
IEEE/CAA Journal of Automatica Sinica
Citation: | Z. J. Zhao, J. Zhang, S. Y. Chen, W. He, and K.-S. Hong, “Neural-network-based adaptive finite-time control for a two-degree-of-freedom helicopter system with an event-triggering mechanism,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 8, pp. 1754–1765, Aug. 2023. doi: 10.1109/JAS.2023.123453 |
Helicopter systems present numerous benefits over fixed-wing aircraft in several fields of application. Developing control schemes for improving the tracking accuracy of such systems is crucial. This paper proposes a neural-network (NN)-based adaptive finite-time control for a two-degree-of-freedom helicopter system. In particular, a radial basis function NN is adopted to solve uncertainty in the helicopter system. Furthermore, an event-triggering mechanism (ETM) with a switching threshold is proposed to alleviate the communication burden on the system. By proposing an adaptive parameter, a bounded estimation, and a smooth function approach, the effect of network measurement errors is effectively compensated for while simultaneously avoiding the Zeno phenomenon. Additionally, the developed adaptive finite-time control technique based on an NN guarantees finite-time convergence of the tracking error, thus enhancing the control accuracy of the system. In addition, the Lyapunov direct method demonstrates that the closed-loop system is semiglobally finite-time stable. Finally, simulation and experimental results show the effectiveness of the control strategy.
HELICOPTERS present numerous benefits over fixed-wing aircraft, including hovering capacity, ability to take off and land vertically, and minimal environmental requirements for takeoff and landing. Consequently, helicopters are typically employed in the military, transportation, and traffic applications [1]. However, a helicopter is a multi-input, multi-output (MIMO) nonlinear system with high degrees of inter-axis coupling, time-varying parameters, and uncertainties [2]–[4], presenting challenges to the design of robust stabilization controllers for helicopter systems. To overcome these hurdles, scholars have studied the stability of helicopter flight control systems.
Recently, numerous algorithms have been proposed for controlling helicopter systems. For instance, a reliable attitude-regulation control approach was developed for a three-degree-of-freedom (3-DOF) helicopter system subjected to external disturbances and uncertainties [5]. In [6], a robust control scheme was proposed to address the estimated tracking control of a 3-DOF helicopter system to allow the helicopter system to achieve desired performance. However, the suggested control stability mentioned above ignored the uncertainty in communication, not to mention the actual helicopter system. Over the past decade, several studies have focused on using neural-network (NN) control for nonlinear systems owing to the rapid advancements in NNs [7]. Among NNs, radial basis function NNs (RBFNNs) have been extensively used to control nonlinear systems owing to their benefits of approximate estimations and fast learning speeds. In [8], the system uncertainties and state constraints of a two-degree-freedom (2-DOF) helicopter system were addressed by developing a reinforcement learning control strategy. Moreover, in [9], for a 2-DOF helicopter system with input backlash and output constraints, adaptive NN control was developed. In [10], in order to handle input saturation and external disturbances in a 2-DOF helicopter system, the authors devised an adaptive NN control system. In [11], an adaptive neural discrete-time fractional control strategy for unmanned aerial vehicles (UAVs) was proposed. Although the aforementioned research on adaptive NN control for helicopters and UAV systems has achieved considerable advances, only limited studies have been conducted on communication restrictions in 2-DOF helicopter systems, which is a driving force motivating future research.
Constraints regarding communication channel bandwidth have always been a problem in practical helicopter systems. Therefore, scholars have suggested an event-triggering mechanism (ETM) to reduce the communication burden and guarantee system control stability [12]–[14]. For example, in [15], an event-triggered feedback control strategy was designed for UAV systems. Furthermore, an event-triggered adaptive dynamic control strategy was proposed for the distributed formation control of multi-UAVs in [16]. In [17], the authors designed an ETM with a relative-threshold policy to reduce the communication load in a multi-intelligent distributed-parameter system. An event-triggered adaptive boundary control strategy was proposed in [18] for the communication load and vibration control of a flexible robotic manipulator. Finally, in [19], the authors designed an event-triggered adaptive asymmetric control strategy to alleviate the communication load on nonlinear systems. Although these aforementioned event-triggered control studies for helicopters or other nonlinear systems are well known, only limited event-triggering-based studies for 2-DOF helicopter systems have been reported. Moreover, in actual helicopter systems, control objectives must be achieved in a finite amount of time to ensure system stability, which paves the way for further research.
When studying the practical applications of helicopter systems, considering the system’s control performance is essential in achieving satisfactory results over a finite time [20], [21]. Consequently, extensive research has been conducted on finite-time control. For example, in [22], the authors proposed finite-time convergent NN control for backlash-like hysteresis nonlinearity in robotic arm systems. To solve the attitude-tracking control of a quadrotor UAV, in [23], the authors presented a finite-time approximation-free control strategy. In [24], an adaptive finite-time control algorithm was developed to address UAV systems with actuator faults. Moreover, in [25], in order to operate a non-strict feedback system with time-varying constraints and input saturation, the authors presented a fixed-time adaptive control strategy. The fixed-time strategy was a finite-time extension. In [26], the stability of tracking control for multi-intelligent systems was investigated using event-triggered adaptive neural finite-time control. In addition, in [27], a global finite-time control strategy based on an ETM was proposed to achieve global finite-time stabilization of nonlinear systems. Finally, for an unknown nonlinear system, a new command filtering composite adaptive neural finite time control was proposed in [28]. Although the above investigations considering finite-time convergence in various nonlinear systems have demonstrated improvements in the field, few studies have focused on finite-time convergence in 2-DOF helicopter systems, which requires further research.
