Abstract
In this paper, advanced interval type-2 fuzzy sliding mode control (AIT2FSMC) for robot manipulator is proposed. The proposed AIT2FSMC is a combination of interval type-2 fuzzy system and sliding mode control. For resembling a feedback linearization (FL) control law, interval type-2 fuzzy system is designed. For compensating the approximation error between the FL control law and interval type-2 fuzzy system, sliding mode controller is designed, respectively. The tuning algorithms are derived in the sense of Lyapunov stability theorem. Two-link rigid robot manipulator with nonlinearity is used to test and the simulation results are presented to show the effectiveness of the proposed method that can control unknown system well.
1. Introduction
In control engineering, the design of robust controller for a class of uncertain nonlinear multiple-input multiple-output (MIMO) systems remains one of the most challenging tasks. When MIMO systems are nonlinear and uncertain, their control problem becomes more challenging.
Conventional control theory is well suited to applications, where the control inputs can be generated based on analytical model [1, 2]. Sliding mode control (SMC), which is based on the theory of variable structure systems (VSS), has been widely applied to robust control of nonlinear systems [3–5]. SMC performs well in trajectory tracking of some nonlinear systems. The SMC employs a discontinuous control law to drive the state trajectory toward a specified sliding surface and maintain its motion along the sliding surface in the state space. Hung et al. [3] have made a comprehensive survey of the VSS theory. The dynamic performance of the SMC system has been confirmed as an effective robust control approach with respect to system uncertainties and unknown disturbance when the system trajectories belong to predetermined sliding surface [4].
Although the SMC performs well in the nonlinear systems, it suffers from some difficulties. First, due to the highly coupled nonlinear and uncertain dynamics, it is generally difficult or even impossible for many physical systems to obtain accurate mathematical models. Secondly, to operate effectively in the sliding surface, the SMC requires instantaneous change of the control input without sacrificing the robustness against the model uncertainties and external disturbances. The discontinuity in the control action becomes the cause of chattering, which is undesirable in most applications [6]. In the practical implementation, the chattering may cause an unnecessarily large control signal as the system uncertainties are large and may damage system components such as actuators. Thus, the chattering has to be eliminated or alleviated as much as possible. Finally, it is difficult to directly extend the SMC design into a multiple-input multiple-output (MIMO) system, especially when the coupling among the subsystems is unknown.
During the last two decades fuzzy logic system (FLS) has been a dominant topic in intelligent systems research or control community. Because the FLS provide a systematic and efficient framework to incorporate linguistic fuzzy information from human expert, it is particularly suitable for those systems with uncertain or complex dynamics. Owing to universal approximation capability [7] of fuzzy system, many FLS schemes have been developed for handling nonlinear systems, especially in the presence of incomplete knowledge of the system [8, 9].
Some researchers applied fuzzy system to sliding mode control to improve the performance of SMC. The fuzzy sliding mode control (FSMC) forms the equivalent control of SMC. By employing the FLS, the set of linearized mathematical model can be integrated into a global model that is equivalent to the nonlinear system [10, 11].
As an extension of the well-known ordinary fuzzy set (type-1 fuzzy sets), the concept of type-2 fuzzy sets (T2FS) was first introduced by Zadeh [12]. The sets are fuzzy sets whose membership grades themselves are type-1 fuzzy sets. They are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set. They are useful for incorporating uncertainties [13].
In this paper, we propose a novel advanced interval type-2 fuzzy sliding mode control (AIT2FSMC) for a class of uncertain nonlinear MIMO systems. To inherit the strength of these two methods, we combine IT2FLS and SMC into one methodology. The AIT2FSMC system is comprised of a fuzzy control design and a hitting control design. For resembling a feedback linearization (FL) control law, IT2 fuzzy system is designed. For compensating the approximation error between the FL control law and IT2 fuzzy system, sliding mode controller is designed, respectively.
The tuning algorithms are derived in the sense of Lyapunov stability theorem. The two-link robot manipulator is used to test the proposed method and the simulation results show the AIT2FSMC can control the unknown system well.
The organization of this paper is as follows. Problem formulation and notation are presented in Section 2. In Section 3, IT2FLS is briefly introduced. Section 4 describes the design process and the stability analysis of AIT2FSMC. In Section 5, the simulation results are presented to show the effectiveness of the proposed control for a two-link robot manipulator. Finally, conclusions are given in Section 6.
