Automatica. Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Mou Chen a,c, Shuzhi Sam Ge b,c,, Beibei Ren c

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1 Automatca 47 (0) Contents lsts avalable at ScenceDrect Automatca ournal homepage: Adaptve trackng control of uncertan MIMO nonlnear systems wth nput constrants Mou Chen a,c, Shuzh Sam Ge b,c,, Bebe Ren c a College of Automaton Engneerng, Nanng Unversty of Aeronautcs and Astronautcs, 006 Nanng, Chna b Insttute of Intellgent Systems and Informaton Technology & Robotcs Insttute, Unversty of Electronc Scence and Technology of Chna, 673 Chengdu, Chna c Department of Electrcal & Computer Engneerng, Natonal Unversty of Sngapore, 7576, Sngapore a r t c l e n f o a b s t r a c t Artcle hstory: Avalable onlne 3 February 0 Keywords: Nonlnear systems Input constrant Command flter Adaptve trackng control Backsteppng control In ths paper, adaptve trackng control s proposed for a class of uncertan mult-nput and mult-output nonlnear systems wth non-symmetrc nput constrants. The auxlary desgn system s ntroduced to analyze the effect of nput constrants, and ts states are used to adaptve trackng control desgn. The spectral radus of the control coeffcent matrx s used to relax the nonsngular assumpton of the control coeffcent matrx. Subsequently, the constraned adaptve control s presented, where command flters are adopted to mplement the emulate of actuator physcal constrants on the control law and vrtual control laws and avod the tedous analytc computatons of tme dervatves of vrtual control laws n the backsteppng procedure. Under the proposed control technques, the closed-loop sem-global unformly ultmate bounded stablty s acheved va Lyapunov synthess. Fnally, smulaton studes are presented to llustrate the effectveness of the proposed adaptve trackng control. 0 Elsever Ltd. All rghts reserved.. Introducton Durng the past several decades, adaptve control of nonlnear systems has receved much attenton for establshng the globally asymptotcal stablty of the closed-loop system (Ge, 996a,b; Ge & Wang, 003; Hung, Tuan, Narkyo, & Apkaran, 008; Krstć & Kokotovć, 995; Luo, Chu, & Lng, 005; Makoud & Radouane, 000; Mrkn & Gutman, 005; Sketnea, Fossen, & Kokotovć, 000; Tang, Tao, & Josh, 007; Yao & Tomzuka, 00; Yu & Sun, 00). In practce, most control plants are nonlnear, uncertan and multvarable n character. It s mportant to nvestgate effectve adaptve control technques for uncertan mult-nput and multoutput (MIMO) nonlnear systems. In Tang et al. (007), drect adaptve control was developed for a class of MIMO nonlnear systems n the presence of uncertan falures of redundant actuators. In Yao and Tomzuka (00), adaptve robust control was proposed for MIMO nonlnear systems n sem-strct feedback forms. Robust adaptve trackng control was developed for the Ths paper was not presented at any IFAC meetng. Ths paper was recommended for publcaton n revsed form by Assocate Edtor Raul Ordóñez under the drecton of Edtor Mroslav Krstc. Correspondng author at: Insttute of Intellgent Systems and Informaton Technology & Robotcs Insttute, Unversty of Electronc Scence and Technology of Chna, 673 Chengdu, Chna. Tel.: ; fax: E-mal addresses: chenmou@nuaa.edu.cn (M. Chen), samge@nus.edu.sg (S.S. Ge), helenren.ac@gmal.com (B. Ren). tme varyng uncertan nonlnear systems wth unknown control coeffcents (Ge & Wang, 003). As an effectve control technology, adaptve control has been successvely used n a varety of practcal control systems. In Ge (996a), adaptve control was proposed for robots wth both dynamc parameter uncertantes and unknown nput scalngs. Adaptve control for flexble ont robots was presented based on sngular perturbaton theory and poston nformaton n Ge (996b). Adaptve recursve desgn was developed for a parametrc uncertan nonlnear plant descrbng the dynamcs of a shp (Sketnea et al., 000). In Luo et al. (005), nverse optmal adaptve control was presented for the atttude trackng of spacecraft. Adaptve control was studed for nonlnearly parameterzed uncertantes n robot manpulators (Hung et al., 008). In the adaptve control of uncertan MIMO nonlnear systems, one man challenge s the possble sngularty of the control coeffcent matrx whch makes the control desgn become more complcated. Exstng research results of adaptve control technques for the MIMO nonlnear system mostly assume that the control coeffcent matrx s known and nonsngular (Kwan & Lews, 000). In ths paper, the spectral radus of the control coeffcent matrx s ntroduced n the control desgn to relax the nonsngular assumpton of the control coeffcent matrx. Snce actuator physcal constrants can severely degrade the closed-loop system performance, control desgn for uncertan MIMO nonlnear systems wth actuator constrants presents a tremendous challenge. Durng the past decades, there has extensve research on the control of mechancal systems wth varous /$ see front matter 0 Elsever Ltd. All rghts reserved. do:0.06/.automatca

2 M. Chen et al. / Automatca 47 (0) constrants. Analyss and desgn of control systems wth nput saturaton constrants have been studed n Cao and Ln (003), Chen, Ge, and Choo (009), Chen, Ge, and How (00), Hu, Ma, and Xe (008), Gao and Selmc (006) and Zhong (005). To handle the physcal lmtaton, constraned adaptve backsteppng control was proposed n whch command flters were used to mplement the emulate of constrants on the control command and the vrtual control laws (Farrell, Polycarpou, & Sharma, 003; Polycarpou, Farrell, & Sharma, 004, 003; Sonneveldt, Chu, & Mulder, 007). In Polycarpou et al. (004), nonlnear approxmaton based backsteppng control was presented for nonlnear dynamcal systems subect to magntude, rate, and bandwdth constrants. The control nput saturaton was nvestgated va on-lne approxmaton based control for uncertan nonlnear systems (Polycarpou et al., 003). Constraned adaptve backsteppng control was presented for fghter arcraft n Sonneveldt et al. (007). In the constraned adaptve control, the key problem s how to analyze the constrant effect of the actuator s physcal constrants. To ths end, we ntroduce an auxlary desgn system to analyze the constrant effect n ths paper. Based on the states of the auxlary desgn system, constraned adaptve control s nvestgated for a class of uncertan MIMO nonlnear systems wth nput constrants usng backsteppng technque. Backsteppng control has became one of the most popular robust adaptve control desgn technques for some specal classes of nonlnear systems (Gong & Yao, 00; Wang & Huang, 005; Zhang, Ge, & Hang, 000). In recent years, the unversal approxmaton ablty of neural network (NN) or fuzzy logcal system (FLS) has been employed to desgn robust adaptve control combng wth backsteppng technque for the uncertan MIMO nonlnear systems, and varous robust adaptve control strateges have been proposed (Chang, 000, 00; Chang & Yen, 005; Ge, 998; Ge & Wang, 004; Ge & Tee, 007; Ge, L, Zhang, & Lee, 004; Ge, Zhang, & Lee, 004; Lee & Lee, 004; Zhang, Ge, & Lee, 005). The proposed robust adaptve control based on NN or FLS s an effcent control approach of MIMO nonlnear systems, but the model-based adaptve control should be wdely developed due to the relatvely easy realzaton (Narendra & Annaswamy, 989; Qu, Dorsey, & Dawson, 994). Furthermore, the adaptve backsteppng control of uncertan MIMO nonlnear systems wth non-symmetrc nput constrants need to be further nvestgated. In ths paper, adaptve trackng control s proposed