Research Article  Open Access
EventTriggered Adaptive Backstepping Control for StrictFeedback Nonlinear Systems with Zero Dynamics
Abstract
This paper focuses on the problem of eventtriggered control for a class of uncertain nonlinear strictfeedback systems with zero dynamics via backstepping technique. In the design procedure, the adaptive controller and the triggering event are designed at the same time to remove the assumption of the inputtostate stability with respect to the measurement errors. Besides, we propose an assumption to deal with the problem of zero dynamics. Three different eventtriggered control strategies are designed, which guarantees that all the closedloop signals are globally bounded. The effectiveness of the proposed methods is illustrated and compared using simulation examples.
1. Introduction
Nowadays, the control problem of uncertain systems draws more and more attention because of the extensive existence in the practical engineering. Due to the fact that vehicles, robots, or UAVs (Unmanned Aerial Vehicles) are required to work more precisely, there have been plenty of researches towards the control problem. For example, Zhou et al. in [1] proposed two adaptive controllers for uncertain nonlinear systems in the presence of input quantization. An eventtriggered outputfeedback controller is discussed in [2], for output feedback in switched linear systems. A constrained optimization issue for a class of strictfeedback nonlinear systems is proposed in [3].
Among these researches, adaptive control strategy [4] has been proved to be an effective and efficient method for handling uncertainties and nonlinearities. It introduces a parameter estimator to eliminate the influence caused by unknown parameters. Therefore, over the past few years, there have been many adaptive control strategies for uncertain nonlinear systems. For instance, an adaptive controller is proposed for switched nonlinear systems with coupled input nonlinearities and state constraints in [5]. Sun et al. study a class of stateconstrained uncertain nonlinear systems which have steadystate behavior and prescribed transient and design a robust adaptive strategy in [6]. In [7], the authors discuss the problems of adaptive outputfeedback tracking control for a class of uncertain nonlinear systems with output constraints, unmodeled dynamics, and quantized input. An extremum seeking controller is proposed based on a fractionalorder sliding mode in [8], which has a faster convergence speed. Further, Li et al. propose two adaptive control approaches to synchronize the neural networks in finite time in [9]. A fractionalorder sliding mode controller is proposed in [10] for robust stabilization of uncertain fractionalorder nonlinear systems. For more applications, see [11â€“18].
In the practical systems, input energy is obviously limited. Therefore, energysaving should also be considered in order to extend the systemsâ€™ lifespan. In the past, the timetriggered scheme is widely used, which means the actuation is executed at every periodic instant even the systems do not need any actuations, and it requires abundant times of experiments to find the most suitable parameters. Therefore, it is impractical for many resourcelimited systems. The eventtriggered control scheme, which is first proposed in [19], can greatly reduce the control execution times and save the computation costs. In eventtriggered systems, actuations are not executed periodically, but only when triggering conditions are satisfied. Furthermore, once using the eventtriggered control scheme, the Zeno behavior must be taken into consideration. It is also of great importance to find a way to avoid the Zeno behavior to ensure that the system will not make any mistakes while running. Recently, some eventtriggered schemes are proposed to handle different problems. For example, an eventtriggered control method is proposed in [20] for parametric strictfeedback nonlinear systems involving backstepping and Lyapunov theory. Xing et al. consider the problems of eventtriggered based adaptive control for uncertain nonlinear systems in [21] and propose three eventtriggered strategies, while Su et al. design an eventtriggered controller by using fuzzy logic method in [22]. In [23], by using inputtostate stable (ISS) condition, a statefeedback eventtriggered scheme was proposed and the closedloop system is guaranteed to be asymptotically stable. Wang et al. design a decentralized eventtriggered scheme in [24] for both linear and nonlinear subsystems considering transmission delays. While in networkedcontrol systems, an eventtriggered scheme for robust set stabilization is proposed in [25]. What is more, eventtriggered schemes also show advantages in consensus and synchronization of multiagent systems in [26â€“29]. Besides, some results on eventtriggered control have also been applied on real systems (see in [30â€“37]). They extend the range of applications of the eventtriggered control scheme.
