International Journal of Systems Science ISSN: 0020-7721 (Print) 1464-5319 (Online) Journal homepage: http://www.tandfonline.com/loi/tsys20 Robust adaptive tracking control of uncertain systems with time-varying input delays Li-Juan Liu, Jing Zhou, Changyun Wen & Xudong Zhao To cite this article: Li-Juan Liu, Jing Zhou, Changyun Wen & Xudong Zhao (2017) Robust adaptive tracking control of uncertain systems with time-varying input delays, International Journal of Systems Science, 48:16, 3440-3449, DOI: 10.1080/00207721.2017.1382604 To link to this article: http://dx.doi.org/10.1080/00207721.2017.1382604 Published online: 23 Oct 2017. Submit your article to this journal Article views: 2 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tsys20 Download by: [UNIVERSITY OF ADELAIDE LIBRARIES] Date: 25 October 2017, At: 05:36 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE, VOL. , NO. , – https://doi.org/./.. Robust adaptive tracking control of uncertain systems with time-varying input delays Li-Juan Liua,b , Jing Zhouc , Changyun Wend and Xudong Zhaoa School of Control Science and Engineering, Dalian University of Technology, Dalian , P. R. China; b School of Software, Dalian Jiaotong University, Dalian , P. R. China; c Department of Engineering Sciences, Faculty of Engineering and Science, University of Agder, Grimstad, Norway; d School of Electronic Engineering, Nanyang Technological University, Singapore, Singapore Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 a ABSTRACT ARTICLE HISTORY In this paper, the problem of robust adaptive tracking control of uncertain systems with time-varying input delays is studied. Under some mild assumptions, a robust adaptive controller is designed by using adaptive backstepping technique such that the system is globally stable and the system output can track a given reference signal. At the same time, a root mean square type of bound is obtained for the tracking error as a function of design parameters and thus can be adjusted. Finally, one numerical example is given to show the effectiveness of the proposed scheme. Received January Accepted September 1. Introduction As is known that the existence of time delays is a common phenomenon in control design. Since time delays usually deteriorate the performance and stability of closed-loop system, the control problem for systems involving time delays has been a major issue and received wide attention (for example, see Das, Ghosh, & Subudhi, 2015; Richard, 2003; Sun, Zhao, & Hill, 2006). However, all these existing results are either only applicable to the systems with time delay in states or applicable to the systems with constant delays. In addition, the delays in practical systems are often time varying, which makes controller design and the stability analysis more complicated and difficult. Adaptive control is potentially an effective approach to use for situations where not only system parameters are unknown, but their ranges are also unavailable. In Xu, Tong, and Li (2015), an adaptive fuzzy decentralised fault-tolerant control is developed for a class of nonlinear large scale systems in strict-feedback form. A simple model reference adaptive control scheme for a class of MIMO linear systems with unknown state delays is developed in Mirkin and Gutman (2009) to achieve output feedback tracking. On the other hand, backstepping technique (Krstic, Knamellakopoulos, & Kokotovic, 1995) has been widely used to design adaptive controllers for uncertain systems (see, e.g. Chen, Liu, Liu, & Chong, 2013; Ge, Hong, & Lee, 2005; Li & Yang, 2016, 2017; Liu, Sun, Wang, Zhou, & Wen, 2016; Tong, Sui, & Li, 2015; Xu et al., 2015 and references therein). In recent years, some fruitful results have been achieved in solving the stabilisation and tracking problems for time delay systems by using adaptive backstepping technique. For instance, a state feedback controller for a class of nonlinear time delay systems is designed with the help of Razumikhin method in Hua, Feng, and Guan (2008) such that the resulting closed-loop system is uniformly ultimately stable in the sense that all the signals are bounded. But except for Zhou, Wen, and Wang (2009) and Liu et al. (2016), all the above-mentioned results are only applicable to systems with CONTACT Xudong Zhao xdzhaohit@gmail.com, xudongzhao@dlut.edu.cn © Informa UK Limited, trading as Taylor & Francis Group KEYWORDS Adaptive control; time-varying delays; backstepping; uncertain systems; unmodelled dynamics state delays. However, when employing backstepping approach with modifications, little attention is paid to systems with control input delays as such systems belong to non-minimum phase systems and the standard backstepping technique has been shown only applicable to minimum phase systems. Although the control scheme in Zhou et al. (2009) can be applied to nonminimum phase systems, the approach is based on the standard backstepping without modifications. In our previous work (Liu et al., 2016), the