Global Stabilization of a Class of Time-Delay Nonlinear Systems with Unknown Control Directions by Nonsmooth Feedback ⁎

Proceedings, Proceedings, 14th 14th IFAC IFAC Workshop Workshop on on Time Time Delay Delay Systems Systems Pesti Vigadó, Hungary, 28-30, 2018 Proceedings, 14th IFAC Workshop onAvailable Time Delay Systems online at www.sciencedirect.com Pesti Vigadó, Budapest, Budapest, Hungary, June June 28-30, 2018 Proceedings, 14th IFAC Workshop on Time Delay Systems Pesti Vigadó, Budapest, Hungary, June 28-30, 2018 Pesti Vigadó, Budapest, Hungary, June 28-30, 2018

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IFAC PapersOnLine 51-14 (2018) 78–83

Global Global Stabilization Stabilization of of a a Class Class of of Time-Delay Time-Delay Global Stabilization of a Class of Time-Delay Nonlinear Systems with Unknown Control Global Stabilization a Class of Time-Delay Nonlinear Systems of with Unknown Control Nonlinear Systems with Unknown Control ⋆⋆ Directions by Nonsmooth Feedback Nonlinear Systems with Unknown Control Directions by Nonsmooth Feedback Directions by Nonsmooth Feedback ⋆⋆ Directions by Nonsmooth Feedback ∗∗ K. Rattanamongkhonkun ∗∗ R. Pongvuthithum ∗∗ Wei Lin ∗∗

K. Rattanamongkhonkun ∗ R. Pongvuthithum ∗ Wei Lin ∗∗ K. Rattanamongkhonkun R. Pongvuthithum Wei Lin ∗ K. Rattanamongkhonkun ∗ R. Pongvuthithum ∗ Wei Lin ∗∗ ∗ Dept. of Mechanical Engineering, Chiang Mai University, Thailand. Dept. of Mechanical Engineering, Chiang Mai University, Thailand. ∗ ∗∗ ∗∗ Dongguan University of Technology, China,Thailand. and Dept. of Mechanical Engineering, ChiangGuangdong, Mai University, ∗ ∗∗ Dongguan University of Technology, Guangdong, China, and Dept. of EECS, Mechanical Chiang Mai University, Thailand. Dept. of CaseEngineering, Western Reserve University, Cleveland, USA Dongguan University of Technology, Guangdong, China, and Dept. of EECS, Case Western Reserve University, Cleveland, USA ∗∗ Dongguan University of Technology, Guangdong,Cleveland, China, and Dept. of EECS, Case Western Reserve University, USA Dept. of EECS, Case Western Reserve University, Cleveland, USA Abstract: Abstract: This This paper paper addresses addresses the the problem problem of of global global stabilization stabilization for for aa class class of of time-delay time-delay systems with inherent nonlinearity and unknown control directions, which may not be controlled Abstract: This paper addresses the problem of global stabilization for a class systems with inherent nonlinearity and unknown control directions, which may not of betime-delay controlled Abstract: This papernonlinearity addresses the problem ofcontrol global stabilization for aand class of time-delay by any smooth feedback. The dynamic gain based approach in Zhang, Lin Lin (2017) and systems with inherent and unknown directions, which may not be controlled by any smooth feedback. The dynamic gain based approach in Zhang, Lin and Lin (2017) and systems with inherent nonlinearity andNussbaum unknown control directions, which be(2017) controlled the ideasmooth of Nussbaum-gain Nussbaum-gain function (1983) are in employed to may deal with inherently by any feedback. Thefunction dynamic gain based approach Zhang, Lin andnot Lin and the idea of Nussbaum (1983) are employed to deal with inherently by feedback. Thefunction dynamic gain based of approach Zhang, Lin and Lin (2017) and nonlinear systems with in presence control With the theany ideasmooth of Nussbaum-gain Nussbaum are in employed to deal with inherently nonlinear systems with time-delay time-delay in the the presence(1983) of unknown unknown control directions. directions. With the help help the idea of Nussbaum-gain function Nussbaum (1983) are employed to deal with inherently of appropriate Lyapunov-Krasovskii functionals, we design a non-smooth delay-independent nonlinear systems with time-delay in the presence of unknown control directions. With the help of appropriate Lyapunov-Krasovskii functionals, we design a non-smooth delay-independent nonlinear systemslaw with time-delay in the presence of directions. With the help feedback control global convergence of system of appropriate functionals, weunknown design acontrol non-smooth feedback controlLyapunov-Krasovskii law that that guarantees guarantees the the global asymptotic asymptotic convergence of the thedelay-independent system state state and and of appropriate Lyapunov-Krasovskii functionals, we design a non-smooth delay-independent global boundedness of theguarantees resulting closed-loop closed-loop system. feedback control lawof that the global asymptotic convergence of the system state and global boundedness the resulting system. feedback control lawof that the global asymptotic convergence of the system state and global boundedness theguarantees resulting closed-loop system. © 2018,boundedness IFAC (International Automatic system. Control) Hosting by Elsevier Ltd. All rights reserved. global of the Federation resulting of closed-loop

