Adaptive tracking control for a class of stochastic switched systems with stochastic input-to-state stable inverse dynamics and input saturation

Systems & Control Letters 134 (2019) 104555

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Adaptive tracking control for a class of stochastic switched systems with stochastic input-to-state stable inverse dynamics and input saturation✩ Liqiang Yao a,b , Weihai Zhang a , a b

∗

College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, Shandong, China School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, China

article

info

Article history: Received 21 September 2018 Received in revised form 20 February 2019 Accepted 6 October 2019 Available online xxxx Keywords: Stochastic switched systems Input saturation SISS inverse dynamics Adaptive backstepping

a b s t r a c t This paper focuses on the adaptive tracking control for a class of stochastic switched systems. Firstly, a new exponentially practical stability criterion in the moment sense is established for stochastic switched systems, which improves the previous stability criteria in existing literature. Secondly, an adaptive tracking controller based on the stochastic backstepping method is constructed for a class of stochastic systems with stochastic input-to-state stable (SISS) inverse dynamics and input saturation under arbitrary switching strategy. The corresponding closed-loop system is fourth moment exponentially practically stable and the tracking error can be regulated small enough by the obtained stability results. Finally, simulation examples verify the feasibility and effectiveness of the obtained results. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Ever since the stochastic backstepping method was first introduced for the fourth-order Lyapunov function in [1], a large number of scholars have been devoting themselves to applying and developing the stochastic backstepping method. Up to now, the stochastic backstepping method has been the most effective controller design method for stochastic nonlinear systems with strict-feedback form and plenty of related research results have been reported such as [2–6]. Switched systems, which are a class of hybrid systems [7], include a family of continuous-time systems and isolated discrete switching events. Many practical systems (such as robot control systems [8] and networked control systems [9,10]) are often modeled as switched systems. During the past few decades, switched systems have received considerable attention in theoretical research and engineering applications, and have become an active and interdisciplinary research field. With the development of stochastic stability theory [2,11,12], many scholars begin ✩ This work was supported by National Natural Science Foundation of China (Nos. 61973198, 61633014), SDUST, China Research Fund (No. 2015TDJH105), the Research Fund for the Taishan Scholar Project of Shandong Province of China, a Project of Shandong Province Higher Educational Science and Technology Program, China (No. J18KA241) and a Postgraduate Technology Innovation Project of Shandong University of Science and Technology, China (No. SDKDYC190365). ∗ Corresponding author. E-mail addresses: [email protected] (L. Yao), [email protected] (W. Zhang). https://doi.org/10.1016/j.sysconle.2019.104555 0167-6911/© 2019 Elsevier B.V. All rights reserved.

to focus their attention on stochastic switched systems, especially stochastic switched systems with strict-feedback form as well as the related studies in stability [13–16], stabilization [17] and adaptive control [18–20] etc. The common Lyapunov function method is often used to stabilize a stochastic switched system and complete the control task of stochastic switched systems under arbitrarily switching strategy, such as [21] and [22]. For stochastic switched systems with a constrained switching strategy, the preferred approach in system analysis and synthesis is the multiple Lyapunov function method; see [6,14,23,24]. Inverse dynamics, which exist in many actual physical systems (such as attitude control systems of aircraft and manipulator systems), may degrade the performance of the controlled system. In recent years, stochastic systems with stochastic inverse dynamics have attracted a great of attention [25–28], but there exist few relevant results for stochastic switched systems including stochastic inverse dynamics. In addition, the structure and performance of the physical components often lead to saturation nonlinear constraints, which may also cause system performance degradation or even destroy the system stability. Input saturations are a class of nonlinear inputs. It can be found that most existing results about input saturations (such as [29] and [30]) are obtained for stochastic non-switched systems by the fuzzy approximation approach, but few results about stochastic switched systems with input saturations appear. With the aid of an auxiliary system proposed for nonlinear systems with input saturations in [31], [32] and [33] recently discussed tracking control for lower triangular stochastic systems by employing

2

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555

the state feedback and output feedback techniques, respectively. Although the relationship between the input deviation (between the ideal input and the saturated input) and the tracking error is given in [32,33], the tracking error cannot be made small enough. The above analysis motivates us to consider the adjustment of the adaptive tracking error for a class of stochastic switched systems with SISS inverse dynamics and input saturations. In engineering, many practical systems have these three features(i.e., stochastic disturbance, inverse dynamics and input saturations), so the studied systems in this paper have wide application. Inspired by [34], we propose a new exponentially practical stability criterion in the moment sense by the uniformly stable function for stochastic systems under arbitrary switching strategy. The new exponentially practical stability criterion weakens the requirement about the time-derivative of Lyapunov function and can be viewed as a generalization of Theorem 1 in [32] and the exponential stability criterion in [34]. We adopt the method presented in [31] to compensate saturated inputs in the considered stochastic switched systems. On the one hand, the constructed adaptive controller in this paper does not require unknown parameters within a known compact set. On the other hand, the introduction of appropriate adjustment parameters overcomes the defect [31–33] that the tracking error cannot be made arbitrary small. Compared with [35] and [36] in which the considered systems do not include unknown parameters, the nonlinear system in this paper is switched stochastic nonlinear system with unknown parameters. Although [37] studied the global output-feedback stabilization for switched stochastic nonlinear time-delay systems under arbitrary switchings, but the drift and diffusion terms only depend on system output. The drift and diffusion terms in this paper are functions with respect to system states and include unknown parameters. The main work can be summarized as follows: (i) Based on the Lyapunov function and the uniformly stable function presented in [34,38], we give a new exponentially practical stability criterion in the moment sense which weakens the constraints for the Lyapunov function. (ii) By introducing an auxiliary system with a special structure and the appropriate adjustment parameters, the adaptive tracking controller based on the stochastic backstepping method is constructed to regulate the system output such that the tracking error can be made small enough. The remainder of this paper is organized as follows. For stochastic switched systems under arbitrary switching strategy, Section 2 establishes a new exponentially practical stability criterion in the moment sense. The constructed adaptive tracking controller for considered stochastic switched systems and the corresponding closed-loop system stability analysis are given in Sections 3 and 4, respectively. Simulation examples are provided in Section 5. Section 6 concludes this paper. Notions: The real n-dimensional space is denoted as Rn . The set of all nonnegative real numbers is denoted as R+ . |x| stands for the Euclidean norm of a vector x. C j represents the family of all functions with continuous jth partial derivative, and C 1,2 ([t0 , ∞) × Rn ; R+ ) stands for the set of all nonnegative functions V (t , x) that are C 1 in t and C 2 in x. γ (t) ∈ K(K∞ ) means that the function γ (t) is a class K function, which is strictly increasing, continuous and γ (0) = 0; if γ (t) ∈ K∞ , in addition, γ (t) is unbounded. If for each fixed t, β (s, t) ∈ K and limt →∞ β (s, t) = 0 with each fixed s, then β (s, t) ∈ KL, that is to say, β (s, t) is a KL function. For q1 , q2 ∈ R, define q1 ∧ q2 = min{q1 , q2 } and q1 ∨ q2 = max{q1 , q2 }.

