Delay-dependent robust stability analysis and stabilization of linear systems using a simple delay-discretization approach

5th International Conference on Advances in Control and 5th 5th International International Conference Conference on on Advances Advances in in Control Control and and Optimization of Systems 5th International Conference on 5th International Conference on Advances Advances in in Control Control and and Optimization of Dynamical Dynamical Systems Optimization of Dynamical Systems Available online at www.sciencedirect.com February 18-22, 2018. Hyderabad, India Optimization of Dynamical Dynamical Systems 5th International Conference on Advances in Control and Optimization of Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India February 18-22, 18-22, 2018. Hyderabad, Hyderabad, India Optimization of Dynamical Systems February 2018. India February 18-22, 2018. Hyderabad, India

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IFAC PapersOnLine 51-1 (2018) 572–579

Delay-dependent robust stability analysis Delay-dependent Delay-dependent robust robust stability stability analysis analysis and stabilization of linear systems using a Delay-dependent robust stability analysis and stabilization of linear systems using a and stabilization of linear systems using a simple delay-discretization approach and stabilization of linear systems using simple delay-discretization approach simple delay-discretization approach a simple delay-discretization approach ∗ Dushmanta Dushmanta Kumar Kumar Das, Das, Member, Member, IEEE IEEE ∗∗

Dushmanta Kumar Das, Member, IEEE ∗ ∗∗ ∗ Dushmanta Kumar Das, IEEE ∗∗ Sandip Ghosh, IEEE Dushmanta Kumar Member, Das, Member, Member, IEEE ∗∗ Sandip Ghosh, Member, IEEE Sandip Ghosh, Member, IEEE ∗∗ ∗ ∗∗∗ ∗∗ Sandip Ghosh, Member, IEEE ∗∗∗ Dushmanta Kumar Das, Member, IEEE Bidyadhar Subudhi, Sr.Member, IEEE SandipSubudhi, Ghosh, Member, IEEE ∗∗∗ Bidyadhar Sr.Member, IEEE Bidyadhar Subudhi, Sr.Member, IEEE ∗∗∗ ∗∗ ∗∗∗ Bidyadhar Subudhi, Sr.Member, IEEE SandipSubudhi, Ghosh, Member, IEEE Bidyadhar Sr.Member, IEEE ∗ Bidyadhar Subudhi, Sr.Member, IEEE ∗∗∗ of ∗ Institute of Technology Nagaland, Department ∗ National National Institute Institute of of Technology Technology Nagaland, Nagaland, Department Department of EEE EEE of ∗ ∗ National National(e-mail: Institute of Technology Nagaland, Department of EEE EEE [email protected]). National Institute of Technology Nagaland, Department of EEE (e-mail: [email protected]). (e-mail: [email protected]). ∗ ∗∗ National(e-mail: [email protected]). ∗∗ Institute of Technology Nagaland, Department of EEE BHU, Department of EEE (e-mail: [email protected]). (e-mail: [email protected]). ∗∗ IIT IIT BHU, Department of EEE (e-mail: [email protected]). IIT BHU, Department of EEE (e-mail: [email protected]). ∗∗ ∗∗∗ ∗∗ IIT BHU, Department of [email protected]). ∗∗∗ [email protected]). National Institute of Technology Rourkela, Department of EEE IIT BHU,(e-mail: Department of EEE EEE (e-mail: (e-mail: [email protected]). ∗∗∗ Institute of Technology Rourkela, Department of EEE National Institute of Technology Rourkela, Department of EEE ∗∗∗ National ∗∗ ∗∗∗ National Institute of Technology Rourkela, Department of EEE IIT BHU, Department of EEE (e-mail: [email protected]). (e-mail: [email protected]). National Institute of Technology Rourkela, Department of EEE (e-mail: [email protected]). (e-mail: [email protected]). ∗∗∗ (e-mail:[email protected]). [email protected]). National Institute Technology Rourkela, Department of EEE (e-mail: (e-mail: [email protected]). Abstract: Abstract: In In this this paper, paper, a a discretization discretization scheme scheme is is proposed proposed for for systems systems with with constant constant delay delay Abstract: In this paper, a discretization scheme is proposed for systems with constant delay Abstract: In this paper, a discretization scheme is proposed for systems with constant delay and the delay is assumed to appear in range. The proposed approach uses simple LyapunovAbstract: In this paper, a discretization scheme is proposed for systems with constant delay and the delay is assumed to appear in range. The proposed approach uses simple Lyapunovand the delay is assumed to appear in range. The proposed approach uses simple Lyapunovand the delay is assumed to appear in range. The proposed approach uses simple LyapunovAbstract: In this paper, a discretization scheme is proposed for systems with constant delay Krasovskii functional to derive the robust stability and stabilization criteria. As a result of which and the delay is assumed to appear in range. The proposed approach uses simple LyapunovKrasovskii functional to derive the robust stability and stabilization criteria. As a result of which Krasovskii functional to derive the robust stability and stabilization criteria. As aa result of which Krasovskii functional to derive the robust stability and stabilization criteria. As result of which and the delay is assumed to appear in range. The proposed approach uses simple Lyapunovthe derived criteria become finite-dimensional. Though the criteria give sufficient condition for Krasovskii functional to derive the robust stability and stabilization criteria. As a result of which the derived derived criteria criteria become become finite-dimensional. finite-dimensional. Though Though the the the criteria criteria give give sufficient sufficient condition condition for for the derived criteria become finite-dimensional. Though the criteria give sufficient condition for Krasovskii functional to derive the robust stability and stabilization criteria. As a result of which stability, but they are less conservative than the existing results. To demonstrate the effectiveness the derived criteria become finite-dimensional. Though the criteria give sufficient condition for stability, but but they they are are less less conservative conservative than than the the existing existing results. results. To To demonstrate demonstrate the the effectiveness effectiveness stability, stability, but they are less than the To the effectiveness the derived finite-dimensional. Though results. the criteria give sufficient for of approach, examples are considered. stability, butcriteria theynumerical arebecome less conservative conservative than the existing existing results. To demonstrate demonstrate thecondition effectiveness of the the approach, numerical examples are considered. of the approach, numerical examples are considered. of the approach, numerical examples are considered. stability, but they are less conservative than the existing results. To demonstrate the effectiveness of the approach, numerical examples are considered. © the 2018,approach, IFAC (International Automatic Control) Hosting by Elsevier Ltd. All rights reserved. of numericalFederation examplesofare considered. Keywords: Keywords: Time-delay Time-delay System, System, Lyapunov-Krasovskii Lyapunov-Krasovskii functional, functional, Delay-dependent Delay-dependent stability stability Keywords: Time-delay System, Lyapunov-Krasovskii functional, Delay-dependent stability Keywords: Time-delay System, Lyapunov-Krasovskii functional, Delay-dependent criteria. Keywords: Time-delay System, Lyapunov-Krasovskii functional, Delay-dependent stability stability criteria. criteria. criteria. Keywords: criteria. Time-delay System, Lyapunov-Krasovskii functional, Delay-dependent stability criteria. 1. INTRODUCTION The objective of the paper is to analyze the 1. The robust stability stability 1. INTRODUCTION INTRODUCTION The objective objective of of the the paper paper is is to to analyze analyze the the robust robust stability 1. INTRODUCTION The objective of the paper is to analyze the robust stability Time-delay is a common phenomenon in almost all the and design a static state feedback controller for the system 1. INTRODUCTION The objective of the paper is to analyze the robust stability Time-delay is aa common phenomenon in almost all the and design aa static state feedback controller for the system Time-delay is common phenomenon in almost all the and design static state feedback controller for the system Time-delay is aaeconomics, common phenomenon in almost all the and design a static state feedback controller for the system 1. INTRODUCTION systems such as economics, biology, engineering and so on. The objective of the paper is to analyze the robust stability with a improved tolerable delay margin. Simple LK funcTime-delay is common phenomenon in almost all the and design a static state feedback controller for the systems such as biology, engineering and so on. with aa improved tolerable delay margin. margin. Simple Simple LK LKsystem funcsystems such as economics, biology, engineering and so on. with improved tolerable delay funcsystems such biology, engineering and so on. Time-delay isas aeconomics, common phenomenon in almost all the with aa improved tolerable delay margin. Simple LK funcThe presence of such phenomenon degrades performance and design a static state feedback controller for the system tionals similar to delay-interval like analysis are defined systems such as economics, biology, engineering and so on. with improved tolerable delay margin. Simple LK funcThe presence of such phenomenon degrades performance tionals similar to delay-interval like analysis are defined The presence of such phenomenon degrades performance tionals similar to delay-interval like analysis are defined The presence of such phenomenon degrades performance systems such as economics, biology, engineering and so on. tionals similar to delay-interval like analysis are defined and/or destabilizes the system Gu et al. (2003); Park with a improved tolerable delay margin. Simple LK funcover arbitrary number of discretized delay intervals and The presence of such phenomenon degrades performance tionals similar to delay-interval like analysis are defined and/or destabilizes the system Gu et al. (2003); Park over arbitrary number of discretized delay intervals and and/or destabilizes the system Gu et al. (2003); Park over arbitrary arbitrary number number of of discretized discretized delay intervals and and/or destabilizes the system Gu et al. (2003); Park The presence of such phenomenon degrades performance over delay intervals and and Kwon (2005b,a). Therefore, stability analysis and tionals similar to delay-interval like analysis are defined it is shown that the criterion for the last/highest interval and/or destabilizes the system Gu et al. (2003); Park over arbitrary number of discretized delay intervals and and Kwon (2005b,a). Therefore, stability analysis and it is shown that the criterion for the last/highest interval and Kwon (2005b,a). Therefore, stability analysis and it is shown that the criterion for the last/highest interval and Kwon (2005b,a). Therefore, stability analysis and and/or destabilizes the system Gu et al.are (2003); Park it is shown that the criterion for the last/highest interval stabilization of such time-delay systems are important over arbitrary number of discretized delay intervals and interval satisfies the stability requirement of all its suband Kwon (2005b,a). Therefore, stability analysis and it is shown that the criterion for the last/highest interval stabilization of such time-delay systems important satisfies the the stability stabilization of of such such time-delay time-delay systems systems are are important important interval interval satisfies stability requirement requirement of of all all its its subsubstabilization and Kwon (2005b,a). Therefore, stability analysis and interval satisfies the stability requirement of all its subissues. According to the dependency on size of the delay, it is shown that the criterion for the last/highest interval intervals leading to a single criterion that satisfies stability stabilization of such time-delay systems are important interval satisfies the stability requirement of all its subissues. According to the dependency on size of the delay, intervals leading to a single criterion that satisfies stability issues. According According to to the the dependency dependency on on size size of of the the delay, delay, intervals leading to a single criterion that satisfies stability issues. stabilization of such time-delay systems areof important intervals leading to single criterion that satisfies stability the above criteria are classified as delay-dependent and interval theaathe stability requirement of all criterion its subrequirement for all the intervals in shot. The issues. According toare the dependency on size the delay, intervals satisfies leading to single criterion satisfies stability the above criteria classified as delay-dependent and for intervals in aaa shot. The criterion the above criteria are classified as delay-dependent and requirement requirement for all all the intervals in that shot. The criterion the above criteria are classified as delay-dependent and issues. According to the dependency on size of the delay, requirement for all the intervals in a shot. The criterion delay-independent criteria. It well known that the former intervals leading to a single criterion that satisfies stability obtained this way is having same number of decision varithe above criteria are classified as delay-dependent and requirement for all the intervals in a shot. The criterion delay-independent delay-independent criteria. criteria. It It well well known known that that the the former former obtained obtained this this way way is is having having same same number number of of decision decision varivaridelay-independent It well that the former the above criteria criteria. are classified asknown delay-dependent and obtained this way is having same number of decision variis less conservative than that of the latter one. requirement for all the intervals in a shot. The criterion ables as that of considering single interval and hence is delay-independent criteria. It well