RESEARCH ARTICLE
Fast smooth second-order sliding mode control for systems with additive colored noises Pengfei Yang1*, Yangwang Fang1, Youli Wu1, Yunxia Liu2, Danxu Zhang1 1 School of Aeronautics and Astronautics Engineering, Air Force Engineering University, Xi’an, Shaanxi, China, 2 College of Education, Hunan University of Science and Technology, Xiangtan, Hunan, China * [email protected]
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OPEN ACCESS Citation: Yang P, Fang Y, Wu Y, Liu Y, Zhang D (2017) Fast smooth second-order sliding mode control for systems with additive colored noises. PLoS ONE 12(5): e0178455. https://doi.org/ 10.1371/journal.pone.0178455
Abstract In this paper, a fast smooth second-order sliding mode control is presented for a class of stochastic systems with enumerable Ornstein-Uhlenbeck colored noises. The finite-time mean-square practical stability and finite-time mean-square practical reachability are first introduced. Instead of treating the noise as bounded disturbance, the stochastic control techniques are incorporated into the design of the controller. The finite-time convergence of the prescribed sliding variable dynamics system is proved by using stochastic Lyapunovlike techniques. Then the proposed sliding mode controller is applied to a second-order nonlinear stochastic system. Simulation results are presented comparing with smooth secondorder sliding mode control to validate the analysis.
Editor: Yilun Shang, Tongji University, CHINA Received: February 22, 2017 Accepted: May 12, 2017
Introduction
Published: May 31, 2017
Sliding mode control (SMC) is well known for its robustness to system parameter variations and external disturbances[1,2]. SMC has extensive applications in practice, such as robots, aircrafts, DC and AC motors, power systems, process control and so on. Recently, using SMC strategy to the nonlinear stochastic systems modeled by the Itoˆ stochastic differential equations with multiplicative noise has been gaining much investigation, see [3–6] and references therein. The existing research findings applying SMC to the stochastic systems always treat the stochastic noise as bounded uncertainties. These methods need to know the upper bound of the noise and they are comparatively more conservative control strategy, which ensure the robustness at the cost of losing control accuracy. Some literatures derived SMC for the stochastic systems described in Itoˆ’s form applying stability in probability[3], which was proved to be unstable under the second moment stability concept[7]. By comparison, mean-square stability is more practical for engineering application. Wu et al.[8] designed SMC guaranteeing the mean-square exponential stability for the continuous-time switched stochastic systems with multiplicative noise. However, the control signal in [8] switches frequently and the results cannot be extended to stochastic systems with additive noise. One disadvantage of classical SMC is that the sliding variable cannot converge to the sliding surface in finite time. Finite-time convergence has been widely investigated in the control
Copyright: © 2017 Yang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper and its Supporting Information files. Funding: This work has been supported by the Major Program of the National Natural Science Foundation of China (No. 61627901). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.
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Fast smooth second-order sliding mode control for systems with additive colored noises
systems. Shang discussed the finite-time state consensus problems for multi-agent systems [9,10], and further investigated the finite-time cluster average consensus in bidirectional networks and the fixed-time group consensus problem for a leader-follower network[11,12]. It is urgent to deduce finite-time convergence sliding mode method for stochastic systems. In addition, traditional SMC has restrictions such as the relative degree constraint and the high frequency control switching that may easily cause chattering effect[13]. Rahmani designed an adaptive neural network to approximate the system uncertainties and unknown disturbances to reduce chattering phenomena, and proposed controllers combining adaptive neural network with sliding mode control methods[14,15]. Ref.[16] designed a fractional order PID controller to a bio-inspired robot manipulator using bat algorithm. Higher-order sliding mode control (HOSM) also mitigates the problems associated with SMC[17–21]. In the past decades, HOSM has found a variety of application in the robust control of uncertain systems[22,23]. But HOSM for the stochastic systems is remaining poorly investigated. Aiming at the defects of the above mentioned research, a smooth control law for a class of nonlinear stochastic systems with Ornstein-Uhlenbeck colored noise is developed in this paper. By using stochastic Lyapunov-like techniques, a sufficient condition of finite-time convergence is derived under the mean-square practical stability concept. Finally, some experimental results are presented to validate the proposed controller.
Materials and methods Problem statement Let α > 0 and σ = const., the following Itoˆ stochastic differential equation _ ZðtÞ ¼
aZðtÞ þ szðtÞ; Zðt0 Þ ¼ 0
ð1Þ
is called Langevin equation, where z(t) is a standard scalar Gaussian white noise. The solution η(t) (t 0) is called Ornstein-Uhlenbeck process, which is a colored noise[24]. Consider single-input single-output (SISO) dynamics with denumerable Ornstein-Uhlenbeck colored noises l X
s_ ¼ f ðtÞ þ gðtÞu þ dðtÞ þ
hi Zi
ð2Þ
i¼1
where hi are constants; f(t), g(t) are given sufficiently smooth function and g(x) 6¼ 0; d(t) presents unmodeled dynamics, parametric uncertainties and external disturbances, which is assumed to be sufficiently smooth; Zi are mutually independent Ornstein-Uhlenbeck colored noises with parameters αi and s i . s can be interpreted as dynamics of the sliding variable s 2 R1 calculated along the system trajectory and s = 0 expresses sliding manifold; u 2 R1 is the control input. In order to prevent the chattering and exploit the benefits of a sliding mode controller in a real-life system, a smooth control, which can provide a finite time convergence s; s_ ! 0, is urgently needed.
