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Extremum Seeking with Drift∗
Preprints, 1st IFAC Conference on Modelling, Identification and Preprints, IFAC Control of 1st Nonlinear Systems on Preprints, 1st IFAC Conference Conference on Modelling, Modelling, Identification Identification and and Preprints, 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems Available online at www.sciencedirect.com June 24-26, 2015. Saint Petersburg, Russia Control of Nonlinear Systems 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) 126–130
Extremum Extremum Extremum
Seeking Seeking Seeking
with with with
Drift Drift Drift
Jan Maximilian Montenbruck, Hans-Bernd D¨ urr, Christian Ebenbauer, Frank Allg¨ ower Jan Maximilian Montenbruck, Hans-Bernd D¨ u rr, Christian Ebenbauer, Frank Allg¨ o wer Jan Maximilian Montenbruck, Hans-Bernd D¨ u rr, Christian Ebenbauer, Frank Allg¨ o Jan Maximilian Montenbruck, Hans-Bernd D¨ urr, Christian Ebenbauer, Frank Allg¨ ower wer Institute for Systems Theory and Automatic Control, University of Stuttgart Institute for Systems Systems Theory and Automatic Control, University of Stuttgart Stuttgart Institute Theory Control, University Pfaffenwaldring 9, Automatic 70550 Stuttgart, Germany Institute for for Systems Theory and and Automatic Control, University of of Stuttgart Pfaffenwaldring 9, 70550 Stuttgart, Germany Pfaffenwaldring mailto:[email protected] Pfaffenwaldring 9, 9, 70550 70550 Stuttgart, Stuttgart, Germany Germany mailto:[email protected] mailto:[email protected] mailto:[email protected] Abstract: We study the convergence properties of extremum seeking controllers when a drift Abstract: We study the convergence properties of such extremum seeking controllers when drift Abstract: We properties of extremum controllers when aaa drift vector field appears in the the convergence closed loop. To cope with issues,seeking we propose a tuning procedure Abstract: We study study properties of such extremum seeking controllers when drift vector field appears appears in the the convergence closed loop. To To cope with issues, we propose a tuning tuning procedure vector field in the closed loop. cope with such issues, we propose a procedure that admits for guaranteeing convergence arbitrarily close to the desired minima despite drift. vector field appears in the closed loop. To cope with such propose a tuning procedure that admits admits for guaranteeing guaranteeing convergence arbitrarily closeissues, to the thewe desired minima despite drift. that for convergence arbitrarily close to desired minima despite drift. that admits for guaranteeing convergence arbitrarily close to the desired minima despite drift. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: extremum seeking; drift; practical stability Keywords: extremum seeking; drift; practical stability Keywords: extremum extremum seeking; seeking; drift; drift; practical practical stability stability Keywords: 1. INTRODUCTION In particular, we study the convergence properties of (4) 1. INTRODUCTION In the convergence properties of (4) 1. INTRODUCTION INTRODUCTION In particular, we study the properties of by particular, introducingwe thestudy auxiliary gradient system with drift 1. In particular, we study the convergence convergence properties of (4) (4) by introducing the auxiliary gradient system with drift by introducing the auxiliary gradient system with drift Extremum seeking controllers are feedbacks of the form by introducingz˙the auxiliary gradient system with drift(5) = f (z) − k∇P (z) =: Z (z) Extremum seeking controllers are feedbacks of the form z˙ = f (z) − k∇P (z) =: Z (z) (5) Extremum seeking seeking controllers controllers are feedbacks of the the form form(1) n = =: Z (5) Extremum feedbacks of →R , uω : R × Rare = ff (z) (z) − − k∇P k∇P (z) =:order Z (z) (z)to let solutions (5) via first findingzz˙˙ sufficiently large(z) k in n n, : R × R → R (1) u ω via first finding sufficiently large k in order to let solutions n, u : R × R → R (1) ω via first finding sufficiently large k in order to let solutions of (5) approach a neighborhood of the minima of P , such × R → R parameterized , (1) via first finding sufficiently large k in order to let solutions uω : Rargument, oscillatory in their second by the (5) approach neighborhood of the minima of P ,, such oscillatory their second parameterized by the of of (5) approach neighborhood of the minima of it was done aaaby Montenbruck (2015), and by oscillatory in their second argument, parameterized by frequency ωin (0, ∞), thatargument, have the purpose of steering of (5) approach neighborhood of et theal. minima of P P , such such oscillatory in∈ their second argument, parameterized by the the as as it was done by Montenbruck et al. (2015), and by frequency ω ∈ (0, ∞), that have the purpose of steering the as it was done by Montenbruck et al. (2015), and by then deriving bounds on the proximity of solutions of (4) frequency ω ∈ (0, ∞), that have the purpose of steering the solutions of the system it deriving was done by Montenbruck et al.of (2015), and by frequencyof ω the ∈ (0,system ∞), that have the purpose of steering the as then bounds on the proximity solutions of (4) solutions then deriving bounds on the proximity of solutions of (4) to solutions of (5) via classical extremum seeking, thus solutions of the system then deriving bounds on the proximity of solutions of (4) solutions of the system x˙ = u (P (x) , t) (2) to solutions of (5) via classical extremum seeking, thus to solutions of classical extremum seeking, thus pursuing a two-step procedure. lets us bring systems x˙˙ = u (P (x) ,, t) (2) solutions of (5) (5) via via classicalThis extremum seeking, thus = (2) pursuing aaarbitrarily two-step procedure. This lets us bring systems x˙minima =u u (P (P (x) (x) , t) t)unknown function P (2): to arbitrarily close to thex of the pursuing two-step procedure. This lets us bring systems with drift close to the minima of a function P pursuing a two-step procedure. This lets us bring systems arbitrarily close to the minima of the unknown function P : n arbitrarily close to the of the unknown function P :: with drift arbitrarily close to the minima of aa function P Rn → R, only with theminima information provided by the values with drift arbitrarily close to the minima of function P arbitrarily close to the minima of the unknown function P only via knowledge of the values of P . with drift arbitrarily close to the minima of a function P R R, only the values R → R,choosing only with with the information information provided by the values via knowledge of the values of P .. ofnnP→ , by sufficiently large ωprovided (we refer by thethe reader to only only via knowledge of the values of P R → R, only with the information provided by the values only via knowledge of the values of P . of P , byand choosing ω (we refer the reader to the manuscript, we assume the twice continof P choosing sufficiently large ω the reader to Ariyur Krsti´csufficiently (2003) for large an introduction topic). of P ,, by byand choosing sufficiently large ω (we (we refer refer to thethe reader to Throughout Throughout the assume continn Ariyur Krsti´ ccextremum (2003) for an introduction to the topic). Throughout the manuscript, manuscript, we assumePthe the twice contingiven uously differentiable potentialwe function : Rtwice Ariyur and Krsti´ (2003) for an introduction to the topic). One challenge in seeking is to establish stability Throughout the manuscript, we assume the twice continn → R Ariyur and Krsti´ cextremum (2003) forseeking an introduction to thestability topic). uously differentiable n → R given function P : R n potential One challenge in is to establish n → R given uously differentiable potential function P : R is an asymptotically stable invariant such that M ⊂ R One challenge in extremum seeking is to establish stability properties of (2), such as practical stability (cf. Tan et al. → R given uously differentiable potential function P : R n One challenge in extremum seeking stability is to establish stability n is an asymptotically stable invariant such that M ⊂ R properties of (2), such as practical (cf. Tan et al. n an asymptotically stable invariant such that M ⊂ R set of (3). Our goal is to consequently find a function (1) properties of (2), such as practical practical stability (cf.approaches Tan et et al. al. such that M ⊂ R is an asymptotically stable invariant (2006) or D¨ urr(2), et al. (2013)). The idea in these properties of such as stability (cf. Tan set of (3). Our goal is to consequently find aa“close” function (1) (2006) or D¨ u rr et al. (2013)). The idea in these approaches set of (3). Our goal is to consequently find function (1) such that one can bring the solutions of (4) to M (2006) or D¨ u rr et al. (2013)). The idea in these approaches is to choose solutions of (2) approximated ofthat (3).one Ourcan goal is tothe consequently find a“close” function (1), ω such (2006) or D¨ uu rr et al.that (2013)). The idea inare these approaches set such bring solutions of (4) to M is to choose u such that solutions of (2) are approximated such that one can bring bring the solutions of (4) (4) “close” “close” to M M ,,, ω as t → ∞, for some given twice continuously differentiable is to choose u such that solutions of (2) are approximated by solutions of the associated gradient system ω such that one can the solutions of to is tosolutions choose uof that solutions of (2)system are approximated as t → ∞, for some given ω such twice continuously differentiable differentiable by the associated gradient as tt → ∞, some drift vector field f : given Rnn →twice Rn . continuously by associated → ∞, for for some twice continuously differentiable by solutions solutions of of the they˙ = associated gradient system −∇P (y)gradient =: Y (y)system . (3) as n → Rn n. drift vector field ff :: given R n n drift vector field R → R . yy˙˙ = −∇P (y) =: Y (y) . (3) Notation. drift