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A Novel Scheme of Non-fragile ControllerDesign for Periodic Piecewise LTV Systems

Xiaochen Xie, Member, IEEE, James Lam, Fellow, IEEE, and Ka-Wai Kwok, Senior Member, IEEE

Abstract—In this paper, a novel non-fragile controller de-sign scheme is developed for a class of periodic piecewisesystems with linear time-varying (LTV) subsystems. Twotypes of norm-bounded controller perturbations, includingadditive and multiplicative ones, are considered and par-tially characterized by periodic piecewise time-varying pa-rameters. Using a new matrix polynomial lemma, the prob-lems of non-fragile controller synthesis for periodic piece-wise time-varying systems (PPTVSs) are made amenable toconvex optimization based on the favorable property of aclass of matrix polynomials. Depending on selectable divi-sions of subintervals, sufficient conditions of the stabilityand non-fragile controller design are proposed for PPTVSs.Case studies based on a multi-input multi-output (MIMO)PPTVS and a mass-spring-damper system show that theproposed control schemes can effectively guarantee theclose-loop stability and accelerate the convergence undercontroller perturbations, with more flexible periodic time-varying controller gains than those obtained by the existingmethods.

Index Terms—Matrix polynomial, non-fragile control, pe-riodic systems, time-varying systems

I. INTRODUCTION

PERODIC characteristics, models and tasks are present orrequired in a multitude of fields, including but not limited

to aerospace, mechatronics, networks and signal processing[1]–[5]. Among them, linear periodic systems play importantroles, given their convenience in tackling linear time-varying(LTV) systems similarly to linear time-invariant (LTI) ones[6]. Aimed at the stability and robust performance of linearperiodic systems, research efforts have been focused on theFloquet-Lyapunov theory for periodic differential equations[7] and lifting-techniques based discrete-time periodic appli-cations [8] over the past decades. Recent studies [9], [10]have revealed the efficiency of periodic piecewise modelsas approximations of continuous-time periodic systems thatdo not necessarily have closed-form expressions. Unlike the

This work was supported in part by the National Natural ScienceFoundation of China (NSFC) under Grant 61973259, the Innovation andTechnology Commission (ITC) of Hong Kong via the Innovation andTechnology Fund (ITF) under Grant UIM/353, and the Research GrantsCouncil (RGC) of Hong Kong under Grants (17206818, 17202317,17200918). (Corresponding author: Ka-Wai Kwok.)

X. Xie, J. Lam and K.W. Kwok are with the Department of MechanicalEngineering, The University of Hong Kong, Pokfulam, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

numerical computational approaches such as the Floquet-Lyapunov transformation and the monodromy matrix, peri-odic piecewise systems are less complicated and capable ofovercoming the difficulties in controller synthesis broughtby continuous-time periodic dynamics, making the relatedproblems more amenable to convex optimization tools.

The investigations on periodic piecewise systems are incip-iently based on the model formulation consisting of severalLTI subsystems, namely, the periodic piecewise linear system(PPLS). Motivated by the decomposition of periodic dynamics[11] and the theory of switched systems [12], the stabilityanalysis and stabilizing controller design of PPLSs have beenachieved by using Lyapunov functions with periodic piecewisematrices [9]. The periodic control scheme for PPLSs has beenimproved in [13] to provide time-varying controller gainsunder the framework of finite-time stability. Further, H∞control and guaranteed cost control schemes are established fortime-delay PPLSs as well as delay-free PPLSs in [14]–[16].A peak-to-peak filter is designed for PPLSs with polytopicuncertainties in [17]. In [18], a matrix polynomial-based time-varying controller is proposed for PPLSs. For PPLSs withpositivity, the stability and L1-gain are analyzed in [19].

Despite the simplicity of PPLSs, the time-invariant subsys-tem formulations may result in a loss of system dynamics.In practice, it is preferable to use time-varying models torepresent practical systems, such as mechanical systems withperiodic time-varying stiffness, loads or motions, and processplants involving periodic variables [20], [21]. Hence, themodel of periodic piecewise time-varying system (PPTVS) isrecommended. Compared with PPLSs, PPTVSs consist of anumber of time-varying subsystems, leading to approximationsthat may be more desirable to preserve periodic dynamics.On the other hand, the time-varying dynamics in PPTVSsalso lead to non-convex conditions especially during controllersynthesis, bringing more difficulties for complicated caseswith uncertainties or perturbations. For many engineeringapplications, controller perturbations commonly exist and leadto the degradation of control performance [22], giving rise tothe need of non-fragile control schemes, which have receivedextensive attention in the related fields including multivariablesystems [23], switched systems [24], [25], time-delay systems[26], [27], sampled-data control [28], [29] and event-triggeredcontrol [30]. In [31], the basic issues of stability and controlare studied for PPTVSs from a polynomial perspective. In[32], the state tracking controller of PPTVSs is developed. In[33], the standard non-fragile control problem is investigatedfor PPTVSs affected by constant time delay. However, the

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previous studies on PPTVSs are with system and controllerstructures sharing the same widths of the subintervals. Inother words, the system and the controller have identicaltime-varying coefficients, which technically reduced the flex-ibility in controller design. If the system and the controllerare constructed over subintervals with different widths, non-identical time-varying coefficients will be incurred, makingthe controller synthesis difficult using the existing methods.In addition, little efforts have been made on non-fragilecontroller synthesis with uncertain periodic piecewise time-varying perturbations, which motivated this study from adifferent perspective.

In this paper, two types of non-fragile periodic controllersare established to deal with norm-bounded additive and multi-plicative perturbations, respectively. A novel lemma on thenegative definiteness of a class of matrix polynomials isproposed, providing a flexible scheme of controller designthat allows time segmentation with non-identical time-varyingcoefficients. Hence, the controller gains can be obtained basedon some selectable parameters of subinterval division, whichdiffers our work from the existing results. The contributionsand novelties are threefold:1) Novel non-fragile time-varying controllers are developed

to resist the controller perturbations partially described byperiodic piecewise time-varying functions.

2) A new matrix polynomial lemma is proposed to avoidthe coupling terms incurred by time-varying dynamics,providing more technical flexibility in controller designwith non-identical time-varying coefficients.

3) The periodic time-varying controller gains can be effi-ciently determined by some selected divisions of PPTVSsubintervals and solved via convex optimization.

