Robust controller design of the integrated direct drive volume control architecture for steering systems

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Robust controller design of the integrated direct drive volume control architecture for steering systems Wei Shen a,n, Yu Pang a, Jihai Jiang b a b

Department of Mechatronics Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150080, China

art ic l e i nf o

a b s t r a c t

Article history: Received 19 January 2017 Received in revised form 19 April 2017 Accepted 12 May 2017

Recently, much effort has been directed toward the large throttling loss and low efficiency of the valve control system widely applied in steering system of ships. This paper presents an Integrated Direct-Drive Volume Control (IDDVC) electro-hydraulic servo system with the advantages of high efficiency and energy conservation. Firstly, the simulation model of IDDVC is improved by software AMESim, including the nonlinear interaction of the motor-pump and the oil supply ignored by traditional transfer function model. Then, by establishing discrete state equations, a controller based on robust sliding control strategy has been designed to enhance the practicality and real-time performance. Finally, the accuracy of the model and the effectiveness of the controller are proved through the experiments which are conducted after constructing the IDDVC prototype. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: IDDVC Hydraulic actuator AMESim Sliding mode controller

1. Introduction The steering gear(SG) hydraulic system is one of the most important marine auxiliary systems which can control ship course and ensure navigation maneuverable. According to the traditional architecture, hydraulic servo valves are often chose as the control units which have fast response and high control precision. However, this system also exists several problems. Firstly, the throttling loss is large and the efficiency is low. This disadvantage is becoming much more serious under the background of energy shortages and environmental problems, recently [1–3]. Secondly, the requirement for the cleanliness of oil is high and its reliability is poor [4]. Besides, the cost of servo valve and integrated system is unaffordable [5–7]. Benefiting from the rapid development of converter technique, a new electro-hydraulic servo system, Direct Drive Volume Control(DDVC), has been investigated during the past 25 years [8,9]. Compared with the valve-controlled system, DDVC has the advantages of high efficiency, easy operation and low cost. Therefore, it provides a promising direction to realize innovations by applying DDVC into SG. DDVC has the similar principle with the Electro-Hydraulic Actuator (EHA) [10]. Both modeling and control algorithm research are investigated in recent years [11]. For modeling, the models among most papers are idealistic after simplifying the dynamic process. Most documents focus on the positive and negative n

Corresponding author. E-mail address: [email protected] (W. Shen).

discontinuity of the Coulomb friction for the actuators. Only a minority of the papers refer to the nonlinear interaction of the motor-pump and oil supply, and fewer documents establish simulation model to simulate this property [12]. Besides, as the modern control theory gradually replaces the classical one, it can be found that there are several researchers focusing on the control algorithm to improve the control performance from both of theory and practical aspects, including back-stepping [13], fuzzy neural networks [14–16], adaptive control or their combination [17,18]. Especially, sliding mode control method is preferred in practical system because it is inherently robust against the system uncertainty and the external disturbance, and has a good transient response [19]. Moreover, some fault diagnosis strategies are also considered for improving the control performance [20,21]. However, there are an ocean of researches aimed at DDVC continuous system which lacks the transformation to the discrete system and limits the application into practical engineering projects. Furthermore, although quite a few control algorithms possess good simulation performance, the calculation is tedious and it is hard to guarantee the real-time performance. Therefore, it is meaningful for applying DDVC on SG hydraulic system to reduce the energy consumption and improve the control performance. However, the precondition is making a precise model for the DDVC system, especially investigating the control algorithm which has quick real-time and good practical performance. In this paper, an Integrated Direct-Drive Volume Control (IDDVC) electro-hydraulic servo system is presented for SG system application, and both of the controller simulation and experiment are proceeded to verify the effectiveness.

http://dx.doi.org/10.1016/j.isatra.2017.05.008 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

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2. Modeling analysis for the IDDVC The equivalent schematic of the SG electro-hydraulic actuator based on IDDVC is shown in Fig. 1. It uses adjustable-speed servo motor to drive bi-direction hydraulic pump, thus replacing the servo valve and servo variable pump used in conventional electrohydraulic system. The direction, speed and position of the hydraulic cylinder can be controlled by changing the direction, speed or running time of the servo motor. By limiting the torque of the servo motor, the maximum pressure of the hydraulic cylinder can be restricted, thus playing the role of the overload protection. It should be noticed that one single-rod double-acting symmetrical hydraulic cylinder is used in this setting. Fig. 2 shows two kinds of single-rod double-acting hydraulic cylinders, the right one has the equal acting area (A1 ¼A2) of the two chambers compared with the left one. Hence, it should reduce the control difficulty because the unbalanced flow rate effect is not needed to consider. For the calculation and simulation of the IDDVC, the transfer function model for the power mechanism of the pump controlled cylinder is adopted. Although it is concise and efficient, quite a few dynamic details of the model are ignored. This paper, first of all, has built an IDDVC transfer function model. And then, after theoretical analysis, simulation model is built under AMESim environment in order to precisely describe the working process of the IDDVC. The model can reflect the nonlinearity of the system, which supports the controller design. 2.1. Transfer function model of the IDDVC The model can be divided into synchronous motor subsystem and hydraulic subsystem. This is a traditional process and this part is omitted. The motor can be simplified as a proportional component after comparing the frequency response between the prototype and servo motor which are 2 Hz and 500 Hz respectively.