stabilisation problem for uncertain systems with time-varying delays and unmodelled dynamics is considered. Furthermore, in all the above-mentioned schemes, combining time-varying delay disturbances and unmodelled dynamics are only considered in Liu et al. (2016). The existence of disturbances caused by time delays and unmodelled dynamics may lead to system instability and also affect the system performance. Especially, the conventional method such as LaSalle-Yoshizawa theorem cited in Zhou et al. (2009) cannot be trivially extended to resolve the problem, which means that it is difficult to compensate for the effects produced by such systems. Except for Mirkin and Gutman (2009), Ge et al. (2005), Tong et al. (2015) and Li and Yang (2016), only stabilisation is considered in all the above-mentioned results, which means that all local outputs track zero local reference signals. Note that the stabilisation methods by using standard backstepping technique cannot be trivially extended to solve the tracking problem in the presence of unmodelled dynamics. This is mainly because the non-zero local reference signals affect the tracking error through unmodelled dynamics. Thus, it is a challenging task to solve the tracking problem for systems with unmodelled dynamics, especially when involving time-varying input delays. Therefore, it is worthwhile to investigate adaptive output feedback tracking for such systems. To our best knowledge, there are still no results reported to address such issues. Motivated by previous works on uncertain systems with unmodelled dynamics and unknown time delays, in this paper, Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE we propose to design the adaptive tracking controllers for systems with unmodelled dynamics and time-varying input delays. It is shown that the controller cannot only globally stabilise such systems but also track a given reference signal. Our results have two significances. On one hand, this result shows that the backstepping controllers with modifications are robust against unmodelled dynamics and time-varying input deleys. On the other hand, the backstepping controllers with modifications can be applied to non-minimum phase systems since systems with input time delay belong to non-minimum phase systems and similar results can be derived. To achieve our results, two key techniques are used in our analysis. First, we transform the unmodelled dynamics with input delays to another dynamics with state delays. Then, we find a statespace realisation with respect to the new dynamics. In this manner, the effects of unmodelled dynamics combining with unknown time-varying input delays are bounded by static functions of state variables of the whole system. The novelties can be summarised in the following three aspects. First, in order to compensate for the effects of reference inputs, a new smooth function is firstly proposed. As sequel, two novel compositive Lyapunov–Krasovskii functions are developed by introducing an exponential term e−l(t−s) , which is different from the general Lyapunov–Krasovskii functions. Second, compared to the conventional backstepping approach in Krstic et al. (1995), two new terms are added in the parameter updating laws to ensure the boundedness of parameter estimators. In controller design, the effects of tracking are taken into consideration in deriving control laws. Third, the root mean square type of the bound for the output tracking error is established as a function of design parameters, which means that the transient output tracking performance of unmodelled dynamic systems can be adjusted by modifying design parameters. This paper is organised as follows. In Section 2, a plant model and some assumptions are presented. In Section 3, the adaptive output feedback tracking controller is designed via backstepping method. The global stability and the transient performance analysis are established in Section 4. Simulation example is given in Section 5. Finally, the paper is concluded in Section 6. Throughout this paper, the following notations are used. denotes the set of real numbers. ei is a column vector in n , whose ith value is 1 and others are 0. For a vector x n , xT represents the transpose of x. σ̂ and σ̃ denote the estimate and estimation error of parameter σ , respectively, and σ̃ = σ − σ̂ . Aj denotes the jth power of matrix A. · denotes the Euclidean norm of vectors and induced norm of matrices. 