Global Finite-Time Partial Stabilization for a Class of Nonholonomic Mobile Robots Subject to Input Saturation

Abstract

In this paper, the global finite-time partial stabilization problem is discussed for a class of nonholonomic mobile wheeled robots with continuous pure state feedback and subject to input saturation. Firstly, for the mobile robot kinematic model, a “3 inputs, 2 chains, 1 generator” nonholonomic chained form systems can be obtained by using a state and input transformation. The continuous, saturated pure state feedback control law is proposed such that the special chained form systems can be stabilized to zero (except an angle variable) in a finite time, i.e., finite-time partial stabilization. Secondly, the rigorous stability analysis of the corresponding closed-loop system is presented by applying Lyapunov theorem combined with the finite-time control theory, and the angle variable can be proved to converge to a constant, moreover, its convergent limit may be accurately estimated in advance. Finally, the simulation results show the correctness and the validity of the proposed controller not only for the chained system but also for the original mobile robots system.

1. Introduction

In recent years, control of nonholonomic mobile robots have attracted considerable attention from the research community because of their practical applications and the theoretical challenges created by the nonholonomic nature of the constraints on it. The stabilization problem is one of the most difficult problems in the nonholonomic control objectives because it has to face a difficult challenge: Brockett's necessary condition for the existence of a smooth time-invariant feedback stabilization is not satisfied [1]. In order to overcome this difficulty, many ingenious feedback stabilization methods have been proposed such as continuous time-varying feedback control laws [2–6], discontinuous feedback control laws [7–11], hybrid feedback control laws [12–13] and optimal control laws [14–17], etc. However, these studies haven't taken into account designing a controller subject to input saturation and such that the closed-loop system converges to zero in a finite time.

Dynamical systems subject to saturation nonlinearity in their input have received significant interest from control system researchers due to their frequent occurrence in various engineering applications [18]. Ignoring saturation nonlinearity in the controller design may lead to performance degradation and even instability of the closed-loop system [19]. Therefore, it is a very important task to design bounded controllers for such nonholonomic mechanical systems, because in practical engineering application, any actuator always imposes limits on the physical inputs. But until now, only a handful of studies have focused on this control issue for stabilizing nonholonomic systems. For a class of type (1,2) nonholonomic mobile robot system, the semiglobal practical stabilizing control scheme has been proposed with saturated inputs in [20]. A model-based controller design method for the global stabilization and the global tracking control with inputs saturation has been discussed by Z. P. Jiang et al in [21] for the traditional nonholonomic mobile robot kinematic systems. Based on which, Hua Chen et al [22–24] have extended the results of [21] to the simple dynamic case. Considering the uncertain nonholonomic mobile robot systems, the global tracking and stabilization control problem of internally damped mobile robots with unknown parameters, and subject to input torque saturation and external disturbances is addressed in [25]. The saturated practical stabilization problem is addressed for a class of nonholonomic mobile robots based on visual servoing feedback with uncertain camera parameters in [26]. And in [27], a saturated output feedback controller is designed for the uncertain nonholonomic mobile robots with the absence of velocity sensors in practice. Besides, the saturated stabilization for the so-called standard chained form systems has been also investigated in [28–29]. However, to our knowledge, almost none of these results mentioned above have involved the convergence rate with continuous control feedback law.

In fact, finite-time stabilization problems have been studied mostly in the contexts of optimality, controllability, and deadbeat control for several decades [30–32]. Compared to the asymptotic stabilization, the finite-time stabilization, which renders the trajectories of the closed-loop systems convergent to the origin in a finite time, has many advantages such as fast response, high tracking precision, and disturbance rejection properties. In the nonholonomic systems control field, some researchers have discussed the finite time stabilization for the (extended) chained form system [33–34], and the finite time tracking control for (multiple) nonholonomic mobile robots [35–36]. Recently, a new concept about the finite-time partial stability has been introduced in [37–39], also the same author has considered the finite-time partially stability of the nonholonomic multichained systems by designing continuous or discontinuous homogeneous state feedback laws to overcome the Brockett's obstruction [40]. However, there is a common flaw among the existing finite-time stabilization methods, that is the input saturation constraint has not been considered simultaneously. Although, we have considered the saturated finite-time stabilization problems with discontinuous switching control strategy in our previous work [26, 41], but many times switching design may cause unexpected oscillatory motion, and thus the continuous, saturated, finite-time stabilizer for nonholonomic systems is all-important.

In this paper, we will consider the global saturated finite-time partial stabilization problem for a class of nonholonomic mobile robots with continuous pure state feedback law. The main contributions can be summarized as the following three respects: 1)A special chained form system is discussed by applying the state and input transformation, for which, the continuous, saturated state feedback controller is proposed such that the closed-loop system can be partially stabilized to zero in a finite time. 2)The stability analysis is presented by Lyapunov criteria and the finite-time control theory, additionally, all the state variables of the closed-loop system (except an angle state) can be proved to converge to zero in a finite time, while the angle variable converges to a constant and its convergent limit can be accurately estimated. 3)Some simulation results show the effectiveness of the controller design not only for the chained system but also for the original mobile robot system. 4) Compared with the existing switching control design method [26], the continuous feedback law may avoid the unexpected chattering phenomenon due to switch methods as many times as desired during the control procedure.

