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Step-by-step implicit discrete super-twisting differentiator for input-output linearizable nonlinear systems
11th 11th IFAC IFAC Symposium Symposium on on Nonlinear Nonlinear Control Control Systems Systems Vienna, Austria, Sept. 4-6, 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 2019 Available online at www.sciencedirect.com 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-16 (2019) 783–788
Step-by-step Step-by-step implicit implicit discrete discrete super-twisting super-twisting Step-by-step implicit discrete super-twisting differentiator for linearizable Step-by-step discrete super-twisting differentiatorimplicit for input-output input-output linearizable differentiator for input-output linearizable nonlinear systems differentiator for input-output nonlinear systems linearizable nonlinear systems nonlinear systems Isaac Chairez ∗∗
Isaac Chairez ∗∗∗ Isaac Chairez ∗ Isaac Chairez ∗ ∗ Bioprocesses Department, UPIBI, Instituto Polit´ cnico Nacional, Nacional, and and the the ∗ eecnico ∗ Bioprocesses Department, UPIBI, Instituto Polit´ ∗ School of Engineering of the Tecnol´ o gico de Monterrey, Campus Bioprocesses Department, UPIBI, Instituto Polit´ e cnico Nacional, and the ∗ School of Engineering of the Tecnol´ ogicoPolit´ de Monterrey, Campus Bioprocesses Department, UPIBI, Instituto cnico Nacional, and the School of Engineering of [email protected] Tecnol´ ogico de eMonterrey, Campus Guadalajara Guadalajara [email protected] School of Engineering of [email protected] Tecnol´ogico de Monterrey, Campus Guadalajara Guadalajara [email protected] Abstract: Abstract: The The aim aim of of this this study study was was to to design design aa step-by-step step-by-step numerical numerical differentiator differentiator for for aa class class of discretized version of a n-th chain-of-integrator system based on the application of the Abstract: The aim of this study was to design a step-by-step numerical differentiator for implicit a class of discretized version of a n-th chain-of-integrator system based on the application of the implicit Abstract: Theof aim of ofthis studychain-of-integrator was to design a step-by-step differentiator for implicit a class discretization the algorithm The reconstructed the of discretized version a n-th basednumerical on differentiator the application of the discretization of the super-twisting super-twisting algorithm (STA). (STA).system The high-order high-order differentiator reconstructed the of discretized version of a n-th chain-of-integrator system based on the application of the implicit non-measurable states of an an input-output input-output linearizable systems. The gains gainsdifferentiator of the the observer observer were calculated calculated discretization ofstates the super-twisting algorithm (STA).systems. The high-order reconstructed the non-measurable of linearizable The of were discretization ofstates the super-twisting algorithm (STA). The high-order differentiator reconstructed the non-measurable of an input-output linearizable systems. The gains of the observer were calculated based on the application of aa class of generalized functions. The convergence analysis showed that based on the application of class of generalized functions. The convergence analysis showed that non-measurable states of an input-output linearizable systems.set The gains of the observer were calculated all the states in the linearized system achieved an invariant with a predefined size. The existence based on the application of a class of generalized functions. The convergence analysis showed that all the on states the linearized an invariant setThe withconvergence a predefinedanalysis size. The existence based the in application of aonsystem class ofachieved generalized functions. showed that of invariant set depends the uniqueness of solution the function. A simple all this the states in the linearized system achieved an the invariant setfor with a generalized predefined size. The existence of this invariant set depends on the uniqueness of the solution for the generalized function. A simple all the states in the linearized system achieved an invariant set with a predefined size. The existence of this invariant set depends on the uniqueness of the solution for the generalized function. A simple interpretation of of aa possible possible exact exact reconstruction reconstruction of of the the non-measurable non-measurable states states of of the the implicit implicit discretized discretized interpretation of thisThe invariant depends onimplicit the uniqueness of the the solution for the states generalized function. A for simple interpretation of asetpossible exact reconstruction of non-measurable of the of implicit discretized form. application of the discretized STA served to design aa class observer form. The application of the implicit discretized STA served to design class of observer for the the interpretation of a possible exact reconstruction of the non-measurable states of the of implicit discretized microalgae culture system. form. The application of the implicit discretized STA served to design a class observer for the microalgae culture system. form. The application of the implicit discretized STA served to design a class of observer for the microalgae culture system. © 2019, IFAC (International microalgae culture system. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Sliding Sliding mode; mode; Super-twisting Super-twisting algorithm; algorithm; Numerical Numerical differentiator; differentiator; Discrete Discrete sliding sliding mode. mode. Keywords: Sliding mode; Super-twisting algorithm; Numerical differentiator; Discrete sliding mode. Keywords: Sliding mode; Super-twisting algorithm; Numerical differentiator; Discrete sliding mode. 1. INTRODUCTION INTRODUCTION 2016). However, However, their their discretized discretized counterparts counterparts have have not not been been 1. 2016). 1. INTRODUCTION 2016). However, their discretized counterparts haveorder not been profoundly developed. The particular class of first difprofoundly developed. The particular class of first order dif2016). However, their discretized counterparts have(STA) not been Sliding mode mode theory theory 1. hasINTRODUCTION contributed to to solve solve the the robust robust control control ferentiator profoundly developed. The particular class of first order difbased on the super-twisting algorithm has Sliding has contributed ferentiator based on the super-twisting algorithm (STA) has profoundly developed. The particular class of first order difSliding modesystems, theory has contributed to solve the robustsystems control proven of uncertain the state observation of uncertain ferentiator based on the super-twisting algorithm (STA) has to be an efficient example of robust and exact differof uncertain systems, the state observation of uncertain systems proven to be an efficient example of robust and exact differSliding mode theory has contributed to solve the robust control ferentiator based on the super-twisting algorithm (STA) has of uncertain systems, the stateestimation observation of uncertain systems and the finite/time parameter (Utkin, 1992). The maproven toThe be an efficient example of of STA robust and exact differentiator. The discretized versions STA appeared recently, and the finite/time parameter estimation (Utkin, 1992). The maentiator. discretized versions appeared recently, of uncertain systems, the statethe observation ofseveral uncertain systems proven to be an efficient example of robust and exact differand the finite/time parameter estimation (Utkin, 1992). The maturity of this theory attracted attention of researchers entiator. The discretized versions of STA appeared recently, mostly presented as explicit discretization forms of the continturity offinite/time this theoryparameter attracted the attention(Utkin, of several researchers mostly presented as explicit discretization forms of therecently, continand the estimation 1992). The maentiator. The discretized versions of STA appeared turity of thistried theory attracted the attentiondiscrete of several researchers who have to introduce consistent forms of the mostly presented as(Salgado explicit discretization forms of theimplicit continuous STA version et al., 2016). Just recently who have tried to introduce consistent discrete forms of the uous STA version (Salgado et al., 2016). Just recently implicit turityhave ofmode thistried theory attracted the attention of several researchers mostly presented as(Salgado explicitof discretization forms of theimplicit continwho to introduce consistent discrete forms of the and sliding algorithms (Utkin et al., 1999). Since the earlier uous STA version et al., 2016). Just recently semi-implicit variants the STA showed better conversliding mode algorithms (Utkin et al., 1999). Since the earlier semi-implicit variants of the STA showed better implicit converwho have tried to introduce consistent discrete forms of the and uous STA version (Salgado et al., 2016). Just recently sliding mode algorithms (Utkin et al., 1999). Since the earlier stages of the sliding mode theory, there were some proposal and semi-implicit variants of the STA showed better convergence properties, properties, including including smaller smaller sizes sizes of of the the invariant invariant zone zone stages the algorithms sliding mode theory, there wereSince somethe proposal sliding of mode (Utkin et al., 1999). earlier gence and semi-implicit variants of theSTA STA showed better converstages of the sliding mode theory, there wereof some proposal showing how to introduce discrete versions discontinuous gence properties, including smaller sizes of the invariant zone around the sliding surface if the plays controller role showing how to introduce discrete versions of discontinuous around the sliding surface if the STA plays the controller role stages of how theThe sliding mode theory, there wereofsome proposal gence properties, including smaller sizes of the the controller invariant zone showing to classical introduce discrete versions discontinuous controllers. applications of sliding modes in disaround the sliding surface if the STA plays role (Brogliato et al., 2018). Some studies showed that the semicontrollers. The classical applications of sliding modes in dis(Brogliato et al., 2018). Some studies showed that the semishowing how to introduce discrete versions of discontinuous around the sliding surface if thestudies STAfirst plays the discretized controller role controllers. The classical applications of sliding modes dis- implicit crete time produced either the exact convergence to aainwell(Brogliato et al., 2018). Some showed that the semiSTA even can estimate the order apcrete time produced either the exact convergence to wellimplicit STA even2018). can estimate the firstshowed order discretized apcontrollers. Thesurface classical applications of sliding modes inwelldis(Brogliato et al., Some studies that the semicrete time produced either the exact convergence to a defined sliding (Utkin et al., 1999) or the convergence to implicit STA even can estimate the first order discretized approximation of a signal derivative exactly (Acary et al., 2012). defined sliding surface (Utkin et al., 1999) or the convergence to proximation of a signal derivative exactly (Acary et al., 2012). crete time produced either the exact convergence to a wellimplicit STA even can estimate the first order discretized apdefined sliding surface (Utkin et al., 1999) or the convergence to an invariant tube circulating the sliding surface (Milosavljevic proximation of a signal derivative exactly (Acary et al., 2012). The implicit realization of STA can offer remarkable characan invariant tube circulating sliding surface (Milosavljevic implicit of realization of STA can offer(Acary remarkable defined sliding surface (Utkinthe et al., 1999) or the convergence to The proximation a signal derivative exactly et This al.,charac2012). an invariant tube circulating the sliding surface (Milosavljevic et al., 2006). The movement of the state trajectories inside the The implicit realization of STA can offer remarkable characteristics to prove the STA properties as controller. study et al., 2006). The movement of the state trajectories inside the to prove the STA as