2. Problem statement We consider the system with multiplicative unmodelled dynamics described as y(t ) = B(s)/A(s)(u(t ) + μ1 1 (s)u(t − d(t ))) + μ2 2 (s)y(t ), 3441 It is noted that μ1 1 (s)u(t − d(t)) denotes the unmodelled dynamics from the time-varying input delays and μ2 2 (s)y(t) denotes the unmodelled dynamics from the system output. System (1) can be represented by T ẋ = Ax + 0(ρ−1)×1 b u, y = eT1 x + μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y(t ), (2) the input where x n , u , y are the system state, and the output, respectively. A = Ā + ag, Ā = 00 In−1 ,a= 0 [an−1 , an − 2 , ..., a0 ]T , g = −eT1 , b = [bm , bm−1 , ..., b0 ], the system parameters ai (i = 0, ..., n − 1) and bj (j = 0, ..., m) are constants but totally unknown and ρ denotes the relative degree. The following assumptions are made on system (1). Assumption 2.1: i (s)(i = 1, 2) are stable, strictly proper and have a unity high frequency gain. Assumption 2.2: The reference signal yr and its first ρ derivatives yr(i) (i = 1, ..., ρ) are piecewise continuous and bounded. Assumption 2.3: B(s) is a Hurwitz polynomial. The sign of bm , the order n and the relative degree ρ are known. Assumption 2.4: The time-varying delay d(t) is uniformly ˙ ) ≤ d¯ < 1 for all time, bounded in time and satisfies inequality d(t ¯ where d is an unknown constant. Remark 2.1: Note that i (s)(i = 1, 2) denote high order unmodelled dynamics. They are neglected for various purposes such as model reduction for simplicity of analysis and designing controllers and thus usually include stable poles corresponding to fast decaying rates. Therefore, Assumption 2.1 is reasonable and practical. For the system parameters, they are totally unknown without any priori knowledge including their ranges which are normally required by robust approaches, except they are constants and bm satisfies Assumption 2.3. Assumption 2.3 is somewhat standard in adaptive control literatures such as in Zhou et al. (2009) and Krstic et al. (1995). The derivative information of time-varying delays for practical systems might be unknown, while it is not an easy job to compute and evaluate the derivative information of time-varying delays. Assumption 2.4 is analogous to the standard assumption in the literature on time delay systems and essentially states that the variance rate of the time delay is not bigger than real-time rate (i.e. | dtd (d(t ))| < dtd t = 1). Basically, it is assumed that Assumption 2.4 holds. For system (1), our goal is to investigate the robustness of the adaptive backstepping tracking controller designed for uncertain systems with regard to the unmodelled dynamics and unknown time-varying input delays. We will also establish the system transient performance of tracking error which can be adjusted by modifying design parameters in some sense. (1) where B(s) = bm sm + + b1 s + b0 , A(s) = sn + an−1 sn − 1 + + a1 s + a0 , d(t) is the time-varying delay, s denotes the differential operator dtd , 1 (s) and 2 (s) are transfer functions of the unmodelled dynamic systems, and μ1 and μ2 are positive scalars indicating the magnitudes of the unmodelled dynamic systems. 3. Design of adaptive backstepping tracking controller In this section, by following backstepping procedure in Krstic et al. (1995), the adaptive tracking controller is summarised as follows. Before presenting the backstepping program, we first design the observer. 3442 L.-J. LIU ET AL. 3.1. Observer design In order to design the observer, the following steps are needed. The detailed procedures for each step are stated as follows. Step 1: First of all, two filters are designed so as to estimate the states of system (2): Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 η̇ = A0 η + en y, λ̇ = A0 λ + en u, (3) (4) where λ and η are the states of the filters. A0 = Ā − keT1 , k = T k1 , ..., kn , and the vector k is chosen in order that the matrix A0 is Hurwitz. Step 2: The state estimate and the state estimation error of the system are given as follows: x̂ = −An0 η − n−1 ai Ai0 η + m i=0 bi Ai0 λ. (5) i=0 Define the state estimation error: = x(t ) − x̂(t ). (6) Combining system (2) and (6), the derivative of state estimation error satisfies ˙ = A0 − (k − a)[μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y(t )]. (7) Step 3: By virtue of filters in (3) and (4), combining system (2) and (6), the system is given as ẏ = bm υm,2 + ξ2 + T θ + 2 + (s + an−1 ) × [μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y(t )], υ̇m,i = υm,i+1 − ki υm,1 , i = 2, ..., ρ − 1, υ̇m,ρ = υm,ρ+1 − kρ υm,1 + u, (9) (10) j 3.2. Backstepping tracking controller In order to compensate for the effects of reference inputs going through the unmodelled dynamics, a useful smooth function and two significant properties are given as follows: s(z1 ) = 1/z12 |z1 | ≥ δ 1/[(δ 2 − z12 )ρ + z12 ] |z1 | < δ , 1 1 1 ˜ ˆ − 0 ) = − l ˜ 2 + l ( − 0 )2 − l (ˆ − 0 )2 l ( 2 2 2 1 1 ≤ − l ˜ 2 + l ( − 0 )2 , (12) 2 2 1 1 1 lθ θ˜ (θˆ − θ 0 ) = − lθ θ˜2 + lθ (θ − θ 0 )2 − lθ (θˆ − θ 0 )2 2 2 2 1 1 ≤ − lθ θ˜2 + lθ (θ − θ 0 )2 , 2 2 (13) where lθ and lϱ are two positive design parameters, θ 0 and ϱ0 are two positive constants, and = 1/bm . In order to obtain the adaptive control law, the backstepping procedure is presented based on (8)–(10). To ensure the robustness against the unmodelled dynamics, certain modifications are made for the output feedback adaptive controller designed with tuning functions. The boundedness of parameter estimations can be warranted by adding two new terms in the parameter updating laws