On Dynamic Output Feedback H

∞

Control for Positive Discrete-time

Delay Systems

Baozhu Du

School of Automation, Nanjing University of Science and Technology, 210094, P. R. China

Keywords:

Bounded Real Lemma, Dynamic Output Feedback, Positive Systems, Time Delay.

Abstract:

This paper is devoted to the H

∞

control design of positive discrete-time systems with multiple delays. Novel

bounded real lemma is presented ﬁrst via linear matrix inequality technique, which reveals that H

∞

norms

of a discrete-time positive system with time delays both in dynamic and output equations are identical to

that of the corresponding delay-free system. Necessary and sufﬁcient conditions for positivity preserving

∞

stabilization via a dynamic output feedback control are established in the forms of matrix equalities, that

guaranteeing the closed-loop system not only to be asymptotically stable and positive, but also to have a

desired H

∞

performance. The proposed results are extended to interval uncertain positive systems with time

delay. Finally, an example is given to illustrate the effectiveness of the obtained design scheme.

1 INTRODUCTION

Positive systems are a class of systems whose state

variables are never negative, for any given nonneg-

ative initial state and nonnegative input. Lots of

stability and stabilization problems for time-delayed

positive systems have been reported in the literature,

see, for instance (Gao et al., 2004b), (Cui et al.,

2018). Necessary and sufﬁcient conditions based on

linear programming technique were given to guaran-

tee the asymptotical stability of discrete-time positive

systems with constant delays in (Liu, 2009), which

proved that the magnitudes of delays have no impact

on system stability. Stability analysis of positive sys-

tems with bounded time-varying delays was studied

in (Liu et al., 2010). Exponential stability of pos-

itive time-delayed systems was investigated in (Zhu

et al., 2012) by the Lyapunov-Krasovskii functional

based method, and diagonal Riccati stability criteria

was presented in (Mason, 2012) by using the separat-

ing hyperplane theorem.

For the controller synthesis, an output feedback

controller has to be used if no full access to the system

states (Wang et al., 2015), (Shu et al., 2012), (Zhang

et al., 2018). There are generally two types of strate-

gies to avoid NP-hard problem (Blondel and Tsitsik-

lis, 1995). One is so-called relaxation, which is easy

to implement, but conservatism may be introduced in

some cases (Gao et al., 2004a). The other strategy is

the local optimization which minimizes the objective

function near the feasible point. Most accurate meth-

ods to static output feedback synthesis involve local

optimization, for instance, the direct iterative proce-

dure (D-K iteration), iterative linear matrix inequality

(ILMI) and the cone complementarity linearization

(CCL) (Geromel et al., 1994). The free-weighting

matrix method proposed in (He et al., 2007) has re-

duced the conservatism in controller synthesis, but

always introduces extra coupling terms among con-

troller gain, Lyapunov matrices and system matri-

ces (Mirkin and Gutman, 2005). To decouple these

cross-product terms, an augmentation approach was

proposed in (Shu and Lam, 2009) provided an equiv-

alent form of the H

∞

stabilization criterion for positive

delay-free systems.

The bounded real lemma (BRL) and Kalman-

Yakubovich-Popov(KYP) lemma for linear positive

systems without time delays was presented by T.

Tanaka and C. Langbort in (Tanaka and Langbort,

2010) , in which the KYP lemma made the condi-

tion of H

∞

controller design be convex and tractable

with the help of the small gain theorem and the hyper-

plane separation theorem. Strict/non-strict inequal-

ity versions of KYP lemma for single-input single-

output discrete-time positive systems without time

delays were developed in (Najson, 2013) where a

quadratic Lyapunov function was formulated by a di-

agonal Lyapunov matrix (Farina and Rinaldi, 2000).

BRL in terms of matrix inequality for continuous-

time positive systems with time delays in states was

760

Du, B.

On Dynamic Output Feedback H Control for Positive Discrete-Time Delay Systems.