The structure of the article is as follows: Section II gives a formulation of the problem considered in this article and contains some preliminary definitions and lemmas. The main results is presented in Section III. Section IV shows some simulation examples. Finally, a conclusion is shown in Section V.

2. Preliminaries and Problem Statement

In this section, the definition of finite-time stabilization and its stability criterion are given, and then our considered problem is introduced.

2.1 Preliminaries

Definition 1 (finite-time stabilization [31, 33]): Consider a time-invariant system in the form of

$$\dot{\xi }=f(\xi )\mathrm{,}f(0)=\mathrm{0,}\xi \in R^n\mathrm{,}$$

(1)

where f: ${\widehat{U}}_0$ of the origin. The equilibrium $\xi =0$ of the system is (locally) finite-time stable if (i) it is asymptotically stable, in $\widehat{U}$ , an open neighborhood of the origin, with $\widehat{U}\subseteq {\widehat{U}}_0$ ; (ii) it is finite-time convergent in $\widehat{U}$ , that is, for any initial condition ${\xi }_0\in \widehat{U}\backslash \{0\}$ , there is a settling time $T>0$ such that every solution $\xi (t\mathrm{,}{\xi }_0)$ of system (1) is defined with $\xi (t\mathrm{,}{\xi }_0)\in \widehat{U}\backslash \{0\}$ for $t\in [\mathrm{0,}T)$ and satisfies ${{\displaystyle \lim }}_{t\to T}\xi (t\mathrm{,}{\xi }_0)=0$ , and $\xi (t\mathrm{,}{\xi }_0)\equiv 0$ if $t\ge T$ . Moreover, if $\widehat{U}=R^n$ , the origin $\xi =0$ , the origin ξ = 0 is globally finite-time stable.

Lemma 1 ([42]): Considering the following system

$$\dot{\overline{x}}=f(\overline{x})\mathrm{,}f(0)=\mathrm{0,}\overline{x}\in R^n\mathrm{,}$$

(2)

suppose there exists a continuous function $\overline{V}(\overline{x}):U\to R$ such that the following conditions hold

• $\overline{V}(\overline{x})$ is positive definite.

• There exist real numbers $\overline{c}>0$ and $\overline{\alpha }\in (\mathrm{0,1})$ and an open neighbourhood U₀ ∊ U of the origin such that $\dot{\overline{V}}(\overline{x})+\overline{c}{\overline{V}}^{\overline{\alpha }}(\overline{x})\le \mathrm{0,}\overline{x}\in U_0\backslash \{0\}$

In that circumstance, the origin of system (2) is finite-time stable. If U = U₀ = Rⁿ, then the origin of system (2) is globally finite-time stable.

For some nonlinear complex control systems, the traditional finite-time stabilization control strategy may be invalid, thus a modified version of finite-time stabilization has been proposed recently, i.e., finite-time partial stabilization [39–40], we thanks to C. Jammazi for having proposed a class of new extended definition of finite-time stabilization for the first time.

Definition 2 (p-finite-time partial stabilization [39–40]): Let

$$y=(y_1\mathrm{,}{y}_2)\in R^{n-p}\times R^p\mapsto Y(y)\in R^n\mathrm{,}$$

be defined and continuous on a neighborhood of (0,0) ∊ R^n−p × R ^p . We assume that

$$Y(y_1\mathrm{,0})=0.$$

One says that (0,0) ∊ R^n−p × R ^p is p-partially stable in finite time for $\dot{y}=Y(y)$ if (i) (0,0) is Lyapunov stable for $\dot{y}=Y(y)$ ; (ii) there exists r > 0 and T = T(y(0))>0, called the settling-time function, such that if $\dot{y}=Y(y)$ , then y₂(t) = 0 for every t ≥ T and y₁(t) is constant for t ≥ T. The solution y(t) starting from the initial condition y(0) is defined and unique in forward time for t ∊ [0, T).

The control system

$$\dot{y}=Y(y\mathrm{,}u)\mathrm{,}Y(y_1\mathrm{,0,0})=0$$

is p-partially stabilizable in finite time if there exists a continuous feedback y → u(y) such that for every y₁ ∊ R^n−p,u(y₁,0) = 0, and such that (0,0) ∊ R^n−p × R ^p is p-partially stable in finite time for the closed-loop system $\dot{y}=Y(y\mathrm{,}u(y))$ .

