A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input

Nonlinear Analysis: Hybrid Systems 29 (2018) 322–332

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input In Seok Park, Nam Kyu Kwon, PooGyeon Park * Department of Electrical Engineering, Pohang University of Science and Technology, Pohang, Gyungbuk 790-784, Republic of Korea

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Article history: Received 25 July 2017 Accepted 2 March 2018

Keywords: Linear programming Positive Markovian jump systems Input saturation State-feedback control

a b s t r a c t This paper proposes a linear programming approach for stabilization of positive Markovian jump systems (PMJSs) with a saturated single input. The proposed approach first derives a sufficient condition for stabilization of PMJSs with input saturation based on the linear co-positive Lyapunov function. By introducing an intermediate scalar whose absolute value is less than the absolute value of product of nonnegative vector of the linear co-positive Lyapunov function and input matrix and constructing a special form of the controller gains, this approach obtains a modified condition applicable for the linear programming. Finally, four numerical examples show that the proposed approach gives the larger domain of attraction than the existing approach based on the quadratic Lyapunov function. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Physical systems often have variables that include nonnegative property: networks of reservoirs, industrial processes involving chemical reactors, the population of human and heat exchangers [1]. Such systems are indicated as positive systems, whose state variables take only nonnegative values for any nonnegative initial condition. More recently, many applications of positive switched systems can be found in practice: virus mutation treatment [2], turbofan engines [3] formation flying [4] and other areas. Consequently, the positive switched systems have become subjects of research interest [5–7]. For positive systems, it has been shown that the linear co-positive Lyapunov function is more valid for discussing the control synthesis than traditional quadratic Lyapunov functions. Accordingly, the linear programming technique is more efficient than the linear matrix inequality (LMI) technique because the number of decision variables in the linear programming conditions is usually far fewer than that in the LMI conditions [8–14]. In the meantime, Markovian jump systems (MJSs) can be modeled by a set of linear systems with mode transition subject to a Markov chain (or Markov stochastic process). Over the past ten years, MJSs have gained a substantial amount of attention due to the fact that they are commonly regarded as suitable mathematical models to describe dynamic systems subject to random abrupt variations in their structure or parameters. For this reason, MJSs have been rapidly developed in many fields: networked control systems [15], economic systems [16], anti-windup design [17] and so on. Moreover, positive Markovian jump systems (PMJSs) are a special class of MJSs which provide a unified framework for mathematical modeling of many dynamic systems such as virus mutation treatment [2], turbofan engines [18,3] and network employing TCP in communication systems [19,20]. The stability analysis of PMJSs also has been studied [8,5,21]. These researches illustrate the necessity of the theoretical findings PMJSs. On the other hand, input saturation often occurs in practical engineering, which gives a clipped control input that is hard-limited by the peak output of an actuator. Since input saturation may deteriorate the performance of systems, it is

*

Corresponding author. E-mail addresses: [email protected] (I.S. Park), [email protected] (N.K. Kwon), [email protected] (P. Park).

https://doi.org/10.1016/j.nahs.2018.03.001 1751-570X/© 2018 Elsevier Ltd. All rights reserved.

