NONLINEAR ADAPTIVE OBSERVER DESIGN IN COMBINED ERROR NONLINEAR ADAPTIVE CONTROL

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1 ХРАНИТЕЛНА НАУКА ТЕХНИКА И ТЕХНОЛОГИИ октомври 3 Пловдив SCIENIFIC WORKS FOOD SCIENCE ENGINEERING AND ECHNOLOGY Otober 3 Plovdiv NONLINEAR ADAPIVE OBSERVER DESIGN IN COMBINED ERROR NONLINEAR ADAPIVE CONROL Rumen Mishkov Valentin Petrov ehnial University Sofia Branh Plovdiv Abstrat he aer deals with the design of nonlinear adative observers and arameter estimators in adative observer anonial form. he losed loo system nonlinear adative ontrol law is designed by the ombined error adative ontrol aroah. he nonlinear adative observer is designed in the adative observer anonial form. he arameter estimator is Lyaunov stable. he roosed aroah is alied to an inverted endulum driven by salient-ole ermanent magnet synhronous motor. he losed loo adative system time resonses are simulated for illustration. Keywords: nonlinear adative observers nonlinear adative systems arameter estimators. Introdution he nonlinear adative state observers with unknown arameter estimator are generally used for adative losed loo system design. he inut-outut signals and the system struture is the only available information for state and arameters estimation. he nonlinear adative state observer design requires an aroriate nonlinear anonial form [ 6 8] being usually the adative observer anonial form (AOCF) [6]. he stringent neessary onditions for transformation into AOCF limit the lass of the transformable nonlinear systems. he known aroahes in this field require ersistent exitation of the system [ 6 7] in order to ahieve exat estimation but this onflits with the losed loo system ontrol goals. In that sense the adative losed loo system has to omensate the arameter estimation errors to ahieve the desired erformane or alternatively an exat arameters estimator without ersisteny of exitation has to be designed. Various methods onsidering losed loo adative ontrol with nonlinear adative observers [3 3] based on lassial methods for nonlinear adative ontrol [9 ] deal with the roblem. he aer resents an aroah for losed loo adative system design with nonlinear adative state and arameter observers alied for nonlinear adative ontrol of a salient-ole ermanent magnet synhronous motors (PMSM) [ ] via a transformation into AOCF. he nonlinear adative traking ontrol design is aomlished by the adative ombined error aroah []. Problem Statement he original objetive nonlinear system is firstly resented in nonlinear observer anonial form η = Aη + F(yu) x ( ) = x (a) y = Cη (b) where the matrix air ( A C) is in single-outut Brunovski anonial form R is the unknown n arameter vetor and F (yu) R. he filtered transformation for seond order systems has the following form z( η) = η m (a) m = λm + βf(y u) m ( ) = m (b) where m R λ = b / b β = [ b / b]. It transforms system () into the adative observer anonial form z = Az + bω z ( ) = z (3a) y = Cz. (3b) he vetor n ω R reads ω = m + CF(y u) and b R. he adative nonlinear observer for the system (3) has the form z ˆ = Azˆ + bωˆ + N(y ŷ) z ˆ() = zˆ (a) ŷ = Cˆ z (b) where the observer gain matrix N makes the matrix Ao = ( A NC) stable. he observer error z = z zˆ has the dynamis z = A o z + bω z () = z. () he adative arameter estimation dynamis ˆ is designed via the diret Lyaunov method. A Lyaunov funtion andidate z Pz V = + Γ o 9