Gsp =

Wp(s ) = Ksp U *(s )

(1)

where Ksp is the driver gain of the servo motor, Ksp ¼ 10π rad/s. The IDDVC can be simplified as the power mechanism of pump controlled cylinder. Then, the IDDVC actuator needs to overcome internal unbalanced force, friction, viscous force, preload and other forces, which has strong time-variant characteristic and uncertainty. Besides, in order to simulate the actual working condition, this project has designed a group of disc-springs as actuator load [22]. Therefore, considering the elastic load, the displacement of the power mechanism of pump-controlled cylinder is expressed as Eq. (2),

Fig. 1. Equivalent Schematic of the SG electro-hydraulic actuator based on IDDVC system.

Fig. 2. Schematic of the two kinds of single-rod double-acting hydraulic cylinders.

Dp

X=

A

ωp −

Ct A2

(

V0 s βe

⎛1 ⎞⎛ 2 ⎜ s + 1⎟⎜ s 2 + ⎝ ωr ⎠⎝ ωh

)

+ 1 FL 2ξh ωh

⎞ + 1⎟ ⎠

where natural frequency

ω0 = ω h 1 +

(2)

ω0 and ζ0 are

K Kh −3 2

Ct ⎛ K⎞ ξ0 = ⎜1 + ⎟ 2A ⎝ Kh ⎠

βeM V0

−1 2

B ⎛ K⎞ + c ⎜1 + ⎟ 2A ⎝ Kh ⎠

V0 βeM

2.2. AMESim model of the IDDVC A multitude of details are ignored according to the modeling method mentioned above. After theoretical analysis, a compound simulation model based on the AMESim (AMESim model for short) is presented. It has two sub-models including volume pump model and pilot operated check valve model. Moreover, it contains the modelling part for the closed volume cavity. 2.2.1. Volume pump model In order to establish the pump model precisely, the leakage should be considered in detail. Firstly, it is a must to distinguish the internal leakage and external leakage for the pump. Then, the loss of flow QL can be divided into laminar flow loss QLμ and turbulence loss QLρ [23]. As the pump works in both directions, it can be assumed that the internal structure with in-out oil ports is symmetric. The leakage laminar flow factors and turbulent factors of the in-out ports are given as Cμi, Cμe, Cρi and Cρe respectively. The average of the pressure is used as reference pressure to amend the calculation leakage flow. Then the leakage of the three oil ports is calculated in Eq. (3). Since the gap leakage between the gear faces accounts for the vast majority of the whole leakage (about 75–80%), the out leakage laminar flow factor Cμe plays a dominant role. Besides, the oil compressibility is amended and is converted into the flow under the 0 MPa reference pressure.

⎧ ⎪ ⎪ Q L12 = ⎪ ⎪ ⎪ ⎪ ⎨QL = ⎪ 13 ⎪ ⎪ ⎪ ⎪ Q L23 = ⎪ ⎩

⎛ ⎞ Δp12 ⎜ Δp12 ⎟ ρ(p12ref ) ⋅sgn(Δp12 )⎟⋅ ⎜ Cμi μ + Cρi ρ(p ρ0 ) 12ref ⎝ ⎠ ⎛ ⎞ Δp13 Δp13 ⎜ ⎟ ρ(p13ref ) Δ + ⋅ ( ) ⋅ C C p sgn ρe 13 ⎟ ⎜ μe μ ρ(p13ref ) ρ0 ⎝ ⎠ ⎛ ⎞ Δp23 Δp23 ⎜ ⎟ ρ(p23ref ) Δ + ⋅ ( ) ⋅ C C sgn p ρe 23 ⎟ ⎜ μe μ ρ(p23ref ) ρ0 ⎝ ⎠

(3)

where QLij is the leakage flow between oil ports; Δpij ¼ pi - pj means the pressure difference between oil ports; pijref represents the reference pressure of QLij and Δpij ¼ (pi þ pj)/2; ρ0 denotes the oil density under 0 MPa reference pressure. Besides, the flow loss ΔQvf should be considered when the pump rotates in high speed due to the insufficient filling of the oil absorption port. A critical speed of rotation ωfl can be further assumed and the oil absorption is regarded as sufficient when the speed is below ωfl. Also, ΔQvf is expressed by using the polynomial

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function as Eq. (4). To sum up, the actual expressions are obtained in Eq. (5).

ΔQ vf = α1dead(ωin ) + α2dead(ωin )⋅ dead(ωin ) + ⋯

(4)

In this equation,

⎧ ωin − ωf1, ωin > ωfl ⎪ dead(ωin ) = ⎨ 0, − ωfl < ωin < ωfl ⎪ω + ω , ω < − ω ⎩ in f1 in fl ρ(p12ref ) ρ(p12ref ) Q v = (Q vt − ΔQ vf )⋅ = (ωin⋅Vp − ΔQ vf (ωin ))⋅ ρ0 ρ0

Fig. 3. The internal structure diagram of the pilot operated check valve.