The paper is organized as follows. Section II gives the prob-lem formulation. Section III provides the close-loop stabilityanalysis and non-fragile controller synthesis. The effectivenessof the designed control schemes are verified through casestudies in Section IV. Section V concludes the paper.Notation: Rn stands for the n-dimensional Euclidean space;N+ denotes the set of positive integers; ‖ · ‖ denotes theEuclidean norm of a vector; I and 0 represent the identitymatrix and zero matrix, respectively. P > 0 (≥ 0) denotesthat P is a real symmetric and positive definite (semi-definite)matrix. PT and P−1 respectively denote the transpose and theinverse of matrix P , sym(P ) = PT + P . diag(·) denotes adiagonal matrix constructed by the given diagonal elements.λ(·), λ(·) refer to the maximum, minimum eigenvalues of areal symmetric matrix. In block symmetric matrices, “∗” isused as an ellipsis for the terms introduced by symmetry.Matrices, if their dimensions are not explicitly stated, areassumed to be compatible for algebraic operations.

II. PROBLEM FORMULATION

Consider a continuous-time PPTVS:

x(t) = A(t)x(t) + B(t)u(t), (1)

where x(t) ∈ Rn and u(t) ∈ Rnu are the state vector andcontrol input, respectively; Tp is the fundamental period of

PPTVS

Tp

t0

t1 t2 tS 1 tS

T1 T2 TS

A1

A2

A3

AS+1=A1

AS

...

... ...

Subsystem 1 Subsystem 2 Subsystem S...

T1

A2

Ā1 Ā2 ĀS

Ā1

...

Subsystem 1

Tp+t1

PPLS

Fig. 1. Evolutions of subsystem matrices in PPTVS (1) and a PPLS

system (1) such that A(t) = A(t + lTp) and B(t) = B(t +lTp) for t ≥ 0, l = 0, 1, . . .. Time interval [lTp, (l + 1)Tp) issupposed to be partitioned into S sub-intervals, namely [lTp+ti−1, lTp + ti) with Ti = ti − ti−1, i ∈ S , 1, 2, . . . , S,ΣSi=1Ti = Tp, t0 = 0 and tS = Tp. For t ∈ Ti , [lTp +ti−1, lTp + ti), the dynamics of ith subsystem is representedby the following LTV matrix functions:

A(t) = Ai(t) = Ai + σi(t)Ai, Ai , Ai+1 −AiB(t) = Bi(t) = Bi + σi(t)Bi, Bi , Bi+1 −Bi

(2)

where LTV coefficients σi(t) =t−lTp−ti−1

Ti∈ [0, 1), i ∈ S;

Ai ∈ Rn×n, Bi ∈ Rn×nu , i ∈ S, are known constant matricesand AS+1 = A1, BS+1 = B1.

Remark 1. For t ∈ Ti, i ∈ S, if Ai(t) = Ai,Bi(t) = Bi, thenPPTVS (1) will reduce to a PPLS, which has been studied in[9]. A sketch comparing the evolutions of subsystem matricesin PPTVS (1) and a PPLS is shown in Fig. 1.

In this paper, one considers a periodic non-fragile time-varying control law:

u(t) = (Ki(t) + ∆Ki(t))x(t), t ∈ Ti, (3)

and two types of controller perturbations as follows.1) Norm-bounded additive perturbations:

∆Ki(t) = HiFi(t)Ei(t), t ∈ Ti. (4)

2) Norm-bounded multiplicative perturbations:

∆Ki(t) = HiFi(t)Ei(t)Ki(t), t ∈ Ti. (5)

In (4) and (5), for t ∈ Ti, i ∈ S, consider time-varyingparameter matrices:

Ei(t) = Ei + σi(t)Ei, Ei , Ei+1 − Ei, (6)

where Hi and Ei, i ∈ S, are known real constant matriceswith appropriate dimensions; Fi(t), i ∈ S, are unknownfunctions that are continuous over the ith subinterval and right-continuous at the switching instants, satisfying

Fi(t) ∈ Φi , Fi(t) | FTi (t)Fi(t) ≤ I, t ∈ Ti, i ∈ S. (7)

Remark 2. In practice, the additive perturbations in (4)are usually due to the external disturbances and/or actuator





perturbations affecting the controller gains in additive ways,such as variations or noises caused by electromagnetic fluctu-ations, changes in the voltage of actuator power, and varyingcurrents in electronic components. On the other hand, themultiplicative perturbations in (5) describe the disturbancesand/or perturbations with effects acting proportionally on thecontroller, which may be incurred by signal distortions.

Based on the previous study [18], the exponential stabilitycharacterized by a general convergence rate α∗ > 0 is appliedto PPTVS (1) under the effects of controller perturbations.

Definition 1. PPTVS (1) with LTV subsystems in (2) and non-fragile control law (3) affected by perturbations (4) or (5) isα∗-exponentially stable if for all Fi(t), i ∈ S, which satisfy(7), there exist constants κ ≥ 1 and α∗ > 0 such that thesystem solution from x(0) satisfies ‖x(t)‖ ≤ κe−α

∗t‖x(0)‖,∀t ≥ 0.

Remark 3. The controller perturbations in (4) and (5) involvesome partially known dynamics described by periodic piece-wise time-varying matrices Ei(t), i ∈ S, which are differentfrom traditional descriptions of parametric uncertainties inprevious studies like [22]–[24], [33]. Such perturbations alsobring more technical challenges to non-fragile controller de-sign, since the time-varying dynamics in (2) and (4) or (5) canresult in non-convex variables that are difficult to decouple inclosed-loop stability analysis.

III. MAIN RESULTS

A. Closed-loop Stability AnalysisFor t ≥ 0, consider a continuous Tp-periodic matrix func-

tion P(t) = Pi(t) > 0, t ∈ Ti, satisfying limt→lTp+ti P(t) =P(lTp + ti). For t ∈ Ti, the upper right Dini derivative ofcontinuous matrix function P(t) is given by

D+P(t) = D+Pi(t) = lim suph→0+

Pi(t+ h)− Pi(t)h

. (8)

Hence, for x(t) 6= 0, construct a continuous periodic quadraticLyapunov function as

V(t) = Vi(t) = xT (t)Pi(t)x(t) > 0, t ∈ Ti. (9)

Based on Lyapunov function (9), a lemma on the generalexponential stability of PPTVSs is given below.

