Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation

2.1. Preliminaries

Article

Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation

Lusong Ding, and Weiwei Sun ^*

¹ Institute of Automation, Qufu Normal University, Qufu 273165, China

^* Correspondence: wwsun@hotmail.com

Received: 31 August 2023

Accepted: 31 October 2023

Published: 21 December 2023

Abstract: This article investigates the predefined-time fuzzy adaptive tracking control problem for a class of nonlinear switched fractional-order systems with input saturation and external disturbances under a nonstrict feedback structure. By combining the backstepping technique and the common Lyapunov function method, a predefined-time switching control method is constructed based on a novel fractional-order auxiliary function. The fuzzy logic system and the adaptive method are introduced to identify unknown compounded continuous functions. Moreover, the issue of calculating explosion and the problem of singularity are tackled through the newly proposed predefined-time and filter-based dynamic surface control. Especially, the construction of a continuous term in the controller eliminates possible chattering. The developed control strategy achieves that the closed-loop system is practically predefined-time stable under arbitrary switchings, where the upper bound of the settling-time can be defined by users in advance. Finally, two simulation examples are illustrated to prove the feasibility and effectiveness of the presented scheme.

1. Introduction

Over the past decades, fractional-order systems have been widely studied in practical industrial areas, such as engineering, physics, biology, and chemistry. The fractional-order systems can depict and reflect the properties of objects more truly and accurately than the integer-order systems, thereby obtaining better control performance. Theoretical and practical research on fractional-order system control has attracted considerable interests, and different control approaches have been successfully applied including the sliding-mode control methods, robust control methods and adaptive control methods [1−3].

Owing to their strong learning and approximation abilities, fuzzy logic systems (FLSs) and neural networks (NNs) have been extensively used in the adaptive control of nonlinear systems to cope with dynamic uncertainties. The control performance of fractional-order systems has been improved by combining several advanced control methods. In [4] and [5], a robust adaptive control scheme was proposed for single-input single-output (SISO) fractional-order nonlinear systems by adopting the backstepping control technique. Such a scheme was extended in [6−8] to study multi-input multi-output (MIMO) fractional-order nonlinear systems. Considering the input nonlinearity and immeasurable states, the observer-based output feedback control scheme was presented in [6] for fractional-order nonlinear systems based on the fractional adaptive type-2 fuzzy technique. The adaptive decentralized control was investigated in [7] for nonlinear interconnected systems with mismatched interconnections. Furthermore, the fully distributed adaptive consensus tracking algorithm was proposed in [8] for nonlinear fractional-order multiagent systems subject to heterogeneous uncertainties.

In a variety of application scenarios, the input saturation is a commonly occurring nonlinearity and is considered to be a major threat to the control system [9−11]. The input saturation may be caused by the physical limitation of the actuator itself, or may be artificially introduced to prevent the excessive impact from destroying the stability of the actuator or even the system. Therefore, designing a suitable control approach for nonlinear systems with input saturation is an essential and necessary issue. By far, there have been two ways to handle the input saturation which include introducing an auxiliary system to the controller [12] or considering the presence of the saturation in the system [13]. Accordingly, the constraint control approach was designed in [14] for nonlinear fractional-order systems by considering the effect of the input saturation and state constraints whose extension to the nontriangular nonlinear systems was presented in [15]. Furthermore, an event-triggered control strategy was proposed in [16] which can dynamically adjust the threshold parameters according to the system output, thereby reducing the communication burden. Note that the above research work all focuses on nonswitched fractional-order systems.

It is generally known that the switched system, as a kind of hybrid system, is composed of exact switching rules and a series of switching subsystems, and has found widespread applications based on different controller switching strategies [17, 18]. Control schemes designed for switched systems are significant and challenging in various engineering control sytems, such as hybrid vehicle systems, robot control systems, and switching power converters [19, 20]. In [21], the common Lyapunov function approach was applied to solve the problem of global stabilization for arbitrary switching systems. The neuro-adaptive control method was proposed in [22] for switching nonlinear systems via the average dwell-time technique, and this method was successfully generalized to nonlinear switching fractional-order systems [23−25]. Nevertheless, the aforementioned work has focused relatively less on the time optimization problem of the control systems.

