ROBUST CONTROLLER DESIGN FOR NEUTRAL TIME-DELAY SYSTEMS∗

ROBUST CONTROLLER DESIGN FOR NEUTRAL TIME-DELAY SYSTEMS∗

Proceedings of the 12th IFAC Workshop on Time Delay Systems Proceedings of the 12th IFAC Workshop on Time Delay Systems June 28-30, 2015. Ann Arbor, U...

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Proceedings of the 12th IFAC Workshop on Time Delay Systems Proceedings of the 12th IFAC Workshop on Time Delay Systems June 28-30, 2015. Ann Arbor, USA Proceedings of the the 12th IFACMI, Workshop on Time Time Delay Delay Systems Systems Proceedings of 12th IFAC Workshop on June 28-30, 2015. Ann Arbor, MI, USA June 28-30, 28-30, 2015. 2015. Ann Ann Arbor, Arbor, MI, MI, USA USAAvailable online at www.sciencedirect.com June

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ROBUST CONTROLLER DESIGN FOR ROBUST CONTROLLER DESIGN FOR ROBUST CONTROLLER DESIGN FOR ROBUST CONTROLLER DESIGN FOR NEUTRAL TIME-DELAY SYSTEMS NEUTRAL TIME-DELAY SYSTEMS NEUTRAL TIME-DELAY TIME-DELAY SYSTEMS SYSTEMS NEUTRAL

