8 : 991 PE, (linear time-varying, LTV),., [9 13]..,,.,,,.,. (System model) n,, [1] M(q) q + V m (q, q) q + F ( q) + G(q) = τ, (1) : q, q, q R n ; M(q)

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1 DOI: /CTA Control Theory & Applications Vol. 30 No. 8 Aug. 013, (, ) :,.,, (RBF),., (PE). PE,.,,.,. : ; ; RBF; : TP73 : A Adaptive neural network control and learning for uncertain robot WU Yu-xiang, WANG Cong (College of Automation Science and Engineering, South China University of Technology, Guangzhou Guangdong , China) Abstract: This paper investigates the adaptive neural network control and learning for the electrically-driven robot with unknown system dynamics. A stable adaptive neural network (NN) controller is first designed, and the radial basis function (RBF) neural-network is used to approximate the unknown closed-loop system dynamics of electrically-driven robot. The stable adaptive tuning laws of network parameters are derived in the sense of the Lyapunov stability theory. Partial persistent excitation (PE) condition of some internal signals in the closed-loop system is satisfied in the control process of tracking a recurrent reference trajectory. Under the PE condition, the proposed adaptive NN controller is rigorously shown to be capable of accurate identification of the uncertain electrically-driven robot dynamics in the stable control process. Subsequently, a novel NN learning control method which effectively utilizes the learned knowledge without re-adapting to the unknown electrically driven robot dynamics is proposed to achieve the closed-loop stability and improve the control performance. Simulation studies are performed to demonstrate the effectiveness of the proposed method. Key words: adaptive NN control; robot; RBF neural network; learning 1 (Introduction),, [1 6]. [1]σ,,. [],. [3],. [5] (robust neural fuzzy network control, RNFNC),,. [6],,,.,., (persistent excitation, PE),,,,,., [7 8] (radial basis function, RBF),, : ; : xyuwu@scut.edu.cn; Tel.: : ( , ); (011A , 01A ); (01ZZ0106).

2 8 : 991 PE, (linear time-varying, LTV),., [9 13]..,,.,,,.,. (System model) n,, [1] M(q) q + V m (q, q) q + F ( q) + G(q) = τ, (1) : q, q, q R n ; M(q) R n n ; V m (q, q) R n n ; F ( q) R n ; G(q) R n ; τ R n. nn.,,. [14] τ(t) = K m i m (t), () : i m R n, K m R n n., i m (t) = i d (t) = K sa u(t), (3) : i d R n, K sa R n n,, u R n. τ(t) = K m K sa u(t) = Ku(t), (4) K = K m K sa. (1)(4) M(q) q + V m (q, q) q + F ( q) + G(q) = Ku, (5) M(q), V m (q, q), F ( q), G(q), K. 1 nn,, K = ki(k > 0), I. [6]. 1 M(q), m m 1 > 0, m 1 I M(q) m I, q R n. x 1 = q, x = q, (5) ẋ 1 = x, ẋ = A(x) + B(x)u, (6) : B(x)=M 1 K, x = [x T 1 x T ] T R n, A(x) = M 1 (V m x + F + G).. 11, B(x) 3 11, Ḃ 1 (x) = K 1 Ṁ, g 3 > 0, Ḃ 1 (x) g 3, x R n., x d Ω d (Ω d ), t 0, x di (i = 1, ) ϕ d (x d (0)). RBF : 1) ; ), ; 3), RBF ; 4),,,. x. 3 (Adaptive neural network control and learning) 3.1 (The design of adaptive neural network controller) (6)ϕ d (x d (0)), x d (t) = [q T d (t) q T d (t)] T, RBF : r: r = ė + λe, (7) : e = x 1 q d, λ. (7) ṙ = ẋ q d +λė = A(x)+B(x)u q d + λė. (8) u = K v r Ŵ T S(Z), (9) : K v = diag{k vi }, k vi > 0, Ŵ, S(Z) = diag{s 1 (Z),, S n (Z)}, Ŵ T S(Z) = [Ŵ T 1 S 1 (Z) Ŵ T n S n (Z)] T. Ŵ T S(Z)

