Electro-Hydraulic Piston Control using Neural MRAC Based on a Modified State Observer

Transcription

1 Aerian Control Conferene on O'Farrell Street, San Franio, CA, USA June 9 - July, Eletro-Hydrauli Piton Control uing Neural MRAC Baed on a Modified State Oberver Y. Yang, S. N. Balakrihnan, L. ang, R.G. Lander Abtrat A new odel referene adaptive ontrol deign ethod uing neural network that iprove both tranient and teady tage perforane i propoed in thi paper. Stable traking of a deired trajetory an be ahieved for nonlinear yte having ignifiant unertaintie. A odified tate oberver truture i deigned to enable deired tranient perforane during unertainty learning. he neural network adaptation rule i derived uing Lyapunov theory, whih guarantee tability of the error dynai and boundedne of the neural network weight. An extra ter i added in the ontroller expreion to introdue a oft withing liding ode that an be ued to adjut traking error. he ethod i applied to ontrol the veloity of an eletro-hydrauli piton, and experiental reult deontrate the deired perforane i ahieved with ooth ontrol effort. Index er Neural network, adaptive ontrol, eletroni-hydrauli yte A I. INROUCION PPLICAIONS of artifiial neural network in the field of ontrol have been developed for deade. Narendra and Parthaarathy [] provided a tability proof for the firt tie, and deontrated the potential of neural network in the identifiation and ontrol of nonlinear yte. Sanner and Slotine [] developed a diret traking ontrol ethod with Gauian radial bai funtion (RBF) network for feedbak ontrol of nonlinear yte. Sine then, variou adaptive ontrol tehnique uing neural network were put forward. Calie et al. [3]-[6] introdued neural network to the dynai inverion tehnique in order to anel the inverion error, and developed odel referene adaptive ontrol (MRAC) baed on Y. Yang (e-ail: yyhw3@t.edu) i a Graduate Student in the epartent of Mehanial and Aeropae Engineering, Miouri Univerity of Siene and ehnology. S. N. Balakrihnan (e-ail: bala@t.edu) i a Profeor in the epartent of Mehanial and Aeropae Engineering, Miouri Univerity of Siene and ehnology. L. ang (e-ail: ltx8d@t.edu) i a Pot otoral Fellow in the epartent of Mehanial and Aeropae Engineering, Miouri Univerity of Siene and ehnology. R. G. Lander (e-ail: landerr@t.edu) i an Aoiate Profeor in the epartent of Mehanial and Aeropae Engineering, Miouri Univerity of Siene and ehnology. neural network etiation. he neural network are trained online uing a Lyapunov-baed approah, iilar to the approah followed in [], [7]. Reently, MRAC ha been applied in olving ontrol proble for yte with athed unodeled dynai [8], [9]. However, although thee developent an be eployed to iprove robutne, traking auray an only be hown to be bounded, and the bound depend on the diturbane itelf. In [], a new MRAC neural network ontroller naed L adaptive ontrol i propoed, and tranient perforane of both the yte input and output ignal are guaranteed. At the ae tie, due to it ipliity and robutne, Sliding Mode Control (SMC) i alo often ued in adaptive ontrol []-[3]. One drawbak of SMC i that unavoidable hattering our when the ontrol ignal withe ign along the liding urfae. A oft-withing liding ode tehnique wa introdued by Lyhevky [4] in order to avoid oillation and ahieve ayptoti tability at the ae tie. In [5], by uing a ethod iilar to SMC, a novel approah obining an adaptive neural network feedforward ontroller with a ontinuou robut integral of ign of error (RISE) feedbak ontroller i introdued. In thi ethod, it i hown uing Lyapunov theory that the traking error i ayptotially table. hi paper develop a new neural network MRAC with guaranteed tranient perforane and ayptoti tability. Baed on the MRAC neural network ontroller, the neural network oberver truture i odified in the anner of [6]. In thi odifiation, intead of introduing additional filter, a fator of the oberver error i added to in the neural network oberver truture. A a reult, thi new ethod enable further inreae in the adaptive gain, leading to better traking perforane. At the ae tie, the odified ter i inative when the neural network etiation i ideal; therefore, etiation auray i guaranteed. In order to ahieve iproved tranient perforane and tability, a oft-withing liding ode odifiation i obined with the neural network adaptive ontroller. It i proven, uing the Lyapunov ethod, that it ideally lead to ayptoti tability intead of UUB and, at the ae tie, i free fro hattering that are oon for typial liding ode adaptive ontroller. In general, the propoed ontroller enable higher //$6. AACC 5