Motivated by previous research, we have noted few studies have been conducted on event triggering and finite time convergence in 2-DOF helicopter systems. Especially, most current studies on event triggering have been based on relative threshold ETM. Therefore, this work presents an adaptive finite-time NN control strategy for a 2-DOF helicopter based on event triggering with switching thresholds. The primary contributions of this study are as follows. 1) For the 2-DOF helicopter system, unlike previous studies considering input and output constraints [8]–[10], this study focuses on the input communication constraints of the 2-DOF helicopter system, and the proposed event triggering technique can alleviate the communication burdens of the system. 2) Unlike [22]–[24], this study considers both finite-time convergence and ETM in the helicopter system, which can save the communication resources of the system while ensuring finite-time convergence. The control strategy designed in this study is general and can be extended to various other MIMO nonlinear systems with wide application prospects. 3) Compared with the ETMs of fixed-threshold or relative-threshold strategies investigated in [13], [14], [17]–[19], the proposed ETM with switching thresholds provides more flexibility to balance the communication constraints and system performance. 4) This study demonstrates that the closed-loop errors of the system can converge in semiglobally finite time using Lyapunov’s stability theorem. Finally, a simulation and experiment confirm the effectiveness and superiority of the control method.
Fig. 1 depicts a sketch of the 2-DOF helicopter structure [29]. According to [30], the following equations for the dynamics of the 2-DOF helicopter system can be obtained:
¨ϑ=−Msgslscosϑ−Dspp˙ϑ−Msl2s˙φ2sinϑcosϑJspp+Msl2s+KsppVp+KspyVyJspp+Msl2s | (1) |
¨φ=−Dsyy˙φ+2Msl2s˙φ˙ϑsinϑcosϑJsyy+Msl2scos2ϑ+KsypVp+KsyyVyJsyy+Msl2scos2ϑ | (2) |
where ϑ and φ denote the pitch and yaw angles, respectively, of the helicopter system. Vp and Vy represent the voltages of the two motors along the pitch and yaw axes, respectively. Jspp and Jsyy denote the moments of inertia along the pitch and yaw axes, respectively. Ms denotes the weight of the helicopter. Kspp, Kspy, Ksyp, and Ksyy denote the thrust torques. gs represents the acceleration due to gravity. Dspp and Dsyy denote viscous friction constants. ls indicates the distance between the center of mass of the body and the fixed frame [29].
We define x=[x1,x2]T, where x1=[ϑ,φ]T and x2=[˙ϑ,˙φ]T. The above helicopter dynamics equations (1) and (2) are transformed into the general MIMO nonlinear equations as follows:
˙x1=x2 | (3) |
˙x2=F(x)+ΔF(x)+Y(x)u | (4) |
y=x1 | (5) |
where ΔF(x)∈R2 is an unknown smooth nonlinear function vector, and u=[Vp,Vy]T represents the control input. Moreover, F(x) and Y(x) are given as follows:
F(x)=[−Msgslscos(x11)−Dsppx21−Msl2sx222sin(x11)cos(x11)Jspp+Msl2s−Dsyyx22+2Msl2sx22x21sin(x11)cos(x11)Jsyy+Msl2scos2(x11)] | (6) |
Y(x)=[KsppJspp+Msl2sKspyJspp+Msl2sKsypJsyy+Msl2scos2(x11)KsyyJsyy+Msl2scos2(x11)]. | (7) |
Assumption 1 [31]: An unknown positive constant ˉY exists such that ‖Y(x)‖≤ˉY.
Remark 1: According to (7), we see that the elements Y11, Y12, Y21, and Y22 of matrix Y(x) are bounded. Therefore, Y(x) is also bounded and satisfies ‖Y(x)‖≤ˉY, where ˉY is an unknown positive constant.
Lemma 1 [32]: For any real variables a and b and any positive constants ϵ, π, and ε, the following inequalities hold:
|a|ϵ|b|π≤ϵϵ+πε|a|ϵ+π+πϵ+πε−ϵπ|b|ϵ+π. | (8) |
Lemma 2 [22]: At ∀t≥0, for any nonlinear system ˙χ= f(χ,x), let the Lyapunov function V(χ) satisfy
˙V(χ)≤−μVȷ(χ)+C,μ>0,0<ȷ<1,C>0. | (9) |
The solution χ(t) to the system ˙χ=f(χ,x) is semiglobally finite-time stable, and the steady time is estimated as follows:
Ta=1(1−ȷ)λμ[V1−ȷ(χ(0))−(C(1−ȷ)λμ)1−ȷȷ] | (10) |
where 0<λ<1, V(χ(0)) denotes the initial value of V(χ(t)). Therefore, Vȷ(χ)≤C(1−ȷ)μ, for ∀t>Ta.
Lemma 3 [33]: For any d0>0 and g0∈R, the following inequality holds:
0≤|g0|−g20√g20+d20<d0. | (11) |
Lemma 4 [34]: For e∈R and d1>0, the tanh(⋅) function holds
|e|−etanh(ed1)≤0.2785d1. | (12) |
Lemma 5 [35]: Here, the RBFNN approximation of an unknown smoothing function is expressed as follows:
O(Z)=ΠTΞ(Z) | (13) |
where Z=[z1,…,zn]T∈Rn and Π=[τ1,…,τj]T∈Rj denote the input and weight vectors, respectively. Ξ(Z)=[ω1(Z),…,ωj(Z)]T denotes the basis function.
