Exponential Stability of Switched Positive Homogeneous Systems

This paper studies the exponential stability of switched positive nonlinear systems defined by cooperative and homogeneous vector fields. In order to capture the decay rate of such systems, we first consider the subsystems. A sufficient condition for exponential stability of subsystemswith time-varying delays is derived. In particular, for the corresponding delay-free systems, we prove that this sufficient condition is also necessary. Then, we present a sufficient condition of exponential stability under minimum dwell time switching for the switched positive nonlinear systems. Some results in the previous literature are extended. Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.


Introduction
Positive systems are those systems whose state variables and output signals are always contained in the first quadrant whenever both the initial conditions and input signals are nonnegative.Many real-word processes are described by positive systems.Examples of this type can be found in areas such as ecology, biology, and chemical engineering [1][2][3].Due to their importance and wide applications, there has been an increasing interest in such systems.
Many important and interesting properties of positive systems have been reported and analyzed.It is well known that the most fundamental one is the stability property.Up until now the stability analysis of positive linear timeinvariant (LTI) systems is well developed [4,5].In [4], it was shown that the delayed positive linear system ẋ () = x() + x( − ) was globally asymptotically stable (GAS) for all  ≥ 0 if the corresponding delay-free system ẋ () = ( + )x() was GAS.This implies that such system is insensitive to certain class of time delays.Moreover, the exponential stability of positive linear systems with constant time delays was investigated by Zhu et al. [5], in which it was also shown that the decay rate depended on the magnitude of delays.Recently, some progress on the positive nonlinear systems called cooperative homogeneous systems has been made [6][7][8][9].For example, [6] derived a necessary and sufficient condition for exponential stability of such positive systems with the homogeneity of a degree of one.
On the other hand, switched systems have attracted much research attention in control theory field [10][11][12][13].Typically, a switched system is a type of hybrid dynamical system consisting of family (either discrete-time or continuous-time) subsystems and a rule that regulates the switching among them.It has been widely applied in many areas, such as chemical processing and traffic control.For such systems, the stability theory is also one of the fundamental problems.The authors in [12] applied the Lyapunov-based theory, such as the multiple Lyapunov function, to study the stability problems.
Furthermore, switched positive systems combining the features of positive systems and switched systems are recently studied [14][15][16][17][18].As switched positive systems are defined on cones rather than on linear spaces, studying such systems is more challenging than that of general switched systems.In recent years, some results concerning the switched positive systems have been reported.For example, the stability problem of switched positive linear systems has been extensively discussed by the approach of linear copositive functions such as [14,15,17].In [19], the author showed a necessary and 2 Complexity sufficient condition for exponential stability of a class of switched positive nonlinear systems under average dwell time switching.
It is worth noting that the results in [7] only dealt with asymptotical stability of positive systems.As is known to us, switched systems may be unstable even if we ensure that all the subsystems are stable.Therefore, it is meaningful to combine the positive systems with switched systems.On the other hand, the author of [19] only studied the switched positive systems with the homogeneity of a degree of one, in which the common exponential decay rate for all subsystems independent of initial conditions can be easily found for all subsystems.However, when the degree is not constrained to be one, the exponential decay rate depends on initial conditions, which would make the problem be more complicated.The state at each switched instant should be considered.In this case, it is worth studying whether it is possible to design suitable switching signals to keep the system exponentially stable.In this paper, by using the approach in which the Lyapunov-Krasovskii functional method is not included, we design appropriate time-dependent switching rules under which the switched positive nonlinear system is exponentially stable with the degree 0 <  ≤ 1.
The main contributions are summarized as follows.We derive a sufficient condition for exponentially stable unswitched positive systems of degree of 0 <  ≤ 1 with bounded time delays.It is shown that the decay rate depends on the initial conditions and the time delays.In particular, for the corresponding delay-free systems, the sufficient condition is also necessary.Then as special switched positive nonlinear systems, we consider the switched positive homogeneous systems with the degree of 0 <  ≤ 1.A sufficient condition for the exponential stability of switched positive homogeneous systems under minimum dwell time (MDT) switching is presented.
The layout of the paper is as follows.In Section 2, we introduce the notation and review some preliminaries.The main results of this paper are stated in Section 3. Section 4 provides a numerical example to show the validity of our results.Finally, concluding remarks are given in Section 5. Definition 4.  vector field g : R  → R  is said to be orderpreserving on R  + \ {0}, if g(x) ≥ g(y) for any x, y ∈ R  + \ {0} such that x ≥ y.

