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IFAC-PapersOnLine 48-11 (2015) 319–326

Adaptive Zooming Strategy in Adaptive Zooming Strategy Adaptive Zooming Strategy in inof Discrete-time Implementation Discrete-time Implementation of Discrete-time Implementation of Sliding-mode Control Sliding-mode Control Sliding-mode Control

Andrievsky ∗,∗∗ Alexey Andrievsky ∗ Iuliia Zaitceva ∗∗∗ ∗,∗∗ ∗,∗∗ Alexey Andrievsky ∗ ∗ Iuliia Zaitceva ∗∗∗ ∗∗∗ Andrievsky Andrievsky Andrievsky ∗,∗∗ Alexey Alexey Andrievsky Andrievsky ∗ Iuliia Iuliia Zaitceva Zaitceva ∗∗∗ ∗ Institute for Problems of Mechanical Engineering, ∗ ∗ Institute for Problems of Mechanical Engineering, ∗ Institute forRussian Problems of Mechanical Mechanical Engineering, Academy of Sciences, Institutethe for Problems of Engineering, the Russian Sciences, the Russian Academy of Sciences, 61 Bolshoy prospekt, V.O.,Academy 199178, of Saint Petersburg, Russia, the Russian Academy of Sciences, 61 Bolshoy prospekt, V.O., 199178, Saint Petersburg, Russia, 61 Bolshoy Saint [email protected] [email protected], Bolshoy prospekt, prospekt, V.O., V.O., 199178, 199178, Saint Petersburg, Petersburg, Russia, Russia, [email protected] ∗∗ [email protected], [email protected], [email protected] Saint-Petersburg State University, 28 Universitetsky prospekt, [email protected], [email protected] ∗∗ ∗∗ Saint-Petersburg State University, 28 Universitetsky prospekt, ∗∗ Saint-Petersburg State 28 198504, Peterhof, Saint Petersburg, Russia prospekt, Saint-Petersburg State University, University, 28 Universitetsky Universitetsky prospekt, 198504, Peterhof, Saint Petersburg, Petersburg, Russia ∗∗∗ 198504, Peterhof, Saint Russia ITMO University,Saint Petersburg, Russia 198504, Peterhof, Saint Petersburg, Russia ∗∗∗ ∗∗∗ ITMO University,Saint Petersburg, Russia ∗∗∗ ITMO University,Saint Petersburg, [email protected] ITMO University,Saint Petersburg, Russia Russia [email protected] [email protected] [email protected] Abstract: The adaptive zooming strategy borrowed from the coding-decoding procedure of Abstract: The strategy borrowed the procedure Abstract: The adaptive adaptive zooming strategy communication borrowed from from channel the coding-decoding coding-decoding procedure of of data transmission over thezooming limited capacity is applied for discrete-time Abstract: The adaptive zooming strategy borrowed from the coding-decoding procedure of data transmission over the limited capacity communication channel is applied for discrete-time data transmission over the limited capacity communication channel is applied for discrete-time sliding mode control design. The capacity proposedcommunication control strategy is numerically studied both for data transmission over the limited channel is applied for discrete-time sliding mode control design. The proposed strategy is numerically studied both for sliding mode control design. The control strategy is studied both illustrative example for control of 3-DOF control laboratory Helicopter benchmark. The simulation sliding mode controland design. The proposed proposed control strategy is numerically numerically studied both for for illustrative example and for control of 3-DOF laboratory Helicopter benchmark. The simulation illustrative example and for control of 3-DOF laboratory The simulation results confirm low-chattering sliding mode behavior andHelicopter efficiency benchmark. of the proposed method. illustrative example and for control of 3-DOF laboratory Helicopter benchmark. The simulation results confirm confirm low-chattering low-chattering sliding sliding mode mode behavior behavior and and efficiency efficiency of of the the proposed proposed method. method. results results low-chattering sliding behavior andHosting efficiency of the Ltd. proposed method. © 2015, confirm IFAC (International Federation of mode Automatic Control) by Elsevier All rights reserved. Keywords: sliding mode, discrete-time, adaptive zooming, chattering, helicopter benchmark Keywords: sliding mode, discrete-time, adaptive zooming, chattering, helicopter benchmark Keywords: Keywords: sliding sliding mode, mode, discrete-time, discrete-time, adaptive adaptive zooming, zooming, chattering, chattering, helicopter helicopter benchmark benchmark 1. INTRODUCTION quantization of the control signal along with its time1. INTRODUCTION INTRODUCTION quantization of the control signal with its time1. quantization of the signal along with its timediscretization et al., 2006,along 2009). The 1. INTRODUCTION quantization of(Fradkov the control control signal along with itswidely timediscretization (Fradkov et al., 2006, 2009). The widely discretization (Fradkov et al., 2006, 2009). The widely approach to ensuring asymptotic stability in the Sliding-mode control is commonly known as an important used discretization (Fradkov et al., 2006, 2009). The widely used approach to ensuring ensuring asymptotic stability in lies the Sliding-mode control is commonly commonly knownhas as an an important to stability in the Sliding-mode control is known as important case ofapproach level quantization andasymptotic discretization on time field of feedback control. This method many appli- used used approach to ensuring asymptotic stability in the Sliding-mode control is commonly known as an important case of level quantization and discretization on time lies field of feedback control. This method has many applicase of level quantization and discretization on time lies field of feedback control. This method has many appliin application of the zooming procedure (see (Brockett cations since a sliding-mode controller ishas robust to appliplant case of level quantization and discretization on time lies field of feedback control. This method many application of theLiberzon, zooming procedure (see cations since sinceuncertainty