This paper presents a second order fixed time sliding mode observer based on an extension of the super-twisting algorithm. This observer can be applied to a class of nonlinear system with a block-wise representation. The block structure provides a straightforward form to the application of the proposed second order sliding mode algorithm, yielding to finite-time convergence with a settling time independent to the system initial conditions. Finally, as numerical simulation example, the case of a linear induction motor is studied, exposing the efficiency and feasibility of the proposal

A. Barzanti

C. Zanelli J

G. Baldi

M. Blosi

M. Dondi

S. Albonetti

BLOSI, MAGDA

ALBONETTI, STEFANIA

C. Zanelli J.

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

“Microwave-assisted synthesis of Pr-ZrSiO4, V-ZrSiO4 and CrYAlO3 ceramic pigments”

This paper presents a second order fixed time sliding mode observer based on an extension of the super-twisting algorithm. This observer can be applied to a class of nonlinear system with a block-wise representation. The block structure provides a straightforward form to the application of the proposed second order sliding mode algorithm, yielding to finite-time convergence with a settling time independent to the system initial conditions. Finally, as numerical simulation example, the case of a linear induction motor is studied, exposing the efficiency and feasibility of the proposal.Consejo Nacional de Ciencia y Tecnologí

Loukianov, Alexander

Sánchez-Torres, Juan D.

