Pseudospectral Model Predictive Control for Exo ...

Paper Int’l J. of Aeronautical & Space Sci. 16(1), 64–76 (2015) DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.64

Pseudospectral Model Predictive Control for Exo-atmospheric Guidance Tawfiqur Rahman*, Hao Zhou**, Liang Yang*** and Wanchun Chen**** School of Astronautics, Beihang University, Beijing, China

Abstract This paper suggests applying pseudospectral model predictive method for exo-atmospheric guidance. The method is a fusion of pseudospectral law and model predictive control, in which a two point boundary value problem is formulated using model predictive approach and solved by applying pseudospectral law. In this work, the method is applied to exo-atmospheric guidance with specific target requirement. The existing exo-atmospheric guidance methods suffice general requirements for guidance, but it cannot ensure specific target constraints; whereas, the presented method is able to do so. The proposed guidance law is assessed through simulation of perturbed cases, and the tests suggest that the method is able to operate semiautonomously under control and thrust vector perturbations. Key words: Exo-atmospheric Guidance, Pseudospectral Model Predictive Control, Gauss Pseudospectral Method, Hypersonic Vehicle Guidance, Model Predictive Control

1. Introduction

by endo-atmospheric guidance system which is generally an open loop method following a preselected profile. Once the altitude is enough to neglect atmospheric effects, the vacuum guidance system takes effect. Here, the transition is made because vacuum guidance methods cannot be applied inside the atmosphere, and the existing endo-atmospheric methods do not offer adequate autonomy so as to ensure the required accuracy. In this paper, we demonstrate the feasibility of a recently proposed pseudospectral model predictive method for exo-atmospheric guidance. This method does not rely on the simplifications of vacuum flight, and yet, it is able to provide on board solution feasible for future application. The method has previously been demonstrated for endo-atmospheric flight, and therefore, it could be a candidate method for the guidance of all phases of global delivery system.

Exo-atmospheric guidance refers to guidance beyond the atmosphere with negligible aerodynamic effects. In vacuum, a vehicle encounters forces from two sources only: its own thrust and gravitation of the planet. Due to the relative simplicity, closed loop optimal or near-optimal solution can be obtained for vacuum guidance. This has been used as an advantage in guidance system of the existing exoatmospheric vehicles. For the past and present missions, vacuum guidance methods have been sufficient. For example, in an orbital injection mission, only the desired orbit and velocity is to be met; for CRV (Crew Return Vehicle), CLV (Crew Launch Vehicle), or CaLV (Cargo Launch Vehicle), the end position is comparatively more important, which the existing vacuum guidance methods can ensure. But, for the missions involving global payload delivery, the desired accuracy is far stringent than that presently expected. For a global payload delivery mission, the flight would comprise of endo, exo, re-entry, and terminal phases of flight which are starkly different from each other. In a typical mission, the first phase is ascending through the atmosphere, controlled

2. Review of Exo-atmospheric Guidance Methods A number of vacuum guidance approaches are available in the existing literature. However, only a few have ever

* Doctoral Candidate Lecturer, Corresponding author: [email protected] ** *** Doctoral Candidate **** Professor

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: November 18, 2014 Revised: December 29, 2014 Accepted: January 11, 2015 Copyright ⓒ The Korean Society for Aeronautical & Space Sciences

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Tawfiqur Rahman Pseudospectral Model Predictive Control for Exo-atmospheric Guidance

constraints is not four, substantial changes to the algorithm will be required. As such, research efforts have continued in expanding the capability of vacuum powered guidance algorithm. The recent on-going efforts to achieve the capabilities of responsive space launch and autonomous space operations have brought renewed interest in both endo and exo-atmospheric guidance. For guidance inside the atmosphere, numerous methods have been developed and studied. Further details can found in the references [6-10]. However, for exo-atmospheric guidance, the PEG approach still remains as the basis for modification and improvement. Some of the researches on PEG include those of Rose M. Benjamin and David K. Gellar [11], Dukeman [12, 13], Lu, P. et al. [7, 14, 15], Lijun Zhang [16], etc. The methods of Lu, P. et al. [7, 14, 15] consider complete ascent profile including coast phases. However, the methods involve analytical multiple shooting method for guidance solution and require iterations. Dukeman [12, 13] solved calculusof-variation two-point-boundary-value- problem starting from vertical rise till main engine cut-off. The proposed method does not involve iteration and is mathematically faster and robust. Here, we suggest pseudospectral model predictive control for exo as well as endo atmospheric guidance. It is demonstrated that the proposed method can guide a vehicle orbital insertion in a rapid manner.

been tested or flown on an actual launch vehicle. The only closed-loop vacuum guidance algorithms applied thus far are as follows: Iterative Guidance Mode (IGM) and Powered Explicit Guidance (PEG). The earliest of these is the Iterative Guidance Mode [1] which was designed for the Saturn class vehicles. The approach was to solve a set of quasi-explicit equations in the near-closed form. However, in this method, the terminal condition was limited to a set, where both the final in-plane velocity components as well as the radial component of the final position coordinates were specified. Compared to IGM, a more general method was developed by G. W. Cherry [2]. His method provided a universal solution to boundaryvalue problems of powered guidance, and derivation of the steering law avoided specialized mathematics. An essential feature of the approach was the E-matrix, which mapped the difference in the boundary conditions into the thrustallocation guidance coefficients. This method was designed to control the final position as well as final components of velocity. It was used for lunar descent and ascent mission of Apollo [3]. A thorough detail of his work can be found in G.W. Cherry’s paper [2]. From IGM, a more capable vector based algorithm called Linear Tangent Guidance (LTG) was developed. It was developed by Ronald F. Jaggers. This method was generalized for usage in the Space Shuttle [4] so that it could handle all exo-atmospheric manoeuvres and requirements, including ascent, ascent aborts, and deorbit. LTG was later renamed as PEG and was chosen for Orion orbit insertion, deorbit, and rendezvous burn guidance, and it was also used in the trade studies for Altair lunar landing guidance. The original theoretical formulation of PEG is given in Jaggers [3]. The basic algorithm derives a closed solution to an optimal control problem by maximizing vehicle’s total mass at terminus. PEG is formulated under the four important assumptions as follows: the aerodynamic forces are negligible; the engine exhaust velocity is constant; either the thrust (mass flow rate) or acceleration is constant; and the target conditions are independent of time and functions, but only of the estimated (navigation) inertial position and velocity states. With the linear gravity model proposed by Jezewski [5], a costate solution is developed. From optimal control theory and the linear gravity model, it is found that the optimal thrust direction, which is required to steer the vehicle to the desired target, is along the direction of the velocity costate vector. The basic PEG algorithm is designed for handling 4 specific orbital insertion constraints of the final radius, velocity, flight path angle, and orbital inclination. For other forms of terminal constraints, modifications are needed on a case-by-case basis. If the number of terminal

3. Pseudospectral Model Predictive Control The pseudospectral model predictive control (PMPC) method, which was first presented by Yang Liang et al. [17], combines MPC (Model Predictive Control) and GPM (Gauss Pseudospectral Method). In this approach, the state and control variables are approximated by using LG (Legendre Gauss) points, and the system dynamics are approximated on the LG collocation points. The linear optimal control problem is modelled to obtain corrections to the nominal control, which will minimize terminal deviation from the nominal state. These corrections are derived from the predicted state of the MPC problem. Mathematical formulation of the method is described in the following, with an introduction to GPM.

3.1 The Gauss Pseudospectral Method GPM is a pseudospectral collocation method where a function is interpolated on LG points. For details on GPM, the readers are referred to references [18, 19]. In this method, the Legendre polynomial LN(τ) is used as the basis function.

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From (2) to andGPM. (4), we see matrix D D that   the .D state dynamics can be defined using differential approximation (5) formulation of the method is described in the following, with an introduction  kl

lk

k

3.1 3.1

and control u‫)כ‬a and thendynamics predict the this method, we assume nominaloftrajectory (with stateAsx‫ כ‬such, which is defined for anya number collocation points. in GPM, system canperturbed be

3.1 The Pseudospectral 3.1 The Gauss Gauss Pseudospectral Method Method The Gauss Pseudospectral Method The Gauss Pseudospectral Method

From (2) and (4), we see that the state dynamics can be defined using differential approximation matrix D

p trajectory = xa‫ כ‬set +forδx) by applying the nominal control u‫ כ‬GPM, to the perturbed system xp. The linear time transformed into ofany algebraic equations, which is As the core concept of this method. which (x is defined number of collocation points. such, in a system dynamics can be

GPM is collocation method where aa function is interpolated on points. For GPMcollocation is aa pseudospectral pseudospectral collocation method where function isalgebraic interpolated on LG points. For ofdetails details GPM is a pseudospectral method where a function is interpolated ona set LG points. For details transformed into of equations, which is the core concept this method. variant perturbation is then defined asLG follows. GPM is a pseudospectral collocation method where a function is interpolated on model LG points. For 3.2 Gauss Pseudospectral Solution todetails PMPC

