Robust Non-Chattering Observer Based Sliding ... - Science Direct

6th IFAC Symposium on Mechatronic Systems The International Federation of Automatic Control April 10-12, 2013. Hangzhou, China

Robust Non-Chattering Observer Based Sliding Control Concept for Electro-Hydraulic Drives Lasse Schmidt ∗ Torben O. Andersen ∗∗ Henrik C. Pedersen ∗∗∗ ∗

Bosch Rexroth A/S, Denmark (e-mail: [email protected]) ∗∗ Department of Energy Technology, Aalborg University, Denmark (e-mail: [email protected]) ∗∗∗ Department of Energy Technology, Aalborg University, Denmark (e-mail: [email protected]) Abstract: This paper presents an observer-based sliding mode control concept with chattering reduction, generally applicable for position tracking control of electro-hydraulic valve-cylinder drives (VCD’s). The proposed control concept requires only common data sheet information and no knowledge on load characteristics. Furthermore the proposed scheme only employ pistonand valve spool positions- and pressure feedback, commonly available in industry. The main target is to overcome problems with linear controllers deteriorating performance due to the inherent nonlinear nature of such systems, without requiring extensive knowledge on system parameters nor advanced control theory. In order to accomplish this task, an integral sliding mode controller designed for the control derivative employing state observation is proposed, based on a generalized reduced order model structure of a VCD with unmatched valve flow- and cylinder asymmetries. It is shown that limited attention can be given to bounds on parameter estimates, that chattering is reduced and the number of tuning parameters is reduced to the level seen in conventional PID schemes. Furthermore, simulation results demonstrate a high level of robustness when subjected to strong perturbations in supply pressure and coulomb friction force, and that tracking accuracy may be reduced to the level of noise. Furthermore, the proposed controller tolerates significant noise levels, while still remaining stable and accurate. Keywords: Electro-Hydraulic Actuator Systems, Sliding Mode Control, Sliding Mode Observer, Chattering Reduction Throughout the paper, the nomenclature by: AA : Piston area on A-side AB : Piston area on B-side Bv : Viscous damping coefficient CL : Leakage coefficient Fext : External disturbance force Ff C : Friction force related to cylinder Ff L : Friction force related to load FG : Force due to gravity KvA : Valve flow gain of flow port A KvB : Valve flow gain of flow port B Meq : Equivalent inertia load PA : Pressure in A-chamber PB : Pressure in B-chamber PS : Supply pressure PT : Tank pressure QA : Flow through flow port A QB : Flow through flow port B uv : Valve control input VA0 : Initial volume of A-chamber VB0 : Initial volume of B-chamber

978-3-902823-31-1/13/$20.00 © 2013 IFAC

applied is given [m2 ] [m2 ] [N s/m] [m3 /s/P a] [N ] [N ] [N ] [N ] √ [m3 /(s√P aV )] [m3 /(s P aV )] [kg] [P a] [P a] [P a] [P a] [m3 /s] [m3 /s] [V ] [m3 ] [m3 ]

99

xP : xv : βe : ωv : ζv :

Piston position Valve spool position Effective oil/hose bulk modulus Valve band width Valve damping ratio

[m] [m] [P a] [rad/s] [−]

Note: For reasons of equation overview, the notation of state dependence of parameters is omitted. 1. INTRODUCTION With the development of reliable proportional flow control components with medium transient performance characteristics to acceptable prices, trends and demands in industry are increasingly to develop and deliver high performance, energy efficient turn-key solutions. The great majority of hydraulic systems developed are produced in a limited number for specialized applications, where budgets are too limited to design professionally engineered model based motion control systems. In these cases stand alone economically feasible digital controllers employing traditional linear control schemes, dedicated to control electro-hydraulic components with ease-of-use interfaces, are widely used. However, limited knowledge on control 10.3182/20130410-3-CN-2034.00041

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

theory often results in poor performance with no indications of stability margins, robustness to perturbations in system parameters due the inherently nonlinear dynamics, and parameters such as viscosity-temperature relations, friction factors, air content, variant inertia load and so forth. Due to this, commissioning of electro-hydraulic motion control systems is often an iterative and hence expensive process making it difficult to comply with tight budgets and delivery deadlines. This, and the fact that performance requirements are constantly increasing, naturally leads to the idea of employing more advanced control schemes in order to improve control performance. Such approaches however, often require extensive system knowledge and full state feedback - such requirements do not comply with the industry in general, as system dynamic characteristics, load characteristics and expected disturbances can rarely be defined, and usually only the piston position and pressures are available from measurements. Hence, in order for industry to capitalize on advanced control strategies, emphasis must be put on limiting the requirements for sensors, reducing the number of tuning parameters, providing adaptability to / robustness towards varying operating conditions and easy commissioning. In literature several research contributions regarding sliding mode control of VCD’s have been reported. The main problem in the application of this theory arises with the introduction of a robust term discontinuous across a sliding constraint (commonly a sliding surface). Physical systems in general, and hydraulic systems in particular, are not able to realize this discontinuous control action due to actuator dynamics, hence chattering of the control signal occur and extremely high activity in e.g. amplifier circuits of the valve and the valve spool position arises. This effect is naturally undesired as this may cause increased wear of the valve due to the resulting load on it, and excitation of un-modeled dynamics and/or may occur and cause damage to the system. A common approach to avoid control chattering is the introduction of a smoothing function, and hence the introduction of a boundary layer on the sliding constraint − hence only a quasi sliding mode may be achieved, as control accuracy is restricted to this boundary layer. Different approaches for smoothing the discontinuous control action have been presented in literature. The most common approach is the application of the saturation function presented in Slotine and Li (1991), among others. Contributions applying this approach include Komsta et al. (2010b), Liu and Handroos (1999) when applied to asymmetric VCD’s actuating non-application specific load systems, in Pi and Wang (2011), Pi and Wang (2010), Guo et al. (2008) and Habibi (1995) when applied to asymmetric actuators of manipulators, and in Yoon and Manurung (2010) Wang and Su (2007) when applied to asymmetric actuators integrated in construction machinery. Other contributions employing this smoothing approach in symmetric electro-hydraulic drives include Fung et al. (1997), Hwang and Lan (1994) and Ghazy (2001), when applied to hydraulic motor control systems, and Chen et al. (2005), Wang et al. (2011) when applied to symmetric VCD’s. Another and rather similar approach to smoothing the discontinuous term is by application of a hyperbolic tangent function. This approach was applied in Hansen et al.