controller. Thischaracstudy an al., invariant tube circulating the sliding (Milosavljevic The implicit realization ofofproperties STA can offer remarkable et 2006). The of the statesurface trajectories inside the teristics referenced tube is known sliding-mode. These variants teristics tothe prove the STA properties as controller. This study considers application such result to develop aa class of referenced tube is movement known as as quasi quasi sliding-mode. These variants considers the application of such result to develop class of et al., 2006). The movement of the state trajectories inside the teristics discretized tothe prove thedifferentiator STAofproperties asoncontroller. This study referenced tubesliding is known as quasi sliding-mode. These variants of the discrete modes used discretized forms of nonlinconsiders application suchbased result tothe develop a class of implicit STA. The develof the discrete sliding modes used discretized forms of nonlinimplicit discretized differentiator based on the STA. The develreferenced tube is known as quasi sliding-mode. These variants considers the application of such result to develop a class ofa of the discrete sliding modes used discretized forms of nonlinear systems based on the the explicit discretization technique. Most opment implicit discretized differentiator based on the STA. The develof the implicit STA was extended for implementing ear systems based on explicit discretization technique. Most opment of the implicit STA was based extended for STA. implementing a of the discrete sliding modes used discretized forms of nonlinimplicit discretized differentiator on the The develear systems based on the explicit discretization technique. Most of these approaches have considered first order sliding modes opment of theorder implicit STA wasSTA-based extended for implementing a class of high step-by-step differentiator. This of these approaches have considered first ordertechnique. sliding modes class of high order step-by-step STA-based differentiator. This ear systems based on the explicit discretization Most opment of theorder implicit STA was extended for implementing a of these approaches have considered first order sliding modes structure only (Azar and Zhu, 2015). class of high step-by-step STA-based differentiator. This was inspired by the works proposed in (Floquet and only (Azar and Zhu, 2015). was order inspired by the works proposed in (FloquetThis and of these approaches have considered first order sliding modes structure class of high step-by-step STA-based differentiator. only (Azar and Zhu, 2015). structure was inspired by the works proposed in (Floquet and Barbot, 2007) and reformulated in (Mart´ ı nez-Fonseca et al., 2007) and reformulated in (Mart´ ınez-Fonseca et and al., In recent years, the only (Azar and Zhu, 2015). of In recent years, the emerging emerging of the the so-called so-called high high order order sliding sliding Barbot, structure was inspired by the works proposed inestimate (Floquet Barbot, 2007) and reformulated inprovided (Mart´ ınez-Fonseca et al., 2014). The proposed differentiator the of the In recent years, the emerging of the so-called high order sliding 2014). The proposed differentiator provided the estimate of the modes provided additional benefits to the classical (first order) modes provided additional benefits to the classical (firstsliding order) invariant Barbot,The 2007) and reformulated in (Mart´ ınez-Fonseca et al., 2014). proposed differentiator provided the estimate of the set centered at the origin for the estimation error. The In recent years, the emerging of the so-called high order modes provided additional benefits to the classical (first order) invariant set centered at the origin for the estimation error. The sliding modes such as reduction of the chattering amplitude sliding modes such as reduction of the chattering amplitude 2014). The proposed differentiator provided the estimate of the invariant centeredset the originon forthe thesolution estimation error. The size of the theset invariant setatdepended depended on the solution of well-posed well-posed modes additional benefits tothe the classicalproperty, (first order) sliding modes surface, such as no reduction chattering amplitude of invariant of on the provided sliding surface, no lose of of the robustness etc size on the sliding lose of the robustness property, etc invariant set centered at the origin for the estimation error. The size of the invariant set depended on the solution of well-posed generalized equations. These solutions were supported on the sliding modes such as reduction chattering amplitude on the sliding loseFridman of of thethe robustness property, etc generalized equations. solutions supported on the (Perruquetti and Barbot, 2002; et However, (Perruquetti andsurface, Barbot,no 2002; et al., al., 2008). 2008). However, size of the invariant set These depended on the were solution of well-posed generalized equations. These solutions were supported on the theory. on the sliding surface, no loseFridman of the robustness property, etc convex-set (Perruquetti and Barbot, 2002; Fridman et al., 2008). However, convex-set theory. the discretization of high-order sliding modes is still a matter the discretization of high-order sliding modes is stillHowever, a matter convex-set generalizedtheory. equations. These solutions were supported on the (Perruquetti and Barbot, 2002; Fridman et al., 2008). theresearching discretization of high-order sliding applications modes is stillas matter This of because their manuscript is of because their potential potential applications asaa on-line on-line convex-set theory. This manuscript is organized organized as as follows. follows. Section Section II II shows shows some some theresearching discretization of controllers high-order sliding modes is still matter of researching because their potential applications as the on-line digital instrumented or state estimators. In parThis manuscript is organized as follows. Section II showswhich some basic mathematical antecedents on convex set theory digital instrumented controllers or state estimators. In the parbasic mathematical antecedents on convex set theory which of researching because their potential applications as satisfies on-line This manuscript is organized as follows. Section IItheory showswhich some digital instrumented controllers or stateto estimators. In the par- were ticular case that the discretized system be observed basic mathematical antecedents on convex set useful in the design of the high order differentiator. ticular case that the discretized system to be observed satisfies useful in the antecedents design of the high order differentiator. digital instrumented controllers orform, stateto estimators. Inobserver the par- were basic mathematical on convex set theory which ticular case that the discretized system be observed satisfies the so-called chain of integrator then the state were useful in the design of the high order differentiator. Section III III details details the the structure structure of of the the high-order high-order differentiator differentiator the so-called chain of integratorsystem form, to then state observer ticular case that thenumerical discretized be the observed satisfies Section were useful instep-by-step the of of thethehigh order differentiator. the so-called chain of integrator form, then the state observer coincides with the high-order differentiator (Levant, Section IIIthe details thedesign structure high-order differentiator including turning-on functions. Section coincides with the numerical high-order differentiator (Levant, including the step-by-step turning-on functions. Section IV IV the so-called of integrator form, then the state observer Section III details the structure of observer the high-order differentiator coincides withchain the numerical high-order differentiator (Levant, contains 1998). including the step-by-step turning-on functions. Section IV a simple analysis of the structure, what are 1998). contains a simple analysis of the observer structure, what are coincides with the numerical high-order differentiator (Levant, including the step-by-step turning-on functions. Section IV 1998). contains a simple analysis of the observer structure, what are the generalized function needed to find the estimated states and generalized function needed to observer find the estimated states Today, 1998). there Today, there are are some some variants variants of of high-order high-order sliding sliding mode mode the contains a simple analysis of the structure, what and are the generalized function needed to find the estimated states and how the step-by-step structure worked. Section V establishes Today, there are variants of systems high-order sliding et mode the step-by-step structure Section V establishes differentiators for continuous time (Chalanga al., differentiators for some continuous time (Chalanga al., how the generalized function needed worked. to find the estimated states and the step-by-step structure worked. Section V establishes Today, there are variants of systems high-order sliding et mode differentiators for some continuous time systems (Chalanga et al., how how the step-by-step structure worked. Section V establishes differentiators for continuous time systems (Chalanga et al., 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 1395 Copyright © under 2019 IFAC IFAC 1395Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 1395 10.1016/j.ifacol.2019.12.058 Copyright © 2019 IFAC 1395
2019 IFAC NOLCOS 784 Vienna, Austria, Sept. 4-6, 2019
Isaac Chairez / IFAC PapersOnLine 52-16 (2019) 783–788
some details on the numerical implementation of the proposed step-by-step differentiator. Section VI presents a numerical realization of the differentiator for a class of micoalgae growth system. Section VI closes the article with some final remarks.
Assumption 2. The function u belongs to the following admissible set: 2
Uadm = u | u(t) ≤ u+ , t ≥ 0, u+ ∈ R+
2. MATHEMATICAL ANTECEDENTS The implicit numerical implementation of the STA requires the application of the convex set theory. The set-valued sign function satisfies the following definition: +1 if z > 0 sign(z) = [−1, +1] if z = 0 (1) −1 if z < 0 The discrete implicit realization of the STA algorithm requires the inclusion of the inverse function for the multiset valued function. This part of the study follows the ideas given in (Acary et al., 2012; Huber et al., 2016). Introduce K ⊆ Rn a closed non empty convex set, which defines its normal cone set. Its normal cone at xK ∈ K satisfies he following definition NK (x) = {w ∈ Rn |w, y − x ≤ 0 ∀y ∈ K} (2) Notice that, for the convex set [−1, +1], one may define R− if s = −1 N[−1,+1] (s) = {0} if s ∈ [−1, +1] R if s = +1
Assumption 1 implies that input u is a stabilizing controller at least. The following assumption is imposed to u:
(3)
The Convex Analysis Theory justifies that N[−1,+1] (s(z)) is the inverse of the set-valued function, sign(z) that is (4) x ∈ sign(z) ⇔ K[−1,+1] (s(z)) Let G ∈ Rn →∈ Rn be a set-valued mapping and f ∈ Rn →∈ Rn be a single-valued vector field, therefore, 0 ∈ f (x) + G(x) (5) is known as the generalized equation. The application of these functions justifies the estimation of the variables that cannot be measured on-line in the discretized version of the chain-ofintegrator systems (Brunovskii form). 3. STEP-BY-STEP OBSERVER IN CONTINUOUS TIME Let consider the nonlinear single-input-single-output (SISO) system in the Brunovskii form: d z1 (t) = z2 (t) dt d z2 (t) = z3 (t) dt (6) .. . d zn (t) = f (z (t)) + g (z (t)) u (t) dt
Here z = [z1 · · · zn ], z ∈ Rn is the state of (6). The variable u ∈ R is an external signal which may represent the feedback control or an open loop injection. Notice that f and gu play the role of internal uncertainties and external perturbations affecting the dynamics of the chain of integrator form. For developing the result of this study, the following assumption is considered valid in this study. Assumption 1. Under the given controller u, all the states of (6) are bounded, that is |zi | ≤ zi+ .
(7)
The nonlinear function f : Rn → R is the uncertain internaldynamics term of (24), which satisfies the local Lipschitz condition: 2 2 |f (za ) − f (zb )| ≤ Lf za − zb with the constant Lf a positive scalar and za , zb ∈ Rn .
