Adaptive Control 2011 Part 15 doc

Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 343 Assumption 4. The approximation error ε is bounded as follows: N ε ε ≤ , (15) where 0 N ε > is an unknown constant. Let ˆ M and ˆ N be the estimates respectively of M and N . Based on these estimates, let ad u be the output of the NN ˆˆ (). TT ad nn uMNx σ = (16) Define ˆ M MM=− % and ˆ NNN = − % , where we use notations: [,] Z diag M N = , [,] Z dia g MN= %%% , ˆˆˆ [,] Z diag M N= for convenience. Then, the following inequality holds: 2 ˆ () T F F F tr Z Z Z Z Z≤− %% % . (17) The Taylor series expansion of () T nn Nx σ for a given nn x can be written as: 2 ˆˆ ()()() () TT TTT nn nn nn nn nn Nx Nx Nx Nx ONx σσσ ′ =+ + %% , (18) with ˆ ˆ :( ) T nn Nx σσ = and ˆ σ ′ denoting its Jacobian, 2 () T nn ON x % the term of order two. In the following, we use notations: :( ) T nn Nx σσ = , :( ) T nn Nx σσ = % % . With the procedure as Appendix A, the approximation error of function can be written as ˆˆ ˆ ˆ ˆˆ ˆ () ()( ) TT TT T T TT nn nn nn nn M Nx M Nx M Nx M Nx σ σσσσω ′′ − =− + + %% , (19) and the disturbance term ω can be bounded as 1 ˆˆ ˆˆ TT nn nn F F NxM M Nx M ωσσ ′′ ≤++ , (20) where the subscript “F” denotes Frobenius norm, and the subscript “1” the 1-norm. Redefine this bound as ˆˆ (,, ) nn M Nx ωω ωρϑ ≤ , (21) Adaptive Control 344 where 1 max{ , , } F MN M ω ρ = and ˆˆ ˆˆ 1 TT nn nn F xM Nx ω ϑσσ ′′ = ++ . Notice that ω ρ is an unknown coefficient, whereas ω ϑ is a known function. 3.2 Parameters update law and stability analysis Substituting (14) and (16) into (13), we have ˆˆ ˆ () () (). TT TT nn nn r nn kM Nx M Nx v x ττ σ σ ψ δε =− + − + − − + & (22) Using(19), the above equation can become ˆˆ ˆ ˆˆ ˆ () . TTTT nn nn r kM Nx MNx v τ τσσ σ ψδωε ′′ =− + − + + − − + + %% & (23) Theorem 1. Consider the nonlinear system represented by Eq. (2) and let Assumption 1-4 hold. If choose the approximation pseudo-control input ˆ ψ as Eq.(12), use the following adaptation laws and robust control law 1 1 ˆˆ ˆˆ () , ˆˆˆ ˆ , (1) ˆˆ (1)tanh (1) ˆ (1)tanh nn T nn r MF Nx kM NRxM kN v ω ω ω ω σσ τ τ στ τ τϑ φ γτϑ λφ α τϑ φϑ α ⎡⎤ ′ =− − ⎣⎦ ⎡⎤ ′ =− ⎣⎦ ⎧ +⎫ ⎡⎤ =+ − ⎨ ⎬ ⎢⎥ ⎣⎦ ⎩⎭ + ⎡⎤ =− + ⎢⎥ ⎣⎦ & & & (24) where 0, 0 TT FF RR=> => are any constant matrices, 1 0k > and 0 γ > are scalar design parameters, ˆ φ is the estimated value of the uncertain disturbance term max( , ) N ω φρ ε = , defining ˆ φ φφ = − % with φ % error of φ , then, guarantee that all signals in the system are uniformly bounded and that the tracking error converges to a neighborhood of the origin. Proof. Consider the following positive define Lyapunov function candidate as 21 112 11 1 1 ()() 22 2 2 TT LtrMFMtrNRN τ γφ −−− =+ + + % %% %% (25) The time derivative of the above equation is given by 111 ()() TT LtrMFMtrNRN τ τ γφφ −−− =+ + + & && %% &%%%% & (26) Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 345 Substituting (23) and the anterior two terms of (24) into (26), after some straightforward manipulations, we obtain 2 111 21 1 21 1 ˆˆ ˆ ˆˆ ˆ [( ) ( ) ] ()() ˆ ˆ () () (). ˆ ˆ () (1) (). TTTT nn nn r TT T r T r Lk M Nx MNx v tr M F M tr N R N kv ktrZZ kv ktrZZ ω τ τσσ σ ψδ ωε γφφ ττψδτ τωεγφφ τ ττψδτ τφϑ γφφ τ −−− − − ′′ =− + − + + − − + + +++ =− + − − + + + + ≤− + − − + + + + &% % & && %% %% %% & %% % & %% % (27) With (4),(6),(12),(16) and the last two equations of (24), the approximation error between actual approximation inverse and ideal control inverse is bounded by 12 3 ˆ , F cc cZ ψδ τ −≤+ + % (28) where 123 ,,cccare positive constants. Using (11) and the last two terms of (24), we obtain 2 1 2 1 (1) ˆ ˆ ()(1)tanh (1) ˆ ˆ (1) (1)tanh () ˆ ˆ ˆ () () T T Lk ktrZZ kktrZZ ω ω ω ωω τϑ ττψδτφϑ α τϑ τφϑ φ τϑ λφ τ α ττψδςφαλφφ τ + ⎡⎤ ≤− + − − + ⎢⎥ ⎣⎦ ⎧+⎫ ⎡⎤ ++− + −+ ⎨⎬ ⎢⎥ ⎣⎦ ⎩⎭ ≤− + − + + + & % % % % (29) Applying (17),(28) , and 2 ˆ φ φφφφ ≤− %% % , after completing square, we have the following inequality 2 212 ()Lkc D D ττ ≤ −− + + & (30) where 22 3 1 11 2 1 1 (), 44 M c k Dc Z D k λ φςφα =+ + = + . Let 2 31221 4( )DDDkcD=+ −+ , thus, as long as 32 [2( )] D kc τ ≥− , and 2 kc> , then 0L ≤ & holds. Now define Adaptive Control 346 {} 13 3 12 11 ,(), . 2( ) ZM F ZZ kZ c D kkc φ τ φφφ ττ ⎧ ⎫⎧ ⎫ ⎪ ⎪⎪ ⎪ Ω= ≤ Ω= ≤ + Ω= ≤ ⎨ ⎬⎨ ⎬ − ⎪ ⎪⎪ ⎪ ⎩⎭⎩⎭ %% %% (31) Since 1 1 2 323 ,,, , , ,, M Z kkDDDcc are positive constants, as long as k is chosen to be big enough, such that 2 kc> holds, we conclude that , Z φ Ω Ω and τ Ω are compact sets. Hence L & is negative outside these compacts set. According to a standard Lyapunov theorem, this demonstrates that , Z φ % % and τ are bounded and will converge to , Z φ ΩΩand τ Ω , respectively. Furthermore, this implies e is bounded and will converge to a neighborhood of the origin and all signals in the system are uniformly bounded. 3.3 Simulation Study In order to validate the performance of the proposed neural network-based adaptive control scheme, we consider a nonlinear plant, which described by the differential equation 12 22322 21 12 12 0.02( ) ( ) ( ) tanh(0.2 ) xx x xxxuxxuud ωω σ = =− − + + + + + + & & (32) where 0.4 ω π = , () (1 )(1 ) uu ue e σ −− =− + and 0.2d = . The desired trajectory 0.1 [sin(2 ) cos( )] d x tt π =− . To show the effectiveness of the proposed method, two controllers are studied for comparison. A fixed-gain PD control law is first used as Polycarpou, (Polycarpou 1996). Then, the adaptive controller based on NN proposed is applied to the system. Input vector of neural network is ˆ [1, , , ] TT nn d xxe ψ = , and number of hidden layer nodes 25. The initial weight of neural network is ˆˆ (0) (0), (0) (0)MN== . The initial condition of controlled plant is (0) [0.1,0.2] T x = . The other parameters are chosen as follows: 1 0.01, 0.1, 0.01, 10k γ λα === = , 2, 8 M F IΛ= = , 5 N R I = , with , M N II corresponding identity matrices. Fig.1, 2, and 3 show the results of comparisons, the PD controller and the adaptive controller based on NN proposed, of tracking errors, output tracking and control input, respectively. These results indicate that the adaptive controller based on NN proposed presents better control performance than that of the PD controller. Fig.4 depicts the results of output of NN, norm values of ˆˆ , M N , respectively, to illustrate the boundedness of the estimates of ˆˆ , M N and the control role of NN. From the results as figures, it can be seen that the learning rate of neural network is rapid, and tracks objective in less than 2 seconds. Moreover, as desired, all signals in system, including control signal, tend to be smooth. Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 347 0 5 10 15 20 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 Tracki ng error time sec Fig. 1. Tracking errors: PD(dot) and NN(solid). 0 5 10 15 20 -0.4 -0.2 0 0.2 0.4 0.6 Output tracki ng time sec Fig. 2. Output tracking: desired (dash), NN(solid) and PD(dot). 0 5 10 15 20 -1.5 -1 -0.5 0 0.5 1 1.5 Control input time sec Fig. 3. Control input: PD (dash), NN(solid) 0 5 10 15 20 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 time sec Fi g . 4. ˆ M ( dash ) , ˆ N ( dot ) , out p ut of NN ( solid ) Adaptive Control 348 4. Decentralized Adaptive Neural Network Control of a Class of Large-Scale Nonlinear Systems with linear function interconnections In the section, the above proposed scheme is extended to large-scale decentralized nonlinear systems, which the subsystems are composed of the class of the above-mentioned non-affine nonlinear functions. Two schemes are proposed, respectively. The first scheme designs a RBFN-based adaptive control scheme with the assumption which the interconnections between subsystems in entire system are bounded linearly by the norms of the tracking filtered error. In another scheme, the interconnection is assumed as stronger nonlinear function. We consider the differential equations in the following form described, and assume the large-scale system is composed of the nonlinear subsystems: 12 23 12 12 1 (, ,, ,) (,,,) 1, 2, , ii ii il i i i ili i i n ii i xx xx x fx x x u gxx x yx in ⎧ = ⎪ = ⎪ ⎪ ⎨ ⎪ =+ ⎪ ⎪ = ⎩ = & & M & LL L (33) where i l i x R∈ is the state vector, 12 [, ,, ] i T iii il x xx x= L , i uR ∈ is the input and i y R∈ is the output of the ith − subsystem. 1 (, ): li iii f xu R R + → is an unknown continuous function and implicit and smooth function with respect to control input i u . Assumption 5. (,)/ 0 iii i fxu u∂∂≠for all (,) ii i x uR ∈ Ω× . 12 (, , , ) in gxx xL is the interconnection term. In according to the distinctness of the interconnection term, two schemes are respectively designed in the following. 4.1 RBFN-based decentralized adaptive control for the class of large-scale nonlinear systems with linear function interconnections Assumption 6. The interconnection effect is bounded by the following function: 12 1 (, , , ) n inijj j gxx x γ τ = ≤ ∑ L , (34) where ij γ are unknown coefficients, j τ is a filtered tracking error to be defined shortly . Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 349 The control objective is: determine a control law, force the output, i y , to follow a given desired output, di x , with an acceptable accuracy, while all signals involved must be bounded. Define the desired trajectory vector 1 [,,, ] i l T di di di di xyy y − = & L and () ,,, i T l di di di di Xyyy ⎡ ⎤ = ⎣ ⎦ & L , tracking error 12 [, , , ] i T iidi ii il exx ee e=− = L , thus, the filter tracking error can be written as (2) (1) ,1 ,2 , 1 [1] ii i ll T iiiiiii ili i ke ke k e e τ − − − =Λ = + + + + & Le , (35) where the coefficients are chosen such that the polynomial (2) ,1 ,2 , 1 i i l ii il kks ks − − +++L (1) i l s − + is Hurwitz. Assumption 7. The desired signal () di x t is bounded, so that di di X X≤ , where di X is a known constant. For an isolated subsystem, without interconnection function, by differentiating (35), the filtered tracking error