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Lyapunov Functions for Continuous and Discontinuous Differentiators
10th IFAC Symposium on Nonlinear Control Systems 10th Symposium on Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems 10th IFAC Symposium on Nonlinear Control Systems August USA August 23-25, 23-25, 2016. 2016. Monterey, Monterey, California, California, USA online at www.sciencedirect.com Available August 23-25, 2016. Monterey, California, USA
ScienceDirect IFAC-PapersOnLine 49-18 (2016) 660–665
Lyapunov Functions for Continuous and Lyapunov Functions for Lyapunov FunctionsDifferentiators for Continuous Continuous and and Discontinuous Discontinuous Differentiators Discontinuous Differentiators ∗ ∗∗
Emmanuel Cruz-Zavala ∗ Jaime A. Moreno ∗∗ ∗∗ Emmanuel Cruz-Zavala ∗∗ Jaime A. Moreno Emmanuel Emmanuel Cruz-Zavala Cruz-Zavala Jaime Jaime A. A. Moreno Moreno ∗∗ ∗ Depto de Ciencias de la Computaci´ on. Centro Universitario de ∗ ∗ Depto de Ciencias de la Computaci´ o n.Universidad Centro Universitario de de Ciencias de la Computaci´ o Centro de ∗ Depto Ciencias Exactas e Ingenier´ ıas (CUCEI), de Guadalajara, Depto de Ciencias de laıas Computaci´ on. n.Universidad Centro Universitario Universitario de Ciencias Exactas e Ingenier´ (CUCEI), de Guadalajara, Ciencias Exactas e Ingenier´ ıas (CUCEI), Universidad de Guadalajara, Guadalajara, Jalisco, Mexico (e-mail: [email protected]) Ciencias Exactas e Ingenier´ ıas (CUCEI), Universidad de Guadalajara, ∗∗ Guadalajara, Jalisco, Mexico Mexico (e-mail:[email protected]) [email protected]) Guadalajara, Jalisco, (e-mail: El´ectrica y Computaci´ on, Instituto Ingenier´ıa, Universidad Guadalajara, Jalisco, Mexico (e-mail:[email protected]) ∗∗ ∗∗ eectrica yynoma Computaci´ o n, Instituto Ingenier´ Universidad El´ ctrica Computaci´ o n, Instituto de Ingenier´ ıa, Universidad ∗∗ El´ Nacional Aut´ o de M´ e xico. Ciudad de M´exico,ıa, Mexico (e-mail: El´ e ctrica y Computaci´ o n, Instituto de Ingenier´ ıa, Universidad Nacional Aut´ o noma de M´ e xico. Ciudad de M´ e xico, Mexico (e-mail: Nacional Aut´ o noma de M´ e xico. Ciudad de M´ e xico, Mexico [email protected]) Nacional Aut´ onoma de M´exico. Ciudad de M´exico, Mexico (e-mail: (e-mail: [email protected]) [email protected]) [email protected]) Abstract: Given a (differentiable) signal it is an important task for many applications to Abstract: (differentiable) it is an important applications to Abstract: Given a (differentiable) signal it important task for many applications to estimate on Given line itsa Somesignal well known algorithms to task solvefor thismany problem include the Abstract: Given aderivatives. (differentiable) signal it is is an an important task for many applications to estimate on line its derivatives. Some well known algorithms to solve this problem include the estimate on line its derivatives. Some well known algorithms to solve this problem include the (continuous) high-gain observersSome and (discontinuous) Levant’s to exact differentiators. In this work estimate on line its derivatives. well known algorithms solve this problem include the (continuous) high-gain observers and and (discontinuous) (discontinuous)encompassing Levant’s exact exact differentiators. In this this work (continuous) high-gain observers Levant’s differentiators. In work we present a family of homogeneous these two algorithms, and we (continuous) high-gain observers anddifferentiators, (discontinuous)encompassing Levant’s exact differentiators. In this work we present a family of homogeneous differentiators, these two algorithms, and we we present a family of homogeneous differentiators, encompassing these two algorithms, and we propose a unified smooth Lyapunov function, that allows a common framework to study their we present a family of homogeneous differentiators, encompassing these two algorithms, and we propose a unified smooth Lyapunov function, that allows aa common framework to study their propose a unified smooth Lyapunov function, that allows common framework to study their convergence and performance analysis. propose a unified smooth Lyapunov function, that allows a common framework to study their convergence and and performance performance analysis. analysis. convergence convergence performance analysis. © 2016, IFAC and (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Observability and Observer Design; Lyapunov Stability Methods; Variable Keywords:Control Observability and Observer Observer Design; Lyapunov Lyapunov Stability Methods; Methods; Variable Keywords: Observability and Structure and Sliding Mode Design; Keywords: Observability and Observer Design; Lyapunov Stability Stability Methods; Variable Variable Structure Control Control and Sliding Sliding Mode Structure and Mode Structure Control and Sliding Mode 1. INTRODUCTION and Khalil (2008); Yang and Lin (2004) and discontinuous 1. INTRODUCTION INTRODUCTION and Khalil Khalil (2008); Yang and and Lin (2004) and and discontinuous 1. and (2008); Yang (2004) Levant (2003) homogeneous differentiators in a unified 1. INTRODUCTION and Khalil (2008); Yang and Lin Lin (2004) and discontinuous discontinuous Levant (2003) homogeneous differentiators inrobustness unified Levant (2003) homogeneous differentiators aa This allows to study the convergence,in Sliding Modes (SM) is a well-known technique to control manner. Levant (2003) homogeneous differentiators inrobustness a unified unified manner. This allows to study the convergence, manner. This allows to study the convergence, robustness Sliding Modes (SM) is a well-known technique to control and performance properties of all the algorithms in the Sliding Modes (SM) is a well-known technique to control uncertain systems, by bringing some variable (sliding varimanner. This allows to study the convergence, robustness Sliding Modes (SM) isbringing a well-known technique to control and performance properties of all the algorithms in the the and performance properties of all the algorithms in uncertain systems, by some variable (sliding varisame framework. The main contribution of this work is uncertain systems, by bringing some variable variable) to zero in finite time and maintaining it (sliding in zero using and performance properties of all the algorithms in the uncertain systems, by bringing some variable (sliding varisame framework. The main contribution of this work is same framework. The main contribution of this work is able) to zero in finite time and maintaining it in zero using to extend the Lyapunov method to the discontinuous able) to zero in finite time and maintaining it in zero using high frequency control switching. For standard SMusing the same framework. The main contribution of this work is able) to zero in finite time and maintaining it in zero to extend the Lyapunov method to the discontinuous to extend the Lyapunov method to the discontinuous high frequency frequency control switching. For Higher-order standard SM SM the the arbitrary first time. high switching. For standard relative degree iscontrol restricted to be one. to extendorder the Levant’s Lyapunovdifferentiator method to for thethe high frequency switching. For Higher-order standard SM slidthe arbitrary arbitrary order Levant’s differentiator for thediscontinuous first time. time. order Levant’s differentiator for the first relative degree iscontrol restricted to berestriction one. slidrelative degree is restricted to be one. Higher-order sliding modes (HOSM) remove this and allow an arbitrary order Levant’s differentiator for the time. The rest of the paper is organized as follows. first In the next relative degree is restricted to berestriction one. Higher-order sliding modes (HOSM) remove this and allow an ing modes (HOSM) remove this restriction and allow an arbitrary relative degree between the sliding variable and The rest of the paper is organized as follows. In the next The rest of the paper is organized as follows. In the next section 2, we give some preliminaries on homogeneous ing modes (HOSM) remove this restriction and allow an The rest of the paper is organized as follows. In the next arbitrary relative degree between the sliding variableis and and arbitrary relative between theSM sliding variable the control. One ofdegree the main uses of and HOSM the section section 2, 2,and we systems. give some some preliminaries on homogeneous homogeneous we give preliminaries on functions In Section 3 we present the main arbitrary relative degree between the sliding variable and section 2, we give some preliminaries on homogeneous the control. One of the main uses of SM and HOSM is the the control. One main HOSM the robust and finite-time andand observation functions andpaper, systems. In Section we present approach the main main and systems. present the result of the and In weSection provide333awe Lyapunov the control. One of of the thedifferentiation main uses uses of of SM SM and HOSM is isLevthe functions functions andpaper, systems. In Section we present approach the main robust and 2003); finite-time differentiation and observation Levrobust and finite-time differentiation and observation Levant (1998, Davila et al. (2005). Convergence proofs result of the and we provide a Lyapunov of the the4.paper, paper, and some we provide provide Lyapunov in Section We give particular cases inapproach Section robust and 2003); finite-time differentiation and observationproofs Lev- result result of and we aa Lyapunov ant (1998, (1998, Davila et al. (2005). (2005). Convergence ant 2003); Davila et al. Convergence proofs for these differentiators have been obtained mainly by in in Section Section 4.with We give give some particular cases inapproach Section 4. We some particular cases in Section 5, together a simulation study to illustrate the ant (1998, 2003); Davila et al. (2005). Convergence proofs in Section 4. We give some particular cases in Section for these these differentiators differentiators have been obtained mainly mainly by 5, together with a simulation study to illustrate for have been obtained by geometric methods and/or using homogeneity properties the 5, together with a simulation study to illustrate differences for the different algorithms. In Section 6 the we for these differentiators have been obtained mainly by 5, together with a simulation study to illustrate the geometric methods and/or using homogeneity properties geometric methods and/or using homogeneity properties Levant (1993, 1998, 2003, 2005). For first order HOSM difdifferences for the different algorithms. In Section 6 we differences for the different algorithms. In Section 6 we draw some conclusions. geometric methods and/or using homogeneity properties for the different algorithms. In Section 6 we Levant (1993, (1993,the 1998, 2003, 2005). 