Revisiting the problem of robust H∞ control with

regional pole location of uncertain discrete-time

systems with delayed states

Marlon H. Teixeira, Valter J. S. Leite, Luis F. P. Silva, Eduardo N. Goncalves

Abstract—In this note we consider the problem ofrobust stabilization of the class of uncertain discrete-time systems with constant and bounded delay inthe state. Such a stabilization is proposed consideringperformance issues such as H∞-guaranteed cost and theregional pole location of the closed loop system. We re-visit this problemwith an approach based on Lyapunov-Krasovskii function, avoiding the classical one which isdirectly based on the system augmentation. For thisproposal, we found an auxiliary system with multipledelays in the state such that if it is Schur-stable then theoriginal state delayed system has a specified regionalpole location. All system matrices are supposed tobelong to a polytopic set with known vertices. Somenumerical examples are given to illustrate the proposal.

I. Introduction

The characterization of the performance of a systemis a quite important issue in both theoretical and prac-tical aspects of control systems. Two quite useful kindsof performance characterization are i) the so called D-stability [1] analysis where a sub-set of the complex planeis certified as containing all poles of the system, andii) the H∞ guaranteed cost between an exogenous inputand the controlled output. Perhaps the main advantageof the characterization via D-stability analysis remainson the connections of this approach with the classicalcontrol theory [2], which provides the user with someuseful engineering insights. In the second case, the H∞

guaranteed cost has been successfully used in a wide rangeof applications and theoretical developments, as can beeasily verified in the literature of control systems. Theconjoint use of these two performance characterizationapproaches may be of interest to improve, for example,the transient performance and also to assure a minimallevel of rejection of exogenous signals in controller designcases [3], [4].

This work was supported by Brazilian Agencies CAPES, CNPqand FAPEMIG.

M. H. Teixeira and V. J. S. Leite are with CEFET–MG/Campus

Divinopolis, R. Alvares Azevedo, 400, 35503−822, Divinopolis, MG,Brazil. marlonhteixeira@gmail.com and valter@ieee.org.

L. F. P. Silva is with PPGEAS/DAS/UFSC. 88040 − 900, Flori-anopolis, SC, Brazil. luisfps@das.ufsc.br.

E. N. Goncalves is with CEFET–MG / Department of ElectricalEngineering, Av. Amazonas 7675, Belo Horizonte, MG, Brazil. ed-uardong@des.cefetmg.br.

In case of systems with delay in the states, D-stabilityanalysis and H∞ guaranteed cost characterization cannotbe directly applied mainly because of the multiple polesintroduced by the delay. Only a few results are availablein the literature to characterize the regional pole loca-tion [5], [6], [3]. In this note we revisit the problems ofrobust D(α, r)-stability [5], [7], [8] analysis and synthesisof controller taking into account the minimization of theH∞ guaranteed cost for the class of linear uncertaintime-invariant discrete-time systems with (constant andbounded) uncertain delay in the states. The D(α, r)-stability is investigated here means a disk region withcenter in (α, 0) and radius r entirely inside the unitarycircle centered at the origin of the complex plane whereall eigenvalues are inside it. We assume that the matricesused to describe the system are subject to polytopicuncertainties. Also we avoid the system augmentation toobtain a delay-free system [2] due the known problems insuch an approach [?] An alternative approach has beenproposed in [7] and has been used in [3], to deal withnon-fragile controller design, and in [5]. The improvedconditions presented in [5], using the same Lyapunov-Krasovskii (L-K) function employed here, usually leads tomore conservative results compared with ours.We revisit the problem of design a robust state feedback

controller by using directly L-K candidate functions andtaking into account some fundamental properties of theaugmented systems. By investigating the formation rule ofthe augmented systems, we are able to provide an equiv-alent system with multiple delays in the state such thatif it is robustly stable the original system with only onedelay is robustly D(α, r)-stable. This equivalent system isobtained thanks to a change of basis that allows to expressthe original problem in more suitable form. In this paperwe use a quite simple L-K functions to emphasize the mainideas of this proposal. Some numerical examples are givento illustrate the proposed conditions and some possibilitiesof research in this area are also presented.Notation: The space of quadratically summable signalsis noted by ℓ2 and the norm-2 of a signal in this space isnoted as ‖ · ‖2. The identity matrix and the null matrixare noted by I and 0, respectively. Here we assume 00 = 1.M > 0means thatM is definite positive matrix andMT isthe transpose of M . The set of reals and natural numbersare noted by IR and IN, respectively.