3 99 30 F G (Z) = B 1 (x)(a(x) + v) = W T S(Z) + ε(z), (10) : v = q d + λė, Z = [x T 1 x T v T ] T, W, W T S(Z) = [W1 T S 1 (Z) Wn T S n (Z)] T, ε(z), ε(z) < ε (ε > 0), Z Ω R 3n (Ω). Ŵ = Γ S(Z)r σγ Ŵ, (11) : Γ, σ. Lyapunov: (1) V = 1 rt B 1 r + 1 W T Γ 1 W. (1) V = r T K v r + r T ε σ W T Ŵ + 1 rt Ḃ 1 r. (13) r T ε ε + k 1 r k 1 ε σ W T Ŵ σ( k 1) W 0 < k 1 <, (13) + k 1 r. (14) k 1 + σ W k 1, (15) V ρ 1 V + δ 1, (16) K v1 = K v 1 diag{k 1} 1 g 3I. K v K v 1 diag{k 1} 1 g 3I > 0. ρ 1 = min{ λ min(k v1 ) λ max (B 1 ), σ( k 1) }, (17) λ max (Γ 1 ) δ 1 = σ W + ε k 1. (18) (6)(7)(9) (11) {ṙ = B(x)[ Kv r W T S(Z) + ε], Ŵ = W = Γ S(Z)r σγ Ŵ, (19) : W T S(Z)=[ W T 1 S 1 (Z) W T n S n (Z)] T, W = Ŵ W. 1 (19), x d (0) Ω d ϕ d (x d (0)), x(0) Ω 0 (Ω 0 ) Ŵ (0) = 0, : i) (19) ; ii), T, eė, t T. 1) (16), t, r W δ 1 λ min (K v1 ), W + ε k1 ( k 1 ) =: W. (0), r W, λr, r, q d, q d, x 1, x. Z, S(Z), u.,. ) LyapunovV r = 1 rt B 1 r, V r r T K v r + r W S(Z) + r ε + 1 rt Ḃ 1 r. (1) r ε k 1 r + ε k 1, r W S(X) k 1 r + W s k 1, : W (0) W, s S(X) ([15]). (1) V r ρ 1 V r + δ 1, () K v1 = K v diag{ k 1 } 1 g 3I. K v K v diag{ k 1 } 1 g 3I > 0. ρ 1 = λ min ( K v1 )/λ max (B 1 ), (3) δ 1 = ( W S + ε )/( k 1 ). (4), k 1 δ 1. () 0 V r (t) < V r (0) exp( ρ 1 t) + δ 1 / ρ 1, (5) 1 λ min(b 1 ) r < V r (0) exp( ρ 1 t) + δ 1 / ρ 1. β > δ 1 /( ρ 1 λ min (B 1 )) T, t T, r r < β. β. K v r, t T.,, eėt [16].. 3. (Learning from adaptive neural network control) RBF, T, ϕ(x(t)) tt, (19)