2005). For For first order order HOSM difdif- differences Levant 1998, 2003, first HOSM ferentiators, super-twisting algorithm, (non-smooth) draw some conclusions. draw some conclusions. Levant (1993,the 1998, 2003, 2005). For first order HOSM dif- draw some conclusions. ferentiators, super-twisting algorithm, (non-smooth) ferentiators, the super-twisting algorithm, (non-smooth) Lyapunov Functions were constructed in Polyakov and ferentiators, the super-twisting algorithm, (non-smooth) Lyapunov(2009); Functions were and constructed in Polyakov Polyakov and Lyapunov Functions were constructed in Poznyak Moreno Osorio (2008, 2012). and For 2. PRELIMINARIES Lyapunov Functions were constructed in Polyakov and Poznyak (2009); (2009); Moreno and Osorio Osorioa (2008, (2008, 2012). LyaFor 2. PRELIMINARIES PRELIMINARIES Poznyak Moreno and 2012). For second-order HOSM differentiators non-smooth 2. Poznyak (2009); Moreno and Osorio (2008, 2012). For 2. PRELIMINARIES second-order HOSM differentiators a non-smooth non-smooth Lyasecond-order HOSM differentiators a Lyapunov function was presented in Moreno (2012). However, second-order HOSM differentiators a non-smooth Lya- An important property of the systems appearing in the punov function was presented in Moreno (2012). However, punov presented Moreno (2012). However, for the function arbitrarywas order HOSMin differentiator the construcAn important important property of of the the we systems appearing in the the property systems in punov function was presented in Moreno (2012). However, paper is the homogeneity, recall appearing briefly. An important property ofthat the we systems in the for the the arbitrary order HOSM differentiator the construcconstruc- An for order HOSM the tion of aarbitrary Lyapunov function is differentiator still an open problem. paper is the the homogeneity, homogeneity, that recall appearing briefly. paper is that we recall briefly. for the arbitrary order HOSM differentiator the construcT n paper the homogeneity, we recall briefly. tion of of a Lyapunov function function is is still still an open open problem. problem. tion a is given vector x = (xthat 1 , ..., xn )T ∈ Rn , the dilation tion of aa Lyapunov Lyapunov is still an anby open problem.algo- For T ∈ Rrnn, the dilation Differentiation can function be also obtained continuous r(x rn1) For a given vector x = (x , ..., x ) 1 For a given vector x = , ..., x ∈ R dilation T1 , ..., n , xthe operator is defined as =∆ r(x x 1:= > 0, n ), ∀ a given vector x , ...,( xrrnn11)x ∈ R rrnn,xthe dilation Differentiation can be be (linear) also obtained obtained by continuous continuous algo- For Differentiation can also by algor x 1:= rithms. In particular, High-Gain observers Vasiloperator is defined as ∆ ( x , ..., ), ∀ >r = 0, 1 n Differentiation can be also obtained by continuous algooperator is defined as ∆ x := ( x , ..., x ), ∀ > 0, r r r where r > 0 are the weights of the coordinates. Let n n i is defined as ∆ x := ( 1 x1 operator , ..., x ), ∀ > = 0, rithms. In particular, (linear) High-Gain observers Vasil1 n rithms. particular, High-Gain jevic andIn Khalil (2008)(linear) have found wide useobservers in outputVasilfeed- (r n r where r > 0 are the weights of the coordinates. Let i rithms. In particular, (linear) High-Gain observers Vasilwhere r > 0 are the weights of the coordinates. Let r = , ..., ) be the vector of weights. A function V : R → R i 1 n > 0 are the weights of the coordinates. Let where r r = jevic and Khalil (2008) have found wide use in output feedn i jevic and Khalil (2008) have found wide use in output feedn →R back control. Continuous and homogeneous differentiators nA n (r ,, ..., rrnn )) be the vector of weights. function V :: R 1 jevic and Khalil (2008) have found wide use in output feed(r ..., be the vector of weights. A function V R n →R (respectively, a vector field f : R → R , or a vector-set 1 vectorfield of weights. V :R →R back control. control. Continuous Continuous andtohomogeneous homogeneous differentiators 1 , ..., rn ) be thevector back and (observers) also be used estimate thedifferentiators derivatives of (r (respectively, Rnnn A→ →function Rnnn ,,ofor ordegree vector-set back control.can Continuous andtohomogeneous differentiators field fff ::: R R aaa vector-set field F (x) ⊂ Raaann )vector is called r-homogeneous m∈R (respectively, vector field R → R , or vector-set can also be be used used estimate the derivativeshas of (respectively, (observers) can also to estimate the derivatives of n a(observers) signal. In Perruquetti et al. (2008) such an algorithm r m r ∈ field Fidentity (x) ⊂ ⊂R RnV))(∆ is called called r-homogeneous of degree degree m ∈= R (observers) can also be used to estimate the derivativeshas of if field (x) is of R theF x) = r-homogeneous m V (x) holds (resp., f (∆m x) field F (x) ⊂ R ) is called r-homogeneous of degree ∈= R a signal. In Perruquetti et al. (2008) such an algorithm r r abeen signal. In Perruquetti et al. (2008) such an algorithm has r x) = l mrV (x) holds (resp., f (∆m r x) presented, with a proof using homogeneous properties l r r if the identity V (∆ a signal. In Perruquetti et al. (2008) such an algorithm has if the identity V (∆ x) = V (x) holds (resp., f (∆ x) = r m r ∆ f (x), or F (∆ x) = ∆ F (x)), Bacciotti and Rosier if the identity V (∆ x) = V (x) holds (resp., f (∆ x) = been presented, with a proof using homogeneous properties l r r l r been presented, with aa proof using homogeneous properties = l ∆r F (x)), Bacciotti and Rosier l ∆r f (x), or F (∆r x) and continuity stability respect to the vector field. (2005); been presented,of with proofwith using homogeneous properties or (∆ l ∆ Bacciotti and F et l ∆r r Levant Bernuau al. (2014). Consider that ∆ ff (x), (x), or F F(2005); (∆r x) x) = = ∆ F (x)), (x)), Bacciotti and Rosier Rosier andLyapunov continuity of stability with respect to the theinvector vector field. and continuity of stability respect to A approach haswith been developed Yangfield. and (2005); r et Levant (2005); Bernuau al. (2014). Consider that and continuity of stability with respect to the vector field. (2005); Levant (2005); Bernuau et al. (2014). Consider that the vector r and dilation ∆ r x are fixed. The Consider homogeneous (2005); Levant (2005); Bernuau et al. (2014). that A Lyapunov approach has been developed in Yang and A Lyapunov approach been developed in Yang r x are 1 Lin (2004); Qian and Linhas (2006); Andrieu et al. (2008)and for the homogeneous fixed. vector rr and dilation ∆ The A Lyapunov approach has been developed in Yang and the vector and dilation ∆ fixed. The p homogeneous x are r p n the vector r and dilation ∆ x are fixed. The homogeneous 1 Lin (2004); Qian and Lin (2006); Andrieu et al. (2008) for Lin (2004); Qian continuous |x | rpi ∈ Rnn , Lin (2004); algorithms. Qian and and Lin Lin (2006); (2006); Andrieu Andrieu et et al. al. (2008) (2008) for for norm is defined by xr, p := n i=1 ppp11 ,, ∀x n |xi | rrp pi continuous algorithms. := norm is defined by x ∀x ∈ i n continuous algorithms. ii | r norm is defined by x ,, ∀x ∈ R Rnn ,,, r, i=1 n |x r, p p := i=1 continuous algorithms. by set xS ∀x for anyis pdefined ≥ 1. The {x ∈ R :|xx 1} ∈is R the i | ir, p = In this paper we provide a smooth Lyapunov function for norm r, p=:= i=1 n n : x any p ≥ 1. The set S = {x ∈ R = 1} 1} is is the the In this this paper we provide aa(linear smooth Lyapunov function for for for any p ≥ 1. The set S = {x ∈ R n : xr, r, p p = In paper we provide smooth Lyapunov function for the family of continuous and nonlinear) Vasiljevic corresponding unit sphere. for any p ≥ 1. The set S = {x ∈ R : x = 1} is the In paper provide a(linear smooth function for corresponding unit sphere. r, p thethis family of we continuous andLyapunov nonlinear) Vasiljevic the family of continuous (linear and nonlinear) Vasiljevic corresponding unit sphere. the family of continuous (linear and nonlinear) Vasiljevic corresponding unit sphere. Copyright © 2016, 2016 IFAC 672Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 672 Copyright ©under 2016 responsibility IFAC 672Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 672 10.1016/j.ifacol.2016.10.241
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Homogeneous functions and vector fields have important properties. For example, for a given family of dilations ∆r x, and any homogeneous functions V1 , V2 (respectively, a vector field f1 ) w.r.t. ∆r x of degrees m1 , m2 (resp., l1 ), we have Bacciotti and Rosier (2005): (i) V1 V2 is homogeneous of degree m1 + m2 . (ii) There exists a m constant c1 > 0 such that V1 ≤ c1 x r, 1p . Moreover, if V1 is positive definite, there exists c2 > 0 such that m V1 ≥ c2 x r, 1p . (iii) ∂V1 (x) /∂xi is homogeneous of degree m1 − ri , with ri being the weight of xi . (iv) Lf V1 (x) is homogeneous of degree m1 + l1 . Homogeneous systems have important properties as e.g. that local stability implies global stability and if the homogeneous degree is negative asymptotic stability implies finite time stability Bacciotti and Rosier (2005); Levant (2005); Bernuau et al. (2014): Assume that the origin of a Filippov Differential Inclusion (DI), x˙ ∈ F (x), is strongly locally Asymptotic Stable and the vector-set field F is r-homogeneous of degree l < 0; then, x = 0 is strongly globally finite-time stable and the settling time is continuous at zero and locally bounded. Along this paper we use the following notation. For a real variable z ∈ R and a real number p ∈ R the symbol zp = |z|p sign(z) is the signed power p of z. We recall the following well-known properties of continuous homogeneous functions Hestenes (1966); Bhat and Bernstein (2005); Andrieu et al. (2008) Lemma 1. Let η : Rn → R and γ : Rn → R+ , that is γ(x) ≥ 0 ∀x, be two continuous homogeneous functions, with weights r = (r1 , ..., rn ) and degrees m, such that {x ∈ Rn \ {0} : γ(x) = 0} ⊆ {x ∈ Rn \ {0} : η(x) < 0}. Then, there exists a real number λ∗ such that, for all λ ≥ λ∗ for all x ∈ Rn \ {0}, and some c > 0, η(x) − m λγ(x) < −c x r, p . Lemma 2. Suppose V1 and V2 are continuous real-valued functions on Rn , r-homogeneous of degrees m1 > 0 and m2 > 0, respectively, and V1 is positive definite. Then, for every x ∈ Rn , with S = {z : V1 (z) = 1} m2 m2 m1 min V2 (z) V1 (x) ≤ V2 (x) ≤ max V2 (z) V1m1 (x) . S
S
3. THE ARBITRARY ORDER DIFFERENTIATOR Let the input signal f (t) to the differentiator be a Lebesgue-measurable function defined on [0, ∞). f (t) is assumed to be decomposed as f (t) = f0 (t) + v(t). The first term is the n-times differentiable unknown base signal f (t), to be differentiated, with a n-th time derivative having a known Lipschitz constant L > 0, while v(t) corresponds to a uniformly bounded noise signal. Defining (i) di the variables ς1 = f0 (t) , . . . , ςi+1 = f0 (t) dt i f0 (t), i = 1, ..., n, a state representation of the base signal is ς˙i = ςi+1 , i = 1, · · · , n − 1, ς˙n =
(n) f0
To estimate the derivatives of the base signal, we consider the following family of homogeneous differentiators x˙ i = −ki x1 − f
ri+1 r1
x˙ n = −kn x1 − f
rn+1 r1
+ xi+1 , i = 1, · · · , n − 1
where ri = ri+1 − d = rn − (n − i) d , i = 1, . . . , n , (we introduce the parameter rn+1 = rn + d for simplicity) and −rn ≤ d ≤ 0. We fix hereafter (without loss of generality) rn = 1. Note that r1 ≥ · · · ≥ ri ≥ · · · ≥ rn = 1 ≥ rn+1 . Since for d = −1 systems (1) and (2) are discontinuous, their solutions are understood in the sense of Filippov Filippov (1988). The family of differentiators (1) is parametrized by −1 ≤ d ≤ 0. For d = 0 we recover a linear differentiator, while for d = −1 we recover Levant’s n-th order differentiator Levant (2003, 2005). Theorem 3. Under the stated assumptions for the signal f (t) there exist appropriate gains (depending on d and the Lipschitz constant of f (t)) such that the family of homogeneous differentiators (1) has the following properties. (1) In the absence of noise (v (t) ≡ 0) and for the class of polynomial signals, satisfying Spn = f (n) (t) ≡ 0 , the estimation xi (t) converges to the derivative f (i−1) (t) as t → ∞ exponentially for d = 0 and in finite time for −1 ≤ d < 0, i = 1, · · · , n. (2) In the absence of noise (v (t) ≡ 0) and of for the class n-Lipschitz signals satisfying SLn = f (n) (t) ≤ L , the differentiation estimation error xi (t) − f (i−1) (t) converges to zero in finite time for d = −1 and it is ultimately uniformly bounded for −1 < d ≤ 0, i = 1, · · · , n. For d = −1 it is required that kn > L. (3) For a uniformly bounded noise (|v (t)| ≤ η) and for the class of n-Lipschitz signals satisfying SLn = f (n) (t) ≤ L , the differentiation estimation error xi (t) − f (i−1) (t) is ultimately uniformly bounded for −1 ≤ d ≤ 0, i = 1, · · · , n. The results of Theorem 3 are well-known, in particular for the linear d = 0, close to linear − < d < 0 Perruquetti et al. (2008) and the discontinuous d = −1 Levant (2003, 2005) cases. The main novelty in this paper is that we provide a unified Lyapunov approach to prove these properties in all cases, the continuous −1 < d ≤ 0 and (in particular) the discontinuous d = −1 one. Beyond the intrinsic beauty of this fact, it opens the possibility of making comparisons and design under the same footing. Note that the discontinuous differentiator d = −1 (Levant’s differentiator) has an astonishing distinguishing feature: in the absence of noise it is exact (i.e. it converges in finite time) to a much larger class of signals SLn than its continuous counterparts, which are only exact for the much thiner set Spn ⊂ SLn for −1 < d < 0, or it only converges exponentially in the linear case d = 0. 4. A LYAPUNOV APPROACH FOR THEOREM 3 (i−1)
, in Defining the differentiation error as ei xi − f0 the absence of noise (v (t) ≡ 0), their dynamics is given by e˙ i = −ki e1
(t) .
