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A Backstepping Sliding Mode Control for a Hydrostatic Transmission with Unknown Disturbances
10th IFAC Symposium on Nonlinear Control Systems 10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems 10th Symposium on Control Systems August 23-25, 2016. Monterey, California, USA Available August USA August 23-25, 23-25, 2016. 2016. Monterey, Monterey, California, California, USA online at www.sciencedirect.com
ScienceDirect IFAC-PapersOnLine 49-18 (2016) 879–884
A Backstepping Sliding Mode Control for a A A Backstepping Backstepping Sliding Sliding Mode Mode Control Control for for a a Hydrostatic Transmission with Unknown Hydrostatic Transmission with Unknown Hydrostatic Transmission with Unknown Disturbances Disturbances Disturbances ∗ ∗ Hao Sun ∗ Harald Aschemann ∗ Hao Sun Harald Aschemann ∗ ∗ Hao Hao Sun Sun ∗ Harald Harald Aschemann Aschemann ∗ ∗ ∗ Chair of Mechatronics, University of Rostock, 18059, Rostock, ∗ Chair of Mechatronics, University of Rostock, 18059, Rostock, ∗ Chair of of (e-mail: Mechatronics, University of Rostock, Rostock, 18059, 18059, Rostock, Rostock, Germany hao.sun, [email protected]). Chair Mechatronics, University of Germany (e-mail: hao.sun, [email protected]). Germany Germany (e-mail: (e-mail: hao.sun, hao.sun, [email protected]). [email protected]).
Abstract: A A robust robust backstepping backstepping control control via via second-order second-order sliding sliding mode mode for for the the speed speed tracking tracking Abstract: Abstract: A robust backstepping control via second-order sliding mode for the speed tracking control of a hydrostatic transmission is presented in this paper. The proposed approach consists Abstract: A robust backstepping control via second-order sliding mode for the speed tracking control of a hydrostatic transmission is presented in this paper. The proposed approach consists control of a hydrostatic transmission is presented in this paper. The proposed approach consists of a stabilising control action designed according to backstepping control techniques and a robust control of a hydrostatic transmission isaccording presentedtoinbackstepping this paper. The proposed approach consists of a control action designed control and of a stabilising stabilising control action designed according to(SOSMC) backstepping control techniques techniques and aaa robust robust control term using a second-order sliding mode approach, which is introduced to of a stabilising control action designed according to backstepping control techniques and robust control term using a second-order sliding mode (SOSMC) approach, which is introduced to control term using a second-order sliding mode (SOSMC) approach, which is introduced to counteract the impact of uncertain system parameters as well as unknown disturbances. For the control term using a second-order sliding mode (SOSMC) approach, which is introduced to counteract the impact of uncertain system parameters as well as unknown disturbances. For the counteract the uncertain system well disturbances. For control implementation implementation on a test test bench bench of aaparameters hydrostaticas transmission, to estimate estimate unmeasurable counteract the impact impact of ofon uncertain system parameters astransmission, well as as unknown unknown disturbances. For the the control a of hydrostatic to unmeasurable control on bench to system implementation states and and two two lumped lumped disturbances present in in transmission, the system, system, the the proposedunmeasurable controller is is control implementation on a a test testdisturbances bench of of aa hydrostatic hydrostatic transmission, to estimate estimate unmeasurable system states present the proposed controller system states and two lumped lumped disturbances present in in the the system, the proposed proposed controllerthe is extended with a sliding mode observer. Both simulations and experimental results highlight system states and two disturbances present system, the controller is extended with a sliding mode observer. Both simulations and experimental results highlight the extended with sliding mode observer. applicability ofa the proposed method. extended withof athe sliding mode method. observer. Both Both simulations simulations and and experimental experimental results results highlight highlight the the applicability proposed applicability applicability of of the the proposed proposed method. method. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear Nonlinear robust robust control, control, second-order second-order sliding sliding mode, mode, hydrostatic hydrostatic transmission, transmission, Keywords: Keywords: Nonlinear robust control, control, second-order second-order sliding sliding mode, mode, hydrostatic hydrostatic transmission, transmission, sliding mode observer. Keywords: Nonlinear robust sliding mode observer. sliding mode observer. sliding mode observer. 1. INTRODUCTION INTRODUCTION input directly directly acts acts on on the the first first time time derivative derivative of of the the sliding sliding 1. input 1. input directly acts on the first time derivative of the sliding manifold, HOSMC is a generalisation of basic SMC and the 1. INTRODUCTION INTRODUCTION input directly acts on the first time derivative of the sliding manifold, HOSMC HOSMC is is aa generalisation generalisation of of basic basic SMC SMC and and the manifold, control input acts on a higher-order time derivative of the manifold, HOSMC is a generalisation of basic SMC and Robustness has become an important issue in control control input input acts acts on on aa higher-order higher-order time time derivative derivative of of the Robustness has has become become an an important important issue issue in in control control control sliding manifold, manifold, see Perruquetti andtime Barbot (2002).of the control input actssee on Perruquetti a higher-order derivative the Robustness systems since sincehas thebecome emergence control theory. sliding and Barbot (2002). Robustness an of important issue To in achieve control sliding systems the emergence of control theory. To achieve manifold, see Perruquetti Barbot (2002). sliding manifold, see Perruquetti and and Barbotcontrol (2002). systems since the emergence of control theory. To achieve a high control performance, not only nonlinearities and a systems since the emergence of control theory. To achieve Backstepping is a Lyapunov-based recursive design high control control performance, performance, not not only only nonlinearities nonlinearities and and aa Backstepping is a Lyapunov-based recursive control design aa high time-varying behaviour mustnot be taken taken into account account at at thea Backstepping is Lyapunov-based recursive control a high controlbehaviour performance, only nonlinearities and method for for the the class of ”strict ”strict feedback” which Backstepping is aaclass Lyapunov-based recursivesystems control design design time-varying must be into the method of feedback” systems which time-varying behaviour must be taken into account at the model-based control design but also model uncertainty and for the class of ”strict feedback” systems which time-varying behaviour must bealso taken intouncertainty account at and the method first appeared in the beginning of the 1990s, see Kokomethod for the class of ”strict feedback” systems which model-based control design but model first appeared in the beginning of the 1990s, see Kokomodel-based control unknown disturbances. disturbances. appeared in the beginning of the 1990s, see Kokomodel-based control design design but but also also model model uncertainty uncertainty and and first tovic (1992). An important advantage of the backstepping first appeared in the beginning of the 1990s, see Kokounknown tovic (1992). (1992). An An important important advantage advantage of of the the backstepping unknown disturbances. unknown disturbances. design(1992). approach is that it it provides provides a systematic systematic procedure tovic An is important advantage of the backstepping backstepping In the the past past decades, aa remarkable remarkable progress progress has has been been tovic design approach that a procedure In decades, design approach is that it provides a systematic procedure to design stabilising controllers, following a step-by-step design approach is that it provides a systematic procedure In the decades, aa remarkable progress has been achieved in the the development of robust control methods to design design stabilising stabilising controllers, controllers, following following aa step-by-step step-by-step In the past past decades, remarkable progress been to achieved in development of robust robust control has methods algorithm. Therefore, controllers, the construction construction of feedback feedback control to design stabilising following a step-by-step achieved in the development of control methods for nonlinear systems. Among all the existing approaches, algorithm. Therefore, the of control achieved in the development of robust control methods for nonlinear nonlinear systems. systems. Among Among all all the the existing existing approaches, approaches, algorithm. Therefore, the construction of feedback control laws and Lyapunov functions is systematic. Backstepping algorithm. Therefore, the construction of feedback control for sliding mode control (SMC) has gained special attention laws and and Lyapunov Lyapunov functions functions is is systematic. systematic. Backstepping Backstepping for nonlinear Amonghas all gained the existing approaches, sliding mode systems. control (SMC) (SMC) special attention laws design hasLyapunov the flexibility flexibility to avoid avoid cancellations of useful useful laws and functions is systematic. Backstepping sliding mode has special attention from practical practical control(SMC) engineers, see David David Young et al. al. design has the to cancellations of sliding mode control control has gained gained special attention from control engineers, see Young et design has the flexibility to avoid cancellations of useful nonlinearities and to achieve stabilisation and tracking. design has the flexibility to avoid cancellations of useful from practical control see Young et (2010). SMC belongs belongs toengineers, variable structure structure control methnonlinearities and and to to achieve achieve stabilisation stabilisation and and tracking. tracking. from practical controlto engineers, see David Davidcontrol Young methet al. al. nonlinearities (2010). SMC variable Moreover, this thisand control method can be be easily nonlinearities to achieve stabilisation and extended tracking. (2010). SMC belongs to variable structure control methods. The control principle is that the SMC brings the states Moreover, control method can easily extended (2010). SMC belongs to variable structure control methods. The The control control principle principle is is that that the the SMC SMC brings brings the the states states Moreover, this control method can be easily extended with adaptive control and sliding mode control without Moreover, this control method can be easily extended ods. towards a sliding surface and forces the states to remain with adaptive adaptive control control and and sliding sliding mode mode control control without without ods. The acontrol principle is that the SMC brings to theremain states with towards sliding surface and forces the states the restriction on matched condition of the uncertainties. withrestriction adaptive on control and condition sliding mode control without towards aa sliding surface forces the to on the the surface surface afterwards. The most most significant property of the matched of the the uncertainties. towards sliding surface and and forcessignificant the states statesproperty to remain remain on afterwards. The of the restriction matched condition of uncertainties. Nowadays, the on application of backstepping control and its its the restriction on matched of condition of thecontrol uncertainties. on the surface afterwards. The most significant property of SMC is its robustness. When the system reaches the sliding Nowadays, the application backstepping and on the surface afterwards. The most significant property of SMC is is its its robustness. robustness. When When the the system system reaches reaches the the sliding sliding Nowadays, the application of backstepping