Disturbance Rejection and Robust Least-Squares ControlAllocation in Flight Control System

Lei Cui∗ and Ying Yang†

Peking University, 100871 Beijing, People’s Republic of China

DOI: 10.2514/1.52234

The problem of disturbance rejection and control allocation with an uncertain control effectiveness matrix is

investigated in this paper for a flight control system. AnH2=H1 feedback controller is designed to produce the three

axismoments and simultaneously suppress disturbance noise. A feedforward controller is used to track the reference

signals.Under the condition of uncertainty included in the control effectivenessmatrix, a robust least-squares scheme

is employed to deal with the problem of distributing the three axis moments to the corresponding control surfaces.

The proposed robust least-squares control allocation is studied for both unstructured and structured uncertainties.

To illustrate the effectiveness of the proposed scheme, a simulation for an experimental satellite launch vehiclemodel

is conducted. Comparisons of robust least-squares control allocation and pseudoinverse control allocation are

presented. Results show that a disturbance is rejected and robust least-squares control allocation is effectively robust

to uncertain control effectiveness matrix.

Nomenclature

A, B, C, C0 = state-space model matricesB, G,M, R, S = linear subspacesd1, d2, d3, d4 = disturbanceIm = identity matrix m �mm = dimension of control inputs, p, q = constant scalarsu = control input vectoru ⩽ u ⩽ �u = upper and lower bounds for vector uumax, umin = upper and lower position constraints_urate = maximal actuator ratesu1, u2, u3, u4 = deflection of the actuators of straponsu5, u6, u7, u8 = engine thrustersv = virtual control moment vectorx = state vectorX > 0�<0� = positive (negative) definite matrix Xy, y0 = control and measurement outputsz1, z2, z3, z4, L,�, �, �, �

= variables in optimization and LMIs

�, � = weighting scalars� = upper bound ofH1 performance index�A, �B = uncertainties in state-space model matrices�t = sampling period�, " = column vectors = upper bound ofH2 performance index = angle of pitch_ = pitch rate� = upper bound of uncertainty� = angle of roll_� = roll rate = angle of yaw_ = yaw rateN

= Kronecker product

L= direct sum

k � k = 2-norm for a vector

Subscripts

d = disturbancer = referencev = virtual control

Superscripts

T = vector or matrix transpose�1 = matrix inverse� = pseudoinverse of matrix

Introduction

M ODERN aircraft often use redundant control inputs fortracking flight paths. A system with more inputs than degrees

of freedom (DOF) is an overactuated system [1–3]. The referencesignals can be tracked with a certain combination of existing controleffectors. Control allocation is able to distribute the virtual controllaw requirements to the control effectors in the best possible mannerwhile accounting for their constraints [4]. Even under the conditionthat control effectors are damaged, the control reallocation can beimplemented without redesigning the control law for fault-tolerantcontrol.

Control allocation is an active research field in the area of over-actuated aerospace control systems. In the past 15 years, voluminousresults have been developed. The general approaches of controlallocation include interior-point algorithms [4], weighted pseudoin-verse (WPI) [5], linear programming [6], quadratic programming [6],sequential least-squares [6], redistributed WPI techniques, daisychain control allocation [7], direct control allocation [8,9], dynamiccontrol allocation [10], and quantized control allocation [11].Generally speaking, most previous works study linear controlallocation by programming algorithms, which can be iterativelyconducted tominimize the error between the commands produced byvirtual control law and the moments produced by practical actuatorcombinations. Furthermore, researchers have intensively consideredthe problem of controller design and control allocation for an aircraftwith redundant actuators [5–7,12]. In [5], an online sliding modecontrol allocation scheme for fault-tolerant control has been pro-posed. Backstepping and control allocation have been investigated in[6]. In [7], the relationship between daisy chain control allocationand the stability of the system is studied. Optimal controller designand control allocation have been proposed in [12].

Received 31 August 2010; revision received 9 May 2011; accepted forpublication 15May 2011. Copyright © 2011 by the authors. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission.Copies of this paper may be made for personal or internal use, on conditionthat the copier pay the $10.00 per-copy fee to theCopyright Clearance Center,Inc., 222RosewoodDrive, Danvers,MA01923; include the code 0731-5090/11 and $10.00 in correspondence with the CCC.

∗Ph.D. Candidate, State Key Laboratory for Turbulence and ComplexSystems, Department of Mechanics and Aerospace Engineering, College ofEngineering; cuilei2007@pku.edu.cn.

†Associate Professor, State Key Laboratory for Turbulence and ComplexSystems, Department of Mechanics and Aerospace Engineering, College ofEngineering; yy@mech.pku.edu.cn (Corresponding Author).

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. 34, No. 6, November–December 2011

1632

The uncertain control effectiveness matrix is not considered inmost of the current literature studying the control allocation problem.In this paper, the main contribution is to provide an effective controlallocation method when the control effectiveness matrix is subject touncertainties. A new methodology of robust least-squares controlallocation (RLSCA) is proposed. When the control effectivenessmatrix is subject to different kinds of uncertainties, the controlallocation is carried out in different manners, which ensure thedistribution of the virtual control law requirements to the controleffectors. At the end of this paper, the proposedmethod is applied to aclosed-loop flight control system.

