Stabilization of positive linear discrete-time systems by using a Brauer's theorem

Research Article Stabilization of Positive Linear Discrete-Time Systems by Using a Brauer’s Theorem Begoña Cantó,1Rafael Cantó,1and Snezhana Kostova2 1Institut de Matem` atica Multidisciplinar, Universitat Polit` ecnica de Val` encia, 46071 Val` encia, Spain 2Institute of System Engineering and Robotics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Correspondence should be addressed to Bego˜ na Cant´ o; bcant[email protected]v.es Received 16 June 2014; Accepted 29 July 2014; Published 11 August 2014 AcademicEditor:R.Sakthivel Copyright © 2014 Bego˜ na Cant´ o et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The stabilization problem of positive linear discrete-time systems (PLDS) by linear state feedback is considered. A method based on a Brauer’s theorem is proposed for solving the problem. It allows us to modify some eigenvalues of the system without changing the rest of them. The problem is studied for the single-input single-output (SISO) and for multi-input multioutput (MIMO) cases and sufficient conditions for stability and positivity of the closed-loop system are proved. The results are illustrated by numerical examples and the proposed method is used in stochastic systems. 1. Introduction Positive systems are used to model many applications fields such as biology, chemistry, ecology, economy, and sociology (see [1] and the references therein). These systems have the peculiar property that any nonnegative input and nonnegative initial state generate a nonnegative state trajectory and a nonnegative output for all time. The positivity of the variables often emerges as the immediate consequence of the nature of modeled process, such as any variable representing a different type of resource like time, money and goods, buffer size and queues, data packets flowing in a network, water and air flows, populations, concentration of any substance, electric charge, and light intensity levels. For a good introduction to the theory of positive systems see [2,3]. Many well-established results for general linear systems cannot be directly applied to positive systems. This feature makes the study of positive systems very interesting and many resultshavebeenobtainedinthisareabothforcontinuous and for discrete-time systems [4,5]. The stability property is fundamental to the correct functioning of any control system and in particular of positive control systems. Stabilization of linear systems using feedback has attracted considerable interest during the last decades. Variousapproacheshavebeenusedtostudytheaspectsofthe stabilization problem, namely, the condition under which the linear system described in the state-space can be stabilized via feedback. For instance, some fundamental results on stabilitypropertyoflinearsystemsaregivenin[6–8]. When