DOI: 10.5220/0007931607600766

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 760-766

ISBN: 978-989-758-380-3

provided in (Zhang et al., 2015), and the criteria for

dynamic output feedback H

∞

stabilizability were also

proposed. To the authors’ knowledge, there are still

no related results to the H

∞

stabilization problem for

discrete-time positive system with discrete delays.

Based on the above observations, this paper is mo-

tivated to present BRL of discrete-time positive sys-

tems with time delays for the ﬁrst time, with further

discussion on H

∞

control by means of dynamic out-

put feedback control strategy. The remaining parts of

this paper are organized as follows. Preliminary is in-

troduced in Section 2 and a novel BRL is established

in Section 3. Necessary and sufﬁcient conditions are

proposed to prove that H

∞

performance of positive

time-delayed systems are independent on the magni-

tudes of delays. In the aspect of controller synthe-

sis, necessary and sufﬁcient conditions are presented

in Section 4 to design dynamic output feedback con-

trollers, which leads to the closed-loop system to pos-

sess asymptotical stability, positivity, and desired H

∞

performance simultaneously. Section 5 extends re-

sults to interval linear discrete-time systems with both

time delays and uncertainties. A numerical example

is given to illustrate the effectiveness of the obtained

results in Section 6.

2 PRELIMINARIES

The notations throughout this paper are fairly stan-

dard. For two matrices A ∈ R

m×n

and B ∈ R

m×n

A ≥ B, A > B, A  ()B, respectively, denote that

i j

≥B

i j

, A

i j

≥B

i j

but A 6= B, A

i j

> (<)B

i j

, for all i =

1, 2, . . . , m, j = 1, 2, . . . , n. For A ∈ R

n×n

, A  0 and

A ≺0 mean that A is a positive semideﬁnite and a neg-

ative deﬁnite matrix, respectively.

X,Y

= trace(XY )

is the inner product on S

. The asterisk “∗” in a matrix

represents a term which can be induced by symmetry.

Moreover,

, R set of positive integers, set of real numbers

set of n-dimensional real vectors

m×n

set of m ×n real matrieces

, R

nonnegative and positive orthants of R

space of n-th order real symmetric matrices

n×n

set of all diagonal positive deﬁnite matrices

vector with 1 in ith position and 0 elsewhere

1, I vector [1, 1, . . . ,1]

, identity matrix

i j

ith component of matrix A

transpose of A

trace(A) trace of matrixA

ρ(A) spectral radius of matrix A

σ(A) maximum singular value of matrixA

D(A) vector composed of diagonal entries of A

Consider a linear discrete-time positive system

with time delays in state and output equations as fol-

lows,

: x(k + 1) = Ax(k) +

∑

i=1

x(k −d

) + Bω(k),

z(k) = Cx(k) +

∑

i=1

x(k −d

) + Dω(k),

x(k) = φ(k), k ∈ [−d, 0],

(1)

where x(k) ∈

, ω(k) ∈

, z(k) ∈

are the

state, exogenous input and output vectors, respec-

tively. A, A

, B,C,C

, D are known real matrices with

appropriate dimensions, d

is a constant time delay,

φ(k) ∈

is the vector-valued initial function on

[−d, 0] with d , max{d

}, i = 1, 2, . . . , q. Some neces-

sary deﬁnitions and lemmas are provided ﬁrst, which

are useful in the subsequent technical development for

linear time-delay positive systems.

Deﬁnition 1. Matrix A ∈ R

n×n

is Schur stable if

ρ(A) < 1.

Lemma 1. ((Berman and Plemmons, 1979)) For two

matrices A ∈ R

n×n

, B ∈ R

n×n

, ρ(A) ≥ ρ(B) if A ≥ B.

Lemma 2. (Liu, 2009) For positive system Σ

, the

following statements hold:

(i) System Σ

is positive if and only if A ≥ 0, B ≥ 0,

C ≥ 0, D ≥ 0, A

≥ 0, C

≥ 0, i = 1, 2, . . . , q;

(ii) System Σ

is asymptotically stable if and only if



A +

∑

i=1



< 1.