control and its extensions to robust control are still frontier subjects Nowadays, the application of backstepping control and its SMC surface, it robustness. is insensitive insensitive to the parameter uncertainties and extensions to robust control are still frontier subjects in in SMC is its When system reaches the sliding surface, it is to parameter uncertainties and extensions to control theory. theory. extensions to robust robust control control are are still still frontier frontier subjects subjects in in surface, it is insensitive to parameter uncertainties and external disturbances if the matching condition is satisfied. control surface, it is insensitive to parameter uncertainties and external disturbances disturbances if if the the matching matching condition condition is is satisfied. satisfied. control theory. control theory. external In comparison with other robust control approaches like external disturbances if the matching condition is satisfied. In this this paper, paper, aa decentralised decentralised control control structure structure is is proposed: proposed: In comparison comparison with with other other robust robust control control approaches approaches like like In In internal model control control (IMC) or aa LMI-based design, SMC this paper, aa decentralised control structure is proposed: In comparison with other robust control approaches like In A backstepping second-order sliding mode control is ememIn this paper, decentralised control structure is proposed: internal model (IMC) or LMI-based design, SMC A backstepping backstepping second-order second-order sliding sliding mode mode control control is is internal model control (IMC) or aa LMI-based design, SMC has a simple control structure and can provide fast dyA eminternal model control (IMC) or LMI-based design, SMC ployed for the tracking control of the motor angular velocbackstepping second-order mode angular control is emhas aa simple simple control control structure structure and and can can provide provide fast fast dydy- A ployed for the the tracking tracking controlsliding of the the motor motor velochas namic responses. Plenty of applications applications canprovide be found found in dythe ployed control velochas a responses. simple control structure and can fast ity of of aafor hydrostatic transmission, whereas flatness-based ployed for the tracking control of of the motoraa angular angular velocnamic Plenty of can be in the ity hydrostatic transmission, whereas flatness-based namic responses. Plenty can be in literatures, see Aschemann Aschemann and Schindele Schindele (2008); Bartolini of whereas flatness-based namic responses. Plenty of of applications applications can be found found in the the ity approach is used used for fortransmission, the tracking tracking control control ofaathe the normalised ity of aa hydrostatic hydrostatic transmission, whereasof flatness-based literatures, see and (2008); Bartolini approach is the normalised literatures, see Aschemann and Schindele (2008); Bartolini et al. (2008); Panchade et al. (2013); Su and Leung (1993). approach is used for the tracking control of the normalised literatures, see Aschemann and Schindele (2008); Bartolini bent axis angle of the hydraulic motor. Furthermore, the approach is used for the tracking control of the normalised et al. (2008); Panchade et al. (2013); Su and Leung (1993). bent axis axis angle angle of of the the hydraulic hydraulic motor. motor. Furthermore, Furthermore, the the et al. (2008); Panchade et al. (2013); Su and Leung (1993). The main drawback of SMC is the so-called chattering bent et al. (2008); Panchade et al. (2013); Su and Leung (1993). unmeasurable system states and unknown disturbances axis anglesystem of the hydraulic Furthermore, the The main main drawback drawback of of SMC SMC is is the the so-called so-called chattering chattering bent unmeasurable states and andmotor. unknown disturbances The effectmain whichdrawback is considered considered as the theismain main obstacle forchattering practical unmeasurable system unknown disturbances The of SMC the obstacle so-calledfor are robustly robustly estimated estimated using aaand sliding mode observer. observer. The unmeasurable system states states unknown disturbances effect which is as practical are using sliding mode The effect which as main for implementation. Chattering usually occurs when a nonrobustly estimated using aa sliding mode The effect which is is considered considered as the the main obstacle obstacle for practical practical combined control structure been validated using both are robustly estimated usinghas sliding mode observer. observer. The implementation. Chattering usually occurs when when non- are combined control structure has been validated validated using both both implementation. Chattering usually occurs aaa nonnegligible, fast dynamics is not considered in the system combined control structure has been using implementation. Chattering usually occurs when nonsimulation and experimental results. combined control structure has been validated using both negligible, fast dynamics is not considered in the system simulation and and experimental experimental results. results. negligible, dynamics not in system model. In In fast digital control,is finite sampling time negligible, fast dynamics isfinite not considered considered in the theintervals system simulation simulation and experimental results. model. digital control, sampling time intervals The rest of the paper is organised as follows: follows: In In Section Section 2, 2, model. In digital control, finite sampling time intervals could cause a switching delay and lead to ”discretisation model. In digital control, finite sampling time intervals The rest of the paper is organised as could cause a switching delay and lead to ”discretisation The rest of the paper is organised as follows: In Section 2, the hydrostatic transmission is briefly introduced. For the could cause aaToswitching delay and lead to ”discretisation The rest of the paper is organised as follows: In Section 2, chattering”. avoid this phenomenon, high-order SMC could cause switching delay and lead to ”discretisation the hydrostatic hydrostatic transmission transmission is is briefly briefly introduced. introduced. For For the the chattering”. To To avoid avoid this this phenomenon, phenomenon, high-order high-order SMC SMC the control design, transmission the nonlinear nonlinearismodelling modelling of the the system system is chattering”. hydrostatic briefly introduced. For the (HOSMC) could could be employed, employed, see Levant Levant (2010); Utkin Utkin chattering”. To avoid this phenomenon, high-order SMC the control design, the of is (HOSMC) be see (2010); control design, the nonlinear modelling of the system is addressed. Based on the state-space model, the tracking (HOSMC) could be employed, see Levant (2010); Utkin control design, the nonlinear modelling of the system is (2014). Different from the basic SMC, where the control (HOSMC) could from be employed, Levant (2010); Utkin addressed. Based on the state-space model, the tracking (2014). Different Different the basic basicsee SMC, where the control control addressed. Based on the state-space model, the tracking (2014). (2014). Different from from the the basic SMC, SMC, where where the the control addressed. Based on the state-space model, the tracking Copyright 2016 IFAC 891 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 891 Copyright 2016 IFAC 891 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 891Control. 10.1016/j.ifacol.2016.10.277
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control design is discussed in details in Section 3. The state and disturbance observer is given in Section 4. In Section 5, the simulation and experimental results are presented. Finally, conclusions are given in Section 6. 2. CONTROL-ORIENTED MODELLING OF A HYDROSTATIC TRANSMISSION A hydrostatic transmission is a hydraulic circuit which usually consists of a variable displacement hydraulic pump and a variable displacement hydraulic motor. It generates and transmits the power by means of a pressurised fluid, and provides a variety of advantages in comparison to pure mechanical transmissions, see Jelali and Kroll (2003). Nowadays, it is commonly considered as a characteristic component of drive chains for the applications in working machines and off-road vehicles. However, hydrostatic transmissions are typical nonlinear systems and characterised by uncertain system parameters and unknown disturbances which limit their applications in highperformance control system. A test bench of a hydrostatic transmission system depicted in Fig. 1, can be found at the Chair of Mechatronics, University of Rostock. From the hydraulic scheme, it can be seen that the test bench consists of a variable displacement
Hydraulic motor
Load motor
Drive motor Hydraulic pump
Volume flow qA Motor angular velocity M and torqueM
Pressure pA
Pump angular velocity
P
M
M
Load motor
Drive motor
Volume flow qB
by two subsystems: one is for the control of the normalised bent axis angle α ˜ M which can be stated as 1 1 α ˜M + uM , (1) α ˜˙ M = − TuM TuM with the time constant TuM and the control input uM . The second subsystem contains the dynamics of the normalised swashplate angle α ˜ P , the pressure difference ∆p and the motor angular velocity ωM given by 1 1 α ˜P + uP , α ˜˙ P = − TuP TuP 2VP ωP tan(αP,max · α ˜P ) CH (2) 2VM ωM sin(αM,max · α qU ˜M ) − − , CH CH dV VM τU ∆p sin(αM,max · α ˜M ) − , ω˙ M = − ωM + JV JV JV and serves for the control of the motor angular velocity ωM . Here, TuP represents the time constant. VP,M are volume displacements resulting from the geometric structure of the pump and motor. αP,M,max denote the maximum tile angles of the pump and motor. CH represents the hydraulic capacitance. The parameters JV and dV denote the moments inertial and damping coefficient on the motor side. The angular velocity of the pump is represented by ωP which is considered here as a constant parameter. Moreover, the unknown disturbances are modelled as lumped parameters: the leakage volume flow qU in the main hydraulic circuit and the disturbance torque τU acting on the hydraulic motor. It is worth to point out that the introduced lumped disturbance parameters contain not only the unknown disturbances existing in the system dynamics, but also the model uncertainty due to uncertain physical parameters, e.g., the bulk modulus of the oil which depends on the working pressure and temperature. Furthermore, the control input is denoted as uP . In Aschemann and Sun (2013), it is also proved that each subsystem is differentially flat w.r.t. the corresponding flat output – the normalised bent axis angle α ˜ M and the motor angular velocity ωM . ∆p˙ =
3. CONTROL DESIGN FOR THE HYDROSTATIC TRANSMISSION
Pressure pB
Fig. 1. Drive train with a closed-circuit hydrostatic transmission. axial piston pump and a variable displacement bent axis motor connected in a closed circuit. Two electric motors are employed to represent the prime mover on the pump side and the load unit on the motor side, respectively. Pressure sensors are mounted on the top of the pump measuring pressures on both high and low pressure side, respectively. Moreover, a volume flow sensor is available on each side of the drive train. This mechatronic system can be split into a hydraulic subsystem and a mechanical subsystem, which are coupled by the torque τM generated by the hydraulic motor. The details of the control-oriented model for the hydrostatic transmission system can be found in Aschemann and Sun (2013). The complete dynamic system is characterised 892
3.1 Decentralised control structure The proposed decentralised control structure is depicted in Fig. 2. The estimated unmeasurable system states – the ˆ˜ P,M – and the estimated unknown normalised tilt angles α ˆ provided disturbances – qˆU and τˆU – form the vector d by the sliding mode observer. In this control structure, the motor angular velocity ωM is directly regulated by the normalised swashplate angle of the pump α ˜ P , and can be further influenced by varying of the normalised bent axis angle of the motor α ˜ M . In the following, the control design for each subsystem will be discussed. 3.2 Flatness-based control design for α ˜M A flatness-based control (FBC) approach is employed for the tracking control of the normalised bent axis angle α ˜M .