I.S. Park et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 322–332

323

necessary to take input saturation into account in stabilization of systems, so that the control synthesis for systems with input saturation have been investigated by many researchers [22–24]. Especially, the authors of [24] considered the LMI approach for stabilization of PMJSs subject to actuator saturation by using the traditional quadratic Lyapunov function. As previously mentioned, the LMI-based strategy may be less effective than the linear programming based strategy. Therefore, it is theoretically meaningful to consider the linear programming approach for stabilization of PMJSs with input saturation by using the linear co-positive Lyapunov function. However, stabilization of PMJSs with input saturation by using the linear co-positive Lyapunov function yields the mutually coupled decision variables, which results in that the condition is not applicable for the linear programming. This difficulty motivates us to carry out this study. This paper proposes a linear programming approach for stabilization of positive Markovian jump systems with a saturated single input. The proposed approach first derives the sufficient conditions for stabilization of PMJSs with input saturation based on the linear co-positive Lyapunov function. As mentioned earlier, the directly obtained stabilization conditions are not applicable for the linear programming. By introducing an intermediate scalar whose absolute value is less than the absolute value of product of nonnegative vector of the linear co-positive Lyapunov function and input matrix and constructing a special form of the controller gains, this approach obtains a modified condition applicable for the linear programming. Finally, four numerical examples show that the proposed approach gives the larger domain of attraction than the existing approach based on the quadratic Lyapunov function [24]. The notations used in this paper are fairly standard. For x ∈ Rn , xT means the transpose of x. In means the n × n identity matrix. N+ r = {1, 2, . . . , r }, where r is positive integer. Given a probability space (Ω , Υ , Θ ), Ω represents the sample space, Υ is the algebra of events, and Θ is the probability measure defined on Υ . A ≻ 0, which indicates that all the elements of A are positive, and A ≻ B, which means that A − B {≻ 0. A matrix A} is called a Metzler matrix if its off-diagonal entries are nonnegative. For a vector v ∈ Rn , define ϵ (v ) = x ∈ Rn xT v < 1 . For a matrix A ∈ Rm×n , (A)pq is used to {indicate the entry in the p}th row and qth column of the matrix A, where p ≤ m and q ≤ n. For a matrix H ∈ Rm×n , + L(H) = x ∈ Rn |hi x| ≤ 1, i ∈ Nm , hi denotes the ith row of H. 2. Problem statement Given a probability space (Ω , Υ , Θ ), consider a continuous-time positive Markovian jump systems with input saturation given by x˙ (t) = A(rt )x(t) + B(rt )sat(u(t)),

(1)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input. {r(t), t ≥ 0} is a continuous-time Markov process on the probability space that takes the values in a finite set N+ N = {1, 2, . . . , N } and has the mode transition probabilities

{ Pr(rt +δ t = j|rt = i) =

πij δ t + o(δ t) 1 + πii δ t + o(δ t)

if j ̸ = i, other w ise,

(2)

where δ t > 0 and limδ t →0 (o(δ t)/δ t) = 0. πij is the transition rate from mode i at time t to mode j at time t + δ t, which ∑N (i) satisfies πij ≥ 0, for j ̸ = i and and B(i) denote the A(r(t) = i) and B(r(t) = i), j=1 πij = 0. To simplify the notation, A respectively. Further, for the vector σ = σ1

[

{ [sat(σ )]k ≜

1

σk −1

···

σm

]T

∈ Rm , the saturation operator sat(·) is defined as

σk ≥ 1, |σk | < 1, σk ≤ −1,

(3)

where [sat(σ )]k is kth element of sat(σ ). Definition 1. System (1) with u(t) ≡ 0 is said to be positive if for any initial condition x0 ⪰ 0, the corresponding trajectory x(t) ⪰ 0 holds for all t > 0. Lemma 1 ([25]). System (1) with u(t) ≡ 0 is positive if and only if A(i) is a Metzler matrix. Lemma 2 (Cao et al. [26] and Hu and Lin [27]). Let u, uv ∈ Rm , u = [u1 u2 · · · um ]T , uv = uv1 uv2 · · · uvm

[

]T

.

(4)

Assume that |eTi uv | ≤ 1 for all i ∈ N+ m , then sat (u) ∈ Co

{

v Dk u + D− k u

k ∈ N+ 2m

}

,

(5) T

where ei is a unit vector with the ith nonzero entry, i.e., ei ≜ [0 · · ·  1 · · · 0] , Dk denotes a diagonal matrix with all possible ith

combinations of 1 and 0 diagonal entries, D− k ≜ I − Dk , and Co is the convex hull.

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The mode-dependent state feedback controllers have the following form: u(t) = K (i) x(t),

(6)

where K = K (r(t) = i), i ∈ NN are the controller gains to be determined. Using (6), the system (1) without saturation can be represented as +

(i)

x˙ (t) = [A(r(t)) + B(r(t))K (r(t))] x(t).

(7)

3. Main results In this section, the stabilization problems of continuous-time positive Markovian jump system with a saturated single input are considered. To utilize Lemma 2 for handling the input saturation, we construct an auxiliary input uv (t) as uv (t) = H (i) x(t).

