, and C. Nataraj

Department of Mechanical Engineering and Center for Nonlinear Dynamics & Control (CENDAC), Villanova University,Villanova, PA 19085, USA

Correspondence should be addressed to DongBin Lee, [email protected]

Received 6 December 2011; Revised 8 March 2012; Accepted 30 March 2012

Academic Editor: Lili Ma

Copyright © 2012 DongBin Lee et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a nonlinear model-based adaptive control approach is proposed for a solenoid-valve system. The challenge is thatsolenoids and butterfly valves have uncertainties in multiple parameters in the nonlinear model; various kinds of physical appear-ance such as size and stroke, dynamic parameters including inertia, damping, and torque coefficients, and operational parametersespecially, pipe diameters and flow velocities. These uncertainties are making the system not only difficult to adjust to the environ-ment, but also further complicated to develop the appropriate control approach for meeting the system objectives. The main con-tribution of this research is the application of adaptive control theory and Lyapunov-type stability approach to design a controllerfor a dynamic model of the solenoid-valve system in the presence of those uncertainties. The control objectives such as set-pointregulation, parameter compensation, and stability are supposed to be simultaneously accomplished. The error signals are first for-mulated based on the nonlinear dynamic models and then the control input is developed using the Lyapunov stability-type analysisto obtain the error bounded while overcoming the uncertainties. The parameter groups are updated by adaptation laws using aprojection algorithm. Numerical simulation results are shown to demonstrate good performance of the proposed nonlinear model-based adaptive approach and to compare the performance of the same solenoid-valve system with a non-adaptive method as well.

1. Introduction

In order to achieve advanced automation [1] in systemssuch as marine vessels or ship-based machinery system [2],solenoid actuators and valves are often used [3] to increasesurvivability and capability. One typical type of actuatordriven by solenoids is shown in Figure 1, which is operated bythe electromagnetic force. The electric-driven solenoid valvesystem [4] and its sophisticated control can provide high lev-els of automation in large systems. The useful function of thesolenoid-valve, once an electrical signal (current or voltage)is applied, is to activate a mechanical motion such as dis-placement or rotation via the solenoid magnetic forces andtorques. The proportional solenoids normally require inte-grated electronics for controlling the plunger to give such asignal. Hydrodynamic torque of a butterfly valve comprisesthe core knowledge of fluid valve system design [5], andit is known that most of the valves in real systems havestrongly nonlinear characteristics between the force and

displacement [6, 7]. The use of an intelligent approach [8, 9]such as adaptive, robust, optimal, or nonlinear control of theactuator-valve machinery systems will benefit a wide spec-trum of nonlinear systems, compensating for nonlinearities[10] and dynamic characteristics. This approach will not onlydecrease the amount of cost and casualties but also improvethe performance of the mechatronic system. To investigatethe particular application, it is important to emphasizethe nonlinear dynamic modeling analysis of such actuator-valve systems because the accuracy and reliability of thesesystems depend highly on the mathematical system modeling[11] and its validation. In [12], the authors developed andanalyzed the nonlinear dynamic model of a solenoid-valvesystem; the reader is also referred to [13, 14] for recent mod-eling and analysis of solenoid actuators.

This paper will focus on model-based nonlinear adaptivecontrol of an actuator-butterfly valve. The solenoid-valvesystem is described based on the exact model knowledgeof the system. Figure 1 shows the integrated system, which

2 Journal of Control Science and Engineering

Pinion

Stem

Disk

Plu

nge

r

Coi

l

Coi

l

Yoke

k B1

V0 DpVJ

x

α ∼ 5

Figure 1: System Configuration.

consists of an electric-driven solenoid and a butterfly valve.The valve operates by solenoids that use a magnetic coilto move a movable plunger connected with the valve stemby means of a gear train and linkage. The control inputis designed by substituting the current signal from themodel of the electromagnetic force, pulling the plunger, andthen controlling the angular position of the butterfly valve.The system has uncertainties in multiple parameters in thedynamic model, which requires the system to continuouslyadjust to the environment and consequently requires adap-tation for sustainability and capability. The integrated systemis highly nonlinear in addition to its parameter uncertainties.Hence, an adaptation law is proposed [8, 9, 17] and anadaptive control method is developed for the solenoid-valvesystem in multiple parametric uncertainties. A closed-loopstable controller is designed for the set-point trajectorytracking by introducing a Lyapunov-based stability analysis[9] based on the error signals of the nonlinear solenoid-valvesystem. The numerical results in the simulation are used forinitial verification and performance evaluation.

