Stochastic Output-Feedback Tracking Control with Second-Order Moment Process

Paper:

Stochastic Output-Feedback Tracking Control with Second-Order Moment Process

Ruitao Wang, Xiaoyu Xu, Wuquan Li

School of Mathematics and Statistics Sciences, Ludong University, Yantai, 264025, China E-mail: [email protected].

This paper investigates the output-feedback tracking control problem of stochastic high-order nonlinear systems perturbed by second-order moment process.

Compared with most of the existing results where on- ly wiener process is considered, a high-gain homoge- neous domination approach is used to construct the output-feedback controller. This controller ensures that the expectation of tracking error can be adjusted to be arbitrarily small and all the states of the closed- loop system are bounded in probability. Finally, a nu- merical example is given to demonstrate the feasibility of the control scheme.

Keywords: Second-order moment process, output- feedback, tracking, stochastic high-order nonlinear sys- tems.

1. Introduction

In recent decades, due to the wide application of s- tochastic nonlinear systems, stochastic nonlinear systems have become the focus of attention, e.g., see [1]-[5]. In order to combine with the reality, people pay more at- tention on stochastic high-order nonlinear systems. [6]

studies the problem of state-feedback stabilization for s- tochastic high-order nonlinear systems with stochastic in- verse dynamics. Then, [7]-[9] discuss the state-feedback stability for stochastic high-order systems. [10] analyses the inverse optimal stabilization. [11] constructs a state- feedback controller and a optimal controller for stochastic nonlinear systems with time-varying powers.

However, in [6]-[10], the corresponding conclusion is obtained by considering rigid power order restriction.

Subsequently, to cancel the rigid power order restric- tion, [12] presents a new stochastic homogeneous dom- ination method for output-feedback stabilization of s- tochastic high-order nonlinear systems whose control in- puts are neither affine nor feedback linearizable. Soon af- ter, this method is extended to adaptive control [13], state- feedback control [14], and time-delay control [15] for s- tochastic high-order nonlinear systems. Under the weaker

. This work is supported by National Natural Science Foundation of Chi- na under Grant (No. 61973150), the Young Taishan Scholars Program of Shandong Province of China under Grant (No. tsqn20161043), Shan- dong Provincial Natural Science Foundation for Distinguished Young Scholars under Grant (No. ZR2019JQ22), and Shandong Province High- er Educational Excellent Youth Innovation team (No. 2019KJN017).

conditions, [16] constructs a stabilizing output-feedback controller. [17] designs a controller via the homogeneous domination method.

A lot of attention has also been paid to the tracking control of stochastic high-order nonlinear systems. [18]

investigates the output tracking problem. In [19], by in- troducing a operator, the output tracking control problem of stochastic high-order system with stationary Marko- vian switching is solved. Distributed cooperative con- trol of multiple stochastic high-order nonlinear system- s is addressed in [20]. It is worth noting that the state- feedback output tracking control for stochastic high-order nonlinear systems [18]-[20] are considered. In fact, states information is not always available in practical applica- tions. For the above-mentioned system, probe into the tracking problem of stochastic system through the output- feedback control scheme is of great significance. [17]

adopts output-feedback control method to deal with the output tracking control for stochastic high-order nonlin- ear systems with unstable linearization. [21] proposes an adaptive output-feedback tracking control method for s- tochastic nonlinear systems with unmeasurable state.

It’s important to note that [17] and [21] discuss the output-feedback tracking problem, which focus on wiener process. However, some engineering system noises can- not be described as white noise, thus, those systems ef- fected by white noise are not suitable for some practical situations. There are some literatures [22]-[26] on the re- lated problems about the second-order moment process.

For example, [22] proposes that, in some circuit system- s, the stationary process (second-order moment process) is used to describe the interference of electronic compo- nents, which can be used in engineering applications to simulate the impact of uneven road on automobile sys- tems. [23] analyzes the stability of nonlinear benchmark system in the oscillating surroundings. [24] discusses the trajectory tracking control problem for Lagrange systems.

[25] investigates the distributed output tracking problem for nonlinear multi-agent systems. [26] solves the prob- lem of cooperative control for multiple nonlinear bench- mark systems. Although the output-feedback tracking control has been studied using a homogeneous domina- tion for several classes of stochastic high-order nonlin- ear systems with unmeasured states, the output-feedback tracking control problem of stochastic high-order nonlin- ear system perturbed by second-order moment process has not been fully discussed.