2. Preliminaries Consider the stochastic switched system described by dx(t) = fσ (t) (x(t), t)dt + gσ (t) (x(t), t)dw (t),

(1)

n

in which the system state is x(t) ∈ R , the independent standard Wiener process w (t) ∈ Rq is defined on the complete filtered probability space (Ω , F , {Ft }t ≥t0 , P) with Ft satisfying the usual conditions. The switching function σ (t) : [t0 , ∞) → P = {1, 2, . . . , N } is piecewise right continuous. Functions fp ∈ Rn and gp ∈ Rn×q are locally Lipschitz in x and piecewise continuous in t ∈ [t0 , ∞), where p ∈ P and fp (t , 0) = 0, gp (t , 0) = 0. The switching rule σ (t) in system (1) is generated by

σ (t) = pk , t ∈ [τk−1 , τk ), k = 1, . . . , k∗ , ∗

(2) ∗

where pk ∈ P with pk ̸ = pk+1 , and k − 1(k ≤ ∞) denotes the maximal number of switches. The switching instant τk−1 (k = 1, 2, . . . , k∗ , τ0 = t0 ) are stopping times and τk∗ , which satisfies τk∗ ≤ ∞, is unknown. If σ (t) = pk (τk−1 ≤ t < τk ), then the state of the pk th subsystem in system (1) is active. According to Theorem 3.15 in [39], the pk th active subsystem has a continuous, Ft -adapted and unique solution on [τk−1 , τk ). By adopting the same recursive procedure in [13], the solution of system (1) is produced by ∗

x(t) =x(t0 ) +

k ∫ ∑ k=1

∗

+

k ∫ ∑ k=1

t ∧τk

fσ (s) (x(s), s)ds

t ∧τk−1

t ∧τk

gσ (s) (x(s), s)dw (s),

(3)

t ∧τk−1

where t belongs to∫ [t0 , τk∗ ) that ∫ t is the maximal existence interval ∑k∗ t ∧τk = t , then (3) is transformed into for x(t). Let k=1 t ∧τ k−1

∫

0

t

fσ (s) (x(s), s)ds +

x(t) = x(t0 ) +

∫

t0

t

gσ (s) (x(s), s)dw (s). t0

Meanwhile, the solution x(t) is a continuous, unique and Ft adapted solution on [t0 , τk∗ ) since the solution of every active subsystem on the corresponding active time interval is continuous, unique and Ft -adapted. The following Lemma 1, which is another version of Lemma 1 of [40], ensures that there is a unique global solution to system (1). Lemma 1. For system (1), if there are constants c0 , d0 > 0 and a function V (t , x) ∈ C 1,2 ([t0 , ∞) × Rn ; R+ ) such that for any positive constant l and t > t0 , lim inf V (t , x) = ∞,

l→∞,|x|>l

E [V (t ∧ ρl , x(t ∧ ρl ))] ≤ d0 ec0 (t −t0 ) ,

(4) (5)

where ρl = inf{t ≥ t0 : |x(t)| ≥ l} with ρ∞ = liml→∞ ρl almost surely (a.s.) and inf φ = ∞, then for any given x0 ∈ Rn \{0}, system (1) has a unique solution on [t0 , ∞). Proof. It follows from Lemma 5 in [13] that τk∗ = ρ∞ a.s. This means that there is a unique solution on [t0 , ρ∞ ) to system (1). Adopting the same way as Theorem 3.19 in [39], together with (4) and (5), we can prove that ρ∞ = ∞ a.s. Thus, system (1) with any given x0 ∈ Rn \{0} has a unique solution on [t0 , ∞). Next, we generalize Definition 1 in [15] and Definition 1 in [41] and give the definition of practical stability in the moment sense for stochastic switched systems.

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555

Definition 1. If there are a constant d ≥ 0 and a function β (·, ·) ∈ KL such that E |x(t)|m ≤ β (|x0 |, t − t0 ) + d, then system (1) is said to be mth moment practically stable. If d = 0, then system (1) is mth moment asymptotically stable. If β (|x0 |, t − t0 ) = d¯ 0 |x0 |m e−λ(t −t0 ) with d¯ 0 > 0 and λ > 0, and d = 0(d > 0), then system (1) is mth moment exponentially (exponentially practically) stable. Definition 2. For any V ∈ C 1,2 ([t0 , ∞) × Rn ; R+ ) associated with system (1), the infinitesimal generator L is defined as Lσ (t) V =

+ ∂∂Vx fσ (t) + 12 Tr {gσT (t) ∂∂ xV2 gσ (t) }, which reduces to Lp whenever σ (t) = p, p ∈ P . ∂V ∂t

2

Inspired by [34], we expect to establish a new exponentially practical stability criterion in the moment sense for stochastic switched systems, so we introduce the following Definition 3 from [38]. Definition 3. For equation χ˙ (t) = µ(t)χ (t) with µ(t) being a piecewise continuous function, if there is a function β¯ (·, ·) ∈ KL such that |χ (t)| ≤ β¯ (|χ (t0 )|, t − t0 ), then µ(t) is called a uniformly stable function. Remark 1. According to the ∫analysis of [38], µ(t) is a uniformly t stable function if and only if t µ(υ )dυ ≤ −ϵ (t − t0 ) + δ , where 0 ϵ > 0 and δ ≥ 0 are constants.

3

Substituting (11) into (10) and taking expectations on both sides of (10), we obtain that EV (t ∧ ρl , x(t ∧ ρl )) ≤E

[∫

t ∧ρl

∫ t ∧ρl

da e

s

µ(υ )dυ

ds

]

t0

[ ] ∫ t ∧ρl µ(s)ds + E V (t0 , x0 )e t0 .

(12)

In view of Remark 1, there are constants ϵ > 0 and δ ≥ 0 such that E

[∫

t ∧ρl

da e

∫ t ∧ρl s

µ(υ )dυ

]

ds ≤

t0

E [V (t0 , x0 )e

∫ t ∧ρl t0

µ(s)ds

da eδ

ϵ

,

(13)

] ≤ V (t0 , x0 )eδ .

(14)

It follows from (13) and (14) that (12) can be changed into EV (t ∧ ρl , x(t ∧ ρl )) ≤ D0 eδ ,

(15)

where D0 = V (t0 , x0 ) + da /ϵ . On the other hand, (6) implies that (4) holds. Thus, system (1) with any given x0 ∈ Rn \{0} has a unique solution on [t0 , ∞) by Lemma 1. From (6) and (12)–(14), we get that a1 E |x(t ∧ ρl )|m ≤ a2 eδ |x0 |m Ee−ϵ (t ∧ρl −t0 ) + Since liml→∞ ρl = ρ∞ turned into a2

da

eδ . (16) ϵ = ∞ a.s., let l → ∞, then (16) can be

eδ |x0 |m e−ϵ (t −t0 ) +

da

eδ .

Theorem 1. For system (1), if there are positive constants a1 , a2 , m, a non-negative constant da , a uniformly stable function µ(t) and a function V (t , x) ∈ C 1,2 ([t0 , ∞) × Rn ; R+ ) such that

E |x(t)|m ≤

a1 |x|m ≤ V (t , x) ≤ a2 |x|m ,

e−ϵ (t −t0 ) , hence system (1) is mth moment exponentially stable by Definition 1.

