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Adaptive nonsingular fast terminal sliding mode control
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International Journal of Systems Science, 2011 1 15 iFirst. Adaptive nonsingular fast terminal sliding mode control for electromechanical actuator. Hao Liab Lihua Douab and Zhong Suc, School of Automation Beijing Institute of Technology Beijing 100081 China bKey Laboratory of Complex System. Intelligent Control and Decision Beijing Institute of Technology Ministry of Education Beijing 100081 China. Key Laboratory of Modern Measurement and Control Technology Beijing Information Science and. Technology University Ministry of Education Beijing 100101 China. Received 1 July 2010 final version received 25 April 2011. An adaptive nonsingular fast terminal sliding mode control scheme consisting of an adaptive control term and a. Downloaded by Beijing Institute of Technology at 00 52 01 December 2011. robust control term for electromechanical actuator is proposed in this article The adaptive control term with an. improved composite adaptive law can estimate the uncertain parameters and compensate for the modelled. dynamical uncertainties While the robust control term which is based on a modified nonsingular fast terminal. sliding mode control method with fast terminal sliding mode TSM reaching law provides fast convergence of. errors and robustifies the design against unmodelled dynamics Furthermore the control method eliminates the. singular problems in conventional TSM control On the basis of the finite time stability theory and the. differential inequality principle it is proved that the resulting closed loop system is stable and the trajectory. tracking error converges to zero in finite time Finally the effectiveness of the proposed method is illustrated by. simulation and experimental study, Keywords nonsingular fast terminal sliding mode control composite adaptive law electromechanical actuator. finite time convergence, 1 Introduction method a stable sliding surface that ensures the desired. With the development of electronic technology the dynamics and a control effort that steers the system. electromechanical actuator EMA is now widely used states to reach and stay on the sliding surface Usually. in various applications For example it is utilised in the sliding surface is a linear hyperplane of system. aeroplanes missiles engine valves injectors brake states and only asymptotic stability is assured on the. systems and so on As an important part of the entire sliding manifold which implies that system errors. system EMA works like a converter or an amplifier cannot converge to zero in finite time. that transfers the electrical signal to the mechanical Terminal SMC TSMC is a variant scheme of. movement Consequently the control of EMA which SMC that can achieve finite time stability Bhat and. is implemented to achieve faster and more precise Bernstein 1998 2000 In Venkataraman and Gulati. regulations of position or velocity has attracted much 1992 the attractor Zak 1989 was adopted in the. attention during the past few decades However the sliding surface and TSMC for second order SISO. performance of EMA is influenced by uncertainties system was first presented By employing the nonlinear. such as parameter variations external disturbances sliding mode TSMC offers a finite time error conver. and unmodelled dynamics Therefore the control gence Inspired by this idea researchers developed. method should circumvent the uncertain problems to TSMC approaches with high order systems Yu and. achieve better static and dynamic performance Man 1996 MIMO linear systems Man and Yu 1997. Sliding mode control SMC which provides and uncertain dynamic systems Wu Yu and Man. invariance to uncertainties once the system dynamics 1998 Nevertheless TSMC cannot deliver the same. are controlled on the sliding mode is an efficient and convergence performance while the system states are. effective robust approach to deal with control prob far away from the equilibrium point To overcome this. lems of uncertain systems It has been widely used in problem Yu and Man 2002 presented the fast TSMC. practical systems such as robot manipulators DC DC FTSMC method that can achieve fast finite time. converters and motors Utkin 1977 1993 Utkin convergence when the states are either far away from. Guldncr and Shi 1999 Young Utkin and Ozguner or near the equilibrium point However singularity. 1999 There are two basic components in the SMC occurs in both TSMC and FTSMC and this issue was. Corresponding author Email lhnewmind yahoo com, ISSN 0020 7721 print ISSN 1464 5319 online.