52nd IEEE Conference on Decision and ControlDecember 10-13, 2013. Florence, Italy

978-1-4673-5717-3/13/$31.00 ©2013 IEEE 922

II. Problem statement

Consider the uncertain discrete time system with de-layed states given by

Υ(β) :

xk+1 = A(β)xk +Ad(β)xk−d

+B(β)uk +Be(β)wk

zk = C(β)xk + Cd(β)xk−d

+D(β)uk +De(β)wk

(1)

where xk ∈ IRn, uk ∈ IRp and wk ∈ IRℓ stand forthe state, input and exogenous inputs, respectively. Thecontrolled output is noted by zk ∈ IRq. The sample timeis indicated by k ∈ IN, the uncertain delay is denotedby d = 1, . . . , τ , where τ maximum allowed value ofthe delay. The uncertain matrices A(β), Ad(β) ∈ IRn×n,B(β) ∈ IRn×p, Be(β) ∈ IRn×ℓ, C(β), Cd(β) ∈ IRq×n,D(β) ∈ IRq×p, De(β) ∈ IRq×ℓ define the system Υ(β) andbelong to a polytope A with N known vertices:

A =

Υ(β) ∈ IR(n+q)×(2n+p+ℓ) : Υ(β) =N∑

v=1

βvΥv, β ∈ Ω

,

(2)

Ω = β ∈ IRn : β ≥ 0,

N∑

v=1

βv = 1, (3)

Υv =

[

Av Adv Bv Bev

Cv Cdv Dv Dev

]

. (4)

We supposed the following state feedback control law

uk = Kxk +Kτxk−τ , (5)

where τ is the maximum value of the delay and the robustgainsK andKτ ∈ IRp×n must be determined. Observe thiscontrol law can be viewed as a simplified version of oneemploying all the states xk, xk−1, . . . , xk−τ . Beyond thetheoretical interest, such a simplification may be useful,for example, in systems with memory restrictions such assatellite applications. The use of this control law in Υ(β),see (1), yields the closed loop system

Υ(β) :

xk+1 = A(β)xk +Ad(β)xk−d

+B(β)Kτxk−τ +Be(β)wk

zk = C(β)xk + Cd(β)xk−d

+D(β)Kτxk−τ +De(β)wk

(6)

where Υ(β) belongs to a polytope described as in (2)-(4) with Υv, Av, and Cv respectively replaced by Υv,Av = Av + BvK, and Cv = Cv + DvK. The H∞ costfor the stable closed-loop system (6) concerning quadraticsummable signals wk and zk, i.e. wk, zk ∈ ℓ2, can becharacterized by the minimal value γ such that

‖zk‖2 ≤ γ‖wk‖2, wk, zk ∈ ℓ2. (7)

Because we are dealing with time-invariant system, thisrelation can also be characterized in terms of the matrixtransfer function between wk and zk, Gzw(e

jω). Mathe-matically, the robust optimal value can be characterizedby

γ∗ =‖ Gzw ‖∞, supω ∈ [−π, π]

β ∈ Ωd = 1, . . . , τ

σ[Gzw(ejω)] (8)

Therefore, all values γ ≥ γ∗ are said H∞-guaranteed costfor system (6).Additionally to the H∞-guaranteed cost, we desire to

assure a regional pole location for the closed-loop system(6), for example to achieve a specified transient response.Thus, we consider a disk region in the complex plane withcenter in (α, 0) and radius r as indicated in Figure 1.See also the discussion in [9] We note such a region as

IRz

IImz

1−1

α

j

− j

R

rD

Fig. 1. D(α, r)-region investigated.