Pseudospectral GPM, the are to [18, 19]. In method, onare GPM, the readers readers are referred referred to references references [18,3.2 19]. In this this method, the Legendre polynomial LLNN((ττ)) is is   x Solution Athe (t )Legendre xtoLPMPC B((τt )) is u polynomial on GPM, the readerson referred to references [18, 19]. In this method, the Gauss Legendre polynomial ) is an algebraic solution is formulated by applying GPM. In on GPM, the readers Int’l areJ.referred to references [18, 19].64–76 In this method,PMPC the Legendre polynomial LNN(τwhere is a predictive control method of Aeronautical & Space Sci. 16(1), (2015) PMPC is a predictive control method where an algebraic solution is formulated by applying GPM. In Where matrices areN defined asthe follows (note superscript ττ)Jacobean of degree has LG points which the used basis A Lmethod, ) LG ofweN Npoints degree has N LGin points which in the u‫‘ )כ‬p’). used as asAthe the basis function. function. A Legendre Legendre polynomial LNthe N degree which exist used as the basis function. Legendre polynomial L (τ) ofpolynomial ‫ כ‬the N((N and in control and then predict the perturbed thishas assume a nominal trajectory state exist xexist has N LG points in (with the(with used as the basis function. A Legendre polynomial LNN(τ) of N degree this method, we assume which a nominalexist trajectory state x‫ כ‬and control u‫ )כ‬and then predict the perturbed p

p

p

p

p p f ( x (time t ),the u (domain tnominal )) t‫)כ‬,to u the (t ))the ‫ כ‬+ δx)onto A Legendre polynomial LN(τ) of N degree has Nonto LG real points c  c x . The linear time trajectory (xtime = by applying u(using system [-1, For problem, the LG points need to [t[t0u0,f‫כ‬, (tottxff]]the p x domain [-1, 1]. 1]. For real problem, the LG points need to be be mapped time domain using the domain [-1, 1]. For domain real problem, the LGreal points need to be mapped ,real tf] using the Aonto (by t ) [t  , B (t ) control ,perturbed A  R sxs ,p.B  Rlinear ‫ כ‬+ δx) (7) 0real trajectory (xmapped = xdomain applying the nominal control perturbed system The time , t ] using the domain [-1, 1]. For real problem, the LG points need to be mapped onto real time domain [t   x u 0 f which exist in the domain [-1, 1]. For real problem, the LG

(6)

(7)

variant perturbation model is then defined as follows.

variant perturbation model is then defined as follows. following affine transformation. following affine transformation. following affine transformation. points need to be mapped onto real time domain [t0,optimal tf] using The control problem can be formed to problem determinecan the control deviation δu (xp, t), such that equation following affine transformation. The optimal control be formed to determine (6)   x A(t ) x  B (t ) u

  x A(t ) x  B (t ) u

the following affine transformation.

the controlthedeviation (6) is satisfied while minimizing functional δu cost(x , t), such that equation (6) is Where the Jacobean matrices are defined as follows (note the superscript ‘p’). (t f  t0 )  (t f  t0 ) ((ttff tt00))  ((ttff tt00)) Where the Jacobean matrices arewhile defined as follows superscript ‘p’). satisfied minimizing the the functional (1) (1)cost (1) (note t  (t f  t0 )  (t f  t0tt) (1) (1) 1f ( x t (t ), u p (t )) T t 22 2 (t )) 1f ( xT (t ), u  p R p  A ( t ) , B ( t ) , A  R , B p p p p 2 J fx( x(tx(ft)),Puf (xt ))(t f )   [(fu(xx (t ),ux)(tQ  u)T (t ) R(7) (u p   u)]dt )) ( x   sx s)  (u c c t p

A(t ) 2 

p

p

x

p

p

0

 , B (t2)

f

s s

u

(6)

cc

, A R

,BR

(8)

(7)

(8)

The optimal control problem can formed tocan determine the control deviationas pseudospectral law, the derivative of state collocated on the LG From law, thecollocated derivative any state From pseudospectral law, thestate derivative of any any state collocated on theapproximated LGbenodes nodes can be approximated approximated asδu (x , t), such that equation From pseudospectralFrom law, the pseudospectral derivative of any onofthe LG nodes can be as be of the system written as From pseudospectral law, the derivative of any state collocated onThe the LG nodes can be approximated asdetermine TheHamiltonian optimal control problem cancan be be formed to the control deviation δu (xp, t), such that equation p

The Hamiltonian of the system can be written as is satisfied while minimizing the functional cost collocated on the LG nodes can be approximated as (6) (similar