Generally, the actual system should be quickly stabilized to a stable operating region within a prescribed time in case of disturbances or uncertainties. Owing to its characteristics of fast response and disturbance rejection [26], the finite-time control strategy is suitable for keeping the control performance of nonlinear systems, and fruitful results have been obtained on the study of finite-time convergence [27, 28]. Note that the upper limited convergence time of the system is a bounded function depending on the initial values [29]. For many practical systems, the initial condition cannot be set arbitrarily to suit the actual needs, which limits the implementation of finite-time control. This motivates the study on fixed-time stability where the settling-time is independent of the initial value [30−32]. A noteworthy point is that both finite-time control and fixed-time control ignore the direct correlation between the bound of the convergence time and the designed system parameters, making it relatively challenging to search for parameters that satisfy the requirement of the convergence time.

To effectively reduce conservatism and easily adjust the convergence time bound, some researchers investigated the predefined-time Lyapunov stability [33−35] and developed practical predefined-time convergent adaptive control schemes for nonlinear systems [36−39]. Such a method further develops the predefined-time control theory. Among them, the adaptive predefined-time control scheme was designed in [36, 37] for nonlinear systems with the convergence time and tracking accuracy set artificially in advance. Furthermore, the new predefined-time filters were designed in [38, 39] to avoid problems of "explosion of complexity" and singularity. The predefined time control has the merit that the settling-time is determined only by the adjustment parameters relative to the finite/fixed time control, which means that the settling-time can be directly specified by adjusting the predefined time parameters. This attractive feature promotes the applications of predefined time control in a wide range of fields, such as tailless aircrafts [39], autonomous surface vehicles [40], and rigid spacecrafts [41]. Besides, a robust controller was designed in [42−44] with predefined-time convergence for fractional-order nonlinear systems, whereas the uncertainties and nonlinearities were not considered. So far, the predefined-time control has not attracted full attention for fractional-order nonlinear systems subject to input nonlinearities including input saturation, asymmetric time-varying constraints, and unknown control directions.

Motivated by the foregoing discussions, this study investigates a new predefined-time control scheme for uncertain MIMO switched fractional-order nonlinear systems with input saturation and external disturbances. Compared with the previous literature, the main contributions are given as follows.

1) In this paper, a novel fractional-order predefined-time control scheme is first developed based on an auxiliary function in a unified framework, where the convergence time explicitly appears as a tuning parameter. This scheme is applied to solve the tracking control problem for a class of switched fractional-order nonlinear systems, while the existing methods in [36−39, 42, 43] are not directly applicable to tackle the considered problem.

2) The presented fuzzy adaptive control scheme guarantees the control performance of closed-loop systems when arbitrary switching and input saturation happen, making it more generalized than the schemes in [6−8, 15, 16, 22].

3) An improved filtering technique is adopted to overcome the effect of repeated derivatives in the backstepping and singularity problems. Note that the chattering phenomenon may arise in the results of [36−39].

2. Problem Formulation and Preliminaries

Definition 1. [45] For a real function , the Riemann-Liouville (RL) fractional integral with fractional-order  is defined as follows:

where  denotes the Gamma function.

Definition 2. [45] The Caputo -order fractional derivative for  can be written as

where  represents the th order Caputo differential operator,  and . This paper only considers the case of  and .

Property 1. [46] Assume that the Caputo fractional derivative  with  is integrable, then

Particularly, if , one has .

Lemma 1. [47] Suppose that  is a smooth and differentiable function vector, then  for .

Lemma 2. [4] For  and a positive constant , one has

Lemma 3. [31] For any positive constants , the following inequality holds:

Lemma 4. [31] For  and , the following inequality holds:

Lemma 5. [31] For a constant  with  and , the following inequality holds:

In the subsequently developed control design scheme, a fuzzy-approximation based approach is employed to approximate the continuous function. Note that FLSs have the following universal approximation property.

Lemma 6. [48] Suppose that  is a continuous arbitrary function defined in a compact set . For any small positive scalar ,

holds where  denotes the ideal constant weights,   is the basis function vector with  representing the number of the fuzzy rules, and  is the fuzzy minimum approximation error.

