research article passive control of switched singular ...allocation process. u s, constructing the...

Research ArticlePassive Control of Switched Singular Systems viaOutput Feedback

Shuhui Shi,1 Guoliang Wang,2 Jianhua Wang,3 and Hong Li4

1Shenyang Institute of Engineering, Shenyang 110136, China2Liaoning Shihua University, Fushun 113001, China3Shenyang University, Shenyang 110004, China4Shenyang Institute of Technology, Fushun 113122, China

Correspondence should be addressed to Guoliang Wang; [email protected]

Received 1 January 2015; Accepted 16 February 2015

Academic Editor: Kun Liu

Copyright © 2015 Shuhui Shi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An instrumental matrix approach to design output feedback passive controller for switched singular systems is proposed inthis paper. The nonlinear inequality condition including Lyapunov inverse matrix and controller gain matrix is decoupled byintroducing additional instrumental matrix variable. Combined withmultiple Lyapunov functionmethod, the nonlinear inequalityis transformed into linear matrix inequality (LMI). An LMI condition is presented for switched singular system to be stableand passive via static output feedback under designed switching signal. Moreover, the conditions proposed do not require thedecomposition of Lyapunov matrix and its inverse matrix or fixing to a special structure. The theoretical results are verified bymeans of an example. The method introduced in the paper can be effectively extended to a single singular system and normalswitched system.

1. Introduction

The switched singular systems arise from, for instance,power systems, economic systems, and complex networks.As pointed out in [1], when the interrelationships amongdifferent industrial sectors are described and the capitaland the demand are variable depending on seasons, thedynamic economic systems are modelled as periodicallyswitched singular systems. In some complex hybrid networks,some algebraic constraints have to be considered. The spe-cial algebraic constraints that, for example, communicationresources are always limited and required to be allocated todifferent levels of privileged users, are needed in the resourceallocation process. Thus, constructing the network modelwith a set of constraints is reasonable and indispensable.The model can be denoted by a class of singular hybridsystems [2]. There has been increasing interest in analysisand synthesis for switched singular systems. The stabilityissues are discussed for continuous-time switched singularsystems [3], discrete-time switched singular systems [4],

linear switched singular systems [5, 6], nonlinear switchedsingular systems [7], and time-delay switched singular sys-tems [8], respectively. In [9, 10], reachability conditions andadmissibility criteria are presented, respectively. Reference[11] studies the initial instantaneous jumps at switchingpoints and a sufficient stability condition is obtained forthe switched singular system with both stable and unstablesubsystems. At arbitrary switching instant, inconsistent statejump for switched singular systems can be compressed byhybrid impulsive controllers in [12]. Filters and observers aredesigned for switched singular systems in [13, 14] and [15],respectively.

It has been shown that passivity is a suitable designapproach in power systems [16], neural networks [17], net-work control [18–20], signal processing [21, 22], Markovianjumping systems [23–25], switched systems [26, 27], and sin-gular systems [28, 29]. In [16], the problem of passive controlis considered for uncertain singular time-delay systems, andthree types of controllers were designed, namely, state feed-back controller, observer-based state feedback controller, and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 573950, 7 pageshttp://dx.doi.org/10.1155/2015/573950

2 Mathematical Problems in Engineering

dynamic output feedback controller. Under these controllers,the closed-loop systems are quadratically stable and passive.Contingent failures are possible for a real system, whichmay cause performance of the system to be degraded andeven hazard [30].The work [17] applies passivity-based fault-tolerant synchronization control to chaotic neural networksagainst actuator faults by using the semi-Markov jumpmodelapproach. The work [27] investigates the problem of robustreliable passive control for a class of uncertain stochasticswitched time-delay systems with actuator failures. Sufficientcondition for the stochastic switched time-delay systems tobe passive and exponentially stable under switching statefeedback controller is derived. The work [29] designs a statefeedback controller such that, for all possible actuator failures,the closed-loop singular system is exponentially stable andpassive.

Up to now, little attention has been paid to passive controlproblem for switched singular systems. This motivates us toinvestigate this problem. Furthermore, considering the oper-ational cost and the reliability of systems and the simplicityof implementation, output feedback is always adopted tostabilize a system.Thus, we study passive control of switchedsingular systems through output feedback.

In this paper, by introducing instrumental matrix vari-able, the nonlinear inequality including Lyapunov inversematrix and controller gain matrix is decoupled, whichmakes the design of output feedback passive controllersfor continuous-time switched singular systems easy. Basedon multiple Lyapunov functions and variable substitutiontechniques, a new and simple sufficient condition is presentedin terms of LMI, by solving which static output feedbackpassive controller can be designed. The novelty of the con-ditions proposed in this paper lies in the following aspect.Decomposition of Lyapunov matrix and its inverse matrix isnot required. Moreover, the Lyapunov inverse matrix is notfixed to a special structure.

