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Journal of Control Science and Engineering
Volume 2012 (2012), Article ID 846458, 13 pages
http://dx.doi.org/10.1155/2012/846458
Research Article

Nonlinear Dynamic Model-Based Adaptive Control of a Solenoid-Valve System

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

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.

Abstract

In this paper, a nonlinear model-based adaptive control approach is proposed for a solenoid-valve system. The challenge is that solenoids and butterfly valves have uncertainties in multiple parameters in the nonlinear model; various kinds of physical appearance such as size and stroke, dynamic parameters including inertia, damping, and torque coefficients, and operational parameters especially, pipe diameters and flow velocities. These uncertainties are making the system not only difficult to adjust to the environment, but also further complicated to develop the appropriate control approach for meeting the system objectives. The main contribution of this research is the application of adaptive control theory and Lyapunov-type stability approach to design a controller for a dynamic model of the solenoid-valve system in the presence of those uncertainties. The control objectives such as set-point regulation, parameter compensation, and stability are supposed to be simultaneously accomplished. The error signals are first formulated based on the nonlinear dynamic models and then the control input is developed using the Lyapunov stability-type analysis to obtain the error bounded while overcoming the uncertainties. The parameter groups are updated by adaptation laws using a projection 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 systems such as marine vessels or ship-based machinery system [2], solenoid actuators and valves are often used [3] to increase survivability and capability. One typical type of actuator driven by solenoids is shown in Figure 1, which is operated by the electromagnetic force. The electric-driven solenoid valve system [4] and its sophisticated control can provide high levels of automation in large systems. The useful function of the solenoid-valve, once an electrical signal (current or voltage) is applied, is to activate a mechanical motion such as displacement or rotation via the solenoid magnetic forces and torques. The proportional solenoids normally require integrated electronics for controlling the plunger to give such a signal. Hydrodynamic torque of a butterfly valve comprises the core knowledge of fluid valve system design [5], and it is known that most of the valves in real systems have strongly 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 the actuator-valve machinery systems will benefit a wide spectrum of nonlinear systems, compensating for nonlinearities [10] and dynamic characteristics. This approach will not only decrease the amount of cost and casualties but also improve the performance of the mechatronic system. To investigate the particular application, it is important to emphasize the nonlinear dynamic modeling analysis of such actuator-valve systems because the accuracy and reliability of these systems depend highly on the mathematical system modeling [11] and its validation. In [12], the authors developed and analyzed the nonlinear dynamic model of a solenoid-valve system; the reader is also referred to [13, 14] for recent modeling and analysis of solenoid actuators.

846458.fig.001
Figure 1: System Configuration.

This paper will focus on model-based nonlinear adaptive control of an actuator-butterfly valve. The solenoid-valve system is described based on the exact model knowledge of the system. Figure 1 shows the integrated system, which consists of an electric-driven solenoid and a butterfly valve. The valve operates by solenoids that use a magnetic coil to move a movable plunger connected with the valve stem by means of a gear train and linkage. The control input is designed by substituting the current signal from the model of the electromagnetic force, pulling the plunger, and then controlling the angular position of the butterfly valve. The system has uncertainties in multiple parameters in the dynamic model, which requires the system to continuously adjust to the environment and consequently requires adaptation for sustainability and capability. The integrated system is highly nonlinear in addition to its parameter uncertainties. Hence, an adaptation law is proposed [8, 9, 17] and an adaptive control method is developed for the solenoid-valve system in multiple parametric uncertainties. A closed-loop stable controller is designed for the set-point trajectory tracking by introducing a Lyapunov-based stability analysis [9] based on the error signals of the nonlinear solenoid-valve system. The numerical results in the simulation are used for initial verification and performance evaluation.