Notably, RBFNNs have strong approximation capabilities and can accurately approximate any continuous function. Thus, we have the following:
O(Z)=Π∗TΞ(Z)+ρ(Z) | (14) |
where Π∗ denotes the optimal weight vector. ρ(Z) is a minimal approximation error satisfying ||ρ(Z)||≤ˉρ, where ˉρ>0 is a small constant.
Property 1 [22]: We assume that the Moore-Penrose pseudoinverse of vector ð∈Rn is ð+. Then, based on the properties of Moore-Penrose, we have the following:
ðT(ðT)+={0,ifð=(0,…,0)T1,otherwise. | (15) |
Inspired by [33], [36], an ETM can be defined as follows:
ui(t)=vi(tik),∀t∈[tik,tik+1),i=1,2 | (16) |
tik+1={inf{t>tik||ζi|≥ϖi|ui|+ηi},if |ui(t)|≤m1inf{t>tik||ζi|≥ϕi},if |ui(t)|>m1 | (17) |
where vi denotes the controller input signal, which is to be designed later. 0<ϖi<1, ηi>0, ϕi>0, and m1 are the design parameters. Moreover, the network measurement error is ζi=vi(t)−ui(t). tik, k∈z+, represents the update time of the ith input signal. When (17) is triggered and the time is recorded as tik+1, v(tik+1) is delivered to helicopter systems (1) and (2).
Remark 2: According to [36], we propose an ETM with a switching threshold. When the control input voltage of the system is relatively small, that is, when |ui(t)|≤m1, the threshold magnitude is linearly proportional to the magnitude of the control input voltage. Therefore, a relative-threshold strategy is used in this case, resulting in accurate control. However, when the control input voltage of the system suddenly increases, that is, when |ui(t)|>m1, a fixed-threshold strategy is required to prevent the control input voltage pulses from becoming too large.
Considering (16) and (17), we conclude the following:
vi(t)−ui(t)={qi(ϖi|ui|+ηi),if |ui(t)|≤m1piϕi,if |ui(t)|>m1 | (18) |
where qi and pi are time-varying scalars satisfying |qi|≤1 and |pi|≤1. According to the definitions of qi and pi, we obtain
vi(t)−ui(t)={qiϖisgn(ui)ui+qiηi,if |ui(t)|≤m1piϕi,if |ui(t)|>m1. | (19) |
From (19), we have the following equation:
u=Hv+L | (20) |
where H=diag{hi(t)}∈R2×2 and L=[l1(t),l2(t)]T∈R2 with
{hi(t)={11+qiϖisgn(ui), if |ui(t)|≤m11, if |ui(t)|>m1li(t)={−qiηi1+qiϖisgn(ui), if |ui(t)|≤m1−piϕi, if |ui(t)|>m1. | (21) |
As 0<ϖi<1, |qi|≤1 and |pi|≤1; then, hi and li are bounded.
Substituting (20) into (4) yields
˙x2=F(x)+ΔF(x)+Y(x)(H(t)v(t)+L(t))=F(x)+ΔF(x)+Y(x)H(t)v(t)+Y(x)L(t). | (22) |
The tracking errors are defined as follows:
e1=x1−xd | (23) |
e2=x2−ð0 | (24) |
where xd is the desired trajectory, and ð0 is the virtual control input signal.
The time derivatives of the tracking errors are calculated as follows:
˙e1=˙x1−˙xd | (25) |
˙e2=˙x2−˙ð0. | (26) |
Substituting (22) into (26) yields
˙e2=F(x)+Td+Y(x)H(t)v(t)+Y(x)L(t) | (27) |
where Td=ΔF(x)−˙ð0. Let us consider the following Lyapunov function:
V1=12eT1e1. | (28) |
Substituting (3), (24), and (25) into the time derivative of V1 yields
˙V1=eT1˙e1=eT1(e2+ð0−xd). | (29) |
According to Property 1, when e1=[0,0]T, we obtain ˙V1=0. When e1≠[0,0]T, the virtual control input signal ð0 is defined as follows:
ð0=−(eT1)+K1(eT1e1)ȷ+xd | (30) |
where K1>0 denotes a design parameter, and 0<ȷ<1 is a constant. Substituting (30) into (29) yields
˙V1=eT1e2−K1(eT1e1)ȷ. | (31) |
The Lyapunov function V2 is constructed as follows:
V2=V1+12eT2e2. | (32) |
Substituting (27) and (31) into the time derivative of V2 yields
˙V2=eT1e2−K1(eT1e1)ȷ+eT2(F(x)+Td+Y(x)H(t)v(t)+Y(x)L(t)). | (33) |
Because Td is unknown, by considering Lemma 5, we have
Td=Π∗TΞ(Z)+ρ(Z) | (34) |
where Z=[xT1,xT2,ðT0]T denotes the input vector of the NN. ρ(Z) is an approximation error with a positive constant ˉρ such that ||ρ(Z)||≤ˉρ. Furthermore, substituting (34) into (33), we obtain
˙V2=eT1e2−K1(eT1e1)ȷ+eT2(F(x)+Π∗TΞ(Z)+Y(x)H(t)v(t)+D) | (35) |
where D=Y(x)L(t)+ρ(Z). According to Assumption 1 and (21), we know that Y(x) and L(t) are bounded, and ||ρ(Z)||≤ˉρ. Thus, D is bounded, satisfying ||D||≤ˉD, where ˉD is an unknown positive constant.
