# H

Authors: Youmei Zhang; Qingling Zhang; Tamaki Tanaka
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

This paper is concerned withH∞ control for linear positive discrete-time systems. Positive systems are characterized by nonnegative restriction on systems’ variables. This restriction results in some remarkable results which are available only for linear positive systems. One of them is the celebrated diagonal positive definite matrix solutions to some existed well-known results for linear systems without nonnegative restriction. We provide an alternative proof for criterion of H∞ norm by using separating hyperplane theorem and Perron-Frobenius theorem for nonnegative matrices. We also consider H∞ control problem for linear positive discrete-time systems via state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions.


---

## Body

## 1. Introduction

Positive systems are widespread in many practical systems, such as economic systems [1], biology systems [2], and age-structured population models [3], whose variables are required to be nonnegative and have no meaning with negative values. The explicit definition of a positive system is that its state and output are always nonnegative for any nonnegative initial state and any nonnegative input. Due to the nonnegative restriction on systems’ variables, positive systems are defined on cones rather than linear space. Hence, there are excellent and remarkable outcomes which are available only for positive systems. One of them is the existence of diagonal positive definite matrix solutions to some celebrated results for linear systems without nonnegative restriction. Therefore, the investigation of positive systems is interesting and challenging and developed a new branch in systems theory. Positive systems have been of great interest to many researchers over several decades. A great number of results have been reported in the literature; see, for instance, [3–16].It is worth noting that convex optimization is a powerful tool for analysis of positive systems. In [9–14], some remarkable results for positive systems are studied using convex optimization. On the other hand, the problem of H∞ control has been a topic of recurring interest for several decades. A great number of results on H∞ control have been obtained, and different approaches have been proposed. In recent years, increasing attention has been paid to H∞ norm analysis for positive systems. In [12], the KYP lemma for linear positive continuous-time systems is proved based on the semidefinite programming duality. The alternative proofs along the line of the rank-one separable property are given to several remarkable and peculiar results for positive systems in [13]. In [14], the KYP lemma for linear positive discrete-time systems is studied using a theorem of alternatives on the feasibility of linear matrix inequalities (LMIs).This paper is organized as follows. Preliminaries are introduced in Section2. Main results on H∞ control for linear positive discrete-time systems are presented and proved in Section 3. Section 4 is devoted to illustrate the effectiveness of the obtained results by numerical examples. Section 5 concludes this paper.


## 2. Preliminaries

In this section, we introduce terminology, positive systems, various other definitions, and lemmas, which will be essentially used for proving our main results.At first, the following notations will be used throughout this paper.ℤ
+ denotes the nonnegative integers. x≥0 denotes the vector x∈ℝn with nonnegative entries. A≥0 denotes the matrix A∈ℝn×m with nonnegative entries. ℝ+n denotes the set of all vectors x∈ℝn with x≥0. ℝ+n×m denotes the set of all matrices A∈ℝn×m with A≥0. Sn denotes the set of all symmetric matrices. D+n×n denotes the set of all diagonal positive definite matrices. A⪰0, A≻0, and A≺0 mean that A is a positive semidefinite, positive definite, and negative definite matrix, respectively. Aij denotes the ijth entry of matrix A. xi denotes the ith entry of vector x. For two matrices A,B∈ℝn×m, A≥B means Aij≥Bij, i=1,2,…,n, j=1,2,…,m. For two vectors x,y∈ℝn, x≥y means xi≥yi, i=1,2,…,n. ρ(A) denotes the spectral radius of matrix A which is defined as ρ(A):=max1≤i≤n|λi|, where λ1,λ2,…,λn are the eigenvalues of A. D(A) denotes the vector which is composed of the diagonal entries of A∈ℝn×n. 〈X,Y〉=trace(XY) is the inner product on Sn.Consider the following linear discrete-time system:(1)x(k+1)=Ax(k)+Bu(k),y(k)=Cx(k)+Du(k),
where x(k)∈ℝn is the state, u(k)∈ℝm is the input, and y(k)∈ℝp is the output. A∈ℝn×n, B∈ℝn×m, C∈ℝp×n, and D∈ℝp×m are known matrices.Definition 1 (see [3]).
System (1) is said to be positive if and only if x(k)≥0, y(k)≥0, k∈ℤ+ for any x(0)≥0 and for any u(k)≥0, k∈ℤ+.Definition 2 (see [3]).
System (1) is said to be asymptotically stable if ρ(A)<1.Definition 3 (see [13]).
For a given matrixH∈Sn with H⪰0, we define h∈ℝ+n by
(2)hi:=Hii,i=1,2,…,n.Lemma 4 (see [3], Perron-Frobenius theorem for nonnegative matrices).
LetA∈ℝ+n×n; then ρ(A) is an eigenvalue of A and A has a nonnegative eigenvector v corresponding to ρ(A).A matrixA∈ℝn×n is called a Metzler matrix if Aij≥0, for all i, j with i≠j.Lemma 5 (see [13]).
For a given Metzler matrixA∈ℝn×n and H∈Sn with H⪰0, the following conditions hold:(i)
(hhT)ii=Hii, (hhT)ij≥Hij, i≠j,(ii)
D(hhTA)≥D(HA),
whereh∈ℝ+n is defined from H as in Definition 3.Lemma 6 (see [3]).
System (1) is positive if and only if A≥0, B≥0, C≥0, and D≥0.Lemma 7 (see [17], separating hyperplane theorem).
Suppose thatC1 and C2 are two convex sets that do not intersect; that is, C1∩C2=⌀. Then, there exist a≠0 and b such that aTx≤b for all x∈C1 and aTx≥b for all x∈C2.ForA∈ℝ+n×n, we consider B:=sI-A. If s>0 and s≥ρ(A), then B is called an M-matrix. If s>ρ(A), then B is a nonsingular M-matrix.Lemma 8 (see [4]).
A nonsingular matrixA∈ℝn×n is an M-matrix if and only if A-1≥0.Lemma 9 (see [18], Schur complement).
Given any real matricesQ, S, and R with Q=QT and R=RT, the following statement holds:
(3)[QSSTR]≺0
if and only if
(4)R<0,Q-SR-1ST<0,
or, equivalently,
(5)Q<0,R-STQ-1S<0.The transfer function matrix of system (1) is given by
(6)G(z)=C(zI-A)-1B+D,
and its H∞ norm is defined as
(7)∥G∥∞:=supθ∈(-π,π]σ-(G(ejθ)),
where σ-(G(ejθ)) denotes the maximum singular value of G(ejθ). In [14], it has been pointed out that ∥G∥∞=∥G(1)∥, where ∥G(1)∥=σ-(G(1)), if system (1) is positive and asymptotically stable.


