Deakin Research Online

Transcription

1 Deakin Research Online his is the pblished version: Herek A. Samir S Steven rinh Hie and Ha Qang P. 9 Performance of first and second-order sliding mode observers for nonlinear systems in AISA 9 : Proceedings of the rd International Workshop on Artificial Intelligence in Science and echnology University of asmania Hobart as. pp. -7. Available from Deakin Research Online: Reprodced with the kind permission of the copyright owner. Copyright : 9 University of asmania

2 Performance of First and Second-Order Sliding Mode Observers for Nonlinear Systems A. Samir Herek Steven S H. rinh and Q. P. Ha Faclty of Engineering and Information echnology University of echnology Sydney PO Box Broadway NSW 7 Astralia School of Engineering Deakin University Geelong 7 Astralia Abstract his paper presents a brief stdy on the design and performance comparison of conventional first-order and sper-twisting second-order sliding mode observers for some nonlinear control systems. Estimation accracy fast response chattering effect peaking phenomenon and robstness are considered for nonlinear systems nder observer-based otpt feedback control and state feedback control.. Introdction he concept of an observer for a dynamic process was introdced in 966 by Lenberger []. Years later many advanced observers have been presented. Dring that time pioneering work in variable strctre systems with sliding modes also emerged in the former Soviet Union []. Sliding mode techniqes have been recognised as a robst control method whereby a variable strctre control is sitably chosen to drive and then constrain the system state to lie within a desired sliding srface. he sliding mode methodology has been applied to the observer design problem to retain the niqe property in that it ensres the accrate convergence of the state estimates while sliding motion is indced for the estimation error dynamics []. Since the earliest work by Slotine et al. in the mid 98s [] sliding mode observers have been widely sed de to their salient advantages sch as robstness against ncertainties finite-time convergence and possibility of estimating nknown inpts e.g. friction [56]. A conventional first-order sliding mode observer (SMO) reqires the relative degree between the system inpts and otpts to be one and qite often involves chattering. o overcome these limitations while preserving SMO advantageos properties higher sliding modes have been proposed for both control and observation (see e.g. [7] [8]). he main difference between conventional and higher order sliding mode is that higher order derivatives of the sliding fnction are sed in place of first-order derivatives. his is also reqired when the time rate of the control signal is sed to achieve the control objective [9]. A new generation of observers based on the secondorder sliding-mode algorithms has been recently developed [5]. hese inclde the se of second-order SMOs for velocity estimation for ncertain nonlinear mechanical systems [] estimation of the absolte orientation of a five-link biped robot [] estimation of road profile [] and vehicle dynamic parameters of the road/vehicle interaction [] or falt detection and isolation in permanent magnet synchronos motors []. Notably higher-order SMOs based on twisting algorithms do not reqire the relative degree to be one and can totally remove chattering [5]. hese merits can lead to exact differentiation of signals which trns ot to be very promising for health monitoring and falt diagnosis in many practical systems as the estimation of errors of the derivatives will be small if the magnitde of noise is small [5]. he se the sper twisting algorithm which is based on an exact and robst sliding mode differentiator of second order [7] facilitates the need for higher order derivatives in higher order SMOs. In [5] the algorithm is sed in designing observers for hybrid systems to avoid chattering and time delays in a class of switched chaotic systems. For the challenging problem of simltaneos estimation of state and nknown inpt for nonlinear systems (see e.g [6]) higher-order SMOs are also expected to be of great potential. his paper motivated by a comparison stdy on advanced state observer design techniqes [7] evalates performance of first- and second-order SMOs and presents advantages of the sper-twisting second-order sliding mode observers over the conventional SMOs. Simlation reslts are given for some nonlinear systems in state feedback and also otpt feedback.