This paper starts with the problem statement. A virtual controlvector is produced by the output of a feedback controller and afeedforward controller. In the next section, the robust least-squaresmethod is employed to deal with the problem of control allocation inthe presence of uncertainties in the control effectiveness matrix. Thevalidity of the method is illustrated through a design example.Conclusions are given at the end of the paper.

Problem Statement

For the overactuatedflight control systems, it is possible to split thecontroller design into two steps [12,13]. Consider a closed-loopflight control system with disturbance and uncertainty, especiallyuncertainty included in the control effectiveness matrix. The twosteps can be described as follows:

Step 1: Virtual Control Law Design

Design an H2=H1 control law specifying the total effort (virtualcontrol) to be produced (net torque, force, etc.). Simultaneously, thiscontroller can suppress disturbances and the closed-loop system isrobust to uncertainty provided that sufficient control authority existsto produce the moments or accelerations commanded by the virtualcontrol law;

Step 2: Robust Least-Squares Control Allocation Design

With consideration of the uncertain control effectiveness matrix,design a RLSCA algorithm that maps the total control demand ontoindividual actuator settings (commanded aerosurfaces deflections,thrust, forces, etc.).

The equivalence between the approach denoted in the above twosteps and that of directly designing the real control law has beenstudied in [12]. Separating the control allocation from controllerdesign has this salient advantage: the computation load can be greatlyreduced by redesigning the control allocation algorithm and keepingthe controller unchanged if a fault occurs in the control effectors.

The overall flight control system is presented in Fig. 1. Thesatellite launch vehicle (SLV) model is the controlled plant. Thefeedforward controller vr � Krefr is designed to track the referencesignal. TheH2=H1 feedback controller and feedforward controllercooperate to produce thevirtual controlmoment v. TheRLSCA is thenew control allocation module, and it distributes the virtual controlmoment v among the multiple control effectors ui�i� 1; . . . ; 8�.

Consider a continuous-time linear SLV system with uncertaintiesand disturbance

_x�t� � �A��A�x�t� � Bv�B��B�u�t� � Bdd�t�y0�t� � C0x�t� y�t� � Cx�t� (1)

where x�t� 2 R6, u�t� 2 R8, d�t� 2 R4, y0�t� 2 R3, and y�t� 2 R3.

The uncertainty �A is assumed to be norm bounded; that is,

�A�HFE

with F 2 Ri�j satisfying

FTF � I

where H and E are appropriate dimensioned matrices.The uncertainty�B is included in the control effectiveness matrix

B and can be categorized into three types: unstructured uncertainty,structured uncertainty, and linear fractional structured uncertainty.

Define the virtual control term v�t� 2 R3 as

v�t� � �B��B�u�t�

which represents the three axis moments. Then, system (1) can berewritten as

_x�t� � �A��A�x�t� � Bvv�t� � Bdd�t�y0�t� � C0x�t�y�t� � Cx�t� (2)

Since control allocation is separated from controller design, twoproblems will be studied in the following:H2=H1 controller designand RLSCA design.

Problem 1,H2=H1 Controller Design: For the linear system (2),design a state feedback control law

v1�t� � Kx�t�

such that the following hold:1) The closed-loop system is asymptotically stable.2) When d�t� is viewed as an energy-limited disturbance, the

transfer function fromd�t� to y�t� isG�s�, theH1 performance of thesystem satisfies

kG�s�k1 < �

3) When d�t� is considered as a white noise with unit powerspectral density, the transfer function from d�t� to y0�t� isG0�s�, theH2 performance of the system satisfies

kG0�s�k2 <

Problem 2, RLSCA Design: Given a control effectiveness matrixB, its uncertainty�B and virtual control v design an optimal RLSCAuRLSCA:

uRLSCA � arg minu⩽u⩽ �u

maxk�Bk1⩽�

k�B��B�u � vk (3)

subject to the following:1) The uncertainty �B is an unknown bounded matrix satisfying

k�Bk1 � �

2) The control vectoru is between the upper bound �u and the lowerbound u: �

�u�min�umax; u��t _urate�u�max�umin; u ��t _urate�

(4)

Controller Design

The virtual control law includes an H2=H1 feedback controllerand a feedforward controller. The aims of an H2=H1 feedbackcontroller are to guarantee stability, suppress disturbances, and to berobust to the uncertainty in matrix A. The main purpose of thefeedforward controller is to compensate for the closed-loop gain fortracking reference signals.Fig. 1 Overall flight control system.

CUI ANDYANG 1633

Feedback Controller Design

The following lemmas are introduced to derive the robustcontroller of the SLV considered later.

Lemma 1 [14]: Let the constant scalar � > 0 be given; then, thereexists a matrix P1 > 0 such that

�A�HFE�TP1 � P1�A�HFE� � ��2P1BBTP1 � CTC < 0

for allF that satisfiesFFT < I if and only if there exists a scalar � > 0such that

ATP1 � P1A� ��2P1BBTP1 � �P1HH

TP1 �1

�ETE� CTC < 0

Lemma 2 [15]: Consider a transfer function that G�s� � C�sI �A��1 B 2 RH2, with A stable, then the following result is obtained:

kGk22 � trace�BTQB� � trace�CP2CT�

where Q and P2 are observability and controllability gramians thatcan be obtained from the following Lyapunov equations:

ATQ�QA� CTC� 0; AP2 � P2AT � BBT � 0

Theorem 1: For the given scalars � > 0, � > 0 and system (2), ifthe following optimal problem is feasible

min�;�;;P;L

�� � �

subject to

AP� PAT � BvL� LTBTv � �HHT Bd PCT PET

��I 0 0

��I 0

��I

2664

3775< 0

(5)

�I C0PPCT0 �P

� �< 0 (6)

HO

P�MO�AP� BvL� �MT

O�AP� BvL�T < 0 (7)

with variables L, P� PT > 0, �, , � 2 R, then v1�t� � LP�1x�t� isthe H2=H1 optimal feedback control law.