The rest of this paper is organized as follows. Problemstatement and preliminaries are given in Section 2. Outputfeedback passive control is studied in Section 3. In Section 4,an example shows the efficiency of main results in the paper.Section 5 concludes this paper.

2. Problem Statement and Preliminaries

Consider the following switched singular system:

𝐸𝑐𝑖�̇� (𝑡) = 𝐴

𝑐𝑖𝑥 (𝑡) + 𝐵

𝑐𝑖𝑢 (𝑡) + 𝐺

𝑐𝑖𝜔 (𝑡)

𝑧 (𝑡) = 𝐻𝑐𝑖𝑥 (𝑡) + 𝐷

𝑐𝑖𝜔 (𝑡)

𝑦 (𝑡) = 𝐶𝑐𝑖𝑥 (𝑡) ,

(1)

where 𝑖 ∈ 𝜒 = {1, 2, . . . , 𝑚} is the switching signal. 𝑥(𝑡) ∈ 𝑅𝑛is the state vector, 𝑢(𝑡) ∈ 𝑅𝑚 is the control input vector,𝜔(𝑡) ∈ 𝐿

𝑟

2[0,∞) is the external disturbance vector, 𝑧(𝑡) ∈

𝑅𝑟 is the control input vector, and 𝑦(𝑡) ∈ 𝑅𝑝 is the mea-

sured controlled output vector. 𝐸𝑐𝑖, 𝐴𝑐𝑖, 𝐵𝑐𝑖, 𝐶𝑐𝑖, 𝐷𝑐𝑖, 𝐺𝑐𝑖, 𝐻𝑐𝑖

are constant matrices with appropriate dimensions. 𝐸𝑐𝑖

∈

𝑅𝑛×𝑛 and rank 𝐸

𝑐𝑖= 𝑟𝑖≤ 𝑛. Without loss of generality, we

assume that 𝐶𝑐𝑖is full row rank.

Let us consider the following static output feedbackcontroller:

𝑢 (𝑡) = 𝐾𝑖𝑦 (𝑡) , (2)

where𝐾𝑖is the controller gain matrix to be designed.

Then, the resulting closed-loop system can be describedas

𝐸𝑐𝑖�̇� (𝑡) = (𝐴

𝑐𝑖+ 𝐵𝑐𝑖𝐾𝑖𝐶𝑐𝑖) 𝑥 (𝑡) + 𝐺

𝑐𝑖𝜔 (𝑡)


𝑐𝑖𝜔 (𝑡)

𝑦 (𝑡) = 𝐶𝑐𝑖𝑥 (𝑡) .

(3)

We are now considering the output feedback passivecontrol problem for system (3). In this paper, we aim todesign output feedback passive controller such that system (3)simultaneously satisfies the following two requirements.

(i) System (3) with 𝜔(𝑡) = 0 is stable.

(ii) For a give scalar 𝜂 > 0, the dissipation inequality

∫

𝑇

0

(𝜔𝑇

𝑧 − 𝜂𝜔𝑇

𝜔) 𝑑𝑡 ≥ 0, ∀𝑇 > 0, (4)

holds for all trajectories with zero initial condition. In thiscase, the closed-loop switched singular system (3) is said tobe stable and passive with dissipation rate 𝜂.

To obtain the main results of this paper, the followingtransformation is introduced.

Since 𝐶𝑐𝑖

is full row rank, there exists a nonsingularmatrix𝑇

𝑖such that𝐶

𝑐𝑖𝑇−1

𝑖= [𝐼𝑝

0]. Using the nonsingular𝑇𝑖

as a similarity transformation for system (3), the closed-loopsystem (3) is equivalent to the following system:

𝐸𝑖�̇� (𝑡) = 𝐴

𝑖𝑥 (𝑡) + 𝐺

𝑖𝜔 (𝑡)

𝑧 (𝑡) = 𝐻𝑖𝑥 (𝑡) + 𝐷

𝑖𝜔 (𝑡)

𝑦 (𝑡) = 𝐶𝑖𝑥 (𝑡) ,

(5)

where 𝐸𝑖

= 𝑇𝑖𝐸𝑐𝑖𝑇−1

𝑖, 𝐴𝑖

= 𝑇𝑖𝐴𝑐𝑖𝑇−1

𝑖, 𝐵𝑖

= 𝑇𝑖𝐵𝑐𝑖, 𝐶𝑖

=

𝐶𝑐𝑖𝑇−1

𝑖= [𝐼𝑝

0], 𝐺𝑖= 𝑇𝑖𝐺𝑐𝑖, 𝐻𝑖= 𝐻𝑐𝑖𝑇−1

𝑖, 𝐷𝑖= 𝐷𝑐𝑖, and

𝐴𝑖= 𝐴𝑖+ 𝐵𝑖𝐾𝑖𝐶𝑖.

