Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
-
Hao Xing
,
Xiangyang Cao
Abstract
This paper considers the guaranteed cost finite-time boundedness of positive switched nonlinear systems with D-perturbation and time-varying delay. Firstly, the definition of guaranteed cost finite-time boundedness is introduced. By using the Lyapunov-Krasovskii functional and average dwell time (ADT) approach, an output feedback controller is designed and sufficient conditions are obtained to ensure the corresponding closed-loop systems to be guaranteed cost finite-time boundedness (GCFTB). Such conditions can be solved by linear programming. Finally, two examples are provided to show the effectiveness of the proposed method.
1 Introduction
The switched system, which comprises a set of subsystems and a switched controller that designating the switching between subsystems, has been studied very well. As a special kind of switched systems, the positive switched systems, whose output and state are nonnegative whenever the initial condition and input are nonnegative, have been applied in many practical systems, such as communication networks [1], viral mutation [2], formation flying [3], and so on. Many remarkable results have been presented, see [4, 5, 6, 7, 8, 9, 10, 11] and references therein.
However, most results mentioned above focus on the Lyapunov asymptotic stability, which reflects the asymptotic behavior of the system within a sufficiently long (in principle infinite) time interval. In many practical circumstances, one is more interested in researching what happens in a finite time interval. [12] firstly defined the definition of finite-time stability (FTS) for linear deterministic systems. Recently, [13] firstly extended the concept of FTS to positive switched systems and gave some FTS conditions of positive switched systems. So far, there have been some meaningful results about FTS of positive switched systems, see [14, 15, 16, 17, 18]. In [14], the problem of finite-time L1 control for a class of positive switched linear systems with time-varying delay was considered and the concept of finite-time L1 boundedness was also proposed. In [15], the problem of finite-time stability analysis and control synthesis of fractional order positive switched systems was considered. In [16], the finite-time stability problem of discrete switched singular positive systems was investigated. In [17], the robust finite-time stability and stabilisation of discrete-time switched positive systems was researched. In [18], the finite-time stability problem of switched positive linear systems was addressed. But, the above results are involved in positive switched linear systems. For positive switched nonlinear systems, only [10] considered local asymptotic stability and [19] studied absolute exponential L1 stability analysis and control synthesis, respectively. Furthermore, there are no results about FTS of positive switched nonlinear systems, though many papers about FTS of switched nonlinear systems (non-positive) have been published [20, 21, 22, 23]. On the other hand, in most practical systems, researchers desire to design the control system which is not only finite-time stable but also guarantees an adequate level of performance. One method to this problem is the so-called guaranteed cost control [24, 25]. In [26], the problem of robust finite-time guaranteed cost control for a class of impulsive switched systems with time-varying delay was studied. Very recently, [27] and [28] considered the guaranteed cost finite-time control for positive switched linear systems, [29] studied the guaranteed cost finite-time control for fractional-order positive switched systems. However, the problem of guaranteed cost finite-time control for positive switched nonlinear systems is still open, because nonlinear system is more complex than the linear system and there is a lack of effective methods. These show that the topic addressed in the paper is interesting but full of challenge.
Moreover, there exists a hardware error, such as fluctuation and modeling error, which will lead to instability of the system and is called D-perturbation. Recently, [30] considered the robust stability of positive switched linear systems with D- perturbation and time-varying delay. Accordingly, for positive switched nonlinear systems, the effect of D-perturbation must be also taken into account in analyzing and implementing guaranteed cost finite-time controller scheme.
Motivated by the above discussion, in this paper, we consider the problem of GCFTB for positive switched nonlinear systems with D-perturbation and time-varying delay. An output feedback controller is designed and some sufficient conditions are obtained to guarantee that the closed-loop system is GCFTB. The main contributions lie in two aspects: 1) This system model is more general, so that the systems dealt with in [2, 4, 5, 7, 14, 19, 27, 28, 30] can be regarded as especial forms of the system. 2) A new nonlinear Lyapunov-Krasovskii functional is constructed and the obtained conditions can be easily solved by linear programming. The remainder of the paper is organized as follows. Section 2 presents the preliminaries and problem statements. Main results are given in Section 3. In Section 4, two examples are provided. Section 5 concludes the paper.
Notations
The representation A ≻ 0 (⪰ 0, ≺ 0, ⪯ 0) means that aij > 0 (≥ 0, < 0, ≤ 0), which is also applying to a vector. A ≻ B (A ⪰ B) means that A − B ≻ 0 (A − B ⪰ 0).
