#### Abstract

This paper investigates the problem of adaptive output feedback stabilization for a class of nonholonomic systems with nonlinear parameterization and strong nonlinear drifts. A parameter separation technique is introduced to transform nonlinearly parameterized system into a linear-like parameterized system. Then, by using the integrator backstepping approach based on observer and parameter estimator, a constructive design procedure for output feedback adaptive control is given. And a switching strategy is developed to eliminate the phenomenon of uncontrollability. It is shown that, under some conditions, the proposed controller can guarantee that all the system states globally converge to the origin, while other signals remain bounded. An illustrative example is also provided to demonstrate the effectiveness of the proposed scheme.

#### 1. 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 [4–20].

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 [21–23] (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 to handle the technical problem of uncontrollability at , 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.

#### 2. 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 are the system state, control input, and system measurable output, , 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, .

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.

*Assumption 1. *For , there exists a smooth nonnegative function such that

For , there are smooth nonnegative function such that

*Remark 2. *Assumptions 1 implies that and , that is, the origin is the equilibrium point of system (1).

Lemma 3 (see [24]). *For any real-valued continuous function , where , there are smooth scalar functions and , such that
*

By Assumption 1 and Lemma 3, we easily obtain the following lemma.

Lemma 4. *For , there are smooth nonnegative function and unknown constant such that
*

#### 3. Adaptive Output Feedback Control Design

In this section, we focus on designing the control input via output feedback provided that . The case where the initial will be treated in Section 4. The special structure of the system (1) suggests that we should design the control inputs and in two separate stages.

Let , where is an estimate of . Assumption 1 leads us to choose the control law as where and are positive design parameters to be specified later.

*Remark 5. *Note the control is an uncertain version of Sontag formula [25]. It is used to stabilize with uncertainties. Because the particular choice of (8), is guaranteed irregardless of the values of . Hence, is well defined.

Consider the Lyapunov function candidate

From (7) and (8) and Assumption 1, we have

Choose the adaptation law as renders

Therefore, and (or, equivalently, ) are bounded; without loss of generality, we assume , so we can choose design parameter in (8) as . Furthermore, we can conclude that as by using LaSalle’s Invariant Theorem [26].

Under the control law (7), the solution of -subsystem can be expressed as which implies where .

Consequently, does not cross zero for all provided that .

From the above analysis, we can see the -state in (1) can be globally regulated to zero via in (7) as . It is troublesome in controlling the* x*-subsystem via the control input in the limit (i.e., ), 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 as in (7), the* x*-system is transformed into
It should be noted that the measurement of state can be obtained if the to-be-designed control is only dependent on output . If for every , the discontinuous state transformation (15) is applicable.

We design the following observer for the system (16): where are design parameters to be determined later.

The estimation error satisfies the dynamical equations: The differential equations (18) can be rewritten into the compact form where 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 .*

*Proof. *The proof can be found in [16] and thus omitted here.

Lemma 7. *By Assumption 1 and Lemma 4, we easily obtain that, for every , there exists smooth nonnegative function such that
**
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 . Define and , where is the estimate of . Consider the Lyapunov function
where is the positive definite solution of the Riccati equation

Then, taking time-derivation of along the solution of (19), we have
By Lemma 7 and the expression of in (20), we have
where is a nonnegative smooth function. Then, for the terms on the right-hand side of (25), by completing the square, we have
where is a positive design parameter to be specified later.

Substituting (27) into (25) leads to
where .

*Step 2. *Take as the Lyapunov function of this step. Then, by (28) we have
with the choice of the virtual controller
we have
where and .

*Step i (2 ≤ ≤ ). *Suppose at step we have designed a set of smooth virtual controllers defined by
with being smooth, such that
where
for .

We intend to establish a similar property for -subsystem. Consider the Lyapunov function Clearly

Since is a smooth function and satisfies therefore there exist some continuous functions such that

After lengthy but simple calculations based on the completion of squares, there is a smooth nonnegative function such that where is a positive design parameter to be specified later.

Using (38) and Lemma 4, we have where is a smooth function.

By Lemma 7, there is a smooth function such that

Based on the completion of squares, it is deduced that there is a smooth function satisfying since for a smooth function .

Substituting (39), (42), and (43) into (36) gives Define Using (34), it is not difficult to verify that for a .

Moreover, (44) can be rewritten as follows: By (46), we have The last inequality from the following relation where are smooth functions and , is a smooth function.

Putting (47) and (48) together, we arrive at where

Now, it easy to see that the smooth virtual controller renders

As , the last step, we can construct explicitly a change of coordinates , a positive-definite and proper Lyapunov function , and a smooth controller such that

Therefore, by choosing the smooth actual control and update law for as we have

In Section 4, we will show that an appropriate choice of the design parameters and guarantees the asymptotic convergence of the state of the original system (1) in closed loop with our control laws (7), (8), (54), and (55) as long as the initial condition is nonzero. It will also be shown in the next section how to handle the singular case where is zero.

#### 4. Switching Controller and Main Result

In the preceding section, we have given controller design for . Without loss of generality, we can assume that . Now, we discuss how to select the control laws and when . Choose as where is given by (7).

Choosing the same Lyapunov function (9), its time derivative is given by which leads to the boundedness of and consequently the bound of as well. Consequently, does not escape and for any given finite . Thus state-scaling for the control design can be carried out.

During the time period , using defined in (57), new control law and new update law 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 . Since at , we can switch the control input and 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 .

Since we have already proven that and is bounded in Section 3, we just need to show that and is bounded. In this case, as a result of (7), (11), (54), and (55), from (56), it holds that
Choose the design parameters and so that
Therefore, we have
which means and are bounded. Furthermore, we can conclude that as by using LaSalle’s Invariant Theorem [26]. This implies that as . From the state scaling transformation (15), we conclude that as . This completes the proof of Theorem 8.

*Remark 9. *As seen from (54) and (7), the control law may exhibit extremely large value when 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 before we appeal to the asymptotically converging controllers (7) and (54).

#### 5. Example

To verify our proposed controller, we consider the following low-dimensional system with parametric uncertainty: where is unknown constant parameter. Define ; obviously, this system satisfies Lemma 4; that is, . The purpose is to design and based on only such that as .

If , controls and are set as in Section 4 in interval , such that ; then we can adopt the controls developed below. Therefore, without loss of generality, we assume that .

For simplicity, in the first subsystem, we can choose the control law and introduce the introduce the state scaling transformation: In new -coordinates, the -subsystem of (62) is rewritten as Design observer to reconstruct and of system (62) as follows: Then, the estimation error satisfies the equation In this case, we have

Solving the matrix equation , we obtain The eigenvalues of are , and thus .

Using the method given in Section 3, we can design the adaptive controller as follows:

The unknown parameter is assumed to be . In simulation, the design parameters are chosen as and . The simulation results for initial conditions and are shown in Figures 1–4. From the figures, it is clear to see all the closed-loop system states are regulated to zero and the parameter estimates are bounded as proven in Theorem 8.

#### 6. 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 linear-like parameterized nonholonomic system. We estimate only , 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper. This work has been supported in part by National Nature Science Foundation of China under Grant no. 61073065 and the Key Program of Science Technology Research of Education Department of Henan Province under Grants nos. 13A120016, 14A520003.