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Backstepping Design for Output Feedback Stabilization for a Class of Uncertain Systems using Dynamic Extension⁎
Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference on Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd IFAC IFAC Conference Conference on Proceedings, 2nd Available online at www.sciencedirect.com Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Proceedings, 2nd IFAC Conference on Modelling, Identification and Control of Modelling, Identification and Control of Nonlinear Nonlinear Systems Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018
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IFAC PapersOnLine 51-13 (2018) 260–265
Backstepping Backstepping Design Design for for Output Output Feedback Feedback Backstepping Design for Output Feedback Stabilization for a Class of Uncertain Backstepping Design for Output Feedback Stabilization for a Class of Uncertain Stabilization for a Class of Uncertain Systems Extension Stabilization forDynamic a Class of Uncertain Systems using using Dynamic Extension Systems using Dynamic Extension Systems using Dynamic Extension Frederic Mazenc ∗∗ Laurent Burlion ∗∗ Michael Malisoff ∗∗∗ ∗∗ ∗∗∗
Frederic Mazenc ∗∗∗ Laurent Burlion ∗∗ Michael Malisoff ∗∗∗ ∗∗ ∗∗∗ Frederic Mazenc Mazenc Laurent Burlion Burlion ∗∗ Michael Malisoff Malisoff ∗∗∗ Frederic Laurent Michael ∗ ∗∗ ∗∗∗ ∗ Frederic Mazenc Laurent Burlion Michael Malisoff Inria, L2S-CNRS-CentraleSup´ e lec, 3 rue Joliot Curie, 91192, ∗ ∗ Inria, L2S-CNRS-CentraleSup´ e lec, 3 rue Joliot Curie, 91192, ∗ ∗ Inria, L2S-CNRS-CentraleSup´ lec, 3 3 rue rue Joliot Curie, Curie, 91192, 91192, Gif-sur-Yvette, (e-mail: eelec, Joliot Gif-sur-Yvette, France, France, (e-mail: ∗ Inria, L2S-CNRS-CentraleSup´ Inria, L2S-CNRS-CentraleSup´ elec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France, (e-mail: [email protected]) Gif-sur-Yvette, France, (e-mail: [email protected]) ∗∗ Gif-sur-Yvette, France, (e-mail: [email protected]) - The French ∗∗ [email protected]) ∗∗ Onera French Aerospace Aerospace Lab, Lab, F-31055 F-31055 Toulouse, Toulouse, France, France, ∗∗ Onera - The [email protected]) ∗∗ Onera -- The The French Aerospace Aerospace Lab, F-31055 F-31055 Toulouse, Toulouse, France, France, (e-mail: [email protected]) Onera French Lab, (e-mail: [email protected]) ∗∗ ∗∗∗ Onera - The (e-mail: French Aerospace Lab, F-31055 Toulouse, France, [email protected]) of Mathematics, Louisiana State University, ∗∗∗ (e-mail: [email protected]) ∗∗∗ Department Department (e-mail: of Mathematics, Louisiana State University, Baton Baton ∗∗∗ [email protected]) ∗∗∗ Department of Mathematics, Louisiana State University, Rouge, LA 70803-4918 (e-mail: [email protected]) of Mathematics, Louisiana State University, Baton Baton Rouge, LA 70803-4918 (e-mail: [email protected]) ∗∗∗ Department Department Louisiana State University, Baton Rouge, LA 70803-4918 [email protected]) Rouge, of LAMathematics, 70803-4918 (e-mail: (e-mail: [email protected]) Rouge, LA 70803-4918 (e-mail: [email protected]) Abstract: We revisit the backstepping approach. We show how bounded globally asymptotAbstract: We revisit the backstepping approach. We show how bounded globally asymptotAbstract: We the backstepping approach. We show how globally asymptotically output feedbacks can constructed aa family of systems. The Abstract: We revisit revisit backstepping approach. We for show how bounded bounded globally asymptotically stabilizing stabilizing outputthe feedbacks can be be constructed for family of nonlinear nonlinear systems. The Abstract: We revisit the backstepping approach. We show how bounded globally asymptotically stabilizing output feedbacks can can be constructed for aaand family of nonlinear nonlinear systems. systems. The approach relies onoutput the introduction of a be dynamic extension a converging-input-convergingically stabilizing feedbacks constructed for family of The approach relies on the introduction of a dynamic extension and a converging-input-convergingically stabilizing feedbackspresents can constructed for aand family of nonlinear The approach relies the of dynamic aa converging-input-convergingstate assumption. The technique severalextension advantages. It provides controlsystems. laws whose approach relies on onoutput the introduction introduction of aa be dynamic extension and converging-input-convergingstate assumption. The technique presents several advantages. It provides control laws whose approach relies on the introduction of a dynamic extension and a converging-input-convergingstate assumption. The technique presents several advantages. It provides control laws whose expressions are simple. It makes it possible to stabilize systems in the presence of certain types state assumption. The It technique presents several advantages. It the provides control laws whose systems in presence of certain types expressions are simple. makes it possible to stabilize state assumption. The technique several advantages. It the provides control laws whose expressions are simple. It makes itpresents possible tothe stabilize systems in the presence of certain certain types of uncertain terms which prevent the use of classical backstepping technique. It applies in expressions are simple. It makes it possible to stabilize systems in presence of types of uncertain terms which prevent the use of the classical backstepping technique. It applies in expressions are simple. It makes it possible to stabilize systems in the presence of certain types of uncertain terms which prevent thestate use of of the classical classical backstepping technique. technique. It It applies applies in notable casesterms wherewhich only part of the variable is measured. of uncertain prevent the use the backstepping in notable cases where only part of the state variable is measured. of uncertain prevent thestate use of the classical backstepping technique. It applies in notable cases where only of variable is notable casesterms wherewhich only part part of the the state variable is measured. measured. © 2018, IFAC (International Federation ofstate Automatic Control) Hosting by Elsevier Ltd. All rights reserved. notable cases where only part of the variable is measured. Keywords: Keywords: backstepping, backstepping, nonlinear, nonlinear, output output feedback, feedback, stabilization stabilization Keywords: backstepping, backstepping, nonlinear, nonlinear, output output feedback, feedback, stabilization stabilization Keywords: Keywords: backstepping, nonlinear, output feedback, stabilization 1. INTRODUCTION that 1. INTRODUCTION that are are used used in in the the controls controls even even when when there there are are no no input input 1. INTRODUCTION INTRODUCTION that are areinused used in the controls controls evenThis whentechnique there are are has no input input delays the original systems. been 1. that in the even when there no delays inused the in original systems. This technique has been The backstepping approach is useful for constructing glob1. INTRODUCTION that are the controls even when there are no delays in in in theMazenc originaland systems. This technique has input been initiated Malisoff (2016) and developed the original systems. This technique has been The backstepping approach is useful for constructing glob- delays initiated in Mazenc and Malisoff (2016) and developed The backstepping approach is useful useful for laws constructing globally backstepping asymptoticallyapproach stabilizing control for nonlinear delays in in theMazenc original systems. This technique has etbeen The is for constructing glob- initiated and Malisoff (2016) and developed and applied in Mazenc et al. (2016) and Mazenc initiated in Mazenc and Malisoff (2016) and developed ally asymptotically stabilizing control laws for nonlinear applied in Mazenc al. (2016) and and Mazenc et al. al. The backstepping approach useful for laws constructing glob- and ally asymptotically stabilizing control laws for nonlinear nonlinear systems in feedback form, isi.e., having a lower trianguinitiated in Mazenc andet Malisoff (2016) developed ally asymptotically stabilizing control for and applied in Mazenc et al. (2016) and Mazenc et al. (2017b), where the case of systems with delays in and applied in Mazenc et al. (2016) and Mazenc et the al. systems in feedback form, i.e., having a lower triangu(2017b), where the case of systems with delays in the ally asymptotically stabilizing control laws for nonlinear systems in feedback form, i.e., having a lower triangular structure; see Dixon et al. (2000). Since the pioneerand applied in Mazenc et al. (2016) and Mazenc et al. systems in feedback form, i.e.,(2000). havingSince a lower triangu(2017b), where the case case and of systems systems withinitial delaysfictitious in the the input is also considered where the (2017b), where the of with delays in lar structure; see Dixon et al. the pioneerinput is also considered and where the initial fictitious systems in feedback form, i.e., having a lower triangu1 lar structure; see in Dixon et and al. (2000). (2000). Since the pioneering contributions Coron Praly (1991) and Tsinias (2017b), where the case of systems with delays in the lar structure; see Dixon et al. Since the pioneerinput is is law alsoisconsidered considered and to where theclass initial fictitious control not be C A also and where initial ing structure; contributions in Coron and Praly (1991) and Tsinias input isconsidered not required required be of ofthe C 1111 ..fictitious A first first lar Dixon et and al. in (2000). Since the ing contributions in Coron and Praly and Tsinias (1997), it has see beenin developed many(1991) contributions and control input is law also andtoto where theclass initial ing contributions Coron Praly (1991) andpioneerTsinias control law is not required to be of class C ..fictitious A adaptation of this technique uncertain outputs has also control law is not required to be of class C A first first (1997), it has been developed in many contributions and adaptation of this technique to uncertain outputs also 1 has ing contributions in Coron and Praly (1991) and Tsinias (1997), it has been developed in many contributions and is successfully used in many applications, as illustrated control law is not required to be of class C . A first (1997), it has been developed in many contributions and adaptation of this technique to uncertain outputs has also been studied in Mazenc et al. (2017a), where, due to the adaptation of in this technique to (2017a), uncertainwhere, outputs has also is successfully used in many applications, as illustrated been studied Mazenc et al. due to the (1997), it has been developed in many contributions and is successfully used in many applications, as illustrated for instance by Jiang and Nijmeijer (1999), Pettersen and adaptation of this technique to uncertain outputs has is successfully used inand many applications, as illustrated beenofstudied studied in Mazenc Mazenc et etonly al. (2017a), (2017a), where, due to to also the use visual information, imprecise measurement of been in al. where, due the for instance by Jiang Nijmeijer (1999), Pettersen and use of visual information, only imprecise measurement of is successfully used in many applications, as illustrated for instance by Jiang Jiang and Nijmeijer Nijmeijer (1999), Pettersen and been Nijmeijer (2002), Smaoui et al. (2006), dePettersen Queiroz and in Mazenc etonly al. (2017a), where, due to the for instance by and (1999), use first ofstudied visual information, only imprecise measurement of the backstepping variable was available. use of visual information, imprecise measurement of Nijmeijer (2002), Smaoui et al. (2006), de Queiroz and the first backstepping variable was available. for instance by Jiang and Nijmeijer (1999), Pettersen and Nijmeijer (2002),and Smaoui et al. al. (2011). (2006), Presentations de Queiroz Queiroz and Dawson (1996) Lee etet of the use of visual information, only was imprecise measurement of Nijmeijer (2002), Smaoui al. (2006), de first backstepping variable available. the first backstepping variable was available. Dawson (1996) and Lee et al. (2011). Presentations of Nijmeijer (2002),and Smaoui al.be (2006), deKhalil Queiroz and paper, we aa new Dawson (1996) and Lee et etetcan al. (2011). Presentations of In the backstepping technique found Presentations in (2002), the firstpresent backstepping was available. Dawson (1996) Lee al. (2011). of In the the present paper,variable we propose propose new backstepping backstepping the backstepping technique can be found in Khalil (2002), In the present paper, we propose a new backstepping backstepping Dawson (1996) and Lee et al. (2011). Presentations of design for globally uniformly asymptotically stabilizing the backstepping technique can be found in Khalil (2002), In the present paper, we propose a new Mazenc and Bowong (2004), and many other research the backstepping technique can and be found inother Khalilresearch (2002), design for globally uniformly asymptotically stabilizing Mazenc and Bowong (2004), many In the for present paper, welinearizable propose a systems; new backstepping design globally uniformly asymptotically stabilizing the backstepping technique can and be found Khalilresearch (2002), control laws for partially see Mazenc and and Bowong (2004), and manyinother other research design for globally uniformly asymptotically stabilizing monographs papers. Mazenc and Bowong (2004), many control laws for partially linearizable systems; see Isidori Isidori monographs and papers. design for globally uniformly asymptotically stabilizing control laws for partially linearizable systems; see Mazenc and Bowong (2004), and many other research (1995) for an introduction to partially linearizable sysmonographs and and papers. papers. control for lawsanforintroduction partially linearizable systems; see