2. Model-Based Nonlinear System

2.1. System Model

The dynamic equations of motion of the plunger and butterfly valve are given by [15] 𝑚̈𝑥+𝐵1̇𝑥+𝑘𝑥=𝐹mag𝐹𝑐,𝐽̈𝑎+𝐵2̇𝛼=𝑟𝑔𝐹𝑐𝑇tot,(1) where 𝑥(𝑡) is the displacement of the solenoid plunger, 𝛼(𝑡) is the angle of butterfly disk, and 𝑇tot(𝑡) is the sum of the hydrodynamic and bearing torques expressed as 𝑇tot=𝑇𝑏+𝑇. The bearing torque is given as 𝑇𝑏=(𝜋/8)𝜇𝐷𝑠𝐷2𝑝Δ𝑃𝑣𝐶𝑅(𝛼), where the bearing toque coefficient, 𝐶𝑅(𝛼)=𝐶2𝐿+𝐶2𝐷=(1.14sin((𝛼/90)3180))2+(3cos𝛼)2, is obtained from the valve modeling (see [5, 15]) and the two subterms, lifting force 𝐶𝐿=1.14sin((𝛼/90)3180) and drag force 𝐶𝐷=3cos𝛼, are nonlinear functions of the valve angle rotation 𝛼(𝑡). The hydrodynamic torque 𝑇 is obtained by reviewing three-dimensional hydrodynamic torque coefficient based on [7, 16] as 𝑇=(8/3𝜋)𝜌𝐷3𝑝𝑉2𝑂𝑇𝑐(𝛼)[(𝑉𝐽/𝑉𝑂)(𝛼)]2, where both 𝑇𝑐(𝛼) and (𝑉𝐽/𝑉𝑂)(𝛼) depend on the closing angle (𝛼) of the butterfly valve and 𝐷3𝑝 term is a nonlinear term according to the pipe size. Solving the two equations in (1) with the contact force, 𝐹𝑐(𝑡), and substituting the magnetic force, 𝐹mag(𝑡), into the equation yields 𝑚+𝐽𝑟2𝑔𝑟𝑔̈𝑥+𝐵1+𝐵2𝑟2𝑔𝑟𝑔̇𝑥+𝑘𝑟𝑔𝑥=𝑟𝑔𝐶2𝑁22𝐶1+𝐶2𝑥2𝑖2𝑇tot,(2) where the magnetic force 𝐹mag=(𝐶2𝑁2/2(𝐶1+𝐶2𝑥)2)𝑖2 used to lift the plunger of the solenoid actuator. The actuator is a current-controlled solenoid [15], proportional to the square of the current 𝑖(𝑡), and 𝐶1, 𝐶2 are reluctances of the magnetic paths, obtained from the geometry of electric actuator [4, 15]. It is assumed that the pinion and the valve are moving at the same speed, that is, the gear ratio is 1 : 1 and 𝑥(𝑡)=𝑟𝑔𝛼 is the simple geometric relationship between the displacement of the pinion and the valve angle. Hence, the current source 𝑖2(𝑡) is substituted for designing the closed-loop control input, 𝑢(𝑡), and then the following equation is obtained: 𝑚+𝐽𝑟2𝑔𝑟𝑔̈𝑥+𝐵1+𝐵2𝑟2𝑔𝑟𝑔̇𝑥+𝑘𝑟𝑔𝑥=𝑟𝑔𝐶2𝑁22𝐶1+𝐶2𝑥2𝑢𝑇tot.(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)̈𝑥+𝐶(𝑥,𝜃)̇𝑥+𝐷(𝑥,𝜃)𝑥=𝑢𝐵𝑇𝑏+𝑇,(4) where 𝜃 describes a lumped expression of parameters obtained from (3) and (4), where each parameter is shown in Table 2 and the substituted terms are defined as follows: 𝑀(𝑥,𝜃)=𝑚+𝐽𝑟2𝑔𝑟𝑔𝐵,𝐵=2𝐶1+𝐶2𝑥2𝑟𝑔𝐶2𝑁2,𝐶(𝑥,𝜃)=𝐵1+𝐵2𝑟2𝑔𝑟𝑔𝐵,𝐷(𝑥,𝜃)=𝑘𝑟𝑔𝐵.(5)

2.2. Error Signals Formulation

The following set-point control approach is used. Let 𝑥𝑑(𝑡) define the set-point trajectory and then the error can be defined as 𝑒𝑥𝑑𝑥,̇𝑒=̇𝑥𝑑̇𝑥,̈𝑒=̈𝑥𝑑̈𝑥,(6) where ̇𝑥𝑑(𝑡) and ̈𝑥𝑑(𝑡) are the first and second time derivatives of 𝑥𝑑(𝑡), which are assumed to be bounded. Premultiplying ̈𝑒(𝑡) in the last error signal of (6) with 𝑀(𝑥,𝜃) yields 𝑀(𝑥,𝜃)̈𝑒=𝑀(𝑥,𝜃)̈𝑥𝑑𝑀(𝑥,𝜃)̈𝑥. Substituting 𝑀(𝑥,𝜃)̈𝑥 in (4) into the above equation produces 𝑀(𝑥,𝜃)̈𝑒=𝑀(𝑥,𝜃)̈𝑥𝑑+𝐶(𝑥,𝜃)̇𝑥+𝐷(𝑥,𝜃)𝑥𝑢+𝐵𝑇𝑏+𝑇.(7)

