On Stochastic Finite-Time Stabilization with Continuous State-Feedback Controllers

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Stochastic Stochastic Finite-Time Finite-Time Stabilization Stabilization Stochastic Finite-Time Stabilization with Continuous State-Feedback Stochastic Finite-Time Stabilization with Continuous State-Feedback with Continuous State-Feedback Controllers with Continuous State-Feedback Controllers Controllers Controllers ∗∗ Kenta Hoshino ∗∗ Yˆ uki Nishimura ∗∗

Kenta u Kenta Hoshino Hoshino ∗ Yˆ Yˆ uki ki Nishimura Nishimura ∗∗ Kenta Hoshino ∗∗ Yˆ uki Nishimura ∗∗ ∗∗ ∗ Kenta Hoshino Yˆ uki Nishimura ∗ University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan ∗ Kyoto Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan ∗ Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan (e-mail: [email protected]). (e-mail: [email protected]). [email protected]). ∗ ∗∗ (e-mail: Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan (e-mail: [email protected]). ∗∗ Kagoshima University, 1-21-40 Korimoto, Kagoshima-city, ∗∗ Kagoshima University, 1-21-40 Korimoto, Kagoshima-city, University, 1-21-40 Korimoto, Kagoshima-city, ∗∗ Kagoshima (e-mail: [email protected]). Kagoshima, Japan (e-mail: [email protected]) Kagoshima University, 1-21-40 Korimoto, Kagoshima-city, Kagoshima, Japan (e-mail: [email protected]) ∗∗ Kagoshima, (e-mail:1-21-40 [email protected]) KagoshimaJapan University, Korimoto, Kagoshima-city, Kagoshima, Japan (e-mail: [email protected]) Kagoshima, Japan (e-mail: [email protected]) Abstract: This paper investigates finite-time stabilization of stochastic systems with homogeAbstract: homogeAbstract: This This paper paper investigates investigates finite-time finite-time stabilization stabilization of of stochastic stochastic systems systems with with homogeneous feedback controllers. In the stabilization problems, chattering-like Abstract: This paper investigates finite-time stabilization of stochastic systems with behavior homogeneous feedback controllers. In the finite-time stabilization problems, chattering-like behavior neous feedback controllers. In the finite-time stabilization problems, chattering-like behavior Abstract: This controllers. paper of stochastic systems with behavior homogeneous feedback In signals the finite-time stabilization problems, usually appears in the theinvestigates input signals due to to stabilization stochastic noises noises when chattering-like homogeneous feedback finite-time usually appears in due stochastic when homogeneous feedback usually appears controllers. in the input input signals due to stabilization stochastic noises when homogeneous feedback neous feedback In the finite-time problems, chattering-like behavior controllers are designed by backstepping-like methods. This study investigates the causes of usually appears in the input signals due to stochastic noises when homogeneous feedback controllers are by methods. This study investigates the of controllers are designed designed by backstepping-like backstepping-like methods. This study investigates the causes causes of usually appears in behavior the input signals dueintothe stochastic noises when homogeneous feedback the chattering-like that appears homogeneous feedback controllers. Moreover, controllers are designed by backstepping-like methods. This study investigates the causes of the chattering-like behavior that appears in the homogeneous feedback controllers. Moreover, the chattering-like behavior that appears in the homogeneous feedback controllers. Moreover, controllers are designed by backstepping-like methods. This study investigates the causes of we present design approaches of controllers for the finite-time stabilization of stochastic systems the chattering-like behavior that appears in the homogeneous feedback controllers. Moreover, we present design approaches of controllers for the finite-time stabilization of stochastic systems we present design approaches of controllers for the finite-time stabilization of stochastic systems the chattering-like behavior that appears in the homogeneous feedback controllers. Moreover, we present design approaches of controllers for the finite-time stabilization of stochastic systems based on on homogeneity, homogeneity, with with the the objective objective of of reducing the the chattering-like chattering-like behavior. behavior. Finally, Finally, the the based based on homogeneity, with the objective offorreducing reducing the chattering-like behavior. Finally, the we present design approaches of controllers the finite-time stabilization of stochastic systems validity the proposed controllers is investigated based numerical experiments. based onof homogeneity, with the objective of reducing theon chattering-like behavior. Finally, the validity of the proposed controllers is investigated based on numerical experiments. validityonofhomogeneity, the proposedwith controllers is investigated based onchattering-like numerical experiments. based the objective of reducing the behavior. Finally, the validity of the proposed controllers is investigated based on numerical experiments. © 2019, IFAC (International Federation isof investigated Automatic Control) Hosting by Elsevier Ltd. All rights reserved. validity of the proposed controllers based on numerical experiments. Keywords: Nonlinear control, Stochastic systems, Feedback stabilization, Finite-time stability, Keywords: Keywords: Nonlinear Nonlinear control, control, Stochastic Stochastic systems, systems, Feedback Feedback stabilization, stabilization, Finite-time Finite-time stability, stability, Keywords: Nonlinear control, Stochastic systems, Feedback stabilization, Finite-time stability, Homogeneity Homogeneity Homogeneity Keywords: Nonlinear control, Stochastic systems, Feedback stabilization, Finite-time stability, Homogeneity Homogeneity 1. methods, chattering-like chattering-like behavior behavior often often arises arises in in input input 1. INTRODUCTION INTRODUCTION methods, 1. methods, chattering-like behavior often arises in input 1. INTRODUCTION INTRODUCTION methods, chattering-like behavior often arisesbe inavoided input signals due to stochastic noises. This should signals due to stochastic noises. This should be avoided signals due to stochastic noises. This should 1. INTRODUCTION methods, often arisesbe input signals duechattering-like toimplementation stochastic behavior noises. This should beinavoided avoided in the actual of controllers. in the the actual actual implementation of controllers. controllers. in implementation of This study investigates the finite-time (FT) stabilization of signals due to stochastic noises. This should be avoided This study study investigates investigates the the finite-time finite-time (FT) (FT) stabilization of of in the actual implementation of controllers. This This paper investigates design approaches for FT stabiThis studysystems investigates finite-time feedback (FT) stabilization stabilization of This stochastic with homogeneous controllers. in thepaper actualinvestigates implementation of approaches controllers. for This paper investigates design approaches for FT FT stabistabidesign stochastic systems withthe homogeneous feedback controllers. stochastic systems with homogeneous feedback controllers. This paper investigates design approaches for FT stabilization of stochastic systems with homogenous feedback This study investigates the finite-time (FT) stabilization of stochastic systems with homogeneous feedback controllers. Moreover, design design approaches approaches for for FT FT stabilizing stabilizing controllers controllers lization lization of stochastic systems with homogenous feedback of stochastic systems with homogenous feedback Moreover, Moreover, design for stabilizing controllers This paper investigates designare approaches for