sliding-mode controller is robust robust to plant in application of zooming procedure (see (Brockett cations aa controller is plant and Liberzon, 2000; Cheng and(Brockett Savkin, parameters and external disturbances, when it in in application of the theLiberzon, zooming 2003; procedure (see (Brockett cations sinceuncertainty a sliding-mode sliding-mode controller is robust to to plant and Liberzon, 2000; 2003; Cheng and Savkin, parameters and external disturbances, when it and Liberzon, 2000; Liberzon, 2003; Cheng and parameters uncertainty and external disturbances, when it 2007; Malyavej and Savkin, 2005; Malyavej et al., 2006; is implemented in the continuous-time domain. However, Liberzon, 2000; Liberzon, 2003; Cheng et andal.,Savkin, Savkin, parameters uncertainty and external disturbances, when it and 2007; Malyavej and Savkin, 2005; Malyavej 2006; is implemented in the continuous-time domain. However, 2007; Malyavej and Savkin, 2005; Malyavej et al., 2006; is implemented in the domain. However, FurtatMalyavej et al., 2014; Bondarko, 2014;Malyavej Fradkov et al., 2015) when a continuous timecontinuous-time sliding controller is implemented 2007; and Savkin, 2005; 2006; is implemented in the continuous-time domain. However, Furtat et al., 2014; Bondarko, 2014; Fradkov et al., 2015) when a continuous time sliding controller is implemented Furtat et al., 2014; Bondarko, 2014; Fradkov et al., 2015) when a continuous time sliding controller is implemented for further references). In the present paper the idea of in discrete time, the controller performance can degrade Furtat et al., 2014; Bondarko, 2014; Fradkov et al., 2015) when a continuous time sliding controller is implemented for further references). In the present paper the idea of in discrete discrete time, time the controller controller performance can degrade for further references). In the present paper the idea of in time, the performance can degrade the adaptive zooming strategy, borrowed from the field due to the finite sampling process of the digital defor further references). In the present paper thethe idea of in discrete time, time the controller performance can degrade the adaptive zooming strategy, borrowed from field due to the finite sampling process of the digital dethe adaptive zooming strategy, borrowed from the field due to the finite time sampling process of the digital deof estimation and control over the digital communication vices (Fradkov and Furuta, 1996; Tang and Misawa, 2000; the adaptive zooming strategy, borrowed from the field due to the finite time sampling process of the digital deof estimation and control control over the digital digital communication vices (Fradkov (Fradkov and Furuta, Furuta, 1996; Tang and and Misawa, Misawa, 2000; estimation and the vices and 1996; Tang 2000; channels, is employed to over the discrete-time sliding mode Bandal and N.Vernekar, 2010; Podivilova al., 2014). It of of estimation and control over the digital communication communication vices (Fradkov and Furuta, 1996; Tang andet 2000; is employed to the discrete-time sliding mode Bandal andnoted N.Vernekar, 2010; Podivilova etMisawa, al., 2014). 2014). It channels, channels, is employed to the discrete-time sliding Bandal and N.Vernekar, 2010; Podivilova et al., It control problem. should be that discrete-time sliding mode control channels, is employed to the discrete-time sliding mode mode Bandal and N.Vernekar, 2010; Podivilova et al., 2014). It control problem. should be noted that discrete-time sliding mode control problem. should be that discrete-time sliding mode control cannot be noted obtained from its continuous time model by control control problem. should be noted that discrete-time sliding mode control cannotofbe besimple obtained from its its continuous timeand model by The rest of the paper is organized as follows. Adaptive cannot obtained from continuous time model by means equivalence (Gao et al., 1995), in some rest of the paper is organized follows. cannot besimple obtained from its continuous timeand model by The The rest of is as follows. Adaptive zooming is briefly recalled inas 2. AnAdaptive illustrameans of equivalence (Gao et al., 1995), in some The rest strategy of the the paper paper is organized organized asSec. follows. Adaptive means of simple equivalence (Gao et al., 1995), and in some cases, the discrete-time model obtained introduces zooming strategy is briefly recalled in Sec. 2. An illustrameans of simple equivalence (Gao et al., 1995), and in some zooming strategy is briefly recalled in Sec. 2. An tive example of sliding-mode discrete-time stabilization of cases, the discrete-time model obtained introduces some zooming strategy is briefly recalled in Sec. stabilization 2. An illustraillustracases, discrete-time model obtained introduces some kind ofthe bounded perturbation. The problem of eliminating tive example of sliding-mode discrete-time of cases, the discrete-time model obtained introduces some tive third example of sliding-mode sliding-mode discrete-time stabilization of the order LTI plant is given in Sec. 3. Application kind of bounded perturbation. The problem of eliminating tive example of discrete-time stabilization of kind of bounded perturbation. The problem of eliminating the chattering effect in discrete-time implementation of the third order LTI plant is given in Sec. 3. Application kind of bounded perturbation. The problem of eliminating the third order LTI plant is given in Sec. 3. Application of the proposed method to control of 3DOF Helicopter the chattering effect in discrete-time implementation of the third order LTI plant is given in Sec. 3. Application the chattering effect discrete-time of sliding-mode addressed inimplementation many researches, proposed method to incontrol 3DOF Helicopter the