The function associated to the function g is bounded as follows 2
0 < g1 ≤ |g(z)| ≤ g2 ,
g1 , g2 ∈ R +
This study centers on developing a nonlinear differentiator in order to recover those non-measurable variables and using the first state as the available information. The observer was designed as a modification of the one presented in (Floquet and Barbot, 2007). The differentiator satisfies the following differential equation d zˆ1 (t) = z˜2 (t) + k11 φ11 (e1 (t)) dt d z˜2 (t) = k12 φ12 (e1 (t)) dt d zˆ2 (t) = E2 [˜ z3 (t) + k21 φ21 (e2 (t))] dt d z˜3 (t) = E2 [k22 φ22 (e2 (t))] dt .. . d zˆn (t) = En [f (ˆ z (t)) + g(ˆ z (t))u(t) + kn1 φn1 (en (t))] dt d z˜n+1 (t) = En [kn2 (t)φn2 (en (t))] dt (8) with: ei = z˜i − zˆi , i ∈ {1, 2, ·, n}. The set of variables zˆi represents the corresponding estimated trajectories of zi . In particular, z˜1 = z1 and zn = 0, which is needed to complete the observer design. The observer aims represented by ki1 , ki2 , i = [1, n] are scalars that must be adjusted to force the convergence of the differentiator trajectories to the states of the uncertain system. Remark 1. The inclusion of the last equation in the differentiator structure (8) has the aim of recovering the uncertain section affecting the nonlinear dynamics 6. The indicator function Ei (t) fulfills the following definition 0 t < Ti∗ Ei (t) = 1 t ≥ Ti∗ The switching time Ti∗ is found as a secondary result of the asymptotic converge obtained for the observer in the following section. The nonlinearfunctions φj1 (ej , αj ) and φj2 (ej ) (j = 1, 2) were designed in agreement to the proposal given in (Levant, 1998). Then, the following structures were considered for the observer design: 1 1/2 φ2j (ej ) = sign(ej ) φ1j (ej ) = |ej | sign(ej ) 2 One must recall that the solution of (8) is understood in the sense of Filippov (Filippov, 1998). In continuous time, the
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differentiator provides the asymptotic stability for the origin of the estimation error (Brogliato et al., 2018). Notice that most of the convergence analysis in high-order sliding modes uses the Lyapunov stability theory Nevertheless, the existing Lyapunov functions for continuous systems cannot be directly used for proving the convergence of the discretized versions of high-order sliding modes because there is not a conclusive evidence that the (continuous-time) Lyapunov function has convex level sets (Brogliato et al., 2018) which is not obvious for the aforementioned Lyapunov functions. Therefore, the analysis developed in this study is not realizing a discrete-time Lyapunov analysis for the proposed differentiator because there is not a continuous Lyapunov function counterpart. 4. THE DISCRETE REALIZATION OF THE STEP-BY-STEP OBSERVER The proposed discretization (implicit) for the observer (8) satisfies: zˆ1,v (tk+1 ) = zˆ1,v (tk ) + T z˜2,v (tk+1 )+ T k11 φ11 (e1 (tk+1 )) z˜2,v (tk+1 ) = z˜2,v (tk ) + T k12 φ12 (e1 (tk+1 )) zˆ2 (tk+1 ) = E2 [ˆ z2 (tk ) + T z˜3 (tk+1 )+ T k21 φ21 (e2 (tk+1 ))] z˜3 (tk+1 ) = E2 [˜ z3 (tk ) + T k22 φ22 (e2 (tk+1 ))] (9) .. . zn (tk ) + T f (ˆ z (tk ))+ zˆn (tk+1 ) = En [ˆ T g(ˆ z (tk ))u(tk )+ T kn1 φn1 (en (tk+1 ))] z˜n+1 (tk+1 ) = En [˜ zn+1 (tk ) + T kn2 φn2 (en (tk+1 ))] Here, T = tk+1 − tk is the sampling time. The proposed discretized observer (9) is not exactly the implicit discretization of (8) because the presence of f (ˆ z (tk )) and g(ˆ z (tk ))u(tk ).
Recalling that sign(0) is a set-valued function, then it is necessary to introduce a generalized equation with the aim of finding e1 (tk+1 ). According to (Polyakov et al., 2018), it is simpler introducing a new auxiliary variable ψ1 (tk+1 ) such that ψ1 (tk+1 ) = φ12 (e1 (tk+1 )). Based on the inverse set-valued function, e1 (tk+1 ) is the solution of the generalized equation given by: h11 (−k11 φ11 (e1 (tk+1 ))) ∈ N[−1,+1] () −1 h11 (r1 ) = h 12 (e1 (tk+1 )) h12 (r1 ) = r1 + c11 |r1 |sign(r1 ) + c12 c11 = T k12 , −c12 = e1,v (tk ) + T δ2,v (tk+1 )
(11)
Complementary, e1 (tk+1 ) can be obtained by implementing the solution of the generalized equation given in: h12 (e1 (tk+1 )) ∈ −k11 φ11 (e1 (tk+1 )))
(12)
According to the result obtained in (Floquet and Barbot, 2007), it is possible to execute a monitoring of the measurable variables e1 and δ2,v to establish a feasible estimation of the switching-on time Tj∗ for Ej , n ≥ j ≥ 2 (Notice here that T1∗ = 0 evidently). In instance, the estimation of the time T2∗ requires the introduction of an indicator function which defines if the first STA in the differentiator design has attained the estimation of z2 . A possible indicator function is e1 δ2,v ≤ Ts,1 e1 δ2,v dt ≤ where > 0 Ts,1 > 0, Ts,1 = M1 T , or t=0 M1 ∈ Z + is a designer choice. This time Ts,1 defines an observation period which serves to characterize if the first STA in the differentiator design is close enough to the estimate of z2 . In consequence, the dynamic system (10) can be transformed to: e1 (tk+1 ) = e1,v (tk ) + T δ2,v (tk+1 ) − T k11 φ11 (e1 (tk+1 )) δ2,v (tk+1 ) ∈ δ2,v (tk ) + T z3 (tk+1 ) − T k12 φ12 (e1 (tk+1 ))
Using the implicit discretization of (8), and estimating the dynamics of δi = zi − z˜1,v , one gets
e2 (tk+1 ) ∈ z˜2,v (tk ) + T k12 φ12 (e1 (tk+1 )) δ3,v (tk+1 ) ∈ δ3,v (tk ) + T z4,v (tk+1 )
e1 (tk+1 ) = e1,v (tk ) + T δ2,v (tk+1 )− T k11 φ11 (e1 (tk+1 ))
e3 (tk+1 ) = 0 .. .
δ2,v (tk+1 ) ∈ δ2,v (tk ) + T z3 (tk+1 )− T k12 φ12 (e1 (tk+1 ))
δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 )
e2 (tk+1 ) = z˜2,v (tk ) + T k12 φ12 (e1 (tk+1 )) − E2 [ˆ z2 (tk ) + T z˜3 (tk+1 ) + T k21 φ21 (e2 (tk+1 ))]
en (tk+1 ) = 0
δ3,v (tk+1 ) ∈ δ3,v (tk ) + T z4,v (tk+1 )− E2 [˜ z3 (tk ) + T k22 φ22 (e2 (tk+1 ))] e3 (tk+1 ) = E2 [˜ z3 (tk ) + T k22 φ22 (e2 (tk+1 ))] − E3 [ˆ z3 (tk ) + T z˜4 (tk+1 ) + T k31 φ31 (e2 (tk+1 ))] .. . δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 )− En−2 [˜ zn−1 (tk )+ T k(n−1)2 φ(n−1)2 (en−1 (tk+1 ))
785
(10)
zn (tk )+ en (tk+1 ) = En−1 [˜ T k(n−1)2 φ(n−1)2 (en−1 (tk+1 )) − zn (tk ) + T f (ˆ z (tk )) + T g(ˆ z (tk ))u(tk )+ En [ˆ T kn1 φn1 (e2 (tk+1 ))]
In (Brogliato et al., 2018), there is a result justifying that the origin is preferably an asymptotic stable equilibrium point for the regular structure of the STA. Notice that the first couple of equations in (10) are the implicit discretization form indeed.
(13)
Notice that the analysis of (13) can follow the ideas proposed in (Acary et al., 2012). However, such results consider a class of semi-implicit discretization. Nevertheless, the semi-implicit form proposed in that study, yields to simpler generalized equation. Instead, this study used the method proposed by (Brogliato et al., 2018). Notice that in those studies, the finitetime convergence of the estimation error can not be ensured. This is still a matter of investigation which deserves more researching efforts. Notice that if there is an unique solution for the generalized equations in (11)-(12), then the first two dynamics in (13) yields to justify that exists a number of sampling steps M1 such that |e1 | ≤ as well as |δ2,v | ≤ . A first result of this study proves that the set of equations proposed in (11) admits at least one solution (unique in the best case). The generalized forms in (11) and (12) have solutions for the given k11 and k12 , e1,v and δ2,v . Based on the generalized functions proposed above, one may argue that the these solutions satisfy: e1,v (tk+1 ) =
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0.5 ∗ −c11 ± c211 − 4(c12 − h12 (r1 )) if c12 < h12 (r1 )), e1,v (tk+1 ) = 0.5 ∗ −c11 ± c211 + 4(c12 − h12 (r1 )) if c12 > h12 (r1 )). The continuation of the analysis developed for the first can be done sequentially for the rest of the steps. This yields to the i − th step. Notice that it is feasible to introduce the corresponding function to obtain the switching-on time Tj∗ for Ej , n ≥ j ≥ i. The corresponding indicator function for the i-th step could be Ts,i ei δi+1,v ≤ or t=0 ei δi+1,v dt ≤ where > 0 Ts,i > 0, Ts,i = Mi T , Mi ∈ Z + is a designer choice. This time Ts,i now corresponds to the proposed observation period, which characterizes if the i-th STA in the observer design is close enough to the estimate of zi . The stability analysis at this i-th step applies on the discrete dynamics represented by: e1 (tk+1 ) ∈ [−, ]
e1 (tk+1 ) ∈ [−, ]
δ2,v (tk+1 ) ∈ [−, ] .. . ei (tk+1 ) ∈ [−, ]
δi+1,v (tk+1 ) ∈ [−, ]
.. . en−1 (tk+1 ) = en−1 (tk ) + T δn,v (tk+1 )− T k(n−1)1 φ(n−1)1 (en−1 (tk+1 )) δn,v (tk+1 ) ∈ δn,v (tk ) − T k(n−1)2 φ(n−1)2 (en−1 (tk+1 )) en (tk+1 ) = en (tk ) − T kn1 φn1 (en (tk+1 ))
In this last step, the existence of the unique solution for the associated generalized equations given by
δ2,v (tk+1 ) ∈ [−, ]
.. . ei (tk+1 ) = ei (tk ) + T δi+1,v (tk+1 )− T ki1 φi1 (ei−1 (tk+1 )) δi+1,v (tk+1 ) ∈ δi+1,v (tk ) + T zi+2,v (tk+1 )− T k(i)2 φ(i)2 (ei (tk+1 ))
(14)
hn1 (−kn1 φn1 (en (tk+1 ))) ∈ N[−1,+1] () −1 hn1 (rn ) = h n2 (en (tk+1 )) hn2 (rn ) = rn + cn1 |rn |sign(rn ) + cn2 cn1 = T kn2 , −cn2 = en,v (tk )
δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 ) en (tk+1 ) = 0
Consider that if there is the corresponding unique solution for the corresponding generalized equations given in (15) and (16), that is hi1 (−ki1 φi1 (ei (tk+1 ))) ∈ N[−1,+1] () −1 hi1 (ri ) = h i2 (ei (tk+1 )) (15) hi2 (ri ) = ri + ci1 |ri |sign(ri ) + ci2 ci1 = T ki2 , −ci2 = ei,v (tk ) + T δi+1,v (tk+1 )
δ2,v (tk+1 ) ∈ [−, ]
.. . ei (tk+1 ) =∈ [−, ] δi+1,v (tk+1 ) ∈ [−, ]
(17)
.. .
δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 ) en (tk+1 ) = 0
In the last step of the convergence analysis, the dynamics governing the estimation error is given by
(20)
In accordance to the ideas developed above, the dynamics for en−1 and δn in (18) justifies the existence of some sampling steps Mn such that |en−1 | ≤ as well as |δn,v | ≤ . Indeed, the expression (18) corresponds to e1 (tk+1 ) ∈ [−, ]
δ2,v (tk+1 ) ∈ [−, ]
.. . ei (tk+1 ) =∈ [−, ] δi+1,v (tk+1 ) ∈ [−, ]
Complementary, ei (tk+1 ) can be obtained by implementing the solution of the generalized equation given in: hi2 (ei (tk+1 )) ∈ −ki1 φi1 (ei (tk+1 ))) (16) Following the ideas developed in the first step, then the corresponding two dynamics for ei and δi+1 in (17) yields to justify that exists a number of sampling steps Mi such that |ei | ≤ as well as |δi+1,v | ≤ . In consequence, the dynamics described by (17) can be simplified to e1 (tk+1 ) ∈ [−, ]
(19)
In correspondence, en (tk+1 ) can be obtained by solving: hn2 (en (tk+1 )) ∈ −kn1 φn1 (en (tk+1 )))
.. .
(18)
(21)
.. . en−1 (tk+1 ) ∈ [−, ] δn,v (tk+1 ) ∈ [−, ]
en (tk+1 ) = en (tk ) − T kn1 φn1 (en (tk+1 )) Notice that the last step in (21) yields to estimate the derivative of en , which may yield to recover the uncertainties of the nonlinear system, that is f + gu. Notice that at this point, we have the way to produce the sequence of z˜ and zˆ which corresponds to the estimated of the non-measurable states of the discretized chain of integrator structure. You may notice that if the indicator functions can be designed such that ∀i ∈ [1, n] can be set to zero, then ei (tk ) = 0 and ei (tk+1 ) = ei (tk ). This implies the equivalence of the finite-time convergence for the class of discretized differentiator proposed in this study. Notice that unless the condition regarding the indicator functions yield to enforce the switching in the differentiator structure with a fixed = 0, then the finite time convergence cannot be proven using the classical control theory. Notice that if there are not enough arguments to justify that ei (tk+1 ) = ei (tk ).
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5. NUMERICAL IMPLEMENTATIONS The numerical realization of the implicit discretized form of the STA-based observer was realized in Matlab. The solution of the implicit relations between used the software proposed by (Huber et al., 2016). The comparison of implicit and explicit Euler discretization was realized with the aim of validating the suggested differentiator. The suggested observation method was evaluated on a regular framework. A continuous nonlinear observable system can be transformed (by the corresponding dipheomorphism) into its equivalent observable form (Brunovskii-like). This discretized form is implicitly discretized and then, based on the proposed step-by-step STA observer, the non-measurable states can be recovered. Once the variables are recovered, the inverse dipheomorphis transform can be used to recover the estimated states of the original nonlinear system. This section describes how the proposed differentiator was applied in this study. The Droop model (Droop, 1968) is a simple and widely used model that can represent the growth of microalgae culture. This model describes the dynamic evolution of biomass, the internal quota of nitrogen and the substrate used to feed microalgae. The continuous time model is formally presented as: x˙ (t) = x (t) (µ (QN (t)) − D) Q˙ N (t) = ρ (s (t)) − µ (Q (22) N (t)) QN (t) s˙ (t) = D sin (t) − s (t) − ρ (s (t)) x (t)
where the time varying parameters ρ (s (t)) and µ (QN (t)) are: KQ s (t) µ (QN (t)) = µ ¯ 1− ρ (s (t)) = ρm s (t) + Ks QN (t) −1 , the The biomass concentration is denoted as x µmol · L −1 internal quota QN µmol · L which is defined as the quantity of intracellular nitrogen per unit of biomass and the substrate concentration is denoted by s µmol · L−1 . The sub−3 −1 strate uptake rate is represented as ρ L · µm while · d ,−1 −3 −1 µ L µm d is the specific growth rate and D d is the constant dilution rate. The control function u corresponds to the substrate injected into the reactor Sin µmolL−1 . The selection of x and sin as output and input respectively, provides the full relative degree equal to 3 according to the procedure described in (Isidori, 1995). Therefore, using the results given in the same reference, there exists a nonlinear transformation z = T (Xb , S, O) z = [z1, z2 , z3 ] given by z1 =y = x KQ −D z2 = x˙ = x µ ¯ 1− Q N µ ¯KQ ρm s (23) z3 = x − µQN + 2 Q s + K s N KQ KQ −D µ ¯ 1− x µ ¯ 1− QN QN such that, the new variables z obey the following dynamics d z1 (t) = z2 (t) dt d (24) z2 (t) = z3 (t) dt d z3 (t) = f (z (t)) + g (z (t)) u (t) dt
The nonlinear function f (z) is given by: µ ¯KQ Q˙ N −ρm sQ˙ N z1 − f (z) = µ ¯KQ QN Q2N (s + Ks ) Q2N KQ ρm s +µ ¯ −1 × +¯ µKQ QN(s + Ks ) Q N ˙ z1 Q N z2 − QN Q2N ˙ z2 µ KQ ¯KQ QN + µ ¯ 1 − − D − + z 3 2 QN Q N ρm sx µ ¯ K Q K s ρm z1 Ds + 2 s + Ks Q2N (s + Ks )
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(25)
Additionally, the input-associated function g (z) satisfies the following structure µ ¯KQ Ks ρm Dz1 g (z) = (26) 2 Q2N (s + Ks )
The non-linear functions QN (z) and s (z) are described as −¯ µKQ β (z) QN (z) Ks s (z) = QN (z) = z1 ρm − βQN (z) −µ ¯+D z2 KQ −D z3 − z2 µ ¯ 1− QN (z) (27) β (z) = + µ ¯KQ z1 QN (z) KQ µ ¯ 1− QN (t) In general, measuring the internal quota and substrate is considered as an expensive and time-consuming procedure. However, the existence of the so-called full relative degree condition allows to use biomass as the sole required information to design an observer or software sensor. Therefore, this study was focused on developing an algorithm that serves as a software sensor to reconstruct the concentrations of QN and s Mart´ınez-Fonseca et al. (2014). As one can understand, microalgae biomass can be straightforwardly estimated by a simple optoelectronic sensor using a red light emitting led and the corresponding photodiode. The application of the proposed implicit discretized sliding-mode differentiator, which is serving here as a class of state observer produced the reconstruction of the non-measurable states as appeared in Figures (1) - (3). These figures show the time evolution of the states considered in the Drop model. For evaluation purposes, the implicit discretization of a high-gain like observer was also considered. The comparison shows the advantages of including the slidingmode sections. Figure (1) demonstrates the evolution of the biomass (the measurable variable) as well as its estimates (STA and high-gain). Both observers-like structures produced equally accurate estimations of this variable. The second figure (2) depicts the estimation of the substrate. Notice that the STA-based produces oscillations with smaller oscillations compared to the high-gain. This is a contribution of the implicit discretization which also helps reducing the region of convergence around the simulated trajectory of the substrate. Figure (3) depicts the variation of the third variable, the nitrogen quota which is an intracellular information. The estimation
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6. CONCLUSIONS
Biomass 20
This paper introduces an implicit discretization of of a class of high-order sliding-mode differentiator-like system. The proposed discretized system served as a robust observer for systems having the so-called Brunovskii form. The proposed differentiator is based on the implicit discretized method. The design method includes the calculus of the STA gains as well as the implicit estimation of the sequence of observer trajectories. The observer has the remarkable property of estimating the convergence region for the estimation error. This characteristic seems to be a step toward a consistent discretized version of the continuous-time variant of the step-by-step observer. The numerical simulation of the state estimation for the variables of the microalgae growth model.
18 16
−1
Biomass (µmol L )
14 12 Modeled biomass Proposed sliding mode observer High−gain observer
10 8 6 4 2 0 0
10
20
30
40
50
Time (day)
REFERENCES
Fig. 1. Time evolution of the microalgae biomass as well as its estimates by STA and high-gain implicit discretized observer-like algorithm. Substrate 110 100 90
−1
Substrate (µmol L )
80 70 60
Modeled QN Proposed sliding mode observer High−gain observer
50 40 30 20 10 0 0
10
20
30
40
50
Time (day)
Fig. 2. Time evolution of the microalgae substrate as well as its estimates by STA and high-gain implicit discretized observer-like algorithm. of this variable also demonstrates the potential benefits of the proposed observer. Internal quota (QN) 9 8.5 Modeled Q 8
QN (µmol L−1)
7.5
N
Proposed sliding mode observer High−gain observer
7 6.5 6 5.5 5 4.5 4 0
10
20
30
40
50
Time (day)
Fig. 3. Time evolution of the microalgae nitrogen quota as well as its estimates by STA and high-gain implicit discretized observer-like algorithm.