FTthestabilization of stochastic systems with homogenous controllers. These controllers developed with stochastic systems with with homogeneous feedback controllers. Moreover, design approaches approaches for FT FT stabilizing controllers based on homogeneity, the objective of reducing the controllers. These controllers controllers are developed withfeedback the aim aim These are developed with the based on on homogeneity, homogeneity, with the the objective of reducing reducing the controllers. based with objective of the lization of stochastic systems with homogenous feedback controllers. These controllers are developed with the aim aim of reducing the chattering-like behavior in the input sigMoreover, design approaches for FT stabilizing controllers based on homogeneity, with the objective of reducing the chattering-like behavior of input signals, are presented. of reducing the chattering-like behavior in the input sigof reducing the chattering-like behavior in the input sigchattering-like behavior behavior of of input input signals, signals, are are presented. presented. chattering-like controllers. These controllers are developed with the aim of reducing the chattering-like behavior in the input signals. Two design approaches are presented in this work. based on homogeneity, with the objective of reducing the chattering-like behavior of input signals, are presented. nals. Two design approaches are presented in this work. nals. Two design approaches are presented in The FT stabilization problem is a recent topic in the of reducing the chattering-like behavior in the inputwork. sigTwo design approaches in this this work. One is a design approach for the FT stabilization of chattering-like behaviorproblem of input is are presented. The FT stabilization stabilization problem issignals, a recent recent topic in in the the nals. The FT a topic One is is design approach forare thepresented FT stabilization stabilization of aa a One aaa design approach for the FT of The FT stabilization problem is a recent topic in the nonlinear control community. Finite-time stability (FTS) nals. Two design approaches are presented in this work. One is design approach for the FT stabilization of a stochastic counterpart of two-dimensional triangular sysnonlinear control control community. community. Finite-time Finite-time stability stability (FTS) (FTS) stochastic nonlinear stochastic counterpart of two-dimensional triangular syscounterpart of two-dimensional triangular sysThe FT stabilization problem is a recent topic in the nonlinear control community. Finite-time stabilityconver(FTS) stochastic is concept of that finite-time One isThe a design approach for the FTwith stabilization of of two-dimensional sys-a tems. other is design approach dynamic feedis aaa concept concept of stability stability that guarantees guarantees finite-time converis of that finite-time convertems. The The counterpart other is is aaa design design approach with with triangular dynamic feedfeedtems. other approach dynamic nonlinear control community. Finite-time stability (FTS) is a concept of stability stability that guarantees guarantees finite-time convergence to an equilibrium (Roxin (1966); Haimo (1986); Bhat stochastic counterpart of atwo-dimensional triangular systems. The other is a design approach with dynamic feedback controllers that is redesign procedure to reduce gence to an equilibrium (Roxin (1966); Haimo (1986); Bhat gence to (Roxin (1966); Haimo (1986); Bhat back controllers controllers that that is is aa redesign redesign procedure procedure to to reduce reduce is a concept of stability that guarantees finite-time convergence to an an equilibrium equilibrium (Roxin (1966); Haimo (1986); Bhat back and Bernstein (2000); Moulay and Perruquetti (2006)). tems. The other is a design approach with dynamic feedback controllers that is a redesign procedure to reduce the chattering-like behavior in the input signals of existing and Bernstein (2000); Moulay and Perruquetti (2006)). and (2000); Moulay and (2006)). the chattering-like chattering-like behavior behavior in in the input input signals signals of of existing existing the genceBernstein to an equilibrium (Roxin Haimo (1986); Bhat and (2000);problem Moulay(1966); and Perruquetti Perruquetti The FT stabilization involves the design of FT backchattering-like controllers that is a redesign procedure toexisting reduce behavior inaa the the input signals of controllers. Moreover, using numerical example we eluThe Bernstein FT stabilization stabilization problem involves the design design(2006)). of FT FT the The FT problem involves the of controllers. Moreover, using numerical example we eluelucontrollers. Moreover, using a numerical example we and Bernstein (2000); Moulay and Perruquetti (2006)). The FT stabilization problem involves the design of FT stabilizing controllers in controlled systems. Because the the chattering-like behavior in the input signals of existing controllers. Moreover, using a numerical example we elucidate that chattering-like behavior is often caused by the stabilizing controllers controllers in in controlled controlled systems. systems. Because Because the the cidate stabilizing cidate that chattering-like behavior is often caused by the that chattering-like behavior is often caused by the The FT stabilization problem involves the in design ofconFT cidate stabilizing controllers in controlled systems. Because the finite-time convergence is a desired property actual controllers. Moreover, using a numerical example we eluthat chattering-like behavior is often caused by the sliding-mode controller-like structure of the FT stabilizing finite-time convergence is a desired property in actual confinite-time convergence is a desired property in actual consliding-mode controller-like structure of the FT stabilizing controller-like structureisofoften the FT stabilizing stabilizing controllers in systems. Because the sliding-mode finite-time convergence is controlled a desired property actual control problems, the FT stabilization has been investigated cidate that chattering-like behavior caused by and the sliding-mode controller-like controllers. This study is continuation Hoshino of theof stabilizing trol problems, problems, the FT FT stabilization stabilization has been beenininvestigated investigated trol the has controllers. This This study is is aaastructure continuation ofFT Hoshino and study continuation of Hoshino and finite-time convergence is a (2002); desiredNakamura property inet actual con- controllers. trol problems, the FT stabilization has been investigated in numerous studies (Hong al. (2011); sliding-mode controller-like structure of the FT stabilizing controllers. This study is a continuation of Hoshino and Nishimura (2018). (2018). In In Hoshino Hoshino and and Nishimura Nishimura (2018), (2018), the the in numerous numerous studies studies (Hong (Hong (2002); (2002); Nakamura Nakamura et et al. al. (2011); Nishimura in Nishimura (2018). In and (2018), the trolnumerous problems, the FT stabilization hasand been investigated in studies (Hong (2002); Nakamura et al. (2011); (2011); Polyakov et al. (2015)). In the analysis synthesis of FT controllers. This study is acontrollers continuation of Hoshino and (2018). In Hoshino Hoshino and Nishimura Nishimura (2018), the existence of FT stabilizing is investigated withPolyakov et al. al. (2015)). In the the analysis and synthesis synthesis of FT FT Nishimura Polyakov et (2015)). In analysis and of existence of FT stabilizing controllers is investigated withof FT controllers is investigated within numerous studies (Hong (2002); Nakamura et al. (2011); Polyakov et al. (2015)). In the analysis and synthesis of (for FT existence stable systems, the homogeneity plays significant roles Nishimura In Hoshino andand Nishimura (2018), the existence of(2018). FT