2. Notation and Problem Formulation
In this section, we present the problem formulation for a class of MIMO nonlinear dynamic systems. Consider the following class of MIMO nonlinear dynamic systems: where is the fully measurable state vector and , is the control input vector, is the output vector, and , are continuous nonlinear functions, and , are continuous nonlinear functions.
Let us denote
Then, system (1) can be rewritten in the following compact form:
The control problem is to design a control law which assures that the system tracks a -dimensional desired vector , which belongs to a class of continuous functions on . In this paper, we make the following assumption.
Assumption 1. The matrix is positive definite; then there exists , such that , with being an identity matrix. In the following may be known or not.
Although this assumption restricts the considered class of MIMO nonlinear systems, many physical systems, such as robotic systems [5], fulfill the above property.
Assumption 2. The desired trajectory , , is a known bounded function of time with bounded known derivatives, and is assumed to be -times differentiable.
Let us define the tracking error asand the sliding surfaces as
The time derivatives of the sliding surfaces can be written aswhere are given as follows:where
Denote
Then, (6) can be written in the compact form
If the nonlinear functions and are known, one can use a sliding mode controller. When the closed loop system is in the sliding mode, it satisfies , and then the traditional sliding mode control law is obtained by the following equation: where is an equivalent control law and is a hitting control law and with for . Using (10) and (11), we can obtain the following equation:
Multiplying to (12) gives
Let us consider the following Lyapunov function candidate:whose time derivative is given by
With (13), (15) can be reexpressed aswhich implies that as Therefore, and all its derivatives up to converge to zero [5].
According to the above analysis, the control law (11) is easily obtained if the nonlinear functions and are known. However, in this paper, these nonlinear functions are assumed to be unknown, so the above design method cannot be applied directly.
3. Interval Type-2 Fuzzy Logic System
The theory and design of interval type-2 fuzzy logic systems (FLS) are presented well in [13–15]. The brief description of the interval type-2 FLS is depicted here. Detailed descriptions can be found in [13–15]. In particular, refer to [13, 15] for more notations and calculations of type-2 fuzzy logic equations.
A T2FS in the universal set is denoted as which is characterized by a type-2 membership function in (17). can be referred to as a secondary membership function (MF) or also referred to as secondary set, which is a type-1 set in . In (17) is a secondary grade, which is the amplitude of a secondary MF; that is, . The domain of a secondary MF is called the primary membership of . In (17), is the primary membership of , where for ; is a fuzzy set in , rather than a crisp point in .
When , , then the secondary MFs are interval sets such that in (17) can be called an interval type-2 MF [13]. Therefore, T2FS can be rewritten as
Also, a Gaussian primary MF with uncertain mean and fixed standard deviation having an interval type-2 secondary MF can be called an interval type-2 Gaussian MF. A 2D interval type-2 Gaussian MF with an uncertain mean in and a fixed standard deviation is shown in Figure 1. It can be expressed as
It is obvious that the T2FS in a region is called a footprint of uncertainty (FOU) and bounded by an upper MF and a lower MF [13], which are denoted as and , respectively. Both of them are type-1 MFs. Hence, (18) can be reexpressed as
A T2FLS is very similar to a T1FLS as shown in Figure 2 [13], the major structure difference being that the defuzzifier block of a T1FLS is replaced by the output processing block in a T2FLS, which consists of type-reduction followed by defuzzification.
There are five main parts in a T2FLS: fuzzifier, rule base, inference engine, type-reducer, and defuzzifier. A T2FLS is a mapping . After fuzzification, fuzzy inference, type-reduction, and defuzzification, a crisp output can be obtained.
Consider a T2FLS having inputs and one output . The type-2 fuzzy rule base consists of a collection of IF-THEN rules. We assume there are rules and the rule of a type-2 relation between the input space and the output space can be expressed aswhere s are antecedent T2FSs () and s are consequent T2FSs.