A filtered error signal and its derivative are defined as 𝑟̇𝑒+𝜆1𝑒,̇𝑟=̈𝑒+𝜆1̇𝑒,(8) where 𝜆1+ is a positive adjustable control gain. Multiplying (8) with 𝑀(𝑥,𝜃) and then substituting for 𝑀(𝑥,𝜃)̈𝑒(𝑡) in (7) yields 𝑀(𝑥,𝜃)̇𝑟=𝑀(𝑥,𝜃)̈𝑥𝑑+𝐶(𝑥,𝜃)̇𝑥+𝐷(𝑥,𝜃)𝑥𝑢+𝐵𝑇𝑏+𝑇+𝑀(𝑥,𝜃)𝜆1̇𝑒+1𝑟2𝑔𝑒1𝑟2𝑔𝑒,(9) where the last term 𝑒(𝑡)/𝑟2𝑔 is added and subtracted for further development of the control design based on Lyapunov’s method.

3. Lyapunov-Based Adaptive Feedback Control

Let 𝑉(𝑡) be a Lyapunov candidate function 𝑉=12𝑟𝑇𝑀𝑟+𝑒𝑇𝑒+𝑒𝑇𝛼𝑒𝛼+Θ𝑇Γ1Θ,(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 estimation error, Θ, is defined as Θ=ΘΘ, where Θ𝑝 is a known constant parameter vector and Θ𝑝 is the estimated constant parameter vector (see Table 1). Differentiating (10) yields ̇𝑉=𝑟𝑇𝑀̇𝑟+12𝑟𝑇̇𝑀𝑟+𝑒𝑇̇𝑒+1𝑟2𝑔𝑒𝑇̇𝑒Θ𝑇Γ1̇Θ,(11) where the time derivative of the inertia matrix is obtained as ̇𝑀=(𝑚+𝐽/𝑟2𝑔)(4(𝐶1+𝐶2𝑥)/𝑁2)̇𝑥, where ̇𝐵=(4(𝐶1+𝐶2𝑥)/𝑟𝑔𝑁2)̇𝑥, 𝑒𝑇𝛼̇𝑒𝛼=(1/𝑟2𝑔)𝑒𝑇̇𝑒 as 𝑒𝛼𝛼𝑑𝛼=𝑥𝑑/𝑟𝑔𝑥/𝑟𝑔=𝑒/𝑟𝑔, ̇𝑒𝛼̇𝛼𝑑̇𝛼=̇𝑥𝑑/𝑟𝑔̇𝑥/𝑟𝑔=̇𝑒/𝑟𝑔, in which the error signals of the valve angle, 𝑒𝛼(𝑡), can be defined using the geometric relationship and ̇Θ(𝑡)=̇Θ comes from the definition of Θ.

tab1
Table 1: Regression and parameter estimation terms.
tab2
Table 2: List of simulation parameters.
3.1. Design of Control Input

