State estimation for short-time switched linear systems under asynchronous switching

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2014; 28:553–561Published online 9 May 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.2408

State estimation for short-time switched linear systems underasynchronous switching

Weiming Xiang 1,2,*,†, Jian Xiao 2 and Muhammad Naveed Iqbal 2

1School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China2School of Electrical Engineering, Southwest Jiaotong University, Chengdu, 610031, China

SUMMARY

In this note, the state estimation problem for a class of short-time switched linear systems is investigated.By incorporating the concept of finite-time stability, an observer is designed to ensure the error dynamicsfinite-time stable in the short-time switching interval under the general asynchronous switching whereas syn-chronous switching mode is its special case. A numerical example is provided to illustrate the effectivenessof our study in this paper. Copyright © 2013 John Wiley & Sons, Ltd.

Received 27 October 2011; Revised 29 January 2013; Accepted 4 April 2013

KEY WORDS: short-time switched systems; state estimation; finite-time stability; asynchronous switching

1. INTRODUCTION

The issue of state estimation has been investigated intensively in both continuous and discretedomains. After Luenberger proposed a method to design an observer for linear time-invariant sys-tems in 1960s [1], numerous results on Luenberger-like observer design were presented. Thereare some results about an observer design for switched systems, and most of them are aboutswitched linear systems and focus on asymptotic observer defined on an infinite time interval [2–8].However, in some practical situations, switching strategies are often designed that switching fre-quently occurs in some certain short-time intervals of interest such as failures and repairs of machinein manufacturing systems, which can be illustrated in Figure 1.

Under this class of switching, the asymptotic stability property is absolutely determined by eachsubsystem, and the boundedness of system state is also of great importance; for example, the statemay reach unacceptable large values due to frequent switching in the short-time interval [9]. Hence,here, we design two types of observers. One class is the asymptotic observer that can be designedindependently for each subsystem. Moreover, another important system property that we are inter-ested in is the boundedness of the error state during the short-time interval, and the second one is theboundedness observer ensuring error state bounded in a prescribed boundary. The generic schemeof the augmented observer is illustrated in Figure 2.

On the other hand, the necessities of considering asynchronous switching have been shown forefficient controller design for practical systems [10, 11], and a few results are available to date forasynchronous switching [12–15]. By introducing the concept of finite-time stability [16–18], themain contribution in this paper is to design boundedness observer for short-time switched systemunder asynchronous switching, whereas the synchronous switching is just a particular case.

*Correspondence to: Weiming Xiang, School of Transportation and Logistics, Southwest Jiaotong University, No.111,Erhuanlu Beiyiduan, Chengdu, 610031, China.

†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

554 W. XIANG, J. XIAO AND M. N. IQBAL

short time interval short time interval

t

i

Figure 1. An illustrative diagram for switching that only occurs in short intervals.

Switched system

State x

Activated subsystem index i

Output y

Asymptotic observerEstimated state 1̂x

Boundness observerEstimated state 2x̂

Input u

1

2

ˆ No switching occursˆ

ˆ In short-time switching intervalx

x

Estimated state x̂x

Figure 2. Generic scheme of augmented observer for short-time switched system.

2. PRELIMINARIES AND PROBLEM FORMULATION

In this note, the switched discrete-time system † is given as follows:

† W x.kC 1/D A�.k/x.k/CB�.k/u.k/ (1a)

y.k/D C�.k/x.k/CD�.k/u.k/ (1b)

where x.k/ 2Rn is the state with x0 as its initial state, u.k/ 2Rm is the input, and y.k/ 2Rp is themeasured output. �.k/ W ZC ! I WD f1, 2, : : : ,N g is the switching signal, which takes a value inindex set I. N > 0 is the number of subsystems. Ai , Bi , Ci , and Di are constant matrices withappropriate dimensions. The time-interval sequence denoted by NS WD

˚�s1,� l1, : : : ,�sn,� ln, : : :

�,

where �sn WD Œkn0 , kn0 C Tn/ represents the short-time interval in which switching occurs, and� ln WD

�Qkn0 , Qkn0 C QTn

�denotes the relatively long interval in which the system maintains in a fixed

mode. It is assumed that kn0 C Tn D Qkn0 , Qkn0 C QTn D knC10 . There exists an upper bound of Tn as T �such that QTn >> T �. In �sn, the switching sequence is defined as Sn WD

˚kn0 , kn1 , : : : , knm, : : : , knM

�where kn0 , knm, and knM denote the initial instant,mth switching instant, and last instant, respectively.Explicitly, the length of interval �sn can be figured out as Tn D knM � kn0 .