(6)

Lp V (t , x) ≤ µ(t)V (t , x) + da , ∀p ∈ P ,

(7)

a1

ϵ a1

(17)

This indicates that system (1) is mth moment exponentially praca tically stable. Particularly, if da = 0, then E |x(t)|m ≤ a2 eδ |x0 |m 1

then system (1) with any given x0 ∈ R \{0} has a unique solution on [t0 , ∞) and is mth moment exponentially practically stable. Particularly, if da = 0, then system (1) is mth moment exponentially stable.

Corollary 1. For system (1), if there are a function V (t , x) ∈ C 1,2 ([t0 , ∞) × Rn ; R+ ), positive constants a1 , a2 , m, µ ¯ and a nonnegative constant da such that a1 |x|m ≤ V (t , x) ≤ a2 |x|m ,

(18)

Proof. From (7) and Definition 2, we get that

Lp V (t , x) ≤ −µ ¯ V (t , x) + da , ∀p ∈ P ,

(19)

n

(

Lp e

−

∫t

t0

µ(s)ds

)

−

V (t , x) ≤ da e

∫t

t0

µ(s)ds

, ∀p ∈ P .

− Meanwhile, we use Itô’s formula for ˜ V = e along system (1), then

∂˜ V

d˜ V = Lσ (t)˜ V dt +

∂x

gσ (t) (x(t), t)dw (t).

(8) ∫t

t0

µ(s)ds

V (t , x)

(9)

Taking integrals from t0 to t ∧ ρl on both sides of (9) results in V (t ∧ ρl , x(t ∧ ρl ))

=V (t0 , x0 )e

∫ t ∧ρl t0

µ(s)ds

+e

∫ t ∧ρl t0

µ(s)ds

∫

t ∧ρl

Lσ (s)˜ V ds t0

+e

∫ t ∧ρl t0

µ(s)ds

t ∧ρl

∫

t0

∑k∗ ∫ t ∧τk

Since k=1 we have

∫

t ∧τk−1

= ∗

t ∧ρl

Lσ (s)˜ V ds =

∂˜ V gσ (s) (x, s)dw (s). ∂x ∫t t0

k ∫ ∑

, together with (2), τk∗ = ρ∞ a.s. and (8), t ∧τk ∧ρl

Lσ (s)˜ V ds

k=1 t ∧τk−1 ∧ρl ∫s t ∧ρl

t0

∫

da e

≤ t0

(10)

−

t0

µ(υ )dυ

ds.

then there is a unique solution on [t0 , ∞) to system (1) with any given x0 ∈ Rn \{0} and system (1) is mth moment exponentially practically stable. Remark 2. (i) Since µ(t) = −µ ¯ is a uniformly stable function with ϵ = µ ¯ > 0 and δ = 0, Corollary 1 holds for system (1) according to Theorem 1 and Remark 1. (ii) Theorem 1 is obtained for stochastic switched system under arbitrary switching, so it is clear that Theorem 1 and Corollary 1 still hold for non-switched stochastic system. Example 1 in Section 5 illustrates that Theorem 1 is also effective for a stochastic non-switched system. (iii) For most existing stability results such as Theorem 4.4.4 in [11], Theorem 5.11 in [12], Theorem 1 in [15], Theorem 1 in [32], the time-derivative of Lyapunov function must be negative definite and this constraint is similar to (19) with da = 0 in Corollary 1. However, Theorem 1 does not restrict the timederivative of Lyapunov function to be negative definite because of the uniformly stable function µ(t). So, Theorem 1 generalizes the aforementioned stability results and some practical stability results (e.g., Lemma 1 in [41]). Lemma 2 ([42]). For functions h1 , h2 , . . . , hn and κ ≥ 1, the following inequality holds.

(11)

|h1 + h2 + · · · + hn |κ ≤ nκ−1 (|h1 |κ + |h2 |κ + · · · + |hn |κ ).

4

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555

Lemma 3 ([43]). For any constants ι1 > 0, ι2 > 0 and any positive definite function φ (χ1 , χ2 ), the following inequality holds.

|χ1 |ι1 |χ2 |ι2 ι2 ι1 φ (χ1 , χ2 )|χ1 |ι1 +ι2 + φ −ι1 /ι2 (χ1 , χ2 )|χ2 |ι1 +ι2 . ≤ ι1 + ι2 ι1 + ι2 3. Adaptive tracking control

3.2. Adaptive tracking controller design

3.1. Problem formulation

Firstly, we construct the following auxiliary system to compensate the effect caused by the saturated input.

Consider the stochastic switched system described by

⎧ dχ ⎪ ⎪ ⎨

dxi

⎪ dx ⎪ ⎩ n y

= f0 (χ , y, t)dt + g0 (χ, y, t)dw(t), = xi+1 dt + fi,σ (t) (χ, x¯ i )dt + gi,σ (t) (χ, x¯ i )dw(t), = u(v )dt + fn,σ (t) (χ, x)dt + gn,σ (t) (χ, x)dw(t), = x1 , i = 1, 2, . . . , n − 1,

{ (20)

v (t) ≤ um ;

um < v (t) < u¯ M ;

(21)

u¯ M ≤ v (t);

in which v (t) ∈ R is the input signal of the saturation controller u(v (t)), u¯ M and um are the known upper bound and lower bound of u, respectively. For system (20) with arbitrary switching strategy, our aim is to construct an adaptive state feedback controller such that the closed-loop system is exponentially practically stable in the moment sense and the system output y(t) can asymptotically track the given trajectory yr (t) satisfying |yr (t)| ∨ |˙yr (t)| < M0 with M0 being a positive constant. For this purpose, we propose the following assumptions: Assumption A1. For stochastic inverse dynamics χ -system, there exist a uniformly stable function µ(t), a function V0 (χ ) ∈ C 2 (Rn , R+ ) and constants q0 > 0, q¯ 0 > 0, m1 ≥ m0 ≥ 4, d01 ≥ 0 such that q |χ|m0 ≤ V0 (χ ) ≤ q¯ 0 |χ|m0 ,

(22)

0

Lp V0 (χ ) ≤ µ(t)V0 (χ ) + d01 |y|

m1

, ∀p ∈ P .