(1)
, 673
661
ri+1 r1
e˙ n = −kn e1 (n)
rn+1 r1
+ ei+1 , i = 1, · · · , n − 1
(2)
+ δ (t) ,
where δ (t) = −f (t) is the n-th order derivative, which is assumed to be bounded, i.e. |δ (t)| ≤ ∆. For δ (t) ≡ 0 this system is homogeneous with homogeneity degree d and weights r = [r1 , · · · , rn ]. Performing the state transformation (for i = 1, · · · , n)
IFAC NOLCOS 2016 662 Emmanuel August 23-25, 2016. Monterey, California, USA Cruz-Zavala et al. / IFAC-PapersOnLine 49-18 (2016) 660–665
e1 e2 ei en , z2 = , · · · , z i = , · · · , zn = , 1 k1 ki−1 kn−1 the dynamics of (2) becomes ri+1 z˙i = −k˜i z1 r1 − zi+1 , i = 1, · · · , n − 1 (3) z1 =
where υn = −βn−1 (p − rn−1 ) |zn | This can be rewritten as
Zi (zi , zi+1 ) =
ri |zi | − zi zi+1 p p p − ri |zi+1 | ri+1 , p
p−ri ri+1
p ri
+
(4)
Note that σi = 0 on the same set as si = 0. The partial derivatives of Zp are
p−ri p−ri ∂Zi (zi , zi+1 ) = zi ri − zi+1 ri+1 = σi ∂zi p−ri −ri+1 p − ri ∂Zi (zi , zi+1 ) =− |zi+1 | ri+1 si , ∂zi+1 ri+1 which are continuous. They become both zero at the points where Zi achieves its minimum.
We prove Theorem 3 by using the following Theorem, which is the main result of the paper: Theorem 4. Each differentiator of the form (1) admits a strong, proper, smooth and r−homogeneous of degree p Lyapunov function of the form 1 p βj Zj (zj , zj+1 ) + βn |zn | V (z) = p j=1 βi > 0 , i = 1, · · · , n .
V (z) is non negative, since it is a positive combination of non negative terms. Moreover, V (z) is positive definite since V (z) = 0 implies that z = 0. Due to homogeneity it is radially unbounded Bhat and Bernstein (2005). For the linear case (d = 0, p = 2) V is a quadratic form.
V˙ (z) =
n−2 j=1
rj+1 βj −k˜j σj z1 r1 − zj+1 +
p−rj −rj+1 rj+2 p − rj |zj+1 | rj+1 sj z1 r1 rj+1 1 −k˜n−1 βn−1 σn−1 z1 r1 − zn − +k˜j+1
rn+1 r1
−k˜n υn z1
n−2 j=1
.
(5)
p−r1 r2 p−r1 z1 r1 − z2 + W = k˜1 β1 z1 r1 − z2 r2
(6)
rj+2 rn+1 k˜j+1 υj+1 z1 r1 − zj+2 + k˜n υn z1 r1
and for j = 1, · · · , n − 2 p−rj −rj+1 p − rj βj |zj+1 | rj+1 sj + βj+1 σj+1 . rj+1 We will show that there exist values of k˜i > 0 such that W (z) > 0. We consider first the continuous case and then the discontinuous one.
4.1 The continuous case: −1 < d ≤ 0 To show that W (z) > 0 we exploit the structure of W . So consider the values of W restricted to some hypersurfaces: for i = 1, · · · , n − 1 ri+1 r2 Zi = z1 r1 = z2 ∧ · · · ∧ z1 r1 = zi+1 .
These sets are related as Zn−1 ⊂ · · · ⊂ Z1 . Note that on Zi the variables σi and si vanish, i.e. σi = si = 0, and therefore they also vanish on Zj , for every j > i. Let Wi = WZi represent the value of W (z) restricted to the the manifold Zi . We can obtain the value of W1 r1 r2 by replacing in W (z) the variable z1 by z1 = z2 , so that W1 becomes a function of (z2 , · · · , zn ). In general, we obtain the value of Wi , for i = 1, . . . , n − 1, by replacing in W (z) the variables (z1 , · · · , zi ) by its values in terms of ri
r1
zi+1 , i.e. z1 = zi+1 ri+1 , · · · , zi = zi+1 ri+1 , so that Wi becomes a function of z¯i+1 (z i+1 , · · · , zn ). For example,
the value of the expression ri+2 zi+1 ri+1 − zi+2 .
Its derivative is
where
p−1
υj+1 = −
which are continuously differentiable, positive semidefinite p p and Zi (zi , zi+1 ) = 0 if and only if zi ri = zi+1 ri+1 . For simplicity we introduce the variables p−ri ri p−ri ri+1 ri σi = zi , si = zi − zi+1 ri+1 . − zi+1
n−1
sn−1 + βn zn
V˙ = −W (z) + υn δ¯ (t) ,
rn+1
z˙n = −k˜n z1 r1 + δ¯ (t) , where for i = 1, · · · , n, f (n) (t) ki k˜i = , k0 = 1 , δ¯ (t) = − . ki−1 kn−1 For n ≥ 2 fix p ≥ r1 + r2 = 2 − (2n − 3) d > 1 and for i = 1, · · · , n − 1 define the homogeneous functions
p−rn−1 −1
z1
ri+2 r1
− zi+2
on Zi is
From (6) it is seen that we can write W (z) as W (z) = k˜1 η1 (z1 , z2 ) + µ1 (z) , where
p−r1 r2 p−r1 η1 (z1 , z2 ) = β1 z1 r1 − z2 r2 z1 r1 − z2 .
η1 (z1 , z2 ) is positive everywhere except on the set Z1 , where it vanishes, i.e. η1 (z1 , z2 ) = 0 for z ∈ Z1 . Note also that µ1 does not depend on the gain k˜1 . According to Lemma 1 there exists a sufficiently large positive value of k˜1 such that W (z) > 0 if the value of µ1 restricted to Z1 , i.e. µ1Z1 = W1 (¯ z2 ), is positive. From (6) it is seen that − zj+2 W1 (¯ z2 ) can be written as z2 ) = k˜2 η2 (z2 , z3 ) + µ2 (¯ z2 ) , W1 (¯ p−r2 p−r2 r3 η2 (z2 , z3 ) = β2 z2 r2 − z3 r3 z2 r2 − z3 .
+ υn δ¯ (t) ,
674
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η2 (z2 , z3 ) is positive on Z1 except on the set Z2 , where z2 ) does not depend on the gain k˜2 . it vanishes, and µ2 (¯ According to Lemma 1 there exists a sufficiently large positive value of k˜2 such that W1 (¯ z2 ) is positive definite on Z1 if µ2Z2 = W2 (¯ z3 ) is positive. This can be done recursively. From (6) it is seen that Wi (¯ zi+1 ), for i = 1, . . . , n − 1, can be written as
V˙ = ∇V · fη (e, η) + ∇V · f (e) + ∇V · f∆ (e, η) . Due to the homogeneity and the H¨older continuity of f (e) and applying Lemma 2 we obtain that ∇V · f∆ = ≤
zi+1 ) = k˜i+1 ηi+1 (zi+1 , zi+2 ) + µi+1 (¯ zi+1 ) , Wi (¯
p−ri+1 ri+1
− zi+2 ηi+1 (·) = βi+1 zi+1 ri+2 zi+1 ri+1 − zi+2 .
