Finite-time stabilization for a class of stochastic ... - Semantic Scholar
ISA Transactions 53 (2014) 709–716
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Research Article
Finite-time stabilization for a class of stochastic nonlinear systems via output feedback Wenting Zha a, Junyong Zhai a,n, Shumin Fei a, Yunji Wang b a b
Key Laboratory of Measurement and Control of CSE, Ministry of Education, School of Automation, Southeast University, Nanjing, Jiangsu 210096, China Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX 78249, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 18 November 2013 Received in revised form 13 January 2014 Accepted 21 January 2014 Available online 14 February 2014 This paper was recommended for publication by Jeff Pieper
This paper investigates the problem of global finite-time stabilization in probability for a class of stochastic nonlinear systems. The drift and diffusion terms satisfy lower-triangular or upper-triangular homogeneous growth conditions. By adding one power integrator technique, an output feedback controller is first designed for the nominal system without perturbing nonlinearities. Based on homogeneous domination approach and stochastic finite-time stability theorem, it is proved that the solution of the closed-loop system will converge to the origin in finite time and stay at the origin thereafter with probability one. Two simulation examples are presented to illustrate the effectiveness of the proposed design procedure. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Stochastic nonlinear systems Finite-time stability Output feedback Homogeneous domination
1. Introduction In this paper, we consider the problem of finite-time stabilization via output feedback for a class of stochastic nonlinear systems described by dx1 ðtÞ ¼ x2 ðtÞ dt þf 1 ðxðtÞ; uðtÞÞ dt þ g T1 ðxðtÞ; uðtÞÞ dωðtÞ; dx2 ðtÞ ¼ x3 ðtÞ dt þf 2 ðxðtÞ; uðtÞÞ dt þ g T2 ðxðtÞ; uðtÞÞ dωðtÞ; ⋮ dxn ðtÞ ¼ uðtÞ dt þ f n ðxðtÞ; uðtÞÞ dt þ g Tn ðxðtÞ; uðtÞÞ dωðtÞ; yðtÞ ¼ x1 ðtÞ T
ð1Þ
where xðtÞ ¼ ðx1 ðtÞ; …; xn ðtÞÞ A Rn , uðtÞ A R and yðtÞ A R are the system states, control input and output, respectively. ωðtÞ is an r-dimensional standard Wiener process defined on a probability space ðΩ; ϝ;ϝt ; PÞ with Ω being a sample space, ϝ being a s-field, ϝt being a filtration and P being a probability measure. The drift terms f i : Rn R-R and the diffusion terms g i : Rn R-Rr , i ¼ 1; …; n, are Borel measurable, continuous in system states and satisfy f i ð0; 0Þ ¼ 0 and g i ð0; 0Þ ¼ 0. In the nonlinear control community, finite-time stabilization is one of the most fundamental and challenging problems. In contrast to the commonly used notion of asymptotic stability, finite-time stability requires essentially that a control system should be stable in the sense of Lyapunov and its trajectories tend n
Corresponding author. E-mail address: [email protected] (J. Zhai).
to zero in finite time. It was demonstrated in [1] that finite-time stable systems might have not only faster convergence but also better robustness and disturbance rejection properties. The work [2] provided a solid foundation for finite-time stability theory of continuous autonomous systems, which gave a judging criterion on finite-time stability. Then, some conditions for finite-time stability have been presented for continuous systems [3] and non-autonomous continuous systems [4]. In the literature, several results on finite-time state feedback stabilization have been achieved in [5–7] and the references therein. However, since finite-time stabilizers are generally not smooth, their design methods are sophisticated, especially when some states are not measurable. Based on a “finite-time separation principle”, global finite-time stabilization via output feedback can be achieved for the double integrator system in [8]. The work [9] has developed a novel systematic design method, namely homogeneous domination approach, which provides us a new perspective to deal with the output feedback control problem for nonlinear systems and leads to several stabilization results [10–12]. By coupling the homogeneous domination approach and finite-time stabilization technique, an output feedback controller was constructed to global stabilize a class of lower-triangular nonlinear systems [13] and upper-triangular ones [14]. Moreover, with unknown output gain, the problem of global finite-time stabilization has been addressed in [15,16]. In spite of these developments, the above-mentioned results cannot be generalized easily to a class of stochastic nonlinear systems. It is well known that stochastic modeling has come to play an important role in many branches of science and industry.
0019-0578/$ - see front matter & 2014 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2014.01.005
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W. Zha et al. / ISA Transactions 53 (2014) 709–716
Florchinger extended the concept of control Lyapunov functions and Sontag's stabilization formula to stochastic setting in [17], which leads to more stabilization results [18–23] and the references therein. However, each of them described the asymptotic behavior of trajectories for a class of stochastic nonlinear systems as time tends to infinity. Recently, the work [24] has presented the concept of finite-time stability in probability for stochastic systems and has proved the stochastic finite-time stability theorem. Subsequently, for a class of stochastic nonlinear systems in strictfeedback form, the work [25] designed a continuous statefeedback controller to guarantee the global finite-time stability in probability and our recent work [26] solved the finite-time stabilization problem by dynamic state-feedback. However, only state feedback was considered, which requires all the system states to be measurable. Immediately, one may ask the following interesting questions: Is it possible to relax the growth conditions for nonlinear functions? Under these weaker conditions, how can one design an output feedback controller to make (1) globally finite-time stable in probability? Motivated by the design of finite-time stabilizer in deterministic cases [13,14], and stochastic finite-time stability theorem proposed in [24], we aim to solve the problem of global finitetime stabilization for a class of stochastic nonlinear systems via output feedback. In order to settle this problem, we first design a homogeneous output feedback controller for the nominal system. Then, a scaling gain is introduced to the controller to dominate the perturbing nonlinearities. By appropriately choosing the scaling gain, the closed-loop system can be rendered globally finite-time stable in probability. Furthermore, we extend the result to a class of upper-triangular stochastic nonlinear systems. The main contributions of this paper are as follows: (i) Compared with deterministic cases [13,14], this paper extends the global finite-time stabilization results to a class of stochastic nonlinear systems according to stochastic finite-time stability theorem. (ii) The uncertain nonlinearities are functions of both measurable and unmeasurable states. Based on the homogeneous observer construction, an output feedback controller guarantees the closed-loop system finite-time stable in probability. Notations: R þ denotes the set of all nonnegative real numbers, þ and Rn denotes the real n-dimensional space. Rodd ≕fq A R : q Z 0 is a ratio of two odd integersg. For a given vector or matrix X, XT represents its transpose; TrfXg represents its trace when X is square; J J denotes the Euclidean norm of a vector X or the Frobenius norm of a matrix X. Ci denotes the set of all functions with continuous ith partial derivatives; K denotes the set of all functions, R þ -R þ , which are continuous, strictly increasing and vanishing at zero; K1 denotes the set of all functions which are of class K and unbounded; a 4 b means the minimum of a and b.
In this section, we present some useful definitions and lemmas which play very important roles in this paper. Consider the following stochastic nonlinear system: xð0Þ ¼ x0 A Rn
ð2Þ
where xðtÞ A R is the system state and ωðtÞ is an r-dimensional standard Wiener process defined on a probability space ðΩ; ϝ;ϝt ; PÞ. The Borel measurable functions f : Rn -Rn and g T : Rn -Rnr are continuous in x that satisfy f ð0Þ ¼ 0 and gð0Þ ¼ 0. n
‖f ðxðtÞÞ‖2 þ ‖gðxðtÞÞ‖2 r Kð1 þ ‖xðtÞ‖2 Þ
ð3Þ
for K 4 0. Then given any x0 independent of ωðtÞ, (2) has a continuous solution with probability one. Definition 2.1 (Khoo et al. [25]). The trivial solution of (2) is said to be finite-time stable in probability if the solution exists for any initial value x0 A Rn , denoted by xðt; x0 Þ. Moreover, the following statements hold: (i) Finite-time attractiveness in probability: For every initial value x0 A Rn \f0g, the first hitting time τx0 ¼ inf ft; xðt; x0 Þ ¼ 0g, which is called the stochastic settling time, is finite almost surely, that is, Pfτx0 o 1g ¼ 1. (ii) Stability in probability: For every pair of ɛ A ð0; 1Þ and r 4 0, there exists a δ ¼ δðɛ; rÞ 4 0 such that Pf J xðt; x0 Þ J or; 8 t Z0g Z 1 ɛ, whenever J x0 J o δ. (iii) The solution xððt þ τx0 Þ; x0 Þ is unique for t Z 0. Definition 2.2 (Florchinger [17]). For any given VðxðtÞÞ A C2 associated with stochastic system (2), theinfinitesimal generator L is defined as LVðxÞ ¼ ð∂V=∂xÞf ðxÞ þ 12 Tr gðxÞð∂2 V=∂x2 Þg T ðxÞ , where 1 2 2 T 2 Tr gðxÞð∂ V =∂x Þg ðxÞ is called as the Hessian term of L. Lemma 2.2 (Khoo et al. [25]). For system (2), if there exist a Lyapunov function V : Rn -R þ , K1 class functions μ1 and μ2, positive real numbers c 4 0 and 0 o γ o1, such that for all x A Rn and t Z 0,
μ1 ð J x J Þ rVðxÞ r μ2 ð J x J Þ;
ð4Þ
LVðxÞ r c ðVðxÞÞγ
ð5Þ
then the trivial solution of (2) is finite-time attractive and stable in probability. Definition 2.3 (Kawski [28]). For real numbers r i 4 0, i ¼ 1; …; n, and fixed coordinates ðx1 ; …; xn Þ A Rn , 8 ɛ 4 0.
the dilation Δɛ ðxÞ is defined by Δɛ ðxÞ ¼ ðɛr x1 ; …; ɛr xn Þ, 8 ɛ 4 0, 1
n
with ri being called as the weights of the coordinates. For simplicity of notation, we define dilation weight Δ ¼ ðr 1 ; …; r n Þ. a function V A CðRn ; RÞ is said to be homogeneous of degree τ if there is a real number τ Z 0 such that 8 x A Rn \f0g, VðΔɛ ðxÞÞ ¼ ɛ τ V ðx1 ; …; xn Þ. a vector field f A CðRn ; Rn Þ is said to be homogeneous of degree τ if there is a real number τ Z min1 r i r n fri g such that for i ¼ 1; …; n, 8 x A Rn \f0g, f i ðΔɛ ðxÞÞ ¼ ɛ τ þ ri f i ðx1 ; …; xn Þ. a homogeneous p-norm is defined as ‖x‖Δ;p ¼ ð∑ni¼ 1 jxi jp=ri Þ1=p , 8 x A Rn , for a constant p Z 1. For simplicity, we choose p ¼2 and write ‖x‖Δ for ‖x‖Δ;2 .