considering the stabilization problem of positive systems, the additional conditions exist on the feedback, ensuring that the closed system remains positive. In this case, it is possible that the generated feedback can be entrywise nonpositive. For example, in biology a nonnegative input means that the species can never be killed, and this situation is not realistic [9]. This problem has been studied by several authors, but it is not completely solved yet. For instance, in [10]itis considered for discrete-time periodic linear systems and in [11] for positive switched systems. The goal of this paper is to propose a new method for stabilization of positive unstable linear discrete-time systems, maintaining its positivity, that is, to use linear state feedback such that the eigenvalues of the closed-loop system have magnitude less than one. For that, we use a Brauer’s theorem to modify some eigenvalues of the state matrix without changing any of the remaining eigenvalues (see [12,13]and the references given there). The obtained results on stability property are applied to stochastic matrices to construct a closed-loop system whose eigenvalues are less than one and it maintains the positivity. Stochastic systems are included in systems theory that deals Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 856356, 6 pages http://dx.doi.org/10.1155/2014/856356 2The Scientific World Journal withdynamicaswellasstaticsystems,whoseprocesses are characterized by probability distributions or spectral measures. These systems can be modeled by a discrete-time system where the matrix 𝐴has a stochastic structure and it canbeused,forexample,tomodeltheevolutionofnondeterministic events. Moreover, it is known that stochastic matrices play an important role in economic models; see [14]. Some results concerning eigenvalues of stochastic matrices and their applications to nonnegative matrices are presented in [15]. The development of the stochastic stability theory is based on Markov parameters and on a Lyapunov approach (see more information in [16] and references therein). The paper is organized as follows. In Section 2 we present the main results for the stability and the positivity of the single-input single-output (SISO) system. In Section 3 we extend the SISO results for a particular multi-input multioutput (MIMO) system. In Section 4 we show an application to stochastic systems. Finally, in Section 5 concluding remarks and some perspectives are given. 