Adaptive Sliding Mode Control of Uncertain High-Order Nonholonomic Systems with Unknown Control Coefficients

This paper investigates the global stabilization problem for a class of high-order nonholonomic systems with unknown control coefficients and uncertain nonlinearities. An adaptive sliding mode control (SMC) law based on a constructive manipulation is proposed by adding a power integrator technique. A switching control strategy is employed in the control scheme to overcome the uncontrollability problem associated with the nonholonomic systems. The designed sliding mode controller could guarantee the attractiveness of the sliding surface S = 0 and achieve the asymptotical convergence of the state as well as the boundedness of the estimated parameters. A simulation example is provided to demonstrate the effectiveness of the proposed scheme.


Introduction
Nonholonomic control system as a particular class of nonlinear systems represents a wide class of mechanical systems with nonholonomic (nonintegrable) constraints.The stabilization control for nonholonomic control systems is a formidable problem in the nonlinear control area [1], because the nonholonomic systems cannot be asymptotically stabilized by means of known linear and nonlinear control methods.The difficulty of controlling this class of systems lies in the fact that nonholonomic systems do not satisfy Brockett's necessary conditions for stability as shown in [2].In the past two decades, many interesting approaches have been proposed to overcome the stabilization obstruction associated with nonholonomic systems, including the smooth timevarying feedback, discontinuous feedback, and nonsmooth time-varying homogeneous feedback; see [1] and references therein.
Variable structure control (VSC) has been shown to be robust in the presence of system variations and external disturbances.During the past few years, the VSC strategy has been also applied to the nonholonomic system control; see [3][4][5][6][7][8][9].It is noted in [5,6]; a combined adaptive/variable structure control approach was presented for a class of uncertain nonlinear systems.The proposed method not only ensures the convergence to the sliding surface  = 0, Ṡ = 0, but also retains the asymptotical stability of the classical backstepping procedure.On the basis of the previously proposed control scheme, the so-called second order sliding mode control was exploited for nonholonomic systems in [7].The work [8] further extended this result to a class of nonholonomic ones with unknown parameters and uncertain nonlinear drifts.In [9], the adaptive SMC problem is studied for a class of perturbed nonholonomic systems which can be transformed into the chained form with uncertain nonlinear drifts.
It should be noted that the results listed previously are all focused on the nonholonomic systems with the affine control variables.High-order nonholonomic systems which include the standard chained form systems as a special case present a new challenge for nonlinear feedback control, since this class of nonholonomic control systems is neither stabilizable by time-invariant continuous state feedback nor affine in the control inputs [10].By means of adding a power integrator developed in [11], the work [12] first investigated a class of high-order nonholonomic systems in the so-called power chained form.As the further development of this approach, [13] studied a class of uncertain nonholonomic control systems with nonlinear drifts.The recent work [14] removed the assumption of the control gain signs.Reference [15] proposed an adaptive stabilization scheme for a class of high-order chained nonholonomic systems without nonlinear drifts in the case of unknown control coefficients.
Motivated by the recent progress in the feedback design for nonholonomic systems, this paper will investigate the global stabilization problem for a class of high order nonholonomic systems with unknown control coefficients and uncertain nonlinearities.A combined backstepping and adaptive SMC methodology developed in [6,16] will be applied to the control design procedure, and adding a power integrator technique is used to deal with the nonaffine control inputs.The tuning function technique presented in [17] is also introduced.We employ a switching control strategy to overcome the stabilization burden associated with the nonholonomic systems.
The paper is organized as follows.Section 2 presents the problem statement.Section 3 gives the SMC design scheme for  0 ( 0 ) ̸ = 0.The switching control strategy and the simulation example are shown in Sections 4 and 5, respectively.Section 6 gives some concluding remarks.
To this end, we make the following assumptions regarding system (1).( Assumption 2. For each 0 ≤  ≤  − 1, the sign of   () is positive, and there are unknown positive constants  1 and  2 such that
Next, we introduce a lemma which is crucial in establishing the main results of this paper.