By adopting output-feedback control scheme, we solve the output tracking control problem for stochastic high- order nonlinear systems perturbed by second-order mo- ment process. The main contributions of this paper in- clude two points:

1. This paper is the first result on the output-feedback tracking problem of stochastic high-order systems under second-order moment process perturbations.

2. Although the high-gain homogeneous domination approach is extended to the stochastic environment, it is important to note that this paper’s system is differen- t from the stochastic systems perturbed by write noise [17], stochastic high-order nonlinear system perturbed by second-order moment process will generate the coupling terms between the second-order moment process and the nonlinear functions. Because of the coupling term’s ef- fect, the original method on stability analysis is no longer applicable,

The organizational structure of this paper is as follows.

Section 2 is problem formulation. Section 3 focuses on the controller design and the stability analysis. Section 4 illustrates the feasibility of the control method through a simulation example. Section 5 is conclusions.

2. Problem formulation

Consider a class of stochastic high-order nonlinear sys- tems perturbed by second-order moment process

˙

xi= x_i+1^pi + fi( ¯xi) + g^T_i( ¯xi)ξ(t), i = 1, · · · , n − 1,

˙

xn= u^pn+ fn( ¯xn) + g^T_n( ¯xn)ξ(t),

y= x1− y0(t), . . . (1) where x= (x1, · · · , xn)^T ∈ Rⁿ, u∈ R, y ∈ R are the sys- tem state, control input, and output, respectively, y₀(t) is the reference signal. x₂, · · · , xn are unmeasurable states.

¯

xi= (x1, · · · , xi)^T, i= 1, · · · , n. pi∈ R^≥1_odd= {q ∈ R: q ≥ 1 and q is a ratio of odd integers}, are the power order of stochastic high-order nonlinear system. ξ(t) ∈ R^m is a standard second-order moment process defined on the complete probability space(Ω, F , Ft, P) with a filtra- tion Ft satisfying the usual conditions (i.e.,it is increas- ing and right continuous while F0 contains all P-null sets). Suppose that the functions fi( ¯xi) : Rⁱ → R and gi( ¯xi) : Rⁱ→ R^mare C¹, vanishing at the origin.

We need the following assumptions.

Assumption 1. Stochastic processξ(t) is Ft-adapted and piecewise continuous, there is a positive constant K such that

sup

t≥t0

E|ξ(t)|²< K.

Assumption 2. For a stochastic processξ(t), positive constantsε,ϕ, there exists T > t0such that for∀t ≥ T

Pn 

1 t− t0

Zt

t₀|ξ(s)|²ds− E|ξ(t)|²  ≥ϕo

≤ε.

Remark 1. In the literatures [16] and [17], the interfer- ence of the system is described as white noise. In terms of

physics, white noise is difficult to achieve, it is more rea- sonable to regard the random perturbation of the system as the second-order moment process. Obviously, Assump- tion 1 indicates stationary processξ(t) is a second-order moment process, which is popular in some practical ap- plications. For Assumption 2, it is important to note that system (1) requires special properties of stochastic pro- cesses.

Assumption 3. Assume that the reference signal y0(t) ∈ R is continuously differentiable, there exists a con- stant M> 0 such that |y0| + | ˙y0| ≤ M.

Assumption 4. For any m≥ 0, there exist two positive constants b and c such that

| fi( ¯xi)| ≤ b(|x1|^(ri+m)/r1+ · · · + |xi|^(ri+m)/ri) + c,

|gi( ¯xi)| ≤ b(|x1|^ri/r1+ · · · + |xi|^ri/ri) + c, where r₁= 1, r_i+1= ^ri+m_p

i > 0. Set r0= max_1≤i≤n{ri} andλi= ^r_r⁰

i, i= 1, · · · , n. At the same time, one of the following relationships needs to be met:

1. rn+ m ≥ ri, ifλi= 1 orλi≥ 2;

2. rn+ m ≥ 2ri, other situations.

In order not to lose generality, let’s assume that m=

2q

2d+1with q and d are integers so that ri∈ R⁺_odd= {q ∈ R : q> 0 and q is a ratio of odd integers}. By referring to the method in [27], select a group of riwith generality.

The requirement for parameter l> 1 is proposed to achieve rn+ m ≥^ri+m_l and r0≥^ri+m_l , i = 1, · · · , n. Based on Assumption 4, chooseσ according to the following conditions: 1.σ= r0, when relationship 1 of Assumption 4 is true.

