1. Introduction

During the past decades, underactuated systems have drawn much attention and some significant results have been obtained in the literature. Wu et al. [1] cope with the problem of fault detection for underactuated manipulators. Lai et al. [2] deal with the problem of the stable control strategy for the planar three-link underactuated mechanical system. Recently, for systems with special structure, i.e. strict-feedback form, many research results have been proposed in [3-5]. In this paper, the underactuated systems in strict-feedback form are investigated and some systematic methodologies are proposed to solve the tracking control problem.

In practice, input saturation is one of the most important input constraints which are usually utilized in many industrial control systems. In addition, because saturation is a potential factor degrading the control system performance, it often gives rise to undesirable inaccuracy, or even affects system stability. In recent years, Chen et al. [6] introduced an adaptive neural-network backstepping control approach for a class of uncertain nonlinear systems in the presence of input saturation. Liu et al. [7] investigated the problem of the observer-based neural control method for MIMO pure-feedback non-linear systems with input saturation and disturbances.

A general sliding-mode control (SMC) is established in this paper for a class of uncertain underactuated systems. As a kind of highly robust variable structural control method, the sliding-mode controller is able to respond quickly, insensitivity to systemic parameters, good transient performance, external disturbance, and so on. Due to the underactuated characteristic of the controlled system, at least two sliding surfaces with specific relation are employed for the controller design. For the aggregated hierarchical structure of a sliding-mode controller, the underactuated system is segmented into several subsystems. For each part, we design a first-layer sliding surface. Then, the first-layer sliding surfaces are utilized to construct the second-layer sliding surface. By theoretical analysis, the conclusion is made that all sliding surfaces of the SMC structure is asymptotically stable.

Fuzzy logic systems are introduced to approximate unknown non-linear functions. An adaptive fuzzy control technique for the problem of time delay is verified by Lyapunov-Krasovskii function. Long et al. [8] investigated the problem of adaptive fuzzy output-feedback control for switched uncertain non-linear systems. In [9], Li et al. introduced an adaptive fuzzy output-feedback stabilization controller to treat the problem for a class of switched nonstrict-feedback nonlinear systems.

The main contribution of this paper is that the proposed hierarchical sliding mode control method deals with the problem of tracking performance and stability for a class of underactuated systems in strict-feedback form with input saturation and time delay. Based on sliding mode control technique and the concept of hierarchical design, an adaptive fuzzy hierarchical sliding mode controller is designed for the underactuated system by using fuzzy logic systems to approximate uncertain functions. The fuzzy state observer is employed to estimate the unavailable state for measurement. Choosing an appropriate Lyapunov-Krasovskii function, it is theoretically ensured that all the signals in the closed-loop system are uniformly ultimately bounded and obtain good tracking performance under our designed controller.

The remainder of the paper is organized as follows. Section II describes the problem formulation and preliminaries together with the definition of fuzzy logic systems (FLSs) are given. In Section III, adaptive fuzzy hierarchical sliding mode controller and observer are designed for a class of uncertain underactuated nonlinear time-delay systems with saturation input and the stability analysis of the whole system is also verified by Lyapunov-Krasovskii functional. Some computer simulation results of a practical example are demonstrated to show the effectiveness of the proposed approach in Section IV. Finally, the conclusion is drawn in Section V.

2. Problem Statement and Preliminaries

2.1. Problem Statement

Consider a class of single-input-multi-output uncertain underactuated nonlinear systems in strict-feedback form with input saturation expressed as follows:

(1)

Eq. (1) can be rewritten as

(2)

where

where is the system state vector which is assumed to be unavailable for measurement, and are input and output of the system, respectively. is the value of time delay. are unknown real continuous nonlinear functions where , for and are unknown external bound disturbances. denotes the nonlinear saturation characteristic. Without loss of generality, the following assumptions are made for the controller design:

Assumption 1: The time delay is a fixed and known constant.

Assumption 2: for all x, where are known positive constants.

In order to deal with the control constraints for convenience, the saturation function can be rewritten as

(3)

where be expressed as

(4)

Assumption 3: ,

where h is an unknown constant.

