This paper focuses on the adaptive neural network (NN) control problem for nonlinear quarter active suspension systems with actuator failure. By using Butterworth low-pass filter (LPF), the second order active suspension system is converted to a fourth order system, which solves the problem of zero dynamics analysis in the second order system. Based on the adaptive backstepping technique, considering the actuator fault of vehicle, the corresponding fault tolerant controller is designed. At the same time, the unknown smooth functions are estimated by the NN. It is proved by stability analysis that all states in active suspension system are bounded. Finally, a simulation example is given to verify the effectiveness of the proposed method in a quarter active suspension system.
Citation: Xing Zhang, Lei Liu, Yan-Jun Liu. Adaptive NN control based on Butterworth low-pass filter for quarter active suspension systems with actuator failure[J]. AIMS Mathematics, 2021, 6(1): 754-771. doi: 10.3934/math.2021046
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Abstract
This paper focuses on the adaptive neural network (NN) control problem for nonlinear quarter active suspension systems with actuator failure. By using Butterworth low-pass filter (LPF), the second order active suspension system is converted to a fourth order system, which solves the problem of zero dynamics analysis in the second order system. Based on the adaptive backstepping technique, considering the actuator fault of vehicle, the corresponding fault tolerant controller is designed. At the same time, the unknown smooth functions are estimated by the NN. It is proved by stability analysis that all states in active suspension system are bounded. Finally, a simulation example is given to verify the effectiveness of the proposed method in a quarter active suspension system.
1.
Introduction
The exponential increase in wireless applications with better quality of service (QoS) in the B5G radio system will demand a rigorous prerequisite, apart from low latency and the ability to handle massive devices and enhance mobile broadband. The requirements of the different sectors will be different and require complex technology to accomplish them. Hence, there will be more challenges to deploying a B5G radio network. There are advanced methods such as m-wave, M-MIMO and reconfigurable intelligent systems (RIS), which can be integrated with B5G to enhance the QoS. The utilization of RIS in M-MIMO will perform accurate analysis of the users with low power consumption and high spectral efficiency, and also reduce the complexity as no A/Ds are required [1]. It is also seen that the channel state information will play an important role in MIMO, and it will become more complex for the single antenna user equipment [2]. The authors in [3] proposed a novel algorithm that can efficiently determine the channel state of the framework with minimal complexity. The utilization of the cellular framework has been tremendously increasing, but the availability of the spectrum is becoming more and more congested. Hence, it is important to explore the new spectrum band. The congestion in the spectrum is also due to the static spectrum sharing regulation of the Federal Communication Commission (FCC) [4]. It is observed that the spectrum is not utilized in an efficient manner, and more than 70% of the spectrum is wasted [5]. The spectrum remains idle in many applications; it is not accessed 24/7 by the users. The idle spectrum cannot be shared with other devices. The low spectral access of the bandwidth can be solved by using cognitive radio (CR) [6]. CR is based on the spectrum sensing (SS) algorithms, which can identify idle bandwidth and allocate the idle spectrum to other devices and subscribers. The CR detects the spectrum from primary users (Pu), also known as licensed users, and shares the spectrum with the secondary user (Su), also known as an unlicensed user [7]. The allocation of spectrum from Pu to Su should take place without any interference, and the quality of service (QoS) should not be compromised. In order to identify idle bandwidth, the Pu should be continuously monitored and analyzed [8]. In the latest decade, it has been seen that CR-based spectrum sensing algorithms such as energy detection (ED) [9], matched filter detection (MF) [10] and cyclostationary detection (CS) [11] have been comprehensively investigated. The primary task of the SS algorithm is to locate idle spectrum. The idle spectrum is identified by estimating the energy of the device and comparing it with a predefined threshold value. If the energy of the device is greater than or equal to the defined threshold, it indicates the availability of the spectrum, and vice versa [12]. In the conventional SS algorithms, the sharing of spectrum from PU to Su takes place in the absence of the Pu. If Pu becomes active, then the spectrum should be reallocated to the Pu user, and Su should also get the spectrum from any idle Pu to continue its applications. Spectrum sensing is the process of identifying the existence or absence of Pu in a certain frequency group. In GSM (Global System for Mobile Communications), spectrum sensing is a crucial