Discrete-Time Exponentially Stabilizing Fuzzy Sliding Mode Control via Lyapunov's Method

Abstract

The exponentially stabilizing state feedback control algorithm is developed by Lyapunov's second method leading to the variable structure system with chattering free sliding modes. Linear time-invariant discrete-time second order plant is considered and the control law is obtained by using a simple fuzzy controller. The analytical structure of the proposed controller is derived and used to prove exponential stability of sliding subspace. Essentially, the control algorithm drives the system from an arbitrary initial state to a prescribed so-called sliding subspace S, in finite time and with prescribed velocity estimate. Inside the sliding subspace S, the system is switched to the sliding mode regime and stays in it forever. The proposed algorithm is tested on the real system in practice, DC servo motor, and simulation and experimental results are given.

Academic Editor:Rustom M. Mamlook

1, Faculty of Mechanical Engineering, University of Belgrade, 16 Kraljice Marije, 11120 Belgrade 35, Serbia

Received 30 July 2014; Revised 7 February 2015; Accepted 8 February 2015; 25 February 2015

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.

1. Introduction

It is well-known that the variable structure control systems (VSCS) theory has been existing for several decades. Almost everything published in the area is related to the VSCS with sliding modes, with known features and advantages. Until the mid-1980s, the results concerned exclusively the continuous time type of these systems. With the development of computer and digital technology, the discrete-time version of the problem has become more important in the academic research and industries. Discrete-time sliding mode control is an attempt to eliminate the problems caused by the discretization of continuous-time controllers. Great attention was directed to the existence of sliding mode regime inside the so-called sliding subspace and finite system state reaching time to the sliding subspace in this type of systems, because it is quite different from its continuous counterpart. Although discrete-time VSCS with sliding mode is characterized by a phenomenon that the actual so-called sliding control is applied inside sliding subspace (see, e.g., [1-3]), the chattering problem was raised. Many different ways were proposed to reduce it or fully eliminate it [4]. On the other side, system convergence to the sliding subspace conditions was dealt with, where analogy to the continuous systems conditions was massively exploited.

Many of the authors treated special classes of the systems, most frequently, linear systems, systems in companion canonical form, and nonlinear systems, with Lyapunov's second method as a powerful design and analysis technique [5, 6]. There are no papers with exponential stability of sliding subspace, except that some authors speak about exponential changes of, in general case, vector variable, usually designated, for example, by [figure omitted; refer to PDF] and used for sliding subspace definition as [figure omitted; refer to PDF] ; see [7], among others. In [8] authors spoke about exponential state vector changes on the way towards zero origin of a ball.

Recently, the integration of fuzzy techniques and conventional control approaches has been an active research focus. Fuzzy controllers are inherently nonlinear controllers, and the major advantage of fuzzy control technology over the traditional control technology is its capability of capturing and utilizing qualitative human experience and knowledge in a quantitative manner through the use of fuzzy sets, fuzzy rules, and fuzzy logic [9]. Based on the differences of fuzzy control rules, approaches to fuzzy logic control can be roughly classified into the conventional fuzzy control and Takagi-Sugeno (T-S) model based fuzzy control. The conventional fuzzy control systems are essentially heuristic and model free and various approaches have been developed for stability analysis. The key idea of these approaches is to regard a fuzzy controller as a nonlinear controller and embed the stability and/or control design problem of fuzzy control systems into conventional nonlinear system stability theory [10, 11]. T-S models are based on using a set of fuzzy rules to describe a global nonlinear system in terms of a set of local linear models which are smoothly connected by fuzzy membership functions, and they provide a basis for development of systematic approaches to stability analysis and controller design of fuzzy control systems in view of powerful conventional control theory and techniques [12-14].

Combining fuzzy logic (FL) and sliding mode control (SMC) theory, so-called fuzzy sliding mode control (FSMC) has the advantages of both SMC and FLC. The analogy between a simple and a sliding-mode controller with a boundary layer is shown in [15]. In [16, 17], for example, fuzzy controllers are designed to satisfy the sliding conditions, and in these approaches, the focus is on the design of sliding surface via the fuzzy logic theory. Some fuzzy control rules to construct reaching control under the assumption that equivalent control already exists is used in [18]. A fuzzy-model-based controller which guarantees the stability of the closed loop controlled system is suggested in [19], where the closed-loop system consists of the TS fuzzy model and the switching type fuzzy-model-based controller. Fuzzy logic has been also utilized to adapt the parameters of SMC to achieve a better performance and especially attenuate the chattering of control input.