1. and Francisco Francisco (2003); (2003); Zhang, Zhang, Zhang Zhang and and Lin Lin (2014); (2014); Zhang, Zhang, Lin Lin and and 1. INTRODUCTION INTRODUCTION and Lin (2017) references therein. 1. INTRODUCTION and (2003); Zhang, Zhang and Lin (2014); Zhang, Lin and Lin Francisco (2017) and and references therein. 1. INTRODUCTION and (2003); Zhang, Zhang and Lin (2014); Zhang, Lin and Lin Francisco (2017) and references therein. Control 1, the In the when θθi = Control of of time-delay time-delay nonlinear nonlinear systems systems is is an an important important yet yet challengchallengthe time-delay time-delay nonlinear nonlinear system system (1) (1) is is in in In the the(2017) the case case Lin andwhen references i = 1,therein. ing problem. It is is often often encountered in many many real-world yet applications Control of time-delay nonlinear systems is an important challenggeneral not stabilizable, even locally, by any smooth state feedback. ing problem. It encountered in real-world applications 1, the time-delay nonlinear system (1) is in In the the case when θi =even general not stabilizable, locally, by any smooth state feedback. that involve time-delay, such as network control, mechanical systems, Control of time-delay nonlinear systems is an important yet challeng3 ing problem. It is often encountered in many real-world applications + d) For instance, the time-delay system x u 1, the time-delay nonlinear system (1) is be in that involve time-delay, such as network control, mechanical systems, In the when θi =even 3smooth general notcase stabilizable, locally, by any state feedback. + x(t x(t − − d) cannot cannot be Forthe instance, the time-delay system x˙˙ = = u biological systems and chemical chemical processes, etc. In this work, work, we first first ing problem. It is often encountered in many real-world applications that involve time-delay, such as network control, mechanical systems, stabilized by any state feedback even when = 0, and hence biological systems and processes, etc. In this we general not stabilizable, even locally, by any feedback. + x(td −state d) cannot be For instance, thesmooth time-delay system x˙ = u3smooth stabilized by any smooth state feedback even when d = 0, and hence consider the following class of time-delay nonlinear systems that involve time-delay, such as control, mechanical systems, biological systems and chemical processes, etc. In this work, we with first it impossible be stabilizable for d ≥ 0. However, + be Foris theto time-delay x˙ = u3 when consider the following class ofnetwork time-delay nonlinear systems with stabilized by any smooth statesystem feedback even =d) 0, and hence it is instance, impossible to be smoothly smoothly stabilizable forx(t dd− ≥ 0. cannot However, unknown control directions biological systems and chemical In this work, we with first consider the following class of processes, time-delayetc. nonlinear systems it is easy to verify that with the aid of the Lyapunov-Krasovskii stabilized by any smooth state feedback even when = 0, hence unknown control directions impossible to be smoothly stabilizable for dd ≥ 0. and However, it is easy to verify that with the aid of the Lyapunov-Krasovskii  consider following class of time-delay nonlinear systems with  tt smoothly unknown the control directions 2 2 (s)ds, impossible to be forglobally d ≥ 0. However, pi x the functional V x it is easy to verify with thestabilizable aidsystem of the is 2 (s)ds, pidirections + that x the system isLyapunov-Krasovskii globally stabilizable stabilizable functional V = = x2 + t−d = + f − d), ·· ·· ·· ,, x − d)), 1 ,, ·· ·· ·· ,, x 1 (t i (x i ,, x i (t unknownx t = θθii x xi+1 + f (x x x (t − d), x (t − d)), x˙˙ iicontrol t−d it is easy to verify that with the the aidsystem of the1/3 Lyapunov-Krasovskii 2 2 (s)ds, 1 1 i i i i+1 0 feedback p + x is globally stabilizable functional V = x  i by non-smooth but C u = −(2x) . Motivated by 1/3 . Motivated t0 feedback x˙ i = θi xi+1 t−d pn + fi (x1 , · · · , xi , x1 (t − d), · · · , xi (t − d)), by non-smooth but C u = −(2x) by this this 2 2 ppin + fn (x, x(t − d)), system1/3 issystem globally stabilizable functional Vand = xthe+under-actuated = θnxu x 0 x (s)ds, the observation mechanical Qian and Lin + (x,1 ,x(t x˙˙x˙n + ffni (x · · ·−, xd)), , x1 (t − d), · · · , xi (t − d)), t−d ni = θn by non-smooth but C feedback u = −(2x) . Motivated by this i ui+1 i observation and the under-actuated mechanical system Qian and Lin pn 1/3in x˙ n = θn u + fn (x, x(t − d)), (2001b) in the time-delay, focus this paper on the by non-smooth but C 0 of feedback u mechanical = we −(2x) . Motivated by this observation andpresence the under-actuated system Qian and Lin x(s) = ζ(s), s ∈ [−d, 0], (1) p (2001b) in the presence of time-delay, we focus in this paper on the n x(s) (1) + [−d, fn (x,0], x(t − d)), θn u s ∈ x˙ n = ζ(s), problem how to control the time-delay nonlinear system (1) by nonobservation and the under-actuated mechanical system Qian and Lin (2001b) in the presence of time-delay, we focus in this paper on the problem how to control the time-delay nonlinear system (1) by nonx(s) = ζ(s), (1) n s ∈ [−d, 0], where x IIR u IR are the system state and input, smooth, rather than smooth, delay-independent state (2001b) in thetopresence of time-delay, we focus in thisfeedback. paper the problem how control the time-delay nonlinear system (1) byonnonwhere x(s) x ∈ ∈= ζ(s), R n sand and u ∈ ∈ ∈ [−d, 0], IR are the system state and input, (1) smooth, rather than smooth, delay-independent state feedback. respectively. The constant d ≥ ≥ 0are is an an unknown time-delay of the the problem how tothan control the time-delay nonlinear state system (1) by nonwhere x ∈ The IR n constant and u ∈d IR 0 theunknown system time-delay state and input, smooth, rather smooth, delay-independent feedback. respectively. is of 2i n 1 mappings Most results obtained so far on 0 odd f ::the IIR → system, where IR and ∈d ≥ IR 0are stateC input, 2isystem Most of of the the results far have have been been concentrated concentrated on timetimesmooth, rather thanobtained smooth, so delay-independent state feedback. respectively. constant is time-delay of the >The 0 are are oddu integers, integers, fi an Runknown → IIR R are are C 1and mappings system,xp pii∈> n i delay nonlinear systems with known control directions, e.g., the signs 2i 1 Most of the results obtained so far have been concentrated on time(0, 0) = 0, and ζ(s) ∈ I R is a continuous function defined with f respectively. The constant d ≥ 0 is an unknown time-delay of the delay nonlinear systems with known control directions, e.g., the signs >= 0 0, areand oddζ(s) integers, → IR are C mappings system, pi 0) i :a IR ∈ IR n fis continuous function defined with fii (0, 2in, are unknown of all coefficients of the chain of integrator are assumed to be known. 1 mappings Most of the results obtained far have been concentrated on timen1f ≤:i IR delay nonlinear systems with so known control directions, e.g., the signs  0, ≤ constants on [−d, 0]. The coefficients θ > 0 are odd integers, → I R are C system, p of all coefficients of the chain of integrator are assumed to be known. i i i (0, 0) = 0, and ζ(s) ∈ I R is a continuous function defined with f i on [−d, 0]. The coefficients θi  0,n1 ≤ i ≤ n, are unknown constants If this crucial information is not available, a new method needs delay nonlinear systems with known control directions, e.g., the signs of all coefficients of the chain of integrator are assumed to be known. bounded by a known constant c ¯ . (0, 0) = 0, and ζ(s) ∈ I R is a continuous function defined with f If this crucial information is not available, a new method needs on [−d,i 0].byThe coefficients θi c¯0, bounded a known constant . 1 ≤ i ≤ n, are unknown constants to be developed control of time-delay nonlinear of offor chain integrator are assumed to besystems. known. If this crucial information isofnot a new method needs on [−d, 0].byThe coefficients θi c¯0, to all becoefficients developed forthethe the control ofavailable, time-delay nonlinear systems. bounded a known constant . 1 ≤ i ≤ n, are unknown constants For and of time-delay systems Gu, Kharitonov Since the sign information of for thethe control input often represents, represents, for instance, instance, If this crucial is input notofavailable, a new method needs to be the developed control time-delay nonlinear systems. For the the analysis analysis and synthesis synthesis bounded by a known constantofc¯. time-delay systems Gu, Kharitonov Since sign of the control often for and Chen (2003); Jankovic (2001); Richard (2003), the Lyapunovmotion directions of systems as robotics modeled to be the developed the control of time-delay systems. For the analysis and synthesis of time-delay Since sign of for themechanical control input often(such represents, for instance, and Chen (2003); Jankovic (2001); Richardsystems (2003), Gu, the Kharitonov Lyapunovmotion directions of mechanical systems (such asnonlinear robotics modeled Krasovskii and Lyapunov-Razumikhin methods are two powerful For the analysis synthesis of time-delay systems Gu, Kharitonov by the thethe Lagrange equation) and mayoften be(such unknown, itfor is instance, certainly Since sign of equation) themechanical controland input represents, and Chen (2003); Jankovic (2001); Richard (2003), the Lyapunovmotion directions of systems as robotics modeled Krasovskii and and Lyapunov-Razumikhin methods are two powerful by Lagrange may be unknown, it is certainly common tools that have been found wide applications. In and Chen (2003); Jankovic (2001); Richard (2003), the Lyapunovinteresting to how to control time delay with motion of mechanical as robotics modeled Krasovskii and Lyapunov-Razumikhin methods are two powerful by the directions Lagrange equation) and may be(such unknown, itsystems is certainly and common tools that have been found wide applications. In the the interesting to investigate investigate how tosystems control time delay systems with literature, research of time-delay systems can be classified Krasovskii and Lyapunov-Razumikhin methods are two primarily powerful and common tools that have been found wide In the unknown control directions. For the time-delay system with by the Lagrange equation) and be time unknown, is unknown certainly interesting to investigate how to control delay it systems with literature, research of time-delay systems can beapplications. classified primarily unknown control directions. For themay time-delay system with unknown into different The first category of study on and common tools categories. that have been found wide In the control (1), stabilization by delay-independent state interesting to investigate to delay with systems with literature, research of time-delay systems can beapplications. classified primarily unknown control directions. For thecontrol time-delay system unknown into three three different categories. The first category of study focuses focuses on control direction direction (1), global globalhow stabilization bytime delay-independent state the in state Gu, and Chen (2003), literature, research ofsystem time-delay systems can be of classified primarily feedback is trivial problem. The are i) the unknown control For the time-delay system unknown into three different categories. The firstKharitonov category study focuses on control (1), global stabilization by delay-independent state the time-delay time-delay in the the system state Gu, Kharitonov and Chen (2003), feedbackdirection is not not a adirections. trivial problem. The difficulties difficulties arewith i) when when the while the second second one is aimed aimedstate at the time-delay in the control input into three different categories. The first category in ofand study focuses on signs of coefficients of a chain of integrators are unknown, the design control direction (1), global stabilization by delay-independent state the time-delay in one the system Gu, Kharitonov Chen (2003), feedback is not a trivial problem. The difficulties are i) when the while the is at the time-delay the control input signs of coefficients of a chain of integrators are unknown, the design Mazenc, and (2003); Mazenc, Mondie Francisco the time-delay in theNiculescu system Gu,time-delay Kharitonov Chen (2003), of controllers intuitive and uncerfeedback is not a trivial problem. The difficulties are as i) the when the while theModie second one is aimedstate at the in and the and control input signs of coefficients ofis chain of integrators areinvolved unknown, design Mazenc, Modie and Niculescu (2003); Mazenc, Mondie and Francisco of virtual virtual controllers isaless less intuitive and more more involved as the uncer(2003);the Liberis and Krstic (2013); Zhang, Boukas, Lui and Baron while secondand oneNiculescu is aimed at theZhang, time-delay in the control input tainties cannot be cancelled directly by a conventional backstepping signs of coefficients of a chain of integrators are unknown, the design Mazenc, Modie (2003); Mazenc, Mondie and Francisco of virtual controllers is less intuitive and more involved as uncer(2003); Liberis and Krstic (2013); Boukas, Lui and Baron tainties cannot be cancelled directly by a conventional backstepping (2010). last category a general case the timeMazenc, Modie Niculescu (2003);Zhang, Mondie andand Francisco design; ii) the of delay makes a delayof virtual controllers is less intuitive and involved as the (2003); Liberis and Krsticaddresses (2013); Boukas, Lui tainties cannot be cancelled directly bynonlinearities amore conventional backstepping (2010). The The lastand category addresses aMazenc, general case where where theBaron timedesign; ii) the presence presence of time time delay nonlinearities makes a uncerdelaydelay is present in both the control input and the system state. For (2003); Liberis and Krstic (2013); Zhang, Boukas, Lui and Baron free, static state feedback law insufficient for mitigating the effects tainties cannot be cancelled directly by a conventional backstepping (2010).is The lastincategory a general casesystem where state. the timedesign; ii) the presence of time delay nonlinearities makes a delaydelay present both theaddresses control input and the For free, static state feedback law insufficient for mitigating the effects each of time-delay nonlinear (2010). last a control general casesystem wheresubstantial the timeof and hence dynamic static state feedback design; ii) the presence time delayinstead nonlinearities a effects delaydelay is The present both theaddresses control input and problems, the state. For free, static state feedback insufficient forof each category category ofincategory time-delay nonlinear control problems, substantial of time-delay, time-delay, and henceofa a law dynamic instead ofmitigating static makes statethe feedback progress has made various results have obtained; delay is present both and the control and thebeen system state. see, For seem to be necessary. free, static state feedback law insufficient for mitigating the effects each category ofintime-delay nonlinear control problems, substantial of time-delay, and hence a dynamic instead of static state feedback progress has been been made and various input results have been obtained; see, seem to be necessary. for Mazenc, Modie and (2003); Mazenc, Mondie each category of time-delay control problems, substantial of time-delay, and hence a dynamic instead of static state feedback progress has been made and nonlinear various results have been obtained; see, seem to be necessary. for instance, instance, Mazenc, Modie and Niculescu Niculescu (2003); Mazenc, Mondie Motivated by progress has been made and various results have been obtained; see, Motivated by the the universal universal control control idea idea Nussbaum Nussbaum (1983); (1983); Lei Lei seem to be necessary. for instance, Mazenc, Modie and Niculescu (2003); Mazenc, Mondie and Lin (2006, 2007) and the recent development Zhang, Lin Motivated by the universal control idea Nussbaum (1983); Lei ⋆ for instance, Mazenc, Modie and Niculescu (2003); Mazenc, Mondie and Lin (2006, 2007) and the recent development Zhang, Lin and and This work was supported in part by the Thailand Research Fund ⋆ This work was supported in part by the Thailand Research Fund Lin Lin (2017), wethe propose a the novel method forNussbaum the construction construction of a Motivated bywe universal control ideafor (1983); Lei and (2006, 2007) and recent development Zhang, Lin and Lin (2017), propose a novel method the of a ⋆ under and Golden Jubilee Program ThisGrant work RSA6080027 was supported in Royal part by the Thailand Research Fund under Grant RSA6080027 and Royal Golden Jubilee Ph.D. Ph.D. Program set of Lyapunov-Krasovskii functionals and delay-free, and Lin (2006, anda the recent development Zhang, dynamic Lin of and Lin (2017), we 2007) propose novel method for a the construction a ⋆ set of Lyapunov-Krasovskii functionals and a delay-free, dynamic (Grant No. PHD/0158/2552), and by the Key Project of NSFC This work was supported in part by the Thailand Research Fund under Grant RSA6080027 and Royal Golden Ph.D. of Program (Grant No. PHD/0158/2552), and by the Jubilee Key Project NSFC state feedback scheme for counteracting the effects of timeLin wecontrol propose a novel method for athe construction of a set of(2017), Lyapunov-Krasovskii functionals and delay-free, dynamic state feedback control scheme for counteracting the effects of timeGrants 61533009, (B08015), Projects under RSA6080027 and Project Royal Golden Ph.D. of Program (Grant No. PHD/0158/2552), and by the Jubilee Key Research Project NSFC under Grant Grants 61533009, 111 111 Project (B08015), Research Projects delay nonlinearities and unknown control directions in set offeedback Lyapunov-Krasovskii functionals and a delay-free, state control scheme for counteracting the effects ofsystem timedelay nonlinearities and unknown control directions in the thedynamic system (JCY20150403161923519) and (KCYKYQD2017005). Correspond(Grant No. PHD/0158/2552), and by(B08015), the Key Research Project ofProjects NSFC under Grants 61533009, 111 (JCY20150403161923519) andProject (KCYKYQD2017005). Correspond(1) simultaneously. With the help of the new dynamic gain-based state feedback control scheme for counteracting the effects ofsystem timedelay nonlinearities and unknown control directions in the (1) simultaneously. With the help of the new dynamic gain-based ing Wei Lin under Grants 111 (B08015), Research Projects (JCY20150403161923519) andProject (KCYKYQD2017005). Corresponding author: author: Wei61533009, Lin ([email protected]) ([email protected]) delay nonlinearities With and unknown directions in the system (1) simultaneously. the help control of the new dynamic gain-based (JCY20150403161923519) and (KCYKYQD2017005). Corresponding author: Wei Lin ([email protected]) (1) simultaneously. With the help of the new dynamic gain-based ing author: Wei Lin ([email protected]) 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 78 Peer review© responsibility of International Federation of Automatic Copyright ©under 2018 IFAC IFAC 78 Control. Copyright © 2018 IFAC 78 10.1016/j.ifacol.2018.07.202 Copyright © 2018 IFAC 78

