Abstract
This work investigates a finite-time observer problem for a class of uncertain switched nonlinear systems in strict-feedback form, preceded by unknown hysteresis. By using a finite-time performance function, a finite-time switched state observer (FTSO) is derived using radial basis function neural networks (RBFNNs) to estimate the unmeasured states. An adaptive feedback neural network tracking control is derived based on the backstepping technique, which guarantees that all the signals of the closed-loop system are bounded, the output tracking error converges to zero, and the observer error converges to a prescribed arbitrarily small region within a finite-time interval. In addition, two simulation studies and an experiment test are provided to verify the feasibility and effectiveness of the theoretical finding in this study.
1. Introduction
In the past decades, great attentions and developments have been gained in finite-time adaptive control design, which developed a great number of typical design approaches in the literature (see, e.g., [1–15]). A variety of different types of complex nonlinear systems have been explored by using neural networks or the fuzzy logic system-based adaptive backstepping technique. For instance, in [1–3], by utilizing the finite-time stability theory, barrier Lyapunov functions, and the adaptive backstepping method, various finite-time adaptive control strategies were proposed with unknown dead-zone, unknown hysteresis, and unknown time-delayed in the nonlinear system, respectively. Considering the full-state constraints in the nonlinear systems, an adaptive fuzzy controller was constructed to address the finite-time tracking control problem for a class of strict-feedback nonlinear systems in [4]. Besides, Wang et al. [5] developed a novel adaptive neural finite-time control strategy which considered quantized problems for single-input and single-output quantized nonlinear systems. Meanwhile, growth condition assumption was removed under the presented control approach. That assumption was also eliminated in [6]. Li et al. [6] presented a finite-time fuzzy adaptive output feedback control for the first time for nonstrict-feedback nonlinear systems, while it overcame the problem of explosion of complexity by using the dynamic surface control technique. Moreover, Yu et al. [7] investigated an adaptive finite-time quantized tracking problem with input and output quantization and asymmetric actuator saturation, and they designed an observer-based adaptive finite-time quantized tracking controller at the same time. In [8], Liu et al. studied a finite-time adaptive fuzzy tracking control problem and proposed a finite-time adaptive state feedback fuzzy tracking controller with a new concept named finite-time performance function. Although a novel assessment standard of the finite-time semiglobal practical stability (SGPS) was found in [1–8], the main innovations of those literatures are designed controllers for nonswitched systems. Sui et al. [9] solved a finite-time switching control issue for nonstrict-feedback nonlinear switched systems, and they also constructed a novel finite-time SGPS for switched systems. Meanwhile, by utilizing a comparison theorem and a mean value theorem of integrals, a significant finite-time stability criterion for stochastic nonlinear systems was first set up in [10]. Chen et al. [11] studied an adaptive finite-time synchronization controller given for multiple robotic manipulators. It was worth noting that there existed some controllers to solve finite-time control problems in [12–15].
As a class of nonlinear inputs, hysteresis widely exists in the industrial system. Besides, the nonlinear characteristics of backlash-like hysteresis could seriously affect tracking performance, and it may cause a severe effect on the whole system. To capture hysteresis dynamics and achieve precise control, many scholars have made efforts to eliminate its effect. The hysteresis models can be roughly divided into two classes in [16, 17]: the operator-based hysteresis model [17–19] and the differential equation-based hysteresis model, such as Bouc–Wen model [20, 21] and backlash-like model [22–26]. For instance, in [18], Mayergoyz described a new approach to the scalar Preisach model of hysteresis which emphasized its phenomenological nature and mathematical generality. Zhang et al. [19] designed a hysteresis compensator to compensate the hysteresis nonlinearity described by a generalized PI hysteresis model, and the Krasnoselskii–Pokrovskii model was applied in [17]. The modeling accuracy can be guaranteed by increasing the number of superposed elementary operators for the operator-based hysteresis model, but also cause the computational burden during implementation. We continue to investigate the backlash-like hysteresis model in this paper since it has fewer parameters and has analytical solution. In [22], Ma et al. dealt with adaptive control for a class of switched nonlinear systems preceded by unknown backlash-like hysteresis, where the hysteresis was modeled by a differential equation. Similarly, Li et al. [23] investigated an adaptive neural output feedback control for a class of nonlinear systems with unknown backlash-like hysteresis of the actuator. Zhang and Lin [24] proposed a robust adaptive DSC control for a class of uncertain perturbed strict-feedback nonlinear systems preceded by unknown backlash-like hysteresis. On one hand, it is an active issue to study the systems with hysteresis nonlinearities which exist widely in the practical systems, such as manipulator system [22, 27], four-motor servomechanism [28, 29], and electrical circuit [30]. On the other hand, to date, there are no systematic methods to achieve a satisfactory result on finite-time switched state estimate of switched nonlinear systems with backlash-like hysteresis, which motivates our research interest. Hence, considering the inevitability of the backlash-like hysteresis constraint in many practical applications, it is worth further studying how to design a finite-time switched state observer (FTSO) for switched strict-feedback nonlinear systems with unknown backlash-like hysteresis.