The inference engine combines rules and gives a mapping from input T2FSs to output T2FSs. To achieve this process, we have to compute unions and intersections of type-2 set, as well as compositions of type-2 relations. The output of inference engine block is a type-2 set. By using the extension principle of type-1 defuzzification method, type-reduction takes us from type-2 output sets of the FLS to a type-1 set called the “type-reduced set.” This set may then be defuzzified to obtain a single crisp value.
In Figure 2, we only consider singleton input fuzzification throughout this paper. Similar to T1FLS, the firing strength in (22) can be obtained by following inference process:where is the meet operation and is the join operation [13].
For Gaussian IT2FS as shown in Figure 1, the upper MF is a subset that has the maximum membership grade and the lower MF is a subset that has the minimum membership grade. The join operation in (22) leads to joining the result from meet operations, which is using maximum value. The result of join operation can be an interval type-1 set [13] aswhere
There are many kinds of type-reduction, such as centroid, height, modified weight, and center-of-sets [13]. The center-of-sets type-reduction will be used in this paper and can be expressed as where is the interval set determined by two end points and , and firing strengths . The interval set should be computed or set first before the computation of . For any value , can be expressed as where is a monotonic increasing function with respect to . Also, in (25) is the minimum associated only with , and in (25) is the maximum associated only with . Note that and depend only on mixture of or values. Hence, left-most point and right-most point can be expressed as [13]
For illustrative purpose, we briefly provide the computation procedure for . Without loss of generality, assume s are arranged in ascending order; that is, .
Step 1. Compute in (27) by initially using for , where and are precomputed by (24); and let .
Step 2. Find such that .
Step 3. Compute in (27) with for and for , and let .
Step 4. If , then go to Step 5. If , then stop and set .
Step 5. Set equal to , and return to Step 2.
This algorithm decides the point to separate two sides by the number , one side using lower firing strengths ’s and another side using upper firing strengths ’s. Hence, in (27) can be reexpressed as
The procedure to compute is similar to computing . In Step 2, it only needs to find , such that . In Step 3, let for , and for . The in (27) can be also rewritten as
The defuzzified crisp output from an IT2FLS is the average of
4. Interval Type-2 Fuzzy Sliding Mode Control
In this section, we propose an adaptive interval type-2 fuzzy sliding mode controller (AIT2FSMC) for nonlinear unknown MIMO systems. Due to unknown functions and in our problem, it is impossible to obtain the control law (11). We use the interval type-2 fuzzy system to approximate unknown functions and . First, let the nonlinear functions and be approximated, over a compact set , by interval type-2 fuzzy systems as follows:where and are fuzzy basis vectors fixed by the designer and and are the corresponding adjustable parameter vectors of each interval type-2 fuzzy system.
Let us define as the optimal parameters of and , respectively. Notice that optimal parameters and are artificial constant quantities introduced only for analytical purpose, and their values are not needed for the implementation. Defineas the parameter estimation errors, andas the minimum fuzzy approximation errors, which correspond to approximation errors obtained when optimal parameters are used.
In this paper, we assume that the used interval type-2 fuzzy systems do not infringe the universal approximation property on the compact set , which is assumed large enough so that state variables remain within under closed loop control. Therefore, it is reasonable to assume that the minimum approximation errors are bounded for all ; that is,where and are given constants.
Denote
From the above analysis, we have
Now, let us consider the control law, , where is a sliding mode control term [4] defined as
The above control term results from (11) by using the adaptive interval type-2 fuzzy approximation and instead of actual functions and , respectively.
The sliding mode control law (38) is not well-defined when the estimated matrix is singular. The matrix is generated online via the estimation of the parameters . In order to implement this controller, additional precautions have to be made to guarantee that remains in a feasible region in which is regular. Therefore, we modify the sliding mode control term (38) as follows [9]:where is a small positive constant.
Within the sliding mode control term (39), we have used the regularized inverse of defined as
In fact, the regularized inverse (40) is well-defined even when is singular, and the sliding mode control term (38) is always well-defined.
Even though the control law (39) is always well-defined, it cannot guarantee alone the stability of the closed loop system. It is due, partly, to the approximation of by the regularized inverse and, partly, to the unavoidable reconstruction errors of the unknown functions and . For these reasons, and hoping for the cancellation of these approximations errors, we append to the controller (39) a robustifying control term [8]
The controller (41) is the sum of two control terms: a modified sliding mode control term, and a robustifying control term, where isand is a design time-varying parameter defined below.
