Closed Loop Near Time Optimal Magnetic Attitude Control Using Dynamic Weighted Neural Network

Transcription

1 Closed Loop Near ime Optimal Magnetic Attitude Control Using Dynamic Weigted Neural Network Ali Heydari, Seid H. Pourtakdoust Abstract e problem of time optimal magnetic attitude control is treated and an open loop solution is first obtained using a variational approac. In order to close te control loop, a neural network wit time varying weigts is proposed as a feedback optimal controller applicable to te time varying nonlinear system. e good robustness and low real-time computational burden of te proposed neuro-controller makes te controller more useful compared to te oter control metods. U I. INRODUCION ILIZAION of magnetic attitude control as te sole control mecanism because of its ligt weigt, low cost and ig reliability as been focused by researcers and engineers over te last two deca especially for small satellites wit low pointing accuracy. A related problem, namely te inability of producing mecanical torque about te local magnetic field vector of te Eart as been considered by several researcers and numerous control metods ave been suggested to its remedy. However, most of tese suggested control metods suc as [1] [4] require a time more tan one or even two orbital periods in order to accomplis an attitude maneuver and terefore, using tem for te satellites wit time critical missions is not very attractive. In [5] te magnetic attitude control as been treated as a time optimal problem and solved troug a software package, RIOS. e resulted open loop optimal control as been sown to accomplis a slew maneuver in a small fraction of an orbital period; terefore, te previous notion of time consuming nature of magnetic attitude control metods as been alleviated. e same autors [5] ave also utilized a model predictive controller in order to close te control loop and te result is a feedback sub-optimal control wic is moderately able to overcome some disturbances and uncertainties. eir metod is based on solving a finite orizon open loop optimal control in eac time step during an online operation using te current states as initial states of te problem and utilizing te first control of te obtained control istory as te applicable control. us, te real time computational burden is relatively ig for most small satellites. Manuscript received February 4, A. Heydari is M.Sc. Student of Aerospace Engineering Dept., Sarif University of ecnology, eran, Iran, ( eydari@ae.sarif.edu). S. H. Pourtakdoust is Professor of Aerospace Engineering Dept., Sarif University of ecnology, eran, Iran, ( pourtak@sarif.edu). Closed loop optimal control using artificial neural networks for nonlinear multi-input multi-output systems as been significantly developed during te last decade. But te developed neuro-controllers do not ave a satisfactory performance for time varying nonlinear systems suc as magnetic attitude control system. For eample te neuro-controller proposed in [6] as been fed by te time togeter wit te states in order to reinforce te time varying nature of te output of te neural network, i.e., te optimal control istory. Altoug tis idea is useful for te simulated time invariant systems, it yields unacceptable performance for te time varying system subject to tis report. In tis paper te problem of magnetic attitude maneuver as a time varying nonlinear multi-input multi-output system is analyzed, for wic an open loop time optimal solution is initially determined troug a variational approac. Subsequently a fairly new neuro-controller structure and training algoritm is proposed for closed loop near time optimal control. e selected neuro-controller is similar to tat proposed in [6] ecept tat a dynamic weigted neural network, i.e., a neural network wit time varying weigts is utilized in order to control te time varying system. Also, te training algoritm of [6] as been modified to manage te time varying weigts. Finally a modification on te training algoritm as been proposed wic causes te controller to become more robust against perturbations. e attained closed loop near time optimal attitude controller as been investigated troug te Monte Carlo analysis for testing te amount of robustness against uncertainties and disturbances. e proposed controller compared to tat proposed in [5] as substantially reduced te real-time computational burden since it as no need for online optimization. us, it can be applicable to satellites wit less computational capabilities. II. SAELLIE DYNAMICS & KINEMAICS MODELING In tis researc an inertial pointing satellite is selected for proof of te proposed concept. e results are also valid and easily applicable for nadir pointing satellites. e satellite dynamics can be represented as [7] dω I = ω Iω + mctrl B+ Ngg + N dist (1) dt were ω,,, Im, ctrl, BN, gg, and N dist refer to body frame, inertial frame, angular velocity of te body frame w.r.t. te