p−ri+1 ri+2
V˙ ≤ −α2 V
Finally, we obtain that we can write zn−1 ) = k˜n−1 ηn−1 (¯ zn−1 ) + µn−1 (¯ zn−1 ) , Wn−2 (¯ p−rn−1 rn−1
− zn ηn−1 = βn−1 zn−1 rn zn−1 rn−1 − zn , µn−1 = k˜n υn zn−1
p−rn−1 rn
n
k˜i µi V
p−ri p
i=1
p+d p
|η|
ri+1 r1
(7)
+ α3 LV
p−1 p
+
n
k˜i µi V
i=1
p−1 r1
where
−k˜n υn z1 ≤ k˜n θn − k˜n βn |z1 | p−r
p−1
≤ −α2 V p + α3 LV p , for some αi > 0, i = 1, 2, 3, and from this, using standard arguments, we conclude that the differentiation error is ISS with respect to δ (t). To prove item 3) we notice that (2), i.e. e˙ = f (e) + bδ (t), when we have a measurement noise η (t) becomes e˙ = fη (e, η) + bδ (t), with fη (e, η) = f (e1 + η, e2 , · · · , en ), and f∆ (e, η) = fη (e, η) − f (e). The derivative of V along the solutions of the noisy system is
675
,
and replacing this expression in υn it 0
cludes the proof of 1) in Theorem 3 in the continuous case. To prove Item 2) in the theorem we observe that W (z) (6) is continuous and r-homogeneous of degree p + r2 − r1 = p + d while vn is r-homogeneous of degree p − 1, whith p − 1 < p + d for −1 < d. From (5) and Lemma 2 we obtain p−1 δ (t) ≤ −W + α1 LW p+d V˙ ≤ −W + |υn | ¯
ri+1 r1
For this case (note that rn+1 = 0) we observe that W (6) is discontinuous and has the same homogeneous degree as vn . We introduce a modification in the last term of W p−1 p−1 to deal with this situation. Since zn = − z1 r1 +
×
which is positive on Zn−1 for every positive values of k˜n and βn . We conclude therefore, that there exist positive values of k˜1 , · · · , k˜n such that W (z) > 0. This con-
|η|
4.2 The discontinuous case: d = −1
p−1
rn+1 rn−1
p−ri p
for some µi > 0. Again, using standard arguments we conclude that the differentiation error is ISS with respect to the noise η (t).
zn + z1 follows that
Note that µn−1 (¯ zn−1 ) does not depend on k˜n−1 . ηn−1 is positive on Zn−2 except on the set Zn−1 , where it vanishes. According to Lemma 1 there exists a sufficiently large value of k˜n−1 such that Wn−2 (¯ zn−1 ) is positive if µn−1, Zn−1 = Wn−1 (zn ) is positive. From (6) we see that p−1 r p−1+rn+1 zn n+1 = k˜n βn |zn | , Wn−1 (zn ) = k˜n βn zn
p+d
ri+1 ri+1 ∂V z1 r1 − z1 + η r1 k˜i ∂zi i=1
n
so that we finally have that ×
Since ηi+1 (zi+1 , zi+2 ) is positive on Zi except on the set Zi+1 , where it vanishes, and µi+1 (¯ zi+1 ) does not depend on the gain k˜i+1 , Lemma 1 assures the existence of a sufficiently large positive value of k˜i+1 such that Wi (¯ zi+1 ) is positive definite on Zi if µi+1,Zi+1 = Wi+1 (¯ zi+2 ) is positive.
663
θn = βn−1 (p − rn−1 ) |zn | n−1 p−1 r p−1 1 +βn z1 − zn ≥ 0 .
−1
p−1 r1
,
|sn−1 | +
Furthermore, the term due to the perturbation in V˙ (5) can be bounded as follows ∆ p−1 υn δ¯ (t) ≤ k˜n θn + βn |z1 | r1 . kn Considering the case with perturbation we can obtain the following bound rn+1 ∆ −k˜n υn z1 r1 + υn δ¯ ≤ k˜n θn 1 + − kn p−1 ∆ |z1 | r1 , k˜n βn 1 − kn and we can write ¯ (z) , V˙ (z) ≤ −W ¯ is where the continuous and homogeneous function W given by p−r r2 p−r ¯ (z) = k˜1 β1 z1 r1 1 − z2 r2 1 z1 r1 − z2 W +
n−2 j=1
rj+2 k˜j+1 υj+1 z1 r1 − zj+2
(8)
p−1 ∆ ∆ + k˜n βn 1 − |z1 | r1 . −k˜n θn 1 + kn kn By the same procedure as for the continuous case it is possible to show that there exist values of k˜i > 0 such
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¯ (z) > 0. In the last step of the analysis performed that W ¯ that before to W we obtain for W ¯ n−2 (¯ W zn−1 ) = k˜n−1 ηn−1 (¯ zn−1 ) + µ ¯n−1 (¯ zn−1 ) , ∆ µ ¯n−1 = −k˜n θn∗ 1 + + kn p−1 ∆ k˜n βn 1 − |zn−1 | rn−1 , kn
g2 (z1 , z2 )
zn−1 ) does not depend on k˜n−1 . ηn−1 Note that µ ¯n−1 (¯ is positive on Zn−2 except on the set Zn−1 , where it vanishes. According to Lemma 1 there exists a sufficiently ¯ n−2 (¯ large value of k˜n−1 such that W zn−1 ) is positive if ¯ n−1 (zn ) is positive. From (8) we see that µ ¯n−1, Zn−1 = W ∆ p−1 ˜ ¯ , |zn | Wn−1 (zn ) = kn βn 1 − kn which is positive on Zn−1 for every positive values of k˜n and βn , and kn > ∆. We conclude therefore, that there ¯ (z) > 0. exist positive values of k˜1 , · · · , k˜n such that W This concludes the proof of items 1) and 2) in Theorem 3 in the discontinuous case. Item 3) in the Theorem follows along the same reasoning as for the continuous case. We have only to modify the last term in (7), since in this case 0 0 for i = n we have z1 − z1 + η = 0 if |z1 | > |η| and it is 2 otherwise. 5. CASE EXAMPLES
The homogeneous first order differentiator is 1
x˙ 1 = −k1 x1 − f 1−d + x2 1+d
x˙ 2 = −k2 x1 − f 1−d , (9) where x2 (t) is an estimation of the first derivative of the signal f (t). The homogeneity degree is −1 ≤ d ≤ 0, with weights r1 = 1 − d, r2 = 1, and r3 = 1 + d. For d = −1 we obtain Levant’s robust and exact differentiator, while for d = 0 we get a linear differentiator. The Lyapunov Function is 2−d 1−d β+1 2−d V (z1 , z2 ) = |z1 | 1−d − z1 z2 + |z2 | , 2−d 2−d which is positive definite for every β > 0 and continuously differentiable. Its derivative V˙ is 2
2 V˙ = −k1 |σ1 | + k˜2 (1 + β) s1 z1 1−d − k˜2 β |z1 | 1−d 2
1+d
s1 − β z2
1−d
1+d
z1 1−d
. 2 |σ1 | g2 (z1 , z2 ) is a homogeneous function of degree zero, it is upper semicontinuous and it has a maximum, that is achieved on the homogeneous sphere.
p−rn−1 −rn p − rn−1 rn |sn−1 | + |zn | θn∗ = βn−1 rn p−rn p−rn +βn zn−1 rn−1 − zn rn .
1+d
2
The first term |σ1 | is non negative and it vanishes when 1 2 z1 1−d = z2 . At these points V˙ ≤ −k˜2 β |z2 | , which is negative. Lemma 1 assures the existence of a value of k1 rendering V˙ negative definite. From V˙ we find that the required value of k1 satisfies k2 k1 = 1 > ω2 max2 g2 (z1 , z2 ) , k2 z∈R k˜2
2
≤ −k1 |σ1 | + k˜2 (1 + β) |s1 | |z1 | 1−d − k˜2 β |z1 | 1−d 1 where we have used the variables σ1 = z1 1−d − z2 , 1−d s1 = z1 − z2 . Note that the right hand side of the last inequality is continuous for all −1 ≤ d ≤ 0. 676
The homogeneous second order differentiator is given by 1−d
x˙ 1 = −k1 x1 − f 1−2d + x2 1
x˙ 2 = −k2 x1 − f 1−2d + x3 x˙ 3 = −k3 x1 − f
(10)
1+d 1−2d
where x2 (t) estimates f (1) (t) and x3 (t) estimates f (2) (t). The homogeneity degree is −1 ≤ d ≤ 0 and the weights are r1 = 1 − 2d, r2 = 1 − d, r3 = 1, r4 = 1 + d. The LF is given by (with p = 2 − 3d)
β3 p |z3 | p which is positive definite for any positive values of β2 > 0 and β3 > 0 and it is C 1 , and the gains that render V˙ < 0 are found solving the inequalities k1 k2 k22 > ω23 , > ω13 , k1 k3 k3 where ω23 depends only on the values of (β2 , β3 , d), while ω13 depends on (β2 , β3 , d, γ23 , ω23 ). V (z) = Z1 (z1 , z2 ) + β2 Z2 (z2 , z3 ) +
Note that these relations are invariant with respect to scaling of the gains, i.e. if we scale the gains k1 → Lk1 , k2 → L2 k2 , k3 → L3 k3 for any L > 0 then the previous relations are still fulfilled. To illustrate the behavior of the differentiators for different values of d we present some simulations results. Consider which the signal f0 (t) = 0.5 sin (0.5t) + 0.5 cos (t), has (3) a non vanishing and bounded derivative f0 (t) ≤ 1. Figure 1 shows the differentiation errors for the linear differentiator (d = 0), a homogeneous differentiator (d = −0.5) and the discontinuous (Levant’s) differentiator (d = −1) in the absence of noise. Figure 2 shows the results with measurement noise v (t) = ε sin (ωt) with ε = 0.001 and ω = 1000. For all differentiators we have selected the √ same gains k1 = 3, k2 = 1.5 3, k3 = 1.1, and we used Euler-method with step size τ = 3 × 10−4 .
Figure 1 shows that in the absence of noise only the discontinuous differentiator (d = −1) can estimate without error (1) (2) both derivatives f0 (t) and f0 (t). When noise is present Figure 2 shows that the discontinuous differentiator is strongly affected by the noise, but its estimate is still better than the ones provided by the continuous differentiators.