Acary, V., Brogliato, B., and Orlov, Y.V. (2012). Chattering-free digital sliding-mode control with state observer and disturbance rejection. IEEE Transactions on Automatic Control, 57(5), 1087–1101. Azar, A.T. and Zhu, Q. (2015). Advances and applications in sliding mode control systems. Springer. Brogliato, B., Polyakov, A., and Efimov, D. (2018). The implicit discretization of the super-twisting sliding-mode control algorithm. In 2018 15th International Workshop on Variable Structure Systems (VSS), 349–353. IEEE. Chalanga, A., Kamal, S., Fridman, L.M., Bandyopadhyay, B., and Moreno, J.A. (2016). Implementation of super-twisting control: Super-twisting and higher order sliding-mode observer-based approaches. IEEE Transactions on Industrial Electronics, 63(6), 3677–3685. Droop, M. (1968). Vitamin b12 and marine ecology iv: The kinetics of uptake growth and inhibition in monochrisis lutheri. Journal of the Marine Biological Association, 48(3), 689–733. Filippov, A. (1998). Differential Equations with Discontinuous Right-Hand Side. Kluwer, Dirdrecht, the Netherlands. Floquet, T. and Barbot, J.P. (2007). Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs. International Journal of Systems Science, 38(10), 803–815. Fridman, L., Shtessel, Y., Edwards, C., and Yan, X.G. (2008). Higherorder sliding-mode observer for state estimation and input reconstruction in nonlinear systems. International Journal of Robust and Nonlinear Control: IFAC-Affiliated Journal, 18(4-5), 399–412. Huber, O., Acary, V., and Brogliato, B. (2016). Lyapunov stability and performance analysis of the implicit discrete sliding mode control. IEEE Trans. Automat. Contr., 61(10), 3016–3030. Isidori, A. (1995). Nonlinear control systems, volume 1 of Communications and control engineering series. Springer-Verlag. Levant, A. (1998). Robust exact differentiation via sliding mode technique. automatica, 34(3), 379–384. Mart´ınez-Fonseca, N., Chairez, I., and Poznyak, A. (2014). Uniform stepby-step observer for aerobic bioreactor based on super-twisting algorithm. Bioprocess and biosystems engineering, 37(12), 2493–2503. Mart´ınez-Fonseca, N., Chairez, I., and Poznyak, A. (2014). Uniform step by step pbserver for aerobic bioreactor based on super-twisting algorithm. Bioprocess and Biosystems Engineering. Milosavljevic, C., Perunicic-Drazenovic, B., Veselic, B., and Mitic, D. (2006). Sampled data quasi-sliding mode control strategies. In Industrial Technology, 2006. ICIT 2006. IEEE International Conference on, 2640–2645. IEEE. Perruquetti, W. and Barbot, J.P. (2002). Sliding mode control in engineering, volume 11. M. Dekker. Polyakov, A., Efimov, D., and Brogliato, B. (2018). Consistent discretization of finite-time stable homogeneous systems. In VSS 2018-15th International Workshop on Variable Structure Systems and Sliding Mode Control. Salgado, I., Kamal, S., Bandyopadhyay, B., Chairez, I., and Fridman, L. (2016). Control of discrete time systems based on recurrent super-twisting-like algorithm. ISA transactions, 64, 47–55. Utkin, V. (1992). Sliding Modes in Control Optimization. Springer Verlag, Berlin. Utkin, V., Guldner, J., and Shi, J. (1999). Sliding Modes in Electromechanical Systems. Taylor&Francis, London.
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Step-by-step Step-by-step implicit implicit discrete discrete super-twisting super-twisting Step-by-step implicit discrete super-twisting differentiator for linearizable Step-by-step discrete super-twisting differentiatorimplicit for input-output input-output linearizable differentiator for input-output linearizable nonlinear systems differentiator for input-output nonlinear systems linearizable nonlinear systems nonlinear systems Isaac Chairez ∗∗
Isaac Chairez ∗∗∗ Isaac Chairez ∗ Isaac Chairez ∗ ∗ Bioprocesses Department, UPIBI, Instituto Polit´ cnico Nacional, Nacional, and and the the ∗ eecnico ∗ Bioprocesses Department, UPIBI, Instituto Polit´ ∗ School of Engineering of the Tecnol´ o gico de Monterrey, Campus Bioprocesses Department, UPIBI, Instituto Polit´ e cnico Nacional, and the ∗ School of Engineering of the Tecnol´ ogicoPolit´ de Monterrey, Campus Bioprocesses Department, UPIBI, Instituto cnico Nacional, and the School of Engineering of [email protected] Tecnol´ ogico de eMonterrey, Campus Guadalajara Guadalajara [email protected] School of Engineering of [email protected] Tecnol´ogico de Monterrey, Campus Guadalajara Guadalajara [email protected] Abstract: Abstract: The The aim aim of of this this study study was was to to design design aa step-by-step step-by-step numerical numerical differentiator differentiator for for aa class class of discretized version of a n-th chain-of-integrator system based on the application of the Abstract: The aim of this study was to design a step-by-step numerical differentiator for implicit a class of discretized version of a n-th chain-of-integrator system based on the application of the implicit Abstract: Theof aim of ofthis studychain-of-integrator was to design a step-by-step differentiator for implicit a class discretization the algorithm The reconstructed the of discretized version a n-th basednumerical on differentiator the application of the discretization of the super-twisting super-twisting algorithm (STA). (STA).system The high-order high-order differentiator reconstructed the of discretized version of a n-th chain-of-integrator system based on the application of the implicit non-measurable states of an an input-output input-output linearizable systems. The gains gainsdifferentiator of the the observer observer were calculated calculated discretization ofstates the super-twisting algorithm (STA).systems. The high-order reconstructed the non-measurable of linearizable The of were discretization ofstates the super-twisting algorithm (STA). The high-order differentiator reconstructed the non-measurable of an input-output linearizable systems. The gains of the observer were calculated based on the application of aa class of generalized functions. The convergence analysis showed that based on the application of class of generalized functions. The convergence analysis showed that non-measurable states of an input-output linearizable systems.set The gains of the observer were calculated all the states in the linearized system achieved an invariant with a predefined size. The existence based on the application of a class of generalized functions. The convergence analysis showed that all the on states the linearized an invariant setThe withconvergence a predefinedanalysis size. The existence based the in application of aonsystem class ofachieved generalized functions. showed that of invariant set depends the uniqueness of solution the function. A simple all this the states in the linearized system achieved an the invariant setfor with a generalized predefined size. The existence of this invariant set depends on the uniqueness of the solution for the generalized function. A simple all the states in the linearized system achieved an invariant set with a predefined size. The existence of this invariant set depends on the uniqueness of the solution for the generalized function. A simple interpretation of of aa possible possible exact exact reconstruction reconstruction of of the the non-measurable non-measurable states states of of the the implicit implicit discretized discretized interpretation of thisThe invariant depends onimplicit the uniqueness of the the solution for the states generalized function. A for simple interpretation of asetpossible exact reconstruction of non-measurable of the of implicit discretized form. application of the discretized STA served to design aa class observer form. The application of the implicit discretized STA served to design class of observer for the the interpretation of a possible exact reconstruction of the non-measurable states of the of implicit discretized microalgae culture system. form. The application of the implicit discretized STA served to design a class observer for the microalgae culture system. form. The application of the implicit discretized STA served to design a class of observer for the microalgae culture system. © 2019, IFAC (International microalgae culture system. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Sliding Sliding mode; mode; Super-twisting Super-twisting algorithm; algorithm; Numerical Numerical differentiator; differentiator; Discrete Discrete sliding sliding mode. mode. Keywords: Sliding mode; Super-twisting algorithm; Numerical differentiator; Discrete sliding mode. Keywords: Sliding mode; Super-twisting algorithm; Numerical differentiator; Discrete sliding mode. 1. INTRODUCTION INTRODUCTION 2016). However, However, their their discretized discretized counterparts counterparts have have not not been been 1. 2016). 1. INTRODUCTION 2016). However, their discretized counterparts haveorder not been profoundly developed. The particular class of first difprofoundly developed. The particular class of first order dif2016). However, their discretized counterparts have(STA) not been Sliding mode mode theory theory 1. hasINTRODUCTION contributed to to solve solve the the robust robust control control ferentiator profoundly developed. The particular class of first order difbased on the super-twisting algorithm has Sliding has contributed ferentiator based on the super-twisting algorithm (STA) has profoundly developed. The particular class of first order difSliding modesystems, theory has contributed to solve the robustsystems control proven of uncertain the state observation of uncertain ferentiator based on the super-twisting algorithm (STA) has to be an efficient example of robust and exact differof uncertain systems, the state observation of uncertain systems proven to be an efficient example of robust and exact differSliding mode theory has contributed to solve the robust control ferentiator based on the super-twisting algorithm (STA) has of uncertain systems, the stateestimation observation of uncertain systems and the finite/time parameter (Utkin, 1992). The maproven toThe be an efficient example of of STA robust and exact differentiator. The discretized versions STA appeared recently, and the finite/time parameter estimation (Utkin, 1992). The maentiator. discretized versions appeared recently, of uncertain systems, the statethe observation ofseveral uncertain systems proven to be an efficient example of robust and exact differand the finite/time parameter estimation (Utkin, 1992). The maturity of this theory attracted attention of researchers entiator. The discretized versions of STA appeared recently, mostly presented as explicit discretization forms of the continturity offinite/time this theoryparameter attracted the attention(Utkin, of several researchers mostly presented as explicit discretization forms of therecently, continand the estimation 1992). The maentiator. The discretized versions of STA appeared turity of thistried theory attracted the attentiondiscrete of several researchers who have to introduce consistent forms of the mostly presented as(Salgado explicit discretization forms of theimplicit continuous STA version et al., 2016). Just recently who have tried to introduce consistent discrete forms of the uous STA version (Salgado et al., 2016). Just recently implicit turityhave ofmode thistried theory attracted the attention of several researchers mostly presented as(Salgado explicitof discretization forms of theimplicit continwho to introduce consistent discrete forms of the and sliding algorithms (Utkin et al., 1999). Since the earlier uous STA version et al., 2016). Just recently semi-implicit variants the STA showed better conversliding mode algorithms (Utkin et al., 1999). Since the earlier semi-implicit variants of the STA showed better implicit converwho have tried to introduce consistent discrete forms of the and uous STA version (Salgado et al., 2016). Just recently sliding mode algorithms (Utkin et al., 1999). Since the earlier stages of the sliding mode theory, there were some proposal and semi-implicit variants of the STA showed better convergence properties, properties, including including smaller smaller sizes sizes of of the the invariant invariant zone zone stages the algorithms sliding mode theory, there wereSince somethe proposal sliding of mode (Utkin et al., 1999). earlier gence and semi-implicit variants of theSTA STA showed better converstages of the sliding mode theory, there wereof some proposal showing how to introduce discrete versions discontinuous gence properties, including smaller sizes of the invariant zone around the sliding surface if the plays controller role showing how to introduce discrete versions of discontinuous around the sliding surface if the STA plays the controller role stages of how theThe sliding mode theory, there wereofsome proposal gence properties, including smaller sizes of the the controller invariant zone showing to classical introduce discrete versions discontinuous controllers. applications of sliding modes in disaround the sliding surface if the STA plays role (Brogliato et al., 2018). Some studies showed that the semicontrollers. The classical applications of sliding modes in dis(Brogliato et al., 2018). Some studies showed that the semishowing how to introduce discrete versions of discontinuous around the sliding surface if thestudies STAfirst plays the discretized controller role controllers. The classical applications of sliding modes dis- implicit crete time produced either the exact convergence to aainwell(Brogliato et al., 2018). Some showed that