Output Feedback Adaptive Stabilization of Uncertain Nonholonomic Systems

and Applied Analysis 3 ̇ P = P(A − μ 0 ( t) L) T + (A − μ 0 ( t) L) P − PC T CP + I, P (0) = P 0 > 0, (7) satisfies pminI ≤ P(t) ≤ pmaxI, t ≥ 0. By Lemma 6 and Assumption 1, we know that there exist smooth functions ω i ≥ 1, and ζ i ≥ 1 such that 󵄨 󵄨 󵄨 󵄨 φ i (t, u 0 (t) , x 0 (t) , x (t) , θ) 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨 󵄨 󵄨 󵄨 x 1 󵄨 󵄨 󵄨 󵄨 ω i (u 0 (t) , x 0 (t) , x 1 (t)) ζ i (θ) . (8) Furthermore, we denote θ = ∑n i=1 ζ i (θ); then it yields 󵄨 󵄨 󵄨 󵄨 φ i (t, u 0 (t) , x 0 (t) , x (t) , θ) 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨 󵄨 󵄨 󵄨 x 1 󵄨 󵄨 󵄨 󵄨 ω i (u 0 (t) , x 0 (t) , x 1 (t)) θ. (9) 3. Output Feedback Adaptive Stabilization Control Design In this paper, we design control lawsu 0 (t) andu 1 (t) separately to globally asymptotically stabilize the system (1). According to the structure of system (1), we can see that when x 0 (t) converges to zero, x i (t) (1 ≤ i ≤ n) will be uncontrollable. A widely used method to design control law u 1 (t) is to introduce a discontinuous input scaling transformation (12). On the other hand, the control directions d i are unknown; then we should employ another coordinate transformation to overcome the obstacle. 3.1. State Coordinate Transformation. Firstly, we design the coordinate transformation as follows: x i ( t) = di−1 x i ( t) , 1 ≤ i ≤ n, (10) where d 0 = 1 and d i−1 = d 1 d 2 ⋅ ⋅ ⋅ d i−1 (1 ≤ i ≤ n + 1). Then, the system (1) can be transformed into ?̇? 0 (t) = d 0 u 0 (t) + φ 0 (t, x 0 (t)) , ̇ x 1 ( t) = u0 ( t) x2 ( t) + φ1 (u 0 ( t) , y (t) , y (t − τ1 )) + φ 1 (t, u 0 (t) , x 0 (t) , x (t) , θ) ,


Introduction
The control and feedback stabilization problems of nonholonomic systems have been widely studied by many researchers.It is well known that control of nonholonomic systems is extremely challenging, largely due to the impossibility of asymptotically stabilizing nonholonomic systems via smooth time-invariant state feedback, a well-recognized fact pointed out in [1,2].In order to overcome this obstruction, a number of approaches have been proposed for the problem, which mainly include discontinuous feedback, time-varying feedback, and hybrid stabilization.The discontinuous feedback stabilization was first proposed by [3], and then further discussion was made in [4][5][6][7]; especially an elegant discontinuous coordinate transformation approach is proposed in [5] for the stabilization problem of nonholonomic systems.Meanwhile, the smooth time-varying feedback control strategies also have drawn much attention [8][9][10][11].
As pointed out in [9], many nonlinear mechanical systems with nonholonomic constraints can be transformed, either locally or globally, to the nonholonomic systems in the so-called chained form.So far, there have been a number of controller design approaches [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] for such chained nonholonomic systems.Recently, adaptive control strategies have been proposed to stabilize the nonholonomic systems.For instance, the problem of adaptive state-feedback control is studied in [15][16][17][18][19], while output feedback controller design in [20][21][22][23][24]. Considering the actual modeling perspective, time delay should be taken into account.The problem of state feedback stabilization is studied for the delayed nonholonomic systems in [25,26].However, the virtual control coefficients and unknown parameter vector are not considered in its system models.Here, an iterative controller design method will be proposed for the output feedback adaptive stabilization of the concerned delayed nonholomic systems.
In this paper, we study a class of chained nonholonomic systems with strong nonlinear drifts, and the problem of adaptive output-feedback stabilization for the concerned nonholonomic systems is investigated.The constructive design method proposed in this note is based on a combined application of the input scaling technique, the backstepping recursive approach, and the novel Lyapunov-Krasovskii functionals.The switching control strategy for the first subsystem is employed to achieve the asymptotic stabilization.
The rest of this paper is organized as follows.In Section 2, the problem formulation and some preliminary knowledge are given.Section 3 presents the controller design procedure and stability analysis.Section 4 gives the switching control strategy.In Section 5, numerical simulations testify to the effectiveness of the proposed method, and Section 6 summarizes the paper.
In this paper, we make the following assumptions on the virtual control directions   and nonlinear functions   ,   in system (1).Assumption 1.  0 is a known constant and the sign of   is known, where ( for all (,  0 (),  0 (), (), ) ∈  + ×  ×  ×   ×   .
in which   and   are known smooth nonnegative nonlinear functions.
Remark 4. Compared with some existing literatures in recent years, the structure of our concerned system (1) is more general.For instance, in [15], it is assumed that not only the virtual control directions   = 1 and the dynamics   satisfy   = φ  , but also the modeled dynamics   do not exist.In [22], the virtual control coefficients and time delays have not been considered, and the expression   = φ   is also required.While   = 1 and   and unknown parameters  are not existent, system (1) degenerates to the one studied in [21].When   = 0, together with   = φ  , system (1) becomes the considered system in [23].
It can be seen that the above inequality condition is used in some existing literatures, such as [20,21], and so on.
Our object of this paper is to design adaptive output feedback control laws under Assumptions 1-3, such that the system states ( 0 (), ()) converge to zero, while other signals of the closed-loop system are bounded.The designed control laws can be expressed in the following form: Next, we list some lemmas which will be applied in the coming controller design.
By Lemma 6 and Assumption 1, we know that there exist smooth functions   ≥ 1, and   ≥ 1 such that