(5)

Therefore, the output flow Qi of each oil port is calculated by Eq. (6)

⎧ Q1 = Q v − Q L − Q L 12 13 ⎪ ⎪ ⎨ Q 2 = − Q v + Q L12 − Q L23 ⎪ ⎪ ⎩ Q 3 = Q L13 + Q L23

Δp12 Vp

2π

2.3. Comparison between the two models After comparison between transfer function model and AMESim, there are two conclusions can be obtained.

(6)

With regard to the torque loss of the IDDVC system, the actual drive torque Tout is calculated by Eq. (7) considering the torque loss,

Tout = TLρ + TLμ + TLp =

circuit cavity in Fig. 4 and combining the pressurized tank and single rod double-acting symmetric cylinder in the IDDVC actuator prototype. (Table 3).

+ Kρωin2 + Kp Δp12

(7)

where TLρ is the resistance torque of the oil inertia; TLμ is caused by the sliding surface viscous friction; TLp is the load of the antifriction bearings. The overall model of the volume pump is constructed in AMESim and the parameters needed in the simulation model can be seen in Table 1. 2.2.2. Model of the pilot operated check valve Choosing ERVE 08021 pilot operated check valve and following the product specification, the model can be established by using AMESim HCD library like Fig. 3. The valve structure is divided into three parts including housing, hydraulic-controlled piston and ball. The HCD model which is equivalent to the Fig. 3 is built in AMESim by using the predetermined block. Moreover, the parameters needed in the simulation model can be seen in Table 2.

(1) The dead-zone nonlinearity of the actuator The amplitude is given as 3000r/min and the frequency of the motor velocity is 1 Hz sinusoidal signal. Displacement curves of two models are shown in Fig. 5. Compared with the transfer function model, AMESim model exists delay. However, the entire tendency is similar. Fig. 6 shows the velocity response. In general, the relationship between actuator velocity and motor speed satisfies linearity. Nevertheless, there is obvious dead-zone nonlinearity in the switchover of the positive and negative motion. This kind of dead-zone nonlinearity is caused not only by insufficient oil supply of the low-velocity pump, but also the combinative result of the low-velocity property, the switchover effect of the check valves in oil supply, the switchover effect of the bidirectional hydraulic locks, and the dynamic properties of the pipes. The pressure response curves of the suction and drain oil ports are depicted in Fig. 7. After the reversion of the motor, the pressure of the suction and drain oil ports declines dramatically while the oil supply has not worked. Hence, the empty of the pump gives rise to the dead-zone nonlinearity. After short delay, the oil supply works and makes the pressure of the suction oil ports return to normal condition. (2) The pressure property of the closed circuit compressed cavity

2.2.3. Simulation of the IDDVC internal cavity In order to model precisely, the internal cavity of the hydraulic system can be divided into V0, V11, V12, V21 and V22, as is shown in Fig. 4. In it, V0 is the cavity of a closed pressurized tank, V11 and V21 are the cavities for the front part of the check valve, which are all constant values. V12 and V22 are the cavities for the back part of the check valve, which vary with the piston position of the hydraulic cylinder. P0, P11, P12, P21, P22 are separately corresponding to the pressure value of the cavity. The positive direction of the defined speed v and the load FΣ is shown in Fig. 4. The model of IDDVC pump controlled cylinder is established in AMESim by using the volume pump and hydraulic controlled check valve sub-model, referring to the distribution of the closed

The actual working process of the actuators can be divided into four quadrants according to the direction of the load and velocity, as is shown in Fig. 8. The Z1 and Z3 regions are positive moving regions, and the Z2 and Z4 regions are negative moving regions. When building the transfer function model, it is assumed that the closed circuit side is compressed and the other side is the pressure of oil supply. Through the division for the cavity of the IDDVC, each cavity pressure can be obtained by simulation as Figs. 9 and 10. From Figs. 9 and 10, it can be found that there is great difference between the positive and negative motion. Take the region Z1 in Fig. 8 as example. When the motor rotates positively, the pilot

Table 1 Simulation parameters of the volume pump sub-model. Parameter Name(unit)

Value

Parameter Name(unit)

Value

The reference oil density ρ0 (kg/m3) The oil dynamic viscosity μ(Pa  s) External leakage coefficient of laminar flow Cμe Internal leakage coefficient of laminar flow Cμi External leakage coefficient of turbulence flow Cρe Internal leakage coefficient of turbulence flow Cρi

849.027 3.887  10  2 5.053  10  4 1.263  10  4 0 3.832

Hydraulic pump displacement Vp (m3/rad) Oil additional inertia torque coefficient Kρ Bearing load additional inertia torque coefficient Kp Insufficient pump-filling starting speed ωfl (rad/s) Insufficient pump-filling flow loss coefficient α1 Insufficient pump-filling flow loss coefficient α2