known that the former obtained this way is having same number of decision variis less conservative than that of the latter one. ables as that of considering single interval and hence is is less conservative than that of the latter one. ables as that of considering single interval and hence is is less conservative than that the latter one. delay-independent criteria. It of well known that the former ables as that of considering single interval and hence is obtained this way is having same number ofresult decision variinvariant to the number of intervals. As a result of which is less conservative than that of the latter one. ables as that of considering single interval and hence is invariant to the number of intervals. As a of which invariant to the number of intervals. As a result of which The existing discretization approaches Gu et al. (2003); The discretization Gu et is lessexisting conservative than thatapproaches of the latter to the of intervals. As aa result of which The existing discretization approaches Guone. et al. al. (2003); (2003); invariant ables as that of number considering single interval and hence is the computational burden does not increase with number invariant to the number of intervals. As result of which the computational burden does not increase with number The existing discretization approaches Gu et al. (2003); the computational computational burden does not increase with number Gouaisbaut and Peaucelle (2006); Han (2009) to derive The existingand discretization approaches Gu et al.to (2003); Gouaisbaut Peaucelle (2006); Han (2009) derive the burden does not increase with number Gouaisbaut and Peaucelle (2006); Han (2009) to derive invariant to the number of intervals. As a result of which of discretization. The resulting criteria become simple and the computational burden does not increase with number Gouaisbaut and Peaucelle (2006); Han (2009) derive The existing discretization approaches Gu etfor al.tosystems (2003); of discretization. discretization. The The resulting resulting criteria criteria become become simple simple and and necessary and sufficient stability criterion for Gouaisbaut and Peaucellestability (2006); Han (2009) derive of necessary and sufficient criterion of The resulting become simple and necessary and sufficient stability criterion fortosystems systems the computational burden doescriteria not increase with number computationally efficient. of discretization. discretization. The resulting criteria become simple and computationally efficient. necessary and sufficient stability criterion for systems Gouaisbaut and Peaucelle (2006); Han (2009) to derive computationally efficient. with single and constant delay uses quadratic Lyapunovnecessary and sufficient stability criterion for systems with single and constant delay uses quadratic Lyapunovcomputationally efficient. with single and constant delay uses quadratic Lyapunovof discretization. The resulting criteria become simple and computationally efficient. 2. PRELIMINARIES with single and constant delay uses quadratic Lyapunovnecessary stability criterion for systems Krasovskii (LK) functional. The use of such functional 2. PRELIMINARIES with singleand and sufficient constant delay uses quadratic Lyapunov2. PRELIMINARIES Krasovskii (LK) functional. The use of such functional Krasovskii (LK) functional. The use of such functional computationally efficient. 2. 2. PRELIMINARIES PRELIMINARIES Krasovskii (LK) functional. The use of functional with single and constant delay uses quadratic Lyapunovresults infinite-dimensional LMI conditions for stability. Let us consider linear time-delay system Krasovskii (LK) functional. The use of such such results infinite-dimensional LMI conditions for stability. time-delay results infinite-dimensional LMI conditions forfunctional stability. Let Let us us consider consider aaaa linear linear time-delay system system 2. PRELIMINARIES results infinite-dimensional LMI conditions for stability. Let us consider linear time-delay system Krasovskii (LK) functional. The use of such functional As result of which the number of matrix variables involved results infinite-dimensional LMI conditions for stability. Let us consider a linear time-delay system x(t) ˙ = f(x(t), x(t − h)), As result of which the number of matrix variables involved (1) As result of which the number of matrix variables involved x(t) ˙ = f(x(t), x(t − h)), (1) x(t) ˙ = f(x(t), x(t − h)), (1) As of number of matrix matrix variables involved results infinite-dimensional LMI conditions fornumber stability. Let us consider a x(t) linear time-delay system are exponentially increases with increase in number of As result result of which which the the number of variables involved ˙ = f(x(t), x(t − h)), (1) n are exponentially increases with increase in of x(t) ˙ = f(x(t), x(t − h)), (1) are exponentially increases with increase in number of n where x(t) ∈ R is the state; h is a constant delay n where x(t) ∈ R is the state; h is a constant delay are exponentially increases with increase in number of As result of which the number of matrix variables involved where x(t) x(t) ∈ ∈ R Rx(t) is the state; h is aa constant delay discretization. Though the derived criteria are able to are exponentially increases with increase in are number of where n ˙ = f(x(t), x(t − h)), (1) discretization. Though the derived criteria able to n is the state; h is constant delay discretization. Though the derived criteria are able to ¯ ˙ where x(t)00 ∈≤ Rh is h, h is a us constant delay = satisfying ≤ h ≤ h, h(t) = 0. Let us define x tt = discretization. Though the derived criteria able to ¯¯¯theh(t) ˙˙˙ state; are exponentially increases with increase in are number of satisfying ≤ = 0. Let define x track the analytical results but they are unable to extend discretization. Though the derived criteria are able to n = satisfying 0 ≤ h ≤ h, h(t) = 0. Let us define x track the analytical results but they are unable to extend t track the the analytical results but they are unable to extend ¯¯¯ ≤ where : x(t)t00∈∈≤ Rh is h condition is a us constant delay ¯the ˙ state; satisfying h, h(t) = 0. Let define x {x(t) [− h 0]}. The initial for system t = = satisfying ≤ h ≤ h, h(t) = 0. Let us define x track analytical results but they are unable to extend t discretization. Though the derived criteria are able to {x(t) : t ∈ [− h 0]}. The initial condition for system for stabilization problems. In spite of the fact that the track the analytical results but they are unable to extend {x(t) : t ∈ [− h 0]}. The initial condition for system for stabilization problems. In spite of the fact that the ¯ ¯ ˙ for stabilization stabilization problems. problems. In In spite spite of of the the fact fact that that the the satisfying ¯ ≤ {x(t) ∈≤order [−hh h 0]}. The initial condition forx system h, The h(t) initial = smooth 0. condition Let so us that define xt = (1), x first differentiably ˙˙˙ 00system exists {x(t) :: is tt0∈ [− 0]}. for 0 for track the analytical results but they are unable to extend (1), x is first order differentiably smooth so that x exists finite dimensional approaches yields sufficient condition 0 for stabilization problems. In spite of the fact that the (1), x is first order differentiably smooth so that x exists finite approaches yields yields sufficient sufficient condition condition and 0 is first order ¯ differentiably finite dimensional dimensional approaches (1), x xcontinuous. differentiably smooth so that that x˙˙ 000system exists {x(t) t ∈ order [−h 0]}. The initial condition forx 0 (1), smooth so exists finite dimensional approaches yields condition 0: is first stabilization problems. In spite the as: fact(i) that the and and continuous. for stability, it has some advantages such as: (i) it uses finite dimensional approaches yieldsofsufficient sufficient condition continuous. for stability, it has some advantages such it uses for stability, it has some advantages such as: (i) it uses and continuous. (1), xcontinuous. ˙ 0 exists 0 is first order differentiably smooth so that x and for stability, it has some advantages such as: (i) it uses finite dimensional approaches yields sufficient condition simple LK functional to obtain the LMI condition, (ii) for stability, it has some advantages such condition, as: (i) it uses for simple LK to the (ii) Also, simple LK functional functional to obtain obtain the LMI LMI condition, (ii) Also,continuous. for quadratic quadratic stability stability analysis analysis of of uncertain uncertain systems, systems, and for quadratic stability analysis of uncertain simple LK to the LMI condition, (ii) for stability, it matrix has some advantages such as: (i)with it uses the number of matrix variables doe not increase with the simple LK functional functional to obtain obtain thenot LMI condition, (ii) Also, Also, for quadratic stability analysis of uncertain systems, systems, the following lemma is well known. the number of variables doe increase the Also, for quadratic stability analysis of uncertain systems, the number of matrix variables doe not increase with the the following lemma is well known. the following lemma is well known. the number of variables increase with the the simple LK functional to obtaindoe thenot LMI condition, (ii) Also, increase in number of discretizations, (iii) the complexity the number of matrix matrix variables doe not increase with the following lemma is well known. for1.quadratic stability analysis of uncertain systems, increase in number of discretizations, (iii) the complexity the following lemma is well known. increase in number of discretizations, (iii) the complexity Lemma (Peterson (1987)). For appropriate dimensional Lemma 1. (Peterson (1987)). For appropriate dimensional increase in number of discretizations, (iii) the complexity the number of matrix variables doe not increase with the Lemma 1. (Peterson (1987)). For appropriate dimensional of the criterion does not increases with number of increase in number of discretizations, (iii) complexity the following lemma is well known. of the criterion does not increases with the number of Lemma 1. 1.X, (Peterson (1987)). matrix For appropriate appropriate dimensional of the the criterion does does not not increases increases with with the the number number of of matrices Y Z following Lemma (Peterson (1987)). For dimensional matrices X, Y and and invertible invertible matrix Z> > 0, 0, the the following of increase in number of discretizations, (iii) complexity X, Y and invertible matrix Z > 0, the following discretization, (iv) most importantly, it can be easily of the criterion criterion does not importantly, increases with the number of matrices discretization, (iv) most it can be easily matrices X, Y and invertible matrix Z > 0, the following Lemma 1. (Peterson (1987)). For appropriate dimensional discretization, (iv) most importantly, it can be easily inequality holds: matrices X, Y and invertible matrix Z > 0, the following inequality holds: holds: discretization, (iv) most importantly, it can be easily of the criterion does not increases with the number of inequality extended for control design. Due to the above facts the discretization, (iv) most importantly, it can be easily extended for control design. Due to the above facts the T T T T −1 inequality holds: matrices X, Y and invertible matrix Z >Z0, the following extended for control design. Due to the above facts the inequality holds: T T T T −1 X Y + Y X ≤ X ZX + Y Y. (2) T T T T −1 extended design. Due the above facts the discretization, (iv)techniques most importantly, it can be easily finite-dimensional techniques have got lot of XT Y Y + +Y Y TX X≤ ≤X X T ZX ZX + +Y Y TZ Z −1 Y. Y. (2) extended for for control control design. Due to to the above facts the inequality holds: X (2) finite-dimensional have got aa lot of attention. attention. finite-dimensional techniques have got a lot of attention. T T T T −1 X Y + Y X ≤ X ZX + Y Z Y. (2) X Y + Y X ≤ X ZX + Y Z Y. (2) finite-dimensional techniques have got a lot of attention. extended for control design. have Due to the above facts the finite-dimensional techniques got a lot of attention. T this work T −1a less conserIn this paper, we have consider linear time delay system The is X T Y + Yof ≤ work X T ZX + Yderive Z aaY.less (2) In this paper, we have consider aa linear delay system The main main objective objective of X this work is to to derive less conserconserfinite-dimensional techniques have got atime lot of attention. In this paper, we have consider a linear time delay system The main objective of this is to derive In paper, we consider time delay system The main objective of this this work is todiscretizing derive aa less less conserwith norm bounded uncertainty in the system matrices. vative stability criterion for (1) by the delay In this thisnorm paper, we have haveuncertainty consider aa linear linear time delay system The main objective of derive with bounded in the system matrices. vative stability criterion forwork (1) is byto discretizing theconserdelay with norm bounded uncertainty in the system matrices. vative stability criterion for (1) by discretizing the delay with bounded in system matrices. vative stability criterion for (1) by the delay In thisnorm paper, we haveuncertainty consider a linear delay system The main objective of this derive a less with norm bounded uncertainty in the thetime system matrices. vative stability criterion forwork (1) is bytodiscretizing discretizing theconserdelay with norm bounded uncertainty in the system matrices. vative stability criterion for (1) by discretizing the delay 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 604 Hosting by Elsevier Ltd. All rights reserved. Copyright © © 2018 2018 IFAC IFAC 604 Copyright 604 Peer review© under responsibility of International Federation of Automatic Copyright 2018 IFAC IFAC 604 Control. Copyright © 2018 604 10.1016/j.ifacol.2018.05.096 Copyright © 2018 IFAC 604

