1. Introduction

In the past two decades, underactuated systems have received much attention and some attractive results have been presented in the literature. Systems are said to be underactuated when the control actuators are fewer than number of degrees of freedom. They have more advantages in case of the reduced weight, cost, energy consumption, and the system structure design than actuated ones. Related problems have been presented in many actual systems such as, the crane system [1], underwater vehicles [2], the four-link manipulator [3], the axisymmetric spacecraft [4], hypersonic vehicles [5], the linear quadratic regulator (LQR) [6], the pendubot [7], the inertia-wheel pendulum (IWP) [8-10], and so on.

Dead-zone input nonlinearity is a nonsmooth function that features certain insensitivity for small control inputs which is often encountered in a variety of practical systems such as the four-link PAAA (Passive-Active-Active-Active) manipulator, the single-link flexible joint manipulator, and so on. The work [11] investigated the position-posture control problem of a planar four-link underactuated manipulator. The problem of fuzzy adaptive control design and discretization for a class of nonlinear uncertain systems was studied in [12]. In [13], an observed-based adaptive fuzzy tracking controller has been presented for switched nonlinear systems with dead-zone.

Fuzzy logic systems (FLSs) with appropriate adaptive laws and projection algorithms are introduced to approximate the unknown nonlinear functions appearing in the structure of the underactuated system. A FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. By using Lyapunov-Krasovskii stability theorem, some adaptive laws are presented for fuzzy model and time delay.

The sliding-mode control technique have some advantages such as insensitivity to system parameter variations, invariance to external disturbances, good transient performance, fast response, and so on. Basically, SMC laws comprise two parts: switching controller design and equivalent controller design. The switching control law is employed to lead the system’s states to a given sliding surface and the equivalent control law guarantees the system’s states to stay on the aforementioned sliding surface and converge to zero along the sliding surface. As a result of the underactuated characteristic of the controlled system, at least two sliding surfaces with specific relation are utilized for the controller design. The sliding-mode controller on the basis of the incremental hierarchical structure and aggregated hierarchical structure were used to deal with the under-actuated system. For the incremental hierarchical structure of a sliding-mode controller, one of the subsystems is defined as the first layer sliding surface. Next, the first-layer sliding surface and one of the left states are used to establish the second-layer sliding surface. The process continues till all the subsystems are included.

The key idea of model reference hierarchical sliding mode control is motivated by the concept of model reference sliding mode control. The model reference hierarchical sliding mode control is to combine or integrate model reference and aforementioned hierarchical sliding mode control in such a way that the advantages of both techniques can be realized. In this paper, incremental hierarchical structure sliding-mode controller (IHSSMC) scheme based on reference model is proposed, which tracks the reference model by itself.

In this paper, an adaptive model reference hierarchical sliding mode control method is developed for a class of uncertain underactuated systems with time delay and dead-zone input. A fuzzy logic system with adaptive laws is introduced to approximate the nonlinear functions appearing in the structure of the underactuated system. Choosing an appropriate Lyapunov-Krasovskii function, it is theoretically verified that all the signals in the closed-loop system are uniformly ultimately bounded under our designed adaptive model reference control method. By introducing a sliding-mode controller with incremental hierarchical structure, the errors between the system outputs and model reference outputs are driven onto the sliding surface and kept on the surface afterward.

The rest of this paper is organized as follows. In Section II, the problem statement and basic preliminaries together with the definition of fuzzy logic systems (FLSs) are given. In Section III, the controller design based on the model reference hierarchical sliding mode control method is proposed to cope with the control problem of a class of uncertain underactuated nonlinear time-delay systems with dead-zone input. The stability analysis of the whole system is also verified by Lyapunov-Krasovskii functional. Simulation results are provided in Section IV to demonstrate the advantages and effectiveness of the proposed approaches. Finally, the concluding remarks are gathered in Section V.

2. Problem Statement and Preliminaries

2.1. System Description

Consider a class of single-input-multi-output uncertain underactuated nonlinear systems with unknown dead-zone input and time delay as follows:

(1)

where is the system state vector which is assumed to be available for measurement, and are input and output of the system output, respectively. is the value of time delay. , and are unknown real continuous nonlinear functions, are unknown external bound disturbances. is the nonlinear input function containing a dead-zone. Without loss of generality, we assume that and the following assumptions are made for the controller design:

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

Assumption 2: , , , for , where and are known constants, and is a set given as follows:

Here is a set of weight, and is a positive constant which denotes all state variables’ boundary. is a fixed point, and is a weighted p-norm, which is defined as

If

If for will denote Euclidean norm .

The non-symmetric dead-zone with input and output as shown in the above Fig. 1. is described by

(2)

where, and are parameters and slopes of the dead-zone, respectively.

Figure 1. Dead-zone model

In order to investigate the key features of the dead-zone in the control problems, the following assumptions should be made:

Assumption 3： The dead-zone output is not available to obtain.

Assumption 4： The coefficients , and are unknown.

Assumption 5： There exist known constants such that the unknown dead-zone parameters , are bounded, i.e. , ,

Based on the above assumptions the expression (2) can be represented as

(3)

where can be calculated form (2) and (3) as

(4)

From Assumption 5, we can conclude that is bounded, and satisfies:

(5)

where is an upper bound, which can be chosen as

(6)

The stable model reference of system is given by

(7)

where is the state vector of reference model, is the bounded reference input. is Hurwitz. Let the model reference output vector as . Because the underactuated system is divided into several different subsystems and for the state variables, there is not obvious differential relationship between these subsystems. As the result, comprises n subsystems with the controllable canonical form for the plant.

