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Output-feedback control for switched linear systemssubject to actuator saturationChang Duan a & Fen Wu aa Department of Mechanical and Aerospace Engineering, North Carolina State University,Raleigh, NC 27695, USA

Available online: 11 Jun 2012

To cite this article: Chang Duan & Fen Wu (2012): Output-feedback control for switched linear systems subject to actuatorsaturation, International Journal of Control, DOI:10.1080/00207179.2012.691611

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International Journal of Control2012, 1–14, iFirst

Output-feedback control for switched linear systems subject to actuator saturation

Chang Duan and Fen Wu*

Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA

(Received 31 August 2011; final version received 3 May 2012)

This article is devoted to the output-feedbackH1 control problem for switched linear systems subject to actuatorsaturation. We consider both continuous- and discrete-time switched systems. Using the minimal switching rule,nonlinear output feedbacks expressed in the form of quasi-linear parameter varying system are designed to satisfya pre-specified disturbance attenuation level defined by the regional L2 (‘2)-gains over a class of energy-boundeddisturbances. The conditions are expressed in bilinear matrix inequalities and can be solved by line search coupledwith linear matrix inequalities optimisation. A spherical inverted pendulum example is used to illustrate theeffectiveness of the proposed approach.

Keywords: switching control; output-feedback control; actuator saturation; linear matrix inequalities

1. Introduction

Switched systems are a subclass of hybrid systems thatinvolve an interaction between continuous time anddiscrete event dynamics (Liberzon 2003). It arises inmany engineering fields, such as communication net-works (Hespanha, Bohacek, Obraczka, and Lee 2001),embedded systems (Zhang and Hu 2008) and powerelectronics (Geyer, Papafotiou, and Morari 2005). Themathematical framework and tools describing them areemerging rapidly due to its theoretical and practicalimportance. For this reason, switched systems havereceived considerable attention in the recent literatureworks (see Liberzon and Morse 1999; DeCarlo,Branicky, Pettersson, and Lennartson 2000;McMclamroch and Kolmanovsky 2000; Sun and Ge2005 and the references therein). However, even theswitching control of multiple linear time-invariant(LTI) plants has been considered to be a non-trivialproblem since it is possible to get instability byswitching between asymptotically stable systems orstabilise unstable systems by switching (Liberzon andMorse 1999; DeCarlo et al. 2000). The stability ofswitched systems in general depends on underlyingswitching logic, which is a rule that determines theswitching between multiple subsystems. Variousswitching rules have been proposed for the stabilisationof different types of switched systems. For multiplestable LTI systems, the existence of a commonLyapunov function provides stability condition ofswitched systems under arbitrary switching sequences(Liberzon 2003). A justification of this approach comes

from the converse Lyapunov theorem for switched

systems, which states that uniform asymptotical sta-

bility of a switched system implies the existence of acommon Lyapunov function (Dayawansa and Martin

1999). However, this type of stability condition is

deemed too conservative when a particular switching

logic is concerned. Since many switched systems inpractice do not posses a common Lyapunov function,

more attention has been paid to stabilisation under

some properly chosen switching logics. For therestricted class of switching signals such as hysteresis

and dwell-time types (Hespanha and Morse 1999;

Hespanha, Liberzon, and Morse 2003), multiple

Lyapunov functions (one associated with each sub-system) have been shown to be very useful tools for

stability analysis. Specifically, non-traditional stability

conditions have been developed using either piecewisecontinuous Lyapunov functions (Peleties and Decarlo

1991; Wicks, Peleties, and deCarlo 1994; Ranzer and

Johansson 2000) or discontinuous Lyapunov functions

(Branicky 1998; Ye, Michel, and Hou 1998; Zhao andHill 2008). Most of the current works, however, are

focused on stability analysis of switched systems. There

are only few works (Xu and Antsaklis 2004; Geromel,Colaneri, and Bolzern 2008; Zhao and Hill 2008;

Blanchini, Miani, and Mesquine 2009) on controls

of switched systems under arbitrary or specific

switching signals. The existence of a switchingcompensator which stabilises the plant under arbitrary

switching is discussed in Blanchini et al. (2009).

Moreover, observer-based all stabilising controller

*Corresponding author. Email: fwu@eos.ncsu.edu

ISSN 0020–7179 print/ISSN 1366–5820 online

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are parametrised under the assumption of having exactknowledge of the plant configuration. Switching opti-mal control is discussed in Xu and Antsaklis (2004)and a two-stage algorithm is proposed to obtain localsolutions and desired switching rule. In Zhao and Hill(2008), a set of necessary and sufficient conditions areproposed for the existence of a stabilising switchinglaw in terms of multiple generalised Lyapunov-likefunctions, though how to explicitly construct suchLyapunov-like functions remains an open problem.Under min-switching rule, the stability conditions forlinear switched systems expressed through Lyapunov–Metzler inequalities are used in Geromel et al. (2008) tocope with a H2 guaranteed cost control designproblem.

On the other hand, saturation is a widely encoun-tered and most dangerous nonlinearity in controlsystems. As a result, actuator saturation problem hasreceived increasing attention from research community(see, e.g. Bernstein and Michel 1995; Tarbouriech andGarcia 1997; Lin 1998; Hu and Lin 2001 and thereferences therein). This literature covers a wide rangeof topics, including stabilisation, output regulation anddisturbance rejection. More recent literature has wit-nessed a shift of focus to exponentially unstable open-loop systems. Indeed many control systems withactuator saturation, such as flight control, are expo-nentially unstable. As such systems under actuatorsaturation are null controllable only in a part of thestate space, the objectives here are to characterise thenull controllable region (Hu and Lin 2001) and todesign feedback laws that work on the entire nullcontrollable region or a large portion of it. Differentsystematic design procedures based on rigorous theo-retical analysis have been proposed through variousframeworks. Nevertheless, most of this literature onexponentially unstable systems pertains to state feed-back. A few exceptions where output feedback are usedinclude Nguyen and Jabbari (1999), Paim,Tarbouriech, Gomes da Silva, and Castelan (2002),Scorletti, Folcher, and El Ghaoui (2002), Hu and Lin(2004), Dai, Hu, Teel, and Zaccarian (2009) and Wu,Zheng, and Lin (2009). In particular, a gain-scheduledsaturation controller in output-feedback form has beendeveloped in Wu et al. (2009) to attenuate the effect ofthe disturbances on the system output in addition toachieving local stability. Nevertheless, the controlsynthesis conditions are in bilinear matrix inequality(BMI) forms and not easy to solve. In Dai et al. (2009),a saturation control synthesis method is proposed forthe construction of output-feedback controllers withan internal deadzone loop. It has been shown that thesynthesis condition for such a controller can beformulated into a linear matrix inequalities (LMIs)form. However, for linear saturated plant, the presence

of saturation/deadzone function makes the algebraicloop harder to deal with.