chatteringcontrollers effect in in is discrete-time of of of the the method of 3DOF benchmark is demonstrated Sec. 4. of Concluding remarks sliding-mode controllers is addressed inimplementation manyand researches, the proposed proposed method to to incontrol control of 3DOF Helicopter Helicopter sliding-mode is addressed in many researches, see e.g. (Xiaocontrollers et al., 2005; Bandyopadhyay Thakar, of benchmark is demonstrated Sec. 4. Concluding remarks sliding-mode controllers is addressed in many researches, benchmark is demonstrated in Sec. 4. Concluding are given in Sec. 5. see e.g. (Xiao et al., 2005; Bandyopadhyay and Thakar, isSec. demonstrated in Sec. 4. Concluding remarks remarks see e.g. (Xiao al., 2005; Bandyopadhyay and Thakar, 2008; Huber etet al., 2013). For example, a promising ap- benchmark are given in 5. see e.g. (Xiao et al., 2005; Bandyopadhyay and Thakar, are given given in in Sec. Sec. 5. 5. 2008; Huber Huber et al., al., the 2013). For example, example, a promising promising ap- are 2008; et 2013). For a approach for avoiding chattering effects due to the time2008; et al., the 2013). For example, a promising approachHuber for avoiding avoiding chattering effects due to the time2. ADAPTIVE ZOOMING STRATEGY proach for the chattering effects due to timediscretization in sliding mode control systems is the proposed proach for avoiding the chattering effects due to the time2. ADAPTIVE ZOOMING STRATEGY 2. discretization in sliding mode control systems is proposed 2. ADAPTIVE ADAPTIVE ZOOMING ZOOMING STRATEGY STRATEGY discretization in sliding mode control systems is proposed by Acary et al.in(2012). is basedison an imdiscretization sliding This modeapproach control systems proposed by Acary et al. (2012). This approach is based on an imConsider the scalar binary quantizers to be a discretized by Acary et al. (2012). This approach is based on an implicit Euler method and the zero-order-hold discretization. by Acary al. (2012). approach is based on an im- Consider scalar quantizers to be aa discretized plicit Euleretmethod method and This the zero-order-hold zero-order-hold discretization. Consider the scalar binary map q : Rthe →R as binary plicit Euler and the discretization. Consider the scalar binary quantizers quantizers to to be be a discretized discretized plicit Euler method and the zero-order-hold discretization. map q : R → R as It should be noted that many similarities may be found map q : R → R as map q : R → R as q(σ) = M sign(σ), (1) It should be noted that many similarities may be found It should be that may found in discrete-time implementation of sliding-mode control, q(σ) = =M M sign(σ), sign(σ), (1) It should be noted noted that many many similarities similarities may be becontrol, found q(σ) (1) in discrete-time implementation of sliding-mode q(σ) = M sign(σ), (1) depending on M as a parameter. M may be referred to as a in discrete-time implementation of sliding-mode sliding-mode control, from one side, and quantized control and estimation over depending on M as a parameter. M may be referred to as in discrete-time implementation of control, a from one side, and quantized control and estimation over depending on M as a parameter. M may be referred to as quantizer range. In (1), σ is a quantizer input signal, sign(·) from one side, and quantized control and estimation dependingrange. on M In as (1), a parameter. M mayinput be referred to as a a the digital communication channel, from the other over one. quantizer from one side, and quantized control and estimation over σ is a quantizer signal, sign(·) theboth digital communication channel, from the the other other one. quantizer range. In (1), σ is aa quantizer signal, sign(·) is the signum function: sign(z) = 1, if yinput ≥ 0, sign(z) = −1, the digital communication channel, from one. quantizer range. In (1), σ is quantizer input signal, sign(·) In cases the control signal is discontinuous on level the digital communication channel, from the other one. function: sign(z) = 1, if y ≥ 0, sign(z) = −1, In both both cases the control control signal is discontinuous discontinuous on level level is is the signum function: if zthe < signum 0. In the signal is on signum function: sign(z) sign(z) = = 1, 1, if if yy ≥ ≥ 0, 0, sign(z) sign(z) = = −1, −1, and time.cases For example, some authors consider the synchroIn both cases the control signal is discontinuous on level is if zzthe < 0. and time. For example, some authors consider the synchroif < 0. and time. For example, some authors consider the synchroif z < 0. nization problem in the case of binary (signum function) and time.problem For example, consider thefunction) synchro- In time-varying quantizers (see, e.g. (Brockett and Libernization in the thesome caseauthors of binary binary (signum nization problem in of (signum function) In time-varying quantizers e.g. (Brockett and Liber In time-varying quantizers (see, e.g. and nization problem in theincase case ofRAS, binary (signum function) zon, 2000; Liberzon, 2003;(see, Tatikonda and Mitter, 2004; The work was performed IPME supported by RSF (grant In time-varying quantizers (see, e.g. (Brockett (Brockett and LiberLiber zon, 2000; Liberzon, 2003; Tatikonda and Mitter, 2004; The work was performed in IPME RAS, supported by RSF (grant zon, 2000; Liberzon, 2003; Tatikonda and Mitter, 2004; Fradkov et al., 2006; Nair et al., 2007)) the range M is 14-29-00142). The work was performed in IPME RAS, supported by RSF (grant The work was performed in IPME RAS, supported by RSF (grant zon, 2000; Liberzon, 2003; Tatikonda and Mitter, 2004; Fradkov et al., 2006; Nair et al., 2007)) the range M is 14-29-00142). Fradkov et al., 2006; Nair et al., 2007)) the range 14-29-00142). Fradkov et al., 2006; Nair et al., 2007)) the range M M is is 14-29-00142). Boris Boris Boris Boris