2.269  10  7 2.280  10  6 1.135  10  8 157.080 0 1.368  10  5

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Table 2 Simulation parameters of the pilot operated check valve sub-model. Parameter Name(unit)

Value

Parameter Name(unit)

Value

Valve nominal diameter Df (mm) Ball valve diameter Db(mm) Hydraulic control piston diameter Dpl(mm) Ball valve core quality Mb(kg) Hydraulic control quality of piston Mpl(kg) bhc1 dead zone volume V1d(cm3) bhc1_1 dead zone volume V2d(cm3) bhc1_2 dead zone volume V3d(cm3)

6 8 14.7 0.02 0.017 0.5 1 0.3

Return-spring stiffness of the hydraulic control piston Kpl(N/mm) Return-spring preload of the hydraulic control piston Fpl(N) Preloading spring stiffness of the ball valve Kb(N/mm) Preloading spring force of the ball valve Fb(N) Damping hole diameter of the cartridge valve oil port D12(mm) Hydraulic-controlled damping hole diameter of the cartridge valve D3(mm) Small damping hole diameter of hydraulic control piston rod Dr(mm) bhc1_3 dead zone volume V4d(cm3)

0.1 5 0.1 5.6 5 2 2 0.2

Fig. 4. The schematic diagram for the closed circuit cavity distribution of the IDDVC pump-controlled cylinder. Fig. 7. The pressure response curve of the suction and drain oil ports. Table 3 Simulation parameters for the power mechanism of the IDDVC. Parameter Name(unit)

Value 5

Front volume V1 of the check valve V11 (m3) Back volume V1 of the check valve V12 (m3)

1.5  10

Front volume V2 of the check valve V21 (m3) Back volume V2 of the check valve V22 (m3) Pressurized tank volume Vo(m3) The preload of pressurized spring Fo(N)

1.5  10  5

8.710  10

5

2.374  10  5 8  10

4

Parameter Name(unit)

Value

Piston diameter of the pressurized tank Do(mm) Pressurized spring stiffness of the pressurized tank Do(N/mm) Piston rod outer diameter D1(mm) Piston rod inner diameter d (mm) Piston inner diameter D (mm)

102 22.698

42 56 70 Fig. 8. The schematic diagram for the four quadrants of the actuators.

408.564

Fig. 9. The front and back pressure response of the check valve V1.

Fig. 5. The displacement response of the IDDVC.

operated check valve V1 opens freely and cavity V11 connects with V12. The oil is compressed by the spring load force. It is obvious that Eq. (8) is true and the pilot operated check valve V2 opens. The cavity V21 connects with V22, which is oil make-up pressure, thus p21 ¼ p22 ¼ pr

p11 = p12 + Δpcv ≫

Fig. 6. The motor input and velocity response curve of the IDDVC.

p22 λ ratio

p11 = p12 + Δpcv ≫

p21 λ ratio

(8)

where Δpcv means the positive opening pressure of the check valve; λratio is the control area ratio of pilot operated check valve Take the Z2 region in Fig. 8 as example. When the pump velocity np o 0, the pressure p11 declines dramatically and the oil supply does not work. When p21 4 p11/λratio, the closed

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between FΣ and pressure p12, p22 can be gained in Eq. (12)

A(p12 − p22 ) = FΣ = Mx¨ + Bẋ + Kx + FL

(12)

Then, Eq. (13) can be gained as follows,

̇ − p22 ̇ ) = FΣ̇ = Mx⃛ + Bx¨ + Kẋ + FL̇ A(p12

(13)

According to the equations above, choose the displacement, velocity and acceleration of the actuator as system state variables, then, Fig. 10. The front and back pressure response of the check valve V2.

pressurized tank begins to supply oil for V11, which means p11 ¼ pr. However, as a result of the spring load, p12 can be maintained at a higher value. Therefore, p21 op12/λratio, the pilot operated check valve V1 cannot open and the closed circuit cannot circulate. As the spring load direction and value do not change, p12 and p22 are elevated at the same time, until p21 4p12/λratio. The pilot operated check valve V1 opens slightly in order to guarantee the circulation of the closed circuit and maintain the proportional relation of p21/ p12 ¼1/λratio. The cavity V12、V21 and V22 are all compressed. Suppose that Δp ¼p12-p22, then the relationship between p12 and p22 can be deduced as Eq. (9). p12 =

λratio λratio λratio 1 Δp − Δpcv p22 = Δp − Δpcv λratio − 1 λratio − 1 λratio − 1 λratio − 1 (9)

From the above analysis, the pressure distribution of each cavity differs in closed circuit when the AMESim model of the IDDVC moves positively and negatively. Therefore, the flow continuity equation needs to be modified and it will be analyzed in the following part.