5th International Conference on Advances in Control and Dushmanta Kumar Das et al. / IFAC PapersOnLine 51-1 (2018) 572–579 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

¯ is divided interval. Further, the tolerable delay range h into N number of δ intervals of equal measure so that one may define   0 for i = 0, hi = iδ for i = 1, 2, . . . , N − 1, h ¯ for i = N.

T

Vi (xt , x˙ t ) = x (t)P x(t) +

(3) ˜ +h

3.1 Robust stability criteria In this section, a robust stability criterion of systems with norm bounded uncertainty in the system matrices is presented. Consider an uncertain system described by (4)

where A0 (t) and A1 (t) time-varying uncertain matrices that can be written as: (5)

where ∆A0 (t) = D1 F (t)E1 and ∆A1 (t) = D2 F (t)E2 are uncertain components of the nominal matrices A0 and A1 respectively. We have considered tuncertain matrices are norm bounded. D1 , D2 , E1 and E2 are appropriate dimensional constant matrices, and F (t) satisfies F T (t)F (t) ≤ I. A robust stability criterion for (4) based on discretizing the total delay interval as per (3) is now presented. Theorem 2. System (4) is stable if there exist matrices P > 0, Qj > 0, j = 1, . . . , 4, Ri > 0 and arbitrary matrices Sl , Mi , Ni , l = 1 . . . 5, i = 1, 2, that satisfy the following LMI: ] [ ¯ ¯ D2 Θk D1 ∗ −ε1 I 0 ∗ ∗ −ε2 I

where Θk =

[

[¯

]

<0

Θ δΦk , k = 1, 2, Φ1 = ∗ −R2

0 0 M2T N2T 0

]T

,δ

¯ h N

k = 1, 2,

[

(6)

0 M1T N1T 0 0

]T

, Φ2 =

¯ = Θ+ ˆ E, ˆ , N is a given positive integer, Θ

ˆ = [Θ ˆ ij ]i,j=1,..,5 Θ ˆ 11 = Θ

3 ∑

i=1

T ˆ Q i − R 1 + S1 A 0 + A T 0 S1 , Θ12 =

T ˆ T T T T ˆ ˆ R1 + A T 0 S2 , Θ13 = A0 S3 + S1 A1 , Θ14 = A0 S4 , Θ15 = P − T,Θ ˆ 22 = −(Q2 − Q4 ) − R1 + δ(M1 + M T ), Θ ˆ 23 = S1 + AT S 0 5 1

ˆ 24 = 0, Θ ˆ 25 = −S2 , Θ ˆ 33 = − δ(−M1 + N1T ) + S2 A1 , Θ

4 ∑

Qi +

i=3

T ˆ δ(−N1 − N1T ) + δ(M2 + M2T ) + S3 A1 + AT 1 S3 , Θ34 = δ(−M2 + T,Θ ˆ 35 = −S3 + AT S T , Θ ˆ 44 = −Q1 + δ(−N2 − N2T ) + AT S 1 4 1 5