Lemma 2.3. Suppose c and d are two positive real numbers. Given any positive number γ 4 0, the following inequality holds:
2. Preliminary results
dxðtÞ ¼ f ðxðtÞÞ dt þg T ðxðtÞÞ dωðtÞ;
Lemma 2.1 (Skorokhod [27]). Suppose that f ðxðtÞÞ and gðxðtÞÞ are continuous with respect to their variables and satisfy the linear growth condition:
jxjc jyjd r
c d c cþd γ jxjc þ d þ γ d jyj : cþd cþd
Lemma 2.4. For x A R, yA R, and p Z1, the following inequalities hold: jx þ yjp r 2p 1 jxp þ yp j; ðjxj þjyjÞ1=p r jxj1=p þ jyj1=p r 2ðp 1Þ=p ðjxj þ jyjÞ1=p :
W. Zha et al. / ISA Transactions 53 (2014) 709–716
with γ k ¼ η^ k þ lk 1 zk 1 , and a set of virtual controllers zn1 ; …; znn defined by
If p Z 1 is an odd integer or a ratio of two odd integers, jx yjp r 2p 1 jxp yp j;
jx1=p y1=p jr 21 1=p jx yj1=p :
ξ1 ¼ zμ1 =r1 zn1μ=r1 ; r i =μ ξi ¼ zμi =ri zni μ=ri ; i ¼ 2; …; n zni ¼ β i 1 ξi 1; zn1 ¼ 0;
3. Main results The objective of this section is to design an output feedback controller for system (1) under the following assumption, which renders the closed-loop system finite-time stable in probability. Assumption 3.1. There exist two positive constants a1 and a2 such that for i ¼ 1; …; n jf i ðx; uÞj r a1 ðjx1 jðri þ τÞ=r1 þ⋯ þ jxi jðri þ τÞ=ri Þ; J g i ðx; uÞ J ra2 ðjx1 jð2ri þ τÞ=2r1 þ ⋯ þ jxi jð2ri þ τÞ=2ri Þ with τ A ð 1=n; 0Þ and a series of parameters r i ¼ ði 1Þτ þ 1;
i ¼ 1; …; n þ 1:
711
ð6Þ
For simplicity, we assume τ ¼ p=q with p being an even integer and q being an odd integer. Based on this, ri will be odd in both denominator and numerator. Remark 3.1. In deterministic cases, the works [13,15] have dealt with the finite-time output feedback stabilization problem for a class of nonlinear systems whose nonlinearities satisfy Assumption 3.1 with a2 ¼ 0. Taking stochastic factors into consideration, we impose lower-triangular homogeneous growth conditions on diffusion terms as well, and therefore, Assumption 3.1 is a more general condition. However, the appearance of Hessian term will produce more nonlinear terms. In what follows, we will present how to handle these terms skillfully.
ð11Þ
one can obtain the global finite-time stabilization result for (7), whose proof is similar to the one introduced in [22] with some modifications. For the sake of space, the detailed proof is omitted here. □ From the construction of U, it can be verified that U is positive definite and proper with respect to Z≔ðz1 ; …; zn ; η^ 2 ; …; η^ n ÞT . First, we consider the following two cases to prove that Wk is positive definite and proper. þ Case 1: If znk r zk , with μ Z 2 max1 r i r n fr i g, μ A Rodd and by Lemma 2.4, one gets Z zk nμ=r Wk ¼ ðsμ=rk zk k Þð4μ rk Þ=μ dsZ ð21 μ=rk Þð4μ rk Þ=μ znk
Z
zk znk
ðs znk Þð4μ rk Þ=rk ds ¼ ð21 μ=rk Þð4μ rk Þ=μ
rk ðz znk Þ4μ=rk : 4μ k
ð12Þ
Case 2: If znk Zzk , (12) can be proved similarly. Next, we aim to prove that Vk is positive definite and proper. With s ¼ srk 1 =ð4μ rk 1 Þ, there exists a positive constant c0 A ½c; c, such that Z Vk ¼
ð4μ r k 1 Þ=r k
zk
ð4μ r k 1 Þ=r k 1
γk
ðsrk 1 =ð4μ rk 1 Þ γ k Þ ds
Z rk 1 =rk 4μ r k 1 z k ðs γ k Þsð4μ 2rk 1 Þ=rk 1 ds rk 1 γk Z zrk 1 =rk k 4μ r k 1 4μ r k 1 r =r ¼ c0 ðs γ k Þ ds ¼ c0 ðzkk 1 k γ k Þ2 rk 1 2r k 1 γk
¼
ð13Þ
The design scheme can be divided into two steps: (i) we first devise a finite-time output feedback controller for the nominal system; and (ii) based on homogeneous domination approach, a scaled output feedback controller is constructed for system (1) and the stability analysis is given to guarantee the finite-time stabilization in probability of the closed-loop system. We first explicitly construct an output feedback stabilizer for the following nominal system:
where c and c represent the infimum and supremum of sð4μ 2rk 1 Þ=rk 1 in ½γ k ; zrkk 1 =rk (or ½zrkk 1 =rk ; γ k , if γ k Z zrkk 1 =rk ), respectively. Therefore, U ¼ ∑nk ¼ 1 W k þ ∑nk ¼ 2 V k Z ∑nk ¼ 1 ð21 μ=rk Þð4μ rk Þ=μ ðr k = r =r 4μÞðzk znk Þ4μ=rk þ ∑nk ¼ 2 ðð4μ r k 1 Þ=2r k 1 Þc0 ðzkk 1 k γ k Þ2 is definite positive and proper with respect to Z. Denoting the dilation weight
dzi ¼ zi þ 1 dt; i ¼ 1; …; n 1;
Δ ¼ ðr 1 ; …; r n ; r1 ; …; rn 1 Þ
T
dzn ¼ v dt;
y ¼ z1 :
ð7Þ
n
where z ¼ ðz1 ; …; zn Þ A R , v A R, yA R are the states, control input and output, respectively. Lemma 3.1. For system (7), there exist positive constants β1 ; …; β n , þ l1 ; …; ln 1 , and μ Z 2 max1 r i r n fr i g, μ A Rodd , such that the following homogeneous output feedback stabilizer
η^ k ¼ lk 1 z^ k ;
z^ k ¼ ðη^ k þ lk 1 z^ k 1 Þrk =ðrk 1 Þ ; k ¼ 2; …; n;
ð8Þ
for z1 ;…;zn
for η^ 2 ;…;η^ n
the closed-loop system (7)–(9) which can be rewritten as dZ ¼ EðZÞ dt ¼ ðz2 ; …; zn ; v; η^_ 2 ; …; η^_ n ÞT dt
v ¼ βn ðz^ n
μ=r μ=r μ=r μ=r μ=r μ þ β n ðz^ n 1 þ ⋯ þ β 3 ðz^ 2 þ β 2 z Þ⋯ÞÞrn þ 1 =μ n1
n1
2
2
1
ɛ rk znk
ð9Þ ¼
k¼1 n
¼ ∑
k¼1 n
þ ∑
k¼2
Z
zk znk
nμ=r k ð4μ r k Þ=μ
ðsμ=rk zk
ð4μ r k 1 Þ=r k 1
γk
γk
Þ
ds
ðsrk 1 =ð4μ rk 1 Þ γ k Þ ds
ðɛr k 1 λ2k 1 r
=ð4μ r k 1 Þ
dλ1 ɛrk 1 γ k Þɛ4μ rk 1 dλ2
ð16Þ
is homogeneous of degree 4μ with respect to Δ, with λ1 ¼ ɛ rk s and λ2 ¼ ɛrk 1 4μ s. By Lemma 1 in [29], there exists a positive constant α1, such that
ð4μ r k 1 Þ=r k
zk
ð4μ r k 1 Þ=rk
zk
ð4μ rk 1 Þ=r k 1
¼ ɛ4μ UðZÞ
k¼2
Z
Z
k¼2
n
ðsrk 1 =ð4μ rk 1 Þ ɛrk 1 γ k Þ ds
λ
n
U ¼ ∑ Wk þ ∑ Vk
ðɛ r k zk Þð4μ rk 1 Þ=rk
ð4μ r k 1 Þ=r k 1 r k ¼ 2 ðɛ k 1 γ k Þ Z zk n μ=r nμ=r ∑ ðɛμ 1 k ɛμ zk k Þð4μ rk Þ=μ ɛ rk k ¼ 1 znk
þ ∑
Proof. By choosing a Lyapunov function U n
Z
n
þ ∑
1
with z^ 1 ¼ z1 , renders the closed-loop system (7)–(9) globally finitetime stable.
