# Nonsingular Global Fixed-Time Stabilization for Nonlinear Systems

Authors: Wei Hu; Zhangyong Zhou; Junjun Tang
Journal: Complexity
 (2021)

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## Abstract

Since existing results about fixed-time stabilization are only applied to strict feedback systems, this paper investigates the nonsingular fixed-time stabilization of more general high-order nonlinear systems. Based on a novel concept named coordinate mapping of time domain, a control method is first proposed to transform the nonsingular fixed-time convergence problem into the finite-time convergence problem of a transformed time-varying system. By extending the existing, adding a power integrator technique into the considered time-varying system, a periodic controller is constructed to stabilize the original system in fixed time. The results of simulations verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

In recent years, more and more attention has been paid to the controller design of high-order nonlinear systems due to its wide application in modeling aerospace craft, rigid robotic systems, and machine systems with underactuation, weak coupling, and instability [1–4]. On the one hand, high-order nonlinear systems are more general forms of strict feedback nonlinear systems, which have been widely studied. More practical systems can be described, including rigid robots, aerospace vehicles, and hydraulic systems. On the other hand, the systems cannot be linearized at the origin or cannot be controlled after linearization when the power of the high-order nonlinear system is not 1, which makes it difficult to control [2].For high-order nonlinear system, the finite-time stabilization is studied from different perspectives in [1–6]. However, the convergence time is decided by the initial state of the system in recent achievements. That is to say, the convergence time cannot be prespecified, since the state can be initialed at any point. Besides, when the initial state of the system tends to infinity, the time tends to infinity as well.The concept of fixed-time stabilization is proposed in [7]. As a special finite-time stabilization method, it requires that the convergence time of the system be bounded and the upper bound be independent of the initial state, which can be set in advance. A primary method for fixed-time stabilization is proposed in [8], which successfully solved the problem of fixed-time stabilization for linear systems with only matching uncertainties. Most of the existing literature is limited to second-order linear systems [9–12]; only [13–15] have studied the fixed-time stabilization for high-order nonlinear systems. Based on the implicit Lyapunov function method, the fixed-time stability of high-order integrators is analyzed in [13]. A nonsmoothed controller is constructed by using the recursive design method to solve the fixed-time stabilization problem of high-order nonlinear interconnected systems [14]. The fixed-time stabilization of strict feedback nonlinear systems with only matching uncertainties is achieved by ingenious state transformation [15]. It should be pointed out that all the systems in [8–15] are merely special forms of higher-order systems. Therefore, it is of great significance to study the fixed-time stabilization of more general high-order nonlinear systems.Based on the above, the nonsingular global fixed-time stabilization of high-order nonlinear systems is proposed. The main difficulty lies in the design complexity caused by various power terms. Particularly, this issue would be intensified if we adopt traditional double-power-term law. The obstacle is partially avoided in this work by using the time-domain mapping, with which the nonsingular fixed-time stabilization problem of the original system is transformed into the finite-time stabilization problem of the corresponding time-varying system; by using the power integration method, the finite-time stabilization problem of the time-varying system is realized. The main innovations are summarized as follows:(1)
A control method based on time-domain mapping is proposed, which transforms the nonsingular fixed-time stabilization problem of the original system into the finite-time stabilization problem of the corresponding time-varying system and provides a new idea for the design of the fixed-time convergence control law. Compared with the traditional double-power-reaching law, the proposed method is designed with a single-power-reaching law, which is more effective and simpler.(2)
In essence, the fixed-time stabilization can be regarded as the optimal control with fixed terminal time, and the design of its control law is easy to produce singularity, while the control law of the proposed method is nonsingular.(3)
The existing method can only solve the problem of fixed-time stabilization for strict feedback systems with only matching uncertainties, while the proposed method can solve the fixed-time stabilization problem of high-order nonlinear systems with unmatched uncertainties. It is noted that the strict feedback systems are special cases of high-order nonlinear systems; the results of this paper greatly extend the research scope of fixed-time stabilization.


## 2. Description

For the convenience of description, we defineR, R+, and Rn as real number, positive real number, and n-dimensional real vectors, respectively, define Ci as i-order continuous differentiable function, and define Qodd as rational number whose numerator and denominator are positive odd integers.Consider the following high-order nonlinear systems:(1)x˙1=x2p1+f1x¯1,t,x˙2=x3p2+f2x¯2,t,⋮x˙n=upn+fnx¯n,t,where x=x1,x2,…,xnT∈Rn and u∈R represent the state of the system and the control input; x¯i is defined as x1,…,xiT∈Ri; pi is a constant, which belongs to Qodd; fi:Ri×0,+∞⟶R is an uncertain nonlinear function that satisfies fi0,t=0, ∀t∈0,+∞. According to the mean value theorem, it means that there exists a function γijx¯i,t, which makes fix¯i,t=∑j=1ixjγijx¯i,t, i=1,…,n. Furthermore, the following assumptions are given.Hypothesis 1.
There exists a functionγ¯ix¯i≥0, which makes fix¯i,t=∑j=1ixjγijx¯i,t, i=1,…,n.Hypothesis 2.
Forpi∈Qodd, i=1,…,n, in formula (1), there exists a constant ηi∈Qodd, i=1,…,n+1, which is larger than 1, and it makes the following formulas hold:(2)0<1η1−p1η2≤1η2−p2η3≤⋯≤1ηn−pnηn+1,(3)piηi+1≤min1η1,1η2,…,1ηi−1.
To accurately describe the concept of fixed-time stability, we give the following definitions.Definition 1.
If any solutionxt,x0 of formula (1) reaches the origin at a certain finite time Tx0 and then remains at the origin, where T:Rn↦R+∪0 is called the convergence time function, then the origin can be called globally finite-time stable. If the origin is globally finite-time stable and the convergence time function Tx0 is bounded, that is, ∃Tmax>0, such that Tx0≤Tmax, ∀x0∈Rn, then the origin is globally fixed-time stable.
The goal of the paper is to design a control input for system (1) to make all closed-loop signals globally bounded and the system state converges to the origin in a fixed time. To facilitate the controller design and stability analysis, the following lemma is introduced from [6, 16].Lemma 1.
The Lyapunov functionVt is assumed to meet the following formula:(4)dVtdt≤−ρVtυ,where ρ is a constant and ρ>0 and 0<υ<1; then, the following formulas can be established:(5)Vt≤V1−υt0−1−υρt−t01/1−υ,t0≤t<t0+V1−υt0ρ1−υ,Vt=0,t≥t0+V1−υt0ρ1−υ.Lemma 2.
For any positive real numberxi, i=1,…,n, 1<b<1, the following inequality is established:(6)x1+⋯+xnb≤x1b+⋯+xnb.
Whenb=p/q≤1 and p>0 and q>0, we can obtain(7)xb−yb≤21−bx−yb.Lemma 3.
For any positive real numberm, n, and function ax,y, the following inequality is established:(8)ax,yxmyn≤cx,yxm+n+nm+n×mm+ncx,ym/nax,ym+n/nym+n,where cx,y>0, x∈R, y∈R.


