Application of linguistic fuzzy neural network to landing control

Abstract

Most aircraft accidents occurred during the final approach. Wind disturbance is one of the significant factors in these accidents. During the landing phase, the Automatic Landing System (ALS) can help aircraft land safely and significantly reduce the pilot’s work loading. Control schemes of the conventional ALS usually use gain-scheduling and traditional PID control techniques. A traditional controller cannot control the aircraft if the weather conditions are beyond the allowed limits. To improve the performance of the landing control, this study applies a linguistic fuzzy neural network (LFNN) to replace the conventional controller of ALS. Adaptive learning rules are proposed to enhance the LFNN control ability. The method used to obtain adaptive learning rules is the Lyapunov stability theory. Moreover, the convergence of the system performance error is proved by the Lyapunov theory. This study also compares previously proposed control schemes in aircraft landing control. Different turbulence strengths are implemented into the flight simulation to make the proposed controller more robust and adaptive to various wind disturbance conditions. The LFNN controller can successfully overcome 75 ft/s wind speed, while the adaptive LFNN can reach 80 ft/s with optimal learning rates. Using optimal convergence theorems, the proposed controller performs better than the controllers trained by a fixed learning rate.

Introduction

Over the past few decades, a considerable number of studies have been conducted on intelligent systems that comprehend neural networks, fuzzy logic systems, and evolutionary algorithms (EAs). These research results have been brought to public attention by the technical literature present. In neural networks, two major classes have become enormously important in recent years: feedforward neural network (FNN) and recurrent neural network (RNN). FNN is a static mapping without the aid of tapped delays. RNN was proposed in 1980, which improved FNN’s static responses to become a dynamic network that uses particular branches composed of unit-delay elements. Williams and Zipser ¹ proposed the real-time recurrent learning (RTRL) algorithm. The algorithm derives its name from the fact that adjustments are made to the weights of a fully connected recurrent neural network in real time. Via an online learning algorithm, RNN has been used for several applications varying from function approximation to nonlinear system identification. One of the notable features of RNN is its powerful approximation ability and fast convergence characteristic, which has been demonstrated in many research studies. In a previous study, ² an RNN controller was successfully applied to an aircraft automatic landing system (ALS). Another popular intelligent system used very often is the fuzzy system, which uses fuzzy sets instead of conventional crisp sets. Fuzzy Sets were published by Lotfi A. Zadeh of the University of California at Berkeley in 1965, ³ which laid out the mathematics of fuzzy set theory and, by extension, fuzzy logic. Researchers were attracted by the fuzzy theory in 1970. After the 1980s, the fuzzy system developed in many research fields, such as engineering and economy. In Pathak and Yadav, ⁴ an intelligent fuzzy logic-based discrete proportional-integral-derivative controller was designed for a battery charging circuit through a maximum power point tracking algorithm. A similar controller was employed for the buck converter to get the constant voltage and constant current for the effective charging of the battery. ⁵ In Zhao et al., ⁶ a self-organized fuzzy partition and fuzzy autoencoder were proposed for a hierarchical fuzzy system that can solve the complexity of interpretable fuzzy systems. A combination of type 2 fuzzy logic and nonsingular fast sliding mode technique was developed to design a robust controller for a robotic system. ⁷ Meanwhile, integrating neural networks and fuzzy logic has given birth to a new research field called fuzzy-neural systems. Fuzzy logic systems have been successfully applied to many control applications. However, the drawback is that the fuzzy system cannot learn experience from the control process. These fuzzy-neural systems can potentially capture the benefits of both fascinating fields into a single framework. So far, many studies have been done on using fuzzy-neural systems in research tasks.^8–12