The transfer function matrix of system Σ

is given

(z) =



C+

∑

i=1

−d



zI −A−

∑

i=1

−d



−1

B+D,

and its H

∞

norm is deﬁned as

||G

∞

= sup

θ∈[0,2π)

σ(G(e

jθ

)).

A sufﬁcient condition to check H

∞

characteristics of

system Σ

with q = 1 has been established in (Gao

et al., 2004a) as follows.

Lemma 3. (Gao et al., 2004a) Positive system Σ

with q = 1 is asymptotically stable and ||G

∞

< γ

if there exist matrices P  0 and Q  0 such that

M + diag

{

Q, −Q, 0

}

≺ 0, (2)

where

M =





PA −P +C

C A

C + A

PA A

C + B

PA B

+ D

D + A

PB +C

PB + D

D −γ





On Dynamic Output Feedback H Control for Positive Discrete-Time Delay Systems

761

3 BOUNDED REAL LEMMA

(BRL)

In this section, we shall point out that H

∞

perfor-

mance of the discrete-time positive linear system Σ

with constant delays is insensitive to the magnitude of

the delays. Our purpose is to give a characterization

on the BRL for system Σ

with multiple time delays.

To this end, we ﬁrst introduce two nominal delay-free

positive systems:

: x(k + 1) =

Ax(k) + Bω(k),

z(k) =

Cx(k) + Dω(k),

(3)

: x(k + 1) = Ax(k) + Bω(k),

z(k) = Cx(k) + Dω(k).

(4)

For simplicity, deﬁne

A = A +

∑

i=1

C =

C +

∑

i=1

. The transfer functions of sys-

tems Σ

and Σ

are, respectively, given by

(z) =



zI −



−1

B + D

(z) = C(zI −A)

−1

B + D

with z = e

jθ

, θ ∈ [0, 2π). It has been pointed out

in (Najson, 2013) that, if system Σ

is positive and

asymptotically stable, ||G

∞

σ(G

(1)). On the ba-

sis of this fact, the following lemma can be obtained

which is useful sequentially.

Lemma 4. If system Σ

is positive, asymptotically

stable and ||G

∞

< γ, then ||G

∞

≤ ||G

∞

< γ.

Proof: If positive system Σ

is asymptotically stable

and ||G

∞

< γ, one has

σ(G

(1)) < γ. Obviously,

||G

∞

σ(G

(1)) =

σ(G

(1)) < γ. It follows from

Lemma 2 that

(I −

−1

∞

∑

k=0

≥

∞

∑

k=0

= (I −A)

−1

≥ 0,

which leads to 0 ≤ G

(1) ≤ G

(1). Accord-

ing to Lemma 1, ||G

∞

σ(G

(1)) ≤

σ(G

(1)) =

||G

∞

< γ is derived. 

Theorem 1. [Single delay] When q = 1. System Σ

is asymptotically stable and ||G

∞

< γ if and only if

there exist P ∈ D

n×n

and Q ∈ D

n×n

such that inequal-

ity (2) holds.

Proof: Necessity. Deﬁne a discrete-time system

: x(k + 1) = Ax(k) + A

x(k −d

) + Bω(k),

z(k) =

Cx(k) +

x(k −d

) +

Dω(k),

x(k) = φ(k), k ∈ [−d

, 0],

(5)

in which

C =

D =

D. Its trans-

fer function matrix is denoted by

(z). If sys-

tem Σ

with q = 1 is positive, asymptotically stable

and ||G

∞

< γ, one has ||

∞

= ||

∞

< 1. At

this point, it turn s to prove that, if system

is posi-

tive, asymptotically stable and ||

∞

< 1, there must

exist P ∈ D

n×n

and Q ∈ D

n×n

satisfying

M + diag

{

Q, −Q, 0

}

< 0, (6)

where

M is M deﬁned in Lemma 3 with C, C

and D,

respectively, replaced by

and

D, and γ = 1.