Proof: The closed-loop linear flight control system with the statefeedback v1�t� � Kx�t� can be rewritten as

_x�t� � �Acx�t� � Bdd�t� y0�t� � C0x�t� y�t� � Cx�t�

where �Ac ≜ Ac �HFE, Ac ≜ A� BvK.1) The transfer function from the disturbance d to control output

y is

G�s� � C�sI � �Ac��1Bd

According to the bounded real lemma [15], kG�s�k1 < � if andonly if there exists X > 0 such that

X �Ac � �ATcX � XBdR�1BTdX� CTC < 0

where R� �2I. Pre- and postmultiplying both sides of the aboveinequality byX�1, and definingX1 � X�1, the following inequality isderived

�A cX1 � X1�ATc � BdR�1BTd � X1C

TCX1 < 0

Using the Schur complement lemma [15] and Lemma 1, letX1 � �P, the above inequality holds if and only if there exist L,� > 0, � > 0, and P� PT > 0 such that

AcP� PATc � �HHT Bd PCT PET

��I 0 0

��I 0

��I

2664

3775< 0

which is equivalent to Eq. (5) with the feedback gain matrixK � LP�1.

2) The transfer function from the disturbance d to measurementoutput y0 is

G0�s� � C0�sI � �Ac��1Bd

The H2 norm of this transfer function satisfies kG0�s�k2 < .Using Lemma 2, the H2 performance index is kG0�s�k2�tracefC0PC

T0 g. Furthermore,

kG0�s�k2 � tracefC0PCT0 g<

where > 0. Thus, Eq. (6) is obtained.3) The poles of the closed-loop system can be assigned to the given

region if Eq. (7) is satisfied [16].Combining Eqs. (5–7), minimizing both theH1 part andH2 part,

which multiply the weighting scalars � and �, respectively, the proofof this theorem is completed. □

From the preceding derivations, an H2=H1 feedback controllerv1�t� � LP�1x�t� satisfying the performance requirements inproblem 1 is designed.

Feedforward Controller Design

The nominal system is represented as

_x� Ax� Bvv y� Cx

Since the tracked signals are square, the following derivation isfeasible. If the outputs have tracked the reference signals, thenominal system must satisfy

_x� Ax� Bvv� 0 y� Cx� r (8)

Since A� BvK is stable, �A� BvK��1 is guaranteed. Bysubstituting v� Krefr� Kx into Eq. (8), one will obtain

C��BvK � A��1Bvvr � r()vr � �C��BvK � A��1Bv��1r

Define that Kref � �C��BvK � A��1Bv��1, then the feedforwardcontroller vr � Krefr is designed.

Finally, the stability and tracking properties of the closed-loopsystem are guaranteed by the control law v� Krefr� Kx.

Robust Least-Squares Control Allocation

The problem of the control allocation with uncertain controleffectiveness matrix is solved by the robust least-squares schemeproposed in [17,18].

As is stated in problem2, the focus of RLSCA is tofind the optimalcontrol vector u by minimizing the worst-case residual, under thecondition of the uncertainty included in control effectiveness matrixB. When an aircraft flies in the atmospheric layer, the aerodynamicparameter variation can be considered as a parameter uncertainty. Inthis paper, the unstructured uncertainty, structured uncertainty, andlinear fractional structured uncertainty in the control effectivenessmatrix are considered. These three kinds of uncertainties in the flightcontrol system are reasonable [19,20]. Accordingly, the RLSCAproblem can be categorized into three types: unstructured RLSCA(URLSCA), structured RLSCA (SRLSCA), and linear fractionalSRLSCA (LFSRLSCA).

Unstructured Robust Least-Squares Control Allocation

If the uncertainty is unstructured, the uncertainty is consideredas �B.

1634 CUI ANDYANG

For the URLSCA problem,

uURLSCA � arg minu⩽u⩽ �u

maxk�Bk1⩽�

k�B��B�u � vk

where �B is an unstructured uncertainty.To solve this problem, the following derives URLSCA to a

second-order cone programming (SOCP) problem.For a variable u in the interval of u; �u�, the worst-case residual is

r�u� � maxk�Bk1��

k�B��B�u � vk

Using the triangle inequality,

r�u� � maxk�Bk1��

�kBu � vk � k�Buk� � kBu � vk

� maxk�Bk1��

k�Buk

Assume that

�B� �

kuk "uT

where

"��

Bu�vkBu�vk ; if Bu ≠ v;any unit norm vector; otherwise

Then, in the direction of ", the worst-case residual is

r�u� � kBu � vk � �kuk (9)

The worst-case residual in Eq. (9) satisfies the following

kBu � vk � �kuk � �

where� is the upper bound of the residual to beminimized byfindingthe optimal u in the interval of u; �u�.