Copyright IFAC 2015 323 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 323 Copyright © IFAC 2015 323 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 323Control. 10.1016/j.ifacol.2015.09.205

MICNON 2015 320 Boris Andrievsky et al. / IFAC-PapersOnLine 48-11 (2015) 319–326 June 24-26, 2015. Saint Petersburg, Russia

updated with time and different values of M are used at each step, M = M [k]. Using such a zooming strategy makes possible to increase the coder accuracy in the steady-state mode and, at the same time, to prevent coder saturation at the beginning of the process. The values of M [k] may be precomputed (the time-based zooming), or alternatively, current quantized measurements may be used at each step for updating M [k] (the event-based zooming). The adaptive zooming strategy realizes the event-based zooming, where range M is updated depending on the current data flow (see (Goodman and Gersho, 1974; Andrievsky, 2007; Gomez-Estern et al., 2007; Fradkov et al., 2010) for further references). In the present work we use the following adaptive zooming procedure of (Andrievsky, 2007; Fradkov et al., 2010): η[k] = (¯ σ [k] + σ ¯ [k− 1] + σ ¯ [k − 2])/3, ρM [k], as |η[k]| ≤ 0.5, (2) M [k + 1] = m + M [k]/ρ, otherwise, η[0] = 0, M [0] = M0 , where σ ¯ [k] = q(σ, M ), m = (1 − ρ)M∞ , M∞ denotes the minimal bound value of M [k], M0 stands for the initial value of M [k].

Fig. 1. Time history of plant output y(t). y(0) = 5 · 10−3 .