## 3.H∞ Control

In this section, we give an alternative proof for the existed result ofH∞ norm for positive discrete-time systems and investigate the H∞ control under state feedback.At first, we propose the following theorem which is helpful for the alternative proof.Theorem 10.
Suppose that system (1) is positive; the following conditions are equivalent:(i)
There exists a nonzeroh∈ℝ+n such that (A-I)h≥0.(ii)
System (1) is not asymptotically stable.Proof.
(i)⇒(ii). Since A≥0, from Perron-Frobenius theorem for nonnegative matrices, it follows that ATv=ρ(A)v≥0, where ρ(A)≥0 is the spectral radius of matrix AT and v≥0 is an eigenvector corresponding to ρ(A). Then it is obtained from condition (i) that
(8)(ρ(A)-1)vTh≥0,
which implies ρ(A)≥1; namely, system (1) is not asymptotically stable.
(ii)⇒(i). System (1) is not asymptotically stable; that is, ρ(A)≥1. From Perron-Frobenius theorem, we immediately obtain the following result:
(9)ρ(A)h=Ah≥h,
where h≥0 is an eigenvector corresponding to ρ(A). This completes the proof.The following theorem was firstly presented and proved in the light of alternatives on the feasibility of LMIs by Federico Najson in the literature [14]. Now we will give another proof using separating hyperplane theorem and Theorem 10.Theorem 11.
Suppose that system (1) is positive; the following conditions are equivalent.(i)
System (1) is asymptotically stable and ∥G∥∞<1.(ii)
There exists a diagonal positive definite matrixP such that
(10)[ATPA-P+CTCATPB+CTDBTPA+DTCBTPB+DTD-I]≺0.(iii)
There exists a diagonal positive definite matrixP such that
(11)[-P0ATPCT0-IBTPDTPAPB-P0CD0-I]≺0.Proof.
We only prove (i)⇒(ii) since the implication (ii)⇒(i) is obvious from the existed criterion for linear discrete-time systems, and the equivalence between (ii) and (iii) is immediately obtained using Schur complement.
To the contrary, suppose that condition (10) does not hold for any diagonal positive definite matrix P. Define the following two sets:
(12)C1:=([ATPA-P+CTCATPB+CTDBTPA+DTCBTPB+DTD-I]∣P∈D+n×n),C2:={Q∣Q≺0,Q∈Sn+m};
then it is easy to check that sets C1 and C2 are nonempty and convex. By the assumption, we have C1∩C2=⌀. Then from the separating hyperplane theorem, there exists a nonzero H∈Sn+m such that
(13)〈H,S〉≥0,∀S∈C1,(14)〈H,S〉≤0,∀S∈C2.
By condition (14), we can conclude that 〈H,S〉=trace(HS)≤0, for all S≺0, from which it is easy to verify that H⪰0. Thus it follows from condition (13) that there exists a nonzero H⪰0 such that
(15)trace(H[ATPA-P+CTCATPB+CTDBTPA+DTCBTPB+DTD-I])≥0,∀P∈D+n×n,
which is equivalent to
(16)trace(H[ATPA-PATPBBTPABTPB])+trace(H[CTCCTDDTCDTD-I])≥0,∀P∈D+n×n.
Let H:=[H11H12H12TH22], H11⪰0, H22⪰0; then the above condition can be rewritten as
(17)trace(H11(ATPA-P)+H12BTPA+H12TATPB+H22BTPB)+trace(H[CTCCTDDTCDTD-I])≥0,∀P∈D+n×n,
or, equivalently,
(18)trace(P(AH11AT-H11+AH12BT+BH12TAT+BH22BT))+trace(H[CTCCTDDTCDTD-I])≥0,∀P∈D+n×n,
which implies that(a)
D(AH11AT-H11+AH12BT+BH12TAT+BH22BT)≥0,(b)
trace(H[CTCCTDDTCDTD-I])≥0.From condition (a), it follows that
(19)D(AH11AT+AH12BT+BH12TAT+BH22BT)=D([AB00][H11H12H12TH22][AT0BT0])≥D((H11000)).
Since system (1) is positive, then
(20)[AB00],[CTCCTDDTCDTD-I]
are nonnegative matrix and Metzler matrix, respectively. Define the nonzero h∈ℝ+n from H as in Definition 3 then from Lemma 5, we obtain
(21)D([AB00]hhT[AT0BT0])≥D([AB00][H11H12H12TH22][AT0BT0])≥D((H11000)),(22)trace(hhT[CTCCTDDTCDTD-I])≥trace(H[CTCCTDDTCDTD-I])≥0.
Set h:=[h1Th2T]T, h1∈ℝ+n, h2∈ℝ+m; conditions (21) and (22) can be rewritten as