aspect of cognitive radio technology, which enables SU to use the frequency when it is not being used by PU. Random sampling is one approach used for spectrum sensing in GSM. In this approach, a secondary user randomly samples the conventional signal and equates it with the threshold to detect the presence or absence of primary users. Random sampling is a simple and effective technique, but it can be affected by noise and fading, which can lead to false alarms or missed detections [13]. Intelligent agent (IA) approaches have been proposed to address the limitations of random sampling. IA-based spectrum sensing uses autonomous agents that can learn from the environment and adapt to changing conditions. The agents can gather information about the spectrum usage patterns and make decisions based on that information. IA-based approaches can improve spectrum sensing performance by reducing false alarms and missed detections [14]. Modeling cognitive radio in GSM scenarios using Petri nets could involve developing a Petri net model that captures the dynamic behavior of the cognitive radio system as it interacts with the GSM network. This could involve exhibiting cognitive radio's sensing and decision-making processes, as well as its interactions with the GSM network's control channels and radio resources. Overall, cognitive radio has the ability to enhance the proficiency and reliability of wireless communication structures. Using Petri nets to model these systems can help researchers better understand the complex dynamics involved and identify opportunities for optimization and improvement [15]. The authors in [16] proposed a hybrid spectrum sensing technique that combines random sampling with a Q-learning algorithm to improve the performance of spectrum sensing in noisy environments. The Q-learning algorithm is used to learn the optimal sampling rate and threshold level for spectrum sensing. The signal was sampled using both homogeneous and random sampling, then it was rebuilt. The article in [17] used a synthetically generated signal to evaluate its condition in the GSM channel. At various SNRs, the outcomes in regards to detection curves are demonstrated. In [18], a Petri net-based model for cognitive radio networks is designed, which includes both the Cr and the PU. The model is then used to evaluate the efficiency of the Cr network in terms of throughput and delay. The authors in [19] designed a colored Petri net-based model for Cr with several Su. The model is used to evaluate the efficiency of the Cr in terms of throughput and collision probability. In [20], Petri net-Cr is implemented to study the spectral performance of the framework. The proposed framework contains both the Cr and the Pu. The simulation outcomes of the work reveal that the proposed algorithm obtained enhanced spectral efficiency with no interference between the users. Due to the high adaptability of the conceptual scheme, additional frameworks are explored for various PU and SU functions under different circumstances. The authors in [21] projected a Petri-Net-centered framework to improve the efficiency of the Su functioning in GSM-900. Further, the framework also decreases the interference between PU and SU. In [22], the dynamic threshold detection-based ED is implemented to enhance the spectral efficiency of the system. In two stages, the ED algorithm, also known as the "double stage ED algorithm", identifies the spectrum. The threshold is set in the first stage by estimating the amount of noise in the signal, and in the second stage, Ed is used to detect the idle spectrum. The numerical outcomes confirmed that the presented Ed performed better than the conventional Ed. The authors designed a novel Ed algorithm to detect idle Pu [23]. The performance of the detector is enhanced by evaluating the effect of noise present in the signal in different scenarios. It is concluded that the presented Ed performed optimally in terms of detection. The authors in [24] designed and investigated the efficiency of ED on fading and non-fading channels. It is noted that the efficiency of Ed depends on the criterion of threshold selection. The simulation outcomes reveal that the choice of an optimal threshold gave the system a high detection performance. The performance of Ed is severely affected due to the unwanted presence of noise variance in the signal. In [25], the authors introduced an ED algorithm where the presence of noise is nullified. The outcome of the work reveals that the presented Ed outperforms the conventional Ed. In [26], the authors implemented a MF based on dynamic threshold selection. As the characteristics of noise are random in nature, it is clear that dynamic threshold selection is one of the most important tasks. The outcome of the work demonstrates that the proposed MF algorithm outperforms the prevailing MF algorithm. The authors in [27] introduced a hybrid algorithm based on the combination of Ed and MF. The experimental outcomes reveal that the presented algorithm successfully reduces the false alarm rate, thus enhancing the accurate detection performance of the MF algorithm. In [28], the article presented a complete description, analysis and use of