2. Model-Based Nonlinear System

2.1. System Model. The dynamic equations of motion of theplunger and butterfly valve are given by [15]

mẍ + B1ẋ + kx = Fmag − Fc,J ä + B2α̇ = rgFc − Ttot,

(1)

where x(t) is the displacement of the solenoid plunger, α(t) isthe angle of butterfly disk, and Ttot(t) is the sum of the hydro-dynamic and bearing torques expressed as Ttot = Tb + Th.The bearing torque is given as Tb = (π/8)μDsD2pΔPvCR(α),where the bearing toque coefficient, CR(α) =

√C2L + C

2D =√

(1.1 4√

sin((α/90)3180))2 + ( 3√

cosα)2, is obtained from the

valve modeling (see [5, 15]) and the two subterms, lifting

force CL = 1.1 4√

sin((α/90)3180) and drag force CD = 3√cosα,are nonlinear functions of the valve angle rotation α(t). Thehydrodynamic torque Th is obtained by reviewing three-dimensional hydrodynamic torque coefficient based on [7,16] as Th = (8/3π)ρD3pV 2OTc(α)[(VJ /VO)(α)]2, where bothTc(α) and (VJ /VO)(α) depend on the closing angle (α) of thebutterfly valve and D3p term is a nonlinear term accordingto the pipe size. Solving the two equations in (1) with thecontact force, Fc(t), and substituting the magnetic force,Fmag(t), into the equation yields

(m +

J

r2g

)rg ẍ +

(B1 +

B2r2g

)rg ẋ + krgx

= rg C2N2

2(C1 + C2x)2 i

2 − Ttot,(2)

where the magnetic force Fmag = (C2N2/2(C1 + C2x)2)i2used to lift the plunger of the solenoid actuator. The actuatoris a current-controlled solenoid [15], proportional to thesquare of the current i(t), and C1, C2 are reluctances ofthe magnetic paths, obtained from the geometry of electricactuator [4, 15]. It is assumed that the pinion and the valveare moving at the same speed, that is, the gear ratio is 1 : 1 andx(t) = rgα is the simple geometric relationship between thedisplacement of the pinion and the valve angle. Hence, thecurrent source i2(t) is substituted for designing the closed-loop control input, u(t), and then the following equation isobtained:

(m +

J

r2g

)rg ẍ +

(B1 +

B2r2g

)rg ẋ + krgx

= rg C2N2

2(C1 + C2x)2 u− Ttot.

(3)

For the subsequent controller design, multiplying (3)with the inverse term of the control input, 2(C1 + C2x)

2/(rgC2N2), yields a compact form of the dynamic equation as

M(x, θ)ẍ + C(x, θ)ẋ + D(x, θ)x = u− B(Tb + Th), (4)

where θ describes a lumped expression of parametersobtained from (3) and (4), where each parameter is shownin Table 2 and the substituted terms are defined as follows:

M(x, θ) =(m +

J

r2g

)rgB, B = 2(C1 + C2x)

2

rgC2N2,

C(x, θ) =(B1 +

B2r2g

)rgB, D(x, θ) = krgB.

(5)

2.2. Error Signals Formulation. The following set-point con-trol approach is used. Let xd(t) define the set-point trajectoryand then the error can be defined as

e ≡ xd − x, ė = ẋd − ẋ, ë = ẍd − ẍ, (6)

Journal of Control Science and Engineering 3

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

80

90

α◦ :

val

ve r

otat

ion

an

gle

(deg

)

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 2: Valve rotation angle α(t) (adaptive approach).

where ẋd(t) and ẍd(t) are the first and second time derivativesof xd(t), which are assumed to be bounded. Premultiplyingë(t) in the last error signal of (6) with M(x, θ) yieldsM(x, θ)ë = M(x, θ)ẍd −M(x, θ)ẍ. Substituting M(x, θ)ẍ in(4) into the above equation produces

M(x, θ)ë = M(x, θ)ẍd + C(x, θ)ẋ + D(x, θ)x − u+ B(Tb + Th).

(7)

A filtered error signal and its derivative are defined as

r ≡ ė + λ1e, ṙ = ë + λ1ė, (8)where λ1 ∈ �+ is a positive adjustable control gain. Multi-plying (8) with M(x, θ) and then substituting for M(x, θ)ë(t)in (7) yields

M(x, θ)ṙ = M(x, θ)ẍd + C(x, θ)ẋ + D(x, θ)x − u

+ B(Tb + Th) + M(x, θ)λ1ė +1r2ge − 1

r2ge,

(9)

where the last term e(t)/r2g is added and subtracted for fur-ther development of the control design based on Lyapunov’smethod.