(23)D(Ah1h1TAT+Ah1h2TBT+Bh2h1TAT+Bh2h2TBT)≥D(h1h1T),(24)h1TCTCh1+h2TDTCh1+h1TCTDh2+h2TDTDh2=(Ch1+Dh2)T(Ch1+Dh2)≥h2Th2.
Since H is nonzero, then we have the following three cases.(1)
Consider thath1=0,h2≠0. By condition (24), h2TDTDh2≥h2Th2, which contradicts ∥G∥∞<1.(2)
Consider thath1≠0,h2=0. We observe from (23) that D(Ah1h1TAT)≥D(h1h1T), which implies that Ah1≥h1.From Theorem10, this is a contradiction.(3)
Consider thath1≠0,h2≠0. From condition (24), it yields that Ch1+Dh2≠0. Define matrix
(25)Δ=h2(Ch1+Dh2)T(Ch1+Dh2)T(Ch1+Dh2),which is well-defined and satisfies σ-(Δ)≤1. Note that h2=Δ(Ch1+Dh2). On the other hand, σ-(D)<1; otherwise, this contradicts ∥G∥∞<1. It is known that for matrices M∈ℂl×m and N∈ℂm×l with σ-(M)≤1 and σ-(N)<1, det(I-MN)≠0. Thus, I-ΔD is invertible and (I-ΔD)-1≥0 since I-ΔD is a nonsingular M-matrix. Therefore, h2=(I-ΔD)-1ΔCh1 holds. Then from (23) we have
(26)D(Ah1h1TAT+Ah1h2TBT+Bh2h1TAT+Bh2h2TBT)=D(Ah1h1TAT+Ah1h1TCTΔT(I-ΔD)-TBT+B(I-ΔD)-1ΔCh1h1TAT+B(I-ΔD)-1ΔCh1h1TCTΔT(I-ΔD)-TBT)=D((A+B(I-ΔD)-1ΔC)h1h1T(A+B(I-ΔD)-1ΔC)T)=D(A~h1h1TA~T)≥D(h1h1T),
where A~:=A+B(I-ΔD)-1ΔC≥0. From Theorem 10, ρ(A~)≥1, which contradicts ∥G∥∞<1.Corollary 12.
Suppose that system (1) is positive; then the following conditions are equivalent.(i)
System (1) is asymptotically stable and ∥G∥∞<γ.(ii)
There exists a diagonal positive definite matrixP such that
(27)[ATPA-P+CTCATPB+CTDBTPA+DTCBTPB+DTD-γ2I]≺0.(iii)
There exists a diagonal positive definite matrixP such that
(28)[-P0ATPCT0-γ2IBTPDTPAPB-P0CD0-I]≺0.Now, our purpose is to design a state feedback controller given by(29)u(k)=Kx(k)+v(k),
where K∈ℝm×n is the controller gain to be designed, and v(k)∈ℝ+n, such that the closed-loop system described as
(30)x(k+1)=(A+BK)x(k)+Bv(k),y(k)=(C+DK)x(k)+Dv(k),
is positive, asymptotically stable, and ∥G-∥∞<1, where G-(z)=(C+DK)(zI-(A+BK))-1+D.At first, we focus on nonnegative control gain, as it has practical importance in many cases. For instance, for a chemical system whose variables represent concentrations of reactants and reaction speed is impacted by concentrations, in order to improve the speed of reaction, it is natural to consider such controller for increasing concentrations.Theorem 13.
For the given positive system (1), there exists a nonnegative controller of the form in (29) such that the closed-loop system (30) is asymptotically stable and ∥G-∥∞<1 if and only if there exist X∈D+n×n and Y≥0 satisfying
(31)[-X0XAT+YTBTXCT+YTDT0-IBTDTAX+BYB-X0CX+DYD0-I]≺0.
Under the above condition, the desired nonnegative controller gain is obtained as
(32)K=YX-1.Proof.
Necessity. From Theorem 11, there exists a diagonal positive definite matrix P such that
(33)[-P0(A+BK)TP(C+DK)T0-IBTPDTP(A+BK)PB-P0C+DKD0-I]≺0.
Multiplying on both sides of inequality (33) by T=diag(P-1,I,P-1,I), it follows that
(34)[-P-10P-1AT+P-1KTBTP-1CT+P-1KTDT0-IBTDTAP-1+BKP-1B-P-10CP-1+DKP-1D0-I]≺0.
By defining X=P-1, Y=KP-1, inequality (31) is immediately obtained. On the other hand, since X∈D+n×n and Y≥0, it is easy to see that K≥0.
Sufficiency. Positivity of the closed-loop system (30) is obvious. From (32), Y=KX; substituting it into inequality (31) leads to
(35)[-X0XAT+XKTBTXCT+XKTDT0-IBTDTAX+BKXB-X0CX+DKXD0-I]≺0.
Multiplying on both sides of inequality (35) by T=diag(X-1,I,X-1,I), we have
(36)[-X-10(A+BK)TX-1(C+DK)T0-IBTX-1DTX-1(A+BK)X-1B-X-10C+DKD0-I]≺0.
Therefore, from Theorem 11, the closed-loop system (30) is asymptotically stable and ∥G-∥∞<1.Remark 14.