3. Controller design As is illustrated above, there exists dead-zone nonlinearity for IDDVC actuator during the switchover of the positive and negative motion. Besides, uncertainty for the load and the structure parameters of the hydraulic system also exist, which imposes great impact on the performance of the IDDVC. In order to improve the performance, the controller based on Discrete Variable Structure Control(DVSC) is designed. What needs to be stressed is that the controller design principle of this section is based on the rapidity of the calculation, reliability and overall consideration of the cost. 3.1. The establishment of the discrete state equation According to the above simulation result for the closed circuit cavity, the flow continuity equation is revised. When the motor moves positively, V11 and V12 are compressed. Then the flow continuity equation of the positive motion can be gained as below, ∙

npDp = Ax +

V11 + V12 + Ax ∙ p12 + CtΔp βe

(10)

It is assumed that V1 = V11 + V12 + Ax , V1 ∈ [0, L ] The load force equilibrium equation of negative motion is the same as the positive motion. However, during the negative motion process, V12, V21, V22 are compressed and flow continuity equation can be shown in Eq. (11).

ωpDp = Aẋ +

V21 + V22 + A(L − x) V + Ax ̇ + 12 ̇ + CtΔp p22 p12 βe βe

Suppose V2 =

1 ⎡λ V λratio − 1 ⎣ ratio 12

⎡ x1̇ ⎤ ⎡ ̇ ⎤ ⎡ ⎤ ⎡0 0 1 0 ⎤⎡ x1 ⎤ ⎡ 0⎤ x ∙ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥⎢ x ⎥ ⎢ ⎥ 0 + + X = ⎢ x2̇ ⎥ = ⎢ x¨ ⎥ = ⎢ 0 u 0 1 0 ⎥ ⎢ ⎥⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ẋ ⎥⎦ ⎣ x⃛ ⎦ ⎣ a3 1 a3 2 a33⎦⎣ x3⎦ ⎣ b ⎦ ⎢⎣ f3 (x, t )⎥⎦ 3 where

+ V21 + V21 + ( λ ratio−1)Ax + AL ⎤⎦, V2 ∈ [0, L ]

From the load force equilibrium equation, the relationship

MV

;

a32 = −

βe MV

(A2 +

VK βe

+ BCt );

a33 = −

(14) B M

−

βeCt V

⎧ V , positive βeCt 1 1 , and g (x, t ) is the f3 (x, t ) = M FL̇ + MV FL + g (x, t ); V = ⎨ ⎩ V2, negative load disturbance item. During the working process of the system, it is hard to measure some parameters accurately including equivalent elasticity coefficient K, viscous coefficient Bc, leakage coefficient Ct, volumetric modulus of elasticity βe and they exist parameter perturbation. In addition, there is also uncertain external load interference in system. Moreover, as the actual systems are all discrete systems, after choosing the sampling time Ts, forward difference method can be used to discretize the system as Eq. (15), ⎪ ⎪

X (k + 1) = (A d + ΔA d )X (k ) + (Bd + ΔBd)u(k ) + f (k )

(15)

where, ⎡ ⎤ ⎡ 1 Ts Tsδ1(k ) 0 ⎤ ⎡ 0⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 Ts ⎥;Bd = ⎢ 0 ⎥; f (k ) = ⎢ Ad = ⎢ 0 Tsδ2(k ) ⎥;A31 = a31Ts;⎢ ⎥ ⎢A ⎥ ⎣⎢ bTs ⎥⎦ ⎣ 31 A32 A33⎦ ⎣ Ts(δ3(k ) + d(k ))⎦ A32 = a32Ts ; A33 = (1 + a33Ts); δi(k ) is approximate error. where, ΔA ∈ R 3 × 3 is the uncertain part of the state-transition matrix; ΔB ∈ R 3 × 3 means the uncertain part of the input coefficient matrix; f represents external disturbance; d ∈ R 3 × 1. 3.2. Sliding mode controller design The controller function is to solve the tracking control problem of the given motion. Assume the given motion needed to be tracked is Xd(k ) ∈ Rn and introduce deviation variable E (k ) = Xd(k ) − X (k ), where n is the order of the state equation for the controlled object. The sliding mode switching function is established as Eq. (16)

s(k ) = C⋅E(k ), C = ⎡⎣ c1, c2, ⋅⋅⋅, cn − 1, 1⎤⎦

(16)

Discrete sliding mode control is quasi-slide switch control and needs to define a switching band to surround the switching surface as the Eq. (17),

SΔ = { x ∈ Rn| − Δ < s(k ) = C⋅E(k ) < Δ}

(17)

Then the stability of the sliding mode and the invariance for the parameter perturbation and disturbance are analyzed. Extend the sliding mode global-reach condition ss ̇ ≤ 0 of the continuous system to the discrete system and choose Lyapunov function as Eq. (18),

V (k ) = (11)

a31 = −

βeCtK

1 2 s (k ) 2

(18)

According to the Lyapunov's stability criterion, as long as the initial state of the arbitrary system satisfies Eq. (19), it will approach sliding mode surface S (k ) = 0 under the control law [24]. s

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(k) ¼ 0 is also globally asymptotically stable.

1 ΔV (k ) = ⎡⎣ s 2(k + 1) − s 2(k )⎤⎦ < 0 2

(19)

If there exists n-dimension row vector A˜ and scalar B˜ , d˜ satisfying equation x(20), the parameter perturbation and disturbance will satisfy the invariant condition of the system. It is obvious that the state equation IDDVC satisfies this condition, then the Eq. (15) can be adapted into Eq. (20).