ˆ 45 = −S4 , Θ ˆ 55 = N2T ), Θ

{ diag ε1 E1T E1 [ T T ˆk = D S D k 1

{(

h(i−1)

}

)2

R1 + δ 2 R2

[

]T

¯k = D ˆk 0 0 ε2 E2T E2 0 0 , D DkT S2T

DkT S3T

DkT S4T

DkT S5T

]

}

∫t−h˜

T

xT (θ)Q4 x(θ)dθ

x (θ)Q3 x(θ)dθ +

(7)

t−h

x˙ T (ϕ)R1 x(ϕ)dϕdθ ˙ +δ

˜ θ t−h

∫t−h˜ ∫t

x˙ T (ϕ)R2 x(ϕ)dϕdθ. ˙

t−hi θ

˜ = h(i−1) . Differentiating Vi with respect to time where h yields

3. ROBUST STABILITY ANALYSIS OF LINEAR TIME-DELAY SYSTEM

A1 (t) = A1 + ∆A1 (t),

∫t ∫t

xT (θ)Qj x(θ)dθ

t−h(i+1−j)

t

t−h

The following section presents robust stability conditions using the above discretization scheme.

A0 (t) = A0 + ∆A0 (t),

∫

∫t

2 ∑ j=1

+

x(t) ˙ = A0 (t)x(t) + A1 (t)x(t − h),

573

ˆ = − S5 − S5T , E

,

, k = 1, 2.

Proof : For the ith delay interval that h ∈ [h(i−1) , hi ], a simple LK functional is defined as: 605

V˙ i (xt , x˙ t ) = 2xT (t)P Ax(t) + 2xT (t)P A1 x(t − h) + Z (8) 3 ∑ T T ˜ ˜ where Z = x (t)Qk x(t) − x (t − h)(Q 2 − Q4 )x(t − h) k=1

−

4 ∑

k=3

xT (t − h)Qk x(t − h) − xT (t − hi )Q1 x(t − hi ) +

} { ∫t ˜ x˙ T (θ)R1 x(θ)dθ ˜ 2 R1 + δ 2 R2 x(t) ˙ −h ˙ x˙ T (t) h ˜ h

−δ

∫h˜

x˙ T (θ)R2 x(θ)dθ. ˙ Instead of replacing x(t) ˙ by di-

t−hi

rectly using (4), we consider in this paper using quadratic formulation of the system dynamics (4) as: { 2 xT (t)S1 + 2xT (t − hi−1 )S2 + 2xT (t − h)S3 } (9) +2xT (t − hi )S4 + 2x˙ T (t)S5 × {−x(t) ˙ + A0 (t)x(t) + A1 (t)x(t − h)} = 0,

where Sk , k = 1, . . . , 5 are arbitrary matrices of appropriate dimensions. Next, the bounds of the uncertain terms in (9) are obtained. Following (5) and Lemma 1, one may write [ ]T 2ξ T (t) S1T S2T S3T S4T S5T [D1 F (t)E1 x(t) D2 F (t)E2 x(t − h)] ≤

2 ∑

T T T ˆT ˆ ε−1 i ξ (t)Dk Dk ξ(t) + ε1 x (t)E1 E1 x(t)

(10)

i=1

+ε2 xT (t − h)E2T E2 x(t − h).

Now, using (10), (9) can be written as: Z1 > 0,

(11)

where [ ] ˜ xT (t − h) xT (t − hi ) x˙ T (t) T , ξ(t) = xT (t) xT (t − h) [ ]T Z1 = 2ξ T (t) S1T S2T S3T S4T S5T × 2 ∑ T ˆT ˆ {−x(t) ˙ + A0 x(t) + A1 x(t − h)} + ε−1 i ξ (t)Dk Dk ξ(t) + i=1

ε1 xT (t)E1T E1 x(t) + ε2 xT (t − h)E2T E2 x(t − h). Adding (11) to (8), one obtains V˙ (t) ≤ Z1 + 2xT (t)P x(t) ˙ + Z.

(12)

Following Lemma 1 of Das et al. (2014), the first integral and second integral terms from (12) is replaced with iequalities, one may write

5th International Conference on Advances in Control and 574 Optimization of Dynamical Systems Dushmanta Kumar Das et al. / IFAC PapersOnLine 51-1 (2018) 572–579 February 18-22, 2018. Hyderabad, India

ˆ+ V˙ i (t) ≤ ξ T (t)(Ψ + E

2 ˆT ˆ ε−1 i Dk Dk + hi−1 Ωi

i=1

+ (1 − ρ)δ 2 Φ2 R2−1 ΦT2 )ξ(t),(13) ˆ 11 , Ψ12 = Θ ˆ 12 , Ψ13 = Θ ˆ 13 , Ψ14 = where Ψ11 = Θ ˆ ˆ ˆ ˆ 24 , Ψ25 = ˆ Θ14 , Ψ15 = Θ15 , Ψ22 = Θ22 , Ψ23 = Θ23 , Ψ24 = Θ ˆ ˆ ˆ ˆ ˆ Θ25 , Ψ33 = Θ{33 , Ψ34 } = Θ34 , Ψ35 = Θ35 , Ψ44 = Θ44 , Ψ45 = ˆ 45 , Ψ55 = δ 2 R2 − S5 − S T , ρ = h−hi−1 , 0 ≤ ρ ≤ Θ 5 δ ] [ 04n×4n 04n×n . 1, Ωi = 0n×4n R1 +ρδ

2

Φ1 R2−1 ΦT1

2 ∑

Therefore, the stability requirement for the ith interval is ˆ+ Ψ+E

2 ∑

2 ˆT ˆ ε−1 i Dk Dk + h(i−1) Ωi

i=1

+ρδ 2 Φ1 R2−1 ΦT1 + (1 − ρ)δ 2 Φ2 R2−1 ΦT2 < 0. (14) Then, (14) can be equivalently written as: 2 ∑ −1 T 2 2 ˆ ˆT ˆ Ψ+E+ ε−1 i Dk Dk +h(i−1) Ωi +δ Φj R2 Φj < 0, j = 1, 2.

ˇ 23 = S2 A1 + R2 , Θ ˇ 24 = Θ ˆ 24 , Θ ˇ 25 = Θ ˆ 25 , Θ ˇ 33 = R1 − R2 , Θ ˇ 34 = AT S T +R2 , Θ ˇ 35 = −(Q3 +Q4 )−2R2 +S3 A1 +AT1 S3T , Θ 1 4 ˇ 44 = −Q1 − R2 , Θ ˇ 45 = Θ ˆ 45 , Θ ˇ 55 = Θ ˆ 55 . ˆ 35 , Θ Θ Proof : Since the last term in (18) is positive definite, one may reduce the stability condition in the form of a single matrix inequalities as: 2 ∑ ¯ + δ 2 Φ1 R−1 ΦT + δ 2 Φ2 R−1 ΦT + ˆT ˆ Θ ε−1 1 2 2 2 i Dk Dk < 0, (19) i=1

Further, following Lemma 1 of Das et al. (2014), substituting the free variables as Mi = Mi T = −Ni = −Ni T = −δ −1 R2 and taking Schur Complement twice, the above stability condition yields (19). Remark 5. Although Corollary 4 may be conservative than that of Theorem 2 in terms of the approximations incorporated, the gap between the two criteria decreases with decreasing the integral limit δ (increasing N ). This characteristic is observed using numerical examples and is discussed further later in this section.