vector field → RRn ,. we mean the unique vector =: ∇Pf ::RRnn → y˙ = = −∇P −∇P (y) =: Y Y (y) (y)of.. how to choose (3) (3) In this paper, we address the(y) question uω Notation. By n n → Rn By ∇P :: R , we mean the unique vector In this paper, we address the question of how to choose u n Notation. By ∇P R → field which satisfies ω In this paper, we question of how to choose u if (2) assumes theaddress form the Notation. By ∇P : R → R Rn ,, we we mean mean the the unique unique vector vector ω In this paper, we address the question of how to choose u ω field which satisfies if (2) assumes the form field which satisfies ∇P (x + hv) − P (x) if (2) (2) assumes assumes the form field which satisfies if the form x˙ = f (x) + u (P (x) , t) =: X (x, t) , (4) ∇P (x) · v = lim ∇P (6) (x + hv) − P (x) ∇P x˙˙ = f (x) + u (P (x) , t) =: X (x, t) ,, (4) h→0 ∇P (x (x + + hv) hv) −P P (x) (x) h − ∇P (x) ·· vv = lim (6) = + =: (x, t) (4) n ,, t) n X ∇P (x) = lim (6) x˙vector = ff (x) (x) +u uf (P (P (x) t) =: X (x, t) , (4) i.e. if a driftx field : R(x) → R appears in the closed ∇P (x) · v = lim (6) h→0 h n h→0 n h , where “·” denotes the dot product. for any x and v ∈ R n → Rn n appears in the closed i.e. if a drift vector field f : R h→0 h n n →solutions n appears i.e. if a drift vector field f : R R in the closed loop, but one still wants to bring of (4) arbitrarily n , where “·” denotes the dot product. for any x and v ∈ R i.e. if but a drift field fto:bring R →solutions R appears in arbitrarily the closed for “·” denotes x R We denote thevv ∈ solution of (4) atdot x0 product. by ϕx : loop, onevector still wants wants of later, (4) where “·” initialized denotes the the dot product. for any any x and and ∈ Rn ,, where loop, one still solutions (4) close but to the of to P .bring As we will seeof this has We denote the solution of (4) initialized at x by ϕxxby:: loop, but one minima still wants to bring solutions of later, (4) arbitrarily arbitrarily 0 We denote the solution of (4) initialized at x by (x , t) → ϕ (x , t), the solution of (3) initialized at 0 0 x 0 0x : close to the minima of P . As we will see this has We, t) denote the solution of (4) initialized at x0 at by yyϕ ϕ close to the minima of .. As we will see this has potential application inP problems where flater, is unknown, (x → ϕ (x , t), the solution of (3) initialized close to the minima of P As we will see later, this has 0 x 0 (x t) → ϕx→(x (xϕ0y,, t), t), the solution of (3) initialized initialized at yy00 by by ϕ (y , t) (y , t), and the solution of (5) initialized 0 ,,: t) y 0 0 potential application in problems where f is unknown, (x → ϕ the solution of (3) at 0 : (y , t) x 0 (y , t), and the solution of (5) initialized 0 by potential application in where unknown, for instance if it is subject to a parametric ϕ →: ϕ ϕ(z potential application in problems problems where ff is isuncertainty unknown, ϕ y 0 y 0 → and the solution of (5) initialized at by ϕ , t) → ϕ (z , t). For a function such as y z:: 0(y 0 ,, t) y0(y 0 ,, t), z z 0 for instance if it is subject to a parametric uncertainty ϕ (y t) → ϕ (y t), and the solution of (5) initialized 0 ϕ : (z y t) 0 → ϕ (z , t). For a function for instance if a parametric uncertainty aty z0 nby (“robust” extremum seeking).to such for instance if it it is is subject subject z z 0 t) → ϕ For aa function such as P :zz00Rnby →ϕ we000 ,,,denote its sublevel sets by UPα = {x as ∈ zR,:: (z z (z 0 ,, t). (“robust” extremum seeking).to a parametric uncertainty at at by ϕ (z t) → ϕ (z t). For function such as α z z 0 (“robust” extremum seeking). n → R, we denote its sublevel setsnby U α = {x ∈ P n:: R (“robust” extremum seeking). P n → α = {xits P R ∈ R, we denote its sublevel sets by U |P (x) ≤ α}. For a set such as M ⊂ R , we denote The problem statement resembles the stabilizing ex- R P P nn:|PR(x)→≤R, weFor denote its sublevel sets by U denote = {xits ∈ n n n P R α}. aa set such ⊂ R M The resembles the exn |P (x) ≤ neighborhood n ,,Rwe R α}. For set such as M ⊂ R we denote its The problem problem statement resembles the stabilizing stabilizing ex- equidistant by Uas = {x ∈ |d (x, M ) ≤ tremum seekingstatement problem for input-affine systems posed R |P (x) ≤ α}. For a set such as M ⊂ R , we denote its M The problem statement resembles the stabilizing ex n n equidistant neighborhood by UM {x ∈ R |d (x, M )) to ≤ tremum seeking for input-affine systems posed = neighborhood by = R M ≤ tremum seeking problem for systems posed }, where, here, d is the infimal Euclidean of by Scheinker and problem Krsti´c (2013a), who proposed a solution equidistant neighborhood by U UM = {x {x ∈ ∈distance Rn |d |d (x, (x, Mx ) to ≤ tremum seeking problem for input-affine input-affine systems posed equidistant M }, where, here, d is the infimal Euclidean distance x n n of by Scheinker and Krsti´ cc (2013a), who proposed a solution }, where, here, d is the infimal Euclidean distance of x to by Scheinker and Krsti´ (2013a), who proposed a solution all points in M . Given a vector field f : R → R and a based on control Lyapunov functions and persistence-of }, where, here, d is the infimal Euclidean distance of x to by Scheinker and Krsti´ c (2013a), who proposed a solution n n n → Rn and a all points in M .. Given aa: vector field ff Lie :: R n based on control Lyapunov functions and persistence-ofn → Rn and all points in M Given vector field R a based on control Lyapunov functions and persistence-ofdifferentiable function P R → R, the derivative of P excitation-type conditions on the control vector fields. Yet, all points in M . Given a: vector field : Rderivative → R and based on control Lyapunov functions and persistence-ofn differentiable function PR, R →∇P R, thef·Lie Lie of P Pa excitation-type conditions on control vector fields. differentiable :: R R, the derivative of excitation-type conditionsand on the the control vectorwe fields. Yet, f is Lf Pfunction : Rnn →P x nn→→ (x) f (x). Throughout in contrast to Scheinker Krsti´ c (2013a), will Yet, not along differentiable function P R → R, the Lie derivative of P excitation-type conditions on the control vector fields. Yet, n → R, x → ∇P (x) · f (x). Throughout along ff is LUff P :: R in contrast Scheinker cc (2013a), we willofnot n → along x (x). Throughout in contrast to Scheinker and Krsti´ we the paper, a R, neighborhood and whenever assume thatto to and (5) Krsti´ approach the minima P along f is is L LUf P Pwill :R Rbe → x → → ∇P ∇P (x) (x)of ·· ffM (x). Throughout in contrast tosolutions Scheinker and Krsti´ c (2013a), (2013a), we will willofnot not the paper, will be aa R, neighborhood of M and whenever assume that solutions to (5) approach the minima P the paper, U will be neighborhood of M and whenever assume that solutions to (5) approach the minima of P we write int U , we refer to the interior of U whilst with exactly. the write paper,int UUwill berefer a neighborhood of M and whenever assume that solutions to (5) approach the minima of P we , we to the interior of U whilst with exactly. we write int U , we refer to the interior of U whilst with exactly. ∂U , we mean its boundary. For two vector fields f , g, we ,write int U ,its weboundary. refer to the interior of U whilst with exactly. All authors thank the German Research Foundation (DFG) for ∂U we mean For two vector fields f ,, g, ∂U , we mean its boundary. For two vector fields f g, [f, g] denotes the Lie bracketFor of vector fields.fields We adopt ∂U , we mean its boundary. two vector f , g, All authors thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in All authors thank the German Research Foundation (DFG) for [f, g] denotes the Lie bracket of vector fields. We adopt All authors thank the German Research Foundation (DFG) for [f, g] denotes the Lie bracket of vector fields. We adopt terminology and results from Bhatia and Szeg˝ o (1970). financial of project within of in [f, g] denotesand the Lie bracket of vectorand fields. We adopt Simulation Technology 310/2) at the the Cluster University of Stuttgart. financial support support of the the(EXC project within the Cluster of Excellence Excellence in terminology from Szeg˝ oo (1970). financial support of the project within the Cluster of Excellence in terminology and results from Bhatia Bhatia and and Szeg˝ (1970). Although weand dealresults with time-dependent vector fields, we Simulation Technology (EXC 310/2) at the University of Stuttgart. terminology results from Bhatia Szeg˝ o (1970). Hans-Bernd D¨ u rr and Christian Ebenbauer are additionally supSimulation Technology (EXC 310/2) at the University of Stuttgart. Simulation Technology (EXC 310/2) at the University of Stuttgart. Although we deal with time-dependent vector fields, we Although we deal with time-dependent vector fields, we Hans-Bernd D¨ u rr and Christian Ebenbauer are additionally supomit the dependence on the initial time t in the solutions 0 ported by the German Research Foundation (DFG) within the Hans-Bernd D¨ u rr and Christian Ebenbauer are additionally supAlthough we deal with time-dependent vector fields, we Hans-Bernd D¨ urr and Christian Ebenbauer are(DFG) additionally supomit the dependence on the initial time t in the solutions 0 ported by the German Research Foundation within the omit the dependence on the initial time t in the solutions of the associated differential equations due to the fact that 0 in Emmy-Noether-Grant “Novel Ways in Control and Computation” ported by the German Research Foundation (DFG) within the omit the dependence on the initial time t the solutions 0 to the fact that ported by the German Research (DFG) within the of the associated differential due Emmy-Noether-Grant “Novel WaysFoundation in Control and Computation” of the associated differential equations due the fact (EB 425/2-1). all results hold uniformly in equations t0 (cf. D¨ urr etto (2013)). Emmy-Noether-Grant “Novel of the associated differential equations due toal. the fact that that Emmy-Noether-Grant “Novel Ways Ways in in Control Control and and Computation” Computation” (EB 425/2-1). all results hold uniformly in t (cf. D¨ u rr et al. (2013)). 0 (EB 425/2-1). all results hold uniformly in t (cf. D¨ u rr et al. (2013)). 0 (EB 425/2-1). all results hold uniformly in t0 (cf. D¨ urr et al. (2013)). Copyright IFAC 2015 130 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 130 Copyright © IFAC 2015 130 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 130Control. 10.1016/j.ifacol.2015.09.171
