Nonlinear output feedback H∞-control of mechanical systems under unilateral constraints

Preprints, 1st IFAC Conference on Modelling, Identification and Preprints, 1st IFAC Conference on Modelling, Identification and Preprints, IFAC Modelling, Identification and Control of 1st Nonlinear Systems on Preprints, IFAC Conference Conference Control of 1st Nonlinear Systems on Modelling, Identification and Control of Nonlinear Systems June 24-26, 2015. Saint Petersburg, Russia Available online at www.sciencedirect.com Control of Nonlinear Systems June 24-26, 2015. Saint Petersburg, Russia June June 24-26, 24-26, 2015. 2015. Saint Saint Petersburg, Petersburg, Russia Russia

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IFAC-PapersOnLine 48-11 (2015) 274–279

Nonlinear output feedback H -control of ∞ Nonlinear output feedback H -control of ∞ Nonlinear output feedback H -control of ∞ Nonlinear output feedback H -control of ∞ mechanical systems under unilateral mechanical systems under unilateral mechanical under mechanical systems systems under unilateral unilateral constraints constraints constraints constraints

1,∗∗, ∗ ∗ ∗∗ O.E. Montano 1,∗∗, ∗ Y. Orlov ∗ Y. Aoustin ∗∗ O.E. Montano 1,∗∗, ∗ Y. Orlov ∗ Y. Aoustin ∗∗ 1,∗∗, ∗ Y. Orlov ∗ Y. Aoustin ∗∗ O.E. Montano O.E. Montano Y. Orlov Y. Aoustin ∗ ∗ Department of Electronics and Telecommunications, Center for ∗ Department of Electronics and Telecommunications, Center for ∗ Department of and Center for Scientific Research and Higher Education at Ensenada, Baja Department of Electronics Electronics and Telecommunications, Telecommunications, Center for Scientific Research and Higher Education at Ensenada, Baja Scientific Research and Higher Education at Ensenada, Baja California, Carretera Ensenada-Tijuana No. 3918, Zona Playitas, Scientific Research and Higher Education at Ensenada, Baja California, Carretera Ensenada-Tijuana No. 3918, Zona Playitas, California, Carretera Ensenada-Tijuana No. C.P. 22860, Ensenada, B.C., Mexico. California, Carretera Ensenada-Tijuana No. 3918, 3918, Zona Zona Playitas, Playitas, C.P. 22860, Ensenada, B.C., Mexico. C.P. 22860, Ensenada, B.C., Mexico. [email protected], C.P. 22860, Ensenada, B.C.,[email protected] Mexico. [email protected], [email protected] ∗∗ [email protected], [email protected] de Recherche en Communications et Cybern´eetique ∗∗ L’UNAM, Institut [email protected], [email protected] ∗∗ L’UNAM, Institut de Recherche en Communications et Cybern´ ∗∗ L’UNAM, Institut de Recherche en Communications et eetique tique de Nantes UMR CNRS 6597, CNRS, Universit´ e de Nantes, Ecole L’UNAM, Institut de Recherche en Communications et Cybern´ Cybern´ tique de Nantes UMR CNRS 6597, CNRS, Universit´ e de Nantes, Ecole de Nantes UMR CNRS 6597, CNRS, Universit´ e de Nantes, Ecole Centrale de Nantes, 1 rue de la No¨ e , 44321 Nantes Cedex 3, France. de Nantes UMR CNRS 6597, CNRS, Universit´ e deCedex Nantes, Ecole Centrale de Nantes, 1 rue de la No¨ e , 44321 Nantes 3, France. Centrale de de {oscar.montano, yannick.aoustin}@irccyn.ec-nantes.fr Centrale de Nantes, Nantes, 1 1 rue rue de la la No¨ No¨ee,, 44321 44321 Nantes Nantes Cedex Cedex 3, 3, France. France. {oscar.montano, yannick.aoustin}@irccyn.ec-nantes.fr {oscar.montano, yannick.aoustin}@irccyn.ec-nantes.fr {oscar.montano, yannick.aoustin}@irccyn.ec-nantes.fr