In some cases, the uncertainty is described in (6) may be decomposed in simple fashion with D1 = D2 = D. In such cases, the above analysis may be less conservative due to individual treatment of uncertain terms in the analysis. 2 For the case D1 = D2 = D, some less conservative benefits ∑ −1 T 2 2 ˆ ˆT ˆ may be extracted by treating two uncertain terms ∆A Ψ+E+ ε−1 i Dk Dk +h(N −1) ΩN +δ Φj R2 Φj < 0, j = 1, 2. and ∆A1 conjugatively. The following corollary utilize this i=1 (16) treatment., Then, (16) can be written as Corollary 6. System (4) is stable if there exist matrices 2 P > 0, Qj > 0, j = 1, . . . , 4, Ri > 0 and arbitrary matrices ∑ ˆ kT D ˆ k < 0, j = 1, 2. (17) Sl , Mi , Ni , l = 1 . . . 5, i = 1, 2, that satisfy the following ¯ + δ 2 Φj R−1 ΦTj + ε−1 D Θ 2 i LMI: i=1 [ ] ¯ ∆k D Taking Schur complement thrice on (17), one obtains (6). < 0, k = 1, 2, (20) ∗ −εI This completes the proof. ] [ ¯ δΦk Remark 3. Unlike conventional delay-discretization ap∆ ¯ ˆ ˜ ˜ proaches of Gu et al. (2003); Han (2009); Meng et al. where ∆k = ∗ −R2 , k = 1, 2, ∆ = Θ + E, E = [ ] (2010), the number of decision variables and size of the ¯T ¯ E ¯= D ¯ = [E1 0 E2 0 0] , D ˆ 0 T, LMI in Theorem 2 does not increase with N . Although it εE E, [ ] ˆ = DT S T DT S T DT S T DT S T DT S T , Φ1 , Φ2 , Θ ˆ are appears that no approximation is used in obtaining (17) D 1 2 3 4 5 previously mentioned in Theorem 2. from (16) to fetch this benefit, the stability criterion is indeed ultimately constrained since the gap in approximating the first integral term in (12) increases due to Proof : Following the same procedure as in Theorem 2. [ ]T increasing h(i−1) with increasing N . This limitation arises 2ξ T (t) S1T S2T S3T S4T S5T DF (t) [E1 x(t) E2 x(t − h)] due to the choice of LK functional and the corresponding ¯ T Eξ(t) ˆ T Dξ(t) ¯ ˆ ≤ εξ T (t)E + ε−1 ξ T (t)D results may be influenced by the approximations of the (21) first integral term. However, one may easily search over N ¯ Adding (21) to (9) can be written as: to obtain the maximum tolerable h. [ ]T 2ξ T (t) S1T S2T S3T S4T S5T The stability criterion developed in Theorem 2 may be (22) conservatively simplified by eliminating the free variables {−x(t) ˙ + A0 x(t) + A1 x(t − h)} and reducing the dimension of the LMI. The following T T −1 T T ¯ Eξ(t) ˆ Dξ(t) ¯ ˆ + ε ξ (t)D >0 +εξ (t)E corollary presents such a result. Corollary 4. System (4) is stable if there exist P > 0, Following the same approach of Theorem 2, one obtains Qk > 0, Rj > 0, k = 1, . . . , 4, j = 1, 2 satisfying the the stability requirement of (4): −1 T 2 ˜ −1 D ˆ T D+h ˆ 2 following LMI condition: Ψ+E+ε j = 1, 2. (N −1) ΩN +δ Φj R2 Φj < 0,   ¯ ˜ ¯ (23) Θ D1 D2  ∗ −ε1 I 0  < 0, (18) Then, (23) can be written as ∗ ∗ −ε2 I ¯ + δ 2 Φj R−1 ΦT + ε−1 D ˆTD ˆ < 0, j = 1, 2. ∆ (24) j 2 ˇ 11 = Θ ˆ 11 , Θ ˇ 12 = Taking Schur complement twice on (24), one obtains (20). ˜ = Θ ˇ + E, ˆ Θ ˇ = [Θ ˇ ij ]i,j=1,..,5 , Θ where Θ ˇ 13 = Θ ˆ 13 , Θ ˇ 14 = Θ ˆ 14 , Θ ˇ 15 = Θ ˆ 15 , Θ ˇ 22 = Q4 − Q2 − This completes the proof. The stability criterion developed ˆ 12 , Θ Θ i=1

(15) It may be noted that, Ωi ≥ 0 and the L.H.S. of (15) is ¯ i.e. the N th interval. maximum when h ∈ [h(N −1) , h],

606

5th International Conference on Advances in Control and Optimization of Dynamical Systems Dushmanta Kumar Das et al. / IFAC PapersOnLine 51-1 (2018) 572–579 February 18-22, 2018. Hyderabad, India

in Corollary 6 may be conservatively simplified by eliminating the free variables and reducing the dimension of the LMI. The following corollary presents such a result. Corollary 7. System (4) is stable if there exist P > 0, Qk > 0, Rj > 0, k = 1, . . . , 4, j = 1, 2 satisfying the following LMI condition: [ ] ˜ D ˆT ∆ < 0, (25) ∗ −εI ˇ 11 = Θ ˆ 11 , ∆ ˇ 12 = ˜ =∆ ˇ + E, ˜ ∆ ˇ = [∆ ˇ ij ]i,j=1,..,5 , ∆ where ∆ ˇ ˆ ˇ ˆ ˇ ˆ ˇ ˆ Θ12 , ∆13 = Θ13 , ∆14 = Θ14 , ∆15 = Θ15 , ∆22 = Q4 − Q2 − ˇ 23 = S2 A1 + R2 , ∆ ˇ 24 = Θ ˆ 24 , ∆ ˇ 25 = Θ ˆ 25 , ∆ ˇ 33 = R1 − R2 , ∆ T T ˇ T T ˇ 35 = −(Q3 +Q4 )−2R2 +S3 A1 +A1 S3 , ∆34 = A1 S4 +R2 , ∆ ˇ ˇ ˆ ˇ ˆ ˆ Θ35 , ∆44 = −Q1 − R2 , ∆45 = Θ45 , ∆55 = Θ55 . Proof : Since the last term in (24) is positive definite, one may reduce the stability condition in the form of a single matrix inequalities as: ¯ + δ 2 Φ1 R−1 ΦT + δ 2 Φ2 R−1 ΦT + ε−1 D ˆTD ˆ < 0, (26) ∆ 2

1

2

2

Further, following Lemma 1 of Das et al. (2014), substituting the free variables as Mi = Mi T = −Ni = −Ni T = −δ −1 R2 and taking Schur Complement once, the above stability condition yields (25). The system of the form (4) without uncertainty may be written as: x(t) ˙ = A0 x(t) + A1 x(t − h), (27) where A0 and A1 constant matrices with appropriate dimension. The stability criterion for (27) may be deduced from Theorem 2. The following corollary presents such a result. Corollary 8. System (27) is stable if there exist matrices P > 0, Qj > 0, j = 1, . . . , 4, Ri > 0 and arbitrary matrices Sl , Mi , Ni , l = 1 . . . 5, i = 1, 2, that satisfy the following LMI: ] [ ˆ δΦk Θ , k = 1, 2, (28) ∗ −R2 ˆ and Φk are mentioned in Theorem 2. Θ The stability criterion developed in Corollary 8 may be conservatively simplified by eliminating the free variables and reducing the dimension of the LMI. The following corollary presents such a result. Corollary 9. System (27) is stable if there exist P > 0, Qk > 0, Rj > 0, k = 1, . . . , 4, j = 1, 2 satisfying the following LMI condition: ˇ < 0, Θ (29) ˇ where Θ is mentioned in Corollary 4. 3.2 Numerical Examples We now consider numerical examples to illustrate the effectiveness of the proposed stability criterion. Example 1: Consider system (27) with [ ] [ ] −1 0 −2 0 . A= , Ad = −1 −1 0 −0.9 ¯ with number of interval N The variation of delay bound h obtained using the Corollary 8 and Corollary 9 is shown in Fig. 1 (as they are responded equally). It can be seen that, ¯ obtained for N = 2 using the present apthe maximum h proach is same as that obtained using available discretized 607

575

interval approaches with two discretizations Han (2009); Briat (2008); Gouaisbaut and Peaucelle (2006); Peaucelle ¯ et al. (2007). With further increase in N , the computed h decreases. Such a behavior can be followed from Remark 5. For N = 2, the delay interval for both the integral inequalities in (12) is halved and so the bounding gap reduces for both, leading to improved result. But with further increase in N , the gap in bounding the first integral term increases as that particular interval increases leading ¯ even though gap in the bounding of to decrease in h the second integral decreases. Therefore, one always gets maximum delay value at N = 2. The same is observed as described in the following. ¯ obtained using Corollary 8 and Corollary The maximum h 9 tabulated in Table I, along with some cursory existing results which shows that the present result considerably ¯ In Chen et al. (2017); Zeng et al. improves the computed h. (2015), the approaches are less conservative than our proposed approach. However, such approaches are difficult to extend for robust stabilization criterion. But the proposed approach can easily be extended for stabilization problem. Example 2: Consider another example of (27) with [ ] [ ] −3 −2.5 1.5 2.5 A= , Ad = . 1 0.5 −0.5 −1.5 ¯ obtained using Corollary 8 For this system, maximum h and Corollary 9 is tabulated in Table I and the variation ¯ with number of interval N is shown in Fig. 1. It can be h observed that in this case also maximum delay bound is obtained at N = 2 as expected. ¯ for Table 1. Comparison of delay bound (h) Example 1 and 2

Example 1

Example 2

Methods Shao and Han (2012) Ariba and Gouaisbaut (2009) Corollary 8 (N = 2) Corollary 9 (N = 2) Chen et al. (2017) Zeng et al. (2015) Shao (2008), Shao (2009) Shao and Han (2012) Corollary 8 (N = 2) Corollary 9 (N = 2)