the CR in cellular framework. The article also discussed the limitations of MF implementation in advanced radio. It is concluded that MF can play a significant role in the utilization of the spectrum in an advanced radio framework. In [29], the authors introduced a novel MF-SS algorithm for the OFDM waveform. The analysis of MF is computed by applying it to OFDM with and without the cyclic prefix (CP). The results show an improvement in Pd and Pfa parameters. The parallel technique-based CS algorithm is implemented to enhance the utilization of the white holes. It should be noted that the presented method outperformed existing standards by 92% [30]. In [31], the availability of Pu is identified by implementing a CS-SS for the FBMC and OFDM frameworks. The authors have considered an AWGN channel for the analysis of the projected CS-SS. When compared to conventional standards, the proposed algorithm obtained a gain of 2 to 3 dB. It is also noted that the use of pilot signals increases the complexity of the framework. In [32], it is seen that the combination of MIMO and OFDM frameworks enhances the throughput and spectrum access of the system. The simulation outcomes demonstrate enhanced detection and performance. In [33], the authors applied an Ed to the OFDM structure, and the different parameters such as Pd, Pfa and BER were estimated. When compared to conventional electric, the proposed work achieved a gain of 2 dB. In [34], the article presented a novel algorithm to enhance the detection performance of the Pu. The proposed algorithm, which is based on the genetic algorithm, improves the framework's throughput. The article in [35] presented a novel ED algorithm based on mean energy estimation. The experimental work demonstrates a gain of 30% as compared with the conventional ED-SS method. In [36], the authors designed a CS-SS algorithm for the generalized frequency division multiplexing (GFDM) waveform. GFDM is considered an ideal candidate for 5G radio. The proposed work detects an idle Pu at a low SNR as compared with the existing approaches. The presented article is based on the principle of PU and SU initial decoding strategies [37]. It is seen that the interference is reduced and detection is achieved at low SNR. In [38], it is noted that an optimal selection of threshold plays a major role in the detection process. The article presented a dynamic threshold-based ED-SS algorithm to identify idle users at low SNR. Finally, the introduction of dynamic threshold in ED-SS yielded a gain of 1 dB to 2 dB when compared to static threshold schemes. In Table 1, we discussed the advantages and disadvantages of the spectrum sensing algorithms.
Table 1.
Merits and demerits of the spectrum sensing algorithms.
S. No
Techniques
Remarks
1
Conventional ED
Easy and straightforward to apply.
Results in interference
When PU is not present, the spectrum is allocated to SUs.
High required SNR.
The detection duration is small.
Low-power, durable detectors.
Spectrum loss
No advance channel information is necessary.
2
Convention CS
The algorithm intricacy is high
Results in interference
When PU is not present, the spectrum is allocated to SUs
Low required SNR.
The detection duration is high.
Intermediate robust detectors
It results in spectrum loss
No advance channel information is necessary.
3
Conventional MF
The algorithm intricacy is high
Results in interference
When PU is not present, the spectrum is allocated to SUs
Low required SNR.
The detection of the signal is superior than the CS but not better tha the ED
Average detectors are required
It results in spectrum loss
No advance channel information is necessary.
4
Proposed method
The proposed algorithm outperforms the conventional algorithm with low intricacy.
The interference between users is reduce by using a SIC method.
The detected spectrum is allocated to the SU in both availability and non-availability of PU.
The proposed algorithms obtained a gain of 2 dB SNR as compared with the conventional methods.
The signal is accurately determined based on dynamic threshold.
● To the best of our knowledge and available literature, the advanced SS algorithms for FBMC and NOMA are presented for the first time.
● The different parameters such as BER, PD, PFA and PSD are estimated and compared with the multi-carrier-waveforms. It is noted that CS outperforms MF and ED.
● We proposed novel ED and CS algorithms based on dynamic threshold detection for the advanced waveforms. It is seen that the spectral efficiency of NOMA is better than that of OFDM and FBMC.
2.
Proposed system model
2.1. Energy Detection (ED)
The conventional ED-SS is based on the principle of static threshold selection. The primary aim of the ED is to identify the spectrum of Pu, which can be allocated to the Su [39]. The Pu spectrum is identified by estimating the predefined threshold and comparing it with the received energy signal of the framework [40]. The hypothesis for the estimation of the spectrum is given by:
h0:z(n)=no(n)
(1)
h1:z(n)=x(n)+no(n)
(2)
The Eqs (1) and (2) indicates the absence and presence of Pu. Where z(n) is the received signal, x(n) is the transmitted signal and no(n) indicate the presence of noise in the signal.