The proof will be given by contradiction. Sup-

pose that, for every P ∈ D

n×n

, there does not exist any

nonzero Q ∈ D

n×n

such that LMI (6) holds. Deﬁne

two sets



M + diag

{

Q, −Q, 0

}



Q ∈ D

n×n



, {R|R ≺ 0, R ∈ S

2n+m

It can be easily veriﬁed that both sets S

and S

are

convex and nonempty. The intersection of sets S

and

is empty means that S

∩S

0. It follows from

the separating hyperplane theorem (Boyd and Balakr-

ishnan, 2004) and convex analysis (see (Rockafellar,

2015)) that two disjoint convex sets can be separated

by a hyperplane, that is, there exists a nonzero matrix

H ∈ S

2n+m

such that

H,Y

≥ 0, ∀Y ∈ S

, (7)

H, X

< 0, ∀X ∈ S

. (8)

From condition (8), that is,

H, X

= trace(HX) < 0,

one can easily verify that H  0. Deﬁning

H =









 0,

inequality (7) yields that, for any Q ∈ D

n×n

trace



M + diag

{

Q, −Q, 0

}

)



=trace(H

M) + trace((H

−H

)Q) ≥ 0.

It follows from Q > 0 that

trace(H

M) ≥ 0, (9)

D(H

−H

) ≥ 0, (10)

otherwise, there must exist nonzero H  0 and Q ∈

n×n

such that trace(H

M) + trace((H

−H

)Q) <

0 for any ﬁxed P ∈ D

n×n

Deﬁne a nonzero vector h ,





∈

2n+m

with h

√

, i = 1, 2, 3. From inequality (9)

and Lemma 1, one has trace(hh

M) ≥ 0 which is

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

762

equivalent to,

Ω ,h

PA −P +

C)h

+ h

PA +

C)h

+ h

PA +

C)h

+ h

PB+

D)h

PB+

D)h

PB+

D−I)h

≥ 0.

(11)

From (10), one has h

≥ h

≥ 0. Note that vector h is

nonzero, and two cases will be discussed as follows.

Case 1: h

= 0. Inequality (11) leads to

PA −P +

C)h

+ h

PA +

C)h

+ h

PB+

D)h

PB+

D−I)h

≥ 0

(12)

which can be rewritten as









≥ 0.

in which

ϒ ,



PA −P +

C A

PB+

PA +

C B

PB+

D−I



It means that there exists a nonzero vector





such that the above inequality holds.

Equivalently, there does not exist P ∈ D

n×n

such that

ϒ ≺ 0. According to the KYP Lemma (in (Rantzer,

2016)) and Lemma 4, one has ||

∞

≥ 1. This is a

contradiction.

Case 2: h

> 0. Due to the fact that h

≥ h

, Ω

deﬁned in condition (11) satisﬁes

Ω ≤







(A + A

)

P(A + A

) −P + (

C +

)

(

C +

)

P(A + A

) +

(

C +

)

(A + A

)

PB+ (

C +

)

PB+

D−I





which is inconsistent with the fact that ||

∞

< 1.

Hence, if system Σ

is positive, asymptotically

stable and

∞

< γ, there must exist P ∈ D

n×n

and

Q ∈ D

n×n

satisfying M + diag

{

Q, −Q, 0

}

≺ 0.

Sufﬁciency condition can be immediately ob-

tained from Lemma 3 and Theorem 1 given in (Wu

et al., 2009). This completes the proof. 

After an algebraic manipulation, a simple equiva-

lent form of (2) in Theorem 1 can be obtained in the

following corollary, in which the matrix variable Q

appearing in Theorem 1 has been removed.

Corollary 1. Positive system Σ

with q = 1 is asymp-

totically stable and

∞

< γ if and only if there exists

P ∈ D

n×n

such that



(A + A

)

P(A + A

) −P + (C +C

)

(C +C

)

P(A + A

) + D

(C +C

)

(A + A

)

PB + (C +C

)

PB + D

D −γ



≺ 0.

