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Reaching law based DSMC with a reference model
11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Control 11th IFAC Symposium on Nonlinear Nonlinear Control Systems Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-16 (2019) 777–782
Reaching Reaching Reaching Reaching
law law law law
based DSMC with a reference based based DSMC DSMC with with a a reference reference model based DSMC with a reference model model model Pawel Latosi´ nski ∗∗∗ Andrzej Bartoszewicz ∗∗∗
Pawe l Latosi´ n ski ∗ Andrzej Bartoszewicz ∗ Pawe n Pawell Latosi´ Latosi´ nski ski Andrzej Andrzej Bartoszewicz Bartoszewicz ∗ ∗ ∗ Pawe l Latosi´ n ski Andrzej Bartoszewicz ∗ Institute of Automatic Control, L ´ od´ d´zz University University of Technology, Technology, ∗ of Automatic Control, L ´ o of Institute of Automatic Control, L ´ o d´ z University of Technology, ∗ Institute Institute of Automatic Control, L ´ od´ z University Technology, B. Stefanowskiego 18/22 St., 90-924 L ´ o d´ zz,, of Poland B. Stefanowskiego 18/22 St., 90-924 L o d´ Poland ∗ B. Stefanowskiego 18/22 St., 90-924 L ´ ´ o d´ zz,, of Poland Institute of Automatic Control, L ´ od´ z University Technology, B. Stefanowskiego 18/22 St., 90-924 L ´ o d´ Poland (e-mail: [email protected], andrzej. [email protected]). (e-mail: [email protected], andrzej. [email protected]). (e-mail: andrzej. B. Stefanowskiego 18/22 St., [email protected]). L ´ od´z, Poland (e-mail: [email protected], [email protected], andrzej. [email protected]). (e-mail: [email protected], andrzej. [email protected]). Abstract: Discrete time sliding mode control (DSMC) strategies are well known to ensure good Abstract: Discrete time sliding sliding mode mode control control (DSMC) (DSMC) strategies strategies are are well well known known to to ensure ensure good good Abstract: Discrete time Abstract: Discrete time sliding mode control (DSMC) strategies are well known to ensure good robustness of the system with respect to perturbations. However, since such algorithms specify robustness of of the the system system with with respect respect to to perturbations. perturbations. However, However, since since such such algorithms algorithms specify specify robustness Abstract: Discrete time mode to control (DSMC) strategies are wellsuch known to ensure good robustness theplant system with respect perturbations. However,is since algorithms specify dynamics of ofof the plant on sliding a step-by-step step-by-step basis, their performance performance is deteriorated by the residual residual dynamics the on a basis, their deteriorated by the dynamics of on a step-by-step basis, their performance is deteriorated by the residual robustness ofthe theplant system with respect tostate. perturbations. However, since such algorithms specify dynamics of the plant on a step-by-step basis, their performance is deteriorated by the residual effect of past disturbance on the system To counteract this effect, in our paper we propose effect of of past past disturbance disturbance on on the the system state. state. To To counteract counteract this this effect, effect, in in our our paper we we propose propose effect dynamics of the plant onreference a step-by-step basis, their performance is deteriorated bytime the residual effect past disturbance on the system system counteract effect, in our paper paper wereaching propose a new approach using model of the plant. In this approach, aa discrete a new new of approach using aa a reference reference modelstate. of the theTo plant. In this this this approach, discrete time reaching a approach using model of plant. In approach, a discrete time reaching effect of past disturbance ondynamics the system state. To counteract effect, in our paper wereaching propose alaw new a reference model of the plant. In this this approach, a discrete time is first used to specify of the disturbance-free model. Then, a secondary control law is approach first used used using to specify specify dynamics of the the disturbance-free model. Then, Then, secondary control law is first to dynamics of disturbance-free model. aaa secondary control a new approach using a reference model of the plant. In this approach, a discrete time reaching law is first used to specify dynamics of the disturbance-free model. Then, secondary control signal is is obtained obtained with with the the aim aim of of driving driving the the state state of of the the actual actual plant plant alongside alongside the the desired desired signal signal is obtained with the aim of driving the state of the actual plant alongside the desired law is first used to specify dynamics ofresult, the the disturbance-free model. Then,alongside a secondary control signal is obtained themodel. aim ofAs state the actual plant theterm desired trajectory provided by its a only the single most recent disturbance has trajectory providedwith by its its model. Asdriving a result, result, only theofsingle single most recent recent disturbance term has trajectory provided by a the most disturbance has signal is obtained with themodel. aim ofAs driving theonly state ofsingle the actual plant performance. alongside theterm desired trajectory provided by its model. As a result, only the most recent disturbance term has an effect of the system state, which effectively enhances its sliding mode an effect effect of of the the system system state, state, which which effectively effectively enhances enhances its its sliding sliding mode mode performance. performance. an trajectory by its model. Aseffectively a result, only the single most recent disturbance term has an effect ofprovided the system state, which enhances its sliding mode performance. © 2019, IFAC (International Federation of Automaticenhances Control) Hosting by Elsevier Ltd. All rights reserved. an effect of the system state, which effectively its sliding mode performance. Keywords: Control theory, sliding mode control, discrete time systems, robust control, Keywords: Control Control theory, theory, sliding sliding mode mode control, control, discrete discrete time time systems, systems, robust robust control, control, Keywords: Keywords: Control theory, sliding mode control, discrete time systems, robust control, model-based control. model-based control. control. model-based Keywords: Control theory, sliding mode control, discrete time systems, robust control, model-based control. model-based control. 1. INTRODUCTION opadhyay and Janardhanan (2006); Bandyopadhyay and 1. INTRODUCTION INTRODUCTION opadhyay and and Janardhanan Janardhanan (2006); (2006); Bandyopadhyay Bandyopadhyay and and 1. opadhyay 1. INTRODUCTION opadhyay and Janardhanan (2006); Bandyopadhyay and Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); 1. INTRODUCTION opadhyay andand Janardhanan (2006); Bandyopadhyay and (2009); and Susy (2010); Niu et al. (2010); Sliding mode control strategies are well known for their Chakrabarty Bandyopadhyay (2016)). Sliding mode mode control control strategies strategies are are well well known known for for their their Fulwani Chakrabarty andMija Bandyopadhyay (2016)). Sliding Chakrabarty and Bandyopadhyay (2016)). Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); Sliding mode control strategies are well known for their Chakrabarty and Bandyopadhyay (2016)). low computational cost and complete insensitivity to a low computational computational cost cost and and complete complete insensitivity insensitivity to to a A significant and often overlooked problem in practical low Sliding mode control strategies are the well so-called known for their A significant significant and and Bandyopadhyay often overlooked overlooked(2016)). problem in in practical low cost and satisfy complete to aa Chakrabarty class of perturbations that matchA and often problem classcomputational of perturbations perturbations that satisfy the insensitivity so-called matchmatchA significantof class of that satisfy the so-called and often time overlooked problem in practical practical application discrete reaching law based sliding low computational cost and complete insensitivity to a application of discrete time reaching law based based sliding class of perturbations that satisfy the so-called matching conditions conditions (Draˇ (Draˇzzenovi´ enovi´cc (1969)). (1969)). Such Such strategies strategies have have A application of discrete time reaching law sliding ing significant and often overlooked problem in practical application of discrete time reaching law based sliding ing conditions (Draˇ z enovi´ c (1969)). Such strategies have mode controllers arises from their reliance on a recursive class of been perturbations that satisfytime the so-called matchmode controllers arises from their reliance on a recursive ing conditions (Draˇ z enovi´ c (1969)). Such strategies have initially used for continuous systems, as seen mode controllers controllers arises from from reliance on recursive initially been been used used for for continuous continuous time time systems, systems, as as seen seen application of discrete time their reaching law on based sliding mode arises reliance aaSince recursive initially function to specify dynamics of the system. new ing conditions (Draˇ zfor enovi´ c (1969)). Suchsystems, strategies function to to specify specify dynamicstheir of the the system. Since new initially been of used continuous seen in the works Emelyanov (1967), Utkin (1977) as or have Ed- mode function dynamics of system. Since new in the the works works of Emelyanov (1967),time Utkin (1977) or Edcontrollers arises from their reliance on a recursive function to specify dynamics of the system. Since new in of Emelyanov (1967), Utkin (1977) or Edvalues of the sliding variable are calculated on a stepinitially been used for continuous time systems, as seen values of the sliding variable are calculated on a stepin the works of Emelyanov (1967), Utkin (1977) or Edwards and Spurgeon (1998), but with the advent of digital values of the sliding variable are calculated on a stepwards and and Spurgeon Spurgeon (1998), (1998), but with with the the advent advent of of digital digital function to dynamics of variable the system. Since new thespecify sliding variable calculated on a stepwards by-step basis, evolution of this is distorted by in the and works of Emelyanov (1967), (1977) Ed- values by-stepofbasis, basis, evolution of this thisarevariable variable is distorted distorted by wards Spurgeon (1998), but but withUtkin the time advent of or digital controllers their equivalents for discrete time plants have by-step evolution of is by controllers their equivalents for discrete plants have values of the sliding variable are calculated on a stepby-step basis, evolution of this variable is distorted by controllers their equivalents for discrete time plants have the effect of disturbance on the system in each step. In wards anddesigned Spurgeon (1998), but with the advent ofc (1985) digital the effect of disturbance on the system in each step. In controllers their equivalents for discrete time plants have also been by authors such as Milosavljevi´ the effect of disturbance on the system in each step. In also been been designed designed by by authors authors such such as as Milosavljevi´ Milosavljevi´cc (1985) (1985) by-step basis, evolution of this variable is each distorted by the effect of disturbance on the system in step. In also particular, this effect can unpredictably affect the length controllers their equivalents for discrete time plants have particular, this effect can unpredictably affect the length also been designed by authors such as Milosavljevi´ c (1985) and Utkin and Drakunov (1989). While continuous time particular, this effect can unpredictably affect the length and Utkin Utkin and and Drakunov Drakunov (1989). (1989). While While continuous continuous time time the effect ofthis disturbance on the system in each step. In effect and canincrease unpredictably affect length and of the the reaching phase and increase state error error in the sliding also been designed by authors such as Milosavljevi´ c (1985) of reaching phase state in the sliding sliding and Utkin andcontrollers Drakunov (1989). time particular, sliding mode drive the system representative of the reaching reaching phase and increase state error in the sliding mode controllers drive theWhile systemcontinuous representative particular, this effect can unpredictably affect the length of the phase and increase state error in sliding sliding mode controllers drive the system representative phase. In order to combat this phenomenon, a very recent and Utkin Drakunov (1989). While continuous time phase. phase. In In order order to to combat combat this this phenomenon, phenomenon, aa very very recent recent sliding mode controllers drive the system representative point onto aand certain hyperplane defined in the state space, point onto onto certain hyperplane defined in the the state space, space, phase. of the reaching phase and increase state error the sliding In order to combat thismodel phenomenon, ainvery recent point aa certain certain hyperplane defined in state approach using the reference of the plant has been sliding mode controllers drive the system representative approach using the reference model of the plant has been point onto a hyperplane defined in the state space, discrete time ones confine it to a specific vicinity of this approach using the reference model of the plant has been discrete time time ones ones confine confine it it to to aa specific specific vicinity vicinity of of this this phase. In order to combat this phenomenon, a very recent approach using the reference model of the plant has been discrete