Intelligent Reference Learning Techniques For Pitch Control Of An Aircraft

Transcription

1 Intelligent Reference Learning Techniques For Pitch Control Of An Aircraft P S Khuntia 1 and Debjani Mitra 2 1 Department of Electronics and Communication Engineering Durgapur Institute of Advanced Technology and Management Rajbandh-12, Durgapur, West Bengal, INDIA parthsarathi_k@yahoocom 2 Department of Electronics and Instrumentation Indian School of Mines, Dhanbad, Jharkhand, INDIA debjani7@yahoocom Abstract: This paper presents a comparative analysis of two intelligent reference learning techniques to achieve better performance of pitch control of an aircraft Fuzzy Model Reference Learning Controller (FMRLC) and Radial Basic Function Neural Controller (RBFNC) are designed for pitch control of a FOXTROT fighter aircraft These controllers utilize a learning mechanism, which observes the plant output and adjusts the configuration in the direct controller, so that the overall system behaves like a "reference model" which characterizes the desired behavior The performance of the pitch control system is demonstrated by simulation for various conditions with change in the aircraft dynamics caused due to change in speed of the aircraft and sensor noise The simulation results establish the superiority of RBFNC over FMRLC with respect to rise time, settling time and overshoot Keywords: pitch control, aircraft, intelligent reference learning I INTRODUCTION The performance of a closed loop system can be improved over time by incorporating a progressively learning mechanism into it as the system is able to use the generated command inputs to the plant and the feedback information from the plant [1] D S Reay and M W Dunnigan investigated and proposed that the extent and importance of learning can be enhanced updating the mechanism of fuzzy logic controller and established similarities with a form of linear high gain robust control [2] Peter J Thomas, Russel J Stonier developed an evolutionary learning algorithm to learn a fuzzy knowledge base for the control of a soccer playing microrobot to hit the ball along the ball to goal line of sight [3]The reference model based learning system is examined to be a popularly used technique in which learning algorithm is designed to make the system response converge to that specified by the model There are several approaches to this type of learning prevalent in the literature The performance of a missile auto pilot was demonstrated with help of FMRLC based learning mechanism [4]A nonlinear fighter aircraft pitch rate tracking control scheme based on a multiple fuzzy reference model adaptive controller was proposed by Kamalasadan and Ghandakly [5] Reinforcement learning is used in this paper to design an RBF Neural Network model based reference controller for pitch control system of an aircraft This learning method unlike supervised learning of neural network adopts a trial and error method mechanism existing in human and animal learning This method emphasizes that an agent obtains a goal by interaction with the environment At first the reinforcement learning agent exploits the environment actively and then exploits the environment results, based on which the controller is modified It can realize unsupervised online learning without a model [6-8]Because of the quickened learning rate and avoidance of local minima RBFNC is immensely fast growing topic of research interest and used in the fields of aircraft, robotics and other fields of control [9-10] II MATHEMATICAL FORMULATION The SH-60F Ocean Hawk Foxtrotis the carrier-based version of the Bravo, as the carrier battle group's primary antisubmarine warfare (ASW) and search and rescue (SAR) aircraft The Foxtrot carries the Mk-46 torpedo and a choice of cabin-mounted machine guns, including the M60D, M240 and GAU-16The pitch control system of the FOXTROT fighter aircraft is considered here as plant The input to the plant is the elevator deflection ( E ) and the output is the pitch angle ( ) The longitudinal dynamics [11] of an aircraft can be represented with following set of equations u Xuu Xww gcos 0 w Z uz wu q gsin 0 Z E u w o E (1) 39

