# Design of aT Factor Based RBFNC for a Flight Control System

Authors: C. S. Mohanty; P. S. Khuntia; D. Mitra
Journal: Advances in Artificial Intelligence
 (2014)

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## Abstract

This paper presents the design of modified radial basic function neural controller (MRBFNC) for the pitch control of an aircraft to obtain the desired pitch angel as required by the pilot while maneuvering an aircraft. In this design, the parameters of radial basis function neural controller (RBFNC) are optimized by implementing a feedback mechanism which is controlled by a tuning factor “α” (T factor). For a given input, the  response of the RBFN controller is tuned by using T factor for better performance of the aircraft pitch control system. The proposed system is demonstrated under different condition (absence and presence of sensor noise). The simulation results show that MRBFNC performs better, in terms of settling time and rise time for both conditions, than the conventional RBFNC. It is also seen that, as the value of the T factor increases, the aircraft pitch control system performs better and settles quickly to its reference trajectory. A comparison between MRBFNC and conventional RBFNC is also established to discuss the superiority of the former techniques.


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## Body

## 1. Introduction

The conventional design methods of a control system often require mathematical models describing the dynamic behavior of the plant to be controlled. When such mathematical models are difficult to obtain due to uncertainty or complexity, the conventional techniques based on mathematical models are not well suited. Artificial neural network (ANN) in last decade has become popular for plant identification and control [1, 2]. An advantage of the ANN is its ability to handle the nonlinear mapping of the input-output space. It is well known that back propagation based ANN suffers from local minima and over fitting problems which is difficult to be implemented in real time due to a large number of neurons in the hidden layer in comparison to the RBFNC [3, 4]. Since early 1990s, radial basis function network with Gaussian function has been widely used as the basic structure of neural network in nonlinear control [5–7]. Locally tuned and overlapping receptive fields have been found in cerebral cortex visual cortex and in other parts of the brain. The concept of localized information processing in the form of receptive fields suggests that such local learning offers alternative computational opportunities to learning with global basis functions [8]. Gomi and Kawato proposed a feedback error learning control strategy, where a Gaussian RBFN is used for online learning of the inverse dynamics of the system [6]. A Radial basic function neural controller (RBFNC) with learning mechanism is used to control the pitch angle of an aircraft [9]. RBFNC is used for UPFC to improve the transient stability performance of a power system [10]. The comparison between multilayer perception network (MLPN) and radial basis function network (RBFN) is done for online identification of the nonlinear dynamics of a synchronous generator [11]. Neural networks are used to build a model of the plant and to construct its “inverse” to approximate a desired model dynamics [12].In this paper, a new modified neural controller is designed to control the pitch angle of an aircraft in various conditions and also in the presence of sensor noise. Due to the presence ofT factor, the weights are updated and the response of the pitch control system is accelerated and settles quickly to its reference value.


## 2. Radial Basic Function Neural Network

Radial Basis function networks are two layers feed forward networks. In radial basis function neural network (RBFNN), the hidden nodes are implementing a set of radial basis functions (e.g., Gaussian functions).In RBFNN, the network training is divided into two stages:(i)
weights from the input to hidden layer are determined;(ii)
weights from the hidden to output layer are also determined.The training/learning in case of RBFNN is very fast and networks are very good at interpolation. A radial basic function neural network is shown in Figure1.Figure 1
RBFNC for aircraft pitch control.The proposed RBFNN model with single neuron outputy presented in Figure 1 consists of three layers [13]. Each input value is assigned to a node and passed directly to the hidden layer without weights. The hidden layer nodes are called radial basic function (RBF) units which are determined by a parameter vector called center and a scalar called width. The Gaussian density function is used as an activation function for the hidden neurons. The RBFNN shown in Figure 1 has inputs xi, i=1,2,3,…,n, and output y=Frbf(x).x=[x1,x2,x3,…,xn]T is the input and Ri(x) is the output of the ith receptive field with strength denoted by bi. Assuming nR receptive fields present in the RBFNN, the output y can be written as
(1)y=Frbf(x,φ)=∑i=1nRbiRi(x),
where φ holds the parameters of the receptive field units which consist of the parameters bi and possibly the parameters of the Ri(x). The Gaussian-shaped functions are preferred for analytical convenience; that is,
(2)Ri(x)=exp⁡[|x-ci|2(σi)2],
where ci=[c1i,c2i,…,cni]T parameterize the locations and σ decides the spreading of the receptive fields in the input space.The weighted average output of the RBFNN can be written as(3)y=Frbf(x,φ)=∑i=1nRbiRi(x)∑i=1nRRi(x).