Abstract: The The work work focuses focuses on on the the output output feedback feedback synthesis synthesis of of hybrid hybrid mechanical mechanical systems systems Abstract: Abstract: The work focuses on the output feedback synthesis of hybrid mechanical systems under unilateral constraints. The problem of robust control of mechanical systems is addressed Abstract: The work focuses on the output feedback synthesis of hybrid mechanical systems under unilateral constraints. The problem of robust control of mechanical systems is addressed under constraints. The problem robust of systems is unilateral constraints constraints by designing nonlinear state feedback feedback H∞ -controller in under unilateral constraints.by The problemaa of of robust control control of mechanical mechanical systems developed is addressed addressed -controller developed in under designing nonlinear state H ∞ -controller developed in under unilateral constraints by designing a nonlinear nonlinear state feedback feedback H ∞the the hybrid setting, covering impact phenomena. Performance issues of developed nonlinear -controller developed in under unilateral constraints by designing a state H ∞the developed nonlinear the hybrid setting, covering impact phenomena. Performance issues of the hybrid setting, covering impact phenomena. Performance issues of the developed nonlinear -tracking controller are illustrated with numerical tests on a mass-spring-damper system H ∞ the hybrid setting, covering impact phenomena. Performance issues of the developed nonlinear controller are illustrated with numerical tests on aa mass-spring-damper system H ∞ -tracking controller are H ∞ -tracking impacting against a barrier. controller are illustrated illustrated with with numerical numerical tests tests on on a mass-spring-damper mass-spring-damper system system H ∞ -tracking impacting against a barrier. impacting against a barrier. impacting against a barrier. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: hybrid hybrid systems, systems, robust robust control, control, tracking, tracking, mechanical Keywords: mechanical systems, systems, output output feedback feedback Keywords: hybrid hybrid systems, systems, robust robust control, control, tracking, tracking, mechanical Keywords: mechanical systems, systems, output output feedback feedback 1. INTRODUCTION unilateral constraints possess nonsmooth solutions, which 1. unilateral constraints possess nonsmooth solutions, which 1. INTRODUCTION INTRODUCTION unilateral constraints possess nonsmooth solutions, which arise due to hitting the constraints, a challenging problem 1. INTRODUCTION unilateral constraints possess nonsmooth solutions, which arise due to hitting the constraints, a challenging problem arise due to hitting the constraints, a challenging problem is to extend the popular nonlinear H approach [Orlov Hybrid dynamical systems have been recently a significant ∞ arise due to hitting the constraints, a challenging problem is to extend the popular nonlinear H [Orlov Hybrid dynamical systems have been a significant ∞ approach to extend popular nonlinear H approach [Orlov Hybrid dynamical systems havebasically, been recently recently significant and 2014] to this kind of dynamic systems. Such and attractive attractive research topic, due aato tosignificant the wide wide is is toAguilar, extend the the popular nonlinear H∞ [Orlov Hybrid dynamical systems have been recently ∞ approach and Aguilar, 2014] to this kind of dynamic systems. Such and research topic, basically, due the and Aguilar, 2014] to this kind of dynamic systems. Such and attractive research topic, basically, due to the wide an extension, presented in the IFAC’2014 world congress variety of applications and the complexity that arises and Aguilar, 2014] to this kind of dynamic systems. Such and attractive research topic, basically, due to the arises wide an extension, presented in the IFAC’2014 world congress variety of applications and the complexity that an extension, presented in the IFAC’2014 world congress variety ofanalysis applications and theof complexity complexity that arises by Montano et al. [2014a] for the state feedback synthesis, from the of this type systems. See, e.g., the an extension, presented in the IFAC’2014 world congress variety of applications and the that arises by Montano et al. [2014a] for the state feedback synthesis, from the analysis of type of See, e.g., by Montano et for the state synthesis, from theworks analysis of this this and typeGrizzle of systems. systems. See, e.g.,etthe the now generalized towards the output relevant by Hamed Hamed [2013],See, Goebel al. is by Montano et al. al. [2014a] [2014a] for the state feedback synthesis, from the analysis of this type of systems. e.g., the is now generalized towards the output feedback synthesis, relevant works by and Grizzle [2013], Goebel et al. is now generalized towards the output feedback synthesis, relevant works by Hamed and Grizzle [2013], Goebel et al. and provides theoretical support to the extension of the [2009], and references quoted therein. Particularly, the is now generalized towards the output feedback synthesis, relevant works by Hamed and Grizzle [2013], Goebel et al. and provides theoretical support to the extension of the [2009], and references quoted therein. Particularly, the and provides theoretical support to the extension of the [2009], and references quoted therein. Particularly, the control approach to mechanical systems with uniH disturbance attenuation problem for hybrid dynamical sys∞ and provides theoretical support to the extension of the [2009], and attenuation references quoted therein. Particularly, the H∞ control approach to mechanical systems with unidisturbance problem for hybrid dynamical syscontrol approach to mechanical systems with uniH disturbance attenuation problem for hybrid dynamical sys∞ control lateral constraints, successfully implemented in [Montano tems has been addressed by Haddad et al. [2006], Neˇ s i´ c approach to mechanical systems with uniH disturbance attenuation problem for hybrid dynamical sys∞ lateral constraints, successfully implemented in [Montano tems has been addressed by et Neˇ lateral constraints, successfully implemented [Montano tems has beenwhere addressed by Haddad Haddad et al. al. [2006], [2006], Neˇsssi´ i´ccc et al., 2014b], for the orbital stabilization of aain gait. et al. al. has [2008], impulsive control inputs inputs were admitlateral constraints, successfully implemented inbiped [Montano tems been addressed by Haddad et al. [2006], Neˇ i´ et al., 2014b], for the orbital stabilization of biped gait. et [2008], where impulsive control were admitet al., 2014b], for the orbital stabilization of a biped gait. et al. [2008], where impulsive control inputs were admitThe effectiveness of this approach is additionally illusted to counteract/compensate disturbances/uncertainties et al.,effectiveness 2014b], for the orbital stabilization of a biped illusgait. et al. [2008], where impulsive control inputs were admit- The of this approach is additionally ted to counteract/compensate disturbances/uncertainties The effectiveness of this approach is additionally illusted to counteract/compensate disturbances/uncertainties trated by means of an illustrative application to periodic at time instants of instantaneous changes of the underThe effectiveness of this approach is additionally illustedtime to counteract/compensate disturbances/uncertainties trated by means of an illustrative application