Abstract
This paper investigates the problem of output feedback adaptive stabilization control design for a class of nonholonomic chained systems with uncertainties, involving virtual control coefficients, unknown nonlinear parameters, and unknown time delays. The objective is to design a robust nonlinear output-feedback switching controller, which can guarantee the stabilization of the closed loop systems. An observer and an estimator are employed for states and parameters estimates, respectively. A constructive controller design procedure is proposed by applying input-state scaling transformation, parameter separation technique, and backstepping recursive approach. Simulation results are provided to show the effectiveness of the proposed method.
1. 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 [4–7]; 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 [8–11].
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 [8–25] 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 [15–19], while output feedback controller design in [20–24]. 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.
2. Problem Formulation and Preliminaries
In this paper, we deal with a class of nonholonomic systems described by where , , and are system states, control input, and measurable output, respectively; is an unknown parameter vector; (known) and (unknown) denote the possible modeling error and neglected dynamics; are known modeled dynamics, which contain output delays; are unknown constants, and referred to the respective virtual control coefficients.
In this paper, we make the following assumptions on the virtual control directions and nonlinear functions in system (1).
Assumption 1. is a known constant and the sign of is known, where .
Assumption 2. There exist known smooth nonnegative functions and such that for all .
Assumption 3. For every , the nonlinear function satisfies inequality 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 and the dynamics satisfy , but also the modeled dynamics do not exist. In [22], the virtual control coefficients and time delays have not been considered, and the expression is also required. While and and unknown parameters are not existent, system (1) degenerates to the one studied in [21]. When , together with , system (1) becomes the considered system in [23].
Remark 5. Note that here we only use the sign of without any knowledge of individual virtual control direction . Moreover, Assumptions 2 and 3 are imposed on the nonlinear functions and , respectively. In fact, if the modeled dynamics do not involve time delays, inequality (3) is reduced into 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 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.
Lemma 6 (see [27]). For any real-valued continuous function , where , there are smooth functions such that
Lemma 7 (see [19]). For any continuous function there exist two strictly positive real numerates and such that the unique solution of the following matrix differential equation: satisfies .
By Lemma 6 and Assumption 1, we know that there exist smooth functions , and such that Furthermore, we denote ; then it yields
3. Output Feedback Adaptive Stabilization Control Design
In this paper, we design control laws and separately to globally asymptotically stabilize the system (1). According to the structure of system (1), we can see that when converges to zero, will be uncontrollable. A widely used method to design control law 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.
3.1. State Coordinate Transformation
Firstly, we design the coordinate transformation as follows: where and . Then, the system (1) can be transformed into
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 and to asymptotically stabilize the states and , respectively. Rewrite system (13) in the compact form where with
In order to obtain the estimation for the nonlinear functions and , the following lemmas are given.
Lemma 8. For every , there exists smooth nonnegative function such that
Lemma 9. For every , there exist smooth nonnegative functions such that
Remark 10. By lemmas and assumptions before, Lemmas 8 and 9 can be derived easily, and then the proof is omitted.
3.2. Observer Design
Define the following filter/estimator: where , , , and are design parameters to be determined later. Let ; then, the estimation error and the newly defined parameter satisfy the dynamical equations
3.3. Control Design
In this section, the intergrator backstepping approach will be used to design the control laws and subject to . The case that the initial condition will be treated in Section 4.
Step 0. At this step, control law will be designed, which is essential to guarantee the effectiveness of the subsequent steps. For the -subsystem, choose the control as follows: where is a constant satisfying . Introduce the Lyapunov function candidate , and the time derivative of satisfies where . This indicates that converges to zero exponentially.
Since is a smooth function, then there exist a constant , such that for . Therefore, the following inequality is true with : which implies that when , the state converges to zero with a rate less than a certain constant . It is which does not become zero in any time instant. Therefore, the adopted input-state scaling discontinuous transformation in (12) is effective.
According to the design of control law in (23), it can be computed that where and − + .
Remark 11. From (26), we know that is a constant and is a function with respect to . Moreover, we can conclude that is smooth because is a nonnegative smooth function.
Denote ; we can choose appropriate design parameters such that is Hurwitz. Then there exists a positive definite matrix satisfying , and is a positive constant.
Step 1. For -subsystem in (13), let , and . Introduce the following Lyapunov functional: where with being positive constants to be designed; , where is an unknown parameter vector to be specified later, and is an estimate of .