can be rewritten as () [0 ] ( , ) i l l T i il di i i iii di x xefxuY τ = −+Λ = + && (36) with () [0 ] i l T di di i i Yx e=− + Λ . Define a continuous function iiidi kY δ τ = −− (37) where i k is a positive constant. With Assumption 5, we know (, ) 0 ii i fxu u ∂ ∂≠, thus, [(, ) ] 0 ii i i fxu u δ ∂−∂≠ . Considering the fact that 0 i i u δ ∂ ∂=, we invoke the implicit function theorem, there exists a continuous ideal control input i u ∗ in a neighborhood of (, ) ii i x uR ∈ Ω× , such that (, ) 0 ii i fxu δ ∗ − = , i.e. (, ) i iii f xu δ ∗ = holds. (, ) i iii f xu δ ∗ = represents ideal control inverse. Adding and subtracting i δ to the right-hand side of (, ) il i i i i i x fxu g = + & of (33), one obtains (,) i il iii i i ii di x fxu g k Y δ τ = +−− − & , (38) and yields Adaptive Control 350 (,) iiiiiiii kfxug τ τδ = −+ +− & . (39) In the same the above-discussed manner as equations (9)-(10) , we can obtain the following equation: ˆ ˆ (,) iiii f xu ψ = . (40) Based on the above conditions, in order to control the system and make it be stable, we design the approximation pseudo-control input ˆ i ψ as follows: ˆ i iidiciri kYuv ψ τ = −−++, (41) where ci u is output of a neural network controller, which adopts a RBFN, ri v is robustifying control term designed in stability analysis. Adding and subtracting ˆ i ψ to the right-hand side of (39), with (, ) iiidiiii kYfxu δτ ∗ =− − = , we have ˆ (,, ) i ii iiii ci i i ri i kxuuu vg ττ ψδ ∗ = −+Δ −+−−+ % & , (42) where (,, ) (, ) (, ) iiii iii iii x uu fxu fxu ∗ ∗ Δ=− % is error between nonlinear function and its ideal control function, we can use the RBFN to approximate it. 4.1.1 Neural network-based approximation Given a multi-input-single-output RBFN, let 1i n and 1i m be node number of input layer and hidden layer, respectively. The active function used in the RBFN is Gaussian function, 2 2 ( ) exp[ 0.5( ) / ] llkki Sz μ σ =− −x , 1 1, , i ln = ⋅⋅⋅ , 1 1, , i km= ⋅⋅⋅ where 1 1 i n i Rz × ∈ is input vector of the RBFN, 11ii nm i R μ × ∈ and 1 1 i m i R σ × ∈ are the center matrix and the width vector. Based on the approximation property of RBFN, (,, ) iiii x uu ∗ Δ % can be written as (,, ) (, , ) () T iiii iii i i ii S x uu W z z μσ ε ∗ Δ= + % , (43) where () ii z ε is approximation error of RBFN, 1 1 i m i WR × ∈ . Assumption 8. The approximation error () nn x ε is bounded by iNi ε ε ≤ , with 0 Ni ε > is an unknown constant. The input of RBFN is chosen as ˆ [,,] TT iiii zx τ ψ = . Moreover, output of RBFN is designed as Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 351 ˆ ˆˆ (, , ). T ci i i i i i SuW z μ σ = (44) Define ˆ ˆˆ ,, iii W μ σ as estimates of ideal ,, iii W μ σ , which are given by the RBFN tuning algorithms. Assumption 9. The ideal values of ,, iii W μ σ satisfy ,, iiM i iM iiM F WW μ μσσ ≤≤≤, (45) where ,, iM iM iM W μ σ are positive constants. F ⋅ and ⋅ denote Frobenius norm and 2- norm, respectively. Define their estimation errors as ˆ ˆˆ ,,. iii iii iii WWW μ μμ σσσ = −=−=− % %% (46) Using the notations: ˆˆ ˆˆ [,,], [,,], [,,] iiiiiiiiiiii ZdiagW ZdiagW ZdiagW μ σμσμσ === %% %% for convenience. The Taylor series expansion for a given i μ and i σ is 2 ˆˆ ˆˆ (, , ) (, , ) ( , ) ii i i ii i i ii ii i i SSSSzz O μσ μ σμσμσμσ ′′ =+++ %%%% (47) where ˆˆ ˆˆ ˆˆ (, , ) , (, , ) ikiiii