In this note, the Luenberger-type observer †o is in the following form:

†o W Ox.kC 1/D A O�.k/ Ox.k/CB O�.k/u.k/CL O�.k/.y.k/�C O�.k/ Ox.k/�D O�.k/u.k// (2)

where Ox 2 Rn is the estimated state, and O�.k/ W ZC! I is the observer switching signal. MatricesLi 2 Rn�q are observer gains to be designed. When O� coincides � exactly, that is, O�.k/ D �.k/,8k 2 ZC, we say that the switching is synchronous between the observer and the system; otherwisethe switching is asynchronous. The corresponding switching sequence of the observer with respect

to �sn is denoted by OSn WDnOkn0 , Okn1 , : : : , Oknm, : : : , OknM

o.

Letting e.k/D x.k/� Ox.k/, error dynamics †e can be stated as

†e W e.kC 1/D .A O�.k/ �L O�.k/C O�.k//e.k/C .A�.k/ �A O�.k/CL O�.k/C O�.k/ �L O�.k/C�.k//x.k/

C .B�.k/ �B O�.k/ �L O�.k/D�.k/CL O�.k/D O�.k//u.k/(3)

Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2014; 28:553–561DOI: 10.1002/acs

STATE ESTIMATION FOR SHORT-TIME SWITCHED LINEAR SYSTEMS 555

If O�.k/ D �.k/, 8k 2 ZC, then one can obtain the error dynamics under synchronousswitching as

e.kC 1/D .A�.k/ �L�.k/C�.k//e.k/ (4)

As well known, frequent switching may lead to large transients [9], so the concept of finite-timestability concerning boundedness of x.k/ is introduced.

Definition 1 ([9, 16])System (1) is said to be finite-time stable with respect to .ı, ",R,M/, where 06 ı < " andM 2 ZC,if xTRx < "2, 8k 2 f1, 2, : : : ,M g whenever xT0 Rx0 < ı

2.If the error state boundedness in �sn WD

�kn0 , knM

�is considered, we can state the following

problem.

Problem 1Given switched system (1) and observer (2), find observer gains Li ensuring the finite-time stabilityof error dynamics (3) with respect to .ı, ",R, Tn/ in �sn.

In practice, the input u is always bounded. Moreover, because state x converges to equilibrium inrelatively long time interval, the state at the initial instant of short-time interval (the same instant atthe end of previous relatively long time interval) is also bounded. Thus, the following assumptionis proposed.

Assumption 1It is assumed that any admissible input u satisfies kuk 6 � , and system state x satisfies

��x �kn0 �� 6�.

3. MAIN RESULTS

Generally, the switching sequence Sn WD˚kn0 , kn1 , : : : , knm, : : : , knM

�and OSn WD

nOkn0 , Okn1 , : : : , Oknm, : : : ,

OknM

ocannot coincide exactly. Without loss of generality, the observer switching is supposed to lag

behind the switching of physical system. Let �nm D Oknm � k

nm > 0, whereas �nm 6 knmC1 � knm. It

means that observer †o identifies the correct mode during a certain period �nm such as in Figure 3.From Figure 3, we can see each interval

�knm, knmC1

�can be classified into two segments. For the

interval�knm, knmC�

nm

�, the system is working in mode M1 with �.k/¤ O�.k/; during the interval�

knmC�nm, knmC1

�, the system is viewed as working in mode M2 with �.k/ D O�.k/. At instant

knm C�nm, the system switches from M1 to M2, so the switching sequence of overall system can

be expressed as QSn WD˚kn0 , kn0 C�

n0 , kn1 , kn1 C�

n1 , : : : , knm, knmC�

nm, : : : , knM , knM C�

nM

�.