The above works are related to continuous systems, and discrete systems were much less considered in the context of FSMC. The design of the fuzzy sliding mode control to meet the requirement of necessary and sufficient reaching conditions of sliding mode control of discrete nonlinear system is considered in [20]. A robust controller based on the sliding mode and the dynamic T-S fuzzy state model for discrete systems is developed in [21].

One motivation for this work stems from the fact that discrete-time type of the problem clearly makes place of microprocessor compensator application in the systems. The main motivation for the paper to be related to the exponential stability of the FSMC, as a higher quality stability property, goes from the fact that, in most of papers, only stability of sliding subspace [figure omitted; refer to PDF] was considered.

Following Hui and Zak's [3] conditions for chattering free sliding mode, in this paper, the exponential stabilizing fuzzy control algorithm is developed by Lyapunov's second method leading to the variable structure system with chattering free sliding modes, for linear time-invariant second order discrete-time system. That is, the exponential stability to be in exact sense related to the sliding subspace as a set, defined over working point distance from the set, during its approaching to the set. In the work, essentially, the objective is to push the system state from an arbitrary initial position to the sliding subspace [figure omitted; refer to PDF] , in finite time, and with estimate of approaching velocity defined by an exponential law. The velocity could not be smaller than the exponential law defines. Once the system reaches [figure omitted; refer to PDF] it stays there forever working in sliding mode regime, which is chattering free, as well, and with the system state asymptotic approaching to the zero equilibrium state. The control law is obtained by using a simple fuzzy controller and the analytical structure of the proposed controller is derived and used in the proof of the theorem on exponential stability of sliding subspace.

2. Problem Statement, Notation, and Some Definitions

Linear time-invariant second order discrete-time system is considered, which is described by its state equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is state vector at time [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is control vector at time [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are real constant matrix and vector of appropriate dimensions, respectively, and [figure omitted; refer to PDF] is assumed to be a controllable pair.

For the system (1), define a hyperplane [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a constant nonzero row vector. The hyperplane is the so-called sliding subspace [figure omitted; refer to PDF] (in further text only [figure omitted; refer to PDF] ). Clearly, for second order system [figure omitted; refer to PDF] is sliding line, described by [figure omitted; refer to PDF] Also, [figure omitted; refer to PDF] is assumed.

The objective of this paper is to develop variable structure type of state feedback control law: [figure omitted; refer to PDF] based on fuzzy logic, which guarantees that the state [figure omitted; refer to PDF] of the system (1) reaches [figure omitted; refer to PDF] in finite time and with velocity whose estimate is defined by an exponential law. Once [figure omitted; refer to PDF] has been reached, the controller is required to keep the state within it thereafter, which means positive invariance of [figure omitted; refer to PDF] relative to the system motion and what is denoted as sliding mode regime . During this regime, inside [figure omitted; refer to PDF] , convergence to zero equilibrium with prescribed mode [figure omitted; refer to PDF] can be guaranteed if the [figure omitted; refer to PDF] has been appropriately chosen. For [figure omitted; refer to PDF] choice, in general case, see [22].

More rigorously, [figure omitted; refer to PDF] is positive invariant relative to the system (1) motion if and only if [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Furthermore, some other notations and definitions are given for the reason of their usage in theorems, which are the main results. Real [figure omitted; refer to PDF] -dimensional state space [figure omitted; refer to PDF] is with Euclidean norm denoted by [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the distance between [figure omitted; refer to PDF] and the point [figure omitted; refer to PDF] .

Definition 1.

The state [figure omitted; refer to PDF] of the system (1), (2) is stable in [figure omitted; refer to PDF] (with respect to [figure omitted; refer to PDF] ) iff [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] exists, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Definition 2.

The state [figure omitted; refer to PDF] of the system (1), (2) is attractive (globally) in [figure omitted; refer to PDF] (with respect to [figure omitted; refer to PDF] ) iff [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] exists, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Definition 3.

