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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 734307, 9 pages
Adaptive Sliding-Mode Tracking Control for a Class of Nonholonomic Mechanical Systems
School of Automation, Southeast University, Nanjing 210096, China
Received 21 April 2013; Accepted 17 June 2013
Academic Editor: Yu Kang
Copyright © 2013 Wei Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper investigates the problem of finite-time tracking control for nonholonomic mechanical systems with affine constraints. The control scheme is provided by flexibly incorporating terminal sliding-mode control with the method of relay switching control and related adaptive technique. The proposed relay switching controller ensures that the output tracking error converges to zero in a finite time. As an application, a boat on a running river is given to show the effectiveness of the control scheme.
In recent decades, sliding-mode control (SMC) has received considerable attention for it is less sensitive to the parameter variations and noise disturbances, and a great number of results have been acquired [1–4]. To get more faster error convergence on the sliding mode, a finite time mechanism, terminal sliding-mode (TSM) control scheme, was presented in [5–7]. Based on it, a series of tracking control problems have been solved in [8–11] and the references therein.
On the other hand, nonholonomic constraints arise in many mechanical systems when there is a rolling or sliding contact, such as wheeled mobile robots, -trailer systems, space robots, underwater vehicles, and multifingered robotic hands. Although considerable effort [11–15] has been made for nonholonomic systems during the last decades, controller design for these systems is still a challenging problem owing to the existence of nonintegrable geometry constraints.
It is worth pointing out that the existing results [12–15] were mainly aimed at the classic nonholonomic linear constraints (i.e., ). However, there are rare results on affine constraints [16, 17] (i.e., ), which are frequently encountered in some mechanical systems, such as a running river with the varying stream, ball on rotating table with invariable angular velocity, and space robot with initial angular momentum. Therefore, researching the tracking problem for such systems is an innovatory and significative work.
This paper, using the terminal sliding-mode technique, investigates the tracking control problem for a class of uncertainty nonholonomic mechanical systems with affine constraints. To achieve the tracking objective, by flexibly using the algebra processing technique, we triumphantly reduce the number of state variables which provide a motion complying with affine constraints. In order to do so with uncertainties, an appropriate adaptive law is established to identify uncertainty parameter. The main contributions of the paper are briefly characterized by the following features.(i) Because of the introduction of affine constraints to mechanical systems, it is difficult to find linearly independent vector fields to cancel the constraint forces in dynamic equation. Hence, a new diffeomorphism transform is presented to deal with it.(ii) Based on the asymptotic tracking idea for uncertain multi-input nonlinear systems, the strategy of terminal sliding-mode control, and related adaptive theory, an adaptive relay switching tracking controller is designed which ensures that the output tracking error converges to zero in a finite time.(iii) As a practical application, a boat on a running river with varying stream is given to illustrate the reasonability of the assumptions and the effectiveness of the control strategy.
The rest of this paper is organized as follows. System description is given in Section 2. The design scheme of the adaptive relay switching controller is addressed in Section 3. Section 4 gives the main results. As for the application, a practical example is considered in Section 5. Section 6 provides some concluding remarks.
2. System Description
2.1. Dynamics Model
In this paper, we consider a class of nonholonomic mechanical systems described by Euler-Lagrangian formulation: where is the generalized coordinates and represent the generalized velocity vector and acceleration vector, respectively; is inertia matrix; represents the vector of centripetal, Coriolis forces; represents the vector of gravitational forces; represents uncertainty of system; denotes the unknown parameter vector; is an input transformation matrix; denotes the vector of constraint forces; is a constraint matrix with full rank; represents the -vector of the generalized control input with ; is a known vector function.
Constraint equation (2) is regarded as affine constraints. When it is imposed on the mechanical system (1), the constraint (generalized reaction) forces are given by where is a Lagrangian multiplier corresponding to nonholonomic affine constraints.
Remark 1. It is worth emphasizing that the system studied in this paper is more general than that in some existing literatures such as [12–15], where dynamic equation satisfies the classical linear constraints. In fact, by taking , (2) transforms to linear constraints, whose tracking problem has been extensively studied in [15, 18–20].
2.2. Reduced Dynamics and State Transformation
This part mainly focuses on reducing the number of state variables which provide motion complying with the affine constraints.
It is easy to find a full-rank matrix satisfying Define ; then (2) can be expressed concisely as Let where satisfies . One can deduce that is a full rank and satisfies According to (5) and (7), we know that there exists an -dimensional vector such that , that is, which implies that . For convenience, define . In view of the relationship (8), the generalized velocity vectors can be written as It is clear that corresponds to the internal state variable.