ΔA d = BdA˜ d ΔBd = BdB˜d f (k ) = Bd f˜ (k )

^ X1 = 3 Y s +

Gf (z ) = (21)

This paper will adopt exponential approach law as Eq. (22)

s(̇ t ) = − ε sgn(s(t )) − qs(t ), ε > 0, q > 0

(22)

It should be noticed that the function of the parameter q is to make the initial state of the arbitrary system approach sliding mode surface with a constant speed and ε is to provide the high frequency switch gain which is necessary for control system to overcome disturbance and parameter uncertainty in the vicinity of the sliding mode surface. Through discretizing the Eq. (21) by the method of forward difference and defining the sampling time Ts, discrete exponential reaching law can be gained as Eq. (23)

s(k + 1) = (1 − qTs)s(k ) − εTs sgn(s(k )), ε > 0, q > 0

(23)

It can be proved that when sampling time is small, the discrete exponential reaching law satisfies the existence of the quasi-sliding mode, reachability condition and stability condition. Besides, the width of the switching band can be obtained as Eq. (24).

2Δ =

2εTs 1 − qTs

(24)

a3 2 s ε

+

a2

a1

ε

ε3

s+ 2

=

R3 (s + R)3

, R=

a + a1z−1 + a2z−2 + ⋅⋅⋅ + a nz−n Y (z ) = 0 X (z ) 1 + b1z−1 + b2z−2 + ⋅⋅⋅ + bnz−n

According to the analysis above, sliding control strategy needs the feedback signals of the position, velocity and acceleration. However, in actual system, there is no velocity sensor and acceleration sensor considering specific conditions and cost. In this paper, a kind of three order integral chained differentiator is designed to estimate velocity and acceleration signals by position signals [25–28]. The noise and disturbance of the differentiator only exist in the highest order state equation and can be effectively restrained layer by layer with the multiple integration. The three order integral chained is Eq. (25) [29]. ⎧ ^̇ ^ ^ ^ ⎪ x1(t ) = f1 (X , t ) = x2(t ) ⎪ ⎪ ^̇ ^ ^ ⎨ x2(t ) = f2 (X , t ) = x^3(t ) ⎪ ⎪ x^ ̇ (t ) = ^f (X^ , t ) = − 1 3 ⎪ 3 ⎩ ε3

{ a ⎡⎣ x^ (t ) − y(t )⎤⎦ + a εx^ (t ) + a ε x^ (t )} 1

1

2

2

2

2

3

^ s^(k ) = C⋅X d(k ) − C⋅X (k ) ≈ s(k )

where x^i(t ) is the estimation to the state variable x i(t ); ε represents perturbed coefficient of the differentiator; y(t) means differential signals; ai are a group of constants which meet s3 þ a3 s2 þ a2 s þ a1 ¼ 0 for the Hurwitz condition. In order to make the design calculation easy, choosing a1 ¼ r3, a2 ¼ 3r2, a3 ¼ 3r. It can be proved that when ε is small enough, x i(t ) → y(i − 1) (t ) is valid [30]. Then operate the Laplace transformation for the highest order state equation from the position observer signal y to position estimation signal x^1 as Eq. (26),

(27)

(28)

Substitute the Eq. (22) into the approaching law Eq. (28) and suppose that the equivalent input of the controller ueq(k) satisfies Eq. (29),

^ ueq (k ) = (1 + B˜d)u(k ) + A˜ d X (k ) + f˜ (k )

(29)

Then the discrete sliding mode control law can be derived as Eq. (30) ⎡ ⎤ ^ u eq(k ) = − (CBd)−1⎢ CA d X d(k + 1) − CA d X (k ) − (1 − qTs)s(k ) − εTssgn(s(k ))⎥ ⎣ ⎦ (30)

From Eq. (29), the moment k needs the order signal Xd (kþ 1) in moment k þ1. Denote the estimation of reference state signal in moment k to the moment k þ1 as Xd (k | kþ 1) and substitute it into Eq. (30). For the sake of the simulation and programming, the extrapolation of the literature [31] is adopted. Supposing that Xd (k-1), Xd (k) and Xd (k | k þ1) are linear distribution. Then the expression of the Xd (k | k þ1) can be shown in Eq. (31)

(31)

So far, the theoretic analysis for the discrete sliding control law with differentiator is completed. 3.4. Design and simulation of the discrete sliding mode controller In summary, the design of the whole discrete sliding mode controller can be concluded into the selection of the sliding mode vector C, approaching speed coefficient q and high-frequency gain ε, in which q and ε both have specific physical meaning and can be adjusted according to the experimental situation. Then, the selection method of the sliding mode vector C is analyzed. The state equation of the IDDVC system is expressed as Eq. (32),

⎧ Ẋ = A X + A X 11 11 12 12 ⎪ 11 ̇ = A X + A X + bu ⎨ X12 21 12 22 22 ⎪ ⎩ s = C1E1 + C2E2 = CX d − C1X11 − C2X12

(25)