This article studies the output-feedback control ofboth continuous-time and discrete-time switched linearsystems subject to actuator saturation. Under min-switching logic, sufficient conditions for the output-feedback control law are derived to stabilise the systemin the presence of actuator saturation and guaranteepre-specified disturbance attenuation level for energy-bounded disturbances. To deal with its non-convexity,the switching control synthesis conditions are furthersimplified to a special case of BMIs and can be solvedby line search coupled with LMI solver. The proposedcontroller also avoids cumbersome algebraic loopwhich facilitates controller implementation.

The notations used in this article are quite stan-dard. R stands for the set of real numbers and Rþ forthe non-negative real numbers. Zþ is the set of non-negative integers. Rm�n is the set of real m� nmatrices.We use Sn�n to denote real, symmetric n� n matrices,and Sn�n

þ for positive-definite matrices. The identitymatrix of any dimension is denoted by I. For twointegers k1, k2, k1< k2, we denote I[k1, k2]¼{k1, k1þ 1, . . . , k2}. The class of Metzler matrices forcontinuous-time systems denoted byM consists of allmatrices �2RN�N with elements �ij, such that �ij� 0for all i, j2 I[1,N] and

PNi¼1 �ij ¼ 0 for all j2 I[1,N].

Another class of Metzler matrices for discrete-timesystems denoted by ~M consists of all matrices~� 2 RN�N with elements ~�ji, such that ~�ji � 0 andPN

j¼1 ~�ji ¼ 1 for all i2 I[1,N]. For real matrices, thehermitian operator He{ � } is defined as He{M}¼MþMT. In large symmetric matrix expressions, termsdenoted as ? will be induced by symmetry. The space ofsquare integrable (summable) functions is denoted byL2 (‘2), that is, for any x2L2 (or ‘2),

kxk2 :¼

Z 10

xTðtÞxðtÞdt

� �12

51 or

kxk2 ¼X1k¼0

xTðkÞxðkÞ

!12

0@

1A:

CoS denotes the convex hull of a set S.

2. Preliminaries

2.1 Regional analysis for saturated linear systems

We consider linear systems subject to saturationnonlinearity. The saturation function is defined as�(ui)¼ sgn(ui)min{ �ui, juij}. Here, we have slightlyabused the notation by using � to denote both thescalar-valued and vector-valued saturation function.Therefore � : Rnu ! Rnu is a vector-valued standard

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saturation function, i.e. �ðuÞ ¼��ðu1Þ �ðu2Þ � � � �ðunuÞ

�T.

The deadzone nonlinearity is closely related to satura-tion function by dz(u)¼ u� �(u). Using deadzonefunctions, the saturated linear systems can be repre-sented as the following general form (Hu, Teel, andZaccarian 2006)

Dx ¼ Axþ B0qþ B1d,

u ¼ C0xþD00qþD01d,

e ¼ C1xþD10qþD11d,

q ¼ dzðuÞ,

8>>><>>>:

ð1Þ

where x 2 Rn, q, u 2 Rnu and d 2 Rnd . The symbol Ddenotes a differentiator for continuous-time systemsand a difference operator for discrete-time systems.Also, let V be the set of nu� nu diagonal matriceswhose diagonal elements are either 1 or 0. There are 2nu

elements in V. Suppose these elements of V are labelledas Dj, j 2 I½1, 2nu �, where for j ¼ z12

nu�1 þ z22nu�2þ

� � � þ znu þ 1 with zi2 {0, 1}, the diagonal elements ofDj will be f1� z1, 1� z2, . . . , 1� znug. DenoteD�j ¼ I� Dj. Clearly, D�j 2 V if Dj2V. In globalanalysis of saturated linear systems, we often use

dzðuÞ 2 Co Dj, j 2 I½1, 2nu �� �

u:

This relation holds for all u 2 Rnu but could beconservative over a local region where the systemoperates. In fact, the saturation and deadzone func-tions will often be sharpened for regional analysis asshown in the following lemma.

Lemma 1 (Hu et al. 2006): Let h(x)¼Hx be a linearmap and suppose ‘Ti Hx 2 � �ui, �ui½ �, where ‘i denotes ithcolumn vector of the identity matrix. For any ui, we have�ðuiÞ 2 Co ui, ‘

Ti Hx

� �and dzðuiÞ ¼ �iðui � ‘

Ti HxÞ for

some �i2 [0, 1].

Using this lemma, it has been shown in Hu et al.(2006) that the original system (1) can be covered by apolytopic differential inclusion as

Dx

e

� 2 Co

ðA j�B0TjHÞ B j

ðC j �D10TjHÞ D j

� x

d

� : j 2 I½1,2nu �

�,

ð2Þ

with

T j ¼ ðI� DjD00Þ�1Dj,

A j ¼ Aþ B0TjC0, B j ¼ B1 þ B0T

jD01,

C j ¼ C1 þD10TjC0, D j ¼ D11 þD10T

jD01:

For the disturbance attenuation problem, we aremainly concerned with a class of energy-boundeddisturbances in continuous time

Ws ¼ d : Rþ ! Rnd ,

Z 1t¼0

dTðtÞdðtÞdt � s2, d 2 L2

�,

or in discrete time

Ws ¼ d : Rþ ! Rnd ,X1k¼0

dTðkÞdðkÞ � s2, d 2 ‘2

( ),

where s is a positive scalar whose value is given. The

level of disturbance attenuation will be measured by

the following performance index:

supxð0Þ¼0,d2Ws

kek2kd k2

:

This performance index specifies regional L2 (‘2)-gain.Note that the L2 (‘2)-gain may not be well defined for

sufficiently large disturbances under the actuator

saturation. With such large disturbances, the state

may go unbounded under any control input. This

motivates us to consider only energy-bounded distur-

bances, and determine the minimum energy amplifica-

tion from disturbance to output.The following lemma provides the L2-gain analysis

condition for saturated continuous-time linear systems.

