# Adaptive Output Feedback Tracking Controller Design of Stochastic Nonlinear Systems with Parameter Uncertainty for Polynomial Function Growth Conditions

Authors: Long-Chuan Guo
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper mainly focuses on output feedback practical tracking controller design for stochastic nonlinear systems with polynomial function growth conditions. Mostly, there are some studies on output feedback tracking control problem for general nonlinear systems with parametric certainty in existing achievements. Moreover, we extend it to stochastic nonlinear systems with parametric uncertainty and system nonlinear terms are assumed to satisfy polynomial function growth conditions which are more relaxed than linear growth conditions or power growth conditions. Due to the presence of unknown parametric uncertainty, an output feedback practical tracking controller with dynamically updated gains is constructed explicitly so that all the states of the closed-loop systems are globally bounded and the tracking error belongs to arbitrarily small interval after some positive finite time. An example illustrates the efficiency of the theoretical results.


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## Body

## 1. Introduction

The problem of global output tracking control for nonlinear systems has been extensively studied by academic researchers and successfully applied in some practical nonlinear systems (see [1–10]). According the existing papers, two types output tracking have been concerned which are asymptotic output tracking and practical tracking (see [11–16]). The asymptotic output tracking focuses on the design of controller that forces the controlled output of system to reach and follow a time-parameterised reference track signal. However, for many practical systems and control design, we can hardly achieve good models, too expensive to derive, and the parameters are not precisely known preventing the error signals from tending to zero, so the practical tracking has been proposed by many researchers.Consider the output feedback practical tracking control problem for a class of nonlinear system which appears in past research papers:(1)dxi=xi+1dt+fixdti=1,⋯,n-1dxn=udt+fnxdty=x1-yrwhere x=x1,....,xnT,u∈R, and y∈R are the system states, input, and output. Correspondingly, yr is given reference trajectory, and the nonlinear term fix satisfies the Lipschitz condition, whose initial value is fi0=0. Some related papers have studied the practical output feedback control problem for nonlinear system (1) (see [17–22]). In [17, 18], the system nonlinear term satisfies lower triangular structure and is limited to the following order of polynomial function growth conditions:(2)c01+x1px1+⋯+xi+c0where p≥1 and c0≥0 are known constants. Similarly, the system nonlinear term satisfies the following growth conditions which have contained known constants c and τ in [22]: (3)cx1iτ+1+x2iτ+1/τ+1+⋯+xiiτ+1/i-1τ+1+cAfter introducing the unknown parameters into the growth condition of system (1), [19] has studied output feedback tracking control research on this kind of uncertain nonlinear system by introducing coordinate transformation, using Nussbaum gain method and adding power integral method so as to obtain a continuous adaptive controller and ensure that system tracking error converges to a small neighborhood of zero after a limited time, while the closed-loop nonlinear system meets the boundedness of the states. In [20], an adaptive output feedback tracking controller is designed by means of general control and dead zone method. It is pointed out that the nonlinear term growth condition of the system in [20] depends on the unmeasurable states and the growth rate is unknown constant; meanwhile, the reference trajectory signal and its first derivative have unknown upper bound.However, all the mentioned references are only limited to the nonlinear systems. Naturally, one may ask an interesting and challenging question:If we consider the tracking control problem for stochastic nonlinear system, how to design a controller? As is well known, stochastic factor is a main resource that contributes significantly to the system complexity. In some practical fields, such as aerospace, biological, and economic systems and health community, the output tracking problem for these stochastic nonlinear systems has a wide range of practical applications. The treatment of stochastic items is the key to solving our problems in the stochastic control system. To our knowledge, the existing research results on the tracking problem for stochastic nonlinear systems are [23–27]. The work [23] has utilized a backstepping-based control for a class of stochastic nonlinear systems where the controller has ensured the overall system stability. The works [24–27] further consider the states feedback tracking controller design problem for high-order stochastic nonlinear system, satisfying a more relax condition. As is known to all, output feedback controller is more practical than state feedback controller in the actual system because states information of actual system is difficult to master. For the stochastic nonlinear system where nonlinear vector fields depend on the unmeasurable states besides the measurable output, how to design an output feedback tracking controller? This paper is focused on the problem of extending the results of [25, 27] which has not yet been studied till now.Meanwhile, some research has studied output feedback practical tracking control problem for a class of stochastic nonlinear system which satisfies the growth condition of polynomial function, the nonlinear term of the system depends on the state of unmeasurable, and the growth condition of the nonlinear term parameters is known (see [17, 18]). This leads to some limitations; since parameter uncertainties and structural uncertainties are common in practical systems such as spaceflight control systems, navigation systems, and bioengineering systems, this is far from sufficient for research on output feedback tracking control problem for stochastic nonlinear system under known parameters conditions.This paper extends it to stochastic nonlinear systems and proposes to set the corresponding parameters of stochastic nonlinear system to be unknown and introduce the parameter uncertainty factors, so as to generalize the output feedback practical tracking control research results in the previous papers. The main contribution of this paper is as follows.Use Ito differential mathematics theory to deal with complex stochastic terms such that a high-order gain observer and trajectory tracking error system are constructed. Then the error system is analyzed by Lyapunov function and the boundedness of system parameters is proved. The output feedback practical tracking controller with dynamically updated gains is constructed explicitly so that all the states of the closed-loop system are bounded and the error converges to any small neighborhood of the origin when the time goes to infinity.The paper is organized as follows. Section2 provides some problem description. In Section 3, an output feedback practical tracking controller is designed and analyzed. A simulation example is provided in Section 4. Section 5 concludes this paper.