2018 IFAC TDS Kanya Rattanamongkhonkun et al. / IFAC PapersOnLine 51-14 (2018) 78–83 Budapest, Hungary, June 28-30, 2018

Lyapunov-Krasovskii functionals, we are able to design a delay free, dynamic state feedback compensator step-by-step, resulting in a solution to the global state regulation of the time-delay system (1) with boundedness. Interestingly, it is worth pointing out that the approach presented in this paper also provides a new yet simpler way for the design a dynamic state compensator that achieves global stabilization of the time-delay nonlinear system (1), in the absence of unknown control direction.

Theorem 6. For the time-delay nonlinear system (1) whose control directions are not known, there exists a delay-free, dynamic state feedback controller of the form L˙ = η(L, k, x), k˙ = h(L, k, x), u = α(L, k, x) (8) with α(L, k, 0) = 0, such that the system state x converges to the origin, while maintaining boundedness of the closed-loop system, where η : IR n−1 × IR n × IR n → IR n−1 , h : IR n−1 × IR n × IR n → IR n and α : IR n−1 × IR n × IR n → IR are C 0 mappings. 

Notations: Denote v¯i = [v1 , · · · , vi ]T ∈ IR i , for i = 1, · · · , n. For ¯i (t−d) = [x1 (t−d), · · · , xi (t−d)]T and instance, x ¯i = [x1 , · · · , xi ]T , x ¯ li = [l1 , · · · , li ]T . A Nussbaum function N (k) = k 2 cos(k), which is obviously an even function, will be used in this work. It is not difficult k to verify the following properties: 1) limk→+∞ sup k1 N (s)ds = +∞; 2) limk→+∞ inf

1 k

k 0

Proof. We apply the adding a power integrator technique Qian and Lin (2001a,b), together with the idea of utilizing the Nussbaum functions Nussbaum (1983) and dynamic gains Zhang, Lin and Lin (2017); Lei and Lin (2006, 2007), to design a delay-free, non-smooth dynamic state compensator (8) that does the job.

0

N (s)ds = −∞.