stabilizing stabilizing controllers investigated without constructive design methods the chattering-like stable systems, systems, the homogeneity homogeneity plays significant significant roles (for stable the plays roles (for out constructive constructive design methods methods andisthe the chattering-like design and chattering-like Polyakov et al. (2015)). In the analysis and synthesis ofand FT out stable systems, the homogeneity plays significant roles (for a review of homogeneity, see Chapter 5 of Bacciotti existence of FT stabilizing controllers is investigated without constructive design methods and the chattering-like behavior of input signals can be reduced using feedback a review of homogeneity, see Chapter 5 of Bacciotti and a review of see Chapter 5 behavior of of input input signals signals can can be be reduced reduced using using feedback feedback stable systems, the homogeneity plays significant roles and (for behavior aRosier review of homogeneity, homogeneity, 5 of of Bacciotti Bacciotti and (2005) and et al. out constructive design methods and the chattering-like behavior of In input signals can reduced using feedback controllers. In this paper, we be present more constructive Rosier (2005) and Bernuau Bernuausee et Chapter al. (2014)). (2014)). Rosier (2005) and et al. controllers. this paper, we present more constructive In this paper, we present more constructive a review of homogeneity, 5 of Bacciotti and controllers. Rosier (2005) and Bernuau Bernuausee et Chapter al. (2014)). (2014)). behavior of In input signals can reduced using feedback controllers. this paper, we be present more constructive design methods for FT stabilizing controllers. FT stability and stabilization can be extended to stochasdesign methods methods for FT FT stabilizing controllers. for stabilizing controllers. Rosier (2005)and andstabilization Bernuau et can al. (2014)). FT stability and stabilization can be extended extended to to stochasstochas- design controllers. In this paper, we present more constructive FT stability be design methods for FT stabilizing controllers. FT stability described and stabilization can bedifferential extended toequations stochas- In the rest of this paper, the next section provides mathetic systems by tic systems systems described by stochastic stochastic differential equations design methods FT stabilizing controllers. tic described by equations In the rest of thisforpaper, paper, the next next section section provides mathemathethe of the FT systems stability and can bedifferential extended stochas- In tic by stochastic stochastic differential equations (SDEs). Yindescribed et stabilization al. (2011) (2011) demonstrated a to LyapunovIn the rest rest of this this paper, the next section provides provides mathematical preliminaries of homogeneity and FTS. The (SDEs). Yin et al. demonstrated a Lyapunov(SDEs). Yin et al. (2011) demonstrated a Lyapunovmatical preliminaries of the homogeneity and FTS. The matical preliminaries of the next homogeneity and FTS. The tic systems byofstochastic differential equations (SDEs). Yindescribed etof al. (2011) demonstrated a Lyapunovtype condition FTS stochastic systems. FT stabiIn the rest of this paper, the sectionofprovides mathematical preliminaries of homogeneity and FTS. The following section discusses the design FT stabilizing the type condition of FTS of stochastic systems. FT stabitype condition of FTS of stochastic systems. FT stabifollowing section discusses the design of FT stabilizing following section discusses the design of FT stabilizing (SDEs). Yin etofal. (2011) demonstrated a studied Lyapunovtype condition lization of stochastic systems has also been by FTS of stochastic systems. FT stabimatical preliminaries of thethe homogeneity FTS. The discusses design of and FT stabilizing controllers for two-dimensional triangular stochastic syslization of stochastic stochastic systems has also also been studied studied by following lization of systems has been by controllerssection for two-dimensional two-dimensional triangular stochastic syscontrollers for triangular stochastic systype condition of FTS of and stochastic systems. FT stabilization of stochastic systems has also been studied by Khoo et al. (2013); Gao Wu (2016); Hoshino and following section discusses the design of FT stabilizing controllers for two-dimensional triangular stochastic systems based on homogeneity, which is developed to reduce Khoo et et al. al. (2013); (2013); Gao Gao and and Wu Wu (2016); (2016); Hoshino Hoshino and and tems Khoo tems based on homogeneity, which is developed to reduce based on homogeneity, which is developed to reduce lization of stochastic systems has also been studied by Khoo et al. (2013); Gao and Wu (2016); Hoshino and Nishimura (2018). (2018). Moreover, Moreover, homogeneity homogeneity can can be be considconsid- tems controllers two-dimensional triangular stochastic sysbased for on homogeneity, is developed to reduce chattering-like behavior. We also discuss the cause of this Nishimura Nishimura Moreover, homogeneity can be chattering-like behavior. We We which also discuss discuss the cause cause of this this chattering-like behavior. also the of Khoofor et stochastic al.(2018). (2013); Gao and Wu (2016); Hoshino and Nishimura (2018). Moreover, homogeneity can be considconsidered systems (Hoshino et al. (2013); Yin tems based on homogeneity, which is developed to reduce chattering-like behavior. We also discuss the cause of this behavior, which is caused by conventional design methods. ered for stochastic systems (Hoshino et al. (2013); Yin ered for stochastic systems (Hoshino et al. (2013); Yin behavior, which is caused by conventional design methods. behavior, which is caused by conventional design methods. Nishimura (2018). Moreover, homogeneity can(2013); beFT considered stochastic systems et al. Yin behavior, et al. (2017); Hoshino et al. (2019)). Therefore, stachattering-like We also discuss the cause of this whichbehavior. isaa caused by conventional design methods. Then, we discuss design approach with dynamic feedback et al. al.for (2017); Hoshino et al. al. (Hoshino (2019)). Therefore, Therefore, FT stastaet (2017); Hoshino et (2019)). FT Then, we we discuss discuss design approach approach with dynamic dynamic feedback aa caused design with feedback ered for stochastic systems (Hoshino et can al. (2013); Yin Then, et al. (2017); Hoshino et al. (2019)). Therefore, FT stabilizing controllers for stochastic systems be designed behavior, which is by conventional design methods. Then, we discuss design approach with dynamic feedback controllers. Finally, the results of numerical experiments bilizing controllers controllers for for stochastic systems systems can can be designed designed controllers. bilizing controllers. Finally, Finally, the the results of of numerical experiments experiments et al. (2017); Hoshino et al. (2019)). Therefore, FT sta- controllers. bilizing controllers for stochastic stochastic can be be methods. designed by employing the homogeneity andsystems backstepping methods. Then, we discuss a design approach with dynamic feedback Finally, the results results of numerical numerical experiments by employing the homogeneity and backstepping by employing the homogeneity and backstepping methods. bilizing controllers for stochastic systems can be methods. designed by employing thecontrollers homogeneity and backstepping However, when are designed based on these controllers. Finally, the results of numerical experiments However, when controllers are designed based on these However, when are designed based on by employing thecontrollers homogeneity backstepping However, when controllers areand designed based methods. on these these However, when controllers are designed based on these 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