Lemma 2 ([40]): For the following nonholonomic chained form system:

$$\begin{array}{c}{\dot{z}}_1={\tilde{u}}_1\mathrm{,}\\ {\dot{z}}_2=z_3{\tilde{u}}_1,\\ ⋮\\ {\dot{z}}_{n-1}=z_n{\tilde{u}}_1,\\ {\dot{z}}_n={\tilde{u}}_2,\end{array}$$

(3)

where $z=[z_1\mathrm{,}{z}_2\mathrm{,\dots ,}{z}_n{]}^T\in R^n\mathrm{,}\tilde{u}=[{\tilde{u}}_1\mathrm{,}{\tilde{u}}_2{]}^T\in R^2$ are the state vector and control input, respectively. For ${\tilde{k}}_1\mathrm{,}{\tilde{k}}_2\mathrm{,\dots ,}{\tilde{k}}_{n-1}>0{\tilde{k}}_1\mathrm{,}{\tilde{k}}_2\mathrm{,\dots ,}{\tilde{k}}_{n-1}>0$ such that the polynomial $s^{n-1}+{\tilde{k}}_{n-1}{s}^{n-2}+\cdots +{\tilde{k}}_{2}s+{\tilde{k}}_1$ is Hurwitz. There exists ${\tilde{\epsilon }}_0\in (\mathrm{0,1})$ such that for every $y_1\in R^{n-p}\mathrm{,}u(y_1\mathrm{,0})=0$ , the origin of (3) is (n−1) -finite-time partially stable under the homogeneous feedback law

$${\tilde{u}}_1={\displaystyle \sum _{i=2}^n}|z_i{|}^{{\tilde{\lambda }}_{i-1}}\mathrm{,}{\tilde{u}}_2=\tilde{u}{\tilde{u}}_1\mathrm{,}$$

where

$$\tilde{u}=-{\tilde{k}}_{1}sgn(z_2)|z_2{|}^{{\tilde{\alpha }}_1}-\mathrm{\dots }-{\tilde{k}}_{n-2}sgn(z_{n-1})|z_{n-1}{|}^{{\tilde{\alpha }}_{n-2}}$$

the design parameters ${\tilde{\alpha }}_1\mathrm{,}{\tilde{\alpha }}_2\mathrm{,\dots ,}{\tilde{\alpha }}_n$ satisfy that

$${\tilde{\alpha }}_{i-1}=\frac{{\tilde{\alpha }}_i{\tilde{\alpha }}_{i+1}}{2{\tilde{\alpha }}_{i+1}-{\tilde{\alpha }}_i}\mathrm{,}(i=\mathrm{2,\dots ,}n-1)$$

with ${\tilde{\alpha }}_n=1$ and ${\tilde{\alpha }}_{n-1}=\tilde{\alpha }$ . Choosing ${\tilde{\lambda }}_i\mathrm{,}(i=\mathrm{1,2,\dots ,}n-1)$ satisfy that

$$\frac{{\tilde{\lambda }}_1}{{\tilde{\alpha }}_1}=\frac{{\tilde{\lambda }}_2}{{\tilde{\alpha }}_2}=\mathrm{\dots }=\frac{{\tilde{\lambda }}_{n-2}}{{\tilde{\alpha }}_{n-2}}=\frac{{\tilde{\lambda }}_{n-1}}{{\tilde{\alpha }}_{n-1}}=\frac{\tilde{\lambda }}{\tilde{\beta }}\mathrm{,}\frac{\tilde{\lambda }}{\tilde{\beta }}+\frac{\tilde{\alpha }-1}{\tilde{\alpha }}<\mathrm{0,}$$

where $\tilde{\beta }\mathrm{,}\tilde{\lambda }>0$ .

Definition 3 (saturation function [24, 41]): A saturation function $Sa{t}_{\epsilon }(\cdot )$ denotes the function that is monotonically increasing and its boundary is less than ε, $|Sa{t}_{\epsilon }(\cdot )|\le \epsilon$ .

Remark 1: Examples of such saturation functions in Definition 3, for instance,

$$Sa{t}_{\epsilon }(\tilde{z})=\epsilon tanh(\tilde{z})\mathrm{,}Sa{t}_{\epsilon }(\tilde{z})=\frac{2\epsilon }{\pi }\arctan (\tilde{z})\mathrm{.}$$

Lemma 3 ([31]): For the double integrator system which is described by

$$\begin{array}{l}{\dot{\eta }}_1={\eta }_2\mathrm{,}\\ {\dot{\eta }}_2=\overline{u}\end{array}$$

(4)

Then the origin of (4) is a globally finite-time-stable equilibrium under the bounded feedback control law

$$\overline{u}=-sa{t}_{\epsilon }\{sgn({\eta }_2)|{\eta }_2{|}^{\alpha }\}-sa{t}_{\epsilon }\{sgn({\overline{\varphi }}_{\alpha })|{\overline{\varphi }}_{\alpha }{|}^{\frac{\alpha }{2-\alpha }}\}\mathrm{,}$$

where $0<\alpha <1$ , and $sa{t}_{\epsilon }(\cdot )$ is the saturation function with its boundboundary less than ε. Lemma 4 ([43]): For ${\tilde{z}}_i\in R\mathrm{,}i=\mathrm{1,2,\dots ,}n,0<p\le 1$ , then

$${({\displaystyle \sum _{i=1}^n}|{\tilde{z}}_i|)}^p\le {\displaystyle \sum _{i=1}^n}|{\tilde{z}}_i{|}^p\le n^{1-p}{({\displaystyle \sum _{i=1}^n}|{\tilde{z}}_i|)}^p\mathrm{.}$$

