Design of Observer-based Adaptive Controller for Nonlinear Systems with Unmodeled Dynamics and Actuator Dead-zone

International Journal of Automation and Computing 8), May, -8 DOI:.7/s633--574-4 Design of Observer-based Adaptive Controller for Nonlinear Systems with Unmodeled Dynamics and Actuator Dead-zone Xue-Li Wu, Xiao-Jing Wu Xiao-Yuan Luo Quan-Min Zhu 3 Department of Electrical Engineering, Yanshan University, Qinhuangdao 664, PRC College of Electrical Engineering and Information, Hebei University of Science and Technology, Shijiazhuang 58, PRC 3 Bristol Institute of Technology, University of the West of England, Coldharbour Lane, Bristol BS6QY, UK Abstract: This paper presents an up-to-date study on the observer-based control problem for nonlinear systems in the presence of unmodeled dynamics and actuator dead-zone. By introducing a dynamic signal to dominate the unmodeled dynamics and using an adaptive nonlinear damping to counter the effects of the nonlinearities and dead-zone input, the proposed observer and controller can ensure that the closed-loop system is asymptotically stable in the sense of uniform ultimate boundedness. Only one adaptive parameter is needed no matter how many unknown parameters there are. The system investigated is more general and there is no need to solve Linear matrix inequality LMI). Moreover, with our method, some assumptions imposed on nonlinear terms and dead-zone input are relaxed. Finally, simulations illustrate the effectiveness of the proposed adaptive control scheme. Keywords: Observer, nonlinear system, unmodelded dynamics, adaptive, actuator dead-zone. Introduction Much progress has been made in the field of adaptive control for nonlinear systems 4. In these researches, adaptive schemes were used online to gather data and adjust the control parameters automatically. However, the results are based on the assumption that system states must be fully accessible, whereas in practice, the assumption is often unreasonable. This has motivated research in observer-based control or output feedback control) for nonlinear systems. Based on state observer, the adaptive control for nonlinear systems was developed in 5. It is easy to see that the nonlinear terms are generally assumed either to satisfy Lipschitz condition or to be bounded by output injection terms or a known function of time). In conclusion, there are some limitations of the proposed method in these literature. Most importantly, the above-mentioned control scheme cannot apply to the nonlinear systems with unmodeled dynamics and actuator dead-zone input. However, it is well-known that unmodeled dynamics exist in almost all applications of observer and controller because it is generally impossible to use exact and detailed models. It was indicated in that the adaptive algorithms without considering unmodeled dynamics may lead the control systems being unstable. It was further indicated in that even the stable unmodeled dynamics may reduce the stable region significantly or cause the controlled systems to be unstable. Therefore, the control of nonlinear systems with unmodeled dynamics has emerged as an active research area recently, e.g., 3. Unfortunately, there are very limited results obtained on the output feedback control of nonlinear systems with unmodeled dynamics 7. In 7, an adaptive backstep- Manuscript received May 4, ; revised August 9, This work was supported by National Natural Science Foundation of China No. 6749). ping state feedback controller was proposed for nonlinear systems with unmodeled dynamics. Moreover, an adaptive output feedback control problem was solved. In 8, a novel combined backstepping and small-gain approach was presented to the global adaptive output feedback control of a class of nonlinear systems with unmodeled dynamics. However, the nonlinear terms were assumed to be bounded by output injection terms in 7, 8. In 9,, the assumptions were more relaxed than that of 7, 8. The nonlinear functions included not only the output, but also the unmeasured state variables. The common drawback of the above-mentioned results is that the actuator dead-zone input was not taken into account. However, actuator dead-zone is common in many practical systems such as mechanical connections, hydraulic servo valves, piezoelectric translators, and electric servomotors. The presence of such nonlinearities usually can be a cause of instability, and even degrades the performance of systems badly. To cope with this inherent problem and further improve the performance of systems, many results have been obtained 5. The inverse dead-zone nonlinearity is employed to minimize the effects of dead-zone, such as,. By modeling the input dead-zone as a combination of a line and a disturbance-like term, Wang et al. 3 designed a state feedback controller for the symmetric dead-zone input case, and the authors of 4, 5 further relaxed the assumption. Nevertheless, the designed controllers were obtained by assuming that all the states can be obtained in 5. To the best of the authors knowledge, to date, there has been no result obtained on the output feedback control of nonlinear systems with unmodeled dynamics and actuator dead-zone input. The two negative factors, as noted above, will bring new challenges for the control of nonlinear systems. The main contribution of this paper is to design an observer-based adaptive controller for nonlinear systems with unmodeled dynamics and actuator dead-zone, by in-