Repositorio Institucional del ITESO

Instituto Tecnológico y de Estudios Superiores de Occidente 2014-06-29 A fixed-time second order sliding modeobserver for a class of nonlinear systems Loukianov, Alexander; Sánchez-Torres, Juan D. A. Loukianov, J. D. Sanchez Torres, “A fixed-time second order sliding mode observer for a class ofnonlinear systems”, in 13th International Workshop on Variable Structure Systems (VSS); Nantes,France, 2014.  Enlace directo al documento: http://hdl.handle.net/11117/3283 Este documento obtenido del Repositorio Institucional del Instituto Tecnológico y de Estudios Superiores deOccidente se pone a disposición general bajo los términos y condiciones de la siguiente licencia:http://quijote.biblio.iteso.mx/licencias/CC-BY-NC-2.5-MX.pdf (El documento empieza en la siguiente página)Repositorio Institucional del ITESO rei.iteso.mxDepartamento de Matemáticas y Física DMAF - Artículos y ponencias con arbitrajeA Fixed-Time Second Order Sliding Mode Observer for a Class ofNonlinear SystemsJuan Diego Sa´nchez-Torres and Alexander G. LoukianovAbstract—This paper presents a second order fixed timesliding mode observer based on an extension of the super-twisting algorithm. This observer can be applied to a class ofnonlinear system with a block-wise representation. The blockstructure provides a straightforward form to the applicationof the proposed second order sliding mode algorithm, yieldingto finite-time convergence with a settling time independent tothe system initial conditions. Finally, as numerical simulationexample, the case of a linear induction motor is studied,exposing the efficiency and feasibility of the proposal.I. INTRODUCTIONThe sliding mode (SM) algorithms are applied withthe idea to drive the dynamics of a system to a slidingmanifold that is an integral manifold with finite reachingtime [1]. Generally, this approach exhibits very interestingand desirable features such as the work with reducedobservation error dynamics, the possibility to decomposethe design problem into two sub problems of the reducedorder, the robustness of the closed-loop system in presenceof parameter variations and external disturbances and, finite-time stability [2]–[4].Considering the observation error as a sliding variable, theSM algorithms can be considered as an effective solutionto the problem of observers design for nonlinear systems[5], specially when finite-time convergence of the observedstates to the real ones is required. An important class of SMobservers use the equivalent control method [6] to obtaininformation of the system by means of continuous equivalentvalues of the discontinuous observer inputs in SM motion[7]. With this idea, several designs have been proposed asthe cascade observers [8], step-by-step observers [9], a SMobserver where the estimation of unknown inputs problemhas been considered [10], fixed time designs [11], amongothers.Another class of SM observers are based on thesecond order SM feature of the super-twisting algorithm(STA) [12]. Those attractive characteristics of the STAalgorithm have been exploited and extended for fixed timeconvergent methods [13]–[15], adaptive controllers [16]–[21], multivariable structures [22], most of them based onthe stability studies presented in [23]–[29]. For the case ofobservers, a design for mechanical systems is presented in[30] being extended to electrical drives [31], more generalThis work was supported by the National Council of Science andTechnology (CONACYT), Mexico, under Grant 129591Juna Diego and Alexander G. Loukianov are with the Automatic ControlLab., CINVESTAV-IPN Gdl, Av. del Bosque 1145, CP 45019, Me´xico,{dsanchez, louk}@gdl.cinvestav.mxforms as in [32] and systems with noisy measurements [33],[34].All these methods present high performance. However,most of them are presented in scalar form. And, themultivariable structure introduced in [22] converges in finitetime but not fixed.Under that consideration, this paper is aimed to present aSM observer for a class of nonlinear systems based on a fixedtime STA with fixed time convergence. This design allowsthe problem to be solved without the individual selection ofeach stabilizing input, instead a multivariable function, basedon the unit control [2], [35], is used. On the other hand, thefixed time stability [13], [36] ensures the existence of a finitetime independent to the initial conditions in which the systemconverges. Thus, the proposed approach have very attractivefeatures as: fixed time convergence to the observed variablesand a fixed parameters number (six for this case), regardlessof the state dimension.The linear induction motor is considered as case study.The effectiveness the proposed observer is demonstrated bymeans of numerical simulation, showing a good performanceof this proposal.This paper is organized as follows: Section II introduces amultivariable fixed time stable STA . Section III describes theproposed observers. The simulations are presented in SectionIV. Finally, in Section V the conclusions are given.II. PRELIMINARY RESULTLet the vectors x1, x2 ∈ Rn. Now, consider the systemx˙1 = −k1x1‖x1‖1/2− k2x1 − k3x1 ‖x1‖1/2 + x2 +∆1x˙2 = −k4x1‖x1‖− k5x1 − k6x1 ‖x1‖1/2+∆2(1)where k1, . . . , k6 > 0, and the disturbances are regarded as‖∆1‖ ≤ δ1 ‖x1‖ and ‖∆2‖ ≤ δ2 with δ1, δ2 > 0.With the Lyapunov functionV = 2k3 ‖x1‖+ k4 ‖x1‖2+12‖x2‖2+ νT ν (2)where ν = k1x1‖x1‖1/2 + k2x1 + k3x1 ‖x1‖1/2− x2, it ispossible to show there exists constants γ1 = γ1(θ), γ2 =γ2(θ) > 0, θ = (k1, k2, k3, k3, k4, k5, k6, δ1, δ2), such thatV˙ ≤ −γ1V1/2 − γ2V3/2. (3)Therefore, from (2)− (3), the system (1) is globally fixedtime stable [36].III. FIXED-TIME SECOND ORDER SLIDING MODEOBSERVERA. Observer DesignConsider the system written in the