to periodic at instants of instantaneous changes of the underby of an application to at time instants of instantaneous instantaneous changes of of theeven undertrajectory tracking mass-spring-barrier lying state. It should should be noted, noted, however, however, that in trated trated by means means of for an aaillustrative illustrative applicationsystem. to periodic periodic at time instants of changes the undertrajectory tracking for mass-spring-barrier system. lying state. It be that even in trajectory tracking for a mass-spring-barrier system. lying state. It should be noted, however, that even in the state feedback design, a pair of independent Riccati trajectory tracking for a mass-spring-barrier system. lying state. It should be noted, however, that even in paper is outlined as follows. Section 22 presents the the state design, a pair of Riccati The the state feedback feedback design, pair continuous of independent independent Riccati paper is outlined as follows. Section the equations, separately comingaa from from and discrete discrete the state feedback design, pair of independent Riccati The The paper is outlined as follows. Section 22 presents presents the hybrid model of interest subject to an unilateral constraint equations, separately coming continuous and The paper is outlined as follows. Section presents the equations, separately coming from continuous and discrete hybrid model of interest subject to an unilateral constraint dynamics, was required to possess a solution that satisfies equations, was separately coming from continuous and satisfies discrete and hybrid model of interest subject to an unilateral constraint the H -control problem is then stated. Section 3 dynamics, required to possess a solution that ∞ -control hybrid model of interest subject to an unilateral constraint dynamics, was required to possess a solution that satisfies and the H problem is then stated. Section 3 both equations, equations, thus yielding yielding aa restrictive on dynamics, was required to possess a solution condition that satisfies and thesufficient H∞ -control problem is then stated. stated. Section 3 ∞ -control derives conditions for a solution of this problem both thus restrictive condition on and the H problem is then Section 3 ∞ both equations, thus yielding a restrictive condition on derives sufficient conditions for a solution of this problem the feasibility feasibility of thus the yielding proposedasynthesis. synthesis. Moreover, the both equations,of restrictiveMoreover, condition the on derives sufficient conditions for aa controller solution of this problem to exist, and an output feedback is synthesized. the the proposed derives sufficient conditions for solution of this problem the feasibility of the the proposed proposed synthesis. Moreover, the exist, and an output feedback controller is synthesized. physical implementation of impulsive impulsive control inputs was was the feasibility of synthesis. Moreover, the to to exist, an output feedback controller is synthesized. Capabilities developed state feedback are physical implementation of control inputs to exist, and and of anthe output feedback controller issynthesis synthesized. physical implementation of impulsive control inputs was Capabilities of the developed state feedback synthesis are impossible in many practical situations. physical implementation of impulsive control inputs was illustrated Capabilities of the developed state feedback synthesis are in Sect. 4 in a numerical study of the tracking of impossible in many practical situations. Capabilitiesinof the 4developed state study feedback synthesis are impossible in in many many practical practical situations. situations. illustrated Sect. in a numerical of the tracking of impossible in Sect. Sect. 44 in in system a numerical numerical study against of the the tracking tracking of aillustrated mass-spring-damper impacting a barrier. Thus motivated, the present investigation introduces a illustrated in a study of of a mass-spring-damper system impacting against a barrier. Thus motivated, the present introduces aa aFinally, mass-spring-damper system impacting against a barrier. Thus motivated, the ensuring present investigation investigation introduces conclusions are presented in Sect. 5. new control strategy, the asymptotic stability of a mass-spring-damper system impacting against a barrier. Thus motivated, the present investigation introduces a Finally, conclusions are presented in Sect. 5. new control ensuring the asymptotic stability of new control strategy, strategy, ensuring the asymptotic stability of Finally, the undisturbed undisturbed hybrid systemthe of interest interest and stability possessing Finally, conclusions conclusions are are presented presented in in Sect. Sect. 5. 5. new control strategy, ensuring asymptotic of the hybrid system of and possessing the undisturbed undisturbed hybrid systemversion of interest interest and possessing -gain of its disturbed to be less than an L 2. PROBLEM STATEMENT 2 the hybrid system of and possessing its disturbed version to be less than an the 2. PROBLEM STATEMENT -gain of of version to the L L22 -gain 2. appropriate attenuation of its its disturbed disturbed versionlevel. to be be less less than than an an the L 2. PROBLEM PROBLEM STATEMENT STATEMENT 2 -gain disturbance appropriate disturbance attenuation level. appropriate disturbance disturbance attenuation attenuation level. level. Given a scalar unilateral constraint F (x ≥ 0, consider appropriate 1 ,, t) The work focuses on impulse hybrid systems, which are Given aa scalar unilateral constraint F (x ≥ 0, consider 1 , t) Given scalar unilateral constraint F (x t) ≥ 0, consider The work focuses on impulse hybrid systems, which are 1 a nonlinear system, evolving within the above constraint, a scalar unilateral constraint Fthe (x1above , t) ≥ 0, consider The on hybrid systems, which are recognized as dynamic systems under unilateral aa nonlinear system, evolving within constraint, The work work focuses focuses on impulse impulse hybrid systems, whichconare Given nonlinear system, evolving within the above constraint, recognized as dynamic systems under unilateral conwhich is governed by continuous dynamics of the a nonlinear system, evolving within the above constraint, recognized as dynamic dynamic systems under unilateral con- which is governed by continuous dynamics of the form straints [Brogliato, 1999]. Since the dynamic systems with form recognized as systems under unilateral conwhich straints [Brogliato, 1999]. Since the dynamic systems with which is is governed governed by by continuous continuous dynamics of of the the form form straints [Brogliato, 1999]. Since the dynamic systems with ˙˙ 1 = x2 dynamics x straints [Brogliato, 1999]. Since the dynamic systems with x x 1 = 2 1 The authors acknowledge the financial support of Campus France (1) ˙ = x x 1 2 1 The authors acknowledge the financial support of Campus France (1) ˙ ˙˙ 2 = Φ(x1 , x2 , t) + Ψ1x(x + Ψ x 1 = 1 ,, x 2 ,, t)w 2 (x 1 ,, x 2 ,, t)u 1 The authors acknowledge the financial support of Campus France (1) = Φ(x , x , t) + Ψ (x x t)w + Ψ (x x t)u x grant Eiffel and CONACYT grant no.165958. 2 1 2 1 1 2 2 1 2 1 (1) ˙x˙ 2 = Φ(x , x , t) + Ψ (x , x , t)w + Ψ (x , x , t)u x TheEiffel authors financial support of Campus France 1 2 1 1 2 2 1 2 grant andacknowledge CONACYTthe grant no.165958. = Φ(x , x , t) + Ψ (x , x , t)w + Ψ (x , x , t)u grant grant Eiffel Eiffel and and CONACYT CONACYT grant grant no.165958. no.165958.