In past decades, some researchers have applied intelligent concepts such as neural networks, fuzzy systems, particle swarm optimization (PSO), GA, and hybrid systems to flight control to increase the controller’s capability to adapt to different environments.^13–17 These articles show that intelligent control systems are better than conventional control systems. Adaptive control for different wind conditions has been demonstrated successfully. The drawback is that the authors only set the wind disturbance as the initial condition. Persistent wind disturbance is not considered. In Jorgensen and Schley, ¹⁶ wind disturbances are included, but the neural controller is trained for a specific wind speed. Robustness for a wide range of wind speeds is not considered. Among the intelligent techniques, neural networks are used most because of their better adaptability and robustness for unmodeled systems. In our previous studies, a linguistic fuzzy neural network (LFNN) controller with a fixed learning rate ¹⁹ and a time-delay neural network ²⁰ have been successfully implemented in automatic landing systems for wind shear environments. Both controllers have better performance than the conventional PID controller. In turbulence conditions, the LFNN controller with a fixed learning rate was also successfully applied to the automatic landing system. ¹⁸ In Juang and Chien, ²¹ a functional fuzzy neural network (FFNN) was proposed for the landing system. An intelligent system includes wind disturbances in controller design, which provides adaptive control ability to the aircraft’s automatic landing system. Both of them showed better performance than the PID controller. This paper presents an adaptive learning scheme that combines linguistic fuzzy rules and neural networks with a modified RNN control scheme for aircraft automatic landing controller design. Adaptive learning rates, which have better control performance than fixed rates, are proposed to adjust the weights of the LFNN parameters in real time. The intelligent controller can overcome environment variations and enhance the capability of control under different turbulence conditions during aircraft landing. The Lyapunov theorem provides stability analyses of aircraft automatic landing control. It proves that the changing direction of the adjustable parameter makes the intelligent controller exponentially stable.

System description

The automatic landing system contains several automatic control systems, which include a localizer and glide path coupler, attitude and airspeed control, and an automatic flare control system. As the purpose of this study is concerned, it is not necessary to discuss ALS in detail. We will limit the discussion to the basic elements of an automatic landing system, consisting of a reference signal, aircraft dynamics, flight controller, and a wind model, as shown in Figure 1.

Figure 1.

Automatic landing system architecture.

Wind turbulence has different levels: Light, passengers may feel slight strain against seat belts, but walking is not difficult. Moderate, passengers feel definite pressures against the seat belts, and walking is difficult. Severe, passengers feel violently against seat belts, objects are unsteady, and walking is impossible. Turbulence intensity depends on air stability and the area where it occurs. According to the airline, an aircraft usually flies at high altitudes and mostly only meets light to moderate wind turbulences, with severe wind turbulence less likely. Nowadays, airlines are armed with extensive information, which allows them to avoid turbulence most of the time or at least have proper warnings. In this study, wind disturbance is considered in aircraft flight paths. A turbulence model called Dryden form ²¹ is implemented in the simulations. The equations of the model are given as follows

$$u_g=u_{\mathrm{gc}}+N(0,1)\sqrt{\frac{1}{\mathrm{\Delta }t}}\left(\frac{{\sigma }_u\sqrt{2a_u}}{s+a_u}\right)$$

(1)

$$w_g=N(0,1)\sqrt{\frac{1}{\mathrm{\Delta }t}}\left(\frac{{\sigma }_w\sqrt{3a_w}(s+b_w)}{{(s+a_w)}^2}\right)$$

(2)

where $u_{\mathrm{gc}}=-u_{510}[1+\frac{\ln (h/510)}{\ln (51)}]$ , $L_w=h$ , $a_u=\frac{U_0}{L_0}$ , $L_u=100h^{\frac{1}{3}}$ for h > 230, $L_u=600$ for h ≤ 230, $a_w=\frac{U_0}{L_w}$ , ${\sigma }_w=0.2\left|u_{\mathrm{gc}}\right|(0.5+0.00098h)$ for $0\le h\le 500$ , ${\sigma }_w=0.2\left|u_{\mathrm{gc}}\right|$ for h > 500, ${\sigma }_u=0.2|u_{\mathrm{gc}}|$ , $b_w=\frac{U_0}{L_w\sqrt{3}}$ . $u_g$ is the longitudinal wind velocity (ft/s), $w_g$ is the vertical wind velocity (ft/s), U₀ is the nominal aircraft speed (ft/s), u₅₁₀ is the wind speed at 510 ft altitude, $L_u$ and $L_w$ are scale lengths (ft), σ_u and σ_w are root mean square values of turbulence velocity (ft/s), Δt is the time interval (s), N(0, 1) is the Gaussian white noise with unity standards deviation and zero mean, u_gc is the constant component of u_g, and h is the aircraft altitude (ft). A turbulence profile with 20 ft/s wind strength initiated at 510 ft above ground is shown in Figure 2.

Figure 2.

A turbulence profile at 20 ft/s.

Aircraft landing consists of two paths: glide path and flare path. By a normal landing process, after the cruise altitude is approximately 1200 feet above the ground, the pilot positions the aircraft so that the aircraft is heading toward the runway centerline. The glide path signal is intercepted when the aircraft approaches the outer airport marker, about four nautical miles from the runway (Figure 3). The aircraft maintains a constant speed along the flight path. Finally, the flare maneuver is executed when the aircraft descends 20–70 feet above the ground. Then, the vertical descent rate is decreased so that the impact of landing gear can be reduced upon landing. The aircraft’s pitch angle is also adjusted, between 0° and 5° for most aircraft, which allows a soft touchdown on the runway surface. ¹⁸

Figure 3.