to handle the nput saturaton and actuator physcal constrants for uncertan MIMO nonlnear systems. The man contrbutons of the paper are as follows: () To the best of our knowledge, t s the frst tme n the lterature that the non-symmetrc nonlnear nput saturaton constrant s consdered for the adaptve trackng control of uncertan MIMO nonlnear systems. () The spectral radus of the control coeffcent matrx s employed n the control desgn to relax the nonsngular assumpton of the control coeffcent matrx. () To handle the non-symmetrc nput saturaton constrant, the auxlary desgn system s ntroduced to analyze the effect of nput constrants, and the states of auxlary desgn system are used to develop adaptve trackng control. (v) command flters are ntroduced to mplement the emulate of actuator physcal constrants on the control command and vrtual control laws, and avod the tedous analytc computatons of tme dervatves of vrtual control laws n the backsteppng procedure. The rest of the paper s organzed n the followng manner. Secton presents the problem formulaton and prelmnares. Adaptve trackng control s nvestgated for uncertan MIMO nonlnear systems wth nput saturaton n Secton 3, followed Fg.. Non-symmetrc nput saturaton constrant. by the constraned adaptve control consderng actuator physcal constrants n Secton 4. The smulaton results are presented to demonstrate the effectveness of proposed adaptve control n Secton 5. Secton 6 contans the concluson. Notatons: denotes for Frobenus norm of matrces and Eucldean norm of vectors,.e., gven a matrx B and a vector ξ, the Frobenus norm and Eucldean norm are gven by B = tr(b T B) =, b and ξ = ξ. x = [x, x,..., x ] T R m stands the vector of partal state varables n the nonlnear system. z For nteger ndces and, we defne Tanh(z ) := dag tanh, ε ε > 0, Ψ = [kε, kε,..., kε m ] T, k = 0.758, ρ ( x ) := dag{ρ ( x )} and Θ = [Θ, Θ,..., Θ m ] T. ˆθ and ˆΘ denote the estmates of uncertan parameter vectors θ and Θ, respectvely, and the estmate errors are defned as θ := ˆθ θ and Θ := ˆΘ Θ, =,,..., n and =,,..., m.. Problem formulaton Consder a class of uncertan MIMO nonlnear systems n the form of ẋ = F ( x )θ (G ( x ) G ( x ))x D ( x, t), =,,..., n... ẋ n = F n ( x n )θ n (G n ( x n ) G n ( x n ))u D n ( x n, t) y = x () where x R m, =,,..., n are the state vectors; θ R q, =,,..., n are the uncertan parameter vectors; F R m q, =,,..., n are known nonlnear functons; G R m m, =,,..., n are known control coeffcent matrces; D R m, =,,..., n are unknown tme-varyng dsturbances; u R m s the control nput vector; y R m s the system output vector; q are postve ntegers and G R m m, =,,..., n are unknown bounded perturbatons of control coeffcent matrces. Consderng actuator non-symmetrc nput constrants as shown n Fg., the control nput u = [u,..., u m ] T s defned by u rmax, f v > v rmax g u = r (v ), f 0 v v rmax () g l (v ), f v lmax v < 0 u lmax, f v < v lmax where v s th element of the desgned control law v = [v, v..., v m ] T, v lmax < 0, and v rmax > 0 are known constants; and g r (v ) and g l (v ) are smooth contnuous known nonlnear functons. To facltate control system desgn, the followng assumptons and lemmas are presented and wll be used n the subsequent developments. Assumpton (Zhang & Ge, 007, 008). There exsts postve constants k l0, k l, k r0 and k r such that

3 454 M. Chen et al. / Automatca 47 (0) < k r0 g r (v ) k r, v [0, v rmax ] (3) 0 < k l0 g l (v ) k l, v [v lmax, 0). (4) Assumpton (Tee & Ge, 006). For the dsturbance terms ( x, t) R m R, D ( x, t), =,,..., n; =,,..., m, there exst known smooth functons ρ ( x ) R, t > t 0 and unknown bounded constants Θ such that D ( x, t) ρ ( x )Θ. (5) Assumpton 3 (Km & Ha, 000). For all known control coeffcent matrces G ( x ), =,,..., n of the uncertan nonlnear system (), there exst known postve constants ζ > 0 such that G ( x ) ζ, x Ω R m wth compact subset Ω contanng the orgn. Assumpton 4. For n, there exst known constants ξ 0 such that G ( x ) ξ. Lemma (Polycarpou & Ioannou, 996). The followng nequalty holds for any ε > 0 and for any η R η 0 η η tanh k p ε (6) ε where k p s a constant that satsfes k p = e (k p),.e, k p = Lemma (Usman, 987; Chen et al., 00). No egenvalue of a matrx A R m m exceeds any of ts norm n absolute value, that s λ A, =,,..., m (7) where λ s a egenvalue of matrx A. Lemma 3 (Chen et al., 00). Consderng a matrx B R m m wth spectral radus ϱ(b), there exsts a postve constant > 0 whch makes matrx B (ϱ(b) )I m m nonsngular. The control obectve s to make x follow a certan desred traectory x d to a compact set n the presence of system uncertantes and dsturbances under the desgned adaptve control law v. Assumpton 5. There exst ε 0 such that for all t, x () d (t) ε 0, =,,..., n. Remark. Assumpton means that the nonlnear functons g r (v ) and g l (v ) of the non-symmetrc nput saturaton are strctly monotonous. Assumpton s reasonable snce the tmedependent component of the dsturbance wth fnte energy s always bounded (Tee & Ge, 006). Assumpton 3 s smlar to the Assumpton A n Km and Ha (000). Assumpton 4 means that perturbatons G ( x ) of control coeffcent matrces G ( x ), =,,..., n are bounded. There are many practcal systems can be expressed as the nonlnear system form as shown n (). For example, rgd robots and motors (Dawson, Carroll, & Schneder, 994), shps (Tee & Ge, 006) and arcraft (Tang et al., 007; Tee, Ge, & Tay, 008). Remark. In ths paper, the matrx spectral radus s employed to desgn adaptve control for uncertan MIMO nonlnear systems (). We do not assume that all control coeffcent matrces G ( x ), =,,..., n are nvertble, but only requre that the norm of control coeffcent matrx s bounded. Ths pont s always vald for a practcal control plant. Consderng Assumpton 3 and Lemma, the spectral radus ϱ(g ) of G ( x ) satsfes ϱ(g ) ζ (Chen et al., 00). Accordng to Lemma 3, we know that G ( x ) (ζ τ )I m m are nonsngular wth τ > 0, =,,..., n. 3. Adaptve control desgn and stablty analyss In ths secton, adaptve control s proposed for the uncertan nonlnear system wth control nput saturaton. The auxlary desgn system s adopted to analyze the nput saturaton constrants. The spectral radus of the control coeffcent matrx s ntroduced to desgn adaptve control and the bounded stablty of all sgnals n the closed-loop system s acheved. Step : Defne error varables z = x x d, and z = x α, where α R m wll be defned. Consderng () and dfferentatng z wth respect to tme, we obtan ż = F (x )θ G (x )(z α ) G (x )x D (x, t) ẋ d. (8) Consder the Lyapunov functon canddate V = zt z. (9) Its dervatve s gven by V = zt F (x )θ z T G (x )(z α ) z T G (x )x z T D (x, t) z T ẋd z T F (x )θ z T G (x )(z α ) ξ z x m z ρ (x )Θ z T ẋd. (0) Invokng Lemma, we have m z ρ (x )Θ m kε z tanh z ε ρ (x )Θ = Ψ T ρ (x )Θ z T Tanh(z )ρ (x )Θ z T Tanh(z )ρ (x )Θ Ψ T ρ (x ) Θ. () Substtutng () nto (0), we have V zt F (x )θ z T G (x )(z α ) ξ z x z T Tanh(z )ρ (x )Θ Ψ T ρ (x ) Θ z T ẋd. () Invokng Lemma 3, choose the followng vrtual control law: α = (G (x ) γ I m m ) (K z F (x )ˆθ Tanh(z )ρ (x ) ˆΘ ẋ d ) (3) where K = K T > 0 and γ = ζ τ. Substtutng (3) nto () yelds V zt K z z T G (x )z z T F (x )θ z T F (x )ˆθ z T Tanh(z )ρ (x )Θ z T Tanh(z )ρ (x ) ˆΘ ξ z x γ z T α Ψ T ρ (x ) Θ = z T K z z T G (x )z z T F (x ) θ z T Tanh(z )ρ (x ) Θ ξ z x γ z T α Ψ T ρ (x ) Θ. (4) Consderng the error sgnals θ and Θ, the augmented Lyapunov functon canddate s wrtten as V = V θ T Λ θ Θ T Λ Θ (5) where Λ = Λ T > 0 and Λ = Λ T > 0.

