Missile Autopilot Design using Artificial Neural Networks

Transcription

1 International Journal o Engineering Trends and Technology (IJETT) Volume 29 umber 6 - ovember 2015 Missile Autopilot Design using Artiicial eural etorks 1 Adel Alsara, 2 Gene Stule 1, 2, Idaho State University Abstract The poer and speed o modern digital computers is truly astounding so that it enables carrying on complex tasks such as aerospace simulation, design and analysis, precisely. In addition to the nature o the guidance problem, the design technique, neural netorks, necessitates cumbersome computations to yield precise and accurate perormance. eural netorks approach the solution o this problem by trying to mimic the structure and unction o the human nervous system. Thereore, this paper is devoted a ne approach using the poer o both computation acilities and neural netorks in the design and analysis o an autopilot or the guidance system. Then, its perormance is ustiied against the classical design approach through the Six degrees o reedom (6DoF) light simulation. I. Introduction The nervous system consists o neurons, hich are connected to each other in a rather complex ay. Each neuron can be thought o as a node and the interconnections beteen them are edges [1]-[4]. Such a structure is called as a directed graph. Further, each edge has a eight associated ith it, hich represents ho much the to interconnected neurons can interact. I the eight is more, then the to neurons can interact much more; and consequently a stronger signal can pass through the edge [5], [6]. Avery simple model and consists o a single trainable neuron. Trainable means that its threshold and input eights are modiiable. Inputs are presented to the neuron and each input has a desired output determined by the user or designer [7]. The threshold and/or input eights can be changed to modiy the output according to the learning algorithm [8]. The output o the perceptron is constrained to Boolean values :( true, alse), (1,0), (1,-1) or hatever [9], [10]. This is not a limitation because i the output o the perceptron ere to be the input or something else, then the output edge could be made to have a eight and consequently the output ould be dependent on this eight [11]. This paper is devoted to the autopilot design or a missile system using the artiicial neural netorks approach. The paper starts ith introduction to the neural netorks, olloed by the eural et-based Guidance and autopilot Design using model reerence neural netork. Then, the designed controller is used ith the system and the simulation results ere analysed. Finally, the conclusions o the paper are discussed. II. Artiicial neural netorks Artiicial eural netorks are composed o simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the netork unction is determined largely by the connections beteen elements. A neural netork can be trained to perorm a particular unction by adusting the values o the connections (eights) beteen elements. Commonly neural netorks are adusted or trained, so that a particular input leads to a speciic desired output, ig. (1).The netork is adusted, based on a comparison o the output and the target, until the netork output matches the target. Typically, many such input/target pairs are used, in this supervised learning, to train a netork. The supervised training methods are commonly used, but other netorks can be obtained rom unsupervised training techniques or rom direct design methods. Unsupervised netorks can be used, or instance, to identiy groups o data. There are a variety o kinds o design and learning techniques that enrich the choices that a user can make. Fig. (1) Idea o the Artiicial eural etork(a) connection eural netorks have been trained to perorm complex unctions in various ields o applications including pattern recognition, identiication, classiication, speech, vision, and control systems [12]. Today, neural netorks can be trained to solve problems that are diicult or conventional computers or human beings. ISS: Page 284

2 International Journal o Engineering Trends and Technology (IJETT) Volume 29 umber 6 - ovember 2015 A. euron model A neuron ith a single scalar input (Simple euron) and no bias is shon in ig. (2-a), here the scalar input p is transmitted through a connection that multiplies its strength by the scalar eight, to orm the product p. The eighted input p is the only argument o the transer unction, hich produces the scalar output a. Hoever, to approach reality, the eighted input (p) isusually corrupted by a bias (b), ig. (2-b). That is, the bias can be vieed as being added to the product p as shon by the summing unction or as shiting the unction to the let by an amount b [13]. The bias is much like a eight, except that it has a constant value. The net input n, again a scalar, is the sum o the eighted input p and the bias b. This sum is the argument o the transer unction. A transer unction is typically a step unction or a sigmoid unction that takes the argument n and produces the output a. ote that and b are both adustable scalar parameters o the neuron [14], [15]. C. euron ith vector input A neuron ith a single R-element input vector is shon in ig. (3). Fig. (3) euron ith vector input In this structure, the individual element inputs p1, p2,,pr are multiplied by eights 1,1, 1,2,...,1,R and the eighted values are ed to the summing unction. Their sum is simply Wp, and it is obtained by the dot product o the matrix W and the vector p. The neuron has a bias b, hich is summed ith the eighted inputs to orm the net input n. This sum, n, is the argument o the transer unction, and it is given by: n 1,1 p p R p R b (1 ) (a) Without Fig. (2) Simple neuron coniguration (b) With bias The central idea o neural netorks is that such parameters can be adusted so that the netork exhibits some desired behavior. Thus, the netork can be trained to carry on a particular ob by adusting the eight or bias parameters, or perhaps the netork itsel can adust these parameters to achieve some desired output. B. Transer unctions The transer unction can be ound in many dierent orms; among them are the hard limit, the linear, and the sigmoid types. The hard limit transer unction is used to limit the output o the neuron to either 0, i the net input argument n is less than 0, or 1, i n is greater than or equal to 0. The linear transer unctionis used to transer the input ith a certain scaling actor. While, the sigmoid transer unctionaccepts the input, hich may have any value beteen plus and minus ininity, and squashes the output into the range rom 0 to 1. D. etork architectures To or more o the neurons shon above may be combined in a layer, and a particular netork might contain one or more o such layers. Single Layer o eurons A one-layer netork ith R input elements and S neurons is shon in ig. (4).In this netork, each element o the input vector p is connected to each neuron input through the eight matrix W. The ith neuron has a summer that gathers its eighted inputs and the bias to orm its on scalar output ni. The various ni taken together orm an S- element net input vector n. Finally, the neuron layer outputs orm a column vector a. ote that it is common or the number o inputs to a layer to be dierent rom the number o neurons. In addition, a layer is not constrained to have the number o its inputs equal to the number o its neurons. A single composite layer o neurons having dierent transer unctions can be created simply by putting to o the netorks shon above in parallel. Both netorks ould have the same inputs, and each netork ould create some o the output elements. The input vector elements are applied to the netork through the eight matrix W, hich has the orm: ISS: Page 285