introduced by Bartoszewicz and Adamiak (2018). In that point onto a certain hyperplane defined in the state space, introduced by Bartoszewicz and Adamiak (2018). In that discrete time ones confine it to a specific vicinity of this hyperplane to ensure good robustness with respect to introduced by Bartoszewicz and Adamiak (2018). In that hyperplane to to ensure ensure good good robustness robustness with with respect respect to to approach using the reaching referenceand model of theet(2018). plant has introduced by Bartoszewicz Adamiak In been that hyperplane work, the seminal law of Gao al. (1995) is discrete time ones confine it to a specific vicinity of this work, the seminal reaching law of Gao et al. (1995) is hyperplane to ensure good robustness with respect to disturbance. Depending on the evolution of the system work, the seminal reaching law of Gao et al. (1995) is disturbance. Depending Depending on on the the evolution evolution of of the the system system introduced by Bartoszewicz Inplant that the reaching and law Adamiak ofof Gao et(2018). al. (1995) is disturbance. applied to aaseminal disturbance-free model the considered hyperplane toDepending ensure two good robustness with respect to work, applied to disturbance-free model of the considered plant disturbance. on the evolution of the system representative point, types of discrete time sliding appliedthe to aseminal disturbance-free model ofGao the considered considered plant representative point, point, two two types types of of discrete discrete time time sliding sliding work, reaching lawis of et al.the (1995) is disturbance-free model of the plant representative and then secondary controller used to drive system disturbance. Depending the evolution of time the system and then thentoaaaasecondary secondary controller is used used to drive drive the the system system representative point, twoontypes of discrete sliding applied mode controllers are considered in literature. Switching and controller is to mode controllers are considered in literature. Switching applied to a disturbance-free model of the considered plant and then a secondary controller is used to drive the system mode controllers controllers are considered considered in discrete literature. Switching representative point close close to to that that of of the the model. model. representative point, twoal.types of time sliding representative point point mode are in literature. Switching type controllers (Gao et (1995)) drive the system reprepresentative close to that that of the thetomodel. model. type controllers controllers (Gao (Gao et al. al. (1995)) (1995)) drive the system system rep- and then a secondary controller is used drive the system representative point close to of type et drive the repmode controllers are considered in literature. Switching type controllers (Gao et al. (1995)) drive the system representative point to cross the system representative point In this paper, aa new discrete time sliding mode control resentative point point to to cross cross the the system system representative representative point point representative point close to that of the model. In this paper, new discrete time sliding mode control control resentative In this paper, a new discrete time sliding mode type controllers (Gao et al. drive the system rep- In resentative point to cross the(1995)) system representative point to cross the sliding hyperplane in each step, while nonthis paper, a new discrete time sliding mode control strategy using a reference model of the plant will be proto cross the sliding hyperplane in each step, while nonstrategy using a reference model of the plant will be proto cross the sliding hyperplane in each step, while nonstrategy using a reference model of the plant will be proresentative point to cross the system representative point to cross the sliding hyperplane in each step, while non- In this Apaper, new discrete time sliding mode control switching type controllers (Bartolini et al. (1995)) confine strategy using aa reference model of the plant will be proposed. non-switching type reaching law will be applied switching type controllers (Bartolini et al. (1995)) confine posed. A non-switching type reaching law will be applied switching type controllers (Bartolini et al. (1995)) confine posed. A non-switching type reaching law will be applied to cross the sliding hyperplane in each step, while nonswitching using a reference modelwith of the plant be protype controllers (Bartolini et of al. the (1995)) confine strategy the representative point to a vicinity hyperplane posed. non-switching type law willwill applied to the disturbance-free model the aim of ensuring the representative representative point to to a vicinity vicinity of the hyperplane to the the A disturbance-free modelreaching with the the aim ofbeensuring ensuring the point of the hyperplane to disturbance-free model with aim of switching type controllers (Bartolini et of al. the (1995)) confine posed. the representative point to aa it. vicinity hyperplane A non-switching type reaching lawaim will beensuring applied without the necessity to cross cross it. to the disturbance-free model with the of a bounded convergence rate of the sliding variable to the without the necessity to a bounded bounded convergence convergence rate rate of of the the sliding sliding variable variable to to the the without the necessity necessity to cross cross it. the representative point to a it. vicinity of the hyperplane to without the to the disturbance-free model with the aim ofvariable ensuring aavicinity bounded convergence rate of the sliding variable to the of zero and asymptotically driving that to vicinity of zero and asymptotically driving that variable to Conventionally, the design process of a sliding mode convicinity of zero and asymptotically driving that variable to without the necessity to cross it. of Conventionally, the design design process of aa sliding sliding mode mode concon- avicinity bounded convergence rate of the slidingfrom variable to the of zero and phase. asymptotically driving that variable to Conventionally, the process zero in the sliding Data obtained the model zero in the sliding phase. Data obtained from the model Conventionally, the design process of a sliding mode controller involves involves stating stating the control signal and analyzing zero in the sliding phase. Data obtained from the model troller the control signal and analyzing ofbezero and asymptotically driving that variable to zero in the theoriginal model troller involves involves stating stating theprocess controlofsignal signal and mode analyzing will then used to design a new controller for the Conventionally, the design a sliding con- vicinity will then then be sliding used to to phase. design Data newobtained controllerfrom for the the original troller control and analyzing its effect on the system. However, using the more recent will be used design aa new new controller for original its effect effect on on the the system. system.the However, using the the more recent zero in the sliding phase. Data obtained from the model will then be used to design a controller for the original its However, using more recent plant. It will be demonstrated that the control scheme troller involves stating control signalthe and analyzing plant. It It will will be be demonstrated demonstrated that that the the control control scheme scheme its effect on the system.the However, using more recent plant. reaching law approach, one can instead a priori specify reaching law approach, one can instead instead a priori priori specify will then used designofa the newplant controller for thethe original plant. It be will be to demonstrated that the control scheme reaching law approach, one can a specify using the reference model eliminates effect its effect onevolution the system. However, using the moreaspecify recent using the reference model of the plant eliminates the effect reaching law approach, one can instead a priori the desired of the sliding variable with recurusing the reference model of the plant eliminates the effect the desired desired evolution evolution of of the the sliding sliding variable variable with with aa recurrecur- plant. It will be demonstrated thatAseliminates the controlthe scheme using the reference model of system. the plant effect the of past perturbations on the a consequence, the reaching law approach, one can instead a priori specify of past perturbations on the system. As a consequence, the the desired evolution of the sliding variable with a recursive function function and and then then use use this this function function to to synthesize synthesize the the using of past perturbations on the system. As a consequence, the sive the reference model of the plant eliminates the effect of past perturbations on the system. As a consequence, the sive function and then use this function to synthesize the proposed strategy ensures a fixed-time convergence rate of the desired evolution ofuse thethis sliding variable with a recurproposed strategy ensures a fixed-time convergence rate of sive function and then function to synthesize the control signal. This approach has first been proposed for proposed strategy ensures ensures a fixed-time fixed-time convergence ratethe of control signal. signal. This This approach approach has has first first been been proposed proposed for for of past perturbations on the system. As a consequence, proposed strategy a convergence rate of control the sliding variable to the vicinity of the sliding hyperplane sive function and then use this function to synthesize the the sliding variable to the vicinity of the sliding hyperplane control signal. This approach has first been proposed for continuous time systems by Gao and Hung (1993), but it the sliding variable to the vicinity of the sliding hyperplane continuous time time systems systems by by Gao Gao and and Hung Hung (1993), (1993), but but it it proposed ensures a fixed-time rate of variable to the vicinity of the sliding hyperplane continuous andsliding drivesstrategy the representative representative point toconvergence a narrower narrower quasicontrol signal.significantly This approach has firstHung beenin proposed and drives the point to quasicontinuous systems by Gao and (1993), butfor it the has received more recognition the field of and drives the representative point to aa narrower narrower quasihas received receivedtime significantly more recognition in the field field of the sliding variable to the vicinity of the sliding hyperplane and drives the representative point to a quasihas significantly more recognition in the of sliding mode band than the conventional reaching law continuous time systems Gaorecognition and Hung (1993), but it sliding sliding mode mode band band than than the the conventional conventional reaching reaching law law has received significantly more the field of discrete time sliding modeby control (Gao et al. al.in (1995); Bardiscrete time sliding mode control (Gao et (1995); Barand drives theband representative point to a narrower quasimode than the conventional reaching law discrete time sliding mode control (Gao et al.in(1995); (1995); Barapproach. has received significantly more recognition the Bandyfield of sliding approach. discrete time sliding mode control (Gao et al. Bartoszewicz (1998); Golo and Milosavljevi´ c (2000); approach. toszewicz (1998); (1998); Golo Golo and and Milosavljevi´ Milosavljevi´cc (2000); (2000); BandyBandy- sliding mode band than the conventional reaching law toszewicz discrete time sliding mode (Gao et al. (1995); Bar- approach. toszewicz (1998); Golo andcontrol Milosavljevi´ c (2000); Bandyapproach. toszewicz Golo and Milosavljevi´ c of (2000); Bandy2405-8963 ©(1998); 2019, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
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2. PROBLEM STATEMENT In this section the considered class of discrete time dynamical systems will be described and a conventional reaching law based sliding mode control strategy for such systems will be presented. Dynamics of the considered plant are expressed with the following equation ˜ ˜ [x(k)] + bup (k) + d(k), xp (k + 1) = Axp (k) + a (1) where xp ∈ Rn is the state vector, up ∈ R is the con˜ ∈ Rn is the model trol signal applied to the plant, a n uncertainty, d˜ ∈ R is the disturbance and A, b are of appropriate dimensions. The objective of the control process is to drive the state of the plant to a constant target value xd . To that end, a reaching law based sliding mode control strategy will be used. The first step in the process of designing such a strategy involves the selection of an appropriate sliding variable and its corresponding switching hyperplane. For the considered plant (1) the hyperplane is specified as σp (k) = cT [xd − xp (k)] = 0, (2)
where c ∈ Rn is selected to ensure that cT b = 0. To guarantee a dead-beat response of the closed-loop system, elements of vector c are selected so that the equality det[ϕIn×n − A + b(cT b)−1 cT A] = ϕn (3) is satisfied for any arbitrary variable ϕ. Indeed, such a choice of c ensures that all poles of the closed-loop system state matrix are placed in zero. In order to make the sliding mode control strategy applicable to the considered plants, we assume that the total effect of disturbance and model uncertainties on the sliding variable is bounded in the following way ˜ ≤ Dmax ˜ [x(k)] + cT d(k) Dmin ≤ D(k) = cT a (4) and we further define the mean effect and its maximum admissible deviation from the mean as Dmax + Dmin Dmax − Dmin , Dδ = . (5) Davg = 2 2 With this in mind, a reaching law based sliding mode control strategy will now be designed for system (1).