2.2 Problem Formulation

As shown in Fig. 1, (x,y) is the position of the mass center P of the Type (1,2) mobile robot moving in the plane, θ denotes its heading angle from the horizontal axis. The orientation of the plane of the wheels with respect to the vertical axis are represented by the angles β₁ and β₂. Positive constant L is a half of the width of the mobile robot, v₁, v₂ and v₃ denote the velocity of castor wheel and two angular velocities of steering wheels, respectively. Then, by introduced in [26, 41, 44–45], the posture kinematic model for the nonholonomic wheeled mobile robot of Type (1,2) can be described by the following differential equations

$$\begin{array}{l}\dot{x}=-L{v}_1[\sin {\beta }_1\sin (\theta +{\beta }_1)+\sin {\beta }_2\sin (\theta +{\beta }_1)]\mathrm{,}\\ \dot{y}=L{v}_1[\sin {\beta }_1\cos (\theta +{\beta }_2)+\sin {\beta }_2\cos (\theta +{\beta }_1)]\mathrm{,}\\ \dot{\theta }=v_1\sin ({\beta }_2-{\beta }_1),\\ {\dot{\beta }}_1=v_2\mathrm{,}\\ {\dot{\beta }}_2=v_3.\end{array}$$

(5)

Figure 1.

Type (1,2) mobile robot

In [26], based on visual servoing model, we presented a saturated switching design method for (5) with uncertain visual parameters such that the closed-loop system is finite-time stable. However, it may cause an unexpected oscillation phenomenon due to too many times switching process.

Therefore, for seeking the continuous state feedback to solve the partial-finite-time stabilization problem, first of all, we take the state transformation and the input transformation for system (5) as follow

$$\begin{array}{l}{x}_0=\theta ,\\ x_1=x\cos \theta +y\sin \theta ,\\ x_2=-x\sin \theta +y\cos \theta -\frac{2L\sin {\beta }_1\sin {\beta }_2}{\sin ({\beta }_2-{\beta }_1)},\\ x_3=x\sin \theta -y\cos \theta \mathrm{,}\\ x_4=x\cos \theta +y\sin \theta -\frac{L\sin ({\beta }_1+{\beta }_2)}{\sin ({\beta }_2-{\beta }_1)}\mathrm{,}\\ u_0=v_1\sin ({\beta }_2-{\beta }_1),\\ u_1=-x_4{v}_1\sin ({\beta }_2-{\beta }_1)-\frac{2L{v}_2{{\displaystyle \sin }}^2{\beta }_2}{{{\displaystyle \sin }}^2({\beta }_2-{\beta }_1)}+\frac{2L{v}_3{{\displaystyle \sin }}^2{\beta }_1}{{{\displaystyle \sin }}^2({\beta }_2-{\beta }_1)},\\ u_2=x_2{v}_1\sin ({\beta }_2-{\beta }_1)-\frac{L{v}_2\sin (2{\beta }_2)}{{{\displaystyle \sin }}^2({\beta }_2-{\beta }_1)}+\frac{L{v}_3\sin (2{\beta }_1)}{{{\displaystyle \sin }}^2({\beta }_2-{\beta }_1)}.\end{array}$$

Remark 2: In this institution, these new state or input variables x_i,(i = 0,1,2,3,4) and u_j,(j = 0,1,2) are also available, because in practical application, engineers usually get the state (x,y) by position sensor, and obtain (θ,β₁,β₂) by angle sensor. Then it's not difficult to achieve x _i ,u _j by calculating online according to the formula above. And this state and input transformation is presented such that the nonholonomic chained systems can be obtained, which is an effective way for controlling nonholonomic mobile robots [47].

Then the so-called “3 inputs, 2 chains, 1 generator” nonholonomic chained form system yields [46]

$$\begin{array}{l}{\dot{x}}_0=u_0,\\ {\dot{x}}_1=x_2{u}_0,\\ {\dot{x}}_2=u_1,\\ {\dot{x}}_3=x_4{u}_0,\\ {\dot{x}}_4=u_2,\end{array}$$

(6)

where x_i,(i = 0,1,2,3,4) and u_j,(j = 0,1,2) are seen as the new states and inputs, respectively. The three inputs are subject to the bounded constraints

$$|u_0|\le u_0^{\max }\mathrm{,}|u_1|\le u_1^{\max }\mathrm{,}|u_2|\le u_2^{\max }\mathrm{,}$$

(7)

where u₀^max, u₁^max and u₂^max are positive constants given in advance.