Glide path and flare path.

A benchmark model of a commercial aircraft that has been applied to many ALS studies is used in the simulations. ¹⁵ The aircraft is assumed to move only in the longitudinal and vertical plane in the simulations. Detailed descriptions can be found in Iiguni et al. ¹⁵ A complete landing process has several steps. Each step has the same fuzzy neural network controller. Turbulence is added to each step during a flight. Figure 4 shows the modified recurrent learning scheme used in Juang and Chio. ²⁰ The FC is the fuzzy neural network controller, and the AM is the aircraft model. Every learning cycle consists of all steps from S₀ to S_k. At each step, C_i is the pitch command of the aircraft. S_i is the state of the aircraft, which includes altitude and velocity. The FC is trained by a modified learning-through-time process. ΔC_k−1 and e_k−1 are obtained from the linearized inverse aircraft model (LIAM). Error signal e_k−1 of each step is calculated by the e_k. The error e_k−1 is used to obtain ΔC_k−2 and e_k−2. The above process is repeated until all necessary ΔC_i and e_i are obtained. The ΔC_k−1 is used to backpropagate through the flight controller (FC) in each step to get the weight change of the fuzzy neural network controller. The error continues to be backpropagated through all k steps, with weight changes computed for the controller at each step. The weight changes from all steps are obtained from the delta learning rule and are added together for the overall update. New commands of the AM are obtained from the FC; then, new flight conditions can be obtained at the AM output. The learning process is repeated in the simulation till the aircraft fulfills its successful landing condition. Inputs of the FC are altitude, altitude command, altitude rate, and altitude rate command. The output of the FC is the pitch command. Detail descriptions can be found in Juang and Chio. ²⁰

Figure 4.

Learning through time process.

Linguistic fuzzy neural networks

In fuzzy neural networks, there are two major fuzzy rules that are most used: one is the linguistic fuzzy rule, and the other is the functional fuzzy rule. The reason why we applied the linguistic fuzzy rule to fuzzy modeling design in this study is because it is simpler than the functional fuzzy rule and is more similar to human knowledge representation. The fuzzy neural network consists of a fuzzy logic system and neural network technique, providing simplicity and nonlinear control ability of fuzzy control but also learning and adaptive capability of the neural network. Therefore, the adaptability and performance of the controlled system are better than the neural network. ²² A time delay network is similar to a multilayer feedforward perceptron. The difference is that inputs consist of outputs (or inputs) of an earlier neuron state. This time delay behavior enables the neural network to produce memory ability, obtaining earlier information into the neural network at a specific time step. It has succeeded well in machine control,^23,24 where the neural network provides dynamic behavior. The fuzzy neural network consists of five layers. In this case, it has four inputs, one output, and two membership functions in each premise. Neuron outputs with a symbol of ∑ are sums of their inputs, and Π are products of their inputs. ²⁰ The network structure is shown in Figure 5.

Figure 5.

Linguistic fuzzy neural network.

Description of feedforward signals

The description of each layer output is given as follows:

Layer 1: Input layer

In this layer, each input value directly relates to the next layer. This layer contains a total of four inputs.

$$O_i^{\left(1\right)}=x_i^{\left(1\right)}i=1,2,3,4$$

(3)

Layer 2: Hidden layer

In this layer, the inputs are $O_i^{\left(1\right)}$ and add another bias; the weights from the input layer to the hidden layer are 1; $\mathrm{Wc}\in R^{4\times 1}$ are the weights from the bias to the hidden layer.

$$O_j^{\left(2\right)}=x_i+w_{\mathrm{cj}}i=1,2,3,4,j=1,2,...,8$$

(4)

Layer 3: Membership function layer

In this layer, each unit performs a membership function. The current work adopts the Sigmoid function as a membership function.

$$\begin{array}{c}\multicolumn{1}{c}{O_j^{\left(3\right)}\left(x_i\right)=\frac{1}{1+\exp \{-w_{\mathrm{gj}}(x_i+w_{\mathrm{ci}})\}}}\\ i=1,2,3,4,j=1,2,...,8\hfill \end{array}$$

(5)

where $W_g\in R^{8\times 1}$ are the weights from the hidden layer to the membership layer. By appropriately initializing the connection weights $w_{\mathrm{ci}}$ and $w_{\mathrm{gj}}$ , the membership function in premises $A_{\mathrm{kl}}$ ( $k$  = 1, 2; $l$  = 1, 2, 3, 4) allocates to the universe of discourse, as shown in Figure 6.