4 8 : 993 {ṙ = B(x)[ Kv r W ξ T S ξ (Z) + ε ξ ], (31), ϕ(x(t)), (6) Ŵ ξ = Wξ = Γ ξ S ξ (Z)r σγ ξ Ŵ ξ, d, ε > 0, ϕ d (x d (0)) Ŵ ξ = W Ω ϕd, [8] ξ = Γ ξs ξ(z)r σγ ξŵ ξ, (7) dist(ϕ(z(t)) tt1, ϕ d (x d (0))) < d T : W ξ S ξ (Z) = [ W 1ξS T T 1ξ (Z) W jξ S jξ (Z)] T, W T S(Z) F G (Z) < ε, (3) S ξ (Z)S(Z), Ŵ ξ, ξϕ(x(t)) tt, ε ξ, ε ξ. (6), ϕ d (x d (0)), x d (0) Ω d, x(0) Ω 0 Ŵ (0) = 0,, : ϕ(x(t)) tt, ŴξWξ, F G (Z)Ŵ T S(Z) W T S(Z), W : W = mean (Ŵ (t)). (8) t [t a,t b ] [t a, t b ](t b > t a > T 1 > T )Ŵ. z = B 1 (x)r, (6) ż = [K v B(x) Ḃ 1 (x)b(x)]z W T ξ S ξ (Z) + ε ξ (Z), W ξ = Γ ξ S ξ (Z)B(x)z σγ ξ Ŵ ξ. (9) [ ] [ ] ż K v B(x) + = W Ḃ 1 (x)b(x) Sξ T (Z) ξ Γ ξ S ξ (Z)B(x) 0 [ ] [ ] z ε ξ +. (30) W ξ σγ ξ Ŵ ξ 1, T, Z(t), S ξ (Z)PE [8]. [17]1(30). (30), [18] 4.6, W ξ T 1 (T 1 > T ),. [13], F G (Z)RBF Ŵ T S(Z) W T S(Z), F G (Z) = Ŵ T ξ S ξ (Z) + Ŵ T ξ S ξ(z) + ε ξ = W T ξ S ξ (Z) + W T ξ S ξ(z) + ε ξ, (31) ε ξ ε ξ, RBF Ŵ T S(Z) W T S(Z)ϕ(x(t)) tt F G (Z).. ε ε. 1, PE,,,.,. 4 (Learning control ) 4.1 (The design of learning controller) (6), (9): u = K v r W T S(Z), (33) : W T S(Z)F G (Z), W (8), K v r (6), (33), x d (0) Ω d ϕ d (x d (0)), x(0) Ω 0 Ŵ (0) = 0, : i) (19) ; ii), T, eė, t T. r ṙ = B(x)[ k v r W T S(Z) + F G (Z)]. (34) LyapunovV x = 1 rt B 1 r. r(f G (Z) W T S(Z)) F G (Z) W T S(Z) + k p r. k p k p > 0, V x ρ V x + δ, (35) K v = K v 1 diag{k p} 1 g 3I. K v K v 1 diag{k p} 1 g 3I > 0, δ = ε /(k p ), ρ = λ min ( K v )/λ max (B 1 ). (36) β 1 = δ /ρ,, ε, δ β 1. (35)

5 r < β 1. (37) 1,. (36), k p, r. T > 0, t T, eė.. 4. (The analysis of system control performance) (9)(11) (33). i) : (9)(11), PE,, W,. eė, k 1 (K v ), δ 1., k 1,. ) : (33),, k p (K v ) δ β 1, eė.,,,,.,,,,.,,,. 5 (Simulation), [5] a + b cos q c + b M(q) = cos q c + b, cos q c V m (q, q) q = b sin q ( q 1 q q ) b sin q, q 1 [ ] d cos q 1 + e cos(q 1 + q ) G(q) =, e cos(q 1 + q ) [ ] 0.4 q 1 F ( q) =, 0.4 q : a = l m + l 1(m 1 + m ), b = l 1 l m, c = l m, d = l 1 (m 1 + m )g, e = l m g. q 1, q, m 1, m, l 1, l g1 ; l 1 = l = 1.0 m, m 1 = 0.8 kg, m =.3 kg; [19] F (q) = [1 q sgn q 1 1 q + 0.5sgn q ] T,, K = diag{10, 10}. q d = (q d11, q d1 ) T = (0.8 sin t, 0.8 cos t) T., : N = 4096, [ 1., 1.] [ 1., 1.] [ 1., 1.] [ 1., 1.] [ 1., 1.] [ 1., 1.], η = 0.85, Γ = diag{10}, λ = diag{30}, K v = diag{50}, σ = 0.001, Ŵ (0) = 0. X(0) = [ ; 0 0]. 1(a) (b),, 3(a) (b), 4(a) (b)., : λ = diag{10}, K v = diag{30}. 1(c) (d), 3(c) (d), 4(c) (d).,,,,. (a) Ŵ T S(X)q 1 (b) Ŵ T S(X)q