2 adaptive gain while iultaneouly providing iproved tranient perforane and ayptoti tability. hi ethod i applied to an eletro-hydrauli tet benh [7] for veloity traking ontrol. he reult illutrate that deired traking perforane i ahieved with ooth (i.e., non-hattering) ontrol. he ret of the paper i organized a follow. In Setion II, the yte and neural network truture are defined. In Setion III, the ontrol olution i propoed. Stability proof of the oberver and tate error ignal are preented in Setion IV. Setion V inlude a deription of the eletro-hydrauli piton yte, and preent the reult and analyi of a erie of veloity traking experient uing the propoed ontrol ethodology. II. PROBLEM ESCRIPION Conider the following ingle input ingle output (SISO) yte x = x x = x3 () x n = b( u f( x)) b >. he yte output i defined a y = x () i a non-zero ontant. he initial ondition i x() = (3) he et of equation in () an be written in a opat for a x () t = Ax() t + B( u() t f( x)) (4) n x i the yte tate vetor, u i the ontrol ignal, A i an n n yte atrix, B i n vetor, ( A, B) i aued to be ontrollable, and f : n i an unknown ontinuou nonlinear funtion. All of the tate are aued to be eaurable. he ontrol objetive i to deign a neural adaptive ontroller whih enure the output trak a deired bounded ontinuou trajetory, denoted rt (), and the yte behavior follow a noinal linear tie-invariant (LI) yte whih i deigned through tandard ethod (e.g., through linear quadrati regulator theory). At the ae tie, the ontroller hould guarantee deired tranient and teady tate perforane in the preene of unertaintie. Aue the following neural network approxiation of f ( x) exit f ( ) φ( ) ε( ) ε( ) ε x = W x + x x < (5) φ( x) i a et of radial bai funtion. Eah eleent of φ ( x) i defined a φ( y) = exp( ( y z) ( y z)/ σ ) (6) z i the enter loation and σ i the width. he vetor W ontain the ideal network weight, ε ( x) i the network approxiation error, and ε i it unifor bound. Further, it i aued that a opat onvex et Ω i known a priori uh that W Ω (7) In order to realize traking ontrol for thi SISO yte, a neural network adaptive ontroller i developed in the next etion. III. CONROL SOLUION he propoed ontroller onit of three part: linear feedbak ontrol K x, neural network adaptive ontrol u e, and oft withing liding ode ontrol μ u = K x + ue + μ (8) K i the loed loop feedbak gain that enure the loed-loop yte atrix A = A BK i Hurwitz. he linear feedbak ontrol enure tability when there i no unertainty. he adaptive ontrol i obtained through the neural network oberver, whih anel the unertainty. he oft withing liding ode ontrol guarantee ayptoti tability in the preene of neural network etiation error. Subtituting (8) into (4) x () t = Ax() t + B( ue()+ t μ f( x)) (9) he following tate oberver truture i defined xˆ () t = A xˆ() ( ()+ ˆ t + B ue t μ f) Kx () t () xˆ( t) repreent the oberver tate at tie t. he initial oberver ondition are xˆ() = () Sine the unertainty and the true neural network weight are unknown, they are repreented a Wˆ φ( x) Ŵ repreent the etiated neural network weight with a proper weight update law. he oberver gain atrix i aued diagonal for onveniene and i expreed a n K= diag( k, k,..., k ). In the oberver truture, ˆf i aued to be aneled perfetly by the neural network ontroller, i.e. ˆ ˆ f = W φ( x). he oberver error i defined a 6

