Decentralized Adaptive Control for Interconnected Nonlinear Systems with Input Quantization

The International International Federation of Automatic Control Control of the 20th World Proceedings The Federation of Congress Automatic Toulouse, France, July 2017 The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com Toulouse, France,Federation July 9-14, 2017 The International of Automatic Control Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 10419–10424 Control for Decentralized Adaptive Decentralized Adaptive Control for Decentralized Adaptive Control for Interconnected Nonlinear Systems with Interconnected Nonlinear Systems with Decentralized Adaptive Control for Interconnected Nonlinear Systems with Input Quantization Input Nonlinear Quantization Interconnected Systems with Input Quantization Input Jing Quantization Zhou ∗∗

Jing Zhou ∗∗ Jing Zhou ∗ Jing Zhou ∗ University of Agder, 4898 ∗ Department of Engineering Department of Engineering Sciences, Sciences, University of Agder, 4898 ∗ ∗ Department of Engineering Sciences,[email protected]) University of Agder, 4898 Grimstad, Norway (e-mail: Grimstad, Norway (e-mail: [email protected]) ∗ Department of Engineering Sciences,[email protected]) University of Agder, 4898 Grimstad, Norway (e-mail: Grimstad, Norway (e-mail: [email protected]) Abstract: Abstract: In In this this paper, paper, a a decentralized decentralized adaptive adaptive control control scheme scheme is is proposed proposed for for aa class class Abstract: this paper, a decentralized adaptive control scheme is proposed for uniform a class of uncertain uncertainInnonlinear nonlinear interconnected systems with input input quantization. A hysteresis hysteresis uniform of interconnected systems with quantization. A Abstract: paper, a decentralized adaptive control scheme is aaproposed for a class of uncertainInnonlinear interconnected systems with quantization. A smooth hysteresis uniform quantization is introduced to reduce In the control design, function is quantization is this introduced to reduce chattering. chattering. In input the control design, smooth function is of uncertain nonlinear interconnected systems with input quantization. A hysteresis uniform quantization is introduced to reduce chattering. In the control design, a smooth function is introduced with backstepping technique to compensate for the effects of interactions. It is shown introduced with backstepping technique to compensate for the effects of interactions. It is shown quantization is introduced to technique reduce chattering. In the design, a smooth function is introduced with backstepping to controllers compensate forcontrol the effects of interactions. Itof shown that decentralized adaptive can ensure global boundedness the that the the proposed proposed decentralized adaptive controllers can ensure global boundedness ofis all all the introduced with backstepping technique to compensate fortracking the effects of interactions. Itof is all shown that thein decentralized adaptive controllers can ensure global boundedness the signals the closed-loop interconnected systems and errors of converge signals inproposed the closed-loop interconnected systems and the the tracking errors of subsystem subsystem converge that theinproposed decentralized adaptive controllers can tracking ensure parameters. global of all the signals the which closed-loop interconnected systems and the errorsboundedness of subsystem converge to can adjusted by suitable design Simulation results to a a residual, residual, which can be be adjusted by choosing choosing suitable design parameters. Simulation results signals in the closed-loop interconnected systems and the tracking errors of subsystem converge to a residual, which can be adjusted by choosing suitable design parameters. Simulation results illustrate the effectiveness of the proposed control scheme. illustrate the effectiveness of the proposed control scheme. to a residual, which can beofadjusted by choosing illustrate the effectiveness the proposed controlsuitable scheme.design parameters. Simulation results © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. illustrate the effectivenesscontrol, of the proposed control scheme. Keywords: Decentralized adaptive control, Keywords: Decentralized control, adaptive control, input input quantization, quantization, backstepping, backstepping, interconnected systems. Keywords: Decentralized control, adaptive control, input quantization, backstepping, interconnected systems. Keywords: Decentralized interconnected systems. control, adaptive control, input quantization, backstepping, interconnected systems. 1. pletely 1. INTRODUCTION INTRODUCTION pletely known. known. In In practice, practice, it it is is often often required required to to consider consider 1. INTRODUCTION pletely known. In practice, it is often required to consider the case where the controlled plant is uncertain. the case where the controlled plant is uncertain. Quantized Quantized 1. INTRODUCTION pletely In practice, itplant is often required consider the caseknown. where the controlled is uncertain. Quantized control of systems with has been studied by control of systems with uncertainties uncertainties has been to studied by Interconnected systems the case where the controlled is uncertain. Quantized of systems with uncertainties has been by using robust approaches, see for De Persis and Interconnected systems have have been been used used to to model model aa wide wide control using robust approaches, see plant for examples, examples, De studied Persis and Interconnected systems have been used to model a wide variety natural, and complex dycontrol of(2010); systems uncertainties has studied by robust approaches, see for examples, Deal. Persis and Mazenc Liu et (2012a,b); Xing et (2016a). variety of of physical, physical, natural, and artificial artificial complex dy- using Mazenc (2010); Liuwith et al. al. (2012a,b); Xingbeen et al. (2016a). Interconnected systems been used to model apower wide variety physical, and artificial complex dy- Mazenc namical systems: such as communications networks, using robust approaches, see for examples, De Persis and (2010); Liu et al. (2012a,b); Xing et al. (2016a). Adaptive control of uncertain systems with input quantinamical of systems: suchnatural, as have communications networks, power Adaptive control of uncertain systems with input quantivariety physical, and etc. artificial complex dynamical systems: suchnatural, as communications power systems and aerospace systems, In the control of Mazenc (2010); Liu et al.in al. (2016a). control of uncertain systemsXing with quantization been Hayakawaa et (2009a); Sun systems of and aerospace systems, etc. Innetworks, the control of Adaptive zation has has been reported reported in(2012a,b); Hayakawaa et al. al.etinput (2009a); Sun namical systems: such as