(similar approximation be control profile as (similar approximation canprofile be used for control profile as well) well) (similar approximation can be used for asfor well) approximation cancontrol be can used forused control profile as well) 1 (6) is satisfied while minimizing the functional p T 1 p cost (similar approximation can be used for control profile as well) p T [ A(t ) x(t )  B(t ) u (t )] H 1 [( xp x)x)(uQp( xp ux))TR(u(up p u) u(t)]) R(u (9) J x x(t ) P  xx()t )Q  ( x [(   u)]dt (9)(8) The Noptimal control is then 22 2 N N 2   t 2   1 1 2   N LG LG N LG LG T T T p p p p N LG   optimal xcontrol Dk xxlThe D is  can x (be t f )written Pf  x(t as x)  (u   u) (t ) R(u   u)]dt ) [( x   x) Q( x  (2) (2) (8) D xxx ((kk)) Dkl xl D x NN ( k )  2  LG The optimal is then klkl x lcontrol  (2) f (2) t l  00   TheD Hamiltonian of the system optimal control is Jthen then  LG  x ( k )   f   0  LG Dkk xll00  D0x0 l k lThe 2 optimal (2)2 is then5 ll11 control   The control l 1  ff kl  H The optimal is then   p T  1   f 0  l 1  u  R (t ) B (t ) (t ) 0  p u (t )  1   The optimal control is Hthen [(uxpcan ) Qwritten ( x   x)as  (u p   u) R(u p   u )]   [ A(t ) x(t )  B(t ) u(t )] (9)  xbe The Hamiltonian of the system  H pp  11  N×N N  N×N N 2H N×N matrices N H     u t  u Rof tLagrange 0  ( ) ))BBTTT((tt)) p  1((Lagrange ((tt)) and are where the differentiation matrices �� and D �� found using the exact derivative where the differentiation D   u t  u 0  ( )  �� and D �� are found using the exact derivative of where the differentiation matrices D  H of where the differentiation matrices D�� N×N and D �� N are found using the exact derivative Lagrange 1 R T (t  u (t ) p   u R t B t t 0  ) ( )  ( ) (10) uuu u  R (t ) B (t ) (t )  u (t )  (10) where the differentiation matrices D�� and D �� are found using the exact derivative of0  Lagrange  1u p p T 1 H found using the exact derivative of LagrangeThe interpolating p  (t ) H u( x (pobtained t)  state and costate equations be (10) 0can x )T  Q  xu5) (uRpas (t )uB)T R(t()u   u )]   T [ A(t ) x(t )  B(t ) u(t )] (9)   [(x  44 4  2u  interpolating polynomials [18]. These related to each other polynomials [18].are These are related eachbyother by 4 to state can obtained The state and costate equations can be obtained as interpolating polynomials [18]. These are related to each other by The The stateand andcostate costate equations equations can bebe obtained as as The state and costate equations can be obtained as The state and equations can be obtained as Hcostate interpolating polynomials [18]. These are related to each other by The state and costate equations  be    N can obtained as   t     t  ( )  ( )  A(t ) x  (t )  B(t ) u (t ) x x 5 LG LG   (3) Dk    Dkl N (3)  H  H       LG LG ((tt)) H  H (3) l N1 xxx((t(t)t)) ((tt)) u((tt)) t )Ax((tt())t)  x (tx) (tA)( uBB x B Dk  These Dkl related to each other by        interpolating polynomials LG [18]. LG are    t    xx ((tt))  A A(t ) xx  ((tt))   B((ttT)) u  ( ) x u (t )    H        (3) H Dk    Dkl  l 1       (xt )(t )x (Bt ()t)A u ((tt))  (t ) (11) (11) Qx p x ((tt ))    x(t )(t) A(t )Q l 1  H     H x        T p  Gauss pseudospectral approximation toalso a state’s derivative  Gauss pseudospectral approximation to a state’s derivative can be made as N H    T pp  ((tt))  (t) (t)Q(t)Qx(t()t) x A  )A T((t )t )Qx  H   (11) (t ()t    LG LG  (((ttt)))Qx   Qx  ((tt))  H  (t )  Q((tt)) xx ((tt))T x   (3) Gauss pseudospectralcan approximation toklas a state’s derivative can also be made as Dk  be  D A AT ((tt)) Qx p also made  (t )   xxx   (t )(t )Q(tQ ) x (t )  A (t )  (t )  Qx p Gauss pseudospectral approximation to a Nstate’s derivative can also be made as l 1  the x  state equation, the TPBVP can be stated as 2  LG †  Substituting the control relationin N x NN ( k )  2   Dkl† xl  LG Dk†† xl N 1  (4) Substituting the controlSubstituting relation in thethe state equation, the TPBVP be stated as the control relation in thecan state equation, LG LG the control relation in the state the TPBVP can be stated as l N 1 x N ( k )   f 2 to D xl  LGDk† xl can (4) equation, 0  Substituting the control relation in the state equation, the TPBVP can be stated as Gauss pseudospectral approximation state’s beSubstituting made as(4) a alsoSubstituting N 1    LG kl† derivative Substituting the control relation in the state equation, the TPBVP can be stated as  1 theT TPBVP can  p as the control relation in the state equation, be stated TPBVP be stated x ( k )   f   0   Dkl xl  Dk xl N 1  (4)as l 1     can t1 ) R T(t ) B (t )  x (t ) p   x  ( t )  ( t )  B ( Bu ( t )  x (t )     (t )  B (t ) R (t ) B (t )   x (t )   Bu (t )   f   0  l 1  (12)       p  ppp  11T matrix D, and their relation [18] is xxxx(t(((()tttt))))  Q where D†† is the adjoint ofNdifferential xx) ((tQx  ()tt)) B B ((tRtT)) RR (t ) B(tT)TT((tt))(xt)(t ( 1t A  2 approximation  N LG †   tt)))Bu Bu  (t )pBu LG † (t((Q (B t )A ()t1)((B(tttT)))B (Qx t )  p(((ttt)))   x  t )  B ( t ) R B ( t )  ( Bu †  x x x ( )  D D    (4) is the adjoint of differential approximation matrix where D    k kl l k l N   1 matrix D, and their relation [18] is     where D † is the adjoint of differential      p pp (12) (12)  f  approximation TAA(tTTT)((tt))   (t)((tt))Qx Qx 0  l 1 Q approximation matrix D, and their relation [18] is  ((((tttt)))) QQ where D is the adjoint of differential A t ) p(((ttt))) Q  A (t )       (t ) (Qx Qx D, and their l relation [18] is (13) xx(t(0t)0 ) 00  † Dkl    .Dlk (5) l Dkl††differential   (5) kl .Dlk approximation matrix D, and their relation [18] is  xx((tt00 )) 000 where D† is the adjoint of (13) (13) Dkl   k .Dlk  x (t00 )  0 (5)   (5)  k (14)  f Pf  x f   T  T      xD    f Pf  xmatrix  From (2) and (4), we see †that the f  f  l state dynamics can be defined using differential approximation     x  TT    Dklthat   the .D (5)  From (2) and (4), we see state dynamics can be defined using differential approximation matrix D lk (14)   TT  f  fff P PPfffxxxfmatrix  (14)  From we see that state dynamics can be approximation    D   ff  From (2) and (4), we see that(2) the state(4), dynamics can the be defined using differential x k and   P    x  be xfxff  which includes the effect of the predicted state deviation ftransversatility f f  condition Here, equation (14) is thedynamics which is defined fordefined any number of collocation points. As such, in GPM, a system can  x   f  using of differential approximation matrix whicha (14) which is defined for any number collocation points. As such, in Dequation GPM, system dynamics can be condition Here, is the transversatility which includes the effect of the predicted state devia which(2) is and defined forissee any number of collocation points. As such, inon GPM, a system dynamics can be From (4), we that the state dynamics can be defined using differential approximation matrix D Here, equation (14) is Here, the transversatility condition whichIfincludes the terminal effect of the predicted state defined for any number of collocation points. As such, Ψ the terminal states x [17]; ν is a Lagrange multiplier. the desired state is restricted to deviation some f equation (14) is the transversatility condition which transformed into a set of algebraic equations, which is the core concept of this method. Here, equation (14) is the transversatility condition which the of predicted state Here, equation (14) is condition which includes the effect of the predicted state devia Here, equation (14)states is the thex transversatility transversatility condition which includes includes the effect effect of the thestate predicted state devia devia transformed into a setin ofGPM, algebraic equations, whichcan is the concept this method. a system dynamics be core transformed into aterminal set Ψ onof the [17]; ν is a Lagrange multiplier. If the desired terminal is restricted to s f includes the effect of the predicted state deviation Ψ on the specific first part the νtransversatility condition is used. the terminal states xf of [17]; iscan a Lagrange multiplier. If the desired terminal state is restricted to some transformed into a for set of algebraic which points. is the core ofGPM, thisvalue, method. which is defined any numberequations, of collocation As concept such,Ψ inon a the system dynamics be of algebraic equations, which is the core concept ofterminal this Ψ on states xxxff [17]; νν is multiplier. If the terminal Ψ on the terminal states [17]; is aa Lagrange Lagrange multiplier. If the desired terminal state is restricted to terminal states xf a[17]; ν is a Lagrange multiplier. If the desired 3.2 Gauss Pseudospectral Solution to PMPC Ψ on the thevalue, terminal is Lagrange multiplier. If used. the desired desired terminal state state is is restricted restricted to to sss f [17]; specific the states first part of theν transversatility condition is method. 