## 3. Design of Nonsingular Fixed-Time Controller

### 3.1. Time-Domain Mapping

In this section, the concept of time-domain mapping is proposed for the first time. The problem of nonsingular fixed-time stabilization of system (1) is transformed into the problem of finite-time stabilization of the time-varying system (12), which simplifies the analysis and design process of fixed-time stabilization.Assume that the upper bound of the convergence time of system (1) is T; when t∈0,T, the following time-domain coordinate mapping is used to extend the finite-time domain to the infinite-time domain:(9)τ=μ+σtantπ/2+arctanμ/σT−arctanμ/σ.The inverse transformation of it is as follows:(10)t=Tarctanτ−μ/σ+arctanμ/σπ/2+arctanμ/σ≜t0τ,where μ∈R and σ>0; according to equation (10), obviously there is t0τ:0,+∞⟶0,T, and find the derivative on the left and right sides of formula; formula (10) can be transformed as follows:(11)dtdτ=Tσπ/2+arctanμ/σ⋅11+τ−μ/σ≜Κτ.Whent∈0,T, multiplying both sides of formula (1) by formula (11) yields(12)dx1dτ=κτx2p1+f˜1x¯1,τ,dx2dτ=κτx3p2+f˜2x¯2,τ,⋮dxndτ=κτupn+f˜nx¯n,τ,where f˜ix¯i,τ=fix¯i,t∈C1; according to Hypothesis 1, the following inequality holds:(13)f˜ix¯i,τ≤γ¯ix¯ix1+⋯+xi,i=1,…,n.Remark 1.
For the converted time-varying system (12), κτ can be regarded as the time-varying control coefficient of the system. Therefore, to stabilize system (12), the control law must contain unbounded gain terms 1/κτ to compensate for the effectiveness loss of the time-varying control factor κτ. This means that if system (12) is asymptotically stabilized in the time domain τ, although system (1) will achieve a fixed-time stabilization in the time domain t, the control input tends to infinity at the terminal time. To overcome the singularity of the control law, a natural method is to achieve finite-time stabilization in time domain τ so that system (1) can achieve fixed-time stabilization at a certain time tf<T. As τ is boundless, the control law is nonsingular.Remark 2.
The coordinate mapping from time domainτ to t is not limited to the form of equation (10). Some coordinate maps such as exponential function and trigonometric function are also feasible when they satisfy the following conditions: (1) infinitely differentiable and monotonically increasing; (2) t00=0 and limτ⟶+∞t0τ=T. On the other hand, by adjusting μ and σ, a relatively smooth and practical control input can be obtained.


### 3.2. Design of Finite-Time Controller in Time Domain

In this section, a finite-time state feedback controller is designed in time domainτ for the time-varying system (12). Firstly, the control parameters are selected. According to Hypothesis 2, a constant νi∈Qodd, i=1,…,n, which is not less than 1 can be chosen to satisfy the following requirements:(14)v1+p1η2=v2+p2η3=⋯=vn+pnηn+1≜ω.Extending the power integral method to the time-varying system (12), the finite-time controller is designed in time domain τ recursively. It should be pointed out that V˙i represents the derivative of Vi to τ.