Finally, solving equation (8) for up , one obtains the reaching law based control signal up (k) = (cT b)−1 (cT xd − cT Axp (k) − f [σp (k)]). (9) In the next subsection, a particular reaching law for discrete time systems with perturbations that do not satisfy the matching conditions will be described. 2.2 Reaching law with limited convergence rate In this paper we consider a reaching law based DSMC strategy first published in Latosi´ nski et al. (2017). The reaching law in question has the following form σp (k + 1) =σp (k) − h[σp (k)]sgn[σp (k)] (10) − D(k) + Davg , where function h(·) = α min{1, | · |/β} (11) β ≥ α > 0 are the design parameters and the sign function is such that sgn(0) = 0. The objective of this reaching law is to guarantee an upper bounded convergence rate of the sliding variable to zero and ensure non-switching type motion in the specified vicinity of the sliding hyperplane. According to relation (9), the control signal obtained from this reaching law is expressed as up (k) = (cT b)−1 (cT xd − cT Axp (k) − σp (k) (12) + h[σp (k)]sgn[σp (k)] − Davg ). Properties of this strategy have already been investigated in existing literature (Latosi´ nski et al. (2017)), which is why they will only be briefly described without proof in the following two theorems. Theorem 1: If the control signal for system (1) is defined by (12), then for any k ≥ 0 the sliding variable rate of change is bounded in the following way |σp (k + 1) − σp (k)| ≤ α + Dδ . (13) Theorem 2: If the control signal for system (1) is defined by (12) and α > Dδ , then the system representative point will approach the quasi-sliding mode band {xp : |cT (xd − xp )| ≤
2.1 Reaching law approach Sliding mode control strategies based on the reaching law approach involve selecting a recursive function, which specifies the desired evolution of the sliding variable (2) on a step-by-step basis. Considering the unpredictable effect of perturbations (4) on the sliding variable, such a function usually has the general form σp (k + 1) = f [σp (k)] − D(k), (6) where f is chosen to ensure stability of the sliding motion. Reaching law (6) can be applied to synthesize the control signal for the considered class of systems. To that end, (1) is first substituted into the right hand side of (2), giving ˜ [x(k)] σp (k + 1) =cT xd − cT Axp (k) − cT a T T ˜ − c bup (k) − c d(k).
(7)
cT xd − cT Axp (k) − cT bup (k) = f [σp (k)].
(8)
Then, taking relation (4) into account, substitution of (7) into the reaching law (6) yields
βDδ } α
(14)
at least asymptotically. Theorems 1 and 2 demonstrate that the proposed reaching law ensures favorable properties of the sliding motion of the system. However, since such control scheme specifies the desired values of the sliding variable on a step-by-step basis, evolution of this variable is affected by perturbations (4) at each time instant and its overall trajectory is distorted by past disturbance terms. In particular, this effect can unpredictably change the length of the reaching phase and increase the quasi-sliding mode band width in the sliding phase. To ensure that the evolution of the sliding variable more closely reflects the one specified in the reaching law, in this paper we propose the application of a reference model of plant (1). This disturbance-free model will be used in conjunction with reaching law (10) to generate the desired values of the sliding variable. Then, a secondary controller will be applied to the original plant with the aim of driving its state alongside the trajectory obtained from the reference model.
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3. REFERENCE MODEL BASED CONTROL In this section we will introduce the perturbation-free reference model of the system and use it to define a new control strategy for the original plant (1). Dynamics of the model are expressed in the state space with the following equation xm (k + 1) = Axm (k) + bum (k), (15) where A, b are the same as in (1) and xm (0) = xp (0). Desired evolution of the state xm will be obtained with the use of a reaching law based sliding mode control strategy. Just like for the conventional strategy described in section 2 of this paper, the process of designing such a controller begins with the selection of the sliding variable and its corresponding switching hyperplane σm (k) = cT [xd − xm (k)] = 0. (16) Vector c in this relation is selected according to (3), which guarantees that all poles of the closed-loop system are placed in zero. In this section, two controllers will be designed. First, a reaching law based control scheme will be used to specify the desired trajectory of the model (15). Then, a secondary controller will be applied to plant (1) with the aim of driving it alongside the trajectory specified by its model. 3.1 Reaching law based strategy for the model Control signal um for the disturbance-free reference model (15) will be obtained from the reaching law (10) with the aim of ensuring non-switching type quasi-sliding motion and limited convergence rate of the sliding variable to zero. Since perturbations are absent in the considered system, reaching law (10) can be expressed as σm (k + 1) = σm (k) − h[σm (k)]sgn[σm (k)], (17) where h is the function specified by (11). According to relation (9), the control signal for the reference model has the following form um (k) =(cT b)−1 (cT xd − cT Axm (k) − σm (k) (18) + h[σm (k)]sgn[σm (k)]). For the considered model, the proposed reaching law will ensure better properties of the sliding motion than the ones obtained for the plant subject to disturbance. Indeed, the following two theorems can be formulated for system (15). Theorem 3: If the control signal for model (15) is defined by (18), then the absolute value of sliding variable (16) will become not greater than β in exactly k ∗ steps, where |σm (0)| − β (19) k∗ = α and · denotes the ceiling function.
Proof: If xm (0) is such a state that |σm (0)| ≤ β, then k ∗ = 0 and assumptions of the theorem are satisfied. Now let us consider any state xm (k) such that |σm (k)| > β. Then, since β > α, relations (11) and (17) imply that |σm (k + 1)| =|σm (k) − αsgn[σm (k)]| (20) =|σm (k)| − α Consequently, since variable σm (k) gets closer to zero by exactly α as long as |σm (k)| > β, the variable is guaranteed to enter the β-vicinity of zero in finite time. Now let k ∗
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be the first time instant for which |σm (k ∗ )| ≤ β. Reaching law (17) is a strictly monotonical function, which implies that |σm (j)| > β for all j = 0, 1, . . . , k ∗ −1. Thus, repeated substitution of the left hand side of relation (20) into its right hand side yields |σm (k ∗ )| =|σm (0)| − αk ∗ . (21) Furthermore, if k ∗ satisfies (19), one gets |σm (0)| − β α (22) ≤|σm (0)| − |σm (0)| + β = β. Thus, sliding variable σm will enter the interval [−β, β] at the time instant specified in relation (19). |σm (k ∗ )| =|σm (0)| − α
It has been demonstrated that the representative point of the reference model will always reach the vicinity of the sliding hyperplane in finite time, which depends on parameters α, β and on the initial conditions of the system. The next theorem will desribe performance of the model in the sliding phase. Theorem 4: If the control signal for model (15) is defined by (18), then for any initial conditions of the system, sliding variable σm will always asymptotically approach zero. Proof: Let xm (k) be such a state that |σm (k)| ≤ β. Existence of this state is ensured by Theorem 3. Reaching law (17) and relation (11) give α σm (k + 1) =σm (k) − |σm (k)|sgn[σm (k)] β (23) α β−α =σm (k) − σm (k) = σm (k). β β Since 0 < β − α < β, relation (23) clearly implies that σm (k) converges to zero as k tends to infinity. According to Theorem 4, in the absence of disturbance the considered reaching law asymptotically drives the system representative point onto the sliding hyperplane, rather than just confining it to a vicinity of this hyperplane. This is an important property of the strategy for the reference model, since it will allow us to design a robust controller for the original plant (1). 3.2 New control strategy for the original plant Reference model introduced in the previous section has been used in conjunction with reaching law (17) to obtain a desired state trajectory. In this section, a secondary controller for the original plant (1) will be designed with the aim of driving the system representative point alongside this trajectory. Such an approach allows one to keep the favorable properties ensured by reaching law (10) while at the same time eliminating the effect of past perturbations on the quasi-sliding motion of the system. The following reaching law is proposed for plant (1) (24) σp (k + 1) = σm (k + 1) − D(k) + Davg , where σm is the value obtained from the reference model. According to relation (9), the control signal obtained from (24) has the following form
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up (k) = (cT b)−1 (cT xd − cT Axp (k) − σm (k + 1) − Davg ).