Step 1: For the x1 —subsystem of the time-delay system (1) with the unknown sign of θ1 , one can regard x2 as a virtual control. Define ξ1 = x1 and construct the Lyapunov function V1 (x1 , l1 ) = 12 (1 + 1 )ξ 2 , where l1 (·) ≥ 1 is a dynamic gain to be designed in Step 2. l1 1 Then, a direct computation gives 1 l˙1 2 1 c|ξ1 ξ2 |+2 |ξ1 f1 (x1 , x1 (t − d))| , (9) V˙ 1 ≤ (1+ )θ1 ξ1 x∗p 2 − 2 ξ1 +2¯ l1 2l1

2. PRELIMINARY To design a non-smooth but C 0 state feedback controller for the time-delay system (1), we introduce several key lemmas to be used throughout this paper. Lemma 1. Qian and Lin (2001a,b) For positive real numbers m, n and a real-valued function π (x, y) > 0, the following inequality holds ∀x, y ∈ IR. m n π (x, y) |x|m+n + π −m/n (x, y) |y|m+n (2) |x|m|y|n≤ m+n m+n

1 where ξ2 = xp21 − x∗p 2 . In view of Lemma 2.5, we have |f1 (·)| ≤ ¯1∗ (x1 (t − d)) |x1 (t − d)| , for some smooth functions γ ¯1 (x1 ) |x1 | + γ ¯1∗ (·) ≥ 0. Hence, γ ¯1 (·) ≥ 0 and γ

2|ξ1 f1 (·)| ≤ 2ξ12 γ ¯1 (x1 ) + ξ12 + ξ12 (t − d) γ ¯1∗2 (x1 (t − d))

Lemma 2. Lin and Qian (2002) For a C 0 function f (x, y), ∃ smooth functions a (x) ≥ 0, b (y) ≥ 0, c (x) ≥ 1 and d (y) ≥ 1, such that |f (x, y)| ≤ a(x) + b(y),

|f (x, y) | ≤ c (x) d (y) .

Use the bound

p−1

|x + y| ≤ 2 (|x| + |y|)

1 p

≤ |x|

1 p

p

p

|x + y | , 1

+ |y| p ≤ 2

p−1 p

1

(|x| + |y|) p .

If p is an odd positive integer, then |x − y|p ≤ 2p−1 |xp − y p | .

(3)

to construct the Lyapunov-Krasovskii functional



t

ξ12 (s) γ ¯1∗2 (x1 (s)) ds. t−d

l˙1 ξ12 1 1 V˙ 1LK ≤−nξ12 +[1+ ]θ1 ξ1 x∗p +ξ12 [2+n+2¯ γ1 (·)+¯ γ1∗2 (·)]+c2 ξ22(11) 2 − l1 2l12 To cope with the unknown sign of θ1 , we use the Nussbaum function (4) Nussbaum (1983) for the design of a virtual controller. Specifically, a virtual controller with the Nussbaum gain can be constructed as 1 = ξ1 N (k1 )(2 + n + 2¯ γ1 (·) + γ1∗2 (·)) x∗p 2 1 k˙ 1 = (1 + )ξ12 β1 (x1 ). l1

Lemma 4. Zhang, Lin and Lin (2017) For a C 0 function f (x, y) and a positive integer k, there exist smooth functions g(x) ≥ 0 and h(y) ≥ 0, such that

:= ξ1 N (k1 )β1 (x1 ) (12)

This, together with l1 (·) ≥ 1, results in l˙1 V˙ 1LK ≤ −nξ12 + (θ1 N (k1 ) + 1)k˙ 1 + c2 ξ22 − 2 ξ22 . 2l1

(6)

C1

function Lemma 5. Zhang, Lin and Lin (2017) For the xi , x ¯i (t − d)) with fi (0, 0) = 0, there exist smooth functions fi (¯ ∗ (x (t − d)) ≥ 0, j = 1, · · · , i, such that ¯ij γ ¯ij (xj ) ≥ 0 and γ j ∗ γij (xj ) |xj | + γ ¯ij (xj (t − d)) |xj (t − d)|) |fi (·)| ≤ Σij=1 (¯

(10)

From (9)-(10), it follows that

(5)

f (x, y) (|x|k + |y|k ) ≤ g (x) |x|k + h (y) |y|k .

γ ¯1∗ (·)

V1LK = V1 (x1 , l1 ) +

Lemma 3. Qian and Lin (2001a,b) Let x, y ∈ IR and p ≥ 1 be an integer. Then, p

79

(13)

Step 2: For the (x1 , x2 )—subsystem of the time-delay system (1) with the unknown sign of θ2 , we construct the Lyapunov-Krasovskii functional   1 ξ12 1 + k12 W2 (·) V2 = V1LK + k12 W2 (·) + l1 l1 l2 2

(7)

3. NONSMOOTH FEEDBACK WITH DYNAMIC GAINS

W2 (k1 , x1 , x2 ) =



x2

1 2−1/p1 (sp1 − x∗p ds, 2 )

(14)

x∗ 2

In this section, we utilize the idea from universal control Nussbaum (1983); Lei and Lin (2006, 2007), coupled with the feedback control strategy in Zhang, Lin and Lin (2017), to design a delay-free, dynamic state compensator that achieves global asymptotic state regulation with boundedness for the time-delay nonlinear system (1), As we shall see, the proposed dynamic compensator is composed of two sets of dynamic state feedback controllers. One of them is capable of mitigating the effects of the unknown control direction, while the other one is able to counteract the time-delay nonlinearities of the system (1). Notably, the idea of utilizing two sets of gain update laws has been explored in the area of adaptive control of nonlinear systems with unknown parameters by output feedback Lei and Lin (2006, 2007). This paper further demonstrates how a similar philosophy can be applied to effectively control the time-delay system (1) with unknown control direction.

where l2 (·) ≥ 1 is a dynamic gain to be designed in the next step. Following the argument in Qian and Lin (2001a,b), one can prove that W2 (k1 , x1 , x2 ) is C 1 and its partial derivatives are ∂W2 2−1/p1 = ξ2 , ∂x2

(15) ∗p

 1  ∂x2 1 ∂W2 =− 2− ∂x1 p1 ∂x1 ∗p

 1  ∂x2 1 ∂W2 =− 2− ∂k1 p1 ∂k1



x2



x2

1 1−1/p1 (sp1 − x∗p ds 2 )

x∗ 2 1 1−1/p1 (sp1 − x∗p ds. 2 )

x∗ 2

Moreover, m2 (x2−x∗2 )2p1 ≤W2 (·)≤(2p1−1)ξ22 , for a constant m2 > 0.

79

2018 IFAC TDS 80 Kanya Rattanamongkhonkun et al. / IFAC PapersOnLine 51-14 (2018) 78–83 Budapest, Hungary, June 28-30, 2018

V˙ 2LK ≤ −nξ12 − (n − 1)k12 ξ22 + (θ1 N (k1 ) + 1)k˙ 1 + c3 k12 ξ32

Since lj ≥ 1, it is deduced from (13) and (15) that l˙1 1 2−1/p1 (1 + )θ2 ξ2 V˙ 2 ≤ −nξ12 + (θ1 N (k1 ) + 1)k˙ 1 + c2 ξ22 − 2 ξ22 + 2l1 l1 l2 k12

k12

 1 2−1/p1 ∗ x3 p2 + k12 ξ22 c2 + c¯2 + n − 1 )θ2 ξ2 l1 l2  l˙2 ξ12 + W2 (·)) +Υ21 (·) + Υ∗21 (·) + Φ21 (·) − (25) ( l1 l22 2

+

(1 +

∂W2 ˙ 2  2 2−1/p1 ∂W2 k1 x˙ 1 + f2 (·) + k ξ l1 1 2 ∂x1 ∂k1  l˙1 l2 + l1 l˙2 ξ12 Similar to Step 1, because of the unknown sign of θ2 , we design the  ξ1 x˙ 1 l˙1 2 + k12 W2 (·))(16) − 2 k1 W2 (·) − +k1 k˙ 1 W2 (·) + ( virtual controller l1 l2 2 l1 l12 l22

2 2 ·(x∗p + xp32 − x∗p 3 3 )+







1 From ξ2 = xp21 − x∗p 2 , (12) and (15), it is not difficult to obtain (by Lemma 2.1 and Lemmas 2.3-2.5),

2k12 l1

2−1/p1 |ξ2 f2 (·)|≤

k12 ξ22 Υ21 (k1 , x1 , x2 )