#### 3.2.1. Choosing theC1 Positive Definite Lyapunov Function

(15)V1x1=1η1v1+1x1η1v1+1≜W1x1.Calculating derivation of equation (15), the following equation can be obtained according to formula (3):(16)V˙1x1=x1η1ν1κτx2p1+f˜1x¯1,τ≤κτx1η1ν1x2p1−x2∗p1+κτx1η1ν1x2∗p1+κτx1η1ν1+p1/η2ρ1x1,where ρ1x1≥x11−p1η1/η2γ¯1x1≥0 is the C1 function, which can be defined as ρ1x1=1+x12γ¯1x1.According to formula (16), the virtual control law x2∗ is defined as follows:(17)x2∗p1=−x1p1η1/η2nΚτ+ρ1x1≜−ξ1p1/η2β1p1x1,τ,where ξ1=x1η1 and β1x1,τ=n/κτ+ρ1x11/p1>0 are C1 functions. Substituting equation (17) into equation (16), the following formula can be obtained:(18)V˙1x1≤−nx1η1ν1+p1/η2+κτx1η1ν1x2p1−x2∗p1.Suppose that, in stepk−1, there exists a positive Lyapunov function, which satisfies the following equation:(19)Vk−1x¯k−1,τ≤2∑i=1k−1ξiνi+1/ηi.Define virtual control law and error as follows:(20)x1∗=0,ξ1=x1η1−x1∗η1,x2∗=−ξ11/η2β1x1,τ,ξ2=x2η2−x2∗η2,⋮⋮xk∗=−ξk−11/ηkβk−1x¯k−1,τ,ξk=xkηk−xk∗ηk,where βix¯i,τ>0 and ξi are C1 functions. Besides, i=1,…,k−1 and(21)V˙k−1x¯k−1,τ≤−n−k+2∑i=1k−1ξiω+κτξk−1νk−1xkpk−1−xk∗pk−1.Asξk−1 and βk−1x¯k−1,τ>0 are C1 functions, ηk≥1; it can be easily obtained that ξk=xkηk+ξk−1βk−1ηkx¯k−1,τ is also a C1 function.DesigningC0 virtual control law xk+1∗, which makes equations (19) and (21) hold in step k, the C1 positive Lyapunov function (equation (22)) is considered.(22)Vkx¯k,τ=Vk−1x¯k−1,τ+∫xk∗xksηk−xk∗ηkνkds.According to equations (7) and (19) and mean value theorem, it can be achieved that Vkx¯k,τ≤2∑i=1kξiνi+1/ηi, which means that equation (19) is right in step k.Defining thatWkx¯k,τ=∫xk∗xksηk−xk∗ηkνkds, calculating derivation of Vkx¯k,τ to τ, equation (23) can be achieved after combining formulas (20) to (22).(23)V˙kx¯k,τ≤−n−k+2∑i=1k−1ξiω+κτξk−1νk−1xkpk−1−xk∗pk−1+κτξkνkxk+1pk−xk+1∗pk+κτξkνkxk+1∗pk+κτξkνkf˜kx¯k,τ+∑i=1k−1∂Wk∂xidxidτ+∂Wk∂τ.Let us estimate each term on the right side of equation (23); according to Lemma 2, it can be obtained that(24)xkpk−1−xk∗pk−1≤21−pk−1/ηkxkηk−xk∗ηkpk−1/ηk≤2ξkpk−1/ηk.According to Lemma3, there exists a C1 function ckx¯k−1≥0 making the following inequality right:(25)κτξk−1νk−1xkpk−1−xk∗pk−1≤14ξk−1ω+κτckx¯k−1ξkω.For the convenience of narration, the following lemmas are given to estimate the residual terms on the right side of equation (23). The proof ideas of Lemmas 4 and 5 are similar to those of inequalities (17) and (18) in [4]. Readers can refer to [4] for proof. For simplicity, this paper omits the proof.Lemma 4.
There exitsC1 function γ˜kx¯k,τ≥0, k=2,…,n, which makes the following inequality hold:(26)κτξkνkf˜kx¯k,τ≤14∑i=1k−1ξiω+κτγ˜kx¯k,τξkω.Lemma 5.
There existsC1 function ρ˜kx¯k,τ≥0, k=2,…,n, which makes the following inequality hold:(27)∑i=1k−1∂Wk∂xidxidτ≤14∑i=1k−1ξiω+κτρ˜kx¯k,τξkω.Lemma 6.
There existsC1 function φ˜kx¯k,τ≥0, φ˜kx¯k,τ≥0, which makes the following inequality hold:(28)∂Wk∂τ≤14∑i=1k−1ξiω+φ˜kx¯k,τξkω.


#### 3.2.2. Demonstration

It is noticed that the time after transformation is only explicitly included inxk∗ηk which belongs to the expression Wkk=2,…,n. So, we use mathematical induction to analyze ∂xk∗ηk/∂τ.According tox2∗η2=−x1β1η2x1,τ, where β1x1,τ>0 is a C1 function, it can be concluded that the partial derivative about τ to x2∗η2 is available and that the following inequality can be obtained:(29)∂x2∗η2∂τ=ξ1∂β1η2x1,τ∂τ≤ξ1φ1x1,τ,where φ1x1,τ≥0 is a designable C1 function.Assuming that whenk=3,…,n, xk−1∗ηk−1 is partially differentiable to τ, there exists a C1 function φk−2x¯k−2,τ, which makes the following equality hold:(30)∂xk−1∗ηk−1∂τ=φk−2x¯k−2,τ∑i=1k−2ξi.It should be noticed thatxk∗ηk=−ξk−1βk−1ηkx¯k−1,τ, where βk−1x¯k−1,τ is a C1 function and ξk−1=xk−1ηk−1−xk−1∗ηk−1; therefore, xk∗ηk is partially differentiable to τ, and(31)∂xk∗ηk∂τ≤ξk−1∂βk−1ηkx¯k−1,τ∂τ+βk−1ηkx¯k−1,τ∂xk−1∗ηk−1∂τ≤ξk−1∂βk−1ηkx¯k−1,τ∂τ+βk−1ηkx¯k−1,τφk−2x¯k−2,τ∑i=1k−2ξi≤φk−1x¯k−1,τ∑i=1k−1ξi,where φk−1x¯k−1,τ≥0 is a designable C1 function.Whenk=2,…,n, Wk is continuous partial differentiable to τ. According to formula (2), we can also know that νk+1/ηk>ω; hence, we have the following:(32)∂Wk∂τ=∂xk∗ηk∂τ∫xk∗xkνksηk−xk∗ηkνk−1ds≤2νkφk−1x¯k−1,τξkvk−1+1/ηk∑i=1k−1ξi≤14∑i=1k−1ξiω+ξkωφ˜kx¯k,τ,where φ˜kx¯k,τ≥0 is a designable C1 function.Substituting formulas (25)–(28) into equation (23), the following inequality can be obtained:(33)V˙kx¯k,τ≤−n−k+1∑i=1k−1ξiω+κτξkνkxk+1pk−xk+1∗pk+κτξkωckx¯k−1+γ˜kx¯k,τ+ρ˜kx¯k,τ+φ˜kx¯k,τκτ+κτξkνkxk+1∗pk.TheC0 virtual control law in step k can be designed as follows:(34)xk+1∗=−ξk1/ηk+1n−k+1+φ˜kx¯k,τκτ+ckx¯k−1+γ˜κx¯k,τ+ρ˜κx¯k,τ1/pκ≜−ξk1/ηk+1βκx¯k,τ,where βkx¯k,τ>0 is a C1 function. Substituting formula (34) into formula (33), the following inequality can be obtained:(35)V˙kx¯k,τ≤−n−k+1∑i=1kξiω+κτξkνkxk+1pk−xk+1∗pk.According to the above derivation process, the followingC0 state feedback controller can be designed in step n:(36)u=xn+1∗=−ξndnβnx,τ,which makes C1 positive Lyapunov function Vnx,τ meet Vnx,τ≤2∑i=1nξiνi+1/ηi, and the following formula is established:(37)V˙nx,τ≤−∑i=1nξiω.Considering that momentk contains the dynamic open set of origin Ω¯=x∈Rn|ξi<1,i=1,…,n, we can choose ε>0, which is small enough to make Ω˜=x∈Rn|Vnx,τ<ε be a subset of Ω¯ in time k. From formulas (2) and (14), we can know that νi+1/ηi>ω, and combining (6) and (37), the following inequality can be achieved:(38)dVdτ≤−Vρ2ρ,∀x∈Ω˜,where ρmini=1,…,nνi+1/ηi=ω.In the time domainτ and, according to equation (37), the solution trajectories of system (12) will enter the set Ω˜ in finite time and the finite time can be marked as τs. It should be noted that τs cannot be calculated but can be obtained online by real-time detection of system errors. Furthermore, by using formula (38) and Lemma 1, the finite convergence time of system (12) in time domain τ can be estimated as follows:(39)τf=τs+2n1−ρ1−ρ.Remark 3.
From formula (37), we can know that x is bounded, and, according to the state feedback control law (36), it can be known that u is also bounded in the closed interval 0,τf of time domain; that is to say, the actual control law (36) is not singular.