(25)
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Remark: It should be noted that, although control signal (25) uses values of σm obtained from the reaching law (17), these values do not necessarily have to be calculated on-line. Indeed, the state of the reference model (15) at any time k depends only on the initial conditions xm (0) and control signals um (0), ..., um (k − 1). Consequently, evolution of the state xm and the sliding variable σm can be calculated in advance knowing only reaching law (17) and the initial state xm (0). Obtained values can then be stored in a look-up table to significantly increase the computational efficiency of strategy (25).
hyperplane. This convergence rate is different from the one ensured by reaching law (17) used for the reference model by at most one time instant. In the next theorem, robustness of the proposed approach in the sliding phase will be investigated.
It will now be demonstrated that the proposed control scheme using the reference model of the plant ensures better properties of the system than the conventional reaching law based approach. In particular, it will be shown that the new method ensures a fixed-time convergence rate of the sliding variable and drives the system representative point to a narrower quasi-sliding mode band than reaching law (10). These properties will be demonstrated in the following two theorems.
Proof: According to Theorem 4, for any initial conditions of the system, sliding variable σm will asymptotically approach zero. Furthermore, since |D(k) − Davg | ≤ Dδ , reaching law (24) gives |σp (k)| =|σm (k) − D(k) + Davg | (31) ≤|σm (k)| + Dδ for all k. Theorem 4 further implies that (32) lim (|σm (k)| + Dδ ) = 0 + Dδ .
Theorem 5: If the control signal for system (1) is defined by (25) and α > Dδ , then for any initial conditions of the system the absolute value of sliding variable σp will become smaller than β not earlier than at time instant k ∗ − 1 and not later than at k ∗ + 1, where k ∗ is specified by (19). Proof: It will first be shown that the system representative point can enter the β-vicinity of the sliding hyperplane not earlier than at time k ∗ − 1. According to Theorem 3, absolute value of variable σm will decrease by α in each time instant and become smaller than β in exactly k ∗ steps. This implies |σm (k ∗ − 2)| > β + α. Consequently, since |D(k) − Davg | ≤ Dδ , reaching law (24) gives either σp (k ∗ − 2) =σm (k ∗ − 2) − D(k) + Davg ≥σm (k ∗ − 2) − Dδ
or
(26)
Theorem 6: If the control signal for system (1) is defined by (25) and α > Dδ , then for any initial conditions of the system, the representative point will at least asymptotically approach the quasi-sliding mode band (30) xp : |cT (xd − xp )| ≤ Dδ .
k→∞
Taking relations (31) and (32) into account, one concludes that the system representative point is guaranteed to approach quasi-sliding mode band (30) at least asymptotically. In Theorem 6 it has been demonstrated that the new approach drives the system representative point to a specified vicinity of the sliding hyperplane at least asymptotically. Furthermore, since β > α > Dδ , quasi-sliding mode band (30) is strictly narrower than the one described in relation (14) for the conventional reaching law approach. It can be concluded that the proposed strategy using the reference model of the plant guarantees better robustness of the system than previously known methods. Properties proven in this section will be further verified in the following simulation example.
>β + α − Dδ > β
σp (k ∗ − 2) ≤σm (k ∗ − 2) + Dδ
4. SIMULATION EXAMPLE (27)
< − β − α + Dδ < −β. Thus, the system representative point will not enter the βvicinity of the sliding hyperplane before the time instant k ∗ −1. Next, it will be demonstrated that this vicinity must be reached at most at time k ∗ + 1. Since |σm (k ∗ )| ≤ β and α > Dδ , reaching laws (17) and (24) imply both σp (k ∗ + 1) =σm (k + 1) − D(k) + Davg α ≥σm (k ∗ ) − σm (k ∗ ) − Dδ (28) β ≥ − β + α − Dδ > −β and α σp (k ∗ + 1) ≤σm (k ∗ ) − σm (k ∗ ) + Dδ β (29) ≤β − α + Dδ < β. As a result, the system representative point is guaranteed to enter the β-vicinity of the sliding hyperplane not later than at time instant k ∗ + 1. Theorem 5 demonstrates that the proposed control scheme (24) guarantees a fixed convergence rate of the system representative point to a specified vicinity of the sliding
The proposed control scheme will now be applied to a particular discrete-time plant subject to perturbations and its performance will be compared with the conventional reaching law based strategy. Dynamics of the system are expressed with the following equation xp (k + 1) = Axp (k) + bup (k) + bd(k), (33) where 1 1 1/2 1/24 1/24 0 1 1 1/2 1/6 A= , b= (34) 0 0 1 1 1/2 0 0 0 1 1 and the disturbance affecting the plant d(k) = (−1) k/25 . (35) The objective of the control process is to drive the state of the plant from its initial conditions xp (0) = [30 0 0 0]T to a narrow vicinity of zero. To that end, the following two strategies will be used: A) The conventional non-switching reaching law based strategy (12) with design parameters α = 3 and β = 4. B) The new control scheme (25) in which the evolution of the reference model state is specified by reaching law (18) with design parameters α = 3 and β = 4.
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Results of the comparison are illustrated in the following three figures showing the evolution of the sliding variable, the first state variable and the control signal, respectively. The blue line in all figures represents the conventional reaching law based strategy (12), while the red line corresponds to the proposed model reference scheme (25).
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the strategy proposed in this paper. On the other hand, for the conventional reaching law based strategy the convergence rate is significantly slower than anticipated due to residual effect of past disturbance terms on the sliding variable. It can be further seen from Figure 1 that the new approach drives the sliding variable to a narrower vicinity of zero than the conventional method. Values to which the sliding variable is driven are consistent with the ones stated in Theorem 2 and Theorem 6, respectively. Figure 2 demonstrates that the new strategy using the reference model guarantees smaller error of the first state variable than the conventional reaching law. This is a direct result of a narrower quasi-sliding mode band achieved by the new strategy. Finally, Figure 3 shows that the control signals obtained for both strategies have similar values at all stages of the control process. 5. CONCLUSIONS In this paper we have proposed a new discrete time sliding mode control strategy using a disturbance-free reference model of the system. The objective of this approach is to eliminate the persistent effect of past disturbance terms on the evolution of the system representative point. This is achieved by first specifying the desired trajectory of the reference model by using a reaching law, and then driving the state of the original plant alongside that trajectory. It has been demonstrated that the proposed method guarantees fixed-time convergence rate of the system representative point to the vicinity of the sliding hyperplane, altered by at most one time instant by the single most recent disturbance term. Furthermore, it has been shown that the new approach ensures a smaller quasisliding mode band width than the conventional reaching law based strategy. This is a significant property, as it clearly shows better robustness of the system with respect to disturbance that does not satisfy matching conditions. With this in mind, in future work the proposed model reference approach can be applied in conjunction with reaching laws using a discrete-time sliding variable with relative degree higher than one, since such reaching laws are typically vulnerable to non-matched uncertainties.
Fig. 1. Sliding variable
Fig. 2. First state variable
REFERENCES
Fig. 3. Control signal Since σp (0) = 30 and parameter α = 3 for both strategies, then according to relation (19) one expects the sliding variable to reach the β-vicinity of zero after 9 time instants. Indeed, as seen from Figure 1, that is the case for
Bandyopadhyay, B. and Fulwani, D. (2009). Highperformance tracking controller for discrete plant using nonlinear sliding surface. IEEE Transactions on Industrial Electronics, 56(9), 3628–3637. Bandyopadhyay, B. and Janardhanan, S. (2006). DiscreteTime Sliding Mode Control. A Multirate Output Feedback Approach. Springer-Verlag, Berlin. Bartolini, G., Ferrara, A., and Utkin, V. (1995). Adaptive sliding mode control in discrete-time systems. Automatica, 31(5), 769–773. Bartoszewicz, A. (1998). Discrete time quasi-sliding mode control strategies. IEEE Transactions on Industrial Electronics, 45(4), 633–637. Bartoszewicz, A. and Adamiak, K. (2018). Model reference discrete-time variable structure control. International Journal of Adaptive Control and Signal Processing, 32, 1440–1452. Chakrabarty, S. and Bandyopadhyay, B. (2016). A generalized reaching law with different convergence rates. Automatica, 63(1), 34–37.