2. Stabilization of SISO PLD System We consider a SISO positive linear discrete-time system 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝑏𝑢(𝑘),𝑘∈Z+,(1) where 𝑥(0)≥0,𝐴=(𝑎𝑖𝑗)∈R𝑛×𝑛 +,𝑏=(𝑏𝑖)∈R𝑛×1 +,𝑥(𝑘)is the state vector, and 𝑢(𝑘)is the control vector. This system can be represented by the pair of matrices (𝐴,𝑏). It is known that the linear discrete-time system (1)is asymptotically stable if and only if the dominant eigenvalue 𝜌=𝜌(𝐴)is smaller than 1.If𝜌>1, then the system is unstable and if 𝜌=1thesystemissaidtobemarginallystable. We consider a positive system (1)with𝜌≥1.Ourgoal is to construct a state feedback vector 𝑓∈R𝑛×1 such that 𝑢(𝑘)=𝑓𝑇𝑥(𝑘)and the closed-loop system 𝑥(𝑘+1)=(𝐴+𝑏𝑓𝑇)𝑥(𝑘)=𝐴𝑐𝑥(𝑘)(2) is positive; that is, 𝐴𝑐≥0and is asymptotically stable; that is, 𝜌(𝐴𝑐)<1. For solving the above problem, we use Brauer’s theorem [17] that shows how to modify one single eigenvalue of 𝐴 using a rank-one perturbation without changing any of the remaining eigenvalues. Theorem 1 (see [12,13,17,18]). Let 𝐴be an 𝑛×𝑛arbitrary matrix with eigenvalues 𝜎(𝐴)={𝜆1,𝜆2,...,𝜆𝑛}.Let𝑥𝑘be an eigenvector of 𝐴associated with the eigenvalue 𝜆𝑘,andlet𝑞 be any 𝑛-dimensional vector. Then the matrix 𝐴+𝑥𝑘𝑞𝑇has eigenvalues {𝜆1,...,𝜆𝑘−1,𝜆𝑘+𝑥𝑇 𝑘𝑞,𝜆𝑘+1,...,𝜆𝑛}. Remark 2. Note that if 𝑥𝑇 𝑘𝑞=0then the eigenvalues of the updated matrix 𝐴+𝑥𝑘𝑞𝑇are the same as the initial matrix 𝐴. In [12,13,18] several applications of the above theorem are presented concerning the stabilization, nonnegative inverse eigenvalue, and pole assignment problems for SISO systems. Proposition 3 (see [12,proposition2.4]). Consider the pair (𝐴,𝑏)that represents a single-input single-output linear time invariant control system. Let 𝜎(𝐴)={𝜆1,𝜆2,...,𝜆𝑛}and let 𝑥𝑘be an eigenvector of 𝐴𝑇associated with 𝜆𝑘.If𝑏𝑇𝑥𝑘= 0, then there exists a vector 𝑓such that 𝜎(𝐴+𝑏𝑓𝑇)= {𝜆1,...,𝜆𝑘−1,𝜆𝑘+𝑥𝑇 𝑘𝑏,𝜆𝑘+1,...,𝜆𝑛}. Remark 4. By Remark 2,if𝑥𝑇 𝑘𝑏=0then the eigenvalues do not change and the closed-loop system with 𝐴𝑐=𝐴+𝑏𝑓𝑇 hasthesamestabilitypropertyastheinitialsystem(1). From the theory of nonnegative matrices [19], the PerronFrobenius theorem states that if 𝐴(the same for 𝐴𝑇)isa nonnegative matrix, then it has a nonnegative eigenvalue 𝜌= 𝜌(𝐴)=𝜌(𝐴𝑇), that is, the Perron root, which is greater than or equal to the modulus of each of the other eigenvalues, and its corresponding eigenvector V=V𝜌∈R𝑛×1,which is referred to as the Perron-Frobenius eigenvector of 𝐴,is also nonnegative. Furthermore, if 𝐴is irreducible then 𝜌is positive and the entries of Vare strictly positive. Next theorem studies the stabilization problem for a SISO PLDsystem(1)givenbythepair(𝐴,𝑏).Notethat𝑏𝑇V=V𝑇𝑏 = 0is a consequence of Remark 4. Theorem 5. Consider the pair (𝐴,𝑏)that represents a SISO PLD system. Let 𝜎(𝐴)={𝜌,𝜆2,...,𝜆𝑛}and suppose that 𝜌≥1 and |𝜆𝑖|<1,𝑖=2,...,𝑛.LetV≥0be an eigenvector of 𝐴𝑇 associated with 𝜌.If (a) 𝑏=𝑏𝑇V=0, (b) for each 𝑎𝑖𝑗 =0the corresponding 𝑏𝑖=0,for𝑖,𝑗= 1,2,...,𝑛, (c) 𝛼∈]𝑀1,𝑀2[where 𝑀1=max {−𝜌 𝑏,max 𝑏𝑖V𝑗=0 {−𝑎𝑖𝑗 𝑏𝑖V𝑗}},𝑀 2=1−𝜌 𝑏,(3) then the control 𝑢(𝑘)=𝑓𝑇𝑥(𝑘)with 𝑓=𝛼Vmakes the closedloop system positive and asymptotically stable. Proof. Let 𝜇=𝜌+𝛼𝑏,with𝑏=𝑏𝑇V=0,betheneweigenvalue of 𝐴𝑐. From condition (c), if 𝛼>𝑀1,itfollowsthat𝛼>−𝜌/𝑏; then 𝜇>−1. Thus, the closed-loop system is