### 3.3. Design of Fixed-Time Controller in Time Domaint

Transform the mapping in (9) into the following compact form:(40)τ=τ0t=μ2+σ2μ+cottπ/2+arctanμ/σ/T.Consider that we have already made controller (36) of system (12) into finite-time stabilization in the last section. Let tf=t0τf, and if the control law in time domain t is designed as (41)ux,t=−ξn1/ηn+1βnx,τ0t,0≤t<tf.Then, any solution trajectory of system (1) will reach the origin in finite time.For the time setktf,ktf+tf, k=1,2,… in time domain t, assume that the following time-domain coordinate mapping is adopted:(42)t=tkτ=t0τ+ktf.The control law can be designed as follows:(43)ux,t=−ξn1/ηn+1βnx,τ0t−ktf,ktf≤t<ktf+tf.The solution trajectory of system (1) will always remain at the origin.In conclusion, the fixed-time stabilization control law in time domaint can be designed as(44)ux,t=−ξn1/ηn+1βnx,τ0modt,tf,where modt,tf denotes modular operation, the result of which is the remainder obtained by t dividing tf.Remark 4.
The proposed control scheme provides a novel perspective of fixed-time stabilization. Compared with the traditional method composed of high-power and low-power terms [8–11], there exist only low-power terms in (44), which simplifies the design and analysis process to some degree. Besides, it is worth noticing that the setting times of [8–11, 17] depend on the control parameter, e.g., control gains and power terms. However, the setting time in this work can be directly specified in advance.


## 3.1. Time-Domain Mapping

In this section, the concept of time-domain mapping is proposed for the first time. The problem of nonsingular fixed-time stabilization of system (1) is transformed into the problem of finite-time stabilization of the time-varying system (12), which simplifies the analysis and design process of fixed-time stabilization.Assume that the upper bound of the convergence time of system (1) is T; when t∈0,T, the following time-domain coordinate mapping is used to extend the finite-time domain to the infinite-time domain:(9)τ=μ+σtantπ/2+arctanμ/σT−arctanμ/σ.The inverse transformation of it is as follows:(10)t=Tarctanτ−μ/σ+arctanμ/σπ/2+arctanμ/σ≜t0τ,where μ∈R and σ>0; according to equation (10), obviously there is t0τ:0,+∞⟶0,T, and find the derivative on the left and right sides of formula; formula (10) can be transformed as follows:(11)dtdτ=Tσπ/2+arctanμ/σ⋅11+τ−μ/σ≜Κτ.Whent∈0,T, multiplying both sides of formula (1) by formula (11) yields(12)dx1dτ=κτx2p1+f˜1x¯1,τ,dx2dτ=κτx3p2+f˜2x¯2,τ,⋮dxndτ=κτupn+f˜nx¯n,τ,where f˜ix¯i,τ=fix¯i,t∈C1; according to Hypothesis 1, the following inequality holds:(13)f˜ix¯i,τ≤γ¯ix¯ix1+⋯+xi,i=1,…,n.Remark 1.
For the converted time-varying system (12), κτ can be regarded as the time-varying control coefficient of the system. Therefore, to stabilize system (12), the control law must contain unbounded gain terms 1/κτ to compensate for the effectiveness loss of the time-varying control factor κτ. This means that if system (12) is asymptotically stabilized in the time domain τ, although system (1) will achieve a fixed-time stabilization in the time domain t, the control input tends to infinity at the terminal time. To overcome the singularity of the control law, a natural method is to achieve finite-time stabilization in time domain τ so that system (1) can achieve fixed-time stabilization at a certain time tf<T. As τ is boundless, the control law is nonsingular.Remark 2.
The coordinate mapping from time domainτ to t is not limited to the form of equation (10). Some coordinate maps such as exponential function and trigonometric function are also feasible when they satisfy the following conditions: (1) infinitely differentiable and monotonically increasing; (2) t00=0 and limτ⟶+∞t0τ=T. On the other hand, by adjusting μ and σ, a relatively smooth and practical control input can be obtained.


## 3.2. Design of Finite-Time Controller in Time Domain

In this section, a finite-time state feedback controller is designed in time domainτ for the time-varying system (12). Firstly, the control parameters are selected. According to Hypothesis 2, a constant νi∈Qodd, i=1,…,n, which is not less than 1 can be chosen to satisfy the following requirements:(14)v1+p1η2=v2+p2η3=⋯=vn+pnηn+1≜ω.Extending the power integral method to the time-varying system (12), the finite-time controller is designed in time domain τ recursively. It should be pointed out that V˙i represents the derivative of Vi to τ.