Lemma 2 (Hu et al. 2006): For any energy-bounded

disturbance d2Ws, if there exists V(x)¼ xTPx with a

positive-definite P matrix satisfying

HefPðA j�B0TjHÞg ? ?

ðB j ÞTP �I ?

C j�D10TjH D j ��2I

264

37550, j 2 I½1, 2nu �,

ð3Þ

�u2is2

‘Ti H

HT‘i P

24

35 � 0, i 2 I½1, nu�: ð4Þ

Then the continuous-time system (1) has its trajectories

contained within �(P, s)¼ { x: xTPx� s2 } and satisfies

kek2<� kdk2 when x(0)¼ 0.

Similarly, the ‘2-gain analysis condition for satu-

rated discrete-time systems is stated in the following

lemma.

Lemma 3: The discrete-time system (1) is regionally

stable and has its ‘2-gain less than � for any energy-

bounded disturbance d2Ws if

P ? ? ?

0 �2I ? ?

A j � B0TjH B j P�1 ?

C j �D10TjH D j 0 I

26664

377754 0, j 2 I½1, 2nu �,

ð5Þ

�u2is2

‘Ti H

HT‘i P

24

35 � 0, i 2 I½1, nu�: ð6Þ

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Proof: First, condition 6 implies s�(P, 1)�L(H ).It guarantees the following polytopic model

Dx

e

� ¼

X2nuj¼1

�j ðAj � B0T

jHÞX2nuj¼1

�jBj

X2nuj¼1

�j ðCj �D10T

jHÞX2nuj¼1

�jDj

2666664

3777775

x

d

� ,

X2nuj¼1

�j ¼ 1, �j � 0

:¼Að�Þ Bð�Þ

Cð�Þ Dð�Þ

" #x

d

�

is a valid description for the discrete-time saturatedsystem (1).

Performing the Schur complement with respect tothe last two rows and columns of (5), we obtain

ATð�ÞPAð�Þ � P ?

BTð�ÞPAð�Þ BTð�ÞPBð�Þ

" #

5�CTð�Þ

DTð�Þ

" #Cð�Þ Dð�Þ� �

þ0 ?

0 �2I

� : ð7Þ

The system is asymptotically stable sinceAT(�)PA(�)�P is negative definite from (7). Using aquadratic Lyapunov function V(x)¼xTPx, we have

Vðxðkþ 1ÞÞ �VðxðkÞÞ

¼ ½xTðkÞATð�Þ þ dTðkÞBTð�Þ�P½Að�ÞxðkÞ þ Bð�ÞdðkÞ�

� xðkÞTPxðkÞ

¼xðkÞ

dðkÞ

� TATð�ÞPAð�Þ �P ?

BTð�ÞPAð�Þ BTð�ÞPBð�Þ

" #xðkÞ

dðkÞ

�

5�eTðkÞeðkÞ þ �2dTðkÞdðkÞ:

ð8Þ

The last inequality follows from (7). Summing upboth sides of (8) from k¼ 0 to 1 with V(x(0))¼ 0,V(x(1))> 0, we get kek22 5 �2kd k22. Thus the regional‘2-gain is less than �. œ

2.2 Stability and performance analysis forswitched linear systems

Consider that the following switched linear systemconsists of N subsystems

Dx ¼ Arxþ Brd,

e ¼ CrxþDrd,

ð9Þ

where the subscript r denotes the active plant index andx(t) is assumed to be available to determine the

switching logic r(t) for all t> 0. Adopting the piecewisequadratic Lyapunov function

vðxÞ ¼ min1�r�N

xTPrx, ð10Þ

where Pr> 0 8r2 I[1,N], it will be shown in the nexttheorem that if these positive-definite matrices satisfysome well-defined conditions, then the min-switchingstrategy

rðtÞ ¼ arg min1�r�N

xðtÞTPrxðtÞ ð11Þ

is globally stabilising the switched systems. At thepoint x where minimum in (10) is not unique, thefunction v(x) is not differentiable and the switchingrule r(t) will choose arbitrarily one of subsystemsswitching to. As a special case that satisfies the hard-to-check non-increasing condition of multipleLyapunov functions, the above-mentioned min-switch-ing strategy (Liberzon and Morse 1999) is easy toutilise.

Theorem 1 (Deaecto and Geromel 2010): For a scalar� > 0, assume that there exist positive-definite matricesPr, r2 I[1,N], a Metzler matrix �2M such that thefollowing Riccati–Metzler inequalities hold for all r

HefATr Prg þ

XNk¼1

�krPk ? ?

BTr Pr �I ?

Cr Dr ��2I

266664

3777755 0, ð12Þ

withPN

k¼1 �kr ¼ 0. Then the min-switching strategy (11)is globally stabilising the continuous-time switched linearsystem (9) and

supd2L2

kek22 � �2kd k22 � 0:

Its proof is provided in Deaecto and Geromel(2010) and will be omitted for space reason. An usefulproperty of the Lyapunov–Metzler inequalities is thatit can be generalised to solve the Hamilton–Jacobi–Bellman inequality which closely relates to optimalcost lower bound calculation (Ranzer and Johansson2000) for switched systems.

Theorem 2 (Deaecto, Geromel, and Daafouz2010): Given a scalar � > 0, if there exist positive-definite matrices Pr, r2 I[1,N], a Metzler matrix~� 2 ~M such that the following Riccati–Metzler inequal-ities hold for all r

Pr ? ? ?

0 �2I ? ?