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Draˇzenovi´c, B. (1969). The invariance conditions in variable structure systems. Automatica, 5(3), 287–295. Edwards, C. and Spurgeon, S. (1998). Sliding Mode Control: Theory and Applications. Taylor & Francis, London. Emelyanov, S.V. (1967). Variable Structure Control Systems. Nauka, Moscow. Gao, W. and Hung, J. (1993). Variable structure control of nonlinear systems: a new approach. IEEE Transactions on Industrial Electronics, 40(1), 45–55. Gao, W., Wang, Y., and Homaifa, A. (1995). Discrete-time variable structure control systems. IEEE Transactions on Industrial Electronics, 42(2), 117–122. ˇ (2000). Robust discreteGolo, G. and Milosavljevi´c, C. time chattering free sliding mode control. Systems & Control Letters, 41(1), 19–28. Latosi´ nski, P., Bartoszewicz, A., and Le´sniewski, P. (2017). A new reaching law based DSMC for inventory management systems. 21st International Conference on System Theory, Control and Computing. Mija, S. and Susy, T. (2010). Reaching law based sliding mode control for discrete MIMO systems. IEEE International Conference on Control, Automation, Robotics & Vision, 1291–1296. ˇ (1985). General conditions for the exisMilosavljevi´c, C. tence of a quasisliding mode on the switching hyperplane in discrete variable structure systems. Automation and Remote Control, 46(3), 307–314. Niu, Y., Ho, DWC., and Wang, Z. (2010). Improved sliding mode control for discrete-time systems via reaching law. IET Control Theory & Applications, 4(11), 2245–2251. Utkin, V. (1977). Variable structure systems with sliding modes. IEEE Transactions on Automatic Control, 22(2), 212–222. Utkin, V. and Drakunov, S.V. (1989). On discrete-time sliding mode control. IFAC Conference on Nonlinear Control, 484–489.
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Reaching Reaching Reaching Reaching
law law law law
based DSMC with a reference based based DSMC DSMC with with a a reference reference model based DSMC with a reference model model model Pawel Latosi´ nski ∗∗∗ Andrzej Bartoszewicz ∗∗∗
Pawe l Latosi´ n ski ∗ Andrzej Bartoszewicz ∗ Pawe n Pawell Latosi´ Latosi´ nski ski Andrzej Andrzej Bartoszewicz Bartoszewicz ∗ ∗ ∗ Pawe l Latosi´ n ski Andrzej Bartoszewicz ∗ Institute of Automatic Control, L ´ od´ d´zz University University of Technology, Technology, ∗ of Automatic Control, L ´ o of Institute of Automatic Control, L ´ o d´ z University of Technology, ∗ Institute Institute of Automatic Control, L ´ od´ z University Technology, B. Stefanowskiego 18/22 St., 90-924 L ´ o d´ zz,, of Poland B. Stefanowskiego 18/22 St., 90-924 L o d´ Poland ∗ B. Stefanowskiego 18/22 St., 90-924 L ´ ´ o d´ zz,, of Poland Institute of Automatic Control, L ´ od´ z University Technology, B. Stefanowskiego 18/22 St., 90-924 L ´ o d´ Poland (e-mail: [email protected], andrzej. [email protected]). (e-mail: [email protected], andrzej. [email protected]). (e-mail: andrzej. B. Stefanowskiego 18/22 St., [email protected]). L ´ od´z, Poland (e-mail: [email protected], [email protected], andrzej. [email protected]). (e-mail: [email protected], andrzej. [email protected]). Abstract: Discrete time sliding mode control (DSMC) strategies are well known to ensure good Abstract: Discrete time sliding sliding mode mode control control (DSMC) (DSMC) strategies strategies are are well well known known to to ensure ensure good good Abstract: Discrete time Abstract: Discrete time sliding mode control (DSMC) strategies are well known to ensure good robustness of the system with respect to perturbations. However, since such algorithms specify robustness of of the the system system with with respect respect to to perturbations. perturbations. However, However, since since such such algorithms algorithms specify specify robustness Abstract: Discrete time mode to control (DSMC) strategies are wellsuch known to ensure good robustness theplant system with respect perturbations. However,is since algorithms specify dynamics of ofof the plant on sliding a step-by-step step-by-step basis, their performance performance is deteriorated by the residual residual dynamics the on a basis, their deteriorated by the dynamics of on a step-by-step basis, their performance is deteriorated by the residual robustness ofthe theplant system with respect tostate. perturbations. However, since such algorithms specify dynamics of the plant on a step-by-step basis, their performance is deteriorated by the residual effect of past disturbance on the system To counteract this effect, in our paper we propose effect of of past past disturbance disturbance on on the the system state. state. To To counteract counteract this this effect, effect, in in our our paper we we propose propose effect dynamics of the plant onreference a step-by-step basis, their performance is deteriorated bytime the residual effect past disturbance on the system system counteract effect, in our paper paper wereaching propose a new approach using model of the plant. In this approach, aa discrete a new new of approach using aa a reference reference modelstate. of the theTo plant. In this this this approach, discrete time reaching a approach using model of plant. In approach, a discrete time reaching effect of past disturbance ondynamics the system state. To counteract effect, in our paper wereaching propose alaw new a reference model of the plant. In this this approach, a discrete time is first used to specify of the disturbance-free model. Then, a secondary control law is approach first used used using to specify specify dynamics of the the disturbance-free model. Then, Then, secondary control law is first to dynamics of disturbance-free model. aaa secondary control a new approach using a reference model of the plant. In this approach, a discrete time reaching law is first used to specify dynamics of the disturbance-free model. Then, secondary control signal is is obtained obtained with with the the aim aim of of driving driving the the state state of of the the actual actual plant plant alongside alongside the the desired desired signal signal is obtained with the aim of driving the state of the actual plant alongside the desired law is first used to specify dynamics ofresult, the the disturbance-free model. Then,alongside a secondary control signal is obtained themodel. aim ofAs state the actual plant theterm desired trajectory provided by its a only the single most recent disturbance has trajectory providedwith by its its model. Asdriving a result, result, only theofsingle single most recent recent disturbance term has trajectory provided by a the most disturbance has signal is obtained with themodel. aim ofAs driving theonly state ofsingle the actual plant performance. alongside theterm desired trajectory provided by its model. As a result, only the most recent disturbance term has an effect of the system state, which effectively enhances its sliding mode an effect effect of of the the system system state, state, which which effectively effectively enhances enhances its its sliding sliding mode mode performance. performance. an trajectory by its model. Aseffectively a result, only the single most recent disturbance term has an effect ofprovided the system state, which enhances its sliding mode performance. © 2019, IFAC (International Federation of Automaticenhances Control) Hosting by Elsevier Ltd. All rights reserved. an effect of the system state, which effectively its sliding mode performance. Keywords: Control theory, sliding mode control, discrete time systems, robust control, Keywords: Control Control theory, theory, sliding sliding mode mode control, control, discrete discrete time time systems, systems, robust robust control, control, Keywords: Keywords: Control theory, sliding mode control, discrete time systems, robust control, model-based control. model-based control. control. model-based Keywords: Control theory, sliding mode control, discrete time systems, robust control, model-based control. model-based control. 1. INTRODUCTION opadhyay and Janardhanan (2006); Bandyopadhyay and 1. INTRODUCTION INTRODUCTION opadhyay and and Janardhanan Janardhanan (2006); (2006); Bandyopadhyay Bandyopadhyay and and 1. opadhyay 1. INTRODUCTION opadhyay and Janardhanan (2006); Bandyopadhyay and Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); 1. INTRODUCTION opadhyay andand Janardhanan (2006); Bandyopadhyay and (2009); and Susy (2010); Niu et al. (2010); Sliding mode control strategies are well known for their Chakrabarty Bandyopadhyay (2016)). Sliding mode mode control control strategies strategies are are well well known known for for their their Fulwani Chakrabarty andMija Bandyopadhyay (2016)). Sliding Chakrabarty and Bandyopadhyay (2016)). Fulwani (2009); Mija and Susy (2010); Niu et al. (2010); Sliding mode control strategies are well known for their Chakrabarty and Bandyopadhyay (2016)). low computational cost and complete insensitivity to a low computational computational cost cost and and complete complete insensitivity insensitivity to to a A significant and often overlooked problem in practical low Sliding mode control strategies are the well so-called known for their A significant significant and and Bandyopadhyay often overlooked overlooked(2016)). problem in in practical low cost and satisfy complete to aa Chakrabarty class of perturbations that matchA and often problem classcomputational of perturbations perturbations that satisfy the insensitivity so-called matchmatchA significantof class of that satisfy the so-called and often time overlooked problem in practical practical application discrete reaching law based sliding low computational cost and complete insensitivity to a application of discrete time reaching law based based sliding class of perturbations that satisfy the so-called matching conditions conditions (Draˇ (Draˇzzenovi´ enovi´cc (1969)). (1969)). Such Such strategies strategies have have A application of discrete time reaching law sliding ing significant and often overlooked problem in practical application of discrete time reaching law based sliding ing conditions (Draˇ z enovi´ c (1969)). Such strategies have mode controllers arises from their reliance on a recursive class of been perturbations that satisfytime the so-called matchmode controllers arises from their reliance on a recursive ing conditions (Draˇ z enovi´ c (1969)). Such strategies have initially used for continuous systems, as seen mode controllers controllers arises from from reliance on recursive initially been been used used for for continuous continuous time time systems, systems, as as seen seen application of discrete time their reaching law on based sliding mode arises reliance aaSince recursive initially function to specify dynamics of the system. new ing conditions (Draˇ zfor enovi´ c (1969)). Suchsystems, strategies function to to specify specify dynamicstheir of the the system. Since new initially been of used continuous seen in the works Emelyanov (1967), Utkin (1977) as or have Ed- mode function dynamics of system. Since new in the the works works of Emelyanov (1967),time Utkin (1977) or Edcontrollers arises from their reliance on a recursive function to specify dynamics of the system. Since new in of Emelyanov (1967), Utkin (1977) or Edvalues of the sliding variable are calculated on a stepinitially been used for continuous time systems, as seen values of the sliding variable are calculated on a stepin the works of Emelyanov (1967), Utkin (1977) or Edwards and Spurgeon (1998), but with the advent of digital values of the sliding variable are calculated on a stepwards and and Spurgeon Spurgeon (1998), (1998), but with with the the advent advent of of digital digital function to dynamics of variable the system. Since new thespecify sliding variable calculated on a stepwards by-step basis, evolution of this is distorted by in the and works of Emelyanov (1967), (1977) Ed- values by-stepofbasis, basis, evolution of this thisarevariable variable is distorted distorted by wards Spurgeon (1998), but but withUtkin the time advent of or digital controllers their equivalents for discrete time plants have by-step evolution of is by controllers their equivalents for discrete plants have values of the sliding variable are calculated on a stepby-step basis, evolution of this variable is distorted by controllers their equivalents for discrete time plants have the effect of disturbance on the system in each step. In wards anddesigned Spurgeon (1998), but with the advent ofc (1985) digital the effect of disturbance on the system in each step. In controllers their equivalents for discrete time plants have also been by authors such as Milosavljevi´ the effect of disturbance on the system in each step. In also been been designed designed by by authors authors such such as as Milosavljevi´ Milosavljevi´cc (1985) (1985) by-step basis, evolution of this variable is each distorted by the effect of disturbance on the system in step. In also particular, this effect can unpredictably affect the length controllers their equivalents for discrete time plants have particular, this effect can unpredictably affect the length also been designed by authors such as Milosavljevi´ c (1985) and Utkin and Drakunov (1989). While continuous time particular, this effect can unpredictably affect the length and Utkin Utkin and and Drakunov Drakunov (1989). (1989). While While continuous continuous time time the effect ofthis disturbance on the system in each step. In effect and canincrease unpredictably affect length and of the the reaching phase and increase state error error in the sliding also been designed by authors such as Milosavljevi´ c (1985) of reaching phase state in the sliding sliding and Utkin andcontrollers Drakunov (1989). time particular, sliding mode drive the system representative of the reaching reaching phase and increase state error in the sliding mode controllers drive theWhile systemcontinuous representative particular, this effect can unpredictably affect the length of the phase and increase state error in sliding sliding mode controllers drive the system representative phase. In order to combat this phenomenon, a very recent and Utkin Drakunov (1989). While continuous time phase. phase. In In order order to to combat combat this this phenomenon, phenomenon, aa very very recent recent sliding mode controllers drive the system representative point onto aand certain hyperplane defined in the state space, point onto onto certain hyperplane defined in the the state space, space, phase. of the reaching phase and increase state error the sliding In order to combat thismodel phenomenon, ainvery recent point aa certain certain hyperplane defined in state approach using the reference of the plant has been sliding mode controllers drive the system representative approach using the reference model of the plant has been point onto a hyperplane defined in the state space, discrete time ones confine it to a specific vicinity of this approach using the reference model of the plant has been discrete time time ones ones confine confine it it to to aa specific specific vicinity vicinity of of this this phase. In order to combat this phenomenon, a very recent approach using the reference model of the plant has been discrete introduced by Bartoszewicz and Adamiak (2018). In that point onto a certain hyperplane defined in the state space, introduced by Bartoszewicz and Adamiak (2018). In that discrete time ones confine it to a specific vicinity of this hyperplane to ensure good robustness with respect to introduced by Bartoszewicz and Adamiak (2018). In that hyperplane to to ensure ensure good good robustness robustness with with respect respect to to approach using the reaching referenceand model of theet(2018). plant has introduced by Bartoszewicz Adamiak In been that hyperplane work, the seminal law of Gao al. (1995) is discrete time ones confine it to a specific vicinity of this work, the seminal reaching law of Gao et al. (1995) is hyperplane to ensure good robustness with respect to disturbance. Depending on the evolution of the system work, the seminal reaching law of Gao et al. (1995) is disturbance. Depending Depending on on the the evolution evolution of of the the system system introduced by Bartoszewicz Inplant that the reaching and law Adamiak ofof Gao et(2018). al. (1995) is disturbance. applied to aaseminal disturbance-free model the considered hyperplane toDepending ensure two good robustness with respect to work, applied to disturbance-free model of the considered plant disturbance. on the evolution of the system representative point, types of discrete time sliding appliedthe to aseminal disturbance-free model ofGao the considered considered plant representative point, point, two two types types of of discrete discrete time time sliding sliding work, reaching lawis of et al.the (1995) is disturbance-free model of the plant representative and then secondary controller used to drive system disturbance. Depending the evolution of time the system and then thentoaaaasecondary secondary controller is used used to drive drive the the system system representative point, twoontypes of discrete sliding applied mode controllers are considered in literature. Switching and controller is to mode controllers are considered in literature. Switching applied to a disturbance-free model of the considered plant and then a secondary controller is used to drive the system mode controllers controllers are considered considered in discrete literature. Switching representative point close close to to that that of of the the model. model. representative point, twoal.types of time sliding representative point point mode are in literature. Switching type controllers (Gao et (1995)) drive the system reprepresentative close to that that of the thetomodel. model. type controllers controllers (Gao (Gao et al. al. (1995)) (1995)) drive the system system rep- and then a secondary controller is used drive the system representative point close to of type et drive the repmode controllers are considered in literature. Switching type controllers (Gao et al. (1995)) drive the system representative point to cross the system representative point In this paper, aa new discrete time sliding mode control resentative point point to to cross cross the the system system representative representative point point representative point close to that of the model. In this paper, new discrete time sliding mode control control resentative In this paper, a new discrete time sliding mode type controllers (Gao et al. drive the system rep- In resentative point to cross the(1995)) system representative point to cross the sliding hyperplane in each step, while nonthis paper, a new discrete time sliding mode control strategy using a reference model of the plant will be proto cross the sliding hyperplane in each step, while nonstrategy using a reference model of the plant will be proto cross the sliding hyperplane in each step, while nonstrategy using a reference model of the plant will be proresentative point to cross the system representative point to cross the sliding hyperplane in each step, while non- In this Apaper, new discrete time sliding mode control switching type controllers (Bartolini et al. (1995)) confine strategy using aa reference model of the plant will be proposed. non-switching type reaching law will be applied switching type controllers (Bartolini et al. (1995)) confine posed. A non-switching type reaching law will be applied switching type controllers (Bartolini et al. (1995)) confine posed. A non-switching type reaching law will be applied to cross the sliding hyperplane in each step, while nonswitching using a reference modelwith of the plant be protype controllers (Bartolini et of al. the (1995)) confine strategy the representative point to a vicinity hyperplane posed. non-switching type law willwill applied to the disturbance-free model the aim of ensuring the representative representative point to to a vicinity vicinity of the hyperplane to the the A disturbance-free modelreaching with the the aim ofbeensuring ensuring the point of the hyperplane to disturbance-free model with aim of switching type controllers (Bartolini et of al. the (1995)) confine posed. the representative point to aa it. vicinity hyperplane A non-switching type reaching lawaim will beensuring applied without the necessity to cross cross it. to the disturbance-free model with the of a bounded convergence rate of the sliding variable to the without the necessity to a bounded bounded convergence convergence rate rate of of the the sliding sliding variable variable to to the the without the necessity necessity to cross cross it. the representative point to a it. vicinity of the hyperplane to without the to the disturbance-free model with the aim ofvariable ensuring aavicinity bounded convergence rate of the sliding variable to the of zero and asymptotically driving that to vicinity of zero and asymptotically driving that variable to Conventionally, the design process of a sliding mode convicinity of zero and asymptotically driving that variable to without the necessity to cross it. of Conventionally, the design design process of aa sliding sliding mode mode concon- avicinity bounded convergence rate of the slidingfrom variable to the of zero and phase. asymptotically driving that variable to Conventionally, the process zero in the sliding Data obtained the model zero in the sliding phase. Data obtained from the model Conventionally, the design process of a sliding mode controller involves involves stating stating the control signal and analyzing zero in the sliding phase. Data obtained from the model troller the control signal and analyzing ofbezero and asymptotically driving that variable to zero in the theoriginal model troller involves involves stating stating theprocess controlofsignal signal and mode analyzing will then used to design a new controller for the Conventionally, the design a sliding con- vicinity will then then be sliding used to to phase. design Data newobtained controllerfrom for the the original troller control and analyzing its effect on the system. However, using the more recent will be used design aa new new controller for original its effect effect on on the the system. system.the However, using the the more recent zero in the sliding phase. Data obtained from the model will then be used to design a controller for the original its However, using more recent plant. It will be demonstrated that the control scheme troller involves stating control signalthe and analyzing plant. It It will will be be demonstrated demonstrated that that the the control control scheme scheme its effect on the system.the However, using more recent plant. reaching law approach, one can instead a priori specify reaching law approach, one can instead instead a priori priori specify will then used designofa the newplant controller for thethe original plant. It be will be to demonstrated that the control scheme reaching law approach, one can a specify using the reference model eliminates effect its effect onevolution the system. However, using the moreaspecify recent using the reference model of the plant eliminates the effect reaching law approach, one