### 3.2.1. Choosing theC1 Positive Definite Lyapunov Function

(15)V1x1=1η1v1+1x1η1v1+1≜W1x1.Calculating derivation of equation (15), the following equation can be obtained according to formula (3):(16)V˙1x1=x1η1ν1κτx2p1+f˜1x¯1,τ≤κτx1η1ν1x2p1−x2∗p1+κτx1η1ν1x2∗p1+κτx1η1ν1+p1/η2ρ1x1,where ρ1x1≥x11−p1η1/η2γ¯1x1≥0 is the C1 function, which can be defined as ρ1x1=1+x12γ¯1x1.According to formula (16), the virtual control law x2∗ is defined as follows:(17)x2∗p1=−x1p1η1/η2nΚτ+ρ1x1≜−ξ1p1/η2β1p1x1,τ,where ξ1=x1η1 and β1x1,τ=n/κτ+ρ1x11/p1>0 are C1 functions. Substituting equation (17) into equation (16), the following formula can be obtained:(18)V˙1x1≤−nx1η1ν1+p1/η2+κτx1η1ν1x2p1−x2∗p1.Suppose that, in stepk−1, there exists a positive Lyapunov function, which satisfies the following equation:(19)Vk−1x¯k−1,τ≤2∑i=1k−1ξiνi+1/ηi.Define virtual control law and error as follows:(20)x1∗=0,ξ1=x1η1−x1∗η1,x2∗=−ξ11/η2β1x1,τ,ξ2=x2η2−x2∗η2,⋮⋮xk∗=−ξk−11/ηkβk−1x¯k−1,τ,ξk=xkηk−xk∗ηk,where βix¯i,τ>0 and ξi are C1 functions. Besides, i=1,…,k−1 and(21)V˙k−1x¯k−1,τ≤−n−k+2∑i=1k−1ξiω+κτξk−1νk−1xkpk−1−xk∗pk−1.Asξk−1 and βk−1x¯k−1,τ>0 are C1 functions, ηk≥1; it can be easily obtained that ξk=xkηk+ξk−1βk−1ηkx¯k−1,τ is also a C1 function.DesigningC0 virtual control law xk+1∗, which makes equations (19) and (21) hold in step k, the C1 positive Lyapunov function (equation (22)) is considered.(22)Vkx¯k,τ=Vk−1x¯k−1,τ+∫xk∗xksηk−xk∗ηkνkds.According to equations (7) and (19) and mean value theorem, it can be achieved that Vkx¯k,τ≤2∑i=1kξiνi+1/ηi, which means that equation (19) is right in step k.Defining thatWkx¯k,τ=∫xk∗xksηk−xk∗ηkνkds, calculating derivation of Vkx¯k,τ to τ, equation (23) can be achieved after combining formulas (20) to (22).(23)V˙kx¯k,τ≤−n−k+2∑i=1k−1ξiω+κτξk−1νk−1xkpk−1−xk∗pk−1+κτξkνkxk+1pk−xk+1∗pk+κτξkνkxk+1∗pk+κτξkνkf˜kx¯k,τ+∑i=1k−1∂Wk∂xidxidτ+∂Wk∂τ.Let us estimate each term on the right side of equation (23); according to Lemma 2, it can be obtained that(24)xkpk−1−xk∗pk−1≤21−pk−1/ηkxkηk−xk∗ηkpk−1/ηk≤2ξkpk−1/ηk.According to Lemma3, there exists a C1 function ckx¯k−1≥0 making the following inequality right:(25)κτξk−1νk−1xkpk−1−xk∗pk−1≤14ξk−1ω+κτckx¯k−1ξkω.For the convenience of narration, the following lemmas are given to estimate the residual terms on the right side of equation (23). The proof ideas of Lemmas 4 and 5 are similar to those of inequalities (17) and (18) in [4]. Readers can refer to [4] for proof. For simplicity, this paper omits the proof.Lemma 4.
There exitsC1 function γ˜kx¯k,τ≥0, k=2,…,n, which makes the following inequality hold:(26)κτξkνkf˜kx¯k,τ≤14∑i=1k−1ξiω+κτγ˜kx¯k,τξkω.Lemma 5.
There existsC1 function ρ˜kx¯k,τ≥0, k=2,…,n, which makes the following inequality hold:(27)∑i=1k−1∂Wk∂xidxidτ≤14∑i=1k−1ξiω+κτρ˜kx¯k,τξkω.Lemma 6.
There existsC1 function φ˜kx¯k,τ≥0, φ˜kx¯k,τ≥0, which makes the following inequality hold:(28)∂Wk∂τ≤14∑i=1k−1ξiω+φ˜kx¯k,τξkω.