¯ h 4.472 5.120 5.717 5.717 6.1719 6.1664 1.9998 2.0050 2.3094 2.3094

Both the examples demonstrate that the proposed method yields quite less conservative results compared to the existing ones. It may also be noted that computational complexity is similar to those existing approaches since the result is based on a simple LMI that corresponds to complexity involved with considering the whole delay as a single interval. Example 3: Consider system (4) with [ ] [ ] −0.5 −2 −0.5 −1 A= , Ad = , Da = Dd = I, 1 −1 0 0.6 Ea = Ed = diag{0.2, 0.2}. ¯ obtained for varying Using Corollary 6, the delay bound h ¯ is found to N is shown in Fig. 2. The delay bound (h) be 0.9021 at N = 2. Numerical comparison with existing results are presented in Table II. Example 4: Consider another example of system (4) with

5th International Conference on Advances in Control and 576 Optimization of Dynamical Systems Dushmanta Kumar Das et al. / IFAC PapersOnLine 51-1 (2018) 572–579 February 18-22, 2018. Hyderabad, India

[

] [ ] −2 0 −1 0 A= , Ad = , Da = Dd = I, 0 −1 −1 −1 [ ] [ ] 1.6 0 0.1 0 Ea = , Ed = . 0 0.05 0 0.3

¯ for different N obtained using Corollary The variation of h 6 is shown in Fig. 2. It is reported in Parlakci (2006) that for this system considering a certain F (= I), the analytical ¯ is 1.3771. The result obtained using Corollary limit of h 6 is 1.3594, which is quite close to this analytical limit. A comparison of the Corollary 6 results with the existing ones is made in Table II. It can be seen that the present result is quite less conservative than the existing ones for both the examples. ¯ for Table 2. Comparison of delay bound (h) Example 3 and 4

Example 3

Example 4

Methods Kim (2001) Moon et al. (2001) Fridman and Shaked (2002) Wu et al. (2004) Parlakci (2006) Corollary 2 Yue and Won (2002), Kim (2001) Moon et al. (2001) Fridman and Shaked (2002), Wu et al. (2004) Corollary 2

where Πk =

[

[¯



] Π δ ϕ¯k ¯ ˇ ¯ ¯ 2 , k = 1, 2, Π = Π + D, ∗ −R ]

¯ = D

[

¯ k = DT λDT βD T γD T αD T , ϕ¯1 = 0 D k k k k k ϕ¯2 =

[

¯T 0 ¯T N 0 0 M 2 2

]T

,δ 

¯ h N 3

(33)

3 ∑

¯T D ¯k, εk D k

k=1 ¯T 0 ¯T N M 1 1

0

]T

,

, N is a given positive inte-

∑

¯i − R ¯ 1 + A0 S¯T + S¯1 AT + Q 1 0 i=1 T T T T T ¯ ¯ ˇ ¯ ¯ ¯ ˇ ¯ B2 Y + Y B2 , Π12 = R1 + λS1 A0 + λY B2 , Π13 = β S1 AT 0 + ˇ 14 = γ S¯1 AT + γ Y¯ T B T , Π ˇ 15 = P¯ − S¯T + αS¯1 AT + β Y¯ T B2T + A1 S¯1 , Π 0 2 1 0 ˇ 22 = Q ¯4 − Q ¯2 − R ¯ 1 + δ(M ¯1 + M ¯ T ), Π ˇ 23 = δ(−M ¯1 + αY¯ T B2T , Π 1 ¯ T ) + λA1 S¯T , Π ˇ 24 = 0, Π ˇ 25 = −λS¯T , Π ˇ 33 = −(Q ¯3 + Q ¯ 4 ) + δ(−N ¯1 − N 1 1 1 T T ˇ ¯ ¯ ¯ ¯ T ¯ N1 ) + δ(M2 + M2 ) + βA1 S¯1T + β S¯1 AT 1 Π34 = γ S1 A1 + δ(−M2 + ¯ T ), Π ˇ 35 = −β S¯T + αS¯1 AT , Π ˇ 44 = −Q ¯ 1 + δ(−N ¯2 − N ¯ T ), Π ˇ 45 = N 2 1 1 2 2 T T 2 ¯1 + δ R ¯2 , E ˇ 55 = −αS¯1 − αS¯ + ˇ1 = h(i−1) R −γ S¯1 , Π 1

ˇ 11 = ˇ = [Π ˇ ij ]i,j=1,..,5 , Π ger, Π

[

¯ h 0.3513 0.5799 0.6812 0.8435 0.8542 0.9021 (N=2) 0.2412

E1 S¯1T 0 0 0 0 0 0

[

]T

,

ˇ2 = 0 0 E2 S¯T 0 0 0 0 E 1

[

]T , ]T

{(

)

}

ˇ3 = E3 Y¯ 0 0 0 0 0 0 , S¯1 = S −1 , P¯ = S¯1 P S¯T , M ¯i = E 1 1 T T T ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ S1 Mi S1 , Ni = S1 Ni S1 , i = 1, 2, Qj = S1 Qj S1 , j = 1, ..4, Y¯ = K S¯1T .

0.7059 1.1490 1.3594 (N=2)

4. ROBUST STABILIZATION OF LINEAR TIME-DELAY SYSTEM In this section, robust stabilization criterion for systems with time-delay is derived for designing static state feedback controller. The stabilization criterion is obtained by extending the stability result derived in the previous section and subsequently linearizing the resulting nonlinear matrix inequality using a LMI solver yields stabilizing control gains. Consider an uncertain system with state delay x(t) ˙ = A0 (t)x(t) + A1 (t)x(t − h) + B2 (t)u(t), (30) where A0 (t), A1 (t) are same as defined in (5), B2 (t) is the matrix with time-varying uncertainty and can be decomposed as B2 (t) = B2 + ∆B2 (t),

ˇ ˇ2 ˇ3  E E k E1 ∗ −ε1 I 0 0  < 0, ∗ ∗ −ε2 I 0 ∗ ∗ ∗ −ε3 I

Π

(31)

where ∆B2 = D3 F (t)E3 is the uncertain component of control input matrix B2 . This uncertain matrix is also norm bounded and D3 and E3 are appropriate dimensional constant matrices. In this section, a static state feedback controller for (30) is considered as: u(t) = Kx(t), (32) where K is the control gain to be designed so that the system can be stabilized. The following stabilization criterion for designing a stabilizing K for system (30) is derived. Theorem 10. System (30) is stable, for arbitrarily chosen ¯ j > 0, λ, β, γ and α, if there exist matrices P¯ > 0, Q ¯ i > 0 and arbitrary matrices S¯l , Mi , Ni , j = 1, . . . , 4, R l = 1 . . . 5, i = 1, 2, that satisfy the following LMI: 608

Proof : With (32), the closed loop system may be represented as: x(t) ˙ = (A0 (t) + B2 (t)K)x(t) + A1 (t)x(t − h) (34) Consider the same Vi as in Theorem 2. Due to the addition of another uncertain term, (11) becomes modified to be [ ]T 2ξ T (t) S1T S2T S3T S4T S5T [ ] D1 F (t)E1 x(t) D2 F (t)E2 x(t − h) D3 F (t)E3 Kx(t) ≤

3 ∑

(35)

ˆ k ξ(t) + ε−1 xT (t)E T E1 x(t) ˆT D εk ξ T (t)D 1 k 1

k=1

−1 T T T T T +ε−1 2 x (t − h)E2 E2 x(t − h) + ε3 x (t)K E3 E3 Kx(t),

Then following the procedure from Theorem 2, one obtains −1 T 2 V˙ i (t) ≤ ξ T (t)(ψ + h2i−1 Ωi + ρδ 2 ϕ1 R2−1 ϕT 1 + (1 − ρ)δ ϕ2 R2 ϕ2 )ξ(t) (36) where

[

]T

ξ(t) = xT (t) xT (t − h(i−1) ) xT (t − h) xT (t − hi ) x˙ T (t) 3 ∑

;