Isidori Isidori monographs (1995) to partially linearizable sysOne of the drawbacks of this technique is the complexity control laws for partially linearizable systems; see Isidori (1995) Its for fundamental an introduction introduction toaspect partially linearizable sysmonographs and papers. tems. new to withlinearizable respect to the for an partially sysOne of the drawbacks of this technique is the complexity (1995) tems. Its fundamental new aspect with respect to the One of formulas the drawbacks drawbacks of the of the it sometimes provides, is notably when it (1995) for fundamental an that introduction partially linearizable One of the of this this technique technique isnotably the complexity complexity tems. Its Its fundamental new toaspect aspect with respect to systhe contributions use artificial delays is that, instead of tems. new with respect to the of the formulas it sometimes provides, when it contributions that use artificial delays is that, instead of One of the drawbacks of this technique is the complexity of the formulas it sometimes provides, notably when it is applied repeatedly and when size constraints on the tems. Its fundamental new aspect with respect to the of the formulas it sometimes provides, notably when it introducing contributions that use artificial delays is that, instead of delays, a finite dimensional dynamic extension contributions that use artificial delays is that, instead of is applied repeatedly and when size constraints on the introducing delays, a finite dimensional dynamic extension of the formulas it sometimes provides, notably when it is applied repeatedly and when size constraints on the control laws have to be respected. This is a limitation contributions that use artificial delays is that, instead of is applied repeatedly and when size constraints on the introducing delays, a finite dimensional dynamic extension is designed, making it possible to obtain feedbacks without introducing delays, a finite dimensional dynamic extension control laws have to be respected. This is a limitation is designed, making it possible to obtain feedbacks without is applied repeatedly and whenbackstepping sizeThis constraints on the control laws have be respected. is limitation of the applicability classical of the introducing delays, dimensional dynamic control laws have to toof be respected. This is aa and limitation is designed, designed, making itfinite possible to obtain obtain feedbacks without delays, which offer athe following advantages: (i)extension they are is making it possible to feedbacks without of the applicability of classical backstepping and of the control laws have toof respected. This is Iggidr a and limitation which offer the following advantages: (i) they are of the applicability applicability ofbe classical backstepping and of the the delays, bounded backstepping results of Mazenc and (2004) is designed, making it possible to obtain feedbacks without of the classical backstepping of delays, which offer the following advantages: (i) are bounded in the cases where bounded feedbacks can be delays, which offer the following advantages: (i) they they are bounded backstepping results of Mazenc and Iggidr (2004) bounded in the cases where bounded feedbacks can be of the applicability of classical backstepping and of the bounded backstepping results of Mazenc and Iggidr (2004) and Mazenc and Bowong (2004). Another limitation of expected delays, which offer the following advantages: (i) they are bounded backstepping results of Mazenc and Iggidr (2004) bounded in the cases where bounded feedbacks can be (which contrasts with Mazenc et al. (2016) where bounded in the cases where bounded feedbacks can be and Mazenc and Bowong (2004). Another limitation of expected (which contrasts with Mazenc et al. (2016) where bounded backstepping results of Mazenc and Iggidr (2004) and Mazenc and Bowong (2004). Another limitation of the approach is due to the fact that in general, it does bounded in the cases where bounded feedbacks can be and Mazenc and Bowong (2004). Another limitation of expected (which contrasts with Mazenc et al. (2016) where the controls obtained for the original systems are not expected (which contrasts with Mazenc etsystems al. (2016) where the approach is due to the fact that in general, it does the controls obtained for the original are not and Mazenc and Bowong (2004). Another limitation of the approach is only due to to the fact fact that in general, general, it does does not apply when a part of the state is measured. In expected (which contrasts etsystems al. (2016) where the approach is due the that in it the controls controls obtained forwith the original systems are not bounded controls), (ii) they areMazenc given by simple formulas, the obtained for the original are not not apply when only a part of the state is measured. In the approach is only due to the fact that general, it does controls), (ii) they are given by simple formulas, not applyitwhen when only part of the the state is measured. measured. In bounded addition, relies on the existence ofstate a in fictitious feedback the controls obtained for the original are not not apply aa part of is In bounded controls), (ii) are given by simple formulas, (iii) they apply in cases where subsystem from which bounded controls), (ii) they they arethe given by systems simple formulas, addition, it relies on the existence of a fictitious feedback k (iii) they apply in cases where the subsystem from which not apply when only a part of the state is measured. In addition, it relies on the existence of a fictitious feedback when the backstepwhich typically has to be of class C bounded controls), (ii) they are given by simple formulas, k addition, it relies on the existence of k a fictitious feedback (iii) they apply in cases where the subsystem from which the backstepping begins contains unknown vector fields, (iii) they apply in begins cases where the unknown subsystemvector from which which typically has to be of class C the backstepthe backstepping contains fields, addition, it relieshas on existence fictitious feedback whichistypically typically has tothe beThe of class class Cofkk awhen when the ping applied times. presence of in theythey apply in begins cases where the unknown subsystem from which the backstepbackstepwhich to be of C the backstepping backstepping begins contains unknown vector fields, and (iv) apply in cases where only aa part of state the contains vector ping istypically applied k khas times. The presence of uncertainties uncertainties in (iii) k when and (iv) they apply in cases where only part of the thefields, state when the backstepwhich to be of class C pingdynamics is applied appliedand k times. times. The presence of uncertainties uncertainties in the the in the output may also be an obstacle. backstepping begins contains unknown vector fields, ping is k The presence of in and (iv) (iv) is they apply in inand cases where onlyparts a part part of the state variable measured where some of the output and they apply cases where only a of the state the dynamics and in the output may also be an obstacle. variable is measured and where some parts of the output ping is applied k times. The presence of uncertainties in the dynamics dynamics and and in in the the output output may may also also be be an an obstacle. obstacle. and (iv)known they apply in cases where only a part of the state the variable is measured and where some parts of the output are not with certainty. The family of partially linvariable is measured and where some parts of the output To overcome these drawbacks, a new technique has been not known with certainty. The family ofofpartially linthe overcome dynamics these and indrawbacks, the outputamay be an obstacle. To new also technique has been are variable is measured and where some parts the output are not known with certainty. The family of partially linearizable systems we consider encompasses many systems not known with The family ofmany partially linTo overcome overcome these based drawbacks, a new newlaws technique hasdelays been are proposed recently, on control in which To these drawbacks, a technique has been earizable systems wecertainty. consider encompasses systems proposed recently, based on control laws in which delays are not known with certainty. The family of partially linearizable systems we consider encompasses many systems that are relevant from an applied point of view, as shown To overcome these drawbacks, a new technique has been earizable systems we consider encompasses many systems proposed recently, based on are control laws in which delays are artificial, meaning there delays in the state values proposed recently, based on control laws in which delays that are relevant from an applied point of view, as shown are artificial, meaning there are delays in the state values earizable systems we consider encompasses many systems thatinstance are relevant relevant from an an applied point of view, view, as to shown for by Spong (1994). Also, with respect the proposed recently, based on are control laws are from applied point of as shown are meaning there delays in the state values are artificial, artificial, meaning there are delays in in thewhich state delays values that for instance by Spong (1994). Also, with respect the are relevant from an applied point of view, as to shown for instance by Spong (1994). Also, with respect to the backstepping with artificial delay methods from the works Supported are artificial, meaning there are Foundation delays in the state values that for instance by Spong (1994). Also, with respect to the by US National Science Grant 1711299. backstepping with artificial delay methods from the works Supported by US National Science Foundation Grant 1711299. for instance by Spong (1994). Also, with respect to the backstepping with artificial delay methods from the works backstepping with artificial delay methods from the works Supported Supported by by US US National National Science Science Foundation Foundation Grant Grant 1711299. 1711299. backstepping with artificial delay methods from the works Supported by US National Science Foundation Grant 1711299.
2405-8963 © 2018, IFACConference (International Proceedings, 2nd IFAC onFederation of Automatic Control) 260 Hosting by Elsevier Ltd. All rights reserved. Proceedings, 2nd IFAC Conference on 260 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference on 260 Proceedings, 2nd IFAC Conference 260 Modelling, Identification and Controlon of Nonlinear 10.1016/j.ifacol.2018.07.288 Systems Proceedings, 2nd IFAC Conference on 260 Modelling, Identification and Control of Nonlinear Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear
2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018
Frederic Mazenc et al. / IFAC PapersOnLine 51-13 (2018) 260–265
Mazenc and Malisoff (2016), Mazenc et al. (2016), and Mazenc et al. (2018, to appear), the main drawback of the main result of the present paper is that it relies on a fictitious feedback of class C 1 (although these prior works did not cover systems with outputs or with uncertain vector fields). Comparisons between the performances of the control laws of Mazenc and Malisoff (2016) and Mazenc et al. (2016) and those of the present paper still have to be made and will be the subject of future studies. Our converging-input-converging-state condition provides a novel and promising approach and was not present in the well known works such as Liu et al. (1996) and Sussmann et al. (1994) for linear systems. This paper is organized as follows. The main result is shown in Section 2. Section 3 provides a technical lemma that can facilitate checking our assumptions from Section 2. An example based on a direct drive manipulator is given in Section 4. Concluding remarks are drawn in Section 5. Our notation will be simplified whenever no confusion can arise from the context. The dimensions of our Euclidean spaces are arbitrary unless other indicated. The Euclidean norm in Ra , and the induced norm of matrices, are denoted by | · |, and | · |∞ is the usual sup norm. Given a constant J > 0, satJ denotes the symmetric saturation function defined by satJ (x) = max{−J, min{J, x}} for all x ∈ R, where J is called the saturation level. We use the usual class of comparison functions K∞ from Khalil (2002). 2. MAIN STABILIZATION RESULT 2.1 The studied system We consider the nonlinear time-varying system x˙ = f (t, x, g1 ) g ˙ 1 = a1,1 g1 + a1,2 g2 g˙ 2 = a2,1 g1 + a2,2 g2 + a2,3 g3 (1) .. . g˙ n = an,1 g1 + an,2 g2 + · · · + an,n gn + u + Ω(Y(t, x), G) where x is valued in Rp , G = (g1 , ..., gn ) is valued in Rn , f is a nonlinear function that is locally Lipschitz with respect to (x, g1 ) and piecewise continuous with respect to t, u is the scalar-valued input, Y : R × Rp → Rl is a continuous function, each ai,j ∈ R is a constant, ai,i+1 = 0 holds for i = 1, 2...n − 1 (with no sign constraint on the aij ’s), and the function Ω is locally Lipschitz. This family of systems has been studied in many papers because systems of this type may result from partial linearization and are frequently encountered in practice. In this section, we assume that the R valued output is (2) Y (t) = (Y(t, x(t)), g1 (t), . . . , gn (t)) , which is realistic in practice. Our first assumption is: Assumption 1. There exist a locally Lipschitz scalar valued function ψ that is bounded by a known constant ψ ≥ 0, and a constant k > 0, such that for any continuous function d : [0, ∞) → R that converges exponentially to the origin, all solutions (ξ, λ1 , ..., λn ) : [0, ∞) → Rp+n of ξ˙ = f (t, ξ, λ1 + d(t)) λ˙ 1 = k[−λ1 + λ2 ] (3) .. . ˙ λn = k[−λn + ψ(t, Y(t, ξ))] l+n
261
261
converge to the origin as t → ∞. Moreover the function Ω in (1) is bounded by a known constant Ω ≥ 0. In terms of the matrices a1,1 a1,2 0 a2,1 a2,2 a2,3 . .. . A = .. . . . an,1 an,2 . . .
... ... .. .
0 0 .. .
..