(23)

Assumption A2. For any p ∈ P , functions fi,p and gi,p satisfy that

{ ∑ |fi,p (χ, x¯ i )| ≤ θp |χ| + θp ij=1 |xj |, ∑ |gi,p (χ , x¯ i )| ≤ θp |χ| + θp ij=1 |xj |,

dνi (t)

= (νi+1 (t) − li νi (t))dt , i = 1, . . . , n − 1, = (∆u(t) − ln νn (t))dt ,

dνn (t)

in which the unmeasurable stochastic inverse dynamics is χ ∈ R, the output is y ∈ R. Functions f0 ∈ R and g0 ∈ R are piecewise continuous in t ∈ [t0 , ∞), and locally Lipschitz in χ and y, respectively. w(t) ∈ R is a standard Wiener process. The switching function σ (t) : [t0 , ∞) → P = {1, 2, . . . , N } is piecewise right continuous. For any p ∈ P , functions fi,p ∈ R and gi,p ∈ R are locally Lipschitz in χ and x¯ i , respectively. Moreover, fi,p (0, 0) = gi,p (0, 0) = 0, x¯ i = (x1 , x2 , . . . , xi )T and x = x¯ n . The saturation input u(v ) is described by

⎧ ⎨ um , u(v (t)) = sat(v (t)) = v (t), ⎩ u¯ M ,

Remark 4. For any p ∈ P , the pth subsystem dx = fp (x, v ) + gp (x, v )dw (t) with v being control input is bounded-input bounded-state (BIBS) a.s., if for any essentially bounded measurable and Ft -adapted input process v with ∥v∥∞ = sups∈[t0 ,∞) infA⊂Ω ,P(A)=0 sup{|v (s, ω)| : ω ∈ Ω \A}, there is a positive constant r¯0 such that P {|x(t)| < r¯0 } = 1.

(24)

where θp (p ∈ P ) are unknown positive constants. Remark 3. For convenience in the controller design, we propose Assumption A2. Assumption A2 means that the drift term fi,p and the diffusion term gi,p can be restrained by a linear function with respect to χ and xj (j = 1, 2, . . . , i). Assumption A2 is often used in many existing literature, for example, Assumption A2 with known θp (p ∈ P ) is equivalent to Assumption 5 in [27] with p = 1 and li1 = li2 = 0. In fact, Assumption A2 can be further weakened, which is discussed in Remarks 6 and 7. Assumption A3. For any p ∈ P , the pth subsystem is BIBS a.s.

(25)

in which l1 , l2 , . . . , ln are positive parameters to be determined, u(v (t)) − v (t) = ∆u(t), and νi (0) = 0(i = 1, . . . , n). Next, we introduce the coordinate transformation: z1 = y − ν1 − yr , zi+1

(26)

= xi+1 − νi+1 − αi (x¯ i , θˆ ), i = 1, 2, . . . , n,

(27)

in which αi (i = 1, . . . , n) are stabilizing functions to be de4 signed, θˆ represents the estimation of θ = max{θ ∗ 3 , θ ∗4 }, θ ∗ = maxp∈P {θp }, zn+1 = 0, xn+1 = u(v ), and νn+1 = ∆u. By Itô’s differentiation rule, it follows from (26) and (27) that dzi =(zi+1 + αi + ϑi,σ (t) + li νi − ϖi−1 −

∂αi−1 ˙ θˆ )dt ∂ θˆ

+ δi,σ (t) dw(t), i = 1, 2, . . . , n, ∑i−1 ∑i−1 ∂αi−1 where ϑi,σ (t) = fi,σ (t) − j=1 j=1 ∂ xj fj,σ (t) , ϖi−1 = ∑i−1 ∂αi−1 δi,σ (t) = gi,σ (t) − j=1 ∂ x gj,σ (t) , α0 = 0.

(28) ∂αi−1 xj+1 , ∂ xj

j

Lastly, we begin to design the adaptive tracking controller by the backstepping recursive procedure. Step 1 Choosing the candidate Lyapunov function V1 = 14 z14 + 1 4 ν + 2λ1θ θ˜ 2 + c0 V0 with θ˜ = θ − θˆ , λθ > 0 and c0 > 0. 4 1 The infinitesimal generator acting on V1 along the pth subsystem satisfies that 3 Lp V1 ≤z13 (z2 + α1 + f1,p + l1 ν1 − y˙ r ) + z12 g12,p 2 1 (29) + ν13 (ν2 − l1 ν1 ) − θ˜ θ˙ˆ + c0 Lp V0 .

λθ

By Assumption A2 and Young’s inequality,

⎧ ⎪ ⎪ ⎪ ⎪ z13 z2 ≤ 34 z14 + 14 z24 , ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ z13 l1 ν1 ≤ 43 l13 z14 + 41 ν14 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ν 3 ν ≤ 3 ν 4 + 1 ν 4 , 1 2

4 1

4 2

(30)

⎪ −1 ⎪ ⎪ −z13 y˙ r ≤ 34 dr 3 z14 + 41 dr M04 , ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ 3 −1 1 ⎪ ⎪ z1 f1,p ≤ 34 (( 2n dχ )− 3 + d113 )z14 θ + 2n dχ |χ|4 + 14 d11 |x1 |4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 3 z 2 g 2 ≤ 9 (nd−1 + d−1 )z 4 θ + 1 d |χ|4 + 1 d |x |4 . 2 1 1,p

2

χ

12

1

2n

χ

2

12

1

Substituting (30) into (29) leads to 1 1 1 ˙ Lp V1 ≤ z24 − ¯l1 ν14 + ν24 + (ωθ 1 z14 − θˆ )θ˜ 4 4 λθ 3 3 −1 3 4 + z13 ( z1 + dr 3 z1 + l13 z1 + ωθ 1 θˆ z1 + α1 ) 4 4 4 1 dr 4 + c0 µ(t)V0 + dχ |χ| + d¯ 1 |x1 |4 + M04 n 4

+ c0 d01 |y|m1 ,

(31)

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555 − 31

1

−1 1 where d¯ 1 = 14 d11 + 21 d12 , ωθ 1 = 34 (( 2n dχ )− 3 + d11 ) + 29 (nd− χ + d12 ), ¯l1 = l1 − 1 > 0. d11 , d12 , dr and dχ are positive constants. Choosing the stabilizing function α1 and the tuning function τ1 as

3

3

4 cθ

4

3

− 13

4 3

α1 = −c1 z1 − z1 − dr z1 − l1 z1 − ωθ 1 θˆ z1 , τ1 = ωθ 1 z14 −

λθ

4

θˆ ,

So, substituting (35) and (37)–(43) into (36) yields that 3 4 3 1 Lp Vi ≤ zi4+1 + zi3 (zi + li3 zi + πiϖ zi + ωθ i θˆ zi 4 4 4

+ α i − λθ

(32)

+ω

4 θ i zi

(33)

where c1 > 0 and cθ > 0. Then, it follows from (31), (32) and (33) that

θ˜ + (τi−1

i−1 ∑

+ λθ

Lp V1 ≤

4

z24

+ +

− cθ

λθ dr 4

c1 z14

−¯ ν + l1 14

1 4

1 ν + (τ1 − θ˙ˆ )θ˜ 4 2

n

+ c0 d01 |y| .

where ωθ i = ωθ i1 + ωθ i2 , d¯ i = d¯ i−1 +

+ 2di2 + di3 ), ¯li = ¯ > 0, diu = 1(i = 2, . . . , n − 1) and d¯ nu = du . li − − Let the tuning function τi = ωθ i zi4 + τi−1 and the stabilizing ¯

3 (diu ) 4

1 2

αi = − ci zi − zi − li3 zi − πiϖ zi − ωθ i θˆ zi

i−1

3

|xj |4 ,

(35)

+ (τi −

+

Lp Vi ≤Lp Vi−1 + zi3 (zi+1 + αi + ϑi,p + li νi − ϖi−1

(36)

According to Assumption A2 and Young’s inequality, we get that

li zi3 νi ≤

νi3 νi+1 ≤ ν ν

3 n n+1

≤

4 3 4 3 4

1 4 3

zi4+1 ,

2

3 4

(39)

4

du ν + 4 n

j=1

j=1

+

cθ

λθ dr 4

i ∑ ∂αj−1 1 ˙ 1 ˙ θˆ )θ˜ + λθ zj3 (τi − θˆ )

λθ

j=2

λθ

∂ θˆ

i θˆ θ˜ + c0 µ(t)V0 + dχ |χ|4 + d¯ i n

i ∑

|xj |4

j=1

M04 + c0 d01 |y|m1 .

