Enhanced trajectory linearization control based advanced guidance and control for hypersonic reentry vehicle with multiple disturbances

Accepted Manuscript Enhanced trajectory linearization control based advanced guidance and control for hypersonic reentry vehicle with multiple disturbances

Shao Xingling, Wang Honglun, Zhang HuiPing

PII: DOI: Reference:

S1270-9638(15)00261-8 http://dx.doi.org/10.1016/j.ast.2015.09.003 AESCTE 3409

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

2 March 2015 10 July 2015 3 September 2015

Please cite this article in press as: X. Shao et al., Enhanced trajectory linearization control based advanced guidance and control for hypersonic reentry vehicle with multiple disturbances, Aerosp. Sci. Technol. (2015), http://dx.doi.org/10.1016/j.ast.2015.09.003

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Enhanced Trajectory Linearization Control based advanced guidance and control for Hypersonic Reentry Vehicle with multiple disturbances SHAO Xingling1,2*, WANG Honglun1,2, ZHANG HuiPing3 (1. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China 2. Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, China 100854, China) *Corresponding author. Tel.: +86-10-82317546.

3. Beijing Aerospace Automatic Control Institute, Beijing

E-mail address: [email protected]

Abstract In this paper, the guidance and control problem for hypersonic reentry vehicle (HRV) in the presence of control constraints and multiple disturbances is handled based on unified enhanced trajectory linearization control (TLC) framework under reference-tracking methodology. First, based on the nominal trajectory and open-loop command generated by Gauss pseudo-spectral method (GPM), a time-varying feedback guidance law with integral action is synthesized to stabilize the tracking error dynamics along the nominal trajectory under the framework of TLC. Second, to improve the robustness of attitude and angular rate loop, variations of various aerodynamic coefficients and external disturbances are considered as lumped uncertainties, reduced-order linear extended state observers (LESO) with given model information are constructed to estimate the lumped uncertainties in each loop, respectively. In addition, comparisons between the estimation efficiency of LESO and reduced-order LESO are carried out. Then augmented with the disturbance estimates and TLC control law, tracking errors of the rotational dynamics can be actively rejected without sacrificing nominal performances. More importantly, fewer control consumption and smooth transient performances are achieved by using nonlinear tracking differentiator (TD) in attitude loop. The stability of the resulting closed-loop system is well established based on Lyapunov stability theory. Finally, the effectiveness of the proposed advanced guidance and control strategy is verified through extensive simulations on the six-degree-of-freedom reentry flight. Keywords: hypersonic reentry vehicle(HRV) ; advanced guidance and control; trajectory linearization control(TLC); reduced-order LESO; smooth transient performance ;multiple disturbances

1. Introduction The advanced guidance and control (AG&C) technology for hypersonic reentry vehicle (HRV) with highly nonlinear dynamics and limits on control authority has received considerable attention due to its promising prospect in military applications. The practical implications of AG&C system for HRV cannot be overemphasized: with a well-designed AG&C system, HRV can fulfill the global strike mission within 2h for targets which are in the range of 9000 nautical miles [1]. Up to now, there’re two prevailing frameworks on dealing with the advanced guidance and control problem for HRV, which are denoted in the literature as predictor-corrector guidance[1-4], and reference-tracking guidance[5-8,11], respectively. For predictor-corrector guidance, the working principles are to fast design of three-degree-of-freedom reentry trajectory subject to all common inequality and equality constraints, and then robust attitude controller should be developed to closely track the guidance commands, as shown in Fig1.(a). The major difficulties of this approach are the burdensome and time-consuming 1

optimization calculation, the lack of effective and broadly applicable means to enforce inequality trajectory constraints, as well as the guaranteed convergence of the numerical process [4]. Major difficulties Real-time reentry trajectory 1 Burdensome optimization calculation with multiple constraints 2 Without inequality trajectory constraints considered satisfied 3 The lack of guaranteed convergence of numerical process

(a)

Robust attitude control design

Predictor-corrector guidance framework Our focus

Main challenges 1 The complicated path constraints 2 Unknown multiple disturbances 3 The AG&C design

Off-line feasible nominal trajectory with all the constraints satisfied

Prestored in the computer

Trajectory tracking law

Robust attitude control design AG&C design

(b)

Reference-trajectory guidance framework

The drawbacks earlier motivate us to investigate into reference-trajectory guidance methodology for a better solution. The reference-tracking guidance consists of two parts: off-line planning of a feasible nominal trajectory that satisfies all the constraints including heating rate and dynamic pressure, as well as aerodynamic load and limits on control authority, and then onboard tracking the nominal trajectory in the presence of severe off-nominal conditions during reentry, as shown in Fig1.(b). Consequently, in order to achieve stable and safe reentry flight, the complicated path constraints, unknown multiple disturbances induced by external disturbances and variations in aerodynamic parameters, together with the inherent nonlinear characteristics of HRV should be handled simultaneously, all of the above issues pose challenges in designing AG&C system from theoretically and practically. Among the existing literature on AG&C design for HRV, the trajectory tracking law and attitude control law are designed separately to simplify and reduce the complexity by using different methods. For the guidance law, many effective efforts have been made, such as feedback linearization [5], proportional navigation-based tracking approach [6], optimal guidance method [7] and state dependent riccati differential equations (SDRE)-based tracking law [8]. Inspired by Legendre pseudo-spectral method (LPM) based optimal feedback control law presented in [9-10], this approach is directly applied to reentry guidance scenario in [11]. It is worth pointing out that the tedious and time-intensive computational process for SDRE can be avoided in LPM-based method. However, LPM-based guidance methods still suffer from the drawback of insufficient capability in rejecting large and multiple uncertainties during reentry. For the aspects of attitude control, a variety of control schemes associated with HRV have been developed in literature, including adaptive nonlinear control[12], continuous sliding mode control with sliding mode disturbance observer [13,14], linear parameter-varying (LPV) approach [15] and trajectory linearization control(TLC) method [16,17]. Among these robust control methodologies, TLC method has been proved to be more effective in solving nonlinear tracking and disturbance attenuation problems. Particularly, it combines an open-loop nonlinear dynamic inversion and a linear time-varying (LTV) feedback stabilization in a novel way, nonlinear tracking and decoupling control by TLC is viewed as the ideal gain-scheduling controller designed at every point on the trajectory [17]. However, based on singular perturbation, theoretical