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α ˜M d α ˜˙ M d
Flatnessbased control
uM ˆ α ˜M
ˆ Sliding mode ωM d d observer ω˙ M d ω ¨ M d ... ω Md Backstepping sliding mode control
∆p ωM
uP
Hydrostatic transmission
[qU τU ]T
Fig. 2. Decentralised control scheme. The FBC contains the inverse dynamics of the corresponding first-order differential equation (1) and a stabilising control input υM , i.e., α ˜ M + υM TuM . (3) uM = kM The stabilising control law represents a combination of a feedforward and feedback control as follows t ˙ ˜ M d + kα0 eα˜ M + kαI · eα˜ M dτ , (4) υM = α 0
˜ M d −α ˜ M as well as positive with the tracking error eα˜ M = α coefficients kα0 > 0 and kαI > 0, which can be determined according to the desired Hurwitz polynomial.
881
Step 2: In the second step, an error variable e2 is considered according to (10) e2 = ω˙ M − α1 = ω˙ M − ω˙ M d + c1 e1 . Given e˙ 1 = ω˙ M − ω˙ M d , this results in (11) e˙ 1 = e2 − c1 e1 . The corresponding error dynamics e˙ 2 can be expressed as ¨M − ω ¨ M d + c1 e˙ 1 . (12) e˙ 2 = ω ¨ M . Introducing Here, the virtual input is chosen as α2 ≈ ω the Lyapunov function 1 V2 = V1 + e22 . (13) 2 Its time derivative results in V˙ 2 = e1 e˙ 1 + e2 e˙ 2 = e1 (e2 − c1 e1 ) + e2 (¨ ωM − ω ¨ M d + c1 e˙ 1 ) 2 = −c1 e1 + e2 (¨ (14) ωM − ω ¨ M d + c1 e˙ 1 + e1 ) , !
=−c2 e2
with the positive coefficient c2 . Therefore, α2 is chosen as ¨ M d − c1 e˙ 1 − e1 − c2 e2 , (15) α2 = ω which leads to V˙ 2 = −c1 e21 − c2 e22 < 0 , when e1 = 0 , e2 = 0 . (16) Step 3: The third error variable e3 is defined as ¨ M − α2 = ω ¨ −ω ¨ M d + c1 e˙ 1 +e1 + c2 e2 . (17) e3 = ω M e˙ 2
This leads to
3.3 Backstepping sliding mode control design for ωM If a nonlinear system is differential flat, the original system dynamics can be differentially parametrised using the flat output. Therefore, the subsystem of the motor angular velocity represented in (2) can be rewritten as ω˙ M = ω˙ M , ω ¨M = ω ¨M , (5) ... ω M = f (ω M , τ , α ˜ M ) + g(˜ α M ) uP , ¨ M ]T and diswith the state vector ω M = [ωM ω˙ M ω T turbance vector τ = [qU τU ] . f and g are continuous functions. Moreover, it is assumed that the dynamics of the disturbances are only slow time-varying, i.e., τ˙ = [q˙U τ˙U ]T = 0. Control design Step 1: The backstepping control design starts with the definition of a quadratic Lyapunov function 1 (6) V1 = e21 , 2 with the tracking error e1 = ωM − ωM d . The first time derivative of V1 results in (7) V˙ 1 = e1 e˙ 1 = e1 (ω˙ M − ω˙ M d ) . !
=−c1 e1
Here, c1 > 0 denotes a constant parameter. The virtual input α1 ≈ ω˙ M is chosen as α1 = ω˙ M d − c1 e1 , (8) which leads to (9) V˙ 1 = −c1 e21 < 0 , when e1 = 0 . 893
e˙ 2 = e3 − e1 − c2 e2 . The error dynamics of e3 can be stated as ... ... e˙ 3 = ω M − ω M d + c1 e¨1 + e˙ 1 + c2 e˙ 2 . Introducing the Lyapunov function 1 V3 = V2 + e23 , 2 it’s first time differentive becomes V˙ 3 = e1 (e2 − c1 e1 ) + e2 (e3 − e1 − c2 e2 ) ... ... + e3 ( ω M − ω M d + c1 e¨1 + e˙ 1 + c2 e˙ 2 ) = −c1 e21 − c2 e22 ... ... + e3 ( ω M − ω M d + c1 e¨1 + e˙ 1 + c2 e˙ 2 + e2 )
(18)
(19) (20) (21)
ψ(e1 ,e2 )
... ... = −c1 e21 − c2 e22 + e3 ( ω M − ω M d + ψ(e1 , e2 )) = −c1 e21 − c2 e22 ... + e3 (f (ω M , τ , α ˜ M ) + g(˜ αM ) uP − ω M d + ψ(e1 , e2 )). The control input uP is designed in such a way that it consists of a feedback control uf b compensating all nonlinearities and disturbances, a stabilising control input υP and a sliding mode control usm , i.e., ... (υP − usm − f (ω M , τ , α ˜ M ) − ψ(e1 , e2 ) + ω M d ) . uP = g(˜ αM ) (22) Introducing a bounded lumped parameter |φ| < φmax , which denotes the residual of an imperfect disturbance compensation, and substituting (22) into (21) results in (23) V˙ 3 = −c1 e21 − c2 e22 + e3 (υP − usm + φ) , with the stabilising control input designed as υP = −c3 e3 , c3 > 0 ,
(24)
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Eq. (23) becomes V˙ 3 = −c1 e21 − c2 e22 − c3 e23 + e3 (φ − usm ) .
(25) The sliding mode control usm is designed by means of a second-order sliding mode approach. For this purpose, the following expression of e3 is considered ¨−x ¨ +c e˙ + e1 + c2 (x˙ − x˙ d +c1 e1 ) e3 = x d 1 1 e¨1
e˙ 1
= e¨1 + (c1 + c2 ) e˙ 1 + (1 + c1 c2 ) e1 . (26) Defining a sliding manifold with a constant parameter α0 > 0 according to (27) s = e˙ 1 + α0 e1 , its first-time derivative becomes (28) s˙ = e¨1 + α0 e˙ 1 . Therefore, the error variable e3 can be represented as (29) e3 = s˙ + β s = e¨1 + (α0 + β) e˙ 1 + α0 β e1 , with (30) α 0 + β = c1 + c 2 , (31) α0 β = 1 + c 1 c 2 . Now, a second-order sliding mode control law is introduced that represents a modification of the one in Levant (2007)
(s˙ + β s) + |s|1/2 sign(s˙ + β s) , (32) |s˙ + β s|+|s|1/2 +η where k, η > 0 are constant parameters. The time derivative of the Lyapunov function V3 can be rewritten as (33) V˙ 3 = −c1 e2 − c2 e2 − c3 e2 usm = k
1
2
3
(s˙ + β s) + |s|1/2 sign(s˙ + β s) + (s˙ + β s) (φ − k ) |s˙ + β s|+|s|1/2 +η = −c1 e21 − c2 e22 − c3 e23
|s˙ + β s|+|s|1/2 φ ) sign(s˙ + β s) − k |s˙ + β s|+|s|1/2 +η To achieve asymptotic stability, V˙ 3 < 0 must be satisfied. This leads to |s˙ + β s|+|s|1/2 φ . (34) sign(s˙ + β s) ≤ k |s˙ + β s|+|s|1/2 +η It is straightforward to derive the following bound |s˙ + β s|+|s|1/2 < 1. (35) |s˙ + β s|+|s|1/2 +η Thereby, the condition for the sliding mode gain becomes (36) k ≥ φ sign(s˙ + β s) ≥ φmax . As a result, the conditions (30), (31) and (36) lead to V˙ 3 < 0, which guarantees the asymptotic stability of the system, i.e., t → 0 ⇒ ei → 0, i ∈ {1, 2, 3} + k |s˙ + β s| (
4. SLIDING MODE OBSERVER The observer design is based on the extended system model with integrators as disturbance model. Following the idea of extended linearisation, the extended system is written in a quasi-linear form with a state-dependent system matrix A(˜ αM , ωM ), i.e., αM , ωM ) xEL + B u , (37) x˙ EL = AE (˜ y m = C xEL . (38) 894
. Here, xEL = [xT1 ..y Tm ]T represents the state vector with the unmeasurable state vector x1 = [α ˜P α ˜ M q U τU ] T and the measurable state vector y m = [∆p ωM ]T . The control input vector is u = [uP uM ]T . The input and output matrices are denoted as B and C, respectively. The observability can be easily confirmed by checking the observability matrix of the quasi-linear system. For the observer design, the state-space representation is written as A11 A12 x1 B1 x˙ 1 = + u. (39) A21 A22 ym B2 y˙ m The observer the form has ˙x ˆ1 A11 A12 x B1 ˆ1 = + ˆm A21 A22 B2 y y ˆ˙ m G1 L − (ˆ ym − ym ) + υ, (40) G2 −I ˆ represents the estimated values. G1,2 denote the where (·) Luenberger-type gain matrices and L denotes a feedback gain matrix. The discontinuous switching part is defined by the vector υ according to M1 sign(∆ˆ p − ∆p) υ= , (41) M2 sign(ˆ ωM − ω M )
where M1,2 denotes the positive constant gains. Considˆ 1 − x1 and eym = y ˆ m − ym ering the definition e1 = x ¯1 = e1 + Leym , the and introducing a new error variable e resulting estimation error dynamics becomes ¯ 11 A ¯ 12 ¯1 A e 0 ¯e˙ 1 = + υ, (42) ¯ 22 −I e ym e˙ ym A21 A where the submatrices are given by ¯ 11 = A11 + L A21 , A ¯ 11 L − G1 + L (A22 − G2 ) , ¯ 12 = A12 − A A ¯ A22 = A22 − G2 − A21 L .