˙˙ Altu˘ g Iftar Altu˘ g Iftar Altu˘ g Iftar ˙˙ Altu˘ g Iftar Department of Electrical and Electronics Engineering, Anadolu Department of Electrical and Electronics Engineering, Department of26470 Electrical and Electronics Engineering, Anadolu Anadolu University, Eski¸ s ehir, Turkey. [email protected] Department of Electrical and Electronics Engineering, Anadolu University, 26470 Eski¸ ssehir, Turkey. [email protected] University, 26470 Eski¸ ehir, Turkey. [email protected] University, 26470 Eski¸sehir, Turkey. [email protected] Abstract: A robustness measure that accounts for the uncertainties in aa neutral time-delay Abstract: A robustness measure that accounts for the uncertainties in neutral time-delay Abstract: A robustness measure that accounts for the uncertainties in a neutral time-delay system is defined. Using this measure, a robust controller design approach, which is based on Abstract: A robustness measure thata accounts for the uncertainties in a which neutral time-delay system is defined. Using this measure, robust controller design approach, is based on aaa system is ismodel, defined. Using this thisThe measure, a robust robust controller designrobust approach, which is abased based on nominal is proposed. proposed approach guarantees stability once condition system defined. Using measure, a controller design approach, which is on a nominal is The proposed approach guarantees robust stability aa condition nominal model, model, is proposed. proposed. The proposed approach guarantees robust stability once once condition depending on the robustness measure is satisfied. An example is also presented to demonstrate nominal model, is proposed. The proposed approach guarantees robust stability once a condition depending on the depending on approach. the robustness robustness measure measure is is satisfied. satisfied. An An example example is is also also presented presented to to demonstrate demonstrate the proposed depending on the robustness measure is satisfied. An example is also presented to demonstrate the proposed approach. the proposed approach. the proposed approach. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2015, IFAC (International Keywords: Neutral Keywords: Neutral Neutral time-delay time-delay systems; systems; robust robust control; control; controller controller design; design; stabilization. stabilization. Keywords: time-delay systems; robust control; controller design; Keywords: Neutral time-delay systems; robust control; controller design; stabilization. stabilization. 1. INTRODUCTION proach. Some concluding remarks are given in the last 1. INTRODUCTION INTRODUCTION proach. Some concluding remarks are given in the last 1. proach. Some Some concluding concluding remarks remarks are are given given in in the the last last section. 1. INTRODUCTION proach. section. section. section. Many physical systems may involve time-delays. The conthe paper, R and C denote the sets of, Many physical physical systems systems may may involve involve time-delays. time-delays. The The concon- Throughout Throughout the paper, R and C denote the sets of, Many troller design problem for a time-delay system is more Throughout the paper, R and andnumbers. C denote denoteForthe thea positive sets of, of, Many physical systems may involve time-delays. The conrespectively, real and complex troller design problem for a time-delay system is more Throughout the paper, R C sets respectively, real and complex numbers. For a positive troller design problem for a time-delay system is more k difficult, compared to a delay-free system, since a timerespectively, real and complex numbers. For a positive troller design problem for a time-delay system is more integer k, R denotes the k-dimensional real vector space. k difficult, compared compared to to aa delay-free delay-free system, since since aa timetime- respectively, real and the complex numbers. For a positive integer k, R denotes k-dimensional real vector space. difficult, delay system is infinite-dimensional (Niculescu integer k, C, Rkk Re(s) denotes the k-dimensional real vector space. difficult, compared to a delay-free system, system, since (2001)). a time- For ss ∈ is the real part of s. II vector denotes the delay system is infinite-dimensional (Niculescu (2001)). integer k, R denotes the k-dimensional real space. For ∈ C, Re(s) is the real part of s. denotes the delay is (Niculescu (2001)). Different approaches, such as operator-based (Curtain and For s ∈ ∈ matrix C, Re(s) Re(s) is the the real real dimensions. part of of s. s. IIσ denotes the delay system system is infinite-dimensional infinite-dimensional (Niculescu (2001)). identity of appropriate ¯ (·), σ(·), and Different approaches, such as operator-based (Curtain and For s C, is part denotes the identity matrix of appropriate dimensions. σ ¯ (·), σ(·), and Different approaches, such as operator-based (Curtain and ¨ Zwart (1995); Foias et al. (1996); Toker and Ozbay (1995)), identity matrix of appropriate dimensions. σ ¯ (·), σ(·), and Different approaches, such as operator-based (Curtain and det(·) respectively denote the maximum singular value, identityrespectively matrix of appropriate σ ¯ (·), σ(·), and Zwart (1995); (1995); Foias et et al. al. (1996); Toker Toker and and Ozbay Ozbay (1995)), det(·) denote the the dimensions. maximum singular singular value, ¨¨¨ (2007)), Zwart (1995)), eigenvalue-based and Niculescu and det(·) respectively denote maximum value, the minimum singular value, the determinant the Zwart (1995); Foias Foias(Michiels et al. (1996); (1996); Toker and Ozbay (1995)), det(·) respectively denote theand maximum singular of value, √ eigenvalue-based (Michiels and Niculescu (2007)), and the minimum singular value, and the determinant of the √−1the eigenvalue-based (Michiels and Niculescu (2007)), and Lyapunov-based (Kolmanovskii et al. (1999)), have so far the minimum singular value, and determinant of the indicated matrix. Finally, j := is the imaginary unit. eigenvalue-based (Michiels and Niculescu (2007)), and √ the minimum singular value, and the determinant of the Lyapunov-based (Kolmanovskii et al. (1999)), have so far indicated matrix. Finally, j := −1 is the imaginary unit. √ Lyapunov-based (Kolmanovskii et (1999)), have so been proposed to design controllers for time-delay systems. Lyapunov-based et al. al. have so far far indicated indicated matrix. matrix. Finally, Finally, jj := := −1 −1 is is the the imaginary imaginary unit. unit. been proposed to to (Kolmanovskii design controllers controllers for (1999)), time-delay systems. been proposed design for time-delay systems. 