Mode M1: During interval�knm, knmC�

nm

�, we assume �.k/ D j , O�.k/ D i . Augmenting the

error dynamics (3) and system model (1), the mode M1 is governed by

�.kC 1/D AM1

i �.k/CBM1

i u.k/ (5)

Figure 3. An illustrative diagram for asynchronous switching. denotes M1 and stands for M2.



where

�.k/D�eT .k/ xT .k/

�T,AM1

i D

�Ai �LiCi Aj �Ai CLiCi �LiCj

0 Aj

, and

BM1

i D

�Bj �Bi �LiDj CLiD

Bj

Mode M2: It is assumed that �.k/ D O�.k/ D i and k 2�knmC�

nm, knmC1

�, because only error

state e.k/ is the point of concern, and the error dynamics is governed by

e.kC 1/D AM2

i e.k/ (6)

where AM2

i D Ai �LiCi .Then, the following necessary definitions are required for our main results.

Definition 2For each short-time interval �sn, the average dwell time �nave is defined by �nave D

�knM � k

n0

�=M D

Tn=M . The minimal dwell time �nmin and the maximal dwell time �nmax are defined by �nmin D

infmD1,2,:::,M˚knm � k

nm�1

�and �nmax D supmD1,2,:::,M

˚knm � k

nm�1

�, respectively.

Now, we are in the position to present the following results.

Theorem 1Consider switched system (1) and Assumption 1 holds. If there exist a set of matrices Pi , Qi , andXi and scalars 1 > 0, 2 > 0, > 0, and � > 1 such that

1

�R 0

� R

<

�Pi 0

� Qi

< 2

�R 0

� R

(7)

��Pi PiAi �XiCi� ��Pi

< 0 (8)

266666664

��Pi 0 0 ATi Pi �CTi X

Ti 0

� ��Qi 0 ATj Pi �ATi Pi CC

Ti X

Ti �C

Tj X

Ti ATj Qi

� � �I BTj Pi �BTi Pi CD

Ti X

Ti �D

Tj X

Ti BTj Qi

� � � �Pi 0

� � � � �Qi

377777775< 0 (9)

.2=1/Tn=�nave�Tn2.ı

2C�2/C �2�.1C ˇ/� 1"2 < 0 (10)

where ˛ D ln.2=1/=�nmin C ln � , ˇ D e˛�nmax.1� e˛M�

nmax/=.1� e˛�

nmax/, � D

.1� ��nmax/=.1� �/ � > 1

�nmax � D 1, and �nmax D supmD0,1,2,:::M f�

nmg, then error dynamics (3) with

Li D P�1i Xi is finite-time stable with respect to .ı, ",R, Tn/.

ProofIn mode M1, we choose VM1

i .�/D �TPi� and Pi D diag.Pi ,Qi / for i th subsystem dynamics.Substituting Li D P�1i Xi and from (9), the following result can be easily derived

VM1

i .kC 1/ < �VM1

i .k/C uT .k/u.k/, k 2�knm, knmC�

nm

�(11)



For mode M2, we choose VM2

i .e/D eTPie, and using (8), we can obtain

VM2

i .kC 1/ < �VM2

i .k/,�knmC�

nm, knmC1

�(12)

As for the overall system, we let

V.k/DX

i2I M1

i .k/VM1

i .�/CX

i2I M2

i .k/VM2

i .e/

DX

i2I�T

M1

i .k/Pi� CX

i2IeT

M2

i .k/Pie (13)

where M1

i .�/ WZC! f0, 1g, M2

i .�/ WZC! f0, 1g, andPi2I

M1

i .k/CPi2I

M2

i .k/D 1.From (11)–(13) and the fact that kuk6 � , it arrives

V.kC 1/ <

(�V.k/C �2 k 2

�knm, knmC�

nm

��V.k/ k 2

�knmC�

nm, knmC1

� (14)

Hence, we deduce

V.kC 1/ <

(�k�k

nmV

�knm�C �2

PklDknm

� l�knm k 2

�knm, knmC�

nm

��k�k

nm��

nmV

�knmC�

nm

�k 2

�knmC�

nm, knmC1

� (15)

For � > 1, we havePklDknm

� l�knm D 1��k�k

nm

1��6 1��

nm

1��and k 2

�knm, knmC�

nm

�, and as �nm 6

�nmax, we obtainPklDknm

� l�knm 6 1��

nmax

1��. If � D 1, we have

PklDknm

� l�knm 6 k � knm 6 �nmax.