¯ 12 = 0 can be achieved by proper choice of the gain A matrix G1 . In the case of υ = 0, asymptotic stability of the error dynamic system (42) can be obtained by choosing the gain matrices L and G2 according to ¯∗ , (43) A11 + L A21 = A 11 ∗ ¯ (44) A22 − G2 − A21 L = A , 22
¯ ∗ and A ¯ ∗ denote asymptotically stable matrices where A 11 22 with the following characteristic polynomials ! ¯∗ ) = det(s I − A (s + sB1 )(s + sB2 )(s + sB3 )(s + sB4 ) , 11 ∗
!
¯ ) = (s + sB5 )(s + sB6 ) . det(s I − A 22
5. SIMULATION AND EXPERIMENTAL RESULTS The proposed control structure is validated using both simulation and experimental results. By exploiting the flatness property, the synchronised desired trajectories of the controlled variables are designed by taking the saturation of tilt angles into account. Fig. 1 illustrates the desired motion of the hydrostatic transmission. All the control parameters are determined based on trial and error and same values will be used in both simulation and experimental studies.
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100
0.6
80
1
60
0
40 20
0 14 28 42 56 70 t in s
BC/BCSMC
eα˜ m
0.7
·10−3
2
−1
0 14 28 42 56 70 t in s
Fig. 3. Desired trajectories of the controlled variables. Simulation results The measurement noise of the pressure sensors as well as the quantisation effects in the encoder signal of the motor angular velocity are introduced in the simulation model in order to obtain realistic results. Moreover, simulated disturbances replace the lumped parameters in (2): The leakage volume flow qU is assumed to be proportional to the pressure difference, i.e., qU = kl ∆p , with a positive leakage coefficient kl . The unknown disturbance torque τU is modelled as the sum of a load torque τL which is a 10% variation of the mass moment inertia and a friction torque τf , i.e., ω Md . (45) τU = 0.1 JV ω˙ M d + 7 tanh 0.1
eωM in rad/s
ωM d in rad/s
α ˜M d
0.8
883
0.8 0.6 0.4 0.2 0 −0.2 −0.4
0
10
20
30 40 t in s
50
60
70
30 40 t in s
50
60
70
BC BCSMC
0
10
20
Fig. 5. Tracking errors of the controlled variables. in comparison with the estimates of a nonlinear reducedorder observer in Aschemann and Sun (2013).
In addition, a parameter uncertainty of the time constant TuP is also introduced where a value of 0.17 is considered in the control and observer implementation, whereas the nominal value 0.13 is applied to the simulation model.
Fig. 5 shows the tracking errors of the controlled variables. In the simulation investigation, a comparison between the backstepping control (BC) and backstepping sliding mode control (BCSMC) is carried out. It can be seen that flatness-based control can guarantee a precise tracking of the desired normalised bent axis angle α ˜ M d . By introducing a switching term to counteract the influence of the model uncertainty, i.e., compensation error and parameter uncertainty in this case, BCSMC leads to a much better tracking performance compared with classical BC.
The estimation results of the sliding mode observer can be found in Fig. 4. It can be concluded that the robust
Experimental results
τL
Simulated Estimated
0.8 α ˜M
α ˜P
0.8
τf
0.6 0.4
The proposed backstepping sliding mode control is implemented on the test bench depicted in Fig. 1 to achieve a robust tracking performance for the motor angular velocity ωM . This test bench is operated by a host-target PC system using the software package Labview. The control approach is validated using the desired trajectories depicted in Fig. 3 under two conditions: one is with a vanishing load torque τL = 0 Nm and the other is with a constant load torque τL = 20 Nm. Moreover, in order to reduce the chattering effect, the sign function in (32) is replaced in the implementation by the tanh function.
0.7 0.6
0 10 20 30 40 50 60 70 ·10−6 t in s Simulated Estimated
5 4 3
0 10 20 30 40 50 60 70 t in s 7.4 τU in Nm
3
qU in m /s
0.2
Simulated Estimated
Simulated Estimated
7.2 7 6.8
2 0 10 20 30 40 50 60 70 t in s
0 10 20 30 40 50 60 70 t in s
Fig. 4. Estimation results of the sliding mode observer. observer provides outstanding estimates of both the unmeasurable system states α ˜ P,M and the unknown disturbances – the leakage volume flow qU and the disturbance torque τU . It is worth pointing out here that the sliding mode observer provides a much better estimation of qU 895
Fig. 6 shows the tracking errors of the controlled variables with a vanishing load torque. A flatness-based control is employed for the tracking control of the normalised bent axis angle α ˜ M in all experiments. Therefore, similar tracking results can be found in Fig. 6 (top). The tracking errors of the motor angular velocity ωM shown in Fig. 6 (bottom) illustrate that both BC and BCSMC can guarantee excellent tracking performance. The comparison between BC and BCSMC indicates that the tracking performance can be improved by extending the BC with a switching control action which can significantly counteract the system uncertainty, e.g., a compensation error. The validation results with a constant load torque τL = 20 Nm can be found in Fig. 7. Similar results can be obtained as in the case
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·10−4
1
It consists of a stabilising control input designed according to the classical backstepping control procedure. Moreover, a robust control term using a second-order sliding mode technique is introduced to counteract the influence of system parameter uncertainty as well as unknown disturbances. To estimate the unmeasurable system states and to achieve a disturbance compensation, a sliding mode observer is introduced in combination with the proposed strategy. From an implementation point of view, two measures are considered to reduce the chattering effect: One is the use of the tanh function instead of the sign function, the other is an online disturbance compensation using observer techniques, which significantly reduces the switching height.
eα˜ M
0.5 0 −0.5
BC BCSMC
−1
−1.5
0
10
20
30 40 t in s
50
60
70
eωM in rad/s
1 0.5 0 −0.5
BC BCSMC
−1
−1.5
The simulation study in comparison with a classical backstepping control (BC) indicates that the proposed control approach guarantees a superior tracking performance. Finally, the experimental implementation highlights the applicability of the robust controller.
0
10
20
30 40 t in s
50
60
70
Fig. 6. Tracking errors of the controlled variables with a vanishing load torque τL = 0 Nm. −4
·10
1
eα˜ M
0.5 0 −0.5
BC BCSMC
−1
−1.5
0
10
20
30 40 t in s
50
60
70
30 40 t in s
50
60
70
eωM in rad/s
1 0.5 0 −0.5
BC BCSMC
−1
−1.5
0
10
20
Fig. 7. Tracking errors of the controlled variables with a constant load torque τL = 20 Nm. τL = 0 Nm. It is justified to point out here that BCSMC still guarantees a better tracking performance than BC. The numerical evaluation using Root-Mean-Square (RMS) errors is stated in Table 1. Table 1. Root-Mean-Square (RMS) errors of the motor angular velocity ωM . eωM ,RM S τL = 0 Nm τL = 20 Nm
BC 0.2481 rad/s 0.3159 rad/s
BCSMC 0.1253 rad/s 0.1694 rad/s
6. CONCLUSIONS In this paper, a backstepping second-order sliding mode approach (BCSMC) for the tracking control of the motor angular velocity of a hydrostatic transmission is proposed. 896
REFERENCES Aschemann, H. and Schindele, D. (2008). Sliding-Mode Control of a High-Speed Linear Axis Driven by Pneumatic Muscle Actuators. IEEE Transactions on Industrial Electronics, 55, 3588–3864. Aschemann, H. and Sun, H. (2013). Decentralised Flatness-Based Control of a Hydrostatic Drive Train Subject to Actuator Uncertainty and Disturbances. In IEEE Intl. Conference on Methods and Models in Automation and Robotics MMAR. Miedzyzdroje, Poland. Bartolini, G., Fridman, L., Pisano, A., and Usai, E. (2008). Modern Sliding Mode Control Theory: New Perspectives and Applications. Springer. ¨ uner, U. (2010). A David Young, K., Utkin, V., and Ozg¨ Control Engineers Guide to Sliding Mode Control. IEEE Transactions on Control Systems Technology, 7, 328– 342. Jelali, M. and Kroll, A. (2003). Hydraulic Servo-Systems: Modelling, Identification and Control. Springer-Verlag, London, UK. Kokotovic, P. (1992). The joy of feedback: nonlinear and adaptive. IEEE Control Systems, 12, 7–17. Levant, A. (2007). Principles of 2-Sliding Mode Design. Automatica, 43, 576–586. Levant, A. (2010). Higher-Order Sliding Modes, Differentiation and Output-Feedback Control. International Journal of Control, 76, 924–941. Panchade, V., Chile, R., and Patre, B. (2013). A Survey on Sliding Mode Control Strategies for Induction Motors. Annual Reviews in Control, 37, 289–307. Perruquetti, W. and Barbot, J. (2002). Sliding Mode Control in Engineering. Marcel Dekker, Inc. Su, C. and Leung, T. (1993). A Sliding Mode Controller with Bound Estimation for Robot Manipulators. IEEE Transactions on Robotics and Automation, 9(2), 208– 214. Utkin, V. (2014). Mechanical Energy-Based Lyapunov Function Design for Twisting and Super-Twisting Sliding Mode Control. Journal of Mathematical Control and Information.