4 M. Chen et al. / Automatca 47 (0) The tme dervatve of V s gven by V z T K z z T G (x )z z T F (x ) θ z T Tanh(z )ρ (x ) Θ ξ z x γ z T α Ψ T ρ (x ) Θ θ T Λ ˆθ Θ T Λ ˆΘ. (6) Consder the adaptve laws for ˆθ and ˆΘ as ˆθ = Λ (F T (x )z ˆθ ) (7) ˆΘ = Λ (ρ (x )Tanh(z )z ˆΘ ) (8) where > 0 and > 0. Substtutng (7) and (8) nto (6), and consderng the followng facts by completon of squares: θ T ˆθ θ θ Θ T ˆΘ Θ Θ we have V z T K z z T G (x )z ξ z x γ z T α Ψ T ρ (x ) θ θ (9) (0) Θ Θ Θ. () The frst term on the rght-hand sde s negatve, and the second term wll be canceled n the next step. The other terms wll be consdered n stablty analyss of the closed-loop system. Step : Defne the error varable z 3 = x 3 α. Consderng () and dfferentatng z wth respect to tme, we obtan ż = F ( x )θ G ( x )x 3 G ( x )x 3 D ( x, t) α. () Consder the Lyapunov functon canddate V = zt z. (3) Consderng Lemma, the dervatve of V s V = zt F ( x )θ z T G ( x )(z 3 α ) z T G ( x )x 3 z T D ( x, t) z T α z T F ( x )θ z T G ( x )(z 3 α ) m ξ z x 3 z ρ ( x )Θ z T α z T F ( x )θ z T G ( x )(z 3 α ) ξ z x 3 z T Tanh(z )ρ ( x )Θ Ψ T ρ ( x ) Θ z T α. (4) Invokng Lemma 3, choose the vrtual control law as α = (G ( x ) γ I m m ) (G T (x )z K z F ( x )ˆθ Tanh(z )ρ ( x ) ˆΘ α ) (5) where K = K T > 0 and γ = ζ τ. Substtutng (5) nto (4), we obtan V zt K z z T G ( x )z 3 z T F ( x )θ z T F ( x )ˆθ z T Tanh(z )ρ ( x )Θ z T Tanh(z )ρ ( x ) ˆΘ z T GT (x )z γ z T α ξ z x 3 Ψ T ρ ( x ) Θ = z T K z z T G ( x )z 3 z T F ( x ) θ z T Tanh(z )ρ ( x ) Θ z T G (x )z γ z T α ξ z x 3 Ψ T ρ ( x ) Θ. (6) Consderng the error sgnal θ and Θ, the augmented Lyapunov functon canddate can be wrtten as V = V V θ T Λ θ Θ T Λ Θ (7) where Λ = Λ T > 0 and Λ = Λ T > 0. Invokng () and (6), the tme dervatve of V s V z T K z γ z T α ξ z x z T G ( x )z 3 z T F ( x ) θ z T Tanh(z )ρ ( x ) Θ Ψ θ T Λ ˆθ Θ T Λ ˆΘ T ρ ( x ) θ θ Θ Θ Θ. (8) Consder the adaptve laws for ˆθ and ˆΘ as ˆθ = Λ (F T ( x )z ˆθ ) (9) ˆΘ = Λ (ρ ( x )Tanh(z )z ˆΘ ) (30) where > 0 and > 0. Substtutng (9) and (30) nto (8), smlar wth (9) and (0) we have V z T K z z T G ( x )z 3 γ z T α ξ z x θ Θ Ψ T ρ ( x ) θ Θ Θ. (3) The frst term on the rght-hand sde s negatve, and the second term wll be canceled n the next step. The other terms wll be consdered n stablty analyss of the closed-loop system. Step ( n ): Defne the error varable z = x α. Consderng () and dfferentatng z wth respect to tme, we have ż = F ( x )θ G ( x )(z α ) G ( x )x D ( x, t) α. (3)

5 456 M. Chen et al. / Automatca 47 (0) Consder the Lyapunov functon canddate V = zt z. (33) Invokng Lemma, the dervatve of V V = z T F ( x )θ z T G ( x )(z α ) z T G ( x )x z T D ( x, t) z T α z T F ( x )θ z T G ( x )(z α ) m ξ z x z ρ ( x )Θ z T α z T F ( x )θ z T G ( x )(z α ) ξ z x z T Tanh(z )ρ ( x )Θ Ψ T ρ ( x ) Θ z T α. (34) Consderng Lemma 3, we choose the followng vrtual control law: α = (G ( x ) γ I m m ) (G T ( x )z K z F ( x )ˆθ Tanh(z )ρ ( x ) ˆΘ α ) (35) where K = K T > 0 and γ = ζ τ. Substtutng (35) nto (34), we obtan V z T K z z T G ( x )z z T F ( x )θ z T F ( x )ˆθ z T Tanh(z )ρ ( x )Θ z T Tanh(z )ρ ( x ) ˆΘ z T GT ( x )z γ z T α ξ z x Ψ T ρ ( x ) = z T K z z T G ( x )z z T F ( x ) θ s Θ z T Tanh(z )ρ ( x ) Θ z T G ( x )z γ z T α ξ z x Ψ T ρ ( x ) Θ. (36) Consderng the error sgnals θ and Θ, the augmented Lyapunov functon canddate can be wrtten as V = V V θ T Λ θ Θ T Λ Θ (37) where Λ = Λ T > 0 and Λ = Λ T > 0. Invokng (3) and (36), the tme dervatve of V s gven by V z T K z γ z T α ξ z x z T G ( x )z z T F ( x ) θ z T Tanh(z )ρ ( x ) Θ Ψ θ T Λ ˆθ Θ T Λ ˆΘ T ρ ( x ) θ θ Θ Θ Θ. (38) Consder the adaptve laws for ˆθ and ˆΘ as ˆθ = Λ (F T ( x )z ˆθ ) (39) ˆΘ = Λ (ρ ( x )Tanh(z )z ˆΘ ) (40) where > 0 and > 0. Substtutng (39) and (40) nto (38), smlar wth (9) and (0) we have V z T K z z T G ( x )z γ z T α ξ z x θ Θ Ψ T ρ ( x ) θ Θ Θ. (4) The frst term on the rght-hand sde s negatve, and the second term wll be canceled n the next step. The other terms wll be consdered n stablty analyss of the closed-loop system. Step n: By dfferentatng z n = x n α wth respect to tme yelds ż n = F n ( x n )θ n G n ( x n )u G n ( x n )u D n ( x n, t) α. (4) Consder the Lyapunov functon canddate V n = zt n z n. (43) Note the fact u U max wth U max > 0. Invokng Lemma, the dervatve of V n s V n = zt n F n( x n )θ n z T n G n( x n )u z T n G nu z T n D n( x n, t) z T n α z T n F n( x n )θ n z T n G n( x n )u ξ n z n u m z n ρ n ( x n )Θ n z T α n z T n F n( x n )θ n z T n G n( x n )u ξ n z n u z T n Tanh(z n)ρ n ( x n )Θ n Ψ T n ρ n( x n ) Θ n z T n α. (44) From (), control nputs u have an upper lmt and a lower lmt. For convenence of nput constrant effect analyss, the auxlary desgn system s gven by K n e ė = e f (u, u, z n, x n )e (45) (G n ( x n ) γ n I m m )(v u), e σ 0, e < σ where f (u, u, z n, x n ) = z T n G n( x n ) u 0.5(γ n ζ n ) u T u γ n z T u ξ n n z n u, u = u v, K n = K T > n 0, γ n = ζ n τ n and e R m s the state of auxlary desgn system. The desgn parameter σ s a postve constant whch should be chosen as an approprate value n accordance wth the requrement of the trackng performance. Defne h(z) = zt n K T n K nz n γ z T α ξ z x Ψ T ρ ( x ) where K n = K T n > 0 and Z = [α, z, x ] T, =,,..., n. (46)

