# Robust Stabilization of Extended Nonholonomic Chained-Form Systems with Dynamic Nonlinear Uncertain Terms by Using Active Disturbance Rejection Control

Authors: Hua Chen; Xiaoying Sun; Shen Xu; Yuxuan Wang
Journal: Complexity
 (2019)

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## Abstract

In this paper, the stabilization problem of nonholonomic chained-form systems is addressed with uncertain constants. In this paper, the active disturbance rejection control (ADRC) is designed to solve this problem. The proposed control strategy combines extended state observer (ESO) and adaptive sliding mode controller. The control of nonholonomic chained-form systems with dynamic nonlinear uncertain terms and uncertain constants is first discussed in this paper. In comparison with existing methods, the proposed method in this paper has better performance. It is proved that, with the application of the proposed control strategy, semiglobal finite-time stabilization of the systems is achieved. An example is given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

The nonholonomic chained-form system was first proposed by Murray and Sastry in [1]. In recent years, more attention has been paid to the finite-time stabilization of the nonholonomic chained-form systems [2–6]. According to Brockett’s necessary conditions [7], there is no smooth-time-invariant static state feedback control law that can stabilize a nonholonomic system. A number of approaches have been proposed to solve the stabilization problem including continuous time-varying feedback control laws, discontinuous time invariant control, and hybrid stabilization [8–14].However, the complexity of understanding complex systems, the inevitable changes in system architecture, and the difficulty of predicting changes in the environment are three key points, leading to the dilemma that uncertainties always exist in the modeling of actual power systems [15]. Plenty of control methods have been developed [16–20] such as adaptive control [21, 22] and robust control [23]. In recent years, the active disturbance rejection control [2] technique has been widely recognized for its abilities to handle with uncertainties and its simplicity in the control structure.Nevertheless, those control algorithms rely significantly on a priori known amplitude of interference. In addition, the finite-time control algorithm has the advantages of fast convergence in the aspect of control performance compared with other algorithms, such as continuous time-varying feedback control laws [24–26].In recent years, more and more studies have been done on the stability of nonholonomic systems [27–34]. Yasir Awais Butt [3] proposed a robust switching controller based on discrete switching logic and ISM. This approach can guarantee the desired performance and robustness properties of the feedback control system. But this method does not take dynamic nonlinear uncertain terms into consideration. Qing Wang [2] designed the active disturbance rejection control (ADRC), which proves that it is an effective method to achieve finite-time stabilization of nonholonomic chained-form systems when the magnitude of the interference is unknown, but it can only be applied to relatively simple chained-form systems. Wang [4] constructed an adaptive output feedback controller by utilizing an adaptive control method and a parameter separation technique to stabilize the whole systems with unknown nonlinear parameters. To the best of the authors’ knowledge, there is no research on stabilizing the dynamic feedback systems with bounded unknown uncertain positive parameters of nonholonomic robots about this issue.In comparison with existing methods, our main contributions can be summarized as the following three respects:(1)
There are no a priori assumptions and it can deal with robust stability effectively in contrast with the existing methods. The reason why a disturbance of lower magnitude has an impact on the overall closed-loop system is that a priori can be estimated and the disturbance can be well compensated.(2)
The proposed controller can be applicable in the nonholonomic systems in chained-form with bounded unknown uncertain positive parameters.(3)
The proposed controllers can achieve the stabilization of extended nonholonomic chained-form systems with dynamic nonlinear uncertain terms. Compared with existing methods, the proposed controllers considered the dynamic nonlinear uncertain terms, resulting in the fact that the proposed controllers become more practical.In this paper, a finite-time switching controller integrates ESO and adaptive sliding mode controller, which is set up to realize stabilization of a class of nonholonomic chained-form systems. Numerical simulation demonstrates the effectiveness of the proposed control method.This paper’s fundamental framework is as follows. Section2 gives a formalization of the problem considered and introduces some preliminaries. In Section 3, we present the proposed switching controller and its stability analysis results. Section 2 gives a formalization of the problem considered and some preliminaries in this paper. In Section 3, we present the proposed switching controller and its stability analysis results. Section 4 states an illustrative example and the validity of the proposed methodology. Section 5 will summarize the full content. Section 6, as the last part of this paper, will introduce the future research direction.