Thus, the URLSCA problem can be rewritten as a SOCP problem:

minu;�;�

�

subject to kBu � vk ⩽ �� ��kuk ⩽ �

u ⩽ u ⩽ �u (10)

with the variables u 2 Rm, �, and � 2 R.Theorem 2: The optimal solution uURLSCA to the URLSCA

problem is given by

uURLSCA ����I � BTB��1BTv; if �≜ �������2

�2��2s > 0

B�v; else

where � � 0, s� k �uk2 � kuk2, �, and � are the optimal solution toproblem (10).

Proof: The dual problem of (10) is

maxz1;z2 ;z3;z4

vTz1 � �uTz3 � uTz4

subject to BTz1 � �z2 � 0

kz1k � 1

kz2k � 1

zT3 � 0

� zT4 � 0 (11)

Both the primal and dual problems are feasible; then, there existsan optimal point for each. According to optimization, the optimalpoint of primal problem is equal to that of the dual problem.

If �� � is at the optimum, then Bu� v and

�� � � �kuk

Under this condition, u� B�v is the optimal solution. It is similar tothe pseudoinverse control allocation (PICA).

If � > � is feasible, the primal and dual optimal objectives areequal:

kBu � vk � �kuk � �� vTz1 � �uTz3 � uTz4 ���Bu � v�Tz1

� uT � �uT uT ��BTz1z3

z4

264

375

The following is derived:

z1 ��Bu � vkBu � vk (12a)

�BTz1z3z4

24

35�� �kuk

kuk2 � k �uk2 � kuk2u� �uu

24

35 (12b)

From Eq. (12b), the following is obtained:

BTz1 ��kuk

kuk2 � k �uk2 � kuk2 u (13)

Substituting Eq. (12a) into Eq. (13) yields

BTBu � vkBu � vk �

�kukkuk2 � k �uk2 � kuk2 u� 0

Define s� k �uk2 � kuk2; the optimal control vector u isrepresented as

u� ��I � BTB��1BTv (14a)

�� ��� ����2

�2 � �2s (14b)

Thus, the conclusion is achieved. □

Structured Robust Least-Squares Control Allocation

When the uncertainty is unstructured, the worst-case residual ofURLSCA may be conservatively estimated [17]. Therefore,SRLSCA problem is proposed.

Definition 1: For the givenB,B1; . . . ; Bq 2 Rn�m,B is the nominalcontrol effectiveness matrix, Bi�i� 1; . . . ; q� are the uncertainmatrices; thus, the structured uncertain control effectiveness matrixcan be represented as

B���≜ B�Xqi�1

�iBi (15)

where �� �1; . . . ; �q�T 2 Rq.For the SRLSCA problem,

uSRLSCA � arg minu<u< �u

maxk�k⩽�;

kB���u � vk

where B��� is defined in Eq. (15).To obtain the optimal solution uSRLSCA to SRLSCA, the following

is given first:

M�u�≜ B1u . . .Bqu�; F�u�≜M�u�TM�u�

g�u�≜M�u�T�Bu � v�; h�u�≜ �Bu � v�T�Bu � v�

CUI ANDYANG 1635

Theorem 3: The SRLSCA problem has an optimal solution��; �; uSRLSCA� if the following matrix inequality is solved:

minu⩽u⩽ �u;�;�

�

subject to� � � � h�u� ��gT�u���g�u� �I � �2F�u�

" #> 0

with the variables u 2 Rm, �, and � > 0.Proof: Since the squared worst-case residual is represented as

r2S�u� � max�T���2

1 �T �h�u� gT�u�g�u� F�u�

" #1

�

" #

� max�0T �0�1

1 �0T �h�u� �gT�u��g�u� �2F�u�

" #1

�0

" #

To ensure r2S�u�< �, it holds

1 �0T � h�u� �gT�u��g�u� �2F�u�

� �1

�0

� �< �

For every �0 � ��, �T� � �2, if and only if there exists a scalar � > 0

such that

1 �0T � � � � � h�u� ��gT�u���g�u� �I � �2F�u�

� �1

�0

� �> 0

Using the fact that � > 0, which is implied by �I > �2F, andrewriting the above inequality as

G ��; ��≜ � � � � h�u� ��gT�u���g�u� �I � �2F�u�

� �> 0 (16)

The proof of this theorem is completed. □

The matrix inequality in Theorem 3 is nonlinear, and it cannot besolved by linear matrix inequality (LMI) solution methods. There-fore, with the definition of F, g, and h given above, the followingtheorem is derived.

Theorem 4: The SRLSCA problem has an optimal solution��; �; uSRLSCA� if the following is solved:

minu;�;�

�

subject to

�� � 0 �Bu � v�T

0 �I �MT�u��Bu � v� �M�u� I

264

375> 0

� �u � u�Tb1 0 0

0 � �u � u�Tb2 0

..

. ... . .

. ...

0 0 � �u � u�Tbm

2666664

3777775> 0

�u � u�Tb1 0 0

0 �u � u�Tb2 0

..

. ... . .

. ...

0 0 �u � u�Tbm

2666664

3777775> 0

with the variables u 2 Rm, �, and � > 0, the control signal isuT � u1; . . . ; um�, the upper and lower bounds of the control signalare �uT � �u1; . . . ; �um� and uT � u1; . . . ; um�, and bi�i� 1; . . . ; m�are unit column vectors and satisfy b1; . . . ; bm� � Im. For the SLVsystem (1) being considered, m� 8.