(8)

Consequently, from Lemma 2, if x(t) ∈ L(H (i) ), then the saturated mode-dependent state feedback controller (6) can be represented as m

(i)

sat(K x(t)) =

2 ∑

) ( ηk D(k) K (i) + D−(k) H (i) x(t),

(9)

k=1

∑2m

η = 1, and ηk ≥ 0. Therefore, the closed-loop system with input saturation (1) can be written as follows: ) ( 2m ∑ ( ) (10) x˙ (t) = A(i) + ηk B(i) D(k) K (i) + B(i) D−(k) H (i) x(t).

where

k=1 k

k=1

In the following theorem, we consider the stability of the closed-loop system (10) by using the linear co-positive Lyapunov function. Theorem 1. For given scalar α (i) and any initial condition x(0) ∈ ∩ni=1 ϵ (v (i) ), the closed-loop system (10) is positive and stochastically stable, if there exist vectors v (i) ∈ Rn , z (i) ∈ Rn×m , and g (i) ∈ Rn×m such that for each i ∈ N+ N either Case 1 or Case 2:

• Case 1 v (i) ≻ 0,

(11) N

A(i)T v (i) +

∑

πij v (j) + D(k) z (i) + D−(k) g (i) ≺ 0, ∀k ∈ N+ , 2m

(12)

j=1

B(i)T v (i) > α (i) > 0,

(13)

−α v ≺g ≺α v , ( (i) (i)T (i) ) A B v + B(i) D(k) z (i)T + B(i) D−(k) g (i)T pq ⪰ 0, (i) (i)

(i)

(i) (i)

+ ∀p, q ∈ N+ n , p ̸ = q, ∀k ∈ N2m .

(14)

(15)

• Case 2 v (i) ≻ 0,

(16)

A(i)T v (i) +

N ∑

πij v (j) + D(k) z (i) + D−(k) g (i) ≺ 0, ∀k ∈ N+ , 2m

(17)

j=1

B(i)T v (i) < α (i) < 0,

α v ( (i)

(i) (i)

≺g

(i)

(18)

≺ −α v , (i) (i)

(19)

A B(i)T v (i) + B(i) D(k) z (i)T + B(i) D−(k) g (i)T

) pq

⪯ 0,

+ ∀p, q ∈ N+ n , p ̸ = q, ∀k ∈ N2m .

(20)

Moreover, the mode-dependent state-feedback controller gains are given by K (i) =

1 B(i)T v (i)

z (i)T , H (i) =

1 B(i)T v (i)

g (i)T .

(21)

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325

Proof. (Case 1) By the condition (13), the condition (14) satisfies

− (B(i)T v (i) )v (i) ≺ g (i) ≺ (B(i)T v (i) )v (i) .

(22)

Since we consider the system with single input, i.e. m = 1, B(i)T v (i) is a scalar, so that the above condition is equivalent to

− v (i) ≺

1 B(i)T

v (i)

g (i) ≺ v (i) .

(23)

Additionally, with the control gains in (21), the condition (23) can be written as

− v (i) ≺ H (i)T ≺ v (i) .

(24)

Then, in view of the condition (24), for any x(t) ∈ Rn satisfying x(t)T v (i) ⪯ 1, we have −1 ⪯ −x(t)T v (i) ⪯ x(t)T H (i)T ⪯ x(t)T v (i) ⪯ 1, which means that ϵ (v (i) ) ⊂ L(H (i) ). ϵ (v (i) ) is the invariant set [3]. Therefore, for any initial condition x(0) ∈ ∩ni=1 ϵ (v (i) ), we have the condition x(t) ∈ L(H (i) ) and can represent the saturated input as (9). In addition, the condition + (15) is equivalent to the following condition: ∀p, q ∈ N+ n , p ̸ = q, ∀k ∈ N2m ,

(

A(i) + B(i) D(k)

1 B(i)T v

z (i)T + B(i) D−(k) (i)

1 B(i)T v

g (i)T (i)

)

⪰ 0,

(25)

pq

which yields that A(i) + B(i) D(k) K (i) + B(i) D−(k) H (i) are Metzler matrices. Therefore, the closed-loop system (10) is positive system by Lemma 1. Now, we choose the linear co-positive Lyapunov function as follows V (x(t), i) = xT (t)v (i) ,