2 Preliminaries and problem statements
Consider the following positive switched nonlinear systems with D-perturbation and time-varying delay:
where x(t) ∈ Rn is the system state, u(t) ∈ Rm and y(t) ∈ Rs represent the control input and output. σ(t) : [0, ∞)→ Ṉ = {1, 2, ⋯, N} is the system switching signal, where N is the number of subsystems; ∀ p ∈ Ṉ, Ap, Adp, Bp, Cp and Gp are constant matrices with suitable dimensions, d(t) ≥ 0 denotes time-varying delay, which satisfies ḋ(t) ≤ h < 1, h is a known positive constants. φ(θ) is the initial condition on [−ι,0], f(x) = (f1(x1), f2(x2), …, fn(xn))T ∈ Rn, perturbations D1 ∈ [D̲1, D1], D2 ∈ [D̲2, D2], D3 ∈ [D̲3, D3] and D4 ∈ [D̲4, D4] for i ∈ Ṉ with D1 ⪰ D̲1 ⪰ 0, D̲2 ⪰ D2 ⪰ 0, D3 ⪰ D̲3 ⪰ 0 and D4 ⪰ D̲4 ⪰ 0, where matrices D̲1, D1, D̲2, D2, D̲3, D3, D̲4, D4 are all diagonal, w(t) ∈ Rl is the exogenous disturbance and is defined as
with a known scalar ζ > 0.
Assumption 2.1
The nonlinear function f(x) lies in sector fields satisfying
for xi ∈ R and i = 1,2,…,n, where 0 ≤ m1 ≤ m2, and fi(0) = 0.
Remark 2.2
The system model (1) is a more general form. Especially, if m1 = m2 = 1 (it means fi(xi) = xi) and D1 = D2 = D3 = D4 = I, then the system (1) is transformed to positive switched linear systems, such as [2, 4, 5, 7, 14, 27, 28]; If m1 = m2 = 1, u(t) = 0 and w(t) = 0, then the system (1) is turned into the model in [30]; If D1 = D2 = D3 = D4 = I, then the system (1) is converted into the model in [19].
Next, we will give some definitions and lemmas for the following positive switched nonlinear systems.
Definition 2.3
([4]). System (4) is said to be positive if for any switching signals σ(t), any initial conditions φ(θ) ⪰ 0, θ ∈ [−ι,0], and any disturbance input w(t) ⪰ 0, the corresponding trajectory satisfies x(t) ⪰ 0 and y(t) ⪰ 0 for all t ≥ 0.
Definition 2.4
([4]). A is called a Metzler matrix if the off-diagonal entries of matrix A are non-negative.
Definition 2.5
([19]). For any switching signal σ(t) and any t2 ≥ t1 ≥ 0, let Nσ(t1, t2) denote the switching numbers, over the interval [t1, t2). For given tα > 0 and n0 > 0, if the inequality
holds, then tα is called an average dwell time, and n0 is called a chatting bounding. Generally, we choose n0 = 0.
Definition 2.6
([14]). Finite-Time Stability (FTS] For a given time Tf and two vectors ς ≻ ρ ≻ 0, positive switched nonlinear system (4) is said to be FTS with respect to (ς, ρ,Tf, σ(t)), if
If the above condition is satisfied for any switching signals σ(t), system (4) is said to be uniformly FTS with respect to(ς, ρ, Tf).
Definition 2.7
([14] Finite-Time Boundedness (FTB)). For a given constant Tf and two vectors ς ≻ ρ ≻ 0, positive switched nonlinear system (4) is said to be FTB with respect to(ς, ρ, Tf, ζ, σ(t)), where w(t) satisfies (2), if
Lemma 2.8
([4]). A matrix A ∈ Rn × n is a Metzler matrix if and only if there exists a positive constant ϱ such that A+ ϱ In ⪰ 0.
Lemma 2.9
([14]). System (4) is positive if and only if Ap, ∀ p ∈ Ṉ are Metzler matrices and ∀ p ∈ Ṉ, Adp ⪰ 0, Bp ⪰ 0, Cp ⪰ 0 and Gp ⪰ 0.
Definition 2.10
([21]). Define the cost function of positive switched nonlinear systems (1) as follows:
where R1 ≻ 0 and R2 ≻ 0 are two given vectors.
Definition 2.11
([21]). (GCFTB) For a given time constant Tf and two vectors ς ≻ ρ ≻ 0, consider positive switched nonlinear systems (1) and cost function (8), if there exist a control law u(t) and a positive scalar J⋆ such that the closed-loop system is FTB with respect to (ς, ρ, Tf, ζ, σ(t)) and the cost function satisfies J ≤ J⋆, then the closed-loop system is called GCFTB, where J⋆ is a guaranteed cost value and u(t) is a guaranteed cost finite-time controller.