(13)

Remark 1. From Lemma 4 and Theorem 1, it is clear

that the exact value of

∞

is given by

σ(G

(1)) if

positive system Σ

with q = 1 is asymptotically stable.

That is, H

∞

norm of system Σ

is equivalent to that of

system Σ

, which is independent of time delays. Due

to this fact, Theorem 1 can be easily extended to the

case of multiple time delays, that is, q > 1, and A

and

being replaced by

∑

i=1

and

∑

i=1

The BRL for positive system Σ

with multiple

time delays can be directly obtained in the following

theorem, in which

A and

C are given in system Σ

Theorem 2. [Multiple delay] Positive system Σ

with

q > 1 is asymptotically stable and

∞

< γ if and

only if there exists a matrix P ∈ D

n×n

such that



A −P +

PB +

A + D

C B

PB + D

D −γ



≺ 0.

(14)

4 DYNAMIC OUTPUT

FEEDBACK H

∞

CONTROL

Due to the fact that full access to the system state is

usually impossible in real plants and often only par-

tial information of the state can be measured, one has

to use a controller based on output measurements.

It becomes necessary to develop H

∞

control theory

via output feedback control signal. On the basis of

the above preparatory work, next an explicit delay-

independent characterization of the positivity preserv-

ing H

∞

control will be developed. Since H

∞

norms of

positive time-delay systems only depend on system

matrices, our attention is restricted to the case of sin-

gle delay (that is, q = 1) and then the derived results

can be easily extended to the case of multiple delays

(that is, q > 1).

Consider a discrete-time positive system with one

constant delay as follows

x(k + 1) = Ax(k) + A

x(k −d) + Bω(k) + B

u(k),

z(k) = Cx(k) +C

x(k −d) + Dω(k) + B

u(k),

y(k) = Fx(k) + Hω(k),

x(k) = φ(k), k ∈ [−d, 0],

(15)

On Dynamic Output Feedback H Control for Positive Discrete-Time Delay Systems

763

where x(k) ∈ R

is the state, ω(k) ∈ R

is the exoge-

nous input, u(k) ∈ R

is the control input, z(k) ∈ R

is the controlled output, y(k) ∈ R

is the measure-

ment. A, A

, B, B

, B

,C,C

, D, F, H are real matrices

with compatible dimensions. A dynamic output feed-

back controller is given by

ξ(k + 1) = A

ξ(k) + B

y(k),

u(k) = C

ξ(k) + D

y(k),

(16)

where ξ(k ) ∈R

is the controller state, A

, B

, D

are the controller gain matrices to be designed. The

following closed-loop system is conducted from sys-

tem (15) via the output feedback controller (16) .



x(k+1)

ξ(k+1)





A+B

F B

F A



x(k)

ξ(k)





0 0



x(k−d)

ξ(k−d)





B+B



ω(k),

z(k)=



C+B

F B





x(k)

ξ(k)









x(k −d)

ξ(k −d)



+(D+B

H)ω(k),

(17)

The validity of the performance-based design lies in

whether the closed-loop performance requirement can

be satisﬁed easily. Let us take an exploration of H

∞

performance-based control design upon the BRL rep-

resentation through an output feedback control, which

allows that the closed-loop system is positive, asymp-

totically stable and

∞

< γ.

Theorem 3. Given positive system (15) and a con-

stant scalar γ > 0, the existence of a dynamic out-

put feedback controller (16) such that the closed-

loop system (17) is positive, asymptotically stable

and

∞

< γ, is equivalent to the existence of ma-

trices P

∈ D

n×n

, P

∈ D

s×s

, Q

∈ D

n×n

, Q

∈ D

s×s

∈R

s×s

, L

∈R

s×r

, L

∈R

l×s

, L

∈R

l×r

satisfying

= I, P

= I, A+B

F ≥0, B

≥0, L

F ≥

0, L

≥ 0, B + B

H ≥ 0, L

H ≥ 0, C + B

F ≥ 0,

≥ 0, D + B

H ≥ 0, and







−P

0 0 A + A

+ B

F B

B + B

∗ −P

0 L

F L

∗ ∗ −I C +C

+ B

F B

D + B

∗ ∗ ∗ −Q

0 0

∗ ∗ ∗ ∗ −Q

∗ ∗ ∗ ∗ ∗ −γ







≺0.