3.2 Gauss Pseudospectral Solution to PMPC specific value, the first part of the transversatility condition is used. terminal state iswerestricted to some value, the first transformed a set of algebraic equations, which is the core concept ofInthis method. order to solve this TPBVP, need to define thespecific derivatives of state and costate on the LG 3.2 Gaussinto Pseudospectral Solution to PMPC specific value, the first part of the transversatility condition is used. specific value, the first part of the transversatility condition is used. specific value, the first part of the transversatility condition is used. part of the transversatility condition is used. PMPC is a predictive control method where an algebraic solution formulated by applying GPM. In need Inis toto solve this TPBVP, we todefine define the derivatives of state costate collocation Thisapplying is done as shown in equation (2)theand (4). For convenience, weand redefine theon the orderpoints. solve TPBVP, we derivatives of state 3.2 Gauss Pseudospectral PMPC is Pseudospectral a predictive control method where anSolution algebraic to solution isInorder formulated by GPM. Inneed Inthis order to solve this toTPBVP, we need to define theand costate on the LG 3.2 PMPC Gauss Solution to PMPC PMPC is a predictive control method where an algebraic solution is formulated by applying GPM. In In order to solve this TPBVP, we need to define the derivatives of state and costate on the In order to solve this TPBVP, we need to define the derivatives of state and costate on the ‫ )כ‬andto In uorder solve TPBVP, we need toequation define of convenience, state and on predict perturbed this method, we assume a nominal trajectory (with state x‫ כ‬and control derivatives ofasas state and on the LG collocation following Npoints. × then N+1 matrices collocation This is the done shown (2) and we redefine collocation Thisthis is done shown in incostate equation (2) the andderivatives (4). (4). For For convenience, we costate redefine thethe is a predictive method where an algebraic control u‫ )כ‬andpoints. then predict the perturbed this method, we assumePMPC a nominal trajectorycontrol (with state x‫ כ‬and ‫ )כ‬and then by points. This is done in equationand (2) and (4). For this PMPC method,iswe assume a control nominal trajectory (with x‫ כ‬and control predict the perturbed a predictive method where an state algebraic solution isuformulated applying GPM. Inas shown collocation points. This is as shown in collocation points. This is done shown in equation equation (2) (2) and (4). (4). For For convenience, convenience, we we redefine redefine is formulated by applying GPM. method, we [ Das ]N time ] collocation This isD]done done as trajectory (xpp = x‫ כ‬+ solution δx) by applying the nominal control u‫כ‬Intothis the perturbed xDpp. [[ The N linear 1 Nshown in equation (2) and (4). For convenience, we redefine following N+1 convenience, matrices following NNpoints. ××system N+1 matrices we redefine the following N × N+1 matrices (15) ‫כ‬ ‫כ‬ trajectory (x p = x + assume δx) by applying the nominal control u to the perturbed system x . The linear time † † † a nominal trajectory (with state u*) [[ D [ Dtime ]N 1 ] ‫ כ‬+ δx) aby ‫כ‬x* N perturbed N trajectory (x we = xassume applying the nominal toand the control perturbed system xDp. The control u‫ )כ‬N then matrices predict the]linear this method, nominal trajectory (with control state x‫ כ‬uand following × Nand × N+1 following × N+1 N+1 matrices matrices variant perturbation model is then defined follows. trajectory (xpfollowing D ]N] N ] ] and then predict theasperturbed = x* + δx)Nby D  [[[[DD]]NN11 [ D [D N N (15) variant perturbation model is then defined as follows. † the following set of algebraic equations. (15) p †transformed TPBVP then be into These include additional D [[DD]†linear [ time D ]†N 1 ] applying the defined nominal control u*control to the perturbed system xp.can †]N  N variant perturbation is then follows. trajectory (xp = x‫ כ‬+model δx) by applying the asnominal u‫ כ‬to theThe perturbed system xD .†The [[ [ D ]  [[ D ] [ D ] ] D [[ D ] [ D ] ] N  1 N  N  N ]N 1 D  [[ D]NN11N  N[ D]NN(6)   x A(t ) x  B (t ) u N The defined †† †† †† (6)   xlinear A(ttime ) x variant B (t ) uperturbation model is then constraints to keep theD boundary state and costate variables under the constraints of dynamic equation via [[ D [ D † †]]N †] D  [[ D [ D ]]NN11]]]following  N N  N into The TPBVP can then be transformed the set of algebraic equations. These include additional D  [[ D ] [ D  (6)   x A ( t )  x  B ( t )  u variant perturbation model is then defined as follows. N  N N 1 as follows. The TPBVP can then be transformed into the following The TPBVP can then be transformed into the following set of algebraic equations. These include additi Where the Jacobean matrices are defined as follows (note the superscript ‘p’). Gauss quadrature rule. Where the Jacobean matrices are defined as follows (note the superscript ‘p’). set of algebraic equations. These include additional constraints to the state into and variablesset under the constraints of dynamic equation via The can then be transformed the algebraic equations. include additi The TPBVP cankeep then beboundary transformed into the following following set of algebraic equations. These include additi (6) costate  x Aare (t )defined x  B (tas ) follows u Where the Jacobean  matrices (notep the superscript ‘p’). (6) The TPBVP TPBVP then transformed into following set of ofunder algebraic equations. These These include additi tocan keep thebeboundary andthe costate variables the constraints of dynamic equation f ( x p (t ), u p (t )) f ( x (t ), u p (t )) constraints constraints tostate keep the boundary state and costate s s cc p p p (7)  A(t ) f ( x p (t ), u  , B ( t ) , A  R , B  R (t )) f ( x p (t u), u p (t )) constraints Gausssquadrature s c  c rule. boundary state and costate variables under the constraints of dynamic equation pJacobean p to keep the state and costate under the of variables under the constraints of dynamic via (7) Where as,constraints follows (note  A (t ) are ,matrices B (t ) (note A  R‘p’). , Bto  Where the Jacobean matrices the the f ( xdefined (tx), u  (as t )) follows are f ( x defined (t ), usuperscript (t )) constraints toRkeep keep the the boundary boundary state and costate variables variables underequation the constraints constraints of dynamic dynamic equation equation s s Gauss x  u (7)  A(t ) , B (t ) , A  Rquadrature , B  R cc rule. Gauss quadrature rule. the superscript‘p’). x u p Gauss quadrature The optimal control problem can be formed to determine the control deviation δu (xrule. , t), such that equation p quadrature rule. Gauss quadrature p f ( x p (t ), u p (t )) to determine f ( x p (the t ), ucontrol (t )) Gauss c rule. The optimal control problem (7)  A(t ) can be formed  , B (t ) , Adeviation  R s s , B δu R c(x p, t), such that equation The optimal control problem can bexformed to determine the u control deviation δu (x , t), such that equation (6) is satisfied while minimizing the functional cost 6 (6) is satisfied while minimizing the functional cost DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.64 (6) isoptimal satisfied while problem minimizing The control canthe be functional formed to cost determine the control deviation δu 66 (xp, t), such that equation 1 T 1 t0 p T T p p p J (8) 1  xT (t f ) Pf  x(t f )  1 ttf0 [( x p   x)T Q( x p   x)  (u p   u)T (t ) R(u p   u)]dt 6 21  x Tthe J Pf  x(t f )  21cost (t f )functional t0 [( x   x) Q( x   x)  (u   u ) (t ) R(u   u )]dt (8) (6) is satisfied while minimizing T T p p p p  t J (8) 2  x (t f ) Pf  x(t f )  2 tf [( x   x) Q( x   x)  (u   u) (t ) R(u   u)]dt 2 2 f 6 The Hamiltonian of the system be written as1 t0 p 1 can T T T p p p J   x t P  x t  x   x Q x   x  u   u t R u   u dt ( ) ( ) [( ) ( ) ( ) ( ) ( )] The Hamiltonian of the system can f be fwritten as  66 (8) f 6 The Hamiltonian of the system 2 can be written as 2 tf t0

