Adaptive Output Feedback Stabilization of Nonholonomic Systems with Nonlinear Parameterization

and Applied Analysis 3 Therefore, x 0 and Θ̃ 0 (or, equivalently, Θ̂ 0 ) are bounded; without loss of generality, we assume |Θ̃ 0 | ≤ M, so we can choose design parameterm 0 in (8) asm 0 > M. Furthermore, we can conclude that x 0 → 0 as t → ∞ by using LaSalle’s Invariant Theorem [26]. Under the control law (7), the solution of x 0 -subsystem can be expressed as x 0 (t) = x 0 (t 0 ) exp(∫ t


Introduction
Control of nonholonomic systems has received a great deal of attention over the last few years.It has been shown in [1] that mechanical systems with nonholonomic constraints such as mobile robots and wheeled vehicle can be either locally or globally converted to the so-called chained form under a coordinate transformation and a control mapping.As a result, the chained form has been used as a canonical form in analysis and control design for nonholonomic systems.Due to the nonsatisfaction of Brockett's necessary condition [2], nonholonomic systems cannot be asymptotically stabilized by stationary continuous state-feedback, although it is controllable.To overcome this difficulty, a number of approaches have been proposed; see the review paper [3] and references therein.The proposed solutions for stabilization nonholonomic systems include discontinuous feedback, time-varying feedback, and hybrid control laws.Using there valid approaches, the robustness issue for the asymptotic and exponential stability properties has been extensively studied [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].
However, it should be noticed that most of these papers were concerned with the systems with linear parameterization.Since nonlinear parameterization is exceptionally difficult to estimate, there are very few reports in literature for adaptive control of nonlinearly parameterized nonholonomic systems.Exceptions include [21][22][23] (and some references therein), where the authors consider that the whole state vector is measurable and adaptive control for chained form systems with nonlinear parameterization using state feedback.
In this paper, the case of only partial state vector being measurable and adaptive control for chained form systems with nonlinear parameterization by output feedback is considered.The contributions of this paper are listed as follows: (i) a new observer design method is proposed, and, based on the observer, the unmeasurable states of the system involved are reconstructed; (ii) using parameter separation technique [24], the parameters nonlinearities are solved and the resulting adaptive regulator is of minimum dimension (1D) independent of the parameter dimension; (iii) output feedback adaptive control based switching strategy is adopted 2 Abstract and Applied Analysis to handle the technical problem of uncontrollability at  0 ( 0 ) = 0, which prevents the finite escape of systems and guarantees that all the states converge to the origin and other signals are bounded.
The rest of this paper is organized as follows.In Section 2 preliminary knowledge and the problem formulation are given.Section 3 presents the state-scaling technique and the backstepping design procedure, while Section 4 provides the switching control strategy and the main result.Section 5 gives a simulation example to illustrate the theoretical finding of this paper.Finally, concluding remarks are proposed in Section 6.

Problem Formulation
In this paper, we present an output feedback adaptive control design procedure for a class of uncertain chained form systems with nonlinear parameterization: where and  ∈  2 are the system state, control input, and system measurable output,   ,  = 0, 1, . . .,  are continuous functions of their arguments, and  ∈   is an unknown constant vector.The function   is called the nonlinear drifts of the system (1).The objective of this paper is to design an output feedback adaptive stabilization control in the form such that all signals of the closed loop system are bounded.Furthermore, the asymptotic regulation of the states is achieved; that is, lim A full characterization of the class of system (1) is given by the following assumption, which will be the base of the coming control design and performance analysis.

