GUO Fuhui and LU Pingli

School of Automation, Beijing Institute of Technology, Beijing 100081, China

Abstract: A fast self-adapting high-order sliding mode (FSHOSM) controller is designed for a class of nonlinear systems with unknown uncertainties. As for uncertainty-free nonlinear system,a new switching condition is introduced into the standard geometric homogeneity. Different from the existing geometric homogeneity method, both state variables and their derivatives are considered to bring a reasonable effective switching condition.As a result, a faster convergence rate of state variables is achieved. Furthermore, based on the integral sliding mode (ISM)and above geometric homogeneity, a self-adapting high-order sliding mode (HOSM) control law is proposed for a class of nonlinear systems with uncertainties. The resulting controller allows the closed-loop system to conduct with the expected properties of strong robustness and fast convergence. Stable analysis of the nonlinear system is also proved based on the Lyapunov approach. The effectiveness of the resulting controller is verified by several simulation results.

Keywords: adaptive control law, geometric homogeneity, highorder sliding mode (HOSM), integral sliding mode (ISM).

1. Introduction

Strong robustness and implementation simplicity are the remarkable advantages of sliding mode control (SMC) for nonlinear system with unknown uncertainties over other nonlinear control methods [1-3]. Therefore, SMC has been widely utilized in a vast amount of real systems,such as flight control system [4-8], robot control system[9-11] and servo control system [12-14]. Under the SMC, the motion of system state variables is divided into two stages, that is, sliding phase and reaching phase [15].Corresponding to these two phases, the structure of the SMC law always consists of equivalent control and switching control. Equivalent control aims at forcing the system to move to origin along the sliding mode surface.While the switching control makes the system state close to the sliding mode surface and move on it, meanwhile the closed-loop system can immune against the uncertainties.

When it comes to SMC, apart from the well-known chattering phenomenon [16-18], the convergence of the system state is also an important issue. In the study of convergence, several modifications of the conventional sliding mode concept, such as terminal sliding mode(TSM), integral sliding mode (ISM) and high-order sliding mode (HOSM) and so on have been developed to improve the control performance, by which the closed-loop system can be stabilized in finite time [19-22]. Asl et al.[19] designed a fast TSM controller for ducted fan engine of thrust vectored aircraft to increase the speed of the convergence of the system states in finite time.However, a well-known drawback of TSM should not be ignored, that is, the singular problem. Therefore non-singular TSM (NTSM) was proposed by Feng et al. [20] for the first time. Thereafter, Ding et al. [21] developed an adaptive NTSM controller. By searching the minimal value of the control gain, both the convergence speed and the chattering are improved. Further research was given by Yi et al. [22], in which the second-order fast NTSM controller was designed to achieve fast convergence. Additionally, ISM is another effective method to improve convergence performance and has been widely available to nonlinear system with disturbance [23,24]. Chen et al.[23] introduced the integral item into the sliding mode surface to improve steady-state error. However, integral saturation cannot be avoided effectively in the presence of large initial error. Seshagiri et al. [24] proposed a conditional ISM controller, by which the effect of the integral item was weakened outside the boundary layer.

Also, the HOSM control method acting as another kind of finite-time control was proposed by Levant [25]. In[26], the HOSM theory was used to design the attitude controller for rigid spacecraft. To a certain extent, HOSM control can cope with both the convergence and the chattering. On the one hand, the HOSM control method can complete finite-time rather than asymptotic convergence[25], which has marked a milestone in the theory development of SMC. On the other hand, the reduction of chattering is realized by setting the discontinuous switching function in the high-order derivative of the sliding variable. Numerous publications on HOSM including theory and application have emerged afterwards [27-31].The HOSM controller is often designed combined with ISM or TSM to deal with uncertainty, while the nominal system is always formulated as an integrator chain system to be studied. The solution to the integrator chain system is always by resorting to the geometric homogeneity technique [32]. Nonetheless, the improvement on the convergence speed is still expected. For instance, the large initial value of state variables leads to a slow convergence. In [33], Li et al. proposed a quasi-optimal finitetime control method, resulting in a faster convergence rate than that in [25]. Furthermore, a global sliding mode scheme was studied in [34,35], by which the reaching phase was removed, resulting in that the large initial value was no longer a problem.

Motivated by the preceding discussion, a fast self-adapting HOSM control law is proposed based on ISM and geometric homogeneity. Specifically, the proposed method possesses the following advantages.