2. σ is selected as any σ ∈ R⁺_odd meeting rn+ m ≥ max1≤i≤n{^ri+m_l , 2ri}, when relationship 2 of Assumption 4 is satisfied.

Remark 2. Assumption 4 is crucial to ensure the exis- tence and uniqueness of the solution.

3. Controller design and stability analysis 3.1. Nominal system analysis

In this subsection, for system (1), the nominal system is as follows:

˙

wi = w_i+1^pi , i = 1, · · · , n − 1,

˙

wn= v^pn,

y= w1. . . (2) A homogeneous observer is designed for the system (2) based on a similar way in [16],

η˙k= −lk−1wˆ_k^pk−1, ˆ

wk= (ηk+ lk−1wˆ_k−1)^rk/rk−1, k = 2, · · · , n, . . (3) the output-feedback controller is designed:

v( ˆw) = −αn( ˆw

σ

nrn +αn−1wˆ

σ rn−1 n−1 + · · · +αn−1αn−2· · ·α2wˆ

r2σ

2 +αn−1αn−2· · ·α1w

r1σ

1 )^rn+1σ ,(4)

where lj, j = 1, · · · , n − 1 are observer gains gen- erated during the design process, wˆ1 = w1, wˆ = (w1, ˆw2, · · · , ˆwn)^T, andα1, · · · ,αnare positive constants.

Define W= (w1, · · · , wn,η2, · · · ,ηn)^T, then the closed- loop system is showed:

W˙ = D(w)

= (w₂^p1, · · · , w^pnⁿ⁻¹, v^pn( ˆw), fn+1, · · · , f2n−1)^T, (5) where

fi= −li−nwˆ_i−n+1^pi−n , i = n + 1, · · · , 2n − 1.

For system (5), choose the appropriate Lyapunov func- tion

V(W ) = V (w1, · · · , wn,η2, · · · ,ηn)

= Vn(w1, · · · , wn) +U(η2, · · · ,ηn), . . (6) where

Vn(w1, · · · , wn)

=

∑

n i=1

Z _wi

w^∗_i (s^riσ− w^∗

σ ri

i )^{4lσ−m−riσ} ds, U(η2, · · · ,ηn)

=

∑

n i=2

Z w

4lσ−m−ri−1 ri i γ

4lσ−m−ri−1 ri−1 i

(s^{4lσ−m−ri−1ri−1} −ηi− li−1wi−1)ds,

w^∗_i = −ζ_i−1^riσ α_i−1^riσ , ζi= w_i^riσ − w^∗_i^riσ.

Then we get the following conclusion based on the sim- ilar method in [16].

Lemma 3.1. For system (5), select a Lyapunov func- tion V(W ) in (6), we have

1. V(W ) is positive definite and homogeneous of de- gree 4lσ− m with the following dilation weight:

△ = (r1, r2, · · · , rn

| {z }

f or w₁, ··· , wn

, r1, r2, · · · , rn−1

| {z }

f orη2, ··· ,ηn

).

2. ∂V

∂WD(W ) ≤ −c₀||W ||^4lσ_△ , where

||W ||_△= (

∑

n i=1

|wi|^2/ri+

∑

n j=2

|ηj|^2/rj−1)¹².

with c₀> 0 is a constant.

3.2. Homogeneous output-feedback controller de- sign

In this subsection, our goal is to construct an output- feedback controller for stochastic system (1) by using the homogeneous domination technology.

Define z1= y,

zi= xi, i = 2, · · · , n. . . (7)

a new form of the system (1) is obtained

˙zi= z_i+1^pi + ˘fi(¯zi, y0) + ˘g^T_i(¯zi, y0)ξ(t), i = 1, · · · , n − 1,

˙zn= u^pn+ ˘fn(¯zn, y0) + ˘g^Tn(¯zn, y0)ξ(t),

y= z1, . . . (8) where

f˘1(¯z1, y0) = f1(z1+ y0) − ˙y0,

˘

g1(¯z1, y0) = g1(z1+ y0),

˘fi(¯zi, y0) = fi(z1+ y0, z2, · · · , zn),

˘

gi(¯zi, y0) = gi(z1+ y0, z2, · · · , zn), i = 2, · · · , n. (9) According to (9) and Assumptions 3 and 4, there are two constants b1> 0 and c1> 0, the following relationship is established

| ˘fi(¯zi, y0)| ≤ b1

∑

i s=1

|zs|^(ri+m)/rs+ c1,

| ˘gi(¯zi, y0)| ≤ b1

∑

i s=1

|zs|^ri/rs+ c1. . . (10) Let

w₁= y,

wi = xi/L^li, i = 2, · · · , n,

v^pn = u^pn/L^ln+1, . . . (11) where l1= 0, li= ^li−1_pi−1⁺¹, and L> 1 is a constant to be designed, a new form of the system (8) is