Proof: Equation (16) can be rewritten as

G��; �� ��� � 0

0 �I

" #�

h�u� �gT�u��g�u� �2F�u�

" #

��� � 0

0 �I

" #��Bu� v�T�Bu� v� ��MT�u��Bu� v��T

�MT�u��Bu� v� �2MT�u�M�u�

" #

��� � 0

0 �I

" #��Bu� v�T

�MT�u�

" # �Bu� v� �M�u� �> 0

Using the Schur complement Lemma [15], the above is rewrittenas

G ��; �� ��� � 0 �Bu � v�T0 �I �MT�u�

�Bu � v� �M�u� I

24

35> 0

Thus, the non-LMI is rewritten as a LMI. To add the constraints tou by LMIs, define that uT � u1; . . . ; um�, �uT � �u1; . . . ; �um�,and uT � u1; . . . ; um�. The identity matrix is written as Im�b1; . . . ; bm�.

Each element ui of control vector u is between its upper and lowerbounds:

u i < ui < �ui�i� 1; . . . ; m� ,

u1u2...

um

26664

37775<

u1u2...

um

26664

37775<

�u1�u2...

�um

26664

37775

and rewrite the constraints on u as two LMIs:

u1

u2

..

.

um

2666664

3777775<

�u1

�u2

..

.

�um

2666664

3777775,

�u1 � u1 0 0

0 �u2 � u2 0

..

. ... . .

. ...

0 0 �um � um

2666664

3777775> 0

,

� �u � u�Tb1 0 0

0 � �u � u�Tb2 0

..

. ... . .

. ...

0 0 � �u � u�Tbm

2666664

3777775> 0

u1

u2

..

.

um

2666664

3777775<

u1

u2

..

.

um

2666664

3777775,

u1 � u1 0 0

0 u2 � u2 0

..

. ... . .

. ...

0 0 um � um

2666664

3777775> 0

,

�u � u�Tb1 0 0

0 �u � u�Tb2 0

..

. ... . .

. ...

0 0 �u � u�Tbm

2666664

3777775> 0

The proof of this theorem is completed. □

Linear Fractional Structured Robust Least-Squares

Control Allocation

In the SRLSCA problem, the norm bound of � may not beconvenient tomeasure the perturbation size. To this end, LFSRLSCAis proposed.

Definition 2: Let M be a subspace of Rp�p, M 2Mp�p,B 2 Rn�m, N 2 Rn�p, and R 2 Rp�m. For every � 2M,

1636 CUI ANDYANG

det�I �M�� ≠ 0 is satisfied, and define the linear fractionalstructured uncertain control effectiveness matrix as

B���≜ B� N��I �M���1R (17)

where � is a full norm bounded matrix and unstructured.For the LFSRLSCA problem,

uLFSRLSCA � arg minu<u< �u

max�2M;k�k1��

kB���u � vk

where B��� is defined in Eq. (17).Introduce the following linear subspaces for the theorem

derivation:

B≜ fB 2 RN�NjB���B for every � 2Mg

S ≜ fS 2 BjS� STg; G≜ fG 2 BjG��GTg

Lemma 3 [17]: Let T1 � TT1 , T2, T3, and T4 be real matrices ofappropriate size. LetM be a subspace ofRN�N and be denoted by S(respectively,G), the set of symmetric (respectively skew-symmetric)matrices that commute with every element of M. We have det�I �T4�� ≠ 0 and

T��� � T1 � T2��I � T4���1T3 � TT3 �I � T4���T�TTT2 > 0

For every� 2M, k�k1 � �, if there existS 2 S andG 2 G suchthat

T1 � �2T2STT2 TT3 � �2T2STT4 � �T2GT3 � �2T4STT2 � �GTT2 S � �GTT4 � �T4G � �2T4STT4

" #> 0;

S > 0

If M�RN�N is satisfied, the condition will be necessary andsufficient.

Theorem 5: The LFSRLSCA problem has an optimal solution��; uLFSRLSCA� if the following optimal problem is solved

minS;G;�;u

�

subject to

�I � �2NSNT Bu � v ��2NSMT � �NG�Bu � v�T � �Ru�T

��2MSNT � �GTNT Ru S� �MG � �GMT � �2MSMT

264

375> 0

� �u � u�Tb1 0 0

0 � �u � u�Tb2 0

..

. ... . .

. ...

0 0 � �u � u�Tbm

2666664

3777775> 0

�u � u�Tb1 0 0

0 �u � u�Tb2 0

..

. ... . .

. ...

0 0 �u � u�Tbm

2666664

3777775> 0

with the variables S 2 S,G 2 G, u 2 Rm, �, and � > 0. For the SLVsystem (1) being considered, m� 8.

Proof: Let � > 0, the inequality � > rM�B; v; �; u� holds if andonly if, for every k�k1 � �, det�I �M�� ≠ 0 is satisfied and

r2M�B; v; �; u� � �Bu � v� N��I �M���1Ru�T

� �Bu � v� N��I �M���1Ru�

With � > rM�B; v; �; u�, the following is derived:

�2>r2M�B;v;�;u�,�I Bu� v�N��I�M���1Ru

�Bu� v�N��I�M���1Ru�T �

� �

>0

and furthermore

�I Bu � v�Bu � v�T �

" #�

N

0

" #��I �M���1 0 Ru �

�0

�Ru�T

" #�I �M���T�T NT 0 �> 0

With Lemma 3, � > rM�B; v; �; u� holds if there exist S 2 S andG 2 G such that

�I��2NSNT Bu� v ��2NSMT��NG�Bu� v�T � �Ru�T

��2MSNT��GTNT Ru S��MG� �GMT ��2MSMT

264

375

> 0

The proof of the constraints on u is similar in Theorem 4; thus, theconclusion is obtained. □

Remark:1) When the conditions that M� 0, N � N1 . . .Np�, RT�R1 . . .Rp�, Bi 0 � � NiRi 0 �, rank�Ni� � rank�Ri��rankBi 0 �, and

���Mpi�1

�iIj�i 2 R; 1< i < p�

are satisfied, the LFSRLSCA can be rewritten as SRLSCA.