(18)

Then the desired controller gain matrices are given by

= L

, B

= L

, D

= L

Proof: Applying Schur complement lemma and set-

ting P

−1

= Q

, P

−1

= Q

, A

= L

, B

= L

, C

, D

= L

, inequality (18) and two equality con-

straints can be derived from Theorem 2. Other in-

equalities used for guaranteeing the positivity of the

closed-loop system (17) can be derived directly from

Lemma 2. The proof is completed. 

5 ROBUST H

∞

CONTROL

It is noted that, for two Schur stable matrices A

≥ 0

and A

≥ 0 with A

≥ A

, we have A

−1

≥ A

−1

. Mo-

tivated by this fact, there is a possible extension of

Theorem 1 to uncertain time-delay positive systems.

In this section, consider an interval uncertain discrete-

time positive system with a time delay as follows

x(k + 1) = A

x(k) + A

x(k −d) + B

ω(k) + B

u(k),

z(k) = C

x(k) +C

x(k −d) + D

ω(k) + B

u(k),

y(k) = Fx(k) + Hω(k),

x(k) = φ(k), k ∈ [−d, 0],

(19)

where A

∈



A, A



, A

∈



, A



, B

∈



B, B



, C

∈



C,C



, C

∈





, D

∈



D, D



, A ≥ 0, A

≥ 0,

B ≥ 0, C ≥ 0, C

≥ 0, D ≥ 0,and A, A, A

, A

, B, B,

C, C, C

, C

, D, D are all constrained in metric space.

Theorem 4. Interval uncertain positive system (19)

is robustly asymptotically stable and

∞

< γ if and

only if there exists a matrix P ∈ D

n×n

satisfying that

(A + A

)

P(A + A

) −P + (C +C

)

(C +C

)

P(A + A

) + D

(C +C

)

(A + A

)

PB + (C +C

)

PB + D

D −γ

≺ 0.

Proof: If system (19) is positive and robustly asymp-

totically stable, ρ(A

) < 1 for any A

∈ [A, A]. It fol-

lows that C

(I −A

−A

)

−1

≤C +C

(I −

A −A

)

−1

B + D. Furthermore, from Lemma 2, one

gets



(I −A

−A

)

−1

+ D



∞

≤



C +C

(I −A −A

)

−1

B + D



∞

< γ.

Therefore, according to Theorem 1, sufﬁciency and

necessity conditions are obvious. 

Next, our objective is to design a dynamic out-

put feedback controller in (16) such that the following

closed-loop system

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

764



x(k+1)

ξ(k+1)





F B

F A



x(k)

ξ(k)





0 0



x(k−d)

ξ(k−d)







ω(k),

z(t)=



F B





x(k)

ξ(k)









x(k −d)

ξ(k −d)







ω(k).

(20)

is positive, robustly asymptotically stable and

||G||

∞

< γ. 

Theorem 5. Given positive system (19) and a scalar

γ > 0, there exists a dynamic output feedback con-

troller (16) such that the closed-loop system (20) is

positive, robustly asymptotically stable and ||G

∞

γ if and only if there exist matrices P

∈ D

n×n

, P

∈

s×s

, Q

∈ D

n×n

, Q

∈ D

s×s

, L

∈ R

s×s

, L

∈ R

s×r

∈ R

l×s

, L

∈ R

l×r

such that A + B

F ≥ 0,

≥0, L

F ≥0, L

≥0, B +B

H ≥0, L

H ≥0,

C +B

F ≥ 0, B

≥0, D +B

H ≥0, P

= I,

= I, and







−P

0 0 A + A

+ B

F B

B + B

∗ −P

0 L

F L

∗ ∗ −I C +C

+ B

F B

D + B

∗ ∗ ∗ −Q

0 0

∗ ∗ ∗ ∗ −Q

∗ ∗ ∗ ∗ ∗ −γ







≺ 0.