Adaptive Output Feedback Control Design
In this section, we focus on designing the control input  1 via output feedback provided that  0 ( 0 ) ̸ = 0.The case where the initial  0 ( 0 ) = 0 will be treated in Section 4. The special structure of the system (1) suggests that we should design the control inputs  0 and  1 in two separate stages.
Let Θ0 = Θ 0 − Θ0 , where Θ0 is an estimate of Θ 0 .Assumption 1 leads us to choose the control law  0 as where  0 and  0 are positive design parameters to be specified later.
Remark 5. Note the control  0 is an uncertain version of Sontag formula [25].It is used to stabilize  0 with uncertainties.Because the particular choice of (8),  0 ( 0 , Θ0 ) ̸ = 0 is guaranteed irregardless of the values of  0 .Hence, 1/ 0 is well defined.
Under the control law (7), the solution of  0 -subsystem can be expressed as which implies where Consequently,  0 does not cross zero for all  ∈ ( 0 , ∞) provided that  0 ( 0 ) ̸ = 0. From the above analysis, we can see the  0 -state in (1) can be globally regulated to zero via  0 in (7) as  → ∞.It is troublesome in controlling the x-subsystem via the control input  1 in the limit (i.e.,  0 = 0), the x-subsystem is uncontrollable.This problem can be avoided by utilizing the following discontinuous state scaling transformation [4]: Under the new z-coordinates, with choice of  0 as in (7), the x-system is transformed into It should be noted that the measurement of state  1 can be obtained if the to-be-designed control  0 is only dependent on output .If  0 () ̸ = 0 for every  ≥ 0, the discontinuous state transformation (15) is applicable.
We design the following observer for the system (16): where  1 , . . .,   are design parameters to be determined later.
The estimation error z =  − ẑ satisfies the dynamical equations: The differential equations ( 18) can be rewritten into the compact form About matrix  defined by (20), there exists the following lemma.
Lemma 6.The eigenvalues of the matrix A defined by (20) can be arbitrarily assigned by a proper selection of the design parameters  1 , . . .,   .
Proof.The proof can be found in [16] and thus omitted here.
In view of (18), (19), and (16), the overall system to be controlled can be expressed as We now turn to the constructive design procedure of the control.
Step 1.This step can be regarded as the initial assignation of the entire design procedure.At this step, we introduce a Lyapunov function for the estimation error z.Define Θ = max{Θ  , Θ 2  } and Θ = Θ − Θ, where Θ is the estimate of Θ.Consider the Lyapunov function where  is the positive definite solution of the Riccati equation Then, taking time-derivation of  0 along the solution of (19), we have By Lemma 7 and the expression of Φ in (20), we have where Φ = ∑  =1  2  ( 0 ,  1 , Θ0 ) is a nonnegative smooth function.Then, for the terms on the right-hand side of (25), by completing the square, we have where  0 is a positive design parameter to be specified later.Substituting (27) into (25) leads to where Step 2. Take  1 =  0 +  2 1 /2 as the Lyapunov function of this step.Then, by (28) we have with the choice of the virtual controller we have where Step i (2 ≤  ≤ ).Suppose at step  − 1 we have designed a set of smooth virtual controllers ẑ * 1 , . . ., ẑ *  defined by with  1 ( 0 ,  1 , Θ) > 0, . . .,  −1 ( 0 ,  1 , . . .,  −1 , Θ) > 0 being smooth, such that where for  ∞  −1 ≥ 0.
Moreover, (44) can be rewritten as follows: By ( 46), we have The last inequality from the following relation where γ =   √ 2 −1 + 1 + ∑  =1 β, ( 0 ,  1 , . . .,   , Θ0 , Θ).Now, it easy to see that the smooth virtual controller renders As  = , the last step, we can construct explicitly a change of coordinates ( 1 , . . .,   ), a positive-definite and proper Lyapunov function   =  −1 +  2  /2, and a smooth controller  * 1 = −    such that Therefore, by choosing the smooth actual control  1 and update law for Θ as we have In Section 4, we will show that an appropriate choice of the design parameters  0 and   guarantees the asymptotic convergence of the state ( 0 , ) of the original system (1) in closed loop with our control laws (7), ( 8), (54), and (55) as long as the initial condition  0 ( 0 ) is nonzero.It will also be shown in the next section how to handle the singular case where  0 ( 0 ) is zero.