(2005) on a hydraulically driven robot with symmetric VCD’s, in Ghazali et al. (2010) on a spring-damper loaded symmetric VCD, in Nguyen et al. (2000) on asymmetric VCD’s of an excavator, and in Hisseine (2005) and Bonchis et al. (2001) on constant inertia- and friction loaded asymmetric VCD’s. A different approach modifying the sign function by dividing the sliding surface by its absolute value plus a small constant has also been presented in several papers. This approach was applied in Fung and Yang (1998) and Dong et al. (2011) to spring-damper loaded asymmetric VCD’s and in Chin-Wen Chuang (2005) and Chern and wu (1991) to symmetric hydraulic motor drives. To compensate for the undesired properties of chattering reduction, different approaches have been reported in literature with application to VCD’s. In Zulfatman et al. (2011), a sliding mode controller which, besides the usual discontinuous term, also includes a term proportional to the surface in order to increase speed of convergence and to compensate for the boundary layer, was presented. This type of reaching law is usually denoted a constant plus proportional reaching law. Similar approaches have been reported in Fung and Yang (1998) and Ghazy (2001) for asymmetric and symmetric rotary actuator systems, respectively. Other approaches introduce a variable boundary layer, made variable according to an appropriate state. Such an approach was presented in Chen et al. (2005). Generally for the mentioned smoothing functions is the introduction of additional tuning parameters, and besides this most sliding mode control approaches require full state feedback and some knowledge on system parameters. These features together with the fact that engineering knowledge on control theory is often not available, make such controllers generally non-applicable in industrial practice. In fact, commonly, only data sheet information together with piston- and valve spool positionand pressure sensors are available together with a rather limited insight into PID control. Hence the problem is to develop a sliding mode control concept that is generally applicable to VCD’s, that does not contain the commonly known boundary layer function, only utilizes the commonly available sensors, contains no more parameters than conventional PID controls and features easy commissioning. Main Contribution In this paper a possible solution to this problem is presented. The focus is placed on valve controlled cylinder drives having unmatched valve flow- and actuator asymmetries, as these may be considered the VCD configuration having the most undesirable features seen from a control point of view. Based on a generalized reduced order VCD model structure which may represent any VCD, a new sliding mode control concept for VCD’s is presented. The novelties of the proposed control concept are related to this type of application, and concerns the way of reducing control chattering without introducing a boundary layer on the control constraint in first order sliding mode controlled drives, and the utilization of online information on valve flows together with a second order sliding mode algorithm for observation of both the piston velocity as

100

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

well as the velocity component due to fluid compression. The proposed observer utilizes parametric data from valveand cylinder data sheets and requires only piston- and valve spool position- and pressure feedback. The control scheme is shown to be asymptotically stable in the sense of Lyapunov, and requires only the tuning of three physically interpretable parameters. These features render the proposed control concept industrially applicable, while maintaining the advantages of sliding mode control. The outline of the paper is as follows. In section 2 a generalized model representing an arbitrary VCD is derived, in section 3 a pressure feedback compensator used to reduce impact of nonlinearities is presented, in sections 4 and 5 the SMC design and observer design are presented, respectively, followed by simulation studies in section 6, in which the proposed control scheme is evaluated. Finally, conclusions are presented in section 7. 2. GENERALIZED REDUCED ORDER VCD MODEL In the following, a reduced order model of a generalized VCD that will cover all possible configurations of such systems is established to the extend that generalizations can be made, i.e. a VCD with unmatched valve flow and cylinder asymmetries (when considering zero lapped valves). The derivation of the model is based on the system sketch depicted in figure 1. The nonlinear system dynamics

Fig. 1. Schematic overview of generalized VCD. are governed by the following equations Merritt (1967) (noting that VA = VA0 + AA xP , VB = VB0 − AB xP , Fad = Ff C + Ff L + FG + Fext ): −1 x ¨P = Meq (PA AA − PB AB − Bv x˙ P − Fad ) βe P˙A = (QA − CL PA + CL PB − AA x˙ P ) VA βe P˙B = (CL PA − CL PB + AB x˙ P − QB ) VB p  KvA xv pPS − PA for xv > 0 QA = KvA xv PA − PT for xv < 0 p  KvB xv pPB − PT for xv > 0 QB = KvB xv PS − PB for xv < 0 x ¨v = ωv2 uv − 2ζv ωv x˙ v − ωv2 xv