compared with the conventional backstepping approach in Krstic et al. (1995). To design the backstepping tracking controller, the following steps are employed. Step 1: As normal, we employ the following change of coordinates: z1 = y − yr , zi = υm,i − αi−1 − y ˆ r(i−1) , i = 2, ..., ρ, (14) (15) (8) where υ j = A0 λ, j = 0, ..., m,ξ = −An0 η, = [0, υm−1,2 , ..., T υ0,2 , 2 − yeT1 ]T , = − An−1 , 0 η, ..., A0 η, η , θ = b a and vi, 2 , ϵ2 , ξ 2 , 2 denote the second entries of v i , , ξ, , respectively. Another property is presented for the following backstepping design and choice: (11) where z1 =y-y_{r} is the output tracking error, δ is a positive design parameter. A significant property is given in the following lemma presented in Zhou and Wen (2008). Lemma 3.1: Function s(z1 ) is (ρ − 1)th order differentiable. where yr is the reference output, zi (i = 1, ..., ρ, ) is the output tracking error, α i−1 (i = 2, ..., ρ) is the virtual control law. Define z(t) = [z1 , z2 , ..., zρ ]T . Clearly, our goal is to regulate z(t) to zero. Step 2: By using backstepping technique, the adaptive control law is obtained as follows: ˆ r(ρ) . u = αρ − υm,ρ+1 + y (16) α1 = ˆ ᾱ1 , (17) ᾱ1 = −c1 z1 − d1 z1 − ξ2 − θ̂ − ls z1 s(z1 )yr , T 2 (18) where ci , di (i = 1, ..., ϱ) and ls are positive design parameters. α2 = −c2 z2 − d2 − ∂α1 ∂ θ̂ ∂α1 ∂y 2 lθ (θ̂ − θ 0 ), z2 − b̂m z1 + β2 + ∂α1 ∂ θ̂ τ 2 (19) INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE where is a positive definite matrix in (n + m + 1) × (n + m + 1) . Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 + i−1 ∂αi−1 ( j) y ( j−1) r j=1 ∂yr + ki υm,1 + m+i−1 j=1 ∂αi−1 ˙ (i−1) + + yr , ˆ i = 2, ..., ρ, ∂ ˆ (20) (21) T where ω = υm,2 , υm−1,2 , ..., υ0,2 , 2 − yeT1 . Define the tuning functions as follows: (22) (23) (24) (25) where γ is a positive constant. Remark 3.1: Different from the traditional backstepping technique in Krstic et al. (1995) and Liu et al. (2016), a crucial term −ls z1 s(z1 )yr 2 in (18) is proposed in order to compensate for the effect of reference inputs going through the unmodelled dynamics. The detailed analysis will be given in Section 4. In addition, the two new terms lθ (θˆ − θ0 ) and γ l (ˆ − 0 ) in parameter updating laws (24) and (25) are introduced to ensure the boundedness of the parameter estimates. Step 3: The tracking error system is characterised by ż = Az (z, t )z + W (z, t )eT2 + Wθ (z, t )T θ̃ − bm (ẏr + ᾱ1 )e1 ˜ + W (z, t )[(s + an−1 )(μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y(t ))], are ∂αρ−1 T 1 , ..., − W = 1, − ∂α , ∂y ∂y Remark 3.2: In Liu et al. (2016) only the stabilisation problem is considered, thus yr = 0 in (14) and (15). Then, there is no term related to ẏr in (26). Note that ẏr is multiplied with unknown parameter estimation error ˜ and handling such a term is challenging. By adding a new term −ˆ ẏr e1 z1 in the tuning function (22), the effect from yr can be compensated. 4. Stability analysis 4.1. Robustness analysis For simplicity, the variable t in X(t), x1 (t), y(t) and yr (t) is neglected if possible in the following analysis, where the definition of X(t) will be given in Lemma 4.1. In order to present the stability analysis clearly and logically, the following steps are given. Step 1: Under a similar transformation as in Krstic et al. (1995), since variable ζ associated with the zero dynamics and variable η of the filter in (3) should be included in the Lyapunov function, two relevant equations are introduced as, η̃˙ = A η̃ + e y, ζ̃˙ = A ζ̃ + b x , where η̇ = A η + e y, η̇r = 0 The parameter update laws are given as ˙ θ̂ = τ ρ − lθ (θ̂ − θ 0 ), ˙ˆ = −γ sgn(bm )(ẏr + ᾱ1 )z1 − γ l (ˆ − 0 ), Wθ (z, t )T ∂ θ̂ ∂αi−1 (−k j λ1 + λ j+1 ) ∂λ j τ 1 = (ω − ( ˆ ẏr + ᾱ1 )e1 )z1 , ∂αi−1 τ i = τ i−1 − ωzi , i = 2, ..., ρ. ∂y and Wθ (z, t )T = W (z, t )ωT − ( ˆ ẏr + ᾱ1 )e1 eT1 , also in the zj ∂αi−1 ∂α j−1 equation, σi j ∂y ω. ∂αi−1 2 ∂αi−1 αi = −zi−1 − ci + di zi + βi + τ i ∂y ∂ θ̂ i−1 ∂α j−1 ∂αi−1 ∂αi−1 z j ω, − lθ (θ̂ − θ 0 ) − ∂y ∂ θ̂ ∂ θ̂ j=2 i = 3, ..., ρ ∂αi−1 ∂αi−1 (ξ2 + ωT θ̂) + (A0 η + en y) βi = ∂y ∂η W n b b 1 0 n = Ab ζ r , with Ab and bb given A0 ηr , ζ̇ = Ab ζ + bb x1 , ζ̇ r −bm−1 /bm Im−1 as, Ab = ,T = Aρb e1 , ..., Ab e1 , Im , bb = ... −b0 /bm 0 ... 0 ρ 0 T(A b − a). The eigenvalues of the matrix Ab m × m are the zeros of m 1 s+b0 the Hurwitz polynomial sn +abmn−1s s+···+b n−1 +···+a s+a . 1 0 Step 2: In order to handle the unmodelled dynamics, two state equations and a lemma are given. We take h1 and h2 as the state vectors associated with i (s)(i = 1, 2), respectively: ḣ1 = A1 h1 + bh1 x1 , 1 (s)x1 = (1, 0, ..., 0)h1 , (28) ḣ2 = A2 h2 + bh2 y, 2 (s)y = (1, 0, ..., 0)h2 , (29) where h1 n , h2 n , A1 n × n , A2 n × n , bh1 n , bh2 n , and A1 and A2 are Hurwitz as i (s)(i = 1, 2) are stable from Assumption 2.1. To prove Theorem 4.1, we need the following Lemma 4.1. Lemma 4.1: The effects of the unmodelled dynamics with timevarying input delays are bounded as follows: (26) 1 (s)x1 (t − d(t ))2 ≤ X(t − d(t ))2 , where the system matrix Az (z, t) is given in (27). ⎡ 3443 −c1 − d1 − ls s(z1 )yr 2 b̂m 0 ∂α1 2 ⎢ −c2 − d2 ( ∂y ) 1 + σ2,3 −b̂m ⎢ ⎢ 2 2 0 −1 − σ2,3 −c3 − d3 ( ∂α ) ⎢ ∂y Az (z, t ) = ⎢ ⎢ 0 −σ2,4 −1 − σ3,4 ⎢ ⎣ ... ... ... 