the semiSTA even can estimate the order apcrete time produced either the exact convergence to wellimplicit STA even2018). can estimate the firstshowed order discretized apcontrollers. Thesurface classical applications of sliding modes inwelldis(Brogliato et al., Some studies that the semicrete time produced either the exact convergence to a defined sliding (Utkin et al., 1999) or the convergence to implicit STA even can estimate the first order discretized approximation of a signal derivative exactly (Acary et al., 2012). defined sliding surface (Utkin et al., 1999) or the convergence to proximation of a signal derivative exactly (Acary et al., 2012). crete time produced either the exact convergence to a wellimplicit STA even can estimate the first order discretized apdefined sliding surface (Utkin et al., 1999) or the convergence to an invariant tube circulating the sliding surface (Milosavljevic proximation of a signal derivative exactly (Acary et al., 2012). The implicit realization of STA can offer remarkable characan invariant tube circulating sliding surface (Milosavljevic implicit of realization of STA can offer(Acary remarkable defined sliding surface (Utkinthe et al., 1999) or the convergence to The proximation a signal derivative exactly et This al.,charac2012). an invariant tube circulating the sliding surface (Milosavljevic et al., 2006). The movement of the state trajectories inside the The implicit realization of STA can offer remarkable characteristics to prove the STA properties as controller. study et al., 2006). The movement of the state trajectories inside the to prove the STA as controller. Thischaracstudy an al., invariant tube circulating the sliding (Milosavljevic The implicit realization ofofproperties STA can offer remarkable et 2006). The of the statesurface trajectories inside the teristics referenced tube is known sliding-mode. These variants teristics tothe prove the STA properties as controller. This study considers application such result to develop aa class of referenced tube is movement known as as quasi quasi sliding-mode. These variants considers the application of such result to develop class of et al., 2006). The movement of the state trajectories inside the teristics discretized tothe prove thedifferentiator STAofproperties asoncontroller. This study referenced tubesliding is known as quasi sliding-mode. These variants of the discrete modes used discretized forms of nonlinconsiders application suchbased result tothe develop a class of implicit STA. The develof the discrete sliding modes used discretized forms of nonlinimplicit discretized differentiator based on the STA. The develreferenced tube is known as quasi sliding-mode. These variants considers the application of such result to develop a class ofa of the discrete sliding modes used discretized forms of nonlinear systems based on the the explicit discretization technique. Most opment implicit discretized differentiator based on the STA. The develof the implicit STA was extended for implementing ear systems based on explicit discretization technique. Most opment of the implicit STA was based extended for STA. implementing a of the discrete sliding modes used discretized forms of nonlinimplicit discretized differentiator on the The develear systems based on the explicit discretization technique. Most of these approaches have considered first order sliding modes opment of theorder implicit STA wasSTA-based extended for implementing a class of high step-by-step differentiator. This of these approaches have considered first ordertechnique. sliding modes class of high order step-by-step STA-based differentiator. This ear systems based on the explicit discretization Most opment of theorder implicit STA was extended for implementing a of these approaches have considered first order sliding modes structure only (Azar and Zhu, 2015). class of high step-by-step STA-based differentiator. This was inspired by the works proposed in (Floquet and only (Azar and Zhu, 2015). was order inspired by the works proposed in (FloquetThis and of these approaches have considered first order sliding modes structure class of high step-by-step STA-based differentiator. only (Azar and Zhu, 2015). structure was inspired by the works proposed in (Floquet and Barbot, 2007) and reformulated in (Mart´ ı nez-Fonseca et al., 2007) and reformulated in (Mart´ ınez-Fonseca et and al., In recent years, the only (Azar and Zhu, 2015). of In recent years, the emerging emerging of the the so-called so-called high high order order sliding sliding Barbot, structure was inspired by the works proposed inestimate (Floquet Barbot, 2007) and reformulated inprovided (Mart´ ınez-Fonseca et al., 2014). The proposed differentiator the of the In recent years, the emerging of the so-called high order sliding 2014). The proposed differentiator provided the estimate of the modes provided additional benefits to the classical (first order) modes provided additional benefits to the classical (firstsliding order) invariant Barbot,The 2007) and reformulated in (Mart´ ınez-Fonseca et al., 2014). proposed differentiator provided the estimate of the set centered at the origin for the estimation error. The In recent years, the emerging of the so-called high order modes provided additional benefits to the classical (first order) invariant set centered at the origin for the estimation error. The sliding modes such as reduction of the chattering amplitude sliding modes such as reduction of the chattering amplitude 2014). The proposed differentiator provided the estimate of the invariant centeredset the originon forthe thesolution estimation error. The size of the theset invariant setatdepended depended on the solution of well-posed well-posed modes additional benefits tothe the classicalproperty, (first order) sliding modes surface, such as no reduction chattering amplitude of invariant of on the provided sliding surface, no lose of of the robustness etc size on the sliding lose of the robustness property, etc invariant set centered at the origin for the estimation error. The size of the invariant set depended on the solution of well-posed generalized equations. These solutions were supported on the sliding modes such as reduction chattering amplitude on the sliding loseFridman of of thethe robustness property, etc generalized equations. solutions supported on the (Perruquetti and Barbot, 2002; et However, (Perruquetti andsurface, Barbot,no 2002; et al., al., 2008). 2008). However, size of the invariant set These depended on the were solution of well-posed generalized equations. These solutions were supported on the theory. on the sliding surface, no loseFridman of the robustness property, etc convex-set (Perruquetti and Barbot, 2002; Fridman et al., 2008). However, convex-set theory. the discretization of high-order sliding modes is still a matter the discretization of high-order sliding modes is stillHowever, a matter convex-set generalizedtheory. equations. These solutions were supported on the (Perruquetti and Barbot, 2002; Fridman et al., 2008). theresearching discretization of high-order sliding applications modes is stillas matter This of because their manuscript is of because their potential potential applications asaa on-line on-line convex-set theory. This manuscript is organized organized as as follows. follows. Section Section II II shows shows some some theresearching discretization of controllers high-order sliding modes is still matter of researching because their potential applications as the on-line digital instrumented or state estimators. In parThis manuscript is organized as follows. Section II showswhich some basic mathematical antecedents on convex set theory digital instrumented controllers or state estimators. In the parbasic mathematical antecedents on convex set theory which of researching because their potential applications as satisfies on-line This manuscript is organized as follows. Section IItheory showswhich some digital instrumented controllers or stateto estimators. In the par- were ticular case that the discretized system be observed basic mathematical antecedents on convex set useful in the design of the high order differentiator. ticular case that the discretized system to be observed satisfies useful in the antecedents design of the high order differentiator. digital instrumented controllers orform, stateto estimators. Inobserver the par- were basic mathematical on convex set theory which ticular case that the discretized system be observed satisfies the so-called chain of integrator then the state were useful in the design of the high order differentiator. Section III III details details the the structure structure of of the the high-order high-order differentiator differentiator the so-called chain of integratorsystem form, to then state observer ticular case that thenumerical discretized be the observed satisfies Section were useful instep-by-step the of of thethehigh order differentiator. the so-called chain of integrator form, then the state observer coincides with the high-order differentiator (Levant, Section IIIthe details thedesign structure high-order differentiator including turning-on functions. Section coincides with the numerical high-order differentiator (Levant, including the step-by-step turning-on functions. Section IV IV the so-called of integrator form, then the state observer Section III details the structure of observer the high-order differentiator coincides withchain the numerical high-order differentiator (Levant, contains 1998). including the step-by-step turning-on functions. Section IV a simple analysis of the structure, what are 1998). contains a simple analysis of the observer structure, what are coincides with the numerical high-order differentiator (Levant, including the step-by-step turning-on functions. Section IV 1998). contains a simple analysis of the observer structure, what are the generalized function needed to find the estimated states and generalized function needed to observer find the estimated states Today, 1998). there Today, there are are some some variants variants of of high-order high-order sliding sliding mode mode the contains a simple analysis of the structure, what and are the generalized function needed to find the estimated states and how the step-by-step structure worked. Section V establishes Today, there are variants of systems high-order sliding et mode the step-by-step structure Section V establishes differentiators for continuous time (Chalanga al., differentiators for some continuous time (Chalanga al., how the generalized function needed worked. to find the estimated states and the step-by-step structure worked. Section V establishes Today, there are variants of systems high-order sliding et mode differentiators for some continuous time systems (Chalanga et al., how how the step-by-step structure worked. Section V establishes differentiators for continuous time systems (Chalanga et al., 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
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some details on the numerical implementation of the proposed step-by-step differentiator. Section VI presents a numerical realization of the differentiator for a class of micoalgae growth system. Section VI closes the article with some final remarks.
Assumption 2. The function u belongs to the following admissible set: 2
Uadm = u | u(t) ≤ u+ , t ≥ 0, u+ ∈ R+
2. MATHEMATICAL ANTECEDENTS The implicit numerical implementation of the STA requires the application of the convex set theory. The set-valued sign function satisfies the following definition: +1 if z > 0 sign(z) = [−1, +1] if z = 0 (1) −1 if z < 0 The discrete implicit realization of the STA algorithm requires the inclusion of the inverse function for the multiset valued function. This part of the study follows the ideas given in (Acary et al., 2012; Huber et al., 2016). Introduce K ⊆ Rn a closed non empty convex set, which defines its normal cone set. Its normal cone at xK ∈ K satisfies he following definition NK (x) = {w ∈ Rn |w, y − x ≤ 0 ∀y ∈ K} (2) Notice that, for the convex set [−1, +1], one may define R− if s = −1 N[−1,+1] (s) = {0} if s ∈ [−1, +1] R if s = +1
Assumption 1 implies that input u is a stabilizing controller at least. The following assumption is imposed to u:
(3)
The Convex Analysis Theory justifies that N[−1,+1] (s(z)) is the inverse of the set-valued function, sign(z) that is (4) x ∈ sign(z) ⇔ K[−1,+1] (s(z)) Let G ∈ Rn →∈ Rn be a set-valued mapping and f ∈ Rn →∈ Rn be a single-valued vector field, therefore, 0 ∈ f (x) + G(x) (5) is known as the generalized equation. The application of these functions justifies the estimation of the variables that cannot be measured on-line in the discretized version of the chain-ofintegrator systems (Brunovskii form). 3. STEP-BY-STEP OBSERVER IN CONTINUOUS TIME Let consider the nonlinear single-input-single-output (SISO) system in the Brunovskii form: d z1 (t) = z2 (t) dt d z2 (t) = z3 (t) dt (6) .. . d zn (t) = f (z (t)) + g (z (t)) u (t) dt
Here z = [z1 · · · zn ], z ∈ Rn is the state of (6). The variable u ∈ R is an external signal which may represent the feedback control or an open loop injection. Notice that f and gu play the role of internal uncertainties and external perturbations affecting the dynamics of the chain of integrator form. For developing the result of this study, the following assumption is considered valid in this study. Assumption 1. Under the given controller u, all the states of (6) are bounded, that is |zi | ≤ zi+ .