2)When the conditions thatN � I,R� I, andM � 0 are satisfied,the LFSRLSCA can be rewritten as URLSCA.

3) For the uncertainty free case, �� 0, the RLSCA problem is theleast-squares control allocation (LSCA) that has been considered inthe previous literature as follows [6]:

uLSCA � arg minu⩽u⩽ �u

kBu � vk

CUI ANDYANG 1637

4) If one actuator saturates, SOCP problem (10) in the URLSCAcan be solved similarly in [6]. The designed algorithm of URLSCAwill optimize in other directions until the required virtual controlvector is obtained. This means that the saturated actuator stays in themaximal or minimal deflection position, other unsaturated actuatorsare used to produce the required yaw (roll or pitch)moment.While inthe other two proposed approaches, since the optimal solutions arereduced to the optimization of a set of LMIs, the saturation casewould not be considered.

Three algorithms of solving RLSCA are proposed as follows:1) Solution to URLSCA problem

Step 1. Solving the SOCP problem (10) by software (YALMIP,etc.) to obtain the optimal � and �.

Step 2. Substituting the optimal � and � in Eq. (14b) toobtain �.

Step 3. Obtaining theURLSCA solution uURLSCA in Theorem 2.2) Solution to SRLSCA problem

Solving the LMIs in Theorem 4 to obtain SRLSCA solutionuSRLSCA.

3) Solution to LFSRLSCA problemSolving the LMIs in Theorem 5 to obtain LFSRLSCA solution

uLFSRLSCA.

Design Example

The aforementioned scheme is applied to a linear SLVmodel. Theproposed RLSCA is compared with PICA to show the effectivenessof the former.

Satellite Launch Vehicle Model

To investigate the effectiveness of the designed robust controllerand RLSCA, a SLV modeled with eight actuators to control fourthrusters is chosen. The control law obtained from the above design isapplied to these eight actuators to control the attitude. Four distur-bances act on the system percentage differential thrust between mainengine thrusters d1: those between strapons d2, center of gravity(CG) offset along yaw axis d3, and cg offset along pitch axis d4. Thesystem is modeled as follows:

_x�t� � �A��A�x�t� � Bv�B��B�u�t� � Bdd�t�y0�t� � C0x�t� y�t� � Cx�t�

where �A and �B are the uncertainties and satisfy k�Ak1 < �,k�Bk1 < �; and x, y, u, and d represent as follows:

x� _ _ � _� �T

y� � �T

u� u1 u2 u3 u4 u5 u6 u7 u8 �T

d� d1 d2 d3 d4 �T

The control signals ui (i� 1; . . . ; 4) are the control inputs to theactuators of strapons, and ui (i� 5; . . . ; 8) are the main enginethrusters, all of which have bounds

u� �5 �5 �5 �5 �8 �8 �8 �8 �T �

180

�u� 5 5 5 5 8 8 8 8 �T �

180

The system matrices are given in the Appendix. Three rows arezero in Bv, denoting that the control surfaces are viewed as puremoment generators and their influence on , �, and is neglected. Asimilar approach has been adopted in [12]. The system matrix�B� Bv B is six rows and eight columns, and the dimension ofinput exceeds the DOF of SLV. This means that the SLV model isinput redundant. The attitude of SLV can be controlled by a certaincombination of eight control signals.

Controller Design

The controller design can be separated into an H2=H1 feedbackcontroller design and a feedforward controller design. The followinggives numerical simulation results.

H2=H1 Feedback Controller Design

According to the H2=H1 feedback controller design inTheorem 1, the control law is v1�t� � Kx�t�:

K � LP�1 ��3:0491 � 10�1 �3:0495 � 10�1 6:9562 � 10�6 1:3311 � 10�5 �1:6990 � 10�5 �3:02761 � 10�5

7:8753 � 10�6 1:3079 � 10�5 �2:6354 � 10�1 �3:4481 � 10�1 2:6135 � 10�5 5:0546 � 10�5

�1:4030 � 10�4 �1:0021 � 10�4 3:6540 � 10�4 5:2867 � 10�4 �6:5355 � 10�2 �1:2602 � 10�1

24

35

Here, a circle region is adopted to assign the poles to the desiredregion with the center ��0:5; 0� and radius r� 0:5. The matrices HandM are chosen as

H � �r qq �r

� �� �0:5 0:5

0:5 �0:5

� �; M� 0 1

0 0

� �

By solving the LMIs in Theorem 1, �� 8:5925 � 10�2, the H2

andH1 performance indices of the closed-loop flight control systemare � 5:7351 � 10�3 and � � 4:0001.