Lemma 1. [18] Consider PPTVS (1) with u(t) = 0. Given ascalar α∗ > 0, if there exist scalars αi, i = 1, 2, . . . , S, andreal symmetric Tp-periodic, continuous and Dini-differentiablematrix function P(t) defined on t ∈ [0,∞) such that, fort ∈ Ti, i ∈ S, P(t) = Pi(t) > 0, the following conditionshold:

sym (Pi(t)Ai(t)) +D+Pi(t) + αiPi(t) < 0, (10)

2α∗Tp −S∑i=1

αiTi ≤ 0, (11)

then the system is α∗-exponentially stable, that is, ‖x(t)‖ ≤κe−α

∗t‖x(0)‖, ∀t ≥ 0, where

κ = eα∗Tp

√λ(P(0))/λ(P(0))

S∏i=1

max(1, eµiTi) ≥ 1,

and constant µi satisfies µi ≥ 12λ(Ai(t) +ATi (t)

)for the ith

subsystem, t ∈ Ti, i ∈ S.

Based on Lemma 1, a criterion of closed-loop stability isobtained for PPTVS (1) with non-fragile controller (3).

Theorem 1. Consider PPTVS (1) with non-fragile controller(3) and norm-bounded additive perturbations in (4). Given ascalar α∗ > 0, the closed-loop system is α∗-exponentiallystable if there exist scalars ξi > 0, αi, i = 1, 2, . . . , S, andreal symmetric Tp-periodic, continuous and Dini-differentiablematrix function P(t) defined on t ∈ [0,∞) such that, fort ∈ Ti, i ∈ S, P(t) = Pi(t) > 0, (11) and the followingcondition, Gi(t) ξiPi(t)Bi(t)Hi ETi (t)

∗ −ξiI 0

∗ ∗ −ξiI

< 0, (12)

where

Gi(t) = sym (Pi(t)Ai(t) + Pi(t)Bi(t)Ki(t))+D+Pi(t) + αiPi(t), (13)

hold for all Fi(t) satisfying (7).

Proof: For PPTVS (1) with non-fragile controller (3) andnorm-bounded additive perturbations in (4), denote the closed-loop system as x(t) = Aci(t)x(t), t ∈ Ti, where Aci(t) ,Ai(t) + Bi(t)(Ki(t) + ∆Ki(t)) = Ai(t) + Bi(t)(Ki(t) +HiFi(t)Ei(t)), i ∈ S. With Lyapunov function (9), one has

D+Vi(t) + αiVi(t)

= xT (t)(sym (Pi(t)Aci(t)) +D+Pi(t) + αiPi(t)

)x(t)

= xT (t)(Gi(t) + sym (Pi(t)Bi(t)HiFi(t)Ei(t))

)x(t).

From (12) and Gi(t) = GTi (t), ξi > 0, i ∈ S , according toSchur complement equivalence, one has

Gi(t) + ξi (Pi(t)Bi(t)Hi) (Pi(t)Bi(t)Hi)T

+ ξ−1i ETi (t)Ei(t) < 0. (14)

For all Fi(t) satisfying (7), based on the fact

sym (Pi(t)Bi(t)HiFi(t)Ei(t))≤ ξi (Pi(t)Bi(t)Hi) (Pi(t)Bi(t)Hi)

T+ ξ−1i E

Ti (t)Ei(t),

one has

Gi(t) + sym (Pi(t)Bi(t)HiFi(t)Ei(t))≤ Gi(t) + ξi (Pi(t)Bi(t)Hi) (Pi(t)Bi(t)Hi)

T

+ ξ−1i ETi (t)Ei(t) < 0, (15)

which implies D+Vi(t)+αiVi(t) < 0, t ∈ Ti. Thus, when (11)and (12) hold, according to Lemma 1 and [31], the closed-loopPPTVS is α∗-exponentially stable.

Theorem 2. Consider PPTVS (1) with non-fragile controller(3) and norm-bounded multiplicative perturbations in (5).Given a scalar α∗ > 0, the closed-loop system is α∗-exponentially stable if there exist scalars ξi > 0, αi, i =1, 2, . . . , S, and real symmetric Tp-periodic, continuous and





Dini-differentiable matrix function P(t) defined on t ∈ [0,∞)such that, for t ∈ Ti, i = 1, 2, . . . , S, P(t) = Pi(t) > 0, (11)and the following condition, Gi(t) ξiPi(t)Bi(t)Hi KTi (t)ETi (t)

∗ −ξiI 0

∗ ∗ −ξiI

< 0, (16)

where Gi(t) is defined in (13), hold for all Fi(t) satisfying (7).

Proof: For PPTVS (1) with non-fragile controller (3) andnorm-bounded multiplicative perturbations in (5), denote theclosed-loop system as x(t) = Aci(t)x(t), t ∈ Ti, whereAci(t) , Ai(t) + Bi(t)(I + HiFi(t)Ei(t))Ki(t). When (11)and (16) hold, by Lyapunov function (9), Schur complementequivalence and following the procedures in the proof ofTheorem 1, it is easy to prove that for all Fi(t) satisfying(7), D+Vi(t) + αiVi(t) < 0, t ∈ Ti. According to Lemma 1,the closed-loop PPTVS is α∗-exponentially stable.

B. Non-fragile Controller SynthesisTo obtain the non-fragile controller gains by convex opti-

mization, one divides each sub-interval of PPTVS (1) into anumber of small segments inspired by [13]. Taking intervalTi for example, with Mi ∈ N+, the interval is assumed tobe divided into Mi segments with equal length δi = Ti/Mi.Denoting θi,m , lTp + ti−1 +mδi for m = 0, 1, . . . ,Mi − 1,and Mi , 0, 1, . . . ,Mi, i ∈ S, a continuous time-varyingmatrix function P(t) is defined by P(t) = Pi(t), t ∈ Ti,where for t ∈ [θi,m, θi,m+1) ∈ Ti,

Pi(t) = Pi,m + εi,m(t)(Pi,m+1 − Pi,m) (17)

with Pi,m > 0, Pi,Mi= Pi+1,0, PS,Mi

= P1,0, i ∈ S, m ∈Mi, and LTV coefficients

εi,m(t) =t− θi,mδi

=Mi(t− θi,m)

Ti∈ [0, 1). (18)

One can observe that periodic matrix function P(t) = P(t+lTp) is continuous at all the switching instants of timesegments, which satisfies the requirements of P(t). Basedon (8) and (17), P(t) is differentiable over each segmentedtime interval but not differentiable for all t ≥ 0. Hence,one considers the upper right Dini derivative of Pi(t) fort ∈ [θi,m, θi,m+1), that is,

D+Pi(t) =Mi(Pi,m+1 − Pi,m)

Ti. (19)

According to the obtained theorems, it is worth noticing that asegment-based P(t) will lead to the coexistence of σi(t) andεi,m(t) in stability criteria, m ∈ Mi, i ∈ S. The followinglemma is first proposed to facilitate the controller design.