During the development of control and applications, there exists a design algorithm which links observer design and hysteresis compensation. In [31], a novel hysteresis compensation method based on extended high-gain observer was presented without any specific hysteresis models. Besides, there exist several seminal observers designed with a system of hysteresis results [30, 32–34]. Huo et al. proposed an MIMO switched fuzzy observer to estimate the system states with hysteresis nonlinearities, which was less conservative than using a common observer for all subsystems. As stated in [33], they addressed the consensus tracking problem of a class of nonlinear multiagent system with hysteresis. And then, they developed a distributed adaptive neural output feedback control scheme proposed by constructing a state observer and using the backstepping technique. In order to improve the functional approximation capability and disturbance compensation ability for a system with unknown backlash-like hysteresis, Wang et al. [34] considered the coupler design between the radial basis function neural network and observer. Wei et al. [30] presented an extremum seeking algorithm to accurately estimate the state-of-power by an electrical circuit incorporating hysteresis effect. However, those contributions in [30–34] do not include finite-time convergence of observers, which further improves the tracking effect of the control strategy.
This study considers the finite-time adaptive switched observer problem for a kind of nonlinear systems with unknown hysteresis from a new perspective. Based on RBFNN approximation and finite-time performance function, the FTSO is proposed, which guarantees that observer error converge to a small neighborhood of the origin point in finite-time. Meanwhile, an adaptive tracking neural network controller is designed by the backstepping technique, which guarantees that all the signals of the switched closed-loop system are bounded and the observation error is converged to a small neighborhood of the origin point in a finite time. Compared with the existing results, the prominent contributions of this work are as follows:(1)This paper proposes design process of a novel finite-time switched state observer which is designed to improve control performances for the switched system with unknown backlash-like hysteresis. In contrast with [1–15] which studies finite-time adaptive tracking controller, the smaller tracking error will be achieved with an FTSO for the same controller. In addition, with the presented control approach, all known state assumptions are removed by using the FTSO.(2)To the best of our knowledge, it is the first time that the finite-time convergence problem of observer error is taken into consideration in a class of uncertain switched nonlinear systems with unknown backlash-like hysteresis, by using a finite-time performance function to obtain a better tracking performance. For switched systems with unknown backlash-like hysteresis, a control strategy with the FTSO can be applied more effectively and exactly.
This paper is organized as follows. Sections 2 presents the problem and the preliminaries. In Section 3, an adaptive finite-time switched state observer is designed. Section 4 draws control design and stability analysis. Simulation studies are conducted in Section 5 to illustrate the effectiveness of the proposed scheme. Finally, the paper is ended by concluding remarks in Section 6.
2. Problem Formulation and Preliminaries
This section begins by presenting the control problem for a class of uncertain switched nonlinear systems with unknown backlash-like hysteresis. Then, it reviews some preliminaries about RBFNNs to facilitate the proposed observer and controller design procedure. Finally, a finite-time performance function is given to ensure observer error finite-time convergence.
2.1. Proposed System Descriptions
Consider a class of switched strict-feedback nonlinear systems with unknown hysteresis in the form ofwhere , , denotes the vectors of system states, is the output of the switched nonlinear system, are smooth unknown nonlinear functions, and are known constants and . expressing the switching signal, which is a piecewise right continuous function. In addition, denotes the subsystem. stands for the bounded external disturbance. is the output signal of backlash-like hysteresis described aswhere indicates the input of the unknown backlash-like hysteresis. , and are unknown constants with .
It follows from the analysis in [22, 23, 35] that the solution of system (2) is expressed aswhere is bounded and satisfieswhere and are the initial values of and , respectively.
Figure 1 shows that a typical backlash-like hysteresis is generated. By (3), rewrite (1) aswith and .
This paper aims to derive an FTSO-based adaptive neural control signal for nonlinear system (1) with unknown backlash-like hysteresis (2), so that the following objectives can be achieved:(1)Observer errors converge to an arbitrarily small specified neighborhood at a finite-time(2)All the variables of the closed-loop system are bounded, and the tracking errors could converge to an arbitrarily small specified neighborhood
Lemma 1 (see [36, 37]). For a nonlinear switching system of , denotes the number of times and switches within a time interval . Ifwhere and are positive constants, then is average dwell time. In order to facilitate calculation, the buffeting bound is generally taken.