T

f

f

f

T

T

tf

0

f

T

(64~76)14-074.indd 66 1

p

T

p

p

T

p

T

T

T

2015-03-30 오후 3:43:28

0 f 0 f 0 11 f 1 T  AT12   AT 1N  I  2 2A1  I 1 t f2  t0 T N 2 N † 2 †  †2  S 0   D D A D2†N t f  t0 2T 21 22 † †  †  † 2 A2 D† S 00 D11†  t f Dt021 DD22 D2D A1T N 1( N 1)  12 1N  2 2    0  t f  t0 T  †  †   0 †  † S  0   D21 D A D D  t t 22 2 2N † f 2( N01)   2 D†  DNN  t f  t0 ANTT DN††1 N† 2  0 †   DNN  2  AN  DN 1  DN 2  00 0  0 0 0 t f  t0 T 0 2† † † † 0 0   D D D AN 0DN ( N 1)  0 0 N1 Nfor 2 NN 0  Tawfiqur Rahman Pseudospectral Model Predictive Control Exo-atmospheric Guidance 2  

0 



1( N 1)   D2(†† N 1)  D2( N 1)      (22) DN†† ( N 1)   DN ( N 1)  I ( N  2)( N  2)  I  ( N  2)( N  2)

0 0 0 t f  t00 t f  t0  I ( N  2)( N  2)  N p N  t  t Qx p  k f k   t f  t0 B1u1p   2 B1u1 2 0 Qxkp  N N k 1 k  N t f ttf t t f  t0 0 tt0  t f  t20   N 2 p 1 T 1 T p p p 0 f  1 T p D x( tk )x  B(ktu) (tk B ( Ak (A )) u (t )) (tB)k R BBkR (Btk ) Dklklklxx(xt((ktt)kk))    D k   t f  tB01u1 p    k  k 1 tQx  f  t0 2 22 ( Akk x(tkk )  BkkR Bkk  (tkk )  Bkku (tkk )) l 0   t2f  t0 B2 u2p   k 1 2 t f t0 Qx1pp  ll00 N B2 u2 t f  t0 t 2  Qx1   t f  t20  t f  t0  NN 2 B2u2p  ; Dkl  (tk )   ttff (tQ x(tk )  AkT  (tTTk )  Qk x p (tk )) pp Qx1p 2  K 00     ((ttkk ))  ((ttkk )) Q )) Dkl  Q Qkkxx ((ttkk )) K x   2  t t  ((Q  xx((ttkk ))  AAkk   2 2  l  0 Dkl f 0 p  K  Qx 2   ; K x  (16) (16)    t t 2    t  t 2 ll00 (23) f 0 K K  ; (23) N p    x  t  t (16) t t f 0 Qx2p 2  Qx2   t tf t t0 BN(16)   xN 1   x0  f tt f 0 NNk ( Ak x(tk )  Bk R 1 BkT (1tk )T Bk u p (tk )) p u Npp  t t 0 2 f 0 2 0 f p      2 B uB u   N xxNN11   xt f  t20f k 01   ( Ak  xx((ttkk ))  BBkpkRR1 BBkTk  ((ttkk ))  BBkkuup ((ttkk )) ))   1  (tk )  Bk u (tk )) Dkl  x(tk)  x00  ( A2k N x( t )  Bkk (RAk BkT   N 2 2 N NN N      t 2f  t02 kkk11 k  t t l 0 p T  t t N N t f t 0  p      N 1  t   (Q  x(tk )  Ak  (tk )  Qk x (tk ))  t f  t0 tpf  t0 Qx p  f t 0 t B N 0 p kuk  t f t2ft0 tkt01 NNk k   f 0 p k k u  2 Bk uB k11   QxNf 2  0 QxNNp   (tk )    f (Q0 x(tk )) Qkk xxpp((ttkk )) D0kl ((ttkk )) Q   k 2 ) k ((AQQkT (txxk ()(ttkkQ))kx pAA(tkTkkT))   2  k  k k   NN11  2 l 0 0 2  2 kk11 k 2  k 1   (16) 2 N f  t0 1 form T p These algebraic equations be texpressed in the matrix as xN 1can Ak x t B R B t B u t These  xalgebraic  (  ( )  ( ) ( ))    can be expressed  equations 0 t t k k k k k k k in the matrix x x   2 ( A kx1 (t )  B R B  (t )  B u (t )) x(t )   D can zz     x  2 expressed These algebraic algebraic equations equations be expressed in the matrix form as N form as These can be in the matrix form as t t    (17) S  z  K  tf  t 0  (Q  x(t )  AT  (t )  Q x p (t ))   z     0 k k N 1 k k k k    ( ) ( ( ) ( ) ( )) D t   Q x t  A t  Q x t  22 k 1 x1   0   xS1  (17)δλ. The elements (24) (17)    0   Kt consisting of (17) SSxvector zz  K Where, ∆z is the column of the state and costate deviations, δx(16) and t x (24) x   1  ;    x(t matrix )  B R Bform  (t as )  B u (t ))  ( Ak These algebraic equations can xbe expressed inthe  x   2x   1   10  2 2     xThe ; S     1  t t x2  and K∆z matrices defined Where, ∆z is the theare column vector consisting of the state and costate deviations, deviations, δx and and δλ. elements of   t ) of A  Q x (t ))and as (Q  x(column (t ) vector Where, isthe consisting of the state   Where, is column vector consisting the state costate δx elements of   (17)δλ. S  z  K  ∆z  xThe  ; S N 1  ( 2 1  ( N 1)1  xN  2)1   N  and costate deviations, δx and δλ. The elements of S and K xx x  xN 1 ( N 1)1 N 1 ( N  2)1 Where, ∆zalgebraic is are the column vector consisting of matrix the state and These equations can be expressed in the form asx costate deviations, δx and δλ. The elements of S    S S K and K matrices defined as x   and K  matrices are defined asare defined N 1  ( (16)  (18)  N 1as  ( Nfollows. S  N 1)1can be reorganized  2)1 From the above definitions, equation matrices asK    ; x S   (17) S z  asSK K  and K matrices are defined Fromequation the above definitions, equation (16) can be From the above definitions, (16) can be reorganized as follows. xx x xx  K  S xx  equation  S x  (16) K Where, ∆z is the column vector deviations, δx and δλ. The elements S xx  ;the state andx KcostateK  SSxxconsisting SSx of x  be reorganized as follows. From theof above definitions, can x  reorganized as follows. x      (18) (25)          S K ; SS  S K     (18) t  t (18)  x   K   f  0 S as   (18)  S x  D1N 0  S   K  and K matrices are defined SSx D 11xS  SS;   A1K  K  D12 K  S S K        S 2 xx x x    S xx  x   S x    K x  S  S  K  t f  t0  Sobtained S the Krelation x      (25) The costate terms can be from above S  K  D ;D  (18) x                as shown. 0 A D   2 2N   S tf  tt21t0 f  tt0  K  22   S D 2   D1N  0  S x   S   K  f A1 0 A D12  11D   D D 0  2 xx D  A D D 0    11 1 12 1 N  1 12 1N 0  (19) S   t11 t  22      t f  t    0 The obtained from the above8relation as shown. D  terms can be  D  D21 A 0t costate DD22  A2 t  D20N t  The costate canabove be obtained the above relation 2 t 0 costate terms from the relationfrom as shown.   2 D t ff  t00  D  f The  can be obtained terms DND  A t t D D DN22NN 0  2  D AA22 0   NN 1 21 00 (19) D 22 N D  S xx    D  D21 0 A 22 2  (19)  as shown. 2 22    8  t f 0 t0 t f  t0 tf  t0  S xx    t f  t0   (19)     (19) D D    8 (19) SSxx   A A   00   N 1 N 2  D   NAN 0 AN  1  1 1 2 2t  NN t   2    x  2 2  ( N 1)( N 1)   D D A 0  2D  1   tt  S    S xx    t f  t0 2 S Kx     t f  t0 t f t0 xx ff  tt00 x    AN D 1   A2 D DA1 t 2t Dt t N   t1  t2 D      1:N       N 1   x f (26) DNN AANN 00 NN       21  A NN11 A2  NN22 A ( N 1)( N 2 1)   S K S      2 2 t f  t0 2 2         x 1:N   x   S x N 1  1 T   N B R B 0 0 0 0       1 ttt2ff  tt001 tt f  tt0 tt  tt   ff 000AANN 11  A  f  0 AA22 0   tt f t 0    T1  R 1 BA 00NN 11 B RB2 0 22t 0  2    1B 11 00 2 t  N1) 1) f 2 0 20 B R 1 BT   2 0  ((N0N1)1)((NThe  0 2 2 cancontrol then beusing usedrelation to get the The obtained costate can then costate be usedvector to get the of equation (10). 2 2  vectorobtained t t t t      1 T 0 0 0 0 Bf R 0BB 2 00 0 0 R B   2 2 2 control using relation of equation (10).  x   2 SS 00 0   0       ttff  tt00  S x  0 0 0 (20) 0  0   B11RR11BB1T1T  t  t0 t t  3.3 Implementation of Pseudospectral Model Predictive Control B 0 00  f 0 00   1 T  B B Rt B 0 20 0 0  0  t  0 0 B R f 0   2 T 2 N N  BN R 1 B 0 0   3.3  2 N  0 t  t 0 Implementation of Pseudospectral Model Pre  t t t t 0 ttftf  tt00 B R11 2BTT 0 t  t    t Bft R tB0 00 1 2 T tB R t Bt f   0 t2  2B 0 0 R B  0 0(20) f 1 2R B 1 T0 T tB 2 t0   0 0 f f f 0 0   dictive Control T T T      1 1 1 B12 2 B2 R B2B BN 0 needs a reference or starting profile in order to get a predicted trajectory. This in 2  00 11 B1 RB1BR BNN RPMPC BN B 0 N Rmethod  2B 2R 2 N  1 2 22 2  ( N 1)( N  2) (20) 2 2 2 2    ( N 1)( N  2) (20) xx  0 S     S 0 t  t 00  t t  t t   f 0 f 0 f 0 control profile may bePMPC needs optimization a reference or or any starting obtained from off-line other profile method. But, the presented me Q 2 Q   N Q 0  method 1 t t  t ff  t00 2 2 2 11in TT order  0 0 0 B R B   00  to get a predicted trajectory. This initial control 0 0 0 BNNR BNN   t f  t0   to be strictly optimal. A rough constant control profile works effective 22 the initial profile does  0 not need profile may t t 0 t t   t  t Q 0  be obtained from off-line optimization or any t t    Qtt f  Q1 T  tt f  Qtt0 0  1 T   tt0 2 t t  f 0   T 1 f 0 B f 0 B R1 BT f 0 B R1 BT   1 BT  2 2 2 R 0 0       0 11 But, the presented method does not need B1t1R tB11 22 B2R B22   NN BNNRother BNN method. 0   2 0 20 running the guidance law. For further board implementation, the following steps are performed: (21) 1)((NN2) 2) on 22 0 7f 0Q  22 0 0 S  x  t  t Q ((NN1) (20) 0 2 (21) (20)  the initial profile to be strictly optimal. A rough constant 2 2    t t      control profile works effectively in running the guidance 0  0 0  Q  S   (21) 2 7  1. Through predictive integration, the terminal error δψ is found from the reference control.  t f   t0  0   law. For further on board implementation, the following   Q 0 7  0   2  t t  steps are performed:  0 0 Q  0  N

kl

l 0

f

k

N

kl

l 0

0

f

k

N 1

T k

k

k

k 1

f

k

k

k

k

k 1

k

xx

x

x

x





p

k

T k

p

k

k

p

k

k

1

k

k

1N

12

f

21

p

k

0

11

k

k

T k

1

k

N

0

f

T k

1

k

N

0

0

0

k

0

f

N 1

k

0

22

2N

2

xx

N1

f

N2

f

0

1

1

f

0

2

0

f

N

2

0

N

0

N

( N 1)( N 1)

T 1

1

1

f

NN

f

0

1

2

T 2

x

f

f

0

1

1

f

1

f

T 1

f

0

1

2

0

1

T 2

N

f

N

2

f

f

0

T N

1

N

N

1

T N

( N 1)( N  2)

0

0

f

x

  

0

2

0

0

0

0

f

0

0

0

 2 0



Pf 

0 Pf 

2. If the terminal errors are within the desired limits i.e., |δψ| < |δψ|tolerance the reference contr

( N  2)( N 1)

1. Through predictive integration, the terminal error δψ is

( N  2)( N 1)

applied t t  for the rest of the flight. t f tt0f T t0 T t f  t0 Tt f  t0 T t f  t0 T I found from the reference control. 1 A1 A12 A2 2  A2N  AN N fI  0 ANT I  1  I  2 2 2 2 2 2    2. If the terminal errors are within the desired limits i.e., t f  tt0 T t   † †  D1†N D1(† N †1)  0 0 D11D† 2 f A1 0 AT D12 3. If terminal not within the desired limits i.e., |δψ| > |δψ|tolerance the guidance la D12†  D1N D1(† the 11 1 N 1)  |δψ|errors < |δψ|are   tolerance the reference control is applied for the 2 t f  t0 T    † † † † 77 (22) S  0  D21 D22 A2 t  D D 2N 2( N 1)  (22)†  rest of the flight.  f  t0 † used uk (complete control profile). S  0 D21 D2†  A2T  D2†N  D2( N 1)to  calculate (22)   22     0 2 3.  If the terminal errors are not within the desired limits    f  t0  D†  t   0 0 DN† 1  DN† 2 ANT DN† ( N1)  NN i.e., |δψ| > |δψ|tolerance the guidance law is used to calculate 2   the new 4. From control profile uk the current control command u0 is passed on to the veh  t t  f  0 0 D† 0 0  0 0  D† 0 D †  I  ( N  2)AT( N  2)D †    0t f

N1

N2

NN

2

N

 t0 t f  t00 0 0 0  B1u1p  Qxkp     k 2 2    k 1   t f  t0   t f  t0  p Qx1p N  t f  t0 t2f Bt20u2 p     2 B1u1  k Qxkp5.      2  K x  K   t f  t0 ;  2 pk 1     Qx2   t t   2   tf  t0    f t f 0BtN0u Np p  p B u Qx1      2 2 2   2  N    2 t f  t0    p  t f  t0 Qx p   B u k k k   ; K x  K N k 1   tf  t0 2 p 2    N



 t t f 0  BN u Np x  2 z    (64~76)14-074.indd 67 N t f  t0

p

   

   

2

 t t

Qx2

N ( N 1)

uk (complete control profile).

 control  I  ( Nsystem.  2)( N  2)

The control profile uk is used in the predictive integration, in order to check the toleranc http://ijass.org (23)67 projected terminal error. (23)

 If the projected terminal error is within limits, this profile replaces the nominal profile 6.   

supplies subsequent control commands.