MICNON 2015 June 24-26, 2015. Saint Petersburg,Jan Maximilian Montenbruck et al. / IFAC-PapersOnLine 48-11 (2015) 126–130 Russia
Structure of the Manuscript. We introduce all terminology and auxiliary results that we require in section 2. Thereafter, i.e. in section 3, we present our main results on how to establish convergence guarantees despite drift. We illustrate these results on the example of practical stabilization of the unit sphere despite drift in section 4. Section 5 concludes the manuscript. 2. PRELIMINARIES The solution that we propose to solve the issue posed in section 1 involves two main ingredients: gradient systems and extremum seeking. We first review some results on gradient systems and consequently repeat the fundamentals of extremum seeking. A key ingredient will be the positive definiteness of P with respect to the desired set M . Definition 1. A continuously differentiable function P : Rn → R is said to be positive definite with respect to M on U , if U ⊂ Rn is a neighborhood of M ⊂ Rn , P is positive on U \ M , zero on M , regular on U \ M , and critical on M . Having this definition at hand, we repeat a fundamental result about gradient systems, that here only serves the purpose of giving the intuition behind the fact that solutions of (5) approach a neighborhood of M . Proposition 2. (cf. (Hirsch et al., 2004, Sections 9.2f) ). If P is positive definite with respect to M on U and M is compact, then M is an asymptotically stable invariant set of (3) and for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, UPα is a subset of the region of attraction of M. Now, guided by the intuition from perturbation theory (cf. Brauer (1966)), we know that solutions of (5) must stay close to solutions of (3) and thus approach a neighborhood of M whose size can be rendered arbitrarily small by appropriate choice of k. This technique was proposed by Montenbruck et al. (2015). Classically, perturbation theory assumes f = constant, whereas, herein, f is allowed to be a vector field. Lemma 3. If P is positive definite with respect to M on U , M is compact, and f is continuous on U , then, for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), UM contains an asymptotically stable invariant set of (5) which is also a uniform attractor, and whose region of attraction is a superset of UPα . Proof. The Lie derivative of P along Z is given by LZ P (z) = ∇P (z) · f (z) − k∇P (z) · ∇P (z). Choose any α ∈ (0, ∞) such that UPα is compact and UPα ⊂ U . As P is positive definite with respect to M on U , for every α ∈ (0, ∞) such that UPα is compact and UPα ⊂ U , for any ∈ (0, d (M, ∂UPα )), there exists δ ∈ (0, α) such that UPδ is a subset of UM . It is then true that UPα \ int UPδ is a compact, nonempty set. As f is continuous and P is continuously differentiable, ∇P · f assumes its maximum on UPα \ int UPδ , which we denote by fδα . It follows that for all z ∈ UPα \ int UPδ , LZ P (z) ≤ fδα − k∇P (z) · ∇P (z). As P is continuously differentiable and positive definite with respect to M on U , for every α ∈ (0, ∞) such that UPα is compact and UPα ⊂ U , ∇P (z) · ∇P (z) assumes 131
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its positive minimum on UPα \ int UPδ , which we denote α δ by pα ≤ fδα − kpα δ. δ . It follows that LZ P UP \ int UP α α Setting kα0 = fδ δ/p δ , we have that for any k ∈ (k0 , δ∞), LZ P UP \ int UP < 0, letting us conclude that UP is an invariant set of (5). Now define a function as being P − δ outside UPδ and to be zero inside UPδ . This function is continuous and its Lie derivative along Z outside UPδ equals LZ P . By Lyapunov’s direct method, it follows that UPδ is an asymptotically set of (5). stable invariant Moreover, as we have LZ P UPα \ int UPδ < 0, we know that UPα is an invariant set. It follows from LaSalle’s invariance principle that UPα is a subset of the region of attraction of UPδ . This concludes the proof. We now repeat some fundamental concepts of extremum seeking, mostly taken from D¨ urr et al. (2013). For doing so, define ξ˙ = b0 (ξ) +
m
√ ωvj (ωt)
(7)
[bj , bk ] (ζ) ηkj
(8)
bj (ξ)
j=1
and ζ˙ = b0 (ζ) +
m
j=1 k=j+1
with
θ 1 T vk (θ) vj (τ ) d τ d θ. (9) T 0 0 Here and henceforth, let ϕξ : (ξ0 , t) → ϕξ (ξ0 , t) denote the solution of (7) initialized at ξ0 and ϕζ : (ζ0 , t) → ϕζ (ζ0 , t) denote the solution of (8) initialized at ζ0 . With these auxiliary systems, we repeat two basic results in extremum seeking. Lemma 4. ((D¨ urr et al., 2013, Theorem 1)). For all i, let vi be T -periodic with zero average. For all i, let bi be twice continuously differentiable. If there exists B ⊂ Rn such that there exists κ ∈ (0, ∞) such that for all ζ0 ∈ B, for all t ∈ [0, ∞), ϕζ (ζ0 , t) < κ, then for every bounded K ⊂ B, for every D ∈ (0, ∞), for every tf ∈ (0, ∞), there exists ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for every ζ0 ∈ K, for all t ∈ [0, tf ], d (ϕζ (ζ0 , t) , ϕξ (ζ0 , t)) < D. Definition 5. A set S is said to be an ω-practically uniformly asymptotically stable set of (7), if, for every ∈ (0, ∞), there exists δ ∈ (0, ∞) and ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for all t ∈ [0, ∞), for all ξ0 ∈ USδ , ϕξ (ξ0 , t) ∈ US , and if there exists δ ∈ (0, ∞) such that for every ∈ (0, ∞), there exists tf ∈ [0, ∞) and ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for all t ∈ [tf , ∞), for all ξ0 ∈ USδ , ϕξ (ξ0 , t) ∈ US . Lemma 6. For all i, let vi be T -periodic with zero average. For all i, let bi be twice continuously differentiable. If a compact set S is an asymptotically stable invariant set of (8), then it is a ω-practically uniformly asymptotically stable set of (7). ηkj =
The lemma resembles (D¨ urr et al., 2013, Theorem 2) and differs from (D¨ urr et al., 2013, Theorem 2) only by its stability definition. Namely, in contrast to the lemma, which presumes S to be asymptotically stable, (D¨ urr et al., 2013, Theorem 2) requires S to be an asymptotically stable uniform attractor.
MICNON 2015 128 Maximilian Montenbruck et al. / IFAC-PapersOnLine 48-11 (2015) 126–130 June 24-26, 2015. Saint Petersburg,Jan Russia
Definition 7. If S is an attractor of (5) and for every δ > 0, for every compact subset K of the region of attraction of S, there exists tf ≥ 0 such that for all t ∈ (tf , ∞), for all z0 ∈ K, ϕz (z0 , t) ∈ USδ , then S is called an uniform attractor of (5).
Next, let k be fixed but greater than the above k0 . We consider (4) under (11) which is
The proof of Lemma 6 thus requires the following lemma. Lemma 8. ((Bhatia and Szeg˝ o, 1970, Theorem V.1.16)). If S is a compact and asymptotically stable invariant set of (5), then S is a uniform attractor of (5).
We now see that (3) can be written in the form (7) by setting m = 2n and identifying b0 = f , b2i−1 = ei P , b2i = 2kei , i = 1, . . . , n. The corresponding Lie bracket system (8) then coincides with (5) which is due to the fact that the frequencies of the perturbations sin and cos are different. Then we have ϕz = ϕζ and ϕx = ϕξ with the property that for all i, vi is T -periodic and has zero average. Now choose any tf ∈ (t f , ∞). As there exists B ⊂ Rn such that for all z0 ∈ B, ϕζ (ζ0 , t) = ϕz (z0 , t) is uniformly bounded on [0, ∞), namely B = UPα , application of Lemma 4 yields ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for every x0 ∈ UPα , for all t ∈ [0, tf ], d (ϕx (x0 , t) , ϕz (x0 , t)) < D. As we had shown before that for all t ∈ (t f , ∞), for all −D z0 ∈ UPα , ϕz (z0 , t) ∈ UM , and as tf > t f , this reveals α that for all x0 ∈ UP , ϕx (x0 , tf ) ∈ UM , which was to be proven.
Proof of Lemma 6. The lemma follows from (D¨ urr et al., 2013, Theorem 2) after application of Lemma 8. 3. MAIN RESULT We solve the problem from section 1 via a two-step procedure. In particular, we first find sufficiently large k such that solutions of (5) approach the minima of P such as solutions of (3) do. This is done by application of Lemma 3. We second use extremum seeking in order to bring (4) to the form of (7) and to hence keep its solutions in proximity of solutions of (5), which has the form of (8) by finding sufficiently large ω. This is done by application of Lemmata 4 and 6. In this spirit, we propose a function u(k,ω) : R × R → Rn ,
(10)
2
parameterized by (k, ω) ∈ (0, ∞) to solve the problem from section 1. More particular, let u(k,ω) (P (x) , t) = n √ √ ei P (x) iω sin (iωt) − 2k iω cos (iωt) (11) i=1
with e1 , . . . , en being orthonormal vectors of Rn that satisfy span{e1 · · · en } = Rn . With this choice of u at hand, we are able to state our main results.
Our first result regards reachability of every (arbitrarily small) proximity of M by choosing sufficiently large k, ω. Theorem 9. If P is positive definite with respect to M on U , M is compact, and f is twice continuously differentiable, then, for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞) and for every tf ∈ (0, ∞) there exists an ω0 ∈ (0, ∞) such that for every ω ∈ (ω0 , ∞) and for every x0 ∈ UPα , ϕx (x0 , tf ) ∈ UM . Proof. For any compact UPα ⊂ U , choose some ∈ (0, d (M, ∂UPα )). Now choose D < . By virtue of Lemma 3, there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), −D UM contains an asymptotically stable invariant set of (5), which we denote by S, whose region of attraction is a superset of UPα . −D is compact, and thus, S Now, as M is compact, U M −D is compact. Define δ = d S, ∂UM (such a δ exists by virtue of the aforementioned compactnesses).