. an−1,n . . . an,n
∈ Rn×n
(4)
for n > 1 and A = a1,1 for n = 1 and B = (0, 0, . . . , 1) ∈ Rn , our second assumption is: Assumption 2. There is a locally Lipschitz function : Rn → R that is bounded by a known constant ≥ 0 such that the origin of the system ˙ Γ(t) = AΓ(t) + B (Γ(t)) (5) is a globally asymptotically and locally exponentially stable equilibrium. 2.2 Statement of and discussion on theorem We are ready to state and prove the following result, where sat is as defined in Section 1: Theorem 1. Let the system (1) satisfy Assumptions 1-2 and let > 0 be any constant. Then there exist a constant matrix L and constants cj for j = 1 to n + 1 such that all solutions (x, G) : [0, ∞) → Rp+n of (1), in closed loop with the dynamic output feedback n cj zj + cn+1 ψ(t, Y(t, x)) u(G, Z, x) = −satZ j=1
+ (G + LZ) − Ω (Y(t, x), G)
z˙1 = k[−z1 + z2 ] .. . z˙n−1 = k[−zn−1 + zn ] z˙n = k[−zn + ψ(t, Y(t, x))] with Z = (z1 , ..., zn ) and the saturation level n+1 Z = ψ(1 + ) |cj |,
(6)
j=1
asymptotically converge to 0 as t → ∞.
Before turning to the proof of Theorem 1, we make several remarks on its motivation and value. 1) The feedback (6) is bounded by the constant Z + +Ω. 2) The formula (6) for the control law does not incorporate the first derivative of ψ(t, Y(t, x(t))), which plays the role of the fictitious control of the classical backstepping approach. Hence, it applies even when ψ(t, Y(t, x)) is not of class C 1 . However, in practice Theorem 1 can only be applied when ψ is of class C 1 because checking that Assumption 1 is satisfied frequently necessitates that ψ be of class C 1 , as we show in the next section. By contrast, in Mazenc et al. (2016) and Mazenc et al. (2018, to appear), fictitious feedbacks of class C 0 can be used. 3) One can use changes of variables and an appropriate choice of feedback to transform the system (1) into a system of the form (3) with d = 0 whose exponential
2018 IFAC MICNON 262 Guadalajara, Mexico, June 20-22, 2018
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stability property is ensured by Assumption 1. However, in a sense, this result is not satisfactory because the feedback obtained that way may possess inappropriate properties. For instance, in general they would be unbounded. This motivates our alternative approach, based on dynamic extensions. 5) None of the assumptions of this paper imply that f has to be known with accuracy. Also, we only require measurements of ψ(t, Y(t, x)) and Ω(Y(t, x), G), instead of Y itself. 2.3 Proof of Theorem 1 A key aspect of the proof consists of making g1 (t) − z1 (t) converge to zero instead of making g1 (t) − ψ(t, Y(t, x(t))) converge to zero, as traditionally done in classical backstepping. To achieve our goal, several changes of coordinates are needed, as follows. We first assume that n > 1, and we explain how these changes of variables can be constructed by induction. Set l1,1 = −1 and r1 = g1 − z1 .
Induction assumption: For all j ∈ {1, ..., i} and 1 < i < n, there are constants lj,m for m = 1 to j such that the variables j lj,m zm (7) rj = gj + m=1
for j = 1 to i satisfy r˙1 = a1,1 r1 + a1,2 r2 r˙2 = a2,1 r1 + a2,2 r2 + a2,3 r3 .. . r˙i−1 = ai−1,1 r1 + ai−1,2 r2 + · · · + ai−1,i ri .
(8)
First step: i = 2. Since r1 = g1 − z1 , we have r˙1 = a1,1 g1 + a1,2 g2 − k[−z1 + z2 ] a1,1 + k k (9) = a1,1 r1 + a1,2 g2 + z1 − z2 . a1,2 a1,2 Then the variable a1,1 + k k r2 = g2 + z1 − z2 (10) a1,2 a1,2 satisfies (11) r˙1 = a1,1 r1 + a1,2 r2 Thus the induction assumption is satisfied at the first step. Step i: Assume that the induction assumption is satisfied at a step i with 1 < i < n. Then since (7) holds for j = 1, 2, . . . , i we have r˙i = ai,1 g1 + ai,2 g2 + ... + ai,i+1 gi+1 i + li,m k[−zm + zm+1 ] m=1
= ai,1 r1 + ai,2 r2 + ... + ai,i ri + ai,i+1 gi+1 +ai,1 z1 − ai,2 (l2,1 z1 + l2,2 z2 ) − ... i i − ai,i li,m zm + li,m k[−zm + zm+1 ].
(12)
we obtain r˙i = ai,1 r1 + ai,2 r2 + ... + ai,i+1 ri+1 ,
(14)
so the induction assumption is satisfied at the step i + 1. Taking the time derivative of (13) with the choice i = n−1, we can then find a linear change of coordinates R = G+LZ with R = (r1 , ..., rn ) that transforms the system (1) into x˙ = f (t, x, z1 + r1 ) = a1,1 r1 + a1,2 r2 r ˙ 1 r˙2 = a2,1 r1 + a2,2 r2 + a2,3 r3 .. . (15) r˙n = an,1 r1 + an,2 r2 + · · · + an,n rn + u n ln,j zj + ln,n+1 ψ(t, Y(t, x)) + j=1 + Ω(Y(t, x), G).
If instead n = 1, then we again obtain (15) with only the dynamics for x and rn present. Let cj = ln,j for j = 1 to n + 1. Then the closed-loop system is x˙ = f (t, x, z1 + r1 ) R˙ = AR + B [(R) n cj zj + cn+1 ψ(t, Y(t, x)) −satZ j=1 n (16) + cj zj + cn+1 ψ(t, Y(t, x)) j=1 = k[−z z ˙ 1 1 + z2 ] . .. z˙n = k[−zn + ψ(t, Y(t, x))]. Consider any solution (x, R, z) : [0, ∞) → Rp+2n of (16).
Since Assumption 1 ensures that ψ is bounded by ψ, it follows from (16) that there is a finite value ta ≥ 0 such that for all t ≥ ta , the inequality n cj zj (t) + cn+1 ψ(t, Y(t, x(t))) ≤ Z (17) j=1
is satisfied. Hence, when t ≥ ta , the closed-loop system is x˙ = f (t, x, z1 + r1 ) ˙ R = AR + B(R) z˙1 = k[−z1 + z2 ] (18) .. . z˙n = k[−zn + ψ(t, Y(t, x))].
Assumption 2 ensures that the R-subsystem of the system (18) is globally asymptotically and locally exponentially stable. Then Assumption 1 allows us to conclude.
m=1
3. TECHNICAL LEMMA
Thus, taking ai,1 ai,2 z1 − ai,i+1 (l2,1 z1 + l2,2 z2 ) − ... ri+1 = gi+1 + ai,i+1 i i (13) ai,i 1 − ai,i+1 li,m zm + ai,i+1 li,m k[−zm + zm+1 ]
Assumption 2 is a simple classical assumption that can often be checked easily; see our illustration below. However, checking Assumption 1 can be nontrivial. In this section, we give conditions ensuring that Assumption 1 holds.
m=1
m=1
m=1
262
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3.1 Assumptions and lemma Consider the system ξ˙ = f (t, ξ, λ1 + d(t)) λ˙ = k[−λi + λi+1 ], 1 ≤ i ≤ n − 1 ˙i λn = k[−λn + ψ(t, yξ )]
263
for all t ≥ 0. Consider the positive definite function n is2i . (27) Q(S) = 12 i=1
(19)
where yξ = Cξ and C ∈ Rq×p is a constant matrix, the constant k > 0 will be further restricted below, and d : [0, ∞) → R exponentially converges to 0. Then (19) is a special case of (3). We add these assumptions on (19): Assumption 3. There is a function κ1 ∈ K∞ such that |ψ(t, yξ )| + |f (t, ξ, w)| ≤ κ1 (|(ξ, w)|) (20) for all t ≥ 0, ξ ∈ Rp and w ∈ R. Also, ψ is of class C 1 and bounded by a known constant ψ. Finally, there are functions κ2 and κ3 of class K∞ , a function V of class C 1 , and a uniformly continuous positive definite W such that κ2 (|ξ|) ≤ V (t, ξ) ≤ κ3 (|ξ|) for all (t, ξ) ∈ [0, ∞)×Rp (21) and such that for any choice of the piecewise continuous function δ, the time derivative of V along all solutions of ˙ = f (t, ξ(t), ψ(t, Cξ(t)) + δ(t)) ξ(t) (22)
Then we can use the triangle inequality to obtain n−1 n−1 1 2 i(si + s2i+1 ) isi si+1 ≤ 2 i=1 i=1 n−1 n 1 2 2 is + (i − 1)si = 2 i=1 i i=2
and then the subadditivity of the square root to conclude that along all solutions of (25) for all t ≥ 0, we have ˙ Q(t) ≤ k
3.2 Proof of Lemma 1 Set si = λi −ψ(t, yξ ) for i = 1, 2, . . . , n and S = (s1 , ..., sn ). Then (19) can be transformed into ˙ ξ = f (t, ξ, ψ(t, yξ ) + s1 + d(t)) ∂ψ (t, yξ ) s˙ 1 = k[−s1 + s2 ] − ∂t ∂ψ (t, yξ )Cf (t, ξ, ψ(t, yξ ) + s1 + d(t)) − ∂yξ (25) .. . ∂ψ (t, yξ ) s˙ n = −ksn − ∂t ∂ψ (t, yξ )Cf (t, ξ, ψ(t, yξ ) + s1 + d(t)) . − ∂yξ From Assumption 3, it follows that V˙ (t) ≤ −W (ξ(t)) + |s1 (t) + d(t)|2 for all t ≥ 0. Assumption 4 gives 2 ∂ψ ∂ψ ∂t (t, yξ )+ ∂yξ (t, yξ )Cf (t, ξ, ψ(t, yξ ) + s1 + d(t)) (26) ≤ bf W (ξ) + bs |s1 + d(t)|2 263
n−1
isi (−si + si+1 ) − kns2n
i=1 n
bf W (ξ) + bs |s1 + d(t)|2 i=1 n−1 n k 2 2 (i − 1)si − kns2n ≤ − is − 2 i=1 i i=2 n + i|si | bf W (ξ) + bs |s1 + d(t)|2 +
satisfies the inequality V˙ (t) ≤ −W (ξ(t)) + |δ(t)|2 for all t ≥ 0, and f is locally Lipschitz. Assumption 4. There are positive constants bf and bs such that for the function W from Assumption 3, the inequality 2 ∂ψ ∂ψ (t, Cξ)Cf (t, ξ, ψ(t, Cξ) + γ) ∂t (t, Cξ) + ∂y ξ (23) ≤ bf W (ξ) + bs |γ|2 holds for all t ≥ 0, ξ ∈ Rp and γ ∈ R. The inclusion of the γ term on the right side of (23) is justified because γ represents uncertainty and overshoot terms involving uncertainties commonly arise, e.g., from input-to-state stability estimates. We prove: Lemma 1. Let the system (19) satisfy Assumptions 3 and 4. Then for any k ≥ k with k = 3n n + n+1 bf + b2s (24) 2 all solutions of (19) converge to 0 as t → ∞.