can instead a priori the desired of the sliding variable with recurusing the reference model of the plant eliminates the effect the desired desired evolution evolution of of the the sliding sliding variable variable with with aa recurrecur- plant. It will be demonstrated thatAseliminates the controlthe scheme using the reference model of system. the plant effect the of past perturbations on the a consequence, the reaching law approach, one can instead a priori specify of past perturbations on the system. As a consequence, the the desired evolution of the sliding variable with a recursive function function and and then then use use this this function function to to synthesize synthesize the the using of past perturbations on the system. As a consequence, the sive the reference model of the plant eliminates the effect of past perturbations on the system. As a consequence, the sive function and then use this function to synthesize the proposed strategy ensures a fixed-time convergence rate of the desired evolution ofuse thethis sliding variable with a recurproposed strategy ensures a fixed-time convergence rate of sive function and then function to synthesize the control signal. This approach has first been proposed for proposed strategy ensures ensures a fixed-time fixed-time convergence ratethe of control signal. signal. This This approach approach has has first first been been proposed proposed for for of past perturbations on the system. As a consequence, proposed strategy a convergence rate of control the sliding variable to the vicinity of the sliding hyperplane sive function and then use this function to synthesize the the sliding variable to the vicinity of the sliding hyperplane control signal. This approach has first been proposed for continuous time systems by Gao and Hung (1993), but it the sliding variable to the vicinity of the sliding hyperplane continuous time time systems systems by by Gao Gao and and Hung Hung (1993), (1993), but but it it proposed ensures a fixed-time rate of variable to the vicinity of the sliding hyperplane continuous andsliding drivesstrategy the representative representative point toconvergence a narrower narrower quasicontrol signal.significantly This approach has firstHung beenin proposed and drives the point to quasicontinuous systems by Gao and (1993), butfor it the has received more recognition the field of and drives the representative point to aa narrower narrower quasihas received receivedtime significantly more recognition in the field field of the sliding variable to the vicinity of the sliding hyperplane and drives the representative point to a quasihas significantly more recognition in the of sliding mode band than the conventional reaching law continuous time systems Gaorecognition and Hung (1993), but it sliding sliding mode mode band band than than the the conventional conventional reaching reaching law law has received significantly more the field of discrete time sliding modeby control (Gao et al. al.in (1995); Bardiscrete time sliding mode control (Gao et (1995); Barand drives theband representative point to a narrower quasimode than the conventional reaching law discrete time sliding mode control (Gao et al.in(1995); (1995); Barapproach. has received significantly more recognition the Bandyfield of sliding approach. discrete time sliding mode control (Gao et al. Bartoszewicz (1998); Golo and Milosavljevi´ c (2000); approach. toszewicz (1998); (1998); Golo Golo and and Milosavljevi´ Milosavljevi´cc (2000); (2000); BandyBandy- sliding mode band than the conventional reaching law toszewicz discrete time sliding mode (Gao et al. (1995); Bar- approach. toszewicz (1998); Golo andcontrol Milosavljevi´ c (2000); Bandyapproach. toszewicz Golo and Milosavljevi´ c of (2000); Bandy2405-8963 ©(1998); 2019, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
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2. PROBLEM STATEMENT In this section the considered class of discrete time dynamical systems will be described and a conventional reaching law based sliding mode control strategy for such systems will be presented. Dynamics of the considered plant are expressed with the following equation ˜ ˜ [x(k)] + bup (k) + d(k), xp (k + 1) = Axp (k) + a (1) where xp ∈ Rn is the state vector, up ∈ R is the con˜ ∈ Rn is the model trol signal applied to the plant, a n uncertainty, d˜ ∈ R is the disturbance and A, b are of appropriate dimensions. The objective of the control process is to drive the state of the plant to a constant target value xd . To that end, a reaching law based sliding mode control strategy will be used. The first step in the process of designing such a strategy involves the selection of an appropriate sliding variable and its corresponding switching hyperplane. For the considered plant (1) the hyperplane is specified as σp (k) = cT [xd − xp (k)] = 0, (2)
where c ∈ Rn is selected to ensure that cT b = 0. To guarantee a dead-beat response of the closed-loop system, elements of vector c are selected so that the equality det[ϕIn×n − A + b(cT b)−1 cT A] = ϕn (3) is satisfied for any arbitrary variable ϕ. Indeed, such a choice of c ensures that all poles of the closed-loop system state matrix are placed in zero. In order to make the sliding mode control strategy applicable to the considered plants, we assume that the total effect of disturbance and model uncertainties on the sliding variable is bounded in the following way ˜ ≤ Dmax ˜ [x(k)] + cT d(k) Dmin ≤ D(k) = cT a (4) and we further define the mean effect and its maximum admissible deviation from the mean as Dmax + Dmin Dmax − Dmin , Dδ = . (5) Davg = 2 2 With this in mind, a reaching law based sliding mode control strategy will now be designed for system (1).
Finally, solving equation (8) for up , one obtains the reaching law based control signal up (k) = (cT b)−1 (cT xd − cT Axp (k) − f [σp (k)]). (9) In the next subsection, a particular reaching law for discrete time systems with perturbations that do not satisfy the matching conditions will be described. 2.2 Reaching law with limited convergence rate In this paper we consider a reaching law based DSMC strategy first published in Latosi´ nski et al. (2017). The reaching law in question has the following form σp (k + 1) =σp (k) − h[σp (k)]sgn[σp (k)] (10) − D(k) + Davg , where function h(·) = α min{1, | · |/β} (11) β ≥ α > 0 are the design parameters and the sign function is such that sgn(0) = 0. The objective of this reaching law is to guarantee an upper bounded convergence rate of the sliding variable to zero and ensure non-switching type motion in the specified vicinity of the sliding hyperplane. According to relation (9), the control signal obtained from this reaching law is expressed as up (k) = (cT b)−1 (cT xd − cT Axp (k) − σp (k) (12) + h[σp (k)]sgn[σp (k)] − Davg ). Properties of this strategy have already been investigated in existing literature (Latosi´ nski et al. (2017)), which is why they will only be briefly described without proof in the following two theorems. Theorem 1: If the control signal for system (1) is defined by (12), then for any k ≥ 0 the sliding variable rate of change is bounded in the following way |σp (k + 1) − σp (k)| ≤ α + Dδ . (13) Theorem 2: If the control signal for system (1) is defined by (12) and α > Dδ , then the system representative point will approach the quasi-sliding mode band {xp : |cT (xd − xp )| ≤
2.1 Reaching law approach Sliding mode control strategies based on the reaching law approach involve selecting a recursive function, which specifies the desired evolution of the sliding variable (2) on a step-by-step basis. Considering the unpredictable effect of perturbations (4) on the sliding variable, such a function usually has the general form σp (k + 1) = f [σp (k)] − D(k), (6) where f is chosen to ensure stability of the sliding motion. Reaching law (6) can be applied to synthesize the control signal for the considered class of systems. To that end, (1) is first substituted into the right hand side of (2), giving ˜ [x(k)] σp (k + 1) =cT xd − cT Axp (k) − cT a T T ˜ − c bup (k) − c d(k).
(7)
cT xd − cT Axp (k) − cT bup (k) = f [σp (k)].
(8)
Then, taking relation (4) into account, substitution of (7) into the reaching law (6) yields
βDδ } α
(14)
at least asymptotically. Theorems 1 and 2 demonstrate that the proposed reaching law ensures favorable properties of the sliding motion of the system. However, since such control scheme specifies the desired values of the sliding variable on a step-by-step basis, evolution of this variable is affected by perturbations (4) at each time instant and its overall trajectory is distorted by past disturbance terms. In particular, this effect can unpredictably change the length of the reaching phase and increase the quasi-sliding mode band width in the sliding phase. To ensure that the evolution of the sliding variable more closely reflects the one specified in the reaching law, in this paper we propose the application of a reference model of plant (1). This disturbance-free model will be used in conjunction with reaching law (10) to generate the desired values of the sliding variable. Then, a secondary controller will be applied to the original plant with the aim of driving its state alongside the trajectory obtained from the reference model.
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3. REFERENCE MODEL BASED CONTROL In this section we will introduce the perturbation-free reference model of the system and use it to define a new control strategy for the original plant (1). Dynamics of the model are expressed in the state space with the following equation xm (k + 1) = Axm (k) + bum (k), (15) where A, b are the same as in (1) and xm (0) = xp (0). Desired evolution of the state xm will be obtained with the use of a reaching law based sliding mode control strategy. Just like for the conventional strategy described in section 2 of this paper, the process of designing such a controller begins with the selection of the sliding variable and its corresponding switching hyperplane σm (k) = cT [xd − xm (k)] = 0. (16) Vector c in this relation is selected according to (3), which guarantees that all poles of the closed-loop system are placed in zero. In this section, two controllers will be designed. First, a reaching law based control scheme will be used to specify the desired trajectory of the model (15). Then, a secondary controller will be applied to plant (1) with the aim of driving it alongside the trajectory specified by its model. 3.1 Reaching law based strategy for the model Control signal um for the disturbance-free reference model (15) will be obtained from the reaching law (10) with the aim of ensuring non-switching type quasi-sliding motion and limited convergence rate of the sliding variable to zero. Since perturbations are absent in the considered system, reaching law (10) can be expressed as σm (k + 1) = σm (k) − h[σm (k)]sgn[σm (k)], (17) where h is the function specified by (11). According to relation (9), the control signal for the reference model has the following form um (k) =(cT b)−1 (cT xd − cT Axm (k) − σm (k) (18) + h[σm (k)]sgn[σm (k)]). For the considered model, the proposed reaching law will ensure better properties of the sliding motion than the ones obtained for the plant subject to disturbance. Indeed, the following two theorems can be formulated for system (15). Theorem 3: If the control signal for model (15) is defined by (18), then the absolute value of sliding variable (16) will become not greater than β in exactly k ∗ steps, where |σm (0)| − β (19) k∗ = α and · denotes the ceiling function.