Figure 6.

Membership function in premise.

Layer 4: Rule layer

The units in this layer are called rule units. This study chooses the fuzzy AND aggregation operation as the PRODUCT operation instead of the MIN operation.

$$I_r^{\left(4\right)}(q)=\underset{l}{\Pi }{A}_{\mathrm{kl}}(q)={\mu }_r(q)r=1,2,\cdots ,16$$

(6)

$$O_r^{\left(4\right)}(q)=\frac{{\mu }_r(q)}{\sum _{k=1}^{16}{\mu }_k(q)}={\hat{\mu }}_r(q)k=1,2,\cdots ,16$$

(7)

where $A_{\mathrm{kl}}$ are fuzzy variables in the premises, ${\mu }_r$ is the truth value of the rth fuzzy rule, and ${\hat{\mu }}_r$ is the normalized value of ${\mu }_r$ .

Layer 5: Output layer

This layer performs the defuzzification operation. The inferred value $y^{*}$ is then obtained as the sum of $w_{\mathrm{fr}}·{\hat{\mu }}_r$ . The network realizes the following inference method:

$$R^r:\mathrm{IF}{x}_1\mathrm{is}{A}_{k1}{x}_2\mathrm{is}{A}_{k2},...,x_4\mathrm{is}{A}_{k4}\mathrm{Then}y\mathrm{is}{B}_r$$

(8)

$$y^{*}(q)=\sum _{r=1}^{16}{\hat{\mu }}_r(q)w_{\mathrm{fr}}(q)$$

(9)

where x_i is the linguistic input variable, y is the linguistic output variable, A_ki is the linguistic value with respect to each input variable, $R^r$ is the $r$ th fuzzy rule, and $B_r$ is a constant. The weight $w_{\mathrm{fr}}$ is the output action strength associated with the r th rule.

Note that the weights $W_c$ , $W_g$ , and $W_f$ are modified to identify fuzzy rules and tune the membership functions in the premises using the following back-propagation algorithm:

$$\mathrm{Output}\mathrm{layer}{\delta }_j^{(n)}=(d_j-O_j^{(n)})\frac{d\phi (I_j^{(n)})}{\mathrm{dI}}$$

(10)

$$\mathrm{Hidden}\mathrm{layer}{\delta }_j^{(n)}=\frac{d\phi (I_j^{(n)})}{\mathrm{dI}}\sum _k{\delta }_k^{(n+1)}{w}_{\mathrm{kj}}^{(n+1)}$$

(11)

where n and j denote the nth layer and jth neuron, respectively. d is the desired target value. I and O are the input and output of the neuron, respectively. $\frac{d\phi (I_j^{(n)})}{\mathrm{dI}}$ denotes the derivative of the output function of the neuron. $w_{\mathrm{kj}}^{(n+1)}$ is the connection weight between the nth layer’s jth neuron and (n + 1)th layer’s kth neuron. The weights $w_c$ and $w_g$ are initialized so that the membership functions in the premises are allocated to the normalized universe of discourse as in Figure 6. The weights $W_f$ are initialized to be zero. The fuzzy neural controller starts learning without any control rule.

Network updating rules

We use the back-propagation algorithm with the updating law as follows:

Layer 5: Output layer

$${\delta }^{\left(5\right)}(q)=(d(q)-y(q))\frac{d\phi (I^{\left(5\right)}(q))}{\mathrm{dI}}=d(q)-y(q)$$

(12)

thus

$$w_{\mathrm{fr}}(q+1)=w_{\mathrm{fr}}\left(q\right)+\eta {\delta }^{\left(5\right)}(q)O^{\left(4\right)}(q)$$

(13)

where $r=1,2,3,\dots ,16$ , $I^{\left(5\right)}$ is the input of the fifth layer, d is the desired output, and $O^{\left(4\right)}$ is the output of the fourth layer.