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for FT stabilization based on the proposed controllers are presented. Notation. In the following, R denotes the set of real numbers, and Rn denotes the n-dimensional Euclidean space. The Euclidean norm is denoted as x for x ∈ Rn . We denote the absolute value of x ∈ R by |x| and the mapping sgn denotes the signum function. For x ∈ R and α ∈ R, xα denotes xα = sgn(x)|x|α . For x ∈ Rn and a 2 sufficiently smooth function V : Rn → R, ∂∂xV2 (x) denotes the Hessian matrix of V (x). The probability space with a filtration is denoted by (Ω, F, Ft , P), where Ω is a sample space, F is σ-algebra, {Ft }t≥0 is a filtration of F, and P is a probability measure on F. 2. MATHEMATICAL PRELIMINARIES 2.1 FTS of stochastic systems This subsection describes the FTS of stochastic systems, followed by an introduction to the analysis of the stability of these systems based on the Lyapunov stability theory (for a review of the Lyapunov stability theory of stochastic systems, see Khasminskii (2012); Mao (2007)). Throughout this subsection, let us consider an Itˆo SDE dx = f (x)dt + σ(x)dw, x(0) = x0 ∈ Rn (1) n n n n n where x ∈ R , f : R → R , σ : R → R , w is a onedimensional standard Wiener process, and x0 is a fixed initial value. For simplicity, we only consider stochastic systems with one-dimensional Wiener processes. In the following, we assume the continuity of f and σ on Rn . Because of the continuity of f and σ, the existence of weak solutions of the SDE (1) is guaranteed (for the definition of the weak solutions and the conditions for their existence, see Ikeda and Watanabe (1989); Karatzas and Shreve (2012)). Moreover, we suppose that f (0) = 0 and σ(0) = 0. The stability and FTS are defined as follows (see Khasminskii (2012), Yin et al. (2011), respectively). Definition 1. The equilibrium x = 0 of (1) is stable in probability if for any  > 0,   lim P sup x(t) >  = 0. x0 →0