(28)
i|si |
i=1 n
k 2 ≤ − s 2 i=1 i n + i|si | bf W (ξ) + bs |s1 + d(t)|2
(29)
i=1
n n k 2 ≤ − s i + bf i|si | W (ξ) 2 i=1 i=1 n n + bs i|si ||s1 | + i|si | bs |d(t)|. i=1
i=1
Consider the candidate Lyapunov function n(n + 1) U (t, ξ, S) = bf V (t, ξ) + Q(S) (30) 2 and any trajectory of (25). Along the trajectories of (25), n(n + 1) bf W (ξ) U˙ (t) ≤ − 2 n(n + 1) + bf |s1 (t) + d(t)|2 2 n n k 2 (31) − s i + bf i|si | W (ξ) 2 i=1 i=1 n n + bs i|si ||s1 | + i|si | bs |d(t)| . i=1
i=1
Using the triangle inequality to obtain 1 bf W (ξ) + i2 s2i , (32) bf i|si | W (ξ) ≤ 2 we deduce that n nbf k 2 n(n + 1) U˙ (t) ≤ − W (ξ) − bf s21 s + 2 2 i=1 i 2 n n 1 2 2 n(n + 1) bf d2 (t) (33) + i s i + bs i|si ||s1 | + 2 i=1 2 i=1 n +n(n + 1)bf s1 d(t) + i|si | bs |d(t)| . Using
√
i=1
bs i|si ||s1 | ≤ 2i s2i + 2i bs s21 , we obtain
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n k 2 nbf ˙ W (ξ) − s U (t) ≤ − 2 2 i=1 i bs bf + s21 +n(n + 1) 2 4 n n(n + 1) bf d2 (t) + n2 s2i + 2 i=1 n +n(n + 1)bf s1 d(t) + i|si | bs |d(t)| .
s1 = λ1 − q1 sin(ξ1 ) +
(34)
1 + ξ12
gives ξ˙1 = ξ2 ξ1 + s1 + d(t) ξ˙2 = −q2 ξ2 − 2 1 + ξ1
(41) 1 ξ2 . s˙ 1 = −ks1 + −q1 cos(ξ1 ) + (1+ξ12 ) 1+ξ12
i=1
From (24), we deduce that n k 2 n(n + 1) nbf W (ξ) − bf d2 (t) s + U˙ (t) ≤ − 2 6 i=1 i 2 n +n(n + 1)bf s1 d(t) + i|si | bs |d(t)| .
ξ1
(35)
i=1
Since each λi (and therefore also each si ) enters a fixed compact set after a finite time (by the boundedness of ψ and the structure of the system (19)), there are constants Λ > 0 and Tl such that for all t ≥ Tl , n k nb f W (ξ) + s2 + d (t) U˙ (t) ≤ − (36) 2 6 i=1 i
with d (t) = n(n+1)bf d2 (t)+Λ|d(t)|. From this inequality, we can conclude from Barbalat’s Lemma (applied to the function in squared brackets in (36), using the fact that d (t) converges exponentially to the origin) that all the solutions of (19) converge to the origin. 4. ILLUSTRATION
Let us choose V (ξ) = 1 + ξ12 − 1 + 12 ξ22 +
with the output Y = (x, g1 ) and the R -valued state x = (x1 , x2 ), where a1,1 > 0 and qi > 0 for i = 1 to 3 are constants. Assumption 2 is satisfied with = 0 and n = 1 (because in this case, the system (5) with = 0 is a linear system that is exponentially stable to 0), and we choose the bounded function Ω(x, g1 ) = −q3 arctan(x2 ). 2
Let us check that Assumption 1 is satisfied with n = 1, under the added assumptions that q 1 q2 q1 q2 (38) < 1 and < . 1 + q1 1 + q1 2 Consider ξ1 ψ(ξ) = q1 sin(ξ1 ) − (39) 1 + ξ12 and any continuous function d : [0, ∞) → R that exponentially converges to 0 and ξ˙1 = ξ2 ξ˙2 = −q1 sin(ξ1 ) − q2 ξ2 + λ1 + d(t) (40) ξ1 ˙ = k −λ + q sin(ξ ) − λ 1 1 1 1 1 + ξ12 with k > 0 being a constant to be specified. The choice
264
√ ξ1
1+ξ12
ξ2 .
(42)
Then the first inequality in (38) and the triangle inequality combine to give q1 ξ ξ 1 2 −q2 4(1 + q1 ) 1 + ξ12 1 ξ12 q1 1 2 2 (43) q ≤ + ξ 4(1 + q1 ) 2 1 + ξ12 2 2 2 q1 q2 ξ12 ≤ + ξ22 , 8(1 + q1 ) 1 + ξ12 8 and so also 2
ξ2 q1 √ 2 V˙ (t) = √ξ1 ξ2 2 + 4(1+q 1 ) (1+ξ 2 ) 1+ξ 1 1 1+ξ1 ξ q1 1 + ξ2 + 4(1+q1 ) √ 2 1+ξ1 ξ 1 × −q2 ξ2 − √ 2 + s1 + d(t) 1+ξ1 q1 2 = −q2 ξ2 + 4(1+q ) √ξ1 ξ2 2 1+ξ1
1
Consider a single-link direct-drive manipulator actuated by a permanent magnet DC brush motor, which produces the following model from Dawson et al. (1994) (after a change of coordinates which removes a constant): x˙ 1 = x2 x˙ 2 = g1 − q1 sin(x1 ) − q2 x2 (37) g˙ 1 = u − q3 arctan(x2 ) − a1,1 g1
q1 4(1+q1 )
ξ12
q1 − 4(1+q 2 1 ) 1+ξ1
+ ξ2 +
+
q1 4(1+q1 )
≤ − 3q42 ξ22 −
ξ22 q1 √ 4(1+q1 ) (1+ξ 2 ) 1+ξ 2 1 1
√ ξ1
ξ12
q1 8(1+q1 ) 1+ξ12
q1 √ ξ1 + 4(1+q 1)
1+ξ12
(s1 + d(t))
1+ξ12
(44)
+ ξ2 (s1 + d(t))
(s1 + d(t)) ξ2
q1 1 ≤ − q42 ξ22 − 16(1+q 2 1 ) 1+ξ1 q 1 (s1 + d(t))2 + 2q12 + 4(1+q 1) ξ2
q1 1 ≤ − q42 ξ22 − 16(1+q 2 1 ) 1+ξ1 q1 2 1 + q2 + 2(1+q1 ) s1 q1 d2 (t), + q12 + 2(1+q ) 1
along all solutions of (41) for all t ≥ 0, where the first inequality in (44) used (38) and (43), the second equality in (44) used the relations ξ2 (s1 + d(t)) ≤ 12 q2 ξ22 + √ ξ1 2 (s1 1+ξ1
+ d(t)) ≤
1 2 2q2 (s1 + d(t)) and 2 ξ 1 2 1 4 1+ξ12 + (s1 + d(t)) ,
and the last inequality in (44) use the inequality (a + 2 2 2 b) √ ≤ 2a + 2b for suitable a and b. Also, the relations 1 + p − 1 ≥ p/(4(1 + p)) for all p ≥ 0, and (43) with q2 replaced by 1, can be used to show that (42) is positive definite, so (42) is proper and positive definite. Next let U (ξ, s1 ) = V (ξ) + 12 s21 . Then (44) gives
2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018 Frederic Mazenc et al. / IFAC PapersOnLine 51-13 (2018) 260–265
2
ξ1 q1 U˙ (t) ≤ − q42 ξ22 − 16(1+q 2 1 ) 1+ξ1 q1 s21 + q12 + 2(1+q 1) q1 d2 (t) − ks21 + q12 + 2(1+q 1) 1√ + −q1 cos(ξ1 ) + s 1 ξ2 2 2 (1+ξ1 )
(45)
1+ξ1
ξ12 q1 16(1+q1 ) 1+ξ12
≤ − q42 ξ22 − + (q1 + 1)|s1 ξ2 | q 1 − k s21 + c∗ d2 , + q12 + 2(1+q 1)
where c∗ = q12 + 12 . Since (q1 +1)|s1 ξ2 | ≤ k4 s21 + k1 (q1 +1)2 ξ22 , it follows that if 2 q1 1 k ≥ max 8(q1q+1) , (46) , 2 + q2 2(1+q1 ) 2 then ξ12 q1 1 2 2 U˙ (t) ≤ − q42 ξ22 − 16(1+q 2 + k (q1 + 1) ξ2 1 ) 1+ξ1 q1 (47) + q12 + 2(1+q − 34 k s21 + c∗ d2 (t) 1) 2 ξ1 q1 k 2 + c∗ d2 (t) . ≤ − q82 ξ22 + 16(1+q 2 + 4 s1 1 ) 1+ξ 1
It follows from integrating (47) and applying Lemma to the function in curly braces in Assumption 1 is satisfied. Hence, Theorem to (40) and provides the dynamic feedback −satZ (−z1 − kψ(x1 )) + q3 arctan(x2 ) with x1 √ z˙1 = k −z1 + q1 sin(x1 ) − 2
Barbalat’s (47) that 1 applies u(Z, x) =
1+x1
with Z = (1 + )(1 + k)(q1 + 1) and the choice (39) of ψ, since the proof of Theorem 1 gives c1 = −1 and c2 = −k. 5. CONCLUSIONS We developed a new backstepping approach using a finite dimensional dynamic extension. Our work is motivated by the ubiquity of engineering applications that produce the required cascade forms. Our variant of backstepping offers possible advantages in terms of the design of output feedback control and boundedness, e.g., since it does not require artificial delays. We hope to develop local versions for systems that are only locally asymptotically stabilizable with prescribed input bounds, and for cases where the ai,j ’s in (1) depend on t, and to allow delays in the input. We will also investigate what family of systems can be stabilized by applying Theorem 1 repeatedly, and the possibility of chattering in the closed loop control. REFERENCES Coron, J.M. and Praly, L. (1991). Adding an integrator for the stabilization problem. Systems and Control Letters, 17, 89–104. Dawson, D., Carroll, J., and Schneider, M. (1994). Integrator backstepping control of a brush DC motor turning a robotic load. IEEE Transactions on Control Systems Technology, 2, 233–244. de Queiroz, M. and Dawson, D. (1996). Nonlinear control of active magnetic bearings: a backstepping approach. IEEE Transactions on Control Systems Technology, 4, 545–552. Dixon, W., Jiang, Z.P., and Dawson, D. (2000). Global exponential setpoint control of wheeled mobile robots: a Lyapunov approach. Automatica, 36, 1741–1746. 265
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Isidori, A. (1995). Nonlinear Control Systems. SpringerVerlag, New York. Jiang, Z.P. and Nijmeijer, H. (1999). A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Transactions on Automatic Control, 44, 265–279. Khalil, H. (2002). Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ. Lee, D.B., Nataraj, C., Burg, T.C., and Dawson, D. (2011). Adaptive tracking control of an underactuated aerial vehicle. In Proceedings of the American Control Conference, 2326–2331. San Francisco, CA. Liu, W., Chitour, Y., and Sontag, E. (1996). On finitegain stabilizability of linear systems subject to input saturation. SIAM Journal on Control and Optimization, 34, 1190–1219. Mazenc, F. and Bowong, S. (2004). Backstepping with bounded feedbacks for time-varying systems. SIAM Journal on Control and Optimization, 43, 856–871. Mazenc, F., Burlion, L., and Gibert, V. (2017a). Stabilization of a system that arises in the context of vision based landing of a civil aircraft. Submitted. Mazenc, F. and Iggidr, A. (2004). Backstepping with bounded feedbacks. Systems and Control Letters, 51, 235–245. Mazenc, F. and Malisoff, M. (2016). New control design for bounded backstepping under input delay. Automatica, 6, 48–55. Mazenc, F., Malisoff, M., Burlion, L., and Gibert, V. (2017b). Bounded backstepping through a dynamic extension with delay. In Proceedings of the 56th IEEE Conference on Decision and Control, 607–611. Melbourne, Australia. Mazenc, F., Malisoff, M., and Weston, J. (2016). New bounded backstepping control designs for time-varying systems under converging input converging state conditions. In Proceedings of the 55th IEEE Conference on Decision and Control, 3167–3171. Las Vegas, NV. Mazenc, F., Malisoff, M., and Weston, J. (2018, to appear). Bounded backstepping control and robustness analysis for time-varying systems under converging-inputconverging-state conditions. European Journal of Control. Pettersen, K. and Nijmeijer, H. (2002). Underactuated ship tracking control; theory and experiments. IEEE Transactions on Automatic Control, 40, 1759–1762. Smaoui, M., Brun, X., and Thomasset, D. (2006). A study on tracking position control of an electropneumatic system using backstepping design. Control Engineering Practice, 14, 923–933. Spong, M. (1994). Partial feedback linearization of underactuated mechanical systems. In Proceedings of the IEEE/RSJ/GI International Conference on Intelligent Robots and Systems, 314–321. Munich, Germany. Sussmann, H.J., Sontag, E., and Yang, Y. (1994). A general result on the stabilization of linear systems using bounded controls. IEEE Transactions on Automatic Control, 39(12), 2411–2425. Tsinias, J. (1997). Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation. ESAIM: Control, Optimisation and Calculus of Variations, 2, 57–85.