3. ILLUSTRATIVE EXAMPLE 3.1 System description Consider the plant model, given by the following transfer function from control signal u to plant output y: s2 + 14s + 100 W (s) = , (3) s(s2 + 2s + 1) where s ∈ C is the Laplace transform variable. Since the plant relative degree n − m = 1 (where n, m are the orders of W (s) denominator and numerator, respectively) and the numerator is the Hurwitz polynomial with positive coefficients, the plant satisfies the strictly-minimumphase condition, see, e.g. (Andrievskii and Fradkov, 2006), and the sign control law may be applied ensuring the slidingmode of the closed-loop system in some finite vicinity of the origin. Let us use the following discrete-time stabilization algorithm: u(tk ) = M (tk ) sign y(tk ), (4) u(t) = u(tk ) as tk−1 ≤ t < tk , tk = kT0 , where y(t) denotes the plant output, T0 stands for the sampling period (the discretization interval), and range M (tk ) is governed by (2). 3.2 Simulation results The simulation results for T0 = 0.05 s, M0 = 1, m = 0, ρ = e−5T0 = 0.7788 and two initial values of y(0) (¨ y (0) = y(0) ˙ = 0) are plotted in Figs. 1–4. Figs. 1, 2 demonstrate time histories of y(t), u(t), M (tk ) for the case of y(0) = 5 · 10−3 . Time histories of the same variables for the case of y(0) = 5·10−2 are shown in Figs. 3, 4. It is seen from the plots that range M (tk ) adaptively adjusted depending on the current system output ensuring system stabilization and suppression the chattering effect.

324

Fig. 2. Time histories of u(t), M (tk ). y(0) = 5 · 10−3 .

Fig. 3. Time history of plant output y(t). y(0) = 5 · 10−2 . 4. CONTROL OF 3DOF HELICOPTER BENCHMARK

4.1 Benchmark description The 3DOF Helicopter setup is manufactured by Quanser Consulting Inc., (Apkarian, 1999). It is widely used as a benchmark for various control design methods and for educational aims as well (see (Kiefer et al., 2005; Peaucelle et al., 2007; Kiefer et al., 2010; R´ıos et al., 2010; Hernandez-Gonzalez et al., 2012; Rajappa et al., 2013; Ruf, 2014) to mention a few). The setup consists of a base on which a long arm is mounted, see Fig. 5. The arm carries the Helicopter body on one end and a counterweight on the other end. The arm can tilt on an elevation axis as well as swivel on a vertical (travel) axis. The Helicopter body, which is mounted at the end of the arm, is free to pitch

MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Boris Andrievsky et al. / IFAC-PapersOnLine 48-11 (2015) 319–326

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Fig. 6. Schematic diagram of the Helicopter. 4.3 Modeling the Helicopter benchmark 1. Non-linear uncertain model of the Helicopter benchmark

Fig. 4. Time histories of u(t), M (tk ). y(0) = 5 · 10−2 . about the pitch axis. Two motors with propellers mounted on the Helicopter body can generate a force proportional to the voltage applied to them. The force, generated by the propellers, causes the Helicopter body to lift off the ground and/or to rotate about the pitch axis.

Based on the results of (Kiefer et al., 2005; Peaucelle et al., 2007; Kiefer et al., 2010; Fradkov et al., 2010; Peaucelle et al., 2011), let us use the following model of Helicopter dynamics:  ϑ x ˙ ϑ¨ = −aω mx ϑ − amx cos ϑ0 sin ϑ    ϑ  +amx sin ϑ0 cos ϑ + aumx cos ε · u(t),  z ˙ aεmz cos ε0 sin ε ε¨ = −aω (5) mz ε−  ε w   +amz sin ε0 cos ε + amz cos ϑ · w(t),   λ ¨ = −aωy λ˙ + aw sin ϑ · w(t). my my

where aij are unknown plant model parameters. The constants ϑ0 and ε0 denote pitch and elevation balance angles. Based on the preliminary identification experiments (Peaucelle et al., 2007), the following bounds of the model (5) parameters may be designated (in SI units): x aϑmx ∈ [0.1, 0.7], aumx ∈ [0.1, 0.4], aω mx ∈ [0.02, 0.15], cos ϑ0 ∈ [0.98, 1], sin ϑ0 ∈ [−0.2, 0.2], cos ε ∈ [0.94,1]; ωz amz ∈ [0.02, 0.15], aεmz ∈ [2, 5], aw mz ∈ [0.1, 0.3], cos ε0 ∈ [0.97, 1], sin ε0 ∈ [−0.26, 0], cos ϑ ∈ [0.5, 1]; 1 w y aω my ∈ [0.01, 0.05], amy ∈ [0.04, 0.1], sin ϑ ∈ [−0.86, 0.86].