## 2. Problem Statement and Preliminaries

Nonholonomic system in extended chained-form [35] can be described by(1)x˙1=k1u1x˙2=k2x3u1⋮x˙n-1=kn-1xnu1x˙n=knu2u˙1=ζ1x,tψ1+f1x,z1,ω1u˙2=ζ2x,tψ2+f2x,z2,ω2z˙1=f01x,z1,ω1z˙2=f02x,z2,ω2y1=x1y2=x2where x=x1,x2,…,xnT∈Rn is a representation of the system state vector. u1,u2T∈R2 can be treated equally as the velocity input for the kinematics model. fi., f0i., and ζi. are some unknown continuously dynamic nonlinear terms, fi0,zi,ωi=0, i=1,2. fix,t∈R and ζix,t≠0∈R for all x,t∈Rn×Ri=1,2 are system dynamics and smooth nonlinear control directions, respectively. zi∈Rmi=1,2 represents nonlinear dynamic auxiliary variable, yi∈R,i=1,2 is the measured output, and ωi∈Rp,i=1,2 are external time-varying uncertain disturbances, assuming that ωi and its derivative ω˙i are continuous and bounded. We donate the practical control input ψ1,ψ2T∈R2 as the formal inputs of force or torque for the extended dynamic model, and ki∈R+i=1,2,…,n are uncertain normal number parameter with bounded unknowns.Remark 1.
System (1) can describe the motion state of multiple (2,0) wheeled mobile robots. The pose of the robot in the inertial coordinate system can be represented by a vector q=x,y,θT. u means the forward speed and steering velocity of the robot. ψ1,ψ2T∈R2 can be donated as the formal inputs of force or torque for the extended dynamic model. As a result, we can control the pose of the robot by means of devising ψ1,ψ2T∈R2. x=x1,x2,…,xnT∈Rn represents the state vectors of i robots. In this case, x1 and x2 are targets, and their motion state can be measured. x3~xii=3,4,… robots follow x1 and x2. In addition, there are dynamic nonlinear uncertainties in the process of motion. According to the constraints of the robot motion and the motion state, it is practicable to establish a model of the nonholonomic motion system. After proper coordinate transformation and input transformation, the model can be converted into a nonholonomic chain system of system (1).
System (1) can be rewritten as(2)x˙1=k1u1u˙1=ζ1x,tψ1+f1x,z1,ω1z˙1=f01x,z1,ω1y1=x1(3)x˙2=k2x3u1⋮x˙n-1=kn-1xnu1x˙n=knu2u˙2=ζ2x,tψ2+f2x,z2,ω2z˙2=f02x,z2,ω2y2=x2To begin with, consider system (2)(4)u˙1=ζ1x,tψ1+f1x,z1,ω1Lemma 2 (see [36, 37]).
Consider a first-order disturbed system:(5)x˙=u+fx,t,where x,u∈R1 are state variable and control input, respectively, and fx,t represents an external disturbance with a known bound 0<c<+∞, satisfying(6)fx,t≤c<+∞,f0,t=0,∀t≥0.Taking a continuous sliding mode control law,(7)u=-c·sgnx,x≥ϖtxϖt,x<ϖtwhere ϖt denotes a continuous, time-variable boundary layer and satisfies that(8)ϖt>0,∫0+∞ϖtdt<σ,where σ is a nonnegative constant. Then, system (1) can be asymptotically stabilized to the zero equilibrium point by (6).Proof.
For the proof, see Yang and Wang (2011).