The zero dynamics, which are well understood and extended to nonlinear finitedimension systems in [38], are the dynamics of the system when choosing a certain input that makes output be identically . It is an expansion of the zeropoint concept in linear systems and it is an intrinsic character of nonlinear systems which describes how internal states act when the output is 0. The stability of zero dynamics in sampleddata nonlinear systems is widely discussed in [39, 40], while a funnel controller is proposed in [41] for linear systems considering zero dynamics. For stochastic systems, a controller is designed in [42, 43] using eventtriggered scheme and backstepping method based on Lyapunov theory. In continuous nonlinear systems, since the zero dynamics are related to the systems dynamics and the minimum interexecution time which is also influenced by the systems dynamics, has to be considered, it increases the complexity to design a controller for such systems. However, to the best of our knowledge, all of the existing eventtriggered control strategies were designed without considering zero dynamics in nonstochastic systems. The existence of unstable zero dynamics may restrict the control performance, such as sensitivity, robustness, and stabilization. And the stabilization of zero dynamics in continuous systems is harder to keep than in discretetime systems [44]. Therefore, it is of great significance to study the stability of zero dynamics for analyzing the internal states of nonstochastic and consensus nonlinear systems.
Inspired by above discussions, an eventtriggered based adaptive control is proposed for strictfeedback nonlinear systems with unknown parameters and zero dynamics. Compared with the existing results, the main contributions are given as follows:
The problem of eventtriggered adaptive control is first considered for a class of continuous nonlinear systems with zero dynamics. The nonlinearity part in the system does not have to be globally Lipschitz.
Three different triggering conditions are designed, and it is proved that the proposed controllers can make the system asymptotically stable. Then, we prove that every actuation is triggered in finite time, which means the Zeno behavior is successfully avoided.
The rest of the paper is organized as follows. Section 2 proposes the problem formulation and gives the objective of this note. Then, three different design schemes and main theoretical results are described in Section 3. Section 4 gives the simulation results. The conclusion is summarized in Section 5.
2. Problem Formulation
Consider a class of strictfeedback nonlinear systems with unknown parameter and zero dynamics:where denotes the zero dynamics of the system. and are known nonlinear functions with respect to . And, are the states of the system, while is the system input. is a vector which contains all the unknown parameters in the system. and are known nonlinear functions with th order () smooth with respect to . For convenience, we define and as and , respectively.
Assumption 1 (see [45]). For the zero dynamics in (1), there exists a smooth and definite positive function and a smooth statefeedback control law , with , such that
Since system (1) contains the zero dynamics and unknown parameters, the existing control approaches are not suitable. Therefore, the control objective of this work is to design an adaptive law and control signal for the system (1), which make the output asymptotically stable, and all the states in the systems are required to be globally bounded.
3. EventTriggered Control Scheme
Similar to [21], three different adaptive controller design strategies are proposed based on different eventtriggered conditions, that is, fixed threshold strategy, relative threshold strategy, and switching threshold strategy. In this research, the considered control system is affected by the zero dynamics which exists widely in many practical applications. For each strategy, we start with system (1) using the backstepping technique which is carried out with the help of coordinates transformation:where denote the virtual control laws and are the dynamics of the new coordinates. The design process is started as follows.
Step 1. From the coordinate transformation , the time derivative of iswhere .
Then, choose a Lyapunov function aswhere is the estimation error, denotes the estimation of the uncertain constant vector , and denotes a positive definite matrix. Therefore, the derivative of iswhere . Next, choose a virtual law aswhere is a positive constant. Further, substituting (7) into (6) results inwhere is a tuning function.
Step 2. The derivative of is given aswhere .
Then, the Lyapunov function is chosen as . The derivative of is
Similar to Step 1, we choose the virtual law aswhere is a positive constant and is a positive constant. Further, substituting (11) into (10) results inwhere is a tuning function.
Step . In this step, we repeat the procedure in a recursive way. The derivative of is given as
Then, the Lyapunov function is chosen as , and the derivative of is
The virtual law is chosen aswhere is a positive constant. Next, substituting (15) into (14) results inwhere is a tuning function.