Fig. 5. Photo of the “LAAS Helicopter Benchmark”.

The pitch angle is mechanically restricted as |ϑ| ≤ 90 deg. In what follows we assume that ϑ(t) is kept within the working area |ϑ| ≤ 60 deg.

4.2 Nomenclature

2. Time-varying linear uncertain model.

Following notation is used through the paper (see Fig. 6): XY Z denotes the body-fixed reference frame of Helicopter; ϑ(t) denotes the pitch angle; ε(t) is the elevation angle; λ(t) is the travel angle; ϑ∗ (t), ε∗ (t), λ∗ (t) are the → reference signals for pitch, elevation, and travel angles; − ω stands for the angular velocity vector of the Helicopter → body; ωx , ωy , ωz are projections − ω to body-fixed axes X, Y , Z; vf (t), vr (t) are the control voltages, applied to the “front” and the “rear” motor (respectively); u(t) denotes the pitch torque control signal; w(t) is the normal force command signal (used for elevation/travel control). The control voltages vf (t) and vr (t) are calculated as vf = 0.5(w + u), vr = 0.5(w − u). 325

Following (Peaucelle et al., 2011), let us find an approximation of model (5) in the form of time-varying linear system with the uncertain parameters and external disturbances. Taking into account the aforementioned bounds on the parameters of (5), one may rewrite (5) as the following linear time-varying model with uncertain parameters:  ω ϑ u  ϑ¨ = −α x ϑ˙ − α ϑ + fϑ (t) + α (t)u(t), mx

mx

mx

ωz ε w ε¨ = −αm ε− ˙ αm ε + fε (t) + αm (t)w(t), z z z  λ ¨ = −αωy λ˙ + αw (t)ϑ(t), my my

(6)

with the following bounds of parameters and unknown disturbances:


ωx ϑ ϑ x αm = aω mx ∈ [0.02, 0.15], αmx = amx ∈ [0.1, 0.7], x

fϑ (t) = aϑmx sin ϑ0 cos ϑ(t), |fϑ (t)| ≤ 0.14, u αm (t) = aumx cos ε(t) ∈ [0.094, 0.4]; x ωz ε ε z αm = aω mz ∈ [0.02, 0.15], αmz = amz ∈ [2, 5], z ε fε (t) = amz sin ε0 cos ε(t) ∈ [−1.3, 0], w αm (t) = aw mz cos ϑ(t) ∈ [0.05, 0.3]; z ωy w w y αm y = a ω ∈ my [0.01, 0.05], αmy (t) = amy w(t) ∈

ϑ w ωx αm ∈ [0.02, 0.15], αm ∈ [0.1, 0.7], ∈ [0.05, 0.3], αm x x z ωy u w αmx ∈ [0.094, 0.4], αmy ∈ [0.01, 0.05], αmy ∈ [0.02, 1] are the plant model parameters (unknown constants).

Equations (9), (10) can be considered as the appropriate approximation of the Helicopter model in the steady-state mode, i.e. for the case of the Helicopter motion with the constant travel rate and constant pitch and elevation angles.

[0.02, 1].

3. Helicopter model decomposition.

4.4 Control of the isolated pitch motion.

Clearly, the simplified system (6) may be represented in the form of two independent linear time-varying subsystems with exogenic disturbances: the elevation and the pitch/travel dynamics models. Elevation dynamics model. It follows from (6) that the elevation dynamics may be described as: ωz ε w ε¨(t) = −αm ε(t)− ˙ αm ε(t) + αm (t)w(t) + fε (t), (7) z z z where w(t) is the elevation control signal, fε (t) ∈ [−1.3, 0] ωz stands for the elevation disturbance, αm ∈ [0.02, 0.15], z ε w αmz ∈ [1.94, 5], αmz (t) ∈ [0.05, 0.3] are the elevation model w parameters. The gain αm (t) depends on the current value z of the pitch angle ϑ(t), which makes model (7) essentially time-dependent. Pitch/travel dynamics model Correspondingly, (6) lead to the following model of pitch/travel dynamics: ϑ u ¨ = −αωx ϑ(t)− ˙ ϑ(t) ϑ(t)+αm (t)u(t) +fϑ (t), αm mx x x (8) w ωy ˙ ¨ λ(t) = −α λ(t)+α (t)ϑ(t), my