Under the assumption that system (1) is positive, it is worth noting that there does not exist any X∈D+n×n or Y≥0 satisfying condition (31) if there exists Aii>1, i=1,2,…,n. It is easy to verify in the light of the following facts.(1)
A linear positive discrete-time system is unstable if at least one diagonal entry of matrixA is greater than 1 which is presented in the literature [3].(2)
ρ(B)≤ρ(A) if 0≤B≤A which has been pointed out in the literature [19].
On the other hand, it is known that the maximal eigenvalueρ(A) of A≥0 belongs to the interval
(37)max{minci,minri}≤ρ(A)≤min{maxcimaxri},
where ci and ri denote the sum of the elements of the ith column and the ith row of matrix A, respectively. Therefore, there also does not exist X∈D+n×n or Y≥0 satisfying condition (31) if max{minci,minri}≥1.Remark 15.
If0≤B≤A, then σ-(A)≥σ-(B) which is obtained immediately due to the fact (2) in Remark 14. Suppose that system (1) is positive, ∥G∥∞≥1, and there exists a nonnegative controller (29) such that system (30) is asymptotically stable; it is obvious that system (1) is also asymptotically stable. Thus,
(38)∥G-∥∞=∥G-(1)∥=∥(C+DK)(I-(A+BK))-1B+D∥=∥(C+DK)(∑i=0∞(A+BK)i)B+D∥=σ-((C+DK)(∑i=0∞(A+BK)i)B+D)≥σ-(C(∑i=0∞Ai)B+D)=∥G∥∞≥1.
Therefore, if system (1) is positive, asymptotically stable, but ∥G∥∞≥1, there does not exist any nonnegative state feedback (29) such that system (30) is asymptotically stable and ∥G-∥∞<1.Corollary 16.
For the given positive system (1), there exists a nonnegative controller of the form in (29) such that the closed-loop system (30) is asymptotically stable and ∥G-∥∞<γ if and only if there exist X∈D+n×n and Y≥0 satisfying
(39)[-X0XAT+YTBTXCT+YTDT0-γ2IBTDTAX+BYB-X0CX+DYD0-I]≺0.
Under the above condition, the desired controller gain is obtained as (32).From Remarks14 and 15, for some positive systems, there is no nonnegative state feedback (29) such that system (30) is asymptotically stable and ∥G-∥∞<1. Hence, we are obligated to pay attention to state feedback without nonnegative restriction. It is also natural to consider such controller. For example, for an ecosystem whose variables represent population of animals in a forest; population cannot exceed the ecological capacity of the forest, otherwise, the ecosystem may be destroyed. Therefore, we must decrease the number of animals by means of harvesting or other methods when population of certain animals exceeds their ecological capacity.Now we are looking for a state feedback without nonnegative restriction having form in (29) such that the closed-loop system (30) is positive, asymptotically stable, and ∥G-∥∞<1.Theorem 17.
For the given positive system (1), there exists a controller of the form in (29) such that the closed-loop system (30) is positive, asymptotically stable, and ∥G-∥∞<1 if and only if there exist X∈D+n×n and Y∈ℝm×n satisfying (39) and
(40)AX+BY≥0,(41)CX+DY≥0.
Under the above conditions, the desired controller gain is given by (32).Proof.
We only prove conditions (40) and (41).
Necessity. The close-loop system (30) is positive then it follows that
(42)A+BK≥0,C+DK≥0.
Since X is a diagonal positive definite matrix, then it is easy to check that
(43)AX+BKX≥0,CX+DKX≥0.
Conditions (40) and (41) are immediately obtained.
Sufficiency. The proof is similar to necessity.Corollary 18.
For the given positive system (1), there exists a controller of the form in (29) such that the closed-loop system (30) is positive, asymptotically stable, and ∥G-∥∞<γ if and only if there exist X∈D+n×n and Y∈ℝm×n satisfying conditions (39), (40), and (41). Under these conditions, the desired controller gain is obtained as (32).