Layer 4: Rule layer

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_j^{\left(4\right)}(q)=\frac{d\phi (I_j^{(4)}(q))}{\mathrm{dI}}\sum _f{\delta }_f^{\left(5\right)}(q)w_{\mathrm{fj}}^{\left(5\right)}(q)}\\ =\frac{1}{\sum {\mu }_r(q)}{\delta }^{\left(5\right)}(q)w_{\mathrm{fj}}(q)\hfill \end{array}$$

(14)

Layer 3: Membership function layer

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_j^{\left(3\right)}(q)=\frac{d\phi (I_j^{(3)}(q))}{\mathrm{dI}}\sum _k{\delta }_k^{\left(4\right)}(q)(\underset{i\ne j}{\Pi }{O}_i^{\left(3\right)}(q))}\\ =O_j^{\left(3\right)}(q)(1-O_j^{\left(3\right)}(q))\sum _k[{{\delta }_k}^{\left(4\right)}(q)\underset{i\ne j}{\Pi }{O}_i^{\left(3\right)}(q)]\hfill \\ =(1-O_j^{\left(3\right)}(q))\sum _k[{{\delta }_k}^{\left(4\right)}(q)\Pi O_i^{\left(3\right)}(q)]\hfill \end{array}$$

(15)

From (6) and (7) we have

$$\begin{array}{l}{\delta }_j^{\left(3\right)}(q)=(1-O_j^{\left(3\right)}(q))\sum _k\left[\frac{{\mu }_k(q)}{\sum {\mu }_r(q)}{\delta }^{\left(5\right)}(q)w_{fk}(q)\right]\\ =(1-O_j^{\left(3\right)}(q))\sum _k[{\widehat{\mu }}_k(q)w_{fk}(q){\delta }^{\left(5\right)}(q)]\end{array}$$

(16)

where $j=1,2,3,\dots ,8$ , and the subscript k denotes the rule unit in connection with the jth unit in layer 3. Thus

$$w_{\mathrm{gj}}(q+1)=w_{\mathrm{gj}}\left(q\right)+\eta {\delta }^{\left(3\right)}(q)O^{\left(2\right)}(q)$$

(17)

Layer 2: Hidden layer

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_j^{\left(2\right)}(q)=\frac{d\phi (I_j^{(2)}(q))}{\mathrm{dI}}\sum _k{\delta }_k^{\left(3\right)}(q)w_{\mathrm{kj}}^{\left(3\right)}(q)}\\ ={\delta }_j^{\left(3\right)}(q)w_g(q)\hfill \end{array}$$

(18)

Thus

$$w_{ci}\left(q+1\right)=w_{ci}\left(q\right)+\eta {\delta }^{\left(2\right)}(q){\stackrel{\wedge }{O}}^{\left(1\right)}(q)$$

(19)

where $i=1,2,3,\dots ,8$ , ${\stackrel{\wedge }{O}}^{\left(1\right)}=1$ .

Adaptive learning rates

• 1. The varying range of the learning rate ( ${\eta }_f$ ) of $W_f$ :

From Chaturvedi et al., ¹⁴ let

$$\begin{array}{c}\multicolumn{1}{c}{P_{\mathrm{Lf}}(q)=\frac{\partial y^{*}(q)}{\partial W_f}}\\ =\frac{\partial (\sum _{r=1}^R{{\mu }^{\wedge }}_r(q)w_{\mathrm{fr}}(q))}{\partial W_f}\hfill \\ ={\left[\begin{array}{cccc}{{\mu }^{\wedge }}_1(q)\hfill & {{\mu }^{\wedge }}_2(q)\hfill & ...\hfill & {{\mu }^{\wedge }}_R(q)\hfill \end{array}\right]}^T\hfill \end{array}$$

(20)

where ${\hat{\mu }}_r$ is the normalized value of ${\mu }_r$ , P_L is a variable with a positive value, and let ${\stackrel{\wedge }{\mu }}_{\max }=\underset{q}{\max }\left|{\widehat{\mu }}_r(q)\right|$ , we have

$$\Vert P_{\mathrm{Lf}}(q)\Vert <\sqrt{R\left({{{\mu }^{\wedge }}^2}_{max}\right)}$$

(21)

where R is the normalized number of ${\mu }_r$ . Since R = 16, we have

$$P_{\mathrm{Lf},max}=\sqrt{16\left({{{\mu }^{\wedge }}^2}_{max}\right)}$$

(22)

Thus

$$0<{\eta }_f<\frac{2}{16\left({{{\mu }^{\wedge }}^2}_{max}\right)}$$

(23)

• 2. The varying range of the learning rate ( ${\eta }_g$ ) of $W_g$ :

Let

$$\begin{array}{c}\multicolumn{1}{c}{P_{\mathrm{Lgj}}(q)=\frac{\partial y^{*}(q)}{\partial w_{\mathrm{gj}}}}\\ =O_j^{\left(2\right)}\left(q\right)(1-O_j^{\left(3\right)}(q))\sum _k[{{\mu }^{\wedge }}_k(q)w_{\mathrm{fk}}(q)]\hfill \end{array}$$