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REFERENCES
f (t)=0.5sin(0.5t)+0.5cos(t), noise=0 0
0.7
2
3
d=0 d=−0.5 d=−1
0.6
d=0 d=−0.5 d=−1
d=0 d=−0.5 d=−1
2.5
1.5 0.5 2 0.4 1 1.5
0.2
e3
e2
e1
0.3 0.5
0.1
1
0.5 0
0 0 −0.1 −0.5 −0.5
−0.2
0
10
20
−1
30
0
10
t
20
−1
30
0
10
t
20
30
t
Fig. 1. Differentiation error for d = 0, d = −0.5, and d = −1 (Levant’s Differentiator) without noise. f0(t)=0.5sin(0.5t)+0.5cos(t), noise=0.001sin(1000t) 0.7
2
3
d=0 d=−0.5 d=−1
0.6
d=0 d=−0.5 d=−1
d=0 d=−0.5 d=−1
2.5
1.5 0.5 2 0.4 1 1.5
0.2
e3
e2
e1
0.3 0.5
0.1
1
0.5 0
0 0 −0.1 −0.5 −0.5
−0.2
0
10
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0
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t
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0
10
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t
Fig. 2. Differentiation error for d = 0, d = −0.5, and d = −1 (Levant’s Differentiator) in presence of noise. 6. CONCLUSIONS For a family of continuous and discontinuous homogeneous differentiators we provide a unified smooth Lyapunov function to study the convergence, robustness and performance properties. Comparison of performance and possible optimization of the behavior of these differentiators is part of ongoing research. ACKNOWLEDGEMENTS The authors would like to thank the financial support from PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigaci´ on e Innovaci´ on Tecnol´ ogica), project IN113614; Fondo de Colaboraci´ on II-FI UNAM, Project IISGBAS100-2015; CONACyT (Consejo Nacional de Ciencia y Tecnolog´ıa), project 241171 and CVU 267513. 677
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ScienceDirect IFAC-PapersOnLine 49-18 (2016) 660–665
Lyapunov Functions for Continuous and Lyapunov Functions for Lyapunov FunctionsDifferentiators for Continuous Continuous and and Discontinuous Discontinuous Differentiators Discontinuous Differentiators ∗ ∗∗
Emmanuel Cruz-Zavala ∗ Jaime A. Moreno ∗∗ ∗∗ Emmanuel Cruz-Zavala ∗∗ Jaime A. Moreno Emmanuel Emmanuel Cruz-Zavala Cruz-Zavala Jaime Jaime A. A. Moreno Moreno ∗∗ ∗ Depto de Ciencias de la Computaci´ on. Centro Universitario de ∗ ∗ Depto de Ciencias de la Computaci´ o n.Universidad Centro Universitario de de Ciencias de la Computaci´ o Centro de ∗ Depto Ciencias Exactas e Ingenier´ ıas (CUCEI), de Guadalajara, Depto de Ciencias de laıas Computaci´ on. n.Universidad Centro Universitario Universitario de Ciencias Exactas e Ingenier´ (CUCEI), de Guadalajara, Ciencias Exactas e Ingenier´ ıas (CUCEI), Universidad de Guadalajara, Guadalajara, Jalisco, Mexico (e-mail: [email protected]) Ciencias Exactas e Ingenier´ ıas (CUCEI), Universidad de Guadalajara, ∗∗ Guadalajara, Jalisco, Mexico Mexico (e-mail:[email protected]) [email protected]) Guadalajara, Jalisco, (e-mail: El´ectrica y Computaci´ on, Instituto Ingenier´ıa, Universidad Guadalajara, Jalisco, Mexico (e-mail:[email protected]) ∗∗ ∗∗ eectrica yynoma Computaci´ o n, Instituto Ingenier´ Universidad El´ ctrica Computaci´ o n, Instituto de Ingenier´ ıa, Universidad ∗∗ El´ Nacional Aut´ o de M´ e xico. Ciudad de M´exico,ıa, Mexico (e-mail: El´ e ctrica y Computaci´ o n, Instituto de Ingenier´ ıa, Universidad Nacional Aut´ o noma de M´ e xico. Ciudad de M´ e xico, Mexico (e-mail: Nacional Aut´ o noma de M´ e xico. Ciudad de M´ e xico, Mexico [email protected]) Nacional Aut´ onoma de M´exico. Ciudad de M´exico, Mexico (e-mail: (e-mail: [email protected]) [email protected]) [email protected]) Abstract: Given a (differentiable) signal it is an important task for many applications to Abstract: (differentiable) it is an important applications to Abstract: Given a (differentiable) signal it important task for many applications to estimate on Given line itsa Somesignal well known algorithms to task solvefor thismany problem include the Abstract: Given aderivatives. (differentiable) signal it is is an an important task for many applications to estimate on line its derivatives. Some well known algorithms to solve this problem include the estimate on line its derivatives. Some well known algorithms to solve this problem include the (continuous) high-gain observersSome and (discontinuous) Levant’s to exact differentiators. In this work estimate on line its derivatives. well known algorithms solve this problem include the (continuous) high-gain observers and and (discontinuous) (discontinuous)encompassing Levant’s exact exact differentiators. In this this work (continuous) high-gain observers Levant’s differentiators. In work we present a family of homogeneous these two algorithms, and we (continuous) high-gain observers anddifferentiators, (discontinuous)encompassing Levant’s exact differentiators. In this work we present a family of homogeneous differentiators, these two algorithms, and we we present a family of homogeneous differentiators, encompassing these two algorithms, and we propose a unified smooth Lyapunov function, that allows a common framework to study their we present a family of homogeneous differentiators, encompassing these two algorithms, and we propose a unified smooth Lyapunov function, that allows aa common framework to study their propose a unified smooth Lyapunov function, that allows common framework to study their convergence and performance analysis. propose a unified smooth Lyapunov function, that allows a common framework to study their convergence and and performance performance analysis. analysis. convergence convergence performance analysis. © 2016, IFAC and (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Observability and Observer Design; Lyapunov Stability Methods; Variable Keywords:Control Observability and Observer Observer Design; Lyapunov Lyapunov Stability Methods; Methods; Variable Keywords: Observability and Structure and Sliding Mode Design; Keywords: Observability and Observer Design; Lyapunov Stability Stability Methods; Variable Variable Structure Control Control and Sliding Sliding Mode Structure and Mode Structure Control and Sliding Mode 1. INTRODUCTION and Khalil (2008); Yang and Lin (2004) and discontinuous 1. INTRODUCTION INTRODUCTION and Khalil Khalil (2008); Yang and and Lin (2004) and and discontinuous 1. and (2008); Yang (2004) Levant (2003) homogeneous differentiators in a unified 1. INTRODUCTION and Khalil (2008); Yang and Lin Lin (2004) and discontinuous discontinuous Levant (2003) homogeneous differentiators inrobustness unified Levant (2003) homogeneous differentiators aa This allows to study the convergence,in Sliding Modes (SM) is a well-known technique to control manner. Levant (2003) homogeneous differentiators inrobustness a unified unified manner. This allows to study the convergence, manner. This allows to study the convergence, robustness Sliding Modes (SM) is a well-known technique to control and performance properties of all the algorithms in the Sliding Modes (SM) is a well-known technique to control uncertain systems, by bringing some variable (sliding varimanner. This allows to study the convergence, robustness Sliding Modes (SM) isbringing a well-known technique to control and performance properties of all the algorithms in the the and performance properties of all the algorithms in uncertain systems, by some variable (sliding varisame framework. The main contribution of this work is uncertain systems, by bringing some variable variable) to zero in finite time and maintaining it (sliding in zero using and performance properties of all the algorithms in the uncertain systems, by bringing some variable (sliding varisame framework. The main contribution of this work is same framework. The main contribution of this work is able) to zero in finite time and maintaining it in zero using to extend the Lyapunov method to the discontinuous able) to zero in finite time and maintaining it in zero using high frequency control switching. For standard SMusing the same framework. The main contribution of this work is able) to zero in finite time and maintaining it in zero to extend the Lyapunov method to the discontinuous to extend the Lyapunov method to the discontinuous high frequency frequency control switching. For Higher-order standard SM SM the the arbitrary first time. high switching. For standard relative degree iscontrol restricted to be one. to extendorder the Levant’s Lyapunovdifferentiator method to for thethe high frequency switching. For Higher-order standard SM slidthe arbitrary arbitrary order Levant’s differentiator for thediscontinuous first time. time. order Levant’s differentiator for the first relative degree iscontrol restricted to berestriction one. slidrelative degree is restricted to be one. Higher-order sliding modes (HOSM) remove this and allow an arbitrary order Levant’s differentiator for the time. The rest of the paper is organized as follows. first In the next relative degree is restricted to berestriction one. Higher-order sliding modes (HOSM) remove this and allow an ing modes (HOSM) remove this restriction and allow an arbitrary relative degree between the sliding variable and The rest of the paper is organized as follows. In the next The rest of the paper is organized as follows. In the next section 2, we give some preliminaries on homogeneous ing modes (HOSM) remove this restriction and allow an The rest of the paper is organized as follows. In the next arbitrary relative degree between the sliding variableis and and arbitrary relative between theSM sliding variable the control. One ofdegree the main uses of and HOSM the section section 2, 2,and we systems. give some some preliminaries on homogeneous homogeneous we give preliminaries on functions In Section 3 we present the main arbitrary relative degree between the sliding variable and section 2, we give some preliminaries on homogeneous the control. One of the main uses of SM and HOSM is the the control. One main HOSM the robust and finite-time andand observation functions andpaper, systems. In Section we present approach the main main and systems. present the result of the and In weSection provide333awe Lyapunov the control. One of of the thedifferentiation main uses uses of of SM SM and HOSM is isLevthe functions functions andpaper, systems. In Section we present approach the main robust and 2003); finite-time differentiation and observation Levrobust and finite-time differentiation and observation Levant (1998, Davila et al. (2005). Convergence proofs result of the and we provide a Lyapunov of the the4.paper, paper, and some we provide provide Lyapunov in Section We give particular cases inapproach Section robust and 2003); finite-time differentiation and observationproofs Lev- result result of and we aa Lyapunov ant (1998, (1998, Davila et al. (2005). (2005). Convergence ant 2003); Davila et al. Convergence proofs for these differentiators have been obtained mainly by in in Section Section 4.with We give give some particular cases inapproach Section 4. We some particular cases in Section 5, together a simulation study to illustrate the ant (1998, 2003); Davila et al. (2005). Convergence proofs in Section 4. We give some particular cases in Section for these these differentiators differentiators have been obtained mainly mainly by 5, together with a simulation study to illustrate for have been obtained by geometric methods and/or using homogeneity properties the 5, together with a simulation study to illustrate differences for the different algorithms. In Section 6 the we for these differentiators have been obtained mainly by 5, together with a simulation study to illustrate the geometric methods and/or using homogeneity properties geometric methods and/or using homogeneity properties Levant (1993, 1998, 2003, 2005). For first order HOSM difdifferences for the different algorithms. In Section 6 we differences for the different algorithms. In Section 6 we draw some conclusions. geometric methods and/or using homogeneity properties for the different algorithms. In Section 6 we Levant (1993, (1993,the 1998, 2003, 2005). 