ð15Þ
is homogeneous of degree τ. Moreover, it can be shown that Z ɛ r k zk n nμ=r UðΔɛ ðZÞÞ ¼ ∑ ðsμ=rk ɛμ zk k Þð4μ rk Þ=μ ds k¼1
μ=r n
ð14Þ
|fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
ð10Þ
∂UðZÞ 4μ þ τ EðZÞ r α1 ‖Z‖Δ : ∂Z
ð17Þ
712
W. Zha et al. / ISA Transactions 53 (2014) 709–716
Under the new coordinates zi ¼
xi Li 1
; i ¼ 1; …; n;
u v¼ n L
ð18Þ
with L Z 1 being a constant to be determined later, system (1) can be rewritten as g T ðÞ f ðÞ dzi ¼ Lzi þ 1 þ ii 1 dt þ ii 1 dω; L L f n ðÞ g Tn ðÞ dzn ¼ Lv þ n 1 dt þ n 1 dω; L L y ¼ z1 ;
i ¼ 1; …; n 1:
ð19Þ
Next, we construct an observer with the scaling gain L
η^_ k ¼ Llk 1 z^ k ;
z^ k ¼ ðη^ k þ lk 1 z^ k 1 Þrk =rk 1 ; k ¼ 2; …; n
ð20Þ
where li, i ¼ 1; …; n 1, are observer gains. With E(Z) defined in (15), it is straightforward to verify that the closed-loop system (19), (20) and (9) can be represented as dZ ¼ ðLEðZÞ þ FðZÞÞ dt þ G ðZÞ dω T
ð21Þ
where FðZÞ ¼ ðf 1 ðÞ; f 2 ðÞ=L; …; f n ðÞ=Ln 1 ; 0; …; 0ÞT and GðZÞ ¼ ðg 1 ðÞ; g 2 ðÞ=L; …; g n ðÞ=Ln 1 ; 0; …; 0Þ. Based on the observer and controller designs, the following theorem gives the finite-time stabilization result of the closedloop system (21) via output feedback. Theorem 3.1. Under Assumption 3.1, there exists an output feedback controller rendering the system (1) finite-time stable in probability. Proof.. According to Definition 2.1, Theorem 3.1 will be proved by four steps. Step 1: We consider the existence of the solution to the closedloop system (21). The construction of observer and controller indicates that the closed-loop system is continuous with respect to its variables. Since τ A ð 1=n; 0Þ, fi and gi satisfy lower-order growth conditions. In addition, if J Z J Z 1, with 0 oðr i þ τÞ=r j o 1, j ¼ 1; …; i, there exists a positive constant K1 such that ‖FðZÞ‖2 r K 1 ‖Z‖2 . If J Z J o 1, one has ‖FðZÞ‖2 rK 2 , with K 2 Z 0. Therefore, by choosing K 3 ¼ maxfK 1 ; K 2 g, one has ‖FðZÞ‖2 r K 3 ð1 þ ‖Z‖2 Þ, which implies that F(Z) satisfies the linear growth condition in Lemma 2.1. Similarly, one can obtain that there is a positive constant K such that ‖LEðZÞ þ FðZÞ‖2 þ‖GðZÞ‖2 r Kð1 þ‖Z‖2 Þ:
ð22Þ
By Lemma 2.1, there exists a continuous solution Z(t) that can be written as Z t Z t ðLEðZðsÞÞ þ FðZðsÞÞÞ ds þ GT ðZðsÞÞ dωðsÞ ð23Þ ZðtÞ ¼ Z 0 þ 0
0
with the initial value Z0. Step 2: Under Assumption 3.1 and new coordinates (18), it can be deduced that, for i ¼ 1; …; n f i ðÞ a1 ðr i þ τ Þ=r 1 þ jLz2 jðri þ τÞ=r2 þ ⋯ þ jLi 1 zi jðri þ τÞ=ri Þ i 1 r i 1 ðjz1 j L L 1 1=r i
r a1 L
ðr i þ τÞ=r 1
ðjz1 j
ri þ τ r a 1 L1 1=ri ‖Z‖Δ
þ ⋯ þ jzi j
ðri þ τÞ=r i
Þ ð24Þ
with a constant a 1 Z 0. Recall that for i ¼ 1; …; n, ∂UðZÞ=∂Z i is homogeneous of degree 4μ r i , then n ∂UðZÞ f ðÞ n ∂UðZÞ 1 1=r i i ρ1i ‖Z‖4Δμ þ τ r ρ1 ‖Z‖4Δμ þ τ ∂Z FðZÞ r ∑ ∂Z i 1 r ∑ L i L i¼1 i¼1 ð25Þ
where ρ1i, i ¼ 1; …; n, and way, for i ¼ 1; …; n,
ρ1 are positive constants. In a similar
gi ðÞ r þ τ=2 1=2 1=2r i ðjz1 jð2ri þ τÞ=2r1 þ ⋯ þ jzi jð2ri þ τÞ=2ri Þ r a 2 L1=2 1=2ri ‖Z‖Δi i 1 r a2 L L
ð26Þ
with a 2 Z 0, which indicates
T
g ðÞ 1 ∂2 UðZÞ T 1 pffiffiffi n ∂2 UðZÞ g i ðÞ j r ρ ‖Z‖4μ þ τ G ðZÞ r r r ∑ Tr GðZÞ 2 Δ 2 i 1 j 1 2 2 ∂Z L L i;j ¼ 1 ∂Z i ∂Z j
ð27Þ for ρ2 Z 0. According to Definition 2.2, the infinitesimal generator L of U along the trajectory of (21) is
∂UðZÞ ∂UðZÞ 1 ∂2 UðZÞ T LEðZÞ þ FðZÞ þ Tr GðZÞ G ðZÞ LUðZÞ r 2 ∂Z ∂Z 2 ∂Z 4μ þ τ r ðLα1 ρ1 ρ2 Þ‖Z‖Δ : ð28Þ By choosing L 4 maxf1; ðρ1 þ ρ2 Þ=α1 g, there exist positive constants α2 and α3, such that 4μ þ τ
LUðZÞ r α2 ‖Z‖Δ
r α3 U ð4μ þ τÞ=4μ :
ð29Þ
By Lemma 2.2, it can be obtained that the solution of the closedloop system (21) is finite-time attractive and stable in probability. Step 3: In this step, we will prove that after the first hitting time τZ0 , the solution Zðt þ τZ0 Þ remains zero almost surely, 8 t Z 0. Define the stopping time τm ¼ inf ft Z τ Z 0 ; J Zðt; Z 0 Þ J Z m; m 4 0g. It is clear that τm is an increasing time sequence. Applying Itô's formula, one has 8 t Z 0 Z ðt þ τZ Þ 4 τm 0 UðZððt þ τZ 0 Þ 4 τm ÞÞ ¼ UðZðτZ 0 4 τm ÞÞ þ LUðZðsÞÞ ds Z þ
ðt þ τZ 0 Þ 4 τ m
τZ 0 4 τm
τZ 0 4 τm
∂UðZðsÞÞ T G ðZðsÞÞ dωðsÞ ∂Z
ð30Þ
which indicates EUðZððt þ τZ 0 Þ 4 τm ÞÞ ¼ EUðZðτZ 0 ÞÞ þE
Z
ðt þ τ Z 0 Þ 4 τm
τZ0
LUðZðsÞÞ ds r 0: ð31Þ
Since U(Z) is positive definite, one can obtain EUðZððt þ τZ 0 Þ 4 τm ÞÞ ¼ 0, which implies that UðZððt þ τZ 0 Þ 4 τ m ÞÞ ¼ 0 almost surely, 8 t Z 0. Letting m- þ 1, since τm - þ 1 almost surely as m- þ1, we get Zðt þ τZ 0 Þ ¼ 0 almost surely, 8 t Z 0. Therefore, the result of finite-time stabilization in probability for the closed-loop system (21) is achieved according to Definition 2.1. Step 4: Since coordinate transformation (18) does not change the properties of the system, then the solution of stochastic nonlinear system (1) is finite-time stable in probability. □ Remark 3.2. In the deterministic work, a C1 Lyapunov function is enough for the design and analysis of the controller. However, for stochastic nonlinear systems, the Hessian term requires the Lyapunov function to be C2 . First, for i; j ¼ 1; …; k 1, ∂W k ð4μ rk Þ=μ ¼ ξk ; ∂zk
nμ=r k
∂W k 4μ r k ∂zk ¼ ∂zi μ ∂zi
Z
zk n
zk
nμ=r k ð3μ rk Þ=μ
ðsμ=rk zk
Þ
ds;
∂2 W k 4μ r k ðμ rk Þ=rk ð3μ rk Þ=μ ¼ zk ξk ; rk ∂z2k nμ=r
∂2 W k ∂2 W k 4μ r k ð3μ rk Þ=μ ∂zk k ξk ¼ ¼ ; ∂zk ∂zi ∂zi ∂zk μ ∂zi nμ=r Z ∂2 W k 4μ r k ∂2 zk k zk μ=rk nμ=r ¼ ðs zk k Þð3μ rk Þ=μ ds; 2 μ ∂zi ∂z2i znk
W. Zha et al. / ISA Transactions 53 (2014) 709–716 nμ=r k
þ
ð4μ r k Þð3μ r k Þ ∂zk ∂zi μ2
!2 Z
zk
znk
nμ=r k ð2μ r k Þ=μ
ðsμ=rk zk
nμ=r k
∂2 W k ∂2 W k ð4μ r k Þð3μ r k Þ ∂zk ¼ ¼ ∂zj ∂zi ∂zi ∂zj ∂zj μ2 Z zk nμ=r k ð2μ r k Þ=μ μ =r k ðs zk Þ ds
Þ
follows: ds;
η^_ 2 ¼ Ll1 z^ 2 ;
nμ=r k
nμ=r k
μ=r
μ=r
μ=r i þ 1
=∂zi ¼ βk k1 ⋯β i
μ=r
ðμ ri Þ=Þr i
ðμ=r i Þzi
nμ=r k
and ∂2 zk
=∂z2j ¼
ðμ 2r Þ=r
which implies that W k ðz1 ; …; zk Þ is C2 . In a similar way, one has ∂V k 4μ r k 1 ð4μ rk 1 rk Þ=rk rk 1 =rk ¼ zk ðzk γ k Þ; ∂zk rk ∂V k ð4μ r k 1 Þ=r k ð4μ rk 1 Þ=r k 1 ¼ lk 1 ðzk γk Þ; ∂zk 1 ∂V k ð4μ r k 1 Þ=rk ð4μ r k 1 Þ=rk 1 ¼ ðzk γk Þ; ∂η^ k ∂η^ k
2
¼
4. Extension to upper-triangular systems In the preceding discussion, it is assumed that drift and diffusion terms satisfy lower-triangular homogeneous growth conditions. In this section, we can extend Assumption 3.1 to the following upper-triangular form. Assumption 4.1. There exist two constants a01 Z 0 and a02 Z0 such that for i ¼ 1; …; n 1 ! jf i ðx; uÞj r a01
4μ r k 1 ð4μ 2rk 1 Þ=rk 1 γk ; rk 1
∂2 V k 4μ r k 1 ð4μ 2rk 1 Þ=rk 1 2 ¼ lk 1 γk ; rk 1 ∂z2k 1 ∂2 V k 4μ r k 1 ð4μ rk 1 rk Þ=rk ¼ zk ; ∂zk ∂η^ k rk
zi ¼
From the fact μ Z 2 max1 r i r n fr i g, it can be verified that Vk is C . Therefore, U ¼ ∑nk ¼ 1 W k þ∑nk ¼ 2 V k is C2 . 2
Remark 3.3. It should be pointed out that the existence of the solution to a class of stochastic nonlinear systems is a precondition of discussing the finite-time stability. According to [30], it is known that only if at least one coefficient does not satisfy the local Lipschitz condition, it is possible to achieve the finite-time stability in probability for a class of stochastic nonlinear systems. In what follows, we use an example to illustrate the effectiveness of the proposed output feedback controller. Example 3.1. Consider the following stochastic nonlinear system:
4=13 5=11 x2
dx2 ¼ u dt þ 15 x1
sin x1 dω;
10=11
dt þ 14 x2
dω:
ð32Þ
In the simulation, we choose τ ¼ 2=13, r 1 ¼ 1 and μ ¼ 3. By Lemma 2.3, it can be verified that 1 f r jx1 j11=13 ; jf jr 1 ðjx1 j9=13 þ jx2 j9=11 Þ; 1 2 4 4 J g 1 J r 13 jx1 j12=13 ;
∑ jxj j
ð2ri þ τÞ=2r n þ 1
þ juj
ð34Þ
j ¼ iþ2
Proof. By introducing the coordinates
∂2 V k 4μ r k 1 ð4μ 2rk 1 Þ=rk 1 ¼ lk 1 γk : ∂zk 1 ∂η^ k rk 1
12=13
! ð2r i þ τ Þ=2rj
Theorem 4.1. Under Assumption 4.1, the problem of finite-time stabilization in probability for system (1) can be solved via output feedback.