2 ХРАНИТЕЛНА НАУКА ТЕХНИКА И ТЕХНОЛОГИИ октомври 3 Пловдив is defined whose total time derivative assuming that the arameters are onstant ( = ) will be V ( ) ( o = z Ao P + PAo z + ωb Pz Γ ˆ). (6) he hoie b = CP (7) where P is the solution of the Lyaunov equation Ao P + PAo = Q (8) and arameter estimator dynamis ˆ = Γω (y ŷ) = Γτ o ˆ() = ˆ (9) redues the derivative (6) into V o = z Qz (ˆ Γτ o ). Hene the observer error z will be asymtotially stable while the arameter errors are only Lyaunov stable. he original system has to be resented in seond order nonlinear omanion form y f ( x) = u () where x is the state vetor and = [ ] with = [ K ]. he task is to stabilize the ombined error e = y y r + λ(y yr ) = y y d with dynamis e = f ( x) + u () y d where y r is the referene trajetory and λ is a dynamis tuning ositive onstant. For that urose a Lyaunov funtion andidate is defined as V = e + + Γ () γ where = / with estimate ˆ and error = ˆ whose total time derivative onsidering () reads f x ˆ V = e( ( ) + u yd) ˆ Γ. (3) γ With the ontrol u = ˆu () ˆ u = y d f ( x) ke () where k > and arameter estimator dynamis ˆ = γ sign( )ue (6) ˆ = Γ f ( x) e = Γ τ (7) the derivative (3) beomes ˆ V ( Γ τ ). (8) he unified Lyaunov funtion reduing the arameter estimates is defined as V = V o + V (9) SCIENIFIC WORKS FOOD SCIENCE ENGINEERING AND ECHNOLOGY Otober 3 Plovdiv whose time derivative reads ˆ V = z Qz (ˆ Γτ ) ke ( Γ τ ). o If the gain matrix Γ is hosen to be Γ Γ = γ then the above derivative transforms to V z Qz (ˆ Γ( τ o + [ τ ] )). () he unified arameter estimator dynamis is ˆ = Γ( τ o + [ τ ] ) () whih redues () to V z Qz. he asymtoti stability of the ombined error and the state observer error follows from the LaSalle- Yoshizawa onvergene theorem. he estimate xˆ is used in all the above deendenies instead of the unknown vetor x. In this way the elimination of the traking error is ahieved when the observer error x = x xˆ onverges to zero. he transformation () is arameter deendent hene the exat estimation of the original state x is ossible only if = but for seond order nonlinear systems in nonlinear omanion form the following deendeny holds y = z + ω = x. he state observer is asymtotially stable and hene ŷ onverges to y asymtotially so ŷ y or ẑ + ω ˆ = xˆ x. hus it is guaranteed that xˆ x whih omletes the nonlinear adative ontrol design. Aliation of the Aroah he aroah is alied to a salient ole PMSM driven inverted endulum system. he PMSM in rotating dqo oordinates ontrolled in urrent mode has a state sae system model x = x () x = x sin y + u u + u 3 where x is the mehanial angle x is the rotor seed u u are the ontrol urrents i d i q. he model arameters are b mgl 3 n ψm (Ld Lq ) = = 3 = J J J 3 n ψm = J where m and l are the endulum mass and length J total moment of inertia g gravity aeleration n number of ole airs ψ m 96

3 ХРАНИТЕЛНА НАУКА ТЕХНИКА И ТЕХНОЛОГИИ октомври 3 Пловдив magnets flux linkages magnitude b visous frition oeffiient and L L are the motor indutanes. he model () is firstly resented in the nonlinear observer anonial form () η = η y η = sin y + 3uu + u via the transformation η x ( η) =. η y Hene the filters () will be m = λ(m y) m = λm sin(y) m 3 = λm3 + uu m = λm + u hen the vetor ω takes the form ω = [ m ym m3m ]. he system in nonlinear adative observer anonial form (3) reads z = z + (m y) + m + m m z = b ((m y) + m + m + m )/ 3 3 b z y = with nonlinear adative state observer () ẑ = ẑ + (m y)ˆ + mˆ + mˆ mˆ + ny = b ((m y)ˆ + m ˆ + mˆ + m ˆ )/b n y ẑ ŷ = ẑ observer error () z = nz + z + (m y) + m + m33 + m z = b((m y) + m + m33 + m)/ b n z and adative arameter estimator dynamis (9) with observer tuning funtion τ o = [m ymm3m ] (y ŷ). he unified arameter estimator dynamis () with adative ontrol tuning funtion τ = [ xˆ sin yuu ] e takes the form ˆ y)y = γ((m e xˆ ) ˆ = γ (my esin y) ˆ 3 = γ3(m3y + euu ) ˆ = γ my ˆ = γ sign( )u e. he omlete adative ontrol law for u = is u = y d + xˆ ˆ + sin yˆ k e = ˆ(y + xˆ ˆ + sin yˆ k e) u d d q SCIENIFIC WORKS FOOD SCIENCE ENGINEERING AND ECHNOLOGY Otober 3 Plovdiv where y d = yr λ(xˆ y r ). he above deendenies are used for dynami simulation of the losed loo nonlinear adative system with nonlinear adative state and arameter observers. Simulation and System ime Resonses he dynami simulation is arried out for Lenze MDSKS7-3 PM synhronous servomotor with arameters: nominal ower P N = kw nominal torque N =. 7 Nm nominal AC voltage U N = 33 V nominal urrent I N =. A torque onstant k =. 37 Nm/A stator resistane R s = 3.Ω moment of inertia J = 6 kg.m indutanes L d = 9. mh L q =. mh and number of ole airs n = 3. he endulum attahed to the shaft is determined by the mass m =. kg link length l =. m and the gravity onstant g = 9. 8 ms. he losed loo nonlinear adative system with nonlinear state observer and arameter estimator resonses are for initial onditions x = [. ] z = [..] z ˆ = [ m = [ ˆ = [ ] ] ] ˆ =. he referene trajetory used is the state of a seond order linear system with double real ole = 3. he observer gain matrix N = [ ] sets a double ole for the observer dynamis. he arameter estimates onvergene gains are 3 γ =.3 γ = γ 3 = γ = γ =. 8 6 x y r x y r