communications networks, power systems and aerospace systems, etc. In the control of interconnected systems, decentralized control strategy is Adaptive control of uncertain systems with input zation has been reported in Hayakawaa et al. (2009a); Sun et al. (2010); Hayakawaa et al. (2009b); Zhou et (2014); interconnected systems, decentralized control strategy is et al. (2010); Hayakawaa et al. (2009b); Zhou et al. al. quanti(2014); systems andand aerospace systems, etc. control In the control of interconnected systems, strategy is et an effective way. decentralized zation reported inIn Hayakawaa et et al.al. (2009a); Sun al. (2010); Hayakawaa al. (2009b); Zhou et (2009a,b), al. (2014); Xing et al. (2015, 2016b). Hayakawaa an efficient efficient and effective decentralized way. In In particular, particular, decentralized Xing ethas al.been (2015, 2016b).et In Hayakawaa et al. (2009a,b), interconnected decentralized control strategy is Xing an efficient and systems, effective way. for In controlling particular, decentralized adaptive control is systems with et al. (2010); Hayakawaa et al. (2009b); Zhou et al. (2014); et al. (2015, 2016b). In Hayakawaa et al. (2009a,b), the stability condition depends on the control signal, which adaptive control is employed employed for controlling systems with the stability condition depends on the control signal, which an efficient and way. for In particular, decentralized adaptive control is employed controlling systems with the large amount of uncertainties. The decentralized adaptive Xing et to al. be (2015, 2016b). In Hayakawaa et al. (2009a,b), stability condition depends on the control signal, which is hard hard to be checked in advance advance as the the control signal is large amount of effective uncertainties. The decentralized adaptive is checked in as control signal is adaptive control is employed for controlling systems with large amount of uncertainties. The decentralized adaptive technique, designed independently for local subsystems the stability condition depends on the control signal, which is hard to be checked in advance as the control signal is only available after the controller is put in operation. In technique, designed independently for local subsystems only available after the controller is put in operation. In large amount of uncertainties. The decentralized adaptive technique, designed independently for feedback local subsystems and locally available signals for propose, is hard checked incontroller advance as the adaptive control signal is available afteraathe is put in operation. In Zhou et al. (2014), backstepping-based control and using using locally available signals for feedback propose, only Zhou et to al.be (2014), backstepping-based adaptive control technique, designed independently for local subsystems and using locally available signals for feedback is an appropriate strategy to be employed, see for only available after the controller is put in operation. In et al. (2014), a backstepping-based adaptive control scheme is presented for uncertain nonlinear systems with is an appropriate strategy to be employed, seepropose, for exex- Zhou scheme is presented for uncertain nonlinear systems with and locally available signals for feedback is anusing appropriate strategy to and be employed, seepropose, for ex- scheme amples Wen (1995); Narendra Oleng (2002); Pagilla Zhou (2014), a for backstepping-based adaptive control isal.presented uncertain nonlinear systems with input et signal by hysteresis hysteresis quantizer. Although the proamples Wen (1995); Narendra and Oleng (2002); Pagilla input signal by quantizer. Although the prois an appropriate strategy to be employed, see for examples Wen (1995); Narendra and Oleng (2002); Pagilla et al. (2007). Since backstepping technique was proposed scheme is presented for uncertain nonlinear systems with input signal by hysteresis quantizer. Although the proposed method can avoid stability conditions depending et al. (2007). Since backstepping technique was proposed posed method can avoid stability conditions depending amples Wen Narendra Oleng (2002); Pagilla et (2007). Since backstepping technique was proposed in al. Krstic et (1995); al. (1995), it has has and been widely used to dede- posed input byinput, hysteresis quantizer. Althoughdepending the promethod can avoid stability conditions on control it the nonlinear functions in Krstic et al. (1995), it been widely used to on the thesignal control input, it requires requires the nonlinear functions et al. (2007). Since backstepping technique was proposed in Krstic et al. (1995), it has been widely used sign adaptive controllers for uncertain systems due its posed method can avoid stability conditions depending the control input, it requires the nonlinear functions to satisfy global Lipschitz condition. This strict condition sign adaptive controllers for uncertain systems duetoto to deits on to satisfy global Lipschitz condition. This strict condition in Krstic etsuch al. (1995), it been widely used sign adaptive advantages as transient performance as on the control input, it requires theet nonlinear functions controllers forhas uncertain systems duetoto deits global Lipschitz condition. This hassatisfy been relaxed recently in Xing Xing et al. strict (2015,condition 2016a). advantages such as improving improving transient performance as to has been relaxed recently in al. (2015, 2016a). sign adaptive controllers for uncertain systems due to its advantages such as improving transient performance as in Zhou et al. (2004, 2007); Wen et al. (2011). Because to satisfy global Lipschitz condition. This strict condition has been relaxed recently in Xing et al. (2015, However in Xing et al. (2015), the control signal is in Zhou et al. (2004, 2007); Wen et al. (2011). Because However in Xing et al. (2015), the control signal2016a). is imimadvantages such(2004, as improving transient performance as However in et al. 2007); Wen et al. (2011). Because of Zhou such advantages, advantages, research on decentralized adaptive has been relaxed recently in Xing et al. (2015, 2016a). in Xing et al. (2015), the control signal is plicitly involved in the proposed control law, where the of such research on decentralized adaptive plicitly involved in the proposed control law, where imthe in Zhou et al.the (2004, 2007); Wen et al. has (2011). Because plicitly of suchusing advantages, research on decentralized control backstepping technique also received However in Xing et al. (2015), the control signal is iminvolved in the proposed control law, where the control signal needs to satisfy the equation resulted from control using the backstepping technique has also adaptive received control signal needs to satisfy the equation resulted from of such advantages, onZhang decentralized control using thesee backstepping also adaptive received great attention, for