0 −σ2,ρ −σ3,ρ ⎤ ... 0 ⎥ ... σ2,ρ ⎥ ⎥ ... σ3,ρ ⎥ ⎥ , (27) ⎥ ... σ4,ρ ⎥ ⎦ ... ... ∂α ... −cρ − dρ ( ∂yρ−1 )2 (30) 3444 L.-J. LIU ET AL. 1 (s)(s + an−1 )x1 (t − d(t ))2 ≤ r1 x1 (t − d(t ))2 + r2 X(t − d(t ))2 , 2 (s)y2 ≤ X2 , 2 (s)(s + an−1 )y2 ≤ r3 X2 + r4 yr 2 , (31) (32) (33) x1 2 ≤ (6 + 3μ22 )X2 + 3μ21 X(t − d(t ))2 + 6yr 2 , (34) Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 where r1 , r2 , r3 and r4 are positive constants depending on an−1 T and i (s)(s = 1, 2), X = ZT , T , η̃T , ζ̃ T , hT1 , hT2 . Proof: See Appendix 1. Remark 4.1: In Liu et al. (2016) tracking is not considered , then yr = 0 in Lemma 4.1 and there is no term related to yr in inequalities (33) and (34). Note that the reference input yr is indispensable in practical closed-loop system and the handling of such a term is nontrivial in stability analysis. By considering a new term related to yr in inequalities (33) and (34), the aim of tracking can be ensured. Step 3: Define a comprehensive Lyapunov function: V = V̄ + U 1 + U 2 , V̄ = Vρ + 1 T 1 η̃ Pη + ζ̃ T Pb ζ̃ kη kζ + q1 hT1 P1 h1 + q2 hT2 P2 h2 , ρ ρ 1 2 1 T −1 |bm | 2 1 T z + θ̃ θ̃ + ˜ + Vρ = P, (35) 2 i 2 2γ d i=1 i=1 i t 1 1 U = r1 r5 μ21 x1T (s)e−l(t−s) Q1 x1 (s)ds, 1 − d¯ t−d(t ) 1 U2 = (r2 r5 μ21 + r6 μ21 + 2r7 μ21 + 3r1 r5 μ41 Q1 ) ¯ 1−d t XT (s)e−l(t−s) Q2 X(s)ds, (36) × t−d(t ) where P, P1 , P2 and Pb satisfy PA0 + AT0 P = −I, P = PT > 0, P1 A1 + AT1 P1 = −I, P2 A2 + AT2 P2 = −I and Pb Ab + ATb Pb = −I, Pb = PTb > 0, respectively, kη , kζ , q1 , q2 and l are positive constants and Q1 , Q2 are two positive definite symmetrical matrix. As sequel, the derivatives of these two Lyapunov– Krasovskii functionals are given: U̇ 1 = 1 r1 r5 μ21 x1T Q1 x1 − ˙ ) 1 − d(t r1 r5 μ21 x1T (t − d(t )) 1 − d¯ Lyapunov–Krasovskii functionals. The objective of such two exponential terms is to generate two negative terms −lU1 and −lU2 in (37) to stabilise the whole system. In this way, effects of time-varying input delays combining with the unmodelled dynamics and output tracking can be compensated. ˙ ) Remark 4.3: If only constant time delays are considered , d(t will not appear in (37), there is no term related to 1−1 d¯ in ˙ ) is multiplied with unknown unmod(36). Note that 1 − d(t elled dynamics parameter μ1 and time delay terms either X(t − d(t))2 or x1 (t − d(t))2 , the handling of such a term is challenging. By adding a new multiplicative term 1−1 d¯ in the Lyapunov–Krasovskii functionals (36), based on Assump˙ ) can be tion 2.4, the effect from time-varying delays 1 − d(t compensated. The following theorem gives the stability conditions for the system (1). Theorem 4.1: Consider the closed-loop adaptive system consisting of system (1) under Assumptions 2.1–2.4, parameter estimates given by (24) and (25), adaptive controllers designed using (16) with virtual control laws (17), (19) and (20) and the filters (3) and (4). Then, there exists a constant ν such that μ1 < ν and μ2 < ν, the system is globally stable in the sense that all signals in the closed-loop system are globally uniformly bounded. Proof: See Appendix 2. Remark 4.4: Theorem 4.1 shows that the designed controller in system (1) is able to stabilise the system with unknown timevarying input delays and unmodelled dynamics whose magnitudes μ1 and μ2 are bounded by parameter ν satisfying (B8). If we have a-priori knowledge of the system on the bounds of μ1 and μ2 and ν, we can choose the design parameter ci , di , ki according to condition (B8). This means that the strengths μ1 and μ2 of unmodeled dynamics can be arbitrarily strong. However in general, it is difficult to obtain a-priori knowledge of the system on the bound ν. Thus, the significance of our result is to show the existence of such a parameter so that any system with μ1 < ν and μ2 < ν can be stabilised. It means that the designed controller possesses certain degree of robustness, similar to the interpretations of the results established for conventional robust adaptive controllers (see, for example Ioannou & Tsakalis, 1986; Middleton, Goodwin, Hill, & Mayne, 1988; Wen & Hill, 1992), where sufficiently small amount of unmodelled dynamics is allowed. Remark 4.5: Equation (B5) is one of the key steps in the stability analysis. Note that this helps cancelling the effects of reference inputs by using the term that ls ≥ r7 + r8 + r4 r5 μ22 + 1 6r r μ2 Q1 . 