6 M. Chen et al. / Automatca 47 (0) Invokng Lemma 3 and consderng the nput saturaton effect, choose the followng control law: v = (G n ( x n ) γ n I m m ) v 0 v 0 = G T ( x )z K n (z n e) F n ( x n )ˆθ n Tanh(z n )ρ n ( x n ) ˆΘ n α z nh(z) ψ z n ψh(z) ψ = ψ z n k vψ, z n l 0, z n < l (47) where k v > 0 and l > 0. The above desgn procedure can be summarzed n the followng theorem, whch contans the results of adaptve control for uncertan MIMO nonlnear systems (). Theorem. Consderng the strct-feedback nonlnear system () wth known coeffcent matrces satsfes Assumptons 5, and gven that the full state nformaton s avalable. Under the control law (47), parameter updated laws (7), (8), (9), (30), (39), (40), (53), (54), and for any bounded ntal condton, there exst desgn parameters σ > 0, K = K T > 0, K n = K T > n 0, > 0, > 0 and k v > 0 such that the overall closed-loop control system s semglobally stable n the sense that all of the closed-loop sgnals e, z, ψ, θ and Θ are bounded, where =,,..., n. Furthermore, the trackng error sgnals z remans wthn the compact sets Ω z defned by Ω z := z R m z D where D = (V n (0) C ) wth C and κ as defned n (55). κ Proof. When e σ, we consder the Lyapunov functon canddate V n = V V n et e θ T n Λ θ n n Θ T n Λ Θ n n ψ (48) where Λ n = Λ T n > 0 and Λ n = Λ T n > 0. Consderng (4) and (44), the tme dervatve of V n s V n z T K z z T G ( x )z n γ z T α ξ z x z T n F n( x n )θ n ξ n z n u z T n G n( x n )(v u) z T n Tanh(z n)ρ n ( x n )Θ n z T α n e T ė θ T n Λ n ˆθ n Θ T n Λ n ˆΘ n Ψ T ρ ( x ) θ θ Θ Θ Θ ψ ψ. (49) Substtutng (45) (47) nto (49), we obtan V n z T K z γ z T α ξ z x γ n z T n u ξ n z n u z T n G n( x n ) u z T n K ne z T n F n( x n ) θ n z T n Tanh(z n)ρ n ( x n ) Θ n e T ė Ψ θ T n Λ n ˆθ n Θ T n Λ n ˆΘ T ρ ( x ) n Θ θ θ Θ Θ z n h(z) ψ z n ψ ψ z T K z e T (K n I m m ) e z T n F n( x n ) θ n z T n Tanh(z n)ρ n ( x n ) Θ n θ T n Λ n ˆθ n Θ T n Λ n ˆΘ n θ θ Θ Θ Θ ψ h(z) ψ z n ψ ψ. (50) Invokng the thrd equaton of (47), we have ψ h(z) ψ z n ψ ψ = k v ψ. (5) Substtutng (5) nto (50) yelds V n z T K z e T (K n I m m ) e z T n F n( x n ) θ n z T n Tanh(z n)ρ n ( x n ) Θ n θ T n Λ n ˆθ n Θ T n Λ n ˆΘ n θ θ Θ Θ Θ k v ψ. (5) Consder the adaptve laws for ˆθ n and ˆΘ n as ˆθ n = Λ n (F T n ( x n)z n n ˆθ n ) (53) ˆΘ n = Λ n (ρ n ( x n )Tanh(z n )z n n ˆΘ n ) (54) where n > 0 and n > 0. Substtutng (53) and (54) nto (5), smlar wth (9) and (0) we have V n z T K z e T (K n I m m ) e k v ψ where = θ Θ = θ Θ Θ κv n C (55)

7 458 M. Chen et al. / Automatca 47 (0) K n λ mn κ := mn C := θ λ max (Λ ),, λ mn (K n I m m ), λ max (Λ ), k v Θ Θ. (56) To ensure that κ > 0, the desgn parameter K n must make K n I m m > 0. From (55), f κ > 0, we can conclude that z converges to a compact set asymptotcally, and therefore the control obectve s reached when the nput saturaton constrant occurs,.e., the desred traectory of MIMO nonlnear system s followed n the presence of parametrc uncertantes and dsturbances under the saturaton constrant. On the other hand, we can conclude that auxlary desgn varables e and ψ, error sgnals z, θ and Θ converge to a compact set asymptotcally. It s worth pontng out that the above proof of Theorem only contans the result when the states of the auxlary desgn system (45) satsfy the condton e σ,.e., there exsts nput saturaton. If e < σ means that there does not exst nput saturaton, we have u = 0,.e., u = v and the control nput u s bounded. Thus, v s bounded. The stablty proof of Theorem can be easly proved by consderng Eqs. (48) (55) when e < σ. The detaled proof s omtted. Ths concludes the proof. Remark 3. In ths secton, the robust adaptve trackng control s proposed for a class of uncertan MIMO nonlnear systems wth non-symmetrc nput saturaton constrants. To handle the nonsymmetrc nput saturaton, the auxlary desgn system (45) s ntroduced to analyze the effect of saturaton constrant, and the auxlary varable e s used to desgn the robust adaptve control law. It s apparent that the constraned control u produced by the desgned control command v can guarantee the closed-loop system stablty. If e σ and ė = 0, t means that there s no saturaton,.e., there s u = v accordng to (45) (Polycarpou et al., 003). It mples that v lmax v v rmax and g r (v ) = g l (v ) = v. Fg.. Confguraton of the command flter, where =,,..., n, α n = v, α 0 are the nomnal vrtual control law or the nomnal control law, α are the vrtual control law or the control law, ξ and ω n are the bandwdth parameters. fltered to provde the magntude, rate and bandwdth lmted vrtual control law α and ts dervatves α whch are wthn the operatng envelope of the system. Such a command flter s shown n Fg. to mplement the emulate of any mechancal or operatng constrants on vrtual control law α 0 (Polycarpou et al., 004). The nomnal vrtual control law α 0 s gven by α 0 = (G (x ) γ I m m ) (K 0 (z ϕ ) F (x )ˆθ Tanh(z )ρ (x ) ˆΘ ẋ d ) (58) where K 0 = K T > 0 0, and ϕ R m s the state vector of auxlary desgn system whch denotes the constrant effect due to the magntude, rate and bandwdth lmtaton of the nomnal vrtual control law. Note the followng facts α ε 0 wth ε 0 > 0, where ε 0 denotes