2005). For For first order order HOSM difdif- differences Levant 1998, 2003, first HOSM ferentiators, super-twisting algorithm, (non-smooth) draw some conclusions. draw some conclusions. Levant (1993,the 1998, 2003, 2005). For first order HOSM dif- draw some conclusions. ferentiators, super-twisting algorithm, (non-smooth) ferentiators, the super-twisting algorithm, (non-smooth) Lyapunov Functions were constructed in Polyakov and ferentiators, the super-twisting algorithm, (non-smooth) Lyapunov(2009); Functions were and constructed in Polyakov Polyakov and Lyapunov Functions were constructed in Poznyak Moreno Osorio (2008, 2012). and For 2. PRELIMINARIES Lyapunov Functions were constructed in Polyakov and Poznyak (2009); (2009); Moreno and Osorio Osorioa (2008, (2008, 2012). LyaFor 2. PRELIMINARIES PRELIMINARIES Poznyak Moreno and 2012). For second-order HOSM differentiators non-smooth 2. Poznyak (2009); Moreno and Osorio (2008, 2012). For 2. PRELIMINARIES second-order HOSM differentiators a non-smooth non-smooth Lyasecond-order HOSM differentiators a Lyapunov function was presented in Moreno (2012). However, second-order HOSM differentiators a non-smooth Lya- An important property of the systems appearing in the punov function was presented in Moreno (2012). However, punov presented Moreno (2012). However, for the function arbitrarywas order HOSMin differentiator the construcAn important important property of of the the we systems appearing in the the property systems in punov function was presented in Moreno (2012). However, paper is the homogeneity, recall appearing briefly. An important property ofthat the we systems in the for the the arbitrary order HOSM differentiator the construcconstruc- An for order HOSM the tion of aarbitrary Lyapunov function is differentiator still an open problem. paper is the the homogeneity, homogeneity, that recall appearing briefly. paper is that we recall briefly. for the arbitrary order HOSM differentiator the construcT n paper the homogeneity, we recall briefly. tion of of a Lyapunov function function is is still still an open open problem. problem. tion a is given vector x = (xthat 1 , ..., xn )T ∈ Rn , the dilation tion of aa Lyapunov Lyapunov is still an anby open problem.algo- For T ∈ Rrnn, the dilation Differentiation can function be also obtained continuous r(x rn1) For a given vector x = (x , ..., x ) 1 For a given vector x = , ..., x ∈ R dilation T1 , ..., n , xthe operator is defined as =∆ r(x x 1:= > 0, n ), ∀ a given vector x , ...,( xrrnn11)x ∈ R rrnn,xthe dilation Differentiation can be be (linear) also obtained obtained by continuous continuous algo- For Differentiation can also by algor x 1:= rithms. In particular, High-Gain observers Vasiloperator is defined as ∆ ( x , ..., ), ∀ >r = 0, 1 n Differentiation can be also obtained by continuous algooperator is defined as ∆ x := ( x , ..., x ), ∀ > 0, r r r where r > 0 are the weights of the coordinates. Let n n i is defined as ∆ x := ( 1 x1 operator , ..., x ), ∀ > = 0, rithms. In particular, (linear) High-Gain observers Vasil1 n rithms. particular, High-Gain jevic andIn Khalil (2008)(linear) have found wide useobservers in outputVasilfeed- (r n r where r > 0 are the weights of the coordinates. Let i rithms. In particular, (linear) High-Gain observers Vasilwhere r > 0 are the weights of the coordinates. Let r = , ..., ) be the vector of weights. A function V : R → R i 1 n > 0 are the weights of the coordinates. Let where r r = jevic and Khalil (2008) have found wide use in output feedn i jevic and Khalil (2008) have found wide use in output feedn →R back control. Continuous and homogeneous differentiators nA n (r ,, ..., rrnn )) be the vector of weights. function V :: R 1 jevic and Khalil (2008) have found wide use in output feed(r ..., be the vector of weights. A function V R n →R (respectively, a vector field f : R → R , or a vector-set 1 vectorfield of weights. V :R →R back control. control. Continuous Continuous andtohomogeneous homogeneous differentiators 1 , ..., rn ) be thevector back and (observers) also be used estimate thedifferentiators derivatives of (r (respectively, Rnnn A→ →function Rnnn ,,ofor ordegree vector-set back control.can Continuous andtohomogeneous differentiators field fff ::: R R aaa vector-set field F (x) ⊂ Raaann )vector is called r-homogeneous m∈R (respectively, vector field R → R , or vector-set can also be be used used estimate the derivativeshas of (respectively, (observers) can also to estimate the derivatives of n a(observers) signal. In Perruquetti et al. (2008) such an algorithm r m r ∈ field Fidentity (x) ⊂ ⊂R RnV))(∆ is called called r-homogeneous of degree degree m ∈= R (observers) can also be used to estimate the derivativeshas of if field (x) is of R theF x) = r-homogeneous m V (x) holds (resp., f (∆m x) field F (x) ⊂ R ) is called r-homogeneous of degree ∈= R a signal. In Perruquetti et al. (2008) such an algorithm r r abeen signal. In Perruquetti et al. (2008) such an algorithm has r x) = l mrV (x) holds (resp., f (∆m r x) presented, with a proof using homogeneous properties l r r if the identity V (∆ a signal. In Perruquetti et al. (2008) such an algorithm has if the identity V (∆ x) = V (x) holds (resp., f (∆ x) = r m r ∆ f (x), or F (∆ x) = ∆ F (x)), Bacciotti and Rosier if the identity V (∆ x) = V (x) holds (resp., f (∆ x) = been presented, with a proof using homogeneous properties l r r l r been presented, with aa proof using homogeneous properties = l ∆r F (x)), Bacciotti and Rosier l ∆r f (x), or F (∆r x) and continuity stability respect to the vector field. (2005); been presented,of with proofwith using homogeneous properties or (∆ l ∆ Bacciotti and F et l ∆r r Levant Bernuau al. (2014). Consider that ∆ ff (x), (x), or F F(2005); (∆r x) x) = = ∆ F (x)), (x)), Bacciotti and Rosier Rosier andLyapunov continuity of stability with respect to the theinvector vector field. and continuity of stability respect to A approach haswith been developed Yangfield. and (2005); r et Levant (2005); Bernuau al. (2014). Consider that and continuity of stability with respect to the vector field. (2005); Levant (2005); Bernuau et al. (2014). Consider that the vector r and dilation ∆ r x are fixed. The Consider homogeneous (2005); Levant (2005); Bernuau et al. (2014). that A Lyapunov approach has been developed in Yang and A Lyapunov approach been developed in Yang r x are 1 Lin (2004); Qian and Linhas (2006); Andrieu et al. (2008)and for the homogeneous fixed. vector rr and dilation ∆ The A Lyapunov approach has been developed in Yang and the vector and dilation ∆ fixed. The p homogeneous x are r p n the vector r and dilation ∆ x are fixed. The homogeneous 1 Lin (2004); Qian and Lin (2006); Andrieu et al. (2008) for Lin (2004); Qian continuous |x | rpi ∈ Rnn , Lin (2004); algorithms. Qian and and Lin Lin (2006); (2006); Andrieu Andrieu et et al. al. (2008) (2008) for for norm is defined by xr, p := n i=1 ppp11 ,, ∀x n |xi | rrp pi continuous algorithms. := norm is defined by x ∀x ∈ i n continuous algorithms. ii | r norm is defined by x ,, ∀x ∈ R Rnn ,,, r, i=1 n |x r, p p := i=1 continuous algorithms. by set xS ∀x for anyis pdefined ≥ 1. The {x ∈ R :|xx 1} ∈is R the i | ir, p = In this paper we provide a smooth Lyapunov function for norm r, p=:= i=1 n n : x any p ≥ 1. The set S = {x ∈ R = 1} 1} is is the the In this this paper we provide aa(linear smooth Lyapunov function for for for any p ≥ 1. The set S = {x ∈ R n : xr, r, p p = In paper we provide smooth Lyapunov function for the family of continuous and nonlinear) Vasiljevic corresponding unit sphere. for any p ≥ 1. The set S = {x ∈ R : x = 1} is the In paper provide a(linear smooth function for corresponding unit sphere. r, p thethis family of we continuous andLyapunov nonlinear) Vasiljevic the family of continuous (linear and nonlinear) Vasiljevic corresponding unit sphere. the family of continuous (linear and nonlinear) Vasiljevic corresponding unit sphere. Copyright © 2016, 2016 IFAC 672Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 672 Copyright ©under 2016 responsibility IFAC 672Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 672 10.1016/j.ifacol.2016.10.241
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Homogeneous functions and vector fields have important properties. For example, for a given family of dilations ∆r x, and any homogeneous functions V1 , V2 (respectively, a vector field f1 ) w.r.t. ∆r x of degrees m1 , m2 (resp., l1 ), we have Bacciotti and Rosier (2005): (i) V1 V2 is homogeneous of degree m1 + m2 . (ii) There exists a m constant c1 > 0 such that V1 ≤ c1 x r, 1p . Moreover, if V1 is positive definite, there exists c2 > 0 such that m V1 ≥ c2 x r, 1p . (iii) ∂V1 (x) /∂xi is homogeneous of degree m1 − ri , with ri being the weight of xi . (iv) Lf V1 (x) is homogeneous of degree m1 + l1 . Homogeneous systems have important properties as e.g. that local stability implies global stability and if the homogeneous degree is negative asymptotic stability implies finite time stability Bacciotti and Rosier (2005); Levant (2005); Bernuau et al. (2014): Assume that the origin of a Filippov Differential Inclusion (DI), x˙ ∈ F (x), is strongly locally Asymptotic Stable and the vector-set field F is r-homogeneous of degree l < 0; then, x = 0 is strongly globally finite-time stable and the settling time is continuous at zero and locally bounded. Along this paper we use the following notation. For a real variable z ∈ R and a real number p ∈ R the symbol zp = |z|p sign(z) is the signed power p of z. We recall the following well-known properties of continuous homogeneous functions Hestenes (1966); Bhat and Bernstein (2005); Andrieu et al. (2008) Lemma 1. Let η : Rn → R and γ : Rn → R+ , that is γ(x) ≥ 0 ∀x, be two continuous homogeneous functions, with weights r = (r1 , ..., rn ) and degrees m, such that {x ∈ Rn \ {0} : γ(x) = 0} ⊆ {x ∈ Rn \ {0} : η(x) < 0}. Then, there exists a real number λ∗ such that, for all λ ≥ λ∗ for all x ∈ Rn \ {0}, and some c > 0, η(x) − m λγ(x) < −c x r, p . Lemma 2. Suppose V1 and V2 are continuous real-valued functions on Rn , r-homogeneous of degrees m1 > 0 and m2 > 0, respectively, and V1 is positive definite. Then, for every x ∈ Rn , with S = {z : V1 (z) = 1} m2 m2 m1 min V2 (z) V1 (x) ≤ V2 (x) ≤ max V2 (z) V1m1 (x) . S
S
3. THE ARBITRARY ORDER DIFFERENTIATOR Let the input signal f (t) to the differentiator be a Lebesgue-measurable function defined on [0, ∞). f (t) is assumed to be decomposed as f (t) = f0 (t) + v(t). The first term is the n-times differentiable unknown base signal f (t), to be differentiated, with a n-th time derivative having a known Lipschitz constant L > 0, while v(t) corresponds to a uniformly bounded noise signal. Defining (i) di the variables ς1 = f0 (t) , . . . , ςi+1 = f0 (t) dt i f0 (t), i = 1, ..., n, a state representation of the base signal is ς˙i = ςi+1 , i = 1, · · · , n − 1, ς˙n =
(n) f0
To estimate the derivatives of the base signal, we consider the following family of homogeneous differentiators x˙ i = −ki x1 − f
ri+1 r1
x˙ n = −kn x1 − f
rn+1 r1
+ xi+1 , i = 1, · · · , n − 1
where ri = ri+1 − d = rn − (n − i) d , i = 1, . . . , n , (we introduce the parameter rn+1 = rn + d for simplicity) and −rn ≤ d ≤ 0. We fix hereafter (without loss of generality) rn = 1. Note that r1 ≥ · · · ≥ ri ≥ · · · ≥ rn = 1 ≥ rn+1 . Since for d = −1 systems (1) and (2) are discontinuous, their solutions are understood in the sense of Filippov Filippov (1988). The family of differentiators (1) is parametrized by −1 ≤ d ≤ 0. For d = 0 we recover a linear differentiator, while for d = −1 we recover Levant’s n-th order differentiator Levant (2003, 2005). Theorem 3. Under the stated assumptions for the signal f (t) there exist appropriate gains (depending on d and the Lipschitz constant of f (t)) such that the family of homogeneous differentiators (1) has the following properties. (1) In the absence of noise (v (t) ≡ 0) and for the class of polynomial signals, satisfying Spn = f (n) (t) ≡ 0 , the estimation xi (t) converges to the derivative f (i−1) (t) as t → ∞ exponentially for d = 0 and in finite time for −1 ≤ d < 0, i = 1, · · · , n. (2) In the absence of noise (v (t) ≡ 0) and of for the class n-Lipschitz signals satisfying SLn = f (n) (t) ≤ L , the differentiation estimation error xi (t) − f (i−1) (t) converges to zero in finite time for d = −1 and it is ultimately uniformly bounded for −1 < d ≤ 0, i = 1, · · · , n. For d = −1 it is required that kn > L. (3) For a uniformly bounded noise (|v (t)| ≤ η) and for the class of n-Lipschitz signals satisfying SLn = f (n) (t) ≤ L , the differentiation estimation error xi (t) − f (i−1) (t) is ultimately uniformly bounded for −1 ≤ d ≤ 0, i = 1, · · · , n. The results of Theorem 3 are well-known, in particular for the linear d = 0, close to linear − < d < 0 Perruquetti et al. (2008) and the discontinuous d = −1 Levant (2003, 2005) cases. The main novelty in this paper is that we provide a unified Lyapunov approach to prove these properties in all cases, the continuous −1 < d ≤ 0 and (in particular) the discontinuous d = −1 one. Beyond the intrinsic beauty of this fact, it opens the possibility of making comparisons and design under the same footing. Note that the discontinuous differentiator d = −1 (Levant’s differentiator) has an astonishing distinguishing feature: in the absence of noise it is exact (i.e. it converges in finite time) to a much larger class of signals SLn than its continuous counterparts, which are only exact for the much thiner set Spn ⊂ SLn for −1 < d < 0, or it only converges exponentially in the linear case d = 0. 4. A LYAPUNOV APPROACH FOR THEOREM 3 (i−1)
, in Defining the differentiation error as ei xi − f0 the absence of noise (v (t) ≡ 0), their dynamics is given by e˙ i = −ki e1
(t) .