∂2 V k 4μ r k 1 ð4μ rk 1 rk Þ=rk ¼ lk 1 zk ; ∂zk ∂zk 1 rk
dt þ 13 x1
nþ1
Remark 4.1. Without diffusion terms, Assumption 4.1 reduces to the one imposed in [14], where an output feedback controller has been designed to make the closed-loop system globally finite-time stable. By appropriately selecting the scaling gain, it will be proved that the proposed controller can still ensure the global finite-time stabilization in probability of system (1).
ð4μ r k 1 Þð4μ r k 1 r k Þ ð4μ 2rk rk 1 Þ=rk zk γk; r 2k
11=13
∑ jxj jðri þ τÞ=rj þ jujðri þ τÞ=rn þ 1 ;
j ¼ iþ2
where xn þ 1 ¼ 0, f n ðx; uÞ ¼ 0, g n ðx; uÞ ¼ 0 and ri's are defined in (6).
∂ V k ð4μ r k 1 Þð4μ r k Þ ð4μ 2rk Þ=rk ¼ zk r 2k ∂z2k
dx1 ¼ x2 dt þ 14 x1
nþ1
J g i ðx; uÞ J r a02
2
ð33Þ
where l1 ¼3.5, β 1 ¼ 2, β2 ¼ 2:6 and L ¼1.8. With initial values x1 ð0Þ ¼ 1, x2 ð0Þ ¼ 0:8 and η^ 2 ð0Þ ¼ 0, Fig. 1 demonstrates the effectiveness of the control scheme.
i i β k k1 ⋯βi i þ 1 ðμðμ r i Þ=r 2i Þzi . From the definition μ Z 2 max1 r i r n fr i g, one has ð2μ ri Þ=μ Z0 and ðμ 2ri Þ=ri Z 0,
∂2 V k
z^ 2 ¼ ðη^ 2 þ l1 z1 Þ11=13 ;
39=11 39=11 3 9=39 u ¼ L2 β2 ðz^ 2 þ β1 z1 Þ
∂zk ∂zi
znk
with ∂zk
713
J g 2 J r 13 jx2 j10=11
satisfy Assumption 3.1 with a1 ¼ 14 and a2 ¼ 13. Therefore, according to Theorem 3.1, there exists an output feedback controller rendering the closed-loop system finite-time stable in probability. Specifically, the output feedback controller can be constructed as
xi
εi 1
; i ¼ 1; …; n;
v¼
u
εn
ð35Þ
and constructing the scaled observer
η^_ k ¼ εlk 1 z^ k ;
z^ k ¼ ðη^ k þ lk 1 z^ k 1 Þrk =rk 1 ; k ¼ 2; …; n
ð36Þ
the closed-loop system can be represented in the following form: T
dZ ¼ ðεEðZÞ þ F ðZÞÞ dt þ G ðZÞ dω
ð37Þ
where 0 o ε o 1 is a constant to be determined later, F ðZÞ ¼ ðf 1 ðÞ; f 2 ðÞ=ε; …; f n 1 ðÞ=εn 2 ; 0; …; 0ÞT , GðZÞ ¼ ðg 1 ðÞ; g 2 ðÞ=ε; …; g n 1 ðÞ=εn 2 ; 0; …; 0Þ and E(Z) is defined in (15). From Assumption 4.1 and coordinates (35), one has, for i ¼ 1; …; n1 f i ðÞ a01 iþ1 ðr i þ τÞ=ri þ 2 þ ⋯ þ jεn 1 zn jðri þ τÞ=rn þ jεn vjðri þ τÞ=rn þ 1 Þ εi 1 r εi 1 ðjε zi þ 2 j
r a01 ε1 þ 1=ri þ 2 ðjzi þ 2 jðri þ τÞ=ri þ 2 þ ⋯ þ jzn jðri þ τÞ=rn þ jvjðri þ τÞ=rn þ 1 Þ r a 01 ε1 þ 1=ri þ 2 ‖Z‖rΔi þ τ under which n1 ∂UðZÞ 1 þ 1=r i þ 2 ρ3i ‖Z‖4Δμ þ τ r ρ3 ε1 þ 1=r3 ‖Z‖4Δμ þ τ ∂Z F ðZÞ r ∑ ε i¼1
ð38Þ
ð39Þ
where a 01 , ρ3i and ρ3 are positive constants. Following the same way, for i ¼ 1; …; n 1, g i ðÞ r þ τ =2 0 1=2 þ 3=2r i þ 2 ‖Z‖Δi ð40Þ i 1 r a2ε
ε
714
W. Zha et al. / ISA Transactions 53 (2014) 709–716
2
6 4
1
2 0
0
−1
−2 −4
−2 −3
−6 0
2
4
6
8
−8
0
2
4 t/s
6
8
2
4 t/s
6
8
t/s 8
30
6
20
4
10
2
0
0
−10
−2
−20
−4
0
2
4
6
8
−30
0
t/s
Fig. 1. The responses of the closed-loop system (32) and (33).
with a 02 Z 0. Then
1 ∂2 UðZÞ T 4μ þ τ Tr GðZÞ G ðZÞ r ρ4 ε1 þ 3=r3 ‖Z‖Δ 2 ∂Z 2
EUðZ m ðτ0m 4 TÞÞ ¼ UðZ 0 Þ þ E
4μ þ τ
r ðεα1 ρ3 ε1 þ 1=r3 ρ4 ε1 þ 3=r3 Þ‖Z‖Δ 4μ þ τ
:
ð42Þ
By choosing a small enough ε, there is a positive constant α4 satisfying LUðZÞ r α4 U ð4μ þ τÞ=4μ . In what follows, we will discuss the existence of the solution to the closed-loop system (37). Since τ A ð 1=n; 0Þ, fi and gi satisfy higher-order growth conditions, which implies that the linear growth condition does not hold for them. Consequently, for m Z 1, define the following truncation functions: 8 þ FðZÞ if J Z J rm; > < εEðZÞ ð43Þ ϕm ðZÞ ¼ εE m Z þ F m Z if J Z J 4m; > : JZ J JZ J
GTm ðZÞ ¼
8 T > ðZÞ : G mJZ J
ð44Þ
It can be verified that ϕm ðZÞ and satisfy the linear growth condition. Based on Lemma 2.1, 8 T 4 0 and 8 t A ½0; T, there exists a continuous solution Zm(t) with the initial value Z0. Then, applying Dynkin's formula, one has GTm ðZÞ
LUðZ m ðsÞÞ ds r UðZ 0 Þ o þ 1 ð45Þ
under which Pðτ0m r TÞ
UðZ m Þ r EUðZ m ðτ0m 4 TÞÞ rUðZ 0 Þ;
inf
J Zm J ¼ m
ð46Þ
τ0m ¼ inf ft; J Z m ðt; Z 0 Þ J Zmg.
Note that limm- þ 1 inf þ 1, which indicates that limm- þ 1 Pðτ0m rTÞ ¼ 0, 8 T 4 0. By letting ZðtÞ ¼ Z m ðtÞ, for t A ½0; τ0m Þ, since τ0m - þ 1 almost surely as m- þ 1, fZðtÞgt Z 0 is the continuous solution of (37). Furthermore, it can be proved that Z(t) remains zero after the hitting time τZ 0 almost surely, which is similar to step 3 in the proof of Theorem 3.1 and therefore is omitted here. □
where
J Z m J ¼ m Uð J Z m J Þ ¼
To make this point clearer, we utilize the following example to show the explicit construction of output feedback controller for upper-triangular cases. Example 4.1. Consider a class of three-order upper-triangular stochastic nonlinear system 4=5
8=5
dx1 ¼ x2 dt þ 14 x3 u dt þ 12 x3 dx2 ¼ x3 dt þ 13 u5=3
if J Z J r m; if J Z J 4 m:
0
ð41Þ
for ρ4 Z0. Hence, the infinitesimal generator L of U along (37) is given by
∂UðZÞ ∂UðZÞ 1 ∂2 UðZÞ T LUðZÞ r εEðZÞ þ F ðZÞ þ Tr GðZÞ G ðZÞ 2 ∂Z ∂Z 2 ∂Z r εðα1 ρ3 ε1=r3 ρ4 ε3=r3 Þ‖Z‖Δ
Z τ0m 4 T
dt þ 12 u2
dω;
dω;
dx3 ¼ u dt; y ¼ x1 :
ð47Þ
r 1 ¼ 1, μ ¼ which implies and r 3 ¼ 59 . Choose τ ¼ According to Lemma 2.3, it is easy to verify that nonlinearities satisfy Assumption 4.1 with a01 ¼ 13 and a02 ¼ 12 . Therefore, by Theorem 4.1, one can obtain the homogeneous observer and 29 ,
7 3,
r 2 ¼ 79
W. Zha et al. / ISA Transactions 53 (2014) 709–716 3
715
control scheme is applicable to both lower-triangular and uppertriangular systems.
2 1
Acknowledgments
0 −1 −2
0
10
20
30
40
50
60
70
80
Time(Sec) 10 5
This work was supported in part by National Natural Science Foundation of China (61104068, 61273119), Research Fund for the Doctoral Program of Higher Education of China (20090092120027, 20110092110021), Natural Science Foundation of Jiangsu Province (BK2010200), China Postdoctoral Science Foundation Funded Project (2012M511176) and the Fundamental Research Funds for the Central Universities.
0
References
−5 −10 −15
0
10
20
30
40
50
60
70
80
Time(Sec)
Fig. 2. State trajectories of the closed-loop system (47) and (48).