PprAr PprBr Ppr ?

Cr Dr 0 I

26664

377754 0, ð13Þ

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where Ppi ¼PN

j¼1 ~�jiPj for all i2 I[1,N]. Then the min-

switching strategy (11) is globally stabilising and the

discrete-time switched linear system (9) satisfies

supd2‘2

kek22 � �2kd k22 � 0:

Its proof can be found in Deaecto et al. (2010) and

the references therein.

3. Output-feedback control design

Given a set of linear plant subject to actuator

saturation,

Dxp

e

y

264

375 ¼ Gr

xp

d

�ðuÞ

264

375, ð14Þ

where

Gr ¼ss

Ap Bp1 Bp2

Cp1 Dp11 Dp12

Cp2 Dp21 0

264

375

r

, r 2 I½1,N�

where xp 2 Rnx is the state, u 2 Rnu is the control input,

e 2 Rne is the controlled output, d 2 Rnd is energy-

bounded disturbance, y 2 Rny is the measurement

output and subscript r denotes the active plant index.We will consider a switching controller consisting

of a set of dynamic output-feedback laws associated

with each subsystem r

Dxc ¼X2nuj¼1

�jAjcr

!xc þ

X2nuj¼1

�jBjcr

!y,

u ¼ Ccrxc þDcry,

8><>: ð15Þ

where xc 2 Rnxc is the controller state. A jcr,B

jcr and Ccr,

Dcr are constant matrices to be determined, subscript r

denotes the active controller index.Note that the values of the parameters �j(xc, y)’s,

which are dependent on xc and y and reflect in a way

the severity of control saturation, are available for real-

time use in gain-scheduling control. For a matrix

Hcr 2 Rnu�nxc , we define the linear subspace

Lð½0 Hcr�Þ

¼ fðx, xcÞ 2 Rnx � Rnxc : j‘Ti Hcrxcj � �ui, i 2 I½1, nu�g,

where ‘i denotes the ith column vector of the identity

matrix. Then one choice of parameters �j’s as

Lipschitzian functions of xc and y is specified as

follows: For each i2 I[1, nu], let

�iðxc, yÞ ¼

0,

if ‘Ti ðCcrxc þDcryÞ ¼ ‘Ti Hcrxc,

dzð‘Ti ðCcrxc þDcryÞÞ

‘Ti ðCcrxc þDcry�HcrxcÞ,

otherwise,

8>>>>><>>>>>:

ð16Þ

and for each j 2 I½1, 2nu �, let zi2 {0, 1} be such that

j ¼ z12nu�1 þ z22

nu�2 þ � � � þ znu þ 1, then a set of suit-

able �j’s will be

�j ðxc, yÞ ¼Ynui¼1

zið1� �iðxc, yÞÞ þ ð1� ziÞ�iðxc, yÞ½ �:

ð17Þ

The system connected to the switched output-

feedback controller is depicted in Figure 1. There are

two feedback loops. The external loop is defined by the

connection of y to u and the internal one is responsible

to activate the selected subsystem and its controller.

Our objective is to design the set of dynamic output-

feedback laws of form (15) and the switching rule

r( y[0,t]) where y[0,t] is the trajectory y(�), � 2 [0, t], inorder to render the origin x¼ 0 of the closed-loop

system asymptotically stable, and achieve disturbance

rejection level � for energy-bounded disturbances.By substituting �(u)¼ u�dz(u), the original plant

(14) can be rewritten in the following form:

Dxp ¼ Aprxp � Bp2rqþ Bp1rdþ Bp2ru,

e ¼ Cp1rxp �Dp12rqþDp11rdþDp12ru,

y ¼ Cp2rxp þDp21rd,

q ¼ dzðuÞ:

8>>><>>>:

ð18Þ

Combining the dynamics of the plant (18) and

the output-feedback law (15), one can obtain the

Figure 1. Closed-loop structure.

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closed-loop system as

Dx

u

e

264

375 ¼

Arð�Þ B0r B1rð�Þ

C0r D00r D01r

C1r D10r D11r

264

375

x

q

d

264

375, ð19Þ

q ¼ dzðuÞ, ð20Þ

where x ¼ xTp xTc� �T

and

Arð�Þ ¼

Apr þ Bp2rDcrCp2r Bp2rCcrX2nuj¼1

�jBjcrCp2r

X2nuj¼1

�jAjcr

264

375,

B0r B1rð�Þ� �

¼

�Bp2r Bp1r þ Bp2rDcrDp21r

0X2nuj¼1

�jBjcrDp21r

264

375,

C0r

C1r

� ¼

DcrCp2r Ccr

Cp1r þDp12rDcrCp2r Dp12rCcr

� ,

D00r D01r

D10r D11r

� ¼

0 DcrDp21r

�Dp12r Dp11r þDp12rDcrDp21r

� :

Since the direct feed-through term D00r from q to u is

zero, it is clear that the closed-loop system does not

contain algebraic loop. Without algebraic loop, the

analysis and implementation of the saturated control

system can be simplified considerably. Specifically, we

have T j¼Dj and the polytopic representation of the rth

closed-loop subsystem becomes

Dx

e

� ¼X2nuj¼1

�j�A jr

�B jr

�C jr

�D jr

" #x

d

� , � 2 �, ð21Þ

with

� ¼

(� 2 R2nu :

X2nuj¼1

�j ¼ 1,

0 � �j � 1, j 2 I½1, 2nu �

),

�A jr

�B jr

�C jr

�D jr

" #¼

Apr 0 Bp1r

0 0 0

Cp1r 0 Dp11r

264

375þ

0 Bp2r

I 0

0 Dp12r

264

375

�X2nuj¼1

�jA j

cr B jcr

D�j Ccr þ DjHcr D�j Dcr

" # !

�0 I 0

Cp2r 0 Dp21r

" #:

The above quasi-linear parameter varying (LPV) form

is a valid representation of the saturated subsystem r

when (xp, xc)2L([0 Hcr]). Moreover, its state-space

matrices are linear functions of the scheduling

parameter �.