2. PROBLEM STATEMENT Although some of these approaches only consider retarded been proposed toofdesign controllers for time-delay systems. 2. PROBLEM STATEMENT Although some these approaches only consider retarded 2. Although of approaches only consider time-delay systems, approaches which specifically consider 2. PROBLEM PROBLEM STATEMENT STATEMENT Although some some of these these approaches only consider retarded retarded time-delay systems, approaches which specifically consider time-delay systems, approaches which specifically consider Consider a linear time-invariant (LTI) neutral time-delay neutral time-delay systems have also been proposed (e.g., time-delay systems, approaches which specifically consider Consider aa linear linear time-invariant time-invariant (LTI) (LTI) neutral neutral time-delay time-delay neutral time-delay time-delay systems systems have have also also been been proposed proposed (e.g., (e.g., Consider neutral system which is described as: Park and Won (1999); Han (2002); Wu et al. (2004); a linear time-invariant (LTI) neutral time-delay neutraland time-delay systems have also been proposed (e.g., Consider system which is described as: Park Won (1999); Han (2002); Wu et al. (2004); as: Park Won Parlak¸ cı (2007)). systemν which which is is described described Park and and Won (1999); (1999); Han Han (2002); (2002); Wu Wu et et al. al. (2004); (2004); system Parlak¸ (2007)). ν as:   Parlak¸ cccııı (2007)). ν ν   Parlak¸ (2007)). ν ν Since any model of any physical system may contain ˙˙ − ττi )) =   ν Di x(t ν (Ai x(t − τi ) + Bi u(t − τi )) (1)   Since any any model model of of any any physical physical system system may may contain contain D − (A x(t − + B u(t − )) (1) i x(t i) = Since D x(t ˙ − τ = (Aii x(t x(t − − τττii ))) + +B Bii u(t u(t − − τττii )) )) (1) (1) uncertainties, any controller designed for such a system i i i=0 i=0 Since any model any physical system maya contain ˙ − τi ) = i=0 (A i i i i uncertainties, any of controller designed for such such system i=0 Di x(t uncertainties, any controller designed for a system must be robust against such uncertainties. In a time-delay i=0 i=0 uncertainties, any controller designed for such a system y(t) = Cx(t) (2) i=0 i=0 must be be robust robust against against such such uncertainties. uncertainties. In In aa time-delay time-delay y(t) = Cx(t) (2) must system, not only the system parameters, but also the timey(t) = Cx(t) (2) n p q must be robust against such uncertainties. In a time-delay system, not not only only the the system system parameters, parameters, but but also also the the timetime- where, x(t) ∈ Ry(t) = Cx(t) (2) ∈ R n , u(t) p , and y(t) ∈ Rq are, respecsystem, , u(t) ∈ R , and y(t) ∈ R are, respecwhere, x(t) ∈ R delays are usually uncertain. In this work, we propose n p q system,are not usually only theuncertain. system parameters, but also the time- tively, n p , and q are, at , u(t) ∈ R y(t) ∈ R respecwhere, x(t) ∈ R delays In this work, we propose the state, the input, and the output vectors time , u(t) ∈ Rand , and y(t) ∈R are, at respecwhere, the x(t)state, ∈ R the delays are usually In we tively, input, the output vectors time aadelays robust design approach for timearecontroller usually uncertain. uncertain. In this this work, work, we propose propose tively, the state, state, the input, and the theconvenience output vectors vectors ati time time robust controller design approach approach for neutral neutral time- t. We use ττ0 := 00 for notational (i.e., = 00 tively, the the input, and output at adelay robust controller design for neutral timet. We use := for notational convenience (i.e., i = systems. The approach uses a frequency-dependent 0 a robust controller design approach for neutral timet. We use τ := 0 for notational convenience (i.e., i = delay systems. The approach uses a frequency-dependent corresponds to the delay-free part). ν is the number of 0 t. We use τ0 to :=the 0 for notational convenience (i.e., i = of 00 delay The approach uses aa frequency-dependent corresponds delay-free part). νν is the number robustness measure accounts the uncertainties in delay systems. systems. The that approach uses for frequency-dependent corresponds to to the delay-free delay-free part). is 0the the number of robustness measure that accounts for the uncertainties in , . . . , τ ≥ are the timeindependent time-delays and τ 1 corresponds the part). ν is number of robustness measure that for the ina independent time-delays and τ1 , . . . , τν ≥ 0 are the timeboth the system parameters and the Such robustness measure that accounts accounts for time-delays. the uncertainties uncertainties are the the timetimeindependent time-delays and ττ1 ,, .. .. .. ,, ττor both the system system parameters and the the time-delays. Such in which may be commensurate incommensurate. ν ≥ 00 are independent time-delays and 1 ν ≥ both the parameters and time-delays. Such aaa delays, delays, which may be commensurate or incommensurate. measure was first used for delay-free large-scale systems both the system parameters and the time-delays. Such delays, which may be commensurate or incommensurate. measure was first used for delay-free large-scale systems D , A , B , i = 0, . . . , ν, and C are appropriately dimeni , Ai , which i delays, may be commensurate or incommensurate. ˙ ¨ uused measure was first for large-scale systems D = 0, ν, and C are appropriately dimeni , Ai , B by Iftar and Ozg¨ ner (1987a,b) and for retarded timemeasure was for delay-free delay-free systems D Bii ,,, iii = = matrices. 