The control task is to design continuous controllers such that the closed-loop system of (6) is 4-finite-time partial stabilization at the zero equilibrium point under the saturation constraints (7), i.e., we will propose the saturated state feedback laws u_j,(j = 0,1,2) such that x_t,(i = 1,2,3,4) can be stabilized to zero in a finite time, furthermore, the convergence rate of the rest state x₀ can be accurately estimated.

3. Main Results

In this section, we will give our main results. Firstly, motivated by the results of C. Jammazi [40], the 4-finite-time partial stabilization for system (6) can be solved with continuous, homogeneous state feedback law.

Theorem 1: Take the following continuous pure state feedback law for the “3 inputs, 2 chains, 1 generator” nonholonomic chained form system (6)

$$u_0={\displaystyle \sum _{i=1}^4}|x_i{|}^{{\tilde{\lambda }}_i}\mathrm{,}{u}_1=x_3{u}_0\mathrm{,}{u}_2=\overline{u}{u}_0\mathrm{,}\mathrm{and}$$

where ${\tilde{k}}_i\mathrm{,}{\tilde{\lambda }}_i\mathrm{,}(i=\mathrm{1,2,3,4})$ are defined as in Lemma 2. Then, the closed-loop system of (6) can be 4-finite-time partially stabilized to zero.

Proof.: Omit.

Remark 3: This conclusion can be obtained by a simple mathematical derivation, substituting u₁ = x₃u₀ into (6), we have

$$\begin{array}{l}{\dot{x}}_0=u_0,\\ {\dot{x}}_1=x_2{u}_0\mathrm{,}\\ {\dot{x}}_2=x_3{u}_0,\\ {\dot{x}}_3=x_4{u}_0\mathrm{,}\\ {\dot{x}}_4=u_2\mathrm{,}\end{array}$$

which has the same structure as it in (3), therefore, Theorem 1 can be seen as a corollary of Lemma 2. However, it's not easy to satisfy the input saturation constraints (7) by applying this design strategy, and thus a new control method should be excavated to address our problem.

Remark 4: Note that in Theorem 1, we present a nonlinear cross feedback law u₁ = x₃u₀ to obtain a 4-finite-time partially stable closed-loop system of (6), it seems like falling into the conclusion of [40] by using a simple mathematical feedback technique without physical interpretation, but if we consider the physical input limitation constraints, it will be invalid. So, we propose a more practical saturated controller in the next results.

Theorem 2: Given a sufficiently large number M₀ > 1, and r _i > 0,(i = 1,2,3,4), M₁₁,M₁₂,M₂₁,M₂₂ > 0 satisfy

$$\begin{array}{l}(M_{11}+M_{12})\frac{u_0^{\max }}{M_0}\le u_1^{\max }\mathrm{,}\\ \max \{M_{11}\mathrm{,}{M}_{12}\}\le \mathrm{1,}\\ (M_{21}+M_{22})\frac{u_0^{\max }}{M_0}\le u_2^{\max }\mathrm{,}\end{array}$$

(8)

for system (6), take the saturated, continuous state feedback law

$$\begin{array}{l}{u}_0=Sa{t}_{\frac{u_0^{\max }}{M_0}}\{k_1|x_1{|}^{{\lambda }_1}+k_2|x_2{|}^{{\lambda }_2}\\ +k_3|x_3{|}^{{\lambda }_3}+k_4|x_4{|}^{{\lambda }_4}\},\\ u_1={\overline{u}}_1{u}_0,\\ u_2={\overline{u}}_2{u}_0,\end{array}$$

(9)

with

$$\begin{array}{l}{\overline{u}}_1=-Sa{t}_{M_{11}}\{sgn(x_2)|x_2{|}^{\alpha }\}\\ -Sa{t}_{M_{12}}\{sgn({\varphi }_{\alpha }(x_1\mathrm{,}{x}_2))|{\varphi }_{\alpha }(x_1\mathrm{,}{x}_2){|}^{\frac{\alpha }{2-\alpha }}\},\\ {\overline{u}}_2=-Sa{t}_{M_{21}}\{sgn(x_4)|x_4{|}^{\beta }\}\\ -Sa{t}_{M_{22}}\{sgn({\varphi }_{\beta }(x_3\mathrm{,}{x}_4))|{\varphi }_{\beta }(x_3\mathrm{,}{x}_4){|}^{\frac{\beta }{2-\beta }}\},\end{array}$$

(10)

where the design parameters $\alpha \mathrm{,}\beta \in (\mathrm{0,1}),\delta =\frac{2}{3-\alpha },0<{\lambda }_i\le \gamma <1-\delta \mathrm{,}(i=\mathrm{1,2,3,4}),k_i\mathrm{,}(i=\mathrm{1,2,3,4})$ satisfying

$$\begin{array}{l}{k}_1\ge {(\frac{2-\alpha }{3-\alpha }+r_1)}^{\gamma }\mathrm{,}{k}_2\ge {\left(\frac{1+r_2}{3-\alpha }+\frac{1+r_1}{2-\alpha }\right)}^{\gamma }\mathrm{,}\\ k_3\ge {(\frac{2-\beta }{3-\beta }+r_3)}^{\gamma }\mathrm{,}{k}_4\ge {\left(\frac{1+r_4}{3-\beta }+\frac{1+r_3}{2-\beta }\right)}^{\gamma }\mathrm{.}\end{array}$$

(11)

Then system (6) is 4-finite-time partially stable, that is the state vector (x₁,x₂,x₃,x₄) can be stabilized to zero in a finite time, and the rest state x₀ converges to a constant.