Substituting (9) into (1), premultiplying on both sides of it, and using , the dynamics of the mechanical system made up by (1) and (2) can be described clearly as where , , , , and . In order to guarantee that all degrees of freedom are actuated independently, we suppose that is full rank.
Remark 2. The aforementioned diffeomorphism transform method differs from the traditional ones in [13, 18–22]. More specifically, when the affine constraints are imposed on the mechanical system, it is difficult to find linearly independent vector fields to proceed with a simple diffeomorphism transformation for canceling the constraint forces in dynamic equations. Hence, we present the forementioned diffeomorphism transform to achieve this goal.
Remark 3. The diffeomorphism transformations consist of (4) and (9), ensure that the transformed system (10) still satisfies constraint equation (2), and possess the practical physical meaning. This can also be confirmed by the practical example in Section 5.
The control objective of this paper can be specified as follows. Given the desired trajectories and , which are assumed to be bounded and should satisfy constraint equation (2), we determine a control law such that all states of the closed-loop system are globally bounded and the output tracking error and its time derivative converge to zero in a finite time.
In order to solve the above tracking problem, we make the following reasonable assumption.
Assumption 1. The matrix is symmetric, positive definite and there exist two known scalars and such that , .
3. Control Design
This subsection will construct a relay switching controller composed of an adaptive TSM controller and an adaptive Pre-TSM controller. For simplicity, sometimes the arguments of functions are dropped in the remainder of the paper.
Step 1 (adaptive TSM controller design). Define ; then system (10) can be expressed as
The following reference model is chosen as that in :
where , , and are constant matrices such that system (12) is stable; is an identity matrix; , , and are measurable and bounded signals. For convenience, define .
Now, we develop the following tracking error system which will be used in the subsequent controller design and stability analysis: We directly get the following equations from (11)~(13): where . To ensure that reaches zero in finite time, one defines a fast switching surface as where , are positive constants and both and are odd and satisfy . Define , . Dynamic equation (16) can be rewritten as
According to the parameter separation technique (Lemma 2.1 in ), for uncertain term in system (1), there exist an unknown constant and a known smooth function , such that . Then, choose a continuously differentiable, positive definite and radially unbounded function as
where is a design parameter and represents parameter estimation error.
According to (9) and the definition of , , we have , where is a known positive smooth function. Taking the time derivative of , using Assumption 1 and substituting (14)–(17) into it result in the following:
Now, if an FTC control law is taken as with the following adaptive law: then the following expression can be obtained from (19):
Remark 4. The FTC controller composed of (20) and (21) may cause the singularity of closed-loop (14)~(21) due to the existence of the term , since may be sufficiently large if is sufficiently small. However, on the sliding surfaces, the singularity does not occur. Since, on the sliding mode, implies , then one can further get This shows that each component of is bounded as is sufficiently small for . Consequently, once the trajectory of arrives on the sliding surfaces, the control law is bounded and does not cause the singularity. However, when moves to the switching surface, singularity may occur. To avoid this phenomenon, we introduce the following controller.
Step 2 (adaptive pre-TSM controller design). Firstly, define as
where is a fixed point on the switching surface and is a sufficiently small constant.
Let us construct an augmented linear system as
where is the controllable gramian matrix with the following form:
Based on linear system theory, under the control law , starting from any initial state vector can be transferred to any given final state at time . Here, let the final state be on the nonlinear switching surface and .
The remaining task is to design the preterminal sliding-mode controller which guarantees that an arbitrary point in the space arrives at in a finite time.
where is a design parameter. In view of Assumption 1, the time derivative of satisfies If we take a pre-terminal controller as
with adaptive law
then (32) reduces to
4. Main Results
Firstly, let us recapitulate how to manipulate the fore-mentioned two adaptive controllers to realize the control objective.(i) According to (29), design a preterminal controller made up by (33) and (34) such that the trajectory of in (14) enters in a given finite time.(ii) Design the TSM controller formed by (20) and (21) such that the trajectory of starting from first reaches switching surface and then moves to zero in a finite time along this surface.
Next, we present the following theorem, which summarizes the main results of this paper.
Theorem 5. Suppose that Assumption 1 holds for the nonholonomic mechanical system described by (1) and (2), then for a desired trajectory satisfying the constraint equation (2), according to the above manipulations (i) and (ii), the following are guaranteed:(a) all states of the closed-loop system are globally bounded;(b) the output tracking errors and converge to zero in a finite time.