Substituting 𝑀(𝑥,𝜃)̇𝑟(𝑡) into (11) yields ̇𝑉=𝑟𝑇𝑀̈𝑥𝑑+𝐶̇𝑥+𝐷𝑥𝑢+𝐵𝑇𝑏+𝑇+𝑀𝜆1̇𝑒+1𝑟2𝑔𝑒(̇𝑒+𝑒)𝑇𝑒𝑟2𝑔+12𝑟𝑇̇𝑀𝑟+𝑒𝑇̇𝑒+1𝑟2𝑔𝑒𝑇̇𝑒Θ𝑇Γ1̇Θ,(12) where the last term in (9) premultiplied by 𝑟(𝑡) came out of the parenthesis in (12) and is used for the definition of 𝑟(𝑡) in (8). Then, combining the parameterized terms in (12) and substituting them into 𝑊Θ yields 𝑊Θ=𝑀̈𝑥𝑑+𝐶̇𝑥+𝐷𝑥+𝐵𝑇𝑏+𝑇+𝑀𝜆1̇𝑒+1𝑟2𝑔𝑒+12̇𝑀𝑟,(13) where 𝑊(̈𝑥𝑑,̇𝑥,𝑥,𝑟,𝛼,̇𝑒,𝑒)1×𝑝 is a known regression vector, which is shown in the left side of Table 1 via the process given later (see (17)) and Θ as the nominal value of the lumped parameter vector. Rearranging (12) produces ̇𝑉=𝑟𝑇(𝑊Θ𝑢)𝑒𝑇𝑒𝑟2𝑔+𝑒𝑇̇𝑒Θ𝑇Γ1̇Θ,(14) where ̇𝑒𝑇𝑒/𝑟2𝑔 is canceled in the last second term in (12), having the opposite sign because they are scalar, ̇𝑒𝑇𝑒=𝑒𝑇̇𝑒.

The control input can be designed based on Lyapunov stability analysis, making ̇𝑉 negative definite to be shown in the end, as 𝑢=𝑊Θ+𝑘1𝑟+𝑒,(15) where 𝑟(𝑡) is a feedback error term, 𝑘1 is a positive constant as the control gain, 𝑒(𝑡) is another feedback error term added to cancel the term having the opposite sign, 𝑒𝑇̇𝑒, outside the parenthesis by utilizing the definition of 𝑟(𝑡) given in (8), and 𝑊Θ captures the uncertainties associated with the elements of 𝑀, 𝐶, 𝐷, 𝐵(𝑇𝑏1+𝑇1), 𝑀𝜆1, 1/𝑟2𝑔, and ̇𝑀, which is defined as 𝑊Θ=𝑀̈𝑥𝑑+𝐶̇𝑥+𝐷𝑥+𝐵̂𝑇𝑏1𝐶𝑅(𝛼)+̂𝑇1𝑇𝑐(𝛼)𝑉𝐽𝑉𝑂(𝛼)2+𝑀𝜆1̇𝑒+1̂𝑟2𝑔𝑒+12̇𝑀𝑟,(16) where the estimated parameter sets are given as 𝑀(𝑥,𝜃)=(𝑚+̂𝐽/̂𝑟2𝑔)̂𝑟𝑔𝐵, where 𝐵=2(𝐶1+𝐶2𝑥)2/̂𝑟𝑔𝐶2𝑁2, 𝐶(𝑥,𝜃)=(𝐵1+𝐵2/̂𝑟2𝑔)̂𝑟𝑔𝐵, 𝐷(𝑥,𝜃)=̂𝑘̂𝑟𝑔𝐵, ̂𝑇𝑏1=(𝜋/8)𝜇𝐷𝑠𝐷2𝑝Δ𝑃𝑣, and ̂𝑇1=(8/3𝜋)̂𝜌𝐷3𝑝𝑉2𝑂. In order to develop the estimate parameter vector Θ in (16), we need to first define the regression and the estimate terms. Thus, the first term, 𝑀̈𝑥𝑑, in (16) can be defined as 𝑀̈𝑥𝑑=𝑚+̂𝐽̂𝑟2𝑔̂𝑟𝑔2𝐶1+𝐶2𝑥2̂𝑟𝑔𝐶2𝑁2̈𝑥𝑑=𝑀𝑠2𝐶21𝐶2𝑁2̈𝑥𝑑+𝑀𝑠4𝐶1𝑁2𝑥̈𝑥𝑑+𝑀𝑠2𝐶2𝑁2𝑥2̈𝑥𝑑=̈𝑥𝑑𝑥̈𝑥𝑑𝑥2̈𝑥𝑑𝑊101𝑊103𝑀𝑠2𝐶21𝐶2𝑁2𝑀𝑠4𝐶1𝑁2𝑀𝑠2𝐶2𝑁2Θ101Θ103,(17) where 𝑀𝑠=(𝑚+̂𝐽/̂𝑟2𝑔),𝐶𝑠=(𝐵1+𝐵2/̂𝑟2𝑔) (this is shown in Table 1), and ̂𝑟𝑔, 𝐶1, or 𝐶2 are canceled. 𝑊101𝑊103 are the measurable regression terms and Θ101Θ103 is the estimated 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