Proof.: First, we prove that (x₁,x₂) and (x₃,x₄) of the following subsystems (12)–(13) can be stabilized to zero by the controller (9) in a finite time.

$${\dot{x}}_1=x_2{u}_0\mathrm{,}{\dot{x}}_2=u_1\mathrm{.}$$

(12)

$${\dot{x}}_3=x_4{u}_0\mathrm{,}{\dot{x}}_4=u_2\mathrm{.}$$

(13)

Substituting u₁ into (12), we have

$$\begin{array}{l}{\dot{x}}_1=x_2{u}_0,\\ {\dot{x}}_2={\overline{u}}_1{u}_0.\end{array}$$

(14)

The corresponding double integrator system is

$$\begin{array}{l}{\dot{x}}_1=x_2,\\ {\dot{x}}_2={\overline{u}}_1.\end{array}$$

(15)

By Lemma 3 and recalling its proof in [31], there exists a finite time T₁ such that when t ≥ T₁, all the states of system (15) can be driven into a region D₁ defined as

$$D_1=\{(x_1\mathrm{,}{x}_2)\in R^2||x_2|\le M_{11}^{\frac{1}{\alpha }}\mathrm{,}|{\varphi }_{\alpha }|\le M_{12}^{\frac{2-\alpha }{\alpha }}\}\mathrm{,}$$

in which, the saturated controller ${\overline{u}}_1$ given in (10) devolves into

$${\overline{u}}_1=-sgn(x_2)|x_2{|}^{\alpha }-sgn({\varphi }_{\alpha })|{\varphi }_{\alpha }{|}^{\frac{\alpha }{2-\alpha }}\mathrm{.}$$

(16)

Take the Lyapunov function for (15) with negative degree of homogeneity −α,

$$V_1(x_1\mathrm{,}{x}_2)=\frac{2-\alpha }{3-\alpha }|{\varphi }_{\alpha }{|}^{\frac{3-\alpha }{2-\alpha }}+r_1{x}_2{\varphi }_{\alpha }+\frac{r_2}{3-\alpha }|x_2{|}^{3-\alpha }\mathrm{,}$$

(17)

as defined in Lemma 3,

$${\varphi }_{\alpha }=x_1+\frac{1}{2-\alpha }sgn(x_2)|x_2{|}^{2-\alpha }\mathrm{.}$$

Substituting (16) into (15), calculating the time derivative along its closed-loop system, there must exist a constant c > 0 such that

$${\varphi }_{\alpha }=x_1+\frac{1}{2-\alpha }sgn(x_2)|x_2{|}^{2-\alpha }\mathrm{.}$$

Next, for system (14), the following two cases are discussed.

1. If $k_1|x_1{|}^{{\lambda }_1}+k_2|x_2{|}^{{\lambda }_2}+k_3|x_3{|}^{{\lambda }_3}+k_4|x_4{|}^{{\lambda }_4}\ge \frac{u_0^{\max }}{M_0}$ , then $u_0=\frac{u_0^{\max }}{M_0}$ , and ${\dot{V}}_1{|}_{(14)}={\dot{V}}_1{|}_{(15)}{u}_0={\dot{V}}_1{|}_{(15)}\frac{u_0^{\max }}{M_0}\le -\frac{c{u}_0^{\max }}{M_0}{V}_1^{\delta }\mathrm{.}$

   According to Lemma 1, in this case, the state (x₁,x₂) of subsystem (12) can be stabilized to zero in a finite time.

2. As for another situation, if $k_1|x_1{|}^{{\lambda }_1}+k_2|x_2{|}^{{\lambda }_2}+k_3|x_3{|}^{{\lambda }_3}+k_4|x_4{|}^{{\lambda }_4}<\frac{u_0^{\max }}{M_0}$ then $u_0=k_1|x_1{|}^{{\lambda }_1}+k_2|x_2{|}^{{\lambda }_2}+k_3|x_3{|}^{{\lambda }_3}+k_4|x_4{|}^{{\lambda }_4}$ . We have ${\dot{V}}_1{|}_{(14)}={\dot{V}}_1{|}_{(15)}{u}_0$

$$\begin{array}{l}\le -c(k_1|x_1{|}^{{\lambda }_1}+k_2|x_2{|}^{{\lambda }_2}+k_3|x_3{|}^{{\lambda }_3}+k_4|x_4{|}^{{\lambda }_4})V_1^{\delta }\\ \le -c(k_1|x_1{|}^{{\lambda }_1}+k_2|x_2{|}^{{\lambda }_2})V_1^{\delta }\end{array}$$