0, ... ... ... ,,, ν, ν, and and Cassumed are appropriately appropriately dimensioned It is that all the inputi i constant by Iftar and first Ozg¨ uused ner (1987a,b) (1987a,b) and large-scale for retarded retarded time- D B 0, C are dimen˙˙˙ ¨¨¨ u i , Ai ,constant i sioned matrices. It is assumed that all the inputby Iftar and Ozg¨ ner and for time˙ delay systems by (2008, 2014). we define by Iftar and Ozg¨ uIftar and In for here, retarded time- sioned sioned constant constant matrices. It is time-delays assumed that that allrepresented the inputinput˙˙ner (1987a,b) output uncertainties and the are delay systems by Iftar (2008, 2014). In here, we define matrices. It is assumed all the output uncertainties and the time-delays are represented delay systems by Iftar (2008, 2014). In here, we define ˙ aadelay similar measure for neutral time-delay systems and output uncertainties and the time-delays are represented systems by Iftar (2008, 2014). In here, we define at the input, so that the output equation (2) is free of any similar measure measure for for neutral neutral time-delay time-delay systems systems and and output uncertainties andoutput the time-delays areisrepresented at the input, so that the equation (2) free of any apropose similar a robust controller design approach using this at the input, so that the output equation (2) is free of any a similar measure for neutral time-delay systems and uncertainties and delays. Thus, C is a known matrix. Howpropose aa robust robust controller controller design design approach approach using using this this at the input, so that the output equation (2) matrix. is free ofHowany uncertainties and delays. Thus, C is a known propose measure. Once this measure is obtained, the proposed uncertainties and delays. delays. Thus, Ci ,is isAaai , known known matrix. Howpropose aOnce robust controller design approach using this uncertainties ever, it is assumed that each of D and B , i = 0, . . . , ν, i measure. this measure is obtained, the proposed and Thus, C matrix. However, it is assumed that each of D A B ,, ii = 0, ν, i ,, and measure. this is obtained, the proposed approach is completely based the nominal of ever, it is is assumed assumed that each each of of Dii ,,, A A and B Biiit = assumed 0, ... ... ... ,,, ν, ν, measure. Once Once this measure measure ison obtained, the model proposed is subject to uncertainties. More precisely, is i , and approach is completely based on the nominal model of ever, it that D , i = 0, i precisely, i iit is assumed is subject to uncertainties. More approach is completely based on the nominal model of n u n u n u the system and satisfying a simple constraint ensures the is subject to uncertainties. More precisely, it is assumed approach is completely based on the nominal model of that D := D + D , A := A + A , and B := B + B i :=to i the system system and and satisfying satisfying aa simple simple constraint constraint ensures ensures the the is subject More precisely, isBassumed inuncertainties. iu in + A iu , and Biit:= in + Biu ,, that D D n + Diu , Ai := An u n i i the i i i i i robust stability of the actual closed-loop system. n thati D D=i := := + Amatrices Bisuperscript := B Bin + +B Biuun,, the system and of satisfying a simple constraint ensures the for 0, .D .. ..in,, + ν, Dwhere the with iu , Ai := i + Au i , and robust stability the actual actual closed-loop system. that A B := i 0, .D isuperscript for i = where the matrices with n i ν, D i , Ai := i + Ai , and i robust stability of the closed-loop system. for iiknown = 0, 0, ..matrices ν, where where the matrices with superscript in n robust stability of the actual closed-loop system. are and the matrices with u for = .. .. ,, ν, the matrices with superscript The problem is formally defined in the next section. The are known matrices and the matrices with superscript u The problem problem is is formally formally defined defined in in the the next next section. section. The The represent are known knownthe matrices and the theThese matrices with superscript u uncertainties. latter matrices are not The are matrices and matrices with superscript u proposed approach is presented in Section 3. Section 4 represent the uncertainties. These latter matrices are not The problem is formally defined in the next section. The proposed approach is presented in Section 3. Section 4 represent the uncertainties. These latter matrices are not known, but are assumed to satisfy proposed approach is to presented in 3. 4 represent theare uncertainties. These latter matrices are not presents example demonstrate the proposed but assumed to satisfy proposed an approach presented in Section Section 3. Section Sectionappresents an exampleis to to demonstrate the proposed proposed ap-4 known, u u to satisfy u known, but are assumed presents an example demonstrate the apknown, but are assumed to satisfy σ ¯ (D ) ≤ δ , σ ¯ (A ) ≤ α σ ¯¯ (B ) ≤ βi , (3) u u i i ,, and i i iu presents an example to demonstrate the proposed apσ ¯¯ (D ¯¯ (A ≤ α σ (B (3) u ) ≤ δi , σ u u ) ≤ βi , i , and iu i) iu u σ (D ) ≤ δ , σ (A ) ≤ α and σ ¯ (B )≤ ≤β βi ,, (3) i ) ≤ δii , σ i ) ≤ αii , and σ i ) σ ¯ (D ¯ (A ¯ (B (3) i i i i  This work is supported by the Scientific Research Projects Comfor some known bounds δ , α , and β , i = 0, . . . , ν. It is i i i  This work is supported by the Scientific Research Projects Comfor some known bounds δδi ,, α and β ii = 0, .. ,, ν. It is  This work for some someassumed known bounds bounds α0ii),,, = and βii ,,, any =D 0,0uu... ...satisfying ν. It It is is further that rank(D n for mission of Anadolu University under grant number is supported supported by the the Scientific Research1204F071. Projects ComComi, α  for known δ and β i = 0, . , ν. This work is by Scientific Research Projects i i i further assumed that rank(D ) = n for any D mission of Anadolu University under grant number 1204F071. u satisfying 0 0 u satisfying further assumed that rank(D ) = n for any D mission of Anadolu University under grant number 1204F071. 0 0 further assumed that rank(D0 ) = n for any D0 satisfying mission of Anadolu University under grant number 1204F071.