examples et al. Jiang plicitly involved in is proposed law, wherefrom the signal needs tonontrivial satisfy theto equation resulted the law. It solve equation to great attention, see forresearch examplestechnique Zhang ethas al. (2000); (2000); Jiang control the control control law. It isthe nontrivial tocontrol solve the the equation to control using the backstepping technique has also received great attention, see for examples Zhang et al. (2000); Jiang (2000); Jain and Khorrami (1997); Wen and Zhou (2007). control signal needs to satisfy the equation resulted from the control law. It is nontrivial to solve the equation to obtain the control signal explicitly. In Xing et al. (2016b), (2000); Jain and Khorrami (1997); Wen and Zhou (2007). obtain the control signal explicitly. In Xing et al. (2016b), great examples al. in (2000); Jiang obtain (2000); Jainbeen andsee Khorrami (1997); Wenetand Zhou Thereattention, has been a for great deal ofZhang attention in the(2007). study the control law. Itsignal is nontrivial to In solve the to the explicitly. et equation al.quantizer (2016b), aa new form of combining aaXing uniform There has a great deal of attention the study new formcontrol of quantizer quantizer combining uniform quantizer (2000); Jain and Khorrami (1997); Wen and Zhou (2007). There has been a great deal of attention in the study of quantized control systems, in which a control system the control signal explicitly. In Xing et al. (2016b), new form of quantizer combining a uniform quantizer and a logarithmic quantizer was introduced. However the of quantized control systems, in which a control system aobtain and a logarithmic quantizer was introduced. However the There has been deal in ofquantization, attention in due thesystem study of quantized control systems, which a control is with information to its aquantization new form of quantizer combining a uniform quantizer a logarithmic quantizer was introduced. However the function in et (2016b) will introduce is interacted interacted witha great information quantization, due to its and quantization function in Xing Xing et al. al. (2016b) will introduce of quantized control systems, in which a control system is interacted with information quantization, due to its theoretical and practical importance in the study of digital and a logarithmic quantizer was introduced. However the quantization function in Xing et al. (2016b) will introduce the discontinuity into the system, which may lead to untheoretical and practical importance in the study of digital the discontinuity into the system, which may lead to unis interacted information quantization, due toand its the theoretical andwith practical importance in the study of digital control, hybrid systems, networked control systems quantization function in Xing et al. (2016b) will introduce discontinuity into the system, which may lead to undesirable switching and chattering phenomenon. control, hybrid systems, networked control systems and desirable switching and chattering phenomenon. theoretical and practical importance in the study of digital control, hybrid systems, networked control systems and desirable so main motivation for input quanthe into system, may lead tononunswitching andthe chattering phenomenon. The decentralized adaptive control of so on. on. The The main motivation for considering considering input quanThe discontinuity decentralized adaptive control which of interconnected interconnected noncontrol, hybrid systems, networked control systems and so on. The main motivation for considering input quantization in control systems is that quantization schemes desirable switching and chattering phenomenon. The decentralized adaptive control of interconnected nonlinear systems preceded by input quantization is a chaltization in control systems is that quantization schemes linear systems preceded by input quantization is a chalso on.sufficient The main motivation considering input quan- linear tization in control systems that quantization schemes have precision and require low The decentralized adaptive control of in interconnected nonsystems preceded input quantization is a challenging task to difficulties considering both have sufficient precision andisfor require low communication communication lenging task due due to the the by difficulties in considering both tization in control systems that quantization schemes have sufficient precision andis require low are communication rate. of stabilization of systems discussed in linear systems preceded by input quantization is a chaltask due to the difficulties in considering both input quantization and uncertain interconnections. In rate. Research Research of stabilization of systems are discussed in lenging input quantization and uncertain interconnections. In this this have sufficient precision require low communication rate. of stabilization of systems discussed in input Ishii and Francis (2002); Tatikonda and Mitter (2004); lenging task due to difficulties considering and uncertain Inboth this paper,quantization we provide solution to interconnections. thisinproblem problem using deIshii Research and Francis (2002);and Tatikonda andare Mitter (2004); paper, we provide aa the solution to this using derate. Research of (2001); stabilization of systems are discussed in paper, Ishii and Mitter Francis (2002);Liu Tatikonda Mitter (2004); Elia and Elia (2004); Nair and input quantization uncertain In this we provide a backstepping solution to interconnections. this problem using decentralized adaptive technique. The control Elia and and Mitter (2001); Liu and Eliaand (2004); Nair and centralized adaptiveand backstepping technique. The control Ishii and Francis (2002);and Tatikonda and Mitter (2004); Elia Mitter Liu and Elia (2004); Nair Evans (2004); De Persis Isidori (2004); Liberzon and paper, provide a backstepping solution to this problem using deadaptiveby technique. The control signal quantized aa hysteresis uniform quantizer. The Evansand (2004); De (2001); Persis and Isidori (2004); Liberzon and centralized signal is iswe quantized by hysteresis uniform quantizer. The Elia and Mitter Liusystems and Elia (2004); Nair Evans (2004); De (2001); Persis Isidori (2004); Liberzon and signal Hespanha (2005), where the considered are centralized adaptivebybackstepping technique. The control is quantized a hysteresis uniform quantizer. The Hespanha (2005), where and the systems considered are comcomEvans (2004); De Persis Isidori (2004); Liberzon and signal is quantized by a hysteresis uniform quantizer. The Hespanha (2005), where and the systems considered are comHespanha (2005), where the systems considered are comCopyright © 2017 IFAC 10906 Copyright © 2017 IFAC 10906 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017, 2017 IFAC 10906 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 10906 10.1016/j.ifacol.2017.08.1969