3 International Journal o Engineering Trends and Technology (IJETT) Volume 29 umber 6 - ovember 2015 W 1,1 2,1 1,2 2, , R 2, R S,1 S,2... S, R ote that the ro indices on the elements o matrix W indicate the destination neuron o the eight and the column indices indicate hich source is the input or that eight. For example, the indices in 1,2 say that the strength o the signal rom the Fig. second (5) A input multi-layer element netork to the irst neuron is 1,2. Fig. (4) A one-layer netork Multiple Layers o eurons A netork can have several layers; each layer has a eight matrix W, a bias vector b, and an output vector a. A three-layer netork is shon in ig. (5) ith the equations ritten belo the igure. This netork has R1 inputs, S1 neurons in the irst layer, S2 neurons in the second layer, etc. It is common or dierent layers to have dierent numbers o neurons and a constant input 1 is ed to the biases or each neuron. ote that the outputs o each intermediate layer are the inputs to the olloing one. Thus, layer 2 can be analysed as a one-layer netork ith S1 inputs, S2 neurons, and an S1S2 eight matrix W2. The input to layer 2 is a1, and the output is a2. The layers o a multilayer netork play dierent roles. In other ords, a layer that produces the netork output is called an output layer, hile all other layers are called hidden layers. That is, the three-layer netork shon in ig. (5) has one output layer (layer 3) and to hidden layers (layer 1 and layer 2). Multiple layer netorks are quite poerul in evaluating complex processes. For instance, a netork o to layers, here the irst layer is sigmoid and the second layer is linear, can be trained to approximate any unction (ith a inite number o discontinuities) arbitrarily ell. E. Learning approaches There are dierent learning approaches and consequently dierent types o Artiicial eural etorks (A) that enable its utiliation ith dierent applications. Among these approaches are [16]: Back-propagation multilayer A,Recurrent type A,Associative type,probabilistic, andadaptive resonance. The Back-propagation is utilied in real time learning controller unction, and consequently it is considered ith autopilot design or the guidance system. Back-propagation as created by generaliing the Widro-Ho learning rule to multiple-layer netorks and nonlinear dierentiable transer unctions. Input vectors and the corresponding output vectors are used to train a netork until it can approximate a unction, associate input vectors ith speciic output vectors, or classiy input vectors in an appropriate ay as deined by the designer. etorks ith biases, a sigmoid layer, and a linear output layer are capable o approximating any unction ith a inite number o discontinuities. Standard back-propagation is a gradient descent algorithm, as is the Widro-Ho learning rule. The term backpropagation reers to the manner in hich the gradient is computed or nonlinear multilayer netorks. There are a number o variations on the basic algorithm, hich are based on other standard optimiation techniques, such as conugate gradient and eton methods. Typically, a ne input ill lead to an output similar to the correct output or input vectors used in training that are similar to the ne input being presented. This generaliation property makes it possible to train a netork on a representative set o input/target pairs and get good results ithout training the netork on all possible input/output pairs [17]. III. eural net-based guidance and control design The application o neural netorks has attracted signiicant attention in several disciplines, ISS: Page 286