Proof: If xm (0) is such a state that |σm (0)| ≤ β, then k ∗ = 0 and assumptions of the theorem are satisfied. Now let us consider any state xm (k) such that |σm (k)| > β. Then, since β > α, relations (11) and (17) imply that |σm (k + 1)| =|σm (k) − αsgn[σm (k)]| (20) =|σm (k)| − α Consequently, since variable σm (k) gets closer to zero by exactly α as long as |σm (k)| > β, the variable is guaranteed to enter the β-vicinity of zero in finite time. Now let k ∗
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be the first time instant for which |σm (k ∗ )| ≤ β. Reaching law (17) is a strictly monotonical function, which implies that |σm (j)| > β for all j = 0, 1, . . . , k ∗ −1. Thus, repeated substitution of the left hand side of relation (20) into its right hand side yields |σm (k ∗ )| =|σm (0)| − αk ∗ . (21) Furthermore, if k ∗ satisfies (19), one gets |σm (0)| − β α (22) ≤|σm (0)| − |σm (0)| + β = β. Thus, sliding variable σm will enter the interval [−β, β] at the time instant specified in relation (19). |σm (k ∗ )| =|σm (0)| − α
It has been demonstrated that the representative point of the reference model will always reach the vicinity of the sliding hyperplane in finite time, which depends on parameters α, β and on the initial conditions of the system. The next theorem will desribe performance of the model in the sliding phase. Theorem 4: If the control signal for model (15) is defined by (18), then for any initial conditions of the system, sliding variable σm will always asymptotically approach zero. Proof: Let xm (k) be such a state that |σm (k)| ≤ β. Existence of this state is ensured by Theorem 3. Reaching law (17) and relation (11) give α σm (k + 1) =σm (k) − |σm (k)|sgn[σm (k)] β (23) α β−α =σm (k) − σm (k) = σm (k). β β Since 0 < β − α < β, relation (23) clearly implies that σm (k) converges to zero as k tends to infinity. According to Theorem 4, in the absence of disturbance the considered reaching law asymptotically drives the system representative point onto the sliding hyperplane, rather than just confining it to a vicinity of this hyperplane. This is an important property of the strategy for the reference model, since it will allow us to design a robust controller for the original plant (1). 3.2 New control strategy for the original plant Reference model introduced in the previous section has been used in conjunction with reaching law (17) to obtain a desired state trajectory. In this section, a secondary controller for the original plant (1) will be designed with the aim of driving the system representative point alongside this trajectory. Such an approach allows one to keep the favorable properties ensured by reaching law (10) while at the same time eliminating the effect of past perturbations on the quasi-sliding motion of the system. The following reaching law is proposed for plant (1) (24) σp (k + 1) = σm (k + 1) − D(k) + Davg , where σm is the value obtained from the reference model. According to relation (9), the control signal obtained from (24) has the following form
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up (k) = (cT b)−1 (cT xd − cT Axp (k) − σm (k + 1) − Davg ).
(25)
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Remark: It should be noted that, although control signal (25) uses values of σm obtained from the reaching law (17), these values do not necessarily have to be calculated on-line. Indeed, the state of the reference model (15) at any time k depends only on the initial conditions xm (0) and control signals um (0), ..., um (k − 1). Consequently, evolution of the state xm and the sliding variable σm can be calculated in advance knowing only reaching law (17) and the initial state xm (0). Obtained values can then be stored in a look-up table to significantly increase the computational efficiency of strategy (25).
hyperplane. This convergence rate is different from the one ensured by reaching law (17) used for the reference model by at most one time instant. In the next theorem, robustness of the proposed approach in the sliding phase will be investigated.
It will now be demonstrated that the proposed control scheme using the reference model of the plant ensures better properties of the system than the conventional reaching law based approach. In particular, it will be shown that the new method ensures a fixed-time convergence rate of the sliding variable and drives the system representative point to a narrower quasi-sliding mode band than reaching law (10). These properties will be demonstrated in the following two theorems.
Proof: According to Theorem 4, for any initial conditions of the system, sliding variable σm will asymptotically approach zero. Furthermore, since |D(k) − Davg | ≤ Dδ , reaching law (24) gives |σp (k)| =|σm (k) − D(k) + Davg | (31) ≤|σm (k)| + Dδ for all k. Theorem 4 further implies that (32) lim (|σm (k)| + Dδ ) = 0 + Dδ .
Theorem 5: If the control signal for system (1) is defined by (25) and α > Dδ , then for any initial conditions of the system the absolute value of sliding variable σp will become smaller than β not earlier than at time instant k ∗ − 1 and not later than at k ∗ + 1, where k ∗ is specified by (19). Proof: It will first be shown that the system representative point can enter the β-vicinity of the sliding hyperplane not earlier than at time k ∗ − 1. According to Theorem 3, absolute value of variable σm will decrease by α in each time instant and become smaller than β in exactly k ∗ steps. This implies |σm (k ∗ − 2)| > β + α. Consequently, since |D(k) − Davg | ≤ Dδ , reaching law (24) gives either σp (k ∗ − 2) =σm (k ∗ − 2) − D(k) + Davg ≥σm (k ∗ − 2) − Dδ
or
(26)
Theorem 6: If the control signal for system (1) is defined by (25) and α > Dδ , then for any initial conditions of the system, the representative point will at least asymptotically approach the quasi-sliding mode band (30) xp : |cT (xd − xp )| ≤ Dδ .
k→∞
Taking relations (31) and (32) into account, one concludes that the system representative point is guaranteed to approach quasi-sliding mode band (30) at least asymptotically. In Theorem 6 it has been demonstrated that the new approach drives the system representative point to a specified vicinity of the sliding hyperplane at least asymptotically. Furthermore, since β > α > Dδ , quasi-sliding mode band (30) is strictly narrower than the one described in relation (14) for the conventional reaching law approach. It can be concluded that the proposed strategy using the reference model of the plant guarantees better robustness of the system than previously known methods. Properties proven in this section will be further verified in the following simulation example.
>β + α − Dδ > β
σp (k ∗ − 2) ≤σm (k ∗ − 2) + Dδ
4. SIMULATION EXAMPLE (27)
< − β − α + Dδ < −β. Thus, the system representative point will not enter the βvicinity of the sliding hyperplane before the time instant k ∗ −1. Next, it will be demonstrated that this vicinity must be reached at most at time k ∗ + 1. Since |σm (k ∗ )| ≤ β and α > Dδ , reaching laws (17) and (24) imply both σp (k ∗ + 1) =σm (k + 1) − D(k) + Davg α ≥σm (k ∗ ) − σm (k ∗ ) − Dδ (28) β ≥ − β + α − Dδ > −β and α σp (k ∗ + 1) ≤σm (k ∗ ) − σm (k ∗ ) + Dδ β (29) ≤β − α + Dδ < β. As a result, the system representative point is guaranteed to enter the β-vicinity of the sliding hyperplane not later than at time instant k ∗ + 1. Theorem 5 demonstrates that the proposed control scheme (24) guarantees a fixed convergence rate of the system representative point to a specified vicinity of the sliding
The proposed control scheme will now be applied to a particular discrete-time plant subject to perturbations and its performance will be compared with the conventional reaching law based strategy. Dynamics of the system are expressed with the following equation xp (k + 1) = Axp (k) + bup (k) + bd(k), (33) where 1 1 1/2 1/24 1/24 0 1 1 1/2 1/6 A= , b= (34) 0 0 1 1 1/2 0 0 0 1 1 and the disturbance affecting the plant d(k) = (−1) k/25 . (35) The objective of the control process is to drive the state of the plant from its initial conditions xp (0) = [30 0 0 0]T to a narrow vicinity of zero. To that end, the following two strategies will be used: A) The conventional non-switching reaching law based strategy (12) with design parameters α = 3 and β = 4. B) The new control scheme (25) in which the evolution of the reference model state is specified by reaching law (18) with design parameters α = 3 and β = 4.
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Results of the comparison are illustrated in the following three figures showing the evolution of the sliding variable, the first state variable and the control signal, respectively. The blue line in all figures represents the conventional reaching law based strategy (12), while the red line corresponds to the proposed model reference scheme (25).
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the strategy proposed in this paper. On the other hand, for the conventional reaching law based strategy the convergence rate is significantly slower than anticipated due to residual effect of past disturbance terms on the sliding variable. It can be further seen from Figure 1 that the new approach drives the sliding variable to a narrower vicinity of zero than the conventional method. Values to which the sliding variable is driven are consistent with the ones stated in Theorem 2 and Theorem 6, respectively. Figure 2 demonstrates that the new strategy using the reference model guarantees smaller error of the first state variable than the conventional reaching law. This is a direct result of a narrower quasi-sliding mode band achieved by the new strategy. Finally, Figure 3 shows that the control signals obtained for both strategies have similar values at all stages of the control process. 5. CONCLUSIONS In this paper we have proposed a new discrete time sliding mode control strategy using a disturbance-free reference model of the system. The objective of this approach is to eliminate the persistent effect of past disturbance terms on the evolution of the system representative point. This is achieved by first specifying the desired trajectory of the reference model by using a reaching law, and then driving the state of the original plant alongside that trajectory. It has been demonstrated that the proposed method guarantees fixed-time convergence rate of the system representative point to the vicinity of the sliding hyperplane, altered by at most one time instant by the single most recent disturbance term. Furthermore, it has been shown that the new approach ensures a smaller quasisliding mode band width than the conventional reaching law based strategy. This is a significant property, as it clearly shows better robustness of the system with respect to disturbance that does not satisfy matching conditions. With this in mind, in future work the proposed model reference approach can be applied in conjunction with reaching laws using a discrete-time sliding variable with relative degree higher than one, since such reaching laws are typically vulnerable to non-matched uncertainties.
Fig. 1. Sliding variable
Fig. 2. First state variable
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Fig. 3. Control signal Since σp (0) = 30 and parameter α = 3 for both strategies, then according to relation (19) one expects the sliding variable to reach the β-vicinity of zero after 9 time instants. Indeed, as seen from Figure 1, that is the case for
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