The state [figure omitted; refer to PDF] of the system (1), (2) is (globally) asymptotically stable in [figure omitted; refer to PDF] (with respect to [figure omitted; refer to PDF] ) iff it is both stable and (globally) attractive in [figure omitted; refer to PDF] .

Definition 4.

The system (1), (2) is stable in [figure omitted; refer to PDF] (with respect to [figure omitted; refer to PDF] ) iff its state [figure omitted; refer to PDF] is globally asymptotically stable in [figure omitted; refer to PDF] .

Definition 5.

[figure omitted; refer to PDF] is exponentially (globally) stable, relative to the system (1), (2), iff [figure omitted; refer to PDF] [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that distance [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] exists and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Remark 6.

Previous definitions are valid and in the general case, for [figure omitted; refer to PDF] .

The problem stated in the objective of the paper can be reformulated as follows: the objective of the paper is to develop the control law (4) such that [figure omitted; refer to PDF] is globally exponentially stable relative to the system (1), (2) and the system (1), (4) is stable in the sliding mode regime with appropriately prescribed modes. Control law (4) will be developed using fuzzy logic controller.

3. Structure of FLC

The key idea of fuzzy sliding mode control is to integrate fuzzy control and sliding mode control in such a way that the advantages of both techniques can be used. One approach is to design conventional fuzzy control systems, and sliding mode controller is used to determine best values for parameters in fuzzy control rules. Thereby, stability is guaranteed and robust performance of the closed-loop control systems is improved. In another approach, the control design is based on sliding mode techniques while the fuzzy controller is used as a complementary controller. Also, sliding mode control law can be directly substituted by a fuzzy controller.

Sliding mode controllers, generally, involve a discontinuous control action which often results in chattering phenomena due to imperfections in switching devices and delays. Commonly used methods for chattering elimination are to replace the relay control by a saturation function and boundary layer technique. In some applications of fuzzy sliding mode control, the continuous switching function of the boundary layer is replaced with equivalent fuzzy switching function.

In this paper, the fuzzy controller is used in the reaching phase. The controller should realize nonlinear control law which will guarantee exponential stability of sliding subspace [figure omitted; refer to PDF] . This section introduces the principal structure of the proposed controller.

The fuzzy logic controller that will be evaluated is one of the simplest, Figure 1. It employs only two input variables, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Constant [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are component of vector [figure omitted; refer to PDF] from (2).

Figure 1: The proposed fuzzy logic controller.

[figure omitted; refer to PDF]

Scaling Factors . The use of normalized domains requires a scale transformation, which maps the physical values of the input variables ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the present study) into a normalized domain. This is called input normalization. Furthermore, output denormalization maps the normalized value of the control output variable ( [figure omitted; refer to PDF] ) into its respective physical domain ( [figure omitted; refer to PDF] ). The relationships between scaling factors [figure omitted; refer to PDF] and the input and output variables are as follows: [figure omitted; refer to PDF]

Fuzzification Module . It converts instantaneous value of a process state variable into a linguistic value with the help of the represented fuzzy set. The parametric functional description of the triangular shaped membership function is the most economic one and hence it is considered here. The membership functions of input variables are shown in Figure 2.

Figure 2: The input membership functions.

[figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be the one crisp input. Then, the fuzzified version of [figure omitted; refer to PDF] after normalization is its degree of membership in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the linguistic values taken by [figure omitted; refer to PDF] . Here, symbols [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have common meanings negative and positive , respectively.

Assumption 7.

The value [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF]

According to the previous assumption, mathematical description of the input membership functions is, respectively, given by [figure omitted; refer to PDF]

Remark 8.

It is noticed that [figure omitted; refer to PDF]

The membership functions for the normalized output ( [figure omitted; refer to PDF] ) are singleton and are shown in Figure 3. In Figures 2 and 3, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two positive constants chosen by the designer, which can be fixed after being determined.

Figure 3: The output membership functions.

[figure omitted; refer to PDF]

Remark 9.

To ensure that Assumption 7 is valid, one must be careful in choosing the scaling factor [figure omitted; refer to PDF] . In practical implementation, the maximum and minimum values of variables [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are known, and the factor [figure omitted; refer to PDF] is chosen so that it satisfies the following conditions: [figure omitted; refer to PDF]

Remark 10.