2.2. Predefined Time Stability Theory

Consider the integer-order system:

where  represents the system states,  denotes the unknown but bounded disturbances, and  stands for the nonlinear function. We assume that the origin is the equilibrium point of system ( ).

Definition 3. [33, 34] For nonlinear system (5), the equilibrium point  is fixed-time convergent if it is finite-time stable and there exists a bounded setting-time function  such that  is bounded for . If there exists a predefined time constant  satisfying , then the origin of system (5) is predefined-time stable.

Lemma 7. [34] For dynamical system (5), if there exists a continuous and differentiable positive-definite function  such that

where , and  is the setting time. Then, the trajectory of system (5) is predefined-time stable.

Theorem 1. For system (5), suppose that there exists a scalar function  such that

where . ,  and  are the condition parameters for the convergence of the system. Then, the origin of system (5) is practically predefined-time stable. Besides, the solution  is bounded by

where  is the settling-time satisfying  with  being the upper bound.

Proof 1. Now, consider the following two parts.

i) Inequality (7) implies

(9) can be rewritten as

for . According to Lemma 7, it is obtained that system (5) is predefined-time stable with , and the state trajectory  is bounded by .

Integrating (10) over [0, ], one has

which is followed by

where  is the initial state during  and  is the state when the system is stable during . Thus, the upper-bound settling time is given by .

ii) It follows from (7) that

Similar to the previous proof, we know that the formula for calculating the convergence region is  with , which completes this proof.

Remark 1. In Lemma 1, there exist two adjustable parameters  and which makes the condition generally available for the actual system. Note that when  and , the result in [34] is a particular case of Lemma 7. Thus, the existence of adjustable parameters  and  provides greater flexibility to the controller design than the fixed case of a=b=1. Further, the values of  and  are the minimum gains of the controller. Compared with the results proposed in [39], the convergence time and convergence region of the system are much easier to be adjusted. Thus, the proposed scheme is suitable for wider application areas.

Remark 2. Note that Theorem 1 is a generalized form of Lemma 7 for the predefined time stability of integer-order systems. Due to the difference between integer-order calculus and fractional-order calculus, the results of predefined time stability of integer-order systems cannot be directly applied to fractional-order systems. Therefore, a novel fractional-order auxiliary function (20) is designed to solve this problem and guarantee the predefined-time stability of Caputo fractional-order systems.

Next, consider the following definition of the predefined-time convergence for fractional-order systems.

Definition 4. [42,43] Suppose that there is a fractional-order nonlinear system , where  is the system state, and  is a continuous function with . Then, the solution will converge to  with the predefined-time , if  and  hold for , where  can be preassigned.

2.3. System Description

Consider a class of nonlinear nonstrict-feedback MIMO fractional-order switched systems composed of  subsystems. The th  subsystem is expressed by

where    is the system state variable, and  is the system output variable.  stands for the switching signal with  being the number of the subsystems, which is assumed to be a known piecewise continuous (from the right) function with respect to the time. When  and , it means that the th subsystem is activated.  denotes an unknown smooth nonlinear function.  denotes the unknown but bounded external disturbance which is bounded by , where  is an unknown constant , .

Besides,  is the input signal of the controller.  is the nonlinear saturation input of the th subsystem, which is the output of the saturation nonlinearity described as follows [12]:

where  and  are bounds of . To deal with non-tiny at sharp corners, a smooth hyperbolic tangent function  of the th system is denoted as follows:

Accordingly,  in (15) can be decomposed as

where . By the mean-value theorem, we can get   and  with .

By choosing , we have

where  and  is an unknown positive constant.

To design the adaptive tracking scheme for system (14), the following two common assumptions are required.

Assumption 1. For , the reference signal  and its -order time-derivatives are continuous and bounded, i.e., , where  is a positive constant.

Control objective: we will design a practically predefined-time tracking control scheme for system (14) without violation of input saturation such that 1) the reference signal can be tracked by the system output in a predefined time and 2) all the closed-loop signals remain bounded.