3. Lyapunov-Based Adaptive Feedback Control

Let V(t) be a Lyapunov candidate function

V = 12

(rTMr + eTe + eTα eα + Θ̃

TΓ−1Θ̃)

, (10)

where the last term of the Lyapunov candidate function, Γ =γIp×p, is a constant diagonal matrix with the gain value γ,Ip×p is a p× p identity matrix, and the parameter estimationerror, Θ̃, is defined as Θ̃ = Θ− Θ̂, where Θ ∈ �p is a known

Table 1: Regression and parameter estimation terms.

No. Terms No. Terms

W101 ẍd W113 CR(α)x

W102 ẍdx W114 Tc(α)[VJ(α)/VO]2x2

W103 ẍdx2 W115 CR(α)x2

W104 ẋ W116 ẋė

W105 ẋx W117 ẋe

W106 ẋx2 W118 xẋė

W107 x W119 xẋe

W108 x2 W120 e

W109 x3 W121 λ1ė

W110 Tc(α)[VJ (α)/VO]2 W122 λ1xė

W111 CR(α) W123 λ1x2ė

W112 Tc(α)[VJ(α)/VO]2x — —

Θ̂101 M̂s2Ĉ21/(Ĉ2N̂2) Θ̂113 T̂b14Ĉ1/(r̂g N̂2)

Θ̂102 M̂s4Ĉ1/N̂2 Θ̂114 T̂h12Ĉ2/(r̂g N̂2)

Θ̂103 M̂s2Ĉ2/N̂2 Θ̂115 T̂b12Ĉ2/(r̂g N̂2)

Θ̂104 Ĉs2Ĉ21/(Ĉ2N̂2) Θ̂116 M̂s2Ĉ1/N̂2

Θ̂105 Ĉs4Ĉ1/N̂2 Θ̂117 M̂s2Ĉ1/N̂2

Θ̂106 Ĉs2Ĉ2/N̂2 Θ̂118 M̂s2Ĉ2/N̂2

Θ̂107 k̂2Ĉ21/(Ĉ2N̂2) Θ̂119 M̂s2Ĉ2/N̂2

Θ̂108 k̂4Ĉ1/N̂2 Θ̂120 1/r2gΘ̂109 k̂2Ĉ2/N̂2 Θ̂121 M̂s2Ĉ21/(Ĉ2N̂2)

Θ̂110 T̂h12Ĉ21/(r̂g Ĉ2N̂2) Θ̂122 M̂s4Ĉ1/N̂2

Θ̂111 T̂b12Ĉ21/(r̂g Ĉ2N̂2) Θ̂123 M̂s2Ĉ2/N̂2

Θ̂112 T̂h14Ĉ1/(r̂g N̂2) — —

Table 2: List of simulation parameters.

m J rg B1 B2 k N ΔPv

0.1 1.04e−6 1e−2 10 20 4e2 8.8e3 0.5

C1 C2 ρ Dp VO μ Ds p

1.57e6 6.32e8 1e3 5′′ ∼ 8′′ 3.7 0.1 0.1 23

constant parameter vector and Θ̂ ∈ �p is the estimatedconstant parameter vector (see Table 1). Differentiating (10)yields

V̇ = rTMṙ + 12rTṀr + eT ė +

1r2geT ė − Θ̃TΓ−1 ˙̂Θ, (11)

where the time derivative of the inertia matrix is obtainedas Ṁ = (m + J/r2g )(4(C1 + C2x)/N2)ẋ, where Ḃ = (4(C1 +C2x)/rgN2)ẋ, eTα ėα = (1/r2g )eT ė as eα ≡ αd − α = xd/rg −x/rg = e/rg , ėα ≡ α̇d − α̇ = ẋd/rg − ẋ/rg = ė/rg , in which theerror signals of the valve angle, eα(t), can be defined using

the geometric relationship and ˙̃Θ(t) = − ˙̂Θ comes from thedefinition of Θ̃.

4 Journal of Control Science and Engineeringx:

plu

nge

r di

spla

cem

ent

(m)

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

00 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 3: Plunger displacement x(t) (adaptive approach).

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

80

90

α◦ :

val

ve r

otat

ion

an

gle

(deg

)

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 4: Valve rotation angle α(t) (no adaptation scheme).

3.1. Design of Control Input. Substituting M(x, θ)ṙ(t) into(11) yields

V̇ = rT(Mẍd + Cẋ + Dx − u + B(Tb + Th) + Mλ1ė + 1

r2ge

)

− (ė + e)Te

r2g+

12rTṀr + eT ė +

1r2geT ė − Θ̃TΓ−1 ˙̂Θ,

(12)

where the last term in (9) premultiplied by r(t) came out ofthe parenthesis in (12) and is used for the definition of r(t)

0.015

0.01

0.005

00 0.5 1 1.5 2 2.5 3 3.5 4

x: p

lun

ger

disp

lace

men

t (m

)

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 5: Plunger displacement x(t) (no adaptation scheme).