### 3.2.2. Demonstration

It is noticed that the time after transformation is only explicitly included inxk∗ηk which belongs to the expression Wkk=2,…,n. So, we use mathematical induction to analyze ∂xk∗ηk/∂τ.According tox2∗η2=−x1β1η2x1,τ, where β1x1,τ>0 is a C1 function, it can be concluded that the partial derivative about τ to x2∗η2 is available and that the following inequality can be obtained:(29)∂x2∗η2∂τ=ξ1∂β1η2x1,τ∂τ≤ξ1φ1x1,τ,where φ1x1,τ≥0 is a designable C1 function.Assuming that whenk=3,…,n, xk−1∗ηk−1 is partially differentiable to τ, there exists a C1 function φk−2x¯k−2,τ, which makes the following equality hold:(30)∂xk−1∗ηk−1∂τ=φk−2x¯k−2,τ∑i=1k−2ξi.It should be noticed thatxk∗ηk=−ξk−1βk−1ηkx¯k−1,τ, where βk−1x¯k−1,τ is a C1 function and ξk−1=xk−1ηk−1−xk−1∗ηk−1; therefore, xk∗ηk is partially differentiable to τ, and(31)∂xk∗ηk∂τ≤ξk−1∂βk−1ηkx¯k−1,τ∂τ+βk−1ηkx¯k−1,τ∂xk−1∗ηk−1∂τ≤ξk−1∂βk−1ηkx¯k−1,τ∂τ+βk−1ηkx¯k−1,τφk−2x¯k−2,τ∑i=1k−2ξi≤φk−1x¯k−1,τ∑i=1k−1ξi,where φk−1x¯k−1,τ≥0 is a designable C1 function.Whenk=2,…,n, Wk is continuous partial differentiable to τ. According to formula (2), we can also know that νk+1/ηk>ω; hence, we have the following:(32)∂Wk∂τ=∂xk∗ηk∂τ∫xk∗xkνksηk−xk∗ηkνk−1ds≤2νkφk−1x¯k−1,τξkvk−1+1/ηk∑i=1k−1ξi≤14∑i=1k−1ξiω+ξkωφ˜kx¯k,τ,where φ˜kx¯k,τ≥0 is a designable C1 function.Substituting formulas (25)–(28) into equation (23), the following inequality can be obtained:(33)V˙kx¯k,τ≤−n−k+1∑i=1k−1ξiω+κτξkνkxk+1pk−xk+1∗pk+κτξkωckx¯k−1+γ˜kx¯k,τ+ρ˜kx¯k,τ+φ˜kx¯k,τκτ+κτξkνkxk+1∗pk.TheC0 virtual control law in step k can be designed as follows:(34)xk+1∗=−ξk1/ηk+1n−k+1+φ˜kx¯k,τκτ+ckx¯k−1+γ˜κx¯k,τ+ρ˜κx¯k,τ1/pκ≜−ξk1/ηk+1βκx¯k,τ,where βkx¯k,τ>0 is a C1 function. Substituting formula (34) into formula (33), the following inequality can be obtained:(35)V˙kx¯k,τ≤−n−k+1∑i=1kξiω+κτξkνkxk+1pk−xk+1∗pk.According to the above derivation process, the followingC0 state feedback controller can be designed in step n:(36)u=xn+1∗=−ξndnβnx,τ,which makes C1 positive Lyapunov function Vnx,τ meet Vnx,τ≤2∑i=1nξiνi+1/ηi, and the following formula is established:(37)V˙nx,τ≤−∑i=1nξiω.Considering that momentk contains the dynamic open set of origin Ω¯=x∈Rn|ξi<1,i=1,…,n, we can choose ε>0, which is small enough to make Ω˜=x∈Rn|Vnx,τ<ε be a subset of Ω¯ in time k. From formulas (2) and (14), we can know that νi+1/ηi>ω, and combining (6) and (37), the following inequality can be achieved:(38)dVdτ≤−Vρ2ρ,∀x∈Ω˜,where ρmini=1,…,nνi+1/ηi=ω.In the time domainτ and, according to equation (37), the solution trajectories of system (12) will enter the set Ω˜ in finite time and the finite time can be marked as τs. It should be noted that τs cannot be calculated but can be obtained online by real-time detection of system errors. Furthermore, by using formula (38) and Lemma 1, the finite convergence time of system (12) in time domain τ can be estimated as follows:(39)τf=τs+2n1−ρ1−ρ.Remark 3.
From formula (37), we can know that x is bounded, and, according to the state feedback control law (36), it can be known that u is also bounded in the closed interval 0,τf of time domain; that is to say, the actual control law (36) is not singular.


## 3.2.1. Choosing theC1 Positive Definite Lyapunov Function

(15)V1x1=1η1v1+1x1η1v1+1≜W1x1.Calculating derivation of equation (15), the following equation can be obtained according to formula (3):(16)V˙1x1=x1η1ν1κτx2p1+f˜1x¯1,τ≤κτx1η1ν1x2p1−x2∗p1+κτx1η1ν1x2∗p1+κτx1η1ν1+p1/η2ρ1x1,where ρ1x1≥x11−p1η1/η2γ¯1x1≥0 is the C1 function, which can be defined as ρ1x1=1+x12γ¯1x1.According to formula (16), the virtual control law x2∗ is defined as follows:(17)x2∗p1=−x1p1η1/η2nΚτ+ρ1x1≜−ξ1p1/η2β1p1x1,τ,where ξ1=x1η1 and β1x1,τ=n/κτ+ρ1x11/p1>0 are C1 functions. Substituting equation (17) into equation (16), the following formula can be obtained:(18)V˙1x1≤−nx1η1ν1+p1/η2+κτx1η1ν1x2p1−x2∗p1.Suppose that, in stepk−1, there exists a positive Lyapunov function, which satisfies the following equation:(19)Vk−1x¯k−1,τ≤2∑i=1k−1ξiνi+1/ηi.Define virtual control law and error as follows:(20)x1∗=0,ξ1=x1η1−x1∗η1,x2∗=−ξ11/η2β1x1,τ,ξ2=x2η2−x2∗η2,⋮⋮xk∗=−ξk−11/ηkβk−1x¯k−1,τ,ξk=xkηk−xk∗ηk,where βix¯i,τ>0 and ξi are C1 functions. Besides, i=1,…,k−1 and(21)V˙k−1x¯k−1,τ≤−n−k+2∑i=1k−1ξiω+κτξk−1νk−1xkpk−1−xk∗pk−1.Asξk−1 and βk−1x¯k−1,τ>0 are C1 functions, ηk≥1; it can be easily obtained that ξk=xkηk+ξk−1βk−1ηkx¯k−1,τ is also a C1 function.DesigningC0 virtual control law xk+1∗, which makes equations (19) and (21) hold in step k, the C1 positive Lyapunov function (equation (22)) is considered.(22)Vkx¯k,τ=Vk−1x¯k−1,τ+∫xk∗xksηk−xk∗ηkνkds.According to equations (7) and (19) and mean value theorem, it can be achieved that Vkx¯k,τ≤2∑i=1kξiνi+1/ηi, which means that equation (19) is right in step k.Defining thatWkx¯k,τ=∫xk∗xksηk−xk∗ηkνkds, calculating derivation of Vkx¯k,τ to τ, equation (23) can be achieved after combining formulas (20) to (22).(23)V˙kx¯k,τ≤−n−k+2∑i=1k−1ξiω+κτξk−1νk−1xkpk−1−xk∗pk−1+κτξkνkxk+1pk−xk+1∗pk+κτξkνkxk+1∗pk+κτξkνkf˜kx¯k,τ+∑i=1k−1∂Wk∂xidxidτ+∂Wk∂τ.Let us estimate each term on the right side of equation (23); according to Lemma 2, it can be obtained that(24)xkpk−1−xk∗pk−1≤21−pk−1/ηkxkηk−xk∗ηkpk−1/ηk≤2ξkpk−1/ηk.According to Lemma3, there exists a C1 function ckx¯k−1≥0 making the following inequality right:(25)κτξk−1νk−1xkpk−1−xk∗pk−1≤14ξk−1ω+κτckx¯k−1ξkω.For the convenience of narration, the following lemmas are given to estimate the residual terms on the right side of equation (23). The proof ideas of Lemmas 4 and 5 are similar to those of inequalities (17) and (18) in [4]. Readers can refer to [4] for proof. For simplicity, this paper omits the proof.Lemma 4.
There exitsC1 function γ˜kx¯k,τ≥0, k=2,…,n, which makes the following inequality hold:(26)κτξkνkf˜kx¯k,τ≤14∑i=1k−1ξiω+κτγ˜kx¯k,τξkω.Lemma 5.
There existsC1 function ρ˜kx¯k,τ≥0, k=2,…,n, which makes the following inequality hold:(27)∑i=1k−1∂Wk∂xidxidτ≤14∑i=1k−1ξiω+κτρ˜kx¯k,τξkω.Lemma 6.
There existsC1 function φ˜kx¯k,τ≥0, φ˜kx¯k,τ≥0, which makes the following inequality hold:(28)∂Wk∂τ≤14∑i=1k−1ξiω+φ˜kx¯k,τξkω.