Letx=u1-1, then system (4) can be rewritten as(9)x˙=u˙1x˙=ζ1x,tψ1+f1x,z1,ω1Our goal is to stabilize system dynamics (9) regardless of external disturbances and uncertainties. Just in this case, system (9) could be straightforwardly stabilized to the zero equilibrium point in finite time by using the first-order continuous sliding mode control law. As a result, u1 could be stabilized to the constant 1. The control signal is designed as(10)ψ1=-cζ1x,t·sgnx,x≥ϖtxϖt,x<ϖtAssumption 3 (see [2]).
The time derivative of thef1x,z1,ω1 is bounded(11)f1x,z1,ω1≤MThere exists some M>0 such that(12)Mq+ε0<E.ε0 is the initial value of the estimation error ε. Then the resulting closed-loop system is stabilized in finite time. Assumption 3 implies that the first control component ψ11 is bounded and the second control component ψ12 is certainly uniformly bounded. Thus, the composed signal ψ1 is uniformly bounded and velocity input u1→1 in finite time.
Letdi=xi+1,i=1,2,…n-1,dn=u2, then system (3) can be rewritten as(13)d˙1=k2d2⋮d˙n-2=kn-1dnd˙n-1=kndnd˙n=ζ2d,tψ2+f2d,z2,ω2z˙2=f02d,z2,ω2y2=d1which are extended nonholonomic chained-form systems.The following assumptions are made for the nonlinear system.Assumption 4.
There exists a unbounded positive definite functionL0z2 such that, ∀d,z2,ω2∈Rn×Rm×Rb,(14)∂L0∂z2z2f02d,z2,ω2≤0,∀z2:z2≥δd,ω2where δd,ω2 is a nonnegative continuous function.Assumption 5.
∀ d , z 2 , ω 2 ∈ R n × R m × R b, ζ2d,t≠0. The sign of ζ2d,t is known.Remark 6.
Assumption3 implies that z˙2=f02d,z2,ω2. Assumption 5 ensures that the control signal always has an effect on the system (13). Since the sign of ζ2d,t is fixed, we assume ζ2d,t>0 and let ζ0d>0 be a nominal model of ζ2d,t.Remark 7.
Set continuous and saturated control lawz˙2=-ksgn⁡z2z2α where k and α are design parameters. For instance, L0z2=z22/2 can satisfy Assumption 4.
Letdn+1=f2d,z2,ω2+ζ2d,t-ζ0dψ2 where the unknown system dynamics f2d,z2,ω2 and the parameter mismatch of control ζ2d,t-ζ0du2 are viewed as an extended state of the system. Assume dn+1 is differentiable with m=d˙n+1, then system (13) can be rewritten as(15)d˙i=ki+1di+1d˙n=dn+1+ζ0dψ2d˙n+1=my2=d1The extended state observer was first proposed by Jingqing Han in [38]. The extended state observer (ESO) is designed as [2, 39, 40](16)d^˙i=ki+1d^i+1+ϕn-igid1-d^1ϕni=1,2,…,n-1d^˙n=d^n+1+gnd1-d^1ϕn+ζ0d^ψ2d^˙n+1=ϕ-1gn+1d1-d^1ϕnwhich is a nonlinear generalization of LESO for gain ϕ and pertinent chosen functions gi., i=1,2,…,n+1. d^1,d^2,…,d^n,d^n+1T∈Rn+1 is the nonlinear extended state observer state and depends on a small positive constant parameter ϕ.Remark 8.
In theory, the value ofϕ is chosen to be arbitrarily small to make the trajectory tracking error as small as possible. However, the existence of noise and sampling constraints in practice are responsible for the restrictions on the values of ϕ.