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−0.01

r(t)

: filt

ered

err

or

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 6: Filtered Error r(t) (adaptive approach).

in (8). Then, combining the parameterized terms in (12) andsubstituting them into WΘ yields

WΘ =Mẍd + Cẋ + Dx + B(Tb + Th) + Mλ1ė + 1r2ge +

12Ṁr,

(13)

where W(ẍd, ẋ, x, r,α, ė, e) ∈ �1×p is a known regressionvector, which is shown in the left side of Table 1 via theprocess given later (see (17)) and Θ as the nominal value ofthe lumped parameter vector. Rearranging (12) produces

V̇ = rT(WΘ− u)− eTe

r2g+ eT ė − Θ̃TΓ−1 ˙̂Θ, (14)


0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−0.01

e(t)

: dis

plac

emen

t er

ror

(m)

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 7: Displacement tracking error e(t) (adaptive approach).

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−0.01

x′: t

he

rate

of

disp

lace

men

t (m

/s)

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 8: The rate of displacement ẋ(t).

where ėTe/r2g is canceled in the last second term in (12),having the opposite sign because they are scalar, ėTe = eT ė.

The control input can be designed based on Lyapunovstability analysis, making V̇ negative definite to be shown inthe end, as

u =WΘ̂ + k1r + e, (15)

where r(t) is a feedback error term, k1 is a positive constantas the control gain, e(t) is another feedback error term addedto cancel the term having the opposite sign, eT ė, outside theparenthesis by utilizing the definition of r(t) given in (8), andWΘ̂ captures the uncertainties associated with the elements

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

DpDp

==

DpDp

0

50

100

150

u(t

): c

ontr

ol in

put

(A2)

Figure 9: Control input u(t) equivalent to square current i2(t)[A2].

0

1

2

3

4

5

6

7

8×104

Fm

ag: e

lect

rom

agn

etic

forc

e (N

)

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 10: Electromagnetic force Fmag(t).

of M, C, D, B(Tb1 +Th1), Mλ1, 1/r2g , and Ṁ, which is definedas

WΘ̂ = M̂ẍd+Ĉẋ+D̂x+B̂(T̂b1CR(α)+T̂h1Tc(α)

[VJVO

(α)]2)

+ M̂λ1ė+1r̂2ge+

12

˙̂Mr,

(16)

where the estimated parameter sets are given as M̂(x, θ) =(m̂ + Ĵ/r̂2g )r̂g B̂, where B̂ = 2(Ĉ1 + Ĉ2x)

2/r̂g Ĉ2N̂2, Ĉ(x, θ) =

(B̂1 + B̂2/r̂2g )r̂g B̂, D̂(x, θ) = k̂r̂g B̂, T̂b1 = (π/8)μ̂D̂sD̂2pΔP̂v,


0

50

100

150

200

250

Tt=Th

+Tb: t

otal

torq

ue

(Nm

)

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 11: Total torque Tt(t).

and T̂h1 = (8/3π)ρ̂D̂3pV̂ 2O. In order to develop the estimateparameter vector Θ̂ in (16), we need to first define theregression and the estimate terms. Thus, the first term, M̂ẍd,in (16) can be defined as

M̂ẍd =(m̂ +

Ĵ

r̂2g

)r̂g

2(Ĉ1 + Ĉ2x

)2

r̂g Ĉ2N̂2ẍd

= M̂s 2Ĉ21

Ĉ2N̂2ẍd + M̂s

4Ĉ1N̂2

xẍd + M̂s2Ĉ2N̂2

x2ẍd

=[ẍd xẍd x2ẍd

]︸︷︷︸

W101∼W103

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

M̂s2Ĉ21Ĉ2N̂2

M̂s4Ĉ1N̂2

M̂s2Ĉ2N̂2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸Θ̂101∼Θ̂103

,

(17)

where M̂s = (m̂ + Ĵ/r̂2g ), Ĉs = (B̂1 + B̂2/r̂2g ) (this is shown inTable 1), and r̂g , Ĉ1, or Ĉ2 are canceled. W101 ∼W103 are themeasurable regression terms and Θ̂101 ∼ Θ̂103 is the estimat-ed parameters, defined in Table 1, respectively. Similarly to(17), the rest of the terms in (16) are also given in Table 1.

3.2. Online Adaptation Laws for Parameter Updates. Thefollowing is constructed to define the known upper andlower bounds but with an unknown parameter of Θ̂(t) in thesense that

Θ̂j≤ Θ̂ j(t) ≤ Θ̂ j , (18)

0

0.005

0.01

0.015

0.02

0.025

0.03

Tc(α

): to

rqu

e co

effici

ents

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 12: Hydrodynamic torque coefficient Tc(α).