International Journal of Automation and Computing 8), May troducing a dynamic signal to dominate the unmodeled dynamics and using an adaptive nonlinear damping to counter the effects of the nonlinearities and dead-zone input. There are some new features in the proposed control scheme: ) The presented control strategy can handle not only the class of nonlinear systems with linear parameterization, but also the nonlinear systems in which the unknown parameters appear nonlinearly. ) Compared with existing results on output feedback control, the conditions imposed on the nonlinear terms in this paper are more relaxed, and the system under investigation is more general. In 5, 6,, Lipschitz conditions were imposed on the systems. In 7, the nonlinear function was bounded by a known function of time. In 8, 9, the nonlinear functions were required to be only related to the output. In 7, 8, nonlinear functions were bounded by output injection terms. As for our work, however, it follows from Assumption that the nonlinear term is bounded by a function that includes not only the output, but also the unmeasured state variables of the system. 3) With the presented method, neither solving linear matrix inequality e.g., 9,, 4, 5) nor estimating several unknown parameters e.g., 4, 5) are needed. No matter how high the order of the system is and how many unknown parameters there are, the observer and controller have only one adaptive parameter updated according to the adaptive law. Moreover, with the help of introducing some unknown parameters, the assumptions imposed in 4 are relaxed. The rest of this paper is organized as follows. In Section, the problem to be tackled is stated and some lemmas and assumptions are introduced. In Section 3, an adaptive controller is proposed by using the measurable signals i.e., the states of the observer and the output of system) and the corresponding stability analysis is made. Some numerical examples are presented to illustrate the effectiveness of the proposed controller in Section 4. Finally, the paper is ended by some conclusions in Section 5. Problem statement Consider the following nonlinear system with unmodeled dynamics: ż = q y, z) ẋ = Ax + Bf x, z, θ) + BΓ u) y = Cx where x R n is the state of the system; u R m is the control input; y R p is the measured output; z represents the unmodeled dynamics; θ is the unknown parameter vector; f x, z, θ) is an unknown nonlinear function; A, B, and C are known constant matrices with appropriate sizes. q y, z) is an unknown nonlinear function. Γ u) is the output of actuator dead-zone and can be expressed in the following form: m r u b r), if u b r Γ u) =, if b l < u < b r ) m l u + b l ), if u b l. The parameters m r and m l stand for the right and the left slope of the dead-zone, respectively. The parameters b r ) and b l represent the breakpoint of the input nonlinearity. The actuator dead-zone is modeled as a combination of a line and a disturbance-like term 3 : with Γ u) = m t) u + p t) 3) m t) = { m l, if u m r, if u > m rb r, if u b r p t) = m t) u, if b l < u < b r m l b l, if u b l. First, the following lemmas are introduced which will be used in our main results. Lemma 7. A C function V w is said to be an exponentially input-to-state practically stable exp-isps) Lyapunov function for system ẇ = f w, υ) if ) there exist functions α and α of class K such that α w ) V w α w ), w R n. 4) ) there exist two constants c >, d, and a class K function γ such that V w f w, υ) cvw + γ υ ) + d 5) w when 5) holds with d =, the function V w is referred to as an exponentially input-to-state stable exp-iss) Lypapunov function. Lemma 7. If V w is an exp-isps Lyapunov function for a control system ẇ = f w, υ), i.e., 4) and 5) hold, then, for any constants c in, c), any initial instant t, any initial condition w = w t ) and w >, for any function γ such that γ υ) γ υ ), there exist a finite T = T c, γ, w ), a nonnegative function D t, t) defined for all t t and a signal described by w = c w + γ υ) + d, w t ) = w 6) such that D t, t) = for all t t + T and V w w) w t) + D t, t) 7) for all t t where the solutions are defined. Throughout the paper, the following assumptions are imposed on system ). Assumption. The z-system in ) has an exp-isps Lyapunov function V z satisfying α z ) V z z) α z ) V z z) q y, z) c V z z) + γ z y ) + d 8) where α ), α ), and γ y ) are functions of class K, c >, d. Without loss of generality, we assume throughout that γ is a smooth function and of the form γ s) = s γ s ) with γ a nonnegative smooth function. Otherwise, using Lemma, it suffices to replace γ in 8) by y γ y ) + ε with ε > being a sufficiently small real number.