Now with the state estimatesd^1,d^2,…,d^n,d^n+1T∈Rn+1, the active disturbance rejection control (ADRC) law, which is based on the output of the ESO (16), can be designed as(17)ψ2nom=-1ζ2d,td^n+1+ψ20d^where -1/ζ2x,td^n+1 is to compensate the total uncertainties and ψ20d^ is to guarantee the stability and performance requirements of the closed-loop system.
In order to protect the system from the peaking in the observer’s transient response caused by the nonzero initial errord1t0-d^1t0,…,dnt0-d^nt0, we design the system that uses a special controller as [39]. The control is modified as (18)ψ2=Wsatϕψ2nomWwhere the function satϕ. is shown by [41](19)satϕeefor0≤e≤1e+e-1ϕ-e2-12ϕfor1<e≤1+ϕ1+ϕ2fore>1+ϕThe function satϕ. is nondecreasing, continuously differentiable. The saturation bound W ensures that the saturation is not invoked in the steady state of the ESO (16).
Set the scaled ESO estimation error as(20)Δit=di-d^i,i=1,2,…,nΔn+1t=f2d,z2,ω2+ζ2d,t-ζ0dWsatϕψ2nomW-d^n+1.For the purpose of getting a compact form of the closed-loop equation for the state estimation error, we design these scaled variables(21)ηi=di-d^iϕn+1-i=Δitϕn+1-i,i=1,2,…,n+1.Then substituting (15) and (16) into (22), the estimation error state dynamics can be written as(22)ϕη˙i=ki+1ηi+1-giη1,i=1,2,…,n-1.ϕη˙n=ηn+1-gnη1ϕη˙n+1=ϕm-gn+1η1Assumption 9 (see [2, 39, 40]).
∀ η = η 1 , η 2 , … , η n + 1 T ∈ R n + 1, there exist constants θi,i=1,2,3,4,γ and positive definite, continuous differentiable functions L1,Q1: Rn+1→R such that(23)iθ1η2≤L1η≤θ2η2,θ3η2≤Q1η≤θ4η2ii∑i=1n-1ki+1ηi+1-giη1∂L1∂ηiη+ηn+1-gnη1∂L1∂ηnη-gn+1η1∂L1∂ηn+1η≤-Q1ηiii∂L1∂ηn+1η≤γηwhere η=η1,η2,…,ηn+1.. denotes the Euclid norm of Rn+1.Assumption 10 (see [2, 41]).
The functionsgiη1≤φiηi for some positive constants φi for all i=1,2,…,n+1. For any d,z2,ω2 belonging to the domain of interest and ∀d^∈Rn, the following inequality holds:(24)ρa≜ζd,t-ζ0dζ0d^<θ3φn+1γRemark 11.
Assumption10 implies that the nominal model ζ0. is close to ζ.. The functions gi.,i=1,2,…,n+1, should be chosen appropriately to make the zero balance of the subsequent system asymptotically stable [40]: (25)ϕη˙i=ki+1ηi+1-giη1,i=1,2,…,n-1.ϕη˙n=ηn+1-gnη1ϕη˙n+1=-gn+1η1Two useful lemmas will be presented in the following section.Lemma 12 (see [2, 42]).
Consider the system(26)d˙i=ki+1di+1,i=1,2,…,n-1d˙n=ω2nomdLet h1,h2,…,hn>0 so that the polynomial θn+hnθn-1+⋯+h2θ+h1 is Hurwitz stable. Then there exists ϕ∈0,1 such that, for any ν∈1-ϕ,1, the origin of (26) is a globally finite-time stable equilibrium under the feedback(27)ω2nomd=-h1sgnd1d1ν1-⋯-hnsgndndnνn,where νn+1=1, νn=ν, νi-1=νiνi+1/2νi+1-νi,i=2,3,…,n.Lemma 13 (see [2, 43]).
If the continuously differentiable, nonnegative functionLj satisfies(28)L˙j+vLj+bLwj≤0,where v,b>0,0<w<1, then j will converge to j=0 in finite time.