D(α, r) and, because of the stability region of discrete-timesystems, it must verifies |α| + r < 1. If the eigenvalues ofthe closed-loop system described by (6) — or simply (6)— belong to D(α, r) for all possible values of β ∈ Ω andd ∈ [1, τ ], then we say that this system is robustly D(α, r)-stable.Therefore, we investigate the following synthesis prob-

lem.Problem 1: Let α and r such that 0 < r < 1 and

|α| + r < 1. Determine robust state feedback gains Kand Kτ such that system (1)-(4) with the control law (5)is robustly D(α, r)-stable with a minimal H∞-guaranteedcost, γ, between the exogenous input wk and the controlledoutput zk, for all β ∈ Ω and any d = 1, . . . , τ .

III. Preliminaires

A basic approach to solve Problem 1 consists in thedesign of a pair of gains K and Kτ that simultaneouslystabilizes a set of τ augmented delay-free systems assuringthe performance requirements of regional pole locationas well as the H∞-guaranteed cost. From (1), such anaugmented system can be described by

Υaugd :

sk+1 = Aaugd sk +Bauguk +Baug

e wk

zk = Caugd sk +Dauguk +Daug

e wk(9)

with sk =[

xTk xT

k−1 xTk−2 · · · xT

k−τ

]T, Daug = D,

Dauge = De, C

augd =

[

C 0 · · · 0 Cd 0 · · · 0]

,

Aaugd =

[[

A 0 · · · 0 Ad 0 · · · 0]

0

Iτn 0τn×n

]

, (10)

Baug =

[

B0τn×p

]

, Bauge =

[

Be

0τn×ℓ

]

, (11)

where Ad and Cd are placed in the (d + 1)-th (block-) column of Aaug

d and Caugd , respectively. Note that the

dependency on the uncertain parameter β ∈ Ω has beenomitted in equations (9)-(11). These equations establish afamily of τ delay-free systems Υaug

d , d = 1, . . . , τ . Each ofthese systems corresponds to a value possibly assumed bythe delay d = 1, . . . , τ . Because we are assuming the con-trol law (5) all augmented systems will have the dynamicmatrix Aaug

d with same size, Aaugd ∈ IRn(τ+1)×n(τ+1), and

923

just the position of the matrix Ad is changed in the aug-mented delay-free system. This same remark is applicableto Caug

d and Cd, respectively. System Υaugd can be handled

by linear techniques available in the literature to assureboth the D(α, r)-stabilization and the H∞-guaranteedcost [10]. Such a representation is used in the sequel toobtain a convex condition for D(α, r)-stabilization withH∞-guaranteed cost of (1) via an auxiliary system withmultiple delays in the state.

A. D(α, r)-stability

Consider each augmented system (9)-(11), d = 1, . . . , τ .Note that the D(α, r)-stability (or mutatis mutandis theD(α, r)-stabilization) of each of these systems can beequivalently addressed by analyzing the Schur-stability(stabilization, repsectvelly) of a transformed system [9]where the respective matrices are noted as follows (omit-ting the dependency on β ∈ Ω):

A(d, α, r) =Aaug

d − αI

r, B(d, α, r) =

1

rBaug,

Be(d, α, r) = Bauge , C(d, α, r) = Caug

d ,

D(d, α, r) = Daug, and De(d, α, r) = Dauge ,

(12)

That is, if A(d, α, r) is Schur-stable, then the originalsystem with delay is D(α, r)-stable.