(26)

When the bandwidth design of the differentiator is large, Eq. (28) is valid,

X d(k|k + 1) = 2Xd (k ) − X d(k − 1) 3.3. The design of the integral chained differentiator

r ε

Noise is a step-down integration from high order to low order in differentiator state equation. In order to prevent the high-frequency noise from mixing into the acceleration signals, the lowpass filtering is operated on estimated acceleration signal and a second order Butterworth low-pass filter is adopted whose cut-off frequency is 20 Hz. Thus, the expression is

(20)

X (k ) = A d X (k ) + Bd[(1 + B˜d)u + A˜ d X + f˜ (k )]

a1 ε3

(32)

T T where X11 = ⎡⎣ x1, x2⎦⎤ ; X12 = ⎣⎡ x3⎦⎤; C1 = ⎣⎡ c1, c2⎦⎤ ; Make C2 = I . As the given movement Xd(t) varies with time, the tracking control system which Eq. (32) expresses is rheonomous. According to [32], the sliding mode switching function of the rheonomous can be designed by adopting the design method of pole assignment on scleronomous. Supposing Xd(t) ¼ 0, the sliding mode switching function becomes Eq. (33),

s = − C1X11 − C2X12 = 0

(33)

Making linear transformation to the state Eq. (32) of IDDVC

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system yields the form of Eq. (34),

⎧ ̇ ⎪ X11 = A11X11 + A12 X12 = ( A11 − A12 C1)X11 ⎨ ⎪ ̇ = C ( A − A C )X ⎩ X12 1 11 12 1 11

(34)

It is obvious that IDDVC system state equation is of great controllability and observability. Therefore, the sub-system [A11, A12] is also of great controllability. By it, the design of the sliding mode vector C becomes the solution of the state feedback pole placement. Given the sub-system [A11, A12] and expected pole λ1 and λ2, the characteristic polynomial of the system is Eq. (35): 3

σ (z ) =

∏ (z − λi) =

zI − A11 + A12 ⋅C1

i=1

Fig. 12. 3D model of the IDDVC.

(35)

The Eq. (36) is to solve the expression of vector C

C = ⎡⎣ λ1λ2, λ1 + λ2, 1⎤⎦

(36)

To sum up, Fig. 11 illustrates the control strategy, which combines the linear integral chained differentiator and the discrete sliding mode controller based on exponential approach law.

4. Experimental analysis The prototype is based on the principle of the integration and control facilitation, which integrates servo motor, hydraulic manifold block, pressurized tanks, single-rod symmetric cylinder and sensors in Figs. 12 and 13. Besides, in order to simulate the change of the disturbance, the disc-spring which is cone-shaped axially and can bear large load is added. Its characteristic of the nonlinear stiffness increases the uncertainty of the actuator external load and can be used to test the anti-interference ability of the actuator. The maximum load of the disc spring group is 13000 N. In order to meet the requirements of the reliability and realtime performance, the hardware architecture of the DSP is adopted as the controller. The integrated development board of the general DSP can be illustrated as Figs. 14 and 15. Its Conroe is the series of TMS320C28x, 32-bit floating-point high-performance CPU of TMS320F28335 type in TI company. Its shortest cycle can reach 6.67 ns. Regarding the software part, the traditional communication mode between up computer and down computer is used and the detail is omitted. Here the flow chat of user-defined program is only listed in Fig. 16. It should be noticed the integral chained differentiator is located after the filter and worked with the Discrete sliding mode controller together. 4.1. The simulation model validation experiments The validity of the simulation model in 2.2 chapter will be proved by IDDVC prototype test rig in this chapter. Traditional PID

Fig. 13. The prototype of the IDDVC with disc spring load.

controller is adopted. By setting the same parameters with the simulation model, kp ¼ 5.1, ki ¼ 1.9 and kd ¼ 0.2, the results of experimental and simulation are compared in Figs. 17 and 18. It can be found that the simulation and the experiment curves are extremely close in both kinds of commands including step and ramp. In addition, Fig. 19 shows the comparison between the experiment and simulation of the sinusoidal position command. From the partial enlarged drawing Fig. 20, both of the measured and simulation position curves present dead-zone nonlinearity and their similarity degree is high. The pressure variation of the cavity is described in the AMESim model by splitting the closed loop cavity when the actuator moves positively or negatively. In order to verify the accuracy of the model, the pressure sensors are installed in V12 and V22 shown in Fig. 4.The pressure simulation and experiment curve of the p12 and p22 for the cavity V12 and V22 can be illustrated as Fig. 21 by

Fig. 11. The schematic diagram of the discrete sliding mode control with differentiator.

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Fig. 17. Tracking response of PID position step command. Fig. 14. TMS320F28335 integrated development board.