Theorem 3 (Continuous-time case): Given a scalar

� > 0, if there exist positive-definite matrices

Rr,S 2 Snx�nxþ , 8r 2 I½1,N�, matrices Trk, r, k2 I[1,N],

r 6¼ k, a Metzler matrix �2M and matrices

ð �A jcr,

�B jcr Þ 2 Rnx�nx � Rnx�ny , j 2 I½1, 2nu �, ð �Ccr, �DcrÞ 2

Rnu�nx � Rnu�ny , �Hcr 2 Rnu�nx , r2 I[1,N] satisfying the

following conditions for r2 I[1,N]

HefAprRr þ Bp2r�U jcrg

þPN

k6¼r¼1 �krTrk

( )? ? ?

�A jcr þ AT

pr þ CTp2r

�V jcr

� TBTp2r HefSApr þ �B j

crCp2rg ? ?

BTp1r þDT

p21r�V jcr

� TBTp2r BT

p1rST þDT

p21r�B jcr

� T�I ?

Cp1rRþDp12r�U jcr Cp1r þDp12r

�V jcrCp2r Dp11r þDp12r

�V jcrDp21r ��

2I

2666666664

37777777755 0, ð22Þ

Trk þ Rr ? ?

Rr Rk ?

I I S

264

375 � 0, k 2 I½1,N�, r 6¼ k, ð23Þ

�u2is2

? ?

�HTcr‘i Rr ?

0 I S

26664

37775 � 0, i 2 I½1, nu�, ð24Þ

Rr I

I S

� 4 0, ð25Þ

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with �U jcr ¼ D�j �Ccr þ Dj

�Hcr and �V jcr ¼ D�j �Dcr. Then the

regional L2-gain of the continuous-time switched closed-

loop system under the min-switching logic (11) is less

than or equal to � for any disturbance d2Ws.Moreover,

the coefficient matrices of the set of the nxth order

output-feedback controllers are given by

A jcr B j

cr

Ccr Dcr

Hcr 0

264

375

¼

N SBp2rD�j SBp2rDj

0 Inu 0

0 0 Inu

264

375�1 �A j

cr�SAprRr�B jcr

�Ccr�Dcr

�Hcr 0

264

375

�MT

r 0

Cp2rRr Iny

" #�1, ð26Þ

where Mr,N 2 Rnx�nx are such that MrNT ¼ Inx � RrS.

Proof: Let

~Ur ¼Rr I

MTr 0

� , U ¼

I S

0 NT

� ,

and specify

~Pr ¼ U ~U�1r ¼S N

NT Xr

� ,

with Xr ¼ �NTRrM

�Tr . We also have

~UTr

~Pr~Ur ¼

Rr II S

� :

It is clear that ~Pr is positive definite if and only if

S4R�1r 4 0 for all r2 I[1,N]. Moreover

~UTr ð

~Pr�Ajr Þ

~Ur¼AprRrþBp2r

�Ujcr AprþBp2r

�VjcrCp2r

�Ajcr SAprþ �Bj

crCp2r

" #,

~UTr ð

~Pr�Bjr Þ¼

Bp1rþBp2r�VjcrDp21r

SBp1rþ �BjcrDp21r

" #,

�Cjr

~Ur¼ Cp1rRrþDp12r�Ujcr Cp1rþDp12r

�VjcrCp2r

� �,

where the transformed controller data relates to the

original ðA jcr, B

jcr, Ccr, Dcr, HcrÞ through

�A jcr

�B jcr

�Ccr�Dcr

�Hcr 0

264

375

¼

SAprRr 0

0 0

0 0

264

375þ

N SBp2rD�j SBp2rDj

0 I 0

0 0 I

264

375

�

A jcr B j

cr

Ccr Dcr

Hcr 0

264

375 MT

r 0

Cp2rRr I

" #: ð27Þ

FromPN

k¼1 �kr ¼ 0, we have �rr ¼ �PN

k6¼r¼1 �kr.Therefore

XNk¼1

�kr ~Pk ¼XNk6¼r¼1

�kr ~Pk þ �rr ~Pr ¼XNk6¼r¼1

�krð ~Pk � ~PrÞ,

and

~UTr

�XNk¼1

�kr ~Pk

�~Ur¼ ~UT

r

XNk6¼r¼1

�krð ~Pk� ~PrÞ

" #~Ur

¼ ~UTr

0 0

0XNk 6¼r¼1

�krðXk� XrÞ

26664

37775 ~Ur

¼

Mr

XNk6¼r¼1

�krðXk� XrÞ

" #MT

r 0

0 0

26664

37775:ð28Þ

Since MrNT¼ I�RrS, we also have Xr ¼

NTðS� R�1r Þ�1N4 0. Through some algebraic manip-

ulations, we obtain

Mr

XNk6¼r¼1

�krðXk� XrÞ

" #MT

r

¼XNk6¼r¼1

�kr S�1�RrþðRr�S�1ÞðRk�S�1Þ�1ðRr�S�1Þ

� �:

ð29Þ

Assuming (23) holds, we apply the Schur complement

to the last two rows and columns and obtain

Trk þ Rr � S�1 Rr � S�1

Rr � S�1 Rk � S�1

24

35 � 0:

Applying the Schur complement to the last row and

column again, and multiplying both sides by �kr andsumming up we have

XNk6¼r¼1

�krTrk

�XNk6¼r¼1

�kr½S�1�RrþðRr�S

�1ÞðRk�S�1Þ�1ðRr�S

�1Þ�:

ð30Þ

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Combining (28), (29) and (30), we conclude that

~UTr

�XNk¼1

�kr ~Pk

�~Ur �

XNk6¼r¼1

�krTrk 0

0 0

2664

3775:

Then multiplying diagf ~UTr , I, Ig to the left and its

transpose from the right of inequality (12), we obtain

inequalities (22) and (25).Note that condition s�(P, 1)L([0 Hcr]) is implied

by

s2 0 ‘Ti Hcr

� �~P�1r

0

HTcr‘i

� � �u2i :

Using the Schur complement, the above inequality can

be rewritten as

�u2is2

0 ‘Ti Hcr

� �0

HTcr‘i

" #~Pr

266664

377775 � 0,

which is equivalent to Equation (24) through a

congruent transformation and ½0 Hcr� ~Ur ¼ ½ �Hcr 0�.