2

analysis regarding TLC is well established in [18], indicating that TLC can only achieve local exponential stability, and thus robustness against larger uncertainties cannot be ensured. To improve the robustness for TLC in the presence of large uncertainties, various modified TLC strategies have been explored in [19-21, 23-26]. The first solutions to enhance TLC by applying adaptive neural network (ANN) technique are reported in [19-21] A novel robust TLC method using model-assisted ESO is proposed in [24, 25] for general nonlinear uncertain system, where internal and external uncertainties are treated as lumped disturbances to be actively rejected. The composite controller developed in [24, 25] has remarkable superiorities over adaptive neural network in ease of parameter tuning and simpler control structure as well as

or improved robustness.

atmosphere density and variations in aerodynamic parameters are not well considered in the previous work. In this paper, we investigate the feasibility of performing desired trajectory tracking and attitude control by exploiting an enhanced TLC framework based AG&C system for HRV with multiple disturbances under control constraints. To be more specific, on the basis of the feasible nominal trajectory profiles generated by Gauss pseudo-spectral method (GPM), the basic TLC scheme with integral action is developed to suppress the tracking error in guidance loop to ensure the reentry flight with safety. Moreover, specified reduced-order LESO is proposed to estimate the lumped disturbance existed in the dynamics of HRV, and thus by augmenting the disturbance estimates with TLC method, enhanced attitude tracking performances can be ensured in spite of

The main

contributions in this paper are summarized as follows. AG&C strategy

GPM

the traditional ESO design principle novel specified reduced-order ESOs are constructed to estimate the lumped disturbance existed in the rotational dynamics of HRV without extending system order, and performance evaluations between

3

2. Mathematical model of HRV and Problem formulation Consider a 6DOF rigid hypersonic reentry vehicle (HRV) described by the following translational and rotational dynamics equations [27]: ­h V sin J ° °I V cos J sin F ° ( RE  h) cos T ° °T V cos J cos F ° ( RE  h) ° D ® 2 °V  m  g sin J  : ( RE  h) cos T (sin J cos T  cos J sin T cos F ) ° : 2 ( RE  h) L cos V g V ° (  ) cos J  2: cos T sin F  cos T (cos J cos T  sin J sin T cos F ) °J V mV V RE  h V ° ° L sin V V : 2 ( RE  h) sin T cos T sin F  cos J sin F tan T  2:(tan J cos T cos F  sin T )  °F V cos J V cos co J ( RE  h) mV ¯

L g cos V cos J ­   wx cos D tan E  wy sin D tan E  wz °D  mV V cos E V ccos E ° Z g sin V cos J °   wx sin D  wy cos D ®E  mV V V ° LL(tan (tan ( E  sin V tan J )  Z cos V tan J g cos V cos J tan E °   wx cos D sec E  wy sin D sec E °V mV V ¯ ­ ° wx ° ° ° ® wy ° ° ° wz ° ¯

(I y  I z ) Ix

wy wz 

Mx Ix

Iz

wy wx 

(2)

(3)

My (I z  I x ) wz wx  Iy Iy (I x  I y )

(1)

Mz Iz

And ªL º «D» « » «¬ Z »¼ ªM x º « » «M y » «M » ¬ z¼

ªcL ,0  cDL D  cLG z G z º « » 2 1 1 UV 2 S ref UV 2 S ref u «cD ,0  cDDD  cDD D 2 » , U U 0 e  kh , 2 2 « E » G G «¬cz E  cz x G x  cz y G y »¼ ª mxE E  mGx x G x  mxG y G y º ª mx º « » 1 « » 1 G UV 2 S ref l u « m y » UV 2 S ref l u «m yE E  mGy x G x  my y G y » 2 2 « D » «m » « mz D  mGz z G z » ¬ z¼ ¬ ¼ ª cL º « u «cD »» «¬cz »¼

h is the altitude, I and T are the longitude and latitude, V is the Earth-relative velocity, J is the flight-path angle of the Earth-relative velocity vector, and F is the heading angle of velocity vector, L , D and Z are the lift, drag and lateral force, g is the gravity ( g P ( RE  h) 2 with P and RE being the Earth gravity constant and the radius of Earth), : is the Earth angular speed, D , E , V are angle of attack, sideslip angle and bank angle respectively. wx , wy , wz are the angular rate of body-fixed reference frame of HRV. G x , G y , G z represent aileron, rudder, and elevator deflections, which are the control actuators of HRV. mx , m y , mz are the rolling, yawing and pitching

where

4

moment coefficients. cL , cz denote the lift and lateral force coefficients. dij (i 1, 2,3, j 1, 2,3) represent the unknown lumped disturbance which includes parameter variations and external disturbances. For simplicity, define

Θ

ªh º «I » , X « » 1 «¬T »¼

ªV º «J » , X « » 2 «¬ F »¼

ªD º «E » , X « » 3 «¬V »¼

ª wx º « » « wy » , U «w » ¬ z¼

ªG x º « » «G y » , di «G » ¬ z¼

ª di1 º « » « di 2 » (i 1..3), Q «¬ di 3 »¼

1 UV 2 2

Then we can rewrite Eq.(1- 3) in the following cascaded and feedback form as ­ °Θ Τ(X1 ) ® ° ¯ X1 = F1 (X1 , X2 , Θ) + d1