## 3. Control Design and Stability Analysis

We design the active disturbance rejection controllers to achieve finite stabilization for a class of systems (3) by integrating extended state observer and adaptive sliding mode controller. The analysis is as follows.Step 1.
According to Lemma13, we can design a controller to achieve the finite-time sliding mode stabilization of system (2). The control signal is designed as (10).Step 2.
Design the active disturbance rejection control to achieve finite-time stabilization for a class of systems (3) by combining extended state observer with adaptive sliding mode controller.
The sliding surface is selected as [2, 44](29)s=dnt-dn0-∫0tω2nomdβdβ.Once the ideal sliding mode s=0 is established, (29) can be rewritten as(30)dnt=dn0-∫0tω2nomdβdβ.Differentiating (30) yields (26), and this implies that system (13) will converge to the origin from any initial condition along the sliding surface s=0 in finite time.
Define an odd continuous and differentiable function(31)πβ,μ,τ=sgnββμ,β≥τμ-1τμ-2sgnβo2+2-μτμ-1β,β<τwhere 0<μ<1, τ is a sufficiently small positive constant. The ADRC law is designed for system (13):(32)ψ2nom=-1ζd,td^n+1-ω^2nomd^+m1s^+n0ς^maxs^+m2πs^,l,τn+1where (33)ω^2nomd^=-h1πd^1,c1,τ1-⋯-hnπd^n,cn,τn,s^=d^nt-d^n0-∫0tω^2nomd^βdβ,where τi,1≤i≤n+1 are sufficiently small positive constants, and 0<l<1, m1,m2,n0>0. Define the estimation of the upper bound of κ12+κ22 as ς^max. κ1 and κ2 will be specified latter. The updating law of ς^max is(34)ς^˙max=m3n0s^2-ς^max,where m3>0.Theorem 14.
Consider the closed-loop system (13) formed of the nonlinear extended observer (16) and active disturbance rejection control law (18) and (32). Suppose Assumptions 3–10 are satisfied, for any d0∈Rn,η0∈Rn [2, 45] (i)
η→0 and dit-d^it→0 as ϕ→0, uniformly in t∈0,∞;(ii)
there existsϕ0>0 such that, for any ϕ∈0,ϕ0,
there existsϕ-dependent Tϕ such that d=0,∀t∈Tϕ,∞.Proof.
Associating (13), (16), and (15), we can compute the derivative of the extended state dn+1 with respect to t in the time interval 0,t1. The derivative of the extended state dn+1 is shown as(35)m=∑i=1n-1ki+1di+1∂f2∂di+ψ2∂ζ2∂di-ψ2∂ζ0∂di+f2d,z2,ω2+ζd,tψ2∂f2∂dn+ψ2∂ζ2∂dn-ψ2∂ζ0∂dn+f20d,z2,ω2∂f2∂z2+f20d,z2,ω2ψ2∂ζ2∂z2+ω˙2∂f2∂ω2+ω˙2ψ2∂ζ2∂ω2+ζ2d,t-ζ0dψ˙2,where(36)ψ˙2=-satϕψ2nomW′1ζd,td^˙n+1+∑i=1nhid^˙idπdi,ci,τidd^i+m1s^˙+m2s˙dπs^,cn,τnds^+n0ς^˙maxs^+n0ς^maxs˙-∑i=1nd^˙i∂ζ0/∂d^iζ02d^dn+1-ω^2nomd^+m1s^+m2πs^,l,τn+1+n0ς^maxs^(37)s^˙=d^˙n-ω^2nomd^We can know that d^i=di-ηiϕn+1-i,i=1,2,…,n+1, from (21). In addition, dt and d^t are continuous in t, and d0 and d^0 and ψ2 are bounded in the time interval 0,t1. Then considering Assumptions 3–10 and substituting (36) and (37) into (35), we can infer that the derivative of the extended state dn+1 with respect to t in the time interval 0,t1 is bounded (38)m≤W0+W1η+ρaφn+1ϕηwhere W0 and W1 are independent positive constants.