According to (21), H(t) is bounded, and inft≥0λmin, where \lambda_{\min}(H(t)) denotes the minimum eigenvalue of H(t) . Thus, we define
\begin{align} \gamma=\underset{t\ge 0}{\text{inf}}\lambda_{\min}(H(t)),\;\; \delta=\frac{1}{\gamma}. \end{align} | (36) |
Further, we define \tilde{\Pi}=\hat{\Pi}-\Pi^* , \tilde{\bar{D}}=\bar{D}-\hat{\bar{D}} , and \tilde{\delta}=\delta-\hat{\delta} , where \hat{\Pi} , \hat{\bar{D}} , and \hat{\delta} are the estimates of \Pi^* , \bar{D} , and δ, respectively.
From Property 1, when e_2=[0,0]^T , we obtain \dot{V}_1= -K_1(e^T_1e_1)^\jmath . When e_2\neq [0,0]^T , the controller input signal v(t) is designed as follows:
\begin{align} v(t)=-Y^{-1}(x)\left(e_2\frac{\hat{\delta}^2\xi^T\xi}{\sqrt{\hat{\delta}^2e^T_2e_2\xi^T\xi+d^2_0}}\right) \end{align} | (37) |
where \xi=F(x)+(e^T_2)^+K_2(e^T_2e_2)^\jmath+\hat{\Pi}^{T}\Xi(Z)+e_1+\hat{\bar{D}}\text{tanh}(\frac{e_2}{d_1}) , and K_2 , d_1 and d_0 are the design parameters. By combining Lemma 3 with (37), we obtain
\begin{split} e^T_2YHv=\;&-e^T_2YHY^{-1}(e_2\frac{\hat{\delta}^2\xi^T\xi}{\sqrt{\hat{\delta}^2e^T_2e_2\xi^T\xi+d^2_0}})\\ \le\;&-\frac{\gamma\hat{\delta}^2e^T_2e_2\xi^T\xi}{\sqrt{\hat{\delta}^2e^T_2e_2\xi^T\xi+d^2_0}} \le\gamma d_0-\gamma\hat{\delta}e^T_2\xi. \end{split} | (38) |
Subsequently, the updating laws \hat{\Pi} , \hat{\bar{D}} , and \hat{\delta} are devised as follows:
\begin{align} \qquad\qquad\dot{\hat{\Pi}}=&\Gamma_1(\Xi(Z)e^T_2-\iota_1\hat{\Pi}) \end{align} | (39) |
\begin{align} \qquad\qquad\dot{\hat{\bar{D}}}=&\Gamma_2(e^T_2\tanh(\frac{e_2}{d_1})-\iota_2\hat{\bar{D}}) \end{align} | (40) |
\begin{align} \qquad\qquad\dot{\hat{\delta}}=&\Gamma_3(e^T_2\xi-\iota_3\hat{\delta}) \end{align} | (41) |
where \Gamma_1=\Gamma^T_1 is a diagonal matrix, and \Gamma_2 , \Gamma_3 , \iota_1 , \iota_2 , and \iota_3 are the design parameters.
Substituting (38) into (35) yields
\begin{split} \dot{V}_2=\;&e^T_1e_2-K_1(e^T_1e_1)^\jmath+e^T_2(\Pi^{*T}\Xi(Z)-\hat{\Pi}^{T}\Xi(Z)\\&+Y(x)H(t)v(t)-(e^T_2)^+K_2(e^T_2e_2)^\jmath-e_1+D\\&-\hat{\bar{D}}\text{tanh}(\frac{e_2}{d_1})+\xi)\\ \le\;&-K_1(e^T_1e_1)^\jmath-K_2(e^T_2e_2)^\jmath-e^T_2\tilde{\Pi}^T\Xi(Z)+\gamma d_0\\&-\gamma\hat{\delta}e^T_2\xi+e^T_2D+e^T_2\tilde{\bar{D}}\text{tanh}(\frac{e_2}{d_1})-e^T_2\bar{D}\text{tanh}(\frac{e_2}{d_1})+e^T_2\xi. \end{split} | (42) |
We consider the following inequality:
\begin{align} e^T_2D\le \bar{D}\sum_{i=1}^2 \vert e_{2i} \vert \end{align} | (43) |
and
\begin{align} \bar{D}e^T_2\text{tanh}(\frac{e_2}{d_1})=\bar{D}\sum_{i=1}^2\Big(e_{2i}\tanh(\frac{e_{2i}}{d_1})\Big). \end{align} | (44) |
By applying Lemma 4, (43), and (44), we obtain
\begin{align} \bar{D}\sum_{i=1}^2 \vert e_{2i} \vert -\bar{D}\sum_{i=1}^2\Big(e_{2i}\tanh(\frac{e_{2i}}{d_1})\Big)\le 0.557d_1\bar{D}. \end{align} | (45) |
Inserting (45) into (42) yields
\begin{split} \dot{V}_2\le\;&-K_1(e^T_1e_1)^\jmath-K_2(e^T_2e_2)^\jmath-e^T_2\tilde{\Pi}^T\Xi(Z)+\gamma d_0\\&-\gamma\hat{\delta}e^T_2\xi+e^T_2\tilde{\bar{D}}\text{tanh}(\frac{e_2}{d_1})+0.557d_1\bar{D}+e^T_2\xi. \end{split} | (46) |
We consider the Lyapunov function as follows:
\begin{align} V_3=V_2+\frac{1}{2}\mathrm{tr}\{\tilde{\Pi}^T\Gamma^{-1}_1\tilde{\Pi}\}+\frac{1}{2\Gamma_2}\tilde{\bar{D}}^2+\frac{\gamma}{2\Gamma_3}\tilde{\delta}^2. \end{align} | (47) |
Substituting (39)−(41), and (46) into the time derivative of V_3 yields