Switching Controller and Main Result
In the preceding section, we have given controller design for  0 ( 0 ) ̸ = 0. Without loss of generality, we can assume that  0 = 0. Now, we discuss how to select the control laws  0 and  1 when  0 (0) = 0. Choose  0 as where  0 is given by (7).
Choosing the same Lyapunov function (9), its time derivative is given by which leads to the boundedness of  0 and consequently the bound of Θ0 as well.Consequently,  0 does not escape and  0 (  ) ̸ = 0 for any given finite   > 0. Thus state-scaling for the control design can be carried out.
During the time period [0,   ), using  0 defined in (57), new control law  1 =  * 1 ( 0 , , ) and new update law μ = ] * ( 0 , , ) can be obtained by following the control procedure described above to the original x-system in (1).Then we can conclude that the x-state of (1) cannot blow up during the time period [0,   ).Since (  ) ̸ = 0 at   , we can switch the control input  0 and  1 to ( 7) and (54), respectively.
We are now ready to state the main theorem of our paper.
Theorem 8.Under Assumption 1, if the above switching control strategy is applied to (1) with an appropriate choice of the design parameters   and    , uncertain system (1) is globally regulated at the origin.Furthermore, the estimated parameters are bounded.
Proof.According to the above analysis, it suffices to prove the statement in the case where  0 (0) ̸ = 0. Since we have already proven that lim  → ∞  0 () = 0 and Θ0 is bounded in Section 3, we just need to show that lim  → ∞ () = 0 and Θ is bounded.In this case, as a result of ( 7), ( 11), (54), and (55), from (56), it holds that Choose the design parameters  0 and     so that Therefore, we have which means  and Θ are bounded.Furthermore, we can conclude that  → 0 as  → ∞ by using LaSalle's Invariant Theorem [26].This implies that  → 0 as  → ∞.From the state scaling transformation (15), we conclude that  → 0 as  → ∞.This completes the proof of Theorem 8.
Remark 9.As seen from ( 54) and ( 7), the control law  1 may exhibit extremely large value when  0 ( 0 ) ̸ = 0 is sufficiently small.This is unacceptable from a practical point of view.It is therefore recommended to apply (57) in order to enlarge the initial value of  0 before we appeal to the asymptotically converging controllers ( 7) and (54).

Example
To verify our proposed controller, we consider the following low-dimensional system with parametric uncertainty: where  is unknown constant parameter.Define Θ = 1 +  2 ; obviously, this system satisfies Lemma 4; that is, The purpose is to design  0 and  1 based on only  = ( 0 ,  1 )  such that ( 0 ,  1 ,  2 )  → 0 as  → ∞.
If (0) = 0, controls  0 and  1 are set as in Section 4 in interval [0,   ), such that (  ) ̸ = 0; then we can adopt the controls developed below.Therefore, without loss of generality, we assume that (0) ̸ = 0.For simplicity, in the first subsystem, we can choose the control law  0 =  0  0 and introduce the introduce the state scaling transformation: In new -coordinates, the ( 1 ,  2 )-subsystem of (62) is rewritten as Design observer to reconstruct  1 and  2 of system (62) as follows: Then, the estimation error z = ( 1 − ẑ1 ,  2 − ẑ2 )  satisfies the equation In this case, we have Solving the matrix equation    +  =  2 , we obtain  = ( 0.500 0.500 0.500 0.100 ) .

Conclusions
In this paper, a constructive adaptive output feedback control strategy is presented for a class of nonlinearly parameterized nonholonomic systems with strong nonlinear drifts.To deal with the nonlinear parameterization problem, a parameter separation technique is introduced to transform the nonlinear parameterized nonholonomic system into a linearlike parameterized nonholonomic system.We estimate only Θ, the bound of the unknown parameter rather than the parameter vector .This, in turn, results in a minimum dimension (1D) parameter update law.The controller design is developed by using the integrator backstepping approach    based on observer and parameter estimator.With a switching strategy eliminating the phenomenon of uncontrollability, the system states asymptotically converge to the origin, while other signals remaining bounded is guaranteed.Simulation results have shown the effectiveness and feasibility of the proposed control strategy.