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the above bound. The time-delays are also assumed to be subject to uncertainties. More precisely, it is assumed that τi := τin + τiu , i = 1, . . . , ν, where τin is the known nominal time-delay and τiu represents its uncertainty, which is assumed to satisfy |τiu | ≤ θi

(4)

for some known bound θi , i = 1, . . . , ν. Furthermore, we also make the following assumptions: Assumption 1: For any Diu , i = 0, . . . , ν, satisfying (3) and any τiu , i = 1, . . . , ν, satisfying (4), µf < 0, where     ν  −sτi =0 (5) Di e µf := sup Re(s) | det

and ¯ n (s) := B

¯ D(s) :=

ν 

¯ Di e−sτi and A(s) :=

i=0

ν 

Ai e−sτi

(6)

i=0

It is known that the system (1) has finitely many modes with real part greater than or equal to µ, for any µ > µf , where µf is given by (5) (e.g., see Michiels and Niculescu (2007)). Therefore, Assumption 1 guarantees that the number of unstable modes of the system (1) is finite for any uncertainties satisfying (3)–(4).

Din x(t ˙

i=0



τin ) =

ν 

(Ani x(t

i=0



τin )

+

Bin u(t



y(t) = Cx(t)

τin )) (7) (8)

so that the actual closed-loop system obtained by applying this controller to the system (1)–(2) is robustly stable for all uncertainties satisfying the bounds (3)–(4). 3. PROPOSED DESIGN APPROACH Note that the transfer function matrix (TFM) of the actual system (1)–(2) is given by  −1 ¯ ¯ ¯ G(s) = C sD(s) − A(s) B(s) (9) ¯ ¯ where D(s) and A(s) are as defined in (6) and ν  ¯ B(s) := Bi e−sτi

(10)

i=0

The TFM of the nominal system (7)–(8), on the other hand, is given by  n  ¯ (s) − A¯n (s) −1 B ¯ n (s) (11) Gn (s) = C sD where

¯ n (s) := D

ν  i=0

n

Din e−sτi ,

A¯n (s) :=

ν 

n

Ai e−sτi

(12)

i=0

181

n

Bin e−sτi

(13)

Let us relate the two TFMs as G(s) = Gn (s) (I + E(s))

(14)

where E(s) is the multiplicative error matrix between the actual TFM, G(s), and the nominal TFM, Gn (s). The next result gives a frequency-dependent upper bound on the norm of E(s): Lemma 1: Let en (ω) := β +

ν 

σ ¯ (Bin ) ρi (ω) + γ(ω)

(15)

i=1

and

  n ¯ (jω) − γ(ω) (16) ed (ω) := σ B ν where β := i=0 βi ,      2 sin |ω|θi , |ω| ≤ π   2 θi , i = 1, . . . , ν, ρi (ω) :=   π  2 , |ω| > θi and   ν  n n γ(ω) := α + δω + σ ¯ (jωDi − Ai ) ρi (ω) σ ¯ (Go (jω)) i=1