However, because matrices A and Ad are uncertain, asdescribed in (2)-(4), then the Cartesian product of theiruncertain domains must be taken, increasing the numberof vertices of the delay-free representation. In such anapproach the main disadvantages are mainly due to i) thenumber of constraints that must be imposed to obtain asingle pair K and Kτ to stabilize all systems in the set,and ii) the exponential growing of the number of verticeswhenever the matrices of the system are uncertain. In whatfollows, the dependency on the uncertain parameter β isomitted to simplify the presentation.We propose an alternative way to investigate Problem 1

where the D(α, r)-stability analysis and controller designis handle by considering an auxiliary system with multipledelays and a L-K function is used to assure the (robust)stability of the closed-loop system. In the sequel, we focuson the transformations over the dynamic matrix becausethe other ones are straightforwardly obtained.The main lines of ours approach can be summarized in

the following steps:1) Obtain the augmented system as in (12): there-

fore the Schur-stability of A(d, α, r) assures the D(α, r)-stability of the original system.2) Found a matrix Q(τ, α, r) such that A(d, α, r) =

Q(τ, α, r)A(d, α, r)Q(τ, α, r)−1 has a structure similar toAaug

d . Thus, Q(τ, α, r) is used to change the basis used torepresent matrix A(d, α, r) in a way to obtain a matrixwith the structure

A(d, α, r) =

[[

A0,d A1,d . . . Aτ−1,d

]

Aτ,d

Iτn 0τn×n

]

(13)

From this changing of basis, we can obtain B(d, α, r) =Q(τ, α, r)B(d, α, r), Be(d, α, r) = Q(τ, α, r)Be(d, α, r),C(d, α, r) = C(d, α, r)Q(τ, α, r)−1, D(d, α, r) = D(d, α, r),and De(d, α, r) = De(d, α, r).3) From A(d, α, r) recover a system with multiple delays

in the state with dynamic equation given by

Υd :

xk+1 =∑τ

m=0 Am,dxk−m + Buk + Bewk

zk =∑τ

m=0 Cm,dxk−m + Duk + Dewk

(14)

with d = 1, . . . , τ , and xk = Q(τ, α, r)sk. Therefore, if(14) is Schur-stable, the original system is D(α, r)-stable.Of course, it is desirable to found Q(τ, α, r) such that

Am,d = 0 for m /∈ 0, τ. However, as we show in thispaper, this is not the case in general.

B. H∞-guaranteed cost

Suppose that system (14) has eigenvalues inside theunitary disk centered on the origin of the complex planeand null initial condition ∀β ∈ Ω. For sake of simplicity,assume the following L-K candidate function such thatV (xk) = 0 for xk = 0 and

V (xk) = xTk P (β)xk +

τ∑

h=1

−1∑

i=−h

xTk+iSh(β)xk+i > 0 (15)

with 0 < P (β) = P (β)T ∈ IRn×n, 0 < Sh(β) = Sh(β)T ∈

IRn×n, and xk 6= 0. Because of the stability of (14), if (15)is a L-K function for this system, then it is also verifiedthat ∆V (xk) < 0. This fact with (7) yields that if

∆V (xk) + zTk zk − µwTk wk < 0 (16)

is verified with µ = γ2, then system (14) is Schur-stablewith an H∞-guaranteed cost γ. Besides, because of thetransformations discussed in III-A, the original system isD(α, r)-stable with an H∞-guaranteed cost γ.Therefore, we propose to address the D(α, r)-stability

with H∞-guaranteed cost of system (1) by investigatingthe Schur-stability with H∞-guaranteed cost of the familyof auxiliary systems Υd, d = 1, . . . , τ , given in (14).To obtain ours main result the following lemma is used.Lemma 1 (Finsler’s Lemma): Consider ω ∈ IRn×n,

Q = QT ∈ IRn×n, and B ∈ IRm×n such that rank(B) < n.Then, ωTQω < 0, ∀ ω : Bω = 0, ω 6= 0 is equivalent to∃ X ∈ IRn×m : Q+ XB + BTX T < 0.