## 2. Problem Description

In this paper, we consider the output feedback practical tracking problem for following stochastic nonlinear system:(4)dxi=xi+1dt+fixdωi=1,⋯,n-1dxn=udt+fnxdωy=x1-yrtwhere x=x1,⋯,xnT∈Rn and u∈R are the system states and input, y is the system output that can be measured, respectively, yr is given reference tracked unmeasured trajectory, similarly, x2t,...,xnt are unmeasurable, and ω is an m-dimensional standard Wiener process defined on the complete probability space Ω,Γ,Ρ with Ω being a sample space, Γ being a filtration, and Ρ being a probability measure. The nonlinear functions fi:R+×Rn→Rm,i=1,⋯,n, are continuous and assumed to satisfy locally Lipschitz condition with fi0=0.According to the above description of system, it leads to the objective of this paper: for any constantδ>0, all the states of stochastic nonlinear system (4) are well defined and bounded. In addition, there exists a finite time T>0 such that, for any t>T, it can get (5)x1t-yrt≤δ,∀t≥TTo achieve the above control performance, the following assumptions are made for the nonlinear term in system (4).Assumption 1.
There exist unknown constantsη1>0, η2>0 and known integer p≥1 such that the following inequality holds:(6)fix≤1+x1pη1x1+⋯+xi+η2Next traced trajectory yr satisfies the following assumption.Assumption 2.
The reference output trajectoryyr of stochastic nonlinear system (4) is continuously differentiable and there exists unknown constant η>0 such that the following inequality holds:(7)yr+y˙r≤ηAssumption 1 can be further extended to the following more general assumption conditions.Assumption 3.
There exist unknown constantsc>0, d>0 (d/c≥1) and known constants p≥1, 0≤μ≤1, such that the following inequality holds:(8)s0f1x+⋯+sn-1fnx≤1+x1pcs0x1+⋯+sn-1xn+dwhere s0≤s≤μ is any constant.This paper will focus on stochastic nonlinear system under more general Assumption3 and then construct an output feedback practical tracking controller such that output of system (4) can be gradually converged to zero.


## 3. System Analysis and Output Feedback Controller Design

The emphasis of this section is to design the global output feedback tracking controller in order to achieve asymptotic stability of system output (4) under Assumptions 2 and 3.Remark 4.
From Assumption3, we can clearly see that the unknown parameter c>0 appears in the product term in the front of unmeasurable states x2,⋯,xn, which leads to the fact that the conventional observer design method difficultly works because the constructed observer contains initial states of the system which contains unknown parameters such that the traditional method is no longer applicable. Inspired by [21, 22], this paper develops a new type of output tracking controller which consists of two dynamic factors, and it will be described in detail below.


### 3.1. Time Varying High-Order Gain Observer Design

First, make a general assumption as follows:(9)dc≥1and for the sake of convenience of formula derivation, the following states transformation is introduced:(10)z1=x1-yrz2=x2⋮zn=xnand then stochastic nonlinear system (4) is converted to(11)dzi=zi+1dt+gizdωi=1,⋯,n-1dzn=udt+gnzdωy=z1where gi is the nonlinear term after conversion and is constructed as follows:(12)g1z=f1z1+yr,z2,⋯,zn-dyrdωgiz=fiz1+yr,z2,⋯,zni=2,⋯,nNext, the output feedback tracking controller design process is initiated. With unmeasurable statesx1,⋯,xn characteristics of system (4), states observer of system (4) is established(13)z^˙i=z^i+1+hiABiz1-z^1i=1,⋯,n-1z^˙n=u+hnABnz1-z^1where z^i is the estimate value of zi and the high-order gain term consists of two variables At and Bt as follows:(14)A˙t=AAB2bz^12+e12-12δ2,A0=A0≥1μB˙t=-μ1B2+μ2B1+x1p2,B0=1where e1=z1-z^1 is error state, hi>0i=1,⋯,n is the coefficient of the Hurwitz polynomial with sn+h1sn-1+···+hn-1s+hn, and the unknown parameters b,μ1,μ2 satisfy 0<b<1/4p,0<μ1<μ2.