my

where u(t) may be referred to as pitch control signal, fϑ (t) denotes the bounded pitch disturbance (|fϑ (t)| ≤ 0.14), ωx ϑ u αm ∈ [0.02, 0.15], αm ∈ [0.1, 0.7], αm (t) ∈ [0.094, 0.4], x x x ωy w αmy ∈ [0.01, 0.05], αmy (t) ∈ [0.02, 1] are the model paw rameters. The gain αm (t) is changing with a high rate y following the control signal w(t). It is worth mentioning that the coupling between elevation and pitch/travel motions exists due to dependence of the u w parameters αm , αm on the angles ε and ϑ (respectively), x z u w but in the present Section, the parameters αm and αm x z are considered as independent variables on time. Note, that as it follows from (Peaucelle et al., 2007), more accurate model should also take into account an influence of the travel rate λ˙ on the elevation motion due to centrifugal force. In the present study it is assumed that the travel rate is low enough for omitting the centrifugal effect. 4. Steady-state LTI model. In the present Section, time-varying properties of systems (7) and (8) are discussed and the following LTI plant models are introduced instead of (7), (8) (respectively): ωz ε w ε¨(t) = −αm ε(t)− ˙ αm ε(t) + αm w(t) + fε (t), (9) z z z ϑ u ˙ ¨ = −αωx ϑ(t)− ϑ(t)+αm u(t) +fϑ (t), αm ϑ(t) mx x x (10) ω w ˙ ¨ = −α y λ(t)+α ϑ(t), λ(t) my my where w(t) is the elevation control signal, u(t) is the pitch control signal; fε (t) ∈ [−1.3, 0] and fϑ (t) are the disturωz ε bances (|fϑ (t)| ≤ 0.14); αm ∈ [0.02, 0.15], αm ∈ [1.94, 5], z z

326

At first consider control of the isolated pitch motion of the Helicopter, assuming that the elevation angle ε in (5) is constant, ε ≡ ε0 . To simplify the exposition let us also assume that pitch balance angle ϑ0 in (5) is zero (its actual value is about 5 deg). These assumptions lead to the following model of the pitch dynamics: ϑ u x ˙ ϑ¨ + aω mx ϑ + amx sin ϑ = amx cos ε0 · u(t).

(11)

To produce the torque control signal u(t) let us apply the following sliding-mode control with the adaptive zooming: eϑ (t) = ϑ∗ (t) − ϑ(t), σu (tk ) = eϑ (tk ) − kωx ωx (tk ), (12) u(tk ) = Mu (tk ) sign σu (tk ) , u(t) = u(tk ) as tk ≤ t < tk+1 , tk = kT0 , where range parameter M (tk ) is governed by (2), T0 stands for the sampling interval, k = 0, 1, . . . , ϑ∗ (t) denotes the reference signal. The control action u(t) is saturated on the level 10 V. Algorithm (12) is intended to ensure the sliding mode motion on the surface σu ≡ 0, which leads to the following closed-loop system dynamics: ˙ + ϑ(t) = ϑ∗ (t). (13) kωx ϑ(t) Consider the following numerical example. Let us pickup the following system parameters (cf. (Peaucelle et al., −1 x 2007)): aω , aϑmx = 0.5 s−2 , aumx = 0.35 s−2 ; mx = 0.1 s kωx = 0.25 s; sampling period T0 = 0.025 s. Zooming algorithm (2) parameters are taken as: M0 = 0.1, m = 0.1, ρ = e−15T0 = 0.6873. The simulation results for “square wave” reference signal ϑ∗ (t) with the magnitude 10 deg and period 10 s are depicted in Figs. 7–9. As is seen from the plots, after the fast transient period, the sliding mode occurs on the surface σu ≡ 0 (see Fig. 7, lower plot) and no chattering appears. The cases when zooming is absent, i.e. M [k] ≡ M0 are illustrated by Figs. 9, 10. The case of M [k] ≡ 10 is demonstrated in Fig. 9, showing chattering of u(t) with a high magnitude. Fig. 10 shows stability loss for a small range M [k] ≡ 0.1. 4.5 Control of the Helicopter position Since the system (5) is an underactuated one, the aim of independent control for pitch and travel angles can not be achieved. Let the control problem in the terms of the desired travel motion be posed, while the pitch angle is


323

Fig. 10. Time histories: ϑ(t), ϑ∗ (t). No zooming, M (tk ) ≡ 0.1.