(4)

­° X2 = F2 (X2 ,Θ) + B2 (X2 )X3 + d 2 ® °¯ X3 = F3 (X3 ,Θ) + B3 (X3 , Θ)U + d3

(5)

where QS ref cL ,0 g cos V cos J ª º   « » cos E cos E mV V » ª F21 ( X 2 ,Θ) º « E « » QS ref cz E g sin V cos J « » F2 (X 2 ,Θ) « F22 ( X 2 ,Θ) » «  » mV V » «¬ F23 ( X 2 ,Θ) »¼ « « QS c (tan E  sin V tan J )  QS c E E cos V tan J g cos V cos J tan E » ref L ,0 ref z « »  mV V «¬ »¼ sin D tan E 1º ª B21 ( X 2 ) º ª  cos D tan E B 2 (X 2 ) «« B22 ( X 2 ) »» «« sin D cos D 0 »» «¬ B23 ( X 2 ) »¼ «¬cos D sec E  sin D sec E 0 »¼ ª (I y  I z ) m E E QSl º wy wz  x « » Ix » « Ix E « (I  I ) m E QSl » » , B3 (X3 , Θ) F3 (X3 ,Θ) « z x wz wx  y Iy « Iy » « » E  I I ( ) m QSl E « x y ww  z » y x «¬ I z »¼ Iz

ª QSlmGx x « « Ix G x « QSlmy « « Iy « « 0 ¬

QSlmGx y Ix QSlmGy y Iy 0

º » » » 0 » » QSlmGz z » » Iz ¼ 0

The main feature of translational dynamic presented by (4) is inherently non-affine due to the virtual control vector X 2 for translational dynamics does not appear linearly in the equations. In addition, system(4) is also underactuated with one virtual control vector X 2 and two output vectors

X1 and Θ , which further adds difficulty in designing guidance law. For the rotational dynamic presented by (5), it is evident that uncertainty dynamics exist both in the attitude and angular rate subsystems, the difference is that the disturbance d 2 satisfies the so-called mismatched condition, i.e.,

d 2 can affect the state vector X 2 directly rather than through the input channel , whereas the disturbance d 3 is matched, indicating that the disturbance exists in the same channel as that of the control input

U.

Next, to achieve reentry flight with sufficient stability and safety, it is crucial to build up the complex path constraints composed by heating rate, dynamic pressure and load factor, which can be formulated as [11]:

5

­q (h0  h1D  h2D 2  h3D 3 )C U NV M d qmax °° 2 ®Q 0.5 U V d Qmax ° L2  D 2 mg d nmax °¯n

(6)

The above inequalities represent heating rate at a stagnation point on the surface of HRV, dynamic pressure and load factor, respectively. The corresponding values of predefined coefficients such as

hi (i 1..3) can be found in [11]. Furthermore, consider the physical limitations on HRV, we give the following assumption. Assumption 1.

(a) For uncertainty dynamics in system (4) and (5), there exist unknown positive

constants d1 , d 2 , d3

such that

d r di d di (i 1..3, r dt r

0,1) , i.e., the disturbances and their

derivatives are all bounded. (b) Due to the physical limitations on actuators, thus there exists bounded set (G i , G i )(i 1, 2,3) for deflection angle U i , where i denotes the i-th component of control vector

U.

The class of disturbances considered here is continuous and subjected to Assumption1.(a) , for square wave disturbance, although its derivative is unbounded, due to the fact that square wave signal can be approximated by infinite sum of sinusoidal signals with different frequencies, in this sense, the assumption is not a restrictive factor and can be met in practice. Based on the above analysis and assumption, the control task can be given as: design an advanced guidance and control law for HRV described by (4) and (5) under the constraints represented by (6) and assumption 1, such that

sup Θ(t) - Θ* (t) d H , sup Xi (t) - X*i (t) d H (i 1..3)

t>t0 , f @

where

H !0

t>t0 ,f @

is a desired arbitrarily small bound between the controlled output Θ(t), Xi (t)(i 1..3)

and its respective reference command Θ* (t), X*i (t)(i 1..3) , in other words, under the designed AG&C law, the tracking errors for guidance and attitude dynamics can all converge to a residue set of zero within a short time. 3. Advanced guidance and control design based on enhanced TLC 3.1 Design of the proposed hierarchical AG&C strategy Based on the time-scale separation and singular perturbation theory, the proposed enhanced AG&C strategy can be divided into the guidance loop, attitude loop as well as angular rate loop, where each loop admits an enhanced TLC configuration as shown in Fig.2. Before introducing the strategy, we remark that the over-bar denotes the nominal command, while the asterisk denotes the reference command. As shown in Fig.2, combined with GPM and TLC method, the guidance law is synthesized to account for suppressing the tracking errors in the presence of off-nominal conditions. Then the attitude loop is designed to follow the reference attitude command X ref with angular rate command 2 (t) X*3 (t) as the virtual control, whereas the angular rate loop is used to regulate the angular rate by acting

on deflection angles with X*3 (t) as reference input. Furthermore, different disturbance rejection 6

strategies are applied in the guidance and control loop under TLC framework, respectively. For guidance loop, the disturbances are passively accommodated by integral action, while for attitude and angular rate loop, the disturbances are actively estimated and compensated for by proposed reduced-order LESO. Multiple disturbances

Complex path constraints& terminal constraints

Trajectory design (off line )

ª X1* º Guidance loop « » « X2 » « Θ* » Guidance law ¬ ¼

Attitude and angular rate loop ref 2

X

Attitude loop controller (TLC& reduced-order LESO, TD )