IV. Auxiliary system

Assuming the family of auxiliary systems (14) with

uk =

τ∑

m=0

Kmxk−m (17)

with Km, m = 0, . . . , τ obtained as discussed in III-A

K(τ, α, r) = K(τ)Q(τ, α, r)−1 =[

K0 · · · Kτ

]

(18)

Using (17)-(18) in (14) we get:

Υd :

xk+1 =∑τ

m=0 Am,dxk−m + Bewk

zk =∑τ

m=0 Cm,dxk−m + Dewk(19)

with m = 0, . . . , τ ,

Am,d = Am,d + BKm, Cm,d = Cm,d + DKm. (20)

Proposition 1: The change basis matrix Q(τ, α, r) thatrelates (19)-(20) with (12) is upper triangular.Proof: It is well known [11] that the change basis matrixthat relates system with matrices in (12) with systemin (19)-(20) is given by Q(τ, α, r) = CC−1, with C andC standing for the controllability matrices of (19)-(20)

924

and (12), respectively. By construction, each of thesecontrollability matrices are upper triangular and thus,Q(τ, α, r) = CC−1 is also upper triangular.Proposition 2: The matrix Q(τ, α, r) that allows the

change of basis described in section III-A is made up byblocks

[Q(τ, α, r)]ij =

ri−1I, i = j,0, i > j,

qi,jri−1αj−iI, i < j.

(21)

where

qi,j =

(−1)i+j , i = j or j = τ + 1,0, i > j,

qi+1,j+1 − qi+1,j , otherwise.(22)

For sake of space the proof is omitted.In addition to Proposition 1, it is interesting to observe

that matrix Q(τ, α, r) is upper triangular with each (block-) column being an homogeneous polynomial in α andr, with degree growing from 0 (leftmost column) to τ(rightmost column) in (21).Using the change basis matrix given in Proposition 2, it

is possible to develop rules for the direct construction ofthe matrices mentioned in step 2) of section III-A, Am,d,Cm,d, and also the transformed gains Km,τ as functionof α, r, and the respective matrices in (1) with (5).As presented in step 2) of section III-A, matrix De,d isalready defined, and additionally considering (21) we havematrices Bd = B(d, α, r) and Be,d = Be(d, α, r).Proposition 3: Consider the discrete-time system with

multiple delays in the state (19)-(20) with null initialconditions, i.e., xk−m = 0,m = 0, . . . , τ , matrices De,d

as defined in step 2) of section III-A, matrices , Bd = Bd

and Be,d = Be,d, and Am,d, Cm,d, and Km as given inTable I (see next page) with

bi,j =

1, i = j or j = τ + 2,0, i > j,

bi+1,j+1 + bi+1,j , otherwise.(23)

Then the following statements are verified:

1) If the auxiliary system (19)-(20) is Schur-stable (withwk = 0) then (6) is D(α, r)-stable.

2) Additionally, if γ is an H∞-guaranteed cost for (19)-(20) then, it is also an H∞-guaranteed cost for (6).

It is worth to note that coefficients bi,j used in Table I canbe obtained from the coefficients of Q(τ + 1, α, r)−1.

V. D(α, r)-stabilization & H∞-guaranteed cost

In the previous development, the change basis matrixQ(τ, α, r) does not depend on the uncertain parameterβ while all other matrices of the proposed systems andof the L-K function (15) do depend. However, such adependency has not been explicitly presented in generalto light the notation. In what follows, because thesematrices depending on β can be written similarly as in(2)-(4), we introduce a subscript v to note the respectivevertex matrix, v = 1, . . . , N . Thus, if we have Mv weget M(β) =

∑N

v=1 Mv, with M standing for any of thosematrices depending on β.