### 3.2. Lyapunov Analysis of Closed-Loop Systems

According to (14) on the definition of e1, it can be generalized to all states; that is, definition of ei=zi-z^ii=1,⋯,n and the available system error equation can be obtained:(15)dei=ei+1dt-hiABie1dt+gizdωden=-hnABne1dt+gnzdωIn order to facilitate the output feedback practical tracking controller design, the transformation of estimated states z^i and error states ei is introduced:(16)αi=eiABb+i-1i=1,⋯,nβi=z^iABb+i-1i=1,⋯,nwhere α=α1,⋯,αnT,β=β1,⋯,βnT,h=h1,⋯,hnT. By (16), stochastic nonlinear systems (11) and (13) can be converted to(17)dα=ABHαdt-A˙B+B˙AABCbαdt+Gz,A,Bdωβ˙=ABHbβ+ABhα1-A˙B+B˙AABCbβwhere(18)H=-h11⋯0⋮⋮⋱⋮-hn-10⋯1-hn0⋯0,Hb=01⋯0⋮⋮⋱⋮00⋯1-h1-h2⋯-hnCb=diagb,b+1,·····,b+n-1Gz,A,B=g1ABb,g2ABb+1,⋯,gnABb+n-1TFrom (14) on the definition of At and Bt, it can be generally obtained that (19)At≥A0≥1α≥1,∀t≥0and defining E=B2, the following can be obtained:(20)E˙≥2μ1E1-E,E0=1According to the comparison lemma in [28], the following can be drawn:(21)Bt≥1,0<1AtBt≤μ,∀t≥0By (16), it can be obtained that (22)zi=ABb+i-1αi+ABb+i-1βiand according to (21)-(22) and Assumptions 2 and 3, the inequality for nonlinear Gz,A,B can be deduced:(23)Gz,A,B=g1ABb2+⋯+gnABb+n-121/2≤1+x1pc∑i=1n1ABb+i-1zi+cη+d≤1+x1pα+β+1cη+dcwhere c-=cη+d/c.Then a systematic Lyapunov analysis is performed. First, the existence and uniqueness of the system solution are given where, since closed-loop stochastic nonlinear system (17) is continuous and satisfies locally Lipschitz condition on the initial point, the corresponding solution α,β,A,B of the system can be considered as existence and uniqueness.Define the following Lyapunov function:(24)V0α,β=αTPα+βTQβand positive definite matrices P,Q satisfy the following relation:(25)HTP+PH≤-Inr1In≤CbP+PCb≤r2InHbTQ+QHbT≤-2Inr3In≤CbQ+QCb≤r4Inwhere rii=1,2,3,4 is positive constant and In is n-dimensional judgment matrix.By (23), the following can be obtained:(26)2αTPG≤2Pcη+dc1+x1pα+1α+β≤1+x1p2α2+β2+η1α2+1where η1=cη+d/c2P2 is unknown constant.Further, by (25) and (26), the trajectory of (24) along the Ito differentiation for nonlinear stochastic system (17) can be obtained as follows:(27)LV0α,β≤-ABα2A˙B+B˙AABαTCbP+PCbα+TrGTPG-2ABβ2+2ABα1βTh-A˙B+B˙AABβTCbQ+QCbβ≤-ABα2-2ABβ2+P1+x1p2α+β+12cη+dc2-A˙B+B˙AABr2α2+r4β2+2αTPG+2ABα1βTQh≤-ABα2-2ABβ2+P1+x1p2α+β+12cη+dc2-A˙B+B˙AABr2α2+r4β2+1+x1p2α2+β2+η1α2+1+ABα2+β2≤-ABα2-2ABβ2+η1α2+1+μ1Br2α2+r4β2-1+x1p2α2+β2+η11+x1p2α+β+12≤-BA-2η1+μ1r2+μ1r4α2+β2+η1and Ito differentiation results after simplifying can be obtained:(28)LV0≤-BA-η2α2+β2+12η2-μ1r2+r4where η2=2η1+μ1r2+r4. Since matrices P,Q are positive, the following can be easily obtained:(29)λ1αTPα+βTQβ≤α2+β2≤λ2αTPα+βTQβwhere 0<λ1≤λ2. Moreover, by (29), (28) be deduced into(30)LV0≤-λ1BA-η2V0+12η2-μ1r2+r4ifA-η2≥0LV0≤-λ2BA-η2V0+12η2-μ1r2+r4ifA-η2<0