2 inertial frame, satellite inertial tensor, magnetic moment produced in te magnetic coils, te local magnetic field vector of te Eart, gravity gradient torque, and disturbance torque respectively. All vectors are represented in te body frame. For simplicity, one can select a satellite wit inertial tensor proportional to identity matri in order for te gravity gradient torque to vanis. Satellite kinematics as in [7] can be implemented using four Euler parameters known as quaternions denoted by{ q }. q0 q1 q0 { q} = = (2) q 2 q q 3 were te vector part, i.e., te last tree elements of te quaternions is denoted by q. e kinematics equation can be written as 1 1 q = ω q0 ω 2 2 q (3) ( ω ) 1 q 0 = 2 q (4) were transpose of ω is denoted by ( ω ). III. OPEN LOOP IME OPIMAL CONROL A. Optimal Control Formulation Using a Variational Approac Coosing te four elements of te quaternions and te tree elements of te angular velocity vector as states and te tree elements of te magnetic moment as controls and discretizing te resulting time-varying nonlinear system for N time steps wit equal sampling time of, will result in te discrete form of te system state equation. [ k + 1] = f [ k], u [ k], k,,,1,..., (5) wit initial condition [0] = 0 (6) e problem in and is a time-optimal terminal-control one; terefore, a suitable cost function can be J = ( [ N ] ) H( [ N ] ) + Wt (7) were is te ired final states of te satellite (attitude + rates), H is 7 7 positive semi-definite weigting matri penalizing te terminal errors in te states, and W t is a positive scalar weigt penalizing te total elapsed time. Adjoining system equation (5) to cost function (7) as an equality constraint using Lagrangian multipliers (co-states) δ [ k ], produces te augmented cost function. ( [ ] ) H( [ ] ) Jaug = N N { W t δ [ k 1] ( f( [ k], u[ k], k, ) [ k 1] )} (8) In order to find te minimal of te augmented cost function one sould differentiate w.r.t. all independent variables and equate tem to zero to derive te required optimality condition. f δ[ k] = δ[ k + 1] [ k ] (9) [ k + 1] = f [ k], u [ k], k, (10) f u[ k ] δ [ k + 1] = 0 ( [ k], [ k], k, f u ) Wt + [ k + 1] = 0 ( N ) (11) δ (12) δ [ N ] = 2 H [ ] (13) Equations (9) and (10) are te co-state and te state equations respectively, equations (11) and (12) are te resulting optimality conditions, wic are valid for,1,..., N 1. e free final time problem is formulated using a fied number of time steps N and a variable sampling time. Hence must be optimized as a parameter as well. B. Numeric Algoritm for Optimal Control e optimal control problem formulated tus far can be solved using an iterative metod suc as first order gradient as outlined below. Coose an initial guess on te control istory and te sampling time and follow te tree steps algoritm given net. 1- Use te system equation (5) and te initial conditions (6) to propagate te states troug time and store te resulting state trajectory. 2- Using te co-state equation (9) and te final conditions (13), back-propagate te co-states toroug time and store te co-states time istory. 3- Update te guess on te control istory and te sampling time according to te following rules: [ k] τ f Δ u = [ k ] u δ [ k + 1] (14) u[ k] u[ k] +Δu [ k] (15) ( [ k], [ k], k, f u ) Δ = τ Wt + [ k + 1] δ (16) +Δ (17) were τ is an arbitrary positive scalar called update rate. e algoritm iterates till te control istory and te sampling time converge to teir (local) optimal values. Of course, divergence is possible for inappropriate initial guesses. Selecting a suitable update rate τ, causes te number of iterations needed for convergence to decrease

3 substantially and tis selection is usually attempted eiter troug a searc algoritm or wit trial and error approac. Finally, in order to take te control saturation constraint into account, one can simply restrict te control values in te admissible interval during te control update process. C. Open Loop ime Optimal Solution o simulate te proposed magnetic attitude controller, one needs a precise model for te magnetic field of te Eart. In tis researc te IGRF2000 wit degree 13 as been utilized [8]. e specification of te orbit and te satellite subject to simulation is mentioned in table I. e weigting matri H is selected in suc a way to penalize te final rates error (wit unit of rad/s) wit factor of 1000 and penalize te final quaternions error wit factor of e elapsed time penalizing weigt W t is selected equal to and te number of time steps N is selected equal to 100. e maneuver conditions used troug te iterative algoritm are tose mentioned in table II. e results of te time optimal maneuver are depicted in fig. 1 and 2. As can be seen in fig. 2, te optimal control is in te form of bang-bang and te optimal time-to-go is about 234s. Since te orbital period is equal to 5829s, te maneuver is accomplised only in a fraction of an orbit, namely in about 4 percent of te orbital period. e maimum final