The following LMI condition allows to design a pair ofrobust control gains K and Kτ that D(α, r)-stabilizes (1)-(4) through the control law (5) with an H∞-guaranteedcost γ. Such a condition is obtained through the auxiliarysystem (19)-(20) which is constructed via Proposition 3.Theorem 1: Consider system (1)-(4). If there exist ma-

trices F ∈ IRn×n, Z ∈ IRp×n, Zτ ∈ IRp×n, 0 < Pv,d =PTv,d ∈ IRn×n, 0 < Sm,v,d = ST

m,v,d ∈ IRn×n, withv = 1, . . . , N , d = 1, . . . , τ , and scalars 0 < θ ≤ 1, andµ > 0 such that Θv,d is given by

Pv,d +∑τ

c=1 Sm,v,d − F − FT A0,v,dFT + (Bv

r)Z

⋆ −Pv,d

⋆ ⋆⋆ ⋆⋆ ⋆

A1×τ 0 (Bev

r)

0 −F CT0,v,d − ZTDT

v 0

−Sτ×τ Cτ×1 0

⋆ −θIq −Dev

⋆ ⋆ −µIℓ

< 0, (24)

is verified with matrices A0,v,d and C0,v,d as in Proposition3, Sτ×τ = block-diagS1,v,d, S2,v,d, . . . , Sτ,v,d, and form = 1, . . . , τ :

A1,m = Am,v,dFT +b2,m+2Bv(α

mZ+Zτδ(m−τ))/rm+1, (25)

Cm,1 = −F CTm,v,d−b2,m+2(α

mZT +ZTτ δ(m−τ))DT

v /rm, (26)

where δ(m− τ) is the discrete version of Dirac’s function,then, system (1)-(4) is robustly D(α, r)-stabilizable withH∞-guaranteed cost equal to γ =

õ by the robust control

law (5) where the robust state feedback gains are given by

K = ZF−T and Kτ = Zτ F−T . (27)

Proof: From block (1, 1) of (24) we get that F is regular.from (27) we replace Z and Zτ by KFT and Kτ F

T ,respectively, and we use (20) and the expressions pre-sented in Proposition 3 to express the terms on theirrespective closed-loop forms. Using the same steps as in[?] and considering the congruence transformation T =block-diagIτ+2 ⊗ F−1, Gq×q, Iℓ we get Ξv,d < 0 whereΞv,d is given in (28).

Γ1 F−1A0,v,d Γ2 0 Bev

⋆ −F−1Pv,dF−T 0 −CT

0,v,dF−T 0

⋆ ⋆ Γ3 Γ4 0

⋆ ⋆ ⋆ G(κ2 − 2κ)GT −Dev

⋆ ⋆ ⋆ ⋆ −µIℓ

(28)

with Γ1 = F−1(Pv,d+∑τ

c=1 Sc,v,d)F−T−F−1−F−T , Γ3 =

block-diag−F−1S1,v,dF−T , · · · ,−F−1Sτ,v,dF

−T ,Γ2 = [F−1A1,v,d F−1A2,v,d · · · F−1Aτ,v,d] and

Γ4 = [−CT1,v,dF

−T −CT2,v,dF

−T · · · −CTτ,v,dF

−T ]T .In the sequel we show that this last inequality is suffi-

cient for the Schur-stability of the system with multipledelays in the state given by (19) with an H∞-guaranteedcost of γ =

õ.

Consider the family of closed-loop system (19), d =

925

TABLE IFormulae for direct constructing of matrices Am,d, Cm,d, and Km,τ .

m Am,d Cm,d Km,τ

0A− (τ + 1)αI

rC K

1, . . . , d− 1b2,m+2Aαm − b1,m+2α

m+1I

rm+1

b2,m+2Cαm

rm b2,m+2Kαm

rmdb2,m+2Aαm − b1,m+2α

m+1I+Aτ

rm+1

b2,m+2Cαm + Cτ

rm

d+ 1, . . . , τ − 1b2,m+2Aαm − b1,m+2α

m+1I+ bd+2,m+2Aτα

m−d

rm+1

b2,m+2Cαm + bd+2,m+2Cταm−d

rm

τAατ +Aτα

τ−d − ατ+1I

rτ+1

Cατ + Cτατ−d

rτ

Kατ +Kτ

rτ

1, . . . , τ , and the candidate L-K function given in (15) with

P (β) =

N∑

v=1

βvPv and Sh(β) =

N∑

v=1

βvShv. (29)

where Pv = PTv > 0, and Shv = ST

hv > 0. Therefore, wecan write (16) as

xTk+1(Pv,d +

τ∑

m=1

Sm,v,d)xk+1 − xTk Pv,dxk

−τ∑

m=1

xTk−mSm,v,dxk−m + zTk zk − µwT

k wk < 0, (30)

where µ = γ2. By using Lemma 1 with

ω =[

xTk+1 xT

k xTk−1 · · · xT

k−3 zTk wTk

]T, (31)