asymptotically stable. Nowwewillprovethepositivenessofthesystem.Thezero entries of 𝐴do not change by condition (b). As 𝛼>𝑀1, 𝛼>max 𝑏𝑖V𝑗=0 {−𝑎𝑖𝑗 𝑏𝑖V𝑗}(4) and this implies that for every ℎand 𝑘such that 𝑏ℎV𝑘=0(and hence they are both positive) we have 𝑎ℎ𝑘 +𝑏ℎ𝛼V𝑥>𝑎ℎ𝑘 +𝑏ℎmax 𝑏𝑖V𝑗=0 {−𝑎𝑖𝑗 𝑏𝑖V𝑗}V𝑘 ≥𝑎ℎ𝑘 +𝑏ℎ{−𝑎ℎ𝑘 𝑏ℎV𝑘}V𝑘=0. (5) The Scientific World Journal 3 Thisprovesthat𝑎ℎ𝑘 +𝑏ℎ𝛼V𝑥is a positive entry because 𝑏ℎ and V𝑘are both positive. On the other hand, if either 𝑏ℎor V𝑘 is zero, then 𝑎ℎ𝑘 +𝑏ℎ𝛼V𝑥=𝑎ℎ𝑘 ≥0. Therefore, the closed-loop system is positive and asymptotically stable. Remark 6. The assumption that 𝑏has a zero entry in every position in which 𝐴hasarowcontainingazeroentrymeans that in this position the matrix 𝐴is not perturbed and its values are not modified. An application is given in [20]. Remark 7. If the matrix 𝐴is irreducible the condition (a) from Theorem 5 is unnecessary since, for V>0and 𝑏≥0, 𝑏=𝑏𝑇Vis always different from 0. By Theorem 5 we introduce the next procedure to obtain the value of 𝛼and the corresponding feedback vector 𝑓=𝛼V. Procedure Input: (𝐴𝑇,𝑏). (1) Obtain the spectral radius 𝜌of 𝐴𝑇and let Vbe a nonnegative eigenvector of 𝐴𝑇associated with 𝜌. (2) If 𝜌<1, then the pair (𝐴,𝑏)is asymptotically stable. END. (3) Otherwise, if 𝑏=𝑏𝑇V=0, then this algorithm cannot be applied to stabilize the system. END. (4) For 𝑖=1,2,...,𝑛,checkif𝑏𝑖=0when there exists an element 𝑎𝑖𝑗 =0for 𝑗=1,...,𝑛.Otherwise,the feedback matrix 𝐴𝑐is not positive. END. (5) Calculate 𝑑𝑖𝑗 =−𝑎𝑖𝑗/𝑏𝑖V𝑗for all (𝑖,𝑗)such that 𝑏𝑖V𝑗=0. 𝐷=max(𝑑𝑖𝑗). (6) Calculate 𝐸=−𝜌/ 𝑏. Consider 𝑀1=max(𝐷,𝐸), 𝑀2=(1−𝜌)/𝑏. (7) If 𝑀1≥𝑀2, then the closed-loop matrix 𝐴𝑐is not positive. END. (8) Otherwise, choose 𝛼∈]𝑀1,𝑀2[. Consider 𝑓=𝛼V and 𝐴𝑐=𝐴+𝑏𝑓𝑇. Example 8. Consider the pair (𝐴,𝑏)that represents a SISO PLD system 𝐴=(0.5 0 0.6 0.60.81.2 0.8 1 0.8), 𝑏=(011). (6) Next, we follow the procedure as follows. (1-2) Consider 𝜎(𝐴) =𝜎(𝐴𝑇)={2.1458,0.3542,−0.4}; then 𝜌=2.1458and an associated eigenvector Vof 𝐴𝑇is given by V=(0.7570,0.7430,1)𝑇.Since𝜌>1, then the pair (𝐴,𝑏)is not asymptotically stable. (3) Consider 𝑏=𝑏𝑇V=1.7430 =0; then the algorithm canbeappliedtostabilizethesystem. (4) For 𝑎12 =0,𝑏1=0. (5-6) Consider 𝑑21 =−0.7926,𝑑22 =−1.0767,𝑑23 =−1.2, 𝑑31 =−1.0568,𝑑32 =−1.3459,𝑑33 =−0.8,𝐷= 𝑑21 =−0.7926,𝐸=−1.2311,𝑀1=−0.7926,and 𝑀2=−0.6574. (7-8) Since 𝑀1<𝑀2,𝛼∈]−0.7926,−0.6574[. Consider 𝑓=𝛼Vand 𝐴𝑐=𝐴+𝑏𝑓𝑇. Now we consider, for instance, 𝛼=−0.7.Inthiscase 𝑓=(−0.5299,−0.5201,−0.7)𝑇(7) and the closed-loop matrix is 𝐴𝑐=(0.5 0 0.6 0.07010.27990.5 0.27010.47990.1)≥0, (8) with 𝜎(𝐴𝑐)={0.9257,0.3542,−0.4}.Then,theclosed-loop system is positive and asymptotically stable. As it is seen from the above theorem and corresponding procedure we can select any value of the parameter 𝛼in the interval ]𝑀1,𝑀2[. The question which arises is how to use this freedom of choice to improve system behavior. In