For a second order system, it is common to define the row vector [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

Fuzzy Control Rules . Using the aforementioned membership functions, the following control rules are established for the fuzzy logic control part: [figure omitted; refer to PDF]

Inference Engine . The basic function of the inference engine is to compute the overall value of control output variable based on the individual contribution of each rule in the rule base. A degree of match for each rule is established by using the defined membership functions.

Here, the antecedent of each rule is evaluated by using the triangular norm ( [figure omitted; refer to PDF] -norm ). The [figure omitted; refer to PDF] -norm used in this paper is intersection (AND function), which is mathematically given as [figure omitted; refer to PDF]

Then, based on this degree of match, the clipped fuzzy set representing the value of the control output variable is determined via Mamdani inference method. Thus, outcomes of fuzzy rules are [figure omitted; refer to PDF] Finally, the clipped values for the control output of each rule, previously denoted by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are aggregated, thus forming the value of the overall control output.

From the rule base it may be noted that the control rules [figure omitted; refer to PDF] and [figure omitted; refer to PDF] generate two memberships [figure omitted; refer to PDF] and [figure omitted; refer to PDF] which have the same output fuzzy set defined by [figure omitted; refer to PDF] . For such situations a combined membership is obtained by using the triangular conorm ( [figure omitted; refer to PDF] -conorm , or [figure omitted; refer to PDF] -norm ). The [figure omitted; refer to PDF] -conorm considered for this study is Lukasiewicz OR [figure omitted; refer to PDF] -conorm : [figure omitted; refer to PDF]

Defuzzification Module . Defuzzification module converts the set of modified control output values into a crisp value. Defuzzification is done using the well-known COS (center of sum) method. According to this method and taking into account the previously analyzed structure of the proposed fuzzy controller and the fact that the output fuzzy sets are singleton, the crisp value of control output is given by [figure omitted; refer to PDF] Since for Lukasiewicz OR [figure omitted; refer to PDF] -conorm it is valid that [figure omitted; refer to PDF] (15) can be written as [figure omitted; refer to PDF]

To obtain analytical expression of the proposed controller, all combinations of input variables must be considered. If Assumption 7 holds, there are eight input combinations (IC's), as shown in Figure 4. The control rules [figure omitted; refer to PDF] in (11) are used to evaluate appropriate control law in each IC region, as in [23]. The results of evaluating the fuzzy control rules [figure omitted; refer to PDF] are given in Table 1.

Table 1: Outcomes of fuzzy rules in IC regions.

IC	Fuzzy rules
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC1	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC2	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC3	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC4	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC5	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC6	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC7	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
IC8	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Figure 4: The regions of the fuzzy controller input values.

[figure omitted; refer to PDF]

Applying (17) to the results from Table 1 and taking into account (6) and (8), analytical structure of the controller is easily obtained, as follows: [figure omitted; refer to PDF]

Defining function [figure omitted; refer to PDF] , [figure omitted; refer to PDF] (18) can be written as [figure omitted; refer to PDF]

It is obvious that the function [figure omitted; refer to PDF] is a nonlinear function, with the following minimal and maximal values: [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . One should not forget that parameter [figure omitted; refer to PDF] must satisfy Assumption 7. The range of the value of function [figure omitted; refer to PDF] is determined by the choice of parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Without loss of generality, it can be assumed that [figure omitted; refer to PDF] , and they are commonly normalized domain boundaries. One example of function [figure omitted; refer to PDF] is shown in Figure 5.

Figure 5: Example of function [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

4. Main Results

In this section, the solution of the problem stated in Section 2 is provided by the following theorems.

Theorem 11.

Applying [figure omitted; refer to PDF] the so-called one step control, to the system (1), outside [figure omitted; refer to PDF] , is necessary and sufficient for [figure omitted; refer to PDF] to reach the [figure omitted; refer to PDF] in one sampling period (one step).

Proof.