ν i=0 αi , δ := i=0 δi , and  n  ¯ (s) − A¯n (s) −1 B ¯ n (s) Go (s) := sD

where α :=

The problem is to design a controller based on the nominal model: ν 

ν  i=0

i=0

Assumption 2: For any Diu , and Aui , i = 0, . . . , ν, satisfying (3) and any τiu , i = 1, . . . , ν, satisfying (4), the number of unstable modes of the system (1) is the same, where so ∈ C is said to bean unstable mode  of the system ¯ o ) = 0, where ¯ o ) − A(s (1) if Re(so ) ≥ 0 and det so D(s

181

Then, assuming that ed (ω) > 0, ∀ω ∈ R, en (ω) σ ¯ (E(jω)) ≤ =: e(ω) , ∀ω ∈ R . ed (ω)

(17)

Proof: By (9) and (11), E(s) in (14) can be chosen to satisfy 

−1 ¯ ¯ ¯ sD(s) − A(s) B(s)   n ¯ n (s) (I + E(s)) ¯ (s) − A¯n (s) −1 B = sD   ¯ ¯ By premultiplying both sides by sD(s) − A(s) and rearranging terms we obtain Q(s) = R(s)E(s), where ν ν   Biu e−sτi + Bin ψi (s) − Γ(s) Q(s) := i=0

i=1

and

¯ n (s) + Γ(s) R(s) := B −sτin where ψi (s) := e −e and  ν  (sDiu − Aui ) e−sτi Γ(s) := −sτi

i=0

+

ν  i=1



(sDin − Ani ) ψi (s) Go (s)

¯ (Γ(jω)) ≤ Note that |ψi (jω)| ≤ ρi (ω), i = 1, . . . , ν, and σ γ(ω), ∀ω ∈ R. The desired result now follows on noting that σ ¯ (Q(jω)) ≤ en (ω) and σ (R(jω)) ≥ ed (ω). 

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We now return to our main problem stated in Section 2, which is to design a controller, based on the nominal model (7)–(8), so that the actual closed-loop system, obtained by applying this controller to the actual system (1)–(2), is robustly stable for all uncertainties satisfying the bounds (3)–(4). To achieve this, we propose to use e(ω), defined by (17), as a robustness measure, and to design a LTI feedback controller with TFM K(s) so that the nominal closed-loop system is stable and the constraint 1 σ ¯ (T n (jω)) < , ∀ω ∈ R , (18) e(ω) is satisfied. Here, the controller is to be applied as U (s) = −K(s)Y (s) ,

where U (s) and Y (s) indicate the Laplace transforms of, respectively, u(t) and y(t). Furthermore, in (18), −1

0.9 0.8 0.7 0.6 0.5 0.4 0.3

(19)

T n (s) := Gn (s)K(s) [I + Gn (s)K(s)]

en, ed, and e

1

0.2 0.1 10-2

10-1

(20)

is the complementary sensitivity matrix for the nominal closed-loop system. The following theorem proves that this controller will then achieve robust stability of the actual closed-loop system. Theorem 1: Suppose that Assumptions 1 and 2 hold, the control law (19) stabilizes the nominal system (7)– (8), and that (18) is satisfied. Then the actual closedloop system, obtained by applying the control law (19) to the actual system (1)–(2), is robustly stable for all uncertainties which satisfy the bounds (3)–(4). Proof: Given that the number of unstable modes of (1) are finite and same as the number of unstable modes of (7), 1 , ∀ω ∈ R , (21) σ ¯ (T n (jω)) < σ ¯ (E(jω)) implies that the actual closed-loop system is stable (e.g., see Zhou et al. (1996)), where E(s) satisfies (14) and T n (s) is given by (20). However, Assumptions 1 and 2 guarantee that the number of unstable modes of (1) are finite and same as the number of unstable modes of (7) for all uncertainties which satisfy the bounds (3)–(4). Furthermore, by (17), (21) is satisfied when the constraint (18) is satisfied. Therefore, the desired result follows.  4. EXAMPLE We consider a system described by (1)–(2) with one state delay and no input delay. The nominal value of the delay is assumed to be unity (i.e., τ1 = 1) with a 10% uncertainty on its actual value (thus, θ1 = 0.1). We assume that the system is modelled so that D0 = D0n = I (thus δ0 = 0) and C = [ 1 0 ]. Furthermore, since there is no input delay, B1 = B1n = 0 (thus β1 = 0). The nominal values of the other matrices appearing in (1)–(2) are assumed to be as follows:       0 0.1 0 1 0 0.1 n n n D1 = , A0 = , A1 = , 0.1 0 1 −2 0.2 0.3   0 . The uncertain parts of these matrices are and B0n = 1 assumed to satisfy (3) with δ1 = α0 = α1 = β0 = 0.05. 182