Fig. 18. Tracking response of PID position ramp command. Fig. 15. Functional components of the integrated development board.

second half of the reverse movement, the deviation gradually increases which results from the error between parameter setting value and actual value of the hydraulic cylinder coulomb friction and viscosity coefficient. 4.2. Discrete sliding mode controller test experiment The testing experiment here is based on the analysis of the DVSC algorithm in chapter 3 which is embedded into the DSP controller. Step, ramp and sinusoidal position tracking experiments were performed on the IDDVC actuator under no-load, load (2600 N and 7800 N) conditions respectively, in order to prove the validity of the DVSC control strategy and make a comparison with PID controller.

Fig. 16. The prototype of the IDDVC with disc spring load.

choosing the same parameters and configuration for the experiments and simulation, setting the actuator track the sinusoidal position instruction y ¼ 0.005sin2πtþ 0.06 (the spring load is about 2600 N at zero-time). It can be seen that the pressure curves of the simulation and experiment have the same tendency that p 11 and p 12 rise together when the actuator moves negatively, which proves the validity of Eq. (9) and the AMESim model. However, in the

4.2.1. Step position tracking The result of the 0.01 m step position tracking experiment for the IDDVC actuator is shown in Fig. 22. By the comparison with PID control during position tracking, it can be found that there exists about 3% of the overshoot and there is little difference between eventual adjusting time and PID control time. The value of the controller sliding mode function s uploaded by DSP processor is shown in Fig. 23. It can be concluded that the sliding mode function quickly approaches 0 and does small-amplitude oscillation in the vicinity of 0, which verifies the accessibility and existence of the quasisliding mode. 4.2.2. Ramp position tracking Fig. 24 illustrates the comparison between DVSC and PID control by setting the tracking rate of the IDDVC actuator as 0.03 m/s and the saturation value of the ramp position command signal as 0.03 m. It can be concluded that the control precision of the former one on the ramp part is obviously higher than the latter one.

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Fig. 19. The comparison between experiment and simulation under PID sinusoidal position command.

Fig. 22. Step position tracking response by using DVSC. Fig. 20. Partial enlarged drawing of Fig. 20.

Moreover, the speed of eliminating the steady state error is also faster than PID control. The estimated speed and acceleration by the differentiator iterative computations are illustrated in Figs. 25 and 26. From the figures, the estimated speed and acceleration do not approach the command value, but to do small-amplitude oscillation with constant frequency in the vicinity of the command value. This is also the result contributed by the pulsation of the pump, which proves the opinions above. 4.2.3. Sinusoidal position tracking Under the no-load condition, making the actuator track the sinusoidal position command signal of 5mm amplitude and 1 Hz frequency, the actual tracking curve is got in Fig. 27. The speed and acceleration estimated by the differentiator are shown in Fig. 28, which also have fine tracking characteristics for command speed and acceleration. Sinusoidal position tracking experiment of the DVSC is carried out separately under no-load and load (2600 N and 7800 N) condition of the actuator. Figs. 29 and 30 illustrate the comparison between different position tracking curves.

Fig. 23. The function value of the controller sliding mode.

Compared with PID control, the phase lag of the DVSC sinusoidal position tracking is smaller and its tracking precision is higher. From the partial enlarged drawing, the speed of eliminating error for DVSC is faster when the actuator reverses motion. Moreover, the inhibiting effect for the dead-zone nonlinearity is stronger.

Fig. 21. The comparison of the experiment and simulation between p12 and p22.

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Fig. 24. Comparison between DVSC and PID control under ramp tracking command.

Fig. 28. The estimated speed and acceleration for the sinusoidal position tracking experiment.

Fig. 25. DVSC estimated speed.

Fig. 26. DVSC estimated acceleration.

5. Conclusion In this paper, a kind of SG system by using IDDVC is investigated. By analyzing the operation mechanism, an integrated model of IDDVC is established. Compared with the conventional transfer function model, they are similar for the rapidity of the system. However, the integrated model can precisely simulate the feature of dead-zone nonlinearity during positive and negative switchover motion. Moreover, there exists pressure variation in closed cavity, which is different from the assumption of transfer function model. Therefore, the result of the experiment proves the validity of composite model. In addition, the feasibility of the sliding mode control strategy with differentiator is proved by position tracking experiment, which has better system rapidity and tracking precision than PID control, and has more obvious compensation effect on IDDVC dead-zone nonlinearity. However, as the frequency of tracking command increases, the nonlinearity

Fig. 27. The sinusoidal position tracking response under DVSC with no-load condition.

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Fig. 29. Tracking responses by using DVSC with different load conditions.

[9]

[10]

[11]

[12]

[13]

[14] Fig. 30. Partial enlarged drawing of Fig. 30. [15]

impact on position tracking is gradually enlarged and the deadzone compensation effect of the sliding mode control. In the future, the controller corresponding to the proposed IDDVC will be tested in the real steering system and the detailed comparison with the traditional system should be proceeded from both of the control performance and energy consumption aspects.

[16]

[17]

[18]

Acknowledgments The authors acknowledge the contribution of National Natural Science Foundation of China (51505289). and the support of Training projects of young teachers from Shanghai universities (ZZsl15024).

[19] [20] [21] [22] [23]

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Please cite this article as: Shen W, et al. Robust controller design of the integrated direct drive volume control architecture for steering systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.008i