### 3.3. Boundedness Analysis of System States and Gain

According to (30), it is easy to observe that if η2 is a known constant, then system (17) can be guaranteed as asymptotically stable by requiring high-gain A to be as large as possible which has been studied by some existing papers. However, in this paper, the parameter η2 is an unknown constant which requires complex and rigorous analysis to determine the boundedness of the system on the maximum time interval 0,T and ensure the asymptotical stability of system tracking output.This section mainly focuses on the boundedness of high-gainsA, B and system states α, β. By (14), the following can be obtained:(31)At≥1μt∈0,TAccordingly, a hypothetical judgment can be drawn:(32)limt→T⁡At∈1μ,+∞To verify the above judgment (32), we make use of the negative proof technology; that is, if conclusion (32) does not hold, there exists a finite time t1∈0,T such that(33)λt=λ1,At≥2η2,∀t∈t1,TBy (30), it can be concluded that, for any time t∈t1,T, the following Ito differential inequality holds:(34)LV0≤-λ12ABV0+12η2-μ1r2+r4According to (34), the inequality with V0α,β on time interval t∈t1,T can be obtained:(35)V0α,β≤η2-μ1r2+r4λ1+V0α,βe-λ1/2t-t1By observing (35), it is easy to infer that V0α,β is bounded on t∈t1,+∞ and continuous on t∈0,t1; then it can be generalized that V0α,β is bounded on t∈0,T. Therefore, two cases of time T need to be discussed.The first case:T<+∞.IfV0α,β=h1>0 when t∈t1,T, then it can obtain the following inequality according to (34):(36)λ1h12∫t1TBsds≤12η2-μ1r2+r4T-t1+Vα,βBy observing (36) and (14), simultaneously, using α1=e1/ABb and β1=z^1/ABb, one has (37)A˙t=AAB2bz^12+e12-δ2≤λ1h12BAccording to (36) and (37) again, one can get (38)AT-A0≤λ1h12∫t1TBsdsAt the moment, by observing (38), it can find that AT-A0=+∞ on the left and λ1h1/2∫t1TBsds is constant on the right, which contradicts (38). Therefore, it is a wrong judgment that conclusion (32) does not hold.IfV0α,β=0 when t∈t1,T, by definition σ as any positive constant and according to limt→TAtBt=+∞, there exists t2≥t1 such that, ∀t∈t2,T, it can obtain (39)AtBt≥η2σλ1In addition, since V0α,β=0, there exists t3≥t2 such that V0αt3,βt3<σ/2. Through the above analysis, (34) can be converted to(40)LV0≤-2η2σV0+12η2-μ1r2+r4,∀t∈t3,TAccording to (40), the inequality with V0α,β on time interval t∈t3,T can be obtained:(41)Vα,β≤σ2+12η2-μ1r2+r4+Vαt3,βt3≤σ+12η2-μ1r2+r4and this proves that limt→TV0α,β=0.Suppose the following inequality holds:(42)A˙t≤ϑ1tBt,t∈0,Twhere ϑ1t=α12+β12-δ2/2AB2b2,t∈0,T. By (4), (14), and (16), the following can be obtained: (43)x1=ABbα1+β1+yrand, correspondingly, one has(44)x1t≤η+ϑ2tABbwhere ϑ2t=α1t+β1t. By (14) and Young’ inequality, one can obtain(45)B˙≤μ2B1+x1p2≤2μ2B1+η+ϑ2tABb2pand after further simplification one has(46)2μ2Bη+ϑ2tABb2p≤22pμ2η2p-1B+22pμ2ϑ22p-1AB2Then (46) can be converted to(47)B˙t≤2μ2+22pμ2η2p-1B+22pμ2ϑ22p-1AB2,t∈0,TNext define the function Ω=ABV0 and then by (34), (42), (47), and Ito differential theorem, one has(48)LΩ≤-AB2η2σV0+12η2-μ1r2+r4≤-AB2η2σ-ϑ1t-22pμ2ϑ22p-1V0+12η2-μ1r2+r4According to (42) and (47), it can be concluded that there exists t4≥t1 such that the following inequality can hold:(49)22pμ2ϑ22p-1≤η2σ-ϑ1Further, it can get(50)LΩ≤-ABη2σΩ-12μ1r2+r4-η2,t≥t4which shows that Ω is bounded. According to the definition of ϑ1t=α12+β12-δ2/2AB2b2,t∈0,T and through inequality amplification, one has(51)AB2be12+z^12≤λ1λ2h12which can prove that limt→Tz^12+e12=0 holds. Thence, there exists t5 such that, for t∈t5,T, it can obtain (52)z^12t+e12t≤λ1λ2h12AB2b≤δ22By (52) and (14), it can get A˙t=0,t∈t5,T which obviously conflicts with description of (14). Therefore, this is a false assertion that conclusion (32) does not hold.The second case:T=+∞.Definitionσ is any positive constant; by limt→TAtBt=+∞ it can be seen that there exist t2≥t1 such that, ∀t∈t2,T, one has(53)AtBt≥2η2σλ1⟹-AtBtλ1≤-2η2σThen (34) can be converted to(54)LV0≤-2η2σV0+12η2-μ1r2+r4,∀t≥t2By (53) and (54), the following can be obtained:(55)V0α,β≤12η2-μ1r2+r4+Vαt2,βt2e-2η2/σt-t2,t≥t2and then there exists t3≥t1 such that the following inequality holds:(56)V0αt2,βt2e-2η2/σt-t2≤12η2-μ1r2+r4,∀t≥t3Accordingly, it can obtain(57)Vα,β≤η2-μ1r2+r4,∀t≥t3Summarize the above analysis and it can be considered that limt→TV0α,β=0, then available A˙t=0, which is also contradictory with (14).Therefore, through the above two cases of analysis, it is a false assertion that conclusion (32) does not hold. Similarly, hypothetical judgment (32) is correct and system gain A is bounded on t∈0,T which satisfies(58)limt→T⁡At∈1μ,+∞Next, we discuss boundedness for statesα,β on 0,T.By (24), (29), and Ito differential theorem, define function (59)V2β=βTQβand then one has(60)LV2≤TrGTPG-2ABβ2+2ABα1βTQh-A˙B+AB˙ABβTCbQ+QCbβ+P1+x1p2α+β+12cη+dc2≤-ABβ2+ABα12-B˙BβTCbQ+QCbβ+P1+x1p2α+β+12cη+dc2≤-ABβ2+ABα12+μ1c3Bβ2-μ2c41+x1p2β2+P1+x1p2α+β+12cη+dc2By (60) and choosing the appropriate parameter μ1, it gets(61)LV2≤-Bβ2+ABα12+P1+x1p2α+β+12cη+dc2≤-BA-2η1+μ1r2+μ1r4β2+η1+ABα12and then by (14), one has(62)α12≤α12+β12-δ22AB2b+δ22AB2b≤α12+β12-δ22AB2b2+δ22+1such that the following inequality can be derived:(63)ABα12≤AA˙+AB1+δ22Thence, the following can be obtained:(64)ABα12≤AA˙+Bμ3where μ3=A1+δ2/2. By (64) and (61), the following can be derived:(65)LV2≤-BA-2η1+μ1r2+μ1r4β2+AA˙+μ3BAccording to the comparison theorem, one has(66)β2≤μ3B+V2β0+A2t-A022η1+μ1r2+μ1r4It has been shown above that A is bounded on 0,T. Further, (66) can prove that β is bounded on 0,T.According to the description of system (17), the state transition is introduced again:(67)Πi=1A-Bi-1+beiwhere A-=2c-2P2/c1μ2.Then the system error equation is converted to(68)Π˙=A-BLΠ+A-BhΠ1-ABΓhΠ1-B˙BCbΠ+G-z,u,A-,Bwhere G-=g1/A-Bb,g2/A-Bb+1,·····,gn/A-Bb+n-1T, Γ=diag1,A/A-,·····,A/A-n-1.Define Lyapunov function(69)V3Π=ΠTPΠand according to Ito's theorem, the trajectory of (69) along the system (68) is(70)LV3≤-A-BΠ2+2Π1A-BhTPΠ+2G-PΠ-2Π1ABhTΓPΠ-B˙BΠTCbP+PCbΠ≤-A-BΠ2+2Π1A-BhTPΠ-2Π1ABhTΓPΠ+2G-TPΠ+μ1c2BΠ2-μ2c11+x1p2Π2Furthermore, by the inequalities(71)2Π1A-BhTPΠ≤A-2BhTP2Π12+BΠ22Π1ABhTΓPΠ≤A2Bh2ΓP2Π12+BΠ2(70) can be converted to(72)LV3≤-BΠ2+lBΠ12+lBβ2+lwhere l is unknown constant. Simultaneously, by (63) and (64), one has(73)LV3≤-λ1BV3+lAA˙+lBβ2+l1+μ3Since the above analysis has proved that β is bounded on 0,T, then (73) can be converted to(74)LV3≤-λ1BV3+lAA˙+l-Bwhere l-=lmax⁡sup⁡β2,1+μ3. Simultaneously, by (16) and (17), one has(75)1λ1Π2≤V3Π0+l-λ1+lA2t-A022,t∈0,TSince the above analysis has proved that A is bounded on 0,T, then it can be inferred that Π is bounded on 0,T, Furthermore, combined with the above conclusions, it can be obtained that α is bounded on 0,T.Finally, we discuss the boundedness of stateB on 0,T.Since it has been concluded that system statesα and β are bounded on 0,T, then it can obtain(76)α1+β1≤kwhere k is positive constant and depends on the state initial value α0,β0. By (14), it can deduce(77)B˙≤-μ1B2+2μ2B+2μ2c-2pBUsing Young's inequality, (77) can be split to get the following relation form:(78)2μ2B≤EB2+C2μ2c-2pB≤FB2+Dwhere C, D, E, F are suitable coefficients or parameters.It is equivalent to(79)B˙≤E-B2+C-where C-, E- are suitable coefficients or parameters again. According to (67), it can be concluded that the state B is bounded on 0,T.In summary, system high-gainsA, B and states α, β are all bounded on 0,T. The boundedness of the system has been judged.