the context of our main goal to improve stability of the system, good idea is to choose parameter 𝛼in such way that it maximizes the stability radii of closed-loop system. The concept of stability radii is developed in [21,22]. The stability radii measure the robustness of stability of the systems under perturbations with structure, defined by the matrices 𝐷and 𝐸, and it is defined by 𝑟𝐾(𝐴)=𝑟𝐾(𝐴,𝐷,𝐸) =inf {‖Δ‖:Δ∈𝐾𝑙×𝑞,𝜌(𝐴+𝐷Δ𝐸)≥1}, (9) where ‖⋅‖is given operator norm and 𝐾=𝑅,𝐶. If complex perturbations are allowed, the complex stability radius is obtained and is denoted by 𝑟𝐶.Ifonly real perturbations are considered the real stability radius is obtained and it is denoted by 𝑟𝑅.Therealandcomplexstability radii are in general distinct and their computation is difficult problem. Fortunately for positive systems the real and complex stability radii coincide (𝑟𝑅=𝑟𝐶=𝑟)and they can be determined via an easy computable formula. Proposition 9 (see [23]). Suppose that (𝐴,𝐷,𝐸)∈R𝑛×𝑛 +× R𝑞×𝑙 +×R𝑞×𝑛 +,𝜌(𝐴)<1,and𝐾𝑙and 𝐾𝑞are provided with monotonic norms, 𝐾=𝑅or 𝐶.Then 𝑟𝐶(𝐴;𝐷;𝐸)=𝑟𝑅(𝐴;𝐷;𝐸) =𝑟(𝐴;𝐷;𝐸)=‖𝐺(1)‖−1,(10) where 𝐺(𝑠)=𝐸(𝑠𝐼−𝐴)−1𝐷and ‖𝐺(1)‖is the operator norm of 𝐺(1):R𝑙→R𝑞. Let J∗denotes the set of all state feedbacks preserving Shur stability and positivity of the closed-loop system with: J∗={𝐹∈R𝑚×𝑛 :𝐴+𝐵𝐹≥0,𝜌(𝐴+𝐵𝐹)<1}. (11) The following proposition is proved in [21]. 4The Scientific World Journal Proposition 10. The map 𝑟(𝐹):R𝑚×𝑛 →R+is continuous and monotone on J∗;thatis,𝐹1≤𝐹2⇒𝑟(𝐹1)≥𝑟(𝐹2). Hencewecanusefreedomtochooseetheparameter𝛼to maximize the stability radii of the system; in other words, the closed-loop system matrix 𝐴𝑐will have minimal sensitivity to theaffinesystemperturbationtype𝐴→𝐴(Δ)=𝐴+𝐷Δ𝐸. From Theorem 5 it is clear that the parameter 𝛼is negative and, according to Proposition 10, if we chose the parameter close to the left bound of the interval, the stability radii will be larger, the sensitivity of the system to perturbation will be better, and the robustness of the system will be guaranteed. 3. Stabilization of MIMO PLD System In this section we consider the MIMO PLD system with only oneeigenvaluegreaterthanorequalto1and we apply the obtained result from SISO systems in this case. Let the MIMO PLD system 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑢(𝑘),𝑘∈Z+,(12) where 𝑥(0)≥0,𝐴=(𝑎𝑖𝑗)∈R𝑛×𝑛 +,and𝐵=(𝑏𝑖𝑗)∈R𝑛×𝑚 +;𝑥(𝑘) is the state vector and 𝑢(𝑘)is the control vector. This system canberepresentedbythepairofmatrices(𝐴,𝐵). We consider a positive system (12) with its spectral radius 𝜌=𝜌(𝐴)≥1. Our goal is to construct a state feedback matrix 𝐹∈R𝑚×𝑛 such that 𝑢(𝑘)=𝐹𝑇𝑥(𝑘)and the closedloop system 𝑥(𝑘+1)=(𝐴+𝐵𝐹𝑇)𝑥(𝑘)=𝐴𝑐𝑥(𝑘)(13) is positive; that is, 𝐴𝑐≥0and is asymptotically stable; that is, 𝜌(𝐴𝑐)<1. From now on, we denote by 𝑒𝑖=(0,...,0,1,0,...,0)𝑇∈ R𝑚×1 the 𝑖th canonical vector with a 1in the 𝑖th coordinate and 0’s