Firstly, for the nominal system, a reasonable and feasible switching condition is introduced into the standard geometric homogeneity in order to improve the convergence. The state and the state change rate are both considered as important indexes in the switching condition.As a result, the adverse effect caused by the large initial state will be alleviated. Meanwhile the finite-time convergence is guaranteed.

Secondly, as for a class of uncertain nonlinear systems,an HOSM controller is designed combined with the above geometric homogeneity method. Additionally, an adaptive gain inspired by [36] is introduced to alleviate chattering.

The rest of this paper is organized as follows. Section 2 gives preliminaries. In Section 3, the fast convergence switching condition is introduced into conventional geometric homogeneity for chain integrator system without uncertainty. Moreover, a self-adapting HOSM control law is presented for a class of nonlinear systems with uncertainties. Simulation results are illustrated in Section 4.Section 5 concludes the paper.

2. Preliminaries

Consider a class of nonlinear systems with uncertainties as follows:

3. Main results

3.1 Motivation

Similar to conventional SMC law design, there are two main parts in the process of high-order SMC design. One is to design the nominal controller in the absence of uncertainty, which aims at forcing the variable σ(x,t) to converge to zero in the sliding phase in finite time. The other is to design a discontinuous controller, aiming at chattering suppression. Firstly, nominal controller design is considered. With absence of uncertainty, the system (4)under equivalent controller (7) can be formulated as

It can be shown clearly in Fig. 2 that the convergence rate is not always larger or smaller than a fixed value such as 1 or some other fixed value. Therefore, the distance fromzi(i=1,2,···,r) to zero is not an enough rigorous condition for adjusting the convergence speed of the state variable.

Fig. 1 Graphical presentation of

Fig. 2 Zoom graphical presentation of

3.2 Fast geometric homogeneous (FGH) controller for integrator chain system

According to the discussion in Subsection 3.1, there are two aspects to be improved for (9) in Lemma 1. Firstly, a reasonable switching condition is proposed, and one can obtain

wherei=1,2,···,r.

Then (9) is expressed as

3.3 Fast self-adapting high-order sliding mode(FSHOSM) controller design

In Subection 3.2, the FGH controller is proposed for system (8). However, the FGH controller performs poorly in the presence of lumped uncertainty. Therefore, a high-order SMC law is designed herein. By combing the ISM and FGH, the robustness of the closed-loop system is also guaranteed. Meanwhile, in order to attenuate the chattering, an adaptive switching gain is introduced [39]. Moreover, with the aid ofvnafin (13), the close-loop system tends to be stable in finite-time more rapidly.

In order to achieve the goal above, an ISM functionsis selected as

Obviously, when the sliding mode is established,z˙r=vnafholds. Furthermore, the controlleruis designed with the similar form as (6) as follows:

where η＞0 . The initial valueMeanwhileThen, one has the following theorem.

Theorem 2Consider the uncertain nonlinear system described in (4) with Assumption 2. Based on ISM manifold (16), the adaptive FSHOSM controller (17) can stabilize the nonlinear system (4) in finite time.

Therefore, by Theorem 2, one can conclude that the sliding mode function (16) will converge tos=0 in finite time. Moreover, the state responsetends to zero in finite time. Meanwhile sliding variables σi(x,t)(i=1,···,r) tend to zero, that is, the outputy(x,t)of thesystem(1) can track the desired trajectoryhd(x,t)in finite-time.□

4. Simulation

Several simulation examples are provided to illustrate the effectiveness of the proposed controller in Theorem 1 and Theorem 2. Firstly, a triple integrator system is considered. With the help of controller in Theorem 1, the system states of the triple integrator system will converge to origin faster than that in [25] and [33]. Then in Subsection 4.2, a nonlinear system with uncertainty is considered, and a kinematic car model is selected as a practical controlled object. By resorting to the controller (17)in Theorem 2, the car motion trajectory will track the desired trajectory quickly even in the face of uncertainties.

4.1 Simulation study on FGH controller for triple integrator system

In this subsection, a triple integrator system, i.e.,r=3 is set in (8), is studied and formulated as

Firstly, control inputvnof triple integrator system (28)is replaced byvn1formulated in (12). Then a comparison with the previous work in [25] is given withz1(0)=40,z2(0)=-50 ,z3(0)=60 ,a3=0.75 andk1=1 ,k2=2,k3=2.

Meanwhile, another comparison with the work in [33]is also given withz1(0)=-6 ,z2(0)=-2 ,z3(0)=15,a3=0.5 andk1=2 ，k2=10 ，k3=20.