attitude error is 3 deg. and te rates error is rad/s. e balance between te attitude and te rates error can be tuned by tuning te elements of te weigting matri H, since most of small magnetic actuated satellites require a pointing accuracy witin ±5 deg., ere te weigting matri is adjusted to eliminate te final angular rates wile meeting te pointing accuracy. For te purpose of decreasing te total error, one sould decrease te value of W t compared to te value of elements of matri H. D. Open Loop Near ime Optimal Solution It as been realized tat by increasing te final time of te maneuver by a small amount, for eample 20%, te robustness of te resulting open loop and te closed loop controller is significantly increased. Since tis increment on te required time-to-go from 4% into 4.8% of an orbital period can be easily ignored compared to te conventional closed loop time consuming controllers; ereafter, te maneuver conditions defined in table II wit fied final time equal to s will be calculated, simulated and called near time optimal solution. ABLE I OR & SAELLIE CHARACERISICS orbit altitude inclination rigt ascension of te ascending node eccentricity orbital period saturation limit of eac magnetic coil satellite inertial tensor Value 630 km s 10 A.m 2 diag(1,1,1) kg.m 2 ABLE II EXAC CONDIIONS MANEUVER SPECIFICAIONS Value initial Euler angles [30,40,50] deg initial angular rates [0,0,0] rad/s error of satellite inertial tensor diag(0%,0%,0%) disturbance torque 0 N.m maneuver startup delay 0 s error of magnetic field elements [0%,0%,0%] ired Euler angles [0,0,0] deg ired angular rates [0,0,0] rad/s Fig. 1. Euler angles trajectory of te time optimal open loop maneuver Fig. 2. Control istory of te time optimal open loop maneuver E. Robustness Analysis For te purpose of comparing te maneuver accomplised wit no disturbance and uncertainty wit tat acieved under some perturbations including uncertainties over te initial states and some modeling errors, te obtained near time optimal solution wic is calculated based on maneuver conditions mentioned in table II is applied to two different maneuvers wit deterministic specifications mentioned in table II and table III and te resulting final errors are presented in table IV. Hereafter, for convenience, we call te maneuver defined in tables II wit eact conditions maneuver and also te maneuver defined in table III wit perturbed conditions maneuver. As can be seen troug table IV, te open loop controller as not been able to manage te perturbed conditions maneuver and te final errors are too large. ABLE III PERURBED CONDIIONS MANEUVER SPECIFICAIONS Value initial Euler angles [31,41,49] deg initial angular rates [-1,1,-1] 10-5 rad/s error of satellite inertial tensor diag(-0.5%,+0.5%,-0.5%) disturbance torque N.m maneuver startup delay 10 s error of mag. field elements [-1%,+1%,-1%] ired Euler angles [0,0,0] deg ired angular rates [0,0,0] rad/s

4 ABLE IV RESULS OF HE OPEN LOOP NEAR IME OPIMAL SOLUION Maneuver Ma. Attitude Ma. Rates Error (deg.) Error (rad/s) Cost eact cond. man perturbed cond. man o ave a measure of robustness of te attained open loop solution we can disturb te slew maneuver conditions using disturbances and uncertainties mentioned in table V. Assigning a Gaussian distribution to te uncertainties and utilizing a Monte Carlo analysis on te value of te cost function (i.e., cost-to-go); te mean cost, as a measure of robustness, based on te selected weigting matri H for te open loop near time optimal solution is determined to be troug 2000 runs of te Monte Carlo simulation. ABLE V ERROR BANDS FOR HE MONE CARLO ANALYSIS Maimum Values initial Euler angles error initial angular rates error satellite inertial tensor error disturbance torque delay of maneuver startup time magnetic field error ±2 deg ± rad/s ±1% ± N.m 0 to 10s ±1% IV. CLOSED LOOP NEAR IME OPIMAL CONROL USING NEURAL NEWORK A. Neural Dynamic Optimization Formulation In implementing a neural network as a time optimal controller, te policy is to calculate te optimal time-to-go for a specified maneuver using available metods suc as te one cribed in part III of tis report as open loop controller, now te problem will be a fied final time one to be minimized for te final errors. e cost function will be ( [ ] ) ( [ ] ) J = N H N (18) Following [6] but using a neural network wit time varying weigts, te problem is to train a multi-layer feedforward neural network wit sigmoid activation function to take te states and te ired states at eac time step and calculate te optimal control needed for feedback into te system. e overall configuration of te controller and te system is presented in fig. 3. Fig. 3. e layout of te neuro-controller and te system e optimal control is someow a function of te magnetic field vector of te Eart represented in te body frame; terefore, te vector is fed into te network in order to produce an implicit relation between te fed magnetic