t≥0

Definition 2. The equilibrium x = 0 of (1) is finite-time stable (FTS) in probability if it is stable in probability and for the settling time τ defined by τ = inf {t | x(t) = 0}, P {τ < ∞} = 1 holds. Before introducing the Lyapunov condition for FTS, we introduce the infinitesimal generator. The infinitesimal generator L of (1) is defined as follows. For a C 2 function V : Rn → R, 1 ∂V ∂2V (x)f (x) + σ(x)T (x)σ(x). (2) LV (x) = ∂x 2 ∂x2 Then, the Lyapunov condition for the FTS is given as follows. Theorem 1. (Yin et al. (2011)). Let us consider the system (1). Let V : Rn → R be C 2 , positive definite, and radially unbounded on Rn . Moreover, assume that the function V satisfies that LV (x) ≤ −cV (x)γ (3) 283

205

for some c > 0 and γ ∈ (0, 1). Then, the origin of (1) is FTS in probability. 2.2 Homogeneous systems This subsection introduces the weighted homogeneity, which is often employed in the analysis and synthesis of FTS systems. Throughout this paper, we refer to the weighted homogeneity as the homogeneity for simplicity. The homogeneity is defined for functions and vector fields. We also introduce the definition of stochastic homogeneous control systems. By following Chapter 5 Bacciotti and Rosier (2005), the definitions of the homogeneity of functions and vector fields are given as follows. Definition 3. A mapping ∆rλ : Rn → Rn with parameters λ ≥ 0 and r = (r1 , r2 . . . , rn ) is a dilation if the mapping ∆rλ is given by T

(4) ∆rλ (x) = (λr1 x1 , λr2 x2 , . . . , λrn xn ) . Definition 4. A function V : Rn → R is an r-homogeneous function (r-HF) of degree m if there exists a dilation ∆rλ such that V (∆rλ (x)) = λm V (x) (5) n holds for any x ∈ R and λ ≥ 0. Definition 5. A vector field f : Rn → Rn is an rhomogeneous vector field (r-HVF) of degree τ if there exists a dilation ∆rλ such that (6) f (∆rλ (x)) = λτ ∆rλ (f (x)) n for any x ∈ R and λ ≥ 0. We introduce stochastic homogeneous control systems, which is a slight modification of Hoshino et al. (2019). Definition 6. Consider the stochastic control system given by the following: dx = f (x, u)dt + σ(x)dw, (7) m n m n n where u ∈ R is the input, f : R × R → R , σ : R → Rn , and w is a one-dimensional standard Wiener process. Then, the system (7) is a stochastic (r, s)-homogeneous control system of degree τ if there exist dilations ∆rλ and ∆sλ such that f (∆rλ (x), ∆sλ (x)) = λτ ∆rλ (f (x)), (8) σ(∆rλ (x)) = λτ /2 ∆rλ (σ(x)) hold for x ∈ Rn and λ ≥ 0. We introduce the homogeneous norms. Definition 7. (Homogeneous norms). The homogeneous norm with respect to a dilation ∆rλ is a mapping given as 1/p  n  p/ri |xi | (9) x{r} := i=1

with sufficiently large p > 0.

We introduce the following results. Proposition 2. Let V : Rn → R be a sufficiently smooth r-HF of degree m and let f : Rn → Rn be an r-HVF of degree τ . Then, ∂V ∂2V (x)f (x), f (x)T (x)f (x) (10) ∂x ∂x2 are r-HF of degrees m + τ , m + 2τ , respectively.

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Proposition 3. (Bhat and Bernstein (2005)). Let V1 , V2 : Rn → R be r-HFs of degrees m1 , m2 , respectively, and suppose that the function V1 is positive definite. Then, there exist c1 , c2 ∈ R such that m2 m1

c1 V1 (x) ≤ V2 (x) ≤ c2 V1 (x) holds for any x ∈ Rn .

m2 m1

(11)

3. FINITE-TIME STABILIZATION OF TWO-DIMENSIONAL SYSTEMS This section considers the FT stabilization of twodimensional stochastic systems for the reduction of the chattering-like behavior of the input signals. A comparison with a conventional design method is provided via numerical experiments. This section considers the FT stabilization of a twodimensional SDE given in the form of (7) with x = (x1 , x2 )T ∈ R2 , u ∈ R, and T

T

f (x, u) = (x2 , u) , σ(x) = (σ1 (x1 ), σ2 (x1 , x2 )) , (12)

where σ1 : R → R and σ2 : R2 → R are continuous, and we assume that σ(0) = 0. Moreover, we put the following assumption. Assumption 1. The system (7) given by (12) is a stochastic (r, s)-homogeneous control system of degree −k for r = (r1 , r2 ) = (1, 1 − k) and s = 1 − 2k, where k > 0 and 1 − 2k > 0 hold. Under Assumption 1, we can design a feedback controllers for the FT stabilization of the two-dimensional systems. Theorem 4. Suppose that Assumption 1 holds. Then, there exist parameters k1 , k2 > 0 such that the feedback controller s s (13) u = −k1 x1  r1 − k2 x2  r2 guarantees the FTS in probability of the origin of (7) with (12). Remark 1. The controller (13) is a modification of the linear feedback controller (14) u = −k1 x1 − k2 x2 , and the controller (13) can be employed for FT stabilization of deterministic integrators as discussed in Bhat and Bernstein (2005). The proof of the stability of Theorem 4 is different from that presented in Bhat and Bernstein (2005) as shown. Proof. We show an outline of the proof. The proof is based on the backstepping design of FT stabilizing feedback controllers by Hong (2002). We first consider the one-dimensional subsystem of the two-dimensional system, which is given as follows: (15) dx1 = x2 dt + σ1 (x1 )dw, 2 and the function V1 : R → R given by 1 V1 (x) = |x1 |β1 +1 , (16) β1 + 1 where the constant β1 satisfies β1 = β with a constant β that satisfies β(1 − k) ≥ 2. By denoting the infinitesimal generator of the subsystem (15) by L1 , we obtain the following: (17) L1 V1 (x) = L0,1 V1 (x) + x1 β1 (x2 − x∗2 ), 284

where L0,1 V1 (x) = x1 β1 x∗2 +

β1 σ1 (x1 )2 |x1 |β1 −1 , 2

(18)

and

x∗2 = −l1 x1 1−k (19) is a virtual feedback controller, which is an HF of degree r2 = 1 − k, with a gain parameter l1 that is determined later. Then, we consider the system (7) with (12). Then, we consider functions V2 , W2 : R2 → R given by  x2   β2 V2 (x) = V1 (x)+W2 (x), W2 (x) = z − x∗2 β2 dz, x∗ 2