Necessity . Let us assume that [figure omitted; refer to PDF] of the system (1) is arbitrary, outside of sliding subspace, [figure omitted; refer to PDF] , and a control pushes it to the sliding subspace in one step (sampling period). It follows that [figure omitted; refer to PDF] . That fact, description of [figure omitted; refer to PDF] , and the system (1) state equation lead to [figure omitted; refer to PDF] and finally [figure omitted; refer to PDF]

Sufficiency . When the system state is outside of sliding subspace, [figure omitted; refer to PDF] , the action of one step control is assumed. State equation of the system (1) and expression of [figure omitted; refer to PDF] give [figure omitted; refer to PDF] which means that [figure omitted; refer to PDF] .

Remark 12.

One step control can be encountered in VSCS papers, in different contexts; see, for example, [3, 4, 13] among others.

Theorem 13.

Applying [figure omitted; refer to PDF] the actual so-called sliding control, to the system (1), inside [figure omitted; refer to PDF] , is necessary and sufficient for [figure omitted; refer to PDF] to be positive invariant relative to the system (1) solution [figure omitted; refer to PDF] .

Proof.

Necessity . Positive invariance of [figure omitted; refer to PDF] relative to the system (1) solution [figure omitted; refer to PDF] is assumed. So, every time [figure omitted; refer to PDF] , [figure omitted; refer to PDF] as well. From the previous sentence, description of [figure omitted; refer to PDF] and state equation of the system (1) follows that [figure omitted; refer to PDF] and further [figure omitted; refer to PDF]

Sufficiency . Let us assume, relative to system (1), [figure omitted; refer to PDF] and control (26) is applied. Then, state equation of the system (1) and expression for [figure omitted; refer to PDF] give [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] .

Remark 14.

One step control and sliding control formally are identical.

The application of actual sliding control [figure omitted; refer to PDF] (henceforth referred to as [figure omitted; refer to PDF] ) to the system (1) inside [figure omitted; refer to PDF] is the phenomenon of discrete-time VSCS with sliding modes. Using this fact and the intention for developing of the variable structure type of the controller for the system (1), the control (4) components could be more precisely specified as [figure omitted; refer to PDF]

Contrary to the [figure omitted; refer to PDF] , which is applied to the system (1) inside [figure omitted; refer to PDF] , let us refer to the control which is applied to the system (1) outside of [figure omitted; refer to PDF] as the outer control, [figure omitted; refer to PDF] . The following theorems are basic for solving the stated problem by Lyapunov's second method.

Theorem 15.

Let the system (1), (30) be considered. [figure omitted; refer to PDF] is globally exponentially stable relative to the system (1), (30) if there exists scalar function [figure omitted; refer to PDF] and numbers [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that

(a) [figure omitted; refer to PDF] ,

(b) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ,

(c) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ,

where [figure omitted; refer to PDF] is the first forward finite difference of [figure omitted; refer to PDF] .

Proof.

The proof starts with the assumption that sufficient conditions are fulfilled, while the exponential stability of [figure omitted; refer to PDF] should be shown.

Condition (b) of the theorem implies [figure omitted; refer to PDF] which with (c) give [figure omitted; refer to PDF]

The last inequality, after solving, gives at first [figure omitted; refer to PDF] and further by means of condition (a), [figure omitted; refer to PDF]

From (a), it is clear that [figure omitted; refer to PDF] which yields [figure omitted; refer to PDF] , and above relationship between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] guarantee that [figure omitted; refer to PDF] . In that way, it is shown that all stated by Definition 5 is fulfilled; that is, the exponential stability of [figure omitted; refer to PDF] is proved.

Remark 16.

Previous theorem is valid and in the general case for [figure omitted; refer to PDF] .

Scalar function [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] as Lyapunov's function candidate and outer control law [figure omitted; refer to PDF] are chosen relative to the system (1). Evidently, such sort of [figure omitted; refer to PDF] together with [figure omitted; refer to PDF] which is unique and already stated represents the control of (30) type. The following theorems are basic for solving of the stated problem by Lyapunov's second method.

Theorem 17.

[figure omitted; refer to PDF] is globally exponentially stable relative to the system (1) if the following control is applied: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] defined by (19), (20) and with [figure omitted; refer to PDF] .

Proof.

