Global Output-Feedback Tracking for Nonlinear Cascade Systems with Unknown Growth Rate and Control Coefficients

Transcription

1 J Syst Sci Complex 05 8: Global Output-Feedback Tracking for Nonlinear Cascade Systems with Unknown Growth Rate and Control Coefficients YAN Xuehua IU Yungang WANG Qingguo DOI: 0.007/s Received: 4 September 0 / Revised: 7 February 03 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 05 Abstract This paper deals with the global practical tracking problem by output-feedback for a class of uncertain cascade systems with zero-dynamics and unmeasured states dependent growth. The systems investigated are substantially different from the closely related works, and have zero-dynamics, unknown growth rate, and unknown time-varying control coefficients. This makes the problem much more difficult to solve. Motivated by the authors recent works, this paper proposes a new adaptive control scheme to achieve the global practical tracking. It is shown that the designed controller guarantees that the state of the resulting closed-loop system is globally bounded and the tracking error converges to a prescribed arbitrarily small neighborhood of the origin after a finite time. This is achieved by combining the methods of universal control and dead zone with backstepping technique, and using the framework of performance analysis in the closely related works. A numerical example demonstrates the effectiveness of the theoretical results. Keywords Adaptive control, dynamic high-gain, global practical tracking, output-feedback, uncertain nonlinear systems. YAN Xuehua School of Electrical Engineering, University of Jinan, Jinan 500, China; School of Control Science and Engineering, Shandong University, Jinan 5006, China. IU Yungang School of Control Science and Engineering, Shandong University, Jinan 5006, China. lygfr@sdu.edu.cn. WANG Qingguo Department of Electrical and Computer Engineering, National University of Singapore, Singapore 960, Singapore. This research was supported by the National Natural Science Foundation of China under Grant Nos , , 63304, and , the Natural Science Foundation for Distinguished Young Scholar of Shandong Province of China under Grant No. JQ0099, the Independent Innovation Foundation of Shandong University under Grant No. 0JC04, and the Doctoral Foundation of Jinan University under Grant No. XBS43. This paper was recommended for publication by Editor HONG Yiguang.

2 TRACKING FOR NONINEAR CASCADE SYSTEMS 3 Introduction Practical tracking control has received much attention over the last two decades for nonlinear systems, see, e.g., [ 3]. Particularly, [ 5] studied various nonlinear systems with unmeasured states dependent growth and proposed effective output-feedback tracking control design schemes. In [4, 5] and [], the system growth rate is a known constant and a known polynomial of output, respectively, and the reference signal to be tracked belongs to a known finite interval. Quite differently, in [], the system growth rate is an unknown constant, and the reference signal cannot be dominated by any known constant. On the other hand, control coefficients are precisely known in [, 4], and unknown in [, 5]. It is worth mentioning that in [,, 4, 5], the systems exclude zero-dynamics and time-varying control coefficients. Although [3] considered the systems with zero-dynamics and the upper bound of the controlcoefficient is not necessarily known, more restrictions are needed on the system and the reference signal. Recently, the output-feedback stabilization was considered for a class of systems with unmeasured states dependent growth, without the precise information on control coefficients and growth rate [4 6]. However, when both control coefficients and growth rate are unknown, the tracking problem is extremely different from the stabilization one and much more difficult to solve, as stated in []. Furthermore, when zero-dynamics exist and obey mild conditions, the tracking problem cannot be solved by trivially extending the corresponding results without zero-dynamics. As further investigation on [4 6] and the application of the analysis pattern in [], this paper is devoted to the tracking control design for the following more general nonlinear systems with zero-dynamics and unknown time-varying control coefficients : Ż = f 0 t, Z, η, u, η i = g i tη i+ + ψ i t, Z, η, u, i =,,,n, η n = g n tu + ψ n t, Z, η, u, y = η y r, where Z R m and η =[η,η,,η n ] T R n jointly define the system state; u R, y R and y r R are the control input, system output and reference signal to be tracked, respectively; g i t,i=,,,n, are nonzero, unknown time-varying functions; and f 0 : R + R m R n R R m, ψ i : R + R m R n R R, i=,,,nare unknown functions but continuous in the first argument and locally ipschitz in the others. In what follows, suppose only the system output y, i.e., the tracking error, is measurable the reasonability of the supposition has been discussed in [4]. The following notations will be used throughout this paper. R denotes the set of all real numbers; R + denotes the set of all nonnegative real numbers; R n denotes the real n-dimensional space. For a vector or matrix X, X T denotes its transpose. For any x R n, x denotes the -norm, i.e., x = x + x + + x n ; x denotes the Euclidean or - norm of vector x, and for the matrix P,weuse P to denote its norm induced by the -norm of the corresponding vector; for any x R n, there always holds x x n x. A continuous function α : R + R + is said to belong to Class K if it is strictly increasing, α0 = 0, and αr + as r +.