### 3.4. Output Feedback Tracking Controller Design

Based on the analysis of above, this section first gives the output feedback tracking controller.Theorem 5.
Consider a class of stochastic nonlinear system (4) whose output is y=x1-yr under Assumptions 2 and 3 and then the following output feedback practical tracking controller can be designed:(80)u=-ABnk1z^1+⋯+ABknz^nwhere A,B are defined by (14); ki>0 is coefficient of Hurwitz polynomial sn+h1sn-1+···+hn-1s+hn with hi=kn-i+1. Then, for any δ>0, there exists a finite time T such that x1-yr≤δ,∀t≥T, and all the states of stochastic nonlinear system (1) and (13) are bounded on 0,T under the action of output feedback practical tracking controller (80).The proof process ofTheorem5 is given.Using the inequality theorem(81)maxx,02-maxy,02≤x-y2It can be easily inferred that(82)BAB4bz12+e12-δ222,x,y∈Ris consistently continuous; that is, A˙ is consistent and continuous.Since system high-gainsA, B and states α, β are all bounded on 0,T, it is easy to conclude that e˙,z^˙,A˙,B˙ are also bounded on 0,T. According to the Barbalat lemma in [28], one has limt→+∞A˙t=0. By the above analysis, it can be deduced that there exists T such that(83)e12t+z^12t1/2≤CONSTANT,t≥TTherefore, it can also be inferred that(84)x1-yr=z1≤e1+z^1≤δ,t≥TThe proof ofTheorem5 is over such that output feedback practical tracking controller design is achieved.


## 3.1. Time Varying High-Order Gain Observer Design

First, make a general assumption as follows:(9)dc≥1and for the sake of convenience of formula derivation, the following states transformation is introduced:(10)z1=x1-yrz2=x2⋮zn=xnand then stochastic nonlinear system (4) is converted to(11)dzi=zi+1dt+gizdωi=1,⋯,n-1dzn=udt+gnzdωy=z1where gi is the nonlinear term after conversion and is constructed as follows:(12)g1z=f1z1+yr,z2,⋯,zn-dyrdωgiz=fiz1+yr,z2,⋯,zni=2,⋯,nNext, the output feedback tracking controller design process is initiated. With unmeasurable statesx1,⋯,xn characteristics of system (4), states observer of system (4) is established(13)z^˙i=z^i+1+hiABiz1-z^1i=1,⋯,n-1z^˙n=u+hnABnz1-z^1where z^i is the estimate value of zi and the high-order gain term consists of two variables At and Bt as follows:(14)A˙t=AAB2bz^12+e12-12δ2,A0=A0≥1μB˙t=-μ1B2+μ2B1+x1p2,B0=1where e1=z1-z^1 is error state, hi>0i=1,⋯,n is the coefficient of the Hurwitz polynomial with sn+h1sn-1+···+hn-1s+hn, and the unknown parameters b,μ1,μ2 satisfy 0<b<1/4p,0<μ1<μ2.