4 ХРАНИТЕЛНА НАУКА ТЕХНИКА И ТЕХНОЛОГИИ октомври 3 Пловдив SCIENIFIC WORKS FOOD SCIENCE ENGINEERING AND ECHNOLOGY Otober 3 Plovdiv 8 [A] 6 xˆ xˆ x u = i d x Figure. Adative system and state observer resonses Figure shows the losed loo adative system trajetory traking and the adative state observer resonses. he time evolution of the arameters and their estimates is given on figure. he torque reating urrent i d and the urrents in the ab frame.... ˆ ˆ ˆ ˆ ˆ 3 3 Figure. Parameter observer dynamis [A] i a i i b - 3 Figure 3. Control urrents dynamis are dislayed on figure 3. he time resonse of the eletromagneti torque is deited on figure. he losed loo adative system traking and ombined errors are shown on figure. he ontrol urrent [Nm] [rad] e 3 Figure. Eletromagneti torque -. 3 e y y r - 3 Figure. raking and ombined errors 98

5 ХРАНИТЕЛНА НАУКА ТЕХНИКА И ТЕХНОЛОГИИ октомври 3 Пловдив u = i q is asymtotially stabilized at zero beause the PMSM oerates at lower than the rated seed ω N = 36 rad / s. he arameter estimates are only Lyaunov stable while the state observer and the ombined error are asymtotially stable. he estimation of the true arameters deends on the referene trajetory exitation aability. he onvergene of the estimates to the true arameters is guaranteed only if the referene trajetory is ersistently exiting. he onvergene of the ombined error deends on the tuning arameters k and λ. All the resonses are for k = and λ =. With these values the traking of the referene trajetory by the losed loo adative system is done with the resribed overdamed erformane seifiations. Conlusions he aer deals with the adative ontrol of a salient-ole PMSM driven inverted endulum with adative state and arameter observers. he task solved is traking of a referene trajetory. he nonlinear adative traking ontrol is aomlished by the adative ombined error aroah. he adative arameter observer is designed via the diret Lyaunov method and ensures Lyaunov stability of the arameter errors only. he nonlinear adative state observer is designed into nonlinear adative observer anonial form and rovides asymtotially stable state traking even with a Lyaunov stable arameter estimator. he losed loo nonlinear adative traking ontrol system with integrated nonlinear adative state and arameter observers ahieves the resribed overdamed traking erformane seifiations by asymtotially stable state traking and Lyaunov stable arameter estimation. Aknowledgement he suort of Researh Fund Projet No. 3PD7-9 from NIS in ehnial University Sofia for this researh work is gratefully aknowledged. Referenes [] Bastin G. Gevers M. Stable Adative Observers for Nonlinear ime-varying Systems IEEE ransations on Automati Control Vol. 33 No SCIENIFIC WORKS FOOD SCIENCE ENGINEERING AND ECHNOLOGY Otober 3 Plovdiv [] Besanon G. Remarks on Nonlinear Adative Observer Design Systems Control Letters Vol. No [3] Kanellakooulos I. Kokotovi P. Middleton R. Observer-based Adative Control of Nonlinear Systems Under Mathing Conditions Amerian Control Conferene [] Kanellakooulos I. Krsti M. Kokotovi P. Interlaed Controller-Observer Design for Adative Nonlinear Control Amerian Control Conferene [] Krsti M. Kanellakooulos I. Kokotovi P. Nonlinear and Adative Control Design John Wiley and Sons In. 99. [6] Marino R. Adative Observers for Single Outut Nonlinear Systems IEEE ransations on Automati Control Vol. 3 No [7] Marino R. omei P. Adative Observers with Arbitrary Exonential Rate of Convergene for Nonlinear Systems IEEE ransations on Automati Control Vol. No [8] Marino R. omei P. Global Adative Observers for Nonlinear Systems via Filtered ransformations IEEE ransations on Automati Control Vol. 37 No [9] Mishkov R. Darmonski S. Adative Nonlinear rajetory raking Control for DC Motor Driven Inverted Pendulum International Conferene Automatis and Informatis'. B-67 B-7. [] Mishkov R. Darmonski S. Adative uning Funtions System Design for Inverted Pendulum International Conferene Engineering ehnologies and Systems ehsys Vol. 6 book [] Mishkov R. Petrov V. Advaned Control Nonlinear State Sae Models of Permanent Magnet Synhronous Motors International Conferene Automatis and Informatis' [] Mishkov R. Petrov V. Stritly Orientated State Sae Models of Permanent-magnet Synhronous Motors for Feedbak Linearization Control International Conferene Engineering ehnologies and Systems ehsys Vol. 6 book [3] Petrov V. Darmonski S. Observer Based Adative Nonlinear Control System Design for a Single Link Maniulator Student Conferene Eletrial Engineering Information and Communiation ehnologies EEIC Brno Czeh Reubli. 6. [] Slotine J.-J. Li W. Alied Nonlinear Control Prentie Hall In