Stochastic fast smooth second-order sliding mode control Problem formulation and definitions. Obviously, system (2) is a stochastic nonlinear system with additive noise, meaning that the system does not have any equilibrium point. This system is unstable under the concept of stability in the sense of Lyapunov, but may also exhibit interesting behavior similar to a conventional stable system near equilibrium[25,26]. That is to say, the desired state is mathematically unstable, but the system may oscillate sufficiently near this state so that the performance is considered acceptable[27]. Motivated by this fact, practical
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Fast smooth second-order sliding mode control for systems with additive colored noises
stability is proposed by LaSalle and Lefschetz[28] and was developed by Martynyuk, Lakshmikantham and Leela et al[29,30]. As a natural extension of the traditional concepts of practical stability, mean-square stability, and finite-time reachability, we shall introduce the concepts of finite-time mean-square practical stability and finite-time mean-square practical reachability. These concepts are concerned with bringing the system trajectory into a bounded neighborhood of a given point or manifold. Consider the following stochastic dynamical system xðtÞ _ ¼ f ðt; xÞ þ hðt; xÞZ
ð3Þ
where f 2 C[R+ × Rn,Rn], h 2 C[R+ × Rn,Rn]; η is 1-dimensional stochastic process. Denote x(t) = x(t,t0,x0) as the solution of (3) under the initial condition (t0,x0). Let s = s(t,x) = 0 be the chosen sliding manifold of the system. Definition 1 (FTMSP): The solution x(t) of system (3) is said to be (S1) mean-square practically stable with respect to (λ,A), if given a pair of positive numbers (λ,A) with A > λ > 0 such that kx0k < λ implies Ekxk2 < A,t t0 for some t0 2 R+; (S2) mean-square uniformly practically stable with respect to (λ,A), if (S1) holds for all t0 2R+; (S3) finite-time mean-square practically stable with respect to (λ,ε), if for every ε, there exist T and λ such that kx0k λ implies Ekxk2 < ε,t t0 + T for some t0 2R+; (S4) finite-time mean-square uniformly practically stable with respect to (λ,ε), if (S3) holds for all t0 2R+; (S5) finite-time mean-square strongly practically stable with respect to (λ,ε), if (S1) and (S3) hold simultaneously; (S6) finite-time mean-square strongly uniformly practically stable with respect to (λ,ε), if (S2) and (S4) hold simultaneously. Remark 1: Unlike definitions in [28,29], which emphasize the boundedness of the system trajectory, the definition we taken here focus far more on the convergence of the system trajectory. Definition 2 (FTMSR): The sliding manifold s=0 is said to be (R1) finite-time mean-square practically reached, if given a pair of positive numbers (λ,ε), λ = λ1 + λ2 and ε = ε1 + ε2, there exists a finite setting time T = T(t0,ε), such that (
2
ksðx0 ; t0 Þk l1 2
k_s ðx0 ; t0 Þk l2 implies Eks(x,t)k2 ε,8t > t0 + T for some t0 2R+; (R2) finite-time mean-square uniformly practically reached, if (R1) holds for all t0 2R+; (R3) second-order finite-time mean-square practically reached, if given a pair of positive numbers (λ,ε), λ = λ1 + λ2 and ε = ε1 + ε2, there exists a finite setting time T = T(t0,ε), such that (
2
ksðx0 ; t0 Þk l1 2
k_s ðx0 ; t0 Þk l2
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Fast smooth second-order sliding mode control for systems with additive colored noises
implies (
2
Eksðx; tÞk ε1
; 8t > t0 þ T
2
Ek_s ðx; tÞk ε2
for some t0 2R+; (R4) second-order finite-time mean-square uniformly practically reached, if (R3) holds for all t0 2R+; Stochastic fast smooth second-order sliding mode control. Consider system Eq (2), denote Zi ¼ hi Zi , si ¼ s i hi and we have Z_ i ¼ hi Z_ i ¼ hi ½ ai Zi ðtÞ þ s i zi ðtÞ ¼
ai Zi ðtÞ þ si zi ðtÞ
ð4Þ
meaning that ηi is a Ornstein-Uhlenbeck noise with parameters αi and σi, so the coefficient hi can be merged by substitute ηi into (2) to get l X
s_ ¼ f ðtÞ þ gðtÞu þ dðtÞ þ
Zi
ð5Þ
i¼1
Consider system Eq (5), the dynamics of the sliding variable is designed as the following form: 8 > > > > < m_ 1 ¼ > > > > :
m 1 k1 jm1 j m sgnðm1 Þ
k2 m1
k3 jm2 jsgnðm1 Þ þ
l X Zi i¼1
m
k4 jm1 j m sgnðm2 Þ
m_ 2 ¼
ð6Þ
2 k5 m2
where μ1 = s; m and ki are positive constants and m > 2; ηi are Ornstein-Uhlenbeck colored noises expressed in (4). Let μ = [μ1, μ2, η1, η2, , η1]T, the following Itoˆ stochastic differential equation can be got by combining (5) and (6) together: 2 2
3
m
3
1
6 k jm j m sgnðm Þ k m k3 jm2 jsgnðm1 Þ þ 1 2 1 6 1 1 6 7 6 6 m_ 7 6 6 27 6 m 2 6 7 6 6 Z_ 1 7 6 k4 jm1 j m sgnðm2 Þ k5 m2 6 7¼6 6 7 6 6 6 .. 7 6 a1 Z1 6 . 7 6 4 5 6 .. 6 Z_ l . 4 m_ 1
l X
2 3 Zi 7 7 0 7 6 7 i¼1 7 6 7 7 607 7 6 7 7 6s 7 7 þ 6 1 7z 7 6 7 7 6 . 7 7 6 .. 7 7 4 5 7 7 sl 5
ð7Þ