(20) where β2 satisfies (β2 + 1)/r2 = β + 1. We can show that  1  β +1 |x2 |β2 +1 + β2 |x∗2 | 2 − x2 x∗2 β2 . W2 (x) = β2 + 1 (21) The function W2 (x) is an r-HF of degree β + 1. We can show that the function W2 (x) is positive when x∗2 = x2 and is zero when x∗2 = x2 by following a similar procedure to the proof of Theorem 3.1 of Hong (2002). Moreover, that implies the positive definiteness of V2 (x). Based on the observation that the function V2 (x) is C 2 on R2 , we obtain that ∂W2 (22) (x)u, L2 V2 (x) = L0,2 V2 (x) + ∂x2 where ∂W2 1 ∂ 2 W2 L0,2 V2 (x) = L1 V1 (x) + (x)x2 + σ(x)T (x)σ(x). ∂x1 2 ∂x2 (23)

Then, it is determined that ∂W2 (x) = x2 β2 − x∗2 β2 , (24) ∂x2 and that (∂W2 )/(∂xi )(x) = 0 for i = 1, 2 when x2 = x∗2 . Therefore, by recalling (17), when x2 = x∗2 , ∂2W 1 L2 V2 (x) = L1 V1 (x) + σ(x)T (x)σ(x) 2 ∂x2 ∂2W β1 σ1 (x)2 1 = −l1 |x1 |β+1−k + |x1 |β−1 + σ(x)T (x)σ(x). 2 2 ∂x2 (25) Given that each term in the last equality of (25) is homogeneous with degree β + 1 − k and is a function with respect to x1 when x2 = x∗2 , according to Proposition 3, we can find the value of l1 > 0 such that the right-hand side of (25) is negative when x2 = x∗2 . Then, we consider a controller given by  r2 −k r2 −k  (26) u = −l2 x2  r2 − x∗2  r2 , where the value of l2 is determined later. Then, we obtain that  r2 −k r2 −k  ∂W2 (x) x2  r2 − x∗2  r2 . L2 V2 (x) = L0,2 V2 (x) − l2 ∂x2 (27) We note that according to (24) and (26), the term  r2 −k r2 −k  ∂W2 < 0 for x2 = x∗2 . (x) x2  r2 − x∗2  r2 −l2 ∂x2 (28)

The function L0,2 V2 (x) and the last term of (27) are HFs of degree β +1−k. Moreover, based on the existence of l1 > 0

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1

x1(t)

−1

s

s

s

(30) u = −l1r1 l2 x1  r1 − l2 x2  r2 . Therefore, the feedback (13) is obtained by setting k1 = s/r l1 1 l2 and k2 = l2 . This completes the proof. We consider an example of a stabilization with the controller (13) with a comparison to a conventional design method. Example 1. We consider the FT stabilization of the system given by dx1 = x2 dt + 0.1sig(x1 )7/8 dw, dx2 = udt, (31) where x = (x1 , x2 ), u ∈ R, and w is a one-dimensional standard Wiener process. By simply applying a slight modification of a conventional backstepping design technique for deterministic systems, as shown in Hong (2002), an FT stabilizing controller of the system (31) is given by  4/21 7/2 u = ϕex1 (x) := −k2 x2 7/2 + k1 x1 21/8 , (32)

where k1 = 1.0 and k2 = 6.7. Fig. 1 shows that a sample path of the state x(t) of (31) with the controller (32) converges to x = 0 in finite-time when x(0) = (1, −1). Then, Fig. 2 shows a sample path of the input u(t) of the controller (32). The time response of the input signal exhibits chattering-like behavior. This behavior arises because the controller (32) shows a behavior similar to sliding-mode controllers. The controller (32) tends to cause the state x(t) converge to the set   S = x ∈ R2 | x2 = −x1 3/4 (33)

in finite time, and the set S can be regarded as a sliding surface. However, the state x(t) frequently crosses the set S when the state x(t) tends to converge to the set S because the stochastic noise does not allow the convergence to the set S. We note that the initial value x(0) = (1, −1) is on the set S of (33). Then, we show a design of a controller based on Theorem 4. The system (31) is a (r, s)-homogeneous control system of degree −k = −1/4 for dilations ∆rλ and ∆sλ with r = (r1 , r2 ) = (1, 1 − k) = (1, 3/4) and s = 1 − 2k = 1/2. That is, the system satisfies Assumption 1. According to Theorem 4, we consider the feedback controller

285

u(t)

2

0

0 −2

0

2

4 time

{r}

6

0

8

2

4 time

6

8

Fig. 1. Time responses of Fig. 2. Time responses of u(t) of (31) with (32) x(t) of (31) with (32) 1

x1(t)

x2(t)

u(t)

0.6 u(t)

x1(t), x2(t)

¯ Then,  by choosing l2 > l2 , L2 V2 (x) is negative on the set x ∈ Rn | x{r} = 1 . In addition, the negative definiteness of L2 V2 (x) on R2 is concluded based on the homogeneity of L2 V2 (x). Given that the functions V2 (x) and L2 V2 (x) are HFs of degree β +1 and β +1−k, there exists a constant c > 0 such that L2 V2 (x) ≤ −cV2 (x)(β+1−k)/(β+1) . Given that (β + 1 − k)/(β + 1) ∈ (0, 1), this implies that the FTS in probability of the closed-loop system with the controller (26) according to Theorem 1. Finally, recalling (19) and r2 −k = s in Assumption 1, the feedback controller (26) is expressed as

x2(t) u(t)

x1(t), x2(t)

such that (25) is negative when x2 = x∗2 , L2 V2 (x) < 0 holds when x2 = x∗2 . Then, because L0,2 V2 (x) < 0 if x2 = x∗2 , the contraposition implies that x2 = x∗2 if L0,2 V2 (x) ≥ 0. Then, we choose the value of l2 > ¯l2 where ¯l2 is given as L0,2 V2 (x) ¯l2 = (29) max  r2 −k r2 −k  . L0,2 V2 (x)≥0, ∂W2 ∗  r2 r2 (x) x  − x 2 2 ∂x x =1 2