1−d¯ 1 5 1 1 − d¯ × e−ld(t ) Q1 x1 (t − d(t )) − lU 1 , 1 (r2 r5 μ21 + r6 μ21 + 2r7 μ21 + 3r1 r5 μ41 Q1 )XT Q2 X U̇ 2 = 1 − d¯ ˙ ) 1 − d(t − (r2 r5 μ21 + r6 μ21 + 2r7 μ21 + 3r1 r5 μ41 Q1 ) 1 − d¯ 4.2. Transient tracking error performance ×XT (t − d(t ))e−ld(t ) Q2 X(t − d(t )) − lU 2 . (37) The transient tracking error performance is characterised in the following theorem. Remark 4.2: The two new compositive Lyapunov–Krasovskii functionals, developed by introducing two exponential terms Theorem 4.2: Given the initial values zi (0) = 0(i = 1, … , e−l(t − s) in (36), are different from the commonly used ρ), η̃(0) = 0, ζ̃(0) = 0, h1 (0) = 0, h2 (0) = 0, U1 (0) = 0 and INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 15 3445 u 10 Control 5 0 −5 −15 0 5 10 15 20 25 t(sec) 30 35 40 45 50 Figure . Control u. U2 (0) = 0, the output tracking error y − yr satisfies that y − yr [0,T ] ≤ l l l|bm | 2 (0) ˜ r5 (0)2P + θ̃(0)2−1 + c0 2c0 2c0 γ 1 |bm | 1 + l ( − 0 )2 + lθ θ − θ 0 2 + F̄ , c0 2 2 (38) where (0)2P = T (0)P(0), θ̃(0)2−1 = θ̃ T (0) −1 θ̃(0). Proof: See Appendix 3. Remark 4.6: Theorem 4.2 gives a quantification of the transient performance by a root mean square type of bound. The transient tracking error performance in y − yr [0, T] depends on the initial estimation errors θ̃(0), (0) ˜ and (0). The closer the initial estimation errors θ̃(0), (0) ˜ and (0) to the true values, the better the transient tracking error performance. Moreover, the transient tracking error performance can be tuned systematically to an arbitrary small value by increasing c0 . 5. Simulation example In this case, we illustrate the developed approach on a relativedegree-two time-varying delay system with unmodelled dynamics as described in (1), where B(s) = b1 s + b0 = 2s + 3, A(s) = s3 + a2 s2 + a1 s + a0 = s3 + 0.5s2 + s + 3, and time-varying delay d(t ) = 1 + 12 sin(t ). Note that the parameters b1 , b0 , a2 , a1 , a0 are unknown and the delay d(t) is uniformly bounded which satisfies Assumption 2.4. Meanwhile, the order n = 3, the sign of b1 and the relative degree ρ = 2 are known which satisfies Assumption 2.3. Further assume that the unmodelled dynamics 1 satisfying Assumption 2.1 and μ1 , μ2 is 1 (s) = 2 (s) = s+1 are μ1 = μ2 = 0.1. The design parameters are chosen as k = [6, 12, 8]T ensuring A0 is Hurwitz, c1 = c2 = 2, d1 = d2 = 0.5, lϱ = ls = lθ = 2, γ = 0.001 and = 0.001 × I5 . The reference signal is yr = sin(t) which satisfies Assumption 2.2. In this example, all the initial values are set as 0 except for y(0)=0.4, (0)=0.2, ˆ ϱ0 = 1 and θ 0 = [1, 1, 1, 1, 1]T . The simulation results are presented in Figures 1–4. From Figures 1 and 3, it is shown that control u and parameter estimates are bounded, even though the unknown time-varying input delays and the unmodelled 3 y yr 2 Tracking performance Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 −10 1 0 −1 −2 −3 0 Figure . The tracking performance. 5 10 15 20 25 t(sec) 30 35 40 45 50 3446 L.-J. LIU ET AL. 0.04 b̂1 b̂0 â2 â1 â0 Parameter estimate θ 0.03 0.02 0.01 0 −0.01 −0.02 −0.04 0 5 10 15 20 25 t(sec) 30 35 40 45 50 Figure . The parameter estimates. 4 output error with different c0 Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 −0.03 c0=2 3 c0=5 2 c0=7 1 0 −1 −2 −3 −4 0 5 10 15 20 25 t(sec) 30 35 40 45 50 Figure . The output tracking error with diﬀerent c . dynamics exist. And from Figure 2, it is clear that the output tracking error maintains within a small range after about 20 seconds. This verifies our theoretical findings. To verify the effectiveness of modifying the design parameter c0 presented in Remark 4.6, we choose c0 = 2, c0 = 5, c0 = 7, respectively, while all the other parameters are the same as above. From Figure 4, it can be observed that the tracking error bound decreases as c0 increases and thus the transient performance is improved. 6. Conclusion In this paper, the robust adaptive output feedback controller has been designed for uncertain systems with time-varying input delays and unmodelled dynamics. It is shown that the designed controller can ensure that the system output tracks a given reference signal besides global stabilisation. Moreover, the root mean square type of bound for the output tracking error has been established as a function of design parameters. Therefore, the transient performance of the system tracking error can be adjusted by modifying design parameters. Finally, a simulation example is given to show the effectiveness of the results. Further research includes the consideration of adaptive fuzzy control (Li & Yang, 2016; Wu, He, & Zhang, 2016), adaptive actuator faults control (Tao, 2014) and adaptive event-triggered control (Li & Yang, 2017). Disclosure statement No potential conflict of interest was reported by the authors. Funding This work was supported in part by the National Natural Science Foundation of China [grant number 61325014], [grant number 61722302], [grant number 61573069]; Liaoning Provincial Natural Science Foundation, China [grant number 201602124]; the Fundamental Research Funds for the Central Universities, China [grant number (DUT16RC(3)033)]. Notes on contributors Li-Juan Liu received the M.S. degree in computer engineering from Liaoning Normal University, China, in 2004. Since July 2004, she was a lecturer in the department of software, Dalian Jiaotong University, China. Since September 2014, she has been pursuing the Ph.D. degree