3. Output Feedback Passive Control

The following theorem provides a sufficient condition underwhich system (5) is stable and passive with dissipation rate 𝜂.

Theorem 1. If there exist simultaneously nonnegative realnumber 𝛽

𝑖𝑗, 𝜂 and matrices 𝑋

𝑖> 0, 𝑋

𝑗> 0 and matrices 𝐾

𝑖

such that for any 𝑖, 𝑗 ∈ 𝜒, 𝑖 ̸= 𝑗,

𝑋𝑇

𝑖𝐸𝑇

𝑖= 𝐸𝑖𝑋𝑖≥ 0 (6)

Mathematical Problems in Engineering 3

and inequality

[[

[

𝑋𝑇

𝑖𝐴𝑇

𝑖+ 𝐴𝑖𝑋𝑖+

𝑚

∑

𝑗=1

𝛽𝑖𝑗(𝐸𝑇

𝑖𝑋𝑖− 𝐸𝑇

𝑗𝑋𝑗) 𝐺

𝑖− 𝑋𝑇

𝑖𝐻𝑇

𝑖

𝐺𝑇

𝑖− 𝐻𝑖𝑋𝑖

2𝜂𝐼 − 𝐷𝑖− 𝐷𝑇

𝑖

]]

]

< 0

(7)

hold, system (5) is stable and passive with dissipation rate 𝜂under static output feedback 𝑢(𝑡) = 𝐾

𝑖𝑦(𝑡) via switching signal

𝑖 = argmax {𝑥𝑇 (𝑡) 𝐸𝑇𝑖𝑋𝑖𝑥 (𝑡) , 𝑖 ∈ 𝜒} . (8)

Proof. When 𝛽𝑖𝑗is simultaneously nonnegative, for 𝑥 ∈

𝑅𝑛

/{𝑥 | ∏𝑚

𝑖=1𝐸𝑖𝑥 = 0}, there must exist a 𝑖 ∈ 𝜒 = {1, . . . , 𝑚},

such that for any 𝑗 ̸= 𝑖, 𝑗 ∈ 𝜒, 𝑥𝑇(𝐸𝑇𝑖𝑋𝑖− 𝐸𝑇

𝑗𝑋𝑗)𝑥 ≥ 0 holds.

Then ∑𝑚𝑗=1

𝑥𝑇

(𝐸𝑇


𝑗𝑋𝑗)𝑥 ≥ 0 holds. Let

Ω𝑖= {𝑥 ∈ 𝑅

𝑛

| 𝑥𝑇

(𝐸𝑇


𝑗𝑋𝑗) 𝑥 ≥ 0,

𝑚

∏

𝑖=1

𝐸𝑖𝑥 ̸= 0, 𝑗 ̸= 𝑖, 𝑗 ∈ 𝜒} .

(9)

Clearly,⋃𝑚𝑖=1

Ω𝑖= 𝑅𝑛

/{𝑥 | ∏𝑚

𝑖=1𝐸𝑖𝑥 = 0}.

Construct Ω1

= Ω1, . . . , Ω

𝑖= Ω𝑖− ⋃𝑖−1

𝑗=1Ω𝑗, . . . , Ω

𝑚=

Ω𝑚− ⋃𝑚−1

𝑗=1Ω𝑗. Obviously, ⋃𝑚

𝑖=1Ω𝑖= 𝑅𝑛

/{𝑥 | ∏𝑚

𝑖=1𝐸𝑖𝑥 = 0}

andΩ𝑖⋂Ω𝑗= 𝜙, 𝑖 ̸= 𝑗, hold.

Design switching signal as

𝑖 = argmax {𝑥𝑇 (𝑡) 𝐸𝑇𝑖𝑋𝑖𝑥 (𝑡) , 𝑖 ∈ 𝜒} . (10)

When 𝑥 ∈ Ω𝑖, choose Lyapunov function as

𝑉 (𝑥 (𝑡)) = 𝑥𝑇

(𝑡) 𝐸𝑇

𝑖𝑋−1

𝑖𝑥 (𝑡) . (11)

Premultiplying 𝑋−𝑇𝑖

and postmultiplying 𝑋−1𝑖

to 𝑋𝑇𝑖𝐸𝑇

𝑖=

𝐸𝑖𝑋𝑖in (6), respectively, one gets 𝐸𝑇

𝑖𝑋−1

𝑖= 𝑋−𝑇

𝑖𝐸𝑖. Then,

the derivation of Lyapunov function𝑉(𝑥(𝑡)) along the closed-loop system (5) is

�̇� (𝑥 (𝑡)) = �̇�𝑇

(𝑡) 𝐸𝑇

𝑖𝑋−1

𝑖𝑥 (𝑡) + 𝑥

𝑇

(𝑡) 𝐸𝑇

𝑖𝑋−1

𝑖�̇� (𝑡)

= [𝐴𝑖𝑥 (𝑡) + 𝐺

𝑖𝜔 (𝑡)]

𝑇

𝑋−1

𝑖𝑥 (𝑡)

+ 𝑥𝑇

(𝑡) 𝑋−𝑇

𝑖[𝐴𝑖𝑥 (𝑡) + 𝐺

𝑖𝜔 (𝑡)] .