(TLC& integral action )

Angular rate loop controller (TLC& reduced-order LESO )

X*3

6-DOF translational and ratational dynamic for HRV (strong nonlinearity& couplings)

U

X3 X2 X1 , Θ

A traditional TLC controller consists of two parts. The first part is an open-loop controller which computes the nominal control using a pseudo-inverse of the plant. The second part is a feedback controller that stabilizes the system tracking error dynamics along the nominal trajectory. For guidance loop, since the pseudo-inverse kinematics cannot be obtained analytically, the TLC method cannot be directly applied. To overcome this issue, the open-loop controller in TLC is substituted by the nominal command X 2 offered by GPM, as can be seen in Fig.3 (a). For rotational dynamics, in a view of disturbance rejection and high accuracy guaranteed, the estimates of lumped disturbance offered by proposed reduced-order LESO are incorporated into TLC to meet the required attitude tracking performances in spite of multiple disturbances, as similar structures can be found in attitude and angular rate loop, the minimal difference is that nonlinear tracking differentiator (TD) is employed in attitude loop to achieve fewer control consumption and smooth transient performances, which can be seen in Fig.3 (b). Gauss pseudospectral method(GPM)

X2

Pseudo-inverse Kinematics Trajectory design (off line )

X ref 2 X1*

Stabilizing Controller

X2

Pole assignment technique

X1

Xref 2

X 2

Nonlinear tracking differentiator

X 2

Guidance loop

Pseudo-inverse Dynamics

X3

Reduced-order LESO

Attitude and angular rate loop

7

X*3

Pseudo-inverse Dynamics

U

U

Stabilizing Controller

Stabilizing Controller Feedback linearization based controller

X2

pseudo differentiator

Feedback linearization based controller

X3

X3

Reduced-order LESO

U

3.2 Closed-loop guidance law based on GPM and TLC technique For the translational dynamics (4), with selected output denoted as X1 , since the states of reentry trajectory are composed by the analytical combination of each state of X1 , in this sense, the problem of tracking the desired trajectory can be transformed to follow the commanded profile X1* (t) produced by GPM. X1 (t) to be controlled is augmented with its integral form, the

augmented state vector and the corresponding tracking error, as well as the controller for stabilizing the tracking error are defined as: X1_ aug

ª X1 dt º «³ » , E1_ aug «¬ X1 »¼

ª X1 dt dt º «³ » «¬ X1 »¼

ª ( X1  X1* )dt º » , X 2 _ aug «³ «¬ X1  X1* »¼

ªD º «V » ¬ ¼

ªD  D º «V  V » ¼ ¬

(7)

*

Based on the nominal control X 2 and reference trajectory profile X1 produced by GPM, by *

linearizing (4) along the nominal trajectory ( X1 , X 2 ) ,the linearized error dynamics for guidance loop can be written as: E11_ aug

ª 03 «0 ¬ 3

ª0 º A1 (t )E11_ aug  B1 (t ) X2 _ aug  « 3 » ¬d1 ¼

where 03

I3 º ª 033*22 º ª0 º E  X  3 A122 (t ) »¼ 11_ aug «¬B12 (t ) »¼ 2 _ aug «¬d1 »¼

(8)

I3

matrices A122 (t )  R

3u3

, B12 (t )  R

3u2

A122 (t )

The coefficient

in (8) can be obtained as follows: wF1 wX1

, B12 (t ) ( X1* ,X2 )

wF1 wX2

(9) ( X1* ,X2 )

where

A122 (t )

B12 (t )

ª U S ref CDV 2  « m « « U S ref CL cos V 1 g   « 2m V 2 ( h  RE ) « « U S C sin V sin F tan T ref L «  2m ( h  RE ) ¬« ª U S ref CDDV 2 «  2m « « V U S ref CLD cos V « 2m « « V U S ref CLD sin V « 2m «¬

g sin J ( V sin J (

g V  ) ( h  RE ) V

U S ref CL sin V 2m

º » » » 0 » » V cos F tan T » » ( h  RE ) ¼» ( X* ,X 0



sin F tan T ) ( h  RE )

1

2)

º 0 » » V U S ref CL sin V »  » 2m » V U S ref CL cos V » » 2m »¼ ( X* ,X ) 1 2

Then for the time-varying error dynamics along the nominal trajectory, to accommodate the uncertainties, the desired closed-loop tracking error dynamics for guidance loop can be designed as:

8

E1_ 1 aug

A1c E1_ aug

I3 º ª 03 «A » E1_ aug A 1c _ 2 ¼ ¬ 1c _1

03 I3 ª º (10) « diag a » E1_ aug a a diag a a a      > @ > @ 111 121 131 112 122 132 ¼ ¬

where the coefficients a1 j1 , a1 j 2 ! 0, j

1..3 are chosen by time-varying PD spectral theory in TLC

method, which is based on trial and error. In this paper, to simplify the tuning process, pole assignment technique is used to regulate the closed-loop error dynamics such that the desired characteristic polynomial of each channel satisfies O 2  a1 j 2O  a1 j1 O 2  2w1 j O  w12j

(O  w1 j )2 , and thus w1 j

becomes the only tuning parameter for each channel in guidance loop. Combined with Eq.(8) and Eq.(10), to stabilize the tracking error in the presence of off-nominal conditions, the proportional-integral (PI) feedback control law is designed as: X 2 _ aug

where the symbol

†

(11)

B1† (t )( A1c  A1 (t ))E1_ aug

denotes the pseudo inverse operator defined as A†



AT AAT



1

.