Control Objective: Design a controller for (1) such that the system output would track the desired output vector , where is the bounded desired signal and contains the finite derivative up to the second order. Define the vector of the output tracking error as

(5)

2.2. Description of Fuzzy Logic Systems

The fuzzy logic system performs a mapping from to. Let where , . The fuzzy rule base consists of a collection of fuzzy IF-THEN rules:

(6)

in which and are the input and output of the fuzzy logic system, and are fuzzy sets in and , respectively. The fuzzifier maps a crisp point into a fuzzy set in . The fuzzy inference engine performs a mapping from fuzzy sets in to fuzzy sets in , based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in to a crisp point in . The fuzzy systems with center-average defuzzifier, product inference and singleton fuzzifier are of the following form:

(7)

where with each variable as the point at which the fuzzy membership function of achieves the maximum value and with each variable as the fuzzy basis function defined as

(8)

where is the membership function of the fuzzy set.

3. Controller Design and Stability Analysis

First, let the unknown nonlinear functions can be approximated, over a compact set , by the fuzzy logic systems as follows:

(9)

where is the fuzzy basis vector, is the corresponding adjustable parameter vectors of each fuzzy logic systems.

Owing to the unavailable states of the system in many practical systems, the fuzzy logic system (9) is not used to control nonlinear systems whose states are not obtained for measurement. Therefore, an observer is employed to estimate the unavailable states. Now, let be defined as the estimates of . Then, we can obtain the following fuzzy logic systems as

(10)

In order to estimate the state, we design the observer for system (2) as follows:

(11)

where

L is the observer gain matrix to guarantee the characteristic polynomial of to be Hurwitz. Let us define the estimation error vector as and , then by (2) and (10), we obtain

(12)

It is assumed that and belong to compact sets , respectively, which are defined as

(13)

(14)

(15)

where , , and are the designed parameters, and is the number of fuzzy inference rules. Let us define the optimal parameter vector as follows:

(16)

where is bounded in the suitable closed set .

The parameter estimation errors can be defined as

(17)

and

(18)

is the minimum approximation errors, which correspond to approximation errors obtained when optimal parameters are used.

Applying (17) and (18) into (12), we obtain

(19)

The output error dynamics of (19) can be expressed as follows:

(20)

where

(21)

and s denotes the complex Laplace transform variable. As has been discussed, we could not obtain all elements of , because not all states of the system are available for measurement. Hence, we could not obtain all elements of . We will employ the state variable filters [11] to cope with this problem. First, we choose a stable filter as the following form:

(22)

And

(23)

Introducing (23) into (20), we can obtain the steady-state equation

(24)

Define a set of state variable filters , thus,

(25)

The corresponding filtered signals are defined as follows:

(26)

(27)

(28)

(29)

and

(30)

Then, Eq. (19) can be rewritten as follows:

(31)

where

We define

(32)

and

(33)

where and are the estimated of and , respectively.where

(34)

Based on Lyapunov stable theorem, we can obtain the robust compensation term and the parameter update laws as follows:

(35)

(36)

(37)

(38)

where are positive constants.

Remark 1: Without loss of generality, the adaptive laws used in this paper are assumed that the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets. If the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, we have to use the projection algorithm to modify the adaptive laws such that the parameter vectors will remain inside of the constraint sets. Readers can refer to reference [10]. The proposed adaptive law (35)-(38) can be modified as the following form:

(39)

where is defined as

(40)

The main result of robust adaptive fuzzy observer scheme is summarized on the following theorem.

Theorem 1: Consider the single-input-multi-output uncertain underactuated system (1). The robust adaptive fuzzy observer is defined by (10) with adaptation laws given by (35)-(38). For the given positive definite matrix , if there exist symmetric positive definite matrix such that the following Lyapunov equation

(41)

is satisfied, then all the closed-loop signals are bound, and the estimation errors converge to a neighborhood of zero.

Proof. Consider the Lyapunov function candidate

(42)

(43)

(44)

By the time derivative of and the facts we obtain

(45)

Applying Assumption 3 and (34) yields

(46)

By employing (35)-(38), we can get

(47)

Then using the robust compensation term as shown in (35), the above equation can be rewritten as

(48)

According to (41), it can be easily shown that

(49)

Therefore, it can be concluded that from (49), and the estimation errors of the closed-loop system converges asymptotically to a neighborhood of zero based on Lyapunov synthesis approach. This completes the proof.