3 3 YAN XUEHUA IU YUNGANG WANG QINGGUO The objective of the paper is to design an adaptive output-feedback tracking controller such that for any prescribed λ>0, the state of the resulting closed-loop systems is well-defined and globally bounded on [0, + ; and for any initial condition, there is a finite time T λ > 0such that sup t Tλ yt =sup t Tλ η t y r t λ. To realize the desired objective, the following assumptions should be imposed on system and reference signal y r. Assumption There exists a continuously differentiable function U 0 Z, such that α Z U 0 Z γ Z, U 0 Z Z f 0t, Z, η, u Z + θ 0 η + θ 0, where α andγ, not necessarily known, are K function and positive constant, respectively, and θ 0 is an unknown positive constant. Assumption There exists an unknown positive constant θ, such that ψ i t, Z, η, u θ Z + θ η + + η i +θ, i =,,,n. Assumption 3 Control coefficients g i t, i=,,,n are continuously differentiable, of known signs and satisfy g g i t g, ġ i t g, where g and g are known positive constants. Assumption 4 The reference signal y r t is continuously differentiable, and moreover, there is an unknown constant M 0 such that sup y r t + ẏ r t M. t 0 The above four assumptions, particularly Assumption 3, make system substantially different from those in the related tracking results [ 5]. Detailedly, compared with [3], Assumption is apparently weaker, Assumption shows that system permits unknown relative degree and lower-order growing unmeasurable states, and Assumption 4 indicates that rather coarser information is required on the reference signal to be tracked. Unlike [, 4, 5], Assumptions and 4 imply that there is no known function or constant to dominate the nonlinearities of the η-subsystem and the reference signal and its derivative. Although [] studied the case of unknown constant growth rate, it excludes zero-dynamics and unknown/time-varying control coefficients, and hence Assumptions 4 make the systems in [] be a special case of this paper. From the foregoing insight, it can be realized that the practical tracking control design for system under Assumptions 4 is much more difficult. Mainly motivated by [, 6], this paper addresses the complete treatment to the tracking control design formulated above, and accomplishes the extension of [4 6], from stabilization to practical tracking. For details, the main contributions of the paper are composed of two respects. First, the system investigated is substantially different from those in [ 5, 4 6, 7] and especially more general than [,4 6],

4 TRACKING FOR NONINEAR CASCADE SYSTEMS 33 due to the presence of zero-dynamics, unknown growth rate, uncertain time-varying control coefficients and unknown reference signal. Second, different from the related tracking results [ 5], a distinct n-dimensional dynamic high-gain observer with a new updating law is constructed to rebuild the unmeasured system states, and based on this, we design an adaptive output-feedback controller to achieve the desired objective of global practical tracking. Controller Design To begin with, we introduce the following scaling transformation: x = gn g n t y, x i = gn g i n t η i, i =, 3,,n, where and whereafter, g i n t = n j=i g jt, i =,,,n, σ i j = j k=i σ k, i j n, and c i j = j k=i c k, i j n. In virtue of the above transformation and Z = Z, system can be transformed as follows: Ż = ft, Z, x, u, ẋ i = x i+ + φ i t, Z, x, u, i =,,,n, 3 ẋ n = g n u + φ n t, Z, x, u, y = gtx, where x =[x,x,,x n ] T, ft, Z, x, u =f 0 t, Z, η, u, gt = g g n n t, φ = gt ψ ẏ r n ġ jt j= g y jt,φ i = gn g ψi i nt n ġ jt j=i g η jt i,i=, 3,,n. By Assumptions 4, it is easy to find a known positive constant g M and an unknown positive constant θ, such that U 0 Z + θy + θ, φ i θ Z + θ x + x + + x i +θ, i =,,,n, gt g M, ġt g M. 4 Then, in terms of emma in [8], for any given positive constant μ<, suitable positive constants a i s can be chosen such that the matrix A is Hurwitz and there exists P = P T > 0 satisfying A T P + PA I and μi + DP + P μi + D 0, 5 where P is a certain symmetric and positive definite matrix, I denotes the n-dimensional identity matrix, D =diag{0,,,n }, and a 0 A = R n n. a n 0 a n 0 0