al Zl then a stochastic system with respect to the state vector μ can be represented as dμ ¼ f ðμÞdt þ gdWðtÞ
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ð8Þ
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Fast smooth second-order sliding mode control for systems with additive colored noises
where 2
3 m 1 l X 6 k jm j m sgnðm Þ k m k3 jm2 jsgnðm1 Þ þ Zi 7 6 1 1 7 1 2 1 6 7 i¼1 6 7 6 7 6 7 m 2 6 7 6 7 m 6 7 sgnðm k jm j Þ k m 4 1 2 5 2 6 7 f ðμÞ ¼ 6 7 6 7 a1 Z1 6 7 6 7 6 7 6 7 .. 6 7 . 6 7 4 5 al Zl g ¼ ½0 0 s1 sl
ð9Þ
T
Let the sliding variable dynamics be of the form (6) and in accordance with the sliding variable system (5), the SFS-SOSM controller is selected as 8 m 1 > > > < u ¼ g 1 ðtÞ½ k jm j m sgnðm Þ k m ^ k3 jm2 jsgnðm1 Þ f ðtÞ dðtÞ 1 1 1 2 1 ð10Þ > m 2 > > : m_ 2 ¼ k4 jm1 j m sgnðm2 Þ k5 m2 ^ is the estimation of uncertain function by means of high-order sliding-mode where dðtÞ observer presented in [22]. Hereafter, FTMSP and FTMSR are employed to analyze the reachability of the sliding manifold. Finite time convergence analysis. Based on the definition proposed above, we give the following theorem: Theorem 1: Consider the stochastic nonlinear system (6) with respect to the sliding variable s, let Pi ¼
k5 kk k5 ; i ¼ 1; 2; ; l þ 2 5 ; Qi ¼ ai ðai þ k2 Þ ai ai þ k2
where m > 2, αi > 0 (i = 1,2, ,l), kj > 0 (j = 1, ,5). Constructing the following matrix 2 3 k5 0 Q1 Q2 Ql 6 7 6 0 k 0 0 0 7 2 6 7 6 7 6 Q1 0 P1 0 0 7 7 16 7 Λ¼ 6 6 7 Q 0 0 P 0 26 2 2 7 6 7 .. .. .. . . 6 .. 7 6 . 7 . . . . 4 5 Ql 0 0 0 Pl
ð11Þ
ð12Þ
ðlþ2Þðlþ2Þ
and assuming that 2 (i) ε ¼ ½1 þ ðk1 þ k2 þ k3 þ lÞ ε and the following inequality holds ε
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g2 lmin ðΛÞg1
ð13Þ
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Fast smooth second-order sliding mode control for systems with additive colored noises
where l kk 1 X g1 ¼ 2 5 ; g2 ¼ ½ 2 i¼1;j¼1;i6¼j lmax ðΛÞ
sffiffiffiffiffiffiffi l 1 1X ðQi þ Qj Þsi sj þ P s2 ai aj 2 i¼1 i i
σi are the parameter of the colored noise mentioned in (4). (ii) Positive number λ satisfies l>
l X s2i 2ai i¼1
g2 lmax ðΛÞg1
ð14Þ
Then the prescribed sliding variable dynamics system (6) is finite-time mean-square practically stable, and the proposed control (10) is an SFS-SOSM control. The sliding manifold s = 0 can be second-order mean-square practically reached in finite time. Proof: According to the definition given before, we want to prove that for the prescribed sliding variable dynamics system (6), if given positive numbers (λ,ε), λ = λ1 + λ2 and ε = ε1+ ε2, there exists a finite setting time T = T(t0,ε), such that (
2
jsðx0 ; t0 Þj l1 2
j_s ðx0 ; t0 Þj l2 implies (
2
Ejsðx; tÞj ε1 2
Ej_s ðx; tÞj ε2
; 8t > t0 þ T
To prove this, aiming at the augmented system (8), we define the Lyapunov-like functional as l l 1 1 1X k5 k2 k5 2 X k5 2 2 V ¼ k5 m1 þ k2 m2 þ Zi þ jm jjZ j þ 2 2 2 i¼1 ai ðai þ k2 Þ ai ai þ k2 1 i i¼1
ð15Þ
Since V(μ) is continuous but not differentiable, a nonsmooth version of Lyapunov’s theory is required, which shows that one can just consider the points where V(μ) is differentiable [28,29]. This argument is valid in all the proofs of this paper. The substitution ξ = [|μ1|,|μ2|,|η1|,|η2|, ,|η1|]T brings the proposed functional (15) to a quadratic form V ¼ ξT Λξ
ð16Þ
where Λ is given in (12). It is obvious that Λ is positive definite since αi > 0 (i = 1,2, ,l), kj > 0 (j = 1, ,5). Note that V(μ) is positive definite and unbounded, the following inequalities can be obtained based on Rayleigh-Ritz Theorem 2
2
lmin ðΛÞEðkξk Þ EV lmax ðΛÞEðkξk Þ
ð17Þ
2
where kξk ¼ m21 þ m22 þ Z21 þ Z22 þ þ Z2l is the Euclidean norm of ξ, λmin(Λ) and λmax(Λ) are minimal and maximal eigenvalues of Λ.
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Fast smooth second-order sliding mode control for systems with additive colored noises
We denote the infinitesimal generator by L. Appling infinitesimal generator along with system (8), we have 2 3 m_ 1 6 7 6 m_ 7 6 2 7 7 @V @V @V @V 6 6 Z_ 1 7 1 LV ¼ ð18Þ 6 7 þ traceðg T DVgÞ @m1 @m2 @Z1 @Zl 6 7 2 6 .. 7 6 . 7 4 5 Z_ l Let 2
3 m_ 1
LV1 ¼
@V @m1
@V @m2
@V @Z1
6 7 6 m_ 7 27 6 7 @V 6 6 Z_ 1 7 6 7; @Zl 6 7 6 .. 7 6 . 7 4 5
1 LV2 ¼ traceðg T DVgÞ 2
Z_ l LV1 can be expanded and the following inequality holds 2
3 T l X 6 k5 m1 þ sgnðm1 Þ Qi Zi 7 6 7 i¼1 6 7 6 7 6 7 6 7 k2 m2 6 7 6 7 6 7 LV1 ¼ 6 P Z þ Q m sgnðZ Þ 7 1 1 1 6 1 1 7 6 7 6 7 6 7 .. 6 7 . 6 7 6 7 4 5 Pl Zl þ Ql m1 sgnðZl Þ
¼
2m 1 k1 k5 m1 m
2
3 m 1 l X 7 6 k m m sgnðm Þ k m k3 jm2 jsgnðm1 Þ þ Zi 7 6 1 1 1 2 1 6 7 i¼1 6 7 6 7 6 7 m 2 6 7 6 7 m 6 7 sgnðm k Þ k m m 4 1 2 5 2 6 7 6 7 6 7 6 7 a1 Z1 6 7 6 7 6 7 .. 6 7 . 6 7 4 5 al Zl l X k3 k5 m1 m2 þ k5 m1 Zi
k2 k5 m21
2
l m 1 X 6 þ ð Qi Zi Þ 4 k1 m1 m
i¼1
3 7 k3 m2 5
X l l X l X k2 m1 Qi Zi þ sgnðm1 Þ Qi Zi Zi
i¼1
i¼1
2m 2 k2 k4 m1 m