(24)

where ${\hat{\mu }}_k$ is the normalized value of ${\mu }_r$ . Let ${\stackrel{\wedge }{\mu }}_{\max }=\underset{q}{\max }\left|{\widehat{\mu }}_r(q)\right|$ , $W_{f,max}=\underset{q}{max}|W_f(q)|$ , ${\stackrel{\wedge }{O}}_{max}^{\left(2\right)}=\underset{q}{max}|O^{\left(2\right)}(q)|$ , and ${\stackrel{\wedge }{O}}_{max}^{\left(3\right)}=\underset{q}{max}\left|(1-O^{\left(3\right)}(q))\right|$ ,

we have

$$\Vert P_{Lg}(q)\Vert <\sqrt{N{\left({\stackrel{\wedge }{O}}_{\max }^{\left(2\right)}{\stackrel{\wedge }{O}}_{\max }^{\left(3\right)}\right)}^2{\left(K·{\stackrel{\wedge }{\mu }}_{\max }{W}_{f,\max }\right)}^2}$$

(25)

where N is the number of $w_{\mathrm{gj}}$ , and K is the number of the rule units in connection with the jth unit in layer 3. In this study, N = 8 and K = 8, we have

$$P_{Lg,\max }=\sqrt{8{(8·{\stackrel{\wedge }{O}}_{\max }^{\left(2\right)}{\stackrel{\wedge }{O}}_{\max }^{\left(3\right)}{\stackrel{\wedge }{\mu }}_{\max }{W}_{f,\max })}^2}$$

(26)

Thus

$$0<{\eta }_g<\frac{2}{8{(8·{\stackrel{\wedge }{O}}_{\max }^{\left(2\right)}{\stackrel{\wedge }{O}}_{\max }^{\left(3\right)}{\stackrel{\wedge }{\mu }}_{\max }{W}_{f,\max })}^2}$$

(27)

• 3. The varying range of the learning rate ( ${\eta }_c$ ) of $W_c$ :

Let

$$\begin{array}{c}\multicolumn{1}{c}{P_{\mathrm{Lcj}}(q)=\frac{\partial y^{*}(q)}{\partial w_{\mathrm{cj}}}}\\ =O_j^{\left(2\right)}(q)w_{\mathrm{gj}}(q){\stackrel{\wedge }{O}}^{\left(1\right)}(q)(1-O_j^{\left(3\right)}(q))\sum _k[{{\mu }^{\wedge }}_k(q)w_{\mathrm{fk}}(q)]\hfill \end{array}$$

(28)

where ${\hat{\mu }}_k$ is the normalized value of ${\mu }_r$ . Let ${\stackrel{\wedge }{\mu }}_{\max }=\underset{q}{\max }\left|{\widehat{\mu }}_k(q)\right|$ , $W_{f,max}=\underset{q}{max}|W_f(q)|$ , ${\stackrel{\wedge }{O}}_{\max }^{\left(2\right)}=\underset{q}{\max }\left|O^{\left(2\right)}(q)\right|$ , and ${\stackrel{\wedge }{O}}_{max}^{\left(3\right)}=\underset{q}{max}\left|(1-O^{\left(3\right)}(q))\right|$ , $W_{g{,}_{max}}=\underset{q}{max}|W_g(q)|$ , and ${\stackrel{\wedge }{O}}^{\left(1\right)}(q)=1$ , we have

$$\Vert P_{Lc}(q)\Vert <\sqrt{N{\left({\stackrel{\wedge }{O}}_{\max }^{\left(2\right)}{\stackrel{\wedge }{O}}_{\max }^{\left(3\right)}{W}_{g,\max }\right)}^2{\left(K·{\stackrel{\wedge }{\mu }}_{\max }{W}_{f,\max }\right)}^2}$$

(29)

where N is the number of $w_{\mathrm{gj}}$ , and K is the number of the rule units in connection with the jth unit in layer 3. We have

$$P_{Lc,\max }(q)=\sqrt{8{(8·{\stackrel{\wedge }{O}}_{\max }^{\left(2\right)}{\stackrel{\wedge }{O}}_{\max }^{\left(3\right)}{W}_{g,\max }{\stackrel{\wedge }{\mu }}_{\max }{W}_{f,\max })}^2}$$

(30)