2 q MuuMwwMw w MqqM E E q Substituting the values of the stability derivatives ( Z w, Mw, M q, M w, U o, Z E, M ) of aircraft [11] for E theflight condition-3 and 4 in (1) the following transfer functions are obtained as follows Flight Condition-3 ( s) ( s) (2) ( s) [1+( i)s] [1+( i)s] E Flight Condition-4 ( s) ( s) (3) ( s) [1+( )s][1+( )s] E The negative sign is not considered as the downward deflection of elevator is considered [12] III FUZZY MODEL REFERENCE LEARNING CONTROLLER (FMRLC) The functional block diagram for the FMRLC is shown in Fig 1 ref is the desired pitch angle and ( kt ) is the actual pitch angle obtained FMRLC has four main parts ie the plant, the fuzzy logic controller to be tuned by changing its rule base, the reference model and the learning mechanism (an adaptation mechanism) The FMRLC uses the learning mechanism to observe ref and ( kt ) from a pitch control system Using these numerical from pitch control system data, it characterizes the fuzzy control system s current performance and automatically synthesizes or adjusts the fuzzy controller so that some given performance objectives are met These performance objectives (closed-loop specifications) are characterized via the reference model shown in Figure 1 The learning mechanism seeks to adjust the fuzzy controller so that the closed-loop system (the map from ref to ( kt )) acts like the given reference model (the map from ref to m ( kt )) The fuzzy control system loop which is the lower part of Fig 1 operates to make ( kt ) to track ref by manipulating ( kt )The upper-level adaptation control loop which is the upper part of Fig 1 seeks to make the output of the plant ( kt ) to track the output of the reference model ref by manipulating the fuzzy controller parameters A The Fuzzy Controller The inputs to the PD (Proportional-Derivative) Fuzzy Controller are generally generated from the plant output and reference input Here inputs to the fuzzy PD controller are the pitch angle error, ekt ( ) and change in pitch angle error, e ( kt ) expressed as c ekt ( ) ref, ec ref ( kt ) ( kt ) The controller output is the elevator angle of the aircraft E ( kt )The fuzzy controller used here has 11 uniformly spaced triangular membership functions (MFs) for each of the controller input ekt ( ) and ec ( kt ) The centers of the input MFs for the ekt ( ) and ec are [-1,-8,-6,-4,-2, 0, 2,4, 6, 8, 1] The centers of the output membership functions for the E ( kt ) are[-1,-8,-6,-3,-1, 0,1, 3,6,8,1] The scaling gains for the error ( g e ), change in error ( g c ) and the controller output ( g u ) are chosen via the design procedure The error e(kt) is restricted not exceed more than 90 degree and the rate of pitch angle is not allowed to exceeds more than 001 rad/sec The elevator also will not have a deflection more than ±20 degree to control the pitch angle These nominal scaling values are considered here For any abrupt change in pitch angle may cause the aircraft to disaster So these designing constraints decides the following values ge (2 ), gc 100, and 2( ) /18 gu T (4) B Fuzzy Inverse Model The inputs to the fuzzy inverse model are the error e ( kt ) and change in error c ( kt ) shown in Fig 1 e( kt ) m( kt ) - ( kt ) c( kt ) ( e( kt )- e ( kt - T ))/ T (5) (6) 40

3 Figure 1 FMRLC for aircraft pitch control For e ( kt ) and c ( kt ) eleven symmetric and triangular-shaped MFs of width 04 are taken as in the case of fuzzy controller Eleven symmetric and triangularshaped MFs of width 04 are considered for output p( kt ) The centers of the input membership functions for the e ( kt ) and c ( kt ) are[-1,-8,-6, -4,-2,0,2,4,6,8,1]The centers of the output membership functions for the p(kt) are [-1,-8,-6,-4,-2,0,2,4,6,8,1] The values of the normalizing gains taken are g e 2/, g c =500 and g p =04 C Reference Model A first order system shown below in Eq (7) is considered here as reference model so that the output ( kt ) tracks a stable and smooth first order response m ( kt )The performance of the overall system is computed with respect to the reference model using learning mechanism The desired performance is met if the learning mechanism forces e ( kt ) to remain very small for all the time no matter what parameters variation occurs or change in the reference input D Knowledge Base Modifier Let b m denotes the center of the MF associated with m th MF Knowledge base modification is performed by shifting the centers of MFs that are associated with the fuzzy controller rules The shifting of centers is contributed by the previous control action E ( kt T ) Let b m (kt) denotes the center of the m th output MF at time instant kt For all rules of Fuzzy Controller(see the Appendix-1 & Appendix-2) bm( kt ) bm( kt T ) p( kt ) (8) IV RADIAL BASIS FUNCTION NEURAL NETWORK (RBFNN) The proposed RBFNN model with single neuron output y is presented in Fig 2 consists of three-layers Each input values are assigned to a node and passed directly to the hidden layer without weights The hidden layer nodes are called radial basic functions (RBF) units are here Gaussian density functions, which are used as activation functions for the hidden neurons The RBFNN is shown in Fig 2 m () s Kr (7) ref () s s ar 41

4 G 1 b 1 x 2 G 2 b 2 y x n b n G n n R Receptive field units Figure 2 Radial basis function network model V DESIGN OF RBF NEURAL CONTROLLER where x [ x1, x2, x3, x ] T n is the input and Ri ( x) is the output of the i th receptive field with strength denoted by bi Assuming n R receptive fields present in the RBFNN, the output y or Frbf (, x ) can be written as n R y Frbf ( x, ) = br i i( x) (9) i1 Here, holds the parameters of the receptive field units which consist of the parameters b i and possibly the parameters of the Ri ( x ) The Gaussian-shaped functions are preferred here for analytical convenience ie i 2 i 2 Ri ( x) exp x c ( ) (10) i i i T where ci [ c1, c2, cn ] parameterize the locations and decides the spreading of the receptive fields in the input space The weighted average output of the RBFNNcan be written as n n R R y Frbf ( x, ) bi Ri ( x) Ri ( x) i1 i1 (11) The RBFNC for aircraft pitch control system is shown in the Fig 3 tracks the desired pitch angle ref The closed loop system has a reference model with input ref and output m ( kt ) As shown in Fig 3 the RBFNC attempts ( kt ) to match the reference model output asymptotically In this reinforcement learning controlthe error between the plant and the reference model outputs is used to adjust the weights of the neural controller ie b i The error ekt ( ) and change of error ec ( kt ) are the inputs to the RBF neural network where ekt ( ) ref (kt) - (kt) ec ( kt ) ekt == ( ) ekt ( T ) T (12) T is the sampling time The output of the RBFNC ( k) is computed by taking ekt ( ) and ec ( kt ) as the argument to the radial basic function, E( kt ) Frbf ( e, ec) (13) 42