in control theory and control engineering at Dalian University of Technology, China. Her research interests include switching systems, time delay systems, positive systems and adaptive control. INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 Jing Zhou received her B.Eng degree from Northwestern Polytechnical University (China) in 2000 and the Ph.D. degree from the Nanyang Technological University (Singapore) in 2006. She is currently a full Professor at the Faculty of Engineering and Science, University of Agder, Norway. She was a Senior Research Scientist at International Research Institute of Stavanger (Norway) from 2009 to 2016 and a Postdoctoral fellow at Norwegian University of Science and Technology (Norway) from 2007 to 2009, respectively. Her research interests include adaptive control, nonliear systems, non-smooth nonlinearities, automatic drilling control, crane control and robotics. She has been actively involved in organizing international conferences playing the roles of Technical Program Committee Chair, Program Committee Member, Invited Session Chair, and so on. She received 1000 Young Talent award in China in 2012. Changyun Wen received the B.Eng degree from Xi’an Jiaotong University, Xi’an, China, in 1983 and the Ph.D. degree from the University of Newcastle, Newcastle, Australia in 1990. From August 1989 to August 1991, he was a Research Associate and then Postdoctoral Fellow at University of Adelaide, Adelaide, Australia. Since August 1991, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a Full Professor. His main research activities are in the areas of control systems and applications, autonomous robotic system, intelligent power management system, smart grids, cyberphysical systems, complex systems and networks, model based online learning and system identification. Prof. Wen is an Associate Editor of a number of journals including Automatica, IEEE Transactions on Industrial Electronics and IEEE Control Systems Magazine. He is the Executive Editor-inChief of Journal of Control and Decision. He served the IEEE Transactions on Automatic Control as an Associate Editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair, General Co-Chair, Technical Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair and so on. He received the IES Prestigious Engineering Achievement Award 2005 from the Institution of Engineers, Singapore (IES) in 2005. He received the Best Paper Award of IEEE Transactions on Industrial Electronics in 2017. He is a Fellow of IEEE, was a member of IEEE Fellow Committee from January 2011 to December 2013 and a Distinguished Lecturer of IEEE Control Systems Society from February 2010 to February 2013. Xudong Zhao was born in Harbin, China, on July. 7. 1982. He received the B.S. degree in Automation from Harbin Institute of Technology in 2005 and the Ph.D. degree from Control Science and Engineering from Space Control and Inertial Technology Center, Harbin Institute of Technology in 2010. Dr. Zhao was a lecturer and an associate professor at the China University of Petroleum, China. From March 2012, he was with Bohai University, China, as a Professor. In 2014, Dr. Zhao worked as a postdoctoral fellow in the Department of Mechanical Engineering, the University of Hong Kong. Since October 2015, he has been with Dalian University of Technology, China, where he is currently a Professor. Dr. Zhao serves as associate editor for IEEE Transactions on Systems, Man and Cybernetics: Systems, Nonlinear Analysis: Hybrid Systems, Neurocomputing, IEEE Access and Intepositive systems, multi-agent systems, fuzzy systems, Hinf control, filtering and their applications. His works have been widely published in international journals and conferences. 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The proof of Lemma 4.1 From (28) and (29), inequalities (30) and (32) can be derived, 1 (s)x1 2 = (1, 0, ..., 0)h1 2 X2 ,1 (s)x1 (t − d(t))2 X(t − d(t))2 , 2 (s)y2 = (1, 0, ..., 0)h2 2 X2 . From system (2), x1 (t) can be obtained, x1 2 = z1 + yr − μ1 1 (s)x1 (t − d(t )) − μ2 2 (s)y2 ≤ (6 + 3μ22 )X2 + 3μ21 X(t − d(t ))2 + 6yr 2 . (A1) This proof is completed. 3448 L.-J. LIU ET AL. Appendix 2. The proof of Theorem 4.1 From the designed controller (14)–(26) and (35) and differentiating (35), the derivative of V̄ is shown as follows: + 1 1 − d¯ 3r1 r5 μ41 Q1 X(t − d(t ))2 + 1 1 − d¯ r1 r5 μ21 ˙ ) 1 − d(t r1 r5 μ21 x1T (t − d(t )) 1 − d¯ 1 × e−ld(t ) Q1 x1 (t − d(t )) + 6r1 r5 μ21 Q1 yr 2 1 − d¯ 1 (r2 r5 μ21 + r6 μ21 + 2r7 μ21 − lU 1 + 1 − d¯ ˙ ) 1 − d(t (r2 r5 μ21 + r6 μ21 + 3r1 r5 μ41 Q1 )XT Q2 X − 1 − d¯ + 2r7 μ21 + 3r1 r5 μ41 Q1 )X T (t − d(t ))e−ld(t ) Q2 × Q1 (6 + 3μ22 )X2 − 1 V̄˙ ≤ − c1 z12 − 2 ρ ci zi2 − i=2 ρ 1 1 1 2 − η̃2 − ζ̃2 2d 2k 4k i η ζ i=1 1 1 |bm | 2 1 l ˜ − lθ θ̃2 − q1 h1 2 − q2 h2 2 + M̄ − 4 2 2 2 Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 − ls z12 s(z1 )yr 2 + ρ 1 d i=1 i μ21 1 (s)(s + an−1 )x1 (t − d(t ))2 × X(t − d(t )) − lU 2 , ρ ρ 1 1 2 2 2 2 + ψ μ1 1 (s)x1 (t − d(t )) + μ2 d d i=1 i i=1 i × 2 (s)(s + an−1 )y2 + where r5 = ρi=1 d1i , r6 = r5 ψ2 , r7 = 4q1 P1 bh1 2 + 4 Pb bb 2 , r8 = 4q2 P2 bh2 2 + k4η Pen 2 , β = min{ 41 c1 , kζ 1 c , ..., 12 cρ , ρi=1 4d1 i , 4k1η , 8k1ζ , q81 , q42 }. If c1 , ci (i = 2, ..., ρ), P, 2 2 Pb , P1 , P2 , lθ , lϱ , Q1 and Q2 are taken as c1 2l, ci l(i = 2, −1 ..., ρ), P ≤ 4l1 I, Pb ≤ 8l1 I, P1 ≤ 8l1 I, P2 ≤ 4l1 I, lθ ≥ I l, l ≥ γl , Q1 eld(t) , Q2 eld(t) , then applying inequality (34) in Lemma 4.1 leads to, ρ 1 ψ2 μ22 2 (s)y2 d i i=1 1 1 2 1 1 − c1 z12 − η̃2 + η̃T Pen z1 − c1 z12 − ζ̃2 8 4kη kη 8 4kζ + 2 T 1 1 ζ̃ Pb bb z1 − c1 z12 − q1 h1 2 + 2q1 hT1 P1 bh1 z1 kζ 8 4 1 1 1 1 − c1 z12 − q2 h2 2 + 2q2 hT2 P2 bh2 z1 − q1 h1 2 8 4 4 1 + 2q1 hT1 P1 bh1 yr − q1 h1 2 − 2q1 hT1 P1 bh1 4 1 × [μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y] − ζ̃2 4kζ + 1 − d¯ 2 T 1 2 ζ̃ Pb bb yr − ζ̃2 − ζ̃ T Pb bb [μ1 1 (s) kζ 4kζ kζ × x1 (t − d(t )) + μ2 2 (s)y] − 1 − q2 h2 2 + 2q2 hT2 P2 bh2 yr . 4 (B2) r1 r5 μ21 x1T Q1 x1 ≤ 1 r1 r5 μ21 (6 + 3μ22 )Q1 X2 1 − d¯ 1 3r1 r5 μ41 Q1 X(t − d(t ))2 + 1 − d¯ 1 6r1 r5 μ21 Q1 yr 2 , + ¯ 1−d (B3) ˙ ))/(1 − d) ¯ ≥ 1, then it folIn view of Assumption 2.4, (1 − d(t lows that 1 2 η̃2 + η̃T Pen yr 4kη kη V̇ + lV ≤ −[β − r3 r5 μ22 − r6 μ22 − 2r7 μ22 − (B1) where ψT = 2(a − k)T P and M̄ = |b2m | l ( − 0 )2 + 12 lθ θ − θ 0 2 . 2 n , Thus, if we choose kη , kζ , q1 and q2 to satisfy, kη ≥ 32Pe c1 kζ ≥ 32Pcb1bb , q1 ≤ 16Pc11bh1 2 , q2 ≤ 16Pc21bh2 2 , then, in view of Lemma 4.1, we get the derivative of V = V̄ + U 1 + U 2 as follows: 2 2 η2 ζ2 1 ρ 1 ρ − − V̇ ≤ − c1 z12 − i=2 ci zi2 − i=1 4 2 4di 4kη 8kζ | q1 q |b 2 m l ˜ 2 − h1 2 − h2 2 − βX2 + M̄ − 8 4 2 1 − lθ θ̃2 − ls z12 s(z1 )yr 2 + (r7 + r8 + r4 r5 μ22 )yr 2 2 + (r3 r5 + 2r7 + r6 )μ22 X2 + (r2 r5 + 2r7 + r6 )μ21 × X(t − d(t ))2 + r1 r5 μ21 x1 (t − d(t ))2 1 1 − d¯ r1 r5 μ21 1 (r2 r5 μ21 + r6 μ21 + 2r7 μ21 1 4 2 6r1 r5 μ21 + 3r1 r5 μ1 Q1 )Q2 ]X + M̄ + 1 − d¯ 2 × Q1 + r7 + r8 + r4 r5 μ2 yr 2 − ls z12 s(z1 )yr 2 . × (6 + 3μ22 )Q1 − 1 − d¯ (B4) Let f = −z12 s(z1 )yr 2 + yr 2 . It can be shown that f is bounded. By taking ls ≥ r7 + r8 + r4 r5 μ22 + 1−1 d¯ 6r1 r5 μ21 Q1 , then from (B4) we have − ls z12 s(z1 )yr 2 + 1 1 − d¯ 6r1 r5 μ21 Q1 + r7 + r8 + r4 r5 μ22 yr 2 = −[ls z12 s(z1 ) − (r7 + r8 + r4 r5 μ22 + 1 1 − d¯ 6r1 r5 μ21 Q1 )]yr 2 ≤ ls f ≤ F̄, (B5) INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE where F̄ is the bound of ls f. By substituting (B5) into (B4) and taking ν = max{μ1 , μ2 }, the derivative of V̇ + lV satisfies V̇ + lV ≤ −[β − r3 r5 ν22 − r6 ν22 − 2r7 ν22 − 1 1 − d¯ r1 r5 ν12 1 (r2 r5 ν12 + r6 ν12 + 2r7 ν12 1 − d¯ + 3r1 r5 ν14 Q1 )Q2 ]X2 + M ∗ , (B6) × (6 + 3ν22 )Q1 − Downloaded by [UNIVERSITY OF ADELAIDE LIBRARIES] at 05:36 25 October 2017 where M ∗ = M̄ + F̄. In order to stabilise the system (1), we need the following condition: β − r3 r5 ν22 − r6 ν22 − 2r7 ν22 − − 1 1 − d¯ 1 1 − d¯ r1 r5 ν12 (6 + 3ν22 )Q1 (r2 r5 ν12 + r6 ν12 + 2r7 ν12 + 3r1 r5 ν14 Q1 )Q2 = 0. (B7) The condition is ensured if we take ν as follows: n22 + 4n1 β − n2 ν= , 2n1 where n1 = 1−1 d¯ 3r1 r5 Q1 + 1−1 d¯ 3r1 r5 Q1 Q2 , r3 r5 + r6 + 2r7 + 1−1 d¯ 6r1 r5 Q1 + 1−1 d¯ r2 r5 Q2 + 1−1 d¯ r6 Q2 + 1−1 d¯ 2r7 Q2 . (B8) n2 = By direct integration of differential inequality (B6), we have V ≤ e−lt V (0) + M∗ M∗ [1 − e−lt ] ≤ V (0) + . l l (B9) This shows that V is uniformly bounded. Thus, zi (i = 1, ..., ρ), , ˆ θ̂, , ζ are bounded. Since z1 is bounded, y is also bounded, variables λ, η, ξ, υ are bounded as A0 is Hurwitz. Moreover, variables h1 , h2 are bounded as A1 , A2 are stable from Assumption 2.1. As 3449 stated in Krstic et al. (1995), the boundedness of all signals is warranted. This proof is thus completed. Appendix 3. The proof of Theorem 4.2 In order to derive a bound for the vector z(t), we first make the following definitions: c0 = min{c1 , ..., cρ }, z[0,T ] = 1 T (z(s))2 ds. Considering (B2), (B5) and (B7), we obtain T 0 z[0,T ] ≤ c10 [ T1 (V (0) − V (T )) + M ∗ ]. On the other hand, ∗ from (B6) we get T1 |V (0) − V (T )| ≤ V T(0) (1 − e−lT ) + MlT (1 − e−lT ) ≤ M ∗ + lV (0). Thus, the bound of z[0, T] is established as follows: l 1 |bm | 1 2 2 z[0,T ] ≤ V (0) + l ( − 0 ) + lθ θ − θ0 + F̄ . c0 c0 2 2 (C1) The initial value of Lyapunov–Krasovskii function is obtained as ρ 1 2 1 |bm | 2 zi (0) + r5 (0)2P + θ̃(0)2−1 + (0) ˜ V (0) = 2 2 2γ i=1 + 1 1 η̃(0)2P + ζ̃(0)2Pb kη kζ + q1 h1 (0)2P1 + q2 h2 (0)2P2 + U 1 (0) + U 2 (0), (C2) ζ̃(0)2Pb = ζ̃ T (0)Pb ζ̃(0), where η̃(0)2P = η̃T (0)Pη̃(0), 2 T q1 h1 (0)P1 = q1 h1 (0)P1 h1 (0), q2 h2 (0)2P2 =q2 hT2 (0)P2 h2 (0). Given the zero initial values η̃(0) = 0, ζ̃(0) = 0, h1 (0) = 0, h2 (0) = 0, U1 (0) = 0, U2 (0) = 0 and from (14) and (15), the initial values zi (0), i = 1, … , ρ can be set to zero. Thus, we 2 ˜ . Note obtain, V (0) = r5 (0)2P + 12 θ̃(0)2−1 + |b2γm | (0) that y − yr [0, T] z[0, T] . Thus, the bound of output tracking error y(t) − yr (t) satisfies inequality (38). This proof is completed.

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