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IFAC PapersOnLine 51-13 (2018) 260–265
Backstepping Backstepping Design Design for for Output Output Feedback Feedback Backstepping Design for Output Feedback Stabilization for a Class of Uncertain Backstepping Design for Output Feedback Stabilization for a Class of Uncertain Stabilization for a Class of Uncertain Systems Extension Stabilization forDynamic a Class of Uncertain Systems using using Dynamic Extension Systems using Dynamic Extension Systems using Dynamic Extension Frederic Mazenc ∗∗ Laurent Burlion ∗∗ Michael Malisoff ∗∗∗ ∗∗ ∗∗∗
Frederic Mazenc ∗∗∗ Laurent Burlion ∗∗ Michael Malisoff ∗∗∗ ∗∗ ∗∗∗ Frederic Mazenc Mazenc Laurent Burlion Burlion ∗∗ Michael Malisoff Malisoff ∗∗∗ Frederic Laurent Michael ∗ ∗∗ ∗∗∗ ∗ Frederic Mazenc Laurent Burlion Michael Malisoff Inria, L2S-CNRS-CentraleSup´ e lec, 3 rue Joliot Curie, 91192, ∗ ∗ Inria, L2S-CNRS-CentraleSup´ e lec, 3 rue Joliot Curie, 91192, ∗ ∗ Inria, L2S-CNRS-CentraleSup´ lec, 3 3 rue rue Joliot Curie, Curie, 91192, 91192, Gif-sur-Yvette, (e-mail: eelec, Joliot Gif-sur-Yvette, France, France, (e-mail: ∗ Inria, L2S-CNRS-CentraleSup´ Inria, L2S-CNRS-CentraleSup´ elec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France, (e-mail: [email protected]) Gif-sur-Yvette, France, (e-mail: [email protected]) ∗∗ Gif-sur-Yvette, France, (e-mail: [email protected]) - The French ∗∗ [email protected]) ∗∗ Onera French Aerospace Aerospace Lab, Lab, F-31055 F-31055 Toulouse, Toulouse, France, France, ∗∗ Onera - The [email protected]) ∗∗ Onera -- The The French Aerospace Aerospace Lab, F-31055 F-31055 Toulouse, Toulouse, France, France, (e-mail: [email protected]) Onera French Lab, (e-mail: [email protected]) ∗∗ ∗∗∗ Onera - The (e-mail: French Aerospace Lab, F-31055 Toulouse, France, [email protected]) of Mathematics, Louisiana State University, ∗∗∗ (e-mail: [email protected]) ∗∗∗ Department Department (e-mail: of Mathematics, Louisiana State University, Baton Baton ∗∗∗ [email protected]) ∗∗∗ Department of Mathematics, Louisiana State University, Rouge, LA 70803-4918 (e-mail: [email protected]) of Mathematics, Louisiana State University, Baton Baton Rouge, LA 70803-4918 (e-mail: [email protected]) ∗∗∗ Department Department Louisiana State University, Baton Rouge, LA 70803-4918 [email protected]) Rouge, of LAMathematics, 70803-4918 (e-mail: (e-mail: [email protected]) Rouge, LA 70803-4918 (e-mail: [email protected]) Abstract: We revisit the backstepping approach. We show how bounded globally asymptotAbstract: We revisit the backstepping approach. We show how bounded globally asymptotAbstract: We the backstepping approach. We show how globally asymptotically output feedbacks can constructed aa family of systems. The Abstract: We revisit revisit backstepping approach. We for show how bounded bounded globally asymptotically stabilizing stabilizing outputthe feedbacks can be be constructed for family of nonlinear nonlinear systems. The Abstract: We revisit the backstepping approach. We show how bounded globally asymptotically stabilizing output feedbacks can can be constructed for aaand family of nonlinear nonlinear systems. systems. The approach relies onoutput the introduction of a be dynamic extension a converging-input-convergingically stabilizing feedbacks constructed for family of The approach relies on the introduction of a dynamic extension and a converging-input-convergingically stabilizing feedbackspresents can constructed for aand family of nonlinear The approach relies the of dynamic aa converging-input-convergingstate assumption. The technique severalextension advantages. It provides controlsystems. laws whose approach relies on onoutput the introduction introduction of aa be dynamic extension and converging-input-convergingstate assumption. The technique presents several advantages. It provides control laws whose approach relies on the introduction of a dynamic extension and a converging-input-convergingstate assumption. The technique presents several advantages. It provides control laws whose expressions are simple. It makes it possible to stabilize systems in the presence of certain types state assumption. The It technique presents several advantages. It the provides control laws whose systems in presence of certain types expressions are simple. makes it possible to stabilize state assumption. The technique several advantages. It the provides control laws whose expressions are simple. It makes itpresents possible tothe stabilize systems in the presence of certain certain types of uncertain terms which prevent the use of classical backstepping technique. It applies in expressions are simple. It makes it possible to stabilize systems in presence of types of uncertain terms which prevent the use of the classical backstepping technique. It applies in expressions are simple. It makes it possible to stabilize systems in the presence of certain types of uncertain terms which prevent thestate use of of the classical classical backstepping technique. technique. It It applies applies in notable casesterms wherewhich only part of the variable is measured. of uncertain prevent the use the backstepping in notable cases where only part of the state variable is measured. of uncertain prevent thestate use of the classical backstepping technique. It applies in notable cases where only of variable is notable casesterms wherewhich only part part of the the state variable is measured. measured. © 2018, IFAC (International Federation ofstate Automatic Control) Hosting by Elsevier Ltd. All rights reserved. notable cases where only part of the variable is measured. Keywords: Keywords: backstepping, backstepping, nonlinear, nonlinear, output output feedback, feedback, stabilization stabilization Keywords: backstepping, backstepping, nonlinear, nonlinear, output output feedback, feedback, stabilization stabilization Keywords: Keywords: backstepping, nonlinear, output feedback, stabilization 1. INTRODUCTION that 1. INTRODUCTION that are are used used in in the the controls controls even even when when there there are are no no input input 1. INTRODUCTION INTRODUCTION that are areinused used in the controls controls evenThis whentechnique there are are has no input input delays the original systems. been 1. that in the even when there no delays inused the in original systems. This technique has been The backstepping approach is useful for constructing glob1. INTRODUCTION that are the controls even when there are no delays in in in theMazenc originaland systems. This technique has input been initiated Malisoff (2016) and developed the original systems. This technique has been The backstepping approach is useful for constructing glob- delays initiated in Mazenc and Malisoff (2016) and developed The backstepping approach is useful useful for laws constructing globally backstepping asymptoticallyapproach stabilizing control for nonlinear delays in in theMazenc original systems. This technique has etbeen The is for constructing glob- initiated and Malisoff (2016) and developed and applied in Mazenc et al. (2016) and Mazenc initiated in Mazenc and Malisoff (2016) and developed ally asymptotically stabilizing control laws for nonlinear applied in Mazenc al. (2016) and and Mazenc et al. al. The backstepping approach useful for laws constructing glob- and ally asymptotically stabilizing control laws for nonlinear nonlinear systems in feedback form, isi.e., having a lower trianguinitiated in Mazenc andet Malisoff (2016) developed ally asymptotically stabilizing control for and applied in Mazenc et al. (2016) and Mazenc et al. (2017b), where the case of systems with delays in and applied in Mazenc et al. (2016) and Mazenc et the al. systems in feedback form, i.e., having a lower triangu(2017b), where the case of systems with delays in the ally asymptotically stabilizing control laws for nonlinear systems in feedback form, i.e., having a lower triangular structure; see Dixon et al. (2000). Since the pioneerand applied in Mazenc et al. (2016) and Mazenc et al. systems in feedback form, i.e.,(2000). havingSince a lower triangu(2017b), where the case case and of systems systems withinitial delaysfictitious in the the input is also considered where the (2017b), where the of with delays in lar structure; see Dixon et al. the pioneerinput is also considered and where the initial fictitious systems in feedback form, i.e., having a lower triangu1 lar structure; see in Dixon et and al. (2000). (2000). Since the pioneering contributions Coron Praly (1991) and Tsinias (2017b), where the case of systems with delays in the lar structure; see Dixon et al. Since the pioneerinput is is law alsoisconsidered considered and to where theclass initial fictitious control not be C A also and where initial ing structure; contributions in Coron and Praly (1991) and Tsinias input isconsidered not required required be of ofthe C 1111 ..fictitious A first first lar Dixon et and al. in (2000). Since the ing contributions in Coron and Praly and Tsinias (1997), it has see beenin developed many(1991) contributions and control input is law also andtoto where theclass initial ing contributions Coron Praly (1991) andpioneerTsinias control law is not required to be of class C ..fictitious A adaptation of this technique uncertain outputs has also control law is not required to be of class C A first first (1997), it has been developed in many contributions and adaptation of this technique to uncertain outputs also 1 has ing contributions in Coron and Praly (1991) and Tsinias (1997), it has been developed in many contributions and is successfully used in many applications, as illustrated control law is not required to be of class C . A first (1997), it has been developed in many contributions and adaptation of this technique to uncertain outputs has also been studied in Mazenc et al. (2017a), where, due to the adaptation of in this technique to (2017a), uncertainwhere, outputs has also is successfully used in many applications, as illustrated been studied Mazenc et al. due to the (1997), it has been developed in many contributions and is successfully used in many applications, as illustrated for instance by Jiang and Nijmeijer (1999), Pettersen and adaptation of this technique to uncertain outputs has is successfully used inand many applications, as illustrated beenofstudied studied in Mazenc Mazenc et etonly al. (2017a), (2017a), where, due to to also the use visual information, imprecise measurement of been in al. where, due the for instance by Jiang Nijmeijer (1999), Pettersen and use of visual information, only imprecise measurement of is successfully used in many applications, as illustrated for instance by Jiang Jiang and Nijmeijer Nijmeijer (1999), Pettersen and been Nijmeijer (2002), Smaoui et al. (2006), dePettersen Queiroz and in Mazenc etonly al. (2017a), where, due to the for instance by and (1999), use first ofstudied visual information, only imprecise measurement of the backstepping variable was available. use of visual information, imprecise measurement of Nijmeijer (2002), Smaoui et al. (2006), de Queiroz and the first backstepping variable was available. for instance by Jiang and Nijmeijer (1999), Pettersen and Nijmeijer (2002),and Smaoui et al. al. (2011). (2006), Presentations de Queiroz Queiroz and Dawson (1996) Lee etet of the use of visual information, only was imprecise measurement of Nijmeijer (2002), Smaoui al. (2006), de first backstepping variable available. the first backstepping variable was available. Dawson (1996) and Lee et al. (2011). Presentations of Nijmeijer (2002),and Smaoui al.be (2006), deKhalil Queiroz and paper, we aa new Dawson (1996) and Lee et etetcan al. (2011). Presentations of In the backstepping technique found Presentations in (2002), the firstpresent backstepping was available. Dawson (1996) Lee al. (2011). of In the the present paper,variable we propose propose new backstepping backstepping the backstepping technique can be found in Khalil (2002), In the present paper, we propose a new backstepping backstepping Dawson (1996) and Lee et al. (2011). Presentations of design for globally uniformly asymptotically stabilizing the backstepping technique can be found in Khalil (2002), In the present paper, we propose a new Mazenc and Bowong (2004), and many other research the backstepping technique can and be found inother Khalilresearch (2002), design for globally uniformly asymptotically stabilizing Mazenc and Bowong (2004), many In the for present paper, welinearizable propose a systems; new backstepping design globally uniformly asymptotically stabilizing the backstepping technique can and be found Khalilresearch (2002), control laws for partially see Mazenc and and Bowong (2004), and manyinother other research design for globally uniformly asymptotically stabilizing monographs papers. Mazenc and Bowong (2004), many control laws for partially linearizable systems; see Isidori Isidori monographs and papers. design for globally uniformly asymptotically stabilizing control laws for partially linearizable systems; see Mazenc and Bowong (2004), and many other research (1995) for an introduction to partially linearizable sysmonographs and and papers. papers. control for lawsanforintroduction partially linearizable systems; see Isidori Isidori monographs (1995) to partially linearizable sysOne of the drawbacks of this technique is the complexity control laws for partially linearizable systems; see Isidori (1995) Its for fundamental an introduction introduction toaspect partially linearizable sysmonographs