Q = block-diagPv,d +

τ∑

c=1

Sc,v,d,−Pv,d,

− S1,v,d, . . . ,−Sτ,v,d, Iq,−µIl, and (32)

B =

[

−In A0,v,d A1,v,d · · · Aτ,v,d 0 Bev

0 C0,v,d C1,v,d · · · Cτ,v,d −Iq Dev,

]

we can apply item 2) with of Lemma 1 with the choice

X =

[

F 0 0 · · · 0 0 0

0 0 0 · · · 0 GT 0

]T

(33)

where F = F−1, what leads to the inequality in (28),d = 1, . . . , τ , with the block inside the box, i.e., block(τ + 3, τ + 3). Choosing G = − 1

kIq such a block can

be simplified as Iq + G + GT [?], [12]. Besides, we verifythat F−1Pv,dF

−T = Pv,d, F−1Sc,v,dF

−T = Sm,v,d, m =1, . . . , τ , recovering the L-K functions. Thus, condition(24) is sufficient for the Schur-stability of the system withmultiple delays (19) and, by Proposition 3, it is assuredthe D(α, r)-stabilization of (1)-(4) with anH∞-guaranteedcost given by γ =

õ by the control law (5) and robust

control gains given by (27).Note that the conditions in Theorem 1 can be used

to search for the minimal value γ such that (7) is veri-fied while the closed-loop system has a specified D(α, r)pole location. This is formalized by the following convexoptimization problem, PH∞

(α, r) ( for v = 1, . . . , N ,m, d = 1, . . . , τ):

minF , Z, Zτ , θ,

Pv,d, Sm,v,d

µ

such that (24),0 < Pv,d, Sm,v,d,0 < θ ≤ 1,

(34)

VI. Examples

The following example illustrate the advantages of thecontroller synthesis conditions proposed in Theorem 1,mainly for uncertain systems.Example 1: In [5] the proposed conditions deals with

systems with norm-bounded uncertainties and two typesof delays: delay in the states and in the input. Becausewe do not consider delay in the input, we are goingto make some comparisons with the conditions in [5,(38)] taking the input delay equal to zero. Thus, considersystem (1) with matrices as follow and τ = 2, i.e., no

uncertainty in the delay, A =

0.47 1.31 −0.31−0.55 −1.27 1.040.39 0.97 −0.11

,

Ad =

0.1 0 −0.12−0.1 0.1 00 0.15 −0.2

, C =

0.7 0.2 0.10.1 0.8 −0.50 0 0.8

,

B =[

1.1 −1 0.8]T

, Be =[

0.3 0.2 0.1]T

, D =[

0.45 0.3 0.5]T

, De =[

0.2 −0.3 −1]T

Precisely knownsystem: In [5, Example 5.3] this is equivalent to assumeG = 0 and d2 = 0. Solving conditions in [5] withD(0.1, 0.78) we get a control law as in (5) withK2 = 0 andan H∞ guaranteed cost γ[5] = 0.77. Using the optimizationproblem PH∞

(α, r) to minimize γ with the same regionalpole specification and with the constraint K2 = 0 we getan H∞ guaranteed cost γT1 = 1.15. Thus, for the preciselyknown system, the conditions in [5, (38)] may give betterresults than ours. However, if we assume some uncertaintyin the system matrices, this may be not the case. This isillustrated in the sequel.Uncertain system: In [5] matrices A, Ad, B, and Be

are supposed to be affected by norm bounded uncer-tainty as follows: A = An + GFE1, A = Adn + GFE2,B = Bn + GFE3, Be = Ben + GFE5, where FTF ≤ I

and An, Adn, Bn, and Ben refer to the nominal matrixvalues given in Example 1. From [5, Example 5.3] wetake E1 =