Then the result follows from Lemma 2 and

Theorem 1. œ

Due to the non-convex nature of the products of

variables in (22), it is often difficult to find a numerical

solution. Alternatively, Theorem 3 can be simplified by

assuming specific forms of �kr, as shown in the

following corollary.

Corollary 1 (Continuous-time case): Assuming there

exist positive-definite matrices S, Rr, symmetric matri-

ces Tk, matrices �A jcr,

�B jcr,

�Ccr, �Dcr, �Hcr, and positive

scalars �, � such that conditions (24), (25) and the

following inequalities

hold for all r2 I[1,N]. Then the result of Theorem 3

remains valid.

Proof: Consider a matrix �2M with the particular

structure � ¼ ��rr ¼PN

k6¼r¼1 �kr 4 0, which obviously

satisfies ��1PN

k 6¼r¼1 �kr ¼ 1 for all r2 I[1,N].

Multiplying inequality (31) by �kr, summing up for

all k2 I[1,N], k 6¼ r and finally, multiplying by ��1, weobtain inequality (22) with Tkr¼Tk for all r2 I[1,N],

r 6¼ k. With the same choice, the inequality (23) reduces

to inequality (32), thus concluding the proof. œ

The corollary states a sufficient condition that is

more suitable for numerical calculations. For each

fixed � > 0, inequalities (24), (25), (31) and (32) are

expressed in terms of LMIs with respect to all variables

including the scalar � > 0. Hence, the determination of

the optimal solution of the following optimisation is

convex.

�ð�Þ ¼ inf�,S,Rr,Tr, �A j

cr, �B jcr, �Ccr, �Dcr, �Hcr2�ð�Þ

�, ð33Þ

where �(�) is the set of all feasible solutions of the

LMIs (24), (25), (31) and (32). The optimisation (33)

enables us to determine, by line search, � such that

�(�)¼ inf�>0 �(�). Afterwards, from Equation (26) the

full order switched output-feedback controllers can be

constructed with the min-switching strategy. It is clear

that the resulting �(�) value is an upper bound

to the optimal performance level, that is,

supd2L2 kek22=kd k

22 � �

2ð�Þ, where the equality is gen-

erally not attained.Parallel to continuous-time result, the switched

control synthesis condition for discrete-time saturated

linear systems is as follows:

Theorem 4 (Discrete-time case): Given a scalar � > 0,

if there exist positive-definite matrices Rr,S 2 Snx�nxþ

8r 2 I½1,N�, matrices Trk, r, k2 I[1,N], r 6¼ k, a Metzler

matrix ~� 2 ~M and matrices ð �A jcr,

�B jcr Þ 2 Rnx�nx�

Rnx�ny , j 2 I½1, 2nu �, ð �Ccr, �DcrÞ 2 Rnu�nx � Rnu�ny ,�Hcr 2 Rnu�nx , Jr, r2 I[1,N] satisfying the following

HefAprRr þ Bp2r�U jcrg þ �Tk ? ? ?

�A jcr þ AT

pr þ CTp2r

�V jcr

� TBTp2r HefSApr þ �B j

crCp2rg ? ?

BTp1r þDT

p21r�V jcr

� TBTp2r BT

p1rST þDT

p21r�B jcr

� T�I ?

Cp1rRþDp12r�U jcr Cp1r þDp12r

�V jcrCp2r Dp11r þDp12r

�V jcrDp21r ��

2I

26666664

377777755 0, ð31Þ

Tk þ Rr ? ?

Rr Rk ?

I I S

264

375 � 0, k 2 I½1,N�, r 6¼ k ð32Þ

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conditions

with �U jcr ¼ D�j �Ccr þ Dj

�Hcr, �V jcr ¼ D�j �Dcr and Vr ¼

Jr þ JTr �PN

k¼1 ~�krTrk. Then the regional ‘2-gainof the discrete-time switched closed-loop system is

less than or equal to � for any disturbance d2Ws.

Moreover, the coefficient matrices of the set of the nxth

order output-feedback controllers are given by

Equation (26).

To prove Theorem 4, we need two lemmas whose

proofs will be omitted for space reason.

Lemma 4 (Deaecto et al. 2010): Assume that the

symmetric matrices Rr, S satisfy (37) and ~� 2 ~M. The

following equalities hold:

XNk¼1

~�krRk ?

I S

� �1 !�1¼

�r ?

I S

� , r 2 I½1,N�,

ð38Þ

where

�r ¼ S�1 þXNk¼1

~�kr Rk � S�1� �1 !�1

, r 2 I½1,N�:

ð39Þ

Lemma 5 (Deaecto et al. 2010): If the positive-definite

matrices Rr, S, Trk and matrices Jr for all r,

k2 I[1,N]� I[1,N] are such that Vr ¼ Jr þ JTr �PNk¼1 ~�krTrk 4 0 and satisfy the LMI (35), then we

have �r>Vr.

The proof of Theorem 4 is as follows:

Proof: We will specify ~Ur, U and ~Pr as in the proof of

Theorem 3.Given (35) and (37), we also have

UT ~P�1pr U ¼XNk¼1

~�kr UT ~P�1k U� �1 !�1

¼�r ?

I S

� 4Vr ?

I S

� ð40Þ

by Lemma 4 and Lemma 5.By multiplying diag( ~UT

r , I, ð~P�1pr UÞ

T, I) to the left

side and its transpose to the right side of Equation (13),

we have

~UTr

~Pr~Ur ? ? ?

0 �2I ? ?

UT �A jr

~Ur UT �B jr UT ~P�1pr U ?