0

20

40

60

80

100

120

140

160

VJ/VO

(α):

th

e ra

tio

of m

ean

flow

an

d je

t ve

loci

ties

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

DpDp

==

DpDp

Figure 13: The ratio of VJ and VO: (VJ /VO)(α).

where Θ̂ j(t) are the estimated parameters as shown in Table 1

and Θ̂j

and Θ̂ j are the lower and upper bounds of the

estimated parameters, respectively, which will be set to the

amount of percentage of their true values. The vector ˙̂Θ j(t)is designed to update using a projection-based algorithm as

˙̂Θ j = Proj

{ΓWTr, Θ̂ j

}, (19)


0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

1.1

1.2

1.3

1.4

1.5×10−5 θ101

8.944

8.946

8.948

8.95

8.952

θ102×10−3

1.8027

1.8027

1.8027

1.8027

1.8027

θ103

20.1068

20.1069

20.107

20.1071θ104

Time (s) Time (s)

Time (s)Time (s)

= 5= 6

78

DpDp

==

DpDp

= 5= 6

78

DpDp

==

DpDp

Figure 14: Parameter estimates: Θ̂101 ∼ Θ̂104.

where Proj{·} is the projection operator [8] and each lumpedparameter is adaptively updated using the adaptation laws[17] for online estimation of unknown parameter as follows:

Proj{

˙̂Θ j

}= Proj

{ΓWTr, Θ̂ j

}

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ΓWTr if Θ̂ j > Θ̂j, Θ̂ j < Θ̂ j ,

ΓWTr if Θ̂ j = Θ̂j, if ΓWTr > 0,

ΓWTr if Θ̂ j = Θ̂ j , if ΓWTr ≤ 0,

0 elsewhere.(20)

Thus, substituting u(t) in (15) into V̇ in (13) yields

V̇ = rTWΘ̃− rTk1r − (ė + e)Te − eTe

r2g+ eT ė − Θ̃TΓ−1 ˙̂Θ

= − rTk1r −(

1 +1r2g

)eTe + Θ̃T

{WTr − Γ−1 ˙̂Θ

}.

(21)

Here, WΘ̃ is defined as

WΘ̃ = M̃(ẍd + λ1ė) + C̃ẋ + D̃x + B̃T̃h1Tc(α) VJVO

(α)

+ B̃T̃b1CR(α) + r̃g e +12˜̇Mr,

(22)

where M̃ = M − M̂, C̃ = C − Ĉ, D̃ = D − D̂, B̃T̃h1 =BTh1 − B̂T̂h1, B̃T̃b1 = BTb1 − B̂T̂b1, r̃g = 1/r2g − 1/r̂2g ,


Time (s)

Time (s)

Time (s)

Time (s)

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 2 4

1.6207

1.6207

1.6207

1.6207

1.6207

×104 θ105

3.2659

3.2659

3.2659

×106 θ106

0.04

0.0401

0.0402

0.0403

0.0404θ107

32.4127

32.4128

32.4128

32.4128

32.4128

θ108

= 5= 6

78

DpDp

==

DpDp

= 5= 6

78

DpDp

==

DpDp


and ˜̇M = Ṁ − ̂̇M, in which T̂h1 = (8/3π)ρ̂D̂3pV̂ 2O and T̂b1 =(π/8)μ̂D̂sD̂2pΔP̂v. Actually, Θ̃(t) can be expressed, for exam-

ple, using the definitions of Θ̂ in (16), as follows: owing to thesubsequent adaptation law, the time derivative of V(t) yieldsa negative definite function except the origin and upperbound by

V̇ ≤ −k1‖r‖2 −(

1 +1r2g

)‖e‖2, (23)

which can be written as

V̇ ≤ −k2‖z‖2, (24)where k2 = min{k1, (1 + 1/r2g )} is a positive constant and

z =[rT , eT

]T. (25)

Using Barbalat’s lemma [18], the set-point tracking error‖z(t)‖ → 0, thus ‖r(t)‖ → 0 and ‖e(t)‖ → 0 as t → ∞.