100

101

102

103

Frequency (rad/sec)

Fig. 1. Plots of en (ω) (in blue - plot with lowest gain), ed (ω) (in red - plot with highest gain), and e(ω) (in cyan - plot with medium gain) vs. ω. We first determine that sup(µf ) = −1.7247, where µf is defined by (5) and the sup is taken over all D1u and τ1u satisfying (3) and (4) respectively. This indicates that Assumption 1 is satisfied. We next verify that, for all uncertainties satisfying (3) and (4), the system (1) has exactly one unstable mode, whose value varies between 0.4019 and 0.5792 depending on the actual values of the uncertainties. This indicates that Assumption 2 is also satisfied. We then calculate en (ω), ed (ω), and e(ω), given in, respectively, (15), (16), and (17). These functions are plotted in Fig. 1. It is verified that ed (ω) > 0 for all ω ∈ R.

Then, to obtain a controller to stabilize the nominal system (7)–(8), we draw the Nyquist plot of Gn (jω) for ω ∈ R, which is shown in Fig. 2. A blow-up of this plot, showing the second left-most real axis crossing, is shown in Fig. 3. These plots indicate that the Nyquist plot of kGn (jω) encircles the −1 point once in the counterclockwise direction if k is chosen in the range: 1.2 =

1 1
(22)

where σ1 = −0.8333 and σ2 = −0.0401 are the first two left-most real axis crossings of the Nyquist plot of Gn (jω). Since the nominal system (7)–(8) has exactly one unstable ¨ mode, by the Nyquist theorem (e.g., see Ozbay (1999)), this implies that a constant gain controller K(s) = k, where k is chosen in the range (22), stabilizes the nominal system. Next, to achieve robust stability of the actual system, we check whether condition (18) is satisfied. For this, we calculate σ ¯ (T n (jω)), where T n (s) is given by (20) with K(s) = k, and compare it with 1/e(ω). Plots of σ ¯ (T n (jω)) together with 1/e(ω) are shown in Figures 4–8 for various values of k in the range (22). These figures indicate that (18) is satisfied if k is chosen in the range: 1.404 < k < 18.7 .

(23)

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Nyquist plot

0.3

imaginary

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0.2

101

0.1

100

0

10-1

-0.1

10-2

-0.2

10-3

-0.3 -1

-0.8

-0.6

-0.4

-0.2

0

10-4 10-2

0.2

real

Complementary sensitivity and 1/e for k=1.2

10-1

100

101

102

103

Frequency (rad/sec)

Fig. 2. Nyquist plot of Gn (jω).

Fig. 4. Plots of σ ¯ (T n (jω)) (in red - plot with lower highfrequency gain) and 1/e(ω) (in blue - plot with higher high-frequency gain) vs. ω for k = 1.2.

Nyquist plot

0.2

183

101

0.15 0.1

Complementary sensitivity and 1/e for k=1.404

100

imaginary

0.05 10-1

0 -0.05

10-2 -0.1 -0.15 -0.2 -0.2

10-3 -0.15

-0.1

-0.05

0

0.05 10-4 10-2

real

10-1

100

101

102

103

Frequency (rad/sec)

n

Fig. 3. Blow-up of the Nyquist plot of G (jω). Therefore, by Theorem 1, a constant gain controller K(s) = k, with k in the range (23), robustly stabilizes the actual system (1)–(2) for all uncertainties satisfying (3)–(4). 5. CONCLUSION Robust controller design problem for neutral time-delay systems has been considered. A frequency-dependent robustness measure, which accounts for the uncertainties in the system parameters and the time-delays, has been defined. A controller design approach, which uses this measure, has then been proposed. By satisfying the constraint (18), this approach guarantees that the actual closed-loop system is robustly stable. Note that, the constraint (18) involves the robustness measure and the complementary sensitivity matrix for the nominal closed-loop system. Thus, once the robustness measure is obtained, the proposed approach is completely based on the nominal model. 183