LetL1η be a positive definite, continuous differentiable function satisfying Assumption 9. The derivative of L1η with respect to t in the time interval 0,t1 satisfies(39)dL1ηdt=1ϕ∑i=1n-1ki+1ηi+1-giη1∂L1∂ηiη+ηn+1-gnη1∂L1∂ηnη+ϕm-gn+1η1∂L1∂ηn+1η≤-1ϕQ1η+m∂L1∂ηn+1η≤-θ3ϕη2+W0+W1η+ρaφn+1ϕηγη≤-1θ2ϕθ3-θ2ρaφn+1γ+ϕW1γθ1L1η+W0γL1ηθ1Considering dL1η/dt=2L1ηdL1η/dt, we can get(40)dL1ηdt≤-12θ2ϕθ3-θ2ρaφn+1γ+ϕW1γθ1L1η+W0γ2θ1Considering (41) and Assumption 9∀ϕ∈0,θ1θ3-θ2ρaφn+1γ/θ2W1γ, the following inequality holds:(41)η≤L1ηθ1This together with (20) yields(42)Δit≤ϕn+1-iηtϕ≤ϕn+1-i×L1η0θ1-W0γθ2ϕθ1θ3-θ2ρaφn+1γ-θ2ϕW1γ×e-1/2θ2ϕθ3-θ2ρaφn+1γ-ϕW1γ/θ1t/ϕ+ϕn+1-i×W0γθ2ϕθ1θ3-θ2ρaφn+1γ-θ2ϕW1γThe right hand side of the inequality (41) converges to 0 in the time interval 0,t1 as ϕ→0, and there exists an ϕ∗>0 such that, for any ϕ∈0,ϕ∗, there exists an ϕ-independent t0∈0,t1 such that Δi=diϕ,t-d^iϕ,t→0,i=1,2,…,n+1,t∈t0,t1 as ϕ→0. What is more, the control will be out of saturation after the transient period of the nonlinear extended observer by appropriately selecting the bound W.
Letκ1 and κ2 be defined as (43)κ1=1n0sζ2ψ2+f2-ω2nom+m1s+m2sgnssl+n0ς^maxs,s≥11n0ζ2ψ2+f2-ω2nom+m1s+m2sgnssl+n0ς^maxsgns+m1sgns,s<1κ2=s^2-s2It can be concluded that both κ1 and κ2 are bounded in the time interval 0,t1. Thus, κ12+κ22 is bounded in the time interval 0,t1.
In the reaching phases≠0, we consider the time derivative of the sliding variable in the time interval t0,t1. In the case s≥1, associating with (13) and (43), we can deduce that(44)s˙=d˙n-ω2nom=f2+ζ2ψ2-ω2nom=n0sκ1-m1s-m2sgnss-n0ς^maxsIn the case s<1, associating (13) with (43), we can get(45)s˙=f2+ζ2ψ2-ω2nom=n0sκ1-m1s-m2sgnss-n0ς^maxsgns-m1sgnsLet ς~max=ςmax-ς^max. Consider the following Lyapunov function:(46)L2=12s2+212m3ς~max2,and differentiate L2 with respect to t in the time interval t0,t1. In the case s≥1, associating with (34), (43), and (44), we can get(47)dL2dt=ss˙+1m3ς^˙max=-m1s2-m2sl+1+n0κ1-ςmaxs2+n0ς~maxςmax-κ2-n0ς~max2.Since κ1,κ2≤ςmax and(48)κmax-κ2ς~max≤12ς~max2+12ςmax-κ22≤12ς~max2+2ςmax2,inequality (46) can be simplified as(49)dL2dt≤-m1s2-n02ς~max2+2n0ςmax2.In the case s<1, associating with (34), (43), and (45), we can get(50)dL2dt=ss˙-1m3ς~maxς^˙max=n0κ1s-m1s2-m2sl+1-n0ς^maxs-m1s-n0ς~maxs2-n0ς^maxκ2+n0ς~maxς^max.Considering (47) and(51)κ1s-ς^maxs-ς~maxs2≤ςmaxs-ς^maxs-ς~maxs2≤14ς~max2+s-s22≤14ς~max2+116,inequality (50) can be simplified as(52)dL2dt≤-m1s2-n04ς~max2+2n0ςmax2+116n0.According to the boundedness theorem, both s and ς~max are bounded in the time interval t0,t1. Assume ς~max≤a.
In order to show the finite-time stability, we consider the Lyapunov function(53)L3=12s2and differentiate L3 with respect to t in the time interval t0,t1.