˙

wi = Lw_i+1^pi + f˘_i(¯zi, y0)

L^li +g˘^T_i(¯zi, y0)

L^li ξ(t), i = 1, · · · , n − 1,

˙

w_n= Lv^pn+ f˘n(¯zn, y0)

L^ln +g˘^T_n(¯zn, y0) L^ln ξ(t),

y= w1. . . (12) Next, by the same way as (3) and (4), construct the fol- lowing homogeneous observer:

η˙k= −Llk−1wˆ^p_k^k−1, ˆ

w_k= (ηk+ l_k−1wˆ_k−1)^rk/rk−1, k = 2, · · · , n, . (13) the output-feedback controller:

v( ˆw) = −αn( ˆw

rnσ

n +αn−1wˆ

σ rn−1 n−1 + · · · +αn−1αn−2· · ·α2wˆ

r2σ

2 +αn−1αn−2· · ·α1w

r1σ

1 )^rn+1σ .(14) Substituting (13) and (14) into (12), the closed-loop sys- tem is

W˙ = LD(W ) + F(W ) + G^T(W )ξ(t), . . . . (15) where

F(W ) = ( ˘f1, f˘2

L^l2, · · · , f˘n

L^ln, 0, · · · , 0)^T, G(W ) = ( ˘g1, g˘2

L^l2, · · · , g˘n

L^ln, 0, · · · , 0). . . (16)

3.3. Stability analysis

The objective of this subsection is to analyze the stabil- ity of the closed-loop system.

The following lemma deal with the gradient terms and coupling terms based on homogeneous theory.

Lemma 3.2 Suppose that Assumptions 3 and 4 are sat- isfied, there exist some positive constants c0, d0, γ0, ˜γ0, β1, ˇβ21andβ22so that:

∂V

∂WF(W ) ≤ (c0L^1−γ0+ n)||W ||^4lσ_△ +β1,

∂V

∂WG(W )ξ(t) ≤ (d0L^{1− ˜γ0}+ n)||W ||^4l_△^σ−m|ξ(t)|

+ ˇβ21+β22|ξ(t)|². . . . (17) Proof: Step 1. Deal with the gradient terms.

For (10), based on [17, Lemma 2.2] and L> 1, one gives

| ˘fi(¯zi, y0)|

L^li ≤ b1L^1−γi

∑

i s=1

|ws|^(ri+m)/rs+ c1L^−li

≤ ¯b1L^1−γi||W ||^r_△^i+m+ c1L^−li, . . (18) whereγiand ¯b1are positive constants.

Through [17, Lemma 2.2] and Lemma 3.1, _∂W^∂V

i is ho- mogeneous with degree 4lσ− m − ri. Based on (18), [17, Lemma 2.1], and [28, Lemma 2.4], then

∂V

∂WF(W ) ≤

∑

n i=1

∂V

∂Wi

˘fi(¯zi, y0) L^li

≤

∑

n i=1

∂V

∂Wi

(¯b1L^1−γi||W ||^r_△^i+m+ c1L^−li)

≤

∑

n i=1

c0||W ||^4lσ_△ ^−m−riL^1−γi||W ||^r_△^i+m

+c1

∑

n i=1

L^−li ∂V

∂Wi

≤ c0L^1−γ0||W ||^4lσ_△ + c1

∑

n i=1

L^−li∂V

∂Wi

, (19) whereγ0= min1≤i≤n{γi}, c0is a positive constant.

According to [28, Lemma 2.4] and [17, Lemma 2.2], it implies that

c1

∑

n i=1

L^−li∂V

∂Wi

≤ ˇc1

∑

n i=1

L^−li||W ||_△^{4lσ−m−ri}

= ˇc1

∑

n i=1

(L^−li/(m+ri))^m+ri||W ||^4lσ−m−r_△ ⁱ

≤ n||W ||^4lσ_△ +

∑

n i=1

βi1L^{−4lσli/(m+ri)}, . . . . (20)

where

βi1= ri+ m 4lσ cˇ

4lσ ri+m 1

 4lσ

4lσ− m − ri

^−(4lσ_ri+m^−m−ri)

with ˇc1> 0 is a constant.