(1)
, 673
661
ri+1 r1
e˙ n = −kn e1 (n)
rn+1 r1
+ ei+1 , i = 1, · · · , n − 1
(2)
+ δ (t) ,
where δ (t) = −f (t) is the n-th order derivative, which is assumed to be bounded, i.e. |δ (t)| ≤ ∆. For δ (t) ≡ 0 this system is homogeneous with homogeneity degree d and weights r = [r1 , · · · , rn ]. Performing the state transformation (for i = 1, · · · , n)
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e1 e2 ei en , z2 = , · · · , z i = , · · · , zn = , 1 k1 ki−1 kn−1 the dynamics of (2) becomes ri+1 z˙i = −k˜i z1 r1 − zi+1 , i = 1, · · · , n − 1 (3) z1 =
where υn = −βn−1 (p − rn−1 ) |zn | This can be rewritten as
Zi (zi , zi+1 ) =
ri |zi | − zi zi+1 p p p − ri |zi+1 | ri+1 , p
p−ri ri+1
p ri
+
(4)
Note that σi = 0 on the same set as si = 0. The partial derivatives of Zp are
p−ri p−ri ∂Zi (zi , zi+1 ) = zi ri − zi+1 ri+1 = σi ∂zi p−ri −ri+1 p − ri ∂Zi (zi , zi+1 ) =− |zi+1 | ri+1 si , ∂zi+1 ri+1 which are continuous. They become both zero at the points where Zi achieves its minimum.
We prove Theorem 3 by using the following Theorem, which is the main result of the paper: Theorem 4. Each differentiator of the form (1) admits a strong, proper, smooth and r−homogeneous of degree p Lyapunov function of the form 1 p βj Zj (zj , zj+1 ) + βn |zn | V (z) = p j=1 βi > 0 , i = 1, · · · , n .
V (z) is non negative, since it is a positive combination of non negative terms. Moreover, V (z) is positive definite since V (z) = 0 implies that z = 0. Due to homogeneity it is radially unbounded Bhat and Bernstein (2005). For the linear case (d = 0, p = 2) V is a quadratic form.
V˙ (z) =
n−2 j=1
rj+1 βj −k˜j σj z1 r1 − zj+1 +
p−rj −rj+1 rj+2 p − rj |zj+1 | rj+1 sj z1 r1 rj+1 1 −k˜n−1 βn−1 σn−1 z1 r1 − zn − +k˜j+1
rn+1 r1
−k˜n υn z1
n−2 j=1
.
(5)
p−r1 r2 p−r1 z1 r1 − z2 + W = k˜1 β1 z1 r1 − z2 r2
(6)
rj+2 rn+1 k˜j+1 υj+1 z1 r1 − zj+2 + k˜n υn z1 r1
and for j = 1, · · · , n − 2 p−rj −rj+1 p − rj βj |zj+1 | rj+1 sj + βj+1 σj+1 . rj+1 We will show that there exist values of k˜i > 0 such that W (z) > 0. We consider first the continuous case and then the discontinuous one.
4.1 The continuous case: −1 < d ≤ 0 To show that W (z) > 0 we exploit the structure of W . So consider the values of W restricted to some hypersurfaces: for i = 1, · · · , n − 1 ri+1 r2 Zi = z1 r1 = z2 ∧ · · · ∧ z1 r1 = zi+1 .
These sets are related as Zn−1 ⊂ · · · ⊂ Z1 . Note that on Zi the variables σi and si vanish, i.e. σi = si = 0, and therefore they also vanish on Zj , for every j > i. Let Wi = WZi represent the value of W (z) restricted to the the manifold Zi . We can obtain the value of W1 r1 r2 by replacing in W (z) the variable z1 by z1 = z2 , so that W1 becomes a function of (z2 , · · · , zn ). In general, we obtain the value of Wi , for i = 1, . . . , n − 1, by replacing in W (z) the variables (z1 , · · · , zi ) by its values in terms of ri
r1
zi+1 , i.e. z1 = zi+1 ri+1 , · · · , zi = zi+1 ri+1 , so that Wi becomes a function of z¯i+1 (z i+1 , · · · , zn ). For example,
the value of the expression ri+2 zi+1 ri+1 − zi+2 .
Its derivative is
where
p−1
υj+1 = −
which are continuously differentiable, positive semidefinite p p and Zi (zi , zi+1 ) = 0 if and only if zi ri = zi+1 ri+1 . For simplicity we introduce the variables p−ri ri p−ri ri+1 ri σi = zi , si = zi − zi+1 ri+1 . − zi+1
n−1
sn−1 + βn zn
V˙ = −W (z) + υn δ¯ (t) ,
rn+1
z˙n = −k˜n z1 r1 + δ¯ (t) , where for i = 1, · · · , n, f (n) (t) ki k˜i = , k0 = 1 , δ¯ (t) = − . ki−1 kn−1 For n ≥ 2 fix p ≥ r1 + r2 = 2 − (2n − 3) d > 1 and for i = 1, · · · , n − 1 define the homogeneous functions
p−rn−1 −1
z1
ri+2 r1
− zi+2
on Zi is
From (6) it is seen that we can write W (z) as W (z) = k˜1 η1 (z1 , z2 ) + µ1 (z) , where
p−r1 r2 p−r1 η1 (z1 , z2 ) = β1 z1 r1 − z2 r2 z1 r1 − z2 .
η1 (z1 , z2 ) is positive everywhere except on the set Z1 , where it vanishes, i.e. η1 (z1 , z2 ) = 0 for z ∈ Z1 . Note also that µ1 does not depend on the gain k˜1 . According to Lemma 1 there exists a sufficiently large positive value of k˜1 such that W (z) > 0 if the value of µ1 restricted to Z1 , i.e. µ1Z1 = W1 (¯ z2 ), is positive. From (6) it is seen that − zj+2 W1 (¯ z2 ) can be written as z2 ) = k˜2 η2 (z2 , z3 ) + µ2 (¯ z2 ) , W1 (¯ p−r2 p−r2 r3 η2 (z2 , z3 ) = β2 z2 r2 − z3 r3 z2 r2 − z3 .
+ υn δ¯ (t) ,
674
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η2 (z2 , z3 ) is positive on Z1 except on the set Z2 , where z2 ) does not depend on the gain k˜2 . it vanishes, and µ2 (¯ According to Lemma 1 there exists a sufficiently large positive value of k˜2 such that W1 (¯ z2 ) is positive definite on Z1 if µ2Z2 = W2 (¯ z3 ) is positive. This can be done recursively. From (6) it is seen that Wi (¯ zi+1 ), for i = 1, . . . , n − 1, can be written as
V˙ = ∇V · fη (e, η) + ∇V · f (e) + ∇V · f∆ (e, η) . Due to the homogeneity and the H¨older continuity of f (e) and applying Lemma 2 we obtain that ∇V · f∆ = ≤
zi+1 ) = k˜i+1 ηi+1 (zi+1 , zi+2 ) + µi+1 (¯ zi+1 ) , Wi (¯
p−ri+1 ri+1
− zi+2 ηi+1 (·) = βi+1 zi+1 ri+2 zi+1 ri+1 − zi+2 .
p−ri+1 ri+2
V˙ ≤ −α2 V
Finally, we obtain that we can write zn−1 ) = k˜n−1 ηn−1 (¯ zn−1 ) + µn−1 (¯ zn−1 ) , Wn−2 (¯ p−rn−1 rn−1
− zn ηn−1 = βn−1 zn−1 rn zn−1 rn−1 − zn , µn−1 = k˜n υn zn−1
p−rn−1 rn
n
k˜i µi V
p−ri p
i=1
p+d p
|η|
ri+1 r1
(7)
+ α3 LV
p−1 p
+
n
k˜i µi V
i=1
p−1 r1
where
−k˜n υn z1 ≤ k˜n θn − k˜n βn |z1 | p−r
p−1
≤ −α2 V p + α3 LV p , for some αi > 0, i = 1, 2, 3, and from this, using standard arguments, we conclude that the differentiation error is ISS with respect to δ (t). To prove item 3) we notice that (2), i.e. e˙ = f (e) + bδ (t), when we have a measurement noise η (t) becomes e˙ = fη (e, η) + bδ (t), with fη (e, η) = f (e1 + η, e2 , · · · , en ), and f∆ (e, η) = fη (e, η) − f (e). The derivative of V along the solutions of the noisy system is
675
,
and replacing this expression in υn it 0
cludes the proof of 1) in Theorem 3 in the continuous case. To prove Item 2) in the theorem we observe that W (z) (6) is continuous and r-homogeneous of degree p + r2 − r1 = p + d while vn is r-homogeneous of degree p − 1, whith p − 1 < p + d for −1 < d. From (5) and Lemma 2 we obtain p−1 δ (t) ≤ −W + α1 LW p+d V˙ ≤ −W + |υn | ¯
ri+1 r1
For this case (note that rn+1 = 0) we observe that W (6) is discontinuous and has the same homogeneous degree as vn . We introduce a modification in the last term of W p−1 p−1 to deal with this situation. Since zn = − z1 r1 +
×
which is positive on Zn−1 for every positive values of k˜n and βn . We conclude therefore, that there exist positive values of k˜1 , · · · , k˜n such that W (z) > 0. This con-
|η|
4.2 The discontinuous case: d = −1
p−1
rn+1 rn−1
p−ri p
for some µi > 0. Again, using standard arguments we conclude that the differentiation error is ISS with respect to the noise η (t).