\begin{split} \dot{V}_3=\;&\dot{V}_2+\mathrm{tr}\{\tilde{\Pi}^T\Gamma^{-1}_1\dot{\hat{\Pi}}\}-\frac{1}{\Gamma_2}\tilde{\bar{D}}\dot{\hat{\bar{D}}}-\frac{\gamma}{\Gamma_3}\tilde{\delta}\dot{\hat{\delta}}\\ \le\;& -K_1(e^T_1e_1)^\jmath-K_2(e^T_2e_2)^\jmath-e^T_2\tilde{\Pi}^T\Xi(Z)+\gamma d_0\\&-\gamma\hat{\delta}e^T_2\xi+e^T_2\tilde{\bar{D}}\text{tanh}(\frac{e_2}{d_1})+0.557d_1\bar{D}+e^T_2\xi\\&+\mathrm{tr}\{\tilde{\Pi}^T(\Xi(Z)e^T_2-\iota_1\hat{\Pi})\}-\tilde{\bar{D}}(e^T_2\tanh(\frac{e_2}{d_1})-\iota_2\hat{\bar{D}})\\&-\gamma\tilde{\delta}(e^T_2\xi-\iota_3\hat{\delta})\\ =\;&-K_1(e^T_1e_1)^\jmath-K_2(e^T_2e_2)^\jmath+\gamma d_0-\iota_1\mathrm{tr}\{\tilde{\Pi}^T\hat{\Pi}\}\\&+\iota_2\tilde{\bar{D}}\bar{D}-\iota_2\tilde{\bar{D}}^2+\gamma\iota_3\tilde{\delta}\delta-\gamma\iota_3\tilde{\delta}^2+0.557d_1\bar{D}. \end{split} | (48) |
Based on Young’s inequality, we derive
\begin{align} -\iota_1\mathrm{tr}\{\tilde{\Pi}^T\hat{\Pi}\}&\le -\frac{\iota_1}{2}||\tilde{\Pi}||^2_{\rm{F}}+\frac{\iota_1}{2}||\Pi^*||^2_{\rm{F}} \end{align} | (49) |
\begin{align} \iota_2\tilde{\bar{D}}\bar{D}&\le \frac{\iota_2}{\sigma_1}\tilde{\bar{D}}^2+\sigma_1\iota_2\bar{D}^2 \end{align} | (50) |
\begin{align} \gamma\iota_3\tilde{\delta}\delta&\le \frac{\gamma \iota_3}{\sigma_2}\tilde{\delta}^2+\sigma_2\gamma\iota_3\delta^2 \end{align} | (51) |
where \sigma_1 and \sigma_2 are positive constants.
By substituting (49)–(51) into (48), the following inequality can be derived:
\begin{split} \dot{V}_3\le\;& -K_1(e^T_1e_1)^\jmath-K_2(e^T_2e_2)^\jmath+\gamma d_0-\frac{\iota_1}{2}||\tilde{\Pi}||^2_{\rm{F}}\\&+\frac{\iota_1}{2}||\Pi^*||^2_F-(1-\frac{1}{\sigma_1})\iota_2\tilde{\bar{D}}^2+\sigma_1\iota_2\bar{D}^2\\&-(1-\frac{1}{\sigma_2})\gamma\iota_3\tilde{\delta}^2+\sigma_2\gamma\iota_3\delta^2+0.557d_1\bar{D}. \end{split} | (52) |
We define a=1 , b=\frac{\iota_1}{2}||\tilde{\Pi}||^2_{\rm{F}}, \epsilon=1-\jmath , \pi=\jmath , and \varepsilon=\jmath^{\frac{\jmath}{1-\jmath}} . Moreover, based on Lemma 1, we have
\begin{align} (\frac{\iota_1}{2}||\tilde{\Pi}||^2_{\rm{F}})^\jmath \le (1-\jmath)\varepsilon+\frac{\iota_1}{2}||\tilde{\Pi}||^2_{\rm{F}}. \end{align} | (53) |
By defining a=1 , b=(1-\frac{1}{\sigma_1})\iota_2\tilde{\bar{D}}^2 , \epsilon=1-\jmath , \pi=\jmath , and \varepsilon=\jmath^{\frac{\jmath}{1-\jmath}} , and applying Lemma 1, we obtain
\begin{align} ((1-\frac{1}{\sigma_1})\iota_2\tilde{\bar{D}}^2)^\jmath \le (1-\jmath)\varepsilon+(1-\frac{1}{\sigma_1})\iota_2\tilde{\bar{D}}^2. \end{align} | (54) |
We define a=1 , b=(1-\frac{1}{\sigma_2})\gamma\iota_3\tilde{\delta}^2 , \epsilon=1-\jmath , \pi=\jmath , and \varepsilon=\jmath^{\frac{\jmath}{1-\jmath}} . By applying Lemma 1, we derive
\begin{align} ((1-\frac{1}{\sigma_2})\gamma\iota_3\tilde{\delta}^2)^\jmath \le (1-\jmath)\varepsilon+(1-\frac{1}{\sigma_2})\gamma\iota_3\tilde{\delta}^2. \end{align} | (55) |
Substituting (53)–(55) into (52), we obtain
\begin{split} \dot{V}_3\le\;& -K_1(e^T_1e_1)^\jmath-K_2(e^T_2e_2)^\jmath-(\frac{\iota_1}{2}||\tilde{\Pi}||^2_{\rm{F}})^\jmath\\&-((1-\frac{1}{\sigma_1})\iota_2\tilde{\bar{D}}^2)^\jmath-((1-\frac{1}{\sigma_2})\gamma\iota_3\tilde{\delta}^2)^\jmath+\frac{\iota_1}{2}||\Pi^*||^2_{\rm{F}}\\&+\sigma_1\iota_2\bar{D}^2+\sigma_2\gamma\iota_3\delta^2+0.557d_1\bar{D}+\gamma d_0+3(1-\jmath)\varepsilon\\\le\; & -\mu V^\jmath_3+C \end{split} | (56) |
where
\begin{split} \mu=\;&\min\biggl\{K_12^\jmath, K_22^\jmath, \frac{\iota^\jmath_1}{(\lambda_{\max}(\Gamma^{-1}_1))^\jmath}, (2(1-\frac{1}{\sigma_1})\iota_2\Gamma_2)^\jmath, \\&(2(1-\frac{1}{\sigma_2})\iota_3\Gamma_3)^\jmath \} \end{split} | (57) |
and
\begin{split} C=\;&\frac{\iota_1}{2}||\Pi^*||^2_F+\sigma_1\iota_2\bar{D}^2+\sigma_2\gamma\iota_3\delta^2+0.557d_1\bar{D}\\&+\gamma d_0+3(1-\jmath)\varepsilon. \end{split} | (58) |
To ensure \mu>0 , the design parameters K_1 and K_2 must satisfy K_1>0 and K_2>0 .