Finally, based on the nominal control produced by GPM and the PI feedback regulator described as (11), the overall reference command to the attitude loop is given by

X2  X2

Xref 2

ª¬D  X2 _ aug (1) 0 V  X2 _ aug (2) ( ) º¼

T

(12)

where X 2 _ aug (i )i 1, 2 denotes the i-th element of the control vector X 2 _ aug . 3.3 Attitude and angular rate controller based on TLC and reduced-order LESO 3.3.1 Reduced-order LESO design In this section, novel reduced-order LESO is investigated to estimate the lumped disturbances existed in attitude and angular rate loop, based on the design principle of reduced-order LESO originally proposed in [30], with given model information, a specified reduced-order LESO for attitude loop is directly given as follows: ­p1 ° ® °d ¯ 2

where d 2 , p1 and

E1 ! 0

 E1p1  E12 X 2  E1 ( F2 (X 2 ,Θ) + B 2 (X2 )X3 ) mod od el inf i f ormation

(13)

p1  E1X 2

are the estimate of the lumped disturbance in attitude loop, the auxiliary

state vector of the observer, and the observer gain to be designed, respectively. Similarly, the model-assisted reduced-order LESO for angular rate loop can be constructed as:

­ °p 2 ® ° ¯d3

 E 2p 2  E 22 X3  E 2 (F3 (X3 ,Θ) + B3 (X3 , Θ)U) p 2  E 2 X3 , E 2 ! 0

(14)

where d 3 denotes the estimate of lumped disturbance d 3 in angular rate loop.

Eo d

max(di ) ,i min( E j )

9

2..3, j 1..2

(15)

where Eo

>eo1

eo 2 @

T

T

ª¬d 2  d 2 d3  d3 º¼ .

Based on Eq.(13) and Eq.(5), the estimation error dynamics for attitude loop can be derived as:

eo1 d 2  d 2

d 2  (p1  E1X 2 ) d 2  E1p1  E12 X 2  E1 (F2 (X2 ,Θ) + B2 (X2 )X3 )  E1X2

d 2  E1p1  E12 X2  E1 (F2 (X2 ,Θ) + B 2 (X2 )X3 )  E1 (F2 (X2 ,Θ) + B 2 (X2 )X3  d 2 ) d 2  E1p1  E12 X 2  E1d 2

d 2  E1d 2  E1d 2

(16)

 E1e o1  d 2

Similarly, the estimation error of disturbances in angular rate loop can also be obtained as:

 E 2e o 2  d 3

eoo22

(17)

Together with Eq.(16-17), the overall estimation error dynamics for system(5) can be rewritten in the following compact form:

Eo where

Ao diag{ E1 ,  E 2 }

, it is evident that

Ao Eo  ª¬d 2

Ao

d3 º¼

T

is Hurwize if

(18)

Ei ! 0, i 1, 2 . Therefore, for any

given positive definite matrix Q1 , there exists a positive definite matrix P such that A To P + P A o = Q1

Let min( Ei ), i 1, 2 denotes the smallest eigenvalue of A o . Define a Lyapunov function as: 1 T Eo P Eo 2

V1

and differentiating

V1

V1

with respect to time yields:

T T 2 1 T T 1 1 Eo (Ao P + P A o )Eo + EoT P ¬ªd 2 d3 ¼º d  ETo Q1Eo  Eo P ¬ªd 2 d 3 ¼º d  Eo Q1  Eo P max(di ) 2 2 2

d  Eo

2

P min( E j )  Eo P max(di ) d  Eo ( Eo P min( E j )  P max( di )), i 2..3, j 1..2

Therefore, within finite time, the norm of the estimation error is bounded by: Eo d

max(di ) ,i min( E j )

2..3, j

1..2

Remark 2: The theoretical result shows that the estimation error for disturbances can be tuned arbitrarily small by increasing the observer gain practitioners, for the special case when d i

Ei ,this

suggests a possible tuning guideline for

0 ,i.e., the magnitude of disturbances in the system is

constant value, with appropriately designed parameter

Ei , the asymptotic convergence for estimation

error dynamics can be achieved. With the convergence of estimation error dynamics established, next, based on time and frequency domain analysis, estimation performances including transient response and noise-tolerant ability are carried out to demonstrate the differences between reduced-order LESO and the conventional LESO used in [26]. In order to explain the basic principle of traditional LESO with clarity, a model-assisted LESO for attitude loop is directly given as follows: 10

z 2  2 E1 ( X 2  z1 )  (F2 (X 2 ,Θ) + B 2 (X 2 )X3 )

­ z1 ® ¯z 2

E12 ( X 2  z1 ), E1 ! 0

where z1 and z 2 stand for the estimation vector of measured output X 2 and the disturbance d 2 in attitude loop, respectively.

E1

is the observer bandwidth, becomes the only tuning parameter of the

observer. To analyze the performance of reduced-order LESO and conventional LESO, frequency and time responses are carried out. To this end, transfer functions between the disturbance d 2 and its estimation generated by reduced-order LESO and conventional LESO can be calculated as: Proposed reduced-order LESO:

d 2 (s) d 2 (s)

E1 s  E1

Conventional LESO:

z 2 ( s) d 2 (s)

E12 s 2  2 E1s  E12

where s denotes the Laplace operator. The gains for the two observers are selected as the same bandwidth being E1 10rad / s , using the transfer functions, frequency response plots as well as time response plots are presented in Fig.4. Step Response

1

System: reduced-order ESO Settling Time (sec): 0.391

Amplitude

0.8

System: conventional ESO Settling Time (sec): 0.583

0.6

0.4

0.2

0

reduced-order ESO conventional ESO 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (sec)

Bode Diagram

Magnitude (dB)

0 -20 -40

reduced-order ESO conventional ESO

-60

Phase (deg)