ScienceDirect IFAC-PapersOnLine 49-18 (2016) 879–884
A Backstepping Sliding Mode Control for a A A Backstepping Backstepping Sliding Sliding Mode Mode Control Control for for a a Hydrostatic Transmission with Unknown Hydrostatic Transmission with Unknown Hydrostatic Transmission with Unknown Disturbances Disturbances Disturbances ∗ ∗ Hao Sun ∗ Harald Aschemann ∗ Hao Sun Harald Aschemann ∗ ∗ Hao Hao Sun Sun ∗ Harald Harald Aschemann Aschemann ∗ ∗ ∗ Chair of Mechatronics, University of Rostock, 18059, Rostock, ∗ Chair of Mechatronics, University of Rostock, 18059, Rostock, ∗ Chair of of (e-mail: Mechatronics, University of Rostock, Rostock, 18059, 18059, Rostock, Rostock, Germany hao.sun, [email protected]). Chair Mechatronics, University of Germany (e-mail: hao.sun, [email protected]). Germany Germany (e-mail: (e-mail: hao.sun, hao.sun, [email protected]). [email protected]).
Abstract: A A robust robust backstepping backstepping control control via via second-order second-order sliding sliding mode mode for for the the speed speed tracking tracking Abstract: Abstract: A robust backstepping control via second-order sliding mode for the speed tracking control of a hydrostatic transmission is presented in this paper. The proposed approach consists Abstract: A robust backstepping control via second-order sliding mode for the speed tracking control of a hydrostatic transmission is presented in this paper. The proposed approach consists control of a hydrostatic transmission is presented in this paper. The proposed approach consists of a stabilising control action designed according to backstepping control techniques and a robust control of a hydrostatic transmission isaccording presentedtoinbackstepping this paper. The proposed approach consists of a control action designed control and of a stabilising stabilising control action designed according to(SOSMC) backstepping control techniques techniques and aaa robust robust control term using a second-order sliding mode approach, which is introduced to of a stabilising control action designed according to backstepping control techniques and robust control term using a second-order sliding mode (SOSMC) approach, which is introduced to control term using a second-order sliding mode (SOSMC) approach, which is introduced to counteract the impact of uncertain system parameters as well as unknown disturbances. For the control term using a second-order sliding mode (SOSMC) approach, which is introduced to counteract the impact of uncertain system parameters as well as unknown disturbances. For the counteract the uncertain system well disturbances. For control implementation implementation on a test test bench bench of aaparameters hydrostaticas transmission, to estimate estimate unmeasurable counteract the impact impact of ofon uncertain system parameters astransmission, well as as unknown unknown disturbances. For the the control a of hydrostatic to unmeasurable control on bench to system implementation states and and two two lumped lumped disturbances present in in transmission, the system, system, the the proposedunmeasurable controller is is control implementation on a a test testdisturbances bench of of aa hydrostatic hydrostatic transmission, to estimate estimate unmeasurable system states present the proposed controller system states and two lumped lumped disturbances present in in the the system, the proposed proposed controllerthe is extended with a sliding mode observer. Both simulations and experimental results highlight system states and two disturbances present system, the controller is extended with a sliding mode observer. Both simulations and experimental results highlight the extended with sliding mode observer. applicability ofa the proposed method. extended withof athe sliding mode method. observer. Both Both simulations simulations and and experimental experimental results results highlight highlight the the applicability proposed applicability applicability of of the the proposed proposed method. method. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear Nonlinear robust robust control, control, second-order second-order sliding sliding mode, mode, hydrostatic hydrostatic transmission, transmission, Keywords: Keywords: Nonlinear robust control, control, second-order second-order sliding sliding mode, mode, hydrostatic hydrostatic transmission, transmission, sliding mode observer. Keywords: Nonlinear robust sliding mode observer. sliding mode observer. sliding mode observer. 1. INTRODUCTION INTRODUCTION input directly directly acts acts on on the the first first time time derivative derivative of of the the sliding sliding 1. input 1. input directly acts on the first time derivative of the sliding manifold, HOSMC is a generalisation of basic SMC and the 1. INTRODUCTION INTRODUCTION input directly acts on the first time derivative of the sliding manifold, HOSMC HOSMC is is aa generalisation generalisation of of basic basic SMC SMC and and the manifold, control input acts on a higher-order time derivative of the manifold, HOSMC is a generalisation of basic SMC and Robustness has become an important issue in control control input input acts acts on on aa higher-order higher-order time time derivative derivative of of the Robustness has has become become an an important important issue issue in in control control control sliding manifold, manifold, see Perruquetti andtime Barbot (2002).of the control input actssee on Perruquetti a higher-order derivative the Robustness systems since sincehas thebecome emergence control theory. sliding and Barbot (2002). Robustness an of important issue To in achieve control sliding systems the emergence of control theory. To achieve manifold, see Perruquetti Barbot (2002). sliding manifold, see Perruquetti and and Barbotcontrol (2002). systems since the emergence of control theory. To achieve a high control performance, not only nonlinearities and a systems since the emergence of control theory. To achieve Backstepping is a Lyapunov-based recursive design high control control performance, performance, not not only only nonlinearities nonlinearities and and aa Backstepping is a Lyapunov-based recursive control design aa high time-varying behaviour mustnot be taken taken into account account at at thea Backstepping is Lyapunov-based recursive control a high controlbehaviour performance, only nonlinearities and method for for the the class of ”strict ”strict feedback” which Backstepping is aaclass Lyapunov-based recursivesystems control design design time-varying must be into the method of feedback” systems which time-varying behaviour must be taken into account at the model-based control design but also model uncertainty and for the class of ”strict feedback” systems which time-varying behaviour must bealso taken intouncertainty account at and the method first appeared in the beginning of the 1990s, see Kokomethod for the class of ”strict feedback” systems which model-based control design but model first appeared in the beginning of the 1990s, see Kokomodel-based control unknown disturbances. disturbances. appeared in the beginning of the 1990s, see Kokomodel-based control design design but but also also model model uncertainty uncertainty and and first tovic (1992). An important advantage of the backstepping first appeared in the beginning of the 1990s, see Kokounknown tovic (1992). (1992). An An important important advantage advantage of of the the backstepping unknown disturbances. unknown disturbances. design(1992). approach is that it it provides provides a systematic systematic procedure tovic An is important advantage of the backstepping backstepping In the the past past decades, aa remarkable remarkable progress progress has has been been tovic design approach that a procedure In decades, design approach is that it provides a systematic procedure to design stabilising controllers, following a step-by-step design approach is that it provides a systematic procedure In the decades, aa remarkable progress has been achieved in the the development of robust control methods to design design stabilising stabilising controllers, controllers, following following aa step-by-step step-by-step In the past past decades, remarkable progress been to achieved in development of robust robust control has methods algorithm. Therefore, controllers, the construction construction of feedback feedback control to design stabilising following a step-by-step achieved in the development of control methods for nonlinear systems. Among all the existing approaches, algorithm. Therefore, the of control achieved in the development of robust control methods for nonlinear nonlinear systems. systems. Among Among all all the the existing existing approaches, approaches, algorithm. Therefore, the construction of feedback control laws and Lyapunov functions is systematic. Backstepping algorithm. Therefore, the construction of feedback control for sliding mode control (SMC) has gained special attention laws and and Lyapunov Lyapunov functions functions is is systematic. systematic. Backstepping Backstepping for nonlinear Amonghas all gained the existing approaches, sliding mode systems. control (SMC) (SMC) special attention laws design hasLyapunov the flexibility flexibility to avoid avoid cancellations of useful useful laws and functions is systematic. Backstepping sliding mode has special attention from practical practical control(SMC) engineers, see David David Young et al. al. design has the to cancellations of sliding mode control control has gained gained special attention from control engineers, see Young et design has the flexibility to avoid cancellations of useful nonlinearities and to achieve stabilisation and tracking. design has the flexibility to avoid cancellations of useful from practical control see Young et (2010). SMC belongs belongs toengineers, variable structure structure control methnonlinearities and and to to achieve achieve stabilisation stabilisation and and tracking. tracking. from practical controlto engineers, see David Davidcontrol Young methet al. al. nonlinearities (2010). SMC variable Moreover, this thisand control method can be be easily nonlinearities to achieve stabilisation and extended tracking. (2010). SMC belongs to variable structure control methods. The control principle is that the SMC brings the states Moreover, control method can easily extended (2010). SMC belongs to variable structure control methods. The The control control principle principle is is that that the the SMC SMC brings brings the the states states Moreover, this control method can be easily extended with adaptive control and sliding mode control without Moreover, this control method can be easily extended ods. towards a sliding surface and forces the states to remain with adaptive adaptive control control and and sliding sliding mode mode control control without without ods. The acontrol principle is that the SMC brings to theremain states with towards sliding surface and forces the states the restriction on matched condition of the uncertainties. withrestriction adaptive on control and condition sliding mode control without towards aa sliding surface forces the to on the the surface surface afterwards. The most most significant property of the matched of the the uncertainties. towards sliding surface and and forcessignificant the states statesproperty to remain remain on afterwards. The of the restriction matched condition of uncertainties. Nowadays, the on application of backstepping control and its its the restriction on matched of condition of thecontrol uncertainties. on the surface afterwards. The most significant property of SMC is its robustness. When the system reaches the sliding Nowadays, the application backstepping and on the surface afterwards. The most significant property of SMC is is its its robustness. robustness. When When the the system system reaches reaches the the sliding sliding Nowadays, the application of backstepping control and its extensions to robust control are still frontier subjects Nowadays, the application of backstepping control and its SMC surface, it robustness. is insensitive insensitive to the parameter uncertainties and extensions to