## 3.2. Lyapunov Analysis of Closed-Loop Systems

According to (14) on the definition of e1, it can be generalized to all states; that is, definition of ei=zi-z^ii=1,⋯,n and the available system error equation can be obtained:(15)dei=ei+1dt-hiABie1dt+gizdωden=-hnABne1dt+gnzdωIn order to facilitate the output feedback practical tracking controller design, the transformation of estimated states z^i and error states ei is introduced:(16)αi=eiABb+i-1i=1,⋯,nβi=z^iABb+i-1i=1,⋯,nwhere α=α1,⋯,αnT,β=β1,⋯,βnT,h=h1,⋯,hnT. By (16), stochastic nonlinear systems (11) and (13) can be converted to(17)dα=ABHαdt-A˙B+B˙AABCbαdt+Gz,A,Bdωβ˙=ABHbβ+ABhα1-A˙B+B˙AABCbβwhere(18)H=-h11⋯0⋮⋮⋱⋮-hn-10⋯1-hn0⋯0,Hb=01⋯0⋮⋮⋱⋮00⋯1-h1-h2⋯-hnCb=diagb,b+1,·····,b+n-1Gz,A,B=g1ABb,g2ABb+1,⋯,gnABb+n-1TFrom (14) on the definition of At and Bt, it can be generally obtained that (19)At≥A0≥1α≥1,∀t≥0and defining E=B2, the following can be obtained:(20)E˙≥2μ1E1-E,E0=1According to the comparison lemma in [28], the following can be drawn:(21)Bt≥1,0<1AtBt≤μ,∀t≥0By (16), it can be obtained that (22)zi=ABb+i-1αi+ABb+i-1βiand according to (21)-(22) and Assumptions 2 and 3, the inequality for nonlinear Gz,A,B can be deduced:(23)Gz,A,B=g1ABb2+⋯+gnABb+n-121/2≤1+x1pc∑i=1n1ABb+i-1zi+cη+d≤1+x1pα+β+1cη+dcwhere c-=cη+d/c.Then a systematic Lyapunov analysis is performed. First, the existence and uniqueness of the system solution are given where, since closed-loop stochastic nonlinear system (17) is continuous and satisfies locally Lipschitz condition on the initial point, the corresponding solution α,β,A,B of the system can be considered as existence and uniqueness.Define the following Lyapunov function:(24)V0α,β=αTPα+βTQβand positive definite matrices P,Q satisfy the following relation:(25)HTP+PH≤-Inr1In≤CbP+PCb≤r2InHbTQ+QHbT≤-2Inr3In≤CbQ+QCb≤r4Inwhere rii=1,2,3,4 is positive constant and In is n-dimensional judgment matrix.By (23), the following can be obtained:(26)2αTPG≤2Pcη+dc1+x1pα+1α+β≤1+x1p2α2+β2+η1α2+1where η1=cη+d/c2P2 is unknown constant.Further, by (25) and (26), the trajectory of (24) along the Ito differentiation for nonlinear stochastic system (17) can be obtained as follows:(27)LV0α,β≤-ABα2A˙B+B˙AABαTCbP+PCbα+TrGTPG-2ABβ2+2ABα1βTh-A˙B+B˙AABβTCbQ+QCbβ≤-ABα2-2ABβ2+P1+x1p2α+β+12cη+dc2-A˙B+B˙AABr2α2+r4β2+2αTPG+2ABα1βTQh≤-ABα2-2ABβ2+P1+x1p2α+β+12cη+dc2-A˙B+B˙AABr2α2+r4β2+1+x1p2α2+β2+η1α2+1+ABα2+β2≤-ABα2-2ABβ2+η1α2+1+μ1Br2α2+r4β2-1+x1p2α2+β2+η11+x1p2α+β+12≤-BA-2η1+μ1r2+μ1r4α2+β2+η1and Ito differentiation results after simplifying can be obtained:(28)LV0≤-BA-η2α2+β2+12η2-μ1r2+r4where η2=2η1+μ1r2+r4. Since matrices P,Q are positive, the following can be easily obtained:(29)λ1αTPα+βTQβ≤α2+β2≤λ2αTPα+βTQβwhere 0<λ1≤λ2. Moreover, by (29), (28) be deduced into(30)LV0≤-λ1BA-η2V0+12η2-μ1r2+r4ifA-η2≥0LV0≤-λ2BA-η2V0+12η2-μ1r2+r4ifA-η2<0