(1) (2) (3) (4) (5) (6)

Hence (1) − (6) constitute a general model of a VCD. Commonly in industrial environments, Meq , Fad , PS and to some extend PT may be considered time variant. Regarding the viscous friction coefficient, this is dependent on both pressure and temperature. According to Andersen and Hansen (2003), as the kinematic viscosity is explicitly dependent on pressure, and implicitly dependent on pressure through fluid density, the kinematic viscosity is less sensitive to pressure rise. This together with fluid temperature control normally present in industrial applications, Bv may be considered fairly constant. These considerations and the model (1) − (6) illustrates the nonlinear and time variant nature of VCD’s, and the difficulties related to designing control schemes for such systems. For reasons of convenience, the model (1) − (6) is approximated by a reduced order representation. It is desirable to represent the system with the least possible number of equations, hence the model (1) − (6) is rewritten in terms of the virtual load pressure. For simplicity, the following coefficients are defined: KvB AB σ= and α = (7) KvA AA Considering the situation of positive flow i.e. xv > 0, the flow equations for the valve ports A and B are defined as: p QA = KvA xv PS − PA and (8) p QB = σKvA xv PB − PT (9) Considering the flows (8), (9) when fluid compression and leakage is absent, the following may be defined: αQA = QB (10) Inserting (8), (9) in (10), and solving for PA and PB , respectively, yield: P S α 2 − σ 2 P B + σ 2 PT PA = and (11) α2 PT σ 2 + α 2 PS − α 2 PA PB = (12) σ2 The virtual load pressure is proportional to the cylinder output force (see e.g. Merritt (1967) and Mohieddine Jelali (2004), among others), and is given by: PL = PA − αPB (13) Inserting (12) into (13) and solving for PA and inserting (11) into (13) and solving for PB , respectively, yield: α3 PS + σ 2 PL + ασ 2 PT PA = and (14) σ 2 + α3 α 2 PS − α 2 PL + σ 2 PT PB = (15) σ 2 + α3 Inserting (14) into (8), and (15) into (9), the valve port flow equations expressed in terms of PL are obtained as: r PS − PL − αPT QA = σKvA xv and (16) σ 2 + α3 r PS − PL − PT α QB = ασKvA xv (17) σ 2 + α3 In the situation of negative flow, i.e. xv < 0, the valve port flows are given by: p QA = KvA xv PA − PT and (18) p QB = σKvA xv PS − PB (19) Carrying out similar calculations as for xv > 0, the valve port flow equations expressed in terms of PL are obtained as:

101

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

r

αPS + PL − PT and (20) QA = σKvA xv σ 2 + α3 r αPS + PL − PT QB = ασKvA xv (21) σ 2 + α3 From (16), (17), (20) and (21) general valve port flow equations may be established as: QA = α−1 QB

(22) p σKvA pPS − PL − αPT for xv > 0 = xv √ 2 3 αPS + PL − PT for xv < 0 σ +α

The flow continuity equations for chamber- and hose/pipe volumes may be expressed as (assuming leakage negligible): βe P˙A = (QA − AA x˙ P ) (23) VA βe P˙B = (AB x˙ P − QB ) (24) VB Introducing the relation ρ = VB /VA , the load pressure derivative is obtained as: βe ρ + α 2 P˙L = Λ(QA − AA x˙ P ), Λ = (25) VA ρ This leads to the derivative of the load force, which is given by: F˙L = P˙L AA (26) (3) xP + F˙ad = Meq xP + (M˙ eq + Bv )¨

Consider the simplified third order model (22), (25), (26). For control design purposes it is appropriate to express the model in SISO form, with the piston velocity as output. Defining xv = H(s)uv with H(s) representing the valve dynamics, one obtains the reduced order model: x˙ P = F + Guv

(27)

With:

√   PS − PL − αPT H(s)σKvA  AA √ G= √ αPS + PL − PT σ 2 + α3    AA (3) Meq xP + Bv x ¨P + F˙ad F =− 2 ΛAA

for xv > 0 (28) for xv < 0 (29)

3. REDUCING EFFECTS OF NONLINEARITIES BY PRESSURE FEEDBACK COMPENSATION Considering the model (27), (28), (29) it is found that the term G may be obtained fairly accurately online from common data sheet information and pressure measurements, whereas the term F may be difficult to determine. Used properly, online estimation of the term G will enable the possibility to account for the strongly nonlinear flow gain of the valve, the forward-backward adjustment due to asymmetries and to some extend account for nonlinear friction phenomena as these are expressed in terms of the load pressure. Furthermore, the flow gain KvA may be determined rather accurately directly from common data sheet information. Disregarding the valve dynamics, an estimate is given by:

p   PS − |PL | − αPT for x > 0 v ˆ vA  σ K AA ˆ =√ p (30) G αPS − |PL | − PT σ 2 + α3    for xv < 0 AA Note; Commonly in properly designed valve cylinder drives, the supply pressure PS is the highest pressure in the system, hence G > 0, i.e. G is invertible. Hence a virtual control signal u ¯v may be defined as: ˆ v u ¯v = Gu (31) Substituting (31) into (27), yield: ˆ −1 u x˙ P = F + Guv = F + GG ¯v (32) Assuming the valve dynamics negligible and assuming ˆ ' G, which is indeed achievable due its parameters and G pressure feedback available, (32) is reduced to: x˙ P = F + u ¯v (33) The model (33) has unity control gain for both positive and negative velocity, and as such the compensated system is remarkably easier to handle compared to the system (27). 4. SLIDING CONTROL DESIGN In order to avoid implementation of a saturation function to account for control chattering (see e.g. Slotine and Li (1991)), a different approach may be applied. Choosing the control error itself as the control target, second order sliding algorithms may be applied for the system (27) i.e. the twisting algorithm-, the super twisting algorithm or the prescribed convergence algorithm Emelyanov et al. (1986a), Emelyanov et al. (1986b), Emelyanov et al. (1990), Levantovsky (1985), Levant (1993), e.g. by deriving a control law for the control derivative. However, applying an integral sliding surface allows for a similar way to reduce chattering when utilizing a first order sliding controller. In order to do this, the derivative of (33) is obtained as: x ¨P = F˙ + u ¯˙ v (34) ˙ ˙ In (34), the function F is given by (assuming Meq negligible compared to remaining components): (3)

Fad + Meq xP + Bv x ¨P F˙ =Λ˙ (35) Λ2 A2A (4) (3) F˙ad + Meq xP + Bv xP − ΛA2A The target for the control system is position tracking, i.e. to get xP to track the reference position xR as closely as possible. Defining the position tracking error e = xR − xP , a time varying integral sliding surface is designed based on the scalar equation proposed in Slotine and Li (1991):  n−1 Z d S(e, t) = +λ edt (36) dt t Here λ is strictly positive and determines the time constant of the system, and n is the system R order. As the model (34) is of order n = 3 relative to t edt, the sliding surface is given by: Z S(e, t) = e˙ + 2λe + λ2 edt (37) t

In order to maintain S-, hence the error states equal to zero, the sliding surface S must be attractive, i.e.: lim S˙ < 0 and lim S˙ > 0 (38)

102

S→0+

S→0−

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

An equivalent criterion is given by S S˙ ≤ −η|S|. Satisfying ˙ this, finite time convergence is obtained. Now obtaining S, yield: S˙ = e¨ + 2λe˙ + λ2 e (39) =x ¨R − x ¨P + 2λe˙ + λ2 e Inserting the model (34) into (40), yield: S˙ = x ¨R − u ¯˙ v − F˙ + 2λe˙ + λ2 e

maximum rate of change in velocity due to fluid compression, hence simulations or physical experiments may provide for off-line estimation of F˙max . 5. SLIDING VELOCITY OBSERVER

(40)

(41) ˙ Solving S = 0 for u˙ v , an equivalent control signal, maintaining S = 0 for S(0) = 0, is obtained Slotine and Li (1991): u ¯˙ eq = γn − F˙ , γn = x ¨R + 2λe˙ + λ2 e (42) The function F˙ may not be exactly determined, and only an estimate may be obtained. This is assumed bounded by: ˙ |Fˆ − F˙ | ≤ F˙max (43) Hence an estimate of the equivalent control may be obtained as: ˆ˙ eq = γn − Fˆ˙ u ¯ (44) ˙ To account for the imprecise Fˆ , a term discontinuous across S is added to (44), whereas the final sliding mode control law is obtained as: ˆ u ¯˙ smc = u ¯˙ eq + Ks sgn(S) (45) ˙ˆ = γn − F + Ks sgn(S) (46) 4.1 Stability of Closed Loop System To define a criterion for asymptotic stability of the closed loop system, consider the Lyapunov-like function: 1 V (S) = S 2 ≥ 0 (47) 2 A sliding condition that guarantees finite time convergence is found by placing the following restriction on the derivative: V˙ (S) = S S˙ ≤ −η|S| (48) Inserting the model (34) into the sliding condition (48), the following is obtained: S S˙ = S[γn − F˙ − u ¯˙ v ] ≤ −η|S| (49) Substituting the control law (46), i.e. u ¯˙ v = u ¯˙ smc , yield: ˙ S S˙ = S[γn − F˙ − (γn − Fˆ + Ks sgn(S))] ˙ = S[γn − F˙ − γn + Fˆ − Ks sgn(S))] ˙ = S[Fˆ − F˙ − Ks sgn(S))] ≤ −η|S| (50) Hence a condition on Ks satisfying (48) may be obtained: ˙ Ks ≥ Fˆ − F˙ + η (51) ˙ ˙ Applying the relation F˙ = Fˆ + (F˙ − Fˆ ) and the bound ˙ (43), then |F˙ | = |Fˆ | + F˙max . Hence (51) may be written as: Ks ≥ η + F˙max (52) The convergence criteria (52) appear rather simple, however, from an industrial point of view F˙max may literally be impossible to determine. However, F˙max represents the