X. L. Wu et al. / Design of Observer-based Adaptive Controller for Nonlinear Systems with 3 In this work, to dominate the effect of the unmodeled dynamics on system, the following dynamic signal is introduced. ṙ = c r + γ y ) + d, r t ) = r > 9) where c, c ). Using Lemmas and and Assumption, it is easy to see that V z z) r t) + D t, t). ) Assumption. There exist unknown constants a i, i =,,, 4, such that f x, z, θ) a + a x ξ y) + a 3ς y) + a 4α z ) ) where ξ y) and ς y) are known nonnegative functions, and α ) is a function of class K. Assumption 3. The pair C, A) is observable. There exist positive-definite Q and P satisfying P A L + A T LP = Q ) B T P = C 3) where A L = A LC is strictly Hurwitz. Assumption 4. The pair A, B) is controllable. There exist positive-definite H and Ω satisfying HA K + A T KH = Ω 4) where A K = A BK is strictly Hurwitz. Assumption 5. Parameters m l, m r, b r, and b l are positive but unknown. There exists enough small unknown parameter χ such that < χ max {m r, m l }. Moreover, the disturbance p t) is bounded by an unknown constant p. Remark. Compared with the literature on output feedback control of nonlinear systems, such as 5, 7,8, the class of nonlinear systems investigated in this paper is more general. It is easy to see that the right side of ) includes not only the output y but also the unmeasured x and z. However, the common assumptions used in the literature are the special cases of Assumption, such as that the unknown nonlinearity satisfies the Lipschitz condition 5, 6,, and the nonlinear functions are only related to the output or a known function 7 9,7, 8. The former condition corresponds to the case when a = a 3 =, ξ y) =, and α z ) = z in ). The latter condition corresponds to the case when a = a 4 = in ). Remark. Unlike 7, 4, the proposed control method can be used in the control of nonlinearly parameterized systems NLP-systems) because a lot of NLP-systems also satisfy the Assumption. The main problem that will be solved in this work is given as follows: Problem. In this paper, the objective is to design an adaptive controller for nonlinear systems with unmodeled dynamics and actuator dead-zone based on observer, which can guarantee the closed-loop system is asymptotically stable in the sense of uniform ultimate boundedness. 3 Combined design of observer and controller In this section, the controller will be proposed based on the signals of observer ˆx and the output of system y. The designed controller can guarantee that the closed-loop system is asymptotically stable in the sense of uniform ultimate boundedness. The observer is designed as follows: { ˆx = Aˆx + L y ŷ) + Bu + ˆβB y ŷ) jm 5) ŷ = C ˆx with j m = + ˆx ξ y) + ξ y) + ς y) + α α r t)) ) + y ŷ. The adaptive law and the control input are given by 6) ˆβ = y ŷ j m σˆβ; ˆβ ) > 7) u = K ˆx ˆβ y ŷ) j m 8) where > and σ > are design constants. The gain matrices L and K in 5) and 8) can be chosen so that A L = A LC and A k = A BK are strictly Hurwitz. Define the error e = x ˆx. From ) and 5), we have ė = ẋ ˆx = A LC) e + Bf x, z, θ) ˆβB y ŷ) j m + B Γ u) u) y e = Ce. 9) Remark 3. It is noted that the control input u of the observer in 5) is different from the control input Γ u) in ) because of the existence of the actuator dead-zone in the original system. This brings some difficulties for the design of observer-based controller. While in most of the observerbased controller design problems, the control input of the observer and the control input of the original system were assumed to be identical, such as 5, 6, 9,. Theorem. If the nonlinear system ) with unmodeled dynamics and dead-zone input satisfies Assumptions 5, the observer and the controller given by 5) 8) ensure that the closed-loop system is asymptotically stable in the sense of uniform ultimate boundedness. Proof. Choose the Lyapunov function candidate with V = π h V + V ) V = e T P e + ) β χˆβ χ V = ˆx T H ˆx ) where β > is a constant, which is the desired value of ˆβ. π h is defined below. Taking the derivative of V with respect to time along the solution of 9) gives V e P T A LC) + A LC) T P e+ e T P Bf x, z, θ) + m ) e T P Bu+ ) e T P Bp t) ˆβe T P B y ŷ) j m ) β χˆβ ˆβ.