the magntude lmt of α whch s decded by the command flter. Let δ = ζ mγ. For convenence of constrant effect analyss (Chen et al., 00), the auxlary desgn system s gven by K ϕ f (z, ϕ, ˆθ, ˆΘ )ϕ ϕ = (G (x ) γ I m m )(α α 0 ), ϕ σ (59) 0, ϕ < σ where f (z, ϕ, ˆθ, ˆΘ ) = φ (z,ˆθ, ˆΘ ), φ ϕ (z, ˆθ, ˆΘ ) = a K 0 z F (x )ˆθ δ ε 0 ζ ε 0 z z F (x )ˆθ Tanh(z )ρ (x ) ˆΘ ẋ d γ z T α z Tanh(z )ρ (x ) ˆΘ, K = K T > 0, a > 0 and σ s a postve desgn parameter. Consder the Lyapunov functon canddate 4. Constraned adaptve control desgn and stablty analyss Although the robust adaptve control for the uncertan MIMO nonlnear system () wth non-symmetrc nput saturaton constrants has been successfully developed n Secton 3, physcs constrants of vrtual control laws have not been consdered, and the analytc computatons of tme dervatves of vrtual control laws α ( =,..., n ) need to be done n the backsteppng procedure. In fact, the vast analytc calculaton of the vrtual control dervatves s a drawback of backsteppng control. Specally, the analytc calculaton of tme dervatves of vrtual control laws s tedous for the MIMO nonlnear systems. In ths secton, we wll nvestgate the constraned robust adaptve control whch consder the mechancal or operatng lmtatons of vrtual control laws and control command, and elmnate the analytc computatons of the vrtual control law dervatves. Therefore, command flters are ntroduced to avod the analytc calculaton of the tme dervatves of the vrtual control laws. Step : Defne error varables z = x x d and z = x α. Consderng () and dfferentatng z wth respect to tme, we obtan ż = F (x )θ (G (x ) G (x ))(z α ) D (x, t) ẋ d (57) where α s a vrtual control law whch s produced by the nomnal vrtual control law α 0. The nomnal vrtual control law α 0 s V = zt z ϕt ϕ. (60) Invokng (57) (59), the tme dervatve of V s V = c z T K 0z z T F (x )θ z T G (x )(z α ) z T G (x )x z T D (x, t) z T ẋd c z T K 0z ϕ T K ϕ f (z, ϕ, ˆθ, ˆΘ ) ϕ ϕ T (G (x ) γ I m m )(α α 0 ) c z T K 0z z T F (x )θ z T Tanh(z )ρ (x )Θ ζ z ζ z ζ ε 0 z ξ z x z ẋ d c K 0 z ϕ T K ϕ f (z, ϕ, θ, Θ ) ϕ δ ϕ δ ε 0 K 0 z 3 K 0 ϕ 3 ϕ F (x )ˆθ Tanh(z )ρ (x ) ˆΘ ẋ d z T F (x )ˆθ z T Tanh(z )ρ (x ) ˆΘ z T F (x )ˆθ z T Tanh(z )ρ (x ) ˆΘ Ψ T ρ (x ) Θ

8 M. Chen et al. / Automatca 47 (0) z T ζ c K 0 I m m z where α s a vrtual control law whch s produced by the nomnal vrtual control law α 0. The nomnal vrtual control law α 0 are ϕ T 3 K0 K 3 δ fltered to provde the magntude, rate and bandwdth lmted I m m ϕ vrtual control law α and ts dervatves α whch are wthn the ζ z ξ z x Ψ T ρ operatng envelope of the system. Such a command flter s smlar (x ) to the frst flter shown n Fg. to mplement any mechancal or operatng constrants on vrtual control law α 0. γ z T α z T F (x ) θ z T Tanh(z )ρ (x ) Θ The nomnal vrtual control law α 0 s gven by (c 0.5) K 0 z (a 0 ) Θ (6) where c > 0, a 0 = a (c 0.5). Consderng the error sgnals θ and Θ, the augmented Lyapunov functon canddate can be wrtten as V = V θ T Λ θ Θ T Λ Θ (6) where Λ = Λ T > 0 and Λ = Λ T > 0. Obvously, we can choose a and c to render a 0 > 0. Invokng (6), the tme dervatve of V s V z T ζ c K 0 I m m z ϕ T 3 K0 K 3 δ I m m ϕ ζ z ξ z x Ψ T ρ (x ) γ z T α z T F (x ) θ z T Tanh(z )ρ (x ) Θ θ T Λ ˆθ Θ T Λ ˆΘ. (63) Consder the adaptve laws for ˆθ and ˆΘ as ˆθ = Λ (F T (x )z ˆθ ) (64) ˆΘ = Λ (ρ (x )Tanh(z )z ˆΘ ) (65) where > 0 and > 0. Substtutng (64) and (65) nto (63), we obtan V z T ζ c K 0 I m m z ϕ T 3 K0 K 3 δ I m m ϕ ζ z ξ z x Ψ T ρ (x ) γ z T α θ θ Θ Θ Θ. (66) The frst term and the second term on the rght-hand sde are negatve f c K 0 ζ I m m > 0 and K 3 K 0 3 δ I m m > 0. The other terms wll be consdered n the next step or the stablty analyss of the closed-loop system. Step ( n ): Defne the error varables z = x α and z = x α. Consderng () and dfferentatng z wth respect to tme, we obtan ż = F ( x )θ G ( x )(z α ) G ( x )x D ( x, t) α (67) α 0 = (G ( x ) γ I m m ) π π = G T ( x )z K 0 (z ϕ ) F ( x )ˆθ Tanh(z )ρ ( x ) ˆΘ α (68) where K 0 = K T > 0 0, and ϕ R m s the state vector of the th auxlary desgn system whch denotes the constrant effect due to the magntude, rate and bandwdth lmtaton of nomnal vrtual control law α 0. In nomnal vrtual control law (68), α need not be computed here whch can be drectly obtaned from the frst command flter n Step. Note the followng facts α ε 0 and α ε wth ε 0 > 0 and ε > 0, where ε 0 and ε denote the magntude lmt of α and the rate lmt of α whch are decded by the command flter. Let δ = ζ mγ. For convenence of constrant effect analyss, the auxlary desgn system s gven by K ϕ f (z, ϕ, ˆθ, ˆΘ )ϕ ϕ = (G ( x ) γ I m m )(α α 0 ), ϕ σ (69) 0, ϕ < σ where f (z, ϕ, ˆθ, ˆΘ ) = φ (z,ˆθ, ˆΘ ), φ ϕ (z, ˆθ, ˆΘ ) = a K 0 z Tanh(z )ρ ( x ) ˆΘ F ( x )ˆθ ζ z ζ ε 0 z z F ( x )ˆθ ( δ )ε z Tanh(z )ρ ( x ) ˆΘ γ z T α, K = K T > 0, a > 0 and σ s a postve desgn parameter. Consder the Lyapunov functon canddate V = zt z ϕt ϕ. (70) Invokng (67) (69), the tme dervatve of V V = c z T K 0z z T F ( x )θ z T G ( x )(z α ) z T G ( x )x z T D ( x, t) z T α c z T K 0z ϕ T K ϕ f (z, ϕ, ˆθ, ˆΘ ) ϕ ϕ T (G ( x ) γ I m m )(α α 0 ) c z T K 0z z T F ( x )θ z T Tanh(z )ρ ( x )Θ ζ z ζ z ζ ε 0 z ξ z x z ε c K 0 z ϕ T K ϕ f (z, ϕ, ˆθ, ˆΘ ) ϕ δ ϕ δ ε 0 ζ ϕ ζ z K 0 z 3 K 0 ϕ 3 ϕ F ( x )ˆθ Tanh(z )ρ ( x ) ˆΘ ε zt F ( x )ˆθ Ψ T ρ ( x ) z T Tanh(z )ρ ( x ) ˆΘ z T F ( x )ˆθ z T Tanh(z )ρ ( x ) ˆΘ Ψ T ρ ( x ) Θ s