2011 Taylor Francis, DOI 10 1080 00207721 2011 601348. http www informaworld com, 2 H Li et al, addressed explicitly in literature Feng Yu and Man effort is less conservative than that of NFTSMC Yu. 2002 Yu Yu Shirinzadeh and Man 2005 Jin Lee et al 2008 The adaptive law in ANFTSMC can. Chang and Choi 2009 where global nonsingular acquire accurate parameter estimates under a weaker. TSMC NTSMC methods with the same convergence condition than that proposed by Barambones and. properties as those of TSMC for uncertain systems Etxebarria 2001 2002 Furthermore the semi global. were proposed To achieve fast finite time convergence singular issue in Barambones and Etxebarria 2001. for NTSMC Yu Du Yu and Xu 2008 introduced 2002 and Zhao Li and Gao 2009 is eliminated as. the nonsingular FTSMC NFTSMC for a class of the NFTSMC approach is adopted and global. n order systems but a high gain control effort was nonsingularity is achieved The main contributions of. adopted to ensure the stability of the close loop this article are as follows 1 a novel architecture of. system ANFTSMC with the combination of composite adap. The aforementioned control approaches i e SMC tive law and NFTSMC that assures fast finite time. TSMC FTSMC and their nonsingular forms are error convergence is provided 2 The stability and. robust control methods that do not have the ability to error convergence analysis of the proposed. learn in the control process and result in a conser ANFTSMC are given. Downloaded by Beijing Institute of Technology at 00 52 01 December 2011. vative design Furthermore the stability of the system This article is organised as follows Model of EMA. is achieved at the cost of performance Song is described and problem formulation is given in. Longman and Mukherjee 1999 While adaptive Section 2 In Section 3 the control design is introduced. SMC ASMC with the integration of adaptive control Stability and error convergence analysis are presented. and SMC may ride this disadvantage and has been in Section 4 Simulation and experimental study are. widely investigated in recent years Man Mike and Yu given in Section 5 and some conclusions are drawn in. 1999 Keleher and Stonier 2002 Zhao Li Gao and Section 6. Zhu 2009 Usually only the tracking error is used in. the adaptive law in ASMC and this brings about slow. parameter convergence and large transient tracking 2 Models and problem formulation. error In Barambones and Etxebarria 2001 2002 a 2 1 Dynamic models of EMA. modified TSMC with the composite adaptive law, EMA with a DC motor driving a gearbox mechanism. Slotine and Li 1989 was presented where the, will be concerned Generally the current dynamics of. parameters were estimated by both the tracking error. the motor is neglected due to the much faster electric. and the prediction error Faster parameter convergence. response in comparison to the mechanical dynamics, and smaller tracking errors were achieved and true.
The model of the actuator can be described as follows. parameter estimates could be acquired if the persistent. Ilyas 2006, excitation PE condition was satisfied To ensure the. boundedness of all signals a nonlinear filtered error d2 d. signal which was switched to zero Barambones and J 2. f ML D K1 u 1, Etxebarria 2001 2002 when the trajectory error. equalled zero was introduced in TSMC However where is the output angle of the gearbox shaft J f. the control effort may become awfully large while the ML K1 and D are equivalent parameters relative to the. trajectory error is close to zero and only semi global actuator shaft total moment of inertia total damping. nonsingularity is obtained coefficient load torque equivalent electrical mechan. In this article the above mentioned problems are ical energy conversion constant and unmodelled. addressed An adaptive NFTSMC ANFTSMC dynamics of the actuator. approach for EMA is proposed The control scheme Usually in servo systems the load is figured as an. comprises an adaptive control term and a robust unknown torque but in some cases the position. control term The adaptive control term uses a novel dependent torque which can be modelled as a spring. composite adaptive law where the integration of load torque is met such as the hinge moment of the. filtered system states are employed to estimate the aerocraft Thus the load torque of EMA can be. parameters and then compensates for the modelled expressed as Wu and Fei 2005. dynamic uncertainties While the robust term which is ML M j MC 2. based on a modified NFTSMC with the fast TSM, reaching law provides fast finite time convergence of where M j is the coefficient of spring load torque and. errors either far away from or near the equilibrium MC is the constant load torque. point In addition it robustifies the design against Regarding the output angle as the system. unmodelled dynamics with a small switching gain duo output y and defining the output angle and angu. to the parameter adaption Consequently the control lar velocity of the actuator as the state variables. International Journal of Systems Science 3, T then the entire system can be. i e x x1 x2 T 3 Controller design, expressed as 3 1 Control law design.