Lemma 2 (see [36, 37]). A Lyapunov function candidate is defined aswhere is switched in accordance with the piecewise constant switching signal . Then, the following properties are obtained:(1)Each in (7) is continuous. There exist constant scalars such that(2)There exist constant scalars such that
2.2. Radial Basis Function Neural Networks
The RBFNN has the ability to approximate the unknown smooth function , and the form of is as follows:where and are the weight vectors of the neural network, are the basis function vector of the neural network, and is approximation error of neural network function, and satisfies that , where is the upper bound of the estimated error.
In the following, is used to represent and is used to represent , where the estimated error of the observer and the controller are expressed as and , respectively.
The RBFNN is an approximator for nonlinear smooth function which is approximated aswhere is the estimation of .
The ideal parameter vectors are defined aswhere and are compact regions for and , respectively. Define the minimum approximation error as
Lemma 3 (see [38]). The following inequality holdsfor any inter , where and .
2.3. Finite-Time Performance Function
Definition 1. Different from a classical prescribed performance function in [39], a smooth function is called finite-time performance function (FTPF) for all . There exists a settling time , such that , which is an arbitrarily small positive number. , in this work, is defined bywhere λ, τ, , and are chosen as positive constants, and let , where p and q are chosen positive constants. Moreover, it has .
3. Finite-Time Switched State Observer Design
In this section, an FTSO is derived, involving the use of FTPF. Stability of the developed observer, estimating the states of switched nonlinear system (1) with unknown hysteresis (2), is proved by Lyapunov function.
For the switched nonlinear system (5), an FTSO is designed as follows:where is the estimation of real system states. and , are parameters to be designed. Furthermore, combining (16), the transfer error of the observer are further expressed as
Further, define the observer error as
From (16) and (18), the differential with respect to t of becomeswhere is the adaptive parameter error.
Define the error transform function aswhere . is a smooth and strictly increasing function and . It also satisfies and . Define a new function aswhich satisfies , when . Combining (20) and (21),
Differentiating (20) with respect to t yields
Then, the dynamic system errors are expressed aswhere .
Step 1. Construct a Lyapunov function candidate asDifferentiating (25),Computing givesThen, is rewritten asComputing givesCombining (27)–(29) into (26),In the light of Young’s inequality and the characteristic of , we haveBy substituting (31) into (30), it follows
Step 2. Consider the Lyapunov function candidate asSimilar to Step 1,We augment the Lyapunov candidate asSynthesizing (32), (34), and (35),Stability analysis is described in the following two parts.
Case 1. and have the same sign (such as both and are positive or negative).
Computing givesSubstituting (37) into (36), one obtains
Case 2. and have opposite signs.
Computing gives
Theorem 1. Consider system (5) with unknown backlash-like hysteresis under Definition 1 and error transform function (20), applying the proposed FTSO mechanism in (16), observer errors (18) converge to a small neighborhood of the origin point in a settling time .
Proof. Substituting (39) into (36), one obtainsFor simplification of the expression, reconfigure the parameters , , and . Futhermore, denote and . The Laypunov function consequently takes the formIntegrating (41) from 0 to t producesTherefore, the observer errors converge to a small neighborhood of the origin point in a finite-time interval , and all of the closed-loop systems signals are bounded.
This completes the proof of Theorem 1.
4. Adaptive Controller Design and Stability Analysis
In this section, an adaptive control scheme is developed for nonlinear system with unknown backlash-like hysteresis by using the above FTSO and the backstepping technique. To facilitate the design procedure, the following errors are defined aswhere are virtual intermediate control signals which will be defined later.
The feedback control laws (44), (45), and adaptive laws (46) are chosen aswhere . , , , , , and are the design parameters.
Step 1. Consider a Lyapunov candidate function in the form of where , with being the estimation of .From (16) and (43), one obtainsFrom (48) and (49), it is obtained that isBased on Young’s inequality and Lemma 3, the following inequalities are obtainedSubstituting (44), (46), and (51) into (50) givesVia Young’s inequality, we havewhere . , , and are bounded.
Thus, (52) is modified aswhere .
Step i (). Similar to Step 1, consider a Lyapunov candidate function in the form ofwhere .From (43) and (45), one obtainsFrom (57) and (58), is given byBased on Young’s inequality and Lemma 3, the following inequalities are obtained:Thus,From Young’s inequality,where and are bounded.