2015-03-30 오후 3:43:29

Int’l J. of Aeronautical & Space Sci. 16(1), 64–76 (2015)

4. Dynamics of Vacuum Flight

4. From the new control profile uk the current control command u0 is passed on to the vehicle control system. 5..  The control profile uk is used in the predictive integration, in order to check the tolerance of projected terminal error. 6.. If the projected terminal error is within limits, this profile replaces the nominal profile and supplies subsequent control commands.

Exo-atmospheric flight ensues after the boundary of dense atmosphere where the guidance system works in a vacuum surrounding. In the exo-atmosphere, the density is significantly low, as to make any aerodynamic effects negligible. Fig. 2 shows the diminishing trend of atmospheric density and air pressure with increasing altitude. In the figure, the dynamic pressure encountered by a point mass vehicle at speeds ranging from 5,000 to 10,000 m/sec is also shown. The plots show negligible dynamic pressure above 100 km altitude where the pressure is around 27~28 kg/(m.sec2) for the specified velocity ranges. Considering the atmospheric condition, vacuum flight can be modelled by completely omitting the aerodynamic effects. The flight mechanical equations can, therefore, be stated as:

In the above calculation for control profile, equation (17) is solved for which the complete code is written and executed in Matlab. The selected number of collocation node was 10 in this collocation type method. Typical collocation methods are time intensive and generally require higher number of nodes for accuracy. However, the proposed method is a predictive method; as such, it is computationally faster and require lesser number of collocation points. The implementation process is shown in Fig. 1.

Fig. 1 Implementation steps of Pseudospectral MPC. 4. Dynamics of Vacuum Flight Exo-atmospheric flight ensues after the boundary of dense atmosphere where the guidance system works in a vacuum surrounding. In the exo-atmosphere, the density is significantly low, as to make any aerodynamic effects negligible. Fig. 2 shows the diminishing trend of atmospheric density and air pressure with increasing altitude. In the figure, the dynamic pressure encountered by a point mass vehicle at speeds ranging from 5,000 to 10,000 m/sec is also shown. The plots show negligible dynamic pressure above 100 Fig. 1.km Implementation stepsthe of Pseudospectral MPC. altitude where pressure is around

27 ~ 28 kg/(m.sec2) for the specified velocity ranges.

Fig. 1 Implementation steps of Pseudospectral MPC. 150

5000 m/s 6000 m/s 7000 m/s 8000 m/s 9000 m/s 10000 m/s

Altitude (km)

4. Dynamics of Vacuum Flight

100 after the boundary of dense atmosphere where the guidance system Exo-atmospheric flight ensues

works in a vacuum surrounding. In the exo-atmosphere, the density is significantly low, as to make any 50 aerodynamic effects negligible. Fig. density and0.1air pressure 0 21 shows 2 0 the diminishing 1 2 trend of 3 atmospheric 4 5 0 0.05 -3

x 10

Air Density (kg/m3)

4

Dynamic Pressure (Kg/m.sec2)

x 10

Air Pressure (kPa)

with increasing altitude. In the figure, the dynamic pressure encountered by a point mass vehicle at speeds Fig. 2. Reduction of atmospheric effect in the exo-atmosphere.

ranging from 5,000 to 10,000 m/sec is also shown. The plots show negligible dynamic pressure above 100 Fig. 2 Reduction of atmospheric effect in the exo-atmosphere.

km altitude where the pressure is around 27 ~ 28 kg/(m.sec2) for the specified velocity ranges. DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.64

Considering the atmospheric condition, vacuum68flight can be modelled by completely omitting the

(64~76)14-074.indd 68

de (km)

150

100

10

5000 m/s 6000 m/s 7000 m/s 8000 m/s 9000 m/s 10000 m/s

2015-03-30 오후 3:43:30

T sin  g v  cos   cos  mv v r T cos  sin  v    cos  cos  tan  mv 2 cos  r

(27)

 

Fig. 4 Launch vehicle model Titan II GLV.

In the above equations, the effect of Earth’s rotation is omitted. Fig. 3 shows the forces acting on a vehicle

Fig. 4 Launch vehicle model Titan II GLV. Rahman Pseudospectral Table Aerodynamic properties coefficient of lift for Titan II [21]. Model Predictive for– Exo-atmospheric Guidance outside the atmosphere, which are thrust and gravitation.Tawfiqur The conventions used for angle of1 attack α andControl aerodynamic effects. The flight mechanical equations can, therefore, be stated as: Table 1 Aerodynamic properties – coefficient of lift for Titan II [21].

pitch angle θp are also shown in the figure. T cos  cos   v

of Machproperties number – coefficient Values Table Range 1. Aerodynamic of liftof forCTitan II [21]. Lα

 g sin 

m

0 ≤ Ma ≤ 0.25 2.8 Range of Mach number Values of CLα 0.25 ≤ Ma ≤(27) 1.1 (27) 2.8 + 0.447 ˣ (Ma – 0.25) 0 ≤ Ma ≤ 0.25 2.8 Fig. 4 Launch vehicle model Titan II GLV. 1.1 ≤ Ma ≤ 1.6 3.18 - 0.660 ˣ (Ma – 1.1) 0.25 ≤ Ma ≤ 1.1 2.8 + 0.447 ˣ (Ma – 0.25) 1.6 ≤ Ma ≤ 3.6 2.85 + 0.350 ˣ (Ma – 1.6) Table 1 Aerodynamic of -lift for Titan 1.1 ≤properties Ma ≤ 1.6 – coefficient 3.18 0.660 ˣ (MaII–[21]. 1.1) Ma ≥ 3.6 3.55 theeffect above the effect of Earth’s ≤ 3.6 In the above equations,Inthe of equations, Earth’s rotation is omitted. Fig. rotation 3 showsisthe forces acting1.6 on≤aMa vehicle 2.85 + 0.350 ˣ (Ma – 1.6) omitted. Fig. 3 shows the forces acting on a vehicle outside Range of Mach number Values of CLα 3.55 Ma ≥ 3.6

T sin  g v    cos   cos  mv v r T cos  sin  v    cos  cos  tan  mv 2 cos  r

outside the atmosphere, which are thrust andare gravitation. Thegravitation. conventionsThe used for angle of attack α and the atmosphere, which thrust and usedthefor angle of attack α and pitchonly). angle θp are Fig. 3 angle Forcesθacting onconventions a flight vehicle exo-atmosphere (Longitudinal pitch are also shown in theinfigure. p

0 ≤ Ma ≤ 0.25 2.8 Table 2 Aerodynamic properties - coefficient of drag for Titan II [21]. Table≤2.Ma Aerodynamic properties - coefficient drag for Titan II [21]. 0.25 ≤ 1.1 2.8 + 0.447 ˣ (Ma of – 0.25)