From Lemma 2 and (56), we obtain the following:
\begin{align} T_a=\frac{1}{(1-\jmath)\lambda \mu}\left[V_3^{1-\jmath}(\chi(0))-(\frac{C}{(1-\jmath)\lambda \mu})^{\frac{1-\jmath}{\jmath}}\right] \end{align} | (59) |
where 0<\lambda<1 and \chi(0)=[e^T_1(0), e^T_2(0), \tilde{\Pi}^T_1(0), \tilde{\Pi}^T_2(0), \tilde{\bar{D}}(0), \tilde{\delta}(0)]^T. From Lemma 2, for \forall t>T_a , V_3^\jmath(\chi)\le \frac{C}{(1-\jmath)\lambda\mu} . Thus, all signals are semiglobally finite-time stable.
Remark 3: According to (16) and (17), the network measurement error \zeta_i cannot be bounded because the magnitude of \zeta_i depends on the variation of v_i . Therefore, we cannot consider \zeta_i as a bounded perturbation. Determining how to deal with the effect of \zeta_i in the system is a major challenge. Therefore, we develop a new expression in (20) that formulates u_i and v_i in a time-varying linear equation. However, in (20), dealing with H is challenging because it is a time-varying matrix multiplied by v_i . We successfully overcome this limitation by using the smooth function in Lemma 1 and the bounded estimation method.
Theorem 1: Considering the presence of uncertainty and an ETM in 2-DOF helicopter systems, we propose an adaptive finite-time NN control strategy. The control can be designed based on (37), and the update rate can be designed using (39)−(41). Under the proposed control strategy, we derive the closed-loop system signal to be semiglobally finite-time stable. When t>T_a , the tracking errors e_1 and e_2 and the approximation errors \tilde{\Pi} , \tilde{\bar{D}} , and \tilde{\delta} converge to the compact sets. Then, we obtain
\begin{align} \Omega_{e_1}&=\left\{e_1\in {\mathbb{R}}^2|\; \Vert e_{1}\Vert \le \sqrt{\Delta}\right\} \end{align} | (60) |
\begin{align} \Omega_{e_2}&=\left\{e_2\in {\mathbb{R}}^2|\; \Vert e_{2}\Vert \le \sqrt{\Delta}\right\} \end{align} | (61) |
\begin{align} \Omega_{\tilde{\Pi}}&=\left\{\tilde{\Pi}\in {\mathbb{R}}^{j\times2}|\; \Vert \tilde{\Pi}\Vert \le \sqrt{\frac{\Delta}{\lambda_{\min}(\Gamma^{-1}_1)}}\right\} \end{align} | (62) |
\begin{align} \Omega_{\tilde{\bar{D}}}&=\left\{\tilde{\bar{D}}\in {\mathbb{R}}|\; \vert \tilde{\bar{D}} \vert \le \sqrt{\Gamma_2\Delta}\right\} \end{align} | (63) |
\begin{align} \Omega_{\tilde{\delta}}&=\left\{\tilde{\delta}\in {\mathbb{R}}|\; \vert \tilde{\delta} \vert \le \sqrt{\Gamma_3\Delta}\right\} \end{align} | (64) |
where \Delta=2(\frac{C}{(1-\jmath)\lambda \mu})^{\frac{1}{\jmath}} .