(7)
The nonlinear function f : Rn → R is the uncertain internaldynamics term of (24), which satisfies the local Lipschitz condition: 2 2 |f (za ) − f (zb )| ≤ Lf za − zb with the constant Lf a positive scalar and za , zb ∈ Rn .
The function associated to the function g is bounded as follows 2
0 < g1 ≤ |g(z)| ≤ g2 ,
g1 , g2 ∈ R +
This study centers on developing a nonlinear differentiator in order to recover those non-measurable variables and using the first state as the available information. The observer was designed as a modification of the one presented in (Floquet and Barbot, 2007). The differentiator satisfies the following differential equation d zˆ1 (t) = z˜2 (t) + k11 φ11 (e1 (t)) dt d z˜2 (t) = k12 φ12 (e1 (t)) dt d zˆ2 (t) = E2 [˜ z3 (t) + k21 φ21 (e2 (t))] dt d z˜3 (t) = E2 [k22 φ22 (e2 (t))] dt .. . d zˆn (t) = En [f (ˆ z (t)) + g(ˆ z (t))u(t) + kn1 φn1 (en (t))] dt d z˜n+1 (t) = En [kn2 (t)φn2 (en (t))] dt (8) with: ei = z˜i − zˆi , i ∈ {1, 2, ·, n}. The set of variables zˆi represents the corresponding estimated trajectories of zi . In particular, z˜1 = z1 and zn = 0, which is needed to complete the observer design. The observer aims represented by ki1 , ki2 , i = [1, n] are scalars that must be adjusted to force the convergence of the differentiator trajectories to the states of the uncertain system. Remark 1. The inclusion of the last equation in the differentiator structure (8) has the aim of recovering the uncertain section affecting the nonlinear dynamics 6. The indicator function Ei (t) fulfills the following definition 0 t < Ti∗ Ei (t) = 1 t ≥ Ti∗ The switching time Ti∗ is found as a secondary result of the asymptotic converge obtained for the observer in the following section. The nonlinearfunctions φj1 (ej , αj ) and φj2 (ej ) (j = 1, 2) were designed in agreement to the proposal given in (Levant, 1998). Then, the following structures were considered for the observer design: 1 1/2 φ2j (ej ) = sign(ej ) φ1j (ej ) = |ej | sign(ej ) 2 One must recall that the solution of (8) is understood in the sense of Filippov (Filippov, 1998). In continuous time, the
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differentiator provides the asymptotic stability for the origin of the estimation error (Brogliato et al., 2018). Notice that most of the convergence analysis in high-order sliding modes uses the Lyapunov stability theory Nevertheless, the existing Lyapunov functions for continuous systems cannot be directly used for proving the convergence of the discretized versions of high-order sliding modes because there is not a conclusive evidence that the (continuous-time) Lyapunov function has convex level sets (Brogliato et al., 2018) which is not obvious for the aforementioned Lyapunov functions. Therefore, the analysis developed in this study is not realizing a discrete-time Lyapunov analysis for the proposed differentiator because there is not a continuous Lyapunov function counterpart. 4. THE DISCRETE REALIZATION OF THE STEP-BY-STEP OBSERVER The proposed discretization (implicit) for the observer (8) satisfies: zˆ1,v (tk+1 ) = zˆ1,v (tk ) + T z˜2,v (tk+1 )+ T k11 φ11 (e1 (tk+1 )) z˜2,v (tk+1 ) = z˜2,v (tk ) + T k12 φ12 (e1 (tk+1 )) zˆ2 (tk+1 ) = E2 [ˆ z2 (tk ) + T z˜3 (tk+1 )+ T k21 φ21 (e2 (tk+1 ))] z˜3 (tk+1 ) = E2 [˜ z3 (tk ) + T k22 φ22 (e2 (tk+1 ))] (9) .. . zn (tk ) + T f (ˆ z (tk ))+ zˆn (tk+1 ) = En [ˆ T g(ˆ z (tk ))u(tk )+ T kn1 φn1 (en (tk+1 ))] z˜n+1 (tk+1 ) = En [˜ zn+1 (tk ) + T kn2 φn2 (en (tk+1 ))] Here, T = tk+1 − tk is the sampling time. The proposed discretized observer (9) is not exactly the implicit discretization of (8) because the presence of f (ˆ z (tk )) and g(ˆ z (tk ))u(tk ).
Recalling that sign(0) is a set-valued function, then it is necessary to introduce a generalized equation with the aim of finding e1 (tk+1 ). According to (Polyakov et al., 2018), it is simpler introducing a new auxiliary variable ψ1 (tk+1 ) such that ψ1 (tk+1 ) = φ12 (e1 (tk+1 )). Based on the inverse set-valued function, e1 (tk+1 ) is the solution of the generalized equation given by: h11 (−k11 φ11 (e1 (tk+1 ))) ∈ N[−1,+1] () −1 h11 (r1 ) = h 12 (e1 (tk+1 )) h12 (r1 ) = r1 + c11 |r1 |sign(r1 ) + c12 c11 = T k12 , −c12 = e1,v (tk ) + T δ2,v (tk+1 )
(11)
Complementary, e1 (tk+1 ) can be obtained by implementing the solution of the generalized equation given in: h12 (e1 (tk+1 )) ∈ −k11 φ11 (e1 (tk+1 )))
(12)
According to the result obtained in (Floquet and Barbot, 2007), it is possible to execute a monitoring of the measurable variables e1 and δ2,v to establish a feasible estimation of the switching-on time Tj∗ for Ej , n ≥ j ≥ 2 (Notice here that T1∗ = 0 evidently). In instance, the estimation of the time T2∗ requires the introduction of an indicator function which defines if the first STA in the differentiator design has attained the estimation of z2 . A possible indicator function is e1 δ2,v ≤ Ts,1 e1 δ2,v dt ≤ where > 0 Ts,1 > 0, Ts,1 = M1 T , or t=0 M1 ∈ Z + is a designer choice. This time Ts,1 defines an observation period which serves to characterize if the first STA in the differentiator design is close enough to the estimate of z2 . In consequence, the dynamic system (10) can be transformed to: e1 (tk+1 ) = e1,v (tk ) + T δ2,v (tk+1 ) − T k11 φ11 (e1 (tk+1 )) δ2,v (tk+1 ) ∈ δ2,v (tk ) + T z3 (tk+1 ) − T k12 φ12 (e1 (tk+1 ))
Using the implicit discretization of (8), and estimating the dynamics of δi = zi − z˜1,v , one gets
e2 (tk+1 ) ∈ z˜2,v (tk ) + T k12 φ12 (e1 (tk+1 )) δ3,v (tk+1 ) ∈ δ3,v (tk ) + T z4,v (tk+1 )
e1 (tk+1 ) = e1,v (tk ) + T δ2,v (tk+1 )− T k11 φ11 (e1 (tk+1 ))
e3 (tk+1 ) = 0 .. .
δ2,v (tk+1 ) ∈ δ2,v (tk ) + T z3 (tk+1 )− T k12 φ12 (e1 (tk+1 ))
δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 )
e2 (tk+1 ) = z˜2,v (tk ) + T k12 φ12 (e1 (tk+1 )) − E2 [ˆ z2 (tk ) + T z˜3 (tk+1 ) + T k21 φ21 (e2 (tk+1 ))]
en (tk+1 ) = 0
δ3,v (tk+1 ) ∈ δ3,v (tk ) + T z4,v (tk+1 )− E2 [˜ z3 (tk ) + T k22 φ22 (e2 (tk+1 ))] e3 (tk+1 ) = E2 [˜ z3 (tk ) + T k22 φ22 (e2 (tk+1 ))] − E3 [ˆ z3 (tk ) + T z˜4 (tk+1 ) + T k31 φ31 (e2 (tk+1 ))] .. . δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 )− En−2 [˜ zn−1 (tk )+ T k(n−1)2 φ(n−1)2 (en−1 (tk+1 ))
785
(10)
zn (tk )+ en (tk+1 ) = En−1 [˜ T k(n−1)2 φ(n−1)2 (en−1 (tk+1 )) − zn (tk ) + T f (ˆ z (tk )) + T g(ˆ z (tk ))u(tk )+ En [ˆ T kn1 φn1 (e2 (tk+1 ))]
In (Brogliato et al., 2018), there is a result justifying that the origin is preferably an asymptotic stable equilibrium point for the regular structure of the STA. Notice that the first couple of equations in (10) are the implicit discretization form indeed.
(13)
Notice that the analysis of (13) can follow the ideas proposed in (Acary et al., 2012). However, such results consider a class of semi-implicit discretization. Nevertheless, the semi-implicit form proposed in that study, yields to simpler generalized equation. Instead, this study used the method proposed by (Brogliato et al., 2018). Notice that in those studies, the finitetime convergence of the estimation error can not be ensured. This is still a matter of investigation which deserves more researching efforts. Notice that if there is an unique solution for the generalized equations in (11)-(12), then the first two dynamics in (13) yields to justify that exists a number of sampling steps M1 such that |e1 | ≤ as well as |δ2,v | ≤ . A first result of this study proves that the set of equations proposed in (11) admits at least one solution (unique in the best case). The generalized forms in (11) and (12) have solutions for the given k11 and k12 , e1,v and δ2,v . Based on the generalized functions proposed above, one may argue that the these solutions satisfy: e1,v (tk+1 ) =
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0.5 ∗ −c11 ± c211 − 4(c12 − h12 (r1 )) if c12 < h12 (r1 )), e1,v (tk+1 ) = 0.5 ∗ −c11 ± c211 + 4(c12 − h12 (r1 )) if c12 > h12 (r1 )). The continuation of the analysis developed for the first can be done sequentially for the rest of the steps. This yields to the i − th step. Notice that it is feasible to introduce the corresponding function to obtain the switching-on time Tj∗ for Ej , n ≥ j ≥ i. The corresponding indicator function for the i-th step could be Ts,i ei δi+1,v ≤ or t=0 ei δi+1,v dt ≤ where > 0 Ts,i > 0, Ts,i = Mi T , Mi ∈ Z + is a designer choice. This time Ts,i now corresponds to the proposed observation period, which characterizes if the i-th STA in the observer design is close enough to the estimate of zi . The stability analysis at this i-th step applies on the discrete dynamics represented by: e1 (tk+1 ) ∈ [−, ]
e1 (tk+1 ) ∈ [−, ]
δ2,v (tk+1 ) ∈ [−, ] .. . ei (tk+1 ) ∈ [−, ]
δi+1,v (tk+1 ) ∈ [−, ]
.. . en−1 (tk+1 ) = en−1 (tk ) + T δn,v (tk+1 )− T k(n−1)1 φ(n−1)1 (en−1 (tk+1 )) δn,v (tk+1 ) ∈ δn,v (tk ) − T k(n−1)2 φ(n−1)2 (en−1 (tk+1 )) en (tk+1 ) = en (tk ) − T kn1 φn1 (en (tk+1 ))
In this last step, the existence of the unique solution for the associated generalized equations given by
δ2,v (tk+1 ) ∈ [−, ]
.. . ei (tk+1 ) = ei (tk ) + T δi+1,v (tk+1 )− T ki1 φi1 (ei−1 (tk+1 )) δi+1,v (tk+1 ) ∈ δi+1,v (tk ) + T zi+2,v (tk+1 )− T k(i)2 φ(i)2 (ei (tk+1 ))
(14)
hn1 (−kn1 φn1 (en (tk+1 ))) ∈ N[−1,+1] () −1 hn1 (rn ) = h n2 (en (tk+1 )) hn2 (rn ) = rn + cn1 |rn |sign(rn ) + cn2 cn1 = T kn2 , −cn2 = en,v (tk )
δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 ) en (tk+1 ) = 0
Consider that if there is the corresponding unique solution for the corresponding generalized equations given in (15) and (16), that is hi1 (−ki1 φi1 (ei (tk+1 ))) ∈ N[−1,+1] () −1 hi1 (ri ) = h i2 (ei (tk+1 )) (15) hi2 (ri ) = ri + ci1 |ri |sign(ri ) + ci2 ci1 = T ki2 , −ci2 = ei,v (tk ) + T δi+1,v (tk+1 )
δ2,v (tk+1 ) ∈ [−, ]
.. . ei (tk+1 ) =∈ [−, ] δi+1,v (tk+1 ) ∈ [−, ]
(17)
.. .