## 3. Mathematical Formulation of an Aircraft Pitch Control System

The pitch angle of the aircraft is generally described by a coordinate system that is fixed to the aircraft. The pitch angle and other forces acting in an aircraft are shown in Figure2.Figure 2
Longitudinal dynamics description of an aircraft, whereδE = deflection of the elevator, θ = pitch angle, and α = angle of attack.The aircraft pitch control model is described by the following equation [14]:
(4)θ…(t)+(1τ1+1τ2)θ..(t)+(1τ1τ2)θ.(t)=kτ1τ2(τ3δ.E(t)+δE(t)).The above equation can be represented as(5)θ(s)δE(s)=k(sτ3+1)s((sτ1+1)(sτ2+1)),
where the values of τ1, τ2, τ3, and k are 0.8995+0.0968i, 0.8995-0.0968i, 1.0824, and −1.1346, respectively, for a Delta aircraft (flight condition 3) [13].


## 4. Design ofT Factor Based RBFNC

The RBFNC for aircraft pitch control system shown in Figure2 tracks the desired pitch angle θ(kT). The system has aT factor that is used to tune the output of the RBFNC.As shown in Figure3, the α, e(kT), ec(kT), and θ(kT) are used to adjust the weights of the neural controller, that is, bi, where
(6)e(kT)=θref(kT)-θ(kT),ec(kT)=e(kT)-e(kT-T)T.Figure 3
Proposed RBFNC controller for pitch control.AndT is the sampling time. The output of the RBF NC δ(k) is computed by taking e(k) and ec(k) as the argument to the radial basic function
(7)δ(k)=Frbf(e(k),ec(k),α,δ(k-1)),
where α=T factor [15] and δ(k-1)= previous output of neural controller.It is decided in the designing of pitch controller that elevator should not to exceed more thanπ/2 radian in either upward or downward direction or the change of error should not be more than 0.01 radian/sec. It concludes that range of error e(kT) and change of error ec(kT) are e(k)∈[-π/2,π/2] and ec(k)∈[-0.01,0.01]. A uniformly grid is created by taking the error and the change of error with the corners of the grid placed at (-π/2,-0.01),(-π/2,0.01), (π/2,0.01), and (π/2,-0.01). Each point on the grid contains a receptive field which is a Gaussian function. The error and change of the error’s spreading (σ) are taken differently as
(8)σc=0.70.02nG,σe=0.7πnR,
where nG is the number of partitions on the grid (nG= 11 here) and  nR is the number of receptive field units in RBFNC which is equal to nR=nG2 (nR= 121). Each center which represents a RBF is represented by circle as shown in Figure 4.Figure 4
Receptive field unit centers.The left most bottom circle(-π/2,-0.01) is counted as 1 and the counting increases by 1 making the left most top circle (-π/2,0.01) numbered as 11. Next counting starts from the bottom circle of the next column with number 12 and the top most circle in that column is represented as 22. So, the right most bottom circle (π/2,0.01) is counted as 111 and top most circle is counted as 121 (π/2,-0.01). The input and output mapping of the radial basis function neural network is shaped by choice of scaling parameters bi (Figure 5). Assume the scaling and summation of the receptive field units with centers at the four dark-shaded circles shown in Figure 1 (the indices here are assumed to be 61, 62, 72, and 73) to be  2R61(e,c)+3R62(e,c)+R72(e,c)+2R73(e,c). The scaling and summing are computed and shown in Figure 4.Figure 5
Scaling and addition of many receptive fields2R61(e,c)+3R62(e,c)+R72(e,c)+2R73(e,c).A single receptive fieldR73(e,c) without scalingis shown in Figure 6.Figure 6
Single receptive fieldR73(e,c).For the receptive field, the parameterσ of the Gaussian function decides the spreading of the Gaussian function. The error and change of the error’s spreading (σ) are different and are taken as
(9)σc=0.70.02nG,σe=0.7πnR,
where nG is number of partitions on each edge of grid (here, nG= 11) and nR is the number of receptive field units in RBF neural controller which is equal to nR=nG2 (here, nR= 121).


## 5. Simulation Results

The reference signal is a step signal and the flight travels with constant speed of 253 m/sec of a Delta aircraft (flight condition-3). A reference pitch angle of 1 degree is given as input to the aircraft and to the reference model simultaneously. The output which is the actual pitch angle follows the reference trajectory of reference model output. The following figures illustrate the pitch angle responses of the aircraft under flight condition-3.


### 5.1. In the Absence of Sensor Noise

(1) Comparison of Closed Loop Response betweenα=0 (withoutT factor) and α=0.9 (withT factor). See Figure 7.Figure 7
Comparison response of pitch angle (deg) betweenα=0 and α=0.9.(2) Comparison of Pitch Angle Errors betweenα=0  (withoutT factor) andα=0.9 (withT factor). See Figure 8.Figure 8
Comparison of response of error betweenα=0 and α=0.9.(3) Output of Neural Controller forα=0 (withoutT factor). See Figure 9.Figure 9
The output of RBFNC to plant forα=0.(4) Output of Neural Controller forα=0.9 (withT factor). See Figure 10.Figure 10
The output of RBFNC to plant forα=0.9.