k2 k5 m21
ð19Þ
k2 k5 m22
k2 k5 m22 l X i¼1
P1 a1 Z21
ai Pi Z2i þ
Q1 a1 m1 Z1
l X ðk5
k2 Qi
i¼1
Pl al Z2l
i¼1
Ql al m1 Zl
ai Qi Þ m1 Zi
i¼1
þ ðQ1 Z1 þ Q2 Z2 þ þ Ql Zl Þð Z1 þ Z2 þ þ Zl Þ
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Fast smooth second-order sliding mode control for systems with additive colored noises
Notice that k5
k2 Qi
ai Qi ¼ k5
ðai þ k2 Þ
k5 ¼ 0; i ¼ 1; 2; ; l ai þ k2
ð20Þ
Then the following inequality can be deduced LV1
k2 k5 m21
l l l X X X ai Pi Z2i þ Qi Z2i þ ðQi þ Qj ÞjZi jjZj j
k2 k5 m22
i¼1
i¼1
X ðai Pi
i¼1;j¼1;i6¼j
k2 k5 m
¼
2 1
2 2
k2 k5 m
ð21Þ
X
l
l
2 i
Qi ÞZ þ
i¼1
ðQi þ Qj ÞjZi jjZj j i¼1;j¼1;i6¼j
Furthermore, ai Pi
Qi ¼ ai
k5 kk þ 2 5 ai ðai þ k2 Þ ai
k5 ¼ k2 k5 ; i ¼ 1; 2; ; l ai þ k2
ð22Þ
then we have LV1
k2 k5 m21
l l X X k2 k5 Z2i þ ðQi þ Qj ÞjZi jjZj j
k2 k5 m22
i¼1
i¼1;j¼1;i6¼j
ð23Þ
X l
¼
2
k2 k5 kξk þ
ðQi þ Qj ÞjZi jjZj j i¼1;j¼1;i6¼j
The inequality about LV2 can be deduced according to the properties of the matrix trace as 1 1 LV2 ¼ traceðg T DVgÞ ¼ g T DVg 2 2 2 2 @ V 6 @m21 6 6 6 6 6 6 6 6 6 6 1 6 ¼ ½ 0 0 s 1 s l 6 6 2 6 6 6 6 6 6 6 6 6 4
3
@2V @m22 @2V @Z21
72 3 7 7 0 76 7 76 7 76 7 76 0 7 76 7 76 7 76 7 76 7 76 s1 7 76 7 76 7 76 7 76 . 7 76 . 7 76 . 7 76 7 .. 74 5 . 7 7 s 7 l 2 @ V5
ð24Þ
@Z2l ¼
l l 1X @2V 1X ðs2i Þ ¼ P s2 2 i¼1 @Z2i 2 i¼1 i i
Substitute(23), (24) into (18) to get LV
2
k2 k5 kξk þ
l X
ðQi þ Qj ÞjZi jjZj j þ i¼1;j¼1;i6¼j
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
l 1X P s2 2 i¼1 i i
ð25Þ
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Fast smooth second-order sliding mode control for systems with additive colored noises
According to Itoˆ’s formula, it follows that 0
ðEVÞ ¼ EðLVÞ l X
2
k2 k5 Eðkξk Þ þ
¼
ðQi þ Qj ÞEjZi jEjZj j þ i¼1;j¼1;i6¼j
l 1X P s2 2 i¼1 i i
ð26Þ
Since ηi are mutually independent, utilizing E[η2 (t)] σ2/2α and Rao inequality[31] to obtain: 2
2
ðEjZi ðtÞjÞ E1EðjZi ðtÞj Þ ¼ E½Z2i ðtÞ
s2i 2ai
ð27Þ
Then inequality (26) can be further represented as 0
ðEVÞ
2
l X
k2 k5 Eðkξk Þ þ
ðQi þ Qj Þ
sffiffiffiffiffiffisffiffiffiffiffiffi s2i s2j
þ
2ai 2aj sffiffiffiffiffiffiffi l l k2 k5 1 X 1 1X ðQi þ Qj Þsi sj þ ½ P s2 EV þ 2 i¼1;j¼1;i6¼j ai aj 2 i¼1 i i lmax ðΛÞ i¼1;j¼1;i6¼j
¼
l 1X P s2 2 i¼1 i i
ð28Þ
g1 EV þ g2
where sffiffiffiffiffiffiffi l l k2 k5 1 X 1 1X g1 ¼ ðQi þ Qj Þsi sj þ ½ P s2 ; g2 ¼ 2 i¼1;j¼1;i6¼j ai aj 2 i¼1 i i lmax ðΛÞ
ð29Þ
It is obvious that γ1,γ2 > 0. Since the solution of the differential equation φ_ ¼
g1 φ þ g2 ; φðt0 Þ ¼ φ0 0
ð30Þ
is given by φðtÞ ¼ ðφ0
g2 Þe g1
g1 ðt t0 Þ
þ
g2 g1
ð31Þ
it follows from the comparison principle[32] that EV(t) φ(t) when EV(t0) φ0. From (31) we can claim that the following inequality holds. g2 Þe g1
EVðtÞ ðEVðt0 Þ
g1 ðt t0 Þ
þ
g2 g1
2
ð32Þ
2
From the initial conditions, we have jsðx0 ; t0 Þj þ j_s ðx0 ; t0 Þj l. So the initial condition of the constructed vector ξ can be got as 2
2
2
l X
Ekξðx0 ; t0 Þk ¼ Ejm1 ðx0 ; t0 Þj þ Ejm2 ðx0 ; t0 Þj þ
EZ2i < l þ
i¼1
l X i¼1
EZ2i l þ
l X s2i 2ai i¼1
ð33Þ
For convenient, we denote ξ0 ¼ ξðx0 ; t0 Þ;
d¼
l X s2i 2ai i¼1
and synthesize the results we have got in (17), (32), (33), the following inequality can be
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Fast smooth second-order sliding mode control for systems with additive colored noises
deduced 2
EkξðtÞk
EVðtÞ 1 f½EV0 lmin ðΛÞ lmin ðΛÞ
g2 e g1
1 2 f½l ðΛÞEkξ0 k lmin ðΛÞ max
Fast smooth second-order sliding mode control for systems with additive colored noises Pengfei Yang1*, Yangwang Fang1, Youli Wu1, Yunxia Liu2, Danxu Zhang1 1 School of Aeronautics and Astronautics Engineering, Air Force Engineering University, Xi’an, Shaanxi, China, 2 College of Education, Hunan University of Science and Technology, Xiangtan, Hunan, China * [email protected]
a1111111111 a1111111111 a1111111111 a1111111111 a1111111111
OPEN ACCESS Citation: Yang P, Fang Y, Wu Y, Liu Y, Zhang D (2017) Fast smooth second-order sliding mode control for systems with additive colored noises. PLoS ONE 12(5): e0178455. https://doi.org/ 10.1371/journal.pone.0178455
Abstract In this paper, a fast smooth second-order sliding mode control is presented for a class of stochastic systems with enumerable Ornstein-Uhlenbeck colored noises. The finite-time mean-square practical stability and finite-time mean-square practical reachability are first introduced. Instead of treating the noise as bounded disturbance, the stochastic control techniques are incorporated into the design of the controller. The finite-time convergence of the prescribed sliding variable dynamics system is proved by using stochastic Lyapunovlike techniques. Then the proposed sliding mode controller is applied to a second-order nonlinear stochastic system. Simulation results are presented comparing with smooth secondorder sliding mode control to validate the analysis.