ikiiii SSSz Sz μσ μ σμ μσσ ′′ ∂∂∂∂ evaluated at ˆ ii μ μ = , ˆ ii σ σ = , 2 (, ) ii O μ σ %% denotes the terms of order two. We use notations: ˆ ˆˆ :(,,), iiiii SSz μ σ = :(,,) iiiii SSz μ σ = % %% , :(,,) iiiii SSz μ σ = . Following the procedure in Appendix B, it can be shown that the following operation. The function approximation error can be written as ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ()()(), TT T T ii ii i i ii ii i ii ii i SS SSWS WS W S W t μσ μσ μσ μσω ′′ ′′ −= −−+ + + % %% (48) The disturbance term () i t ω is given by ˆˆ ˆˆ ˆ ˆ ˆˆ ()()()() TT T iiiiiiiiiiiiii SS SStWSS W W μσ μσ ω μσ μσ ′′ ′′ =−+ +− + (49) Then, the upper bound of () i t ω can be written as 1 ˆˆ ˆ ˆˆˆ ˆˆ () ( ) 2 TT iiiiiiiiiiiiiii F FF FF SS S StW W W W μ σμ σ ωω ω μσ μ σ ρϑ ′′ ′ ′ ≤++ ++≤ (50) Adaptive Control 352 where 1 max( , , , 2 ) iiiii F WW ω ρμσ = , ˆˆ ˆ ˆˆˆ ˆˆ 1 TT iii ii ii ii FF FF SS S SWW ωμ σ μ σ ϑμ σ ′′ ′ ′ = ++ + + , with 1 ⋅ 1 norm. Notice that i ω ρ is an unknown coefficient, whereas i ω ϑ is a known function. 4.1.2 Controller design and stability analysis Substituting (43) and (44) into (42), we have ˆ ˆ ˆ () TT iiiiiiiiiriiii kWSWS v g z ττ ψδ ε =− + − + − − + + & , (51) using (48), the above equation can become ˆˆ ˆˆ ˆ ˆ ˆˆ ()() ˆ () (). TT iiiiiiiiiiiiii iiri iii i SS SSkWS W vg z t μσ μσ τ τμσμσ ψδ ε ω ′′ ′′ =− + − − + + +−−++ + % &%% (52) Theorem 2. Consider the nonlinear subsystems represented by Eq. (33) and let assumptions hold. If choose the pseudo-control input ˆ i ψ as Eq.(41), and use the following adaptation laws and robust control law ˆˆ ˆ ˆˆ ˆˆ () iii ii iiiWiii SSWFS W μσ μ στ γ τ ′′ ⎡ ⎤ =−− − ⎣ ⎦ & , (53) ˆ ˆ ˆˆ T iiiiiWiii SGW μ μ τγμτ ′ ⎡ ⎤ =− ⎣ ⎦ & , (54) ˆ ˆ ˆˆ T iiiiiWiii SHW σ σ τγστ ′ ⎡ ⎤ =− ⎣ ⎦ & , (55) * * ˆˆ tanh( ) ii iiii iii i ω φω φ τϑ φ γτϑ λφτ α ⎡ ⎤ =− ⎢ ⎥ ⎣ ⎦ & , (56) 2 ˆˆ () idii diii dd γ τλτ =− & , (57) * * ˆ ˆ tanh( ) ii ri i i i i i vd ω ω τϑ φ ϑτ α =+, (58) [...]... 10 time sec Fig 6 Control input of two subsystems: 1(solid), 2(dot) 15 20 Adaptive Control 358 0.6 x1 d1 1 ,x 1 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0 5 10 time sec Fig 7 Comparison of the tracking of subsystem 1: 15 20 x11 (solid) and xd 11 (dot) 0.6 x1 d1 2 ,x 2 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0 5 10 time sec Fig 8 Comparison of the tracking of subsystem 2: 15 20 x21 (solid) and xd 21 (dot) 15 10 5 0 -5 -10... Fig 9 Subsystem 1: Output of RBFN (solid), norms of dot) 15 20 ˆ ˆ ˆ W (dash), μ (dot), σ (dash- Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 359 15 10 5 0 -5 0 5 10 time sec 15 20 ˆ ˆ ˆ W (dash), μ (dot), σ (dash- Fig 10 Subsystem 2: Output of RBFN (solid), norms of dot) 4.2 RBFN-based decentralized adaptive control for the class of large-scale nonlinear systems with... results of control inputs, after shortly shocking, they tend to be smoother, and this is because neural networks are unknown for objective in initial stages As desired, though the system is complex, the whole running process is well Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 3 u1 u2 2.5 2 1.5 u ,u 1 2 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0 5 10 time sec 15 20 Fig 12 Control. .. error (98) can be rewritten as & ˆ Li ≤ −kiτ i2 + τ i (δ i −ψ i ) − vriτ i + φi (τ i2 + τ i ) %& % % ˆ % ˆ + λφ−i1φiφi + γ WiWiTWi τ i + γ giWgiTWgi τ i Applying the adaptive law (56) and robust control term (58), we have (99) Adaptive Control 364 ˆ & ˆ Li ≤ − kiτ i2 + τ i (δ i −ψ i ) − φiτ i ( τ i + 1) tanh(τ i α i ) + φi τ i ( τ i + 1) % %ˆ % ˆ % ˆ − φiτ i ( τ i + 1) tanh(τ i α i ) + γ WiWiT Wi τ... ˆ = f i ( xi , ui ) holds Based on the above conditions, in order to control the system and make it be stable, we ˆ design the approximation pseudo -control input ψ i as follows: ˆT ˆ ψ i = −kiτ i − Ydi − uci − Wgi S gi (| τ i |)τ i − vri , where uci (82) is output of a neural network controller, which adopts a RBFN, robustifying control term designed in stability analysis, ˆ W S gi (| τ i |) T gi vri... τ&i = −kiτ i + Δ i ( xi , ui , ui ∗ ) − uci − Wgi S gi (| τ i |)τ i + δ i −ψ i − vri + gi , (83) Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks where % Δ i ( xi , ui , ui ∗ ) = fi ( xi , ui ) − f i ( xi , ui ∗ ) is 361 error between the nonlinear function and its ideal control function, we can use the RBFN to approximate it 4.2.1 Neural network-based approximation... Wi (86) 4.2.2 Controller design and stability analysis Substituting (84) and (85) into (83), we have % ˆT ˆ τ&i = − kiτ i + WiT Si + δ i −ψ i − vri + gi − Wgi S gi (| τ i |)τ i + ε i ( zi ) (87) Theorem 3 Consider the nonlinear subsystems represented by Eq (33) and let assumptions ˆ hold If choose the pseudo -control input ψ i as Eq.(82), and use the following adaptation laws and robust control law &... bounded Furthermore, the learning rate of neural network controller is rapid, and can track the desired trajectory in about 1 second From the results of control inputs, after shortly shocking, they tend to be smoother, and this is because neural networks are unknown for objective in initial stages 0.4 e1 2 1 ,e 1 0.2 0 -0.2 -0.4 0 5 10 time sec 15 20 Fig 5 Tracking error of two subsystems: 1(solid),... )σ (u1 ) + 0.2 + sin(0.2 x21 ) ⎩ (74) & ⎧ x21 = x22 ⎪ 2 2 & 2 : ⎨ x22 = x21 + 0.1(1 + x22 )u2 + tanh(0.1u2 ) ⎪ + 0.15u23 + tanh(0.1x11 ) ⎩ (75) where ω = 0.4π , σ (u1 ) = (1 − e−u1 ) (1 + e−u1 ) The desired trajectory xd 11 = 0.1π [sin(2t ) − cos(t )] , xd 21 = 0.1π cos(2t ) Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 357 Input vectors of neural networks are zi... function: gi ( x1 , x2 ,L , xn ) ≤ ∑ j =1 ξij (| τ j |) , n where ξij (| τ j |) are unknown smooth nonlinear function, (76) τ j are filtered tracking errors to be defined shortly The control objective is: determine a control law, force the output, desired output, xdi yi , to follow a given , with an acceptable accuracy, while all signals involved must be bounded ( & X di = [ ydi , ydi ,L , ydili ) . comparisons, the PD controller and the adaptive controller based on NN proposed, of tracking errors, output tracking and control input, respectively. These results indicate that the adaptive controller. of the proposed method, two controllers are studied for comparison. A fixed-gain PD control law is first used as Polycarpou, (Polycarpou 1996). Then, the adaptive controller based on NN proposed. (dash), NN(solid) and PD(dot). 0 5 10 15 20 -1.5 -1 -0.5 0 0.5 1 1.5 Control input time sec Fig. 3. Control input: PD (dash), NN(solid) 0 5 10 15 20 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 time