following block-wiseform:x˙1 = B1(x1)x2 + f1(x1, u)x˙2 = B2(x1)x2 + f2(x1) (4)y = x1where x = [x1 x2]T and the vectors x1, x2 ∈ Rn. Thematrix B1(x1) is considered to be invertible.Based on the system (4), the following observer isproposed in order to provide a uniform finite estimation ofthe state x:˙ˆx1 = B1(x1)xˆ2 + f1(x1, u) + φ1(x˜1)˙ˆx2 = B2(x1)xˆ2 + f2(x1) +B−11 (x1)φ2(x˜1) (5)where xˆ1 and xˆ2 are the estimates of x1 and x2, respectivelyand, the observer errors are given by x˜1 = xˆ1 − x1 andx˜2 = xˆ2 − x2. The observer inputs φ1(x˜1), and φ2(x˜1) aredefined asφ1(x˜1) =k1x˜1‖x˜1‖1/2+ k2x˜1 + k3x˜1 ‖x˜1‖1/2φ2(x˜1) =k4x˜1‖x˜1‖+ k5x˜1 + k6x˜1 ‖x˜1‖1/2(6)where k1, . . . , k6 > 0.B. Convergence AnalysisTo analyze the observer convergence, consider thedynamics of the errors x˜1 and x˜2. From (4) and (5) it follows˙˜x1 = B1(x1)x˜2 − φ1(x˜1)˙˜x2 = B2(x1)x˜2 −B−11 (x1)φ2(x˜1). (7)Defining q = B1(x1)x˜2, the system (7) is transformed to˙˜x1 = φ1(x˜1) + qq˙ = φ2(x˜1) +B(x1)q (8)where B(x1) =[B˙1(x1) +B1(x1)B2(x1)]B−11 (x1).Considering ‖B(x1)q‖ < δ where δ > 0, with a suitablechoice of the gains k1, . . . , k6, it follows from (2)− (3) thatthe system (8) is globally fixed time stable. Therefore, theobserver variables converges to the real ones in fixed time.IV. NUMERICAL SIMULATION RESULTSThis section shows numerical simulations results of theproposed observer for a linear induction motor. The measuredvariables are the velocity and the currents. The observedvariables are the flux and the load torque, introduced as stepform. This is motivated, due the difficulty of the flux andtorque direct measurement [37].The model for the induction can be described by equationsfor the stator current and rotor fluxes in stationary referenceframe αβ as follows:dΘdt= d1 (λαriβs − λβriαs)− d2Γ− d3Θdλαrdt= −η1λαr + η2Θλβr + η3iαsdλβrdt= −η1λβr − η2Θλαr + η3iβs (9)diαsdt= −η4iαs + η5λαr − η6Θλβr + η7vαsdiβsdt= −η8iβs + η9λβr + η10Θλαr + η11vβswhere λαr and λβr are the rotor magnetic-flux-linkagecomponents, respectively; iαs and iβs are the stator currentcomponents, respectively, vαs and vβs are the voltage of αand β axes in the stator, respectively.For the three-phase linear induction motor in αβ frame,the voltages are presented of the formvαs = vs sin(ωt) (10)vβs = −vs sin(ωt). (11)Thus, for this case, the parameters are: η1 =RrLr,η2 = np(piτ), η3 =RrLmLr, η4 =Rs(Ls2Lr−LsLm2LsLr) +1−(LsLr−Lm2LsLr)(LsLr−Lm2LsLr)RrLr, η5 =LmRr(LsLr−Lm2LsLr)LsLr2, η6 =np(piτ)Lm(LsLr−Lm2LsLr)LsLr, η7 =1(LsLr−Lm2LsLr)Ls, η8 = η4,η9 = η5, η10 = η6, η11 = η7, d1 =3nppiLm2LrτM, d2 =1M ,d3 =DM where Rs and Ls are the resistance and inductanceof the stator, respectively. τ is the pole pitch, M is the totalmass of the moving element, D is viscous friction, Θ = υis the linear velocity and Γ = FL is the external force.For the observer design, the availability of continuousmeasurements of motor speed and currents is assumed.In addition the mechanic load Γ is considered asan unknown and slowly-varying perturbation to beestimated, that is Γ˙ = 0. Thus, the system (9)can be and, the blocks are x1 = [Θ iαs iβs]Tand, x2 = [λαr λβr Γ]T , with u = [vαs vβs]T .Here B1(x1) =d1iβs d1iαs −d2η5 −η6Θ 0η10Θ η9 0, f1(x1, u) =−d3Θη7vαs − η4iαsη10Θ, B2(x1) =−η1 η2Θ 0−η2Θ −η1 00 0 0 and,f2(x1) =η3iαsη3iβs0.B−11 (x1) is the inverse of the matrixB1(x1) and is given by B−11 (x1) =0 η9η6η10Θ2+η5η9η6Θη6η10Θ2+η5η90 − η10Θη6η10Θ2+η5η9η5η6η10Θ2+η5η9− 1d2d1(η9iβs+η10iαsΘ)d2(η6η10Θ2+η5η9)−d1(η5iαs−η6iβsΘ)d2(η6η10Θ2+η5η9) .For three-phase linear induction motor the parameter arepresented as [38]:Three-phase linearH.P. 4 Vs 180 (V )f 60 (Hz) np 2Rs 5.3685 (Ω) Rr 3.5315 (Ω)Ls 0.02846(H) Lr 0.02846 (H)Lm 0.02419 (H) M 2.78 (kg)D 36.0455 (Kg/s) τ 0.027 (m)Imax 14.2 (A)µ1 1 µ2 1m11 640 m12 640m13 45 m21 64000m22 64000 m23 20The observer gains are chosen as k1 = 5, k2 = 10, k3 = 2,k4 = 10, k5 = 5 and k6 = 1. The simulation results areshown in the following figures:0 1 2 3 4 5−1−0.500.510 0.05−1−0.500.510 1 2 3 4 5−1−0.500.510 0.05−1−0.500.51λ˜αrλ˜βrTime [Sec] Time [Sec]Fig. 1. Error of rotor flux λ˜αr and λ˜βr of TLIM.0 1 2 3 4 5−2−1012345  0 0.05 0.1−2−1012345  0 1 2 3 4 5−2−1.5−1−0.500.511.520 0.05 0.1−2−1.5−1−0.500.511.52TLTLestTLandTˆLT˜LTime [Sec] Time [Sec]Fig. 2. Load torque estimated TˆL and error of load torque T˜L of TLIM.In Figure 1 the time evolution of the rotor flux λ˜αr and λ˜βrerrors of induction motors are shown, while Fig. 2 presentsthe time evolutions of the estimated load torque TˆL and theload estimation error T˜L of induction motors cases.100 101 102 103 104 105024681012141618  GSTAAsymptoteSTConvergenceTime[s]Initial Condition (Logarithmic Scale)Fig. 3. Convergence time of both observers by growing initial conditionnorm.The Figure 3 presents a comparison of the proposedobserver with another which uses the mutivariable STA[22], that means fixing k3 = 0 and k6 = 0 in (6).Here is highlighted that the convergence time for themultivariable STA grows unboundedly with the norm of theinitial condition, while the convergence time of the proposedobserver is asymptotically bounded by a constant for growinginitial condition’s norm.V. CONCLUSIONSIn this work a fixed time convergent observer wasproposed. The scheme was applied to the model on thestationary frame αβ for induction motors. The flux andload torque were estimated, all of them are shown to giveappreciable results in order of convergence time to estimatethe rotor flux and the load torque.REFERENCES[1] S. V. Drakunov and V. I. Utkin, “Sliding mode control in dynamicsystems,” International Journal of Control, vol. 55, pp. 1029–1037,1992.[2] V. I. 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A fixed-time second order sliding mode observer for a class of nonlinear systems