Feedforward Controller Design

The feedforward gain matrix Kref is

Kref � �C��BvK � A��1Bv��1

�1:8132 � 10�1 �6:9591 � 10�6 1:6990 � 10�5

�7:8396 � 10�6 1:7889 � 10�1 �2:6135 � 10�5

5:9587 � 10�5 �2:7425 � 10�4 6:5355 � 10�2

264

375

Thus, the virtual controller v� Krefr� Kx is designed.

Robust Least-Squares Control Allocation Implementation

Choose the SLV stated above as a simulation model. The mainpurpose of the simulation is to show that the proposed RLSCA iseffectively robust to an uncertain control effectiveness matrix.

At the beginning of the simulation results description, fourpreliminary points need to be known as follows:

1) The upper bound of uncertainty in the control effectivenessmatrix is adopted as �� 0:1, and the disturbance is chosen asd� 1 1 1 1 �T .

2) The PICA without uncertainty is applied in the experimentalSLV model, the actuator deflections of which serve as referencedeflections to compare with three RLSCA approaches.

3) The simulation result of PICAwith uncertainty is only presentedin Fig. 2 for illustrative purposes.

4) All of the three proposed RLSCA approaches use the virtualcontrol signals produced by the same controller.

1638 CUI ANDYANG

0 5 10 15−1

0

1

2

3

4

5Pitch output

Time(s)P

itch(

deg)

refPICAPICAUURLSCA

0 5 10 15−1

0

1

2

3

4

5Yaw output

Time(s)

Yaw

(deg

)

refPICAPICAUURLSCA

0 5 10 15−1

0

1

2

3

4

5Roll output

Time(s)

Rol

l(deg

)

refPICAPICAUURLSCA

0 5 10 15

−5

0

5

u1 vs Time

Time(s)

u 1(deg

)

PICAPICAUURLSCAbound

0 5 10 15

−5

0

5

u2 vs Time

Time(s)

u 2(deg

)

PICAPICAUURLSCAbound

0 5 10 15

−5

0

5

u3 vs Time

Time(s)

u 3(deg

)PICAPICAUURLSCAbound

0 5 10 15−5

0

5

u4 vs Time

Time(s)

u 4(deg

)

PICAPICAUURLSCAbound

0 5 10 15

−5

0

5

u5 vs Time

Time(s)

u 5(deg

)

PICAPICAUURLSCAbound

0 5 10 15

−5

0

5

u6 vs Time

Time(s)

u 6(deg

)

PICAPICAUURLSCAbound

0 5 10 15

−5

0

5

u7 vs Time

Time(s)

u 7(deg

)

PICAPICAUURLSCAbound

0 5 10 15

−5

0

5

u8 vs Time

Time(s)

u 8(deg

)

PICAPICAUURLSCAbound

Fig. 2 Simulation results of PICA without uncertainty, pseudoinverse control allocation with uncertainty (PICAU), and URLSCA.

CUI ANDYANG 1639

0 5 10 15

0

2

4

Pitch output

Time(s)

Pitc

h(de

g)

refPICASRLSCA

0 5 10 15−1

0

1

2

3

4

5Yaw output

Time(s)

Yaw

(deg

)

refPICASRLSCA

0 5 10 15

0

2

4

Roll output

Time(s)

Rol

l(deg

)

refPICASRLSCA

0 5 10 15

−5

0

5

u1 vs Time

Time(s)

u 1(d

eg)

PICASRLSCAbound

0 5 10 15

−5

0

5

u2 vs Time

Time(s)

u2(

deg)

PICASRLSCAbound

0 5 10 15

−5

0

5

u3 vs Time

Time(s)

u3(

deg)

PICASRLSCAbound

0 5 10 15

−5

0

5

u4 vs Time

Time(s)

u4(

deg)

PICASRLSCAbound

0 5 10 15

−5

0

5

u5 vs Time

Time(s)

u 5(d

eg)

PICASRLSCAbound

0 5 10 15

−5

0

5

u6 vs Time

Time(s)

u6(

deg)

PICASRLSCAbound

0 5 10 15

−5

0

5

u7 vs Time

Time(s)

u 7(d

eg)

PICASRLSCAbound

0 5 10 15

−5

0

5

u8 vs Time

Time(s)

u 8(d

eg)

PICASRLSCAbound

Fig. 3 Simulation results of PICA without uncertainty and SRLSCA.

1640 CUI ANDYANG

0 5 10 15−1

0

1

2

3

4

5Pitch output

Time(s)P

itch(

deg)

refPICALFSRLSCA

0 5 10 15−1

0

1

2

3

4

5Yaw output

Time(s)

Yaw

(deg

)

refPICALFSRLSCA

0 5 10 15−1

0

1

2

3

4

5Roll output

Time(s)

Rol

l(deg

)

refPICALFSRLSCA

0 5 10 15

−5

0

5

u1 vs Time

Time(s)

u 1(d

eg)

PICALFSRLSCAbound

0 5 10 15

−5

0

5

u2 vs Time

Time(s)

u2(

deg)

PICALFSRLSCAbound

0 5 10 15

−5

0

5

u3 vs Time

Time(s)

u3(

deg)

PICALFSRLSCAbound

0 5 10 15

−5

0

5

u4 vs Time

Time(s)

u4(

deg)

PICALFSRLSCAbound

0 5 10 15

−5

0

5

u5 vs Time

Time(s)

u 5(d

eg)

PICALFSRLSCAbound

0 5 10 15

−5

0

5

u6 vs Time

Time(s)

u 6(d

eg)

PICALFSRLSCAbound

0 5 10 15

−5

0

5

u7 vs Time

Time(s)

u7(

deg)

PICALFSRLSCAbound

0 5 10 15

−5

0

5

u8 vs Time

Time(s)

u 8(d

eg)

PICALFSRLSCAbound

Fig. 4 Simulation results of PICA without uncertainty and LFSRLSCA.