4 International Journal of Automation and Computing 8), May Substituting ) and ) into ) results in e a T 3 P B ς y) τ a 3 + τ e T P B ς y) 33) V e T Qe + a e T P B + a e T P B x ξ y) + a3 e T P B ς y) + a 4 e T P B α z ) ˆβe T P B y ŷ) j m β χˆβ) ˆβ + m ) e T P Bu+ e T P B p. 3) Since α is K function, α is an increasing function, by 8) and ), we have z α r t) + D t, t)). 4) Then, taking into consideration that α is K function, one further has the following relation: α z ) α α r t) + D t, t)) ). 5) It is easy to see the fact that r t) + D t, t) max {r t), D t, t)}. 6) Hence, following the above fact and 5), one has α z ) α α r t)) ) + α α D t, t)) ). 7) Thus, V e T Qe + a e T P B + a e T P B x ξ y) + a3 e T P B ς y) + a 4 e T P B sup α α D t, t)) ) + Denoting ā gives a 4 e T P B α α r t)) ) β χˆβ) ˆβ+ ˆβe T P B y ŷ) j m m ) e T P Bu + e T P B p. 8) = a 4 sup α α D t, t)) ) + a + p, it V e T Qe + ā e T P B + a e T P B x ξ y) + a 3 e T P B ς y) + a 4 e T P B α α r t)) ) ˆβe T P B y ŷ) j m β χˆβ) ˆβ+ m ) e T P Bu. Substituting control input u into 9), we have V e T Qe + ā e T P B + a e T P B x ξ y) + a3 e T P B ς y) + a 4 e T P B α α r t)) ) β χˆβ) ˆβ m ) e T P BK ˆx mˆβe T P B y ŷ) j m. 9) 3) It follows from the fact Young s inequality) that e ā T P B τ ā + τ e T P B 3) a e T P B x ξ y) = a e T P B ˆx ξ y) + a e T P B e ξ y) τ a + τ e T P B ˆx ξ y) + a ρ e + a ρ e T P B ξ y) 3) a 4 e T P B α α r t)) ) τ a 4 + τ e T P B α α r t)) ) 34) m ) e T P BK ˆx τ m ) e T P B + τ ˆx T K T K ˆx τι m ) 4 + τι e T P B 4 + τ ˆx T K T K ˆx where τ, ρ, and ι are the positive constants. One has V e T Qe + a ρ e + τ e T P B + τ e T P B ˆx ξ y) + a ρ e T P B ξ y) + τ e T P B ς y) + 35) τ e T P B α α r t)) ) + τι e T P B 4 + τ ˆx T K T K ˆx + τ ā + τ a + τ a 3 + τ a 4 + τι m ) 4 mˆβe T P B y ŷ) j m β χˆβ) ˆβ. 36) a Choosing ρ >, e.g., a λ min Q) ρ λmin Q), and let β = max {a ρ/, τ, τι/}, 36) can be rewritten as V et Qe + β e T P B + ˆx ξ y) + ξ y) + ς y) + α α r t)) ) + y ŷ + N mˆβe T P B y ŷ) j m β χˆβ) ˆβ+ τ ˆx T K T K ˆx 37) where N = τ ā + τ a + τ a 3 + τ a 4 + τι m ) 4. Substituting definition of j m 6) and adaptive law 7) into 37), V et Qe + β e T P B j m mˆβe T P B ) y ŷ) j m β χˆβ y ŷ j m+ σ β χˆβ) ˆβ + N + τ ˆx T K T K ˆx. 38) There exists an enough small parameter χ such that < χ max {m r, m l } due to Assumption 5. Furthermore, it follows from adaptive law 7) that the solution satisfy ˆβ >, and one has mˆβe T P B y ŷ) j m χˆβe T P B y ŷ) j m. Therefore, V et Qe σ β χˆβ) + τ ˆx T K T K ˆx + M η ē + τ ˆx T K T K ˆx + M 39) where M = N + σβ, η = min {λ min Q/), σ} ; ē = T e β χˆβ. Taking the derivative of V with respect to time along

X. L. Wu et al. / Design of Observer-based Adaptive Controller for Nonlinear Systems with 5 the solution of 5) gives V = ˆx T H A BK) + A BK) T H ˆx T HLCe ˆx T Ωˆx + HLC ˆx e η c ˆx + π ˆx e η c ˆx + π ˆx ē ˆx+ 4) where π = HLC and η c = λ min Ω). Based on the above analysis, the derivative of V becomes V π h η ē + τ ˆx T K T K ˆx + M ) η c ˆx + π ˆx ē π h η ē + π h τ ˆx T K T K ˆx+ π h M η c ˆx + π ˆx ē π h η ē + π ˆx ē + π h M ηc π h τ λ max K T K )) ˆx = π h η ē + π ˆx ē η ˆx + π h M 4) where parameter τ can be selected to be large enough to let η = η c π h τ λ max K T K ) >. If choosing π h > π, 4) can be rewritten as η η V π h η ē η ˆx + π h η η ˆx ē + π h M. 4) With the following inequality πh η η ˆx ē π hη the derivative of V is transformed to ē + η ˆx 43) V π hη ē η ˆx + π h M π hη e π hη β χˆβ η ˆx + π h M µv + π h M. { } 44) where