Proof: For t>T_a , we obtain
\begin{align} V_1<V_2<V_3<(\frac{C}{(1-\jmath)\lambda \mu})^{\frac{1}{\jmath}} \end{align} | (65) |
and we further have the following inequalities:
\begin{align} \Vert e_{1}\Vert &\le \sqrt{\Delta}, \;\;\Vert e_{2}\Vert \le \sqrt{\Delta} \end{align} | (66) |
\begin{align} \Vert \tilde{\Pi}\Vert &\le \sqrt{\frac{\Delta}{\lambda_{\min}(\Gamma^{-1}_1)}},\; \vert \tilde{\bar{D}} \vert \le \sqrt{\Gamma_2\Delta},\;\vert \tilde{\delta} \vert \le \sqrt{\Gamma_3\Delta}. \end{align} | (67) |
Subsequently, we explore the effectiveness of the proposed ETM to demonstrate that the Zeno effect can be avoided. Suppose that there exists a positive constant \psi_i such that t^i_{k+1}-t^i_k>\psi_i, k\in {\textit{z}}^+ . Based on the following definitions \zeta_i(t)= v_i(t)-u_i(t) and u_i(t)=v_i(t^i_k) , \forall t\in[t^i_k, t^i_{k+1}) , we have:
\begin{align} \frac{d}{dt}|\zeta_i|=\frac{d}{dt}(\zeta_i\times\zeta_i)^{\frac{1}{2}}=\text{sgn}(\zeta_i)\zeta_i\le|\dot{v}_i|. \end{align} | (68) |
Because all the signals in \dot{v}_i are bounded, there exists a positive constant \varsigma_i such that |\dot{v}_i|<\varsigma_i . Here, we note that \zeta_i(t^i_k)=0 and \mathop {{\rm{lim}}}\nolimits_ {t\to t^i_{k+1}}|\zeta_i(t)|\ge \min\{\eta_i, \phi_i\}. Next, we determine that the minimum sampling interval for event triggering satisfies \psi_i>\frac{1}{\varsigma_i}\min\{\eta_i, \phi_i\} . Hence, the Zeno phenomenon is successfully avoided.
In this section, we demonstrate the effectiveness of the proposed NN-based adaptive finite-time control strategy through simulations.
Note that the design parameters of the 2-DOF helicopter system are as follows: K_{spp} = 0.022\;{\rm{(N{\cdot} m)/V}}, K_{spy} = 0.0221\;({\rm{N}}{\cdot} {\rm{m}})/ {\rm{V}}, K_{syp}\;=\;-0.0227\,{\rm{(N{\cdot} m)/V}}, K_{syy}\;=\;0.0022\,{\rm{(N{\cdot} m)/V}}, J_{spp}= 0.0215\; {\rm{kg{\cdot} m}}, J_{syy}=0.0237\;{\rm{kg{\cdot} m}}, g_s = 9.8\;{\rm{m/s}}2, M_s = 1.0750\;{\rm{kg}}, l_s=0.002\;{\rm{m}}, D_{spp}=0.0071\;{\rm{N/V}}, and D_{syy}=0.0220\;{\rm{N/V}}.
The initial value of the output variable is x_1(0) = [0, 0]^T\; \text{(rad/s)}, and the reference trajectory is x_d = [\frac{\pi}{10}\sin(t),\frac{\pi}{15}\sin(t)]^T \text{(rad/s)}. For the controller (28), the RBFNN nodes are 64 and the centers of \Xi_j(Z) are uniformly distributed in the domain of [−1, 1]×[−1, 1]×[−1, 1]×[−1, 1]×[−1, 1]×[−1, 1]. The variance of the RBFNN is 4. The initial weight \hat{\Pi}_{1,j} = \hat{\Pi}_{2,j} = 0, j=1, 2, \ldots, 64 . The design parameters are K_1=25 , K_2=30 , \jmath=0.97 , \Gamma_1=256I_{64\times64}, \Gamma_2=1 , \Gamma_3=2 , \iota_1=0.8 , \iota_2=0.5 , \iota_3= 0.8 , d_0=0.1 , and d_1=0.35 . The parameters of the ETM are selected as \varpi_i=0.2 , \eta_i=0.1 , \phi_i=1.5 , i=1,2 , and m_1=3.5 .
Relevant simulation results are depicted in Figs. 2(a)–2(e). In particular, Fig. 2(a) presents the response of the output variable tracking the desired signal. Fig. 2(b) depicts the tracking error. The error results indicate that the output variable tracks the desired signal well. The input signal response of the controller is depicted in Fig. 2(c), while Fig. 2(d) displays the change in the control input. Finally, Fig. 2(e) represents the event-trigger interval, with the number of triggers being 268 and 332. The effectiveness of the control strategy proposed in this study has been demonstrated in the simulation.
Note that we introduced PD control for comparison purposes to clearly demonstrate the superiority of the proposed controller. The controller is defined as follows:
\begin{align} v=-k_{pp}e_1-k_{dd}\dot{e}_1 \end{align} | (69) |
where k_{pp} and k_{dd} denote proportional and differential gains, respectively. After reasonable parameter adjustments, the proportional and differential gains are selected as k_{pp}=\text{diag}\{30, 30\} and k_{dd}=\text{diag}\{80, 80\} , respectively.
Fig. 2(a) indicates that the output of the system follows the trajectory of the desired signal. The tracking error of the system is depicted in Fig. 2(b). Moreover, Figs. 2(c) and 2(d) illustrate the controller input and control input response, respectively. Fig. 2(f) depicts the trigger interval of the controller under the PD control strategy, where the number of triggers are 852 and 619, respectively.
Compared to Case 1, the tracking error e_1 is larger under the PD control strategy, illustrating that the output error does not converge in finite time, thereby reducing system stability. In addition, the number of triggers in the ETM of the conversion threshold is greater under the PD control strategy than under the proposed control strategy, implying that its ability to reduce the communication load of the control input is weaker.
In addition, we quantitatively compare the errors and controller inputs of these two methods to evaluate the control performance of Cases 1 and 2 intuitively. The results obtained are shown in Table I. In particular, we conclude that the error and controller input in Case 1 are smaller than those in Case 2, indicating stronger finite-time convergence and control stability of the system under the control strategy of Case 1.