Proceedings of the 20th IFAC World Congress 10420 Jing Zhou / IFAC PapersOnLine 50-1 (2017) 10419–10424 Toulouse, France, July 9-14, 2017

hysteresis uniform quantization scheme can reduce the chattering phenomenon caused by the discontinuity of the uniform and logarithmic quantization. The quantization error of the hysteresis uniform quantization is shown to be bounded. By using the property of the quantizer and introducing a smooth function in the controller design, the effects of the input quantization and interconnections are effectively compensated. With the proposed schemes, the nonlinear functions in the system are not required to satisfy the Lipschitz conditions. It is established that the designed controller can ensure the boundedness of all signals in the closed loop interconnected systems. The tracking errors can asymptotically converge to a residual, which can be made arbitrarily small by choosing suitable design parameters. Simulation results illustrate the effectiveness of the proposed scheme.

¯ i,j (¯ xk,j ) are known smooth and bounded functions, where h ¯ i,j such that that is, there exist positive constants H ¯ ¯ xk,j )| ≤ Hi,j . |hi,j (¯

2. PROBLEM STATEMENT

In this paper, a hysteresis uniform quantizer is defined as follows:

2.1 System Model In this paper, a class of interconnected nonlinear systems is considered and described by the following strict-feedback form as in Krstic et al. (1995); Marino and Tomei (1995). xi,1 ) + hi,1 (¯ x1,1 , x ¯2,1 , · · · , x ¯N,1 ) x˙ i,1 = xi,2 + θiT ϕi,1 (¯ x˙ i,2 = xi,3 + θiT ϕi,2 (¯ xi,2 ) + hi,2 (¯ x1,2 , x ¯2,2 , · · · , x ¯N,2 ) .. .. . .

xi,ni −1 ) x˙ i,ni −1 = xi,ni + θiT ϕi,ni −1 (¯ x1,ni −1 , x ¯2,ni −1 , · · · , x ¯N,ni −1 ) +hi,ni −1 (¯

x˙ i,ni = qi (ui (t)) + θiT ϕi,n (xi ) + hi,ni (x1 , x2 , · · · , xN ) yi = xi,1 ,

(1)