Substituting (20) into (19) yields

∂V

∂WF(W ) ≤ (c0L^1−γ0+ n)||W ||^4l_△^σ+β1, . . (21) where

β1=

∑

n i=1

βi1L^{−4lσli/(m+ri)}. . . (22) Step 2. Estimate the coupling terms.

By (10), it obtains

| ˘gi(¯zi, y0)|

L^li ≤ b1L^{1− ˜γi}

∑

i s=1

|ws|^ri/rs+ c1L^−li

≤ ˜b1L^{1− ˜γi}||W ||^r_△ⁱ + c1L^−li, . . . (23) where ˜b1, ˜γiare positive constants.

Similar to the proof method of (19), we get

∂V

∂WG(W ) ≤

∑

n i=1

∂V

∂W_i

˘ gi(¯zi, y0)

L^li

≤

∑

n i=1

∂V

∂Wi

(˜b1L^{1− ˜γi}||W ||^r_△ⁱ + c1L^−li)

≤

∑

n i=1

d0||W ||^4lσ−m−r_△ ⁱL^{1− ˜γi}||W ||^r_△ⁱ

+c1

∑

n i=1

L^−li∂V

∂Wi

≤ d0L^{1− ˜γ0}||W ||^4lσ−m_△ + c1

∑

n i=1

L^−li∂V

∂W_i,(24) where ˜γ0= min1≤i≤n{ ˜γi}, d0≤ 0 is a constant.

Similar to (20), we have c₁

∑

n i=1

L^−li ∂V

∂Wi

≤ n||W ||^4lσ_△ ^−m+

∑

n i=1

βi2L^{−li(4lσ−m)/ri}, (25)

where

βi2= ri

4lσ− mcˆ

4lσ−m ri 1

 4lσ− m 4lσ− m − ri

^−(4lσ_ri^−m−ri)

with ˆc₁is a positive constant.

Substituting (25) into (24) yields

∂V

∂WG(W )

≤ (d0L^{1− ˜γ0}+ n)||W ||^4lσ_△ ^−m+

∑

n i=1

βi2L^−li(4l

σ−m) ri . (26) Therefore, according to (26) and ab≤εa²+_4ε¹b², it has

∂V

∂WG(W ) ·ξ(t) ≤

(d0L^{1− ˜γ0}+ n)||W ||^4lσ_△ ^−m +

∑

n i=1

βi2L^{−li(4lσ−m)/ri}

|ξ(t)|

≤ (d0L^{1− ˜γ0}+ n)||W ||^4lσ−m_△ |ξ(t)|

+ ˇβ21+β22|ξ(t)|², . . . (27)

where

βˇ21= 1 4β22

 ⁿ

i=1

∑

βi2L^{−li(4lσ−m)/ri}2

withβ21is arbitrary positive constant.

Remark 3. On the one hand, through the proof of Lem- ma 3.2, it can be found that the system is different from the deterministic system in [29]. On the other hand, com- pared with [17], the stochastic system is considered sub- jected to the second-order moment process. Stochastic systems will generate some coupling terms because of the effect of second-order moment process, it is impossible to separate second-order moment process from nonlinear function. In order to give stability analysis, the coupling term satisfy (27).

Next, we are in a position to state the main result of this paper.

Theorem 4.1 Consider the system (1) described above, with coordinates (11), the observer (13), and the con- troller (14). If Assumptions 1 to 4 hold, the following conclusions hold:

1). system (15) has an unique solution on[t0,∞);

2). each state of system (15) remains bounded in prob- ability;

3). the expectation of tracking error can be made arbi- trarily small, i.e. for each givenε> 0 and initial value x(t0), there exists a finite-time T (x(t0),ε) such that

E|x1(t) − y0(t)| <ε, ∀t > T.

Proof. The proof is based on Lemma 3.1 and 3.2. The whole proof process is tedious, so it is omitted here.

4. A Simulation Example

In this section, a numerical example with the following form is given:

˙

x1= x⁷₂^/3+ x1x2+ (1

4sin x1)ξ(t),

˙

x2= u³+ x1cos x2+1

2x⁴₂^/3ξ(t),

y= x1− sint, . . . . (28) where p1=⁷₃and p2= 3. Choose m =⁴₃and y0(t) = sint, obviously, it’s easy to verify that the Assumptions 3 and 4 are satisfied.