In order to realize the control scheme (45) under normal conditions for industrial VCD’s (i.e. only piston- and valve spool position- and pressure measurements available and a noisy environment), the piston velocity need to be derived online through some algorithm. It is well known that differentiation of signals containing noise may lead to erroneous results due to the possible strong gradients obtained through this operation. Also, in order to compensate for the noise impact, a filter may be applied however this introduces a phase shift that may deteriorate the overall system performance. Generally for industrial VCD’s, especially due to unknown properties of disturbances, the signal structure is not exactly known. Hence this operation may be carried by application of sliding modes, as properties similar to sliding mode controllers are obtained, i.e. being highly robust towards uncertainties. Robust differentiation of signals via second order sliding modes in terms of the super twisting algorithm was proposed in Levant (1998), and successfully applied in Damiano et al. (2004) and Davila et al. (2005). An important and convenient property of this approach is that it satisfies the separation principle Levant (2003). Hence the control design may be carried out separately, while preserving the main features of the combined observer- and feedback controller. 5.1 Differentiation of Position Signal via Second Order Sliding Algorithm Let the velocity observation error be defined by (53): x ˜˙ P = x ˆ˙ P − x˙ P (53) The objective is then ideally to maintain x ˜˙ P ≡ 0. Consider the velocity estimate x ˆ˙ P based on the super twisting algorithm (robust differentiation): x ˆ˙ P = −k1 |ˆ xP − xP |1/2 sgn(ˆ xP − xP ) + z (54) z˙ = −k2 sgn(ˆ xP − xP ) In order to consider the conditions under which the constraint x ˜˙ P ≡ 0 may be maintained, the derivative is utilized, assuming the acceleration bounded: ¨ˆP − x ¨˜P = x x ¨P , |¨ xP | ≤ ξ (55) ¨ˆP from (54) (at some intersection x Obtaining x ˜P 6= 0), yield: ˜˙ P ¨ˆP = − k1 x x − k2 sgn(˜ xP ) (56) 2 |˜ xP |1/2 Substituting into (55), this satisfies the differential inclusion: ˜˙ P ¨˜P ∈ [−ξ, ξ] − k1 x x − k2 sgn(˜ xP ) (57) 2 |˜ xP |1/2 Choosing ξ < δ, finite time convergence to a second order sliding mode x ˜˙ P = x ˜P = 0 is provided for if parameters are chosen (according to Levant (2003)): √ k1 = 1.5 δ ∧ k2 = 1.1δ (58)

103

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

Table 1. Parameters of the simulated system.

Once a second order sliding mode is achieved, then ideally x ˆ˙ P = x˙ P . However, the ability of the algorithm (54) to track fast transients in the derivative and at the same time achieve noise robustness will always be a compromise, particularly if noise frequency components close to system frequencies are present. 5.2 Velocity Observation via Available Measurements, Valve Information and a Second Order Sliding Algorithm In order to allow the possibility to reduce impact of noise in the observed velocity further, combining an online estimate of the velocity with the super twisting algorithm is proposed here.

ˆ v − x˙ P , the super twisting algorithm is Defining ∆v = Gu utilized to obtain Fˆ , by: 1/2 Z Z  (∆v + Fˆ )dt + γ Fˆ = −c1 (∆v + Fˆ )dt sgn t t Z  γ˙ = −c2 sgn (∆v + Fˆ )dt (60) t

˙ v | ≤ ϕ, the following inclusion is satisfied: Assuming |∆ ˙ ¨ x ˜P ∈ [−ϕ, ϕ] + Fˆ (61) ˙ Choosing ϕ < ε, a second order sliding mode x ˜P = x ˜P = 0 is provided for, with parameters chosen: √ (62) c1 = 1.5 ε ∧ c2 = 1.1ε Evidently this approach allows for online observation of the piston velocity x˙ P , and the velocity component due to fluid compression F , explicitly. Having the component F available online represents information on the load dynamics, and may be used for e.g. diagnosis and/or fault detection purposes. It is apparent that the proposed observer does not require any additional system information apart from those already introduced in the control design. 6. SIMULATION RESULTS Having elaborated the combined robust sliding controland observer concept, its features are evaluated via simulation studies. This is chosen here, rather than physical experiments, in order to verify the properties of the

Value

Unit

AA AB Bv Ff C + Ff L Meq PS PT VA0 VB0 βe Stroke(xP )

0.0154 0.0090 22000 1900 2000 115e5 5e5 0.0121 0.0089 8000e5 1

[m2 ] [m2 ] [kg/s] [N ] [kg] [P a] [P a] [m3 ] [m3 ] [P a] [m]

Table 2. Perturbation of parameters. PS and Ff C + Ff L refer to the nominal values of the verified model.

The approach applied here is somewhat similar to that of Komsta et al. (2010a), in which the acceleration was successfully observed in the presence of noise based on a rough acceleration estimate. Here it is proposed to include an estimate of the velocity utilizing online system information already available from the control, and observe only the velocity estimation error. Consider again the system (27). The fairly accurate estiˆ used in the control scheme may be applied here mate G as well. F essentially represents the velocity component ˆ v, resulting from fluid compression (and leakage), and Gu the velocity component due to the valve flow. Combining ˆ v with the super twisting algorithm allow for obtaining Gu the estimate Fˆ . Substituting an estimate of the model (27) into (53), this may be written: ˆ v + Fˆ − x˙ P x ˜˙ P = Gu (59)

Parameter

Test No.