5 34 YAN XUEHUA IU YUNGANG WANG QINGGUO Next, for any prescribed λ>0, mainly motivated by [] and [4], by combining the traditional backstepping technique, with the ideas of universal control and dead-zone, we construct the following adaptive output-feedback controller based on the high-gain observer: x i = x i+ i a i x, i =,,,n, x n = g n u n a n x, { } 6 y λ = max μ + ξn, 0, with 0 =, and u = g n n+μ α n, 7 where ξ n is defined as ξ = μ y, ξ i = x i i +μ α i, y α = signgc μ, α i = c i ξ i, i =, 3,,n, 8 with c i s being some positive constants to be determined. Similar to those in [6], c i s in the paper are required to satisfy the following inequalities: c + Pa + gm, c i + i + 3 g M σ i i + a i i + a j c j j= i + c i g M + 3 g Mc 3 i +3 i 3 j +c j c j i j= +9 i 4 gm c i +i gm c4 c i, i =, 3,,n, 9 where a =[a,a,,a n ] T,and σ = + c, μ 4 { σ i = c i +max μ 4, c j i i c, 4σ j c j+ k j + σ j+ i σ k=j+ j+ k } j =,,,i, i =, 3,,n. 0 From 9 and 0, it is not difficult to see that there always exist the suitable c i s. In fact, as stated in [6], for known positive constants g M and μ, and design parameters a i s satisfying 5,

6 TRACKING FOR NONINEAR CASCADE SYSTEMS 35 c i s can be chosen along the following rule: first, select c by 9, and σ by 0; then, from i =toi = n, select c i by 9 under the chosen c,c,,c i and σ i, and subsequently select σ i by 0; finally, select c n by 9 under the chosen c,c,,c n and σ n. Remark. Different from the tracking results [ 5,8],adistinctn-dimensional observer based on a new dynamic high-gain, is introduced, which successfully overcomes the technical obstacles caused by unknowns/uncertainties in the system and the reference signal. Specifically, in contrast to [, 4, 5], the high-gain is dynamic, to effectively deal with the unknown system growth rate. The introduced observer given in 6 has lower dimension than that adopted in []. Besides, the observer in this paper is not driven by error y x, but is driven by the input, and hence distinguish from [] and [3]. Remark. Unlike [,, 4, 5, 8, 4 6], a sufficiently small positive parameter μ is introduced in the updating law, in order to deal with zero-dynamics satisfying Assumption. In fact, when without zero-dynamics, μ can be taken value since it is not necessarily small enough. On the other hand, it is easy to see that compared with [4, 6], although zero-dynamics are considered in the paper, the corresponding stabilizing controller is easier to implement due to the relatively simpler structure of the dynamic high-gain given in 6 when λ = 0. Itisworth pointing out that the expression of, is crucial to guarantee the boundedness of, as will be seen in the proof of Theorem Main Results In the subsequent section, let ε i = x i x i i +μ, z i = x i i +μ, i =,,,n, and denote x = [ x, x,, x n ] T,ξ = [ξ,ξ,,ξ n ] T,ε = [ε,ε,,ε n ] T,z = [z,z,,z n ] T. By, 6 and 7, one can verify that the closed-loop system is continuous and locally ipschitz in t and Z, η, x,, respectively, in a small open neighborhood of the initial condition, and hence the closed-loop system has a unique solution on a small interval [0, t s see Theorem 3., Page 8 of [9]. et [0, t f be its maximal interval on which a unique solution exists, where 0 <t f + see Theorem., Page 7 of [9]. Before stating the main results of the paper, we first give two fundamental propositions, whose detailed proofs are provided in Appendix for the sake of compactness. Specifically, Proposition 3. and Proposition 3., which play a key role in the later performance analysis, reveal the dynamic behavior of the closed-loop system via a yapunov candidate function, and the intrinsic relationship between the high-gain and the other variables, respectively. Proposition 3. For the resulting closed-loop system with c i s satisfying 9, thereexists a continuously differentiable and nonnegative function V Z, ε, ξ such that V Z μ c μ θ ε + ξ +θ holds on [0, t f,wherec is a known positive constant, and θ is an unknown positive constant.