Remark 1. Assumption 2 implies that the effect of the nonlinear interaction to the ith local subsystem from the jth subsystem is bounded by a nonlinear higher order function of the states xj of the jth subsystem. This assumption is more relaxed than the existing one in Panagi and Polycarpou (2011); Zhou et al. (2012); Wen and Zhou (2007), where the interconnected functions are outputdependent functions. 2.2 Quantizer

qi (ui (t)) =  δi δi   Qi,j sgn(ui ), Qi,j − + βi < |ui | ≤ Qi,j + + βi   2 2    and u˙ i > 0, or    δi δi   Qi,j − − βi < |ui | ≤ Qi,j + − βi   2 2 and u˙ i < 0 ,   δi δi   + βi and u˙ i > 0, or 0, − + βi < u i ≤   2 2   δi δi    − βi and u˙ i < 0 − − βi < u i ≤   2 2  qi (ui (t− )), u˙ i = 0 (3)

where Qi,0 = 0, Qi,j = Qi,j−1 + δi , δi > 0, j = 1, 2, . . . , m, βi = δi pi is the the hysteresis width constant, 0 < pi < 0.5 is the hysteresis percentage, and δi +2βi is the quantization interval length. qi (ui ) is in the set U = {0, ± Qi,j }. The map of the hysteresis uniform quantizer qi (ui (t)) for ui > 0 is shown in Figure 1.

q(u)

where xi (t) = [xi,1 (t), xi,2 (t), . . . , xi,ni (t)]T ∈ ℜni are the states, qi (ui (t)) ∈ ℜ1 and yi (t) ∈ ℜ1 are input and output of the system, respectively, the vector θi ∈ ℜri is constant and unknown, ϕi,j ∈ ℜri , j = 1, . . . , ni are known nonlinear functions and differentiable, x ¯i,j (t) = Hysteresis uniform quantizer 4 [xi,1 (t), xi,2 (t), . . . , xi,j (t)]T ∈ ℜj , hi,j (¯ x1,j , x ¯2,j , . . . , x ¯N,j ) ∈ ℜ1 denotes the nonlinear interactions from the jth sub3.5 system to the ith subsystem for j ̸= i or a nonlinear unmodeled part of the ith subsystem for j = i. The input 3 qi (ui (t)) represents the quantization and takes the quan2.5 tized values and ui (t) ∈ ℜ1 is the control input signal to be quantized at the encoder side. For this class of nonlinear 2 interconnected systems, we assume that the existence and uniqueness of solution are satisfied. 1.5 The control objective is to design a totally decentralized adaptive controller for system (1) such that the closed1 loop system is stable and the output yi (t) can track a 0.5 given reference signal yri (t) as close as possible with the following assumptions. 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Assumption 1. The reference signal yri (t) and its ni th u order derivatives are known and bounded. Assumption 2. The nonlinear interaction functions satFig. 1. The map of hysteresis uniform quantizer q(u) with isfy δi = 1 and pi = 0.3 x1,j , x ¯2,j , · · · , x ¯N,j )|2 ≤ |hi,j (¯

N ∑

k=1

¯ i,j (¯ xk,j ), h

(2)

Lemma 1. The quantization error for the hysteresis uniform quantizer (3) is bounded as follows.

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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Jing Zhou / IFAC PapersOnLine 50-1 (2017) 10419–10424

δi + βi , (4) 2 where δi > 0 and βi > 0 are quantization parameters. Remark 2. The hysteresis characteristics ensure that for the given time intervals the solution of the quantized system has finite switching times. Thus, the chattering phenomenon caused by discontinuities of the uniform or logarithmic quantizer can be effectively avoided, at the price of requiring more quantization levels to maintain the same error bound. Compared with the hysteresis logarithmic quantizer in De Persis and Mazenc (2010); Ceragiolia et al. (2011); Zhou et al. (2014), the quantization error of the quantizer (3) is bounded as (4). |qi (ui ) − ui | ≤

Zhou et al. (2004). Step 1: Considering a Lyapunov function 1 1 2 ˜ Vi,1 = zi,1 + θ˜iT Γ−1 i θi , 2 2 where θ˜i = θi − θˆi . Then its derivative follows that

zi,1 = xi,1 − yri

(5)

2 ∗ 2 +zi,1 hi,1 − li zi,1 − li,1 zi,1 si,1 (zi,1 )

(7) j−1 ∑ ∂αi,k−1

T ˆ θi + αi,j = −ci,j zi,j − zi,j−1 − ωi,j

+

j−1 ∑ ∂αi,k−1

k=2

∂ θˆi (

k=1

∂xi,j−1

where by using the property (2) in Assumption 2 and Young’s inequality as follows. −li (zi,1 )2 + zi,1 hi,1 ≤

Vi,j =

xi,k+1

ωi,j = ϕi,j −

j ∑

V˙ i,j ≤ −

)

+

k=1

j ∑

(16)

k=1

( ) 2 ˆ˙ + zi,j zi,j+1 + θ˜iT τi,j − Γ−1 ci,k zi,k i θi

(j−1 ∑ ∂αi,k k=1

+

(8) (9) (10)