(46)

In the last step (i = n), let the update law

θ˙ˆ = λθ τn , τn = ωθ n zn4 + τn−1 ,

(47)

3

3

4

v = − cn zn − zn − ln3 zn − πnϖ zn − ωθ n θˆ zn 4

νi4 + νi4+1 (i = 2, . . . , n − 1),

zi2 δi2,p ≤ ωθ i2 zi4 θ +

− zi3 ϖi−1 ≤

(38)

4 1

− 13

i

and the actual control law

4

νi4 + li3 zi4 ,

zi3 ϑi,p ≤ ωθ i1 zi4 θ + 3

(37)

i

∑ ∑ 1 ¯lj νj4 + 1 d¯ iu νi4+1 Lp Vi ≤ zi4+1 − cj zj4 − 4 4

j=1

∂αi−1 ˙ 3 θˆ ) + zi2 δi2,p + νi3 (νi+1 − li νi ). 2 ∂ θˆ

(45)

where ci > 0. Then, (44) can be turned into

satisfies (35). From (25) and (28), we can deduce that

4 1

∑ ∂αj−1 ∂αi−1 τi + λθ ωθ i zi zj3 , ∂ θˆ ∂ θˆ j=2

1 ˙ dr cθ θˆ )θ˜ + θˆ θ˜ + c0 µ(t)V0 + M04 λθ 4 i−1 ∑

4

i−1

+ λθ

λθ

dχ |χ |4 + d¯ i−1

zi4 +

3

4

4

where Vi−1 = Vi−2 + 41 zi4−1 + 14 νi4−1 , ¯lj > 0, cj > 0, and d¯ i−1 > 0. In the following, we prove that Vi = Vi−1 + 14 zi4 + 14 νi4 also

3

1 (di1 4

j=1

+ (τi−1 −

zi3 zi+1 ≤

− 31

j=1

∂αj−1 1 ˙ θˆ ) + c0 d01 |y|m1 zj3 (τi−1 − λθ ∂ θˆ

(44)

4

function

j=2

−

λθ

i−1

i−1 ∑

n

λθ

i−1

∑ ∑ 1 ¯lj νj4 + 1 νi4 Lp Vi−1 ≤ zi4 − cj zj4 − 4 4

+

j=1

i 1 ˙ ∂αj−1 θˆ ) + dχ |χ|4 (τi−1 − ˆ λ n θ ∂θ

zj3

j=1

(34)

Step i (i = 2, 3, . . . , n) Assume that we have designed a series of smooth functions αj , τj (1 ≤ j ≤ i − 1) such that

+ λθ

j=1

1 cθ − θ˙ˆ )θ˜ + θˆ θ˜ + c0 µ(t)V0

∂αi−1 1 ˙ 1 (τi − θˆ ) + d¯ iu νi4+1 ˆ λθ 4 ∂θ i ∑ d r + d¯ i |xj |4 + M04 + c0 d01 |y|m1 ,

1

m1

i

+ λθ zi3

λθ

θˆ θ˜ + c0 µ(t)V0 + dχ |χ|4 + d¯ 1 |x1 |4 M04

i−1

∑ ∑ ∂αi−1 ¯lj νj4 τi ) − cj zj4 − ˆ ∂θ

j=2

1

5

1 4 1

2n 1 2n 1

ν

du n4+1

dχ |χ|4 +

dχ |χ|4 +

πiϖ zi4 + di3 4

,

4 1 2

di1

i ∑

|xs |4 ,

(41)

s=1

∑

|xs |4 ,

(48)

j=2

Lp Vn ≤ − p¯ 0

n (∑ 1 j=1

i

di2

∑ ∂αj−1 ∂αn−1 τn + λθ ωθ n zn zj3 , ∂ θˆ ∂ θˆ

then, together with zn+1 = 0, νn+1 = ∆u and d¯ nu = du , we have

|xs |4 ,

(42)

s=1

i

∑

n−1

+ λθ

(40) 1

4

(43)

s=1

where positive constants di1 , di2 , di3 and du are to be selected. The proof of (41)–(43) are given in Appendix.

4

zj4 +

n ∑ 1 j=1

4

νj4 +

1 2λθ

θ˜ 2

)

+ c0 µ(t)V0 + dχ |χ|4 + H(θ, ∆u, x), (49) ∑n 4 4 2 where H(θ, ∆u, x) = cθ θ /2λθ + d¯ n j=1 |xj | + dr M0 /4 + c0 d01 |y|m1 + du |∆u|4 /4 and p¯ 0 = min{4c1 , . . . , 4cn , 4¯l1 , . . . , 4¯ln , cθ }. By Lemma 3 and Assumption A1, it is certain that dχ |χ|4 ≤

4dχ m0 q

0

V0 +

m0 − 4 m0

dχ .

(50)

6

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555

Substitute (50) into (49), then we obtain that n

n

Lp Vn ≤ − p¯ 0 (

∑1 4

j=1

zj4 +

+µ ¯ (t)V0 +

∑1 4

j=1

m0 − 4 m0

1

νj4 +

2λθ

θ˜ 2 )

dχ + H(θ, ∆u, x),

(51)

where µ ¯ (t) = c0 µ(t) + 4dχ /m0 q . According to Remark 1, µ(t) is a 0

uniformly stable function, so we have

∫t t0

µ(s)ds ≤ −ϵ (t − t0 ) + δ

with ϵ > 0 and δ ≥ 0. Choose dχ = (n − 1)ϵ c0 m0 q /4n, then 0

µ ¯ (t) = c0 (µ(t) + ϵ ) ≜ c0 µ0 (t) and ∫ t ∫ t n−1 1 µ(s)ds + µ0 (s)ds = ϵ (t − t0 ) ≤ − ϵ (t − t0 ) + δ, n−1 n

n

t0

t0

n

which means that µ0 (t) is also a uniformly stable function by Remark 1. For t ∈ [t0 , ∞), we define if µ0 (t) ≤ −¯p0

{ −¯p0 µ ˜ (t) = µ0 (t)

(52)

if − p¯ 0 ≤ µ0 (t),

then µ ˜ (t) is a uniformly stable function and t0 ) + δ with p0 = min{ 1n ϵ, p¯ 0 }. By (52), (51) can be rewritten as

∫t t0

µ ˜ (s)ds ≤ −p0 (t −

and (48), ∆u = u(v ) − v is bounded a.s., so there exists a positive constant Mu such that P {|∆u|4 < Mu } = 1. From the analysis above and (53), we get that (54)