(26)

where v (i) is positive by the condition (13). Then, the weak infinitesimal operator Γ of the stochastic process x(t) acting on V (x(t), i) is given as

Γ V (x(t), i) = lim [E {V (x(t + ∆t), r(t + ∆t) | x(t), r(t) = i)} − V (x(t), r(t) = i)] ∆t →0

m

( T

2 ∑

(i)

= x (t) A +

ηk B D K

(

(i)

(k)

(i)

(i)

−(k)

+B D

H

(i)

)T )

v (i)

k=1

+ xT (t)

N ∑

πij v (j) .

(27)

j=1

From the fact

∑2m

η = 1, Γ V (x(t), i) < 0 holds if and only if the following condition holds for all k ∈ N+ , 2m

k=1 k

A(i)T v (i) + K (i)T D(k) B(i)T v (i) + H (i)T D−(k) B(i)T v (i) +

N ∑

πij v (j) ≺ 0.

(28)

j=1

Then, noting that K (i) and H (i) in (21), and D(k) is 0 or 1, we have K (i)T D(k) B(i)T v (i) = D(k) z (i) and H (i)T D(k) B(i)T v (i) = D(k) g (i) . Consequently, by x(t) ⪰ 0, the condition (12) yields Γ V (x(t), i) < 0. Therefore, we can conclude that the closed-loop system (10) is stochastically stable. (Case 2) The proof is very similar to that of (Case 1) so is omitted. □ Remark 1. Satisfying the set invariant condition (24) and the negativity of the weak infinitesimal of linear co-positive Lyapunov function (28), simultaneously, yields the mutually coupled decision variables, which results in that the conditions are not applicable for the linear programming. By introducing the controller gains as in (21) and the intermediate scalar α (i) as in (18) and (13), the conditions are converted into the form that can be solved by linear programming. In the following corollary, we give the stabilization condition without the input saturation. The closed-loop system without the input saturation can be written as x˙ (t) = A(i) + B(i) K (i) x(t).

(

)

(29)

Corollary 1. For given vector p(i) ∈ Rm , the closed-loop system (10) is positive and stochastically stable, if there exist vectors v (i) ∈ Rn , z (i) ∈ Rn×m , and g (i) ∈ Rn×m such that for each i ∈ N+ N either Case 1 or Case 2:

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• Case 1 v (i) ≻ 0,

(30) N

A(i)T v (i) +

∑

πij v (j) + z (i) ≺ 0,

(31)

j=1

p(i)T B(i)T v (i) > 0,

(

(i) (i)T

A p

(i)T

B

(32)

v +B p z (i)

(i) (i) (i)T

) pq

⪰ 0, ∀p, q ∈ Nn , p ̸= q. +

(33)

• Case 2 v (i) ≻ 0,

(34) N

A(i)T v (i) +

∑

πij v (j) + z (i) ≺ 0,

(35)

j=1

p(i)T B(i)T v (i) < 0, A(i) p(i)T B(i)T v (i) + B(i) p(i) z (i)T

(

(36)

) pq

⪯ 0, ∀p, q ∈ N+ n , p ̸ = q.

(37)

Moreover, the mode-dependent state-feedback controller gain is given by K (i) =

1 p(i)T B(i)T v (i)

p(i) z (i)T .

(38)

Proof. The proof is similar to Theorem 1 so is omitted. □ Remark 2. Differently from the approach [8] which considers only the case of B(i) ≻ 0, the proposed approaches can deal with the case of B(i) ̸ ≻ 0. Remark 3. The Corollary 1 considers the unsaturated multiple input by introducing the given vector p(i) . However, it is hard to extend Theorem 1 to the saturated multiple input case because of the virtual input issue. Therefore, it can be said that the linear programming approach for the stabilization of PMJSs with saturated multiple input is a challenging future work. In the following steps, we try to enlarge the domain of attraction for the closed-loop system (10). The set ∩ni=1 ϵ (v (i) ) can be served as an estimate of domain of attraction [3]. To enlarge the domain of attraction, thus, v (i) need to be minimized, so that we suggest the following optimization problem. min