Editor: Yilun Shang, Tongji University, CHINA Received: February 22, 2017 Accepted: May 12, 2017
Introduction
Published: May 31, 2017
Sliding mode control (SMC) is well known for its robustness to system parameter variations and external disturbances[1,2]. SMC has extensive applications in practice, such as robots, aircrafts, DC and AC motors, power systems, process control and so on. Recently, using SMC strategy to the nonlinear stochastic systems modeled by the Itoˆ stochastic differential equations with multiplicative noise has been gaining much investigation, see [3–6] and references therein. The existing research findings applying SMC to the stochastic systems always treat the stochastic noise as bounded uncertainties. These methods need to know the upper bound of the noise and they are comparatively more conservative control strategy, which ensure the robustness at the cost of losing control accuracy. Some literatures derived SMC for the stochastic systems described in Itoˆ’s form applying stability in probability[3], which was proved to be unstable under the second moment stability concept[7]. By comparison, mean-square stability is more practical for engineering application. Wu et al.[8] designed SMC guaranteeing the mean-square exponential stability for the continuous-time switched stochastic systems with multiplicative noise. However, the control signal in [8] switches frequently and the results cannot be extended to stochastic systems with additive noise. One disadvantage of classical SMC is that the sliding variable cannot converge to the sliding surface in finite time. Finite-time convergence has been widely investigated in the control
Copyright: © 2017 Yang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper and its Supporting Information files. Funding: This work has been supported by the Major Program of the National Natural Science Foundation of China (No. 61627901). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.
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Fast smooth second-order sliding mode control for systems with additive colored noises
systems. Shang discussed the finite-time state consensus problems for multi-agent systems [9,10], and further investigated the finite-time cluster average consensus in bidirectional networks and the fixed-time group consensus problem for a leader-follower network[11,12]. It is urgent to deduce finite-time convergence sliding mode method for stochastic systems. In addition, traditional SMC has restrictions such as the relative degree constraint and the high frequency control switching that may easily cause chattering effect[13]. Rahmani designed an adaptive neural network to approximate the system uncertainties and unknown disturbances to reduce chattering phenomena, and proposed controllers combining adaptive neural network with sliding mode control methods[14,15]. Ref.[16] designed a fractional order PID controller to a bio-inspired robot manipulator using bat algorithm. Higher-order sliding mode control (HOSM) also mitigates the problems associated with SMC[17–21]. In the past decades, HOSM has found a variety of application in the robust control of uncertain systems[22,23]. But HOSM for the stochastic systems is remaining poorly investigated. Aiming at the defects of the above mentioned research, a smooth control law for a class of nonlinear stochastic systems with Ornstein-Uhlenbeck colored noise is developed in this paper. By using stochastic Lyapunov-like techniques, a sufficient condition of finite-time convergence is derived under the mean-square practical stability concept. Finally, some experimental results are presented to validate the proposed controller.
Materials and methods Problem statement Let α > 0 and σ = const., the following Itoˆ stochastic differential equation _ ZðtÞ ¼
aZðtÞ þ szðtÞ; Zðt0 Þ ¼ 0
ð1Þ
is called Langevin equation, where z(t) is a standard scalar Gaussian white noise. The solution η(t) (t 0) is called Ornstein-Uhlenbeck process, which is a colored noise[24]. Consider single-input single-output (SISO) dynamics with denumerable Ornstein-Uhlenbeck colored noises l X
s_ ¼ f ðtÞ þ gðtÞu þ dðtÞ þ
hi Zi
ð2Þ
i¼1
where hi are constants; f(t), g(t) are given sufficiently smooth function and g(x) 6¼ 0; d(t) presents unmodeled dynamics, parametric uncertainties and external disturbances, which is assumed to be sufficiently smooth; Zi are mutually independent Ornstein-Uhlenbeck colored noises with parameters αi and s i . s can be interpreted as dynamics of the sliding variable s 2 R1 calculated along the system trajectory and s = 0 expresses sliding manifold; u 2 R1 is the control input. In order to prevent the chattering and exploit the benefits of a sliding mode controller in a real-life system, a smooth control, which can provide a finite time convergence s; s_ ! 0, is urgently needed.