Previously introduced Lyapunov's function candidate fulfills evidently the conditions (a) of Theorem 15. Let us show that conditions (b) and (c) are fulfilled. Related to the condition (b) and taking into account that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a constant and [figure omitted; refer to PDF] is distance between [figure omitted; refer to PDF] and hyperplane [figure omitted; refer to PDF] , it follows that [figure omitted; refer to PDF] and further [figure omitted; refer to PDF]

State of system (1) at time [figure omitted; refer to PDF] is adopted to be out of [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] in which way the proof is finished.

According to Theorem 17, [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , which leads to the following equation: [figure omitted; refer to PDF]

From nonlinearity of the control low it is obvious that control gain is high when the system state is far from sliding surfaces and as small as possible in neighborhood of the sliding subspace. Also, it may be noted that [figure omitted; refer to PDF] increments, from step to step, that is, absolute value of [figure omitted; refer to PDF] , are getting smaller over time.

Function [figure omitted; refer to PDF] is a decreasing function along motion of the system, but it will never be zero. Obviously, the system (1) state never gets into [figure omitted; refer to PDF] . Moreover, from (42), it is obvious that as the working point approaches the [figure omitted; refer to PDF] , the value of the function [figure omitted; refer to PDF] tends to its maximum value [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -neighborhood of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . To be guaranteed for [figure omitted; refer to PDF] to enter [figure omitted; refer to PDF] , settling time [figure omitted; refer to PDF] is necessary to elapse, where [figure omitted; refer to PDF] is the biggest integer from the denoted segment. After the state reached [figure omitted; refer to PDF] , one step control, [figure omitted; refer to PDF] , should be applied to the system (1) in which way finite reaching time in [figure omitted; refer to PDF] for [figure omitted; refer to PDF] is realized. The previous facts are summarized by the following theorem.

Theorem 18.

[figure omitted; refer to PDF] is globally exponentially stable relative to the system (1) with finite reaching time in [figure omitted; refer to PDF] , if the following control is applied: [figure omitted; refer to PDF]

Proof.

Compared to the proof of Theorem 17, here the proof is the same.

5. Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DC servo motor with the gear and load, whose linear mathematical model (without considering the major nonlinear effects: the speed dependent friction, dead zone, and backlash) is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the angular position of the load shaft, [figure omitted; refer to PDF] is the total moment of inertia reflecting to the output shaft, [figure omitted; refer to PDF] is the viscous friction coefficient, and [figure omitted; refer to PDF] is the motor input voltage. [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are, respectively, the motor armature resistance, the motor torque constant, the motor efficiency, the gearbox efficiency, and the total gear ratio.

In the previous model, it is assumed that the motor inductance is much smaller than the resistance, so it is ignored, as the case with used servo motor is. The experiments were performed with the Quanser rotary servo motor, SRV02. This model is equipped with the optical encoder and tachometer, for motor position and speed measuring, respectively. Q8-USB data acquisition board for real-time data acquisition and control was used, with Matlab/Simulink software and QUARC@, real-time control. The parameters numerical values are as follows: [figure omitted; refer to PDF] (kgm² ), [figure omitted; refer to PDF] (Nms/rad), [figure omitted; refer to PDF] (Ω), [figure omitted; refer to PDF] (Nm/A), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Choosing [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as state variables and [figure omitted; refer to PDF] as control, at first, the continuous time state equation was obtained, and finally after applying an accurate discretization procedure with sampling period [figure omitted; refer to PDF] (s), the discrete-time state equation of the system was obtained as follows: [figure omitted; refer to PDF]

The above mathematical model of real DC servo motor was derived as adequate discrete-time state equation. In the way the system matrix parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are available. Control algorithm from Theorem 18 is used to control the motor. For simulation and experiment, parameter [figure omitted; refer to PDF] is adopted which defines very narrow σ -neighborhood of sliding subspace [figure omitted; refer to PDF] . Also, sampling time [figure omitted; refer to PDF] (s) and vector [figure omitted; refer to PDF] , determined so as the system is stable in sliding mode regime, are adopted which gives [figure omitted; refer to PDF] . Fuzzy logic controller is defined by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , which implies [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Taking into account estimate relationships, on one hand, between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and, on another hand, between [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are chosen, giving parameters of exponential low [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . For initial state [figure omitted; refer to PDF] , after calculation, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and settling time [figure omitted; refer to PDF] are obtained.