robust control are still frontier subjects in in SMC is its When system reaches the sliding surface, it is to parameter uncertainties and extensions to control theory. theory. extensions to robust robust control control are are still still frontier frontier subjects subjects in in surface, it is insensitive to parameter uncertainties and external disturbances if the matching condition is satisfied. control surface, it is insensitive to parameter uncertainties and external disturbances disturbances if if the the matching matching condition condition is is satisfied. satisfied. control theory. control theory. external In comparison with other robust control approaches like external disturbances if the matching condition is satisfied. In this this paper, paper, aa decentralised decentralised control control structure structure is is proposed: proposed: In comparison comparison with with other other robust robust control control approaches approaches like like In In internal model control control (IMC) or aa LMI-based design, SMC this paper, aa decentralised control structure is proposed: In comparison with other robust control approaches like In A backstepping second-order sliding mode control is ememIn this paper, decentralised control structure is proposed: internal model (IMC) or LMI-based design, SMC A backstepping backstepping second-order second-order sliding sliding mode mode control control is is internal model control (IMC) or aa LMI-based design, SMC has a simple control structure and can provide fast dyA eminternal model control (IMC) or LMI-based design, SMC ployed for the tracking control of the motor angular velocbackstepping second-order mode angular control is emhas aa simple simple control control structure structure and and can can provide provide fast fast dydy- A ployed for the the tracking tracking controlsliding of the the motor motor velochas namic responses. Plenty of applications applications canprovide be found found in dythe ployed control velochas a responses. simple control structure and can fast ity of of aafor hydrostatic transmission, whereas flatness-based ployed for the tracking control of of the motoraa angular angular velocnamic Plenty of can be in the ity hydrostatic transmission, whereas flatness-based namic responses. Plenty can be in literatures, see Aschemann Aschemann and Schindele Schindele (2008); Bartolini of whereas flatness-based namic responses. Plenty of of applications applications can be found found in the the ity approach is used used for fortransmission, the tracking tracking control control ofaathe the normalised ity of aa hydrostatic hydrostatic transmission, whereasof flatness-based literatures, see and (2008); Bartolini approach is the normalised literatures, see Aschemann and Schindele (2008); Bartolini et al. (2008); Panchade et al. (2013); Su and Leung (1993). approach is used for the tracking control of the normalised literatures, see Aschemann and Schindele (2008); Bartolini bent axis angle of the hydraulic motor. Furthermore, the approach is used for the tracking control of the normalised et al. (2008); Panchade et al. (2013); Su and Leung (1993). bent axis axis angle angle of of the the hydraulic hydraulic motor. motor. Furthermore, Furthermore, the the et al. (2008); Panchade et al. (2013); Su and Leung (1993). The main drawback of SMC is the so-called chattering bent et al. (2008); Panchade et al. (2013); Su and Leung (1993). unmeasurable system states and unknown disturbances axis anglesystem of the hydraulic Furthermore, the The main main drawback drawback of of SMC SMC is is the the so-called so-called chattering chattering bent unmeasurable states and andmotor. unknown disturbances The effectmain whichdrawback is considered considered as the theismain main obstacle forchattering practical unmeasurable system unknown disturbances The of SMC the obstacle so-calledfor are robustly robustly estimated estimated using aaand sliding mode observer. observer. The unmeasurable system states states unknown disturbances effect which is as practical are using sliding mode The effect which as main for implementation. Chattering usually occurs when a nonrobustly estimated using aa sliding mode The effect which is is considered considered as the the main obstacle obstacle for practical practical combined control structure been validated using both are robustly estimated usinghas sliding mode observer. observer. The implementation. Chattering usually occurs when when non- are combined control structure has been validated validated using both both implementation. Chattering usually occurs aaa nonnegligible, fast dynamics is not considered in the system combined control structure has been using implementation. Chattering usually occurs when nonsimulation and experimental results. combined control structure has been validated using both negligible, fast dynamics is not considered in the system simulation and and experimental experimental results. results. negligible, dynamics not in system model. In In fast digital control,is finite sampling time negligible, fast dynamics isfinite not considered considered in the theintervals system simulation simulation and experimental results. model. digital control, sampling time intervals The rest of the paper is organised as follows: follows: In In Section Section 2, 2, model. In digital control, finite sampling time intervals could cause a switching delay and lead to ”discretisation model. In digital control, finite sampling time intervals The rest of the paper is organised as could cause a switching delay and lead to ”discretisation The rest of the paper is organised as follows: In Section 2, the hydrostatic transmission is briefly introduced. For the could cause aaToswitching delay and lead to ”discretisation The rest of the paper is organised as follows: In Section 2, chattering”. avoid this phenomenon, high-order SMC could cause switching delay and lead to ”discretisation the hydrostatic hydrostatic transmission transmission is is briefly briefly introduced. introduced. For For the the chattering”. To To avoid avoid this this phenomenon, phenomenon, high-order high-order SMC SMC the control design, transmission the nonlinear nonlinearismodelling modelling of the the system system is chattering”. hydrostatic briefly introduced. For the (HOSMC) could could be employed, employed, see Levant Levant (2010); Utkin Utkin chattering”. To avoid this phenomenon, high-order SMC the control design, the of is (HOSMC) be see (2010); control design, the nonlinear modelling of the system is addressed. Based on the state-space model, the tracking (HOSMC) could be employed, see Levant (2010); Utkin control design, the nonlinear modelling of the system is (2014). Different from the basic SMC, where the control (HOSMC) could from be employed, Levant (2010); Utkin addressed. Based on the state-space model, the tracking (2014). Different Different the basic basicsee SMC, where the control control addressed. Based on the state-space model, the tracking (2014). (2014). Different from from the the basic SMC, SMC, where where the the control addressed. Based on the state-space model, the tracking Copyright 2016 IFAC 891 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 891 Copyright 2016 IFAC 891 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 891Control. 10.1016/j.ifacol.2016.10.277
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control design is discussed in details in Section 3. The state and disturbance observer is given in Section 4. In Section 5, the simulation and experimental results are presented. Finally, conclusions are given in Section 6. 2. CONTROL-ORIENTED MODELLING OF A HYDROSTATIC TRANSMISSION A hydrostatic transmission is a hydraulic circuit which usually consists of a variable displacement hydraulic pump and a variable displacement hydraulic motor. It generates and transmits the power by means of a pressurised fluid, and provides a variety of advantages in comparison to pure mechanical transmissions, see Jelali and Kroll (2003). Nowadays, it is commonly considered as a characteristic component of drive chains for the applications in working machines and off-road vehicles. However, hydrostatic transmissions are typical nonlinear systems and characterised by uncertain system parameters and unknown disturbances which limit their applications in highperformance control system. A test bench of a hydrostatic transmission system depicted in Fig. 1, can be found at the Chair of Mechatronics, University of Rostock. From the hydraulic scheme, it can be seen that the test bench consists of a variable displacement
Hydraulic motor
Load motor
Drive motor Hydraulic pump
Volume flow qA Motor angular velocity M and torqueM
Pressure pA
Pump angular velocity
P
M
M
Load motor
Drive motor
Volume flow qB
by two subsystems: one is for the control of the normalised bent axis angle α ˜ M which can be stated as 1 1 α ˜M + uM , (1) α ˜˙ M = − TuM TuM with the time constant TuM and the control input uM . The second subsystem contains the dynamics of the normalised swashplate angle α ˜ P , the pressure difference ∆p and the motor angular velocity ωM given by 1 1 α ˜P + uP , α ˜˙ P = − TuP TuP 2VP ωP tan(αP,max · α ˜P ) CH (2) 2VM ωM sin(αM,max · α qU ˜M ) − − , CH CH dV VM τU ∆p sin(αM,max · α ˜M ) − , ω˙ M = − ωM + JV JV JV and serves for the control of the motor angular velocity ωM . Here, TuP represents the time constant. VP,M are volume displacements resulting from the geometric structure of the pump and motor. αP,M,max denote the maximum tile angles of the pump and motor. CH represents the hydraulic capacitance. The parameters JV and dV denote the moments inertial and damping coefficient on the motor side. The angular velocity of the pump is represented by ωP which is considered here as a constant parameter. Moreover, the unknown disturbances are modelled as lumped parameters: the leakage volume flow qU in the main hydraulic circuit and the disturbance torque τU acting on the hydraulic motor. It is worth to point out that the introduced lumped disturbance parameters contain not only the unknown disturbances existing in the system dynamics, but also the model uncertainty due to uncertain physical parameters, e.g., the bulk modulus of the oil which depends on the working pressure and temperature. Furthermore, the control input is denoted as uP . In Aschemann and Sun (2013), it is also proved that each subsystem is differentially flat w.r.t. the corresponding flat output – the normalised bent axis angle α ˜ M and the motor angular velocity ωM . ∆p˙ =
3. CONTROL DESIGN FOR THE HYDROSTATIC TRANSMISSION
Pressure pB
Fig. 1. Drive train with a closed-circuit hydrostatic transmission. axial piston pump and a variable displacement bent axis motor connected in a closed circuit. Two electric motors are employed to represent the prime mover on the pump side and the load unit on the motor side, respectively. Pressure sensors are mounted on the top of the pump measuring pressures on both high and low pressure side, respectively. Moreover, a volume flow sensor is available on each side of the drive train. This mechatronic system can be split into a hydraulic subsystem and a mechanical subsystem, which are coupled by the torque τM generated by the hydraulic motor. The details of the control-oriented model for the hydrostatic transmission system can be found in Aschemann and Sun (2013). The complete dynamic system is characterised 892
3.1 Decentralised control structure The proposed decentralised control structure is depicted in Fig. 2. The estimated unmeasurable system states – the ˆ˜ P,M – and the estimated unknown normalised tilt angles α ˆ provided disturbances – qˆU and τˆU – form the vector d by the sliding mode observer. In this control structure, the motor angular velocity ωM is directly regulated by the normalised swashplate angle of the pump α ˜ P , and can be further influenced by varying of the normalised bent axis angle of the motor α ˜ M . In the following, the control design for each subsystem will be discussed. 3.2 Flatness-based control design for α ˜M A flatness-based control (FBC) approach is employed for the tracking control of the normalised bent axis angle α ˜M .