As UPα is compact, and as it is moreover a superset of the region of attraction of S, it follows from Lemma 8, that there exists t f such that for all t ∈ (t f , ∞), for all z0 ∈ UPα , ϕz (z0 , t) ∈ USδ . By our very choice of δ, we moreover have −D . that USδ ⊂ UM
132
x˙ = f (x) +
n i=1
ei P (x)
√ √ iω sin (iωt) − 2kei iω cos (iωt) .
Our second result states that every (arbitrarily small) neighborhood of M contains ω-practically uniformly asymptotically stable sets when choosing sufficiently large k. Theorem 10. If P is positive definite with respect to M on U , M is compact, and f is twice continuously differentiable, then, for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), UM contains an ω-practically uniformly asymptotically stable set of (4). Proof. Application of Lemma 3 reveals that for every compact UPα ⊂ U , for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), UM contains an asymptotically stable invariant set of (5), which we denote by S. We now see that (3) can be written in the form (7) by setting m = 2n and identifying b0 = f , b2i−1 = ei P , b2i = 2kei , i = 1, . . . , n. The corresponding Lie bracket system (8) then coincides with (5) which is due to the fact that the frequencies of the perturbations sin and cos are different. Then we have ϕz = ϕζ and ϕx = ϕξ with the property that for all i, vi is T -periodic and has zero average. By virtue of Lemma 6, S is an ω-practically uniformly asymptotically stable set of (4). This concludes the proof. In both results, we rely on the existence of some positive α such that UPα ⊂ U and UPα is compact. The results of Wilson (1967) would shed light on the existence of such α for the case that M be a compact submanifold of Rn . For the sake of self-containedness, we yet also include such an existence result here for rather general compact M . Proposition 11. (cf. (Bhatia and Szeg˝o, 1970, Theorem VIII.2.5)). Let P : U → R be continuously differentiable with U ⊂ Rn open and let M ⊂ U be compact. Let P be positive definite with respect to M on U . Then, there exists δ0 ⊂ U and for all δ ∈ (0, δ0 ), there a δ0 > 0 such that int UM δ0 δ exists an α > 0 such that (UPα ∩ UM ) ⊂ UM . Moreover, if δ0 n α U = R , then UP ∩ UM is a compact, isolated component of UPα .
MICNON 2015 June 24-26, 2015. Saint Petersburg,Jan Maximilian Montenbruck et al. / IFAC-PapersOnLine 48-11 (2015) 126–130 Russia
δ0 Proof. First, choose some δ0 > 0 such that UM ⊂ U, which exists due to the fact that M ⊂ U , M is compact and U is open.
Second, we show that for all δ ∈ (0, δ0 ) there exists an δ0 δ α > 0 such that UPα ∩ UM ⊂ UM . Suppose for the sake of contradiction that there exists a δ ∈ (0, δ0 ) such that for δ0 δ such that x ∈ UM . all α > 0 there exists an x ∈ UPα ∩ UM
Then define a sequence (αn )n∈N such that αn > 0 and δ0 αn → 0. For each of the αn there exists a xn ∈ UPαn ∩ UM δ δ such that xn ∈ UM . Now, since UM0 is bounded, by the Bolzano-Weierstrass theorem, there exists a subsequence δ0 (xnk )k∈N such that xnk → x∞ for some x∞ ∈ UM . However, by continuity of P , we have on the one hand that (12) lim P (xnk ) = P lim xnk = P (x∞ ) k→∞
k→∞
and on the other hand lim P (xnk ) ≤ lim αnk = 0, k→∞
k→∞
(13)
thus x∞ ∈ M . By convergence, there exists a k0 ∈ N such that for all k ≥ k0 we have that xnk − x∞ ≤ 2δ , which δ/2 leads to the contradiction xnk ∈ UM , thus proving the claim. Third, let U = Rn . We observe that since P is continuous, UPα is closed for all α > 0 (Rudin, 1964, Corollary of δ0 is Theorem 4.8). Now, since δ < δ0 we have that UPα ∩ UM bounded and hence compact. In particular, since δ < δ0 , there exists a δ1 ∈ (δ, δ0 ) such that (δ −δ) δ0 , (14) U 1α δ0 ⊂ UPα ∩ UM UP ∩UM
δ0 UM
i.e., UPα ∩ is an isolated component of UPα . This was the last statement to be proven.
Together with this latter proposition, our main results endow one with the ability to choose k and ω appropriately in order to not only let solutions of (4) under (11) reach arbitrarily small neighborhoods of M , but also to remain there in a practically stable fashion. This solves the control problem from section 1. Our approach has potential application in problems where f is unknown, but parametrized by a bounded parameter, which we term “robust” extremum seeking. In particular, assume that f is subject to a parametric uncertainty, i.e. that f is parameterized via a parameter µ ∈ Rm , f (x) = f (x, µ). If f is continuous in µ and µ is restricted to a compact set ∆ ⊂ Rm , then it is possible to replace fδα in the proof of Lemma 3 by α max ∇P (z) · f (z, µ) =: fδ,∆ . (15) α δ z∈UP \int UP µ∈∆
This lets one obtain an overestimate α fδ,∆ k0 = α pδ which is valid for any µ ∈ ∆.
(16)
Let ϕx,µ : (x0 , t) → ϕx,µ (x0 , t) denote the solution of x˙ = f (x, µ) + u (P (x) , t) , (17) initialized at x0 , for some particular µ ∈ P . 133
129
In this setting, it is possible to recast our main results for the robust extremum seeking problem. Following the course of Theorem 9, we infer that for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), for every tf ∈ (0, ∞), and every µ ∈ ∆, there exists an ω0 ∈ (0, ∞) such that for every ω ∈ (ω0 , ∞) and for every x0 ∈ UPα , ϕx,µ (x0 , tf ) ∈ UM .
Following the course of Theorem 10, we further have that for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), for every µ ∈ ∆, UM contains an ω-practically uniformly asymptotically stable set of (17). 4. EXAMPLE: THE UNIT CIRCLE In this example, we apply our above approach to practical stabilization of the unit sphere S1 = x ∈ R2 | x = 1 (18)
(i.e. n = 2 and M = S1 ) despite drift. Stabilization of the unit sphere is, for instance, relevant in artificial pattern generators. To apply our findings to this problem, we need to define a potential function which is positive definite with respect to S1 on R2 \ {0}, for instance 1 1 1 2 3 P : x → − x + x + . (19) 2 3 6 The function P is plotted in Fig. 1. In this example, we shall be concerned with the exemplary drift vector field x1 x1 x − x2 f: → (20) x2 x2 x + x1
under which S1 is unstable for u = 0 and which will turn out to be particularly suited for illustrating the two-step tuning procedure that we proposed, i.e. that there exists a k0 such that for any k ∈ (k0 , ∞) there exists an ω0 such that for any ω ∈ (ω0 , ∞), solutions approach the desired neighborhood US1 , but that it is not in general true that for any k, there exists an ω0 such that for any ω ∈ (ω0 , ∞), solutions approach the desired neighborhood US1 (simply said, the two parameters can not be tuned independently). Please note that this choice of f is not twice continuously differentiable (one would have to exclude the origin to obtain this property). We refer to Scheinker and Krsti´c (2013b) for the extension of extremum seeking to such vector fields and omit the technical discussion here. We solved the differential equation (4) under the extremum seeking feedback (11) numerically in Matlab using ode45 for different values of k and ω and depict the resulting numerical approximation of ϕx for x0 = [2 2] in Fig. 2. The simulation reveal that for k = 2, increasing ω results in a decrease of , as expected. For k = 1, however, can not be rendered small by choice of ω. This illustrates that first a sufficiently large k (here it is k ∈ (1, ∞)) must be found before ω can be adjusted in order to decrease as desired; it is yet not true that the latter tuning of ω is feasible for any choice of k (here e.g. not for k = 1).
MICNON 2015 130 Maximilian Montenbruck et al. / IFAC-PapersOnLine 48-11 (2015) 126–130 June 24-26, 2015. Saint Petersburg,Jan Russia
5. CONCLUSION
P
0.2
We studied convergence properties of extremum seeking controllers which are subject to drift. In order to cope with such issues, we presented a framework in which we could bring the solutions of the controlled system arbitrarily close to the minima of a given potential function despite the drift vector field. Our approach can be applied to robust extremum seeking problems in which the drift vector field is unknown but contained in a compact set, for instance when the drift vector field contains a uncertain parameter. We illustrated our findings on a numerical example in which we practically stabilized the unit sphere.
0.1
0 1
1
0
0 −1 −1
x2
x1
REFERENCES
Fig. 1. Plot of the function P as in (19), which is positive definite with respect to S1 on R2 \ {0} k = 2, ω = 50 x0
ϕx
x2
S1 US 1
0
0
S1 US 1
0
2
0
2
x1
x1
k = 2, ω = 400
k = 1, ω = 400
x0
ϕx
x2
S
2
1
x2
2
x0
ϕx
2 x2
2
k = 2, ω = 100
US 1
0
x0 S1
ϕx
0
−2 0
2 x1
−2
0 x1
2
Fig. 2. Numerical approximations of the solutions ϕx to (4) under the extremum seeking feedback (11) for different choices of k and ω for M being the unit sphere S1
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Ariyur, K.B. and Krsti´c, M. (2003). Real-time optimization by extremum-seeking control. John Wiley & Sons. Bhatia, N.P. and Szeg˝o, G.P. (1970). Stability Theory of Dynamical Systems. Springer. Brauer, F. (1966). Perturbations of nonlinear systems of differential equations. Journal of Mathematical Analysis and Applications, 14, 198–206. D¨ urr, H.B., Stankovi´c, M., Ebenbauer, C., and Johansson, K.H. (2013). Lie bracket approximation of extremum seeking systems. Automatica, 49, 1538–1552. Hirsch, M.W., Smale, S., and Devaney, R.L. (2004). Differential Equations, Dynamical Systems, and an Introduction to Chaos. Elsevier. Montenbruck, J.M., B¨ urger, M., and Allg¨ower, F. (2015). Compensating drift vector fields with gradient vector fields for asymptotic submanifold stabilization. IEEE Transactions on Automatic Control. Submitted. Rudin, W. (1964). Principles of Mathematical Analysis. McGraw-Hill. Scheinker, A. and Krsti´c, M. (2013a). Minimum-seeking for CLFs: Universal semiglobally stabilizing feedback under unknown control directions. IEEE Transactions on Automatic Control, 58, 1107–1122. Scheinker, A. and Krsti´c, M. (2013b). Non-C2 Lie bracket averaging for nonsmooth extremum seekers. Journal of Dynamic Systems, Measurement, and Control, 136, 011010. Tan, Y., Neˇsi´c, D., and Mareels, I. (2006). On nonlocal stability properties of extremum seeking control. Automatica, 42, 889–903. Wilson, F. (1967). The structure of the level surfaces of a Lyapunov function. Journal of Differential Equations, 3, 323–329.