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3
0
10
20
30
40
50
60
70
80
Fig. 3. The input curve of the closed-loop system (47) and (48).
controller
η^_ 2 ¼ εl1 z^ 2 ;
z^ 2 ¼ ðη^ 2 þ l1 z1 Þ7=9 ;
η^_ 3 ¼ εl2 z^ 3 ;
z^ 3 ¼ ðη^ 3 þ l2 z^ 2 Þ5=7 ;
u ¼ ε3 β 3 ðz^ 3
21=5
þ β2
21=5
ðz^ 2 þ β1 z1 ÞÞ1=7 3
3 7=3
ð48Þ
with l1 ¼ 3.2, l2 ¼ 5.2, β1 ¼ 0:9283, β2 ¼ 1:0834 and ε ¼ 0:38. Simulation results are shown in Figs. 2 and 3 with initial conditions x1 ð0Þ ¼ 1:5, x2 ð0Þ ¼ 0:5, x3 ð0Þ ¼ 0:2, η^ 2 ð0Þ ¼ 0 and η^ 3 ð0Þ ¼ 0. 5. Conclusion In this paper, the problem of global finite-time stabilization in probability via output feedback has been solved for a class of stochastic nonlinear system. By employing adding one power integrator technique, homogeneous domination approach and stochastic finite-time stability theorem, a systematic design method has been presented to ensure that the closed-loop system is globally finite-time stable in probability. Moreover, the proposed
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Research Article
Finite-time stabilization for a class of stochastic nonlinear systems via output feedback Wenting Zha a, Junyong Zhai a,n, Shumin Fei a, Yunji Wang b a b
Key Laboratory of Measurement and Control of CSE, Ministry of Education, School of Automation, Southeast University, Nanjing, Jiangsu 210096, China Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX 78249, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 18 November 2013 Received in revised form 13 January 2014 Accepted 21 January 2014 Available online 14 February 2014 This paper was recommended for publication by Jeff Pieper
This paper investigates the problem of global finite-time stabilization in probability for a class of stochastic nonlinear systems. The drift and diffusion terms satisfy lower-triangular or upper-triangular homogeneous growth conditions. By adding one power integrator technique, an output feedback controller is first designed for the nominal system without perturbing nonlinearities. Based on homogeneous domination approach and stochastic finite-time stability theorem, it is proved that the solution of the closed-loop system will converge to the origin in finite time and stay at the origin thereafter with probability one. Two simulation examples are presented to illustrate the effectiveness of the proposed design procedure. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Stochastic nonlinear systems Finite-time stability Output feedback Homogeneous domination
1. Introduction In this paper, we consider the problem of finite-time stabilization via output feedback for a class of stochastic nonlinear systems described by dx1 ðtÞ ¼ x2 ðtÞ dt þf 1 ðxðtÞ; uðtÞÞ dt þ g T1 ðxðtÞ; uðtÞÞ dωðtÞ; dx2 ðtÞ ¼ x3 ðtÞ dt þf 2 ðxðtÞ; uðtÞÞ dt þ g T2 ðxðtÞ; uðtÞÞ dωðtÞ; ⋮ dxn ðtÞ ¼ uðtÞ dt þ f n ðxðtÞ; uðtÞÞ dt þ g Tn ðxðtÞ; uðtÞÞ dωðtÞ; yðtÞ ¼ x1 ðtÞ T
ð1Þ
where xðtÞ ¼ ðx1 ðtÞ; …; xn ðtÞÞ A Rn , uðtÞ A R and yðtÞ A R are the system states, control input and output, respectively. ωðtÞ is an r-dimensional standard Wiener process defined on a probability space ðΩ; ϝ;ϝt ; PÞ with Ω being a sample space, ϝ being a s-field, ϝt being a filtration and P being a probability measure. The drift terms f i : Rn R-R and the diffusion terms g i : Rn R-Rr , i ¼ 1; …; n, are Borel measurable, continuous in system states and satisfy f i ð0; 0Þ ¼ 0 and g i ð0; 0Þ ¼ 0. In the nonlinear control community, finite-time stabilization is one of the most fundamental and challenging problems. In contrast to the commonly used notion of asymptotic stability, finite-time stability requires essentially that a control system should be stable in the sense of Lyapunov and its trajectories tend n
Corresponding author. E-mail address: [email protected] (J. Zhai).
to zero in finite time. It was demonstrated in [1] that finite-time stable systems might have not only faster convergence but also better robustness and disturbance rejection properties. The work [2] provided a solid foundation for finite-time stability theory of continuous autonomous systems, which gave a judging criterion on finite-time stability. Then, some conditions for finite-time stability have been presented for continuous systems [3] and non-autonomous continuous systems [4]. In the literature, several results on finite-time state feedback stabilization have been achieved in [5–7] and the references therein. However, since finite-time stabilizers are generally not smooth, their design methods are sophisticated, especially when some states are not measurable. Based on a “finite-time separation principle”, global finite-time stabilization via output feedback can be achieved for the double integrator system in [8]. The work [9] has developed a novel systematic design method, namely homogeneous domination approach, which provides us a new perspective to deal with the output feedback control problem for nonlinear systems and leads to several stabilization results [10–12]. By coupling the homogeneous domination approach and finite-time stabilization technique, an output feedback controller was constructed to global stabilize a class of lower-triangular nonlinear systems [13] and upper-triangular ones [14]. Moreover, with unknown output gain, the problem of global finite-time stabilization has been addressed in [15,16]. In spite of these developments, the above-mentioned results cannot be generalized easily to a class of stochastic nonlinear systems. It is well known that stochastic modeling has come to play an important role in many branches of science and industry.
0019-0578/$ - see front matter & 2014 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2014.01.005
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W. Zha et al. / ISA Transactions 53 (2014) 709–716
Florchinger extended the concept of control Lyapunov functions and Sontag's stabilization formula to stochastic setting in [17], which leads to more stabilization results [18–23] and the references therein. However, each of them described the asymptotic behavior of trajectories for a class of stochastic nonlinear systems as time tends to infinity. Recently, the work [24] has presented the concept of finite-time stability in probability for stochastic systems and has proved the stochastic finite-time stability theorem. Subsequently, for a class of stochastic nonlinear systems in strictfeedback form, the work [25] designed a continuous statefeedback controller to guarantee the global finite-time stability in probability and our recent work [26] solved the finite-time stabilization problem by dynamic state-feedback. However, only state feedback was considered, which requires all the system states to be measurable. Immediately, one may ask the following interesting questions: Is it possible to relax the growth conditions for nonlinear functions? Under these weaker conditions, how can one design an output feedback controller to make (1) globally finite-time stable in probability? Motivated by the design of finite-time stabilizer in deterministic cases [13,14], and stochastic finite-time stability theorem proposed in [24], we aim to solve the problem of global finitetime stabilization for a class of stochastic nonlinear systems via output feedback. In order to settle this problem, we first design a homogeneous output feedback controller for the nominal system. Then, a scaling gain is introduced to the controller to dominate the perturbing nonlinearities. By appropriately choosing the scaling gain, the closed-loop system can be rendered globally finite-time stable in probability. Furthermore, we extend the result to a class of upper-triangular stochastic nonlinear systems. The main contributions of this paper are as follows: (i) Compared with deterministic cases [13,14], this paper extends the global finite-time stabilization results to a class of stochastic nonlinear systems according to stochastic finite-time stability theorem. (ii) The uncertain nonlinearities are functions of both measurable and unmeasurable states. Based on the homogeneous observer construction, an output feedback controller guarantees the closed-loop system finite-time stable in probability. Notations: R þ denotes the set of all nonnegative real numbers, þ and Rn denotes the real n-dimensional space. Rodd ≕fq A R : q Z 0 is a ratio of two odd integersg. For a given vector or matrix X, XT represents its transpose; TrfXg represents its trace when X is square; J J denotes the Euclidean norm of a vector X or the Frobenius norm of a matrix X. Ci denotes the set of all functions with continuous ith partial derivatives; K denotes the set of all functions, R þ -R þ , which are continuous, strictly increasing and vanishing at zero; K1 denotes the set of all functions which are of class K and unbounded; a 4 b means the minimum of a and b.
In this section, we present some useful definitions and lemmas which play very important roles in this paper. Consider the following stochastic nonlinear system: xð0Þ ¼ x0 A Rn
ð2Þ
where xðtÞ A R is the system state and ωðtÞ is an r-dimensional standard Wiener process defined on a probability space ðΩ; ϝ;ϝt ; PÞ. The Borel measurable functions f : Rn -Rn and g T : Rn -Rnr are continuous in x that satisfy f ð0Þ ¼ 0 and gð0Þ ¼ 0. n
‖f ðxðtÞÞ‖2 þ ‖gðxðtÞÞ‖2 r Kð1 þ ‖xðtÞ‖2 Þ
ð3Þ
for K 4 0. Then given any x0 independent of ωðtÞ, (2) has a continuous solution with probability one. Definition 2.1 (Khoo et al. [25]). The trivial solution of (2) is said to be finite-time stable in probability if the solution exists for any initial value x0 A Rn , denoted by xðt; x0 Þ. Moreover, the following statements hold: (i) Finite-time attractiveness in probability: For every initial value x0 A Rn \f0g, the first hitting time τx0 ¼ inf ft; xðt; x0 Þ ¼ 0g, which is called the stochastic settling time, is finite almost surely, that is, Pfτx0 o 1g ¼ 1. (ii) Stability in probability: For every pair of ɛ A ð0; 1Þ and r 4 0, there exists a δ ¼ δðɛ; rÞ 4 0 such that Pf J xðt; x0 Þ J or; 8 t Z0g Z 1 ɛ, whenever J x0 J o δ. (iii) The solution xððt þ τx0 Þ; x0 Þ is unique for t Z 0. Definition 2.2 (Florchinger [17]). For any given VðxðtÞÞ A C2 associated with stochastic system (2), theinfinitesimal generator L is defined as LVðxÞ ¼ ð∂V=∂xÞf ðxÞ þ 12 Tr gðxÞð∂2 V=∂x2 Þg T ðxÞ , where 1 2 2 T 2 Tr gðxÞð∂ V =∂x Þg ðxÞ is called as the Hessian term of L. Lemma 2.2 (Khoo et al. [25]). For system (2), if there exist a Lyapunov function V : Rn -R þ , K1 class functions μ1 and μ2, positive real numbers c 4 0 and 0 o γ o1, such that for all x A Rn and t Z 0,
μ1 ð J x J Þ rVðxÞ r μ2 ð J x J Þ;
ð4Þ
LVðxÞ r c ðVðxÞÞγ
ð5Þ
then the trivial solution of (2) is finite-time attractive and stable in probability. Definition 2.3 (Kawski [28]). For real numbers r i 4 0, i ¼ 1; …; n, and fixed coordinates ðx1 ; …; xn Þ A Rn , 8 ɛ 4 0.
the dilation Δɛ ðxÞ is defined by Δɛ ðxÞ ¼ ðɛr x1 ; …; ɛr xn Þ, 8 ɛ 4 0, 1
n
with ri being called as the weights of the coordinates. For simplicity of notation, we define dilation weight Δ ¼ ðr 1 ; …; r n Þ. a function V A CðRn ; RÞ is said to be homogeneous of degree τ if there is a real number τ Z 0 such that 8 x A Rn \f0g, VðΔɛ ðxÞÞ ¼ ɛ τ V ðx1 ; …; xn Þ. a vector field f A CðRn ; Rn Þ is said to be homogeneous of degree τ if there is a real number τ Z min1 r i r n fri g such that for i ¼ 1; …; n, 8 x A Rn \f0g, f i ðΔɛ ðxÞÞ ¼ ɛ τ þ ri f i ðx1 ; …; xn Þ. a homogeneous p-norm is defined as ‖x‖Δ;p ¼ ð∑ni¼ 1 jxi jp=ri Þ1=p , 8 x A Rn , for a constant p Z 1. For simplicity, we choose p ¼2 and write ‖x‖Δ for ‖x‖Δ;2 .