1/p1

2 = l1 N (k2 )ξ2 x∗p 3

:= l1 N (k2 )(ξ2 β2 (k1 , x1 , x2 ))1/p1 1 1/p k˙ 2 = (1 + )ξ22 β2 1 (k1 , x1 , x2 ), k2 (0) = 1. (26) l2 with the Nussbaum gain k2 that is updated dynamically. Clearly, the dynamic compensator (26) leads to

1 + ξ12 Υ22 (k1 , x1 ) l1

1 2 ξ (t − d)Υ∗22 (k1 (t − d), x1 (t − d)) l1 1

+

[¯ c2 + n + Υ21 (·) + Υ∗21 (·) + Φ21 (·)]

+ξ22 (t − d)Υ∗21 (k1 (t − d), x1 (t − d), x2 (t − d)), (17)

  2k12  ∂W2 ∂W2 ˙  2 1 k1  + k1 k˙ 1 W2 (·) + x˙ 1 + ξ1 x˙ 1  l1

∂x1

∂k1

≤ k12 ξ22 Φ21 (k1 , x1 , x2 )+

V˙ 2LK ≤ −(n − 1)(ξ12 + k12 ξ22 ) + (θ1 N (k1 ) + 1)k˙ 1 l˙2 ξ12 + W2 (·)). (27) +c3 k12 ξ32 + (θ2 N (k2 ) + 1)k12 k˙ 2 − ( l1 l22 2

l1 l1 l2 1 2 1 ξ1 Φ22 (k1 , x1 )+ ξ12 (t − d)Φ∗2 (x1 (t − d)), l1 l1

Inductive Step: At step i − 1, assume that there are a LyapunovKrasovskii functional V(i−1)LK , a set of dynamic gains lj (·) ≥ 1, j = 1, · · · , i − 1, updated by

where Υ2j (·) ≥ 0, Υ∗2j (·) ≥ 0, Φ2j (·) ≥ 0 and Φ∗2 (·) ≥ 0, j = 1, 2, are smooth functions. Using the bounds Υ∗2j (·) and Φ∗2 (·) thus obtained, one can construct the Lyapunov-Krasovskii functional V2LK = V2 +



t

l˙1 = max{−l12 + l1 ρ1 (k1 , x1 ), 0},

ξ22 (s)Υ∗21 (k1 (s), x1 (s), x2 (s))ds

l˙2 = max{−α2 l22 + l2 ρ2 (l1 , k1 , k2 , x1 , x2 ), 0},

t−d

+



t

t−d

.. .

1 2 ξ (s)[Υ∗22 (k1 (s), x1 (s)) + Φ∗2 (x1 (s))]ds (18) l1 (s) 1

Then, it is deduced from (16) and (18) that V˙ 2LK ≤−nξ12

− (n −

1)k12 ξ22

with αj = 1/(2p1 ···pj−1 − 1), and a set of non-smooth but C 0 virtual controllers x∗1 , · · · , x∗i , with the Nussbaum gains, given by

1  l˙1 + (θ1 N (k1 ) + 1)k˙ 1− 2 ξ22+ ξ12 Φ∗2 (x1 ) 2l1 l1

 k12

+Υ22 (k1 , x1 )+Υ∗22 (k1 , x1 )+Φ22 (k1 , x1 ) +

(1 +

x∗1 x∗2 p1 k˙ 1

=0 ξ1 = x1 − x∗1 = ξ1 N (k1 )β1 (x1 ) ξ2 = xp21 − x∗2 p1 1 2 = (1 + )ξ1 β1 (·) l1 . . .. .. ∗ p1 ···pi−1 p1 ···pi−2 = (l1 · · · li−2 N (ki−1 )) ξi = xi p1 ···pi−1 xi ¯i−2 , x ·ξi−1 βi−1 (¯ li−3 , k ¯i−1 ) −x∗i p1 ···pi−1 1 1/p ···p k˙ i−1 = (1 + )ξ 2 β 1 i−2 (·) li−1 i−1 i−1

1 2−1/p1 ∗p2 )θ2 ξ2 x3 l2

l1  2¯ c 2 2−1/p1 p2 ∗p2 2 2 (x3 − x3 )| + k1 ξ2 c2 + (n − 1) + Υ21 (k1 , x1 , x2 ) + k1 |ξ2 l1  l˙2 ξ12 +Υ∗21 (k1 , x1 , x2 ) + Φ21 (k1 , x1 , x2 ) − + W2 (·)). ( (19) l1 l22 2

The inequality above is derived by neglecting the negative terms that are related to l˙1 and using the facts that −k12 W2 (·) ≤ −W2 (·) and 1 1 − l (t−d) ≤ 0 (see (22)). From (19), it is not difficult to show that l1 1 the dynamic state compensator l˙1 = max{−l12 + l1 ρ1 (k1 , x1 ), 0}, l1 (0) = 1,



ρ1 (k1 , x1 ) = 2 Υ22 (·) +

Υ∗22 (·)

(28)

2 ¯i−2 x l˙i−2 = max{−αi−2 li−2 + li−2 ρi−2 (¯ li−3 , k ¯i−2 ), 0},

+ Φ22 (·) +



Φ∗2 (·)

(20)

with ρj (·) > 0 and βj (·) > 0 being smooth functions, such that

V˙ (i−1)LK ≤ −(n − i + 2)

+

i−1  

j−1



(θj N (kj ) + 1)

j=1

0 ≤ l˙1 ≤ l1 ρ1 (·), l˙1 ≥ −l12 + l1 ρ1 (·), l1 ≥ l1 (t − d) ≥ 1



xp31 p2

−

1 p2 x∗p , 3



(30)

 l˙i−1 ( ξ12 + Σi−1 Wj (¯lj−2 , k¯j−1 , x¯j )) j=2 2

2 ˙ kj − km

2 l1 · · · li−2 li−1

(23) 2 k12 · · · ki−1

1  ξ12 l1 · · · li−1 l1 · · · li 2  i−1 2 2 ¯ ¯ ¯ ¯ ¯j ) + k1 · · · ki−1 Wi (li−2 , ki−1 , x ¯i ) , (31) +Σj=2 Wj (lj−2 , kj−1 , x

Vi = V(i−1)LK +

2¯ c2 2  2−1/p1 p2 2  k ξ (x3 − x∗p ¯2 k12 ξ22 + c3 k12 ξ32 , 3 ) ≤ c l1 1 2

m=0

We claim that (30) also holds at Step i. To prove this claim, consider the Lyapunov-Krasovskii functional

Moreover,

where ξ3 =

m=0



2 2 2 km ξj + ci k12 · · · ki−2 ξi2

where ci > 0 is a constant and k0 = 1. Clearly, (30) reduces to (27) when i = 3.

(22)

As a consequence, l˙1 1 2 − 2 ξ12 ≤ ξ12 − ξ ρ1 (k1 , x1 ) 2l1 2l1 1

i−1    j−1 j=1

(21)

can counteract the effect of the time-delay nonlinearity. In fact, by construction the gain l1 satisfies

(29)

(24)

Wi =

c¯2 and c3 are positive constants.



xi

x∗ i



¯i−1 , x Wi (¯ li−2 , k ¯i ) +

∗p1 ···pi−1 2−1/( p1 ···pi−1 )

sp1 ···pi−1 − xi



where li (·) ≥ 1 is a dynamic gain to be designed.