and papers. tems. new to withlinearizable respect to the for an partially sysOne of the drawbacks of this technique is the complexity (1995) tems. Its fundamental new aspect with respect to the One of formulas the drawbacks drawbacks of the of the it sometimes provides, is notably when it (1995) for fundamental an that introduction partially linearizable One of the of this this technique technique isnotably the complexity complexity tems. Its Its fundamental new toaspect aspect with respect to systhe contributions use artificial delays is that, instead of tems. new with respect to the of the formulas it sometimes provides, when it contributions that use artificial delays is that, instead of One of the drawbacks of this technique is the complexity of the formulas it sometimes provides, notably when it is applied repeatedly and when size constraints on the tems. Its fundamental new aspect with respect to the of the formulas it sometimes provides, notably when it introducing contributions that use artificial delays is that, instead of delays, a finite dimensional dynamic extension contributions that use artificial delays is that, instead of is applied repeatedly and when size constraints on the introducing delays, a finite dimensional dynamic extension of the formulas it sometimes provides, notably when it is applied repeatedly and when size constraints on the control laws have to be respected. This is a limitation contributions that use artificial delays is that, instead of is applied repeatedly and when size constraints on the introducing delays, a finite dimensional dynamic extension is designed, making it possible to obtain feedbacks without introducing delays, a finite dimensional dynamic extension control laws have to be respected. This is a limitation is designed, making it possible to obtain feedbacks without is applied repeatedly and whenbackstepping sizeThis constraints on the control laws have be respected. is limitation of the applicability classical of the introducing delays, dimensional dynamic control laws have to toof be respected. This is aa and limitation is designed, designed, making itfinite possible to obtain obtain feedbacks without delays, which offer athe following advantages: (i)extension they are is making it possible to feedbacks without of the applicability of classical backstepping and of the control laws have toof respected. This is Iggidr a and limitation which offer the following advantages: (i) they are of the applicability applicability ofbe classical backstepping and of the the delays, bounded backstepping results of Mazenc and (2004) is designed, making it possible to obtain feedbacks without of the classical backstepping of delays, which offer the following advantages: (i) are bounded in the cases where bounded feedbacks can be delays, which offer the following advantages: (i) they they are bounded backstepping results of Mazenc and Iggidr (2004) bounded in the cases where bounded feedbacks can be of the applicability of classical backstepping and of the bounded backstepping results of Mazenc and Iggidr (2004) and Mazenc and Bowong (2004). Another limitation of expected delays, which offer the following advantages: (i) they are bounded backstepping results of Mazenc and Iggidr (2004) bounded in the cases where bounded feedbacks can be (which contrasts with Mazenc et al. (2016) where bounded in the cases where bounded feedbacks can be and Mazenc and Bowong (2004). Another limitation of expected (which contrasts with Mazenc et al. (2016) where bounded backstepping results of Mazenc and Iggidr (2004) and Mazenc and Bowong (2004). Another limitation of the approach is due to the fact that in general, it does bounded in the cases where bounded feedbacks can be and Mazenc and Bowong (2004). Another limitation of expected (which contrasts with Mazenc et al. (2016) where the controls obtained for the original systems are not expected (which contrasts with Mazenc etsystems al. (2016) where the approach is due to the fact that in general, it does the controls obtained for the original are not and Mazenc and Bowong (2004). Another limitation of the approach is only due to to the fact fact that in general, general, it does does not apply when a part of the state is measured. In expected (which contrasts etsystems al. (2016) where the approach is due the that in it the controls controls obtained forwith the original systems are not bounded controls), (ii) they areMazenc given by simple formulas, the obtained for the original are not not apply when only a part of the state is measured. In the approach is only due to the fact that general, it does controls), (ii) they are given by simple formulas, not applyitwhen when only part of the the state is measured. measured. In bounded addition, relies on the existence ofstate a in fictitious feedback the controls obtained for the original are not not apply aa part of is In bounded controls), (ii) are given by simple formulas, (iii) they apply in cases where subsystem from which bounded controls), (ii) they they arethe given by systems simple formulas, addition, it relies on the existence of a fictitious feedback k (iii) they apply in cases where the subsystem from which not apply when only a part of the state is measured. In addition, it relies on the existence of a fictitious feedback when the backstepwhich typically has to be of class C bounded controls), (ii) they are given by simple formulas, k addition, it relies on the existence of k a fictitious feedback (iii) they apply in cases where the subsystem from which the backstepping begins contains unknown vector fields, (iii) they apply in begins cases where the unknown subsystemvector from which which typically has to be of class C the backstepthe backstepping contains fields, addition, it relieshas on existence fictitious feedback whichistypically typically has tothe beThe of class class Cofkk awhen when the ping applied times. presence of in theythey apply in begins cases where the unknown subsystem from which the backstepbackstepwhich to be of C the backstepping backstepping begins contains unknown vector fields, and (iv) apply in cases where only aa part of state the contains vector ping istypically applied k khas times. The presence of uncertainties uncertainties in (iii) k when and (iv) they apply in cases where only part of the thefields, state when the backstepwhich to be of class C pingdynamics is applied appliedand k times. times. The presence of uncertainties uncertainties in the the in the output may also be an obstacle. backstepping begins contains unknown vector fields, ping is k The presence of in and (iv) (iv) is they apply in inand cases where onlyparts a part part of the state variable measured where some of the output and they apply cases where only a of the state the dynamics and in the output may also be an obstacle. variable is measured and where some parts of the output ping is applied k times. The presence of uncertainties in the dynamics dynamics and and in in the the output output may may also also be be an an obstacle. obstacle. and (iv)known they apply in cases where only a part of the state the variable is measured and where some parts of the output are not with certainty. The family of partially linvariable is measured and where some parts of the output To overcome these drawbacks, a new technique has been not known with certainty. The family ofofpartially linthe overcome dynamics these and indrawbacks, the outputamay be an obstacle. To new also technique has been are variable is measured and where some parts the output are not known with certainty. The family of partially linearizable systems we consider encompasses many systems not known with The family ofmany partially linTo overcome overcome these based drawbacks, a new newlaws technique hasdelays been are proposed recently, on control in which To these drawbacks, a technique has been earizable systems wecertainty. consider encompasses systems proposed recently, based on control laws in which delays are not known with certainty. The family of partially linearizable systems we consider encompasses many systems that are relevant from an applied point of view, as shown To overcome these drawbacks, a new technique has been earizable systems we consider encompasses many systems proposed recently, based on are control laws in which delays are artificial, meaning there delays in the state values proposed recently, based on control laws in which delays that are relevant from an applied point of view, as shown are artificial, meaning there are delays in the state values earizable systems we consider encompasses many systems thatinstance are relevant relevant from an an applied point of view, view, as to shown for by Spong (1994). Also, with respect the proposed recently, based on are control laws are from applied point of as shown are meaning there delays in the state values are artificial, artificial, meaning there are delays in in thewhich state delays values that for instance by Spong (1994). Also, with respect the are relevant from an applied point of view, as to shown for instance by Spong (1994). Also, with respect to the backstepping with artificial delay methods from the works Supported are artificial, meaning there are Foundation delays in the state values that for instance by Spong (1994). Also, with respect to the by US National Science Grant 1711299. backstepping with artificial delay methods from the works Supported by US National Science Foundation Grant 1711299. for instance by Spong (1994). Also, with respect to the backstepping with artificial delay methods from the works backstepping with artificial delay methods from the works Supported Supported by by US US National National Science Science Foundation Foundation Grant Grant 1711299. 1711299. backstepping with artificial delay methods from the works Supported by US National Science Foundation Grant 1711299.
2405-8963 © 2018, IFACConference (International Proceedings, 2nd IFAC onFederation of Automatic Control) 260 Hosting by Elsevier Ltd. All rights reserved. Proceedings, 2nd IFAC Conference on 260 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference on 260 Proceedings, 2nd IFAC Conference 260 Modelling, Identification and Controlon of Nonlinear 10.1016/j.ifacol.2018.07.288 Systems Proceedings, 2nd IFAC Conference on 260 Modelling, Identification and Control of Nonlinear Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear
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Mazenc and Malisoff (2016), Mazenc et al. (2016), and Mazenc et al. (2018, to appear), the main drawback of the main result of the present paper is that it relies on a fictitious feedback of class C 1 (although these prior works did not cover systems with outputs or with uncertain vector fields). Comparisons between the performances of the control laws of Mazenc and Malisoff (2016) and Mazenc et al. (2016) and those of the present paper still have to be made and will be the subject of future studies. Our converging-input-converging-state condition provides a novel and promising approach and was not present in the well known works such as Liu et al. (1996) and Sussmann et al. (1994) for linear systems. This paper is organized as follows. The main result is shown in Section 2. Section 3 provides a technical lemma that can facilitate checking our assumptions from Section 2. An example based on a direct drive manipulator is given in Section 4. Concluding remarks are drawn in Section 5. Our notation will be simplified whenever no confusion can arise from the context. The dimensions of our Euclidean spaces are arbitrary unless other indicated. The Euclidean norm in Ra , and the induced norm of matrices, are denoted by | · |, and | · |∞ is the usual sup norm. Given a constant J > 0, satJ denotes the symmetric saturation function defined by satJ (x) = max{−J, min{J, x}} for all x ∈ R, where J is called the saturation level. We use the usual class of comparison functions K∞ from Khalil (2002). 2. MAIN STABILIZATION RESULT 2.1 The studied system We consider the nonlinear time-varying system x˙ = f (t, x, g1 ) g ˙ 1 = a1,1 g1 + a1,2 g2 g˙ 2 = a2,1 g1 + a2,2 g2 + a2,3 g3 (1) .. . g˙ n = an,1 g1 + an,2 g2 + · · · + an,n gn + u + Ω(Y(t, x), G) where x is valued in Rp , G = (g1 , ..., gn ) is valued in Rn , f is a nonlinear function that is locally Lipschitz with respect to (x, g1 ) and piecewise continuous with respect to t, u is the scalar-valued input, Y : R × Rp → Rl is a continuous function, each ai,j ∈ R is a constant, ai,i+1 = 0 holds for i = 1, 2...n − 1 (with no sign constraint on the aij ’s), and the function Ω is locally Lipschitz. This family of systems has been studied in many papers because systems of this type may result from partial linearization and are frequently encountered in practice. In this section, we assume that the R valued output is (2) Y (t) = (Y(t, x(t)), g1 (t), . . . , gn (t)) , which is realistic in practice. Our first assumption is: Assumption 1. There exist a locally Lipschitz scalar valued function ψ that is bounded by a known constant ψ ≥ 0, and a constant k > 0, such that for any continuous function d : [0, ∞) → R that converges exponentially to the origin, all solutions (ξ, λ1 , ..., λn ) : [0, ∞) → Rp+n of ξ˙ = f (t, ξ, λ1 + d(t)) λ˙ 1 = k[−λ1 + λ2 ] (3) .. . ˙ λn = k[−λn + ψ(t, Y(t, ξ))] l+n
261
261
converge to the origin as t → ∞. Moreover the function Ω in (1) is bounded by a known constant Ω ≥ 0. In terms of the matrices a1,1 a1,2 0 a2,1 a2,2 a2,3 . .. . A = .. . . . an,1 an,2 . . .
... ... .. .
0 0 .. .
..