In order to meet the control objectives, the adaptive parameters , , and the design parameter are updated by the following adaptive laws:where .
Then, we can prove the following theorem.
Theorem 3 (consider system (1)). Suppose that Assumptions 1 and 2 are satisfied. Then the control law defined by (41) and (42), with adaptation laws given by (45)–(47), guarantees the following properties:(3.1)All signals in the closed loop system are bounded.(3.2)The tracking errors and its derivatives decrease asymptotically to zero; that is, as for and .
Proof. Using the control law (51), (10) can be rewritten asBy introducing the control term (42)–(48), we obtainHere, we have used the fact thatFrom (37), one can write (49) asMultiplying to (51) givesLet us now consider the following Lyapunov function candidate:whose time derivative is given byWith (52), (53) can be expressed as whereSubstituting the parameter adaptive laws (45) and (46) into (56) gives Using (43), we can writeHere, we have used the inequalitywhich is true because is assumed positive definite and satisfies .
Equation (57) can be bounded as follows:With (60), (61) becomesUsing (47) in (62) yieldsFrom (58) and (63), it follows thatBy Barbalat’s lemma [5], it can conclude that as . In spite of the demonstrated properties of the controller, the hitting control law leads to the well-known chattering phenomenon. In order to overcome the undesirable chattering effects, the sign function is replaced with the saturation function [5].
5. Simulation Results
In this section, we test the AIT2FSMC design on the tracking control of a two-link robot. Consider a two-link rigid robot manipulator moving a horizontal plant in Figure 3. The first link is mounted on a rigid base by means of frictionless hinges and the second is mounted at the end of first link by means of a frictionless ball bearing. The dynamic equations of this MIMO system are given by [5] wherewith
In the simulation, the following parameter values are used:
Let , , , andand then, the robot system (65) can be described as follows:which is the input-output form given by (3). Since the matrix is positive definite [5], then it is always regular and is positive definite.
The control objective is to force the system outputs and to track the sinusoidal desired trajectories and . In order to analyze the performance of the AIT2FSMC, we compared the AIT2FSMC with the A-Fuzzy Sliding Mode Controller (AFSMC) which used the type-1 FLS to approximate the nonlinear and . The external disturbances are added to system (65). Since the components of and are assumed unknown, two fuzzy systems in the form of (30) are used to approximate the elements of , and four are used to approximate the elements of . In the AIT2FSMC and the AFSMC, the sliding surface is selected as with and the design parameters used in this simulation are chosen as follows: , , , for and the initial conditions of robot are selected as . The fuzzy systems used to describe have , , , and as inputs. The input membership functions and parameters for the AIT2FSMC and AFSMC are shown in Table 1.
As shown in Figures 4 and 5, the AIT2FSMC shows the better performance than the AFSMC. In the AFSMC, a type-1 FLS, which is not able to handle rule uncertainties, is used for control of unknown nonlinear MIMO system. Therefore, the system performance is deteriorated by the disturbance. Meanwhile, the proposed AIT2FSMC utilizes the interval type-2 FLS. The simulation results show that the interval type-2 FLS is able to handle rule uncertainties, and thus the system performance is compensated by the interval type-2 FLS [13].
(a) Tracking curve of
(b) Tracking curve of
(c) Tracking error of
(d) Tracking error of
(a) Tracking curve of
(b) Tracking curve of
(c) Tracking error of
(d) Tracking error of
6. Discussion
In this paper, we propose a novel advanced interval type-2 fuzzy sliding mode control (AIT2FSMC) for a class of uncertain nonlinear MIMO systems with external disturbances. The parameters of the proposed AIT2FSMC system, as well as the approximation error bound, are tuned online. The control laws are obtained in the Lyapunov sense to ensure the stability of the control system.
Unlike the conventional SMCs, the design of the proposed AIT2FSMC is independent of the mathematical model of the system and can be applied to both unknown and uncertain nonlinear MIMO systems. Furthermore, the uncertainty bound is not needed to be available beforehand. Simulation results performed on a two-link robot manipulator demonstrate the feasibility of the proposed control system.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Incheon National University Research Grant in 2013.