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Copyright © 2015, IFAC 2015 278 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © IFAC 2015 278 Copyright © IFAC 2015 278 Peer review under responsibility of International Federation of Automatic Copyright © IFAC 2015 278Control. 10.1016/j.ifacol.2015.09.197

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MICNON 2015 June 24-26, 2015. Saint Petersburg, RussiaO.E. Montano et al. / IFAC-PapersOnLine 48-11 (2015) 274–279

z = h1 (x1 , x2 , t) + k12 (x1 , x2 , t)u

(2)

y = h2 (x1 , x2 , t) + k21 (x1 , x2 , t)w (3) beyond the surface F (x1 , t) = 0 when the constraint is inactive, and by the algebraic relations − x1 (t+ i ) = x1 (ti ) − − i x2 (t+ i ) = µ0 (x1 (ti ), x2 (ti ), ti ) + ω(x1 (ti ), x2 (ti ), ti )wd (4) + zd i = x2 (ti )

(5)

at a priori unknown collision time instants t = ti , i = 1, 2, . . . , when the system trajectory hits the surface  2n F (x1 , t) = 0. In the above relations, x = [x 1 , x2 ] ∈ R n represents the state vector with components x1 ∈ R and x2 ∈ Rn ; u ∈ Rn is the control input of dimension n; w ∈ Rl and wdi ∈ Rq collect exogenous signals affecting the motion of the system (external forces, including impulsive ones, as well as model imperfections). The variable y ∈ Rp is the only available measurement of the state of the system whereas the variable z ∈ Rs represents a continuous time component of the system output to be controlled whereas the post-impact value of the only state component x2 (t) subjected to the instantaneous change is pre-specified as a discrete component zd i of the tobe-controlled output. The overall system in the closedloop should be dissipative with respect to the output thus specified. Throughout, the functions Φ, Ψ1 , Ψ2 , h1 , k12 , h2 , k21 , F, µ0 , and ω are of appropriate dimensions, which are continuously differentiable in their arguments and uniformly bounded in t. Admitting these functions to be time-varying is particularly invoked to deal with tracking problems where the plant description is given in terms of the state deviation from the reference trajectory to follow. The origin is assumed to be an equilibrium of the unforced system (1)-(5), Φ(0, 0, t) = 0, h1 (0, 0, t) = 0, h2 (0, 0, t) = 0 for all t and µ0 (0, 0, 0) = 0. For later use, consider a causal dynamic feedback controller to be constructed of the same structure ξ˙ 1 = ξ 2 , ξ˙ 2 = η(ξ1 , ξ 2 , y, t) − ξ 1 (t+ j ) = ξ 1 (tj ), u = θ(ξ, t)

− ξ2 (t+ j ) = ν(ξ 1 (tj ), ξ 2 (tj ), tj )

(6)

as that of the plant and with the internal state ξ = [ξ1 , ξ 2 ] ∈ R2n of the same dimension, with the time instants t = ti , i = 1, 2, . . . , which coincide with the impact time instants in the plant equations (1)-(5), and with some piece-wise continuous and uniformly bounded in t functions η(ξ, y, t), ν(ξ, t), and θ(ξ, t) of class C 2 in the variables ξ and y such that η(0, 0, t) = 0, ν(0, t) = 0, and θ(0, t) = 0 for all t. The disturbance attenuation problem of interest consists in finding a local controller (6) (if any) such that the L2 -gain of the disturbed closed-loop system (1)–(6) is less than a certain γ > 0, further referred to as the disturbance attenuation level. In other words, the inequality  T  T NT  2 2 2 2 z(t) dt + zd  ≤ γ w(t) dt+ i t0

t0

i=1

γ2

NT 

i=1

2 wid 

+

NT 

(7)