δn−1,v (tk+1 ) ∈ δn−1,v (tk ) + T zn,v (tk+1 ) en (tk+1 ) = 0
In the last step of the convergence analysis, the dynamics governing the estimation error is given by
(20)
In accordance to the ideas developed above, the dynamics for en−1 and δn in (18) justifies the existence of some sampling steps Mn such that |en−1 | ≤ as well as |δn,v | ≤ . Indeed, the expression (18) corresponds to e1 (tk+1 ) ∈ [−, ]
δ2,v (tk+1 ) ∈ [−, ]
.. . ei (tk+1 ) =∈ [−, ] δi+1,v (tk+1 ) ∈ [−, ]
Complementary, ei (tk+1 ) can be obtained by implementing the solution of the generalized equation given in: hi2 (ei (tk+1 )) ∈ −ki1 φi1 (ei (tk+1 ))) (16) Following the ideas developed in the first step, then the corresponding two dynamics for ei and δi+1 in (17) yields to justify that exists a number of sampling steps Mi such that |ei | ≤ as well as |δi+1,v | ≤ . In consequence, the dynamics described by (17) can be simplified to e1 (tk+1 ) ∈ [−, ]
(19)
In correspondence, en (tk+1 ) can be obtained by solving: hn2 (en (tk+1 )) ∈ −kn1 φn1 (en (tk+1 )))
.. .
(18)
(21)
.. . en−1 (tk+1 ) ∈ [−, ] δn,v (tk+1 ) ∈ [−, ]
en (tk+1 ) = en (tk ) − T kn1 φn1 (en (tk+1 )) Notice that the last step in (21) yields to estimate the derivative of en , which may yield to recover the uncertainties of the nonlinear system, that is f + gu. Notice that at this point, we have the way to produce the sequence of z˜ and zˆ which corresponds to the estimated of the non-measurable states of the discretized chain of integrator structure. You may notice that if the indicator functions can be designed such that ∀i ∈ [1, n] can be set to zero, then ei (tk ) = 0 and ei (tk+1 ) = ei (tk ). This implies the equivalence of the finite-time convergence for the class of discretized differentiator proposed in this study. Notice that unless the condition regarding the indicator functions yield to enforce the switching in the differentiator structure with a fixed = 0, then the finite time convergence cannot be proven using the classical control theory. Notice that if there are not enough arguments to justify that ei (tk+1 ) = ei (tk ).
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5. NUMERICAL IMPLEMENTATIONS The numerical realization of the implicit discretized form of the STA-based observer was realized in Matlab. The solution of the implicit relations between used the software proposed by (Huber et al., 2016). The comparison of implicit and explicit Euler discretization was realized with the aim of validating the suggested differentiator. The suggested observation method was evaluated on a regular framework. A continuous nonlinear observable system can be transformed (by the corresponding dipheomorphism) into its equivalent observable form (Brunovskii-like). This discretized form is implicitly discretized and then, based on the proposed step-by-step STA observer, the non-measurable states can be recovered. Once the variables are recovered, the inverse dipheomorphis transform can be used to recover the estimated states of the original nonlinear system. This section describes how the proposed differentiator was applied in this study. The Droop model (Droop, 1968) is a simple and widely used model that can represent the growth of microalgae culture. This model describes the dynamic evolution of biomass, the internal quota of nitrogen and the substrate used to feed microalgae. The continuous time model is formally presented as: x˙ (t) = x (t) (µ (QN (t)) − D) Q˙ N (t) = ρ (s (t)) − µ (Q (22) N (t)) QN (t) s˙ (t) = D sin (t) − s (t) − ρ (s (t)) x (t)
where the time varying parameters ρ (s (t)) and µ (QN (t)) are: KQ s (t) µ (QN (t)) = µ ¯ 1− ρ (s (t)) = ρm s (t) + Ks QN (t) −1 , the The biomass concentration is denoted as x µmol · L −1 internal quota QN µmol · L which is defined as the quantity of intracellular nitrogen per unit of biomass and the substrate concentration is denoted by s µmol · L−1 . The sub−3 −1 strate uptake rate is represented as ρ L · µm while · d ,−1 −3 −1 µ L µm d is the specific growth rate and D d is the constant dilution rate. The control function u corresponds to the substrate injected into the reactor Sin µmolL−1 . The selection of x and sin as output and input respectively, provides the full relative degree equal to 3 according to the procedure described in (Isidori, 1995). Therefore, using the results given in the same reference, there exists a nonlinear transformation z = T (Xb , S, O) z = [z1, z2 , z3 ] given by z1 =y = x KQ −D z2 = x˙ = x µ ¯ 1− Q N µ ¯KQ ρm s (23) z3 = x − µQN + 2 Q s + K s N KQ KQ −D µ ¯ 1− x µ ¯ 1− QN QN such that, the new variables z obey the following dynamics d z1 (t) = z2 (t) dt d (24) z2 (t) = z3 (t) dt d z3 (t) = f (z (t)) + g (z (t)) u (t) dt
The nonlinear function f (z) is given by: µ ¯KQ Q˙ N −ρm sQ˙ N z1 − f (z) = µ ¯KQ QN Q2N (s + Ks ) Q2N KQ ρm s +µ ¯ −1 × +¯ µKQ QN(s + Ks ) Q N ˙ z1 Q N z2 − QN Q2N ˙ z2 µ KQ ¯KQ QN + µ ¯ 1 − − D − + z 3 2 QN Q N ρm sx µ ¯ K Q K s ρm z1 Ds + 2 s + Ks Q2N (s + Ks )
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(25)
Additionally, the input-associated function g (z) satisfies the following structure µ ¯KQ Ks ρm Dz1 g (z) = (26) 2 Q2N (s + Ks )
The non-linear functions QN (z) and s (z) are described as −¯ µKQ β (z) QN (z) Ks s (z) = QN (z) = z1 ρm − βQN (z) −µ ¯+D z2 KQ −D z3 − z2 µ ¯ 1− QN (z) (27) β (z) = + µ ¯KQ z1 QN (z) KQ µ ¯ 1− QN (t) In general, measuring the internal quota and substrate is considered as an expensive and time-consuming procedure. However, the existence of the so-called full relative degree condition allows to use biomass as the sole required information to design an observer or software sensor. Therefore, this study was focused on developing an algorithm that serves as a software sensor to reconstruct the concentrations of QN and s Mart´ınez-Fonseca et al. (2014). As one can understand, microalgae biomass can be straightforwardly estimated by a simple optoelectronic sensor using a red light emitting led and the corresponding photodiode. The application of the proposed implicit discretized sliding-mode differentiator, which is serving here as a class of state observer produced the reconstruction of the non-measurable states as appeared in Figures (1) - (3). These figures show the time evolution of the states considered in the Drop model. For evaluation purposes, the implicit discretization of a high-gain like observer was also considered. The comparison shows the advantages of including the slidingmode sections. Figure (1) demonstrates the evolution of the biomass (the measurable variable) as well as its estimates (STA and high-gain). Both observers-like structures produced equally accurate estimations of this variable. The second figure (2) depicts the estimation of the substrate. Notice that the STA-based produces oscillations with smaller oscillations compared to the high-gain. This is a contribution of the implicit discretization which also helps reducing the region of convergence around the simulated trajectory of the substrate. Figure (3) depicts the variation of the third variable, the nitrogen quota which is an intracellular information. The estimation
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2019 IFAC NOLCOS 788 Vienna, Austria, Sept. 4-6, 2019
Isaac Chairez / IFAC PapersOnLine 52-16 (2019) 783–788
6. CONCLUSIONS
Biomass 20
This paper introduces an implicit discretization of of a class of high-order sliding-mode differentiator-like system. The proposed discretized system served as a robust observer for systems having the so-called Brunovskii form. The proposed differentiator is based on the implicit discretized method. The design method includes the calculus of the STA gains as well as the implicit estimation of the sequence of observer trajectories. The observer has the remarkable property of estimating the convergence region for the estimation error. This characteristic seems to be a step toward a consistent discretized version of the continuous-time variant of the step-by-step observer. The numerical simulation of the state estimation for the variables of the microalgae growth model.
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REFERENCES
Fig. 1. Time evolution of the microalgae biomass as well as its estimates by STA and high-gain implicit discretized observer-like algorithm. Substrate 110 100 90
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Fig. 2. Time evolution of the microalgae substrate as well as its estimates by STA and high-gain implicit discretized observer-like algorithm. of this variable also demonstrates the potential benefits of the proposed observer. Internal quota (QN) 9 8.5 Modeled Q 8
QN (µmol L−1)
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Fig. 3. Time evolution of the microalgae nitrogen quota as well as its estimates by STA and high-gain implicit discretized observer-like algorithm.
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