3 . Sliding Mode Observers. First order SMO: Consider a n-th order continos time system with n p state vector x( t) R inpt ( t) R and measrable m otpt y( t) R : x & = Ax f ( x ) (a) y = Cx (b) where (AC) is an observable pair f ( x ) is a nonlinear vector fnction locally Lipschitz in x with a Lipchitz constant γ : f ( x ) f ( x ) γ x x ( x x ) () and γ > is a positive scalar. he first-order sliding mode observer can be obtained as [8]: x & ˆ = Axˆ f ( xˆ ) L ( y Cxˆ) S( xˆ y) () where L is a constant gain matrix obtained by assigning the observer desired dynamics i.e. stable eigenvales eig( A LC) S ˆ ( x y) = P C sign( Cex ) P C Cex ε Ce Ce x x > ε ε () with e x = x xˆ or Ce x = y Cxˆ ε > is the amplitde of a bondary layer and P is a symmetric positive definite matrix solving for the following algebraic Riccati eqation: ( A LC) P P( A LC) I γ PP = Q (5) for a given symmetric positive definite matrix Q >. It can be shown that the error system of observer () nder the variable strctre term () asymptotically converges to zero or remains bonded in the case of noisy measrements [8]. Remark : Given a Lipschitz constant γ > one can frther relax (5) into an ineqation in order to conveniently compte P by sing a linear matrix ineqality (LMI) formlation: ( A LC) P P( A LC) I γp > (6) γp -I where γ = in the case of linear systems.. Second-order SMO: For the design of a higher-order sliding mode observer let s consider system () in a canonical form as follows: x& = x x& = x x ) ξ( x x ) where f ( (7) n [ x ( t) x ( t)] R is the system state n ( t) R is the control y ( t) = x( t) is the measrable otpt and the system nonlinear fnction f x x ) is known and pertrbed by ncertainty ( ( x x ξ ) where f and ξ are Lebesgemeasrable in any compact region of the state space ( x x ). As physical states of a practical system are bonded it is assmed the existence of a finite constant f > sch that for any ( x ˆ x x) and for any (t) : f x x ) f ( x xˆ ) ξ ( x x ) f. (8) ( Remark : he model (7) widely sed as the dynamic eqation for most mechanical systems and robotic maniplators [58] has a the relative degree of two from the control (t) to the otpt y (t) which makes the design of a first-order SMO infeasible. Using the sper-twisting algorithm a second order sliding mode observer for system (7) can be constrcted as follows []: x& ˆ ˆ = x z x& ˆ = f ( x xˆ ) z (9) where the correction variables ( z z) are otpt injections of the form: z = λ x xˆ z = αsign( x / sign( x xˆ ). xˆ ) () By denoting estimation errors e = x ˆx and e = x ˆx and taking into accont condition (8) one can obtain the following differential inclsion: e& = e λ e e& [ f f / sign( e ) ] αsign( e ) () which is nderstood in the Filippov sense [9] i.e. the right hand is enlarged in some points to satisfy the pper semi-continity reqirement.

4 It was proven in [] that if design parameters α and λ are selected to satisfy the following condition: α > f () ( α f ) λ > q q > α f then the observer states ( x ˆ x ˆ ) in (9) will converge to x ) in finite time. ( x. Illstrative examples In the following performance of first- and secondorder SMOs will be evalated for two nonlinear systems nder linearization state feedback () and also otpt feedback control (OFB).. Example : Consider the first and the second order sliding mode observers for following nonlinear system: x & = x x & = x () x y =. o avoid the nonlinearity feedback linearisation is sed to place the closed-loop poles at ( ± j ) /. he state feedback controller for this system is constrcted as follows: x x = x. () he otpt feedback controller is designed as: = xˆ. (5) y xˆ he first-order sliding mode observer can be constrcted as: x & ˆ = xˆ L ( y xˆ ) Ksign( y xˆ ) (6a) x & = y xˆ L ( y xˆ ) K sign( y ˆ ) (6b) ˆ x L = L L are chosen according to the desired eigenvales of s Ls L = and K = P C = [ K K ] with positive definite matrix P solving for the following Lyapnov eqation: where the observer gains [ ] ( A LC) P P( A LC) = I in which = A and = [ ] C. x x OFB poles at - OFB poles at - OFB poles at Fig. : State feedback and otpt feedback responses with different pole locations of a first-order SMO. x x OFB poles at - OFB poles at - OFB poles at Fig. : First-order SMO responses with satrated otpt feedback Figre shows the otpt responses of the firstorder SMO nder otpt feedback and linearization state feedback respectively. As the poles decrease the error increases. Large negative poles often lead to faster estimation. It can also be seen that the control action is greatly increased in magnitde with large negative poles. When the observer has a pole located at -5 a finite escape time of arond.7 second de to the peaking phenomenon [] can be observed. With satrated otpt feedback. 5 the peaking effect can be improved as shown in Fig.. he time responses nder otpt feedback in this case also approach the responses nder state feedback when observer poles move towards the negative direction along the real axis. Even with poles larger than -5 the observer-based responses still show a convergence. Let s now consider the proposed sper twisting second-order SMO with the same control as in (5):