elsewhere. The next theorem follows directly from Theorem 5. Theorem 11. Consider the pair (𝐴,𝐵)that represents a MIMO PLD system. Let 𝜎(𝐴)={𝜌,𝜆2,...,𝜆𝑛}and suppose that 𝜌≥1 and |𝜆𝑖|<1,𝑖=2,...,𝑛.Let𝑥≥0be an eigenvector of 𝐴𝑇 associated with 𝜌.Let𝐵𝑖be the 𝑖th column of 𝐵,𝑖=1,2,...,𝑚. If there exists at least one column 𝐵𝑘of 𝐵such that (a) 𝐵𝑘=𝐵𝑇 𝑘V=0, (b) for each 𝑎𝑖𝑗 =0the corresponding 𝑏𝑖𝑘 =0,for𝑖,𝑗= 1,2,...,𝑛, (c) 𝛼𝑘∈]𝑀1,𝑀2[,where 𝑀1=max {−𝜌 𝐵𝑘,max 𝑏𝑘𝑖V𝑗=0 {−𝑎𝑖𝑗 𝑏𝑘𝑖V𝑗}}, 𝑀2=1−𝜌 𝐵𝑘,(14) then the control 𝑢(𝑘)=𝐹𝑇𝑥(𝑘)with 𝐹=𝛼𝑘V𝑒𝑇 𝑘,makesthe closed-loop system positive and asymptotically stable. Example 12. Consider the pair (𝐴,𝐵)that represents a MIMO PLD system 𝐴=(0.5 0 0.6 0.60.81.2 0.8 1 0.8), 𝐵=(010 100.5 110.8). (15) Note that 𝐴is the same as in Example 8.Now,wecheck the conditions of the theorem for each column of 𝐵. (i) For 𝐵1we consider the results obtained in Example 8. Then 𝛼1∈]−0.7926,−0.6574[: 𝐹=𝛼1V𝑒𝑇 1=𝛼1(0.7570 0.7430 1)(1,0,0) =𝛼1(0.757000 0.743000 100 )(16) and 𝐴𝑐=𝐴+𝐵𝐹𝑇,with𝐴𝑐≥0and 𝜌(𝐴𝑐)<1. (ii) For 𝐵2,since𝑎12 =0but 𝑏12 =0, then the procedure cannot be applied to stabilize the system using this column of 𝐵. (iii) For 𝐵3, using the procedure for SISO PLD systems we obtain that 𝛼3∈]−1,−0.9781[. For this example, we conclude that there exist two columns of 𝐵such that the conditions of Theorem 11 are satisfied and different feedback matrices can be obtained for stabilization of the system in such way that positiveness is guaranteed. 4. Application: Stochastic Systems Stochastic systems are becoming extensively used as realistic models of physical phenomena. They are at the core of a number of disciplines in engineering, social systems, markets, management actions, molecular biology, and epidemiology; see, for example, [24]. The theory of Markov processes comprises the largest and the most important part of the theory of stochastic processes. This is well recognized in the theory of queues and in branching processes [25]. Different problems concerning modeling, analysis, synthesis, and simulation of stochastic systems can be found in the literature. On stability property, in this section we apply the obtained results in previous sections to stochastic systems in order to obtain a closed-loop system asymptotically stable and positive. For that, we consider the system (1)where𝐴is a nonnegative left-stochastic matrix. A left stochastic matrix 𝐴is square with nonnegative elements satisfying 𝑛 ∑ 𝑖=1𝑎𝑖𝑗 =1, 𝑗=1,2,...,𝑛. (17) The Scientific World Journal 5 Note that the system is not asymptotically stable, because 𝜌=1is an eigenvalue of 𝐴.Thus,𝜎(𝐴)=𝜎(𝐴𝑇)={1,𝜆𝑖}, being |𝜆𝑖|<1,𝑖=2,3,...,𝑛,andV=(1,1,...,1)𝑇is an eigenvector of 𝐴𝑇associated with 𝜌=1. The following result is a consequence of Theorem 5 where the condition (a) is fulfilled because 𝑏=𝑏𝑇V=𝑏1+𝑏2+⋅⋅⋅+ 𝑏𝑛=0. Proposition 13. Consider the pair (𝐴,𝑏)that represents a