The estimation of the threshold (ETh) is given by:
ETh=∑Mn=1(x[n])2
(3)
Further, the threshold for h0andh1 is evaluated with mean (µ), noise variance (σno) and transmitted power signal σx, given by:
ETh=M(Mσ2no,2σ4no):h0
(4)
ETh=M(M(σ2x+σ2no,2N(σ2x+σ2no)2):h1
(5)
According to Eqs (4) and (5), a signal is sometimes misrepresented as a detected signal, which is known as a false alarm (PFA). The Pd and Pfa are important characteristics to define the performance of ED, given by:
Pfa=Prob(ETh>λTh):h0
(6)
Pd=Prob(ETh>λTh):h1
(7)
Further, the Eqs (6) and (7) can be express as:
Pfa=Q(λTh−Mσ2no√2σ4no)
(8)
Pd=Q(M(σ2x+σ2no)√2N(σ2x+σ2no)2)
(9)
Where Q represent a gaussian function. Considering Eq (8), the threshold is given by:
It is one of the most significant spectrum sensing algorithms that can be considered for advanced radio. The identification of spectrum at low SNR with independence from noise makes it a promising and effective algorithm. It exploits the periodicity properties of the signal by estimating the mean and autocorrelation of the signal. Another substantial characteristic of CS is the identification of Pu without a significant interference between Pu and Su. In recent years, the CS algorithm has been applied to detect the spectrum in various conditions [41]. The CS signals are estimated by utilizing the cyclic autocorrelation and spectrum correlation density functions. The first step in CS is to transform the signal into second-order CS by utilizing several operations such as sampling, filtering and encoding, defined as [42]:
E{y(+)}=E{y(t+to)}
(11)
Ry(t,τ)=Ry(t+to,τ)
(12)
From Eq (11), it is noted the signal is periodic in nature with fundamental period tO. The Eq (12) can be expressed with the cyclic frequency (β):
Ry(t,τ)=∑βRβx(τ)exp(i2πβt)
(13)
The Fourier coefficient of Eq (13) can be written as:
Rβy(τ)≜limT→∞1T∫T2−T2Ry(t,τ)τ)exp(−i2πβt)dt
(14)
The Rβy(τ) is represented as cyclic auto-corelation function at:
β={M/To}
(15)
The spectral correlation density function of Eq (15) is estimated as:
Sβy(f)≜limT→∞∫∞−∞Rβy(t−τ)exp(−i2πβt)
(16)
3.
Simulation results
We evaluated the performance of the CS and ED spectrum sensing algorithms for 5G and beyond waveforms in the proposed article. The computer simulation is used to estimate the performance of the CR algorithms. The parameters used in the simulation are listed in Table 2.
The accurate detection capabilities of the cognitive radio are analyzed for the NOMA waveform shown in Figure 2. It is seen that the CS-SS method achieved a detection at -4.8 dB as compared with the 5 and 6.7 dB detection obtained by the MF and ED algorithms, respectively. So, we can say that the CS method is better than the MF and ED methods. This makes the NOMA a good 5G spectral access waveform.
In Figure 3, the detection of signal characteristics is analyzed for the OFDM waveform. The CS achieved a detection at an SNR of 5.1 dB, compared to the MF's 6.3 dB and the ED's 10.1 dB. Hence, it is concluded that the CS obtained a gain of 1.2 and 5 dB as compared with the MF and ED.
The detection performance of the SS algorithm for FBMC is given in Figure 4. The CS algorithm obtained a detection at an SNR of 4.1 dB as compared with 5.2 dB (MF) and 10.1 dB (ED). Hence, CS outperforms the MF and ED by the gains of 1.1 and 6 dB, respectively, as compared with the MF and ED.
In Figures 2–4, it is seen that the CS obtained the best performance for OFDM, FBMC and NOMA waveforms. The CS also offers effective performance in noisy environments as compared with the existing SS algorithms. The primary objective of the CR is to detect the presence of an idle spectrum in an efficient manner. However, the noise is sometimes detected as a signal, which is referred to as PFA. In this work, we have evaluated the performance of SS algorithms under false alarm conditions. The performance of false alarms for the OFDM waveform is given in Figure 5. The pd and Pfa are estimated by determining the presence or absence of the signal for different threshold values. It is seen that ED misrepresents noise as the signal at an early stage (pfa = 0.4 and Pd = 1). In the same scenario, however, MF and CS detected at pfa = 0.2 and 0.1, respectively. Hence, it is concluded that CS gave a robust performance as compared with the prevailing standard.