5 xˆ & ˆ ˆ ( ˆ = x λ y x sign y x ) (7a) x & = y xˆ α sign( y ˆ ) (7b) ˆ x x..5 OFB poles at - OFB poles at - OFB poles at -5 With feedback linearisation law () condition () will be less conservative for the selection of λ α. Here these parameters play the role of the variable strctre control K K in the first order SMO (6). Figre shows the control responses nder nonsatrated otpt feedback. hey are faster bt tend to exhibit the peaking phenomenon as the vales of λ and α increase. A finite time escape will occr when the pole location is at arond -5. his effect is redced when sing otpt feedback with satrated control (. 5) as shown in Fig.. For performance comparison responses sing observer-based otpt feedback (OFB) are shown in Fig. 5 with conventional SMO and higher-order sliding mode observer (HOSMO) where satration is imposed on the control signal. With observer poles located at - it is noticeable that the control action in the case of twisting HOSMO is higher in magnitde than the in the case of SMO bt the responses are mch faster reslting in more accrately following the responses. As shown in Fig. 5 the peaking phenomenon has been limited by the satration of the control signal. o frther evalate the performance of HOSMO verss SMO a step response of. for the first state is considered in an open loop test. he responses shown in Fig. 6 indicate that HOSMO is not only faster bt also more accrate lower estimation error than SMO. Moreover as depicted in the zooming-in figre it is observed that there is no chattering for the HOSMO response while chattering does affects the state response in the case of SMO. x Fig. : State feedback and satrated otpt feedback with different poles for second order SMO. x x OFB HOSMO polea at - OFB SMO polea at Fig. 5: Responses of satrated otpt feedback controller based on HOSMO and SMO with poles at -. x. OFB poles at - OFB poles at - OFB poles at -5 Actal SMO at - pole vale HOSMO at - pole vale x Extreme zoom in x x Fig. : State feedback and otpt feedback responses with different pole locations for second-order SMO. Fig. 6: Chattering effect of HOSMO and SMO with poles taken at -.

6 . Example : (a) Let s now consider a pendlm system described by: ml & θ mg l sinθ k l & θ = ( t). (8) heta OFB poleas at OFB poleas at.5-5 (b) OFB poleas at - he state space eqation is: heta & θ = ω (9a) g k & ω = sinθ & θ ml l m (9b) where the measrable otpt is angle θ. If the continos sliding mode state feedback () controller is designed []: ω μθ = [ asinθ ( b μ) ω] k sat (a) c ε heta (c) Fig. 7: SMO position estimates with (a) nominal parameters (b) actal plant and (c) linear high gain observer. (a) then the observer-based otpt feedback control is given by: Velocity ˆ ω μθ = [ aˆ sinθ ( bˆ μ) ˆ] ω k sat (b) cˆ ε where a = g / l b = k / m and c = / ml are the system constants depending on its parameters (mass m damping k length l gravitational acceleration g ) k is the variable strctre control gain with bondary layer ε and μ > is the slope of the sliding srface ω μθ =. he sliding mode observer (SMO) in () can be constrcted as: ˆ& θ = ˆ ω L ( θ ˆ) θ Ksign( θ ˆ) θ & ˆ ω = ( c ˆ aˆ sinθ bˆ ˆ) ω L ( θ ˆ) θ Ksign( θ ˆ) θ () Nominal Plant L = L L are chosen according to the desired eigenvales of s ( L μ ) s L = and K = P C = [ K K ] with positive definite matrix P solving for LMI (6) in which A = C = [ ] and Lipschitz constant μ γ = a. In the simlation the actal plant parameters were taken as a = 9. 9 c = b= with their nominal estimates a ˆ = 9. 8 c ˆ = b ˆ = μ =. We also considered the third case when â = c ˆ = in () i.e. (a linear high-gain observer). he responses are sed as a benchmark for all cases. where the observer gains [ ] Velocity Velocity OFB poleas at.5 - (b) OFB poleas at - OFB poleas at (c) Fig. 8: SMO velocity estimates with (a) nominal parameters (b) actal plant and (c) linear high gain observer. As shown in Figs. 7 and 8 respectively for estimates of the angle θ and anglar velocity ω the otpt feedback responses tend to approach the state feedback responses at higher observer poles. In the figres we sed three different sets for observer () with actal plant parameters (a) nominal plant parameters (b) and linear high gain observer (c). When poles are relatively large the nominal plant apparently does not affect mch on the improvement of the estimation performance. his is expected becase increasing the vales of poles will yield higher robstness with against the system ncertainty. From (9-) the sper-twisting higher-order sliding mode observer (HOSMO) for system (9) is: & ˆ ˆ ˆ θ = ω λ θ θ sign( θ ˆ) θ ( c ˆ aˆ sinθ bˆ ˆ ω) α ( θ ˆ θ ) ˆ & ω = sign ()