Apparently, the dynamic process of statesz1,z2,z3demonstrates that the proposed control law (12) brings a quicker convergence rate and shorter convergence time with the help of the proposed switching condition in (11)(seen from Fig. 3 and Fig. 4).

Fig. 3 Comparison results of states z1,z2,z3 in system (28)between controller v n1 and the controller in [25]

Fig. 4 Comparison results of states z1,z2,z3 in system (28)between controller v n1 and the controller in [33]

Secondly, control inputvnof triple integrator system(28) is replaced byvn1formulated in (12) andvnformulated in (13), respectively. Parameters are set asz1(0)=By resorting to control inputvnaf, the convergence rate of statesz1,z2,z3performs faster (seen from Fig. 5), and the control inputvnafperforms steadily and quickly (seen fromFig.6).Besides, Fig.7 showsthetime-varying parametersHowever, thecontrol law(13)fails to tackle the external disturbance and uncertainties(see more details in [40]). For example, adding some disturbances to the right-hand side of triple integrator system (28), one obtainsz˙r=vnaf+5sin(0.8t+2). Simulation results are shown in Fig. 8. Clearly, system statesz1,z2,z3fail to converge to zero.

Fig. 5 Comparison results of states z1,z2,z3 in system (28) under vn1 and vnaf

Fig. 6 Control input vnaf

Fig. 7 Adaptive gai

Fig. 8 States z1,z2,z3 of the system (28) under controller v naf with uncertainties

4.2 Application to car control with uncertainties via FSHOSM controller

In practical engineering, uncertainty is inevitable. Therefore, in this subsection, a kinematic car model (seen from Fig. 9) is chosen as a practical controlled object.

Fig. 9 Kinematic car model

Mathematical model of the kinematic car [33,41-44] is as follows:

wherex1andx2denote the Cartesian coordinates of the rear-axle middle point.x3andx4denote the orientation angle and the steering angle respectively.uis the control input. Longitudinal velocity and distance between the two axles are represented byVcandL, respectively.

Corresponding to (1), one has

wherex=[x1,x2,x3,x4]T, and

Vc,LandHare represented as follows:Vc=Vc0+ΔVc,L=L0+ΔL,H=H0+ΔH.Vc0,L0andH0are the nominal values. Measured errors of speed and longitude are denoted by ΔVcand ΔL, respectively. Uncertainty in control inputuformulated in (30) is expressed as ΔH.

Then one gets the form with a chain of integrator of kinematic car model (29) as follows:

Hence one has φn(x,t), Δφ(x,t), γn(x,t) andΔγ(x,t)expressed specifically as

From (33), it is obvious that the control inputuappears explicitly in the third time derivative of the sliding variable σ(x,t) for the first time. Therefore, the relative degree of kinematic car model (29) is 3, then combined with Assumption 1,r=3 holds. According to Theorem 2,one has the following controller:

Simulation results are illustrated in Fig. 10-Fig. 13.The actual trajectoryx2and the desired trajectoryx2care shown in Fig. 10. It is obvious that actual trajectoryx2can track the desired trajectoryx2crapidly and exactly although uncertainties exist. As for the convergence ofz1,z2,z3, it is depicted in Fig. 11(a).z1,z2,z3succeed in converging to a vicinity of zero in finite time.

Fig. 10 Trajectories of x2 and x2c under controller (35)

Fig. 11 States z1,z2,z3 in system (33) under controller (35) and sliding variable s

Fig. 13 Adaptive gain and

Furthermore, Fig. 14 and Fig. 15 show the comparison results of statesz1,z2,z3in system (33) with three different controllers proposed in [25], [33] and the controller(35) proposed in this paper. Both illustrate the superiority of the controller (35) in convergence speed.

Fig. 14 Comparison results of states z1,z2,z3 in system (33)between our controller (35) and the method in [25]

Fig. 15 Comparison results of states z1,z2,z3 in system (33)between our controller (35) and the method in [33]

5. Conclusions

This paper focuses on a fast adaptive HOSM control for a class of nonlinear systems with uncertainties. The main contribution of this paper differs from others in two aspects. Firstly, it provides a reasonable switching condition into a standard geometric homogeneous controller,resulting in a faster convergence rate. Secondly, it offers an adaptive HOSM control based on ISM and the geometric homogeneous controller. As a result, both fast convergence rate and robustness are guaranteed for such uncertain nonlinear systems. Simulation results and comparison studies confirm the efficiency of the proposed control algorithm.