In Figure 6, it is seen that the CS is detecting the idle spectrum at a higher Pfa as compared with the MF and ED for the FBMC waveform. For the MF, ED and CS, the Pd is 1 for different values of Pfa. However, the MF and CS show a similar characteristic, which is better than the ED method. Hence, it is concluded that the detection of signal performance is gracefully degraded for the ED scheme.
The characteristics of the false alarm for the NOMA are given in Figure 7. It is seen that the noise is easily misrepresented as a signal for the ED method. The CS and MF performances, on the other hand, are stable and efficient. It is concluded that the probability of a signal being missed in CS and MF is low when compared to current standards. It is noted that when the Pd is 1, the signal is detected, and when the Pfa is 1, the noise is misrepresented as a signal. Hence, it is noted that the CS and MF for NOMA and FBMC gave an approximately similar detection performance for different PFAs, outperforming the OFDM system.
It is important to analyze the throughput of the system for spectrum sensing techniques. The BER curves for the NOMA are given in Figure 8. The BER of 10-3 is obtained at SNRs of 2.8 dB for CS, 3.3 dB for MF and 6 dB for ED. It is seen that the CS algorithm enhances the throughput of the system and obtains a gain of 0.5 and 3.2 dB for a BER of 10-3 as compared with the MF and ED.
The BER curves for the FBMC are evaluated in Figure 9. The BER of 10-3 is obtained at an SNR of 3.8 dB for CS, 4.2 dB for MF and 7 dB for ED. CS performed admirably, achieving gains of 0.4 and 3.2 decibels. Hence, it is concluded that CS outperforms the MF and ED methods.
The BER of OFDM is given in Figure 10. It is noted that CS, MF and ED obtained a BER of 10-3, respectively. At SNRs of 5, 6.1 and 9 dB. Hence, it is noted that the CS outperforms the prevailing standard by obtaining a gain of 1.1 and 4 dB.
The BER characteristics of OFDM, FBMC and NOMA indicate that the performance of the CS is better than the MF and ED algorithms. It is also noted that the performance of MF is very close to that of CS, obtaining a gain approximately equivalent to CS in most of the cases. It is also concluded that the NOMA is most compatible with the SS algorithms as compared with the FBMC and OFDM.
In Figure 11, we have analyzed the performance of PSD for the OFDM, FBMC and NOMA. It is seen that the bandwidth leakage of OFDM is -100, FBMC is -79 and NOMA is -150. As a result, when compared to the FMBC and OFDM waveforms, NOMA obtained an efficient spectral access.
In this part, the computational complexity of the sensing techniques is given in Figure 12. The complexity is the quantity of operations—like addition and multiplication—necessary to get the best possible detection. The proposed algorithm, ED, MF and CS complexity in the work provided is given by NM, 2KN, 2Nn(k+1+n) and 2Nn(2k+Nn−1) respectively.
In this work, we presented a CS algorithm for waveforms used in 5G and beyond, including NOMA, FBMC and OFDM. By applying CS, MF and ED algorithms to the 5G waveforms, metrics including Pd, Pfa, BER and PSD are assessed and comprehensively analyzed. The static results from the Matlab-2016 simulation on the Rayleigh channel are shown to assess how well the Cr algorithms perform for various multi-carrier waveforms. The proposed CS, MF and ED obtained a detection at an SNR of 5.1, 6.3 and 10.1 dB respectively. It is noted that the CS achieved a gain of 1.2 and 5 dB as compared with the MF and ED. Further, it is seen that the intricacy of the proposed algorithm is lower than that of the ED and MF. The proposed CS had good detection and throughput performance even at low SNR. It is noted that the proposed CS outperforms the conventional spectrum sensing algorithms. Further, in the future, the proposed algorithm can be used for spectrum sensing in channels with non-flat characteristics.
Acknowledgments
This work was supported by Suranaree University of Technology (SUT), Thailand Science Research and Innovation (TSRI) and National Science, Research and Innovation Fund (NSRF).
Conflict of interest
The authors declare there is no conflict of interest.
References
[1]
J. A. Tamboli, S. G. Joshi, Optimum design of a passive suspension system of a vehicle subjected to actual random road excitations, J. Sound Vib., 219 (1999), 193-205. doi: 10.1006/jsvi.1998.1882
[2]
G. Pepe, A. Carcaterra, VFC-variational feedback controller and its application to semi-active suspensions, Mech. Syst. Signal Pr., 76-77 (2016), 72-79. doi: 10.1016/j.ymssp.2016.01.002
[3]
H. Zhang, J. Liu, E. Wang, S. Rakheja, C. Su, Nonlinear dynamic analysis of a skyhookbased semi-active suspension system with magneto-rheological damper, IEEE T. Veh. Technol., 67 (2018), 10466-10456.