(18)

Note that once the state (x₁,x₂) of system (15) has been driven into D₁ and in which $|x_2|\le \mathrm{1,}|{\varphi }_{\alpha }|\le \mathrm{1,0}<\gamma <1-\delta \mathrm{,1}<\frac{3-\alpha }{2-\alpha }<2$ , we have

$$\le {\left(\frac{2-\alpha }{3-\alpha }|{\varphi }_{\alpha }|+r_1|{\varphi }_{\alpha }|+\frac{r_2}{3-\alpha }|x_2|\right)}^{\gamma }$$

(19)

Because

$$|{\varphi }_{\alpha }|\le |x_1|+\frac{1}{2-\alpha }|x_2{|}^{2-\alpha }\le |x_1|+\frac{1}{2-\alpha }|x_2|\mathrm{,}$$

from (19), one can obtain

$$\begin{array}{l}{V}_1^{\gamma }\le {\left((\frac{2-\alpha }{3-\alpha }+r_1)(|x_1|+\frac{1}{2-\alpha }|x_2|)+\frac{r_2}{3-\alpha }|x_2|\right)}^{\gamma }\\ ={\left((\frac{2-\alpha }{3-\alpha }+r_1)|x_1|+(\frac{1+r_2}{3-\alpha }+\frac{r_1}{2-\alpha })|x_2|\right)}^{\gamma }\end{array}$$

By applying Lemma 4, we have

$$V_1^{\gamma }\le {\left(\frac{2-\alpha }{3-\alpha }+r_1{)}^{\gamma }|x_1{|}^{\gamma }+(\frac{1+r_2}{3-\alpha }+\frac{1+r_1}{2-\alpha }\right)}^{\gamma }|x_2{|}^{\gamma }\mathrm{.}$$

(20)

From (11), we have

$$k_1\ge {\left(\frac{2-\alpha }{3-\alpha }+r_1\right)}^{\gamma }\mathrm{,}{k}_2\ge {\left(\frac{1+r_2}{3-\alpha }+\frac{1+r_1}{2-\alpha }\right)}^{\gamma }\mathrm{,}$$

and because 0 < Λ _i ≤ γ < 1 − δ, (i = 1,2), then

$$k_1|x_1{|}^{{\lambda }_1}+k_2|x_2{|}^{{\lambda }_2}\ge V_1^{\gamma }\mathrm{.}$$

Substituting the inequality above into (18), we have

$${\dot{V}}_1{|}_{(14)}\le -c{V}_1^{\gamma +\delta }\mathrm{,}$$

note 0 < γ + δ < 1, then according to the criterion of finite-time stability in Lemma 1, controllers (9)–(10) can stabilize the subsystem (12) in a finite time.

Similarly, two other states x₃ and x₄ can be proved to converge to zero in a finite time, furthermore, in this control process, according to the structure characteristic of subsystem (12)–(13), the state (x₁,x₂) remains at the origin equilibrium point. After that, from (9), we have u₀ ≡ 0, it means the state x₀ finally converges to a constant, i.e., system (6) is 4-finite-time partially stable.

On the other hand, it's not difficult to prove that the controllers (9)–(10) satisfy the saturation constants (7) by choosing proper design parameters (8)–(11) given in advance. This completes the proof of this theorem.

Remark 5: The p-finite-time partially stabilization is important to engineering practice, for example, state x₀ denotes the angle θ, it may only to be regulated to a constant angle while other position must be driven to a specified point such as the origin equilibrium point. But also, it's very critical to estimate the convergent limit of the angle x₀.

Remark 6: Then design parameters can be selected one by one according to Theorem 2, for example, given M₀ > 1, u_i^max > 0,(i = 1,2,3), from (8), we can choose M₂₁,M₂₂ > 0 satisfying $M_{21}+M_{22}\le \frac{M_0{u}_2^{\max }}{u_0^{\max }}$ . Then select $0<M_{11}\le \min \{\mathrm{1,}\frac{M_0{u}_1^{\max }}{u_0^{\max }}\}$ and next, M₁₂ can be chosen as $M_{12}=\min \{\mathrm{1,}\frac{M_0{u}_1^{\max }}{u_0^{\max }}-M_{11}\}$ . As for k_j,(j = 1,2,3,4), it is easy to be chosen from (11).

Theorem 3: Assuming | x(0) | ≤ ε/2 for any given ε > 0, where

$$x(0)=(x_0(0)\mathrm{,}{x}_1(0)\mathrm{,}{x}_2(0)\mathrm{,}{x}_3(0)\mathrm{,}{x}_4{(0))}^T\in R^5\mathrm{.}$$

Then (0,0) ∊ R × R⁴ is 4-partially stable in finite time for the closed-loop system (6) with the saturated, continuous pure state feedback u₀, u₁ and u₂ given by (9)–(10). Moreover, | x₀(t) | ≤ ε, ∀ t ≥ 0.