Fig. 7. Time histories: ϑ(t), ϑ∗ (t) (upper plot); σ(t) (lower plot). Adaptive zooming.

considered as an inner state variable. This is the typical case of a back-stepping problem, which naturally arise in cascade systems. The error eλ (t) between the prescribed λ∗ (t) and the actual λ(t) travel angles, eλ (t) = λ∗ (t) − λ(t), should be minimized, cf. (Andrievsky et al., 2007; Peaucelle et al., 2011). Computation of torque action u(t). Torque control signal u(t) is computed in accordance with the adaptive zooming algorithm as follows: eλ (t) = λ∗ (t) − λ(t), σu (tk ) = eλ (tk ) − kωy ωy (tk ) −kωx ωx (tk ) − kϑ ϑ(tk ), (14) u(tk ) = Mu (tk ) sign σu (tk ) , u(t) = u(tk ) as tk ≤ t < tk+1 , tk = kT0 ,

where range parameter M (tk ) is governed by (2), T0 stands for the time discretization (sampling) interval, k = 0, 1, . . . Algorithm (14) is intended to produce the sliding mode motion on the surface σu ≡ 0, which leads to fulfillment of the following “model” motion of tracking travel angle λ∗ (t): ˙ (15) kω λ(t) + λ(t) = λ∗ (t) + f λ (t), y

Fig. 8. Time histories: u(t) (upper plot); M (tk ) (lower plot). Adaptive zooming.

ϑ

˙ where = −kωx ϑ(t)−k ϑ ϑ(t) is a bounded disturbance, acting to travel channel from the motion of pitch. fϑλ (t)

Computation of lifting force action w(t). Lifting force control signal w(t) is computed in accordance with the following discrete-time control law: eε (t) = ε∗ (t) − ε(t), η˙ ε (t) = eε (t), σw (tk ) = eε (t) − kσ,ωz ωz(tk ) (16) w(tk ) = Mw sign σu (tk ) + kI,w ηε (tk ) +kP,w ηε (tk ) − kωz ωz (tk ) − kϑ ϑ(tk ), w(t) = w(tk ) as tk ≤ t < tk+1 , tk = kT0 .

Algorithm (16) resets the steady-state error eε (t) due to presence of the integral term. Due to the sign component, it ensures high accuracy in the vicinity of the desired position, robustness with respect to external disturbances and parameter variations.

The control actions u(t), w(t) are saturated on the level 10 V. Simulation parameters. Fig. 9. Time histories: ϑ(t), ϑ∗ (t) (upper plot); u(t) (upper plot) (lower plot). No zooming, M (tk ) ≡ 10. 327

The simulation results are plotted in Figs. 11–16. The following parameter values have been taken for the simulations:


−1 z x , aϑmx = 0.5 s−2 , aumx = 0.35 s−2 , aω aω mz = mx = 0.1 s ωy −1 ε −2 v −2 0.1 s , amz = 3 s , amz = 0.15 s , amy = 0.03 s−1 , −2 aw ; my = 0.4 s

kωx = 0.25 s, kωy = 1.0 s, kωz = 15 s, kσ,ωz = 0.5 s, kI,ε = 25 s−1 , kP,ε = 25; sampling period T0 = 0.025 s. Zooming algorithm (2) parameters are taken as: M0 = 15, m = 0.1, ρ = e−15T0 = 0.6873. Travel reference signal λ∗ (t) has been taken linearly changing on time to assign double-turn of the Helicopter around the vertical axis, see Fig. 13. The elevation reference signal ε∗ (t) has been assigned as the function of the current position λ(t) to produce the desired trajectory (take-off, travel with the constant altitude, landing), see Tab. 1. Table 1. Elevation reference signal ε∗ λ(t) . λ, deg ε∗ , deg

0 0

30 30

680 30

700 15

720 2

Fig. 14. Time histories of σu (t), σw (t).

Fig. 11. Time history of pitch angle ϑ(t). Fig. 15. Time history of Mu (tk ).

Fig. 12. Time histories of elevation angle ε(t) and elevation reference signal ε∗ (t).

Fig. 16. Time histories of torque and lifting force control signals u(t), w(t). 5. CONCLUSIONS Fig. 13. Time histories of travel angle λ(t) and travel reference signal λ∗ (t). The simulation results demonstrate high tracking accuracy and absence of chattering in the sliding mode discrete-time control of Helicopter motion. 328

The adaptive zooming strategy, borrowed from the field of estimation and control over the digital communication channels, is employed to the discrete-time sliding mode control problem. The proposed control strategy is numerically studied both for illustrative example and for


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