Remark 1. According to the analysis from (11) to (25) ofV̇ , the property of V , and the control law of (15) with theparameter updates of (19) and the projection-based methodof update laws of (20), it is straightforward to derive aconclusion that the tracking error vector z(t) in (25) is drivento zero. Thus, the set-point errors r(t), e(t), and eα(t) alsovanish and the parameter estimation error vector Θ̃ in (10)is bounded where Θ̂, defined after (10), is bounded due tothe projection-based update method and the constant knownparameter, Θ. Owing to the bounds of r(t), e(t) in (8), ė(t)is bounded and then x(t), resulting in α(t), and ẋ(t) arebounded, respectively, where all desired trajectories such asxd(t) and ẋd(t) are assumed to be bounded. M(·), B(·), C(·),and D(·) matrices in (4) are bounded because θ (due toΘ) and x(t) are bounded, and Ḃ and Ṁ after (11) are alsobounded owing to the bounds of ẋ(t). Ttot is thus boundedbecause α(t) is bounded. Hence, W(·) and the control inputu(t) are bounded and, thus, the current is bounded. Thisleads to the boundedness of ẍ(t) in the dynamic model given


0 1 2 3 4

0 1 2 3 4 0 2 4

0 1 2 3 4

6531.5313

6531.5313

6531.5313

6531.5313

×10−8

θ109 θ110

θ111 θ112

Time (s) Time (s)

Time (s) Time (s)

= 5= 6

78

DpDp

==

DpDp

= 5= 6

78

DpDp

==

DpDp

0.2

0.4

0.6

0.8

1

0.5

1

1.5

2

2.5

0

200

400

600

800


in (4), which enables the boundedness of ë(t) in (6), and thenthe set-point tracking error dynamics ṙ(t) in (8) is bounded.Therefore, we can conclude that all signals are bounded.

Remark 2. WΘ̂ in (16) where the known regression W(ẍd, ẋ,x, r,α, ė, e) ∈ �1×23 terms and the parameter estimates Θ̂ ∈�23 are given in this system as in Table 1.

4. Simulation Results

Based on the dynamic model in (4), the numerical sim-ulation is performed to verify the proposed controllerswith consistently changing parameter values. The parametervalues can be divided into two categories: operational valuessuch as Dp and VO and uncertain values such as B1, B2,and μ. After the flow velocity VO is kept on 3.7 [m/s], thepipe diameter Dp shown in Table 2 is used to vary from 5inches to 8 inches as Dp = {5.0, 6.0, 7.0, 8.0} [in], whereDp = {0.1270, 0.1524, 0.1778, 0.2032}[m], by assuming thatthe pipe or transmission lines are different according totheir applications. With each Dp size, the variations of all

the parameters are set to 30% for the simulation resultsprovided here. The determined parameter vectors Tc(α) and[VJ /VO](α) of the butterfly valve model are borrowed from[16] for the simulation given as look-up tables from theexperimental data. The amount of upper and lower variationof the unknown parameter sets is 30% of their real values.The control gains were chosen selectively as γ = 10, k1 =250, and λ1 = 1.0 for all cases. MATLAB and Simulink areused for the simulation. The desired set-point distance of thesolenoid, xd(t), is given as A · (1 − e−Bt), where A = 0.0148and B = 5.

A typical parameter set for this simulation is given byTable 2. Figures 2 and 3 show the rotation of the valve angleand the actual displacement of the plunger, respectively. Thisadaptive control approach shows better results compared tothe results obtained from the previous research [15] usingnonadaptive method, which are shown in Figures 4 and 5 forthe displacement of the plunger and the angle of the butterflyvalve. The developed mathematical model is the same asthat of adaptive method based on the nonlinear models andthe new specific approach in this paper is that the error


0

0.5

1

1.5

2

0

0.5

1

1.5

2

1

2

3

4

4.4725

4.473

4.4735

4.474

0 1 2 3 4

0 1 2 3 4 0 2 4

0 1 2 3 4

×10−5 ×105

×10−3 ×10−3

θ113 θ114

θ115 θ116

Time (s) Time (s)

Time (s) Time (s)

= 5= 6

78

DpDp

==

DpDp

= 5= 6

78

DpDp

==

DpDp


signals are formulated by introducing the desired trajectoryto reducing the displacement error in the adaptive methodwhile overcoming the complicated parametric uncertainties.And also the response of the adaptive method shows that themotion process is more consistent in both variables, showingoverdamped phenomena and short rising times, and thenit quickly reaches the steady state. The actual movement issmoother and faster while all the parameters are varying 30%in each flow velocity.

Figures 6 and 7 are the filtered error r(t), defined in (8),and set-point error e(t), given in (6), respectively. Figure 8shows the rate of the displacement to Figure 3. From Figure 8,all figures are presented without the notation of the adaptiveapproach. Figure 9 shows the control input u(t) of the sole-noid actuator for each pipe diameter, Dp, designed in (15)with the nonlinear adaptive controller by substituting thesquare of the current, i2(t). The electromagnetic force Fmag(t)given in (2) and the total torque Tt(t) given in (1) by sum-ming up the hydrodynamic and bearing torques are plotted in Figures 10 and 11, respectively. The torque coefficient Tc(t)

shown in Figure 12 and the ratio of input and output jetvelocities shown in Figure 13 are changed for every 5◦ of thebutterfly angle. As the strokes are increased by the controlinput, the angles get larger and then accordingly the valueof the inlet jet velocity increases, which affects the slowermotion of the strokes and angles but the variables (strokesand angles) reach the steady state and the ratio as well.