�C jr

~Ur�D jr 0 I

266664

3777754 0: ð41Þ

Bringing in the value of ~UTr

~Pr~Ur, UT �A j

r~Ur,

UT �B jr ,

�C jr

~Ur as calculated in Theorem 3 and replacing

UT ~P�1pr U with its lower bound (right-hand side of

Equation (40), we arrive at Equation (34). Thus, from

the derivations above, we conclude Equations (34),

(35) and (37) guarantee Equation (13).Note that condition s�(P, 1)L([0 Hcr]) is guar-

anteed by

s2 0 ‘Ti Hcr

� �~P�1r

0HT

cr‘i

� � �u2i :

Rr I ? ? ? ?

I S ? ? ? ?

0 0 �2I ? ? ?

AprRr þ Bp2r�U jcr Apr þ Bp2r

�V jcrCp2r Bp1r þ Bp2r

�V jcrDp21r Vr ? ?

�A jcr SApr þ �B j

crCp2r SBp1r þ �B jcrDp21r I S ?

Cp1rRþDp12r�U jcr Cp1r þDp12r

�V jcrCp2r Dp11r þDp12r

�V jcrDp21r 0 0 I

2666666664

37777777754 0, ð34Þ

Trk ? ?

Jr Rk ?

I I S

264

375 � 0, k 2 I½1,N�, r 6¼ k, ð35Þ

�u2is2

? ?

�HTcr‘i Rr ?

0 I S

26664

37775 � 0, i 2 I½1, nu�, ð36Þ

Rr I

I S

� 4 0, ð37Þ

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Using Schur complements, it can be rewritten as

�u2is2

0 ‘Ti Hcr

� �0

HTcr‘i

� ~Pr

2664

3775 � 0,

which is equivalent to (36) through a congruenttransformation and ½0 Hcr� ~Ur ¼ ½ �Hcr 0�. Then it fol-lows from Lemma 3 and Theorem 2 that the proof ofTheorem 4 is accomplished. œ

Unfortunately, the synthesis condition in Theorem4 is non-convex due to the products of variables in Vr.As an alternative, Theorem 4 can be simplified byassuming specific forms of ~�kr.

Corollary 2: For any scalar 0� � < 1, the result ofTheorem 4 remains valid whenever Vr ¼ Jr þ JTr �ð�Tr þ ð1� �ÞTkÞ for all r 6¼ k2 I[1,N]� I[1,N] andTrk¼Tk for all r, k2 I[1,N]� I[1,N].

Proof: Choosing ~� 2 ~M with a special structurecomposed by the same element � on the main diagonal,that is, ~�rr ¼ � for all r2 I[1,N], the remaining elementssatisfy ð1� �Þ�1

PNk6¼r¼1 ~�kr ¼ 1 for all r2 I[1,N]. As

~�kr � 0 for all rk2 I[1,N]� I[1,N], we have

XNk¼1

~�krTrk ¼ �Tr þXNk 6¼r¼1

~�krTk

¼ ð1� �Þ�1XNk 6¼r¼1

~�krð�Tr þ ð1� �ÞTkÞ ð42Þ

and the proof is concluded. œ

Similarly to the continuous-time case, inequalities(34), (35), (36) and (37) are expressed in terms of LMIswith respect to all variables including the scalar � > 0for each fixed 0� � < 1. Hence, the optimal perfor-mance � can be solved by the following convexoptimisation:

�ð�Þ ¼ inf�,S,Rr,Tr, �A j

cr, �B jcr, �Ccr, �Dcr, �Hcr2�ð�Þ

�, ð43Þ

where �(�) is the set of all feasible solutions of theLMIs (34), (35), (36) and (37). Afterwards, fromEquation (26) the gains of switched output-feedbackcontrollers can be constructed.

4. Example

In this section, we consider the switching control ofspherical inverted pendulum (SIP) in Zhang, Hu, andAbate (2012). As shown in Figure 2, a sphericalpendulum is mounted on a cart through an unactuatedjoint (�, ). The cart can move in the x� y plane by

two orthogonal planar forces ux and uy. The dynamics

of the SIP can be described by the following set of

normalised differential equations

€x ¼ ux,

€y ¼ uy,

€� ¼ sinð�Þ cosð�Þ � sinð�Þ cosð�Þ _2

� cosð�Þux þ sinð�Þ sinðÞuy,

€ ¼1

cos2ð�Þðcos � sinð�Þ þ 2 sinð�Þ cosð�Þ _� _

� cosð�Þ cosðÞuyÞ,

8>>>>>>>>>><>>>>>>>>>>:

ð44Þ

where (x, y) is the normalised planar location of the

center of mass of the cart, and angles and � denote

rotation about the x-axis and y-axis, respectively.

We denote the state of the pendulum as q ¼

ðx, _x, y, _y, �, _�,, _Þ. The control input is composed by

the exclusive actuation forces ux, uy in x, y directions.

The control goal is to (locally) regulate the pendulum

to the unstable equilibrium point q¼ 0 and satisfy the

disturbance attenuation criterion described in previous

section. The SIP is a widely used example to test

advanced control strategies and is also a reasonable

model for rocket propelled body (Zhang et al. 2012).

Instead of treating the cart as a point mass that can be

propelled along arbitrary planar direction as in most

literature, we assume at any time only one of the two

forces ux and uy can be applied to the cart and there are

bounds on the forces, i.e. they are subjected to actuator

saturation. As a result, the nonlinear SIP dynamics can

be described by the switching of two subsystems

_q 2 f f1ðq, uÞ, f2ðq, uÞg with

f1ðq, uÞ ¼ f ðq, uÞjuy¼0 and f2ðq, uÞ ¼ f ðq, uÞjux¼0,

ð45Þ

Figure 2. Spherical inverted pendulum (Zhang et al. 2012).

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where indices 1, 2 represent the SIP under x- and

y-axes actuation separately. The switched nonlinear

system has only one continuous input variable u,

denoting the composed force magnitude, while the

force direction is determined by the active subsystem

index.