-80 0 -45

phase lag

-90 -135 -180

-1

10

0

1

10

2

10

10

3

10

Frequency (rad/sec)

d2 From Fig.4.(a), it can be seen that under the same bandwidth, step responses of the considered observers can both track the step signal accurately without any overshoot. And the transient response of reduced-order LESO is faster with settling time being 0.4s. In addition, it can be noted from Fig4.(b) that reduced-order LESO yields less phase lag within the low frequency range, implying a better estimation performance can be achieved. Regarding the peaking phenomenon which is a common issue for linear differentiators or observers, since the order of proposed observer is identical to one, peaking phenomenon during initial phase can be totally eliminated for reduced-order LESO, which can be viewed as a unique advantage over the existing linear observers, including the conventional LESO, the slight drawback of reduced-order LESO is that it gives lower attenuation within high frequency range, 11

implying a poor noise-tolerant performance , which can also be observed from Fig.4.(b). Moreover, to validate the above statements, performance comparisons between reduced-order LESO and conventional LESO are tested under two different cases with the same bandwidth. For the first case, the real disturbance is selected as sine function without considering measurement noise. For the second case, a more aggressive condition with a random noise of variance being 0.1 adding to the original sine function is taken into consideration, as shown in Fig.5. The simulation results show that a higher accuracy of estimation performance can be achieved by reduced-order LESO in the absence of noisy measurement, whereas better noise suppression ability is revealed for conventional LESO, which

estimation performance

estimation performance

is consistent with the previous statements. 1 0.5 0 -0.5 -1

0

2

4

6

8

10

12

14

16

18

20

real disturbance estimate by reduced-order LESO estimate by traditional LESO

5

0

-5

0

2

4

6

8

time(s)

estimation error

estimation error

12

0.1 0 -0.1

2

4

6

8

10

16

18

20

0.2 0 -0.2 -0.4

0

14

reduced-order LESO conventional LESO

0.4

0.2

-0.2

10

time(s)

12

14

16

18

20

0

2

4

6

time(s)

8

10

12

14

16

18

20

time(s)

Remark 3: Based on the above analysis, the estimation performance of reduced-order LESO can be significantly improved by increasing the observer gain when measurement noise is not considered. However, the observer gain is usually limited by the measurement noises in practice. Hence, the observer tuning should always be a compromise between estimation quality and noise sensitivity. 3.3.2 Nonlinear controller design for attitude and angular rate loop To implement the nominal controller for attitude loop, the derivative of Xref is required and can 2 be calculated by differentiators. Among the existing differentiators, Therefore,

* ­ fh fhan( X*2 (k )  Xref 2 ( k ), X 2 ( k ), r , h) ° * * * ® X 2 (k  1) X 2 (k )  h ˜ X 2 (k ) ° * * ¯ X 2 (k  1) X 2 (k )  h ˜ fh

X*2 and X*2 denote attitude desired transient profile and its derivative.

12



the nominal control law with the aim of tracking

X*2 can be obtained by the following pseudo-inversion: X3 = B 2 (X*2 )1 ( X*2  F2 (X*2 ,Θ* )) where X 3 denotes the nominal controller for attitude loop, and

(24)

Θ*

.

E2 = X2 - X*2

*

trajectory ( X 2 , X 3 ) , define the stabilizing law as X3

X3  X3 , then

E2 = A 2 (t)E2 + B 2 (t)X3 + d 2 A 2 (t) (

wF2 wB 2  X3 ) wX2 wX2

( X*2 ,X3 )

, B 2 (t) = B 2 (X2 , t)

( X*2 ,X3 )

d2 To achieve asymptotically stability along the nominal trajectory, define the desired closed-loop tracking error dynamics for attitude loop as:

E2 A 2c

A 22cc E2 diag > k21 k22 k23 @ , k2i ! 0(i 1..3) T

(26)

where k 2i denotes the parameter to be tuned in attitude loop. Simultaneously, with the estimation vector d 2 of lumped disturbance

d2

X3

B 2 (t)1 (  A 2 (t)E2  d 2  A 2 c E2 )

(27)

The reference angular rate command to the angular rate loop is synthesized as

X*3

X3  X3

(28)

Since similar control structure is used in angular rate loop, in the lines of controller design principles of attitude loop, we directly give the required deflection angle vector to the HRV as:

U

UU

B3 (X*3 , Θ* )1 ( X*3  F3 (X*3 , Θ* ))  B3 (t)1 (  A3 (t)E3  d3  A3c E3 )

13

(29)

where

U denotes the nominal controller for angular rate loop. X*3 denotes the derivative of X*3 ,

which is calculated by passing through pseudo-differentiator. And E3 = X3 - X*3

A3c is the desired closed-loop tracking error dynamics for angular rate loop. 4. Stability analysis of closed-loop dynamics

Consider the tracking error dynamics of guidance loop, attitude and angular rate loop, if

Omin (P ) , where O (P), O (P) denote min max 2Omax ( P)

there exists a definite positive matrix P such that P d

the minimum and maximum eigenvalues of P, respectively. Then, under the proposed control law, the tracking error for the overall closed-loop HRV system can converge to a residual set of the origin in spite of multiple disturbances. Proof: First, consider the linearized error dynamics for guidance loop presented by (8), substitute the PI feedback law denoted as (11) into (8) yields E1_ 1 aug

A1c E1_ aug

I 3 º ª ³ ( X1  X1* )dt º ª03 º ª 03 »« » «A »« * ¬ 1c _1 A1c _ 2 ¼ «¬ X1  X1 »¼ ¬d1 ¼

I 3 º ª ³ E1dt º ª03 º ª 03 »« » «A »« A d 1c _ 2 ¼ « E ¬ 1c _1 ¬ 1 »¼ ¬ 1 ¼

(30)