− βk (x(t− k ), ξ(tk ), tk )

k=0

279

275

should hold for some positive definite functions βk (x, ξ, t), k = 0, . . . , NT , for all segments [t0 , T ], for a natural NT such that tNT ≤ T < tNT +1 , for all piecewise continuous disturbances w(t) and discrete ones wid , i = 1, 2, . . . , for which the state trajectory (x(t), ξ(t)) of the closedloop system, starting from an initial point (x(t0 ), ξ(t0 )) = (x0 , ξ 0 ) ∈ U within some neighborhood U ⊂ R4n of the origin, remains in U for all t ∈ [t0 , T ]. The positive functions βk , k = 1, 2, . . . are included to consider nonzero re-initializations at impact instants. Such a controller is said to be a nonlinear H∞ controller if the closed-loop system is in addition internally asymptotically stable. 3. NONLINEAR H∞ -CONTROL SYNTHESIS For later use, the continuous dynamics (1) are rewritten in the form x˙ = f (x, t) + g1 (x, t)w + g2 (x, t)u (8) whereas the restitution rule is represented as follows − − i x(t+ i ) = µ(x(ti ), ti ) + Ω(x(ti ), ti )wd , i = 1, 2, . . . (9)      with x = [x 1 , x2 ] , f (x, t) = [x2 , Φ (x, t)] , g1 (x, t) =     [0, Ψ1 (x, t)] , g2 (x, t) = [0, Ψ2 (x, t)] , µ(x, t) =    [x 1 , µ0 (x, t)] , and Ω(x, t) = [0, ω(x, t)] . In order to simplify the synthesis to be developed and to provide reasonable expressions for the controller design, the following assumptions h1  k12 = 0, k12  k12 = I, (10) k21 g1  = 0, k21 k21  = I, which are standard in the literature (see, e.g., Orlov [2009]) are made. Relaxing these assumptions is indeed possible, but it would substantially complicate the formulas to be worked out. 3.1 State-space solution Below we list the hypotheses under which a solution to the problem in question is derived. Given γ > 0, in a domain x ∈ Bδ2n , ξ ∈ Bδ2n , t ∈ R, where Bδ2n ∈ R2n is a ball of radius δ > 0, centered around the origin, H1) the √norm of the matrix function ω is upper bounded by 22 γ, i.e., √ 2 γ. (11) ω(x, t) ≤ 2 H2) there exist a smooth, positive definite, decrescent function V (x, t) and a positive definite function R(x) such that if computed along the trajectories of the system (1)-(5) with initial conditions within Bδ2n , for all t ∈ (ti−1 , ti ), i = 1, 2, . . . with t0 being the initial time, and ti the collision time instants of the disturbed system (1)-(5), the Hamilton–Jacobi–Isaacs inequality ∂V ∂V + (f (x, t) + g1 (x, t)α1 + g2 (x, t)α2 ) + ∂t ∂x h1  h1 + α2  α2 − γ 2 α1  α1 ≤ −R(x) (12) holds with   1 ∂V α1 (x, t) = 2 g1 (x, t) 2γ ∂x   1  ∂V α2 (x, t) = − g2 (x, t) ; 2 ∂x

MICNON 2015 276 June 24-26, 2015. Saint Petersburg, RussiaO.E. Montano et al. / IFAC-PapersOnLine 48-11 (2015) 274–279

H3) there exist a continuous uniformly bounded function G(t), and a positive definite function Q(x, ξ) subject to Q(0, ξ) is positive definite, and a smooth, positive semidefinite, decrescent function W (x, ξ, t) subject to W (0, ξ, t) is positive definite, and such that computed along the trajectories of system (1)-(5) with initial conditions within (x(t0 ), ξ(t0 )) ∈ Bδ2n , for all t ∈ (ti−1 , ti ), the Hamilton-Jacobi-Isaacs inequality   ∂W ∂W ∂W + fe (x, ξ, t) + he  he ∂x ∂ξ ∂t (13) −γ 2 ψ  ψ ≤ −Q(x, ξ) holds with fe (x, ξ, t) =     f (x, t) g1 (x, t)α1 (x, t) + g2 (ξ, t)α2 (ξ, t) f (ξ, t) + g1 (ξ, t)α1 (ξ, t)   g2 (x, t)α2 (ξ, t) + , G(t)(h2 (x, t) − h2 (ξ, t))

he (x, ξ, t) = α2 (x, t) − α2 (ξ, t),    ∂W   ∂x 1 ψ(x, ξ, t) = 2 ge  (x, t)     ,  2γ ∂W ∂ξ   g1 (x, t) ; ge (x, t) = G(t)k21 (x, t) H4) Hypotheses H2) and H3) are satisfied with the functions V (x, t) and W (x, ξ, t) which decrease along the direction µ in the sense that the inequalities V (x, t) ≥ V (µ(x, t), t), (14) W (x, ξ, t) ≥ W (µ(x, t), µ(ξ,t), t) hold in the domains of V and W . The first main result of the present work is given below. Theorem 3.1. Consider system (1)-(5), and given γ > 0, suppose Hypotheses H1)-H3) be satisfied in a domain x ∈ Bδ2n , ξ ∈ Bδ2n , t ∈ R. Then, the closed-loop system (1)-(5), driven by the dynamic controller ξ˙ = f (ξ, t) + g1 (ξ, t)α1 (ξ, t) + g2 (ξ, t)α2 (ξ, t)+ G(t)(y(x, t) − h2 (ξ, t)) (15) + − + ξ1 (ti ) = ξ 1 (ti ), ξ 2 (ti ) = µ0 (ξ 1 (ti ), ξ 2 (t− i ), ti ) u = α2 (ξ, t), locally possesses a L2 -gain less than γ. Moreover, the disturbance-free closed-loop system (1)-(5), (15) is uniformly asymptotically stable provided that Hypothesis H4) is satisfied as well. Proof. Since the proof follows the same line of reasoning as that used in Orlov [2009] for the impact-free case here we provide only a sketch. Similar to the proof of [Orlov, 2009, Theorem 7.1], let us consider the function U (x, ξ, t) = V (x, t) + W (x, ξ, t) (16) whose time derivative along the disturbed closed-loop system (1)-(5) between collision time instants is estimated as follows [Orlov, 2009, p.138]: dU ≤ −z(t)2 + γ 2 w2 − R(x) − Q(x, ξ) dt (17) −γ 2 w − α1 (x, t) − ψ(x, ξ, t)2 , t ∈ (tk , tk+1 ), k = 0, 1, . . . . 280