9 460 M. Chen et al. / Automatca 47 (0) z T ζ c K 0 I m m z ϕ T 3 K0 K ζ 3 δ θ Θ Θ I m m ϕ Θ γ z T ζ α z ξ z x z T F ( x ) θ z T Tanh(z )ρ ( x ) Θ γ z T α Ψ Ψ T T ρ ( x ) ρ ( x ) ξ z x. (76) (c 0.5) K 0 z (a 0 ) Θ The frst four terms are negatve f c (7) K 0 ζ I m m > ζ where c > 0, a 0 = a (c 0.5). 0, c K 0 ζ I m m > 0 ( =,..., ), K 3 K 0 Consderng the error sgnals θ and Θ, the augmented Lyapunov functon canddate can be wrtten as ( =,..., ). The other terms wll be consdered n the next 3 δ 3 K0 I m m and K ζ 3 δ I m m step. V = V V θ T Λ θ Θ T Λ Θ (7) where Λ = Λ T > 0 and Λ = Λ T > 0. Smlarly, we can choose a and c to render a 0 > 0. Invokng (66) and (7), the tme dervatve of V s V z T ζ c K 0 I m m z z T ζ c K 0 ζ I m m z = ϕ T 3 K0 K 3 δ I m m ϕ ϕ T K K ϕ ζ z z T F ( x ) θ = z T Tanh(z )ρ ( x ) Θ θ T Λ ˆθ Θ T Λ θ θ Θ Θ γ z T α ξ z x where K = Ψ T ρ ( x ) 3 K0 ζ 3 δ I m m. Consder the adaptve laws for ˆθ and ˆΘ as ˆΘ Θ (73) ˆθ = Λ (F T ( x )z ˆθ ) (74) ˆΘ = Λ (ρ ( x )Tanh(z )z ˆΘ ) (75) where > 0 and > 0. Substtutng (74) and (75) nto (73), we obtan V z T ζ c K 0 I m m z z T ζ c K 0 ζ I m m z = ϕ T 3 K0 K 3 δ I m m ϕ ϕ T K K ϕ ζ z θ = Step n: By dfferentatng z n = x n α wth respect to tme yelds ż n = F n ( x n )θ n G n ( x n )u G n ( x n )u D n ( x n, t) α (77) where u s a control law whch s produced by the nomnal control law v. The nomnal control law v s fltered to provde the magntude, rate and bandwdth lmted vrtual control law u and ts dervatves u whch are wthn the operatng envelope of the system. Such a command flter s smlar to the frst flter shown n Fg. to mplement the mechancal or operatng constrants on vrtual control law v. Here, t s requred that the command flter can mplement the same poston constrants on adaptve control v as shown n (). Defne h(z) = zt n K T n0 K nz n0 ξ z x γ z T α Ψ T ρ ( x ) where K n0 = K T n0 > 0 and Z = [α, z, x ] T, =,,..., n. The nomnal vrtual control law v s gven by v = (G n ( x n ) γ n I m m ) v 0 v 0 (78) = G T ( x )z K n0 (z n ϕ n ) F n ( x n )ˆθ n Tanh(z n )ρ n ( x n ) ˆΘ n α z nh(z) (79) ψ z n ψh(z) ψ = ψ z n k vψ, z n l (80) 0, z n < l where ϕ n R m s the state vector of the auxlary desgn system whch denotes the constrant effect due to the magntude, rate and bandwdth lmtaton of nomnal vrtual control law. In nomnal control law (79), α need not be computed here whch can be drectly obtaned from the command flter n Step n. Note the followng facts u ε n0 and α ε n wth ε n0 > 0 and ε n > 0, where ε n0 and ε n denote the magntude lmt of u and the rate lmt of α whch are decded by the command flter. Let δ n = ζ n mγ n. For convenence of constrant effect analyss, the auxlary desgn system s gven by K n ϕ n f n (u, z n, ϕ n, ˆθ n, ˆΘ n )ϕ n ϕ n = (G n ( x n ) γ n I m m )(u v), ϕ n σ n (8) 0, ϕ n < σ n where f n (u, z n, ϕ n, ˆθ n, ˆΘ n ) = φ n(u,z n,ˆθ n, ˆΘ n ) ϕ n, φ n (u, z n, ˆθ n, ˆΘ n ) = a n K n0 z n ζ n u F n( x n )ˆθ n Tanh(z n)ρ n ( x n ) ˆΘ n

10 ζ z z n F n ( x n )ˆθ n z n Tanh(z n )ρ n ( x n ) ˆΘ n δ nε n0 ε n γ nz T n u ξ n z n u zt n K T n0 K nz n0 z n h(z) ψ z n, K n = K T n > 0, a n > 0 and σ n s a postve desgn parameter. Consder the Lyapunov functon canddate V n = zt n z n ϕt n ϕ n. (8) Invokng (77), (79) and (8), the tme dervatve of V n s V n = c nz T n K n0z n z T n F n( x n )θ n z T n G n( x n )u z T n G n( x n )u z T n D n( x n, t) z T n α c n z T n K n0z n ϕ T n K nϕ n f n (z n, ϕ n, ˆθ n, ˆΘ n ) ϕ n ϕ T n (G n( x n ) γ n I m m )(u v) M. Chen et al. / Automatca 47 (0) = ϕ T K K ϕ z T n F n( x n ) θ n z T n Tanh(z n)ρ n ( x n ) Θ n θ T n Λ n ˆθ n Θ T n Λ n ˆΘ n θ θ Θ Θ Θ ψ h(z) ψ z n ψ ψ. (85) Consder the adaptve laws for ˆθ n and ˆΘ n as c n z T n K n0z n z T n F n( x n )θ n ζ n z n z T n Tanh(z n)ρ n ( x n )Θ n γ n z T n u ζ n u ξ n z n u z n ε n c n K n0 z n ϕ T n K nϕ n f n (z n, ϕ n, ˆθ n, ˆΘ n ) ϕ n δ n ϕ n δ nε n0 ζ ϕ n ζ z K n0 z n 3 K n0 ϕ n 3 ϕ n F n( x n )ˆθ n Tanh(z n)ρ n ( x n ) ˆΘ n ε n zt n F n( x n )ˆθ n z T n Tanh(z n)ρ n ( x n ) ˆΘ n z T n F n( x n )ˆθ n z T n Tanh(z n)ρ n ( x n ) ˆΘ n Ψ T ρ n n( x n ) Θ n zt n K T n0 K nz n0 z T ζn n c n K n0 I m m z n ϕ T n Kn Kn ϕn zt n K T n0 K nz n0 z T n F n( x n ) θ n z T n Tanh(z n)ρ n ( x n ) Θ n (c n 0.5) K n0 z n (a n0 ) Θ n Ψ T ρ n n( x n ) where Kn = 3 Kn0 ζ δ n 3 z n h(z) ψ z n (83) I m m, c n > 0 and a n0 = a n (c n 0.5). Consderng the error sgnals θ n and Θ n, the augmented Lyapunov functon canddate can be wrtten as V n = V V n θ T n Λ θ n n Θ T n Λ Θ n n ψ (84) where Λ n = Λ T > n 0 and Λ n = Λ T > n 0. Obvously, we can choose a n and c n to render a n0 > 0. Invokng (76), (78), (79) and (83), the tme dervatve of V n s V n z T ζ c K 0 I m m z z T ζ c K 0 ζ I m m z = ϕ T 3 K0 K 3 δ I m m ϕ ˆθ n = Λ n (F T n ( x n)z n n ˆθ n ) (86) ˆΘ n = Λ n (ρ n ( x n )Tanh(z n )z n n ˆΘ n ) (87) where n > 0 and n > 0. Substtutng (78), (86) and (87) nto (85), we obtan V n z T ζ c K 0 I m m z z T ζ c K 0 ζ I m m z = ϕ T 3 K0 K 3 δ I m m ϕ ϕ T K K ϕ θ = = θ Θ k v ψ Θ = Θ κv n C (88) where λ mn (Q 0 ), λ mn (Q ), λ mn (Q ), λ mn (Q 3 ), κ := mn λ max (Λ ), λ max (Λ ), k, v C := θ Θ Θ, ζ Q 0 = c K 0 I m m, ζ Q = c K 0 ζ I m m, = 3 K0 Q = K 3 δ I m m, Q 3 = (K K ), =,..., n. (89) = To ensure the closed-loop stablty, we can choose correspondng desgn parameters to make Q 0 > 0, Q > 0, Q > 0 and Q 3 > 0. The above desgn procedure can be summarzed n the followng theorem, whch contans the results for the constraned adaptve control of an uncertan nonlnear system.