In this section the nonsingular fast terminal sliding. 1 x 2 u 2 x1 3 x2 4 D0 3 mode NFTSM for EMA is first introduced Then the. control law comprising an adaptive control term and a. robust control term is designed, where D0 D K1 V and i i 1 2 3 4 are given as NFTSM Yu et al 2008 for model 3 can be. follows described as, 1 V rad s 2 1, 2 1 j 1 c 1 j2 sign 1 c 1. K1 4 where 40 c40 z1 z2 05z15z2 z1 and z2 are, f odd integers c are parameters to be designed The. V rad s 1 1, Downloaded by Beijing Institute of Technology at 00 52 01 December 2011. K1 signum function sign for the scalar is defined as. sign 0 0 6, The first derivative of 2 is as follows Yu et al 2005.
2 2 Assumptions and problem formulation 2008, For simplicity the following notations will be used i 2 1 j 1 c 1 j1 1 c 1 7. for the ith component of the vector for the estimate. of min for the minimum value of and max for the Remark 3 1 In Barambones and Etxebarria 2001. maximum value of k k is the Euclidean norm of 2002 and Zhao et al 2009 a nonlinear filtered error. min and max are the minimum eigenvalue and named as er is used in sliding mode design er is. maximum eigenvalue of the matrix respectively The switched to zero Barambones and Etxebarria 2001. operation or for two vectors is performed in 2002 or a constant value Zhao et al 2009 when the. terms of the corresponding elements trajectory error e equals zero and all the internal. In general the parameters of the model cannot be signals are bounded when e 0 However er may. accurately determined but we assume that the uncer become awfully large as e 0 and it does not achieve. tain parameters lie in some previously known intervals global nonsingularity The details can be seen in. as shown in Assumptions 1 and 2 In addition Appendix A In 7 as 51 1 40 then 2. Assumption 3 is for the desired trajectories Zhang and 2 are bounded when 1 c 1 0 Thus the. Chen and Li 2010 singular problems in classic TSMC and FTSMC are. Assumption 1 min 4 4 max min and max are avoided and global nonsingularity can be achieved. known with min 1 min 4 min T max 1 max Yu et al 2008. 4 max T Moreover 1 min40 which conforms to Noting 3 and 7 we obtain the following. the physical point of view dynamics, Assumption 2 The unmodelled dynamics is bounded 1 2 1 1 j 1 c 1 j1 1 1 1 c 1. i e kD0 k where 40 is known, 1 1 j 1 c 1 j1 u 2 x1 3 x2 4. Assumption 3 The desired trajectory xd is continuous. D0 1 x d 1 c 1 8, and its first order derivative x d and second order deriv. ative x d are bounded and available Then the control law is as follows. Consider model 3 which has unknown parame u ua us 9. ters disturbances and unmodelled dynamics the con, trol problem of this article can be formulated as where ua denotes the adaptive control term in 10 and.
follows Given the desired motion trajectory xd the us is the robust control term in 13 Let. object is to synthesise a control action u such that the. system tracking error e1 x1 xd converges to zero ua 1 x d c 1 1 2 x1 3 x2 4 T 1. while maintaining all signals in the system bounded 10. 4 H Li et al, where 1 4 T i is the estimate of i i 1 2 where. 3 4 1 and 1 are given by, 2 1 c 2 0 x d c 1 0 x1 0 x2 0 T. 1 x d c 1 1 x1 x2 1 T 11, The adaptive law for can be chosen as. 1 c proj 1 2 2 1 2 ef 1 3 sig ef r 17, 1 j 1 c 1 j sign 1 c 1 1 c 1. 12 where i diag i1 i2 i3 i4 ij40 i 1 2 3, j 1 2 3 4 The projection operator is defined as.
The robust control term contains two parts proj proj 1 1 proj 4 4 T with Zhang. 8 Chen and Li 2009 Zhang et al 2010, us us 1 us2, us 1 k1 2 k2 j 2 jr sign 2 13 0 i i max and i 4 0. us 2 k3 sign 2 proj i i 0 i i min and i 5 0 18, Downloaded by Beijing Institute of Technology at 00 52 01 December 2011. dynamical uncertainties While the robust control term which is based on a modified nonsingular fast terminal sliding mode control method with fast terminal sliding mode TSM reaching law provides fast convergence of errors and robustifies the design against unmodelled dynamics Furthermore the control method eliminates the

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