Then integrating (17) from tk to tk+1 , k = 0, 1, . . . , yields  tk+1  tk+1 2 2 2 [γ w − z(t) ]dt ≥ R(x(t))dt + +γ 2





tk tk+1

Q(x(t), ξ(t))dt + tk

tk+1 tk



tk

tk+1

tk

dU (x(t), ξ(t), t) dt dt

w(t) − α1 (x(t), t) − ψ(x(t), ξ(t), t)2 dt > 0.

(18) Manupulating the third term on the right side of (18) according to (16), and skipping the remaining positive terms in the right-hand side of the latter inequality, it follows that  T (γ 2 w2 − z(t)2 )dt ≥ U (x(T ), ξ(T ), T ) t0

+

NT  i=1

+

NT  i=1

+ [V (x(t− i ), ti ) − V (x(ti ), ti )]

(19)

− + + [W (x(t− i ), ξ(ti ), ti ) − W (x(ti ), ξ(ti ), ti )]

−U (x(t0 ), ξ(t0 ), t0 ). Since the functions V and W are smooth by Hypotheses H2) and H3) the following relations + − + V |V (x(t− i ), ti ) − V (x(ti ), ti )| ≤ Li |x(ti ) − x(ti )| − + − |W (x(t− i ), ξ(ti ), ti ) − W (x(ti ), ξ(ti ), ti )| ≤ (20) − + − + LW i [|x(ti ) − x(ti )| + |ξ(ti ) − ξ(ti )|]

hold true with LVi > 0 and LW i > 0 being local Lipschitz constants of V and W in the domain Bδ2n ∈ R2n . Relations (19) and (20), coupled together, result in  T (γ 2 w2 − z(t)2 )dt t0

≥−

NT  i=1

− − W [2(LVi + LW i )x(ti ) + 2Li ξ(ti )]

(21)

−U (x(t0 ), ξ(t0 ), t0 )

Apart from this, inequality NT NT   2 d 2 zi  = x2 (t+ i ) ≤ i=1

2

NT  i=1

i=1

2 µ0 (x(t− i ), ti ) + 2

2

NT  i=1

NT  i=1

d 2 ω(x(t− i ), ti )wi  ≤ (22)

2 2 µ0 (x(t− i ), ti ) + γ

NT  i=1

wid 2

is ensured by Hypothesis H1). Thus, combining (21)-(22), one derives  T NT  2 2 zd z(t) dt + i  ≤ U (x(t0 ), ξ(t0 ), t0 )+ γ

2



t0

i=1

T

t0

2

w(t) dt + +

NT  i=1

NT 

i=1

2 wid 



+2

NT  i=1

2 µ0 (x(t− i ), ti )

− − W [(2LVi + 2LW i )x(ti ) + 2Li ξ(ti )],

(23)

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277

i.e., the disturbance attenuation inequality (7) is established with the positive definite functions β0 (x(t0 ), ξ(t0 ), t0 ) = U (x(t0 ), ξ(t0 ), t0 ), − βi (x(ti ), ξ(ti ), ti ) = (2LVi + 2LW i )x(ti )

(24) − − 2 +2LW i ξ(ti ) + 2µ0 (x(ti ), ti ) , i = 1, . . . , N. To complete the proof it remains to establish the asymptotic stability of the undisturbed version of the closedloop system (1)-(5),(15). In order to do that, we can use [Haddad et al., 2006, Theorem 2.4] specified to the present case with x1 = x and x2 = t. Indeed, according to this result, Hypothesis H4 and the negative definiteness (16) of the time derivative of the Lyapunov function U (x, ξ, t) between the collision time instants ensure that the system is uniformly asymptotically stable. If in addition, Hypotheses H2-H3 hold globally with the radially unbounded function U (x, ξ, t) then the results of the theorem hold globally. Theorem 3.1 is proved.  To circumvent the difficulty of solving the Hamilton– Jacobi–Isaacs PDIs (12), (13) their solutions are further approximated by those to the corresponding Riccati equations that appear in solving the H∞ control problem for the linearized system which is given by x˙ = A(t)x + B1 (t)w + B2 (t)u, (25) (26) z = C1 (t)x + D12 (t)u, y = C2 (t)x + D21 (t)w, (27) within impact-free time intervals (ti−1 , ti ) where t0 is the initial time instant and ti , i = 1, 2, . . . are the collision time instants whereas  ∂f  , B1 (t) = g1 (0, t), A(t) = ∂x x=0  ∂h  B2 (t) = g2 (0, t), C(t) = , D12 (t) = k12 (0, t). ∂x x=0 By the time-varying strict bounded real lemma [Orlov and Aguilar, 2014, p.46], the following conditions are necessary and sufficient for the linear H∞ control problem (25)-(27) to possess a solution: given γ > 0,