11 46 M. Chen et al. / Automatca 47 (0) Theorem. Consderng the uncertan MIMO nonlnear system () satsfes Assumptons 5, and gven that full state nformaton s avalable. The control law s produced by nomnal control law (79) usng the command flter. Under the parameter adaptaton laws (64), (65), (74), (75), (86), (87) and for any bounded ntal condton, the closed-loop sgnals z, ϕ, ψ, θ and Θ ( n) are sem-globally stable n the sense that all of the closed-loop sgnals are bounded, where =,,..., n. The trackng error z asymptotcally converges to a compact set Ω z defned by Ω z := z R m z D where D = (V n (0) C ) wth C and κ as defned n (89). κ It s apparent that the Theorem can be easly proved accordng to (84) and (88). Remark 4. In the proposed constraned adaptve control, we can see that the satsfactory closed-loop stablty wth sutable transent performance can be acheved by properly adustng desgn parameters K 0, K,,, Λ, and k, =,,..., n. For example, the trackng error could be decreased by ncreasng the value of K 0, but that ncrease would also ncrease the control sgnal, and could excte unmodeled dynamcs. Therefore, cauton must be exercsed n the choce of these parameters, due to the fact that there s some trade-off between the control performance and other ssues. Remark 5. In the developed constraned adaptve control, f ϕ = 0 and ϕ = 0, there are α 0 = α and u = v accordng to (59), (69) and (8),.e., there are no constrants. At the same tme, the nomnal vrtual control law (58), (67) and nomnal control law (79) are the same as the vrtual control law (3), (35) and control law (47) of the proposed adaptve control n Secton 3. It should be ponted out that we do not drectly consder the nput saturaton constrant (). However, the command flter can not only mplement the same poston and also rate constrants can be consdered on the adaptve control v by choosng the approprate desgn parameter. Remark 6. In practce, t s apparent that the magntude of the actual/vrtual control nput, as well as ther dervatons should be bounded due to the physcal lmtaton. Thus, the command flter could be presented accordng to mechancal and operatng constrants of actuator. Magntude lmt functon and rate lmt functon can be chosen as conservatve common saturaton functon or other lmt functons. If lmt functons are chosen as conservatve common saturaton functons, the relatonshp between the nput and the output of the command flter can be found n Farrell et al. (003). 5. Smulaton results Consder the uncertan MIMO nonlnear system wth nput saturaton n the form of Chen et al. (00) ẋ = F (x )θ (G (x ) G (x ))x D (x, t) ẋ = F ( x )θ (G ( x ) G ( x ))u D ( x, t) y = x (90) where x = [x, x ] T, x = [x, x ] T, 0. sn(x ) cos(x F (x ) = ), 0.x x g (x) G (x ) =, 5 g (x) g (x) =. cos(x ) sn(x ), g (x) =.3 cos(x ) sn(x ), 0. sn(x ) 0 G (x ) =, 0 0. cos(x ) x x F ( x ) = 0 0 x x cos(x ) sn(x G ( x ) = ) sn(x ), sn(x ) cos(x ) 0. sn(x x G ( x ) = ) 0. cos(x x ), 0.5 cos(x x ) 0.3 sn(x x ) 0.(cos(x )) D (x, t) = 0.04 sn(0.3x t), 0.(sn(x )) 0.03 sn(0.x t) 0.3(sn(x )) D ( x, t) = 0.05 sn(0.x t). 0.(cos(x )) 0. sn(0.3x t) For smulaton purposes, parameter values are set to θ =, θ = 0.5, u max = u mn = 3.0, u max = u mn =.0, g r (v ) = g l (v ) = v, γ = 3.0 and γ =.0. Now, the control obectve s to desgn adaptve control and constrant adaptve control for system (90) such that the system output y = x follows the desred traectory x d, where the desred traectores are taken as x d = 0.5[sn(.5t) sn(0.5t)] and x d = 0.8 sn(t) 0.5 sn(0.5t). The adaptve control s desgned as follows: α = (G (x ) γ I m m ) K z F (x )ˆθ Tanh(z )ρ (x ) ˆΘ ẋ d K e ė = e f (u, u, z, x )e (G ( x ) γ I )(v u), e σ 0, e < σ v = (G ( x ) γ I m m ) v 0 v 0 = G T (x )z K (z e) F ( x )ˆθ Tanh(z )ρ ( x ) ˆΘ α z h(z) ψ z ψh(z) ψ = ψ z kψ, z l 0, z < l where k > 0, K = K T > 0, K = K T > 0, K = K T > 0, f (u, u, z, x ) = z T G ( x ) u 0.5(γ ζ ) u T u and σ = 0.. The adaptve laws for ˆθ and ˆΘ are chosen as (7) and (8). The adaptve laws of ˆθ and ˆΘ are chosen as (53) and (54). The desgn parameters of the control are chosen as K = dag{8.0, 8.0}, K = dag{0.0, 80.0}, K = dag{0.0, 0.0} and Λ = Λ = Λ = Λ = dag{0.0, 0.0}. The smulaton results of the trackng output are shown n Fgs. 3 and 4 wth ntal states x =.0 and x = 0.0. It can be observed that the system output x and x follow the desred traectory x d and x d well despte the unknown parameters, perturbaton of the control coeffcent matrces and nput saturaton. From Fgs. 5 and 6, we can see that the control nputs are saturated n the ntalzaton transent phase. These smulaton results show that good trackng performance can be obtaned under the proposed adaptve control. To llustrate the effectveness of the proposed constraned adaptve control, the nomnal vrtual control law and the control command are desgned based on (58) and (79). Then, the nomnal vrtual control law α 0 and the nomnal control command v are used to produce the vrtual control law α and the system control