N n ∑ ∑

vl(i)

(39)

l=1 i=1

s.t. Either Case 1 or Case 2 in Theorem 1 holds. To solve the optimization problem (39), we provide an algorithm: Algorithm 1 : A(v (i) ) is area of the estimate of domain of attraction ∩ni=1 ϵ (v (i) ). λ is step size of each iteration for α (i) . α (i) and α (i) are upper and lower bounds of α (i) , i.e. α (i) ∈ [α (i) , α (i) ]. (i) Set adequately large upper bound α (i) , small lower bound α (i) and large step size λ. (ii) For the given α (i) ∈ [α (i) , α (i) ] with step size λ, iteratively solve the optimization problem (39), and calculate the A(v (i) ): (i) (i) (i) 1: for α = α : λ : α 2: solve the optimization problem (39) 3: calculate A(v (i) ) 4: end (i) (iii) Choose αop as α (i) which gives the largest A(v (i) ). (i) (iv) Reset α , α (i) , and λ: 5: 6: 7:

(i) α (i) ← αop +λ (i) (i) α ← αop −λ λ ← λ/10

(v) For the selected α (i) , α (i) , and λ in (iv), repeat (ii)-(iv) until A(v (i) ) becomes reasonably large.

I.S. Park et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 322–332

(1)

327

(2)

Fig. 1. Set α (i) = 10.1, α (i) = 0.1 for all i ∈ {1, 2}, and λ = 0.1: αop = 0.4, αop = 0.2.

4. Numerical examples In this section, three numerical examples will be provided to illustrate the effectiveness of the proposed method. The transition rate matrix used in the following Examples 1–3 is as follows:

[ −5 10

]

5 . −10

Example 1. In this example, consider the PMJS (1) with input saturation which has the following system matrices: A(1) = B(1) =

0.0852 0.8810

[

0.3232 −1.0765 , A(2) = −0.7841 1.0306

]

[

0.3275 , 0.6521

]

1 0.1 , B(2) = . 0.1 1

[

]

[

]

To enlarge the domain of attraction (DoA), we perform the Algorithm 1: First, set α (i) = 10.1, α (i) = 0.1 for all i ∈ {1, 2} (1) (2) and λ = 0.1, which results in αop = 0.4, αop = 0.2 as shown in Fig. 1. Second, with the results in Fig. 1, reset α (1) = 0.5, (2) (1) (2) α = 0.3, α = 0.3, α = 0.1, and λ = 0.01, so that Fig. 2 shows the simulation results. Consequently, α (i) is selected as α (1) = 0.31, α (2) = 0.21. With the selected α (i) , Theorem 1 offers the following values:

v (1) = [0.2935 0.1652]T , v (2) = [0.2767 0.1823]T , z (1) = [−2.1436 − 0.0772]T , z (2) = [−0.1373 − 0.3442]T , g (1) = [−0.0890 − 0.0512]T , g (2) = [−0.0581 − 0.0379]T . Then, the resulting controller gains are obtained as follows H (1) = [−0.2872 − 0.1652], H (2) = [−0.2767 − 0.1803], K (1) = [−6.9148 − 0.2492], K (2) = [−0.6539 − 1.6392]. Based on the above controller gains, the estimate of domain of attraction and the simulation results are shown in Fig. 3. The domain of attraction obtained from the proposed linear programming (LP) approach covers that obtained from the LMI approach [24], which means that the proposed approach based on the linear co-positive Lyapunov function provides the larger domain of attraction than the LMI approach based on the quadratic Lyapunov function [24]. The state trajectory starting from the initial state x0 = [2.5 0.7]T tends to zero. Especially, the chosen initial state is not covered by the domain of attraction obtained from LMI approach. Remark 4. In order to show that the estimate of DoA given by the proposed approach is larger than that given by the LMI approach [24], we select the initial condition which belongs to the region included in the estimate of DoA given by the proposed approach but not included in the estimate of DoA given by the LMI approach [24].

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(1)

(2)

Fig. 2. Set α (1) = 0.5, α (1) = 0.3, α (1) = 0.3, α (1) = 0.1, and λ = 0.01: αop = 0.31, αop = 0.21.