C1) there exists a positive constant ε0 such that the differential Riccati equation ˙ ε (t) = Pε (t)A(t) + A (t)Pε (t) + C1  (t)C1 (t) −P 1 +Pε (t)[ 2 B1 B1  − B2 B2  ](t)Pε (t) + εI γ (28) has a uniformly bounded symmetric positive definite solution Pε (t) for each ε ∈ (0, ε0 ); C2) while being coupled to (28), the differential Riccati equation  Z˙ ε (t) = Aε (t)Zε (t) + Zε (t)A ε (t) + B1 (t)B1 (t) 1 +Zε (t)[ 2 Pε B2 B2  Pε − C2  C2 ](t)Zε (t) + εI, γ (29) has an uniformly bounded symmetric positive definite solution Zε (t) with Aε (t) = A(t) + γ12 B1 B1  Pε (t). Our second main result asserts that these conditions, if coupled to a certain monotonicity condition, are also 281

Fig. 1. Mass-spring-damper-barrier system sufficient for a local solution to the nonlinear H∞ control problem to exist under unilateral constraints. Theorem 3.2. Let conditions C1) and C2) be satisfied with some γ > 0. Then Hypotheses H2) and H3) hold locally around the equilibrium (x, ξ) = (0, 0) of the nonlinear system (1)-(5) with V (x, t) = x Pε (t)x (30) ε 2 (31) R(x) = x 2 (32) W (x, ξ, t) = γ 2 (x − ξ) Z−1 ε (t)(x − ξ) ε 2 −1 2 2 Q(x, ξ) = γ min1 Zε (t) x − ξ (33) 2 t∈R (34) G(t) = Zε (t)C2  (t) and the closed-loop system driven by the output feedback ξ˙ = f (ξ, t) + G(t)[y − h2 (ξ, t)]   1 + 2 g1 (ξ, t)g1  (ξ, t) − g2 (ξ, t)g2  (ξ, t) Pε (t)ξ γ (35) − + − ξ 1 (t+ ) = ξ (t ), ξ (t ) = µ (ξ (t ), ξ (t ), t ) i 1 i 2 i 0 1 i 2 i i 

u = −g2 (ξ, t) Pε (t)ξ locally possesses a L2 -gain less than γ provided that Hypothesis H1 holds as well. Moreover, the disturbance-free closed-loop system (1)-(5), (35) is uniformly asymptotically stable provided that Hypothesis H4) is satisfied with the quadratic functions (30) and (32). Proof. Due to [Orlov and Aguilar, 2014, Theorem 24], Hypotheses H2) and H3) locally hold with (30)-(34). Then by applying Theorem 3.1, the validity of Theorem 3.2 is concluded. 4. ILLUSTRATIVE EXAMPLE: PERIODIC POSITION FEEDBACK TRACKING OF A MASS-SPRING-DAMPER-BARRIER SYSTEM The objective of this section is to demonstrate the effectiveness of the proposed control synthesis on a simple model of a mass-spring-damper-barrier system, that captures all the essential features of the general treatment. 4.1 Mass-spring-damper-barrier model Theorem 3.2 will be applied to a simple mass-springdamper-barrier system depicted in Fig. 1 where m represents the mass, k the spring constant, b the damping constant, τ is the applied control force, and q represents the position. The objective is to follow a periodic motion q r (t) that bounces against the wall located at q = 0, considering that only the position is available for measurements. For the free-motion dynamics (q > 0), the plant equation reads b 1 1 k q¨ = − q − q˙ + τ + w1 (36) m m m m

MICNON 2015 278 June 24-26, 2015. Saint Petersburg, RussiaO.E. Montano et al. / IFAC-PapersOnLine 48-11 (2015) 274–279

whereas for the transition phase (q = 0), one has the restitution rule q + = q − , q˙+ = −eq˙− + wid (37)

− The notation f + (f − ) is equivalent to f (t+ i ) (f (ti )). d The variables w1 and wi were introduced to account for model inadequacies and non-modeled external forces such as friction. Since the objective is to track a reference trajectory q r , let’s define the tracking error variables x1 = q − q r , and x2 = q˙ − q˙r . Also let us define the pre-feedback control (38) τ = m¨ q r + kq r + bq˙r + u, composed of a controller a trajectory compensator, and the disturbance attenuator u to be designed using the results of Section 3.1. Then, setting x = (x1 , x2 ) , w = (w1 , w2 ) , where w2 represents the measurement disturbance, and rewriting the system (36)-(38) in terms of the tracking error variables, one derives:

Free-motion phase error system (F (x, t) = x1 + q r (t) > 0)       0 1 0 0 0 b x+ 1 k w+ 1 u (39) x˙ = 0 − − m   m     m m  B1

A

B2

Transition phase error system (F (x, t) = x1 + q r (t) = 0)     1 0 0 + − x = x + wid . (40) 0 −e 1     µ0

ω

zid

to be controlled and the Finally, the variables z, measured variable y are specified with     1 0 0 d (41) z = ρp 0 x + 0 u, zid = −ex− 2 + wi 0 ρv 0       C1

D12

y = [1 0]x + [0 1]w       C2

(42)

D21

thus focusing the present investigation on the position measurements. 4.2 Reference Trajectory In simulation runs, the periodic trajectory to follow was generated by a Van der Pol oscillator bouncing against a surface with a restitution coefficient of 0.5. The model used was: Free-motion phase (q r > 0) q¨r = c(1 − q r 2 )q˙r − q r Transition phase (q r = 0)

(43)

r − r − q r (t+ q˙r (t+ (44) i ) = −eq˙ (ti ) i ) = q (ti ), r r where q represented the desired position and q˙ the velocity. The parameters used for this oscillator were c = 1, e = 0.5, q r (0) = 0 and q˙r (0) = 1.0126. A numerical analysis of the Poincar´e map as suggested by Grizzle et al. [1999], was made in order to ensure that this reference system generated a locally asymptotically stable hybrid periodic orbit.