3 Main results
3.1 Guaranteed cost finite-time boundedness analysis
In this subsection, we will focus on the problem of GCFTB for positive switched nonlinear system (4). Firstly, we present the following lemma which is essential for our later development.
Lemma 3.1
([14]). Consider the positive switched nonlinear system (4), for a given time constant Tf and vectors ς ≻ ρ ≻ 0, if there exist a set of positive vectors νp, υp, ϑp, p ∈ Ṉ and positive constants ϕ1, ϕ2, ϕ3, ϕ4, λ and γ, and such that the following inequalities hold:
where
apr(adpr, d1r, d2r, d4r) represents the rth column vector of the matrix Ap(Adp, D1, D2, D4), νp = [νp1, νp2,…, νpn]T, υp = [υp1, υp2,…, υpn]T, and ϑp = [ϑp1, ϑp2,…, ϑpn]T, νpr, υpr and ϑpr represent the rth elements of the vectors νp, υp and θp, respectively, then under the following ADT scheme
the system (4) is FTB with respect to (ς, ρ, Tf, ζ, σ(t)), where μ ≥ 1 satisfies
Proof
Construct the Lyapunov-Krasovskii functional for the system (4) as follows:
where νp, υp and ϑp ∈
Along the trajectory of the system (4), we have
From (3), (10), (11), (15) and (16), we have
Substituting (9) into (17) yields
Integrating both sides of (18) during the period [tk, t) for t ∈ [tk, tk+1) leads to
On the other hand, from (14) and (15), one can easily obtain
Let N be the switching number of σ(t) over [0, Tf), and denote t1, ⋯, tk as the switching instants over the interval [0, Tf). From (19), we have
Substituting (13) into (24), one has
According to Definition 2.7, we conclude that the system (4) is FTB with respect to (ς, ρ, Tf, ζ, σ(t)). □
Remark 3.2
For positive switched nonlinear system (4), a new nonlinear Lyapunov-Krasovskii functional (15) is constructed, which plays an important role in the proof procedure.
The following theorem gives sufficient conditions of guaranteed cost finite-time boundedness for system (4) with ADT.
Theorem 3.3
Consider the positive switched nonlinear system (4), for a given time constant Tf and vectors ς ≻ ρ ≻ 0 and R1 ≻ 0, if there exist a set of positive vectors νp, υp, ϑp, p ∈ Ṉ and positive constants ϕ1, ϕ2, ϕ3, ϕ4, λ and γ, and such that the following inequalities hold:
where
apr(adpr, d1r, d2r, d4r) represents the rth column vector of the matrix Ap(Adp, D1, D2, D4), R1r denotes the rth element of the vector R1, νp = [νp1, νp2,…, νpn]T, υp = [υp1, υp2,…, υpn]T, and ϑp = [ϑp1, ϑp2,…, ϑpn]T, νpr, υpr and ϑpr represent the rth elements of the vectors νp, υp and θp, respectively, then under the following ADT scheme
the system (4) is GCFTB with respect to (ς, ρ, Tf, ζ, σ(t)), where μ ≥ 1 satisfies
and the guaranteed cost value of system (4) is given by
Proof
Adding xT(t)R1 to both sides of (17), we get
Substituting (26) into (33) yields
It implies
From lemma 3.1, we conclude the system (4) is FTB. Next, we will give the guaranteed cost value of system (4).
Denoting ▽(t) = γ∥w(t)∥−xT(t)R1 and integrating both sides of (34) from [tk,t), for t ∈ [tk,tk+1) it gives rise to
where
Similarly to the proof process of (21), for any t ∈ [0,Tf], we can obtain
From (37), we can get
Multiplying both sides of (38) by μ−Nσ(t)(0,t) leads to
Noting that
Let t = Tf, then multiplying both sides of (40) by e−λ Tf leads to
Substituting (2) into (41) yields
which can be rewritten as
Substituting (23) into (43), the guaranteed cost value of system (4) is given by
Therefore, according to Definition 2.11, we can conclude that the system (4) is GCGTB. Thus, the proof is completed. □
Remark 3.4
From D1 ∈ [D̲1, D1], D2 ∈ [D̲2, D2] and D4 ∈ [D̲4, D4], which means D1Ap ∈ [D̲1Ap, D1Ap], D2Adp ∈ [D̲2Adp, D2Adp] and D4Bp ∈ [D̲4Bp, D4Bp], the obtained results in Theorem 3.3 should be extended to positive switched nonlinear systems with interval uncertainties.
3.2 Guaranteed cost finite-time controller design
In this subsection, we are concerned with the guaranteed cost finite-time controller design of positive switched nonlinear system (1). Under the controller u(t) = Kσ(t)y(t) the corresponding closed-loop system is given by
By Lemma 2.8, to guarantee the positivity of system (45), D1Ap+D3GpKpCp should be Metzler matrices, ∀ p ∈ Ṉ. The following Theorem 3.5 gives some sufficient conditions to guarantee that closed-loop system (45) is GCFTB.