Lemma 2.3. Suppose c and d are two positive real numbers. Given any positive number γ 4 0, the following inequality holds:
2. Preliminary results
dxðtÞ ¼ f ðxðtÞÞ dt þg T ðxðtÞÞ dωðtÞ;
Lemma 2.1 (Skorokhod [27]). Suppose that f ðxðtÞÞ and gðxðtÞÞ are continuous with respect to their variables and satisfy the linear growth condition:
jxjc jyjd r
c d c cþd γ jxjc þ d þ γ d jyj : cþd cþd
Lemma 2.4. For x A R, yA R, and p Z1, the following inequalities hold: jx þ yjp r 2p 1 jxp þ yp j; ðjxj þjyjÞ1=p r jxj1=p þ jyj1=p r 2ðp 1Þ=p ðjxj þ jyjÞ1=p :
W. Zha et al. / ISA Transactions 53 (2014) 709–716
with γ k ¼ η^ k þ lk 1 zk 1 , and a set of virtual controllers zn1 ; …; znn defined by
If p Z 1 is an odd integer or a ratio of two odd integers, jx yjp r 2p 1 jxp yp j;
jx1=p y1=p jr 21 1=p jx yj1=p :
ξ1 ¼ zμ1 =r1 zn1μ=r1 ; r i =μ ξi ¼ zμi =ri zni μ=ri ; i ¼ 2; …; n zni ¼ β i 1 ξi 1; zn1 ¼ 0;
3. Main results The objective of this section is to design an output feedback controller for system (1) under the following assumption, which renders the closed-loop system finite-time stable in probability. Assumption 3.1. There exist two positive constants a1 and a2 such that for i ¼ 1; …; n jf i ðx; uÞj r a1 ðjx1 jðri þ τÞ=r1 þ⋯ þ jxi jðri þ τÞ=ri Þ; J g i ðx; uÞ J ra2 ðjx1 jð2ri þ τÞ=2r1 þ ⋯ þ jxi jð2ri þ τÞ=2ri Þ with τ A ð 1=n; 0Þ and a series of parameters r i ¼ ði 1Þτ þ 1;
i ¼ 1; …; n þ 1:
711
ð6Þ
For simplicity, we assume τ ¼ p=q with p being an even integer and q being an odd integer. Based on this, ri will be odd in both denominator and numerator. Remark 3.1. In deterministic cases, the works [13,15] have dealt with the finite-time output feedback stabilization problem for a class of nonlinear systems whose nonlinearities satisfy Assumption 3.1 with a2 ¼ 0. Taking stochastic factors into consideration, we impose lower-triangular homogeneous growth conditions on diffusion terms as well, and therefore, Assumption 3.1 is a more general condition. However, the appearance of Hessian term will produce more nonlinear terms. In what follows, we will present how to handle these terms skillfully.
ð11Þ
one can obtain the global finite-time stabilization result for (7), whose proof is similar to the one introduced in [22] with some modifications. For the sake of space, the detailed proof is omitted here. □ From the construction of U, it can be verified that U is positive definite and proper with respect to Z≔ðz1 ; …; zn ; η^ 2 ; …; η^ n ÞT . First, we consider the following two cases to prove that Wk is positive definite and proper. þ Case 1: If znk r zk , with μ Z 2 max1 r i r n fr i g, μ A Rodd and by Lemma 2.4, one gets Z zk nμ=r Wk ¼ ðsμ=rk zk k Þð4μ rk Þ=μ dsZ ð21 μ=rk Þð4μ rk Þ=μ znk
Z
zk znk
ðs znk Þð4μ rk Þ=rk ds ¼ ð21 μ=rk Þð4μ rk Þ=μ
rk ðz znk Þ4μ=rk : 4μ k
ð12Þ
Case 2: If znk Zzk , (12) can be proved similarly. Next, we aim to prove that Vk is positive definite and proper. With s ¼ srk 1 =ð4μ rk 1 Þ, there exists a positive constant c0 A ½c; c, such that Z Vk ¼
ð4μ r k 1 Þ=r k
zk
ð4μ r k 1 Þ=r k 1
γk
ðsrk 1 =ð4μ rk 1 Þ γ k Þ ds
Z rk 1 =rk 4μ r k 1 z k ðs γ k Þsð4μ 2rk 1 Þ=rk 1 ds rk 1 γk Z zrk 1 =rk k 4μ r k 1 4μ r k 1 r =r ¼ c0 ðs γ k Þ ds ¼ c0 ðzkk 1 k γ k Þ2 rk 1 2r k 1 γk
¼
ð13Þ
The design scheme can be divided into two steps: (i) we first devise a finite-time output feedback controller for the nominal system; and (ii) based on homogeneous domination approach, a scaled output feedback controller is constructed for system (1) and the stability analysis is given to guarantee the finite-time stabilization in probability of the closed-loop system. We first explicitly construct an output feedback stabilizer for the following nominal system:
where c and c represent the infimum and supremum of sð4μ 2rk 1 Þ=rk 1 in ½γ k ; zrkk 1 =rk (or ½zrkk 1 =rk ; γ k , if γ k Z zrkk 1 =rk ), respectively. Therefore, U ¼ ∑nk ¼ 1 W k þ ∑nk ¼ 2 V k Z ∑nk ¼ 1 ð21 μ=rk Þð4μ rk Þ=μ ðr k = r =r 4μÞðzk znk Þ4μ=rk þ ∑nk ¼ 2 ðð4μ r k 1 Þ=2r k 1 Þc0 ðzkk 1 k γ k Þ2 is definite positive and proper with respect to Z. Denoting the dilation weight
dzi ¼ zi þ 1 dt; i ¼ 1; …; n 1;
Δ ¼ ðr 1 ; …; r n ; r1 ; …; rn 1 Þ
T
dzn ¼ v dt;
y ¼ z1 :
ð7Þ
n
where z ¼ ðz1 ; …; zn Þ A R , v A R, yA R are the states, control input and output, respectively. Lemma 3.1. For system (7), there exist positive constants β1 ; …; β n , þ l1 ; …; ln 1 , and μ Z 2 max1 r i r n fr i g, μ A Rodd , such that the following homogeneous output feedback stabilizer
η^ k ¼ lk 1 z^ k ;
z^ k ¼ ðη^ k þ lk 1 z^ k 1 Þrk =ðrk 1 Þ ; k ¼ 2; …; n;
ð8Þ
for z1 ;…;zn
for η^ 2 ;…;η^ n
the closed-loop system (7)–(9) which can be rewritten as dZ ¼ EðZÞ dt ¼ ðz2 ; …; zn ; v; η^_ 2 ; …; η^_ n ÞT dt
v ¼ βn ðz^ n
μ=r μ=r μ=r μ=r μ=r μ þ β n ðz^ n 1 þ ⋯ þ β 3 ðz^ 2 þ β 2 z Þ⋯ÞÞrn þ 1 =μ n1
n1
2
2
1
ɛ rk znk
ð9Þ ¼
k¼1 n
¼ ∑
k¼1 n
þ ∑
k¼2
Z
zk znk
nμ=r k ð4μ r k Þ=μ
ðsμ=rk zk
ð4μ r k 1 Þ=r k 1
γk
γk
Þ
ds
ðsrk 1 =ð4μ rk 1 Þ γ k Þ ds
ðɛr k 1 λ2k 1 r
=ð4μ r k 1 Þ
dλ1 ɛrk 1 γ k Þɛ4μ rk 1 dλ2
ð16Þ
is homogeneous of degree 4μ with respect to Δ, with λ1 ¼ ɛ rk s and λ2 ¼ ɛrk 1 4μ s. By Lemma 1 in [29], there exists a positive constant α1, such that
ð4μ r k 1 Þ=r k
zk
ð4μ r k 1 Þ=rk
zk
ð4μ rk 1 Þ=r k 1
¼ ɛ4μ UðZÞ
k¼2
Z
Z
k¼2
n
ðsrk 1 =ð4μ rk 1 Þ ɛrk 1 γ k Þ ds
λ
n
U ¼ ∑ Wk þ ∑ Vk
ðɛ r k zk Þð4μ rk 1 Þ=rk
ð4μ r k 1 Þ=r k 1 r k ¼ 2 ðɛ k 1 γ k Þ Z zk n μ=r nμ=r ∑ ðɛμ 1 k ɛμ zk k Þð4μ rk Þ=μ ɛ rk k ¼ 1 znk
þ ∑
Proof. By choosing a Lyapunov function U n
Z
n
þ ∑
1
with z^ 1 ¼ z1 , renders the closed-loop system (7)–(9) globally finitetime stable.