In the last step , , which means the actual control input appears and it is at our disposal. The objective is now to design an appropriate controller and the triggering event.
3.1. Fixed Threshold Strategy
Under this strategy, the adaptive controller is proposed asand the triggering event is given aswhere denotes the error between the current control signal and last control signal. , and are all positive constants with , is the update time of the controller, which means when (20) is triggered, the time will jump to , and the control value will be applied to the system. In the period between and , the control value will keep being and will not be changed.
Now we analyze the stability of the eventtriggered controller in system (1), and the design process to avoid the Zeno behavior is also showed. The theorem and proof are proposed as follows.
Theorem 2. Consider a certain of nonlinear closedloop system (1) and adaptive controller based on the eventtriggered scheme (17)(20) under Assumption 1. The system is globally stable; that is, all the signals are bounded, and the system can be adjusted by selecting proper parameters. In addition, there must be a time such that is greater than or equal to the execution period , , which means the Zeno behavior is successfully avoided.
Proof. The Lyapunov function is chosen aswhere . During the execution interval , we have . Assuming that there exists a continuous timevarying parameter , which satisfies , , and , such that . For convenience, and are defined as and . With this in mind, we haveBy recalling , substituting (17) into (22) givesNotice that . Then, satisfiesBased on the property [46], the hyperbolic tangent function haswhere and . As above, we design as . Based on the Assumption 1, its derivative is given bywhere is selected based on [47]. Let , where is the maximum eigenvalue of . And with the help of Youngâ€™s inequalities in [37], Finally, let Then, we getBased on the LaSalleâ€™s invariance principle in [48] and the result in [46], , and are proved to be bounded. Therefore, it can be concluded that the control signal is also bounded. That is, all the closedloop signals are globally bounded. Then, assuming that there exists a such that , and noticing , the following can be verifiedFrom (17), we get that . Since and are at least th order smooth functions, is continuous. Next, since is a function of and , and all the closedloop signals are globally bounded, there must be a constant such that . Noticing and , there exists a lower bound of which satisfies ; that is, the Zeno behavior is successfully avoided.
3.2. Relative Threshold Strategy
In practical systems, when considering a stabilization problem, the value of the control signal always requires to be considered. In the fixed threshold strategy, no matter how big the control magnitude is, the error is always bounded by a given constant, which may not be applicable to all practical systems. Based on this consideration, a varying threshold comes to mind. Therefore, the following relative threshold control strategy is proposed:and the triggering event is given aswhere and are all positive parameters. is added specially to guarantee the Zeno behavior will not happen, which will be elaborated in the following proof. Using the control strategy, it reaches the following theorem.
Theorem 3. Consider a certain of nonlinear closedloop system (1) and adaptive controller based on the eventtriggered scheme (29)â€“(32) under Assumption 1. The system is globally stable; that is, all the signals are bounded, and the system can be adjusted by selecting proper parameters. In addition, there must be a time such that is greater than or equal to the execution period , , which means the Zeno behavior is successfully avoided.
Proof. From (32), we can get that during the interval . Similar to the fixed threshold strategy, and are timevarying parameters which satisfy and Therefore, it reaches that . For convenience, , , and are defined as , , and . Then, we getSince and , , it can be obtained that . Notice that and . Therefore, similar to (24), substituting (29) into (33) givesThen, substituting (15) with into (34) givesSimilar to (27), it reacheswhere . As the proof in Theorem 2, all the closedloop signals are globally bounded, and the interexecution time , which means the Zeno behavior is successfully avoided.
3.3. Switching Threshold Strategy
In the above, the relative threshold strategy shows its advantages. For example, when the value of control signal is large, the triggering threshold is large too, which ensures that the system can get longer update intervals, while when the is close to zero, the system can get precise control. Nevertheless, once the value of control signal is excessively big, the system will get excessively large errors of control signal. Therefore, the control signal will jump suddenly when an event is triggered, which will give the system a large impulse and affect the performance of the system. Hence, with these considerations, we give the switching threshold strategy as follows:where is a positive constant parameter designed by users and , and are same parameters which are designed before. This strategy is a combination of first two strategies. From (38), it can be seen that when the value of control signal is less than , the relative strategy is applied; therefore the system will obtain a more accurate control. While when the value of is high, the control method switches to the fixed threshold strategy so that the error will ensure the system performance. Apparently, the advantages of the fixed threshold strategy and the relative strategy are brought to this strategy.