The following is constructed to define the known upper and lower bounds but with an unknown parameter of Θ(𝑡) in the sense that Θ_𝑗Θ𝑗(𝑡)Θ𝑗,(18) where Θ𝑗(𝑡) are the estimated parameters as shown in Table 1 and Θ_𝑗 and Θ𝑗 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 ̇Θ𝑗(𝑡) is designed to update using a projection-based algorithm as ̇Θ𝑗=ProjΓ𝑊𝑇𝑟,Θ𝑗,(19) where Proj{} is the projection operator [8] and each lumped parameter is adaptively updated using the adaptation laws [17] for online estimation of unknown parameter as follows: Proj̇Θ𝑗=ProjΓ𝑊𝑇𝑟,Θ𝑗=Γ𝑊𝑇𝑟ifΘ𝑗>Θ_𝑗,Θ𝑗<Θ𝑗,Γ𝑊𝑇𝑟ifΘ𝑗=Θ_𝑗,ifΓ𝑊𝑇𝑟>0,Γ𝑊𝑇𝑟ifΘ𝑗=Θ𝑗,ifΓ𝑊𝑇𝑟0,0elsewhere.(20) Thus, substituting 𝑢(𝑡) in (15) into ̇𝑉 in (13) yields ̇𝑉=𝑟𝑇𝑊Θ𝑟𝑇𝑘1𝑟(̇𝑒+𝑒)𝑇𝑒𝑒𝑇𝑒𝑟2𝑔+𝑒𝑇̇𝑒Θ𝑇Γ1̇Θ=𝑟𝑇𝑘1𝑟1+1𝑟2𝑔𝑒𝑇𝑒+Θ𝑇𝑊𝑇𝑟Γ1̇Θ.(21) Here, 𝑊Θ is defined as 𝑊Θ=𝑀̈𝑥𝑑+𝜆1̇𝑒+𝐶̇𝑥+𝐷𝑥+𝐵̃𝑇1𝑇𝑐(𝛼)𝑉𝐽𝑉𝑂(𝛼)+𝐵̃𝑇𝑏1𝐶𝑅(𝛼)+̃𝑟𝑔𝑒+12̇𝑀𝑟,(22) where 𝑀=𝑀𝑀, 𝐶=𝐶𝐶, 𝐷=𝐷𝐷, 𝐵̃𝑇1=𝐵𝑇1𝐵̂𝑇1, 𝐵̃𝑇𝑏1=𝐵𝑇𝑏1𝐵̂𝑇𝑏1, ̃𝑟𝑔=1/𝑟2𝑔1/̂𝑟2𝑔, and ̇𝑀=̇𝑀̇𝑀, in which ̂𝑇1=(8/3𝜋)̂𝜌𝐷3𝑝𝑉2𝑂 and ̂𝑇𝑏1=(𝜋/8)𝜇𝐷𝑠𝐷2𝑝Δ𝑃𝑣. Actually, Θ(𝑡) can be expressed, for example, using the definitions of Θ in (16), as follows: owing to the subsequent adaptation law, the time derivative of 𝑉(𝑡) yields a negative definite function except the origin and upper bound by ̇𝑉𝑘1𝑟21+1𝑟2𝑔𝑒2,(23) which can be written as ̇𝑉𝑘2𝑧2,(24) where 𝑘2=min{𝑘1,(1+1/𝑟2𝑔)} is a positive constant and 𝑧=𝑟𝑇,𝑒𝑇𝑇.(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) of ̇𝑉, the property of 𝑉, and the control law of (15) with the parameter updates of (19) and the projection-based method of update laws of (20), it is straightforward to derive a conclusion that the tracking error vector 𝑧(𝑡) in (25) is driven to zero. Thus, the set-point errors 𝑟(𝑡), 𝑒(𝑡), and 𝑒𝛼(𝑡) also vanish and the parameter estimation error vector Θ in (10) is bounded where Θ, defined after (10), is bounded due to the projection-based update method and the constant known parameter, Θ. Owing to the bounds of 𝑟(𝑡), 𝑒(𝑡) in (8), ̇𝑒(𝑡) is bounded and then 𝑥(𝑡), resulting in 𝛼(𝑡), and ̇𝑥(𝑡) are bounded, respectively, where all desired trajectories such as 𝑥𝑑(𝑡) and ̇𝑥𝑑(𝑡) are assumed to be bounded. 𝑀(), 𝐵(), 𝐶(), and 𝐷() matrices in (4) are bounded because 𝜃 (due to Θ) and 𝑥(𝑡) are bounded, and ̇𝐵 and ̇𝑀 after (11) are also bounded owing to the bounds of ̇𝑥(𝑡). 𝑇tot is thus bounded because 𝛼(𝑡) is bounded. Hence, 𝑊() and the control input 𝑢(𝑡) are bounded and, thus, the current is bounded. This leads to the boundedness of ̈𝑥(𝑡) in the dynamic model given in (4), which enables the boundedness of ̈𝑒(𝑡) in (6), and then the set-point tracking error dynamics ̇𝑟(𝑡) in (8) is bounded. Therefore, we can conclude that all signals are bounded.