4.1 Continuous-time case

Linearising the system around its vertical equilibrium

(q, u)¼ 0 yields the continuous-time switched system

(14) with

G1 ¼ss

Ap Bp1 Bp2

Cp1 Dp11 Dp12

Cp2 Dp21 0

264

375

1

, G2 ¼ss

Ap Bp1 Bp2

Cp1 Dp11 Dp12

Cp2 Dp21 0

264

375

2

,

ð46Þ

where

½Ap�1 ¼ ½Ap�2 ¼A11 0

0 A22

� ,

A11 ¼

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

26664

37775, A22 ¼

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

26664

37775,

½Bp1�1 ¼ ½Bp1�2 ¼ 0 0 0 0 0:5 0 0:2 0� �T

,

½Bp2�1 ¼ 0 1 0 0 0 �1 0 0� �T

,

½Bp2�2 ¼ 0 0 0 1 0 0 0 �1� �T

,

½Cp1�1 ¼ ½Cp1�2 ¼ ½Cp2�1 ¼ ½Cp2�2

¼

1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0

26664

37775,

0 5 10 15−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2(a) (b)

(c) (d)

t (second)

u x(t) 0.866 0.867 0.868 0.869

0

0.5

1

0 5 10 15−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t (second)

u y(t) 0.866 0.867 0.868 0.869

0

0.5

1

0 5 10 15−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

t (second)

angl

e (r

adia

nt)

θφ

0 5 10 150.8

1

1.2

1.4

1.6

1.8

2

2.2

t (second)

activ

e sy

stem

inde

x

0.866 0.867 0.868 0.869

1

1.2

1.4

1.6

1.8

2

Figure 3. Continuous-time switched control: (a) control input ux(t), (b) control input uy(t), (c) controlled output and (d) activesubsystem index.

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and Dp11, Dp12, Dp21 are zero matrices with appropriatedimension. Note that Bp1 are picked arbitrarily tointroduce disturbance and (x, y, �,) are measuredoutput. Using Corollary 1, we perform line search andobtain �¼ 100 to achieve the suboptimal L2-gain� ¼ 1.5980 for the disturbance level s¼ 0.1. With animpulse disturbance of magnitude 0.5 and starting from0.5 s and lasting for 0.7 s, we carry out the simulationsusing the designed continuous-time switchingcontrollers.

Figure 3(a) and (b) shows the trajectories of thecontrol signal ux(t) and uy(t), respectively. Saturation atmagnitude 1 and �1 can be observed from 0.5 to 2 s.Controlled output trajectory is shown in Figure 3(c).Figure 3(d) shows the active subsystem index between6 s to 7 s. It is observed that the proposed controllerindeed stabilised the system andmet the specification ondisturbance attenuation when subjected to actuatorsaturation. However, frequent chattering can be seenfrom the zoomed windows in Figure 3(a) and (b) andactive subsystem index in Figure 3(c) due to thecontroller switching behaviour. The chattering in theswitching, though is always stable (Geromel andColaneri 2006), is undesirable from actual

implementation aspect. Interested reader can refer toDuan and Wu (submitted) for possible approaches toreduce chattering. Another way for chattering reduc-tion is to use discrete-time control and this will bediscussed next.

4.2 Discrete-time case

By discretising (46) with a sampling time Ts¼ 0.2 s,we obtain a discrete-time switching system (14) with

G1¼ss

Ap Bp1 Bp2

Cp1 Dp11 Dp12

Cp2 Dp21 0

264

375

1

, G2¼ss

Ap Bp1 Bp2

Cp1 Dp11 Dp12

Cp2 Dp21 0

264

375

2

,

ð47Þ

where

½Ap�1 ¼ ½Ap�2 ¼A11 0

0 A22

� ,

A11 ¼

1 0:2 0 0

0 0 1 0

0 0 1 0:2

0 0 0 1

26664

37775,

0 5 10 15 20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

k

u x(k

)

0 5 10 15 20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

k

u y(k

)

0 5 10 15 20−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

k

angl

e (r

adia

nt)

θ(k)φ(k)

0 5 10 15 200.8

1

1.2

1.4

1.6

1.8

2

2.2

k

activ

e su

bsys

tem

inde

x

(a)(b)

(c) (d)

Figure 4. Discrete-time switched control: (a) control input ux(t), (b) control input uy(t), (c) controlled output and (d) activesubsystem index.

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A22 ¼

1:02 0:2013 0 0

0:2013 1:02 0 0

0 0 1:02 0:2013

0 0 0:2013 1:02

26664

37775,

½Bp1�1 ¼ ½Bp1�2

¼ 0 0 0 0 0:1007 0:01 0:0403 0:004� �T

,

½Bp2�1 ¼ 0:02 0:2 0 0 �0:0201 �0:2013 0 0� �T

,

½Bp2�2 ¼ 0 0 0:02 0:2 0 0 �0:0201 �0:2013� �T

,

½Cp1�1 ¼ ½Cp1�2 ¼ ½Cp2�1 ¼ ½Cp2�2

¼

1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0

26664

37775,

and Dp11, Dp12, Dp21 are zero matrices with appropriatedimension.

Using Corollary 2, we perform line search andobtain �¼ 0.9 to achieve the suboptimal ‘2-gain� ¼ 3.3396 for the disturbance level s¼ 0.1. With animpulse disturbance of magnitude 0.3 and starting from0.6 s and lasting for 0.4 s, we carry out the simulationsusing our designed discrete-time switching controllers.

Figure 4(a) and (b) shows the trajectories of thecontrol signal ux(k) and uy(k), respectively. Saturationcan be observed for both control signals. Controlledoutput trajectory is shown in Figure 4(c). Figure 4(d)shows the active subsystem index. It is observed thatthe proposed controller indeed stabilised the systemand met the specification on disturbance attenuationwhen subjected to actuator saturation. Moreover, thediscrete-time switching controller is easier to imple-ment and has less chattering.

5. Conclusions

In this article, we jointly designed output-feedbacklaws with a min-switching rule for switched linearsaturated systems and provided control synthesisconditions. Taking into consideration actuator satura-tions, the minimal switching rule renders the switchedclosed-loop system stable with suboptimal L2 perfor-mance. The specified quasi-LPV switching controllerstructure also avoids cumbersome algebraic loop andfacilitates controller implementation. The effectivenessof the proposed design approach is illustrated throughan SIP example.

Acknowledgements

This work is supported in part by NASA GrantNNX07AC40A and NSF Grant 1200242.

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