7 36 YAN XUEHUA IU YUNGANG WANG QINGGUO Proof See Appendix A., where the emphasis is focused on showing that the following nonnegative function V is a suitable one for : V Z, ε, ξ =nσ n +U 0 Z+V n ε, ξ, V ε, ξ =ε T Pε+ ξ, V k ε, ξ =σ k V k + ξ k, k =, 3,,n. The proof is finished. Proposition 3. For the resulting closed-loop system, if is bounded on [0, t f,thenz, z, andε are bounded on [0, t f as well. Proof See Appendix A.. Now, we state the main theorem in this paper. Theorem 3.3 Suppose Assumptions 4 hold. Then, there exist design parameters, a i s and c i s, such that the closed-loop system defined by, 6 and 7 achieves the global practical tracking. Proof The theorem is proven by invoking the new framework of performance analysis proposed in []. Before proving the tracking performance, we will first show that the state of closed-loop system is well-defined and bounded on [0, + : Part I see below verifies that t f =+, i.e., the closed-loop solution exists on [0, +. Part II shows that all closed-loop signals are bounded on [0, +. To this end, it suffices to prove the boundedness of on [0, +, which together with Proposition 3. yields the boundedness of Z, ε, z, and hence that of all closed-loop signals by the definitions of x i s, ε i s and z i s, and the boundedness of g i s and y r. Part I t f =+ The proof of this part proceeds by contradiction. Suppose that t f is finite, then there must exist at least one of the following two situations: Case. is bounded on [0, t f ; Case. is unbounded on [0, t f. By the same reason as in [], we can conclude that Case is impossible to occur. In Case, by the continuousness and monotone nondecreasing property of, there exists a finite time t s 0, t f, such that t s = θ+ μ c and t θ+ μ c, t [t s,t f. Then, on [t s,t f, becomes V Z ε + ξ +θ. 3 In view of Theorem 4.8 in [0], from 3, it is easy to see that ξ is bounded on [t s,t f, and therefore, + = t f t s = tf t s tdt tf t s ξ t+ξ n t dt < +. This contradiction shows that Case also does not occur.

8 TRACKING FOR NONINEAR CASCADE SYSTEMS 37 Part II Boundedness of all closed-loop signals on [0, + From the above discussion and Proposition 3., we know that the key is to prove the boundedness of on [0, + for proving the boundedness of all closed-loop signals. Suppose for contradiction that is unbound on [0, +, i.e., lim t + t =+. Then, it is easy to see that 3 still holds on [t s, +. This shows that Z is bounded on [t s, +,andon[0, + as well. Therefore, by letting k = n ina8,wehaveon[t s, + V n σ n n+ θ ε ε σ n n θ n ξ n σ i n n i θ i ξi + μ + μ nσ n Z i= c θ ε + ξ + μ +nσ n sup Zt. t t s Similar to 3, for any constant δ, there must exist a finite time t δ t s, +, such that V n δv n + μ +nσ n sup Zt t t s { holds on [t δ, +, where is chosen such that t c θ + δ max λmax P σ n, }, t [tδ, +. Along the line as the proof of emma in Appendix A3 of [], the σ n boundedness of can be similarly established on [t s, +, and then, by the nondecreasing property of, we easily know is bounded on [0, +. By Proposition 3., it is concluded immediately that Z, z, andε are bounded on [0, +. Furthermore, by the definitions of z and ε, and the boundedness of, z and ε, it is not difficult to obtain that x is bounded on [0, +. From and Assumptions 3-4, it follows that η is bounded on [0, +. By the definition of u, we conclude that u is also bounded on [0, +. Up to now, we have proved that the state of the closed-loop system is well-defined and bounded on the maximal interval [0, +. Next, let us show the property of practical tracking, i.e., for any given λ>0, there exists a finite time T λ, such that yt λ, t T λ. It is easy to see that is continuously differentiable and lim t + t exists by virtue of the boundedness and the nondecreasing property of on [0, +. Using 3, 6, a9 with k = n, and the boundedness of y, x,,z,ξ,zon [0, +, we conclude that ẏ,, and ξ n are bounded on [0, +, and hence y λ +ξ μ n in has a bounded derivative on [0, +, denoting by N this upper bound, so one can obtain that is uniformly continuous on [0, +. In fact, ε >0, taking δ 0 = ε N, for any t,t [0, +, if only t t <δ 0,then t t y t λ μ y t λ t μ + ξ t nt ξnt N t t <ε.