Substituting (62) and (63) into (61) giveswhere .
Step n. Similar to Step i, consider a Lyapunov candidate function in the form of where .
Similar to Step i, one obtainsMoreover, the time-derivative of is given byBased on Young’s inequality and Lemma 3, the following inequalities are obtained:Substituting (45), (67), and (68) givesBy using Young’s inequality,where and are bounded.
Thus, (69) is modified to bewhere .
A Lyapunov function is chosen asThe differentiation of equation (73) isBased on Young’s inequality, it is obtained thatwith .
Thus, equation (74) is modified asFurthermore, denote and . The Laypunov function consequently takes the form of
Theorem 2. For switched strict-feedback nonlinear systems with unknown hysteresis (1), FTSO (16), feedback control laws (44) and (45), and adaptive laws (46), if the average dwell time of the switching signal satisfies the condition , the control scheme based on FTSO ensures that all variables of the closed-loop system are bounded. By properly selecting the parameters, the controller and observer error converge to origin, such that and .
Proof. There is a piecewise differentiable function in any interval . Then, its time differentiation on isFrom (76) and (78),Then, noting that at the switching point from Lemma 2, where , it follows thatLet and , then equation (80) is further expressed asSince and Lemma 1, for any ,Substituting (82) from (81),By Lemma 2 and (83), one hasTherefore, from (84), it follows thatSimilarly, one obtainsThat is, using (85) and (86), observation errors and tracking errors converge to zero.
5. Simulation Results
In this section, three examples are considered to test the proposed control method. Consider the following second-order nonlinear system:where , , , , , , , , , , , , , and .
The selection guidelines of the design parameters in the proposed control design method are summarized in the following: Step 1. During selection of design parameters of observer (16), the parameter are selected before . Then, the values are usually chosen to ensure . Step 2. Determine the parameters of finite-time performance function (15) in FTSO (16). In this step, one needs to choose finite-time , the initial values , and the final value . Then, and are chosen. Moreover, it is obtained that . Step 3. Select controller parameters of , , and to ensure , which is defined in (76). Step 4. Initial values of the system states and observer are set according to the proof of Theorem 1. The appropriate initial conditions are given in order to make the contrast effect of control and observation more clear. In this simulation, and .
For better illustration, we consider such two cases: (1) the proposed adaptive controller scheme with FTSO and (2) the proposed adaptive controller scheme with the observer in [12]. The reference signals are selected for and . The basis function vector is chosen as the form of the Gaussian function:with five nodes, the width and , and the centers are equally distributed between .
The chosen control parameters are , , , , , , , and and , , and . The FTSO parameters are selected as , , , and , and the other observer parameters are , , , and .
Example 1. .
Switching signal is shown in Figure 2, and the observer errors with the FTSO and the observer in [12] are shown in Figure 3. The tracking performance of the switched system are shown in Figure 4, while the tracking errors and the control input of two schemes are shown in Figures 5 and 6, respectively.
Example 2. .
The tracking responses and errors with the FTSO and the contrastive observer are depicted in Figure 7. Furthermore, in order to further analyze the performance of the tracking effect with the same controller in Figure 7, the performance evaluation, including average tracking error and average fluctuation of tracking error, is summarized in Table 1.
Figures 8 and 9 show the observer error compared and the control input for two cases, respectively.
Example 3. In order to provide the convictive proof of the proposed control approach, a test is proceeded on a servo test rig (as shown in Figure 10), which is given as follows:where and denote the displacement of motor subsystem and the load subsystem, respectively. is the transmitted torque. is the driving torque of the system, and is the disturbance torque at the driven gear. , , , and .
The reference signal , where parameters are selected as , , , and . The initial state of the switched system and observer are , , , and .
The trajectory shown in Figure 11 is the estimation of output and reference signal and the tracking error. Figure 12 shows the trajectories of system output and and the error between system output and estimation of output.
6. Conclusion
In this paper, the adaptive switched control with a finite-time switched state observer has been investigated for a class of uncertain nonlinear systems with unknown backlash-like hysteresis in a strict-feedback form. A finite-time switched state observer has been adopted to estimate the unmeasurable states and obtain a better tracking performance, with the help of finite-time performance function and neural network technologies. By transforming the tracking error into a virtual variable and based on the backstepping technique, an adaptive control method has been proposed. It has been proved that the observer error is bounded and finite time is convergent with the proposed control scheme. Moreover, the signals of the resulting system are bounded and the output tracking error is convergent to a small compact set of the origin.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant nos. 61703224, 61620304, and 21606141.