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IFAC-PapersOnLine 48-11 (2015) 126–130
Extremum Extremum Extremum
Seeking Seeking Seeking
with with with
Drift Drift Drift
Jan Maximilian Montenbruck, Hans-Bernd D¨ urr, Christian Ebenbauer, Frank Allg¨ ower Jan Maximilian Montenbruck, Hans-Bernd D¨ u rr, Christian Ebenbauer, Frank Allg¨ o wer Jan Maximilian Montenbruck, Hans-Bernd D¨ u rr, Christian Ebenbauer, Frank Allg¨ o Jan Maximilian Montenbruck, Hans-Bernd D¨ urr, Christian Ebenbauer, Frank Allg¨ ower wer Institute for Systems Theory and Automatic Control, University of Stuttgart Institute for Systems Systems Theory and Automatic Control, University of Stuttgart Stuttgart Institute Theory Control, University Pfaffenwaldring 9, Automatic 70550 Stuttgart, Germany Institute for for Systems Theory and and Automatic Control, University of of Stuttgart Pfaffenwaldring 9, 70550 Stuttgart, Germany Pfaffenwaldring mailto:[email protected] Pfaffenwaldring 9, 9, 70550 70550 Stuttgart, Stuttgart, Germany Germany mailto:[email protected] mailto:[email protected] mailto:[email protected] Abstract: We study the convergence properties of extremum seeking controllers when a drift Abstract: We study the convergence properties of such extremum seeking controllers when drift Abstract: We properties of extremum controllers when aaa drift vector field appears in the the convergence closed loop. To cope with issues,seeking we propose a tuning procedure Abstract: We study study properties of such extremum seeking controllers when drift vector field appears appears in the the convergence closed loop. To To cope with issues, we propose a tuning tuning procedure vector field in the closed loop. cope with such issues, we propose a procedure that admits for guaranteeing convergence arbitrarily close to the desired minima despite drift. vector field appears in the closed loop. To cope with such propose a tuning procedure that admits admits for guaranteeing guaranteeing convergence arbitrarily closeissues, to the thewe desired minima despite drift. that for convergence arbitrarily close to desired minima despite drift. that admits for guaranteeing convergence arbitrarily close to the desired minima despite drift. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: extremum seeking; drift; practical stability Keywords: extremum seeking; drift; practical stability Keywords: extremum extremum seeking; seeking; drift; drift; practical practical stability stability Keywords: 1. INTRODUCTION In particular, we study the convergence properties of (4) 1. INTRODUCTION In the convergence properties of (4) 1. INTRODUCTION INTRODUCTION In particular, we study the properties of by particular, introducingwe thestudy auxiliary gradient system with drift 1. In particular, we study the convergence convergence properties of (4) (4) by introducing the auxiliary gradient system with drift by introducing the auxiliary gradient system with drift Extremum seeking controllers are feedbacks of the form by introducingz˙the auxiliary gradient system with drift(5) = f (z) − k∇P (z) =: Z (z) Extremum seeking controllers are feedbacks of the form z˙ = f (z) − k∇P (z) =: Z (z) (5) Extremum seeking seeking controllers controllers are feedbacks of the the form form(1) n = =: Z (5) Extremum feedbacks of →R , uω : R × Rare = ff (z) (z) − − k∇P k∇P (z) =:order Z (z) (z)to let solutions (5) via first findingzz˙˙ sufficiently large(z) k in n n, : R × R → R (1) u ω via first finding sufficiently large k in order to let solutions n, u : R × R → R (1) ω via first finding sufficiently large k in order to let solutions of (5) approach a neighborhood of the minima of P , such × R → R parameterized , (1) via first finding sufficiently large k in order to let solutions uω : Rargument, oscillatory in their second by the (5) approach neighborhood of the minima of P ,, such oscillatory their second parameterized by the of of (5) approach neighborhood of the minima of it was done aaaby Montenbruck (2015), and by oscillatory in their second argument, parameterized by frequency ωin (0, ∞), thatargument, have the purpose of steering of (5) approach neighborhood of et theal. minima of P P , such such oscillatory in∈ their second argument, parameterized by the the as as it was done by Montenbruck et al. (2015), and by frequency ω ∈ (0, ∞), that have the purpose of steering the as it was done by Montenbruck et al. (2015), and by then deriving bounds on the proximity of solutions of (4) frequency ω ∈ (0, ∞), that have the purpose of steering the solutions of the system it deriving was done by Montenbruck et al.of (2015), and by frequencyof ω the ∈ (0,system ∞), that have the purpose of steering the as then bounds on the proximity solutions of (4) solutions then deriving bounds on the proximity of solutions of (4) to solutions of (5) via classical extremum seeking, thus solutions of the system then deriving bounds on the proximity of solutions of (4) solutions of the system x˙ = u (P (x) , t) (2) to solutions of (5) via classical extremum seeking, thus to solutions of classical extremum seeking, thus pursuing a two-step procedure. lets us bring systems x˙˙ = u (P (x) ,, t) (2) solutions of (5) (5) via via classicalThis extremum seeking, thus = (2) pursuing aaarbitrarily two-step procedure. This lets us bring systems x˙minima =u u (P (P (x) (x) , t) t)unknown function P (2): to arbitrarily close to thex of the pursuing two-step procedure. This lets us bring systems with drift close to the minima of a function P pursuing a two-step procedure. This lets us bring systems arbitrarily close to the minima of the unknown function P : n arbitrarily close to the of the unknown function P :: with drift arbitrarily close to the minima of aa function P Rn → R, only with theminima information provided by the values with drift arbitrarily close to the minima of function P arbitrarily close to the minima of the unknown function P only via knowledge of the values of P . with drift arbitrarily close to the minima of a function P R R, only the values R → R,choosing only with with the information information provided by the values via knowledge of the values of P .. ofnnP→ , by sufficiently large ωprovided (we refer by thethe reader to only only via knowledge of the values of P R → R, only with the information provided by the values only via knowledge of the values of P . of P , byand choosing ω (we refer the reader to the manuscript, we assume the twice continof P choosing sufficiently large ω the reader to Ariyur Krsti´csufficiently (2003) for large an introduction topic). of P ,, by byand choosing sufficiently large ω (we (we refer refer to thethe reader to Throughout Throughout the assume continn Ariyur Krsti´ ccextremum (2003) for an introduction to the topic). Throughout the manuscript, manuscript, we assumePthe the twice contingiven uously differentiable potentialwe function : Rtwice Ariyur and Krsti´ (2003) for an introduction to the topic). One challenge in seeking is to establish stability Throughout the manuscript, we assume the twice continn → R Ariyur and Krsti´ cextremum (2003) forseeking an introduction to thestability topic). uously differentiable n → R given function P : R n potential One challenge in is to establish n → R given uously differentiable potential function P : R is an asymptotically stable invariant such that M ⊂ R One challenge in extremum seeking is to establish stability properties of (2), such as practical stability (cf. Tan et al. → R given uously differentiable potential function P : R n One challenge in extremum seeking stability is to establish stability n is an asymptotically stable invariant such that M ⊂ R properties of (2), such as practical (cf. Tan et al. n an asymptotically stable invariant such that M ⊂ R set of (3). Our goal is to consequently find a function (1) properties of (2), such as practical practical stability (cf.approaches Tan et et al. al. such that M ⊂ R is an asymptotically stable invariant (2006) or D¨ urr(2), et al. (2013)). The idea in these properties of such as stability (cf. Tan set of (3). Our goal is to consequently find aa“close” function (1) (2006) or D¨ u rr et al. (2013)). The idea in these approaches set of (3). Our goal is to consequently find function (1) such that one can bring the solutions of (4) to M (2006) or D¨ u rr et al. (2013)). The idea in these approaches is to choose solutions of (2) approximated ofthat (3).one Ourcan goal is tothe consequently find a“close” function (1), ω such (2006) or D¨ uu rr et al.that (2013)). The idea inare these approaches set such bring solutions of (4) to M is to choose u such that solutions of (2) are approximated such that one can bring bring the solutions of (4) (4) “close” “close” to M M ,,, ω as t → ∞, for some given twice continuously differentiable is to choose u such that solutions of (2) are approximated by solutions of the associated gradient system ω such that one can the solutions of to is tosolutions choose uof that solutions of (2)system are approximated as t → ∞, for some given ω such twice continuously differentiable differentiable by the associated gradient as tt → ∞, some drift vector field f : given Rnn →twice Rn . continuously by associated → ∞, for for some twice continuously differentiable by solutions solutions of of the they˙ = associated gradient system −∇P (y)gradient =: Y (y)system . (3) as n → Rn n. drift vector field ff :: given R n n drift vector field R → R . yy˙˙ = −∇P (y) =: Y (y) . (3) Notation. drift vector field → RRn ,. we mean the unique vector =: ∇Pf ::RRnn → y˙ = = −∇P −∇P (y) =: Y Y (y) (y)of.. how to choose (3) (3) In this paper, we address the(y) question uω Notation. By n n → Rn By ∇P :: R , we mean the unique vector In this paper, we address the question of how to choose u n Notation. By ∇P R → field which satisfies ω In this paper, we question of how to choose u if (2) assumes theaddress form the Notation. By ∇P : R → R Rn ,, we we mean mean the the unique unique vector vector ω In this paper, we address the question of how to choose u ω field which satisfies if (2) assumes the form field which satisfies ∇P (x + hv) − P (x) if (2) (2) assumes assumes the form field which satisfies if the form x˙ = f (x) + u (P (x) , t) =: X (x, t) , (4) ∇P (x) · v = lim ∇P (6) (x + hv) − P (x) ∇P x˙˙ = f (x) + u (P (x) , t) =: X (x, t) ,, (4) h→0 ∇P (x (x + + hv) hv) −P P (x) (x) h − ∇P (x) ·· vv = lim (6) = + =: (x, t) (4) n ,, t) n X ∇P (x) = lim (6) x˙vector = ff (x) (x) +u uf (P (P (x) t) =: X (x, t) , (4) i.e. if a driftx field : R(x) → R appears in the closed ∇P (x) · v = lim (6) h→0 h n h→0 n h , where “·” denotes the dot product. for any x and v ∈ R n → Rn n appears in the closed i.e. if a drift vector field f : R h→0 h n n →solutions n appears i.e. if a drift vector field f : R R in the closed loop, but one still wants to bring of (4) arbitrarily n , where “·” denotes the dot product. for any x and v ∈ R i.e. if but a drift field fto:bring R →solutions R appears in arbitrarily the closed for “·” denotes x R We denote thevv ∈ solution of (4) atdot x0 product. by ϕx : loop, onevector still wants wants of later, (4) where “·” initialized denotes the the dot product. for any any x and and ∈ Rn ,, where loop, one still solutions (4) close but to the of to P .bring As we will seeof this has We denote the solution of (4) initialized at x by ϕxxby:: loop, but one minima still wants to bring solutions of later, (4) arbitrarily arbitrarily 0 We denote the solution of (4) initialized at x by (x , t) → ϕ (x , t), the solution of (3) initialized at 0 0 x 0 0x : close to the minima of P . As we will see this has We, t) denote the solution of (4) initialized at x0 at by yyϕ ϕ close to the minima of .. As we will see this has potential application inP problems where flater, is unknown, (x → ϕ (x , t), the solution of (3) initialized close to the minima of P As we will see later, this has 0 x 0 (x t) → ϕx→(x (xϕ0y,, t), t), the solution of (3) initialized initialized at yy00 by by ϕ (y , t) (y , t), and the solution of (5) initialized 0 ,,: t) y 0 0 potential application in problems where f is unknown, (x → ϕ the solution of (3) at 0 : (y , t) x 0 (y , t), and the solution of (5) initialized 0 by potential application in where unknown, for instance if it is subject to a parametric ϕ →: ϕ ϕ(z potential application in problems problems where ff is isuncertainty unknown, ϕ y 0 y 0 → and the solution of (5) initialized at by ϕ , t) → ϕ (z , t). For a function such as y z:: 0(y 0 ,, t) y0(y 0 ,, t), z z 0 for instance if it is subject to a parametric uncertainty ϕ (y t) → ϕ (y t), and the solution of (5) initialized 0 ϕ : (z y t) 0 → ϕ (z , t). For a function for instance if a parametric uncertainty aty z0 nby (“robust” extremum seeking).to such for instance if it it is is subject subject z z 0 t) → ϕ For aa function such as P :zz00Rnby →ϕ we000 ,,,denote its sublevel sets by UPα = {x as ∈ zR,:: (z z (z 0 ,, t). (“robust” extremum seeking).to a parametric uncertainty at at by ϕ (z t) → ϕ (z t). For function such as α z z 0 (“robust” extremum seeking). n → R, we denote its sublevel setsnby U α = {x ∈ P n:: R (“robust” extremum seeking). P n → α = {xits P R ∈ R, we denote its sublevel sets by U |P (x) ≤ α}. For a set such as M ⊂ R , we denote The problem statement resembles the stabilizing ex- R P P nn:|PR(x)→≤R, weFor denote its sublevel sets by U denote = {xits ∈ n n n P R α}. aa set such ⊂ R M The resembles the exn |P (x) ≤ neighborhood n ,,Rwe R α}. For set such as M ⊂ R we denote its The problem problem statement resembles the stabilizing stabilizing ex- equidistant by Uas = {x ∈ |d (x, M ) ≤ tremum seekingstatement problem for input-affine systems posed R |P (x) ≤ α}. For a set such as M ⊂ R , we denote its M The problem statement resembles the stabilizing ex n n equidistant neighborhood by UM {x ∈ R |d (x, M )) to ≤ tremum seeking for input-affine systems posed = neighborhood by = R M ≤ tremum seeking problem for systems posed }, where, here, d is the infimal Euclidean of by Scheinker and problem Krsti´c (2013a), who proposed a solution equidistant neighborhood by U UM = {x {x ∈ ∈distance Rn |d |d (x, (x, Mx ) to ≤ tremum seeking problem for input-affine input-affine systems posed equidistant M }, where, here, d is the infimal Euclidean distance x n n of by Scheinker and Krsti´ cc (2013a), who proposed a solution }, where, here, d is the infimal Euclidean distance of x to by Scheinker and Krsti´ (2013a), who proposed a solution all points in M . Given a vector field f : R → R and a based on control Lyapunov functions and persistence-of }, where, here, d is the infimal Euclidean distance of x to by Scheinker and Krsti´ c (2013a), who proposed a solution n n n → Rn and a all points in M .. Given aa: vector field ff Lie :: R n based on control Lyapunov functions and persistence-ofn → Rn and all points in M Given vector field R a based on control Lyapunov functions and persistence-ofdifferentiable function P R → R, the derivative of P excitation-type conditions on the control vector fields. Yet, all points in M . Given a: vector field : Rderivative → R and based on control Lyapunov functions and persistence-ofn differentiable function PR, R →∇P R, thef·Lie Lie of P Pa excitation-type conditions on control vector fields. differentiable :: R R, the derivative of excitation-type conditionsand on the the control vectorwe fields. Yet, f is Lf Pfunction : Rnn →P x nn→→ (x) f (x). Throughout in contrast to Scheinker Krsti´ c (2013a), will Yet, not along differentiable function P R → R, the Lie derivative of P excitation-type conditions on the control vector fields. Yet, n → R, x → ∇P (x) · f (x). Throughout along ff is LUff P :: R in contrast Scheinker cc (2013a), we willofnot n → along x (x). Throughout in contrast to Scheinker and Krsti´ we the paper, a R, neighborhood and whenever assume thatto to and (5) Krsti´ approach the minima P along f is is L LUf P Pwill :R Rbe → x → → ∇P ∇P (x) (x)of ·· ffM (x). Throughout in contrast tosolutions Scheinker and Krsti´ c (2013a), (2013a), we will willofnot not the paper, will be aa R, neighborhood of M and whenever assume that solutions to (5) approach the minima P the paper, U will be neighborhood of M and whenever assume that solutions to (5) approach the minima of P we write int U , we refer to the interior of U whilst with exactly. the write paper,int UUwill berefer a neighborhood of M and whenever assume that solutions to (5) approach the minima of P we , we to the interior of U whilst with exactly. we write int U , we refer to the interior of U whilst with exactly. ∂U , we mean its boundary. For two vector fields f , g, we ,write int U ,its weboundary. refer to the interior of U whilst with exactly. All authors thank the German Research Foundation (DFG) for ∂U we mean For two vector fields f ,, g, ∂U , we mean its boundary. For two vector fields f g, [f, g] denotes the Lie bracketFor of vector fields.fields We adopt ∂U , we mean its boundary. two vector f , g, All authors thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in All authors thank the German Research Foundation (DFG) for [f, g] denotes the Lie bracket of vector fields. We adopt All authors thank the German Research Foundation (DFG) for [f, g] denotes the Lie bracket of vector fields. We adopt terminology and results from Bhatia and Szeg˝ o (1970). financial of project within of in [f, g] denotesand the Lie bracket of vectorand fields. We adopt Simulation Technology 310/2) at the the Cluster University of Stuttgart. financial support support of the the(EXC project within the Cluster of Excellence Excellence in terminology from Szeg˝ oo (1970). financial support of the project within the Cluster of Excellence in terminology and results from Bhatia Bhatia and and Szeg˝ (1970). Although weand dealresults with time-dependent vector fields, we Simulation Technology (EXC 310/2) at the University of Stuttgart. terminology results from Bhatia Szeg˝ o (1970). Hans-Bernd D¨ u rr and Christian Ebenbauer are additionally supSimulation Technology (EXC 310/2) at the University of Stuttgart. Simulation Technology (EXC 310/2) at the University of Stuttgart. Although we deal with time-dependent vector fields, we Although we deal with time-dependent vector fields, we Hans-Bernd D¨ u rr and Christian Ebenbauer are additionally supomit the dependence on the initial time t in the solutions 0 ported by the German Research Foundation (DFG) within the Hans-Bernd D¨ u rr and Christian Ebenbauer are additionally supAlthough we deal with time-dependent vector fields, we Hans-Bernd D¨ urr and Christian Ebenbauer are(DFG) additionally supomit the dependence on the initial time t in the solutions 0 ported by the German Research Foundation within the omit the dependence on the initial time t in the solutions of the associated differential equations due to the fact that 0 in Emmy-Noether-Grant “Novel Ways in Control and Computation” ported by the German Research Foundation (DFG) within the omit the dependence on the initial time t the solutions 0 to the fact that ported by the German Research (DFG) within the of the associated differential due Emmy-Noether-Grant “Novel WaysFoundation in Control and Computation” of the associated differential equations due the fact (EB 425/2-1). all results hold uniformly in equations t0 (cf. D¨ urr etto (2013)). Emmy-Noether-Grant “Novel of the associated differential equations due toal. the fact that that Emmy-Noether-Grant “Novel Ways Ways in in Control Control and and Computation” Computation” (EB 425/2-1). all results hold uniformly in t (cf. D¨ u rr et al. (2013)). 0 (EB 425/2-1). all results hold uniformly in t (cf. D¨ u rr et al. (2013)). 0 (EB 425/2-1). all results hold uniformly in t0 (cf. D¨ urr et al. (2013)). Copyright IFAC 2015 130 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 130 Copyright © IFAC 2015 130 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 130Control. 10.1016/j.ifacol.2015.09.171