The parallel simulation and experimental results are shown, in Figures 6-11 related to the algorithm from Theorem 18. Figures 6 and 7 show the system state variables. The control signal that drives the state onto the sliding subspace is shown in Figure 9. The sliding function and distance are shown in Figures 8 and 10, respectively. From the figures, it could be seen that the system state from initial position approaches the sliding subspace [figure omitted; refer to PDF] but not slower than the exponential law defined by the parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . At instant [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] sec) the state for the first time enters the [figure omitted; refer to PDF] -neighborhood of [figure omitted; refer to PDF] . If it continues in the same way, it will never reach [figure omitted; refer to PDF] . To reach it, one step control is applied at [figure omitted; refer to PDF] driving state in one step to [figure omitted; refer to PDF] , so the character of the system approaching [figure omitted; refer to PDF] changes at [figure omitted; refer to PDF] . At [figure omitted; refer to PDF] the system state is in [figure omitted; refer to PDF] staying in it thereafter forever. Inside [figure omitted; refer to PDF] the system is stable too, so the system state asymptotically approaches the zero state. Simulation results fully confirm the theoretical results and meet the expectations of Theorems 15, 17, and 18.

Figure 6: The state variable [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 7: The state variable [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 8: The sliding function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 9: The control variable.

[figure omitted; refer to PDF]

Figure 10: The distance.

[figure omitted; refer to PDF]

Figure 11: Augmented part of Figure 10.

[figure omitted; refer to PDF]

The experimental results have demonstrated that the proposed control scheme is valid and effective for the real applications. It is shown that there is no chattering in sliding mode. However, in practice, errors could not be caused only by computation but also by measurement accuracy (resolution, sensitivity), noise existence, nonlinearity, and inadequate description of the real system by linear model as well. All this could cause not entering the [figure omitted; refer to PDF] - neighborhood of sliding subspace exactly in step [figure omitted; refer to PDF] than at some later instant, which is the case here in the paper experimental application example. After entering the [figure omitted; refer to PDF] -neighborhood of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] or not and applying one step control at [figure omitted; refer to PDF] , a little bit of sliding subspace missing occurs. Here, that missing is a little bit more visible at the beginning of reaching the sliding subspace, but very shortly, after the like transition period expires, it is again invisible like in the mathematical case simulation. This is illustrated by Figure 11, which is an enlarged segment of Figure 10 for [figure omitted; refer to PDF] between 0.15 and 0.5 (sec). Evasion of the occurrence is achieved by appropriate choice of the sliding subspace and [figure omitted; refer to PDF] - neighborhood parameters through vector [figure omitted; refer to PDF] and parameter [figure omitted; refer to PDF] , as well as the state space area, where the linear model is very adequate.

It has already been pointed out that exponential stability is a higher quality stability property compared to "common" stability in Lyapunov's sense (in this case it refers to global asymptotic stability of a set-sliding subspace [figure omitted; refer to PDF] ). For that, it is natural that the sufficient conditions of Theorem 15 are more rigorous compared to the conditions of "common" stability. The conditions of Theorem 15 are indirectly applicable in the control of real-world processes through control law of Theorems 17 and 18, for these control laws were derived from the conditions of Theorem 15. Real experiment with control of DC servo motor is control of real-world process. In the way it is confirmed that the stability conditions of Theorem 15 are applicable in the control processes in practice.

To resume, in both ideal simulation case and in experimental application, the proposed new developed algorithm, in the paper, guarantees exponential stability of [figure omitted; refer to PDF] , and no chattering in sliding mode regime. Despite the existence of all upper stated nonidealities, in the case of real application, by the proper parameters choice, the system motion is such that distance of working point from [figure omitted; refer to PDF] is always under proposed exponential envelope, as it could be seen in provided experiment.

6. Conclusion

In the paper, discrete-time type of VSCS with sliding modes is considered, related to the linear time-invariant plant of second order. The fuzzy exponential stabilizing state feedback control algorithm, so with prescribed approaching to the sliding subspace velocity estimate, is developed strictly by Lyapunov's second method and with chattering free sliding mode. Also, application of the proposed ideal cases algorithm, in practice, is considered and discussed through the experimental example. It shows that experiment is quite consistent with mathematical ideal cases.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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