3 x () t xˆ () t x() t () he adaptive weight update law i defined a ˆ () Proj( ˆ W t =Γ W(), t φ( x) x () t PB) (3) P i found by olving AP+ PA = Q, Q i a poitive definite atrix and Γ i the neural network learning rate. he projetion operator property guarantee the boundedne of the neural network weight error WW W (4) W ˆ 4 W, W W W[8]. With the neural W Ω network weight, the adaptive ontrol expreion beoe () ˆ u = k r t + W φ( x) (5) e k g g (6) CA B i the referene yte open loop gain. Subtrating (9) fro (), and ubtituting (5) into the reulting equation, the oberver error dynai are x () t = ( A K ) x + B(W φ( x) ε ) (7) By uing the Lyapunov ethod [9], it will be hown that the neural network etiation error and the oberver error are bounded. By introduing the oberver gain K, the learning proe i oothed, and the odified ter dereae a x dereae; therefore, learning auray i guaranteed. A a reult, the odified oberver truture enable inreaing adaptation gain. IV. SABILIY ANALYSIS In thi etion, the Lyapunov ethod i ued to prove the boundedne of the oberver error dynai. In order to aure ayptoti onvergene of the referene error, the oft-withing liding ode ontroller i derived. etail of the proof are provided in the following ubetion. A. Oberver error o derive the error bound for the neural network oberver, onider the Lyapunov funtion: V( xw, ) = x Px +Γ WW. ifferentiating V V = x Px + x Px +Γ ( WW + WW ) (8) Subtituting the weight update law in (3) and the oberver dynai in (7), (8) beoe V = x Qx x KPx x PB( W φ( x) ε( x)) Proj( ˆ + W W, φ( x) x PB) PBε x λ ( Q) + λ ( K P) x (9) [ ] in in therefore V when Pbε x () λin ( Q) + λin ( KP) A a reult, V( xw, ) x Px +Γ W and PBε λ +Γ λin ( Q) + λin ( KP) W () V( xw, ) x Px λin x () Equation () and () lead to Pbε λ +Γ W λin ( Q) + λin ( KP) x λin (3) In (3), by inreaing the adaptation gain Γ and the oberver gain K, x an be ade arbitrarily all; therefore, preie unertainty etiation uing an online neural network i guaranteed. B. Referene error Note that with adaptive ontrol and linear feedbak ontrol alone (i.e., μ = ), the ontroller i able to trak the referene yte. However, it i only able to do o with bounded traking error. With the addition of a oft-withing liding ode ontroller, the traking error an be ade ayptoti table. he referene LI yte dynai are haraterized by x r = Axr + b( ur + kgr W φ( xr)) (4) ur W φ( xr) i the referene ontroller, whih anel the unertainty. By ubtrating the referene dynai (4) fro the atual yte dynai (9), the traking error dynai are expreed a e x () t x r () t e ( ( Wˆ (5) = A + b μ+ W ) φ( x) ε( x)) Realling the definition of yte dynai a given in (), (5) an be written a e = e e = e3... en = bke+ b( μ+ W (6) φ( x) ε( x)) he liding urfae i defined a n ( n p ) pe λ (7) 7

4 λ p >, p =,,..., n. In ot ae, the deigner an et λ =. For exaple, when n = 3, the liding anifold i = e + λ e + λ e. 3 With the Lyapunov funtionv n ( n p) p =, it derivative i V = = ( λ e + bk e + b( μ+ )) (8) W φ( x) ε( x) (9) Realling the neural network approxiation property in (5) and the error boundedne of the weight in (4), the bound for i = W φ( x) ε( x) W + ε (3) Now the oft withing liding anifold ontrol ter i forulated a n ( n p) μ = λpe / b Ke β tanh( α) (3) Subtituting (3) into (8) V = b( β tanh( α) + ) (3) When > V b herefore, V only when ( β tanh( α ) + ) (33) A long a α > and β >, the liding anifold will reain bounded. By inreaing α and β, the bound of the liding anifold will onverge to. he loed-loop yte i ayptotially table when γ =. With (8), (5), and (3), the final expreion for the propoed ontroller i n ˆ ( n p) u = Kxr + kgr() t + W φ( x) λpe / b β tanh( α) (4) Note that ine no diontinuou funtion i introdued, thi ontroller i ooth and apable of driving the traking error ayptotially to zero. V. EXPERIMENAL RESULS he tet bed for the propoed ontrol ethod i a Caterpillar Eletro-Hydrauli et Benh, whih wa a gift fro Caterpillar to the Miouri Univerity of Siene and ehnology a part of a laboratory dediated to eletro-hydrauli and ehatroni. he tet benh onit of five ditint phyial oponent whih affet the yte operation and dynai: ontrol eletroni, pilot olenoid valve, pool valve, piton, and enor. Additionally, peialized oputer hardware and oftware interfae for atuator and enor provide for real-tie ontrol. A yte diagra i hown in Fig.. tanh( α) / β (34) whih lead to + β < ln (35) α β When < ( )( β tanh( α ) + ) (36) V b herefore, V only when Pilot Valve I Eletroni Control Module x Spool Valve P P Return Supply tanh( α) / > β (37) whih lead to β > ln (38) α + β Fro (35) and (38), it an be oberved that the bound for the liding anifold i + β < ln γ (39) α β Fig.. Eletro-Hydrauli Syte iagra. he eletro-hydrauli yte onidered in thi tudy annot be well deribed by a linear, tie-invariant odel ine the harateriti uh a nonlinear frition, dead band, and nonlinear valve gain annot be negleted. he yte inlude preure enor that eaure the preure in both haber and an enoder that eaure the piton diplaeent. In thi yte, the pool valve i ontained in a ealed houing with no integrated enor; therefore, it i ipoible to eaure it poition either in real-tie or offline. Additionally, it i ubjet to ignifiant and unpreditable tition effet and flow fore, o it poition annot be aurately predited baed olely on the ontrol input and eaured tate. 8