5 Figure3 RBFNC for aircraft pitch control It is decided in the designing of pitch controller that elevator should not to exceed more than 2 radian in both upward or downward direction and the change of error should not be more than 001 radian/sec It concludes the range of ekt ( ) and ec are ekt ( ) [ 2, 2] and ec [ 001,001] The purpose of reinforcement function is to modify the field strength (b i ) as shown in RBFNN The reinforcement function [13] is defined below as J R ( e( kt ), c( kt )) ( ee( kt ) cc( kt )) where Adoption gain e Adoption gain ek ( ) c Adoption gainfor ec ( k ) (14) e and c are adjusted to indicate the performance of tracking and change in trackingwhose values are 065 and 110 respectively The smaller value of indicates slower and larger value of indicates faster rate of adoption of RBFNC The value of the considered here is equals to 1The threshold value beyond which the adoption takes place is chosen here 0001First b i is initialized to be zero for all values ofi to indicate the neural network knows little how to control the pitch of the aircraft and later it is modified with following equation bi( kt ) bi( kt T ) J R( kt ) Ri( kt T ) (15) where Ri is the output of the i th receptive field unit VI SIMULATION OF PITCH CONTROL SYSTEM The general form of pitch control system of the aircraft is given below () s K(1 3s) E () s (1 1s)(1 2s) (16) k 3 k Let a ( ), b, c, d Then Eq (16) can be represented as () s ( cs d) () 3 2 E s ( s as bs) Inverse Laplace Transform of Eq (17) results (17) () t a () t b () t c () t de () t (18) x i is so chosen that fi depends only on x i and for i 1, 2, 3 so that x 3 () t () t c E () t, 2 () () (19) x 1() t x2() t (19) x 1 () t x 2 () t f 1 ( x(), t E ()) t (20) x 2() t x3() t c E () t (21) 43

6 x 3 () t a () t b () t de ()) t (22) Now substituting the value of () t and () t x 3 () t ax () t bx () t ( acd) () t (23) 3 2 E These equations with initial conditions (0) (0) (0) 0 are ready for simulation using Runge-Kutta method VII SIMULATION RESULTS The aircraft is simulated for the following conditions 1 From t=0 to 15 seconds the flight travels with flight condition 3 (350 m/sec) and after 15 seconds the flight travels with flight condition 4 (650 m/sec) 2 The reference signal is a pulse of duration 40 seconds of amplitude 5 degree 3 The sensor measuring the pitch angle is added with a random noise ie 001 (2 rand -1) 180 A FMRLC The simulation results for pitch control system employing FMRLC are shown in Fig 4 The Fig 4(a) shows the trajectory of the actual pitch angle ( kt ) and m ( kt ) Fig (b) shows the fuzzy controller output E ( kt ) (input to the plant) is continuous because the adoption takes place continuously due to continuous presence of random noise in sensor The continuous output E ( kt ) tries ( kt ) to follow the reference pitch angle ref It is observed that E ( kt ) is significant where there is a transition of ref and change in speed of the aircraft from one flight condition to another The fuzzy inverse output, p(kt) is shown in Fig 4 (c) tries to modify the rule base of fuzzy controller B RBFNC The ( kt ) and ref is plotted in Fig 5(a) In Fig 5(b) neural controller output E ( kt ) is found to be continuous because the adoption takes place continuously due to continuous random noise present in the sensor The continuous output present here tries ( kt ) to follow the reference pitch angle ref reducing error to zero The reinforcement signal J R ( kt ) is shown in Fig 5(c) establishes the adoption takes place where it is non zero VII COMPARATIVE ANALYSIS OF FMRLC AND RBFNC The result of FMRLC and RBFNC are listed in Table 1 for comparatively analysis, which establishes the superiority of RBFNC FMRLC and RBFNC are used for aircraft pitch control and the simulation study shows the adaptive nature of the controllers in the way in which it can tune the controller The simulation results show to what extent the controller is capable of adaptation in the face of change in aircraft dynamics due to change in reference signal, change in aircraft speed and sensor noise The non-zero value of the output of FMRLC and RBFNC exhibits its adaptive nature when the error in the desired pitch angle occurs due to change in reference signal, change in speed of the aircraft and presence of sensor noise The learning strategies are strong enough to make the output of FMRLC and RBFNC least sensitive to the effects of sensor noise The controller output continuously changes to nullify the effect of this noise The settling time and overshoot of pitch control system with controllers are calculated and shown in the Table 1 The results establish the superiority of RBFNC over FMRLC and PID controller with respect to rise time, settling time and overshoot The settling time for RBFNC is moderately reduced to 0654 second where as in the case of FMRLC it is 1012 second The overshoot in the case of RBFNC is slightly higher by an amount of 25% which is a minor trade off As the aircraft is a fast dynamic system the less settling time utmost important for the pitch angle to settle In the case of RBFNC the overshoot is higher by an amount of 25% than FMRLC, which is tolerable 44