Perturbation Parameter

Impact Time

Size of Variation

1 1 3 3

PS PS Ff C + Ff L Ff C + Ff L

3 [s] 20 [s] 8 [s] 17 [s]

−0.2PS +0.3PS +10.0(Ff C + Ff L ) −5.0(Ff C + Ff L )

proposed concept in general. Hence perturbations can be applied to specific parameters, and the system response related to a given perturbation may be evaluated explicitly. Furthermore the ability of the observer to actually track the true piston velocity can be verified exactly. 6.1 Simulated System The system model used for simulation is a verified model of an asymmetric VCD implemented in an ethanol production plant. The VCD resembles that of figure 1, however horizontally mounted. The flow control valve applied is a Bosch Rexroth 4WRTE 16 V1 200L (two stage) proportional valve. Remaining parameters of the system are presented in table 1. The proposed controller (hereafter denoted PCSM-C) is evaluated with parameters ε = 0.4, λ = 1.26, Ks = 0.02 and is compared to a constant gain feed-forward plus PI controller (PI-FWR), given by: Z uc = KF (x˙ R + Kp e + Ti−1 edt), t  149 for xv > 0 KF = , Kp = 5 , Ti = 0.0106 201 for xv < 0 6.2 Parametric Perturbations & Noise In VCD’s designed for industrial environments, variations are often profound in supply pressure due to multiple VCD’s supplied by a single pump, and in the combined friction of the drive itself and the attached load. Hence the controller is evaluated according to such parametric perturbations. Furthermore in order to evaluate the proposed control concept under what may be considered worst case situations, the perturbations are applied as steps in time. The parameter variations and time of impact are outlined in table 2. The reference trajectory applied for evaluation resembles that of the physical system, with blends between constant velocity segments being fifth order polynomials.

104

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

Furthermore the trajectory covers 94.5 % of the piston range, meaning that the natural nonlinearities of the VCD will influence the control performance. The reference trajectory is shown in figure 2.

a maximum amplitude of ∼ 5 [µm]. The observer response to the noisy signal is shown in figure 4 compared with the robust differentiator and the position differentiated by use of the Forward Euler rule. From figure 4 it is

500 400 300

Position [mm]

200 100 0 -100 -200 -300

Fig. 4. Observer performance in the presence of position measurement noise (∼ 5 µm amplitude).

-400 -500 0

5

10

Time [s]

15

20

25

Fig. 2. Reference trajectory applied for controller validation. In industrial environments zero mean noise is usually seen in the range 50 − 2000 [Hz] (besides quantization noise), and for robustness evaluation toward signal noise, random noise of seven different frequencies in this range is applied, having a combined maximum amplitude of ∼ 1 [mm], i.e. 0.1 % of the position range, which may be considered an extreme noise level for closed loop control applications in the industry.

found that both the robust differentiator as well as the proposed velocity observer pertains the ability to track transient- and steady state piston velocities in the presence of noise, with significantly reduced noise impact compared with the Forward Euler differentiation approach. Also it is found that the proposed velocity observer exhibits slightly reduced noise impact compared to the robust differentiator. It should be noted that if fluid compression dominate the velocity, the required gains of the velocity observer needed to guarantee convergence, may cause noise impact on the level of the robust differentiator. In any case, the results of figure 4 illustrates the superiority in noise robustness of the robust differentiator and the proposed observer, compared to the conventional differentiation approach.

6.3 Observer Performance 6.4 General Controller Performance The performance of the proposed observer is evaluated separately in order to emphasize its properties. The phase portrait shown in figure 3 illustrates the convergence of the super twisting algorithm when applied for differentiation of the piston position. Clearly, the error states encircles the origin in a twisting manner, and finally converges in the vicinity of the origin, as would be expected.

The performance of the proposed controller is evaluated generally by considering its ability to handle the nonlinear dynamics of the system. The convergence to the constraint S = 0 is evaluated with an initial offset e = xR − xP = 100 [mm]. It is evident from figure 5 that the proposed controller preserves the features of conventional first order sliding control design, in switching about S = 0 after reaching this. Furthermore figure 6 shows that chattering reduction is achieved without introducing any boundary layer. 6.5 Controller Robustness Towards Perturbations Having demonstrated the chattering free property of the proposed controller, its robustness towards the parameter perturbations described above is considered.

Fig. 3. Phase portrait of position- and velocity errors, showing the convergence of the differentiated position signal.

Perturbations in Friction It is found from figure 7 that the PCSM-C- and PI-FWR controlled systems differ significantly from each other, when subjected to variations in coulomb friction. The PCSM-C exhibits significantly lesser oscillations while converging to e = 0 than the PIFWR, and significantly smaller maximum errors at impact of perturbations.

A position signal is fed to the observer with the addition of zero mean noise in the frequency range 50−2000 [Hz], with

Also from figure 8 it is found that control chattering is limited.

105

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

60

0.3

0.25

40

PCSM-C control signal PFWR-PI control signal

0.2

Control Signal [%]

Sliding Surface [-]

20 0.15

0.1

-20

0.05

-40

0

-0.05 0

5

10

Time [s]

15

20

-60 0

25

Fig. 5. Convergence of the sliding surface S.

20 Control Signal [%]

10

Time [s]

15

20

25

PI-FWR will in this case be off, and the PI terms must compensate for this. Furthermore figure 10 show that control chattering also here is limited.