MICNON 2015 June 24-26, 2015. Saint Petersburg,Jan Maximilian Montenbruck et al. / IFAC-PapersOnLine 48-11 (2015) 126–130 Russia
Structure of the Manuscript. We introduce all terminology and auxiliary results that we require in section 2. Thereafter, i.e. in section 3, we present our main results on how to establish convergence guarantees despite drift. We illustrate these results on the example of practical stabilization of the unit sphere despite drift in section 4. Section 5 concludes the manuscript. 2. PRELIMINARIES The solution that we propose to solve the issue posed in section 1 involves two main ingredients: gradient systems and extremum seeking. We first review some results on gradient systems and consequently repeat the fundamentals of extremum seeking. A key ingredient will be the positive definiteness of P with respect to the desired set M . Definition 1. A continuously differentiable function P : Rn → R is said to be positive definite with respect to M on U , if U ⊂ Rn is a neighborhood of M ⊂ Rn , P is positive on U \ M , zero on M , regular on U \ M , and critical on M . Having this definition at hand, we repeat a fundamental result about gradient systems, that here only serves the purpose of giving the intuition behind the fact that solutions of (5) approach a neighborhood of M . Proposition 2. (cf. (Hirsch et al., 2004, Sections 9.2f) ). If P is positive definite with respect to M on U and M is compact, then M is an asymptotically stable invariant set of (3) and for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, UPα is a subset of the region of attraction of M. Now, guided by the intuition from perturbation theory (cf. Brauer (1966)), we know that solutions of (5) must stay close to solutions of (3) and thus approach a neighborhood of M whose size can be rendered arbitrarily small by appropriate choice of k. This technique was proposed by Montenbruck et al. (2015). Classically, perturbation theory assumes f = constant, whereas, herein, f is allowed to be a vector field. Lemma 3. If P is positive definite with respect to M on U , M is compact, and f is continuous on U , then, for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), UM contains an asymptotically stable invariant set of (5) which is also a uniform attractor, and whose region of attraction is a superset of UPα . Proof. The Lie derivative of P along Z is given by LZ P (z) = ∇P (z) · f (z) − k∇P (z) · ∇P (z). Choose any α ∈ (0, ∞) such that UPα is compact and UPα ⊂ U . As P is positive definite with respect to M on U , for every α ∈ (0, ∞) such that UPα is compact and UPα ⊂ U , for any ∈ (0, d (M, ∂UPα )), there exists δ ∈ (0, α) such that UPδ is a subset of UM . It is then true that UPα \ int UPδ is a compact, nonempty set. As f is continuous and P is continuously differentiable, ∇P · f assumes its maximum on UPα \ int UPδ , which we denote by fδα . It follows that for all z ∈ UPα \ int UPδ , LZ P (z) ≤ fδα − k∇P (z) · ∇P (z). As P is continuously differentiable and positive definite with respect to M on U , for every α ∈ (0, ∞) such that UPα is compact and UPα ⊂ U , ∇P (z) · ∇P (z) assumes 131
127
its positive minimum on UPα \ int UPδ , which we denote α δ by pα ≤ fδα − kpα δ. δ . It follows that LZ P UP \ int UP α α Setting kα0 = fδ δ/p δ , we have that for any k ∈ (k0 , δ∞), LZ P UP \ int UP < 0, letting us conclude that UP is an invariant set of (5). Now define a function as being P − δ outside UPδ and to be zero inside UPδ . This function is continuous and its Lie derivative along Z outside UPδ equals LZ P . By Lyapunov’s direct method, it follows that UPδ is an asymptotically set of (5). stable invariant Moreover, as we have LZ P UPα \ int UPδ < 0, we know that UPα is an invariant set. It follows from LaSalle’s invariance principle that UPα is a subset of the region of attraction of UPδ . This concludes the proof. We now repeat some fundamental concepts of extremum seeking, mostly taken from D¨ urr et al. (2013). For doing so, define ξ˙ = b0 (ξ) +
m
√ ωvj (ωt)
(7)
[bj , bk ] (ζ) ηkj
(8)
bj (ξ)
j=1
and ζ˙ = b0 (ζ) +
m
j=1 k=j+1
with
θ 1 T vk (θ) vj (τ ) d τ d θ. (9) T 0 0 Here and henceforth, let ϕξ : (ξ0 , t) → ϕξ (ξ0 , t) denote the solution of (7) initialized at ξ0 and ϕζ : (ζ0 , t) → ϕζ (ζ0 , t) denote the solution of (8) initialized at ζ0 . With these auxiliary systems, we repeat two basic results in extremum seeking. Lemma 4. ((D¨ urr et al., 2013, Theorem 1)). For all i, let vi be T -periodic with zero average. For all i, let bi be twice continuously differentiable. If there exists B ⊂ Rn such that there exists κ ∈ (0, ∞) such that for all ζ0 ∈ B, for all t ∈ [0, ∞), ϕζ (ζ0 , t) < κ, then for every bounded K ⊂ B, for every D ∈ (0, ∞), for every tf ∈ (0, ∞), there exists ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for every ζ0 ∈ K, for all t ∈ [0, tf ], d (ϕζ (ζ0 , t) , ϕξ (ζ0 , t)) < D. Definition 5. A set S is said to be an ω-practically uniformly asymptotically stable set of (7), if, for every ∈ (0, ∞), there exists δ ∈ (0, ∞) and ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for all t ∈ [0, ∞), for all ξ0 ∈ USδ , ϕξ (ξ0 , t) ∈ US , and if there exists δ ∈ (0, ∞) such that for every ∈ (0, ∞), there exists tf ∈ [0, ∞) and ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for all t ∈ [tf , ∞), for all ξ0 ∈ USδ , ϕξ (ξ0 , t) ∈ US . Lemma 6. For all i, let vi be T -periodic with zero average. For all i, let bi be twice continuously differentiable. If a compact set S is an asymptotically stable invariant set of (8), then it is a ω-practically uniformly asymptotically stable set of (7). ηkj =
The lemma resembles (D¨ urr et al., 2013, Theorem 2) and differs from (D¨ urr et al., 2013, Theorem 2) only by its stability definition. Namely, in contrast to the lemma, which presumes S to be asymptotically stable, (D¨ urr et al., 2013, Theorem 2) requires S to be an asymptotically stable uniform attractor.
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Definition 7. If S is an attractor of (5) and for every δ > 0, for every compact subset K of the region of attraction of S, there exists tf ≥ 0 such that for all t ∈ (tf , ∞), for all z0 ∈ K, ϕz (z0 , t) ∈ USδ , then S is called an uniform attractor of (5).
Next, let k be fixed but greater than the above k0 . We consider (4) under (11) which is
The proof of Lemma 6 thus requires the following lemma. Lemma 8. ((Bhatia and Szeg˝ o, 1970, Theorem V.1.16)). If S is a compact and asymptotically stable invariant set of (5), then S is a uniform attractor of (5).
We now see that (3) can be written in the form (7) by setting m = 2n and identifying b0 = f , b2i−1 = ei P , b2i = 2kei , i = 1, . . . , n. The corresponding Lie bracket system (8) then coincides with (5) which is due to the fact that the frequencies of the perturbations sin and cos are different. Then we have ϕz = ϕζ and ϕx = ϕξ with the property that for all i, vi is T -periodic and has zero average. Now choose any tf ∈ (t f , ∞). As there exists B ⊂ Rn such that for all z0 ∈ B, ϕζ (ζ0 , t) = ϕz (z0 , t) is uniformly bounded on [0, ∞), namely B = UPα , application of Lemma 4 yields ω0 ∈ (0, ∞) such that for all ω ∈ (ω0 , ∞), for every x0 ∈ UPα , for all t ∈ [0, tf ], d (ϕx (x0 , t) , ϕz (x0 , t)) < D. As we had shown before that for all t ∈ (t f , ∞), for all −D z0 ∈ UPα , ϕz (z0 , t) ∈ UM , and as tf > t f , this reveals α that for all x0 ∈ UP , ϕx (x0 , tf ) ∈ UM , which was to be proven.
Proof of Lemma 6. The lemma follows from (D¨ urr et al., 2013, Theorem 2) after application of Lemma 8. 3. MAIN RESULT We solve the problem from section 1 via a two-step procedure. In particular, we first find sufficiently large k such that solutions of (5) approach the minima of P such as solutions of (3) do. This is done by application of Lemma 3. We second use extremum seeking in order to bring (4) to the form of (7) and to hence keep its solutions in proximity of solutions of (5), which has the form of (8) by finding sufficiently large ω. This is done by application of Lemmata 4 and 6. In this spirit, we propose a function u(k,ω) : R × R → Rn ,
(10)
2
parameterized by (k, ω) ∈ (0, ∞) to solve the problem from section 1. More particular, let u(k,ω) (P (x) , t) = n √ √ ei P (x) iω sin (iωt) − 2k iω cos (iωt) (11) i=1
with e1 , . . . , en being orthonormal vectors of Rn that satisfy span{e1 · · · en } = Rn . With this choice of u at hand, we are able to state our main results.