The challenge is that most parameter terms in Table 1 arecombined and lumped together due to the dynamic modelin the presence of parameter uncertainty and the model iscomplicated owing to the control objectives, set-point regu-lation, and parametric adaptation. As given in the right sideof Table 1, the unknown bounded parameter estimate vectorΘ̂ ∈ �23 is shown in Figures 14, 15, 16, 17, 18, and 19. It canbe seen that the parameters such as 101, 110, 111, 113, 116,118, and 121 are quickly updated in the form of premultipli-cation by its regression term, W and go to steady state. Thus,they are more parametric-centric terms and related to thefiltered error r(t). The parameters such as 102, 103, 104, 105,117, 119, and 122 are updated according to the motion of


4.473

4.473

4.473

4.4731

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1

1

1

1

1

0 2 4 0 1 2 3 4

0 1 2 3 40 1 2 3 4

×104

×10−3 θ117 θ118

θ119 θ120

Time (s) Time (s)

Time (s) Time (s)

= 5= 6

78

DpDp

==

DpDp

= 5= 6

78

DpDp

==

DpDp


the angle and stroke of the solenoid-butterfly valve system.Some parameters such as 106, 112, 114, and 115 are notupdated but other parameters such as 107, 108, and 120 keepgetting updated. Any parameters changing their values mustbe related to the control objective because these terms areincorporated into the control input in the form of the esti-mates.

5. Conclusion and Future Work

For developing advanced automation systems such as ship-based hydraulic systems, a typical solenoid-butterfly valve,which is driven by electromagnetic, fluid mechanics, andhydrodynamic forces and torques, is chosen as a continuationof previous research and an adaptive stable control approachwith adaptation laws is developed accounting for uncertain-ties in multiple parameters on the nonlinear dynamic model.A stable adaptive controller of the solenoid-valve system isdesigned positioning the angle of the butterfly valve via aLyapunov-based approach. The approach yields bounded

error while adapting to the environment in the presence ofcomplex uncertainties such as different physical appearances,uncertain parameters, operational characteristics, and para-metric nonlinear dynamic models.

The parameter estimation for the unknown boundedparameters is performed using a projection algorithm whoseoutput yields the upper and lower bounds. Numericalsimulation is used to verify the performance of the proposedapproach to show its effectiveness by comparing to thesame dynamic model without adaptation from the previousresearch; when compared to the nonadaptive method, theresponses of the plunger displacement and the rotatingangle are steadier, smoother, and faster. Future work willbe focused on demonstrating the results of hardware-in-the-loop or experiments for the nonlinear solenoid valve systemas well as applying the suggested adaptive method based onLyapunov-based control approach to the real-world system.Further research on developing control techniques usingrobust or optimal method would be continued to overcomenonlinearities such as hysteresis or nonlinear dynamics.


1.1

1.2

1.3

1.4

1.5

8.9459

8.9459

8.9459

8.9459

8.9459

1.8027

1.8027

1.8027

1.8027

1.8027

0 2 4 0 1 2 3 4

0 0.5 1 1.5 2 2.5 3 3.5 4

×10−3×10−5 θ121 θ122

θ123

Time (s) Time (s)

Time (s)

= 5= 6

78

DpDp

==

DpDp


Nomenclature

B1, B2: Damping coefficients of thesolenoid and butterfly valve,respectively (Ns/m)

C1, C2: Reluctances of the magnetic pathsobtained from plunger geometry ofsolenoid actuator

CR(α): Bearing torque coefficient as afunction of the valve angle (α)

Ds: Stem diameter (m)Dp: Pipe diameter (inch or m)Fmag: Magnetic force (N)Fc: Contact force or resultant force

Fr(N)i: Current of solenoid actuator (A)J : Inertia moment (kgm2)k: Spring stiffness (N/m)m: Mass of solenoid plunger (kg)μ: Friction coefficient of bearing areaN : Number of turns of the coilΔPv: Valve differential pressure (psi)p: Number of estimatesrg : Radius of pinion gear (m)ρ: Fluid mass density (kg/m3)u: Control input (A2)

Tc(α): Hydrodynamic torque coefficient as afunction of the valve angle (α)

Tb, Th: Bearing and hydrodynamic torques,respectively (Nm)

VJ , VO, (VJ /VO)(α): Jet velocity, mean flow velocity, andtheir ratio as a function of the valveangle (α), respectively.