[

−0.2 0.15 0.13]

, E2 =[

0.2 −0.3 0.1]

,

E3 = 0.2, E5 = −0.8, and assuming G = 0.4[

1 2 1]T

,

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the conditions (38) in [5] yield a static state feedback gainthat assures an H∞-guaranteed gain of 9.57.If this same system is represented in a polytopic do-

main, we get two vertices obtained at the minimal andmaximal values of F , i.e., F = −1 and F = 1, re-spectively. In this case, minimizing the H∞-guaranteedcost trough the convex optimization problem PH∞

(α, r)with Kτ = 0 we get K =

[

−0.5647 −1.3354 0.7204]

assuring an H∞-guaranteed cost of γ = 1.43. Thus,the cost trough (38) in [5] is about 669% greater thanthe obtained by ours condition. Besides, if we allowto feedback xk−2, the optimization problem PH∞

(α, r)yields K =

[

−0.5762 −1.3362 0.7444]

and Kτ =[

−0.0407 −0.0088 0.0539]

with anH∞-guaranteed costof 1.39, showing that the feedback of the delayed statecan, as expected, reduces the H∞-guaranteed cost of theclosed-loop system.The following example is used to illustrate how the poles

location is changed with different values of the delay d, andto give an idea of the behavior of the H∞-guaranteed costwith the region specification D(α, r).Example 2: Consider the system used in Example 1

with τ = 2 and that a larger set of uncertainty is affectingonly matrices A, Ad, and B as follows: A(ρ) = (1 + ρ)A,Ad(η) = (1 + η)Ad and B(σ) = (1 + σ)B, where |ρ| ≤ 0.3,|η| ≤ 0.06, and |σ| ≤ 0.04. This leads to a polytopicrepresentation with 8 vertices obtained by the combinationof the maximal and minimal values of ρ, η, and σ. Assum-ing a region D(0.1, 0.78) we can obtain the robust statefeedback gains K =

[

−0.7019 −1.5421 1.0279]

andKτ =

[

−0.0429 0.0009 0.0674]

that used in control law(5) assure to the closed-loop system an H∞-guaranteedcost of γ = 4.665. In Figure 2 it is shown the poles of thesystem for d = 1 and d = 2. In both cases, we can seethat the poles are inside the specified region, D(0.1, 0.78),delimited by a circle with solid line. Also, we can note thatthe greater is the delay, the bigger is the dispersion of theeigenvalues. The stability region of discrete-time systemsis identified by the unitary circle with dashed line.

−1

0

1

−1

0

1

0

1

2

d

IRzIImz

Fig. 2. Cloud of eigenvalues for closed-loop system (6).

With this case, we have investigated theH∞-guaranteedcost for different pairs (α, r), α > 0. This is a typical regionof interest to avoid, for example, oscillations in discrete-time systems. For each value of r, we have swept the values

of α, and evaluated optimization problem PH∞(α, r) given

in (34).

VII. Conclusions

We presented an alternative approach to solve the prob-lem of robust D(α, r)-stabilization with H∞-guaranteedcost of uncertain discrete-time systems with delayedstates. In ours proposal, a Lyapunov-Krasovskii functionis used conjointly with an auxiliary discrete-time systemwith multiple delays in the state to achieve the controlobjective: if the auxiliary system is Schur-stable with anH∞ guaranteed cost, then we shown that the original sys-tem is D(α, r)-stable with the same H∞ guaranteed cost.Although we have based our presentation on a quite sim-ple Lyapunov-Krasovskii candidate function, the resultsare interesting and better than others in the literature.Moreover, the proposed approach can be improved withmore complete Lyapunov-Krasovskii functions leading toless conservative conditions, allowing even more stringentregional pole specifications. Some examples were presentedwhere we shown the potentiality of ours proposal.

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