field vector and te generated control. As will be sown, tis can improve te robustness of te controller against te errors of te magnetic field used during te training process. ese errors can appear from two sources, first from te inerent errors of te magnetic field model, and second from te delay on actual maneuver startup wic causes te satellite to start te maneuver in a different point in its orbit and observe a fairly different magnetic field compared to tat used in te training process. Denoting te neural network wit g (.) and its time varying weigts wit W [ k ], te control equation will be u[ k] = g [ k],, B[ k], W [ k] (19) e control saturation wile using a neural network to calculate te control can be easily implemented by scaling te output of te last layer s activation functions of te neural network to ave a saturation limit equal to te ired control limit. Augmenting te state equation (5) using te co-states δ [ ] and control equation (19) using te co-inputs δ [ ] k as equality constraints to te cost function (18), yields te augmented cost function: ( [ ] ) H( [ ] ) Jaug = N N { δ [ 1] ( ( [ ], [ ],, ) [ 1] )} k f k u k k k + { δ [ ]( ( [ ],, [ ], [ ]) [ ])} u k gk Bk Wk uk u k (20) By setting te first derivative of te augmented cost function w.r.t. uδ,,, δ, and W equal to zero and a little u algebraic manipulation, we will arrive to te network training algoritm. Here te sampling time is not to be optimized so te derivative of te augmented cost function w.r.t. te sampling time will not be utilized. f( [ k ], u[ k], k, δ[ k] = δ[ k + 1] [ k ] g( [ k],, B[ k], W[ k] ) + δu [ k ] [ k ] f δu[ k] = δ[ k + 1] [ k ] u (21) (22) [ k + 1] = f [ k], u [ k], k, (23) u[ k] = g [ k],, B[ k], W [ k] (24) ( [ k],, [ k], [ k] ) g B W W[ k ] ( N ) δu [ k ] = 0 (25) δ [ N ] = 2 H [ ] (26) Equations (21), (22), (23), (24) and (25) are te co-state

5 equation, te co-input equation, te state equation, te control equation, and te resulting optimality condition respectively. e optimality condition (25) attained based on te dynamic weigted neural network is fairly different wit its equivalent condition for time-invariant weigted neural network [6] tat is ( [ k],, [ k], ) g B W δu [ k ] = 0 W B. Numeric Algoritm for Neural Dynamic Optimization (27) Similar to te first order gradient algoritm cribed in part III of tis report, but using equations (21) to (25) along wit final co-states condition (26), te learning algoritm will be as follows: Coose an initial guess on te network weigts and repeat te following tree steps until te cost converges: 1- Randomly coose initial conditions of te states from a ired interval of initial conditions uncertainty, and use system equation (23) and control equation (24) to propagate te states troug time based on a (near) optimal final time already calculated troug an iterative metod, and store te resulting state trajectory. 2- Using te co-input equation (22), te co-state equation (21), and te co-states final condition (26); back-propagate te co-inputs and te co-states troug time and store teir resulting time istory. 3- Update te guess on te neural network weigts according to te following relation: ( [ k],, [ k], [ k] ) [ k] τ g B W Δ W = δu [ k] (28) [ k ] W W[ k] W[ k] +ΔW [ k] (29) In case of divergence, one sould cange te initial guesstimates and follow te algoritm steps again. C. Closed Loop Near ime Optimal Solution Coosing a neural network wit ten inputs, one idden layer containing only one neuron, te output layer containing tree neurons wit te saturation limit equal to te satellite maimum control, and wit number of discretizing time steps equal to 100; te network is trained to acieve te eact conditions maneuver mentioned in table II. According to te algoritm, te initial states of te maneuver in eac iteration of te algoritm is selected troug an interval of uncertainty mentioned in table VI. As te algoritm is based on a fied final time, we used te obtained time-to-go from te open loop near time optimal solution for te same eact maneuver. D. Robustness Analysis Using te trained neuro-controller for te eact and te perturbed conditions maneuvers defined in tables II and III respectively, te final results are given in table VII. e robustness of te controller against te perturbations as been improved and te final errors are decreased. initial Euler angles error initial angular rates error ABLE VI INIIAL SAES UNCERAINY Maimum Values ±2 deg ± rad/s ABLE VII RESULS OF HE CLOSED LOOP NEAR IME OPIMAL CONROLLER Maneuver Ma. Attitude Ma. Rates Error (deg.) Error (rad/s) Cost eact cond. man perturbed cond. man Applying te Monte Carlo analysis on te trained neurocontroller wit uncertainties mentioned in table V, now te mean cost to go will be , wic is less tan te mean cost of te open loop controller by a factor of ten. E. A Modification for Additional Robustness Using a small modification on te mentioned algoritm for training te neural network, one