Output Feedback Adaptive Stabilization Control Design
In this paper, we design control laws  0 () and  1 () separately to globally asymptotically stabilize the system (1).According to the structure of system (1), we can see that when  0 () converges to zero,   () (1 ≤  ≤ ) will be uncontrollable.
A widely used method to design control law  1 () is to introduce a discontinuous input scaling transformation (12).On the other hand, the control directions   are unknown; then we should employ another coordinate transformation to overcome the obstacle.
Next, the following input-state scaling discontinuous transformation is introduced: Under the new ()-coordinates, the ()-subsystem (10) is changed into Next, we can design the control laws  0 () and  1 () to asymptotically stabilize the states  0 () and (), respectively.Rewrite system (13) in the compact form where with In order to obtain the estimation for the nonlinear functions Ψ  and Φ  , the following lemmas are given.Lemma 8.For every 1 ≤  ≤ , there exists smooth nonnegative function ω ( 0 (),  0 (), Remark 10.By lemmas and assumptions before, Lemmas 8 and 9 can be derived easily, and then the proof is omitted.

Observer Design.
Define the following filter/estimator: where 3.3.Control Design.In this section, the intergrator backstepping approach will be used to design the control laws  0 () and  1 () subject to  0 ( 0 ) ̸ = 0.The case that the initial condition  0 ( 0 ) = 0 will be treated in Section 4.
0. At this step, control law  0 () will be designed, which is essential to guarantee the effectiveness of the subsequent steps.For the  0 ()-subsystem, choose the control  0 () as follows: where  0 is a constant satisfying  0  0 > 1. Introduce the Lyapunov function candidate  0 = (1/2) 2 0 (), and the time derivative of  0 satisfies where  0 =  0  0 > 1.This indicates that  0 () converges to zero exponentially.
According to the design of control law  0 () in ( 23), it can be computed that where Remark 11.From (26), we know that  is a constant and φ0 ( 0 ()) is a function with respect to  0 ().Moreover, we can conclude that φ0 ( 0 ()) is smooth because  0 ( 0 ()) is a nonnegative smooth function.
Denote  1 =  0 −  − ; we can choose appropriate design parameters   (1 ≤  ≤ ) such that  1 is Hurwitz.Then there exists a positive definite matrix  satisfying  1 +    1  = −, and  is a positive constant.
1.For  1 ()-subsystem in (13), let  1 () =  1 (), and  2 () =  2 () −  1 .Introduce the following Lyapunov functional: where with ℓ 1 ,  2 being positive constants to be designed; Θ1 = Θ 1 − Θ1 , where Θ 1 is an unknown parameter vector to be specified later, and Θ1 is an estimate of Θ 1 .Associated with ( 22) and ( 27), the time derivatives of  1 and Ṽ1 can be calculated, respectively, that For some terms on the right-hand side of (30), the following estimations (32)-(34) should be conducted.Firstly, by Lemma 8 and Young's inequality, we can obtain that there exist positive constants ℓ 1 ,  1 to make the following inequalities hold: where Next, employ Lemma 9 and Young's inequality, and we have where , and  2 is a positive constant.By completing the square, the following estimations are also true: where and Choose the virtual control function  1 and the adaptation law of Θ1 as follows: 2. Introduce the new variable  3 () =  3 ()− 2 , where  2 is regarded as the virtual control input, and take the Lyapunov functional as where Θ2 = Θ 2 − Θ2 , Θ 2 is an unknown parameter vector to be defined later, and Θ2 is an estimate of Θ 2 .Then, combined with ( 20), (37), and (39), we have Using Lemmas 8 and 9 and Young's inequality, the following inequalities hold: By the above inequalities, we get where 1  2 (),  2 ()/4, −( 1 / 1 ) 2 ()]  .By taking the adaptation law Θ2 = Υ 2  2 () and the virtual control function  2 as Abstract and Applied Analysis we can obtain 3. Define that  4 () =  4 () −  3 , where  3 is the virtual control input, and consider the following Lyapunov functional: The time derivative of  3 along the estimator system (20) satisfies By similar conduction method in (42), we have where ℓ 3 > 0 is a scalar.Based on (48), it yields where 1  3 (), −( 2 / 1 ) 2 ()]  .Choose the tuning function  3 Υ 3  3 (), and the virtual control function  3 as follows: Under the virtual control function  3 and the tuning function  3 defined above, the derivative of  3 becomes that Step i (4 ≤  ≤ ).Assume that, at Step i−1, a virtual control function  −1 , a tuning function  −1 , and a Lyapunov functional  −1 have been designed in such a way that Let  +1 () =  +1 ()−  , where   is regarded as the virtual control input, and choose Lyapunov functional as Based on (52), the time derivative of   satisfies Next, we estimate the following terms in the right-hand side of (53) by Lemmas 8 and 9 and Young's inequality as follows: Choosing the virtual control function   as and the tuning function 1   (), −( −1 / 1 ) 2 ()]  .Then, we can show that At the last step ( = ), the true input  1 () will be designed on the basis of the virtual control     and the Lyapunov function  −1 introduced before.
Since  0 (),  1 (),  0 () and the system parameters are all bounded, then G1 , G2 in (63) are also bounded.Employing the convergence of  0 (),  1 (),  1 (), we can get that ()system is globally asymptotically convergent.From the introduced transformations before, it can be deduced that system (1) is also asymptotically convergent.Now, we can express the following theorem.
Theorem 12.For system (1), under Assumptions 1-3, if the control strategies (23) and (58) are applied with an appropriate choice of the design parameters, the global asymptotic stabilization of the closed loop system is achieved for  0 ( 0 ) ̸ = 0.
In the next section, we will deal with the stability analysis of the closed loop as long as the initial condition  0 ( 0 ) is zero.

Simulation Example
In this section, a numerical example will be given to illustrate that the proposed systematic control law design method is effective.Consider the following system: where  0 ,   output.We assume that  0 ( 0 ) ̸ = 0 and make the following estimation for some nonlinear terms in system (67): where  =  (1/2)ln 2  1 + | 2 |.Firstly, we introduce the following transformation: and then the system (67) can be rewritten as ẋ 0 () =  0  0 () +  0 () where  2 =  1  2 , and assume that the sign of  2 is known.Define the invariable that  1 () =  1 (),  2 () =  2 () −  1 .According to the iterative procedure in Section 3, we can design the virtual control function and controller  1 () as

Conclusion
The output-feedback adaptive stabilization was investigated for a class of nonholonomic systems with unknown virtual control coefficients, nonlinear uncertainties, and unknown time delays.In order to overcome the difficulties, we introduce suitable transformation and novel Lyapunov-Krasovskii functionals, and then a recursive technique is given to design the adaptive controller.To make the input-state scaling transformation effective, the switching control strategy is employed to achieve the asymptotic stabilization.