−

˙ θˆi = Γi τi,ni − Γi lθi θˆi . (11) The function si,j (zi,j ) is continuous and defined as follows:  1   |zi,j | ≥ σi  (zi,j )2 si,j (zi,j ) = (12) 1   |z | < σ i,j i  2 (σi − (zi,j )2 )ni −j+2 + (zi,j )2 where σi i = 1, . . . , ni is a positive design parameter. It is shown that si,j (zi,j ) is ni − j + 1th order differentiable in

zi,k+1

)

j N ∑ ∑ ni − k + 1

4li

j N ∑ ∑

(

˙ Γi τi,j − θˆi − Γi lθi θˆi

)

hi,k (¯ xm,k )

2 ∗ ¯ m,k (¯ si,k (zi,k )h xi,k ). li,k zi,k

(17)

m=1 k=1

Step ni . We choose a Lyapunov function as

∗ where ci,j , li , li,j , lθi are positive constants, Γi is a positive (j) definite matrix, yri represents the jth derivative of yri , θˆi

is the estimate of θi . The parameter adaptation law for θˆi is designed as

∂ θˆi

m=1 k=1

m=1

∂αi,j−1 ϕi,k , j = 1, . . . , ni ∂xi,k

1 2 ˜ zi,k + θ˜iT Γ−1 i θi . 2 2

Then its derivative follows that

k=1

τi,j = τi,j−1 + ωi,j zi,j

j ∑ 1

k=1

(j)

¯ m,j (¯ xi,j ) h

N 1 1 ∑¯ |hi,1 |2 ≤ xm,1 )(15) hi,1 (¯ 4li 4li m=1

Step j, (j = 2, . . . , ni −1): We choose a Lyapunov function

∂αi,j−1 Γi τi,j − Γi lθi θˆi + ∂ θˆi )2 j−1 ( ∑ ∂αi,j−1 zi,j −li zi,j − li ∂xi,k ∗ zi,j si,j (zi,j ) −li,j

¯ m,1 (¯ xi,1 ) h

m=1

(14)

Γi ωi,j zi,k + yri

N ∑

N ∑

( ) 2 ˆ˙ ≤ −ci,1 zi,1 + zi,1 zi,2 + θ˜iT τi,1 − Γ−1 i θi ) N ( ∑ 1 ¯ ∗ 2 ¯ + xm,1 ) − li,1 zi,1 si,1 (zi,1 )hm,1 (¯ xi,1 ) , hi,1 (¯ 4li m=1

zi,j = xi,j − αi,j−1 , j = 2, 3, . . . , ni , i = 1, . . . , N (6) where αi,j are virtual controllers. The final controls and the virtual laws are designed as follows. ui = αi,ni − ki zi,ni

(13)

( ) 2 ˆ˙ V˙ i,1 ≤ −ci,1 zi,1 + zi,1 zi,2 + θ˜iT τi,1 − Γ−1 i θi

3. STATE FEEDBACK CONTROL In this section, adaptive decentralized feedback controllers are constructed by using backstepping design technique. To this end, we begin by introducing the change of coordinates

10421

Vi,ni =

ni ∑ 1

k=1

2

1 2 ˜ zi,k + θ˜iT Γ−1 i θi . 2

(18)

Then the derivative of Vi,ni satisfies ( V˙ i,ni = V˙ i,ni −1 + zi,ni qi (ui ) + θiT ϕi,ni + hi,ni (n )

−yri i −

n∑ i −1 k=1

∂αi,ni −1 (xi,k+1 + θiT ϕi,k + hi,k ) ∂xi,k

∂αi,ni −1 ˆ˙ ) − θi . ∂ θˆi

(19)

Using the property of the hysteresis uniform quantizer in (4), the following inequality is derived.

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Proceedings of the 20th IFAC World Congress 10422 Jing Zhou / IFAC PapersOnLine 50-1 (2017) 10419–10424 Toulouse, France, July 9-14, 2017

zi,ni qi (ui ) ≤ zi,ni ui + |zi,ni |Di ,

(20)

δi 2

where Di is the bound of + βi . The following inequality is derived using Young’s inequality.

¯ i,k (σi , li , H ¯ i,k ) by using the property of the function M hi,j in Assumption 2. In conclusion, Mi,m,k (zi,k , x ¯m,k ) ≤ ¯ i,k ). ¯ i,k (σi , li , H M Then the derivative of V in (24) satisfies

1 1 (21) lθi θ˜T θˆi ≤ − lθi ∥ θ˜i ∥2 + lθi ∥ θi ∥2 . 2 2 Then the derivative of Vi,ni in (19) is derived by using (7), (11), (17), (20), and (21) V˙ i,ni ≤ − +

ni ∑

2 ci,k zi,k

k=1

+

1 D2 1 − lθi ∥ θ˜i ∥2 + i + lθi ∥ θi ∥2 2 4ki 2

k=1

−

∗ 2 ¯ m,k (¯ li,k zi,k si,k (zi,k )h xi,k ),

(22)

m=1 k=1 D2

where −ki (zi,ni )2 + |zi,ni Di | ≤ 4kii is used. Now we construct a global Lyapunov function of the overall system as follows: V =