Exponential Stability of Unswitched Positive Homogeneous
Systems.We first consider positive system with time-varying delays: where x() ∈ R  is the state variable and f, g : In addition, we assume that system (3) satisfies the following conditions.
(i) f and g are homogeneous of degree of  > 0.
(ii) f is cooperative and g is order-preserving on R  + \ {0}.
Assumption 6.The time-varying delay () : R + → R + is continuous and satisfies Here () is not necessarily continuously differentiable and no restriction on its derivative such as τ () < 1 is imposed.
Complexity 3 Remark 7. Condition (i) of Assumption 5 implies that f(0) = g(0) = 0. Furthermore, as f is cooperative and g is orderpreserving on R  + , one has It has been shown in [7, Proposition 3.2] that (5) implies that system (3) is positive for any initial condition (⋅) ∈ R  + .Throughout this paper, we always assume that the initial conditions are nonnegative.
Before giving the sufficient condition for the exponential stability of system (3), a lemma that provides a precise estimation of the upper bound of system ( 3) is proposed first.
Lemma 8 (see [7]).Consider system (3) under Assumptions 5 and 6.If there exists a vector  ≫ 0 such that f() + g() ≪ 0, then the solution x() satisfies where Definition 9. System (3) is said to be exponentially stable if there exist two constants  > 0 and  > 0 such that where ‖ ⋅ ‖ is some norm in R  .
The following theorem states a sufficient condition for exponential stability of system (3) with 0 <  ≤ 1.
Theorem 10.For system (3) with the degree of 0 <  ≤ 1, suppose that Assumptions 5 and 6 hold.If there exists a vector  ≫ 0 such that then system (3) is exponentially stable for any nonnegative initial conditions.In particular, every solution of system (3) satisfies where 0 <  < min 1≤≤   and   is the unique positive solution to Proof.Note that (10) has two parameters: the maximum delay  max and   .So, naturally for fixed  max ≥ 0, define a set of functions with respect to : It now follows from (10) that Furthermore, it can be easily checked that   () is strictly monotonically increasing in  > 0, which implies that So if  ∈ (0, min 1≤≤   ), we have Now, inspired by the method developed in [6], let Then it follows from Lemma 8 that It is immediate to see that, for any  ∈ {1, 2, . . ., }, In order to prove exponential stability of system (3), we still need to prove that, for all  and all  ≥  max ,   () ≤ 0.
Obviously,   ()| = max ≤ 0 for all .As   () is continuous function, we claim that it is also holds for all  ≥  max and all .By contradiction, suppose that this is not always true.
Then there exist an index  ∈ {1, 2, . . ., } and a time  * ∈ [ max , +∞) such that From ( 18) and ( 19), we have Then it follows from cooperativity and homogeneity of f that

Complexity
Note that  * − ( * ) ∈ (0,  * ], so we obtain Furthermore, as g is order-preserving and homogeneous, it in turn implies that The upper-right Dini derivative of   () along the trajectories of system (3) at  =  * is given by where we have used ( 22) and (24) to get the first inequality.Note also that 0 ≤ () ≤  max and   ()/  +  max    ()/  < 0, so the second and third inequalities are true, and the last inequality is from ( 14).Now we arrive at a contradiction with (20).Hence,   () ≤ 0; for all  ≥ 0,  ∈ {1, 2, . . ., }, which means that This completes the proof.
Remark 11.Theorem 10 provides an estimate on how the decay rate depends on initial condition with the degree of 0 <  ≤ 1, which is fundamental to study the switched positive systems.
Next, consider the corresponding delay-free system: In the following, we give a sufficient and necessary condition for the exponential stability of system (27).
(2) System ( 27) is exponentially stable and every solution x() satisfies where 0 <  < min 1≤≤   with   being the unique positive solution to Proof.
Remark 13.From (29), one can verify that We now consider two cases.