Lemma 2. Let f : [0, 1]2 → R be a bounded matrixpolynomial function defined as f(η1, η2) = Ω0 + η1Ω1 +η2Ω2 + η1η2Ω12, where scalars η1 ∈ [0, 1], η2 ∈ [0, 1], andΩ0,Ω1,Ω2,Ω12 are real symmetric matrices. Matrix polyno-mial f(η1, η2) < 0 if and only if

Ω0 < 0, (20)Ω0 + Ω1 < 0, (21)

00.5

1 0

0.5

1

−4

−3

−2

−1

η2

η1

f(η

1,η

2)

Fig. 2. Variations of function f(η1, η2) = 1− 2η1 − 3η2 + 5η1η2

Ω0 + Ω2 < 0, (22)Ω0 + Ω1 + Ω2 + Ω12 < 0. (23)

Proof: Necessity: With f(η1, η2) < 0, η1 ∈ [0, 1] andη2 ∈ [0, 1], it is easy to obtain (20)–(23) by letting (η1, η2) =(0, 0), (1, 0), (0, 1), (1, 1).

Sufficiency: With Ω0 < 0, Ω0 + Ω1 < 0 and η1 ∈ [0, 1],one has

Ω0 + η1Ω1 = (1− η1)Ω0 + η1(Ω0 + Ω1) < 0. (24)

Similarly, with Ω0 + Ω2 < 0, Ω0 + Ω1 + Ω2 + Ω12 < 0 andη1 ∈ [0, 1], it follows that

(Ω0 + Ω2) + η1(Ω1 + Ω12)

= (1− η1)(Ω0 + Ω2) + η1(Ω0 + Ω1 + Ω2 + Ω12) < 0,

which can be rewritten as

(Ω0 + η1Ω1) + (Ω2 + η1Ω12) < 0. (25)

With η2 ∈ [0, 1], from (24) and (25), it holds that

f(η1, η2) = (Ω0 + η1Ω1) + η2(Ω2 + η1Ω12)

= (1− η2)(Ω0 + η1Ω1)

+ η2(Ω0 + η1Ω1 + Ω2 + η1Ω12)

< 0. (26)

The proof is complete.

Remark 4. The property in Lemma 2 can be schematicallydemonstrated by imposing Ω0,Ω1,Ω2 and Ω12 in f(η1, η2) asscalars. For instance, let f(η1, η2) = 1−2η1−3η2+5η1η2 withscalar parameters satisfying Lemma 2. For η1 ∈ [0, 1], η2 ∈[0, 1], Fig. 2 shows the variations of f(η1, η2), which can befound strictly less than zero. If one considers the boundarysituation with Ω0 + Ω1 + Ω2 + Ω12 = 0 by letting f(η1, η2) =1−2η1−3η2+6η1η2, Fig. 3 shows that the bound of f(η1, η2)will reach zero. If one considers f(η1, η2) > 0, by replacing“<” to “>” in (20)–(23), the corresponding result is also anequivalent condition.

Using Lemma 2, two tractable criteria for non-fragile time-varying controller design are provided for PPTVSs.

Theorem 3. Consider PPTVS (1) with non-fragile controller(3) and norm-bounded additive perturbations in (4). Given ascalar α∗ > 0, if there exist scalars ξi > 0 and αi, matrices





00.5

1 0

0.5

1

−4

−3

−2

−1

0

η2

η1

f(η

1,η

2)

Fig. 3. Variations of function f(η1, η2) = 1− 2η1 − 3η2 + 6η1η2

Xi,m > 0 and Yi,m, m ∈ Mi, i ∈ S, (11) and the followingconditions,

Ξi,m,0 < 0, (27)Ξi,m,0 + Ξi,m,1 < 0, (28)Ξi,m,0 + Ξi,m,2 < 0, (29)

Ξi,m,0 + Ξi,m,1 + Ξi,m,2 + Ξi,m,3 < 0, (30)Xi,Mi

= Xi+1,0, XS,Mi= X1,0, (31)

where

Ξi,m,0 =

sym (AiXi,m +BiYi,m)

−MiTiXi,m + αiXi,m

ξiBiHi Xi,mETi

∗ −ξiI 0

∗ ∗ −ξiI

,

Ξi,m,1 =

sym(AiXi,m + BiYi,m

)ξiBiHi Xi,mE

Ti

∗ 0 0

∗ ∗ 0

,

Ξi,m,2 =

sym

(AiXi,m +BiYi,m

)+αiXi,m

0 Xi,mETi

∗ 0 0

∗ ∗ 0

,

Ξi,m,3 =

sym(AiXi,m + BiYi,m

)0 Xi,mE

Ti

∗ 0 0

∗ ∗ 0

,hold for all Fi(t) satisfying (7), then the closed-loop system

is α∗-exponentially stable. The periodic non-fragile controllergains are given as

K(t) = Ki(t) = Yi(t)X−1i (t), t ∈ Ti, (32)

with time-varying matrix functions Xi(t) and Yi(t) for t ∈[θi,m, θi,m+1) determined by

Xi(t) = Xi,m + εi,m(t)Xi,m, (33)

Yi(t) = Yi,m + εi,m(t)Yi,m, (34)

where Xi,m , Xi,m+1 − Xi,m, Yi,m , Yi,m+1 − Yi,m,εi,m(t) =

Mi(t−θi,m)Ti

∈ [0, 1), m ∈Mi, i ∈ S.