can decrease te mean cost of te Monte Carlo analysis ever furter and reinforce te robustness property of te proposed controller. e modification is as follows: In eac iteration during te training process, wen a new initial states is to be selected from an interval of uncertainty, modify te inertial tensor of te satellite in suc a way tat te new one belongs to an admissible interval of uncertainty. Also, randomly select a disturbance torque via an allowable interval and apply te torque on te satellite during tat iteration; in addition, select a new startup time witin its admissible delay interval; and finally, perturb te magnetic field model witin its acceptable error band. Add all of tese perturbations simultaneously before eac iteration and run te algoritm. is modified algoritm causes te network to eperience te various disturbances and uncertainties wic eist in te real world; tus, elps te network to produce an output more strongly based on te fed inputs, i.e., te current states and te observed magnetic field vector, not based on a memorized istory. F. Modified Closed Loop Near ime Optimal Solution e structure of te network used for training is as before, and te maneuver conditions in eac iteration are randomly selected troug te intervals of uncertainties mentioned in table V. G. Robustness Analysis e results of simulation of te neuro-controller wit modified training algoritm for te eact and te perturbed conditions maneuvers are mentioned in table VIII. e Euler angles and te angular rates trajectory and te control istory of te perturbed conditions maneuver are sown in fig. 4, 5, and 6 respectively. Comparing te errors of te perturbed conditions maneuver managed by te modified closed loop controller wit tose of te open loop

6 controller sows a substantial improvement in disturbance rejection and robustness against uncertainties in te modified closed loop controller especially due to less final angular rates. Finally, applying te Monte Carlo analysis to te trained neuro-controller wit modified algoritm for te similar uncertainties of table V, te mean cost-to-go will be wic is about 2% of te mean cost of te open loop controller, and it is an indication for good performance of te proposed controller. ABLE VIII RESULS OF HE MODIFIED CLOSED LOOP NEAR IME OPIMAL CONROLLER Maneuver Ma. Attitude Ma. Rates Error (deg.) Error (rad/s) Cost eact cond. man perturbed cond. man Fig. 4. Euler angles trajectory of te perturbed conditions maneuver controlled by te modified closed loop controller be seen te mean cost of te modified closed loop solution is about 2% of te mean cost of te open loop solution. e real-time computational burden of te neuro-controller corresponds to tat of only one feed forward calculation troug a neural network wit four neurons; terefore, te proposed near time optimal closed loop controller of slew maneuvers can be te best coice for time critical missions of small satellites wit low computational capabilities. ABLE IX HE SUMMARY OF HE MONE CARLO ANALYSIS OF HE DISCUSSED CONROL MEHODS Controller Mean Cost open loop controller closed loop controller modified closed loop controller REFERENCES [1] M. Lovera and A. Astolfi, Global attitude regulation using magnetic control, in Proc. 40t IEEE Conf. on Decision and Control, Orlando, Florida USA, 2001, pp [2] R. Wisniewski, Linear time varying approac to satellite attitude control using only electromagnetic actuation, in Proc. AIAA Guidance, Navigation and Control. Conf., vol. 23, AIAA, New Orleans, 1997, pp [3] R. Wisniewski, Sliding mode attitude control for magnetic actuated satellite, in Proc. 14t IFAC Symposium on Automatic Control in Aerospace, Seoul, [4] M. L. Psiaki, Magnetic torquer attitude control via asymptotic periodic linear quadratic regulation, Journal of Guidance, Control, and Dynamics, vol. 24, no. 2, pp , Marc April [5] J. Liang, R. Fullmer, and Y. Cen, ime-optimal magnetic attitude control for small spacecraft, in Proc. 43rd IEEE Conf. on Decision and Control, Atlantis, Paradise Island, Baamas, 2004, pp [6] C. Seong and B. Widrow, Neural dynamic optimization for control systems Part II: eory, IEEE ransactions on Systems, Man, and Cybernetics-Part B: Cybernetics, vol. 31, no. 4, pp , August [7] J. R. Wertz, Spacecraft Attitude Determination and Control, Reidel, [8] J. Davis, Matematical Modeling of Eart s Magnetic Field, ecnical Note, Virginia ec., Blacksburg, Fig. 5. Angular rates trajectory of te perturbed conditions maneuver controlled by te modified closed loop controller Fig. 6. Control istory of te perturbed conditions maneuver controlled by te modified closed loop controller V. CONCLUSION ree controllers for te purpose of rest to rest maneuver attitude control of magnetic actuated satellites ave been proposed and te robustness of te metods against several uncertainties as been investigated troug te Monte Carlo analysis and te summary is mentioned in table IX. As can