Based on the same deduction principle, the tracking error dynamics for attitude and angular rate loop in the presence of multiple disturbances can be given as

E2

A 2c E2  e01 , E3

A3c E3  e02

(31)

E ª¬E1_ aug E2 E3 º¼

T

(30) and (31) can be shown as I3 03 03 º ª ³ E1dt º ª03 º ª 03 « » « » «A A 03 03 »» « E » «d1 » 1  E « 1c _11 1c _ 2 « 03 03 A 2 c 03 » « E » «e01 » 2 » « » « »« 03 03 A33cc ¼ « E » ¬e02 ¼ ¬ 03 ¬ 3 ¼ : d

: A

With

appropriately

tuned

parameters

(32)

in

translational

w1i ! 0, k2i ! 0, k3i ! 0,(i 1..3) , it is obvious that the matrix

A

and

rotational

loop

such

as

is Hurwitz such that the stability of the

overall closed-loop system without considering uncertainties can be ensured. Let P be the positive-definite solution of the following Lyapunov equation:

AT P  PA Let

I12

(33)

Omin (P), Omax (P) be the minimum and maximum eigenvalues of P, respectively. Consider the

14

following Lyapunov candidate function V as:

1 T E PE 2

V

(34)

The time derivative of V can be calculated as: 1 T 1 T T 1 2 ªE ( A P  PA )E  2dT PE º¼ d  E  d ˜ P ˜ E V (E PE  ET PE) 2 2 2¬ 2

P ˜V 1 2 1 V V 2 2 d  P ˜ E  d d   d 4 Omax (P) Omax (P) (1 2)Omin (P) 4 2

d (

P 1 1 )V  d  4 Omax (P) (1 2)Omin m (P)

2

KV 

(35)

2

1 2 1 1 2 d d  KOmin (P) E  d 4 4 2

2

: Kc

: K

It can be concluded from (35) that the closed-loop system is input-to-state stable with regarding to bounded uncertainties and arbitrarily small estimation errors generated by reduced-order ESO. Moreover, it also yields t

t

t0

t0

³ VddW V (t )  V (t0 ) d K c³ E dW  2

t

t

t

1 1 2 2 2 d dW d K c³ Ei dW  ³ d dW ³ 4 t0 4 t0 t0

(36)

Consequently, one has t

³

2

Ei dW d

t0

t

V (t0 ) 1 2  d dW , i 1..3 ³ c c 4K t0 K

Obviously, the overall closed-loop stability can be guaranteed under the condition that

(37)

K c ! 0 , to

attenuate the adverse effects of the disturbances on the closed-loop dynamics, we can increase

Kc

by

adjusting the tunable parameters in the guidance law and control law. Then, it leaves (35) as 1§ · V d KV  ¨ sup d ¸ 4 © t0 dW dt ¹

2

(38)

Using the comparison theorem, the definite integral of above inequality becomes 2

V (t ) d V (t0 )e K (t t0 )  Ei d

1 § · sup d , 4K ¨© t0 dW dt ¸¹

2V (t0 ) 2V d e Omin (P) Omin (P)

K  ( t  t0 ) 2



(39) 1 sup d , i 1..3 4KOmin (P) t0 dW dt

Based on (39), it can be concluded that the closed-loop tracking error can converge to an arbitrarily small neighborhood of zero with the proposed AG&C design. Furthermore, as shown in (32), the tracking error dynamics are decoupled with each loop, implying that the controller parameters in each loop can be regulated separately. In addition, it is obvious that by placing the controller and observer gain appropriately, the bounded-input-bounded-output stability can be assured. A special case arises under the nominal condition, i.e., the estimation error and the disturbance are zero, in this case, the error dynamics is asymptotically stable with respect to the origin. 5. Simulation results and comparison

15

design reentry trajectory prior to flight to make it satisfy all the path constraints. The reference trajectory profiles and nominal controller X 2 for guidance loop optimized by GPM developed in Ref [11] are shown in Fig.6-7, where multiple constrained reentry trajectory optimization problem has been well handled with upper bounds for path constraints being qmax 200slug / ft 2 , Qmax 300Btu / ft 2 / s, nmax 2 . It can be obviously observed from Fig.6 that the states of the trajectory vary smoothly with all the path constraints satisfied. Moreover, the nominal controls for guidance loop are illustrated in Fig.7, implying that by interpolating the values of nominal command and then integrating the translational dynamical equations (4) in the absence of uncertainties, the same results on the nominal trajectory profile as shown in Fig.6 can be obtained. However, in the presence of uncertainties, additional controls produced by guidance law should be added to the nominal control to form the reference profile so as to eliminate the undesired tracking error, which will be further explained in the following simulations. 4

500

1000

1500

2000

time(s) V*

6000 4000 2000 0

0

500

1000

time(s)

X: 1634 1500 Y: 762

2000

200 150

upper bound

100 50 0

0

500

1000

1500

0

500

1000

-4

-6

0

500

1000

200

upper bound

100 0

0

500

*

T 0

0

500

1000

1000

1500

2000

16 600

800

1000

1200

1400

1600

1800

time(s)

2000

F

*

80 60 40 20 0

0

500

1000

1500

2000

time(s) 2 1.5

upper bound

1 0.5 0

0

500

1000

1500

2000

time(s)

The nominal value for bank angle

0

18

1500

100

time(s)

bank angle(deg)

angle of attack(deg)

X: 1634 Y: -5 2000

300

20

400

10

time(s)

The nominal value for angle of attack

200

1500

20

time(s)

*

-2

22

0

2000

J

0

time(s)

14

1500

time(s)

2000

*

heading angle(deg)

0

latitude(deg)

I 0

heating rate(BTU/ft2/s)

speed(m/s)

20

loading factor

X: 1634 Y: 2.438e+004

8000

dynamics pressure(slug/ft2)

longtitude(deg)