zn + z1 follows that
Note that µn−1 (¯ zn−1 ) does not depend on k˜n−1 . ηn−1 is positive on Zn−2 except on the set Zn−1 , where it vanishes. According to Lemma 1 there exists a sufficiently large value of k˜n−1 such that Wn−2 (¯ zn−1 ) is positive if µn−1, Zn−1 = Wn−1 (zn ) is positive. From (6) we see that p−1 r p−1+rn+1 zn n+1 = k˜n βn |zn | , Wn−1 (zn ) = k˜n βn zn
p+d
ri+1 ri+1 ∂V z1 r1 − z1 + η r1 k˜i ∂zi i=1
n
so that we finally have that ×
Since ηi+1 (zi+1 , zi+2 ) is positive on Zi except on the set Zi+1 , where it vanishes, and µi+1 (¯ zi+1 ) does not depend on the gain k˜i+1 , Lemma 1 assures the existence of a sufficiently large positive value of k˜i+1 such that Wi (¯ zi+1 ) is positive definite on Zi if µi+1,Zi+1 = Wi+1 (¯ zi+2 ) is positive.
663
θn = βn−1 (p − rn−1 ) |zn | n−1 p−1 r p−1 1 +βn z1 − zn ≥ 0 .
−1
p−1 r1
,
|sn−1 | +
Furthermore, the term due to the perturbation in V˙ (5) can be bounded as follows ∆ p−1 υn δ¯ (t) ≤ k˜n θn + βn |z1 | r1 . kn Considering the case with perturbation we can obtain the following bound rn+1 ∆ −k˜n υn z1 r1 + υn δ¯ ≤ k˜n θn 1 + − kn p−1 ∆ |z1 | r1 , k˜n βn 1 − kn and we can write ¯ (z) , V˙ (z) ≤ −W ¯ is where the continuous and homogeneous function W given by p−r r2 p−r ¯ (z) = k˜1 β1 z1 r1 1 − z2 r2 1 z1 r1 − z2 W +
n−2 j=1
rj+2 k˜j+1 υj+1 z1 r1 − zj+2
(8)
p−1 ∆ ∆ + k˜n βn 1 − |z1 | r1 . −k˜n θn 1 + kn kn By the same procedure as for the continuous case it is possible to show that there exist values of k˜i > 0 such
IFAC NOLCOS 2016 664 Emmanuel August 23-25, 2016. Monterey, California, USA Cruz-Zavala et al. / IFAC-PapersOnLine 49-18 (2016) 660–665
¯ (z) > 0. In the last step of the analysis performed that W ¯ that before to W we obtain for W ¯ n−2 (¯ W zn−1 ) = k˜n−1 ηn−1 (¯ zn−1 ) + µ ¯n−1 (¯ zn−1 ) , ∆ µ ¯n−1 = −k˜n θn∗ 1 + + kn p−1 ∆ k˜n βn 1 − |zn−1 | rn−1 , kn
g2 (z1 , z2 )
zn−1 ) does not depend on k˜n−1 . ηn−1 Note that µ ¯n−1 (¯ is positive on Zn−2 except on the set Zn−1 , where it vanishes. According to Lemma 1 there exists a sufficiently ¯ n−2 (¯ large value of k˜n−1 such that W zn−1 ) is positive if ¯ n−1 (zn ) is positive. From (8) we see that µ ¯n−1, Zn−1 = W ∆ p−1 ˜ ¯ , |zn | Wn−1 (zn ) = kn βn 1 − kn which is positive on Zn−1 for every positive values of k˜n and βn , and kn > ∆. We conclude therefore, that there ¯ (z) > 0. exist positive values of k˜1 , · · · , k˜n such that W This concludes the proof of items 1) and 2) in Theorem 3 in the discontinuous case. Item 3) in the Theorem follows along the same reasoning as for the continuous case. We have only to modify the last term in (7), since in this case 0 0 for i = n we have z1 − z1 + η = 0 if |z1 | > |η| and it is 2 otherwise. 5. CASE EXAMPLES
The homogeneous first order differentiator is 1
x˙ 1 = −k1 x1 − f 1−d + x2 1+d
x˙ 2 = −k2 x1 − f 1−d , (9) where x2 (t) is an estimation of the first derivative of the signal f (t). The homogeneity degree is −1 ≤ d ≤ 0, with weights r1 = 1 − d, r2 = 1, and r3 = 1 + d. For d = −1 we obtain Levant’s robust and exact differentiator, while for d = 0 we get a linear differentiator. The Lyapunov Function is 2−d 1−d β+1 2−d V (z1 , z2 ) = |z1 | 1−d − z1 z2 + |z2 | , 2−d 2−d which is positive definite for every β > 0 and continuously differentiable. Its derivative V˙ is 2
2 V˙ = −k1 |σ1 | + k˜2 (1 + β) s1 z1 1−d − k˜2 β |z1 | 1−d 2
1+d
s1 − β z2
1−d
1+d
z1 1−d
. 2 |σ1 | g2 (z1 , z2 ) is a homogeneous function of degree zero, it is upper semicontinuous and it has a maximum, that is achieved on the homogeneous sphere.
p−rn−1 −rn p − rn−1 rn |sn−1 | + |zn | θn∗ = βn−1 rn p−rn p−rn +βn zn−1 rn−1 − zn rn .
1+d
2
The first term |σ1 | is non negative and it vanishes when 1 2 z1 1−d = z2 . At these points V˙ ≤ −k˜2 β |z2 | , which is negative. Lemma 1 assures the existence of a value of k1 rendering V˙ negative definite. From V˙ we find that the required value of k1 satisfies k2 k1 = 1 > ω2 max2 g2 (z1 , z2 ) , k2 z∈R k˜2
2
≤ −k1 |σ1 | + k˜2 (1 + β) |s1 | |z1 | 1−d − k˜2 β |z1 | 1−d 1 where we have used the variables σ1 = z1 1−d − z2 , 1−d s1 = z1 − z2 . Note that the right hand side of the last inequality is continuous for all −1 ≤ d ≤ 0. 676
The homogeneous second order differentiator is given by 1−d
x˙ 1 = −k1 x1 − f 1−2d + x2 1
x˙ 2 = −k2 x1 − f 1−2d + x3 x˙ 3 = −k3 x1 − f
(10)
1+d 1−2d
where x2 (t) estimates f (1) (t) and x3 (t) estimates f (2) (t). The homogeneity degree is −1 ≤ d ≤ 0 and the weights are r1 = 1 − 2d, r2 = 1 − d, r3 = 1, r4 = 1 + d. The LF is given by (with p = 2 − 3d)
β3 p |z3 | p which is positive definite for any positive values of β2 > 0 and β3 > 0 and it is C 1 , and the gains that render V˙ < 0 are found solving the inequalities k1 k2 k22 > ω23 , > ω13 , k1 k3 k3 where ω23 depends only on the values of (β2 , β3 , d), while ω13 depends on (β2 , β3 , d, γ23 , ω23 ). V (z) = Z1 (z1 , z2 ) + β2 Z2 (z2 , z3 ) +
Note that these relations are invariant with respect to scaling of the gains, i.e. if we scale the gains k1 → Lk1 , k2 → L2 k2 , k3 → L3 k3 for any L > 0 then the previous relations are still fulfilled. To illustrate the behavior of the differentiators for different values of d we present some simulations results. Consider which the signal f0 (t) = 0.5 sin (0.5t) + 0.5 cos (t), has (3) a non vanishing and bounded derivative f0 (t) ≤ 1. Figure 1 shows the differentiation errors for the linear differentiator (d = 0), a homogeneous differentiator (d = −0.5) and the discontinuous (Levant’s) differentiator (d = −1) in the absence of noise. Figure 2 shows the results with measurement noise v (t) = ε sin (ωt) with ε = 0.001 and ω = 1000. For all differentiators we have selected the √ same gains k1 = 3, k2 = 1.5 3, k3 = 1.1, and we used Euler-method with step size τ = 3 × 10−4 .
Figure 1 shows that in the absence of noise only the discontinuous differentiator (d = −1) can estimate without error (1) (2) both derivatives f0 (t) and f0 (t). When noise is present Figure 2 shows that the discontinuous differentiator is strongly affected by the noise, but its estimate is still better than the ones provided by the continuous differentiators.
IFAC NOLCOS 2016 August 23-25, 2016. Monterey, California, Emmanuel USA Cruz-Zavala et al. / IFAC-PapersOnLine 49-18 (2016) 660–665
REFERENCES
f (t)=0.5sin(0.5t)+0.5cos(t), noise=0 0
0.7
2
3
d=0 d=−0.5 d=−1
0.6
d=0 d=−0.5 d=−1
d=0 d=−0.5 d=−1
2.5
1.5 0.5 2 0.4 1 1.5
0.2
e3
e2
e1
0.3 0.5
0.1
1
0.5 0
0 0 −0.1 −0.5 −0.5
−0.2
0
10
20
−1
30
0
10
t
20
−1
30
0
10
t
20
30
t
Fig. 1. Differentiation error for d = 0, d = −0.5, and d = −1 (Levant’s Differentiator) without noise. f0(t)=0.5sin(0.5t)+0.5cos(t), noise=0.001sin(1000t) 0.7
2
3
d=0 d=−0.5 d=−1
0.6
d=0 d=−0.5 d=−1
d=0 d=−0.5 d=−1
2.5
1.5 0.5 2 0.4 1 1.5
0.2
e3
e2
e1
0.3 0.5
0.1
1
0.5 0
0 0 −0.1 −0.5 −0.5
−0.2
0
10
20
t
30
−1
0
10
20
30
−1
t
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0
10
20
30
t
Fig. 2. Differentiation error for d = 0, d = −0.5, and d = −1 (Levant’s Differentiator) in presence of noise. 6. CONCLUSIONS For a family of continuous and discontinuous homogeneous differentiators we provide a unified smooth Lyapunov function to study the convergence, robustness and performance properties. Comparison of performance and possible optimization of the behavior of these differentiators is part of ongoing research. ACKNOWLEDGEMENTS The authors would like to thank the financial support from PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigaci´ on e Innovaci´ on Tecnol´ ogica), project IN113614; Fondo de Colaboraci´ on II-FI UNAM, Project IISGBAS100-2015; CONACyT (Consejo Nacional de Ciencia y Tecnolog´ıa), project 241171 and CVU 267513. 677
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