¯ + du Mu /4, M ¯ = cθ θ 2 /2λθ + d¯ n R4 + dr M 4 /4 + where M = M 0 0 m1 c0 d01 R0 + (m0 − 4)(n − 1)ϵ c0 q /4n. 0

Theorem 2. For system (20) with Assumptions A1–A3, there is a state feedback controller (48) such that the closed-loop system including (20), (25)–(27) and (47)–(48) under arbitrary switching strategy (i) has a unique solution on [t0 , ∞) for any given x0 ∈ Rn \{0}. (ii) is fourth moment exponentially practically stable. (iii) has the following tracking performance

p0

8du p0

(Mu + E( sup |∆u(s)|4 )), t0 ≤s≤t

where e(t) = y(t) − yr (t) is the tracking error. Moreover, the tracking performance can be improved as much as possible by the parameter adjustment technique. Proof. For system (20) satisfying Assumptions A1–A3, we choose the Lyapunov function V = Vn =

n 1∑

4

k=1

zk4 +

n 1∑

4

k=1

νk4 +

1 2λθ

θ˜ 2 + c0 V0 .

(55)

(56)

Next, we analyze limt →∞ E∑ |ν1 (t)|4 . For system (25), choosing n 1 4 the Lyapunov function Vν = 4 k=1 νk and taking the design procedure of the adaptive controller in Section 3 into consideration, we have n−1 ∑ 3 k=1

+

1 3 −1 ( νk4 + νk4+1 − lk νk4 ) + du 3 νn4 4 4 4

1 4

du |∆u|4 − ln νn4 +

n 1∑

4

νk4

k=1

1 ≤ − ¯l0 Vν + du |∆u|4 ,

(58)

4

where du is a positive design parameter, ¯l0 = min1≤k≤n {4¯lk } with 1 3 ¯l1 = l1 −1, ¯lk = lk − 5 (k = 2, . . . , n−1) and ¯ln = ln − 1 − 3 d− > 0. 4 2 4 u Further, we have the following inequality from (58): ¯

EVν ≤ Vν |t =0 e−l0 (t −t0 ) +

du 4¯l0

E( sup |∆u(s)|4 ), t0 ≤s≤t

which signifies that E |ν1 (t)|4 ≤ 4Vν |t =0 e−l0 (t −t0 ) +

du

¯l0

E( sup |∆u(s)|4 ). t0 ≤s≤t

Let t → ∞, then lim E |ν1 (t)|4 ≤

t →∞

4. Stability analysis

t →∞

eδ .

¯

L p Vn ≤ µ ˜ (t)Vn + M , a.s.,

eδ +

M p0

It follows from (56) that limt →∞ E |z1 (t)|4 ≤ 4M eδ , together p0 with (26) and Lemma 2, the tracking error e(t) = y(t) − yr (t) satisfies 32M δ lim E |e(t)|4 ≤ e + lim 8E |ν1 (t)|4 . (57) t →∞ t →∞ p0

(53)

¯ θ , ∆u, x) = H(θ, ∆u, x) + (m0 − 4)(n − 1)ϵ c0 q /4n. where H( 0 Since each subsystem input u(v ) in switched system (20) is subject to a saturation type nonlinearity as defined in (21) and Assumption A3 holds for each subsystem of switched system (20), the solution of every subsystem is bounded a.s. by Remark 4. The solution of switched system (1) is generated by (3), so there exists a positive constant R0 such that the solution of the switched system (20) satisfies P {|x(t)| < R0 } = 1, which, together with ∑n ∑n 4 4 4 ¯ 4 ¯ j=1 |xj | < dn R0 } = 1. By (21) j=1 |xj | ≤ |x| , leads to P {dn

¯ 32M

EV ≤ V |t0 e−p0 (t −t0 )+δ +

V˙ ν ≤

¯ θ , ∆u, x), L p Vn ≤ µ ˜ (t)Vn + H(

lim E |e(t)|4 ≤

It is clear that for system (20), liml→∞ inf|x|>l V (x) = ∞ and Lp V ≤ µ ˜ (t)V + M , ∀p ∈ P , a.s., where the uniformly stable function µ ˜ (t) and M are as defined in (52) and (54), respectively. So, for any given x0 ∈ Rn \{0}, the closed-loop system containing (20), (25)–(27) and (47)–(48) has a unique solution and is exponentially practically stable in fourth moment sense by Theorem 1. Furthermore, the Lyapunov function V has the following property

du

¯l0

E( sup |∆u(s)|4 ).

(59)

t0 ≤s≤t

¯ + Mu /4, yields Substituting (59) into (57), together with M = M that lim E |e(t)|4 ≤

t →∞

¯ 32M p0

eδ +

8du p0

(Mu + E( sup |∆u(s)|4 )), t0 ≤s≤t

which implies that the tracking error can be regulated small enough by choosing appropriate adjustment constants. Remark 5. (i) If ∆u = 0 (i.e., u(v ) = v ), which indicates that the subsystem input is not subject to a saturation type nonlinearity, ¯ δ /p0 . then the tracking error satisfies limt →∞ E |e(t)|4 ≤ 32Me m1 4 4 2 ¯ ¯ Note that M = cθ θ /2λθ + dn R0 + dr M0 /4 + c0 d01 R0 + (m0 − 4)(n − 1)ϵ c0 q /4n, we can make c0 , d11 , d12 , dij (i = 2, . . . , n; j = 1, 2, 3) 0

¯ can be and dr small enough and λθ large enough such that M adjusted arbitrarily small at the cost of a larger controller gain. (ii) If limt →∞ ∆u(t) = 0, then the tracking error e(t) can also be made small enough by adopting the same way as item (i). ¯ δ /p0 arbitrarily small by (iii) If ∆u ̸ = 0, we can make 32Me the same parameter regulation approach as item (i) and select du small enough, then the tracking error e(t) can be regulated

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555

7

∫t

µ1 (s)ds ≤ −4(t − t0 ) + 2 holds, which implies that µ1 (t) is a t0 uniformly stable function by Remark 1. According to Theorem 1, the solution to system (60) is exponentially stable in fourth moment sense. Fig. 1 shows the state trajectory of system (60) with ϕ (x) = −2.5x and x0 = −1.5 in a solid line. 1 case 2: If ϕ (x) = −3x + 23 x 3 , let the Lyapunov function V = x4 , then ∫we deduce that LV ≤ µ2 (t)V + 1 with µ2 (t) = 2t cos t 2 − 1 t and t µ2 (s)ds ≤ −(t − t0 ) + 2(t ≥ t0 ≥ 0). So, µ2 (t) is also 0 a uniformly stable function and the solution to system (60) is exponentially practically stable in fourth moment sense. The state 1 trajectory of system (60) with ϕ (x) = −3x + 32 x 3 and x0 = 2 is given as the dotted line in Fig. 1. Fig. 1. The state trajectory of system (60) in different case.