## 3.3. Boundedness Analysis of System States and Gain

According to (30), it is easy to observe that if η2 is a known constant, then system (17) can be guaranteed as asymptotically stable by requiring high-gain A to be as large as possible which has been studied by some existing papers. However, in this paper, the parameter η2 is an unknown constant which requires complex and rigorous analysis to determine the boundedness of the system on the maximum time interval 0,T and ensure the asymptotical stability of system tracking output.This section mainly focuses on the boundedness of high-gainsA, B and system states α, β. By (14), the following can be obtained:(31)At≥1μt∈0,TAccordingly, a hypothetical judgment can be drawn:(32)limt→T⁡At∈1μ,+∞To verify the above judgment (32), we make use of the negative proof technology; that is, if conclusion (32) does not hold, there exists a finite time t1∈0,T such that(33)λt=λ1,At≥2η2,∀t∈t1,TBy (30), it can be concluded that, for any time t∈t1,T, the following Ito differential inequality holds:(34)LV0≤-λ12ABV0+12η2-μ1r2+r4According to (34), the inequality with V0α,β on time interval t∈t1,T can be obtained:(35)V0α,β≤η2-μ1r2+r4λ1+V0α,βe-λ1/2t-t1By observing (35), it is easy to infer that V0α,β is bounded on t∈t1,+∞ and continuous on t∈0,t1; then it can be generalized that V0α,β is bounded on t∈0,T. Therefore, two cases of time T need to be discussed.The first case:T<+∞.IfV0α,β=h1>0 when t∈t1,T, then it can obtain the following inequality according to (34):(36)λ1h12∫t1TBsds≤12η2-μ1r2+r4T-t1+Vα,βBy observing (36) and (14), simultaneously, using α1=e1/ABb and β1=z^1/ABb, one has (37)A˙t=AAB2bz^12+e12-δ2≤λ1h12BAccording to (36) and (37) again, one can get (38)AT-A0≤λ1h12∫t1TBsdsAt the moment, by observing (38), it can find that AT-A0=+∞ on the left and λ1h1/2∫t1TBsds is constant on the right, which contradicts (38). Therefore, it is a wrong judgment that conclusion (32) does not hold.IfV0α,β=0 when t∈t1,T, by definition σ as any positive constant and according to limt→TAtBt=+∞, there exists t2≥t1 such that, ∀t∈t2,T, it can obtain (39)AtBt≥η2σλ1In addition, since V0α,β=0, there exists t3≥t2 such that V0αt3,βt3<σ/2. Through the above analysis, (34) can be converted to(40)LV0≤-2η2σV0+12η2-μ1r2+r4,∀t∈t3,TAccording to (40), the inequality with V0α,β on time interval t∈t3,T can be obtained:(41)Vα,β≤σ2+12η2-μ1r2+r4+Vαt3,βt3≤σ+12η2-μ1r2+r4and this proves that limt→TV0α,β=0.Suppose the following inequality holds:(42)A˙t≤ϑ1tBt,t∈0,Twhere ϑ1t=α12+β12-δ2/2AB2b2,t∈0,T. By (4), (14), and (16), the following can be obtained: (43)x1=ABbα1+β1+yrand, correspondingly, one has(44)x1t≤η+ϑ2tABbwhere ϑ2t=α1t+β1t. By (14) and Young’ inequality, one can obtain(45)B˙≤μ2B1+x1p2≤2μ2B1+η+ϑ2tABb2pand after further simplification one has(46)2μ2Bη+ϑ2tABb2p≤22pμ2η2p-1B+22pμ2ϑ22p-1AB2Then (46) can be converted to(47)B˙t≤2μ2+22pμ2η2p-1B+22pμ2ϑ22p-1AB2,t∈0,TNext define the function Ω=ABV0 and then by (34), (42), (47), and Ito differential theorem, one has(48)LΩ≤-AB2η2σV0+12η2-μ1r2+r4≤-AB2η2σ-ϑ1t-22pμ2ϑ22p-1V0+12η2-μ1r2+r4According to (42) and (47), it can be concluded that there exists t4≥t1 such that the following inequality can hold:(49)22pμ2ϑ22p-1≤η2σ-ϑ1Further, it can get(50)LΩ≤-ABη2σΩ-12μ1r2+r4-η2,t≥t4which shows that Ω is bounded. According to the definition of ϑ1t=α12+β12-δ2/2AB2b2,t∈0,T and through inequality amplification, one has(51)AB2be12+z^12≤λ1λ2h12which can prove that limt→Tz^12+e12=0 holds. Thence, there exists t5 such that, for t∈t5,T, it can obtain (52)z^12t+e12t≤λ1λ2h12AB2b≤δ22By (52) and (14), it can get A˙t=0,t∈t5,T which obviously conflicts with description of (14). Therefore, this is a false assertion that conclusion (32) does not hold.The second case:T=+∞.Definitionσ is any positive constant; by limt→TAtBt=+∞ it can be seen that there exist t2≥t1 such that, ∀t∈t2,T, one has(53)AtBt≥2η2σλ1⟹-AtBtλ1≤-2η2σThen (34) can be converted to(54)LV0≤-2η2σV0+12η2-μ1r2+r4,∀t≥t2By (53) and (54), the following can be obtained:(55)V0α,β≤12η2-μ1r2+r4+Vαt2,βt2e-2η2/σt-t2,t≥t2and then there exists t3≥t1 such that the following inequality holds:(56)V0αt2,βt2e-2η2/σt-t2≤12η2-μ1r2+r4,∀t≥t3Accordingly, it can obtain(57)Vα,β≤η2-μ1r2+r4,∀t≥t3Summarize the above analysis and it can be considered that limt→TV0α,β=0, then available A˙t=0, which is also contradictory with (14).Therefore, through the above two cases of analysis, it is a false assertion that conclusion (32) does not hold. Similarly, hypothetical judgment (32) is correct and system gain A is bounded on t∈0,T which satisfies(58)limt→T⁡At∈1μ,+∞Next, we discuss boundedness for statesα,β on 0,T.By (24), (29), and Ito differential theorem, define function (59)V2β=βTQβand then one has(60)LV2≤TrGTPG-2ABβ2+2ABα1βTQh-A˙B+AB˙ABβTCbQ+QCbβ+P1+x1p2α+β+12cη+dc2≤-ABβ2+ABα12-B˙BβTCbQ+QCbβ+P1+x1p2α+β+12cη+dc2≤-ABβ2+ABα12+μ1c3Bβ2-μ2c41+x1p2β2+P1+x1p2α+β+12cη+dc2By (60) and choosing the appropriate parameter μ1, it gets(61)LV2≤-Bβ2+ABα12+P1+x1p2α+β+12cη+dc2≤-BA-2η1+μ1r2+μ1r4β2+η1+ABα12and then by (14), one has(62)α12≤α12+β12-δ22AB2b+δ22AB2b≤α12+β12-δ22AB2b2+δ22+1such that the following inequality can be derived:(63)ABα12≤AA˙+AB1+δ22Thence, the following can be obtained:(64)ABα12≤AA˙+Bμ3where μ3=A1+δ2/2. By (64) and (61), the following can be derived:(65)LV2≤-BA-2η1+μ1r2+μ1r4β2+AA˙+μ3BAccording to the comparison theorem, one has(66)β2≤μ3B+V2β0+A2t-A022η1+μ1r2+μ1r4It has been shown above that A is bounded on 0,T. Further, (66) can prove that β is bounded on 0,T.According to the description of system (17), the state transition is introduced again:(67)Πi=1A-Bi-1+beiwhere A-=2c-2P2/c1μ2.Then the system error equation is converted to(68)Π˙=A-BLΠ+A-BhΠ1-ABΓhΠ1-B˙BCbΠ+G-z,u,A-,Bwhere G-=g1/A-Bb,g2/A-Bb+1,·····,gn/A-Bb+n-1T, Γ=diag1,A/A-,·····,A/A-n-1.Define Lyapunov function(69)V3Π=ΠTPΠand according to Ito's theorem, the trajectory of (69) along the system (68) is(70)LV3≤-A-BΠ2+2Π1A-BhTPΠ+2G-PΠ-2Π1ABhTΓPΠ-B˙BΠTCbP+PCbΠ≤-A-BΠ2+2Π1A-BhTPΠ-2Π1ABhTΓPΠ+2G-TPΠ+μ1c2BΠ2-μ2c11+x1p2Π2Furthermore, by the inequalities(71)2Π1A-BhTPΠ≤A-2BhTP2Π12+BΠ22Π1ABhTΓPΠ≤A2Bh2ΓP2Π12+BΠ2(70) can be converted to(72)LV3≤-BΠ2+lBΠ12+lBβ2+lwhere l is unknown constant. Simultaneously, by (63) and (64), one has(73)LV3≤-λ1BV3+lAA˙+lBβ2+l1+μ3Since the above analysis has proved that β is bounded on 0,T, then (73) can be converted to(74)LV3≤-λ1BV3+lAA˙+l-Bwhere l-=lmax⁡sup⁡β2,1+μ3. Simultaneously, by (16) and (17), one has(75)1λ1Π2≤V3Π0+l-λ1+lA2t-A022,t∈0,TSince the above analysis has proved that A is bounded on 0,T, then it can be inferred that Π is bounded on 0,T, Furthermore, combined with the above conclusions, it can be obtained that α is bounded on 0,T.Finally, we discuss the boundedness of stateB on 0,T.Since it has been concluded that system statesα and β are bounded on 0,T, then it can obtain(76)α1+β1≤kwhere k is positive constant and depends on the state initial value α0,β0. By (14), it can deduce(77)B˙≤-μ1B2+2μ2B+2μ2c-2pBUsing Young's inequality, (77) can be split to get the following relation form:(78)2μ2B≤EB2+C2μ2c-2pB≤FB2+Dwhere C, D, E, F are suitable coefficients or parameters.It is equivalent to(79)B˙≤E-B2+C-where C-, E- are suitable coefficients or parameters again. According to (67), it can be concluded that the state B is bounded on 0,T.In summary, system high-gainsA, B and states α, β are all bounded on 0,T. The boundedness of the system has been judged.