Theorem 3.5
([14]). Consider the positive switched nonlinear system (45), for a given time constant Tf, vectors ς ≻ ρ ≻ 0, R1 ≻ 0 and R2 ≻ 0, if there exist a set of positive vectors νp, υp, ϑp, p ∈ Ṉ and positive constants ϕ1, ϕ2, ϕ3, ϕ4, λ and γ, such that (27)-(29) and the following inequalities hold:
where
Proof
From Lemma 2.8 and (46), D̲1Ap+ D̲3GpKpCp ⪯ D1Ap+D3GpKpCp ⪯ D1Ap+ D3GpKpCp are held, for each p ∈ Ṉ. It means that D1Ap+D3GpKpCp is Metzler matrice. According to Lemma 2.9, the system (45) is positive. Replacing D1Ap in (26) with D1Ap+D3GpKpCp and letting
The proof is completed. □
Remark 3.6
In Theorem 3.5, the gain matrix Kp ⪰ 0, ∀ p ∈ Ṉ is used. Naturally, when Kp ⪯ 0, we only replace (46) by the following condition
Following the proof line of Theorem 3.5, we can also conclude that the closed-loop system (45) is GCFTB with respect to (ς, ρ, Tf}, ζ, σ(t)).
Next, an algorithm is presented to obtain the feedback gain matrices Kp, p ∈ Ṉ.
Algorithm 3.7
Step 1. By adjusting the parameters λp and solving (27)-(29), (31) and (47) via linear programming, positive vectors νp, υp, ϑp and fp can be obtained.
Step 2. Substituting vp and
Step 3. If the gain Kp satisfy (46) or (49), then Kp are admissible. Otherwise, return to Step 1.
4 Numerical example
Example 4.1
Consider the positive switched nonlinear systems (1) with the parameters as follows:
Let d(t) = 0.1+0.1cos(t), then we get
By
It is easy to confirm that (46) is satisfied. Then, according to (30), we get
The simulation results are shown in Figs. 1-3, where the initial conditions of system (1) are x(0) = [0.2, 0.1]T, which meet the condition xT(t) ρ < 1. The state trajectory of the closed-loop system is shown in Fig. 1. The switching signal σ(t) is depicted in Fig. 2. Fig. 3 plots the evolution of x(t) ρ, which implies that the corresponding closed-loop system is GCFTB with respect to (ς, ρ, Tf, ζ, σ(t)), and the cost value J⋆ = 20.8, which can be obtained by (48).
State trajectories of closed-loop system (1).
Switching signal of system (1) with ADT.
The evolution of xT(t) ρ of system (1).
Example 4.2
In [30], a price dynamic model described by positive switched linear systems was presented. But, in fact, the demand and supply functions are nonlinear. So, it is more suitable to describe the price dynamic model by positive switched nonlinear systems. Consider the parameters as follows:
Let d(t) = 0.1+0.1 sin2t, then we get ι = 0.2, h = 0.1.
By
It is easy to confirm that (46) is satisfied. Then, according to (30), we get
The simulation results are shown in Figs. 4-6, where the initial conditions of system (1) are x(0) = [0.2, 0.1]T, which meet the condition xT(t) ρ < 1. The state trajectory of the closed-loop system is shown in Fig. 4. The switching signal σ(t) is depicted in Fig. 5. Fig. 6 plots the evolution of x(t) ρ, which implies that the corresponding closed-loop system is GCFTB with respect to (ς, ρ, Tf, ζ, σ(t)), and the cost value J⋆ = 12.6, which can be obtained by (48).
State trajectories of closed-loop system (1).
Switching signal of system (1) with ADT.
The evolution of xT(t) ρ of system (1).
5 Conclusions
In this paper, we have considered the issue of guaranteed cost finite-time control for positive switched nonlinear systems with D-perturbation and time-varying delay. Based on the ADT approach, an output feedback controller is constructed to guarantee that the closed-loop system is GCFTB. Finally, two examples are given to illustrate the effectiveness of the proposed method.
It is worth noting that there are some interesting yet challenging issues like mode-dependent average dwell time approach (which is more applicable and less conservative than ADT), asynchronously switching approach and cyclic switching approach. So, how to apply these approaches to positive switched nonlinear systems is our further work.
-
Conflict of interest
Conflict of interests: The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgement
The authors are thankful for the supports of the National Natural Science Foundation of China under grants U1404610, 61473115, and 61374077, young key teachers plan of Henan province (2016GGJS-056).
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