### 5.2. In Presence of Sensor Noise

(1) Comparison of closed loop responses betweenα=0  (withoutT factor) and α=0.9  (withT factor). See Figure 11.Figure 11
Comparison response of pitch angle (deg) for aircraft pitch control system betweenα=0  and  α=0.9.(2) Comparison of pitch angle errors betweenα=0 (withoutT factor) and α=0.9 (withT factor). See Figure 12.Figure 12
Comparison of response of error betweenα=0 and α=0.9.(3) Output of Neural Controller forα=0 (withoutT factor). See Figure 13.Figure 13
The output of RBFNC to plant forα=0.(4) Output of Neural Controller forα=0.9 (withT factor). See Figure 14.Figure 14
The output of RBFNC to plant forα=0.9.


### 5.3. Comparison of Pitch Control System Responses for Different Values ofT Factor (α=0, α=0.2, α=0.6andα=0.9)

See Figure15.Figure 15
Comparison of closed loop responses of aircraft for different values ofT factors.


## 5.1. In the Absence of Sensor Noise

(1) Comparison of Closed Loop Response betweenα=0 (withoutT factor) and α=0.9 (withT factor). See Figure 7.Figure 7
Comparison response of pitch angle (deg) betweenα=0 and α=0.9.(2) Comparison of Pitch Angle Errors betweenα=0  (withoutT factor) andα=0.9 (withT factor). See Figure 8.Figure 8
Comparison of response of error betweenα=0 and α=0.9.(3) Output of Neural Controller forα=0 (withoutT factor). See Figure 9.Figure 9
The output of RBFNC to plant forα=0.(4) Output of Neural Controller forα=0.9 (withT factor). See Figure 10.Figure 10
The output of RBFNC to plant forα=0.9.


## 5.2. In Presence of Sensor Noise

(1) Comparison of closed loop responses betweenα=0  (withoutT factor) and α=0.9  (withT factor). See Figure 11.Figure 11
Comparison response of pitch angle (deg) for aircraft pitch control system betweenα=0  and  α=0.9.(2) Comparison of pitch angle errors betweenα=0 (withoutT factor) and α=0.9 (withT factor). See Figure 12.Figure 12
Comparison of response of error betweenα=0 and α=0.9.(3) Output of Neural Controller forα=0 (withoutT factor). See Figure 13.Figure 13
The output of RBFNC to plant forα=0.(4) Output of Neural Controller forα=0.9 (withT factor). See Figure 14.Figure 14
The output of RBFNC to plant forα=0.9.


## 5.3. Comparison of Pitch Control System Responses for Different Values ofT Factor (α=0, α=0.2, α=0.6andα=0.9)

See Figure15.Figure 15
Comparison of closed loop responses of aircraft for different values ofT factors.


## 6. Conclusion

The nonzero value of the MRBFNC output exhibits its adaptive nature whenever the actual pitch angle differs from its reference value and at the time of transition of the reference signal. When the speed of the aircraft is changed, the control signal to the pitch control system also changes to cope up with the speed change. From Table1, it is clear that, for the value of “α = 0.9,” the settling time and rise time of pitch control system are much better than the RBFNC based pitch control system. It is shown from Figures 10 and 14 thatT factor tunes the output of the RBFNC in both normal and sensor noise conditions. Figure 7 shows comparative analysis of closed loop response between conventional and MRBFNC. TheT factor based system has a better response and settles earlier to its references than the conventional RBFNC. Figure 15 shows the responses of the pitch control system that varies according to the value of theT factor (α). It is concluded that, to increase the value of α, the response θ(k) becomes settled earlier to the reference value. It is also shown in this simulation that the sensor noise does not affect the output of the RBFNC because the controller output continuously changes to nullify the effect of this noise. Considering α beyond 1 results in distorted value and does not follow the trajectory. This work can be further extended using a complex-valued real-time recurrent learning (CRTRL) algorithm for pitch control system. It can also be realized as fully connected recurrent neural networks [16] and can be compared with theT factor based RBFNC. The current problem discussed here can also be analyzed using quaternion domain [17].Table 1
Comparison of time domain specifications of pitch control system for different values ofα.
Sl. number
T factor (α)
Settling time (second)
Rise time(second)
Overshoot(percentage)
1
0
15.84
6.556
0
2
0.2
13.02
6.191
0
3
0.6
10.46
5.401
0
4
0.9
9.065
4.875
0



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*Source: 897691-2014-04-24.xml*