(12)

Therefore

�̇� (𝑥 (𝑡)) − 2𝜔𝑇

(𝑡) 𝑧 (𝑡) + 2𝜂𝜔𝑇

(𝑡) 𝜔 (𝑡)

= [𝐴𝑖𝑥 (𝑡) + 𝐺

𝑖𝜔 (𝑡)]

𝑇

𝑋−1

𝑖𝑥 (𝑡)

+ 𝑥𝑇

(𝑡) 𝑋−𝑇

𝑖[𝐴𝑖𝑥 (𝑡) + 𝐺

𝑖𝜔 (𝑡)]

− 2𝜔𝑇

(𝑡) [𝐻𝑖𝑥 (𝑡) + 𝐷

𝑖𝜔 (𝑡)] + 2𝜂𝜔

𝑇

(𝑡) 𝜔 (𝑡)

= [

𝑥(𝑡)

𝜔(𝑡)]

𝑇

[

[

𝐴𝑇

𝑖𝑋−1

𝑖+ 𝑋−𝑇

𝑖𝐴𝑖

𝑋−𝑇

𝑖𝐺𝑖− 𝐻𝑇

𝑖

𝐺𝑇

𝑖𝑋−1

𝑖− 𝐻𝑖


𝑖

]

]

[

𝑥 (𝑡)

𝜔 (𝑡)]

= 𝜁𝑇

(𝑡) Ξ𝑖𝜁 (𝑡) ,

(13)

where 𝜁(𝑡) = [ 𝑥(𝑡)𝜔(𝑡)

], Ξ𝑖

= [𝐴

𝑇

𝑖𝑋−1

𝑖+𝑋−𝑇

𝑖𝐴𝑖 𝑋

−𝑇

𝑖𝐺𝑖−𝐻

𝑇

𝑖

𝐺𝑇

𝑖𝑋−1

𝑖−𝐻𝑖 2𝜂𝐼−𝐷𝑖−𝐷

𝑇

𝑖

]. Pre-

and postmultiplying Ξ𝑖by [𝑋𝑇𝑖 00 𝐼

] and [𝑋𝑖 00 𝐼

], respectively, weobtain

[𝑋𝑇

𝑖0

0 𝐼

]Ξ𝑖[

𝑋𝑖0

0 𝐼] = Φ

𝑖, (14)

whereΦ𝑖= [𝑋𝑇

𝑖𝐴

𝑇

𝑖+𝐴𝑖𝑋𝑖 𝐺𝑖−𝑋

𝑇

𝑖𝐻𝑇

𝑖

𝐺𝑇

𝑖−𝐻𝑖𝑋𝑖 2𝜂𝐼−𝐷𝑖−𝐷

𝑇

𝑖

]. Then, Ξ𝑖< 0 is equivalent

to Φ𝑖< 0. Next, we prove thatΦ

𝑖< 0 holds.

Suppose that (7) holds. Inequality (7) can be rewritten as

Φ𝑖+[[

[

𝑚

∑

𝑗=1



𝑗𝑋𝑗) 0

0 0

]]

]

< 0. (15)

According to switching signal (8), Φ𝑖< 0 when (15) holds,

and by (13), we get that the following inequality holds:

�̇� (𝑥 (𝑡)) − 2𝜔𝑇

(𝑡) 𝑧 (𝑡) + 2𝜂𝜔𝑇

(𝑡) 𝜔 (𝑡) ≤ 0. (16)

Taking the integral on the two sides of (16) from 0 to 𝑇, weobtain

∫

𝑇

0

[�̇� (𝑥 (𝑡)) − 2𝜔𝑇

(𝑡) 𝑧 (𝑡) + 2𝜂𝜔𝑇

(𝑡) 𝜔 (𝑡)] 𝑑𝑡 ≤ 0. (17)

To get the result, we introduce

𝑆 (𝑇) = ∫

𝑇

0

[−2𝜔𝑇

(𝑡) 𝑧 (𝑡) + 2𝜂𝜔𝑇

(𝑡) 𝜔 (𝑡)] 𝑑𝑡, (18)

where 𝑇 > 0. Noting the zero initial condition, it can beshown that, for any 𝑇 > 0,

𝑆 (𝑇) ≤ ∫

𝑇

0

[�̇� (𝑥 (𝑡)) − 2𝜔𝑇

(𝑡) 𝑧 (𝑡) + 2𝜂𝜔𝑇

(𝑡) 𝜔 (𝑡)] 𝑑𝑡.