Remark 2. 𝑊Θ in (16) where the known regression 𝑊(̈𝑥𝑑,̇𝑥,𝑥,𝑟,𝛼,̇𝑒,𝑒)1×23 terms and the parameter estimates Θ23 are given in this system as in Table 1.

4. Simulation Results

Based on the dynamic model in (4), the numerical simulation is performed to verify the proposed controllers with consistently changing parameter values. The parameter values can be divided into two categories: operational values such 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], the pipe diameter 𝐷𝑝 shown in Table 2 is used to vary from 5 inches to 8 inches as 𝐷𝑝={5.0,6.0,7.0,8.0}[in], where 𝐷𝑝={0.1270,0.1524,0.1778,0.2032}[m], by assuming that the pipe or transmission lines are different according to their applications. With each 𝐷𝑝 size, the variations of all the parameters are set to 30% for the simulation results provided 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 the experimental data. The amount of upper and lower variation of 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 are used for the simulation. The desired set-point distance of the solenoid, 𝑥𝑑(𝑡), is given as 𝐴(1𝑒𝐵𝑡), where 𝐴=0.0148 and 𝐵=5.

A typical parameter set for this simulation is given by Table 2. Figures 2 and 3 show the rotation of the valve angle and the actual displacement of the plunger, respectively. This adaptive control approach shows better results compared to the results obtained from the previous research [15] using nonadaptive method, which are shown in Figures 4 and 5 for the displacement of the plunger and the angle of the butterfly valve. The developed mathematical model is the same as that of adaptive method based on the nonlinear models and the new specific approach in this paper is that the error signals are formulated by introducing the desired trajectory to reducing the displacement error in the adaptive method while overcoming the complicated parametric uncertainties. And also the response of the adaptive method shows that the motion process is more consistent in both variables, showing overdamped phenomena and short rising times, and then it quickly reaches the steady state. The actual movement is smoother and faster while all the parameters are varying 30% in each flow velocity.

846458.fig.002
Figure 2: Valve rotation angle 𝛼(𝑡) (adaptive approach).
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Figure 3: Plunger displacement 𝑥(𝑡) (adaptive approach).
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Figure 4: Valve rotation angle 𝛼(𝑡) (no adaptation scheme).
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Figure 5: Plunger displacement 𝑥(𝑡) (no adaptation scheme).

Figures 6 and 7 are the filtered error 𝑟(𝑡), defined in (8), and set-point error 𝑒(𝑡), given in (6), respectively. Figure 8 shows the rate of the displacement to Figure 3. From Figure 8, all figures are presented without the notation of the adaptive approach. Figure 9 shows the control input 𝑢(𝑡) of the solenoid actuator for each pipe diameter, 𝐷𝑝, designed in (15) with the nonlinear adaptive controller by substituting the square of the current, 𝑖2(𝑡). The electromagnetic force 𝐹mag(𝑡) given in (2) and the total torque 𝑇𝑡(𝑡) given in (1) by summing 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 jet velocities shown in Figure 13 are changed for every 5 of the butterfly angle. As the strokes are increased by the control input, the angles get larger and then accordingly the value of the inlet jet velocity increases, which affects the slower motion of the strokes and angles but the variables (strokes and angles) reach the steady state and the ratio as well.