ð15Þ
is homogeneous of degree τ. Moreover, it can be shown that Z ɛ r k zk n nμ=r UðΔɛ ðZÞÞ ¼ ∑ ðsμ=rk ɛμ zk k Þð4μ rk Þ=μ ds k¼1
μ=r n
ð14Þ
|fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
ð10Þ
∂UðZÞ 4μ þ τ EðZÞ r α1 ‖Z‖Δ : ∂Z
ð17Þ
712
W. Zha et al. / ISA Transactions 53 (2014) 709–716
Under the new coordinates zi ¼
xi Li 1
; i ¼ 1; …; n;
u v¼ n L
ð18Þ
with L Z 1 being a constant to be determined later, system (1) can be rewritten as g T ðÞ f ðÞ dzi ¼ Lzi þ 1 þ ii 1 dt þ ii 1 dω; L L f n ðÞ g Tn ðÞ dzn ¼ Lv þ n 1 dt þ n 1 dω; L L y ¼ z1 ;
i ¼ 1; …; n 1:
ð19Þ
Next, we construct an observer with the scaling gain L
η^_ k ¼ Llk 1 z^ k ;
z^ k ¼ ðη^ k þ lk 1 z^ k 1 Þrk =rk 1 ; k ¼ 2; …; n
ð20Þ
where li, i ¼ 1; …; n 1, are observer gains. With E(Z) defined in (15), it is straightforward to verify that the closed-loop system (19), (20) and (9) can be represented as dZ ¼ ðLEðZÞ þ FðZÞÞ dt þ G ðZÞ dω T
ð21Þ
where FðZÞ ¼ ðf 1 ðÞ; f 2 ðÞ=L; …; f n ðÞ=Ln 1 ; 0; …; 0ÞT and GðZÞ ¼ ðg 1 ðÞ; g 2 ðÞ=L; …; g n ðÞ=Ln 1 ; 0; …; 0Þ. Based on the observer and controller designs, the following theorem gives the finite-time stabilization result of the closedloop system (21) via output feedback. Theorem 3.1. Under Assumption 3.1, there exists an output feedback controller rendering the system (1) finite-time stable in probability. Proof.. According to Definition 2.1, Theorem 3.1 will be proved by four steps. Step 1: We consider the existence of the solution to the closedloop system (21). The construction of observer and controller indicates that the closed-loop system is continuous with respect to its variables. Since τ A ð 1=n; 0Þ, fi and gi satisfy lower-order growth conditions. In addition, if J Z J Z 1, with 0 oðr i þ τÞ=r j o 1, j ¼ 1; …; i, there exists a positive constant K1 such that ‖FðZÞ‖2 r K 1 ‖Z‖2 . If J Z J o 1, one has ‖FðZÞ‖2 rK 2 , with K 2 Z 0. Therefore, by choosing K 3 ¼ maxfK 1 ; K 2 g, one has ‖FðZÞ‖2 r K 3 ð1 þ ‖Z‖2 Þ, which implies that F(Z) satisfies the linear growth condition in Lemma 2.1. Similarly, one can obtain that there is a positive constant K such that ‖LEðZÞ þ FðZÞ‖2 þ‖GðZÞ‖2 r Kð1 þ‖Z‖2 Þ:
ð22Þ
By Lemma 2.1, there exists a continuous solution Z(t) that can be written as Z t Z t ðLEðZðsÞÞ þ FðZðsÞÞÞ ds þ GT ðZðsÞÞ dωðsÞ ð23Þ ZðtÞ ¼ Z 0 þ 0
0
with the initial value Z0. Step 2: Under Assumption 3.1 and new coordinates (18), it can be deduced that, for i ¼ 1; …; n f i ðÞ a1 ðr i þ τ Þ=r 1 þ jLz2 jðri þ τÞ=r2 þ ⋯ þ jLi 1 zi jðri þ τÞ=ri Þ i 1 r i 1 ðjz1 j L L 1 1=r i
r a1 L
ðr i þ τÞ=r 1
ðjz1 j
ri þ τ r a 1 L1 1=ri ‖Z‖Δ
þ ⋯ þ jzi j
ðri þ τÞ=r i
Þ ð24Þ
with a constant a 1 Z 0. Recall that for i ¼ 1; …; n, ∂UðZÞ=∂Z i is homogeneous of degree 4μ r i , then n ∂UðZÞ f ðÞ n ∂UðZÞ 1 1=r i i ρ1i ‖Z‖4Δμ þ τ r ρ1 ‖Z‖4Δμ þ τ ∂Z FðZÞ r ∑ ∂Z i 1 r ∑ L i L i¼1 i¼1 ð25Þ
where ρ1i, i ¼ 1; …; n, and way, for i ¼ 1; …; n,
ρ1 are positive constants. In a similar
gi ðÞ r þ τ=2 1=2 1=2r i ðjz1 jð2ri þ τÞ=2r1 þ ⋯ þ jzi jð2ri þ τÞ=2ri Þ r a 2 L1=2 1=2ri ‖Z‖Δi i 1 r a2 L L
ð26Þ
with a 2 Z 0, which indicates
T
g ðÞ 1 ∂2 UðZÞ T 1 pffiffiffi n ∂2 UðZÞ g i ðÞ j r ρ ‖Z‖4μ þ τ G ðZÞ r r r ∑ Tr GðZÞ 2 Δ 2 i 1 j 1 2 2 ∂Z L L i;j ¼ 1 ∂Z i ∂Z j
ð27Þ for ρ2 Z 0. According to Definition 2.2, the infinitesimal generator L of U along the trajectory of (21) is
∂UðZÞ ∂UðZÞ 1 ∂2 UðZÞ T LEðZÞ þ FðZÞ þ Tr GðZÞ G ðZÞ LUðZÞ r 2 ∂Z ∂Z 2 ∂Z 4μ þ τ r ðLα1 ρ1 ρ2 Þ‖Z‖Δ : ð28Þ By choosing L 4 maxf1; ðρ1 þ ρ2 Þ=α1 g, there exist positive constants α2 and α3, such that 4μ þ τ
LUðZÞ r α2 ‖Z‖Δ
r α3 U ð4μ þ τÞ=4μ :
ð29Þ
By Lemma 2.2, it can be obtained that the solution of the closedloop system (21) is finite-time attractive and stable in probability. Step 3: In this step, we will prove that after the first hitting time τZ0 , the solution Zðt þ τZ0 Þ remains zero almost surely, 8 t Z 0. Define the stopping time τm ¼ inf ft Z τ Z 0 ; J Zðt; Z 0 Þ J Z m; m 4 0g. It is clear that τm is an increasing time sequence. Applying Itô's formula, one has 8 t Z 0 Z ðt þ τZ Þ 4 τm 0 UðZððt þ τZ 0 Þ 4 τm ÞÞ ¼ UðZðτZ 0 4 τm ÞÞ þ LUðZðsÞÞ ds Z þ
ðt þ τZ 0 Þ 4 τ m
τZ 0 4 τm
τZ 0 4 τm
∂UðZðsÞÞ T G ðZðsÞÞ dωðsÞ ∂Z
ð30Þ
which indicates EUðZððt þ τZ 0 Þ 4 τm ÞÞ ¼ EUðZðτZ 0 ÞÞ þE
Z
ðt þ τ Z 0 Þ 4 τm
τZ0
LUðZðsÞÞ ds r 0: ð31Þ
Since U(Z) is positive definite, one can obtain EUðZððt þ τZ 0 Þ 4 τm ÞÞ ¼ 0, which implies that UðZððt þ τZ 0 Þ 4 τ m ÞÞ ¼ 0 almost surely, 8 t Z 0. Letting m- þ 1, since τm - þ 1 almost surely as m- þ1, we get Zðt þ τZ 0 Þ ¼ 0 almost surely, 8 t Z 0. Therefore, the result of finite-time stabilization in probability for the closed-loop system (21) is achieved according to Definition 2.1. Step 4: Since coordinate transformation (18) does not change the properties of the system, then the solution of stochastic nonlinear system (1) is finite-time stable in probability. □ Remark 3.2. In the deterministic work, a C1 Lyapunov function is enough for the design and analysis of the controller. However, for stochastic nonlinear systems, the Hessian term requires the Lyapunov function to be C2 . First, for i; j ¼ 1; …; k 1, ∂W k ð4μ rk Þ=μ ¼ ξk ; ∂zk
nμ=r k
∂W k 4μ r k ∂zk ¼ ∂zi μ ∂zi
Z
zk n
zk
nμ=r k ð3μ rk Þ=μ
ðsμ=rk zk
Þ
ds;
∂2 W k 4μ r k ðμ rk Þ=rk ð3μ rk Þ=μ ¼ zk ξk ; rk ∂z2k nμ=r
∂2 W k ∂2 W k 4μ r k ð3μ rk Þ=μ ∂zk k ξk ¼ ¼ ; ∂zk ∂zi ∂zi ∂zk μ ∂zi nμ=r Z ∂2 W k 4μ r k ∂2 zk k zk μ=rk nμ=r ¼ ðs zk k Þð3μ rk Þ=μ ds; 2 μ ∂zi ∂z2i znk
W. Zha et al. / ISA Transactions 53 (2014) 709–716 nμ=r k
þ
ð4μ r k Þð3μ r k Þ ∂zk ∂zi μ2
!2 Z
zk
znk
nμ=r k ð2μ r k Þ=μ
ðsμ=rk zk
nμ=r k
∂2 W k ∂2 W k ð4μ r k Þð3μ r k Þ ∂zk ¼ ¼ ∂zj ∂zi ∂zi ∂zj ∂zj μ2 Z zk nμ=r k ð2μ r k Þ=μ μ =r k ðs zk Þ ds
Þ
follows: ds;
η^_ 2 ¼ Ll1 z^ 2 ;
nμ=r k
nμ=r k
μ=r
μ=r
μ=r i þ 1
=∂zi ¼ βk k1 ⋯β i
μ=r
ðμ ri Þ=Þr i
ðμ=r i Þzi
nμ=r k
and ∂2 zk
=∂z2j ¼
ðμ 2r Þ=r
which implies that W k ðz1 ; …; zk Þ is C2 . In a similar way, one has ∂V k 4μ r k 1 ð4μ rk 1 rk Þ=rk rk 1 =rk ¼ zk ðzk γ k Þ; ∂zk rk ∂V k ð4μ r k 1 Þ=r k ð4μ rk 1 Þ=r k 1 ¼ lk 1 ðzk γk Þ; ∂zk 1 ∂V k ð4μ r k 1 Þ=rk ð4μ r k 1 Þ=rk 1 ¼ ðzk γk Þ; ∂η^ k ∂η^ k
2
¼
4. Extension to upper-triangular systems In the preceding discussion, it is assumed that drift and diffusion terms satisfy lower-triangular homogeneous growth conditions. In this section, we can extend Assumption 3.1 to the following upper-triangular form. Assumption 4.1. There exist two constants a01 Z 0 and a02 Z0 such that for i ¼ 1; …; n 1 ! jf i ðx; uÞj r a01
4μ r k 1 ð4μ 2rk 1 Þ=rk 1 γk ; rk 1
∂2 V k 4μ r k 1 ð4μ 2rk 1 Þ=rk 1 2 ¼ lk 1 γk ; rk 1 ∂z2k 1 ∂2 V k 4μ r k 1 ð4μ rk 1 rk Þ=rk ¼ zk ; ∂zk ∂η^ k rk
zi ¼
From the fact μ Z 2 max1 r i r n fr i g, it can be verified that Vk is C . Therefore, U ¼ ∑nk ¼ 1 W k þ∑nk ¼ 2 V k is C2 . 2
Remark 3.3. It should be pointed out that the existence of the solution to a class of stochastic nonlinear systems is a precondition of discussing the finite-time stability. According to [30], it is known that only if at least one coefficient does not satisfy the local Lipschitz condition, it is possible to achieve the finite-time stability in probability for a class of stochastic nonlinear systems. In what follows, we use an example to illustrate the effectiveness of the proposed output feedback controller. Example 3.1. Consider the following stochastic nonlinear system:
4=13 5=11 x2
dx2 ¼ u dt þ 15 x1
sin x1 dω;
10=11
dt þ 14 x2
dω:
ð32Þ
In the simulation, we choose τ ¼ 2=13, r 1 ¼ 1 and μ ¼ 3. By Lemma 2.3, it can be verified that 1 f r jx1 j11=13 ; jf jr 1 ðjx1 j9=13 þ jx2 j9=11 Þ; 1 2 4 4 J g 1 J r 13 jx1 j12=13 ;
∑ jxj j
ð2ri þ τÞ=2r n þ 1
þ juj
ð34Þ
j ¼ iþ2
Proof. By introducing the coordinates
∂2 V k 4μ r k 1 ð4μ 2rk 1 Þ=rk 1 ¼ lk 1 γk : ∂zk 1 ∂η^ k rk 1
12=13
! ð2r i þ τ Þ=2rj
Theorem 4.1. Under Assumption 4.1, the problem of finite-time stabilization in probability for system (1) can be solved via output feedback.