4 International Journal o Engineering Trends and Technology (IJETT) Volume 29 umber 6 - ovember 2015 such as signal processing, identiication and control. The success o neural netorks is mainly attributed to their unique eatures such as: Parallel structures ith distributed storage and processing o massive amounts o inormation, and Learning ability made possible by adusting the netork interconnection eights and biases based on certain learning algorithms. The irst eature enables neural netorks to process large amounts o dimensional inormation in real-time. The implication o the second eature is that the non-linear dynamics o a system can be learned and identiied directly by an artiicial neural netork. In addition, the netork can adapt to changes in the environment and make decisions despite uncertainty in operating conditions. Thereore, neural netorks are implemented in aerospace applications and consequently the guidance system or enhancing its perormance. Most neural netorks can be represented by a standard (+1) layer eed orard netork. In this netork, the input is 0 y hile the output is n. The input and output are related by the olloing recursive relationship: net and net i W (net net W 1 ) 1 V V, 1,2,... here the eights W and V are o the appropriate dimensions. V is the connection o the eight vector to the bias node. The activation unction vectors (.), = 1, 2,..., 1 are usually chosen as some kind o sigmoid, but they may be simple identity gains. The activation unction o the output layer nodes is generally an identity unction. The neural netork can, thus, be succinctly expressed as (y;w,v) (W 1 (W 3) 1 2 W 1 2) ( 3 ) here i i 2 (net (k)) i net (k) 1 e 1, 4) herei denotes the ith element o and λ is the learning constant. For netork training, error back propagation is one o the standard methods used to adust the eights o neural netorks [18]. A. eural netork ith model reerence control In this control structure, the desired perormance o the closed-loop system is speciied through a stable reerence model, hich is deined by its input-output pair {r(t), yre(t)}, ig. (6) [19]. This igure (shos that the control system attempts to make the plant output y(t) match the reerence model output yre(t), asymptotically. Thus, the error beteen the plant and the reerence model outputs is used to adust the eights o the neural netork controller [20]. Fig. (6) Model reerence control B. Autopilot design using model reerence A hybrid model reerence adaptive control scheme is implemented ith the guidance system. In this system, a neural netork is placed in parallel ith a linear ixed-gain independently regulated autopilot as shon in ig. (7). The linear autopilot is chosen so as to stabilie the plant over the operating range and provide approximate control, hile the neural controller is used to enhance the perormance o the linear autopilot hen perormance becomes poor by adusting its eights. A suitable reerence model is chosen to deine the desired closed-loop autopilot pre yre responses and across the light envelope. These outputs are then compared ith the actual outputs o the lateral autopilot Fig. (7) Block Diagram o acceleration control system using model reerence controller yielding an error measurement vector [ p and y p rerror ISS: Page 287

5 International Journal o Engineering Trends and Technology (IJETT) Volume 29 umber 6 - ovember 2015 y rerror ]T. This error is used in conunction ith an adaptive rule to adust the eights o the neural netork so that the tracking error is minimied. A direct eect o this approach is to suppress the inluence resulting rom roll rate coupling. The neural netork model and controller are designed using the Matlab neural netork toolbox. A to layer netork is designed ith sigmoid transer unction olloed by a linear one or both the plant and the controller. This structure is shon in ig. (8), hich shos the connection The netork is trained oline ith a step reerence signal yielding the system response shon in ig. (9). This igure shos a stable system, but ith distorted transients. This neural The netork is trained oline ith a step reerence signal yielding the system response shon in ig. (9). This igure shos a stable system, but ith distorted transients. This neural netork autopilot is implemented ith the Six degrees o reedom (6DoF) simulation and the same engagement scenario o [21]. The obtained miss distance is reduced to only 5%. beteen the to netorks in Simulink point o vie. Fig. (8) The connection beteen the to netorks in Simulink point o vie. [m] and the time o light is 8.27 seconds. Using the modiied neural netork controller yields the engagement scenario shon in ig. (11-b), here the miss distance is about 3[m], and the light time is 8.15 seconds. That is, it yields to save 2% in the light time and to reduce 94% in the miss distance, compared to the previous design. It is clear that, the ne system is much aster than the original one, and ith less miss distance o about 93% o the lead netork and 85% Fig. (9) Acceleration step response ith neural controller at 6 sec For more enhancements in the system perormance, the netork is retrained but ith reerence signal adusted to cope ith the values obtained rom the previous 6DOF simulations. Then, the ne autopilot is implemented yielding aster response, ig. (10), and higher relative stability compared ith the previous one and also that obtained ith classical control in [21]. For more ustiication o this ne autopilot, it is implemented ithin the 6DOF simulation, hich is conducted ith target initial position o [6 1 2] Km, initial velocity o [ ]. This target experienced a manoeuvre o [ ] [m/sec2], i.e g ater 5 seconds rom the instance o missile launch, and lasted or 2 seconds. The missile-target light path ith a lead netork is shon in ig. (11-a) here the miss distance is 47.8 Fig. (10) Acceleration step response ith modiied neural controller at 6 sec Fig. (11) Missile and Target traectory (a) ith original autopilot (b) modiied controller It is clear that, the ne system is much aster than the original one, and ith less miss distance o about 93% o the lead netork and 85% less than the classical PID controller. The three ISS: Page 288