Our first result regards reachability of every (arbitrarily small) proximity of M by choosing sufficiently large k, ω. Theorem 9. If P is positive definite with respect to M on U , M is compact, and f is twice continuously differentiable, then, for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞) and for every tf ∈ (0, ∞) there exists an ω0 ∈ (0, ∞) such that for every ω ∈ (ω0 , ∞) and for every x0 ∈ UPα , ϕx (x0 , tf ) ∈ UM . Proof. For any compact UPα ⊂ U , choose some ∈ (0, d (M, ∂UPα )). Now choose D < . By virtue of Lemma 3, there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), −D UM contains an asymptotically stable invariant set of (5), which we denote by S, whose region of attraction is a superset of UPα . −D is compact, and thus, S Now, as M is compact, U M −D is compact. Define δ = d S, ∂UM (such a δ exists by virtue of the aforementioned compactnesses).
As UPα is compact, and as it is moreover a superset of the region of attraction of S, it follows from Lemma 8, that there exists t f such that for all t ∈ (t f , ∞), for all z0 ∈ UPα , ϕz (z0 , t) ∈ USδ . By our very choice of δ, we moreover have −D . that USδ ⊂ UM
132
x˙ = f (x) +
n i=1
ei P (x)
√ √ iω sin (iωt) − 2kei iω cos (iωt) .
Our second result states that every (arbitrarily small) neighborhood of M contains ω-practically uniformly asymptotically stable sets when choosing sufficiently large k. Theorem 10. If P is positive definite with respect to M on U , M is compact, and f is twice continuously differentiable, then, for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), UM contains an ω-practically uniformly asymptotically stable set of (4). Proof. Application of Lemma 3 reveals that for every compact UPα ⊂ U , for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), UM contains an asymptotically stable invariant set of (5), which we denote by S. We now see that (3) can be written in the form (7) by setting m = 2n and identifying b0 = f , b2i−1 = ei P , b2i = 2kei , i = 1, . . . , n. The corresponding Lie bracket system (8) then coincides with (5) which is due to the fact that the frequencies of the perturbations sin and cos are different. Then we have ϕz = ϕζ and ϕx = ϕξ with the property that for all i, vi is T -periodic and has zero average. By virtue of Lemma 6, S is an ω-practically uniformly asymptotically stable set of (4). This concludes the proof. In both results, we rely on the existence of some positive α such that UPα ⊂ U and UPα is compact. The results of Wilson (1967) would shed light on the existence of such α for the case that M be a compact submanifold of Rn . For the sake of self-containedness, we yet also include such an existence result here for rather general compact M . Proposition 11. (cf. (Bhatia and Szeg˝o, 1970, Theorem VIII.2.5)). Let P : U → R be continuously differentiable with U ⊂ Rn open and let M ⊂ U be compact. Let P be positive definite with respect to M on U . Then, there exists δ0 ⊂ U and for all δ ∈ (0, δ0 ), there a δ0 > 0 such that int UM δ0 δ exists an α > 0 such that (UPα ∩ UM ) ⊂ UM . Moreover, if δ0 n α U = R , then UP ∩ UM is a compact, isolated component of UPα .
MICNON 2015 June 24-26, 2015. Saint Petersburg,Jan Maximilian Montenbruck et al. / IFAC-PapersOnLine 48-11 (2015) 126–130 Russia
δ0 Proof. First, choose some δ0 > 0 such that UM ⊂ U, which exists due to the fact that M ⊂ U , M is compact and U is open.
Second, we show that for all δ ∈ (0, δ0 ) there exists an δ0 δ α > 0 such that UPα ∩ UM ⊂ UM . Suppose for the sake of contradiction that there exists a δ ∈ (0, δ0 ) such that for δ0 δ such that x ∈ UM . all α > 0 there exists an x ∈ UPα ∩ UM
Then define a sequence (αn )n∈N such that αn > 0 and δ0 αn → 0. For each of the αn there exists a xn ∈ UPαn ∩ UM δ δ such that xn ∈ UM . Now, since UM0 is bounded, by the Bolzano-Weierstrass theorem, there exists a subsequence δ0 (xnk )k∈N such that xnk → x∞ for some x∞ ∈ UM . However, by continuity of P , we have on the one hand that (12) lim P (xnk ) = P lim xnk = P (x∞ ) k→∞
k→∞
and on the other hand lim P (xnk ) ≤ lim αnk = 0, k→∞
k→∞
(13)
thus x∞ ∈ M . By convergence, there exists a k0 ∈ N such that for all k ≥ k0 we have that xnk − x∞ ≤ 2δ , which δ/2 leads to the contradiction xnk ∈ UM , thus proving the claim. Third, let U = Rn . We observe that since P is continuous, UPα is closed for all α > 0 (Rudin, 1964, Corollary of δ0 is Theorem 4.8). Now, since δ < δ0 we have that UPα ∩ UM bounded and hence compact. In particular, since δ < δ0 , there exists a δ1 ∈ (δ, δ0 ) such that (δ −δ) δ0 , (14) U 1α δ0 ⊂ UPα ∩ UM UP ∩UM
δ0 UM
i.e., UPα ∩ is an isolated component of UPα . This was the last statement to be proven.
Together with this latter proposition, our main results endow one with the ability to choose k and ω appropriately in order to not only let solutions of (4) under (11) reach arbitrarily small neighborhoods of M , but also to remain there in a practically stable fashion. This solves the control problem from section 1. Our approach has potential application in problems where f is unknown, but parametrized by a bounded parameter, which we term “robust” extremum seeking. In particular, assume that f is subject to a parametric uncertainty, i.e. that f is parameterized via a parameter µ ∈ Rm , f (x) = f (x, µ). If f is continuous in µ and µ is restricted to a compact set ∆ ⊂ Rm , then it is possible to replace fδα in the proof of Lemma 3 by α max ∇P (z) · f (z, µ) =: fδ,∆ . (15) α δ z∈UP \int UP µ∈∆
This lets one obtain an overestimate α fδ,∆ k0 = α pδ which is valid for any µ ∈ ∆.
(16)
Let ϕx,µ : (x0 , t) → ϕx,µ (x0 , t) denote the solution of x˙ = f (x, µ) + u (P (x) , t) , (17) initialized at x0 , for some particular µ ∈ P . 133
129
In this setting, it is possible to recast our main results for the robust extremum seeking problem. Following the course of Theorem 9, we infer that for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), for every tf ∈ (0, ∞), and every µ ∈ ∆, there exists an ω0 ∈ (0, ∞) such that for every ω ∈ (ω0 , ∞) and for every x0 ∈ UPα , ϕx,µ (x0 , tf ) ∈ UM .
Following the course of Theorem 10, we further have that for every α ∈ (0, ∞) such that UPα ⊂ U and UPα is compact, for every ∈ (0, d (M, ∂UPα )), there exists a k0 ∈ (0, ∞) such that for every k ∈ (k0 , ∞), for every µ ∈ ∆, UM contains an ω-practically uniformly asymptotically stable set of (17). 4. EXAMPLE: THE UNIT CIRCLE In this example, we apply our above approach to practical stabilization of the unit sphere S1 = x ∈ R2 | x = 1 (18)
(i.e. n = 2 and M = S1 ) despite drift. Stabilization of the unit sphere is, for instance, relevant in artificial pattern generators. To apply our findings to this problem, we need to define a potential function which is positive definite with respect to S1 on R2 \ {0}, for instance 1 1 1 2 3 P : x → − x + x + . (19) 2 3 6 The function P is plotted in Fig. 1. In this example, we shall be concerned with the exemplary drift vector field x1 x1 x − x2 f: → (20) x2 x2 x + x1
under which S1 is unstable for u = 0 and which will turn out to be particularly suited for illustrating the two-step tuning procedure that we proposed, i.e. that there exists a k0 such that for any k ∈ (k0 , ∞) there exists an ω0 such that for any ω ∈ (ω0 , ∞), solutions approach the desired neighborhood US1 , but that it is not in general true that for any k, there exists an ω0 such that for any ω ∈ (ω0 , ∞), solutions approach the desired neighborhood US1 (simply said, the two parameters can not be tuned independently). Please note that this choice of f is not twice continuously differentiable (one would have to exclude the origin to obtain this property). We refer to Scheinker and Krsti´c (2013b) for the extension of extremum seeking to such vector fields and omit the technical discussion here. We solved the differential equation (4) under the extremum seeking feedback (11) numerically in Matlab using ode45 for different values of k and ω and depict the resulting numerical approximation of ϕx for x0 = [2 2] in Fig. 2. The simulation reveal that for k = 2, increasing ω results in a decrease of , as expected. For k = 1, however, can not be rendered small by choice of ω. This illustrates that first a sufficiently large k (here it is k ∈ (1, ∞)) must be found before ω can be adjusted in order to decrease as desired; it is yet not true that the latter tuning of ω is feasible for any choice of k (here e.g. not for k = 1).
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5. CONCLUSION
P
0.2
We studied convergence properties of extremum seeking controllers which are subject to drift. In order to cope with such issues, we presented a framework in which we could bring the solutions of the controlled system arbitrarily close to the minima of a given potential function despite the drift vector field. Our approach can be applied to robust extremum seeking problems in which the drift vector field is unknown but contained in a compact set, for instance when the drift vector field contains a uncertain parameter. We illustrated our findings on a numerical example in which we practically stabilized the unit sphere.
0.1
0 1
1
0
0 −1 −1
x2
x1
REFERENCES
Fig. 1. Plot of the function P as in (19), which is positive definite with respect to S1 on R2 \ {0} k = 2, ω = 50 x0
ϕx
x2
S1 US 1
0
0
S1 US 1
0
2
0
2
x1
x1
k = 2, ω = 400
k = 1, ω = 400
x0
ϕx
x2
S
2
1
x2
2
x0
ϕx
2 x2
2
k = 2, ω = 100
US 1
0
x0 S1
ϕx
0
−2 0
2 x1
−2
0 x1
2
Fig. 2. Numerical approximations of the solutions ϕx to (4) under the extremum seeking feedback (11) for different choices of k and ω for M being the unit sphere S1
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