Thus

$$0<{\eta }_c<\frac{2}{8{(8·{\stackrel{\wedge }{O}}_{\max }^{\left(2\right)}{\stackrel{\wedge }{O}}_{\max }^{\left(3\right)}{W}_{g,\max }{\stackrel{\wedge }{\mu }}_{\max }{W}_{f,\max })}^2}$$

(31)

Stability analysis

In the LFNN, we choose a Lyapunov function that can be expressed as:

$$V(q)=\frac{1}{2}{e}^2(q)$$

(32)

Thus, the change in the Lyapunov function is obtained by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }V(q)=V(q+1)-V(q)}\\ =\frac{1}{2}[e^2(q+1)-e^2(q)]\hfill \\ =\mathrm{\Delta }e(q)[e(q)+\frac{1}{2}\mathrm{\Delta }e(q)]\hfill \end{array}$$

(33)

The error difference due to learning can be represented by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }e(q)=e(q+1)-e(q)}\\ =\frac{1}{2}[e^2(q+1)-e^2(q)]\hfill \\ =\mathrm{\Delta }e(q)[e(q)+\frac{1}{2}\mathrm{\Delta }e(q)]\hfill \\ =\left[\left(\frac{\partial e(q)}{\partial W_f}\right)\left(\frac{\partial e(q)}{\partial W_g}\right)\left(\frac{\partial e(q)}{\partial W_c}\right)\right]\left[\begin{array}{c}\mathrm{\Delta }{W}_f(q)\hfill \\ \multicolumn{1}{c}{\mathrm{\Delta }{W}_g(q)}\\ \multicolumn{1}{c}{\mathrm{\Delta }{W}_c(q)}\end{array}\right]\hfill \end{array}$$

(34)

Using the gradient descent method, we have

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{W}_L(q)=-{\eta }_L\frac{\partial E(q)}{\partial W_L}}\\ ={\eta }_{L}e(q)\frac{\partial y(q)}{\partial W_L}\hfill \\ =e(q)\left[\begin{array}{ccc}{\eta }_f\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\eta }_g\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & {\eta }_c\hfill \end{array}\right]\left[\begin{array}{c}\frac{\partial y(q)}{\partial W_g}\hfill \\ \multicolumn{1}{c}{\frac{\partial y(q)}{\partial W_g}}\\ \multicolumn{1}{c}{\frac{\partial y(q)}{\partial W_c}}\end{array}\right]\hfill \end{array}$$

(35)

From Juang et al., ¹⁹ let

$$\begin{array}{c}\multicolumn{1}{c}{P_{L,max}\equiv {\left[\begin{array}{ccc}{P}_{\mathrm{Lf},max}\hfill & P_{\mathrm{Lg},max}\hfill & P_{\mathrm{Lc},max}\hfill \end{array}\right]}^T}\\ ={\left[\begin{array}{ccc}\underset{q}{max}\Vert \frac{\partial y(q)}{\partial W_f}\Vert \hfill & \underset{q}{max}\Vert \frac{\partial y(q)}{\partial W_g}\Vert \hfill & \underset{q}{max}\Vert \frac{\partial y(q)}{\partial W_c}\Vert \hfill \end{array}\right]}^T\hfill \end{array}$$

(36)

Then we know the asymptotic convergence is achieved if ${\eta }_i$ are chosen to satisfy

$$0<{\eta }_i<\frac{2}{{\left(P_{\mathrm{Li},max}\right)}^2}i=f,g,c.$$

(37)

From (23), (27), and (31) we know the learning rates satisfy the above condition. Thus $\mathrm{\Delta }V<0$ and $\mathrm{\Delta }V$ are negative definite. The tracking error e(q) is asymptotically stable, guaranteeing convergence of the learning process.

Simulation results

The simulations quote^{2,16,20,21,25} results. The inputs of the LFNN controller are altitude, altitude command, altitude rate, and altitude rate command of the aircraft. The output of the controller is the pitch command. After training, the LFNN controller replaces the PID controller as the pitch autopilot. Successful touchdown conditions are defined as follows ¹⁶ :