T T T T Qi − R1 + S1 A0 + AT 0 S1 + S1 B2 K + K B2 S1 + i=1 T ε1 S1 D1 D1T S1T + ε2 S1 D2 D2T S1T + ε3 S1 D3 D3T S1T + ε−1 1 E1 E 1 + T E T E K, ψ T S T + K T B T S T + ε S D DT S T + ε−1 K = R + A 12 1 1 1 1 1 2 3 3 0 2 2 2 3 T T T T ε2 S1 D2 D2T S2T + ε3 S1 D3 D3T S2T , ψ13 = AT 0 S3 + K B2 S3 + S1 A1 + T ε1 S1 D1 D1T S3T + ε2 S1 D2 D2T S3T + ε3 S1 D3 D3T S3T , ψ14 = AT 0 S4 + T T T T T T T T T K B2 S4 + ε1 S1 D1 D1 S4 + ε2 S1 D2 D2 S4 + ε3 S1 D3 D3 S4 , ψ15 = T T T T T T T T P − S + AT 0 S 5 + K B 2 S 5 + ε 1 S1 D 1 D 1 S 5 + ε 2 S1 D 2 D 2 S 5 + ε3 S1 D3 D3T S5T , ψ22 = Q4 −Q2 −R1 +δ(M1 +M1T )+ε1 S2 D1 D1T S2T + ε2 S2 D2 D2T S2T + ε3 S2 D3 D3T S2T , ψ23 = δ(−M1 + N1T ) + S2 A1 + ε1 S2 D1 D1T S3T + ε2 S2 D2 D2T S3T + ε3 S2 D3 D3T S3T , ψ24 = ε1 S2 D1 D1T S4T + ε2 S2 D2 D2T S4T + ε3 S2 D3 D3T S4T , ψ25 = ε1 S2 D1 D1T S5T + ε2 S2 D2 D2T S5T + ε3 S2 D3 D3T S5T − S2 , T ψ33 = −(Q3 + Q4 ) + δ(−N1 − N1T ) + δ(M2 + M2T ) + S3 A1 + AT 1 S3 + −1 T T T T T T T ε1 S3 D1 D1 S3 + ε2 S3 D2 D2 S3 + ε3 S3 D3 D3 S3 + ε2 E2 E2 , ψ34 = T T T T T T AT 1 S4 + δ(−M2 + N2 ) + ε1 S3 D1 D1 S4 + ε2 S3 D2 D2 S4 + ε3 S3 D3 D3T S4T , T T T T T T T ψ35 = −S3 +AT 1 S5 +ε1 S3 D1 D1 S5 +ε2 S3 D2 D2 S5 +ε3 S3 D3 D3 S5 ,

ψ11 =

5th International Conference on Advances in Control and Optimization of Dynamical Systems Dushmanta Kumar Das et al. / IFAC PapersOnLine 51-1 (2018) 572–579 February 18-22, 2018. Hyderabad, India ψ44 = −Q1 + δ(−N2 − N2T ) + ε1 S4 D1 D1T S4T + ε2 S4 D2 D2T S4T + ε3 S4 D3 D3T S4T , ψ45 = −S4 + ε1 S4 D1 D1T S5T + ε2 S4 D2 D2T S5T + ε3 S4 D3 D3T S5T , ψ55 = −S5 − S5T + δ 2 R2 + ε1 S5 D1 D1T S5T h−h

i−1 + ε2 S5 D2 D2T S5T + ε3 S5 D3 D3T S5T , ρ = , δ [ ] 04n×4n 04n×n . Therefore, the stability 0n×4n R1

the i

0 ≤ ρ ≤ 1; Ωi =

requirement for

th

interval is ψ + h2(i−1) Ωi + δ 2 ϕj R2−1 ϕTj < 0,

j = 1, 2.

(37)

To this end, note that, Ωi ≥ 0 and is maximum when ¯ the N th interval. Therefore, irrespective h ∈ [h(N −1) , h], of h lies in any of the intervals, the following condition always ensures stability of (5): ψ + h2(N −1) ΩN + δ 2 ϕj R2−1 ϕTj < 0,

j = 1, 2.

(38)

Since the third term in (38) are positive definite, one may approximate them in a reduced LMI form as: [ ] ψ¯ δϕk < 0, k = 1, 2 (39) ∗ −R2 where ψ¯ = ψ + h2 Ωi . (i−1)

For linearization, considering S2 , S3 , S4 and S5 as: S2 = λS1 , S3 = βS1 , S4 = γS1 , S5 = αS1 . and then, pre- and post-multiplying (39) by } { diag S1 −1 S1 −1 S1 −1 S1 −1 S1 −1 S1 −1 and its transpose respectively, and subsequently adopting the change of variables ¯ i = S¯1 Mi S¯T , N ¯i = S¯1 Ni S¯T , S¯1 = S1−1 , P¯ = S¯1 P S¯1T , M 1 1 ¯ ¯ T T ¯ ¯ ¯ i = 1, 2, Qj = S1 Qj S , j = 1, ..4, Y = K S . 1

Ξ1 =

Ξ21

[

Ξ3 =

[

(40) ]

Ξ21 03n×3n ε1 S1 E1 E1 S1 05n×5n , Ξ2 = , 03n×3n 03n×3n 05n×n 0n×5n

02n×2n 02n×n = ¯1 E T E2 S¯T S 0n×2n ε−1 2 1 2

]

,

]

¯T T ¯ ε−1 3 Y E3 E3 Y 05n×5n . 05n×n 0n×5n

Applying Schur Complement thrice on (40), one obtains (33). The following Corollary presents a simpler criterion by eliminating variables from Theorem 10. Corollary 11. System (30) is stable if there exist P¯ > 0, ¯ k > 0, R ¯ j > 0, k = 1, . . . , 4, j = 1, 2 satisfying the Q following LMI condition:   ˜ Π

ˇ0 ˇ1 ˇ2 E E E −ε1 I 0 0  < 0, ∗ −ε2 I 0  ∗ ∗ ∗ −ε3 I

∗ ∗

(41)

¯ + D, ¯ Π ¯ = [Π ¯ ij ]i,j=1,..,5 , Π ¯ 11 = Π ˇ 11 , Π ¯ 12 = Π ˇ 12 , Π ¯ 13 ˜ 11 = Π Π ˇ 13 , Π ¯ 14 = Π ˇ 14 , Π ¯ 15 = Π ˇ 15 , Π ¯ 22 = Q ¯4 − Q ¯2 − R ¯1 − R ¯2 , Π ¯ 23 Π ¯2 , Π ¯ 24 = Π ˇ 24 , Π ¯ 25 = Π ˇ 25 , Π ¯ 33 = −(Q ¯3 + Q ¯ 4 ) − 2R ¯2 λA1 S¯1T + R ¯ 34 = γ S¯1 AT + R ¯2 , Π ¯ 35 = Π ˇ 35 , Π ¯ 44 = −Q ¯1 βA1 S¯1T + β S¯1 AT , Π 1 1 ¯2 , Π ¯ 45 = Π ˇ 45 , Π ¯ 55 = Π ˇ 55 . R

Proof :The proof is similar to the proof of Corollary 4. The uncertain system of type (30) can be modified only by considering uncertainty in the system matrices not in the input matrices. This can be represented as x(t) ˙ = A0 (t)x(t) + A1 (t)x(t − h) + B2 u(t), (42) The uncertainties can be defined in simple fashion by assuming D1 = D2 = D and E1 = E2 = E. Using such assumption, the following corollary is developed. Corollary 12. System (42) is stable if there exist matrices ¯ j > 0, j = 1, . . . , 4, R ¯ i > 0 and arbitrary matrices P¯ > 0, Q ¯ i, N ¯i , l = 1 . . . 5, i = 1, 2, that satisfy the following S¯l , M LMI: ] [ ¯1 ∇k E <0 (43) ∗ −εI where ] [ ¯ δ ϕ¯k ∇ ¯ ˇ ˜ ˜ ˆT ˆ ∇k = ¯ 2 , k = 1, 2, ∇ = Π + D, D = εD D, ∗ −R ] [ ˆ = DT λDT βDT γDT αDT , D [ ] ˜1 = E S¯T 0 E S¯T 0 0 T , E 1 1 ˇ is mentioned in Theorem 10. Π Proof : Following Theorem 10, modifying (33) as [ ]T 2ξ T (t) S1T S2T S3T S4T S5T DF (t) [Ex(t) E2 x(t − h)] ˇ T Dξ(t) ` T Eξ(t), ˇ ` ≤ εξ T (t)D + ε−1 ξ T (t)E (44) where [ ] ˇ = DT S T DT S T DT S T DT S T DT S T T , D 1 2 3 4 5

1

After linearization, one may write Πk + Ξ1 + Ξ2 + Ξ3 < 0, where [ ] [ −1 ¯ T ¯T

577

= = + −

609

` = [E 0 E 0 0]T , E Then following the procedure from Theorem 10, one obtains (43). For further simplification and variable elimination, Corollary 12 can be reduced to subsequent corollary. Corollary 13. System (42) is stable if there exist P > 0, Qk > 0, Rj > 0, k = 1, . . . , 4, j = 1, 2 satisfying the following LMI condition: [ ] ˜ E ˜ ∇ <0 (45) ∗ −εI ˜ =∇ ˇ + D, ˜ ∇ ˇ = [∇ ˇ ij ]i,j=1,..,5 , ∇ ˇ 11 = Π ˇ 11 , ∇ ˇ 12 = where ∇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ¯ ¯2 − ˇ Π12 , ∇13 = Π13 , ∇14 = Π14 , ∇15 = Π15 , ∇22 = Q4 − Q ¯1 −R ¯2, ∇ ˇ 23 = λA1 S¯T + R ¯2, ∇ ˇ 24 = Π ˇ 24 , ∇ ˇ 25 = Π ˇ 25 , ∇ ˇ 33 = R 1 ¯ 4 ) − 2R ¯ 2 + βA1 S¯T + β S¯1 AT , ∇ ˇ 34 = γ S¯1 AT + ¯3 + Q −(Q 1 1 1 ˇ ˇ ˇ ¯ ¯ ˇ ˇ ˇ 55 = Π ˇ 55 . ¯ R2 , ∇35 = Π35 , ∇44 = −Q1 − R2 , ∇45 = Π45 , ∇ Proof : The proof is straight forward. Hence, omitted. The above criteria are robust stabilization criteria derived for (30) and (42). As it is said earlier that the criteria for uncertain system can be easily reduced for nominal system. The following equation represents the equation for nominal system reduced from (30) by without considering the uncertainty. It can be written as: x(t) ˙ = A0 x(t) + A1 x(t − h) + B2 u(t). (46) Subsequently, the stabilizing criteria for (46) is derived by using a static state feedback controller of the form (32). Corollary 14. System (46) is stable if there exist matrices ¯ j > 0, j = 1, . . . , 4, R ¯ i > 0 and arbitrary matrices P¯ > 0, Q