Proof: For t ≥ 0, consider a continuous periodic piecewisetime-varying matrix function X (t) = Xi(t), t ∈ Ti, where forXi(t) is in form of (33) with symmetric matrices Xi,m > 0satisfying (31) and Xi,m = Xi,m+1 −Xi,m, m ∈Mi, i ∈ S,

and εi,m(t) =Mi(t−θi,m)

Ti∈ [0, 1). Based on (8), the upper

right Dini derivative of Xi(t) exists and is given by

D+Xi(t) =Mi

TiXi,m, t ∈ [θi,m, θi,m+1). (35)

According to Lemma 2, conditions (27)–(34) imply that thefollowing polynomial matrix inequality holds:

Ξi,m,0+σi(t)Ξi,m,1+εi,m(t)Ξi,m,2+σi(t)εi,m(t)Ξi,m,3 < 0,(36)

which can be rewritten assym

(Ai(t)Xi(t)

+Bi(t)Yi(t))

−D+Xi(t) + αiXi(t)ξiBi(t)Hi Xi(t)ETi (t)

ξiHTi BTi (t) −ξiI 0

Ei(t)Xi(t) 0 −ξiI

< 0,

(37)where

sym (Ai(t)Xi(t) + Bi(t)Yi(t))−D+Xi(t) + αiXi(t)

= sym (AiXi,m +BiYi,m)− Mi

TiXi,m + αiXi,m

+ σi(t)sym(AiXi,m + BiYi,m

)+ εi,m(t)

(sym

(AiXi,m +BiYi,m

)+ αiXi,m

)+ σi(t)εi,m(t)sym

(AiXi,m + BiYi,m

),

ξiBi(t)Hi = ξi(Bi + σi(t)Bi)Hi = ξiBiHi + σi(t)ξiBiHi,

Xi(t)ETi (t) = Xi,mETi + σi(t)Xi,mE

Ti + εi,m(t)Xi,mE

Ti

+ σi(t)εi,m(t)Xi,mETi .

From Xi,m > 0, m ∈ Mi, i ∈ S, one has Xi(t) > 0and X−1i (t) > 0 for t ≥ 0. Define a quadratic Lyapunovfunction V(t) = xT (t)Q(t)x(t), where continuous matrixfunction Q(t) = Qi(t) = X−1i (t), t ∈ Ti. Based on (32)and the fact that D+X−1i (t) = −X−1i (t)D+Xi(t)X−1i (t),t ∈ Ti, by multiplying both sides of inequality (37) withdiag(Qi(t), I, I), one has

sym(Qi(t)Ai(t)

+Qi(t)Bi(t)Ki(t))

+D+Qi(t) + αiQi(t)ξiQi(t)Bi(t)Hi ETi (t)

ξiHTi BTi (t)Qi(t) −ξiI 0

Ei(t) 0 −ξiI

< 0,

which can be rewritten by (12). Thus, when conditions (11),(27)–(31) hold, from Theorem 1 one can conclude that theclosed-loop PPTVS with non-fragile controller (3) and norm-bounded additive perturbations in (4) is α∗-exponentiallystable for all Fi(t) satisfying (7). The proof is complete.

Similarly, a sufficient condition of designing non-fragilecontroller that concerns multiplicative perturbations is pro-vided in the following theorem.

Theorem 4. Consider PPTVS (1) with non-fragile controller(3) and norm-bounded multiplicative perturbations in (5).Given a scalar α∗ > 0, if there exist scalars ξi > 0 and





αi, matrices Xi,m > 0 and Yi,m, m ∈ Mi, i ∈ S , (11) andthe following conditions,

Θi,m,0 < 0, (38)Θi,m,0 + Θi,m,1 < 0, (39)Θi,m,0 + Θi,m,2 < 0, (40)

Θi,m,0 + Θi,m,1 + Θi,m,2 + Θi,m,3 < 0, (41)Xi,Mi

= Xi+1,0, XS,Mi= X1,0, (42)

where

Θi,m,0 =

sym (AiXi,m +BiYi,m)

−MiTiXi,m + αiXi,m

ξiBiHi Y Ti,mE

Ti

∗ −ξiI 0

∗ ∗ −ξiI

,

Θi,m,1 =

sym(AiXi,m + BiYi,m

)ξiBiHi Y T

i,mETi

∗ 0 0

∗ ∗ 0

,

Θi,m,2 =

sym

(AiXi,m +BiYi,m

)+αiXi,m

0 Y Ti,mE

Ti

∗ 0 0

∗ ∗ 0

,

Θi,m,3 =

sym(AiXi,m + BiYi,m

)0 Y T

i,mETi

∗ 0 0

∗ ∗ 0

,hold for all Fi(t) satisfying (7), then the closed-loop system

is α∗-exponentially stable. The periodic non-fragile controllergains are obtained by (32)–(34).

Based on Lemma 2, one can prove Theorem 4 in a similarway by combining the relevant deductions in the proofs ofTheorem 2 and Theorem 3. Hence, the proof is omitted.

In practice, there may be cases where the controller gainsare more favorable to be continuous at all the switchinginstants. In this way, two continuous time-varying non-fragilecontrol schemes are given in the following corollaries, whichcan be derived based on Theorem 3 and Theorem 4 by lettingYi,1 = Yi, Yi,2 = Yi+1, i ∈ S.

Corollary 1. Consider PPTVS (1) with non-fragile controller(3) and norm-bounded additive perturbations in (4). Given ascalar α∗ > 0, if there exist scalars ξi > 0 and αi, matricesXi,m > 0 and Yi,m, m ∈ Mi, i ∈ S, conditions (11), (27)–(31) and

Yi,Mi= Yi+1,0, YS,Mi

= Y1,0 (43)

hold for all Fi(t) satisfying (7), then the closed-loop systemis α∗-exponentially stable. The periodic non-fragile controllergains are obtained from (32)–(34).

Corollary 2. Consider PPTVS (1) with non-fragile controller(3) and norm-bounded multiplicative perturbations in (5).Given a scalar α∗ > 0, if there exist scalars ξi > 0 andαi, matrices Xi,m > 0 and Yi,m, m ∈Mi, i ∈ S, conditions(11), (38)–(42) and (43) hold for all Fi(t) satisfying (7), thenthe closed-loop system is α∗-exponentially stable. The periodicnon-fragile controller gains are obtained from (32)–(34).