An innovative two-steps route was successfuly applied to the microwave-assisted synthesis of Pr-ZrSiO4, V-ZrSiO4 and Cr-YAlO3 ceramic pigments. It is able to reduce the total synthesis time from many hours down to a few minutes. Industrial batches were pelletized and underwent a short time pre-heating in conventional electric furnace (from 300 up to 1000?C) prior dielectric heating in microwave oven (2.45 GHz, 800 W continuous or paused). Pre-heating is necessary to reach a thermal level at which pigment precursors are activated under microwave irradiation (high enough tan delta of dielectric properties). Activation kinetics was followed by measuring (optical pyrometer) the external temperature of crucible during MW irradiation and the internal temperature at the end of heating process. Pigments were characterized by colourimetry (CIE Lab) and XRPD with Rietveld refinement. The V-ZrSiO4 system is easily activated, already at pre-heating of 300?C, with good reaction yield and colour performance; nevertheless it is difficult to keep under control, as temperatures exceeding 1200?C are rapidly reached. The Pr-ZrSiO4 system is activated after pre-heating over 600?C and exhibits a slow kinetics, resulting in a low reaction yield and unsatisfactory colour. The Cr-YAlO3 pigment is formed with colour and reaction yield close to those of the correspondent industrial product, even though after pre-heating at 800?C or more. A peculiar behaviour of silica phases was observed in zircon systems, particularly after lower pre-heating temperatures. It consists in an early transformation of quartz (present as precursor) into cristobalite and abundant amorphous phase, before the SiO2 + ZrO2 -> ZrSiO4 transformation occurred

Blosi, Magda

Dondi, Michele

Albonetti, Stefania

Baldi, Giovanni

Barzanti, Andrea

Zanelli, Chiara

PUblication MAnagement

Microwave-assisted synthesis of Pr-ZrSiO4, V-ZrSiO4 and Cr-YAlO3 ceramic pigments

Juan Diego Sanchez-Torres

Alexander G. Loukianov

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A fixed-time second order sliding mode observer for a class of nonlinear systems

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