Comparative performance | The tracking error and the controller action indexes under Case 1 | The tracking error and the controller action indexes under Case 2 |
\sum_{\kappa=1}^{1250}[e_{11}(\kappa)]^2 | 0.0060002 | 0.014709 |
\sum_{\kappa=1}^{1250}[e_{12}(\kappa)]^2 | 0.0004923 | 0.805241 |
\sum_{\kappa=1}^{1250}[v_1(\kappa)]^2 | 174.31798 | 865.2022 |
\sum_{i=1}^{1250}[v_2(\kappa)]^2 | 1945.8023 | 2059.7359 |
To demonstrate the advantages of finite-time techniques, we design a set of comparative simulations. In this simulation, we do not consider finite-time convergence and then design a practical control algorithm using the design method of the controller in Case 1. Some of the design parameters are K_1=25 and K_2=30 . The simulation results of the control scheme created in this subsection are shown in Figs. 3(a)–3(d). The responses of the output variables are shown in Fig. 3(a). Fig. 3(b) represents the tracking error response. Figs. 3(c) and 3(d) represent the controller and control input signals, respectively.
Compared to Case 1, Figs. 3(a)–3(d) show that in the algorithm, without considering the finite-time technique, the error of the system is always in a fluctuating state and cannot converge rapidly to a minimal domain, decreasing the stability of the system. In addition, both the controller and the control input of the system are relatively large, which can cause a large overload on the input signal of the system during the control process, which can degrade the control stability of the system.
In this section, we present experiments conducted to verify the practical application of the proposed NN-based adaptive finite-time control strategy. We experimentally verified the proposed approach on a 2-DOF helicopter laboratory platform from Quanser, as depicted in Fig. 4, where the input voltage range of the platform was [-24\; {\rm{V}}, +24\; {\rm{V}}] .
The parameters for the ETM were selected as \varpi_i=0.1 , \eta_i=5 , \phi_i=5 , i=1,2 , and m_1=15 . The design parameters were K_1=22 and K_2=20 . Fig. 5(a) indicates the tracking response of the output angle. Moreover, Fig. 5(b) represents the tracking error response, and the errors are semiglobally finite-time convergent. Fig. 5(c) illustrates the control inputs of the system. Finally, the event-trigger interval is depicted in Fig. 5(d), and the number of events triggered is 205 and 492. As shown in Figs. 5(a)–5(d), the effectiveness of the proposed adaptive neural finite-time control strategy has been verified in the experiment.
In this scheme, we only considered the NN-based adaptive finite-time control with a relative-threshold ETM. The parameters of the ETM were set as \varpi_i=0.1 and \eta_i=5 , i=1,2 . After several parameter adjustments, the design parameters were chosen as K_1=24 and K_2=22 . Figs. 6(a)–6(c) present comparisons between the system output variables, tracking errors, and control input responses obtained using Schemes 1 and 2, respectively. The tracking errors converge rapidly in finite time. Fig. 6(d) presents the event-trigger interval, with the number of triggers as 231 and 593. As illustrated in Fig. 6(d), the number of triggers for the ETM considering relative thresholds is higher than the number of triggers for the ETM considering the switching threshold strategy. These results indicate that the system input-communication burden is reduced when considering the switching threshold strategy, thus effectively protecting the helicopter system components.
The superiority of the event-triggering strategy with switching thresholds considered in Scheme 1 was further verified by considering an event-triggering strategy with fixed thresholds. The parameter for the ETM with fixed thresholds was set as \phi_i=5 , i=1,2 . The design parameters were K_1=22 and K_2=23 . Figs. 7(a)–7(c) present comparisons between the response of the system output, tracking error, and control input signals in the ETM for the switching threshold and fixed-threshold strategies. Fig. 7(d) presents the event-trigger interval, and the number of triggers is 254 and 642. In particular, Fig. 7(d) reveals that the fixed-threshold strategy presents more triggers than the switching threshold strategy, indicating that the fixed-threshold strategy is less capable of reducing the input-communication burden, rendering the system less stable in terms of control. Furthermore, the system components are not effectively protected.
In this study, an adaptive finite-time event-triggered control strategy based on NNs was proposed to address the communication constraint problem in control networks and the uncertainty terms. The RBFNN was employed to estimate the uncertainty of the system. An ETM with a switching threshold was constructed by combining fixed-threshold and relative-threshold strategies. The effect of the network measurement error in the ETM was eliminated by introducing adaptive parameters, smooth functions, and bounded estimation methods. The proposed control strategy enabled the convergence of the tracking error of the system to a small zero domain in finite time. Both simulations and experiments verified the effectiveness and superiority of the proposed control strategy. Motivated by [37], [38], in the future, we will focus on the quantitative control of NNs for discrete-time nonlinear systems with event-triggered mechanisms.
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Comparative performance | The tracking error and the controller action indexes under Case 1 | The tracking error and the controller action indexes under Case 2 |
\sum_{\kappa=1}^{1250}[e_{11}(\kappa)]^2 | 0.0060002 | 0.014709 |
\sum_{\kappa=1}^{1250}[e_{12}(\kappa)]^2 | 0.0004923 | 0.805241 |
\sum_{\kappa=1}^{1250}[v_1(\kappa)]^2 | 174.31798 | 865.2022 |
\sum_{i=1}^{1250}[v_2(\kappa)]^2 | 1945.8023 | 2059.7359 |