3. Main Results

3.1. Controller Design

In this subsection, on account of the backstepping technology, the adaptive fuzzy predefined-time controller of system (14) is constructed for the th activated subsystem.

For  and , define the following coordinates transformation:

where  is a reference signal, and  is the output of the first-order filter in connection with the virtual controller . The filtering error is defined by .

Besides, we introduce the following novel auxiliary function:

where  and .  is a predefined parameter, and  the ratio of two positive odd integers satisfying , which implies that  and . Note that we assume the existence of a nonlinear function  such that there exist strict positive constants  satisfying  and  ( , ). Then, we can deduce that the virtual control function  for  is

where .  and  for .  and  are the design constants.  and . Other parameters will be defined later.

Step 1: When , according to (14) and (19), one can obtain the derivative of  as

Putting  into  and substituting the error dynamics (19) into (23), one has

The candidate Lyapunov function is selected as

where  and  are positive design parameters.  is the parameter estimation error.  is the estimate of . Define  as the parameter error with  being the estimate of . Among them,  and .

By Lemma 6, an FLS  is used to approximate . Based on (23) and Lemma 1, the time derivative of  can be calculated as

By means of Young’s inequality and the property of the fuzzy basis function , one has that

where  and  are positive constants to be designed.

Substituting (27) into (26) yields

The parameter adaptive law is designed as

where , and , .

Recall the definition of  in (20), we have

According to Lemma 2, substituting (21), (29)−(31) into (28) results in

where .

Motivated by the work in [4], in order to reduce the computational complexity inherently embedded in the backstepping procedure, we propose a new predefined-time filter for fractional-order systems. Let  pass through the following predefined-time filter with a constant  and a new state variable . The dynamics of  can be expressed as

where  and  denote positive parameters to be designed. The -th order derivative of the boundary layer errors  is

where  is a continuous function of the variables , , ,  and . Meanwhile, there exists a positive constant  such that  is a given compact set .

Step   According to the coordination transformation (19) and (20), one obtains

by taking the time derivative of .

To complete the induction, at the -th step, we construct the following Lyapunov function:

where  and  are parameters to be designed. Define  and . Among them,  and  represent the parameter estimation errors with  and  being the estimates of  and , respectively.

By (34) and (35), the derivative of  is

Using a similar procedure in Step 1, it can be verified that

where  and  is a positive constant.

Putting together (37), (38) and (39) yields

Design the parameter adaptation functions as

where ,  and .

Similar to the previous step, by the definition of , we have

Substituting (21), (41)−(43) into (40) results in

where .

Step : In what follows, the derivative of  can be formulated by

Consider the Lyapunov function candidate as follows:

where  and  are positive constants.

The following inequalities hold:

where  and  is a positive parameter.

By differentiating the Lyapunov function and using (47)−(50), it can be obtained that

where  and  is a positive constant.

According to (51), the adaptive updated laws are designed as follows:

where ,  and .

From the definition of , substituting (22), (52)−(53) into (51), one has

where .

3.2. Stability Analysis

In this subsection, the stability of the considered system is readily verified. The major result of the proposed design is stated in the following theorem.

Theorem 2. Consider switched fractional-order system (14) with input saturation (15) under Assumption 1. Given predefined-time filter (33), actual controller (22), intermediate control functions (21), and parameter adaptation functions (29), (30), (41), (42) and (52) and (53), then all the closed-loop signals remain bounded and closed-loop system (14) is practically predefined-time stable. Furthermore, the tracking error can tend to a small region of the origin within a predefined time, and the impact of the input saturation is compensated simultaneously.

Proof 2. Applying Young’s inequality to (54) yields , and . Substituting the above two inequalities into (54), one has

where .

Furthermore, applying Lemma 3 to the above inequality, we have

where  and .

Consequently, with the help of Lemma 4, we have

By further substituting (56)-(59) into (55), it can be deduced that

where  .

Let  and . Then, according to Lemma 5, one can easily obtain

According to Theorem 1, it is known that for any , the solution of system (14) will converge to the following compact set:

where the settling-time is given by .