Fig. 3. The simulation results: initial condition x0 = [2.5 0.7]T .

Example 2. In this numerical example, the system matrix B(i) has both positive and negative elements, which cannot be considered by the approach in [8]. The system matrices are same to that of Example 1 except for the sign of B(i) : (1)

B

1 −0.1 = , B(2) = . −0.1 1

[

]

[

]

Firstly, we consider the systems without input saturation. Then, Corollary 1 with p(i) = 1, ∀i ∈ {1, 2} gives

v (1) = [93.2968 93.6966]T , v (2) = [98.9959 92.6144]T ,

(40)

z (1) = [−181.9564 6.8463]T , z (2) = [−7.2775 − 161.6466]T ,

(41)

K

(1)

= [−2.1680 0.0816], K

(2)

= [−0.0880 − 1.9543].

(42)

By applying the obtained control gains, Fig. 4 shows the state trajectory, the control input, and the mode evolution, which yields that Corollary 1 effectively stabilizes the PMJSs with the system matrix B(i) ̸ ≻ 0. Now, let us consider the systems with input saturation. We select the α (i) with a same procedure as in the previous example, which results in α (1) = 0.33, α (2) = 0.18. Then, Theorem 1 provides

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329

Fig. 4. The simulation results: initial condition x0 = [3 1]T .

v (1) = [0.3496 0.1960]T , v (2) = [0.3304 0.2170]T , z (1) = [−202.7427 − 0.0861]T , z (2) = [−0.1246 − 254.7042]T , g (1) = [−0.1145 − 0.0647]T , g (2) = [−0.0595 − 0.0391]T . Then, the corresponding controller gains are given as follows: H (1) = [−0.3469 − 0.1960], H (2) = [−0.3233 − 0.2123], K (1) = [−614.3717 − 0.2610], K (2) = [−0.7000 − 1384.3000]. With the above controller gains, Fig. 5 shows the estimate of domain of attraction and the simulation results. The estimate of domain of attraction obtained from the proposed approach includes that obtained from the LMI approach [24]. The states converge to zero from the initial state x0 = [2.3 0.5]T which is out of the estimate of the domain of attraction obtained from the LMI approach. It is obvious that the domain of attraction obtained from the proposed approach based on the linear co-positive Lyapunov function is larger than that obtained from the LMI approach based on quadratic Lyapunov function [24]. Furthermore, since the input matrix B(i) has both positive and negative elements, the theorem in [8] which has similar structure the proposed controller gains cannot give the solution for this example. Example 3. In this numerical example, the system matrix B(i) has only negative elements, which can be considered by the Case 2 in Theorem 1. The two-mode SMJSs (1) with the input saturation are given as follows: (1)

A

B(1)

] [ −1.0 1.2 −1.0 (2) = ,A = 2.5 −2.0 1.0 [ ] [ ] −0.1 −0.2 = , B(2) = . −0.2 −0.3 [

1.6 , −1.2

]

We select the α (i) with a same procedure as in the previous example, which results in α (1) = −0.2420, α (2) = −0.4160. Then, Theorem 1 provides

v (1) = [1.0732 0.6736]T , v (2) = [1.0029 0.7180]T , z (1) = [−1.8280 − 1.6863]T , z (2) = [−0.9077 − 2.0344]T , g (1) = [−0.2597 − 0.1630]T , g (2) = [−0.4172 − 0.2985]T . Then, the corresponding controller gains are given as follows: H (1) = [1.0730 0.6735], H (2) = [1.0029 0.7176], K (1) = [7.5527 6.9675], K (2) = [2.1819 4.8903]. With the above controller gains, Fig. 6 shows the estimate of domain of attraction and the simulation results. The estimate of domain of attraction obtained from the proposed linear programming approach involves that obtained from the LMI approach [24]. The state trajectory goes to zero from the initial state x0 = [0.2 1]T , where the initial state is out of the estimate of the domain of attraction obtained from the LMI approach, which means that the domain of attraction obtained from the proposed approach based on the linear co-positive Lyapunov function is larger than that obtained from the LMI approach based on quadratic Lyapunov function [24].