also shown in the figure. Aerodynamic properties - coefficient of drag for Titan II [21]. 1.1 Table ≤ Ma ≤2 1.6 3.18 - 0.660 ˣ (Ma – 1.1) Because of the absence of atmosphere, flight in exoof Mach number Values of CDα Because of the absence of atmosphere, flight in exo-atmosphere is considered Range to have no path 1.6 ≤ Ma ≤ 3.6 atmosphere is considered to have no path constraints, and 2.85 + 0.350 ˣ (Ma – 1.6) 0 ≤ Ma ≤ 0.8 0.29 only physical constraint of control parameter is considered. constraints, and only physical constraint of control parameter is considered. Moreover, flightnumber is of Mach MaRange ≥ 3.6the 3.55 Values of CDα aerodynamic effects. TheMoreover, flight mechanical equations can, therefore, be stated as: the flight is constrained by end constraints which 0.8 ≤ Ma ≤ 1.068 Ma – 0.51 0 ≤ Ma ≤ 0.8 0.29 constrained by end constraints whichas: are represented as: are represented T cos  cos  ≥ 1.068 0.091 Ma -1 0.8Ma ≤ Ma ≤ 1.068 Ma+ –0.5 0.51   g sin  d v Table 2 Aerodynamic properties - coefficient of drag for Titan II [21]. rf  m r d  0,  f    0,  f d  0 (28) (28) T sin  d g v  v d  0, Ma(27) ≥ 1.068 0.091 + 0.5 Ma -1 f  d  0    f    0,cos  v f cos  mv v r atop a launch vehicle, release fromof launch vehicle, re-entry, aerodynamic model the Range ofSimplified Mach number Values of C Dα hypersonic vehicle is represented through t cos  sin  T v Fig. 3 Forces acting on a flight vehiclein the exo-atmosphere (Longitudinal only).  cos cos tan     and finally, reaching the terminus. We assume that Titan II 5. Vehicle Models mv 2 cos  r 0 ≤Simplified Ma ≤(GLV) 0.8C 0.29  C  C  aerodynamic model of the hypersonic vehicle is represented through t Gemini is used for launch. This is a liquid propelled L L0 L1 2 5. Vehicle Models vehicle aDtotal of 154,200 which carries a payload C  Because of the atmosphere, flight invehicle exo-atmosphere is forces considered to noCmass 0.8 ≤ Ma ≤have Ma –kg, 0.51 0  kC L Dpath Guidance method presented here is for a hypersonic promptthe global delivery mission. The In the above equations, theabsence effect ofofEarth’s rotation is omitted. Fig. in 3 shows acting onwith a1.068 vehicle  CLkg CL121]. Detailed mass breakdown and L 0 [20, of maximumC 3,580 -1 2 constraints, and only physical constraint ofofcontrol parameter considered. Moreover, the flight iskC Ma ≥attack 1.068 0.091 +in 0.5Fig. C  C Guidance method presented here is forvehicle a ishypersonic = from -0.01, CL1 = 0.018, CMa 0.01, and k = 3.4. Where, C D 0 are Lshown complete mission scenario involves launching the hypersonic atop launch vehicle, release propulsion properties aerodynamic outside the atmosphere, which are thrust and gravitation. The conventions used afor angle of αL0Dand D04.=The vehicle in prompt global delivery mission. The complete properties are tabulated in Table 1 and 2 [21]. constrained by end constraints which are represented as: We assume that Titan II Gemini = -0.01, CL1 = 0.018, = 0.01, and vehicle k = 3.4. Where, re-entry, and finally, the terminus. (GLV) is used pitchlaunch angle θvehicle, shown in the figure.reaching L0 D0 hypersonic p are also mission scenario involves launching of the hypersonic vehicle 6. Exo C Atmospheric Ascent Guidance Simplified aerodynamic model of C the 12 Simplified aerodynamic model of the hypersonic vehicle is represented through the follow d d d is represented through the following relations. rf  r  0,  f    0,  f   0

for launch. This is a liquid propelled vehicle with a total mass of 154,200 kg, which carries a payload of (28) 6. Exo Atmospheric Ascent Guidance  f   d  0, v f  v d  0, f  d  0 C  CL 0flight  CL1profile considered here requires strict positioning 12 of the vehicle at e L The maximum 3,580 kg [20, 21]. Detailed mass breakdown and propulsion properties are shown in Fig. 4. The (29) 2

5. Vehicle Models aerodynamic properties are tabulated in Table 1 and 2 [21].

C  CD 0  kCL D Therefore, we need to ensure very accurate end positions which calls for a robust guida

CL1deviation = 0.018, 0.01, k = 3.4. without significant performan Where, CL0 =C-0.01, Where, autonomously C CD0 =Cand 0.01, and and k = vehicle 3.4. D0 = steer L0 = -0.01, L1 = 0.018, isolate the Guidance method presented here is for a hypersonic vehicle in prompt global delivery mission. The a reference 11 6. Exoevaluation, Atmospheric Ascenttrajectory Guidanceconsisting of two boost and one coast phase is genera complete mission scenario involves launching of the hypersonic vehicle atop a launch6. vehicle, release from Exo Atmospheric Ascent Guidance12

quadratic programming (SQP). For this Matlab command ‘fmincon’ was used. Fig. Fig. 3. Forces acting on a flight vehicle in the exo-atmosphere (LongiFig. 3launch Forces acting on re-entry, a flight vehicle in the exo-atmosphere tudinal only). vehicle, and finally, reaching the (Longitudinal terminus. Weonly). assume that Titan II Gemini (GLV) is used

The flight profile considered here forrequires strict flight are listed in Table reference profile. The boundary conditions exo-atmospheric positioning of the vehicle at each phase of flight. Therefore, for launch. This is a liquid propelled vehicle with a total mass of 154,200 kg, which carries a payload of

Because of the absence of atmosphere, flight in exo-atmosphere is considered to have no path

Table 3 Boundary conditions for exo-atmospheric flight.

Table 3. Boundary for exo-atmospheric flight. maximum [20, 21].constraint Detailed mass breakdown and propulsion properties are shown in Fig. The constraints, and3,580 onlykgphysical of control parameter is considered. Moreover, the flight4.conditions is

aerodynamic tabulated in Table 1 and constrained by endproperties constraintsarewhich are represented as: 2 [21]. rf  r d  0,

 f   d  0,

 f d  0

 f    0,

d

f   0

d

v f  v  0,

11 d

5. Vehicle Models

Parameter

Initial condition

Altitude

118.64 km

144 km

-0.00043°

1.9858°

Longitude(28) Latitude

2.044°

2.0379°

Velocity

5418.45 m/sec

6733 m/sec

Flight path angle

9.4767°

3.3968°

Azimuth angle

-0.036473°

-0.107°

Vehicle mass

17428.82 kg

10900 kg

Guidance method presented here is for a hypersonic vehicle in prompt global delivery mission. The Fig. 4. Launch vehicle model Titan GLV. Fig. 4IILaunch vehicle model Titan II GLV.

Terminal condition

complete mission scenario involves launching of the hypersonic vehicle atop a launch vehicle, release from Table 1 Aerodynamic properties – coefficient of lift for Titan II [21].

aunch vehicle, re-entry, and finally, reaching the terminus. We assume that Titan II Gemini (GLV) is used 69 Range of Mach number of CLα for launch. This is a liquid propelled vehicle with a Values total mass of 154,200 kg, which carries a payload of 0 ≤ Ma ≤ 0.25

http://ijass.org

2.8

150

1.1 ≤ Ma ≤ 1.6

3.18 - 0.660 ˣ (Ma – 1.1)

aerodynamic properties are tabulated in Table 1 and 2 [21]. ≤ Ma (64~76)14-074.indd1.669

≤ 3.6

2.85 + 0.350 ˣ (Ma – 1.6)

de (Km)

maximum 3,580 kg [20, 21]. propulsion properties are shown in Fig. 4. The 0.25Detailed ≤ Ma ≤ 1.1mass breakdown 2.8 + 0.447and ˣ (Ma – 0.25) 100

Exo-atmospheric Flight

Coast

2015-03-30 오후 3:43:36

actuators or power supplies, or it may be due to software issues regarding the improper commands bei

sent to the actuators. These problems could manifest themselves by actuators failing to hard over (maximu

angle), failing in place, or failing to the null position. Typical actuator failures include the followings:

lock in place (LIP); 2) hard over failure (HOF); 3) float; and 4) loss of effectiveness (LOE). In the case Int’l J. of Aeronautical & Space Sci. 16(1), 64–76 (2015)

LIP failures, the actuator “freezes” at certain condition and does not respond to subsequent commands. HO

is characterized by the actuator moving to and remaining at the upper or lower position limit, regardless of the command. The speed of response is limited by the we need to ensure very accurate end positions which calls the command. The speed response limitedoccurs by the when actuator rate limit. Float failure occurs when actuator rateoflimit. Floatisfailure the actuator for a robust guidance method that can autonomously contributes zero moment to the control authority. Loss of isolate deviation and steer the vehicle without significant actuator contributes zero moment to the control authority. Loss of effectiveness is characterized by loweri effectiveness is characterized by lowering the actuator gain performance degradation. For evaluation, a reference the actuator gain with respect to nominal its nominal value.These Theseactuator actuator problems problems can be simulated by using t with respect to its value. trajectory consisting of two boost and one coast phase is can be simulated by using the following parameterizations generated using sequential quadratic programming (SQP). following parameterizations [23]. [23]. For this Matlab command ‘fmincon’ was used. Fig. 5 below shows the reference profile. The boundary conditions for for all t  0 No failure uc (t ) exo-atmospheric flight are listed in Table 3. k (t )u (t ) for all t  t LOEof flight. F  c ofc the vehicle at each phase The flight profile considered here requires strict positioning u (t )  0

for all t  tF

Float

for all t  t

HOF

 (t ) 6.1 Off nominal cases for all t  tmethod LIP, uthat Therefore, we need to ensure very accurate end positions which ucalls for a robust guidance can c F c = constant  u

or

u

(30)