## 3.4. Output Feedback Tracking Controller Design

Based on the analysis of above, this section first gives the output feedback tracking controller.Theorem 5.
Consider a class of stochastic nonlinear system (4) whose output is y=x1-yr under Assumptions 2 and 3 and then the following output feedback practical tracking controller can be designed:(80)u=-ABnk1z^1+⋯+ABknz^nwhere A,B are defined by (14); ki>0 is coefficient of Hurwitz polynomial sn+h1sn-1+···+hn-1s+hn with hi=kn-i+1. Then, for any δ>0, there exists a finite time T such that x1-yr≤δ,∀t≥T, and all the states of stochastic nonlinear system (1) and (13) are bounded on 0,T under the action of output feedback practical tracking controller (80).The proof process ofTheorem5 is given.Using the inequality theorem(81)maxx,02-maxy,02≤x-y2It can be easily inferred that(82)BAB4bz12+e12-δ222,x,y∈Ris consistently continuous; that is, A˙ is consistent and continuous.Since system high-gainsA, B and states α, β are all bounded on 0,T, it is easy to conclude that e˙,z^˙,A˙,B˙ are also bounded on 0,T. According to the Barbalat lemma in [28], one has limt→+∞A˙t=0. By the above analysis, it can be deduced that there exists T such that(83)e12t+z^12t1/2≤CONSTANT,t≥TTherefore, it can also be inferred that(84)x1-yr=z1≤e1+z^1≤δ,t≥TThe proof ofTheorem5 is over such that output feedback practical tracking controller design is achieved.


## 4. Simulation Example

This section considers two-dimensional stochastic nonlinear systems as described below which are used extensively in offshore engineering valve models to verify the output feedback practical tracking controller designed in Section3:(85)dx1=x2dtdx2=udt-σ11+x12x2σ2dωy=x1-yrwhere σ1, σ2 are unknown parameters. System (85) satisfies Assumptions 2 and 3 with the corresponding parameters η1=σ1σ2, η2=σ11-σ2, p=2. ω is the m-dimensional standard Brownian motion defined on the probability space Ω,Γ,Ρ, which Ω is the sample space, Γ is the algebra, and Ρ is the probability measure. The given tracking trajectory is yrt=1.5sin⁡2t1-e-0.05t3.In numerical simulation, definef1x=0,f2x=-σ11+x12x2σ2 and then, using inequality transform, it can get(86)f2≤1+x12σ1σ2x2+σ11-σ2Next start the system simulation and select the initial value and related parameters:(87)x10=0.035,x20=-2.5,z^10=0,z^20=0,δ=0.5,h1=2,h2=5,k1=45,k2=55,μ1=10,μ2=20,Figures 1–3 show the response characteristics of the closed-loop system (85). By observing the simulation graph, the practical states of the two-dimensional system (85) are bounded and gradually converged. Simultaneously, it can be observed that the output tracking error of the system gradually converges after 2 s and finally adjusts to near zero. The results verify the effectiveness of the designed output feedback practical tracking controller.Figure 1
The trajectory of system statex1.Figure 2
The trajectory of system statex2.Figure 3
The trajectory of system outputy.


## 5. Conclusion and Future Prospects

In this paper, we study the tracking problem of output feedback for a class of stochastic nonlinear systems which remove the more stringent growth conditions and satisfy polynomial function growth conditions. The parameters of the output polynomial function growth conditions are unknown which leads to the uncertainty of the parameters and generalizes the previous output feedback practical tracking control research results. Finally, an output feedback practical tracking controller with dynamically updated gains is constructed explicitly so that all the states of the closed-loop systems are globally bounded and the tracking error belongs to arbitrarily small interval after some positive finite time. A numerical example illustrates the efficiency of the theoretical results.This paper has made some valuable research results, but also produced some further in-depth research of problem. We consider whether it can generalize it to stochastic nonlinear systems with uncertain control coefficients and expand it to time-delay systems. How to design the output feedback practical tracking controller for this type of extended system? It is worth studying the problem.



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