Acknowledgments

This research is supported by the Office of Naval Research(N00014-08-1-0435), which the authors gratefully acknowl-edge. Thanks are in particular due to Mr. Anthony SemanIII. The authors would also like to thank Dr. Stephen Mastroand Mr. Frank Ferrese of Naval Surface Warfare Center(NSWC, Philadelphia) for help with many aspects of thepaper. They are grateful to the anonymous reviewer forcritical comments, which led to substantial improvement ofthe paper.

References

[1] A. Seman, “Adaptive automation for machinery control,” inProceedings of the Office of Naval Research (ONR) Presentation,2007.


[2] R. Hughes, S. Balestrini, K. Kelly, N. Weston, and D. Mavris,“Modeling of an integrated reconfigurable intelligent system(IRIS) for ship design,” in Proceedings of the ASNE Ships & ShipSystems Symposium, 2006.

[3] P. Tich, V. Mark, P. Vrba, and M. Pechoucek, “Chilled watersystem automation,” Rockwell Automation Case Study Report,2005.

[4] J. R. Brauer, Magnetic Actuators and Sensors, Wiley IEEE Press,Hoboken, NY, USA, 2006.

[5] Z. Leutwyier and H. Dalton, “A CFD study of the flow field,resultant force, and aerodynamic torque on a symmetric diskbutterfly valve in a compressible fluid,” Journal of PressureVessel Technology, vol. 130, Article ID 021302, pp. 1–10, 2008.

[6] T. Sarpkaya, “Oblique impact of a bounded stream on a planelamina,” Journal of the Franklin Institute, vol. 267, no. 3, pp.229–242, 1959.

[7] T. Sarpkaya, “Torque and cavitation characteristics of butterflyvalves,” Journal of Applied Mechanics, vol. 29, pp. 511–518,1961.

[8] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinearand Adaptive Control Design, Series on Adaptive and LearningSystems for Signal Processing, John Wiley & Sons, New York,NY, USA, 1st edition, 1995.

[9] D. B. Lee, C. Nataraj, and P. Naseradinmousavi, “Nonlinearmodel-based adaptive control of a solenoid-valve system,” inProceedings of the ASME Dynamic Systems and Control Con-ference (DSCC ’10), pp. 1–8, Boston, Mass, USA, September2010.

[10] J. Pohl, M. Sethson, P. Krus, and J.-O. Palmberg, “Modellingand validation of a fast switching valve intended for combus-tion engine valve trains,” Proceedings of the Institution of Mech-anical Engineers. Part I: Journal of Systems and Control Engi-neering, vol. 216, no. 2, pp. 105–116, 2002.

[11] C. Nataraj, Vibration of Mechanical Systems, Cengage, 1st edi-tion, 2011.

[12] P. Naseradinmousavi and C. Nataraj, “Nonlinear mathemati-cal modeling of butterfly valves driven by solenoid actuators,”Applied Mathematical Modelling, vol. 35, no. 5, pp. 2324–2335,2011.

[13] M. K. Zavarehi, P. D. Lawrence, and F. Sassani, “Nonlinearmodeling and validation of solenoid-controlled pilot-operat-ed servovalves,” IEEE/ASME Transactions on Mechatronics, vol.4, no. 3, pp. 324–334, 1999.

[14] R. R. Chladny, C. R. Koch, and A. F. Lynch, “Modeling auto-motive gas-exchange solenoid valve actuators,” IEEE Transac-tions on Magnetics, vol. 41, no. 3, pp. 1155–1162, 2005.

[15] C. Nataraj and P. Mousavi, “Nonlinear analysis of solenoidactuators and butterfly valve systems,” in Proceedings of the14th International Ship Control Systems Symposium, pp. 1–8,Ottawa, Canada, September 2009.

[16] J. Y. Park and M. K. Chung, “Study on hydrodynamic torqueof a butterfly valve,” Journal of Fluids Engineering, Transactionsof the ASME, vol. 128, no. 1, pp. 190–195, 2006.

[17] J. B. Pomet and L. Praly, “Adaptive nonlinear regulation: esti-mation from the Lyapunov equation,” IEEE Transactions onAutomatic Control, vol. 37, no. 6, pp. 729–740, 1992.

[18] H. K. Khalil, Nonlinear Systems, Prentice Hall, Upper SaddleRiver, NJ, USA, 3rd edition, 2002.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


nonlineardynamicmodel-basedadaptivecontrolof asolenoid ... · 2 journal of control science and...

Documents