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Fig. 5. The simulation results: initial condition x0 = [2.3 0.5]T .

Example 4 (Practical Example). This example shows that the proposed method can be effectively applied to the practical problem, the virus mutation treatment [2]. The dynamics can be represented as follows: x˙ (t) = (R(i) − δ I + ζ M)x(t) + B(i) sat(u(t)),

(43) +

where the state vector x(t) ∈ R2 stands for two different viral genotypes, i ∈ N2 denotes a Markovian process with two different modes, ζ indicates the mutation rate, δ is the death or decay rate, M stands for the system matrices, (M)mn ∈ {0, 1} represents the genetic connections between genotypes, i.e. (M)mn = 1 if and only if it is possible that genotype n mutates into genotype m, and u(t) indicates the control input. Then, the PMJSs with two modes are given as follows: R(1) = B(1) =

[

0.05 0

0 0.06 , R(2) = 0.25 0

]

[

]

[

0 0 ,M= 0.26 1

]

1 , 0

[ ] [ ] −0.05 0.3 , B(2) = , δ = 0.27, ζ = 0.1. 0.1 −0.04

The transition rate matrix is assumed to be:

[ −1.5 0.9

1.5 . −0.9

]

We choose the α (i) with a similar method to the previous examples as α (1) = 0.031, α (2) = 0.031. Then, Theorem 1 gives the following parameters:

v (1) = [0.1591 0.3896]T , v (2) = [0.1598 0.3910]T , z (1) = [−0.0185 − 206.1035]T , z (2) = [−210.6566 − 0.0108]T , g (1) = [−0.0049 − 0.0111]T , g (2) = [−0.0050 − 0.0108]T , H (1) = [−0.1591 − 0.3567], H (2) = [−0.1534 − 0.3333], K (1) = [−0.6000 − 6648.5000], K (2) = [−6522.5000 − 0.3000]. With the above parameters, Fig. 7 shows the estimate of domain of attraction and the simulation results. The estimate of domain of attraction provided from the proposed linear programming approach covers that provided from the LMI approach [24]. The state trajectory goes to zero from the initial state x0 = [3 1]T , where the initial state is out of the estimate of the domain of attraction provided from the LMI approach, which means that the domain of attraction provided from the proposed approach based on the linear co-positive Lyapunov function is larger than that provided from the LMI approach based on quadratic Lyapunov function’ [24]. 5. Conclusion This paper proposed a linear programming approach for stabilization of positive Markovian jump systems (PMJSs) with a saturated single input. The proposed approach first derived a sufficient condition for stabilization of PMJSs with input saturation based on the linear co-positive Lyapunov function. By introducing an intermediate scalar whose absolute value was less than the absolute value of product of nonnegative vector of the linear co-positive Lyapunov function and input

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Fig. 6. The simulation results: initial condition x0 = [0.2 1]T .

Fig. 7. The simulation results: initial condition x0 = [3 1]T .

matrix and constructing a special form of the controller gains, this approach obtained a modified condition applicable for the linear programming. Finally, four numerical examples showed that the proposed approach provided the larger domain of attraction than the existing approach based on the quadratic Lyapunov function. Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (2017R1D1A1A09000787). This research was also supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the ICT Consilience Creative Program (IITP-R0346-16-1007) supervised by the IITP (Institute for Information & Communications Technology Promotion). This research was supported by Korea Electric Power Corporation (Grant number: R18XA01). References [1] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications, Vol. 50, John Wiley & Sons, 2011. [2] E. Hernandez-Vargas, P. Colaneri, R. Middleton, F. Blanchini, Discrete-time control for switched positive systems with application to mitigating viral escape, Internat. J. Robust Nonlinear Control 21 (10) (2011) 1093–1111. [3] J. Wang, J. Zhao, Stabilisation of switched positive systems with actuator saturation, IET Control Theory Appl. 10 (6) (2016) 717–723. [4] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control 48 (6) (2003) 988–1001. [5] M. Xiang, Z. Xiang, Stability, L1-gain and control synthesis for positive switched systems with time-varying delay, Nonlinear Anal. Hybrid Syst. 9 (2013) 9–17.

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