(3

i max F  i min Launch vehicle ascent generally employs an open-loop autonomously isolate deviation and steer the vehicle without significant performance degradation. For method for endo atmospheric guidance and a closed-loop In the above parameterization, uc control, is the kcommanded In the above parameterization, uc is the commanded c is the ratio by which the actuator los law for vacuum flight.a Here, we consider theconsisting launch vehicle evaluation, reference trajectory of two boost and one coast phase is generated using control, kc is the ratio by which thesequential actuator loses using thrust vector for controlling flight. Therefore,effectiveness, the and tF is the moment the actuator problem starts. In the case of LIP problem, t effectiveness, and tF isfrom the when moment from when the actuator quadratic programming (SQP).from Forproblem this Matlab considered perturbations are resulting with command ‘fmincon’ was used. Fig. 5 below shows the problem starts. case LIP problem, thewill uc(t) will uc(t) will remain constant; andInforthe LOE, theofcommanded control lose its effectiveness defined by t the thrust vector control or the engine nozzle. remain constant; and for LOE, the commanded control will reference profile. The boundary conditions for exo-atmospheric flight are listed in Table 3. ratio kc. Generally, are multiple thrustbyvector nozzles for controlling lose its there effectiveness defined the ratio kc. Generally, therepitch, yaw and roll motion 6.1.1 Thrust vector control problem are multiple thrust vector nozzles for controlling pitch, yaw Table 3 Boundary conditions for exo-atmosphericlaunch flight.vehicle. Therefore, simulating the actuator problems require detailed modelling of the thrust vec Thrust vector control (TVC) problems [22-24] may be and roll motion of launch vehicle. Therefore, simulating the caused by hardware issues regarding the actuators or power 14 actuator problems require detailed modelling of the thrust supplies, or it may be due to software issues regarding the vector nozzles. In our evaluation, we only apply loss of Parameter Initial condition Terminal condition improper commands being sent to the actuators. These control effectiveness. nozzles. In our Altitude 118.64bykmactuators 144 km evaluation, we only apply loss of control effectiveness. problems could manifest themselves failing to hard over (maximum in place, or failing Longitude angle), failing -0.00043° 1.9858° 6.1.2damage Engine nozzle damage 6.1.2 Engine nozzle to the null position. Latitude Typical actuator 2.044° failures include 2.0379° the followings:Velocity 1) lock in place 5418.45 (LIP); 2)m/sec hard over failure 6733 m/sec Engine nozzle damage could occur during flight, Engine damageincould occur during flight, control resulting in theorloss of performance, control iss (HOF); 3) Flight float;path andangle 4) loss of effectiveness (LOE). 3.3968° In nozzle resulting the loss of performance, issues, both. 9.4767° the case of LIP failures, the actuator “freezes” at certain This can be represented by the following parameterization Azimuth angle -0.036473° -0.107° both. This can be represented by the following parameterization of thrust. condition and does not respond to subsequent commands. of thrust. Vehicle mass 17428.82 kg 10900 kg HOF is characterized by the actuator moving to and  T (t ) kT Tc (t ) for all t  t F (31) remaining at the upper or lower position limit, regardless

6.2

Simulation Results Gauss PMPC method for ascent guidance is evaluated through perturbed cases, with thrust and

150

Altitude (Km)

errors.Exo-atmospheric Guided trajectories forCoast thrust perturbation are compared to unguided trajectories, and the te Flight

100

errors are compared. The first set of results is for vector nozzle problem. Here, simulation results for of thrust effect loss are shown. Control and initial state perturbations are not included here. The

50

0 5

calculated from the guidanceEndo-atmospheric law is seen to effectively steer the vehicle on to the reference tra Flight 4

3

Effectiveness of the guidance law can be understood1 from Table 4. The terminal altitude error is redu 2

Latitude (deg)

-2

-1

0

about 10 times the unguided error in vehicle mass is maintained to less than 1 k -3 -3 terminal altitude. The 1 x 10 0

-4

-5

Longitude (deg)

Fig. 5. Ascent trajectory of launch vehicle.

Fig. 5 Ascent trajectory of launch vehicle. DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.64

70 13

(64~76)14-074.indd 70

2015-03-30 오후 3:43:37

Tawfiqur Rahman Pseudospectral Model Predictive Control for Exo-atmospheric Guidance

6.2 Simulation Results

simulation results for 6 cases of thrust effect loss are shown. Control and initial state perturbations are not included here. The control calculated from the guidance law is seen to effectively steer the vehicle on to the reference trajectory. Effectiveness of the guidance law can be understood from Table 4. The terminal altitude error is reduced by about 10 times the unguided terminal altitude. The error in vehicle

Gauss PMPC method for ascent guidance is evaluated through perturbed cases, with thrust and control errors. Guided trajectories for thrust perturbation are compared to unguided trajectories, and the terminal errors are compared. The first set of results is for vector nozzle problem. Here,

Longitude (deg)

Latitude (deg)

110 0 2.042

2.0382.036 0 Flight Path Angle (deg)

Flight Path Angle (deg)

2.036 0

10

10

10

8

8

6

6

4

4

2 0

2 0

10

10 4

x 10 10

2

Mass (Kg)

20 Time 20 Time

30 30

20 Time 20 Time

4

Mass (Kg)

Velocity (m/sec)

2.042.038

40 40

40

x 10 2 1.5

1.5

1

1 0.5 0 0.5 0

10 10

20 Time 20 Time

30 30

1

1

0

0

-1 0

-1 0 8000

10 10

40 40

20 Time 20 Time

30 30

40 40

80007000 70006000 60005000 0 0

10 10

20 Time 20 Time

30 30

40 40

0-0.05

-0.05 -0.1 -0.1-0.15

-0.2 40 -0.15 0

30 30

2

5000 0

Pitch Angle (deg)

Latitude (deg)

2.042 2.04

2

Velocity (m/sec)

120 1100

3

Azimuth Angle (deg)

130 120

Guided trajectory Unguided trajectory Nominal trajectory Guided trajectory Unguided trajectory 10 20 trajectory 30 40 Nominal Time 10 20 30 40 Time

Azimuth Angle (deg)

140 130

-0.2 0 Pitch Angle (deg)

Altitude (Km)

150 140

Longitude (deg)

3

Altitude (Km)

150

-16

10 10

-18 -20

-22 0

30

40

20 30 40 Time Commanded Control Nominal Control Commanded Control Nominal Control

-16 -18

-20 -22 0

20 Time

10 10

20 Time 20 Time

30 30

40 40

Fig. 6 Guided and unguided trajectory for thrust perturbation (unit of time is seconds). Fig. 6. Guided and unguided trajectory for thrust perturbation (unit of time is seconds).

Fig. 6 Guided and unguided trajectory for thrust perturbation (unit of time is seconds). Table 4 Terminal error for guided and unguided trajectory for thrust perturbation. Table 4. Terminal error for guided and unguided trajectory for thrust perturbation. Table 4 Terminal error for guided and unguided trajectory for thrust perturbation. |∆hf| (m)

T = 0.85 × T T = 0.90 × T T = ×0.95 T ×T Unguided T = 0.85 T = ×1.05 T = 0.90 T ×T T = ×1.10 T = 0.95 T ×T Unguided T = 1.05 T ×T T = ×1.15 T = 1.10 × T T = 0.85 × T T = 1.15 T ×T T = ×0.90 T = ×0.95 T ×T Guided T = 0.85 T = ×1.05 T = 0.90 T ×T T = ×1.10 T = 0.95 T ×T Guided T = 1.05 T ×T T = ×1.15 T = 1.10 × T T = 1.15 × T

|∆hf|862.2 (m)

584.01 296.76 862.2 306.73 584.01 623.93 296.76 306.73 952.15 623.93 87.4 952.15 58.86 87.429.72 58.8630.26 29.7261.04 30.2692.3 61.04 92.3

|∆ીf|

|∆૎f|

(deg)

|∆ીf|0.037

(deg)

0.025 0.0370.013 0.0250.012 0.0130.026 0.0120.039 0.026 9.08×10-5 0.039 6.17×10-5 -5 -5 3.15×10 9.08×10 -5 -5 3.28×10 6.17×10 -5 -5 6.65×10 3.15×10 -5 3.28×10 1.02×10-5 6.65×10-5

|∆૎ -5 f| 6.85×10 (deg) 4.62×10-5 -5 -5 2.33×10 6.85×10 -5 -5 2.38×10 4.62×10 -5 -5 4.82×10 2.33×10 -5 2.38×10 7.32×10-5 4.82×10-5 -7 1.68×10 -5 7.32×10 1.14×10-7 -8 -7 5.82×10 1.68×10 -8 -7 6.04×10 1.14×10 -7 -8 1.23×10 5.82×10 -8 6.04×10 1.88×10-7 1.23×10-7

1.02×10-5

1.88×10-7

(deg)

71

16

(m/sec)

|∆vf|

(deg)

|∆઻f|

|∆mf|

| |∆vf257.20

|∆઻f| 0.68

| |∆mf999.7

(m/sec)

173.88 88.21 257.20 90.89 173.88 184.63 88.21 90.89 281.42 184.63 0.58 281.420.39 0.58 0.20 0.39 0.22 0.20 0.45 0.22 0.70 0.45 0.70

(deg)

0.46 0.68 0.23 0.46 0.23 0.23 0.47 0.23 0.72 0.47 0.0017 0.720.0015 0.0009 0.0017 0.0013 0.0015 0.0031 0.0009 0.0013 0.0055 0.0031 0.0055

(kg)

(kg)

666.5 999.7333.2 333.23 666.5 666.47 333.2 333.23 999.6 666.47