9 38 YAN XUEHUA IU YUNGANG WANG QINGGUO Thus, by Barbălat s emma, we have lim t + t = 0. That is, for any initial condition Z0, η0, x0, there exists a finite time T λ > 0 such that for all t>t λ, y t λ μ t + ξn t t holds, which implies λ μ t yt = η t y r t λ, t>t λ. The proof of the theorem is completed. Remark 3.4 Note that it is necessary to introduce a suitable positive parameter μ in the updating law, since it can be seen from that 0 <μ< guarantees that sufficiently large can eliminate the influence of unknown positive constant θ. Although a similar approach has been developed in [], it cannot solve the problem of the paper by the existence of zero dynamics and unknown control coefficients. In addition, compared with [], the system investigated is general, and the observer adopted is not driven by the error. However, similar property such as, can be still obtained by choosing the suitable yapunov function. 4 An Illustrative Example In the section, a numerical example is given to illustrate the correctness of the theoretical results. Consider the following second-order nonlinear system: Ż = Z + θ 0 η, η = g tη + θ 0 η sinη +θ 0 +sintz, η = g tu + θ 0 sinuz, y = η y r, where θ 0 is an unknown constant, g i,i =, are unknown time-varying functions, and y r is the signal to be tracked. Suppose that control coefficients g and g satisfy 9 g i t 0, ġ i t 0, i=,. Then g =9, g = 0, g M canbechosenasg M =.47. Then a direct application of our proposed control method yields an adaptive output-feedback controller as follows: x = x a x, i =,,,n, x = g u a x, { } y λ = max μ + ξ, 0, 0 =, and u = g +μ c ξ, ξ = x +μ + c signg y μ. Barbălat s emma Suppose that ω : [0, + R is a continuously differentiable function, and lim t + ωt exists and is finite. If ωt,t [0, + is uniformly continuous, then lim t + ωt =0. For more basic and alterative forms of Barbălat s emma, refer the reader to [].

10 TRACKING FOR NONINEAR CASCADE SYSTEMS 39 Suppose that θ 0 =,g t =0 e t,g t = cost, y r =sint, and choose design parameters a =0,a =. It can be verified that such design parameters are suitable. In fact, by solving the matrix inequalities 5, one gets P =, and furthermore, in view of 9 and 0, we have c =7.6, σ =5.599, and then, let us choose c = et the tracking accuracy be λ =0., μ = 4, the initial conditions be Z0 =, η 0 =, η 0 = 0, x 0 =, x 0 = 7.5, we obtain the following Figures 6 by numerical simulation. To show the transient behavior more clearly, the Y -coordinates in Figures 6 have been shifted backward 0.s. From these figures, all the signals in the closed-loop system are bounded and the global practical tracking is achieved..5 6 η η 4 ^x ^x Time Sec Time Sec Figure The trajectories of state η,η T Figure The trajectories of observer x, x T Time Sec Time Sec Figure 3 The trajectory of tracking error y Figure 4 The trajectory of high-gain