N ∑

Vi,ni =

i=1

ni N ∑ ∑ 1 i=1 k=1

2

2 zi,k +

N ∑ 1 i=1

and its derivative is given as (n N i ∑ ∑

1 2 ci,k zi,k + lθi ∥ θ˜i ∥2 V˙ ≤ − 2 i=1 k=1 ) N ( 2 ∑ 1 Di + + lθi ∥ θi ∥2 4ki 2 i=1 +

ni N ∑ N ∑ ∑ ni − k + 1

4li

i=1 m=1 k=1

− ≤

ni N ∑ N ∑ ∑

2

˜ θ˜iT Γ−1 i θi ,

)

4li

−

(

ni N ∑ ∑

2 ∗ zi,k si,k (zi,k ) li,k

)

)

(zi,k )2 si,k (zi,k ) = 1, 2

∀|zi,k | ≥ σi ,

ni N ∑ N ∑ ∑

)

+D

¯ i,k ) ¯ i,k (σi , li , H M

4. SIMULATION RESULTS

(25)

∀|zi,k | < σi . (26) 0 < (zi,k ) si,k (zi,k ) < 1, ) ( ∗ 2 ¯ i,k (¯ − li,k zi,k si,k h ¯m,k ) = ni −k+1 xm,k ). Let Mi,m,k (zi,k , x 4li By choosing ni − k + 1 , (27) 4li we have Mi,m,k (zi,k , x ¯m,k ) = 0 for |zi,k | ≥ σi . For |zi,k | < σi , Mi,m,k (zi,k , x ¯m,k ) is bounded by a positive constant ∗ li,k =

k=1

1 + lθi ∥ θ˜i ∥2 2

≤ −CV + F, (28) where Ci = min{2ci,1 , 2ci,2 , ..., 2ci,ni , lθi λmin (Γi )}, C = min{Ci }, i = 1, . . . , N , λmin (Γi ) is the minimum eigen∑ N ∑ N ∑ ni ¯ value of Γi , and F = D + i=1 m=1 k=1 Mi,k is a positive constant. Finally, the main results are stated in the following theorem. Theorem 1. Consider the closed-loop systems consisting of plant (1) with a hysteresis uniform quantizer (3), the decentralized adaptive backstepping controllers (7) with virtual control laws (8)-(10), parameter estimators with updating laws (11). The global boundedness of all the signals in the interconnected systems is ensured. Furthermore, the tracking error is ultimately bounded as follows: } { 2F 2 . (29) ∥ e(t) ∥ ≤ max 2V (0), C Proof 1. By direct integrations of the differential inequality (28), we have

¯ i,k h

1 2 ci,k zi,k + lθi ∥ θ˜i ∥2 + D (24) 2 i=1 k=1 ) ∑N ( D 2 where D = i=1 4kii + 12 lθi ∥ θi ∥2 . From the definition of si,k (zi,k ) in (12), it is shown that −

i=1

2 ci,k zi,k

F V ≤ V (0)e−Ct + (1 − e−Ct ), , (30) C ∑N ∑ni 1 2 −1 ˜ 1 ˜T where V (0) = j=1 2 zi,j (0) + 2 θi (0)Γi θi (0). It i=1 shows that V is uniformly bounded, yielding boundedness of all closed-loop signals such that xi,j , θˆi , αi,j , ui for i = 1, . . . , N and j = 1, . . . , ni are all bounded. We define z(t) = [z1 , ..., zN ]T , zi = [zi,1 , ..., zi,ni ]T {, i = 1, . . .}, N . From (30), it obtains that ∥ z(t) ∥2 ≤ max 2V (0), 2F C . It implies that the tracking errors will converge to a compact set.

¯ i,k (¯ xm,k ) h

i=1 m=1 k=1

i=1 m=1 k=1

(23)

2 ∗ ¯ m,k (¯ zi,k si,k (zi,k )h xi,k ) li,k

ni ( N ∑ N ∑ ∑ ni − k + 1

(n N i ∑ ∑

i=1 m=1 k=1

ni N ∑ ∑ ni − k + 1 ¯ xm,k ) hi,k (¯ 4li m=1 ni N ∑ ∑

V˙ ≤ −

In this section, a nonlinear interconnected uncertain system with a hysteresis uniform quantized input is considered as follows. x ¨1 + θ1 x˙ 1 + tanh(x1 ) + x˙ 21 + h1 = q1 (u1 ) x ¨2 + θ2 x˙ 2 + x22 + h2 = q2 (u2 ) (31) where q1 (u1 ) and q2 (u2 ) represent the quantizer in (3), parameters θ1 and θ2 are unknown constants, the interconnections are choosen as h1 = sin(x1 ) + sin(x˙ 1 ) + sin(x2 ) + sin(x˙ 2 ) and h2 = 0.5 cos(x1 ) + 0.5 sin(x2 ) which are unknown functions. The objective is to design a decentralized control for ui to make the outputs yi = xi to track the reference signals yr1 (t) = 0.5 sin(t) and yr2 (t) = 1−0.2 cos(t). In the simulation, the actual values of parameters are θ1 = θ2 = 1 for simulations, the quantization parameters are chosen as δ1 = δ2 = 0.2 and β1 = β2 = 0.3. The initials of states and parameter are set as x1 (0) = 0.3, x2 (0) =