5 he pilot valve input will deterine the diret input into the forward and revere valve. he relationhip between the variable i If = I + If, Ir =, if I > If =, Ir = I + If, if I < If = Ir =, if I = (4) I (A) i the input urrent, I f (A) i the input urrent to the forward valve, I r (A) i the input urrent to the revere valve and If = Ir =.4 A are the etiated dead band value for the forward and revere diretion, repetively. Fro previou work [7], a iple input-output odel for the piton repone wa developed baed on experiental data. In order to reove noie fro the enoder, a low-pa filter i utilized. he filtered piton poition i nuerially differentiated to alulate the (approxiate) piton veloity. It i done online uing a firt order bakward finite differene hee. he relationhip between the etiated veloity and the enoder poition output i ve = x f = 5xf + 5xe (4) v e (/) i the etiated veloity, x f () i the filtered poition, and x e () i the enoder poition eaureent. A a reult, intead of uing a high order nonlinear yte, in the following experient a iple linear odel with athed unertaintie i ued v = ( Bv+ b ( I f ( x, v, P, P, I )) (43) x () i the piton diplaeent, v (/) i the piton veloity, B = (kg/) i the etiated value of the viou frition oeffiient, and b = (N/A) i the etiated value of the ontrol gain. he paraeter P (kpa) and P (kpa) are the eaured preure fro the firt and eond haber, repetively, f i the unknown nonlinear dynai, and = 3.85 kg i the eaured piton a. he aple period i.. With the feedbak ontrol gain K, the loed loop referene veloity dynai are piked a v r = ( 3 vr + 3 r ) (44) 3.85 A erie of open loop experient are onduted to enure referene yte i realizable. he ontrol law i ˆ I = Kv + kgr+ W φ Ke β tanh( α ) (45) e = v v r. he radial bai funtion φ ued for neural network truture i [ φ φ ] φ= φ(), v (), x φ3, φ 4, 5( I ), (46) () ( ) /, ( ) /, v z v z ( v z3) / v e σ σ σ e e φ = ( 4) / σ ( ) x z φ x = e ( P z5) / σ3 ( P z6) / σ3 ( P z7) / σ3 φ 3 = e, e, e ( P z8) / σ4 ( P z9) / σ4 ( P z) / σ6 φ 4 = e, e, e ( ) φ5 I = e ( I z) / σ 7 Here z = /, z = 5 /, z 3 = 5 /, z 4 = /, z5 = z8 = kpa, z6 = z9 = 4 kpa, z7 = z = 4 kpa, z = A, σ =, σ =, σ 3 = σ4 = σ5 = σ6 =, and σ 7 =.5. he enter and width of the Radial Bai Funtion (RBF) are eleted o that the neural network an etiate unertainty over the entire working region of the yte with iilar enitivity. Notie that a long a f i a ontinuou funtion of x, v, P, P, and I, the neural network approxiation auption (3) i valid. While frition i not ontinuou at origin of veloity, two eparate network an be ued for poitive and negative diretion to enable preie approxiation. he learning rate for the adaptive ontroller i eleted a Γ =, the oberver gain i K =, and the liding urfae i = e= v vr ine the odeled yte i firt order. he oft withing liding ode ontrol paraeter are α =5 and β =5. Inreaing Γ aue larger overhoot, while dereaing it inreae the error bound. Inreaing K inreae the error bound, while dereaing it inreae overhoot. he paraeter are tuned in order to dereae the teady tate error bound and obtain the bet poible tranient perforane. Inreaingα will inreae the feedbak ontroller enitivity to the traking error, and inreaing β an inreae the ontroller repone peed; however, when it i too large there will be ignifiant overhoot. Neural network weight are updated by the adaptive law (), with P =. o verify the feaibility of the propoed ontroller, take the oand input a r = 8 ign(in( π / 3) t) / (47) he reult, with oparion to previou work uing a well tuned PI ontroller, are hown in Fig.. 9