IFAC NOLCOS 2016 August 23-25, 2016. Monterey, California, USA Hao Sun et al. / IFAC-PapersOnLine 49-18 (2016) 879–884
α ˜M d α ˜˙ M d
Flatnessbased control
uM ˆ α ˜M
ˆ Sliding mode ωM d d observer ω˙ M d ω ¨ M d ... ω Md Backstepping sliding mode control
∆p ωM
uP
Hydrostatic transmission
[qU τU ]T
Fig. 2. Decentralised control scheme. The FBC contains the inverse dynamics of the corresponding first-order differential equation (1) and a stabilising control input υM , i.e., α ˜ M + υM TuM . (3) uM = kM The stabilising control law represents a combination of a feedforward and feedback control as follows t ˙ ˜ M d + kα0 eα˜ M + kαI · eα˜ M dτ , (4) υM = α 0
˜ M d −α ˜ M as well as positive with the tracking error eα˜ M = α coefficients kα0 > 0 and kαI > 0, which can be determined according to the desired Hurwitz polynomial.
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Step 2: In the second step, an error variable e2 is considered according to (10) e2 = ω˙ M − α1 = ω˙ M − ω˙ M d + c1 e1 . Given e˙ 1 = ω˙ M − ω˙ M d , this results in (11) e˙ 1 = e2 − c1 e1 . The corresponding error dynamics e˙ 2 can be expressed as ¨M − ω ¨ M d + c1 e˙ 1 . (12) e˙ 2 = ω ¨ M . Introducing Here, the virtual input is chosen as α2 ≈ ω the Lyapunov function 1 V2 = V1 + e22 . (13) 2 Its time derivative results in V˙ 2 = e1 e˙ 1 + e2 e˙ 2 = e1 (e2 − c1 e1 ) + e2 (¨ ωM − ω ¨ M d + c1 e˙ 1 ) 2 = −c1 e1 + e2 (¨ (14) ωM − ω ¨ M d + c1 e˙ 1 + e1 ) , !
=−c2 e2
with the positive coefficient c2 . Therefore, α2 is chosen as ¨ M d − c1 e˙ 1 − e1 − c2 e2 , (15) α2 = ω which leads to V˙ 2 = −c1 e21 − c2 e22 < 0 , when e1 = 0 , e2 = 0 . (16) Step 3: The third error variable e3 is defined as ¨ M − α2 = ω ¨ −ω ¨ M d + c1 e˙ 1 +e1 + c2 e2 . (17) e3 = ω M e˙ 2
This leads to
3.3 Backstepping sliding mode control design for ωM If a nonlinear system is differential flat, the original system dynamics can be differentially parametrised using the flat output. Therefore, the subsystem of the motor angular velocity represented in (2) can be rewritten as ω˙ M = ω˙ M , ω ¨M = ω ¨M , (5) ... ω M = f (ω M , τ , α ˜ M ) + g(˜ α M ) uP , ¨ M ]T and diswith the state vector ω M = [ωM ω˙ M ω T turbance vector τ = [qU τU ] . f and g are continuous functions. Moreover, it is assumed that the dynamics of the disturbances are only slow time-varying, i.e., τ˙ = [q˙U τ˙U ]T = 0. Control design Step 1: The backstepping control design starts with the definition of a quadratic Lyapunov function 1 (6) V1 = e21 , 2 with the tracking error e1 = ωM − ωM d . The first time derivative of V1 results in (7) V˙ 1 = e1 e˙ 1 = e1 (ω˙ M − ω˙ M d ) . !
=−c1 e1
Here, c1 > 0 denotes a constant parameter. The virtual input α1 ≈ ω˙ M is chosen as α1 = ω˙ M d − c1 e1 , (8) which leads to (9) V˙ 1 = −c1 e21 < 0 , when e1 = 0 . 893
e˙ 2 = e3 − e1 − c2 e2 . The error dynamics of e3 can be stated as ... ... e˙ 3 = ω M − ω M d + c1 e¨1 + e˙ 1 + c2 e˙ 2 . Introducing the Lyapunov function 1 V3 = V2 + e23 , 2 it’s first time differentive becomes V˙ 3 = e1 (e2 − c1 e1 ) + e2 (e3 − e1 − c2 e2 ) ... ... + e3 ( ω M − ω M d + c1 e¨1 + e˙ 1 + c2 e˙ 2 ) = −c1 e21 − c2 e22 ... ... + e3 ( ω M − ω M d + c1 e¨1 + e˙ 1 + c2 e˙ 2 + e2 )
(18)
(19) (20) (21)
ψ(e1 ,e2 )
... ... = −c1 e21 − c2 e22 + e3 ( ω M − ω M d + ψ(e1 , e2 )) = −c1 e21 − c2 e22 ... + e3 (f (ω M , τ , α ˜ M ) + g(˜ αM ) uP − ω M d + ψ(e1 , e2 )). The control input uP is designed in such a way that it consists of a feedback control uf b compensating all nonlinearities and disturbances, a stabilising control input υP and a sliding mode control usm , i.e., ... (υP − usm − f (ω M , τ , α ˜ M ) − ψ(e1 , e2 ) + ω M d ) . uP = g(˜ αM ) (22) Introducing a bounded lumped parameter |φ| < φmax , which denotes the residual of an imperfect disturbance compensation, and substituting (22) into (21) results in (23) V˙ 3 = −c1 e21 − c2 e22 + e3 (υP − usm + φ) , with the stabilising control input designed as υP = −c3 e3 , c3 > 0 ,
(24)
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Eq. (23) becomes V˙ 3 = −c1 e21 − c2 e22 − c3 e23 + e3 (φ − usm ) .
(25) The sliding mode control usm is designed by means of a second-order sliding mode approach. For this purpose, the following expression of e3 is considered ¨−x ¨ +c e˙ + e1 + c2 (x˙ − x˙ d +c1 e1 ) e3 = x d 1 1 e¨1
e˙ 1
= e¨1 + (c1 + c2 ) e˙ 1 + (1 + c1 c2 ) e1 . (26) Defining a sliding manifold with a constant parameter α0 > 0 according to (27) s = e˙ 1 + α0 e1 , its first-time derivative becomes (28) s˙ = e¨1 + α0 e˙ 1 . Therefore, the error variable e3 can be represented as (29) e3 = s˙ + β s = e¨1 + (α0 + β) e˙ 1 + α0 β e1 , with (30) α 0 + β = c1 + c 2 , (31) α0 β = 1 + c 1 c 2 . Now, a second-order sliding mode control law is introduced that represents a modification of the one in Levant (2007)
(s˙ + β s) + |s|1/2 sign(s˙ + β s) , (32) |s˙ + β s|+|s|1/2 +η where k, η > 0 are constant parameters. The time derivative of the Lyapunov function V3 can be rewritten as (33) V˙ 3 = −c1 e2 − c2 e2 − c3 e2 usm = k
1
2
3
(s˙ + β s) + |s|1/2 sign(s˙ + β s) + (s˙ + β s) (φ − k ) |s˙ + β s|+|s|1/2 +η = −c1 e21 − c2 e22 − c3 e23
|s˙ + β s|+|s|1/2 φ ) sign(s˙ + β s) − k |s˙ + β s|+|s|1/2 +η To achieve asymptotic stability, V˙ 3 < 0 must be satisfied. This leads to |s˙ + β s|+|s|1/2 φ . (34) sign(s˙ + β s) ≤ k |s˙ + β s|+|s|1/2 +η It is straightforward to derive the following bound |s˙ + β s|+|s|1/2 < 1. (35) |s˙ + β s|+|s|1/2 +η Thereby, the condition for the sliding mode gain becomes (36) k ≥ φ sign(s˙ + β s) ≥ φmax . As a result, the conditions (30), (31) and (36) lead to V˙ 3 < 0, which guarantees the asymptotic stability of the system, i.e., t → 0 ⇒ ei → 0, i ∈ {1, 2, 3} + k |s˙ + β s| (
4. SLIDING MODE OBSERVER The observer design is based on the extended system model with integrators as disturbance model. Following the idea of extended linearisation, the extended system is written in a quasi-linear form with a state-dependent system matrix A(˜ αM , ωM ), i.e., αM , ωM ) xEL + B u , (37) x˙ EL = AE (˜ y m = C xEL . (38) 894
. Here, xEL = [xT1 ..y Tm ]T represents the state vector with the unmeasurable state vector x1 = [α ˜P α ˜ M q U τU ] T and the measurable state vector y m = [∆p ωM ]T . The control input vector is u = [uP uM ]T . The input and output matrices are denoted as B and C, respectively. The observability can be easily confirmed by checking the observability matrix of the quasi-linear system. For the observer design, the state-space representation is written as A11 A12 x1 B1 x˙ 1 = + u. (39) A21 A22 ym B2 y˙ m The observer the form has ˙x ˆ1 A11 A12 x B1 ˆ1 = + ˆm A21 A22 B2 y y ˆ˙ m G1 L − (ˆ ym − ym ) + υ, (40) G2 −I ˆ represents the estimated values. G1,2 denote the where (·) Luenberger-type gain matrices and L denotes a feedback gain matrix. The discontinuous switching part is defined by the vector υ according to M1 sign(∆ˆ p − ∆p) υ= , (41) M2 sign(ˆ ωM − ω M )
where M1,2 denotes the positive constant gains. Considˆ 1 − x1 and eym = y ˆ m − ym ering the definition e1 = x ¯1 = e1 + Leym , the and introducing a new error variable e resulting estimation error dynamics becomes ¯ 11 A ¯ 12 ¯1 A e 0 ¯e˙ 1 = + υ, (42) ¯ 22 −I e ym e˙ ym A21 A where the submatrices are given by ¯ 11 = A11 + L A21 , A ¯ 11 L − G1 + L (A22 − G2 ) , ¯ 12 = A12 − A A ¯ A22 = A22 − G2 − A21 L .