According to the definition of , it is concluded from (61) that the auxiliary function , adaptive parameter errors  and  ( , ), and filtering errors  ( ) reach the boundary of  after a predefined time. Taking into account  and , the boundedness of  and  can be guaranteed. Then, the virtual control functions  and  also remain bounded. Furthermore, from the definitions of the auxiliary function and the coordinate transformation, it can be concluded that all closed-loop signals of system (14) remain bounded. In other words, when the error variables are located at the auxiliary function, we have  for . When , the following equation holds:

Taking the  derivative and using the properties of the RL derivative, the following equation holds:

Based on Lemma 7, we can prove that  is predefined-time convergent. When the auxiliary function satisfies ,  will converge to zero after a predefined-time which is bounded by . From the initial time , the tracking error  is achieved by the predefined-time, and the settling-time upper boundary is predetermined as  by designers. The proof is completed.

Remark 3. In fact, compared with the first-order linear filter commonly used in the dynamic surface control techniques for integer-order systems, the proposed predefined-time filter is designed for fractional-order systems and is able to directly set the convergence time of the filtering error. Such a filter also eliminates the complexity explosion problem caused by repeatedly deriving the intermediate control law during the backstepping recursive design procedure. Owing to the existence of the term , the derivative of the filtering error tends to infinity when tends to zero, and the singularity problem is avoided by using the function in this paper.

Remark 4. is the -order RL derivative. From Definition 1-2 and Property 1, it is known that in (23) coincides with the -th order RL derivative other than the -th order Caputo derivative, i.e. . According to the relationship between the Caputo and RL derivatives, if , we have . It is clear from the above description that, if , one can immediately have . Indeed, the term will monotonically decrease towards as time goes to infinity.

Remark 5. In Eqs. (20), (31) and (43), we suppose that the coefficient is greater than the maximum of the absolute error , which implies that the error , , is a priori bounded. This fact is usually possible since the solution of a fractional-order system is restricted to a bounded domain. Meanwhile, the effective determination of the parameter deserves attention in practice, and relative results can be found in many available applications. For example, the trial-and-error method can be successfully used to determine the actual value of the parameter [32].

Remark 6. So far, most of the results on predefined time control have designed their controllers based on the sign function [39−41, 43]. Although the fast convergence behavior is achieved, the chattering phenomenon still exists. Therefore, similar to [4], we introduce a function to avoid the possible chattering phenomenon. Note that cannot be too small; otherwise, the existence of the term will lead to the occurrence of the chattering phenomenon.

Remark 7. For switched nonlinear systems, the proposed method uses the common Lyapunov function approach (see [17,21, 23−25]) to construct the adaptive switching control law instead of using the multiple Lyapunov function approach and the average dwell time theory [18, 22]. As a result, the switching signal is no longer required to satisfy the predefined dwell time.

Remark 8. In fact, the radius of the tracking error region is determined by the control parameters. To guarantee a small tracking error, it is necessary to increase the values of parameters , and , or decrease the values of parameters , , , , and . Note that if the controller's predetermined time or the saturation value of the selected actuator is not reasonable, then the required stable time exceeds the actuator's capability.

4. Simulations

This section illustrates the applicability and effectiveness of the proposed control strategy by two simulation examples.

Example 1: Consider the following MIMO switched nonstrict feedback nonlinear system [23]:

where  denotes the switching signal. , ,  and . ,   and . We chose the external disturbances as , ,  and . The switching signal is selected as  in ; , otherwise.

The design parameters of this example are chosen as , , , , , , , , , , ,  and . The choice of parameters  depends on the maximum values of the errors. In anticipation, we set . The nonlinear functions  are chosen as . The conditions  are well satisfied. In this example, the Gaussian membership functions are defined as  with ,  and  which are uniformly distributed on . The initial condition is chosen as ,  and . The input saturation limits are set as  and .