(19)


It follows from (19) that 𝑆(𝑇) ≤ 0, and hence

∫

𝑇

0

[−2𝜔𝑇

(𝑡) 𝑧 (𝑡) + 2𝜂𝜔𝑇

(𝑡) 𝜔 (𝑡)] 𝑑𝑡 ≤ 0 (20)

is satisfied for any 𝑇 > 0. The system is passive withdissipation 𝜂.

Next, we prove the stability of system (5) with 𝜔(𝑡) = 0.Following the similar procedures as used above, we have

�̇� (𝑥 (𝑡)) = 𝑥𝑇

(𝑡) (𝐴𝑇

𝑖𝑋−1

𝑖+ 𝑋−𝑇

𝑖𝐴𝑖) 𝑥 (𝑡) . (21)

Note that (7) implies𝐴𝑇𝑖𝑋−1

𝑖+𝑋−𝑇

𝑖𝐴𝑖< 0. Therefore, we have

�̇�(𝑥(𝑡)) < 0 when (7) holds. According to Lyapunov theory,system (5) is stable. This completes the proof.

Condition (7) inTheorem 1 is not linearmatrix inequality,which cannot be solved by MATLAB.The controller gains𝐾

𝑖

are also difficult to be computed from these conditions. Inorder to solve output feedback passive controller, we inducethe following important lemma.

Lemma 2. For any 𝑖, 𝑗 ∈ 𝜒, inequality (7) holds if there existsan instrumental matrix 𝑌

𝑖such that

[[[

[

Ψ𝑖

𝑋𝑇

𝑖− 𝑌𝑖+ 𝐴𝑖𝑌𝑇

𝑖𝐺𝑖− 𝑋𝑇

𝑖𝐻𝑇

𝑖

𝑋𝑖− 𝑌𝑇

𝑖+ 𝑌𝑖𝐴𝑇

𝑖−𝑌𝑖− 𝑌𝑇

𝑖0

𝐺𝑇


0 2𝜂𝐼 − 𝐷𝑖− 𝐷𝑇

𝑖

]]]

]

< 0,

(22)

where Ψ𝑖= 𝑌𝑖𝐴𝑇

𝑖+ 𝐴𝑖𝑌𝑇

𝑖+ ∑𝑚

𝑗=1𝛽𝑖𝑗(𝐸𝑇


𝑗𝑋𝑗).

Proof. Let Λ𝑖= [ 𝐼 𝐴𝑖 00 0 𝐼

]; then Λ𝑖is full row rank. By setting

Ψ𝑖= 𝑌𝑖𝐴𝑇

𝑖+ 𝐴𝑖𝑌𝑇

𝑖+ ∑𝑚



𝑗𝑋𝑗), the result of

Lemma 2 can be evidently derived from the fact that

Λ𝑖⋅

[[[

[

Ψ𝑖

𝑋𝑇


𝑖𝐺𝑖− 𝑋𝑇

𝑖𝐻𝑇

𝑖

𝑋𝑖− 𝑌𝑇

𝑖+ 𝑌𝑖𝐴𝑇

𝑖−𝑌𝑖− 𝑌𝑇

𝑖0

𝐺𝑇


0 2𝜂𝐼 − 𝐷𝑖− 𝐷𝑇

𝑖

]]]

]

⋅ Λ𝑇

𝑖

=[[

[

𝑋𝑇

𝑖𝐴𝑇

𝑖+ 𝐴𝑖𝑋𝑖+

𝑚

∑

𝑗=1



𝑗𝑋𝑗) 𝐺

𝑖− 𝑋𝑇

𝑖𝐻𝑇

𝑖

𝐺𝑇



𝑖

]]

]

.

(23)

Remark 3. The instrumental matrix variable 𝑌𝑖is introduced

to decouple Lyapunov inverse matrix 𝑋𝑖and controller gain

matrix𝐾𝑖, whichmakes the design of output feedback passive

controllers feasible, and, at the same time,𝑋𝑖and𝑋−1

𝑖do not

require decomposition or fixing to a certain structure.

Based on the above lemma, an LMI condition is pre-sented, under which system (5) is stable and passive in thefollowing theorem.