Stochastic fast smooth second-order sliding mode control Problem formulation and definitions. Obviously, system (2) is a stochastic nonlinear system with additive noise, meaning that the system does not have any equilibrium point. This system is unstable under the concept of stability in the sense of Lyapunov, but may also exhibit interesting behavior similar to a conventional stable system near equilibrium[25,26]. That is to say, the desired state is mathematically unstable, but the system may oscillate sufficiently near this state so that the performance is considered acceptable[27]. Motivated by this fact, practical
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
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Fast smooth second-order sliding mode control for systems with additive colored noises
stability is proposed by LaSalle and Lefschetz[28] and was developed by Martynyuk, Lakshmikantham and Leela et al[29,30]. As a natural extension of the traditional concepts of practical stability, mean-square stability, and finite-time reachability, we shall introduce the concepts of finite-time mean-square practical stability and finite-time mean-square practical reachability. These concepts are concerned with bringing the system trajectory into a bounded neighborhood of a given point or manifold. Consider the following stochastic dynamical system xðtÞ _ ¼ f ðt; xÞ þ hðt; xÞZ
ð3Þ
where f 2 C[R+ × Rn,Rn], h 2 C[R+ × Rn,Rn]; η is 1-dimensional stochastic process. Denote x(t) = x(t,t0,x0) as the solution of (3) under the initial condition (t0,x0). Let s = s(t,x) = 0 be the chosen sliding manifold of the system. Definition 1 (FTMSP): The solution x(t) of system (3) is said to be (S1) mean-square practically stable with respect to (λ,A), if given a pair of positive numbers (λ,A) with A > λ > 0 such that kx0k < λ implies Ekxk2 < A,t t0 for some t0 2 R+; (S2) mean-square uniformly practically stable with respect to (λ,A), if (S1) holds for all t0 2R+; (S3) finite-time mean-square practically stable with respect to (λ,ε), if for every ε, there exist T and λ such that kx0k λ implies Ekxk2 < ε,t t0 + T for some t0 2R+; (S4) finite-time mean-square uniformly practically stable with respect to (λ,ε), if (S3) holds for all t0 2R+; (S5) finite-time mean-square strongly practically stable with respect to (λ,ε), if (S1) and (S3) hold simultaneously; (S6) finite-time mean-square strongly uniformly practically stable with respect to (λ,ε), if (S2) and (S4) hold simultaneously. Remark 1: Unlike definitions in [28,29], which emphasize the boundedness of the system trajectory, the definition we taken here focus far more on the convergence of the system trajectory. Definition 2 (FTMSR): The sliding manifold s=0 is said to be (R1) finite-time mean-square practically reached, if given a pair of positive numbers (λ,ε), λ = λ1 + λ2 and ε = ε1 + ε2, there exists a finite setting time T = T(t0,ε), such that (
2
ksðx0 ; t0 Þk l1 2
k_s ðx0 ; t0 Þk l2 implies Eks(x,t)k2 ε,8t > t0 + T for some t0 2R+; (R2) finite-time mean-square uniformly practically reached, if (R1) holds for all t0 2R+; (R3) second-order finite-time mean-square practically reached, if given a pair of positive numbers (λ,ε), λ = λ1 + λ2 and ε = ε1 + ε2, there exists a finite setting time T = T(t0,ε), such that (
2
ksðx0 ; t0 Þk l1 2
k_s ðx0 ; t0 Þk l2
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
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Fast smooth second-order sliding mode control for systems with additive colored noises
implies (
2
Eksðx; tÞk ε1
; 8t > t0 þ T
2
Ek_s ðx; tÞk ε2
for some t0 2R+; (R4) second-order finite-time mean-square uniformly practically reached, if (R3) holds for all t0 2R+; Stochastic fast smooth second-order sliding mode control. Consider system Eq (2), denote Zi ¼ hi Zi , si ¼ s i hi and we have Z_ i ¼ hi Z_ i ¼ hi ½ ai Zi ðtÞ þ s i zi ðtÞ ¼
ai Zi ðtÞ þ si zi ðtÞ
ð4Þ
meaning that ηi is a Ornstein-Uhlenbeck noise with parameters αi and σi, so the coefficient hi can be merged by substitute ηi into (2) to get l X
s_ ¼ f ðtÞ þ gðtÞu þ dðtÞ þ
Zi
ð5Þ
i¼1
Consider system Eq (5), the dynamics of the sliding variable is designed as the following form: 8 > > > > < m_ 1 ¼ > > > > :
m 1 k1 jm1 j m sgnðm1 Þ
k2 m1
k3 jm2 jsgnðm1 Þ þ
l X Zi i¼1
m
k4 jm1 j m sgnðm2 Þ
m_ 2 ¼
ð6Þ
2 k5 m2