. an−1,n . . . an,n
∈ Rn×n
(4)
for n > 1 and A = a1,1 for n = 1 and B = (0, 0, . . . , 1) ∈ Rn , our second assumption is: Assumption 2. There is a locally Lipschitz function : Rn → R that is bounded by a known constant ≥ 0 such that the origin of the system ˙ Γ(t) = AΓ(t) + B (Γ(t)) (5) is a globally asymptotically and locally exponentially stable equilibrium. 2.2 Statement of and discussion on theorem We are ready to state and prove the following result, where sat is as defined in Section 1: Theorem 1. Let the system (1) satisfy Assumptions 1-2 and let > 0 be any constant. Then there exist a constant matrix L and constants cj for j = 1 to n + 1 such that all solutions (x, G) : [0, ∞) → Rp+n of (1), in closed loop with the dynamic output feedback n cj zj + cn+1 ψ(t, Y(t, x)) u(G, Z, x) = −satZ j=1
+ (G + LZ) − Ω (Y(t, x), G)
z˙1 = k[−z1 + z2 ] .. . z˙n−1 = k[−zn−1 + zn ] z˙n = k[−zn + ψ(t, Y(t, x))] with Z = (z1 , ..., zn ) and the saturation level n+1 Z = ψ(1 + ) |cj |,
(6)
j=1
asymptotically converge to 0 as t → ∞.
Before turning to the proof of Theorem 1, we make several remarks on its motivation and value. 1) The feedback (6) is bounded by the constant Z + +Ω. 2) The formula (6) for the control law does not incorporate the first derivative of ψ(t, Y(t, x(t))), which plays the role of the fictitious control of the classical backstepping approach. Hence, it applies even when ψ(t, Y(t, x)) is not of class C 1 . However, in practice Theorem 1 can only be applied when ψ is of class C 1 because checking that Assumption 1 is satisfied frequently necessitates that ψ be of class C 1 , as we show in the next section. By contrast, in Mazenc et al. (2016) and Mazenc et al. (2018, to appear), fictitious feedbacks of class C 0 can be used. 3) One can use changes of variables and an appropriate choice of feedback to transform the system (1) into a system of the form (3) with d = 0 whose exponential
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stability property is ensured by Assumption 1. However, in a sense, this result is not satisfactory because the feedback obtained that way may possess inappropriate properties. For instance, in general they would be unbounded. This motivates our alternative approach, based on dynamic extensions. 5) None of the assumptions of this paper imply that f has to be known with accuracy. Also, we only require measurements of ψ(t, Y(t, x)) and Ω(Y(t, x), G), instead of Y itself. 2.3 Proof of Theorem 1 A key aspect of the proof consists of making g1 (t) − z1 (t) converge to zero instead of making g1 (t) − ψ(t, Y(t, x(t))) converge to zero, as traditionally done in classical backstepping. To achieve our goal, several changes of coordinates are needed, as follows. We first assume that n > 1, and we explain how these changes of variables can be constructed by induction. Set l1,1 = −1 and r1 = g1 − z1 .
Induction assumption: For all j ∈ {1, ..., i} and 1 < i < n, there are constants lj,m for m = 1 to j such that the variables j lj,m zm (7) rj = gj + m=1
for j = 1 to i satisfy r˙1 = a1,1 r1 + a1,2 r2 r˙2 = a2,1 r1 + a2,2 r2 + a2,3 r3 .. . r˙i−1 = ai−1,1 r1 + ai−1,2 r2 + · · · + ai−1,i ri .
(8)
First step: i = 2. Since r1 = g1 − z1 , we have r˙1 = a1,1 g1 + a1,2 g2 − k[−z1 + z2 ] a1,1 + k k (9) = a1,1 r1 + a1,2 g2 + z1 − z2 . a1,2 a1,2 Then the variable a1,1 + k k r2 = g2 + z1 − z2 (10) a1,2 a1,2 satisfies (11) r˙1 = a1,1 r1 + a1,2 r2 Thus the induction assumption is satisfied at the first step. Step i: Assume that the induction assumption is satisfied at a step i with 1 < i < n. Then since (7) holds for j = 1, 2, . . . , i we have r˙i = ai,1 g1 + ai,2 g2 + ... + ai,i+1 gi+1 i + li,m k[−zm + zm+1 ] m=1
= ai,1 r1 + ai,2 r2 + ... + ai,i ri + ai,i+1 gi+1 +ai,1 z1 − ai,2 (l2,1 z1 + l2,2 z2 ) − ... i i − ai,i li,m zm + li,m k[−zm + zm+1 ].
(12)
we obtain r˙i = ai,1 r1 + ai,2 r2 + ... + ai,i+1 ri+1 ,
(14)
so the induction assumption is satisfied at the step i + 1. Taking the time derivative of (13) with the choice i = n−1, we can then find a linear change of coordinates R = G+LZ with R = (r1 , ..., rn ) that transforms the system (1) into x˙ = f (t, x, z1 + r1 ) = a1,1 r1 + a1,2 r2 r ˙ 1 r˙2 = a2,1 r1 + a2,2 r2 + a2,3 r3 .. . (15) r˙n = an,1 r1 + an,2 r2 + · · · + an,n rn + u n ln,j zj + ln,n+1 ψ(t, Y(t, x)) + j=1 + Ω(Y(t, x), G).
If instead n = 1, then we again obtain (15) with only the dynamics for x and rn present. Let cj = ln,j for j = 1 to n + 1. Then the closed-loop system is x˙ = f (t, x, z1 + r1 ) R˙ = AR + B [(R) n cj zj + cn+1 ψ(t, Y(t, x)) −satZ j=1 n (16) + cj zj + cn+1 ψ(t, Y(t, x)) j=1 = k[−z z ˙ 1 1 + z2 ] . .. z˙n = k[−zn + ψ(t, Y(t, x))]. Consider any solution (x, R, z) : [0, ∞) → Rp+2n of (16).
Since Assumption 1 ensures that ψ is bounded by ψ, it follows from (16) that there is a finite value ta ≥ 0 such that for all t ≥ ta , the inequality n cj zj (t) + cn+1 ψ(t, Y(t, x(t))) ≤ Z (17) j=1
is satisfied. Hence, when t ≥ ta , the closed-loop system is x˙ = f (t, x, z1 + r1 ) ˙ R = AR + B(R) z˙1 = k[−z1 + z2 ] (18) .. . z˙n = k[−zn + ψ(t, Y(t, x))].
Assumption 2 ensures that the R-subsystem of the system (18) is globally asymptotically and locally exponentially stable. Then Assumption 1 allows us to conclude.
m=1
3. TECHNICAL LEMMA
Thus, taking ai,1 ai,2 z1 − ai,i+1 (l2,1 z1 + l2,2 z2 ) − ... ri+1 = gi+1 + ai,i+1 i i (13) ai,i 1 − ai,i+1 li,m zm + ai,i+1 li,m k[−zm + zm+1 ]
Assumption 2 is a simple classical assumption that can often be checked easily; see our illustration below. However, checking Assumption 1 can be nontrivial. In this section, we give conditions ensuring that Assumption 1 holds.
m=1
m=1
m=1
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3.1 Assumptions and lemma Consider the system ξ˙ = f (t, ξ, λ1 + d(t)) λ˙ = k[−λi + λi+1 ], 1 ≤ i ≤ n − 1 ˙i λn = k[−λn + ψ(t, yξ )]
263
for all t ≥ 0. Consider the positive definite function n is2i . (27) Q(S) = 12 i=1
(19)
where yξ = Cξ and C ∈ Rq×p is a constant matrix, the constant k > 0 will be further restricted below, and d : [0, ∞) → R exponentially converges to 0. Then (19) is a special case of (3). We add these assumptions on (19): Assumption 3. There is a function κ1 ∈ K∞ such that |ψ(t, yξ )| + |f (t, ξ, w)| ≤ κ1 (|(ξ, w)|) (20) for all t ≥ 0, ξ ∈ Rp and w ∈ R. Also, ψ is of class C 1 and bounded by a known constant ψ. Finally, there are functions κ2 and κ3 of class K∞ , a function V of class C 1 , and a uniformly continuous positive definite W such that κ2 (|ξ|) ≤ V (t, ξ) ≤ κ3 (|ξ|) for all (t, ξ) ∈ [0, ∞)×Rp (21) and such that for any choice of the piecewise continuous function δ, the time derivative of V along all solutions of ˙ = f (t, ξ(t), ψ(t, Cξ(t)) + δ(t)) ξ(t) (22)
Then we can use the triangle inequality to obtain n−1 n−1 1 2 i(si + s2i+1 ) isi si+1 ≤ 2 i=1 i=1 n−1 n 1 2 2 is + (i − 1)si = 2 i=1 i i=2
and then the subadditivity of the square root to conclude that along all solutions of (25) for all t ≥ 0, we have ˙ Q(t) ≤ k
3.2 Proof of Lemma 1 Set si = λi −ψ(t, yξ ) for i = 1, 2, . . . , n and S = (s1 , ..., sn ). Then (19) can be transformed into ˙ ξ = f (t, ξ, ψ(t, yξ ) + s1 + d(t)) ∂ψ (t, yξ ) s˙ 1 = k[−s1 + s2 ] − ∂t ∂ψ (t, yξ )Cf (t, ξ, ψ(t, yξ ) + s1 + d(t)) − ∂yξ (25) .. . ∂ψ (t, yξ ) s˙ n = −ksn − ∂t ∂ψ (t, yξ )Cf (t, ξ, ψ(t, yξ ) + s1 + d(t)) . − ∂yξ From Assumption 3, it follows that V˙ (t) ≤ −W (ξ(t)) + |s1 (t) + d(t)|2 for all t ≥ 0. Assumption 4 gives 2 ∂ψ ∂ψ ∂t (t, yξ )+ ∂yξ (t, yξ )Cf (t, ξ, ψ(t, yξ ) + s1 + d(t)) (26) ≤ bf W (ξ) + bs |s1 + d(t)|2 263
n−1
isi (−si + si+1 ) − kns2n
i=1 n
bf W (ξ) + bs |s1 + d(t)|2 i=1 n−1 n k 2 2 (i − 1)si − kns2n ≤ − is − 2 i=1 i i=2 n + i|si | bf W (ξ) + bs |s1 + d(t)|2 +
satisfies the inequality V˙ (t) ≤ −W (ξ(t)) + |δ(t)|2 for all t ≥ 0, and f is locally Lipschitz. Assumption 4. There are positive constants bf and bs such that for the function W from Assumption 3, the inequality 2 ∂ψ ∂ψ (t, Cξ)Cf (t, ξ, ψ(t, Cξ) + γ) ∂t (t, Cξ) + ∂y ξ (23) ≤ bf W (ξ) + bs |γ|2 holds for all t ≥ 0, ξ ∈ Rp and γ ∈ R. The inclusion of the γ term on the right side of (23) is justified because γ represents uncertainty and overshoot terms involving uncertainties commonly arise, e.g., from input-to-state stability estimates. We prove: Lemma 1. Let the system (19) satisfy Assumptions 3 and 4. Then for any k ≥ k with k = 3n n + n+1 bf + b2s (24) 2 all solutions of (19) converge to 0 as t → ∞.
(28)
i|si |
i=1 n
k 2 ≤ − s 2 i=1 i n + i|si | bf W (ξ) + bs |s1 + d(t)|2
(29)
i=1
n n k 2 ≤ − s i + bf i|si | W (ξ) 2 i=1 i=1 n n + bs i|si ||s1 | + i|si | bs |d(t)|. i=1
i=1
Consider the candidate Lyapunov function n(n + 1) U (t, ξ, S) = bf V (t, ξ) + Q(S) (30) 2 and any trajectory of (25). Along the trajectories of (25), n(n + 1) bf W (ξ) U˙ (t) ≤ − 2 n(n + 1) + bf |s1 (t) + d(t)|2 2 n n k 2 (31) − s i + bf i|si | W (ξ) 2 i=1 i=1 n n + bs i|si ||s1 | + i|si | bs |d(t)| . i=1
i=1
Using the triangle inequality to obtain 1 bf W (ξ) + i2 s2i , (32) bf i|si | W (ξ) ≤ 2 we deduce that n nbf k 2 n(n + 1) U˙ (t) ≤ − W (ξ) − bf s21 s + 2 2 i=1 i 2 n n 1 2 2 n(n + 1) bf d2 (t) (33) + i s i + bs i|si ||s1 | + 2 i=1 2 i=1 n +n(n + 1)bf s1 d(t) + i|si | bs |d(t)| . Using
√
i=1
bs i|si ||s1 | ≤ 2i s2i + 2i bs s21 , we obtain
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Frederic Mazenc et al. / IFAC PapersOnLine 51-13 (2018) 260–265
n k 2 nbf ˙ W (ξ) − s U (t) ≤ − 2 2 i=1 i bs bf + s21 +n(n + 1) 2 4 n n(n + 1) bf d2 (t) + n2 s2i + 2 i=1 n +n(n + 1)bf s1 d(t) + i|si | bs |d(t)| .
s1 = λ1 − q1 sin(ξ1 ) +
(34)
1 + ξ12
gives ξ˙1 = ξ2 ξ1 + s1 + d(t) ξ˙2 = −q2 ξ2 − 2 1 + ξ1
(41) 1 ξ2 . s˙ 1 = −ks1 + −q1 cos(ξ1 ) + (1+ξ12 ) 1+ξ12
i=1
From (24), we deduce that n k 2 n(n + 1) nbf W (ξ) − bf d2 (t) s + U˙ (t) ≤ − 2 6 i=1 i 2 n +n(n + 1)bf s1 d(t) + i|si | bs |d(t)| .