SISO PLD system, where 𝐴is a left-stochastic matrix. Let 𝜎(𝐴)={1,𝜆2,...,𝜆𝑛}and let V=(1,1,...,1)𝑇be an eigenvector of 𝐴𝑇associated with 𝜌=1.If (b󸀠)for each 𝑎𝑖𝑗 =0the corresponding 𝑏𝑖=0,for𝑖,𝑗= 1,2,...,𝑛, (c󸀠)𝛼∈]𝑀,0[,where 𝑀=max {−2 𝑏,max 𝑏𝑖=0 {−𝑎𝑖𝑗 𝑏𝑖}}, (18) then the feedback law 𝑢(𝑘)=𝑓𝑇𝑥(𝑘)with 𝑓=𝛼Vmakes the closed-loop system positive and asymptotically stable. Note that the above proposition is useful when 𝐴is also a doubly stochastic matrix. Corollary 14. For a MIMO PLD system (12),where𝐴is a left-stochastic matrix or 𝐴is a double stochastic matrix with only one eigenvalue equal to 1,wecombinetheresultsgivenin Theorem 11 and Proposition 13 as the following examples show. Example 15. Consider the pair (𝐴,𝑏)that represents a SISO PLD system, where 𝐴and 𝑏are given by 𝐴=(0.10.20.30.4 0.7 0 0 0.1 0.20.80.20.3 0 0 0.50.2), 𝑏=(0.1 0 0.3 0). (19) In this case 𝐴is a left-stochastic matrix and 𝜎(𝐴)=𝜎(𝐴𝑇)= {1,−0.3454,−0.0773+0.4131𝑖,−0.0773−0.4131𝑖}. Note that 𝑎22 =𝑎23 =0and 𝑏2=0;and𝑎41 =𝑎42 =0and 𝑏4=0.Thatis,thecondition(b 󸀠)ofProposition 13 is satisfied. Moreover, 𝑏=0.4,−2/𝑏=−5and max{−𝑎𝑖𝑗/𝑏𝑖}=−𝑎31/𝑏3= −𝑎33/𝑏3=−2/3and −23<𝛼<0. (20) If, for instance, 𝛼=−0.5,then𝑓=−0.5𝑒, 𝐴𝑐=𝐴+𝑏𝑓𝑇=(0.050.150.250.35 0.7 0 0 0.1 0.050.650.050.15 0 0 0.5 0.2)≥0, (21) and 𝜎(𝐴𝑐) = {0.8,−0.3454,−0.0773+0.4131𝑖,−0.0773− 0.4131𝑖}. Then the closed-loop system is positive and asymptotically stable. Now, we consider the system (1)wherethedoublystochastic matrix 𝐴and the vector 𝑏are given by 𝐴=(0.50.20.3 0.50.40.1 0 0.40.6), 𝑏=(0.1 0.3 0). (22) We have 𝜎(𝐴)=𝜎(𝐴𝑇)={1,0.2500+0.1936𝑖,0.2500− 0.1936𝑖}.Then,thecondition(b 󸀠)ofProposition 13 is satisfied. In this case, 𝑏=0.4,−2/𝑏=−5,andmax{−𝑎𝑖𝑗/𝑏𝑖}= −𝑎23/𝑏2=−1/3.Then−1/3<𝛼<0. If we consider, for instance, 𝛼=−0.1,then𝑓=−0.1𝑒, 𝐴𝑐=(0.490.190.29 0.470.370.07 0 0.4 0.6)≥0, (23) and 𝜎(𝐴𝑐)={0.96,0.2500+0.1936𝑖,0.2500−0.1936𝑖}.Then the closed-loop system is positive and asymptotically stable. 5. Conclusions Motivated by results obtained in [12] concerning application of Brauer’s theorem for solving several control problems, we have considered a stabilization problem for both SISO and MIMO positive linear discrete time system by using linear statefeedback.Wehavegivensimplesufficientconditionsfor the system matrices ensuring solvability of the stabilization problem. In this paper we consider the case when the only spectral radius is greater or equal to 1.Wehaveshownthat several choices for feedback matrix are possible according to the parameter 𝛼. That fact gives us possibility to satisfy other requirements on system behavior. In the future the same problems will be solved for the case when more than oneeigenvalueisgreaterthanorequalto1.InthiscaseRado’s theorem will be used, which is extension of Brauer’s theorem. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. 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