µ = min πh η, π hη χ η λ maxp, ) λ max{h}. From 44), the following relations are obtained: V t) e µt V ) π hm µ ) + π hm µ lim V t) π 45) hm t µ. On the other hand, π h M/µ can be reduced by choosing an appropriately large τ, an enough large ι, and a sufficiently small σ. Therefore, the signals e, ˆx, and β χˆβ are uniformly ultimately bounded. Then, following x = e + ˆx, the signal x is also uniformly ultimately bounded. Remark 4. The works of 9, relaxed the assumptions imposed in 7, 8 that the nonlinear functions were bounded by the output injection terms. Also, the proposed scheme in 5 did not need the condition imposed in 4 that the maximum and minimum values of the slopes should be available. But solving linear matrix inequality LMI) and choosing some constants were needed when designing the controller. With our method, however, neither solving LMI nor choosing the parameters is needed, although some parameters are introduced, such as ρ, τ, and π h. Therefore, our proposed method is simpler. Remark 5. In the proof of Theorem, parameters ρ, τ, π h, are introduced. With the help of these parameters, the effectiveness of the designed adaptive controller is proved. However, the observer, controller, and adaptive law given in 5) 8) are independent of these constants. Thus, it is not necessary for the system designer to know or to choose these constants. Moreover, the unknown parameter χ is introduced for the purpose of proving the Theorem, but this parameter is not required to be known for controller design. Unlike 4, it follows from the Assumption 5 that it need not have any knowledge on actuator dead-zone input by using our method. In fact, the proposed observer-based adaptive controller can adjust automatically to counter the destabilizing effects of the unknown parameters, unmodeled dynamics, and actuator dead-zone input. Remark 6. In this paper, adaptive output feedback control is investigated for nonlinear systems with unmodeled dynamics and actuator dead-zone input. Obviously, it is easy to design an observer-based adaptive controller for the nonlinear systems not only with unmodeled dynamics and actuator dead-zone, but also with the external disturbance. 4 Simulations In this section, three examples are presented to show the effectiveness of the proposed controller design method. Example. Consider a nonlinearly parameterized system with unmodeled dynamics and actuator dead-zone: ż = z + y 3 3 ẋ = x + θ 5z sin θ 6z) + Γ u)} y = x { θx x θ + θ 3e θ 4x + 46) where θ =, θ =, θ 3 =.5, θ 4 =, θ 5 =, and θ 6 =. To dominate the effects of the unmodeled dynamics, the dynamic signal r t) is introduced as ṙ t) =.6 r t) +.5 y 4 +.65. 47) Choosing L = 4, 4 T and K =, 5, the observer and the controller are designed as 5) and 8), where j m = + y 4 + ˆx y 4 +4r + y ŷ and ˆβ ) = y ŷ j m σˆβ. The dead-zone input parameters are m r =.7, m l =, b r =, and b l = 3. Initial values are selected as x ) =, T, ˆx ) =, T, z ) =, r ) =, and ˆβ ) = 8. With the proposed controller in Theorem, the trajectories of x and x are shown in Fig., from which we can see that the constructed observer and controller can render the closed-loop system to have good transient and steady-state performances in the presence of the unmodeled dynamics and dead-zone input. Moreover, if the unmodeled dynamics are not considered in the design scheme eliminating the term 4r in j m), the curves of states are shown in Fig.. It can be seen that the transient responses of the closed-loop system are worse.