7 heta heta heta (a) OFB poles at - OFB poles at OFB poles at.5- (b) (c) Fig. 9: HOSMO position estimates with (a) nominal parameters (b) actal plant and (c) linear high gain observer. w w w (a) (b) (c) OFB poles at - OFB poles at -5 OFB poles at Fig. : HOSMO velocity estimates with (a) nominal parameters (b) actal plant and (c) linear high gain observer. where parameters λ α are chosen according to (). Figres 9 and show the estimates of the first and second states namely the pendlm s position and anglar speed respectively. We can see in all cases (a) (b) and (c) that the otpt feedback responses approach the state feedback responses in finite time. It can be expected that robstness of the estimation against ncertainty on the nominal plant is improved as the poles increase. By sing larger poles (-) OFB responses are almost coincident with responses in face of a wide range of system s parametric variations. Notably a close comparison between SMO and HOSMO responses shown respectively in Figs 78 and Figs. 9 also indicates the otperformance of the higher-order sliding mode observer over the conventional one in terms of estimation accracy.. Conclsion his paper has presented a refinement of the design of first-order and higher-order sliding mode observers for nonlinear systems. A comparison stdy is given to illstrate the estimation performance of these observers for two nonlinear systems sing state feedback and observer-based otpt feedback control. Some advantages of the sper-twisting second-order sliding mode observers over the conventional ones are shown in the simlation reslts. Acknowledgments his work is spported in part by the Centre of Excellence programme fnded by the Astralian Research Concil (ARC) and the New Soth Wales State Government. he second and forth athors wold like to grateflly acknowledge US research grant 688. References [] D. Lenberger "Observers for mlti variable systems" IEEE rans. Atom. Control Vol. pp [] V. Utkin "Variable strctre systems with Sliding Modes" IEEE rans. Atom. Control Vol. pp. -. [] S. Sprgeon Sliding Mode Observers: a Srvey Int. J. Systems Science Vol. 9 No. 8 pp [] J.J.E. Slotine J.K. Hedrick and E.A. Misawa On sliding observers for nonlinear systems Proc. of the American Control Conference pp Seattle USA 986. [5] M. Saif W. Chen and Q. W Higher order sliding mode observers and differentiators- application to falt diagnosis problem in Modern Sliding Mode Control heory Bartolini et al. (Eds.) LNCIS Springer-Verlag Berlin 8 pp. -. [6] Q.P. Ha A. Bonchis D.C. Rye and H.F. Drrant- Whyte Variable strctre systems approach to friction estimation and compensation Proc. of the Int. Conference on Robotics and Atomation California USA pp [7] A. Levant Higher-order sliding modes differentiation and otpt-feedback control Int. J. Control Vol. 76 No. 8 pp [8] G. Bartolini A. Pisano E. Pnta and E. Usai A srvey of applications of second-order sliding mode control to mechanical systems Int. J. Control Vol. 76 No. 9/ pp [9] H. Dalvand H.. Ngyen and Q.P. Ha Design of second-order sliding mode controllers for MR damper-

8 embedded smart strctres Proc. Int. Symp. Atomation and Robotics in Constrction Astin exas US pp [] J. Davila L. Fridman and A. Levant Second Order Sliding Mode Observer for Mechanical Systems IEEE rans. Atomatic Control Vol. 5 No. pp [] V. Lebastard Y. Aostin F. Plestan and L. Fridman Absolte Orientation Estimation Based on Higher Order Sliding Mode Observer for Five Link Walking Biped Robot Proc. International Workshop on Variables Strctre Systems Alghero Italy pp Jne 6. [] A. Rabhi N.K. M sidiri L. Fridman and Y. Delanne Second Order Sliding Mode Observer for Estimation of Road Profile Proc. International Workshop on Variables Strctre Systems Alghero Italy pp Jne 6. [] H. Sharim M. Oladsine and L. Fridman Vehicle parameter and state estimation via sliding mode observers in Modern Sliding Mode Control heory Bartolini et al. (Eds.) LNCIS Springer-Verlag Berlin 8 pp [] Y. Hangf W. Li and R. Ma Permanent magnet synchronos motor falt detection and isolation sing second order sliding mode observer Proc. rd IEEE Conference on Indstrial Electronics and Applications Singapore pp [5] H. Saadaoi M. Djemai N. Manamanni. Floqet and J.P. Barbot Exact Differentiation via Sliding Mode Observer for Switched Systems Proc. nd IFAC Conference on Analysis and Design of Hybrid Systems Alghero Italy pp. -9 Jne 6. [6] Q.P. Ha and H. rinh Simltaneos state and inpt estimation for a class of nonlinear systems Atomatica Vol. pp [7] W. Wang and Z. Gao A comparison stdy of advanced states observer design techniqes Proceedings of the American Control Conference Denver Colorado pp [8] A. Alessandri Sliding-mode estimators for a class of nonlinear systems affected by bonded distrbances Int. J. Control Vol. 76 No. pp [9] A.F. Fillippov Differential Eqations with Discontinos Right-Hand Sides Dordrecht he Netherlands: Klwer 988. [] H.K. Khalil Nonlinear Systems d Edition Prentice- Hall New Jersey.