In the cases≥1, associating with (44) and (45), we can get(54)dL3dt=ss˙=n0κ1s2-m1s2-m2sl+1-n0ς^maxs2≤-2m1-n0ς~maxL3-2l+1/2m2L3l+1/2.In the case s<1, associating with (42) and (46), we can get(55)dL3dt=n0κ1s2-m1s2-m2sl+1-n0ς^maxs-m1s≤-2m1-n0ς~maxL3-2l+1/2m2L3l+1/2+sn0ς~max1-s-m1.Choose m1 and n0 satisfying m1>n0a, and then together with (54) and (55), we can get(56)dL3dt≤-2m1-n0ς~maxL3-2l+1/2m2L3l+1/2.According to Lemma 13, the sliding surface s will converge to zero in finite time tl1. Besides, on the basis of Lemma 12, di=0,i=1,2,…n will be arrived in finite tl2(here one can select t1>tl2).
Next, it can be illustrated that system (13) will stay at the origin for all t>tl2. We can get that d=0 in the time interval tl2,t1 in the first step. Considering d is continuous in t, d is bounded in the time interval 0,2t1. Then running the analysis above, we can get d=0 in the time interval tl2,2t1, and then d is bounded in the time interval 0,3t1. We can get d=0 in the time interval tl2,3t1 similarly.
Finally, it can be summarized that there existsϕ0>0 such that, for any ϕ∈0,ϕ0, there exists ϕ-dependent Tϕ such that d=0,∀t∈Tϕ,∞. As a result, inequality (41) holds in the time interval 0,∞, and consequently η→0 as ϕ→0 uniformly in t∈0,∞.Remark 15.
We can get that the closed-loop can converge to 0 only whenϕ→0 according to the analysis above [36–44]. However, the condition ϕ=0 cannot be met in practice. What is more, reducing the value of will increase the high-frequency oscillations. In this paper, the proposed control law can guarantee the closed-loop converge to 0 asymptotically and in finite time, without relying on the condition ϕ=0.Step 3.
Rethink system (1) and design the ψ1 so that the sliding surface s=0 will be reached in finite time and the nonholonomic system in extended chained-form (2) will converge to the origin in finite time as system (3).
Letn=2, x1=x11,u1=x21; system (2) can be rewritten as(57)x˙11=k1x21x˙21=ζ1x,tψ1+f1x,z1,ω1z˙1=f01x,z1,ω1y1=x11In accordance with the state estimations of the nonlinear extended state observer (15), the output of the ESO (16), and control input (17), the ADRC control law of the system (57) can be designed as (17) and the control injected into the system (57) is modified as (18).
On the basis of Lemmas12 and 13, the sliding surface s=0 will be reached in finite time, and system (59) will converge to the origin in finite time.
Associating with system (11) and (57), we can get a conclusion that, under the condition (15), (16), and (17), the nonholonomic system in extended chained-form (2) and (3) will converge to the origin in finite time. Finally, we can get a conclusion that there exists ϕ2>0 such that, for any ϕ∈0,ϕ2, there exists ϕ-dependent Tϕ such that x=x1,x2,…,xnT=0∀t∈Tϕ,∞, and consequently η→0 as ϕ→0 uniformly in t∈0,∞.