6 第8期吴玉香等: 不确定机器人的自适应神经网络控制与学习 (c) 使用W T S(X)时q1 的跟踪误差 (d) 使用W T S(X)时q 的跟踪误差图 1 状态跟踪误差 Fig. 1 The status tracking error (a) 部分权值W 1ξ 收敛 (a) 函数FG1 (Z)逼近 (b) 函数FG (Z)逼近 (c) 函数FG1 (Z)逼近 (b) 部分权值W ξ 收敛 (d) 函数FG (Z)逼近图部分权值收敛曲线 Fig. The curve of partial parameter convergence 图 3 函数逼近曲线 Fig. 3 The curve of function approximation 995

7 (Conclusions). RBF, PE,..,,, (a) Ŵ T S(X)u 1. References): [1] LEWIS F L, YEGILDIREK A, LIU K. Multilayer neural-net robot controller with guaranteed tracking performance [J]. IEEE Transactions on on Neural Networks, 1996, 1(): [],,,. [J]., 008, 5(1): (YU Jiancheng, LI Qiang, ZHANG Aiqun, et al. Neural network adaptive control for underwater vehicles[j]. Control Theory & Applications, 008, 5(1): 9 13.) (b) Ŵ T S(X)u [3],,. [J]., 011, 8(9): (ZHANG Wenhui, QI Nailiang, YIN Hongliang. Neural-network tracking control of space robot based on sliding-mode variable structure [J]. Control Theory & Applications, 011, 8(9): ) [4] ZUO Y, WANG Y N, AL E. Neural network robust Htracking control strategy for robot manipulator [J]. Applied Mathematical Modelling, 010, 34(7): [5] WAI R J, CHEN P C. Robust neural-fuzzy-network control for robot manipulator including actuator dynamics [J]. IEEE Transactions on Industrial Electronics, 006, 53(4): [6] CHANG Y C, YEN H M. Robust tracking control for a class of uncertain electrically driven robots [J]. IET Control Theory & Applications, 009, 3(5): [7] WANG C, HILL D J. Learning form neural control [J]. IEEE Transactions on Neural Networks, 006, 17(1): (c) W T S(X)u 1 [8] WANG C, HILL D J. Deterministic Learning Theory for Identification, Recognition and Control [M]. Boca Raton: CRC Press, 009. [9] WANG C, CHEN T R. Rapid detection of small oscillation faults via deterministic learning [J]. IEEE Transactions on Neural Networks, 011, (8): [10] WANG C, WANG M, LIU T F, et al. Learning from ISS-Modular adaptive NN control of nonlinear strict-feedback systems[j]. IEEE Transactions on Neural Networks and Learning Systems, 01, 3(10): [11] DAI S L, WANG C, LUO F. Identification and learning control of ocean surface ship using neural networks [J]. IEEE Transations Industria Informatics, 01, 8(4), (d) W T S(X)u 4 Fig. 4 The curve of control input [1],. [J]., 013, 39(6): (WU Yuxiang, Wang Cong. Deterministic learning based adaptive network control of robot in task space [J]. Acta Automation Sinica, 013, 39(6): ) [13],,. [J]., 01, 7(11):

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28 31,,., [11] ; [12 14],,,,.,,,..,,,,., :,,, (extended state observer, ESO),,.,,,,.,, ;,,., Lyapunov,. 2 (Problem description) 1,,. ( ) θ L (t) = ω L

28 31,,., [11] ; [12 14],,,,.,,,..,,,,., :,,, (extended state observer, ESO),,.,,,,.,, ;,,., Lyapunov,. 2 (Problem description) 1,,. ( ) θ L (t) = ω L 31 1 2014 1 DOI: 10.7641/CTA.2014.30591 Control Theory & Applications Vol. 31 No. 1 Jan. 2014,,, (, 210094) :,., :, ;,., ; Lyapunov ;,,. : ; ; ; ; : TP273 : A Fault diagnosis and fault tolerant control