207

0

0.4 0.2

−1

0.0

0

2

4 time

6

0

8

2

4 time

6

8

Fig. 3. Time responses of Fig. 4. Time responses of u(t) of (31) with (34) x(t) of (31) with (34) 1

2

u = −k1 x1  2 − k2 x2  3 . (34) The values of the feedback gains k1 and k2 are chosen as k1 = k2 = 4.0 using a numerical method to solve the optimization problem given by (29). Figs. 3 and 4 show sample paths of x(t) and u(t) of (31) with the feedback controller (34) when x(0) = (1, −1). Compared to the time responses shown in Fig. 2, the chattering-like behavior is reduced in the time responses of the input signal with (34) shown in Fig. 4. This is because the feedback controller (34) does not have the sliding surface, while the feedback controller (32) tends to cause the state to converge to the sliding surface given by (33). Currently, it is challenging to extend the feedback controller of the form (13) and its design to n-dimensional cases. In the next section, we consider another approach for the design of FT stabilizing feedback controllers for more general cases with the aim of the reduction of chatteringlike behavior. 4. FINITE-TIME STABILIZING CONTROLLERS BY DYNAMIC FEEDBACK CONTROLLERS This section considers another design of FT stabilizing controllers for n-dimensional systems, which also aims to reduce the chattering-like behavior of the input signals. To discuss the reduction of the chattering-like behavior, we consider a re-design of FT stabilizing controllers and introduce dynamic feedback controllers. To this end, we consider an extended system of the system (7), which is given by dx = f (x, u)dt + σ(x)dw, (35a) du = vdt, (35b) where x ∈ Rn , u ∈ R, f , and σ are the same as in (7), and v ∈ R.

We then introduce the following assumption: Assumption 2. Let u of the system (35a) be a onedimensional input. Moreover, the system (35a) is a (r, s)-

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homogeneous control system of degree −k < 0 where r = (r1 , . . . , rn ) and s > 0 that satisfy r1 = 1 and ri+1 = ri − k for i = 1, . . . , n − 1, s = rn − k,

(36)

with the parameter k satisfies k > 0 and s − k > 0. Moreover, there exists a controller u = ϕ(x) such that the origin x = 0 of the system (35a) with the controller u = ϕ(x) is FTS in probability and the function ϕ(x) is a r-HF of degree s for r satisfying (36). Further, there exists a Lyapunov function V (x) of the closed-loop system such that this function is C 2 and is a r-HF of degree m. For the function V (x), consider V : Rn × R → R given by V (x, u) = V (x) + W (x, u),

(37)

where V (x) is the homogeneous Lyapunov function and W (x, u) is given by  u  β  z − ϕ(x)β dz, (38) W (x, u) = ϕ(x)

with β satisfying (β + 1)/s = m. Then, by letting Lext be the infinitesimal generator of the extended system (35), Lext V (x, u) < 0 when u = φ(x), x = 0.

(39)

In addition, the constant β > 0 satisfies that ϕ(x)β is C 2. An example of the system that satisfies Assumption 2 is the system (31) (see also discussions in Example 2 below). Assumption 2 supposes that we consider the reduction of the chattering-like behavior by redesigning a FT stabilizing controller u = ϕ(x) using a conventional design method. Under Assumption 2 and the preceding settings, we can present a re-design method, which can be stated as follows. Theorem 5. Consider the system (35a). Suppose that Assumption 2 holds. Then, with the feedback controller ϕ(x) in Assumption 2, there exists l > 0 such that the controller given as follows:   s−k s−k v = −l u s − ϕ(x) s . (40)

stabilizes the origin of (35) with the controller (40) in the sense of the FTS in probability. Proof. We present an outline of the proof, which is similar to that of Theorem 4.

Consider the function V (x, u) given by (37) in Assumption 2. We can show that the V (x, u) is C 2 , positive definite, and radially unbounded with respect to x and u. Moreover, under Assumption 2, we can also show that the function V (x, u) is a (r, s)-HF of degree m, i.e., there exist dilations ∆rλ and ∆sλ such that V (∆rλ (x), ∆sλ (u)) = λm V (x, u) holds. Then, we demonstrate the existence of the value of the feedback gain l > 0 of (40), which guarantees the FTS. By recalling the infinitesimal generator Lext of the extended system (35), it holds that

where

∂W (x, u) Lext V (x, u) = L0,ext V (x, u) − l ∂u   s−k s−k × u s − ϕ(x) s ,

L0,ext V (x, u) = LV (x) +

n  ∂W

∂xi

(x, u)fi (x, u)

(42) 1 T∂ W (x, u)σ(x). + σ(x) 2 ∂x2 The last term in the right-hand side of (41) can be found to be negative whenever u = ϕ(x). When u = ϕ(x), we can ∂W show that ∂W ∂xi (x, u) = 0 for i = 1, . . . , n, ∂u (x, u) = 0, and L0,ext V (x, u) is negative under Assumption 2. Then, by a similar procedure used in the proof of Theorem 4, we choose the value of the feedback gain l as l > l where l is given by L V (x, u) ¯l =  (43)  0,exts−k max s−k Lext V (x,u)≥0, ∂W (x, u) u s − ϕ(x) s ∂u i=1 2

(x,u){r,s} =1

with a homogeneous norm (x, u){r,s} , which is given in the same manner as x{r} of (9). Then, by choosing the feedback gain l > l, we can show the negative definiteness of Lext V (x, u) with respect to x and u. Moreover, because it can be determined that Lext V (x, u) of (41) is a (r, s)-HF of degree m − k, according to Proposition 3, we can obtain a constant c > 0 such that m−k