[4]
T. H. S. Li, Y. P. Kuo, Evolutionary algorithms for passive suspension systems, JSME Int. J. Ser. C, 43 (2000), 537-544. doi: 10.1299/jsmec.43.537
[5]
H. Gao, J. Lam, C. Wang, Multi-objective control of vehicle active suspension systems via loaddependent controllers, J. Sound Vib., 290 (2006), 654-675. doi: 10.1016/j.jsv.2005.04.007
[6]
H. Pan, W. Sun, Nonlinear output feedback finite-time control for vehicle active suspension systems, IEEE T. Ind. Inform., 15 (2019), 2073-2082. doi: 10.1109/TII.2018.2866518
[7]
H. Pan, X. J. Jing, W. Sun, H. J. Gao, A bio-inspired dynamics-based adaptive tracking control for nonlinear suspension systems, IEEE T. Contr. Syst. T., 26 (2018), 903-914. doi: 10.1109/TCST.2017.2699158
[8]
H. Pan, X. Jing, W. Sun, Z. C. Li, Analysis and design of a bio-inspired vibration sensor system in noisy environment, IEEE-ASME T. Mech., 23 (2018), 845-855. doi: 10.1109/TMECH.2018.2803284
[9]
Y. Liu, Q. Zeng, S. Tong, C. L. P. Chen, L. Liu, Adaptive neural network control for active suspension systems with time-varying vertical displacement and speed constraints, IEEE T. Ind. Electron., 66 (2019), 9458-9466. doi: 10.1109/TIE.2019.2893847
[10]
Q. Zeng, Y. Liu, L. Liu, Adaptive vehicle stability control of half-car active suspension systems with partial performance constraints, IEEE T. Syst. Man. Cy. A, (2019), 1-11.
[11]
N. Yagiz, Y, Hacioglu, Backstepping control of a vehicle with active suspensions, Control Eng. Pract., 16 (2018), 1457-1467.
[12]
Z. Wang, Y. Xu, R. Lu, H. Peng, Finite-time state estimation for coupled Markovian neural networks with sensor nonlinearities, IEEE T. Neur. Net. Lear., 28 (2017), 630-638. doi: 10.1109/TNNLS.2015.2490168
[13]
T. Zhang, S. S. Ge, Adaptive neural network tracking control of MIMO nonlinear systems with unknown dead zones and control directions, IEEE T. Neural Networ., 20 (2019), 483-497.
[14]
Y. J. Liu, S. Tong, Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems, Automatica, 76 (2017), 143-152. doi: 10.1016/j.automatica.2016.10.011
[15]
T. Li, R. Zhao, C. L. P. Chen, L. Fang, C. Liu, Finite-time formation control of under-actuated ships using nonlinear sliding mode control, IEEE T. Cybernetics, 48 (2018), 3243-3253. doi: 10.1109/TCYB.2018.2794968
[16]
J. Lan, Y. J. Liu, L. Liu, S. C. Tong, Adaptive output feedback tracking control for a class of nonlinear time-varying state constrained systems with fuzzy dead-zone input, IEEE T. Fuzzy Syst., (2020), 1-1.
[17]
M. Zapateiro, N. Luo, H. R. Karimi, J. Vehi, Vibration control of a class of semi active suspension system using neural network and backstepping techniques, Mech. Syst. Signal Pr., 23 (2009), 1946- 1953. doi: 10.1016/j.ymssp.2008.10.003
[18]
L. Liu, X. Li, Event-triggered tracking control for active seat suspension systems with time-varying full-state constraints, IEEE Transactions on Systems, Man, and Cybernetics: Systems, (2020), 1-9.