¯ 12 = 0 can be achieved by proper choice of the gain A matrix G1 . In the case of υ = 0, asymptotic stability of the error dynamic system (42) can be obtained by choosing the gain matrices L and G2 according to ¯∗ , (43) A11 + L A21 = A 11 ∗ ¯ (44) A22 − G2 − A21 L = A , 22
¯ ∗ and A ¯ ∗ denote asymptotically stable matrices where A 11 22 with the following characteristic polynomials ! ¯∗ ) = det(s I − A (s + sB1 )(s + sB2 )(s + sB3 )(s + sB4 ) , 11 ∗
!
¯ ) = (s + sB5 )(s + sB6 ) . det(s I − A 22
5. SIMULATION AND EXPERIMENTAL RESULTS The proposed control structure is validated using both simulation and experimental results. By exploiting the flatness property, the synchronised desired trajectories of the controlled variables are designed by taking the saturation of tilt angles into account. Fig. 1 illustrates the desired motion of the hydrostatic transmission. All the control parameters are determined based on trial and error and same values will be used in both simulation and experimental studies.
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100
0.6
80
1
60
0
40 20
0 14 28 42 56 70 t in s
BC/BCSMC
eα˜ m
0.7
·10−3
2
−1
0 14 28 42 56 70 t in s
Fig. 3. Desired trajectories of the controlled variables. Simulation results The measurement noise of the pressure sensors as well as the quantisation effects in the encoder signal of the motor angular velocity are introduced in the simulation model in order to obtain realistic results. Moreover, simulated disturbances replace the lumped parameters in (2): The leakage volume flow qU is assumed to be proportional to the pressure difference, i.e., qU = kl ∆p , with a positive leakage coefficient kl . The unknown disturbance torque τU is modelled as the sum of a load torque τL which is a 10% variation of the mass moment inertia and a friction torque τf , i.e., ω Md . (45) τU = 0.1 JV ω˙ M d + 7 tanh 0.1
eωM in rad/s
ωM d in rad/s
α ˜M d
0.8
883
0.8 0.6 0.4 0.2 0 −0.2 −0.4
0
10
20
30 40 t in s
50
60
70
30 40 t in s
50
60
70
BC BCSMC
0
10
20
Fig. 5. Tracking errors of the controlled variables. in comparison with the estimates of a nonlinear reducedorder observer in Aschemann and Sun (2013).
In addition, a parameter uncertainty of the time constant TuP is also introduced where a value of 0.17 is considered in the control and observer implementation, whereas the nominal value 0.13 is applied to the simulation model.
Fig. 5 shows the tracking errors of the controlled variables. In the simulation investigation, a comparison between the backstepping control (BC) and backstepping sliding mode control (BCSMC) is carried out. It can be seen that flatness-based control can guarantee a precise tracking of the desired normalised bent axis angle α ˜ M d . By introducing a switching term to counteract the influence of the model uncertainty, i.e., compensation error and parameter uncertainty in this case, BCSMC leads to a much better tracking performance compared with classical BC.
The estimation results of the sliding mode observer can be found in Fig. 4. It can be concluded that the robust
Experimental results
τL
Simulated Estimated
0.8 α ˜M
α ˜P
0.8
τf
0.6 0.4
The proposed backstepping sliding mode control is implemented on the test bench depicted in Fig. 1 to achieve a robust tracking performance for the motor angular velocity ωM . This test bench is operated by a host-target PC system using the software package Labview. The control approach is validated using the desired trajectories depicted in Fig. 3 under two conditions: one is with a vanishing load torque τL = 0 Nm and the other is with a constant load torque τL = 20 Nm. Moreover, in order to reduce the chattering effect, the sign function in (32) is replaced in the implementation by the tanh function.
0.7 0.6
0 10 20 30 40 50 60 70 ·10−6 t in s Simulated Estimated
5 4 3
0 10 20 30 40 50 60 70 t in s 7.4 τU in Nm
3
qU in m /s
0.2
Simulated Estimated
Simulated Estimated
7.2 7 6.8
2 0 10 20 30 40 50 60 70 t in s
0 10 20 30 40 50 60 70 t in s
Fig. 4. Estimation results of the sliding mode observer. observer provides outstanding estimates of both the unmeasurable system states α ˜ P,M and the unknown disturbances – the leakage volume flow qU and the disturbance torque τU . It is worth pointing out here that the sliding mode observer provides a much better estimation of qU 895
Fig. 6 shows the tracking errors of the controlled variables with a vanishing load torque. A flatness-based control is employed for the tracking control of the normalised bent axis angle α ˜ M in all experiments. Therefore, similar tracking results can be found in Fig. 6 (top). The tracking errors of the motor angular velocity ωM shown in Fig. 6 (bottom) illustrate that both BC and BCSMC can guarantee excellent tracking performance. The comparison between BC and BCSMC indicates that the tracking performance can be improved by extending the BC with a switching control action which can significantly counteract the system uncertainty, e.g., a compensation error. The validation results with a constant load torque τL = 20 Nm can be found in Fig. 7. Similar results can be obtained as in the case
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·10−4
1
It consists of a stabilising control input designed according to the classical backstepping control procedure. Moreover, a robust control term using a second-order sliding mode technique is introduced to counteract the influence of system parameter uncertainty as well as unknown disturbances. To estimate the unmeasurable system states and to achieve a disturbance compensation, a sliding mode observer is introduced in combination with the proposed strategy. From an implementation point of view, two measures are considered to reduce the chattering effect: One is the use of the tanh function instead of the sign function, the other is an online disturbance compensation using observer techniques, which significantly reduces the switching height.
eα˜ M
0.5 0 −0.5
BC BCSMC
−1
−1.5
0
10
20
30 40 t in s
50
60
70
eωM in rad/s
1 0.5 0 −0.5
BC BCSMC
−1
−1.5
The simulation study in comparison with a classical backstepping control (BC) indicates that the proposed control approach guarantees a superior tracking performance. Finally, the experimental implementation highlights the applicability of the robust controller.
0
10
20
30 40 t in s
50
60
70
Fig. 6. Tracking errors of the controlled variables with a vanishing load torque τL = 0 Nm. −4
·10
1
eα˜ M
0.5 0 −0.5
BC BCSMC
−1
−1.5
0
10
20
30 40 t in s
50
60
70
30 40 t in s
50
60
70
eωM in rad/s
1 0.5 0 −0.5
BC BCSMC
−1
−1.5
0
10
20
Fig. 7. Tracking errors of the controlled variables with a constant load torque τL = 20 Nm. τL = 0 Nm. It is justified to point out here that BCSMC still guarantees a better tracking performance than BC. The numerical evaluation using Root-Mean-Square (RMS) errors is stated in Table 1. Table 1. Root-Mean-Square (RMS) errors of the motor angular velocity ωM . eωM ,RM S τL = 0 Nm τL = 20 Nm
BC 0.2481 rad/s 0.3159 rad/s
BCSMC 0.1253 rad/s 0.1694 rad/s
6. CONCLUSIONS In this paper, a backstepping second-order sliding mode approach (BCSMC) for the tracking control of the motor angular velocity of a hydrostatic transmission is proposed. 896
REFERENCES Aschemann, H. and Schindele, D. (2008). Sliding-Mode Control of a High-Speed Linear Axis Driven by Pneumatic Muscle Actuators. IEEE Transactions on Industrial Electronics, 55, 3588–3864. Aschemann, H. and Sun, H. (2013). Decentralised Flatness-Based Control of a Hydrostatic Drive Train Subject to Actuator Uncertainty and Disturbances. In IEEE Intl. Conference on Methods and Models in Automation and Robotics MMAR. Miedzyzdroje, Poland. Bartolini, G., Fridman, L., Pisano, A., and Usai, E. (2008). Modern Sliding Mode Control Theory: New Perspectives and Applications. Springer. ¨ uner, U. (2010). A David Young, K., Utkin, V., and Ozg¨ Control Engineers Guide to Sliding Mode Control. IEEE Transactions on Control Systems Technology, 7, 328– 342. Jelali, M. and Kroll, A. (2003). Hydraulic Servo-Systems: Modelling, Identification and Control. Springer-Verlag, London, UK. Kokotovic, P. (1992). The joy of feedback: nonlinear and adaptive. IEEE Control Systems, 12, 7–17. Levant, A. (2007). Principles of 2-Sliding Mode Design. Automatica, 43, 576–586. Levant, A. (2010). Higher-Order Sliding Modes, Differentiation and Output-Feedback Control. International Journal of Control, 76, 924–941. Panchade, V., Chile, R., and Patre, B. (2013). A Survey on Sliding Mode Control Strategies for Induction Motors. Annual Reviews in Control, 37, 289–307. Perruquetti, W. and Barbot, J. (2002). Sliding Mode Control in Engineering. Marcel Dekker, Inc. Su, C. and Leung, T. (1993). A Sliding Mode Controller with Bound Estimation for Robot Manipulators. IEEE Transactions on Robotics and Automation, 9(2), 208– 214. Utkin, V. (2014). Mechanical Energy-Based Lyapunov Function Design for Twisting and Super-Twisting Sliding Mode Control. Journal of Mathematical Control and Information.