Substituting (23) and (24) into (19), we arrive at

80

ds,

2018 IFAC TDS Kanya Rattanamongkhonkun et al. / IFAC PapersOnLine 51-14 (2018) 78–83 Budapest, Hungary, June 28-30, 2018

Similar to the argument in Step 2, one can show that Wi (·) = ¯i−1 , x li−2 , k ¯i ) is C 1 . Moreover, Wi (¯

 2 ˙ Σi−1 j=1 2kj kj l1 · · · li−1 

∂Wi 2−1/(p1 ···pi−1 ) = ξi ∂xi ∂Wi =− ∂xj



∗p ···p ∂xi 1 i−1

∂Wi =− ∂kj





∂xi

∂Wi =− ∂lj





∗p ···p ∂xi 1 i−1

1 2− p1 · · · pi−1 1 2− p1 · · · pi−1 1 p1 · · · pi−1

∂xj

¯i−1 , x Ui (¯ li−2 , k ¯i )

 xi 

sp1 ···pi−1 −

x∗ i

2 2k12 · · · ki−1

l1 · · · li−1

∗p1 ···pi−1

¯i−1 , x Ui (¯ li−2 , k ¯i )

∂kj ∂lj

∗p ···p xi 1 i−1

j ≤ i − 1 and a positive constant mi .

  j−1

1− (p

1 1 ···pi−1

(32)

j−1







∗ · ξ12 + Σi−1 j=2 xj − xj



2 ˙ km kj

m=0





j · Σi−1 j=2 Σm=1

−

∂Wj ˙ ∂Wj ˙ ∂Wj km + Σj−2 lm x˙ m + Σj−1 m=1 m=1 ∂xm ∂km ∂lm

+ ξ1 x˙ 1

ξ2 l˙i ( 1 + Σij=1 Wj (·)). 2 l1 · · · li−1 li 2

¯i−1 (t − d), x li−2 (t − d), k ¯i (t − d)) + ·Υ∗i1 (¯ ∗ · ξ12 + Σi−1 j=2 xj − xj



t−d

l1 · · · li−1

∗ +Σi−1 j=2 xj − xj

+



From (33)-(38) and the fact that l ···l 1 (t) ≤ l (t−d)···l1 (t−d) 1 1 i−1 i−1 and ki ≥ 1, i = 1, . . . , i−1, A straightforward but tedious calculation gives j−1 j−1  

V˙ iLK ≤−(n − i + 2)Σi−1 j=1 [



2

− (34)

2 2 km ξj ]+Σi−1 j=1 (θj N (kj )+1)

2p1 ···pj−1 

2 l1 · · · li−2 li−1

+



2 ˙ km kj

m=0

ξ12

+

Σi−1 j=2



xj −

2p1 ···pj−1 x∗j



l1 · · · li−1



¯i−1 , x ¯i−1 , x ¯i−1 , x · Υi2 (¯ li−2 , k ¯i−1 )+Υ∗i2 (¯ li−2 , k ¯i−1 )+Φi2 (¯ li−2 , k ¯i−1 )



¯i−1 , x ¯i−1 , x ¯i−1 , x +Φ∗i2 (¯ li−2 , k ¯i−1 )+Ψi (¯ li−2 , k ¯i−1 )+Ψ∗i (¯ li−2 , k ¯i−1 ) +

2 k12 · · · ki−1

l1 · · · li−1

(1 +

1 2−1/(p1 ···pi−1 ) ∗pi 2 2 ξi+1 )θi ξi xi+1 + ci+1 k12 · · · ki−1 li



2 ¯i−1 , x ¯i−1 , x +k12 · · · ki−1 ξi2 1 + ci + c¯i +Υi1 (¯ li−2 , k ¯i )+Υ∗i1 (¯ li−2 , k ¯i )

 l˙i [ ξ12 + Σi Wj (·)] j=2 2

¯i−1 , x ¯i−2 , x +Φi1 (¯ li−2 , k ¯i )+ωi (¯ li−3 , k ¯i−1 ) −

l1 · · · li−1 li2

(40)

Based on (40), one can design the delay-free gain update law

ξ1

¯i−1 , x Φi2 (¯ li−2 , k ¯i−1 )

ξ l˙i−1 ( 21 + Σi−1 j=2 Wj (·))





¯i−1 , x Υi2 (¯ li−2 , k ¯i−1 )

l1 · · · li−1



¯i−1 (s), x li−2 (s), k ¯i−1 (s)) ds +Ψ∗i (¯

1 l1 · · · li−1

   i−1 ∂Wi  i−1 ∂Wi ˙ i−2 ∂Wi ˙  Σj=1 kj + Σj=1 lj x˙ j + Σj=1  ∂xj ∂kj ∂lj   2 1 2

(39)

 2  2p1 ···pj−1 1 ∗ ξ1 (s) + Σi−1 j=2 xj (s) − xj (s) l1 (s) · · · li−1 (s)



2p1 ···pj−1 

2 ¯i−1 , x ≤ k12 · · · ki−1 ξi Φi1 (¯ li−2 , k ¯i ) +



¯i−1 (s), x ξi2 (s)Υ∗i1 (¯ li−2 (s), k ¯i (s))ds +

¯i−1 (s), x ¯i−1 (s), x · Υ∗i2 (¯ li−2 (s), k ¯i−1 (s)) + Φ∗i2 (¯ li−2 (s), k ¯i−1 (s))



i2

2 2k12 · · · ki−1

t

t−d

t

 i−1  2p1 ···pj−1 1 Σ xj (t − d) − x∗j (t − d) l1 · · · li−1 j=2  ¯i−1 (t − d), x +ξ 2 (t − d) Υ∗ (¯ li−2 (t − d), k ¯i−1 (t − d)), 1

(38)

2p1 ···pj−1 

m=0

  ξ 2−1/(p1 ···pi−1 ) f  ≤ k2 · · · k2 ξ 2 i i 1 i−1 i i

+

2 + k12 · · · ki−1 ξi2

With the help of the bounds Υ∗ij (·), Φ∗ij (·) and Ψ∗i (·) thus obtained, which are related to the delay terms, we construct the LyapunovKrasovskii functional

(33)

l1 · · · li−1 ¯i−1 , x ·Υi1 (¯ li−2 , k ¯i ) + ξ 2 (t − d)



2p1 ···pj−1 



ViLK = Vi +

Using an argument similar to Zhang, Lin and Lin (2017), we obtain the estimations (34)-(38) (see the appendix for details): 2 2k12 · · · ki−1

(37)

where Υij (·) ≥ 0, Υ∗ij (·) ≥ 0, Φij (·) ≥ 0, Φ∗ij (·) ≥ 0, Ψi (·) ≥ 0 and Ψ∗i (·) ≥ 0, ωi (·) ≥ 0 j = 1, 2, are smooth functions.



1 2 km ) Wi (·) + l1 · · · li







  1   2− (p1 ···p1 i−1 ) 2− (p ···p 1 i−1 ) i  + ξi fi (·) + θi ξi (x∗p i+1   l1 · · · li−1    i−1 ∂Wi  ∗pi pi i−1 ∂Wi ˙ i−2 ∂Wi ˙   −xi+1 + xi+1 ) + Σj=1 x˙ j + Σj=1 kj + Σj=1 lj ∂xj ∂kj ∂lj    i−1  m=1 mj

∗p

∗ · ξ12 (t − d) + Σi−1 j=2 xj (t − d) − xj (t − d)

1 (1 + ) li

˙ Σi−1 j=1 (kj kj

p

i i (xi+1 − xi+1 )|

1 ¯i−1 (t − d), x li−2 (t − d), k ¯i−1 (t − d)) Ψ∗ (¯ + l1 · · · li−1 i

  ξ2 l˙i−1 i−1 1 2 ¯ ¯j−1 , x + Σ W ( l , k ¯ ) +ci k12 · · · ki−2 ξi2 − j j−2 j j=2 2 2 l1 · · · li−2 li−1

2 + l1 · · · li−1

1 2− p ···p 1 i−1

|θi ξi

(36)



) ds for 1 ≤

2 2 km ξj +Σi−1 j=1 (θj N (kj )+1)

m=0

2 k12 · · · ki−1

Wi (·)

 j ∂Wj ∂Wj ˙ 1  x˙ m + Σj−1 km ξ1 x˙ 1 + Σi−1 m=1 j=2 Σm=1 l1 · · · li ∂xm ∂km  ∂Wj ˙  1 ¯i−1 , x +Σj−2 Ψi (¯ li−2 , k ¯i−1 ) lm  ≤ m=1 ∂lm l1 · · · li−1

Analogous to the derivation of (19), using the facts that lj ≥ 1 2 W (·) ≤ −W (·), we deduce from (30)-(32) that (by and −k12 · · · ki−1 i i neglecting the negative terms which are related to l˙j , j = 1, . . . , i−1) V˙ i ≤ −(n − i + 2)Σi−1 j=1

m=1 mj



2 2 ), (¯ ci ξi2 + .ci+1 ξi+1 ≤ k12 · · · ki−1

¯i−1 , x Ui (¯ li−2 , k ¯i )

mi (xi − x∗i )2p1 ···pi−1 ≤ Wi (·) ≤ (2p1 ···pi−1 − 1)ξi2 where Ui =

2 km

2 ¯i−2 , x ≤ k12 · · · ki−1 ξi2 ωi (¯ li−3 , k ¯i−1 ),



2−

i−1 

81

2 ¯i−1 , x l˙i−1 =max{−αi−1 li−1 +li−1 ρi−1 (¯ li−2 , k ¯i−1 ), 0}, li−1 (0) = 1

(35)

 i−1  2p1 ···pj−1 1 Σ xj (t − d) − x∗j (t − d) l1 · · · li−1 j=2

ρi−1 (·) =

¯i−1 (t − d), x +ξ12 (t − d) Φ∗i2 (¯ li−2 (t − d), k ¯i−1 (t − d)),



 1  Υi2 (·)+Υ∗i2 (·)+Φi2 (·)+Φ∗i2 (·)+Ψi (·)+Ψ∗i (·) , Mi−1

where αi−1 =

81

1 p ···pi−2 (2 1 −1)

and Mi−1 = min{ 21 , m2 , · · · , mi−1 }

(41)