11 40 YAN XUEHUA IU YUNGANG WANG QINGGUO Time Sec Time Sec Figure 5 The trajectory of control law u Figure 6 The trajectory of state Z 5 Concluding Remarks In this paper, different from the related tracking results [ 5], the global practical tracking problem has been studied for a class of uncertain nonlinear systems with the unmeasured state dependent growth in the presence of zero-dynamics, unknown growth rate and uncertain timevarying control coefficients. By flexibly using the idea of dead zone, the techniques in universal control theory, and backstepping approach, we have successfully constructed a new dynamic high-gain with an appropriate positive parameter, and based on this, designed an adaptive output-feedback controller to achieve the prescribed tracking objective. The future work will be directed at extending the control design scheme to more general nonlinear systems, such as the systems with output dependent growth rate. Acknowledgements The first author wishes to thank Department of Electrical and Computer Engineering, National University of Singapore, for all the encouragement and support during her visiting NUS. References [] Shang F, iu Y G, and Zhang C H, New results on adaptive tracking by output feedback for a class of uncertain nonlinear systems, Control Theory & Applications, 00, 76: in Chinese. [] Yan X H and iu Y G, Global practical tracking by output-feedback for nonlinear systems with unknown growth rate, Science China: Information Sciences, 0, 540: [3] Bullinger E and Allgöwer F, Adaptive λ-tracking for nonlinear higher relative degree systems, Automatica, 005, 47: [4] Gong Q and Qian C J, Global practical tracking of a class of nonlinear systems by output feedback, Automatica, 007, 43:

12 TRACKING FOR NONINEAR CASCADE SYSTEMS 4 [5] Zhai J Y and Fei S M, Global practical tracking control for a class of uncertain nonlinear systems, IET Control Theory and Applications, 0, 5: [6] Ilchmann A and Ryan E P, Universal λ-tracking for nonlinearly-perturbed systems in the presence of noise, Automatica, 994, 30: [7] Ryan E P, A nonlinear universal servomechanism, IEEE Transactions on Automatic Control, 994, 394: [8] Ye X D, Universal λ-tracking for nonlinearly-perturbed systems without restrictions on the relative degree, Automatica, 999, 35: [9] Ye X D and Ding Z T, Robust tracking control of uncertain nonlinear systems with unknown control directions, Systems & Control etters, 00, 4: 0. [0] Qian C J and in W, Practical output tracking of nonlinear systems with uncontrollable unstable linearization, IEEE Transactions on Automatic Control, 00, 47: 36. [] in W and Pongvuthithum R, Adaptive output tracking of inherently nonlinear systems with nonlinear parameterization, IEEE Transactions on Automatic Control, 003, 480: [] Sun Z Y and iu Y G, Adaptive practical output tracking control for high-order nonlinear uncertain systems, Acta Automatica Sinica, 008, 348: [3] Yan X H and iu Y G, Global practical tracking for high-order uncertain nonlinear systems with unknown control directions, SIAM Journal on Control and Optimization, 00, 487: [4] Yan X H and iu Y G, Global output-feedback asymptotic stabilization for a class of uncertain nonlinear systems with unknown growth rate, Proceedings of the 30th Chinese Control Conference, Yantai, China, 0. [5] Shang F and iu Y G, Adaptive output-feedback stabilization for a class of uncertain nonlinear systems, Acta Automatica Sinica, 00, 36: [6] Yan X H and iu Y G, New results on global output-feedback stabilization for nonlinear systems with unknown growth rate, Journal of Control Theory and Applications, 03, 3: [7] Chen Z Y and Huang J, Global output feedback stabilization for uncertain nonlinear systems with output dependent incremental rate, Proceedings of the 004 American Control Conference, Boston, MA, USA, 004. [8] Praly and Jiang Z P, inear output feedback with dynamic high gain for nonlinear systems, Systems & Control etters, 004, 53: [9] Hale J K, Ordinary Differential Equations, nd ed., Huntington, New York, 980. [0] Khalil H K, Nonlinear Systems, 3rd ed., Upper Saddle River, New Jersey, 00. [] Min Y Y and iu Y G, Barbălat lemma and its application in analysis of system stability, Journal of Shandong University Engineering Science, 007, 37: 5 55 in Chinese. Appendix A. The Proof of Proposition 3. With the help of 3, 6, 7, and the definitions of ε i s and z i s, the whole control systems can be represented in the following compact form:

13 4 YAN XUEHUA IU YUNGANG WANG QINGGUO Ż = ft, Z, x, u, ε = Aε + Φt, Z, x, u, + μ ax μi + Dε, ż = Az e n c n ξ n μi + Dz, =max { y λ μ + ξn, 0 }, 0 =, a where Φ = [ μ φ, +μ φ,, n +μ φ n ] T,anden =[0, 0,, 0, ] T. First, let V ε = ε T Pε. Noticing that 0, and, and in view of 5, along the solutions of a, the time derivative of V ε satisfies V ε ε +ε T P Φ +ε T P μ ax. a For the last two terms on the right-hand side of a, noticing that 0 <μ<,weobtain ε T P Φ μ Z + θ ε ε + 4 n i= z i + ξ + σ n n + μ, ε T P μ ax =ε T Pa gt ξ ε + Pa ξ, a3 where θ ε =4n 3 θ P +n nθ P +3n θ P + σ n n +n θ P. Thus, from a and a3, it follows that V ε θ ε ε + + Pa ξ + n 4 i= + σ n n + μ + μ Z. z i a4 When k =, in view of, a and a4, we have V θ ε ε + + Pa ξ + 4 n zi + σ n n + μ. a5 + μ Z + ξ gtε + gtz + gt μ φ + ġt gt ξ μ ξ By 4, 8, and completing the square, there holds ξ gtε 4 ε + gmξ, gt μ ξ φ μ Z + θgm ++ 4 θ gm n +σ n ġt gt g M, i= ξ + σ n n + μ,

14 TRACKING FOR NONINEAR CASCADE SYSTEMS 43 4 z c ξ + ξ. Besides, by 8, and the definition of z,noting gt, we have gtξ z gtξ ξ c ξ. a6 a7 Then, by 9, substituting a6 and a7 into a5 yields that V 4 θ ε ε θ ξ μ ξ + n zi 4 i=3 + σ n n + μ + ξ + gtξ ξ + μ Z, where θ = θg M ++ 4 θ g Mn +σ n + + g M + c. When k =, 3,,n, we will prove the following conclusion by induction that for any k =, 3,,n, the virtual controller 7 can guarantee that the time derivative of the augmented function V k given by satisfies V k σ k k+ θ ε ε σ k k θ k k σ i+ k σ i μ c k c i i+ j + 4 where i= k + + σ k n n + μ + σ k σ j=i+ i+ j ξ k i= σ i k k i θ i ξi ξ i μ ξ k + σ k 4 ξ k+ + ξ kξ k+ + μ kσ k Z, n i=k+ θ k = θ k, + θ c k + g M, c θ i = i + σ i + σ i n θ gm n +c i 4 + g M, i =, 3,,k. In fact, it is assumed that V k satisfies similar properties to a8. Noticing that ξ k =ż k α k = z k+ a k z k +μ ξ k + c k z k a k z + ξ k z i a8 +c k z k a k z + ξ k + +signgc gtε + gtz + gt μ φ + ġt gt ξ + ξ, a9 differentiating V k yields that V k = σ k Vk + ξ k ξk

15 44 YAN XUEHUA IU YUNGANG WANG QINGGUO σ k k θ ε ε σ k k σ k σ i+ k σ i μ c i 4 i= + n 4 σ k zi k + i=k+ + μ k σ k Z + ξ k +c k + z k a k z + ξ k + c k k k θ k, ξ σ i k k i θ i i= k c i+ j ξ i σ k μ ξ k σ j=i+ i+ j σ k n n + μ + σ k ξk + σ k ξ k ξ k z k+ a k z k +μ ξ k z k a k z + ξ k + +signgc gtε + gtz + gt μ φ + ġt gt ξ + ξ. a0 According to, 4, 8, the definitions of ε and z i s, and the fact that σ i >, i=,,,k, by the method of completing square, we have σ k ξ k ξ k 6 σ k ξk + 3 σ k ξk, k a k a i c i z ξ k k k σ k ξ a +k k + a i c i ξk i= + k+ σ k ε + k ξk, c k z k ξ k c k + 3 c3 k ξk + σ k 6 ξ k, k k c i k z i+ ξ k 3 k 3 i c i k +c i ξk i= i= + σ k 6 ξ k + k i=3 i= σ i k k i ξ i + σ k 3 k 3 ξ, k signgc k ξ ξ k + c i k ξ iξ k k c i k 4 i= i= signgc k gtε ξ k σ k k+ ε + k gmc k ξ k, 3 signgc k gtz ξ k k gmc k 4 + c ξ i +k ξ k, ξ k + σ k k ξ + σ k 6 k 3 ξ, gt signgc k μ φ θ ξ k σ k μ Z + σ k c k ξ σ k + + θ g M n +σ k n c k ξk 4 + σ k n n + μ, ġt gt ξ ξ k σ k g M ξ + g M ξk, 4 σ k zk+ σ k c k ξ k + σ k ξk+. a ξ i