Theorem 4. If there exist simultaneously nonnegative realnumber 𝛽

𝑖𝑗, 𝜂 and matrices 𝑍

𝑖> 0, 𝑍

𝑗> 0, 𝑌

𝑖,𝑊𝑖, 𝑈𝑖, 𝑈𝑗such

that for any 𝑖, 𝑗 ∈ 𝜒, 𝑖 ̸= 𝑗,

[[[

[

Θ𝑖11

Θ𝑖12

Θ𝑖13

Θ𝑇

𝑖12−𝑌𝑖− 𝑌𝑇

𝑖0

Θ𝑇

𝑖130 2𝜂𝐼 − 𝐷

𝑖− 𝐷𝑇

𝑖

]]]

]

< 0, (24)

whereΘ𝑖11

= 𝑌𝑖𝐴𝑇

𝑖+𝐴𝑖𝑌𝑇

𝑖+𝐵𝑖𝑊𝑖+𝑊𝑇

𝑖𝐵𝑇

𝑖+∑𝑚


𝑖𝑍𝑖𝐸𝑇

𝑖+

𝐸𝑇

𝑖𝐿𝑖𝑈𝑖− 𝐸𝑇

𝑗𝑍𝑗𝐸𝑇

𝑗− 𝐸𝑇

𝑗𝐿𝑗𝑈𝑗), Θ𝑖12

= 𝐸𝑖𝑍𝑇

𝑖+ 𝑈𝑇

𝑖𝐿𝑇

𝑖− 𝑌𝑖+

𝐴𝑖𝑌𝑇

𝑖+𝐵𝑖𝑊𝑖,Θ𝑖13

= 𝐺𝑖−𝐸𝑖𝑍𝑇

𝑖𝐻𝑇

𝑖−𝑈𝑇

𝑖𝐿𝑇

𝑖𝐻𝑇

𝑖, 𝑌𝑖= [𝑌𝑖11 𝑌𝑖12

0 𝑌𝑖22

],and 𝑊

𝑖= [𝑊𝑖1

0], then, system (5) is stable and passive withdissipation rate 𝜂 under static output feedback

𝐾𝑖= 𝑊𝑖1𝑌−𝑇

𝑖11(25)

via switching signal

𝑖 = argmax {𝑥𝑇 (𝑡) 𝐸𝑇𝑖(𝑍𝑖𝐸𝑇

𝑖+ 𝐿𝑖𝑈𝑖) 𝑥 (𝑡) , 𝑖 ∈ 𝜒} . (26)

Proof. Since𝐶𝑖= [𝐼𝑝

0] in𝐴𝑖= 𝐴𝑖+𝐵𝑖𝐾𝑖𝐶𝑖, then𝐵

𝑖𝐾𝑖𝐶𝑖𝑌𝑇

𝑖=

𝐵𝑖[𝐾𝑖

0]𝑌𝑇

𝑖. Letting 𝑌

𝑖= [𝑌𝑖11 𝑌𝑖12

0 𝑌𝑖22

], one gets 𝑌𝑇𝑖

= [𝑌𝑇

𝑖110

𝑌𝑇

𝑖12𝑌𝑇

𝑖22

].Thus, 𝐵

𝑖[𝐾𝑖

0]𝑌𝑇

𝑖= 𝐵𝑖[𝐾𝑖𝑌𝑇

𝑖110]. Let 𝑊

𝑖= [𝑊𝑖1

0], 𝑊𝑖1

=

𝐾𝑖𝑌𝑇

𝑖11. A static output feedback gain is denoted by 𝐾

𝑖=

𝑊𝑖1𝑌−𝑇

𝑖11.

By making use of the existent methods (e.g., [31, 32]), weintroduce the matrix 𝐿

𝑖∈ 𝑅𝑛×(𝑛−𝑟𝑖) with rank 𝐿

𝑖= 𝑛 − 𝑟

𝑖sat-

isfying 𝐸𝑖𝐿𝑖= 0. By setting 𝑋

𝑖= 𝑍𝑖𝐸𝑇

𝑖+ 𝐿𝑖𝑈𝑖, one obtains

that 𝑋−𝑇𝑖

𝐸𝑇

𝑖= 𝐸𝑖𝑋𝑖≥ 0 holds when 𝑍

𝑖> 0. Then, accord-

ing to Lemma 2, inequality (6) and (7) can be rewritten asinequality (24). Inequality (24) is a linear matrix inequality.This completes the proof.

Remark 5. If 𝛽𝑖𝑗is nonpositive simultaneously, Theorem 4

also holds through choosing the switching signal as 𝑖 =argmin{𝑥𝑇(𝑡)𝐸𝑇

𝑖(𝑍𝑖𝐸𝑇

𝑖+ 𝐿𝑖𝑈𝑖)𝑥(𝑡), 𝑖 ∈ 𝜒}.

Remark 6. When switched singular system reduces to a singlesingular system (i.e., no switching), Theorem 4 also holds bysetting 𝐴

𝑖= 𝐴, 𝐵

𝑖= 𝐵, 𝐺

𝑖= 𝐺, and so forth. This will

be illustrated in Corollary 7. When switched singular systemreduces to a normal switched system (i.e., 𝐸

𝑐𝑖= 𝐼), the

method introduced in this paper is also available, and, at thesame time, Theorem 4 is rewritten as Corollary 8.