2018 IFAC TDS 82 Kanya Rattanamongkhonkun et al. / IFAC PapersOnLine 51-14 (2018) 78–83 Budapest, Hungary, June 28-30, 2018

 +∞

By construction, the gain thus constructed satisfies 0 ≤ l˙i−1 ≤ li−1 ρi−1 (·),

ξi2 ds ≤ ki (+∞) − ki (0) = c. On the other li (t) ≥ 1. Hence, 0 hand, (47) and the boundedness of ki (t), 1 ≤ i ≤ n, imply that

2 l˙i−1 ≥ −αi−1 li−1 + li−1 ρi−1 (·)(42)

Using (32) and (42), it is not difficult to prove that

VnLK (t) ≤ Σn j=1

 ξ2 −l˙i−1 i−1 2 ( 1 + Σi−1 )ξj2 ] km j=2 Wj (·)) ≤ Σj=1 [( 2 l1 · · · li−2 li−1 2

≤ c 1 Σn j=1

m=0



l1 · · · li−1

j−1





2 2 km ξj ]+Σi−1 j=1 [(θj N (kj )+1)

2 ˙ km kj

m=0

that x1 and





ξ2 l˙i 2 2 ξi+1 . ( 1 + Σij=2 Wj (·)) + ci+1 k12 · · · ki−1 2 l1 · · · li−1 li 2

(44)

In view of (44), one can design the non-smooth virtual controller with the Nussbaum gain 1/(p1 ···pi−1 )

+Υi1 (·) +

Υ∗i1 (·)



2 + ci + c¯i + n − i

+ Φi1 (·) + ωi (·)

(45)

Using the claim for i = n + 1 with u = xn+1 = x∗n+1 , we conclude that the dynamic state feedback controller that is composed of (28) with i = n + 1 and

2 ¯n−1 , x) k˙ n = ξn βn (¯ ln−2 , k

¯n−1 , x) ξn βn (¯ ln−2 , k



1 (p1 ···pn−1 )

 (p

1 1 ···pn )

2 2 k1 ···ki−1

l1 ···li−1

(xi −x∗i )2pi ···pi , i = 2, · · · , n, are also bounded.

(46) i + φi (x, x(t − d), t), x˙ i = θi xpi+1

j−1

j−1





m=0

2 2 k1 ···ki−1 Wi (·), l1 ···li−1

Theorem 7. Consider a family of uncertain time-delay systems with unknown control directions:

is such that V˙ nLK ≤ −Σn j=1 [(

(48)

Because the proposed nonsmooth control scheme is based on the Lyapunov-Krasovskii functional method, it is not surprising that Theorem 6 is robust with respect to the uncertainty. With this observation in mind, Theorem 6 can be extended to a larger family of uncertain time-delay systems dominated by a homogeneous system with time-delay. In fact, the following more general result also holds.

Substituting (45) into (44), we can show that (30) holds at Step i.

1

k˙ j (s)ds + c2 ≤ C.

To prove the convergence of the system we observe that  +∞state, ξ˙i , i = 1, · · · , n are also bounded and ξi2 (t)dt < +∞. By the 0 Barbalat’s lemma, it is concluded that ξi , i = 1, · · · , n converge to zero. This, in view of the coordinate transformation (29), implies that all the states x1 (t), · · · , xn (t) converge to zero as well, thus completing the proof of Theorem 3.1. 



¯i−1 , x := l1 · · · li−1 N (ki )(ξi βi (¯ li−2 , k ¯i ))1/(p1 ···pi−1 ) 1 k˙ i = (1 + )ξi2 βi (·)1/(p1 ···pi−1 ) li

u = (l1 · · · ln−1 N (kn )) pn

t

Because x1 and k1 are bounded and the gain l1 (·) given by (52)(21) is monotone non-deceasing, then l1 (·) must be bounded. If not, limt→+∞ l1 (t) = +∞. By continuity of ρ1 (·), ρ1 (k1 , x1 ) is bounded. Consequently, there is a time instant T > 0 such that −l12 + l1 ρ1 (k1 , x1 ) ≤ 0 on [T, +∞). This, together with (22), yields l˙1 = 0 on [T, +∞), which contradicts to the unboundedness of l1 (·). In conclusion, l1 (·) is bounded. The boundedness of l1 (·) and k1 implies the boundedness of x∗2 as well as x2 −x∗2 . As a such, x2 is also bounded. Similarly, one can prove the boundedness of li (·) and xi in the following recursive manner: x2 → l2 → x3 → · · · → ln−1 → xn , by the boundedness of ki (·), i = 1, · · · , n, (28) and the estimation (32). Therefore, all the signals of the closed-loop system (1)-(46)-(28) are bounded ∀t ∈ [0, +∞).

1 2−1/(p1 ···pi−1 ) ∗pi 2 (1 + )ξi xi+1 + k12 · · · ki−1 ξi2 li

∗pi xi+1 = l1 · · · li−1 N (ki )ξi



VnLK (·) on [0, +∞) implies the boundedness of x1 ,

· 2 + ci + c¯i + n − i + Υi1 (·) + Υ∗i1 (·) + Φi1 (·) + ωi (·) −

2 km (s))k˙ j (s)ds + VnLK (0)

m=0

i = 2, · · · , n. Using the estimation of Wi (·) in (32), one concludes

j−1

m=0

+



In view of (39) and (31), it is clear that the boundedness of

Substituting (41) into (40) yields

2 k12 · · · ki−1

|θj N (kj (s))+1|(

0

   Mi−1 ρi−1 (·)  2 ∗ 2p1 ···pj−1 ξ1 + Σi−1 (43) j=2 xj − xj l1 · · · li−1

V˙ iLK ≤−[n − i + 1]Σij=1

j−1

t

0

j−1

−



2 km )ξj2 ] + Σn j=1 [(θj N (kj ) + 1)(

i = 1, · · · , n,

(49)

where xn+1 = u and φi : IR n × IR n × IR → IR, is a continuous mapping. Assume that the uncertain function φi , i = 1, · · · , n, satisfies the homogeneous growth condition

2 ˙ km )kj ].

m=0

(47)

4. STATE REGULATION WITH BOUNDEDNESS



1

1

1

xi , x ¯i (t − d)) |x1 | p1 ···pi−1 +|x2 | p2 ···pi−1 +· · ·+|xi−1 | pi−1 |φi (·)|≤γi (¯ 1

1



+|xi |+|x1 (t − d)| p1 ···pi−1 +· · ·+|xi−1 (t − d)| pi−1 + |xi (t − d)| (50) We now use the inequality (47) to complete the proof of Theorem 6. In particular, it is shown that the proposed dynamic state feedback controller (46) and (28) can regulate the system state to the origin, while maintaining the boundedness of the closed-loop system.

xi , x ¯i (t − d)) ≥ 0 being a known smooth function. Then, with γi (¯ there is a delay-free, nonsmooth but C 0 dynamic state feedback (8) that steers the state x to zero and keeps the boundedness of the closed-loop system (8)-(49). 

First of all, we can establish, based on the Lyapunov inequalities (30) and (47), the following lemma.

Under the homogeneous growth condition (50), the proof of Theorem 7 can be carried out, with some subtle modifications, by means of an argument analogue to that of Theorem 6. For this reason, the details are left to the reader as an exercise.

Lemma 4.1. The Nussbaum gains ki (t), i = 1 · · · n, given by (45) are bounded ∀t ∈ [0, +∞).