CUI ANDYANG 1641

Figure 2 presents the comparisons between the PICA and theproposed URLSCA with uncertainties in both cases, where theactuator deflections using PICA without uncertainty serve asreference deflections. In the PICA, it is shown that if the uncertaintyis in the control effectiveness matrix, the actuator does not deflect inaccord with reference deflections. However, our proposed URLSCAshows that the actuators deflect in accord with the referencedeflections. Thus, the proposed URLSCA is effectively robust to theunstructured uncertainties in the control effectiveness matrix. Thereference signals (pitch, yaw, and roll) are tracked well, anddisturbance is also suppressed.

Figure 3 shows that the reference signals (pitch, yaw, and roll) aretracked well, the actuators deflect in accord with the referencedeflections, and the disturbance is also suppressed. Thus, theproposed SRLSCA is effectively robust to the structured uncertaintyin the control effectiveness matrix.

Figure 4 shows that the reference signals (pitch, yaw, and roll) aretracked well, the actuators deflect in accord with the referencedeflections, and the disturbance is also suppressed. Thus, theproposed LFSRLSCA is effectively robust to the linear fractionalstructured uncertainty in control effectiveness matrix, but the resultsare not better than those obtained by the two previously proposedmethods.

In summary, three necessary statements are presented as follows:1) In the PICA scheme, the control surfaces’deflection sees drastic

fluctuations when the reference signals either jump from one stable

value to another one or otherwise. The appearance of this fluctuationresults from not considering the dynamics of the actuator.

2) By comparing the PICA andRLSCA, onefinds that theRLSCAis able to distribute the three total moments to control surfaces whenuncertainty exists in control effectivenessmatrix. The unknown inputdisturbance is suppressed by the designed controller, and theRLSCAis robust to uncertainty in the control effectiveness matrixsimultaneously.

3) Themain purpose of this paper is to provide an effective controlallocation method when the control effectiveness matrix is subject touncertainties.When the actuator saturation occurs, the stability of thewhole system is a challenging problem [21].

Conclusions

In this paper, the problem of disturbance rejection and controlallocation in the presence of uncertainties in the control effectiveness

matrix has been investigated. An H2=H1 feedback controller isdesigned to guarantee the stability and simultaneously suppress thedisturbance in the closed-loop SLV model, and a feedforwardcontroller is designed for tracking the reference signals. Both thefeedback and feedforward controllers cooperate to produce three axismoments. The robust least-squares method is introduced to solve theproblem of control allocationwith the uncertain control effectivenessmatrix while maintaining the control inputs within their requiredbounds. Three RLSCA methods are proposed when the controleffectiveness matrix is subject to unstructured, structured, and linearfractional structured uncertainties.

According to the simulation results, it is concluded that the controleffectors can deflect smoothly to produce the required virtual controlmoments by use of the proposed RLSCA. The RLSCA is robust touncertainty in the control effectiveness matrix. Disturbance isrejected by the feedback controller, and the reference signals (pitch,yaw, and roll) are well tracked by the feedforward controller. Theinfluence of RLSCA’s online computation on the stability of thewhole flight control system and the stability in presence of actuatorsaturation are beyond the scope of this paper.

Appendix: System Matrices of Satellite Launch Vehicle

The system matrices of the laboratory experimental SLV modelare given by

A�

0 1 0 0 0 0

0:7066 0 1:87 � 10�5 0 0 0

0 0 0 1 0 0

2:71 � 10�5 0 0:4379 0 0 0

0 0 0 0 0 1

5:71 � 10�4 0 5:468 � 10�4 0 0 0

266666666664

377777777775

Bd �

0 0 0 0

7:82 � 10�4 �6:84 � 10�6 �4:34 � 10�4 �8:43 � 10�8

0 0 0 0

3 � 10�8 7:61 � 10�3 �1:665 � 10�8 9:376 � 10�5

0 0 0 0

�6:32 � 10�7 9:506 � 10�6 3:509 � 10�7 1:171 � 10�7

266666666664

377777777775

�B� Bv B B�0:2851 �0:2851 �0:2851 0:2851 0:0968 0 �0:0968 0

�0:2851 �0:2851 0:2851 0:2851 0 0:0968 0 �0:0968�0:2851 0:2851 �0:2851 0:2851 0 0:0968 0 0:0968

264

375

Bv �

0 0 0

5:7508 2:191 � 10�4 �1:495 � 10�3

0 0 0

2:207 � 10�4 5:173 2:077 � 10�3

0 0 0

�4:65 � 10�3 2:17 � 10�2 14:15276

266666666664

377777777775; C0 � C�

1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

264

375

Acknowledgments

This work is supported byNational Natural Science Foundation ofChina under grants 60874011 and 90916003. The authors gratefullyacknowledge Frank Morgan’s help revising this paper and Li Gao’shelp providing optimization theory.

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1642 CUI ANDYANG

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CUI ANDYANG 1643