12 M. Chen et al. / Automatca 47 (0) Fg. 3. Output x (sold lne) follows desred traectory x d (dashed lne). Fg. 7. Output x (sold lne) follows desred traectory x d (dashed lne) for Case. Fg. 4. Output x (sold lne) follows desred traectory x d (dashed lne). Fg. 8. Output x (sold lne) follows desred traectory x d (dashed lne) for Case. Fg. 5. Control sgnal u. Fg. 9. Control sgnal u for Case. Fg. 6. Control sgnal u. law u usng the command flter as shown n Fg.. The desgn parameters of the flters are chosen as ω n = 0, ξ = ξ = and ω n = 00. To observer the varety of closed-loop system control performance for the dfferent desgn parameters under the constraned adaptve trackng control, the followng two cases are consdered: Case : K 0 and K 0 are chosen as K 0 = dag{8.0, 8.0} and K 0 = dag{0.0, 80.0}. Other desgn parameters are chosen as the same desgn parameters as the correspondng desgn parameters n the adaptve trackng control. Fg. 0. Control sgnal u for Case. Case : The desgn parameters K 0 and K 0 are chosen as K 0 = dag{0.0, 0.0} and K 0 = dag{0.0, 0.0}. Other desgn parameters are chosen as the same desgn parameters as the correspondng desgn parameters n the adaptve trackng control. Under ntal states are x =.0 and x = 0.0, the trackng results of the Case are shown n Fgs. 7 and 8. It can be observed that the outputs x and x of Case stll follow the desred traectory x d and x d when the actuator constrants are consdered. In accordance wth Fgs. 9 and 0, t s observed that the control nputs

13 464 M. Chen et al. / Automatca 47 (0) Concluson Fg.. Case. Output x (sold lne) follows desred traectory x d (dashed lne) for Model-based adaptve control has been nvestgated for the uncertan MIMO nonlnear systems wth nput constrants n ths paper. Consderng actuator physcal constrants, the adaptve control and the constraned adaptve control n combnaton wth the backsteppng technque and Lyapunov synthess have been proposed. In the development of adaptve control, the auxlary desgn system has been ntroduced to analyze the effect of actuator physcal constrant, and states of auxlary desgn system are used to develop adaptve control. The cascade property of the studed systems has been fully utlzed n developng the control structure and parameter adaptve laws. It has proved that both the proposed adaptve control and the constraned adaptve control are able to guarantee the asymptotcal stablty of all sgnals n the closedloop system. 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Automatca, 4(), Mou Chen receved hs B.Sc. degree n materal scence and engneerng at Nanng Unversty of Aeronautcs & Astronautcs, Nanng, Chna, n 998, the M.Sc. and the Ph.D. degree n automatcal control engneerng at Nanng Unversty of Aeronautcs & Astronautcs, Nanng, Chna, n 004. He s currently an assocaton professor n automaton college at Nanng Unversty of Aeronautcs & Astronautcs, Chna. From June 008 to Step 009, he was a research fellow n the Department of Electrcal and Computer Engneerng, the Natonal Unversty of Sngapore. Hs research nterests nclude nonlnear control, artfcal ntellgence, magne processng and pattern recognton, and flght control. Shuzh Sam Ge IEEE Fellow, IFAC Fellow, IET Fellow, P.Eng, s foundng Drector of Socal Robotcs Lab, Interactve Dgtal Meda Insttute and Full Professor n the Department of Electrcal and Computer Engneerng, the Natonal Unversty of Sngapore and the Insttute of Intellgent Systems and Informaton Technology, Unversty of Electronc Scence and Technology of Chna, Chengdu, Chna. He receved hs BSc degree from Beng Unversty of Aeronautcs and Astronautcs (BUAA), and the Ph.D. degree and the Dploma of Imperal College (DIC) from Imperal College of Scence, Technology and Medcne. He has (co)-authored three books and over 300 nternatonal ournal and conference papers. He has served/been servng as an Assocate Edtor for a number of flagshp ournals ncludng IEEE Transactons on Automatc Control, IEEE Transactons on Control Systems Technology, IEEE Transactons on Neural Networks, and Automatca. He also serves as an Edtor of the Taylor & Francs Automaton and Control Engneerng Seres. He s an elected member of Board of Governors, IEEE Control Systems Socety. He provdes techncal consultancy to ndustral and government agences. He s the Edtor-n-Chef of the Internatonal Journal of Socal Robotcs. Hs current research nterests nclude socal robotcs, multmeda fuson, adaptve control, and ntellgent systems. Bebe Ren receved the B.E. degree n the Mechancal & Electronc Engneerng and the M.E. degree n Automaton from Xdan Unversty, X an, Chna, n 00 and n 004, respectvely, and the Ph.D. degree n the Electrcal and Computer Engneerng from the Natonal Unversty of Sngapore, Sngapore, n 00. Currently, she s workng as a postdoctoral scholar n the Department of Mechancal and Aerospace Engneerng, Unversty of Calforna, San Dego. Her current research nterests nclude nonlnear system control and ts applcatons.