Lext V (x, u) ≤ −cV (x, u) m . (44) Given that (m − k)/m ∈ (0, 1), Theorem 1 implies the FTS in probability of the origin of (35) with (40). This completes the proof. Example 2. We consider an example of an application of Theorem 5. We also recall that the system (31) is a stochastic (r, s)-homogeneous control system of degree −1/4 with r = (1, 3/4) and s = 1/2. Then, the system (31) satisfies Assumption 2 because the feedback u = ϕex1 (x) of (32) is a FT stabilizing controller and is a r-HF of degree s = 1/2 and there exists a Lyapunov function of the closedloop system (31) with (32), which is given by V (x) = c1 x61 + c2 x1 21/4 x2 + c3 x82 , (45) where c1 = 7/8, c2 = 1, and c3 = 1/8. The function V (x) is constructed using a backstepping method based on the result by Hong (2002). The function V (x) of (45) is also a r-HF of degree six. Moreover, β = 11 satisfies Assumption 2. Then, based on Theorem 5, we can design the dynamic feedback controller given by   v = −l u2/3 − ϕex1 (x)2/3 , (46)

where ϕex1 (x) is given by (32). The value of l is determined by following a procedure given in the proof of Theorem 5, and we choose the value l as l = 15. Figs. 5 and 6 illustrate sample paths of x(t) and u(t) of (31) with the controller (46) when x(0) = (1, −1) and u(0) = 0. Fig. 5 shows the finite-time convergence of x(t) to the origin. Compared with the time response of u(t) in Fig. 2, the chattering-like behavior of the input signal u(t) is reduced as shown in Fig. 6 using the dynamic feedback controller (46). 5. CONCLUSIONS

(41)

286

This paper discussed design approaches for FT stabilizing controllers of stochastic systems with the primary objective of reducing the chattering-like behavior in input

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x1(t)

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0

−1

0

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4 time

6

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Fig. 5. Time responses of x(t) of (31) with (46) u(t)

u(t)

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6

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Fig. 6. Time responses of u(t) of (31) with (46) signals. We also presented two design approaches. One provides FT stabilizing controllers for two-dimensional triangular systems and the other provides FT stabilizing controllers in the dynamic feedback form. Finally, the effects of the reduction of the chattering-like behavior were discussed based on the results of numerical experiments. REFERENCES Bacciotti, A. and Rosier, L. (2005). Liapunov functions and stability in control theory. Springer-Verlag Berlin Heidelberg. Bernuau, E., Efimov, D., Perruquetti, W., and Polyakov, A. (2014). On homogeneity and its application in sliding mode control. Journal of the Franklin Institute, 351(4), 1866–1901. Bhat, S.P. and Bernstein, D.S. (2000). Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 38(3), 751–766. Bhat, S.P. and Bernstein, D.S. (2005). Geometric homogeneity with applications to finite-time stability. Mathematics of Control, Signals, and Systems (MCSS), 17(2), 101–127. Gao, F. and Wu, Y. (2016). Global finite-time stabilisation for a class of stochastic high-order time-varying nonlinear systems. International Journal of Control, 89(12), 2453–2465. Haimo, V.T. (1986). Finite time controllers. SIAM Journal on Control and Optimization, 24(4), 760–770. Hong, Y. (2002). Finite-time stabilization and stabilizability of a class of controllable systems. Systems & control letters, 46(4), 231–236. Hoshino, K. and Nishimura, Y. (2018). On design of homogeneous feedback controllers for finite-time stabilization of stochastic systems. In Proceedings of The 50th ISCIE International Symposium on Stochastic Systems Theory and Its Applications. Hoshino, K., Yamashita, Y., Nishimura, Y., and Tsubakino, D. (2013). Stability and stabilization of homo287

209

geneous stochastic sytems. In Proceedings of the 52th IEEE Conference on Decision and Control, 1229–1234. Hoshino, K., Nishimura, Y., and Yamashita, Y. (2019). Convergence rates of stochastic homogeneous systems. Systems & Control Letters, 124, 33–39. Ikeda, N. and Watanabe, S. (1989). Stochastic differential equations and diffusion processes. North-Holland, Amsterdam, 2nd edition. Karatzas, I. and Shreve, S. (2012). Brownian motion and stochastic calculus, volume 113. Springer Science & Business Media. Khasminskii, R. (2012). Stochastic stability of differential equations. Springer-Verlag Berlin Heidelberg, second edition. Khoo, S., Yin, J., Man, Z., and Yu, X. (2013). Finite-time stabilization of stochastic nonlinear systems in strictfeedback form. Automatica, 49(5), 1403 – 1410. Mao, X. (2007). Stochastic differential equations and applications. Woodhead Publishing, Cambridge. Moulay, E. and Perruquetti, W. (2006). Finite time stability and stabilization of a class of continuous systems. Journal of Mathematical analysis and applications, 323(2), 1430–1443. Nakamura, N., Nakamura, H., and Nishitani, H. (2011). Global inverse optimal control with guaranteed convergence rates of input affine nonlinear systems. IEEE Transactions on Automatic Control, 56(2), 358–369. Polyakov, A., Efimov, D., and Perruquetti, W. (2015). Finite-time and fixed-time stabilization: Implicit Lyapunov function approach. Automatica, 51, 332–340. Roxin, E. (1966). On finite stability in control systems. Rendiconti del Circolo Matematico di Palermo, 15(3), 273–282. Yin, J., Khoo, S., and Man, Z. (2017). Finite-time stability theorems of homogeneous stochastic nonlinear systems. Systems & Control Letters, 100, 6–13. Yin, J., Khoo, S., Man, Z., and Yu, X. (2011). Finite-time stability and instability of stochastic nonlinear systems. Automatica, 47(12), 2671–2677.