16 TRACKING FOR NONINEAR CASCADE SYSTEMS 45 Then, noticing that σ i >, i=,,,k, from 9, a0, and a, with the help of the definitions of θ k and θ k, it is not difficult to verify that a8 holds. Thus, by completing the square, from 4, 0, and a8 with k = n, differentiating yapunov function V =nσ n +U 0 + V n yields that ε σ n V Z σ n n+ θ ε n μ ξ σ i n n i θ i i= Z σ n μ n+ μ θ ε n θ + nσ n ξ + + θ + nσ n. i= ξ i + n θ n θ + nσ n μ + θ + nσ n ε σ n μ σ i n μ n i μ θ i ξ i n μ θ n a Moreover, one easily verifies a becomes with c = n+, θ =max { σ n θ ε,σ n θ n +θ+nσ n, σ n θ,,σ n θ n,θ n, +θ+nσ n } under the design parameters σ i s satisfying 0. A. The Proof of Proposition 3. In this section, we will prove the boundedness of Z, z and ε on [0, t f under the premise that is bounded on [0, t f denote the upper bound of on [0, t f by. We first prove the boundedness of Z on [0, t f. With the aid of the definition of, by Assumption, we have U 0 Z + θ μ ξ + θ γ U 0 + θ μ + θ λ +, from which, it follows that U 0 U θ μ + μ+ + θγ λ +. From this and Assumption, we easily obtain the boundedness of Z on [0, t f. Next, we prove that z is bounded on [0, t f. Consider the yapunov function V z = z T Pz for the z-dynamic system of a. By 5, a direct computation yields that V z = z T A T P + PAz c n z T Pe n ξ n zt μi + DP + P μi + D z z +c n P ξ n z +c n P + c n P λ, from which, it follows that zt z T 0Pz0 + c n λ min P P +λ max P c n P λ, t [0, t f.

17 46 YAN XUEHUA IU YUNGANG WANG QINGGUO Finally, let us establish the boundedness of ε on [0, t f. To this end, let ε i = where is a constant satisfying x i x i, i =,,,n, i +μ max {, θ ε +4 }. a3 Then, the error dynamics a can be rewritten as ε = Aε + aε Γ aε + μ Γ ax + Φ t, Z, x, u, a4 { where ε =[ε, ε,, ε n ] T, Γ =diag,,, n }, Γ =diag{ μ,, n +μ }, [ ] and Φ = φ φ, φ T. μ,, n +μ n +μ For system a4, consider the function V ε = ε T P ε. Then, along the solutions of a4, by a simple calculation, on [0, t f, we have V ε ε +ε a T P ε ε a T Γ P ε +x μ a T Γ P ε +Φ T P ε. According to 4, a3 and the definitions of ε and ε, by the method of completing the square, the following inequalities hold ε a T P ε +ε a T Γ P ε a T P + a T Γ P ε + ε a T P + a P ξ + z+ ε, x μ a T Γ P ε a T Γ P μ x + ε a P ξ + ε, Φ T P ε Z + θ ε ε + n zi + 4 ξ +, i= and then, by a3, noticing that, we get V ε Z θ ε 3 ε + θ εz z + θ εz ξ + λ max P V ε + θ εz + θεz sup zt + sup Zt + λ θ εz +, t [0,t f t [0,t f a5 where θ εz = + a P + a T P + a P. From a5, it is not difficult to get εt θ εz sup zt + sup λ min P t [0,t f +ε T 0P ε0 + θ εz, t [0, t f. t [0,t f Zt + λ θ εz + λ max P Furthermore, it follows from the definitions of ε i and ε i, and a3 that ε is bounded on [0, t f. The proof is completed.