4

40

flight-path angle(deg)

altitude(m)

h*

6

2

30

60

x 10

8

-20 -40 -60 -80

0

200

400

600

800

1000

1200

1400

1600

1800

time(s)

Next, to demonstrate the tracking performance and anti-disturbance capability of the proposed AG&C scheme,

16

Initial conditions and simulation parameters. Θ(0) [70100m;0deg;0deg] X2 (0) [0;0; 62]deg

X1 (0) [7315m / s;0deg;0deg]

X3 (0) [0;0;0]deg s 1

(G i , G i ) (23,23)deg(i 1,2,3)

w1

h

0.005I 3

0.02s, r

E1 = 8, E 2

0.06 rad s 2

25

A 2 c = 2 I 3 , A3c = 12 I 3

wdiff

15

flight-path angle response oscillates along the reference command. For enhanced TLC based guidance law, since the disturbances in the translational dynamics is suppressed through the designed time-varying feedback control law with PI form, the proposed guidance design is less sensitive to the adverse effects caused by disturbances. From Fig.8.(b), the magnitude of angle of attack in closed-loop form is obviously different from the open-loop command, i.e., additional control is generated to modulate the open-loop command to form the closed-loop command to null the tracking error. In addition, the complex path constraints during reentry are given in Fig.9. It can be also noted that with the proposed guidance law, all the complex path constraints including dynamics pressure, load factor as well as heating rate can be guaranteed in spite of disturbances, implying that the reentry flight with sufficient stability and safety can be met. On the other hand, the responses of open-loop controller all surpass the specified upper bounds. Hence, the proposed guidance law exhibits prominent superiority in steering HRV to track the reference trajectory with all the path constraints satisfied in the presence of off-nominal conditions.

17

4

4

x 10

8

6

7

x 10

60 50

5

40 450

500

550

600

I (deg)

h(m)

6 5

30

4

20

3

10

2

0

0

500

1000

1500

2000

55

50 1320 1340 1360 1380 0

500

1000

time(s)

26 30 25

2000

4300

25 24

4250 1400

1410

6000

1420

4200

V(m/s)

20 15

880

4000

890

900

910

10 2000

5 0

0

500

1000

1500

0

2000

0

500

1000

time(s)

closed-loop guidance law

open-loop controller

0

80

20

-1

60

10

-2

F(deg)

J(deg)

2000

100

1

0.4 0.2 0 -0.2 -0.4

-3 -4 -5

1500

time(s)

Reference command

20 500

0

1520154015601580160016201640

40

600 500

700 1000

0 1500

2000

0

500

1000

Reference command

1500

2000

time(s)

time(s)

closed-loop guidance law

23

open-loop controller

0

22 -20

V (deg)

21

D (deg)

T(deg)

1500

time(s) 8000

20 19

-60

18 17

-40

0

500

1000

1500

-80

2000

time(s)

closed-loop command

0

500

open-loop command

18

1000

time(s)

1500

2000

dynamics pressure(slug/ft2)

800 600 400 200 0

upper bound 0

200

400

600

800

1000

1200

1400

1600

1000

1200

1400

1600

1800

time(s)

load factor

4 3 2 1 0

upper bound 0

200

400

600

800

1800

time(s) heating rate(BTU/ft2/s)

300 200

upper bound

100 0

0

200

400

600

800

1000

1200

1400

1600

1800

time(s)

closed-loop guidance law

50

0

0

40 20 0 -20

0

0 1

2

V (deg)

50

E(deg)

D (deg)

50

open-loop controller

3

-50

2000

0.903 0.902 0

874.5 875 1000

500

875.5 876 1500

2000

0 -2

0

0

500

1000

500

20 0 -20

0

2

4

4

6 1000

1500

2000

-40

0

500

1000

4

1500

4

6

0

500

1000

1500

0

0 200

10 400

20 30 600 800

0

2

4

6

0

2000

-40

0

500

1000

1500

SHLNN-TLC method

1200

1400

1600

0

20

40

200

0

5 400

10 15 20 600 800

1000

1200

1400

1600

1800

0.005 0

32

-0.1 0

200

400

600

800

1000

1200

1400

1600

1800

-0.005 -0.01

0

0.2 0 2 -0.2 -0.4 1

200

400

600

800

1000

1200

1400

1600

1800

800

1000

1200

1400

1600

1800

3

1 d (rad/s 2)

0.5

33

23

0 -0.5 -1

0

0.01

-0.05

-0.15

0

-0.01

1800

0 -0.05 -0.1

0

0.05

0

-0.05 1000

200

400

600

800

1000

1200

1400

1600

1800

0

0

200

19

2

400

600

time(s)

time(s)

Estimate

1

0 -1

0

2000

20 0 -20

time(s)

31

1.5 1 0.5 0

0.5

2000

8

0.01 d (rad/s 2)

21

d (rad/s)

1500

Proposed method

1

0.05

22

2

time(s)

1.5

d (rad/s)

3

-20

8

Reference command

d (rad/s)

6

2 1000

8 6 4 2 0

20 0

1

0 -5

2000

0

500

40

time(s)

0

1500

-20 2

0

5

20

-30

2000

2

0

-20

1500

10

40

0 -10 -20

1000

4

10

-10

500

d (rad/s 2)

Gx (deg)

y

x

0.9025

-5

-10

w (deg/s)

0

Gy (deg)

w (deg/s)

5

-200 0

z

4 1500

w (deg/s)

2 3 1000

500

Gz (deg)

1 0

400 200 0 -200

-100 -150

-50 -50

-50

Actual

2000

6. Conclusion

predictor-corrector predictor-corrector based guidance

̄ ̄

20

̄

–

21

22