6 /e A /e veloity 3 Referene Propoed ethod PI e Control e Veloity erorr 5-5 Propoed ethod Propoed ethod e Fig.. Experiental reult for r = 8ign(in( π / 3) t) /. here i an initial inevitable delay (approxiately.8 ) for eah experient, due to the flow filling proe of the tet benh. Negleting the firt peak, the ontroller keep the veloity error bounded within 4.5 / during both teady and tranient tage. iregarding the firt tep, the 5% ettling tie for the eond, third and fourth tep are.5,.7, and.5 repetively. he differene i due to the nonlinearity and ayetry of the yte. A oparion, for PI ontroller, the teady tage error are greater than / for all tep, whih ean it doen t ettle under 5%. uring the tranient tage, the traking error inreae to a peak value of 4. / for the forward diretion and 4.4 / for the revere diretion, while for PI they are 4.7 / and 4.8 /. o u up, propoed ethod iproved repone peed, and provide better traking perforane during both tranient and teady tage. VI. SUMMARY AN CONCLUSION A new odel referene adaptive ontrol ethod ha been reated in thi paper. Bound on the tranient repone error have been derived. A novel liding ode ter ha been added, reulting in guaranteed ayptoti tability of the error, a oppoed to upper bound guarantee.. he ontroller i deigned and applied for veloity traking of an eletro hydrauli yte. Experiental reult illutrate that preie traking of the referene odel output i realized uing the adaptive ontroller for different ae. At the expene of relatively oplex truture, the ontroller ake it poible to ahieve ayptoti tability. Future reearh will fou on further inreaing robutne under ignifiant diturbane, and ore onvenient tuning tehnique. PI PI REFERENCES [] K. S. Narendra and K. Parthaarathy, Identifiation and ontrol of dynaial yte uing neural network, IEEE ran. Neural Netw., vol., no., pp. 4 7, Marh 99. [] R. M. Sanner and J. J. Slotine, Gauian network for diret adaptive ontrol, IEEE ran. Neural Netw., vol. 3, no. 6, pp , Noveber 99. [3] B. S Ki and A. J. Calie, Nonlinear flight ontrol uing neural network, AIAA J. Guidane Control, ynai, vol., no., pp. 6 33, eeber 997. [4] J. Leitner, A. Calie, and J. V. R. Praad, Analyi of adaptive neural network for heliopter flight ontrol, AIAA J. Guidane Control ynai, vol., no. 5, pp , Septeber 997. [5] M. B. MFarland, R.. Rydyk, and A. J. Calie, Robut adaptive ontrol uing ingle-hidden-layer feed-forward neural network, Pro. Aerian Control Conf., pp , 999. [6] J. A. Mue and A. J. Calie, H adaptive flight ontrol of the generi tranport odel, AIAA Infoteh@Aeropae, AIAA -333,. [7] F. L. Lewi, A. Yeildirek, and K. Liu, Multilayer neural net robot ontroller with guaranteed traking perforane, IEEE ran. Neural Netw., vol. 7, no., pp , Marh 996. [8] Z.. ydek and A. M. Annaway, Adaptive ontrol of quadrotor UAV in the preene of atuator unertaintie, AIAA Infoteh@Aeropae, AIAA -346,. [9]. E. Gibon and A. M. Annaway, Adaptive ontrol of hyperoni vehile in the preene of thrut and atuator unertaintie, AIAA Guidane, Navigation and Control Conferene and Exhibit, AIAA 8-696, 8. [] C. Cao and N. Hovakiyan, Novel L neural network adaptive ontrol arhiteture with guaranteed tranient perforane, IEEE ran. Neural Netw., vol. 8, no. 4, pp. 6-7, July 7. [] K. K.. Young, Controller deign for a anipulator uing the theory of variable truture yte, IEEE ran. on Syt. Man Cyber., vol. 8, no., pp. -9, February 978. [] J. J. E. Slotine, he Robut Control of Robot Manipulator, Int. J. Roboti Reearh, vol. 4, no., pp , June 985. [3] V. I. Utkin, Variable truture yte with liding ode, IEEE ran, on Autoati and Control, vol., no., pp. -, April 977. [4] S. E. Lyhevki, Control Syte heory with Engineering Appliation, Birkhäuer,. [5] Z. Cai, M. S. de Queiroz, and. M. awon, Robut adaptive ayptoti traking of nonlinear yte with additive diturbane, IEEE ran. Autoati Control, vol. 5, no. 3, pp , Marh 6. [6] R. Padhi, N. Unnikrihnan and S. N. Balakrihnan, Model-following neuro-adaptive ontrol deign for non-quare, non-affine nonlinear yte, IE Control heory Appl., vol., no. 6, pp , Noveber 7. [7]. G. Fenteraher, K. Krihnaurthy, R. G. Lander, and J.. Patel, evelopent of a novel eletro hydrauli laboratory, ASME International Mehanial Engineering Congre and Exhibition, Chiago, Illinoi, Noveber 5, 6. [8] J. B. Poet and L. Praly, Adaptive nonlinear regulation: Etiation fro the Lyapunov equation, IEEE ran. Auto. Control, vol. 37, no. 6, pp , June 99. 3