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Fig. 2. Hybrid Van der Pol Oscillator used as reference trajectory. Left: Position and velocity profiles. Right: Generated periodic orbit. 4.3 Numerical results In order to ensure robust tracking of the desired trajectory, governed by (43)-(44), Theorem 3.2 was applied to the mass-spring-damper-barrier error system (39)(42)driven by the control force (38). The linearizing terms A, B1 , B2 , C1 , C2 , which were required to specify the Riccati equations (28)-(29) and which turned out to be time-invariant, were identified from (39)-(42). A constant positive semidefinite solution of the corresponding timeinvariant system (28)-(29) subject to ε = 0 was then obtained by iterating on γ in order to approach the infimal achievable level γmin ≈ 1.01. The value γ = 2 was however selected to avoid an undesirable high-gain controller design that would appear for a value of γ close to the optimum γmin ≈ 1.01. With γ = 2, the value ε = 0.01 was obtained so the corresponding perturbed Riccati equations (28)(29) possessed constant positive definite solutions. Since the unit mass was used in the simulations, this value of γ = 2 was straightforwardly verified to meet Hypothesis 1 H1) with ω = m = 1, corresponding to the present investigation. In addition, Hypothesis H4) was numerically verified in the simulations. Thus, Theorem 3.2 proved to be applicable to the controller design (35) of robust tracking of the reference output q r (t). Table 1. Simulation parameters Param k b m e ρp q(to ) ξ1 (to )

Value 10 N/m 1 N/m/s 1 kg 0.5 1 0.2 m 0m

Param ρv ε wid w1 w2 q(t ˙ 0) ξ2 (t0 )

Value 1 0.01 0.2q2 m/s 0.1q2 + 0.1sign(q2 ) N 0.1 sin(1.5t) m 0.8 m/s 0 m/s

The simulation results shown in Figs. 3-5 were performed using the parameters from Table 1. From Figs. 3 and 4, a good performance is concluded for the periodic tracking synthesis despite the disturbances affecting the freemotion (friction) and transition phases (imperfect knowledge of the restitution coefficient). Figure 4 shows the peaking phenomena reported by Biemond et al. [2013]; this phenomena occurs in both the disturbed and undisturbed cases due to the asymptotic nature of the synthesized controller, so the plant dynamics does not reach the reference velocity before the impact and the plant velocity jumps do not match the reference velocity jumps. Despite this difference, the performance shown by Fig. 3 shows a good

MICNON 2015 June 24-26, 2015. Saint Petersburg, RussiaO.E. Montano et al. / IFAC-PapersOnLine 48-11 (2015) 274–279

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5. CONCLUSION

Fig. 3. Periodic trajectory tracking for the disturbance attenuation level γ = 2. 2 Plant Velocity Ref. Velocity

q˙ [cm/s]

1 0 −1 −2

x2 = q˙ − q˙r [cm/s]

−3

0

5

0

5

10

15

10

15

3 2 1

In this paper, the output feedback H∞ -control problem for orbital stabilization of fully actuated mechanical systems subject to unilateral constraints is solved. Sufficient conditions for the existence of a local solution of the tracking problem in question are carried out in terms of three coupled inequalities: two standard Hamilton-Jacobi-Isaacs inequality (perturbed differential Riccati equations) for the continuous dynamics, coming from the state-feedback and output injection designs respectively, and a novel inequality, imposed on the corresponding solutions of the Hamilton-Jacobi-Isaacs inequality (perturbed differential Riccati equations) at the impact time instants. The proposed robust synthesis constitutes the contribution of the paper to the existing literature. Effectiveness of the resulting design procedure is supported by numerical tests made for a mass-spring-damper system, impacting against a barrier, to track a periodic reference trajectory, generated by a nonlinear Van der Pol oscillator subject to an unilateral constraint. The desired disturbance attenuation is obtained in the presence of both disturbances in the free motion phase and uncertainties in the impact phase.

0

−1 −2

REFERENCES

t [sec]

Fig. 4. Velocity error for the disturbed system. The peaks shown are due to the des-synchronization of the reference trajectory with the plant trajectory.

Fig. 5. Behavior of the velocity filter . behavior. It should be noted that due to the asymptotic convergence of the closed loop system to the reference trajectory, the so-called Zeno behavior (accumulation of an infinite number of impacts at a finite instant) is not presented in the resulting dynamics. Finally, Fig. 5 shows the velocity behavior of the filter both in the presence of disturbances in the continuous dynamics of the plant and in the measurements, and under uncertainties in the restitution law. The observed velocity q˙obs := ξ2 + q˙r and the observation error x ˆ2 := x2 − ξ2 for the disturbed and undisturbed case are compared. If disturbances are not applied, the filter perfectly tracks the system velocity, whereas in the disturbed case, a reasonably small observation error persists so that a good tracking performance is achieved, as is in Fig. 3. 283

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