6 International Journal of Automation and Computing 8), May term 4r in j m), the curves of states are shown in Fig. 5. It can be seen that performances of the closed-loop system are degraded. Fig. The curves of x and x with proposed controller Fig. 3 The chaotic system Fig. The simulation results without considering unmodeled dynamics Example. Consider a chaotic system 6 with unmodeled dynamics and actuator dead-zone: ż = z + y ẋ = y =.43 6 x x + f x, µ, z) + Γ u) 48) where f x, µ, z) = µx 3 + θ z sin θ z), µ =, θ =, and θ =. The dynamic signal r t) introduced is the same as Example. The matrices L and K are designed as L =.43,.67,.559 T and K =,, 5. The dead-zone input parameters are m r =.7, m l =, b r =, and b l = 3. Initial values are selected as x ) =,, 3 T, and ˆx ) =,, T, z ) =, r ) =, and ˆβ ) = 8. The simulation results are shown in Figs. 3 5. The behavior of the chaotic system without unmodeled dynamics is shown in Fig. 3. Fig. 4 displays the trajectories of states x and x by using the proposed observer and controller as 5) 8). Moreover, if the unmodeled dynamics are not considered in the design scheme eliminating the Fig. 4 The curves of x, x, and x 3 with proposed controller Fig. 5 The simulation results without considering unmodeled dynamics

X. L. Wu et al. / Design of Observer-based Adaptive Controller for Nonlinear Systems with 7 Example 3. In order to illustrate that our method make the closed-loop system possess a better performance compared with the method proposed in 6, a simple example is given in the following. Consider the following system without unmodeled dynamics and actuator dead-zone: 3 3 ẋ = x + y = x. u + x sin x ) + x sin x ) ) 49) By using the method in 6, the observer and controller are designed as ˆx = 3 3 ˆx + u = 5 u + x ˆx ) ˆθ 5) ˆx ˆθ y C ˆx) 5) Fig. 6 The response of state x ˆθ = x ˆx. 5) Then, employing our proposed method, the observer and controller are designed as 3 3 ˆx = ˆx + u + ˆβ ) y C ˆx) j m + 53) 7 y C ˆx) u = 5 ˆx ˆβ y C ˆx) j m 54) j m = + ˆx 55) ˆβ = y ŷ j m.ˆβ. 56) In the simulations, the initial values are designed as x ) = 3, T, ˆx ) =, T, ˆθ ) =, and ˆβ ) =. The state curves of the closed-loop system are shown in Figs. 6 and 7. The dashed lines are the response curves of the closed-loop system by using 5) and 5). The solid lines are the response curves of the closed-loop system by using 53) and 54). From Figs. 6 and 7, it can be seen that our proposed method is more effective than the method proposed in 6. 5 Conclusions In this paper, a new adaptive controller is presented based on observer, which can be used for the nonlinearly parameterized systems with unmodeled dynamics and actuator dead-zone. With the proposed control scheme, the assumptions imposed on the nonlinear terms are more general than some of the existing results. Moreover, neither solving LMI nor employing several adaptive parameters is needed. By choosing appropriate design parameters, the controller can guarantee that the closed-loop system is uniformly ultimately bounded in the presence of unmodeled dynamics and dead-zone nonlinearity. Simulation results have illustrated the effectiveness of the proposed observer and controller. References Fig. 7 The response of state x S. S. Ge, F. Hong, T. H. Lee. Robust adaptive control of nonlinear systems with unknown time delays. Automatica, vol. 4, no. 7, pp. 8 9, 5. C. C. Hua, G. Feng, X. P. Guan. Robust controller design of a class of nonlinear time delays via backstepping method. Automatica, vol. 44, no., pp. 567 573, 8. 