Exponential Stability of Switched Positive Homogeneous
Systems.In the following, we consider the following switched positive nonlinear system: where x() ∈ R  is the state vector and () : [0, +∞) →  denotes the switching signal.The finite set  = {1, 2, . . ., } is an index set and stands for the collection of subsystems.For all  ∈ , f p and g p : R  → R  are continuously differentiable on R  \ {0}.
Remark 14.It should be pointed that, for every  ∈ , as f p is defined to be cooperative and homogeneous and g p is orderpreserving and homogeneous, system (35) is positive under arbitrary switching laws.This implies that, for any initial condition x(0) ∈ R  + , the corresponding state trajectory x() ∈ R  + for all  ≥ 0.
Lemma 15 (see [19]).Assume that x ∈ R  ; then, for any ,  ∈ , where Based on Theorem 10 and Lemma 15, we next establish a sufficient condition for exponential stability of switched system (35) and the symbols that are defined in Lemma 15 continue to be used.In addition, let Theorem 16.For system (35), suppose that, for every  ∈ , f p , g p satisfy Assumption 5 with 0 <  ≤ 1.If there exists a vector  p ≫ 0 such that then system (35) is exponentially stable for any given set Q under MDT switching signal  satisfying where 0 <  < min{, } with  = min ∈ min 1≤≤   and  = min ∈ min 1≤≤  *  and   and  *  are the solutions to the following equations: Proof.Consider a switching sequence From (39), we have Δ  =  +1 −   ≥  > ln /,  ∈ N 0 .
The proof now proceeds in two steps.
Step 1. First, for any x(0) ∈ , we show that Note that, for every  ∈ , there exists  p ≫ 0 such that f  (  )+g  (  ) ≪ 0, so, from Corollary 12, for the first interval [ 0 ,  1 ), we have where we have used Δ 1 =  1 ≥  > ln / to get the second inequality.Furthermore, we obtain Next, consider the second interval [ 1 ,  2 ).It follows from Corollary 12 that From Remark 13, we have 0 < where  ∈ (0, min{, }). (  ) = 0. (57) Then we can prove The rest of the proof is similar to the one for the case of  = 2 and thus is omitted.
According to the previous arguments, we conclude that (43) hold for each  ∈ N, which implies that for all subsystems we find the common exponential decay rate .In addition, We also prove that ‖x(  )‖ ∞ under minimum dwell time switching, where x() is the initial condition of the ( + 1)th subsystem.
Step 2. We prove that system (35) is exponentially stable.For any  ∈ [0, +∞), assume that  ∈ [  ,  +1 ) on which () = .Then, from Step 1, we have Moreover, applying Lemma 15, we can get where we have used the MDT to obtain the last inequality.On the other hand, note that ‖x In summary, system (35) is exponentially stable for any given set  under MDT switching.This is the end of the proof.

Numerical Simulation
Consider the switched nonlinear positive system consisting of two subsystems given by Let  = { ∈ R  + |   ≤ 8,  = 1, 2}; then  = 1.It follows from Theorem 16 that the corresponding switched nonlinear system is exponentially stable under minimum dwell time  > 2.2314.Figure 1 shows the actual rate and the upper bound which agree with the implication of Theorem 16.

Conclusion
In this paper, we derive a sufficient condition for the exponential stability of switched positive homogeneous systems with degree of 0 <  ≤ 1.Some results in literature [19] are extended.In addition, a sufficient condition is presented for the exponential stability of the unswitched positive systems with time-varying delays.A numerical example is given to demonstrate the main result.While the stability problem of such system with delays is the subject of ongoing work, we hope to report the result of which in the future.

Figure 1 :
Figure 1: The simulation results of the actual decay rate ‖x()‖  ∞ and the guaranteed upper bound we calculated with the initial condition x(0) = [7, 8]  and the MDT  = 3.

x > y, if x ≥ y and x ̸ = y; x ≫ y, if
>   1 ≤  ≤ .  is used to denote the th component of x.Similarly,   denotes the th coordinate of vector x p .In[7], given a vector  ≫ 0, the weighted  ∞ norm is defined by × , let   denote its entry in row  and column .A real  ×  matrix  = (  ) × is Metzler if and only if its off-diagonal entries   ,  ̸ =  are nonnegative.For a real interval [, ], let ([, ], R  ) be the space of all continuous functions on [, ] taking values in R  .The upper-right Dini derivative of a continuous function ℎ : R → R is denoted by  + ℎ(⋅).
Definition 2. A continuous vector field f : R  → R  which is continuously differentiable on R  \{0} is said to be cooperative if the Jacobian matrix (f/x)(a) is Metzler for all a ∈ R  + \{0}.It follows from [1, Remark 3.1] that the cooperative systems satisfy the following property.Proposition 3. Let f : R  → R  be cooperative.For any two vectors x and y in R  + with x ≥ y and   =   , one has   (x) ≥   (y).