Associated with (22) and (27), the time derivatives of and 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 to make the following inequalities hold: where . Next, employ Lemma 9 and Young’s inequality, and we have where , and is a positive constant.
By completing the square, the following estimations are also true:
Substitute (31)–(34) into , it yields where , and with
Choose the virtual control function and the adaptation law of as follows:
Notice that , then it follows from (35)–(38) that
Step 2. Introduce the new variable , where is regarded as the virtual control input, and take the Lyapunov functional as where , is an unknown parameter vector to be defined later, and is an estimate of . 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 and . By taking the adaptation law and the virtual control function as we can obtain
Step 3. Define that , where is the virtual control input, and consider the following Lyapunov functional:
The time derivative of along the estimator system (20) satisfies
By similar conduction method in (42), we have where is a scalar. Based on (48), it yields where and . Choose the tuning function , and the virtual control function as follows:
Under the virtual control function and the tuning function defined above, the derivative of becomes that
Step i (). Assume that, at Step i−1, a virtual control function , a tuning function , and a Lyapunov functional have been designed in such a way that
Let , 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 with . Then, we can show that
At the last step , the true input will be designed on the basis of the virtual control and the Lyapunov function introduced before.
The actual control input can be designed as and the update law with and . Eventually, it can be achieved that
3.4. Stability Analysis
Notice that tends to zero as converges to origin, and in (59) are positive design parameters. Therefore, by an appropriate parameter choice, there exist positive constants such that
It can be seen that are bounded. Since and are unknown bounded parameters, are bounded. According to estimator equations (19)–(21), it can be deduced that the boundedness of guarantees the boundedness of , and then and are also bounded. By similar analysis, we can conclude that all signals of the closed loop system are bounded.
By LaSalle invariant Theorem, it further achieves that as . By the controller design procedure, we get that asymptotically tend to zero. Then, the definitions and show the asymptotical convergence of and . Finally, from the transformations (10) and (12), we know , which indicates that the states asymptotically converge to zero with the initial condition .
For purposes of analysis, we can rewrite the system (14) as follows:
To solve the above differential equation, we have
Notice that is , and tends to zero as , then by Lemmas 8 and 9, there exist constants such that where is a nonnegative smooth function of , and is a nonnegative smooth function of .
Since and the system parameters are all bounded, then in (63) are also bounded. Employing the convergence of , 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 .
In the next section, we will deal with the stability analysis of the closed loop as long as the initial condition is zero.
4. Switching Controller
Several switching controllers have been proposed in some existing literatures. As well known, the choice of a constant feedback for may lead to a finite escape. In this note, the following switching category can be designed for the stabilization of system (1) with the initial sate . Choosing controller as where and are constants.
Since , then with can be deduced then during the initial small time period, is increasing and satisfies .
Choose that satisfy
Obviously, is increasing when . When , choose the controller , and the controller can be designed according to the simple nonlinear backstepping iterative approach. Since , at , we switch the control laws and into (23) and (58), respectively.
Theorem 13. For system (1), under Assumptions 1–3, if above switching control strategy is applied with an appropriate choice of the design parameters, then the closed-loop system is globally asymptotic regulated at the origin for .
5. 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 are virtual control directions with unknown and known, and the sign of is also known. are unknown bounded parameters. Next, we consider to design the controller and to asymptotically stabilize system (67) by the measurable output. We assume that and make the following estimation for some nonlinear terms in system (67): where .
Firstly, we introduce the following transformation: and then the system (67) can be rewritten as where , and assume that the sign of is known.
Next, make the following input scaling transformation for -system: and then the transformed system is where
Design the following controller : and then can be calculated as follows:
For system (72), constructing the following estimator: where , , and . The design of can guarantee that is . It is further achieved that there exists plosive definite matrix satisfying , in which is a constant. Denote and , and then the observation error and parameter invariable satisfy
Define the invariable that . According to the iterative procedure in Section 3, we can design the virtual control function and controller as where
The adaption laws of the parameter invariable in controller are chosen as
For simulation use, we pick the unknown parameters . In addition, we take the other controller design parameters as , , , , , , , . Moreover, The initial state condition is . Simulation results are shown in Figures 1, 2, 3, and 4. It is obvious that the states and control input converge to zero, and the parameters estimation invariable tend to constants.
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6. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publishing of this paper.
Acknowledgments
This work is partially supported by National Natural Science Foundation (U1304620,61273091,61374079), Basic and Frontier Technologies Research Program of Henan Province (122300410279), and Doctoral Fund of Zhengzhou University of Light Industry (201BSJJ006).