Example 2. Consider the stochastic switched system: arbitrarily small. It should be pointed out that these adjusting parameters are independent of each other and have nothing to do with the value of p0 . (iiii) According to the definition of ∆u and Theorem 2, the fluctuation of the input saturation upper and lower bounds do not destroy the stability of the considered system, but the controller parameters need to be readjusted to ensure the tracking error small enough. Remark 6. From the design procedure of the adaptive controller in this section, if Assumption A2 is relaxed to Assumption A2′ (i.e., For any p ∈ P , there exist positive-definite functions f¯i,p (x¯ i ) and g¯i,p (x¯ i ) satisfying |fi,p (χ, x¯ i )| ≤ θp |χ | + θp f¯i,p (x¯ i ) and |gi,p (χ, x¯ i )| ≤ θp |χ| + θp g¯i,p (x¯ i ) with θp being unknown positive constants.), then Theorem 2 still holds. In fact, since system (20) satisfies Assumption A3 and u(v ) satisfies (21), functions f¯i,p (x¯ i ) and g¯i,p (x¯ i ) are bounded a.s. The technique that is used to deal with can also be employed to cope with f¯i,p (x¯ i ) and g¯i,p (x¯ i ).

∑i

j=1

|xj |

Remark 7. If system (20) does not contain SISS inverse dynamics, i.e.,

⎧ ⎨dxi

dxn

⎩

y

= xi+1 dt + fi,σ (t) (x¯ i )dt + gi,σ (t) (x¯ i )dw(t), = u(v )dt + fn,σ (t) (x)dt + gn,σ (t) (x)dw(t), = x1 , i = 1, 2, . . . , n − 1,

and only satisfies Assumptions A3 and A2′′ (i.e., For any p ∈ P and i ∈ {1, 2, . . . , n}, there are functions f˜i,p (x¯ i ) and g˜i,p (x¯ i ) satisfying fi,p (x¯ i ) = θp f˜i,p (x¯ i ) and gi,p (x¯ i ) = θp g˜i,p (x¯ i ) with θp being unknown constants.). Adopting the same controller design procedure proposed in this section, we can construct an adaptive tracking controller to track a given trajectory as close as possible under arbitrary switching strategy. By Corollary 1, the corresponding closed-loop system is fourth moment exponentially practically stable. This means that the presented adaptive tracking controller scheme improves the conservatism of the existing results [31–33]. Here, we omit the detailed controller design procedure, which is similar to the steps shown above and we only give Example 2 in Section 5 to illustrate this point. 5. Simulation examples Example 1. Consider the stochastic system: t cos t 2

x + ϕ (x))dt + xdw (t), (60) 2 where ϕ (x) is a locally Lipschitz function and w (t) ∈ R is a standard Wiener process. case 1: If ϕ (x) = −2.5x, then we choose the Lyapunov function V = x4 . After a series of computations, we have LV ≤ µ1 (t)V with µ1 (t) = 2t cos t 2 − 4. Further, for any t ≥ t0 ≥ 0,

dx(t) = (

⎧ ⎨dx1 = x2 dt + f1,σ (t) (x1 )dt + g1,σ (t) (x1 )dw(t), dx = u(v )dt + f2,σ (t) (x)dt + g2,σ (t) (x)dw (t), ⎩ 2 y = x1 ,

(61)

where w (t) ∈ R is a standard Wiener process and the switching function σ (t) = p, p ∈ {1, 2, 3}, for any t ∈ [0, ∞). 2 f1,1 (x1 ) = θ1 x21 , g1,1 (x1 ) = θ1 x1 sin(x21 ), f2,1 (x) = θ1 e−x2 , g2,1 (x) = 2 θ1 x1 sin(x1 x2 ), f1,2 (x1 ) = θ2 x1 , g1,2 (x1 ) = θ2 x1 , f2,2 (x) = θ2 x1 x2 , g2,2 (x) = θ2 x21 sin x2 , f1,3 (x1 ) = θ3 x31 , g1,3 (x1 ) = θ3 x1 cos x1 , f2,3 (x) = θ3 x2 , g2,3 (x) = θ3 x1 cos x2 , where θ1 , θ2 and θ3 are unknown parameters. It is clear that Assumptions A2" and A3 hold for system (61). By Remark 7, we can design the adaptive controller to track the given reference trajectory yr (t) = cos t + sin t. The controller design procedure: Step 1: Since there are no SISS inverse dynamics in system (61), we choose dχ = 0, c0 = 0 and obtain that z13 f1,p ≤ d12 z1 θ +

3 −1 4 4

3 d g4 , 4 12 1,p g1,2 x21

˜

1

3 −3 4 d z 4 11 1

where f˜1,1

θ + 14 d11 f˜14,p and 23 z12 g12,p ≤ = x21 , f˜1,2 = x1 , f˜1,3 = x31 , g˜1,1 =

, g˜1,3 = x1 cos x1 . Then we design the stabi4 −1 lizing function α1 = −(c1 + 43 + 34 dr 3 + 43 l13 + ωθ 1 θˆ )z1 and the 1 −1 3 ˆ θ with ωθ 1 = 3 (d− tuning function τ1 = ωθ 1 z14 − cθ θ/λ 11 + d12 ).

x1 sin(x21 )

,˜

=

Step 2: Using Young’s inequality, we get that z23 ϑ2,p ≤ 3 2

z22 δ22,p ≤

−z23 ϖ1 ≤

3 4 3 4 3 4

−1

1

−1

4 1

d213 z24 θ + d223 z24 θ + −1

d233 z24 +

1 4

4

4

d21 (f˜2,p − η1,1 f˜1,p )4 , d22 (g˜2,p − η1,1 g˜1,p )4 ,

d23 (η1,1 x2 )4 ,

2

where f˜2,1 (x) = e−x2 , f˜2,2 (x) = x1 x2 , f˜2,3 (x) = x2 , g˜2,1 (x) = x1 sin(x1 x2 ), g˜2,2 (x) = x21 sin x2 , g˜2,3 (x) = x1 cos x2 . Let the update

˙

law θˆ = λθ (ωθ 2 z24 +τ1 ) ≜ λθ τ2 with ωθ 2 =

1

3 −3 d 4 21

−1

+ 34 d223 , then the 4

actual control law can be designed as v = −c2 z2 − z2 − 43 l23 z2 − 1

3 −3 d z 4 23 2

1 τ . − ωθ 2 θˆ z2 + λθ ∂α ∂ θˆ 2

The choice of design parameters: In the simulation, we choose c1 = c2 = 12, l1 = 1.2, l2 = 1.8, cθ = 1, λθ = 8, dr = 0.5, du = 0.5, d11 = d12 = d21 = d22 = d23 = 5; the initial value x1 (0) = 0.8, x2 (0) = −0.6, θˆ (0) = −0.5, ν1 (0) = 0, ν2 (0) = 0; the true values of θp (p = 1, 2, 3) are selected as θ1 = 1, θ2 = −0.8 and θ3 = 0.8; the saturation control input u(v ) = sat(v ) with u¯ M = 20 and u¯ m = −20. Fig. 2 shows the trajectory of the given switching strategy σ (t), under which Fig. 3 illustrates the responses of the corresponding closed-loop system. Simulation results demonstrate the validity of the presented controller design method.