7 Figure 4Result of FMRLC Figure 5Results of RBFNC 45

8 Controllers Rise Time (Sec) Table 1 Settling Time (sec) REFERENCES Overshoot (%) RBFNC FMRLC [1] J R Layne and K M Passino, Fuzzy Model Reference Learning Control, Journal of Intelligent and Fuzzy Systems, vol 4, no 1, 1996, pp [2] D S Reay and M W Dunnigan, Learning Issues in Model Reference Based Fuzzy Control, IEE Proceedings on Control Theory Applications, vol 144, no 6, 1997, pp [3] P J Thomas and R J Stonier, Fuzzy Control in Robot-Soccer, Evolutionary Learning in the First Layer of Control, Journal of Systemics, Cybernetics and Informatics, vol 1, no 4, 2003, pp75-80 [4] Y Yuan, Y Feng, W Gu, Fuzzy Model Reference Learning Control for Aircraft Pitch Autopilot Design, 8th International Conference on Control, Automation, Robotics and Vision, 2004, pp [5] S Kamalasadan and A AGhandakly, Nonlinear Fighter Aircraft Pitch-Rate Tracking Using a Multiple Fuzzy Reference Model Adaptive Controller, IEEE International Conference on Computational Intelligence for Measurement Systems and Applications, 2005, pp [6] S Kamalasadan, A A Ghandakly, Multiple Fuzzy Reference Model Adaptive Controller Design for Pitch- Rate Tracking, IEEE Transactions on Instrumentation and Measurement, vol 56, no 5, 2007, pp [6] M Sedighizadeh and A Rezazadeh, Adaptive PID Controller based on Reinforcement Learning for Wind Turbine Control, Proceedings of World Academy of Science, Engineering and Technology, vol 27, 2008, pp [7] X Wang, Y Cheng, and W Sun, Q Learning Based on Self-Organizing Fuzzy Radial Basis Function Network, Springer Berlin/Heidelberg, vol 3971, 2006, pp [8] W Xue-song, C Yu-Hu, and Sun Wei, A Proposal for Adaptive PID Controller Based on Reinforcement Learning, China University of Mining and Technology, vol 17, 2007, pp [9] Z Ming-Guang, W Xing-Gui, and L Man-Qiang, Adaptive PID Control Based on RBF Neural Network Identification, Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence, 2005, pp [10] Y Li, N Sundararajan, and P Saratchandran, Neuro-Controller Design for Nonlinear Fighter Aircraft Maneuver using Fully Tuned RBF Networks, Automatica, vol 37, no 8, 2001, pp [11] M Donald, Automatic Flight Control Systems, Prentice Hall International, 1990 [12] O Momcilovic, Discrete Time Variable Structure Controller for Aircraft Elevator Control, Journal of Electrical Engineering, vol 59, no 2, 2008, pp [13] K M Passino, Biomimicry for Optimization, Control and Optimization, Springer Verlag, London,

9 Appendix-1 Linguistic Variables and their Mathematical Representations Linguistic variables Linguistic representation Numerical representation Center of The triangular membership function NVL Negative Very Large -5-1 NL Negative Large NM Negative Medium NS Negative Small Z Negative Zero Z Zero 0 0 +Z Positive Zero 1 01 PS Positive Small 2 03 PM Positive Medium 3 06 PL Positive Large 4 08 PVL Positive Very Large 5 1 The numerical values (111 numbers) within the rule base matrix for fuzzy controller which shown in Appenidx-1above denote the centers of the triangular membership functions For an example if ekt ( ) is positive large (4) and ec ( kt ) is negative small (-2) then E ( kt ) is negative small (-03) Appendix-2 Rule Base Matrix for Fuzzy PD Controller NVL NL NM NS -Z Z Z+ PS PM PL PVL e c e