Proof . From (35), one knows that is monotonically decreasing, that is, , for all , which results in and . Therefore, it follows that is bounded from (25)~(28). Moreover, the boundedness of , shows that and are bounded, so are in (33) and . In addition, integrating on (35) from to and using the boundedness of , we have . The boundedness of means that is uniformly continuous. According to the Barbalat lemma , one has and . Therefore, there exists a finite time such that , . If one selects , then
This shows that under the action of the pre-terminal control law (33), starting from any initial state vector enters the small neighborhood of after time . On the other hand, (18) and (22) imply that the trajectory of the system globally asymptotically converges to . In the following, we will talk about the convergence rate of dynamic surface . (18) and (22) result in It is equivalent to or Integrating on both sides of (21) and according to the definition of , one has This ensures by appropriately choosing . Hence, one further gets . means that ; without loss of generality, suppose . and can conclude and , respectively. In both cases, one can deduce that tends to zero in a finite time on account of the convergence of . In view of the above analysis, one knows that tends to zero in finite time; that is, the error dynamic reaches the nonlinear switching surface in a finite time.
Next, we prove that the singularity cannot occur by using the above proposed relay switching controller.
Specifically, given a specified initial value , the trajectory of error dynamics can be divided into two stages in which it firstly moves to the switching surface in a certain time interval and then slides along to the origin at a finite time. According to Remark 4, once error arrives to the sliding surface, the singularity may not occur. In the following, one can prove that under control law composed of (20) and (21), the trajectory initiating from the open set cannot escape to infinity. For this purpose, we assume that at some time instant , . By (18) and (22), can be close to zero in a finite time. Since , and is sufficiently small; is very small, which guarantees that is sufficiently smal1. Therefore, by (17), one has
Consequently, is far away from the origin when is away from the origin. This shows that the initial value is away from zero. By solving the inequalities (22) and (41) in time interval and by considering the initial values and , we conclude that reaches zero firstly before becomes very small. As shown previously, on the switching surface , the control signal is bounded. This illustrates that the control law given in (20) and (21) is bounded if the starting state of the trajectories is in a sufficiently small neighborhood of , and in this sense, the singularity is avoided.
Altogether, we consider the trajectories of (14) in time intervals and . In view of the above analysis, as is fixed, is a finite one. Equations (31) and (32) guarantee that arrives at and does not escape to infinite in . When , the controller is switched to the TSM controller under which the trajectory arrives at the switching surface first and then moves along this surface to the origin in a finite time. The TSM controller has been proved to be bounded in time interval and all the signals in the closed loop are bounded on the switching surface . Till now, the theorem is proved completely.
Remark 6. If one adopts a general finite time controller in , the control signals may tend to infinity before the state of the error system reaches the switching surface. For instance, the term in controller (20) is infinity, if are all sufficiently small for . Therefore, one proposed a relay switching control scheme to avoid this phenomenon.
Consider a boat on a running river (see Figure 1). The -axis and -axis denote the transverse direction and the downstream direction of the river, respectively. According to the motion of the boat on the river, one can get the following kinematic equations: where and denote the stream of the river and the speed of the boat, respectively. After some simple calculations, the affine constraints can be obtained as follows: From the above equation, the following can be obtained: The dynamics model of the boat on a running river can be expressed as where is the mass of the boat and is the inertia of the boat; and denote the external resistance, where and are unknown. In this simulation, let , and choose
It follows from the procedure of the aforementioned diffeomorphism transformation that
Then, the original dynamics system can be converted into the following form:
For the given , , and , the desired trajectory satisfies kinematic constraint and diffeomorphism transform with . The control objective, based on the proposed scheme, is to determine an adaptive relay switching control law such that the trajectory follows . According to , the reference model is chosen as
In this simulation, the parameters in the system are selected as , , , , , , , , , , , and . In order to eliminate the effects of the chattering, we replace by . The results of the simulation are shown in Figures 2–9. Specifically, Figures 2 and 3 show that output tracking errors go to at time instant under the action of adaptive Pre-TSM controller. Figures 4 and 5 show that both the adaptive pre-TSM controller and state are bounded. On the other hand, the satisfactory results of TSM design scheme after are presented in Figures 6–9. Figures 6 and 7 show that the errors become zero at approximate time instant . Figures 8 and 9 show that the adaptive TSM controller and tend to the steady state and both of them are bounded. Thereby, the practical simulation example confirms the validity of the proposed algorithm.
This paper studies tracking problem for a class of uncertain nonholonomic mechanical systems based on the idea of terminal sliding-mode control. The adaptive relay switching tracking controller guarantees that output tracking error converges to zero in a finite time and avoids the singularity problem. A practical mechanical model is constructed to confirm the reasonability of the assumption and the effectiveness of the control scheme.
This work was supported by the National Natural Science Foundation of China under Grant 61273091, the Shandong Provincial Natural Science Foundation of China under Grant ZR2011FM033, and the Fundamental Research Funds for the Central Universities under Grant CXLX12_0096.
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