Thus, (15) becomes

V.kC 1/ <

(�k�k

nmV

�knm�C �2� k 2

�knm, knmC�

nm

��k�k

nm��

nmV

�knmC�

nm

�k 2

�knmC�

nm, knmC1

� (16)

where �D

.1� ��

nmax/=.1� �/ � > 1

�nmax � D 1. Then, assuming k 2

�knm, knmC�

nm

�(the results when

k 2�knmC�

nm, knmC1

�can be derived similarly), we can obtain

V.k/ < .2=1/Tn=�nave�k

nM�kn0 V

�kn0�C �2�

�1C

XM

lD1.2=1/

l�knM�knM�l

�(17)

By Definition 2, we have .2=1/l < .2=1/.knM�knM�l/=�

nmin , and by some manipulations, we see

XM

lD1.2=1/

l�knM�knM�l < ˇ

where ˛ D ln.2=1/=�nminCln � and ˇ D e˛�nmax�1� e˛M�

nmax�=�1� e˛�

nmax�. Therefore, (17) arrives

V.k/ < .2=1/Tn=�nave�k

nM�kn0 V

�kn0�C �2�.1C ˇ/ (18)

When k 2�knm, knmC�

nm

�, we can derive

V.k/D �TPi� D �TR1=2QiR1=2�T > infi2I fmin.Qi /g �TR�T

> infi2I fmin.Qi /g eT .t/Re.t/

where Qi DR�1=2PiR�1=2 and RD diag.R, I /. Also,

V�kn0�6 supi2Ifmax.Qi /g �T

�kn0�R�

�kn0�



Using the fact��x �kn0 �� 6 �, we obtain �T

�kn0�R�

�kn0�D eT

�kn0�Re

�kn0�C xT

�kn0�x�kn0�6

ı2C�2, which implies that

V�kn0�D �T

�kn0�Pi�

�kn0�6 supi2Ifmax.Qi /g .ı2C�2/ (19)

Altogether (18), (19), and M < Tn=�ave, Tn D knM � kn0 , the following inequality can be derived

eT .k/Re.k/ <.2=1/

Tn=�nave�Tn supi2I fmax.Qi /g .ı2C�2/infi2I fmin.Qi /g

C�2�.1C ˇ/

infi2I fmin.Qi /g,8k 2 �sn

(20)

Because of (7), we have 1R < Pi < 2R) 1I <R�1=2PiR�1=2 < 2I ) 1I <Qi < 2I ,which implies supi2I fmax.Qi /g 6 2 and infi2I fmin.Qi /g > 1. Therefore, by (10), we haveeT .k/Re.k/ < "2. �

If we consider synchronous switching, that is, �nmax D 0, we have � D 0. Then, the error dynam-ics is governed by M2 regardless of system state x. Thus, we can choose �D 0 in (10). Theorem 1is then reduced to the following corollary.

Corollary 1Consider switched system (1) with average dwell time �nave. If there exist a set of matrices Pi and Xiand scalars 1 > 0, 2 > 0 and � > 1 such that

1R < Pi < 2R (21)

��Pi PiAi �XiCi� ��Pi

< 0 (22)

.2=1/Tn=�aveC1ı2�Tn � "2 < 0 (23)

then error dynamics (4) with Li D P�1i Xi is finite-time stable with respect to .ı, ",R, Tn/.