Thus,

(8)

where for are positive real constants to be chosen.

(9)

is the known and real constant matrix.

Define the dynamic of the tracking error:

(10)

Control objective: Design a controller for (1) such that the system output would track the reference model output vector . Define the vector of the output tracking error as

(11)

where the model reference output vector is defined as

(12)

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:

(13)

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:

(14)

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

(15)

where is the membership function of the fuzzy set.

3. Controller Design and Stability Analysis

From (7), the model reference of system can be rewritten as follows:

(16)

Then the dynamic equation of tracking error can be described by

(17)

Substituting (1) and (16) into (17) yields

(18)

Then the change of output tracking error vector can be expressed as

(19)

Let the suitable sliding surfaces be defined as follows:

(20)

where are positive constants.

Differentiating with respect to time, we have

(21)

where

Assumption 6:

where are unknown positive smooth continuous functions.

According to the equivalent control method, the equivalent control law of the systems can be obtained as:

(22)

Figure 2. Hierarchical structure of sliding surfaces

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

(23)

(24)

(25)

where and are the fuzzy basis vector, and for are the corresponding adjustable parameter vector of each fuzzy logic systems. It is assumed that and belong to compact sets , respectively, which are defined as:

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

where for are bounded in the suitable closed sets , respectively. The parameter estimation errors can be defined as:

(26)

(27)

(28)

and

(29)

where

(30)

(31)

(32)

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

Secondly, we define:

(33)

(34)

where is an estimate of , which is defined as , and be as the estimate of .

Based on the fuzzy logic systems, the equation (22) can be replaced as the following controller:

(35)

Next, the ith-layer sliding surface and its control law can be defined as follows.

(36)

(37)

where for are positive constants; . From the recursive formulas (36), we have

(38)

Here for a given is a constant, and is the switching control of sliding surface can be chosen as.

(39)

where K is a positive constant, and the parameter update laws as follows:

(40)

(41)

(42)

(43)

(44)

where for and 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 [14]. The proposed adaptive laws (40)-(44) can be modified as the following form:

(45)

where is defined as

(46)

(47)

where is defined as

(48)

(49)

where is defined as

(50)

The main result of robust adaptive tracking control scheme is summarized on the following theorem.

Theorem 1: Consider the single-input-multi-output uncertain underactuated system (1). If Assumptions 1-6 are satisfied, then the proposed model reference sliding-mode controller defined by (37) with adaptive laws (40)-(44) guarantees that all signals of closed-loop system are bounded.

Proof: Consider the Lyapunov-Krasovskii function candidate

(51)

Differentiating the Lyapunov function with respect to time, and by the fact we can obtain.

According to (38), and (21), we have

(52)

Substituting (21), and Assumption 6 into (52) yields

(53)

By applying the fuzzy logic system (13) and (26)-(28) into (53), we obtain

(54)

By employing the adaptive laws (40), (42), and (3), we can get

According to (27), the above equation can be rewritten as

(55)

By using the adaptive law (41) and (5) we have

According to (29)-(33) and (43), we obtain

(56)

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

(57)

Differentiating the with respect to time, we can obtain

According to (34) (37) and (44), we have

Using the switching control laws (37), the above equation can be rewritten as

Therefore, the hierarchical sliding surface S is stable, and the all signals of the closed-loop system are bounded based on the proposed controller. This completes the proof.

4. An Example and Simulation Results

In this section, a mass-spring-damper system [15] in the presence of uncertain parameter and exogenous disturbances is considered as our simulation example Fig. 3. The corresponding mathematical model is described as follows:

Figure 3. The mass-spring-damper system

where is the displacement of the mass, is the velocity of the mass, are the spring force, are the friction force, is the body mass, and is the applied force. The structures of spring force and friction force are assumed to be known. The exogenous disturbance is assumed to be , . sec is time delay. In the implementation, six fuzzy sets are defined over interval [-3,3] for and , with labels NB,NM,NS,PS,PM, and PB, and their membership functions are

,

We apply the robust model reference sliding mode control approach in Section 3 to deal with control problem. The system matrices of reference model are given as follows:

and the reference input .

The control object is to maintain the system output to follow the reference model . In the case, the first level sliding surface and , where , the hierarchical sliding surface is constructed as , where . The initial values are chosen as

and the boundary layer

The simulation results are shown in Figs. 4-6. Fig.4 and Fig.5 reveal that the state trajectories, respectively. The control signal is shown in Figs. 6. The simulation results verify the usefulness of the proposed adaptive model reference hierarchical sliding-mode controller.

Figure 4. The trajectories of state and state of reference model

Figure 5. The trajectories of state and state of reference model

Figure 6. The control signal u

5. Conclusions

This paper proposes an adaptive model reference hierarchical sliding mode control scheme for a class of uncertain underactuated systems that guarantees the closed-loop stability in presence of unknown input dead-zone and time delay. Fuzzy logic systems are used to approximate the unknown nonlinear functions by some adaptive laws. Based on the Lyapunov-Krasovskii stability theorem, the proposed incremental hierarchical structure sliding-mode controller not only guarantees the stability of the uncertain nonlinear state time-delay systems with input dead-zone, but also ensures good tracking performance. Finally, some simulation results of a practical example are illustrated to show the effectiveness of the proposed control method.