Compared with the previous studies [13], [31], it can beseen that without Lemma 2, it is difficult to derive the tractable

Algorithm 1 (Algorithm for Non-fragile Controller Design)Input: n, nu, Tp, S, Ti, Ai, Bi, Ei, Hi, Mi, αi, i ∈ S , α∗,

flagp, flagK .Output: ξi > 0, Xi,m > 0, Yi,m, m ∈Mi, i ∈ S.

1: Check the feasibility of condition (11). If TRUE, continue.Otherwise, break and give warning.

2: if (flagp,flagK) = (1, 1) then3: Based on Theorem 3, solve (27)–(31).4: else if (flagp,flagK) = (1, 0) then5: Based on Corollary 1, solve (27)–(31) and (43).6: else if (flagp,flagK) = (0, 1) then7: Based on Theorem 4, solve (38)–(42).8: else9: Based on Corollary 2, solve (38)–(42) and (43).

10: end if11: Compute the controller gains using (32)–(34).

criteria for controller design due to the different LTV coeffi-cients σi(t) and εi,m(t) in Theorems 1 and 2, i ∈ S,m ∈Mi.Using the selectable parameters Mi, i ∈ S, the controller gainscan become more flexible to increase the feasibility during thesolution process via convex optimization. It should be notedthat the number of conditions will be raised by larger valuesof Mi, i ∈ S. To ensure a reasonable computational cost, Mi

should be selected based on the balance of control effects andcomputing resources in practice.

Remark 5. Different from the corresponding constraints forPPLSs, the signs of αi, i ∈ S, are independent of the stabilityof time-varying subsystems in PPTVSs, while one only needsto guarantee α∗ > 0 and (11). To ensure the convexity of theproposed criteria, αi, i ∈ S, may be either given based onnumerical experience, or iteratively generated and tested bysome searching approaches like genetic algorithms [34].

Based on Theorems 3, 4 and Corollaries 1, 2, the pro-posed non-fragile controller design scheme is summarizedin Algorithm 1, where Boolean operators flagp and flagKrespectively indicate the type of perturbations (1 for additive, 0for multiplicative) and the continuity of K(t) at the switchinginstants (1 for discontinuous, 0 for continuous), with defaultvalues as 1.

IV. CASE STUDIES

To validate the effectiveness of the proposed criteria, thenon-fragile control of a numerical example and a mass-spring-damper benchmark system are considered in this section.Using Algorithm 1 with the MATLAB solver SeDuMi, theresults obtained under different cases are compared to illustratethe effectiveness of the proposed control scheme.

A. Numerical MIMO PPTVSFirst, one considers the non-fragile controller design for

a MIMO PPTVS consisting of three LTV subsystems de-scribed by (2) with Tp = 3.5, T1 = 1, T2 = 1.5 andT3 = 1 in appropriate time unit. Periodic time-varyingmatrix functions A(t), B(t) and E(t) are based on constant





TABLE IPARAMETER MATRICES AT THE SWITCHING INSTANTS

i Ai Bi Ei

1

2 1 0

0.5 1 1

1 1.5 1

0 0

0 1

1 0

[0 0.1 0.2

1 0.8 1

]

2

1.5 1 0

0 0.6 1

0.5 1 0.9

0.3 0

0 1

1 0.5

[0.1 0.3 0.5

0.8 0.6 1.2

]

3

1 1 0

0.5 1 1

1 3 1

0.2 0

0 1

2 0

[0.2 0 0.1

1.2 −0.5 0.9

]

matrices Ai, Bi, Ei, i = 1, 2, 3, as given in Table I. Consideradditive controller perturbations in (4), where H1 = I ,H2 = 1.5I , H3 = 0.5I , and uncertain continuous time-varying functions Fi(t) = diag

(a1 sin(2πt/Tp)+a2 sin(πt)+

a3 sin(2πt), b1 cos(2πt/Tp) + b2 sin(πt) + b3 cos(2πt))

withrandomly generated nonnegative scalars ai, bi, i = 1, 2, 3,satisfying

∑3i=1 ai ≤ 1,

∑3i=1 bi ≤ 1. With x(0) = [3, 2, 0]T ,

the open-loop state trajectory is shown in Fig. 4, from whichit can be seen that the open-loop PPTVS is unstable.

Let α1 = 0.8, α2 = 0.9, α3 = 1, and α∗ =∑3i=1 αiTi/2Tp = 0.45. Consider the non-fragile controllers

of the following cases (i = 1, 2, 3):

• Case 1: Mi = 5 with discontinuous Y(t) and K(t)• Case 2: Mi = 1 with discontinuous Y(t) and K(t)• Case 3: Mi = 5 with continuous Y(t) and K(t)• Case 4: Mi = 1 with continuous Y(t) and K(t)

Based on Theorem 3 (for Case 1, Case 2) and Corollary 1 (forCase 3, Case 4), one can obtain the solutions of Xi,m, Yi,m,m ∈ Mi, i = 1, 2, 3. The time-varying controller gainscan be computed via (32)–(34). The closed-loop system statetrajectories and variations of ‖X (t)‖, ‖Y(t)‖ and ‖K(t)‖ overone period for the four cases are illustrated in Fig. 5 andFig. 6, where the continuity of X (t), Y(t) and K(t) at theswitching instants can be clearly reflected by the variationsof their norms. For comparison, the closed-loop system statetrajectories and the corresponding parameters obtained underthe PPTVS controllers obtained by Theorem 2 and Corollary1 in [31] are also respectively shown in the figures.