Next, we design the observer-based incremental sliding mode controller. Define the sliding surfaces as follows:

(50)

where are positive constants.

Differentiating with respect to time and from (5), we have

(51)

Then, the second-level sliding surface can be defined as follows:

(52)

where , for i=1,2,…,n, are sliding mode parameters which maybe remain constant or change according to different conditions.

Based on the fuzzy logic systems and the states of the system are not obtained for measurement, the equivalent control law can be obtained as

(53)

and can be obtained by backward from .

Then we define:

(54)

where

(55)

for a positive constant .

Theorem 2: Consider the single-input-multi-output underactuated system (1). If Assumptions 1-3 are satisfied, then the proposed adaptive fuzzy sliding mode controller defined by (54) with adaptive laws (35)-(38) guarantees that all signals of the closed-loop system are bounded, and the output tracking errors converge to a neighborhood of zero. If the choice of , which ensures that , satisfies

(56)

and such that , for i=1,2,…,n-1 and are the same sign. Then, the first-level sliding surfaces are also asymptotically stable.

Proof:

Consider the Lyapunov function candidate Differentiating the and with respect to time, we can obtain

(57)

(58)

With the use of for scalars a and b, we obtain

(59)

According to (58), (59) and , we have

(60)

By introducing (3) into (60), we obtain

(61)

since according to the density property of a real number [12], there exists a constant that satisfies

(62)

(63)

Then substituting the equivalent control in (53) into (63), we have

(64)

Using the switching control law (53), one can obtain

(65)

Therefore, it can be concluded that from (65), and all the signals of the closed-loop system converge asymptotically to a neighborhood of zero based on the Lyapunov synthesis approach. This completes the proof.

4. An Example and Simulation Results

In this section, a mass-spring-damper system [8] in the presence of uncertain parameter and exogenous disturbances is considered as our simulation example as shown in Fig. 1, and described by the following form:

(66)

and are the displacement of the mass, are the velocity of the mass, and are the spring force, and are the friction force. are the body mass, and is the force of saturation. The disturbances are assumed to be and .

Figure 1. The mass-spring-damper system

In the implementation, six fuzzy sets are defined over interval [-3, 3] for , with labels NB, NM, NS, PS, PM, and PB, and their membership functions are

In this case, the control objective is to maintain the system output to follow the reference signal and . First, we select the observer gain matrix as . The sliding surfaces are selected as and , where, the second-level sliding surface is constructed as , where = 1.1 and = 2. The initial values are chosen as ,, , and . The other parameters are selected as The simulation results are shown in Figs. 2-10. Fig. 2 and Fig. 3 show the trajectories of the system states and desired outputs. Figs. 4-5 show the trajectories of the system states and their estimation states. The performances of sliding dynamics are shown in Figs. 6-8. Figs. 9-10 demonstrate the trajectories of the system functions. From the simulation results, the tracking error and the estimation error converge asymptotically to a neighborhood of zero, respectively. The simulation results verify the usefulness of the proposed robust adaptive fuzzy sliding mode controller. Moreover, the unknown system functions can be approximated well based on the proposed adaptive laws.

Remark 2: In this paper, the time delay, the state estimator, and input saturation are considered simultaneously, which are different from the previous literature [6] and [8].

Figure 2. The outputs

Figure 3. The outputs

Figure 4. The trajectories of

Figure 5. The trajectories of

Figure 6. The first-level sliding surface

Figure 7. The first-level sliding surface

Figure 8. The second-level sliding surface

Figure 9. The trajectories of

Figure 10. The trajectories of

5. Conclusions

The main contribution of this paper is to develop the hierarchical sliding mode control method to tackle the problem of tracking performance and stability analysis for a class of uncertain underactuated systems in strict-feedback form with input saturation and time delay. Based on sliding mode control technique and the concept of hierarchical design, the unknown system functions can be approximated by the fuzzy logic systems with some adaptive laws. Also, the fuzzy state observer is employed to estimate the unavailable state for measurement. Moreover, choosing an appropriate Lyapunov-Krasovskii function, it is theoretically ensured that all the signals in the closed-loop system are uniformly ultimately bounded and the tracking performance can be achieved under our proposed control approach.