## 3.2.2. Demonstration

It is noticed that the time after transformation is only explicitly included inxk∗ηk which belongs to the expression Wkk=2,…,n. So, we use mathematical induction to analyze ∂xk∗ηk/∂τ.According tox2∗η2=−x1β1η2x1,τ, where β1x1,τ>0 is a C1 function, it can be concluded that the partial derivative about τ to x2∗η2 is available and that the following inequality can be obtained:(29)∂x2∗η2∂τ=ξ1∂β1η2x1,τ∂τ≤ξ1φ1x1,τ,where φ1x1,τ≥0 is a designable C1 function.Assuming that whenk=3,…,n, xk−1∗ηk−1 is partially differentiable to τ, there exists a C1 function φk−2x¯k−2,τ, which makes the following equality hold:(30)∂xk−1∗ηk−1∂τ=φk−2x¯k−2,τ∑i=1k−2ξi.It should be noticed thatxk∗ηk=−ξk−1βk−1ηkx¯k−1,τ, where βk−1x¯k−1,τ is a C1 function and ξk−1=xk−1ηk−1−xk−1∗ηk−1; therefore, xk∗ηk is partially differentiable to τ, and(31)∂xk∗ηk∂τ≤ξk−1∂βk−1ηkx¯k−1,τ∂τ+βk−1ηkx¯k−1,τ∂xk−1∗ηk−1∂τ≤ξk−1∂βk−1ηkx¯k−1,τ∂τ+βk−1ηkx¯k−1,τφk−2x¯k−2,τ∑i=1k−2ξi≤φk−1x¯k−1,τ∑i=1k−1ξi,where φk−1x¯k−1,τ≥0 is a designable C1 function.Whenk=2,…,n, Wk is continuous partial differentiable to τ. According to formula (2), we can also know that νk+1/ηk>ω; hence, we have the following:(32)∂Wk∂τ=∂xk∗ηk∂τ∫xk∗xkνksηk−xk∗ηkνk−1ds≤2νkφk−1x¯k−1,τξkvk−1+1/ηk∑i=1k−1ξi≤14∑i=1k−1ξiω+ξkωφ˜kx¯k,τ,where φ˜kx¯k,τ≥0 is a designable C1 function.Substituting formulas (25)–(28) into equation (23), the following inequality can be obtained:(33)V˙kx¯k,τ≤−n−k+1∑i=1k−1ξiω+κτξkνkxk+1pk−xk+1∗pk+κτξkωckx¯k−1+γ˜kx¯k,τ+ρ˜kx¯k,τ+φ˜kx¯k,τκτ+κτξkνkxk+1∗pk.TheC0 virtual control law in step k can be designed as follows:(34)xk+1∗=−ξk1/ηk+1n−k+1+φ˜kx¯k,τκτ+ckx¯k−1+γ˜κx¯k,τ+ρ˜κx¯k,τ1/pκ≜−ξk1/ηk+1βκx¯k,τ,where βkx¯k,τ>0 is a C1 function. Substituting formula (34) into formula (33), the following inequality can be obtained:(35)V˙kx¯k,τ≤−n−k+1∑i=1kξiω+κτξkνkxk+1pk−xk+1∗pk.According to the above derivation process, the followingC0 state feedback controller can be designed in step n:(36)u=xn+1∗=−ξndnβnx,τ,which makes C1 positive Lyapunov function Vnx,τ meet Vnx,τ≤2∑i=1nξiνi+1/ηi, and the following formula is established:(37)V˙nx,τ≤−∑i=1nξiω.Considering that momentk contains the dynamic open set of origin Ω¯=x∈Rn|ξi<1,i=1,…,n, we can choose ε>0, which is small enough to make Ω˜=x∈Rn|Vnx,τ<ε be a subset of Ω¯ in time k. From formulas (2) and (14), we can know that νi+1/ηi>ω, and combining (6) and (37), the following inequality can be achieved:(38)dVdτ≤−Vρ2ρ,∀x∈Ω˜,where ρmini=1,…,nνi+1/ηi=ω.In the time domainτ and, according to equation (37), the solution trajectories of system (12) will enter the set Ω˜ in finite time and the finite time can be marked as τs. It should be noted that τs cannot be calculated but can be obtained online by real-time detection of system errors. Furthermore, by using formula (38) and Lemma 1, the finite convergence time of system (12) in time domain τ can be estimated as follows:(39)τf=τs+2n1−ρ1−ρ.Remark 3.
From formula (37), we can know that x is bounded, and, according to the state feedback control law (36), it can be known that u is also bounded in the closed interval 0,τf of time domain; that is to say, the actual control law (36) is not singular.