[19]
T. Li, Z. Li, D. Wang, C. L. P. Chen, Output-feedback adaptive neural control for stochastic nonlinear time-varying delay systems with unknown control directions, IEEE T. Neur. Net. Lear., 26 (2015), 1188-1201. doi: 10.1109/TNNLS.2014.2334638
[20]
S. Tong, Y. Li, Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones, IEEE T. Fuzzy Syst., 20 (2012), 168-180. doi: 10.1109/TFUZZ.2011.2171189
[21]
S. Tong, X. Min, Y. Li, Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions, IEEE T. Cybernetics, 50 (2020), 3903- 3913. doi: 10.1109/TCYB.2020.2977175
[22]
B. Chen, X. Liu, C. Lin, Observer and adaptive fuzzy control design for nonlinear strict-feedback systems with unknown virtual control coefficients, IEEE T. Fuzzy Syst., 26 (2018), 1732-1743. doi: 10.1109/TFUZZ.2017.2750619
[23]
L. Liu, Y. J. Liu, S. Tong, Fuzzy based multi-error constraint control for switched nonlinear systems and its applications, IEEE T. Fuzzy Syst., 27 (2019), 1519-1531. doi: 10.1109/TFUZZ.2018.2882173
[24]
X. Yang, X. Li, Q. Xi, P. Duan, Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Eng., 15 (2018), 1495-1515. doi: 10.3934/mbe.2018069
[25]
X. Li, X. Yang, T. Huang, Persistence of delayed cooperative models: impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.
[26]
B. Niu, L. Lu, Adaptive backstepping-based neural tracking control for MIMO nonlinear switched systems subject to input delays, IEEE T. Neur. Net. Lear., 29 (2018), 2638-2644. doi: 10.1109/TNNLS.2017.2690465
[27]
B. Niu, H. R. Karimi, H. Wang, Y. Liu, Adaptive output-feedback controller design for switched nonlinear stochastic systems with a modified average dwell-time method, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 1371-1382. doi: 10.1109/TSMC.2016.2597305
[28]
T. Gao, Y. J. Liu, L. Liu, D. Li, Adaptive neural network-based control for a class of nonlinear pure-feedback systems with time-varying full state constraints, IEEE-CAA J. Automatic., 5 (2018), 923-933.
[29]
Z. Wang, H. Xue, Y. Pan, H. Liang, Adaptive neural networks event-triggered fault-tolerant consensus control for a class of nonlinear multi-agent systems, AIMS Mathematics, 5 (2020), 2780- 2800. doi: 10.3934/math.2020179
[30]
G. Wang, C. Chen, S. Yu, Finite-time sliding mode tracking control for active suspension systems via extended super-twisting observer, P. I. Mech. Eng. I-J. Sys., 231 (2017), 459-470.
[31]
Y. Zhang, Y. Liu, L. Liu, Adaptive Finite-Time NN Control for 3-DOF Active Suspension Systems With Displacement Constraints, IEEE Access, 7 (2019), 13577-13588. doi: 10.1109/ACCESS.2019.2891724
[32]
B. Liu, M. Saif, H. Fan, Adaptive fault tolerant control of a half-car active suspension systems subject to random actuator failures, IEEE-ASME T. Mech., 21 (2016), 2847-2857. doi: 10.1109/TMECH.2016.2587159
[33]
W. Sun, H. Gao, O. Kaynak, Vibration isolation for active suspensions with performance constraints and actuator saturation, IEEE-ASME T. Mech., 20 (2015), 675-683. doi: 10.1109/TMECH.2014.2319355
[34]
F. Zhao, S. S. Ge, F. Tu, Y. Qin, M. Dong, Adaptive neural network control for active suspension system with actuator saturation, IET Control Theory A., 10 (2016), 1696-1705. doi: 10.1049/iet-cta.2015.1317
[35]
H. Pan, W. Sun, X. Jing, H. Gao, J. Yao, Adaptive tracking control for active suspension systems with non-ideal actuators, J. Sound Vib., 399 (2017), 2-20. doi: 10.1016/j.jsv.2017.03.011
[36]
V. S. Deshpande, P. D. Shendge, S. B. Phadke, Nonlinear control for dual objective active suspension systems, IEEE T. Intell. Transp., 18 (2017), 656-665. doi: 10.1109/TITS.2016.2585343
[37]
A. M. Zou, Z. G. Hou, M. Tan, Adaptive control of a class of nonlinear pure-feedback systems using fuzzy Backstepping approach, IEEE T. Fuzzy Syst., 16 (2008), 886-897. doi: 10.1109/TFUZZ.2008.917301
[38]
W. He, Y. Chen, Z. Yin, Adaptive neural network control of an uncertain robot with full-state constraints, IEEE T. Cybernetics, 46 (2016), 620-629. doi: 10.1109/TCYB.2015.2411285
[39]
L. Liu, Y. J. Liu, S. C. Tong, C. L. P. Chen, Integral barrier Lyapunov function based adaptive control for switched nonlinear systems, Sci. China Inf. Sci., 60 (2020), 1-14.