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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Jing Zhou / IFAC PapersOnLine 50-1 (2017) 10419–10424

0.5, x˙ i (0) = 0 and θˆi (0) = 0.8. The control parameters are c1,j = 5 and c2,j = 10. Figures 2, 3 and 4 show the trajectory outputs yi , the control inputs ui and the quantization inputs qi for system (31). The results show that the outputs can track the reference signals. Figures 5, 6 and 7 show the outputs, the control inputs and the quantization inputs by changing the control parameters to c1,j = 1 and c2,j = 2. Figures 2 and 5 show that the tracking error can be made smaller by increasing ci,j . The comparisons with Figures 3 and 6 show that one can improve the tracking performance by paying more control efforts. In conclusion, the simulation results verify the theoretical results and show the effectiveness of the proposed control scheme.

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1 0.5

y1

0 y

-0.5 -1

y

0

1

2

3

4

5

6

7

8

1 r1

9

10

t(sec)

y2

1.5

1 y

2

y

r2

0.5

0

1

2

3

4

5

6

7

8

9

10

t(sec)

Fig. 2. Outputs y1 and y2 with c1,j = 5 and c2,j = 10

5. CONCLUSION

2

Input u 1

In this paper, a totally decentralized adaptive control scheme is developed for a class of uncertain nonlinear interconnected systems with input quantization and interconnections. A hysteresis uniform quantizer is incorporated to reduce chattering. In the control design, a smooth function is introduced to compensate for the effects of interactions. It is shown that the designed decentralized adaptive controllers can ensure the stability of the overall interconnected systems. The tracking error performance is established and can be adjusted by choosing suitable design parameters. The effectiveness of the proposed decentralized control scheme is illustrated with a numerical example.

0 -2 -4 -6

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

t(sec)

Input u 2

50

0

-50

0

1

2

3

4

5

t(sec)

Fig. 3. Inputs u1 & u2 with c1,j = 5 and c2,j = 10 2

Input q 1 (u 1 )

Ceragiolia, F., De Persisb, C., and Frascaa, P. (2011). Discontinuities and hysteresis in quantized average consensus. Automatica, 47(9), 1916–1928. De Persis, C. and Isidori, A. (2004). Stabilizability by state feedback implies stabilizability by encoded state feedback. Systems and Control Letters, 53, 249–258. De Persis, C. and Mazenc, F. (2010). Stability of quantized time-delay nonlinear systems: A Lyapunov-Krasowskii functional approach. Mathematics of Control, Signals, and Systems, 21, 4337–370. Elia, N. and Mitter, S. (2001). Stabilization of linear systems with limited information. IEEE Transactions on Automatic Control, 46, 1384–1400. Hayakawaa, T., Ishii, H., and Tsumurac, K. (2009a). Adaptive quantized control for linear uncertain discretetime systems. Automatica, 45, 692–700. Hayakawaa, T., Ishii, H., and Tsumurac, K. (2009b). Adaptive quantized control for nonlinear uncertain systems. Systems and Control Letters, 58, 625–632. Ishii, H. and Francis, B. (2002). Limited Data Rate in Control Systems with Network. Springer, Berlin, Germany. Jain, S. and Khorrami, F. (1997). Decentralized adaptive output feedback design for large-scale nonlinear systems. IEEE Transactions on Automatic Control, 42, 729–735. Jiang, Z.P. (2000). Decentralized and adaptive nonlinear tracking of large-scale systems via output feedback. IEEE Transactions on Automatic Control, 45, 2122– 2128.

0 -2 -4 -6

0

1

2

3

4

5

6

7

8

9

10

t(sec) 50 -2

Input q 2 (u 2 )

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0

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t(sec) -50

0

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Proceedings of the 20th IFAC World Congress 10424 Jing Zhou / IFAC PapersOnLine 50-1 (2017) 10419–10424 Toulouse, France, July 9-14, 2017

1

y1

0.5 0

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1

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-0.5

0

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0.5

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Fig. 5. Outputs y1 and y2 with c1,j = 1 and c2,j = 2

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1

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4

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7

8

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10

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Input u 2

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t(sec)

Fig. 6. Inputs u1 & u2 with c1,j = 1 and c2,j = 2

Input q 1 (u 1 )

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