∂2 V k 4μ r k 1 ð4μ rk 1 rk Þ=rk ¼ lk 1 zk ; ∂zk ∂zk 1 rk
dt þ 13 x1
nþ1
Remark 4.1. Without diffusion terms, Assumption 4.1 reduces to the one imposed in [14], where an output feedback controller has been designed to make the closed-loop system globally finite-time stable. By appropriately selecting the scaling gain, it will be proved that the proposed controller can still ensure the global finite-time stabilization in probability of system (1).
ð4μ r k 1 Þð4μ r k 1 r k Þ ð4μ 2rk rk 1 Þ=rk zk γk; r 2k
11=13
∑ jxj jðri þ τÞ=rj þ jujðri þ τÞ=rn þ 1 ;
j ¼ iþ2
where xn þ 1 ¼ 0, f n ðx; uÞ ¼ 0, g n ðx; uÞ ¼ 0 and ri's are defined in (6).
∂ V k ð4μ r k 1 Þð4μ r k Þ ð4μ 2rk Þ=rk ¼ zk r 2k ∂z2k
dx1 ¼ x2 dt þ 14 x1
nþ1
J g i ðx; uÞ J r a02
2
ð33Þ
where l1 ¼3.5, β 1 ¼ 2, β2 ¼ 2:6 and L ¼1.8. With initial values x1 ð0Þ ¼ 1, x2 ð0Þ ¼ 0:8 and η^ 2 ð0Þ ¼ 0, Fig. 1 demonstrates the effectiveness of the control scheme.
i i β k k1 ⋯βi i þ 1 ðμðμ r i Þ=r 2i Þzi . From the definition μ Z 2 max1 r i r n fr i g, one has ð2μ ri Þ=μ Z0 and ðμ 2ri Þ=ri Z 0,
∂2 V k
z^ 2 ¼ ðη^ 2 þ l1 z1 Þ11=13 ;
39=11 39=11 3 9=39 u ¼ L2 β2 ðz^ 2 þ β1 z1 Þ
∂zk ∂zi
znk
with ∂zk
713
J g 2 J r 13 jx2 j10=11
satisfy Assumption 3.1 with a1 ¼ 14 and a2 ¼ 13. Therefore, according to Theorem 3.1, there exists an output feedback controller rendering the closed-loop system finite-time stable in probability. Specifically, the output feedback controller can be constructed as
xi
εi 1
; i ¼ 1; …; n;
v¼
u
εn
ð35Þ
and constructing the scaled observer
η^_ k ¼ εlk 1 z^ k ;
z^ k ¼ ðη^ k þ lk 1 z^ k 1 Þrk =rk 1 ; k ¼ 2; …; n
ð36Þ
the closed-loop system can be represented in the following form: T
dZ ¼ ðεEðZÞ þ F ðZÞÞ dt þ G ðZÞ dω
ð37Þ
where 0 o ε o 1 is a constant to be determined later, F ðZÞ ¼ ðf 1 ðÞ; f 2 ðÞ=ε; …; f n 1 ðÞ=εn 2 ; 0; …; 0ÞT , GðZÞ ¼ ðg 1 ðÞ; g 2 ðÞ=ε; …; g n 1 ðÞ=εn 2 ; 0; …; 0Þ and E(Z) is defined in (15). From Assumption 4.1 and coordinates (35), one has, for i ¼ 1; …; n1 f i ðÞ a01 iþ1 ðr i þ τÞ=ri þ 2 þ ⋯ þ jεn 1 zn jðri þ τÞ=rn þ jεn vjðri þ τÞ=rn þ 1 Þ εi 1 r εi 1 ðjε zi þ 2 j
r a01 ε1 þ 1=ri þ 2 ðjzi þ 2 jðri þ τÞ=ri þ 2 þ ⋯ þ jzn jðri þ τÞ=rn þ jvjðri þ τÞ=rn þ 1 Þ r a 01 ε1 þ 1=ri þ 2 ‖Z‖rΔi þ τ under which n1 ∂UðZÞ 1 þ 1=r i þ 2 ρ3i ‖Z‖4Δμ þ τ r ρ3 ε1 þ 1=r3 ‖Z‖4Δμ þ τ ∂Z F ðZÞ r ∑ ε i¼1
ð38Þ
ð39Þ
where a 01 , ρ3i and ρ3 are positive constants. Following the same way, for i ¼ 1; …; n 1, g i ðÞ r þ τ =2 0 1=2 þ 3=2r i þ 2 ‖Z‖Δi ð40Þ i 1 r a2ε
ε
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W. Zha et al. / ISA Transactions 53 (2014) 709–716
2
6 4
1
2 0
0
−1
−2 −4
−2 −3
−6 0
2
4
6
8
−8
0
2
4 t/s
6
8
2
4 t/s
6
8
t/s 8
30
6
20
4
10
2
0
0
−10
−2
−20
−4
0
2
4
6
8
−30
0
t/s
Fig. 1. The responses of the closed-loop system (32) and (33).
with a 02 Z 0. Then
1 ∂2 UðZÞ T 4μ þ τ Tr GðZÞ G ðZÞ r ρ4 ε1 þ 3=r3 ‖Z‖Δ 2 ∂Z 2
EUðZ m ðτ0m 4 TÞÞ ¼ UðZ 0 Þ þ E
4μ þ τ
r ðεα1 ρ3 ε1 þ 1=r3 ρ4 ε1 þ 3=r3 Þ‖Z‖Δ 4μ þ τ
:
ð42Þ
By choosing a small enough ε, there is a positive constant α4 satisfying LUðZÞ r α4 U ð4μ þ τÞ=4μ . In what follows, we will discuss the existence of the solution to the closed-loop system (37). Since τ A ð 1=n; 0Þ, fi and gi satisfy higher-order growth conditions, which implies that the linear growth condition does not hold for them. Consequently, for m Z 1, define the following truncation functions: 8 þ FðZÞ if J Z J rm; > < εEðZÞ ð43Þ ϕm ðZÞ ¼ εE m Z þ F m Z if J Z J 4m; > : JZ J JZ J
GTm ðZÞ ¼
8 T > ðZÞ : G mJZ J
ð44Þ
It can be verified that ϕm ðZÞ and satisfy the linear growth condition. Based on Lemma 2.1, 8 T 4 0 and 8 t A ½0; T, there exists a continuous solution Zm(t) with the initial value Z0. Then, applying Dynkin's formula, one has GTm ðZÞ
LUðZ m ðsÞÞ ds r UðZ 0 Þ o þ 1 ð45Þ
under which Pðτ0m r TÞ
UðZ m Þ r EUðZ m ðτ0m 4 TÞÞ rUðZ 0 Þ;
inf
J Zm J ¼ m
ð46Þ
τ0m ¼ inf ft; J Z m ðt; Z 0 Þ J Zmg.
Note that limm- þ 1 inf þ 1, which indicates that limm- þ 1 Pðτ0m rTÞ ¼ 0, 8 T 4 0. By letting ZðtÞ ¼ Z m ðtÞ, for t A ½0; τ0m Þ, since τ0m - þ 1 almost surely as m- þ 1, fZðtÞgt Z 0 is the continuous solution of (37). Furthermore, it can be proved that Z(t) remains zero after the hitting time τZ 0 almost surely, which is similar to step 3 in the proof of Theorem 3.1 and therefore is omitted here. □
where
J Z m J ¼ m Uð J Z m J Þ ¼
To make this point clearer, we utilize the following example to show the explicit construction of output feedback controller for upper-triangular cases. Example 4.1. Consider a class of three-order upper-triangular stochastic nonlinear system 4=5
8=5
dx1 ¼ x2 dt þ 14 x3 u dt þ 12 x3 dx2 ¼ x3 dt þ 13 u5=3
if J Z J r m; if J Z J 4 m:
0
ð41Þ
for ρ4 Z0. Hence, the infinitesimal generator L of U along (37) is given by
∂UðZÞ ∂UðZÞ 1 ∂2 UðZÞ T LUðZÞ r εEðZÞ þ F ðZÞ þ Tr GðZÞ G ðZÞ 2 ∂Z ∂Z 2 ∂Z r εðα1 ρ3 ε1=r3 ρ4 ε3=r3 Þ‖Z‖Δ
Z τ0m 4 T
dt þ 12 u2
dω;
dω;
dx3 ¼ u dt; y ¼ x1 :
ð47Þ
r 1 ¼ 1, μ ¼ which implies and r 3 ¼ 59 . Choose τ ¼ According to Lemma 2.3, it is easy to verify that nonlinearities satisfy Assumption 4.1 with a01 ¼ 13 and a02 ¼ 12 . Therefore, by Theorem 4.1, one can obtain the homogeneous observer and 29 ,
7 3,
r 2 ¼ 79
W. Zha et al. / ISA Transactions 53 (2014) 709–716 3
715
control scheme is applicable to both lower-triangular and uppertriangular systems.
2 1
Acknowledgments
0 −1 −2
0
10
20
30
40
50
60
70
80
Time(Sec) 10 5
This work was supported in part by National Natural Science Foundation of China (61104068, 61273119), Research Fund for the Doctoral Program of Higher Education of China (20090092120027, 20110092110021), Natural Science Foundation of Jiangsu Province (BK2010200), China Postdoctoral Science Foundation Funded Project (2012M511176) and the Fundamental Research Funds for the Central Universities.
0
References
−5 −10 −15
0
10
20
30
40
50
60
70
80
Time(Sec)
Fig. 2. State trajectories of the closed-loop system (47) and (48).
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3
0
10
20
30
40
50
60
70
80
Fig. 3. The input curve of the closed-loop system (47) and (48).
controller
η^_ 2 ¼ εl1 z^ 2 ;
z^ 2 ¼ ðη^ 2 þ l1 z1 Þ7=9 ;
η^_ 3 ¼ εl2 z^ 3 ;
z^ 3 ¼ ðη^ 3 þ l2 z^ 2 Þ5=7 ;
u ¼ ε3 β 3 ðz^ 3
21=5
þ β2
21=5
ðz^ 2 þ β1 z1 ÞÞ1=7 3
3 7=3
ð48Þ
with l1 ¼ 3.2, l2 ¼ 5.2, β1 ¼ 0:9283, β2 ¼ 1:0834 and ε ¼ 0:38. Simulation results are shown in Figs. 2 and 3 with initial conditions x1 ð0Þ ¼ 1:5, x2 ð0Þ ¼ 0:5, x3 ð0Þ ¼ 0:2, η^ 2 ð0Þ ¼ 0 and η^ 3 ð0Þ ¼ 0. 5. Conclusion In this paper, the problem of global finite-time stabilization in probability via output feedback has been solved for a class of stochastic nonlinear system. By employing adding one power integrator technique, homogeneous domination approach and stochastic finite-time stability theorem, a systematic design method has been presented to ensure that the closed-loop system is globally finite-time stable in probability. Moreover, the proposed
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