[40]
L. Tang, D. Ma, J. Zhao, Adaptive neural control for switched non-linear systems with multiple tracking error constraints, IET Signal Process., 13 (2019), 330-337. doi: 10.1049/iet-spr.2018.5077
[41]
D. Li, C. L. P. Chen, Y. J. Liu, S. Tong, Neural network controller design for a class of nonlinear delayed systems with time-varying full-state constraints, IEEE T. Neur. Net. Lear., 30 (2019), 2625- 2636. doi: 10.1109/TNNLS.2018.2886023
[42]
Z. Wang, P. Shi, C. C. Lim, H-/H∞ fault detection observer in finite frequency domain for linear parameter-varying descriptor systems, Automatica, 86 (2017), 38-45. doi: 10.1016/j.automatica.2017.08.021
[43]
Z. Liu, J. Liu, W. He, Modeling and vibration control of a flexible aerial refueling hose with variable lengths and input constraint, Automatica, 77 (2017), 302-310. doi: 10.1016/j.automatica.2016.11.002
[44]
L. B. Wu, J. H. Park, Adaptive fault-tolerant control of uncertain switched nonaffine nonlinear systems with actuator faults and time delays, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50 (2020), 3470-3480. doi: 10.1109/TSMC.2019.2894750
[45]
L. Liu, Y. J. Liu, D. Li, S. Tong, Z. Wang, Barrier Lyapunov function based adaptive fuzzy FTC for switched systems and its applications to resistance inductance capacitance circuit system, IEEE T. Cybernetics, 50 (2020), 3491-3502. doi: 10.1109/TCYB.2019.2931770
[46]
H. Gao, Y. Song, C. Wen, Backstepping design of adaptive neural fault-tolerant control for MIMO nonlinear systems, IEEE T. Neur. Net. Lear., 28 (2017), 2605-2613. doi: 10.1109/TNNLS.2016.2599009
[47]
Y. Pan, G. Yang, Event-triggered fault detection filter design for nonlinear networked systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1851-1862. doi: 10.1109/TSMC.2017.2719629
[48]
X. Tang, G. Tao, S. M. Joshi, Adaptive actuator failure compensation for parametric strict feedback systems and an aircraft application, Automatica, 39 (2003), 1975-1982. doi: 10.1016/S0005-1098(03)00219-X
[49]
J. Qiu, M. Ren, Y. Zhao, Y. Guo, Active fault-tolerant control for vehicle active suspension systems in finite-frequency domain, IET Control theory A., 5 (2011), 1544-1550. doi: 10.1049/iet-cta.2010.0519
[50]
H. Li, H. Liu, H. Gao, P. Shi, Reliable fuzzy control for active suspension systems with actuator delay and fault, IEEE T. Fuzzy Syst., 20 (2012), 342-357. doi: 10.1109/TFUZZ.2011.2174244
[51]
B. Liu, M. Saif, H. Fan, Adaptive fault tolerant control of a half-car active suspension systems subject to random actuator failures, IEEE-ASME T. Mech., 21 (2016), 2847-2857. doi: 10.1109/TMECH.2016.2587159
[52]
M. Moradi, A. Fekih, Adaptive PID-sliding-mode fault-tolerant control approach for vehicle suspension systems subject to actuator faults, IEEE T. Veh. Technol., 63 (2014), 1041-1054. doi: 10.1109/TVT.2013.2282956
[53]
W. Wang, C. Wen, Adaptive compensation for infinite number of actuator failures or faults, Automatica, 47 (2011), 2197-2210. doi: 10.1016/j.automatica.2011.08.022
[54]
H. B. Ji, H. S. Xi, Adaptive output-feedback tracking of stochastic nonlinear systems, IEEE T. Automat. Contr., 51 (2006), 355-360. doi: 10.1109/TAC.2005.863501
[55]
Y. Li, S. Tong, L. Liu, G. Feng, Adaptive output-feedback control design with prescribed performance for switched nonlinear systems, Automatica, 80 (2017), 225-231. doi: 10.1016/j.automatica.2017.02.005
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