Remark 1As a critical step to implement the augmented observer composed of asymptotic and boundednessobservers, the classification of time intervals into short-time intervals and relatively long intervalsis required. Thus, the start and end of corresponding short-time and long intervals have to be deter-mined so as to switch on/off the corresponding observers. In practice, parameter Tn may be difficultto obtain. But, it is easy to see that we can choose the upper bound T � to satisfy Theorem 1 orCorollary 1. Thus, if Tn D T �, we can determine kn0 , which is the start time of �sn, and othertimings can also be figured out. In synchronous case, the timings can be exactly determined. Fur-thermore, when asynchronous switching is considered, because there exists a mode-mismatchedtime unit that is always much smaller than T �, the upper bound T � can be enhanced slightly asT �C�max, where �max D sup

˚�nmax

�, to ascertain other timings.

Once the state bound " is not ascertained, we will be interested in observer gains with minimalvalue "min. If we denote 1 D 1 and 2 D �, (7) and (10) become�

R 0

� R

<

�Pi 0

� Qi

< �

�R 0

� R

(24)

"2 > �1CTn=�nave�Tn.ı2C�2/C �2�.1C ˇ/ (25)

Then, the following optimization problem can be constructed

min� s.t. (8), (9), and (24) (26)



with optimized gains Li D P�1i Xi and "min D ıp�1CTn=�

nave�Tn.ı2C�2/C �2�.1C ˇ/. On the

basis of (26), we construct the parameter search algorithm as follows:

Algorithm 1

Step 1 Initialize � D 1, set a variation value �� > 0 and termination value N� .Step 2 Setting � D � C�� , solve (26) with a fixed � .Step 3 When (26) is solvable for the first time, record � as �min. Then, if � > �minC N� , terminate

procedure; otherwise, record the parameters .� , "min/ pair-wisely and go back to Step 2.Step 4 Select Q� with smallest "min recorded in Step 3.Step 5 Obtain the local optimal observer gains and "�min near Q� by an unconstrained nonlinear

optimization approach.

Besides Algorithm 1, according to augmented observer idea, the asymptotic observer for eachsystem has to be designed, which can be easily obtained by linear system theory.

4. NUMERICAL EXAMPLES

Consider a switched system with two subsystems without input as follows:

A1 D

��0.6051 0.2615

0.0253 0.7435

,A2 D

��0.1620 0.8613

0.4357 0.0641

,

C T1 D

��1.7273

0.7160

,C T2 D

�0.0366�0.1849

Here, the short-time interval is Œ0, 20�, and the switching sequence is S1 WD f0, 8, 16, 20g.

Let ı Dp2, R D I , Tn D 20, and � D 1 to obtain L1 D

�0.3527 0.1400

�Tand

L2 D��4.6492 0.1320

�Twith "�min D 1.4144 by Corollary 1. The simulation under syn-

chronous switching is shown in Figure 4. Now, we assume the switching signal of observer delaysfor two steps, and the simulation is in Figure 5.

Explicitly, although the design result works efficiently in synchronous case, it is not obtain-able under asynchronous case anymore, because kek is not bounded in the boundary "�min D

1.4144. So, we resort to Theorem 1 to obtain L1 D�

0.1956 0.2733�T

and L2 D��0.1298 0.4493

�Twith "�min D 5.0954, and the simulation result is given in Figure 6.

Figure 4. kek under synchronous switching.



Figure 5. kek under asynchronous switching.

Figure 6. kek under asynchronous switching.

We can observe that Theorem 1 concerned with asynchronous switching is more general and hasobvious advantages. Moreover, in this example, we choose the system without input; actually if weconsider input u.k/, it is easy to see that although the design result works efficiently in synchronouscase, it is not satisfied under asynchronous case because we can choose u.k/ large enough to breakthe specified boundary in M1, and similar results can be easily obtained.

5. CONCLUSIONS

In this paper, the state estimation problem for short-time switched systems is addressed. By inves-tigating the general asynchronous switching, the finite-time observer is designed. Moreover, thesynchronous switching is only a particular case covered by our approach. An LMI-based designalgorithm is proposed. A numerical example is provided to illustrate our results.

ACKNOWLEDGEMENT

This work is supported by the National Natural Science Foundation of China (51177137 and 61134001).



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State estimation for short-time switched linear systems under asynchronous switching