## 4. Simulations

In this section, we demonstrate the effectiveness of the proposed control strategy for the following nonholonomic systems in extended chained-form (1) through a series of simulations, n=3 for (1). The control signal ψ1 is designed as (10).The ESO is designed as(58)x^˙2=2x^3+2ϕx1-x^1,x^˙3=4x^4+6ϕ2x3-x^3+ζ0ψ2,x^˙4=4ϕ3x4-x^4,In this example, we assume x0=0.2,-0.6,1T, x^0=0.1,-1,1.2T, q=0.01, r0=0, E=2, ζ200=15, and ς^max0=0. The control parameters are selected as m1=300, m2=1, m3=1, n0=0.01, ν1=1/2, ν2=1/4, h1=9, h2=6, h3=3, and W=30.Figures1~4 are the simulation results of the three steps. Figure 1 shows that the velocity input u1 for the kinematics model can converge to 1 in a finite time t≤6s in Step 1 and keep it in Step 2 until it is driven to zero in the last step. Figure 2 shows that the sliding surface s=0 is reached in a finite time t≤6s. Figures 3–7 show the time histories of x,x^ and ηi. These figures suggest that the system state vector x=x1,x2,…,xnT is well estimated by the ESO and finally the state x scaled ESO estimation error ηi converge to zero in finite time. In addition, Figure 8 suggests that the control signal ψ2 converges to zero in a finite time t≤8s.Figure 1
The velocity inputu1 for the kinematics model.Figure 2
The sliding surfaces.Figure 3
System states and ESO outputs ofx1.Figure 4
System states and ESO outputs ofx2.Figure 5
System states and ESO outputs ofx3.Figure 6
System states and ESO outputs ofx4.Figure 7
The convergence of the scaled ESO estimation errorηi.Figure 8
The control signalψ2.What is more, in [46], a finite-time tracking control law is designed for the nonholonomic mobile robot. The control law in [46] also used the switching control method and the simulation results are depicted in Figures 9 and 10. We can see that the state x converges to zero in finite time t≤10s and the tracking distance is stabilized to a constant in finite time t≤15s. From Figure 10 we can get that the controller proposed in this paper is more smooth than the controller in [46] in switching control. We can see that the controller proposed in this paper has faster convergence speed and more stable performance than that obtained in [46].Figure 9
The convergence of tracking distanced with respect of time.Figure 10
The stability of distance errord∗ with respect of time.Remark 16.
For your convenience review, we make Table1 to explain how to choose the design parameters.Table 1 Parameters Source of each parameter m 1 , m 2 , m 3 , n 0 > 0 From (28) and (29) ν n + 1 = 1 ,   ν n = ν ,   ν i - 1 = ν i ν i + 1 2 ν i + 1 - ν i ,   i = 2,3 , … , n From (26) and (27) h 1 , h 2 , … h n > 0 From Lemma2 W > 0 From (18) and (19)Lemma12 shows that h1,h2,…,hn>0 ensure that the polynomial θn+hnθn-1+⋯+h2θ+h1 is Hurwitz stable. Then there exists ϕ∈0,1 such that, for any ν∈1-ϕ,1, the origin of (22) is a globally finite-time stable equilibrium under the feedback (27), where νn+1=1, νn=ν, νi-1=νiνi+1/2νi+1-νi,i=2,3,…,n. In addition, m1,m2,m3,n0>0 allow four terms -ω^2nomd^+m1s^+n0ς^maxs^+m2πs^,l,τn+1 to guarantee the stability and performance requirements of the closed-loop system. The saturation bound W is chosen so that the saturation is not invoked in the steady state of the ESO. Practically, we can choose a group of available parameters m1=300, m2=1, m3=1, n0=0.01, ν1=1/2, ν2=1/4, h1=9, h2=6, h3=3, and W=30 in the simulation section.Remark 17.
By comparing the performance of the controller proposed in this paper with the performance of the controller proposed in [47, 48], we can know that the fixed and predefined-time controllers have better performance for nonholonomic systems. The fixed and predefined-time controllers predetermine the time, so the operation of the controller is independent of the initial value of the nonholonomic systems. However, for the nonholonomic chained-form systems with dynamic nonlinear uncertain terms considered in this paper, it is difficult to estimate the time in advance due to the existence of dynamic nonlinear uncertain terms. Achieving fixed-time control of nonholonomic chained-form systems with dynamic nonlinear uncertain terms is one of our future research directions.


## 5. Conclusion

In this paper, finite-time switching controllers are put forward in order to address the stabilization problem of nonholonomic chained-form systems with uncertain parameters and external perturbations. The proposed control strategy is able to guarantee the semiglobal finite-time stabilization of the extended nonholonomic chained-form systems. The simulation results of the numerical example show that the method is effective.


## 6. Future Research Directions and Prospects

We consider the application of the finite-time switching controllers proposed in the theory to the anti-interference of the robot in the source seeking work as our future research direction. It is very practical for realistic engineering. We will conduct more research and experiments in practical application.



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