8

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555

Fig. 2. The trajectory of switching strategy σ (t). Fig. 4. The trajectory of the switching strategy σ (t).

Example 3. Consider the following stochastic switched systems with SISS inverse dynamics:

⎧ dχ ⎪ ⎪ ⎨

dx1

⎪ dx ⎪ ⎩ 2 y

= f0 (χ , y, t)dt + g0 (χ, y, t)dw(t), = x2 dt + f1,σ (t) (χ, x1 )dt + g1,σ (t) (χ, x1 )dw(t), = u(v )dt + f2,σ (t) (χ, x)dt + g2,σ (t) (χ, x)dw(t), = x1 ,

(62)

where x = (x1 , x2 )T and the switching law σ (t) : [0, ∞) → P = {1, 2}. w (t) ∈ R is a√standard Wiener process. f0 = tcost χ − 3.5χ + 0.5y, g0 = 2χ , f1,1 = 0, g1,1 = θ1 x1 , f2,1 = 8(1+t)

θ1 χ cos x1 , g2,1 = θ1 x2 sin x2 , f1,2 = 0, g1,2 = θ2 x1 sin x1 , f2,2 = θ2 χ sin x2 , g2,2 = θ2 x2 cos x1 , where θ1 and θ2 are unknown positive parameters. For χ -system, let the Lyapunov function V0 = χ 4 , then for tcost − 21 ∀p ∈ P , we have Lp V0 ≤ µ3 (t)V0 + 0.5y4 with µ3 (t) = 2(1 +t)

Fig. 5. The trajectory of the saturation control input.

∫t

and t µ3 (s)ds ≤ −0.5(t − t0 ) + 2(t ≥ t0 ≥ 0). So, χ -system 0 satisfies Assumption A1 with q = q¯ 0 = 1, ϵ = 0.5, δ = 2, 0

d01 = 0.5. Besides, we can verify that Assumptions A2 and A3 hold for system (62). In the simulation, the reference signal yr (t) = 2(sin t − sin 0.5t), the true parameter values are set as θ1 = 0.8 and θ2 = 1. We select the design parameters c1 = 1, c2 = 1, l1 = 1.2, l2 = 1.5, cθ = 1, λθ = 5, dr = 0.1, du = 0.1, c0 = 1, dχ = 0.25, d12 = d22 = d23 = 1.5; the initial value χ (0) = 0.1, x1 (0) = 0.01, x2 (0) = −0.02, θˆ (0) = −0.1, ν1 (0) = 0, ν2 (0) = 0. For a given switching strategy σ (t) shown in Fig. 4, we simulate the corresponding tracking performance for system (62) with u(v ) = sat(v ), u¯ M = 15 and u¯ m = −15. The trajectory of

the saturation control input is given in Figs. 5, and 6 provides the responses of the closed-loop system. The simulation results illustrate that the presented controller design procedure in this paper is valid. 6. Conclusions For a class of stochastic switched systems with SISS inverse dynamics and input saturation, the adaptive tracking control has been studied in this paper. By a new established exponentially practical stability criterion and the stochastic backstepping

Fig. 3. The responses of the closed-loop system.

L. Yao and W. Zhang / Systems & Control Letters 134 (2019) 104555

9

Fig. 6. The responses of the closed-loop system.

method, an adaptive tracking controller is constructed to guarantee that the closed-loop system is exponentially practically stable and the system output can track a given reference trajectory as close as possible in fourth moment sense. Some examples illustrate that the proposed scheme is feasible and effective.

where πiχ 2 = 3(i + 1) + 3(i − 1) max{3(i + 1) + 3(i − 1) 1)

Appendix

≜ ηi−1,j (1 ≤ j ≤ i − 1, 2 ≤ i ≤ n), then it

follows from (28) and Assumption A2 that i ∑

1, . . . ,

|xs |),

(63)

∑ 1 |ηi−1,j | + 1, πix1 = max{ ij− =1 |ηi−1,j | + |η | + 1 , 1 } . i − 1 , j j=i−1 j=1

For (63), the following inequalities with positive parameters dχ and di1 can be obtained by Lemma 2 and Young’s inequality.

|zi |3 θ ∗ πiχ 1 |χ| ≤ |zi |3 θ ∗ πix1

|xs | ≤

s=1

3

2

4 3

1

πiχ 1 ( dχ )− 3 zi4 θ +

4

n

3i 4

4 3

− 31

πix1 di1 zi4 θ +

1 2n

i di1 ∑

4

dχ |χ|4 ,

|xs |4 .

(65)

s=1

where ωθ i1 =

3 4

4 3

1 2n

dχ |χ|4 + − 31

πiχ 1 ( 2n dχ )

+

1 4 3i 4

di1

i ∑

|xs |4 ,

−1

πix1 di1 3 .

Proof of (42). By Assumption A2, Lemma 2 and (28), we get that 3 2

zi2 δi2,p ≤ zi2 θ ∗2 (πiχ 2 |χ|2 + πix2

i ∑ s=1

|xs |2 ),

, 3(i + 1)}. n 2 i 2

1

1 4 πi2χ 2 d− χ zi θ +

2n 1

dχ |χ|4 ,

2 −1 4 πix2 di2 zi θ + di2

i ∑

2

(67)

|xs |4 ,

(68)

s=1

2

zi2 δi2,p ≤ ωθ i2 zi4 θ + n 2

1 2n

dχ |χ|4 +

1 2

di2

i ∑

|xs |4 ,

s=1

i 2 −1 1 πi2χ 2 d− χ + 2 πix2 di2 .

Proof of (43). For i = 2, . . . , n, by Young’s inequality, we obtain that

−zi3 ϖi−1 ≤

≤

i−1 1∑

4 1 4

di3j |xj+1 |4 +

j=1

di3

i ∑ s=1

3 4

zi4

i−1 ∑

−1

4

di3j3 |ηi−1,j | 3

j=1

3

|xs |4 + πiϖ zi4 , 4

where di3 = max{di31 , . . . , di3(i−1) } and πiϖ = with di3j > 0(j = 1, . . . , i − 1).

∑i−1

j=1

−1

4

di3j3 |ηi−1,j | 3

References

s=1 4 3

+ 1)η

j=1 (j

where dχ and di2 are positive constants. In view of (67) and (68), the following inequality can be deduced from (66).

(64)

From (64) and (65), (63) can be changed into zi3 ϑi,p ≤ ωθ i1 zi4 θ +

|xs |2 ≤

s=1

∑i−1

∑i−1

i ∑

i ∑

where ωθ i2 =

s=1

where πiχ 1 =

zi2 θ ∗2 πix2

3

zi3 ϑi,p ≤ |zi |3 θ ∗ (πiχ 1 |χ| + πix1

+ 1)η

zi2 θ ∗2 πiχ 2 |χ|2 ≤

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Proof of (41). Let

j=i−1 (j

2 i−1,j

+ 1)ηi2−1,j , πix2 = , . . . , 3(i + 1) + 3(i −

j=1 (j 2 i−1,j

Using Young’s inequality, we have inequalities

Declaration of competing interest

∂αi−1 ∂ xj

∑i−1

∑i−1

∑i−1

(66)

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