(21)

If the above conditions hold, then the desired con-

troller gain matrices are given by

= L

, B

= L

, D

= L

Proof: It is obvious that A

+ B

F ≤ A

+ B

for any A

∈ [A, A]. If A + B

F ≥ 0 holds, then

+ B

F ≥ 0. Setting P

−1

= Q

, P

−1

= Q

, A

, B

= L

, C

= L

, D

= L

, and taking a similar

line as the proof of Theorem 3, the detailed proof is

omitted. 

6 NUMERICAL EXAMPLE

This section presents one numerical example to illus-

trate the effectiveness of the proposed results. Con-

sider a discrete-time interval uncertain positive sys-

tem in (19) with one delay in the system state, and

system matrices given as follows:

A =





0.3648 0.3986 0.2695

0.3178 0.4146 0.4423

0.4812 0.1218 0.3757





, A =





0.3931 0.4181 0.2914

0.3398 0.4527 0.4718

0.5054 0.1470 0.3990









0.0221 0.0982 0.0322

0.0313 0.1001 0.0271

0.0182 0.0235 0.0283





, A





0.0323 0.1002 0.0385

0.0348 0.1320 0.0334

0.0293 0.0264 0.0379





B =





0.1560

0.1820

0.141





, B =





0.1835

0.2273

0.1705





, B





0.2318

0.4836

0.1931





, B

= 0.2358,



0.2170 0.1911 0.2143



,C =



0.2321 0.2233 0.3097





0.0243 0.0435 0.0219



, C



0.0339 0.0500 0.0254



F =



0.1408 0.1619 0.2045



, D = 0.2970, D = 0.3102.

It can be veriﬁed that this system is not robustly sta-

ble. We now apply the proposed approach to ﬁnd

a reduced-order dynamic output feedback controller

in (16) with r = 2 such that the closed-loop system is

positive, robustly asymptotically stable and ||G||

∞

1. One group of feasible solutions of the constrained

conditions in Theorem 3 is obtained as follows,





0.9972 0 0

0 1.2481 0

0 0 0.8515





, Q





1.0028 0 0

0 0.8012 0

0 0 1.1744







0.9733 0

0 0.9733



, Q



1.0275 0

0 1.0275





0.1724 0.1724



, L



0.0423





0.0363 0.0363



, L

= −3.8956.

The desired controller gain matrices A

, B

, C

and

are given by L

, L

, respectively. Figure 1

gives the maximal singular value plots of the closed-

loop system when d = 2, d = 50, and d = 0. It shows

clearly in this example that H

∞

norm of discrete-time

interval uncertain positive system with time delays is

independent of the delay magnitude.

−10 0 10

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Theta

Maximal singular value

d=2

−10 0 10

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

d=50

−10 0 10

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

d=0

Figure 1: Maximal singular value plots of the closed-loop

system.

On Dynamic Output Feedback H Control for Positive Discrete-Time Delay Systems

765

7 CONCLUSIONS

This paper has established the BRL for discrete-

time positive linear system with multiple time delays.

The proposed delay-independent criteria results re-

veal that H

∞

performance of positive systems with

time delays in state and output equations is equivalent

to the characterization of the corresponding delay-

free systems. The necessary and sufﬁcient conditions

in the forms of matrix (in)equalities are established

for the H

∞

control problem via dynamic output feed-

back controls, which can be easily solved by using

Matlab toolbox, although the proposed approach is

not guaranteed to ﬁnd a feasible solution even it ex-

ists.

ACKNOWLEDGEMENTS

This work is supported by the Alexander von Hum-

boldt Foundation of Germany.

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