Remark 8. When the nonlinear system (1) or (49) has multiple delays, the design of a delay-independent controller remains almost same, except that multiple Lyapunov-Krasovskii functionals with different time-delays need  t to be used. Specifically, the K(s)ds should be replaced by Lyapunov-Krasovskii functional

The proof of Lemma 4.1 requires delicate and tedious analyses and can be carried out in a fashion similar to the one in the appendix of Pongvuthithum, Rattanamongkhonkun and Lin (2018). The details are omitted due to the limited space. With the aid of the boundedness of ki (t), 1 ≤ i ≤ n, we deduce from (45) that ξi2 (t) ≤ k˙ i (t) because, by construction, βi (·) ≥ 1 and

t

t−di

82

t−d

K(s)ds in the recursive design. Of course, a similar philosophy

2018 IFAC TDS Kanya Rattanamongkhonkun et al. / IFAC PapersOnLine 51-14 (2018) 78–83 Budapest, Hungary, June 28-30, 2018

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can be employed to handle the general case when every subsystem of (1) involves different time delays.

5. CONCLUDING REMARKS

Remark 9. The assumption that the bound C of unknown coefficients θi , 1 ≤ i ≤ n is known is used for a technical convenience and can indeed be removed. When the bound C is unknown, a similar design procedure can be carried out with slightly different estimations of the right hand side of V˙ iLK in (44) so that the term (θj N (kj )+1) is replaced by (θj N (kj )+Cj ), where Cj is an unknown constant. Due to the characteristics of the Nussbaum function and the monotone property of the adaptive gains kj , 1 ≤ j ≤ n, the same argument in Appendix B can also be used for the stability proof.

In this paper, we have presented a delay-free, non-smooth dynamic state feedback scheme to control a family of uncertain time-delay systems with strong nonlinearities and unknown control directions. To cope with the effects of time-delay nonlinearities and unknown control directions, we have introduced, respectively, two sets of gains that need to be updated online, in a dynamic manner. One of them is the Nussbaum-type gains from universal control Nussbaum (1983), making it possible to mitigate the effect of unknown control directions, while the other one is borrowed the idea from the dynamic state feedback control method Zhang, Lin and Lin (2017), which is able to counteract the time-delay effects via a delay-free nonsmooth controller. Global asymptotic state regulation with boundedness of the closed-loop system has been proved to be possible, thanks to the construction of a set of new Lyapunov-Krasovskii functionals that are different from the previous ones in the literature, due to the involvement of the Nussbaum gains.

Finally, we present a simple but nontrivial example that demonstrates how Nussbaum gains need to be introduced to handle the problem of unknown control directions. Example 10. Consider a time-delay system in the plane, with strong nonlinearity and unknown control directions, of the form 1 3 x (t − d), (51) 2 2 where θ1 , θ2  0 are unknown constants whose signs are also unknown (either positive or negative), and represents unknown directions of the actuator. Note that the time-delay system under consideration involves not only an unknown control direction but also strong nonlinearities. The latter requires the use of a nonsmooth rather than smooth feedback control strategy. In fact, even in the case when control directions are known (e.g., θ1 = θ2 = 1) and no time delay is involved (i.e., d = 0), it is known that the planar system cannot be controlled by any smooth state feedback, even locally, and a nonsmooth feedback must be employed. x˙ 1 = θ1 x32 + x1 ,

x˙ 2 = θ2 u +

REFERENCES N. Bekiaris-Liberis and M. Krstic, “Compensation of statedependent input delay for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 58, 275-289, (2013). K. Gu, V. Kharitonov and J. Chen, Stability of time-delay systems. Boston: Birkhauser, (2003). C. Hua, X. Liu and X. Guan, “Backstepping control for nonlinear systems with time delays and applications to chemical reactor systems,” IEEE Trans. Ind. Electron., vol. 56, 3723-3732 (2009). M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilizationof time delay systems,” IEEE Trans. Automat. Contr., Vol. 46, 1048-1060 (2001). H. Lei and W. Lin, “Universal adaptive control of nonlinear systems with unknown growth rate by output feedback,”Automatica, vol. 42, pp. 1783-1789 (2006). H. Lei and W. Lin, “Adaptive regulation of uncertain nonlinear systems by output feedback: a universal control approach, ” Syst. Contr. Lett., vol. 56, 529-537 (2007). W. Lin and C. Qian, “Adaptive control of nonlinearly parameterized systems: a nonsmooth feedback framework," IEEE Trans. Automat. Contr., pp. 757-774 (2002). F. Mazenc, S. Mondie and S. I. Niculescu, “Global asymptotic stabilization for chains of integrators with a delay in the input,” IEEE Trans. Auto. Contr., vol. 48, pp. 57-63 (2003). F. Mazenc, S. Mondie and R. Francisco, “Global asymptotic stabilization of feedforward systems with delay in the input,” Proc. of the 42nd IEEE CDC, Maui, Hawaii, 4020-4025 (2003). R. D. Nussbaum, “Some remarks on a conjecture in parameter adaptive control,” Syst. & Contr. Lett., Vol. 3, pp. 243-246 (1983). C. Qian and W. Lin, “Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization,” Syst. Control Lett., vol. 42, 185-200 (2001). C. Qian and W. Lin, “A continuous feedback approach to global strong stabilization of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 46, 1061-1079, (2001). R. Pongvuthithum, K. Rattanamongkhonkun and W. Lin, “Asymptotic regulation of time-delay nonlinear systems with unknown control directions,” IEEE Trans. Auto. Contr., Vol. 63, 5 (2018). J. P. Richard, “Time-delay systems: an overview of some recent advances and open problems,”Automatica, vol. 39, 1667-94 (2003). X. Ye, “Asymptotic regulation of time-varying uncertain nonlinear systems with unknown control directions,” Automatica, vol. 35, pp. 929-935, (1999). X. Zhang and Y. Lin, “Global stabilization of high-order time-delay systems by state feedback,” Sys.Contr. Lett., vol. 65, 89-95 (2014). X. Zhang, W. Lin and Y. Lin, "Nonsmooth feedback control of timedelay systems: a dynamic gain based approach," IEEE Trans. Automat. Contr., Vol. 62, pp. 438-444 (2017). X. Zhang, E. K. Boukas, Y. Lui and L. Baron, “Asymptotic stabilization of high-order feedforward systems with delays in the input, ” Int. J. Robust Nonlinear Control, vol. 20, 1395-1406, (2010).

Following the control scheme proposed in section 3, we first consider the Lyapunov function V1 (x1 , l1 ) = 12 (1 + l1 )ξ12 , where ξ1 = x1 and 1 the gain l1 is updated by l˙1 = max{−l12 + l1 ρ(k1 , x1 ), 0},

l1 (0) = 1,

(52)

with ρ1 (·) ≥ 0 being a smooth function to be determined later on. For the x1 -subsystem, it is clear that the nonsmooth virtual control 1 2 ˙ law x∗3 2 = 2x1 N (k1 ), with k1 = 2(1 + l1 )x1 , globally asymptotically regulates it. 3 Define ξ2 = x32 − x∗3 2 = x2 − 2x1 N (k1 ). From (52), it is easy to see that l1 (·) ≥ 1 and l˙1 ≥ −l12 + l1 ρ1 (k1 , x1 ). Moreover,

1 2 V˙ 1 ≤ −2x21 + (θ1 N (k1 ) + 1)k˙ 1 − ξ ρ1 (k1 , x1 ). 2l1 1 Then, consider the Lyapunov-Krasovskii functional

V2LK =V1 (·)+

k12 l1

x2

1

2− 3 (s3 − x∗3 ds+ 2 )

x∗ 2

t 

ξ26 (s)+2(k16 x31 (s))2 l1 (s)

(53)



ds. (54)

t−d

Following the design procedure in Step 2, one can find a dynamic state compensator that consists of (52) and 1 10 (1 + )x21 + l1 + ξ24 3 l1   1 1 10 (1 + )x21 + l1 + ξ24 k˙ 2 = ξ22 2(k12 + 2k1 )2 + l1 3 l1 1/3

u = N (k2 )ξ2



2(k12 + 2k1 )2 +

with ρ1 (k1 , x1 ) = 2(2x41 k112 + 4k16 + N (k2 ) = k22 cos(k2 ), such that

5 (1 3

+

1 )x21 k12 ) l1

 (55)

in (52) and

(θ2 N (k2 ) + 1)k12 k˙ 2 , V˙ 2LK ≤ −x21 − k12 ξ22 + (θ1 N (k1 ) + 1)k˙ 1 + l1 from which it is deduced, as shown in Section 4, that the delayfree controller (55) and (52) achieves asymptotic state regulation and maintains the boundedness of the closed-loop system (51), (52) and (55), without the information of the sign of the parameter θi , i = 1, 2.

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