5th International Conference on Advances in Control and 578 Optimization of Dynamical Systems Dushmanta Kumar Das et al. / IFAC PapersOnLine 51-1 (2018) 572–579 February 18-22, 2018. Hyderabad, India

¯ i, N ¯i , l = 1 . . . 5, i = 1, 2, that satisfy the following S¯l , M LMI: ] ˇ δ ϕ¯j Π ¯ 2 < 0, j = 1, 2, ∗ −R ˇ and ϕ¯j are given in Theorem 10. where Π [

(47)

Proof : The proof is similar to the proof of the Theorem 10 for robust stabilization criterion. So it is omitted. Again, the obtained stabilization criterion is simplified by eliminating the matrix variables. Hereinafter, the simplified stabilization criteria are presented in the form of a corollary. Corollary 15. System (46) is stable if there exist matrices ¯ j > 0, j = 1, . . . , 4, R ¯ i > 0 and arbitrary matrices P¯ > 0, Q ¯ Sl , l = 1 . . . 5, i = 1, 2, that satisfy the following LMI: ¯ < 0, Π (48) ¯ is given in Corollary 11. where Π Proof : The proof is omitted. The following section presents some numerical examples to endorse the effectiveness of the above derived criteria. 4.1 Numerical Examples Some academic existing examples are considered for validation. Example 3 Consider a system of the form (46) with [ ] [ ] [ ] 0 0 −1 −1 0 , A1 = , B2 = . A0 = 0 1 0 −0.9 1 From the previous analysis, one always gets maximum delay value at N = 2. For less conservative synthesis, a static state feedback controller is designed by setting the number of discretization to two. A comparison of maximum tolerable delay bound for this system is done in the Table III which is shown below. From the comparison, it is clear that the proposed discretization approach gives very less conservative result than the existing results. To verify this, the designed controller using Corollary 14 is used to get simulation result of the closed loop system with initial condition x(t) = [2, −2], t ∈ [−20, 0] is shown in Fig. 3. From the simulation, it is seen that the states of the closed loop system are stable. ¯ Table 3. Comparison of delay bound (h) Methods Fridman and Shaked (2002) Zhang et al. (2005) Parlakci (2006) Corollary 14

¯ h 1.51 6 8 20

Controller gain [-58.31 -294.935] [-70.18 -77.67] [-65.4058 -76.7778] [-2852.3 -2959.7]

Example 4 Let us consider a linearized model of a realtime aircraft control system Parlakci (2006), which is in the form of (46) with −0.0366 0.0271 0.0188 −0.4555 A0 = 

0.0482 −1.0100 0.0024 −4.0208 , 0.1002 0.3681 −0.7070 1.4200 0 0 1.0000 0



A1 = 0.3A0 , B2 = 



0.4422 0.1761 3.5446 −7.5922 . −5.5200 4.4900 0 0

610

The state variables of this aircraft system is represented by [ ]T x = xT1 xT2 xT3 xT4 , where x1 and x2 are position (m) and velocity (m/s) of center of mass in spatial coordinates respectively, x3 is rotation matrix (rad) of the body axes relative to the spatial axes and x4 is body angular velocity vector (rad/s). For this case also, the controller is designed by setting N = 2. A comparison of maximum tolerable delay bounds using different approaches for this system is given in Table IV below. From the comparison, it is clear that the proposed decomposition approach customarily gives less conservative result than the existing results. The obtained controller gain using Corollary 14 ¯ = 79) is used for maximum tolerable delay bound (h to simulate the closed loop system with initial condition x(t) = [2, −4, 3, −5], t ∈ [−79, 0]. The variation of norm of the state vector with respect to time is shown in Fig. 4. From the simulation result, it is seen that the states of the closed loop system are stable. ¯ Table 4. Comparison of delay bound (h) Methods

¯ h

Li and souza (1997)

1.4142

Parlakci (2006)

6

Corollary 79.4851 14

Controller gain ] 13.6188 1.8680 0.7661 −8.0951 21.9119 2.7268 −0.1298 −14.7952

[

[

[

]

−0.0458 0.1447 0.5490 0.2080 −0.0187 0.1331 0.2516 −0.4175

19.6354 2.4332 0.9949 −10.9001 29.7580 3.4891 0.3834 −18.1027

]

Example 5 Consider a uncertain system of the form (42) with [ ] [ ] ] −1 −1 0 0 0 , B2 = , , A1 = A0 = 0 −0.9 1 0 1 [

D1 = D2 = 0.2I, D3 = 0, E1 = E2 = I, E3 = 0. Using the proposed discretization technique, one always gets maximum delay value at N = 2. At N = 2, comparison of maximum tolerable delay bound for this system is made in the Table V which is shown below. ¯ Table 5. Comparison of delay bound (h) Methods Parlakci (2006) Theorem 10 Corollary 12

¯ h 1.3 1.2498 1.3870

Controller gain [-2.1485 -5.6948] [-2246.5 -5142.5] [-370.3688 -867.3798]

Example 6 Consider another academic example of form (42) with A0 =

[

] [ ] [ ] 0 0 −2 −0.5 0 , A1 = , B2 = , 0 1 0 −1 1

D1 = D2 = 0.2I, D3 = 0, E1 = E2 = I, E3 = 0. For this case also, the controller is designed by setting N = 2. A comparison of maximum tolerable delay bound for this system is done in the Table VI. From the comparison, it is clear that the proposed discretization approach gives less conservative result also for uncertain systems.

5th International Conference on Advances in Control and Optimization of Dynamical Systems Dushmanta Kumar Das et al. / IFAC PapersOnLine 51-1 (2018) 572–579 February 18-22, 2018. Hyderabad, India

¯ Table 6. Comparison of delay bound (h) Methods Moon et al. (2001) Fridman and Shaked (2002) Parlakci (2006) Theorem 10 Corollary 12

¯ h 0.4500 0.5865 0.6900 0.6905 0.7605

Controller gain [-4.8122 -7.7129] [-0.3155 -4.4417] [-23.2572 -26.1488] [6541.9 -5451.4] [-4228.8 -3048.3]

1.5

6 Example 1 Example 2

5

Example 3 Example 4

1.4 1.3 1.2

¯ h

¯ h

4

1.1 3

1 0.9

2 0

1

10

10

N

10

2

0.8 0 10

3

10

1

2

10

10

N

10

3

¯ with N Fig. 2. Variation of h ¯ with N Fig. 1. Variation of h 140

2

x1 x2

1.5

120 100

0.5

x(t)

System states

1

0

80 60

−0.5 40 −1 20

−1.5 −2 0

50

100

150

200

250

0 0

Time

100

200

Time

300

400

500

Fig. 3. System states for Ex.Fig. 4. Norm of system states 3 for Ex. 4

5. CONCLUSIONS AND FUTURE SCOPE In this paper, discretization approach is proposed to obtain new less conservative stability and stabilization criteria for systems with single, constant delay. Most importantly, the proposed approach uses simple LK functional to derive LMI condition. As a result of which the LMI conditions become finite-dimensional and the number of matrix variables does not increase with number of discretizations. In the future scope of the paper, the performance of the proposed approach can be studied when the actual delay ¯ h is significantly different from the estimated h. REFERENCES Ariba, Y. and Gouaisbaut, F. (2009). An augmented model for robust stability analysis of time-varying delay systems. International Journal of Control, 82, 1616– 1626. Briat, C. (2008). Robust control and observation of LPV time-delay systems. Ph.D. thesis, INP-Grenoble. France. Chen, J., Xu, S., and Zhang, B. (2017). Single/multiple integral inequalities with applications to stability analysis of time-delay systems. IEEE Transactions on Automatic Control, 62(7), 3488–3493. Das, D.K., Ghosh, S., and Subudhi, B. (2014). Stability analysis of linear systems with two delays of overlapping ranges. Applied Mathematics and Computation, 243, 83– 90. Fridman, E. and Shaked, U. (2002). An improved stabilization method for linear time delay systems. IEEE Transactions on Automatic Control, 47, 1931–1937. 611

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