With the same parameters of systems and αi, i = 1, 2, 3,from Fig. 5 and Fig. 6 it can be seen that the proposed non-fragile controllers can achieve better control effects and fasterconvergences against perturbations than those obtained by theexisting methods for PPTVSs in [31]. On the other hand,benefiting from the divisions of subintervals during controllerdesign, the differences in control effects of Theorem 3 andCorollary 1 are not obvious despite their different constraintsin the continuity of K(t) at the switching instants. Moreover,from the variations of ‖X (t)‖, ‖Y(t)‖ and ‖K(t)‖ over oneperiod of the four cases, one can observe that a larger Mi canresult in more flexible time-varying controller gains especiallyfor the cases imposing the continuity of controller gains atthe switching instants, which explains the similar non-fragilecontrol performances of the cases.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

9x 10

5

t

x(t)

x1(t)

x2(t)

x3(t)

0 0.5 10

10

20

30

40

Fig. 4. Open-loop system state trajectory

0 1 2 3 4 5−2

0

2

4

t

x1(t)

Case 1 Case 2 [27]

0 1 2 3 4 5−15

−10

−5

0

5

t

x2(t)

0 1 2 3 4 5

−5

0

5

t

x3(t)

0 1 2 30.5

1

1.5

‖X(t)‖

t

3

3.5

4

‖X(t)‖

[27]

0 1 2 32

2.5

3

3.5

‖Y(t)‖

t

4

8

12

16

‖Y(t)‖

[27]

0 1 2 30

50

100

150

t

‖K(t)‖

Fig. 5. Closed-loop system state trajectories and variations of ‖X (t)‖,‖Y(t)‖, ‖K(t)‖ over one period for Case 1, Case 2 based on Theorem3 and the method in [31]

B. Mass-spring-damper System

In this example, the non-fragile controller design is consid-ered for a mass-spring-damper system described by a PPTVSwith four subsystems. The mass-spring-damper system can beused as a benchmark model for many practical vibration sys-tems. For example, it provides a firm foundation for modellingsome engineering objects with complex material propertieslike beams [35], which may exhibit time-periodic stiffnessand/or damping properties due to periodic time-varying lengths[36] or time-periodic modulations [37]. Inspired by [36], [38],the system is supposed to involve two masses, two springelements and two damping elements, as shown in Fig. 7, wheresome of the parameters are modulated to be time-periodic. Thetwo system inputs are the force u1(t) applied to m1 and theforce u2(t) applied to m2, while the displacement x1(t) ofm1 and the velocity x2(t) of m2 are chosen as the system





0 1 2 3 4 5−2

0

2

4

t

x1(t)

Case 3 Case 4 [27]

0 1 2 3 4 5−15

−10

−5

0

5

t

x2(t)

0 1 2 3 4 5

−5

0

5

t

x3(t)

0 1 2 30.5

1

1.5

‖X(t)‖

t

3

3.5

4

‖X(t)‖

[27]

0 1 2 32

2.5

3

3.5

‖Y(t)‖

t

4

8

12

16

‖Y(t)‖

[27]

0 1 2 30

50

100

150

t

‖K(t)‖

Fig. 6. Closed-loop system state trajectories and variations of ‖X (t)‖,‖Y(t)‖, ‖K(t)‖ over one period for Case 3, Case 4 based on Corollary1 and the method in [31]

outputs. The parameters of the system are given in Table II.For t ≥ 0, based on the equations of motion, that is,[

m1 0

0 m2

] [x1(t)

x2(t)

]+

[c1 + c2(t) −c2(t)

−c2(t) c2(t)

] [x1(t)

x2(t)

]+

[k1(t) −k1(t)

−k1(t) k1(t) + k2

] [x1(t)

x2(t)

]=

[u1(t)

u2(t)

].

A state-space representation of the system is derived as

x(t) =

0 0 1 0

0 0 0 1

−k1(t)m1

k1(t)m1

− c1+c2(t)m1

c2(t)m1

k1(t)m2

−k1(t)+k2m2

c2(t)m2

− c2(t)m2

x(t)

+

[0 0 1

m10

0 0 0 1m2

]Tu(t),

y(t) =

[1 0 0 0

0 0 0 1

]x(t) (44)

with state vector x(t) = [x1(t), x2(t), x1(t), x2(t)]T , controlinput vector u(t) = [u1(t), u2(t)]T , output vector y(t) =[x1(t), x2(t)]T , and periodic piecewise time-varying k1(t) =k1(t + lTp), c2(t) = c2(t + lTp), l = 0, 1, . . .. Consider afundamental period Tp = 15s with T1 = 4s, T2 = 3s, T3 = 5sand T4 = 3s, the variations of k1(t) and c2(t) over one periodare shown in Fig. 8. Hence, system (44) is a PPTVS. Theobjective here is to design a non-fragile controller for vibrationattenuation of the masses.

For PPTVS (44), consider M1 = 4, M2 = 3, M3 = 5,M4 = 3, αi = 0.5, i = 1, 2, 3, 4, α∗ =

∑4i=1 αiTi/2Tp =

0.25 and multiplicative controller perturbations in (5), wherefor i = 1, 2, 3, 4, Hi = 1.2I , Fi(t) = diag(sin(t), 0.5 sin(t))and Ei(t) satisfying (6) with

E1 =

[0.3 0.2

0 0.1

], E2 =

[0.1 0.4

0.3 0.2

],

m1 m2

k1(t)

c2(t)

k2

c1

u1(t) u2(t)

x1(t) x2(t)

Fig. 7. Sketch of the considered mass-spring-damper system

TABLE IIPARAMETERS OF THE MASS-SPRING-DAMPER SYSTEM

Parameter Meaning Value Unitm1 Mass of the left mass 10 kgm2 Mass of the right mass 4 kgk1(t) Time-periodic stiffness between

m1 and m2

[10, 15] N/m

k2 Constant stiffness between m2

and the fixed world5 N/m

c1 Constant viscous damping be-tween m1 and the fixed world

0.2 N s/m

c2(t) Time-periodic viscous dampingbetween m1 and m2

[0.2, 0.35] N s/m

E3 =

[0 0.1

0.1 0.3

], E4 =

[0.1 0.1

0.2 0.2

].

For convenience of application, a non-fragile control schemewith periodic time-varying controller gains continuous at theswitching instants is obtained by Corollary 2, which is devel-oped from Theorem 4. One obtains ξ1 = 1.8, ξ2 = 1.8452,ξ3 = 1.496, ξ4 = 1.6729 and the relevant matrix solutions.Denote yo(t) = [yo1(t), yo2(t)]T = [xo1(t), xo2(t)]T andyc(t) = [yc1(t), yc2(t)]T = [xc1(t), xc2(t)]T as the open-loop system output and the closed-loop system output, re-spectively, which are obtained under the designed controllerwith multiplicative perturbations and x(0) = [0.5, 0.1, 0, 0]T .Fig. 9 shows the variations of open-loop and closed-loopsystem outputs. It can be seen that the closed-loop system isstable against multiplicative controller perturbations, and theconcerned vibrations are attenuated. Therefore, the advantagesof the proposed control scheme can be summarized as the su-perior technical flexibility in controller than previous studies,as well as the effectiveness in achieving resilience and fastconvergence under controller perturbations.