PCSM-C control signal

30

1.5

10

PCSM-C controlled system PFWR-PI controlled system

1

0 0.5

-10 Position [mm]

-20 -30 -40 -50 0

0 -0.5 -1

5

10

Time [s]

15

20

25

-1.5 -2

Fig. 6. Control signal of the proposed controller.

-2.5 0

1.5

5

10

Time [s]

15

20

25

PCSM-C controlled system PFWR-PI controlled system

1

Fig. 9. Tracking errors of controllers when subjected to perturbations in supply pressure.

0.5

Position [mm]

5

Fig. 8. Output signals of controllers when subjected to perturbations in coulomb friction.

50 40

0

60 0

-1 Control Signal [%]

20

-1.5

-2 0

PCSM-C control signal PFWR-PI control signal

40

-0.5

5

10

Time [s]

15

20

25

0

-20

Fig. 7. Tracking errors of controllers when subjected to perturbations in coulomb friction.

-40

Perturbations in Supply Pressure Figure 9 show the tracking error when decreasing- and increasing the nominal supply pressure. It is found that the PCSM-C rapidly adjusts the control gain, resulting from the perturbed supply pressure. However, the feed forward control of the

-60 0

5

10

Time [s]

15

20

25

Fig. 10. Output signals of controllers when subjected to perturbations in supply pressure.

106

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

50

Table 3. RMS- and maximum error measures of controllers, when applied to parameter perturbations.

40

PCSM-C control signal

30

Error

PS pert.

Ff C + Ff L pert.

PCSM-C RMS PCSM-C MAX PI-FWR RMS PI-FWR MAX

0.0733 0.9078 0.2676 2.3428

0.0779 1.1275 0.2242 1.9149

20

[mm] [mm] [mm] [mm]

Control Signal [%]

[mm] [mm] [mm] [mm]

In order to have some measures of the PCSM-C performance compared to the PI-FWR, RMS- and maximum errors are considered. These are shown in table 3. Also from table 3 it is evident that the proposed controller (PCSM-C) demonstrates improved performance over the conventional PI-FWR.

2

PCSM-C controlled system

1.5

Position [mm]

1

0.5

-10

-30 -40 -50 0

5

10

Time [s]

15

20

25

Fig. 12. Control signal to valve when subjected to 1 [mm] amplitude noisy position signal. deriving a control law for the control signal derivative was applied, with the model being a second order system representation. An integral sliding surface was established requiring the piston velocity, and the closed loop system was proven stable. Due to the lack of a velocity measurement, a velocity observer was designed based on the same control gain estimate as that of the control law, and was used to provide a rough estimate of the velocity. In order to obtain the velocity estimation error, the second order super twisting algorithm was applied, introducing only a single additional parameter. From evaluation of the proposed control concept it was found that within a broad range of values for the tuning parameters, excellent performance may be achieved, i.e. the closed loop system is not particularly sensitive to its parameters, making the proposed concept highly applicable in industry.

0

-0.5

-1

-1.5 0

0

-20

6.6 Controller Robustness Towards Measurement Noise From figure 11 it is found that the position tracking error expands to slightly above the level of noise for the proposed controller. From figure 12 it is seen that the control signal is significantly affected, but still only limited chattering is present due to the integral action on the control signal.

10

5

10

Time [s]

15

20

25

Fig. 11. Tracking error when subjected to 1 [mm] amplitude noisy position signal.

The proposed control concept provides excellent tracking performance, and proves highly robust to parametric perturbations commonly seen in industry - namely supply pressure and friction. Furthermore the control concept indicate strong robustness toward extreme noise levels in the piston position measurement.

7. CONCLUSION

REFERENCES

In this paper a robust sliding mode control concept was presented with the aim of reducing control chattering, maintaining the number of tuning parameters to the level of tradition linear PID schemes being generally applicable to VCD’s operating under industrial conditions (i.e. relying only on common data sheet information and utilizing only piston- and valve spool positions- and pressure feedback).

Andersen, T.O. and Hansen, M.R. (2003). Fluid Power Systems - Modelling and Analysis. Aalborg University, Denmark, 2nd Edition. Bonchis, A., Corke, P.I., Rye, D.C., and Ha., Q.P. (2001). Variable structure methods in hydraulic servo systems control. Automatica, 37, pp. 589–595. Chen, H.M., Renn, J.C., and Su, J.P. (2005). Sliding mode control with varying boundary layers for an electrohydraulic position servo system. International Journal of Advanced Manufacturing Technology, 26, 117–123. Chern, T. and wu, Y. (1991). Design of integral variable structure controller and application to electrohydraulic velocity servosystems. IEE Proceedings-D, 138(5). Chin-Wen Chuang, C.L.H. (2005). Applying discrete dynamic integral sliding surface control to hydraulic position control. IEEE International Conference on Industrial Technology (ICIT 2005).