Corollary 7. If there exist a real number 𝜂 > 0 and matrices𝑍 > 0, 𝑌,𝑊,𝑈 such that

[[[

[

Θ11

Θ12

Θ13

Θ𝑇

12−𝑌 − 𝑌

𝑇

0

Θ𝑇

130 2𝜂𝐼 − 𝐷 − 𝐷

𝑇

]]]

]

< 0, (27)

whereΘ11

= 𝑌𝐴𝑇

+𝐴𝑌𝑇

+𝐵𝑊+𝑊𝑇

𝐵𝑇,Θ12

= 𝐸𝑍𝑇

+𝑈𝑇

𝐿𝑇

−

𝑌+𝐴𝑌𝑇

+𝐵𝑊,Θ13

= 𝐺−𝐸𝑍𝑇

𝐻𝑇

−𝑈𝑇

𝐿𝑇

𝐻𝑇, 𝑌 = [ 𝑌11 𝑌12

0 𝑌22

],and 𝑊 = [𝑊

10], then, the corresponding singular system is


stable and passive with dissipation rate 𝜂 under static outputfeedback

𝐾 = 𝑊1𝑌−𝑇

11. (28)

Corollary 8. If there exist simultaneously nonnegative realnumber 𝛽

𝑖𝑗, 𝜂 and matrices 𝑋

𝑖> 0, 𝑋

𝑗> 0, 𝑌

𝑖,𝑊𝑖such that

for any 𝑖, 𝑗 ∈ 𝜒, 𝑖 ̸= 𝑗,

[[[

[

Θ𝑖11

Θ𝑖12

Θ𝑖13

Θ𝑇

𝑖12−𝑌𝑖− 𝑌𝑇

𝑖0

Θ𝑇

𝑖130 2𝜂𝐼 − 𝐷

𝑖− 𝐷𝑇

𝑖

]]]

]

< 0, (29)

whereΘ𝑖11

= 𝑌𝑖𝐴𝑇

𝑖+𝐴𝑖𝑌𝑇

𝑖+𝐵𝑖𝑊𝑖+𝑊𝑇

𝑖𝐵𝑇

𝑖+∑𝑚

𝑗=1𝛽𝑖𝑗(𝑋𝑖−𝑋𝑗),

Θ𝑖12

= 𝑋𝑇


𝑖+ 𝐵𝑖𝑊𝑖, Θ𝑖13

= 𝐺𝑖− 𝑋𝑇

𝑖𝐻𝑇

𝑖, 𝑌𝑖=

[𝑌𝑖11 𝑌𝑖12

0 𝑌𝑖22

], and𝑊𝑖= [𝑊𝑖1

0], then, the corresponding switchedsystem is stable and passive with dissipation rate 𝜂 under staticoutput feedback

𝐾𝑖= 𝑊𝑖1𝑌−𝑇

𝑖11(30)

via switching signal

𝑖 = argmax {𝑥𝑇 (𝑡) 𝑋𝑖𝑥 (𝑡) , 𝑖 ∈ 𝜒} . (31)

4. Example

Consider the switched singular system composed of twosubsystems

𝐸𝑐𝑖�̇� (𝑡) = 𝐴

𝑐𝑖𝑥 (𝑡) + 𝐵

𝑐𝑖𝑢 (𝑡) + 𝐺

𝑐𝑖𝜔 (𝑡)


𝑐𝑖𝜔 (𝑡)

𝑦 (𝑡) = 𝐶𝑐𝑖𝑥 (𝑡)

𝑖 = 1, 2,

(32)

where

𝐸𝑐1

= [

1 0

0 0] , 𝐸

𝑐2= [

1 0

0 0] ,

𝐴𝑐1

= [

5 −2

10 −15] , 𝐴

𝑐2= [

−6 −1

−1 1.5] ,

𝐵𝑐1

= [

−3

2.5] , 𝐵

𝑐2= [

−4

2] ,

𝐶𝑐1

= [2 1] , 𝐶𝑐2

= [−1 2] ,

𝐺𝑐1

= [

0

1] , 𝐺

𝑐2= [

0

1] ,

𝐷𝑐1

= 2, 𝐷𝑐2

= 1,

𝐻𝑐1

= [1 1] , 𝐻𝑐2

= [1 2] .