• −3 $\le \stackrel{·}{h}(T)$ ft/s ≤0,

• 200 $\le \stackrel{·}{x}(T)$ ft/s ≤270

• −300 $\le x(T)$ ft ≤1000,

• −10 $\le \theta (T)$ degree ≤5

where T is the time at touchdown. Initial flight conditions are: h(0) = 500 ft, $\stackrel{·}{x}(0)$  = 235 ft/s, $x(0)$  = 9240 ft, and ${\gamma }_o$  = −3°. The current work trains the LFNN controller to replace the PID controller in this simulation. We choose the varying learning rates as ${\eta }_i^{*}=\frac{1}{P_{L,max}^2}$ to train the adaptive linguistic fuzzy neural network controller. Table 1 shows the results using different wind turbulence speeds. The proposed adaptive LFNN controller can successfully guide an aircraft to overcome 80 ft/s turbulence. Figures 7 to 10 show the results using the adaptive LFNN controller at 80 ft/s wind speed. The turbulence profile is generated randomly. At each trial, the shape of the profile is different. Since the strength of wind disturbance at each time instance is different and unpredictable, the aircraft’s real pitch angle and vertical velocity did not follow the commands well. But in the final period, the aircraft is controlled to follow the command. A plot of the commanded elevator angle is shown in Figure 11. This is in contrast to some adaptive schemes that provide extremely good results by driving the control surfaces at high rates that can never be achieved in actual systems. The proposed control scheme shows an acceptable rate for real actuators. Comparison from using different controllers is shown in Table 2, where BPNN is the backpropagation neural network, FFNN is the functional fuzzy neural network, RNN is the recurrent neural network, ² and ARAN is the adaptive resource allocating network. ²⁵ The conventional PID controller only overcomes about 30 ft/s wind speed. ¹⁶ The adaptive FFNN controller with optimal learning rates can overcome 60 ft/s. With a fixed rate, it can only reach 55 ft/s. ²¹ The LFNN controller can successfully overcome 75 ft/s, ²⁰ while the adaptive LFNN can reach 80 ft/s with optimal learning rates. Using optimal convergence theorems, the proposed controller performs better than the controllers trained by the fixed learning rate.

Table 1.

The results from using an adaptive linguistic fuzzy neural network controller.

Wind speed	Landing point (ft)	Aircraft vertical speed (ft/s)	Pitch angle (°)
30	762.56	−1.75	−0.06
40	218.24	−2.81	0.06
50	213.51	−2.64	0.14
60	633.10	−2.87	−0.46
70	806.20	−2.76	−0.47
80	886.86	−2.25	−0.65

Figure 7.

Turbulence profile (80 ft/s).

Figure 8.

Aircraft pitch and pitch command.

Figure 9.

Aircraft vertical velocity and command.

Figure 10.

Aircraft altitude and command.

Figure 11.

Commanded elevator angle.

Table 2.

Comparison of using different controllers.

Type of controller	PID 16	BPNN 16	FFNN 21	Adaptive FFNN 21	RNN 2	ARAN 25	LFNN 20	Adaptive LFNN
Maximum wind speed	30	40	55	60	65	75	75	80

In each wind disturbance condition, 100 tests were executed in this study. The turbulence profiles were different since the turbulence was randomly generated from equations (1) and (2). The simulation results of the 100 tests were different, although they were at the same wind speed. From the simulations, most of the failure cases were over the landing point. The limit of the landing point is 1000 ft. Since the pattern of the disturbance is unpredictable, for a significant value disturbance, the flight controller will try to overcome the interrupt first and then control the pitch angle of the aircraft. And the process takes time. Thus, the aircraft will fly over the predefined landing point. A better controller can shorten the process time and make the aircraft land within the landing point.

Conclusions

In this study, the back-propagation algorithm is applied as the updating law. However, it only converges to a local region. The variation of adjustable parameters of the linguistic fuzzy neural network depends on the gradient descent and learning rate. The gradient descent method provides the changing direction of variation, and the learning rate decides the size of the variation. The convergence theorem for selecting appropriate learning rates is developed in this study. We use the discrete-time Lyapunov function to analyze the stability of the linguistic fuzzy neural network controller and obtain appropriate varying learning rates to attain an optimal stable state. The convergence theorem derives optimal learning rates. Finally, from theory analysis and simulation results, the proposed adaptive linguistic fuzzy neural network controller can enable the aircraft to adapt to a wide range of turbulence and guide the aircraft to a safe landing. The contributions of this study are summarized in points form as follows. (1) Adaptive learning rates are derived, and these time-varying rates are more suitable for unpredictable turbulence conditions. (2) Convergence of the system performance error is guaranteed by the Lyapunov stability analysis. (3) The proposed controller can overcome turbulence to 80 ft/s which is better than previous studies. Although the results of software simulations are good, hardware systems still have a long way to go. Hardware simulations will be constructed in the future. In addition, this study only considers the longitudinal plane and the pitch control system. Future directions of this study will include the horizontal plane and the yaw control system to the control process.