The control design was carried out based on a generalized reduced order model representation of a VCD constituted by an asymmetric cylinder and an asymmetric proportional valve, with unmatched asymmetries as commonly seen in industry. In order to reduce chattering without introducing additional parameters, and removing the well known, and commonly applied, smoothing function of the robust discontinuous term, a design approach focusing on

107

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

Damiano, A., Gatto, G.L., Marongiu, I., and Pisano, A. (2004). Second-order sliding-mode control of dc drives. IEEE Transactions on Industrial Electronics, 51(2). Davila, J., Fridman, L., and Levant, A. (2005). Secondorder sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control, 50(11). Dong, C., Lu, J., and Meng, Q. (2011). Position control of an electro-hydraulic servo system based on improved smith predictor. International Conference on Electronic and Mechanical Engineering and Information Technology. Emelyanov, S., Korovin, S., and Levantovsky, L. (1986a). Higher order sliding modes in binary control systems algorithms, in russian. Soviet Physics, Doklady, 31(4), pp. 291–293. Emelyanov, S., Korovin, S., and Levantovsky, L. (1986b). Second order sliding modes in controlling uncertain systems, in russian. Soviet Journal of Computing and Systems Science, 24(4), pp. 63–68. Emelyanov, S., Korovin, S., and Levantovsky, L. (1990). New class of second order sliding algorithms. Mathematical Modelling, 2(3), pp. 89–100. Fung, R.F., Wang, Y.C., Yang, R.T., and Huang, H.H. (1997). A variable structure control with proportional and integral compensators for electrohydraulic position servo control system. Mechatronics, 7(1), pp. 67–81. Fung, R.F. and Yang, R.T. (1998). Application of vsc in position control of a nonlinear electrohydraulic servo system. 66(4), pp. 365–372. Ghazali, R., Sam, Y.M., Rahmat, M.F., Hashim, A.W.I.M., and Zulfatman (2010). Position tracking control of an electro-hydraulic servo system using sliding mode control. Proceedings of 2010 IEEE Student Conference on Research and Development (SCOReD 2010), Putrajaya, Malaysia. Ghazy, M.A. (2001). Variable structure control for electrohydraulic position servo system. 27th Annual Conference of the IEEE Industrial Electronics Society. Guo, H., Liu, Y., Liu, G., and Li, H. (2008). Cascade control of a hydraulically driven 6-dof parallel robot manipulator based on a sliding mode. Control Engineering Practice, 16, pp. 1055–1068. Habibi, S.R. (1995). Sliding mode control of a hydraulic industrial robot. Proceedings of the American Control Conference, Seattle, Washington. Hansen, M.R., Andersen, T.O., and Pedersen, H.C. (2005). Robust control of a hydraulically actuated manipulator using sliding mode control. Proceedings of The Sixth International Conference on Fluid Power Transmission and Control. Hisseine, D. (2005). Robust tracking control for a hydraulic actuation system. Proceedings of the 2005 IEEE Conference on Control Applications Toronto, Canada. Hwang, C. and Lan, C. (1994). The position control of electrohydraulic servomechanism via a novel variable structure control. 4(4), pp. 369–391. Komsta, J., Adamy, J., and Antoszkiewicz, P. (2010a). Input-output linearization and integral sliding mode disturbance compensation for electro-hydraulic drives. 11th International Workshop on Variable Structure Systems, Mexico City, Mexico. Komsta, J., Antoszkiewicz, P., Heeg, T., and Adamy, J. (2010b). New nonlinear robust control concept for

electro-hydraulic drives. 7th International Fluid Power Conference, Aachen, Germany. Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(3), pp. 1247–1263. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34(3), pp. 379–384. Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, 76(9/10), pp. 924–941. Levantovsky, L. (1985). Second order sliding algorithms: Their realization, in russian. Dynamics of Heterogeneuos Systems. Material of the Seminar (Moscow: The Institute for System Studies), in Russian, pp. 32– 43. Liu, Y. and Handroos, H. (1999). Sliding mode control for a class of hydraulic position servo. Mechatronics, 9, pp. 111–123. Merritt, H.E. (1967). Hydraulic Control Systems. New York: Wiley. Mohieddine Jelali, A.K. (2004). Hydraulic servo systems. Springer. ISBN: 1-85233-692-7. Nguyen, Q., Ha, Q., Rye, D., and Durrant-Whyte, H. (2000). Force/position tracking for electrohydraulic systems of a robotic excavator. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia. Pi, Y. and Wang, X. (2010). Observer-based cascade control of a 6-dof parallel hydraulic manipulator of a 6-dof hydraulic parallel robot manipulatorin joint space coordinate. Mechatronics, 20, pp. 648–655. Pi, Y. and Wang, X. (2011). Trajectory tracking control of a 6-dof hydraulic parallel robot manipulator with uncertain load disturbances. Control Engineering Practice, 19, pp. 185–193. Slotine, J.J.E. and Li, W. (1991). Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. Wang, S., Burton, R., and Habibi, S. (2011). Sliding mode controller and filter applied to an electrohydraulic actuator system. Journal of Dynamic Systems, Measurement, and Control, 133. Wang, X. and Su, X. (2007). Modeling and sliding mode control of the upper arm of a shotcrete robot with hydraulic actuator. Proceedings of the IEEE International Conference on Integration Technology, Shenzhen, China. Yoon, J. and Manurung, A. (2010). Development of an intuitive user interface for a hydraulic backhoe. Automation in Construction, 19, pp. 779–790. Zulfatman, Rahmat, M.F., Husain, A.R., Ghazali, R., and Rozali, S.M. (2011). Smooth control action of sliding mode for a class of electro-hydraulic actuator. 4th International Conference on Mechatronics (ICOM11), Kuala Lumpur, Malaysia.

108