In this strategy, it can be obtained that
Combining the switching threshold strategy above and (39), the following results are stated.
Theorem 4. Consider a certain of nonlinear closedloop system (1) under Assumption 1 and a switching adaptive controller between the fixed threshold and relative threshold with a userdesigned parameter with . The system is globally stable; that is, all the signals are bounded. And the system can be adjusted by selecting proper parameters. In addition, there must be a time such that is greater than or equal to the execution period , , which means the Zeno behavior is successfully avoided.
Proof. Since the same control law of the first two strategies are adopted by the switching threshold strategy, the system is also globally stable, which means all the signals are bounded. And by letting , the interexecution satisfies . That completes the proof.
4. Simulation Results
Example 1. In this section, based on the system given in [49], we combine zero dynamics with it. A stabilization case is illustrated with three proposed eventtriggered strategies and timetriggered strategy, and each strategyâ€™s performance is compared. Consider the following systems:where denotes an unknown parameter. Our control goal is now to make the system output asymptotically back to zero. We use the above four strategies to balance the performance of the system and the times of triggering events. The design constants are chosen as follows: Fixed threshold strategy: . Relative threshold strategy: . Switching threshold strategy: . Timetriggered strategy: interexecution time: . The other parameters are chosen as And since and are known, we define that . The Lyapunov function is chosen as ; therefore we get
Let Thereforeand is chosen as
Figure 1 shows the stabilization performance of the output signal in three eventtriggered strategies and timetriggered strategy, and the control signals and are shown in Figure 2. The times of triggering events is shown in Table 1, while state signal and the parameter estimate are, respectively, shown in Figures 3 and 4. It can be seen that the fixed threshold strategy has the best stabilization performance due to its relatively small threshold, but it has the most triggering events. On the contrary, the relative threshold strategyâ€™s performance is not relatively good, while it has the fewest triggering events. And the switching threshold strategy gives a flexibility between the performance of system and times of triggering events. For timetriggered strategy, though it has the best performance at last, it also has the most triggering times, which may result in unnecessary high workloads in practical systems. Besides, the system does not perform well at the beginning.

Example 2. Consider a higher order networkedcontrol system based on [45] with zero dynamicswhere denotes an unknown parameter. The above three eventtriggered strategies are applied to the system. The design constants are chosen as follows: Fixed threshold strategy: . Relative threshold strategy: . Switching threshold strategy: . The other parameters are chosen as And and are defined as .
Figure 5 shows the stabilization performance of the output signal in three eventtriggered strategies, and the control signals and are shown in Figure 6. The time of triggering events is shown in Table 2, while state signals and and the parameter estimate are, respectively, shown in Figures 7, 8, and 9. In this simulation example, the switching threshold strategy provides the smallest triggering times, but its performance is not better than the other strategies, while the fixed threshold strategy provides the best performance with the most triggering events. And the relative threshold strategy balances the performance and times of triggering events. It can be seen that the parameters can be designed by users to find the most suitable strategy for different practical systems under different situations.

5. Conclusion
This paper has considered an eventtriggered adaptive backstepping control for strictfeedback nonlinear systems with zero dynamics. An adaptive control method is utilized to overcome the unknown system parameters, by which all the closedloop signals are guaranteed to be uniformly bounded. Besides, to overcome the need of ISS assumption, an adaptive controller and eventtriggered technique are designed at the same time. And three different eventtriggered control strategies are also given. It is shown from the simulation results that all the strategies guarantee that the stability error approaches to a small neighborhood 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 in part by the Taishan Scholar Project of Shandong Province of China under Grant 2015162 and Grant tsqn201812093.
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