3 X. H. Jiao, T. L. Shen. Adaptive feedback control of nonlinear time-delay systems: The LaSalle-Razumikhin-based approach. IEEE Transactions on Automatic Control, vol. 5, no., pp. 99 93, 5. 4 X. H. Jiao, T. L. Shen, Y. Z. Sun. Further results on robust stabilization for uncertain nonlinear time-delay systems. Acta Automatica Scinica, vol. 33, no., pp. 64 69, 7. 5 C. C. Hua, X. P. Guan, P. Shi. Robust backstepping control for a class of time delayed systems. IEEE Transactions on Automatic Control, vol. 5, no. 6, pp. 894 899, 5. 6 C. C. Hua, F. L. Li, X. P. Guan. Observer-based adaptive control for uncertain time-delay systems. Information Sciences, vol. 76, no., pp. 4, 6.

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In Proceedings of the 35th Conference on Decision and Control, IEEE, Kobe, Japan, vol. 4, pp. 438-4386, 996. 3 X. S. Wang, C. Y. Su, H. Hong. Robust adaptive control of a class of nonlinear systems with unknown dead-zone. Automatica, vol. 4, no. 3, pp. 47 43, 4. 4 S. Ibrir, W. F. Xie, C. Y. Su. Adaptive tracking of nonlinear systems with non-symmetric dead-zone input. Automatica, vol. 43, no. 3, pp. 5 53, 7. 5 C. C. Hua, Q. G. Wang, X. P. Guan. Adaptive tracking controller design of nonlinear systems with time-delays and unknown dead-zone input. IEEE Transactions on Automatic Control, vol. 53, no. 7, pp. 753 759, 8. 6 S. Bowong, J. J. Tewa. Unknown inputs adaptive observer for a class of chaotic systems with uncertainties. Mathematical and Computer Modelling, vol. 48, no., pp. 86 839, 8. Xue-Li Wu received the B. Sc. and M. Sc. degrees from Yanshan University, PRC in 983 and 988, respectively. He received the Ph. D. degree from Huazhong University of Science and Technology in 5. He is currently a professor and dean of the Department of Electronic Engineering and Information, Hebei University of Science and Technology, PRC. His research interests include nonlinear control systems and intelligent control. E-mail: xlwu3@63.com Xiao-Jing Wu received the B. Sc. and M. Sc. degrees from Yanshan University, PRC in 6 and 9, respectively. Now, she is a Ph.D. candidate in control theory and control engineering, Yanshan University, PRC. Her research interests include nonlinear control, adaptive control, and fault-tolerant control. E-mail: wuxiaojing3@63.com Corresponding author) Xiao-Yuan Luo received the M. Sc and Ph. D. degrees from the Department of Electrical Engineering, Yanshan University, PRC in and 5, respectively. He is currently an associate professor at the Department of Electrical Engineering, Yanshan University. His research interests include fault detection and fault tolerant control, multi-agent, and networked control systems. E-mail: xyluo@ysu.edu.cn Quan-Min Zhu received the M. Sc. degree from Harbin Institute of Technology, PRC in 983, and Ph. D. from Faculty of Engineering, University of Warwick, UK in 989. Now, he is a professor in control systems at Bristol Institute of Technology BIT), University of the West of England UWE), Bristol, UK. His research interests include nonlinear system modelling, identification, and control. E-mail: Quan.Zhu@uwe.ac.uk