where μ1 = s; m and ki are positive constants and m > 2; ηi are Ornstein-Uhlenbeck colored noises expressed in (4). Let μ = [μ1, μ2, η1, η2, , η1]T, the following Itoˆ stochastic differential equation can be got by combining (5) and (6) together: 2 2
3
m
3
1
6 k jm j m sgnðm Þ k m k3 jm2 jsgnðm1 Þ þ 1 2 1 6 1 1 6 7 6 6 m_ 7 6 6 27 6 m 2 6 7 6 6 Z_ 1 7 6 k4 jm1 j m sgnðm2 Þ k5 m2 6 7¼6 6 7 6 6 6 .. 7 6 a1 Z1 6 . 7 6 4 5 6 .. 6 Z_ l . 4 m_ 1
l X
2 3 Zi 7 7 0 7 6 7 i¼1 7 6 7 7 607 7 6 7 7 6s 7 7 þ 6 1 7z 7 6 7 7 6 . 7 7 6 .. 7 7 4 5 7 7 sl 5
ð7Þ
al Zl then a stochastic system with respect to the state vector μ can be represented as dμ ¼ f ðμÞdt þ gdWðtÞ
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
ð8Þ
4 / 22
Fast smooth second-order sliding mode control for systems with additive colored noises
where 2
3 m 1 l X 6 k jm j m sgnðm Þ k m k3 jm2 jsgnðm1 Þ þ Zi 7 6 1 1 7 1 2 1 6 7 i¼1 6 7 6 7 6 7 m 2 6 7 6 7 m 6 7 sgnðm k jm j Þ k m 4 1 2 5 2 6 7 f ðμÞ ¼ 6 7 6 7 a1 Z1 6 7 6 7 6 7 6 7 .. 6 7 . 6 7 4 5 al Zl g ¼ ½0 0 s1 sl
ð9Þ
T
Let the sliding variable dynamics be of the form (6) and in accordance with the sliding variable system (5), the SFS-SOSM controller is selected as 8 m 1 > > > < u ¼ g 1 ðtÞ½ k jm j m sgnðm Þ k m ^ k3 jm2 jsgnðm1 Þ f ðtÞ dðtÞ 1 1 1 2 1 ð10Þ > m 2 > > : m_ 2 ¼ k4 jm1 j m sgnðm2 Þ k5 m2 ^ is the estimation of uncertain function by means of high-order sliding-mode where dðtÞ observer presented in [22]. Hereafter, FTMSP and FTMSR are employed to analyze the reachability of the sliding manifold. Finite time convergence analysis. Based on the definition proposed above, we give the following theorem: Theorem 1: Consider the stochastic nonlinear system (6) with respect to the sliding variable s, let Pi ¼
k5 kk k5 ; i ¼ 1; 2; ; l þ 2 5 ; Qi ¼ ai ðai þ k2 Þ ai ai þ k2
where m > 2, αi > 0 (i = 1,2, ,l), kj > 0 (j = 1, ,5). Constructing the following matrix 2 3 k5 0 Q1 Q2 Ql 6 7 6 0 k 0 0 0 7 2 6 7 6 7 6 Q1 0 P1 0 0 7 7 16 7 Λ¼ 6 6 7 Q 0 0 P 0 26 2 2 7 6 7 .. .. .. . . 6 .. 7 6 . 7 . . . . 4 5 Ql 0 0 0 Pl
ð11Þ
ð12Þ
ðlþ2Þðlþ2Þ
and assuming that 2 (i) ε ¼ ½1 þ ðk1 þ k2 þ k3 þ lÞ ε and the following inequality holds ε
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
g2 lmin ðΛÞg1
ð13Þ
5 / 22
Fast smooth second-order sliding mode control for systems with additive colored noises
where l kk 1 X g1 ¼ 2 5 ; g2 ¼ ½ 2 i¼1;j¼1;i6¼j lmax ðΛÞ
sffiffiffiffiffiffiffi l 1 1X ðQi þ Qj Þsi sj þ P s2 ai aj 2 i¼1 i i
σi are the parameter of the colored noise mentioned in (4). (ii) Positive number λ satisfies l>
l X s2i 2ai i¼1
g2 lmax ðΛÞg1
ð14Þ
Then the prescribed sliding variable dynamics system (6) is finite-time mean-square practically stable, and the proposed control (10) is an SFS-SOSM control. The sliding manifold s = 0 can be second-order mean-square practically reached in finite time. Proof: According to the definition given before, we want to prove that for the prescribed sliding variable dynamics system (6), if given positive numbers (λ,ε), λ = λ1 + λ2 and ε = ε1+ ε2, there exists a finite setting time T = T(t0,ε), such that (
2
jsðx0 ; t0 Þj l1 2
j_s ðx0 ; t0 Þj l2 implies (
2
Ejsðx; tÞj ε1 2
Ej_s ðx; tÞj ε2
; 8t > t0 þ T
To prove this, aiming at the augmented system (8), we define the Lyapunov-like functional as l l 1 1 1X k5 k2 k5 2 X k5 2 2 V ¼ k5 m1 þ k2 m2 þ Zi þ jm jjZ j þ 2 2 2 i¼1 ai ðai þ k2 Þ ai ai þ k2 1 i i¼1
ð15Þ
Since V(μ) is continuous but not differentiable, a nonsmooth version of Lyapunov’s theory is required, which shows that one can just consider the points where V(μ) is differentiable [28,29]. This argument is valid in all the proofs of this paper. The substitution ξ = [|μ1|,|μ2|,|η1|,|η2|, ,|η1|]T brings the proposed functional (15) to a quadratic form V ¼ ξT Λξ
ð16Þ
where Λ is given in (12). It is obvious that Λ is positive definite since αi > 0 (i = 1,2, ,l), kj > 0 (j = 1, ,5). Note that V(μ) is positive definite and unbounded, the following inequalities can be obtained based on Rayleigh-Ritz Theorem 2
2
lmin ðΛÞEðkξk Þ EV lmax ðΛÞEðkξk Þ
ð17Þ
2
where kξk ¼ m21 þ m22 þ Z21 þ Z22 þ þ Z2l is the Euclidean norm of ξ, λmin(Λ) and λmax(Λ) are minimal and maximal eigenvalues of Λ.
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
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Fast smooth second-order sliding mode control for systems with additive colored noises