According to the analysis of the previous sections, choose l= 1 andσ = 1, one gets

η˙2= −Ll1wˆ^7/3₂ , ˆ

w₂=η2+ l1(x1− sint),

u= −10.8( ˆw₂+ 2w1)^7/9. . . (29) Choose parameters l1= 11, L = 3.5, and a set of initial values(x1(0), x2(0),η2(0))^T = (−3, 1, −0.01)^T. Through the actual simulation, it’s obtained that the system re- sponses of the tracking trajectory, output tracking error, and controller are shown in Fig. 1-3. Fig. 2 shows that the tracking error|y| < 0.05, ∀t > T = 2s and Fig. 3 shows

0 2 4 6 8 10 12

−4

−3

−2

−1 0 1 2 3

Time (seconds)

Tracking

x1 y₀

Figure 1. Response of tracking.

0 2 4 6 8 10 12

−3

−2

−1 0 1 2 3

Time (seconds)

Tracking error

x1−y 0

Figure 2. Response of tracking error.

that the controller is bounded, which intuitively shows the feasibility of the design scheme proposed in this paper.

Remark 3. Different from the white noise in [17], the noise in system (28) is a second-order moment process, which is a kind of color noise. Therefore, the results ob- tained are more practical.

5. Conclusions

The output-feedback tracking control problem for s- tochastic nonlinear systems with second-order moment process is investigated in this paper. The output-feedback tracking controller is constructed by the high-gain homo- geneous domination design approach. The obtained con- troller ensures that the expectation of tracking error can be adjusted to be arbitrarily small, and all the states of the closed-loop control system are bounded in probability.

There are many questions worth studying in the future.

For example, generalizing the results in this paper to more general stochastic system [30]-[34].

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0 2 4 6 8 10 12

−500

−400

−300

−200

−100 0 100 200 300 400 500

Time (seconds)

Control

u

Figure 3. Response of controller.

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[13] W. Q. Li, Y. W. Jing, S. Y. Zhang. Adaptive state-feedback stabi- lization for a large class of high-order stochastic nonlinear systems, Automatica. 47: 819-828, 2011.

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[17] W. Q. Li, L. Liu, G. Feng. Output tracking of stochastic nonlinear systems with unstable linearization, International Journal of Robust and Nonlinear Control. 28: 466-477, 2018.

[18] X. J. Xie, N. Duan. Output tracking of high-order stochastic non- linear systems with application to benchmark mechanical system, IEEE Transactions on Automatic Control. 55: 1197-1202, 2010.

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[24] Z. J. Wu, H. R. Karimi, P. Shi. Practical trajectory tracking of ran- dom Lagrange systems, Automatica. 105: 314-322, 2019.

[25] H. Wang, W. Q. Li, M. Q. Tang. Distributed output tracking of non- linear multi-agent systems perturbed by second-order moment pro- cesses, Neurocomputing. DOI.org/10.1016/j.neucom.2019.10.120, 2020.

[26] W. Q. Li, L. Liu, G. Feng. Cooperative control of multiple nonlinear benchmark systems perturbed by second-order moment processes, IEEE Transactions on Cybernetics. 50: 902-910, 2020.

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[30] W. Q. Li, M. Krstic. Stochastic adaptive nonlinear control with filterless least-squares, IEEE Transactions on Automatic Control.

DOI:10.1109/TAC. 2020.3027650, scheduled in 66(9), 2021.

[31] W. Q. Li, M. Krstic. Mean-nonovershooting control of stochas- tic nonlinear systems, IEEE Transactions on Automatic Control.

DOI:10. 1109/TAC.2020.3042454, scheduled in 66(12), 2021.

[32] W. Q. Li, M. Krstic. Stochastic nonlinear prescribed-time stabiliza- tion and inverse optimality, IEEE Transactions on Automatic Con- trol. DOI: 10.1109/TAC.2021.3061646, scheduled in 67(3), 2022.

[33] W. Q. Li, M. Krstic. Prescribed-time output-feedback control of s- tochastic nonlinear systems, IEEE Transactions on Automatic Con- trol. Conditionally accepted as a Full Paper, 2021.

[34] W. Q. Li, L. Liu, G. Feng. Distributed output-feedback tracking of multiple nonlinear systems with unmeasurable states, IEEE Trans- actions on Systems, Man, and Cybernetics: Systems. 51(1): 477- 486, 2021.