SMC Applied to Backstepping Design
In this part, we first assume that  0 ( 0 ) ̸ = 0 for the system (1).The case of  0 ( 0 ) = 0 will be discussed in the subsequent section.
Step 1. Choose the candidate Lyapunov function Θ2 for this step, where Θ = Θ − Θ is the parameter estimation error, and  1 > 0 is the design gain.Then, along the trajectories of system (8), we have Denote , and a direct substitution leads to Choose the virtual control law and one gets Step i (2 ≤  < ).Suppose that at Step  − 1, there is a positive definite and proper Lyapunov function Let Then   is a  2 , positive definite and proper function such that With the help of Young's Inequality, we can obtain where  1 and   ( = 2, . . ., ;  = 1, . . ., 5) given later are smooth nonnegative functions.From (18), we get In view of it can be derived that By Lemma 5, we can obtain × Substituting ( 20)-( 24) into (19), we have Denote Substituting ( 26) into (25), we have According to (26), there holds As a result, (27) can be further transformed into Choose the th virtual control law and a direct substitution into (29) results in Step n.Let where 2 > 0 is an arbitrary constant, Λ = Λ − Λ with Λ is the estimate of the unknown constant Λ = max{ 2 , }. Choose the parameter update law as and then According to Lemma 4, (9), and (10), we get By Lemma 4, it can be concluded that With the same strategy as previously, it follows mentioned that In terms of (34), it can be seen that with  4 ( 0 , , Θ) = 0. Denote As a result, (40) can be further transformed into Inspired by [6,16], the error variable can be viewed as an adaptive sliding surface in the -coordinate.Denote the sliding manifold by  =   , then the adaptive sliding mode control and the parameter update law can be chosen as where  > 0 is a constant and sgn(⋅) is the sign function.Accordingly, we obtain Since   is a positive definite and proper Lyapunov function, (45) guarantees the convergence of the system trajectory to the origin of the sliding surface  =   in finite time.In addition, the convergence properties of the states and the parameter estimate remain unchanged with respect to those of the standard backstepping design procedure.That is, the states ( 0 (), ()) of system (1) converge to zero asymptotically and the parameter estimates Θ(), Λ() are bounded.

Switching Control Strategy
In this section, we consider the case of  0 ( 0 ) = 0. Without loss of generality, we assume that  0 = 0.As to the case of  0 (0) ̸ = 0, we have given the controller ( 5) and (43) for  0 and  of system (1), respectively.Now, we turn to how to select control laws while  0 (0) = 0.
We consider the constant control if | 0 | is small enough.When  0 (0) = 0, we choose  0 as follows: In accordance with the first subsystem of (1), we know that So, for any small enough constant  * 0 , there exists an instant  *  > 0 such that  0 ( *  ) =  * 0 .During the finite time interval [0,  *  ), a new adaptive sliding mode control  =  * ( 0 , ) using  0 as defined in (46) and two new parameter update laws can be obtained following the procedure described in the previous section.Combining with  0 ( *  ) =  * 0 ̸ = 0, we can switch the control inputs  0 and  into (5) and (43), respectively.Now we state the main result of this paper.
Theorem 6. Suppose that Assumptions 1-3 are satisfied.If the proposed control design procedure together with the previous switching control strategy is applied to system (1), then, for any initial conditions in the state space Ω = {( 0 , ) ∈ R +1 }, system (1) will be asymptotically stabilized at the equilibrium, and specifically, the states are regulated to the sliding surface  = 0 in finite time while keeping the estimated parameters bounded.

Conclusions
In this paper, the global stabilization problem is considered for a class of uncertain high-order nonholonomic systems with unknown control coefficients.By adding a power integrator technique, we propose a combined adaptive/sliding mode control scheme.To get around the stabilization burden associated with the nonholonomic control systems, a switching control strategy is exploited in this procedure.The designed adaptive sliding mode controller guarantees the global asymptotical stabilization of the closed-loop system as well as the boundedness of the parameter estimates.Moreover, the attractiveness of the sliding surface  = 0 in finite time is also obtained with the help of the proposed control approach.

Figure 1 :
Figure 1: States of the closed-loop system.