846458.fig.006
Figure 6: Filtered Error 𝑟(𝑡) (adaptive approach).
846458.fig.007
Figure 7: Displacement tracking error 𝑒(𝑡) (adaptive approach).
846458.fig.008
Figure 8: The rate of displacement ̇𝑥(𝑡).
846458.fig.009
Figure 9: Control input 𝑢(𝑡) equivalent to square current 𝑖2(𝑡)[A2].
846458.fig.0010
Figure 10: Electromagnetic force 𝐹mag(𝑡).
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Figure 11: Total torque 𝑇𝑡(𝑡).
846458.fig.0012
Figure 12: Hydrodynamic torque coefficient 𝑇𝑐(𝛼).
846458.fig.0013
Figure 13: The ratio of 𝑉𝐽 and 𝑉𝑂: (𝑉𝐽/𝑉𝑂)(𝛼).

The challenge is that most parameter terms in Table 1 are combined and lumped together due to the dynamic model in the presence of parameter uncertainty and the model is complicated owing to the control objectives, set-point regulation, and parametric adaptation. As given in the right side of Table 1, the unknown bounded parameter estimate vector Θ23 is shown in Figures 14, 15, 16, 17, 18, and 19. It can be seen that the parameters such as 101, 110, 111, 113, 116, 118, and 121 are quickly updated in the form of premultiplication by its regression term, 𝑊 and go to steady state. Thus, they are more parametric-centric terms and related to the filtered error 𝑟(𝑡). The parameters such as 102, 103, 104, 105, 117, 119, and 122 are updated according to the motion of the angle and stroke of the solenoid-butterfly valve system. Some parameters such as 106, 112, 114, and 115 are not updated but other parameters such as 107, 108, and 120 keep getting updated. Any parameters changing their values must be related to the control objective because these terms are incorporated into the control input in the form of the estimates.

846458.fig.0014
Figure 14: Parameter estimates: Θ101Θ104.
846458.fig.0015
Figure 15: Parameter estimates: Θ105Θ108.
846458.fig.0016
Figure 16: Parameter estimates: Θ109Θ112.
846458.fig.0017
Figure 17: Parameter estimates: Θ113Θ116.
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Figure 18: Parameter estimates: Θ117Θ120.
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Figure 19: Parameter estimates: Θ121Θ123.

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, and hydrodynamic forces and torques, is chosen as a continuation of previous research and an adaptive stable control approach with adaptation laws is developed accounting for uncertainties in multiple parameters on the nonlinear dynamic model. A stable adaptive controller of the solenoid-valve system is designed positioning the angle of the butterfly valve via a Lyapunov-based approach. The approach yields bounded error while adapting to the environment in the presence of complex uncertainties such as different physical appearances, uncertain parameters, operational characteristics, and parametric nonlinear dynamic models.

The parameter estimation for the unknown bounded parameters is performed using a projection algorithm whose output yields the upper and lower bounds. Numerical simulation is used to verify the performance of the proposed approach to show its effectiveness by comparing to the same dynamic model without adaptation from the previous research; when compared to the nonadaptive method, the responses of the plunger displacement and the rotating angle are steadier, smoother, and faster. Future work will be focused on demonstrating the results of hardware-in-the-loop or experiments for the nonlinear solenoid valve system as well as applying the suggested adaptive method based on Lyapunov-based control approach to the real-world system. Further research on developing control techniques using robust or optimal method would be continued to overcome nonlinearities such as hysteresis or nonlinear dynamics.

Nomenclature

𝐵1, 𝐵2:Damping coefficients of the solenoid and butterfly valve, respectively (Ns/m)
𝐶1, C2:Reluctances of the magnetic paths obtained from plunger geometry of solenoid actuator
𝐶𝑅(𝛼):Bearing torque coefficient as a function 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 area
𝑁:Number of turns of the coil
Δ𝑃𝑣:Valve differential pressure (psi)
𝑝:Number of estimates
𝑟𝑔:Radius of pinion gear (m)
𝜌:Fluid mass density (kg/m3)
𝑢:Control input (A2)
𝑇𝑐(𝛼):Hydrodynamic torque coefficient as a function of the valve angle (𝛼)
𝑇𝑏, 𝑇:Bearing and hydrodynamic torques, respectively (Nm)
𝑉𝐽, 𝑉𝑂, (𝑉𝐽/𝑉𝑂)(𝛼)Jet velocity, mean flow velocity, and their ratio as a function of the valve angle (𝛼), respectively.

Acknowledgments

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

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