The simulation results are shown in Figures 1−6. Figure 1 depicts the tracking performance of the presented control strategy. It is observed that the trajectories of the reference signals  (  = 1, 2) can be tracked rapidly and precisely, illustrating the rationality of the designed filter. Figure 2 shows the trajectories of state variables . The adaptive parameters  and  (  = 1, 2; =1,2) are presented in Figure 3. Figure 4 displays the curves of controllers  and , which are constrained by input saturation with  and . As depicted in these figures, it is obvious that the practical settling time is less than the predefined time . The practical convergence time of the tracking error is much less than . Moreover, for the case that the switching signal occurs before the practical settling time, it follows from Figures 5 and 6 that, the tracking errors and filtering errors at different settling times ,  and  can all be retained to zeros within a small region, but variations exist in the convergence rate. According to Figure 6, we can see that the new filtering algorithm proposed in this paper can directly set the convergence time of the filtering error. Hence, it can be concluded that the proposed control method achieves the control objective successfully.

Figure 1. Example 1: Trajectories of system outputand the reference signal(=1, 2).

Figure 2. Example 1: Trajectories of state variablesand.

Figure 3. Example 1: Trajectories of the adaptive parametersand(=1, 2;=1, 2).

Figure 4. Example 1: Trajectories of the saturation inputand.

Figure 5. Example 1: Trajectories of the tracking errorsandbased on the different settling time.

Figure 6. Example 1: Trajectories of the filtering errorsandbased on the different settling time.

Remark 9. It is seen in Figures 1 and 4 that, achieving fast convergence at the same time implies that the control input needs to be large enough. Therefore, the trade-off between better tracking performance and less control energy is a constant consideration for researchers, where the consideration of the input saturation in system (14) can be treated as a reasonable solution. However, as shown from Figures 4 and 9, the obvious deficiency is that, the control input results are not smooth enough due to switching. Theoretically, the term is similar to the sign function if is too small, which obviously results in the chattering phenomenon.

Example 2: To further demonstrate the feasibility and effectiveness of the proposed scheme in this article, we consider a practically predefined-time control problem [49] of a permanent magnet synchronous motor model with a smooth air gap.

where ,  and  stand for the rotor angular velocity, the current of the  axis and the current of the  axis, respectively.  and  denote the  axis voltage. Assume that only two controllers  and  are applied to system (65). The system parameters are set as , ,  and . In this simulation, we denote ,  and . Then, the fractional-order dynamic equation of this mechanical system can be transformed to a typical form (64) where , , , , , , , , , , , and .

The fractional order is selected as  and the reference signals are assumed to be  and . The design parameters of this example are chosen as , , , , , ,  and . Other parameters are selected the same with Example 1.

Figures 7−11 depict the corresponding simulation results. The qualitative analysis results of this example are identical to that of Example 1. From these figures, we can observe that the tracking error of the presented method can be tuned to a smaller neighborhood of the origin. Moreover, the given upper boundary on the settling-time provides the user with more practical metrics to understand and tune the system performance.

Figure 7. Example 2: Trajectories of system outputand the reference signal,= 1, 2.

Figure 8. Example 2: Trajectories of the adaptive parametersand(=1, 2;=1, 2).

Figure 9. Example 2: Trajectories of the saturation inputand.

Figure 10. Example 2: Trajectories of the tracking errorsandbased on the different settling time.

Figure 11. Example 2: Trajectories of the filtering errorsbased on the different settling time.

5. Conclusion

In this article, the fuzzy adaptive predefined-time tracking control problem has been investigated for a class of switched fractional-order nonlinear systems in the presence of input saturation and external disturbances. A new predefined-time auxiliary function has been constructed to guarantee the convergence of the tracking error to a bounded compact set within a user-defined time. Furthermore, to deal with the complex calculation problem and the singularity problem, a novel predefined-time filter has been designed in the design process of an adaptive controller in order to obtain better tracking performance. In the end, the simulations have been provided to show the effectiveness of the proposed method. In the future, we will extend the proposed approach to deal with the practical prescribed-time convergence problem of the fractional-order nonlinear systems subject to cyber-attacks.

Author Contributions: All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Lusong Ding and Weiwei Sun. The first draft of the manuscript was written by Lusong Ding and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Funding: This work was supported in part by the National Natural Science Foundation of China under Grant 62073189 and Grant 62173207, in part by the Taishan Scholar Project of Shandong Province under Grant tsqn 202211129.

Conflicts of Interest: The authors declare no conflict of interest.