We denote the infinitesimal generator by L. Appling infinitesimal generator along with system (8), we have 2 3 m_ 1 6 7 6 m_ 7 6 2 7 7 @V @V @V @V 6 6 Z_ 1 7 1 LV ¼ ð18Þ 6 7 þ traceðg T DVgÞ @m1 @m2 @Z1 @Zl 6 7 2 6 .. 7 6 . 7 4 5 Z_ l Let 2
3 m_ 1
LV1 ¼
@V @m1
@V @m2
@V @Z1
6 7 6 m_ 7 27 6 7 @V 6 6 Z_ 1 7 6 7; @Zl 6 7 6 .. 7 6 . 7 4 5
1 LV2 ¼ traceðg T DVgÞ 2
Z_ l LV1 can be expanded and the following inequality holds 2
3 T l X 6 k5 m1 þ sgnðm1 Þ Qi Zi 7 6 7 i¼1 6 7 6 7 6 7 6 7 k2 m2 6 7 6 7 6 7 LV1 ¼ 6 P Z þ Q m sgnðZ Þ 7 1 1 1 6 1 1 7 6 7 6 7 6 7 .. 6 7 . 6 7 6 7 4 5 Pl Zl þ Ql m1 sgnðZl Þ
¼
2m 1 k1 k5 m1 m
2
3 m 1 l X 7 6 k m m sgnðm Þ k m k3 jm2 jsgnðm1 Þ þ Zi 7 6 1 1 1 2 1 6 7 i¼1 6 7 6 7 6 7 m 2 6 7 6 7 m 6 7 sgnðm k Þ k m m 4 1 2 5 2 6 7 6 7 6 7 6 7 a1 Z1 6 7 6 7 6 7 .. 6 7 . 6 7 4 5 al Zl l X k3 k5 m1 m2 þ k5 m1 Zi
k2 k5 m21
2
l m 1 X 6 þ ð Qi Zi Þ 4 k1 m1 m
i¼1
3 7 k3 m2 5
X l l X l X k2 m1 Qi Zi þ sgnðm1 Þ Qi Zi Zi
i¼1
i¼1
2m 2 k2 k4 m1 m
k2 k5 m21
ð19Þ
k2 k5 m22
k2 k5 m22 l X i¼1
P1 a1 Z21
ai Pi Z2i þ
Q1 a1 m1 Z1
l X ðk5
k2 Qi
i¼1
Pl al Z2l
i¼1
Ql al m1 Zl
ai Qi Þ m1 Zi
i¼1
þ ðQ1 Z1 þ Q2 Z2 þ þ Ql Zl Þð Z1 þ Z2 þ þ Zl Þ
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
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Fast smooth second-order sliding mode control for systems with additive colored noises
Notice that k5
k2 Qi
ai Qi ¼ k5
ðai þ k2 Þ
k5 ¼ 0; i ¼ 1; 2; ; l ai þ k2
ð20Þ
Then the following inequality can be deduced LV1
k2 k5 m21
l l l X X X ai Pi Z2i þ Qi Z2i þ ðQi þ Qj ÞjZi jjZj j
k2 k5 m22
i¼1
i¼1
X ðai Pi
i¼1;j¼1;i6¼j
k2 k5 m
¼
2 1
2 2
k2 k5 m
ð21Þ
X
l
l
2 i
Qi ÞZ þ
i¼1
ðQi þ Qj ÞjZi jjZj j i¼1;j¼1;i6¼j
Furthermore, ai Pi
Qi ¼ ai
k5 kk þ 2 5 ai ðai þ k2 Þ ai
k5 ¼ k2 k5 ; i ¼ 1; 2; ; l ai þ k2
ð22Þ
then we have LV1
k2 k5 m21
l l X X k2 k5 Z2i þ ðQi þ Qj ÞjZi jjZj j
k2 k5 m22
i¼1
i¼1;j¼1;i6¼j
ð23Þ
X l
¼
2
k2 k5 kξk þ
ðQi þ Qj ÞjZi jjZj j i¼1;j¼1;i6¼j
The inequality about LV2 can be deduced according to the properties of the matrix trace as 1 1 LV2 ¼ traceðg T DVgÞ ¼ g T DVg 2 2 2 2 @ V 6 @m21 6 6 6 6 6 6 6 6 6 6 1 6 ¼ ½ 0 0 s 1 s l 6 6 2 6 6 6 6 6 6 6 6 6 4
3
@2V @m22 @2V @Z21
72 3 7 7 0 76 7 76 7 76 7 76 0 7 76 7 76 7 76 7 76 7 76 s1 7 76 7 76 7 76 7 76 . 7 76 . 7 76 . 7 76 7 .. 74 5 . 7 7 s 7 l 2 @ V5
ð24Þ
@Z2l ¼
l l 1X @2V 1X ðs2i Þ ¼ P s2 2 i¼1 @Z2i 2 i¼1 i i
Substitute(23), (24) into (18) to get LV
2
k2 k5 kξk þ
l X
ðQi þ Qj ÞjZi jjZj j þ i¼1;j¼1;i6¼j
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
l 1X P s2 2 i¼1 i i
ð25Þ
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Fast smooth second-order sliding mode control for systems with additive colored noises
According to Itoˆ’s formula, it follows that 0
ðEVÞ ¼ EðLVÞ l X
2
k2 k5 Eðkξk Þ þ
¼
ðQi þ Qj ÞEjZi jEjZj j þ i¼1;j¼1;i6¼j
l 1X P s2 2 i¼1 i i
ð26Þ
Since ηi are mutually independent, utilizing E[η2 (t)] σ2/2α and Rao inequality[31] to obtain: 2
2
ðEjZi ðtÞjÞ E1EðjZi ðtÞj Þ ¼ E½Z2i ðtÞ
s2i 2ai
ð27Þ
Then inequality (26) can be further represented as 0
ðEVÞ
2
l X
k2 k5 Eðkξk Þ þ
ðQi þ Qj Þ
sffiffiffiffiffiffisffiffiffiffiffiffi s2i s2j
þ
2ai 2aj sffiffiffiffiffiffiffi l l k2 k5 1 X 1 1X ðQi þ Qj Þsi sj þ ½ P s2 EV þ 2 i¼1;j¼1;i6¼j ai aj 2 i¼1 i i lmax ðΛÞ i¼1;j¼1;i6¼j
¼
l 1X P s2 2 i¼1 i i
ð28Þ
g1 EV þ g2
where sffiffiffiffiffiffiffi l l k2 k5 1 X 1 1X g1 ¼ ðQi þ Qj Þsi sj þ ½ P s2 ; g2 ¼ 2 i¼1;j¼1;i6¼j ai aj 2 i¼1 i i lmax ðΛÞ
ð29Þ
It is obvious that γ1,γ2 > 0. Since the solution of the differential equation φ_ ¼
g1 φ þ g2 ; φðt0 Þ ¼ φ0 0
ð30Þ
is given by φðtÞ ¼ ðφ0
g2 Þe g1
g1 ðt t0 Þ
þ
g2 g1
ð31Þ
it follows from the comparison principle[32] that EV(t) φ(t) when EV(t0) φ0. From (31) we can claim that the following inequality holds. g2 Þe g1
EVðtÞ ðEVðt0 Þ
g1 ðt t0 Þ
þ
g2 g1
2
ð32Þ
2
From the initial conditions, we have jsðx0 ; t0 Þj þ j_s ðx0 ; t0 Þj l. So the initial condition of the constructed vector ξ can be got as 2
2
2
l X
Ekξðx0 ; t0 Þk ¼ Ejm1 ðx0 ; t0 Þj þ Ejm2 ðx0 ; t0 Þj þ
EZ2i < l þ
i¼1
l X i¼1
EZ2i l þ
l X s2i 2ai i¼1
ð33Þ
For convenient, we denote ξ0 ¼ ξðx0 ; t0 Þ;
d¼
l X s2i 2ai i¼1
and synthesize the results we have got in (17), (32), (33), the following inequality can be
PLOS ONE | https://doi.org/10.1371/journal.pone.0178455 May 31, 2017
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Fast smooth second-order sliding mode control for systems with additive colored noises
deduced 2
EkξðtÞk
EVðtÞ 1 f½EV0 lmin ðΛÞ lmin ðΛÞ
g2 e g1
1 2 f½l ðΛÞEkξ0 k lmin ðΛÞ max