(33)

Choose similarity transformation matrices as

𝑇1= [

2 1

−1 −1] , 𝑇

2= [

−1 2

1 −1] . (34)

Then, we get an equivalent state-space description of theabove system:

𝐸𝑖�̇� (𝑡) = 𝐴

𝑖𝑥 (𝑡) + 𝐵

𝑖𝑢 (𝑡) + 𝐺

𝑖𝜔 (𝑡)

𝑧 (𝑡) = 𝐻𝑖𝑥 (𝑡) + 𝐷

𝑖𝜔 (𝑡)

𝑦 (𝑡) = 𝐶𝑖𝑥 (𝑡) ,

𝑖 = 1, 2,

(35)

where

𝐸1= 𝑇1𝐸𝑐1𝑇−1

1= [

2 2

−1 −1] ,

𝐸2= 𝑇2𝐸𝑐2𝑇−1

2= [

−1 −2

1 2] ,

𝐴1= 𝑇1𝐴𝑐1𝑇−1

1= [

39 58

−32 −49] ,

𝐴2= 𝑇2𝐴𝑐2𝑇−1

2= [

8 12

−7.5 −12.5] ,

𝐵1= 𝑇1𝐵𝑐1

= [

−3.5

0.5] ,

𝐵2= 𝑇2𝐵𝑐2

= [

8

−6] ,

𝐶1= 𝐶𝑐1𝑇−1

1= [

1

0]

𝑇

,

𝐶2= 𝐶𝑐2𝑇−1

2= [

1

0]

𝑇

,

𝐷1= 𝐷𝑐1

= 2,

𝐷2= 𝐷𝑐2

= 1,

𝐺1= 𝑇1𝐺𝑐1

= [

1

−1] ,

𝐺2= 𝑇2𝐺𝑐2

= [

2

−1] ,

𝐻1= 𝐻𝑐1𝑇−1

1= [0 −1] ,

𝐻2= 𝐻𝑐2𝑇−1

2= [3 4] .

(36)


Choose 𝛽12

= 𝛽21

= 0.5, 𝐿1= [1

−1], 𝐿2= [−2

1]. By solving

linear matrix inequality (24) in Theorem 4, we can obtain

𝑍1= [

6.1107 1.6528

1.6528 0.9175] ,

𝑍2= [

2.0404 −1.1790

−1.1790 0.9889] ,

𝑌1= [

0.6280 −0.0791

0 0.0364] ,

𝑌2= [

2.7362 −1.3902

0 0.5073] ,

𝑊1= [10.0425 0] ,

𝑊2= [−2.2475 0] ,

𝑋1= [

19.9846 −6.8114

0.6831 −3.5225] ,

𝑋2= [

7.5201 −3.3638

−4.4000 2.3218] ,

𝑋−1

1= [

0.0536 −0.1036

0.0104 −0.3040] ,

𝑋−1

2= [

0.8729 1.2647

1.6543 2.8273] .

(37)

By linear matrix inequality (24), output feedback passivecontrol gains can be obtained:

𝐾1= 15.9905, 𝐾

2= −0.8214. (38)

The dissipation rate can be obtained:

𝜂 = 0.5. (39)

The corresponding switching signal is chosen as

𝑖 =

{{

{{

{

1, 𝑥 (𝑡) ∈ Ω1

2, 𝑥 (𝑡) ∈Ω2

Ω1

,

Ω1= {𝑥 ∈ 𝑅

𝑛

| 𝑥𝑇

(𝐸𝑇

1𝑋1− 𝐸𝑇

2𝑋2) 𝑥 ≥ 0,

𝐸1𝑥 ̸= 0, 𝐸

2𝑥 ̸= 0}

Ω2= {𝑥 ∈ 𝑅

𝑛

| 𝑥𝑇

(𝐸𝑇

2𝑋2− 𝐸𝑇

1𝑋1) 𝑥 ≥ 0,

𝐸1𝑥 ̸= 0, 𝐸

2𝑥 ̸= 0} .

(40)

Under the switching signal, the response curve of the abovesystem is exhibited in Figure 1, where 𝜔(𝑡) = sin 𝑡𝑒−0.1𝑡. FromFigure 1, it is obvious that the resulting closed-loop system isstable.

0 0.05 0.1 0.20.15 0.25−1

0

1

2

3

4

5

6

7

8

9

t

x

Figure 1: State response of the corresponding closed-loop system.

5. Conclusion

In this paper, the output feedback passive control problem fora class of switched singular systems is investigated. A novelmethod is proposed to solve static output feedback passivecontrollers. Sufficient linear matrix inequality condition ispresented by means of introducing instrumental matrixvariable 𝑌

𝑖and multiple Lyapunov function technique under

designed switching law. An example is given to verify the LMIcondition proposed for the resulting closed-loop system to bestable and passive with a lower dissipation rate 𝜂. Passivityis a suitable design approach in network control. How toextend the results of this paper to network-based control isan interesting problem.This problemdeserves a further studyand it remains as our future work.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grants nos. 61203001, 61104066,and 61473140, Liaoning Educational Committee Foundationunder Contract L2014525, and the Natural Science Founda-tion of Liaoning Province under Grant no. 2014020106.

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