ξ1
(35)
i=1
Since each λi (and therefore also each si ) enters a fixed compact set after a finite time (by the boundedness of ψ and the structure of the system (19)), there are constants Λ > 0 and Tl such that for all t ≥ Tl , n k nb f W (ξ) + s2 + d (t) U˙ (t) ≤ − (36) 2 6 i=1 i
with d (t) = n(n+1)bf d2 (t)+Λ|d(t)|. From this inequality, we can conclude from Barbalat’s Lemma (applied to the function in squared brackets in (36), using the fact that d (t) converges exponentially to the origin) that all the solutions of (19) converge to the origin. 4. ILLUSTRATION
Let us choose V (ξ) = 1 + ξ12 − 1 + 12 ξ22 +
with the output Y = (x, g1 ) and the R -valued state x = (x1 , x2 ), where a1,1 > 0 and qi > 0 for i = 1 to 3 are constants. Assumption 2 is satisfied with = 0 and n = 1 (because in this case, the system (5) with = 0 is a linear system that is exponentially stable to 0), and we choose the bounded function Ω(x, g1 ) = −q3 arctan(x2 ). 2
Let us check that Assumption 1 is satisfied with n = 1, under the added assumptions that q 1 q2 q1 q2 (38) < 1 and < . 1 + q1 1 + q1 2 Consider ξ1 ψ(ξ) = q1 sin(ξ1 ) − (39) 1 + ξ12 and any continuous function d : [0, ∞) → R that exponentially converges to 0 and ξ˙1 = ξ2 ξ˙2 = −q1 sin(ξ1 ) − q2 ξ2 + λ1 + d(t) (40) ξ1 ˙ = k −λ + q sin(ξ ) − λ 1 1 1 1 1 + ξ12 with k > 0 being a constant to be specified. The choice
264
√ ξ1
1+ξ12
ξ2 .
(42)
Then the first inequality in (38) and the triangle inequality combine to give q1 ξ ξ 1 2 −q2 4(1 + q1 ) 1 + ξ12 1 ξ12 q1 1 2 2 (43) q ≤ + ξ 4(1 + q1 ) 2 1 + ξ12 2 2 2 q1 q2 ξ12 ≤ + ξ22 , 8(1 + q1 ) 1 + ξ12 8 and so also 2
ξ2 q1 √ 2 V˙ (t) = √ξ1 ξ2 2 + 4(1+q 1 ) (1+ξ 2 ) 1+ξ 1 1 1+ξ1 ξ q1 1 + ξ2 + 4(1+q1 ) √ 2 1+ξ1 ξ 1 × −q2 ξ2 − √ 2 + s1 + d(t) 1+ξ1 q1 2 = −q2 ξ2 + 4(1+q ) √ξ1 ξ2 2 1+ξ1
1
Consider a single-link direct-drive manipulator actuated by a permanent magnet DC brush motor, which produces the following model from Dawson et al. (1994) (after a change of coordinates which removes a constant): x˙ 1 = x2 x˙ 2 = g1 − q1 sin(x1 ) − q2 x2 (37) g˙ 1 = u − q3 arctan(x2 ) − a1,1 g1
q1 4(1+q1 )
ξ12
q1 − 4(1+q 2 1 ) 1+ξ1
+ ξ2 +
+
q1 4(1+q1 )
≤ − 3q42 ξ22 −
ξ22 q1 √ 4(1+q1 ) (1+ξ 2 ) 1+ξ 2 1 1
√ ξ1
ξ12
q1 8(1+q1 ) 1+ξ12
q1 √ ξ1 + 4(1+q 1)
1+ξ12
(s1 + d(t))
1+ξ12
(44)
+ ξ2 (s1 + d(t))
(s1 + d(t)) ξ2
q1 1 ≤ − q42 ξ22 − 16(1+q 2 1 ) 1+ξ1 q 1 (s1 + d(t))2 + 2q12 + 4(1+q 1) ξ2
q1 1 ≤ − q42 ξ22 − 16(1+q 2 1 ) 1+ξ1 q1 2 1 + q2 + 2(1+q1 ) s1 q1 d2 (t), + q12 + 2(1+q ) 1
along all solutions of (41) for all t ≥ 0, where the first inequality in (44) used (38) and (43), the second equality in (44) used the relations ξ2 (s1 + d(t)) ≤ 12 q2 ξ22 + √ ξ1 2 (s1 1+ξ1
+ d(t)) ≤
1 2 2q2 (s1 + d(t)) and 2 ξ 1 2 1 4 1+ξ12 + (s1 + d(t)) ,
and the last inequality in (44) use the inequality (a + 2 2 2 b) √ ≤ 2a + 2b for suitable a and b. Also, the relations 1 + p − 1 ≥ p/(4(1 + p)) for all p ≥ 0, and (43) with q2 replaced by 1, can be used to show that (42) is positive definite, so (42) is proper and positive definite. Next let U (ξ, s1 ) = V (ξ) + 12 s21 . Then (44) gives
2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018 Frederic Mazenc et al. / IFAC PapersOnLine 51-13 (2018) 260–265
2
ξ1 q1 U˙ (t) ≤ − q42 ξ22 − 16(1+q 2 1 ) 1+ξ1 q1 s21 + q12 + 2(1+q 1) q1 d2 (t) − ks21 + q12 + 2(1+q 1) 1√ + −q1 cos(ξ1 ) + s 1 ξ2 2 2 (1+ξ1 )
(45)
1+ξ1
ξ12 q1 16(1+q1 ) 1+ξ12
≤ − q42 ξ22 − + (q1 + 1)|s1 ξ2 | q 1 − k s21 + c∗ d2 , + q12 + 2(1+q 1)
where c∗ = q12 + 12 . Since (q1 +1)|s1 ξ2 | ≤ k4 s21 + k1 (q1 +1)2 ξ22 , it follows that if 2 q1 1 k ≥ max 8(q1q+1) , (46) , 2 + q2 2(1+q1 ) 2 then ξ12 q1 1 2 2 U˙ (t) ≤ − q42 ξ22 − 16(1+q 2 + k (q1 + 1) ξ2 1 ) 1+ξ1 q1 (47) + q12 + 2(1+q − 34 k s21 + c∗ d2 (t) 1) 2 ξ1 q1 k 2 + c∗ d2 (t) . ≤ − q82 ξ22 + 16(1+q 2 + 4 s1 1 ) 1+ξ 1
It follows from integrating (47) and applying Lemma to the function in curly braces in Assumption 1 is satisfied. Hence, Theorem to (40) and provides the dynamic feedback −satZ (−z1 − kψ(x1 )) + q3 arctan(x2 ) with x1 √ z˙1 = k −z1 + q1 sin(x1 ) − 2
Barbalat’s (47) that 1 applies u(Z, x) =
1+x1
with Z = (1 + )(1 + k)(q1 + 1) and the choice (39) of ψ, since the proof of Theorem 1 gives c1 = −1 and c2 = −k. 5. CONCLUSIONS We developed a new backstepping approach using a finite dimensional dynamic extension. Our work is motivated by the ubiquity of engineering applications that produce the required cascade forms. Our variant of backstepping offers possible advantages in terms of the design of output feedback control and boundedness, e.g., since it does not require artificial delays. We hope to develop local versions for systems that are only locally asymptotically stabilizable with prescribed input bounds, and for cases where the ai,j ’s in (1) depend on t, and to allow delays in the input. We will also investigate what family of systems can be stabilized by applying Theorem 1 repeatedly, and the possibility of chattering in the closed loop control. REFERENCES Coron, J.M. and Praly, L. (1991). Adding an integrator for the stabilization problem. Systems and Control Letters, 17, 89–104. Dawson, D., Carroll, J., and Schneider, M. (1994). Integrator backstepping control of a brush DC motor turning a robotic load. IEEE Transactions on Control Systems Technology, 2, 233–244. de Queiroz, M. and Dawson, D. (1996). Nonlinear control of active magnetic bearings: a backstepping approach. IEEE Transactions on Control Systems Technology, 4, 545–552. Dixon, W., Jiang, Z.P., and Dawson, D. (2000). Global exponential setpoint control of wheeled mobile robots: a Lyapunov approach. Automatica, 36, 1741–1746. 265
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Isidori, A. (1995). Nonlinear Control Systems. SpringerVerlag, New York. Jiang, Z.P. and Nijmeijer, H. (1999). A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Transactions on Automatic Control, 44, 265–279. Khalil, H. (2002). Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ. Lee, D.B., Nataraj, C., Burg, T.C., and Dawson, D. (2011). Adaptive tracking control of an underactuated aerial vehicle. In Proceedings of the American Control Conference, 2326–2331. San Francisco, CA. Liu, W., Chitour, Y., and Sontag, E. (1996). On finitegain stabilizability of linear systems subject to input saturation. SIAM Journal on Control and Optimization, 34, 1190–1219. Mazenc, F. and Bowong, S. (2004). Backstepping with bounded feedbacks for time-varying systems. SIAM Journal on Control and Optimization, 43, 856–871. Mazenc, F., Burlion, L., and Gibert, V. (2017a). Stabilization of a system that arises in the context of vision based landing of a civil aircraft. Submitted. Mazenc, F. and Iggidr, A. (2004). Backstepping with bounded feedbacks. Systems and Control Letters, 51, 235–245. Mazenc, F. and Malisoff, M. (2016). New control design for bounded backstepping under input delay. Automatica, 6, 48–55. Mazenc, F., Malisoff, M., Burlion, L., and Gibert, V. (2017b). Bounded backstepping through a dynamic extension with delay. In Proceedings of the 56th IEEE Conference on Decision and Control, 607–611. Melbourne, Australia. Mazenc, F., Malisoff, M., and Weston, J. (2016). New bounded backstepping control designs for time-varying systems under converging input converging state conditions. In Proceedings of the 55th IEEE Conference on Decision and Control, 3167–3171. Las Vegas, NV. Mazenc, F., Malisoff, M., and Weston, J. (2018, to appear). Bounded backstepping control and robustness analysis for time-varying systems under converging-inputconverging-state conditions. European Journal of Control. Pettersen, K. and Nijmeijer, H. (2002). Underactuated ship tracking control; theory and experiments. IEEE Transactions on Automatic Control, 40, 1759–1762. Smaoui, M., Brun, X., and Thomasset, D. (2006). A study on tracking position control of an electropneumatic system using backstepping design. Control Engineering Practice, 14, 923–933. Spong, M. (1994). Partial feedback linearization of underactuated mechanical systems. In Proceedings of the IEEE/RSJ/GI International Conference on Intelligent Robots and Systems, 314–321. Munich, Germany. Sussmann, H.J., Sontag, E., and Yang, Y. (1994). A general result on the stabilization of linear systems using bounded controls. IEEE Transactions on Automatic Control, 39(12), 2411–2425. Tsinias, J. (1997). Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation. ESAIM: Control, Optimisation and Calculus of Variations, 2, 57–85.