V. CONCLUSION

This paper has developed the non-fragile controller designscheme for a class of PPTVSs with LTV subsystems usinga new matrix polynomial lemma. Additive and multiplicativeperturbations, which can be partially characterized by periodicpiecewise time-varying parameters, have been considered. Theproposed lemma has provided an alternative approach to makethe non-fragile controller design of PPTVSs amenable to con-vex optimization, and the controller gains can be more flexibledue to some selectable parameters of subinterval division.Sufficient conditions on stability analysis and tractable non-fragile controller synthesis have been established based on aperiodic Lyapunov function in time interpolative form. Case





0 5 10 1510

12

14

16

t (s)

k1(t)(N

/m)

0 5 10 150.2

0.25

0.3

0.35

0.4

t (s)

c2(t)(N

s/m)

Fig. 8. k1(t) and c2(t) over one period

0 5 10 15 20 25 30−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t (s)

Out

puts

yo1(t)

yo2(t)

yc1(t)

yc2(t)

Fig. 9. Open-loop and closed-loop system outputs

studies on a numerical MIMO PPTVS model and a mass-spring-damper system have demonstrated the effectiveness ofthe designed controllers. Compared with the existing controlmethod of PPTVSs, the proposed non-fragile control schemenot only performs well in guaranteeing the stability, but alsocan achieve faster convergence under the impacts of uncer-tain controller perturbations. In future work, research effortswill be extended to the input-output performance analysis,especially for the cases involving more complicated peri-odic time-varying uncertainties, faults and industry-orientedrequirements [39], [40].

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[40] J. Liu, J. Lam, and Z. Shu, “Positivity-preserving consensus of ho-mogeneous multi-agent systems,” IEEE Trans. Autom. Control, DOI10.1109/TAC.2019.2946205, in press.

Xiaochen Xie (S’13–M’18) received the B.E.degree in automation and the M.E. degree incontrol science and engineering from HarbinInstitute of Technology, Harbin, China, in 2012and 2014, respectively. She obtained the Ph.D.degree in control engineering from The Uni-versity of Hong Kong in 2018. She was anawardee of the Hong Kong PhD FellowshipScheme (HKPFS) 2014/15, which supported herdoctoral study and academic exchanges. She iscurrently a Post-doctoral Fellow at Department

of Mechanical Engineering, The University of Hong Kong. Her researchinterests include robust control and filtering, periodic systems, switchedsystems, intelligent systems and process monitoring.

James Lam (SM’99–F’12) received a B.Sc. (1st

Hons.) degree in mechanical engineering fromthe University of Manchester, and was awardedthe Ashbury Scholarship, the A.H. Gibson Prize,and the H. Wright Baker Prize for his academicperformance. He obtained the M.Phil. and Ph.D.degrees from the University of Cambridge. He isa Croucher Scholar, Croucher Fellow, and Dis-tinguished Visiting Fellow of the Royal Academyof Engineering. Prior to joining The Universityof Hong Kong in 1993 where he is now Chair

Professor of Control Engineering, he was lecturer at the City Universityof Hong Kong and the University of Melbourne.

Prof. Lam is a Chartered Mathematician, Chartered Scientist, Char-tered Engineer, Fellow of Institute of Electrical and Electronic Engineers,Fellow of Institution of Engineering and Technology, Fellow of Institute ofMathematics and Its Applications, and Fellow of Institution of MechanicalEngineers. He is Editor-in-Chief of IET Control Theory and Applicationsand Journal of The Franklin Institute, Subject Editor of Journal ofSound and Vibration, Editor of Asian Journal of Control, Senior Editorof Cogent Engineering, Associate Editor of Automatica, InternationalJournal of Systems Science, Multidimensional Systems and SignalProcessing, and Proc. IMechE Part I: Journal of Systems and ControlEngineering. He is a member of the IFAC Technical Committee onNetworked Systems, and Engineering Panel (Joint Research Schemes),Research Grant Council, HKSAR. His research interests include modelreduction, robust synthesis, delay, singular systems, stochastic systems,multidimensional systems, positive systems, networked control systemsand vibration control. He is a Highly Cited Researcher in Engineering(2014, 2015, 2016, 2017, 2018, 2019) and Computer Science (2015).

Ka-Wai Kwok (SM’18) has served as AssistantProfessor in Department of Mechanical Engi-neering, The University of Hong Kong (HKU),since 2014. He obtained the Ph.D. degree atHamlyn Centre for Robotic Surgery, Depart-ment of Computing, Imperial College London in2012, where he continued research on surgicalrobotics as a postdoctoral fellow. In 2013, he wasawarded the Croucher Foundation Fellowship,which supported his research jointly supervisedby advisors in The University of Georgia, and

Brigham and Women’s Hospital – Harvard Medical School. His researchinterests focus on surgical robotics, intra-operative medical image pro-cessing, and their uses of high-performance computing techniques.He has involved in various designs of surgical robotic devices andinterfaces for endoscopy, laparoscopy, stereotactic and intra-cardiaccatheter interventions. To date, he has co-authored with over 40 clinicalfellows and over 80 engineering scientists.

Dr. Kwok’s multidisciplinary work has been recognized by variousscientific communities through many international conference/journalpaper awards, e.g. the Best Conference Paper Award of ICRA’18,which is the largest conference ranked top in the field of robotics,as well as TPEL’18, RCAR’17, ICRA’19, ICRA’17, ICRA’14, IROS’13,FCCM’11, Hamlyn’12, Hamlyn’08 and Surgical Robot Challenge’16. Healso became the recipient of the Early Career Awards 2015/16 offered byResearch Grants Council (RGC) of Hong Kong. He serves as AssociateEditor for IROS’17–19, ICRA’19, Frontier in Robotics and AI, and Annalsof Biomedical Engineering. He is a principal investigator of Group forInterventional Robotic and Imaging Systems (IRIS) at HKU.


http://dx.doi.org/10.1109/TAC.2019.2946205

http://dx.doi.org/10.1109/TAC.2019.2946205

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