Fig. 5. Plots of σ ¯ (T n (jω)) (in red - plot with lower gain) and 1/e(ω) (in blue - plot with higher gain) vs. ω for k = 1.404. REFERENCES Curtain, R.F. and Zwart, H. (1995). An Introduction to Infinite-Dimensional Linear Systems Theory. SpringerVerlag, New York. ¨ Foias, C., Ozbay, H., and Tannenbaum, A. (1996). Robust Control of Infinite Dimensional Systems: Frequency Domain Methods, Lecture Notes in Control and Information Sciences, No. 209. Springer-Verlag, London. Han, Q.L. (2002). Robust stability of uncertain delaydifferential systems of neutral type. Automatica, 38, 719–723. ˙ Iftar, A. (2008). Decentralized robust control of large-scale time-delay systems. In Proceedings of the 17th IFAC Word Congress, 9332–9337. Seoul, Korea. ˙ Iftar, A. (2014). A robust controller design approach for systems with distributed time-delay. In Proceedings of the European Control Conference, 85–90. Strasbourg,

IFAC TDS 2015 June 28-30, 2015. Ann Arbor, MI, USA 184

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Complementary sensitivity and 1/e for k=10

102

100

101

10-1

100

10-2

10-1

10-3

10-2

10-4 10-2

10-1

100

101

102

10-3 10-2

103

Frequency (rad/sec)

10

10-1

10-2

10-1

100

101

100

101

102

103

102

Fig. 8. Plots of σ ¯ (T n (jω)) (in red - plot with lower highfrequency gain) and 1/e(ω) (in blue - plot with higher high-frequency gain) vs. ω for k = 24.91. ¨ Ozbay, H. (1999). Introduction to Feedback Control Theory. CRC Press, Boca Raton. Park, J.H. and Won, S. (1999). Asymptotic stability of neutral systems with multiple delays. Journal of Optimization Theory and Applications, 103, 183–200. Parlak¸cı, M.N.A. (2007). Delay-dependent robust stability criteria for uncertain neutral systems with mixed timevarying dicrete and neutral delays. Asian Journal of Control, 9, 411–421. ¨ Toker, O. and Ozbay, H. (1995). H∞ optimal and suboptimal controllers for infinite dimensional SISO plants. IEEE Transactions on Automatic Control, 40, 751–755. Wu, M., He, Y., and She, J.H. (2004). New delaydependent stability criteria and stabilizing method for neutral systems. IEEE Transactions on Automatic Control, 49, 2266–2271. Zhou, K., Doyle, J.C., and Glover, K. (1996). Robust and Optimal Control. Prentice Hall, Englewood Cliffs.

Complementary sensitivity and 1/e for k=18.7

100

10-3 10-2

10-1

Frequency (rad/sec)

Fig. 6. Plots of σ ¯ (T n (jω)) (in red - plot with lower gain) and 1/e(ω) (in blue - plot with higher gain) vs. ω for k = 10. 1

Complementary sensitivity and 1/e for k=24.91

103

Frequency (rad/sec)

Fig. 7. Plots of σ ¯ (T n (jω)) (in red - plot with lower gain) and 1/e(ω) (in blue - plot with higher gain) vs. ω for k = 18.7. France. ˙ ¨ uner, U. ¨ (1987a). Iftar, A. and Ozg¨ Decentralized LQG/LTR controller design for interconnected systems. In Proceedings of the American Control Conference, 1682–1687. Minneapolis, MN, U.S.A. ˙ ¨ uner, U. ¨ (1987b). Local LQG/LTR Iftar, A. and Ozg¨ controller design for decentralized systems. IEEE Transactions on Automatic Control, AC–32, 926–930. Kolmanovskii, V., Niculescu, S.I., and Richard, J.P. (1999). On the Liapunov-Krasovskii functionals for stability analysis of linear delay systems. International Journal of Control, 72, 374–384. Michiels, W. and Niculescu, S.I. (2007). Stability and Stabilization of Time-Delay Systems. SIAM, Philadelphia. Niculescu, S.I. (2001). Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, No. 269. Springer-Verlag, London. 184