## 3.3. Design of Fixed-Time Controller in Time Domaint

Transform the mapping in (9) into the following compact form:(40)τ=τ0t=μ2+σ2μ+cottπ/2+arctanμ/σ/T.Consider that we have already made controller (36) of system (12) into finite-time stabilization in the last section. Let tf=t0τf, and if the control law in time domain t is designed as (41)ux,t=−ξn1/ηn+1βnx,τ0t,0≤t<tf.Then, any solution trajectory of system (1) will reach the origin in finite time.For the time setktf,ktf+tf, k=1,2,… in time domain t, assume that the following time-domain coordinate mapping is adopted:(42)t=tkτ=t0τ+ktf.The control law can be designed as follows:(43)ux,t=−ξn1/ηn+1βnx,τ0t−ktf,ktf≤t<ktf+tf.The solution trajectory of system (1) will always remain at the origin.In conclusion, the fixed-time stabilization control law in time domaint can be designed as(44)ux,t=−ξn1/ηn+1βnx,τ0modt,tf,where modt,tf denotes modular operation, the result of which is the remainder obtained by t dividing tf.Remark 4.
The proposed control scheme provides a novel perspective of fixed-time stabilization. Compared with the traditional method composed of high-power and low-power terms [8–11], there exist only low-power terms in (44), which simplifies the design and analysis process to some degree. Besides, it is worth noticing that the setting times of [8–11, 17] depend on the control parameter, e.g., control gains and power terms. However, the setting time in this work can be directly specified in advance.


## 4. Simulation Experiment

### 4.1. System Scheme

To verify the effectiveness of the proposed control law, a practical example simulation is used to compare the proposed control method with recent literature.To the best of our knowledge, existing fixed-time results consider at most so-called normal form systems [15], which is the trivial case of the high-order systems. Therefore, consider the following single link manipulator system:(45)mlφ¨+flφ˙+mgsinφ=u,where φ∈−π/2,π/2 is the rotation angular displacement, u is the external force acting on the manipulator, m is the mass of the manipulator, l is the distance from the center of mass of the manipulator to the rotating shaft, f is the unknown friction coefficient whose upper bound is known, and g is the acceleration of gravity. Assuming that the equivalent angular displacement is x1=mlφ and the equivalent angular acceleration is x2=mlφ˙, equation (45) can be written in the following form:(46)x˙1=x2,x˙2=u−mgsinx1ml−fx2m.In the above formula, all physical quantities are in Si basic units, and the values of parameters in the simulation process are selected asl=g and m=f¯=1/g.For the fixed-time controller (44) designed in this paper, we can choose the parameters as follows: η1=13/11, η2=11/9, η3=9/7, ν1=1, ν2=103/99, and μ=σ=1. For the controller designed in [15], we can choose corresponding parameters as follows: λ=0.1, k1=1, and k=θ=2.Considering the value range ofφ, the three following groups of initial conditions are selected in the simulation: x10=0.5,1,1.5 and x20=0,0,0. Figures 1(a) and 2(a) show the state changes of system (44) with the control law (46) proposed in this paper and with the control law proposed in [15]. It can be seen that, no matter what the initial conditions are, the said two methods achieve stabilization in a fixed time.Figure 1
Response of system (46) to control law (44). (a) Equivalent angular displacement x1. (b) Control input u.
(a)(b)Figure 2
Response of system (46) to control law in [15]. (a) Equivalent angular displacement x1. (b) Control input u.
(a)(b)It can be seen from Figures1(b) and 2(b) that the control input in this paper is nonsingular in the whole process; however, that of [15] diverges at the terminal moment. Therefore, from this point of view, the method proposed in this paper is more acceptable.


## 4.1. System Scheme

To verify the effectiveness of the proposed control law, a practical example simulation is used to compare the proposed control method with recent literature.To the best of our knowledge, existing fixed-time results consider at most so-called normal form systems [15], which is the trivial case of the high-order systems. Therefore, consider the following single link manipulator system:(45)mlφ¨+flφ˙+mgsinφ=u,where φ∈−π/2,π/2 is the rotation angular displacement, u is the external force acting on the manipulator, m is the mass of the manipulator, l is the distance from the center of mass of the manipulator to the rotating shaft, f is the unknown friction coefficient whose upper bound is known, and g is the acceleration of gravity. Assuming that the equivalent angular displacement is x1=mlφ and the equivalent angular acceleration is x2=mlφ˙, equation (45) can be written in the following form:(46)x˙1=x2,x˙2=u−mgsinx1ml−fx2m.In the above formula, all physical quantities are in Si basic units, and the values of parameters in the simulation process are selected asl=g and m=f¯=1/g.For the fixed-time controller (44) designed in this paper, we can choose the parameters as follows: η1=13/11, η2=11/9, η3=9/7, ν1=1, ν2=103/99, and μ=σ=1. For the controller designed in [15], we can choose corresponding parameters as follows: λ=0.1, k1=1, and k=θ=2.Considering the value range ofφ, the three following groups of initial conditions are selected in the simulation: x10=0.5,1,1.5 and x20=0,0,0. Figures 1(a) and 2(a) show the state changes of system (44) with the control law (46) proposed in this paper and with the control law proposed in [15]. It can be seen that, no matter what the initial conditions are, the said two methods achieve stabilization in a fixed time.Figure 1
Response of system (46) to control law (44). (a) Equivalent angular displacement x1. (b) Control input u.
(a)(b)Figure 2
Response of system (46) to control law in [15]. (a) Equivalent angular displacement x1. (b) Control input u.
(a)(b)It can be seen from Figures1(b) and 2(b) that the control input in this paper is nonsingular in the whole process; however, that of [15] diverges at the terminal moment. Therefore, from this point of view, the method proposed in this paper is more acceptable.


## 5. Conclusion

The problem of fixed-time stabilization for high-order nonlinear systems is studied in the paper. A control method based on time-domain mapping is firstly introduced. Compared with the existing literature, the paper proposes a new idea to realize fixed-time stabilization based on time-domain mapping, which greatly expands the research scope of fixed-time stabilization. Note that all the states should be accessible in this work; an interesting problem is the observer design in the case of partially unknown state, which will be considered in our further work.



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