6 International Journal o Engineering Trends and Technology (IJETT) Volume 29 umber 6 - ovember 2015 engagement scenarios are plotted together ith ooming to clariy the dierence beteen them as shon in ig. (12). this igure clariies ho the neural netork achieved a smooth and ast approach to the interception ith minimum miss distance. IV. Conclusions A neural netork based adaptive inverting autopilot design is developed and implemented or a guided missile system. This design approach as superior to the original and designed classical approaches rom the point o vie o miss distance and demanded acceleration. That is, the neural netork proved its robustness ith such a stochastic non-linear system provided it is careully trained. Fig. (12) Missile-target engagement scenarios ith lead, PID and neural netorks Reerences [1] Calise A., and R. Rysdyk; onlinear Adaptive Flight Control Using eural etorks, Control Systems Magaine, December [2] McFarland M., A.J. Calise; eural-adaptive onlinear Autopilot Design or an Agile Anti-Air Missile, AIAA Guidance, avigation and Control Conerence, San Diego, CA, AIAA , July 29-31, [3] McFarland, M.; Adaptive onlinear Control o Missiles Using eural etorks, Ph.D. Thesis, Georgia Institute o Technology, [4] Michael B. McFarland and Shaheen M. Hoque; Robu onlinear Missile Autopilot Designed Using Dynamic AIAA [5] Chao A., M. Athans, and G. Stein; Stability Robustness to Un-structured Uncertainty or onlinear Systems Under Feedback Lineariation, 53rd IEEE Conerence on Decision and Control, IEEE Publications, Piscataay, J, pp , [6] Hornik K., M. Stinchombe, and H. White; Multilayer Feedorard etorks are Universal Approximators, eural etorks, Vol. 52, [7] Chun-Liang Lin and Huai-Wen Su; Intelligent Control Theory in Guidance and Control System Design, an Overvie, (Invited Revie Paper), Proc. atl. Sci, Counc. ROC (A), Vol. 24, o. 1, pp , [8] Cronvich L.L.; Aerodynamic Considerations or Autopilot Design, AIAA, pp3-42, [9] Fraoli E., M.A. Dahleh, E. Feron; Robust Hybrid Control or Autonomous Systems Motion Planning, Technical report LIDS-P-2468, Laboratory or Inormation and Decision Systems, Massachusetts Institute o Technology, Cambridge, MA, [10] Garnell P., and East D. J.; Guided Weapon Control Systems, Pergamon Press, Oxord, England, [11] Monaco J., D. Ward, A. Barto; Automatic Aircrat Recovery via Reinorcement Learning, Initial Experiments, eural Inormation Processing Systems Conerence, Denver, CO, [12] Jiang T.; Combined Model and Rule-based Controller Synthesis With Application to Helicopter, Flight Control. Ph.D. Thesis, Georgia Institute o Technology, [13] Kim B. S., and A.J. Calise; onlinear Flight Control Using eural etorks, AIAA Journal o Guidance, Control, and Dynamics, Vol. 80, o. 1, [14] Kim B., and A. Calise; onlinear Flight Control Using eural etorks, Journal o Guidance, Control, and Dynamics, Vol. 80, o. 1, [15] Leis F., S. Jagannathan, and A. Yesildirek; eural etork Control o Robot Manipulators and onlinear Systems, Taylor and Fancis, London, [16] McFarland M., A.J. Calise; Multilayer eural etorks and Adaptive onlinear Control o Agile Anti-Air Missiles, AIAA Guidance, avigation and Control Conerence, AIAA , e Orleans, L.A., August [17] esline F. W., B. H. Wells, and P. Zarchan; A Combined Optimal/Classical Approach to Robust Missile Autopilot Design, AIAA Guidance and Control Conerence, AIAA, e York, pp , [18] Rovithakis G.; onlinear Adaptive Control in the Presence o Unmodelled Dynamics using eural etorks. Proceedings o the Conerence on Decision and Control, [19] Steinberg M.; A Comparison o Intelligent, Adaptive, and onlinear Control Las, Proceedings o the AIAA Guidance, avigation, and Control Conerence, [20] Chun-Liang Lin and Huai-Wen Su; Intelligent Control Theory in Guidance and Control System Design, Proc. ational Science Council ROC(A) Vol. 24, o. 1, pp , [21] Chun-Liang Lin and Huai-Wen Su; Intelligent Control Theory in Guidance and Control System Design, Proc. ational Science Council ROC(A) Vol. 24, o. 1, pp , ISS: Page 289