Proof.: Clearly, under the bounded feedback (9)–(10) and form Theorem 2, the system (6) is 4-partially stable in finite time. Therefore, there exists a finite time T₀ such that x₁(t) = x₂(t) = x₃(t) = x₄(t) = 0 for all t ≥ T₀.

By integration, the state x₀ verifies

$$x_0(t)=x_0(0)+{\displaystyle {\int }_0^{T_0}}{u}_0(x(s))ds\mathrm{.}$$

by using the triangular inequality, we have

$$|x_0(t)|\le |x_0(0)|+{\displaystyle {\int }_0^{T_0}}|u_0(x(s))|ds$$

Since M₀ is large enough given in Theorem 2, then $\frac{T_0{u}_0^{\max }}{M_0}\le \frac{\epsilon }{2}$ , and we get

$$|x_0(t)|\le \frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon \mathrm{,}\forall t\ge 0.$$

Hence, state x₀ is constant and large enough for t, then the limit of x₀ is denoted by $\tilde{c}(x(0))$ and satisfies $|\tilde{c}(x(0))|\le \epsilon$ .

4. Simulations

In this section, the continuous, bounded, pure state feedback controller proposed in Theorem 2 is used to show how to stabilize the nonholonomic chained form system (6) in a finite time. We will demonstrate the effectiveness of our methods by a numerical example.

For system (6) under the saturated constraints (7), we suppose that u₀^max = 3.0, u₁^max = 3.5, u₂^max=3.0. The initial conditions are x₀(0)=0.3, x₁(0) = −1.5, x₂(0) = −0.6, x₃(0) = 1.2, x₄(0) = −2.0. Given ε = 15, from Theorem 2, the control parameters are chosen as M₀=10, M₁₁ = M₁₂ = M₂₁=M₂₂=1.0, r_i = 1.0,(i = 1,2,3,4), k_j = 2.0,(j = 1,2,3). The saturation function is selected as $Sa{t}_{\tilde{\epsilon }}(s)=\tilde{\epsilon }tanh(s)$ in our simulations.

Figs. 3–4 show that under the controllers (9)–(10) proposed in Theorem 2, the state variable (x₁,x₂) and (x₃,x₄) are stabilized to zero in a finite time T ≤ 10s, that is, as t ≥ 10s, x _i = 0(i = 1,2,3,4), and remain unchanged. From Fig. 2, we can observe that x₀ converges to a constant c(x(0)) satisfying 5<c(x(0)) < 6 < ε = 15, as expressed by Theorem 3.

Figs. 5–7 show that at any time, the bounds of all the control inputs U₀,u₁ and u₂ are not less than the saturation level given in advance. Particularly, since the first input u₀ depends on the state (x₁,x₂,x₃,x₄), and from Fig. 5, we can observe that (x₁,x₂,x₃,x₄) = 0 as t ≥ 9s, thus u₀ = 0, as ≥ 9s.

Figure 2.

State variable x₀ of the transformed chained system (6)

Figure 3.

State variable (x₁, x₂) of the transformed chained system (6)

Figure 4.

State variable (x₃, x₄) of the transformed chained system (6)

Figure 5.

Response of bounded input u₀ with respect to time

Compared with the existing control methods for Type (1,2) nonholonomic mobile robots, the best advantage of our controller design is that control inputs are subject to the saturation constraints. Next, we also presents some simulation results with the same design parameters and initial conditions as in this paper by the control strategy of the reference [48].

Figure 6.

Response of bounded input u₁ with respect to time

Figure 7.

Response of bounded input u₂ with respect to time

Figure 8.

Response of state variable x₀, x₁ with respect to time when using controller of [48]

Figs. 8–9 show the convergence of the state variable (x₀, …, x₄) when using the controller of [48], while from Fig. 10, we observe that the bounded constraints | u₀ | ≤ 3.0, | u₁ | ≤ 3.5, | u₂ | ≤ 3.0 were broken, it may exceed the limitation of actuators at some time, in more detail,

Figure 9.

Response of state variable x₂, x₃, x₄ with respect to time when using controller of [48]

Figure 10.

Response of input variable u₀, u₁, u₂ with respect to time when using controller of [48]

we can see that | u₁ |, | u₂ | exceed the maximum limitation of u_i^max,i = 0,1,2 frequently at about t < 3.8s in Fig. 10, which means that the practical constraints | u_i | ≤u_i^max can not always be guaranteed at any time by the design method in [48].

5. Conclusion

In this paper, the saturated, continuous, pure state feedback controller is designed for a class of nonholonomic mobile robots for the first time, such that its’ chained form systems can be stabilized to zero in a finite time except the angle state variable. Moreover, the angle state can be driven to a constant, and the convergence limit constant may be accurately estimated in advance.