## 4. Examples

In this section, we give some examples to illustrate the validity of the obtained results.Example 1.
Consider a linear positive discrete-time system with the following parameter matrices:(44)A=[0.604800.08610.4187],B=[0.095200.04570.1813],C=[1001],D=[0.0921000.1286].
Solving the conditions in Theorem 13, one feasible solution is obtained as
(45)X=[0.3033000.4477],Y=[0.08540.11420.07430.1150].
Then the desired controller gain is given by
(46)K=[0.28160.25510.25510.2551].Example 2.
Consider a linear positive discrete-time system with the following parameter matrices:(47)A=[1.214800.08610.9187],B=[0.17020.0457],C=[0.8160],D=0.3157.
We observe thatA11>1 and check the condition in Theorem 13; as pointed out in Remark 14, there is no feasible solution.
Then solving the conditions in Theorem17, one feasible solution is obtained as
(48)X=[2.174700810.9260],Y=[-4.00625.0132].
The desired controller gain is
(49)K=[-1.84210.0062].Example 3.
Consider a linear positive discrete-time system with the following parameter matrices:(50)A=[0.91870.12950.08610.9187],B=[0.17020.1570],C=[0.57000.3850],D=0.2103.By calculating,max{minci,minri}=1.0048. We check the condition in Theorem 13; there is no feasible solution.Solving the conditions in Theorem17, one feasible solution is obtained as
(51)X=[0.4315001.6751],Y=[-0.2126-1.2241].Then the controller gain is given by(52)K=[-0.4926-0.7308].


## 5. Conclusion

In this paper, we are interested inH∞ control for linear positive discrete-time systems. We present a necessary and sufficient condition to check the stability of linear positive discrete-time systems using Perron-Frobenius theorem for nonnegative matrices, which is the key point for the alternative proof. We believe that it is useful for checking the existence of diagonal positive definite matrices to some other results given for linear discrete-time systems without nonnegative restriction. The alternative proof is along the line of separating hyperplane theorem and Theorem 10 given in this paper. In addition, we investigate the H∞ control for such systems under state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions. However, in this paper, we have restricted our attention to the case of state feedback, but in practice, it is not always possible to have access to all of the state variables. The case of static and dynamic output feedback is left for future research. Robust H∞ analysis and synthesis for positive uncertain systems are also open problems.



---

*Source: 576379-2013-09-11.xml*